A Novel Multi-Population Based Chaotic JAYA Algorithm with Application in Solving Economic Load Dispatch Problems

: The economic load dispatch (ELD) problem is an optimization problem of minimizing the total fuel cost of generators while satisfying power balance constraints, operating capacity limits, ramp-rate limits and prohibited operating zones. In this paper, a novel multi-population based chaotic JAYA algorithm (MP-CJAYA) is proposed to solve the ELD problem by applying the multi-population method (MP) and chaotic optimization algorithm (COA) on the original JAYA algorithm to guarantee the best solution of the problem. MP-CJAYA is a modiﬁed version where the total population is divided into a certain number of sub-populations to control the exploration and exploitation rates, at the same time a chaos perturbation is implemented on each sub-population during every iteration to keep on searching for the global optima. The proposed MP-CJAYA has been adopted to ELD cases and the results obtained have been compared with other well-known algorithms reported in the literature. The comparisons have indicated that MP-CJAYA outperforms all the other algorithms, achieving the best performance in all the cases, which indicates that MP-CJAYA is a promising alternative approach for solving ELD problems.


Introduction
With the issues of global warming and depletion of classical fossil fuels, saving energy and reducing the operational cost have become the key topics in power systems nowadays.The economic load dispatch problem (ELD) is a crucial issue of power system operation that minimizes the operational cost while satisfying a set of physical and operational constraints imposed by generators and system limitations [1].A large number of conventional optimization methods have been applied successfully for solving the ELD problem such as gradient method [2], lambda iteration method [3], semi-definite programming [4], quadratic programming [5], dynamic programming [6], Lagrangian relaxation method [7] and linear programming [8].However, they suffer from difficulties when dealing with problems with nonconvex objective function and complex constraints, which tends to exhibit highly non-linear, non-convex and non-smooth characteristics with a number of local optima [9].
To overcome these drawbacks, meta-heuristic methods are proposed, such as genetic algorithm (GA) [10], particle swarm optimization (PSO) [11], tabu search (TS) [12], artificial bee colony algorithm Based on the descriptions above, a novel multi-population based chaotic JAYA algorithm (MP-CJAYA) is proposed in this paper.It is a modified version of JAYA algorithm where the total population is divided into sub-populations by the MP method to control the exploration and exploitation rates, meanwhile a chaos perturbation is implemented on each sub-population during every iteration to keep on searching for the global optima.The MP-CJAYA algorithm is applied for solving the ELD cases with constraints including valve-point effects, power balance constraints, operating capacity limits, ramp-rate limits and prohibited operating zones.In all the experimented ELD cases, the proposed MP-CJAYA has produced the most competitive results.
The rest of this paper is arranged as follows: In Section 2, the problem formulation of ELD problem is constructed.The basic JAYA, the compared CJAYA and the proposed MP-CJAYA algorithms are described in Section 3. The experimental results and comparisons of MP-CJAYA with other algorithms are presented and analyzed in Section 4. Finally, the conclusions and future work are given in Section 5.

Problem Formulation
The ELD problem is described as an objective function to minimize the total fuel cost while satisfying different constraints, we adopt the problem formulation described in [16,42].

Objective Function
The objective function is to sum up all the costs of committed generators as expressed below: where n is the total generator number in power systems, F i (P i ) is the cost function of ith generator with output P i .
Approximately, the cost function can be expressed as a quadratic polynomial by the following equation: where a i , b i , c i are the cost coefficients of ith generator, which are constants.
In reality, a higher-order non-linearity rectified sinusoid contribution is usually added to the cost function to model the valve-point effect, which is given below: where e i and f i are cost coefficients of ith generator due to valve-point effect, while P i min is the minimum output for generator i.
According to the discussion above, the objective function of ELD problem considering the valve-point effect can be represented as: 2.2.Constrained Functions

Power Losses
The total power generated by available units must equal to the summation of the demanded power and the system power loss, which can be formulated as: n ∑ i = 1 P i = P demand + P loss (5) where P demand and P loss is the value of the demanded power and the whole power loss in the system respectively.P loss is calculated by Kron's formula: B i0 P i + B 00 (6) where B ij , B i0 , B 00 are the loss coefficients that generally can be assumed to be constants under a normal operating condition.

Generating Capacity
The real output P i generated by a available unit must be ranged between its minimum limit and maximum limit: where P min i and P max i are the minimum and maximum limits of ith generator.

Ramp Rate Limit
In practical circumstances, the output power P i can not be adjusted immediately, the operating range is restricted by the ramp-rate limit constraint expressed below: where P i is the present power output, P i 0 is the previous power output, UR i and DR i is the up-ramp and down-ramp limit of generator i respectively.

Prohibited Operating Zones
For generator with prohibited operating zones (POZs), which are the sets of output power ranges where the generator can not work, the feasible operating zones are as discontinuous as follows: where j is the index of POZs, n i is the total number of POZs where j ∈ [1, n i ], P i,j lower and P i,j upper are the lower and upper bounds of the jth POZ of the ith unit, respectively.

The Proposed MP-CJAYA Algorithm
Since the proposed MP-CJAYA algorithm is a hybrid of the basic JAYA, COA and MP methods, it is quite necessary to observe the relative strength of each constituent when solving the ELD problem, so three different algorithms are studied: (1) The basic JAYA algorithm: The classical JAYA algorithm with standard parameters; it is selected to compare its performance at solving different ELD cases with the other two algorithms.
(2) The compared CJAYA algorithm: The basic JAYA algorithm combined by COA but without the MP method.(3) The proposed MP-CJAYA algorithm: The basic JAYA algorithm integrated with both the COA and MP methods.

The Basic JAYA Algorithm
The JAYA algorithm is a powerful heuristic algorithm proposed by Rao for solving optimization problems.It always attempts to get success to reach the best solution as well as move far away from the worst solution.Different from most of the other heuristic algorithms, JAYA is free from algorithm-specific parameters, only two common parameters named the population size N pop and the number of iterations N iter are required [21].
Suppose the objective function is F(X) which is required to be minimized or maximized.Let F(X) best and F(X) worst represent the best value and the worst value of F(X) among the entire candidate solutions during each iteration.Let X j,k,i be the value of the jth variable for the kth candidate during the ith iteration, then the new modified value X j,k,i by JAYA algorithm is calculated by: where X j,k,i is the updated value of X j,k,i .X j,best,i and X j,worst,i are the values of the jth variable for F(X) best and F(X) worst during the ith iteration respectively.r 1,j,i and r 2,j,i are two random numbers ranged in [0, 1].The term 'r 1,j,i × (X j,best,i − |X j,k,i |)' indicates the tendency of the solution to move closer to the best solution and the term 'r 2,j,i × (X j,worst,i − X j,k,i )' indicates the tendency of the solution to avoid the worst solution.Suppose F(X) is the modified value of F(X), if F(X) provides better value than F(X), then X j,k,i is replaced by X j,k,i and F(X) is replaced by F(X) ; otherwise, keep the old value.All the values of new obtained X j,k,i and F(X) at the end of every iteration are maintained and become the inputs to the next iteration [21].
The procedure for the basic JAYA algorithm to solve ELD problem is described as follows: Step 1: Set parameters.Common parameters of JAYA are initialized in this step.The first one is the population size (N pop ) which represents how many solutions will be generated; the second one is the maximum iteration number (N J AYA_iter ) which indicates the stopping condition during the calculation; the last one is the total number of generators (N gen ) for N gen -units system.
Set the iteration counter as iter.
Step 2: Initialize the solution.A set of initial solutions are randomly generated as follows: where min and X j max are the lower and upper limits of jth generator given by generating capacity limits in Equation (7).
Step 4: Evaluate the solution.Calculate the objective function (cost function) by using Equation (3) with considering the valve-point effect or Equation (2) without considering the valve-point effect to obtain the initial value F(X).
Step 5: Determine the best and worst.Choose X j,best,i and X j,worst,i according to the value of F(X) best and F(X) worst , which means the lowest and highest value among all the populations.
Step 6: Generate new solution.Generate new output X j,k,i by Equation (10).
Step 8: Evaluate the new solution.Calculate the new objective function value F(X) by Equation (3) with considering the valve-point effect or Equation (2) without considering the valve-point effect.
Step 9: Compare.The new F(X) is compared with the old F(X), the values are updated as follows: If F(X) < F(X) then F(X) = F(X) and X j,k,i = X j,k,i ; Otherwise, keep the old value.
Step 10: Check the stopping condition.If the current iteration number iter < N J AYA_iter , then iter = iter + 1 and return to Step 5. Otherwise, stop the procedure.

The Compared CJAYA Algorithm
In this chapter, the Chaos Optimization Algorithm (COA) is combined with the basic JAYA algorithm to form the compared CJAYA algorithm.COA has used chaotic map for new search surface during every iteration, which is a discrete-time dynamical system running in chaotic state: A widely used logistic map which appears in nonlinear dynamics of biological population evidencing chaotic behavior is shown below [43].
where i is the serial number of chaotic variables, k is the iteration number.The initial value of the ith chaotic variable is Z i (0) where Z i (0) / ∈ {0.0, 0.25, 0.5, 0.75, 1.0}.α = 4 is used in this paper.It is obvious that Z i (k + 1) ∈ (0, 1) under the conditions of Z i (0) ∈ (0, 1).
The procedure for the CJAYA algorithm to solve ELD problem is provided here, the symbol * denotes a new added step compared with the basic JAYA: Step 1: Set parameters.Common parameters of CJAYA are initialized in this step.The population size (N pop ), the maximum iteration number (N J AYA_iter ) and the total number of generators (N gen ) are as the same as basic JAYA.However, one more parameter (N COA_iter ) is introduced which represents the maximum iteration number by COA.
Set the iteration counter as iter.
Step 2 * : Generate chaotic sequence.The chaotic sequence Z j,k,q is generated by Logistic map in this step, where j denoting the number of generators of the system, k denoting the population number and q denoting the number of iteration by COA, which is shown in the following equation: Here Step 3: Initialize the solution.By the carrier wave method, the set of initial variable X j,k,i can be transformed to chaos variables by: where X j min and X j max are the lower and upper limits of jth generator given by generating capacity limits in Equation (7).
Step 4: Apply constraints.As the same as Step 3 in Section 3.1.
Step 5: Evaluate the solution.As the same as Step 4 in Section 3.1.
Step 6: Determine the best and worst.As the same as Step 5 in Section 3.1.
Step 7: Generate new solution.As the same as Step 6 in Section 3.1.
Step 8: Apply constraints.As the same as Step 7 in Section 3.1.
Step 9: Evaluate the new solution.As the same as Step 8 in Section 3.1.
Step 10: Compare.As the same as Step 9 in Section 3.1.
Step 11 * : Apply COA.In the former step we have obtained the best set of solutions X j,k,i up to now, then the second carrier wave method can be performed by: where R is a constant, R × Z j,k,q generates chaotic states with small ergodic ranges around current X j,k,i to seek further for improving the quality of current solutions.Then the generated neighborhood solutions will be compared with current solutions to check if they give better objective function values by the following steps: (1) Apply constraints.As the same as Step 7 in Section 3.1.
(2) Evaluate the new solution.As the same as Step 8 in Section 3.1.
(3) Compare.As the same as Step 9 in Section 3.1.
Step 12: Check the stopping condition.If the current iteration number iter < N J AYA_iter , then iter = iter + 1 and return to Step 6.Otherwise, stop the procedure.

The Proposed MP-CJAYA Algorithm
In this section, Multi-population based optimization method (MP) is combined with CJAYA algorithm to form the proposed MP-CJAYA algorithm.Figure 1 presents the flowchart of the proposed MP-CJAYA algorithm, the pseudo code of the proposed MP-CJAYA is described in Algorithm 1.The whole steps of MP-CJAYA to solve ELD problem is described as follows, the symbol * denotes a newly added step compared with CJAYA: Step 1: Set parameters.Common parameters of MP-CJAYA are initialized in this step.The population size (N pop ), the maximum iteration number (N J AYA_iter ), the total number of generators (N gen ) and the maximum COA iteration number (N COA_iter ) are as the same as basic JAYA and CJAYA.However, another important parameter (K) is introduced which represents the divided number of sub-populations, so the population size of the sub-populations (N sub_pop ) is: Set the iteration counter as iter.
Step 2: Generate chaotic sequence.As the same as Step 2 in Section 3.2.
Step 3: Initialize the solution.As the same as Step 3 in Section 3.2.
Step 4: Apply constraints.As the same as Step 3 in Section 3.1.
Step 5: Evaluate the solution.As the same as Step 4 in Section 3.1.
Step 6 * : Divide the population.The entire population is divided into K sub-populations with population size of N sub_pop by Equation (17).It is noted that the solutions in the whole population are randomly assigned to a sub-population, each sub-population is arranged to explore a different area of the whole search space.
The following steps are performed on each sub-population: Step 7: Determine the best and worst.As the same as Step 5 in Section 3.1.
Step 8: Generate new solution.As the same as Step 6 in Section 3.1.
Step 9: Apply constraints.As the same as Step 7 in Section 3.1.
Step 10: Evaluate the new solution.As the same as Step 8 in Section 3.1.
Step 11: Compare.As the same as Step 9 in Section 3.1.
Step 12: Apply COA.As the same as Step 11 in Section 3.2.
Step 13: Check the stopping condition.If the current iteration number iter reaches N J AYA_iter , stop the loop and report the best solution; otherwise follow the next step and set iter = iter + 1.
Step 14 * : Merge the sub-populations.All the sub-populations are merged together to form one population, then for re-divide the population go to Step 6. Step 14 * : Merge the sub-populations.All the sub-populations are merged together to form one population, then for re-divide the population go to Step 6.

Begin
Initialize N pop , N J AYA_iter , N gen , N COA_iter and K; Generate initial solution X j,k,i by chaotic sequence; Calculate objective function value F(X); Set iter = 1 While iter < N J AYA_iter do Divide the whole population P into K sub-populations by Equation ( 17) randomly P 1 , P 2 , ..., P K−1 , P K For m = 1 → K do Confirm X j,best,i and X j,worst,i within P m For k = 1 → N sub_pop do Generate new solution X j,k,i by Equation ( 10)

End if End for
For k = 1 → N sub_pop do Generate new solution X j,k,i by Equation ( 16)

End if End for End for
Merge the sub-populations (P 1 , P 2 , ..., P K−1 , P K ) into P iter = iter + 1 End while

Experimental Results and Analysis
In this section, the basic JAYA, the compared CJAYA and the proposed MP-CJAYA algorithms are applied on the following ELD cases to test their performances: Since for meta-heuristic algorithms, parameter setting is critical for the quality of their performances, so the parameters used in the cases above are all listed below.All the cases are run in MATLAB 2016 under windows 7 on Intel(R) Core(TM) i5-6500 CPU 3.20 GHz, with 8 GB RAM.

Case I: 3-Units System for Load Demand of 850 MW
All detailed data are provided in [44].The common parameters and constraint conditions are given in Table 1.The cost value of F mean and F best obtained by JAYA, CJAYA and MP-CJAYA are compared with GA [45], EP [45], EP-SQP [45], PSO [45], PSO-SQP [45], CPSO [46] and CPSO-SQP [46] in Table 2.The best cost are highlighted in bold font.Obviously, all the compared algorithms give the same best cost of 8234.07 $/h, except for GA who did not meet the load demand.However, JAYA, CJAYA and MP-CJAYA are able to give continuously decreasing values of F best and MP-CJAYA achieves the minimum value of 8223.29 $/h, as well as the minimum value of F mean which is 8232.06$/h.To observe the cost convergence characteristics more visually, Figure 2 depicts one randomly chosen convergence curve from 20 times of independent runs (N runs ).We can see that JAYA has been trapped into local optimum at about 320 iterations and CJAYA has also settled down at around 230 iterations, but MP-CJAYA has showed extraordinary fast convergence ability at the beginning of 10 iterations and reached global optimum at approximately 200 iterations.It reveals that MP-CJAYA has faster convergence rate compared with JAYA and CJAYA due to its strong searching ability.Figure 3 shows the distribution outlines of F best at each independent run time.In case of MP-CJAYA, the value of F best after each run remains more or less steady, whereas in CJAYA the value of F best varies much more than MP-CJAYA, while JAYA shows the worst stability of F best with maximum cost as much as 8800 $/h.This indicates that MP-CJAYA is more consistent and robust than CJAYA and JAYA.

Case II: 13-Units System for Load Demand of 2520 MW
As the same as case I, all detailed data are provided in [44].Since the increasing number of generators causes more non-linearity and complexity, Npop, NJAYA_iter and Nruns have all increased in this case, which are given in Table 1.The best individual of dispatched outputs obtained by different methods including GA [47], SA [47], HSS [47], EP-SQP [45], PSO-SQP [45], CPSO [46], CPSO-SQP [46], JAYA, CJAYA and MP-CJAYA are reported in Table 3.The best cost are highlighted in bold font.It is observed that the minimum value of Fmean and Fbest are both achieved by MP-CJAYA, which is 24,228.1331$/h and 24,175.5444$/h respectively.In Figure 4 the convergence curve of MP-CJAYA is compared with JAYA and CJAYA, it can be observed that JAYA has been trapped into a local optimum in about 1300 iterations, while CJAYA has the same problem at around 1500 iterations.However, the proposed MP-CJAYA has greatly accelerated the convergence rate and reached the best value within only 750 iterations.Figure 5 is the distribution outlines of Fbest at each run time.Once again, it can be easily compared that MP-CJAYA shows the most robust characteristic among the three versions of JAYA due to most of its independent runs have achieved getting close to the best individual.All the comparisons above real that MP-CJAYA has greatly improved the best cost, the mean cost, the convergence rate and the consistency of the solution.

Case II: 13-Units System for Load Demand of 2520 MW
As the same as case I, all detailed data are provided in [44].Since the increasing number of generators causes more non-linearity and complexity, Npop, NJAYA_iter and Nruns have all increased in this case, which are given in Table 1.The best individual of dispatched outputs obtained by different methods including GA [47], SA [47], HSS [47], EP-SQP [45], PSO-SQP [45], CPSO [46], CPSO-SQP [46], JAYA, CJAYA and MP-CJAYA are reported in Table 3.The best cost are highlighted in bold font.It is observed that the minimum value of Fmean and Fbest are both achieved by MP-CJAYA, which is 24,228.1331$/h and 24,175.5444$/h respectively.In Figure 4 the convergence curve of MP-CJAYA is compared with JAYA and CJAYA, it can be observed that JAYA has been trapped into a local optimum in about 1300 iterations, while CJAYA has the same problem at around 1500 iterations.However, the proposed MP-CJAYA has greatly accelerated the convergence rate and reached the best value within only 750 iterations.Figure 5 is the distribution outlines of Fbest at each run time.Once again, it can be easily compared that MP-CJAYA shows the most robust characteristic among the three versions of JAYA due to most of its independent runs have achieved getting close to the best individual.All the comparisons above real that MP-CJAYA has greatly improved the best cost, the mean cost, the convergence rate and the consistency of the solution.

Case II: 13-Units System for Load Demand of 2520 MW
As the same as case I, all detailed data are provided in [44].Since the increasing number of generators causes more non-linearity and complexity, N pop , N JAYA_iter and N runs have all increased in this case, which are given in Table 1.The best individual of dispatched outputs obtained by different methods including GA [47], SA [47], HSS [47], EP-SQP [45], PSO-SQP [45], CPSO [46], CPSO-SQP [46], JAYA, CJAYA and MP-CJAYA are reported in Table 3.The best cost are highlighted in bold font.It is observed that the minimum value of F mean and F best are both achieved by MP-CJAYA, which is 24,228.1331$/h and 24,175.5444$/h respectively.In Figure 4 the convergence curve of MP-CJAYA is compared with JAYA and CJAYA, it can be observed that JAYA has been trapped into a local optimum in about 1300 iterations, while CJAYA has the same problem at around 1500 iterations.However, the proposed MP-CJAYA has greatly accelerated the convergence rate and reached the best value within only 750 iterations.Figure 5 is the distribution outlines of F best at each run time.Once again, it can be easily compared that MP-CJAYA shows the most robust characteristic among the three versions of JAYA due to most of its independent runs have achieved getting close to the best individual.All the comparisons above real that MP-CJAYA has greatly improved the best cost, the mean cost, the convergence rate and the consistency of the solution.

Case III: 40-Units System for Load Demand of 10,500 MW
In order to investigate the effectiveness of MP-CJAYA for larger scale power system, it is further evaluated by 40 generating units with load demand of 10,500 MW, which is the largest system of ELD problem considering the valve-point effect in the available literature.Considering the increased number of generators and the much more complex solution space, Npop, NJAYA_iter, NCOA_iter, Nsub_pop and Nruns have all increased, as shown in Table 1.The results comparison from methods PSO-LRS [48], NPSO [48], NPSO-LRS [48], SPSO [49], PC-PSO [49], SOH-PSO [49], JAYA, CJAYA and MP-CJAYA are shown in Table 4.The minimum value of Fmean and Fbest are highlighted in bold font.It is observed that MP-CJAYA has achieved the minimum value of Fbest among all the values by above-mentioned methods, which is 121,480.10$/h.What's more, the minimum value of Fmean is also achieved by MP-CJAYA, which is 121,861.08$/h.In Figure 6 the convergence curve of MP-CJAYA is compared with JAYA and CJAYA, it can easily be observed that CJAYA performs better than JAYA due to the local searching ability provided by COA, while MP-CJAYA shows superiority over CJAYA due to the extra searching diversification provided by MP method.

Case III: 40-Units System for Load Demand of 10,500 MW
In order to investigate the effectiveness of MP-CJAYA for larger scale power system, it is further evaluated by 40 generating units with load demand of 10,500 MW, which is the largest system of ELD problem considering the valve-point effect in the available literature.Considering the increased number of generators and the much more complex solution space, Npop, NJAYA_iter, NCOA_iter, Nsub_pop and Nruns have all increased, as shown in Table 1.The results comparison from methods PSO-LRS [48], NPSO [48], NPSO-LRS [48], SPSO [49], PC-PSO [49], SOH-PSO [49], JAYA, CJAYA and MP-CJAYA are shown in Table 4.The minimum value of Fmean and Fbest are highlighted in bold font.It is observed that MP-CJAYA has achieved the minimum value of Fbest among all the values by above-mentioned methods, which is 121,480.10$/h.What's more, the minimum value of Fmean is also achieved by MP-CJAYA, which is 121,861.08$/h.In Figure 6 the convergence curve of MP-CJAYA is compared with JAYA and CJAYA, it can easily be observed that CJAYA performs better than JAYA due to the local searching ability provided by COA, while MP-CJAYA shows superiority over CJAYA due to the extra searching diversification provided by MP method.In order to investigate the effectiveness of MP-CJAYA for larger scale power system, it is further evaluated by 40 generating units with load demand of 10,500 MW, which is the largest system of ELD problem considering the valve-point effect in the available literature.Considering the increased number of generators and the much more complex solution space, N pop , N JAYA_iter , N COA_iter , N sub_pop and N runs have all increased, as shown in Table 1.The results comparison from methods PSO-LRS [48], NPSO [48], NPSO-LRS [48], SPSO [49], PC-PSO [49], SOH-PSO [49], JAYA, CJAYA and MP-CJAYA are shown in Table 4.The minimum value of F mean and F best are highlighted in bold font.It is observed that MP-CJAYA has achieved the minimum value of F best among all the values by above-mentioned methods, which is 121,480.10$/h.What's more, the minimum value of F mean is also achieved by MP-CJAYA, which is 121,861.08$/h.In Figure 6 the convergence curve of MP-CJAYA is compared with JAYA and CJAYA, it can easily be observed that CJAYA performs better than JAYA due to the local searching ability provided by COA, while MP-CJAYA shows superiority over CJAYA due to the extra searching diversification provided by MP method.Figure 7 is the distribution outlines of Fbest within 50 times of independent runs.Once again, it can be observed that MP-CJAYA shows the most robust characteristic among the three versions of JAYA because most of the Fbest value keeps steady and very close to the best individual.The comparisons have verified that MP-CJAYA get better results than all of the other algorithms in best cost, mean cost, convergence rate and consistency when dealing with larger scale power system.

Case IV: 6-Units System for Load Demand of 1263 MW
In this case, the three versions of JAYA are applied to 6-units system with constraints of ramp rate limit, prohibited operating zones (POZs) and transmission loss ( loss P ), as shown in Table 1.The generator data and B-coefficients have been taken from [50].For every generator it has two POZs , this problem causes challenging complexity to find the global optima because of increasing number of non-convex decision spaces.
The best individual achieved by MP-CJAYA, as well the other algorithms such as SA [51], GA [51], TS [51], PSO [51], MTS [51], PSO-LRS [48], NPSO [48], NPSO-LRS [48], JAYA and CJAYA have been recorded in Table 5.It can be observed that MP-CJAYA provides the lowest Fbest among all the methods as 15,446.17$/h, while CJAYA and JAYA provide the second and third lowest Fbest as 15,446.71$/h and 15,447.09$/h.Furthermore, the best cost Fbest, the worst cost Fworst and the mean cost Fmean of the three version of JAYA algorithms are also compared with those above-mentioned methods Figure 7 is the distribution outlines of F best within 50 times of independent runs.Once again, it can be observed that MP-CJAYA shows the most robust characteristic among the three versions of JAYA because most of the F best value keeps steady and very close to the best individual.The comparisons have verified that MP-CJAYA get better results than all of the other algorithms in best cost, mean cost, convergence rate and consistency when dealing with larger scale power system.Figure 7 is the distribution outlines of Fbest within 50 times of independent runs.Once again, it can be observed that MP-CJAYA shows the most robust characteristic among the three versions of JAYA because most of the Fbest value keeps steady and very close to the best individual.The comparisons have verified that MP-CJAYA get better results than all of the other algorithms in best cost, mean cost, convergence rate and consistency when dealing with larger scale power system.

Case IV: 6-Units System for Load Demand of 1263 MW
In this case, the three versions of JAYA are applied to 6-units system with constraints of ramp rate limit, prohibited operating zones (POZs) and transmission loss ( loss P ), as shown in Table 1.The generator data and B-coefficients have been taken from [50].For every generator it has two POZs , this problem causes challenging complexity to find the global optima because of increasing number of non-convex decision spaces.
The best individual achieved by MP-CJAYA, as well the other algorithms such as SA [51], GA [51], TS [51], PSO [51], MTS [51], PSO-LRS [48], NPSO [48], NPSO-LRS [48], JAYA and CJAYA have been recorded in Table 5.It can be observed that MP-CJAYA provides the lowest Fbest among all the methods as 15,446.17$/h, while CJAYA and JAYA provide the second and third lowest Fbest as 15,446.71$/h and 15,447.09$/h.Furthermore, the best cost Fbest, the worst cost Fworst and the mean cost Fmean of the three version of JAYA algorithms are also compared with those above-mentioned methods  In this case, the three versions of JAYA are applied to 6-units system with constraints of ramp rate limit, prohibited operating zones (POZs) and transmission loss (P loss ), as shown in Table 1.The generator data and B-coefficients have been taken from [50].For every generator it has two POZs, this problem causes challenging complexity to find the global optima because of increasing number of non-convex decision spaces.
The best individual achieved by MP-CJAYA, as well the other algorithms such as SA [51], GA [51], TS [51], PSO [51], MTS [51], PSO-LRS [48], NPSO [48], NPSO-LRS [48], JAYA and CJAYA have been recorded in Table 5.It can be observed that MP-CJAYA provides the lowest F best among all the methods as 15,446.17$/h, while CJAYA and JAYA provide the second and third lowest F best as 15,446.71$/h and 15,447.09$/h.Furthermore, the best cost F best , the worst cost F worst and the mean cost F mean of the three version of JAYA algorithms are also compared with those above-mentioned methods and summarized in Table 6.It can be found that MP-CJAYA is superior to all the other compared methods and achieves the minimum value of F best , F worst and F mean at the same time, which are highlighted in bold font.Figure 8 is the distribution outlines of F bes , it can be noticed that MP-CJAYA shows the most robust characteristic and the value keeps almost steady within 20 independent runs, which has greatly surpassed JAYA and a little surpassed CJAYA.One randomly chosen convergence curve of fuel cost is shown in Figure 9, from which we can see that MP-CJAYA is extraordinary fast in convergence rate and approaches global optimum within only about 60 iterations.It all demonstrates that MP-CJAYA has the strongest capabilities of handling ELD problems with different constraint conditions.In the last case, the three versions of JAYA are applied to a larger 15-units system with the same constraints as in case 4, the system data and B-coefficients have been taken from [50].There are 4 generators having POZs.Generators 2, 5 and 6 have three POZs and generator 12 has two POZs.Considering that these POZs result in non-convex decision spaces consisting of 192 convex sub-spaces, the value of N pop , N J AYA_iter , N COA_iter , N sub_pop and N runs are all increased compared to Case IV to cope with the challenges.
The best outputs from JAYA, CJAYA, MP-CJAYA and other algorithms including SA [51], GA [51], TS [51], PSO [51], MTS [51], TSA [52], DSPSO-TSA [52] and AIS [53] are summarized in Table 7. From the table we can observe that DSPSO-TSA has provided lower F best than JAYA, but it is not as lowest as CJAYA and MP-CJAYA, which obtains 32,710.0768$/h and 32,706.5158$/h respectively and ranks the second and first best value among all the algorithms.Furthermore, in addition to the best cost F best , the worst cost F worst and the mean cost F mean of the three version of JAYA algorithms are also compared with those above-mentioned methods in Table 8.It can be found that MP-CJAYA achieves the minimum value of F best , F worst and F mean at the same time, which are highlighted in bold font.Figure 10 is the distribution outlines of F best , we can notice that MP-CJAYA exhibits the best consistency in achieving minimum F best within 50 independent runs.One randomly chosen convergence curve is shown in Figure 11, from which we can see that CJAYA has improved the convergence rate and accuracy of basic JAYA, while MP-CJAYA has made further improvements of CJAYA in the rate of approaching the lowest cost.From the analysis above, it can be concluded that MP-CJAYA has the strongest capabilities of handling larger size of ELD problems with different constraint conditions.

Discussion and Conclusions
A novel multi-population based chaotic JAYA algorithm (MP-CJAYA) is proposed in this paper.By introducing the MP method and chaotic map to the basic JAYA algorithm, both the global exploration capability and the local searching capability have been greatly improved.MP-CJAYA is employed in five typical ELD cases to compare the performances with other well-established algorithms in terms of best solutions, convergence rate and robustness.The results have proved that MP-CJAYA algorithm has outstanding superiority to all the other compared algorithms in all cases.
It is noteworthy that for most of the meta-heuristic algorithms, parameter setting is critical for the quality of their results.But for MP-CJAYA, it does not require for specific algorithm parameters except for common parameters.What's more, it is observed that the common parameter population size (N pop ) does not affect the performance of its final optimal solution significantly, as shown in Figure 12.With increased N pop of 30, 50, 100 and 200 under the same circumstances, a slightly steady improvement of the convergence rate can be observed at initial part of the iteration.However, after about 5000 iterations, the differences among those curves become difficult to be observed and they all have reached the same best solution, which has proved that MP-CJAYA algorithm is not highly dependent on the common parameter N pop .

Discussion and Conclusions
A novel multi-population based chaotic JAYA algorithm (MP-CJAYA) is proposed in this paper.By introducing the MP method and chaotic map to the basic JAYA algorithm, both the global exploration capability and the local searching capability have been greatly improved.MP-CJAYA is employed in five typical ELD cases to compare the performances with other well-established algorithms in terms of best solutions, convergence rate and robustness.The results have proved that MP-CJAYA algorithm has outstanding superiority to all the other compared algorithms in all cases.
It is noteworthy that for most of the meta-heuristic algorithms, parameter setting is critical for the quality of their results.But for MP-CJAYA, it does not require for specific algorithm parameters except for common parameters.What's more, it is observed that the common parameter population size (Npop) does not affect the performance of its final optimal solution significantly, as shown in Figure 12.With increased Npop of 30, 50, 100 and 200 under the same circumstances, a slightly steady improvement of the convergence rate can be observed at initial part of the iteration.However, after about 5000 iterations, the differences among those curves become difficult to be observed and they all have reached the same best solution, which has proved that MP-CJAYA algorithm is not highly dependent on the common parameter Npop.As a newly proposed meta-heuristic algorithm, even though MP-CJAYA has gained the most outstanding superiority in this paper, it still has not been used for solving other optimization issues, except for the ELD problem.Hence, authors are planning to apply it to different kinds of optimization issues in the future to broaden its applications, such as multiple fuel options, micro grid power dispatch problems and multi-objective scheduling optimization problems.

Step 13 :
Check the stopping condition.If the current iteration number iter reaches iter JAYA N _ , stop the loop and report the best solution; otherwise follow the next step and set 1 + = iter iter .

Figure
Figure Fuel cost convergence characteristic of 15-units system (P D = 2630 MW).

Figure 12 .
Figure 12.Convergence characteristics of MP-CJAYA with varying population sizes for case V.

Table 1 .
Parameters and constraint conditions of the ELD cases.
NA indicates the cost value is not found.
Fuel cost convergence characteristic of 13-units system (P D = 2520 MW).
Fuel cost for 20 independent runs of 6-units system (P D = 1263 MW).