# Optimal Reactive Power Flow for Large-Scale Power Systems Using an Effective Metaheuristic Algorithm

Authors: Thanh Long Duong; Minh Quan Duong; Van-Duc Phan; Thang Trung Nguyen
Journal: Journal of Electrical and Computer Engineering
 (2020)

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## Abstract

In this paper, stochastic fractal search method (SFS) is employed for solving the optimal reactive power flow (ORPF) problem with a target of optimizing total active power losses (TPL), voltage deviation (VD), and voltage stability index (VSI). SFS is an effective metaheuristic algorithm consisting of diffusion process and two update processes. ORPF is a complex problem giving challenges to applied algorithms by taking into account many complex constraints such as operating voltage from generators and loads, active and reactive power generation of generators, limit of capacitors, apparent power limit from branches, and tap setting of transformers. For verifying the performance, solutions of IEEE 30 and 118-bus system with TPL, VD, and VSI objectives are found by the SFS method with different control parameter settings. Result comparisons indicate that SFS is more favorable than other methods about finding effective solutions and having faster speed. As a result, it is suggested that SFS should be used for ORPF problem, and modifications performed on SFS are encouraged for better results.


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## Body

## 1. Introduction

In the power system, optimal reactive power flow (ORPF) is not only one of the best famous optimization problems but also a very complex problem. In the ORPF problem, two variables need to be considered such as control variables and dependent variables. Control variables are voltage of generation buses, on load tap-changer setting of transformers and generated reactive power of capacitor banks, while dependent variables are voltage of load buses, apparent power flow of transmission lines, and reactive power of generators. So, the major objectives of such ORPF problem is to find control variable so that others have values falling into a permitted operating range [1, 2]. Traditionally, the ORPF problem concentrates on reducing three individual objectives such as power losses of transmission lines, voltage deviation, and voltage stability index. So, a power system economically and stably operates when these goals are fully achieved.In the last decades of the 20th century, the ORPF problem has been successfully addressed by many conventional methodologies called deterministic methodologies such as the Newton method [3], linear programming [4–7], interior point method [8, 9], quadratic programming method [10, 11], and dynamic programming method [12]. With appearance of the mentioned methods, they proved their strong points in dealing with the ORPF problem having linear constraints and differentiable functions for application, but a large system or more complicated constraints and their applicability must be stopped to make rooms for new methods which have a promising ability.Luckily, developing computer science supported researchers much in creating new population-based methods to handle drawbacks of conventional methods. These methods have been successfully and widely applied to solving the ORPF problem, consisting many original methods, improve/modified methods, or combined/hybrid methods. They have been constantly developed and have become a big method family such as particle swarm optimization (PSO) family [13–17], differential evolution (DE) family [18–21], and genetic algorithm (GA) family [22–25], while many standard methods have been also applied in [26–34]. Sahli et al. [16] presented a combination between particle swarm optimization and tabu search (PSO-TS) by incorporating the best search function of PSO and TS. It was capable of finding the global solution and avoiding to fall into local optimum. By the way of evaluation of results attained from the standard IEEE 30-bus system with objective of power loss minimization, PSO-TS has seen better solution quality than other methods as conventional DE, PSO, and TS. Furthermore, a modified version of PSO called modified pseudogradient search-particle swarm optimization (MPG-PSO) has been proposed in [17]. MPG-PSO has the most powerful ability in the PSO family due to applying pseudogradient theory for determining the best velocity direction. As a result, the method has overtaken other PSO methods involving PSO using the time-varying inertia weight factor (PSO-TVIW), PSO using time-varying acceleration coefficients (PSO-TVAC), self-organizing particle swarm optimization using time-varying acceleration coefficients (SPSO-TVAC), PSO using constriction factor (PSO-CF), pseudogradient-based PSO (PG-PSO), PSO using stochastic weight trade-off factor (SWT-PSO), and SWT-PSO using pseudogradient method (PGSWT-PSO). Differential evolution (DE) family has been offered for ORPF-like traditional differential evolution (DE), hybrid ant system and differential evolution method (HAS-DE) [20], and hybrid double differential evolution technique and modified teaching learning technique (DDET-MTLT) [21]. In [20], Huang and Huang have replaced the selection operation of DE by the ant system to enhance the global search capability and avoid falling into local minima and decrease computational time. DDET-MTLT [21] was a combination of double differential evolution technique (DDET) and modified teaching learning technique (MTLT). The obtained results of DDET-MTLT have been compared to some methods on IEEE-30 and IEEE 118-bus systems. Besides, variants of genetic algorithm have been applied for ORPF such as genetic algorithm (GA) [22], enhanced genetic algorithm (EGA) [23], modified NSGA-II (MNSGA-II) [24], self-adaptive real coded genetic algorithm (SARCGA) [25], and hybrid evolutionary programming technique HEP [25].In addition to the three above method family, other standard methodologies have been also applied for solving ORPF problem such as gravitational search algorithm (GSA) [26], ant lion optimizer (ALO) [27], quasi-oppositional teaching learning based optimization (QOTLBO) [28], teaching learning based optimization (TLBO) [28], Pooled-neighbor swarm intelligence algorithm (PNSA) [29], hybrid Nelder–Mead simplex-based firefly algorithm (HFA-NMS) [30], chaotic krill herd algorithm (CKHA) [31], artificial bee colony algorithm (ABC) [32], exchange market algorithm (EMA) [33], backtracking search algorithm (BTSA) [34], and harmony search algorithm (HSA) [35]. In summary, all methods have demonstrated their qualification for addressing almost constraints of ORPF problem with acceptable solutions.In this article, we present a standard stochastic fractal search (SFS) with the goal determining minimization of three different individual objectives of ORPF problem such as minimizing TPL, reducing of VD, and enhancing VSI. The standard stochastic fractal search [36] was developed in 2014 by Salimi. It has been applied for addressing twenty-three standard benchmark functions and has proven its proficiency in finding optimal solutions better than many methods available in this literature. In [37], Tran at al. have applied an improved SFS (ISFS) method and SFS method for dealing with the ORPF problem. Only the IEEE 30-bus system with three objectives consisting of total power losses, voltage deviation and L-index have been employed for comparing and evaluating the real performance of the SFS method and ISFS method. From the minimum, average, and maximum fitness functions obtained from the three objectives, ISFS has been considered to be more effective than the SFS method. However, Tran at al. [37] have not taken the setting of control parameters into account. In fact, SFS has three important control parameters consisting of walk factor, population size, and the maximum number of iterations. Among the three parameters, the walk factor has high impact on working performance of the diffusion process; meanwhile, population size has high contribution to the first update and the second update processes. So, in this paper, we focus on the setting of control parameters to overcome such mentioned shortcoming of the work in [37]. Furthermore, we also expand study cases by considering both IEEE 30-bus system and IEEE 118-bus system with the three mentioned objectives. As a result, the novelty and the main contribution of the paper are as follows:(i)
Finding optimal solutions for IEEE 30-bus and 118-bus transmission power networks of ORPF problem by using the SFS method(ii)
Testing the real performance of the SFS method with the change of population size and iterations(iii)
Tuning the best walk factor for determining more appropriate equation for diffusion of the SFS method(iv)
Illustrating the fluctuations of search process of SFS method with different settings;the simulation can support to evaluate the real performance of SFS and the impact of each parameter on the real performance of the SFS method(v)
Demonstrating the effectiveness and robustness of the SFS method by comparing total power losses, load bus voltage, and enhancement of voltage profileApart from the introduction, other parts of the paper are as follows: single objective functions and constraints of electric components are mathematically formulated in Section2. The structure of the SFS method consisting of diffusion and the first and the second update techniques are described in detail in Section 3. Computation steps of solving ORPF problem by using the SFS method are shown in Section 4. The results obtained by SFS and other methods from two standard IEEE transmission power networks with 30 buses and 118 buses are compared and discussed in Section 5. The whole work of the paper is summarized in and concluded in Section 6.


## 2. Formulation of ORPF Problem

The ORPF problem is constructed by considering minimization of total active power losses (Ploss), reduction of load bus voltage deviation (VD), and enhancement of voltage stability as objectives and taking into account operating voltage of generators and loads, apparent power limit of branches, reactive power generation of generators, and active and reactive power balance equations. The structure of ORPF problem is described in detail as follows.


### 2.1. Objectives of ORPF Problem

Three main objectives of the ORPF problem considered in the paper consist of minimization of total active power losses, minimization of voltage deviation at load buses, and minimization of L-index in which minimization of L-index is corresponding to the enhancement of voltage stability. The three objectives can be seen in formulas (1)–(3) as follows:(1)Minimize∑Ploss=∑i=1Nbus∑j=1j≠iNbusGijVi2+Vj2−2ViVjcosβi−βj,(2)Minimize  VD=∑i=1NloadVloadi−Vref,(3)Minimize L−index=maxLj;j=1,...,Nbus,where Gij is the conductance of conductor ij; βi and βj are the phases of voltage at buses i and j, respectively; Vloadi is the voltage of load bus i; Vref is expected to be the voltage equaling 1.0 pu; and Lj is called L-index of-bus j [16].Basically, L-index is within the range from 0 to 1 in which 0 is the best value and 1 is the worst value. The power system is considered to be working stably when L-index is close to 0, and it is working unstably or it will be collapsed in some seconds if L-index is close to 1. So, the main task to keep the power system working stably and economically is to reduce L-index close to 0. However, it is hard to tune control parameter for obtaining 0 value for L-index.


### 2.2. Constraints of ORPF Problem

ORPF are constrained by equality and inequality constraints covering the whole transmission power network. The equality constraints are considering active and reactive power balance while the inequality constraints are upper and lower limitations of electricity components and working parameters of power network.The two equality constraints are as follows:(4)PGi−Pdi=Vi∑j=1NbusVjgijcosβi−βj+bijsinβi−βj,(5)QGi+Qci−Qdi=Vi∑j=1NbusVjgijsinβi−βj−bijcosβi−βj.where gij and bij are real and unreal terms of admittance of conductor ij, respectively. The inequality constraints are limitations of reactive power output and voltage of generators, the reactive power output of capacitors, tap values of transformers, voltage of load buses, and apparent power of lines. All the inequality constraints are as follows:(6)QGi,min≤QGi≤QGi,max;i=1,...,NG,(7)VGi,min≤VGi≤VGi,max;i=1,...,NG,(8)Qci,min≤Qci≤Qci,max;i=1,...,Nc,(9)Ti,min<Ti<Ti,max;i=1,...,Nt,(10)Vloadi,min≤Vloadi≤Vloadi,max;i=1,...,Nload,(11)Sl≤Sl,max;l=1,...,Nbranch.


## 2.1. Objectives of ORPF Problem

Three main objectives of the ORPF problem considered in the paper consist of minimization of total active power losses, minimization of voltage deviation at load buses, and minimization of L-index in which minimization of L-index is corresponding to the enhancement of voltage stability. The three objectives can be seen in formulas (1)–(3) as follows:(1)Minimize∑Ploss=∑i=1Nbus∑j=1j≠iNbusGijVi2+Vj2−2ViVjcosβi−βj,(2)Minimize  VD=∑i=1NloadVloadi−Vref,(3)Minimize L−index=maxLj;j=1,...,Nbus,where Gij is the conductance of conductor ij; βi and βj are the phases of voltage at buses i and j, respectively; Vloadi is the voltage of load bus i; Vref is expected to be the voltage equaling 1.0 pu; and Lj is called L-index of-bus j [16].Basically, L-index is within the range from 0 to 1 in which 0 is the best value and 1 is the worst value. The power system is considered to be working stably when L-index is close to 0, and it is working unstably or it will be collapsed in some seconds if L-index is close to 1. So, the main task to keep the power system working stably and economically is to reduce L-index close to 0. However, it is hard to tune control parameter for obtaining 0 value for L-index.


## 2.2. Constraints of ORPF Problem

ORPF are constrained by equality and inequality constraints covering the whole transmission power network. The equality constraints are considering active and reactive power balance while the inequality constraints are upper and lower limitations of electricity components and working parameters of power network.The two equality constraints are as follows:(4)PGi−Pdi=Vi∑j=1NbusVjgijcosβi−βj+bijsinβi−βj,(5)QGi+Qci−Qdi=Vi∑j=1NbusVjgijsinβi−βj−bijcosβi−βj.where gij and bij are real and unreal terms of admittance of conductor ij, respectively. The inequality constraints are limitations of reactive power output and voltage of generators, the reactive power output of capacitors, tap values of transformers, voltage of load buses, and apparent power of lines. All the inequality constraints are as follows:(6)QGi,min≤QGi≤QGi,max;i=1,...,NG,(7)VGi,min≤VGi≤VGi,max;i=1,...,NG,(8)Qci,min≤Qci≤Qci,max;i=1,...,Nc,(9)Ti,min<Ti<Ti,max;i=1,...,Nt,(10)Vloadi,min≤Vloadi≤Vloadi,max;i=1,...,Nload,(11)Sl≤Sl,max;l=1,...,Nbranch.


## 3. Stochastic Fractal Search Algorithm

### 3.1. Diffusion Technique

SFS is an improved version of fractal search and was developed by Salimi in 2014 [36]. SFS is constructed by three main processes including diffusion and two different update processes. Consequently, SFS has three new solution generations in each iteration in which the diffusion process plays the most important role. The diffusion process uses Gaussian random walk for generating new solutions as follows:(12)Xs,new1=GaussianGbest,σ+rand×Gbest−Xs,(13)Xs,new1=GaussianXs,σ,(14)σ=logCICI×Xs−Gbest,where Gbest is the best solution among the set of points. CI is the current iteration.As seen from equations (12) and (13), the diffusion process can be accomplished by using either equations (12) or (13) meanwhile the two equations have a major difference. Equation (12) uses Gaussian random walk around the best solution and an updated step by using (Gbest − Xs) while equation (13) only employs random walk around Xi. Due to the difference, SFS must propose one more control parameter, called walk factor (WF) to control the use of either equations (12) or (13). The walk factor is compared to a random number and the result will lead to a decision of used equation. If walk factor is higher than the random number, equation (12) is employed for producing new solution for the sth solution. Otherwise, equation (13) is selected. The diffusion technique can be performed by using Algorithm 1.Algorithm 1: Diffusion technique for the first generation of new solutions.
Set a value to WF
Fors = 1 to Npop
Randomly produce a random numberλs for the sth solution
If WF is higher thanλs
Using equation (12) for updating the sth new solution
Else
Using equation (13) for updating the sth new solution
End
end


### 3.2. Second Update Technique

The first update is the second new solution generation and is performed by the following equation:(15)Xs,new2=Xrandom1−rand×Xrandom2−Xs,where Xrandom1 and Xrandom2 are solutions chosen randomly in the current set of solutions and Xs,new2 is the new solution of solution Xs.


### 3.3. Third Update Technique

After implementing the first update, the second update equivalent to the third new solutions is carried out for updating solutions of the second generation and is expressed by(16)Xs,new3=Xs,new2−rand×Xrandom1−Gbest,forRNs≤0.5Xs,new2+rand×Xrandom2−Xrandom3,forRNs>0.5.where Xrandom1, Xrandom2, and Xrandom3 are three random solutions chosen from the current set and RNs is a random number ranging from 0 to 1 for solution s.


## 3.1. Diffusion Technique

SFS is an improved version of fractal search and was developed by Salimi in 2014 [36]. SFS is constructed by three main processes including diffusion and two different update processes. Consequently, SFS has three new solution generations in each iteration in which the diffusion process plays the most important role. The diffusion process uses Gaussian random walk for generating new solutions as follows:(12)Xs,new1=GaussianGbest,σ+rand×Gbest−Xs,(13)Xs,new1=GaussianXs,σ,(14)σ=logCICI×Xs−Gbest,where Gbest is the best solution among the set of points. CI is the current iteration.As seen from equations (12) and (13), the diffusion process can be accomplished by using either equations (12) or (13) meanwhile the two equations have a major difference. Equation (12) uses Gaussian random walk around the best solution and an updated step by using (Gbest − Xs) while equation (13) only employs random walk around Xi. Due to the difference, SFS must propose one more control parameter, called walk factor (WF) to control the use of either equations (12) or (13). The walk factor is compared to a random number and the result will lead to a decision of used equation. If walk factor is higher than the random number, equation (12) is employed for producing new solution for the sth solution. Otherwise, equation (13) is selected. The diffusion technique can be performed by using Algorithm 1.Algorithm 1: Diffusion technique for the first generation of new solutions.
Set a value to WF
Fors = 1 to Npop
Randomly produce a random numberλs for the sth solution
If WF is higher thanλs
Using equation (12) for updating the sth new solution
Else
Using equation (13) for updating the sth new solution
End
end


## 3.2. Second Update Technique

The first update is the second new solution generation and is performed by the following equation:(15)Xs,new2=Xrandom1−rand×Xrandom2−Xs,where Xrandom1 and Xrandom2 are solutions chosen randomly in the current set of solutions and Xs,new2 is the new solution of solution Xs.


## 3.3. Third Update Technique

After implementing the first update, the second update equivalent to the third new solutions is carried out for updating solutions of the second generation and is expressed by(16)Xs,new3=Xs,new2−rand×Xrandom1−Gbest,forRNs≤0.5Xs,new2+rand×Xrandom2−Xrandom3,forRNs>0.5.where Xrandom1, Xrandom2, and Xrandom3 are three random solutions chosen from the current set and RNs is a random number ranging from 0 to 1 for solution s.


## 4. Implementation of SFS for ORPF Problem

### 4.1. Initialization

The ORPF problem can be solved by using mathpower 4.1 programming after a set of control variables is predetermined by the SFS problem. In fact, the control variable set that has to be inserted into the program is the voltage of thermal generators (VG1, …, VGNG), tap values of transformers (T1, …, TNt), and reactive power output of capacitor banks (Qc1, …, QcNc). Thus, each point (corresponding to each solution) of the SFS method must represent all the variables and is randomly produced as follows:(17)Xs=Xmin+rand×Xmax−Xmin;s=1,...,Npop,where Xmin and Xmax are the lowest and highest values of such control variables.


### 4.2. Fitness Function

As a result, the fitness function is constructed equaling the sum of objective function and penalty terms [17]. For three single objective functions including total power loss, voltage deviation, and L-index, three corresponding fitness functions are formulated as follows:(18)FFs=∑Ploss+VPF⋅ΔQGi+ΔVloadi+ΔSl2,FFs=VD+VPF⋅ΔQGi+ΔVloadi+ΔSl2,FFs=L−index+VPF⋅ΔQGi+ΔVloadi+ΔSl2.where ∑Ploss, VD and L-index are obtained by using equations (1)–(3), VPF is the penalty factor for the violation of the dependent variable, and ΔQGi,ΔVloadi, and ΔSl are penalty terms corresponding to the violation of QGi in constraint (6), Vloadi in constraint (10), and Sl in constraint (11).The penalty terms are determined by(19)ΔQGi=0,ifQGi,min<QGi<QGi,max,QGi,min−QGi,ifQGi,min≥QGi,QGi−QGi,max2,ifQGi,max≤QGi,ΔVloadi=0,ifVloadi,min<Vloadi<Vloadi,max,Vloadi,min−Vloadi,ifVloadi,min≥Vloadi,Vloadi−Vloadi,max2,ifVloadi,max≤Vloadi,ΔSl=0,ifSl≤Sl,max,Sl−Sl,max,else..


### 4.3. Termination Condition

The three main processes of the SFS method will be terminated when current iteration (CI) is equal to the maximum iteration (MI), which is predetermined initially.


## 4.1. Initialization

The ORPF problem can be solved by using mathpower 4.1 programming after a set of control variables is predetermined by the SFS problem. In fact, the control variable set that has to be inserted into the program is the voltage of thermal generators (VG1, …, VGNG), tap values of transformers (T1, …, TNt), and reactive power output of capacitor banks (Qc1, …, QcNc). Thus, each point (corresponding to each solution) of the SFS method must represent all the variables and is randomly produced as follows:(17)Xs=Xmin+rand×Xmax−Xmin;s=1,...,Npop,where Xmin and Xmax are the lowest and highest values of such control variables.


## 4.2. Fitness Function

As a result, the fitness function is constructed equaling the sum of objective function and penalty terms [17]. For three single objective functions including total power loss, voltage deviation, and L-index, three corresponding fitness functions are formulated as follows:(18)FFs=∑Ploss+VPF⋅ΔQGi+ΔVloadi+ΔSl2,FFs=VD+VPF⋅ΔQGi+ΔVloadi+ΔSl2,FFs=L−index+VPF⋅ΔQGi+ΔVloadi+ΔSl2.where ∑Ploss, VD and L-index are obtained by using equations (1)–(3), VPF is the penalty factor for the violation of the dependent variable, and ΔQGi,ΔVloadi, and ΔSl are penalty terms corresponding to the violation of QGi in constraint (6), Vloadi in constraint (10), and Sl in constraint (11).The penalty terms are determined by(19)ΔQGi=0,ifQGi,min<QGi<QGi,max,QGi,min−QGi,ifQGi,min≥QGi,QGi−QGi,max2,ifQGi,max≤QGi,ΔVloadi=0,ifVloadi,min<Vloadi<Vloadi,max,Vloadi,min−Vloadi,ifVloadi,min≥Vloadi,Vloadi−Vloadi,max2,ifVloadi,max≤Vloadi,ΔSl=0,ifSl≤Sl,max,Sl−Sl,max,else..


## 4.3. Termination Condition

The three main processes of the SFS method will be terminated when current iteration (CI) is equal to the maximum iteration (MI), which is predetermined initially.


## 5. Numerical Results

In the section, we test the performance of the SFS method for the ORPF problem with two systems having 30 buses and 118 buses under considering three objectives such as power loss, voltage deviation, and L-index. The method is executed on Matlab program language and a computer with the processor of Core i7, 2.4 GHz, and 4 GB of RAM.


### 5.1. Results Obtained on IEEE 30-Bus System

In the section, we implement the SFS method for optimizing total power losses, voltage deviation, and L-index of the IEEE 30-bus system by setting different values to control parameters such as population size, maximum iteration, and walk factor. The IEEE 30-bus system consists of 6 generators, 24 loads, and 41 branches, 9 VAR compensators, and 4 transformers [38]. In order to indicate the impact of control parameters on results, we perform three testing cases as follows:(1)
The first testing case: tune different values for population size while fixing the maximum iteration at value of 200(2)
The second testing case: tune different values for the maximum iteration while fixing population size to the best value obtained by the first testing case(3)
The third testing case: tune different values for walk factor while fixing population size and the maximum iteration at the best value obtained by the first testing case and the second testing caseThe three testing cases are presented in Section5.1.1, while the comparison of results from SFS method and other ones is shown in Section 5.1.2 and Section 5.2.


#### 5.1.1. Analysis of Control Parameters on Results Obtained by SFS Method

The results in terms of the best, mean, worst power losses, standard deviation, and simulation time for each run (ST) are shown in Tables1–3 corresponding to the first, second, and third testing cases. The results from Table 1 indicate that population size of 10 is high enough for finding the best solution while increasing the population size to 15 and 20 cannot find a better solution but average and maximum power losses can be improved whereas simulation time is increased. Figure 1 also gives the same evaluation since the curve in blue has high number of solutions with better fitness than other curves.Table 1
Results obtained by setting different values toNpop, and fixing MI = 200 and WF = 1.0.
Population size
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
5
5.0708
5.2901
6.3250
0.1847
10.2
10
4.5275
4.6732
5.1844
0.1399
19.7
15
4.5275
4.6639
5.1688
0.1370
29.4
20
4.5275
4.6499
5.1430
0.1352
38.2Table 2
Results obtained by setting different values to MI, and fixingNpop = 10 and WF = 1.0.
MI
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
50
6.3838
7.1500
8.1395
0.2252
6.2
100
6.2561
7.0070
7.9767
0.2207
12.1
150
4.7539
6.3063
7.2588
0.1876
18.6
200
4.5275
4.6732
5.1844
0.1399
23.9
250
4.5275
4.6078
5.0029
0.1321
30.0Table 3
Results obtained by setting different values to WF and fixingNpop = 10 and MI = 200.
Walk factor (WF)
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
0
6.9724
7.1687
7.6936
0.2154
20.2
0.2
5.9854
6.6453
7.6325
0.2129
20.1
0.4
5.5236
6.1780
7.2167
0.1707
19.7
0.6
4.7539
4.7900
5.3659
0.1560
19.6
0.8
4.5366
4.7293
5.7132
0.1430
19.8
1.0
4.5275
4.6732
5.1844
0.1399
19.5Figure 1
The impact of population size on obtained fitness values of 50 trial runs.Table2 show the results obtained by setting different values to the maximum iteration while keeping the population size at 10. The maximum iteration is increased from 50 to 250 with a step of 50. The best power loss implies that SFS method can find the best solution at different values of the maximum iteration such as 200 and 250; meanwhile, smaller number of iterations cannot reach the same best solution. The best optimal solution of MI = 200 and MI = 250 has the same quality; meanwhile, the mean fitness function value of 50 runs from MI = 250 is smaller. In spite of the advantage of MI = 250, SFS should adopt the most appropriate number of iterations is 200 because it can find as good as solution but it can reduce simulation time. Figure 2 shows that the curve in blue has many solutions with the same fitness as those from curve in black.Figure 2
The impact of the maximum iteration on obtained fitness values of 50 trial runs.From Tables1 and 2 as well as from Figures 1 and 2, it can result in the selection of population size and the maximum iteration in which the best value of the comer is 10 while the most appropriate value of the later is 200. By using the two values, walk factor’s impact on performance of the SFS method is tried by setting it to 0, 0.2, 0.4, 0.6, 0.8, and 1. Table 3 and Figure 3 show that the walk factor to be 1 is the best value for reaching the best solution, whereas other values cannot reach the same best solution. As observing from the best power loss, the walk factor with higher value can support the SFS method find superior solutions. In fact, power loss is the highest at WF = 0, and it decreases since the factor approaches to 1. So, it can conclude that diffusion technique becomes more effective if equation (12) is used to replace equation (13).Figure 3
The impact of walk factor on obtained fitness values of 50 trial runs.


#### 5.1.2. Result Comparisons for IEEE 30-Bus System

The results in terms of minimum, average, maximum, and standard deviation accompany with control parameters including MI andNpop from SFS and other methods are reported in Tables 4–6 for power loss, voltage deviation, and L-index, respectively. Besides, saving percentage (%) of the SFS method compared with each one is also calculated and reported in the tables for further comparisons. Saving percentage from these tables can see that SFS outperforms most methods excluding PSO-TS [16] and ISFS [37] for power loss objective and QOTLBO [28] for voltage deviation objective. Saving values show that these methods get improvement over SFS by 0.137%, 0.295%, and 2.45%; however, only ISFS [37] has found better solution than SFS, meanwhile PSO-TS [16] has not reported MI and Npop for comparison of convergence speed and recalculated minimum of QOTLBO is 0.1031, which is much higher than reported value of 0.0856. Clearly, QOTLBO is less effective than SFS. SFS can get improvement over other methods for power loss optimization from 0.7% to 8.365%, for voltage deviation optimization from 1.68% to 57.51%, and for L-index from 13.26% to 32.82%. Clearly, the improvement is significant and optimal solution of SFS is much better than those from other methods. Furthermore, convergence speed of SFS is also faster or approximate with other compared methods since SFS uses MI = 200, Npop = 10, while those from others are 100 and 50, 200 and 20, or MI = 30,000.Table 4
Comparisons for TPL of the IEEE 30-bus system.
Method
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
MI
N
pop
Saving percentage (%)
PSO-TS [16]
4.5213
—
—
—
—
—
−0.137
TS [16]
4.9203
—
—
—
—
—
7.983
PSO [16]
4.6862
—
—
—
—
—
3.387
ALO [27]
4.59
100
—
1.362
QOTLBO [28]
4.5594
4.5601
4.5617
0.037
100
50
0.700
TLBO [28]
4.5629
4.5695
4.5748
0.0564
100
50
0.776
SGA [35]
4.9408
5.0378
5.1651
—
30,000
—
8.365
PSO [35]
4.9239
4.972
5.0576
—
30,000
—
8.051
HSA [35]
4.9059
4.924
4.9653
—
30,000
—
7.713
SFS [37]
4.5777
—
—
1.05
100
20
1.097
ISFS [37]
4.5142
—
—
0.012
—
—
−0.295
SFS
4.5275
4.6732
5.1844
0.1399
200
10Table 5
Comparisons for VD of the IEEE 30-bus system.
Method
The best VD (pu)
Mean VD (pu)
The worst VD (pu)
Std. dev. VD (pu)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
0.1038
0.1597
0.5791
0.1112
200
20
15.51
PSO-TVAC [17]
0.2064
0.2376
0.5796
0.0153
200
20
57.51
SPSO-TVAC [17]
0.1354
0.1558
0.1833
0.0103
200
20
35.23
PSO-CF [17]
0.1287
0.1557
0.4041
0.0404
200
20
31.86
PG-PSO [17]
0.1202
0.144
0.2593
0.0222
200
20
27.04
SWT-PSO [17]
0.1614
0.1814
0.2296
0.133
200
20
45.66
PGSWT-PSO [17]
0.1539
0.2189
0.5532
0.0656
200
20
43.01
MPG-PSO [17]
0.0892
0.1078
0.2518
0.0298
200
20
1.68
QOTLBO [28]
0.0856
0.0872
0.0907
0.0314
200
10
−2.45
TLBO [28]
0.0913
0.0934
0.0988
0.0403
100
50
3.94
SFS [37]
0.122
—
—
0.016
100
20
28.115
ISFS [37]
0.089
—
—
0.0031
100
20
1.461
SFS
0.0877
0.1207
0.1577
0.0155
200
10
—Table 6
Comparisons for VSI of the IEEE 30-bus system.
Method
The best L-index (pu)
Mean L-index (pu)
The worst L-index (pu)
Std. dev. L-index (pu)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
0.1258
0.127
0.1289
0.0008
200
20
19.95
PSO-TVAC [17]
0.1499
0.1513
0.1544
0.0009
200
20
32.82
SPSO-TVAC [17]
0.1271
0.1285
0.1297
0.0006
200
20
20.77
PSO-CF [17]
0.1261
0.1279
0.1295
0.0008
200
20
20.14
PG-PSO [17]
0.1264
0.1297
0.1313
0.0008
200
20
20.33
SWT-PSO [17]
0.1488
0.1634
0.1806
0.0074
200
20
32.33
PGSWT-PSO [17]
0.1394
0.1537
0.1749
0.0081
200
20
27.76
MPG-PSO [17]
0.1241
0.1266
0.1298
0.001
200
20
18.86
BA [27]
0.1191
—
—
—
100
40
15.45
GWO [27]
0.118
—
—
—
100
40
14.66
ABC [27]
0.1161
—
—
—
100
40
13.26
ALO [27]
0.1161
—
—
—
100
40
13.26
QOTLBO [28]
0.1242
0.1245
0.1247
0.0452
100
50
18.92
TLBO [28]
0.1252
0.1254
0.1258
0.0454
100
50
19.57
SFS [37]
0.1252
—
—
0.021
100
20
19.569
ISFS [37]
0.1245
—
—
0.004
100
20
19.116
SFS
0.1007
0.1043
0.1138
0.0026
200
10
—Optimal solutions of three optimization cases are given in Table7.Table 7
Optimal solutions of the IEEE 30-bus system.
Variable
TPL objective
VD objective
L-index objective
V
G1
1.1000
1.0055
1.0807
V
G2
1.0952
1.0011
1.0538
V
G5
1.0750
1.0173
1.0739
V
G8
1.0765
1.0120
1.0089
V
G11
1.0873
1.0320
1.0801
V
G13
1.0999
1.0234
1.0851
T
1
1.0500
1.0500
0.9000
T
2
0.9200
0.9000
0.9000
T
3
1.0100
1.0000
0.9000
T
4
0.9800
0.9700
0.9000
Q
c1
5.0000
4.0000
5.0000
Q
c2
5.0000
2.0000
5.0000
Q
c3
5.0000
4.0000
0.0000
Q
c4
5.0000
3.0000
0.0000
Q
c5
3.0000
5.0000
5.0000
Q
c6
5.0000
3.0000
3.0000
Q
c7
3.0000
5.0000
5.0000
Q
c8
4.0000
5.0000
5.0000
Q
c9
2.0000
3.0000
1.0000Table 8
Optimal solution of the IEEE 118-bus transmission power network.
V
G1 (pu)
1.0081
V
G62
1.0223
V
G113
1.005
V
G4
1.0439
V
G65
1.0563
V
G116
1.0379
V
G6
1.0376
V
G66
1.051
T
8 (pu)
0.9762
V
G8
1.0081
V
G69
1.0574
T
32
1.0192
V
G10
1.0267
V
G70
1.0308
T
36
1.0031
V
G12
1.0309
V
G72
1.0204
T
51
0.9785
V
G15
1.004
V
G73
1.0374
T
93
0.9981
V
G18
1.0021
V
G74
1.0043
T
95
1.0081
V
G19
0.9992
V
G76
0.989
T
102
1.0132
V
G24
1.0069
V
G77
1.0298
T
107
0.9963
V
G25
1.0436
V
G80
1.0471
T
127
0.9821
V
G26
1.0356
V
G85
1.0349
Q
C5 (MVAR)
−2.349
V
G27
1.0366
V
G87
0.9944
Q
C34
7.4927
V
G31
1.023
V
G89
1.0764
Q
C37
−16.55
V
G32
1.0212
V
G90
1.0575
Q
C44
6.8138
V
G34
1.0266
V
G91
1.0464
Q
C45
3.8515
V
G36
1.0244
V
G92
1.0515
Q
C46
5.0318
V
G40
1.0371
V
G99
1.0285
Q
C48
7.1142
V
G42
1.0388
V
G100
1.0516
Q
C74
5.985
V
G46
1.0173
V
G103
1.0361
Q
C79
10.574
V
G49
1.0469
V
G104
1.0148
Q
C82
14.366
V
G54
1.0366
V
G105
1.0133
Q
C83
6.8917
V
G55
1.0321
V
G107
1.0265
Q
C105
6.4333
V
G56
1.0328
V
G110
1.0201
Q
C107
2.3679
V
G59
1.0388
V
G111
1.0197
Q
C110
5.3222
V
G61
1.0221
V
G112
1.0261


### 5.2. Results Obtained on IEEE 118-Bus System

This section uses the IEEE 118-bus system consisting of 54 generator buses, 64 load buses, and 186 transmission lines, 14 VAR compensators, and 9 transformers [37] (Table 8). We implement the SFS method for optimizing total power losses of the system by setting different values to population size and the maximum iteration. In the first trial, we increase the population size from 10 to 100 with a change of 10 and fixing the maximum iteration to 150 and 200. In the second trial, population size is selected to be 50 and 75; meanwhile, the maximum iteration is increased from 50 to 400 with a step of 50. The optimal solution obtained by SFS method is given in Table 9. In addition, the results in terms of the best power loss, mean power loss, and success rate (SR) in percentage are, respectively, shown in Tables 9 and 10. From the two tables, it is clear that SFS can improve search ability since the population size and the maximum iteration are increased but simulation time is also increased. 113.021 MW in Table 9 indicates SFS can find the best solution with the setting of Npop = 60, 70, 80, 90, and 100, and MI = 150, and Npop = 50, 60, 70, 80, 90, and 100 and MI = 200. Clearly, Npop = 60 and MI = 150 are the best selection for the case of tuning population size and fixing the maximum iteration. SFS cannot improve result better if the population size is set to higher than 60 for MI = 150 and 70 for MI = 200. Similarly, 113.021 MW in Table 10 indicates that Npop = 50 and MI = 200 and Npop = 75 and MI = 150 are the best selection for finding the best performance of the SFS method. Improvement of the best solution fails if increasing MI to higher than 200 for Npop = 50 and to higher than 150 for Npop = 75. So, it can conclude that the best selection for population size and the maximum iterations are Npop = 60 and MI = 150. The results from the setting are reported for comparisons with other methods shown in Table 11. The saving percentage of SFS compared with other methods is from 0.09% to 9.1%, while SFS is only less effective than QOTLBO [28] by −0.66%. The comparison indication can conclude that optimal solution of SFS is the second best among all compared methods. Furthermore, search speed of SFS is in the fastest group once MI and Npop are, respectively, 150 and 60, while those from others are 200 and 40, 300 and 15, and 100 and 50. In summary, SFS can find optimal solutions with higher quality than most compared methods; however, compared with some best methods, SFS is also less potential. Consequently, we suggest SFS should be used as an optimization tool for the ORPF problem and it is more promising if SFS is improved by proposing modifications.Table 9
Result obtained by setting MI = 150, MI = 200 and different values for population size.
N
pop
MI = 150
MI = 200
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
10
161.620
206.233
15
46
156.772
193.034
18.69
61
20
154.186
196.747
25
90
149.560
184.155
31.15
119
30
145.552
187.696
47
126
141.185
175.684
54
163
40
135.945
179.062
54
181
131.867
167.602
61
239
50
122.894
159.423
65
226
113.021
149.220
75
302
60
113.021
134.793
88
275
113.021
126.166
89
354
70
113.021
130.075
92
317
113.021
121.750
94
422
80
113.021
125.783
95
359
113.021
124.651
96
477
90
113.021
121.758
94
402
113.021
120.662
98
534
100
113.021
117.618
95
449
113.021
116.559
98
572Table 10
Result obtained by settingNpop = 50 and Npop = 75 and different values for the maximum iterations.
MI
N
pop = 50
N
pop = 75
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
50
154.308
212.230
12
72
124.323
198.647
25
107
100
125.454
202.467
54
147
119.124
189.510
69
221
150
119.011
193.154
77
226
113.021
130.221
82
340
200
113.021
131.027
91
293
113.021
121.887
90
437
250
113.021
129.590
93
365
113.021
121.107
92
545
300
113.021
126.856
95
438
113.021
120.545
94
656
350
113.021
123.036
97
511
113.021
114.952
96
768
400
113.021
119.578
98
583
113.021
114.648
96
876Table 11
Comparisons for TPL of the IEEE 118-bus system.
Method
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
116.8976
118.234
126.6222
1.6009
200
40
3.32
PSO-TVAC [17]
124.3335
129.749
134.1254
2.156
200
40
9.10
SPSO-TVAC [17]
116.2026
117.355
118.139
0.4696
200
40
2.74
PSO-CF [17]
115.6469
116.986
119.8378
0.8655
200
40
2.27
PG-PSO [17]
116.6075
119.396
127.0772
2.107
200
40
3.08
SWT-PSO [17]
124.1476
129.371
141.6147
3.309
200
40
8.96
PGSWTPSO [17]
119.427
122.781
125.762
1.2455
200
40
5.36
MPG-PSO [17]
115.06
116.462
118.35
0.528
200
40
1.77
SARCGA [25]
113.12
113.968
—
0.0002
300
15
0.09
HEP [25]
115.58
115.8
—
0.0018
300
15
2.21
QOTLBO [28]
112.2789
113.769
115.4516
0.0244
100
50
−0.66
TLBO [28]
116.4003
121.390
118.4427
0.0482
100
50
2.90
SFS
113.0213
134.793
233.4538
2.91894
150
60


## 5.1. Results Obtained on IEEE 30-Bus System

In the section, we implement the SFS method for optimizing total power losses, voltage deviation, and L-index of the IEEE 30-bus system by setting different values to control parameters such as population size, maximum iteration, and walk factor. The IEEE 30-bus system consists of 6 generators, 24 loads, and 41 branches, 9 VAR compensators, and 4 transformers [38]. In order to indicate the impact of control parameters on results, we perform three testing cases as follows:(1)
The first testing case: tune different values for population size while fixing the maximum iteration at value of 200(2)
The second testing case: tune different values for the maximum iteration while fixing population size to the best value obtained by the first testing case(3)
The third testing case: tune different values for walk factor while fixing population size and the maximum iteration at the best value obtained by the first testing case and the second testing caseThe three testing cases are presented in Section5.1.1, while the comparison of results from SFS method and other ones is shown in Section 5.1.2 and Section 5.2.


### 5.1.1. Analysis of Control Parameters on Results Obtained by SFS Method

The results in terms of the best, mean, worst power losses, standard deviation, and simulation time for each run (ST) are shown in Tables1–3 corresponding to the first, second, and third testing cases. The results from Table 1 indicate that population size of 10 is high enough for finding the best solution while increasing the population size to 15 and 20 cannot find a better solution but average and maximum power losses can be improved whereas simulation time is increased. Figure 1 also gives the same evaluation since the curve in blue has high number of solutions with better fitness than other curves.Table 1
Results obtained by setting different values toNpop, and fixing MI = 200 and WF = 1.0.
Population size
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
5
5.0708
5.2901
6.3250
0.1847
10.2
10
4.5275
4.6732
5.1844
0.1399
19.7
15
4.5275
4.6639
5.1688
0.1370
29.4
20
4.5275
4.6499
5.1430
0.1352
38.2Table 2
Results obtained by setting different values to MI, and fixingNpop = 10 and WF = 1.0.
MI
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
50
6.3838
7.1500
8.1395
0.2252
6.2
100
6.2561
7.0070
7.9767
0.2207
12.1
150
4.7539
6.3063
7.2588
0.1876
18.6
200
4.5275
4.6732
5.1844
0.1399
23.9
250
4.5275
4.6078
5.0029
0.1321
30.0Table 3
Results obtained by setting different values to WF and fixingNpop = 10 and MI = 200.
Walk factor (WF)
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
0
6.9724
7.1687
7.6936
0.2154
20.2
0.2
5.9854
6.6453
7.6325
0.2129
20.1
0.4
5.5236
6.1780
7.2167
0.1707
19.7
0.6
4.7539
4.7900
5.3659
0.1560
19.6
0.8
4.5366
4.7293
5.7132
0.1430
19.8
1.0
4.5275
4.6732
5.1844
0.1399
19.5Figure 1
The impact of population size on obtained fitness values of 50 trial runs.Table2 show the results obtained by setting different values to the maximum iteration while keeping the population size at 10. The maximum iteration is increased from 50 to 250 with a step of 50. The best power loss implies that SFS method can find the best solution at different values of the maximum iteration such as 200 and 250; meanwhile, smaller number of iterations cannot reach the same best solution. The best optimal solution of MI = 200 and MI = 250 has the same quality; meanwhile, the mean fitness function value of 50 runs from MI = 250 is smaller. In spite of the advantage of MI = 250, SFS should adopt the most appropriate number of iterations is 200 because it can find as good as solution but it can reduce simulation time. Figure 2 shows that the curve in blue has many solutions with the same fitness as those from curve in black.Figure 2
The impact of the maximum iteration on obtained fitness values of 50 trial runs.From Tables1 and 2 as well as from Figures 1 and 2, it can result in the selection of population size and the maximum iteration in which the best value of the comer is 10 while the most appropriate value of the later is 200. By using the two values, walk factor’s impact on performance of the SFS method is tried by setting it to 0, 0.2, 0.4, 0.6, 0.8, and 1. Table 3 and Figure 3 show that the walk factor to be 1 is the best value for reaching the best solution, whereas other values cannot reach the same best solution. As observing from the best power loss, the walk factor with higher value can support the SFS method find superior solutions. In fact, power loss is the highest at WF = 0, and it decreases since the factor approaches to 1. So, it can conclude that diffusion technique becomes more effective if equation (12) is used to replace equation (13).Figure 3
The impact of walk factor on obtained fitness values of 50 trial runs.


### 5.1.2. Result Comparisons for IEEE 30-Bus System

The results in terms of minimum, average, maximum, and standard deviation accompany with control parameters including MI andNpop from SFS and other methods are reported in Tables 4–6 for power loss, voltage deviation, and L-index, respectively. Besides, saving percentage (%) of the SFS method compared with each one is also calculated and reported in the tables for further comparisons. Saving percentage from these tables can see that SFS outperforms most methods excluding PSO-TS [16] and ISFS [37] for power loss objective and QOTLBO [28] for voltage deviation objective. Saving values show that these methods get improvement over SFS by 0.137%, 0.295%, and 2.45%; however, only ISFS [37] has found better solution than SFS, meanwhile PSO-TS [16] has not reported MI and Npop for comparison of convergence speed and recalculated minimum of QOTLBO is 0.1031, which is much higher than reported value of 0.0856. Clearly, QOTLBO is less effective than SFS. SFS can get improvement over other methods for power loss optimization from 0.7% to 8.365%, for voltage deviation optimization from 1.68% to 57.51%, and for L-index from 13.26% to 32.82%. Clearly, the improvement is significant and optimal solution of SFS is much better than those from other methods. Furthermore, convergence speed of SFS is also faster or approximate with other compared methods since SFS uses MI = 200, Npop = 10, while those from others are 100 and 50, 200 and 20, or MI = 30,000.Table 4
Comparisons for TPL of the IEEE 30-bus system.
Method
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
MI
N
pop
Saving percentage (%)
PSO-TS [16]
4.5213
—
—
—
—
—
−0.137
TS [16]
4.9203
—
—
—
—
—
7.983
PSO [16]
4.6862
—
—
—
—
—
3.387
ALO [27]
4.59
100
—
1.362
QOTLBO [28]
4.5594
4.5601
4.5617
0.037
100
50
0.700
TLBO [28]
4.5629
4.5695
4.5748
0.0564
100
50
0.776
SGA [35]
4.9408
5.0378
5.1651
—
30,000
—
8.365
PSO [35]
4.9239
4.972
5.0576
—
30,000
—
8.051
HSA [35]
4.9059
4.924
4.9653
—
30,000
—
7.713
SFS [37]
4.5777
—
—
1.05
100
20
1.097
ISFS [37]
4.5142
—
—
0.012
—
—
−0.295
SFS
4.5275
4.6732
5.1844
0.1399
200
10Table 5
Comparisons for VD of the IEEE 30-bus system.
Method
The best VD (pu)
Mean VD (pu)
The worst VD (pu)
Std. dev. VD (pu)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
0.1038
0.1597
0.5791
0.1112
200
20
15.51
PSO-TVAC [17]
0.2064
0.2376
0.5796
0.0153
200
20
57.51
SPSO-TVAC [17]
0.1354
0.1558
0.1833
0.0103
200
20
35.23
PSO-CF [17]
0.1287
0.1557
0.4041
0.0404
200
20
31.86
PG-PSO [17]
0.1202
0.144
0.2593
0.0222
200
20
27.04
SWT-PSO [17]
0.1614
0.1814
0.2296
0.133
200
20
45.66
PGSWT-PSO [17]
0.1539
0.2189
0.5532
0.0656
200
20
43.01
MPG-PSO [17]
0.0892
0.1078
0.2518
0.0298
200
20
1.68
QOTLBO [28]
0.0856
0.0872
0.0907
0.0314
200
10
−2.45
TLBO [28]
0.0913
0.0934
0.0988
0.0403
100
50
3.94
SFS [37]
0.122
—
—
0.016
100
20
28.115
ISFS [37]
0.089
—
—
0.0031
100
20
1.461
SFS
0.0877
0.1207
0.1577
0.0155
200
10
—Table 6
Comparisons for VSI of the IEEE 30-bus system.
Method
The best L-index (pu)
Mean L-index (pu)
The worst L-index (pu)
Std. dev. L-index (pu)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
0.1258
0.127
0.1289
0.0008
200
20
19.95
PSO-TVAC [17]
0.1499
0.1513
0.1544
0.0009
200
20
32.82
SPSO-TVAC [17]
0.1271
0.1285
0.1297
0.0006
200
20
20.77
PSO-CF [17]
0.1261
0.1279
0.1295
0.0008
200
20
20.14
PG-PSO [17]
0.1264
0.1297
0.1313
0.0008
200
20
20.33
SWT-PSO [17]
0.1488
0.1634
0.1806
0.0074
200
20
32.33
PGSWT-PSO [17]
0.1394
0.1537
0.1749
0.0081
200
20
27.76
MPG-PSO [17]
0.1241
0.1266
0.1298
0.001
200
20
18.86
BA [27]
0.1191
—
—
—
100
40
15.45
GWO [27]
0.118
—
—
—
100
40
14.66
ABC [27]
0.1161
—
—
—
100
40
13.26
ALO [27]
0.1161
—
—
—
100
40
13.26
QOTLBO [28]
0.1242
0.1245
0.1247
0.0452
100
50
18.92
TLBO [28]
0.1252
0.1254
0.1258
0.0454
100
50
19.57
SFS [37]
0.1252
—
—
0.021
100
20
19.569
ISFS [37]
0.1245
—
—
0.004
100
20
19.116
SFS
0.1007
0.1043
0.1138
0.0026
200
10
—Optimal solutions of three optimization cases are given in Table7.Table 7
Optimal solutions of the IEEE 30-bus system.
Variable
TPL objective
VD objective
L-index objective
V
G1
1.1000
1.0055
1.0807
V
G2
1.0952
1.0011
1.0538
V
G5
1.0750
1.0173
1.0739
V
G8
1.0765
1.0120
1.0089
V
G11
1.0873
1.0320
1.0801
V
G13
1.0999
1.0234
1.0851
T
1
1.0500
1.0500
0.9000
T
2
0.9200
0.9000
0.9000
T
3
1.0100
1.0000
0.9000
T
4
0.9800
0.9700
0.9000
Q
c1
5.0000
4.0000
5.0000
Q
c2
5.0000
2.0000
5.0000
Q
c3
5.0000
4.0000
0.0000
Q
c4
5.0000
3.0000
0.0000
Q
c5
3.0000
5.0000
5.0000
Q
c6
5.0000
3.0000
3.0000
Q
c7
3.0000
5.0000
5.0000
Q
c8
4.0000
5.0000
5.0000
Q
c9
2.0000
3.0000
1.0000Table 8
Optimal solution of the IEEE 118-bus transmission power network.
V
G1 (pu)
1.0081
V
G62
1.0223
V
G113
1.005
V
G4
1.0439
V
G65
1.0563
V
G116
1.0379
V
G6
1.0376
V
G66
1.051
T
8 (pu)
0.9762
V
G8
1.0081
V
G69
1.0574
T
32
1.0192
V
G10
1.0267
V
G70
1.0308
T
36
1.0031
V
G12
1.0309
V
G72
1.0204
T
51
0.9785
V
G15
1.004
V
G73
1.0374
T
93
0.9981
V
G18
1.0021
V
G74
1.0043
T
95
1.0081
V
G19
0.9992
V
G76
0.989
T
102
1.0132
V
G24
1.0069
V
G77
1.0298
T
107
0.9963
V
G25
1.0436
V
G80
1.0471
T
127
0.9821
V
G26
1.0356
V
G85
1.0349
Q
C5 (MVAR)
−2.349
V
G27
1.0366
V
G87
0.9944
Q
C34
7.4927
V
G31
1.023
V
G89
1.0764
Q
C37
−16.55
V
G32
1.0212
V
G90
1.0575
Q
C44
6.8138
V
G34
1.0266
V
G91
1.0464
Q
C45
3.8515
V
G36
1.0244
V
G92
1.0515
Q
C46
5.0318
V
G40
1.0371
V
G99
1.0285
Q
C48
7.1142
V
G42
1.0388
V
G100
1.0516
Q
C74
5.985
V
G46
1.0173
V
G103
1.0361
Q
C79
10.574
V
G49
1.0469
V
G104
1.0148
Q
C82
14.366
V
G54
1.0366
V
G105
1.0133
Q
C83
6.8917
V
G55
1.0321
V
G107
1.0265
Q
C105
6.4333
V
G56
1.0328
V
G110
1.0201
Q
C107
2.3679
V
G59
1.0388
V
G111
1.0197
Q
C110
5.3222
V
G61
1.0221
V
G112
1.0261


## 5.1.1. Analysis of Control Parameters on Results Obtained by SFS Method

The results in terms of the best, mean, worst power losses, standard deviation, and simulation time for each run (ST) are shown in Tables1–3 corresponding to the first, second, and third testing cases. The results from Table 1 indicate that population size of 10 is high enough for finding the best solution while increasing the population size to 15 and 20 cannot find a better solution but average and maximum power losses can be improved whereas simulation time is increased. Figure 1 also gives the same evaluation since the curve in blue has high number of solutions with better fitness than other curves.Table 1
Results obtained by setting different values toNpop, and fixing MI = 200 and WF = 1.0.
Population size
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
5
5.0708
5.2901
6.3250
0.1847
10.2
10
4.5275
4.6732
5.1844
0.1399
19.7
15
4.5275
4.6639
5.1688
0.1370
29.4
20
4.5275
4.6499
5.1430
0.1352
38.2Table 2
Results obtained by setting different values to MI, and fixingNpop = 10 and WF = 1.0.
MI
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
50
6.3838
7.1500
8.1395
0.2252
6.2
100
6.2561
7.0070
7.9767
0.2207
12.1
150
4.7539
6.3063
7.2588
0.1876
18.6
200
4.5275
4.6732
5.1844
0.1399
23.9
250
4.5275
4.6078
5.0029
0.1321
30.0Table 3
Results obtained by setting different values to WF and fixingNpop = 10 and MI = 200.
Walk factor (WF)
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
ST (s)
0
6.9724
7.1687
7.6936
0.2154
20.2
0.2
5.9854
6.6453
7.6325
0.2129
20.1
0.4
5.5236
6.1780
7.2167
0.1707
19.7
0.6
4.7539
4.7900
5.3659
0.1560
19.6
0.8
4.5366
4.7293
5.7132
0.1430
19.8
1.0
4.5275
4.6732
5.1844
0.1399
19.5Figure 1
The impact of population size on obtained fitness values of 50 trial runs.Table2 show the results obtained by setting different values to the maximum iteration while keeping the population size at 10. The maximum iteration is increased from 50 to 250 with a step of 50. The best power loss implies that SFS method can find the best solution at different values of the maximum iteration such as 200 and 250; meanwhile, smaller number of iterations cannot reach the same best solution. The best optimal solution of MI = 200 and MI = 250 has the same quality; meanwhile, the mean fitness function value of 50 runs from MI = 250 is smaller. In spite of the advantage of MI = 250, SFS should adopt the most appropriate number of iterations is 200 because it can find as good as solution but it can reduce simulation time. Figure 2 shows that the curve in blue has many solutions with the same fitness as those from curve in black.Figure 2
The impact of the maximum iteration on obtained fitness values of 50 trial runs.From Tables1 and 2 as well as from Figures 1 and 2, it can result in the selection of population size and the maximum iteration in which the best value of the comer is 10 while the most appropriate value of the later is 200. By using the two values, walk factor’s impact on performance of the SFS method is tried by setting it to 0, 0.2, 0.4, 0.6, 0.8, and 1. Table 3 and Figure 3 show that the walk factor to be 1 is the best value for reaching the best solution, whereas other values cannot reach the same best solution. As observing from the best power loss, the walk factor with higher value can support the SFS method find superior solutions. In fact, power loss is the highest at WF = 0, and it decreases since the factor approaches to 1. So, it can conclude that diffusion technique becomes more effective if equation (12) is used to replace equation (13).Figure 3
The impact of walk factor on obtained fitness values of 50 trial runs.


## 5.1.2. Result Comparisons for IEEE 30-Bus System

The results in terms of minimum, average, maximum, and standard deviation accompany with control parameters including MI andNpop from SFS and other methods are reported in Tables 4–6 for power loss, voltage deviation, and L-index, respectively. Besides, saving percentage (%) of the SFS method compared with each one is also calculated and reported in the tables for further comparisons. Saving percentage from these tables can see that SFS outperforms most methods excluding PSO-TS [16] and ISFS [37] for power loss objective and QOTLBO [28] for voltage deviation objective. Saving values show that these methods get improvement over SFS by 0.137%, 0.295%, and 2.45%; however, only ISFS [37] has found better solution than SFS, meanwhile PSO-TS [16] has not reported MI and Npop for comparison of convergence speed and recalculated minimum of QOTLBO is 0.1031, which is much higher than reported value of 0.0856. Clearly, QOTLBO is less effective than SFS. SFS can get improvement over other methods for power loss optimization from 0.7% to 8.365%, for voltage deviation optimization from 1.68% to 57.51%, and for L-index from 13.26% to 32.82%. Clearly, the improvement is significant and optimal solution of SFS is much better than those from other methods. Furthermore, convergence speed of SFS is also faster or approximate with other compared methods since SFS uses MI = 200, Npop = 10, while those from others are 100 and 50, 200 and 20, or MI = 30,000.Table 4
Comparisons for TPL of the IEEE 30-bus system.
Method
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
MI
N
pop
Saving percentage (%)
PSO-TS [16]
4.5213
—
—
—
—
—
−0.137
TS [16]
4.9203
—
—
—
—
—
7.983
PSO [16]
4.6862
—
—
—
—
—
3.387
ALO [27]
4.59
100
—
1.362
QOTLBO [28]
4.5594
4.5601
4.5617
0.037
100
50
0.700
TLBO [28]
4.5629
4.5695
4.5748
0.0564
100
50
0.776
SGA [35]
4.9408
5.0378
5.1651
—
30,000
—
8.365
PSO [35]
4.9239
4.972
5.0576
—
30,000
—
8.051
HSA [35]
4.9059
4.924
4.9653
—
30,000
—
7.713
SFS [37]
4.5777
—
—
1.05
100
20
1.097
ISFS [37]
4.5142
—
—
0.012
—
—
−0.295
SFS
4.5275
4.6732
5.1844
0.1399
200
10Table 5
Comparisons for VD of the IEEE 30-bus system.
Method
The best VD (pu)
Mean VD (pu)
The worst VD (pu)
Std. dev. VD (pu)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
0.1038
0.1597
0.5791
0.1112
200
20
15.51
PSO-TVAC [17]
0.2064
0.2376
0.5796
0.0153
200
20
57.51
SPSO-TVAC [17]
0.1354
0.1558
0.1833
0.0103
200
20
35.23
PSO-CF [17]
0.1287
0.1557
0.4041
0.0404
200
20
31.86
PG-PSO [17]
0.1202
0.144
0.2593
0.0222
200
20
27.04
SWT-PSO [17]
0.1614
0.1814
0.2296
0.133
200
20
45.66
PGSWT-PSO [17]
0.1539
0.2189
0.5532
0.0656
200
20
43.01
MPG-PSO [17]
0.0892
0.1078
0.2518
0.0298
200
20
1.68
QOTLBO [28]
0.0856
0.0872
0.0907
0.0314
200
10
−2.45
TLBO [28]
0.0913
0.0934
0.0988
0.0403
100
50
3.94
SFS [37]
0.122
—
—
0.016
100
20
28.115
ISFS [37]
0.089
—
—
0.0031
100
20
1.461
SFS
0.0877
0.1207
0.1577
0.0155
200
10
—Table 6
Comparisons for VSI of the IEEE 30-bus system.
Method
The best L-index (pu)
Mean L-index (pu)
The worst L-index (pu)
Std. dev. L-index (pu)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
0.1258
0.127
0.1289
0.0008
200
20
19.95
PSO-TVAC [17]
0.1499
0.1513
0.1544
0.0009
200
20
32.82
SPSO-TVAC [17]
0.1271
0.1285
0.1297
0.0006
200
20
20.77
PSO-CF [17]
0.1261
0.1279
0.1295
0.0008
200
20
20.14
PG-PSO [17]
0.1264
0.1297
0.1313
0.0008
200
20
20.33
SWT-PSO [17]
0.1488
0.1634
0.1806
0.0074
200
20
32.33
PGSWT-PSO [17]
0.1394
0.1537
0.1749
0.0081
200
20
27.76
MPG-PSO [17]
0.1241
0.1266
0.1298
0.001
200
20
18.86
BA [27]
0.1191
—
—
—
100
40
15.45
GWO [27]
0.118
—
—
—
100
40
14.66
ABC [27]
0.1161
—
—
—
100
40
13.26
ALO [27]
0.1161
—
—
—
100
40
13.26
QOTLBO [28]
0.1242
0.1245
0.1247
0.0452
100
50
18.92
TLBO [28]
0.1252
0.1254
0.1258
0.0454
100
50
19.57
SFS [37]
0.1252
—
—
0.021
100
20
19.569
ISFS [37]
0.1245
—
—
0.004
100
20
19.116
SFS
0.1007
0.1043
0.1138
0.0026
200
10
—Optimal solutions of three optimization cases are given in Table7.Table 7
Optimal solutions of the IEEE 30-bus system.
Variable
TPL objective
VD objective
L-index objective
V
G1
1.1000
1.0055
1.0807
V
G2
1.0952
1.0011
1.0538
V
G5
1.0750
1.0173
1.0739
V
G8
1.0765
1.0120
1.0089
V
G11
1.0873
1.0320
1.0801
V
G13
1.0999
1.0234
1.0851
T
1
1.0500
1.0500
0.9000
T
2
0.9200
0.9000
0.9000
T
3
1.0100
1.0000
0.9000
T
4
0.9800
0.9700
0.9000
Q
c1
5.0000
4.0000
5.0000
Q
c2
5.0000
2.0000
5.0000
Q
c3
5.0000
4.0000
0.0000
Q
c4
5.0000
3.0000
0.0000
Q
c5
3.0000
5.0000
5.0000
Q
c6
5.0000
3.0000
3.0000
Q
c7
3.0000
5.0000
5.0000
Q
c8
4.0000
5.0000
5.0000
Q
c9
2.0000
3.0000
1.0000Table 8
Optimal solution of the IEEE 118-bus transmission power network.
V
G1 (pu)
1.0081
V
G62
1.0223
V
G113
1.005
V
G4
1.0439
V
G65
1.0563
V
G116
1.0379
V
G6
1.0376
V
G66
1.051
T
8 (pu)
0.9762
V
G8
1.0081
V
G69
1.0574
T
32
1.0192
V
G10
1.0267
V
G70
1.0308
T
36
1.0031
V
G12
1.0309
V
G72
1.0204
T
51
0.9785
V
G15
1.004
V
G73
1.0374
T
93
0.9981
V
G18
1.0021
V
G74
1.0043
T
95
1.0081
V
G19
0.9992
V
G76
0.989
T
102
1.0132
V
G24
1.0069
V
G77
1.0298
T
107
0.9963
V
G25
1.0436
V
G80
1.0471
T
127
0.9821
V
G26
1.0356
V
G85
1.0349
Q
C5 (MVAR)
−2.349
V
G27
1.0366
V
G87
0.9944
Q
C34
7.4927
V
G31
1.023
V
G89
1.0764
Q
C37
−16.55
V
G32
1.0212
V
G90
1.0575
Q
C44
6.8138
V
G34
1.0266
V
G91
1.0464
Q
C45
3.8515
V
G36
1.0244
V
G92
1.0515
Q
C46
5.0318
V
G40
1.0371
V
G99
1.0285
Q
C48
7.1142
V
G42
1.0388
V
G100
1.0516
Q
C74
5.985
V
G46
1.0173
V
G103
1.0361
Q
C79
10.574
V
G49
1.0469
V
G104
1.0148
Q
C82
14.366
V
G54
1.0366
V
G105
1.0133
Q
C83
6.8917
V
G55
1.0321
V
G107
1.0265
Q
C105
6.4333
V
G56
1.0328
V
G110
1.0201
Q
C107
2.3679
V
G59
1.0388
V
G111
1.0197
Q
C110
5.3222
V
G61
1.0221
V
G112
1.0261


## 5.2. Results Obtained on IEEE 118-Bus System

This section uses the IEEE 118-bus system consisting of 54 generator buses, 64 load buses, and 186 transmission lines, 14 VAR compensators, and 9 transformers [37] (Table 8). We implement the SFS method for optimizing total power losses of the system by setting different values to population size and the maximum iteration. In the first trial, we increase the population size from 10 to 100 with a change of 10 and fixing the maximum iteration to 150 and 200. In the second trial, population size is selected to be 50 and 75; meanwhile, the maximum iteration is increased from 50 to 400 with a step of 50. The optimal solution obtained by SFS method is given in Table 9. In addition, the results in terms of the best power loss, mean power loss, and success rate (SR) in percentage are, respectively, shown in Tables 9 and 10. From the two tables, it is clear that SFS can improve search ability since the population size and the maximum iteration are increased but simulation time is also increased. 113.021 MW in Table 9 indicates SFS can find the best solution with the setting of Npop = 60, 70, 80, 90, and 100, and MI = 150, and Npop = 50, 60, 70, 80, 90, and 100 and MI = 200. Clearly, Npop = 60 and MI = 150 are the best selection for the case of tuning population size and fixing the maximum iteration. SFS cannot improve result better if the population size is set to higher than 60 for MI = 150 and 70 for MI = 200. Similarly, 113.021 MW in Table 10 indicates that Npop = 50 and MI = 200 and Npop = 75 and MI = 150 are the best selection for finding the best performance of the SFS method. Improvement of the best solution fails if increasing MI to higher than 200 for Npop = 50 and to higher than 150 for Npop = 75. So, it can conclude that the best selection for population size and the maximum iterations are Npop = 60 and MI = 150. The results from the setting are reported for comparisons with other methods shown in Table 11. The saving percentage of SFS compared with other methods is from 0.09% to 9.1%, while SFS is only less effective than QOTLBO [28] by −0.66%. The comparison indication can conclude that optimal solution of SFS is the second best among all compared methods. Furthermore, search speed of SFS is in the fastest group once MI and Npop are, respectively, 150 and 60, while those from others are 200 and 40, 300 and 15, and 100 and 50. In summary, SFS can find optimal solutions with higher quality than most compared methods; however, compared with some best methods, SFS is also less potential. Consequently, we suggest SFS should be used as an optimization tool for the ORPF problem and it is more promising if SFS is improved by proposing modifications.Table 9
Result obtained by setting MI = 150, MI = 200 and different values for population size.
N
pop
MI = 150
MI = 200
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
10
161.620
206.233
15
46
156.772
193.034
18.69
61
20
154.186
196.747
25
90
149.560
184.155
31.15
119
30
145.552
187.696
47
126
141.185
175.684
54
163
40
135.945
179.062
54
181
131.867
167.602
61
239
50
122.894
159.423
65
226
113.021
149.220
75
302
60
113.021
134.793
88
275
113.021
126.166
89
354
70
113.021
130.075
92
317
113.021
121.750
94
422
80
113.021
125.783
95
359
113.021
124.651
96
477
90
113.021
121.758
94
402
113.021
120.662
98
534
100
113.021
117.618
95
449
113.021
116.559
98
572Table 10
Result obtained by settingNpop = 50 and Npop = 75 and different values for the maximum iterations.
MI
N
pop = 50
N
pop = 75
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
The best TPL (MW)
Mean TPL (MW)
SR (%)
ST (s)
50
154.308
212.230
12
72
124.323
198.647
25
107
100
125.454
202.467
54
147
119.124
189.510
69
221
150
119.011
193.154
77
226
113.021
130.221
82
340
200
113.021
131.027
91
293
113.021
121.887
90
437
250
113.021
129.590
93
365
113.021
121.107
92
545
300
113.021
126.856
95
438
113.021
120.545
94
656
350
113.021
123.036
97
511
113.021
114.952
96
768
400
113.021
119.578
98
583
113.021
114.648
96
876Table 11
Comparisons for TPL of the IEEE 118-bus system.
Method
The best TPL (MW)
Mean TPL (MW)
The worst TPL (MW)
Std. dev. TPL (MW)
MI
N
pop
Saving percentage (%)
PSO-TVIW [17]
116.8976
118.234
126.6222
1.6009
200
40
3.32
PSO-TVAC [17]
124.3335
129.749
134.1254
2.156
200
40
9.10
SPSO-TVAC [17]
116.2026
117.355
118.139
0.4696
200
40
2.74
PSO-CF [17]
115.6469
116.986
119.8378
0.8655
200
40
2.27
PG-PSO [17]
116.6075
119.396
127.0772
2.107
200
40
3.08
SWT-PSO [17]
124.1476
129.371
141.6147
3.309
200
40
8.96
PGSWTPSO [17]
119.427
122.781
125.762
1.2455
200
40
5.36
MPG-PSO [17]
115.06
116.462
118.35
0.528
200
40
1.77
SARCGA [25]
113.12
113.968
—
0.0002
300
15
0.09
HEP [25]
115.58
115.8
—
0.0018
300
15
2.21
QOTLBO [28]
112.2789
113.769
115.4516
0.0244
100
50
−0.66
TLBO [28]
116.4003
121.390
118.4427
0.0482
100
50
2.90
SFS
113.0213
134.793
233.4538
2.91894
150
60


## 6. Conclusion

In the paper, we apply the SFS method for finding optimal solutions of the ORPF problem for different objectives consisting of power loss, voltage deviation, and voltage stability index. Two different power systems with 30 buses and 118 buses are employed for running the SFS method and results found by SFS together with control parameters are compared with those from other methods. As a result, SFS becomes one of the best methods searching the best optimal solutions for each case and its search speed is also faster than most methods. SFS can own the outstanding points, thanks to its construction consisting of three new solution generations, diffusion process, first update process, and the second update process. However, the performance of SFS still copes with constriction, leading to worse results than several methods about solution and speed. Thus, we suggest SFS should be used for finding solutions of the ORPF problem but modifications should be performed on the conventional SFS for improving the search ability.



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*Source: 6382507-2020-03-01.xml*