energy converison published

Energy Conversion and Management 45 (2004) 2495–2523www.elsevier.com/locate/enconman

Integration of an iterative methodology for exergoeconomicimprovement of thermal systems with a process simulator

Leonardo S. Vieira a, Jo~ao L. Donatelli b, Manuel E. Cruz c,*

a CEPEL, Av. Hum s/n, Cidade Universit�aria, CP 68007, Rio de Janeiro, RJ 21944-970, Brazilb UFES, Department of Mechanical Engineering, CP 019011, Vit�oria, ES 29060-970, Brazil

c COPPE/UFRJ, Department of Mechanical Engineering, CP 68503, CT,

Cidade Universitaria, Rio de Janeiro, RJ 21945-970, Brazil

Received 7 August 2003; accepted 12 November 2003

Available online 24 January 2004

Abstract

In this paper, we present the development and automated implementation of an iterative methodology

for exergoeconomic improvement of thermal systems integrated with a process simulator, so as to be

applicable to real, complex plants. The methodology combines recent available exergoeconomic techniques

with new qualitative and quantitative criteria for the following tasks: (i) identification of decision variablesthat affect system total cost and exergetic efficiency; (ii) hierarchical classification of components; (iii)

identification of predominant terms in the component total cost; and (iv) choice of main decision variables

in the iterative process. To show the strengths and potential advantages of the proposed methodology, it is

here applied to the benchmark CGAM cogeneration system. The results obtained are presented and dis-

cussed in detail and are compared to those reached using a mathematical optimization procedure.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Exergoeconomic improvement; Thermal systems; Process simulator; Optimization; Exergy

1. Introduction

The growing concern about conservation of energy and environmental preservation [1] has led tothe development and intense application of techniques based on the second law of thermo-dynamics, such as exergoeconomic analysis––an exergy analysis combined with an economic eva-luation. Today, exergy is seen as a confluence of energy, environment and sustainable development

* Corresponding author. Tel.: + 55-21-2562-8403; fax: + 55-21-2562-8383.

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

0196-8904/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2003.11.007

mail to: [email protected]

Nomenclature

Bk constant in cost Eq. (24) for kth componentBk1 constant in cost Eq. (24) for kth componentBk2 constant in cost Eq. (24) for kth componentc specific exergetic cost_C exergetic cost flow rateCRF capital recovery factor_E exergy flow ratef exergoeconomic factori interest rateLHV lower heating value of fuel_m mass flow ratemk exponent in cost Eq. (24) for kth componentn useful system lifenk exponent in cost Eq. (24) for kth componentNF number of system fuelsNK number of system componentsNP number of system productsp parameter for sensitivity analysis, expressions (4) and (5)P pressurePEC purchased equipment costq parameter for choosing main decision variablesr relative cost differenceRPc compressor compression ratios standard deviationt parameter for hierarchical classification of componentsT temperatureTCI total capital investmentx decision variable_Z investment cost flow rate

Greek letters

a user prescribed tolerance for the iterative process, Eq. (1)b multiplying factor for purchased equipment costd relative change in a variableD absolute change in a variablee exergetic efficiencyc maintenance factorg isentropic efficiencyr1 modified structural bond coefficient, Eq. (11)r2 modified structural bond coefficient, Eq. (12)wdrk;xi

sensitivity coefficient, Eq. (16)

2496 L.S. Vieira et al. / Energy Conversion and Management 45 (2004) 2495–2523

wdek;xisensitivity coefficient, Eq. (17)

s yearly plant operating hours

Subscriptsa airAC air compressorD exergy destructionF fuelg gasGT gas turbinei ith decision variable, ith system exergy flow rateiter iterationk kth plant componentl lower limiting valueL lossNK number of componentsP products steamtotal totalu upper limiting value

Superscript

OPT optimum

L.S. Vieira et al. / Energy Conversion and Management 45 (2004) 2495–2523 2497

[2], and exergoeconomic analysis can address environmental issues, reveal the cost formationprocess of system products and aid system optimization. A critical review of relevant publicationsbetween the 1970s and 1990s regarding exergy and exergoeconomic analysis is found in an articleby Tsatsaronis [3]. Recent important contributions to this field are due to Frangopoulos [4], vonSpakovsky [5], Lozano and Valero [6], Tsatsaronis [7], Hua et al. [8], Kim et al. [9], Zhang et al.[10], Kwon et al. [11], Sciubba [12] and Rosen and Dincer [13].

According to El-Sayed and Gaggioli [14,15], exergoeconomic methods can be grouped in twoclasses: algebraic methods and calculus methods. The algebraic methods use algebraic balanceequations, always require auxiliary cost equations for each component, focus essentially on thecost formation process and determine average costs. On the other hand, calculus methods usedifferential equations, such that the system cost flows are obtained in conjunction with optimi-zation procedures based on the method of Lagrange multipliers, and determine marginal costs.

Algebraic methods (e.g., [6,7,9,11,13,16]) are considered subjective with regard to the definitionof auxiliary equations. The theory of the exergetic cost by Lozano and Valero [6] is a matrixformulation of the productive structure of the system, adequate for computer implementation.Tsatsaronis [7] introduces an iterative exergoeconomic optimization procedure based on exer-goeconomic variables (relative cost difference, exergoeconomic factor and exergetic efficiency) andon minimization of the product cost of each system component treated in thermoeconomicisolation. Lazzaretto and Tsatsaronis [16] present the Specific Exergy Costing/Average Cost


(SPECO/AVCO) approach, seeking to reduce the subjectivity of fuel and product definitions andcost partitioning. Subsequent improvements of the SPECO/AVCO methodology by Tsatsaronisand co-workers [17,18] involve combination with fuzzy inference systems for a more preciseexergoeconomic evaluation of plant components [17] and focus only on the unavoidable exergydestruction and investment costs [18]. However, such improvements are still not free from sub-jective decisions. In principle, the SPECO/AVCO methodology, as well as others (MOPSA [9],[11], EXCEM [13]), can be applied to the analysis of any system, but they lack some objectivecriteria for automated computer optimization of complex thermal systems.

Calculus methods (e.g. [4,5]) are generally based on the Lagrange multipliers technique and areconsidered subjective with regard to the mathematical description of the function of each com-ponent in the system. A particular difficulty in the application of calculus methods to complexsystems is the fact that the Lagrange multipliers vary from iteration to iteration when componentthermoeconomic isolation [19] is not achieved. This problem has led to the development of theThermoeconomic Functional Analysis (TFA) [4]. TFA introduces the concept of the functionaldiagram, and assigns only one function and one product to each component, such that auxiliarycost equations are not needed. With the application of the second law in TFA, the Lagrangemultipliers show relatively smaller variations in the optimization process. Despite later simplifi-cations of the TFA [5], construction of the functional diagram remains subjective, and conver-gence difficulties are encountered as the system size increases.

Additionally, decomposition strategies (e.g. [8,20,21]) based on second law reasoning have beenproposed to reduce complexity in the optimization of complete systems. Hua et al. [8] suggest thatan energy system be decomposed into a main subsystem and a recovery subsystem, with a reversedexergy costing procedure for the recoverable exergy. El-Sayed [20] proposes the decomposition ofthe Lagrangean of the system, together with the identification of local and global variables.Benelmir and Feidt [21] present the IEEB method, in which an expression of the optimal unit costis developed for each component of the system through exergy and cost balance equations. Themethod is, in fact, a consequence of the application of the Lagrange multipliers technique. Yet,these diverse strategies rest on problem dependent, subjective system decomposition.

Most exergoeconomic optimization theories have been applied to relatively simple systemsonly. Mathematical optimization, exergoeconomic or not, of real thermal systems are large scaleproblems due to their complicated nonlinear characteristics and because the mass, energy andexergy (or entropy) balance equations must be introduced in the problem as restrictions [22]. Tooptimize complex systems efficiently, it is, thus, desired, if not required, to integrate optimizationalgorithms with a professional thermodynamic process simulator, such that the optimization taskdoes not have to deal with the mass, energy and exergy balance equations. Furthermore,according to Jaluria [22], it is important to focus on the dominant decision variables, rather thanmanipulating simultaneously all variables that might affect the solution.

In this paper, we present the development and automated implementation of a new approachfor design improvement of complex thermal systems based on the integration of an iterativemethodology for exergoeconomic improvement with a professional process simulator. As such,the proposed methodology is algebraic and should be easily assimilated and applied by practisingengineers of the industrial community. The iterative algorithm efficiently combines well knownexergoeconomic techniques [6,7] with new qualitative and quantitative criteria for the follow-ing tasks: (i) identification of variables that affect system total cost and exergetic efficiency;

combustor

feedwater

gas turbine

compressor

1

23

4

air

567

natural gas

HRSG

8 9

12

11

10

11

steam

preheater

Fig. 1. Schematic of the CGAM cogeneration system (HRSG is the heat recovery steam generator).


(ii) hierarchical classification of components; (iii) identification of predominant terms in thecomponent total cost; and (iv) choice of �main� decision variables in the iterative process. Withregard to task (iv), the proposed method selects in every iteration which of the decision variablesare to be modified through an auxiliary mathematical optimization procedure; these variables arecalled main decision variables. Two alternatives to select the main decision variables are identifiedand implemented. In Alternative 1, the choice is based on modified structural bond coefficients.Structural bond coefficients were defined originally by Kotas [23]. In Alternative 2, the choice isbased on the sensitivity of two exergoeconomic variables––relative cost difference and exergeticefficiency––with respect to changes in the decision variables. For comparison and evaluationpurposes, the proposed method, designed to treat complex systems with no user interference, isapplied in this paper to the benchmark CGAM cogeneration system [24,25], shown schematicallyin Fig. 1. We remark that, by necessity––see Section 4––the physical, thermodynamic and eco-nomic models used in this paper are different from those of the original CGAM problem. Theresults obtained are compared to those reached using a mathematical optimization procedure.The main difference between the proposed method and a mathematical method is that the formerdoes not manipulate all of the components and decision variables in each iteration. This com-putational advantage is due to the knowledge gained from a preceding exergoeconomic analysis ofthe system.

2. Proposed iterative methodology for exergoeconomic improvement

To start, it is assumed that the system physical, thermodynamic and economic models areknown, as well as the design variables and their respective initial values for the improvementprocess. The proposed method consists of six steps, and requires no user interference. The stepscombine and complement the iterative procedure originally developed by Tsatsaronis [3,7,25], the


definitions of fuel and product and auxiliary cost equations by Lazzaretto and Tsatsaronis [26],and the cost formulation of the theory of the exergetic cost by Lozano and Valero [6]. Iterationstops when changes in the decision variables do not further reduce the objective function,according to the equation

_Ctotal;iterþ1 � _Ctotal;iter

_Ctotal;iter

< a; ð1Þ

where _Ctotal is the system total cost, i.e. the sum of the cost flows of the system products, subscriptsiter + 1 and iter represent consecutive iterations and a is a user prescribed tolerance.

2.1. Step 1

First, perform an exergoeconomic system analysis to determine all exergy flow rates ( _Ei), exergydestruction flow rates ( _ED;k), component exergetic efficiencies (ek), exergetic cost flow rates ( _Ci),component product (cP;k) and fuel (cF;k) specific exergetic costs, exergy destruction cost flow rates( _CD;k), component investment cost flow rates ( _Zk), and system total cost ( _Ctotal). Also, for eachcomponent k, determine two exergoeconomic variables, the relative cost difference (rk) and theexergoeconomic factor (fk). Detailed information on how to calculate these variables can be foundin the article by Tsatsaronis [7] and in the book by Bejan et al. [25]. Definitions of fuel and productand auxiliary cost equations are the same as those proposed by Lazzaretto and Tsatsaronis [26].Cost balance equations are implemented according to the theory of the exergetic cost by Lozanoand Valero [6].

Next, define the values of the following parameters to be used in the course of the iterativeprocess: lower (xi;l) and upper (xi;u) limiting values for each decision variable (xi), increments (Dx)for numerical evaluation of the sensitivity of the system total cost and system exergetic efficiencywith respect to the decision variables, parameter for identification of variables that affect total cost(p), parameter for hierarchical classification of components (t), parameter for choosing maindecision variables (q) and the tolerance a. Finally, some additional economic parameters mustalso be defined: the multiplying factor for the purchased equipment cost (b), the useful system life(n), the interest rate (i), the yearly plant operating hours (s) and the maintenance factor (c).

The objective function is the specific total cost of the products, expressed by

minXNP

i¼1

cPi ¼ min

PNF

i¼1 cFi_EFi þ

PNK

k¼1_ZkPNP

i¼1_EPi

; ð2Þ

where NP, NF, and NK are the total number of product flows, fuel flows and system components,respectively. It is observed that when there is just one fuel and the product is constant, as in theCGAM problem, Eq. (2) reduces to [25]

_Ctotal ¼Xk

_Zk þ _CF ¼P

kðCRFþ cÞTCIk� �

sþ cF _mFLHV; ð3Þ

where CRF is the capital recovery factor (see Eq. (20), here not applied to the operation andmaintenance cost rates), TCI is the total capital investment (see Eq. (18)), and LHV is the lowerheating value of the fuel.


2.2. Step 2

Perform a numerical sensitivity analysis of the effect of each decision variable (xi) on the systemtotal cost ( _Ctotal) and system exergetic efficiency (e). Decision variables are thus grouped into fourdifferent classes: (1) those that affect _Ctotal only, (2) those that affect e only, (3) those that affectboth _Ctotal and e and (4) those that affect neither _Ctotal nor e. Class-1 variables are modified at theend of the iteration. Class-4 variables are not modified in the iteration. The sensitivity analysis isperformed for each xi according to the following expressions:

ifxiDxi

Dee> p ) xi affects exergetic efficiency; ð4Þ

ifxiDxi

D _Ctotal

_Ctotal

> p ) xi affects system total cost: ð5Þ

The left-hand sides of expressions (4) and (5) are numerically evaluated and compared to thevalue of the parameter p defined in Step 1.

2.3. Step 3

Identify the �most important� components, i.e. those that exert the larger influence on thesystem total cost. Three tests are performed to categorize the components hierarchically intomain, secondary and remainder components:

if _Zk þ _CD;k >1

NK

XNK

i¼1

ð _Zi þ _CD;iÞ þ tsð _Ziþ _CD;iÞ ) component k is ‘main’; ð6Þ

if _Zk þ _CD;k >1

NK

XNK

i¼1

ð _Zi þ _CD;iÞ � tsð _Ziþ _CD;iÞ and

_Zk þ _CD;k <1

NK

XNKi¼1

ð _Zi þ _CD;iÞ þ tsð _Ziþ _CD;iÞ ) component k is ‘secondary’; ð7Þ

if _Zk þ _CD;k <1

NK

XNK

i¼1

ð _Zi þ _CD;iÞ � tsð _Ziþ _CD;iÞ ) component k is ‘remainder’; ð8Þ

where the parameter t is defined in Step 1. Components are ranked in decreasing order in terms ofthe sum of the investment and exergy destruction cost rates ( _Zk þ _CD;k). The main componentsexert the largest influences on the system total cost and are considered first in the improvementprocess. The remainder components exert the smallest effects on the system total cost and are notconsidered in the iteration. It is observed that the rank of a component may vary in the course ofthe improvement process. For example, a main component in one iteration can become secondaryin the next iteration or vice-versa.


2.4. Step 4

Identify, for each main and secondary component, the cost element (investment or exergydestruction) that exerts the larger influence on the component total cost, according to the fol-lowing tests:

if_CD;k

ð _Zk þ _CD;kÞ> q ) _CD;k is predominant; ð9Þ

if_Zk

ð _Zk þ _CD;kÞ> q ) _Zk is predominant; ð10Þ

where the parameter q, defined in Step 1, is chosen in the range 1 > qP 0:5.

2.5. Step 5

In this step, two distinct alternatives are identified for the choice of the main decision variablesfor each component.

Alternative 1 is based on modified structural bond coefficients, defined as

r1k;xi ¼ðD _CD;total=DxiÞðD _CD;k=DxiÞ

; ð11Þ

r2k;xi ¼ðDð _Z þ _CDÞtotal=DxiÞðDð _Z þ _CDÞk=DxiÞ

: ð12Þ

The coefficients r1k;xi and r2k;xi are modified relative to the original structural bond coefficientsdefined by Kotas [23]. From Eqs. (11) and (12), it is seen that the modified coefficients r1k;xi andr2k;xi are based, respectively, on the cost of exergy destruction and on the total cost (investmentcost plus exergy destruction cost) of component k. Positive values of r1k;xi (respectively, r2k;xi)indicate that a change in the decision variable xi that causes a reduction in the exergy destruction(respectively, total) cost of component k also leads to a reduction in the exergy destruction(respectively, total) cost of the entire system. The coefficients are used for selecting decisionvariables: when the component exergy destruction cost is far superior to its investment cost (Eq.(9) is satisfied), decision variables are selected for positive values of r1k;xi ; otherwise, decisionvariables are selected for positive values of r2k;xi . Alternative 1 has the useful feature of being ableto handle component cost equations in any analytical form.

Alternative 2 requires that component cost equations be in the analytical form

PECk ¼ Bkek

1� ek

� �nk_EmkP;k; ð13Þ

where PECk is the purchased equipment cost. Eq. (13) imposes that the PECk should depend onthe component exergetic efficiency, ek, and the product exergetic flow rate, _EP;k. In Eq. (13), Bk, nkand mk are component specific constants, which may be derived by curve fitting to appropriatecost data. Similarly to the methodology by Tsatsaronis [7], Alternative 2 is based on the relative


deviations between the actual and the optimal values of exergetic efficiency and relative costdifference for each main and secondary component:

dek ¼ek � eOPT

k

eOPTk

100; ð14Þ

drk ¼rk � rOPT

k

rOPTk

100: ð15Þ

The optimal values eOPTk and rOPT

k in Eqs. (14) and (15) are determined considering component kisolated from the system (see Section 4.2 for details) and expressing PECk by an equation of theform of Eq. (13). It is proposed here that these equations be applied to the main and secondarycomponents only, as defined in Step 3.

In Alternative 2, the choice of main decision variables is based on negative values for thesensitivity coefficients wdek;xi

and wdrk;xigiven by

wdek;xi¼ Ddek

dek

xiDxi

; ð16Þ

wdrk;xi¼ Ddrk

drk

xiDxi

: ð17Þ

The value of wdek;xiis used when Eq. (9) is satisfied (i.e. when the exergy destruction cost is

predominant), otherwise the value of wdrk;xiis used. The proposed Eqs. (16) and (17) permit

automation of the methodology.

2.6. Step 6

For the main and secondary components, perform a mathematical optimization for the selecteddecision variables that affect the system total cost and/or exergetic efficiency (Step 2). Any con-ventional mathematical optimization method can be chosen at this point [27]. The objectivefunction is the specific total cost of the system.

Consider first the decision variables of the main components and then those of the secondarycomponents. Finally, consider the decision variables that affect the system total cost only. In theoptimization process, decision variables are bounded, and they are modified within the limits ofvalidity of the component cost equations.

3. Integration with a process simulator

In order that the proposed iterative methodology for exergoecomomic improvement beapplicable to complex systems, it must exploit the computational power of a process simulator.Here, the IPSEpro simulator [28] has been chosen. The IPSEpro contains the basic thermalsimulator module PSE, which can be coupled to the optional modules Model DevelopmentKit (MDK) and PSE-Excel. The integration of the methodology with the IPSEpro is depicted inFig. 2.

MS-EXCEL IPSEpro

sending

running

receiving

decisionvariables

all variables

Macros

Worksheets

Visual Basicroutines

PSE

MDK

simulationenvironment

PSE-Excel(interface)

Fig. 2. Integration of the improvement routine with the IPSEpro simulator program.


In this work, both modules MDK and PSE-Excel are used in addition to the AdvancedPower Library (APP-Lib), which contains models of the components of thermal systems. TheMDK module permits programming and modification of component models in the APP-Liblibrary. With MDK, it is possible to incorporate the exergy property calculation into thesimulator. The PSE-Excel module permits integration of the improvement routine with thePSE simulator through the Dynamic Data Exchange (DDE) protocol. Also, through Excelworksheets, it is possible to exchange variables between IPSEpro and the improvement routine,written in the VBA (Visual Basic) language. Control of data exchange can be effected intwo different ways: first, through the additional menu commands that become availableduring installation of the PSE-Excel integration module and second, through Excel macros––series of special commands to exchange input and output data and run IPSEpro––also avail-able during the installation of PSE-Excel. The latter procedure has been chosen in thiswork because it is necessary to control the data exchange from within the improvement rou-tine.

The simulator determines all mass, energy and exergy flow rates of the system and is called bythe improvement (VBA) routine each time a modification of any decision variable is necessary. Toprevent failure of the VBA routine due to IPSEpro�s internal error messages caused by selection ofunfeasible thermodynamic data, a penalty function is used, whose value is added to the objectivefunction. Note that the improvement routine does not have to deal with the thermodynamicbalance equations as restrictions.

As previously mentioned, the proposed method (Alternatives 1 and 2) requires that an auxiliarymathematical optimization procedure be used to modify the main decision variables in the ite-rative process. In principle, any mathematical optimization method will do. In this work, theflexible polyhedron method developed by Nelder and Mead [27] has been selected and modified tocomply with the bounded intervals for the decision variables.


4. Application to the CGAM cogeneration system

The application of the proposed exergoeconomic improvement procedure to the benchmarkCGAM cogeneration system [24,25] is now described. Because the CGAM cogeneration system isrelatively simple, it is possible to compare the results obtained with the proposed method(Alternatives 1 and 2) to those reached using a mathematical optimization procedure. It isimportant to remark that the physical, thermodynamic and economic models used in this paperare different from those of the original CGAM problem: here, the physical and thermodynamicmodels are solved with the IPSEpro program, not through simplified balance and propertyequations, and the economic model consists of component cost equations derived from fits to datapoints obtained from the original CGAM cost equations [24]. To compare Alternative 1 withAlternative 2, investment data are determined using component cost equations in the form of Eq.(13) (required by Alternative 2 only). In the next three subsections, the modification of the originalcost equations, the exergoeconomic analysis and the application of the improvement procedure tothe CGAM system are reported.

4.1. Cost equations

The principal costs of a thermal system are the capital investment, the operation and main-tenance, and the fuel costs. The economic analysis performed here is a simplification of therevenue requirement method [25]. A simplified economic model is assumed, based on the capitalrecovery factor (CRF), and considering that the total capital investment (TCI) in a plant is givenby the sum of all the purchased equipment costs (PEC) multiplied by a constant factor b [25]. Thismultiplying factor supposedly incorporates all direct and indirect costs of the plant. The totalcapital investment in a plant is, thus, given by

TCI ¼Xk

TCIk ¼Xk

bPECk ¼ bXk

PECk: ð18Þ

The investment cost rate for each component, _ZINk , is expressed by

_ZINk ¼ TCIkCRF

s; ð19Þ

CRF ¼ ið1þ iÞn

ð1þ iÞn � 1; ð20Þ

where CRF is the capital recovery factor, i is the interest rate, and s is the yearly plant operatinghours. Consistently with the CGAM problem, the operation and maintenance cost rate for eachcomponent, _ZOM

k , is now expressed by

_ZOMk ¼ cCRF TCIk

s; ð21Þ

where c is the maintenance factor, here assumed constant.The system total cost rate, excluding fuel costs, _Z, is the summation over all components of _ZIN

k ,Eq. (19), and _ZOM

k , Eq. (21),

Table

Limit

impro

Com

Com

Com

Tur

HR


_Z ¼Xk

_Zk ¼Xk

ð _ZINk þ _ZOM

k Þ ¼ ðP

k CRFð1þ cÞTCIkÞs

: ð22Þ

The system total cost, including fuel costs, _Ctotal, is expressed by

_Ctotal ¼Xk

_Zk þ _CF ¼ ðP

k CRFð1þ cÞTCIkÞs

þ cF _mFLHV; ð23Þ

where cF, LHV and _mF are, respectively, the specific cost, lower heating value and mass flow rateof the fuel. Because, for the CGAM problem, the system products are constant, the cost _Ctotal asgiven by Eq. (23) is the objective function to be optimized, i.e. minimized.

As previously mentioned, to permit comparison between results from Alternatives 1 and 2, thecomponent cost equations in the form of Eq. (13) are derived from fits to data points obtainedfrom the original CGAM economic model [24]. However, for the turbine and combustionchamber, to evaluate correctly the strong effect of the combustion chamber outlet temperature T4on component costs, Eq. (13) has been slightly modified to read

PECk ¼ Bk expðBk1T4 þ Bk2Þek

1� ek

� �nk_EmkP;k: ð24Þ

The modified cost equations are determined by simulating the turbine and combustion chamberindividually with IPSEpro and by calculating their exergetic efficiencies and product flow rates fordifferent input operating conditions. Table 1 shows the limits for the input data of all componentsof the CGAM cogeneration system, which are also applicable to the values of the decisionvariables in the improvement process. The values of the parameters Bk, Bk1, Bk2, nk and mk havebeen determined through efficient nonlinear estimation procedures available with the Statistica 5.0

1

s for the input data of all components of the CGAM cogeneration system, applicable to the cost equations and

vement process

ponent Input variable Value

Minimum Maximum

pressor RPc 7 27

gAC 0.700 0.900

_ma (kg/s) 25 425

bustor _ma (kg/s) 25 425

T4 (�C) 800 1400

DP (%) 5 5

bine _mg (kg/s) 25 425

gGT 0.700 0.900

T4 (�C) 600 1400

RPc 6.4 24.5

SG _ms (kg/s) 10 50

_mg (kg/s) 125 425


program [29], and are shown in Table 2 together with the correlation index (equal to 1 for aperfect fit).

4.2. Exergoeconomic analysis

An exergoeconomic analysis of the thermal system is performed in Step 1 of the methodology.Here, we analyse the CGAM cogeneration system, seen in Fig. 1 to have five components and 12flow streams. The definitions of fuel and product, presented in Table 3, and auxiliary cost

Table 2

Values of parameters for the purchased equipment cost equations

Component Number of

cases

Bk Bk1 Bk2 nk mk Correlation

index

Compressor 65 51.9 – – 2.499 1.002 0.999

Combustor 29 299.9 0.014 )19.898 1.038 1.002 0.996

Turbine 98 181.3 0.035 )53.799 1.450 1.004 0.999

HRSG 37 98645.7 – – 0.783 0.649 0.971

Preheater 24 44839.1 – – 0.917 0.371 0.992

Table 3

Definitions of fuel and product flow rates for the system components

Component Product Fuel

Compressor _E2 � _E1_E12

Combustor _E4_E10 þ _E3

Turbine _E11 þ _E12_E4 � _E5

HRSG _E9 � _E8_E6 � _E7

Preheater _E3 � _E2_E5 � _E6

Table 4

Matrix representation of system costs; the numeric entries form the 12 · 12 matrix A

Stream Shaft

1 2 3 4 5 6 7 8 9 10 11 12

Compressor 1 )1 0 0 0 0 0 0 0 0 1 0

Combustor 0 0 1 )1 0 0 0 0 0 1 0 0

Turbine 0 0 0 1 )1 0 0 0 0 0 )1 )1HRSG 0 0 0 0 0 1 )1 1 )1 0 0 0

Preheater 0 1 )1 0 1 )1 0 0 0 0 0 0

Air 1 0 0 0 0 0 0 0 0 0 0 0

Fuel 0 0 0 0 0 0 0 0 0 1 0 0

Water 0 0 0 0 0 0 0 1 0 0 0 0

Turbine 0 0 0 1= _E4 �1= _E5 0 0 0 0 0 0 0

Turbine 0 0 0 0 0 0 0 0 0 0 1= _E11 �1= _E12

HRSG 0 0 0 0 0 1= _E6 �1= _E7 0 0 0 0 0

Preheater 0 0 0 0 1= _E5 �1= _E6 0 0 0 0 0 0


equations are the same as those proposed by Lazzaretto and Tsatsaronis [26]. Cost balanceequations are implemented according to the theory of the exergetic cost by Lozano and Valero [6],such that the cost flow rates are determined by solving the linear system A � C ¼ Z. The matrix Arepresents the cost balance of the system and is shown in Table 4. The last four rows of A cor-respond to the auxiliary cost equations. The vector C ¼ f _Ci; i ¼ 1; . . . ; 12g contains all the costflow rates, and the vector Z ¼ f _Zi; i ¼ 1; . . . ; 12g contains all the external economic assignmentsfor the system (component investment, operation and maintenance and fuel cost rates).

Table 5

Parameters (Par.) used in the iterative process

Par. Value Equation Description/function

Dxi=xi 1% (4) and (5) Increments on decision variables (sensitivity analysis)

p 0.1 Dxi=xi (4) and (5) Identification of variables which affect total cost and

system exergetic efficiency

t 0.5 (6)–(8) Classification of components

q 2/3 (9) and (10) Identification of predominant term in total cost (invest-

ment or exergy destruction)

a 0.00001 (1) End of iterative process

b 6.32 (18) Multiplying factor for purchased equipment cost

n 10 (20) Useful system life (years)

i 12.7 (20) Interest rate (%)

s 8000 (19) Yearly plant operating hours

c 0.06 (21) Maintenance factor


Exergy destruction flow rates ( _ED;k), component exergetic efficiencies (ek), component product(cP;k) and fuel (cF;k) specific exergetic costs, exergy destruction cost flow rates ( _CD;k), componentinvestment cost flow rates ( _Zk) and relative cost differences (rk) are, respectively, calculated by thefollowing equations:

Table

Defini

Dec

RPc

T3 (

T4 (

gAC

gGT

Init

fu

Table

Final

Dec

RPc

T3 (

T4 (

gAC

gGT

Num

Fin

fu

_ED;k ¼ _EF;k � _EP;k � _EL;k; ð25Þ

6

tion of Cases 1–4, corresponding to four different sets of initial values for the decision variables

ision variable Case 1 Case 2 Case 3 Case 4

8.000 20.000 18.000 12.000

�C) 626.85 525.00 500.00 500.00

�C) 1126.85 1225.00 1200.00 1000.00

0.800 0.800 0.900 0.700

0.800 0.800 0.900 0.700

ial value of the objective

nction (US$/h)

1870.62 2132.76 2080.97 13863.39

7

values for the decision variables obtained using a mathematical optimization procedure

ision variable Case 1 Case 2 Case 3 Case 4

7.000 8.195 8.201 8.201

�C) 634.36 608.02 607.71 609.01

�C) 1226.98 1226.98 1226.98 1226.98

0.869 0.873 0.873 0.873

0.842 0.837 0.837 0.838

ber of function calls 1509 1414 3235 2414

al value of the objective

nction (US$/h)

1650.93 1647.01 1647.01 1647.01

1600

1650

1700

1750

1800

0 250 500 750 1000 1250 1500 1750Number of function calls

Tota

l cos

t (U

S$/h

)

Math. (571) Alt. 2 (1329) Alt. 1 (706)

Fig. 3. Optimization (Math.) and improvement (Alternatives 1 and 2) results for Case 1.


ek ¼_EP;k

_EF;k

¼ 1�_ED;k þ _EL;k

_EP;k

; ð26Þ

cP;k ¼_CP;k

_EP;k

; ð27Þ

1600

1650

1700

1750

1800

0 250 500 750 1000 1250 1500Number of function calls

Tota

l cos

t (U

S$/h

)

Math. (659) Alt. 2 (830) Alt. 1 (568)


1600

1650

1700

1750

1800

0 500 1000 1500 2000 2500 3000Number of function calls

Tota

l cos

t (U

S$/h

)

Math. (630) Alt. 2 (683) Alt. 1 (628)

c


1600

1650

1700

1750

1800

0 500 1000 1500 2000 2500 3000 3500Number of function calls

Tota

l cos

t (U

S$/h

)

Math. (1704) Alt. 2 (1772) Alt. 1 (2561)


Table

Appli

of iter

Iter

Com

Com

Tur

HR

Preh

Iter

Com

Com

Tur

HR

Preh

Iter

Com

Com

Tur

HR

Preh


cF;k ¼_CF;k

_EF;k

; ð28Þ

_CD;k ¼ cF;k _ED;k; ð29Þ

_Zk ¼ _ZINk þ _ZOM

k ¼ CRFð1þ cÞTCIks

; ð30Þ

rk ¼cP;k � cF;k

cF;k¼ 1� ek

ekþ

_Zk

cF;k _EP;k

: ð31Þ

The exergetic cost flow rate _Ci is given by the product of the respective specific exergetic costsand the exergy flow rates.

Optimal values of exergetic efficiency and relative cost difference for each main and secondarycomponent are determined considering the component thermoeconomically isolated from thesystem and are given by the following equations [25]:

eOPTk ¼ 1

1þ Fk; ð32Þ

Fk ¼CRFð1þ ckÞbBknk

scF;k _E1�mkP;k

24

35

1=ðnkþ1Þ

; ð33Þ

8

cation of Alternative 1 to Case 1: cost rates resulting from an exergoeconomic analysis performed at the beginning

ations 1–3

_CP (US$/h) _CF (US$/h) _Z (US$/h) _CD (US$/h) _Z þ _CD (US$/h)

ation 1

pressor 831.78 1160.35 67.61 122.56 190.17

bustor 3161.93 3126.52 35.40 508.50 543.91

bine 1924.52 1841.95 82.57 135.23 217.80

SG 567.96 433.78 134.18 145.52 279.70

eater 907.15 743.89 163.26 81.59 244.84

ation 2

pressor 1110.07 931.35 70.41 80.66 151.08

bustor 2898.54 2876.99 21.56 518.76 540.32

bine 1971.01 1754.73 216.28 56.91 273.19

SG 695.97 541.25 154.71 160.31 315.03

eater 652.13 562.62 89.50 87.94 177.44

ation 3

pressor 1069.94 952.60 79.62 81.01 160.63

bustor 2724.59 2703.75 20.84 495.94 516.78

bine 1942.92 1729.63 213.29 57.93 271.22

SG 671.56 522.76 148.80 160.31 309.12

eater 516.28 432.04 84.25 66.42 150.67

Table

Appli

begin

Iter

Com

Com

Tur

HR

Pre

Iter

Com

Com

Tur

HR

Pre

Iter

Com

Com

Tur

HR

Pre


rOPTk ¼ nk þ 1

nkFk: ð34Þ

4.3. Improvement procedure

The proposed methodology (Alternatives 1 and 2) described in Section 3 requires an auxiliarymathematical optimization procedure to modify the main decision variables in the iterativeprocess. While any conventional mathematical method could be chosen, the easy to implementflexible polyhedron method [27] has been used in this work, with slight modifications in order tocomply with the allowable range of values of the decision variables imposed by the cost equations.Table 5 shows the values of the parameters used for the selection of main components and maindecision variables in the proposed iterative process, and also, it indicates the values of the eco-nomic parameters used in all our calculations. We point out that, differently from the CGAMproblem [24], the value of the parameter b is 6.32, as recommended by Bejan et al. [25] for a newsystem.

Finally, the CGAM system is simple enough to permit mathematical optimization. To comparethe proposed improvement procedure with a mathematical optimization method, we have alsochosen the flexible polyhedron method to act on the same decision variables to optimize the whole

9

cation of Alternative 1 to Case 1: modified structural bond coefficients r1 (Eq. (11)) and r2 (Eq. (12)) at the

ning of iterations 1–3

Decision variable

RPc T3 (�C) T4 (�C) gGT gAC

r1 r2 r1 r2 r1 r2 r1 r2 r1 r2

ation 1

pressor )18.83 )0.11 )1.54 1.04 )0.28 0.34 1.79 0.47 0.13 0.16

bustor 1.06 0.19 1.24 )0.74 3.19 )17.63 1.95 0.67 )7.13 )1.50bine )1.20 )0.30 )1.55 1.13 )0.29 0.60 0.43 0.92 )9.02 )0.05SG 0.90 0.38 0.28 )1.79 0.13 )0.81 0.69 )5.67 0.56 0.06

heater 0.17 0.81 0.36 0.12 0.14 0.26 0.58 3.39 )0.55 0.95

ation 2

pressor 9.54 )0.38 )2.11 0.30 )0.81 )0.02 )0.48 20.26 0.00 0.23

bustor 1.46 0.55 2.11 )0.21 3.86 0.14 )1.26 3.72 0.01 0.94

bine )2.27 3.95 )1.98 0.75 )0.78 0.26 0.11 0.22 0.01 )1.47SG 0.58 1.12 0.24 )0.30 0.21 0.04 0.19 4.20 )0.01 1.53

heater 0.30 0.34 1.68 0.04 0.49 )0.04 0.72 )31.29 0.00 0.91

ation 3

pressor 9.20 )0.28 )2.11 0.43 )0.72 0.02 )0.60 )22.24 0.00 0.23

bustor 1.35 0.48 1.98 )0.27 3.68 )0.12 )1.77 4.51 0.00 0.99

bine )1.97 4.60 )1.98 0.98 )0.69 )0.34 0.12 0.22 0.00 )1.51SG 0.64 0.91 0.25 )0.34 0.21 )0.03 0.23 5.20 0.00 1.66

heater 0.25 0.29 2.10 0.04 0.42 0.02 0.67 756.64 0.00 0.90


CGAM system. We remark that the mathematical strategy does not make use of a precedingexergoeconomic analysis of the system.

5. Results and discussion

The results obtained with the application of the proposed exergoeconomic improvement pro-cedure, Alternatives 1 and 2, to the CGAM system are now presented and discussed in detail andare compared with those reached using a mathematical optimization method. As remarked inSection 4.1, to establish the same basis for comparison, investment costs are determined withequations in the form of Eqs. (13) and (24). Because the physical, thermodynamic and economicmodels used in this paper are different from those of the original CGAM problem, the presentresults for the optimized costs and values of the decision variables should not be compared tothose in Ref. [24].

Table 10

Application of Alternative 1 to Case 1: results at the end of iteration 1

RPc T3 (�C) T4 (�C) gGT gAC Total

cost

(US$/h)

NFa

Initial value 8.000 626.85 1126.85 0.800 0.800 1870.62 0

Effect of decision

variable on system

No effect Only

efficiency

Cost and

efficiency

Cost and

efficiency

Cost and

efficiency

Main components Combustor_CD is

predominant

Selection of decision

variables

p p p

Improved results 8.000 602.56 1226.95 0.900 0.800 1683.30 129

Secondary component HRSG


variables

p


Secondary component Preheater


variables

p p p p


Remainder components Turbine

Compressor

Last improvement


variables

Improved results

Final value 8.000 585.65 1226.97 0.900 0.838 1667.26 290p––selected variables.aNF––cumulative number of function calls.


The chosen decision variables are: compressor compression ratio (RPc), compressor outlettemperature (T3), turbine inlet temperature (or combustor outlet temperature, T4), compressorisentropic efficiency (gAC) and turbine isentropic efficiency (gGT). The sophisticated built inthermodynamic balance equations solvers of the simulator permit reduction of the number ofdecision variables of the optimization problem to only five. Furthermore, in the proposedimprovement procedure, these five decision variables are not simultaneously modified throughoutthe iterative process. The number of function calls, i.e. objective function evaluations, is equal tothe number of calls to the simulator. It is observed that a conventional mathematical approachapplied directly to this problem, without the use of a simulator and adopting simplified mass andenergy balances as restrictions to the optimization problem, would require the choice of morethan 25 decision variables. Integration of the mathematical method to the simulator permitsreduction of the number of decision variables to five; however, all five of them must be simul-taneously modified by the method throughout the entire process.

Table 11



cost

(US$/h)

NFa

Initial value 8.000 585.65 1226.97 0.900 0.838 1667.26 290

Effect of decision

variable on system

Only

efficiency

Only

efficiency

Cost and

efficiency

Cost and

efficiency

Only

efficiency


predominant


variables

p p p p




variables

p p p p




variables

p p p p



Compressor

Last improvement


variables

Improved results

Final value 9.287 566.04 1226.98 0.900 0.833 1664.33 614p––selected variables.aNF––Cumulative number of function calls.


Results are presented for four different cases, indicated in Table 6. Each case possesses its ownset of initial values for the decision variables and a corresponding initial value for the objectivefunction. For each case, the improvement process is executed using Alternative 1 and Alternative2, and an optimization is effected with the chosen mathematical method. In all cases, to reducesignificantly the possibility of achieving local minima, the iterative process is restarted with aninitial polyhedron with a different configuration from that used in the preceeding iterations. Notethat the high initial value of the objective function for Case 4 is indicative that the initial values ofthe decision variables are thermodynamically inconsistent. Therefore, the chosen set of values isconsidered unfeasible for the optimization problem. A penalty function is added to the objectivefunction whenever unfeasibility is observed.

Table 7 shows the final results obtained using the mathematical method of the flexible poly-hedron integrated with the IPSEpro program. The objective function minimum is 1647.01 US$/h,obtained for Cases 2, 3 and 4. By comparing Cases 1 and 2, one observes that large deviations inRPc may correspond to small variations of the objective function. The graphs in Figs. 3–6 show

Table 12



cost

(US$/h)

NFa

Initial value 9.287 566.04 1226.98 0.900 0.833 1664.33 614

Effect of decision

variable on system

No effect Only

efficiency

Only

cost

Cost and

efficiency

Only

efficiency


predominant


variables

p


Secondary component HRSGp


variables

p


Secondary component Turbine_Z is

predominant


variables

p p


Remainder

components

Preheater

Compressor

Last improvement


variables

p




the evolution of the objective function as the optimizing process progresses. Also, the legendsindicate the number of function calls before restart of the iterative process for the mathematicalmethod (and for both alternatives as well). It is seen in the graphs that the objective function issubstantially reduced after about 700 function calls for all cases. Furthermore, the restart providesadditional significant objective function reduction (more than 5%) for Case 3. It is verified that theamount of time consumed by the simulator is the major portion of time of the entire optimizingprocess. Even for the simple CGAM system, the mathematical approach may fail to reach theminimum if the restart is not performed, as evidenced by Case 3.

The results obtained with application of the proposed method to the CGAM system are pre-sented below in great detail for Case 1, first for Alternative 1 and second for Alternative 2. Be-cause similar operations and considerations apply to the other cases, only the final results areshown for Cases 2–4.

Application of Alternative 1 to Case 1 requires four iterations. Table 8 shows the values of thecost rates _CP, _CF, _Z, _CD and _Z þ _CD resulting from an exergoeconomic analysis performed at thebeginning of the first three iterations. The fourth iteration corresponds to the restart, and because,in this case, it does not provide further objective function reduction, it is not shown in the tables.Table 9 displays the calculated modified structural bond coefficients r1k;xi and r2k;xi for eachcomponent and decision variable at the beginning of the three iterations. Tables 10–12 show,respectively, for iterations 1–3, the evolution of the hierarchical classification of components, theselection of main decision variables and the improved results. It is seen that for Case 1, thecombustor is always the main component, the HRSG is a secondary component and the com-pressor is a remainder component. The preheater is secondary in the first and second iterations,

Table 13

Application of Alternative 2 to Case 1: cost rates resulting from an exergoeconomic analysis performed at the beginning

of iterations 1, 2 and 3

_CP (US$/h) _CF (US$/h) _Z (US$/h) _CD (US$/h) _Z þ _CD

Iteration 1

Compressor 831.78 1160.35 67.61 122.56 190.17

Combustor 3161.93 3126.52 35.40 508.50 543.91

Turbine 1924.52 1841.95 82.57 135.23 217.80

HRSG 567.96 433.78 134.18 145.52 279.70

Preheater 907.15 743.89 163.26 81.59 244.84

Iteration 2

Compressor 1063.15 892.41 83.37 73.23 156.60

Combustor 2859.50 2838.42 21.08 533.00 554.08

Turbine 1872.18 1705.46 166.72 66.12 232.84

HRSG 689.18 561.79 127.39 196.86 324.25

Preheater 582.48 509.86 72.62 93.50 166.12

Iteration 3

Compressor 1046.31 925.20 75.24 80.00 155.25

Combustor 2941.93 2917.87 24.06 505.91 529.97

Turbine 1896.27 1744.88 151.39 73.15 224.54

HRSG 672.84 520.64 152.20 156.49 308.69

Preheater 731.25 627.21 104.05 90.34 194.38


and remainder in the third. The turbine is remainder in the first two iterations, and secondary inthe third. It is worth noting that the selected decision variables associated with each main andsecondary component change between iterations. In each iteration, not all of the decision vari-ables are selected simultaneously. Also, it is observed that towards the end of the iterative process(Table 12), just a few decision variables are selected. The minimum obtained is 1664.32 US$/h,only 0.81% higher than 1650.93 US$/h (see Table 7), the value obtained using a mathematicalprocedure beginning from the same initial values for the decision variables.

Application of Alternative 2 to Case 1 requires five iterations, only the first three of which arerelevant to be shown here. Table 13 shows the values of the cost rates _CP, _CF, _Z, _CD, and _Z þ _CD

Table 14

Application of Alternative 2 to Case 1: sensitivity coefficients wdrk;xi(Eq. (16)) and wdek;xi

(Eq. (17)) at the beginning of

iterations 1, 2 and 3

Iteration 1 Iteration 2 Iteration 3

wdrk;xi(%) wdek;xi

(%) wdrk;xi(%) wdek;xi

(%) wdrk;xi(%) wdek;xi

(%)

Variable RPc

Compressor 0.91 4.21 )3.41 1.12 )6.41 1.39

Combustor )0.39 )0.17 )0.26 )0.12 )0.27 )0.13Turbine 4.35 2.31 )0.62 )0.16 )0.71 )0.21HRSG )3.05 )1.48 )3.18 )1.61 )10.06 )5.22Preheater 9.93 4.55 )12.42 )6.47 116.97 47.02

Variable T3Compressor 0.31 )1.48 )0.56 )0.32 )1.20 )0.55Combustor )1.85 )1.57 )1.12 )0.99 )1.30 )1.18Turbine 8.40 4.04 )0.58 )0.34 )0.93 )0.54HRSG )32.63 )18.43 )12.91 )7.07 )47.52 )28.20Preheater 28.74 10.61 )20.41 )12.05 290.08 93.91

Variable T4Compressor )4.25 )1.63 16.30 )0.50 30.01 )0.53Combustor 1.54 0.89 0.64 0.30 0.87 0.49

Turbine )24.86 )13.78 3.01 0.92 3.59 1.21

HRSG 43.02 19.90 17.74 8.75 69.30 30.89

Preheater )27.58 )13.04 28.42 14.47 )96.86 )117.99

Variable gGT

Compressor )4.48 )4.30 17.52 )2.87 32.15 )3.16Combustor 1.23 0.26 1.46 0.25 1.50 0.28

Turbine )98.08 )86.49 34.23 10.11 36.63 11.55

HRSG )36.34 )21.03 )17.96 )10.45 )61.15 )38.75Preheater 17.74 7.50 )12.03 )6.81 151.60 57.24

Variable gAC

Compressor )5.46 39.39 25.53 15.19 43.34 17.89

Combustor 1.37 0.29 1.40 0.24 1.41 0.26

Turbine 27.09 12.43 )3.04 )1.83 )3.41 )2.01HRSG )18.33 )10.37 )5.74 )3.69 )26.77 )15.34Preheater )5.13 )2.80 14.67 6.65 )66.20 )42.06


resulting from an exergoeconomic analysis performed at the beginning of the first three iterations.Table 14 displays the calculated sensitivity coefficients wdek;xi

and wdrk;xifor each component and

decision variable at the beginning of the iterations. Tables 15–17 show, respectively, for iterations1–3, the evolution of the hierarchical classification of components, the selection of decisionvariables and the improved results. It is seen that, for Case 1, similarly to Alternative 1, thecombustor is always the main component, the HRSG is a secondary component, and the com-pressor is a remainder component. The preheater is secondary in the first iteration, and the turbineis secondary in the second and third iterations. The selected decision variables associated witheach main and secondary component also change between iterations. In each iteration, not all ofthe decision variables are selected simultaneously. It is observed that for all iterations (Tables 15–17), just a few decision variables are selected. The minimum obtained is 1647.16 US$/h, 0.23%lower than 1650.93 US$/h (see Table 7).

Table 15



cost

(US$/h)

NFa

Initial value 8.000 626.85 1126.85 0.800 0.800 1870.62 0

Effect of decision

variable on system

No effect Only

efficiency

Cost and

efficiency

Cost and

efficiency

Cost and

efficiency


predominant


variables

p




variables

p p p




variables

p p



Compressor

Last improvement


variables

Improved results

Final value 8.000 557.73 1226.98 0.881 0.847 1663.98 350p––selected variables.aNF––Cumulative number of function calls.

Table 16



cost

(US$/h)

NFa

Initial value 8.000 557.73 1226.98 0.881 0.847 1663.98 350

Effect of decision

variable on system

No effect Cost and

efficiency

Only

cost

Only

efficiency

Only

efficiency


predominant


variables

p




variables

p p p



predominant


variables

p p


Remainder components Preheater

Compressor

Last improvement


variables

p




Figs. 3–6 show, respectively, for Cases 1–4, a comparison between the results obtained using themathematical optimization procedure and the proposed method, Alternatives 1 and 2. In generalterms, with one exception or another, we can make the following remarks. First, significantimprovement of the objective function is reached with no more than 700 function calls. Second,Alternative 1 exhibits faster improvement than Alternative 2 in the early stages of the iterationprocess, and third, the mathematical optimization method reduces the objective function at afaster rate than Alternatives 1 and 2. Nevertheless, the initial faster reduction does not mean asmaller number of calls to the simulator to reach the minimum system total cost, and also, it doesnot mean smaller final values for the objective function.

Table 18 summarizes the final values for the decision variables and the objective function, thetotal number of function calls and the relative difference between each final value of the objectivefunction and the �reference value�. The reference value is the smallest minimum achieved, obtainedwith the mathematical approach applied to Case 4 (1647.01 US$/h). Based on the total number of

Table 17

Application of Alternative 2 to Case 1: Results at the end of iteration 3


cost

(US$/h)

NFa

Initial value 8.000 611.60 1226.98 0.872 0.838 1647.24 586

Effect of decision

variable on system

Only

efficiency

Only

efficiency

Cost and

efficiency

Cost and

efficiency

Only

efficiency


predominant


variables

p p




variables

p p p p



predominant


variables

p p p


Remainder components Preheater

Compressor

Last improvement


variables

p

Improved results



function calls, Alternative 1 is the best for Cases 1, 2 and 3; relative to the mathematical opti-mization, Alternative 2 is better for Cases 2 and 3. It is verified in Table 18 that all final values ofthe objective function match the reference value to within a discrepancy below 1.4%. Consideringa 90% confidence interval, it is estimated that deviations between the final value of the objectivefunction and the reference value will be less than 2.5% for both alternatives.

On physical grounds, it is observed that large variations in the final values of the RPc areobtained. This is due to the fact that RPc has a relatively small effect near the minimum of theobjective function. On the other hand, no variation is obtained for the combustion chamber outlettemperature (T4). In fact, high outlet temperatures imply less exergy destruction in the combustor.Because the combustor is always classified as the main component in all iterations, for bothAlternatives 1 and 2, no difference should really be expected for the final values of T4.

Table 18

Final results obtained using the mathematical method (Math.), Alternative 1 (Alt. 1) and Alternative 2 (Alt. 2) applied

to Cases 1–4

Case Method Values of decision variables Objective

function

(US$/h)

Number

of calls

Relative

difference

(%)RPc T3 (�C) T4 (�C) gGT gAC

1 Initial 8.000 626.85 1126.85 0.800 0.800 1870.62 –

Math. 7.000 634.36 1226.98 0.869 0.842 1650.93 1509 0.24

Alt. 1 9.287 566.04 1226.98 0.900 0.834 1664.32 798 1.05

Alt. 2 8.142 610.01 1226.98 0.873 0.838 1647.16 1700 0.01

2 Initial 20.000 525.00 1225.00 0.800 0.800 2132.76

Math 8.195 608.02 1226.98 0.873 0.837 1647.01 1414 0.00

Alt. 1 9.779 561.64 1226.98 0.900 0.832 1664.58 666 1.07

Alt. 2 9.779 560.37 1226.98 0.900 0.833 1664.57 1005 1.07

3 Initial 18.000 500.00 1200.00 0.900 0.900 2080.97

Math. 8.201 607.71 1226.98 0.873 0.837 1647.01 3235 0.00

Alt. 1 11.387 548.88 1226.98 0.900 0.825 1669.58 688 1.37

Alt. 2 11.416 547.39 1226.98 0.900 0.828 1669.56 844 1.37

4 Initial 12.000 500.00 1000.00 0.700 0.700 13863.39

Math. 8.201 609.01 1226.98 0.873 0.838 1647.01 2414 0.00

Alt. 1 8.193 608.46 1226.98 0.873 0.838 1647.01 2913 0.00

Alt. 2 7.000 634.42 1226.98 0.869 0.842 1650.78 3424 0.23


6. Conclusions

Thermodynamic process simulation programs are widely used in complex thermal systemsprojects, but most of them do not incorporate second law optimization techniques. Likewise,available optimization techniques do not usually contemplate integration with simulation pro-grams. In this paper, a new iterative methodology for exergoeconomic improvement of thermalsystems integrated with a process simulator is proposed, so as to be applicable to complex sys-tems. For automated implementation of the method, quantitative and qualitative criteria areproposed for an adequate choice of decision variables for system improvement. Two alternativesare identified and implemented: Alternative 1, based on modified structural bond coefficients, andAlternative 2, based on sensitivity coefficients for two exergoeconomic variables, the relative costdifference and exergetic efficiency.

The proposed methodology is applied to the benchmark CGAM cogeneration system, and theresults obtained are compared to those reached using the mathematical optimization procedure ofthe flexible polyhedron method. The results of both Alternatives 1 and 2 closely match those ofthe mathematical optimization, with differences in the values of the objective function below 1.4%.Relative to the mathematical optimization, the proposed method has the important advantage ofnot manipulating all decision variables simultaneously, thus leading, in general, to a smallernumber of function calls.

Comparison between Alternatives 1 and 2 reveals that Alternative 1 exhibits a fasterimprovement in the early stages of the iteration process and a reduced number of function calls in


the majority of cases. However, it is impossible to ascertain, in general, which of the alternativescalls the simulator less. Alternative 2 reveals less average deviations from the reference value butrequires the use of cost equations in the form of Eq. (13), in which cost is expressed as a functionof component product exergy flow and component exergetic efficiency. Therefore, Alternative 1 isbetter suited to situations for which the cost equations have an arbitrary form.

The principal difference between the proposed method and the mathematical method is that inthe proposed method, some system components and decision variables are not considered in eachiteration, due to the knowledge gained from a preceding exergoeconomic analysis of the system.Despite the fact that not all decision variables are modified at the same time in the improvementprocess, the final values of the objective function are very close to the minimum, and sometimeseven better than those obtained with the mathematical approach. The proposed method exploitsthe computational power of a process simulator and is, therefore, suitable for application tocomplex systems with a large number of decision variables. In fact, the proposed method does notrequire user intervention and does not enlarge the optimization problem unnecessarily withrestrictions and dependent variables to solve the thermodynamic balance equations of the system.

Acknowledgements

The authors would like to gratefully acknowledge the support of CNPq (Grant 500086/2003-6,521002/97-4) and CAPES (Grant PICDT/UFES 23068.003437/98-85).

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