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

Abstract

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.

1. 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 (ABC) [13], firefly algorithm [14], harmony search (HS) [15] and teaching-learning-based optimization (TLBO) [16]. Additionally, hybrid meta-heuristic optimization approaches built by the combination between conventional methods and meta-heuristic methods or among the meta-heuristic methods have also been reported to deal with the ELD problem, such as DE-PSO method [17], HS-DE method [18], GA-PS-SQP algorithm [19] and Quantum-PSO method [20]. Even though hybrid methods offer much faster convergence rates, the combination may lead to increased numbers of parameters which causes more difficulties in selecting the proper value for each one. Hence, a new method with strong searching ability and less number of control parameters is needed.

The JAYA algorithm is a newly developed yet advanced heuristic algorithm for solving constrained and unconstrained optimization problems [21]. Different from other algorithms requiring for algorithm-specific parameters in addition to common parameters, the JAYA algorithm does not require any algorithm-specific parameters except for two common parameters named the population size (N_pop) and the number of iteration (N_iter). This significant benefit makes it popular in various real-world optimization problems such as optimum power flow [22], heat exchangers [23], photovoltaic models [24], thermal devices [25], MPPT of PV system [26], constrained mechanical design optimization [27], modern machining processes [28] and PV-DSTATCOM [29]. However, as a newly developed algorithm, the JAYA algorithm still has some disadvantages even though the number of parameters is less and the convergence rate is accelerated. Since there is only guidance as approach to get close to the best solution and get away from the worst solution, the population diversity may not be maintained efficiently, easily leading to local optimal solutions.

The multi-population based optimization method (MP) is applied for improving the search diversity by dividing the whole population into a certain number of sub-populations and distributing them throughout the search area so that the problem changes can be monitored more effectively. The MP method is aimed at maintaining population diversity during the search period by distributing different sub-populations to different search spaces. Each population is used to either intensify or diversifying the search process [30,31]. The interaction among the sub-populations occurs by dividing and merging process as long as a change in the solution is detected. Branke proposed a multi-population evolutionary algorithm in [32]. Turky and Abdullah proposed a multi-population electromagnetic algorithm and a multi-population harmony search algorithm in [33,34]. Nseef proposed a multi-population artificial bee colony algorithm in [35]. The published literature have demonstrated that employing MP method is useful for maintaining the population diversity when dealing with various problem changes.

Its worthy to be noted that the MP optimization method has superior behaviors because [36]:

(1)

By dividing the whole population into sub-populations, population diversity can be maintained since the sub-populations are located in different regions of the problem landscape.

(2)

With the ability to search various regions simultaneously, it is able to track the movement of optimum value more effectively.

(3)

Population-based optimization algorithms can be easily integrated with MP method.

At the same time the chaotic optimization algorithm (COA) which adopts chaotic sequences instead of random sequences is also employed here. Due to the non-repetitive characteristics of chaotic sequences, the COA can execute with shorter execution time and more robust mechanisms than stochastic ergodic searches that depending on random probabilities. It also has the feature of easy implementation in meta-heuristic algorithms, such as chaotic evolutionary algorithms [37], chaotic ant swarm optimization [38], chaotic harmony search algorithm [39], chaotic particle swarm optimization [40], chaotic firefly algorithm [41]. The choice of chaotic sequences is justified theoretically by their unpredictability, i.e., by their spread-spectrum characteristic, non-periodic, complex temporal behavior and ergodic properties. Simulation results from the abovementioned literature have demonstrated that the application of deterministic chaotic signals to meta-heuristic algorithms is a promising strategy in engineering applications. In this paper, COA has been applied twice:

(1)

During the initialization step, chaotic sequences generated by a chaotic map are used to initialize the initial solutions.

(2)

During the iteration step, COA is conducted to search further around the solution obtained by former algorithm to enhance the global convergence and to prevent to be trapped on local optima.

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.

2. 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].

2.1. Objective Function

The objective function is to sum up all the costs of committed generators as expressed below:

$$\min F={\displaystyle \sum _{i=1}^n{F}_i(P_i)}$$

(1)

where $n$ is the total generator number in power systems, $F_i(P_i)$ is the cost function of $i\mathrm{th}$ generator with output $P_i$.

Approximately, the cost function can be expressed as a quadratic polynomial by the following equation:

$$F_i(P_i)=a_i{P}_i{}^2+b_i{P}_i+c_i$$

(2)

where $a_i$, $b_i$, $c_i$ are the cost coefficients of $i\mathrm{th}$ 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:

$$F_i(P_i)=a_i{P}_i{}^2+b_i{P}_i+c_i+\left|e_i\times \sin (f_i\times (P_i{}^{\min }-P_i))\right|$$

(3)

where $e_i$ and $f_i$ are cost coefficients of $i\mathrm{th}$ 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:

$$\min F={\displaystyle \sum _{i=1}^n(a_i{P}_i{}^2+b_i{P}_i+c_i+\left|e_i\times \sin (f_i\times (P_i{}^{\min }-P_i))\right|)}$$

(4)

2.2. Constrained Functions

2.2.1. 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:

$${\displaystyle \sum _{i=1}^n{P}_i=P_{demand}+P_{loss}}$$

(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:

$$P_{loss}={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{P}_i{B}_{ij}{P}_j}}+{\displaystyle \sum _{i=1}^n{B}_{i0}}{P}_i+B_{00}$$

(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.

2.2.2. Generating Capacity

The real output $P_i$ generated by a available unit must be ranged between its minimum limit and maximum limit:

$$P_i^{\min }\le P_i\le P_i\le P_i^{\max }$$

(7)

where $P_i^{\min }$ and $P_i^{\max }$ are the minimum and maximum limits of $i\mathrm{th}$ generator.

2.2.3. 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:

$$\max (P_i{}^{\min },P_i{}^0-D{R}_i)\le P_i\le \min (P_i{}^{\max },P_i{}^0+U{R}_i)$$

(8)

where $P_i$ is the present power output, $P_i{}^0$ is the previous power output, $U{R}_i$ and $D{R}_i$ is the up-ramp and down-ramp limit of generator $i$ respectively.

2.2.4. 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:

$$\begin{array}{l}{P}_i{}^{\min }\le P_i\le P_{i,1}{}^{lower}\\ P_{i,j-1}{}^{upper}\le P_i\le P_{i,j}{}^{lower}\\ P_{i,n_i}{}^{upper}\le P_i\le P_i{}^{\max }\end{array}$$

(9)

where $j$ is the index of POZs, $n_i$ is the total number of POZs where $j\in [1,n_i]$, $P_{i,j}{}^{lower}$ and $P_{i,j}{}^{upper}$ are the lower and upper bounds of the $j\mathrm{th}$ POZ of the $i\mathrm{th}$ unit, respectively.

3. 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.

3.1. 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 $j\mathrm{th}$ variable for the $k\mathrm{th}$ candidate during the $i\mathrm{th}$ iteration, then the new modified value $X_{j,k,i}\prime$ by JAYA algorithm is calculated by:

$${X^{\prime }}_{j,k,i}=X_{j,k,i}+r_{1,j,i}\times (X_{j,best,i}-\left|X_{j,k,i}\right|)-r_{2,j,i}\times (X_{j,worst,i}-\left|X_{j,k,i}\right|)$$

(10)

where $X_{j,k,i}\prime$ is the updated value of $X_{j,k,i}$. $X_{j,best,i}$ and $X_{j,worst,i}$ are the values of the $j\mathrm{th}$ variable for $F{(X)}_{best}$ and $F{(X)}_{worst}$ during the $i\mathrm{th}$ iteration respectively. $r_{1,j,i}$ and $r_{2,j,i}$ are two random numbers ranged in [0, 1]. The term ‘$r_{1,j,i}\times (X_{j,best,i}-\left|X_{j,k,i}\right|)$’ indicates the tendency of the solution to move closer to the best solution and the term ‘$r_{2,j,i}\times (X_{j,worst,i}-\left|X_{j,k,i}\right|)$’ indicates the tendency of the solution to avoid the worst solution. Suppose $F(X)\prime$ is the modified value of $F(X)$, if $F(X)\prime$ provides better value than $F(X)$, then $X_{j,k,i}$ is replaced by $X_{j,k,i}\prime$ and $F(X)$ is replaced by $F(X)\prime$; 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 ($N_{JAYA\_iter}$) 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:

$$X_{j,k,i}=X_j{}^{\min }+(X_j{}^{\max }-X_j{}^{\min }).*rand(N_{pop},N_{gen})$$

(11)

where $j\in [1,N_{gen}],$ $k\in [1,N_{pop}]$, $i\in [1,N_{JAYA\_iter}]$, $X_j{}^{\min }$ and $X_j{}^{\max }$ are the lower and upper limits of $j\mathrm{th}$ generator given by generating capacity limits in Equation (7).

Step 3: Apply constraints. Apply the constraints in Section 2.2 by using Equations (5)–(9).

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 $F(X)$.

Set iter = 1.

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 $X_{j,k,i}\prime$ by Equation (10).

Step 7: Apply constraints. Apply the constraints in Section 2.2 by using Equations (5)–(9).

Step 8: Evaluate the new solution. Calculate the new objective function value $F(X)\prime$ by Equation (3) with considering the valve-point effect or Equation (2) without considering the valve-point effect.

Step 9: Compare. The new $F(X)\prime$ is compared with the old $F(X)$, the values are updated as follows:

If $F(X)\prime <F(X)$

then $F(X)=F(X)\prime$ and $X_{j,k,i}=X_{j,k,i}\prime$;

Otherwise, keep the old value.

Step 10: Check the stopping condition. If the current iteration number $iter<N_{JAYA\_iter}$, then $iter=iter+1$ and return to Step 5. Otherwise, stop the procedure.

3.2. 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:

$$Z(k+1)=f(Z(k)) (k=0,1,2,3,...)$$

(12)

A widely used logistic map which appears in nonlinear dynamics of biological population evidencing chaotic behavior is shown below [43].

$$Z_i(k+1)=\alpha \times Z_i(k)(1-Z_i(k))$$

(13)

where $i$ is the serial number of chaotic variables, $k$ is the iteration number. The initial value of the $i\mathrm{th}$ chaotic variable is $Z_i(0)$ where $Z_i(0)\notin \left\{0.0,0.25,0.5,0.75,1.0\right\}$. $\alpha =4$ is used in this paper. It is obvious that $Z_i(k+1)\in (0,1)$ under the conditions of $Z_i(0)\in (0,1)$.

The procedure for the CJAYA algorithm to solve ELD problem is provided here, the symbol $\ast$ 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 ($N_{JAYA\_iter}$) 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$\ast$: 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:

$$Z_{j,k,q}=4\times Z_{j,k-1,q}(1-Z_{j,k-1,q})$$

(14)

Here $j\in [1,N_{gen}]$, $k\in [1,N_{pop}]$, $q\in [1,N_{COA\_iter}]$.

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:

$$X_{j,k,i}=X_j{}^{\min }+(X_j{}^{\max }-X_j{}^{\min }).*Z_{j,k,q}$$

(15)

where $X_j{}^{\min }$ and $X_j{}^{\max }$ are the lower and upper limits of $j\mathrm{th}$ 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$\ast$: 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:

$$X_{j,k,i}\prime =X_{j,k,i}+R\times Z_{j,k,q}$$

(16)

where $R$ is a constant, $R\times Z_{j,k,q}$ 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 $iter<N_{JAYA\_iter}$, then $iter=iter+1$ and return to Step 6. Otherwise, stop the procedure.

3.3. 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 $\ast$ 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 ($N_{JAYA\_iter}$), 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:

$$N_{sub\_pop}=N_{pop}/K$$

(17)

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$\ast$: 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 $N_{JAYA\_iter}$, stop the loop and report the best solution; otherwise follow the next step and set $iter=iter+1$.

Step 14$\ast$: 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.

Algorithm 1 Pseudo code of the MP-CJAYA Algorithm

Begin

Initialize $N_{pop}$,$N_{JAYA\_iter}$,$N_{gen}$,$N_{COA\_iter}$ and $K$;

Generate initial solution $X_{j,k,i}$ by chaotic sequence;

Calculate objective function value $F(X)$;

Set $iter=1$

While$iter<N_{JAYA\_iter}$do

Divide the whole population $P$ into $K$ sub-populations by Equation (17) randomly

$P_1,P_2,...,P_{K-1},P_K$

For$m=1\to K$do

Confirm $X_{j,best,i}$ and $X_{j,\mathrm{w}orst,i}$ within $P_m$

For$k=1\to N_{sub\_pop}$do

Generate new solution $X_{j,k,i}\prime$ by Equation (10)

End for

If$F(X_{j,k,i}\prime )$ is better than $F(X_{j,k,i})$ then

$X_{j,k,i}=X_{j,k,i}\prime$

$F(X_{j,k,i})=F(X_{j,k,i}\prime )$

Else

Keep the old value

End if

For$k=1\to N_{sub\_pop}$do

Generate new solution $X_{j,k,i}\prime$ by Equation (16)

If$F(X_{j,k,i}\prime )$ is better than $F(X_{j,k,i})$ then

$X_{j,k,i}=X_{j,k,i}\prime$

$F(X_{j,k,i})=F(X_{j,k,i}\prime )$

Else

Keep the old value

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

4. 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:

Case I.

3-units system for load demand of 850 MW.

Case II.

13-units system for load demand of 2520 MW.

Case III.

40-units system for load demand of 10500 MW.

Case IV.

6-units system for load demand of 1263 MW.

Case V.

15-units system for load demand of 2630 MW.

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.

4.1. 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. Parameters and constraint conditions of the ELD cases.

	Case I			Case II			Case III			Case IV			Case V
	JAYA	CJAYA	MP-CJAYA	JAYA	CJAYA	MP-CJAYA	JAYA	CJAYA	MP-CJAYA	JAYA	CJAYA	MP-CJAYA	JAYA	CJAYA	MP-CJAYA
$N_{pop}$	20	20	20	50	50	50	100	100	100	20	20	20	100	100	100
$N_{JAYA\_iter}$	500	500	500	3000	3000	3000	5000	5000	5000	1000	1000	1000	5000	5000	5000
$N_{COA\_iter}$	-	20	20	-	20	20	-	30	30	-	20	20	-	30	30
$N_{sub\_pop}$	-	-	10	-	-	10	-	-	20	-	-	10	-	-	20
$N_{runs}$	20	20	20	30	30	30	50	50	50	20	20	20	50	50	50
Valve-point effect	●			●			●			-			-
Ramp-rate limit	-			-			-			●			●
POZ	-			-			-			●			●
$P_{loss}$				-			-			●			●

. 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. Best outputs for 3-units system ($P_D$ = 850 MW).

Unit	GA [45]	EP [45]	EP-SQP [45]	PSO [45]	PSO-SQP [45]	CPSO [46]	CPSO-SQP [46]	JAYA	CJAYA	MP-CJAYA
1	398.700	300.264	300.267	300.268	300.267	300.267	300.266	350.3314	350.0254	350.2464
2	399.600	400.000	400.000	400.000	400.000	400.000	400.000	400.0000	400.0000	400.0000
3	50.100	149.736	149.733	149.732	149.733	149.733	149.734	99.6453	99.9511	99.7576
$P_{total}$(MW)	848.400	850.000	850.000	850.000	850.000	850.000	850.000	849.977	849.977	850.004
$F_{mean}$($/h)	8234.72	8234.16	8234.09	8234.72	8234.07	NA	NA	8382.10	8289.41	8232.06
$F_{best}$($/h)	8222.07	8234.07	8234.07	8234.07	8234.07	8234.07	8234.07	8230.23	8226.18	8223.29

NA indicates the cost value is not found.

. 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.

4.2. 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. Best outputs for 13-units system ($P_D$ = 2520 MW).

Unit	GA [47]	SA [47]	HSS [47]	EP-SQP [45]	PSO-SQP [45]	CPSO [46]	CPSO-SQP [46]	JAYA	CJAYA	MP-CJAYA
1	628.32	668.40	628.23	628.3136	628.3205	628.32	628.31	628.3185	628.3185	628.3183
2	356.49	359.78	299.22	299.1715	299.0524	299.83	299.83	299.2009	299.1992	299.0170
3	359.43	358.20	299.17	299.0474	298.9681	299.17	299.16	306.9105	299.1993	299.1428
4	159.73	104.28	159.12	159.6399	159.4680	159.70	159.73	159.7339	159.7330	159.5714
5	109.86	60.36	159.95	159.6560	159.1429	159.64	159.73	159.7337	159.7331	159.6930
6	159.73	110.64	158.85	158.4831	159.2724	159.67	159.73	159.7338	159.7331	159.6801
7	159.63	162.12	157.26	159.6749	159.5371	159.64	159.73	109.8673	159.7330	159.7270
8	159.73	163.03	159.93	159.7265	158.8522	159.65	159.73	159.7342	159.7330	159.7328
9	159.73	161.52	159.86	159.6653	159.7845	159.78	159.73	159.7340	159.7331	159.5119
10	77.31	117.09	110.78	114.0334	110.9618	112.46	109.07	114.8012	110.0403	111.0288
11	75.00	75.00	75.00	75.00	75.00	74.00	77.40	114.8001	114.7994	77.1661
12	60.00	60.00	60.00	60.00	60.00	56.50	55.00	92.4018	55.0000	55.0014
13	55.00	119.58	92.62	87.5884	91.6401	91.64	92.85	55.0027	55.0000	92.3862
$P_{total}$(MW)	2520	2520	2520	2520	2520	2520	2520	2519.97	2519.96	2519.98
$F_{mean}$($/h)	NA	NA	NA	NA	NA	NA	NA	24,476.5247	24,385.7604	24,228.1331
$F_{best}$($/h)	24,398.23	24,970.91	24,275.71	24,266.44	24,261.05	24,211.56	24,190.97	24,220.7529	24,178.8040	24,175.5444

NA indicates the cost value is not found.

. 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.

4.3. 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, 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. Best outputs for 40-units system ($P_D$ = 10,500 MW).

Unit	PSO-LRS [48]	NPSO [48]	NPSO-LRS [48]	SPSO [49]	PC-PSO [49]	SOH-PSO [49]	JAYA	CJAYA	MP-CJAYA
1	111.9858	113.9891	113.9761	113.97	113.98	110.80	114.0000	113.5264	114.0000
2	110.5273	113.6334	113.9986	114.00	114.00	110.80	111.6651	110.7998	110.7998
3	98.5560	97.5500	97.4141	109.19	97.26	97.40	119.9876	120.0000	97.3999
4	182.9266	180.0059	179.7327	179.77	179.51	179.73	188.2606	179.7331	179.7331
5	87.7254	97.0000	89.6511	97.00	89.38	87.80	96.9763	97.0000	93.1276
6	139.9933	140.0000	105.4044	91.01	105.20	140.00	139.9488	140.0000	140.0000
7	259.6628	300.0000	259.7502	259.87	259.55	259.60	264.0949	300.0000	300.0000
8	297.7912	300.0000	288.4534	286.99	286.90	284.60	299.9814	284.5997	284.5997
9	284.8459	284.5797	284.6460	284.09	284.71	284.60	284.9042	284.5997	284.5997
10	130.0000	130.0517	204.8120	204.05	206.24	130.00	130.0908	130.0000	130.0000
11	94.6741	243.7131	168.8311	168.40	166.52	94.00	94.0011	94.0000	94.0000
12	94.3734	169.0104	94.00	94.00	94.00	94.00	94.0000	94.0000	94.0000
13	214.7369	125.0000	214.7663	212.30	214.56	304.52	125.1028	125.0000	125.0000
14	394.1370	393.9662	394.2852	393.76	392.76	304.52	394.2529	394.2794	394.2794
15	483.1816	304.7586	304.5187	303.62	306.24	394.28	484.1262	394.2794	394.2794
16	304.5381	304.5120	394.2811	392.05	394.88	394.28	304.5950	394.2794	394.2794
17	489.2139	489.6024	489.2807	489.49	489.26	489.28	490.8265	489.2794	489.2794
18	489.6154	489.6087	489.2832	489.35	489.82	489.28	489.3438	489.2794	489.2794
19	511.1782	511.7903	511.2845	512.39	510.62	511.28	511.3775	511.2794	511.2794
20	511.7336	511.2624	511.3049	511.21	511.68	511.27	512.1395	511.2794	511.2794
21	523.4072	523.3274	523.2916	522.61	523.52	523.28	523.6621	523.2794	523.2794
22	523.4599	523.2196	523.2853	523.65	523.26	523.28	523.3534	523.2794	523.2794
23	523.4756	523.4707	523.2797	523.06	523.98	523.28	524.9677	523.2794	523.2794
24	523.7032	523.0661	523.2994	520.72	523.21	523.28	524.2850	523.2794	523.2794
25	523.7854	523.3978	523.2865	524.86	523.54	523.28	522.9279	523.2794	523.2794
26	523.2757	523.2897	523.2936	525.22	523.10	523.28	523.2298	523.2794	523.2794
27	10.0000	10.0208	10.0000	10.00	10.00	10.00	10.0000	10.0000	10.0000
28	10.6251	10.0927	10.0000	10.00	10.00	10.00	10.0047	10.0000	10.0000
29	10.0727	10.0621	10.0000	10.00	10.00	10.00	10.0000	10.0000	10.0000
30	51.3321	88.9456	89.0139	87.64	89.05	97.00	97.0000	97.0000	87.7999
31	189.8048	189.9951	190.0000	190.00	190.00	190.00	190.0000	190.0000	190.0000
32	189.7386	190.0000	190.0000	190.00	190.00	190.00	189.9503	190.0000	190.0000
33	189.9122	190.0000	190.0000	190.00	190.00	190.00	190.0000	190.0000	190.0000
34	199.3258	165.9825	199.9998	200.00	200.00	185.20	169.8860	164.7998	200.0000
35	199.3065	172.4153	165.1397	167.18	164.78	164.80	199.8549	200.0000	200.0000
36	192.8977	191.2978	172.0275	172.12	172.89	200.00	199.9896	200.0000	200.0000
37	110.0000	109.9893	110.0000	110.00	110.00	110.00	109.9712	110.0000	110.0000
38	109.8628	109.9521	110.0000	110.00	110.00	110.00	109.9977	110.0000	110.0000
39	92.8751	109.8733	93.0962	95.58	94.24	110.00	109.9871	110.0000	110.0000
40	511.6883	511.5671	511.2996	510.85	511.36	511.28	511.2250	511.2794	511.2794
$P_{total}$(MW)	10,499.9452	10,499.9989	10,499.9871	10,500	10,500	10,500	10,499.97	10,499.97	10,499.97
$F_{mean}$($/h)	122,558.4565	122,221.3697	122,209.3185	NA	NA	121,853.57	122,581.85	121,926.77	121,861.08
$F_{best}$($/h)	122,035.7946	121,704.7391	121,664.43	122,049.66	121,767.89	121,501.14	121,799.88	121,516.97	121,480.10

NA indicates the cost value is not found.

. 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 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.

4.4. 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 ($P_{loss}$), 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. Best outputs for 6-units system ($P_D$ = 1263 MW).

Generator	SA [51]	GA [51]	TS [51]	PSO [51]	MTS [51]	PSO-LRS [48]	NPSO [48]	NPSO-LRS [48]	JAYA	CJAYA	MP-CJAYA
1	478.1258	462.0444	459.0753	447.5823	448.1277	447.4440	447.4734	446.96	457.9858	452.3884	444.7000
2	163.0249	189.4456	185.0675	172.8387	172.8082	173.3430	173.1012	173.3944	176.8785	162.1065	171.1458
3	261.7146	254.8535	264.2094	261.3300	262.5932	263.3646	262.6804	262.3436	250.0717	256.4885	253.8111
4	125.7665	127.4296	138.1222	138.6812	136.9605	139.1279	139.4156	139.5120	129.3748	142.1863	134.8118
5	153.7056	151.5388	154.4716	169.6781	168.2031	165.5076	165.3002	164.7089	172.8886	170.7924	175.4557
6	93.7965	90.7150	74.9900	85.8963	87.3304	87.1698	87.9761	89.0162	88.4618	91.5015	95.6913
$P_{total}$ (MW)	1276.1339	1276.0270	1275.94	1276.0066	1276.0232	1275.95	1275.96	1275.94	1275.6611	1275.4637	1275.6158
$P_{loss}$ (MW)	13.1317	13.0268	12.9422	13.0066	13.0205	12.9571	12.9470	12.9361	12.6665	12.4444	12.6030
$F_{best}$ ($/h)	15,461.10	15,457.96	15,454.89	15,450.14	15,450.06	15,450.00	15,450.00	15,450.00	15,447.09	15,446.71	15,446.17

. 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. Results comparison of 6-units system ($P_D$ = 1263 MW).

	${\mathit{F}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}(\mathbf{\$}/\mathbf{h})$	${\mathit{F}}_{\mathit{w}\mathit{o}\mathit{r}\mathit{s}\mathit{t}}(\mathbf{\$}/\mathbf{h})$	${\mathit{F}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}}(\mathbf{\$}/\mathbf{h})$
SA [51]	15,461.10	15,545.50	15,488.98
GA [51]	15,457.96	15,524.69	15,477.71
TS [51]	15,454.89	15,498.05	15,472.56
PSO [51]	15,450.14	15,491.71	15,465.83
MTS [51]	15,450.06	15,453.64	15,451.17
PSO-LRS [48]	15,450.00	15,455.00	15,454.00
NPSO [48]	15,450.00	15,454.00	15,452.00
NPSO-LRS [48]	15,450.00	15,452.00	15,450.50
JAYA	15,447.09	15,622.16	15,500.11
CJAYA	15,446.71	15,484.34	15,461.62
MP-CJAYA	15,446.17	15,451.68	15,449.23

. 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.

4.5. Case V: 15-Units System for Load Demand of 2630 MW

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 $N_{pop}$, $N_{JAYA\_iter}$, $N_{COA\_iter}$, $N_{sub\_pop}$ 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. Best outputs for 15-units system ($P_D$ = 2630 MW).

Unit	SA [51]	GA [51]	TS [51]	PSO [51]	MTS [51]	TSA [52]	DSPSO-TSA [52]	AIS [53]	JAYA	CJAYA	MP-CJAYA
1	453.6646	445.5619	453.5374	454.7167	453.9922	440.500	453.627	441.159	455.0000	455.0000	455.0000
2	377.6091	380.0000	371.9761	376.2002	379.7434	346.800	379.895	409.587	379.9848	380.0000	380.0000
3	120.3744	129.0605	129.7823	129.5547	130.0000	110.880	129.482	117.298	130.0000	130.0000	130.0000
4	126.2668	129.5250	129.3411	129.7083	129.9232	122.460	129.923	131.258	129.9821	130.0000	130.0000
5	165.3048	169.9659	169.5950	169.4407	168.0877	177.740	168.956	151.011	169.6535	170.0000	170.0000
6	459.2455	458.7544	457.9928	458.8153	460.0000	459.110	459.907	466.258	460.0000	460.0000	460.0000
7	422.8619	417.9041	426.8879	427.5733	429.2253	406.410	429.971	423.368	429.0688	430.0000	430.0000
8	126.4025	97.8230	95.1680	67.2834	104.3097	107.550	103.673	99.948	81.7235	106.1556	71.8662
9	54.4742	54.2933	76.8439	75.2673	35.0358	107.270	34.909	110.684	51.3258	25.0000	58.9683
10	149.0879	144.2214	133.5044	155.5899	155.8829	140.560	154.593	100.229	146.6714	160.0000	160.0000
11	77.9594	77.3002	68.3087	79.9522	79.8994	78.470	79.559	32.057	79.1805	80.0000	80.0000
12	93.9489	77.0371	79.6815	79.8947	79.9037	74.170	79.388	78.815	80.0000	80.0000	80.0000
13	25.0022	31.1537	28.3082	25.2744	25.0220	31.950	25.487	23.568	25.0000	25.0000	25.0000
14	16.0636	15.0233	17.7661	16.7318	15.2586	37.380	15.952	40.258	27.7503	15.0000	15.0000
15	15.0196	33.6125	22.8446	15.1967	15.0796	22.470	15.640	36.906	15.0000	15.0000	15.0000
$P_{total}$ (MW)	2663.29	2661.23	2661.53	2661.19	2661.36	2663.70	2660.96	2662.04	2660.3408	2661.1556	2660.8346
$P_{loss}$ (MW)	33.2737	31.2363	31.4100	31.1697	31.3523	33.8110	30.9520	32.4075	30.3442	31.1643	30.8346
$F_{best}$ ($/h)	32,786.40	32,779.81	32,762.12	32,724.17	32,716.87	32,918.00	32,715.06	32,854.00	32,716.8706	32,710.0768	32,706.5158

. 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. Results comparison of 15-units system ($P_D$ = 2630 MW).

	${\mathit{F}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}(\mathbf{\$}/\mathbf{h})$	${\mathit{F}}_{\mathit{w}\mathit{o}\mathit{r}\mathit{s}\mathit{t}}(\mathbf{\$}/\mathbf{h})$	${\mathit{F}}_{\mathit{m}\mathit{e}\mathit{a}\mathit{n}}(\mathbf{\$}/\mathbf{h})$
SA [51]	32,786.40	33,028.95	32,869.51
GA [51]	32,779.81	33,041.64	32,841.21
TS [51]	32,762.12	32,942.71	32,822.84
PSO [51]	32,724.17	32,841.38	32,807.45
MTS [51]	32,716.87	32,796.15	32,767.21
TSA [52]	32,917.87	33,245.54	33,066.76
DSPSO-TSA [52]	32,715.06	32,730.39	32,724.63
AIS [53]	32,854.00	32,892.00	32,873.25
JAYA	32,716.8706	32,967.8314	32,789.1472
CJAYA	32,710.0768	32,828.6554	32,740.0719
MP-CJAYA	32,706.5158	32,708.8736	32,706.7150

. 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.

5. 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.

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.