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A new approach for optimization of thermal power plant based

on the exergoeconomic analysis and structural optimization

method: Application to the CGAM problem

Seyyed Masoud Seyyedi *, Hossein Ajam, Said Farahat

Department of Mechanical Engineering, University of Sistan and Baluchestan, Zahedan 98164, Iran

a r t i c l e i n f o

Article history:

Received 28 July 2009

Accepted 21 March 2010

Available online 24 April 2010

Keywords:

Optimization

Exergoeconomic analysis

Structural method

a b s t r a c t

In large thermal systems, which have many design variables, conventional mathematical optimization

methods are not efficient. Thus, exergoeconomic analysis can be used to assist optimization in these

systems. In this paper a new iterative approach for optimization of large thermal systems is suggested.

The proposed methodology uses exergoeconomic analysis, sensitivity analysis, and structural optimiza-

tion method which are applied to determine sum of the investment and exergy destruction cost flow

rates for each component, the importance of each decision variable and minimization of the total cost

flow rate, respectively. Applicability to the large real complex thermal systems and rapid convergency

are characteristics of this new iterative methodology. The proposed methodology is applied to the bench-

mark CGAM cogeneration system to show how it minimizes the total cost flow rate of operation for the

installation. Results are compared with original CGAM problem.

Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The development of design techniques for an energy system

with minimized costs is a necessity in a world with finite natural

resources and the increase of the energy demand in developing

countries [1]. Optimization has always been one of the most inter-

ested and essential subjective in the design of energy systems.

Usually we are interested to know optimum conditions of thermal

systems. Thus we need methods for optimization of such systems.

In large complex thermal systems, which have many design vari-

ables, conventional mathematical optimization methods are not

efficient. Thus, exergoeconomic analysis can be used to assist opti-

mization in these systems. On the other hand, complex thermal

systems cannot always be optimized using mathematical optimi-

zation techniques. The reasons include incomplete models, system

complexity and structural changes [2].

Exergoeconomic (Thermoeconomic) is the branch of engineer-

ing that combines exergy analysis with economic constraints to

provide the system designer with information not available

through conventional energy analysis and economic evaluation

[3]. The objective of a thermoeconomic analysis might be: (a) to

calculate separately the cost of each product generated by a system

having more than one product; (b) to understand the cost forma-

tion process and the flow of costs in the system; (c) to optimize

specific variables in a single component; or (d) to optimize the

overall system [2]. A thermodynamic optimization aims at mini-mizing the thermodynamic inefficiencies: exergy destruction and

exergy loss. The objective of a thermoeconomic optimization, how-

ever, is to minimize costs, including costs owing to thermodynamic

inefficiencies [4].

In 1994, a cogeneration plant, known as the CGAM problem,

was defined as a test case by a group of concerned specialists in

the filed of exergoeconomic, in order to compare their different

thermoeconomic methodologies [5–9]. Exergoeconomic methods

can be grouped in two classes: the algebraic methods and the cal-

culus methods [10,11]. All of these methods are based on an exer-

goeconomic model, which basically consists of an interposed set of

linear exergy equations that define the productive objective of

each component of the plant [3]. Some of the algebraic methods

are: exergetic cost theory (ECT) [12], average cost theory (ACT)

[4], specific cost exergy costing method (SPECO) [13] and modified

productive structural analysis (MOPSA) [14,15]. Furthermore,

some of the calculus methods are: thermoeconomical functional

analysis (TFA) [16,17] and engineering functional analysis (EFA)

[18]. Then, in 1992, Erlach et al. [19] developed a common mathe-

matical language for exergoeconomics, called the structural theory

of thermoeconomics. Furthermore, Hua et al. [20], El-Sayed [21],

Benelmir and Feidt [22] have proposed decomposition strategies

based on second law reasoning to reduce complexity in the

optimization of complete systems. A critical review of relevant

publications regarding exergy and exergoeconomic analysis can

be found in articles by Leonardo et al. [23], Sahoo [3] and Zhang

et al. [24]. In 1997, Tsatsaronis and Moran [2], showed how certain

0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.enconman.2010.03.014

* Corresponding author. Tel.: +98 541 2426206; fax: +98 541 2447092.

E-mail address: [email protected](S.M. Seyyedi).

Energy Conversion and Management 51 (2010) 2202–2211

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exergy-related variables can be used to minimize the cost of a ther-

mal system. They applied this iterative optimization technique to

the benchmark CGAM problem. In 2004, Leonardo et al. [23] pre-

sented the development and automated implementation of an iter-

ative methodology for exergoeconomic improvement of thermal

systems integrated with a process simulator, so as to be applicable

to real, complex plants. Also, see Refs. [25,26]. Most exergoeco-nomic optimization theories have been applied to relatively simple

systems only. Conventional mathematical optimization, exergo-

economic or not, of real thermal systems are large scale problems,

due to their complicated nonlinear characteristics and because the

mass, energy and exergy (or entropy) balance equations must be

introduced in the problem as restrictions [23].

In this paper, a new iterative method for the optimization of

thermal systems is developed using exergoeconomic analysis, sen-

sitivity analysis, and structural optimization method. Exergo-

economoic analysis is used to determine sum of the investment

and exergy destruction cost flow rates for each component. A

numerical sensitivity analysis is performed in order to determine

the importance of each decision variable. Finally, the total cost flow

rate is minimized and the optimum vector of decision variables is

determined by using structural optimization method. The advanta-

ges of this new iterative method are: (1) it can be applied to the

real complex large thermal systems; (2) the procedure of optimiza-

tion is performed without user interface, i.e. there is no to the deci-

sion of designer in each iteration, and (3) since it uses a numerical

sensitivity analysis, convergency is improved. In order to represent

how this new methodology can be used for optimization of real

complex large thermal systems, it is applied to the benchmark

CGAM cogeneration system as a test case and results are compared

with the original CGAM problem.

2. CGAM problem

In 1990, a group of concerned specialists in the filed of exergo-economic (C. Frangopoulos, G. Tsatsaronis, A. Valero, and M. von

Spakovsky) decided to compare their methodologies by solving a

predefined and simple problem of optimization: the CGAM prob-

lem, which was named after the first initials of the participating

investigators. The objective of the CGAM problem was to show

how the methodologies were applied, what concepts were used

and what numbers were obtained in a simple and specific problem.

In the final analysis, the aim of CGAM problem was the unificationof exergoeconomic methodologies [3]. The CGAM system refers to

a cogeneration plant which delivers 30 MW of electricity and

14 kg sÀ1 of saturated steam at 20 bar. A schematic of cogeneration

plant is shown in Fig. 1. The system consists of an air compressor

(AC), an air preheater (APH), a combustion chamber (CC), a gas-tur-

bine (GT) and a heat-recovery steam generator (HRSG). The envi-

ronment conditions are defined as T 0 = 298.15 K and P 0 = 1.013 bar.

The objective function is the total cost flow rate of operation for

the installation that is obtained from

_C T ¼ _mf c f LHV þX5

i¼1

_ Z i ð1Þ

where C ˙ T ($/s) is the total cost flow rate of fuel and equipment and_

Z i in ($/s) is the cost flow rate associated with capital investmentand the maintenance cost for the ith component (i = AC, CC, GT,

APH, HRSG).

Also, exergetic efficiency of the cycle (gII) is defined as:

gII ¼_W net þ _msðe9 À e8Þ

_mf ef

ð2Þ

The key design variables, (the decision variables), for the cogen-

eration system are the compressor pressure ratio PR, the isentropic

compressor efficiency gAC, the isentropic turbine efficiency gGT, the

temperature of the air entering the combustion chamber T 3, and

the temperature of the combustion products entering the gas tur-

bine T 4. The objective is to minimize Eq. (1) subject to the con-

straints imposed by the physical, thermodynamic and cost

models of the installation. For more details see Appendix A andRef. [5].

Nomenclature

c cost per exergy unit ($/kJ)C ˙ exergetic cost flow rates ($/s)com componentCRF capital recovery factorE ˙ exergy flow rate (kW)

i ith plant component_I irreversibility rate (kW)

j jth decision variableLHV lower heating value of fuel (kJ/kg)_m mass flow rate

N number of the hours of plant operation per year (h/year) p parameter for sensitivity analysis, expressions (5) and

(6)PR pressure ratioT temperature (K)_W net net work of the cycle (kW)

x decision variable X vector of decision variables Z purchase costs of the ith component ($),_ Z investment cost flow rate ($/s)

Greek lettersa user – prescribed tolerance for the iterative process, Eq.

(7)e component exergetic efficienciesf capital cost coefficient

g isentropic efficiencygII exergetic efficiency of the cyclel defined in Eq. (8)r coefficient of structural bondsu maintenance factor

Subscripts0 index for environment (reference state)AC air compressorAPH air preHeaterCC combustion chamberD destructionf fuelGT gas-turbineHRSG heat-recovery steam generatorIn inletIter iterationk kth plant componentL lowerOPT optimumOut outletP productS steamT totalU upper

S.M. Seyyedi et al. / Energy Conversion and Management 51 (2010) 2202–2211 2203

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3. Structural optimization method

The purpose of this optimization is to determine the capital cost

of a selected component (system element) corresponding to the

minimum annual operating cost of the plant with a given plant

output and thus, by implication, corresponding to the minimum

unit cost of the product [27]. The following relation for the kth

component must be satisfied until the total cost flow rate of oper-

ation for the installation, C ˙ T , be minimized:

@ _I k@ xi

!OPT

¼ À1

c Ik;i

@ _ Z k@ xi

!ð3Þ

In the proposed methodology, Eq. (3) is numerically calculated

using Newton’s finite difference formula:

_I kð xi þ 1Þ À _I kð xiÞ

D xi ! ¼ À

1

c Ik;i

_ Z kð xiþ1Þ À _ Z kð xiÞ

D xi ! ð4Þ

where D xi ¼ xi þ 1 À xi . For more details see Appendix B and Ref.

[27].

4. Proposed iterative methodology

In this paper, we have proposed a new approach for optimiza-

tion of complex thermal power plant. It is worth to mention that

structural optimization method (see Section 3 and Appendix B) is

a method of optimization for one component of the system, so that

the total cost flow rate of the system to be minimized while we

have applied it for optimization of complex thermal power plant

using exergoeconomic analysis (here, ACT method [4]) and numer-

ical sensitivity analysis.

4.1. Numerical sensitivity analysis

A numerical sensitivity analysis is performed for each design

variable (decision variable) in order to determine the importance

of each decision variable. The sensitivity analysis is performed

for each xi according to the following procedure:

if xiD xi

DgII

gII

> p; xi affects exergetic efficiency; and ð5Þ

if xiD xi

D _C T_C T

> p; xi affects total cost flow rate of the system

ð6Þ

Thus, each decision variable will be settled in one of the follow-

ing groups:

Group 1: if decision variable affects both C ˙ T and gII.

Group 2: if decision variable affects on the C ˙ T only.

Group 3: if decision variable affects on the gII only.

Group 4: if decision variable affects neither C ˙ T nor gII.

The left-hand sides of expressions (5) and (6) are numerically

evaluated and compared to the value of the parameter p where

p ¼ 0:1 xi=D xi.

4.2. Algorithm for the proposed iterative methodology

Fig. 2 represents a general flow diagram of the proposed itera-

tive methodology. More details of the general flow diagram arepresented as follows:

Step 1. Selecting vector X = [ x1, x2, . . . , xn] where n is the number

of decision variables. Then, determining the lower limit

value ( xi,L ) and the upper limit value ( xi,U) for each decision

variable ( xi).

Step 2. Thermodynamic analysing and checking for whether X is a

feasible solution or not. If X is a feasible solution, iter = 1. If

X is infeasible, go to step 1.

Step 3. Exergoeconomic analysing using ACT method [4].

Step 4. Sorting components with decreasing order in terms of the

sum of the investment and exergy destruction cost flow

rates (C ˙ D,k + _ Z D,k). Setting F CZ = [com1, com2, . . . , comm]

where m is the number of components and F CZ is a vector

that includes names of components corresponding to

(C ˙ D,k + _ Z D,k) in decreasing order.

Step 5. Numerical sensitivity analysing and classifying each x j of

the vector X as:

X Ã ¼ xÃ1; x

Ã2; :::; x

Ã g |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

Group1

; xÃ g þ1; x

Ã g þ2; . . . ; xÃ

l |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} Group2

; xÃlþ1; x

Ãlþ2; . . . xÃ

k |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} Group3

; xÃkþ1; x

Ãkþ2; . . . xÃ

n |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Group4

264

375

where xÃ j is the same as x j that set in groups 1, 2, 3 or 4.

Step 6. Performing structural optimization method:

for i = 1 to m

for j = 1 to n

Fig. 1. Schematic of CGAM problem.

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6.i.j. For the ith element of F CZ (comi) and the jth element

of vector X Ãð xÃ j Þ , structural optimization is implemented. Thus,

xÃ j is updated and replaced by this value in vector X Ã.

end

Vector X Ã is updated which is called X com,i.

end

Step 7. Checking for convergency:

7.1. X Ã is named X iter and total cost flow rate (C ˙ T) is calculated

and named C ˙ T,iter.

7.2. The final vector of X (here, X com,m) is named X iter+1 and total

cost flow rate (C ˙ T) is calculated and named C ˙ T,iter+1.

7.3. Checking the following inequality:

_C T ; iterþ1 À _C T ; iter

_C T ; iter

< a ð7Þ

If Eq. (7) is satisfied, X iter+1 is the optimum solution otherwise, X is

replaced by X iter+1 and the procedure is repeated from step 3 and set

iter = iter + 1.

In Eq. (7), a is a small positive value.

It should be noted that step 5 is preformed in order to assist ra-

pid convergency.

4.3. Application to the CGAM problem

In order to represent how this proposed methodology optimizesthermal power plant, CGAM problem is selected as a test case. All

codes for calculations were developed in MATLAB. For each deci-

sion variable xi, the lower ( xi,L ) and the upper ( xi,U) limiting values

should be determined with predefined steps that have been pre-

sented in Table 1. The following steps are performed respectively:

Step 1. A vector of decision variables is randomly selected.

X ¼ ½PR; T 3; T 4;g AC ;gGT �

For each decision variable ( xi), the lower limit value ( xi,L ) and the

upper limit value ( xi,U) should be determined.

Step 2. A Thermodynamic analysis is performed to determine

whether X is a feasible solution or not. If X is a feasiblesolution iter = 1. If X is infeasible, go to step 1.

Start

Select X = [ x 1, x 2 , …. x n]

Evaluate objective function, i.e. total cost flow rate (Ċ T) by X

Exergoeconomic analysis (ACT method)

& Evalue (Ċ D,k + Ż D,k) for each component

Evaluate objective function, i.e. total cost flow rate ( T) by X new

Numerical sensitivity analysis and classify each x j in vector X that

],...,,,...,,,...,,,,...,,[

4

**2

*1

3

**2

*1

2

**2

*1

1

**2

*1

*

Group

nk k

Group

k ll

Group

Lgg

Group

g x x x x x x x x x x x x X ++++++=

STOP

NO

Yes

ConvergenceCriteria?

Components are ranked in decreasing order in terms of the sum of the investment andexergy destruction cost flow rates (Ċ D,k + Ż D,k), i.e. F CZ = [com1, com2,…,comm]

Print optimum vector ( X OPT ) & optimum total cost flow rate Ċ T,OPT

For all components of Vector F CZ & all decision variables of vector X * Perform Structural Optimization

X replaced by Xnew

Fig. 2. General flow diagram of the proposed iterative methodology.

Table 1

The lower and the upper limiting values and steps for the decision variables of the

CGAM cogeneration system.

Variable Value Step

Minimum Maximum

PR 5 25 0.01

T 3 (K) 500 1200 0.5

T 4 (K) 1200 1800 0.5

gAC 0.7 0.9 0.0001

gGT 0.7 0.92 0.0001




each case. In addition, the value of the objective function

(system total cost flow rate) in each case has been represented. lin the last column shows the percent of relative error of the

optimum solution results from Ref. [5] and the total cost flow rate

(C ˙ T) from the present work that is calculated by the followingequation:

l ¼_C T À _C T;OPT

_C T;OPT

!Â 100 ð8Þ

where_

C T ;OPT is the optimum solution from Ref. [5] (see Tables 4 and5). Table 3 also shows that the number of iterations increases when

Fig. 3. Sum of the investment cost flow rate ( _ Z T), fuel cost flow rate (C ˙ F) and total cost flow rate (C ˙ T) with respect to iteration.

Fig. 4. Total cost flow rate (C ˙ T) with respect to (a) PR, (b) T 3, (c) T 4 and (d) gGT.




both T 3 and T 4 in the initial vector are more than those of real valuesin the optimum solution (e.g. see cases 1 and 7). Also, the table

shows that the number of iterations decreases when T 3 in the initialvector is near to its real value in the optimum conditions (e.g. see

Table 3

Some test cases to optimum solution.

Case X C ˙ T ($/s) No. iteration l (%)

PR T 3 (K) T 4 (K) gAC gGT

1 Initial 10 1000 1720 0.88 0.78 4.936762 – 1264

Final 8.37 920.5 1493.5 0.8447 0.8763 0.362089 24 0.0202 Initial 18.5 860 1720 0.89 0.88 15.623946 – 4215

Final 8.52 912.5 1491 0.8464 0.8794 0.362033 22 0.003

3 Initial 15 850 1682 0.83 0.80 2.099394 – 480

Final 8.5 917.5 1492 0.8432 0.8777 0.362080 13 0.016

4 Initial 10 900 1550 0.88 0.83 0.411631 – 13.7

Final 8.64 914 1493 0.8456 0.8783 0.362037 6 0.004

5 Initial 8 680 1532 0.88 0.88 0.457613 – 26.4

Final 8.57 912 1493 0.8459 0.8803 0.362051 6 0.008

6 Initial 14.24 820.18 1630.55 0.8280 0.8002 0.733198 – 102.5

Final 8.29 924 1495 0.8442 0.8750 0.362168 8 0.041

7 Initial 10.5 1100 1693 0.76 0.77 2.483640 – 586

Final 8.3 921.5 1492.5 0.8446 0.8758 0.362136 28 0.032

8 Initial 6.1240 702.35 1522.54 0.8365 0.8216 0.531029 – 46.68

Final 8.43 911.5 1491.5 0.8480 0.8803 0.362053 9 0.009

9 Initial 12.5 900 1590 0.86 0.81 0.471676 – 30.29

Final 8.61 914.5 1492.5 0.8460 0.8778 0.362037 5 0.005

10 Initial 12.5 815 1392 0.75 0.75 0.797919 – 120

Final 8.52 914 1492 0.8470 0.8782 0.362033 11 0.003

Table 4

Variables for optimum solution from Ref. [5].

Variable PR T 3 (K) T 4 (K) gAC gGT

Value 8.5234 914.28 1492.63 0.8468 0.8786

Table 5

Sum of the investment cost flow rate ( _ Z T), fuel cost flow rate (C ˙ F), total cost flow rate

(C ˙ T) and exergetic efficiency (gII) corresponding to optimum solution from Ref [5].

_ Z T ($/s) C ˙ F ($/s) C ˙ T ($/s) gII (%)

0.036555 0.325465 0.362021 50.66

Fig. 5. Exergetic efficiency (gII) with respect to total cost flow rate ( C ˙ T) (numbers represent iteration).




cases 4 and 9). Table 4 shows vector of decision variables for the

optimum conditions from Ref. [5]. Table 5 shows sum of the invest-

ment cost flow rate ( _ Z T), fuel cost flow rate (C ˙ F), total cost flow rate

(C ˙ T) and exergetic efficiency (gII) corresponding to the optimum

conditions from Ref. [5] (see Appendix A).

6. Conclusions

A new approach has been developed for the optimization of

thermal power plant based on the exergoeconomic analysis and

structural optimization. In the real complex large thermal systems

– that conventional mathematical optimization methods are not

efficient – the proposed iterative methodology is efficient for

optimization. This methodology uses exergoeconomic analysis,

sensitivity analysis, and structural optimization method. The

advantages of this new iterative methodology are: (1) it can be ap-

plied to the real complex large thermal systems; (2) the procedure

of optimization is performed without user interface, i.e. it dose not

need to the decision of designer in each iteration; and (3) since it

uses from a numerical sensitivity analysis, its convergency is rapid.

The benefits of the present work with respect to Refs. [2,23] can be

summarized as: (1) for each new iteration, new design variablesare calculated by structural optimization sub-routine (perfectly de-

scribed in Appendix B), while in Ref. [2] new design variables are

selected (and not calculated) by quality analysis of the component

with current design variable; (2) in large complex thermal systems,

(which have many design variables), it is difficult (or it is impossi-

ble) to determine which decision variable remains constant, in-

creases or decreases for the next iteration. Even if we overcome

on this problem, it is too difficult to determine the increased or de-

creased value of each decision variable; (3) since the present work

does not need any user-supplied data, this optimization can be

used widely while Ref. [2] needs to specialists of exergoeconomic

analysis for optimization; (4) a numerical sensitivity analysis is

used in the present work to achieve rapid convergency; (5) Ref.

[23] uses polyhedron method (a conventional mathematical opti-mization method) for optimization in each iteration, while present

method uses structural optimization method which is more consis-

tent with thermal systems, because of the irreversibility rate and

investment cost rate terms in Eqs. (3); and (6) although numerical

sensitivity analysis in the present work and Ref. [23] is the same,

however, the aim of the making use of it is not the same. Compar-

ison of the present results with results of original CGAM problem

test shows suitable performance and good accuracy of the pro-

posed methodology.

Appendix A. Purchase costs of the component

When evaluating the costs of a plant, it is necessary to consider

the annual cost of fuel and the annual cost associated with owning

and operating each plant component. The expressions for obtain-

ing the purchase costs of the component ( Z ) are presented in Tables

A1 and A2. Based on the costs, the general equation for the cost

rate ( _ Z i in $/s) associated with capital investment and the mainte-

nance cost for the ith component is:

_ Z i ¼ Z iCRF u=ðN Â 3600Þ ðA:1Þ

Here Z i is the purchase costs of the ith component ($), CRF is the

annual capital recovery factor (CRF = 18.2%), N represents the num-

ber of the hours of plant operation per year (N = 8000 h), and u is

the maintenance factor (u = 1.06).

In the CGAM problem the objective function is total cost flow

rate (C ˙ T) that is sum of the investment cost flow rate (_

Z T) and fuelcost flow rate (C ˙ F), i.e.

_C T ¼ _C F þ _ Z T ðA:2Þ

where

_C F ¼ _mf c f LHV ðA:2:aÞ

and

_ Z T ¼Xmk¼1

_ Z k ðA:2:bÞ

where m is the number of components.

Appendix B. Develop structural optimization method for using

in the present new approach

B.1. Derive governing equation for structural optimization method

The purpose of this optimization is to determine for a selected

component (system element) the capital cost corresponding tothe minimum annual operating cost of the plant for a given plant

output and thus, by implication, to the minimum unit cost of the

product.

Assume that there is a plant parameter xi affecting the perfor-

mance of the kth element of the system and thus, in most case, also

indirectly affecting the performance of the system. Any variation in

xi will also, in general, cause changes in the irreversibility rates of

the other elements of the system, and necessitate changes in the

capital costs of the different elements. The exergy balance for the

system as a whole can be written:

_I T ð xiÞ ¼ _E INð xiÞ À _E OUT ðB:1Þ

As shown, term E ˙ OUT which represents the joint exergy of the

plant products is taken to be independent of xi. The irreversibilityrate _I T( xi) may be looked upon as the consumption of exergy in

Table A1

Equations for calculating the purchase cost ( Z ) for the components.

Compressor Z AC ¼ C 11 _ma

C 12ÀgAC

P 2P 1

ln P 2

P 1

Combustion

chamber Z CC ¼ C 21 _ma

C 22ÀP 4P 3

!1 þ expðC 23T 4 À C 24Þ½ �

Turbine Z GT ¼C 31 _m g

C 32 ÀgGT ln P 4

P 5 1 þ expðC 33T 4 À C 34Þ½ �

Air preheater Z APH ¼ C 41_m g ðh5 Àh6 ÞU ðDTLMÞ

0:6

Heat-recovery

steam

generator

Z HRSG ¼ C 51_Q PH

ðDTLMÞPH

0:8þ

_Q EV

ðDTLMÞEV

0:8

þ C 52 _mst þ C 53 _m1:2g

_ma; _m g ; _mst are the mass flow rates of air, gas and steam respectively; h5 and h6 are

the specific enthalpies of streams 5 and 6; DTLM is the log mean temperature

difference; _Q PH and _Q EV represents the rate of heat transfer in the preheater

(economizer) and evaporator, respectively.

Table A2

Constants used in the equations of Table A1 for the purchase cost of the components

(Table A1).

Compressor C 11 ¼ 39:5 $=ðkg=sÞ C 12 ¼ 0:9

Combustion chamber C 21 ¼ 25:6 $=ðkg=sÞ C 22 ¼ 0:995

C 23 ¼ 0:018 ðKÀ1Þ C 24 ¼ 26:4

Turbine C 31 ¼ 266:3 $=ðkg=sÞ C 32 ¼ 0:92

C 33 ¼ 0:036 ðKÀ1Þ C 34 ¼ 54:4

Air preheater C 41 ¼ 2290 $=ðm1:2Þ U ¼ 0:018 kW=ðm2 KÞ

Heat-recovery steam

generatorC 51 ¼ 3650 $=ðkW=KÞ0:8

C 52 ¼ 11820 $=ðkg=sÞ

C 53 ¼ 658 $=ðkg=sÞ1:2


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the system, necessary to generate the product exergy EOUT. An in-

crease in exergy consumption will necessitate corresponding to

additional exergy input, DE ˙ IN( xi).

The nature of the techniquerequires that the exergy input to the

plant should have a single fixed unit cost. This condition can be sat-

isfied by a single form of exergy input of invariable quality, e.g. fuel

or electric energy. Alternatively, theinputcouldbe made up of more

than one form of exergy of invariable quality in fixed proportions.The objective function is the total cost flow rate of operation for

the installation that is obtained from:

_C Tð xiÞ ¼ c IN_E INð xiÞ þ

Xml¼1

_ Z lð xiÞ ðB:2Þ

where C ˙ T is the total cost flow rate of fuel and equipment ($/s) and_Zl in ($/s) is the cost flow rate associated with capital investment

and the maintenance cost for the ith component.

Subject to the usual mathematical conditions being fulfilled, the

objective will be differentiated with respect to xi. From (B.1):

@ _E IN

@ xi¼

@ _I T@ xi

ðB:3Þ

So:

@ _C T@ xi

¼ c IN@ _I T@ xi

þXml¼1

@ _ Z l@ xi

ðB:4Þ

The second term on the RHS of (B.4) may be rearranged conve-

niently as:

Xml¼1

@ _ Z l@ xi

¼Xml0¼1

@ _ Z l0

@ xiþ

@ _ Z k@ xi

ðB:5Þ

where l0–k , i.e. subscript l

0 marks any of the element of the system

except that one which is subject to the optimization. Also, it will be

convenient to make the rearrangement:

Xml0 ¼1

@ _ Z l0

@ xi¼ @ _I k

@ xi

Xml0¼1

@ _ Z l0

@ _I k

!¼ @ _I k

@ xifk;i ðB:6Þ

where

fk;i ¼Xml0¼1

@ _ Z l0

@ _I k

! xi¼var; l0–k

ðB:7Þ

fk,i is the capital cost coefficient.

The coefficient of structural bonds (CSB) is defined by:

rk;i ¼

@ _I T@ xi

@ _I k@ xi

ðB:8:aÞ

Alternatively

rk;i ¼@ _I T

@ _I k

! xi¼var

ðB:8:bÞ

From (B.8-a)

@ _I T@ xi

¼ rk;i

@ _I k@ xi

!ðB:9Þ

Using (B.5)–(B.9), Eq. (B.4) can be modified as:

@ _C T@ xi

¼ c Ik;i@ _I T@ xi

þ@ _ Z k@ xi

ðB:10Þ

where

c I k;i ¼ c INrk;i þ fk;i ðB:11Þ

For optimization, set equation (B.10) equal zero. Thus:

@ _I k@ xi

!OPT

¼ À1

c Ik;i

@ _ Z k@ xi

!ðB:12Þ

B.2. Numerical solution of Eq. (B.12)

In this section, it has been described how Eq. (B.12) can be ap-

plied for structural optimization. Consider kth component and ith

decision variable. xi is ith decision variable that sets between xiL < -

xi < xiU with a step size D xi

Eq. (B.12) is numerically calculated using Newton’s finite differ-

ence formula as:

_I kð xiþ1Þ À _I kð xiÞ

D xi

!¼ À

1

c Ik;i


D xi

!ðB:13Þ

where D xi ¼ xiþ1 À xi. Consider:

f 1 ¼_I kð xiþ1Þ À _I kð xiÞ

D xi

!ðB:14Þ

and

f 2 ¼ À1

c Ik;i


D xi

!ðB:15Þ

Thus, when Eq. (B.13) is numerically evaluated, it can be written as:

f ¼ j f 1 À f 2j ðB:16Þ

If f ffi 0, then xi is obtained. Note that c Ik;i is numerically calcu-

lated for each xi. Fig. B1 shows functions f 1 and f 2 with respect to

xi (see Ref. [27]).

References

[1] Silveira JL, Tuna CE. Thermoeconomic analysis method for optimization of combined heat and power systems. Part I Prog Energy Combust Sci2003;29:479–85.

[2] Tsatsaronis G, Moran M. Exergy-aided cost minimization. Energy ConversManage 1997;38(15):1535–42.

[3] Sahoo PK. Exergoeconomic analysis and optimization of a cogeneration systemusing evolutionary programming. Appl Therm Eng 2008;28:1580–8.

[4] Bejan A, Tsatsaronis G, Moran M. Thermal design and optimization. NewYork: Wiley; 1996.

f 2 f 1

xi x

Fig. B1. Functions f 1 and f 2 with respect to x.


seyyedi_2010_optimization of tpp based on ex ergo economic analysis and structural optimization

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