Differential evolution based on truncated Lévy-type flights and population diversity measure to solve economic load dispatch problems

L e ra PR, Brazil b Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil cDepartment of Mechanical Engineering, Pontifical Cath a r t i c l e i n f o Article history: Received 25 November 2012 gradient method, Lagrange relaxation approximate, and Newton method can solve the ELD problems if the cost function is piecewise linear and monotonically increasing. However, ELD problem with constraints such as valve-point effects, multi-fuel cost, transmission losses, and prohibited zones are non-smooth ry algorithms [7– 4] have received eir ability to find heuristics olutions in time, they often provide a fast and reasonable solution. One of the evolutionary algorithms that have shown po and good perspective for the solution of various optimizatio lems [15–19] is differential evolution (DE). DE was proposed by Storn and Price [15,16] and is now widely used by researchers in many scientific and engineering fields [17–19] as it is a robust and fast stochastic function optimizer. DE is particularly simple to work with, having only a few control parameters. However, their associated control parameters, namely the mutation factor (MF) and the crossover rate (CR) may significantly influence the search- ing accuracy and convergence speed of the DE. Choosing suitable ⇑ Corresponding author at: Electrical Engineering Graduate Program (PPGEE), Department of Electrical Engineering, Federal University of Parana (UFPR), Poly- technic Center, CP 19011, 81531-980 Curitiba, PR, Brazil. E-mail addresses: [email protected] (L.S. Coelho), [email protected] Electrical Power and Energy Systems 57 (2014) 178–188 Contents lists availab Electrical Power an .e l (T.C. Bora), [email protected] (V.C. Mariani). The economic load dispatch (ELD) problem plays a key role in operation planning of modern power systems. The main objective of ELD problem is to reduce the total generation cost, subject to load demand and other equality and inequality constraints. Classical optimization algorithms such as lambda iteration, approaches, metaheuristics based on evolutiona 10] and swarm intelligence approaches [11–1 much attention by many researchers due to th potential solutions. However, although the meta always guarantee discovering globally optimal s 0142-0615/$ - see front matter � 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.11.024 do not finite tential n prob- consisting of 13 and 140 thermal units. Simulation results reveal that, compared with the classical DE and those other methods reported in literatures recently, the proposed DEL is capable of obtaining better quality solutions with higher efficiency. � 2013 Elsevier Ltd. All rights reserved. 1. Introduction optimization problems, in which finding the global optimum is a challenge [1–6]. As an alternative to the conventional optimization Received in revised form 4 October 2013 Accepted 12 November 2013 Keywords: Power systems Economic load dispatch Thermal generators Optimization methods Evolutionary algorithms olic University of Parana (PUCPR), Imaculada Conceição, 1155, 80215-901 Curitiba, PR, Brazil a b s t r a c t Economic load dispatch (ELD) is an important constrained optimization task addressing this vital concern for power system operations. ELD problem is the process of allocating generation among available committed generating units such that cost of generation is optimum subject to several equality and inequality constraints. The conventional optimization methods are mainly classical mathematical methods, which include gradient method and Lagrange relaxation method. In recent years, different types of evolutionary algorithms have been used to solve ELD problems. Among the existing evolutionary algorithms, a well-known branch is the differential evolution (DE). The mutation operation of DE applies vector differentials between existing population members for determining both the degree and the direc- tion applied to the individual subject of the mutation operation. With an eye to improve the performance of classical DE, in this paper, a DE algorithm combined with truncated Lévy flight random walks and a population diversity measure (DEL) to improve the crossover and mutation operations is designed to help avoiding premature convergence effectively. A Lévy flight random walks (a sequence of displacements) in which the increments are distributed according to a heavy-tailed probability distribution form the a-sta- ble distribution family. The effectiveness of the proposed DEL is demonstrated for two benchmark ELD problems. In order to evaluate the performance of the proposed DEL, it is applied to benchmark systems a Electrical Engineering Graduate Program (PPGEE), Department of Electrical Engineering, Federal University of Parana (UFPR), Polytechnic Center, CP 19011, 81531-980 Curitiba, Differential evolution based on truncated and population diversity measure to solv dispatch problems Leandro dos Santos Coelho a,b,⇑, Teodoro Cardoso Bo journal homepage: www évy-type flights economic load a, Viviana Cocco Mariani c le at ScienceDirect d Energy Systems sevier .com/locate / i jepes Pow control parameter values is, frequently, a problem dependent task and requires previous experience of the user. To successfully solve a specific optimization problem at hand, it is generally required to perform a time-consuming trial-and-error search to tune its asso- ciated control parameter values. Inappropriate choice of parame- ters may lead to premature convergence or stagnation. Adapting the DE’s control parameters is one possible improvement. In this context, researchers are working on DE and further improving it through the application of adaptive or self-adaptive techniques to the control parameters [20–26]. In [20], a version of the DE with adaptive control parameters in which uses fuzzy systems to adapt the search parameters for the mutation operation and crossover operation was proposed. The control inputs incorporate the relative objective function values and individuals of the successive generations. Salman et al. [21] proposed a self-adaptive DE algorithm with the mutation factor tuning generated by normal distribution. This self-adaptive DE was tested on nine benchmark functions where it generally out- performed other well-known versions of DE. Nobakhti and Wang [22] proposed a self-adaptive approach called randomized adap- tive DE with adaptation of the mutation factor. Das et al. [23] lin- early reduced the mutation factor with increasing generation count from a maximum to a minimum value, or randomly varied in the range [0.5, 1]. They also employed a uniform distribution be- tween 0.5 and 1.5 (with a mean value of 1) to obtain a hybrid DE approach. In [24], an improved self-adaptive DE algorithm with multiple strategies using a different search strategy and a parallel evolution mechanism was proposed and validated. Brest et al. [25] proposed a DE with a parameter control technique based on the self-adaptation of mutation factor and crossover rate. Zaharie [26] proposed to transform the mutation factor into a Gaussian random variable. In this paper, in order to solve the ELD problem of units with valve-point effects effectively, a solution methodology integrating DE, a population diversity measure and Lévy flights random walks (DEL) to improve the crossover and mutation rates tuning has been proposed. Lévy flights are Markov processes and have infinite var- iance, and their increments are distributed by the a-stable Levy laws of index 0 < a < 2 (Levy distributions). The probability distri- bution of truncated Lévy flight increments is a slightly deformed Lévy distribution. Details about the Levy flights implementation can be found in [27–30]. Mantegna and Stantely [27] introduced a class of stochastic process called truncated Lévy flight in which the arbitrarily large steps of a Lévy flight are eliminated. The pro- posed method is useful to solve the problem of infinite variance, in which the probability of taking a step is abruptly cut to zero at a certain critical step size. Gupta and Campanha [28] extended the gradually truncated Lévy flight combining statistical distribu- tion factor and a controlling mechanism. The authors discussed also the variation of controlling parameters with the increase of time difference between successive observations. This allows us to generate a number of theoretical curves for the same system by varying the time difference between successive observations. In [29], a version of an exponentially damped Lévy flight is evalu- ated. Grothe and Schmidt [30] described the scaling behavior of such Lévy-Student processes and the parameters of its marginal distributions by a simple analytical scaling law. This deformation must change the variance of the resulting distribution from the infinite to the finite, and consequently, according to the generalized central limit theorem, the resulting distribution belongs to the Gaussian basin of attraction. In this paper, to show the advantages of the proposed DEL algo- rithm, it has been applied for solving two benchmark ELD prob- Leandro dos Santos Coelho et al. / Electrical lems with 13 and 140 generators with valve-point loading are adopted to demonstrate the performance of the proposed DEL algorithm. Simulation results obtained through the DEL approach are analyzed and compared with the classical DE method and those reported in the recent literature. The next section of this paper contains the description of ELD problem, while Section 3 explains the classical DE and the pro- posed DEL algorithm. Section 4 briefly describes the procedure of constraint handling in DE and DEL algorithms. Section 5 presents and discusses the optimization results. Finally, the conclusion is drawn in Section 6. 2. Description of the ELD problem The increasing energy demand and decreasing energy resources have requested the optimum use of available resources. The basic objective of the ELD is to determine the allocation of output pow- ers of generators so as to meet the power demand at minimum operating cost under various system and operating constraints. Improvements in scheduling of the unit power outputs can lead to significant cost savings. The ELD can be formulated mathemati- cally as an optimization problem (minimization) with an objective function and constraints. The equality constraint given by the power balance equation is expressed as:Xn i¼1 Pi � PL � PD ¼ 0: ð1Þ The inequality constraints related to real power operating limits are represented by Eq. (2) given by Pmini 6 Pi 6 P max i : ð2Þ In the power balance criterion, an equality constraint must be satisfied, as shown in Eq. (1). The generated power should be the same as the total load demand plus total line losses. The generating power of each generator should lie between maximum and mini- mum limits represented by Eq. (2), where Pi is the power of gener- ator i (in MW); n is the number of generators in the system; PD is the system load demand (in MW); PL represents the total line losses (in MW) and Pmini and P max i are, respectively, the minimum and maximum power outputs of the ith generating unit (in MW). The total fuel cost function OF is formulated as follows: min OF ¼ Xn i¼1 FiðPiÞ ð3Þ where Fi is the total fuel cost for the generator unity i (in $/h). To simplify the optimization problem and facilitate the application of classical techniques, cost functions of generation units are typically modeled by smooth quadratic function form given as FiðPiÞ ¼ aiP2i þ biPi þ ci ð4Þ where ai, bi and ci are cost coefficients of generator i. Generally, the input–output characteristics of modern power generating units are inherently high nonlinear because of valve-point loading effects, multi-fuel effects, among others. To take these effects into consider- ation, the ELD problem can be represented as a non-smooth optimi- zation problem. The incremental fuel cost function of the generators in this paper including valve-point loading effects can be repre- sented as followseF iðPiÞ ¼ FðPiÞ þ jei sinðfiðPmini � PiÞÞj or ð5ÞeF iðPiÞ ¼ aiP2i þ biPi þ ci þ jei sinðfiðPmini � PiÞÞj ð6Þ where ei and fi are non-smooth fuel cost coefficients of generator i. In other words, to take account for the valve-point effects in the ELD problem, sinusoidal functions are added to the quadratic cost func- er and Energy Systems 57 (2014) 178–188 179 tions given by Eq. (4). Hence, the total fuel cost that must be minimized, according to Eq. (3), is modified to: Pow min OF ¼ Xn i¼1 eF iðPiÞ ð7Þ where OF is the objective function to be minimized, and eF i is the cost function of generator i (in $/h) defined by Eq. (6). In the case studies presented here, transmission losses are neglected, i.e., PL = 0. The problem formulation mentioned until here in this section considers the valve point loading effects. This formulation is re- lated to the 13 generators benchmark problem, the first case study adopted in this paper. However, a second case study was adopted related to the power system of Korea using 140 generators is con- sidered with ramp rates and prohibited operating zones beyond the valve point loading effects. When the generator ramp rate limits are considered, the oper- ating limits of the ith generating unit are modified as follows Pi � P0i 6 URi P0i � Pi 6 DRi ( ð8Þ where URi and DRi are ramp-up and ramp-down rate limits of ith unit, respectively and are expressed in MW/h. In other words, the inclusion of ramp rate limits modifies the generator operation con- straints (2) as follows: maxðPmini ;URi � PiÞ 6 Pi 6minðPmaxi ; P0i � DRiÞ ð9Þ A generator with prohibited operating zones has discontinuous fuel-cost characteristics. The generator is constrained since there occurs the physical limitations or instability. The feasible operating zones of ith generator can be described as follows Pi 2 Pmini 6 Pi 6 P LB1 i PUBk�1i 6 Pi 6 P LBk i ; k ¼ 2;3; . . . ;NPZi PUBki 6 Pi 6 P max i ; k ¼ NPZi i ¼ 2;3; . . . ;NGPZ 8>>>: ð10Þ where PLBki and P UBk i are the lower and upper boundaries of prohib- ited operating zone k of ith generator in MW; NPZi is the number of prohibited operating zones of ith generator; and NGPZ is the number of generators with prohibited operating zones. 3. Differential evolution approaches This section describes the proposed DEL algorithm. First, an overview of the classical DE is presented, and then the DEL is detailed. 3.1. Classical DE algorithm DE is a simple strategy of stochastic optimization. It uses a gree- dy and less stochastic approach with floating point coding in prob- lem solving rather than the other evolutionary algorithms. DE algorithm uses arithmetical operators to evolve from a ran- domly generated starting population to a final solution. Basically, the weighted difference between two individuals is added to a third individual in population. This way no separate probability distribution has to be used, which makes the scheme completely self-organizing. There are several variant strategies of DE [15–19]. In general expressions, first part represents a vector to be perturbed. The first part is either ‘rand’ (randomly chosen vector) or ‘best’ (best vector of current population). The second part is the number of difference vectors (one or two vectors) chosen for perturbation of first part, and the last part denotes the type of crossover being used. The type of crossover can be ‘bin’ (binomial) or ‘exp’ (exponential). The var- 180 Leandro dos Santos Coelho et al. / Electrical iant implemented here was the DE/rand/1/bin, i.e., the random selection, one difference vector and binomial crossover which in- volved the following steps: Step 1: Initialization of the parameter setup: The user must choose the key parameters that control the DE, i.e., population size (N), boundary constraints of optimization variables, muta- tion factor (MF), crossover rate (CR), and the stopping criterion (kmax). Step 2: Initialize the initial population of individuals: Set genera- tion k = 1. Initialize a population of individuals (solution vectors) with random values generated according to a uniform probability distribution in the n-dimensional problem space. This population is successively improved by the mutation, crossover and selection operators. Step 3: Evaluate the objective function value: For each individual, evaluate its objective function (fitness) value. Step 4: Mutation operation (or differential operation): In this phase, DE generates a mutant vector zi(k + 1), by adding a weighted difference of two population vectors to a third vector using the following equation: ziðkþ 1Þ ¼ xi1 ðkÞ þMF � ½xi2 ðkÞ � xi3 ðkÞ� ð11Þ In the Eq. (11), i = 1, 2, . . . , N is the individual’s index of population; k is generation; xiðkÞ ¼ ½xi1 ðkÞ; xi2 ðkÞ; . . . ; xin ðkÞ�T stands for the posi- tion of the ith individual of population of N real-valued n-dimen- sional vectors; ziðkÞ ¼ ½zi1 ðkÞ; zi2 ðkÞ; . . . ; zin ðkÞ�T stands for the position of the ith individual of a mutant vector; MF > 0 is a real parameter, called mutation factor (scaling factor or amplification factor), which controls the amplification of the difference between two individuals so as to avoid search stagnation. The mutation operation randomly select the target vector xi1 ðkÞ, with i– i1. Then, two individuals xi2 ðkÞ and xi3 ðkÞ are randomly selected with i1– i2 - – i3– i, and the difference vector xi2 � xi3 is calculated. Unlike other evolutionary algorithms, where perturbation occurs in accordance with a random quantity, DE uses weighted difference between solution vectors (target vectors) to perturb the population at each generation as expressed in Eq. (8). Step 5: Crossover (recombination) operation: The next task after mutation is crossover, to increase the diversity of the perturbed parameter vectors. For each mutant vector, zi(k + 1), an index rnbr(i) e {1, 2, . . . , n} is randomly chosen using a uniform distri- bution, and a trial vector, uiðkþ 1Þ ¼ ½ui1 ðkþ 1Þ;ui2 ðkþ 1Þ; . . . ; uin ðkþ 1Þ�T, is generated with uij ðkþ 1Þ ¼ zij ðkþ 1Þ if randbðjÞ 6 CR or j ¼ rnbrðiÞ; xij ðkÞ otherwise; ( ð12Þ where randb(j) is the jth evaluation of a uniform random number generation with [0, 1], and CR is a crossover rate or probability of crossover in the range [0,1]. The CR has a direct influence on the diversity of DE. The higher the probability of recombination, the more variation is introduced in the new population, thereby increasing diversity and explora- tion. Increasing CR often results in faster convergence, while decreasing CR increases search robustness. Step 6: Selection operation: The selection operator compares the fitness of the trial vector and the objective function of the cor- responding target vector, and selects the one which performs better. In this context, to decide whether or not the vector ui(k + 1) should be a member of the population comprising the next generation, it is compared to the corresponding vector xi(k). Thus, if f denotes the objective function under minimization, then u ðkþ 1Þ if f ðu ðkþ 1ÞÞ < f ðx ðkÞÞ;� er and Energy Systems 57 (2014) 178–188 xiðkþ 1Þ ¼ i i i xiðkÞ otherwise: ð13Þ Pow Step 7: Verification of the stopping criterion: Update generation k = k + 1. Loop to Step 3 until a stopping criterion is met, usually a maximum number of generations (iterations) kmax. In this paper, DE is run for a fixed kmax generations. 3.2. The proposed DEL approach In classical DE approaches, the MF and CR have constant values. Suitable tuning of MF and CR parameters may result in faster con- vergence of the DE and alleviate the risk of settling in one of the local minima. Recently, Rönkkönen et al. [31] suggested using MF values in range [0.4,0.95], and the CR value in range [0,0.2] for sep- arable functions while CR in range [0.9,1] for dependent functions. However, the rules for choosing the control parameters of the DE algorithm are quite different as their validity is restricted to the problems and parameters values considered in the respective investigation. In this paper, the proposed DEL uses Lévy flight random walks to improve the MF and CR tuning mechanisms. Fur- thermore, a population diversity measure is adopted to acted as trade-off operators to balance the rate of convergence and popula- tion diversity. Lévy processes are a class of stochastic processes based on the Lévy-stable distribution. Such processes are tightly related to anomalous diffusion and fractal statistics. Lévy processes are com- monly described by Lévy flight random walk models. These sorting of movements describes several natural behaviors [32–37]. The stable random variables laws, also called a-stable were introduced by Levy in 1925 during his investigations of the behav- ior of sums of independent random variables [38]. The main fact is that when two different stable random variables taken from two different stable distributions, but with the same a parameter, have their sum as an a-stable distribution. The central limit theorem is a special case of stable distribution. The most common and convenient way to define a-stable ran- dom variables is through their characteristic function [39], which is the Fourier transform of the probability density function. In the context of this paper, the generation of Lévy flight truncated in range [0,1], adopted in the DEL approach, was based on proce- dures based on Chambers–Mallows–Stuck method [39] for simu- lating a-stable variables and the boundary adaptation method called here planimetric mapped space method. Details about the Lévy flight can see found in [39–43]. The proposed DEL presents the Steps 1–3 and Steps 6–7 equal to classical DE. However, the following procedure is adopted to the Steps 4 and 5 in DEL: Step 4: Mutation operation based on Lévy flights and a population diversity measure: In this phase, DE generates a mutant vector zi(k + 1): ziðkþ 1Þ ¼ xi1 ðkÞ þ SM � ½xi2 ðkÞ � xi3 ðkÞ� ð14Þ In Eq. (14), SM is a scale vector (1 � n) where n is the dimension of the optimization problem. The SM is tuning for the dth dimension of the solutions vector using a Lévy flight and a diversity measure of the population given by SMd ¼ ð1� divdðkÞÞ � ln 1u � � ð15Þ where divd(k) is the normalized diversity for the dth dimension of the solutions vector and u is a truncated Levy flight value generated in range [0.05,2]. In this work, the diversity measure divd(k) adopted in DEL is determined by the Eq. (16),ffiffiffip Leandro dos Santos Coelho et al. / Electrical divdðkÞ ¼ 1NP �maxðEÞ Xn d¼1 E divdð1Þ ; ð16Þ with E ¼ Xn j¼1 ðxij � �xjÞ2 ð17Þ where divd (1) is the population diversity in the first generation (k = 1). The diversity measure is useful to control the mutation step size using population diversity. Step 5: Crossover operation using Lévy flights: For each mutant vector, zi(k + 1), an index rnbr(i) 2 {1, 2, . . . , n} is randomly cho- sen using a uniform distribution, and a trial vector, uiðkþ 1Þ ¼ ½ui1 ðkþ 1Þ;ui2 ðkþ 1Þ; . . . ;uin ðkþ 1Þ�T, is generated with uij ðkþ 1Þ ¼ zij ðkþ 1Þ if randbðjÞ 6 CRL or j ¼ rnbrðiÞ; xij ðkÞ otherwise; ( ð18Þ where CRL is a crossover rate generated using a Lévy flight in the range [0.1,0.3]. 4. Constraints handling The key factor in solving an ELD is how to handle the con- straints relating to the problem. Over the last few decades, kinds of approaches had been proposed to handle the constraints. They can be grouped into four categories: ideas that preserve the feasi- bility of solutions, penalty-based approaches, methods that clearly distinguish between feasible and unfeasible solutions, and hybrid techniques [44,45]. However, in this work is adopted a repair pro- cedure to the equality constraints based on [46] instead of penaliz- ing infeasible solutions (see details in Fig. 1). The procedure mentioned in Fig. 1 is employed to the first case study with 13 generators. However, the second case study presents ramp rate limits and prohibited operating zones too. In this con- text, the objective in the second case study of ELD with 140 gener- ators is to minimize the Eq. (7) subjecting to the constraints (8) and (9). For this purpose, it was adopted the following penalized fuel cost function: min OF ¼ Xn i¼1 eF iðPiÞ þw1 Xn i¼1 Ri " # þw2 Xn i¼1 Zi " # ð19Þ where w1 and w2 are the penalty factors for ramp rate limits and prohibited operating zone constraints, respectively; Ri and Zi are indicators of falling into the ramp rate limits constraint (given by Eq. (9)) and prohibited operating zone, respectively. The penalty factorsw1 andw2 are used to penalize the fuel cost function propor- tional to the amount of constraint violations. If there are no prohib- ited zones and ramp rate limits are attended, w1 and w2 are set to zero. 5. Case studies and results analysis In order to verify the effectiveness of the DEL for solving ELD problems, we assess the performance of the proposed algorithm described in Section 3 using computer simulations. The proposed algorithm has been applied to ELD problems in two test cases. The first case study is a 13 generating units with quadratic cost functions considering valve-point effects (details in [47]). Taking the valve-point effects into account will increase the number of lo- cal minimum points in the fuel cost function and make the prob- lem of finding the global optimum more difficult. The second one is a simplified Korean power system with 140 thermal generating units (description in [48]). er and Energy Systems 57 (2014) 178–188 181 The proposed algorithm has been implemented in Matlab 7 un- der Microsoft Windows XP and executed on QuadCore processor, 2.66 GHz personal computer with 8 GB RAM to solve the ELD Pow 182 Leandro dos Santos Coelho et al. / Electrical problems. In order to eliminate stochastic discrepancy, in each case study, 30 independent runs were made for each of the optimiza- tion methods involving 30 different initial trial solutions for each optimization method. For all runs in two ELD case studies, the parameters of DE and DEL approaches were adjusted to: population size equal to 25 (first case) and 20 (second case), the maximum number of iterations (generations) allowed was set to 9000 and 1200, respectively, for the two case studies. In this context, for each DE and DEL algo- rithm, 24,000 and 225,000 evaluations of objective function are realized, respectively, for the first and second case studies in each run. The strategy adopted in the simulations using DE and DEL was the best/1/exp. In order to check the performance of DE and DEL on the second case study, many trials with different w1 and w2 values were car- ried out to test the consistency of the optimization procedure. After, we choice w1 = 107 and w2 = 104. Fig. 1. Equality constraints handling Table 1 Data of ELD with 13 thermal generators. Generator a ($/MW h2) b ($/MW h) c ($/h) 1 0.00028 8.10 550 2 0.00056 8.10 309 3 0.00056 8.10 307 4 0.00324 7.74 240 5 0.00324 7.74 240 6 0.00324 7.74 240 7 0.00324 7.74 240 8 0.00324 7.74 240 9 0.00324 7.74 240 10 0.00284 8.60 126 11 0.00284 8.60 126 12 0.00284 8.60 126 13 0.00284 8.60 126 Table 2 Results of DE and DEL in terms of objective function for the first case study in 30 runs. A Optimization method MF CR Minimum ($/h) DE 0.5 0.2 17,960.6861 DE 0.5 0.8 18,054.0224 DE 0.5 0 18,733.4683 DE 0.5 1 18,733.4683 DE Uniform 0.2 17,960.4676 DE Uniform 0.8 18,026.1589 DE Uniform 0 18,733.4683 DE Uniform 1 18,733.4683 DEL Lévy Lévy 17,960.3661 Note: Uniform in MF uses numbers generated randomly in range [0,1]. er and Energy Systems 57 (2014) 178–188 5.1. First case study: ELD using 13 generators considering valve-point effects This case study consisted of 13 thermal units of generation with the effects of valve-point loading, as given in Table 1. The data shown in Table 1 are also available in [47]. The load demand of the system expected to be satisfied, PD, totalizes 1800 MW. Optimization results related to DE and DEL are presented in Ta- ble 2. In Table 2, various standard statistical measures including maximum, mean, minimum, and standard deviation are carried out to compare the objective function values obtained through the DE and DEL algorithms in 30 runs. From Table 2, in terms of statistics criteria, the DEL presented the best result in all evaluated indices. The best results obtained for solution vector using DEL are given in Table 3. To show the advantages of the proposed DEL, the results ob- tained from the DEL are compared with those of other methods using DE and DEL algorithms. e ($/h) f Pmini ðMWÞ P max i ðMWÞ 300 0.035 0 680 200 0.042 0 360 150 0.042 0 360 150 0.063 60 180 150 0.063 60 180 150 0.063 60 180 150 0.063 60 180 150 0.063 60 180 150 0.063 60 180 100 0.084 40 120 100 0.084 40 120 100 0.084 55 120 100 0.084 55 120 result with Boldface means the best value found. Mean ($/h) Maximum ($/h) Standard deviation ($/h) 18,000.4577 18,066.6786 30.1025 18,159.3293 18,340.3734 73.3535 18,974.1298 19,124.6329 101.0102 18,971.1401 19,123.0936 100.9019 17,993.1617 18,062.9389 26.6631 18,174.2414 18,384.6416 83.5723 18,984.3534 19,094.6190 88.03021 18,984.3497 19,093.4995 87.9990 17,966.1306 17,975.4109 4.7219 available in the literature in Table 4. It can be seen that the DEL is superior to several algorithms in terms of minimum value of objec- tive function. The best result using DEL is equal to differential harmony search (DHS) [59], cultural self-organizing migrating strategy (CSOMA) [60], and modified group search optimizer method (MGSO) [61] in terms of the objective function value. However, an analysis related to the number of the objective function evalu- ations can reveal that DEL employed only 24,000 evaluations of the objective function in each run. The DHS [59], CSOMA [60] and MGSO [61] adopted 60,000, 25,000 and 40,000 evaluations of the objective function in each run, respectively. 5.2. Second case study: Power system of Korea using 140 generators The system consists of 140 thermal generating units with ramp rate limits where the hydro and pump storage plants are not con- sidered. The input data are given in Table 5. The total demand is set to 49,342 MW. It is assumed that 12 generators have the cost func- tion with valve-point effects and 4 generators are considered the prohibited operating zones [48] (see Tables 6 and 7). Optimization results related to DE and DEL are presented in Table 8. From Table 8, in terms of statistics criteria, the DEL got bet- ter results that the other tested DE approaches in most indices. However, in terms of maximum values, the results using DE with uniformMF andMF = 0.5 were slightly worse than the results using DEL. In terms of results, the DEL presented better result with OF = 1,657,962.7166, but it is a very closed to the best result obtained by a particle swarm optimization approach [48] with OF = 1,657,962.73 and by a group search optimizer [63] with Table 3 Best result after 30 runs for the first case study using DEL. Power Generation (MW) P1 628.3185269 P2 149.5996490 P3 222.7490774 P4 109.8665501 P5 109.8665479 P6 109.8665504 P7 109.8665483 P8 60.0000000 P9 109.8665496 P10 40.0000000 P11 40.0000000 P12 55.0000000 P13 55.0000000P13 i¼1Pi 1800 Table 4 Comparison of results for fuel costs presented in previous publications. A result with Boldface means the best value found. Optimization method Minimum fuel costs ($/h) Mean fuel costs ($/h) Pattern search method [49] 17,969.17 18,088.84 Genetic algorithm with pattern search and quadratic programming [50] 17,964.25 18,199 Improved genetic algorithm [51] 17,963.98 – Cultural differential evolution [52] 17,963.94 Self-adaptive genetic algorithm using Taguchi method [53] 17,963.94 17,974.31 Quantum particle swarm optimization [54] 17,969.01 – Accelerated biogeography-based optimization [55] 17,963.8521 17,967.3560 Multi-strategy ensemble biogeography-based optimization [56] 17,963.8292 17,964.0468 Improved harmony search [57] 17,963.83 17,976.475 Adaptive hybrid particle swarm optimization [58] 17,963.84 17,963.9577 Differential harmony search [59] 17,960.3661 17,961.1226 Cultural self-organizing migrating strategy [60] 17,960.3661 17,967.8708 62] Leandro dos Santos Coelho et al. / Electrical Power and Energy Systems 57 (2014) 178–188 183 Modified group search optimizer method [61] Improved coordinated aggregation-based particle swarm optimization [ Best result of this paper using DE with CR = 0.2 Table 5 Data of EDP with 140 generators (second case study). Generator a ($/MW h2) b ($/MW h) c ($/h) COAL#01 0.032888 61.242 1220.645 COAL#02 0.008280 41.095 1315.118 COAL#03 0.003849 46.310 874.288 COAL#04 0.003849 46.310 874.288 COAL#05 0.042468 54.242 1976.469 COAL#06 0.014992 61.215 1338.087 COAL#07 0.007039 11.791 1818.299 COAL#08 0.003079 15.055 1133.978 COAL#09 0.005063 13.226 1320.636 COAL#10 0.005063 13.226 1320.636 COAL#11 0.005063 13.226 1320.636 COAL#12 0.003552 14.498 1106.539 COAL#13 0.003901 14.651 1176.504 COAL#14 0.003901 14.651 1176.504 COAL#15 0.003901 14.651 1176.504 COAL#16 0.003901 14.651 1176.504 COAL#17 0.002393 15.669 1017.406 COAL#18 0.002393 15.669 1017.406 COAL#19 0.003684 14.656 1229.131 17,960.3661 – 17,960.60 17,967.94 17,960.3661 17,966.1306 Pmin (MW) Pmax (MW) UR DR P0 71 119 30 120 98.4 120 189 30 120 134.0 125 190 60 60 141.5 125 190 60 60 183.3 90 190 150 150 125.0 90 190 150 150 91.3 280 490 180 300 401.1 280 490 180 300 329.5 260 496 300 510 386.1 260 496 300 510 427.3 260 496 300 510 412.2 260 496 300 510 370.1 260 506 600 600 301.8 260 509 600 600 368.0 260 506 600 600 301.9 260 505 600 600 476.4 260 506 600 600 283.1 260 506 600 600 414.1 260 505 600 600 328.0 (continued on next page) Table 5 (continued) Generator a ($/MW h2) b ($/MW h) c ($/h) Pmin (MW) Pmax (MW) UR DR P0 COAL#20 0.003684 14.656 1229.131 260 505 600 600 389.4 COAL#21 0.003684 14.656 1229.131 260 505 600 600 354.7 COAL#22 0.003684 14.656 1229.131 260 505 600 600 262.0 COAL#23 0.004004 14.378 1267.894 260 505 600 600 461.5 COAL#24 0.003684 14.656 1229.131 260 505 600 600 371.6 COAL#25 0.001619 16.261 975.926 280 537 300 300 462.6 COAL#26 0.005093 13.362 1532.093 280 537 300 300 379.2 COAL#27 0.000993 17.203 641.989 280 549 360 360 530.8 COAL#28 0.000993 17.203 641.989 280 549 360 360 391.9 COAL#29 0.002473 15.274 911.533 260 501 180 180 480.1 COAL#30 0.002547 15.212 910.533 260 501 180 180 319.0 COAL#31 0.003542 15.033 1074.810 260 506 600 600 329.5 COAL#32 0.003542 15.033 1074.810 260 506 600 600 333.8 COAL#33 0.003542 15.033 1074.810 260 506 600 600 390.0 COAL#34 0.003542 15.033 1074.810 260 506 600 600 432.0 COAL#35 0.003132 13.992 1278.460 260 500 660 660 402.0 COAL#36 0.001323 15.679 861.742 260 500 900 900 428.0 COAL#37 0.002950 16.542 408.834 120 241 180 180 178.4 COAL#38 0.002950 16.542 408.834 120 241 180 180 194.1 COAL#39 0.000991 16.518 1288.815 423 774 600 600 474.0 COAL#40 0.001581 15.815 1436.251 423 769 600 600 609.8 LNG#01 0.902360 75.464 669.988 3 19 210 210 17.8 LNG#02 0.110295 129.544 134.544 3 28 366 366 6.9 LNG_CC#01 0.024493 56.613 3427.912 160 250 702 702 224.3 LNG_CC#02 0.029156 54.451 3751.772 160 250 702 702 210.0 LNG_CC#03 0.024667 54.736 3918.780 160 250 702 702 212.0 LNG_CC#04 0.016517 58.034 3379.580 160 250 702 702 200.8 LNG_CC#05 0.026584 55.981 3345.296 160 250 702 702 220.0 LNG_CC#06 0.007540 61.520 3138.754 160 250 702 702 232.9 LNG_CC#07 0.016430 58.635 3453.050 160 250 702 702 168.0 LNG_CC#08 0.045934 44.647 5119.300 160 250 702 702 208.4 LNG_CC#09 0.000044 71.584 1898.415 165 504 1350 1350 443.9 LNG_CC#10 0.000044 71.584 1898.415 165 504 1350 1350 426.0 LNG_CC#11 0.000044 71.584 1898.415 165 504 1350 1350 434.1 LNG_CC#12 0.000044 71.584 1898.415 165 504 1350 1350 402.5 LNG_CC#13 0.002528 85.120 2473.390 180 471 1350 1350 357.4 LNG_CC#14 0.000131 87.682 2781.705 180 561 720 720 423.0 LNG_CC#15 0.010372 69.532 5515.508 103 341 720 720 220.0 LNG_CC#16 0.007627 78.339 3478.300 198 617 2700 2700 369.4 LNG_CC#17 0.012464 58.172 6240.909 100 312 1500 1500 273.5 LNG_CC#18 0.039441 46.636 9960.110 153 471 1656 1656 336.0 LNG_CC#19 0.007278 76.947 3671.997 163 500 2160 2160 432.0 LNG_CC#20 0.000044 80.761 1837.383 95 302 900 900 220.0 LNG_CC#21 0.000044 70.136 3108.395 160 511 1200 1200 410.6 LNG_CC#22 0.000044 70.136 3108.395 160 511 1200 1200 422.7 LNG_CC#23 0.018827 49.840 7095.484 196 490 1014 1014 351.0 LNG_CC#24 0.010852 65.404 3392.732 196 490 1014 1014 296.0 LNG_CC#25 0.018827 49.840 7095.484 196 490 1014 1014 411.1 LNG_CC#26 0.018827 49.840 7095.484 196 490 1014 1014 263.2 LNG_CC#27 0.034560 66.465 4288.320 130 432 1350 1350 370.3 LNG_CC#28 0.081540 22.941 13,813.001 130 432 1350 1350 418.7 LNG_CC#29 0.023534 64.314 4435.493 137 455 1350 1350 409.6 LNG_CC#30 0.035475 45.017 9750.750 137 455 1350 1350 412.0 LNG_CC#31 0.000915 70.644 1042.366 195 541 780 780 423.2 LNG_CC#32 0.000044 70.959 1159.895 175 536 1650 1650 428.0 LNG_CC#33 0.000044 70.959 1159.895 175 540 1650 1650 436.0 LNG_CC#34 0.001307 70.302 1303.990 175 538 1650 1650 428.0 LNG_CC#35 0.000392 70.662 1156.193 175 540 1650 1650 425.0 LNG_CC#36 0.000087 71.101 2118.968 330 574 1620 1620 497.2 LNG_CC#37 0.000521 37.854 779.519 160 531 1482 1482 510.0 LNG_CC#38 0.000498 37.768 829.888 160 531 1482 1482 470.0 LNG_CC#39 0.001046 67.983 2333.690 200 542 1668 1668 464.1 LNG_CC#40 0.132050 77.838 2028.945 56 132 120 120 118.1 LNG_CC#41 0.096968 63.671 4412.017 115 245 180 180 141.3 LNG_CC#42 0.054868 79.458 2982.219 115 245 120 180 132.0 LNG_CC#43 0.054868 79.458 2982.219 115 245 120 180 135.0 LNG_CC#44 0.014382 93.966 3174.939 207 307 120 180 252.0 LNG_CC#45 0.013161 94.723 3218.359 207 307 120 180 221.0 LNG_CC#46 0.016033 66.919 3723.822 175 345 318 318 245.9 LNG_CC#47 0.013653 68.185 3551.405 175 345 318 318 247.9 LNG_CC#48 0.028148 60.821 4322.615 175 345 318 318 183.6 LNG_CC#49 0.013470 68.551 3493.739 175 345 318 318 288.0 NUCLEAR#01 0.000064 2.842 226.799 360 580 18 18 557.4 NUCLEAR#02 0.000252 2.946 382.932 415 645 18 18 529.5 NUCLEAR#03 0.000022 3.096 156.987 795 984 36 36 800.8 184 Leandro dos Santos Coelho et al. / Electrical Power and Energy Systems 57 (2014) 178–188 Pmin (MW) Pmax (MW) UR DR P0 NUCLEAR#04 0.000022 3.040 154.484 795 978 36 36 801.5 578 682 138 204 582.7 615 720 144 216 680.7 612 718 144 216 670.7 Power and Energy Systems 57 (2014) 178–188 185 NUCLEAR#05 0.000203 1.709 332.834 NUCLEAR#06 0.000198 1.668 326.599 NUCLEAR#07 0.000215 1.789 345.306 Table 5 (continued) Generator a ($/MW h2) b ($/MW h) c ($/h) Leandro dos Santos Coelho et al. / Electrical OF = 1,657,962.727. The best result using DEL is presented in Table 9. 6. Conclusion and future work DE is a powerful heuristic method for solving nonlinear non-dif- ferentiable and multi-modal optimization problem. It is based on a particular way of constructing so-called mutant vectors by using Table 6 Unit data with valve-point loading of Korea system. Generator a ($/MW h2) b ($/MW h) c ($/h) e f COAL#05 0.042468 54.242 1976.469 700 0.080 COAL#10 0.005063 13.226 1320.636 600 0.055 COAL#15 0.003901 14.651 1176.504 800 0.060 COAL#22 0.003684 14.656 1229.131 600 0.050 COAL#33 0.003542 15.033 1074.810 600 0.043 COAL#40 0.001581 15.815 1436.251 600 0.043 LNG_CC#10 0.000044 71.584 1898.415 1100 0.043 LNG_CC#28 0.081540 22.941 13813.001 1200 0.030 LNG_CC#30 0.035475 45.017 9750.750 1000 0.050 LNG_CC#42 0.054868 79.458 2982.219 1000 0.050 OIL#08 0.001580 81.805 2290.381 600 0.070 OIL#10 0.076810 46.665 6743.302 1200 0.043 NUCLEAR#08 0.000218 1.815 350.372 612 720 144 216 651.7 NUCLEAR#09 0.000193 2.726 370.377 758 964 48 48 921.0 NUCLEAR#10 0.000197 2.732 367.067 755 958 48 48 916.8 NUCLEAR#11 0.000324 2.651 124.875 750 1007 36 54 911.9 NUCLEAR#12 0.000344 2.798 130.785 750 1006 36 54 898.0 NUCLEAR#13 0.000690 1.595 878.746 713 1013 30 30 905.0 NUCLEAR#14 0.000650 1.503 827.959 718 1020 30 30 846.5 NUCLEAR#15 0.000233 2.425 432.007 791 954 30 30 850.9 NUCLEAR#16 0.000239 2.499 445.606 786 952 30 30 843.7 NUCLEAR#17 0.000261 2.674 467.223 795 1006 36 36 841.4 NUCLEAR#18 0.000259 2.692 475.940 795 1013 36 36 835.7 NUCLEAR#19 0.000707 1.633 899.462 795 1021 36 36 828.8 NUCLEAR#20 0.000786 1.816 1000.367 795 1015 36 36 846.0 OIL#01 0.014355 89.830 1269.132 94 203 120 120 179.0 OIL#02 0.014355 89.830 1269.132 94 203 120 120 120.8 OIL#03 0.014355 89.830 1269.132 94 203 120 120 121.0 OIL#04 0.030266 64.125 4965.124 244 379 480 480 317.4 OIL#05 0.030266 64.125 4965.124 244 379 480 480 318.4 OIL#06 0.030266 64.125 4965.124 244 379 480 480 335.8 OIL#07 0.024027 76.129 2243.185 95 190 240 240 151.0 OIL#08 0.001580 81.805 2290.381 95 189 240 240 129.5 OIL#09 0.022095 81.140 1681.533 116 194 120 120 130.0 OIL#10 0.076810 46.665 6743.302 175 321 180 180 218.9 OIL#11 0.953443 78.412 394.398 2 19 90 90 5.4 OIL#12 0.000044 112.088 1243.165 4 59 90 90 45.0 OIL#13 0.072468 90.871 1454.740 15 83 300 300 20.0 OIL#14 0.000448 97.116 1011.051 9 53 162 162 16.3 OIL#15 0.599112 83.244 909.269 12 37 114 114 20.0 OIL#16 0.244706 95.665 689.378 10 34 120 120 22.1; OIL#17 0.000042 91.202 1443.792 112 373 1080 1080 125.0 OIL#18 0.085145 104.501 535.553 4 20 60 60 10.0 OIL#19 0.524718 83.015 617.734 5 38 66 66 13.0 OIL#20 0.176515 127.795 90.966 5 19 12 6 7.5 OIL#21 0.063414 77.929 974.447 50 98 300 300 53.2 OIL#22 2.740485 92.779 263.810 5 10 6 6 6.4 OIL#23 0.112438 80.950 1335.594 42 74 60 60 69.1 OIL#24 0.041529 89.073 1033.871 42 74 60 60 49.9 OIL#25 0.000911 161.288 1391.325 41 105 528 528 91.0 OIL#26 0.005245 161.829 4477.110 17 51 300 300 41.0 OIL#27 0.234787 84.972 57.794 7 19 18 30 13.7 OIL#28 0.234787 84.972 57.794 7 19 18 30 7.4 OIL#29 1.111878 16.087 1258.437 26 40 72 120 28.6 Table 7 Prohibit zones of units of Korea system. Generator Zone 1 Zone 2 Zone 3 COAL#08 [250,280] [305,335] [420,450] COAL#32 [220,250] [320,350] [390,420] LNG_CC#32 [230,255] [365,395] [430,455] OIL#25 [50,75] [85,95] – uns Pow Table 8 Results of DE and DEL in terms of objective function for the second case study in 30 r Optimization method MF CR Minimum ($/h) DE 0.5 0.2 1,657,962.7170 186 Leandro dos Santos Coelho et al. / Electrical differences between randomly selected elements from the current population. Several advantages of DE can be mentioned such as: (i) DE is easy to implement, and there are few parameters to adjust; (ii) DE can be more efficient than genetic algorithms, simulated annealing, particle swarm optimization, evolutionary DE 0.5 0.8 1,658,110.9710 DE 0.5 0 2.15 � 1049 DE 0.5 1 2.16 � 1049 DE Uniform 0.2 1,657,963.2570 DE Uniform 0.8 1,658,145.1721 DE Uniform 0 1.99 � 1049 DE Uniform 1 1.98 � 1049 DEL Lévy Lévy 1,657,962.7166 Table 9 Best result after 30 runs for the second case study using DEL. Power Generation (MW) Power P1 119.000000 P49 P2 164.000000 P50 P3 190.000000 P51 P4 190.000000 P52 P5 168.539816 P53 P6 190.000000 P54 P7 490.000000 P55 P8 490.000000 P56 P9 496.000000 P57 P10 496.000000 P58 P11 496.000000 P59 P12 496.000000 P60 P13 506.000000 P61 P14 509.000000 P62 P15 506.000000 P63 P16 505.000000 P65 P17 506.000000 P66 P18 506.000000 P67 P19 505.000000 P68 P20 505.000000 P69 P21 505.000000 P70 P22 505.000000 P71 P23 505.000000 P72 P24 505.000000 P73 P25 537.000000 P74 P26 537.000000 P75 P27 549.000000 P76 P28 549.000000 P77 P29 501.000000 P78 P30 499.000000 P79 P31 506.000000 P80 P32 506.000000 P81 P33 506.000000 P82 P34 506.000000 P83 P35 500.000000 P84 P36 500.000000 P85 P37 241.000000 P86 P38 241.000000 P87 P39 774.000000 P88 P40 769.000000 P89 P41 3.000000 P90 P42 3.000000 P91 P43 250.000000 P92 P44 250.000000 P93 P45 250.000000 P94 P46 250.000000 P95 P47 250.000000 P96 P48 249.999999 P97 . A result with Boldface means the best value found. Mean ($/h) Maximum ($/h) Standard deviation ($/h) 1,658,073.7663 1,658,330.5785 115.3365 er and Energy Systems 57 (2014) 178–188 programming, and tabu search; that is, DE often finds the solutions with fewer objective function evaluations than are re- quired by mentioned optimization metaheuristics; and (iii) the variants of DE continued to secure front ranks in the IEEE CEC (Conference on Evolutionary Computation) competitions (see comments in [18]). 2,700,668.5829 32,863,018.2320 5,696,758.8208 3.36 � 1049 5.61 � 1049 8.95 � 1048 3.39 � 1049 5.60 � 1049 8.97 � 1048 1,883,740.2569 5,749,273.5368 875,4276.6467 3,402,539.2880 24,739,557.7504 5,518,107.9500 3.34 � 1049 5.58 � 1049 8.95 � 1048 3.34 � 1049 5.57 � 1049 8.94 � 1048 1,658,001.7003 1,651,518.6719 57.9836 Generation (MW) Power Generation (MW) 250.000000 P98 718.000000 250.000000 P99 720.000000 165.000000 P100 964.000000 165.000000 P101 958.000000 165.000000 P102 947.900000 165.000000 P103 934.000000 180.000000 P104 935.000000 180.000000 P106 876.500000 103.000001 P107 880.900000 198.000000 P108 873.700000 312.000000 P109 877.400000 308.565459 P110 871.700000 163.000000 P111 882.000000 95.000000 P112 94.000000 511.000000 P113 94.000000 511.000000 P114 94.000000 490.000000 P115 244.000000 256.740281 P116 244.000000 490.000000 P117 244.000000 490.000000 P118 95.000000 130.000000 P119 95.000000 141.545478 P120 116.000000 388.327412 P121 175.000000 195.000000 P122 2.000000 196.293426 P123 4.000000 196.261416 P124 15.000000 257.943689 P125 9.000000 400.852472 P126 12.000000 330.000000 P127 10.000000 531.000000 P128 112.000000 531.000000 P129 4.000000 541.999999 P130 5.000000 56.000000 P131 5.000000 115.000000 P132 50.000000 115.000000 P133 5.000000 115.000000 P134 42.000000 207.000000 P135 42.000000 207.000000 P136 41.000000 175.000000 P137 17.000000 175.000000 P138 7.000000 180.391040 P139 7.000000 175.000002 P140 26.000000 575.400000 547.500000 836.800000 837.500000 682.000000 720.000000 P140 i¼1Pi 49,342 Pow To find quality solution with good computational efficiency dif- ferent DE approaches have been developed and proposed in the recent literature. However, the choice of the control parameters, such as CR and MF is often critical for the performance of DE. In this paper, a DEL approach is proposed and validated in two ELD problems consisting of 13 and 140 thermal units whose fuel cost function is calculated by taking account of the effect of valve-point loading. In comparison to other optimization tech- niques, the simulation results reveal that the DEL is a promising approach capable of obtaining better quality solutions with good efficiency for ELD. Also, the equality and inequality constraints treatment method detailed in Section 4 have always provided the solutions satisfying the constraints without disturbing the opti- mum process of the DEL. However, without forgetting the No Free Lunch theorem [64,65], one may note that each of the DEL has its own advantages and disadvantages and none is suitable for tack- ling all kinds of optimization problems that appear in real world. 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Remarks on a recent paper on the ‘No Free Lunch’ theorems. IEEE Trans Evol Comput 2001;5(3):295–6. 188 Leandro dos Santos Coelho et al. / Electrical Power and Energy Systems 57 (2014) 178–188 Differential evolution based on truncated Lévy-type flights and population diversity measure to solve economic load dispatch problems 1 Introduction 2 Description of the ELD problem 3 Differential evolution approaches 3.1 Classical DE algorithm 3.2 The proposed DEL approach 4 Constraints handling 5 Case studies and results analysis 5.1 First case study: ELD using 13 generators considering valve-point effects 5.2 Second case study: Power system of Korea using 140 generators 6 Conclusion and future work Acknowledgments References