Exergoeconomic Cost Evaluation based on Irreversibility ... - circe

<strong>Exergoeconomic</strong> <strong>Cost</strong> <strong>Evaluation</strong> <strong>based</strong> on Irreversibility
Decomposition Analysis
César Torres and Antonio Valero
CIRCE. Centro de Investigación de Recursos y Consumos Energéticos
University of Zaragoza, Spain
e-mail: ctorrescuadra@endesa.es
ABSTRACT: The objectives of the paper are to review the concepts of the avoidable
endogenous/exogenous part of the irreversibilities or exergy destruction from
the viewpoint of thermoeconomic diagnosis methodology and, as result of this analysis,
to introduce a general technique to decompose the exergy and exergoeconomic
cost into its unavoidable, endogenous and exogenous parts.
The proposed method provides a detailed analysis of the cost formation process and
helps to analyze the potential improvement and local optimization of the system
components.
Keywords: <strong>Exergoeconomic</strong>s, Irreversibility analysis.
NOMENCLATURE
n Number of components
E Exergy rate [kW]
F Fuel exergy of a component [kW]
P Product exergy of a component [kW]
I Irreversibility of a component [kW]
Z Investment cost rate [$/h]
C <strong>Cost</strong> rate associated with an exergy
flow [$/h]
c Unit exergoeconomic cost [c/kWh]
k Component unit exergy consumption
Matrices and Vectors
ω System outputs exergy vector (n × 1)
u Unity vector (n × 1)
UD
Identity Matrix (n × n)
KD Diagonal matrix which contains the
unit consumption of each component
(n × n)
〈KP〉 Unit consumption matrix (n × n)
〈KP〉 Residue ratios Matrix (n × n)
|P〉 Product operator matrix (n × n)
|I〉 Irreversibility operator matrix (n × n)
|R〉 Residue operator matrix (n × n)
Greek Letters
κ Unit exergy consumption
τ Technical exergy saving
Subscripts and superscripts
F Related to Fuel
P Related to product
e Related to external resources
z Related to investment costs
r Related to residues
t Transpose Matrix
−1 Inverse Matrix
0 Reference state
UN Unavoidable
AV Avoidable
EN Endogenous
EX Exogenous
Abbreviations
MAI Maximum Avoidable Irreversibility
AICS Avoidable Irreversibility <strong>Cost</strong> Saving
261

1. INTRODUCTION
In practice, when attempting to achieve
effective energy savings in energy systems,
only a part of the exergy destruction or irreversibility
can be avoided by using the best
current technology.
Valero and co-workers [1, 2] stated in the
early 80’s that for any decision level of an energy
system: operation, maintenance, design
or synthesis, there exists a relationship called
Law of Saving-Investment which relates the
potential energy saving ∆I and its required
investment ∆Z following a saturation curve
of the type:
∆I = τ 1 − exp(−ε∆Z)
(1)
This equation notes two important facts.
First, the existence of an energy saving limit
for each level of decision, called Technical
exergy saving [4] and represented by τ,
which is defined as the maximum irreversibility
that it is possible to save relating to a
reference condition defined by the decision
level. And second, a saturation trend given
by the elasticity parameter ε.
Independently, Tsataronis and coworkers
[3] also states that the exergy destruction
of an energy system component can be decomposed
into its avoidable and unavoidable
parts:
I = I UN + I AV
(2)
In an energy analysis, the avoidable irreversibility
IAV gives a realistic picture of the
potential for improving the thermodynamic
effectiveness of each component. The unavoidable
irreversibility IUN cannot be reduced
due to technological limitations (e.g.
availability and cost of materials and manufacturing
methods)
Theoretically, the investment cost per production
unit could be expressed as function
of the unit consumption k:
α k
Z = Z0 + β P
k − k0
γ
262
where k0 is the unit consumption at an avoidable
state x0, such as the investment cost Z
of a system component verifies:
lim Z(k) = ∞
k→k0
this state is called Maximum Avoidable Irreversibility
(MAI) state. In practical applications,
the MAI state could be determined by
selecting the most important thermodynamic
parameters to obtain its maximum avoided
irreversibility per production unit, such a
component will have a very large investment
cost.
Then, the avoidable exergy destruction
could be considered as a particular instance
of the more general concept of technical exergy
saving applied to the design level. The
avoidable irreversibility is the difference between
the current and the MAI state, and applying
eq. (1), we get:
I AV = ∆I = τ (3)
The technical exergy saving is the base
idea of the thermoeconomic diagnosis
methodology [5], developed by Valero and
co-workers from the beginning of the 90’s
and formerly called Perturbations Theory
[6]. Thermoeconomic diagnosis [7]
states that the variation of the irreversibility
respect to a reference state {x 0 } could be
decomposing into two parts:
∆I = P(x 0 ) ∆k + k(x) ∆P (4)
The first term represents the malfunction
or endogenous irreversibility due to the
degradation of its efficiency regarding to a
reference state and the second term represents
the irreversibility variation caused by
the production conditions, which origin are
the inefficiencies of the remaining components
of the system, called dysfunction or exogenous
irreversibilities.
The Advanced exergoeconomic evaluation
(AEE) [8, 9] applies these concepts to
the evaluation of thermal system cost effectiveness,
<strong>based</strong> on exergy destruction decomposition
analysis. The avoidable part of
the exergy destruction could be decomposed
into the endogenous part of the exergy destruction
in a system component I EN , which

is associated only with the irreversibilities
occurring in the component when all other
components operate in an ideal way, and the
exogenous part of the exergy destruction I EX
caused in the component by the irreversibilities
that occur in the remaining components,
therefore:
I AV = I EN + I EX
(5)
The notion of ideal or theoretical system
is associated with the MAI state. The
method is applied to refrigeration cycles,
which defines a theoretical and a real system
and compare them.
Therefore, the aim of this paper is to analyze
the AEE methodology from the perspective
of the Thermoeconomic Diagnosis
in order to clarify these concepts and to
take advantage of the contributions of both
methodologies to improve them and to integrate
into a more general theory of energy
saving, which fundamentals are exposed in
ref. [2].
7
8 9
1
5
HRSG
2
6
Air
Preheater
4
12
2
Combustion
Chamber
11
3
Air
Gas
Compressor
Turbine
Figure 1: Flow Diagram of CGAM System.
The well-known CGAM problem [10] is
used to illustrate the theoretical results. It is a
cogeneration plant, which purpose is to generate
a net electric power of 30 MW and to
provide saturated steam at 20 bar. It consists
of a gas turbine system and a heat recovery
steam generator. The flow diagram of the
plant is depicted in Fig. 1. The productive
structure and the corresponding exergy flow
values are shown in Fig. 2.
The MAI state definition of the problem is
made according reference [3]. Table I shows
1
3
5
4
10
the values of the system parameters applied
to the current and the MAI state. The thermoeconomic
model, which is <strong>based</strong> on EES
and TAESS, used to make the calculation
could be found in reference [11].
Table I: CGAM Parameters.
Parameter MAI state CUR state
∆Pcc 0.005 0.05
ηAC 0.9 0.84
ηGT 0.92 0.88
ηAP 0.98 0.82
T4 [ o C] 1300 1200
∆TP [ o C] 2 8
m9 [kg/s] 13.83 7.90
2. COST DECOMPOSITION
Symbolic <strong>Exergoeconomic</strong>s [12] allows
relating the production and other thermoeconomic
variables of the plant as a function
of the efficiency of each component and the
system outputs, by means of the production
operator 1 :
|P〉 ≡ (UD − 〈KP〉) −1
(6)
where 〈KP〉 is the unit consumption matrix
which elements are defined as κi j ≡ Ei j/P j.
According to ref. [13] is possible to decompose
the unit production cost into three
types of contributions:
cP = c e P
+ cz
P + cr P
(7)
The contribution caused by external fuel resources:
t e
cP ≡ t |P〉 κe
(8)
where κe,i = C0,i/Pi is the cost of the external
resources consumed in a component per
production unit.
The contribution due to the investment
and maintenance cost of the equipment is:
t z
cP ≡ t |P〉 zP
(9)
where zP,i = Zi/Pi is the equipment cost per
production unit.
1 It is equivalent, in terms of the economic Input-
Output theory, to the Leontief inverse matrix.
263

E12
1
E1
E4 − E3
Air-Gases
E2 − E1
E5 − E4
E3 − E2
E5 − E6
E6 − E7
E7
2
3
4
5
E11
E10 + E11
E9 − E8
Work
E10
E9
E8
Water-Steam
Flow# MAI state CUR state
1 0.0 0.0
2 20321.0 28151.3
3 39681.5 46824.8
4 84735.8 103406.5
5 31921.6 40161.2
6 11295.9 19277.5
7 1939.5 2095.7
8 0.0 0.0
9 6968.2 12427.1
10 30000.0 30000.0
11 21455.2 30591.5
12 63405.1 82073.4
Figure 2: Flow Exergies [KW] and Fuel Product Diagram of CGAM System.
Finally, the production cost due to the
residues could be computed as:
c r t |R〉
P =
UD − t
e
cP + c
|R〉
z
P (10)
where |R〉 ≡ 〈KR〉 |P〉 is the residue operator,
introduced in [14].
In the case of a sequential system where
the product of a process is the fuel of the next
one, the product exergy cost of the i–th process
is equal to its exergy plus the sum of all
irreversibilities of the processes required to
obtain it. This fact could be generalized to
any system no matter how complex is, see
ref. [15], by mean of the equation:
P ∗ n
i = Pi +
264
j=1
πi j I j
where πi j are the exergy cost distribution ratios.
This equation could be written in terms
of unit exergy cost as:
k ∗ P,i = 1 +
n
j=1
φ ji
where φ jiPi = πi jI j, and could be written in
matrix format as:
t k ∗ P = t u (UD + |I〉) (11)
where |I〉 ≡ (KD − UD) |P〉 is the irreversibility
operator, that satisfies:
I = |I〉 ω (12)
If there is only one external fuel with a
known unit price ce the exergoeconomic cost
is simply cP,i = ce · k∗ P,i . Moreover, equa-
tion (11) could be extended to compute the
exergoeconomic cost for a general case, with
several external flows, by means of the equation:
t cP = t c ∗ e (UD + |I〉) (13)
where c ∗ e represents the unit price of the external
resources that are used for each component,
and it is evaluated as:
(UD − 〈PF〉) c ∗ e = ce
(14)
where 〈PF〉 is the junction ratios matrix,
see [16], which elements are qi j ≡ Ei j/F j
and satisfy the relationship:
κi j = qi j ki
(15)
The cost decomposition, according to
eqs. (7) and (13), is illustrated in Figure 4.
It shows how the unit production cost of
the CGAM components is decomposed into
their different contributions: fuel price, investment
and residues costs, and the irreversibilities
costs. It also proves that the
main contributions are the irreversibilities
on the combustion chamber and the HRSG.
These results will be reviewed later.

The parameter c ∗ e plays an important role,
not only to generalize the irreversibility cost
relationship, but also to determine the price
of the external resources used in each component
and to find alternatives to reduce it.
3. IRREVERSIBILITY
DECOMPOSITION
Now, the question is to evaluate how
much irreversibility could be saving if the
system is moving from the current state to
the MAI state, and how much it does cost.
P
P0
I UN
I EX
MAI State
I EN
1 k0 k
CUR State
Figure 3: Irreversibility decomposition.
According to the ideas discussed in previous
section, the irreversibility of each system
component could be decomposed into
its unavoidable and avoidable irreversibility
and moreover its avoidable part could be decomposed
into its endogenous and exogenous
parts, on the following way:
I = (k 0 − 1)P 0 + (k − k 0 )P
+ (k 0 − 1)∆P
(16)
where k 0 and P 0 are the unit consumption
and production in the MAI state, meanwhile
k and P represents these values in the current
state. The first term, denoted as I UN , represents
the unavoidable irreversibility. The
second term I EN represents the avoidable
endogenous irreversibility and the last term
I EX is the avoidable exogenous irreversibility.
Figure 3 shows graphically this decomposition.
Using Symbolic <strong>Exergoeconomic</strong> notation
the unavoidable irreversibility could be
written as a function of the production demand
at the MAI state as:
I UN = |I〉 UN ω 0 where
|I〉 UN ≡ K 0
UN
D − UD |P〉
the avoidable irreversibility as:
I EN = |I〉 EN ω 0 where
|I〉 EN ≡ KD − K 0
UN
D |P〉
(17)
(18)
and the exogenous irreversibility that also
depends on the system output variation:
I EX = |I〉 EX ω 0 + |I〉 ∆ω where
|I〉 EX ≡ |I〉 ∆ 〈KP〉 |P〉 UN
(19)
Therefore the total irreversibility of each
component verifies:
and
I = I UN + I EN + I EX
|I〉 = |I〉 UN + |I〉 EN + |I〉 EX
(20)
(21)
Table II shows the decomposition of the irreversibilities
of each component into their
parts. In case of the combustor chamber the
endogenous irreversibilities are lower than
10%, meanwhile the air-preheater represents
near 50%.
Table II: Irreversibility Decomposition for
CGAM system [kW].
Device I I UN I EN I EX
CC 25492 18351 1947 5193
AC 2440 1134 627 679
GT 2654 1359 895 400
APH 2210 1265 1026 -81
HRSG 4755 2388 278 2089
Total 37551 24497 4773 8280
In the same way that the malfunction location
is the key for thermoeconomic diagnosis,
the endogenous exergy destruction is
the appropriate measure to locate the potential
improvement of the system components,
instead of the total irreversibility.
265

266
CC
AC
GT
AP
HRSG
STK
CC
AC
GT
AP
HRSG
STK
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
<strong>Cost</strong> [c/kWh]
Figure 4: <strong>Cost</strong> decomposition of CGAM System.
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
<strong>Cost</strong> [c/kWh]
Figure 5: Avoidable/Unavoidable cost decomposition of CGAM System.
c ∗ e
I1
I2
I3
I4
I5
Z
R
c ∗ e
I UN
I AV
1
I AV
2
I AV
3
I AV
4
I AV
5
Z
R

4. IRREVERSIBILITY AND COST
DECOMPOSITION
The combination of previous eq. (13),
with the irreversibility operator decomposition
formula (21) permits to decompose the
unit production cost into the different irreversibility
cost causes.
cP = c UN
P
+ cAV
P
where the unavoidable cost could is defined
as:
t UN
cP = t c ∗
e UD + |I〉 UN
(22)
and the avoidable cost as:
EN EX
|I〉 + |I〉
t c AV
P = t c ∗ e
(23)
In practice, it is only interesting to know
the part of the avoidable cost of a component
caused by its own irreversibilities and
the cost due to the avoidable irreversibilities
both endogenous and exogenous of the rest
of components:
c AV
P,i = c∗ e,i φ AV
ii +
i j
c ∗ e, j φ AV
ji
(24)
Considering the relationship between the
irreversibility and the coefficients of the irreversibility
operator: φAV ji Pi = πi jI AV
j , previous
equation could be written in terms of
cost rate as:
C AV
P,i = πii c ∗ e,i I AV
i +
ji
πi j c ∗ e, j I AV
j
(25)
The proposed cost decomposition, introduced
here, it is an important enhance in the
knowledge of the cost formation process, because
it isolates the unavoidable cost, i.e. the
cost is not possible to save with the current
design. It calculates the contribution of the
avoidable irreversibility of each component
to the production costs. This information
could be applied to take decisions in order to
make investments that improve the efficiency
and to reduce the endogenous irreversibilities
and as consequence the production costs.
Table III shows the unit production cost
decomposition into the different concepts explained
here. Figure 5 presents the same
graph depicted in Figure 4, but now it is split
into the contribution of the unavoidable and
avoidable irreversibilities. Here, it is demonstrated
that only a small part of cost is caused
by the avoidable irreversibility of combustor
and HRSG. The main contribution is now
the unavoidable irreversibility. Moreover,
the contributions due to the investment and
residue costs are, in most of the cases, higher
than the cost originated by the avoidable irreversibility,
and the cost saving margin is low.
5. AVOIDABLE IRREVERSIBILITY
COST SAVING
As stated by the thermoeconomic diagnosis
methodology, the cost of a component
malfunction is the sum of its malfunction
(endogenous irreversibilities) plus the sum
of all dysfunctions (exogenous irreversibilities)
generated in the rest of components.
The fuel impact formula [7] asserts that the
exergy technical saving of a system is the
sum of the malfunction cost of all components.
This formula could be generalized to
compute the Total fuel cost saving in monetary
terms, and it could be written as:
∆C e T =
n
c0 j ∆E0 j =
(26)
where:
j
C e
0 =
i=0
c e P,s ∆ωs
s
∆C e i
(27)
is the cost saving [$/h] due to the output variation
(production demand and residues generation),
and:
∆C e i =
⎛
c e P, j ∆κ ⎞
ji (28)
⎜⎝ c0i ∆κ0i +
j
⎟⎠ P0i
is the cost that we can save for each component
by reducing its local unit consumption
in ∆κ ji. If ∆Zi is the required investment,
per unit of time, required to obtain a
, then the cost benefit ra-
cost saving of ∆C e i
tio CBRi = ∆Zi/∆C e i
is a good index for the
feasibility estimation of a local investment.
This approach could be easily applied to
compute the Avoidable Irreversibility <strong>Cost</strong>
267

Table III: Unit exergoeconomic cost decomposition [c/kWh].
Device cF cP c UN
P
c EN
P
c EX
P c r P c z
P
CC 1.4400 2.1477 2.0265 0.0622 0.0000 0.0527 0.0063
AC 2.7915 3.2017 2.3509 0.2182 0.0806 0.1749 0.3770
GT 2.6009 2.7915 2.2266 0.1646 0.0472 0.1125 0.2407
APH 2.6009 3.0683 2.3111 0.2212 0.0803 0.1732 0.2826
HRSG 2.6009 3.8293 2.9128 0.2400 0.0771 0.1490 0.4504
Stack 2.6009 2.6009 2.1693 0.1360 0.0308 0.1077 0.1571
Saving (AICS) i.e. the cost, in monetary
terms, of the total avoidable irreversibility:
C AV
T =
n
C AV
i
(29)
i=0
Applying eq. (15) to eq. (28), the avoidable
cost saving for each individual component is
written as:
C AV
i
= ce F,i IEN
i
+
c e P, j ∆q ji F 0
i
j
(30)
because the fact j ∆q ji = 0, it is possible to
estimate CAV i ce F,i IEN
i as the cost of the en-
dogenous irreversibility.
The exergoeconomic factor, see [17],
compares the sources contributing to the
cost increase between cF and cP and shows
the relation between the investment cost and
the cost of exergy destruction. This factor
is used as measure of the importance of a
component from the cost viewpoint within
the overall system. The lower the value is
the more feasible the potential cost saving
investment is. Variations of this factor, using
the avoidable and endogenous irreversibilities,
are introduced in [9]. According to
eq. (28), the exergoeconomic factor could be
redefined as:
f AV
i
=
C AV
i
Z AV
i
+ ZAV
i
(31)
where ZAV i is the part of investment cost that
depends on the component production. This
definition takes in consideration the cost of
the endogenous irreversibility of the compo-
268
nent instead its total irreversibility. More-
over the expression C AV
k
+ ZAV
k could be used
as cost function for local optimization to improve
the quality of the solution and to increase
the convergence speed.
Table IV shows the values of the avoidable
cost saving and proposed exergoeconomic
factors. These value ratios suggest
an investment to improve the component efficiency
could be made in the combustor, but
the investment in other components as the
GT, AC and HRSG could be rejected.
Table IV: Avoidable irreversibility cost saving
and exergoeconomic factors.
Device
Z
[$/h]
C AV
[$/h]
fZ
[%]
f AV
Z
[%]
CC 3.542 28.044 0.96 11.22
AC 32.507 15.297 32.30 68.00
GT 46.470 17.254 40.24 72.92
AP 19.960 22.553 25.77 46.95
HRSG 28.986 5.848 18.99 83.21
6. CONCLUSIONS
The concepts of unavoidable/avoidable
and endogenous/exogenous part of the irreversibility
or exergy destruction in energy
system are reviewed on this paper.
The first aim of the paper has been to
demonstrate that the Advanced <strong>Exergoeconomic</strong>
<strong>Evaluation</strong> methodology could be
explained as a particular application of the
Thermoeconomic Diagnosis methodology.
Both methodologies have a common objective:
to evaluate the cost effectiveness of an

energy system and to estimate the profitability
of potential improvements.
They have got the same conclusions: improvement
efforts should be focus on the
avoidable endogenous irreversibilities. The
AEE has been applied to theoretical cycles,
but it is difficult to apply to real plants.
Thermoeconomic Diagnosis (TD), as a
part of the Exergy <strong>Cost</strong> Theory (ECT), could
be applied to any system no matter how complex.
The more difficult task in this case is
to determine the maximun avoidable irreversibility
state that fulfills the production demand
of the system. Once the MAI state is
correctly defined, the TD formulation could
be applied to get the values of endogenous
and exogenous part of the avoidable irreversibility
and compute the associated cost saving
for each component.
The previous ECT works provided a
method to analyze the cost formation process,
<strong>based</strong> on computing the production
costs as the sum of the irreversibilities of
the processes required to obtain a product.
The main practical contribution of the paper
is shown how to decompose the production
cost into its unavoidable and avoidable parts.
By means of this fact it is possible to determine
which part of the cost could not be
reduced because of technological and thermodynamic
limitations, which part of the
cost is due to the investment costs of the system
or due to the formation and abatement
cost of the residues and finally which part
is caused by the avoidable irreversibilities
of the own component and how much they
contribute to the cost of the rest of components.
The results allow taking a step ahead
in the identification of the process of cost
formation of products and residues.
This approach could be generalized to
other decision levels than the design, such as
operation and maintenance, by decomposing
the technical exergy savings into the contribution
of the free variable for the decision
level, according to the Quantitative Causality
Analysis [18].
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