Stochastic fractal search based method for economic load dispatch

This paper presents a nature-inspired meta-heuristic, called a stochastic fractal search based method (SFS) for coping with complex economic load dispatch (ELD) problem. Two SFS methods are introduced in the paper by employing two different random walk generators for diffusion process in which SFS with Gaussian random walk is called SFS-Gauss and SFS with Levy Flight random walk is called SFS-Levy. The performance of the two applied methods is investigated comparing results obtained from three test system. These systems with 6, 10, and 20 units with different objective function forms and different constraints are inspected. Numerical result comparison can confirm that the applied approach has better solution quality and fast convergence time when compared with some recently published standard, modified, and hybrid methods. This elucidates that the two SFS methods are very favorable for solving the ELD problem.


Introduction
Economic load dispatch (ELD) problem is becoming more important in power system operation and control. The prime objective of the ELD problem is to minimize the total fuel cost by economically distributing power of generating units to electric load. In addition, load demand, all physical and operational constraints are required to be within predetermined bounds. In traditional ELD problem, a fuel cost function of generators is considered as the single quadratic cost function with linear constraint [1]. In practical, realistic ELD problem must take complex and nonlinear characteristic with many equality and inequality constraints into account to provide the completeness for the ELD problem formulation. Thus, fuel cost curve of thermal units should be presented as non-smooth presented form, a piecewise function when thermal units are supplied by multi-fuel sources like coal, natural gas, and oil [2]. Also, to get more precise cost model, the valve-point effects and the prohibited zones must also be taken consideration [3,4]. The complexity of the problem dramatically increases once both multi-fuel option and valve-point effects are considered simultaneously. Generally, the ELD problem on later is more and more difficult when taking many power systems and generator constraints into account. Over the past decades, there were many applied methods with the task of solving ELD problem such as Lambda Iteration method [5], Dynamic Programming (DP) [6], Gradient Method [7], Lagrangian Relaxation algorithm [8], and Hopfield neural network based numerical method (HNNNM) [9]. For the classical methods above, parameters of these algorithms are surveyed and selected after many trial run times, which helps to find a global solution in a short time. However, the process of setting parameters takes much time. As complex ELD problem has non-convex features and various nonlinear constraints, the mentioned classical methods cannot afford to handle and result in low convergence. A series of novel methodologies have been born called meta-heuristic method to deal with these disadvantages such as Genetic algorithm (GA) [10], Firefly algorithm [11], Particle Swarm Optimization (PSO) [12], Differential Evolution (DE) algorithm [13], Anti-predatory particle swarm optimization (APPSO) [14], Biogeography-Based Optimization (BBO) [15], and Ant Lion algorithm (ALO) [16]. Because of their outstanding characteristics, such meta-heuristic methods proved their efficiency for solving the aforementioned difficulties. Consequently, meta-heuristic methods have been received much more curiosity by researchers. Besides, a large number of scientists in many engineering fields have been constantly strived and selected the strong points of methods to modify/improve them into the promising methods such as Colonial Competitive Differential Evolution (CCDE) [17], Efficient Real-Coded Genetic algorithm (ERCGA) [18], Improved Real-Coded Genetic algorithm (IRCGA) [19], and Modified Cuckoo Search algorithm (MCSA) [20]. As known, obtained results of the meta-heuristic family are better than that of the standard methods although they may still exist some weaknesses. Hence, improving meta-heuristic ones is the expectation of researchers with the goal of finding the best solution quality in exploring and exploiting search space effectively.
In this paper, a nature-inspired Stochastic Fractal Search (SFS) algorithm is applied to determine the minimum cost of the ELD problem. SFS was first recommended by Salimi [31] and applied to optimize twenty-three benchmark functions with a quite good solution quality. In the paper, our purpose is to investigate the efficacy and robustness of the SFS method on various standard IEEE systems through using two different random walk generators for diffusion process. Firstly, SFS with Gaussian random walk is called SFS-Gauss and secondly, SFS with Levy Flight random walk is called SFS-Levy. In addition, the achievement of SFS method has also competed against other ones available in the literature.

Problem Formulation 2.1. Formulation of the Smooth ELD Problem
The traditional fuel cost function is often represented as a single quadratic polynomial function polynomial function in (1):  (1) can indicate that the fuel cost for each MWh is different for different power output of thermal generating unit. Thus, the major target of the ELD problem is to reduce the total fuel cost of all thermal generating units and it can be described as the following model:

Formulation of the Non-smooth ELD Problem 2.2.1. ELD Problem Considering Valve-point Effects
In the practical power system, thermal units often use many valve for adjusting their power output. This makes the fuel cost function become discontinuous form as shown in (3).

ELD Problem Considering Multi-fuel Options
Since the generators are supplied by various fuel sources such as coal, natural gas, oil etc., the total fuel cost function of each unit can be represented by a piecewise quadratic cost function as follows:

ELD Problem Considering Both Valve-point Effects and Multiple Fuel Options
The ELD problem will be practical and more accurate if both valve-point effects and multiple fuel options are considered as the following [22]. 2 min min max sin(q ( )) , for fuel 1, P

Constrains 2.3.1. Generating Capacity Limit
A real power output of units is generated that must be lied in the range of their lower and their upper limit as:

Power Balance Constraint
The formula of the generator power balance constraint with considering the total transmission power losses are presented by: where PL is calculated by the Kron's loss formula expressed as:

Stochastic Fractal Search
The SFS algorithm that was formulated by Salimi in 2014, is a variant of Fractal Search by adding two update processes. So, the structure of SFS comprises of three update phases such as diffusion phase, the first update phase and the second update phase. As results, three new solution generations are created by SFS in each iteration. In SFS, the task of diffusion phase is to find solutions in small search space whilst the task of two update phases is to search solutions in large search space. Basically, SFS has a population corresponding to the number of points where each point Yd is represented as an optimal solution d (d=1, Np). At the beginning, all the points are randomly created and their fitness function are calculated to find the best solution Ybest among all solutions in the population. Then, SFS continues to perform the iterative search process with three phases above. The detail of three phase is described as follows:

Diffusion Phase
Based on the previous points, the first new solutions are produced by using one of two random walks as Levy flight and Gaussian. In this phase, each solution (point) Yd diffuses around its position into a number of new diffusion solutions Ydi where di=1,..,Ndf. The diffusion can be mathematically formulated as follows:

Diffusion Phase with Levy Flight
The equation of the diffusion phase using Levy flight random walk is performed in (9): where α > 0 is scale factor; ε is a normally distributed random numbers restricted to (0,1); Yd denoted the d th solution in the current population and ∆ is described by [20]: where randx and randy are two normally distributed stochastic variables.

Diffusion Phase with Gaussian Walk
The process of creating solution following Gaussian random walk is performed by: where bd is a binary number (0 or 1) dependent on comparison of a random number randd and walk factor w (0≤w≤1) as follows: and one out of 1 and 2 is used to create solutions, described (14): where is uniformly distributed random numbers; is the standard deviation. At the end of creating the new solutions, all new solutions are appraised by computing the fitness function and then the evaluation between each old solution and Ndf new solutions at each point is done in order to retain a better solution with the best fitness, named Yd.

The First Update Phase
First of all, all current points are assigned to a value of probability Pad, which is determined by: according to (16), the point with the best fitness has the highest probability and ranks at the last position otherwise stands at the first position. After ranking for all points, each point Yd in group is updated by the comparison of the probability Pad and a random number α1 (0<α1<1).
If Pad < α1, the d th point is updated like (17), otherwise it doesn't remain changed.
where 1 the new modified position of Yd; Y1 and Y2 is symbolize randomly selected points in the group. Through the first update phase, it is easily seen that all of points with a bad quality are often updated while the points with a better quality have low possibility to be newly updated.
After performing the second generation, once again, mechanism of the comparison is re-performed to select the better solution between old solution and new solution at each point, named Yd.

The Second Update Phase
Similar to the first update stage, the first step in the second update stage is also to determine rankd and Pad for each solution d and then Pad is compared to a random number within 0 and 1 for determining if the solution is newly updated. In case that considered solution d is accepted to be newly updated, there are two models to be used as follows: where random selected points Y3, Y4 and the best point Ybest are obtained from the first phase; ε' is random number in the range (0,1).

Implementation of SFS for Solving ELD Problem 4.1. Constraint Violation Handling Technique
A punishment function technique is employed in the ELD problem to deal with constraint violations by using two variable types such as dependent variable and control variable. From (7), P1d corresponding to power output of the 1 st unit of the d th solution is selected to be a dependent variable. Other variables from Pd2 to PdN are control variables and included in each solution d. Thus, these control variables are supposed to be given and then the dependent variable needs to be determined. The value of P1d obtained has no assurance that satisfies its limit power as (6). Therefore, the violation of the dependent variable must be penalized if one occurs and is calculated by: where PUNd is the violation of solution d Thus, in order to optimize the ELD problem related to the cost function (1)-(5), punishment function PUNd must be considered in fitness function in the case of occurring violation of value of P1d. Accordingly, the fitness function is determined as the following (21): where K is factor for handling constraint violation.

The Detail of SFS's Proceduref the ELD Problem
Step 1: Set parameters including the number of population Np, the number of diffusion population Ndf and the maximum number of iterations MI. Step 7: Save the best point Ybest for the current iteration.

Numberical Results
In this section, we present two issues as follows: 1) Analysis of the efficiency of the SFS method based on the simulation results applying Levy Flight or Gauss walk for the diffusion phase; 2) Comparing results from three various standard IEEE test systems with 6 units, 10 units and 20 units to evaluate performance of SFS methods. Three test systems are solved by running SFS on Matlab 2016 a and a computer with 2.4 GHz processor and 4 GB of RAM.

Appling Levy Flight or Gaussian Random Walk for the Diffusion Phase
In [31], author described that the diffusion phase of SFS could use Levy Flight or Gaussian random walk. However, all applications for solving benchmark functions, Gaussian random walk was only selected. In order to fully investigate the characteristics of SFS, we examine its characteristics via applying Levy Flight or Gauss walk in the diffusion phase via three test systems. They comprise of 6-unit test system considering with and without line transmission losses, 10-unit test system with both multi fuels and valve point effect, and 20 unit test system with transmission losses. Such investigation process of using Levy Flight or Gauss walk has a very important role because it helps researchers easily to know the effectiveness of SFS to application for different systems. This investigation will be implemented based on two cases including the influence of walk factor ꞷ on the obtained results of SFS_Gauss and impact of the number of population and the number of iterations on the results of SFS_Gauss and SFS_Levy.

Survey 1: the Influence of the Walk Factor
For the first case, walk factor of SFS_Gauss is set from 0 to 1 with a step of 0.25 to analyze its impact on the tested results from convex or non-convex test systems. If is select to 0, (15) is used to create the new solutions in the diffusion phase. If is select to 1, (14) is employed to produce the new solutions. Otherwise, both. (14)(15) are utilized for the solution creating process. To see the changes clearly, some parameters like the number of populations and number of iterations need to be established suitably for different test systems. Particularly, the number of populations and number of iterations are respectively set to 5 and 30 for 6-unit system, 10 and 50 for 20-unit system, and 20 and 500 for 10-unit system. The obtained results from these systems are summarized in Tables 1, 2 and 3.
As shown in these tables, when the value of is varied from 0 to 1 with the step size of 0. 25

Survey 2: the Impact of the Number of Population and Number of Iterations on Obtained Results
For the second survey, we scrutinize the impact of Np and MI on the results of SFS_Gauss and SFS_Levy for 10-unit and 20-unit test system. Np is fixed and chosen to be 20 for 10-unit system and 10 for 20-unit system while MI is altered from 100 to 550 for the comer, and from 50 to 250 for the latter. Moreover, according to the survey 1, SFS_Gauss had good solutions if only (14) is used for producing new solutions. So, we only use (14) corresponding to ω=1. And statistical results from 10-unit system and 20-unit system are shown in Tables 4 and 5.
In accordance with Tables 4 and 5, the minimum cost of SFS_Gauss and SFS_Levy are more and more changeful when MI is changed. For the case with non-smooth objective function, the best cost of SFS-Gauss and SFS-Flevy are 623.8252 $/h and 623.8235 $/h, respectively. For the case with smooth objective function, those of SFS-Gauss and SFS-Flevy are 62456.6331 $/h and 62456.6338 ($/h), respectively. It is clearly recognized that the best cost of SFS-Gauss is always better than that of SFS-Flevy at the same number of iterations for 20-unit system. In contrast to the case above, the best cost of SFS-Flevy outperforms than that of SFS-Gauss with the same manner for 10-unit system. This implies that SFS_Levy is suitable for solving non-convex economic load dispatch problem with many local optimum solutions because its strong characteristic is to search solutions in large space, while SFS_Gauss is appropriate for disentangling convex one as it is powerfully capable for finding solutions in small space.

Comparison and Discussion
In section, the SFS method performance is evaluated by comparing the minimum costs with other available methods. For fair comparison, some parameters such as Np and MI along with the number of function evaluations Fes are also reported in tables.

Case Study 1: 6-Unit Test System
This study solves 6-generating unit taking with or without line transmission losses into account. Load demand level of 600, 700, and 800 MVA in turn for both test circumstances are scrutinized. Problem data for various load demand levels of the first test system can be reached in Moustafa et al. [11]. In such study, we set Np to 10 and MI to 50 for testing all the cases with or without transmission losses. Tables 6 and 7 report the numerical results achieved by FFA [11], MFA [11], VSSFA [11], MFFA [11], SFS_Gauss and SFS_Levy.
From Table 6, it can be seen that minimum fuel and standard cost values attained by SFS_Gauss and SFS_Levy are much lower than those of other methods. In addition, SFS_Gauss, and SFS_Levy only use Np=10, MI=50 and Fes=1500 while other ones need to employ Np=50 MI=150 and Fes=3750. It easily confirms that SFS_Gauss and SFS_Levy are faster than those of variants of FA. Even as with the case including transmission losses  Table 7, SFS_Gauss and SFS_Levy still display its supremacy about the best cost with the shortest execution time. Consequently, it can clinch that SFS_Gauss and SFS_Levy are the best methods for these cases.

Case Study 2: 10-unit Test System
This portion applied 10-unit system with valve-point loading, multiple fuel options and without transmission to size up the real performance of the SFS on non-convex problem. The data such as upper and lower powers of the units and fuel cost coefficients are come from the previous study as in [17,23]. In such study, the load of all thermal units is 2700 MW. Table 8 describe the comparison of results performed by SFS method and other thirteen ones in terms of minimum cost, population, the maximum iterations, and the number of function evaluations. As seen from the table, the best fuel cost of DWPSO [29] is 622.74 $/h and is better than those of other methods. After rechecking this value by substituting the optimum dispatch solutions of DWPSO into function objective, the exact cost is 624.23 $/h. It is clearly seen that such cost is higher than that reported in Samir Sayah [29]. For this aspect, 623.8252 $/h and 623.8240 $/h are the best costs of SFS-Gauss and SFS-Levy and these values are approximate or smaller than those of other methods. If considering the number of function evaluations, we see that the Fes value of almost all methods is completely different. The Fes value of SFS_Gauss and SFS_Levy is 30.000 which is smaller than that of CSO [26] and DWPSO [29] and higher than those of remaining ones. Moreover, the convergence features of SFS_Gauss and SFS_Levy illustrated in Figure 1 reveal that SFS-Gauss converges to local optimal solutions faster than SFS-Levy from the 1 st iteration to the 150 th iteration, and then its

Case Study 3: 20-Unit Test System
In the study, the twenty generating units with transmission line losses has been deliberated to value the effectiveness of SFS_Gauss and SFS_Levy for clinching the ELD problem. The input data for the case is accessible from [16] and the total load demand is 2500 MW. The result comparison summary is presented in Table 9. The best fuel cost achieved by SFS-Gauss is 62456.633 $/h, which is better than that of SFS-Levy. In Table 9, it is also seen that SFS_Gauss and SFS_Levy completely devastate other ones such as BBO [15] and aBBOmDE [25] but share the standing position with ALO [16] and MCSA [20]. Besides, SFS-Gauss and SFS-Levy only use Fes=4500 and the value is smaller than that of the four mentioned methods. So, this underlines that SFS_Gauss and SFS_Levy are very favorable tool for this case. Figure 2 delineates the convergence features of SFS algorithms. As perceived in the figure, SFS-Gauss can improve the fuel cost significant from the 115 th iteration to the last one but SFS_Levy just reduce gradually.

Conclusions
In this paper, efficient optimum solutions of the ELD problem have been uncovered by executing two forms of the stochastic fractal search algorithm. The focal contributions presented in the paper can be encapsulated as the following aspects: − Analyze the performance of the SFS method when applying two distribution random walk in the diffusion phase which is one out of three basic mechanisms of the original SFS method. − Point out the selection of the most effective formula for producing new SFS solutions using Gauss walk. − Reveal the most suitable form of the SFS for applying the non-smooth or smooth ELD problem. Moreover, the SFS methods have been investigated via three case studies with different objective function forms and different constraints. The results obtained by the SFS method has been compared to those of other existing techniques. These comparative results show that the SFS method is an effective optimization tool for addressing convex or non-convex ELD problem in a power system.