Outer‐approximation method for security constrained unit commitment
2013; Institution of Engineering and Technology; Volume: 7; Issue: 11 Linguagem: Inglês
10.1049/iet-gtd.2012.0311
ISSN1751-8695
AutoresJuan P. Ruiz, Jianhui Wang, Cong Liu, Gengyang Sun,
Tópico(s)Advanced Numerical Methods in Computational Mathematics
ResumoIET Generation, Transmission & DistributionVolume 7, Issue 11 p. 1210-1218 ArticleFree Access Outer-approximation method for security constrained unit commitment Juan P. Ruiz, Juan P. Ruiz Department of Chemical Engineering, University of Texas, Austin, TX, 78712 USASearch for more papers by this authorJianhui Wang, Corresponding Author Jianhui Wang [email protected] Division of Information and Science, Argonne National Laboratory, Lemont, IL, 60439 USASearch for more papers by this authorCong Liu, Cong Liu Division of Information and Science, Argonne National Laboratory, Lemont, IL, 60439 USASearch for more papers by this authorGengyang Sun, Gengyang Sun Division of Information and Science, Argonne National Laboratory, Lemont, IL, 60439 USASearch for more papers by this author Juan P. Ruiz, Juan P. Ruiz Department of Chemical Engineering, University of Texas, Austin, TX, 78712 USASearch for more papers by this authorJianhui Wang, Corresponding Author Jianhui Wang [email protected] Division of Information and Science, Argonne National Laboratory, Lemont, IL, 60439 USASearch for more papers by this authorCong Liu, Cong Liu Division of Information and Science, Argonne National Laboratory, Lemont, IL, 60439 USASearch for more papers by this authorGengyang Sun, Gengyang Sun Division of Information and Science, Argonne National Laboratory, Lemont, IL, 60439 USASearch for more papers by this author First published: 01 November 2013 https://doi.org/10.1049/iet-gtd.2012.0311Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract In this study, the authors present an outer-approximation method to solve the mixed-integer non-linear security constrained unit commitment problem. The main idea lies in solving sequentially a set of mixed-integer linear programs (MILP) to obtain lower bounds (LBs) of the global optimum and perform function evaluations on the incumbent solution of the MILP problem to obtain upper bounds (UBs). The algorithm stops when the LB and UB are sufficiently close. The authors also present a heuristic method that builds on the proposed framework to improve the quality of the solution obtained from the piecewise linear method. The authors show through a set of numerical examples the performance of this approach. Nomenclature Index i index of bus j index of load t index of hour m, n index pair for line between buses Variables and functions I status indicator of generating units Z, Y startup/shutdown indicator of generating units P generation dispatch of a unit SR spinning reserve of a unit OR operating reserve of a unit QSC quick start capacity of a unit STC, DSC startup and shutdown cost of a unit PLmn power flow in transmission line from m to n maximum potential generation of a unit Constants Pmax,i maximum generation capacity of unit i Pmin,i minimum generation capacity of unit i PLmax, mn maximum power flow in a transmission line C0I, C1I, C2I coefficients for quadratic fuel cost function RU, RD maximum ramp up/down rates SU, SD startup and shutdown costs of a unit RSU, RSD startup/shutdown ramp rate MSR maximum sustained ramp rate of a unit (MW/min) QSC quick start capacity of a unit LD load SRD spinning reserve requirement ORD operating reserve requirement UT, DT minimum on/off time of a unit Abbreviations SCUC security constrained unit commitment MIP mixed-integer programming MINLP mixed-integer non-linear programming GDP generalised disjunctive programming (DP) CDP convex DP LDP linear DP tol maximum relative gap allowed in the solution ctol maximum tolerance for cut addition LB lower bound on the global optimum UB upper bound on the global optimum 1 Introduction In an electricity market, the independent system operator (ISO) executes the security constrained unit commitment (SCUC) to plan a secure and economical hourly generation schedule for the day-ahead market [1]. With the fast development of renewable energy, solar power [2] and wind generation [3] have also been considered in SCUC problems. In general, the commitment of a generating unit at a given hour can be represented by using discrete variables while continuous variables can be used to describe the power generated by each unit. The unit commitment and economic dispatch heavily depend on the cost of power generation which, in general, can be represented by using second-order polynomials. In order to fully describe the system many constraints should be also considered such as minimum on/off time of units and load balances. As a result, the SCUC problem can be modelled as a mixed-integer non-linear programming (MINLP) problem. The problem itself is NP-hard and difficult to solve. Many approaches have been proposed in the literature to solve the problem more effectively and efficiently (e.g. mixed-integer linear programming (MILP)-based methods, Lagrangian relaxation methods [4], modified branch-and-bound method with quadratic programming [5], augmented Lagrangian relaxation [6], linear programming [7], semi-definite programming-based method [8] and Benders decomposition methods [9]). Although it is known that the solution of MILP may require long computational times and large memory availability for large-scale programs, recent advances in MILP solution methods [10], have shown great improvement in performance, which is confirmed by most ISOs already using MILP formulations. In this regard, a MILP formulation is proposed in [11] to model the self-scheduling problem by a power producer in an electricity market and a computationally efficient MILP formulation is investigated in [12, 13] where only one integer variable is used to model the status of the unit and startup/shutdown. However, the reformulation of the original MINLP problem to a MILP program is in general achieved by replacing the non-linear quadratic objective function with a linear step-wise approximation. Hence, accuracy in the results and the optimality might be lost. The use of generic traditional non-linear solvers, on the other hand, often leads to significant computational time to find the solution since they do not take advantage of the particular characteristics of the problem. As shown in this work, the outer-approximation approach we propose combined with an efficient initialisation step can overcome these issues effectively. The general outer-approximation method has been used to solve a wide class of MINLP programs [14]. For the particular case of SCUC, Frangioni et al. [15] proposed the use of perspective functions to generate cuts that can be used in a general outer-approximation framework. In comparison, in this paper, we combine the benefits of the piece-wise linear method to find local solutions with an efficient outer-approximation method that uses strong cuts obtained from disjunctive programming (DP) theory. It is important to note that this method is able to guarantee the global optimality within a given tolerance while still taking full advantage of high performance mixed-integer linear solvers. As shown in the numerical tests (Section 5), these low computational times, compared with other methods make this method appealing for large-scale systems. The paper is organised as follows. In Section 2, we introduce the MINLP formulation of the SCUC problem we aim at solving. In Section 3, we present the theoretical foundation of the proposed method. The convergence properties are discussed in Section 4. The performance of the method is then assessed in Section 5. A number of test systems are solved. Finally, we describe in Section 6 a fast local method that can be developed by slightly modifying the proposed global framework. Section 7 concludes the paper. 2 Formulation SCUC model The unit commitment problem is formulated as a MINLP problem as shown in (1)–(21) where all the equations are linear except the objective function. The objective function in (1) is to minimise the total operating costs. The startup and shutdown costs of generating units are represented by the constraints in (2) and (3). The power balance constraint is described in (4) and transmission constraints are described in (5)–(7). Spinning reserve and operating reserve constraints are represented by (8) and (9). The constraints in (10)–(21) represent operation characteristics of individual units such as startup/shutdown, capacity constraints, ramp rate and minimum on/off time [12]. Objective function (1)Startup and shutdown cost (2) (3)Power balance (4)Transmission constraints (5) (6) (7)Reserve constraints in the system (8) (9)Coupling constraints of individual unit status, startup and shutdown indicators (10) (11)Capacity constraints (12) (13) (14)Individual unit balance and reserve constraints (15) (16)Individual unit ramping constraints (17) (18) (19)Individual unit minimum on/off time (20) (21)Note that if no generation unit exists in bus i, the variables Pit, Iit, Zit, Yit and constraints (2)–(4), (10)–(21) are eliminated from the formulation. 3 Reformulation with DP As presented in section 2, the SCUC problem can be modelled as a MINLP program. The general structure of the SCUC is presented below Note that without losing generality we have removed the subscript that corresponds to the time period. Clearly, the variables xi and yi represent the power injected and the commitment or de-commitment of a unit at node i, respectively. Also, fi(xi) represents the non-linear part of the objective function, which in this case is convex quadratic and ciyi is the linear part associated with the fixed cost of the unit. An alternative approach to representing discrete and continuous optimisation problems like the one above is by using models consisting of algebraic constraints, logic disjunctions and logic propositions [16–19]. This approach not only facilitates the development of the models by making the formulation process intuitive, but it also keeps in the model the underlying logic structure of the problem that can be exploited to find the solution more efficiently. A particular case of these models is DP which uses a set of disjunctions to represent logic relationships that can be described through the OR operator. Each disjunction is formed by a set of terms or disjuncts which describe different conditions in the solution space of the problem. For instance, the selection or no selection of a unit can be easily represented by a disjunction of two terms. DP has been successfully applied to process design and planning and scheduling problems [20]. In this section, the problem is reformulated as an equivalent convex disjunctive program (CDP) in order to exploit the wealth of theory of DP to infer strong lower bounds (LBs) of the optimal solution that will lead to a more efficient solution method. Below we present a brief review of DP. Disjunctive convex programming deals with the optimisation over the intersection of the union of closed convex sets [21, 22] and its general structure is presented in (CDP) where f: Rn → R1, is a convex function of the continuous variables x in the objective function g: Rn → Rm, belongs to the set of global constraints, the disjunctions h in H are composed of a number of terms j in Dh connected by the OR operator. Each term is associated with a set of convex constraints defined by the convex functions rjh: Rn → Rt. For the particular case in which f, g and r are linear functions, a linear disjunctive program (LDP) is obtained as shown below where cT, ajh, B and Ajh are the vectors and matrices of suitable dimensions. Balas [23, 24] showed that LDP can be systematically transformed into an equivalent MILP whose continuous relaxation is often tight, hence, facilitating the search for the solution. The equivalent (MILP) proposed is [23] In the proposed framework in this paper, the underlying logic of (MINLPSCUC) can be clearly captured by reformulating it as an equivalent CDP as follows [21] Note that the selection of a unit at node i enforces the capacity constraints of that unit and its cost in the objective function, whereas, it enforces the injected power to be zero if the unit is not selected. This logic is easily represented through the disjunctions in (CDPSCUC). The model (CDPSCUC) is equivalent to (MINLPSCUC) because every point (z, x, y) in the feasible region of (MINLPSCUC) has a one to one correspondence with every point (z, x, y) in (CDPSCUC) and that the optimal value of the objective function in (CDPSCUC) is the same as the optimal value of the objective function in (MINLPSCUC). The CDP (CDPSCUC) in turn is equivalent to the following LDP, where represents the supporting hyperplane of the function fi(xi) evaluated at a given point . (LDPSCUC) formulation is not of practical use considering that the number of supporting hyperplanes necessary to define is infinite fi(xi). However, if we only select a subset JS ⊂ J then the solution of the formulation (RLDPSCUC) is a valid LB of the solution of (LDPSCUC). Hence, it is also a LB to the solution of (MINLPSCUC). By following the same transformation from (LDP) to (DPMIP) and after simplifying redundant equations and variables, (RLDPSCUC) can be equivalently represented as the following linear mixed-integer program The main question that arises is how to pick the subset JS in such a way that it will lead to a solution (LB) close enough to the actual optimum, which will be discussed in Section 4. Another important aspect of the solution method is its capability to obtain good upper bounds (UBs) of the optimal solution. The following proposition forms the basis for a strategy that deals with this regard. Proposition 1.Any solution to DPMIPSCUC is feasible to MINLPSCUC. Proof.The proof is trivial considering that the feasible region of DPMIPSCUC is in the feasible region of MINLPSCUC. Note that the feasible region of DPMIPSCUC is described by the set of constraints of MINLPSCUC plus the set of constraints obtained by considering the supporting hyperplanes of fi(xi) Hence, from Proposition 1, UBs of MINLPSCUC can be obtained by evaluating the objective function at any solution of DPMIPSCUC. An alternative way to obtain UBs of the global optimum is to use the traditional piecewise linear approximation method. In this method the problem (MINLPSCUC) is approximated by a piecewise linearisation of the objective function as follows where the set K represents the points used for the linearisation.As shown in Fig. 1, the set of constraints always over-estimate the objective function, generating a significant gap between the real objective function and the approximation. Note that the gap can be reduced by increasing the number of intervals. However, this implies an increase in the number of constraints added. Fig. 1Open in figure viewerPowerPoint Piecewise linearisation of quadratic terms 4 Solution method and convergence properties Solution method In this section, we present the proposed solution method, which builds on the work by Duran and Grossmann [14] to solve general MINLP programs and uses the theory developed in Section 3 to obtain strong LBs. The general framework is presented in Fig. 2. Fig. 2Open in figure viewerPowerPoint Block-diagram of the proposed solution method The master problem is the MIP problem (DPMIPSCUC) where the objective function is given by and the set of constraints (2)–(21) with the set JS given by the following constraints (22) (23) (24)where represents the incumbent solution obtained by solving a piecewise linear approximation of SCUC (i.e. LMINLPSCUC). represents any point in the range (Pmin, i ≤ Pit ≤ Pmax,i) used to initialise the Master problem and represents the incumbent solution of the Master problem at iteration 'iter' if or the zero value, otherwise. Also note that (22)–(24) are obtained by considering the supporting hyperplanes of the non-linear term of the objective function and that is the reason why the method is called outer-approximation. Note that ctol ensures that no cuts are added if there is no significant improvement in the relaxation of the master problem. Also note that if UB-LB/UB > tol then the method ensures that there will be at least one that is greater than ctol as shown in Proposition 2 below. This plays a very important role in the convergence properties of the method. The parameter ctol is further studied in Section 4.2. The proposed approach first obtains a good solution from the piecewise linear approximation where a set of initial outer-approximation cuts are generated (22). This solution is then improved by solving the Master problem iteratively to obtain LBs of the global optimum (note that this LB is guaranteed since the (22) and (23) always underestimate the square terms in the objective function as mentioned in Section 3) and evaluating the objective function (1) at the incumbent solution to obtain UBs. If the best UB is not sufficiently close to the LB then a set of cuts are generated (24) and added to the Master problem. Clearly, as the iterations proceed, these cuts obtain a better and better approximation of the objective function (see Fig. 3). As a result, the algorithm converges to the global solution. Fig. 3Open in figure viewerPowerPoint Illustration of the outer-approximation method Convergence properties When 'ctol = 0', the method borrows the convergence properties of the general outer-approximation method proposed by Duran and Grossmann [14]. However, for 'ctol > 0', the convergence is not guaranteed in general. On the other hand, if the outer-approximation cuts are close enough to the real quadratic curve at the point Pit, adding an outer-approximation cut at this point in the next iteration will only increase the size of the Master problem without leading to a significant improvement in the LB (see Fig. 4 for an illustration). Clearly, finding a suitable value for 'ctol' is of main importance. Fig. 4Open in figure viewerPowerPoint No cuts added when Δ ≤ ctol We here provide an UB of 'ctol' for which the convergence properties are not altered in the following proposition: Proposition 2.An UB of 'ctol' for which the convergence properties of the method are not altered is given by where |I| is the number of generators, |T| is the number of time intervals and LB∗ is the first LB obtained by the Master problem. Proof.We focus on the convergence criteria of the method when the problem is feasible since infeasibility is certainly not affected by 'ctol'. The algorithm stops when where UB and LB represent the upper bound and lower bound predicted by the algorithm. It is clear that the convergence of the method will not be affected by the selection of ctol if the following relationship holds If , then Defining UBL as the value of the objective function at a given iteration and considering that Then we have Hence and this concludes the proof. 5 Numerical examples In order to assess the performance of the method we consider two main indicators, namely, the accuracy of the solutions obtained and the time consumed to find the solution. We compare the performance with traditional approaches proposed in literature and commercial solvers. The models were implemented in GAMS [25] and solved in a Pentium(R) CPU 1.66 GHz and 1 GB of RAM using CPLEX 12.3 [10]. The relative tolerance for termination of the iterations was set to 0.0001 in all cases and the number of intervals was set to 10. The characteristics of the instances tested are shown in the following Table 1: Table 1. Size and characteristic of the test instances Constraints Continuous variables Binary variables 6-bus problema 1201 745 72 57-bus problema 11 641 5977 360 118-bus problema 45 505 24 241 2160 aThe 6-bus problem is a modified version of the test case in [28], where cost parameters have been changed to put more emphasis on the quadratic terms. The 57- and 118-bus problems are a modified version of examples in http://www.ee.washington.edu/research/pstca (the 118-bus problem can also be found in http://motor.ece.iit.edu/data/SCUC_118test.xls). The modification is made to include necessary parameters for unit commitment such as Quick Start Capacity (QSC), Maximum Sustained Ramp Rate (MSR) and Maximum Ramp Up/Down Rates (RU/RD), which are missing in the original test cases. Also, in order to increase the complexity of the problems we have increased the number of generators and loads. Please contact the authors if the readers are interested in the data. The data for the 6-bus system can be found in the Appendix. Accuracy: Based on a specified tolerance level, the proposed algorithm converges to a solution that is within this given tolerance level in reference to the global solution. This is done by providing rigorous LB and UB. This differs from linear piecewise approximation methods where only UBs are given. Table 2 compares the solutions obtained by the algorithm with piecewise approximation methods using different number of intervals (n) in four instances of a 6-bus system using different parameters for the units. It can be seen that the proposed method outperforms the piecewise linearisation method by being closer to or equal to the optimum in all cases. Table 2. Accuracy of piecewise-linear approximation and proposed approach Piecewise linearisation method Instance Optimum, $ n = 1 n = 4 n = 10 Proposed 1 123 264 123 569 123 271 123 268 123 264 2 124 753 125 159 124 762 124 760 124 754 3 130 773 132 738 130 858 130 836 130 777 4 168 997 186 060 169 575 169 119 169 000 Convergence speed: By adding the first set of initial cuts (22) and (23), to the master problem we are able to start with a tight relaxation predicting a strong initial LB. This LB is then refined until the solution is found by the addition of the cuts (24). As we show in Table 3, significant improvements in the computational speed when compared with commercial solvers could be found. Table 3. Computational performance of CPLEX and proposed approach Optimum Proposed approach CPLEX Cutsa Iter Time, s Time, s 6-bus sys 168 997 57.5 3 2 2 57-bus sys 37 232 48.6 4 652 >10 000 118-bus sys 525 832 0 1 1790 >10 000 aThe number of cuts reported is the average per iteration In the column 'CPLEX' we report the computational time for solving the original quadratic program MINLPSCUC using the MIQCQP solver in CPLEX, which is one of the most widely used commercial solvers for this type of problem [26, 27]. Fig. 5 and Table 4 show the UBs and LBs the algorithm predicts as well as the cuts (24) that are added in the different iterations for the different test instances. Table 4. LB and UB and added cuts at each iteration 6-bus sys 57-bus sys 118-bus sys Iter LB UB Cuts LB UB Cuts LB UB Cuts 1 168935.23 169024.00 – 37151.88 37268.50 – 525801.26 525832.40 – 2 168981.51 169005.40 60 37204.65 37241.83 50 3 168993.25 168999.89 55 37223.10 37232.62 51 4 37288.65 37231.69 45 Fig. 5Open in figure viewerPowerPoint LBs and UBs as a function of the iterations To illustrate the effect of 'ctol' in the convergence of the algorithm we will apply the method to a 6 bus system using 'ctol' above and below the maximum allowed as shown in Fig. 6. As can be seen the convergence to the global optimum can not be guaranteed when ctol > ctolmax. Fig. 6Open in figure viewerPowerPoint Convergence with different tolerance levels 6 Extension Improving piecewise linear methods to obtain better local solutions In this section, we propose a heuristic method that builds on the previous framework to improve the quality of the solution obtained from the piecewise linear method. The main idea includes the same steps as discussed but now fixing the binary variables to be equal to the incumbent of the piecewise approximation method. Note that this implies that the Master problem is an LP at every iteration, hence, the algorithm converges quickly to a local solution that is better than the initial one. It is important to remark, although, that this variation does not guarantee the global solution. Table 5 shows the comparison of the quality of the solutions obtained using different methods. Table 5. Comparison of solutions with different methods Piecewise method Proposed method Global Obj Time, s Obj Time, s 6 bus system 168 997 169 119 1 169 000 1 57 bus system 37 232 37 283 82 37 234 84 118 bus system 525 832 525 845 1204 525 832 1208 Note that the objective function using the piecewise approximation method is clearly improved by the proposed local method without incurring in a significant increase in the computational time. Also, it is important to remark that the solution of the proposed approach guarantees to be at least as accurate as the one obtained by the piecewise method. 7 Conclusions and remarks In this paper, we have presented an outer-approximation method to solve the MINL SCUC problem. We have shown through a set of numerical examples how more accurate solutions can be achieved in comparison with traditional piecewise linear approximation methodologies while still solving linear programs. A significant improvement in the computational time can be obtained when compared with the black-box commercial solvers such as CPLEX. Moreover a variation of the proposed method is presented to obtain better local solutions. Even though the method presented can be very useful for those SCUC systems that can be modelled with DC power flow constraints, for those cases where the use of AC constraints is mandatory, a more sophisticated technique will be necessary. AC constraints are in general non-convex which present a challenge for NLP solvers. However, having an approximated method that can be used as a starting point to obtain close optimal solutions could potentially help the NLP solvers to find the global solution more efficiently. With this in mind, in future work we are aiming at tackling the AC-based SCUC problems by integrating a robust starting point strategy. Moreover, we think that the method we developed can inspire the development of similar techniques for the AC case by further exploiting the DP theory. Appendix 9 See Tables 6–9. Table 6. Data for the 6-bus system – power generators Parameters Generators G1 G2 G3 C0, $ 50 150 150 C1, $/MWh 19.96 17 20.14 C2, $/MW2h 0.24 0.15 0.18 max generation, MW 120 200 160 min generation, MW 50 50 10 ramp up rates, MW/h 50 50 50 ramp down rates, MW/h 40 50 60 MSR, MW/min 4 2.5 1.5 start-up costs, $ 90 90 40 shut-down costs, $ 100 100 120 start-up ramp rate, MW/h 100 100 120 shut-down ramp rate, MW/h 30 30 20 quick start capacity, MW 13 14 8 minimum on time, h 3 4 2 minimum off time, h 3 4 2 Table 7. Data for the 6-bus system – line characteristics Line (bus to bus) x Pmax, MW 1–2 0.17 140 1–4 0.258 110 2–3 0.037 150 2–4 0.197 140 3–6 0.018 130 4–5 0.037 50 5–6 0.14 140 Table 8. Data for the 6-bus system – loads and reserves Hour/bus Load, MW Spinning reserves, MW Operating reserves, MW 3 4 5 1 35.038 70.076 70.076 2.63 12.26 2 33.03 66.06 66.06 2.48 11.56 3 31.734 63.468 63.468 2.38 11.1 4 30.946 61.892 61.892 2.32 10.83 5 31.012 62.024 62.024 2.33 10.85 6 32.096 64.192 64.192 2.4 11.23 7 34.678 69.356 69.356 2.6 12.14 8 35.52 71.04 71.04 2.85 13.33 9 37.362 74.724 74.724 3.09 14.39 10 41.392 82.784 82.784 3.26 15.2 11 45.722 91.444 91.444 3.43 16 12 47.22 94.44 94.44 3.54 16.52 13 48.436 96.872 96.872 3.63 16.95 14 48.72 97.44 97.44 3.66 17.05 15 49.772 99.544 99.544 3.73 17.42 16 51.158 102.316 102.316 3.84 17.91 17 51.2 102.4 102.4 3.84 17.92 18 49.348 98.696 98.696 3.7 17.27 19 49.194 98.388 98.388 3.69 17.22 20 47.47 94.94 94.94 3.56 16.62 21 47.47 94.94 94.94 3.56 16.62 22 46.534 93.068 93.068 3.41 15.9 23 39.186 78.372 78.372 3.02 14.07 24 39.12 78.24 78.24 2.95 13.78 Table 9. Data for the 6-bus system – allocated power Power, MW Hour G1 G2 G3 1 50.00 72.07 53.13 2 50.00 66.88 48.28 3 50.00 63.09 45.59 4 50.00 62.92 41.81 5 50.00 63.14 41.92 6 50.00 66.88 43.61 7 50.00 70.45 52.95 8 50.00 74.48 53.13 9 50.00 78.58 58.23 10 50.88 89.11 66.98 11 56.13 100.24 72.25 12 57.88 102.60 75.63 13 59.09 104.38 78.72 14 59.63 104.55 79.43 15 59.97 108.13 80.77 16 62.47 110.20 83.13 17 62.63 110.25 83.13 18 59.63 106.38 80.74 19 59.63 106.38 79.97 20 57.88 103.59 75.89 21 57.88 103.59 75.89 22 55.71 101.34 75.63 23 50.00 84.53 61.40 24 50.00 84.36 61.24 8 References 1Shahidehpour, M., Yamin, H., Li, Z.: ' Market operations in electric power systems' (Wiley, New York, NY, 2002) 2Chandrasekaran, K., Sishaj, P.S.: 'Demand response scheduling in SCUC problem for solar integrated thermal system using firefly algorithm'. IET Conf. 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