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Fuzzy thermoeconomic optimization of energy-transforming systems
Applied Energy 84 (2007) 749–762
APPLIED ENERGY www.elsevier.com/locate/apenergy
Fuzzy thermoeconomic optimization of energy-transforming systems V. Mazur
*
Department of Thermodynamics, 1/3 Dvoryanskaya Str., Academy of Refrigeration, Odessa 65026, Ukraine Available online 30 March 2007
Abstract We have developed a new approach for thermoeconomic analysis of energy-transforming systems based on the sequential uncertainty account to make decisions that simultaneously meet thermodynamic and economic goals. Thermoeconomic optimization has been considered as a fuzzy non-linear programming problem in which local criteria: maximum energy (exergy) efficiency and minimum total cost rate as well as different constraints in an ill-structured situation can be represented by fuzzy sets. The trade-off or the Pareto domain, where the value of a thermodynamic criterion cannot be improved without the value of economic criterion being worsened, has been considered as a first step of optimization strategy. The Bellman–Zadeh model, as the intersection of all fuzzy criteria and constraints, has been used for a final decision-making. Case studies of fuzzy thermoeconomic analysis application have been presented. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Thermoeconomic optimization; Pareto domain; Fuzzy sets; Cogeneration; Refrigerant selection
1. Introduction A proper choice of the efficiency criteria predetermines the sustainability of energy transforming system (ETS) design. There are many criteria of efficiency to be taken into account and the extreme values are desirable for each. Usually, three main goals are involved in the design process: thermodynamic, economic and environmental. The holistic criterion is represented by the vector K, including the local criterion Ki as the components *
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0306-2619/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.01.006
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Nomenclature C COP d e G GWP K Ki LCC M m P Pr q0 R w W X Z
cost rate coefficient of performance admissible displacement specific exergy vector of constraints global-warming potential global criterion local criterion life-cycle cost model of system efficiency (property) mass-flow rate properties of working fluid condenser/evaporator pressure-ratio cooling capacity technological operator weight of criterion work vector of decision variable purchase-cost function
Greek symbols k decision variable l membership function n exergetic efficiency Subscripts C compromise cond condenser ec economic ev evaporator fuel fuel L lower boundary NET net Opt optimum P Pareto steam steam th thermodynamic U upper boundary water feed water Superscripts DB database Æ rate with respect to time 0 ‘‘ideal’’ value max maximum min minimum
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and mapping the multitude of requirements imposed on the ETS. A dilemma exists for design criteria: if energy efficiency is attained, the economic or environmentally friendly solution may differ from the desired goal. The attainment of an optimum solution corresponds to a compromise among different criteria and reflects a sustainability of engineering decisions. Various methodologies have been suggested to reach a compromise between thermodynamic and economic criteria and to merge the local criteria into a single objective function, e.g. El-Sayed and Evans [1], Valero [2], Bejan et al. [3]. Design objectives usually contradict each other, so that is difficult to find a solution, which simultaneously satisfies both of them. Meaningful analysis of this ill-structured situation should include uncertainty conception. For multicriteria problems, the local criteria usually have a different physical meaning, and consequently, incomparable dimensions. These complicate the solution of a multicriteria problem and make it necessary to introduce the procedure of normalizing criteria or making these criteria dimensionless. There is no unique method for criteria normalizing and a choice of method depends on the problem statement having a subjective nature. The main idea of conventional thermoeconomic analysis is the introduction of exergetic or exergo-environmental costs to normalize monetary and energy units. The weak point of this idea is an implicit assumption about the concordance of economic and energetic interests that are contrary to the real situation. The aim of the present work is to include uncertainty into the thermoeconomic analysis in order to find solutions that simultaneously satisfy thermodynamic and economic goals. 2. Statement of problem 2.1. Models of uncertainty The last two decades are characterized by growing comprehension of the fact that uncertainty should not be neglected in thermal design and optimization of energy-transforming systems. Since one of the meanings of uncertainty is randomness, a conventional approach is utilized via the theory of probability and random processes. However, as was recognized recently, the probabilistic methods are accompanied with serious troubles during the implementation process in engineering practice. Different considerations led to the development of the broader concept of uncertainty. There are three main complementary models of uncertainty described in the literature: Anti-optimization or convex models of uncertainty, arising from non-specificity of sets of equally-possible alternatives, which were developed by Ben-Haim and Elishakoff [4] as extensions of interval analysis [5]. Game-theoretic models of uncertainty deriving from conflict among the different goals (in our case, it is a conflict between economic and thermodynamic interests). Verbal models of uncertainty deriving from vagueness, i.e. not clearly or precisely expressed or stated linguistic terms when quantitative or qualitative goals of system are described. Fuzzy-set theory, first propounded by Zadeh [6], emerged as a new paradigm in which linguistic uncertainty could be modelled systematically. Three uncontaminated models of uncertainty (uncertainty triangle [7]) cannot exist one by one in a design process and mathematical tools should reveal this fact. Conventional
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thermoeconomic analysis is an example of lop-sided vision of a multicriteria decision-making problem where only a single set of design variables, strongly dependent on decision making experience, is recommended. Attempts to construct a single thermoeconomic criterion narrow considerably an opportunity to find a balanced solution. Here we attempt to develop a flexible model of thermoeconomic analysis, taking into consideration the multicriteria nature of decision-making process, and to minimize the uncertainties arising from conflict and vagueness sources in the thermal design. 2.2. Problem formulation The thermoeconomic optimization is considered as a fuzzy non-linear programming problem with n non-compatible criteria (economic and thermodynamic), m decision variables, and k non-linear constraints: Optimize K½K th ðXÞ; K ec ðXÞ
ð1Þ
subject to GLi 6 Gi ðXÞ 6 GUi ; i ¼ 1; 2; . . . ; k; X Li 6 X i 6 X Ui ; i ¼ 1; 2; . . . ; m;
ð2Þ ð3Þ
where Kth(X) and Kec(X) represent the fuzzy local-criteria of thermodynamic and economic efficiency; X(X1, X2, . . . , Xm) is a vector of decision variables; GLi and GUi, are respectively the lower and upper limits for the constraints Gi(X), and XLi and XUi are respectively the lower and upper bounds for the decision variables. We assume that K j ðX Þ ¼ kK 0j ; M j ðXÞk is a ‘‘distance’’ between the desired (ideal) efficiency of system K Oj and that for its real model Mj. For the thermodynamic criterion, Kth, the value K Oj corresponds to the theoretical maximum of the efficiency objective function, e.g. efficiency of the Carnot cycle. Solving the multicriteria problem involves finding a compromise among all criteria and constraints and can be formulated as follows: to construct the function K ¼ K 1 \ K 2 \ \ K n:
ð4Þ
The formal solution of the problem is added to determine the optimum vector Xopt such that jK(Xopt)j jK(X)j for any X 6¼ Xopt. The model parameters Xopt identify a trade-off decision possessing the desired efficiency criteria. There are several methods of finding ‘‘good’’ solutions to the above problem in thermoeconomic analysis based on scalar optimization. However, as an example, the attempts to resolve the CGAM problem [8] via a single objective paradigm illustrate a conflict among different approaches [8–11] and a lack of a compromise decision. The multicriteria approach is based on synergetic combination of formal and informal decision-making procedures to select a trade-off solution of the problem. There are no entirely formal mathematical tools to resolve a multicriteria problem and additional exogenous information is needed. In the present study, the next sequence of decision-making steps in fuzzy thermoeconomic analysis of energy transforming system is proposed. Determination of the Pareto optimal (or compromise, or trade-off) set XP as the formal solution of the multicriteria problem to minimize a conflict source of uncertainty;
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Fuzzification of goals as well as constraints to represent an ill-structured situation; Informal selection of a convolution scheme to switch over a vector criterion K [Kth(X), Kec(X)] into a scalar combination of the Kth(X) and Kec(X); Evaluation of the final decision vector Xopt 2 XP to minimize the vagueness source of uncertainty. 2.3. Pareto optimal-set In the framework of a multicriteria non-linear programming problem, only the interval of decision variables can be determined, where the value of one of the local criteria cannot be improved without the values of the others to be worsened. This set of parameter values is called the Pareto set [12]. The Pareto optimality concept is a formal solution of the multicriteria problem. Geometrical interpretation of the Pareto set (AB-line) for the case of two criteria Kth = K1, Kec = K2 and two decision variables is given in Fig. 1. Here K min is interpreted as a minimum deviation of the thermodynamic-efficiency model 1 from the ideal solution (e.g., exergetic efficiency n = 1 is an example of ideal solution), K min 2 is a minimum of the economic efficiency model (e.g., the minimum of the cost objective function). The best result from thermodynamic-efficiency reasons usually corresponds to the worst decision for economic consideration. The conventional paradigm of thermoeconomic analysis is a simple approximation of the Pareto domain by the straight line. It is obvious that this assumption is wide of the mark and direct calculations of the Pareto set confirm this fact. There are several methods of finding the Pareto set. Let us assume that we compare two obtained solutions, X* and X0, for which: KðX 0 Þ 6 KðX Þ:
ð5Þ X *;
to be exact the It is evident, that the selection of X0 is preferable in comparison with problem of minimization is considered. Thus, all X* vectors satisfying (5) may be excluded from the following analysis. It make sense to analyze only those X* vectors, for which there is no such X0 when for all the criteria the inequality (5) is satisfied. The set of all such values XP = X* is the Pareto set, and the vector XP is the unimprovable vector of the results, if from Ki(X0) 6 Ki(XP) for any i, it follows that K(X0) = K(XP). A1
K1
K1 - const
A
min
K1 = K 1
B K2 - const min
K 2 = K2
B
min K1
min
K2
A
K2
A2
Fig. 1. 2D – Pareto set in criteria (K1–K2) and decision variable (x1–x2) spaces.
x2
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The approximate construction of the Pareto set is referred to the sphere of insufficiently explored problems in thermal and refrigeration system design and usually it is reduced to the solution of the sequence of mathematical programming problems [13] or the application of evolutionary multi-objective optimization technique [14]. With the help of such approach, the versions regarded as beforehand unacceptable can be rejected, narrowing, in this way, the domain of the optimal solution being searched for. The main defect of this method is the impossibility of finding multiple solutions and the heuristic nature of K 0i boundary guessing. The same defects accompany the weighting methods, which play a central role in the theory of multi-objective optimality. There are several methods available in the literature for solving multi-objective optimization problems as mathematical programming models, via goal programming [15], weightedsum method, goals as requirement, goal attainment, and the iso-resource-cost solution method [16]. All drawbacks of weighting coefficients are more of a pragmatic than theoretical character. Strengths and weakness of these methods are described in the literature [17,18]. Recently, evolutionary algorithms were found useful for solving multicriteria problems [19]. Evolutionary algorithms have some advantages over traditional mathematical programming techniques. For example, considerations for convexity, concavity, and/or continuity of functions are not necessary in this approach. Although these algorithms are successful, they have not proved to be the best solution. At present, there is no wellaccepted method of the Pareto-set determination, that will produce an appropriate set of solutions for all problems. This motivates the further development of reliable approaches to the multicriteria problem. The alternate normal-boundary intersection (NBI) method for generating of the Pareto surface was proposed by Das and Dennis [20]. This algorithm is independent of the relative scales of the functions and is successful in producing an evenly-distributed set of points in the Pareto set. A public domain MATLAB implementation of the NBI algorithm is available [20]. For thermoeconomic optimization problems, this method seems preferable and is applied for finding the Pareto optimal points in the present study. In the Pareto domain, there is no single optimal solution, but rather a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior when all objectives are simultaneously considered. Pareto optimality should be regarded as a tool for generating alternatives from which the decision maker can select the final decision.
2.4. Crisp convolution schemes The next step consists in determining a parameter set from the Pareto set using additional exogenous information and reducing the vector criterion to a scalar one. This step is in fact a problem of decision-making and cannot be fully formalized. There are many ways how a vector criterion can be transformed into a scalar criterion. A general concept as origin of the final decision selection that satisfies the Pareto ranking is based on striving for uncertainty minimization. Two basic tendencies (isolationistic and cooperative) usually are considered to aggregate a vector of local criteria into global (or generalized) scalar criterion. The isolationistic convolution schemes are additive (the global criterion is represented as a weighted sum of local criteria) and entropy (the global
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criterion is represented as a sum of local criteria logarithms). If behaviour of each criterion complies with a common decision to minimize some general (cooperative) criterion, then a convolution scheme can be presented in the form K C ðX Þ ¼ min½wi ðK i ðX Þ K 0i Þ;
1 6 i 6 n; X 2 X P ;
ð6Þ
where wi are the weight coefficients, K 0i – the desired result (the ideal point) that is acceptable for decision maker remaining in the coalition, KC is a global trade-off criterion. If it is possible to come to an agreement about preference (weight) for each criterion then final decision can be found as the solution of scalar non-linear programming problem: n X K C ðX Þ ¼ min jwi ðK i ðX Þ K 0i Þj; X 2 X P : ð7Þ i¼1
If no concordance exists among decision makers concerning weight choice, then an arbitration network is preferable. A classical arbitration-scheme was derived rigorously by Nash, but very often criticized as it opposed common sense [21]: K C ðX Þ ¼ min
n Y
jK i ðX Þ K 0i j;
X 2 X P:
ð8Þ
i¼1
All crisp convolution schemes under discussion try to reduce the uncertainty deriving from conflict among the different criteria in the Pareto domain. The next step is extenuation of uncertainty driving from the vagueness. 2.5. Fuzzy convolution scheme The theory of fuzzy sets was put forward by Zadeh [6] with explicit reference to the vagueness of natural language, when describing quantitative or qualitative goals. Here we assume that local criteria, maximum thermodynamic efficiency (or minimum deviation of real thermodynamic efficiency from ideal one) and maximum profit per unit of production (or minimum total cost rate) as well as different constraints in an ill-structured situation, can be represented by fuzzy sets. A final decision is defined by the Bellman and Zadeh model [22] as the intersection of all fuzzy criteria and constraints and is represented by its membership function l(X) as follows: M C ðXÞ ¼ lEth ðXÞ \ lEec ðXÞ \ lGi ðXÞÞ;
i ¼ 1; 2; . . . ; k; X 2 X P :
ð9Þ
The membership function of the objectives and constraints, linear or nonlinear, can be chosen depending on the context of problem. One of possible fuzzy convolution schemes is presented below. Initial approximation X-vector is chosen. Maximum (minimum) values for each criterion Ki are established via scalar maximization (minimization). Results are denoted as ‘‘ideal’’ points fX 0j ; j ¼ 1; . . . ; mg. The matrix table {T}, where the diagonal elements are ‘‘ideal’’ points, is defined as follows:
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2
K 1 ðX 01 Þ
6 6 K 1 ðX 02 Þ 6 fT g ¼ 6 . 6 .. 4 K 1 ðX 0m Þ
K 2 ðX 01 Þ
K n ðX 01 Þ
K 2 ðX 02 Þ .. .
K n ðX 02 Þ .. .
K 2 ðX 0m Þ K n ðX 0m Þ
3 7 7 7 7: 7 5
ð10Þ
Maximum and minimum bounds for the criteria are defined: K min ¼ min K j ðX 0j Þ ¼ K i ðX 0i Þ; i j
¼ max K j ðX 0j Þ; K min i j
i ¼ 1 . . . n; ð11Þ
i ¼ 1 . . . n:
The membership functions are assumed for all fuzzy goals as follows: 8 ; 0; if K i ðX Þ > K max > i > < K max K i min max i lKi ðX Þ ¼ K max K min ; if K i < K i 6 K i ; i i > > : 1; if K i ðX Þ 6 K min i :
ð12Þ
Fuzzy constraints are formulated: Gj ðX Þ 6 Gmax þ d j; j
j ¼ 1; 2; . . . ; k;
ð13Þ
where dj is a subjective parameter that denotes a distance of admissible displacement for the bound Gmax of the j-constraint. Corresponding membership functions are defined in j following manner: 8 ; 0; if Gj ðX Þ > Gmax > j > < Gj ðX ÞGmax max i ð14Þ lGj ðX Þ ¼ 1 ; if Gj < Gj ðX Þ 6 Gmax þ d j; j dj > > : 1; if G ðX Þ 6 Gmax : j
j
A final decision is determined as the intersection of all fuzzy criteria and constraints represented by its membership functions. This problem is reduced to the standard nonlinear programming problems: to find the such values of X and k that maximize k subject to k 6 lK i ðX Þ;
i ¼ 1; 2; . . . ; n;
k 6 lGj ðX Þ;
j ¼ 1; 2; . . . ; k:
ð15Þ
The solution of the multicriteria problem discloses the meaning of the optimality operator (1) and depends on the decision-maker’s experience and problem understanding. 3. Results and discussion 3.1. Optimization of cogeneration plant The CGAM problem [8] has been investigated using different performance criteria, such as exergetic efficiency, energy-utilization factor, artificial thermal-efficiency, exergetic cost,
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etc. [8–11,23,29]. Lack of concordance for alternative performance criteria raises doubts in the effectiveness of conventional thermoeconomic-optimization algorithms. Fuzzy thermoeconomic optimization of cogeneration plant obviates this difficulty for the decision maker. Let us consider an extended thermoeconomic model of cogeneration plant, developed by Toffollo and Lazzaretto [14], with two criteria to be minimized: thermodynamic criterion as deviation exergetic-efficiency of the cogeneration plant from ideal value and economic criterion as the total cost rate of operation: K th ðX Þ ¼ 1
W_ NET þ m_ steam ðesteam ewater Þ ; m_ fuel efuel
K ec ðX Þ ¼ C_ fuel þ
X
Z_ i ;
ð16Þ
i
where W_ NET is net electrical-power, m_ steamðfuelÞ is steam (fuel) mass-flow rate, esteam, ewater, efuel are process steam, feed water, and fuel specific exergy, respectively. Definition of the decision variables X {X1 = compressor pressure-ratio, X2 = compressor isentropic-efficiency, X3 = gas-turbine inlet-temperature, X4 = gas-turbine isentropic efficiency, X5 = air-preheater effectiveness), physical constraints, and the purchase cost functions for each plant component Z_ i are taken from the comprehensive study [14] on the CGAM problem. Fuzzification of the goals leads to the following membership functions: lK th ðX Þ ¼
8 0; > > < K max K > > :
th ðX Þ th K mac K max th th
1;
if K th ðX Þ > K max th ; ; if
K min th
< K th 6
K max th ;
lK ec ðX Þ ¼
if K th ðX Þ 6 K min th ;
8 0; > < max > :
if K ec ðX Þ > K max ec ;
K ec K ec ðX Þ min ; K max ec K ec
max if K min ec < K ec 6 K ec ;
1;
if K ec ðX Þ 6 K min ec ;
ð17Þ where matrix table {T} is defined as " # K min K max th ðX Þ ¼ 0:455 th ðX Þ ¼ 0:4935 fT g ¼ : $ $ K max K min ec ðX Þ ¼ 3:000 s ec ðX Þ ¼ 0:361 s The unit cost of fuel was chosen to be 0.004$/MJ from [14]. To visualize the decision making process, the intersection of fuzzy criteria is depicted for the single decision variable X3 – gas-turbine inlet temperature (Fig. 2). The Pareto line for conflicting thermodynamic and economic criteria is illustrated in Fig. 3. It is clear the self-consistent relationship between economic and thermodynamic criteria is essentially non-linear and a standard statement about the exergetic equivalent of the capital expenditure is a dubious conjecture. Existence of the Pareto set is evidence of the fact that no universal correlation between thermodynamic and economic objectives and we need to find a compromise for each problem of concordance of thermodynamic and economic interests. 3.2. Working-fluid selection in vapour-compression refrigeration cycle The other case study involves an example of refrigerant selection problem to replace the ozone-depletion substances in the vapour-compression refrigeration cycle. Replacement of artificial refrigerants that are incompatible with nature can eliminate or block a pathway of ozone harmful substances to the biosphere. Thermoeconomic and environmental audit of refrigeration or heat-pump cycles is a first step for refrigerant selection among a wide
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1.0
0.8
μ
0.6
μ (K th)
μ (K ec)
0.4
0.2
0.0
1440
1480
1520
1560
1600
X 3, K Fig. 2. Intersection of membership functions.
variety of ozone-safe working fluids. The accuracy of prognosis for experimentally observable thermodynamic and design characteristics narrows the area of search in the space of competitive economic, environmental and technological criteria. We consider here only such criteria, which are linked by certain relations R to the properties of working fluids P, i.e. the system is defined by a function fK; R; Pg. The relation R is a kind of technological operator and its structure can be determined via the equations of mass, momentum and energy balance, supplemented with the characteristic equationof-state.
Fig. 3. Thermoeconomic Pareto line.
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The estimation of properties P, via information characteristics X, is required to generate the property models Mj(Pj, X). The accuracy of employed estimation-techniques is critical to the reliability of a molecular-design strategy. The operation of a refrigeration system is simulated by the reverse Rankine cycle. The key processes in the single-stage vapour-compression cycle include isentropic compression, isobaric cooling + condensation + subcooling, throttling, and isobaric cooling + evaporation + superheating. The main target properties are as follows: coefficient of performance – COP, evaporator and condenser temperatures – Tev(cond), net refrigerating effect – q0, condenser/evaporator pressure ratio – Pr, compressor displacement – CD, ozone-depletion potential – ODP, globalwarming potential – GWP and flammability index – KF. This list could be increased, but its main feature is the relationship between descriptors of molecular structure, which identifies a working fluid and effectiveness indexes, which include design, operational, ecologic and other indexes. For example, flammability index is correlated to atomic species by simple ratio of fluoride to hydrogen atoms KF = nF/(nF nH). If this ratio exceeds 0.7, then this substance is not flammable. General expressions for the ODP of CFCs with one or two carbon atoms are expressed in a similar manner [24]. The following design specifications are chosen: evaporator and condenser temperatures, T 0ev ¼ 10 C; T 0cond ¼ 35 C; cooling capacity – q00 and condenser/evaporator pressure ratio – Pr < 10. The entire set of design indices includes: economic (life-cycle cost), thermodynamic (specific refrigerating effect, volumetric capacity, specific adiabatic work, condenser/evaporator pressure-ratio, coefficient of performance, adiabatic power), and environmental (flammability index and GWP) criteria. The environmental criterion constraint is chosen as GWP < 400. The class of substances under consideration is presented by the possible alternative refrigerants for R12 and R502 replacements proposed by manufacturers (R32, R401A, R406A, R410A, etc.) [25]. The database for critical parameters of concurrent refrigerants was chosen from [25]. LCC calculations have been provided by algorithms from [26]. Thermodynamic properties of refrigerants and their blends, together with appropriate design specifications, are simulated by the one-fluid Peng–Robinson [27] model of EoS. This type of EoS is chosen because of relationships between their model parameters and critical constants derived from critical conditions. Calculations of target properties via EoS are provided by known engineering thermodynamics expressions for COP, q0 and other indexes. Hence, the chain, starting from descriptors of molecular structure to target property of refrigeration system, should be constructed through critical parameters of the working fluid and EoS model, as a result. Critical parameters of refrigerants are their information characteristics which generate a set of target properties for a refrigeration system. The set of parameters Xopt defines the optimum critical-constants opt opt X opt ðX 1opt ¼ T opt c ; X 2opt ¼ P c ; X 3opt ¼ V c Þ
for the hypothetical ‘‘tailored’’ refrigerant [13,30]. To identify a molecular structure of the real refrigerant, it is possible to organize a direct search in the critical property database (DB) with the selection criterion j1 T opt =T DB þ 1 P opt =P DB þ 1 V opt =V DB ! min ð18Þ c
c
c
c
c
c
to find a solution of the inverse ‘‘structure–critical property’’ relationship problem. Generally, a direct search in a database is quite enough to identify a refrigerant with a desirable combination of target properties. But more sound considerations should be
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guided by quantitative ‘‘structure–property’’ relationships (QSPR). The basic idea of QSPR is to find a relationship between the structure of a compound, expressed in terms of constitutional, geometrical, topological, and different quantum-chemistry descriptors and target property of interest. Solution of the inverse QSPR problem identifies chemical formulae of molecules (in our case, from refrigerant critical property). The QSPR employs two databases – the critical property database and structure database. The correlations between databases is established in the form of the property model – M(P), the parameters of which are determined by minimizing the ‘‘distance’’ between the experimental property Pj and its model Mj. For most refrigerants, experimental data are available via on-line databanks e.g., Chemsafe, Beilstein, Gmelin, etc. If direct measurements are not an option, thermodynamic models from process simulators like ASPEN PLUS, REFPROP, CoolPack can be partly used to estimate the missing properties. There are many group contribution methods to estimate critical properties of industrially-important compounds from molecular structure and the success of any model depends on the amount of data used in determining the contribution of independent variables (molecular descriptors), as compared with the number of the selected variables. The start point of the group-contribution technique is a decomposition of the molecular structure into particular groups and the counting of atoms in those groups. Increments are assigned to the groups by the regression of known experimental-data for the chosen property. The molecular structure can be retrieved by summation of the contributions of all groups. The least sophisticated atom-count technique, suitable for refrigerant selection, was proposed by Joback [28]. Thermodynamic properties and vapour-compression cycle performance characteristics are calculated from the equation of state. As an example, the image of the property surface (condenser/evaporator pressure ratio was chosen as the target property) in the space of information characteristics (molecular masses and critical temperatures) is given in Fig. 4. The fuzzy thermoeconomic optimization algorithm is realized by the following way: Thermodynamic properties and design characteristics of the vapour-compression cycle are calculated for specified external conditions. The best set of design characteristics K 0i is chosen for each criterion among all concurrent refrigerants. The ‘‘ideal’’ indexes K 0i are presented by the vector criterion K, which is calculated via thermodynamic properties.
Fig. 4. ‘‘Information Characteristics versus Pressure Ratio’’ relationship.
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The membership functions li for each thermodynamic and economic index are defined by relations (10)–(14). Environmental criteria cannot be expressed directly by information characteristics in the terms of critical parameters of the substances. Hence, environmental requirements should be taken into account after thermoeconomic considerations for a small set of the competitive alternatives. The fuzzy thermoeconomic criterion of each refrigerant is written as an intersection of membership functions lC = l[COP(X)] \ l[LCC(X)]. Maximum value of lC – fuzzy criterion corresponds to the best refrigerant among competitive working-fluids from a thermoeconomic point of view. A final decision takes into account a ranking of the environmental criteria. The parameters Xopt 2 XP identify an optimum working medium having the desired complex of properties , so called a ‘‘tailored’’ working-fluid [13,30]. The trade-off values of the decision variables Xopt as a result of the fuzzy optimization were obtained by the direct enumeration of information characteristics for all substances from the refrigerant database [25] and consequent ranking of their membership numbers lC. No substances with targeted cooling capacity q00 ¼ 0:5 kW and GWP < 400 were found that can replace existing refrigerants that are non-compatible with nature. There are significant environmental problems associated with prevalent refrigerant R32, most notable a GWP of 650, which exceeds the EU F gas regulation threshold of 150 by a considerable margin. 4. Conclusions The combination of the Pareto optimality and fuzzy set concepts allows the decision maker to conduct a comprehensive study of the energy-transforming systems considering various combinations of economic goals and thermodynamic constraints. Fuzzy non-linear mathematical programming method that attempts to minimize all kinds of uncertainties is a flexible and transparent tool for the economic and thermodynamic optimization of thermal and refrigeration systems. This study is one of the first attempts to apply the methodology of multicriteria makingdecision to select the trade-off working fluids in engineering practice. Tailored working fluid conception is a powerful tool to achieve a compromise among energy efficiency, economic expediency, and environmental constraints of working media in a conceptual thermal design and optimization of energy-transforming systems. References [1] El-Sayed Y, Evans RB. Thermoeconomics and design of heat systems. J Eng Power 1970;92(27):27–34. [2] Valero A. Thermoeconomics: the meeting point of thermodynamics, economics and ecology. In: Proceedings of the second-law analysis of energy systems: towards the 21st century, CIRCUS-Roma, 1995, p. 179–88. [3] Bejan A, Tsatsaronis G, Moran M. In: Moran M, Sciubba E, editors. Thermal design and optimization. New York: Wiley; 1996. [4] Ben-Haim Y, Elishakoff I. Convex models of uncertainty in applied mechanics. Amsterdam: Elsevier; 1990. [5] Moore RE. Interval analysis. Englewood Cliffs, NJ: Prentice-Hall; 1966. [6] Zadeh L. Fuzzy sets. Inform Control 1965;8:338–53. [7] Elishakov I. On the uncertainty triangle. The shock and vibration digest 1990;22(10):1. [8] Valero A, Lozano MA, Serra L, Tsatsaronis G, Pisa J, Frangopoulos CA, von Spakovsky MR. CGAM problem: definition and conventional solution. Energy 1994;19(3):279–86.
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[9] Tsatsaronis G, Pisa J. Exergoeconomic evaluation and optimization of energy systems: application to the CGAM problem. Energy 1994;19(3):287–321. [10] von Spakovsky MR. Application of engineering functional-analysis and optimization of the CGAM problem. Energy 1994;19(3):343–64. [11] Valero A, Serra L, Lozano MA, Torres C. Application of the exergetic cost theory to the CGAM problem. Energy 1994;19(3):365–81. [12] Pareto V. Cours d’economie politique. Laussane, Switzerland: Rouge; 1896. [13] Mazur V, Clodic D, Shamray A. Optimal refrigerant selection. Multicriteria approach. In: Proceedings international conference. CFCs, The Day After. Padova, 21–23 September 1994, p. 57–64. [14] Toffolo A, Lazzaretto A. Evolutionary algorithms for multi-objective energetic and economic optimization in thermal-system design. Energy 2002;27:549–67. [15] Charnes A, Cooper W. Management models and industrial applications of linear programming, vol. 1. New York: John Wiley; 1969. [16] Zeleny M. Multiple-criteria decision-making: eight concepts of optimality. Human Syst Manage 1998;17:97–107. [17] Goicoechea A, Hansen D, Duckstein L. Multi-objective decision-analysis with engineering and business applications. New York: Wiley; 1990. [18] Statnikov R, Matusov J. Multicriteria optimization and engineering. New York: Chapman& Hall; 1995. [19] Coello C. A comprehensive survey of evolutionary-based multi-objective optimization-techniques. Knowledge Inform Syst 1999;1(3):269–308. [20] Das I, Dennis J. Normal boundary-intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems, 1996. Available from: www.owlnet.rice.edu/~indra/ NBIhomepage.html. [21] Germejer YB. Non-contrary interest games. Moscow: Nauka; 1976 [in Russian]. [22] Bellman RE, Zadeh LA. Decision-making in a fuzzy environment. Manage Sci 1970;17:141–64. [23] Feng X, Cai Y, Qian L. A new performance-criterion for cogeneration systems. Energy Convers Manage 1998;39:1607–9. [24] Duvedi A, Achenie E. Designing environmentally-safe refrigerants using mathematical programming. Chem Eng Sci 1996;51(15):3727–39. [25] Calm JM, Hourahan GC. Refrigerant data summary. Eng Syst 2001;18(11):74–88. [26] http://www.colpack.dtu. [27] Peng D, Robinson DB. A new two-constant equation-of-state. Ind Eng Chem Fundam 1976;15:59–64. [28] Joback K. Knowledge bases for computerized physical property estimation. Fluid Phase Equilib 2001;185:45–52. [29] Frangopulos CA. Application of thermoeconomic-optimization methods to the CGAM problem. Energy 1994;19(3):323–42. [30] Mazur V. Optimum refrigerant-selection for low-temperature engineering. In: Kakac S, editor. Low Ttemperature and Cryogenic Refrigeration. Kluwer Academic Publishers; 2003. p. 101–18.
APPLIED ENERGY www.elsevier.com/locate/apenergy
Fuzzy thermoeconomic optimization of energy-transforming systems V. Mazur
*
Department of Thermodynamics, 1/3 Dvoryanskaya Str., Academy of Refrigeration, Odessa 65026, Ukraine Available online 30 March 2007
Abstract We have developed a new approach for thermoeconomic analysis of energy-transforming systems based on the sequential uncertainty account to make decisions that simultaneously meet thermodynamic and economic goals. Thermoeconomic optimization has been considered as a fuzzy non-linear programming problem in which local criteria: maximum energy (exergy) efficiency and minimum total cost rate as well as different constraints in an ill-structured situation can be represented by fuzzy sets. The trade-off or the Pareto domain, where the value of a thermodynamic criterion cannot be improved without the value of economic criterion being worsened, has been considered as a first step of optimization strategy. The Bellman–Zadeh model, as the intersection of all fuzzy criteria and constraints, has been used for a final decision-making. Case studies of fuzzy thermoeconomic analysis application have been presented. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Thermoeconomic optimization; Pareto domain; Fuzzy sets; Cogeneration; Refrigerant selection
1. Introduction A proper choice of the efficiency criteria predetermines the sustainability of energy transforming system (ETS) design. There are many criteria of efficiency to be taken into account and the extreme values are desirable for each. Usually, three main goals are involved in the design process: thermodynamic, economic and environmental. The holistic criterion is represented by the vector K, including the local criterion Ki as the components *
Tel.: +38 0482 209169; fax: +38 0482 238931. E-mail address: [email protected]
0306-2619/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2007.01.006
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Nomenclature C COP d e G GWP K Ki LCC M m P Pr q0 R w W X Z
cost rate coefficient of performance admissible displacement specific exergy vector of constraints global-warming potential global criterion local criterion life-cycle cost model of system efficiency (property) mass-flow rate properties of working fluid condenser/evaporator pressure-ratio cooling capacity technological operator weight of criterion work vector of decision variable purchase-cost function
Greek symbols k decision variable l membership function n exergetic efficiency Subscripts C compromise cond condenser ec economic ev evaporator fuel fuel L lower boundary NET net Opt optimum P Pareto steam steam th thermodynamic U upper boundary water feed water Superscripts DB database Æ rate with respect to time 0 ‘‘ideal’’ value max maximum min minimum
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and mapping the multitude of requirements imposed on the ETS. A dilemma exists for design criteria: if energy efficiency is attained, the economic or environmentally friendly solution may differ from the desired goal. The attainment of an optimum solution corresponds to a compromise among different criteria and reflects a sustainability of engineering decisions. Various methodologies have been suggested to reach a compromise between thermodynamic and economic criteria and to merge the local criteria into a single objective function, e.g. El-Sayed and Evans [1], Valero [2], Bejan et al. [3]. Design objectives usually contradict each other, so that is difficult to find a solution, which simultaneously satisfies both of them. Meaningful analysis of this ill-structured situation should include uncertainty conception. For multicriteria problems, the local criteria usually have a different physical meaning, and consequently, incomparable dimensions. These complicate the solution of a multicriteria problem and make it necessary to introduce the procedure of normalizing criteria or making these criteria dimensionless. There is no unique method for criteria normalizing and a choice of method depends on the problem statement having a subjective nature. The main idea of conventional thermoeconomic analysis is the introduction of exergetic or exergo-environmental costs to normalize monetary and energy units. The weak point of this idea is an implicit assumption about the concordance of economic and energetic interests that are contrary to the real situation. The aim of the present work is to include uncertainty into the thermoeconomic analysis in order to find solutions that simultaneously satisfy thermodynamic and economic goals. 2. Statement of problem 2.1. Models of uncertainty The last two decades are characterized by growing comprehension of the fact that uncertainty should not be neglected in thermal design and optimization of energy-transforming systems. Since one of the meanings of uncertainty is randomness, a conventional approach is utilized via the theory of probability and random processes. However, as was recognized recently, the probabilistic methods are accompanied with serious troubles during the implementation process in engineering practice. Different considerations led to the development of the broader concept of uncertainty. There are three main complementary models of uncertainty described in the literature: Anti-optimization or convex models of uncertainty, arising from non-specificity of sets of equally-possible alternatives, which were developed by Ben-Haim and Elishakoff [4] as extensions of interval analysis [5]. Game-theoretic models of uncertainty deriving from conflict among the different goals (in our case, it is a conflict between economic and thermodynamic interests). Verbal models of uncertainty deriving from vagueness, i.e. not clearly or precisely expressed or stated linguistic terms when quantitative or qualitative goals of system are described. Fuzzy-set theory, first propounded by Zadeh [6], emerged as a new paradigm in which linguistic uncertainty could be modelled systematically. Three uncontaminated models of uncertainty (uncertainty triangle [7]) cannot exist one by one in a design process and mathematical tools should reveal this fact. Conventional
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thermoeconomic analysis is an example of lop-sided vision of a multicriteria decision-making problem where only a single set of design variables, strongly dependent on decision making experience, is recommended. Attempts to construct a single thermoeconomic criterion narrow considerably an opportunity to find a balanced solution. Here we attempt to develop a flexible model of thermoeconomic analysis, taking into consideration the multicriteria nature of decision-making process, and to minimize the uncertainties arising from conflict and vagueness sources in the thermal design. 2.2. Problem formulation The thermoeconomic optimization is considered as a fuzzy non-linear programming problem with n non-compatible criteria (economic and thermodynamic), m decision variables, and k non-linear constraints: Optimize K½K th ðXÞ; K ec ðXÞ
ð1Þ
subject to GLi 6 Gi ðXÞ 6 GUi ; i ¼ 1; 2; . . . ; k; X Li 6 X i 6 X Ui ; i ¼ 1; 2; . . . ; m;
ð2Þ ð3Þ
where Kth(X) and Kec(X) represent the fuzzy local-criteria of thermodynamic and economic efficiency; X(X1, X2, . . . , Xm) is a vector of decision variables; GLi and GUi, are respectively the lower and upper limits for the constraints Gi(X), and XLi and XUi are respectively the lower and upper bounds for the decision variables. We assume that K j ðX Þ ¼ kK 0j ; M j ðXÞk is a ‘‘distance’’ between the desired (ideal) efficiency of system K Oj and that for its real model Mj. For the thermodynamic criterion, Kth, the value K Oj corresponds to the theoretical maximum of the efficiency objective function, e.g. efficiency of the Carnot cycle. Solving the multicriteria problem involves finding a compromise among all criteria and constraints and can be formulated as follows: to construct the function K ¼ K 1 \ K 2 \ \ K n:
ð4Þ
The formal solution of the problem is added to determine the optimum vector Xopt such that jK(Xopt)j jK(X)j for any X 6¼ Xopt. The model parameters Xopt identify a trade-off decision possessing the desired efficiency criteria. There are several methods of finding ‘‘good’’ solutions to the above problem in thermoeconomic analysis based on scalar optimization. However, as an example, the attempts to resolve the CGAM problem [8] via a single objective paradigm illustrate a conflict among different approaches [8–11] and a lack of a compromise decision. The multicriteria approach is based on synergetic combination of formal and informal decision-making procedures to select a trade-off solution of the problem. There are no entirely formal mathematical tools to resolve a multicriteria problem and additional exogenous information is needed. In the present study, the next sequence of decision-making steps in fuzzy thermoeconomic analysis of energy transforming system is proposed. Determination of the Pareto optimal (or compromise, or trade-off) set XP as the formal solution of the multicriteria problem to minimize a conflict source of uncertainty;
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Fuzzification of goals as well as constraints to represent an ill-structured situation; Informal selection of a convolution scheme to switch over a vector criterion K [Kth(X), Kec(X)] into a scalar combination of the Kth(X) and Kec(X); Evaluation of the final decision vector Xopt 2 XP to minimize the vagueness source of uncertainty. 2.3. Pareto optimal-set In the framework of a multicriteria non-linear programming problem, only the interval of decision variables can be determined, where the value of one of the local criteria cannot be improved without the values of the others to be worsened. This set of parameter values is called the Pareto set [12]. The Pareto optimality concept is a formal solution of the multicriteria problem. Geometrical interpretation of the Pareto set (AB-line) for the case of two criteria Kth = K1, Kec = K2 and two decision variables is given in Fig. 1. Here K min is interpreted as a minimum deviation of the thermodynamic-efficiency model 1 from the ideal solution (e.g., exergetic efficiency n = 1 is an example of ideal solution), K min 2 is a minimum of the economic efficiency model (e.g., the minimum of the cost objective function). The best result from thermodynamic-efficiency reasons usually corresponds to the worst decision for economic consideration. The conventional paradigm of thermoeconomic analysis is a simple approximation of the Pareto domain by the straight line. It is obvious that this assumption is wide of the mark and direct calculations of the Pareto set confirm this fact. There are several methods of finding the Pareto set. Let us assume that we compare two obtained solutions, X* and X0, for which: KðX 0 Þ 6 KðX Þ:
ð5Þ X *;
to be exact the It is evident, that the selection of X0 is preferable in comparison with problem of minimization is considered. Thus, all X* vectors satisfying (5) may be excluded from the following analysis. It make sense to analyze only those X* vectors, for which there is no such X0 when for all the criteria the inequality (5) is satisfied. The set of all such values XP = X* is the Pareto set, and the vector XP is the unimprovable vector of the results, if from Ki(X0) 6 Ki(XP) for any i, it follows that K(X0) = K(XP). A1
K1
K1 - const
A
min
K1 = K 1
B K2 - const min
K 2 = K2
B
min K1
min
K2
A
K2
A2
Fig. 1. 2D – Pareto set in criteria (K1–K2) and decision variable (x1–x2) spaces.
x2
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The approximate construction of the Pareto set is referred to the sphere of insufficiently explored problems in thermal and refrigeration system design and usually it is reduced to the solution of the sequence of mathematical programming problems [13] or the application of evolutionary multi-objective optimization technique [14]. With the help of such approach, the versions regarded as beforehand unacceptable can be rejected, narrowing, in this way, the domain of the optimal solution being searched for. The main defect of this method is the impossibility of finding multiple solutions and the heuristic nature of K 0i boundary guessing. The same defects accompany the weighting methods, which play a central role in the theory of multi-objective optimality. There are several methods available in the literature for solving multi-objective optimization problems as mathematical programming models, via goal programming [15], weightedsum method, goals as requirement, goal attainment, and the iso-resource-cost solution method [16]. All drawbacks of weighting coefficients are more of a pragmatic than theoretical character. Strengths and weakness of these methods are described in the literature [17,18]. Recently, evolutionary algorithms were found useful for solving multicriteria problems [19]. Evolutionary algorithms have some advantages over traditional mathematical programming techniques. For example, considerations for convexity, concavity, and/or continuity of functions are not necessary in this approach. Although these algorithms are successful, they have not proved to be the best solution. At present, there is no wellaccepted method of the Pareto-set determination, that will produce an appropriate set of solutions for all problems. This motivates the further development of reliable approaches to the multicriteria problem. The alternate normal-boundary intersection (NBI) method for generating of the Pareto surface was proposed by Das and Dennis [20]. This algorithm is independent of the relative scales of the functions and is successful in producing an evenly-distributed set of points in the Pareto set. A public domain MATLAB implementation of the NBI algorithm is available [20]. For thermoeconomic optimization problems, this method seems preferable and is applied for finding the Pareto optimal points in the present study. In the Pareto domain, there is no single optimal solution, but rather a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior when all objectives are simultaneously considered. Pareto optimality should be regarded as a tool for generating alternatives from which the decision maker can select the final decision.
2.4. Crisp convolution schemes The next step consists in determining a parameter set from the Pareto set using additional exogenous information and reducing the vector criterion to a scalar one. This step is in fact a problem of decision-making and cannot be fully formalized. There are many ways how a vector criterion can be transformed into a scalar criterion. A general concept as origin of the final decision selection that satisfies the Pareto ranking is based on striving for uncertainty minimization. Two basic tendencies (isolationistic and cooperative) usually are considered to aggregate a vector of local criteria into global (or generalized) scalar criterion. The isolationistic convolution schemes are additive (the global criterion is represented as a weighted sum of local criteria) and entropy (the global
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criterion is represented as a sum of local criteria logarithms). If behaviour of each criterion complies with a common decision to minimize some general (cooperative) criterion, then a convolution scheme can be presented in the form K C ðX Þ ¼ min½wi ðK i ðX Þ K 0i Þ;
1 6 i 6 n; X 2 X P ;
ð6Þ
where wi are the weight coefficients, K 0i – the desired result (the ideal point) that is acceptable for decision maker remaining in the coalition, KC is a global trade-off criterion. If it is possible to come to an agreement about preference (weight) for each criterion then final decision can be found as the solution of scalar non-linear programming problem: n X K C ðX Þ ¼ min jwi ðK i ðX Þ K 0i Þj; X 2 X P : ð7Þ i¼1
If no concordance exists among decision makers concerning weight choice, then an arbitration network is preferable. A classical arbitration-scheme was derived rigorously by Nash, but very often criticized as it opposed common sense [21]: K C ðX Þ ¼ min
n Y
jK i ðX Þ K 0i j;
X 2 X P:
ð8Þ
i¼1
All crisp convolution schemes under discussion try to reduce the uncertainty deriving from conflict among the different criteria in the Pareto domain. The next step is extenuation of uncertainty driving from the vagueness. 2.5. Fuzzy convolution scheme The theory of fuzzy sets was put forward by Zadeh [6] with explicit reference to the vagueness of natural language, when describing quantitative or qualitative goals. Here we assume that local criteria, maximum thermodynamic efficiency (or minimum deviation of real thermodynamic efficiency from ideal one) and maximum profit per unit of production (or minimum total cost rate) as well as different constraints in an ill-structured situation, can be represented by fuzzy sets. A final decision is defined by the Bellman and Zadeh model [22] as the intersection of all fuzzy criteria and constraints and is represented by its membership function l(X) as follows: M C ðXÞ ¼ lEth ðXÞ \ lEec ðXÞ \ lGi ðXÞÞ;
i ¼ 1; 2; . . . ; k; X 2 X P :
ð9Þ
The membership function of the objectives and constraints, linear or nonlinear, can be chosen depending on the context of problem. One of possible fuzzy convolution schemes is presented below. Initial approximation X-vector is chosen. Maximum (minimum) values for each criterion Ki are established via scalar maximization (minimization). Results are denoted as ‘‘ideal’’ points fX 0j ; j ¼ 1; . . . ; mg. The matrix table {T}, where the diagonal elements are ‘‘ideal’’ points, is defined as follows:
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2
K 1 ðX 01 Þ
6 6 K 1 ðX 02 Þ 6 fT g ¼ 6 . 6 .. 4 K 1 ðX 0m Þ
K 2 ðX 01 Þ
K n ðX 01 Þ
K 2 ðX 02 Þ .. .
K n ðX 02 Þ .. .
K 2 ðX 0m Þ K n ðX 0m Þ
3 7 7 7 7: 7 5
ð10Þ
Maximum and minimum bounds for the criteria are defined: K min ¼ min K j ðX 0j Þ ¼ K i ðX 0i Þ; i j
¼ max K j ðX 0j Þ; K min i j
i ¼ 1 . . . n; ð11Þ
i ¼ 1 . . . n:
The membership functions are assumed for all fuzzy goals as follows: 8 ; 0; if K i ðX Þ > K max > i > < K max K i min max i lKi ðX Þ ¼ K max K min ; if K i < K i 6 K i ; i i > > : 1; if K i ðX Þ 6 K min i :
ð12Þ
Fuzzy constraints are formulated: Gj ðX Þ 6 Gmax þ d j; j
j ¼ 1; 2; . . . ; k;
ð13Þ
where dj is a subjective parameter that denotes a distance of admissible displacement for the bound Gmax of the j-constraint. Corresponding membership functions are defined in j following manner: 8 ; 0; if Gj ðX Þ > Gmax > j > < Gj ðX ÞGmax max i ð14Þ lGj ðX Þ ¼ 1 ; if Gj < Gj ðX Þ 6 Gmax þ d j; j dj > > : 1; if G ðX Þ 6 Gmax : j
j
A final decision is determined as the intersection of all fuzzy criteria and constraints represented by its membership functions. This problem is reduced to the standard nonlinear programming problems: to find the such values of X and k that maximize k subject to k 6 lK i ðX Þ;
i ¼ 1; 2; . . . ; n;
k 6 lGj ðX Þ;
j ¼ 1; 2; . . . ; k:
ð15Þ
The solution of the multicriteria problem discloses the meaning of the optimality operator (1) and depends on the decision-maker’s experience and problem understanding. 3. Results and discussion 3.1. Optimization of cogeneration plant The CGAM problem [8] has been investigated using different performance criteria, such as exergetic efficiency, energy-utilization factor, artificial thermal-efficiency, exergetic cost,
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etc. [8–11,23,29]. Lack of concordance for alternative performance criteria raises doubts in the effectiveness of conventional thermoeconomic-optimization algorithms. Fuzzy thermoeconomic optimization of cogeneration plant obviates this difficulty for the decision maker. Let us consider an extended thermoeconomic model of cogeneration plant, developed by Toffollo and Lazzaretto [14], with two criteria to be minimized: thermodynamic criterion as deviation exergetic-efficiency of the cogeneration plant from ideal value and economic criterion as the total cost rate of operation: K th ðX Þ ¼ 1
W_ NET þ m_ steam ðesteam ewater Þ ; m_ fuel efuel
K ec ðX Þ ¼ C_ fuel þ
X
Z_ i ;
ð16Þ
i
where W_ NET is net electrical-power, m_ steamðfuelÞ is steam (fuel) mass-flow rate, esteam, ewater, efuel are process steam, feed water, and fuel specific exergy, respectively. Definition of the decision variables X {X1 = compressor pressure-ratio, X2 = compressor isentropic-efficiency, X3 = gas-turbine inlet-temperature, X4 = gas-turbine isentropic efficiency, X5 = air-preheater effectiveness), physical constraints, and the purchase cost functions for each plant component Z_ i are taken from the comprehensive study [14] on the CGAM problem. Fuzzification of the goals leads to the following membership functions: lK th ðX Þ ¼
8 0; > > < K max K > > :
th ðX Þ th K mac K max th th
1;
if K th ðX Þ > K max th ; ; if
K min th
< K th 6
K max th ;
lK ec ðX Þ ¼
if K th ðX Þ 6 K min th ;
8 0; > < max > :
if K ec ðX Þ > K max ec ;
K ec K ec ðX Þ min ; K max ec K ec
max if K min ec < K ec 6 K ec ;
1;
if K ec ðX Þ 6 K min ec ;
ð17Þ where matrix table {T} is defined as " # K min K max th ðX Þ ¼ 0:455 th ðX Þ ¼ 0:4935 fT g ¼ : $ $ K max K min ec ðX Þ ¼ 3:000 s ec ðX Þ ¼ 0:361 s The unit cost of fuel was chosen to be 0.004$/MJ from [14]. To visualize the decision making process, the intersection of fuzzy criteria is depicted for the single decision variable X3 – gas-turbine inlet temperature (Fig. 2). The Pareto line for conflicting thermodynamic and economic criteria is illustrated in Fig. 3. It is clear the self-consistent relationship between economic and thermodynamic criteria is essentially non-linear and a standard statement about the exergetic equivalent of the capital expenditure is a dubious conjecture. Existence of the Pareto set is evidence of the fact that no universal correlation between thermodynamic and economic objectives and we need to find a compromise for each problem of concordance of thermodynamic and economic interests. 3.2. Working-fluid selection in vapour-compression refrigeration cycle The other case study involves an example of refrigerant selection problem to replace the ozone-depletion substances in the vapour-compression refrigeration cycle. Replacement of artificial refrigerants that are incompatible with nature can eliminate or block a pathway of ozone harmful substances to the biosphere. Thermoeconomic and environmental audit of refrigeration or heat-pump cycles is a first step for refrigerant selection among a wide
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1.0
0.8
μ
0.6
μ (K th)
μ (K ec)
0.4
0.2
0.0
1440
1480
1520
1560
1600
X 3, K Fig. 2. Intersection of membership functions.
variety of ozone-safe working fluids. The accuracy of prognosis for experimentally observable thermodynamic and design characteristics narrows the area of search in the space of competitive economic, environmental and technological criteria. We consider here only such criteria, which are linked by certain relations R to the properties of working fluids P, i.e. the system is defined by a function fK; R; Pg. The relation R is a kind of technological operator and its structure can be determined via the equations of mass, momentum and energy balance, supplemented with the characteristic equationof-state.
Fig. 3. Thermoeconomic Pareto line.
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The estimation of properties P, via information characteristics X, is required to generate the property models Mj(Pj, X). The accuracy of employed estimation-techniques is critical to the reliability of a molecular-design strategy. The operation of a refrigeration system is simulated by the reverse Rankine cycle. The key processes in the single-stage vapour-compression cycle include isentropic compression, isobaric cooling + condensation + subcooling, throttling, and isobaric cooling + evaporation + superheating. The main target properties are as follows: coefficient of performance – COP, evaporator and condenser temperatures – Tev(cond), net refrigerating effect – q0, condenser/evaporator pressure ratio – Pr, compressor displacement – CD, ozone-depletion potential – ODP, globalwarming potential – GWP and flammability index – KF. This list could be increased, but its main feature is the relationship between descriptors of molecular structure, which identifies a working fluid and effectiveness indexes, which include design, operational, ecologic and other indexes. For example, flammability index is correlated to atomic species by simple ratio of fluoride to hydrogen atoms KF = nF/(nF nH). If this ratio exceeds 0.7, then this substance is not flammable. General expressions for the ODP of CFCs with one or two carbon atoms are expressed in a similar manner [24]. The following design specifications are chosen: evaporator and condenser temperatures, T 0ev ¼ 10 C; T 0cond ¼ 35 C; cooling capacity – q00 and condenser/evaporator pressure ratio – Pr < 10. The entire set of design indices includes: economic (life-cycle cost), thermodynamic (specific refrigerating effect, volumetric capacity, specific adiabatic work, condenser/evaporator pressure-ratio, coefficient of performance, adiabatic power), and environmental (flammability index and GWP) criteria. The environmental criterion constraint is chosen as GWP < 400. The class of substances under consideration is presented by the possible alternative refrigerants for R12 and R502 replacements proposed by manufacturers (R32, R401A, R406A, R410A, etc.) [25]. The database for critical parameters of concurrent refrigerants was chosen from [25]. LCC calculations have been provided by algorithms from [26]. Thermodynamic properties of refrigerants and their blends, together with appropriate design specifications, are simulated by the one-fluid Peng–Robinson [27] model of EoS. This type of EoS is chosen because of relationships between their model parameters and critical constants derived from critical conditions. Calculations of target properties via EoS are provided by known engineering thermodynamics expressions for COP, q0 and other indexes. Hence, the chain, starting from descriptors of molecular structure to target property of refrigeration system, should be constructed through critical parameters of the working fluid and EoS model, as a result. Critical parameters of refrigerants are their information characteristics which generate a set of target properties for a refrigeration system. The set of parameters Xopt defines the optimum critical-constants opt opt X opt ðX 1opt ¼ T opt c ; X 2opt ¼ P c ; X 3opt ¼ V c Þ
for the hypothetical ‘‘tailored’’ refrigerant [13,30]. To identify a molecular structure of the real refrigerant, it is possible to organize a direct search in the critical property database (DB) with the selection criterion j1 T opt =T DB þ 1 P opt =P DB þ 1 V opt =V DB ! min ð18Þ c
c
c
c
c
c
to find a solution of the inverse ‘‘structure–critical property’’ relationship problem. Generally, a direct search in a database is quite enough to identify a refrigerant with a desirable combination of target properties. But more sound considerations should be
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guided by quantitative ‘‘structure–property’’ relationships (QSPR). The basic idea of QSPR is to find a relationship between the structure of a compound, expressed in terms of constitutional, geometrical, topological, and different quantum-chemistry descriptors and target property of interest. Solution of the inverse QSPR problem identifies chemical formulae of molecules (in our case, from refrigerant critical property). The QSPR employs two databases – the critical property database and structure database. The correlations between databases is established in the form of the property model – M(P), the parameters of which are determined by minimizing the ‘‘distance’’ between the experimental property Pj and its model Mj. For most refrigerants, experimental data are available via on-line databanks e.g., Chemsafe, Beilstein, Gmelin, etc. If direct measurements are not an option, thermodynamic models from process simulators like ASPEN PLUS, REFPROP, CoolPack can be partly used to estimate the missing properties. There are many group contribution methods to estimate critical properties of industrially-important compounds from molecular structure and the success of any model depends on the amount of data used in determining the contribution of independent variables (molecular descriptors), as compared with the number of the selected variables. The start point of the group-contribution technique is a decomposition of the molecular structure into particular groups and the counting of atoms in those groups. Increments are assigned to the groups by the regression of known experimental-data for the chosen property. The molecular structure can be retrieved by summation of the contributions of all groups. The least sophisticated atom-count technique, suitable for refrigerant selection, was proposed by Joback [28]. Thermodynamic properties and vapour-compression cycle performance characteristics are calculated from the equation of state. As an example, the image of the property surface (condenser/evaporator pressure ratio was chosen as the target property) in the space of information characteristics (molecular masses and critical temperatures) is given in Fig. 4. The fuzzy thermoeconomic optimization algorithm is realized by the following way: Thermodynamic properties and design characteristics of the vapour-compression cycle are calculated for specified external conditions. The best set of design characteristics K 0i is chosen for each criterion among all concurrent refrigerants. The ‘‘ideal’’ indexes K 0i are presented by the vector criterion K, which is calculated via thermodynamic properties.
Fig. 4. ‘‘Information Characteristics versus Pressure Ratio’’ relationship.
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The membership functions li for each thermodynamic and economic index are defined by relations (10)–(14). Environmental criteria cannot be expressed directly by information characteristics in the terms of critical parameters of the substances. Hence, environmental requirements should be taken into account after thermoeconomic considerations for a small set of the competitive alternatives. The fuzzy thermoeconomic criterion of each refrigerant is written as an intersection of membership functions lC = l[COP(X)] \ l[LCC(X)]. Maximum value of lC – fuzzy criterion corresponds to the best refrigerant among competitive working-fluids from a thermoeconomic point of view. A final decision takes into account a ranking of the environmental criteria. The parameters Xopt 2 XP identify an optimum working medium having the desired complex of properties , so called a ‘‘tailored’’ working-fluid [13,30]. The trade-off values of the decision variables Xopt as a result of the fuzzy optimization were obtained by the direct enumeration of information characteristics for all substances from the refrigerant database [25] and consequent ranking of their membership numbers lC. No substances with targeted cooling capacity q00 ¼ 0:5 kW and GWP < 400 were found that can replace existing refrigerants that are non-compatible with nature. There are significant environmental problems associated with prevalent refrigerant R32, most notable a GWP of 650, which exceeds the EU F gas regulation threshold of 150 by a considerable margin. 4. Conclusions The combination of the Pareto optimality and fuzzy set concepts allows the decision maker to conduct a comprehensive study of the energy-transforming systems considering various combinations of economic goals and thermodynamic constraints. Fuzzy non-linear mathematical programming method that attempts to minimize all kinds of uncertainties is a flexible and transparent tool for the economic and thermodynamic optimization of thermal and refrigeration systems. This study is one of the first attempts to apply the methodology of multicriteria makingdecision to select the trade-off working fluids in engineering practice. Tailored working fluid conception is a powerful tool to achieve a compromise among energy efficiency, economic expediency, and environmental constraints of working media in a conceptual thermal design and optimization of energy-transforming systems. References [1] El-Sayed Y, Evans RB. Thermoeconomics and design of heat systems. J Eng Power 1970;92(27):27–34. [2] Valero A. Thermoeconomics: the meeting point of thermodynamics, economics and ecology. In: Proceedings of the second-law analysis of energy systems: towards the 21st century, CIRCUS-Roma, 1995, p. 179–88. [3] Bejan A, Tsatsaronis G, Moran M. In: Moran M, Sciubba E, editors. Thermal design and optimization. New York: Wiley; 1996. [4] Ben-Haim Y, Elishakoff I. Convex models of uncertainty in applied mechanics. Amsterdam: Elsevier; 1990. [5] Moore RE. Interval analysis. Englewood Cliffs, NJ: Prentice-Hall; 1966. [6] Zadeh L. Fuzzy sets. Inform Control 1965;8:338–53. [7] Elishakov I. On the uncertainty triangle. The shock and vibration digest 1990;22(10):1. [8] Valero A, Lozano MA, Serra L, Tsatsaronis G, Pisa J, Frangopoulos CA, von Spakovsky MR. CGAM problem: definition and conventional solution. Energy 1994;19(3):279–86.
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[9] Tsatsaronis G, Pisa J. Exergoeconomic evaluation and optimization of energy systems: application to the CGAM problem. Energy 1994;19(3):287–321. [10] von Spakovsky MR. Application of engineering functional-analysis and optimization of the CGAM problem. Energy 1994;19(3):343–64. [11] Valero A, Serra L, Lozano MA, Torres C. Application of the exergetic cost theory to the CGAM problem. Energy 1994;19(3):365–81. [12] Pareto V. Cours d’economie politique. Laussane, Switzerland: Rouge; 1896. [13] Mazur V, Clodic D, Shamray A. Optimal refrigerant selection. Multicriteria approach. In: Proceedings international conference. CFCs, The Day After. Padova, 21–23 September 1994, p. 57–64. [14] Toffolo A, Lazzaretto A. Evolutionary algorithms for multi-objective energetic and economic optimization in thermal-system design. Energy 2002;27:549–67. [15] Charnes A, Cooper W. Management models and industrial applications of linear programming, vol. 1. New York: John Wiley; 1969. [16] Zeleny M. Multiple-criteria decision-making: eight concepts of optimality. Human Syst Manage 1998;17:97–107. [17] Goicoechea A, Hansen D, Duckstein L. Multi-objective decision-analysis with engineering and business applications. New York: Wiley; 1990. [18] Statnikov R, Matusov J. Multicriteria optimization and engineering. New York: Chapman& Hall; 1995. [19] Coello C. A comprehensive survey of evolutionary-based multi-objective optimization-techniques. Knowledge Inform Syst 1999;1(3):269–308. [20] Das I, Dennis J. Normal boundary-intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems, 1996. Available from: www.owlnet.rice.edu/~indra/ NBIhomepage.html. [21] Germejer YB. Non-contrary interest games. Moscow: Nauka; 1976 [in Russian]. [22] Bellman RE, Zadeh LA. Decision-making in a fuzzy environment. Manage Sci 1970;17:141–64. [23] Feng X, Cai Y, Qian L. A new performance-criterion for cogeneration systems. Energy Convers Manage 1998;39:1607–9. [24] Duvedi A, Achenie E. Designing environmentally-safe refrigerants using mathematical programming. Chem Eng Sci 1996;51(15):3727–39. [25] Calm JM, Hourahan GC. Refrigerant data summary. Eng Syst 2001;18(11):74–88. [26] http://www.colpack.dtu. [27] Peng D, Robinson DB. A new two-constant equation-of-state. Ind Eng Chem Fundam 1976;15:59–64. [28] Joback K. Knowledge bases for computerized physical property estimation. Fluid Phase Equilib 2001;185:45–52. [29] Frangopulos CA. Application of thermoeconomic-optimization methods to the CGAM problem. Energy 1994;19(3):323–42. [30] Mazur V. Optimum refrigerant-selection for low-temperature engineering. In: Kakac S, editor. Low Ttemperature and Cryogenic Refrigeration. Kluwer Academic Publishers; 2003. p. 101–18.