GPS-X Technical Reference

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Optimizer 386 Portmanteau Sub-Section Portmanteau test on weighted residuals (ON - OFF): This switch turns the Portmanteau test on or off. The Portmanteau test is used to detect trends in the weighted residuals. The Portmanteau test is designed for data taken in sequence (e.g. time or space). If trends are present, the residuals are not independent. This violates one of the assumptions of the maximum likelihood method and indicates that the model does not account for all of the non-random variability in the data. An appropriate message is printed depending upon the outcome of the test. This test is only reported when the maximum likelihood objective function is used. It does not affect the outcome of an optimization run. Maximum number of lags used in portmanteau statistic: By default, the Portmanteau statistic involves autocorrelations at lags up to half the length of the time series. This setting can be used to impose further restriction on the number of autocorrelations taken into account. A large value effectively disables this option. This setting does not affect the outcome of an optimization run. OPTIMIZATION STRATEGIES Before using the optimizer tool to estimate optimal parameter values or optimize operating conditions, it is usually best to experiment with manual adjustment of the optimization variables. By conducting interactive simulations you can observe the effects of the model parameters on the response variables of interest. With this information you will be able to make better judgments on appropriate variables to use in a parameter estimation or optimization run. For example, you can set up an interactive simulation with slider controls for the parameters and then try adjusting these variables to try and achieve a visually acceptable fit. You can plot actual data along with the target variables so that you can compare simulation and actual data. This approach is useful for generating starting guesses for parameter estimation and process optimization runs. GPS-X Technical Reference
387 Optimizer APPENDIX A: MAXIMUM LIKELIHOOD METHOD Introduction In this appendix the maximum likelihood method is discussed in more detail. First a general introduction on parameter estimation is given based on material found in Bard (1974). This is followed by a detailed discussion of the maximum likelihood method as implemented in GPS-X. Finally, a brief description of the sum of squares objective function is given. Parameter Estimation Parameter estimation is the procedure of fitting a mathematical model of a process to measured data by calculating optimal estimates of the model parameters. Parameter estimation differs from simple curve fitting in that the criterion used to judge the best fit is not arbitrary but is based on statistical considerations. In addition, the model structure is based on theoretical principles and the model parameters often have physical significance. In curve fitting, the choice of model structure is more arbitrary and is often chosen to simplify the computations. The aim of parameter estimation is to not only fit a model to data but to calculate parameter values that are good estimates of the true values of the physical quantities. Parameter estimation techniques can be applied to empirical models, but the statistical properties of the estimates may not be as meaningful in a physical sense. In addition, empirical models are not as well suited for extrapolation as mechanistic models. Parameter estimation is an important step in the development of mathematical process models. Process models contain parameters with physical significance that may vary significantly from plant to plant. To develop process models that can be used for predictive purposes, it is important to estimate the unknown process parameters using measured process data. Using literature values for the unknown parameters will often result in a model that is not very useful for predicting actual plant behavior. Maximum Likelihood Method GPS-X uses the maximum likelihood method for parameter estimation. In the maximum likelihood method, the optimal parameter estimates are obtained by maximizing the joint probability density function of the measurements. This joint probability density function is a function of the parameters and is known as the likelihood function. The form of the likelihood function depends on the structure of the experimental error. GPS-X Technical Reference
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GPS-X Technical Reference GPS-X Ver
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Table of Contents iii Table of Cont
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v Table of Contents Sedimentation M
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vii Table of Contents Real Time Clo
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ix Table of Contents Figure 6-6 - P
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xi Table of Contents Figure 10-8 -
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xiii Table of Contents Figure 15-6
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xv Table of Contents Table 6-2 - Mo
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17 Modelling Fundamentals Experimen
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19 Modelling Fundamentals The proce
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21 Modelling Fundamentals Overview
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23 Modelling Fundamentals Primary a
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25 Modelling Fundamentals Vesilin
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27 GPS-X Objects CHAPTER 2 GPS-X Ob
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29 GPS- X Objects The mass balance
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31 GPS- X Objects Combining Equatio
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33 GPS-X State Variable Libraries S
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35 GPS-X State Variable Libraries C
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37 GPS-X State Variable Libraries C
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39 GPS-X Composite Variable Librari
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41 GPS-X Composite Variable Calcula
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55 Composite Variable Calculations
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61 Influent Models CHAPTER 5 Influe
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63 Influent Models Figure 5-3 - Inf
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65 Influent Models The three influe
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67 Influent Models Total runoff (Q
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69 Influent Models These inputs are
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71 Influent Models CODFractions COD
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73 Influent Models TSSCOD Figure 5-
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75 Influent Models Figure 5-13 - CN
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77 Influent Models CN and CNIP Libr
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83 Influent Models INFLUENT OBJECTS
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85 Influent Models Figure 5-22 - In
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87 Influent Models Chemical Dosage
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89 Influent Models Figure 5-29 - Ac
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91 Influent Models Figure 5-33 - Se
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93 Influent Models Figure 5-36 - Se
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95 Suspended Growth Models where: V
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97 Suspended Growth Models The baro
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99 Suspended Growth Models Oxygen M
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101 Suspended Growth Models In the
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103 Suspended Growth Models The SOT
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105 Suspended Growth Models Standar
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107 Suspended Growth Models The reg
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109 Suspended Growth Models SUMMARY
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111 Suspended Growth Models Figure
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113 Suspended Growth Models General
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115 Suspended Growth Models Number
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117 Suspended Growth Models Elevati
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121 Suspended Growth Models AERATIO
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123 Suspended Growth Models The ent
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125 Suspended Growth Models The org
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127 Suspended Growth Models Process
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129 Suspended Growth Models It i
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131 Suspended Growth Models Nitroge
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133 Suspended Growth Models Include
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137 Suspended Growth Models Althoug
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139 Suspended Growth Models Particu
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141 Suspended Growth Models 9. Grow
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143 Suspended Growth Models 21. Sto
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145 Suspended Growth Models 33. Gro
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147 Suspended Growth Models To mode
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149 Suspended Growth Models Algebra
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151 Suspended Growth Models The sto
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153 Suspended Growth Models Decay o
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155 Suspended Growth Models SUGGEST
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157 Suspended Growth Models Special
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159 Suspended Growth Models Table 6
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161 Suspended Growth Models The GPS
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163 Suspended Growth Models where:
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165 Suspended Growth Models The max
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169 Suspended Growth Models ANAEROB
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171 Suspended Growth Models Simple
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173 Suspended Growth Models With th
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175 Suspended Growth Models Manual
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177 Suspended Growth Models Equatio
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179 Suspended Growth Models The met
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181 Suspended Growth Models Pond Mo
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183 Suspended Growth Models Special
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187 Suspended Growth Models aerated
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193 Suspended Growth Models Calcula
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195 Suspended Growth Models The vap
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209 Suspended Growth Models MODELLI
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211 Suspended Growth Models TEMPERA
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213 Suspended Growth Models Table 6
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215 Suspended Growth Models The PAC
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217 Suspended Growth Models CHAPTER
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219 Attached-Growth Models Each lay
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221 Attached-Growth Models Equation
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223 Attached-Growth Models The mode
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225 Attached-Growth Models The phys
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227 Attached-Growth Models Liquid F
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233 Attached-Growth Models The SBC
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235 Attached-Growth Models Based on
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239 Attached-Growth Models Hydrauli
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241 Attached-Growth Models ADVANCED
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243 Attached-Growth Models Filtrati
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245 Attached-Growth Models Operatio
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247 Attached-Growth Models This for
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249 Attached-Growth Models HYBRID S
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251 Attached-Growth Models Figure 7
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253 Sedimentation and Flotation Mod
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267 Sand Filtration Models Floatati
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271 Sand Filtration Models The mass
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273 Sand Filtration Models Figure 9
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275 Digestion Models Figure 10-1 -
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277 Digestion Models Seven yield co
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279 Digestion Models Based on the a
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281 Digestion Models Combining the
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283 Digestion Models The Operationa
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285 Digestion Models Kinetic and St
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287 Digestion Models The rest of th
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291 Digestion Models ADM1 State Var
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293 Digestion Models AEROBIC DIGEST
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297 Other Models CHAPTER 11 Other M
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299 Others Models High volume for p
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301 Others Models The sludge pretre
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311 Others Models DISINFECTION UNIT
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317 Others Models DISC AND BELT MIC
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319 Others Models Figure 11-12 - Pl
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321 Others Models Physical Setup Fo
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323 Others Models Exponential const
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325 Others Models BLACK BOX The Bla
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327 Tools Object Figure 11-19 - Spe
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329 Tools Object Figure 11-21 - Inp
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331 Tools Object Typical Model Outp
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333 Tools Object The variable speed
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Page 361 and 362: 361 Operating Cost Models CHAPTER 1
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Page 405 and 406: 405 Optimizer To check for violatio
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Page 409 and 410: 409 Optimizer APPENDIX C: NOMENCLAT
Page 411 and 412: 411 Optimizer APPENDIX D: REFERENCE
Page 413 and 414: 413 Miscellaneous CHAPTER 15 Miscel
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Page 433 and 434: Appendix A - Petersen Matrices A-1
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ASM2d in CNPLIB Appendix A - Peters
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15 Lysis of XBP 16 Lysis of XPP 17
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Appendix A - Petersen Matrices A-9
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NewGeneral in CNPLIB Appendix A - P
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NewGeneral in CNPLIB Appendix A - P
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Component i Process Rates Appendix
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Appendix A - Petersen Matrices A-17
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2 Appendix A - Prefermenter Peterse
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D-2 Appendix D - References Brockwe
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D-4 Appendix D - References Hur, D.
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D-6 Appendix D - References Reilly,
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D-8 Appendix D - References Wuhrman
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