Comparative analysis of causal diagnosis methods of ... - circe

PhD Thesis
Comparative analysis of causal
diagnosis methods of malfunctions
in power cycles
Sergio Usón Gil
Directed by: Antonio Valero Capilla, Luis Carlos Correas Usón
Department of Mechanical Engineering.
University of Zaragoza.
January, 2008

Antonio Valero Capilla, PhD, professor of the Department of Mechanical
Engineering of the University of Zaragoza and Luis Carlos Correas Usón, PhD,
acknowledge that the PhD thesis ‘Comparative analysis of causal diagnosis methods of
malfunctions in power cycles’ written by Sergio Usón Gil, has been developed under
their supervision.
Zaragoza, January 2008
Antonio Valero Capilla Luis Carlos Correas Usón

I would like to thank a lot of people and institutions for their help in the development
of this thesis.
My supervisors, Antonio Valero and Luis Correas, for their valuable comments,
their words to encourage me to start, develop and finish this thesis, and their interest in
my professional development. Their complementary vision of the diagnosis problem has
become essential in the development of this work.
My workmates of CIRCE, for their help in a lot of tasks and, above all, for their
friendship. The list of names would be endless; so, please feel all of you included.
Part of this work was developed at Politecnico di Torino, with the group of Prof.
Vittorio Verda. All people I met in those three months made me feel like at home.
The Regional Government of Aragón and European Social Found for the grant
B120/2003, which provided me the financial support to develop this work; and the
personnel of Results Office at Teruel Power plant (Endesa Generación) for the
information and valuable advises about the ‘practical side’ of the diagnosis problem.
My first pupils and the personnel of the Department of Mechanical Engineering, for
these nice first steps with the chalk.
Last but not least, my friends and, above all, my family.
Thanks very much to all of you.

To my parents, Tomás and
Conchita, and my sister, Natalia.

Table of contents
Table of contents............................................................................................................ ix
List of tables ................................................................................................................. xiii
List of figures ................................................................................................................ xv
Nomenclature............................................................................................................. xxiii
1. INTRODUCTION ...................................................................................................... 1
1.1Justification, objectives and content of the thesis ............................................... 2
2. STATE OF THE ART OF THERMOECONOMIC DIAGNOSIS. ...................... 5
2.1 Thermoeconomic approach to diagnosis. ............................................................ 5
2.1.1 Thermoeconomic model.................................................................................... 5
2.1.2 Exergetic cost. ................................................................................................... 7
2.1.3 The fuel impact formula .................................................................................... 8
2.1.4. Cost formation process of wastes (or residues).............................................. 10
2.1.5. Other issues. ................................................................................................... 11
2.2 Thermoeconomic diagnosis methodologies. ...................................................... 13
2.2.1 The filtration of the effects induced by the control system. ............................ 14
2.2.2 Other methods based on the filtration of induced effects................................ 17
2.2.3 Representation of the malfunctions in the h-s plane........................................ 19
2.2.4 Methods directly based on the thermodynamic representation of the system 22
2.2.5. Concept of the diagnosis algorithm and mathemathical formulation............. 23
2.2.6. Other examples............................................................................................... 25
2.3. State of the art of systems for diagnosis and other related issues. ................. 26
2.3.1. Diagnosis systems in general.......................................................................... 27
2.3.2. Diagnosis centres and manufacturer-operator collaboration.......................... 28
2.3.3. Monitoring, diagnosis and optimization of boilers......................................... 29
2.3.4. Other issues. ................................................................................................... 30
2.4. Conclusion ........................................................................................................... 31
3. QUANTITATIVE CAUSALITY ANALYSIS ....................................................... 33
3.1. Re-formulation of the diagnosis problem......................................................... 33
3.1.1. Example.......................................................................................................... 35
3.2. Influence of measurement errors. ..................................................................... 40
3.2.1 Example........................................................................................................... 42
3.3. Analysis of non-linearities.................................................................................. 45
3.3.1. Evaluation of the influence of linearization errors in the diagnosis result ..... 45
3.3.2. Strategies to reduce the errors of the methodology due to non-linearities ..... 47
3.4. Free diagnosis variables ..................................................................................... 51
3.4.1 Mathematical conditions required to the set of free diagnosis variables......... 51
3.4.2. Influence of a modification in the free diagnosis variable choice on the
diagnosis result. .............................................................................................................. 52
3.4.3. Choice of the free diagnosis variables............................................................ 53
3.5. Causality chains .................................................................................................. 55
ix

3.5.1. General formulation of causality chains......................................................... 56
3.5.2. Causality chains based on the dependence among free diagnosis variables. . 58
3.5.3. Causality chains based on the dependence of free diagnosis variables with
other variables. ............................................................................................................... 59
3.6. Calculation of fuel impact on a time span. ....................................................... 60
3.7. Conclusion ........................................................................................................... 61
4. QUANTITATIVE CAUSALITY ANALYSIS AND OTHER DIAGNOSIS
METHODS.................................................................................................................... 63
4.1. Quantitative causality analysis and the fuel impact formula ......................... 63
4.1.1. From free diagnosis variables to unit exergy cost and final product.............. 64
4.1.2. Quantification of intrinsic and induced malfunctions .................................... 66
4.1.3. Determination of the suitability of a productive structure.............................. 73
4.1.4. Conclusion...................................................................................................... 75
4.2. Application of neural networks and linear regression to the diagnosis of
energy systems. ............................................................................................................. 75
4.2.1. Linear regression fundamentals...................................................................... 76
4.2.2. Neural network fundamentals......................................................................... 77
4.2.2.1. Neurons, layers and networks. .............................................................................. 78
4.2.2.2. Feedforward networks and backpropagation......................................................... 80
4.2.3. Application for diagnosis................................................................................ 82
4.2.3.1. Diagnosis of the complete system......................................................................... 82
4.2.3.2. Diagnosis of sub-systems. Hybrid configurations................................................. 83
4.2.3.3. Other issues. .......................................................................................................... 84
4.3. Conclusión ........................................................................................................... 84
5 CASE STUDY: 3X 350 MW COAL-FIRED POWER PLANT............................ 85
5.1 Teruel power plant description. ......................................................................... 85
5.1.1 Steam cycle description................................................................................... 86
5.1.2 Cooling system ................................................................................................ 89
5.1.3 Boiler ............................................................................................................... 90
5.2 Diagnosis model for a pulverized-coal power plant. ........................................ 95
5.2.1 Steam cycle...................................................................................................... 96
5.2.1.1 Steam turbine.......................................................................................................... 96
5.2.1.2 Steam turbine seals and other secondary flows...................................................... 99
5.2.1.3 Alternator ............................................................................................................. 100
5.2.1.4 Feedwater heaters................................................................................................. 100
5.2.1.5 Pumps................................................................................................................... 102
5.2.1.6 Pressure losses...................................................................................................... 103
5.2.2 Cooling system .............................................................................................. 104
5.2.2.1 Condenser............................................................................................................. 105
5.2.2.2 Cooling tower....................................................................................................... 107
5.2.2.3 Pumps and ducts................................................................................................... 108
5.2.3 Boiler ............................................................................................................. 108
5.2.3.1 Boiler efficiency and coal consumption............................................................... 109
5.2.3.2 Combustion .......................................................................................................... 114
5.2.3.3 Heat transfer in boilers ......................................................................................... 115
5.2.3.4 Air preheaters ....................................................................................................... 120
5.2.3.5 Ancillary devices.................................................................................................. 122
5.3 Monitoring and diagnosis model for Teruel power plant.............................. 123
5.3.1 Measurement review. .................................................................................... 123
5.3.2 Free diagnosis variables ................................................................................ 127
5.3.3 Global efficiency indicators. ......................................................................... 131
5.4 Conclusion .......................................................................................................... 132
x

6 APPLICATION OF QUANTITATIVE CAUSALITY ANALYSIS................... 133
6.1 Introduction ....................................................................................................... 133
6.2 Evolution of efficiency indicators..................................................................... 136
6.3 Diagnosis results analysis.................................................................................. 139
6.3.1 Ambient conditions. ...................................................................................... 139
6.3.2 Fuel quality.................................................................................................... 146
6.3.3 Steam cycle set points. .................................................................................. 158
6.3.4 Boiler set points............................................................................................. 164
6.3.5 Steam cycle component parameters. ............................................................. 175
6.3.6 Cooling system component parameters......................................................... 192
6.3.7 Boiler component parameters........................................................................ 196
6.3.8 Summary of results........................................................................................ 203
6.4 Residual term analysis. ..................................................................................... 209
6.4.1 Residual term distribution. ............................................................................ 209
6.4.2 Non-dimensional residual term distribution.................................................. 212
6.4.3 Conclusion..................................................................................................... 214
6.5 Impacts aggregation .......................................................................................... 214
6.5.1 Cycle heat rate ............................................................................................... 215
6.5.2 Boiler efficiency ............................................................................................ 218
6.5.3 Unit heat rate ................................................................................................. 223
6.5.4 Conclusion..................................................................................................... 225
6.6 Causality chains ................................................................................................. 226
6.6.1 Determination of a correlation for cooling tower effectiveness.................... 227
6.6.2 Decomposition of the impact of cooling tower effectiveness. ...................... 228
6.6.3 Aggregated impacts ....................................................................................... 230
6.7 Conclusion .......................................................................................................... 232
6.7.1 Conclusions on the method ........................................................................... 232
6.7.2 Practical results about the working example............................................ 233
6.7.2.1 Steam cycle .......................................................................................................... 233
6.7.2.2 Boiler.................................................................................................................... 234
7 APPLICATION OF QUANTITATIVE CAUSALITY ANALYSIS TO THE
QUANTIFICATION OF INTRINSIC AND INDUCED MALFUNCTIONS. ..... 235
7.1 Description of the productive structure. ......................................................... 235
7.1.1 Simplified physical structure......................................................................... 235
7.1.2 Productive structure....................................................................................... 239
7.2 Example of thermoeconomic diagnosis ........................................................... 243
7.2.1 Preliminary results......................................................................................... 243
7.2.2 Decomposition of unit exergy consumptions and plant products.................. 248
7.2.3 Decomposition of malfunctions..................................................................... 253
7.2.4 Malfunction cost decomposition. MFI and MFD tables. .............................. 255
7.3 General quantification of intrinsic and induced effects................................. 263
7.3.1 Decomposition of unit exergy consumptions and plant products.................. 263
7.3.2 Decomposition of malfunctions..................................................................... 266
7.3.3 Decomposition of malfunction costs. ............................................................ 268
7.4 Conclusion .......................................................................................................... 272
8 APPLICATION OF LINEAR REGRESSION AND NEURAL NETWORKS. 273
8.1 Linear regression ............................................................................................... 274
8.1.1 Variable change ............................................................................................. 274
8.1.2 Results ........................................................................................................... 276
8.2 Neural networks................................................................................................. 284
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8.2.1 Neural network definition and training ......................................................... 285
8.2.2 Results ........................................................................................................... 289
8.3 Conclusion .......................................................................................................... 296
9 CONCLUSION ........................................................................................................ 299
9.1 Synthesis ............................................................................................................. 299
9.2 Contributions ..................................................................................................... 301
9.3 Perspectives ........................................................................................................ 304
REFERENCES ........................................................................................................... 307
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List of tables
Table 3.1. Properties of streams in the cycle.................................................................. 36
Table 4.1: Table of malfunctions and free diagnosis variables (MFD).......................... 69
Table 4.2: Table of intrinsic and induced malfunctions (MFI). ..................................... 73
Table 5.1. Main characteristics of the Teruel Power plant steam cycle. (Zaleta, 1997). 88
Table 5.2. Condenser characteristics (Lazaro, 2001) ..................................................... 89
Table 5.3. Cooling water design characteristics (Lazaro, 2001). ................................... 90
Table 5.4. Teruel power plant boilers, nominal production (Díez, 2002). ..................... 90
Table 5.5. Exchange area of the boiler heat exchangers. (Díez, 2002) .......................... 94
Table 5.6.A. List of plant measurements used. ............................................................ 124
Table 5.6.B. List of plant measurements used.............................................................. 125
Table 5.6.C. List of plant measurements used.............................................................. 126
Table 5.7.A. Free diagnosis variables. ......................................................................... 128
Table 5.7.B. Free diagnosis variables........................................................................... 129
Table 5.7.C. Free diagnosis variables.......................................................................... 130
Table 5.8. Free diagnosis variables with constant value. ............................................. 131
Table 6.1.A. Summary of diagnosis results.................................................................. 204
Table 6.1.B. Summary of diagnosis results.................................................................. 205
Table 6.1.C. Summary of diagnosis results.................................................................. 206
Table 6.1.D. Summary of diagnosis results.................................................................. 207
Table 6.1.E. Summary of diagnosis results. ................................................................. 208
Table 7.1. A. Description of flows of the simplified physical structure. ..................... 236
Table 7.1. B. Description of flows of the simplified physical structure....................... 237
Table 7.2. Description of components of the physical structure. ................................. 239
Table 7.3. Components of the productive structure ..................................................... 240
Table 7.4. FPR Table.................................................................................................... 241
Table 7.5.A. Free diagnosis variables in the actual and reference states. .................... 244
Table 7.5.B. Free diagnosis variables in the actual and reference state. ...................... 245
Table 7.6. Exergy and exergy costs of the productive flows (kW). ............................. 246
Table 7.7. Exergy and exergy cost of waste flows (kW).............................................. 247
Table 7.8. Unit exergy costs. ........................................................................................ 247
Table 7.9. Unit exergy consumptions........................................................................... 248
Table 7.10. Unit exergy consumptions associated with the residues. .......................... 248
Table 7.11. Table of intrinsic and induced malfunctions (MFI). ................................. 258
Table 7.12.A. Table of malfunctions induced by the free diagnosis variables (MFD). 259
Table 7.12.B. Table of malfunctions induced by the free diagnosis variables (MFD). 260
Table 7.12.C. Table of malfunctions induced by the free diagnosis variables (MFD). 261
Table 7.12.D. Table of malfunctions induced by the free diagnosis variables (MFD) 262
Table 8.1. Pairs of variables highly correlated. ............................................................ 274
Table 8.2.A. Impact factors obtained by linear regression........................................... 277
Table 8.2.B. Impact factors obtained by linear regression. .......................................... 278
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Table 8.2.C. Impact factors obtained by linear regression. .......................................... 279
Table 8.2.D. Impact factors obtained by linear regression........................................... 280
Table 8.2.E. Impact factors obtained by linear regression. .......................................... 281
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List of figures
Figure 2.1. Example of a productive structure. ................................................................ 6
Figure 2.2. RPS representation in the ω, σ, MFR space................................................. 20
Figure 2.3: Dissipation temperature. .............................................................................. 21
Figure 3.1. Scheme of the example cycle....................................................................... 35
Figure 4.1. Single neuron. Source: Demuth and Beale (2002)....................................... 78
Figure 4.2. Hard-lim transfer function. Source: Demuth and Beale (2002)................... 78
Figure 4.3. Linear transfer function. Source: Demuth and Beale (2002) ....................... 79
Figure 4.4. Log-sigmoid transfer function. Source: Demuth and Beale (2002)............. 79
Figure 4.5. Tan-sigmoid transfer function. Source: Demuth and Beale (2002) ............. 79
Figure 4.6. Layer of a neural network. Source: Demuth and Beale (2002).................... 80
Figure 5.1. Steam cycle. ................................................................................................. 87
Figure 5.2. Fans and preheaters...................................................................................... 91
Figure 5.3. Schematic side view of the boiler. ............................................................... 93
Figure 5.4. Temperature profile in a feedwater heater. ................................................ 101
Figure 6.1. Unit heat rate evolution.............................................................................. 137
Figure 6.2. Steam cycle heat rate evolution. ................................................................ 138
Figure 6.3. Boiler efficiency evolution......................................................................... 139
Figure 6.4. Ambient temperature evolution.................................................................. 140
Figure 6.5. Cycle heat rate impact due to ambient temperature................................... 141
Figure 6.6. Impact of ambient temperature on boiler efficiency.................................. 142
Figure 6.7. Unit heat rate impact due to ambient temperature. .................................... 142
Figure 6.8. Ambient temperature variation and its impact on cycle heat rate.............. 143
Figure 6.9. Cycle heat rate impact due to relative humidity......................................... 144
Figure 6.10. Relative humidity variation and its impact on cycle heat rate. ................ 144
Figure 6.11. Relative humidity variation and its impact on boiler efficiency.............. 145
Figure 6.12. Wind speed evolution............................................................................... 146
Figure 6.13. Coal high heating value evolution. .......................................................... 147
Figure 6.14. Impact on boiler efficiency due to coal HHV. ......................................... 147
Figure 6.15. Coal HHV variation and its impact on boiler efficiency.......................... 148
Figure 6.16. Impact on boiler efficiency due to carbon in coal.................................... 149
Figure 6.17. Carbon in coal variation and its impact on boiler efficiency. .................. 150
Figure 6.18. Impact on boiler efficiency due to hydrogen in coal................................ 150
Figure 6.19. Hydrogen in coal and its impact on boiler efficiency. ............................. 151
Figure 6.20. Impact on boiler efficiency due to coal moisture..................................... 152
Figure 6.21. Coal moisture variation and its impact on boiler efficiency. ................... 153
Figure 6.22. Impact on boiler efficiency due to ash in coal. ........................................ 153
Figure 6.23. Ash in coal variation and its impact on boiler efficiency......................... 154
Figure 6.24. Impact on boiler efficiency due to sulphur in coal................................... 155
Figure 6.25. Sulphur in coal variation and its impact on boiler efficiency. ................. 156
Figure 6.26. Natural gas fraction evolution.................................................................. 156
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Figure 6.27. Natural gas fraction variation and its impact on boiler efficiency........... 157
Figure 6.28. Live steam temperature evolution............................................................ 158
Figure 6.29. Live steam temperature variation and its impact on cycle heat rate. ....... 159
Figure 6.30. Reheated steam temperature evolution. ................................................... 160
Figure 6.31. Reheated steam temperature variation and its impact on cycle heat rate. 160
Figure 6.32. Live steam pressure evolution.................................................................. 161
Figure 6.33. Live steam pressure and its impact on cycle heat rate. ............................ 161
Figure 6.34. Gross electric power evolution................................................................. 162
Figure 6.35. Gross electric power variation and its impact on cycle heat rate............. 163
Figure 6.36. Gross electric power variation and its impact on boiler efficiency.......... 163
Figure 6.37. Oxygen set-point evolution...................................................................... 164
Figure 6.38. Oxygen set point variation and its impact on boiler efficiency. .............. 165
Figure 6.39. Evolution of average cold-side temperature in secondary air preheaters. 166
Figure 6.40. Average cold-side temperature in secondary air preheaters variation and its
impact on unit heat rate. ............................................................................................... 167
Figure 6.41. Average cold-side temperature in primary air preheaters evolution........ 167
Figure 6.42. Average cold-side temperature in primary air preheaters variation and its
impact on unit heat rate. ............................................................................................... 168
Figure 6.43. Sootblowing steam evolution................................................................... 169
Figure 6.44. Sootblowing steam variation and its impact on unit heat rate. ................ 170
Figure 6.45. Air for flue gas desulfuration unit evolution............................................ 171
Figure 6.46. Air for flue gas desulfuration unit variation and its impact on boiler
efficiency. ..................................................................................................................... 171
Figure 6.47. Relation of tempering and primary air evolution..................................... 172
Figure 6.48. Variation of the relation of tempering and primary air and its impact on
boiler efficiency............................................................................................................ 173
Figure 6.49. Primary air-coal ratio and its impact on boiler efficiency........................ 173
Figure 6.50 Impact on boiler efficiency due to primary air-coal ratio. ........................ 174
Figure 6.51. Evolution of temperature difference of flue gases entering air preheaters.
...................................................................................................................................... 175
Figure 6.52. Evolution of high pressure steam turbine isoentropic efficiency............. 176
Figure 6.53. High pressure steam turbine isoentropic efficiency variation and its impact
on cycle heat rate. ......................................................................................................... 177
Figure 6.54. Medium pressure 1 steam turbine isoentropic efficiency evolution. ....... 177
Figure 6.55. Medium pressure 1 steam turbine isoentropic efficiency variation and its
impact on cycle heat rate. ............................................................................................. 178
Figure 6.56. Medium pressure 2 steam turbine isoentropic efficiency evolution. ....... 179
Figure 6.57. Medium pressure 2 steam turbine isoentropic efficiency variation and its
impact on unit heat rate. ............................................................................................... 179
Figure 6.58. Impact on cycle heat rate due to low pressure steam turbine isoentropic
efficiency. ..................................................................................................................... 180
Figure 6.59. Low pressure steam turbine isoentropic efficiency variation and its impact
on unit heat rate. ........................................................................................................... 181
Figure 6.60. Impact on unit heat rate due to flow coefficient of medium pressure steam
turbine 1........................................................................................................................ 182
Figure 6.61. Flow coefficient of medium pressure steam turbine 1 variation and its
impact on cycle heat rate. ............................................................................................. 182
Figure 6.62. Hot side temperature difference (TTD) of 6th preheater evolution. ........ 183
Figure 6.63. TTD of 6th preheater variation and its impact on cycle heat rate............ 184
Figure 6.64. Hot side temperature difference of 5th water preheater evolution........... 185
xvi

Figure 6.65. Hot side temperature difference (TTD) of 5th water preheater variation and
its impact on cycle heat rate. ........................................................................................ 186
Figure 6.66. Hot side temperature difference (TTD) of 3rd water preheater evolution.
...................................................................................................................................... 186
Figure 6.67 Pressure drop in reheater evolution........................................................... 188
Figure 6.68. Pressure drop in preheater variation and its impact on cycle heat rate. ... 189
Figure 6.69. Pressure increment in turbo-pump evolution. .......................................... 189
Figure 6.70. Pressure increment in turbo-pump variation and its impact on cycle heat
rate. ............................................................................................................................... 190
Figure 6.71. Water losses evolution. ............................................................................ 191
Figure 6.72. Water losses variation and its impact on unit heat rate............................ 191
Figure 6.73. Condenser effectiveness evolution........................................................... 192
Figure 6.74. Condenser effectiveness variation and its impact on cycle heat rate....... 193
Figure 6.75. Cooling water flow rate evolution............................................................ 193
Figure 6.76. Cooling water flow rate variation and its impact on cycle heat rate........ 194
Figure 6.77. Cooling tower effectiveness evolution..................................................... 195
Figure 6.78. Cooling tower effectiveness variation and its impact on cycle heat rate. 195
Figure 6.79. Secondary air preheater effectiveness evolution...................................... 196
Figure 6.80. Secondary air preheater effectiveness variation and its impact on boiler
efficiency. ..................................................................................................................... 197
Figure 6.81. Primary air preheater effectiveness evolution.......................................... 198
Figure 6.82. Primary air preheater effectiveness variation and its impact on boiler
efficiency. ..................................................................................................................... 198
Figure 6.83. Aggregated boiler effectiveness evolution............................................... 199
Figure 6.84. Aggregated boiler effectiveness variation and its impact on boiler
efficiency. ..................................................................................................................... 200
Figure 6.85. Air infiltration in preheaters evolution..................................................... 200
Figure 6.86. Air infiltration in preheaters and its impact on boiler efficiency............. 201
Figure 6.87. Carbon in ashes evolution. ....................................................................... 202
Figure 6.88. Carbon in ashes variation and its impact on boiler efficiency. ................ 202
Figure 6.89. Distribution of residuals of cycle heat rate. ............................................. 210
Figure 6.90. Distribution of residuals of boiler efficiency. .......................................... 211
Figure 6.91. Distribution of residuals of unit heat rate................................................. 211
Figure 6.92. Distribution of non-dimensional residuals of cycle heat rate................... 212
Figure 6.93. Distribution of non-dimensional residuals of boiler efficiency ............... 213
Figure 6.94. Distribution of non-dimensional residuals of unit heat rate..................... 213
Figure 6.95. Impact on cycle heat rate due to ambient conditions. .............................. 216
Figure 6.96. Impact on cycle heat rate due to cycle set-points..................................... 216
Figure 6.97. Impact on cycle heat rate due to boiler. ................................................... 217
Figure 5.98. Impact on cycle heat rate due to cooling system...................................... 217
Figure 6.99. Impact on cycle heat rate due to cycle components................................. 218
Figure 6.100. Impact on boiler efficiency due to ambient conditions.......................... 219
Figure 6.101. Impact on boiler efficiency caused by fuel variation............................. 219
Figure 6.102. Impact on boiler efficiency caused by cycle and cooling system. ......... 220
Figure 6.103. Impact on boiler efficiency caused by components and set-points of the
boiler............................................................................................................................. 221
Figure 6.104. Impact on boiler efficiency caused by boiler set-points......................... 222
Figure 6.105. Impact on boiler efficiency caused by boiler components..................... 222
Figure 6.106. Impact on unit heat rate caused by ambient conditions. ........................ 223
Figure 6.106. Impact on unit heat rate caused by fuel.................................................. 224
xvii

Figure 6.107. Impact on unit heat rate caused by set-points. ....................................... 224
Figure 6.108. Impact on unit heat rate caused by components..................................... 225
Figure 6.109. Dependence of cooling tower effectiveness with ambient temperature. 228
Figure 6.110. Impact of cooling tower effectiveness on cycle heat rate, after filtering
effects induced by ambient temperature....................................................................... 229
Figure 6.111. Impact on cycle heat rate due to cooling tower effectiveness variation
induced by ambient temperature................................................................................... 229
Figure 6.112. Total impact on cycle heat rate due to ambient temperature, including
induced effect on cooling tower effectiveness. ............................................................ 230
Figure 6.113. Impact caused by cooling system on cycle heat rate after filtering effects
induced by ambient temperature on cooling tower effectiveness................................. 231
Figure 6.114. Impact caused by plant components on unit heat rate, after filtering effects
induced by ambient temperature on cooling tower effectiveness................................. 231
Figure 7.1. Simplified physical diagram of the cycle................................................... 238
Figure 7.2. Simplified physical diagram of the boiler.................................................. 238
Figure 7.3. Productive structure. .................................................................................. 242
Figure 7.4. Decomposition of unit exergy consumptions............................................. 250
Figure 7.5. Normalized decomposition of unit exergy consumptions.......................... 250
Figure 7.6. Decomposition of the unit exergy consumptions of boiler. ....................... 251
Figure 7.7. Decomposition of unit exergy consumptions associated with wastes. ...... 252
Figure 7.8. Decomposition of plant products. .............................................................. 252
Figure 7.9. Decomposition of malfunctions. ................................................................ 253
Figure 7.10. Decomposition of malfunctions corresponding to the boiler................... 254
Figure 7.11. Decomposition of malfunctions associated with the residues.................. 254
Figure 7.12. Decomposition of malfunction cost. ........................................................ 256
Figure 7.13. Decomposition of malfunction cost corresponding to boiler................... 256
Figure 7.14. Decomposition of malfunction cost associated with wastes.................... 257
Figure 7.15 Decomposition of unit exergy consumptions............................................ 264
Figure 7.16. Decomposition of unit exergy consumptions........................................... 264
Figure 7.17. Decomposition of unit exergy consumption associated with the boiler. . 265
Figure 7.18. Decomposition of unit exergy consumptions associated with the residues.
...................................................................................................................................... 265
Figure 7.19. Decomposition of plant product............................................................... 266
Figure 7.20. Decompositions of malfunctions. ............................................................ 267
Figure 7.21. Decomposition of malfunctions associated with the boiler. .................... 267
Figure 7.22. Decomposition of malfunctions associated with the residues.................. 268
Figure 7.23. Decomposition of malfunction costs........................................................ 269
Figure 7.24. Decomposition of malfunction cost associated with the boiler ............... 269
Figure 7.25. Decomposition of malfunction costs associated with the residues. ......... 270
Figure 7.26. Fuel impact of free diagnosis variables (1).............................................. 271
Figure 7.27. Fuel impact of free diagnosis variables (2).............................................. 271
Figure 8.1. Error in the value of impact factors and impact standard deviation .......... 283
in boiler efficiency........................................................................................................ 283
Figure 8.2. Error in the value of impact factors and impact standard deviation .......... 283
in cycle heat rate........................................................................................................... 283
Figure 8.3. Error in the value of impact factors and impact standard deviation .......... 284
in unit heat rate. ............................................................................................................ 284
Figure 8.4. Training process for boiler efficiency neural network............................... 286
Figure 8.5. Training process for cycle heat rate neural network. ................................. 286
Figure 8.6. Training process for unit heat rate neural network. ................................... 287
xviii

Figure 8.7. Comparison of real and calculated values of boiler efficiency.................. 287
Figure 8.8. Comparison of real and calculated cycle heat rate..................................... 288
Figure 8.9. Comparison of real and calculated unit heat rate....................................... 288
Figure 8.10. Variable increment and impact for the six variables most influent on boiler
efficiency: coal HHV, aggregated boiler effectiveness, ambient temperature, moisture in
coal, hydrogen in coal and carbon in ashes. ................................................................. 290
Figure 8.11. Relation between variable increment and its impact on boiler efficiency for
several variables: tempering and primary air relation, air infiltration in preheaters,
average cold-side temperature in primary air preheaters, primary air-coal ratio, sulphur
mass fraction in coal and low pressure isoentropic efficiency. .................................... 291
Figure 8.12. Variable increment and impact for the six variables most influent on cycle
heat rate: ambient temperature, low pressure turbine isoentropic efficiency, cooling
tower effectiveness, condenser effectiveness, cooling water flow rate and high pressure
turbine isoentropic efficiency. ...................................................................................... 292
Figure 8.13. Relation between variable increment and its impact on cycle heat rate for
several variables: relative humidity, intermediate pressure 1 isoentropic efficiency,
intermediate pressure 2 isoentropic efficiency, gross electric power, coal high heating
value and carbon mass fraction in coal......................................................................... 293
Figure 8.14. Variable increment and impact for the six variables most influent on unit
heat rate: low pressure turbine isoentropic efficiency, ambient temperature, coal high
heating value, cooling tower effectiveness moisture mass fraction in coal and condenser
effectiveness. ................................................................................................................ 294
Figure 8.15. Relation between variable increment and its impact on unit heat rate for
several variables: carbon mass fraction in coal, relative humidity, sootblowing steam
flow rate, intermediate pressure 3 turbine isoentropic efficiency, hot side temperature
difference in 6th water heater and average cold-side temperature in primary air
preheaters...................................................................................................................... 295
xix

Nomenclature
a Generic point. Output of a neuron. Sound speed. Area per unit of volume
A Area. Ambient
b Generic point. Bias.
B Boiler incomes different from fuel.
Bo Boltzman number
c Specific heat. Generic point. Speed.
C Heat capacity rate
CFD Computer fluid dynamics
CP Component parameter
d Differential variation
D Diameter.
DF Dysfunction
e Intensive magnitude, Global efficiency indicator
ee Error in the calculation of e from x
E Exergy flow in a productive structure, Extensive magnitude. Energy entering the
boiler
E * Exergetic cost of a flow
EP Electrostatic precipitator
EZ1 Primary economizar
EZ2 Secondary economizer
f Global efficiency indicator. Function. Fraction.
F Fuel. Factor. Fouling thermal resistence. View factor
FDF Forced draft fan
FT
Fuel entering the plant
g Intensive magnitude. Gravity constant.
G Extensive magnitude. Irradiation
h Enthalpy. Convection coefficient
H Height. Enthalpy
HR Heat rate
i Coefficient of increment of intrinsic malfunctions.
I Irreversibility. Impact
IDF Induced draft fan
k Constant. Thermal conductivity
k * Unit exergy cost. Marginal unit exergy cost
*
k Average unit exergy cost.
*
k R Unit exergy cost associated with wastes
L Losses
LMTD Log-mean temperature difference
m Flow rate
xxi

M Coefficient
MFR Mass flow ratio
MF Malfunction
MF * Malfunction cost
MR Malfunction associated with wastes
MR * Malfunction cost associated with wastes
n Components in the thermoeconomic model. Number of observations in linear
regression. Rotation speed. Exponent. Intermediate value in a neuron.
NTU Number of transfer units
N Dimension of predictor in linear regression
P Product,
p Pressure. Number of terms in linear regression
PAF Primary air fan
PAH Primary air heater
PF Power factor
q Flow rate
Q Heat. Volumetric flow rate.
r Number of governing parameters. Dimension of the input vector in a neuron.
Relation
RH Reheater
RPS Reference performance state
s Entropy
S Number of neurons in a layer. Surface.
SAH Secondary air preheater
SH1 Primary superheater
SH2 Secondary superheater
SH3 Final superheater
SP Set point
T Temperature
Td
xxii
Dissipation temperature
TDCA Drain cooling approach temperature
TTD Terminal temperature difference
U Global heat transfer coefficient. Utile heat.
UE Used enthalpy
umb Umburned carbon related to coal
umbres Umburned carbon related to ashes
W Mass fraction of moisture
W Power
x Thermodynamic variable. Quality. Molar fraction
ˆx Corrected variable
y Response in linear regression
z Teeth number in a seal.
Z Ash in coal
Matrices and arrays
a Output of a layer of neurons
AD Matrix of correction factors with free diagnosis variables
AX Matrix of correction factors with all thermodynamic variables
b Bias of a layer or neurons
bd Independent terms vector for correction with free diagnosis variables

DD Matrix to connect free diagnosis variables of first and second level
DE Matrix of impacts on e due to two levels of causality
DP Matrix to connect measured variables and free diagnosis variables
e Error vector
EA Matrix of impacts induced by other free diagnosis variables
EAD Matrix of sensitivity for impacts induced by other free diagnosis variables
ed Vector of sensitivity coefficients of e related to xc
ED Diagonal matrix of sensitivity coefficientys of e related to xc
EDD Matrix of sensitivity coefficients of e related to two levels of causality
EP Matrix to connect measured variables and impacts of free diagnosis variables.
ep Vector to connect measured variables and efficiency indicator e.
er Vector to connect residues and indicator e
ex Vector of sensitivity coefficientys of e related to x
g Restrictions. Gradient
H Hessian matrix
J Jacobian matrix
JD General matrix of the diagnosis problem
JP General matrix of the performance test problem
JR Jacobian matrix of restrictions
JRD Jacobian matrix of restrictions of the diagnosis problem
JRP Jacobian matrix of restrictions of the performance test problem
k *
Unit exergy costs
kd Vector to connect κ and xd
KD Matrix to connect κ and xd
Matrix of unit exergy consumptions
kx Vector to connect κ and x
M Generic matrix
N Generic matrix
p Input vector in a neuron.
P Product
Ps
Product of the system
P> Product operator
R Restrictions. Correlation matrix
RD Restrictions of the diagnosis problem
rs Residues
RP Restrictions for the performance test problem
tx Vector to connect θ and x
td Vector to connect θ and xd
TD Matrix to connect θ and xd
UD
Unit matrix
VD Matrix of diagnosis variables
VP Matrix of measured variables
W Weight matrix
wx Vector to connect ω and x
wd Vector to connect ω and xd
x Thermodynamic variables. Predictor in linear regression
X Diagonal matrix of thermodynamic variables. Matrix of observations of the
predictor in linear regression.
y Vector of observations of response in linear regression.
xxiii

Greek
α Coefficient. Learning rate. Absorptivity
β Coefficient. Mass transfer coefficient. Angle
β Vector of coefficients
ˆβ Coefficient estimator in linear regression
ˆ
β Vector of coefficient estimator in linear regression
Δ Increment
ε Heat exchanger effectiveness. Error. Emisivity. Residual.
ε Vector of errors in linear regression.
η Isoentropic efficiency
θ Unit exergy consumption associated with residues. Internal parameter
κ Unit exergy consumption
ˆκ Unit exegy consumption corrected by variations in operation point
κe Unit exergy consumption of external resources
μ Parameter in Levenberg-Marquardt algorithm
ρ Density. Ratio of entropy increment.
σ Entropy variation. Standard deviation. Stefan-Boltzman constant
σ 2 Uncertainty vector
τ Set of thermodynamic variables
υ Specific volume
φ Factor
Φ Flow coefficient. Relative humidity.
χ Internal or endogenous variables
ω Product of a component leaving the system. Enthalpy variation
Subscripts
a Air
ac Adiabatic combustion
amb Ambient
b Boiler
B Radiation and convection
c Cycle. Condenser. Cold. Corrected.
calc Calculated
CO Carbon monoxide
conv Convection
crs Cold reheated steam
dep Dependent
d Diagnosis. Design. Sensible heat in ashes
dp Drum purge
e Initial point
ext External
f Fuel
F Furnace
fdf Forced draft fan
fw Feeding water
g Final point. Gases
gen Generator
go Gases output
gs Sensible heat in flue gases
xxiv

h Hot
H Hydrogen
hrs Hot reheated steam
ind Induced
indep Independent
int Intrinsic. Internal
in Input
l Linealization
lm Log-mean
LP Low pressure
mA Air moisture
mat Material
max Maximum
mec Mechanical
mf Coal moisture
min Minimum
ms Main steam
n Net
nd Non dimensional
ng Natural gas
o Output
ol Other loses
p Pump. Measured in plant. Pressure. Point.
P Product
paf Primary air fan
r Control system. Restrictions
rad Radiation
rd Restrictions of the diagnosis problem
rp Restrictions for performance test problem
res Residual
ref Reference
rs Residual
s Isoentropic. Steam
sat Saturation
sb Sootblowing
sl Saturated liquid
sru Air to sulphur removing unit.
ss Saturated steam
st Static
t Thermodynamic model. Turbine. Per unit of time.
tot Total
tp Turbo-pump
tr Transmission
tw Tempering water
u Unit
uc Unburned carbon in ashes
ul Uncontrolled leakages
v Volumetric. Valve
var Variables
w Water. Wall
xxv

wb Wet bulb
0 Environment, reference example
1 First causality level
1ch Primary air coil heater
2 Second causality level
2ch Secondary air coil heater
Superscripts
ac Ambient conditions
av Average
fq Fuel quality
int Intrinsic
lr Linear regression
oc Other components
r Residual
rep Representative
sp Set points
t Transpose
0 Reference
1 Actual situation
xxvi

1 Introduction
The word diagnosis refers to the activities aimed at the detection of anomalies in any
system. In general, the interest of locating these anomalies appears because they can
lead, if not solved, to catastrophic effects.
In energy systems, besides this kind of failures that potentially originates
component breaking and, perhaps, plant shut off, there are other anomalies which are of
interest. These situations might never originate a catastrophic failure but may entail
huge economic expenses due to the additional fuel consumption they cause.
In this context, it is interesting not only to detect anomalies but also to quantify the
effect of each one of them in the variation of fuel consumption. This quantification is
crucial in order to decide whether or not to repair or replace a component. This task is
the objective of thermoeconomic diagnosis.
Thermoeconomics can be considered as the science of energy saving and emerges
from the combination of the Second Principle of Thermodynamics and Economics. It
provides several tools and concepts to deal with the diagnosis problem. The main of
them is the fuel impact formula, which links the variations of the unit exergy
consumptions of the plant components (the indicators of their individual behaviour) to
the fuel increment they cause.
Thermoeconomic analysis provides a solid conceptual basis for the diagnosis, but in
the practical application to real examples, difficulties appear because of the problem of
induced effects. The presence of these effects makes it more difficult to identify the
components where anomalies take place, and the development of methods to eliminate
them is a wide field of research. (Valero et al., 2004a,b).
Due to the difficulty to develop methods to deal successfully with induced effects,
some authors have developed diagnosis methodologies which avoid the use of
Thermoeconomic analysis and are directly based on a thermodynamic representation of
1

Chapter 1
the studied system. Among these proposals, the diagnosis algorithm developed by
Correas (2001, 2004), constitutes a systematic approach, which provide good results in
the application to real systems.
Finally, there are other methods, based on techniques such as neural networks,
which look directly for the relations among the variables, with the use of neither
thermoeconomic nor thermodynamic models.
1.1 Justification, objectives and content of the thesis
In this framework, the diagnosis algorithm presents characteristics of rigor,
completeness and suitability for practical application which makes it an interesting tool
for the diagnosis of whatever energy system. However, their capabilities should be
incremented and their drawbacks should be minimized. Besides, it is necessary to
compare the different diagnosis methods available, in order to identify the strengths and
weaknesses of each one, to state their common points and differences and to look for
possible ways of integration. Last but not least, it should be kept in mind that, despite of
the interest of rigorous fundamentals, the aim of these techniques is to deal with real
situations; so that, the application of the methodologies to an actual power plant is
crucial.
The objectives of this thesis are:
2
• The development of a complete diagnosis methodology (quantitative
causality analysis); starting from the diagnosis algorithm, but introducing
new features and concepts: effect of measurement errors, influence of the
choice of the diagnosis variables, causality chains…
• Practical application of the methodology in a real-life example. The analysis
has to be made for a time span of several years, in order to be able to
appreciate the effect of the degradation of the components.
• Demonstration that the potential disadvantage of the method (the
linealization of equations) has negligible influence of results.
• Connection of quantitative causality analysis and thermoeconomic analysis:
development of the theoretical basis and application of the working example.

Introduction
• Analysis of the applicability of neural networks and linear regression to the
diagnosis problem. Comparison of the results provided by these methods and
quantitative causality analysis.
To accomplish these objectives, the thesis is developed in 7 chapters plus
introduction and conclusion. The aim of chapter 2 is to review methodologies for the
diagnosis of thermal systems; above all, the fundamentals of thermoeconomic diagnosis
and the diagnosis algorithm. Quantitative Causality Analysis is developed in chapter 3:
formulation, influence of measurement error, analysis of the effect of linearization,
choice of suitable free diagnosis variables and theory of causality chains. Chapter 4 has
two parts. In the first one, quantitative causality analysis and thermoeconomic diagnosis
are connected. In the second, the applicability of linear regression and neural networks
to the diagnosis problem is analyzed.
The thermal system chosen as a test-bench is a 3x350 MW conventional coal-fired
power plant. It is presented in Chapter 5 as an example of the methodology to choose
the suitable free diagnosis variables according to the phenomena to be diagnosed and
information available. The study is focused on the steam cycles, while the analysis of
the boilers has been substantially simplified. Results of the anamnesis (or the repeated
diagnosis), of the three units of the working example during more than 6 years are
presented on Chapter 6. An exhaustive analysis of all free diagnosis variables is made,
demonstrating the applicability and coherence of the approach. Besides, it is proved that
the error provided by the method is negligible in practical applications. In chapter 7, the
theory linking quantitative causality analysis and thermoeconomic diagnosis, developed
in Chapter 4, is successfully applied to the working example. Finally, neural networks
and linear regression are applied in Chapter 8, and results compared with those obtained
with the quantitative causality analysis.

Chapter 1
4

2. State of the art of thermoeconomic
diagnosis.
In this chapter, the main approaches to thermoeconomic diagnosis of thermal
systems are revised. First of all, the main concepts and tools provided by
Thermoeconomics are presented. Then, several proposals for applying these tools are
described. Finally, some diagnosis systems are presented.
2.1 Thermoeconomic approach to diagnosis.
Exergy analysis is based on the application of this magnitude to the study of thermal
systems (Kotas, 1985). By using this approach, the analyst can identify not only the
losses of energy quantity but also the losses in energy quality (irreversibilities).
However, according to the Principle of non-equivalence of the irreversibilities, the
same variation of the irreversibility implies different additional consumption of
resources depending on the component of the plant in which it takes place.
Thermoeconomic analysis goes a step forward than exergy analysis by taking into
account this fact by introducing the concept of cost. In a few words, the cost of a flow
can be defined as the amount of resources needed to produce that flow.
2.1.1 Thermoeconomic model.
Thermoeconomic analysis is based on the representation of a thermal system by
using a thermoeconomic model. This is done by describing the system by means of a
productive structure. The productive structure (also “functional diagram” or “structural
scheme”) is a graph which shows a set of relations defining the interaction among
components themselves and the environment. The nodes of the graph are:
• Control volumes which can be physical devices or groups of them.
5

Chapter 2
• Junctions and branches of flows required to define the interaction between
components. They can be either physical devices appearing in the plant or
fictious elements introduced to build the productive structure (Frangopoulos,
1983).
The flows E of the productive structure are described by exergy (Valero et al.,
1986a), sometimes split into thermal and mechanical components (Tsatsaronis et al.,
1990) or using also negentropy flows (Frangopoulos, 1983; Frangopoulos, 1986; Von
Spakovsky and Evans, 1987; Von Spakovsky and Evans, 1990). The name of this
structure comes from the concept of purpose. Each subsystem has one or more entering
flows that represent the resources or fuel and exiting flows (product). Sometimes, byproducts
or residues can be also considered (Lozano and Valero, 1993a; Torres et al.,
2003). Flows are named by using the notation Eij, and it means that the flow comes from
the component i and goes to the component j. In Figure 2.1, an example of productive
structure is shown.
6
E01
2
E25
1
E15
E50
5
E32
Figure 2.1. Example of a productive structure.
Due to irreversibilities, in every component, product is smaller than fuel. So that, the
amount of resources per unit of exergy increases throughout the plant. This leads us to
the concept of cost.
E53
E54
3
4
E30
E40

State of the art of thermoeconomic diagnosis
2.1.2 Exergetic cost.
The cost of a flow Eij (E * ij) is defined as the amount of external resources in terms of
exergy that are required to produce the flow Eij. The unit exergy cost is the relation
between the cost of a flow and its exergy:
k
E
= 2.1
*
* ij
ij
Eij
Given a thermoeconomic model of a system, the cost of every flow can be calculated
by the cost assessment rules provided by the exergy cost theory (Valero et al., 1986a;
Lozano and Valero, 1993b; Valero, 2006a):
P1: The cost of the external resources is equal to its exergy:
E = E
2.2
*
0i 0i
P2: In each component, the cost is conserved, so that, the cost of the fuel is equal to
the cost of the product.
where:
P = F
2.3
* *
i i
n
* *
i ij
j=
0
P = ∑ E
2.4
n
* *
i ji
j=
0
F = ∑ E
2.5
P3: The exergy cost of the flows produced in a component is proportional to their
exergy, so that they have the same unit exergy cost:
k = k
2.6
* *
ij P, i
The unit exergy consumption is defined as the number of units of exergy that each
component requires from the other components to obtain a unit of its product:
E
ij
κ ij = 2.7
Pj
The sum of all unit exergy consumptions in a component is the inverse of the exergy
efficiency of that component:
k
F
∑ 2.8
n
j = κij
=
i= 0
j
Pj
7

Chapter 2
According to the structural theory (Valero et al., 1992, 1993), the productive
structure can be built considering that each product includes only one flow. So that, the
characteristic equation of the Thermoeconomic model can be written as follows:
8
i i0 ij j
j=
1
n
P = B + ∑ κ ⋅P
2.9
Or, in matrix notation:
P= PS+ KP P 2.10
where Ps is a (n x 1) vector representing the contribution of each component to the
product of the whole system, and KP is a (n x n) matrix whose elements are the unit
exergy consumption κij. The previous expression allows to obtain the product of each
component from the final product of the system.
P = P P 2.11
where
S
( ) 1 −
D
P ≡ U − KP 2.12
The total resources of the plant may be obtained by:
F = κ P P 2.13
t
T e s
t
where κ ( κ ,..., κ )
≡ is a (n x 1) vector whose elements contain the unit
e 01 0n
consumption of external resources. Finally, the unit cost of the flows can be obtained by
using the following equation:
* t
P = e
k P κ 2.14
2.1.3 The fuel impact formula
This equation was suggested by Valero et al. (1990, 1999) and developed by Reini
(1994), Lozano et al. (1994) and Torres et al. (1999). It is very important for
Thermoeconomic diagnosis because it relates the variation of fuel consumption of a
system with the variation of unit exergy consumptions of the components of that system
and to the variation of production:
t t * t *
Δ F = Δ κ + k ( x) Δ KP P x + k ΔP
2.15
( ) ( 0 )
T e P P s
In scalar format:
n ⎛ n
⎞
Δ FT = kP j x Δ jiPi x + kP i x Δ i
i= 1⎝ j=
0
⎠
* *
∑∑ ⎜ , ( ) κ ( 0 ) , ( ) ϖ ⎟
2.16

State of the art of thermoeconomic diagnosis
where ωi is the part of the product of the plant coming from the component i. When
the unit exergy consumption of a component increases Δ κ ji , the irreversibility in this
component also increases in a quantity which is called malfunction (Torres et al., 1999;
Lerch et al., 1999; Torres et al., 2002; Valero et al., 2002).
ji ji i
( )
MF =Δ κ P x
2.17
( )
i i i 0
ji
j=
0
0
n
MF =Δ k P x =∑ MF
2.18
This malfunction implies an additional amount of fuel called the malfunction cost.
( )
MF = k x MF
2.19
* *
ji P, j ji
n
* *
i ji
j=
0
MF = ∑ MF
2.20
If MF0 is defined as:
n
* *
0 ,
i=
1
∑ P i( ) i
2.21
MF = k x Δϖ
it can be proved that:
n
*
T i
i=
0
Δ F =∑ MF
2.22
There are two types of malfunctions. When the unit exergy consumption of a
component increases due to a degradation of this component, it is called intrinsic
malfuncion. When an intrinsic malfunction occurs, the operation point of the other
components varies. So that, since the efficiency curves of the components are usually
non-flat, variations in the specific consumption of other components appear, which
leads to induced malfunctions. The irreversibility of a component can also vary due to a
change in its product. This is called dysfunction.
( ( ) 1)
DF = k x − Δ P
2.23
i i i
The problem of induced malfunction is the main drawback of using the
Thermoeconomic approach to diagnose energy systems, because it makes very difficult
to identify the real components which do not work properly. To overcome this problem,
several researchers have proposed techniques, most of them based in the filtering of
these induced effects. These procedures, among others, are revised in the section 2.2.
9

Chapter 2
More details on thermoeconomic analysis can be seen in Valero and Torres (2006a).
For further information on thermoeconomic diagnosis see Valero and Torres (2006b)
2.1.4. Cost formation process of wastes (or residues).
In previous sections, only productive flows have been considered. However, there
are other flows (such as gases leaving the stack or heat dissipated in a condenser) which
are necessary for the operation of the thermal system but do not have a productive
purpose. These flows are called wastes or residues (Torres et al., 2006; Rangel, 2005).
The cost of the wastes has to be charged to the component where they have been
produced; for example, the cost of gases leaving the stack should be charged to the
combustion process, and the cost of heat dissipated in a condenser is usually shared
according to the entropy produced in the different components.
Consequently it is necessary to enlarge the productive structure to include these
issues. A waste flow produced in a component i and charged to the component j is
represented as Rij. So that, it is possible to define unit exergy consumptions associated
with the wastes:
10
R
ij
θ ij = 2.24
Pj
And to calculate cost associated with residues:
k
R
= 2.25
*
* ij
Rij
Rij
The variation of these unit exergy consumptions, entails malfunctions associated
with the wastes:
ji ji i
( )
MR =Δ θ P x
2.26
0
( )
i i i 0
ji
j=
0
n
MR =Δ θ P x =∑ MR
2.27
These malfunctions entail their corresponding malfunction cost:
( )
MR = k x MR
2.28
* *
ji PR, j ji
n
* *
i ji
j=
0
MR = ∑ MR
2.29
These malfunction costs associated with the residues have to be considered in the
expression of fuel impact as a summation of malfunction costs:

n
*
n
*
T i i
i= 0 i=
1
Δ F = MF + MR
State of the art of thermoeconomic diagnosis
∑ ∑ 2.30
Finally, the fuel impact formula has to be complemented with the variation of unit
exergy consumption associated with the residues:
n ⎛ n
* * * ⎞
Δ FT = ∑∑ ⎜ ( kP, j( x) Δ κ jiPi( x0 ) + kPR, j( x) Δ θ jiPi( x0 ) ) + kP, i( x)
Δϖi⎟
2.31
i= 1⎝ j=
0
⎠
2.1.5. Other issues.
In the previous sections, a brief review of Thermoeconomics has been presented in
order to explain the basic concepts and the fuel impact formula. However, there are
other interesting issues, perhaps not so related straightforward by thermoeconomic
diagnosis but worth to be presented. This is the aim of this section.
The definition of cost as a quotient between fuel entering the plant and a given flow
introduced previously corresponds to an average cost ( *
k ) and it is suitable for cost
accounting methods. For optimization methods, what it is interesting is to know the
impact of fuel entailed by an additional unit of a given flow; in other words, a marginal
* cost, k (Serra et al., 1995):
k
E
= 2.32
E
* 0
i
* ⎛∂E⎞ 0 k = ⎜ ⎟
2.33
⎝ ∂Ei
⎠conditions
In an attempt to unify average and marginal cost and, subsequently,
thermoeconomic theories based on cost accounting and aimed at system optimization,
the structural theory was developed. This theory is based on four premises:
1.- Every energy system can be structurally described as a number of physical or
logical components, n, connected to one another and to their environment though a set
of m relations (mass, energy, money, information).
2.- Every relation among component must be defined completely using an extensive
magnitude which reckons, at the same time, its quantity and quality:
Ei = qi⋅ ei
or Gj = qj⋅ g j
2.34
3.- Each component, l, needs to have as many characteristic equations as relations
which affect (enter) it. Characteristic equation is defined as that which relates the
magnitude of a flow entering the component, Ei, to the flows leaving it, G1, G2,…, Gj,
11

Chapter 2
and a set of internal variables or endogenous variables: {χl} which describe the
behaviour of the component, l:
E = Eˆ G , G ,..., G , χ
2.35
12
( 1 2 )
i i j l
4.- The characteristic equation is a homogeneous function of first order in each
production interval, concerning the subset {Gl} of independent variables:
( 1, 2,..., , ) ( 1, 2,...,
, )
λE = Eˆ λG λG λG χ = λEˆ G G G χ
2.36
i i j l i j l
Under the previous hypotheses, average and marginal cost coincide. Thanks to this
result, structural theory can serve as a common formalism to unify different
thermoeconomic theories previously presented. For more details see: Valero et al.,
1992; Valero et al., 1993; Serra et al., 1995; Erlach et al., 1998; Serra and Torres, 2006.
An interesting point of structural theory is that it is general, so that it is possible to
use other thermodynamic functions different from exergy to characterize flows and
different sets of parameters to characterize the behaviour of the component. Royo and
Valero (1995) propose to use internal parameters as universal indicators to characterize
simple thermal processes:
θ
eg
h − h
=
s − s
e g
e g
Where h and s are enthalpy and entropy and e and g are the initial and the final
points of the process. The use of these parameters allows to simplify characteristics
equations of thermal systems (Valero et al., 1995).
Although the use of exergy is widely accepted, it has several limitations (Valero,
2006b), which have lead to the proposal of other magnitudes such as processable exergy
(Valero and Guallar, 1991) or relative free energy (Valero and Royo, 1992).
Thermoeconomic analysis provides numerical results very valuable for the analysis,
synthesis, diagnosis and optimization of complex energy systems. However, if the
concepts of exergy cost theory are combined with symbolic manipulators, its
capabilities increase. This line of research is called symbolic exergoeconomics, and
allows to obtain explicit formulae relating the efficiency of the whole system with
efficiencies of components forming it (Valero and Torres, 1988; Valero and Torres,
1990; Torres, 2006). The application of symbolic exergoeconomics for thermal system
simulation can be seen in Torres et al. (1989) and an analysis of how this theory can
contribute to the unification of different thermoeconomic theories is made in Valero et
al. (1989). Valero et al. (1991) describe a program for symbolic computation of
2.37

State of the art of thermoeconomic diagnosis
exergoeconomic variables. A detailed description of the requirements for the
development of software for thermoeconomic analysis can be seen in Torres et al.
(2007).
Although Thermoeconomics has an academic origin, its final aim is to serve as
useful tool for improving efficiency in both design and operation stages. So that,
practical applications have been developed in parallel with theoretical research. The
concept of exergy has been applied to analyze steam power plants: simulation
(Alconchel, 1988; Alconchel et al., 1989) and monitoring (Valero et al., 1986b).
Thermoeconomic analysis has been applied to a steam boiler (Lozano and Valero,
1987) cogeneration plants (Valero et al., 1987; Muñoz and Valero, 1989), a sugar
factory (Guallar and Valero, 1988) an air conditioning system with cogeneration (Tozer
et al., 1996) fuel cell systems (Álvarez et al., 2006) a pressurized fluidised bed
combustion power plant (Schwarz et al., 1997) and analysis and optimization of dualpurpose
power and desalination plants (Uche et al., 2001; Uche et al., 2006).
Plant monitoring has also been tackled by Thermoeconomics (Valero et al., 1996;
Valero et al., 1999; Correas et al., 1999).
Finally, it should be noted that the field of application of Thermoeconomics is not
limited to artificial energy systems but can be a valuable tool to analyze and quantify
natural resources (Valero, 1995). Furthermore, it has been proposed to connect
Thermoconomics with the Aristotelian concept of cause (Valero and Carreras, 1990).
2.2 Thermoeconomic diagnosis methodologies.
To deal with the problem of induced malfunctions, several methods have been
proposed. Some of them are directly based on the ideas proposed above, combined with
the use of several types of filtration techniques. Others, however, use different
approaches.
To compare these methodologies, the TADEUS (Thermoeconomic Approach to the
Diagnosis of Energy Utility Systems) initiative was launched. In the first paper of this
series (Valero et al., 2004a), the problem of thermoeconomic diagnosis is presented and
an example problem is proposed in order to encourage researchers interested in the topic
to try their methodologies and then compare results. This example is a combined cycle
power plant made up of two 125 MW gas turbines, two heat recovery steam generators
13

Chapter 2
and a 110 MW steam turbine, where several malfunctions have been simulated. In a
second paper (Valero et al., 2004b), concepts of productive structure, cost, fuel impact
formula and malfunctions and dysfunctions are revised. The third paper (Verda et al.,
2003) focuses on the characteristics that a diagnosis system installed in a power plant
should have and on the concept of reference condition. In other papers, several authors
explain their methodologies and test them with the example proposed: (Toffolo and
Lazaretto, 2004; Zaleta et al., 2004a; Reini and Taccani, 2004; Verda, 2004, and
Correas, 2004). Valero et al. (2004c) analyzes the importance of the fuel impact formula
and describes the working example. Lazzaretto et al. (2006) compare results provided
by several approaches.
2.2.1 The filtration of the effects induced by the control system.
When a malfunction occurs in a component of an energy system, the operation
condition tends to move to a point that does not accomplish the set-points fixed by the
operators (temperatures, power produced…). So that the control system intervenes in
order to restore the values of these set-points. For example, the set points of a gas
turbine are the power produced and the temperature of the gas entering the turbine. To
maintain these two values, the control system modifies the angle of the guide vanes and
the amount of fuel introduced in the combustion chamber. If the isoentropic efficiency
of the turbine decreases (e. g. due to blade erosion), the power decreases. In this point,
the control system intervenes in order to increase the fuel entering the combustion
chamber. Due to the increment in fuel, temperature of gas entering the turbine increases,
so the inlet guide vanes open in order to increase the air mass. As a result, the
intervention of the control system allows to keep the set-points but modifies the
operation points of all the parts of the plant, not only the ones where an anomaly takes
place. In conclusion, the control system intervention produces induced anomalies that
makes more difficult the identification of the actual malfunctioning component.
The filtration of effects induced by the control system is the origin of a
thermoeconomic diagnosis methodology developed by Verda (Verda et al., 2005a;
Verda et al., 2005b; Verda 2004, Verda et. al. 2004, Verda et. al. 2002a, Verda et. al.
2002b). This author defines the free condition of a malfunctioning thermal system as the
state of that system characterized by the same position of the governing parameters as
that for the reference condition, but containing the anomalies occurring at the actual
operating condition. In the example of the gas turbine, the free condition would have the
14

State of the art of thermoeconomic diagnosis
same isoentropic efficiency as the malfunctioning condition but the same amount of fuel
and guide vanes angle as the reference condition.
The free condition is a fictitious state of the system, so that it can not be determined
by any measurement. It can only be calculated mathematically, for example, with a
simulator. If the anomalies are low enough, the operation condition is close to the
reference condition. So that, the values of the exergy flows in the free condition can be
obtained by using a linear approximation:
( )
∂E
E E ∑ x x
2.38
i =
free i +
op
r
i
j= 1 ∂x
j
j −
ref jop
where xj is one of the r governing parameters of the system. The partial derivatives can
be obtained by using r different particular working conditions. The idea of control
system intervention leads us to two new concepts: Malfunction induced by the control
system and cost of the control system (Verda et. al. 2004). Considering the effect of the
control system, the fuel impact can be divided into three kinds of malfunctions:
1. The intrinsic malfunction, which corresponds to the component where the
anomaly takes place. Assuming that Δκgh is the intrinsic effect in the h th
component:
( κ κ )
MF = − ⋅ P
2.39
int ghfree ghref href
2. The malfunction induced by variations in the component productions is
expressed by:
( κ κ ) free ref ref int
n n
⎛ ⎞
MFind = ∑∑ ⎜ ij − ij ⎟⋅Pj−MF
2.40
j= 1⎝ i=
0
⎠
3. Malfunction induced by the control system intervention can be obtained as
follows:
( κ κ )
n n
⎛ ⎞
MFr = ∑∑ ⎜ ij −
op ij P
free ⎟⋅
j
2.41
ref
j= 1⎝ i=
0
⎠
The cost of control system intervention is defined as:
k
F − F
* Top Tfree
r =
Pext − P
op ext free
2.42
where FT is the total fuel entering the system and Pext is the product of the plant.
This parameter is an indicator of the influence of the control system intervention in the
15

Chapter 2
efficiency of the plant. If this cost is greater than the cost of the product, it means that
the control system intervention induces a reduction in the efficiency of the system.
The comparison of free condition and reference condition has been used to diagnose
a gas-turbine based cogeneration system (Verda et. al. 2004). This approach has been
suitable to detect single anomalies, above all compared to the direct comparison of
operation condition and reference condition. However, when several anomalies appear,
the filtration of the influence of the control system is not enough.
Also in the free condition, the operation point of the components of the system is not
the same as in the reference condition. So that, due to the fact that efficiency curves are,
in general, non flat, induced malfunctions appear. To overcome this problem, the
behaviour of the components depending on the product should be introduced. One
possibility is to consider the variation of unit exergy consumption (Verda et. al. 2002a,
2002b):
16
n ⎛∂κ⎞ ji
ˆ κ ji = κ ji + E
ref ∑ ⎜ ⋅Δ
⎜ jl ⎟
l= 0 ∂E
⎟
⎝ jl ⎠
2.43
where the derivatives can be obtained by using a set of operation conditions, and
ΔEjl is calculated by comparing free and reference condition. The value ˆ k ji takes into
account the variation of unit exergy consumption due to the variation of operation point,
which should be taken into account in order to identify the actual malfunctioning
component.
( )
free ref
Δ κ = κ −κ −Δ ˆ κ
2.44
jiint ji ji ji
Another possibility to filtrate this effect is to consider the dependence of the product
on each resource (Verda 2004):
n
ˆ
⎛ ∂E
⎞
i
Ei = Ei + E
ref ∑ ⎜ ⋅Δ li⎟´
2.45
l= 0 ⎝∂Eli ⎠
and then to calculate ˆ k ji
Eji E
ˆ
free jiref
Δ k ji = − 2.46
Eˆ
P
i
iref
The filtration of induced effects due to control system intervention and to non-flat
efficiency curves allow to highlight intrinsic anomalies that could not be detected by the
direct comparison of operation and reference conditions. This approach is applied by to
diagnose a combined cycle (Verda, 2004; Verda et al., 2005a, 2004b and 2005b). In the

State of the art of thermoeconomic diagnosis
first two papers, a zooming strategy is proposed in order to first locate the macrocomponent
where the anomaly occurs and then to identify the specific malfunctioning
component inside the macro-component. In the two last papers, the effects of splitting
the exergy into its thermal and mechanical components in order to diagnose a
malfunction in a heat recovery steam generator are studied.
Although the ideas presented above can be used to identify the malfunctioning
components, sometimes they are not enough, mainly when there are several anomalies.
The procedure can be improved by using anamnesis, which is the analysis of the history
of the system (Verda, 2004). This technique is based on the diagnosis of the plant
several times: malfunctions corresponding to the components where anomalies occur
are expected to increase while induced malfunctions are expected to have a random
evolution. To overcome the limitations of a linear model, the use of neural-networks
based thermoeconomic model has also been proposed. (Verda 2004).
Once the malfunctioning components have been identified, the fuel impact of each
anomaly should be determined. Since this figure corresponds to the expected fuel
savings if the anomaly is removed, this step can be named as prognosis (Verda, 2004).
This information is very important to decide whether it is worth to repair the component
or not. In theory, this could be determined by appliying the fuel impact formula
previously presented. However, due to induced effects, this is not so easy. If this
formula is rearranged, a corrected fuel impact can be defined.
n
*
n
*
Tcorr T
i= 1
piei i=
1
i
∑ ∑ 2.47
Δ F =ΔF − k ⋅Δ P = MF
Some of the malfunctions are intrinsic while the other are induced. If there are r
intrinsic malfunctions, the impact of induced malfunctions should be split among the
intrinsic ones.
r
*
∑ int ( 1 )
2.48
Δ F = MF ⋅ + i
Tcorr l l
l=
1
The coefficients il can be adjusted by using the information available for the
prognosis. In (Verda 2004) the complete approach including filtering of control system
effects, filtering of non-flat efficiency curves effects and prognosis is applied to the
Tadeus problem. Prognosis results are not as good as expected due to non-linearities.
17

Chapter 2
2.2.2 Other methods based on the filtration of induced effects
Since the filtration of the intervention of the control system is usually not enough to
detect the malfunctioning components, some authors propose methods based in the
filtration of induced effects in only one step.
Reini and Taccani (2004) propose a method based in the calculation of the cost of
malfunction induced by the fuel variation. First, they calculate the variation of specific
consumption due to the variation of the product of the component:
18
∂kij
∂ Pk
ij =
2.49
∂E
i
This derivative can be obtained by using historical data. Then it is used to calculate
the cost of malfunction induced by the variation of product:
*
*
( MF ) ∑ k ∂
i
P
= j
j
P
k
ij
E
i
2.50
Finally, intrinsic malfunction can be detected by analyzing the values of the cost of
malfunction (
*
MF i ) and the cost of malfunction induced by product variation ( ( MF i ) P
*
and using the following rules:
a) If
b) If
malfunction.
c) If
*
MF i is negative, it is probably an effect of induced malfunction.
*
MFi is very close to/smaller than ( MF i ) P
*
, it is the effect of induced
*
MF i related to a junction is not equal to zero, it is the effect of induced
malfunction because inside junctions control volumes there are no exergy losses.
This methodology is applied to the TADEUS problem. It is able to highlight the
malfunctions separately, but when there is more than one at the same time it is not so
useful.
Toffolo and Lazzaretto (2004) analyse the use of several indicators to identify the
component where the anomaly takes place. They consider that the use of exergetic or
thermoeconomic indicators is not suitable to identify malfunctions, because anomalies
can be detected due a variation of specific curve of the components and not due a
variation of specific consumption. They propose to isolate the component and to
calculate an indicator based on the irreversibility variation but filtering the effect of
thermodynamic variables.
h ∂I
n
h
I res,
h I h
τ h,
i
i 1 τ h
Δ ⋅
= Δ − ∑
= ∂
2.51
)

State of the art of thermoeconomic diagnosis
where Ires,h is the residual irreversibility of the h th component that is due to intrinsic
malfunctions, ΔIh is the irreversibility variation and τh is the set of nh thermodynamic
variables that influence the behaviour of component h. The indicator proposed is the
non-dimensional value of Ires,h that is obtained by dividing it into the absolute value of
ΔIh. This approach has been applied to the TADEUS problem; it is able to highlight
malfunctioning components but it not provided the fuel impact originated by each one
of them. These authors analyze several thermoeconomic diagnosis methodologies and
conclude that the true independent causes of malfunctions are the variation of
parameters of the thermodynamic model, and this information has to be included in the
thermoeconomic diagnosis (Lazzaretto and Toffolo, 2006).
Valero et al. (1999) propose to use a simulator of the thermal system, in order to
determine the effect of the variation of an operating parameter xr on a unit exergy
consumption:
0 0
( x ) ( )
Δ κ = κ x +Δ −κ
x 2.52
r
ij ij r ij
The application of the previous equation allows to decompose malfunctions in two
terms, intrinsic and induced, depending on whether the operating parameters are
associated to the studied component or not. It should be noted that these operating
parameters are not only associated with the analyzed component, but parameters
characterizing the behavior of all the components of the plant. The approach has been
successfully applied to a steam cycle of a power plant (Valero et al., 1999). Zhang et al.
(2007) apply a similar approach.
2.2.3 Representation of the malfunctions in the h-s plane
It has been shown that the direct use of classical Thermoeconomic analysis and the
fuel impact formula is not always useful due to induced effects. For this reason, several
authors have proposed diagnosis methodologies based on the filtration of these effects.
Other authors consider that Thermoeconomic analysis is a good option for cost
calculation but the diagnosis problem deals with the analysis of deviations around the
reference operation point, so that the evolution of thermodynamic properties when a
malfunction occurs should be the starting point of a diagnosis methodology. In this
sense, several methodologies based on the trajectory in the h-s plane of the points when
a malfunction occurs have been proposed.
19

Chapter 2
Zaleta et al. (2004b) proposes a thermo-characterization of power system
components based on the representation in the ω, σ, MFR space. ω is defined as the
enthalpy of the entering flow minus the enthalpy of the exiting flow, while σ is the
entropy difference. The mass flow ratio (MFR) is the mass flow divided into the
maximum mass flow. In this 3D space, the Reference Performance State (RPS) can be
represented (Figure 2.2). RPS can be defined as “The range of operation of both
intensive and extensive thermodynamic conditions (at full or partial load) guaranteed by
the manufacturer, when neither intrinsic nor induced malfunctions arise in the
component”.
20
MFR
ω
RPS
Figure 2.2. RPS representation in the ω, σ, MFR space.
Once the reference behaviour of a component has been represented by using RPS,
malfunctions can be detected by comparing the actual values of ω and σ with the
reference values for the same MFR. Examples of RPS and the impact of several types of
malfunctions in ω and σ can be seen in Zaleta et. al. (2004b) for several components of
a cycle and in Zaleta (1997) for several types of steam turbines.
Once the deviations in ω and σ have been determined (the malfunctions have been
detected), Zaleta et. al. (2004b) provide expressions to relate Δω and Δσ to the variation
they produce in the heat rate of a steam cycle.
An important parameter related to the analysis of malfunctions is the dissipation
temperature (Royo et al., 1997). It is defined as the quotient between the increment in
enthalpy and the increment in entropy of the exiting flow of a component when a
differential malfunction occurs:
σ

T
d j
⎛∂h⎞ out j
= ⎜ ⎟
⎜∂s⎟ ⎝ out j ⎠ Malfunction
State of the art of thermoeconomic diagnosis
This derivative is the tangent of the trajectory described by the state of the exiting
point of a malfunctioning component when a malfunction occurs. For example, in
Figure 2.3, an intrinsic malfunction in a turbine has been represented in the h-s plane,
where the state of the exiting flow moves from 2 to 2’. The value of Td is determined by
the behaviour of the system interacting with the malfunctioning component. So that, it
can have two main origins: i) environmental condition, such as ambient pressure in the
discharge of a gas turbine and ii) control system, such as temperature of the gas leaving
a combustion chamber.
Figure 2.3: Dissipation temperature.
By using the concept of dissipation temperature, information about the behaviour of
a thermal system when a malfunction occurs can be introduced in the analysis, which
allow to develop more accurate diagnosis methodologies. Zaleta et. al. (2003) have
related the variation of the specific exergy consumption of a turbine in a steam cycle to
the variation of the specific exergy consumption that it induced in the whole cycle. The
following equation shows this relation for a low pressure turbine:
⎛ ⎞
⎛P⎞ ⎜
1
⎟
dK ≅ ⎜ ⎟⎜ ⎟dk
⎜ ⎟
bT ,
LP
⎝ PT
⎠ T0
⎜
1−kLP
−
TdLP
,
⎟
LP
h
⎝ ⎠
1
2
2' atn Td
2.53
2.54
where Kb,T is the unit exergy consumption of the cycle, PLP and PT are the power
produced by the low pressure turbine and the whole cycle, kLP is the unit exergy
s
21

Chapter 2
consumption of the low pressure turbine, T0 is the reference temperature and Td,LP is the
dissipation temperature of the low turbine pressure.
These expressions have been applied to the diagnosis of a conventional steam cycle
power plant of 158 MW (Zaleta et al. 2003). When they are compared with a
simulation, they provided very good results. In the same paper, the fuel impact formula
and a diagnosis system proposed by ASME (1996) have also been tested. Their results
are not so good.
The methodology proposed by ASME is based on the work developed by Cotton
and Wescott (1960) and related the variation in isoentropic efficiency of a turbine to the
impact in heat rate that it produces. For example, the heat rate impact produced by an
anomaly in the low pressure steam turbine can be calculated as follows:
22
⎛ UEhp ⋅ m
hp ⎞
Δ HR%
=Δηlp % ⎜1− ⎟
2.55
⎝ W
Gen ⎠
where ΔHR% and Δη% are the percentage variation of heat rate and isoentropic
efficiency, UE is the used energy (enthalpy drop), m is the mass flow and W is the
power produced by the cycle. Expressions for high and medium pressure steam turbines
are more complex. This comparison should be made at constant throttle flow.
2.2.4 Methods directly based on the thermodynamic representation
of the system
As it have been seen, methods based on exergetic and thermoeconomic indicators
are general but often need the use of not only some kind of filtration but also the
intervention of an analyst that may introduce subjectivity. Other methods like the ones
proposed in the previous section provide quite good results but can only be used to
diagnose a restricted type of components (e.g. turbines). In this context, the
development of diagnosis systems that can be applied to real-life power plants need
other methods that should be general (capable of diagnose all types of malfunctions)
and reliable (the solution should not be probabilistic and should not need the
intervention of an analyst). To achieve this, some authors consider that thermoeconomic
tools are not the only solution and prefer to develop diagnosis techniques directly based
in the thermodynamic description of the system.
Zaleta et al. (2004a) propose a reconciliation in heat rate and power methodology
based in the use of a simulator. Variables needed to describe thermodynamically the
model can be classified in either dependent or independent. Dependent variables are

State of the art of thermoeconomic diagnosis
external (ambient conditions and fuel quality) or internal (indicators of equipment
efficiency and set points of the control system). A simulator is able to obtain all the
variables (the complete description of the system) only from the value of the
independent ones. The procedure consists in calculating the test operation condition of
the plant and moving towards the reference operation condition by modifying the value
of one independent variable in each step until all independent variables have the value
corresponding to the reference condition. The increment in heat rate and power obtained
in each step correspond to the impact due to the variable modified in this step. Authors
remark that the method is not based in any linearization. On the other hand, a simulator
is needed and results might vary depending on the order of variable modification. The
method has been successfully applied to the TADEUS problem and it is the core of the
diagnosis systems installed in more than eight combined cycle power plants in Mexico.
Remiro and Lozano (2007) consider that fuel impact formula and the use of a
simulator are complementary methods and propose to use both of them.
Correas (2001, 2004) proposes a diagnosis algorithm also aimed at sharing the
variation in efficiency among the set of thermodynamic independent variables (“free
diagnosis variables”). The main advantage of this method is that it does not need a
tuned simulator. It has been applied to solve the TADEUS problem (Correas, 2004) and
is used in a diagnosis system at Elcogas IGCC power plant in Puertollano, Spain
(García-Peña et al., 2000; García-Peña et al., 2001, and Correas 2001). This
methodology is applied and improved in this thesis, so that it is explained in detail in the
next section.
2.2.5. Concept of the diagnosis algorithm and mathemathical
formulation.
The aim of this section is to describe the diagnosis algorithm proposed and applied
by Correas (2001, 2004), which is the origin of the quantitative causality analysis
developed in this thesis.
The algorithm is based on a thermodynamic description of the energy system. The
plant is represented as a graph, whose nodes are components where processes take
place. The edges linking these nodes are material or energy streams. Every stream is
characterized by using an extensive property (flow rate or power) and several intensive
properties such as pressure, temperature or quality. The components are characterized
by using parameters like efficiencies, pressure losses and so on.
23

Chapter 2
The value of these variables has been previously determined by another calculation
method different from the diagnosis for the two states to be compared by the diagnosis
methodology. Usually, one of these states is the reference one. This previous calculation
method can be a simulator or a “performance test code” that solves the mass and energy
balances from field instrumentation.
All the variables (properties of streams and parameters of components) can be
classified as dependent and independent. Usually, the independent variables are
environmental conditions, fuel quality, set points and component parameters. There is
also an indicator of the global efficiency of the whole system. The aim of the algorithm
is to determine the influence of the variations of the independent variables in the
variation of the global efficiency indicator.
Let’s consider a plant that is represented by a set of variables x. It is also a global
efficiency indicator that, obviously, depends of this set. The variation of this indicator
between two plant states 0 and 1 is expressed by:
24
1 0 ( ) ( )
Δ f = f x − f x 2.56
The aim of the diagnosis algorithm is to share this efficiency variation in a
summation of impacts, each one due to the variation of an independent variable:
n n
indep indep
0 1 * 0 1
∑ ( , , ) ∑ cos , ( , )
Δf≅ I Δ x x x = k x x ⋅Δx
2.57
f i t i i
i= 1 i=
1
So that, the purpose of the algorithm is to obtain the set of impact factors
k that
relate the variation of the indicator to the variation to the independent variables. This
task is not direct because the indicator depends not only on the values of the
independent variables but also on the dependent ones:
( ) = ( dep, indep )
*
cos t,
i
f x f x x 2.58
The variation of the indicator can be related to the variation of x by using a Taylor
series:
n
n
n
var
indep
dep
∂f
∂f
∂f
Δf ≅ ∑ ⋅ Δxi
= ∑ ⋅ Δxindep,
i + ∑
i= 1 ∂xi
i= 1 ∂xindep,
i
i= 1 ∂xdep,
i
⋅ Δx
*
This relation can be expressed in matrix notation by defining k f i = ∂f
/ ∂xi
:
* T * T ⎡Δxindep ⎤
Δf≅kf ⋅Δ x = kf
⋅⎢
Δ
⎥
⎣ xdep
⎦
dep,
i
,
2.59
2.60

State of the art of thermoeconomic diagnosis
At this point, the variation of f has been related to the variation of all the variables of
the system. To relate f to the independent variables only, the ndep restrictions (g(x) = 0)
of the problem that relate dependent and independent variables should be introduced. If
the function g is expanded in a Taylor series and terms of higher order are neglected, it
can be written:
1 0 ( ) ( ) ( ) 0
g x ≅ g x + J x ⋅Δx
2.61
x
Both vectors x 1 and x 0 satisfy the set of constraints because they are solutions of
g(x) = 0. So that, the previous equation can be simplified to:
J( x) 0 ⋅Δx ≅0
2.62
x
The size of the previous equation system is (ndep x nvar), so that it must be completed
to be solved:
( ) 0 xindep
⎡J x
⎢
⎢⎣ U D
⎤ ⎡Δ x
⎥⋅⎢ 0 ⎥ ⎣ Δxdep ⎦
⎤ ⎡ 0 ⎤
⎥ ≅ ⎢
Δ
⎥
⎦ ⎣ xindep
⎦
2.63
If the matrix is inverted, the relationship between dependent and independent
variables is obtained:
⎡Δxindep ⎤ ⎡ 0
⎢ ⎥ ≅ ⎢
⎣ Δxdep⎦ ⎣M U ⎤ ⎡ 0 ⎤
⎥⋅⎢
Δ
⎥
N⎦
⎣ xindep
⎦
2.64
Δxdep ≅ N⋅Δx indep
2.65
Once this relation is known, the relation of variation of the indicator to variation of
independent variables is determined.
The method presented above is approximate because only first order terms in the
series expansion have been considered. It would be exact if both the variation in the
indicator as well as the constraints were linear. The error induced by linearization is
studied in this thesis.
2.2.6. Other examples.
Rusinowski et al. (2007) have developed a diagnosis method of a boiler based on a
neural network which links temperature of gases leaving the boiler with seven
independent variables.
A method based on neural networks has also been used by Gluch et al. (1998). First
of all, symptoms of failure (difference between measured and reference parameters) are
detected. Afterwards, a two step approach is applied: first of all, a module for searching
25

Chapter 2
defected apparatuses; second, a set of modules to diagnose the causes of degradation of
individual apparatuses.
Sciubba (2004) proposes a hybrid system based on both qualitative and quantitative
approaches. It combines reconciled data and physical modelling to extract a limited
number of numerical coefficients that introduce a sufficient degree of quantification.
The potentialities of a qualitative method for the analysis of system operating
condition by means of an expert system with fuzzy rules have been explored by Toffolo
and Lazzaretto (2007). The knowledge base of the expert system is obtained by a given
list of malfunctioning conditions. A multi-objective evolutionary algorithm is used in
order to accomplish two conflicting objectives: the number of correct predictions (to be
maximized) and the complexity of the set of rules (to be minimized). Two approaches
are considered: global approach for the overall system and local approach by splitting
the system into subsystems.
El-Sayed (2007) proposes to analyze the fingerprint of malfunctions. The idea is to
use a simulator to obtain fingerprint tables which show the variation of thermal system
parameters when a malfunction in a component appears; the comparison of the
operating situation with the different fingerprints allows to determine the actual
malfunctioning component.
26
2.3. State of the art of systems for diagnosis and other related
issues.
After the revision of the methodologies for thermoeconomic diagnosis, the aim of
this section is to present the systems developed to perform this task. Obviously, a
diagnosis system has to be based on a diagnosis method, so that, this section is strongly
linked to the previous one; in other words, the separation is mainly due to the
organization of the text and should not be considered as a strict rule.
Diagnosis systems are seldom offered as independent items, but they are usually
part of packages covering other issues related to plant management which use
information available: performance monitoring, mechanical condition monitoring, online
optimization (set-points, sootblowing, compressor washing in gas turbines…). In
this framework, the aim of this section is to review the diagnosis systems and methods
commercially available and to show how the thermoeconomic diagnosis task is a part of

State of the art of thermoeconomic diagnosis
the total plant management. In order to organize the revision, information is classified
into several sub-sections: diagnosis systems in general, diagnosis centres and
collaboration, diagnosis of boilers, and other issues. Again, this classification should not
be seen as a strict rule.
2.3.1. Diagnosis systems in general.
Plant equipment manufacturers usually develop the plant control system and are the
people who best known the plant characteristics. In this situation, they have developed
monitoring and diagnosis systems which are sold in the same package of the plant or,
sometimes, can be added afterwards. It is usual that these systems have a modular
structure based on a core system which can be optionally complemented with several
packages with different aims.
GE Energy (2005a) has developed the System 1® Optimization and Diagnostic
Platform in order to organize and distribute plant information. This system can include
the ‘Machine Performance. Condition Monitoring and Diagnosis’ package (GE Energy,
2005b) which can monitor thermodynamic monitoring of gas and steam turbines and
allow the user to combine this information with mechanical condition monitoring, in
order to take decisions on maintenance. Modules for modelling and detection of
inconsistent measurements are also included.
Alstom have developed the ALSPA P320 flexible automation system. It can include
an integrated plant management system OPTIPLANT (Alstom, 2007).
ABB provides the Optimax suite, which includes several packages for plant
monitoring and optimization: maintenance and lifecycle management, optimization of
power and desalination plants or optimization of compressor washing (ABB, 2005a-d;
Morton, 2002). According this company, plant operation optimization needs: process
models to determine fuel consumption, stress models to compute lifetime cost,
operation constraints and an optimisation routine (Weisenstein et al., 2002). Finally,
Antoine (2005) describes how parameters indicative of gas turbine degradation can be
estimated from measurements, applies the idea of probability of failure and presents a
compressor washing optimiser.
Siemens offers the SPPA-P3000 system process optimization which includes
packages for optimizing different items such as: boiler start-up, optimum set-points,
sootblowing, combustion… (Siemens, 2006a-h).
27

Chapter 2
Besides these proposals of plant manufacturers, other companies have developed
programs aimed at monitoring and diagnose power plants. SOLCEP is a software
implemented in power plants of Endesa (Pablo Moreno S.A., 2006). This software is
based on a modular structure easy to configure by the plant staff. It not only calculates
the plant heat rate, but also compares it with the expected one, sharing this deviation
first into sub-systems (boiler, steam cycle and ancillary devices) and then into main
components. It has other features such as data validation, historian retrieval and
distribution of information among users according to different levels.
An on-line supervision system of energetic efficiency has been installed in
Compostilla power station, a coal-fired power plant owned by Endesa (Gómez-Yagüe et
al., 1999). This system is able to calculate global and individual efficiencies and
evaluate their deviations with respect to current plant objectives and to dynamic
references established by the operator.
DADIC is a system to diagnose combined cycle power plants (Arranz et al., 2007).
The diagnosis is performed by using a multi-agent system: the detection is based on
agents that use neural-networks based models, and the diagnosis is performed by agents
based on an expert-system structure.
Kim and Joo (2005) describe a diagnosis system based on EfficiencyMap from GE
Energy applied in two Korean combined cycles. Corrected gas turbine power is
calculated by a simulator, and the comparison of this parameter with the actual power is
an indicator of turbine degradation. Compressor wash advisor and a module to
optimally allocate load among the different turbines of the two plants are also included.
Barbucci et al. (2002) present a system for monitoring and diagnosis of two gas
turbines owned by Enel. The system features includes: gas path analysis to determine
the thermodynamic parameters of turbine components, a neural network based system
to analyze the pattern of flue gas temperatures, vibration monitoring and emissions and
stability analysis of the combustion system. A system to diagnose a gas turbines from
Ansando Energia based on the use of a tuned simulator is presented by Crosa and
Torelli (2005).
Lopez et al. (1999) describe an expert system based on decision trees and fuzzy
logic to determine causes of power losses in Cofrentes Nuclear Power Plant (Spain).
The problem of lost electric output in nuclear power plants has also been tackled by Heo
et al. (2005) and Heo and Chang (2005), which propose a system based on statistical
regression whose coefficients have been determined by using a simulator.
28

State of the art of thermoeconomic diagnosis
2.3.2. Diagnosis centres and manufacturer-operator collaboration.
Manufacturers of power plant equipment sometimes propose collaboration
agreements with their customers in order to sell not only the product and sometimes its
maintenance but also a monitoring and diagnosis service which can be performed
remotely in one or more centres by using information technologies widespread available
today. Besides, it should be noted that, although it is not usually acknowledged
explicitly, manufacturers are also interested in knowing the true behaviour of their
products in service. Examples of these centres are the Ansaldo Diagnosis Centre
(Rebizzo et al., 2002; Rebizzo et al., 2003), the diagnostic centres of Siemens in
Orlando and Erlangen (Scheidel et al., 2004; Bauch and Killich, 2005), and the Alstom
centre (Decoussemaeker et al., 2005). Collaboration of manufacturers and users is also
proposed by the engine manufacturer Wärtsilä (Klimnstra, 2004). The idea of common
diagnosis centres for several power plants has also been applied by utilities such as
Iberdrola (Mendivil and Álvarez, 2002) and RWE (Grimwade and Gornm, 2005).
2.3.3. Monitoring, diagnosis and optimization of boilers.
Coal-fired boilers are complex systems whose operation can be improved in several
ways: combustion in burners, sootblowing optimization, fuel or start-up.
Several tools have been developed to optimize operation of burners in coal boilers:
suitable amount of primary and secondary air, choice of burners used… They are
proposed as affordable tools to reduce NOx while boiler efficiency is improved. The use
of additional instrumentation is usually needed in order to determine coal and air mass
flows to burners. Sometimes, systems to measure carbon in ash are also used. Suitable
models (often empirical, built by tools such as neural networks) are developed capable
to predict the consequence of modification of operation. Usually, an optimization
module is used to decide the optimum operation, and sometimes the use of closed-loop
operation is possible. Examples of these systems are: Jarc and Lang, 1998; Otero et al.,
1999; Ray and Howard, 1999; Copado and Rodríguez, 2002; Thulen and Peper, 2003;
Reismann et al., 2005; Siemens 2006a.
Sootblowing of heat transfer surfaces is a key point to achieve high efficiency and
operation without problems of coal boilers. However, this operation needs the use of
steam, which implies an additional consumption of coal. In this context, there is a big
interest in substituting the traditional sootblowing programming at regular time intervals
by intelligent systems able to determine the reduction in the heat transfer in the different
29

Chapter 2
surfaces of the boiler, to predict its future evolution and to use this information to
decide the optimum moment for sootblowing. This is a complex task because both
complex monitoring techniques and models of fouling and slagging are needed: Cortés
et al., 1989; Cortés et al., 1993; Bella et al., 1994; Valero and Cortés, 1996; Bartels et
al., 2003; Teruel et al., 2005; Siemens, 2006b; Romeo and Gareta, 2006.
A detailed monitoring of boilers is a difficult task due to the complexity of
combustion and heat transfer processes and the lack of instrumentation. Cortés et al.
(1998) and Díez et al. (1998, 2001) have studied this task.
A computer system including sootblowing optimization, combustion set-points
optimization, and on-line monitoring of heat transfer and air-coal system has been
developed to the boiler of the power plant studied in this thesis (Carrasco et al., 2001).
Boilers have been designed to burn a determined type of fuel, so that, its
modification has to be carefully analyzed. An application of a program to perform this
task is described by Gálvez et al. (2004). Systems for adapting burners operation to coal
properties have also been developed (Siemens, 2006c).
Boiler start-up is an operation which has to be done carefully and progressively in
order to avoid excess of thermal stress which can produce damage. However, the
necessity to adapt the operation to competitive markets and to reduce fuel consumption
recommends to perform this operation in a time span as short as possible. Several
systems have been developed to conjugate these contradictory requirements (ABB,
2005a; Siemens 2006d).
One important cause of reduction in boilers’ efficiency is air leakages in preheaters.
Drobnic et al. (2006) propose to use a numerical model to determine sealing failure by
using information from air and gases temperatures.
2.3.4. Other issues.
Ioanilli et al. (1998) describe a method to detect the state of feedwater heaters in
order to optimise their maintenance. The goal is reduce the additional fuel consumption
caused by tube plugging or heater’s by-pass while keeping low the associated
maintenance operations.
GE Energy has developed a system for closed-loop optimal control of set-points in
combined cycle and co-generation plants (Davari and de Boer, 2004).
An indicator of mechanical failure in turbomachinery is the presence of abnormal
vibrations. Although vibrations above a certain level should be always detected for
30

State of the art of thermoeconomic diagnosis
safety reasons, the detection of anomalies in an early stage is very convenient in order to
avoid further damage of the turbine and additional fuel consumption. (Herbert and
Davies, 2005; Parenti et al., 1998).
Malfunctions in the combustion chamber of gas turbines can lead not only to
additional fuel consumption but also to damages in the turbine. Verhage et al. (1999)
present a system to detect instabilities in combustion, while Siemens (2007) has
developed a system to optimize the operation of this component.
The analysis of previous examples of component degradation and their symptoms
can serve as valuable experience to diagnose new failures. Tirone et al. (1998) present
some examples of steam turbines owned by Enel, and Kubiak et al. (2002) analyse
degradation of gas turbine components.
2.4. Conclusion
The main concepts and tools provides by Thermoeconomics for diagnosis have been
presented, as well as other techniques to filter induced effects. Proposals which avoid
the use of Thermoeconomics have also been reviewed, especially the diagnosis
algorithm which is further developed in next chapter. Finally, systems for monitoring
and diagnose thermal systems have been presented. This allows to see the capabilities
and requirements of industrial applications as well as other related issues which can
result in synergies (e.g. mechanical condition monitoring, on-line optimization…).
31

Chapter 2
32

3. Quantitative causality analysis
In this chapter the diagnosis methodology applied in this thesis is presented. This
method is based on the diagnosis algorithm proposed by Correas (2001), which is the
core of a diagnosis system applied to an IGCC power plant (Correas 2001, García-Peña
et. al., 2000 and 2001)and has also been successfully applied to the TADEUS problem
(Correas, 2004). It has been presented in section 2.2.5.
The idea of the method is to share the variation of an indicator of the efficiency of
the whole system (e.g. efficiency, heat rate…) in the summation of several impact terms
each one due to the variation of an independent variable. These independent diagnosis
variables are called free diagnosis variables and can be ambient conditions, fuel quality
parameters, set-points or indicators of the state of the components of the energy system.
The original diagnosis algorithm is improved by developing several points. First, the
nomenclature is adapted in order to easily introduce the improvements. Then, the
methodology is extended to include the influence of measurement errors. Afterwards,
the problem of non-linearities is analyzed. Finally, the concept of causality chains is
introduced. A very simple system is used as an example to explain the concepts.
3.1. Re-formulation of the diagnosis problem
In this paragraph, a new nomenclature for the diagnosis problem variables is
introduced in order to clarify the notation. This task is important because new problems,
such as the influence of error measurement, are going to be considered.
A thermal system described by a set of nt thermodynamic variables (x) is considered.
nd of these variables are the free diagnosis variables. There are a set of nr restrictions
33

Chapter 3
that link the variables. These restrictions are matter and energy balances and definitions
of parameters of equipments. So that, they are general restrictions that are common to
the problems of diagnosis, performance test, simulation and optimization.
R( x ) = 0
3.1
34
There may be also other specific restrictions for the diagnosis problem:
RD( x ) = 0
3.2
The number of these restrictions nrd should accomplish the following relation:
nrd = nt−nr− nd
3.3
For small deviations, both restrictions can be expanded in a first order Taylor series:
1 0 ( ) ( ) ( ) 0
R x ≅ R x + JR x ⋅Δx
3.4
1 0
( ) ( ) ( ) 0
x
RD x ≅ RD x + JRD x ⋅Δx
3.5
x
Where JR and JRD are the Jacobian matrices corresponding to restrictions R and
1 0
RD and Δ x = x − x . Since both x 0 and x 1 are solutions of the systems, they accomplish
the restrictions. So that:
⎡ JR ⎤
⎢ ⎥⋅Δx
≅0
3.6
⎣JRD⎦ To obtain a resoluble system, the variation of the free diagnosis variables should be
introduced.
⎡ JR ⎤
⎢ ⎥ ⎡ 0 ⎤
⎢
JRD
⎥
⋅Δ x = JD ⋅Δx ≅ ⎢
Δ
⎥
⎣ xd
⎢ ⎥
⎦
⎣ VD ⎦
Where JD is a nt x nt matrix including JR, JRD and VD. VD is a nd x nt matrix
defined as:
VDij = 1 if the j th element of xt is the ith element of xd.
VDij = 0 in other case.
The use of the matrix VD instead of an identity and a null matrices, makes
unnecessary that the vector x is ordered. This is very important when the same vector x
is used for other problems different from the diagnosis. If x were ordered, VD would be
composed of an identity and a null matrices. The VD matrix allows to obtain the xd
vector from the x vector:
d = ⋅ x VD x d
3.7
Δ x = VD⋅Δx 3.8
By inverting the JD matrix, the variation of x is related to the variation of xd.

Quantitative causality analysis
−1 ⎡ 0 ⎤ −1
⎡ 0 ⎤
Δx ≅ JD ⋅ ⎢ ⎥ = JD ⋅⎢ ⎥⋅Δxd
3.9
⎣Δxd⎦ ⎣UD⎦ An indicator of the global efficiency of the system e(x) is considered. Its variation
can be expressed by using a Taylor series again:
∂e
∑ ex x 3.10
n
t
Δe≅ ⋅Δ xi=
⋅Δ
i= 1 ∂xi
Finally, variation of e can be related to the variation of the free diagnosis variables:
t −1 ⎡0⎤ t
Δe≅ex ⋅JD ⋅⎢ ⋅Δ d = ⋅Δ d
U
⎥ x ed x
⎣ ⎦
Where ed is the vector containing the sensitivity of the efficiency indicator e respect
to the free diagnosis variables. For convenience for further developments, it is
interesting to express ed t as a product of an unit vector times a diagonal matrix ED:
t
Δe≅ ⋅ ⋅Δ d
u ED x 3.11
3.1.1. Example
To clarify the concepts and nomenclature presented above, an example consisting of
a very simple steam cycle is proposed. It is going to be used to explain other ideas in
this chapter.
2
1
6
Wb
Qc
Figure 3.1. Scheme of the example cycle.
5
3
4
Wt
35

Chapter 3
The steam cycle is represented in Figure 3.1. It is composed of a pump, a boiler, a
turbine and a condenser. There are two flows: the flow describing the water-steam cycle
that comprises streams 1 to 4 and the cooling water flow that includes streams 5 and 6.
The properties of the streams are summarized in Table 3.1.
POINT m (kg/s) p (bar) T (ºC) x (-) h (kJ/kg) s
(kJ/kg·K)
36
1 97.882 0.07 39.0 0 163.4 0.559
2 97.882 40 39.3 168.1 0.561
3 97.882 35 540.0 3541.8 7.272
4 97.882 0.07 39.0 0.9767 2515.4 8.094
5 10000 1 20 83.9 0.296
6 10000 1 25.5 106.9 0.374
Table 3.1. Properties of streams in the cycle.
Other important values related to the example cycle are: heat provided by the boiler,
heat rejected in the condenser, power produced in the turbine, power consumed by the
pump, net power produced, cycle efficiency, condenser effectiveness and isoentropic
efficiencies of pump and turbine:
Q boiler = 330,220 kW
Q condenser = 230,220 kW
W turbine = 100,463 kW
W pump = 463 kW
W net = 100,000 kW
ηcycle = 0.303
εcondenser = 0.29
ηpump = 0.85 tpu
ηturbine = 0.8 tpu
To represent the cycle, 18 variables are considered (nt = 18): pressure and
temperature of points 2 and 3, condensing pressure and temperature, quality of points 1
and 4, temperatures of points 5 and 6 (cooling water is considered as incompressible
liquid with constant specific heat), two mass flows (steam and cooling water), two

Quantitative causality analysis
isoentropic efficiencies (pump and turbine), one pressure drop (in the boiler), condenser
efficiency, the power and cycle efficiency. It should be noted that the cycle could be
represented by using more variables, e. g. six mass flows instead of only two. In this
case, four mass balances should be introduced in the set of restrictions. So that, the x
vector containing the n variables necessary to describe the system is:
x
t
( psat p2 p3 TsatT2 T3 T5 T6 x1 x4 m1 m5 W n εcηcηpηtpb) = Δ
3.12
These 18 variables are linked by 9 restrictions:
• Net power calculation
W−m ⋅ h p , x − h p , T + h p , T − h p , x = 0 3.13
( ( ) ( ) ( ) ( ) )
n 1 1 sat 1 2 2 2 3 3 3 4 sat 4
• Condenser energy balance
( ) ( ( ) ( ) )
m ⋅c ⋅ T −T −m ⋅ h p , x − h p , x = 0
3.14
5 pw , 6 5 1 4 sat 4 1 sat 1
• Pressure drop in boiler
Δp − p + p = 3.15
b
2 3 0
• Condenser effectiveness
( )
ε ⋅ − − + = 3.16
c T5 Tsat Tsat T5
0
• Cycle efficiency
( ( ) ( ) )
η ⋅m ⋅ h p , T − h p , T + W=
0
3.17
c 1 3 3 3 2 2 2 n
• Pump isoentropic efficiency
( h2( p2 T2) h1( p x1) ) h2 ( p2 T2 p x1) h1( p x1)
η ⋅ , − , − , , , + , = 0 3.18
p sat s sat sat
• Steam turbine isoentropic efficiency:
( h3( p3 T3) h4 ( p x4 p3 T3) ) h3( p3 T3) h4( p x4)
η ⋅ , − , , , − , + , = 0 3.19
t s sat sat
• Saturated liquid in point 1:
x = 0
3.20
1
• Saturation conditions:
sat sat sat
( ) 0
T − T P = 3.21
37

Chapter 3
These restrictions are common to the problems of diagnosis, performance test,
simulation and optimization; so that, they all corresponds to the equation R(x) = 0. By
deriving the operator R(x) in the reference point, the JR matrix can be obtained:
38
⎡ − ⋅ −
⎢
⎢ 67940 0.000242 −0.00025 −0.000115 8.60⋅10 2.20⋅10 ⎢ 0 −1
1 0 0 0
⎢
−5
⎢−2.20⋅10
0 0 0.6698 0 0
−5
JR = ⎢ −0.52336 −2.637 −28.757 −1.70⋅10 −123.26
67.052
⎢
−6
⎢ −1081.3 −0.024588 1.00⋅100 3.547 0
⎢
−1074.
7 0 3.68 0 0 −1.1446
⎢
⎢ 0 0 0 0 0 0
⎢ −9
⎣ −304.01 −2.00⋅10 0 1 0 0
1
0
0
0
−1
0
0
0
0
−
67942 8.6096 93.889
−5
5.60 10 402.43 218.92
−5 −5
0.
000102
− 20915
0
0.
3302
3.
10 ⋅10
1.
00 ⋅10
0
0
0
−5
−6
0.
003037
− 0.
012514
0
16.
168
− 0.
00093
2.
70 ⋅10
0
0
0
−5
6.
60E
− 5
−
20915
0
−1
2.
00 ⋅10
1.
00 ⋅10
0
0
0
0.
006642
0.
00426
0
0
3.
26 ⋅10
5.
90 ⋅10
0
0
0
5
−5
−5
−6
−
2.
33⋅10
2.
33⋅10
−
0
0
0.
000623
−
2055.
8
0
1
0
0.
002393
0.
018231
0
0
− 0.
000733
4.
7251
0
0
0
5
5
2.
33⋅10
−
−
2.
33⋅10
0
0
0.
000641
1.
80 ⋅10
2415
0
0
0.
002543
− 0.
013099
0
0
− 0.
000779
2.
20 ⋅10
1302,
1
0
0
−5
5
5
−5
−1036.
9
−
2348.
6
0
0
1036.
9
0
0
0
0
0.
000407 ⎤
0.
00107
⎥
⎥
1 ⎥
⎥
0 ⎥
− 0.
000125⎥
4.
00 ⋅10
0
0
0
−6
⎥
⎥
⎥
⎥
⎥
⎥
⎦
0
45.
299
9 free diagnosis variables have been chosen: the two isoentropic efficiencies,
temperature and pressure at inlet of steam turbine, pressure drop in the boiler, condenser
0
0
0
0
0
0
0
3.22

Quantitative causality analysis
efficiency, temperature and mass flow of the cooling water and net power. So that, the
xd vector can be defined as:
( p3 T3 T5 m5 W ε η η p )
x = Δ
3.23
t
d n c p t b
And the VD matrix that connects the vectors xd and x:
⎡0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎤
⎢
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
⎥
⎢ ⎥
⎢0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0⎥
⎢ ⎥
⎢0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0⎥
VD = ⎢0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0⎥
3.24
⎢ ⎥
⎢0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0⎥
⎢0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0⎥
⎢ ⎥
⎢0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0⎥
⎢
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
⎥
⎣ ⎦
There are 18 total variables, 9 restrictions corresponding to R(X) = 0 and 9
diagnosis variables. So that, no additional conditions specific for the diagnosis problem
are needed (the dimension of RD(x) = 0, nrd, is equal to zero) and the matrix JD is only
formed by JR and VD.
Finally, the indicator for the efficiency of the global system (e) belongs to the set of
variables x; so that, the vector ex has a very easy expression composed of only one 1
and zeros:
( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0)
t
ex =
3.25
By using the matrices calculated above and the expressions obtained in the previous
section, the vector that relates the variations of the free diagnosis variables (Δxd) to the
variations of the efficiency indicator (Δe) can be obtained:
ed
t −6 −7
= ( 0.0009514 0.00014003 −0.0013413 4.3373⋅10 −2.1687⋅10 0.
032378
0.
0012553
0.
42385
)
−5
− 2.
6692 ⋅10
3.26
According to this result, when turbine inlet pressure increases one bar, efficiency
increases 0.0009514 units and if turbine inlet temperature increases one degree,
efficiency increases 0.00014 units. When cooling water temperature increases one
degree, efficiency decreases 0.0013413 units. If cooling water flow rate increases one
kilogram per second, efficiency increases 4.3373·10 -6 units. If power increases one
kilowatt, efficiency decreases 2.1687·10 -7 . If the condenser efficiency increases one
39

Chapter 3
unit, the efficiency of the plant increases 0.032378. The increment of isoentropic
efficiencies of pump and turbine makes efficiency to increase, although the effect of the
turbine is 0.42385 units of cycle efficiency per unit of turbine efficiency while the effect
of the pump is only 0.0012553. Finally, an increment of one bar in the boiler pressure
drop produces an efficiency decrement of only 2.6692·10 -5 units. As it can be seen, the
information contained in this vector is very important because it allows to connect the
variation of the free diagnosis variables to the variation of the efficiency indicator,
which is the purpose of the diagnosis algorithm. It should be noted that this vector is not
dimensionless.
40
3.2. Influence of measurement errors.
The methodology presented above is able to perform a quite accurate diagnosis
starting from two sets of variables x 1 and x 0 corresponding to the actual and reference
states of the plant. However, these plant states (at least the actual one) are determined
by a performance test. This test is not exact because it relies on information provided by
instruments that present errors as well as hypotheses made to overcome the lack of
information. These errors in the performance test may affect not only the deviation of
the efficiency indicator but also the share of this value in the free diagnosis variables.
Measurements affecting the value of the efficiency indicator are usually a few, but
measurements affecting the diagnosis results are a lot. For example, in the case of the
cycle, temperature of the cooling water entering the condenser does not affect the value
of the cycle efficiency, but it modifies the contribution of the condenser efficiency and
the water mass flow. Another example is the value of a temperature between two stages
of a turbine, which does not affect the cycle efficiency but it modifies the isoentropic
efficiency of the stages upstream and downstream the measurement.
The impact of a measurement error in the value of the efficiency or heat rate has
been studied and it is widely applied to determine the uncertainty of the calculated
efficiency. In this section, a method to determine the impact of a measurement error in
the diagnosis result is explained. The methodology is very similar as that of the
diagnosis.
The thermal system to be analyzed is described by means of a set of nt
thermodynamic variables (x). These variables correspond to properties of streams (mass

Quantitative causality analysis
flow, pressure, temperature) and parameters of equipments (isoentropic efficiencies,
pressure drops). It should be noted that this vector x is the same as that used for the
diagnosis problem, so that, there are the same nr restrictions (R(x) = 0). A set of np
variables from x (xp) is measured by the field instrumentation. So that, a VP matrix (np
x nt) can be defined as follows
VPij = 1 if the variable i of xp is the variable j of x
VPij = 0 in other case
Depending of the problem a set of nrp additional restriction may be needed:
RP ( x ) = 0
3.27
The value of nrp should accomplish the following rule.
nt = nr + nrp + np
3.28
Our goal is to relate the variation of the set of variables x to the variation to the
measured ones xp. This can be done in the same way as in the diagnosis problem, but
considering that now the free variables are the measured ones. So that:
⎡ JR ⎤
⎢ ⎥ ⎡ 0 ⎤
⎢
JRP
⎥
⋅Δ x = JP ⋅Δx ≅ ⎢
Δ
⎥
⎣ x p
⎢ ⎥
⎦
⎣ VP ⎦
3.29
where JR and JRP are the Jacobian matrices of the restrictions R(x) = 0 and RP(x)
= 0. The relation we were looking appears by inverting the JP matrix:
0 −1 ⎡ ⎤ −1
⎡0⎤ Δx ≅ JP ⋅ ⎢ ⎥ = JP ⋅⎢ ⋅Δ p
Δ p U
⎥ x
3.30
⎣ x ⎦ ⎣ ⎦
The introduction of the VD matrix allows to relate the variation of the diagnosis
variables to the variation of the measured ones:
−1 ⎡0⎤ Δxd ≅VD⋅JP ⋅⎢ ⋅Δ p = ⋅Δ p
U
⎥ x DP x 3.31
⎣ ⎦
DP is a nd x np matrix whose elements DPij are the sensitivity of the diagnosis
variable i to the measured variable j. It is very important because it shows the impact of
a measurement error in the values of the free diagnosis variables. Finally, the sensitivity
of the diagnosis to the measurements can be obtained by combining equations 3.11 and
3.31:
Δe≅ u ⋅ED⋅Δ x = u ⋅ED⋅DP⋅Δ x = u ⋅EP⋅Δx 3.32
t t t
d p p
41

Chapter 3
42
where EP is a nd x np matrix whose elements EPij are the sensitivity of the impact
due to the diagnosis variable i to the measured variable j. This matrix allows to
understand how an error in a measurement can affect the diagnosis result.
Matrices DP and EP link the problems of performance test and diagnosis. They are
very useful to determine the diagnosis uncertainty due to measurement uncertainty and
to choose the right instrumentation for a correct diagnosis system. In fact, diagnosis
uncertainty of each diagnosis variables can be obtained by:
2 2 2
, = ∑ ⋅ ,
= 1
p n
xd i DPij
xp j
j
σ σ 3.33
where
σ is a nd dimensional vector containing the uncertainties of the diagnosis
2
xd
2
variables and σ is the np dimensional vector of the instrumentation uncertainties. The
xp
uncertainties of the impacts due to the measurement can be obtained as follows:
2 2
=
, ∑ ⋅ ,
=
p n
2
e EP
xdi ij xp j
j i
σ σ 3.34
2
where σ is a nd dimensional vector containing the uncertainties of the impacts
exd
due to the instrumentation errors. It should be highlighted that this impact only
considers the effect of the measurements. There are other errors due to the diagnosis
methodology and the hypothesis of linearization that should be added. The effect of
linearization is studied in a following section.
Finally, a vector ep can be defined that directly links the variation of the efficiency
indicator to the variation of the measurements:
Δe ≅u ⋅EP⋅Δ x = ep ⋅Δx
3.35
t t
p p
Each position of this vector is the summation of a row of the EP matrix, and
represents the sensibility of the efficiency indicator to the variation of the
measurements. These sensitivities are usually used to determine the uncertainty of the
indicator in a performance test:
2
np n
⎛ ⎞
p
2 ∂e
2 2 2
e = ∑⎜ ⎟ ⋅ = ⋅
⎜ ⎟ x , ∑ep
p i i xp,
i
i= 1 ∂xpi
,
i=
1
σ σ σ 3.36
⎝ ⎠
3.2.1 Example
In this section the example presented in 3.1.1 is continued in order to clarify the
concepts introduced above.

Quantitative causality analysis
9 from the 18 variables of the thermodynamic model of the cycle are measured:
pressure and temperature in points 2 and 3, net power, condenser pressure, inlet and
outlet cooling water temperature and inlet boiler water mass flow. As it can be seen,
some of the measured variables are the same as the free diagnosis ones while other does
not. So that, the xp vector and the VP matrix are:
( p p2 p3 T2 T3 T5 T6 m1 W)
x =
3.37
p sat n
⎛1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎞
⎜ ⎟
⎜
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
⎟
⎜0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0⎟
⎜ ⎟
⎜0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0⎟
VP = ⎜0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0⎟
3.38
⎜ ⎟
⎜0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0⎟
⎜0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0⎟
⎜ ⎟
⎜0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0⎟
⎜
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
⎟
⎝ ⎠
Since n = 18, nr = 9 and np= 9, additional restrictions are not needed (nrp = 0). So
that, the DP and EP matrices can be calculated:
⎡ 0 0 1 0 0
⎢
⎢
0
⎢ 0
⎢
⎢−0.076898
DP
= ⎢ 0
⎢
⎢ −12.594
⎢ 228.84
⎢
⎢ 0.28433
⎢
⎣ 0
0
0
−0.1910 0
0
0.0052028
−5
6.86⋅10 1
0
0
−2.0726 0
0
−7 6.45⋅10 −0.002079 −1
0
0
−8.8836
0
0
−0.75067 0.0032048
0
1
0
4.8328
0
0
−9
3.01⋅10 −0.000864
0
43

Chapter 3
44
0 0 0 0 ⎤
0 0 0 0
⎥
⎥
1 0 0 0 ⎥
⎥
461.71 −461.72 74.738 −0.022076⎥ 0 0 0 1 ⎥⎥
−0.020424 0.061852 0 0 ⎥
−8 −7 −8
−9.49⋅10 −5.65⋅10 6.11⋅10 0 ⎥
⎥
−10 −9 −6
3.18⋅10 2.02⋅10 −0.008258 7.96⋅10⎥ 0 0 0 0
⎥⎦
The first row of the DP matrix has a 1 in the third position and zeros in the other
positions, because the first diagnosis variable is the turbine inlet pressure, which
corresponds to the third measured variable. A similar situation occurs in the second,
third and fifth rows (turbine inlet temperature, cooling water temperature and power
produced). The fourth diagnosis variable is water cooling mass flow that depends of all
the variables because quality at the exit turbine is not measured. The same occurs with
the isoentropic efficiency of the turbine (eighth variable), although its dependence on
cooling water temperatures is negligible and due to numerical errors. The sixth variable
is the condenser efficiency that depends of the cooling water temperatures and the
condensing pressure. The seventh variable is the pump isoentropic efficiency that only
depends on pressure and temperature in point 2 and condensing pressure. Finally, the
ninth diagnosis variable is the pressure drop in the boiler, so that in this row only
appears one 1 and one –1.
EP matrix is obtained by multiplying each row of the ED matrix times the impact
factor of the diagnosis variable corresponding to that row:
⎡ 0 0 0.0009514 0 0
⎢
⎢
0 0 0 0 0.00014
⎢ 0 0 0 0 0
⎢ −7 −7 −6 −5 −5
⎢−3.34⋅10
−8.25⋅10 −8.99⋅10 −3.85⋅10 2.096⋅10 EP = ⎢ 0 0 0 0 0
⎢
⎢ −0.40778
0 0 0 0
⎢
−6 −10
0.28726 6.531⋅10 8.097⋅10−0.000942 0
⎢
−5
⎢ 0.12051 2.906⋅10 −0.000881
0.0013584
⎢
−5 −5
⎣ 0 −2.67⋅10 2.669⋅10 0
3.39

0 0 0 0 ⎤
0 0 0 0
⎥
⎥
−0.001341
0 0 0 ⎥
−8
⎥
0.0020026 −0.002003 0.0003242 −9.575⋅10 ⎥
−7
0 0 0 −2.169⋅10 ⎥
⎥
−0.000661 0.0020026 0 0 ⎥
−10 −10
−1.19⋅10 −7.09⋅10 0 0 ⎥
⎥
−10 −10 −6
1.349⋅10 8.568⋅10 −0.0035 3.375⋅10 ⎥
0 0 0 0
⎥⎦
Quantitative causality analysis
The interpretation of this matrix is very similar as that of the previous one. An
important detail is that measured variables can affect in all the diagnosis variables,
although they are diagnosis variables themselves. For example, temperature in point 3 is
a diagnosis variable (so that the first row has only one 1), but it also affects other
diagnosis variables such as the isoentropic efficiency of steam turbine. Finally, ep
vector can be obtained:
ep
t −6 −6 −5
(
= 2.10⋅10 8.08⋅10 8.81⋅10 0.00037752 −0.0002054
)
3.40
−10 −6
⋅ − ⋅ 3.41
0 2.73 10 0.003176 3.06 10
The analysis of these two matrices allows to understand how a variation in a
measurement modifies the results of the diagnosis. For example, a variation in the
measured value of the entering cooling water temperature (sixth diagnosis variable)
does not affect the calculated value of the cycle efficiency but it modifies the diagnosis
result. If the measured value of this temperature increases 1 degree, the impact in cycle
efficiency due to the cooling water temperature decreases 0.001341 units, the impact
due to condenser mass flow increases 0.0020026 units and the impact due to condenser
efficiency decreases 0.000661 units. As it can be seen, an error in this measurement
makes the diagnosis results to vary substantially, so that an effort to ensure good quality
instrumentation in this point should be made. It should be highlighted that matrices DP,
EP and vector ep are not dimensionless, so that they not only depend on the analysed
plant ant the variable choice but also on the units of the variables of e, xd and xp.
45

Chapter 3
46
3.3. Analysis of non-linearities
In the previous section, the problems of diagnosis and performance test have been
connected and the influence of the measurement errors in the results of diagnosis has
been studied. Another source of diagnosis inaccuracy is related to the equation
linearization, which is studied here.
3.3.1. Evaluation of the influence of linearization errors in the
diagnosis result
The diagnosis methodology presented above is based on the linearization of the sets
of restrictions R(x) = 0 and RD(x) = 0. If non-linear behaviour is considered, equations
3.4 and 3.5 become:
( )
( )
( )
( )
( )
( )
⎡ R x ⎤ ⎡ R x ⎤ ⎡ JR x ⎤
= + ⋅Δ x+ rs
1 0
⎢
1 ⎢
⎣
RD x
⎥
⎥
⎦
⎢
0 ⎢
⎣
RD x
⎥
⎥
⎦
0
x
⎢ ⎥
⎢ ⎥ 0 ⎣
JRD x x ⎦
l
3.42
where rsl is a (nr + nrd) x 1 vector that has been introduced in order to make exact
that approximate equation. The previous equation can be reordered and expanded in
order to consider also the diagnosis variables:
⎡ rs ⎤
JD ⋅Δ x = ⎢
Δx
⎥
3.43
⎣ d ⎦
where rs is a (nr + nrd) x 1 vector that takes into account the errors due to
linearization and to the non-exact restriction accomplishment of vectors x0 and x1.
( )
( )
( )
( )
0 1
⎡ ⎤ ⎡ ⎤
R x R x
rs = rs ⎢ ⎥ ⎢ ⎥
l + −
0 1
⎢ ⎥ ⎢ ⎥
⎣
RD x
⎦ ⎣
RD x
⎦
Since JD, Δx and the residues are known, vectors rs and rsl can be determined by
using the previous equations. It should be noted that equations corresponding to the
lower part of the system are identities (Δxd,i = Δxd,i), so that they are not affected by
linearization errors. This error due to restrictions (rs) is propagated when it is multiplied
by JD -1 . So that, the value of Δx that can be calculated is the summation of two terms,
the first one obtained from the causalization of the diagnosis variables and the other
corresponds to the error of restrictions.
⎡ 0 ⎤ ⎡rs⎤ ⎣Δ⎦ ⎣ ⎦
−1 −1
Δ x = JD ⋅ ⎢ ⎥+ JD ⋅ ⎢ =Δ calc +Δ rs
d 0
⎥ x x
x
3.44
3.45

Quantitative causality analysis
There is another linearization in the diagnosis algorithm, which corresponds to the
dependence of the indicator e of the set of values x:
t
Δ e= ex ⋅Δ x + ee
3.46
Since ee is the only unknown quantity in the previous equation, it can be easily
determined by substituting Δx:
Δ e= ex ⋅Δ x + ex ⋅Δ x + ee
3.47
t t
calc rs
if we define er as:
t t −1 ⎡U D ⎤
er = ex ⋅JD ⋅⎢⎥ ⎣ 0 ⎦
3.48
the exact equation for Δe becomes:
Δ e= ed ⋅Δ x + er ⋅ rs + ee= ed ⋅Δ x + ε
3.49
t t t
d d
The previous equation is very important because it demonstrates that the efficiency
variation is due to three contributions: the first one is the summation of impacts
calculated by the diagnosis algorithm, the second is an error due to the linearization of
the restrictions and inaccuracy of the vectors x 0 and x 1 and the last one is the error due
to the linearization of the expression of the efficiency indicator. If there were no
linearization error, terms two and three would be null (ε = 0) and the method would be
exact. However, this is not true in general, so that errors appear mainly when the two
situations to be compared are not close enough. Methods to reduce these errors are
presented in the next section.
3.3.2. Strategies to reduce the errors of the methodology due to nonlinearities
Once the error sources of the diagnosis algorithm have been studied, the analysis of
strategies to reduce this error can be tackled.
According to the expressions presented above, matrices JR and JRD and vector ex
are calculated at reference conditions (x = x 0 ). However, results may improve if these
derivatives are evaluated in a point x m between x 0 and x 1 . In fact, the Mean Value
Theorem states that:
Let f be a function continuous on a closed interval [a, b] and differentiable in an
open interval (a, b), then there is at least one point c in (a, b) where:
( ) − ( )
f b f a
f '(
c)
=
b−a One possibility would be to evaluate the derivatives at point x av defined as:
3.50
47

Chapter 3
0 1
av +
=
2
x x
x 3.51
Another option is to calculate the matrices in points x 0 and x 1 and then use average
matrices:
48
( ( ) 0 ( ) 1
x x )
1
JR = ⋅ JR x + JR x 3.52
2
( ( ) 0 ( ) 1
x x )
1
JRD = ⋅ JRD x + JRD x 3.53
2
( ( ) 0 ( ) 1
x x )
1
ex = ⋅ ex x + ex x 3.54
2
This strategy can be implemented with no big difficulties and allows obtaining quite
good results, as it is shown in Chapter 6.
To improve the results, a second order approach might be considered. The goal
would be to obtain an expression linking the variation of the efficiency indicator to the
variation of the free diagnosis variables that has quadratic terms:
Δ e= n
⋅Δ x +
n
⋅ Δ x +
n i−1
⎛
⋅Δx ⋅Δx
⎞
i= 1 i= 1 i= 1⎝ j=
1
⎠
d d d
2
∑αi d, i ∑βii ( d, i) ∑⎜∑ βij
d, i d, j⎟
3.54
where αi and βij are the sensitivity coefficients to be determined by the diagnosis
procedure.
This second order approach has not only the problem of calculation complexity but
also the interpretation of the crossed terms. A diagnosis procedure without crossed
terms provides a more or less exact solution consisting in the share of a deviation of a
global efficiency indicator into several elements each one corresponding to an
independent “free” diagnosis variable. In this way, each impact represents the
improvement of the efficiency indicator e that would be obtained if the corresponding
anomaly were repaired. However, when crossed terms appear, this is not so simple,
because the removal of two anomalies does not have exactly the same effect as the
summation of the effects obtained by the removal of each anomaly separately. As an
example, a very simple system with only two free diagnosis variables (x1and x2) can be
considered. To fix ideas, the system could be a steam cycle, whose efficiency is related
to the efficiency of a turbine and to the ambient temperature. In the reference state
x = 0.7 ,
0
1
0
x 2 = 10 and 0
e = 0.372 and in the operation state 1
1
x = 0.68,
1
e = 0.364 The equation linking the variations of these three variables is:
15 x = and
1
2

( ) ( )
2 2
Quantitative causality analysis
Δ e= 0.15⋅Δx−0.0002⋅Δ y+ 2.5⋅ Δx −0.00016⋅ Δ y + 0.01⋅Δx⋅Δ
y 3.55
Application of the previous equation shows that if x1 is restored to the previous
value (increases 0.02), efficiency increases 0.003. If x2 is restored after restoring x1,
efficiency increases 0.005, so that it reaches the reference value. On the other hand, if x2
is restored first, efficiency increases 0.006, and if x1 is then restored, efficiency
increases 0.002 additional points. As it can be seen, the removal of anomalies does not
have exactly the same effect depending on whether the starting point is the plant with all
the anomalies or if some anomalies have previously been eliminated. However, this
difference is negligible in most cases, and only has a conceptual interest.
This situation leads to consider that the use of crossed terms should be avoided in
most cases, because the additional difficulty in analysis results does not balance the
accuracy increment. If these developments are to be used, this fact should be clarified.
In the text above the interest of using crossed terms has been discussed starting from
the hypothesis that there is a diagnosis methodology that provides the values of α and β.
However, a methodology to obtain this information has to be developed. One way to do
this might be to perform a development similar to the one used above for the linear
methodology. This approach has several problems. First is that the second derivatives of
every restriction should be introduced, which means a very important effort in the
development of the diagnosis tool. Second, the diagnosis procedure would be more
complex and the calculation time would increase.
Another option that is not much more complex and can provide very interesting
advantages is explored here.
First of all, it should be highlighted that the methodology based on the linealization
is exact if Δxd tends to zero. So that, it is possible to write:
( )
0 0
t
0
= ⋅ d
de ed dx
3.56
where
0
de and
in the reference condition.
0
dx d are a differential variation of these variables around their value
the derivatives also in the reference condition:
( ( ) ) ( )
0
ed means that this vector has been calculated by evaluating
( )
t −1 0
⎡ 0 ⎤
ed = ex x 0 ⋅ JD x 0 ⋅ x x ⎢ ⎥
⎣U ⎦
3.57
The same relation can be obtained around the point x 1 :
( )
1 1
t
1
= ⋅ d
de ed dx
3.58
49

Chapter 3
50
( ( ) ) ( )
( )
t −1 1
⎡ 0 ⎤
ed = ex x 1 ⋅ JD x 1 ⋅ x x ⎢ ⎥
⎣U ⎦
3.59
Vectors ed represents the derivatives of the function e related to the set of
independent variables xd. This information is known, as well as the values of e 1 , e 0 , x 1
and x 0 and can be used to obtain a relation that links Δe to Δxd.
A first idea is to look for quadratic expressions including crossed terms. However,
this is not possible because more information would be needed. For example, a function
depending on only two variables x1 and x2 can be considered. The goal is to obtain an
expression like the following:
2 2
( ) ( ) ( ) ( ) ( ) ( )
Δ e= e−e ≅α ⋅ x − x + α ⋅ x − x + α ⋅ x − x + α ⋅ x − x + α ⋅ x −x ⋅ x − x
0 0 0 0 0 0 0
1 1 1 2 2 2 1,1 1 1 2,2 2 2 1,2 1 1 2 2
To obtain the values of the coefficients α, this equation can be derived:
∂e
∂x
∂e
∂x
∂e
∂x
0 0
x1 , x2
1
1 1
x1, x2
1
2
∂e
∂x
2
2
1
0 0
x1 , x2
1 1
x1, x2
= α
1
1 0 1 0
( x x ) ( x x )
= α + 2⋅α
⋅ − + α ⋅ −
= α
1 1,1 1 1 1,2 2 2
2
2
∂ e
= 2⋅α1,1
∂x
2
∂ e
= 2⋅α
∂x
2
2
2
∂ e
∂x∂x 1 2
1 0 1 0
( x x ) ( x x )
= α + 2⋅α
⋅ − + α ⋅ −
2 2,2 2 2 1,2 1 1
2,2
= 2⋅α
1,2
3.60
3.61
3.62
3.63
3.64
3.65
3.66
3.67
From the previous set of equations, only the values of first-order derivatives and the
values of x1 and x2 in points 0 and 1 are known. So that, seven equations are available to
determine the five αs and the three second-order derivatives. As it can be seen, more
information is needed. One solution of this problem is not to consider the crossed term.
This also allows to overcome the conceptual problem related to crossed terms presented
above. The expression of the variation of e becomes:

0
nd
i=
1
αi i
0
i βi
i
0
i
( ) ( ) 2
Quantitative causality analysis
e−e ≅ ⎡ ⋅ x − x + ⋅ x −x
⎤
∑ ⎢⎣ ⎥
3.68
⎦
If this equation is derived, 3·nd conditions can be obtained:
∂e
∂x
i
∂e
∂x
i
= α
0
x= x i
1
x= x
2
∂ e
= ⋅β
∂x
2 2 i
i
1 0 ( x x )
= α + 2⋅β
⋅ −
i i i i
By solving the previous system, the values for α, β and the second order derivatives
can be obtained. Values of alpha are direct and the others are:
∂
∂e ∂e
−
∂ −
0
2
x= x x= x
e ∂xi ∂xi
= 2⋅
β
2 i = 1 0
xi xi xi
By substituting, the expression of e depending of the value of x can be obtained:
⎡ ∂e ∂e
⎤
1 0
n ⎢ −
d
x= x x= x ⎥
0 ∂e
0 ∂xi ∂xi
0
2
e− e =
⎢
0
x x ( xi xi )
1 0 ( xi x
⎥
∑ ⋅ − + ⋅ −
=
i )
3.73
⎢ i= 1 ∂xi 2⋅(
xi −xi
)
⎥
⎢ ⎥
⎢⎣ ⎥⎦
If the previous function is evaluated in x = x 1 , it yields:
∂e ∂e
1 +
n 0
d x= x x= x
1 0 ∂xi ∂xi
1 0
Δ e= e − e = ∑ ⋅( xi −xi
)
3.74
i=
1 2
As it can be seen, when crossed terms are non considered, evaluation of derivatives
in points 1 and 0 allows obtaining a second order approximation by using a linear
approach. The difference between Equation 3.74 and Equations 3.52, 3.53 and 3.54 is
that in the first one the average is performed after calculating ed (at the end of the
process) and in the other approach the average is made at the beginning and then ed is
computed.
3.69
3.70
3.71
3.72
51

Chapter 3
52
3.4. Free diagnosis variables
The diagnosis methodology presented above is able to relate the variation of an
indicator of the global efficiency of the system to the variation of a set of free diagnosis
variables. So that, the diagnosis result depends not only on the accuracy of the
methodology and the quality of the measurements but also on the choice of the
efficiency indicator e and mainly the free diagnosis variables xd. The problem of free
diagnosis variable election is tackled in this section. First, the mathematical conditions
that these variables must fulfill are expressed. Then, some characteristic that the
variables should accomplish are presented.
3.4.1 Mathematical conditions required to the set of free diagnosis
variables
The conditions that the free diagnosis variables must accomplish are imposed by the
diagnosis methodology. This methodology is based on the inversion of the JD matrix,
so that, the free diagnosis variables set must be chosen in order to accomplish this
condition. First, the dimension of vector xd (nd) has to be the difference of the total
number of variables nt minus the dimension of the independent restrictions nr and nrd.
nd = nt −nr − nrd
3.75
Second, the rows of matrix JD have to be independent. This means that both general
restrictions (R(x) = 0) and diagnosis restrictions (RD(x) = 0) have to be independent,
which is obvious. Second, these restrictions and the fixation of the values of the free
diagnosis variables should be independent. As an example, the pressure drop in a heat
exchanger cannot be chosen as diagnosis variable if its value is simulated in the
restrictions of the problem. In this case, there are three possibilities.
The first one is to consider the pressure drop as an independent free diagnosis
variable.
The second one is to introduce an equation in the restrictions that fix the value of
this variable or introduces their relation to other variables of the problem:
2
⎛ m
⎞
Δ p=Δp0⋅⎜ ⎟
m
0
⎝ ⎠
where 0 p Δ and m 0 are not additional variables but constants.
A third possibility is to use the previous equation but adding a factor that becomes
the free diagnosis variable:
3.76

2
Quantitative causality analysis
⎛ m
⎞
Δ p= ϕ ⋅ p0⋅⎜ ⎟
3.77
⎝m0⎠ The choice of one of the options above depends on the preferences of the analyst,
and influences the results of the diagnosis. This is the objective of the following
paragraphs. It should be highlighted that if second option is chosen, the equation should
be accomplished by the two sets of variables entering the diagnosis procedure (x 0 and
x 1 ). If this is not true (e.g. both pressures are measured in the performance test), errors
appears because restriction are not accomplished (see paragraph 3.3.1).
3.4.2. Influence of a modification in the free diagnosis variable choice
on the diagnosis result.
In this paragraph, it is shown how the variation of only a few free diagnosis
variables can affect the impact of all the free diagnosis variables.
The impact in the efficiency of a system for a certain set of free diagnosis variables
is given by the following equation:
e
⎡ JR ⎤
ex
⎢
JRD
⎥
−1
⎡ 0 ⎤
t
Δ ≅ ⋅
⎢ ⎥
⋅⎢ Δx
⎥
d
⎢⎣ VD ⎥⎦
⎣ ⎦
If the set of free diagnosis variables is modified, VD and xd vary (VD’ and xd’) and
the other terms remain constant:
−1
⎡ JR ⎤
t
e
⎢ ⎥ ⎡ 0 ⎤
Δ ≅ex ⋅
⎢
JRD
⎥
⋅⎢ ,
Δ
⎥
d
⎢ ' ⎥
⎣ x ⎦
⎣VD ⎦
The previous equation can be expanded:
−1 −1
⎡ JR ⎤ ⎡ JR ⎤
0 0
t ⎡ ⎤ t ⎡ ⎤
Δ e = ex ⋅
⎢ ⎥ ⎢ ⎥
⎢
JRD
⎥
⋅ ⎢ ⎥+ ex ⋅
⎢
JRD
⎥
⋅⎢
,
Δ
⎥
d Δ d −Δ d
⎢ '⎥ ⎣ x ⎦
⎢ '⎥
⎣ x x ⎦
⎣VD ⎦ ⎣VD ⎦
3.78
3.79
3.80
−1 −1 −1 −1
⎧ ⎫
⎡ JR ⎤ ⎛⎡ JR ⎤ ⎡ JR ⎤ ⎞ ⎡ JR ⎤
0 0 0
t ⎪ ⎡ ⎤ ⎜ ⎟ ⎡ ⎤ ⎡ ⎤⎪
⎪
Δ e = ex ⋅
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎨⎢ JRD
⎥
⋅ ⎢ ⎥+ ⎜⎢ JRD
⎥
−
⎢
JRD
⎥ ⎟⋅
⎢ ⎥+ ⎢
JRD
⎥
⋅⎢
, ⎥⎬
⎪ ⎣Δxd⎦ ⎜ Δ d Δ d −Δ d
⎢ ⎥ ⎢ '⎥ ⎢ ⎥ ⎟ ⎣ x ⎦
⎢ '⎥
⎣ x x ⎦⎪
⎪⎩⎣ VD ⎦ ⎝⎣VD ⎦ ⎣ VD ⎦ ⎠ ⎣VD ⎦
⎪⎭
3.81
53

Chapter 3
The first term of the right-hand side is the impact of the initial free diagnosis
variables. The last term represents the direct impact due to the modification of the free
diagnosis variables. The second term represent how the original variables xd have
additional impact due to the modification of the variables. This additional term confirms
that the modification of one or more variables can affect the whole diagnosis result.
3.4.3. Choice of the free diagnosis variables.
In the previous paragraphs the conditions that free diagnosis variables have to
accomplish have been presented. However, there are several possibilities so that the
analyst can choose the best option depending on his preferences. It has also been shown
that depending on the choice of the variables the result of the diagnosis may vary
substantially.
Free diagnosis variables are classified into: i) ambient conditions and fuel quality
(external variables), ii) set-points and iii) parameters of component efficiency. Usually,
there are no problems in the choice of diagnosis variables for external causes and set
points, because they are universally accepted and each variable corresponds to a
measured quantity: temperature, pressure, mass flow, composition, heating value and so
on.
However, a right choice of a diagnosis variable to represent efficiency of a
component is usually more complex. The main reason is that component efficiencies are
usually determined by indicators including several variables of the flows entering and
exiting the component. For example: isoentropic efficiency of a turbine, efficiency of a
heat exchanger, pressure drop in a pipe and so on. The direct use of these indicators as
free diagnosis variables has two advantages: i) they can be calculated directly from the
properties of the streams in the plant and ii) the have a widely accepted definition.
On the other hand, it should be remembered that the degradation of the components
behaviour comes from a microscopic level (fouling, erosion). Although this degradation
implies a variation of component efficiency indicators, this variation may be also caused
by variations of the operation point of the component (induced malfunctions). For
example, pressure drop in a heat exchanger can vary due to fouling but also due to a
variation of the flow crossing this component. If the evolution of this parameter is
plotted, an oscillating tendency could be seen. This problem can be overcome by
introducing information of the component behaviour:
54

2
Quantitative causality analysis
⎛ m
⎞
Δ p= ϕ ⋅Δp0⋅⎜ ⎟
3.77
⎝m0⎠ Now the diagnosis variable is the factor φ that substitutes the pressure drop. Since a
new equation and a new variable have been introduced, the mathematical conditions for
the free diagnosis variables can be fulfilled. The use of the factor φ allows to filtrate the
pressure drop variations due to the modification of the operation point of the heat
exchanger. So that if the evolution of this factor is plotted, the graph would be
composed of long slowly increasing zones showing the fouling increment separated by
dramatic drops corresponding to the cleaning. Another option would be to use a factor
which is added instead of multiplied:
2
⎛ m
⎞
Δ p=Δ pint +Δp0⋅⎜ ⎟
3.82
⎝m0⎠ As it can be seen, the use of models of the component and factors instead of
indicators does not imply additional complexity in the formulation of the diagnosis
problem: only the addition of other equation and other variable. However, this solution
has two important problems. First, the behaviour models of the component have to be
implemented (which can be complex) and perhaps periodically revised because due to
degradation the behaviour of the plant may vary. Second, the interpretation of results
may become more difficult, because there is not a universal acceptation of these kinds
of factors. So that, the decision of the free variables definitions should be made by
taking into account the preferences of the people that are going to use the diagnosis tool.
In general, the best option is to use conventional parameters that accept a widely
accepted definition, which can be sometimes substituted by other variables by using
behaviour models of the components. Another possibility for some cases may be the use
of causality chains, which is presented below.
3.5. Causality chains
In the previous section, it has been explained how the analyst need to choose
suitable definitions for the free diagnosis variables. This task may be difficult in some
parameters describing components, because an equilibrium between an easy definition
widely accepted and a good reproduction of the physical behaviour is required.
55

Chapter 3
If there is no definition which accomplish these two conditions at the same time, it
may be interesting to use two sets of free diagnosis variables, one corresponding to
parameters close to the real behaviour (first level of causality) and other with parameters
with easy and widely accepted definition (second level of causality). It should be noted
that most variables would be the same in both sets (ambient conditions, fuel quality
indicators, set-points and some indicators of components). Besides, an interesting way
to obtain an indicator close to the physical behaviour is by considering a simple
indicator but with some corrections to take into account the deviation from the design
point.
The objective of this section is to develop a methodology to use these two levels of
free diagnosis variables. The idea is to make a double decomposition. First, the
variation of the global efficiency indicator is divided into a summation of terms, each
one caused by a free diagnosis variable of the second level. Second, each one of these
terms is again separated into a summation of terms corresponding to the free diagnosis
variables of the first level. For example, part of the variation of the efficiency of a
steam cycle is caused by a variation in the condenser effectiveness, but this variation is
partly caused by degradation of the component, and partly due to variation in other
parameters (ambient temperature, cooling water mass flow…).
It should be noted that the definition of the free diagnosis variables of first level is a
problem of modelling. If the accuracy required is not very high, perhaps the use of only
one level of causality is enough. If this requirement increases, the use of two levels and
a model may be necessary. The accuracy of the result depends on the accuracy of the
model. However, the formulation of causality chains theory, which is tackled in this
section, is independent of the complexity of the models. First, a general formulation is
made, which allows to introduce any sets of free diagnosis variables. Afterwards, a
formulation based on models which relate only free diagnosis variables among them is
presented. Finally, a formulation which considers models which relate the free diagnosis
variables with all variables of the thermodynamic model is presented.
3.5.1. General formulation of causality chains.
The thermal system considered is represented by a set of nt thermodynamic variables
(x). These variables can be temperatures, pressures, mass flows, indicators of
component’s behaviour and so on. They are linked by nr restrictions such as mass and
energy balances, and definitions of parameters:
56

Quantitative causality analysis
Rx ( )= 0
3.1
Besides these general restrictions, there may be other nrd, specific for the diagnosis
problem:
RD( x )= 0
3.2
If the previous equations are linealized, it is possible to write:
⎡ JR ⎤
⎢ ⎥⋅Δx
≅0
3.6
⎣JRD⎦ where JR and JRD are the Jacobian matrices of sets of equations R and RD.
Besides, there are two sets of nd free diagnosis variables, xd1 and xd2. Associated with
each one of these sets, there is one VD matrix defined as:
VDij = 1 if variable j of vector x is variable i of vector xd
VDij = 0 in other case.
Which allows to calculate the vector of free diagnosis variables xd from the vector of
variables x:
x = VD ⋅x
3.83
d1
1
x = VD ⋅x
3.84
d2
2
In this situation, it is possible to write:
⎡ JR ⎤
⎢ ⎥ ⎡ 0 ⎤
⎢
JRD
⎥
⋅Δ x = JD1 ⋅Δx ≅ ⎢
Δ
⎥
⎢ ⎥
⎣ xd
1 ⎦
⎣VD1 ⎦
⎡ JR ⎤
⎢ ⎥ ⎡ 0 ⎤
⎢
JRD
⎥
⋅Δ x = JD2 ⋅Δx ≅ ⎢
Δ
⎥
⎢ ⎥
⎣ xd2
⎦
⎣VD2 ⎦
By inverting JD matrices, it is possible to relate the variation of all the variables
with the variation of the free diagnosis variables.
−1
1
3.85
3.86
Δx≅ JD ⋅Δx d1
3.87
−1
Δ ≅ 2 ⋅Δ d2
x JD x 3.88
At this point, it is possible to connect the variations of both sets of free diagnosis
variables:
Δ x = VD ⋅Δx ≅VD ⋅JD ⋅Δ x = DD ⋅Δx
3.89
d2 2 2
−1
1 d1 d1
57

Chapter 3
where DD is the matrix which connects the variations of the two sets of free
diagnosis variables. Each element DDij indicates the variation of the i th component of
xd2 caused by an increment of one unit of the value of the j th component of xd1.
A global efficiency indicator e is also considered. Since this indicator is a function
of the set of thermodynamic variables x, there is a vector ex which relates their
variations:
58
t
Δe≅ ⋅Δ
ex x 3.10
The variation of this global efficiency indicator can be related to the variations of
both sets of free diagnosis variables:
Δe≅ex ⋅JD ⋅Δ x = ed ⋅Δx
3.90
t −1
t
1 d1 1 d1
Δe≅ex ⋅JD ⋅Δ x = ed ⋅Δx
3.91
t −1
t
2 d2 2 d2
For convenience, matrices ed can also be expressed as a product of a unit vector and
a diagonal matrix:
t
Δe≅ ⋅ 1⋅Δ d1
u ED x 3.92
t
Δe≅ ⋅ 2⋅Δ d 2
u ED x 3.93
The last point in the development is to connect the variation of the global efficiency
indicator and the variation of xd1 through the variation of xd2, by using the matrix DD:
Δe≅ u ⋅ED ⋅Δx ≅ u ⋅ED ⋅DD⋅Δ x = u ⋅EDD⋅Δx 3.94
t t t
2 d2 2 d1 d1
Matrix EDD is calculated by multiplying matrix ED2 times matrix DD. It is very
important because each element EDDij represents the impact on e due to an increment of
a unit of the j th variable of xd1 through the i th variable of xd2.
Finally, if the increment of xd1 is expressed as the product of a diagonal matrix times
a unit vector, it is possible to write:
Δe≅u ⋅EDD⋅Δ x = u ⋅EDD⋅ΔX ⋅ u= u ⋅DE⋅u 3.95
t t t
d1 d1
Where DE is the product of EDD and ΔXd1. Each element DEij represents the impact
on e due to the j th variable of xd1 through the i th variable of xd2. This is the double
decomposition we were looking for.
3.5.2. Causality chains based on the dependence among free diagnosis
variables.
Theory previously developed can be used for any pair of sets of free diagnosis
variables. However, in the practical application, such wide range of application is

Quantitative causality analysis
usually not needed. It should be kept in mind that the main purpose of the use of
causality chains is to use free diagnosis variables which have widely accepted
definitions while induced effects are minimized. For this purpose, to introduce
correlations linking free diagnosis variables among themselves is often enough. For
example, to consider the influence of power produced or ambient conditions on the
efficiency of a component. This is the task tackled in this section.
Given a thermal system, described by a thermodynamic model, the decomposition
of the variation of a global efficiency indicator e into a summation of terms
corresponding to a set of free diagnosis variables xd is given by the following
expression:
t
Δe≅ u ⋅ED⋅Δx 3.11
d
Besides, it is possible to decompose the free diagnosis into an intrinsic and an
induced term, by using a model which connects each one of the free diagnosis variables
with the others:
x = x + AD⋅ x + bd 3.96
d d, int
d
where AD is a matrix of correction factors which allows to take into account the
crossed influence among free diagnosis variables (for example, influence of load on the
effectiveness of a heat exchanger). It can be determined by modelling or, performing
regression from plant data. The role of bd is explained later. It should be noted that most
values of AD and bd are usually zero, because, in most cases crossed effects are not
considered. Besides, elements of the diagonal of AD have to be zero.
If increments are considered, the previous equation becomes:
Δ x =Δ x + AD⋅Δx 3.97
d d, int
d
which can be substituted in equation 3.11:
Δe ≅ u ⋅ED⋅Δ x + u ⋅ED⋅AD⋅Δ x = u ⋅ED⋅Δ x + u ⋅EAD⋅Δx 3.98
t t t t
d, int d d, int
d
where each element EADij represents the variation of e induced by an increment of
one unit of the free diagnosis variable j through the free diagnosis variable i.
If the vector of the variation of the free diagnosis variables is expressed as a
product of a diagonal matrix and a unit vector, it is possible to write:
Δe≅ u ⋅ED⋅Δ x + u ⋅EAD⋅ΔX ⋅ u= u ⋅Δ e + u ⋅EA⋅u 3.99
t t t t
d, int d
int
The previous equations show how the variation of e is decomposed into two terms;
the first one is intrinsic, and the second is induced by the other free diagnosis variables.
Each element EAij indicates the variation of e induced by the free diagnosis variable j
through the free diagnosis variable i.
59

Chapter 3
The last point to be dealt with in this section is the determination of vector bd. It
has been seen how it does not influence the diagnosis because it does not affect
increments. However, it may be important in the interpretation because it determines the
value of the vector of intrinsic free diagnosis variables (xd,int). It is suggested to
calculate it by using the following expression:
60
bd =−AD⋅x d,ref
3.100
where the subscript ref means reference point, which can be an average point or
the reference state of the diagnosis. This choice makes that, for a reference value of the
free diagnosis variables, the value of induced effects is zero.
Finally it should be noted that, although the development has been made for a
linear dependence among free diagnosis variables, it is possible to consider any function
linealized by applying Taylor series.
3.5.3. Causality chains based on the dependence of free diagnosis
variables with other variables.
The possibility of introducing dependence among free diagnosis variables,
developed in the previous section, can be very interesting in some cases. However, to
introduce models of the components, it is preferable to have the possibility to take into
account the dependence of the free diagnosis variables with all the other variables. This
task can be performed by a quite similar approach, as it is explained in this section.
If there is a set of models which connect free diagnosis variables which the set of
variables, there is a matrix AX which contains the partial derivatives of free diagnosis
variables in relation to all the variables (For example, the dependence of condenser
effectiveness with cooling water mass flow and heat to be exchanged)
Δxd≅ AX ⋅Δx
3.101
It should be noted that matrix AX has to have zeros in the positions AXij if the i th free
diagnosis variable is the j th variable of vector x. Besides, increment of x can be related
−1
to the increment of xd ( Δx≅ JD ⋅Δx d ). So, matrix AD, which contains the crossed
influence among free diagnosis variables, can be calculated as:
−1
AD= AX ⋅JD
3.102
Once the matrix AD has been calculated, the procedure is exactly the same
explained for causality chains based on the dependence among free diagnosis variables.

3.6. Calculation of fuel impact on a time span.
Quantitative causality analysis
Diagnosis can be defined as the comparison of an operation situation with a situation
considered as reference; in other words, this is a static comparison. However,
degradation phenomena are dynamic (although usually very slow), and not always is
enough the comparison of two situations. Fortunately, the diagnosis method can be
easily extended to deal with these situations.
First, the global efficiency has to represent resources consumption per unit of time
of the system (et). Given a free diagnosis variable i, its impact during a given time span
from t1 to t2 can be calculated by integration:
t2 t2
∫ ∫ 3.103
Δ e = Δe⋅dt ≅ ed ⋅Δx⋅dt t1−t2, i t, i t, i d, i
t1 t1
The previous integral can be calculated numerically by dividing the total time span.
The detail of this division would depend on the information available.
This approach can be useful for dynamic situations such as progressive degradation
of a component (long time) or short time situations such as fouling.
It can be directly used to evaluate the total impact due to a given cause during a past
period. Besides, it can be applied to predict future situations if a degradation model is
available. Neural networks can be very useful for this task.
3.7. Conclusion
In this chapter, the quantitative causality analysis for the diagnosis of energy
systems has been developed. Although the method starts from a diagnosis algorithm,
several features have been developed and analyses have been made in order to develop a
complete diagnosis methodology.
First of all, a new nomenclature has been introduced in order to clarify the notation
and the concepts. Second, a methodology to quantify the influence of measurement
errors on the diagnosis results has been developed. Then, the error due to linearization
made in the method is analyzed, and techniques to reduce it are commented.
Afterwards, some ideas on the choice of the free diagnosis variables are presented. If it
is difficult to choose a definition for a free diagnosis variable which has wide
acceptation and reproduces the physical behaviour of the system at the same time, it is
61

Chapter 3
possible to apply causality chains theory developed here. Finally, the idea of fuel
impact during a time span has been introduced.
A simple example has been used to clarify the basic ideas presented. Besides, most
of the topics developed here are applied to a real example in Chapter 6.
62

4. Quantitative causality analysis and other
diagnosis methods
In the previous chapter, a complete diagnosis methodology has been presented. This
methodology is based on the linearization of equations of the thermodynamic
representation of the thermal system to be diagnosed. In next chapters, it is going to be
applied to the diagnosis of a three units coal fired power plant. However, as it has been
seen in chapter two, there are other methodologies, several of them based on the
thermoeconomic representation of the system. It would be interesting to analyze
connections of both formulations. This is the aim of the first section of this chapter.
In the second section, the capabilities of linear regression and neural networks for
the diagnosis problem are explored
4.1. Quantitative causality analysis and the fuel impact
formula
The fuel impact formula was developed to relate the variation of the exergy
resources entering the plant to the variations of the independent variables of the
thermoeconomic model (unit exergy consumptions and final products). Problems in the
application of this formula appear because unit exergy consumptions are not actually
independent variables, so that induced effects are present.
On the other hand, the quantitative causality analysis method is able to relate a set of
independent variables to a user-defined efficiency indicator. However, this flexibility to
be adapted to the real thermodynamic behaviour of the system makes impossible to
obtain analytical formulas, so numerical calculations are used.
63

Chapter 4
The idea proposed here is to link both approaches by considering the independent
variables of the thermoeconomic analysis (unit exergy cost and plant product) as the
dependent efficiency indicators of quantitative causality approach. This allows to
determine the value of both intrinsic and induced effects.
4.1.1. From free diagnosis variables to unit exergy cost and final
product.
Given a thermodynamic description of an energy system, where a set of free
diagnosis variables consistent with a set of restrictions has been chosen. The
quantitative causality algorithm allows to relate the variation of all the variables
describing the system to the variations of the free diagnosis variables.
64
0
Δx ≅ JD ⋅⎢ ⎥
−1 ⎡ ⎤
⎣Δxd⎦ Once an efficiency indicator e has been chosen, the variation of this indicator can be
related to the variation of the free diagnosis variables by using the vector ex containing
the partial derivatives of e respect to the set of thermodynamic variables x:
t t −1 ⎡ 0 ⎤
Δe≅ex ⋅Δx≅ex ⋅JD ⋅⎢ Δ
⎥
4.2
⎣ xd
⎦
Variable e is defined as a kind of efficiency indicator only for convenience, because
of this is the interest of the diagnosis methodology. It is possible to relate other
variables depending on the vector x only by replacing the vector ex by other vector
containing the partial derivatives of the new variable to the components of the vector x.
The same system can be represented by a thermoeconomic model. Once the model
has been described and the purpose of each susbsystem introduced, the magnitudes of
this representation depend only on the set of thermodynamic variables x. So that, the
variation of this magnitudes can be related to Δx only by multiplying times a suitable
vector containing the corresponding partial derivatives. Thermoeconomic representation
of a system includes exergy of flows, exergy and unit exergy costs of flows, and unit
exergy consumptions of components. However, when the fuel impact formula is applied
(for small deviations), the only free variables are unit exergy consumptions and exergy
flows leaving the component (final products).
n ⎛ n
* 1
0 * 1 ⎞
Δ FT = ∑∑ ⎜ kP, j( x ) Δ κ jiPi( x ) + kP,
i( x ) Δϖi⎟
4.3
i= 1⎝ j=
0
⎠
4.1

Quantitative causality analysis and other diagnosis methods
It should be noted that in this section the set of thermodynamic variables
representing actual and reference states are represented as x1 and x 0 (nomenclature used
in the quantitative causality analysis) instead on x and x0 (usual in thermoeconomic
analysis).
In this situation, it is interesting to consider κij and ωi as the indicators depending on
x. Each specific consumption κij depends on x, so that:
ij
ij, t
nt
l
∑ kxij l=
1
xl
Δκ≅kx ⋅Δ x = ⋅Δ
4.4
Where kx ij is an nt x 1 vector containing the partial derivatives of the unit exergy
consumption κij related to the set of nt variables of the thermodynamic model of the
system. By using the vector xd, the variation of the unit exergy consumption can be
finally related to the variation of the free diagnosis variables:
nd
ij, T −1
⎡0⎤ r l r
Δ κij = kx ⋅JD ⋅⎢ ⎥⋅Δ
xd
+ κij = ∑ kdij ⋅Δx
d , l + κij
4.5
⎣U⎦ l=
1
Where the term r
κ ij is the residual term, introduced to keep the equal sign. The
previous relation can be extended to all the unit exergy cost of the system, including the
κ :
t
unit exergy costs of flows coming from the outside = ( , k ,..., k )
e
k01 02 0n
t n
⎡ d Δ ⎤ e l r
⎢ ⎥ = ⋅Δ xdl
, +
⎣Δ⎦ l=
1
∑
k
KD KD
4.6
KP
Where each one of the nd matrices KD is a ((n + 1) x n) matrix containing the
elements l
ij
r
k or k :
ij
l l l
⎡kd01 kd02 ... kd ⎤ 0n
⎢ l l l ⎥
l kd11 kd12 ... kd0n
KD = ⎢ ⎥
4.7
⎢ . . ... . ⎥
⎢ l l l ⎥
⎢⎣kdn1 kdn2 ... kdnn⎥⎦
r r r
⎡kd01 kd02 ... kd ⎤ 0n
⎢ r r r ⎥
r kd11 kd12 ... kd0n
KD = ⎢ ⎥
4.8
⎢ . . ... . ⎥
⎢ r r r ⎥
⎢⎣kdn1 kdn2 ... kdnn⎥⎦
A similar development can be made for each one of the products of the plant:
it ,
Δωi≅ ⋅Δ
wx x 4.9
nd nd
it , −1
⎡0⎤ r l r l r
Δ ωi = wx ⋅JD ⋅⎢ ⎥⋅Δ
xd
+ ωi = ∑wdi ⋅Δ xd,
l + ωi = ∑Δ
ωi + ωi
⎣U⎦ l= 1 l=
1
4.10
65

Chapter 4
66
Finally, the previous equation can be applied for all the products leaving the plant:
nd ∑
l
xdl
,
r
nd
∑
l r
l= 1 l=
1
Δ ω = ω ⋅Δ + ω = Δ ω + ω 4.11
If wastes (or residues) are considered in the thermoeconomic model, the fuel impact
formula becomes:
n n n
⎛ * 1 0 * 1 0 * 1 ⎞
Δ FT = ∑∑ ⎜ kP, j( x ) Δ κ jiPi( x ) + ∑kPR,
j( x ) Δ θ jiPi( x ) + kP,
i( x ) Δϖi⎟
4.12
i= 1⎝ j= 0 j=
1
⎠
In this situation, θij are also independent variables of the thermoeconomic model,
which have to be decomposed:
nd
ij, t −1
⎡0⎤ r l r
Δ θij = tx ⋅JD ⋅⎢ ⎥⋅Δ
xd
+ θij = ∑tdij⋅Δx
d , l + θij
⎣U⎦ l=
1
If all elements θij are considered together:
r or θ ij .
nd
l=
1
4.13
l r
Δ Θ = ∑TD ⋅Δ xdl
, + TD
4.14
l
where each one of the nd matrices TD is a (n x n) matrix containing the elements θ ij
Equations presented above allow to relate the independent variables of the
thermoeconomic model of a system to the free diagnosis variables. This result is very
useful to determine the effects induced by the variation of ambient conditions and set
points and to malfunctions appearing in other components.
4.1.2. Quantification of intrinsic and induced malfunctions
According to the fuel impact formula, variations of specific exergy consumptions
and final products are the source of the variation of fuel entering the plant. Besides,
these variations are also the source of malfunctions and malfunction costs. In this
section, expressions obtained previously for Δ κij
, Δ ωi
and Δ θij
are substituted in the
fuel impact formula and in the definitions of malfunctions and malfunction costs, in
order to definitely link the diagnosis approach based on the thermoeconomic model and
quantitative causality analysis.
If the decomposition of Δ κij
, Δ ωi
and Δ θij
into elements corresponding to the free
diagnosis variables is substituted in the fuel impact formula, it yields:

Quantitative causality analysis and other diagnosis methods
T
n ⎡ n
∑∑ ⎢
nd * 1 ⎛
P, j ( x ) ⎜∑ l
ji
r ⎞
ji ⎟
0
i ( x )
n
nd * 1 ⎛
∑ PR, j ( x ) ⎜∑ l
θ ji
r ⎞
θ ji⎟ 0
i( x )
nd
* 1
P, i( x ) ∑(
l
ωi ⎤ r
ωi
) ⎥
i= 1 j= 0 l= 1 j= 1 l= 1 l=
1
Δ F = k ⋅ Δ k + k ⋅ P + k ⋅ Δ + ⋅ P + k ⋅ Δ +
⎣ ⎝ ⎠ ⎝ ⎠
⎦
4.15
In the previous equation, the impact in fuel due to each variation of unit exergy
consumption and each fuel variation is decomposed into impacts due the free diagnosis
variables. It can be rearranged by introducing elements kd, wd and td:
Δ F =
⎡
⎢⎣ ⎛
⎝
k ⋅kd ⋅ P + k ⋅td ⋅ P + k ⋅wd ⎞⎤
⋅Δ x
⎠⎥⎦
4.16
+ F
The previous equation shows how the fuel increment can be shared into several
nd n n n
* 1 l 0
l
* 1 l 0 * l r
T ∑∑∑ ⎢ ⎜ P, j( x ) ji i( x ) , ( ) ( )
i ∑ PR j x ji i x P, i i ⎟⎥
d, l T
l= 1 i= 1 j= 0 j=
1
components each one due to an independent free diagnosis variable.
term, calculated as:
r
F T is a residual
n n n
r ⎡ * 1 r 0 * 1 r 0 * 1 r⎤
FT = ∑∑ ⎢ kP, j( x ) ⋅kji⋅ Pi( x ) + ∑kPR,
j( x ) ⋅θji⋅ Pi( x ) + kP,
i( x ) ⋅ωi
⎥ 4.17
i= 1 ⎣ j= 0 j=
1
⎦
Malfunctions can also be decomposed in a summation of several terms each one
corresponding to an impact due to a free diagnosis variable:
nd nd
0 ⎛ l r ⎞ 0
l r
MFji =Δkji ⋅ Pi ( x ) = ⎜∑Δ k ji + k ji ⎟⋅
Pi ( x ) = ∑MFji
+ MFji
4.18
⎝ l= 1 ⎠
l=
1
n n nd nd
⎛ l r ⎞ 0
l r
MFi = ∑MFji = ∑⎜∑Δ k ji + k ji⎟⋅ Pi( x ) = ∑MFi
+ MFi
4.19
j= 0 j= 0⎝ l= 1 ⎠
l=
1
where
l
MF ji and
to MF ji and MF i :
l l
ji ji i
0 ( )
l
MF i are respectively the contribution of the l th diagnosis variable
MF =Δk ⋅P x 4.20
n n
0
∑ ( ) ∑
MF = Δk ⋅ P x = MF
4.21
l l l
i ji i ji
j= 0 j=
0
r
MF ji and
r
MF ji are the residual term of malfunctions. The same decomposition can
be made for malfunction associated with residues:
nd nd
0 ⎛ l r ⎞ 0
l r
MRji =Δθ ji ⋅ Pi( x ) = ⎜∑Δ θ ji + θ ji ⎟⋅
Pi( x ) = ∑MRji
+ MRji4.22
⎝ l= 1 ⎠
l=
1
n n nd nd
⎛ l r ⎞ 0
l r
MRi = ∑MRji = ∑⎜∑Δ θ ji + θ ji⎟⋅ Pi( x ) = ∑MRi
+ MRi
4.23
j= 1 j= 1⎝ l= 1 ⎠
l=
1
67

Chapter 4
Malfunction cost can also be decomposed in the summation of several components,
each one due to a variation of a free diagnosis variable.
68
*
ji =
* 1
P, j ( x ) ⋅
nd ji = ∑
* 1
P, j ( x ) ⋅Δ
l
ji ⋅
0
i ( x ) +
* 1
P, j ( x ) ⋅
r
ji ⋅
nd
0
i ( x ) = ∑
*, l
ji +
*, r
ji
l= 1 l=
1
MF k MF k k P k k P MF MF
4.24
n
*
i = ∑
n nd *
ji = ∑∑
* 1
P, j( x ) ⋅Δ
l
ji⋅ n
0 * 1
i( x ) + ∑ P, j( x ) ⋅
r
ji⋅ nd
0
i( x ) = ∑
*, l
i +
*, r
i
j= 0 j= 0 l= 1 j= 0 l=
1
MF MF k k P k k P MF MF
where
l
MF ji
*, and
l
MF i
*, are the contributions of x d , l
( ) ( )
*, l * 1 l 0
ji P, j ji i
Δ to
MF and MF :
*
ji
*
i
4.25
MF = k x ⋅Δk⋅Px 4.26
n n
∑ , ( ) ( ) ∑
MF = k x ⋅Δk ⋅ P x = MF
4.27
*, l * 1 l 0 *, l
i P j ji i ji
j= 0 j=
0
*,r
MF ji and
*,r
MF i are the residual term of malfunction costs. The malfunction cost
due to the variation of the product
variation of free diagnosis variables:
*
MF 0 can also be shared into impacts due to the
n n ndnd * * 1 ⎛ * 1 l * 1 r⎞ *, l *, r
MF0 = ∑kPi , ( x ) ⋅Δωi≅∑⎜∑kPi , ( x ) ⋅Δ ωi+ kPi , ( x ) ⋅ ωi⎟=
∑MF0
+ MF0
i= 1 i= 1⎝ l= 1 ⎠ l=
1
4.28
Where
l
MF *,
0 is the contribution of xd,l to
*
MF 0 and
MF is the residual.
Finally, the decomposition can also be extended to the malfunction cost associated
with the residues:
*
ji =
* 1
PR, j ( x ) ⋅
nd ji = ∑
* 1 l
PR, j ( x ) ⋅Δθ ji ⋅
0
i ( x ) +
* 1 r
PR, j ( x ) ⋅θ ji ⋅
nd
0
i ( x ) = ∑
*, l
ji +
*, r
ji
l= 1 l = 1
MR k MR k P k P MR MR
4.29
n
*
i = ∑
n nd *
ji= ∑∑
* 1 l
PR, j( x ) ⋅Δθ ji⋅ n
0 * 1 r
i( x ) + ∑ PR, j( x ) ⋅θ ji⋅ nd
0
i( x ) = ∑
*, l
i +
*, r
i
j= 0 j= 0 l= 1 j= 0 l=
1
MR MR k P k P MR MR
4.30
The impact in fuel is the summation of the malfunction costs of all the components,
included
*
MF0 and malfunctions associated with wastes:
*, r
0

n
*
n
*
T i i
i= 0 i=
1
Δ F = MF + MR
Quantitative causality analysis and other diagnosis methods
∑ ∑ 4.31
The previous equation can be expanded by introducing
l
MF i
*, and
*,l
MR i :
Δ F = MF + MR =
⎛
MF + MF
⎞
+
⎛
MR + MR
⎞
= MF + MF + MR + MR
⎝ ⎠ ⎝ ⎠
T
n
∑
*
i
n
∑
*
i
n nd ∑⎜∑ *, l
i
*, r
i ⎟
n nd ∑⎜∑ *, l
i
*, r
i ⎟
nd ∑
*, l *, r
nd
∑
*, l *, r
i= 0 i= 1 i= 0 l= 1 i= 1 l= 1 l= 1 l=
1
l
Where MF *, *,l
and MR are the contribution of the fuel consumption variation due
to the free diagnosis variable xd,l, associated with productive flows and to residues:
MF
*, l
=
n
∑
i=
0
MF
*, l
i
n
*, l *, l
i
i=
1
4.32
4.33
MR = ∑ MR
4.34
MF
MF
*, 1
0
*, 1
1
n
MF … d MF *,
*, 2
0
n
MF … d MF *,
*, 2
1
0
1
MF
MF
*, r
0
*, r
1
*
MF 0
*
MF 1
… … … … … …
MF
*, 1
n
*,1
MR 1
n
MF … d MF *,
*, 2
n
*,2
*,
MR …
1
1 d n
MR
n
MF
MR
*, r
1
*, r
1
*
MF n
*
MR 1
… … … … … …
*,1
MR n
*,1 *,1
MF + MR
*,2
*, n
MR … d
n
MR n
*,2 *,2
MF + MR
*, n *,
… d nd
MF + MR
*,r
MR n
*
MR n
… T F Δ
Table 4.1: Table of malfunctions and free diagnosis variables (MFD).
As it is shown by the previous equations, the impact on fuel can be obtained by a
double summation of the cost of malfunctions in each component due to each free
diagnosis variable. So that,
l
MF i
*, and
*,l
MRi elements can be represented in a table,
where each column correspond to a free diagnosis variable xd,l and each component can
have one or two rows depending on whether there are waste flows associated with it.
The summation of elements in a column is
l
MF *, *,l
+ MR and the summation of
69

Chapter 4
elements in a row is MF or
70
*
i
*
MR i . Finally, the total summation corresponds to ΔFT.
This table shows how the variation of fuel consumption is due to the malfunctions
induced by the free diagnosis variables in all the components and in the variation of
final product. So that, it can be named table of malfunctions induced by the free
diagnosis variables (MFD).
The previous analysis has been done by decomposing the variation of unit exergy
consumptions into contributions due to each free diagnosis variable. This study is
interesting but when the system is big and there are more than twenty or thirty variables
this analysis may become rather complex. So that, it may be better to do a similar study
but separating the variation of unit exergy consumptions into intrinsic variations (int)
and variations induced by other components (oc), ambient conditions (ac), fuel quality
(fq) and set points (sp). It should be noted that variations induced by ambient
conditions, fuel quality and set points are calculated simply by a summation of the
variations induced by the free diagnosis variables related to these categories. However,
separation of intrinsic and induced effect depends not only on the free diagnosis
variables but also on the component. Each free diagnosis variable corresponding to a
parameter of a component originates an intrinsic variation of κij and θij in that
component and induced variations in the other components.
Δ κ =Δ κ +Δ κ +Δ κ +Δ κ +Δ κ +Δ κ
4.35
int oc ac fq sp r
ij ij ij ij ij ij ij
Δ θ =Δ θ +Δ θ +Δ θ +Δ θ +Δ θ +Δ θ
4.36
int oc ac fq sp r
ij ij ij ij ij ij ij
Δ ω =Δ ω +Δ ω +Δ ω +Δ ω +Δ ω +Δ ω
4.37
int oc ac fq sp r
i i i i i i i
The fuel impact formula can be modified to consider separately intrinsic and
induced effects:
n n ⎡ * 1 int oc ac fq sp r
0 ⎤
Δ FT = ∑∑ ⎢ kP, j( x ) ⋅( Δ κ ji +Δ κ ji +Δ κ ji +Δ κ ji +Δ κ ji + κ ji) ⋅ Pi(
x ) ⎥+
i= 1 ⎣ j=
0
⎦
n ⎡ n
* 1 int oc ac fq sp r
0 ⎤
+ ∑∑ ⎢ kPR, j ( x ) ⋅( Δ θ ji +Δ θ ji +Δ θ ji +Δ θ ji +Δ θ ji + θ ji ) ⋅ Pi(
x ) ⎥+
i= 1 ⎣ j=
1
⎦
oc ac fq sp r
( ) ( )
n
* 1 int
∑ ⎡kPi , ωiωiωiωiωiω ⎤
⎣
x i ⎦
4.38
i=
1
+ ⋅ Δ +Δ +Δ +Δ +Δ +
Malfunctions can also be decomposed into intrinsic and induced:
0 int oc ac fq sp r
0
( ) ( ) ( )
MF =Δκ ⋅ P x = Δ κ +Δ κ +Δ κ +Δ κ +Δ κ + κ ⋅P
x 4.39
ji ji i ji ji ji ji ji ji i

int oc ac fq sp r
ji ji ji ji ji ji ji
Quantitative causality analysis and other diagnosis methods
MF = MF + MF + MF + MF + MF + MF
4.40
n n
int oc ac fq sp r
∑ ∑ ( ) 4.41
MF = MF = MF + MF + MF + MF + MF + MF
i ji ji ji ji ji ji ji
j= 0 j=
0
MF = MF + MF + MF + MF + MF + MF
4.42
Where
int oc ac fq sp r
i i i i i i i
x
MFi and
x
MF ji (x = int, oc, ac, fq and sp) are malfunctions intrinsic and
induced by other components, ambient conditions, fuel quality and set points:
x x
ji ji i
0 ( )
MF =Δk ⋅P x 4.43
n n
0
∑ ( ) ∑
MF = Δk ⋅ P x = MF
4.44
x x x
i ji i ji
j= 0 j=
0
A similar development can be made with malfunctions associated with residues:
0 int oc ac fq sp r
0
( ) ( ) ( )
MR =Δθ ⋅ P x = Δ θ +Δ θ +Δ θ +Δ θ +Δ θ + θ ⋅P
x 4.45
ji ji i ji ji ji ji ji ji i
MR = MR + MR + MR + MR + MR + MR
4.46
int oc ac fq sp r
ji ji ji ji ji ji ji
n n
int oc ac fq sp r
∑ ∑ ( ) 4.47
MR = MR = MR + MR + MR + MR + MR + MR
i ji ji ji ji ji ji ji
j= 0 j=
0
MR = MR + MR + MR + MR + MR + MR
4.48
int oc ac fq sp r
i i i i i i i
The decomposition in intrinsic and induced effects can be extended to malfunction
costs:
oc ac fq sp r
( x ) ( x ) ( ) ( x )
MF = k ⋅ MF = k ⋅ Δ k +Δ k +Δ k +Δ k +Δ k + k ⋅P
* * 1 * 1 int 0
ji P, j ji P, j ji ji ji ji ji ji i
* *,int *, oc *, ac *, fq *, sp *, r
ji ji ji ji ji ji ji
4.49
MF = MF + MF + MF + MF + MF + MF
4.50
n n
oc ac fq sp r
∑ ∑ ( ) 4.51
MF = MF = MF + MF + MF + MF + MF + MF
* * *,int *, *, *, *, *,
i ji ji ji ji ji ji ji
j= 0 j=
0
MF = MF + MF + MF + MF + MF + MF
4.52
Where
* *,int *, oc *, ac *, fq *, sp *, r
i i i i i i i
x
MF i
*, and
MF *, (x = int, oc, ac, fq and sp) are the costs of malfunctions
x
ji
intrinsic and induced by other components, ambient conditions, fuel quality and set
points:
( ) ( )
MF = k x ⋅Δk⋅Px 4.53
*, x * 1 x 0
ji P, j ji i
n n
∑ , ( ) ∑
MF = k Δk ⋅ P x = MF
4.54
*, x * x 0 *, x
i P j ji i ji
j= 0 j=
0
71

Chapter 4
Again, a parallel distribution is made for the malfunction costs associated with the
residues:
72
oc ac fq sp r
( x ) ( x ) ( θ θ θ θ θ θ ) ( x )
MR = k ⋅ MR = k ⋅ Δ +Δ +Δ +Δ +Δ + ⋅P
* * 1 * 1 int 0
ji PR, j ji PR, j ji ji ji ji ji ji i
* *,int *, oc *, ac *, fq *, sp *, r
ji ji ji ji ji ji ji
4.55
MR = MR + MR + MR + MR + MR + MR
4.56
n n
oc ac fq sp r
∑ ∑ ( ) 4.57
MR = MR = MR + MR + MR + MR + MR + MR
* * *,int *, *, *, *, *,
i ji ji ji ji ji ji ji
j= 0 j=
0
MR = MR + MR + MR + MR + MR + MR
4.58
* *,int *, oc *, ac *, fq *, sp *, r
i i i i i i i
Finally, fuel impact due to the variation of the final product can be separated into
induced and intrinsic effects:
n n
oc ac fq sp r
∑ P i( x ) ωi ∑ P i( x ) ( ωi ωi ωi ωi ωi ωi
)
MF = k ⋅Δ = k ⋅ Δ +Δ +Δ +Δ +Δ +
* * 1 * 1 int
0 , ,
i= 1 i=
1
* *,int *, oc *, ac *, fq *, sp *, r
0 0 0 0 0 0 0
4.59
MF = MF + MF + MF + MF + MF + MF
4.60
Where
x
MF *,
0 can be the fuel impact caused by the variation of plant product due to
intrinsic causes and causes induced by other components, ambient conditions, fuel
quality and set points.
Finally, the fuel impact can be expressed by a summation either of malfunction costs
of each component or malfunction costs intrinsic and induced by other components,
ambient conditions, fuel quality and set points:
n n n
* * *,int *, oc *, ac *, fq *, sp *, r
∑ ∑ ∑ ( )
Δ F = MF + MR = MF + MF + MF + MF + MF + MF +
T i i i i i i i i
i= 0 i= 1 i=
0
n
i=
1
*,int *, oc *, ac *, fq *, sp *, r
( MRi MRi MRi MRi MRi MRi
)
+ ∑ + + + + +
4.61
*,int *, oc *, ac *, fq *, sp *, r
Δ FT= MF + MF + MF + MF + MF + MF +
*,int *, oc *, ac *, fq *, sp *, r
+ MR + MR + MR + MR + MR + MR
4.62
x
Where MF *, *,x
and MR are the fuel impacts due to effect intrinsic and induced by
other components, ambient conditions, fuel quality and set points.
MF
*, x
=
n
∑
i=
1
MF
*, x
i
n
*, x *, x
i
i=
1
4.63
MR = ∑ MR
4.64

Quantitative causality analysis and other diagnosis methods
As it can be seen, the separation of malfunctions into intrinsic and induced has been
made in a parallel way as the separation of malfunctions into the different free diagnosis
variables. So that, result can also be summarized in a table of intrinsic and induced
malfunctions (MFI). In this table, elements
x
MF i
*, and
*,x
MRi are placed. Each column
corresponds to a type of malfunction while each component has two rows: for
malfunction associated with productive flows and with wastes. The summations of
columns are
x
MF *, *,x
+ MR , and the summations of rows are elements
Total summation is the fuel impact ΔFT.
MF
MF
MF
*, int
0
*, int
1
oc
MF *,
0
oc
MF *,
1
ac
MF *,
0
ac
MF *,
1
fq
MF *,
0
fq
MF *,
1
sp
MF *,
0
sp
MF *,
1
MF
MF
*, r
0
*, r
1
MF or MR .
… … … … … … …
*, int
n
*,int
MR 1
oc
MF n
*,
MR
*, oc
1
ac
MF n
*,
MR
*, ac
1
fq
MF n
*,
MR
*, fq
1
sp
MF n
*,
MR
*, sp
1
*,r
MF n
… … … … … … …
*,int
MR n
*, int
MF
+ MR
*,int
*,oc
MR n
oc
MF *,
+ MR
*,oc
*,ac
MR n
ac
MF *,
+ MR
*,ac
*, fq
MR n
fq
MF *,
+ MR
*, fq
*,sp
MR n
sp
MF *,
+ MR
*,sp
MR
*, r
1
*,r
MR n
*,r
MF
+ MR
Table 4.2: Table of intrinsic and induced malfunctions (MFI).
*,r
*
i
*
MF 0
*
MF 1
*
MF n
*
MR 1
*
MR n
4.1.3. Determination of the suitability of a productive structure.
The main source of inaccuracy in the application of thermoeconomic analysis to the
diagnosis problem is due to the induced effects. To overcome it, several filtering
strategies as well as variations of the productive structure have been proposed. The
suitability of these techniques is usually determined by testing them in specific
problems; so that, it is difficult to obtain general conclusions. In this way, when a new
problem is tackled, it is very difficult to assure that a specific thermoeconomic model is
suitable to diagnose it and to determine the error produced. Ideas presented in previous
sections can help in this task. Equations developed to link the variations of free
diagnosis variables and the variations of thermoeconomic parameters have been
developed, but considering an example of diagnosis (a comparison of an actual and a
Δ FT
*
i
73

Chapter 4
reference situation). So, they are not suitable to draw general conclusions about the
connection of physical and thermoeconomic models. However, this general picture can
be obtained by considering an artificial diagnosis example, where free diagnosis
variations have a representative value. This is the goal of this section.
First of all, KD and TD matrices presented above can be very useful to determine the
influence of induced effects in the unit exergy consumptions of a component due to:
ambient conditions, fuel quality, set points and anomalies in other components. Each
value
74
l
l kd ij or td ij represents the sensibility of the unit exergy consumption κ ij or θ ij to
the free diagnosis variable l. So that, the direct observation of KD l and TD l matrices
allows to determine whether a variation in a free diagnosis variable induces a variation
in an unit exergy consumption. However, the direct observation of
l kd and θ d values
is not very useful for comparison because they are expressed in different units (free
diagnosis variables have different units). This problem can be solved by multiplying
them times ‘representative’ increments of the corresponding free diagnosis variables. If
a large amount of plant data is available, a representative variation of the value of a
given free diagnosis variable can be its standard deviation. If not, it can be calculated by
dividing the difference of maximum minus minimum value into two.
Δ x = σ Δ
4.65
rep
di , xi
Δ κ = kd ⋅Δ x
4.66
lrep ,
l rep
ij ij d , l
rep
ij
nd l, rep
ij
nd
l
kdij rep
xd
, l
l= 1 l=
1
∑ ∑ 4.67
Δ κ = Δ κ = ⋅Δ
Δ θ = td ⋅Δ x
4.68
lrep , l rep
ij ij d , l
rep
ij
nd l, rep
ij
nd
l
tdij rep
xd
, l
l= 1 l=
1
∑ ∑ 4.69
Δ θ = Δ θ = ⋅Δ
Δ ω = wd ⋅Δ x
4.70
lrep ,
l rep
ij ij d , l
rep
ij
nd l, rep
ij
nd
l
wdij rep
xd
, l
l= 1 l=
1
∑ ∑ 4.71
Δ ω = Δ ω = ⋅Δ
It should be noted that the previous decomposition can be considered as exact
(without the need of any residual term), because this is a fictitious situation. It is also
possible to make the same decomposition into intrinsic variations (int) and variations
induced by other components (oc), ambient conditions (ac), fuel quality (fq) and set
points (sp):
l
ij
ij

Quantitative causality analysis and other diagnosis methods
rep
nd
l, rep int, rep oc, rep ac, rep fq, rep sp, rep
ij
l=
1
ij ij ij ij ij ij
Δ κ = ∑ Δ κ =Δ κ +Δ κ +Δ κ +Δ κ +Δκ
4.72
rep
nd
l, rep int, rep oc, rep ac, rep fq, rep sp, rep
ij
l=
1
ij ij ij ij ij ij
Δ θ = ∑ Δ θ =Δ θ +Δ θ +Δ θ +Δ θ +Δθ
4.73
rep
nd
l, rep int, rep oc, rep ac, rep fq, rep sp, rep
i i i i i i i
l=
1
Δ ω = ∑ Δ ω =Δ ω +Δ ω +Δ ω +Δ ω +Δω
4.74
By using the previous decompositions of κij, θij and ωi, it is possible to calculate
representative values of malfunctions and malfunctions costs, by using expressions
developed in section 4.1.2.
4.1.4. Conclusion
The aim of thermoeconomic diagnosis is to determine intrinsic malfunctions, so that,
a good thermoeconomic model should be able to eliminate induced effects.
Malfunctions induced by ambient conditions and fuel quality are usually not present
because the reference condition is usually chosen with the same external conditions.
Malfunctions induced by the set points are usually also eliminated, except in some
approaches that are based in the filtration of effects induced by the control system
(Verda et al., 2004). Finally, malfunctions induced by anomalies in other components
are the most difficult to eliminate, so they constitute the main research topic in the
application of the thermoeconomic approach to the diagnosis of energy systems. The
formulation proposed here is able to quantify the importance of this induced terms, and
thus to evaluate the ability of a given productive structure to reproduce the
thermodynamic behaviour of the system. It is applied in Chapter 7.
4.2. Application of neural networks and linear regression to
the diagnosis of energy systems.
Diagnosis methods previously presented are based on a representation of the thermal
system analyzed by using either thermodynamic or thermoeconomic models, or the
combination of both. However, when operation data are available, it is possible to use
this information to build black-box models which can help to the diagnosis task. This
idea is explored in this section and applied in Chapter 8 and in Section 6.6.
75

Chapter 4
First of all, fundamentals of linear regression and neural networks are sumarized.
Afterwards, the possibilities of these tools to perform diagnosis of energy systems are
presented.
4.2.1. Linear regression fundamentals.
In general, a linear regression model represents the relationship between a
continuous response (y) and a continuous or categorical predictor x (x1,…xN) in the
form:
y = β ⋅ f ( x) + β ⋅ f ( x) + + β ⋅ f ( x ) + ε
4.75
76
1 1 2 2 ... p p
The response is modelled as a linear combination of (not necessarily linear)
functions of the predictor, plus a random error ε. This error is assumed to be
uncorrelated and distributed with mean 0 and constant (but unknown) variance. The
expressions fj(x) (j=1,…,p) are the terms of the model. The βj (j=1,…,p) are the
coefficients.
The simplest examples of linear regression are linear additive models:
y = β ⋅ x + β ⋅ x + + β ⋅ x + β + ε
4.76
1 1 2 2 ... N N N+
1
Given n independent observations (x 1 , y 1 ), (x 2 , y 2 ), …, (x n ,y n ) of the predictor x and
the response y, the aim of regression is to estimate the model parameters βj (j=1,…,p).
( ) ( )
1
1 1
⎛ y ⎞ ⎛ f1x fpx ⎞ ⎛ β ⎞ 1 ⎛ε1⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟= ⎜ ⎟⋅
⎜ ⎟+ ⎜
⎟
⎜ n
y ⎟ ⎜ n n ⎟
⎜ f
1 ( x ) fp( x
⎜ ) ⎟ β ⎟ ⎜
p ε ⎟
⎝ ⎠ ⎝ n
⎝
⎠⎠
⎝⎠
y
ε
X
β
4.77
In general, n>p and the problem is overdetermined, so that coefficients ˆ
β are
calculated as estimators of β to minimise the sum of squares of the errors. ˆ
β is
unbiased with expected value β, and it has minimum variance among all unbiased
estimators formed from linear combinations of the response data. Geometrically, to
minimize the sum of squares of the errors,
y−X ⋅ ˆ β must be perpendicular to the
column space of X, which contains all the linear combinations of the model terms. This
is summarized in the normal equations:
( )
t
X ⋅ y−X ⋅ ˆ β =0
4.78
which allow to calculate ˆ
β from X and y:

( ) 1
Quantitative causality analysis and other diagnosis methods
ˆ t
−
t
β = X ⋅X ⋅X ⋅ y 4.79
It should be noted that normal equations are often bad conditioned, so that it is usual
not to solve them directly but by using a QR (orthogonal, triangular) decomposition of
X.
The previous calculation of ˆ β as an estimator of β relies on the independence of
the model terms. When terms are correlated and the columns of the design matrix X
have an approximate linear dependence, the matrix ( ) 1
T
−
⋅
X X becomes close to
singular and the least squares estimate ˆ
β becomes highly sensitive to random errors in
the observed response y, producing a large variance. This situation of multicollinearity
can arise, for example, when data are collected without an experimental design, which is
quite usual in diagnosis of real operating systems. Multicollinearity can be detected by
observing the correlation matrix R (p x p) defined as:
R
n
l l
∑(
xi −xi) ⋅( xj −xj)
l=
1
ij =
n n
1/2
⎡ l
2
l
2⎤
⎢∑( xi −xi) ⋅∑( xj −xj)
⎥
⎣ l= 1 l=
1 ⎦
where x is the mean value:
x
i
=
4.80
n
∑
l
xi
l=
1
4.81
n
R is symmetrical and contains ones in the diagonal. The other values vary between -
1 and 1. An absolute value near to 1 outside the diagonal indicates high correlation
between the variables.
There are several methods to deal with multicollinearity. When a variable is highly
linear dependent with others, it might be possible to eliminate it (assuming that
information is lost) or substitute it by a combination of variables. Other option is the use
of more advanced methods such as ridge regression, which are out of the scope of this
work.
For more details see MathWorks (2007), Montgomery and Peck (1992) and Jang et
al. (1997).
77

Chapter 4
4.2.2. Neural network fundamentals.
Neural networks are composed of simple elements (neurons) which operate in
parallel. As in biological nervous systems, connections between neurons make possible
to achieve complex behaviour of the whole system. A neural network can be trained to
perform a function by adjusting the values of the connections (weights) between
elements.
4.2.2.1. Neurons, layers and networks.
A neuron is represented in Figure 4.1. The input vector (with dimension r) is
multiplied times the (single row) weight matrix W and then the bias (b) is added to form
the value n:
78
n = W ⋅ p +b=w ⋅ p + w ⋅ p + ... + w ⋅ p + b
4.82
1,1 1 1,2 2 1, R R
The value of n is modified by a transfer function f producing the output of the
neuron (a).
( ) ( )
a = f n = f W ⋅ p + b
4.83
The behaviour of the neuron depends on the value of the parameters W and p, and of
the type of function f. The value of W and p for the different neurons in a given neural
network are calculated in a training process. The transfer function f influences strongly
the behaviour of the neurons.
Figure 4.1. Single neuron. Source: Demuth and Beale (2002)
There are several types of transfer functions. Figure 4.2. shows a hard-limit transfer
function, which limits the output of the neuron to either 0, if n is less than 0; or 1, if n is
greater than or equal to 0. This neuron allows to make classification decisions and is
used in Perceptron networks.

Quantitative causality analysis and other diagnosis methods
Figure 4.2. Hard-lim transfer function. Source: Demuth and Beale (2002)
Figure 4.3 (Equation 4.84) represents a linear transfer function, which does not
change the value of the input n. Figure 4.4 and 4.5 (and Equations 4.85 and 4.86)
represent log-sigmoid and tan-sigmoid transfer functions, respectively.
a = n
4.84
1
a
1 e − =
+
n
Figure 4.3. Linear transfer function. Source: Demuth and Beale (2002)
Figure 4.4. Log-sigmoid transfer function. Source: Demuth and Beale (2002)
2
a = −1
2n
( 1 e ) −
+
4.85
4.86
79

Chapter 4
80
Figure 4.5. Tan-sigmoid transfer function. Source: Demuth and Beale (2002)
Two or more neurons can be combined in a layer. A layer with S neurons is
represented in Figure 4.6. It should be noted that now W is a S x R matrix and b and a
are S x 1 vectors.
Figure 4.6. Layer of a neural network. Source: Demuth and Beale (2002)
A network is formed by the connexion of several layers. There are several types of
networks according to the number, type and connexion of neurons and layers. It is also
possible to include feedback or delays: a neural network containing one or more of
those elements is called dynamic; in other case it is called static.
Once a network has been created (its structure has been defined), it has to be trained.
Training consists in determining the value of weights and biases in order to obtain a
network with a given behaviour by using a set of known inputs with its corresponding
target outputs. In incremental training the weights and biases are updated each time an

Quantitative causality analysis and other diagnosis methods
input is presented to the network. In batch training the weights and biases are only
updated after all of the inputs are presented. It is more common to apply incremental
training for dynamic networks and batch training to static networks, although other
possibilities are also possible.
4.2.2.2. Feedforward networks and backpropagation.
A network composed of several layers connected in a way that the output of one is
the input of the other is called feedforward; because information entering with the
inputs is transformed in each layer and advances forward up to the network output. A
feedforward network with a sigmoid layer and a linear output layer is capable of
approximating any function with a finite number of discontinuities.
Feedforward networks are sometimes called backpropagation because this is a
family of training algorithms used for them. The idea is to determine weight and biases
to minimise the mean squared error between the network outputs and the target outputs.
This is an exaple of batch training and is done iteratively. The simplest implementation
of backpropagation learning updates the network weights and biases in the negative
direction of the gradient. One iteration of this algorithm can be written as:
x x α g 4.87
k+ 1 k k k
= − ⋅
Where xk is a vector of current weights and biases, gk is the current gradient, and αk
is the learning rate.
Other more complex training algorithms have been developed in order to avoid
problems appearing in different types of networks and to achieve faster convergence.
For example, the Levenberg-Marquardt algorithm was designed to approach secondorder
training speed without the need to compute the Hessian matrix. If the performance
function has the form of a sum of squares, the Hessian matrix can be approximated as:
t
H ≅ J ⋅J
4.88
And the gradient can be computed as:
t
g = J ⋅e
4.89
Where J is the Jacobian matrix that contains first derivatives of the network errors
with respect to the weights and biases, and e is a vector of network errors. One iteration
of Levenberg-Marquardt algorithm can be written as:
T
1
T
k 1 k μ −
x ⎡ ⎤
+ = x −⎣J ⋅ J + I⎦ ⋅J ⋅e
4.90
81

Chapter 4
μ is an scalar which can vary during the iterative process. Note that when μ is large,
the method becomes gradient descendent with small step size.
One of the problems that occur during neural network training is called overfitting.
The error on the training set is driven to a very small value, but when new data is
presented to the network the error is large. If the number of parameters in the network is
much smaller than the total number of points in the training set, there is little chance of
overfitting. Fortunately, this is the common situation in diagnosis problems. In any case,
it is advisable to use for training a fraction of the set of points available, and keeping the
other points for checking.
More details on neural networks can be seen in Demuth and Beale (2002) and in
Jang et al. (1997).
4.2.3. Application for diagnosis.
Once the fundamentals of linear regression and neural networks have been
presented, their capabilities for diagnosis are analyzed in this section. It should be noted
that a sufficient high amount of plant data is needed to determine the coefficients of the
regression model and the weights and biases of the network. It might be possible to
obtain this information from simulation; but in most cases, if a simulator is available, it
is more useful to use the simulator itself or quantitative causality analysis instead of
these methods.
4.2.3.1. Diagnosis of the complete system.
One possibility is to consider the whole system to be diagnosed as a black box, and
to build a pure experimental model of it. Inputs of these models should be the free
diagnosis variables and the output is the global efficiency indicator.
If linear regression with linear additive models is used, coefficients β are directly the
elements of vector ed. These factors are constant, so that this method is not able to deal
with non-linear behaviour. It should be noted that this point could be solved by
including quadratic terms, but this possibility is not considered in this work.
When a complex system is analyzed as a whole, the problem of multicollinearity
often appears. To avoid it, it is useful to perform a variable change in order to eliminate
the most important dependences. If two variables xi and xj are correlated:
82
x ≅ a + a ⋅x
4.91
i i0ij j

The first of them is substituted by:
i i 0i
ij j
Quantitative causality analysis and other diagnosis methods
xˆ = x −a −a ⋅x
4.92
Then, linear regression is applied to the new sets of variables (which now are
independent). Finally, inverse variable change is used in order to obtain the impact
factors corresponding to the initial variables, which is the objective of the procedure.
If neural networks are used, it is advisable to use a feedforward net with two layers:
one hidden layer with sigmoid neurons and one output layer with linear neurons. Since
the functions to be approximated have few singularities, it is convenient to use few
neurons in the hidden layer.
Once the network is trained, impacts on the global efficiency indicator due to
variations of free diagnosis variables can only be obtained by simulation with the
network.
Linear regression (with linear additive models) has the advantage of providing
directly the impact factors ed. On the other hand, it is only suitable for linear relations
and variable change is often needed. Neural networks are able to deal with non-linear
dependences, but they do not provide information explicitly. It should be kept in mind
that both techniques are based on the minimisation of the error in the determination of
the global efficiency indicator, so that they tend to quantify precisely only the effect of
variables with high impact. This effect can be seen in the practical application
developed in Chapter 8.
4.2.3.2. Diagnosis of sub-systems. Hybrid configurations.
Sometimes, applicability of black box based approaches for complete systems with a
lot of free diagnosis variables is limited because they are able to quantify precisely only
those with high influence.
One possiblity would be a tree approach. First of all, the whole system should be
decomposed into subsystems, each one with several input variables (free diagnosis
variables) and one or more output variables. So, it is possible to build local models of
the subsystems. Afterwards, a global model has to be developed with the output
variables of the local models (and, perhaps, some free diagnosis variables) as input
variables, and the global efficiency indicator as the output variable. This approach can
provide good results but suitable decomposition is needed, which it is not
straightforward.
83

Chapter 4
Another option is to use a hybrid approach, combining quantitative causality
analysis and linear regression and/or neural networks. Causality chains theory provides
the possibility to introduce local models on quantitative causality analysis. These
models can be built by using linear regressions or neural networks. This approach can
be very convenient because the use of experimental models is limited to those parts of
the analysis which are most difficult to study analytically. It is applied in Section 6.6 to
filter the influence of ambient temperature in a cooling tower.
4.2.3.3. Other issues.
In Section 3.6 it has been introduced the idea of diagnosis of a time span, what needs
a determination of the degradation along time. This evolution can be very complex, so
that neural networks can be useful.
4.2.4. Conclusion
Fundamentals of linear regression and neural networks as well as some possibilities
of application of them for diagnosis purposes have been presented in this section. A
practical application can be seen in Chapter 8 and in Section 6.6.
These approaches can be useful when plant information is available. They are able
to quantify the main diagnosis causes of a complete system degradation. Besides, they
can help to diagnose complex sub-systems and could be integrated in a diagnosis
structure based on quantitative causality analysis by using causality chains.
84

5 Case study: 3x 350 MW coal-fired power
plant.
The methodology previously described has been applied to the diagnosis of a
pulverized coal-fired power plant. Although diagnosis comprises the boiler, steam cycle
and cooling system; the aim of the system is to diagnose the cycle and the main
parameters of the boiler, so that complex issues related to heat transfer have been
simplified.
In the fist part, the plant is described. Then the plant components’ are reviewed,
presenting the models and parameters proposed to analyze their behaviour as well as
their degradation mechanisms, which are the aspects to be diagnosed, and the choice of
suitable diagnosis variables. Afterwards, available plant measurements are presented,
which determines the depth of the achievable diagnosis; lack of information should be
substituted by suitable models. According to the components’ characteristics and the
available plant data, the list of diagnosis variables can be built.
5.1 Teruel power plant description.
The Teruel power plant has been chosen to apply the diagnosis methodologies
presented in Chapter 3 and Chapter 4. It is a conventional pulverised coal-fired plant
that consists of three units of 350 MW each one, built around 1980. It is located in
Andorra, in the Teruel province (Spain) and owned by Endesa Generación S.A. In this
section, the main characteristics of this power plant are presented: the cycle, the cooling
system and the boiler.
85

Chapter 5
5.1.1 Steam cycle description
The steam cycle of the plant was built by Mitsubishi and is depicted in Figure 5.1,
along with the cooling system. Main steam leaving the boiler at 160 bar and 540 ºC
enters the high pressure turbine where it is expanded down to 40 bar. Then it goes back
to the boiler (actually to the reheater) in order to increase its temperature up to 540 º C
again. Finally, it is expanded in two intermediate pressure turbines and in the low
pressure turbine. These turbines are separated in two casings: one for high and medium
pressure ones and the other for the low pressure. Due to the high volumetric flow, the
low pressure turbine is double flow type, and has four stages (in each side). At the end
of each turbine stage, part of the steam is extracted for the feedwater heaters, the
deareator and the turbo-pump.
Steam leaving the low pressure turbine is condensed in the condenser. Water leaving
the condenser is pumped by an electric pump up to a pressure of about 8 bars. Then it
goes trough two components of minor importance that are not depicted in the diagram:
ejector and sealing steam condenser, where it is slightly heated. Afterwards, the feeding
water is heated in the low pressure feedwater heaters, by using bleed steam from the low
pressure turbine. These heaters are shell and tubes-type, with the water flowing inside
the tubes and the steam condensing outside. Then, the water enters the deaerator, which
is a direct contact heater (the fourth heater). Its aim is not only to heat the water but
mainly to lead it to the saturation point, in order to release incondensable gases
dissolved in it. Water leaving the deaerator is pumped again up to 180 bar. This second
pump is not electrically-driven but it has its own steam turbine that uses bleed steam
from the end of the second medium pressure turbine. Finally, the water is heated in two
high pressure heaters (5 and 6). It should be noted that once it has condensed, the steam
used in each heater is fed into the previous one, except the 4 th that does not have
drainage and the 1 st that drains in the condenser.
There are a lot of secondary flows that have not been included in figure 5.1, most of
them corresponding to the steam seals system. Besides, there are other flows entering
the deaerator, such as the condensed water from the steam coil heaters.
Some details on the cycle are included in Table 5.1.
86

Case study: 3x350 MW coal-fired power plant
RH
HP IP 1 IP 2 LP
COND
CT
TP
1
2
3
5
6
4
BOILER
Figure 5.1. Steam cycle.
87

Chapter 5
Parameter Characteristics
Type Mitsubishi ‘Reheat Condensing Two Casing Double Flow’
Inlet regulation by eight sequential valves
Number of casings Three: high pressure (HP), intermediate pressure (IP 1 and 2) and low pressure
(LP)
Peak power 360 000 kW (maximum in generator)
Nominal power 350 000 kW (optimum)
Load levels 350 MW; 262.5 MW; 175 MW; 87.5 MW
Rotation 3000 rpm, clockwise sense (from governing stage towards LP)
Technical limits: 48.5 Hz < frequency < 50.5 Hz
Cycle Water-steam Feeding water (boiler inlet):
254 º C
conditions
Steam pressure prior to HP closing valve:
162 kg/cm
Steam temperature prior to HP closing valve:
Steam temperature prior to IP closing valve:
Absolute condenser pressure
2
538 ºC
538 ºC
0.069 kg/cm 2
Steam extractions: 1
maximum values and
turbine section
st :
2 nd :
3 rd :
BFPT:
4 th :
5 th :
6 th 26.2 kg/s, 354 ºC, 10
18.6 kg/s, 445 ºC,
17.2 kg/s, 318 ºC,
11.9 kg/s, 318 ºC,
15.4 kg/s, 207 ºC,
9.94 kg/s, 94 ºC,
:
8.81 kg/s, 70 ºC,
th stage,
15 th stage,
20 th stage,
20 th stage,
22 nd stage,
24 th stage,
25 th 10
stage,
th (HP)
5 th (IP-1)
10 th (IP-2)
10 th (IP-2)
2 nd (LP-1)
4 th (LP-2)
5 th (LP-3)
Pressure losses Reheater
< 10%
Extractions (HP)
< 3 %
Other extractions
< 5 %
Stages in turbine High pressure
Curtis double impulsion wheel
Reaction: 9
Intermediate pressure Reaction: 10
Low pressure
Reaction: 6 x 2 (double flow)
Low pressure last Blade length
850.9 mm
wheel
Pitch diameter
2552.7 mm
Peripheral velocity 534.0 m/s
Mechanical losses < 1000 kW
Radial clearance Blading fins: > 1.0 mm Gland fins: > 0.5 mm
Generator Hydrogen cooling system
0.987 design efficiency
88
Table 5.1. Main characteristics of the Teruel Power plant steam cycle. (Zaleta, 1997).

Case study: 3x350 MW coal-fired power plant
5.1.2 Cooling system
The cooling system is a closed loop type, based on a natural draft cooling tower. It is
composed of condenser, cooling tower, pumps and water ducts.
The condenser is located just below the low pressure turbine. Steam leaving the
pressure goes downwards becoming condensed by releasing heat to the cooling water
that flows inside tubes that cross the condenser. As it is usual when cooling towers are
used, the condenser has two pass, which means that cooling water crosses it twice.
Some design characteristics of the condenser are summarized in table 5.2.
Condenser pressure 0.069 kg/cm 2 absolute
Steam flow rate 202.8 kg/s
Cooling water flow rate 9.78 m 3 /s
Inlet cooling water 24.4 ºC
Outlet cooling water 35.3 ºC
Number of tubes 23700
Cooling surface 17650 m 2
Table 5.2. Condenser characteristics (Lazaro, 2001)
Heated cooling water leaving the condenser goes to a cooling tower through a duct
of 2.5 meters of diameter. There, water is dispersed over a fill media located in order to
favour the matter and heat transfer between the water going downwards and the air
going upwards. The evaporation of part of the water causes the cooling of the rest of the
water. The air flow is originated by density difference (natural-draft type). Some
characteristics of the tower appear in Table 5.3.
Finally, water from the bottom of the tower is again pumped towards the condenser.
Water is added in order to compensate the amount evaporated.
89

Chapter 5
90
Cooling water flow rate 10.56 m 3 /s
Hot water temperature 35.3 ºC
Cold water temperature 24.4 ºC
Ambient wet bulb temperature 12.9 ºC
Relative humidity 79 %
Tower height 107.3 m
Bottom diameter 81.22 m
Top diameter 50.70 m
Fill volume 84000 m 3
Table 5.3. Cooling water design characteristics (Lazaro, 2001).
5.1.3 Boiler
Boilers at Teruel power plant are pulverized coal-fired, front-fired, natural
circulation and balanced draft. They have three superheating stages and one reheating
stage. The main nominal characteristics of each one of them are summarized in table
5.4.
Main steam flow rate 302.8 kg/s
Main steam temperature 540 ºC
Main steam pressure 169 kg/cm 2
Reheated steam flow rate 266.7 kg/s
Reheated steam temperature 540 ºC
Reheated steam pressure 40 kg/cm 2
Table 5.4. Teruel power plant boilers, nominal production (Díez, 2002).
Air-gases circuit is composed of two parallel lines, called ‘side A’ and ‘side B’.
Each one of them is depicted in Figure 5.2. Air entering the boiler is pressurized by the
forced draft fan (FDF) and then is divided into two flows: secondary and primary air.
Secondary air goes to the secondary air preheater (SAH) where it increases its
temperature by using sensible heat from the flue gases leaving the boiler. These heaters
are regenerative-type, which means that they work based on heating and cooling of
heating elements, due to an alternative contact with hot and cold flow. There are two

Case study: 3x350 MW coal-fired power plant
ways to achieve this: by rotation of the basket where heating elements are located
(Ljungstrom), or by rotation of the bells that conduct the air and gas flows through fixed
baskets (Rothemuhle). In 1999, Rothemuhle type secondary air preheaters at Teruel
power plant were substituted by Ljungstrom type. One important consequence of a
working principle based on moving parts is the presence of air leakages, which
constitute an important efficiency reduction cause. Heated secondary air goes to the
wind box and then to the furnace through tangential holes located at the burners.
TO BURNERS
TO FGD
TO MILLS
FROM EZ1 FROM BY-PASS
SAH
EP
IDF
PAH
SCH1
Figure 5.2. Fans and preheaters.
Pressure of primary air is further increased by the primary air fan (PAF). Then, this
flow is heated by the primary air heaters (PAH). These preheaters were originally
regenerative-type but in 1999 they were substituted by tubular-recuperative heaters. In
these heat exchangers, hot flue gases flow inside tubes that are surrounded by the cold
air. They do not present leakages, or at least they can be neglected compared to the
recuperative-type ones. Heated primary air goes to the mills where it is used to dry and
transport the pulverized coal.
Both primary and secondary air pre-heaters can be damaged due to acid
condensation if surface temperature is too low. To prevent this situation, manufacturer
advises to keep the average cold side temperatures (mean of air inlet and flue gases
outlet temperatures) above a certain limit (about 110 ºC). To assure that this condition is
fulfilled also in cold weather, steam coil air heaters are located before the pre-heaters in
SCH2
PAF
FDF
91

Chapter 5
both primary and secondary air flows. These heaters work by condensing steam
extracted from the drum, whose pressure has been previously reduced to 16 bar.
A part of the primary air is devised after the primary air fan an goes to the mill
without being heated. This tempering air is used to control the temperature of primary
air entering the mills. There is also a derivation of heated primary air that can be fed in
the secondary air flow before the preheater, in order to increase its temperature without
using steam in the coil heaters. Finally, it is possible to extract hot air from both primary
and secondary flows to heat-up flue gases leaving the sulphur removal unit in order to
get suitable stack entering temperature. It should be noted that only primary air is
usually used for this issue.
Figure 5.3 shows a schematic side view of the boiler, excluded the air pre-heaters
area previously commented. There are 6 mills, each one corresponding to a row of
burners and named with a letter from A to F. The flow of primary air and pulverised
coal goes from each mill to the four burners of each row where secondary air is also
introduced and coal is burnt. Sometimes, natural gas is also burnt to assure good flame
characteristics in part load or start-up processes.
The union of up to 24 flames forms a fire ball where a great amount of heat is
released, mainly by radiation. This radiation is mainly used to boil the water in the walls
of the furnace: water goes downwards from the drum through external tubes and
evaporates while going upwards back to the drum through tubes covering all the furnace
walls. Density difference of water and steam is enough to assure enough flow rate
(natural circulation). Furnace radiation also contributes notably to heat transfer in the
second superheater (also called radiative superheater).
Hot gases leaving the furnace are derived by the nose and goes trough the secondary
and tertiary superheaters. Then, they arrive to the plenum and are separated in two
flows: one flow going to the reheater and one part of the first economizer and other
goint through primary superheater, second economizer and the second part of the first
economizer. This flow distribution is controlled by dampers and is adjusted in order to
maintain the reheated temperature set-point. Heat exchanger surfaces are cleaned by a
sootblowing system that uses steam taken from the steam cycle in the boiler, before the
final superheater. Exchange surface of heat exchangers located in the boiler are
summarized in table 5.5.
92

D
C
E
B
F
A
SH 2
FURNACE
Case study: 3x350 MW coal-fired power plant
SH 3
SH 1
EZ 2
Figure 5.3. Schematic side view of the boiler.
EZ 1
RH
93

Chapter 5
94
Heat exchanger Exchange area (m 2 )
Primary economizer (EZ1) 3360
Secondary economizer (EZ2) 1970
Primary superheater (SH1) 2545
Secondary superheater (SH2) 1296
Final superheater (SH3) 1668
Reheater (RH) 8440
Table 5.5. Exchange area of the boiler heat exchangers. (Díez, 2002)
Boiler can also be explained from the water-steam point of view. Water coming
from the cycle (actually from the 6 th feedwater heater), is first heated in the primary and
secondary economizers and goes into the drum. As it has been previously said, there is a
secondary water-steam loop in which water goes downwards from the drum and goes
back to it after being evaporated in the furnace walls. Dry steam is taken from the upper
part of the drum and goes to the plenum, boiler ceiling and division walls of the
convective area where increases its temperature. Afterwards, it goes to the primary,
radiant and final superheaters, where it reaches the suitable temperature of 540 ºC. This
set-point, as well as the temperature of steam exiting the radiant superheater, are
controlled by introducing water at the entering of the final and radiant superheaters,
respectively. This water is taken from the impulsion of the turbo-pump (see figure 5.1).
Flue gases flow leaving the first economizer goes to the secondary and primary air
preheaters (Figure 5.2). Part of flue gases leaving the reheater can be deviated directly
to the primary air preheater in order to increase the temperature of gases entering this
heat exchanger (by-pass of economizer). Gases leaving the primary and secondary air
preheaters go to the electrostatic precipitator (EP) that removes the particulates that the
gas flow has, and to the induced draft fan (IDF) which moves them towards the sulphur
removing unit and the stack.
More details on this boiler can be seen in Diez (2002).

Case study: 3x350 MW coal-fired power plant
5.2 Diagnosis model for a pulverized-coal power plant.
A main task of this section is to review the models and parameters available to
describe the behaviour of the elements constituting a conventional coal-fired power
plant. The problem of modelling and the choice of suitable diagnosis variables are
related topics: a diagnosis variable close to the observable behaviour of a system that
can be calculated from data directly measured or obtained by a performance test does
not need any modelling; on the other hand, it is possible to choose a diagnosis variable
closer to the physical state and degradation level of the system, which probably will
need a more or less complex model to be related to the observable parameters. In
general, the previous option will be preferred, and the influence of the operation point
on the variable will be filtered by using methods such as anamnesis (repeated diagnosis
along time) or the use of causality chains which allow to integrate crossed dependence
and models.
Besides, modelling is sometimes needed to include in the performance tests flows
and systems for which measurement is not available (e.g. seals flows). Of course, it may
be possible to include simulation as an additional capability for what-if analyses of a
diagnosis system installed in a power plant, by using almost the same set of equations of
the diagnosis but substituting identities corresponding to diagnosis variables by a
suitable model.
Another important issue tackled in this section is a review of the main anomalies or
causes of efficiency decrease of the different plant components. Although the diagnosis
system is based on a thermodynamic representation of the system and the
characterization of its components by using parameters relating pressures, temperatures
and so on, it should be kept in mind that the final goal is to detect physical degradations
in components and to quantify their effects on the global system.
These two aspects, modelling and diagnosis variables, and degradation mechanisms
are studied for the main plant components, ordered by plant zones: steam cycle, cooling
system and boiler. The idea is to start with the general ideas and then to descend to the
case of study. However, the most specific points on the implementation at Teruel power
plant are summarized later, after having presented the available plant data.
95

Chapter 5
5.2.1 Steam cycle
The main components of this area are steam turbines. Then, the diagnosis model for
regenerative water heaters is explained. Finally, other components such as pumps,
generator and steam seals are considered.
5.2.1.1 Steam turbine
A detailed study of the steam turbine would strictly require an aerodynamic model
of each stage based on the speed triangle (see Salisbury, 1974). However, this approach
is neither practical (due to the lack of information) nor necessary, because for the
purpose of this work, turbines can be considered as black boxes. So that, only two
relations are needed for each turbine section. In the case study there are seven turbine
sections (one high pressure, two intermediate pressure and four low pressure), because
the two flows of the low pressure are symmetrical and considered as only one. These
two parameters are isoentropic efficiency and flow coefficient.
Isoentropic efficiency of a turbine is defined as the quotient between the enthalpy
drop and the enthalpy drop to the same isobar if the expansion were isoentropic.
96
η
h − h
=
in out
st ,
hin−hout, s
Where hin and hout are the enthalpy of steam entering and exiting the turbine and
hout,s is the enthalpy of a fictitious point that has the same entropy of the inlet and the
same pressure of the outlet: hout,s = h(sin, pout). Once known the inlet conditions and the
discharge pressure of a turbine section, isoentropic efficiency allows to determine the
temperature of the exiting flow (or its quality, if it is located inside the vapor dome).
Spencer et al. (1974), provide a methodology to calculate the isoentropic efficiency
of steam turbines of above 16500 kW working with or without reheating. These
relations are essentially empirical, but they are supported by several cases of practical
application. For a steam cycle with reheating, these authors propose the use of two
expansion lines. The first one corresponds to the regulation stage and the high pressure
turbine. The isoentropic efficiency of this section is calculated by considering a
reference value that depends on the turbine characteristics and two correction factors
depending on the volumetric flows and the pressure ratio.
The second expansion line comprises the two intermediate pressure and the low
pressure turbine. It is calculated by modifying a base value by using three correcting
factors related to volumetric flow and temperature and pressure in the inlet of this
5.1

Case study: 3x350 MW coal-fired power plant
section. This expansion line should be corrected for the last section, taking into account
the exhaust losses originated by the kinetic energy of the steam leaving the turbine. This
correction depends on volumetric flow and quality of steam leaving the turbine.
It may be possible to choose the reference values of these correlations as free
diagnosis variables. They would be quite close to physical behaviour of turbines and
might be more suitable to filter malfunctions induced by variable operation points.
However, they would be more difficult to calculate and to interpret, because they could
not be directly calculated from properties of fluids entering and exiting the equipment.
The other parameter needed to determine the steam turbines behaviour is the flow
coefficient, which is defined as:
m
φ =
5.2
pin
υ
in
where m is the flow rate and pin and υ in are the pressure and specific volume of
steam entering the turbine stage. Under the hypothesis of ideal gas, the previous
equation can be expressed as:
m⋅T φ =
p
in
in
(ideal gas) 5.3
where Tin is the temperature of steam entering the turbine. According to the Stodola
ellipse applied to a group of stages (see Cooke, 1985), for very high pressure ratio sonic
flow is reached and the flow coefficient is constant. In general, actual and design flow
coefficients are related by:
φ = φ ⋅
d
1−
r
1−
r
2
p
2
pd ,
where the sub-index d means design conditions and rp is the pressure relation:
r
p
out
p = 5.5
pin
By using these equations, it is possible to determine the value of the pressure along
all the expansion line, except in two points: the condenser and the inlet of the high
pressure turbine. Stodola ellipse refers to turbine inlet, so that it cannot be applied to the
condenser, whose pressure appears as an equilibrium between the heat to be removed to
condense the steam leaving the turbine and the capacity of evacuate this heat, which is
5.4
97

Chapter 5
determined by the cooling system (condenser, pumps and cooling tower) and mainly by
the ambient conditions (temperature and relative humidity). If the high pressure turbine
would operate at sliding pressure, the Stodola ellipse should be applied. However, it
operates at constant pressure that is maintained by opening and closing the regulation
valves. This variation of the regulation system makes the model useless, because steam
pressure is directly a set-point.
Zaleta (1997) shows a review of the causes of turbine degradation, which he calls
intrinsic malfunction. They can be classified into i) erosion and breakings, ii) sediments,
iii) rugosity, iv) damage in steam way and seals and v) control valves.
Erosion and breakings is one of the most common malfunctions, and it is mainly
due to the presence of solid particles in the steam. Erosion is a function of particle size,
impact angle and particle speed. Breaking is similar to erosion in the sense that it entails
a lost of material. However its effect is more severe and can produce high vibrations and
lead to turbine shut-off.
Effects produced by erosion are: i) increment of inlet areas (and reduction of inlet
pressure), ii) changes in incidence angles and iii) efficiency losses and changes in steam
properties. Corrective methods (after erosion) are very costly and often entail the
replacement of wheels of blades. For damages not very severe, material injection by
using spray plasma technology may be used. Some preventive methods are the use of
filters to remove solid particulates, special water treatment and the use of anti-oxidant
materials.
Sediments are caused by inadequate water treatment and for high particulate content
in water. They occur usually in turbine areas where speed is relatively low. Sediments
produce increment in rugosity and reduction in effective areas. Preventive methods are
the same used for avoiding erosions. Corrective methods use conventional cleaning
processes based on air jet and particulates.
Some parts of a turbine usually have a special surface treatment in order to assure
low rugosity. However, due to the flow of steam and particulates, surfaces increase its
rugosity and losses increase.
Damage in steam way is caused by an excess of rugosity and the erosion of the
initial and final parts of the blade: ‘Leading Edge Erosion’ and ‘Trailing Edge Erosion’
Besides, a damage in seals will produce a derivation of a fraction of steam (‘By-pass
leakage’). Seals are studied later.
98

Case study: 3x350 MW coal-fired power plant
Control valves, when are completely open, have pressure losses of 1-2 %. However,
this figure can also be substantially increased due to erosion and sediments. Besides,
damages in valve seats origin steam leakages.
Along with these intrinsic malfunctions, Zaleta (1997) refers to induced
malfunctions: changes in admission temperature, outlet pressure and extraction flow
rate.
5.2.1.2 Steam turbine seals and other secondary flows.
The presence of holes to hold the turbine axis connecting several turbine sections at
different pressure among themselves and with the atmosphere requires the use of a
sealing system to avoid steam leakages and, in the low pressure sections, air infiltration.
Sealing in steam turbines is achieved by using labyrinth seals through which a small
amount of steam flows in a controlled manner. In the middle of these seals, chambers
working at suitable pressure are located in order to keep reduced these leakages.
Pressure of these chambers are controlled by quite complex systems that also comprise
a device called seals steam regulator and a seals steam condenser, as well as a grid of
ducts that connects suitably all these parts. In this condenser, steam containing
infiltrated air is condensed while this air is released.
To calculate the amount of steam flowing through each seal, the Martin equation can
be applied (Salisbury, 1974):
p
in
i = ⋅ t ⋅ ⋅
5.6
ν in
m k A β
Where p and υ are pressure and specific volume of the flow entering the seal, At is
the area of the clearance and β is a geometric factor depending on the teeth number of
the seal (z):
β =
2
1−
rp
z−ln r
p
k is a constant determined by the shape of teeth and rp is the pressure ratio:
r
p
out
p = 5.5
pin
If geometric information is not available, it is possible to adjust a constant
comprising k·At·β by using plant design information, neglecting the small variation of
pressure ratio. Hussain (1984) proposes another expression to be used when critical
conditions are not reached:
5.7
99

Chapter 5
100
2 2
pin − pout
m i = At
⋅
5.8
z⋅p ⋅ν
in in
In the case of study, not only leakages had to be calculated, but also flows
connecting turbine seals, seal steam regulator and seal steam condenser. So that design
information available for 100, 75 and 50 % load (Auxiesa, 1980) has been used to adjust
second order polynomials that relate these flows to the load or to the nearest main flow.
As it has been commented previously, damages in seals can originate an increment
of leakages that implies additional losses. However, these flows are not measured and
are not considered as diagnosis variables.
The same approach (use of polynomials adjusted to design and not consideration as
diagnosis variable) is applied to other secondary flows of the cycle: steam flow rate to
ejector and seal flows in turbo-pump.
5.2.1.3 Alternator
Walsh and Fletcher (1998) propose curves to determine the efficiency of a generator
depending on its active power and its power factor. These curves can be summarized in
the following expression, which relates the actual to the maximum efficiency:
k2+ k3⋅W gen
( 1 ( 0.85) ) ( 1 )
ηgen = ηgen
max ⋅ + k ⋅ PF − ⋅ −e
, 1
where PF is the power factor, Wgen is the electric power and k1, k2 and k3 are
constants. In the case of study, only information for three load levels were available, so
that, a second order polynomial relation has been adjusted, considering not only the
generator efficiency but also mechanical losses in the shaft. Since mechanical power
provided to the generator cannot be measured nor determined, alternator efficiency is
not considered as a diagnosis variable.
5.2.1.4 Feedwater heaters
The aim of these components is to increase the temperature of water before it enters
the boiler by condensing steam that is extracted from the turbines at several pressures
(bleed steam). These heaters are shell and tube type; water to be heated flows inside the
tubes while steam is cooled down to the saturation temperature, condensed, and finally
it is sub cooled. This process is shown in Figure 5.4. It should be noted that in the first
heaters, corresponding to the last turbine stages, steam extracted is saturated, so that the
first stage of steam cooling does not take place.
5.9

T
TTD
Case study: 3x350 MW coal-fired power plant
TDCA
Figure 5.4. Temperature profile in a feedwater heater.
Thermal behaviour of regenerative heaters is characterized by two parameters:
terminal temperature difference (TTD) and drain cooling approach temperature
(TDCA). TTD is the difference between the saturation temperature of the extracted
steam and the temperature of water leaving the heater. It should be noted that it can be
positive, zero or negative, except when steam is saturated, and it can be only positive.
TDCA is the difference between the temperatures of water entering the heater and
drainage (water resulting from the condensing of steam). This value can only be
positive.
To model these heat exchangers, it is possible to use the classical approach based on
the global heat transfer coefficient U, combined with the use of methods such as ε-NTU
or Log-mean temperature difference (LMTD) (Mills, 1992). The problem with these
approaches is that U coefficient varies in the three zones of the exchanger: steam
cooling, condensing and subcooling.
In this work, TTD and TDCA have been chosen as diagnosis variables. They can
also be supposed constants for simulation purposes.
Anomalies that may appear in these heaters are mainly fouling and duct blocking,
which originate a decrease in the heat transfer capability and an increase in the value of
TTD. Sometimes, due to the necessity of plant operation and maintenance, a heater
should be by-passed. This means that the equipment is isolated by closing the steam
inlet and drainage and by connecting directly water inlet and outlet. This situation is
correctly diagnosed by the TTD parameters, which increases dramatically because water
at the exit has the same temperature as at the inlet.
x
101

Chapter 5
Deaerator can also be considered as a water heater, because it heats the water.
However, its main purpose is to release gases dissolved in the water. For this reason,
water and steam flows are not separated but they are mixed, so that no drainage is
present. Besides, water leaves the deaerator as saturated liquid (and at the extraction
pressure), so that TTD equals zero. Due to these fixed operation conditions determined
for the deaerator working principle, no diagnosis variable is defined for this component.
5.2.1.5 Pumps
Pump manufacturers provide curves that relate the pressure increment provided by
the pump, and its volumetric efficiency to the flow rate that is pumped:
102
out in Δp
( )
p − p =Δ p= f m
5.10
( )
η v = fη m
5.11
The first curve, called characteristic curve of the pump, should be compared with the
characteristic curve of the installation to determine the operation point of the system
(flow rate and pressure increment). Power required by the pump can be calculated by
the following expression:
m ν p
W
⋅ ⋅Δ
=
5.12
η ⋅η
mec v
Where W is the mechanical power required by the pump, ν is the specific volume of
the liquid, ηmec is the mechanical efficiency and ηv is volumetric efficiency. Equations
presented above constitute the model of a pump working at constant speed. To simulate
a pump working at different speed (e. g. when a variable speed hydraulic coupling
system is used), relations provided by dimensional analysis can be used. According this,
flow rate relation is proportional to speed relation, pressure increment is proportional to
the square of speed relation, and power varies with the cube:
m n n
1 1 =
5.13
m n
n2
2
2
n1
= ⎜
1
⎟
n2
2
Δp ⎛ n ⎞
Δp ⎝n⎠ W n
W n
3
n ⎛ ⎞
1 1 = ⎜ ⎟
n2
⎝ 2 ⎠
5.14
5.15

Case study: 3x350 MW coal-fired power plant
The parameter suitable to diagnose the pumps depends on the type of pump and
plant measurement available. In the case of electric pumps, a good option is to use a
global efficiency comprising electric, mechanical and volumetric efficiencies:
m⋅ν⋅Δp ηp = ηe⋅ηmec⋅ ηv
=
W
e
5.16
where ηe and We are respectively the efficiency of the electric motor and the
electric power consumed by it.
Big pumps for feeding water in power plants are usually driven by a small turbine
fed with extraction steam, coupled to a hydraulic transmission. The group of turbine,
transmission and pump is called turbo-pump. This system can be diagnosed by
considering separately its three components and defining the transmission efficiency as:
W
η tr =
W
p
t
5.17
where p W and t W are respectively the power delivered to the pump and the power
provided by the turbine. The problem of this approach is that these two values are
seldom available. For this reason, it is preferable to define a global efficiency
comprising turbine, transmission and pump. By combining equations of isoentropic
efficiency and energy balance of a turbine, efficiency of a transmission and efficiency of
a pump, it yields:
mp⋅ν⋅Δp ηtp = ηt⋅ηtr ⋅ ηp=
m⋅ h −h
( , , , )
t in t s out t
The previous equation shows that turbo-pump efficiency is the quotient between the
minimum power to pump the water and the maximum power obtained by expanding the
steam, considering ideal processes in both cases. This efficiency is very suitable for
diagnosis purposes. However, it varies with load, and this dependence should be taken
into account for simulation (Alconchel, 1988).
5.2.1.6 Pressure losses
Pressure losses in water-steam circuit appear in turbine inlet valves, cross-over pipe,
regenerative water heaters, steam extraction and heat exchangers and ducts in the boiler.
Zaleta (1997) provides expressions for pressure drops in turbine inlet valves:
5.18
103

Chapter 5
104
v
Sa = m ⋅
a
5.20
Where m is the steam flow rate, ν is the specific volume and a is the sound speed for
steam in those conditions, k varies from 0.3 and 2.5 and D is the valve diameter.
For the cross-over pipe:
2
c
Δ px−over=
0.25⋅
5.21
ν
Where c is the steam speed and ν is the specific volume. Alconchel (1988) proposes
the following expression for pressure losses in ducts connecting turbine and feedwater
heaters:
Δ p
= 0.03 if extraction comes from the final of a turbine 5.22
p
Δ p
= 0.05 if extraction comes from an intermediate point of a turbine 5.23
p
These values have obtained from the values observed in different plants. For a given
unit, it is possible to adjust them by using design plant data.
Pressure losses in regenerative water heaters, reheater and other parts of the boiler
can be calculated by potential or polynomial equations adjusted to plant balance:
n
Δ p = k⋅m 5.24
Δ p = a+ b⋅ m+ c⋅m 2 5.25
Where k, a, b and c are generic coefficients and n is an exponent that can vary from
1.6 to 2. Alconchel (1988) has determined that, for the case of study, pressure losses are
almost constant in reheater and linear with the load in the other heat exchangers.
Although the previous equations should be used for simulation purposes, pressure
drops can be used directly for diagnosis purposes. In order to simplify the modelling
and diagnosis, pressure losses in ducts can be supposed to occur in one of the
components that the duct connects.
5.2.2 Cooling system
The cooling system is composed of a condenser, a cooling tower and pumps and
ducts system that is responsible for the water circulation among them.

Case study: 3x350 MW coal-fired power plant
5.2.2.1 Condenser
To model the condenser, a transfer equation should be included in order to take into
account the capacity of the equipment to condense the steam while heating the cooling
water. This equation can be based on the mean logarithmic temperature:
Q = F⋅U⋅A⋅ΔT 5.26
lm
Where Q is the amount of heat exchanged, U is the global transfer coefficient, A is
the area and F is a factor which takes into account the heat exchanger configuration (if
tubes are parallel or not). When one of the fluids changes its phase, the value of F is 1.
ΔTlm is the log-mean temperature difference:
Δ T =
lm
( Th, in −Tc, out ) −( Th, out −Tc,
in )
⎛T −T
⎞
ln
h, in c, out
⎜ ⎟
Th, out T ⎟
⎝ − c, in ⎠
Where subscripts h and c means hot and cold, respectively. For the particular
example of a condenser, where there are a flow of water (w) and condensing steam that
enters and exits at the same temperature (s), the previous expression becomes:
Tw, in −Tw,
out
Δ Tlm,
c =
5.28
⎛Ts−T ⎞ w, out
ln ⎜ ⎟
Ts T ⎟
⎝ − w, in ⎠
Another way to introduce the characteristic equation is given by the ε-NTU method.
The effectiveness of a heat exchanger (ε), relates the heat transferred to the maximum
heat that could be achievable in an ideal infinite exchanger:
Q = ε ⋅Q
5.29
Where
max
( hin cin)
max min , ,
5.27
Q = C ⋅ T −T
5.30
( h p h c p c)
C = min m ⋅c , m ⋅c
5.31
min , ,
The previous equations show that the maximum heat would correspond to a situation
when one of the flows exits the heat exchanger at the same temperature at which the
other enter. That flow would be the one for which the product of flow rate and specific
heat capacity is minimum. In university texts on heat transfer (Mills, 1992) are available
equations that relate the efficiency to two parameters, for several heat exchangers
configurations. These two parameters are heat capacity rates relation, Cr, and the
number of transfer units, NTU:
105

Chapter 5
106
C
C
min
r = 5.32
Cmax
U⋅A NTU = 5.33
C
min
For heat exchangers where one of the flows suffers a change of phase (Cr tends to
zero), effectiveness equation becomes quite simple:
− NTU
ε = 1−
e
5.34
This second method is preferable when numerical methods are used to solve the set
of equations in simulation because the equation to calculate ΔTlm may difficult
convergence when cold and hot side temperatures are quite closed, as occurs in a
condenser. In other situations, the log-mean temperature difference method has the
advantage that it is not needed to asses previously which one of the two flows has lower
value of C.
In both methods, it is necessary to obtain the global transfer coefficient, which can
be calculated by thermal resistances composition, taking into account convection and
fouling in the two sides of each tube, and conduction in the tube wall.
1 1 Aext Aext
1
= ⋅ + Fint + + Fext
+
5.35
U D
ext hint Aint ext
2⋅π⋅kln hext
w ⋅
D
int
Where h the convection coefficient, A is area, F is fouling resistance and kw is
thermal conductivity of the wall. Subscripts int and ext stand for interior and exterior. It
should be noted that the value of U calculated should be multiplied by the exterior area
in equations 5.26 and 5.33. It would be equivalent to calculate the value of U related to
the interior by modifying slightly the equation and then to multiply it times the interior
area.
The convection coefficient is calculated from the dimensionless number of Nusselt
(Nu):
Nu ⋅k
h = 5.36
D
Where k is the thermal conductivity of the substance and D is the inner or outer
diameter. Suitable correlations for calculating the value of Nu depending of the situation
can be found in university textbooks on Heat Transfer such as Mills (1992).
As a diagnosis variable to summarize the condenser behaviour, effectiveness has
been used, because of its universality and easiness to be calculated from plant data:

ε
( )
( )
Q
m ⋅c ⋅ T −T T −T
c w p, w w, out w, in w, out w, in
c =
Q = =
c,max mw⋅cp, w⋅ Tsat −Tw,
in Tsat −Tw,
in
Case study: 3x350 MW coal-fired power plant
5.2.2.2 Cooling tower
To model the behaviour of a cooling tower, it is possible to use a polynomial adjust
of curves provided by tower manufacturer. These curves provide the tower outlet
temperature as a function of the cooling water flow rate, temperature difference between
water entering and exiting the tower, ambient temperature and relative humidity. An
expression quadratic for the flow rate and linear for the other variables is enough.
The most extended theoretical model for cooling towers is based on the Merkel
equation. It is derived by considering that the heat exchange in the tower depends on the
enthalpy difference of saturated air and air in the conditions of the tower. By
considering a differential of height in a section of the tower, the following equation
applies:
( )
air, sat air w p, w w
5.37
dq = β ⋅ h −h ⋅a⋅ dV = m ⋅c ⋅dT
5.38
where β is a mass transfer coefficient and a is the area of tower filling per unit of
volume. By integrating and rearranging the previous equation, the Merkel number
appears, which is a parameter associated to the tower design.
Twin
,
β ⋅ A c ⋅ dT
Me = =
m∫ h − h
pw , w
w Twout
, air, sat air
where A is the total area of the tower filling. To calculate the previous integral, the
Chebyshev method is usually used (Perry, 1984):
( )
c ⋅ T −T
5.39
Tw,in
cpw , ⋅ dTwpw
, win , wout , ⎛ 1 1 1 1 ⎞
∫ ≈ ⋅ ⎜ + + + ⎟
5.40
h
, , 4
Twout
air sat −hair ⎝Δh1 Δh2 Δh3 Δh4
⎠
Δh1 is the value of hair,sat – hair at a temperature [Tw,out + 0.1·(Tw,in – Tw,out)]
Δh2 is the value of hair,sat – hair at a temperature [Tw,out + 0.4·(Tw,in – Tw,out)]
Δh3 is the value of hair,sat – hair at a temperature [Tw,out + 0.6·(Tw,in – Tw,out)]
Δh4 is the value of hair,sat – hair at a temperature [Tw,out + 0.9·(Tw,in – Tw,out)]
Finally, in a natural-draft tower, the amount of air should accomplish the flotability
equation, which states that the pressure drop of air crossing the tower is equal to the
force experimented by the air due to the density differences (Muñoz et al., 1986):
( )
Δ p = g⋅H⋅ ρ − ρ
5.41
air, tower air, in air, out
107

Chapter 5
Where g is the gravity constant, H is the tower height and ρair is the air density.
Better results can be obtained by using more complex models. For example, Al-
Waked and Behnia (2006) have developed a CFD model of cooling towers which
allows to consider even influence of wind.
The previous models can be used to simulate the tower. However, neither the
Merkel number nor the cooled water temperature are suitable free diagnosis variables.
The first one is not commonly used by plant operators and does not have a simple
definition. The second is simple, but vary at least an order of magnitude more for
ambient conditions than for tower degradation. The parameter proposed is a tower
effectiveness related to ambient conditions, which consider not only air temperature but
also its humidity.
108
ε
T −T
=
5.42
w, in w, out
tower
Twin , −Twb
Where Tw is the temperature of water entering or exiting the tower and Twb is the
wet-bulb temperature of air.
5.2.2.3 Pumps and ducts
Expressions to model the pumps have been presented previously, in the paragraph
corresponding to the cycle. Besides, the pressure drop in ducts can be modelled by
parabolic equations adjusted to plant data or by equations available in textbooks on
Fluid Mechanics (White, 1979). However, to diagnose separately pump and ducts it
would be necessary the value of water pressure at pump inlet and outlet, which is not
usually available. Besides, pumps operation strategy is to work at full load in order to
keep the water mass flow at design conditions. For these reasons, the diagnosis variable
chosen is the cooling water flow rate. This parameter can be calculated by using only
water temperatures (and heat rejected by the cycle), without any pressure losses model
of the system.
5.2.3 Boiler
Once the steam cycle and cooling system have been presented, this last section
describes the models and diagnosis variables of the boiler. First, the separated lossesmethod
to calculate the boiler efficiency and coal demand is described. Then, the
combustion model is presented. Afterwards, heat transfer models for the boiler and air
heaters are described, along with some comments on the blowing system. It should be

Case study: 3x350 MW coal-fired power plant
noted that heat transfer in a boiler is a very complex issue that is going to be simplified
in this work. Finally, some comments on ancillary devices are included.
5.2.3.1 Boiler efficiency and coal consumption
The method to calculate coal consumption and boiler efficiency has been adapted
from that implemented in a monitoring system installed in the plant (Uche, 2001). This
fact allows plant staff to compare easily results provided by the two systems. It is based
in the ASME PTC 4.1 standard for calculating the thermal efficiency of boilers (ASME,
1964).
The main differences come from the different definition of the system boundary,
which is a key issue in the calculation of the efficiency, because it affects strongly to the
flows to be used to calculate this parameter. ASME PTC 4.1 standard suggest that
forced draft fans, recycling air fans, electrostatic precipitators or other ash elimination
devices downstream the air heaters, induced draft fans, economizers for using residual
gases heat (downstream the induced draft fans) and stack should be located outside the
control volume. So that, the only ancillary systems whose consumption is considered by
the standard are: air-gas preheaters, primary air fans (if there are one per each mill, they
are located after the preheaters), mills, fans for recycling hot gases towards the furnace
and pumps to assist water circulation in the furnace. This criterion is not strictly adopted
here. First, induced draft fans and electrostatic precipitators are included in the control
volume, because reliable temperature measurement is located in induced draft fans
impulsion. Second, primary air fans and steam coil heaters and mills are considered
outside. More details on these differences can be found in Uche (2001) and in Rangel
(2005).
The calculation of coal consumption is done by considering the energy balance of
the boiler:
E = U + L
5.43
Where E is the total amount of energy entering the boiler, U is the utile heat
delivered to the cycle and other parts of the power plant and L are the loses. Efficiency
is calculated by:
U L
η boiler = = 1−
5.44
E E
According to the previous expression, there are two ways to calculate the efficiency.
The first consists directly in dividing the utile energy into the energy entering the boiler
109

Chapter 5
(direct method), which is not possible because the amount of fuel is not measured. On
the other hand, is also possible to determine the efficiency by considering the losses
(indirect method). This second method has been widely used traditionally because it is
easy to implement by expressing losses in energy units per kilogram of coal. When the
calculation is integrated in a global system to calculate all the plant, both energy balance
and a form of the efficiency equation are solved simultaneously, so that both methods
are actually merged.
Energy entering the boiler is composed of two terms, the enthalpy of the fuel and
other incomes:
110
E = H f + B
5.45
Energy of the fuel includes the coal and, sometimes, other fuels such as natural gas.
It should be noted that the amount of gas can be easily measured, so that the amount of
coal can be calculated by difference. Fuel energy can be approximated by the product of
flow rate and high heating value (Díez, 2001):
H = m ⋅ HHV + m ⋅HHV
5.46
f f f ng ng
Other incomes are due to forced draft fan, primary air fan, secondary and primary
coil heaters:
B = B + B + B + B
5.47
fdf paf 1ch 2ch
Each one of the terms of the previous equation is calculated by multiplying the
amount of air times the enthalpy increment suffered by it:
( )
B = m ⋅ h −h
5.48
i air, i air, out air, in
The previous equation shows that only the effective energy transferred to the air is
considered, without taking into account losses produced for example in the electric
motor. This definition allows to keep the energy balance of the control volume, but is
not suitable to diagnose malfunctions in these ancillary devices.
Utile heat provided by the boiler is divided in two terms: utile heat to the cycle and
utile heat to ancillaries:
U = U + U
5.49
boiler −cycle boiler −ancillaries
Heat delivered by the boiler to the steam cycle should take into account the
following flows: main steam, reheated steam (cold and hot), boiler feeding water and
tempering water:
Uboiler− cycle = mms ⋅hms−mcrs ⋅ hcrs + mhrs ⋅hhrs −m fw ⋅hfw−m tw ⋅htw
5.50

Case study: 3x350 MW coal-fired power plant
Heat to ancillaries includes sootblowing, uncontrolled leakages, drum purge, and
primary and secondary coil heaters:
U − = U + U + U + U + U
5.51
boiler ancillaries sb ul dp 1ch 2ch
Terms of the previous equations can be calculated by using the following
expressions:
( ( ) )
( ( ) ( ) )
( ) ( )
U = m ⋅ h −h
T
5.52
sb sb sb sl ref
U = m ⋅ h p −h
T
5.53
ul ul ss drum sl ref
( )
( ( ) ( ) )
( ( ) ( ) )
U = m ⋅ h p −h
T
5.54
dp dp sl drum sl ref
U = m ⋅ h p −h
T
5.55
1ch 1ch
ss drum sl ref
U = m ⋅ h p −h
T
5.56
2ch 2chss
drum sl ref
Where hsl and hss are the enthalpies of saturated liquid and saturated steam,
respectively, pdrum is the drum pressure and Tref is the reference temperature (ambient
temperature).
After considering the boiler incomes and the utile heat, only boiler losses have to be
studied prior to finishing the paragraph. Losses considered are due to: unburned carbon
in ashes (uc), coal moisture (mf), hydrogen (H), air moisture (mA), CO in gases (CO),
radiation and convection (B), sensible heat in ashes (d), sensible heat in dry flue gases
(gs), air to sulphur removing unit (sru) unit and other losses (ol):
L= Luc + Lmf+ LH+ LmA+ LCO+ LB+ Ld+ Lgs+ Lsru + Lol
5.57
Losses due to unburned carbon in ashes appears due to incomplete combustion that
leads to residual carbon present in ashes and slag:
L = 4.1868⋅8055.7⋅m ⋅unb
[kW] 5.58
uc f
Where 8055.7 kcal/kg is the energy released in the combustion of one kilogram of
carbon, 4.1868 is the conversion factor from kcal to kJ, m f is the coal flow rate and unb
is the number of kilograms of unburned carbon in residues per kilogram of coal entering
the boiler:
unbres
unb = Z ⋅
1−
unbres
5.59
unbres = fashes ⋅ unbresashes + fslag ⋅ unbresslag
5.60
111

Chapter 5
Where Z is the mass fraction of ash in coal, unbres is the number of kilograms of
carbon per kilogram of residues, which is calculated by taking into account the fraction
of residues which is ashes and the fraction which is slag.
Losses for fuel moisture are calculated by:
112
( ( , , ) ( ) )
2
L = W ⋅m ⋅ h T p −h
T
5.61
mf f s go g H O sl ref
Where w is the mass fraction of moisture in fuel and Tgo and pg,H2O are the
temperature and partial pressure of steam in the gases leaving the control volume.
Reference enthalpy corresponds to saturated liquid at reference temperature, because
high heating value is used for the fuel.
Losses due to hydrogen in fuel correspond to the water formed in the combustion of
the hydrogen contained in the fuel, so that they are calculated in a parallel way as fuel
moisture losses:
( )
2
( ) ( , ) ( )
L = 8.936 ⋅ H⋅ m + H ⋅m ⋅ h T , P −h
T
5.62
H f ng ng s go g H O sl ref
Where 8.936 are the kilograms of water produced in the combustion of one kilogram
of hydrogen and H and Hng are the mass fractions of hydrogen in coal and natural gas,
respectively.
Humidity of air also contributes to the losses. However, saturated steam is
considered as reference because this water enters the boiler as steam:
( ( , ) ( ) )
2
L = W ⋅m ⋅ h T p −h
T
5.62
mA mA a, dry s go g, H O ss ref
Where WmA is the number of kilograms of water in air per kilogram of dry air. It can
be calculated from ambient temperature and pressure and relative humidity (Φ):
W
mA
( )
( )
psat Tamb
= 0.622⋅Φ⋅
p −Φ⋅p
T
amb sat amb
Where 0.622 is the relation between the molar masses of water and dry air.
Losses per unburned gases take into account the effect produced by the fraction of
carbon that has been partially oxidized to CO:
CO CO, g
5.63
L = 23631⋅12⋅n [kW] 5.64
Where 23631 kJ/kg is the difference of energy released by burning 1 kilogram of
carbon producing CO2 and producing CO, 12 is the carbon molar mass and nCO,g is the
number of kmol of CO per second produced in the combustion.

Case study: 3x350 MW coal-fired power plant
Although the standard ASME PTC 4.1 (ASME, 1964) provides curves to calculate
boiler losses due to convection and radiation, for the case of study there were empirical
correlations available (Uche, 2001), which have been used here:
4 4
( ( ) ( ) )
Q = S⋅ ⋅ ⋅ ⋅T − − ⋅T − [kcal/h] 5.65
−9
rad 4.48286 10 1.8 wall 491.68 1.8 amb 491.68
( ( ) )
Q = S⋅4.8816⋅ 1+ 0.73818⋅v
⋅ T − T [kcal/h] 5.66
conv wind wall amb
4.1868
LB= ⋅ ( Qrad + Qconv
) [kW] 5.67
3600
Where S is the boiler outer surface in square meters (4158 m 2 ), Tamb and Twall are
respectively temperatures of ambient and boiler wall in ºC (the second one has been
fixed to 40ºC) and vwind is the wind speed in m/s.
Losses in ashes and slag are calculated as:
( ( ) ( ) )
( , ) , ( )
L = f ⋅Z⋅m ⋅ c T ⋅T −c T ⋅T
5.68
d ash f ash go go ash ref ref
( )
L = f ⋅Z⋅m ⋅ c T ⋅T −c T ⋅T
5.69
p slag f slag slag o slag o slag ref ref
Where cash and cslag are the specific heat of ash and slag, respectively, and Tslag,o is
the temperature of slag leaving the furnace. It is considered that flying ashes have the
same temperature as the gas.
The main contribution to the losses corresponds to the sensible heat of dry flue gases
leaving the boiler:
( ( ) ( ) )
L = m ⋅ h T −h
T
5.70
gs gdry , gdry , go gdry , ref
A flow of hot air is extracted from the boiler to heat-up flue gases leaving the flue
gas desulfuration unit. This energy flow leaving the control volume is also considered
as losses:
( ( ) ( ) )
L = m ⋅ h T −h
T
5.71
fgd air, fgd air fgd air ref
Finally, a 0.7 % corresponding to other losses is also added, which takes into
account other issues not considered previously.
L = 0.007⋅
E
5.72
ol
It should be noted that sootblowing steam has not been considered a loss but a useful
product. Due to this choice, variation in this magnitude will not affect boiler efficiency.
113

Chapter 5
5.2.3.2 Combustion
The aim of this paragraph is to explain the diagnosis variables, equations and models
related to combustion.
To determine the composition of gases leaving the furnace, first it is necessary to
consider the fuel composition. Coal contains carbon, hydrogen, nitrogen, oxygen,
sulphur, ashes and moisture. All these mass fractions minus one (oxygen, which is
calculated by difference) are degrees of freedom and, subsequently, free diagnosis
variables. Natural gas contains CH4, C2H6, C3H8, CO2 and other components. Again, the
molar fractions of all components minus one are degrees of freedom and can be free
diagnosis variables. Besides, relation between the amounts of coal and natural gas
should be specified. A good variable to this relation is the fraction of energy provided
by natural gas. High heating values of coal and gas are also free diagnosis variables.
Once the fuel composition is determined, gas composition is calculated by matter
balances. If complete and stoichiometric combustion is achieved, gas contains: CO2,
H2O, SO2, N2 and Ar. The amount and composition of gases and the amount of air
needed are calculated by performing matter balances for carbon, hydrogen, sulphur,
oxygen, nitrogen and argon.
In practical applications, stoichiometric combustion is not possible, so that an
additional amount of air is introduced (air excess), which implies an additional degree
of freedom. The variable chosen is the oxygen concentration of gases leaving the boiler
(before air preheaters), which is usually a set point to control the air introduced in the
boiler.
Besides, real combustion is not complete, so that unburned carbon appears in ash
and CO is present in gases, which adds two more degrees of freedom and diagnosis
variables: mass fraction of carbon in ashes and molar fraction of CO in flue gases. Flue
gases have minor amounts of other components such as NOx, which are of great
importance from the environmental point of view, but its influence of energy efficiency
is negligible, so that they are not considered. It should be noted that all these variables
are not actually independent, because they depend strongly of the type of coal and boiler
operation (excess air, choice of burners rows…). However, due to the difficulty to
determine these relations, they are going to be considered as independent.
A lot of research has been done in order to predict unburned coal in ashes, some of
them applied to the case of study of this thesis. Influence of CO in boiler efficiency is
about one order of magnitude lower than effect of carbon in ashes (Díez, 2002).
114

Case study: 3x350 MW coal-fired power plant
A method to predict unburned carbon in ashes has to calculate the combustion time
of coal particles in the furnace, which is the time necessary to burn all the carbon in the
particle. If it is lower than the residence time, all particle is burned; if not, an amount of
residual carbon appears. Combustion of a carbon particle in a utility boiler can be
divided in three stages (Spalding, 1983):
1. Devolatilisation: is the phase when volatiles contained in the particle are released.
2. Homogeneous volatile combustion.
3. Heterogeneous combustion of the carbonaceous residue (fixed carbon + ashes).
Time necessary for the first and second stages are very reduced compared to the
time for the last one, so that this third stage is going to determine the combustion time.
Two models have been proposed to simulate it. The first one is the progressive
conversion model, which considers that the gas surrounding the particle enters inside it
and all particle reacts simultaneously.
The unreacted core shrinking model considers that the reaction starts in the outer
part of the particle and goes towards the particle centre. Díez (2002) applies this
methodology to one boiler of Teruel power plant.
CFD techniques have also been applied to simulate combustion in this boiler (Iranzo
et al., 1999, 2001).
5.2.3.3 Heat transfer in boilers
To model heat transfer in a boiler, it has been proposed to decompose it in a set of
heat exchangers and to apply to each one of them either the ε-NTU or the ΔTlm methods
presented in the condenser section.
An important issue of heat transfer in the boiler that was negligible in the condenser
is radiation. If radiation absorbed by a heat exchanger is considered and heat emission is
neglected (which is a good approximation due to the low temperature of the surfaces
and the high temperature of the gases or the flame (Díez, 2002)), the energy balance
becomes:
Q + α ⋅G⋅ A = Q
5.73
h w h c
where h Q and c Q are the heat corresponding to hot and cold sides, αw is the wall
absorptivity, G is the irradiation and Ah is the area of the hot side of the heat exchanger.
The problem of the radiative term appears when the transference equation is written,
because two options are possible:
FUA ⋅ ⋅Δ T + α ⋅GA ⋅ = Q
5.74a
lm w h c
115

Chapter 5
116
Q + α ⋅G⋅ A = F⋅UA⋅Δ T
5.74b
h w h lm
There is no reason for choosing one or the other, and this choice can lead to different
solutions. A method has been developed to solve this problem (Díez, 2002). The
mathematical development is similar to the one used to develop the ΔTlm equation: to
consider a section of differential length and constant temperature of both hot and cold
fluids. However, the contribution of the radiation of this section is also considered. By
integrating that balance, the following equation is obtained for a counterflow heat
exchanger with radiation from outside:
⎛Th, out − Tc, in + k ⎞ ⎛Tc, out −Tc, in Th, in −Th,
out ⎞
ln ⎜ = UA⋅
−
⎜ ⎟ ⎜ ⎟
Th, in − Tc, out + k ⎟ Q ⎝ c Q
5.75
⎝ ⎠
h ⎠
Where k is a factor with temperature dimensions that take into account the effect of
radiation:
α w⋅G⋅Ah αw⋅G
k = +
5.76
⎛C⎞ h
h
h
UA⋅⎜ −1⎟
⎝Cc⎠ where hh is the convection coefficient of the hot side and Cc and Ch are the relations
between the energy and temperature variations of cold and hot fluids:
( , , )
Q = C ⋅ T −T
5.78
c c c out c in
( , , )
Q = C ⋅ T −T
5.79
h h h in h out
The transference equation obtained can be completed by adding a factor F
multiplying U·A, in order to take into account effect due to non parallel fluxes. The use
of this factor in this equation is an approximation, but at least it can be made from an
excact equation. As it can be seen, when radiation is not present, the value of k is zero
and the transfer equation becomes the classical ΔTlm expression.
Once the transfer equation has been presented, the global heat transfer coefficient
should be calculated, by using the thermal resistances composition equation previously
presented (5.2.2.1).
Due to the high temperatures of the hot fluid (combustion gases), convection
coefficient in the hot side should take into account not only convective but also radiant
contributions:
hh= hconv + hrad
5.80

( )
h g wall
Case study: 3x350 MW coal-fired power plant
hconv can be calculated by using suitable correlations available in bibliography
(Mills, 1992) The apparent radiation coefficient hrad is defined according to the heat
transferred by radiation from the gases inside the heat exchanger to the tubes:
Q
rad , g→wall hrad
=
5.81
A ⋅ T −T
It should be noted that the apparent radiation coefficient takes into account the
radiation from the gas located inside the exchanger (next to the tubes) and does not
affect the energy balance of the heat exchanger. Radiation from other areas of the boiler
(mainly the furnace and the plenum of gases) is introduced by the flux G which has
originated the modification of the transference equation. For example, radiation from
the furnace can be calculated by:
4 4
( )
Q = F ⋅A ⋅σ ⋅ε ⋅ T −T
5.82
rad , furnace ij i gas gas, furnace dwall
Where Fij is the view factor between two surfaces, which represents the fraction of
radiation leaving the furnace (surface i) that arrives at the heat exchanger (surface j),
and depends on the geometry, σ is the Stefan-Boltzman constant and Tdwall is the
temperature of “dirty wall” of the tubes. This expression can also be used to calculate
radiation from the plenum, by considering the suitable temperatures and view factors.
To model the furnace, Kakaç (1991) cites the standard method developed by Blokh
(1984). It uses experimental coefficients obtained in test performed in different boiler
types, mainly in China and the former USSR, where this method was adopted as a
standard for boiler manufacturers. The basic equation of this method is the Gurvich
formula, which provides the value of the temperature of gases leaving the furnace:
T = T ⋅
Bo
0.6
go, furnace g, ac 0.6 0.6
Bo + M ⋅εF
Where Tg,ac is the adiabatic combustion temperature, M is a coefficient which takes
into account temperature distribution in the furnace, εF is the average emissivity of the
furnace and Bo is the Boltzman number, which represents the relation between heat
transferred by enthalpy variation and by radiation. Actually, the previous equation
represents a global energy balance in the furnace.
Ideas previously presented summarized a heat transfer model developed by Díez
(2002) for Teruel power plant. Boiler has been discretized in: furnace, primary and
secondary economizers, reheaters, primary, secondary and radiant superheaters, gases
5.83
117

Chapter 5
plenum and boiler walls next to the radiant superheater and the plenum. The following
radiation flows have been considered: furnace to secondary and tertiary superheaters
and to walls, secondary superheater to walls and tertiary superheater, and plenum to
tertiary superheater, primary superheater, reheater and walls.
This model has been simplified and included in this work. By using a simulator,
relations between radiation flows and heat exchangers effectiveness with load have been
obtained. This simplification may be strong but it is justified because interest is focused
in the global behaviour of the boiler, and models for every individual heat exchanger are
considered of secondary importance.
The diagnosis variable proposed for heat exchangers is its effectiveness. However,
due to the lack of information, it is not possible to diagnose individually all heat
exchangers. The aim is to determine whether the boiler is able to use efficiently the heat
of flue gases, so that a good parameter may be the average temperature of gases leaving
the boiler (prior to air preheaters):
118
T
go, av
m ⋅ T + m ⋅T
=
m + m
g, bypass g , bypass go, ez1 go, ez1
g, bypass go, ez1
In the previous equation, flow of gases leaving the primary economizer and by pass
gases extracted after the reheater have been considered. Specific heat variation has been
neglected. This variable can be transformed into an effectiveness by considering
reference temperatures for adiabatic combustion temperature and water entering the
boiler:
ε
=
T −T
5.84
5.85
g, ac, ref go, av
boiler
Tg, ac, ref −Tw,
in, ref
where Tg,ac,ref and Tw,in,ref are reference values for temperatures of adiabatic
combustion and water entering the boiler. They have been fixed to 2000 and 200 ºC,
although it may be also possible to use the specific values of these temperatures instead
of these constant values.
The standard method for furnaces calculation allows easy implementation and fast
response that allows to implement the heat transfer models for real time monitoring.
However, it entails a strong simplification. So that, it has been proposed to use a more
sophisticated method to model the furnace (Díez, 2002; Díez et al., 2005; Cortés et al.,
2001). This method is based on the use of the Hottel method to modelise radiation,
which implies the discretization of the furnace in several sections. Fluid flow and heat

Case study: 3x350 MW coal-fired power plant
release patterns are needed, so that a combustion model and a CFD simulation are used.
CFD calculations are time expensive, so that they are not suitable for some important
applications such as real time monitoring. These authors propose to pre-calculate the
fluid flow for several representative scenarios, and then use this information in real-time
calculations. Hottel method has also been applied to the gases plenum.
The main problem related to heat transfer degradation in boilers is fouling, or
mineral matter deposits in heat transfer surfaces. Fouling in a pulverized coal-fired
boiler can be classified in slagging and fouling (Valero and Cortés, 1996):
-Slagging, referring to the deposition taking place in sections of the boiler where
radiant heat transfer is dominant. Most commonly, slagging means ash fouling in the
water walls making up the furnace. Also deposits in radiant platens and first rows of
pendant superheaters are occasionally included in this category.
-Fouling, defined as deposits on the surface of convective (i.e. not seen from the
flame) tube bundles.
These problems can increase when design fuel is modified. To clean the surfaces,
sootblowing is used. However, when slagging has started, sootblowing is not effective
because slagging produces a reduction in heat transfer with leads to higher temperature
at walls surface and, subsequently, more slagging. Due to this situation, load should be
reduced in order to decrease the radiation and to enable a solidification of deposits,
which allows its removal. The problem of slagging in coal-fired boilers, and specifically
at Teruel power station has been tackled by Cortés et. al. (1989).
Sootblowing usually succeeds in fouling cleaning, but inadequate use of this
technique has serious disadvantages. First, if steam is applied to a clean surface, it can
be damaged. Second, steam consumption originates an important reduction of power
plant efficiency. For this reason, it is very important to use sootblowing in an optimum
way.
The first step is to monitor the heat transfer in the boiler, which can be used by
applying models such as those previously presented. Fouling monitoring can lead to
improve sootblowing: do not blow according to pre-fixed schedules but taking into
account the state of the surfaces. The next step is sootblowing optimization, which looks
for an equilibrium between the efficiency losses produced by sootblowing and
efficiency losses caused by heat transfer reduction. To perform this optimization,
predictive models should be used, capable to determine the probability of blow success
for a given fouling conditions and to predict the evolution of the fouling along time.
119

Chapter 5
Neural networks are a suitable tools to build-up these models. More details on these
topics and its application to Teruel power plant can be seen in: Cortés et al. (1993),
Bella et al. (1994), and Teruel et al. (2005).
In this thesis, the amount of steam has been considered as independent free
diagnosis variable. This will allow to see the high influence of it on unit efficiency.
5.2.3.4 Air preheaters
As it has been commented in the plant description, at Teruel power plant there are
two pairs of preheaters: primary air preheaters (recuperative type) and secondary air
preheaters (regenerative type). In this section, aspects on modelling, working and
suitable free diagnosis definitions are commented for these plant components.
The main factors defining a right operation of a preheater are fouling and air
leakages. As in boiler heat exchangers, fouling appears when flying ashes are deposed
on heat exchanger surfaces reducing the heat exchange capability. In regenerative type
preheaters, this effect may lead to significant reduction of channel areas with important
pressure losses. Besides, chemical fouling may appear if gases reach temperatures lower
than the acid condensation point. To avoid this, cold side temperature should be kept
above a value fixed by the preheater manufacturer.
Air leakages are caused by pressure difference between air and gases circuits.
Although they can appear in all types of preheaters, they are relevant in regenerative
preheaters of rotating type, where the working principle based on a rotary basket makes
it difficult to achieve a good sealing. An excessive value of air leakages implies a
reduction in preheater efficiency and additional work of forced and induced fans.
Air leakages also affect the definition of thermal efficiency:
120
ε
preheater
( ) ( )
( ) ( )
m ⋅h T −m ⋅h
T
=
m ⋅h T −m ⋅h
T
a, o a a, o a, in a a, in
ao , a gin , ain , a ain ,
For simulation purposes, the calculation of this parameter depends on the type of
preheaters.
For recuperative preheaters, the conventional ε-NTU method is usually applied. The
case of regenerative heat exchangers is a bit different, because these types of heat
exchangers do not work in stationary state, but in a cyclic operation: an intermediate
heating matrix is alternatively heated by a flow of hot fluid and cooled by the cold
stream. To simulate this type of heat exchangers, the ε – NTU0 method can be used.
5.86

Case study: 3x350 MW coal-fired power plant
It is supposed that conduction along the matrix axis is negligible in the matrix and in
circulating fluids, and transversal conductivity in the matrix is considered to be very
high. These assumptions are equivalent to suppose that matrix temperature is constant in
each normal section considered. With these hypotheses it is possible to write
expressions similar to those of the ε – NTU method but considering the exchange
periods and the heat capacity of the matrix material:
⎛ ma cp, a Pa hc Pa C ⎞
mat
ε preheater ε NTU0 , , ,
mgcpg , PghhPgmacpa , Pa
⋅ ⋅
⋅
= ⎜
⎟
⋅ ⋅ ⋅ ⋅ ⋅ ⎟
⎝
⎠
5.87
Where Pa and Pg are the exchange periods for air and gases, hc and hh are the
convection coefficients for cold and hot sides, Cmat is the thermal capacity of the
material of the matrix and NTU0 is the modified number of transfer units:
NTU
0
⎛ ⎞
1
⎜
1
⎟
= ⋅⎜
⎟
m 1 1
a⋅cp, a ⎜ ⎟
⎜
+
hh⋅Ah hc⋅A ⎟
⎝ c ⎠
It should be noted that heat exchanger configuration does not appear, because it is
supposed that regenerative preheaters always work at counterflow. Besides, matrix
thermal conductivity is not present because it has been supposed to be zero in
longitudinal direction and infinite in transversal direction. These previous equations
have been developed for static preheaters. However, they can be adapted to rotary type
exchangers by discretizing them in angular sectors. More details on the simulation of
regenerative preheaters can be seen in Díez (2002).
The parameter proposed as free diagnosis variable is a simplified expression of
efficiency:
ε
preheater
T −T
=
T −T
ao , ain ,
g, in a, in
It should be noted that this equation is the same as that one presented previously if
infiltration is neglected and specific heat capacity of air is considered as constant. This
simplified equation has been used because it can be calculated directly from plant data
and is used by plant staff. For a simple modelling of heat exchangers, it is possible to
correlate it with plant load.
The other aspect to be considered in preheaters is infiltration. The diagnosis variable
to take into account this phenomenon is air leakages. They can be quantified by
measuring oxygen molar fraction in flue gases entering and exiting the preheater:
5.88
5.89
121

Chapter 5
122
r − r
O2, o O2, in
a, leakage = g, in ⋅
5.90
0.21−
rO2,
o
m m
where rO2 are the oxygen molar fractions in flue gases entering and exiting the
preheater.
The previous equations allow to determine leakages in a preheater if oxygen fraction
is known at the outside of it. However, in some cases, such as the working example,
oxygen content at outside is only known in one point where flows leaving all the
preheaters. In this case, equation 5.90 can be applied but considering total gas flow and
total leakages. To determine leakages separately in all preheaters, additional
assumptions should be made, such as neglecting leakages in recuperative preheaters.
For these reasons, chosen diagnosis variable is the relation between total leakages and
total gas entering all preheaters, which can be decomposed in two terms corresponding
to primary and secondary preheaters:
m m + m
a, leakages, total a, leakages,1 ph a, leakages,2ph Leakages = = = Leakages1ph + Leakages2ph
mgas, in, total mgas,
in, total
5.2.3.5 Ancillary devices
Equipments included in this last section are fans, mills and electrostatic precipitator.
Fans behaviour are represented in characteristic curves provided by the
manufacturer. These curves, usually relate the static pressure increment to the
volumetric flow and an angle:
st st v
( , )
5.91
Δ p =Δp Q β
5.92
Where β is the angle of the director blades in radial fans, and the angle of fan blades
in axial fans. In radial fans, a set of curves indicates the power consumed in function of
the same parameters:
W = W Q β
5.93
(
v,
)
In axial fans, curves provide the efficiency as function of the operation point:
( Q , p )
η = η Δ
5.94
fan fan v st
Operation point depends on both the fan and the installation, and is determined by
crossing their characteristic curves (such as in pumps). ηfan, power consumed and
operation point are related by the following expression:

Case study: 3x350 MW coal-fired power plant
Δptot ⋅Q
v
η fan =
5.95
W
Ancillary devices have not been studied in this work, because gross electric power
and efficiency are considered. The impact in fuel of each additional kW consumed by
them is directly the gross heat rate of the unit. These extra consumptions can be caused
not only by a malfunction in the component itself but also by a malfunction in other
parts of the plant that make these devices to work more. For example, additional
electricity consumptions by fans can be due to a degradation in the fans or to a higher
pressure loss in the boiler. In the same way, increment in electricity consumed by
electrostatic precipitators can be caused not only by a failure in them but also to an
increment in flying ashes in flue gases. Finally, electricity consumed by mills depends
on both internal causes and coal properties.
Analysis of these relations requires analysis of the behaviour and modelling of these
components. This problem has been tackled in general and specifically for Teruel
problem plant by Arauzo et al. (Arauzo et al., 1993; Arauzo and Cortés 1995a, 1997).
Díez (2002) has adapted the methodology of fan modelling for the new primary air fans.
Relation between ancillary devices operation and boiler has also been studied by
applying a symptom-problem scheme (Arauzo and Cortés, 1995b).
5.3 Monitoring and diagnosis model for Teruel power plant.
In the first part of this section, used measurements are reviewed, because they are
going to determine the depth of the diagnosis achievable. According to this information
and to the guidelines presented in the previous paragraph, free diagnosis variables are
presented. Finally, global efficiency indicators defined are presented.
5.3.1 Measurement review.
In table 5.6, a list of measurements used to monitor the power plant is presented.
They are ordered by areas: ambient conditions, water-steam circuit (according to fluid
flow and included cooling system) and boiler.
123

Chapter 5
Number Description
124
1 Ambient temperature
2 Dew point
3 Ambient pressure
4 Wind speed
5 Main steam temperature right side
6 Main steam temperature left side
7 Main steam pressure right side
8 Main steam pressure left side
9 Cold reheated steam temperature right side
10 Cold reheated steam temperature left side
11 Cold reheated steam pressure right side
12 Cold reheated steam pressure left side
13 Hot reheated steam temperature right side
14 Hot reheated steam temperature left side
15 Hot reheated steam pressure right side
16 Hot reheated steam pressure left side
17 Temperature of steam extraction for 5 th heater
18 Pressure of steam extraction for 5 th heater
19 Temperature of steam extraction for deaerator
20 Pressure of steam extraction for deaerator
21 Gross power
22 Condenser absolute pressure
23 Temperature of cold cooling water right side
24 Temperature of cold cooling water left side
25 Temperature of hot cooling water right side
26 Temperature of hot cooling water left side
27 Temperature of water entering the deaerator
28 Flow rate of water entering the deaerator
Table 5.6.A. List of plant measurements used.

Number Description
29 Mass flow of water from dribbling tank to deaerator
30 Pressure of water entering the turbo-pump
31 Pressure of water leaving the turbo-pump
32 Temperature of the drain of 5 th heater
33 Temperature of water leaving the 5 th heater
34 Temperature of the drain of 6 th heater
35 Temperature of water leaving the 6 th heater
36 Steam for sootblowing flow rate
37 Temperature of air entering primary preheater side A
38 Temperature of air entering primary preheater side B
39 Temperature of air leaving primary preheater side A
40 Temperature of air leaving primary preheater side B
41 Temperature of gases entering primary preheater side A
42 Temperature of gases entering primary preheater side B
43 Temperature of gases leaving primary preheater side A
44 Temperature of gases leaving primary preheater side B
45 Temperature of air leaving secondary coil heater side A
46 Temperature of air leaving secondary coil heater side B
47 Temperature of air leaving secondary preheater side A
48 Temperature of air leaving secondary preheater side B
Case study: 3x350 MW coal-fired power plant
49 Temperature of gases entering secondary preheater side A
50 Temperature of gases entering secondary preheater side B
51 Temperature of gases leaving secondary preheater side A
52 Temperature of gases leaving secondary preheater side B
53 Temperature of gases leaving induced draft fans
54 Primary air mass flow side A
55 Primary air mass flow side B
56 Secondary air mass flow side A
Table 5.6.B. List of plant measurements used.
125

Chapter 5
Number Description
126
57 Secondary air mass flow side B
58 Tempering air mass flow side A
59 Tempering air mass flow side B
60 Air to flue gas desulfuration mass flow side A
61 Air to flue gas desulfuration mass flow side B
62 Oxygen in flue gases leaving the boiler side A
63 Oxygen in flue gases leaving the boiler side B
64 Oxygen in flue gases leaving induced draft fans
65 Natural gas flow
66 Coal high heating value
67 Carbon mass fraction in coal
68 Hydrogen mass fraction in coal
69 Oxygen mass fraction in coal
70 Nitrogen mass fraction in coal
71 Sulphur mass fraction in coal
72 Ash mass fraction in coal
73 Moisture mass fraction in coal
74 Carbon in ash
Table 5.6.C. List of plant measurements used.
As it can be seen, there is an important lack of information in the low pressure zone
of the steam cycle and in the boiler. This fact is going to limit the depth of the diagnosis
in these areas, and is going to lead to the use of aggregated variables and models. On the
other hand, there are local redundancies in some areas: temperatures in air heaters, mass
flows of primary, secondary and tempering air (one is redundant), and regenerative
water heaters. Although several techniques are available to solve this fact, for example
by minimizing a global uncertainty function (Correas, 2001), here, these variables have
been used to pre-calculate several parameters that are going to be introduced in the
resolution of the plant: for example, instead of introducing the flows of primary and
secondary air, the relation between them is used, thus eliminating one variable.

Case study: 3x350 MW coal-fired power plant
All these variables are measured by plant instruments and are available in the plant
information system, except results coming from laboratory (coal and ash analysis),
which corresponds to daily analyses.
5.3.2 Free diagnosis variables
The aim of this section is to present the diagnosis variables used and how they are
calculated in the performance test and in the simulation problems, in order to clarify the
differences and common points between the three problems.
Free diagnosis variables appear in table 5.7. They are ordered by categories: ambient
conditions (A), fuel (F), set points (SP) and component parameters (CP). It should be
noted that this classification is only indicative, and it is subject to discussion. For
example, the amount of natural gas is fixed by a set-point but is included in fuel, and air
for desulfuration unit is not strictly a set point because the set point is the temperature of
flue gases to stack.
The column of “calculation in performance test” summarizes the differences
between the diagnosis and performance test problems. Most of variables are
measurements or can be calculated directly from them. In this case, the equation is the
same for diagnosis and performance test problems. It should be noted that prior to the
main equations system, a module to pre-calculate parameters directly from plant signals
is used. On the other hand, there are other variables that cannot be easily obtained from
plant data. In this case, another parameter is fixed instead of the value of the diagnosis
variable considered. For simulation purposes, equations are the same as for diagnosis,
but the value of free diagnosis variables corresponding to components parameters is
determined by suitable correlations that should be tuned to the component
characteristics. Formally, this is a minor difference, but in practice this is the more
complex and time consuming issue.
In table 5.8, more diagnosis variables are included. They are formulated in the same
way, however, due to the lack of information, they have a constant value, so that they
do not provide any impact.
127

Chapter 5
Num. Description units Calculation in
performance
test
128
Category
1 Ambient temperature ºC Direct A
2 Relative humidity % Direct A
3 Wind speed m/s Direct A
4 Coal high heating value kJ/kg Direct F
5 Carbon mass fraction in coal % Direct F
6 Hydrogen mass fraction in coal % Direct F
7 Moisture mass fraction in coal % Direct F
8 Ash mass fraction in coal % Direct F
9 Sulphur mass fraction in coal % Direct F
10 Nitrogen mass fraction in coal % Direct F
11 Energy provided by natural gas % Natural gas
flow rate
12 Main steam temperature ºC Direct SP
13 Reheated steam temperature ºC Direct SP
14 Main steam pressure bar Direct SP
15 Gross electric power MW Direct SP
16 Oxygen in flue gases leaving the boiler % Direct SP
17 Average cold-side temperature in
secondary air preheaters
18 Average cold-side temperature in
primary air preheaters
ºC Direct SP
ºC Direct SP
19 Sootblowing steam flow rate kg/s Direct SP
20 Air for flue gas desulfuration unit kg/s Direct SP
21 Tempering and primary air relation kg/kg Direct SP
22 Primary air-coal ratio kg/kg Primary and
secondary air
relation
Table 5.7.A. Free diagnosis variables.
F
SP

Case study: 3x350 MW coal-fired power plant
Num. Description units Calculation in
performance
test
23 Temperature difference of flue gases
entering preheaters
Category
ºC Direct SP
24 Fraction of flue gases through PAH - Relation of
flue gases
temperature
drop in PAH
and SAH
25 High pressure turbine isoentropic
efficiency
26 Intermediate pressure 1 isoentropic
efficiency
27 Intermediate pressure 2 isoentropic
efficiency
SP
% Direct CP
% Direct CP
% Direct CP
28 Low pressure isoentropic efficiency % Water entering
the deaerator
flow rate.
29 Intermediate
coefficient
pressure 1 flow
30 Intermediate
coefficient
pressure 2 flow
kg 1/2 ·m 3/2 ·
s -1 ·bar -1/2
kg 1/2 ·m 3/2 ·
s -1 ·bar -1/2
31 Low pressure flow coefficient kg 1/2 ·m 3/2 ·
s -1 ·bar -1/2
CP
Pressure CP
Pressure CP
Pressure CP
32 TTD 6 th water heater ºC Direct CP
33 TTD 5 th water heater ºC Direct CP
34 TTD 3 rd water heater ºC Direct. CP
Table 5.7.B. Free diagnosis variables.
129

Chapter 5
Num. Description units Calculation in
performance
test
130
Category
35 TDCA 6 th water heater ºC Direct CP
36 TDCA 5 th water heater ºC Direct CP
37 Pressure drop in reheater bar Direct CP
38 Pressure increment in turbo-pump bar Direct. CP
39 Water losses kg/s Direct CP
40 Condenser effectiveness - Condenser CP
41 Cooling water flow rate kg/s pressure, CP
42 Cooling tower effectiveness - Cooling water
inlet and outlet
temperatures
CP
43 Secondary air preheater effectiveness - Direct CP
44 Primary air preheater effectiveness - Direct CP
45 Aggregated boiler effectiveness - Temperature
of flue gases
leaving IDF
46 Air infiltration in preheaters % Oxygen in flue
gases leaving
IDF
47 Carbon in ashes % Direct CP
Table 5.7.C. Free diagnosis variables.
CP
CP

Case study: 3x350 MW coal-fired power plant
Num. Description Value Units Category
48 Natural gas high heating value 43488 kJ/Nm 3 F
49 Methane in natural gas 91.54 % F
50 Ethane in natural gas 5.27 % F
51 Propane in natural gas 1.99 % F
52 Temperature of steam leaving
secondary superheater
53 Air from the output of primary
heater entering secondary heater
480 ºC SP
0 kg/s SP
54 Air leakage in primary air heater 0 kg/s CP
55 CO in flue gases leaving the boiler 130 ppm CP
Table 5.8. Free diagnosis variables with constant value.
5.3.3 Global efficiency indicators.
Although the theoretical development has been made for only one global efficiency
indicator, it is possible to define as much indicators as the analyst considers. In this
work, three parameters have been considered, according to the usual practice in the
power plant: gross unit heat rate, gross cycle heat rate and boiler efficiency.
Gross unit heat rate is the relation of energy of fuel entering the plant and power
produced by the alternator:
H
f
HRu =
W
e
Gross cycle heat rate is the relation of energy provided to the cycle and electric
power produced:
HR
c
U + U
=
W
boiler− cycle others
e
where Uothers takes into account other minor energy sources: water from dribbling
tank and vent from blow down tank entering the deaerator, and make-up water entering
the condenser. Finally, boiler efficiency is calculated as:
5.96
5.97
U L
η boiler = = 1−
5.44
E E
131

Chapter 5
although it is usually expressed in %. More details can be seen in section 5.2.3,
Rangel (2005) or Uche (2001).
5.4 Conclusion
This chapter should be considered as an example of methodology to be followed in
order to define a diagnosis system. First of all, the power plant has to be analyzed.
Second, information regarding models, degradation mechanisms and operational
problems has to be collected for the main plant components. Afterwards, available
measurements should be considered, in order to determine the depth of diagnosis
achievable. Finally, all this information has to be used to define free diagnosis variables
and global efficiency indicator(s).
It should be noted that the process is not finished until the system is installed and
tested with plant data. The detail of the diagnosis required is not only determined by
previous experiences and theory but also by the actual situation of the system to be
analyzed, which is not known until the diagnosis is performed. In other words, this
design is not usually straightforward but revisions are needed.
Information on the data retrieval system and instrumentation accuracy can be seen in
Rangel (2005). Results of the application of a diagnosis system in the working example
are reported in next chapter.
132

6 Application of quantitative causality
analysis
In this chapter, results of anamnesis or the repeated diagnosis of the Teruel power
plant are presented. This technique allows to see the evolution of degradation indicators
with its dispersion, and helps to overcome errors induced by instrumentation uncertainty
and the diagnosis method itself. After a brief introduction to present the main
assumptions made, evolution of the three efficiency indicators considered (unit and
cycle heat rate and boiler efficiency) is shown. Then, the influence of all free diagnosis
variables is analyzed, demonstrating the capability of quantitative causality analysis to
quantify the impact caused by each one of them. Afterwards, error produced by the
method is studied by analysing the residual term (the part of the deviation in the global
efficiency indicator which is not assigned to free diagnosis variables variations). Results
show that this term is negligible. Finally, techniques such as impact addition and
causality chain analysis are applied to facilitate the interpretation of the diagnosis results
and to filtrate crossed relation between free diagnosis variables.
6.1 Introduction
The diagnosis system presented in the previous chapter has been used to diagnose
the three units of Teruel power plant for a time span of more than six years: from 1 st
January 1999 to 20 th March 2005. Information needed to perform the diagnosis has been
obtained from the data base of the plant information system, where average daily
measurements classified by load levels were available. Only full load level (more than
320 MW) has been considered. In total, 4570 operation days have been studied: 1539 of
unit 1, 1495 of unit 2 and 1536 of unit 3.
133

Chapter 6
Since the interest of the study is the analysis of the evolution of free diagnosis
variables and their impacts, the choice of a reference state is a secondary issue, which
depends on the preferences of the analyst. In fact, results obtained by using one
reference state or another, would very similar but the graphs of impacts would be
biased. When a new plant is studied, design conditions are usually used as reference.
However, the age of the plant considered advises to use a more recent reference, such as
one of the days analyzed. According to plant staff and to the results obtained by
preliminary analyses, unit 3 during September 2003 represents a good period to be used
as reference, because most free diagnosis variables have a value which can be
considered as representative of the plant, neither too high nor too low. For this reason,
the state of this unit on 17 th September 2003 has been chosen.
For each one of the 47 free diagnosis variables considered, seven graphs may be
studied: variable evolution, the evolution of the impact on the three efficiency indicators
considered and relation between these three impacts and the variation of the free
variable. In each one of these graphs, results for the three groups are plotted by using
different colours. This multiplicity of graphs makes non practical to include all of them
in this document. Information contained in them can be summarized by using several
parameters.
Besides the use of indicators to quantify the variability of the free diagnosis
variables and their impacts is needed. Standard deviations are proposed for this task.
They allow to resume the variation of free diagnosis variables and the importance of the
impacts caused by them. An impact with high standard deviation indicates that its
corresponding free diagnosis variable has influenced strongly the evolution of the
efficiency indicator during the time span studied. The use of these parameters has the
advantage of avoiding the influence of the reference state chosen. They are defined as
follows:
134
σ
σ
xdi
,
Ii
, ηb
=
=
n
p
j
∑(
xdi , − xdi
, )
j=
1
n
n
p
p
j
∑(
Ii, η − I , )
b iηb
j=
1
n
p
2
2
6.1
6.2

σ
σ
IiHR
, c
IiHR
, u
=
=
n
p
j
∑(
IiHR , − I , )
c iHRc
j=
1
n
n
p
j
∑(
IiHR , − I , )
u iHRu
j=
1
n
p
p
2
2
Application of quantitative causality analysis
where σ is the standard deviation of a free diagnosis variable ( x di , ), or of its impacts
on boiler efficiency ( Ii, η ), steam cycle heat rate ( I
b
iHR , ) or unit heat rate (
c
iHR , u
6.3
6.4
I ), which
are calculated by using the individual (j) and average values of these parameters in a set
of n p points. These sets can be the points of a unit, or all the points of the three units
together. When no unit is specified, parameters are calculated by using all the units of
the plant.
Other interesting aspect is the relation between the variation of a free diagnosis
variable ( Δ xdi
, ) and the impact which it produces on the boiler efficiency and cycle and
unit heat rates ( Ii, η , I
b iHR , and
c iHR , u
correspond to the elements of vector ed.
I
edi,
η b x
= Δ
ed
ed
iHR , c
iHR , u
i,
ηb
di ,
I
=
Δx
iHR , c
I
=
Δx
di ,
iHR , u
di ,
I ). For a simple diagnosis, these impact factors
These factors are very important because they constitute the relation between free
diagnosis variable increment and impact, which is the key of the diagnosis
methodology. However, they are associated to only one point. Since now there are
several points, it is preferable to calculate average values by obtaining the slopes of the
graphs which represent impacts versus variation of free diagnosis variables. These
slopes are obtained by calculating the linear regression of the graph. To remark this fact,
6.5
6.6
6.7
135

Chapter 6
av av
av
the superscript av can be added. In this way, edi, η , ed
b iHR , and
c
iHR , u
136
ed stand for
calculated slope of the curves that relate the impacts on boiler efficiency and cycle and
unit heat rates produced by the variation of the free diagnosis variable x di , to the
increment of this variable.
In graphs relating impacts to free variables increments, not only the slope presented
above has to be considered. Besides, the degree of non linearity or dispersion of the
points is also important. To take into account this fact, the linear regression coefficients
(R 2 ) are also considered. These parameters reach the value 1 when the correlation is
perfectly lineal, and decrease when the graphs curve or when dispersion appears. If all
of them were 1, relations would be perfectly known and constant values of ed could be
used. However, these relations are not always so perfect, and it is worth to use the
diagnosis methodology to calculate ed for each point. This task is detailed in section
6.4.
Once both assumptions made for performing the anamnesis and definitions and
importance of some parameters have been presented, in the next section the evolution of
the three efficiency indicators is shown.
6.2 Evolution of efficiency indicators
The three efficiency indicators considered are unit and steam cycle gross heat rate
and boiler efficiency. Units of heat rate are sometimes kcal/kWh, although in this work
are kJ/kJ (non-dimensional). Boiler efficiency is expressed in %.
Figure 6.1 shows the evolution of unit gross heat rates. During the first three
months, unit heat rates are high. Then they decrease dramatically and then increase
steadily for three years and stand more or less flat for other three. Finally, they decrease
sharply again and tend to increase. Apart from these common trends, some differences
among the three units can be observed. During the first two years, unit two has higher
heat rate. Then, in the last three years, unit three is the best and then comes unit two.
However, in the last months, unit one heat rate is reduced substantially, so that the
values of the three units become quite close. Although some general points can be
observed, the graph is characterized by a high short-term dispersion (width of the band
of points) of about 0.05, which makes difficult to detect tendencies. Besides, some

Application of quantitative causality analysis
points too high or too low (outliers) are present. A first step towards a suitable
decomposition of unit heat rate evolution is the use of cycle heat rate and boiler
efficiency.
3.10
3.05
3.00
2.95
2.90
2.85
2.80
2.75
Unit heat rate
2.70
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.1. Unit heat rate evolution.
Unit 1
Unit 2
Unit 3
Evolution of steam cycle gross heat rate is shown in figure 6.2. This variable
oscillates with yearly period around the value of 2.4 due to the effect of ambient
temperature. In summer, due to higher temperatures condenser pressure increases and
cycle heat rate also does. Besides this stationary oscillation of about 0.1, there is a short
term dispersion of less than 0.05. Some outliers are also present, probably caused by
measurement errors or corresponding to days in which the unit affected has operated in
the full load range only for a short time. An example of these out of range values are the
points of unit 2 in the summer of 1999. Fortunately, due to the big amount of days
available, the presence of these points does not constitute a big problem.
Apart from this stationary oscillation and dispersion, a long time evolution can be
observed. During the first months, the heat rate is quite high, and then efficiency
improves suddenly. Afterwards, due to component degradation, heat rate increases for
two years and tends to remain constant for other two. Finally, around the end of 2003, it
decreases again. Some differences among the three units can also be observed. During
137

Chapter 6
the first three years, all units show similar efficiency. However, in the last three years it
can be seen how unit three works better and unit one works worse.
138
2.60
2.55
2.50
2.45
2.40
2.35
2.30
2.25
Steam cycle heat rate
2.20
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.2. Steam cycle heat rate evolution.
Unit 1
Unit 2
Unit 3
In figure 6.3, the evolution of boiler efficiency for the time span considered is shown
for the three units of the power plant. It can be seen how the graph oscillates around
83% with amplitude of almost 2 points. This oscillation corresponds to the seasons of
summer and winter: efficiency is higher in summer because losses associated with hot
gases leaving the boiler decrease. There is also a short-term dispersion with amplitude
of around 1 point. This dispersion makes difficult to identify whether one unit works
better than the others. It seems that all boilers work worse in the first two years, but,
without a diagnosis procedure it is difficult to assure this fact and, of course, to detect
its cause.
As it has been seen, evolution of the global efficiency indicators is complex and, due
to crossed influence of the variation in ambient conditions, sometimes is difficult even
to determine whether a unit tends to work worse or to remain steadily. The use of cycle
heat rate and boiler efficiency entails a first improvement compared to unit heat rate, but
obviously they are not enough. In conclusion, a diagnosis methodology is needed in
order to decompose these complex patterns into the causes that originate them. This is
the task of the next section.

%
87
86
85
84
83
82
81
80
6.3 Diagnosis results analysis
Boiler efficiency
Figure 6.3. Boiler efficiency evolution.
Application of quantitative causality analysis
79
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
In this section, evolution of the free diagnosis variables and their impacts on the
three global efficiency indicators is analyzed, as well as relations among free variable
increments and the impacts they produce. Due to the multiplicity of graphs available (7
per each one of the 47 free diagnosis variables), not all of them are shown. Instead, the
value of the indicators presented in the previous sections is often given.
First, free diagnosis variables related to ambient conditions are shown. Second,
variables related to fuel quality. Afterwards, importance of set points is studied, first in
steam cycle and second in boiler. Finally, variables related to equipment efficiency are
presented: steam cycle, cooling system and boiler. At the end, indicators corresponding
to all the variables are summarized in a table.
6.3.1 Ambient conditions.
Ambient conditions are temperature, relative humidity and wind speed. Their main
characteristic is that they are not controllable.
139

Chapter 6
The main ambient condition is ambient temperature. It affects both steam cycle
(through cooling system and condenser pressure) and boiler (due to the variation of
losses associated with flue gases). It has another important effect related to the steam
consumption of the coil heaters that preheat air entering the preheaters: in cold weather,
more steam is needed to keep the temperature set point.
Figure 6.4 shows the evolution of this variable for the three units, as well as the
value of ambient temperature at the reference condition. The climatic characteristics of
the plant emplacement makes this variable to oscillate from less than 5 degrees in winter
to more than 30 in summer, which is a strong variation above all taking into account
that we are working with daily averages. The average value of this variable is 15.6 ºC
and its standard deviation is 7.30 ºC.
ºC
140
35
30
25
20
15
10
5
Ambient temperature evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.4. Ambient temperature evolution.
Reference
Unit 1
Unit 2
Unit 3
Variation of ambient temperature induces a strong impact on steam cycle heat rate,
as it can be seen in figure 6.5. Importance of this impact is summarized in a standard
deviation of 0.0528, which is a quite high value, as it will be seen later when standard
deviations of impacts caused by other variables are analysed. At first sight, it may seem
strange that this impact graph, like many others, is not centred. The vertical position of
an impact graph depends on the average value of the free diagnosis variable and its

Application of quantitative causality analysis
value for the reference state. If they are the same, the graph is centred, if not, it is
displaced upwards or downwards.
0.10
0.05
-0.05
-0.10
-0.15
-0.20
Cycle heat rate impact due to ambient temperature
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.5. Cycle heat rate impact due to ambient temperature.
Unit 1
Unit 2
Unit 3
The influence on boiler efficiency is also important as it can be seen in figure 6.6.
Standard deviation of this impact is 0.402 (%).
Finally, the impact on unit heat rate also shows the same cyclic behaviour but with
less amplitude than the cycle (figure 6.7), because effects on boiler partially compensate
effects of condenser pressure. Besides, the use of steam for pre-heating the air entering
the preheaters implies an additional increment in unit heat rate at low temperatures.
Standard deviation of this impact is 0.0448, which is lower than that of the cycle heat
rate.
141

Chapter 6
%
142
0.10
0.05
-0.10
-0.15
1.00
0.50
-0.50
-1.00
-1.50
Boiler efficiency impact due to ambient temperature
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.6. Impact of ambient temperature on boiler efficiency.
Unit heat rate impact due to ambient temperature
0.00
24/07/1998 0:00
-0.05
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00 Unit 1
Unit 2
Unit 3
Figure 6.7. Unit heat rate impact due to ambient temperature.
Unit 1
Unit 2
Unit 3

Impact
Ambient temperature and its impact on cycle heat rate
Application of quantitative causality analysis
0
-25 -20 -15 -10 -5 0 5 10 15
Figure 6.8. Ambient temperature variation and its impact on cycle heat rate.
Figure 6.8 shows the relation betwen the variation of ambient temperature and the
impact produced in cycle heat rate. The slope of the graph (ed av ) is 0.00723 ºC -1 , and the
regression coefficient (R 2 ) is 0.9998. This very high value of R 2 corresponds to a graph
where the points are aligned, such as the one seen here. In the graph of impact on boiler
efficiency, the slope is 0.0552 %/ºC and R 2 = 0.9957, which means a bit more
dispersion. Finally, the impact on unit heat rate has a slope of 0.00614 ºC -1 and also a
very high value of R 2 (0.999). These high values of R 2 indicate that constant values of
the impact factor might be used with low error, which is not true in general, as it will be
seen later.
Another weather-related free diagnosis variable is relative humidity. This variable
has an average of 63.4 % and a standard deviation of 7.08 (%). It has also a stationary
oscillation (more humidity in summer), but with more short term dispersion than
temperature. This behaviour can be seen in the evolution of the impact on cycle heat
rate, plotted in figure 6.9. It should be noted that dispersion is lower in the first two
years because wet bulb temperature measurement was not reliable and a validation in
relation to ambient temperature was used. The standard deviation of this impact is
0.00784, which is seven times lower than that of the ambient temperature. Impact of this
0.1
0.05
-0.05
-0.1
-0.15
-0.2
Ambient temperature (ºC)
Unit 1
Unit 2
Unit 3
143

Chapter 6
variable on boiler efficiency is reduced, and has a standard deviation of 0.0108 %. For
this reason, impact on unit heat rate is almost parallel to that on cycle heat rate; it has a
standard deviation of 0.00931.
144
0.04
0.03
0.02
0.01
-0.01
-0.02
-0.03
Impact on cycle heat rate due to relative humidity
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Impact
Figure 6.9. Cycle heat rate impact due to relative humidity.
Relative humidity and its impact on cycle heat rate
0.03
0.02
0.01
0
-20 -15 -10 -5 0 5 10 15 20 25 30
-0.01
-0.02
-0.03
Relative humidity (%)
Figure 6.10. Relative humidity variation and its impact on cycle heat rate.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
Figure 6.10 shows the relation between relative humidity variation and its impact on
cycle heat rate, where it can be seen how the graph curves slightly. It should be noted
that this non linear shape could not be obtained if all the derivatives were calculated
only in the reference state. Slope of the curve is 0.00107 (1/%) and R 2 = 0.992.
The graph relating relative humidity to its impact on unit heat rate is quite similar.
Slope is 0.00127 (1/%) and R 2 = 0.992. Relation to impact on boiler efficiency is more
disperse, with a slope of -0.00113 (%/%) and R 2 = 0.900 (Figure 6.11). Physically, this
dispersion means that the variation in humidity has different impact depending on other
factors, mainly the ambient temperature. As it will be seen later, this dispersion appears
in other variables that have low influence.
Impact (%)
Relative humidity and its impact on boiler efficiency
0.06
0.04
0.02
0
-20 -15 -10 -5 0 5 10 15 20 25 30
-0.02
-0.04
-0.06
Relative humidity variation (%)
Figure 6.11. Relative humidity variation and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
The last ambient condition considered is wind speed, which affects the losses to the
ambient in the boiler. The average value of this variable is 4.55 m/s and its standard
deviation is 2.82 m/s. In figure 6.12, the evolution of this variable is shown.
145

Chapter 6
Wind speed (m/s)
146
25
20
15
10
5
-5
Wind speed evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.12. Wind speed evolution.
Reference
Unit 1
Unit 2
Unit 3
Impact of wind speed on boiler efficiency has a standard deviation of 0.115%, and
the graph relating variable and impact has a slope of -0.0406 (%·s/m), with R 2 = 0.985.
For the impact on unit heat rate, standard deviation is 0.00412, and the graph has a slope
of 0.00145 s/m with R 2 = 0.984. Influence on cycle heat rate is practically zero, with an
impact standard deviation of 8.04·10 -5 .
6.3.2 Fuel quality
Free diagnosis variables related to fuel quality are coal high heating value (HHV),
coal composition (carbon, hydrogen, nitrogen, sulphur, moisture and ashes), and amount
of natural gas introduced. Oxygen in coal has not been considered because one of all the
coal components is not free in order to accomplish that the summation of all the mass
fractions should be 100%. The amount of natural gas is expressed in terms of
percentage of power introduced by the fuel. Gas properties variation has not been
considered.
Figure 6.13 shows the evolution of coal HHV. This variable has an average of
16975 kJ/kg and a standard deviation of 639 kJ/kg, which comes from both long term
variability and short-term dispersion. It should be noted that fuel properties are
determined by daily analysis performed in the plant laboratory. Although these analyses

Application of quantitative causality analysis
are performed according to standard procedures, the sampling method induces a non
negligible error which causes this dispersion.
Coal HHV (kJ/kg)
%
19000
18000
17000
16000
15000
Coal HHV evolution
14000
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
1.00
0.50
-0.50
-1.00
-1.50
-2.00
-2.50
-3.00
-3.50
Figure 6.13. Coal high heating value evolution.
Impact on boiler efficiency due to coal HHV
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.14. Impact on boiler efficiency due to coal HHV.
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
147

Chapter 6
Impact (%)
148
Coal HHV and its impact on boiler efficiency
0
-3500 -3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500
-0.5
HHV variation (kJ/kg)
Figure 6.15. Coal HHV variation and its impact on boiler efficiency.
Variations in coal HHV induce strong variations in boiler efficiency, which can be
seen in figure 6.14. Standard deviation of this impact is 0.574 %, the curve relating
variable and impact has a slope of 0.000889 (%·kg/kJ) and it is very linear (R 2 = 0.999),
as it can be seen in figure 6.15. This influence on boiler efficiency is obviously traduced
in impacts on unit heat rate. Standard deviation of these impacts is 0.0237, and relation
between variable and impact is -3.64·10 -5 kg/kJ (R 2 = 0.998). What is more surprising is
that this magnitude affects also the cycle: standard deviation of cycle heat rate impact is
0.00262 and the slope of the graph which relates variable variation and impact is -
4.02·10 -6 kg/kJ (R 2 = 0.996). This influence of fuel-related variables on cycle heat rate
can be explained by considering that fuel properties affect strongly the amount of air
and gases flowing through the boiler, which, in turn, affects both tempering flows and
the amount of steam consumed at coil heaters. So that, although steam cycle works in
the same way, flows entering and exiting this part of the plant are affected, and heat rate
varies. This example demonstrated that an analysis based on the decomposition into
blocks is not suitable if a precision of 5% or lower is required.
1
0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
Unit 1
Unit 2
Unit 3

%
1.20
1.00
0.80
0.60
0.40
0.20
-0.20
-0.40
Impact on boiler efficiency due to carbon in coal
Application of quantitative causality analysis
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.16. Impact on boiler efficiency due to carbon in coal.
Unit 1
Unit 2
Unit 3
One of the coal components that more affects plant behaviour is carbon. Its average
value is 42.5 % and its standard deviation is 1.61 (%). In figure 6.16, impact of this
variable on boiler efficiency is shown, where it can be observed both long term and
short time variation. Standard deviation of this impact is 0.190 (%), and relation
between variable and impact is -0.116 %/%, with R 2 = 0.997 (Figure 6.17). This
negative value indicates that when carbon increases boiler efficiency decreases. It
occurs because fuel HHV has been considered as an independent variable, which
implies that carbon varies without modifying HHV. In this situation, an increment in
carbon originates an increment of air to burn it and, subsequently, more losses. A
similar situation will appear in other carbon components like hydrogen. This result
appears because HHV is not actually an independent variable, because it is linked to
fuel composition. It is further analyzed in Section 6.5.
This impact on boiler efficiency originates an impact on unit heat rate. The standard
deviation of this impact is 0.00978 and the slope of the graph impact-variable is
0.00596 (1/%) (R 2 = 0.997). The standard deviation of the impact on cycle heat rate is
149

Chapter 6
0.00224 which comes from of a relation between variable and impact of 0.00137 (1/%),
(R 2 = 0.997).
Impact (%)
%
150
Carbon in coal and its impact on boiler efficiency
0
-10 -8 -6 -4 -2 0 2 4
1.5
1.0
0.5
-0.5
-1.0
-1.5
Carbon in coal variation (%)
Figure 6.17. Carbon in coal variation and its impact on boiler efficiency.
Figure 6.18. Impact on boiler efficiency due to hydrogen in coal.
1.2
1
0.8
0.6
0.4
0.2
-0.2
-0.4
Impact on boiler efficiency due to hydrogen in coal
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
Another important fuel component is hydrogen. This variable has an average value
of 2.74 % and a standard deviation of 0.173 (%). Impact on boiler efficiency has a
standard deviation of 0.310 and its evolution is plotted on figure 6.18, where it can be
appreciated a long term evolution parallel to those seen in HHV and carbon.
Impact (%)
Hydrogen in coal and its impact on boiler efficiency
1.5
1
0.5
0
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-0.5
-1
-1.5
-2
Hydrogen in coal variation (%)
Figure 6.19. Hydrogen in coal and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
Figure 6.19 shows the relation of hydrogen in coal variation and its impact on boiler
efficiency. Its slope is -1.80 (%/%) and its R 2 is 0.999. This variable shows the same
effect as carbon: it has a negative slope because HHV has been defined as an
independent variable. Hydrogen in fuel has also an important impact on unit heat rate,
with a standard deviation of 0.01189 and a sensitivity coefficient of 0.0688 (1/%), (R 2 =
0.999). Finally, the impact on cycle heat rate has a standard deviation of 0.000864 due
to a sensitivity coefficient of 0.00499 (1/%), (R 2 = 0.997).
151

Chapter 6
Coal moisture has an average value of 18.9 % and a standard deviation of 1.62 %.
Its impact on boiler efficiency has a standard deviation of 0.325 % and its evolution is
plotted on figure 6.20. Although there is important point dispersion, it can be
appreciated how moisture has been higher during the first years. It can be seen also
peaks corresponding to summer months, when, because of the high ambient
temperature, coal becomes dryer.
%
152
1.0
0.5
-0.5
-1.0
-1.5
Impact on boiler efficiency due to coal humidity
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.20. Impact on boiler efficiency due to coal moisture.
Unit 1
Unit 2
Unit 3
Figure 6.21 shows the relation between the variation of moisture and its impact on
boiler efficiency. Its slope is -0.201 (%/%), and R 2 is 0.999. Impact on unit heat rate has
a standard deviation of 0.0126 and an impact factor of 0.00776 (1/%) with a very linear
behaviour (R 2 = 0.999). Finally, moisture has a small impact on cycle heat rate, with a
standard deviation of 0.000918 and a relation between impact and variable variation of
0.000564 (1/%), with R 2 = 0.996.

Impact (%)
%
Coal humidity and its impact on boiler efficiency
1
0.5
-0.5
-1
-1.5
Application of quantitative causality analysis
0
-6 -4 -2 0 2 4 6 8
0.6
0.4
0.2
-0.2
-0.4
-0.6
Coal humidity variation (%)
Figure 6.21. Coal moisture variation and its impact on boiler efficiency.
Impact on boiler efficiency due to ash in coal
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
d
Figure 6.22. Impact on boiler efficiency due to ash in coal.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
153

Chapter 6
Figure 6.22 shows the impact on boiler efficiency due to ashes variation, which has
a standard deviation of 0.134. It can be seen how the ash content of coal was lower
during the first two years, which implied higher efficiency. The average ash content in
coal is 23.3 % and the standard deviation is 1.84 %. The relation of ash variation and its
impact on boiler efficiency is -0.0726 (%/%) with R 2 = 0.996 (Figure 6.23). It may seem
that this impact due to ashes is quite high, considering that this component is an inert
that only affects losses related to heat of ashes and slag. However, it affects also to
losses due to carbon in ashes (the independent variable is a percentage) and mainly it
makes to vary the oxygen component in coal (which is not an independent variable)
and, subsequently, the amount of air needed for combustion. This effect of variation of
oxygen in coal of course appears in all the composition variables, but it is stronger in
those which vary more, such as ashes and moisture.
Impact (%)
154
Ash in coal and its impact on boiler efficiency
0
-8 -6 -4 -2 0 2 4 6
0.6
0.4
0.2
-0.2
-0.4
-0.6
Ash in coal variation (%)
Figure 6.23. Ash in coal variation and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
Impact on unit heat rate has a standard deviation of 0.00566 and a relation between
impact and variable increment of 0.00307 (1/%) with R 2 = 0.996. In the case of the
impact on cycle heat rate, standard deviation is 0.000723, with an impact factor of
0.000392 (1/%) and R 2 = 0.996.

Application of quantitative causality analysis
Another significant coal property is sulphur content, which has also a strong
importance from the environmental point of view. However, the operation of the plant
for removing SO2 from flue gases is not considered in this work, apart from the amount
of preheated gases extracted. Average value of sulphur in coal is 4.84 % and standard
deviation is 0.4118 %. The impact of sulphur on boiler efficiency is shown in figure
6.24, and its standard deviation is 0.0217 %. During the first year, coal had low sulphur
content, so that boiler efficiency was higher. However, this variable has an important
dispersion.
%
0.06
0.04
0.02
-0.02
-0.04
-0.06
-0.08
-0.10
Boiler efficiency impact due to sulphur in coal
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.24. Impact on boiler efficiency due to sulphur in coal.
Unit 1
Unit 2
Unit 3
Relation of sulphur variation and its impact on boiler efficiency is -0.0519 (%/%)
with R 2 = 0.996 (Figure 6.25). Impact of sulphur on unit heat rate has a standard
deviation of 0.00113, and an impact factor of 0.00272 (1/%) with R 2 = 0.996. Finally,
sulphur has a negligible impact on cycle heat rate, with a standard deviation of
0.000279 and a relation between impact and variable of 0.000672 (1/%), R 2 = 0.997.
The last coal component is nitrogen, and has a negligible effect. For example, the
standard deviation of its impact on boiler efficiency is 0.00211, which is ten times lower
than that of sulphur.
155

Chapter 6
Impact (%)
%
156
Sulphur in coal and its impact on boiler efficiency
0
-1 -0.5 0 0.5 1 1.5 2
10
8
6
4
2
-2
0.06
0.04
0.02
-0.02
-0.04
-0.06
-0.08
-0.1
Sulphur in coal variation (%)
Figure 6.25. Sulphur in coal variation and its impact on boiler efficiency.
Natural gas fraction evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.26. Natural gas fraction evolution.
Unit 1
Unit 2
Unit 3
Reference
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
The last free diagnosis variable related to fuel is the fraction of energy due to natural
gas. This variable has an average value of 0.576 % and a standard deviation of 1.10 %.
This high variability appears because this variable has a discontinuous evolution
varying from 0 to almost 10% (Figure 6.26). The impact on boiler efficiency due to gas
contribution to energy in fuel is reduced with a standard deviation of 0.00683 % and an
averaged impact factor of -0.00542 (%/%). However, the main feature of this
dependence is its variability with R 2 = 0.766, which can be seen in figure 6.27. This
means that the introduction of gas natural can have different effect depending on the
quality of the coal. In most cases, the use of natural gas makes the boiler efficiency to
decrease, mainly due to the increment of losses related to hydrogen in fuel. It should be
noted that effects such as influence of natural gas in boiler heat transfer are not
considered, because these aspects are included in other free diagnosis variables.
Impact (%)
0.04
0.02
-0.02
-0.04
-0.06
-0.08
-0.1
Natural gas fraction and its impact on boiler efficiency
0
-2 0 2 4 6 8 10
Natural gas fraction variation (%)
Figure 6.27. Natural gas fraction variation and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
Impact of natural gas fraction on unit heat rate has a standard deviation of 0.000163
and a relation between impact and variable of 0.0000979 (1/%), with high dispersion
(R 2 = 0.437). Impact on cycle heat rate has a standard deviation of 0.0000852 and an
impact factor of -0.0000767 (1/%). Surprisingly, it has low dispersion (R 2 = 0.985),
which can be explained by considering that the use of natural gas always reduces the
157

Chapter 6
amount of air needed for combustion and, subsequently, the steam needed by coil
preheaters.
6.3.3 Steam cycle set points.
These set points are four: temperature and pressure of live steam entering the high
pressure turbine, temperature of reheated steam entering the first medium pressure
turbine and gross power produced. Possibility of regenerative pre-heaters by-pass could
also be considered as operator-controlled degrees of freedom, but these situations are
considered in the temperature difference of preheaters (indicators of equipment
efficiency).
Figure 6.28 shows the evolution of live steam temperature. It can be seen how this
set point is fixed around 539 ºC, although it is sometimes modified for example during
the first two years in unit 1. Besides, several points appear below these fixed points, due
to the control based on tempering. Average temperature is 537.9 ºC and standard
deviation is 1.28 ºC.
ºC
158
544
542
540
538
536
534
532
530
Live steam temperature evolution
528
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.28. Live steam temperature evolution.
Reference
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
Variation of live steam temperature has an impact on cycle heat rate. Standard
deviation of this impact is 0.00101 and the relation between variation and impact is -
0.00787 1/ºC with very linear behaviour (R 2 = 0.999), as it can be seen in figure 6.29.
This impact on cycle heat rate entails an impact on unit heat rate, with a standard
deviation of 0.01090 and an impact factor of -0.000850 1/ºC (R 2 = 0.999). Impact on
boiler efficiency is negligible, with a standard deviation of 0.000357%.
Impact
Live steam temperature and its impact on cycle heat rate
Figure 6.29. Live steam temperature variation and its impact on cycle heat rate.
Reheated steam temperature has an average value of 538.2 ºC and a standard
deviation of 2.10 ºC, and its evolution is plotted in figure 6.30. It can be seen how this
set point is almost constant in unit 3 but has been set in different levels in unit 1. This
set-point variation entails an impact on steam cycle heat rate which has a standard
deviation of 0.00129, and an impact factor of -0.000623 (1/ºC) with R 2 = 0.999, as it can
be seen in figure 6.31. This steam cycle heat rate variation implies a variation of unit
heat rate of a standard deviation of 0.00132 and a relation between impact and variable
of -0.000625 (1/ºC) with R 2 = 0.999. Impact on boiler efficiency is again negligible,
with a standard deviation of 0.000475 %.
0.008
0.006
0.004
0.002
0
-10 -8 -6 -4 -2 0 2 4 6
-0.002
-0.004
Live steam temperature variation (ºC)
Unit 1
Unit 2
Unit 3
159

Chapter 6
160
ºC
Impact
550
545
540
535
530
525
Reheated steam temperature evolution
520
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.30. Reheated steam temperature evolution.
Reheated steam temperature and its impact on cycle heat rate
0.015
0
-20 -15 -10 -5 0 5 10 15
0.01
0.005
-0.005
-0.01
Reheated steam temperature variation (ºC)
Reference
Unit 1
Unit 2
Unit 3
Figure 6.31. Reheated steam temperature variation and its impact on cycle heat rate.
Unit 1
Unit 2
Unit 3
Another set-point related to cycle operation is live steam pressure, which has an
average value of 157.5 bar and a standard deviation of 1.35 bar. Its evolution is shown
in figure 3.32, where it can be appreciated how this parameter is usually high in unit 2
and low in unit 3.

ar
Impact
164
162
160
158
156
154
152
Live steam pressure evolution
Figure 6.32. Live steam pressure evolution.
Application of quantitative causality analysis
150
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Live steam pressure and its impact on cycle heat rate
0.008
0.006
0.004
0.002
0
-6 -4 -2 0 2 4 6
-0.002
-0.004
-0.006
-0.008
Live steam pressure variation (bar)
Figure 6.33. Live steam pressure and its impact on cycle heat rate.
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
Relation between steam pressure and its impact on cycle heat rate is -0.00123 1/bar
and is very linear (R 2 = 0.998) as it can be seen in figure 6.33. Standard deviation of this
161

Chapter 6
impact is 0.00166. Impact on unit heat rate has a standard deviation of 0.00183 and an
impact factor of -0.00136 1/bar (R 2 = 0.999). Live steam pressure has a negligible
impact on boiler efficiency, with a standard deviation of 0.000502 %.
The last free diagnosis variable considered in this paragraph is gross power
produced by the alternator, although it obviously affects not only the cycle but also the
boiler. For the operation points studied (full load, or above 320 MW), this variable has
an average value of 340.007 MW and a standard deviation of 7.00 MW (Figure 6.34).
This high deviation appears because the units are not operated in fixed values but with
load regulation.
MW
162
380
370
360
350
340
330
320
Gross electric power evolution
310
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.34. Gross electric power evolution.
Reference
Unit 1
Unit 2
Unit 3
Relation of electric power variation and its impact on cycle heat rate is shown in
figure 6.35. Its slope has a value of 0.000542 1/MW (R 2 = 0.992). It may be surprising
this positive value, which indicates that when load increases heat rate also does.
However, it should be noted that condenser pressure is not a free diagnosis variable, and
when load increases this pressure should also increase because the same cooling system
has more heat to reject. Impact on boiler efficiency has a standard deviation of 0.0184
(%) and a relation between variable and impact of 0.00197 (%/MW). The main
characteristic of this relation is its high dispersion, which can be appreciated in Figure

Application of quantitative causality analysis
6.36 (R 2 = 0.666). This dispersion also appears in the case of unit heat rate (R 2 = 0.613),
which has an impact factor of 0.0000724 1/MW and a standard deviation of 0.000733.
This low influence is due to the compensation of the influence negative in cycle and
positive in boiler that a load increment has.
Impact
Impact (%)
Gross electric power and its impact on cycle heat rate
0.025
0.02
0.015
0.01
0.005
0
-20 -10 0 10 20 30 40
-0.005
-0.01
-0.015
Figure 6.35. Gross electric power variation and its impact on cycle heat rate.
Gross electric power and its impact on boiler efficiency
0.15
0.1
0.05
0
-20 -10 0 10 20 30 40
-0.05
-0.1
Gross electric power variation (MW)
Gross electric power variation (MW)
Figure 6.36. Gross electric power variation and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
163

Chapter 6
6.3.4 Boiler set points.
In this section, operation set-points associated with the boiler are studied. The main
free diagnosis variables belonging to this group are: oxygen content in flue gases
leaving the boiler, cold-side average temperature of air preheaters and steam for
blowing. There are other variables related to the distribution of air and flue gases in the
pre-heaters area: (i) relation of tempering and primary air, (ii) relation of primary air
and coal mass flows, (iii) amount of air extracted for desulfuration, (iv) temperature
difference of flue gases entering the preheaters and (v) flue gases distribution among
primary and secondary air preheaters.
Figure 6.37 shows the evolution of the set-point of the oxygen in combustion gases
at the exit of the boiler (before air preheaters). It can be seen how this variable has been
modified throughout time following diverse operation strategies. First, it was quite high,
then it decreased and finally it reached a stable value for the last three years. Differences
among units can also be observed; for example, oxygen in unit 2 is usually higher than
in unit 3. Average value of this variable is 2.05 % and it has a standard deviation of
0.259 %.
%
164
3.5
3
2.5
2
1.5
Oxygen set-point evolution
1
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.37. Oxygen set-point evolution.
Reference
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
Each increment of 1 point in oxygen in gases implies a reduction of 0.265 points in
boiler efficiency. This relation is very linear (R 2 = 0.997) as it can be seen in figure
6.38. Standard deviation of the impacts on boiler efficiency is 0.0683%.
Impact (%)
Oxygen set-point and its impact on boiler efficiency
0.2
0.15
0.1
0.05
0
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
Oxygen set-point variation (%)
Figure 6.38. Oxygen set point variation and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
This impact on boiler efficiency originates an impact on unit heat rate, which has a
standard deviation of 0.00339. Relation between this impact and oxygen variation is
0.0131 (1/%) with R 2 = 0.997. Impact on cycle efficiency is reduced, with a standard
deviation of 0.000747 and an impact factor of 0.00289 (1/%), R 2 = 0.995.
The aim of average cold-side temperature on primary and secondary air preheaters is
to avoid low temperatures in these heat exchangers, than can lead to acid condensation
and corrosion. Each one of these parameters is maintained by preheating the air entering
each preheater, by using a coil heater that consumes steam from the drum. Due to this
fact, this set point should be high enough to avoid corrosion but not too high in order to
reduce the steam consumption. It should be noted that these set points are minimum
values, so that, if outside temperature is high enough it may happen that the amount of
steam needed reaches the value of zero, and the average cold-side temperature increases
above the set point. In these cases, to be strict, impact corresponding to this last part
should be associated to ambient temperature.
165

Chapter 6
Average cold-side temperature on secondary air preheaters has a mean value of
109.3 ºC and a standard deviation of 6.67 ºC. Figure 6.39 shows how this variable has
been modified along time. During some months of the first years, it has reached low
values because part of the heat exchangers had been replaced by using corrosion
resistant alloy. However, then it was decided to maintain higher values. Besides, some
peaks in summer can be seen, corresponding to situations in which steam extracted for
heating the air has become zero.
ºC
166
140
130
120
110
100
90
80
70
Average cold-side temperature in secondary air preheaters
60
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Reference
Unit 1
Unit 2
Unit 3
Figure 6.39. Evolution of average cold-side temperature in secondary air preheaters.
Due to the consumption of steam, this variable affects directly to unit heat rate, with
an impact standard deviation of 0.00683. As it can be seen in figure 6.40, each
additional degree implies an increment of 0.00102 in unit heat rate (R 2 = 0.999). Impact
on boiler efficiency has a standard deviation of 0.174%, with an impact factor of -
0.0262 %/ºC and R 2 = 0.999. This important impact appears because when the average
cold-side temperature increases, air entering the preheater is hotter and this heat
exchanger works worse. Impact on cycle heat rate is negligible, with a standard
deviation of 0.000276.

Impact
ºC
Average cold-side temperatue in secondary air preheater
and its impact on unit heat rate
-0.01
-0.02
-0.03
-0.04
Application of quantitative causality analysis
0
-40 -30 -20 -10 0 10 20 30
0.03
0.02
0.01
Average temperature variation (ºC)
Figure 6.40. Average cold-side temperature in secondary air preheaters variation and its
impact on unit heat rate.
150
140
130
120
110
100
90
Average cold-side temperature in primary air preheaters
80
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.41. Average cold-side temperature in primary air preheaters evolution.
Unit 1
Unit 2
Unit 3
Reference
Unit 1
Unit 2
Unit 3
167

Chapter 6
Figure 6.41 shows the evolution of the average cold-side temperature in primary air
preheaters. It can be seen how this variable was very high during the first months and
then decreased. It decreased again at the end of 2000. The effect commented for
secondary air preheaters and consisting of quite high values of this average temperature
appearing when high ambient temperature or operation conditions make the steam
required to be equal to zero, appears now in more points. Mean value of this variable is
111.6 ºC and its standard deviation is 7.25 ºC.
Impact of this variable on unit heat rate has a standard deviation of 0.00208, and the
impact factor is 0.000281 1/ºC (R 2 = 0.990). As it can be seen in figure 6.42, relation of
variable increment and its impact is not homogeneous but it present two tendencies: a
main one and another with fewer points. This secondary tendency corresponds to the
first months when the old preheaters were working. As it will be seen later, these heat
exchangers presented important air infiltration; this fact, in turn, originated an additional
amount of air to be heated and subsequently more steam consumption and more impact.
Impact
168
Average cold-side temperature in primary air preheaters
and its impact on unit heat rate
0.01
0.008
0.006
0.004
0.002
0
-30 -20 -10 0 10 20 30 40
-0.002
-0.004
-0.006
-0.008
Average temperature variation (ºC)
Figure 6.42. Average cold-side temperature in primary air preheaters variation and its
impact on unit heat rate.
Unit 1
Unit 2
Unit 3
Impact on boiler efficiency has a standard deviation of 0.0534 % and an impact
factor of -0.00724 %/ºC (R 2 = 0.994). In the graph relating variable and this impact, a

Application of quantitative causality analysis
secondary tendency similar to that on figure 6.42 also appears. It should be noted that
impact factors corresponding to average temperature at primary air preheaters are
around three times and a half smaller than those of secondary preheaters, which
approximately correspond to the relation of air mass flows. Finally, impact on cycle
heat rate is negligible, with a standard deviation of 0.0000877.
Blowing steam has an average value of 1.40 kg/s and a standard deviation of 0.564
kg/s. Its evolution is plotted in figure 4.63, where it can be seen a tendency to increase
throughout time and a quite high dispersion. It should be noted that figures considered
are values averaged along every day from a continuous variable that usually is equal to
zero and in some moments reaches values higher than those shown here.
kg/s
6
5
4
3
2
1
-1
Blowing steam evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.43. Sootblowing steam evolution.
Reference
Unit 1
Unit 2
Unit 3
Steam consumption for sootblowing originates an important impact on unit heat rate,
whose standard deviation is 0.00455. Relation of steam flow variation and its impact is
0.00804 s/kg with R 2 = 0.999 (Figure 6.44). Impacts on cycle heat rate and boiler
efficiency are lower, with standard deviations of 0.000941 and 0.00123%, respectively.
It should be noted that, in the case of boiler efficiency, effects on flow pattern and heat
transfer while blowing are not considered, which provides an impact lower than the real
169

Chapter 6
one. However, the main effect of blowing, which is steam consumption, has been
considered.
Impact
170
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
-0.01
-0.015
Blowing steam and its impact on unit heat rate
0
-2 -1 0 1 2 3 4 5
-0.005
Blowing steam variation (kg/s)
Figure 6.44. Sootblowing steam variation and its impact on unit heat rate.
Unit 1
Unit 2
Unit 3
The main set point related to air and flue gases distribution in the preheaters area is
the amount of hot air that is extracted from the boiler to heat the air leaving the flue gas
desulfuration unit before it enters the stack. This variable has an average value of 4.86
kg/s and a standard deviation of 8.79 kg/s. This high dispersion appears because this
variable has usually low values that sometimes increase dramatically, as it can be seen
in Figure 6.45.
Extraction of this hot air originates an important impact on boiler efficiency, with a
standard deviation of 0.140 % and an impact factor of -0.0160 %·s/kg with R 2 = 0.998
(Figure 6.46). This boiler efficiency variation originates an impact on unit heat rate with
a standard deviation 0.00540 and a relation between impact and variable of 0.000613
s/kg (R 2 = 0.996). Standard deviation of impact on cycle heat rate is negligible
(0.000214).

kg/s
Impact (%)
70
60
50
40
30
20
10
-10
Air for SO 2 removing unit evolution
Application of quantitative causality analysis
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
0.2
-0.2
-0.4
-0.6
-0.8
-1
-1.2
Figure 6.45. Air for flue gas desulfuration unit evolution.
Air for SO 2 removing unit and its impact on boiler efficiency
0
-10 0 10 20 30 40 50 60 70
Air mass flow variation (kg/s)
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
Figure 6.46. Air for flue gas desulfuration unit variation and its impact on boiler efficiency.
171

Chapter 6
Figure 6.47 shows the evolution of the relation of tempering and primary air (%).
Actually, the most suitable diagnosis variable should have been the set point of
temperature of primary air leaving the mills; however, this information was not
available, so that, this mass flow relation was chosen. In the graph it can be seen two
different zones. The second one presents more dispersion and corresponds to the actual
measured flows, while in the first one (the older periods) information about flows was
not available and a correlation with ambient temperature has been used. Evolution of
this variable shows a cyclic tendency: in summer, the amount of tempering air should be
increased in order to keep the temperature set point. Average value of this variable is
33.8% and its standard deviation is 7.06 %.
%
172
60
50
40
30
20
10
Tempering and primary air relation
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.47. Relation of tempering and primary air evolution.
Reference
Unit 1
Unit 2
Unit 3
This variable originates an impact on boiler efficiency with a standard deviation of
0.103 %. Relation between variable increment and its impact is -0.0148 (%/%) with R 2
= 0.991 (Figure 6.48). The negative sign appears because when more primary air is bypassed,
primary air preheaters work worse. Impact on unit heat rate has a standard
deviation of 0.00345 and an impact factor of 0.000497 (1/%), with R 2 = 0.991. Effect on
cycle heat rate is negligible.

Impact (%)
Impact (%)
Application of quantitative causality analysis
Relation of tempering and primary air and its impact on boiler efficiency
0.4
0.3
0.2
0.1
0
-25 -20 -15 -10 -5 0 5 10 15 20 25
-0.1
-0.2
-0.3
Flows relation variation (%)
Figure 6.48. Variation of the relation of tempering and primary air and its impact on boiler
efficiency.
Primary air-coal ratio and its impact on boiler efficiency
0.1
0.05
0
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
-0.4
Primary air-coal ratio variation (kg/kg)
Figure 6.49. Primary air-coal ratio and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
The variable that controls air distribution among primary and secondary preheaters
is the primary air to coal ratio (kg/kg). This variable has an average value of 1.97 kg/kg
173

Chapter 6
and a standard deviation of 0.143 kg/kg. Relation between impact on boiler efficiency
and variable increment is on average -0.347 % but it has a little dispersion, with R 2 =
0.970 (Figure 6.49). When the amount of primary air increases, also the amount of
tempering air does (relation of tempering and primary air is a free diagnosis variable),
so that efficiency decreases; however, dispersion appears due to the crossed influence of
preheaters efficiency variation. Impact on boiler efficiency has a standard deviation of
0.0530 % and is plotted in figure 6.50. During the first three years, point dispersion is
lower because information on flow measurement was not available, and relation of
primary and secondary air was fixed. In the last three years, it can be seen how this
impact affects more negatively to unit 1 than unit 3.
Impact on unit heat rate has a standard deviation of 0.00176 and an impact factor of
0.0114 kg/kg (R 2 = 0.964), while effect on cycle heat rate is negligible.
%
174
0.10
0.05
-0.10
-0.15
-0.20
-0.25
-0.30
-0.35
-0.40
Impact on boiler efficiency due to primary air-coal ratio
0.00
24/07/1998 0:00
-0.05
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.50 Impact on boiler efficiency due to primary air-coal ratio.
Unit 1
Unit 2
Unit 3
Flue gases distribution among primary and secondary air preheaters is controlled by
the percentage of flue gases entering primary air preheaters. This variable has an
average value of 24.5 % and a standard deviation of 3.22 %. It has a very low influence
on both boiler and unit, with impact standard deviations of 0.0157 % and 0.000646.
Average values of impact factors are -0.00102 %/% and 0.0000523 1/%, but these

Application of quantitative causality analysis
relations are characterized with high dispersion (R 2 = 0.080 and 0.123). Effect on cycle
heat rate is negligible.
The last variable in this section is temperature difference between flue gases
entering primary air preheaters and flue gases entering secondary air preheaters. This
difference is due to the effect of the by-pass of the first economizer. Average value of
this variable is 7.50 ºC and it has a standard deviation of 14.8 ºC. Impact of this variable
is negligible, with a standard deviation of 0.00855 % in the boiler and 0.000526 in the
unit. However, its evolution is shown in figure 6.51 because it shows a difference in
operation strategy that induces a difference in aggregated boiler effectiveness, as it will
be seen later. In this graph, it can be seen how during the first two years temperature
difference was higher in winter, probably due to the opening of the by-pass in order to
reduce the amount of steam consumed by the coil heaters.
ºC
80
70
60
50
40
30
20
10
-20
-30
Temperature difference of flue gases entering preheaters
0
24/07/1998 0:00
-10
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Reference
Unit 1
Unit 2
Unit 3
Figure 6.51. Evolution of temperature difference of flue gases entering air preheaters.
6.3.5 Steam cycle component parameters.
Free diagnosis variables corresponding to this group are turbine isoentropic
efficiencies, turbine flow coefficients (except in the high pressure one) and temperature
differences in regenerative water heaters. Pressure drop in reheater and pressure
increment provided by turbo-pump are also included because, although its physical
175

Chapter 6
causes are mainly located in the boiler, they affect above all the steam cycle. Finally,
water losses are also considered.
Figure 6.52 shows the evolution of high pressure steam turbine isoentropic
efficiency. It can be seen how this variable presents an oscillating tendency: it has
higher efficiency in summer than in winter. This effect appears because of the variations
of pressure ratio and the effect of the regulation system: since the load is almost
constant, in winter, with lower condenser pressure, governing valves should be more
closed. Units 1 and 3 present quite constant values, but the former has higher efficiency
than the latter. Efficiency of unit 2 tends to decrease but it is recovered again after being
repaired in april 2004. Considered all units together, this variable has an average value
of 79.5% with a standard deviation of 1.81 %.
%
176
85
83
81
79
77
75
High pressure steam turbine isoentropic efficiency evolution
73
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.52. Evolution of high pressure steam turbine isoentropic efficiency.
Reference
Unit 1
Unit 2
Unit 3
High pressure steam turbine variation implies an important impact on cycle heat
rate. Standard deviation of this impact is 0.00905 and the relation between variable
increment and impact is -0.00499 1/%, with a very linear behaviour (R 2 = 0.999), as it
can be seen in figure 6.53. This impact on cycle heat rate causes an effect on unit heat
rate, with an impact standard deviation of 0.00993 and an impact factor of -0.00548 1/%
(R 2 = 0.999). Impact on boiler efficiency is very reduced; it has a standard deviation of
0.00357 %.

Impact
%
Application of quantitative causality analysis
Figure 6.53. High pressure steam turbine isoentropic efficiency variation and its impact on
cycle heat rate.
90
88
86
84
82
80
78
76
74
High pressure steam turbine isoentropic efficiency
and its impact on cycle heat rate
0
-8 -6 -4 -2 0 2 4
Efficiency variation (%)
Medium pressure 1 steam turbine isoentropic efficiency evolution
72
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.54. Medium pressure 1 steam turbine isoentropic efficiency evolution.
Reference
Unit 1
Unit 2
Unit 3
Evolution of isoentropic efficiency of first medium pressure steam turbine is plotted
in figure 6.54. It can be seen how units 1 and 3 remain quite stable for a long period and
0.05
0.04
0.03
0.02
0.01
-0.01
-0.02
-0.03
Unit 1
Unit 2
Unit 3
177

Chapter 6
increase their efficiency suddenly in may 2004 and may 2003 respectively, probably
due to the maintenance works performed in these components. Efficiency of turbine in
unit 2 does not present clear efficiency variations along the period considered, although
it presents some dispersion during the second and third year. It should be noted that a
quite important dispersion appears in all units during 1999. Average value of this
variable is 80.9 % and it has a standard deviation of 2.10 %.
Relation between this efficiency and its impact on cycle heat rate is -0.00244 1/%,
and originates an impact standard deviation of 0.00513 (both values approximately half
of those of the high pressure turbine). This relation is also quite linear (R 2 = 0.998), as it
can be seen in figure 6.55. Impact on unit heat rate has a standard deviation of 0.00579
and an impact factor of -0.00276 1/% (R 2 = 0.998). Standard deviation of impact on
boiler efficiency is only 0.00222 %.
Impact
178
Medium pressure 1 steam turbine isoentropic efficiency
and its impact on cycle heat rate
0
-12 -10 -8 -6 -4 -2 0 2 4 6 8
-0.005
-0.015
Figure 6.55. Medium pressure 1 steam turbine isoentropic efficiency variation and its impact
on cycle heat rate.
Isoentropic efficiency of the second medium pressure steam turbine has an average
value of 88.2 % and a standard deviation of 1.26 %. Evolution of this variable is plotted
in figure 6.56, where it can be appreciated a steadily degradation of the turbine on unit
2. It can also be seen a strong degradation in unit 3 in may 2003, so that part of the
0.03
0.025
0.02
0.015
0.01
0.005
-0.01
-0.02
Efficiency variation (%)
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
improvement observed in this month in the first medium pressure turbine may be
induced by a deviation in instrumentation.
%
Impact
92
91
90
89
88
87
86
85
Medium pressure 2 steam turbine isoentropic efficiency evolution
84
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.56. Medium pressure 2 steam turbine isoentropic efficiency evolution.
Medium pressure 2 steam turbine isoentropic efficiency
and its impact on cycle heat rate
0.015
0.01
0.005
0
-4 -2 0 2 4 6 8
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
Efficiency variation(%)
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
Figure 6.57. Medium pressure 2 steam turbine isoentropic efficiency variation and its impact
on cycle heat rate.
179

Chapter 6
Impact on cycle heat rate due to variation of this variable has a standard deviation of
0.0383 (lower that the medium pressure 1 turbine) and an impact factor of -0.00303 1/%
(a little bit more than the other medium pressure turbine). Relation between variable and
impact is again linear (R 2 = 0.998), as figure 6.57 shows. Impact on unit heat rate has a
standard deviation of 0.00433 and an impact factor of -0.00343 1/% (R 2 = 0.999). Effect
on boiler efficiency is again negligible.
The last isoentropic efficiency corresponds to the low pressure turbine, and has the
important characteristic that it is not directly calculated from measurements, but it is has
the value suitable to obtain the cycle heat rate calculated, for a given value of the other
boundary conditions. For this reasons, this value is affected not by measurement errors
of a few variables but by errors of all the other variables affecting the cycle and errors
induced by all hypothesis made (efficiency of the turbo-pump, behaviour of the low
pressure preheaters…). As a result, this variable has a big dispersion: it has a standard
deviation of 3.60 % and an average value of 80.9%.
Impact
180
0.15
0.10
0.05
-0.05
-0.10
-0.15
Impact on cycle heat rate due to low pressure steam turbine
isoentropic efficiency
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.58. Impact on cycle heat rate due to low pressure steam turbine isoentropic
efficiency.
Unit 1
Unit 2
Unit 3
Since, as explained above, the value of this variable includes effects not only due to
the low pressure turbine itself but also to the hypothesis made in its calculation, it is

Application of quantitative causality analysis
perhaps better to analyze its impact on cycle heat rate than the variable. This impact has
a standard deviation of 0.0435 and its evolution is plotted on figure 6.58. In this figure it
can be seen how impacts in units 1 and 2 increase steadily and then decreases suddenly
in May 2004 and in April 2003 respectively. Impact on unit 3 has a quite high
dispersion and it is the lowest of the three units.
Relation of variable increment and its impact on cycle heat rate is -0.0120 1/% with
R 2 = 0.999. Impact on unit heat rate has a standard deviation of 0.0493 and an impact
factor of -0.0136 1/% with R 2 = 0.999 (Figure 6.59). Finally, effect on boiler efficiency
is very low, with an impact standard deviation of 0.0158 %.
Impact
Low pressure steam turbine isoentropic efficiency and its impact
on unit heat rate
0.2
0.15
0.1
0.05
0
-15 -10 -5 0 5 10 15
-0.05
-0.1
-0.15
-0.2
Efficiency variation (%)
Unit 1
Unit 2
Unit 3
Figure 6.59. Low pressure steam turbine isoentropic efficiency variation and its impact on
unit heat rate.
Apart from the isoentropic efficiency, the other parameter used to characterize steam
turbines behaviour is flow coefficient, except in the cases of the high pressure one,
which works in controlled inlet pressure. They determine the value of steam pressure in
turbine inlets, which affects not only the distribution of enthalpy drops between turbines
but the properties of steam extracted to water heaters. However, it should be noted that
influence of flow coefficients on cycle heat rate is about one order of magnitude lower
than effects of isoentropic efficiency.
181

Chapter 6
Impact
Impact
182
0.002
0.001
-0.001
-0.002
-0.003
-0.004
Impact on unit heat rate due to flow coefficient in medium
pressure steam turbine 1
0.000
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.60. Impact on unit heat rate due to flow coefficient of medium pressure steam
turbine 1.
Flow coefficient of medium pressure steam turbine 1 and its
impact on cycle heat rate
0
-1.5 -1 -0.5 0 0.5
-0.001
-0.002
-0.003
-0.004
Flow coefficient variation (kg 1/2 m 3/2 s -1 bar -1/2 )
Unit 1
Unit 2
Unit 3
Figure 6.61. Flow coefficient of medium pressure steam turbine 1 variation and its impact
on cycle heat rate.
0.002
0.001
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
Flow coefficient of the first medium pressure steam turbine has an average value of
13.1 kg 1/2 ·m 3/2 ·s -1 ·bar -1/2 and a standard deviation of 0.234 kg 1/2 ·m 3/2 ·s -1 ·bar -1/2 . Its impact
on cycle heat rate has a standard deviation of 0.000743 and its evolution is plotted in
figure 6.60. It can be seen how this parameter remains quite constant; the only
remarkable variation is an increment in June 2003 in unit 3.
Relation between this variable and its impact on cycle heat rate is shown in figure
6.61. Average slope of the curve is 0.00331 kg -1/2 ·m -3/2 ·s 1 ·bar 1/2 and it presents a little
dispersion (R 2 = 0.978). Impact on unit heat rate has a standard deviation of 0.00123
and an impact factor of 0.00546 kg -1/2 ·m -3/2 ·s 1 ·bar 1/2 (R 2 = 0.990).
Flow coefficients of the second medium pressure and low pressure steam turbines
have lower influence on cycle heat rate, with standard deviations of 0.000394 and
0.000370 respectively. The first has an impact factor of 0.000796 although dispersion is
quite high (R 2 = 0.898). The value of the impact factor of the second is not significant,
because it has a very high dispersion (R 2 = 0.00403).
ºC
15
10
5
-5
-10
Hot side temperature difference of 6 th water preheater evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.62. Hot side temperature difference (TTD) of 6 th preheater evolution.
Reference
Unit 1
Unit 2
Unit 3
183

Chapter 6
Impact
184
Hot side temperature difference of 6 th water preheater and its
impact on cycle heat rate
0.015
0.01
0.005
0
-10 -5 0 5 10 15
-0.005
-0.01
Temperature difference variation (ºC)
Figure 6.63. TTD of 6 th preheater variation and its impact on cycle heat rate.
Unit 1
Unit 2
Unit 3
After analysing variables related to turbines, parameters characterizing regenerative
water heaters are going to be studied. These parameters are temperature differences in
hot and cold sides: terminal temperature difference (TTD) and drain cooling approach
temperature (TDCA). The first ones have an important influence on cycle and unit heat
rate, while the others have low impact.
Figure 6.62 shows the evolution of TTD of 6 th water preheater. This difference is
calculated by substracting the saturation temperature at the pressure of steam exiting the
turbine minus temperature of hot water leaving the preheater. Since steam is
superheated, this difference can reach also negative values. The fact that pressure is
considered at the exit of turbine is important because if a pressure drop occurs in the
duct or in the preheater entry, it is diagnosed as an increment in temperature difference
in the preheater. This phenomenon occurs during the first year in unit 3 and during other
periods in unit 2, when temperature difference can reach values of 10 and 5 ºC
respectively. A small oscillating tendency with temperatures slightly higher in summer
than in winter can also be appreciated. Average value of this variable is -0.76 ºC and it
has a standard deviation of 2.27 ºC.

Application of quantitative causality analysis
Relation of temperature difference and its impact on cycle heat rate is 0.000710
1/ºC, with R 2 = 0.998 (Figure 6.63). This impact has a standard deviation of 0.00162.
Influence on cycle heat rate originates an impact on unit heat rate, characterized by an
impact factor of 0.00103 1/ºC (R 2 = 0.999) and a standard deviation of 0.00236.
Evolution of TTD of 5 th water preheater is plotted in figure 6.64. It can be
appreciated how this variable reaches values near 10 ºC during the first three years in
unit 2, probably due to pressure drop in the steam duct. There are other periods in this
unit in which temperature difference reaches almost around 40 ºC. In these situations,
water leaves the preheater at the same temperature at which it enters, which means that
the heat exchanger is by-passed. This situation demonstrated how the diagnosis method
is able to deal with discrete changes by using continuous variables chosen in a suitable
way. Seasonal oscillation is also present but is difficult to see due to the high range of
the y axis. Average value of this variable is 1.78 ºC and it has a standard deviation of
6.70 ºC.
ºC
60
50
40
30
20
10
-10
Hot side temperature difference of 5 th water preheater evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.64. Hot side temperature difference of 5 th water preheater evolution.
Reference
Unit 1
Unit 2
Unit 3
Relation between variable increment and its impact on cycle heat rate is 0.000397
1/ºC, with R 2 = 0.996, as it can be seen in figure 6.64. Standard deviation of this impact
is 0.00265. Effect on unit heat rate is characterized by an impact factor of 0.000455 1/ºC
(R 2 = 0.9962) and a standard deviation of 0.00304.
185

Chapter 6
Impact
ºC
186
0.025
0.02
0.015
0.01
0.005
-0.005
Hot side temperature difference of 5 th water preheater and its impact
on cycle heat rate
0
-10 0 10 20 30 40 50 60
Temperature difference variation (%)
Figure 6.65. Hot side temperature difference (TTD) of 5 th water preheater variation and its
impact on cycle heat rate.
10
8
6
4
2
-4
-6
-8
-10
Hot side temperature difference of 3 rd water preheater evolution
0
24/07/1998 0:00
-2
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.66. Hot side temperature difference (TTD) of 3 rd water preheater evolution.
Unit 1
Unit 2
Unit 3
Reference
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
The last hot side temperature considered is that corresponding to third preheater,
because of the lack of instrumentation of low pressure area. Its evolution is plotted in
figure 6.66. In this graph, it can be seen the cyclic evolution commented in the other
preheaters, as well as some differences among units (variable value is bigger in unit 3).
However, no important differences due to by-passes or pressure drops in steam ducts
are observed. Average value of this temperature difference is 0.11 ºC and it has a
standard deviation of 2.02 ºC. Due to this low variability, this variable has low influence
on cycle heat rate. Its impact has a standard deviation of 0.00223 and an impact factor
of 0.00111 1/ºC, with very linear behaviour (R 2 = 0.999). This impact on cycle
originates an impact on unit heat rate, with an impact factor of 0.00125 1/ºC (R 2 =
0.999) and a standard deviation of 0.00252.
The other parameters used to characterize water preheaters behaviour are
temperature differences at cold side (drain cooling approach temperature, TDCA).
These differences are calculated by substracting drain water temperature minus entering
water temperature. However, influence of these parameters is lower.
Cold side temperature difference of water of preheater 6 has an average value of
10.9 ºC and a standard deviation of 5.40 ºC. Its impacts on cycle and unit heat rates
have standard deviations of 0.000256 and 0.000293 respectively, which are about eight
times smaller than those of hot side temperature difference in the same preheaters.
Impact factors are 0.0000470 1/ºC (R 2 = 0.985) and 0.0000537 1/ºC (R 2 = 0.985).
In the case of 5 th preheater, cold side temperature difference has an average value of
8.06 ºC and a standard deviation of 17.5 ºC. This variability appears mainly because of
difference among units: values in unit 2 are around 25 or 30 ºC, in unit 3 around 10
degrees and in unit 1 in the middle. Impact factors are around twice as high as those of
6 th preheater: 0.000106 1/ºC (R 2 = 0.998) and 0.000120 1/ºC (R 2 = 0.998). The
combination of high variable deviation and impact factors is responsible for impact
standard deviations in cycle and unit heat rates more than three times higher than in 6 th
preheater (0.000850 and 0.000961).
Pressure loses in the water-steam circuit are summarized in two variables: pressure
drop in reheater and pressure increment provided by the turbo-pump. Although these
parameters vary also due to degradation of components located in the boiler, they affect
mainly to the cycle heat rate, so they are presented here.
Figure 6.67 shows the evolution of pressure drop in reheater. It can be seen how unit
1 presents an oscillating tendency, and in unit 3 a jump is produced in June 2003. In unit
187

Chapter 6
2, some points indicate a very low pressure drop during the first months of 2004;
however, in this period 6 th water preheater was by-passed, so that pressure indication
was not very reliable. Average value of this variable is 1.52 bar and its standard
deviation is 0.669 bar. It should be noted that, for convenience, this variable is defined
as the pressure difference of steam entering the 6 th preheater and reheated steam
entering medium pressure turbine. So that, it comprises the pressure drop in the reheater
and ducts minus the pressure drop in the duct connecting high pressure turbine
extraction with 6 th water preheater. This fact justifies the low values of this variable.
bar
188
4
3.5
3
2.5
2
1.5
1
0.5
-1
Pressure drop in reheater evolution
0
24/07/1998 0:00
-0.5
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.67 Pressure drop in reheater evolution.
Reference
Unit 1
Unit 2
Unit 3
This pressure drop variation originates an important impact on cycle heat rate.
Relation of variable increment and its impact has an average value of 0.00524 1/bar (R 2
= 0.999) and is shown in figure 6.68, while impact factor for unit heat rate is 0.00535
1/bar (R 2 = 0.999). Standard deviations are 0.00350 and 0.00358, respectively. Impact
on boiler efficiency has a standard deviation of only 0.000972 %.
Pressure increment provided by turbo-pump is plotted in figure 6.69. It can be seen
how units 2 and 3 have quite constant values, while unit 1 presents two peaks during the
first months of 2000 and 2004. Average value of this variable is 177.8 bar and it has a
standard deviation of 2.76 bar.

Impact
bar
Pressure drop in preheater and its impact on cycle heat rate
0.015
0.01
0.005
-0.005
-0.01
-0.015
Application of quantitative causality analysis
0
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
190
185
180
175
170
165
Pressure drop variation (bar)
Figure 6.68. Pressure drop in preheater variation and its impact on cycle heat rate.
Pressure increment in turbo-pump evolution
160
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.69. Pressure increment in turbo-pump evolution.
Unit 1
Unit 2
Unit 3
Reference
Unit 1
Unit 2
Unit 3
Influence of this variable on cycle heat rate is characterized by an impact factor of
0.000333 1/bar (R 2 = 0.990) and by its slightly curved shape, as it can be seen in figure
189

Chapter 6
6.70. The impact has a standard deviation of 0.00104. This variation of cycle heat rate
originates an impact on unit heat rate with an impact factor of 0.000376 1/bar (R 2 =
0.991) and a standard deviation of 0.00117.
Impact
190
Pressure increment in turbo-pump and its effect on cycle heat rate
0.008
0.006
0.004
0.002
0
-10 -5 0 5 10 15 20
-0.002
-0.004
Pressure increment variation (bar)
Unit 1
Unit 2
Unit 3
Figure 6.70. Pressure increment in turbo-pump variation and its impact on cycle heat rate.
The last variable to be considered in this section is uncontrolled water losses. These
losses are distributed along cycle and boiler, and conventionally are supposed to be
located in the drum. However, they mainly affect the unit heat rate and also the cycle
heat rate, so that they are presented here. Evolution of this variable is plotted in Figure
6.71. It has an average value of 2.52 kg/s and a standard deviation of 1.69 kg/s. This
variable is calculated by mass balance of the water-steam circuit, and it is affected by
some error sources, mainly tank levels. So that it has a lot of dispersion and some
outliers with high values appear. Besides, assumption on the location of these loses also
introduces an important uncertainty in the calculation of its impact.
Water losses have an important effect on unit heat rate, because they entail a loss of
heated water, apart from the cost of demineralised water that is not considered in this
work. Relation between variable increment and its impact on unit heat rate is 0.00339
s/kg (R 2 = 0.999) and it is plotted on figure 6.72. Standard deviation of this impact is
0.00576. Impact on cycle heat rate is more reduced, with a standard deviation of

Application of quantitative causality analysis
0.00282 and an impact factor of -0.00167 s/kg (R 2 = 0.999). It may seem strange the
negative value of this factor, but it should be noted that water loses are located at the
drum, so that, when they increases, the amount of water entering the boiler (which has a
negative sign in the calculation of the heat given to the cycle) also does.
kg/s
Impact
16
14
12
10
8
6
4
2
-2
Water loses evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.71. Water losses evolution.
Water loses and its impact on unit heat rate
0.04
0.03
0.02
0.01
0
-6 -4 -2 0 2 4 6 8 10 12
-0.01
-0.02
Water loses variation (kg/s)
Figure 6.72. Water losses variation and its impact on unit heat rate.
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
191

Chapter 6
6.3.6 Cooling system component parameters
In this section, influence of condenser effectiveness, cooling water flow rate and
cooling tower effectiveness is analyzed. These parameters characterize the behaviour of
the cooling system, whose degradation implies higher condenser pressure and higher
cycle heat rate for the same ambient conditions. In their determination, temperatures of
cooling water entering and exiting the condenser are used. These values are very
difficult to measure, because there are only two thermocouples in each section, so that
big uncertainty appears in these three free diagnosis variables.
192
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
Condenser effectiveness evolution
0.5
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.73. Condenser effectiveness evolution.
Reference
Unit 1
Unit 2
Unit 3
Evolution of condenser effectiveness is plotted in figure 6.73. Although high
dispersion is present, it can be seen how this effectiveness is higher in unit 1. This
variable has an average value of 0.689 and a standard deviation of 0.0622. Relation
between condenser effectiveness and its impact on cycle heat rate is shown in figure
6.74; this curved graph demonstrates the capability of the method to deal with nonlinear
behaviour. Impact factor is -0.177 (R 2 = 0.983) and impact standard deviation is
0.0110. Relation between variable increment and its impact on unit heat rate presents

Application of quantitative causality analysis
the same curved shape. Its average slope is -0.200 (R 2 = 0.982), and the impact has a
standard deviation of 0.0125. Effect on boiler efficiency is negligible.
Impact
kg/s
Condenser effectiveness and its impact on cycle heat rate
0
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
13000
12000
11000
10000
9000
8000
0.05
0.04
0.03
0.02
0.01
-0.01
-0.02
-0.03
-0.04
Effectiveness variation
Figure 6.74. Condenser effectiveness variation and its impact on cycle heat rate.
Cooling water mass flow evolution
7000
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.75. Cooling water flow rate evolution.
Unit 1
Unit 2
Unit 3
Reference
Unit 1
Unit 2
Unit 3
193

Chapter 6
Cooling water flow rate has an average value of 10699 kg/s and a standard deviation
of 553 kg/s. Figure 6.75 shows how evolution of this variable is characterized by a big
dispersion, above all in unit 3. Although not very clearly, it can be seen how unit 1 had
a sudden flow reduction in April 2003 and how flows in units 2 and 3 have tended to
increase along time. These variations in flow rate cause an important impact on cycle
heat rate, characterized by an impact factor of -0.0000190 s/kg with R 2 = 0.997 (Figure
6.76) and a standard deviation of 0.0105. In the case of unit heat rate, relation between
variable increment and impact is -0.0000216 s/kg (R 2 = 0.997), which originates an
impact standard deviation of 0.0119.
Impact
194
Cooling water mass flow and its impact on cycle heat rate
0
-3000 -2500 -2000 -1500 -1000 -500 0 500 1000 1500
Mass flow variation (kg/s)
Figure 6.76. Cooling water flow rate variation and its impact on cycle heat rate.
Finally, figure 6.77 shows the evolution of cooling tower effectiveness. The main
characteristic of this graph is its cyclic evolution, which clearly reveals that this variable
is not actually independent but it is strongly linked to ambient conditions, mainly
ambient temperature. This problem is tackled in a following section. Besides this
oscillating behaviour, it can be seen how effectiveness tends to decrease along time,
which indicates a progressive degradation. Average value of this variable is 0.4580 and
it has a standard deviation of 0.05860.
0.06
0.05
0.04
0.03
0.02
0.01
-0.01
-0.02
-0.03
Unit 1
Unit 2
Unit 3

Cooling tower effectiveness evolution
Figure 6.77. Cooling tower effectiveness evolution.
Application of quantitative causality analysis
0.3
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Impact
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
Cooling tower effectiveness and its impact on cycle heat rate
0
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Reference
Unit 1
Unit 2
Unit 3
Figure 6.78. Cooling tower effectiveness variation and its impact on cycle heat rate.
This variable has a strong influence on both cycle and unit heat rates, with impact
factors of -0.329 (R 2 = 0.998) and -0.372 (R 2 = 0.998) respectively (Figure 6.78). Impact
0.1
0.08
0.06
0.04
0.02
-0.02
-0.04
Tower effectiveness variation
Unit 1
Unit 2
Unit 3
195

Chapter 6
standard deviations are 0.0195 and 0.0221. Impact on boiler efficiency has a standard
deviation of only 0.0101 %.
6.3.7 Boiler component parameters
In this last section are included indicators related to heat transfer, such as preheaters
effectiveness and aggregated boiler effectiveness, air infiltration in preheaters and
residual carbon in ashes. This last variable is an indicator of the quality of the
combustion, and depends mainly on coal quality, oxygen in flue gases set point and
burners operation. However, since this dependence is difficult to be modelled precisely,
it is considered as an uncontrollable variable.
Evolution of secondary air preheater effectiveness is shown in figure 6.79. Point
dispersion appears because effectiveness is not a pure independent variable but it
depends on flow rates and temperatures of flows entering the heat exchanger. However,
thanks to the use of a lot of points, tendencies can be appreciated. It can be seen how
units 1 and 2 have an important improvement in May 2001, while unit 3 remains more
constant. This variable has an average value of 0.838 and a standard deviation of
0.0324.
Preheater effectiveness
196
0.95
0.9
0.85
0.8
0.75
Secondary air preheater effectiveness evolution
0.7
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.79. Secondary air preheater effectiveness evolution.
Reference
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
Impact on boiler efficiency due to secondary air preheater effectiveness has a
standard deviation of 0.249 % and relation between variable increment and impact is
7.71 %, with R 2 = 0.997 (Figure 6.80). This variation of boiler efficiency entails an
important effect on unit heat rate, characterized by an impact factor of -0.262 (R 2 =
0.997) and a standard deviation of 0.00844.
Impact (%)
Secondary air preheater effectiveness and its impact on boiler efficiency
0
-0.15 -0.1 -0.05 0 0.05 0.1
-0.2
Figure 6.80. Secondary air preheater effectiveness variation and its impact on boiler
efficiency.
Primary air preheater effectiveness has an average value of 0.873 and a standard
deviation of 0.0225. In figure 6.81, it can be seen how effectiveness has decreased
steadily or remained constant except two improvements in December 1999 and May
2001. Effectiveness of unit 1 is lower from the last months of 2002 to the repairing in
May 2004.
Impact of this variable is lower than that of secondary air preheater effectiveness,
because the amount of primary air is also lower. Effect on boiler efficiency is
characterized by an impact factor of 2.16 % with R 2 = 0.991 (Figure 6.82) and a
standard deviation of 0.0490 %. Impact on unit heat rate has a standard deviation of
0.00167 and relation between variable and impact is -0.0733 (R 2 = 0.989). It should be
0.8
0.6
0.4
0.2
-0.4
-0.6
-0.8
-1
-1.2
Effectiveness variation
Unit 1
Unit 2
Unit 3
197

Chapter 6
noted that these correlation coefficients are not as high as in other variables due to
crossed effects caused by air distribution variation.
Preheater effectiveness
Impact (%)
198
1
0.95
0.9
0.85
0.8
Primary air preheater effectiveness evolution
0.75
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.81. Primary air preheater effectiveness evolution.
Primary air preheater effectiveness and its impact on boiler efficiency
0
-0.15 -0.1 -0.05 0 0.05 0.1
-0.05
0.2
0.15
0.1
0.05
-0.1
-0.15
-0.2
-0.25
-0.3
Effectiveness variation
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
Figure 6.82. Primary air preheater effectiveness variation and its impact on boiler efficiency.

Application of quantitative causality analysis
Heat transfer effectiveness of all heat exchangers in the boiler different from air
preheaters is summarized in only one variable called aggregated boiler effectiveness.
Evolution of this variable is plotted in figure 6.83, where two parts can be appreciated.
During the first years, an oscillating tendency appears, with higher values in summer
than in winter. Afterwards, this behaviour disappears and effectiveness remains quite
constant. This behaviour can be justified by considering the evolution of the
temperature difference between gases entering primary and secondary air preheaters
(Figure 6.51), which follows the same pattern. During the first years, when the gases
by-pass was opened in winter, not only temperature difference but also average gas
temperature increased, so that aggregated boiler effectiveness decreased. The effect in
summer was the opposite. Aggregated boiler effectiveness has an average value of
0.861 and a standard deviation of 0.0182.
Boiler effectiveness
0.95
0.93
0.91
0.89
0.87
0.85
0.83
0.81
0.79
0.77
Aggregated boiler effectiveness evolution
0.75
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.83. Aggregated boiler effectiveness evolution.
Reference
Unit 1
Unit 2
Unit 3
This variable affects strongly the boiler efficiency and hence the unit heat rate.
Impact standard deviations are 0.424 % and 0.0114 respectively. Relation between
aggregated boiler effectiveness and its impact on boiler efficiency is 23.4 %, with R 2 =
0.998 (Figure 6.84). In the case of unit heat rate, this impact factor is -0.627 (R 2 =
0.998).
199

Chapter 6
Impact
Air infiltration (%)
200
Aggregated boiler effectiveness and its impact on boiler efficiency
2.5
2
1.5
1
0.5
0
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
-0.5
-1
-1.5
-2
-2.5
Effectiveness variation
Figure 6.84. Aggregated boiler effectiveness variation and its impact on boiler efficiency.
45
40
35
30
25
20
15
10
5
Air infiltration in preheaters evolution
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.85. Air infiltration in preheaters evolution.
Unit 1
Unit 2
Unit 3
Reference
Unit 1
Unit 2
Unit 3
Figure 6.85 shows the evolution of air infiltration in preheaters, expressed in mass
percentage of flue gases. It can be seen how during the first months, infiltration was
very high. Then, preheaters were replaced, so that infiltration decreased dramatically.

Application of quantitative causality analysis
Then, infiltration tends to increase along time, except in some points when preheaters
are repaired: May 2002 for units 2 and 3, May 2003 for unit 3 and May 2004 for units 1
and 2. Average value of infiltration is 13.0 % and it has a standard deviation of 3.56 %.
Each point in air infiltration entails a reduction of 0.0278 points in boiler efficiency,
with R 2 = 0.998 (Figure 6.86), and an increment of 0.00107 in unit heat rate (R 2 =
0.996). Impact standard deviations are 0.0997 % and 0.00387, respectively. Effect on
cycle heat rate is negligible.
Impact (%)
Air infiltration in preheaters and its impact on boiler efficiency
0.4
0.2
0
-10 -5 0 5 10 15 20 25 30 35
-0.2
-0.4
-0.6
-0.8
-1
Air infiltration variation (%)
Figure 6.86. Air infiltration in preheaters and its impact on boiler efficiency.
Unit 1
Unit 2
Unit 3
The last variable considered is carbon percentage in ashes. It should be noted that
this variable is not actually independent, because it depends on fuel and boiler
operation. This dependence may be included by using a suitable combustion model
(which is a complex task which falls out of the scope of this work) and causality chains
theory. Carbon in ashes has an average value of 0.955 % and a standard deviation of
0.556 %, and is plotted in figure 6.87. This variable has quite dispersion, but it can be
seen how it was low during the first two years and then increased.
Relation between carbon percentage and its impact on boiler efficiency is -0.452
%/%, with R 2 = 0.997 (Figure 6.88). This impact has a standard deviation of 0.252 %.
Effect on unit heat rate is characterized by an impact factor of 0.0162 1/% (R 2 = 0.997)
201

Chapter 6
and a standard deviation of 0.0901. Impact on cycle heat rate has a standard deviation of
only 0.000178.
Carbon %
Impact (%)
202
4.5
4
3.5
3
2.5
2
1.5
1
0.5
Carbon in ashes evolution
0
24/07/1998 0:00
-0.5
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.87. Carbon in ashes evolution.
Carbon in ashes and its impact on boiler efficiency
1
0.5
0
-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
-0.5
-1
-1.5
-2
Carbon in ashes variation (%)
Figure 6.88. Carbon in ashes variation and its impact on boiler efficiency.
Reference
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis
6.3.8 Summary of results.
In the previous sections, a lot of results corresponding to almost all diagnosis
variables have been presented. Here, all of them are collected in order to summarize
information and to facilitate comparison among them. To achieve this, indicators
defined and applied previously are shown in Table 6.1: average and standard deviation
of the free diagnosis variables, and impact factors (with their R 2 values) and standard
deviation of their impacts on the three global efficiency indicators. These standard
deviations quantify the importance of each free diagnosis variables. Units indicated in
the third column correspond to the variable average and standard deviation. Units for
impacts are % in boiler efficiency and dimensionless for cycle and unit heat rates. R 2
values are dimensionless. Finally, units for impact factors are the inverse of those
indicated in the unit column for impacts on cycle and unit heat rate, and these inverse
values multiplied times % for impacts on boiler efficiency.
In order to clearly distinguish the most important variables from those which have
negligible impact, they have been classified according to the standard deviation of their
impacts on boiler efficiency, cycle and unit heat rates. Variables whose impact standard
deviation is above 20% of the maximum are labelled pink, those from 10 to 20% are
labelled orange, those from 5 to 10% are yellow and variables from 1 to 5% are bluegreen.
The other variables remain white.
As it can be seen, most important variables regarding boiler efficiency are ambient
temperature, wind speed, most variables related to coal, average cold side temperature
in secondary air preheaters, air extracted to flue gas desulfuration, aggregated boiler
effectiveness, preheaters effectiveness and carbon in ashes. On the other hand, influence
of variables related to cycle is negligible.
Cycle heat rate varies mainly due to ambient temperature and humidity, cooling
system condition (condenser, tower and cooling water flow rate), and also turbines
isoentropic efficiencies.
Finally, unit heat rate is influenced by variables related to both the boiler and the
steam cycle. Due to this fact, the number of variables whose influence is lower than 1%
of the maximum is only 5, compared to 22 and 20 in boiler efficiency and cycle heat
rate respectively.
203

Chapter 6
Impact on cycle heat rate Impact on unit heat rate
Variable Impact on boiler
Num. Description Units
effficiency
Stand.
R 2
Impact
dev.
factor
R 2 Stand.
dev.
0.9991 4.484
6.137
0.9998 5.276
Mean Stand. Impact R
dev. factor
2 Stand. Impact
dev. factor
1 Ambient temperature ºC 15.5 7.30 0.05524 0.9958 0.4017 7.234
·10 -2
·10 -3
·10 -2
·10 -3
0.9920 9.314
1.273
0.9923 7.844
0.8996 0.01081 1.073
2 Relative humidity % 63.4 7.08 -1.132
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
0.9838 4.117
1.455
0.9827 8.039
0.9854 0.1150 2.837
3 Wind speed m/s 4.55 2.82 -4.065
·10 -3
·10 -3
·10 -5
·10 -5
·10 -2
0.9977 2.367
-3.644
0.9960 2.618
0.9989 0.5744 -4.020
4 Coal high heating value kJ/kg 16975 639 8.889
·10 -2
·10 -5
·10 -3
·10 -6
·10 -4
0.9969 9.782
5.959
0.9971 2.238
5 Carbon mass fraction in coal % 42.53 1.61 -0.1164 0.9969 0.1898 1.365
·10 -3
·10 -3
·10 -3
·10 -3
0.9988 1.189
6.877
0.9967 8.645
6 Hydrogen mass fraction in coal % 2.74 0.173 -1.796 0.9993 0.3102 4.989
·10 -2
·10 -2
·10 -4
·10 -3
0.9988 1.257
7.757
0.9964 9.181
7 Moisture mass fraction in coal % 18.87 1.62 -0.2007 0.9994 0.3249 5.641
·10 -2
·10 -3
·10 -4
·10 -4
0.9956 5.657
3.066
0.9963 7.228
0.9958 0.1339 3.921
8 Ash mass fraction in coal % 23.16 1.84 -7.263
·10 -3
·10 -3
·10 -4
·10 -4
·10 -2
0.9962 1.134
2.717
0.9972 2.791
6.721
0.9962 2.169
9 Sulphur mass fraction in coal % 4.84 0.412 -5.191
·10 -3
·10 -3
·10 -4
·10 -4
·10 -2
·10 -2
0.9969 1.040
1.755
0.9969 2.221
3.735
0.9973 2.109
-3.558
10 Nitrogen mass fraction in coal % 0.66 5.94
·10 -4
·10 -3
·10 -5
·10 -4
·10 -3
·10 -2
·10 -2
Table 6.1.A. Summary of diagnosis results.
204

Application of quantitative causality analysis
Impact on cycle heat rate Impact on unit heat rate
Variable Impact on boiler
Num. Description Units
effficiency
Stand.
R 2
Impact
R 2 Stand.
dev.
Impact
R 2 Stand.
dev.
Impact
dev.
factor
factor
factor
Mean Stand.
dev.
0.4373 1.633
9.792
0.9854 8.518
-7.669
0.7648 6.829
11 Energy provided by natural gas % 0.576 1.10 -5.415
·10 -4
·10 -5
·10 -5
·10 -5
·10 -3
·10 -3
0.9991 1.090
-8.504
0.9991 1.009
-7.871
0.5825 3.566
12 Live steam temperature ºC 537.9 1.28 -2.038
·10 -3
·10 -4
·10 -3
·10 -4
·10 -4
·10 -4
0.9997 1.310
-6.239
0.9997 1.308
-6.239
0.6481 4.750
13 Reheated steam temperature ºC 538.2 2.10 -1.163
·10 -3
·10 -4
·10 -3
·10 -4
·10 -4
·10 -4
0.9987 1.830
-1.351
0.9984 1.664
-1.228
0.5918 5.025
14 Live steam pressure bar 157.5 1.35 -2.832
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
·10 -4
0.6125 7.331
7.245
0.9923 3.804
5.415
0.6664 1.843
15 Gross electric power MW 340.01 7.003 1.971
·10 -4
·10 -5
·10 -3
·10 -4
·10 -2
·10 -3
0.9967 3.388
1.310
0.9951 7.474
2.887
% 2.05 0.259 -0.2645 0.9967 6.832
16 Oxygen in flue gases leaving the
·10 -3
·10 -2
·10 -4
·10 -3
·10 -2
boiler
0.9992 6.830
1.025
0.9854 2.762
0.9989 0.1741 4.140
109.3 6.67 -2.617
ºC
17 Average cold-side temperature
·10 -3
·10 -3
·10 -4
·10 -5
·10 -2
in secondary air preheaters
0.9897 2.082
2.809
0.9811 8.771
1.174
0.9942 5.3412
111.6 7.24 -7.243
ºC
18 Average cold-side temperature
·10 -3
·10 -4
·10 -5
·10 -5
·10 -2
·10 -3
in primary air preheaters
0.9992 4.550
8.041
0.9986 9.412
-1.662
0.5924 1.232
19 Sootblowing steam flow rate kg/s 1.40 0.564 1.579
·10 -3
·10 -3
·10 -4
·10 -3
·10 -3
·10 -3
0.9962 5.396
6.134
0.9919 2.141
0.9977 0.1403 2.425
4.86 8.79 -1.595
kg/s
20 Air for flue gas desulfuration
·10 -3
·10 -4
·10 -4
·10 -5
·10 -2
unit
Table 6.1.B. Summary of diagnosis results.
205

Chapter 6
Impact on cycle heat rate Impact on unit heat rate
Variable Impact on boiler
Num. Description Units
effficiency
Stand.
R 2
Impact
dev.
factor
R 2 Stand.
dev.
0.9912 3.455
4.964
0.5181 2.314
·10 -3
·10 -4
·10 -5
0.9637 1.764
1.142
0.6548 2.216
Mean Stand. Impact R
dev. factor
2 Stand. Impact
dev. factor
21 Tempering and primary air kg/kg 33.8 7.06 -1.477
relation
·10 -2
0.9909 0.1028 2.276
·10 -6
22 Primary air-coal ratio kg/kg 1.97 0.143 -0.3467 0.9704 5.299 9.733
·10 -3
·10 -2
·10 -5
·10 -5
·10 -2
0.6739 5.261
2.796
0.9869 2.365
1.594
8.549
8.093
7.50 14.8 -1.460
ºC
23 Temperature difference of flue
·10 -4
·10 -5
·10 -4
·10 -5
·10 -3
·10 -2
·10 -4
gases entering preheaters
0.1231 6.461
5.233·1
0.4478 4.786
7.759
1.568
8.001
24.5 3.22 -1.021
-
24 Fraction of flue gases through
·10 -4
0 -5
·10 -5
·10 -6
·10 -2
·10 -2
·10 -3
PAH
0.9994 9.927
-5.478
0.9994 9.051
-4.994
0.7024 3.567
79.5 1.81 -1.434
%
25 High pressure turbine
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
isoentropic efficiency
0.9985 5.793
-2.761
0.9982 5.127
-2.443
0.6685 2.219
80.9 2.10 -7.000
%
26 Intermediate pressure 1
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
isoentropic efficiency
0.9986 4.327
-3.428
0.9982 3.830
-3.032
0.6378 1.719
88.2 1.26 -8.515
%
27 Intermediate pressure 2
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
isoentropic efficiency
0.9990 4.925
-1.359
0.9989 4.351
-1.199
0.5902 1.577
80.9 3.60 -3.130
%
28 Low pressure isoentropic
·10 -2
·10 -2
·10 -2
·10 -2
·10 -2
·10 -3
efficiency
0.9898 1.233
5.462
0.9783 7.434
3.314
0.6561 5.467
13.1 0.233 1.301
kg 1/2 ·m 3/2
29 Intermediate pressure 1 flow
·10 -3
·10 -3
·10 -4
·10 -3
·10 -4
·10 -3
·s -1 ·bar -1/2
coefficient
Table 6.1.C. Summary of diagnosis results.
206

Application of quantitative causality analysis
Impact on cycle heat rate Impact on unit heat rate
Variable Impact on boiler
Num. Description Units
effficiency
Stand.
R 2
Impact
R 2 Stand.
dev.
Impact
dev.
factor
factor
R 2 Stand.
dev.
0.9016 4.509
9.126
0.8985 3.943
7.956
0.5958 1.462
Mean Stand. Impact
dev. factor
24.7 0.439 2.321
kg 1/2 ·m 3/2
30 Intermediate pressure 2 flow
·10 -4
·10 -4
·10 -4
·10 -4
·10 -4
·10 -4
·s -1 ·bar -1/2
coefficient
4.230
1.418
2.323
3.695
4.032
-1.091
1.278
9.476
51.4 0.964 5.734
31 Low pressure flow coefficient kg 1/2 ·m 3/2
·10 -4
·10 -2
·10 -5
·10 -4
·10 -3
·10 -5
·10 -4
·10 -3
·10 -6
·s -1 ·bar -1/2
0.9985 2.356
1.034
0.9983 1.616
7.093
0.6635 8.663
32 TTD 6 th water heater ºC -0.76 2.27 2.933
·10 -3
·10 -3
·10 -3
·10 -4
·10 -4
·10 -4
0.9962 3.040
4.548
0.9959 2.652
3.966
0.7828 1.057
33 TTD 5 th water heater ºC 1.78 6.70 1.371
·10 -3
·10 -4
·10 -3
·10 -4
·10 -3
·10 -4
0.9996 2.523
1.253
·10 -3
0.9997 2.226
1.105
0.6505 7.914
34 TTD 3 rd water heater ºC 0.11 2.02 3.128
·10 -3
·10 -3
·10 -3
·10 -4
·10 -4
0.9854 2.932
5.373
0.9847 2.564
4.697
0.6576 8.936
35 TDCA 6 th water heater ºC 10.9 5.44 1.326
·10 -4
·10 -5
·10 -4
·10 -5
·10 -5
·10 -5
0.9978 9.612
1.202
0.9977 8.501
1.063
0.6466 3.026
36 TDCA 5 th water heater ºC 17.5 8.06 2.819
·10 -4
·10 -4
·10 -4
·10 -4
·10 -4
·10 -5
0.9990 3.579
5.352
0.9989 3.503
5.238
0.6757 9.723
37 Pressure drop in reheater bar 1.52 0.669 1.193
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
·10 -3
0.9907 1.165
3.756
0.9898 1.039
3.333
0.6467 5.117
177.8 2.76 8.762
bar
38 Pressure increment in turbo-
·10 -3
·10 -4
·10 -3
·10 -4
·10 -4
·10 -5
pump
Table 6.1.D. Summary of diagnosis results.
207

Chapter 6
Impact on cycle heat rate Impact on unit heat rate
Variable Impact on boiler
Num. Description Units
effficiency
Stand.
R 2
Impact
R 2 Stand.
dev.
Impact
R 2 Stand.
dev.
Impact
dev.
factor
factor
factor
Mean Stand.
dev.
0.9989 5.756
3.391
0.9987 2.823
-1.671
0.6215 1.870
39 Water losses kg/s 2.52 1.69 7.813
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
-0.2002 0.9825 1.246
-0.1771 0.9826 1.102
0.6620 3.505
-4.579
40 Condenser effectiveness - 0.689 6.22
·10 -2
·10 -2
·10 -3
·10 -2
·10 -2
0.9967 1.190
-2.155
0.9970 1.050
-1.903
0.64620 3.910
41 Cooling water flow rate kg/s 10699 553 -5.757
·10 -2
·10 -5
·10 -2
·10 -5
·10 -3
·10 -6
-0.3724 0.9982 2.206
-0.3286 0.9983 1.946
-0.1202 0.7997 1.009·
42 Cooling tower effectiveness - 0.458 5.86
·10 -2
·10 -2
10 -2
·10 -2
-0.2615 0.9973 8.441
0.8510 6.703
7.707 0.9972 0.2488 -1.880
0.838 3.24
-
43 Secondary air preheater
·10 -3
·10 -5
·10 -3
·10 -2
effectiveness
0.9894 1.668
-7.330
0.5778 1.403
-4.135
2.163 0.9912 4.904
0.873 2.25
-
44 Primary air preheater
·10 -3
·10 -2
·10 -5
·10 -4
·10 -2
·10 -2
effectiveness
-0.6269 0.9976 1.140
23.38 0.9984 0.4243 0.1144 0.9891 2.070
45 Aggregated boiler effectiveness - 0.861 1.82
·10 -2
·10 -3
·10 -2
0.9956 3.869
1.070
0.9889 1.593
4.404
0.9984 9.969
46 Air infiltration in preheaters % 13.0 3.56 -2.776
·10 -3
·10 -3
·10 -4
·10 -5
·10 -2
·10 -2
0.9965 9.010
1.618
0.9936 1.775
47 Carbon in ashes % 0.955 0.556 -0.4522 0.9974 0.2518 3.182
·10 -3
·10 -2
·10 -4
·10 -4
Table 6.1.E. Summary of diagnosis results.
208

6.4 Residual term analysis.
Application of quantitative causality analysis.
The idea of the diagnosis procedure is to share the variation of an efficiency
indicator into several terms, each one corresponding to an independent free diagnosis
variable. Graphs shown in the previous section relating the variations of the free
variables and their impacts prove that the method provides coherent results. However,
to demonstrate the accuracy of the methodology, it should be checked that the residual
term is low enough. This residual term ε is the difference between the variation of the
efficiency indicator and the summation of all impacts originated by the free diagnosis
variables considered (section 3.3.1):
Δ = ed ⋅Δ x +
6.8
t
e d ε
On the other hand, the good correlation among variables and impacts apparently
shows that it would be enough to consider constant impact factors (ed av ), without
applying the whole procedure for each period.
In this section, residual term ε is studied for both the diagnosis method proposed
(variable impact factors) and for a hypothetical approximated method which would use
constant impact factors ed av (those appearing in table 6.1). Histograms are used to
represent the distribution of the residual terms, for cycle heat rate, boiler efficiency and
unit heat rate. First, the values of residuals are used directly. Then, non-dimensional
values (calculated by dividing the residual term into the maximum impact are used).
Results show that the diagnosis method used provides very low residuals, above all
compared to residuals appearing when constant impact factors are used. In other words,
the quantitative causality analysis provides good relation between accuracy and effort
required.
6.4.1 Residual term distribution.
Figure 6.89 shows a histogram with the distribution of the residual term of cycle
heat rate. Blue bars show the distribution when the proposed methodology is used
(variable impact factors), while red bars show the residuals corresponding to constant
impact factors. As it can be seen, residuals distribution with variable impact factor is
more concentrated around the zero than the distribution with constant impact factor. For
example, in the first case, more than 63% of points have residuals between -0.0015 and
209

Chapter 6
0.0015, while in the second this figure is lower than 40%. To put in context these
figures, it should be taken into account that cycle heat rate has a variation range of
almost 0.2 (Figure 6.1). Another interesting characteristic of these graphs is that they
are bell-shaped butt not necessarily symmetrical. This lack of symmetry appears
because the state selected as reference does not correspond to the average values of all
free diagnosis variables.
Fraction of points
210
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
< -0.0065
-0.0065 -0.0055
-0.0055 -0.0045
-0.0045 -0.0035
-0.0035 -0.0025
-0.0025 -0.0015
-0.0015 -0.0005
-0.0005 0.0005
0.0005 0.0015
0.0015 0.0025
0.0025 0.0035
0.0035 0.0045
0.0045 0.0055
0.0055 0.0065
0.0065 0.0075
0.0075 0.0085
0.0085 0.0095
0.0095 0.0105
0.0105 0.0115
0.0115 0.0125
0.0125 0.0135
0.0135 0.0145
0.0145 0.0155
0.0155 0.0165
Residual of cycle heat rate
0.0165 0.0175
0.0175 0.0185
> 0.0185
Figure 6.89. Distribution of residuals of cycle heat rate.
Variable impact factors
Constant impact factors
The distribution of residuals of boiler efficiency appears in figure 6.90. Again, this
distribution is appreciably more concentrated for variable impact factors: more than
67% between -0.03% and 0.03% for variable impact factors and less than 40% between
-0.01% and 0.05% for constant impact factors. The range of variation of boiler
efficiency is about 4% (see Figure 6.2). Finally, the centre of the distribution
corresponding to constant factor is slightly displaced. This effect appears because the
factors correspond to the average slopes considering all points, which can be slightly
different from the slope around the reference state.
Finally, figure 6.91 shows the residual distribution for unit heat rate. By using
variable impact factors, 56% of points have residuals between -0.015 and 0.015. This

Application of quantitative causality analysis.
percentage is only 26% when constant impact factors are used. As it can be seen in
Figure 6.3, the range of variation of unit heat rate is about 0.2.
Fraction of points
Fraction of points
0.3
0.25
0.2
0.15
0.1
0.05
0
0.25
0.2
0.15
0.1
0.05
0
< -0.27
-0.27 -0.25
-0.25 -0.23
-0.23 -0.21
-0.21 -0.19
-0.19 -0.17
-0.17 -0.15
-0.15 -0.13
-0.13 -0.11
-0.11 -0.09
-0.09 -0.07
-0.07 -0.05
-0.05 -0.03
-0.03 -0.01
-0.01 0.01
0.01 0.03
0.03 0.05
0.05 0.07
0.07 0.09
0.09 0.11
0.11 0.13
0.13 0.15
0.15 0.17
0.17 0.19
0.19 0.21
0.21 0.23
> 0.23
Residual of boiler efficiency (%)
Figure 6.90. Distribution of residuals of boiler efficiency.
< 0.0115
-0.0105 -0.0095
-0.0085 -0.0075
-0.0065 -0.0055
-0.0045 -0.0035
-0.0025 -0.0015
-0.0005 0.0005
0.0015 0.0025
0.0035 0.0045
0.0055 0.0065
0.0075 0.0085
0.0095 0.0105
0.0115 0.0125
0.0135 0.0145
0.0155 0.0165
0.0175 0.0185
Residual of unit heat rate
0.0195 0.0205
0.0215 0.0225
> 0.0235
Figure 6.91. Distribution of residuals of unit heat rate.
Variable impact factors
Constant impact factors
Variable impact factors
Constant impact factors
211

Chapter 6
6.4.2 Non-dimensional residual term distribution.
Graphs presented previously demonstrated that residual terms are low in absolute
values. However, it should be noted that the residual admitted in a diagnosis may vary
according to the magnitude of the impacts. So that, it is interesting to use nondimensional
residual terms, obtained by dividing each one of these residuals into the
biggest impact of its diagnosis case.
212
ε
ε = 6.8
nd , i
i
max j
i
j
Where
( ) I
i
I j is the impact originated by the j th free diagnosis variable in the i th
diagnosis test.
Figure 6.92 shows the non-dimensional impact distribution of cycle heat rate. By
using variable impact factors, 73% of points have residuals between -3 and 3% of the
maximum impact, which can be considered as a very narrow range. The amount of
points whose residuals are out of the ±10% range is negligible. In the case of constant
impact factors, only 47% of points below to the ±3% range.
Fraction of points
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.19
Non-dimensional residual of cycle heat rate
Figure 6.92. Distribution of non-dimensional residuals of cycle heat rate.
Variable impact factors
Constant impact factors
The distribution of residuals in boiler efficiency is also concentrated but less than in
the previous case, as it can be seen in figure 6.93. The fraction of points with residuals
in the ±3% range is 69%, but almost all cases (94%) are in the ±9% range, which can be

Application of quantitative causality analysis.
considered as an error low enough. Regarding the constant value approach, 29% of the
points belong to the -5 to 1% range, and 70% to the -13 to 5% range.
Fraction of points
Fraction of points
0.25
0.2
0.15
0.1
0.05
0
0.3
0.25
0.2
0.15
0.1
0.05
0
< 0.25
-0.25 -0.23
-0.23 -0.21
-0.21 -0.19
-0.19 -0.17
-0.17 -0.15
-0.15 -0.13
-0.13 -0.11
-0.11 -0.09
-0.09 -0.07
-0.07 -0.05
-0.05 -0.03
-0.03 -0.01
-0.01 0.01
0.01 0.03
0.03 0.05
0.05 0.07
0.07 0.09
0.09 0.11
0.11 0.13
0.13 0.15
0.15 0.17
0.17 0.19
0.19 0.21
0.21 0.23
0.23 0.25
0.25 0.27
0.27 0.29
0.29 0.31
0.31 0.33
> 0.33
Non-dimensional residual of boiler efficiency
Figure 6.93. Distribution of non-dimensional residuals of boiler efficiency
0.25
Non-dimensional residual of unit heat rate
Figure 6.94. Distribution of non-dimensional residuals of unit heat rate
Variable impact factors
Constant impact factors
Variable impact factors
Constant impact factors
213

Chapter 6
Figure 6.94 represents the distribution of non-dimensional residuals of unit heat rate.
When variable impact factors are used, 73% of the points are in the ±3% range, and
more than 99% in the ±9% range. With the approximation of constant impact factors,
these fractions decrease to 47 and 86% respectively.
6.4.3 Conclusion
Graphs previously presented show that, when the proposed diagnosis method is
used, about 70% of points have a non-dimensional residual term very reduced (in the
±3% range). This value decreases to less than 50% is constant impact factors is used.
These results are very important because prove that the quantitative causality
analysis proposed presents a good compromise between accuracy and implementation
cost. The use of a simulator would provide an unappreciable accuracy increment but it
entails a substantial increment in computation and implementation cost. On the other
hand, accuracy decreases with the use of constant values. It should be noted that this
difference would be higher if data of all load range would have been used. Besides, the
use of constant value requires the assessment of the validity of these coefficients along
time, while the method proposed is self-adjustable.
6.5 Impacts aggregation
Results shown in previous sections demonstrate that the quantitative causality
diagnosis method is able to decompose the variations of boiler efficiency and cycle and
unit heat rate into a summation of terms each one corresponding to an independent free
diagnosis variable with a negligible residual term. The method provides an exhaustive
analysis taking into account a large amount of independent causes. This fact constitutes
and advantage but, due to the large amount of information provided, it sometimes
makes it difficult to analyse the results.
In this context, it can be remembered that impacts are additive, so that it is possible
to group them in families. This allows to follow a zooming approach in the
interpretation of results: first consider the evolution of the global efficiency indicators,
then analyse the evolution of impacts due to a family of causes, and finally to analyse
the impacts of individual causes, if necessary. It should be noted that this approach is
not obliged by the methodology: the method provides in one step the impacts provided
214

Application of quantitative causality analysis.
by each one of the free diagnosis variables, and the impact of a group of variables can
be calculated later by adding the individual impacts. Obviously, in order to facilitate the
analyst interpretation, results can be shown in a zooming manner: first, impacts of
groups of variables, and then individual impacts.
The choice of the groups of variables depends on the interest of the analyst. A
reasonable option for the working example is: ambient variables, fuel properties, setpoints
for boiler and cycle and components efficiency indicators for boiler, cycle and
cooling system. The distribution of free diagnosis variables into categories is detailed in
section 5.3. Further aggregation can be made: for example, all boiler related variables
when cycle heat rate is considered, or all set points. This strategy has been applied to the
three global efficiency indicators of the working example: cycle heat rate, boiler
efficiency and unit heat rate.
6.5.1 Cycle heat rate
Figure 6.95 shows the evolution of impact on cycle heat rate due to ambient
conditions. The graph has a cyclic evolution originated by the variation of ambient
conditions throughout the year. This impact has the highest standard deviation of all the
variables groups considered: 0.0470.
Influence of cycle set-points is represented in Figure 6.96, and has a standard
deviation of only 0.01616. Although there is an important short-time dispersion, it can
be appreciated some evolution along time and some differences among units.
Finally, impact of variables related to boiler (fuel, boiler set-points and boiler
components) are represented in Figure 6.97. It can be appreciated the higher influence
during the first two years, caused by the variation of aggregated boiler effectiveness
induced by the different operation of the economizer by-pass (see Fig. 6.83). In any
case, the he standard deviation of this impact is very reduced: 0.003410.
215

Chapter 6
216
Impact
Impact
0.10
0.05
-0.05
-0.10
-0.15
Cycle heat rate impact due to ambient conditions
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
0.025
0.020
0.015
0.010
0.005
-0.005
-0.010
-0.015
-0.020
Figure 6.95. Impact on cycle heat rate due to ambient conditions.
Cycle heat rate impact due to cycle set-points
0.000
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.96. Impact on cycle heat rate due to cycle set-points.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3

Impact
Impact
0.020
0.015
0.010
0.005
-0.005
-0.010
Cycle heat rate impact due to boiler
Application of quantitative causality analysis.
0.000
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
0.10
0.08
0.06
0.04
0.02
-0.02
-0.04
-0.06
Figure 6.97. Impact on cycle heat rate due to boiler.
Cycle heat rate impact due to cooling system
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 5.98. Impact on cycle heat rate due to cooling system.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
217

Chapter 6
Impact
218
0.15
0.10
0.05
-0.05
-0.10
-0.15
Cycle heat rate impact due to cycle components
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.99. Impact on cycle heat rate due to cycle components.
Unit 1
Unit 2
Unit 3
After having removed effects of ambient condition, set-points and boiler, only the
components of cooling system and cycle itself remain. Cooling system components
appear in Figure 6.98. The main characteristic of this graph is the oscillating evolution,
originated by the dependence of cooling tower effectiveness with ambient conditions
(see Figure 6.77). To filtrate this dependence, causality chains theory is used later.
Besides this cyclic behaviour, a slight tendency to increase along time (due to
components degradation) can be observed. Standard deviation of this impact is 0.01917.
Figure 6.99 shows the evolution of impact caused by cycle components. Although
there is a lot of dispersion, it can be appreciated how impact tends to increase along
time due to components degradation except in the springs of 1999 and 2004. Besides,
some differences among units can be observed. This impact has a standard deviation of
0.04315.
6.5.2 Boiler efficiency
Variables affecting boiler efficiency can be classified into: ambient conditions, fuel,
boiler set-points, boiler components and cycle influence. Figure 6.100 shows the impact
on boiler efficiency due to ambient conditions. It can be appreciated the cyclic evolution

Application of quantitative causality analysis.
corresponding to the seasonal weather variation. This impact has a standard deviation of
0.4579 %.
Impact (%)
Impact (%)
1.0
0.5
-0.5
-1.0
-1.5
-2.0
Boiler efficiency impact due to ambient conditions
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
2
1
-1
-2
-3
-4
Figure 6.100. Impact on boiler efficiency due to ambient conditions.
Boiler efficiency impact due to fuel
0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.101. Impact on boiler efficiency caused by fuel variation.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
219

Chapter 6
Figure 6.101 represents the evolution of the impact on boiler efficiency due to fuel
variation. It can be appreciated a long term evolution along with a short term dispersion
with an amplitude of about 1%. This impact has a quite high standard deviation (0.5515
%). However this figure is slightly lower than the standard deviation of the impacts
caused by the coal HHV (0.5744%). This is result can be explained because there is a
dependence among coal properties that compensate their impacts; when carbon or
hydrogen content increase (and boiler efficiency decrease) HHV also increase (and
boiler efficiency increase). All units consume the same coal, so that differences among
them are negligible (only caused by the possible use of natural gas).
Effect on boiler efficiency caused by cycle and cooling system (including
components and set-points) is negligible, with a standard deviation of only 0.0244%.
Evolution of this impact is plotted in Figure 6.102, where some peaks caused by the
dependence on cooling tower effectiveness with ambient conditions.
Impact (%)
220
0.20
0.15
0.10
0.05
-0.05
-0.10
-0.15
Boiler efficiency impact due to cycle
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.102. Impact on boiler efficiency caused by cycle and cooling system.
Unit 1
Unit 2
Unit 3
Figure 6.103 shows the impact caused by components and set-points of the boiler.
This aggregated impact has a standard deviation of 0.4968 %, which is higher than the
deviation of ambient conditions and lower than that of fuel. However, the graph is

Application of quantitative causality analysis.
characterized by point dispersion, which makes difficult to appreciate tendencies. Only
the lower efficiency during the first months, before the substitution of air preheaters,
and a seasonal oscillation during the first years are clearly visible. It should be noted
that heat transfer reduction is mainly caused by fouling, which has short time
characteristic times, compared to degradation of other components such as turbines.
Impact (%)
2.0
1.5
1.0
0.5
-0.5
-1.0
-1.5
-2.0
-2.5
Boiler efficiency impact due to components and set-points
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
Figure 6.103. Impact on boiler efficiency caused by components and set-points of the boiler.
Finally, in order to clarify the picture provided by Figure 6.103, it is advisable to
consider separately set-points and boiler components. Impact of set points has a
standard deviation of 0.3020 % and its evolution is plotted in Figure 6.104. In this
graph, point dispersion is low and a cyclic evolution can be appreciated, with better
efficiency in summer than in winter. This phenomenon appears because the set-points
variation in order to keep the average cold-side temperature of air preheaters above the
condensation point. An improvement in boiler operation through the six years can also
be appreciated.
Figure 6.105 shows the evolution of impact caused by boiler components. The main
characteristic of this graph is the seasonal oscillation appearing during the first three
years, which corresponds to the boiler aggregated efficiency variation when economizer
by-pass was used. Besides, there is big point dispersion, caused by short time fouling
221

Chapter 6
variation and responsible for dispersion appearing in Figure 6.103. This impact has a
standard deviation of 0.4672 %.
222
Impact (%)
Impact (%)
1.5
1.0
0.5
-0.5
-1.0
-1.5
Boiler efficiency impact due to set points
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
2.0
1.5
1.0
0.5
-0.5
-1.0
-1.5
-2.0
-2.5
Figure 6.104. Impact on boiler efficiency caused by boiler set-points.
Boiler efficiency impact due to boiler components
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.105. Impact on boiler efficiency caused by boiler components.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis.
6.5.3 Unit heat rate
After considering cycle heat rate and boiler efficiency separately, in this paragraph
unit heat rate, which comprises the other two, is studied. Since the influence of setpoints
and components separated by cycle, boiler and cooling system has been studied
previously, in this part a more general vision is shown and impact of set-points and
components efficiencies of all plants will be considered together.
The characteristic cyclic evolution of impact on unit heat rate caused by ambient
conditions is shown in Figure 6.106. It has a standard deviation of 0.03737, which is
lower than the standard deviation of cycle heat rate because impact on boiler efficiency
partially compensates impact on cycle.
Impact
0.08
0.06
0.04
0.02
-0.04
-0.06
-0.08
-0.10
-0.12
Unit heat rate impact due to ambient conditions
0.00
24/07/1998 0:00
-0.02
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.106. Impact on unit heat rate caused by ambient conditions.
Unit 1
Unit 2
Unit 3
Figure 6.107 represents the impact on unit heat rate caused by fuel variation. Since
the main influence of fuel is on the boiler, this graph is similar to Figure 6.101 but
inverted and with a scale change. Standard deviation of this impact is 0.02078. Impact
caused by set-points is plotted in Figure 6.108. It can be appreciated a slight cyclic
evolution and a decreasing tendency. However, importance of this family of variables is
not high, with a standard deviation of 0.01203.
223

Chapter 6
Impact
Impact
224
0.14
0.12
0.10
0.08
0.06
0.04
0.02
-0.04
-0.06
-0.08
Unit heat rate impact due to fuel
0.00
24/07/1998 0:00
-0.02
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
0.06
0.05
0.04
0.03
0.02
0.01
-0.02
-0.03
-0.04
-0.05
Figure 6.106. Impact on unit heat rate caused by fuel.
Unit heat rate impact due to set-points
0.00
24/07/1998 0:00
-0.01
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.107. Impact on unit heat rate caused by set-points.
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3

Application of quantitative causality analysis.
After removing the effects of ambient conditions, fuel and set points, only impact
due to components remains (neglecting residual term). Evolution of this impact is
plotted on Figure 6.108. In the first year, a big heat rate reduction appears due to
improvements made on boilers (preheaters replacement). Then, heat rate tends to
increase due to components degradation, although there is a reduction around 2003 or
2004. Besides, differences among units can be appreciated: unit 3 is the best in most
cases and unit 1 is the worst from 2002 to 2004 included. However, there is an
oscillating behaviour which makes difficult to see clearly the results. This effect appears
because of the dependence of the efficiency indicator chosen for the cooling tower with
the ambient conditions. Standard deviation of this impact is 0.05756.
Impact
0.3
0.2
0.1
-0.1
-0.2
-0.3
Unit heat rate impact due to components
0.0
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.108. Impact on unit heat rate caused by components.
Unit 1
Unit 2
Unit 3
6.5.4 Conclusion
In this section, it has been illustrated how the aggregation of impacts can be very
useful to provide a first decomposition of the variation of the global efficiency
indicators. Then, it is possible to decompose further up to all individual free diagnosis
variables. This zooming strategy can be made because impacts are additive and helps to
clarify the analysis, but it is not mandatory for the application of the diagnosis
procedure, because the methodology provides directly the impacts corresponding to all
225

Chapter 6
free diagnosis variables. In this point, the task of dividing the variation of the efficiency
indicators is finished: an exhaustive analysis of all free diagnosis variables and its
impact has been done, residual terms have been checked and proved to be low enough,
and impacts has been grouped in order to clarify the analysis. However, a problem
remains unsolved because there is a dependence of cooling tower efficiency with
ambient conditions that introduces oscillation. This question is tackled in the next
section.
6.6 Causality chains
In previous sections, it has been detected that a dependence of efficiency indicators
with other variables, such as ambient conditions, is an issue that makes the diagnosis
more difficult. The most relevant example of this is the dependence of cooling tower
effectiveness with ambient conditions. It should be highlighted that problem is not
caused by the diagnosis method itself but by the choice of the free diagnosis variables.
If another variable more linked to the physical behaviour of the equipment had been
chosen, this effect would have been avoided, but this variable would not correspond to a
widely accepted definition. A possibility to solve this task is the use of causality chains,
which is applied in this paragraph. Of course, the case of the cooling tower is not the
only example of crossed effects, however, is the only which makes more difficult the
clear interpretation of results. Obviously, the same solution could be made to other
situations in order to increase the precision of the analysis.
The concept of causality chains was developed in point 3.5. The idea is to share the
impact caused by a diagnosis variable into several impacts, one corresponding to the
intrinsic variation of this variable, and others induced by some of the rest of the free
diagnosis variables. To obtain this decomposition, a relation linking the variations of the
variable whose impact has to be decomposed to the other free diagnosis variables (or to
the other variables of the thermal representation of the system) is needed. This analysis
provides a causality chain because, after the decomposition of the global efficiency
indicators into a summation of impacts, each one corresponding to a free diagnosis
variable, each impact can be decomposed again into an intrinsic component and a
summation of induced components. This analysis is very useful to solve the dilemma
appearing in the definition of free diagnosis variables corresponding to efficiency
226

Application of quantitative causality analysis.
components. On one hand, it is preferable to use a definition widely accepted, easy to be
calculated from properties of flows entering and exiting the component and without the
need of tuning of complex models. On the other hand, it is interesting to choose a
variable very close to the physical behaviour of the components, in order to avoid
induced effects. When it is difficult to choose a variable able to accomplish with the two
conditions, the use of causality chains analysis can help to the diagnosis. The idea is to
choose a variable with widely accepted definition, easy to be calculated directly from
properties of flows entering and exiting the component, and then to use a more or less
sophisticated model to decompose this impact into intrinsic and induced components.
The choice of the complexity level of the model depends on the difficulty of the
modelling and the precision required in the task of filtering induced effects. In general, a
theoretical or experimental correlation linking the variable with other variables (mainly
ambient conditions) is enough. However, the developed theory allows to introduce a
model linking the efficiency indicator of the component to the variables of the
thermodynamic representation of the system, so that it is suitable to be used with any
model, whatever its complexity.
In this section, the approach is used to practically eliminate the effects induced by
ambient conditions in the cooling tower effectiveness. The model used is very simple
and links directly the cooling tower effectiveness to the ambient temperature. First, this
experimental correlation is determined by using the large amount of operation points
available. Then, the impact of the cooling tower effectiveness is decomposed in an
intrinsic term and in a term induced by ambient temperature. Finally, the evolution of
some aggregated impacts taking into account this new decomposition is shown.
6.6.1 Determination of a correlation for cooling tower effectiveness.
Figure 6.109 shows the close relation between cooling tower effectiveness and
ambient temperature for the three units, which is responsible for the seasonal oscillation
detected in the evolution of the tower efficiency indicators. Correlations for this
dependence for each one of the three units are also shown. As it can be seen, linear
regression is enough to achieve a good fitting of the points. Since lines adjusting the
three units are quite close, it is possible to use one correlation for all of them:
ε cooling _ tower = 0.007096⋅ Tamb
+ 0.3476 R 2 = 0.7821 6.9
227

Chapter 6
Cooling tower effectiveness
228
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
y = 0.0072x + 0.3421
R 2 = 0.8016
Cooling tower effectiveness and ambient temperature
y = 0.0069x + 0.3499
R 2 = 0.7735
y = 0.0073x + 0.3498
R 2 = 0.7875
0.25
0 5 10 15 20 25 30 35
Ambient temperature (ºC)
Unit 1
Unit 2
Unit 3
Lineal (Unit 1)
Lineal (Unit 2)
Lineal (Unit 3)
Figure 6.109. Dependence of cooling tower effectiveness with ambient temperature.
6.6.2 Decomposition of the impact of cooling tower effectiveness.
By using the correlation obtained above and the causality chains theory, the impact
caused by cooling tower can be decomposed into two terms: one intrinsic and the other
induced by ambient temperature. Figure 6.110 shows the intrinsic impact on cycle heat
rate due to cooling tower effectiveness. Although high dispersion is present due to the
high uncertainty in cooling water temperature measurements, it can be seen how the
oscillating behaviour has disappeared. Consequently, standard deviation of this impact
has dropped to 0.008906.
Impact on cycle heat rate caused by ambient temperature through cooling tower (in
other words, the induced part of the impact caused by cooling tower effectiveness), is
plotted in Figure 6.111. It can be appreciated the cyclic behaviour that made difficult the
diagnosis of the cooling tower by using directly its effectiveness. This impact has a
standard deviation of 0.01671.
Finally, figure 6.112 represents the total impact on cycle heat rate caused by ambient
temperature. It is calculated by adding the direct impact of ambient temperature plus the
impact of this variable through cooling tower effectiveness. This total impact has a
standard deviation of 0.03609.

Impact
0.06
0.05
0.04
0.03
0.02
0.01
-0.01
-0.02
Intrinsic cycle heat rate impact due to cooling tower
Application of quantitative causality analysis.
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Figure 6.110. Impact of cooling tower effectiveness on cycle heat rate, after filtering effects
induced by ambient temperature.
Impact
0.06
0.05
0.04
0.03
0.02
0.01
-0.01
-0.02
-0.03
Cycle heat rate impact due to ambient temperature through cooling tower
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
Unit 1
Unit 2
Unit 3
Figure 6.111. Impact on cycle heat rate due to cooling tower effectiveness variation induced
by ambient temperature.
229

Chapter 6
Impact
230
0.08
0.06
0.04
0.02
-0.04
-0.06
-0.08
-0.10
-0.12
Total cycle heat rate impact due to ambient temperature
0.00
24/07/1998 0:00
-0.02
06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
Figure 6.112. Total impact on cycle heat rate due to ambient temperature, including induced
effect on cooling tower effectiveness.
Although it is not illustrated here, this new decomposition also affects unit heat rate.
Intrinsic and induced impacts of cooling tower effectiveness on unit heat rate have
standard deviations of 0.01008 and 0.01895, respectively. Combined impact on unit
heat rate due to ambient temperature (both directly and through cooling tower
effectiveness) has a standard deviation of 0.02600.
6.6.3 Aggregated impacts
After the decomposition of impact caused by cooling tower in two terms, all
aggregated impacts including cooling tower effectiveness or ambient temperature vary.
Two examples are shown here. Figure 6.113 shows the aggregated impacts of cooling
system components on cycle heat rate, but using only the intrinsic component of cooling
tower effectiveness. Although high point dispersion is present, the oscillation appearing
in figure 6.98 is not present. Impact tends to increase during the first two years, then
decreases suddenly and tend to increase slowly again. Standard deviation of this
aggregated impact is 0.009020.

Impact
Impact
0.06
0.05
0.04
0.03
0.02
0.01
-0.01
-0.02
Intrinsic ycle heat rate impact due to cooling system
Application of quantitative causality analysis.
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
Figure 6.113. Impact caused by cooling system on cycle heat rate after filtering effects
induced by ambient temperature on cooling tower effectiveness.
0.25
0.20
0.15
0.10
0.05
-0.05
-0.10
-0.15
-0.20
Intrinsic unit heat rate impact due to components
0.00
24/07/1998 0:00 06/12/1999 0:00 19/04/2001 0:00 01/09/2002 0:00 14/01/2004 0:00 28/05/2005 0:00
Unit 1
Unit 2
Unit 3
Figure 6.114. Impact caused by plant components on unit heat rate, after filtering effects
induced by ambient temperature on cooling tower effectiveness.
231

Chapter 6
Aggregated impact on unit heat rate due to components but eliminating the induced
effect of ambient temperature on cooling tower effectiveness is plotted in figure 6.114.
Again, oscillations previously present (Figure 6.108) have been substantially reduced.
During the first year, a big reduction appears due to pre-heaters replacement. Then, a
steady increment is present due to component degradation, with reductions in spring
2003 and 2004, corresponding to maintenance operations. Standard deviation of this
aggregated impact is 0.04801.
6.7 Conclusion
6.7.1 Conclusions on the method
In this chapter, results obtained in the application of quantitative causality analysis
to the diagnosis of 6 years of operation of a three unit coal fired power plant have been
shown. First, evolution of the three global efficiency indicators selected (boiler
efficiency and cycle and unit heat rate) have been plotted. Then, all free diagnosis
variables have been studied: evolution of the variable and its impacts and relation
between variable and impact. This exhaustive exposition has demonstrated the
capability of the approach and its consistency. Results have been summarized by using
suitable parameters such as standard deviation of the impacts and impact factors.
Afterwards, residual term appearing in the decomposition of global efficiency
indicators into impacts has been studied. This indicator has demonstrated to be low
enough to ensure good accuracy of the procedure. Besides, its value is substantially
lower than the residual obtained by using constant impact factor, which justifies the
small additional computational effort required for the approach.
The use of aggregated impacts facilitates the interpretation of results by using a
zooming approach: from the aggregation of groups of variables to the detailed analysis.
It should be noted that this approach is not mandatory for the application of the
methodology, but an option to improve its usability.
Finally, the use of causality chains allows to decompose an impact into an intrinsic
term and a summation of terms induced by other free diagnosis variables. This
232

Application of quantitative causality analysis.
procedure has been successfully applied for the filtration of effects induced by ambient
conditions on cooling tower effectiveness.
To sum up, results provided in this chapter clearly demonstrate the applicability of
the diagnosis procedure proposed for the analysis of complex energy systems, using real
operation data available in every power plant.
6.7.2 Practical results about the working example
Besides its interest regarding the procedure, this chapter constitutes a true example
of diagnosis of a thermal system. So that, valuable information has been obtained,
which can be very useful for plant operators. The aim of this last section is to
summarize the main results presented throughout the chapter.
Unit heat rate was higher in the first months and then suddenly decreased. During
the first three years, it was higher in unit 2. In the two following years, the highest heat
rate was in unit 1 and the lowest in unit 3. Finally, the heat rate of unit 1 decreased
substantially in the last year (Figure 6.1). Evolution of cycle heat rate (Figure 6.2) has a
cyclic tendency, with higher values in summer and lower in winter. Besides, this
parameter shows the same differences among units as unit heat rate. Boiler efficiency
has also stationary evolution (higher values in summer) but in this parameter it is
difficult to appreciate differences among units.
6.7.2.1 Steam cycle
Ambient temperature originates variations of cycle heat rate as big as 0.15 units, if
effects induced on cooling tower are included (Figure 6.112). Relative humidity causes
a variation of more than 0.03 units (Figure 6.9). Influence of cycle components causes
variations of cycle heat rate of more than 0.10 units (Figure 6.99); impact increases
during the first years and decreases in springs of 1999 and 2004.
Isoentropic efficiency of high pressure of unit 1 is about 3 points higher than in unit
3, which implies an impact on cycle heat rate of -0.015 units (Figure 6.52). Maintenance
operations in the first medium pressure steam turbine in units 1 and 3 have originated
increments of 3 points in their isoentropic efficiency, which implies almost 0.0075
points of reduction of cycle heat rate (Figure 6.54). The turbine with highest influence
on cycle heat rate is the low pressure one, with differences as big as 0.15 points (Figure
6.58). Due to degradation, impact increases during the first years; then, unit 1 has the
highest impact and unit 3 the lowest. Afterwards, impacts of unit 2 and 1 decrease as
much as 0.05 units due to maintenance actions. It should be noted that, due to the lack
233

Chapter 6
of measurements, this variable does not include only the low pressure turbine itself but
also other aspects which have not been included in other variables. It would be very
interesting to have more information in order to further decompose these impacts.
Influence of set-points can originate a variation in cycle heat rate of almost 0.02
points, although no clear tendencies can be appreciated (Figure 6.96). A by-pass of 5 th
water heater of unit 2 was performed in 2004, which implied an impact on cycle heat
rate of 0.015 units.
6.7.2.2 Boiler
Ambient conditions (mainly ambient temperature) can originate a variation in boiler
efficiency of 2 points (Figure 6.100). In winter, losses increase and efficiency decreases,
and in summer losses decrease and efficiency increases. Variation of fuel quality causes
an impact with short term variation of 1 point and long term variation of 2 points
(Figure 6.101).
Substitution of air preheaters in the first months of 1999 originated a reduction in air
infiltration of around 20%, which caused an increment of boiler efficiency of more than
0.5% (Figure 6.86 and 6.87). Carbon in ashes ranges from almost 0 to more than 2 %,
and was lower during the first two years (Figure 6.87). Each point of carbon in ashes
represents almost 0.5 points of reduction of boiler efficiency (Figure 6.88). However, it
should be kept in mind that this variable actually depends on fuel quality and boiler
operation. For example, the set-point of oxygen in flue gases leaving the boiler, which
has varied from 1.5 to 2.5 %, and was higher during the first years (Figure 6.37). Each
additional point of oxygen content entails a reduction of 0.25 points in boiler efficiency
(Figure 6.38).
Average cold-side temperatures in preheaters vary from less than 100 to more than
130 ºC (Figures 6.39 and 6.41). Each degree causes an increment of unit heat rate of
0.001 units (secondary air preheaters, Figure 6.40) or 0.0003 units (primary air
preheaters, Figure 6.42), due to the use of high pressure steam. The amount of steam
used for sootblowing can reach up to 3 kg/s (Figure 6.43) and each kg/s causes an
increment of unit heat rate of 0.08 units (Figure 6.44). It should be noted that this flowrate
of steam is an average value which results of periods without sootblowing and
periods with high flow rate.
234

7 Application of quantitative causality
analysis to the quantification of intrinsic
and induced malfunctions.
In the previous chapter, the quantitative causality analysis method has proved its
ability to diagnose an actual power plant, with coherent results and negligible error. On
the other hand, thermoeconomic analysis provides tools and concepts which can be
more homogeneous in their formulation but do not always guarantee good results due to
the problem of induced malfunctions. In Chapter 4, a connection of the two approaches
has been formulated, based on the application of the quantitative causality analysis to
the decomposition of the unit energy consumptions and final plant products, which are
the free variables of the thermoeconomic model. The aim of this chapter is to prove the
practical application of these theoretical developments.
First, the chosen productive structure for the case of work is described. Afterwards,
the methodology is applied to an example of diagnosis. Finally, the correspondence of
the physical and thermoeconomic representations of the system is checked.
7.1 Description of the productive structure.
7.1.1 Simplified physical structure
The first step to define a productive structure is to have a physical representation,
which should be complete enough to take into account all the components of interest
and simple enough to avoid too complex definitions of fuels and products. This
simplified physical structure is represented in figure 7.1, while the flows are listed in
Table 7.1 and the components in Table 7.2. Some of these components are macro-
235

Chapter 7
components which include several physical devices. For example, the furnace and all
the heat exchangers of the boiler are included in only one component, and the low
pressure feeding water heaters, the deaerator and the turbo-pump are included in another
one. Besides, all flows of minor importance, such as steam for seals have not been
considered. Finally, since all the analysis has been made by using the gross power
production and gross heat rate, the electricity consumption of all ancillary devices
(pumps, fans and mills) has not been considered.
Flow Description
236
1 Air entering forced draft fans
2 Air exiting primary air fans
3 Air entering primary air coil heaters
4 Tempering air
5 Air from primary air coil heaters to primary air pre-heaters
6 Air exiting primary air pre-heaters
7 Hot air to sulphur removing unit
8 Primary air to boiler
9 Air from forced draft fans to secondary air coil heaters
10 Air from secondary air coil heaters to secondary air pre-heaters
11 Secondary air to boiler
12 Flue gases entering primary air pre-heaters
13 Flue gases exiting primary air pre-heaters
14 Flue gases entering secondary air pre-heaters
15 Flue gases exiting secondary air pre-heaters
16 Flue gases exiting induced draft fans
17 Steam to primary air coil heaters
18 Condensed steam exiting primary air coil heaters
19 Steam to secondary air coil heaters
20 Condensed steam exiting secondary air coil heaters
21 Flows leaving the boiler to deaerator
22 Coal entering the boiler
23 Natural gas entering the boiler
Table 7.1. A. Description of flows of the simplified physical structure.

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
Flow Description
24 Steam entering high pressure turbine
25 Steam exiting high pressure turbine
26 Steam to 6 th feeding water heater
27 Steam entering the reheater
28 Steam from the reheater to the 1 st intermediate pressure turbine
29 Steam exiting 1 st intermediate pressure turbine
30 Steam to 5 th feeding water heater
31 Steam entering 2 nd intermediate pressure turbine
32 Steam exiting 2 nd intermediate pressure turbine
33 Steam to turbo-pump
34 Steam to deaerator
35 Steam entering low pressure turbine
36 Steam to 3 rd feeding water heater
37 Steam to 2 nd feeding water heater
38 Steam to 1 st feeding water heater
39 Steam entering the condenser
40 Condensed water exiting the condenser
41 Water exiting the turbo-pump
42 Tempering water
43 Water entering 5 th water heater
44 Water from 5 th to the 6 th water heater
45 Water from 6 th water heater to boiler
46 Condensed water exiting 6 th water heater
47 Condensed water exiting 5 th water heater
48 Condensed water to condenser
49 Cooling water entering the condenser
50 Cooling water exiting the condenser
51 Power produced by high pressure turbine
52 Power produced by 1 st intermediate pressure turbine
53 Power produced by 2 nd intermediate pressure turbine
54 Power produced by low pressure turbine
Table 7.1. B. Description of flows of the simplified physical structure.
237

Chapter 7
238
24
27
28
H
26
51
45 44
O
46
52
53
I J K
25 29 31
32
30 33 34 36 37 38
42
N
43
47
41
Figure 7.1. Simplified physical diagram of the cycle.
22
23
7
8
6
11
D
13
12
G
F
Figure 7.2. Simplified physical diagram of the boiler.
35
M
18 20 21
10 9
E C
16
14
15
5
17
18
19
20
21
24
28
27
41
44
3 2
B A
4
1
L
39
40
54
48
50
49

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
Component Description
A Forced draft fans and primary air fans
B Primary air coil heaters
C Secondary air coil heaters
D Primary air pre-heaters
E Secondary air pre-heaters
F Boiler
G Induced draft fans
H High pressure steam turbine
I 1 st intermediate pressure steam turbine
J 2 nd intermediate pressure steam turbine
K Low pressure steam turbine
L Condenser
M Low pressure water heaters, deaerator and turbo-pump
N 5 th feeding water heater
O 6 th feeding water heater
Table 7.2. Description of components of the physical structure.
Some coments should be made in order to explain this physical structure. First, the
effects of induced draft fans on flue gases have been artificially translated to primary
and secondary air pre-heaters, so that component G is actually only a mixer. Second,
mechanical and electrical losses produced in the generator have been included in the
low pressure steam turbine, which means that the flow 54 is calculated by substracting
the power produced by that turbine minus mechanical and electrical losses
corresponding to all turbines.
Another important point is the reference for exergy calculation. It has been chosen
an ambient temperature, which obviously may vary from one example to another. The
first consequence of this decision is that exergy of flow 1 is always equal to zero. Other
consequences are commented in subsequent sections.
7.1.2 Productive structure
From the previous physical structure, a productive structure has been built. Since
forced and primary air fans are out of the scope of the diagnosis, the component A has
239

Chapter 7
been divided into two parts (primary and secondary air) which have been merged with
components B and C. As a result, 15 components are obtained (taking into account the
environment). They are described in Table 7.3.
There are two waste flows: flue gases leaving the 6 th component and exergy
increment of the cooling water. The cost of the first one is imputed to the boiler, while
the cost of the second is shared among feeding water heaters, boiler and turbines,
according to ratios proportional to the specific entropy generated in each component:
240
Δsi
ρi
=
Δs
tot
Flow 7 (hot air to desulfuration) may be considered as a product of the plant or as a
residue. However, since the aim of the sulphur removing unit is to reduce the SO2
emissions which are produced in the boiler, and its amount is low compared to other
residues, it has been considered as a productive flow which goes to the boiler.
The FPR table appears in Table 7.4, and the productive structure is plotted in Figure
7.3.
Component Description
0 Ambient
1 Primary air coil heaters (includes primary air fan and forced draft fan)
2 Secondary air coil heaters (includes forced draft fan)
3 Primary air pre-heaters (includes induced draft fan)
4 Secondary air pre-heaters (includes induced draft fan)
5 Boiler
6 Mixer of flue gases
7 High pressure steam turbine
8 1 st intermediate pressure steam turbine
9 2 nd intermediate pressure steam turbine
10 Low pressure steam turbine
11 Condenser
12 Low pressure feeding water heaters, deaerator and turbo-pump
13 5 th feeding water heater
14 6 th feeding water heater
Table 7.3. Components of the productive structure
7.1

F0 F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 R5 R7 R8 R9 R10 R12 R13 R14 TOTAL
P0 E22+E23 E22+E23
P1 E4+E5 E4+E5
P2 E10 E10
E7+E8
P3 E7+E8
-E5
-E5
P4 E11-E10 E11-E10
E26-
E30+E46
E39+E48
E31-
E28-
E24-
E13
E40
E14-
E12-E13
E19-
P5 E17-
(3)
E46
-E47
(2)
-E40
(1)
E23-
E30-
E26-
+E15
E15
E20
E18
E34-
E31
E27
E35
P6 E16 E16
P7 E51 E51
P8 E52 E52
P9 E53 E53
P10 E54 E54
P11 ρ5P11 ρ7P11 ρ8P11 ρ9P11 ρ10P11 ρ12P11 ρ13P11 ρ14P11 E50-E49
P12 E42+E43-
E42+E43-
E40
E40
P13 E44+E43 E44+E43
P14 E45+E44 E45+E44
ρ7P11 ρ8P11 ρ9P11 ρ10P11 ρ12P11 ρ13P11 ρ14P11
E16+
E26-
E30+E46
E39+E48
E31-
E28-
E13+E15 E24-
E12+E13 E14-
E19-
E17-
TOTAL E51+E52+
ρ5P11
E46
-E47
(2)
-E40
(1)
E23-
E30-
E26-
(4)
E15
E20
E18
E53+E54
E34-
E31
E27
E35
(1) = E35-E36- E37-E38- E39; (2) = E18+E20+E21+E33+E34+E36+E37+E38+ E47-E48; (3) = E12+E14+E17+E19+ E21+E24-E27+E28; (4) = E7+E8+E11+E22+ E23+E42 +E45
Table 7.4. FPR Table.
241

Chapter 7
242
B 0,5
B 5,1
B 5,2
B 5,3
B 5,4
B 5,6
1
2
3
4
5
6
B 1,5
B 2,5
B 3,5
B 4,5
R 6,5
B 5,11
11
B 5,7
Figure 7.3. Productive structure.
R 11,7
B 5,8
R 11,8
B 5,9
R 11,9
B 5,10
R 11,10
B 5,12
R 11,12
B 5,13
R 11,13
B 5,14
R 11,14
R 11,5
7
8
9
10
12
13
14
B 7,0
B 8,0
B 9,0
B 10,0
B 12,5
B 13,5
B 14,5

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
7.2 Example of thermoeconomic diagnosis
In this section, the productive structure described in section 7.1 and the methodology
developed in chapter 4 are applied to diagnose a real situation of the working example.
First of all, some preliminary results are reported (free diagnosis variables, exergy flows
and costs and unit exergy costs). Afterwards, the decomposition of unit exergy
consumption increment, final plant product variations and malfunctions into intrinsic
and induced is presented. Finally, malfunction costs are analyzed.
7.2.1 Preliminary results
Operation of unit 1 on April 26 th of 2004 has been selected as an example to be
diagnosed in comparison with the reference point used along all this work (unit 3 on
September 17 th 2003). This choice provides a diagnosis example where a lot of free
diagnosis variables change. The values of all these variables in the actual and reference
cases and their variations appear in Table 7.5.
Table 7.6 shows the values of the exergy and exergy costs of the productive flows,
both in the actual and in the reference state. Exergy and exergy costs of residues flows
appear in Table 7.7.
Unit exergy costs are listed in Table 7.8. It is worth to highlight the high value of the
products of components 1 and 2, due to the high irreversibility caused by using a high
quality flow (steam) to slightly increase the temperature of the air entering the
preheaters.
Finally, unit exergy consumptions appear in Table 7.9 (associated with productive
flows) and Table 7.10 (associated with residues).
243

Chapter 7
244
Description units Actual Reference Increment
1 Ambient temperature ºC 17.6 22.2 -4.6
2 Relative humidity % 60.1 58.1 2.0
3 Wind speed m/s 3.65 3.72 -0.07
4 Coal high heating value kJ/kg 16912 17587 675
5 Carbon mass fraction in coal % 42.09 44.55 -2.46
6 Hydrogen mass fraction in coal % 2.53 2.79 -0.26
7 Moisture mass fraction in coal % 19.65 17.53 2.12
8 Ash mass fraction in coal % 23.16 23.57 -0.41
9 Sulphur mass fraction in coal % 5.01 4.45 0.56
10 Nitrogen mass fraction in coal % 0.67 0.70 0.03
11 Energy provided by natural gas % 0.016 0.545 -0.529
12 Main steam temperature ºC 538.06 538.53 -0.47
13 Reheated steam temperature ºC 537.33 539.78 -2.45
14 Main steam pressure bar 157.51 156.91 0.60
15 Gross electric power MW 351.560 334.420 17.140
16 Oxygen in flue gases leaving the
boiler
17 Average cold-side temperature in
secondary air preheaters
18 Average cold-side temperature in
primary air preheaters
% 1.87 1.90 -0.03
ºC 109.9 116.8 -6.9
ºC 105.0 116.0 -11.0
19 Sootblowing steam flow rate kg/s 1.800 1.115 0.685
20 Air for flue gas desulfuration unit kg/s 0 3.228 -3.228
21 Tempering and primary air relation kg/kg 0.2786 0.3653 -0.0867
22 Primary air-coal ratio kg/kg 2.268 1.800 0.468
23 Temperature difference of flue
gases entering preheaters
ºC -0.28 -0.54 0.26
24 Fraction of flue gases through PAH - 0.252 0.216 0.036
25 High pressure turbine isoentropic
efficiency
% 81.46 81.50 -0.04
Table 7.5.A. Free diagnosis variables in the actual and reference states.

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
Description Unit Actual Reference Increment
26 Intermediate pressure 1 isoentropic
efficiency
27 Intermediate pressure 2 isoentropic
efficiency
% 85.15 83.48 1.67
% 82.84 86.56 -3.72
28 Low pressure isoentropic efficiency % 81.18 83.23 -2.05
29 Intermediate
coefficient
pressure 1 flow
30 Intermediate
coefficient
pressure 2 flow
kg 1/2 ·m 3/2 ·
s -1 ·bar -1/2
kg 1/2 ·m 3/2 ·
s -1 ·bar -1/2
31 Low pressure flow coefficient kg 1/2 ·m 3/2 ·
s -1 ·bar -1/2
12.623 13.553 -0.930
24.643 25.034 -0.391
52.672 53.358 -0.686
32 TTD 6 th water heater ºC -2.19 -1.76 -0.43
33 TTD 5 th water heater ºC 1.22 -0.64 1.86
34 TTD 3 rd water heater ºC -1.00 1.09 -2.09
35 TDCA 6 th water heater ºC 13.65 10.15 3.50
36 TDCA 5 th water heater ºC 0.00 0.00 0.00
37 Pressure drop in reheater bar 0.80 1.57 -0.77
38 Pressure increment in turbo-pump bar 181.83 171.48 10.35
39 Water losses kg/s 1.634 3.752 -2.118
40 Condenser effectiveness - 0.7293 0.6720 0.0573
41 Cooling water flow rate kg/s 10147.7 10496.5 -348.8
42 Cooling tower effectiveness - 0.4858 0.5463 -0.0605
43 Secondary
effectiveness
air preheater
- 0.8816 0.8305 0.0511
44 Primary air preheater effectiveness - 0.8449 0.8918 -0.0468
45 Aggregated boiler effectiveness - 0.8441 0.8515 -0.0074
46 Air infiltration in preheaters kg/kg 0.1493 0.1047 0.0446
47 Carbon in ashes % 2.29 0.90 1.39
Table 7.5.B. Free diagnosis variables in the actual and reference state.
245

Chapter 7
Flow
246
Actual Reference
Exergy Exergy cost Exergy Exergy cost
B0,5 1014825 1014825 951236 951236
B1,5 426.48 4164.5 236.37 2094.0
B2,5 295.28 3667.5 641.32 6785.4
B3,5 17539 50843 12630 36856
B4,5 41665 138813 40189 130434
B5,1 1710.7 4164.5 862.42 2094.0
B5,2 1506.5 3667.5 2794.6 6785.4
B5,3 20885 50843 15179 36856
B5,4 57021 138813 53719 130434
B5,5 2344.9 5708.4 1880.0 4564.8
B5,6 37914 92298 37432 90889
B5,7 108293 263630 106854 259448
B5,8 66722 162429 57306 139143
B5,9 64546 157132 62562 151904
B5,10 170480 415022 159118 186350
B5,11 62937 153215 59074 143437
B5,12 59951 145947 52577 127661
B5,13 23772 57870 22042 53518
B5,14 31736 77260 24874 60397
B7,0 96022 266212 94669 262096
B8,0 62414 163573 53209 14234
B9,0 58759 158867 58319 153188
B10,0 134365 426173 128223 395718
B12,5 52927 175597 46609 154728
B13,5 22446 66270 20727 61683
B14,5 29908 86284 23547 67923
Table 7.6. Exergy and exergy costs of the productive flows (kW).

Flow
Application of quantitative causality analysis to determine intrinsic and induced malfunctions
Actual Reference
Exergy Exergy cost Exergy Exergy cost
R6,5 37728 92298 37373 90889
R11,5 28637 89529 27504 86287
R11,7 825.81 2581.8 844.19 2648.5
R11,8 365.75 1143.5 347.91 1091.5
R11,9 555.04 1735.2 409.32 1284.2
R11,10 3567.1 11152 2986.0 9368.0
R11,12 9483.9 29650 8627.5 27067
R11,13 2686.8 8399.6 2602.5 8164.7
R11,14 2886.6 9024.5 2398.9 7526.2
Table 7.7. Exergy and exergy cost of waste flows (kW).
Unit cost Actual Reference Unit cost Actual Reference
*
k 1 1 *
P,0
P,8
*
k 9.7648 8.8590 *
P,1
P,9
*
k 12.420 10.580 *
P,2
P,10
*
k 2.8989 2.9181 *
P,3
P,11
*
k 3.3317 3.2455 *
P,4
P,12
*
k 2.4344 2.4281 *
P,5
P,13
*
k 2.4464 2.4319 *
P,6
P,14
k 2.7724 2.7685
*
P,7
Table 7.8. Unit exergy costs.
k 2.6208 2.6356
k 2.7037 2.6267
k 3.1718 3.0862
k 3.1263 3.1373
k 3.3177 3.3197
k 2.9525 2.9760
k 2.8849 2.8846
247

Chapter 7
248
Actual Reference Increment Actual Reference Increment
κ 1.4297 1.4494 -1.9751·10
0,5
-4
κ 6.0083·10
1,5
-4
3.6017·10 -4 2.4066·10 -4
κ 4.1599·10
2,5
-4 9.7722·10 -4 -5.6122·10 -4
κ 2.4709·10
3,5
-2 1.9245·10 -2 5.4635·10 -3
κ 5.8698·10
4,5
-2 6.1238·10 -2 -2.5400·10 -3
κ 4.0112 3.6486 3.6255·10
5,1
-1
κ 5.1020 4.3575 7.4446·10
5,2
-1
κ 1.1908 1.2018 -1.1045·10
5,3
-2
κ 1.3686 1.3367 3.1896·10
5,4
-2
κ 3.3035·10
5,5
-3 2.8647·10 -3 4.3878·10 -4
κ 1.0049 1.0016 3.3164·10
5,6
-3
κ 1.1278 1.1287 -9.1996·10
5,7
-4
κ 1.0690 1.0770 -7.9846·10
5,8
-3
κ 1.0985 1.0727 2.5738·10
5,9
-2
κ 1.2688 1.2409 2.7842·10
5,10
-2
κ 1.2842 1.2921 -7.8836·10
5,11
-3
κ 1.1327 1.1280 4.6674·10
5,12
-3
κ 1.0591 1.0634 -4.3422·10
5,13
-3
κ 1.0611 1.0564 4.7226·10
5,14
-3
κ 7.4565·10
12,5
-2 7.1021·10 -2 3.5439·10 -3
κ 3.1622·10
13,5
-2 3.1583·10 -2 3.8682·10 -5
κ 4.2135·10
14,5
-2 3.5879·10 -2 6.2561·10 -3
Table 7.9. Unit exergy consumptions.
Actual Reference Increment Actual Reference Increment
θ 5.3152·10
6,5
-2 5.6947·10 -2 -3.7946·10 -3
θ 4.0345·10
11,5
-2
4.1909·10 -2 -1.5640·10 -3
θ 8.6002·10
11,7
-3 8.9172·10 -3 -3.1699·10 -4
θ 5.8601·10
11,8
-3 6.5387·10 -3 -6.7855·10 -4
θ 9.4461·10
11,9
-3 7.0187·10 -3 2.4274·10 -3
θ 2.6548·10
11,10
-2 2.3288·10 -2 3.2603·10 -3
θ 1.7919·10
11,12
-1 1.8510·10 -1 -5.9175·10 -3
θ 1.1970·10
11,13
-1 1.2556·10 -1 -5.8574·10 -3
θ 9.6516·10
11,14
-2 1.0188·10 -1 -5.3653·10 -3
Table 7.10. Unit exergy consumptions associated with the residues.
7.2.2 Decomposition of unit exergy consumptions and plant
products.
Once some basic results of thermoeconomic analysis have been shown, the aim of
this section is to apply quantitative causality analysis to decompose the variations of the
independent variables of the thermoeconomic model (unit exergy consumptions and

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
final plant products) into a summation of terms corresponding to the free diagnosis
variables of the thermodynamic model.
The complete set of results including the effect of the 47 free diagnosis variables
would be too detailed, so that impacts are grouped into intrinsic and induced by ambient
conditions, fuel quality, set points and other components. The classification of the free
diagnosis variables into ambient conditions, fuel quality, set-points and component
parameters is clear and was made in the definition of the free diagnosis variables in
Chapter 5. However, the classification of component parameters into intrinsic and
induced depends on the component, so that it should be specified here.
Steam coil-heaters (components 1 and 2) have no free diagnosis variables considered
as intrinsic. In the case of air-preheaters (3 and 4), intrinsic variables are their
effectiveness and air infiltration. Variables associated with the boiler (5) are: aggregated
boiler effectiveness, unburned carbon in ashes, pressure drop in reheater, water losses
and pressure increment in turbo-pump. It should be noted that the last two variables
could be associated with more than one component, but for simplicity are assigned to
the boiler. There are no component variables associated with the flue gas mixer (6).
Free diagnosis variables corresponding to the turbines (7 to 10) are isoentropic
efficiencies and flow coefficients. The condenser (11) includes not only its effectiveness
but also the flow rate of cooling water and the effectiveness of the cooling tower.
Finally, the variables associated to the feeding water heaters (12 to 14) are the
corresponding temperature differences (TTD and TDCA).
Decomposition of the unit exergy consumptions of all the components is plotted in
Figure 7.4. Due to the high variation of unit exergy consumption of components 1 and
2, most of decompositions are difficult to see, so that a normalized version of Figure 7.4
is plotted in Figure 7.5. It should be noted that a residual term (‘error’) is included in
order to take into account the error produced by the quantitative causality analysis in the
decomposition. This term is negligible in most cases; perhaps, it may seem that error in
the normalized variation of κ5,6 is not very low, but it should be noted that the absolute
increment of this variable is negligible. Components where intrinsic effects can be
appreciated more clearly are the turbines (except the high pressure one, which has a
very low variation). The condenser can also be diagnosed easily but effect induced by
ambient conditions has to be filtered. The effect of ambient temperature is also
important in the coil heaters. Influence of fuel quality is important in all components
related to the boiler (1 to 6). Set-points have high influence in high pressure turbine,
249

Chapter 7
boiler, flue gas mixer and preheaters. Finally, the role of effects induced by other
components is important in feeding water heaters, high pressure turbine, boiler, flue gas
mixer and steam coil heaters.
Component
Component
250
k5_14
k5_13
k5_12
k5_11
k5_10
k5_9
k5_8
k5_7
k5_6
k_5
k5_4
k5_3
k5_2
k5_1
k ij decomposition
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Variation
Figure 7.4. Decomposition of unit exergy consumptions.
k ij decomposition
k5_14
k5_13
k5_12
k5_11
k5_10
k5_9
k5_8
k5_7
k5_6
k_5
k5_4
k5_3
k5_2
k5_1
-100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100%
Normalized variation
Figure 7.5. Normalized decomposition of unit exergy consumptions.
ambient
fuel
set-points
intrinsic
induced
error
ambient
fuel
set-points
intrinsic
induced
error

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
The detailed decomposition of all the unit exergy consumptions of the boiler (the
only component which has several productive fuels) appears in Figure 7.6. It can be
appreciated the high importance of induced effects, which can be justified by the low
detail of the thermoeconomic model of this section of the plant.
Figure 7.7 shows the decomposition of the unit exergy consumptions associated with
the residues. Most of the variation is due to ambient conditions and induced effects.
Importance of ambient conditions is due by their influence on cooling system and on the
exergy of flue gases (which would be higher if constant reference value were used).
Induced effects appear because residual flows not only depend on the component where
they are charged, but mainly on the component from which they come (condenser or
flue gases mixer).
Finally, plant product decomposition is plotted on Figure 7.8. It should be noted that
only the set point of gross power produced affects the total product of the plant
(summation of the product of the four turbines), while the other variables only affect the
distribution of this total power among the turbines. For example, the variation of
ambient conditions makes the power produced by the low pressure turbine to increase
and the power produced by the other turbines to decrease exactly the same amount.
Component
k ij decomposition
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02
Variation
k14_5
k13_5
k12_5
Figure 7.6. Decomposition of the unit exergy consumptions of boiler.
k5_5
k4_5
k3_5
k2_5
k1_5
k0_5
ambient
fuel
set-points
intrinsic
induced
error
251

Chapter 7
Component
Component
252
theta ij decomposition
theta11_14
theta11_13
theta11_12
theta11_10
theta11_9
theta11_8
theta11_7
theta11_5
theta6_5
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
Variation
Figure 7.7. Decomposition of unit exergy consumptions associated with wastes.
E10_0
E9_0
E8_0
E7_0
Plant product decomposition
-6000 -4000 -2000 0 2000 4000 6000 8000 10000 12000
Variation (kW)
Figure 7.8. Decomposition of plant products.
ambient
fuel
set-points
intrinsic
induced
error
ambient
fuel
set-points
intrinsic
induced
error

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
7.2.3 Decomposition of malfunctions.
The decomposition of unit exergy consumptions analyzed in the previous section is
the origin of the decomposition of malfunctions. It should be noted that, due to the
malfunction definition, the distribution of each malfunction among the different free
diagnosis causes, is exactly the same as the distribution of the corresponding unit
exergy consumption. However, the magnitude of malfunctions varies because of the
different value of the component’s product.
Figure 7.9 shows how the highest malfunction takes place in the boiler, and there are
also significant malfunctions in preheaters, in condenser and in some turbines. Besides,
induced effects are only important in the boiler. All malfunctions associated with the
boiler are detailed in Figure 7.10. Finally, Figure 7.11 shows the importance of ambient
conditions and induced effects in malfunctions associated with wastes; which justifies
the separate accounting of them.
Component
Malfunction decomposition
MF14
MF13
MF12
MF11
MF10
MF9
MF8
MF7
MF6
MF5
MF4
MF3
MF2
MF1
-20000 -15000 -10000 -5000 0 5000 10000 15000
Malfunction (kW)
Figure 7.9. Decomposition of malfunctions.
ambient
fuel
set-points
intrinsic
induced
error
253

Chapter 7
Component
254
Component
Malfunction decomposition (boiler)
MF14_5
MF13_5
MF12_5
MF5_5
MF4_5
MF3_5
MF2_5
MF1_5
MF0_5
-30000 -25000 -20000 -15000 -10000 -5000 0 5000 10000
Malfunction (kW)
Figure 7.10. Decomposition of malfunctions corresponding to the boiler.
Malfunction decomposition (residues)
MR11_14
MR11_13
MR11_12
MR11_10
MR11_9
MR11_8
MR11_7
MR11_5
MR6_5
-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000
Malfunction (kW)
Figure 7.11. Decomposition of malfunctions associated with the residues.
ambient
fuel
set-points
intrinsic
induced
error
ambient
fuel
set-points
intrinsic
induced
error

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
7.2.4 Malfunction cost decomposition. MFI and MFD tables.
Finally, malfunctions originate malfunction costs or, in other words, impact on fuel.
This magnitude is decomposed in Figure 7.12. In this graph, the fuel impact due to plant
product variation ( MF ) has also been considered, and it can be seen how this term is
*
0
the highest contribution to fuel demand increment. Again the decomposition into causes
is identical to the decomposition of unit exergy consumptions, except in the components
with more than one productive fuel (only the boiler). The decomposition of malfunction
cost of the boiler can be seen in Figure 7.13, and malfunction costs associated with the
residues are represented in Figure 7.14.
Information contained in Figure 7.12 and 7.14 can be seen numerically in the Table
of Intrinsic and Induced Malfunctions (MFI) (Table 7.11). Columns of this table
correspond to malfunction costs induced by ambient conditions, fuel quality and set
points, intrinsic malfunction costs, malfunction costs induced by other components and
malfunction costs associated with the error of the decomposition of unit exergy
consumptions variation. The rows correspond to the component; first, there are one row
for each component (included the environment), to take into account malfunction costs
associated with productive flows, and then, there are several rows to take into account
malfunction costs associated with the wastes.
The total summation of the table corresponds to the increment in the fuel of the
plant. Since the fuel impact formula is exact, due to the use of the column
corresponding to the error, the fuel impact obtained in the MFI table is also exact. This
fact can be easily checked by comparing the obtained result with the increment of fuel
(B05) which can be calculated from Table 7.6 (1014825 – 951236 = 63589 kW).
More detailed information can be seen in the Table of Malfunctions caused by the
Free Diagnosis variables (MFD), which takes into account the influence of the 47 free
diagnosis variables considered (Table 7.12). It should be noted that this decomposition
has to be performed also for the unit exergy consumptions, plant products and
malfunctions, but results have not been detailed in the text of this section.
255

Chapter 7
Component
Component
256
MF*14
MF*13
MF*12
MF*11
MF*10
MF*9
MF*8
MF*7
MF*6
MF*5
MF*4
MF*3
MF*2
MF*1
MF*0
Malfunction cost decomposition
-20000 -10000 0 10000 20000 30000 40000 50000 60000 70000
Malfunction cost (kW)
Figure 7.12. Decomposition of malfunction cost.
Malfunction cost decomposition (boiler)
MF*14_5
MF*13_5
MF*12_5
MF*5_5
MF*4_5
MF*3_5
MF*2_5
MF*1_5
MF*0_5
-30000 -25000 -20000 -15000 -10000 -5000 0 5000 10000 15000 20000
Malfunction cost (kW)
Figure 7.13. Decomposition of malfunction cost corresponding to boiler.
ambient
fuel
set-points
intrinsic
induced
error
ambient
fuel
set-points
intrinsic
induced
error

Component
Application of quantitative causality analysis to determine intrinsic and induced malfunctions
Malfunction cost decomposition (residues)
MR*11_14
MR*11_13
MR*11_12
MR*11_10
MR*11_9
MR*11_8
MR*11_7
MR*11_5
MR*6_5
-15000 -10000 -5000 0 5000 10000 15000
Malfunction cost (kW)
Figure 7.14. Decomposition of malfunction cost associated with wastes.
ambient
fuel
set-points
intrinsic
induced
error
257

Chapter 7
258
Ambient Fuel Set-points Intrinsic Induced Error Total
*
MF 1223.14 -22.659 49533.3 9650.78 -11836.4 -2.5492 48545.6
0
*
MF 222.653 -96.836 -35.981 0 120.975 -2.188 208.623
1
*
MF 1035.71 67.2392 166.959 0 -116.14 8.51615 1162.28
2
*
MF -3.6009 -1015.8 -1265.4 1814.81 120.651 9.79102 -339.605
3
*
MF 19.1812 2918.05 6252.61 -6613 407.857 135.862 3120.58
4
*
MF 2952.33 254.984 -6893 3276.61 8641.55 950.207 9182.66
5
*
MF -0.8266 91.0295 244.101 0 -130.852 98.2792 301.731
6
*
MF 48.5634 -0.3782 -163.21 65.3341 -171.924 9.59065 -212.019
7
*
MF -0.3597 0.87055 25.0896 -970.14 -55.5988 -34.133 -1034.27
8
*
MF -6.3476 0.59917 29.3314 3432.97 84.1649 113.402 3654.12
9
*
MF 82.5177 9.03462 -108.73 8768.67 96.2304 -157.01 8690.71
10
*
MF 7387.96 2.63096 41.8375 -7606.6 -343.452 -359.85 -877.461
11
*
MF 379.023 -5.3794 239.811 -660.8 601.261 -24.316 529.596
12
*
MF 2.05732 10.6732 -67.333 42.307 -204.739 -2.0657 -219.1
13
*
MF -7.8586 2.26523 23.019 1.88763 253.555 -2.1588 270.709
14
*
MR -11997.3 620.007 -1394.76 1714.93 2677.62 -921.647 -9301.17
5
*
MR -375.99 -2.8782 23.9363 6.61792 246.967 7.53239 -93.8177
7
*
MR -152.38 -0.643 27.752 -126.26 128.89 9.76867 -112.874
8
*
MR -222.81 -0.4341 43.3401 425.302 190.28 6.88719 442.566
9
*
MR -1416.2 -1.8208 916.273 467.459 1437.15 -95.897 1306.93
10
*
MR -5114.4 23.3432 752.908 -55.497 3346.96 184.464 -862.268
12
*
MR -1141 -38.768 3.03088 47.1471 723.605 26.3784 -379.557
13
*
MR -1045.4 -35.944 2.33185 -15.851 675.817 24.1149 -394.96
14
Total -8131.4 2779.15 48397.2 13666.7 6894.39 -17.024 63589
Table 7.11. Table of intrinsic and induced malfunctions (MFI).

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
1 2 3 4 5 6 7 8 9 10 11 12
MF 1304.656 -81.3984 -0.11719 181.5281 -211.646 -87.8132 79.66839 -10.6376 23.57984 -0.70412 3.366404 12.52364
MF 222.6118 0.042635 -0.00149 2.145792 -94.903 -30.0754 27.24925 -4.31793 10.9898 -0.31858 -7.6056 0.00714
MF 1035.894 -0.19438 0.00678 -10.4247 78.99232 17.97042 -16.2873 3.56597 -10.3787 0.264946 3.536239 -0.03255
MF -3.92788 0.338766 -0.01182 16.64351 -988.771 -314.126 284.6037 -44.908 114.1737 -3.31304 -80.1475 0.056729
MF 19.6353 -0.47047 0.016409 -21.9582 2970.643 602.0518 -545.464 131.3773 -395.927 9.943224 167.3882 -0.07878
MF 2983.942 -5.28968 -26.3177 3806.129 -3105.17 -5576.91 5077.544 -541.922 962.2613 -26.6761 -340.272 85.4972
MF -0.79334 -0.03448 0.001203 -1.73837 90.15011 20.12212 -18.2314 3.440256 -8.97071 0.242391 6.015195 -0.00577
MF 51.5227 -2.95721 -0.0021 3.250444 -3.76013 -1.57024 1.424599 -0.1892 0.417655 -0.01251 0.061131 6.048637
MF -0.41234 0.047956 0.004674 -7.24017 8.405663 3.499798 -3.17519 0.422738 -0.93496 0.027963 -0.1353 -0.10013
MF -6.76778 0.417618 0.002585 -4.00448 4.775935 1.944898 -1.7645 0.239274 -0.53667 0.015893 -0.07118 -0.02184
MF 87.93334 -5.45626 0.040631 -62.9369 74.64949 30.53738 -27.7049 3.742832 -8.37112 0.248393 -1.13061 -6.1898
MF 7876.316 -488.364 0.0098 -15.1803 18.4923 7.400846 -6.71433 0.923733 -2.09417 0.061549 -0.2587 -0.41862
MF 405.3078 -26.2784 -0.00599 9.270373 -15.3591 -4.81416 4.367227 -0.73919 1.90572 -0.05125 0.041031 1.461923
MF 1.976032 0.024608 0.056674 -87.7888 102.0474 42.44509 -38.5083 5.13125 -11.3561 0.33949 -1.63685 -1.71643
*
0
*
1
*
2
*
3
*
4
*
5
*
6
*
7
*
8
*
9
*
10
*
11
*
12
*
13
*
14
*
5
*
7
*
8
MF -8.45582 0.584989 0.01224 -18.9597 21.99616 9.163727 -8.31379 1.106343 -2.44596 0.073175 -0.35475 -4.875
MR -12796.9 801.9109 -2.32584 3606.894 -3014.69 -129.206 119.2644 0.979112 -237.156 5.221115 268.7008 30.73837
MR -400.844 24.86334 -0.01189 18.41892 -22.0991 -8.95526 8.124593 -1.10623 2.488774 -0.07354 0.323622 1.859251
MR -162.472 10.0912 -0.00202 3.127595 -3.91872 -1.53268 1.390494 -0.195 0.448227 -0.01305 0.050171 0.090507
MR -237.576 14.76712 -0.0003 0.468611 -0.95113 -0.25601 0.232229 -0.04489 0.12327 -0.00318 -0.00295 0.197701
MR -1510.07 93.82978 0.0013 -2.01559 0.116428 0.813217 -0.73799 0.021942 0.082511 0.000312 -0.10162 3.306296
MR -5442.63 328.3106 -0.11961 185.3054 -165.962 -86.0118 78.03864 -8.70231 16.34891 -0.55045 4.877077 4.870434
MR -1215.31 74.55702 -0.19923 308.6056 -360.075 -149.306 135.4572 -18.0959 40.12789 -1.19794 5.715356 4.722556
MR -1113.55 68.30965 -0.1846 285.9528 -333.667 -138.348 125.5156 -16.7686 37.1859 -1.11008 5.295162 4.382153
*
9
*
11
*
12
*
13
*
14
Total -8909.93 807.6526 -29.1475 8195.493 -4950.7 -5792.97 5275.979 -496.676 531.9625 -17.5854 33.65363 142.3236
Table 7.12.A. Table of malfunctions induced by the free diagnosis variables (MFD).
259

Chapter 7
13 14 15 16 17 18 19 20 21 22 23 24
*
MF -42.0355 -23.8973 49588.01 -5.30926 21.74434 9.913834 -32.8979 5.901595 -21.9988 18.03994 0.252517 3.072405
0
*
MF 0.026378 -0.01539 2.463902 -2.42385 -0.11868 119.1511 0.090475 59.99041 -93.448 -390.695 0.324015 268.6661
1
*
MF -0.12026 0.070171 -11.2333 2.076838 80.27113 0.241327 -0.41249 0.148208 0.390187 165.9861 -0.13199 -70.2948
2
*
MF 0.209588 -0.12229 19.57739 -25.2085 -0.94301 570.9798 0.718888 658.7732 -1188.48 -4108.78 1.721405 2806.082
3
*
MF -0.29107 0.169837 -27.1883 78.25897 1856.005 0.584089 -0.99836 0.358712 0.944379 7466.796 -1.96794 -3119.98
4
*
MF 373.4518 -229.607 -774.065 -177.842 -2850.22 -1290.75 1988.493 -2294.78 1563.445 -2896.81 -0.63453 -389.193
5
*
MF -0.02133 0.012449 -1.99286 1.972558 -23.5978 58.61209 -0.07318 -41.9616 75.74553 328.3632 -0.27526 -152.677
6
*
MF -3.75002 15.13221 -180.636 -0.09425 0.465197 0.211869 -0.69289 0.126315 -0.43111 0.345412 0.004502 0.065508
7
*
MF 24.06949 0.011674 1.926221 0.210775 -0.95883 -0.43688 0.597043 -0.2603 0.922323 -0.74654 -0.01005 -0.13521
8
*
MF 22.18058 -0.03826 6.419347 0.120062 -0.20551 -0.09452 1.028783 -0.05557 0.350807 -0.31702 -0.00564 -0.02984
9
*
MF 39.55719 12.73926 -151.637 1.875653 -4.28527 -1.96349 2.297685 -1.16061 6.031169 -5.2941 -0.08839 -0.61504
10
*
MF 79.70012 0.599622 -48.7041 0.465783 0.213486 0.091267 9.893751 0.059468 0.843001 -0.90879 -0.02164 0.024205
11
*
MF -23.5673 -2.49211 251.4586 -0.39615 -10.5436 -4.77231 30.89781 -2.87038 4.59297 -2.51895 0.015926 -1.45559
12
*
MF -28.4439 0.341022 -28.682 2.559178 -11.3017 -5.15038 7.777932 -3.06795 11.02429 -8.95628 -0.12192 -1.59458
13
*
MF 1.279835 -6.30119 35.12431 0.551524 -2.5508 -1.16214 1.562717 -0.69252 2.43486 -1.96675 -0.0263 -0.3596
14
*
MR -79.2753 -53.4627 314.2242 15.62484 -1786.54 -792.222 -54.6493 767.3913 -1608.6 2067.027 -0.95624 -214.067
5
*
MR -2.42945 1.695775 26.49096 -0.55586 0.607801 0.281886 -4.205 0.163761 -1.44804 1.358567 0.026037 0.090569
7
*
MR 1.979683 -0.40591 28.01453 -0.09895 -0.32248 -0.14495 -1.30139 -0.08805 -0.03708 0.105028 0.004532 -0.04352
8
*
MR 3.890206 -0.44495 42.84743 -0.02483 -0.98046 -0.44392 -1.49184 -0.26688 0.45167 -0.25943 0.000921 -0.1355
9
*
MR 169.3666 -7.60226 763.9547 -0.00241 -5.96152 -2.70094 -2.79989 -1.6223 3.05001 -1.88885 -0.00132 -0.82556
10
*
MR -18.0079 -8.82312 637.6214 -4.04369 150.4698 68.22023 -110.349 40.93513 -85.3755 56.2809 0.224417 20.88462
12
*
MR -2.68788 -0.91305 -19.07 -9.0333 36.28284 16.54437 -31.623 9.846951 -37.0636 30.46691 0.429467 5.128653
13
*
MR -3.46628 -0.85009 -16.9174 -8.37085 33.56043 15.30317 -29.4584 9.108059 -34.314 28.21307 0.397958 4.744011
14
Total 511.6153 -304.203 50458 -129.688 -2518.91 -1239.71 1772.406 -794.024 -1400.96 2743.842 -0.83952 -842.652
Table 7.12.B. Table of malfunctions induced by the free diagnosis variables (MFD).
260

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
25 26 27 28 29 30 31 32 33 34 35 36
*
MF 6.273276 -434.192 889.8358 -939.518 -1498.4 69.78665 360.9716 -15.579 121.4205 190.0653 24.815 0
0
*
MF 0.004045 -0.08998 0.233577 0.484826 -0.11091 0.000574 0.003017 -0.00886 0.018564 -0.04422 0.003794 0
1
*
MF -0.01844 0.410234 -1.06492 -2.2104 0.505657 -0.00261 -0.01375 0.040383 -0.08464 0.20162 -0.0173 0
2
*
MF 0.032138 -0.71495 1.855927 3.852272 -0.88126 0.004557 0.023972 -0.07038 0.147507 -0.35138 0.030146 0
3
*
MF -0.04463 0.9929 -2.57744 -5.34989 1.223853 -0.00633 -0.03329 0.09774 -0.20485 0.487985 -0.04187 0
4
*
MF 1.724259 13.76189 -43.2129 -173.263 1242.975 34.63315 106.7058 136.868 -79.8802 40.3699 -2.70955 0
5
*
MF -0.00327 0.072778 -0.18892 -0.39214 0.089706 -0.00046 -0.00244 0.007164 -0.01502 0.035769 -0.00307 0
6
*
MF 65.3341 7.004463 -17.3257 -35.5445 -158.086 -3.31513 -0.75549 0.179674 5.688694 3.090142 1.162612 0
7
*
MF -0.0013 -1168.28 -0.82887 0.376965 198.1439 -41.3691 -2.67998 0.019077 -9.45699 -0.79757 -1.93275 0
8
*
MF 0.012709 21.0578 3409.263 1.23515 101.7835 23.70569 -36.4824 0.063217 -4.78133 -10.4949 -0.97717 0
9
*
MF -3.3483 112.5414 -302.978 8922.126 355.3157 5.062669 -153.452 7.150255 -28.2666 -142.379 -5.77692 0
10
*
MF -0.15116 81.22939 -231.745 -518.419 372.6457 7.21368 -10.0845 0.194641 -18.3256 2.487351 -3.74524 0
11
*
MF 0.69124 -43.9034 126.8996 40.07616 -130.146 -4.28096 3.614106 0.487232 14.31681 -660.804 2.925962 0
12
*
MF -0.11648 -32.6147 -27.9251 -5.69511 -159.077 99.84244 -66.8793 -0.90604 42.30697 -18.4991 -5.56701 0
13
*
MF 0.833511 4.023239 2.175287 6.909105 245.9394 -47.9819 -3.01856 -31.2331 103.7325 -1.50129 33.12075 0
14
*
MR -0.0825 -180.34 480.9833 1013.749 -2763.5 -48.615 -13.504 -202.914 145.9169 -97.2432 29.82139 0
5
*
MR 6.61792 -6.26146 16.49369 34.50674 -79.6634 -0.89133 -0.1364 -2.72778 3.326594 -3.22088 0.679864 0
7
*
MR 0.092597 -136.463 7.858294 16.52252 10.20462 -4.73568 -0.30417 -0.1243 3.092182 -1.66428 0.631957 0
8
*
MR 0.118034 1.118525 426.1643 24.46143 15.57404 -0.86244 -5.22309 -0.11813 -0.43154 -3.82089 -0.0882 0
9
*
MR 2.023727 104.9565 -311.954 496.0681 674.7338 14.1166 -28.609 -2.8156 -25.005 -54.1859 -5.11033 0
10
*
MR 2.112628 -79.2926 205.1851 471.1814 -459.946 -0.71841 -55.6019 -31.9603 19.81495 -55.4975 4.049629 0
12
*
MR 0.198116 -11.0485 33.74628 81.39292 -136.372 -20.3892 -15.062 -11.3096 47.14715 -12.1739 1.164193 0
13
*
MR 0.18518 -12.7568 37.28025 78.64327 -203.423 -15.7952 -1.13587 -17.6908 37.25888 -7.56157 1.839978 0
14
Total 82.4874 -1758.79 4698.173 9511.194 -2370.47 65.40217 78.3398 -172.351 377.7364 -833.502 74.27587 0
Table 7.12.C. Table of malfunctions induced by the free diagnosis variables (MFD).
261

Chapter 7
37 38 39 40 41 42 43 44 45 46 47 Error Total
*
MF -108.784 -282.609 100.4448 411.0331 -272.868 -753.927 -52.031 17.28126 -37.6365 -12.5834 30.53384 -2.54920 48545.6
0
*
MF -0.07332 0.075883 -0.11489 -0.20379 0.135285 0.373788 -0.23687 98.75705 21.29866 0.080876 0.387442 -2.18796 208.623
1
*
MF 0.334281 -0.34596 0.5238 0.929092 -0.61678 -1.70416 -84.4591 -0.32514 27.92661 -54.3816 -1.76641 8.51615 1162.28
2
*
MF -0.58258 0.60294 -0.91287 -1.61921 1.074928 2.970003 -1.88213 1814.808 113.3507 0.642614 3.078493 9.79102 -339.605
3
*
MF 0.809067 -0.83734 1.267765 2.248702 -1.49282 -4.12462 -7738.93 -0.78695 420.5048 1125.941 -4.27529 135.862 3120.58
4
*
MF -1668.49 667.9024 -3108 222.2448 -147.539 -407.647 8148.206 -2283.06 528.1301 1831.378 6857.057 950.207 9182.66
5
*
MF 0.059303 -0.06138 0.092925 0.164826 -0.10942 -0.30233 -38.3438 -65.7777 -1.46109 -24.4 -0.31337 98.2792 301.731
6
*
MF 53.2862 -6.03813 2.118769 14.91682 -9.90265 -27.3608 -1.01303 0.337877 -0.64359 -0.27155 0.546736 9.59065 -212.019
7
*
MF -0.02789 0.226712 -1.79539 -0.27639 0.183481 0.506953 2.173464 -0.72359 1.464501 0.5577 -1.21782 -34.1328 -1034.27
8
*
MF -0.47473 12.24468 -3.15484 -2.12357 1.40975 3.895107 0.853662 -0.27842 0.939889 0.110455 -0.67358 113.402 3654.12
9
*
MF 61.16528 69.26361 -6.70373 27.19185 -18.0515 -49.876 14.5489 -4.77152 14.34987 2.379321 -10.5864 -157.013 8690.71
10
*
MF 3.083112 -0.05152 -30.4926 -2893.12 -199.868 -4513.61 2.17127 -0.68327 3.959853 -0.18634 -2.55347 -359.851 -877.461
11
*
MF -17.792 894.4639 -95.4524 132.4601 -87.9347 -242.962 9.845576 -3.48734 -6.58229 6.461725 1.559772 -24.3156 529.596
12
*
MF 1.819416 6.460572 -23.4371 -0.5452 0.361934 1.000015 26.00581 -8.65206 17.88714 6.564521 -14.7664 -2.06573 -219.100
13
*
MF -65.4128 2.063589 -4.69888 -3.03869 2.017262 5.57365 5.734457 -1.90982 3.819091 1.484785 -3.18909 -2.1588 270.709
14
*
MR 221.8016 106.0253 366.8655 -873.315 1827.436 7380.008 -4374.55 1365.797 414.24 -1012.03 605.9958 -921-647 -9301.17
5
*
MR 14.95033 2.560436 12.88528 -28.2537 57.15216 229.6389 -3.56446 1.154072 -4.45804 -0.30233 3.098206 -1278.51 -93.8177
7
*
MR -1.88286 1.464764 4.007028 -12.4797 23.47555 93.24106 -0.14857 0.036347 -0.92722 0.208163 0.526103 356.862 -112.874
8
*
MR -2.51035 3.417351 4.616602 -18.5553 34.43957 136.482 0.977766 -0.34408 -0.51167 0.599428 0.078863 7.53239 442.566
9
*
MR -40.9752 85.52718 8.717301 -99.9451 214.0292 867.2455 6.714348 -2.33672 -1.86946 3.626693 -0.33881 6.88719 1306.93
10
*
MR 33.72449 -129.898 338.6576 -663.796 832.634 3032.801 -190.728 65.73884 12.87438 -91.0408 31.1642 -95.8971 -862.268
12
*
MR 25.4445 6.487937 95.65031 -77.5644 168.982 686.3822 -87.7217 29.1226 -64.257 -20.9757 51.90887 184.464 -379.557
13
*
MR 42.47164 4.43736 89.11387 -71.5708 154.9518 628.8398 -81.2192 26.96272 -59.564 -19.3999 48.09856 26.3784 -394.960
14
Total -1448.05 1443.384 -2249.8 -3935.21 2579.9 7067.449 -4437.59 1046.856 1402.835 1744.465 7594.353 24.1149 63588.99
Table 7.12.D. Table of malfunctions induced by the free diagnosis variables (MFD)
262

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
7.3 General quantification of intrinsic and induced effects.
In the previous section, the capability of the approach to perform a detailed
thermoeconomic diagnosis of a thermal system has been demonstrated in a practical
example.
Besides, the importance of induced effects has been quantified, which is a very
important result for assessing the suitability of a given thermoeconomic model to be
used for diagnosis. However, it should be noted that it was only an example of only one
situation; so that, results can only be seen as indicative. In order to test the accuracy of
the chosen productive structure, information of a set of situations as wide as possible
should be used.
In this section, a ‘representative’ example has been developed in order to serve as a
general test. This example is a fictitious situation where the reference state is the same
as in all previous analysis, and the work state is built by adding to the reference state an
increment of all free diagnosis variables equal to their standard deviation (which can be
seen in Table 6.1). This approach enables to take into account the typical variations of
all variables. Since it is a fictitious situation, the derivatives are calculated in the
reference state and it has no sense to consider residuals.
7.3.1 Decomposition of unit exergy consumptions and plant
products.
Decomposition of unit exergy consumptions is plotted in Figure 7.15, where it can
be seen how the highest variations correspond to the coil heaters, characterized by the
high influence of ambient conditions. In order to see the decomposition of the other unit
exergy consumptions, these two cases have been removed from Figure 7.16: ambient
conditions have an important influence on the condenser, turbines have negligible
induced effects, fuel impact is relevant in the boiler and in the preheaters and set points
affect mainly the secondary air preheater.
A decomposition of all unit exergy consumptions of the boiler is plotted in figure
7.17, where it can be seen how they are affected by all groups of variables. Figure 7.18
shows how the unit exergy consumptions associated with the residues vary mainly due
to ambient conditions and induced effects. Finally, Figure 7.19 shows the
decomposition of the different components of plant product. Although the total
263

Chapter 7
summation only depends on the corresponding set-point, the distribution among the four
turbines is affected by all variables. At this point it should be reminded that the analysis
developed in this work only considers the full load range of the plant operation (> 320
MW); in other situation, this variation would be higher.
Component
Component
264
k ij decomposition
-1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
Impact
k5_14
k5_13
k5_12
k5_11
k5_10
k5_9
k5_8
k5_7
k5_6
k5_4
k5_3
k5_2
k5_1
Figure 7.15 Decomposition of unit exergy consumptions.
k ij decomposition
-0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06
Impact
k5_14
k5_13
k5_12
k5_11
k5_10
k5_9
k5_8
k5_7
k5_6
k5_4
k5_3
Figure 7.16. Decomposition of unit exergy consumptions.
k_5
k_5
ambient
fuel
set points
intrinsic
induced
ambient
fuel
set points
intrinsic
induced

Component
Component
Application of quantitative causality analysis to determine intrinsic and induced malfunctions
k14_5
k13_5
k12_5
k5_5
k4_5
k3_5
k2_5
k1_5
k0_5
k ij decomposition (boiler)
-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Figure 7.17. Decomposition of unit exergy consumption associated with the boiler.
theta11_14
theta11_13
theta11_12
theta11_10
theta11_9
theta11_8
theta11_7
theta11_5
theta6_5
Variation
theta ij decomposition
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Variation
ambient
fuel
set points
intrinsic
induced
ambient
fuel
set points
intrinsic
induced
Figure 7.18. Decomposition of unit exergy consumptions associated with the residues.
265

Chapter 7
Component
266
Plant product decomposition
E10_0
E9_0
E8_0
E7_0
-6000 -4000 -2000 0 2000 4000 6000 8000
Variation (kW)
Figure 7.19. Decomposition of plant product.
ambient
fuel
set points
intrinsic
induced
7.3.2 Decomposition of malfunctions.
Figure 7.20 shows the decomposition of malfunctions and how the high variation of
unit exergy consumptions of the coil heaters has become a low malfunction, due to the
reduced value of the product of these components. On the other hand, malfunction of
the boiler is clearly the highest. Other components with high value of the malfunctions
are the preheaters, the low pressure turbine and the condenser. It should be reminded
that the decomposition of malfunctions is, by definition, exactly the same as that of the
unit exergy consumptions.
Separate analysis of the different contributions of the boiler malfunctions is
performed in Figure 7.21. Besides, Figure 7.22 shows the malfunctions associated with
the residues, caused mainly by ambient conditions and induced by other components.

Component
Component
Application of quantitative causality analysis to determine intrinsic and induced malfunctions
MF14
MF13
MF12
MF11
MF10
MF9
MF8
MF7
MF6
MF5
MF4
MF3
MF2
MF1
Malfunction decomposition
-10000 -5000 0 5000 10000 15000 20000
Figure 7.20. Decompositions of malfunctions.
MF14_5
MF13_5
MF12_5
Malfunction (kW)
Malfunction decomposition (boiler)
MF5_5
MF4_5
MF3_5
MF2_5
MF1_5
MF0_5
-10000 -5000 0 5000 10000 15000 20000
Malfunction (kW)
Figure 7.21. Decomposition of malfunctions associated with the boiler.
ambient
fuel
set points
intrinsic
induced
ambient
fuel
set points
intrinsic
induced
267

Chapter 7
268
Component
MR11_14
MR11_13
MR11_12
MR11_10
MR11_9
MR11_8
MR11_7
MR11_5
MR6_5
Malfunction decomposition (residues)
-4000 -2000 0 2000 4000 6000 8000 10000
Malfunction (kW)
Figure 7.22. Decomposition of malfunctions associated with the residues.
7.3.3 Decomposition of malfunction costs.
Malfunctions presented in the previous sections originate malfunction costs, or
impacts on fuel consumed by the plant. These malfunction costs are summarized in
Figure 7.23, where the impact caused by variation of plant product ( MF ) has also been
included. The main malfunction costs are caused by variations in final product, boiler,
secondary air preheater, low pressure turbine and condenser. Effects induced by other
components are low, except in the boiler and in the variation of plant product.
Figure 7.24 shows the detailed decomposition of malfunction cost associated with
the boiler, and Figure 7.25 plots the decomposition of malfunction cost associated with
the residues. Importance of induced effects in the last figure justifies the use of the
residues analysis to reduce induced effects in thermoeconomic diagnosis.
*
0
ambient
fuel
set points
intrinsic
induced

Component
Component
Application of quantitative causality analysis to determine intrinsic and induced malfunctions
Malfunction cost decomposition
MF*14
MF*13
MF*12
MF*11
MF*10
MF*9
MF*8
MF*7
MF*6
MF*5
MF*4
MF*3
MF*2
MF*1
MF*0
-30000 -20000 -10000 0 10000 20000 30000 40000
Malfunction cost (kW)
Figure 7.23. Decomposition of malfunction costs.
Malfunction cost decomposition (boiler)
MF*14_5
MF*13_5
MF*12_5
MF*5_5
MF*4_5
MF*3_5
MF*2_5
MF*1_5
MF*0_5
-25000 -20000 -15000 -10000 -5000 0 5000 10000 15000 20000 25000
Malfunction cost (kW)
Figure 7.24. Decomposition of malfunction cost associated with the boiler
ambient
fuel
set points
intrinsic
induced
ambient
fuel
set points
intrinsic
induced
269

Chapter 7
component
270
Malfunction cost decomposition (residues)
MR*11_14
MR*11_13
MR*11_12
MR*11_10
MR*11_9
MR*11_8
MR*11_7
MR*11_5
MR*6_5
-15000 -10000 -5000 0 5000 10000 15000 20000 25000 30000
Malfunction cost (kW)
Figure 7.25. Decomposition of malfunction costs associated with the residues.
ambient
fuel
set points
intrinsic
induced
Results presented up to now show the decomposition of malfunction cost into terms
caused by the several groups of free diagnosis variables (ambient, fuel, set-points,
intrinsic and induced). This has allowed us to determine the relative importance of all of
them and to evaluate the suitability of the thermoeconomic model to keep reduced
induced effects. In other words, we have analyzed the MFD and MFI tables ‘row by
row’.
However, to have a complete picture, it is interesting to analyze these tables also ‘by
columns’; that is, to analyze the influence of a given free diagnosis variable
characterizing the behaviour of a component on the malfunction cost of that component
and of malfunction costs of the other components. Productive and residual flows should
be also considered separately. This task is performed in Figures 7.26 and 7.27.
In the case of isoentropic efficiencies of the turbines, impact assigned to the
corresponding components match fairly well the total impact. In the case of flow
coefficients and parameters of feeding water pre-heaters this correspondence is not very
good, although influence of this variables is low (Figure 7.26). Pressure variation,
uncontrolled water loses and unburned carbon show good behaviour, whereas other

Application of quantitative causality analysis to determine intrinsic and induced malfunctions
variables do not. In variables associated with the cooling system, the importance of
induced effects associated with the residues justifies the separate analysis of these
flows.
Fuel impact (kW)
Impact (kW)
5000
0
-5000
-10000
-15000
-20000
-25000
10000
5000
0
-5000
-10000
-15000
delta P rec.
Fuel impact of free diagnosis variables
eff. HP eff. IP1 eff. IP2 eff. LP phi IP1 phi IP2 phi LP TDCA
6
delta P TP
Free diagnosis variable
TDCA
5
TDCA
4
TTD 5 TTD 6
Figure 7.26. Fuel impact of free diagnosis variables (1)
unc. losses
Fuel impact of free diagnosis variables
eff. cond.
m. cooling
eff. tower
eff. SAH
eff. PAH
Free diagnosis variable
eff. boiler
infiltration
unburned
Figure 7.27. Fuel impact of free diagnosis variables (2).
induced residues
induced productive
intrinsic residues
intrinsic productive
induced residues
induced productive
intrinsic residues
intrinsic productive
271

Chapter 7
7.4 Conclusion
In this chapter, theory developed in chapter 4 which connects quantitative causality
analysis and thermoeconomic diagnosis has been successfully applied to the case of
work.
Besides the demonstration of the practical applicability of the approach, results
provide valuable information on the suitability of the used productive structure to
successfully reduce induced effects, by precisely quantifying the influence of ambient
conditions, fuel quality, set-points and other components. For example, turbines are
diagnosed correctly, whereas the analysis of the boiler should be improved.
Furthermore, the separate analysis of residues flows has proved to reduce induced
effects, so that, to clarify the diagnosis results.
In conclusion, the formulation built shows that the thermoeconomic diagnosis can be
understood as the highest level of causalization in the quantitative causality analysis.
Besides, it opens the way towards an exhaustive evaluation of the suitability not only of
productive structures but also of functions to describe flows different from exergy.
272

8 Application of linear regression and
neural networks.
Once the suitability of quantitative causality analysis for the diagnosis of a real
example has been demonstrated in chapter 6, the aim of this chapter is to analyze the
possibility of applying linear regression and neural networks to the same task.
The theoretical development of these approaches has been presented in Chapter 4.
The idea is the same: to relate the variations of the free diagnosis variables to the impact
that they have on the global efficiency indicators. In the quantitative causality method,
this relation is related by using only a thermodynamic representation of the system. The
other two approaches do not use a description of the system (they consider it as a black
box); instead they need a large amount of operation data to experimentally determine
these relations.
In this chapter, these two approaches are used to determine the relations between
free diagnosis variables and their impacts. First, linear regression (linear additive
model) is applied and then neural networks are used. To adjust the linear regression
coefficients and to train the neural networks, the large amount of operation data
available is used. Since the temporal evolution of variables and impacts has been deeply
presented previously, to validate the approaches and to compare them with the
quantitative causality method, only relations between variables and impacts are studied.
Results show that the methods based on experimental relations are capable to correctly
quantify impacts caused by the most influent variables, but provide inaccurate results in
the case of less relevant variables. Calculations have been performed by using Matlab
(MathWorks, 2007; Demuth and Beale, 2002).
273

Chapter 8
8.1 Linear regression
Relation between free diagnosis increments and their impacts is, in most cases, quite
linear. So that, it is possible to determine impact factors that relate variations and
impacts. In Chapter 6, these factors have been calculated by adjusting a line to the
points of the graph variable increment versus impact obtained by the quantitative
causality method. The aim of this paragraph is to apply lineal regression (linear additive
models) to determine these coefficients directly by using plant data.
A problem appears because regression is applicable under the hypothesis of
independence of the free variables, which is not true in an example like this, in which a
lot of variables are present. To solve this question, a variable change is performed in
order to eliminate the most important dependences.
8.1.1 Variable change
The correlation matrix indicates the degree of linear correlation between the free
diagnosis variables. The analysis of this matrix shows that there are 12 examples of high
correlation (absolute value above 0.6). They are summarized in Table 8.1.
Variable 1 Variable 2 Correlation
Coefficient
2 Relative humidity 1 Ambient temperature -0.7735
4 Coal HHV 5 Carbon mass fraction in coal 0.951
4 Coal HHV 9 Sulphur mass fraction in coal 0.6354
9 Sulphur mass fraction in coal 8 Ash mass fraction in coal 0.6604
10 Nitrogen mass fraction in coal 5 Carbon mass fraction in coal 0.6302
21 Tempering and primary air relation 1 Ambient temperature 0.6374
24 Fraction of flue gases through PAH 20 Air for flue gas desulfuration 0.684
36 TDCA 5 th water heater 35 TDCA 6 th water heater 0.7163
37 Pressure drop in reheater 29 Intermediate pressure 1 flow
coefficient
0.6773
38 Pressure increment in turbo-pump 14 Live steam pressure 0.6437
42 Cooling tower effectiveness 1 Ambient temperature 0.8844
42 Cooling tower effectiveness 2 Relative humidity -0.6288
274
Table 8.1. Pairs of variables highly correlated.

Application of linear regression and neural networks
To avoid this situation, it is possible to make a variable change. The firs step is to
determine experimental correlations linking variable 1 and variable 2. Then, these
correlations are used to calculate theoretical values of variables 1 for each point.
Afterwards, variables 1 are substituted by the difference between its actual value minus
the theoretical value calculated by the correlations.
With these new independent variables, it is possible to apply linear regression
without problems. Finally, an inverse variable change should be made in order to obtain
the coefficients corresponding to the initial set of variables.
Correlations obtained to link variables 1 and variables 2 are detailed below:
x = 75.04 −0.7494⋅ x
R 2 = 0.5983 8.1
dc,2 d ,1
x = 886.7 + 378.3⋅
x
R 2 = 0.9043 8.2
dc,5 d ,5
x = 1.408 + 0.1480⋅
x
R 2 = 0.4361 8.3
dc,9 d ,8
x =− 0.3324 + 2.329⋅10 ⋅ x R 2 = 0.3971 8.4
−2
dc,10 d ,5
x = 24.22 + 0.6158⋅
x
R 2 = 0.4063 8.5
dc,21 d ,1
x = 23.26 + 0.2508⋅
x
R 2 = 0.4678 8.6
dc,24 d ,20
x = 5.861+ 1.062⋅
x
R 2 = 0.5131 8.7
dc,36 d ,35
x =− 23.91+ 1.940⋅
x
R 2 = 0.4587 8.8
dc,37 d ,29
x =− 30.11+ 1.320⋅
x
R 2 = 0.4144 8.9
dc,38 d ,14
x = 0.3476 + 7.096⋅10 ⋅ x R 2 = 0.7821 8.10
−3
dc,42 d ,1
It should be noted that relations between coal high heating value and coal sulphur
mass fraction, and between cooling tower effectiveness and relative humidity have not
been used. This is due because coal high heating value and cooling tower effectiveness
have been already transformed by filtering their dependence with carbon mass fraction
and ambient temperature, respectively.
275

Chapter 8
8.1.2 Results
Initial variables have been transformed by using the relations obtained in the
previous section; then, linear regression has been applied and finally the inverse of the
transformation has been used to obtain the correlation coefficients corresponding to the
original variables. Results are presented in Table 8.2. This table contains the value of
the impact factors calculated by using linear regression, as well as the error of these
impact factors when compared to the average impact factors obtained by the
quantitative causality method.
276
error ed
error ed
error ed
Where
ed − ed
= 8.11
lr av
_ i, ηb
i, ηb av
edi,
ηb
i,
ηb
ed − ed
= 8.12
lr av
_ iHR , c
iHR , c
av
ediHR
, c
iHR , c
ed − ed
= 8.13
lr av
_ iHR , u
iHR , u
av
ediHR
, u
iHR , u
av
edi is the average impact factor for the variable i obtained by applying the
quantitative causality method and
lr
ed i is the impact factor calculated by using linear
regression, both of them for boiler efficiency, cycle heat rate and unit heat rate. It should
be noted that this error is calculated by using absolute value.
Besides impact factor and its error, the standard deviations of the impacts are
included, in order to show which ones are the most relevant variables. These standard
deviations are calculated by using the quantitative causality method, so that they are the
same as those shown in Table 6.1. The same colour scheme proposed in that table is
conserved: variables whose impact standard deviation is above 20% of the maximum
are labelled pink, those from 10 to 20% are labelled orange, those from 5 to 10% are
yellow and variables from 1 to 5% are blue-green. The rest of the variables remain
white.

Application of linear regression and neural networks
Impact on boiler effficiency Impact on cycle heat rate Impact on unit heat rate
Num. Description Units
Error Stand.
dev.
Impact
Error Stand.de
v.
Impact
factor
factor
0.2338 4.484
4.702
0.1535 5.276
Error Stand. Impact
dev. factor
6.223 0.4017 6.124
1 Ambient temperature ºC 5.868
·10 -2
·10 -3
·10 -2
·10 -3
·10 -2
·10 -2
0.3673 9.314
8.055
0.3465 7.844
0.7914 0.01081 7.010
2 Relative humidity % -2.361
·10 -3
·10 -4
·10 -3
·10 -4
·10 -4
0.4098 4.117
2.051
8.302 8.039
0.2376 0.1150 2.639
3 Wind speed m/s -5.030
·10 -3
·10 -3
·10 -5
·10 -4
·10 -2
2.367
1.960
-3.572
0.1722 2.618
0.5744 -3.327
1.546
4 Coal high heating value kJ/kg 9.026
·10 -2
·10 -2
·10 -5
·10 -3
·10 -6
·10 -2
·10 -4
0.02687 9.782
5.799
0.5499 2.238
5 Carbon mass fraction in coal % -0.1356 0.1654 0.1898 6.146
·10 -3
·10 -3
·10 -3
·10 -4
1.189
2.543
6.894
8.645
6.259
0.3102 5.301
6 Hydrogen mass fraction in coal % -1.775 1.153
·10 -2
·10 -3
·10 -2
·10 -4
·10 -2
·10 -3
·10 -2
1.257
5.671
7.317
0.8467 9.181
0.3249 8.647
7 Moisture mass fraction in coal % -0.2029 1.097
·10 -2
·10 -2
·10 -3
·10 -4
·10 -5
·10 -2
5.657
1.874
3.124
0.9526 7.228
0.1695 0.1339 1.857
8 Ash mass fraction in coal % -8.494
·10 -3
·10 -2
·10 -3
·10 -4
·10 -5
·10 -2
0.3624 1.134
1.732
0.8729 2.791
8.544
0.1759 2.169
9 Sulphur mass fraction in coal % -4.278
·10 -3
·10 -3
·10 -4
·10 -5
·10 -2
·10 -2
5.819 1.040
1.197
16.86 2.221
6.670
1.186 2.109
10 Nitrogen mass fraction in coal % -7.777
·10 -4
·10 -2
·10 -5
·10 -3
·10 -3
·10 -2
Table 8.2.A. Impact factors obtained by linear regression.
277

Chapter 8
Impact on boiler effficiency Impact on cycle heat rate Impact on unit heat rate
Num. Description Units
Error Stand.
dev.
Impact
Error Stand.de
v.
Impact
Error Stand.
dev.
Impact
factor
factor
factor
0.8093 1.633
1.772
1.579 8.518
4.443
0.4481 6.829
11 Energy provided by natural gas % -2.988
·10 -4
·10 -4
·10 -5
·10 -5
·10 -3
·10 -3
1.090
5.458
-8.968
1.009
9.640
-7.112
17.46 3.566
12 Live steam temperature ºC 3.355
·10 -3
·10 -2
·10 -4
·10 -3
·10 -2
·10 -4
·10 -4
·10 -3
0.3803 1.310
-3.873
0.2319 1.308
-4.785
8.633 4.750
13 Reheated steam temperature ºC -1.571
·10 -3
·10 -4
·10 -3
·10 -4
·10 -4
·10 -3
0.1997 1.830
-1.081
1.664
3.847
-1.223
20.82 5.025
14 Live steam pressure bar -6.181
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
·10 -3
0.8529 7.331
1.066
0.1654 3.804
4.520
0.6367 1.843
15 Gross electric power MW 7.159
·10 -4
·10 -5
·10 -3
·10 -4
·10 -2
·10 -4
3.388
1.520
1.330
0.3161 7.474
1.975
% -0.2978 0.1261 6.832
16 Oxygen in flue gases leaving the
·10 -3
·10 -2
·10 -2
·10 -4
·10 -3
·10 -2
boiler
6.830
8.566
1.112
0.5678 2.762
0.1741 6.490
3.519
-2.709
ºC
17 Average cold-side temperature
·10 -3
·10 -2
·10 -3
·10 -4
·10 -5
·10 -2
·10 -2
in secondary air preheaters
0.2910 2.082
3.627
0.6911 8.771
1.985
0.2049 5.3412
-8.727
ºC
18 Average cold-side temperature
·10 -3
·10 -4
·10 -5
·10 -5
·10 -2
·10 -3
in primary air preheaters
4.550
5.789
8.506
0.1223 9.412
-1.1459
1.480 1.232
19 Sootblowing steam flow rate kg/s -7.574
·10 -3
·10 -2
·10 -3
·10 -4
·10 -3
·10 -3
·10 -4
0.1665 5.396
5.112
0.5867 2.141
0.1233 0.1403 1.002
-1.398
kg/s
20 Air for flue gas desulfuration
·10 -3
·10 -4
·10 -4
·10 -5
·10 -2
unit
Table 8.2.B. Impact factors obtained by linear regression.
278

Application of linear regression and neural networks
Impact on boiler effficiency Impact on cycle heat rate Impact on unit heat rate
Num. Description Units
Error Stand.
dev.
Impact
Error Stand.de
v.
Impact
Error Stand.
dev.
Impact
factor
factor
factor
0.2389 3.455
6.151
14.90 2.314
·10 -3
·10 -4
·10 -5
0.4996 1.764
5.715
8.327 2.216
21 Tempering and primary air kg/kg -1.670
relation
·10 -2
0.1305 0.1028 3.618
·10 -5
22 Primary air-coal ratio kg/kg -0.2010 0.4202 5.299 9.078
·10 -3
·10 -3
·10 -5
·10 -4
·10 -2
0.3081 5.261
3.657
0.4848 2.365
8.212
3.600 8.549
-6.716
ºC
23 Temperature difference of flue
·10 -4
·10 -5
·10 -4
·10 -6
·10 -3
·10 -4
gases entering preheaters
10.12 6.461
5.821
11.36 4.786
9.591
9.942 1.568
-1.118
-
24 Fraction of flue gases through
·10 -4
·10 -4
·10 -5
·10 -5
·10 -2
·10 -2
PAH
0.1018 9.927
-4.920
9.051
8.491
-4.570
1.198 3.567
-3.153
%
25 High pressure turbine
·10 -3
·10 -3
·10 -3
·10 -2
·10 -3
·10 -3
·10 -3
isoentropic efficiency
5.793
7.764
-2.546
5.127
8.200
-2.243
0.8888 2.219
-7.782
%
26 Intermediate pressure 1
·10 -3
·10 -2
·10 -3
·10 -3
·10 -2
·10 -3
·10 -3
·10 -5
isoentropic efficiency
4.327
3.766
-3.299
3.830
5.301
-2.871
0.7077 1.719
-2.489
%
27 Intermediate pressure 2
·10 -3
·10 -2
·10 -3
·10 -3
·10 -2
·10 -3
·10 -3
·10 -4
isoentropic efficiency
4.925
3.989
-1.304
4.351
2.828
-1.165
0.8452 1.577
-5.774
%
28 Low pressure isoentropic
·10 -2
·10 -2
·10 -2
·10 -2
·10 -2
·10 -2
·10 -2
·10 -3
efficiency
0.7393 1.233
1.424
1.506 7.434
-1.678
6.051 5.467
-6.569
kg 1/2 ·m 3/2
29 Intermediate pressure 1 flow
·10 -3
·10 -3
·10 -4
·10 -3
·10 -4
·10 -3
·s -1 ·bar -1/2
coefficient
Table 8.2.C. Impact factors obtained by linear regression.
279

Chapter 8
Impact on boiler effficiency Impact on cycle heat rate Impact on unit heat rate
Num. Description Units
Error Stand.
dev.
Impact
Error Stand.de
v.
Impact
Error Stand.
dev.
Impact
factor
factor
factor
2.956 4.509
3.610
2.972 3.943
3.160
60.10 1.462
-1.371
kg 1/2 ·m 3/2
30 Intermediate pressure 2 flow
·10 -4
·10 -3
·10 -4
·10 -3
·10 -4
·10 -2
·s -1 ·bar -1/2
coefficient
43.76 4.230
1.040
74.18 3.695
7.987
930.0 1.278
-5.327
31 Low pressure flow coefficient kg 1/2 ·m 3/2
·10 -4
·10 -3
·10 -4
·10 -4
·10 -4
·10 -3
·s -1 ·bar -1/2
2.356
2.600
1.061
0.1189 1.616
7.937
8.376 8.663
32 TTD 6 th water heater ºC 2.750
·10 -3
·10 -2
·10 -3
·10 -3
·10 -4
·10 -4
·10 -3
0.1463 3.040
3.882
2.652
9.141
3.604
5.995 1.057
33 TTD 5 th water heater ºC 9.586
·10 -3
·10 -4
·10 -3
·10 -2
·10 -4
·10 -3
·10 -4
0.3537 2.523
1.696
0.2874 2.226
1.422
10.93 7.914
34 TTD 3 rd water heater ºC -3.105
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
·10 -3
1.748 2.932
1.476
0.6409 2.564
7.707
132.7 8.936
35 TDCA 6 th water heater ºC -1.747
·10 -4
·10 -4
·10 -4
·10 -5
·10 -5
·10 -3
0.1387 9.612
1.035
0.1654 8.501
8.874
1.372 3.026
36 TDCA 5 th water heater ºC 6.686
·10 -4
·10 -4
·10 -4
·10 -5
·10 -4
·10 -5
0.1955 3.579
4.306
0.1344 3.503
4.534
0.7979 9.723
37 Pressure drop in reheater bar 2.145
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
·10 -3
1.165
7.535
3.473
0.2787 1.039
4.261
28.55 5.117
2.589
bar
38 Pressure increment in turbo-
·10 -3
·10 -2
·10 -4
·10 -3
·10 -4
·10 -4
·10 -3
pump
Table 8.2.D. Impact factors obtained by linear regression.
280

Application of linear regression and neural networks
Impact on boiler effficiency Impact on cycle heat rate Impact on unit heat rate
Num. Description Units
Error Stand.
dev.
Impact
Error Stand.de
v.
Impact
Error Stand.
dev.
Impact
factor
factor
factor
0.1022 5.756
3.734
0.1506 2.823
-1.419
0.5539 1.870
39 Water losses kg/s 3.485
·10 -3
·10 -3
·10 -3
·10 -3
·10 -3
·10 -4
-0.1390 0.3055 1.246
-0.1296 0.2685 1.102
40 Condenser effectiveness - -0.2262 3.941 3.505
·10 -2
·10 -2
·10 -3
0.3546 1.190
-1.391
0.3228 1.050
-1.288
2.640 3.910
41 Cooling water flow rate kg/s -2.096
·10 -2
·10 -5
·10 -2
·10 -5
·10 -3
·10 -5
-0.2671 0.2878 2.206
-0.2477 0.2462 1.946
42 Cooling tower effectiveness - -0.4303 2.580 1.009·
·10 -2
·10 -2
10 -2
-0.2015 0.2296 8.441
11.36 6.703
- 6.961 0.09682 0.2488 1.948
43 Secondary air preheater
·10 -3
·10 -5
·10 -2
effectiveness
-0.1289 0.7586 1.668
34.75 1.403
-1.478
- 3.113 0.4388 4.904
44 Primary air preheater
·10 -3
·10 -5
·10 -2
·10 -2
effectiveness
1.140
-0.6308 6.177
2.070
0.4243 0.1179 3.074
45 Aggregated boiler effectiveness - 23.95 2.443
·10 -2
·10 -3
·10 -3
·10 -2
·10 -2
0.2648 3.869
1.353
0.6955 1.593
7.467
0.2338 9.969
46 Air infiltration in preheaters % -3.424
·10 -3
·10 -3
·10 -4
·10 -5
·10 -2
·10 -2
9.010
2.924
1.665
0.3095 1.775
0.2518 2.197
47 Carbon in ashes % -0.4726 4.507
·10 -3
·10 -2
·10 -2
·10 -4
·10 -4
·10 -2
Table 8.2.E. Impact factors obtained by linear regression.
281

Chapter 8
The most important conclusion obtained from Table 8.2 is that variables with high
impact (high value of standard deviation) presents low error while low impact variables
presents a high error. In conclusion, the method is suitable to analyze only major
impacts.
This fact should not be surprising if the concept of the method is considered. The
approach is based on the determination of sensitivity coefficients which multiply the
independent variables by minimising the error in the approximation of a dependent
variable. So that, variables which have high impact are determined precisely, while the
other do not.
This relation between error and standard deviation can be seen more clearly by using
graphs. Figure 8.1 shows the case of boiler efficiency, Figure 8.2 corresponds to cycle
heat rate and Figure 8.3 shows results of unit heat rate. X axis represents the normalized
value of the standard deviations, calculated by dividing each one of them into the
maximum one. Errors are represented on axis y, where logarithmic scale has been used
due to the wide range of this variable. Pink points represent the errors obtained by using
linear regression combined with variable change to reduce problems induced by nonindependence
of variables, while blue points correspond to errors produced when linear
regression is directly used, without applying variable change. Since variable change
only affects a few variables, in most cases points pink and blue are coincident.
In the three graphs, it can be seen how errors are low for the most important
variables. In boiler efficiency, error remains lower than 0.1 for variables whose impact
is above 40% of the maximum. In unit and cycle heat rate, error is lower than 0.4 for
variables with impact above 10% of the maximum. In all cases, error increases
dramatically in variables with low influence.
These graphs demonstrate that variable change performed to reduce linear
dependence between variables originates a good improvement in the results. It is
expected that the use of this technique for more variables (establishing a maximum level
lower than the 60% used here) would lead to better results.
282

Error
Error
1000
100
10
0.1
0.01
Application of linear regression and neural networks
Error and impact in boiler efficiency
1
0 0.2 0.4 0.6 0.8 1 1.2
100
10
0.1
0.01
0.001
Normalised impact standard deviation
Direct
Variable change
Figure 8.1. Error in the value of impact factors and impact standard deviation
in boiler efficiency.
Error and impact in cycle heat rate
1
0 0.2 0.4 0.6 0.8 1 1.2
Normalzed impact standard deviation
Direct
Variable change
Figure 8.2. Error in the value of impact factors and impact standard deviation
in cycle heat rate.
283

Chapter 8
Error
284
100
10
0.1
0.01
0.001
Figure 8.3. Error in the value of impact factors and impact standard deviation
in unit heat rate.
8.2 Neural networks
Error and impact in unit heat rate
1
0 0.2 0.4 0.6 0.8 1 1.2
Normalized impact standard deviation
Direct
Variable change
An approach similar as that one presented in the previous section is applied here, but
using neural networks instead of linear regression. The idea is to use a large amount of
plant data to train a neural network whose inputs are the free diagnosis variables and the
output is the global efficiency indicator. Since, in our case of work, there are three
global efficiency indicators, three neural networks are needed.
Once the networks are trained, they can be used to calculate the impact caused by
each one of the free variables by substracting the value provided by the net when the
corresponding free diagnosis variable has the value of the actual point, minus the value
provided by the net when this free variable takes the value of the reference points.
Values of the other variables should be the same for the two points. They can be those
of the reference point, those of the actual point or an intermediate value.
Three neural networks, each one corresponding to a global efficiency indicator have
been built and trained by using plant information available from the working example.

Application of linear regression and neural networks
Afterwards, the capability of these networks for the diagnosis is analyzed by comparing
the sensitivity for a change in an input shown by them with the results provided by the
quantitative causality method and linear regression.
8.2.1 Neural network definition and training
As previously said, three networks have been created and trained, one for each
global efficiency indicator. All of them have the same architecture.
There are 47 inputs for the free diagnosis variables and only one output. The number
of neurons in the hidden layer has been fixed in 3, because the form of the function
required is not complex. As it has been seen in chapter 6, when impact is plotted against
free diagnosis variable increment, a straight or slightly curved shape appears in most
cases. If more neurons were used, results provided by the net would be fluctuating. The
output function selected for these neurons is sigmoid. Each one of them has 47 inputs
corresponding to all inputs of the net. Outputs of these 3 neurons are the inputs of the
output neuron. The output function of this output neuron is linear.
The three networks have been trained by using 70% of the operation points from the
case study already used in chapter 6 and in section 8.1. Evolution of the residual during
the training process is plotted in Figures 8.4. 8.5 and 8.6. It should be noted that this
residual corresponds not to the absolute value of the objective function but to its
normalized value.
Since the transformation of the free diagnosis variables has provided good results in
the application of linear regression, this technique has also been applied in the neural
networks. So that, the nets have not been trained using the original set of variables, but
using the transformed set. Consequently, this transformation should be used as well
when nets are used for simulation.
Finally, Figures 8.7, 8.8 and 8.9 compare the actual and the simulated value for all
the working points available. Pink points correspond to training points and blue points
are the other points, used only for test. Graphs show that the degree of accuracy of both
point families is the same, so that the nets have not been over-trained.
285

Chapter 8
286
Error
Error
10 0
10 -1
10 -2
10 -3
10 -4
10 1
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
Performance is 5.47821e-005
0 10 20 30 40 50
100 Epochs
60 70 80 90 100
Figure 8.4. Training process for boiler efficiency neural network.
Performance is 6.11472e-006
0 10 20 30 40 50
100 Epochs
60 70 80 90 100
Figure 8.5. Training process for cycle heat rate neural network.

Calculated boiler efficiency (%)
87
86
85
84
83
82
81
80
Error
10 0
10 -1
10 -2
10 -3
10 -4
Application of linear regression and neural networks
Performance is 2.05167e-005
0 10 20 30 40 50
100 Epochs
60 70 80 90 100
Figure 8.6. Training process for unit heat rate neural network.
Real and calculated boiler efficiency
79
79 80 81 82 83 84 85 86 87
Real boiler efficiency (%)
Figure 8.7. Comparison of real and calculated values of boiler efficiency.
Train
Test
287

Chapter 8
Calculated cycle heat rate
Calculated unit heat rate
288
2.7
2.6
2.5
2.4
2.3
2.2
Real and calculated cycle heat rate
2.1
2.1 2.2 2.3 2.4
Real cycle heat rate
2.5 2.6 2.7
3.2
3.1
3
2.9
2.8
2.7
Figure 8.8. Comparison of real and calculated cycle heat rate.
Real and calculated unit heat rate
2.6
2.6 2.7 2.8 2.9
Real unit heat rate
3 3.1 3.2
Figure 8.9. Comparison of real and calculated unit heat rate.
Train
Test
Train
Test

Application of linear regression and neural networks
8.2.2 Results
To evaluate the suitability of the neural networks for the diagnosis problem,
sensitivity of them when the free diagnosis variables are varied is considered and
compared to results provided by the quantitative causality method and by linear
regression.
Since 47 free diagnosis variables multiplied times three global efficiency indicators
constitute a too large amount of graphs, only a fraction of them are considered. For each
global efficiency indicator, the 6 most influent free diagnosis variables are analyzed
(Figures 8.10, 8.12 and 8.14). Besides, 6 more variables are studied in order to take into
account the capability of the approach to take into account the impact of less influent
variables (Figures 8.11, 8.13 and 8.15). They have been chosen according to the
following criteria: two variables whose influence (standard deviation of its impact) is
just below 20% of the maximum, two below 10% of the maximum and two below 5%
of the maximum. In the case of cycle heat rate, only 4 variables have an influence
higher than 20% of the maximum, and only 3 variables have an influence between 10%
and 20% of the maximum. So that, in Figure 8.13 are plotted the variable which has an
impact just above 20% of the maximum, three variables whose influence is just below
10% of the maximum and two variables with influence just below 5%.
Results shown in these graphs demonstrate that neural networks (and also linear
regression) are suitable for the more influent variables, but error increases when
influence of variables decreases. It should be noted that divergence increases strongly
for variables with influence around 1%, as it can be expected from figures 8.1 to 8.3.
Besides, although the use of variable change entails a good improvement, variables
which have dependence with others present worse results than the others. For example,
ambient temperature in unit heat rate.
Variables such as condenser effectiveness and cooling tower effectiveness and
cooling water flow rate demonstrate the capability of neural networks and quantitative
analysis quantification methods to deal with non-linear behaviours, in opposition to
approaches based on constant impact factors.
Finally, it should be highlighted that, if more neurons had been included in the
networks, oscillating tendency would appear in the variables with the lowest influence.
289

Chapter 8
Impact (%)
Impact (%)
Impact (%)
290
-4000 -3000 -2000
1
0.5
0
-1000-0.5 0
-1
-1.5
-2
-2.5
-3
-3.5
-4
1000 2000
HHV variation (kJ/kg)
0.5
0
-30 -20 -10 0 10 20
-0.5
1
-1
-1.5
Ambient temperature variation (ºC)
1.5
1
0.5
-1 -0.5
0
-0.5
0 0.5 1
-1
-1.5
-2
Hydrogen in coal variation (%)
0
-6 -4 -2 0 2 4 6 8
-0.5
Figure 8.10. Variable increment and impact for the six variables most influent on boiler
efficiency: coal HHV, aggregated boiler effectiveness, ambient temperature, moisture in
coal, hydrogen in coal and carbon in ashes.
Impact (%)
Impact (%)
Impact (%)
-0.1 -0.05
2.5
2
1.5
1
0.5
0
-0.5 0
-1
-1.5
-2
-2.5
0.05 0.1
Aggregated effectiveness variation
1
0.5
-1
-1.5
Coal moisture variation (%)
-2 -1
0
0
-0.5
1 2 3 4
Unit 1 Unit 2 Unit 3 Linear regression Neural network
1
0.5
-1
-1.5
-2
Carbon in ashes variation (%)

Impact (%)
Impact (%)
Impact (%)
0.4
0.3
0.2
0.1
-30 -20 -10
0
-0.1 0 10 20 30
-0.2
-0.3
-0.4
Tempering and primary air relation variation (%)
0.2
0.15
0.1
0.05
-30 -20
0
-10-0.05 0 10 20 30 40
-0.1
-0.15
-0.2
-0.25
Average cold side temp. in PAP variation (ºC)
0.06
0.04
0.02
-1
0
-0.5-0.02 0 0.5 1 1.5 2
-0.04
-0.06
-0.08
-0.1
Sulphur in coal variation (%)
Application of linear regression and neural networks
-10
0
-0.2
0 10 20 30 40
Figure 8.11. Relation between variable increment and its impact on boiler efficiency for
several variables: tempering and primary air relation, air infiltration in preheaters, average
cold-side temperature in primary air preheaters, primary air-coal ratio, sulphur mass fraction
in coal and low pressure isoentropic efficiency.
Impact (%)
Impact (%)
0.4
0.2
-0.4
-0.6
-0.8
-1
Air infiltration variation (%)
-0.5
0.15
0.1
0.05
0
-0.05 0
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35
-0.4
0.5 1
Primary air-coal ratio variation (kg/kg)
0.15
0.1
0.05
-25 -15
0
-0.05
-5 5 15 25
-0.1
-0.15
-0.2
Low pressure turbine efficiency variation (%)
Unit 1 Unit 2 Unit 3 Linear regression Neural network
Impact (%)
291

Chapter 8
Impact
Impact
Impact
292
0.1
0.05
-30 -20 -10
0
0
-0.05
10 20
-0.1
-0.15
-0.2
Ambient temperature variation (ºC)
0.12
0.1
0.08
0.06
0.04
0.02
-0.3 -0.2 -0.1
0
-0.02 0 0.1 0.2
-0.04
Cooling tower effectiveness variation
0.06
0.05
0.04
0.03
0.02
0.01
-3000 -2000
0
-1000 -0.01 0 1000 2000
-0.02
-0.03
Cooling water flow rate variation (kg/s)
-20 -10
0
-0.05 0 10 20
Figure 8.12. Variable increment and impact for the six variables most influent on cycle heat
rate: ambient temperature, low pressure turbine isoentropic efficiency, cooling tower
effectiveness, condenser effectiveness, cooling water flow rate and high pressure turbine
isoentropic efficiency.
Impact
Impact
Impact
0.2
0.15
0.1
0.05
-0.1
-0.15
-0.2
Low pressure turbine efficiency variation (%)
0.05
0.04
0.03
0.02
0.01
-0.2 -0.1
0
-0.01 0 0.1 0.2 0.3
-0.02
-0.03
-0.04
Condenser effectiveness variation
0.05
0.04
0.03
0.02
0.01
-8 -6 -4
0
-2 -0.01 0 2 4
-0.02
-0.03
High pressure turbine efficiency variation (%)
Unit 1 Unit 2 Unit 3 Linear regression Neural network

Impact
Impact
Impact
0.03
0.02
0.01
-20 -10
0
0
-0.01
10 20 30
-0.02
-0.03
Relative humidity variation (%)
0.02
0.01
0
-4 -2 0 2 4 6 8
-0.01
-0.02
-0.03
IP 2 turbine efficiency variation (%)
0.015
0.01
0.005
0
-3000 -2000 -1000 0 1000 2000
-0.005
Coal HHV variation (kJ/kg)
Application of linear regression and neural networks
0
-12 -7 -2
-0.01
3 8
Figure 8.13. Relation between variable increment and its impact on cycle heat rate for
several variables: relative humidity, intermediate pressure 1 isoentropic efficiency,
intermediate pressure 2 isoentropic efficiency, gross electric power, coal high heating value
and carbon mass fraction in coal.
Impact
Impact
Impact
0.03
0.02
0.01
-0.02
IP 1 turbine efficiency variation (%)
0.025
0.02
0.015
0.01
0.005
-20
0
-10 -0.005 0 10 20 30
-0.01
-0.015
Gross electric power variation (MW)
0.005
0
-10 -8 -6 -4 -2 0 2 4
-0.005
-0.01
-0.015
Carbon mass fraction variation (%)
Unit 1 Unit 2 Unit 3 Linear regression Neural network
293

Chapter 8
Impact
Impact
Impact
294
0.3
0.2
0.1
-20 -10
0
0
-0.1
10 20
-0.2
-0.3
Low pressure turbine efficiency variation (%)
0.12
0.1
0.08
0.06
0.04
0.02
-3000 -2000
0
-1000 -0.02 0 1000 2000
-0.04
Coal HHV variation (kJ/kg)
0.06
0.04
0.02
0
-6 -4 -2
-0.02
0 2 4 6 8
-0.04
Moisture mass fraction in coal variation (%)
0
-30 -20 -10 0 10 20
-0.05
Figure 8.14. Variable increment and impact for the six variables most influent on unit heat
rate: low pressure turbine isoentropic efficiency, ambient temperature, coal high heating
value, cooling tower effectiveness moisture mass fraction in coal and condenser
effectiveness.
Impact
Impact
Impact
0.1
0.05
-0.1
-0.15
Ambient temperature variation (ºC)
0.1
0.08
0.06
0.04
0.02
-0.3 -0.2 -0.1
0
-0.02 0 0.1 0.2
-0.04
-0.06
Cooling tower effectiveness variation
0.06
0.04
0.02
0
-0.2 -0.1 0 0.1 0.2 0.3
-0.02
-0.04
Condenser effectiveness variation
Unit 1 Unit 2 Unit 3 Linear regression Neural network

Impact
Impact
Impact
0.02
0
-10 -8 -6 -4 -2 0 2 4
-0.02
-0.04
-0.06
Carbon in coal variation (%)
0.04
0.03
0.02
0.01
-3 -2
0
-1 0
-0.01
1 2 3 4 5
-0.02
Blowing steam variation (kg/s)
0.02
0.015
0.01
0.005
-10 -5
0
0
-0.005
5 10 15
-0.01
6 th heater hot side temp. diff. variation (ºC)
Application of linear regression and neural networks
-20 -10
0
-0.01 0 10 20 30
Figure 8.15. Relation between variable increment and its impact on unit heat rate for several
variables: carbon mass fraction in coal, relative humidity, sootblowing steam flow rate,
intermediate pressure 3 turbine isoentropic efficiency, hot side temperature difference in 6 th
-0.02
-0.03
water heater and average cold-side temperature in primary air preheaters.
Impact
Impact
Impact
0.05
0.04
0.03
0.02
0.01
Relative humidity variation (%)
0.03
0.02
0.01
-4
0
-2
-0.01
0 2 4 6 8
-0.02
-0.03
-0.04
IP 2 turbine efficiency variation (%)
0.01
0.005
0
-30 -20 -10 0 10 20 30 40
-0.005
-0.01
Average cold-side temp. in PAP variation (ºC)
Unit 1 Unit 2 Unit 3 Linear regression Neural network
295

Chapter 8
8.3 Conclusion
In this chapter, two diagnosis methods based respectively on linear regression and
neural networks have been applied to the working example in order to test their
suitability to the diagnosis of an actual example.
These two methods are based on a purely experimental determination of relations
between free diagnosis variables and global efficiency indicators, using a large amount
of operation data but without introducing any information about the system under study.
It has been seen that the two approaches provide good result for the more influent
variables, but error increases substantially for low influence variables. It should be
highlighted that the interest of a variable not only depends of its influence but also of its
easiness of correction. For example, a set-point may have a small impact, but it is
interesting to quantify it in order to advise plant operators to take care about it. This
capability to determine the impact of all variables is an important advantage of the
quantitative causality method.
The problem of variable independence is a key issue. It has been demonstrated how
the variable change helps to improve results when some degree of lineal dependence
among free diagnosis variables is present. If the level for considering two variables as
not independent had been fixed below 60%, accuracy of results for variables with low
influence would have increased, but more work would have been needed to determine
more correlations. So that, equilibrium should be reach in this point.
Although linear regression and the use of neural networks have several points in
common, they are differenced by the capability of the second of them to deal with non
linear behaviour. This advantage is irrelevant in most variables, but can be decisive in
others. It should be noted that this fact is due because linear additive models are used in
linear regression. It might be possible to develop more complex models based on linear
regression but considering other functions (eg. 2 nd order polynomials), which would be
able to deal with non-linear behaviour but would also entail more complexity of
calculation and interpretation).
Finally, it should be noted that here it has been compared the capability of the three
approaches for the direct diagnosis of all the system considered. Probably, the best
solution is to apply hybrid configurations based on a general diagnosis structure based
on the quantitative causality analysis, where linear regression and/or neural networks
are included for diagnose specific parts. Causality chain theory provides the
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Application of linear regression and neural networks
methodological structure to build this kind of complex diagnosis systems. A first step
towards this solution has been made in point 6.6, where linear regression has been
applied to determine the relation between cooling tower effectiveness and ambient
temperature.
297

Chapter 8
298

9.1 Synthesis
9 Conclusion
Chapter 2 is a review of several methods and systems for diagnosis of power plants.
It starts with a short exposition of thermoeconomic concepts and tools; especially
malfunction, malfunction cost and the fuel impact formula. Afterwards, proposals of
several authors to overcome the problem of induced effects (the main drawback of the
application of thermoeconomic analysis to diagnosis) are presented. Besides, other
authors have developed diagnosis methodologies which avoid the use of
Thermoeconomics and are based only on thermodynamic representations of the
systems. Among these proposals, the diagnosis algorithm of Correas (2001) is
explained with more detail, because it is the basis of the diagnosis method developed in
the next chapter. Finally, some systems for plant monitoring and diagnosis are
presented.
The quantitative causality analysis is developed in Chapter 3. First of all, the
diagnosis methodology and the nomenclature are presented. Afterwards, a method to
determine the influence of measurement errors on the diagnosis results is explained.
Since the methodology is based on the linearization of equations, the effect of nonlinear
behaviour is also analysed. The aim of the method is to compare two situations in
order to share the variation of a global efficiency indicator of the thermal system into a
summation of terms each one due to a different free diagnosis variable; so that,
guidelines to the choice of these free variables are also given. Finally, the fundamentals
of the application of causality chains are stated, in order to introduce several sets of free
diagnosis variables. A simple example is developed in order to clarify the main
concepts and methods explained.
299

Chapter 9
In Chapter 4, the quantitative causality analysis is related and compared to other
diagnosis methods. In the first part, this analysis is connected to thermoeconomic
diagnosis. The key point is to apply quantitative causality analysis to decompose the
independent variables of the thermoeconomic model (unit exergy consumptions and
plant product) into a summation of terms corresponding to the free diagnosis variables
of the thermodynamic model. The combination of this approach and the concept of
malfunction cost and the fuel impact formula allows to perform a double decomposition
of the fuel impact: by components of the thermoeconomic model and by free diagnosis
variables. This double summation leads to the table of intrinsic and induced
malfunctions (MFI) and the table of malfunctions induced by the free diagnosis
variables (MFD).
The second part is focused on the application of diagnosis methods based on linear
regression and neural networks. The idea is also to look for the connection between the
variation of a free diagnosis variable and its impact on the global efficiency indicator,
but this search is developed by using only experimental results and without any model
of the system, neither thermodynamic nor thermoeconomic. These empirical techniques
may constitute all the diagnosis method, or can be used to simulate a part of the system
and integrated with the quantitative causality analysis, thus constituting a hybrid system.
The test bench to the methodologies presented in the previous chapters is presented
in Chapter 5. This is a conventional pulverized coal-fired power plant, composed by
three units of 350 MW. Along with the plant description, the main problems affecting
their components are reviewed, in order to define suitable free diagnosis variables.
Although all the plant is considered, attention is focused on the steam cycle, while the
boiler is analyzed with low detail. 47 free diagnosis variables have been defined,
including ambient conditions, parameters of fuel quality, set-points and indicators of
components behaviour. This chapter constitutes a demonstration of the methodology to
be followed to define a diagnosis system, starting from the physical description of the
power plant to be diagnosed and the identification of possible malfunctions, up to the
definition of free diagnosis variables and global efficiency indicators.
In Chapter 6, the anamnesis, or the repeated diagnosis, of the three units of the
working example during 6 years is presented. The evolution of all free diagnosis
variables and their impact, as well as relation between free variables and impacts is
systematically analyzed in order to demonstrate the capability of the approach to deal
with a real system. Residual term, or the error produced due to the non-linear behaviour,
300

Conclusion
is proved to be low enough. Finally, causality chains are applied in order to filter the
dependence of cooling tower effectiveness with ambient temperature.
The theory developed in Chapter 4 to connect quantitative causality analysis and
thermoeonomic diagnosis is applied to the working example in Chapter 7. First, the
chosen productive structure is described. Afterwards, the method is applied to a real
example of diagnosis. Finally, a fictitious general example which considers an average
variation of all the free diagnosis is developed in order to test the suitability of the
productive structure to diagnose the system. Results show how effects induced by
ambient conditions, fuel, set-points and other components are negligible in turbines, but
not so low in the boiler. Besides, the use of separate analysis for the wastes also allows
to clarify results.
The aim of Chapter 8 is to test the suitability of linear regression and neural
networks to diagnose the working case, and to compare these empirical approaches and
quantitative causality analysis. Results show that linear regression and neural network
quantify correctly the effects of the most influent variables, but fail when dealing with
variables of secondary importance. Besides, neural networks and quantitative causality
analysis are able to deal with non-linear behaviour, while linear regression based on
linear additive models does not. Finally, it is convenient to apply variable change to
improve results when interdependence among free diagnosis variables appears.
9.2 Contributions
In Chapter 2, a revision of the diagnosis methods for thermal systems has been
made. Thermoeconomics fundamentals and its application to diagnosis has been
reviewed, as well as the methodologies developed to overcome the problem of induced
malfunctions. Besides, other methods with the same purpose but not based on
Thermoeconomics are analyzed. As a result, a general picture on diagnosis of thermal
systems is obtained, in order to identify the main diagnosis tendencies and to serve as a
basis to improve them or to develop new ones.
Starting from an existing diagnosis algorithm (Correas, 2001), a complete diagnosis
methodology is developed in Chapter 3. First of all, a new nomenclature is introduced,
which clarify the formulation in order to include new features of the diagnosis method.
Besides, this nomenclature shows clearly that, the diagnosis problem has its own
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Chapter 9
mathematical entity, but its implementation has a lot of common points with other
problems such as simulation, optimization and performance test (Correas, 2001); this
point contributes to reduce the marginal effort needed to develop a diagnosis system
from a performance test code.
Equations to relate the deviation in measurements with the variation in diagnosis
results have been developed. Due to the practical interest of the diagnosis problem, this
result is very important because it allows to quantify the reliability of a diagnosis result
depending on the uncertainty of available measurements. Besides, this methodology
gives valuable information which can help to improve plant instrumentation depending
on the accuracy required for the diagnosis. There is another source of error, which is
due to the linealization of equations made in the methods. Proposals are made in order
to reduce this error, which will prove to be successful in Chapter 6.
The choice of the definition of free diagnosis variables should represent a
equilibrium between the need of a widely accepted definition and the need to use
parameters as close as possible to the physical behaviour of the components. Guidelines
are listed to choose properly these definitions, which complement the analysis of the
equipment parameters analyzed for the working example in Chapter 5. For situations
when it is not easy to find a variable with these advisable properties, a theory of
causality chains has been developed. This formulation allows to use simultaneously
more than one level of free diagnosis variables, and to trace quantitatively the path of
causality from the root causes, through intermediate indicators up to the global
efficiency indicator of the whole plant.
Finally, the idea of total fuel impact over a period of time has been stated. It extends
the concept of diagnosis (comparison of only two situations) to a defined time span (e.g.
period between plant revisions or between steam blowing), while introducing models of
the evolution of component degradation.
In the first part of Chapter 4, the quantitative causality analysis has been connected
with thermoeconomic diagnosis. This integration is very important from the conceptual
point of view, because thermoeconomic analysis can be seen in the highest level of
causality in the analysis of thermal systems, while quantitative causality analysis is able
to determine the true causes of the malfunctions pinpointed by thermoeconomic
indicators. It should be noted that the use of thermodynamic variables to complement
thermoeconomic diagnosis in order to eliminate induced effects has been previously
used by several authors (Toffolo and Lazzaretto, 2004; Valero et al., 1999). However,
302

Conclusion
the formulation developed here constitutes a systematic approach which takes into
account the crossed influence of all variables of the thermal system analyzed. On the
other hand, the understanding of these relations allows to determine precisely the effects
induced in each components by the influence of ambient conditions, fuel quality, setpoints
and other components. This analysis is not only interesting for the diagnosis of
particular situations, but represents a systematic tool to asses the suitability of a given
productive structure for the analysis of a thermal system. In other words, it enables to
perform the physical diagnosis of the thermoeconomic diagnosis.
In the second part of Chapter 4, the capabilities of linear regression and neural
networks for the diagnosis problem have been explored. These techniques constitute a
method to link the variation of the free diagnosis variables and the global efficiency
indicator based on operational experience, without the need of building a model. These
relations can be direct, or the system can be decomposed in several blocks. Besides,
these methods can be combined with quantitative causality analysis by using hybrid
approaches.
Chapter 5 is an example of the methodology to be followed in the specification of a
diagnosis system. The components of the power plant have to be analyzed: possible
malfunction and degradation mechanisms, models and indicators of their behaviour.
Measurements available are an important point because they limit the depth of the
diagnosis achievable. Finally, the level of detail required in the analysis has to be
considered. With this information, the thermodynamic model can be built and the free
diagnosis variables and the global efficiency indicator (or indicators), can be
determined.
Quantitative causality analysis is applied to the diagnosis of the three units of the
example power plant during more than 6 years (Chapter 6). This exhaustive analysis of
a real example during such long time span allows to prove the capability of the
approach to quantify the influence of seasonal variations, changes in fuel, modification
in operations and component degradation and repairing. It should be noted that all free
diagnosis variables considered have been analyzed in order to prove that the method is
able to deal with all of them, not only the most influent ones. Besides, graphs
representing the relation of variation of the free diagnosis variables and its impact
demonstrate the coherence of the approach. Global indicators have also been proposed
and applied in order to summarize the results.
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Chapter 9
Impacts on global efficiency indicators are additive, which allows to group them in
families in order to summarize results. This fact enables the use of a zooming strategy
which starts with the evolution of the global efficiency indicators, continues with
families of causes and finalizes with the individual free diagnosis variables. However, it
should be noted that this approach is not mandatory for the application of the diagnosis
method; it is only a property of it which can facilitate the interpretation of results.
Another feature of the method which can improve the quality of the results is the use of
causality chains. This technique has been applied to separate the variation of cooling
tower effectiveness into an intrinsic term and other term induced by ambient
temperature.
Finally, errors produced by the method due to linealization have been analyzed for
this large amount of real examples. Results prove that this error is very low in a real
application.
The formulation developed in Chapter 4 to connect quantitative causality analysis
and thermoeconomic analysis is applied to the working example in Chapter 7. Results
show exhaustively the influence of the free diagnosis variables on the malfuncions, and
demonstrate the capability of the approach to precisely quantify the effects intrinsic and
induced by ambient conditions, fuel quality, set-points and other components. The
approach has been applied not only to a diagnosis example but also to determine the
general influence of all the variables, taking into account their typical deviation. It has
been demonstrated that thermoeconomic analysis can be integrated with quantitative
causality analysis, as the highest level of causality.
Finally, linear regression and neural networks have been applied to the diagnosis of
thermal systems in Chapter 8. Advantages and drawbacks of these two methods and
quantitative causality analysis have been evaluated on the basis of the analysis of a real
system. Results show that pure empirical approaches are able to identify precisely the
influence of the main variables, but do not guarantee accuracy for the others.
9.3 Perspectives
In this thesis, the diagnosis algorithm has been improved up to the quantitative
causality analysis. This method has been applied to a working example, connected with
thermoeconomic analysis and compared with other approaches such as linear regression
304

Conclusion
and neural networks. However, some points remain open and more additional research
lines have appeared.
The idea of causality chains has been formulated but its application has been
reduced to a simple example of the cooling tower. This formulation opens the way to
integrate models of the components in order to improve the diagnosis. Like in the
problem of simulation, more exact models lead to better results, but there would be a
point after which it is not worth to look for more accuracy. This point is probably
determined by the error provided by the method and error determined by available
measurements.
Regarding measurements availability and accuracy, it should be noted that the
availability of a large amount of plant data has been a key point in the development of
this thesis. However, it should be noted that if information with more quality would be
available (e.g. in a new power plant), point dispersion would be lower. Besides, it would
be interesting to analyze the possibility to perform real-time diagnosis. In this situation,
oscillating tendencies (caused, for example, by tanks), would be eliminated by
introducing suitable free diagnosis variables (e.g. the levels of those tanks). Finally, the
formulation developed to quantify the influence of measurement errors on diagnosis
results can be applied to calculate the diagnosis uncertainty and to suggest
improvements in instrumentation.
Up to now, diagnosis compares two situations with the final objective of taking
decisions on maintenance and repairing. Information on remaining life of components
and models of degradation would improve the usefulness of diagnosis results. This idea
could be extended not only to components (long term), but also to optimize operation in
the short term (e.g. steam sootblowing).
A diagnosis system needs the development of a thermodynamic representation of the
system, including mass and energy balances, definitions of components effectiveness
and other ancillary equations, as well as the corresponding Jacobian matrix. Automation
of this process would reduce substantially the cost of development of these systems, and
thus, their economic profitability.
An original formulation has been developed to connect quantitative causality
analysis and thermoeconomic diagnosis. This method opens a promising way to
systematically evaluate the suitability of different productive structures (and even
magnitudes different from exergy) to the diagnosis of a given system. In the long term,
305

Chapter 9
this connection perhaps would lead to develop methodologies to build structures able to
reproduce the physical behaviour of any thermal system.
Linear regression and neural networks have proved to be able to quantify the
influence of the main variables. In order to improve these results, decomposition of the
structure in several blocks should be made. The use of blocks modelled empirically
would also be introduced in the quantitative causality analysis in order to build hybrid
systems (It should be noted that a first step in this direction has been made in this work
by applying linear regression to build a simple model of the cooling tower). With this
point, and with the connection of thermoeconomic analysis, it can be seen how
quantitative causality analysis can not only be seen as an alternative diagnosis method
but also as a global method which can be used to connect others. Finally, neural
networks are very suitable to model complex time-dependent functions, such as
degradation of components or evolution of fouling, which opens a promising way to
introduce these features in a diagnosis system.
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