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economic dispatch problem
Energy 53 (2013) 14e21
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Energy journal homepage: www.elsevier.com/locate/energy
Multiobjective scatter search approach with new combination scheme applied to solve environmental/economic dispatch problem Marsil de Athayde Costa e Silva a, Carlos Eduardo Klein b, Viviana Cocco Mariani c, d, Leandro dos Santos Coelho b, c, * a
Mechatronics Engineering Undergraduate Program, Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil c Department of Electrical Engineering, Federal University of Parana, 81531-980 Curitiba, PR, Brazil d Mechanical Engineering Graduate Program (PPGEM), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 February 2012 Received in revised form 18 February 2013 Accepted 19 February 2013 Available online 1 April 2013
The environmental/economic dispatch (EED) is an important daily optimization task in the operation of many power systems. It involves the simultaneous optimization of fuel cost and emission objectives which are conflicting ones. The EED problem can be formulated as a large-scale highly constrained nonlinear multiobjective optimization problem. In recent years, many metaheuristic optimization approaches have been reported in the literature to solve the multiobjective EED. In terms of metaheuristics, recently, scatter search approaches are receiving increasing attention, because of their potential to effectively explore a wide range of complex optimization problems. This paper proposes an improved scatter search (ISS) to deal with multiobjective EED problems based on concepts of Pareto dominance and crowding distance and a new scheme for the combination method. In this paper, we have considered the standard IEEE (Institute of Electrical and Electronics Engineers) 30-bus system with 6-generators and the results obtained by proposed ISS algorithm are compared with the other recently reported results in the literature. Simulation results demonstrate that the proposed ISS algorithm is a capable candidate in solving the multiobjective EED problems. 2013 Elsevier Ltd. All rights reserved.
Keywords: Environmental/economic dispatch Evolutionary algorithms Scatter search
1. Introduction Environmental/economic dispatch (EED) problem is one of the most important problems in power systems management. It consists of scaling the outputs of thermoelectric generators, in a thermal power plant, such that a power demand is supplied while satisfying equality and inequality constraints. Each generation unit has particular aspects in the cost of the fuel spent and the pollutants emitted. Fossil-fired electric power plants cause pollution emission while they operate. The emission control results from the requirement for power utilities to reduce their pollutant levels below the annual * Corresponding author. Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil. E-mail addresses: [email protected] (M. de Athayde Costa e Silva), carlos.klein@ volvo.com (C.E. Klein), [email protected] (V.C. Mariani), leandro.coelho@ pucpr.br (L. dos Santos Coelho). 0360-5442/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.02.045
emission allowances assigned for the affected fossil units. The total emission can be reduced by minimizing the major pollutants; this is one objective of the EED problem. There are two objectives to be optimized in this kind of EED problem: the cost of the generation and the total emission. Both objectives must be minimized, however they are conflicting, i.e. if there is an optimal solution it is impossible to optimize an objective without worsen other objective. Thus, there is not only one solution but a set of solutions that form the Pareto optimal set. Pareto optimality is a measure of efficiency in multi-criteria and multi-objective situations. A state A (a set of object parameters) is said to be Pareto optimal, if there is no other state B dominating the state A with respect to a set of objective functions. A state A dominates a state B, if A is better than B in at least one objective function and not worse with respect to all other objective functions. Many researchers have tackled this problem in the past. The first use of multiobjective programming with power systems has been addressed by Cheong and Dillon [1]. However, it is realized that
M. de Athayde Costa e Silva et al. / Energy 53 (2013) 14e21
conventional mathematical techniques, such as gradient method, linear programming algorithm, quadratic programming, Lagrange relaxation algorithm, become very complicated when dealing with increasingly complex dispatch problems, and are further limited by their lack of robustness and efficiency in a number of practical applications [2]. Therefore, because of its particularities, an efficient technique must be used to provide good solutions for the EED optimization problems. In this context, more and more often, metaheuristics are used to find solutions of complex EED optimization problems. Metaheuristics are approximate, heuristic or general-purpose algorithms for solving complex optimization problems, with continuous and/or discrete variables. The use of metaheuristic search algorithms have gained attention in the recent decades due to the failure of conventional methods to solve NP-hard (Non-Polynomial hard) combinatorial and difficult continuous optimization problems. In general, metaheuristics are suitable for solving hard and/or largesize instances of an optimization problem for which there is no efficient exact algorithm available. Over the last few years, metaheuristics such as genetic algorithms [3], differential evolution [4,5], particle swarm optimization [6,7], bacterial foraging algorithm [8], shuffled frog leaping algorithm [9], seeker optimization algorithm [10], biogeography-based optimization algorithm [11], chaotic ant swarm optimization [12], artificial bee colony optimization [13], harmony search [14,15], selforganizing migrating strategy [16], quantum-inspired evolutionary algorithm [17], artificial immune systems [18], imperialist competitive algorithm [19], charged system search algorithm [20], fuzzy adaptive chaotic ant swarm optimization [12], have been used to solve economic dispatch problems. In Ref. [21], a summary of EED algorithms is also given. On the other hand, multiobjective EED optimization involves the simultaneous optimization of several incommensurable and often competing objectives. Recently, the research focus has shifted towards handling both the objectives simultaneously. Over the past decade, this option has received much interest due to the development of a number of multiobjective search strategies based on metaheuristics. Many examples of metaheuristic optimization approaches applied to multiobjective EED optimization problems are given in Refs. [22e36]. In this context, a metaheuristic method called scatter search (SS) can be useful. The scatter search methodology was first introduced in 1977 [37e39] as a heuristic for integer programming; it is a population-based metaheuristic method that uses a set of reference solutions to create new improved solutions by intelligently combining and improving others in a set of good and disperse solutions. The reference set is formed based on quality and diversity of the solutions. Initially, the best solutions are added to the reference set, and then the most diversified solutions, based on the distance to the already added solutions, are included in the set. The algorithm combines these solutions to create new ones and achieve a desirable result. From the standpoint of metaheuristic classification, scatter search may be viewed as an evolutionary algorithm [40] because it builds, maintains and evolves a set of solutions throughout the search. However, scatter search differs from other populationbased evolutionary heuristics like genetic algorithms mainly in its emphasis on generating new elements of the population mostly by deterministic combinations of previous members of the population as opposed to the more extensive use of randomization. The two principles that govern SS are i) intensification, and ii) diversification. Intensification refers to the role of isolating the best performing solutions from the populations in order to obtain a group of good solutions. Diversification in turn isolates the solutions which are the furthest from the best solutions and combine
15
them with the best solutions. This new pool of solutions is the reference set where crossover occurs in order to create solutions from new solution regions by the combination of the intensified solutions and diversified solutions. Intensification and diversification are commonly termed as adaptive memory programming [41]. These principles help to increase the exploration and the exploitation capabilities of the algorithm in finding solutions, which is very desirable in complex problems such as EED problem because the search space is generally wide and there are several local optimal points that can prevent the algorithm to advance. Nevertheless, the original approach does not consider multiple objectives, so some modifications must be done in order to apply it to solve the EED problem. Our interest here is to adapt the wellknown scatter search template to multiobjective optimization and the proposition of improvements in the scatter search design. Scatter search has been found to be successful in a wide variety of optimization problems [42e45]. Moreover, recently it had been extended to deal with multiobjective optimization problems (see examples in Refs. [46e49]). Scatter search uses improvement strategies to efficiently produce the local tuning of the solutions, and a remarkable aspect concerning scatter search is the trade-off between the exploration abilities of the combination method and the exploitation capacity of the improvement mechanism. This paper proposes an improved scatter search (ISS) methodology to deal with multiobjective EED problems based on concepts of Pareto dominance and crowding distance [50] and a new scheme for the combination method. The results of the original SS and proposed ISS approaches are compared to other techniques presented in the recent literature about EED. The mentioned algorithms have been implemented on standard IEEE (Institute of Electrical and Electronics Engineers) 30bus six-generators system in order to obtain the trade-off between the cost and emission. Results show that the modifications proposed increased the performance of the scatter search algorithm. The remainder of the paper is organized as follows: Section 2 presents the concepts of scatter search. Section 3 provides the description of the scatter search algorithm and the proposed approach. The description of the formulation of the optimization problem is mentioned in Section 4. Finally, Sections 5 and 6 present the results and conclusions, respectively.
2. Fundamentals of the scatter search Scatter search is a metaheuristic algorithm that can be considered an evolutionary algorithm in the sense that it incorporates the concept of population. However, SS approaches usually avoid using typical evolutionary operators such as mutation or crossover operators. SS derives its foundations from earlier strategies for combining decision rules and constraints. Historically, the antecedent strategies for combining decision rules were introduced in the context of scheduling methods to obtain improved local decision rules for job shop scheduling problems. New rules were generated by creating numerically weighted combinations of existing rules, suitably restructured so that their evaluations embodied a common metric [38]. The original SS algorithm has five procedures: i) a diversification generation method to randomly generate diversified trial solutions; ii) an improvement method to converge the current solutions toward the optimum; iii) reference set update method to renew the set of reference solutions with the best ones; iv) a subset generation method to create subsets for the combination method; and v) a combination method to combine the reference solutions and generate new offspring. Next subsections describe each procedure.
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2.1. Diversification generation method This method generates solutions so that they are uniformly distributed in the search space. Each dimension of the search space is divided in 4 regions of equal size and the number of times each region is used should be almost the same to guarantee diversity in the population and uniformity in the distribution. Table 1 shows an example of a diversified population generated with this procedure. We have randomly generated 100 solutions of the form x ¼ [x1 x2 x3 x4] with each dimension lying between 0 and 1 and using uniform distribution. In this case each dimension of the solutions should appear approximately 25 times in each range. The values are the number of times that each variable appeared in each range. This procedure is carried out until a reasonable set of solutions is found considering the frequencies of appearance in the four ranges. Also, this procedure do not consider the objective function values, it is focused on the distribution of the population in the search space. 2.2. Improvement method In order to increase the exploitation capability of scatter search an improvement method is used and its goal is to transform a solution in a better one in terms of objective function or feasibility. Typically this method is a local search procedure which is carried out until no improvement is detected in the solution. However, since each solution is improved using this procedure, it needs to be a very fast and effective method in order to not increase the computational demand. The possible solution improvement methods, applicable to the solutions, range from the simplest local searches to specialized procedures. 2.3. Reference set update method The reference set is updated during the search process by using criteria based on comparisons and measures of the diversity between the new solutions and the existing solutions. The reference set contains high quality solutions, in terms of objective function, and diverse solutions. Its size is equal to b ¼ b1 þ b2, where b1 is the number of high quality solutions and b2 is the number of diversified solutions. Initially, we remove the best b1 solutions from the population and add them to the reference set. Then, the b2 diverse solutions must be added to the reference set and removed from the population. The minimum value of the Euclidian distance between each solution and the reference solutions is used as criterion of solutions diversification, i.e. given a solution in the population, the distances between the solution and each reference solution is calculated and the minimum value found is assigned to that solution. This idea prevents the algorithm to being trapped in local optima points. These two steps for adding solutions to the reference set increase the exploration capability of the algorithm and help to avoid being trapped in local minima. 2.4. Subset generation method Scatter search starts by generating a reference set from the population of solutions. Then subsets are iteratively selected from this reference set and are used in the combination method in order Table 1 Example of a diversified population. Range
x1
x2
x3
x4
0.00e0.25 0.25e0.50 0.50e0.75 0.75e1.00
23 24 28 25
23 26 22 29
27 24 22 27
23 22 24 31
to create a pool of new solutions. In this paper subsets of size equal to 2 were considered. All possible subsets are generated; however, a subset already used is discarded because it will recreate solutions already used in the algorithm. 2.5. Combination method The combination procedure tries to intelligently combine good characteristics of the subset of solutions to yield new high quality solutions. The combination method is one of the most important steps of the algorithm. It uses the subsets created in the subset generation method to combine solutions and create new ones. Consider two reference solutions x1and x2, three new solutions are created using the following:
C 1 ¼ x1 d C 2 ¼ x1 þ d C 3 ¼ x2 þ d d ¼ r
ðx1 x2 Þ 2
(1)
(2)
where r is a random number with uniform distribution generated in the range between 0 and 1. 3. Algorithm of the scatter search and the proposed improved version The algorithm just uses the methods described above in the same sequence. An initial random population is created using the diversification generation method and each element is improved using the improvement method. Then, the reference set is constructed with the best and the most diversified solutions with the reference set update method. Note that, at this point, the reference set is empty, so there is no need to compare the solutions. The reference set is used to create new solutions using the combination method; the new solutions are then improved and the reference set is updated with the best and the most diverse solutions of the population. If there are no new solutions, the reference set is rebuilt using the diversification generation method. This process is carried out until a maximum number of iterations to be achieved. Fig. 1 shows the main steps of SS algorithm. The variable called “New solutions” is a flag to notify whether new solutions were created or not. After the application of the combination method, all solutions are evaluated and if there is at least one solution better than any reference solution (according to the objective function value or the distance value) then the flag “New solutions” is set to true, otherwise is set to false. Thus, the reference set is updated or rebuilt according to the flag’s value. In this paper we propose a new scheme for the combination method in the ISS design. Instead of using Equations (1) and (2) of classical SS approach, we generate four new solutions using the following expressions:
8 C > > < 1 C2 C > > : 3 C4
¼ ¼ ¼ ¼
a1 x1 þ ð1 a1 Þx2 a2 x1 þ ð1 a2 Þx2 a3 x1 þ ð1 a3 Þx2 a4 x1 þ ð1 a4 Þx2
(3)
where ai is a random number with uniform distribution generated in the range between 0.2 and 0.8 with different values for each dimension. This scheme creates solutions more diversified because it uses different random numbers for each solution created. Also, it creates four new solutions instead of three as the original method.
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This increases the capability of explore the search space of the algorithm, which is important in problems such as the EED that has a wide search space. A change in the improvement method is also necessary because the original method do not deal with multiple objectives. In this paper we suggest a simple but efficient modification which is described by the following equations.
X1;I ¼ YI þ dYI
(4)
X2;:I ¼ Y:I þ dY:I
(5)
X3;I ¼ YI þ dYI X3;:I ¼ Y:I dY:I
(6)
X4;:I ¼ Y:I þ dY:I X4;I ¼ YI dY
17
(7)
where Y is the solution to be improved, d is a constant number, I is a random logical vector which indicates the dimension to be modified, :I is the negation of I and Xi is the i-th solution constructed. The algorithm creates these 4 solutions from Y and the best between them (according to all objectives) substitutes Y. This improvement method is used in both original SS and the proposed ISS approaches. Compare solutions considering more than one objective function is a problem itself; so, in this paper, to deal with multiple objectives, is used the concepts of Pareto dominance and crowding ! distance [50]. Given two vectors u ¼ ðu1 ; .; uk Þ and ! ! v ¼ ðv1 ; .; vÞ ,u dominates v if and only if ci˛f1; .; kg; u ! ! ! v ^di˛f1; .; kg : u < v . In other words, in at least one dimension u is less than v and in no other dimension v is less than u. The objective vectors are evaluated using this concept and a solution that dominates another is preferred. The solutions that are not dominated by any other are the Pareto optimal solutions and constitute the Pareto set of non-dominated solutions. In multiobjective optimization this set is the desired set of solutions and generally is not known. The population of solutions is analyzed on this aspect and the first front is constructed with the non-dominated solutions of the population, then these solutions are stored and removed from the population, the second front is then constructed with the remaining non-dominated solutions and so on until all solutions are stored in a front. The first front is the set of best solutions in the population and its solutions are in the same level, so, to decide the best solution between two of the same front, the concept of crowding distance is used. Each solution has a crowding distance value which is calculated using the following expression for a solution i:
Di ¼
m X fk ði þ 1Þ fk ði 1Þ fkmax fkmin k¼1
(8)
where Di is the distance of the i-th solution, fk is the k-th objective function’s value of the i-th solution, and fkmax and fkmin are the maximum and minimum values of the k-th objective function, respectively. The solutions of the boundaries of the front have their distance set to infinite and the solution with the great crowding distance is preferred. These two concepts are used when constructing or updating the reference set; the solutions are sorted by the Pareto fronts and by the crowding distances. And when there are new solutions attempting to be inserted in the reference set these two concepts are used to decide which of any two solutions is the best. 4. Formulation of the multiobjective EED problem
Fig. 1. Pseudo code of SS.
The EED problem is formulated as a multiobjective optimization one, where one objective is to minimize the total emission of pollutants and the other is to minimize the cost of generation related to the fuel used. These two objectives should be optimized while satisfying equality and inequality constraints. The problem evaluated in this paper is the standard IEEE 30-bus system with six-generators. The IEEE 30-bus test case represents a portion of the American Electric Power System (in the Midwestern US) as of December, 1961 with 41 transmission lines and 21 load buses (see topology of the IEEE 30-bus system in Fig. 2). Details and data of IEEE 30-bus are presented in Refs. [22,51e53]. The data for the optimization problem are given by Tables 2e4.The system base is 100 MVA and the total system demand for the 21 load buses is
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M. de Athayde Costa e Silva et al. / Energy 53 (2013) 14e21 Table 3 Data for the objective functions. Variable
G1
Cost ($/h) a 10 b 200 c 100 Emission (ton/h) a 4.091 b 5.554 g 6.490 z 2.0e-4 l 2.857
n X
Fig. 2. Topoloy of the IEEE 30-bus system.
4.1. Objective functions The first objective of EED is the classical economic dispatch problem of finding the optimal combination of power generation which minimizes the total cost of generation while satisfying the required demand. Considering a plant with n generation units, the total fuel cost in $/h can be defined as:
i¼1
ai þ bi Pi þ ci Pi2
G5
G6
10 150 120
20 180 40
10 100 60
20 180 40
10 150 100
2.543 6.047 5.638 5.0e-4 3.333
4.258 5.094 4.586 1.0e-6 8.000
5.326 3.550 3.380 2.0e-3 2.000
4.258 5.094 4.586 1.0e-6 8.000
6.131 5.555 5.151 1.0e-5 6.667
(11)
The power losses are modeled as a function of the outputs of generators, as in (12). In this case,
PL ¼
n X n X
Pi Bij Pj þ
n X
B0i Pi þ B00
(12)
i¼1
where B0i and B00 are the loss coefficients on transmission lines, and Bij is the loss coefficient between the i-th and j-th generator. Moreover, the power generated by each unit is constrained between its minimum and maximum limits (Pmin and Pmax). Then, the multiobjective environmental/economic dispatch optimization problem is formulated as a minimization of the Equations (9) and (10) subject to constraints of power balance and generation capacity (limits).
(9) 4.3. Constraint handling
where Pi is the output of the i-th generator and ai, bi and ci are the cost coefficients of the i-th generator. The second objective considered in the EED problem is the minimization of the atmospheric pollutants caused by fossil-fueled thermal units. The emission function, in ton/h, can be modeled as a sum of a quadratic and an exponential function given by:
Femission ¼
G4
Pi ¼ PD þ PL
i¼1 j¼1
n X
G3
i¼1
2.834 p.u. (per unit). Transmission losses are considered in this problem and are estimated using the coefficients presented in Table 4.
Ffuel ¼
G2
n X 102 ai þ bi Pi þ gi Pi2 þ zi $expðli Pi Þ
(10)
i¼1
where ai, bi, gi, zi and li are the emission coefficients of the i-th generator. 4.2. Constraints of the problem There are two types of constraints in the environmental/economic dispatch problem: the power balance and the generation capacity of the units. The first one is an equality constraint which relates to the demand and the transmission losses, that is, the power generated must cover the total load demand PD and the total power transmission loss PL (in p.u. values, in this paper) as follows:
In order to guarantee feasibility in all solutions, it is used a simple method that obligate solutions to satisfy the constraint (11) and the lower and upper boundaries constraint. First, the solution is multiplied by a factor which considers the transmission loss and the load demand, as follows:
C B BPD þ PL0 C 0 CP P〞 ¼ B C B X n @ 0 A Pi
Power (p.u.)
G1
G2
G3
G4
G5
G6
0.05 0.5
0.05 0.6
0.05 1.0
0.05 1.2
0.05 1.0
0.05 0.6
(13)
i¼1
where P0 is the solution array to be adjusted, P00 is the new solution array, PD is the load demand and PL0 is the transmission loss for the solution P0. However, this does not guarantee that the new solution is satisfying (11) because the power loss is dependent of the solution. Thus the mentioned procedure is carried out using the new yielded solution until a certain number of iterations is achieved or it Table 4 B-loss coefficients matrix. B
Values
B0 B00
0.1382 0.0299 0.0044 0.0022 0.0010 0.0008 0.0535 9.8573e-4
Table 2 Lower and upper boundaries of each generator.
Pmin Pmax
1
0
0.0299 0.0487 0.0025 0.0004 0.0016 0.0041 0.0030
0.0044 0.0025 0.0182 0.0070 0.0066 0.0066 0.0085
0.0022 0.0004 0.0070 0.0137 0.0050 0.0033 0.0004
0.0010 0.0016 0.0066 0.0050 0.0109 0.0005 0.0001
0.0008 0.0041 0.0066 0.0033 0.0005 0.0244 0.0015
M. de Athayde Costa e Silva et al. / Energy 53 (2013) 14e21
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Table 5 Best results for the fuel cost. Technique
Reference
Cost ($/h)
Emission (ton/h)
G1 (p.u.)
G2 (p.u.)
G3 (p.u.)
G4 (p.u.)
G5 (p.u.)
G6 (p.u.)
SS ISS MO-DE/PSO MODE
This paper This paper [48] [25]
603.6846 603.5888 606.0073 606.1260
0.2165 0.2159 0.2209 0.2195
0.1666 0.1630 0.1220 0.1332
0.3182 0.2832 0.2843 0.2727
0.5347 0.5813 0.5857 0.6018
0.9589 0.9322 0.9962 0.9747
0.5243 0.5387 0.5149 0.5146
0.3448 0.3473 0.3566 0.3617
Table 6 Best results for the emission function. Technique
Reference
Cost ($/h)
Emission (ton/h)
G1 (p.u.)
G2 (p.u.)
G3 (p.u.)
G4 (p.u.)
G5 (p.u.)
G6 (p.u.)
SS ISS MO-DE/PSO MODE
This paper This paper [48] [25]
632.4107 633.3926 646.0243 642.8490
0.194559 0.194469 0.194179 0.194200
0.3806 0.3716 0.4118 0.39266
0.4269 0.4436 0.4616 0.46256
0.5532 0.5597 0.5435 0.56311
0.4486 0.4381 0.3922 0.40309
0.5522 0.5418 0.5454 0.5676
0.4821 0.4890 0.5148 0.47826
stops in case of the constraint (11) being satisfied. In the simulations we have experimentally choose 5 iterations for this method. The lower and upper boundaries constraint is controlled within the same loop simply by replacing the elements of the solution that are outside the boundaries with the boundaries’ values. Also, it is used a penalty factor to penalize solutions which do not satisfy the constraints. There are two ways for calculating the penalty factor. The first one is a weighting of the total violation of the constraints and the second is an adaptive procedure which considers the current iteration of the algorithm [54]. The first approach is used in the first half of total quantity of iterations of the algorithm and the second is used in the remaining generations. Equations (14) and (15) describe both methods as follows,
pðXÞ ¼ a1
m X
fi ðXÞ2
(14)
i¼1
pðXÞ ¼ ðCtÞa2
m X
jfi ðXÞjb
(15)
i¼1
where fi(X) is the violation of the i-th constraint of the solution X, a1 ¼ 1000, a2 ¼ 2, C ¼ 0.5, b ¼ 2 and t is the current iteration. Note that the choice of the parameters is an optimization problem itself, thus these values were chosen experimentally. 5. Simulation results Simulations were carried out using MatLab environment (MathWorks). In order to eliminate stochastic discrepancies, 30 independent runs were carried out using different initial populations of solutions. The number of initial solutions in the population of SS and ISS approaches is 20 and the number of reference solutions (b) is equal to 10, being the first 5 the bests (b1) and the last 5 the more diversified (b2). The parameter d ¼ 0.02 was used for both SS and ISS approaches. Results presented here are compared to a hybrid algorithm based on particle swarm optimization and differential evolution (MO-DE/PSO) presented in Ref. [51] and a
multiobjective differential evolution (MODE) presented in Ref. [28]. The best values found, in terms of the fuel cost and the emission, are bolded in Tables 5 and 6, respectively. The best solutions considering the costs of generation obtained by ISS are compared to SS, MO-DE/PSO and MODE and are given in Table 5. It is observed that ISS finds better minimum fuel cost than the other mentioned optimization algorithms. In Table 6 are presented the best results related to emission. In this case, the best result for the emission was found by MO-DE/PSO, however all results are very near. Table 7 presents a statistical comparison between the proposed and the original SS approaches. In this case, three metrics are considered for comparing the approaches: the mean Euclidian distance to the origin (in this case an optimum point), the spacing index and the number of Pareto solutions. Spacing metric measures how evenly the points in the approximation set are distributed in the objective space. A value of zero for this metric indicates all members of the Pareto front currently available are equidistantly spaced. In this context, based on Table 7 with respect to spacing, SS performed worse than the ISS. Since the nondominated vectors found by the ISS are clustered together, the spacing metric evidences the better results found by this method. For the metrics calculations, all data collected during the 30 runs were stored in an external repository and a new Pareto front, for each technique, was generated using all this data. The main objective of the external repository is to keep a historical record of the nondominated vectors found along the search process. Fig. 3 shows a good diversity in the nondominated solutions obtained by ISS after 20 iterations (in 30 runs) when compared with the original SS.
Table 7 Comparison of SS and the proposed ISS. Technique
Spacing (f1, f2)
Mean Euclidian distance to origin (f1, f2)
Pareto solutions
SS ISS
0.1815 0.1673
614.7549 614.4093
139 156 Fig. 3. Pareto fronts found by the SS and ISS approaches.
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6. Concluding remarks Most decision problems in the real world require the simultaneous optimization of more than one criterion. EED can be approached as a multiobjective problem having conflicting objectives, as the minimization of emission is contrary to the maintenance of fuel cost economy. The optimization of multiple conflictive objectives is a hard problem. Effective stochastic optimization methods based on metaheuristics have been developed to solve multiobjective EED problems. A metaheuristic is a set of concepts that can be used to define heuristic methods that can be applied to a wide range of different problems. In other words, a metaheuristic can be seen as a general algorithmic framework which can be applied to different optimization problems with relatively few modifications to make them adapted to a specific problem [55]. In general, metaheuristics provide quite effective strategies to find near optimal solutions for complex problems. Scatter search is a newly emerging population-based evolutionary method that, unlike genetic algorithm, searches for global optima or satisfactory solutions by operating on a small data collection of intensification and diversification, and making much use of various systematic sub-methods and limited use of randomization. This paper evaluated the performance of a classical SS method and an ISS approach when dealing with an EED multiobjective problem. In this context, some changes were necessary for the study, and a modification of the original combination method was introduced. The proposed ISS approach performed better than the original and showed promising results for a standard IEEE 30-bus with six-generators test system. In order to make a comparison against competing solution methods, other results for the IEEE 30bus with 6-generators system presented in the recent literature were showed in Tables 5 and 6. In this context, the results using the proposed ISS based on some comparison metrics were competitive. In order to explore the abilities of the proposed methodology, our future research will test the ISS approach in other multiobjective EED problems. We will also need to carry out more formal performance measurements on the ISS algorithm, for example using the ManneWhitney rank-sum test.
Acknowledgments This work was supported by the National Council of Scientific and Technologic Development of Brazil e CNPq e under Grants 303963/ 2009-3/PQ and 478158/2009-3, and ‘Fundação Araucária’ under Grant: 14/2008-416/09-15149. The first author, also, would like to thanks the Pontifical Catholic University of Parana e PUCPR e and the National Council of Scientific and Technologic Development of Brazile CNPq e for the financial support provided through the Institutional Program for Scientific Initiation Scholarships e PIBIC.
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Energy journal homepage: www.elsevier.com/locate/energy
Multiobjective scatter search approach with new combination scheme applied to solve environmental/economic dispatch problem Marsil de Athayde Costa e Silva a, Carlos Eduardo Klein b, Viviana Cocco Mariani c, d, Leandro dos Santos Coelho b, c, * a
Mechatronics Engineering Undergraduate Program, Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil c Department of Electrical Engineering, Federal University of Parana, 81531-980 Curitiba, PR, Brazil d Mechanical Engineering Graduate Program (PPGEM), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 February 2012 Received in revised form 18 February 2013 Accepted 19 February 2013 Available online 1 April 2013
The environmental/economic dispatch (EED) is an important daily optimization task in the operation of many power systems. It involves the simultaneous optimization of fuel cost and emission objectives which are conflicting ones. The EED problem can be formulated as a large-scale highly constrained nonlinear multiobjective optimization problem. In recent years, many metaheuristic optimization approaches have been reported in the literature to solve the multiobjective EED. In terms of metaheuristics, recently, scatter search approaches are receiving increasing attention, because of their potential to effectively explore a wide range of complex optimization problems. This paper proposes an improved scatter search (ISS) to deal with multiobjective EED problems based on concepts of Pareto dominance and crowding distance and a new scheme for the combination method. In this paper, we have considered the standard IEEE (Institute of Electrical and Electronics Engineers) 30-bus system with 6-generators and the results obtained by proposed ISS algorithm are compared with the other recently reported results in the literature. Simulation results demonstrate that the proposed ISS algorithm is a capable candidate in solving the multiobjective EED problems. 2013 Elsevier Ltd. All rights reserved.
Keywords: Environmental/economic dispatch Evolutionary algorithms Scatter search
1. Introduction Environmental/economic dispatch (EED) problem is one of the most important problems in power systems management. It consists of scaling the outputs of thermoelectric generators, in a thermal power plant, such that a power demand is supplied while satisfying equality and inequality constraints. Each generation unit has particular aspects in the cost of the fuel spent and the pollutants emitted. Fossil-fired electric power plants cause pollution emission while they operate. The emission control results from the requirement for power utilities to reduce their pollutant levels below the annual * Corresponding author. Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil. E-mail addresses: [email protected] (M. de Athayde Costa e Silva), carlos.klein@ volvo.com (C.E. Klein), [email protected] (V.C. Mariani), leandro.coelho@ pucpr.br (L. dos Santos Coelho). 0360-5442/$ e see front matter 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.energy.2013.02.045
emission allowances assigned for the affected fossil units. The total emission can be reduced by minimizing the major pollutants; this is one objective of the EED problem. There are two objectives to be optimized in this kind of EED problem: the cost of the generation and the total emission. Both objectives must be minimized, however they are conflicting, i.e. if there is an optimal solution it is impossible to optimize an objective without worsen other objective. Thus, there is not only one solution but a set of solutions that form the Pareto optimal set. Pareto optimality is a measure of efficiency in multi-criteria and multi-objective situations. A state A (a set of object parameters) is said to be Pareto optimal, if there is no other state B dominating the state A with respect to a set of objective functions. A state A dominates a state B, if A is better than B in at least one objective function and not worse with respect to all other objective functions. Many researchers have tackled this problem in the past. The first use of multiobjective programming with power systems has been addressed by Cheong and Dillon [1]. However, it is realized that
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conventional mathematical techniques, such as gradient method, linear programming algorithm, quadratic programming, Lagrange relaxation algorithm, become very complicated when dealing with increasingly complex dispatch problems, and are further limited by their lack of robustness and efficiency in a number of practical applications [2]. Therefore, because of its particularities, an efficient technique must be used to provide good solutions for the EED optimization problems. In this context, more and more often, metaheuristics are used to find solutions of complex EED optimization problems. Metaheuristics are approximate, heuristic or general-purpose algorithms for solving complex optimization problems, with continuous and/or discrete variables. The use of metaheuristic search algorithms have gained attention in the recent decades due to the failure of conventional methods to solve NP-hard (Non-Polynomial hard) combinatorial and difficult continuous optimization problems. In general, metaheuristics are suitable for solving hard and/or largesize instances of an optimization problem for which there is no efficient exact algorithm available. Over the last few years, metaheuristics such as genetic algorithms [3], differential evolution [4,5], particle swarm optimization [6,7], bacterial foraging algorithm [8], shuffled frog leaping algorithm [9], seeker optimization algorithm [10], biogeography-based optimization algorithm [11], chaotic ant swarm optimization [12], artificial bee colony optimization [13], harmony search [14,15], selforganizing migrating strategy [16], quantum-inspired evolutionary algorithm [17], artificial immune systems [18], imperialist competitive algorithm [19], charged system search algorithm [20], fuzzy adaptive chaotic ant swarm optimization [12], have been used to solve economic dispatch problems. In Ref. [21], a summary of EED algorithms is also given. On the other hand, multiobjective EED optimization involves the simultaneous optimization of several incommensurable and often competing objectives. Recently, the research focus has shifted towards handling both the objectives simultaneously. Over the past decade, this option has received much interest due to the development of a number of multiobjective search strategies based on metaheuristics. Many examples of metaheuristic optimization approaches applied to multiobjective EED optimization problems are given in Refs. [22e36]. In this context, a metaheuristic method called scatter search (SS) can be useful. The scatter search methodology was first introduced in 1977 [37e39] as a heuristic for integer programming; it is a population-based metaheuristic method that uses a set of reference solutions to create new improved solutions by intelligently combining and improving others in a set of good and disperse solutions. The reference set is formed based on quality and diversity of the solutions. Initially, the best solutions are added to the reference set, and then the most diversified solutions, based on the distance to the already added solutions, are included in the set. The algorithm combines these solutions to create new ones and achieve a desirable result. From the standpoint of metaheuristic classification, scatter search may be viewed as an evolutionary algorithm [40] because it builds, maintains and evolves a set of solutions throughout the search. However, scatter search differs from other populationbased evolutionary heuristics like genetic algorithms mainly in its emphasis on generating new elements of the population mostly by deterministic combinations of previous members of the population as opposed to the more extensive use of randomization. The two principles that govern SS are i) intensification, and ii) diversification. Intensification refers to the role of isolating the best performing solutions from the populations in order to obtain a group of good solutions. Diversification in turn isolates the solutions which are the furthest from the best solutions and combine
15
them with the best solutions. This new pool of solutions is the reference set where crossover occurs in order to create solutions from new solution regions by the combination of the intensified solutions and diversified solutions. Intensification and diversification are commonly termed as adaptive memory programming [41]. These principles help to increase the exploration and the exploitation capabilities of the algorithm in finding solutions, which is very desirable in complex problems such as EED problem because the search space is generally wide and there are several local optimal points that can prevent the algorithm to advance. Nevertheless, the original approach does not consider multiple objectives, so some modifications must be done in order to apply it to solve the EED problem. Our interest here is to adapt the wellknown scatter search template to multiobjective optimization and the proposition of improvements in the scatter search design. Scatter search has been found to be successful in a wide variety of optimization problems [42e45]. Moreover, recently it had been extended to deal with multiobjective optimization problems (see examples in Refs. [46e49]). Scatter search uses improvement strategies to efficiently produce the local tuning of the solutions, and a remarkable aspect concerning scatter search is the trade-off between the exploration abilities of the combination method and the exploitation capacity of the improvement mechanism. This paper proposes an improved scatter search (ISS) methodology to deal with multiobjective EED problems based on concepts of Pareto dominance and crowding distance [50] and a new scheme for the combination method. The results of the original SS and proposed ISS approaches are compared to other techniques presented in the recent literature about EED. The mentioned algorithms have been implemented on standard IEEE (Institute of Electrical and Electronics Engineers) 30bus six-generators system in order to obtain the trade-off between the cost and emission. Results show that the modifications proposed increased the performance of the scatter search algorithm. The remainder of the paper is organized as follows: Section 2 presents the concepts of scatter search. Section 3 provides the description of the scatter search algorithm and the proposed approach. The description of the formulation of the optimization problem is mentioned in Section 4. Finally, Sections 5 and 6 present the results and conclusions, respectively.
2. Fundamentals of the scatter search Scatter search is a metaheuristic algorithm that can be considered an evolutionary algorithm in the sense that it incorporates the concept of population. However, SS approaches usually avoid using typical evolutionary operators such as mutation or crossover operators. SS derives its foundations from earlier strategies for combining decision rules and constraints. Historically, the antecedent strategies for combining decision rules were introduced in the context of scheduling methods to obtain improved local decision rules for job shop scheduling problems. New rules were generated by creating numerically weighted combinations of existing rules, suitably restructured so that their evaluations embodied a common metric [38]. The original SS algorithm has five procedures: i) a diversification generation method to randomly generate diversified trial solutions; ii) an improvement method to converge the current solutions toward the optimum; iii) reference set update method to renew the set of reference solutions with the best ones; iv) a subset generation method to create subsets for the combination method; and v) a combination method to combine the reference solutions and generate new offspring. Next subsections describe each procedure.
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2.1. Diversification generation method This method generates solutions so that they are uniformly distributed in the search space. Each dimension of the search space is divided in 4 regions of equal size and the number of times each region is used should be almost the same to guarantee diversity in the population and uniformity in the distribution. Table 1 shows an example of a diversified population generated with this procedure. We have randomly generated 100 solutions of the form x ¼ [x1 x2 x3 x4] with each dimension lying between 0 and 1 and using uniform distribution. In this case each dimension of the solutions should appear approximately 25 times in each range. The values are the number of times that each variable appeared in each range. This procedure is carried out until a reasonable set of solutions is found considering the frequencies of appearance in the four ranges. Also, this procedure do not consider the objective function values, it is focused on the distribution of the population in the search space. 2.2. Improvement method In order to increase the exploitation capability of scatter search an improvement method is used and its goal is to transform a solution in a better one in terms of objective function or feasibility. Typically this method is a local search procedure which is carried out until no improvement is detected in the solution. However, since each solution is improved using this procedure, it needs to be a very fast and effective method in order to not increase the computational demand. The possible solution improvement methods, applicable to the solutions, range from the simplest local searches to specialized procedures. 2.3. Reference set update method The reference set is updated during the search process by using criteria based on comparisons and measures of the diversity between the new solutions and the existing solutions. The reference set contains high quality solutions, in terms of objective function, and diverse solutions. Its size is equal to b ¼ b1 þ b2, where b1 is the number of high quality solutions and b2 is the number of diversified solutions. Initially, we remove the best b1 solutions from the population and add them to the reference set. Then, the b2 diverse solutions must be added to the reference set and removed from the population. The minimum value of the Euclidian distance between each solution and the reference solutions is used as criterion of solutions diversification, i.e. given a solution in the population, the distances between the solution and each reference solution is calculated and the minimum value found is assigned to that solution. This idea prevents the algorithm to being trapped in local optima points. These two steps for adding solutions to the reference set increase the exploration capability of the algorithm and help to avoid being trapped in local minima. 2.4. Subset generation method Scatter search starts by generating a reference set from the population of solutions. Then subsets are iteratively selected from this reference set and are used in the combination method in order Table 1 Example of a diversified population. Range
x1
x2
x3
x4
0.00e0.25 0.25e0.50 0.50e0.75 0.75e1.00
23 24 28 25
23 26 22 29
27 24 22 27
23 22 24 31
to create a pool of new solutions. In this paper subsets of size equal to 2 were considered. All possible subsets are generated; however, a subset already used is discarded because it will recreate solutions already used in the algorithm. 2.5. Combination method The combination procedure tries to intelligently combine good characteristics of the subset of solutions to yield new high quality solutions. The combination method is one of the most important steps of the algorithm. It uses the subsets created in the subset generation method to combine solutions and create new ones. Consider two reference solutions x1and x2, three new solutions are created using the following:
C 1 ¼ x1 d C 2 ¼ x1 þ d C 3 ¼ x2 þ d d ¼ r
ðx1 x2 Þ 2
(1)
(2)
where r is a random number with uniform distribution generated in the range between 0 and 1. 3. Algorithm of the scatter search and the proposed improved version The algorithm just uses the methods described above in the same sequence. An initial random population is created using the diversification generation method and each element is improved using the improvement method. Then, the reference set is constructed with the best and the most diversified solutions with the reference set update method. Note that, at this point, the reference set is empty, so there is no need to compare the solutions. The reference set is used to create new solutions using the combination method; the new solutions are then improved and the reference set is updated with the best and the most diverse solutions of the population. If there are no new solutions, the reference set is rebuilt using the diversification generation method. This process is carried out until a maximum number of iterations to be achieved. Fig. 1 shows the main steps of SS algorithm. The variable called “New solutions” is a flag to notify whether new solutions were created or not. After the application of the combination method, all solutions are evaluated and if there is at least one solution better than any reference solution (according to the objective function value or the distance value) then the flag “New solutions” is set to true, otherwise is set to false. Thus, the reference set is updated or rebuilt according to the flag’s value. In this paper we propose a new scheme for the combination method in the ISS design. Instead of using Equations (1) and (2) of classical SS approach, we generate four new solutions using the following expressions:
8 C > > < 1 C2 C > > : 3 C4
¼ ¼ ¼ ¼
a1 x1 þ ð1 a1 Þx2 a2 x1 þ ð1 a2 Þx2 a3 x1 þ ð1 a3 Þx2 a4 x1 þ ð1 a4 Þx2
(3)
where ai is a random number with uniform distribution generated in the range between 0.2 and 0.8 with different values for each dimension. This scheme creates solutions more diversified because it uses different random numbers for each solution created. Also, it creates four new solutions instead of three as the original method.
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This increases the capability of explore the search space of the algorithm, which is important in problems such as the EED that has a wide search space. A change in the improvement method is also necessary because the original method do not deal with multiple objectives. In this paper we suggest a simple but efficient modification which is described by the following equations.
X1;I ¼ YI þ dYI
(4)
X2;:I ¼ Y:I þ dY:I
(5)
X3;I ¼ YI þ dYI X3;:I ¼ Y:I dY:I
(6)
X4;:I ¼ Y:I þ dY:I X4;I ¼ YI dY
17
(7)
where Y is the solution to be improved, d is a constant number, I is a random logical vector which indicates the dimension to be modified, :I is the negation of I and Xi is the i-th solution constructed. The algorithm creates these 4 solutions from Y and the best between them (according to all objectives) substitutes Y. This improvement method is used in both original SS and the proposed ISS approaches. Compare solutions considering more than one objective function is a problem itself; so, in this paper, to deal with multiple objectives, is used the concepts of Pareto dominance and crowding ! distance [50]. Given two vectors u ¼ ðu1 ; .; uk Þ and ! ! v ¼ ðv1 ; .; vÞ ,u dominates v if and only if ci˛f1; .; kg; u ! ! ! v ^di˛f1; .; kg : u < v . In other words, in at least one dimension u is less than v and in no other dimension v is less than u. The objective vectors are evaluated using this concept and a solution that dominates another is preferred. The solutions that are not dominated by any other are the Pareto optimal solutions and constitute the Pareto set of non-dominated solutions. In multiobjective optimization this set is the desired set of solutions and generally is not known. The population of solutions is analyzed on this aspect and the first front is constructed with the non-dominated solutions of the population, then these solutions are stored and removed from the population, the second front is then constructed with the remaining non-dominated solutions and so on until all solutions are stored in a front. The first front is the set of best solutions in the population and its solutions are in the same level, so, to decide the best solution between two of the same front, the concept of crowding distance is used. Each solution has a crowding distance value which is calculated using the following expression for a solution i:
Di ¼
m X fk ði þ 1Þ fk ði 1Þ fkmax fkmin k¼1
(8)
where Di is the distance of the i-th solution, fk is the k-th objective function’s value of the i-th solution, and fkmax and fkmin are the maximum and minimum values of the k-th objective function, respectively. The solutions of the boundaries of the front have their distance set to infinite and the solution with the great crowding distance is preferred. These two concepts are used when constructing or updating the reference set; the solutions are sorted by the Pareto fronts and by the crowding distances. And when there are new solutions attempting to be inserted in the reference set these two concepts are used to decide which of any two solutions is the best. 4. Formulation of the multiobjective EED problem
Fig. 1. Pseudo code of SS.
The EED problem is formulated as a multiobjective optimization one, where one objective is to minimize the total emission of pollutants and the other is to minimize the cost of generation related to the fuel used. These two objectives should be optimized while satisfying equality and inequality constraints. The problem evaluated in this paper is the standard IEEE 30-bus system with six-generators. The IEEE 30-bus test case represents a portion of the American Electric Power System (in the Midwestern US) as of December, 1961 with 41 transmission lines and 21 load buses (see topology of the IEEE 30-bus system in Fig. 2). Details and data of IEEE 30-bus are presented in Refs. [22,51e53]. The data for the optimization problem are given by Tables 2e4.The system base is 100 MVA and the total system demand for the 21 load buses is
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M. de Athayde Costa e Silva et al. / Energy 53 (2013) 14e21 Table 3 Data for the objective functions. Variable
G1
Cost ($/h) a 10 b 200 c 100 Emission (ton/h) a 4.091 b 5.554 g 6.490 z 2.0e-4 l 2.857
n X
Fig. 2. Topoloy of the IEEE 30-bus system.
4.1. Objective functions The first objective of EED is the classical economic dispatch problem of finding the optimal combination of power generation which minimizes the total cost of generation while satisfying the required demand. Considering a plant with n generation units, the total fuel cost in $/h can be defined as:
i¼1
ai þ bi Pi þ ci Pi2
G5
G6
10 150 120
20 180 40
10 100 60
20 180 40
10 150 100
2.543 6.047 5.638 5.0e-4 3.333
4.258 5.094 4.586 1.0e-6 8.000
5.326 3.550 3.380 2.0e-3 2.000
4.258 5.094 4.586 1.0e-6 8.000
6.131 5.555 5.151 1.0e-5 6.667
(11)
The power losses are modeled as a function of the outputs of generators, as in (12). In this case,
PL ¼
n X n X
Pi Bij Pj þ
n X
B0i Pi þ B00
(12)
i¼1
where B0i and B00 are the loss coefficients on transmission lines, and Bij is the loss coefficient between the i-th and j-th generator. Moreover, the power generated by each unit is constrained between its minimum and maximum limits (Pmin and Pmax). Then, the multiobjective environmental/economic dispatch optimization problem is formulated as a minimization of the Equations (9) and (10) subject to constraints of power balance and generation capacity (limits).
(9) 4.3. Constraint handling
where Pi is the output of the i-th generator and ai, bi and ci are the cost coefficients of the i-th generator. The second objective considered in the EED problem is the minimization of the atmospheric pollutants caused by fossil-fueled thermal units. The emission function, in ton/h, can be modeled as a sum of a quadratic and an exponential function given by:
Femission ¼
G4
Pi ¼ PD þ PL
i¼1 j¼1
n X
G3
i¼1
2.834 p.u. (per unit). Transmission losses are considered in this problem and are estimated using the coefficients presented in Table 4.
Ffuel ¼
G2
n X 102 ai þ bi Pi þ gi Pi2 þ zi $expðli Pi Þ
(10)
i¼1
where ai, bi, gi, zi and li are the emission coefficients of the i-th generator. 4.2. Constraints of the problem There are two types of constraints in the environmental/economic dispatch problem: the power balance and the generation capacity of the units. The first one is an equality constraint which relates to the demand and the transmission losses, that is, the power generated must cover the total load demand PD and the total power transmission loss PL (in p.u. values, in this paper) as follows:
In order to guarantee feasibility in all solutions, it is used a simple method that obligate solutions to satisfy the constraint (11) and the lower and upper boundaries constraint. First, the solution is multiplied by a factor which considers the transmission loss and the load demand, as follows:
C B BPD þ PL0 C 0 CP P〞 ¼ B C B X n @ 0 A Pi
Power (p.u.)
G1
G2
G3
G4
G5
G6
0.05 0.5
0.05 0.6
0.05 1.0
0.05 1.2
0.05 1.0
0.05 0.6
(13)
i¼1
where P0 is the solution array to be adjusted, P00 is the new solution array, PD is the load demand and PL0 is the transmission loss for the solution P0. However, this does not guarantee that the new solution is satisfying (11) because the power loss is dependent of the solution. Thus the mentioned procedure is carried out using the new yielded solution until a certain number of iterations is achieved or it Table 4 B-loss coefficients matrix. B
Values
B0 B00
0.1382 0.0299 0.0044 0.0022 0.0010 0.0008 0.0535 9.8573e-4
Table 2 Lower and upper boundaries of each generator.
Pmin Pmax
1
0
0.0299 0.0487 0.0025 0.0004 0.0016 0.0041 0.0030
0.0044 0.0025 0.0182 0.0070 0.0066 0.0066 0.0085
0.0022 0.0004 0.0070 0.0137 0.0050 0.0033 0.0004
0.0010 0.0016 0.0066 0.0050 0.0109 0.0005 0.0001
0.0008 0.0041 0.0066 0.0033 0.0005 0.0244 0.0015
M. de Athayde Costa e Silva et al. / Energy 53 (2013) 14e21
19
Table 5 Best results for the fuel cost. Technique
Reference
Cost ($/h)
Emission (ton/h)
G1 (p.u.)
G2 (p.u.)
G3 (p.u.)
G4 (p.u.)
G5 (p.u.)
G6 (p.u.)
SS ISS MO-DE/PSO MODE
This paper This paper [48] [25]
603.6846 603.5888 606.0073 606.1260
0.2165 0.2159 0.2209 0.2195
0.1666 0.1630 0.1220 0.1332
0.3182 0.2832 0.2843 0.2727
0.5347 0.5813 0.5857 0.6018
0.9589 0.9322 0.9962 0.9747
0.5243 0.5387 0.5149 0.5146
0.3448 0.3473 0.3566 0.3617
Table 6 Best results for the emission function. Technique
Reference
Cost ($/h)
Emission (ton/h)
G1 (p.u.)
G2 (p.u.)
G3 (p.u.)
G4 (p.u.)
G5 (p.u.)
G6 (p.u.)
SS ISS MO-DE/PSO MODE
This paper This paper [48] [25]
632.4107 633.3926 646.0243 642.8490
0.194559 0.194469 0.194179 0.194200
0.3806 0.3716 0.4118 0.39266
0.4269 0.4436 0.4616 0.46256
0.5532 0.5597 0.5435 0.56311
0.4486 0.4381 0.3922 0.40309
0.5522 0.5418 0.5454 0.5676
0.4821 0.4890 0.5148 0.47826
stops in case of the constraint (11) being satisfied. In the simulations we have experimentally choose 5 iterations for this method. The lower and upper boundaries constraint is controlled within the same loop simply by replacing the elements of the solution that are outside the boundaries with the boundaries’ values. Also, it is used a penalty factor to penalize solutions which do not satisfy the constraints. There are two ways for calculating the penalty factor. The first one is a weighting of the total violation of the constraints and the second is an adaptive procedure which considers the current iteration of the algorithm [54]. The first approach is used in the first half of total quantity of iterations of the algorithm and the second is used in the remaining generations. Equations (14) and (15) describe both methods as follows,
pðXÞ ¼ a1
m X
fi ðXÞ2
(14)
i¼1
pðXÞ ¼ ðCtÞa2
m X
jfi ðXÞjb
(15)
i¼1
where fi(X) is the violation of the i-th constraint of the solution X, a1 ¼ 1000, a2 ¼ 2, C ¼ 0.5, b ¼ 2 and t is the current iteration. Note that the choice of the parameters is an optimization problem itself, thus these values were chosen experimentally. 5. Simulation results Simulations were carried out using MatLab environment (MathWorks). In order to eliminate stochastic discrepancies, 30 independent runs were carried out using different initial populations of solutions. The number of initial solutions in the population of SS and ISS approaches is 20 and the number of reference solutions (b) is equal to 10, being the first 5 the bests (b1) and the last 5 the more diversified (b2). The parameter d ¼ 0.02 was used for both SS and ISS approaches. Results presented here are compared to a hybrid algorithm based on particle swarm optimization and differential evolution (MO-DE/PSO) presented in Ref. [51] and a
multiobjective differential evolution (MODE) presented in Ref. [28]. The best values found, in terms of the fuel cost and the emission, are bolded in Tables 5 and 6, respectively. The best solutions considering the costs of generation obtained by ISS are compared to SS, MO-DE/PSO and MODE and are given in Table 5. It is observed that ISS finds better minimum fuel cost than the other mentioned optimization algorithms. In Table 6 are presented the best results related to emission. In this case, the best result for the emission was found by MO-DE/PSO, however all results are very near. Table 7 presents a statistical comparison between the proposed and the original SS approaches. In this case, three metrics are considered for comparing the approaches: the mean Euclidian distance to the origin (in this case an optimum point), the spacing index and the number of Pareto solutions. Spacing metric measures how evenly the points in the approximation set are distributed in the objective space. A value of zero for this metric indicates all members of the Pareto front currently available are equidistantly spaced. In this context, based on Table 7 with respect to spacing, SS performed worse than the ISS. Since the nondominated vectors found by the ISS are clustered together, the spacing metric evidences the better results found by this method. For the metrics calculations, all data collected during the 30 runs were stored in an external repository and a new Pareto front, for each technique, was generated using all this data. The main objective of the external repository is to keep a historical record of the nondominated vectors found along the search process. Fig. 3 shows a good diversity in the nondominated solutions obtained by ISS after 20 iterations (in 30 runs) when compared with the original SS.
Table 7 Comparison of SS and the proposed ISS. Technique
Spacing (f1, f2)
Mean Euclidian distance to origin (f1, f2)
Pareto solutions
SS ISS
0.1815 0.1673
614.7549 614.4093
139 156 Fig. 3. Pareto fronts found by the SS and ISS approaches.
20
M. de Athayde Costa e Silva et al. / Energy 53 (2013) 14e21
6. Concluding remarks Most decision problems in the real world require the simultaneous optimization of more than one criterion. EED can be approached as a multiobjective problem having conflicting objectives, as the minimization of emission is contrary to the maintenance of fuel cost economy. The optimization of multiple conflictive objectives is a hard problem. Effective stochastic optimization methods based on metaheuristics have been developed to solve multiobjective EED problems. A metaheuristic is a set of concepts that can be used to define heuristic methods that can be applied to a wide range of different problems. In other words, a metaheuristic can be seen as a general algorithmic framework which can be applied to different optimization problems with relatively few modifications to make them adapted to a specific problem [55]. In general, metaheuristics provide quite effective strategies to find near optimal solutions for complex problems. Scatter search is a newly emerging population-based evolutionary method that, unlike genetic algorithm, searches for global optima or satisfactory solutions by operating on a small data collection of intensification and diversification, and making much use of various systematic sub-methods and limited use of randomization. This paper evaluated the performance of a classical SS method and an ISS approach when dealing with an EED multiobjective problem. In this context, some changes were necessary for the study, and a modification of the original combination method was introduced. The proposed ISS approach performed better than the original and showed promising results for a standard IEEE 30-bus with six-generators test system. In order to make a comparison against competing solution methods, other results for the IEEE 30bus with 6-generators system presented in the recent literature were showed in Tables 5 and 6. In this context, the results using the proposed ISS based on some comparison metrics were competitive. In order to explore the abilities of the proposed methodology, our future research will test the ISS approach in other multiobjective EED problems. We will also need to carry out more formal performance measurements on the ISS algorithm, for example using the ManneWhitney rank-sum test.
Acknowledgments This work was supported by the National Council of Scientific and Technologic Development of Brazil e CNPq e under Grants 303963/ 2009-3/PQ and 478158/2009-3, and ‘Fundação Araucária’ under Grant: 14/2008-416/09-15149. The first author, also, would like to thanks the Pontifical Catholic University of Parana e PUCPR e and the National Council of Scientific and Technologic Development of Brazile CNPq e for the financial support provided through the Institutional Program for Scientific Initiation Scholarships e PIBIC.
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