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Optimal design of energy supply systems based on relative robustness criterion
Energy Conversion and Management 43 (2002) 499±514
www.elsevier.com/locate/enconman
Optimal design of energy supply systems based on relative robustness criterion Ryohei Yokoyama *, Koichi Ito Department of Energy Systems Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan Received 10 October 2000; accepted 19 February 2001
Abstract A robust optimal design method, based on the relative robustness criterion, is proposed to conduct the unit sizing of energy supply systems, so that they are robust economically under uncertain energy demands. The values of design variables or equipment capacities, as well as those of operation variables or utility contract demands and energy ¯ow rates, are determined to minimize the maximum normalized regret or the maximum regret rate in the annual total cost and satisfy all the possible energy demands. This optimization problem is formulated as a multi-level nonlinear programming problem, and its solution is obtained by repeatedly evaluating the upper and lower bounds for the optimal value of the maximum regret rate by means of the fractional, the bi-level and the linear programming. Through a case study on a cogeneration system, features of the robust optimal design, based on the relative robustness criterion, are clari®ed. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Energy supply systems; Unit sizing; Operational planning; Robust design; Optimization; Relative robustness; Worst case analysis; Multi-level programming; Fractional programming
1. Introduction For the purpose of improving economic and energy saving characteristics, distributed energy supply systems, such as cogeneration ones, have been installed increasingly into districts and buildings in recent years. To attain the highest economic and energy saving characteristics, design and operation are important issues. In designing energy supply systems, the estimation of energy demands is an important work. This is because planners are requested to determine rationally what types, numbers and capacities *
Corresponding author. Tel.: +81-722-56-2030/54-9232; fax: +81-722-56-2031/54-9904. E-mail address: [email protected] (R. Yokoyama).
0196-8904/02/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 0 2 7 - 9
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of equipment should be installed in consideration of equipment operational strategies for seasonal and hourly variations in estimated energy demands, which signi®cantly aect the economic and energy saving characteristics. Related with this issue, an optimal unit sizing method has been proposed, by which equipment capacities and utility contract demands can be determined in order to minimize the annual total cost and primary energy consumption while considering equipment operational strategies for average energy demands estimated on multiple representative days, and to satisfy peak energy demands in summer and winter [1±3]. This method can be a useful tool for unit sizing, if we can estimate energy demands precisely. However, many conditions under which energy demands are estimated have some uncertainty at the design stage, which makes it impossible to estimate energy demands precisely. If the unit sizing is conducted by considering that estimated energy demands are certain, the economic and energy saving characteristics expected may not be attained, and de®cits in energy supply may occur. This is because energy demands, which occur at the operation stage, dier from those estimated at the design stage. Therefore, planners should consider that energy demands have some uncertainty, evaluate the robustness in economic and energy saving characteristics under uncertain energy demands and design energy supply systems rationally in consideration of the robustness. An optimal unit sizing method of energy supply systems in consideration of the robustness under uncertain energy demands has been proposed [4] based on the minimax regret criterion [5]. This method has been applied to the evaluation of the economic robustness of some types of gas turbine cogeneration systems, and features of the robustness have been clari®ed [6]. According to the method, the maximum regret in an objective function is adopted as the evaluation criterion for the robustness, and the values of design and operation variables are determined to minimize the maximum regret. However, in the case that the value of the objective function varies signi®cantly with energy demands, a relative value may be preferable, as the evaluation criterion, to an absolute one, such as the maximum regret. The ®rst objective of this paper is to propose an optimal unit sizing method of energy supply systems in consideration of the robustness under uncertain energy demands based on the relative robustness criterion. The second objective is to clarify features of the robustness through a case study. First, the concept and formulation of an optimal design problem based on the relative robustness criterion is described. Second, a solution method of the robust optimal design problem is proposed. Finally, a case study on a gas turbine cogeneration system for district energy supply is performed to investigate the validity and eectiveness of the proposed method and clarify features of the robust optimal design. 2. Optimal design based on relative robustness criterion 2.1. Basic concept In designing an energy supply system under uncertain energy demands, two items should be considered: ¯exibility and robustness [7]. The former means the feasibility in energy supply for all the possible values of uncertain energy demands and is related with constraints. The latter means the sensitivity of performance for all the possible values of uncertain energy demands and is
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Fig. 1. Concept of robust optimal design based on relative robustness criterion.
related with objective functions. In this paper, an optimal design method is proposed, by which the robustness is maximized while the ¯exibility is satis®ed for all the possible values of uncertain energy demands. The relative robustness criterion is adopted here [5]. The concept of the robust optimal design, based on this criterion, is shown in Fig. 1. The normalized regret or the regret rate is de®ned as the dierence in an objective function between nonoptimal and optimal designs divided by the optimal value of the objective function for some values of uncertain parameters. The relative robustness criterion means that the maximum regret rate is selected as the evaluation criterion for the robustness among the regret rates for all the possible values of uncertain parameters, and that the values of decision variables are determined to minimize the maximum regret rate, or maximize the robustness. Therefore, if this criterion is adopted, the relative dierences in the objective function between the resultant and optimal designs can be small for all the possible values of uncertain parameters. 2.2. Formulation of robust optimal design problem According to the aforementioned basic concept, a robust optimal design problem for the unit sizing of an energy supply system is described as follows: the values of design variables or equipment capacities x, as well as those of operation variables or utility contract demands and energy ¯ow rates z, are determined to minimize the maximum regret rate in an objective function f and satisfy all the possible values of uncertain energy demands y. As shown in Fig. 2, it should be noted that although the values of design variables x must be determined at the design stage when energy demands y are uncertain, those of operation variables z can be adjusted for energy demands y which become certain at the operation stage. Here, it is assumed that the objective function f and all the constraints are expressed as linear functions with respect to x, y and z. First, the problem is formulated without considering the operation variables z and the ¯exibility. The ordinary optimal design problem in which equipment capacities x0 are determined to minimize the objective function f under certain energy demands y is expressed by f x0 ; y min 0 x 2X
1
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Fig. 2. Hierarchical relationships among design variables, uncertain parameters and operation variables.
where X is the set for all the possible values of x0 . Therefore, the robust optimal design problem in which equipment capacities x are determined to minimize the maximum regret rate in the objective function f under uncertain energy demands y is expressed by 0 f x ; y 1 2 min max f x; y min 0 x2X
y2Y
x 2X
where Y is the set for all the possible values of y. Next, the incorporation of the operation variables z, which were neglected previously, into the formulation results in 0 0 min f x ; y; z 1 3 min max min f x; y; z min 0 0 0 x2X
z2Z
y2Y
x 2X z 2Z
0
where Z and Z are the sets for all the possible values of z and z0 , which depend on the values of x; y and x0 ; y, respectively. This is because utility contract demands and energy ¯ow rates must be determined to satisfy capacities and performance characteristics of equipment, as well as energy balance and supply±demand relationships. Finally, the ¯exibility, or the feasibility, in energy supply, which was neglected previously, is incorporated into the formulation. To satisfy the ¯exibility for all the possible values of uncertain energy demands y, an objective function which expresses the infeasibility in energy supply is introduced [8], and equipment capacities x are determined to minimize or make zero the maximum of this objective function for all the possible values of y. This idea is applied to the ordinary and robust optimal designs, and the corresponding optimization problems are expressed by max min p x0 ; y00 ; z00 min 0 00 00 00 x 2X y 2Y z 2Z
4
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503
and min max min p x; y000 ; z000 000 000 000
5
x2X y 2Y z 2Z
respectively, where p is the objective function for the infeasibility in energy supply. In addition, Z 00 and Z 000 are the sets for all the possible values of z00 and z000 , which depend on the values of x0 ; y00 and x; y000 , respectively. To take account of Eqs. (4) and (5) prior to Eq. (3), they are added to Eq. (3) as penalty terms. As a result, the robust optimal design problem is formulated as 0 0 0 00 00 min f x ; y; z P max min p x ; y ; z 1 min max min f x; y; z min 0 0 0 00 00 00 x2X
y2Y
z2Z
x 2X
000 000 P max min p x; y ; z 000 000 000
z 2Z
y 2Y z 2Z
y 2Y z 2Z
6
where P is the coecient for penalty terms, and should be given a suciently large value. 2.3. Solution of robust optimal design problem The optimization problem of Eq. (6) is formulated as a multi-level programming one with the hierarchical operations of minimization and maximization [9]. It is also a nonlinear programming one with the objective function in both the numerator and denominator. Since the penalty terms can be treated in the way similar to the one used in the robust optimal design method based on the minimax regret criterion [4], a solution method for only the optimization problem of Eq. (3), without the penalty terms, is described in the following. On the assumption that f x0 ; y; z0 > 0 for any values of x0 , y and z0 , the operations of minimization with respect to x0 and z0 are moved forward to reformulate Eq. (3) as 0 0 7 min 0 max 0 0 min f x; y; z f x ; y; z 1 x2X x 2X ; y2Y ; z 2Z
z2Z
2.3.1. Evaluation of upper bound Appropriate values of x are assumed in Eq. (7), and the following optimization problem is considered: 0 0 8 max 0 0 min f x; y; z f x ; y; z 1 0 x 2X ; y2Y ; z 2Z
z2Z
The value of the maximum regret rate corresponding to the optimal solution of this problem gives an upper bound for that of the original problem of Eq. (7). The optimization problem of Eq. (8) is a bi-level nonlinear programming one with the hierarchical operations of maximization and minimization. First, on the assumption that f x0 ; y; z0 > 0 for any values of x0 , y and z0 , this problem is transformed into a bi-level linear programming one [10] by means of the fractional programming [11]. Namely, a positive variable q is de®ned as q 1=f x0 ; y; z0
9
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and the problem of Eq. (8) is transformed into min f x; y; zq 1 max0 0 0 x 2X ; y2Y ; z 2Z ; q>0
s:t:
z2Z 0
10
0
f x ; y; z q 1
In addition, x0 , y, z and z0 are multiplied by q to be replaced by u0 , v, w and w0 , respectively. The value of x assumed is also multiplied by q to be replaced by u. This replacement transforms the problem of Eq. (10) into min f u; v; w; q 1 max 0 0 0 u2U ; u 2U ; v2V ; w 2W ; q>0
s:t:
w2W
f u0 ; v; w0 ; q 1
11
u xq where U, V, W and W 0 are the sets for all the possible values of u and u0 , v, w and w0 , respectively. The argument q in the objective function f means that constant terms in f are transformed into the products of the corresponding constants and q. Similarly, U, V, W and W 0 may include q. This is because constant terms in X, Y, Z and Z 0 are also transformed into the products of the corresponding constants and q. Next, the bi-level linear programming problem of Eq. (11) can be reformulated as an ordinary single level optimization one by applying the Kuhn-Tucker optimality condition to its lower level. This reformulation produces the complementarity condition which is the inner product of the inequality constraint and the corresponding Lagrange multiplier vectors. To avoid the nonlinearity due to this complementarity condition, binary variables are introduced to linearize the nonlinear term, and the problem is reduced to a mixed-integer linear programming one [12]. The combination of the branch and bound and the dual simplex algorithms is adopted to solve this problem [13]. The optimal values of x0 , y, z and z0 are obtained by dividing those of u0 , v, w and w0 , respectively, by that of q. 2.3.2. Evaluation of lower bound On the other hand, the values of x0 , y and z0 are assumed to be selected only from the ones obtained by solving Eq. (8), and the following optimization problem is considered in place of Eq. (7): min f x; y; z f x0 ; y; z0 1 12 min 0 max0 x2X x ; y; z 2A
z2Z
where A is the set for combinations of values of x0 , y and z0 . The value of the maximum regret rate corresponding to the optimal solution of this problem gives a lower bound for that of the original problem of Eq. (7). The optimization problem of Eq. (12) is a three level nonlinear programming one, with the hierarchical operations of minimization and maximization, and seems to be dicult to solve. However, the operation of maximization, is only with respect to x0 , y and z0 , which is conducted by selecting the values of x0 , y and z0 from their ®nite number of candidates in the set A. Therefore, the introduction of variable g for the maximum with respect to x0 , y and z0 and the corresponding
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inequality constraints changes the problem of Eq. (12) into the following ordinary linear programming one: min
x2X ; z2Z
s:t:
g g P f x; y; z=f x0 ; y; z0
1
13
8 x0 ; y; z0 2 A This problem can be solved easily by the simplex algorithm. According to the aforementioned procedure, the calculation of upper and lower bounds for the optimal value of the maximum regret rate in Eq. (7) is repeated. If the upper and lower bounds coincide with each other, it is judged that the optimal solution of Eq. (7) is obtained, and the calculation is stopped. 3. Case study To investigate the validity and eectiveness of the robust optimal design method proposed here and to clarify features of the robust optimal design, a case study is performed on a gas turbine cogeneration system for district energy supply. 3.1. System con®guration Fig. 3 shows the con®guration of the system studied, which is composed of a gas turbine generator (GT), a waste heat recovery boiler (BW), a gas ®red auxiliary boiler (BG) and a device for receiving electricity (EP). Electricity is supplied to customers by operating the gas turbine generator and purchasing electricity from an outside electric power company. Exhaust heat generated by the gas turbine is recovered by the waste heat recovery boiler, and it is used for steam supply. An excess of exhaust heat is disposed of through an exhaust gas dumper. A shortage of
Fig. 3. Con®guration of gas turbine cogeneration system.
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steam is supplemented by the gas ®red auxiliary boiler. Steam is used for space cooling/heating and hot water supply. Here, steam demand is estimated on the assumption that steam absorption refrigerators are installed to produce cold water for space cooling. As shown in Fig. 3, since there are relationships between the supplies and demands of electricity and steam, it is impossible to determine equipment capacities by simply using the maxima of electricity and steam demands. Here, the unit sizing is conducted using the aforementioned robust optimal design method, and features of the economic robustness are investigated. The symbols which are used as energy ¯ow rates in the following formulation are included in Fig. 3. 3.2. Formulation of robust optimal design problem The symbols used for the formulation are de®ned, and constraints and an objective function are formulated with the symbols. To take account of variations in energy demands, a typical year is divided into T periods. A quantity corresponding to each period is identi®ed by the subscript t, i.e. t 1; 2; . . . ; T . The duration per year of each period is denoted by dt . As constraints, the performance characteristics of the equipment are formulated as linear equations. For example, for the gas turbine generator, the relationships among the natural gas consumption FGT , the electric power generated EGT , the heat ¯ow rate of exhaust gas QeGT and the power generating capacity EGT are expressed as follows: EGTt aGT FGTt QeGTt bGT FGTt 0 6 EGTt 6 EGT
t 1; 2; . . . ; T
14
where a and b are performance characteristic values, and denotes an upper limit for an energy ¯ow rate, i.e. an equipment capacity or a utility contract demand. Similarly, the performance characteristics of the waste heat recovery and gas ®red auxiliary boilers are expressed by QsBWt aBW QeBWt s 0 6 QsBWt 6 QBW
t 1; 2; . . . ; T
15
and QsBGt aBG FBGt s 0 6 QsBGt 6 QBG
t 1; 2; . . . ; T
16
respectively. In addition, the electricity and natural gas consumptions, Eelec and Fgas , and their contract demands, Eelec and F gas , are related by 0 6 Eelect 6 Eelec 0 6 Fgast 6 F gas
t 1; 2; . . . ; T
17
respectively. Here, the contract demand of purchased electricity Eelec is assumed to be smaller than the capacity of the device for receiving electricity EEP as follows: Eelec 6 EEP
18
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507
Furthermore, energy balance and supply±demand relationships are expressed by Fgast FGTt FBGt Eelect EGTt Edemt QeGTt QeBWt Qedispt QsBWt QsBGt Qsdemt
t 1; 2; . . . ; T
19
From the economic viewpoint, the annual total cost is adopted as the objective function to be minimized and is evaluated by f fR 1
T X t1
s
s
s rsg CEP EEP CGT EGT CBW QBW CBG QBG 12 welec Eelec wgas F gas uelec Eelect ugas Fgast dt
20
where C is the capital unit cost of each piece of equipment, R is the capital recovery factor, s is the ratio of salvage value to capital cost of equipment, r is the interest rate, w is the monthly unit cost for demand charge of each utility and u is the unit cost for energy charge of each utility. The design variable vector composed of equipment capacities x, the uncertain parameter vector composed of energy demands y and the operation variable vector composed of utility contract demands and energy ¯ow rates z are de®ned as s
s
x EEP ; EGT ; QBW ; QBG T
21
yt y
Edemt ; Qsdemt t 1; 2; . . . ; T y1 ; y2 ; . . . ; yT T
zt
EGTt ; QeGTt ; FGTt ; QsBWt ; QeBWt ; QsBGt ; FBGt ; Qedispt ; Eelect ; Fgast t 1; 2; . . . ; T
z
Eelec ; F gas ; z1 ; z2 ; . . . ; zT
22
and T
23
respectively, where the superscript T means the transposition of a vector. As the feasible region X for x, the nonnegative condition xP0
24
is considered. Average and peak energy demands, yA and yP , respectively, are considered as y, and as the possible region Y for y, the yA are assumed to be bounded by lower and upper limits estimated in advance as follows: 1
yA a~ yA 6 yA 6 1 a~
25
where ~ yA is a reference value for yA , and a is a parameter for the uncertainty in energy demands. In addition, the following relationship between yP and yA is considered as Y: yP 1 byA
26
where b is a parameter for the increase rate of peak to average energy demands. As the feasible region Z for z, Eqs. (14)±(19) are considered. The robust optimal design problem is described as follows: equipment capacities x of Eq. (21), as well as utility contract demands and energy ¯ow rates z of Eq. (23), are determined to minimize
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the maximum regret rate in the annual total cost f of Eq. (20), subject to the constraints of Eqs. (14)±(19) and Eq. (24) under the uncertain energy demands y of Eq. (22), constrained by Eqs. (25) and (26). 3.3. Input data To conduct a fundamental study, a single representative day is considered in a typical year, and the day is divided into six periods, each of which has four hours per day. Reference values for average electricity and steam demands ~ yA are estimated for each period. As mentioned previously, average electricity and steam demands yA are assumed to vary within 1 a~yA ; 1 a~yA for each period (t 1; 2; . . . ; 6), and peak electricity and steam demands yP are assumed to be equal to 1 byA for each period (t 7; 8; . . . ; 12; T 12). As an example, Fig. 4(a) and (b) show the uncertainty in electricity and steam demands, respectively, in the case that a b 0:2. Here, the in¯uence of the parameters a and b on the unit sizing is investigated. Other input data are given in
Fig. 4. Uncertainity in energy demands (a b 0:2): (a) electricity and (b) steam.
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Table 1 Input data Item
Value
Duration of period
d1 ±d6 1460 h/year d7 ±d12 0 h/year
Performance characteristic value of equipment
aGT 3:23 kW/(m3 /h) bGT 6:71 kW/(m3 /h) aBW 0:78 kW/kW aBG 10:40 kW/(m3 /h)
Capital unit cost of equipment
CEP 56:0 103 yen/kW CGT 230:0 103 yen/kW CBW 9:6 103 yen/kW CBG 6:9 103 yen/kW
Unit cost of utility
welec 1:74 103 yen/(kW month) wgas 2:37 103 yen/(m3 /h month) uelec 11:0 yen/kWh ugas 31:0 yen/m3
Parameter for annual total cost
R 0:132 r 0:10 s 0:0 P 1:0 103 yen/kWh
Penalty coecient
Table 1. All values of costs are stated in yen, which is equivalent to about 9:3 10 recent exchange rate.
3
dollars on the
3.4. Results and discussion As an example of the convergence characteristics of the solution method, Fig. 5 shows the relationships between the number of iterations for the optimization calculation, and the values of the upper and lower bounds for the optimal value of the maximum regret rate in the annual total cost in the case that a b 0:2. At the ®rst iteration, a large dierence between the values of the upper and lower bounds is found. However, they coincide with each other after several iterations, and the optimal solution is obtained. Similar convergence characteristics are found in other cases. This result shows the validity and eectiveness of the solution method proposed to solve the robust optimal design problem formulated as a multi-level nonlinear programming one. Fig. 6 shows the relationships between the power generating capacity of the gas turbine generator and the maximum regret rate in the annual total cost with a as parameter in the case that b 0:2. The optimal solutions are denoted by the black points. The solutions on the left and right sides of the optimal ones, denoted by white points, are obtained by adding the constraints which set upper and lower limits for the power generating capacity at the corresponding values, respectively. The curve in the case that the uncertainty in energy demands is not considered, i.e. a 0, is also obtained by an ordinary optimization calculation. According to Fig. 6, when the uncertainty in energy demands is not considered, the curve for the maximum regret rate is ¯at in the vicinity of the optimal solution, and the in¯uence of the
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Fig. 5. Convergence characteristics of solution method (a b 0:2).
Fig. 6. Relationships between power generating capacity of gas turbine generator and maximum regret rate in annual total cost (b 0:2).
deviation of the power generating capacity from its optimal value on the maximum regret rate is small. On the contrary, with an increase of the uncertainty in energy demands, the slopes of the curve in the vicinity of the optimal solution increase, and the in¯uence of the deviation of the
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power generating capacity from its optimal value on the maximum regret rate becomes large. However, the maximum regret rate for the optimal solution remains small, even in the case of a large value of a, as compared with that for the solution of the conventional system with zero power generating capacity. Therefore, it becomes more important to determine equipment capacities appropriately with an increase of the uncertainty in energy demands. Fig. 7 shows the relationship between the uncertainty in energy demands a and the optimal values of equipment capacities in the case that b 0:2. According to this ®gure, with an increase of the value of a, the capacities of the gas turbine generator and waste heat recovery boiler decrease slightly, and those of the device for receiving electricity and the gas ®red auxiliary boiler increase dramatically. This is because the robust optimal design avoids an increase of the economic regret due to small energy demands or unbalanced electricity and steam demands, which tend to occur with an increase of the uncertainty in energy demands. Fig. 8 shows the in¯uence of the uncertainty in energy demands a and the increase rate of peak to average energy demands b on the maximum regret rate in the annual total cost. The maximum regret rate naturally increases with the value of a, and its increase rate also tends to increase. On the other hand, the maximum regret rate tends to decrease with an increase of the value of b. Fig. 9 shows the in¯uence of the uncertainty in energy demands a and the increase rate of peak to average energy demands b on the optimal values of the capacities of the gas turbine generator and the device for receiving electricity. The capacity of the gas turbine generator decreases slightly, and that of the device for receiving electricity increases dramatically with an increase of the value of a for all the values of b. The capacity of the gas turbine generator remains almost constant, regardless of the value of b, and that of the device for receiving electricity increases with an increase of the value of b. This result means that the optimal capacity of the gas turbine generator depends on average energy demands rather than peak ones.
Fig. 7. Relationships between uncertainty in energy demands and optimal values of equipment capacities (b 0:2).
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Fig. 8. In¯uence of uncertainty in energy demands and increase rate of peak to average energy demands on optimal values of maximum regret rate in annual total cost.
Fig. 9. In¯uence of uncertainty in energy demands and increase rate of peak to average energy demands on optimal values of equipment capacities.
4. Conclusions A robust optimal design method, based on the relative robustness criterion, has been proposed for the unit sizing of energy supply systems under uncertain energy demands. Equipment capacities, as well as utility contract demands and energy ¯ow rates, have been determined to
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minimize the maximum regret rate in the annual total cost and satisfy all the possible energy demands. This robust optimal design problem has been formulated as a multi-level nonlinear programming one, and its solution has been derived by repeatedly evaluating the upper and lower bounds for the optimal value of the maximum regret rate by means of the fractional, the bi-level, and the linear programming. A case study has been performed on a gas turbine cogeneration system for district energy supply to evaluate its economic robustness under uncertain energy demands. The following are the main results obtained here: (1) The robust optimal design method can rationally assess a relative value for the robustness of energy supply systems. (2) The solution method can eciently derive the optimal solution of the robust optimal design problem. (3) With an increase of the uncertainty in energy demands, the in¯uence of equipment capacities on the maximum regret rate in the annual total cost increases, and it becomes more important to conduct the unit sizing appropriately. (4) With an increase of the uncertainty in energy demands, the optimal capacity of the cogeneration unit decreases slightly, and those of the auxiliary equipment increase dramatically, to avoid the economic regret. (5) The optimal capacity of the cogeneration unit depends on average energy demands rather than peak ones.
Acknowledgements A part of this work was ®nancially supported by the ``Research for the Future'' Program of the Japan Society for the Promotion of Science (JSPS-RFTF 97P01002).
References [1] Yokoyama R, Ito K, Matsumoto Y. Optimal sizing of a gas turbine cogeneration plant in consideration of its operational strategy. Trans ASME J Eng Gas Turb Power 1994;116(1):32±8. [2] Yokoyama R, Ito K. Multi-objective optimization in unit sizing of a gas turbine cogeneration plant. Trans ASME J Eng Gas Turb Power 1995;117(1):53±9. [3] Yokoyama R, Ito K, Matsumoto Y. Optimal multistage expansion planning of a gas turbine cogeneration plant. Trans ASME J Eng Gas Turb Power 1996;118(4):803±9. [4] Yokoyama R, Ito K. Robust optimal design of a gas turbine cogeneration plant based on minimax regret criterion. 44th ASME Gas Turbine and Aeroengine Technical Congress, Exposition and Users Symposium 1999, paper no. 99-GT-128:1±8. [5] Kouvelis P, Yu G. Robust discrete optimization and its applications. Dordrecht: Kluwer Academic Publishers; 1997. [6] Yokoyama R, Ito K, Sato K. Evaluation of economic robustness of gas turbine cogeneration plants with variable heat to power ratios. Proceedings of the International Gas Turbine Congress 1999 Kobe, vol. I, 1999. p. 381±8. [7] Parkinson A, Sorensen C, Pourhassan N. A general approach for robust optimal design. Trans ASME J Mech Des 1993;115(1):74±80.
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Optimal design of energy supply systems based on relative robustness criterion Ryohei Yokoyama *, Koichi Ito Department of Energy Systems Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, Osaka 599-8531, Japan Received 10 October 2000; accepted 19 February 2001
Abstract A robust optimal design method, based on the relative robustness criterion, is proposed to conduct the unit sizing of energy supply systems, so that they are robust economically under uncertain energy demands. The values of design variables or equipment capacities, as well as those of operation variables or utility contract demands and energy ¯ow rates, are determined to minimize the maximum normalized regret or the maximum regret rate in the annual total cost and satisfy all the possible energy demands. This optimization problem is formulated as a multi-level nonlinear programming problem, and its solution is obtained by repeatedly evaluating the upper and lower bounds for the optimal value of the maximum regret rate by means of the fractional, the bi-level and the linear programming. Through a case study on a cogeneration system, features of the robust optimal design, based on the relative robustness criterion, are clari®ed. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Energy supply systems; Unit sizing; Operational planning; Robust design; Optimization; Relative robustness; Worst case analysis; Multi-level programming; Fractional programming
1. Introduction For the purpose of improving economic and energy saving characteristics, distributed energy supply systems, such as cogeneration ones, have been installed increasingly into districts and buildings in recent years. To attain the highest economic and energy saving characteristics, design and operation are important issues. In designing energy supply systems, the estimation of energy demands is an important work. This is because planners are requested to determine rationally what types, numbers and capacities *
Corresponding author. Tel.: +81-722-56-2030/54-9232; fax: +81-722-56-2031/54-9904. E-mail address: [email protected] (R. Yokoyama).
0196-8904/02/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 0 1 ) 0 0 0 2 7 - 9
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of equipment should be installed in consideration of equipment operational strategies for seasonal and hourly variations in estimated energy demands, which signi®cantly aect the economic and energy saving characteristics. Related with this issue, an optimal unit sizing method has been proposed, by which equipment capacities and utility contract demands can be determined in order to minimize the annual total cost and primary energy consumption while considering equipment operational strategies for average energy demands estimated on multiple representative days, and to satisfy peak energy demands in summer and winter [1±3]. This method can be a useful tool for unit sizing, if we can estimate energy demands precisely. However, many conditions under which energy demands are estimated have some uncertainty at the design stage, which makes it impossible to estimate energy demands precisely. If the unit sizing is conducted by considering that estimated energy demands are certain, the economic and energy saving characteristics expected may not be attained, and de®cits in energy supply may occur. This is because energy demands, which occur at the operation stage, dier from those estimated at the design stage. Therefore, planners should consider that energy demands have some uncertainty, evaluate the robustness in economic and energy saving characteristics under uncertain energy demands and design energy supply systems rationally in consideration of the robustness. An optimal unit sizing method of energy supply systems in consideration of the robustness under uncertain energy demands has been proposed [4] based on the minimax regret criterion [5]. This method has been applied to the evaluation of the economic robustness of some types of gas turbine cogeneration systems, and features of the robustness have been clari®ed [6]. According to the method, the maximum regret in an objective function is adopted as the evaluation criterion for the robustness, and the values of design and operation variables are determined to minimize the maximum regret. However, in the case that the value of the objective function varies signi®cantly with energy demands, a relative value may be preferable, as the evaluation criterion, to an absolute one, such as the maximum regret. The ®rst objective of this paper is to propose an optimal unit sizing method of energy supply systems in consideration of the robustness under uncertain energy demands based on the relative robustness criterion. The second objective is to clarify features of the robustness through a case study. First, the concept and formulation of an optimal design problem based on the relative robustness criterion is described. Second, a solution method of the robust optimal design problem is proposed. Finally, a case study on a gas turbine cogeneration system for district energy supply is performed to investigate the validity and eectiveness of the proposed method and clarify features of the robust optimal design. 2. Optimal design based on relative robustness criterion 2.1. Basic concept In designing an energy supply system under uncertain energy demands, two items should be considered: ¯exibility and robustness [7]. The former means the feasibility in energy supply for all the possible values of uncertain energy demands and is related with constraints. The latter means the sensitivity of performance for all the possible values of uncertain energy demands and is
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Fig. 1. Concept of robust optimal design based on relative robustness criterion.
related with objective functions. In this paper, an optimal design method is proposed, by which the robustness is maximized while the ¯exibility is satis®ed for all the possible values of uncertain energy demands. The relative robustness criterion is adopted here [5]. The concept of the robust optimal design, based on this criterion, is shown in Fig. 1. The normalized regret or the regret rate is de®ned as the dierence in an objective function between nonoptimal and optimal designs divided by the optimal value of the objective function for some values of uncertain parameters. The relative robustness criterion means that the maximum regret rate is selected as the evaluation criterion for the robustness among the regret rates for all the possible values of uncertain parameters, and that the values of decision variables are determined to minimize the maximum regret rate, or maximize the robustness. Therefore, if this criterion is adopted, the relative dierences in the objective function between the resultant and optimal designs can be small for all the possible values of uncertain parameters. 2.2. Formulation of robust optimal design problem According to the aforementioned basic concept, a robust optimal design problem for the unit sizing of an energy supply system is described as follows: the values of design variables or equipment capacities x, as well as those of operation variables or utility contract demands and energy ¯ow rates z, are determined to minimize the maximum regret rate in an objective function f and satisfy all the possible values of uncertain energy demands y. As shown in Fig. 2, it should be noted that although the values of design variables x must be determined at the design stage when energy demands y are uncertain, those of operation variables z can be adjusted for energy demands y which become certain at the operation stage. Here, it is assumed that the objective function f and all the constraints are expressed as linear functions with respect to x, y and z. First, the problem is formulated without considering the operation variables z and the ¯exibility. The ordinary optimal design problem in which equipment capacities x0 are determined to minimize the objective function f under certain energy demands y is expressed by f x0 ; y min 0 x 2X
1
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Fig. 2. Hierarchical relationships among design variables, uncertain parameters and operation variables.
where X is the set for all the possible values of x0 . Therefore, the robust optimal design problem in which equipment capacities x are determined to minimize the maximum regret rate in the objective function f under uncertain energy demands y is expressed by 0 f x ; y 1 2 min max f x; y min 0 x2X
y2Y
x 2X
where Y is the set for all the possible values of y. Next, the incorporation of the operation variables z, which were neglected previously, into the formulation results in 0 0 min f x ; y; z 1 3 min max min f x; y; z min 0 0 0 x2X
z2Z
y2Y
x 2X z 2Z
0
where Z and Z are the sets for all the possible values of z and z0 , which depend on the values of x; y and x0 ; y, respectively. This is because utility contract demands and energy ¯ow rates must be determined to satisfy capacities and performance characteristics of equipment, as well as energy balance and supply±demand relationships. Finally, the ¯exibility, or the feasibility, in energy supply, which was neglected previously, is incorporated into the formulation. To satisfy the ¯exibility for all the possible values of uncertain energy demands y, an objective function which expresses the infeasibility in energy supply is introduced [8], and equipment capacities x are determined to minimize or make zero the maximum of this objective function for all the possible values of y. This idea is applied to the ordinary and robust optimal designs, and the corresponding optimization problems are expressed by max min p x0 ; y00 ; z00 min 0 00 00 00 x 2X y 2Y z 2Z
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503
and min max min p x; y000 ; z000 000 000 000
5
x2X y 2Y z 2Z
respectively, where p is the objective function for the infeasibility in energy supply. In addition, Z 00 and Z 000 are the sets for all the possible values of z00 and z000 , which depend on the values of x0 ; y00 and x; y000 , respectively. To take account of Eqs. (4) and (5) prior to Eq. (3), they are added to Eq. (3) as penalty terms. As a result, the robust optimal design problem is formulated as 0 0 0 00 00 min f x ; y; z P max min p x ; y ; z 1 min max min f x; y; z min 0 0 0 00 00 00 x2X
y2Y
z2Z
x 2X
000 000 P max min p x; y ; z 000 000 000
z 2Z
y 2Y z 2Z
y 2Y z 2Z
6
where P is the coecient for penalty terms, and should be given a suciently large value. 2.3. Solution of robust optimal design problem The optimization problem of Eq. (6) is formulated as a multi-level programming one with the hierarchical operations of minimization and maximization [9]. It is also a nonlinear programming one with the objective function in both the numerator and denominator. Since the penalty terms can be treated in the way similar to the one used in the robust optimal design method based on the minimax regret criterion [4], a solution method for only the optimization problem of Eq. (3), without the penalty terms, is described in the following. On the assumption that f x0 ; y; z0 > 0 for any values of x0 , y and z0 , the operations of minimization with respect to x0 and z0 are moved forward to reformulate Eq. (3) as 0 0 7 min 0 max 0 0 min f x; y; z f x ; y; z 1 x2X x 2X ; y2Y ; z 2Z
z2Z
2.3.1. Evaluation of upper bound Appropriate values of x are assumed in Eq. (7), and the following optimization problem is considered: 0 0 8 max 0 0 min f x; y; z f x ; y; z 1 0 x 2X ; y2Y ; z 2Z
z2Z
The value of the maximum regret rate corresponding to the optimal solution of this problem gives an upper bound for that of the original problem of Eq. (7). The optimization problem of Eq. (8) is a bi-level nonlinear programming one with the hierarchical operations of maximization and minimization. First, on the assumption that f x0 ; y; z0 > 0 for any values of x0 , y and z0 , this problem is transformed into a bi-level linear programming one [10] by means of the fractional programming [11]. Namely, a positive variable q is de®ned as q 1=f x0 ; y; z0
9
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and the problem of Eq. (8) is transformed into min f x; y; zq 1 max0 0 0 x 2X ; y2Y ; z 2Z ; q>0
s:t:
z2Z 0
10
0
f x ; y; z q 1
In addition, x0 , y, z and z0 are multiplied by q to be replaced by u0 , v, w and w0 , respectively. The value of x assumed is also multiplied by q to be replaced by u. This replacement transforms the problem of Eq. (10) into min f u; v; w; q 1 max 0 0 0 u2U ; u 2U ; v2V ; w 2W ; q>0
s:t:
w2W
f u0 ; v; w0 ; q 1
11
u xq where U, V, W and W 0 are the sets for all the possible values of u and u0 , v, w and w0 , respectively. The argument q in the objective function f means that constant terms in f are transformed into the products of the corresponding constants and q. Similarly, U, V, W and W 0 may include q. This is because constant terms in X, Y, Z and Z 0 are also transformed into the products of the corresponding constants and q. Next, the bi-level linear programming problem of Eq. (11) can be reformulated as an ordinary single level optimization one by applying the Kuhn-Tucker optimality condition to its lower level. This reformulation produces the complementarity condition which is the inner product of the inequality constraint and the corresponding Lagrange multiplier vectors. To avoid the nonlinearity due to this complementarity condition, binary variables are introduced to linearize the nonlinear term, and the problem is reduced to a mixed-integer linear programming one [12]. The combination of the branch and bound and the dual simplex algorithms is adopted to solve this problem [13]. The optimal values of x0 , y, z and z0 are obtained by dividing those of u0 , v, w and w0 , respectively, by that of q. 2.3.2. Evaluation of lower bound On the other hand, the values of x0 , y and z0 are assumed to be selected only from the ones obtained by solving Eq. (8), and the following optimization problem is considered in place of Eq. (7): min f x; y; z f x0 ; y; z0 1 12 min 0 max0 x2X x ; y; z 2A
z2Z
where A is the set for combinations of values of x0 , y and z0 . The value of the maximum regret rate corresponding to the optimal solution of this problem gives a lower bound for that of the original problem of Eq. (7). The optimization problem of Eq. (12) is a three level nonlinear programming one, with the hierarchical operations of minimization and maximization, and seems to be dicult to solve. However, the operation of maximization, is only with respect to x0 , y and z0 , which is conducted by selecting the values of x0 , y and z0 from their ®nite number of candidates in the set A. Therefore, the introduction of variable g for the maximum with respect to x0 , y and z0 and the corresponding
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inequality constraints changes the problem of Eq. (12) into the following ordinary linear programming one: min
x2X ; z2Z
s:t:
g g P f x; y; z=f x0 ; y; z0
1
13
8 x0 ; y; z0 2 A This problem can be solved easily by the simplex algorithm. According to the aforementioned procedure, the calculation of upper and lower bounds for the optimal value of the maximum regret rate in Eq. (7) is repeated. If the upper and lower bounds coincide with each other, it is judged that the optimal solution of Eq. (7) is obtained, and the calculation is stopped. 3. Case study To investigate the validity and eectiveness of the robust optimal design method proposed here and to clarify features of the robust optimal design, a case study is performed on a gas turbine cogeneration system for district energy supply. 3.1. System con®guration Fig. 3 shows the con®guration of the system studied, which is composed of a gas turbine generator (GT), a waste heat recovery boiler (BW), a gas ®red auxiliary boiler (BG) and a device for receiving electricity (EP). Electricity is supplied to customers by operating the gas turbine generator and purchasing electricity from an outside electric power company. Exhaust heat generated by the gas turbine is recovered by the waste heat recovery boiler, and it is used for steam supply. An excess of exhaust heat is disposed of through an exhaust gas dumper. A shortage of
Fig. 3. Con®guration of gas turbine cogeneration system.
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steam is supplemented by the gas ®red auxiliary boiler. Steam is used for space cooling/heating and hot water supply. Here, steam demand is estimated on the assumption that steam absorption refrigerators are installed to produce cold water for space cooling. As shown in Fig. 3, since there are relationships between the supplies and demands of electricity and steam, it is impossible to determine equipment capacities by simply using the maxima of electricity and steam demands. Here, the unit sizing is conducted using the aforementioned robust optimal design method, and features of the economic robustness are investigated. The symbols which are used as energy ¯ow rates in the following formulation are included in Fig. 3. 3.2. Formulation of robust optimal design problem The symbols used for the formulation are de®ned, and constraints and an objective function are formulated with the symbols. To take account of variations in energy demands, a typical year is divided into T periods. A quantity corresponding to each period is identi®ed by the subscript t, i.e. t 1; 2; . . . ; T . The duration per year of each period is denoted by dt . As constraints, the performance characteristics of the equipment are formulated as linear equations. For example, for the gas turbine generator, the relationships among the natural gas consumption FGT , the electric power generated EGT , the heat ¯ow rate of exhaust gas QeGT and the power generating capacity EGT are expressed as follows: EGTt aGT FGTt QeGTt bGT FGTt 0 6 EGTt 6 EGT
t 1; 2; . . . ; T
14
where a and b are performance characteristic values, and denotes an upper limit for an energy ¯ow rate, i.e. an equipment capacity or a utility contract demand. Similarly, the performance characteristics of the waste heat recovery and gas ®red auxiliary boilers are expressed by QsBWt aBW QeBWt s 0 6 QsBWt 6 QBW
t 1; 2; . . . ; T
15
and QsBGt aBG FBGt s 0 6 QsBGt 6 QBG
t 1; 2; . . . ; T
16
respectively. In addition, the electricity and natural gas consumptions, Eelec and Fgas , and their contract demands, Eelec and F gas , are related by 0 6 Eelect 6 Eelec 0 6 Fgast 6 F gas
t 1; 2; . . . ; T
17
respectively. Here, the contract demand of purchased electricity Eelec is assumed to be smaller than the capacity of the device for receiving electricity EEP as follows: Eelec 6 EEP
18
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Furthermore, energy balance and supply±demand relationships are expressed by Fgast FGTt FBGt Eelect EGTt Edemt QeGTt QeBWt Qedispt QsBWt QsBGt Qsdemt
t 1; 2; . . . ; T
19
From the economic viewpoint, the annual total cost is adopted as the objective function to be minimized and is evaluated by f fR 1
T X t1
s
s
s rsg CEP EEP CGT EGT CBW QBW CBG QBG 12 welec Eelec wgas F gas uelec Eelect ugas Fgast dt
20
where C is the capital unit cost of each piece of equipment, R is the capital recovery factor, s is the ratio of salvage value to capital cost of equipment, r is the interest rate, w is the monthly unit cost for demand charge of each utility and u is the unit cost for energy charge of each utility. The design variable vector composed of equipment capacities x, the uncertain parameter vector composed of energy demands y and the operation variable vector composed of utility contract demands and energy ¯ow rates z are de®ned as s
s
x EEP ; EGT ; QBW ; QBG T
21
yt y
Edemt ; Qsdemt t 1; 2; . . . ; T y1 ; y2 ; . . . ; yT T
zt
EGTt ; QeGTt ; FGTt ; QsBWt ; QeBWt ; QsBGt ; FBGt ; Qedispt ; Eelect ; Fgast t 1; 2; . . . ; T
z
Eelec ; F gas ; z1 ; z2 ; . . . ; zT
22
and T
23
respectively, where the superscript T means the transposition of a vector. As the feasible region X for x, the nonnegative condition xP0
24
is considered. Average and peak energy demands, yA and yP , respectively, are considered as y, and as the possible region Y for y, the yA are assumed to be bounded by lower and upper limits estimated in advance as follows: 1
yA a~ yA 6 yA 6 1 a~
25
where ~ yA is a reference value for yA , and a is a parameter for the uncertainty in energy demands. In addition, the following relationship between yP and yA is considered as Y: yP 1 byA
26
where b is a parameter for the increase rate of peak to average energy demands. As the feasible region Z for z, Eqs. (14)±(19) are considered. The robust optimal design problem is described as follows: equipment capacities x of Eq. (21), as well as utility contract demands and energy ¯ow rates z of Eq. (23), are determined to minimize
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the maximum regret rate in the annual total cost f of Eq. (20), subject to the constraints of Eqs. (14)±(19) and Eq. (24) under the uncertain energy demands y of Eq. (22), constrained by Eqs. (25) and (26). 3.3. Input data To conduct a fundamental study, a single representative day is considered in a typical year, and the day is divided into six periods, each of which has four hours per day. Reference values for average electricity and steam demands ~ yA are estimated for each period. As mentioned previously, average electricity and steam demands yA are assumed to vary within 1 a~yA ; 1 a~yA for each period (t 1; 2; . . . ; 6), and peak electricity and steam demands yP are assumed to be equal to 1 byA for each period (t 7; 8; . . . ; 12; T 12). As an example, Fig. 4(a) and (b) show the uncertainty in electricity and steam demands, respectively, in the case that a b 0:2. Here, the in¯uence of the parameters a and b on the unit sizing is investigated. Other input data are given in
Fig. 4. Uncertainity in energy demands (a b 0:2): (a) electricity and (b) steam.
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Table 1 Input data Item
Value
Duration of period
d1 ±d6 1460 h/year d7 ±d12 0 h/year
Performance characteristic value of equipment
aGT 3:23 kW/(m3 /h) bGT 6:71 kW/(m3 /h) aBW 0:78 kW/kW aBG 10:40 kW/(m3 /h)
Capital unit cost of equipment
CEP 56:0 103 yen/kW CGT 230:0 103 yen/kW CBW 9:6 103 yen/kW CBG 6:9 103 yen/kW
Unit cost of utility
welec 1:74 103 yen/(kW month) wgas 2:37 103 yen/(m3 /h month) uelec 11:0 yen/kWh ugas 31:0 yen/m3
Parameter for annual total cost
R 0:132 r 0:10 s 0:0 P 1:0 103 yen/kWh
Penalty coecient
Table 1. All values of costs are stated in yen, which is equivalent to about 9:3 10 recent exchange rate.
3
dollars on the
3.4. Results and discussion As an example of the convergence characteristics of the solution method, Fig. 5 shows the relationships between the number of iterations for the optimization calculation, and the values of the upper and lower bounds for the optimal value of the maximum regret rate in the annual total cost in the case that a b 0:2. At the ®rst iteration, a large dierence between the values of the upper and lower bounds is found. However, they coincide with each other after several iterations, and the optimal solution is obtained. Similar convergence characteristics are found in other cases. This result shows the validity and eectiveness of the solution method proposed to solve the robust optimal design problem formulated as a multi-level nonlinear programming one. Fig. 6 shows the relationships between the power generating capacity of the gas turbine generator and the maximum regret rate in the annual total cost with a as parameter in the case that b 0:2. The optimal solutions are denoted by the black points. The solutions on the left and right sides of the optimal ones, denoted by white points, are obtained by adding the constraints which set upper and lower limits for the power generating capacity at the corresponding values, respectively. The curve in the case that the uncertainty in energy demands is not considered, i.e. a 0, is also obtained by an ordinary optimization calculation. According to Fig. 6, when the uncertainty in energy demands is not considered, the curve for the maximum regret rate is ¯at in the vicinity of the optimal solution, and the in¯uence of the
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Fig. 5. Convergence characteristics of solution method (a b 0:2).
Fig. 6. Relationships between power generating capacity of gas turbine generator and maximum regret rate in annual total cost (b 0:2).
deviation of the power generating capacity from its optimal value on the maximum regret rate is small. On the contrary, with an increase of the uncertainty in energy demands, the slopes of the curve in the vicinity of the optimal solution increase, and the in¯uence of the deviation of the
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power generating capacity from its optimal value on the maximum regret rate becomes large. However, the maximum regret rate for the optimal solution remains small, even in the case of a large value of a, as compared with that for the solution of the conventional system with zero power generating capacity. Therefore, it becomes more important to determine equipment capacities appropriately with an increase of the uncertainty in energy demands. Fig. 7 shows the relationship between the uncertainty in energy demands a and the optimal values of equipment capacities in the case that b 0:2. According to this ®gure, with an increase of the value of a, the capacities of the gas turbine generator and waste heat recovery boiler decrease slightly, and those of the device for receiving electricity and the gas ®red auxiliary boiler increase dramatically. This is because the robust optimal design avoids an increase of the economic regret due to small energy demands or unbalanced electricity and steam demands, which tend to occur with an increase of the uncertainty in energy demands. Fig. 8 shows the in¯uence of the uncertainty in energy demands a and the increase rate of peak to average energy demands b on the maximum regret rate in the annual total cost. The maximum regret rate naturally increases with the value of a, and its increase rate also tends to increase. On the other hand, the maximum regret rate tends to decrease with an increase of the value of b. Fig. 9 shows the in¯uence of the uncertainty in energy demands a and the increase rate of peak to average energy demands b on the optimal values of the capacities of the gas turbine generator and the device for receiving electricity. The capacity of the gas turbine generator decreases slightly, and that of the device for receiving electricity increases dramatically with an increase of the value of a for all the values of b. The capacity of the gas turbine generator remains almost constant, regardless of the value of b, and that of the device for receiving electricity increases with an increase of the value of b. This result means that the optimal capacity of the gas turbine generator depends on average energy demands rather than peak ones.
Fig. 7. Relationships between uncertainty in energy demands and optimal values of equipment capacities (b 0:2).
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Fig. 8. In¯uence of uncertainty in energy demands and increase rate of peak to average energy demands on optimal values of maximum regret rate in annual total cost.
Fig. 9. In¯uence of uncertainty in energy demands and increase rate of peak to average energy demands on optimal values of equipment capacities.
4. Conclusions A robust optimal design method, based on the relative robustness criterion, has been proposed for the unit sizing of energy supply systems under uncertain energy demands. Equipment capacities, as well as utility contract demands and energy ¯ow rates, have been determined to
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minimize the maximum regret rate in the annual total cost and satisfy all the possible energy demands. This robust optimal design problem has been formulated as a multi-level nonlinear programming one, and its solution has been derived by repeatedly evaluating the upper and lower bounds for the optimal value of the maximum regret rate by means of the fractional, the bi-level, and the linear programming. A case study has been performed on a gas turbine cogeneration system for district energy supply to evaluate its economic robustness under uncertain energy demands. The following are the main results obtained here: (1) The robust optimal design method can rationally assess a relative value for the robustness of energy supply systems. (2) The solution method can eciently derive the optimal solution of the robust optimal design problem. (3) With an increase of the uncertainty in energy demands, the in¯uence of equipment capacities on the maximum regret rate in the annual total cost increases, and it becomes more important to conduct the unit sizing appropriately. (4) With an increase of the uncertainty in energy demands, the optimal capacity of the cogeneration unit decreases slightly, and those of the auxiliary equipment increase dramatically, to avoid the economic regret. (5) The optimal capacity of the cogeneration unit depends on average energy demands rather than peak ones.
Acknowledgements A part of this work was ®nancially supported by the ``Research for the Future'' Program of the Japan Society for the Promotion of Science (JSPS-RFTF 97P01002).
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