Accepted Manuscript Solution of combined economic and emission dispatch problem using a novel chaotic improved harmony search algorithm H. Rezaie, M.H. Kazemi-Rahbar, B. Vahidi, H. Rastegar PII: DOI: Reference: S2288-4300(18)30125-8 https://doi.org/10.1016/j.jcde.2018.08.001 JCDE 160 To appear in: Journal of Computational Design and Engineering Received Date: Revised Date: Accepted Date: 23 May 2018 9 July 2018 16 August 2018 Please cite this article as: H. Rezaie, M.H. Kazemi-Rahbar, B. Vahidi, H. Rastegar, Solution of combined economic and emission dispatch problem using a novel chaotic improved harmony search algorithm, Journal of Computational Design and Engineering (2018), doi: https://doi.org/10.1016/j.jcde.2018.08.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Solution of combined economic and emission dispatch problem using a novel chaotic improved harmony search algorithm H. Rezaie1, M. H. Kazemi-Rahbar2, B. Vahidi1,*, H. Rastegar1 1 Department of Electrical Engineering, Amirkabir University of Technology (AUT), 424 Hafez Ave, Tehran, Iran 2 Department of Electrical Engineering, Shahed University, Persian Gulf Freeway, Tehran, Iran Emails: h.rezaie@aut.ac.ir, mhkazemirahbar@gmail.com, vahidi@aut.ac.ir, rastegar@aut.ac.ir *Corresponding author Abstract This paper presents a new optimization technique developed based on harmony search algorithm (HSA), called chaotic improved harmony search algorithm (CIHSA). In the proposed algorithm, the original HSA is improved using several innovative modifications in the optimization procedure such as using chaotic patterns instead of uniform distribution to generate random numbers, dynamically tuning the algorithm parameters, and employing virtual harmony memories. Also, a novel type of local optimization is introduced and employed in the algorithm procedure. Applying these modifications to HSA has resulted in enhancing the robustness, accuracy and search efficiency of the algorithm, and significantly reducing the iterations number required to achieve the optimal solution. To validate the effectiveness of CIHSA, it is used to solve the combined economic emission dispatch (CEED) problem, which practically is a complex high-dimensional non-convex optimization task with several equality and inequality constraints. Six test systems having 6, 10, 13, 14, 40, and 140 generators are investigated in this study, and the valve-point loading effects, ramp rate limits and power transmission losses are also taken into account. The results obtained by CIHSA are compared with the results reported in a large number of other research works. Furthermore, the statistical data regarding the CIHSA performance in all test systems is presented. The numerical and statistical results confirm the high quality of the solutions found by CIHSA and its superiority compared to other existing techniques employed in solving CEED problems. Keywords: Economic emission dispatch, environmental/economic dispatch, improved harmony search algorithm, chaotic harmony search algorithm, evolutionary algorithm, valve-point loading effect 1 Introduction Optimization has always been an integral part of engineering design and decision making procedures to attain the maximum benefit at the minimum cost. Over the last few decades, a large number of metaheuristic algorithms have been introduced to remedy the computational drawbacks of traditional optimization techniques which predominantly are based on numerical linear and nonlinear programming methods. Some of the most popular and well-known metaheuristic algorithms are genetic algorithm (GA), particle swarm optimization (PSO), differential evolution (DE), harmony search algorithm (HSA), tabu search (TS), simulated annealing (SA), ant colony optimization (ACO), artificial bee colony (ABC), cuckoo search (CS), and krill herd algorithm (KHA) which is younger than others but has attracted a lot of attention over a few past years. Also, a lot of research works can be found in the literature, which have proposed modified variants of these algorithms or have combined them with other algorithms to enhance their search efficiency. For instance, some of the outstanding modified algorithms suggested based on PSO, HSA, and KHA can be found in [1-8], [9-16], and [17-25], respectively. This paper introduces a new HSA-based optimization method called chaotic improved harmony search algorithm (CIHSA). This algorithm is actually formed by combining two algorithms proposed in this paper, called chaotic harmony search algorithm (CHSA) and improved harmony search algorithm (IHSA). In CHSA, to avoid being trapped into local optima and better covering the search space, the chaotic pattern is used instead of uniform distribution to generate random numbers in the optimization procedure, which leads to the improving the algorithm robustness. In IHSA, some innovative modifications are incorporated into the original algorithm procedure that significantly enhances the accuracy and efficiency of HSA. These modifications are explained in details in section 4.1. According to the results and statistical data obtained in this work, CIHSA can be considered as one of the most successful evolutions of the original HSA performance. To validate the effectiveness of CIHSA, it is employed to solve the combined economic emission dispatch (CEED) problem, which is one of the most important issues in optimizing the operation of electric power systems. In practice, the CEED problem is a high-dimensional non-convex optimization problem with several equality and inequality constraints, and it is one of the most complicated optimization problems in electrical power engineering. Economic load dispatch (ELD) problem is a computational process to determine the generation contribution of each unit at minimum fuel cost while satisfying the total load demand and all operational constraints [26], [27]. The electricity generation from fossil fuels is exacerbating the atmospheric pollution which has become one of the most important concerns in recent years. So, besides moving toward generating electricity from clean and renewable energies [28], pollutant emissions released by traditional generation units should also be considered in the generation scheduling of the system. Emission constrained dispatch (ECD) is similar to ELD, except that its goal is minimizing emissions instead of fuel cost. However, due to the fact that fuel cost and emission are in conflict with each other (minimizing one increases the other), system operation with either minimum fuel cost or minimum emissions will be unworkable. To deal with this problem, a load dispatching technique to simultaneously minimize both fuel cost and emissions, called combined economic emission dispatch (CEED), has been developed. It is clear that the determination of a single optimal solution for the bi-objective CEED problem requires that the weight of both objective functions be specified [11]. Generally, there are three main approaches to solve the CEED problems; The first approach is to convert the bi-objective CEED problem into a single-objective optimization task by defining a price penalty factor (pf) or normalizing the fuel cost and emissions. This approach has been practiced in several previous research works using different methods such as particle swarm optimization algorithm (PSO), gravitational search algorithm (GSA) and gravitational acceleration enhanced PSO algorithm (GAEPSO) [29], moth swarm algorithm (MSA) [30], real coded chemical reaction algorithm (RCCRO) [31], spiral optimization algorithm (SOA) [32], opposition-based harmony search algorithm [11], biogeography based optimization algorithm (BBO) [33], and hybrid PSO and GSA (PSOGSA) [34]. The second method involves simultaneous minimization of fuel cost and emissions as a bi-objective optimization problem. This approach has been practiced in several previous papers using different methods such as multi-objective particle swarm optimization (MOPSO) [35], multi-objective differential evolution (MODE) [36], multi-objective harmony search (MOHS) [37], tribe-modified differential evolution (Tribe-MDE) [38], non-dominated sorting bacterial foraging (NSBF) and fuzzy dominance sorting bacterial foraging (FSBF) [39], and elitist non-dominated sorting genetic algorithm (NSGA-II) [40]. However, these methods are computationally involved and time-consuming and might produce only sub-optimal solutions. The third approach is to consider the emission as a constant within the permitted limits, in which the problem can be solved through a single-objective optimization task. However, this approach is unable to provide any information regarding trade-offs between fuel cost and emissions. For instance, DavidonFletcher-Powell’s method of optimization has been used in [41] for solving the CEED problem while the emission amount has been considered as a constant in the allowed limits. In this paper, to verify the effectiveness of the proposed CIHSA, it is applied to solve the CEED problem in six test systems having 6, 10, 13, 14, 40, and 140 generators considering valve-point loadings effect. The bi-objective CEED problem is converted into a single-objective optimization task by defining a modiﬁed price penalty factor. In addition to solving the CEED problem, the ELD and ECD problem also have been solved, and the obtained results are compared with a large number of other existing algorithms. Also, it is taken into account that the selected papers for comparison, as much as possible, be among the latest published articles in this field. Thereby, the section of numerical results in this paper is able to clearly confirm the effectiveness and superiority of the proposed method. In addition, the statistical results of the proposed algorithm are provided for all test systems which demonstrate that the proposed algorithm not only has high ability to find the most appropriate solution, but also has high robustness and reliability in its performance. The rest of this paper is organized as follow: The ELD, ECD and CEED problems formulation is expressed in section 2. A brief description of HSA is provided in section 3. The proposed CIHSA is explained in details in section 4. Finally, the effectiveness of the proposed algorithm is verified by applying CIHSA to six test systems, in section 5. 2 2.1 Problem formulation Economic load dispatch (ELD) The objective function of the ELD problem is minimizing the fuel cost for a specified load demand while satisfying various system and unit constraints. The fuel cost of thermal power plants can be approximately modeled as a quadratic function of the generators output power as given in (1). FC ai Pi 2 bi Pi ci n i 1 (1) where FC is the total fuel cost of generations ($/hr), ai , bi and ci are the fuel cost coefficients of the i-th unit, Pi is the output power of the i-th unit, and n is the number of generation units. To obtain a more practical model for the cost function of thermal power plants, the effect of valve-point loading should be also considered. The fuel cost function considering valve-point loadings can be expressed as (2) [42]: FC ai Pi 2 bi Pi ci ei sin fi Pi min Pi i 1 n (2) where FC is the total fuel cost ($/hr) considering valve-point loadings, ei and fi are the fuel cost coefficients of the i-th unit that reflect the valve-point effect. a) Power output constraints There is a practical range for the minimum and maximum of the electrical output power of each unit as shown in (3). Pi min Pi Pi max (3) b) System power balance constraint In load dispatching, the system power balance, which is presented in (4), should be satisfied. n Pi PD PL (4) i 1 where PD is the total load demand, and PL is the total power transmission losses which can be expressed as a function of the units output power and B-loss coefﬁcients as presented in (5). n n n PL PB i ij Pj B0i Pi B00 i 1 j 1 i 1 (5) where Bij is the ij-th element of the loss coefﬁcients square matrix, B0i is the i-th element of the loss coefﬁcients vector and B00 is the loss coefﬁcient constant. c) Ramp rate limits constraint In practice, the output power of the units cannot vary instantaneously. So, the operational range of the output power of each unit is limited by its ramp up-/down-rate limits as given in (6). max Pi min , Pi 0 - DR i Pi min Pi max , Pi 0 UR i (6) where Pi0 is the previous operating output power of i-th unit (MW); DRi and URi are the down-rate and up-rate limits of i-th unit (MW/h), respectively. 2.2 Emission constrained dispatch (ECD) Total emissions released by thermal power plants also can be approximated as a quadratic function of the output active power of the units. The emission constrained dispatch (ECD) problem can be expressed as an optimization task to minimize the total amount of emissions, which is defined by (7) [38]: E 102 i Pi 2 i Pi i i exp i Pi n i 1 (7) where E is the total amount of emissions (lb/hr), αi ,βi ,γi, ζi and λi are the emission coefficients of the i-th unit. 2.3 Combined economic emission dispatch (CEED) The bi-objective CEED problem can be converted into a single objective function using a modiﬁed price penalty factor (pf) approach as (8): PTC FC pf E (8) where PTC is the pure total cost of the system operation. For providing a tradeoff between minimization of the fuel cost and emission, (8) can be rewritten as follow: TC w FC (1 w) pf E (9) where w is weight factor and specifies the optimization type; if w=1, then the problem is ELD; if w=0, then the problem is ECD; if w=0.5, then the problem is CEED. By the following steps pf for a specified load demand can be calculated: Step 1: Calculate hi for each unit according to (10): FC Pi max E Pi max hi i 1, 2,..., n $ / kg (10) Step 2: Sort hi values in an ascending order. Step 3: Add maximum output power of each unit one at a time starting from the unit with smallest hi until Pi max PD Step 4: hi associated with the last unit is pf for the given load demand [33]. 3 Harmony Search Algorithm (HSA) Harmony search algorithm (HSA) is a derivative-free meta-heuristic algorithm which is inspired by the music improvisation process. In comparison to other algorithms, the desirable feature of HSA is that it generates a new solution vector (Harmony) after considering all the existing solution vectors in the harmony memory (HM) matrix. This feature leads to the increasing the exploration power of HSA to achieve better solutions. Furthermore, some modified variants of HSA have been introduced that have employed a stochastic derivative in the optimization procedure and are applicable to the problems in which the differential derivative cannot be used due to the problem characteristics, for instance, design problems with discrete ranges of the control variables [15], [16]. There are five steps in the optimization procedure of HSA, as follows [13]: Step 1. Definition of the optimization problem and HSA parameters initialization The optimization problem is defined as follow: Minimize f(x) subject to xi X i , i 1,, N where f(x) is the objective function, x is the set of each decision variable (xi), Xi is the set of the possible values range for each design variable that xi.l ≤ Xi ≤ xi,u , where xi,l and xi,u are the lower and upper bounds for each decision variable, and N is the number of design variables. Also, in this step, the required HSA parameters are specified. The harmony memory size (HMS) which determines the number of solution vectors in HM matrix, harmony memory considering rate (HMCR), pitch adjusting rate (PAR), and the termination criteria are selected in this step. HMCR and PAR are the parameters required to improve the solution vector and they are defined in Step 3. Step 2. Harmony memory (HM) initialization In this step, the harmony memory (HM) matrix, given in (11), is filled with randomly generated solution vectors and is sorted based on the objective function f(x) values. x11 2 x HM 1 HMS x1 x12 x22 x2HMS x1N xN2 HMS ... xN ... ... (11) Step 3. Improvising a new harmony from the HM A new harmony vector, x´ = (x´1, x´2,…, x´N), is generated based on memory consideration, pitch adjustment, and random selection. According to memory consideration, the value of the first decision variable x´1 for the new vector is selected from any value in the specified HM range (x11 - xHMS1). Values of the other design variables (x´2,…, x´N) can be chosen with the same method. The HMCR varies between 0 to 1, and determines the probability of choosing one value from the historical values stored in the HM, while (1-HMCR) is the probability of randomly selecting one value from the possible range of values. x {x1i , x i2 ,..., x iHMS } with probability HMCR xi i with probability 1 HMCR xi X i (12) Then, every component obtained by the memory consideration is examined to determine whether it should be pitch-adjusted. This operation uses the PAR parameter, which is the rate of pitch adjustment as follows: Yes Pitch adjusting decision for xi NO with probability PAR with probability 1 PAR (13) The value of (1-PAR) determines the probability of doing nothing. If the pitch adjustment decision for x´i is Yes, then x´i is replaced according to (14). xi xi r bw (14) where r is a random number generated using uniform distribution between 0 and 1, and BW is an arbitrary distance bandwidth. All of the mentioned considerations are applied to each variable of the new harmony vector. Step 4. Updating the HM According to objective function value, if the new generated harmony vector, x´ = (x´1, x´2,…, x´N), is better than the worst harmony in the HM, the new harmony will replace it in the HM. Step 5. Repeating Steps 3 and 4 and checking the termination criteria Steps 3 and 4 will be repeated until the termination criterion has been satisfied. 4 The chaotic improved harmony search algorithm (CIHSA) HSA has a high ability to find the high-performance regions of the solution space at a reasonable time, but gets into trouble in performing a local search for numerical applications. So, adding an extra local optimization method to the optimization procedure of HSA can improve its performance in optimization tasks. Also, in order to improve the fine-tuning characteristic of HSA, some modifications can be applied to set the PAR and BW parameters of HSA. Moreover, to enhance the convergence rate of HSA and reduce the required number of iterations to achieve the optimum solution, HSA can use more than one HM in each iteration. Applying the mentioned modifications to the classic HSA leads to the creating a new optimization algorithm with a more suitable performance called improved harmony search algorithm (IHSA). The IHSA will be explained in section 4.1 in details. Furthermore, to enrich the searching behavior of HSA and also to avoid being trapped into local optimu m solutions, a chaotic pattern can be used instead of uniform distribution for generating the random numbers in steps of HM initialization and improvising new harmonies in the HSA procedure. Using chaotic pattern in HSA leads to the better coverage of the search space and increases the robustness of HSA. HSA with a chaotic pattern for generating the random numbers in the optimization procedure called chaotic harmony search algorithm (CHSA) and will be explained in section 4.2. The combination of IHSA and CHSA results in creating a fast, accurate and robust optimization algorithm which has the advantages of both of them and called chaotic improved harmony search algorithm (CIHSA). The optimization procedures of CIHSA is explained in section 4.3 in details. 4.1 The Improved Harmony Search Algorithm (IHSA) The improved harmony search algorithm (IHSA) has three main differences with the classic HSA; 1- The PAR and BW parameters are changed dynamically according to the number of iteration, 2- Three virtual HMs are employed in the optimization procedure, 3- An additional local optimization step after each iteration is added to the optimization procedure. In the classic HSA, PAR and BW parameters are adjusted in the initialization step as fixed values and cannot be changed during iterations. While to make the HSA performance more suitable, these parameters should be changed dynamically according to the number of iteration. For example, a small value of BW in final stages, which harmonies are close to the optimal solution, increases the fine-tuning of solutions, but in early stages, the value of BW should be relatively large to enforce the algorithm to increase the diversity of solutions. A large value of PAR with a small value of BW usually leads to the improving the best-found solutions in final stages. A small value of PAR with a large value of BW can cause a poor performance of HSA and a significant increase in the required number of iterations to achieve the optimum solution. So, the PAR and BW parameters should be adapted with the optimization procedure. For suitable setting the values of PAR and BW parameters, they are updated in each iteration according to (15) and (16). Notice that the used equations for updating the PAR and BW parameters have been previously used in some other papers such as [12]. PARmax PARmin PAR(it ) PARmin NI it it BWmin BW (it ) BWmax exp Ln BWmax NI (15) (16) where PARmax , PARmin , BWmax and BWmin are the maximum pitch adjusting rate, minimum pitch adjusting rate, maximum bandwidth and minimum bandwidth, respectively. it and NI are the iteration number and the specified maximum number of iterations, respectively. To enhance the convergence rate and reduce the required number of iterations, in each iteration, based on existing HM, four HMs are generated and the “improvising new harmonies” procedure is performed on all of them. In other words, IHSA uses from three virtual HMs. After updating the HMs, their harmonies are merged and sorted, and a new HM is provided with the best solutions found among all solutions, and the extra solutions are discarded. The four HMs in the next iteration, are generated according to this new HM. After providing new HM in each iteration, a local optimization procedure is performed to increase the quality of the solutions. In the local optimization step, two decision variables are selected randomly and their values are replaced with the best possible values. This procedure is repeated N times for each harmony. The local optimization can significantly decrease the required number of iterations to achieve the optimal solution. The details of the local optimization step are explained in section 4.3. 4.2 The Chaotic Harmony Search Algorithm (CHSA) Recently, the idea of using chaotic patterns instead of random sequences is taken into consideration in several fields and suitable results have been shown in many applications such as optimization tasks. Chaos is mathematically defined as generation randomly by simple deterministic systems. Generally, the sensitive dependence on initial conditions, the semi-stochastic property and ergodicity are three main dynamic properties of the chaos. The investigations show that often using chaotic patterns in heuristic optimization techniques leads to the enhancing the searching behavior and it can avoid being trapped into local optimum solutions [14], [43]. There are several one-dimensional chaotic maps such as Logistic map, Tent map, Bernoulli shift map, Liebovitch map, Intermittency map and so on. Among these maps, the logistic map is relatively the simplest one and its average computational time is less than others [43]. In the proposed chaotic algorithms, the logistic map is used to generate the initial population and to improvise new harmonies. The mathematical expression of the logistic map is given in (17) [43]. Yit 1 p Yit 1 Yit for 0 p4 (17) where ‘it’ is the number of iteration and ‘Y’ is the chaotic variable. In the chaotic pattern used in the proposed method, the value of ‘p’ is considered 4, and by defining the initial condition as Yn ∈ (0,1) and Yn ∉ {0,0.25,0.5,0.75,1}, the value of chaotic variables Yn will be distributed between [0, 1]. Figure 1 shows the variation of chaotic variables for 150 iterations with starting value of 0.91. Equation (17) is used for HM initialization, but in the procedure of improvising new harmonies, random numbers between -1 to +1 are needed. So, (17) is modified as (18) to obtain chaotic random numbers between -1 to +1. Yn1 2 4 Yn 1 Yn 0.5 (18) Figure 2 shows the variation of chaotic variables according to Eq. (18) for 150 iterations with starting value of 0.91. 4.3 Combination of IHSA and CHSA The combination of IHSA and CHSA leads to the forming of an optimization technique which benefits from the advantages of both of them and called chaotic improved harmony search algorithm (CIHSA). For example, IHSA has a high accuracy and a very suitable convergence rate. On the other hand, CHSA has a high robustness in the solutions found in different runs. So, CIHSA can provide high-quality solutions with a high accuracy and robustness. In the following, the computational steps of the proposed CIHSA are explained and the flowchart of the CIHSA optimization procedure is presented in Fig. 3. Step 1: Read the system data. In this step, the system data including fuel cost and emission coefficients, generation limits, and total load demand, and also the algorithm parameters including HMS, HMCR, PAR min , PARmax , BWmin , BWmax , NI (specified maximum number of Iterations) and the number of decision variables N (number of generating units) should be read and initialized. Step 2: Initialize the HM according to chaotic selection between P min and Pmax In this step, each harmony of HM should be initialized according to the following equation: xi Li Yn 1 Ui Li (19) where xi is i-th decision variable (generator) of the harmony, and Yn+1 is a chaotic number distributed between 0 and 1 using (17), Ui and Li denote the upper and lower limit of the i-th decision variable, respectively. Since the total generation should be equal to total load demand, after HM initialization, this condition should be checked and satisfied by modifying the decision variables value while the generation limits are not violated. The strategy employed to meet the system power balance constraint is shown in Fig. 4. In this figure, Δ determines our precision in satisfying this constraint. Step 3: Improvise new harmonies In this step, first the PAR and BW value should be updated according to (15) and (16), and four HMs should be generated based on the existing HM. Then, each harmony of any of four HMs should be updated according to the procedure shown in Fig. 3. In the procedure of “improvise new harmonies” shown in Fig. 3, r1 and r2 are two 1×N matrices with random numbers between 0 to 1. If r1(i) (i ϵ {1,2,..,N}) is less than HMCR, then one of the other harmonies will be randomly selected and the value of its i-th decision variable will be considered as the new value of x(i) (x'(i)). If r1(i) is more than HMCR, then x'(i) will be calculated according to (19). If r2(i) (i ϵ {1,2,..,N}) is less than PAR, then the new value of x'(i) (x"(i)) will be calculated according to (20). If r2(i) is more than PAR, then the value of x'(i) will be remained unchanged (x'(i) = x"(i)). x ''i x 'i Yn1 BW (20) where the chaotic variable Yn+1 is obtained from (18). Note that the chaotic patterns used in updating of each HM are different in initial value. After updating each decision variable in each harmony, this variable should be checked to have a value between its minimum and maximum limit. If the decision variable is out of limit, its value will be fixed with the limit. Also, after updating all decision variables in each harmony, the system power balance constraint should be checked and met according to the flowchart given in Fig. 4. After updating the HMs, their harmonies are merged and sorted, and a new HM is provided with the best solutions found among all solutions, and the extra solutions are discarded. The four HMs in the next iteration are generated according to this new HM. Step 4: Local optimization The following procedure will be repeated for N times for each harmony, where N is considered equal to the number of the decision variables (generation units): The r-th and t-th decision variables of the harmony will be selected randomly and by using a local optimization, their value will be replaced with the best possible values according to the procedure represented in Fig. 3. In this figure, ɛ determines the resolution of the search space. Two variables A and B are defined for two reasons; first, to avoid infringement of the generation limits, second, to determine the shortest possible period of search space and reduce the process time consequently. TCba denotes the total cost calculated by fuel cost and emission coefficients related to the a-th generator with the output power of ‘b’ MW. After local optimization in each harmony, the system power balance constraint will be checked again, and the harmony will be modified if it has violated from this constraint. Step 5: Check stopping criteria The above procedure will be repeated from step3 until one of the termination criteria is reached. Termination criteria are: the maximum number of iterations (NI) or no improvement in TC for a speciﬁed number of iterations (SNI). 5 Numerical results and discussions In order to assess the effectiveness of CIHSA, it is applied to solve the CEED problem on six test systems having 6, 10, 13, 14, 40, and 140 generators considering valve-point loading effect. The algorithm has been implemented on a PC with the detailed settings as follow: Hardware: CPU: Intel® Core™ i5-4690, Frequency: 3.50 GHz, RAM: 4.0 GB, Hard drive: 500 GB Software: Operating system: Windows 7 Ultimate, Language: MATLAB 8.1 (R2013a) The results obtained by CIHSA are compared with the classic HSA, CHSA, IHSA and a large number of other optimization algorithms which have been practiced in previous works. In addition, the statistical results regarding the proposed algorithm performance are provided for each test system. The provided comparisons and the presented statistical data clearly indicate the high efficiency, robustness, and accuracy of the proposed algorithm. In the tables provided in this section, fuel cost, emission, total generation, total cost, pure total cost, power transmission losses, and standard deviation have been represented by FC, E, TG, TC, PTC, PL, and SD abbreviations, respectively. 5.1 Test system 1: 6-generator i) In this case, the test system is the standard IEEE 30-bus six-generator system. This power system consists of 21 load buses with 283.40 MW (2.834 per-unit) total load demand and 41 interconnected transmission lines. The generation limits of the units and the fuel cost and the emission coefﬁcients are given in the appendix. To compare the results obtained by the proposed CIHSA with the results reported in previous works, the system is considered lossless. Table 1 presents the best solutions found by CIHSA, including generation output of each unit for ELD, ECD and CEED problem. In this case, the achieved results using four algorithms HSA, CHSA, IHSA, and CIHSA are identical. Also, to investigate whether w=0.5 leads to the obtaining the minimum PTC, the PTC value obtained by CIHSA for different values of w for 20 trial runs is presented in Table 2. According to this table, the minimum PTC is achieved in w=0.5 which has been considered as the CEED case in this work. A comparison between the achieved solutions by CIHSA and by several recently published methods for minimum FC in ELD, and minimum E in ECD problem, is performed in Table 3. According to this table, the results achieved by CIHSA for the minimum fuel cost is less than many of those algorithms and is similar to some of them. Only the Opposition-based Harmony Search algorithm (OHS) [11] has achieved a lower fuel cost. But, according to the reported generation outputs in [11] and the fuel cost coefficients, the fuel cost is calculated as 600.0703416. Also, the reported total generation in [11] for this case is 2.8338 per-unit which is less than the total load demand (2.834 per-unit) and has led to the obtaining a lower fuel cost compared to other algorithms. Also, the achieved results using CIHSA for the minimum emission is the least of all methods with the exception of SOA [32]. However, according to the reported generation outputs in [32] and the emission coefficients, the emission is calculated as 0.195978 which is sufficiently more than the reported value in [32]. It should be noted that the minimum, average and maximum value of the objective functions obtained by the four algorithms for 20 trial runs are equal to each other. It shows that the algorithms have suitable performance and they are absolutely robust in this case. In Table 4, a statistical comparison between the results obtained by CIHSA and by some other algorithms implemented in [29] and [38] is provided. Figure 5 shows the convergence rate of the four algorithms for the fuel cost minimization. According to this figure, the suggested modifications greatly affect the convergence rate of the classic HSA. Figure 6 presents the optimal generation outputs found by CIHSA for ELD, ECD, and CEED, which provides useful information about the generation units. For instance, the units with higher generation in ELD rather than ECD can be considered by approximation as units with low generation cost and high pollution. Those units with lower generation in ELD than ECD can be approximately considered as units with high generation cost and low pollution. ii) In this case, the ELD problem is only considered. The generation limits of the units and the coefﬁcients of the fuel cost and the transmission loss matrices are given in the appendix. Total load demand for this test system is 700 MW. Table 5 gives the optimal solutions found by CIHSA and some other algorithms. It is clear that CIHSA is superior to the algorithms presented in [49], [50] and [51] as the minimum fuel cost achieved by CIHSA is significantly less than the values obtained by those algorithms. It is worth mentioning that the CIHSA performance in this test system is absolutely robust and the minimum fuel cost achieved for 20 trial runs are the same as what is given in Table 5. 5.2 Test system 2: 10-generator This case study consists of ten generation units considering the valve-point effect. The fuel cost and emission coefﬁcients, the generation limit constraints and the transmission loss coefﬁcients matrix are given in the appendix. The generation outputs of the most appropriate solutions for ELD, ECD and CEED problem for 2000 MW load demand are listed in Table 6. A comparison between the solutions found by CIHSA and the results obtained by other algorithms for ELD and ECD is provided in Table 7. According to this table, in ELD, the minimum fuel cost obtained by CIHSA is the least of all other methods, and in ECD, the achieved minimum emission is more than DE [52] and is similar to TLBO [53], QOTLBO [53] and RCCRO [31]. But, it should be mentioned that according to the generation outputs reported in [52] and the emission coefficients, the emission is calculated as 3932.417288 which is more than the value reported in [52] and the minimum emission obtained by CIHSA. As a comparison regarding the CEED solution found by CIHSA, the minimum total cost obtained by FPA [51] is 321,322.571, while the total cost of the optimal solution found by CIHSA is 320,948.532910. For this case, the minimum, maximum, and average total cost achieved by CIHSA for 20 trail runs are 320,948.532910, 320,975.241456, and 320,953.658421, respectively. Thus, even the worth solution found by CIHSA is significantly better than the best solution achieved by FPA [51]. 5.3 Test system 3: 13-generator In this case, the test system includes thirteen thermal power plants considering valve-point loading effect. The system data is given in the appendix. For this test system, in the first case, the total load demand is assumed 1800 MW, for which the best solutions found by CIHSA for ELD, ECD, and CEED problem are listed in Table 8. In the second case, the total load demand is assumed 2520 MW, and the power transmission losses are also considered. The transmission loss matrices are presented in appendix too. The best solution found by CIHSA for ELD problem in this case is also included in Table 8. According to the results, for 2520 MW load demand, the best FC found by CIHSA is 24,512.432933 $/h, which is lower than the values found by other algorithms reported in the previous works such as OIWO (24,514.83 $/h) [44], ORCCRO (24,513.91 $/h) [45], and SDE (24,514.88 $/h) [46]. The PTC value obtained by CIHSA for different values of w for 20 trial runs is given in Table 9, which shows that w=0.5 leads to the obtaining the minimum PTC value. Table 10 shows the solutions of ELD problem for 1800 MW load demand using CIHSA and other algorithms. According to this table, the best fuel cost found by CIHSA is the least among all others. Furthermore, the CIHSA performance is absolutely robust in this case and the worth solution found by CIHSA is better than the best solutions found by other algorithms. Also, as a comparison between the ECD solutions obtained by CIHSA and other algorithms, the minimum emission value found by BBO [33] and RCCRO [31] is reported as 58.241 and 58.2407 ton/h, respectively. In addition, Table 11 presents the statistical data with respect to the results obtained by the four algorithms for 20 trial runs. As it is clear from this table, all the algorithms have suitable performance especially CIHSA which is completely robust in this case. Furthermore, a statistical comparison between the results obtained by CIHSA and by several other algorithms in ELD for this case study has been provided in Table 12, which clearly confirms the superb performance of CIHSA. Figure 7 shows the convergence rate of the four algorithms in fuel cost minimization. In this figure, the suitable effect of the suggested modifications on the convergence rate of the classic HSA is obvious. Figure 8 shows the distribution of the fuel cost achieved by IHSA and CIHSA for this test system for 20 trial runs. According to Fig. 8, employing the chaotic pattern in the optimization procedure of IHSA leads to the enhancing the performance and robustness of the algorithm. 5.4 Test system 4: 14-generator This test system includes fourteen generation units considering valve-point loadings effect and power transmission losses. The system data can be found in the appendix, total load demand is 2000 MW, and only the CEED problem is solved for this case. Table 13 presents the optimal solution found by CIHSA and by some other algorithms. According to this table, the minimum total cost obtained by CIHSA is significantly lower than the others. Also, it is worth mentioning that while in the solutions found by other algorithms, the sum of total load demand and power transmission losses is not exactly equal to the total generation, the system power balance constraint has been met precisely in the solution found by CIHSA that demonstrate the effective constraint handling strategy employed in this algorithm, which leads to the obtaining high quality solutions. For this case, the minimum, maximum, and average total cost achieved by CIHSA for 20 trail runs are 14,454.202397, 14462.352354, and 14456.759621, respectively. These results show the robustness of the proposed algorithm as all solutions found are so close to each other. In this case, the worth solution found by CIHSA is better than the best solutions found by GA, PSO, BBO [63], and DE [64]. 5.5 Test system 5: 40-generator In this test system, the four algorithms are applied to minimize the total cost in a power system with forty thermal power plants considering valve-point loading effects. The system data including the generation limits and the fuel cost and the emission coefﬁcients is given in the appendix, and the total load demand is assumed 10500 MW. Since this is a larger system with more nonlinear elements, it has more local minima, thereby it will be more difﬁcult to achieve the global solution. Table 14 presents the most optimal solutions found by CIHSA for ELD, ECD and CEED problems. The minimum fuel cost in ELD and the minimum emission in ECD is 121,412.536561 $/hr and 176,682.264680 ton/h, respectively. The fuel cost and emission obtained in CEED are 128,726.248081 $/h and 178,577.661404 ton/h, respectively. It is seen that according to the explanations mentioned in section 2, the CEED provides a trade-off between minimum fuel cost and emission. Note that all of the units satisfy the generation limit constraints. Also, the PTC value obtained by CIHSA for different values of w for 20 trial runs is presented in Table 15, which again shows that the minimum PTC value is achieved in w=0.5. The solution found by CIHSA for ELD problem is compared with the results obtained by several other algorithms in Table 16. According to this table, the best fuel cost found by CIHSA is the least among all of the other methods. Also, as a comparison between the ECD solutions obtained by CIHSA and other algorithms, the minimum emission value found by MDE, Tribe-DE, Tribe-MDE [38], NSGA-II and MODE [75] is reported as 176,719.22153, 176,825.6902, 176,682.2646796, 176,691.9677 and 176,683.2718 ton/h, respectively. The statistical data of the results obtained by the four algorithms for 20 trial runs is given in Table 17, and Table 18 presents a statistical comparison between the results obtained by CIHSA and by several other optimization techniques in ELD for this test system, which demonstrates the superiority of the proposed CIHSA compared to those optimization techniques. Figure 9 presents the convergence rate of the four algorithms in ELD. According to Fig. 9, the proposed modifications have great effects on the HSA performance and lead to the improving the efficiency, accuracy and convergence rate of the algorithm. The distribution of the fuel cost achieved by IHSA and CIHSA for this test system for 20 trial runs is shown in Fig. 10. As it is clear in this figure, due to the use of chaotic pattern, CIHSA has more efficiency and robustness in comparison to IHSA. 5.6 Test system 6: 140-generator To better show the effectiveness of CIHSA in large-scale test systems, it is tested for solving the ELD problem in Korean power system which consists of 140 generators. The system data can be found in the appendix. The minimum, maximum, and average value of the solutions found for 20 trial runs are equal to each other and less than the reported results in the previous works, which clearly demonstrates the highquality performance and high robustness of the proposed CIHSA in large-scale test systems. The results obtained by CIHSA and some other algorithms for this test system are given in Table 19. In addition, Table 20 presents the scheduled output power for each generation unit in the optimal solution found by CIHSA for this test system. 6 Conclusion This paper presented a novel optimization technique for solving the CEED problems with several operational considerations of the power system. In the proposed algorithm, the classic HSA has been modified by several innovative modifications that result in enhancing the efficiency, accuracy, and robustness of the algorithm and decreasing the required number of iterations to achieve the optimal solution. The proposed CIHSA was employed in solving generation scheduling problem in six test systems to minimize the system fuel cost and pollutant emissions. The solutions found by CIHSA were compared with the results reported in several other research works. The numerical and statistical results clearly demonstrated the high efficiency of CIHSA and its superiority compared to other existing optimization techniques in solving CEED problems. References [1] Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2008). Self-organizing hierarchical particle swarm optimization for nonconvex economic dispatch. IEEE transactions on power systems, 23(3), 1079-1087. [2] Elsayed, W. T., Hegazy, Y. G., El-bages, M. S., & Bendary, F. M. (2017). Improved random drift particle swarm optimization with self-adaptive mechanism for solving the power economic dispatch problem. IEEE Transactions on Industrial Informatics, 13(3), 1017-1026. [3] Hosseinnezhad, V., Rafiee, M., Ahmadian, M., & Ameli, M. T. (2014). Species-based quantum particle swarm optimization for economic load dispatch. International Journal of Electrical Power & Energy Systems, 63, 311-322. [4] Niknam, T. (2010). A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, nonsmooth and non-convex economic dispatch problem. Applied Energy, 87(1), 327-339. [5] Selvakumar, A. I., & Thanushkodi, K. (2007). A new particle swarm optimization solution to nonconvex economic dispatch problems. IEEE transactions on power systems, 22(1), 42-51. [6] Wang, G. G., Gandomi, A. H., Alavi, A. H., & Deb, S. (2016). A hybrid method based on krill herd and quantum-behaved particle swarm optimization. Neural Computing and Applications, 27(4), 989-1006. [7] Wang, G. G., Hossein Gandomi, A., Yang, X. S., & Hossein Alavi, A. (2014). A novel improved accelerated particle swarm optimization algorithm for global numerical optimization. Engineering Computations, 31(7), 11981220. [8] Wang, G. G., Hossein Gandomi, A., & Hossein Alavi, A. (2013). A chaotic particle-swarm krill herd algorithm for global numerical optimization. Kybernetes, 42(6), 962-978. [9] Wang, G. G., Gandomi, A. H., Zhao, X., & Chu, H. C. E. (2016). Hybridizing harmony search algorithm with cuckoo search for global numerical optimization. Soft Computing, 20(1), 273-285. [10] Wang, G., Guo, L., Duan, H., Wang, H., Liu, L., & Shao, M. (2013). Hybridizing harmony search with biogeography based optimization for global numerical optimization. Journal of Computational and Theoretical Nanoscience, 10(10), 2312-2322. [11] Chatterjee, A., Ghoshal, S. P., & Mukherjee, V. (2012). Solution of combined economic and emission dispatch problems of power systems by an opposition-based harmony search algorithm. International Journal of Electrical Power & Energy Systems, 39(1), 9-20. [12] Mahdavi, M., Fesanghary, M., & Damangir, E. (2007). An improved harmony search algorithm for solving optimization problems. Applied mathematics and computation, 188(2), 1567-1579. [13] dos Santos Coelho, L., & Mariani, V. C. (2009). An improved harmony search algorithm for power economic load dispatch. Energy Conversion and Management, 50(10), 2522-2526. [14] Arul, R., Ravi, G., & Velusami, S. (2013). Chaotic self-adaptive differential harmony search algorithm based dynamic economic dispatch. International Journal of Electrical Power & Energy Systems, 50, 85-96. [15] Geem, Z. W. (2008). Novel derivative of harmony search algorithm for discrete design variables. Applied mathematics and computation, 199(1), 223-230. [16] Geem, Z. W. (2010). Stochastic co-derivative of harmony search algorithm. International Journal of Mathematical Modelling and Numerical Optimisation, 2(1), 1-12. [17] Wang, G. G., Guo, L., Gandomi, A. H., Hao, G. S., & Wang, H. (2014). Chaotic krill herd algorithm. Information Sciences, 274, 17-34. [18] Wang, G. G., Gandomi, A. H., & Alavi, A. H. (2014). An effective krill herd algorithm with migration operator in biogeography-based optimization. Applied Mathematical Modelling, 38(9-10), 2454-2462. [19] Wang, G. G., Gandomi, A. H., & Alavi, A. H. (2014). Stud krill herd algorithm. Neurocomputing, 128, 363370. [20] Wang, G., Guo, L., Wang, H., Duan, H., Liu, L., & Li, J. (2014). Incorporating mutation scheme into krill herd algorithm for global numerical optimization. Neural Computing and Applications, 24(3-4), 853-871. [21] Wang, H., & Yi, J. H. (2018). An improved optimization method based on krill herd and artificial bee colony with information exchange. Memetic Computing, 10(2), 177-198. [22] Wang, G. G., Gandomi, A. H., Yang, X. S., & Alavi, A. H. (2016). A new hybrid method based on krill herd and cuckoo search for global optimisation tasks. International Journal of Bio-Inspired Computation, 8(5), 286-299. [23] Wang, G. G., Deb, S., Gandomi, A. H., & Alavi, A. H. (2016). Opposition-based krill herd algorithm with Cauchy mutation and position clamping. Neurocomputing, 177, 147-157. [24] Wang, G. G., Gandomi, A. H., Alavi, A. H., & Hao, G. S. (2014). Hybrid krill herd algorithm with differential evolution for global numerical optimization. Neural Computing and Applications, 25(2), 297-308. [25] Guo, L., Wang, G. G., Gandomi, A. H., Alavi, A. H., & Duan, H. (2014). A new improved krill herd algorithm for global numerical optimization. Neurocomputing, 138, 392-402. [26] Kim, J., Kim, C. S., & Geem, Z. W. (2014). A memetic approach for improving minimum cost of economic load dispatch problems. Mathematical Problems in Engineering, 2014. [27] Geem, Z. W. (2013). Economic dispatch using parameter-setting-free harmony search. Journal of Applied Mathematics, 2013. [28] Rezaie, H., Moosavy Chashmi, S. M., Mirsalim, M., & Rastegar, H. (2017). Enhancing LVRT Capability and Smoothing Power Fluctuations of a DFIG-based Wind Farm in a DC Microgrid. Electric Power Components and Systems, 45(10), 1080-1090. [29] Jiang, S., Ji, Z., & Wang, Y. (2015). A novel gravitational acceleration enhanced particle swarm optimization algorithm for wind–thermal economic emission dispatch problem considering wind power availability. International Journal of Electrical Power & Energy Systems, 73, 1035-1050. [30] Jevtic, M., Jovanovic, N., Radosavljevic, J., & Klimenta, D. (2017). Moth Swarm Algorithm for Solving Combined Economic and Emission Dispatch Problem. Elektronika ir Elektrotechnika, 23(5), 21-28. [31] Bhattacharjee, K., Bhattacharya, A., & nee Dey, S. H. (2014). Solution of economic emission load dispatch problems of power systems by real coded chemical reaction algorithm. International Journal of Electrical Power & Energy Systems, 59, 176-187. [32] Benasla, L., Belmadani, A., & Rahli, M. (2014). Spiral optimization algorithm for solving combined economic and emission dispatch. International Journal of Electrical Power & Energy Systems, 62, 163-174. [33] Rajasomashekar, S., & Aravindhababu, P. (2012). Biogeography based optimization technique for best compromise solution of economic emission dispatch. Swarm and Evolutionary Computation, 7, 47-57. [34] Radosavljević, J. (2016). A solution to the combined economic and emission dispatch using hybrid PSOGSA algorithm. Applied Artificial Intelligence, 30(5), 445-474. [35] Abido, M. A. (2009). Multiobjective particle swarm optimization for environmental/economic dispatch problem. Electric Power Systems Research, 79(7), 1105-1113. [36] Abido, M. A. (2006). Multiobjective evolutionary algorithms for electric power dispatch problem. IEEE transactions on evolutionary computation, 10(3), 315-329. [37] Sivasubramani, S., & Swarup, K. S. (2011). Environmental/economic dispatch using multi-objective harmony search algorithm. Electric power systems research, 81(9), 1778-1785. [38] Niknam, T., Mojarrad, H. D., & Firouzi, B. B. (2013). A new optimization algorithm for multi-objective economic/emission dispatch. International Journal of Electrical Power & Energy Systems, 46, 283-293. [39] Panigrahi, B. K., Pandi, V. R., Das, S., & Das, S. (2010). Multiobjective fuzzy dominance based bacterial foraging algorithm to solve economic emission dispatch problem. Energy, 35(12), 4761-4770. [40] King, R. T. A., Rughooputh, H. C., & Deb, K. (2005, March). Evolutionary multi-objective environmental/economic dispatch: Stochastic versus deterministic approaches. In International Conference on Evolutionary Multi-Criterion Optimization (pp. 677-691). Springer, Berlin, Heidelberg. [41] Arya, L. D., Choube, S. C., & Kothari, D. P. (1997). Emission constrained secure economic dispatch. International journal of electrical power & energy systems, 19(5), 279-285. [42] Güvenç, U., Sönmez, Y., Duman, S., & Yörükeren, N. (2012). Combined economic and emission dispatch solution using gravitational search algorithm. Scientia Iranica, 19(6), 1754-1762. [43] Tavazoei, M. S., & Haeri, M. (2007). Comparison of different one-dimensional maps as chaotic search pattern in chaos optimization algorithms. Applied Mathematics and Computation, 187(2), 1076-1085. [44] Barisal, A. K., & Prusty, R. C. (2015). Large scale economic dispatch of power systems using oppositional invasive weed optimization. Applied Soft Computing, 29, 122-137. [45] Bhattacharjee, K., Bhattacharya, A., & nee Dey, S. H. (2014). Oppositional real coded chemical reaction optimization for different economic dispatch problems. International Journal of Electrical Power & Energy Systems, 55, 378-391. [46] Reddy, A. S., & Vaisakh, K. (2013). Shuffled differential evolution for large scale economic dispatch. Electric Power Systems Research, 96, 237-245. [47] Niknam, T., Mojarrad, H. D., & Meymand, H. Z. (2011). A novel hybrid particle swarm optimization for economic dispatch with valve-point loading effects. Energy Conversion and Management, 52(4), 1800-1809. [48] Gong, D. W., Zhang, Y., & Qi, C. L. (2010). Environmental/economic power dispatch using a hybrid multiobjective optimization algorithm. International Journal of Electrical Power & Energy Systems, 32(6), 607-614. [49] Serapião, A. B. (2013). Cuckoo search for solving economic dispatch load problem. Intelligent Control and Automation, 4(04), 385. [50] Serapião, A. B. D. S. (2009). Fundamentos de otimização por inteligência de enxames: uma visão geral. Sba: Controle & Automação Sociedade Brasileira de Automatica, 20(3), 271-304. [51] Abdelaziz, A. Y., Ali, E. S., & Elazim, S. A. (2016). Combined economic and emission dispatch solution using flower pollination algorithm. International Journal of Electrical Power & Energy Systems, 80, 264-274. [52] Basu, M. (2011). Economic environmental dispatch using multi-objective differential evolution. Applied soft computing, 11(2), 2845-2853. [53] Roy, P. K., & Bhui, S. (2013). Multi-objective quasi-oppositional teaching learning based optimization for economic emission load dispatch problem. International Journal of Electrical Power & Energy Systems, 53, 937948. [54] Roy, P. K., Bhui, S., & Paul, C. (2014). Solution of economic load dispatch using hybrid chemical reaction optimization approach. Applied Soft Computing, 24, 109-125. [55] Mohammadi-Ivatloo, B., Rabiee, A., Soroudi, A., & Ehsan, M. (2012). Iteration PSO with time varying acceleration coefficients for solving non-convex economic dispatch problems. International journal of electrical power & energy systems, 42(1), 508-516. [56] Hemamalini, S., & Simon, S. P. (2009). Maclaurin series-based Lagrangian method for economic dispatch with valve-point effect. IET generation, transmission & distribution, 3(9), 859-871. [57] Kumar, R., Sharma, D., & Sadu, A. (2011). A hybrid multi-agent based particle swarm optimization algorithm for economic power dispatch. International journal of electrical power & energy systems, 33(1), 115-123. [58] Al-Sumait, J. S., Sykulski, J. K., & Al-Othman, A. K. (2008). Solution of different types of economic load dispatch problems using a pattern search method. Electric Power Components and Systems, 36(3), 250-265. [59] Meng, K., Wang, H. G., Dong, Z., & Wong, K. P. (2010). Quantum-inspired particle swarm optimization for valve-point economic load dispatch. IEEE transactions on power systems, 25(1), 215-222. [60] Wang, S. K., Chiou, J. P., & Liu, C. W. (2007). Non-smooth/non-convex economic dispatch by a novel hybrid differential evolution algorithm. IET Generation, Transmission & Distribution, 1(5), 793-803. [61] Chaturvedi, K. T., Pandit, M., & Srivastava, L. (2009). Particle swarm optimization with time varying acceleration coefficients for non-convex economic power dispatch. International Journal of Electrical Power & Energy Systems, 31(6), 249-257. [62] He, D., Wang, F., & Mao, Z. (2008). A hybrid genetic algorithm approach based on differential evolution for economic dispatch with valve-point effect. International journal of electrical power & energy systems, 30(1), 31-38. [63] Roy, P. K., Ghoshal, S. P., & Thakur, S. S. (2010). Combined economic and emission dispatch problems using biogeography-based optimization. Electrical Engineering, 92(4-5), 173-184. [64] Mandal, K. K., & Chakraborty, N. (2008). Effect of control parameters on differential evolution based combined economic emission dispatch with valve-point loading and transmission loss. International Journal of Emerging Electric Power Systems, 9(4). [65] Selvakumar, A. I., & Thanushkodi, K. (2009). Optimization using civilized swarm: solution to economic dispatch with multiple minima. Electric Power Systems Research, 79(1), 8-16. [66] Victoire, T. A. A., & Jeyakumar, A. E. (2004). Hybrid PSO–SQP for economic dispatch with valve-point effect. Electric Power Systems Research, 71(1), 51-59. [67] Subathra, M. S. P., Selvan, S. E., Victoire, T. A. A., Christinal, A. H., & Amato, U. (2015). A hybrid with cross-entropy method and sequential quadratic programming to solve economic load dispatch problem. IEEE Systems Journal, 9(3), 1031-1044. [68] Murali, K., & Jayabarathi, T. (2014). Solution to economic dispatch problem with valve-point loading effect by using catfish PSO algorithm. Frontiers in Energy, 8(3), 290-296. [69] Binetti, G., Davoudi, A., Naso, D., Turchiano, B., & Lewis, F. L. (2014). A distributed auction-based algorithm for the nonconvex economic dispatch problem. IEEE Transactions on Industrial Informatics, 10(2), 1124-1132. [70] Bhattacharya, A., & Chattopadhyay, P. K. (2010). Biogeography-based optimization for different economic load dispatch problems. IEEE transactions on power systems, 25(2), 1064-1077. [71] Elsayed, W. T., Hegazy, Y. G., Bendary, F. M., & El-bages, M. S. (2016). Modified social spider algorithm for solving the economic dispatch problem. Engineering science and technology, an international journal, 19(4), 16721681. [72] Elsayed, W. T., & El-Saadany, E. F. (2015). A fully decentralized approach for solving the economic dispatch problem. IEEE Transactions on Power Systems, 30(4), 2179-2189. [73] Moradi-Dalvand, M., Mohammadi-Ivatloo, B., Najafi, A., & Rabiee, A. (2012). Continuous quick group search optimizer for solving non-convex economic dispatch problems. Electric Power Systems Research, 93, 93-105. [74] Adarsh, B. R., Raghunathan, T., Jayabarathi, T., & Yang, X. S. (2016). Economic dispatch using chaotic bat algorithm. Energy, 96, 666-675. [75] Sharma, R., Samantaray, P., Mohanty, D. P., & Rout, P. K. (2011, December). Environmental economic load dispatch using multi-objective differential evolution algorithm. In Energy, Automation, and Signal (ICEAS), 2011 International Conference on (pp. 1-7). IEEE. [76] Park, J. B., Jeong, Y. W., Shin, J. R., & Lee, K. Y. (2010). An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Transactions on Power Systems, 25(1), 156-166. Table1. Best achieved solutions for test system 1-i ELD ECD CEED P1 0.10971930 0.40607387 0.26832394 P2 0.29976607 0.45906893 0.37942508 P3 0.52429825 0.53793856 0.53956257 P4 1.01619883 0.38295303 0.67117460 P5 0.52429826 0.53793855 0.53956256 P6 0.35971928 0.51002706 0.43595125 FC 600.111408 638.273440 611.130692 E 0.222145 0.194203 0.199906 TC 600.111408 317.940559 469.204431 Table 2. The PTC values obtained by CIHSA for different values of W for test system 1-i W 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Min PTC 956.213999 952.941891 949.986200 947.350846 945.040522 943.060847 941.418539 Max PTC 956.213999 952.941893 949.986204 947.350848 945.040523 943.060848 941.418540 Avg PTC 956.213999 952.941892 949.986202 947.350847 945.040522 943.060847 941.418539 W 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Min PTC 940.121623 939.179684 938.604188 938.408863 938.610208 939.228140 940.286850 Max PTC 940.121623 939.179685 938.604188 938.408863 938.610208 939.228141 940.286850 Avg PTC 940.121623 939.179685 938.604188 938.408863 938.610208 939.228140 940.286850 W 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Min PTC 941.815950 943.852064 946.441037 949.641147 953.527864 958.201170 963.797487 Max PTC 941.815952 943.852065 946.441039 949.641149 953.527865 958.201171 963.797487 Avg PTC 941.815951 943.852065 946.441038 949.641148 953.527864 958.201170 963.797487 Table 3. Comparison of the minimum FC in ELD, and minimum E in ECD, obtained by different algorithms for test system 1-i MOHS Method GSA [29] DE [38] SOA [32] NPGA GAEPSO NSBF SPEA MDE [36] [29] [39] [36] [38] NSGA [36] [37] FCELD 602.2311 601.3428 600.986 600.6909 600.34 600.31 600.2978 600.2704 600.22 600.173 EECD 0.1954 0.194217 0.18729* 0.1947 0.1946 0.1943 0.1942 0.1944 0.1942 0.194208 NSGA-II MOPSO PSOGSA FSBF Tribe- [34] [39] MDE [38] MOMethod DE/PSO [40] MSA [30] [35] OHS [11] CIHSA [48] FCELD 600.155 600.12 600.115 600.11141 600.11141 600.1141 600.1114 600* 600.111408 EECD 0.1942 0.1942 0.194203 0.19420 0.194203 0.1942 0.194202 0.1942 0.194203 * The calculated fuel cost according to the reported generation outputs in [11] and the fuel cost coefficients is equal to 600.0703416. * The calculated emission according to the reported generation outputs in [32] and the emission coefficients is equal to 0.195978. Table 4. Statistical comparison between the results obtained by different algorithms in ELD and ECD problem for test system 1-i ELD ECD Method Min FC Avg FC Max FC SD Min E Avg E Max E SD CIHSA 600.111408 600.111408 600.111408 0 0.194203 0.194203 0.194203 0 Tribe-MDE [38] 600.1114 600.1114 600.1114 0 0.194202 0.194202 0.194202 0 MDE [38] 600.173 600.186 600.2214 2.3481 0.194206 0.194212 0.194219 4.25 GAEPSO [29] 600.2978 600.2997 600.3090 2.5601e-3 0.1942 0.1942 0.1942 4.0891e-9 PSO [29] 600.5692 603.1347 607.1275 1.5618 0.1947 0.1964 0.1990 1.1037e-3 DE [38] 601.3428 601.7951 602.1126 6.8624 0.194217 0.194225 0.194241 9.781 GSA [29] 602.2311 609.0541 615.2996 3.4189 0.1954 0.2004 0.2057 2.6205e-3 Table 5. Best solutions found for test system 1-ii in ELD by different algorithms Unit P1 P2 P3 P4 P5 P6 PL FC BFO [50] 222.26 58.777 150.395 106.963 101.601 72.559 11.73 8428.69 SFL [50] 287.392 67.637 140.933 98.357 64.052 53.15 11.59 8419.78 PSO [50] 288.653 82.753 132.988 50 99.565 57.768 11.73 8401.45 HS [49] FA [49] NA NA NA NA NA NA NA 8398.06 293.312 79.546 123.334 69.7 79.546 63.778 11.44 8388.45 ABC [49] 323.043 54.965 147.354 50 85.815 50.233 11.4 8372.27 CS [49] 324.113 76.859 158.094 50 51.963 50 11.03 8356.06 FPA [51] 323.995 76.846 158.2 50 51.983 50 11.024 8356.05 CIHSA 303.521476 113.817531 143.442499 50.000296 50.004526 50.000000 10.786326 8313.222466 Table 6. Best solutions found by CIHSA for test system 2 Unit P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 TL TG FC E TC ELD 55.000000 80.000000 106.934727 100.600317 81.476793 83.026871 300.000000 340.000000 470.000000 470.000000 87.038709 2087.038709 111,497.630981 4572.276303 - ECD 55.000000 80.000000 81.149904 81.359769 160.000000 240.000000 294.507931 297.268922 396.720288 395.587840 81.594656 2081.594656 116,412.565528 3932.243301 - CEED 55.000000 80.000000 81.081501 80.930292 160.000000 240.000000 290.800949 296.689692 398.842744 398.331226 81.676404 2081.676404 116,390.278321 3932.447341 320,948.532910 Table 7. Comparison of the solutions found by different algorithms for ELD problem Method DE [52] TLBO [53] FCELD EECD 111,500 3923.4* 111,500 3932.2 QOTLBO [53] 111,498 3932.2 RCCRO [31] 111,497.6319 3932.2433 CIHSA 111497.630981 3932.243301 * The calculated emission according to the reported generation outputs in [52] and the emission coefficients is equal to 3932.417288. Table 8. Best solutions found by CIHSA for test system 3 Demand 2520MW Demand 1800MW ELD ELD ECD CEED P1 628.318526 628.318531 80.640745 89.759790 P2 299.199291 149.599650 166.328710 149.599650 P3 297.367318 222.749069 166.328709 158.133171 P4 159.733097 109.866550 154.733174 159.733100 P5 159.733094 109.866550 154.733172 159.733100 P6 159.733094 60.000000 154.733174 159.733100 P7 159.733099 109.866550 154.733174 159.733100 P8 159.733095 109.866550 154.733174 159.733100 P9 159.733094 109.866550 154.733173 159.733100 P10 77.399902 40.000000 119.963737 114.799825 P11 114.799811 40.000000 119.963737 114.799825 P12 92.399910 55.000000 109.187661 94.509138 P13 92.399902 55.000000 109.187660 120.000000 PL 40.283232 0.000000 0.000000 0.000000 FC 24512.432933 17960.366122 19113.256777 18376.521665 E 526.302229 461.480560 58.240712 58.737659 TC 24512.432933 17960.366122 16779.772577 17649.734958 Table 9. The PTC values obtained by CIHSA for different values of W for test system 3 w 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Min PTC 35893.029326 35763.443948 35613.244855 35459.469547 35388.708514 35350.704459 35348.552378 Max PTC 35893.029408 35763.444003 35613.244896 35459.469587 35388.708531 35350.704462 35348.552380 Avg PTC 35893.029386 35763.443974 35613.244874 35459.469571 35388.708520 35350.704461 35348.552379 W 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Min PTC 35344.922472 35300.685247 35299.884610 35299.469975 35300.193306 35300.358787 35300.358836 Max PTC 35344.922482 35300.688353 35299.899802 35299.469980 35300.193306 35300.358845 35300.358845 Avg PTC 35344.922477 35300.686124 35299.887725 35299.469977 35300.193306 35300.358810 35300.358843 w 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Min PTC 35300.358845 35300.358845 35819.693512 35819.693514 35819.693819 35819.694193 150917.855090 Max PTC 35300.358845 35300.358902 35819.693512 35819.694166 35819.694190 35819.694193 150917.855137 Avg PTC 35300.358845 35300.358866 35819.693512 35819.693805 35819.694047 35819.694193 150917.855099 Table 10. The minimum fuel costs obtained by different algorithms in ELD problem for test system 3 with 1800 load demand PSO-TVAC Method MSL [56] HMAPSO[57] PSM [58] QPSO [59] SHDE [60] HGA [62] SDE [46] 17,963.83 17,963.83 [61] FC 18,158.68 17,969.31 RCCRO FAPSO-VDE Method FC [31] [47] 17963.8292 17,963.82 17,969.17 17,969.01 DE [54] CRO [54] 17,961.6108 17,961.0703 17,963.89 17,963.879 HCRO-DE IPSO- [54] TVAC [55] 17,960.3820 17,960.3703 CIHSA 17960.366122 Table 11. Statistical data regarding the solutions found by the four algorithms for test system 3 HSA CHSA IHSA CIHSA Min FC 17960.510699 17960.366729 17960.366148 17960.366122 Avg FC 17976.978559 17972.844383 17960.366153 17960.366122 Max FC 17986.249765 17985.848755 17960.366159 17960.366122 SD 11.248992 9.172839 0.000003 0.000000 Min E 58.240712 58.240712 58.240712 58.240712 Avg E 58.240712 58.240712 58.240712 58.240712 Max E 58.240712 58.240712 58.240712 58.240712 SD 0.000000 0.000000 0.000000 0.000000 Min TC 17649.734948 17649.734944 17649.735008 17649.734958 Avg TC 17649.760961 17649.756915 17649.735015 17649.734983 Max TC 17650.015798 17649.916931 17649.735024 17649.734990 SD 0.061504 0.041899 0.000009 0.000004 ELD ECD CEED Table 12. Statistical comparison between the results obtained by different algorithms in ELD problem for test system 3 Method MSL [56] HMAPSO [57] PSM [58] QPSO [59] SHDE [60] PSO-TVAC [61] HGA [62] Min FC 18,158.68 17,969.31 17,969.17 17,969.01 17,963.89 17,963.88 17,963.83 Avg FC NA 17,969.31 18,088.84 18,075.11 18,046.38 18,154.56 17,988.04 Max FC NA 17,969.31 18,233.52 NA NA 18,358.31 NA NA NA SD NA NA NA NA NA Method FAPSO-VDE [47] DE [54] CRO [54] HCRO-DE [54] IPSO-TVAC [55] CIHSA Min FC 17,963.82 17,961.61 17,961.07 17,960.38 17,960.37 17,960.366122 Avg FC 17,963.83 17,963.83 17,962.77 17,960.59 17,960.64 17,960.366122 Max FC 17,963.83 17,980.35 17,974.82 17,961.04 17,961.27 17,960.366122 SD NA 1.44 1.18 0.069 NA 0 Table 13. Best solutions found by different algorithms for test system 4 Method P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 TG TL FC E TC GA [63] 264.0103 150.0000 102.7986 119.6997 200.0000 277.2228 238.0072 157.7048 124.4686 136.7911 65.0000 79.7628 62.5000 52.3999 2030.4 30.3658 9889.3 3351.8 15017.3 PSO [63] 239.7598 150.0000 129.9988 119.7331 150.0000 281.3344 184.8666 159.7331 161.9994 160.0000 79.9986 80.0000 85.0000 52.3999 2034.8 34.8236 10166.0 2990.6 14941.0 DE [64] 239.76 150.01 95.21 119.75 199.86 284.59 234.86 159.73 124.89 137.32 66.65 79.95 84.97 52.43 2030.0 29.98 9592.4 3232.2 14537.0 BBO [63] 239.7594 150.0000 126.7956 119.7331 150.0000 284.5995 184.8665 159.7331 161.8835 160.0000 80.0000 80.0000 85.0000 52.3999 2034.8 34.7706 9874.3 2999.0 14462.5 CIHSA 239.751073 150.000001 128.900000 119.733137 199.800031 234.730936 184.866516 159.726441 162.000000 160.000000 80.000000 80.000000 85.000000 52.400000 2036.908134 36.908134 9809.081794 3036.134581 14454.202397 Table 14. Best solutions found by CIHSA for test system 5 Unit ELD ECD CEED Unit ELD ECD CEED P1 110.799824 114.000000 114.000000 P21 523.279369 439.446403 437.467746 P2 110.799821 114.000000 114.000000 P22 523.279368 439.446404 437.467748 P3 97.399908 120.000000 120.000000 P23 523.279370 439.772065 437.976522 P4 179.733098 169.368008 178.225888 P24 523.279370 439.772067 437.976521 P5 87.799902 97.000000 97.000000 P25 523.279368 440.111764 437.759425 P6 140.000000 124.257413 129.443556 P26 523.279366 440.111764 437.759419 P7 259.599646 299.711393 300.000000 P27 10.000000 28.993702 19.531198 P8 284.599639 297.914857 299.540696 P28 10.000000 28.993704 19.531198 P9 284.599647 297.260102 298.627148 P29 10.000000 28.993703 19.531201 P10 130.000000 130.000000 130.000000 P30 87.799899 97.000000 97.000000 P11 94.000000 298.410143 307.573033 P31 190.000000 172.331904 175.807155 P12 94.000000 298.026013 307.001570 P32 190.000000 172.331904 175.807156 P13 214.759789 433.557638 433.807799 P33 190.000000 172.331903 175.807158 P14 394.279365 421.728406 408.955374 P34 164.799820 200.000000 200.000000 P15 394.279361 422.779651 411.450301 P35 194.397895 200.000000 200.000000 P16 394.279360 422.779650 411.450298 P36 199.999989 200.000000 200.000000 P17 489.279365 439.412855 452.156805 P37 109.999993 100.838377 104.255068 P18 489.279366 439.402887 452.179572 P38 110.000000 100.838377 104.255066 P19 511.279371 439.412855 437.466768 P39 109.999999 100.838377 104.255067 P20 511.279370 439.412854 437.466768 P40 511.279366 439.412857 437.466772 ELD ECD CEED FC 121,412.536561 129,995.271365 128,726.248081 E 359,901.367106 176,682.264680 178,577.661404 TC - - 95,790.897555 Table 15. The PTC values obtained by CIHSA for different values of W for test system 5 w 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Min PTC 192183.670224 192121.792709 192059.937736 191995.546995 191928.509235 191859.977989 191790.486531 Max PTC 192183.929224 192121.996468 192060.278195 191995.624038 191928.739017 191860.062080 191790.682177 Avg PTC 192183.781361 192121.914493 192060.128025 191995.584579 191928.623352 191860.029794 191790.573188 W 0.35 0.40 0.45 0.50 0.55 0.60 0.65 Min PTC 191722.075166 191658.196138 191605.964745 191581.795179 191624.175129 191798.972333 192141.399370 Max PTC 191722.151665 191658.263990 191605.995518 191581.795266 191624.233307 191799.061231 192141.650122 Avg PTC 191722.108907 191658.222692 191605.973925 191581.795234 191624.199095 191799.009589 192141.563575 w 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Min PTC 192989.513462 193985.622434 194468.657787 195083.252153 201072.283619 243632.698809 248090.164140 Max PTC 192989.982299 193986.387450 194468.740024 195083.254160 201072.292447 243632.715432 248090.172770 Avg PTC 192989.772353 193985.858881 194468.717847 195083.253166 201072.288762 243632.707784 248090.169535 Table 16. The minimum fuel costs obtained by different algorithms in ELD problem for test system 5 Method SCA [65] EP-SQP [66] PSO-SQP [67] PSO [68] AA (Dist.) [69] FAPSO [4] Catfish PSO [68] FC 122,713.6828 122,323.97 122,094.67 121,818.04 121,788.7 121,712.4 121,683.70 Method NPSO-LRS [5] SOH-PSO [1] CSO [65] QPSO [3] BBO [70] FAPSO-NM [4] MSSA [71] FC 121,664.4308 121,501.14 121,461.6707 121,448.21 121,426.953 121,418.3 121,413.4686 Method CE-SQP [67] DE [72] Tribe-MDE [38] SQPSO [3] CQGSO [73] CBA [74] CIHSA FC 12,1412.88 121,412.68 121,412.5704 121,412.57 121,412.5512 121,412.5468 121,412.536561 Table 17. Statistical data regarding the solutions found by the four algorithms for test system 5 HSA CHSA IHSA CIHSA Min FC 121,748.313700 121,747.486373 121,412.537474 121,412.536561 Avg FC 121,823.084480 121,782.368421 121,417.134018 121,413.373697 Max FC 122,192.942093 122,185.721380 121,420.895493 121,420.896252 SD 157.536309 97.133565 4.266395 2.572547 Min E 176,688.401224 176,684.339661 176,682.264787 176,682.264680 Avg E 176,697.526418 176,687.732416 176,682.264843 176,682.264680 Max E 176,710.689257 176,701.074282 176,682.264899 176,682.264680 SD 5.400849 3.613488 3.255279 0.000000 Min TC 95,791.739657 95,791.184559 95,790.897599 95,790.897555 Avg TC 95,794.250667 95,792.056144 95,790.897612 95,790.897555 Max TC 95,812.516821 95,793.345271 95,790.897623 95,790.897555 SD 4.447689 0.669487 0.000009 0.000000 ELD ECD CEED Table 18. Statistical comparison between the results obtained by different algorithms in ELD problem for test system 5 Method SCA [65] EP-SQP [66] PSO-SQP [67] NPSO-LRS [5] SOH-PSO [1] CSO [65] QPSO [3] Min FC 122,713.6828 122,323.97 122,094.67 121,664.4308 121,501.14 121,461.6707 121,448.21 Avg FC 125,235.1288 122,379.63 122,245.25 122,209.3185 121,853.57 121,936.1926 122,225.07 Max FC 130,918.3914 NA NA 122,981.5913 122,446.3 NA 121,994.0267 SD NA NA NA NA NA 32 114.08 Method BBO [70] CE-SQP [67] DE [72] SQPSO [3] CBA [74] CIHSA Min FC 121,426.953 12,1412.88 121,412.68 121,412.57 121,412.5468 121,412.536561 Avg FC 121,508.0325 121,423.65 121,439.89 121,455.7 121,418.9826 121,413.373697 Max FC 121,688.6634 NA 121,479.63 121,709.5582 121,436.15 121,420.896252 SD NA NA NA 49.8076 NA 2.572547 Table 19. Statistical comparison between the results obtained by different algorithms in ELD problem for test system 6 CCPSO [76] CTPSO [76] GSO [73] CQGSO [73] CIHSA Min FC 1,655,685 1,655,685 1,734,405.7923 1,655,679.426 1,655,679.425866 Max FC 1,655,685 1,655,685 1,745,315.0894 1,655,679.430 1,655,679.425866 Avg FC 1,655,685 1,655,685 1,739,400.0802 1,655,679.428 1,655,679.425866 Table 20. The optimal generation outputs determined by CIHSA in ELD problem for test system 6 Unit Power Unit Power Unit Power Unit Power Unit Power P1 119.000000 P29 501.000000 P57 103.000000 P85 115.000000 P113 94.000000 P2 164.000000 P30 499.000000 P58 198.000000 P86 207.000000 P114 94.000000 P3 190.000000 P31 506.000000 P59 312.000000 P87 207.000000 P115 244.000000 P4 190.000000 P32 506.000000 P60 308.589343 P88 175.000000 P116 244.000000 P5 190.000000 P33 506.000000 P61 163.000000 P89 175.000000 P117 244.000000 P6 190.000000 P34 506.000000 P62 95.000000 P90 180.423911 P118 95.000000 P7 490.000000 P35 500.000000 P63 511.000000 P91 175.000000 P119 95.000000 P8 490.000000 P36 500.000000 P64 511.000000 P92 575.400000 P120 116.000000 P9 496.000000 P37 241.000000 P65 490.000000 P93 547.500000 P121 175.000000 P10 496.000000 P38 241.000000 P66 256.825727 P94 836.800000 P122 2.000000 P11 496.000000 P39 774.000000 P67 490.000000 P95 837.500000 P123 4.000000 P12 496.000000 P40 769.000000 P68 490.000000 P96 682.000000 P124 15.000000 P13 506.000000 P41 3.000000 P69 130.000000 P97 720.000000 P125 9.000000 P14 509.000000 P42 3.000000 P70 294.561866 P98 718.000000 P126 12.000000 P15 506.000000 P43 250.000000 P71 141.585409 P99 720.000000 P127 10.000000 P16 505.000000 P44 250.000000 P72 365.907593 P100 964.000000 P128 112.000000 P17 506.000000 P45 250.000000 P73 195.000000 P101 958.000000 P129 4.000000 P18 506.000000 P46 250.000000 P74 217.548960 P102 947.900000 P130 5.000000 P19 505.000000 P47 250.000000 P75 217.549207 P103 934.000000 P131 5.000000 P20 505.000000 P48 250.000000 P76 258.662735 P104 935.000000 P132 50.000000 P21 505.000000 P49 250.000000 P77 403.245249 P105 876.500000 P133 5.000000 P22 505.000000 P50 250.000000 P78 330.000000 P106 880.900000 P134 42.000000 P23 505.000000 P51 165.000000 P79 531.000000 P107 873.700000 P135 42.000000 P24 505.000000 P52 165.000000 P80 531.000000 P108 877.400000 P136 41.000000 P25 537.000000 P53 165.000000 P81 542.000000 P109 871.700000 P137 17.000000 P26 537.000000 P54 165.000000 P82 56.000000 P110 864.800000 P138 7.000000 P27 549.000000 P55 180.000000 P83 115.000000 P111 882.000000 P139 7.000000 P28 549.000000 P56 180.000000 P84 115.000000 P112 94.000000 P140 26.000000 Fig. 1. Logistic map with the initial value of 0.91 for generating random numbers between 0 to 1. Fig. 2. Modified logistic map with the initial value of 0.91 for generating random numbers between -1 to 1. for it = 1 to NI Generate four new HMs according to the existing HM PAR = PAR(it) BW = BW (it) C == 0 No for j = 1 to HMS x i = xki Yes k is a random int eger number between 1 to HMS r1(i) < HMCR Yes r2(i) < PAR Yes C = C+1 for i = 1 to N No TC(it) == TC(it+1) x i = Li + Yn+1×(Ui - Li) x i = x i + Yn+1×BW No if TC (Xjnew) < TC (Xjworst ) then Xjworst = Xjnew HM HMS×1 = {The best harmonies among all harmonies in the four HMs} Run local optimization for i = 1 to HMS It == NI No Yes for j = 1 to N C == SNI Yes Stop Select r-th and t-th decision variable randomly V = xir + xit & TCold = TC (xir) + TC (xit) No max min min max max min min max A = max (V - Pr B = max (V - Pt , Pt ) to min (V - Pr , Pr ) to min (V - Pt , Pt Length(A) > Length(B) , Pr H=A d1 = t & d2 = r ) TCold = TCnew xid1 = k & xid2 = V-k H=B d1 = r & d2 = t for k = min(H) : Ɛ : max(H) No Yes TCnew = TC (k) + TC (V-k) j == N Yes ) Yes TCnew < TCold No Fig. 3. Flowchart of the CIHSA optimization procedure. N PM = Pi - PD +PL i=1 PM < Δ YES NO Select j-th decision variable randomly Fix it with the limit YES Output power is out of limits NO Fig. 4. The handling strategy for the system power balance constraint. Fig. 5. Convergence characteristic for fuel cost minimization obtained by four algorithms for test system 1. Fig. 6. Optimal generation outputs found by CIHSA for test system 1. Fig. 7. Convergence characteristic for fuel cost minimization obtained by four algorithms for test system 3. Fig. 8. Distribution of the fuel cost achieved by IHSA and CIHSA for test system 3 for 20 trial runs. Fig. 9. Convergence characteristic for fuel cost minimization obtained by four algorithms for test system 5. Fig. 10. Distribution of the fuel cost achieved by IHSA and CIHSA for test system 5 for 20 trial runs. Appendix The data of all test systems including the fuel cost and emission coefficients and transmission losses matrices are given in this section. This data is collected from [32], [33], [44], [49], [52], [64], and [76]. Test system 1-i Unit Pmin Pmax a b c α β γ ξ λ 1 5 150 100 200 10 6.490 −5.554 4.091 2e-4 2.857 2 5 150 120 150 10 5.638 −6.047 2.543 5e-4 3.333 3 5 150 40 180 20 4.586 −5.094 4.258 1e-6 8.000 4 5 150 60 100 10 3.380 −3.550 5.326 2e-3 2.000 5 5 150 40 180 20 4.586 −5.094 4.258 1e-6 8.000 6 5 150 100 150 10 5.151 −5.555 6.131 1e-5 6.667 Test system 1-ii Unit Pmin Pmax a b c 1 100 500 0.007 7 240 2 50 200 0.005 10 200 3 80 300 0.009 8.5 220 4 50 150 0.009 11 200 5 50 200 0.008 10.5 220 6 50 120 0.0075 12 120 0.0014 0.0017 0.0015 Bij = 0.0019 0.0026 0.0022 0.0017 0.0060 0.0013 0.0016 0.0015 0.0020 B0i = -0.0003908 0.0015 0.0013 0.0065 0.0017 0.0024 0.0019 -0.0001297 0.0019 0.0016 0.0017 0.0071 0.0030 0.0025 0.0022 0.0020 0.0019 0.0025 0.0032 0.0085 0.0026 0.0015 0.0024 0.0030 0.0069 0.0032 0.0007047 0.0000591 -0.0006635 0.0002161 B00 =0.00056 Test system 2 Unit Pmin Pmax a b c e f α β γ ξ λ 1 2 3 4 5 6 7 8 9 10 10 20 47 20 50 70 60 70 135 150 55 80 120 130 160 240 300 340 470 470 0.12951 0.10908 0.12511 0.12111 0.15247 0.10587 0.03546 0.02803 0.02111 0.01799 40.5407 39.5804 36.5104 39.5104 38.539 46.1592 38.3055 40.3965 36.3278 38.2704 1000.403 950.606 900.705 800.705 756.799 451.325 1243.531 1049.998 1658.569 1356.659 33 25 32 30 30 20 20 30 60 40 0.0174 0.0178 0.0162 0.0168 0.0148 0.0163 0.0152 0.0128 0.0136 0.0141 4.702 4.652 4.652 4.652 0.420 0.420 0.680 0.680 0.460 0.460 −398.64 −395.24 −390.23 −390.23 +032.77 +032.77 −054.55 −054.55 −051.12 −051.12 36000.12 35000.56 33000.56 33000.56 1385.93 1385.93 4026.69 4026.69 4289.55 4289.55 0.25475 0.25475 0.25163 0.25163 0.2497 0.2497 0.248 0.2499 0.2547 0.2547 0.01234 0.01234 0.01215 0.01215 0.012 0.012 0.0129 0.01203 0.01234 0.01234 0.0049 0.0014 0.0015 0.0015 0.0016 Bij = 0.0017 0.0017 0.0018 0.0019 0.0020 0.0014 0.0045 0.0016 0.0016 0.0017 0.0015 0.0015 0.0016 0.0018 0.0018 0.0015 0.0016 0.0039 0.0010 0.0012 0.0012 0.0014 0.0014 0.0016 0.0016 0.0015 0.0016 0.0010 0.0040 0.0014 0.0010 0.0011 0.0012 0.0014 0.0015 0.0016 0.0017 0.0012 0.0014 0.0035 0.0011 0.0013 0.0013 0.0015 0.0016 0.0017 0.0015 0.0012 0.0010 0.0011 0.0036 0.0012 0.0012 0.0014 0.0015 0.0017 0.0015 0.0014 0.0011 0.0013 0.0012 0.0038 0.0016 0.0016 0.0018 0.0018 0.0016 0.0014 0.0012 0.0013 0.0012 0.0016 0.0040 0.0015 0.0016 0.0019 0.0018 0.0016 0.0014 0.0015 0.0014 0.0016 0.0015 0.0042 0.0019 0.0020 0.0018 0.0016 0.0015 0.0016 0.0015 0.0018 0.0016 0.0019 0.0044 Test system 3 Unit Pmin Pmax a b c e f α β γ ξ λ 1 0 680 0.00028 8.10 550 300 0.035 0.06320 −2.434 40 0.855 0.0087 2 0 360 0.00056 8.10 309 200 0.042 0.03480 −3.630 50 0.623 0.0068 3 0 360 0.00056 8.10 307 150 0.042 0.03480 −3.630 50 0.623 0.0068 4 60 180 0.00324 7.74 240 150 0.063 0.04376 −5.271 40 0.312 0.0085 5 60 180 0.00324 7.74 240 150 0.063 0.04376 −5.271 40 0.312 0.0085 6 60 180 0.00324 7.74 240 150 0.063 0.04376 −5.271 40 0.312 0.0085 7 60 180 0.00324 7.74 240 150 0.063 0.04376 −5.271 40 0.312 0.0085 8 60 180 0.00324 7.74 240 150 0.063 0.04376 −5.271 40 0.312 0.0085 9 60 180 0.00324 7.74 240 150 0.063 0.04376 −5.271 40 0.312 0.0085 10 40 120 0.00284 8.60 126 100 0.084 0.05710 −4.852 100 0.424 0.0052 11 40 120 0.00284 8.60 126 100 0.084 0.05710 −4.852 100 0.424 0.0052 12 55 120 0.00284 8.60 126 100 0.084 0.05710 −4.343 100 1.130 0.0055 13 55 120 0.00284 8.60 126 100 0.084 0.05710 −4.343 100 1.130 0.0055 +0.14 +0.12 +0.07 0.01 0.03 0.01 -2 Bij =10 0.01 0.01 0.03 0.05 0.03 0.02 +0.04 +0.12 +0.15 +0.13 0.000 0.05 0.02 0.000 +0.01 0.02 0.04 0.04 0.000 +0.04 +0.07 +0.13 +0.76 0.01 0.13 0.09 0.01 0.000 0.08 0.12 0.17 0.000 0.26 0.01 0.000 0.01 0.34 0.07 0.04 0.11 0.50 0.29 0.32 0.11 0.000 0.01 0.03 0.05 0.13 0.07 0.90 0.14 0.03 0.12 0.10 0.13 0.07 0.02 0.02 0.01 0.02 0.09 0.04 0.14 0.16 0.000 0.06 0.05 0.08 0.11 0.01 0.02 0.01 0.000 0.01 0.11 0.03 0.000 0.15 0.17 0.15 0.09 0.05 0.07 0.000 0.01 0.01 0.000 0.50 0.12 0.06 0.17 1.68 0.82 0.79 0.23 0.36 0.01 0.03 0.02 0.08 0.29 0.10 0.05 0.15 0.82 1.29 1.16 0.21 0.25 0.07 0.05 0.04 0.12 0.32 0.13 0.08 0.09 0.79 1.16 2.00 0.27 0.34 0.09 0.03 0.04 0.17 0.11 0.07 0.11 0.05 0.23 0.21 0.27 1.40 0.01 0.04 0.02 0.000 0.000 0.000 0.02 0.01 0.07 0.36 0.25 0.34 0.01 0.54 0.01 +0.04 +0.04 0.26 +0.01 0.02 0.02 0.000 +0.01 +0.07 +0.09 +0.04 0.01 +1.03 B0i = 0.0001 0.0002 0.0028 0.0001 0.0001 0.0003 0.0002 0.0002 0.0006 0.0039 0.0017 0.0000 0.0032 B00 = +0.0055 Test system 4 Unit Pmin Pmax a b c e f α β γ 1 150 455 0.0050 1.89 150 300 0.035 1.6 −150.0 2333.3 2 150 455 0.0055 2.00 115 200 0.042 3.1 −182.0 2102.2 3 20 130 0.0060 3.50 40 200 0.042 1.3 −124.9 2205.0 4 20 130 0.0050 3.15 122 150 0.063 1.2 −135.5 2298.3 5 150 470 0.0050 3.05 125 150 0.063 2.0 −190.0 2131.3 6 135 460 0.0070 2.75 120 150 0.063 0.7 +80.50 2190.0 7 135 465 0.0070 3.45 70 150 0.063 1.5 −140.0 2300.1 8 60 300 0.0070 3.45 70 150 0.063 1.8 −180.0 2400.3 9 25 162 0.0050 2.45 130 150 0.063 1.9 −200.0 2512.1 10 25 160 0.0050 2.45 130 100 0.084 1.2 −136.0 2299.0 11 20 80 0.0055 2.35 135 100 0.084 3.3 −210.0 2701.0 12 20 80 0.0045 1.60 200 100 0.084 1.8 −180.0 2510.1 13 25 85 0.0070 3.45 70 100 0.084 1.8 −181.0 2431.3 14 15 55 0.0060 3.89 45 100 0.084 3.0 −192.1 2711.9 Test system 5 Unit Pmin Pmax a b c d e α β γ ξ λ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 36 36 60 80 47 68 110 135 135 130 94 94 125 125 125 125 220 220 242 242 254 254 254 254 254 254 10 10 10 47 60 60 60 90 90 90 25 25 25 114 114 120 190 97 140 300 300 300 300 375 375 500 500 500 500 500 500 550 550 550 550 550 550 550 550 150 150 150 97 190 190 190 200 200 200 110 110 110 0.00690 0.00690 0.02028 0.00942 0.01140 0.01142 0.00357 0.00492 0.00573 0.00605 0.00515 0.00569 0.00421 0.00752 0.00708 0.00708 0.00313 0.00313 0.00313 0.00313 0.00298 0.00298 0.00284 0.00284 0.00277 0.00277 0.52124 0.52124 0.52124 0.01140 0.00160 0.00160 0.00160 0.00010 0.00010 0.00010 0.01610 0.01610 0.01610 6.73 6.73 7.07 8.18 5.35 8.05 8.03 6.99 6.60 12.9 12.9 12.8 12.5 8.84 9.15 9.15 7.97 7.95 7.97 7.97 6.63 6.63 6.66 6.66 7.10 7.10 3.33 3.33 3.33 5.35 6.43 6.43 6.43 8.95 8.62 8.62 5.88 5.88 5.88 94.705 94.705 309.54 369.03 148.89 222.33 287.71 391.98 455.76 722.82 635.20 654.69 913.40 1760.4 1728.3 1728.3 647.85 649.69 647.83 647.81 785.96 785.96 794.53 794.53 801.32 801.32 1055.1 1055.1 1055.1 148.89 222.92 222.92 222.92 107.87 116.58 116.58 307.45 307.45 307.45 100 100 100 150 120 100 200 200 200 200 200 200 300 300 300 300 300 300 300 300 300 300 300 300 300 300 120 120 120 120 150 150 150 200 200 200 80 80 80 0.084 0.084 0.084 0.063 0.077 0.084 0.042 0.042 0.042 0.042 0.042 0.042 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.077 0.077 0.077 0.077 0.063 0.063 0.063 0.042 0.042 0.042 0.098 0.098 0.098 4.800 4.800 7.620 5.400 8.500 8.540 2.420 3.100 3.350 42.50 3.220 3.380 2.960 5.120 4.960 4.960 1.510 1.510 1.510 1.510 1.450 1.450 1.380 1.380 1.320 1.320 184.2 184.2 184.2 8.500 1.210 1.210 1.210 0.120 0.120 0.120 9.500 9.500 9.500 −222 −222 −236 −314 −189 −308 −306 −232 −211 −434 −434 −428 −418 −334 −355 −355 −268 −266 −268 −268 −222 −222 −226 −226 −242 −242 −111 −111 −111 −189 −208 −208 −208 −348 −324 −324 −198 −198 −198 6000 6000 10000 12000 5000 8000 10000 13000 15000 28000 22000 22500 30000 52000 51000 51000 22000 22200 22000 22000 29000 28500 29500 29500 31000 31000 36000 36000 36000 5000 8000 8000 8000 6500 7000 7000 10000 10000 10000 1.3100 1.3100 1.3100 0.9142 0.9936 1.3100 0.6550 0.6550 0.6550 0.6550 0.6550 0.6550 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.9936 0.9936 0.9936 0.9936 0.9142 0.9142 0.9142 0.6550 0.6550 0.6550 1.4200 1.4200 1.4200 0.05690 0.05690 0.05690 0.04540 0.04060 0.05690 0.02846 0.02846 0.02846 0.02846 0.02846 0.02846 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.04060 0.04060 0.04060 0.04060 0.04540 0.04540 0.04540 0.02846 0.02846 0.02846 0.06770 0.06770 0.06770 40 242 550 0.00313 7.97 647.83 300 0.035 1.510 −268 22000 0.5035 0.02075 Test system 6 Unit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Pmin 71 120 125 125 90 90 280 280 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 260 280 280 280 280 260 260 260 260 260 260 260 260 120 120 423 423 3 3 160 160 160 160 160 160 160 160 165 165 165 165 180 180 103 198 100 153 163 95 160 160 196 196 196 196 130 130 Pmax 119 189 190 190 190 190 490 490 496 496 496 496 506 509 506 505 506 506 505 505 505 505 505 505 537 537 549 549 501 501 506 506 506 506 500 500 241 241 774 769 19 28 250 250 250 250 250 250 250 250 504 504 504 504 471 561 341 617 312 471 500 302 511 511 490 490 490 490 432 432 a 0.032888 0.008280 0.003849 0.003849 0.042468 0.014992 0.007039 0.003079 0.005063 0.005063 0.005063 0.003552 0.003901 0.003901 0.003901 0.003901 0.002393 0.002393 0.003684 0.003684 0.003684 0.003684 0.004004 0.003684 0.001619 0.005093 0.000993 0.000993 0.002473 0.002547 0.003542 0.003542 0.003542 0.003542 0.003132 0.001323 0.002950 0.002950 0.000991 0.001581 0.902360 0.110295 0.024493 0.029156 0.024667 0.016517 0.026584 0.007540 0.016430 0.045934 0.000044 0.000044 0.000044 0.000044 0.002528 0.000131 0.010372 0.007627 0.012464 0.039441 0.007278 0.000044 0.000044 0.000044 0.018827 0.010852 0.018827 0.018827 0.034560 0.081540 b 61.242 41.095 46.310 46.310 54.242 61.215 11.791 15.055 13.226 13.226 13.226 14.498 14.651 14.651 14.651 14.651 15.669 15.669 14.656 14.656 14.656 14.656 14.378 14.656 16.261 13.362 17.203 17.203 15.274 15.212 15.033 15.033 15.033 15.033 13.992 15.679 16.542 16.542 16.518 15.815 75.464 129.544 56.613 54.451 54.736 58.034 55.981 61.520 58.635 44.647 71.584 71.584 71.584 71.584 85.120 87.682 69.532 78.339 58.172 46.636 76.947 80.761 70.136 70.136 49.840 65.404 49.840 49.840 66.465 22.941 c 1220.645 1315.118 874.288 874.288 1976.469 1338.087 1818.299 1133.978 1320.636 1320.636 1320.636 1106.539 1176.504 1176.504 1176.504 1176.504 1017.406 1017.406 1229.131 1229.131 1229.131 1229.131 1267.894 1229.131 975.926 1532.093 641.989 641.989 911.533 910.533 1074.810 1074.810 1074.810 1074.810 1278.460 861.742 408.834 408.834 1288.815 1436.251 669.988 134.544 3427.912 3751.772 3918.780 3379.580 3345.296 3138.754 3453.050 5119.300 1898.415 1898.415 1898.415 1898.415 2473.390 2781.705 5515.508 3478.300 6240.909 9960.110 3671.997 1837.383 3108.395 3108.395 7095.484 3392.732 7095.484 7095.484 4288.320 13813.001 UR 30 30 60 60 150 150 180 180 300 300 300 300 600 600 600 600 600 600 600 600 600 600 600 600 300 300 360 360 180 180 600 600 600 600 660 900 180 180 600 600 210 366 702 702 702 702 702 702 702 702 1350 1350 1350 1350 1350 720 720 2700 1500 1656 2160 900 1200 1200 1014 1014 1014 1014 1350 1350 DR 120 120 60 60 150 150 300 300 510 510 510 510 600 600 600 600 600 600 600 600 600 600 600 600 300 300 360 360 180 180 600 600 600 600 660 900 180 180 600 600 210 366 702 702 702 702 702 702 702 702 1350 1350 1350 1350 1350 720 720 2700 1500 1656 2160 900 1200 1200 1014 1014 1014 1014 1350 1350 P0 98.4 134 141.5 183.3 125 91.3 401.1 329.5 386.1 427.3 412.2 370.1 301.8 368 301.9 476.4 283.1 414.1 328 389.4 354.7 262 461.5 371.6 462.6 379.2 530.8 391.9 480.1 319 329.5 333.8 390 432 402 428 178.4 194.1 474 609.8 17.8 6.9 224.3 210 212 200.8 220 232.9 168 208.4 443.9 426 434.1 402.5 357.4 423 220 369.4 273.5 336 432 220 410.6 422.7 351 296 411.1 263.2 370.3 418.7 Unit 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 Pmin 137 137 195 175 175 175 175 330 160 160 200 56 115 115 115 207 207 175 175 175 175 360 415 795 795 578 615 612 612 758 755 750 750 713 718 791 786 795 795 795 795 94 94 94 244 244 244 95 95 116 175 2 4 15 9 12 10 112 4 5 5 50 5 42 42 41 17 7 7 26 Pmax 455 455 541 536 540 538 540 574 531 531 542 132 245 245 245 307 307 345 345 345 345 580 645 984 978 682 720 718 720 964 958 1007 1006 1013 1020 954 952 1006 1013 1021 1015 203 203 203 379 379 379 190 189 194 321 19 59 83 53 37 34 373 20 38 19 98 10 74 74 105 51 19 19 40 a 0.023534 0.035475 0.000915 0.000044 0.000044 0.001307 0.000392 0.000087 0.000521 0.000498 0.001046 0.132050 0.096968 0.054868 0.054868 0.014382 0.013161 0.016033 0.013653 0.028148 0.013470 0.000064 0.000252 0.000022 0.000022 0.000203 0.000198 0.000215 0.000218 0.000193 0.000197 0.000324 0.000344 0.000690 0.000650 0.000233 0.000239 0.000261 0.000259 0.000707 0.000786 0.014355 0.014355 0.014355 0.030266 0.030266 0.030266 0.024027 0.001580 0.022095 0.076810 0.953443 0.000044 0.072468 0.000448 0.599112 0.244706 0.000042 0.085145 0.524718 0.176515 0.063414 2.740485 0.112438 0.041529 0.000911 0.005245 0.234787 0.234787 1.111878 b 64.314 45.017 70.644 70.959 70.959 70.302 70.662 71.101 37.854 37.768 67.983 77.838 63.671 79.458 79.458 93.966 94.723 66.919 68.185 60.821 68.551 2.842 2.946 3.096 3.040 1.709 1.668 1.789 1.815 2.726 2.732 2.651 2.798 1.595 1.503 2.425 2.499 2.674 2.692 1.633 1.816 89.830 89.830 89.830 64.125 64.125 64.125 76.129 81.805 81.140 46.665 78.412 112.088 90.871 97.116 83.244 95.665 91.202 104.501 83.015 127.795 77.929 92.779 80.950 89.073 161.288 161.829 84.972 84.972 16.087 c 4435.493 9750.750 1042.366 1159.895 1159.895 1303.990 1156.193 2118.968 779.519 829.888 2333.690 2028.954 4412.017 2982.219 2982.219 3174.939 3218.359 3723.822 3551.405 4322.615 3493.739 226.799 382.932 156.987 154.484 332.834 326.599 345.306 350.372 370.377 367.067 124.875 130.785 878.746 827.959 432.007 445.606 467.223 475.940 899.462 1000.367 1269.132 1269.132 1269.132 4965.124 4965.124 4965.124 2243.185 2290.381 1681.533 6743.302 394.398 1243.165 1454.740 1011.051 909.269 689.378 1443.792 535.553 617.734 90.966 974.447 263.810 1335.594 1033.871 1391.325 4477.110 57.794 57.794 1258.437 UR 1350 1350 780 1650 1650 1650 1650 1620 1482 1482 1668 120 180 120 120 120 120 318 318 318 318 18 18 36 36 138 144 144 144 48 48 36 36 30 30 30 30 36 36 36 36 120 120 120 480 480 480 240 240 120 180 90 90 300 162 114 120 1080 60 66 12 300 6 60 60 528 300 18 18 72 DR 1350 1350 780 1650 1650 1650 1650 1620 1482 1482 1668 120 180 180 180 180 180 318 318 318 318 18 18 36 36 204 216 216 216 48 48 54 54 30 30 30 30 36 36 36 36 120 120 120 480 480 480 240 240 120 180 90 90 300 162 114 120 1080 60 66 6 300 6 60 60 528 300 30 30 120 P0 409.6 412 423.2 428 436 428 425 497.2 510 470 464.1 118.1 141.3 132 135 252 221 245.9 247.9 183.6 288 557.4 529.5 800.8 801.5 582.7 680.7 670.7 651.7 921 916.8 911.9 898 905 846.5 850.9 843.7 841.4 835.7 828.8 846 179 120.8 121 317.4 318.4 335.8 151 129.5 130 218.8 5.4 45 20 16.3 20 22.1 125 10 13 7.5 53.2 6.4 69.1 49.9 91 41 13.7 7.4 28.6 Highlights: An innovative and strong optimization technique based on harmony search is proposed The proposed algorithm is tested on solving economic emission dispatch problem It has the potential to be applied to many other engineering optimization problems Four test systems considering valve point effect and transmission loss are studied High quality solutions are obtained and compared with a large number of other methods Conflict of Interest: Authors have no conflict of interest.

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