# Optimal Placement of Wind Power Plants in Transmission Power Networks by Applying an Effectively Proposed Metaheuristic Algorithm

Authors: Minh Quan Duong; Thang Trung Nguyen; Thuan Thanh Nguyen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a modified equilibrium algorithm (MEA) is proposed for optimally determining the position and capacity of wind power plants added in a transmission power network with 30 nodes and effectively selecting operation parameters for other electric components of the network. Two single objectives are separately optimized, including generation cost and active power loss for the case of placing one wind power plant (WPP) and two wind power plants (WPPs) at predetermined nodes and unknown nodes. In addition to the proposed MEA, the conventional equilibrium algorithm (CEA), heap-based optimizer (HBO), forensic-based investigation (FBI), and modified social group optimization (MSGO) are also implemented for the cases. Result comparisons indicate that the generation cost and power loss can be reduced effectively, thanks to the suitable location selection and appropriate power determination for WPPs. In addition, the generation cost and loss of the proposed MEA are also less than those from other compared methods. Thus, it is recommended that WPPs should be placed in power systems to reduce cost and loss, and MEA is a powerful method for the placement of wind power plants in power systems.


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## Body

## 1. Introduction

Solving optimal power flow problem (OPF) to have the steady and effective states of power systems is considered as the leading priority in operation of power systems. Specifically, the steady state is represented as a state vector and regarded as a set of variables, such as output of active and reactive power from power plants, voltage of generators in power plants, output of reactive power from shunt capacitors, transformers’ tap, voltage of loads, and operating current of transmission lines [1–3]. Generally, during the whole process of solving the OPF problem to determine the steady state in power system operation, the mentioned variables are separated into control variables and dependent variables [4, 5]. The output of the reactive power from power plants (QG), output of the active power of power plants at slack node (PGs), voltage of loads (VL), and current of lines (Il) are grouped in a dependent variable set [6–10], whereas other remaining variables including tap changer of transformers (TapT), output of the active power from the generators excluding that at slack node (PGs), and output of the reactive power supplied by capacitor banks (QCap) are put in a control variable set [11–15]. These control variables are utilized as the input of the Mathpower programme to find the dependent variables. The Mathpower programme is a calculating tool developed based on the Newton–Raphson method to deal with power flow. After having the dependent variable set, it is checked and penalized based on previously known upper bound and lower bound. The violation of the bounds will be considered for the quality of both control and dependent variable sets [16–20]. These violations are converted into penalty terms and added to objective functions, such as electrical power generation cost (Coste), active power loss (Ploss), polluted emission (Em), and load voltage stability index (IDsl).Recently, the presence of renewable energies has been considered in power systems when the percentages of wind power and solar energy joining into the process of generating electricity become more and more. In that situation, the OPF problem was modified and became more complex than ever. The conventional version of the OPF problem only considers thermal power plants (THPs) as the main source [21–24]. Other modified versions of the OPF problem, both THPs and renewable energies, are power sources. The modified OPF problem is outlined in Figure 1 in which the conventional OPF problem is a part of the figure without variables regarding renewable energies, such as output of active and reactive power of wind power plant (Pw, Qw), output of active and reactive power of photovoltaic power plants (PVPs) (Ppv, Qpv), and location of WPPs and PVPs (Lw, Lpv). There are large number of studies proposed to handle the modified OPF problems. These studies can be classified into three main groups. Specifically, the first group solves the OPF problem considering wind power source injecting both active and reactive power into grid. The second group considers the assumption that wind energy sources just generate active power only. The third group considers both wind and solar energies in the process of solving the OPF problem. The applied methods, test systems, objective functions, placed renewable power plants, and compared methods regarding modified OPF problems are summarized in Table 1. All the studies in the table have focused on the placement of wind and photovoltaic power plants to cut electricity generation fuel cost for THPs, and the results were mainly compared to base systems without the contribution of the renewable plants. In addition, other research directions of optimal power flow are without renewable power plants but using reactive power dispatch [50, 51] and using VSC (voltage source converter) based on HVDC (high-voltage direct current) [52, 53]. These studies also achieved the reduction of cost and improved the quality of voltage as expected. If the combination of both using renewable energies and optimal dispatch of reactive power or the combination of using both renewable energies and these converters can be implemented, expected results such as the reduction of cost and power loss and the voltage enhancement can be significantly better.Figure 1
Configuration of the modified OPF problem in the presence of renewable energies.Table 1
The summary of studies proposed to solve the modified OPF problem considering renewable energies.
ReferenceMethodApplied systemRenewable energyCompared methods[25]BFA30 nodesWind (P, Q)GA[26]MBFA30 nodesWind (P, Q)ACA[27]HABC30 nodesWind (P, Q)ABA[28]MCS30, 57 nodesWind (P, Q)MPSO[29]HA30 nodesWind (P, Q)PSO[30]MFO30 nodesWind, solar (P, Q)GWA, MVA, IMA[31]AFAPA30, 75 nodesWind (P)APO, BA[32]KHA30, 57 nodesWind (P)ALPSO, DE, RCGA[33]GSO300 nodesWind (P)NSGA-II[34]MHGSPSO30, 57 nodesWind (P, Q)MSA, GWA, WA[35]BSA30 nodesWind, solar (P)—[36]IMVA30 nodesWind, solar (P, Q)PSO, MVA, NSGA-II[37]PSO39 nodesWind, solar (P)PSO variants[38]NSGA-II30, 118 nodesWind, solar (P)—[39]MFA30 nodesWind, solar (P, Q)MDE[40]GWO30, 57 nodesWind, solar (P)GA, PSO, CSA1, MDE, ABA[41]BWOA, ALO, PSO30 nodesWind, solar (P, Q)—GSA, MFA, BMA[42]FPAIEEE 30-busWind, solar (P)—[43]APDEIEEE 30-busWind, solar (P, Q)—[44]HGTPEA30 nodesWind, solar (P)—[45]HGNIPA118 nodesWind, solar (P, Q)—[46]NDSGWA30 nodesWind, solar (P)MDE[47]JYA30 nodesWind, solar (P)—[48]HSQTIICA30 nodesDG (P, Q)IICA[49]MJYA30, 118 nodesWind (P)MSA, ABA, CSA, GWA, BSAIn recent years, metaheuristic algorithms have been developed widely and applied successfully for optimization problems in engineering. One of the most well-known algorithms is the conventional equilibrium algorithm (CEA) [54], which was introduced in the early 2020. The conventional version was demonstrated more effective than PSO, GWA, GA, GSA, and SSA for a set of fifty-eight mathematical functions with a different number of variables and types. Over the past year and this year, CEA was widely replicated for different optimization problems such as AC/DC power grids [55], loss reduction of distribution networks [56], component design for vehicles [57], and multidisciplinary problem design [58]. However, the performance of CEA is not the most effective among utilized methods for the same problems. Consequently, CEA had been indicated to be effective for large-scale problems, and it needs more improvements [59–62]. Thus, we proposed another version of CEA, called the modified equilibrium algorithm (MEA), and also applied four other metaheuristic algorithms for checking the performance of MEA.In this paper, the authors solve a modified OPF (MOPF) problem with the placement of wind power plants in an IEEE 30-bus transmission power network. About the number of wind power plants located in the system, two cases are, respectively, one wind power plant (WPP) and two WPPs. About the locations of the WPPs, simple cases are referred to the previous study [47] and other more complicated cases are to determine suitable buses in the system by applying metaheuristic algorithms. It is noted that the study in [47] has only studied the placement of one WPP, and it has indicated the most suitable bus as bus 30 and the most ineffective bus as bus 3. In this paper, we have employed buses 3 and 30 for two separated cases to check the indication of the study [47]. The results indicated that the placement of one WPP at bus 30 can reach smaller power loss and smaller fuel cost than at bus 3. In addition, the paper also investigated the effectiveness of locations by applying MEA and four other metaheuristic algorithms to determine the location. As a result, placing one WPP at bus 30 has reached the smallest power loss and the smallest total fuel cost. For the case of placing two WPPs, buses 30 and 3 could not result in the smallest fuel cost and the smallest power loss. Buses 30 and 5 are the best locations for the minimization of fuel cost, while buses 30 and 24 are the best locations for the minimization of power loss. Therefore, the main contribution of the study regarding the electrical field is determining the best locations for the best power loss and the best total cost.All study cases explained above are implemented by the proposed MEA and four existing algorithms published in 2020, including conventional equilibrium algorithm (CEA) [54], heap-based optimizer (HBO) [63], forensic-based investigation (FBI) [64], and modified social group optimization (MSGO) [65]. As a result, the best locations leading to the smallest cost and smallest loss are obtained by MEA. Thus, the applications of the four recent algorithms and the proposed MEA aim to introduce a new algorithm and show their effectiveness to readers in solving the MOPF problem. Readers can give evaluations and decide if the algorithms are used for their own optimization problems, which maybe in electrical engineering or other fields. The major contributions of the paper are summarized again as follows:(1)
Find the best locations for placing wind power plants in the IEEE 30-bus transmission power gird.(2)
The added wind power plants and other found parameters of the system found by MEA can reach the smallest power loss and smallest total cost.(3)
Introduce four existing algorithms developed in 2020 and a proposed MEA. In addition, the performance of these optimization tools is shown to readers for deciding if these tools are used for their applications.(4)
Provide MEA, the most effective algorithm among five applied optimization tools for the MOPF problem.The organization of the paper is as follows. Two single objectives and a considered constraint set are presented in Section2. The configuration of CEA for solving a sample optimization problem and then modified points of MEA are clarified in detail in Section 3. Section 4 summarizes the computation steps for solving the modified OPF problem by using MEA. Section 5 presents results obtained by the proposed MEA and other methods such as JYA, FBI, HBO, and MSGO. Finally, conclusions are given in Section 6 for stating achievements in the paper.


## 2. Objective Functions and Constraints of the Modified OPF Problem

### 2.1. Objective Functions

#### 2.1.1. Minimization of Electricity Generation Cost

In this research, the first single objective is considered to be electricity generation cost of all thermal generators. At generator nodes, where thermal units are working, the cost is the most important factor in optimal operation of the distribution power networks, and it should be low reasonably as the following model. The total cost is formulated by(1)EGC=∑i=1NTGFETGiPTGi,where FETGi is the fuel cost of the ith thermal unit and calculated as follows:(2)FETGiPTGi=μ1i+μ2iPTGi+μ3iPTGi2.


#### 2.1.2. Minimization of Active Power Loss

Minimizing active power loss (APL) is a highly important target in transmission line operation. In general, reactive power loss of transmission power networks is very significant due to a high number of transmission lines with high operating current. If the loss can be minimized, the energy loss and the energy loss cost are also reduced accordingly. The loss can be obtained by different ways as follows:(3)APL=∑q=1NBr3.Iq2.Rq,APL=∑i=1NTGPTGi−∑i=1NnPrqi,APL=∑x=1Nn∑y=1,x≠yNnYxyUx2+Vy2−2UxUycosφx−φy.


### 2.2. Constraints

#### 2.2.1. Physical Constraints regarding Thermal Generators

In the operating process of thermal generators, three main constraints need to be supervised strictly consisting of the limitation of real power output, the limitation of reactive power output, and the limitation of the voltage magnitude. The violation of any limitations as mentioned will cause damage and insecure status in whole system substantially. Thus, the following constraints should be satisfied all the time:(4)PTGimin≤PTGi≤PTGimax,withi=1,…,NTG,QTGimin≤QTGi≤QTGimax,withi=1,…,NTG,UTGimin≤UTGi≤UTGimax,withi=1,…,NTG.


#### 2.2.2. The Power Balance Constraint

The power balance constraint is the relationship between source side and consumption side in which sources are TUs and renewable energies, and consumption side is comprised of loads and loss on lines. The balance status is established when the amount of power supplied by thermal generators equals to the amount of power required by load plus the loss.Active power equation at each nodex is formulated as follows:(5)PTGx−Prqx=Ux∑y=1NnUyYxycosφx−φy+Xxysinφx−φy.For the case that wind turbines supply electricity at nodex, the balance of the active power is as follows:(6)PTGx+PWindx−Prqx=Ux∑y=1NnUyYxycosφx−φy+Xxysinφx−φy,where PWindx is the power generation of wind turbines at node x and limited by the following constraint:(7)PWindmin≤PWindx≤PWindmax.Similarly, reactive power is also balanced at nodex as the following model:(8)QTGx+QComx−Qrqx=Ux∑y=1NnUyYxysinφx−φy−Xxycosφx−φy,where(9)QComxmin≤QComx≤QComxmax,withi=1,…,NCom.For the case that wind turbines are placed at nodex, the reactive power is also supplied by the turbine as the role of thermal generators. As a result, the reactive power balance is as follows:(10)QTGx+QWindx+QComx−Qrqx=Ux∑y=1NnUyYxysinφx−φy−Xxycosφx−φy,∈where QWindx is the reactive power generation of wind turbines at node x and is subject to the following constraint:(11)QWindmin≤QWindx≤QWindmax.


#### 2.2.3. Other Inequality Constraints

These constraints are related to operating limits of electric components such as lines, loads, and transformers. Lines and loads are dependent on other operating parameters of other components like TUs, wind turbines, shunt capacitors, and transformers. However, operating values of lines and loads are very important for a stable operating status of networks. If the components are working beyond their allowable range, networks are working unstably, and fault can occur in the next phenomenon. Thus, the operating parameters of loads and lines must be satisfied as shown in the following models:(12)ULNmin≤ULNt≤ULNmax,witht=1,…,NLN,SBrq≤SBrmax,withq=1,…,NBr.In addition, transformers located at some nodes need to be tuned for supplying standard voltage within a working range. The voltage regulation is performed by setting tap of transformers satisfying the following constraint:(13)Tapmin≤Tapi≤Tapmax,withi=1,…,NT.


## 2.1. Objective Functions

### 2.1.1. Minimization of Electricity Generation Cost

In this research, the first single objective is considered to be electricity generation cost of all thermal generators. At generator nodes, where thermal units are working, the cost is the most important factor in optimal operation of the distribution power networks, and it should be low reasonably as the following model. The total cost is formulated by(1)EGC=∑i=1NTGFETGiPTGi,where FETGi is the fuel cost of the ith thermal unit and calculated as follows:(2)FETGiPTGi=μ1i+μ2iPTGi+μ3iPTGi2.


### 2.1.2. Minimization of Active Power Loss

Minimizing active power loss (APL) is a highly important target in transmission line operation. In general, reactive power loss of transmission power networks is very significant due to a high number of transmission lines with high operating current. If the loss can be minimized, the energy loss and the energy loss cost are also reduced accordingly. The loss can be obtained by different ways as follows:(3)APL=∑q=1NBr3.Iq2.Rq,APL=∑i=1NTGPTGi−∑i=1NnPrqi,APL=∑x=1Nn∑y=1,x≠yNnYxyUx2+Vy2−2UxUycosφx−φy.


## 2.1.1. Minimization of Electricity Generation Cost

In this research, the first single objective is considered to be electricity generation cost of all thermal generators. At generator nodes, where thermal units are working, the cost is the most important factor in optimal operation of the distribution power networks, and it should be low reasonably as the following model. The total cost is formulated by(1)EGC=∑i=1NTGFETGiPTGi,where FETGi is the fuel cost of the ith thermal unit and calculated as follows:(2)FETGiPTGi=μ1i+μ2iPTGi+μ3iPTGi2.


## 2.1.2. Minimization of Active Power Loss

Minimizing active power loss (APL) is a highly important target in transmission line operation. In general, reactive power loss of transmission power networks is very significant due to a high number of transmission lines with high operating current. If the loss can be minimized, the energy loss and the energy loss cost are also reduced accordingly. The loss can be obtained by different ways as follows:(3)APL=∑q=1NBr3.Iq2.Rq,APL=∑i=1NTGPTGi−∑i=1NnPrqi,APL=∑x=1Nn∑y=1,x≠yNnYxyUx2+Vy2−2UxUycosφx−φy.


## 2.2. Constraints

### 2.2.1. Physical Constraints regarding Thermal Generators

In the operating process of thermal generators, three main constraints need to be supervised strictly consisting of the limitation of real power output, the limitation of reactive power output, and the limitation of the voltage magnitude. The violation of any limitations as mentioned will cause damage and insecure status in whole system substantially. Thus, the following constraints should be satisfied all the time:(4)PTGimin≤PTGi≤PTGimax,withi=1,…,NTG,QTGimin≤QTGi≤QTGimax,withi=1,…,NTG,UTGimin≤UTGi≤UTGimax,withi=1,…,NTG.


### 2.2.2. The Power Balance Constraint

The power balance constraint is the relationship between source side and consumption side in which sources are TUs and renewable energies, and consumption side is comprised of loads and loss on lines. The balance status is established when the amount of power supplied by thermal generators equals to the amount of power required by load plus the loss.Active power equation at each nodex is formulated as follows:(5)PTGx−Prqx=Ux∑y=1NnUyYxycosφx−φy+Xxysinφx−φy.For the case that wind turbines supply electricity at nodex, the balance of the active power is as follows:(6)PTGx+PWindx−Prqx=Ux∑y=1NnUyYxycosφx−φy+Xxysinφx−φy,where PWindx is the power generation of wind turbines at node x and limited by the following constraint:(7)PWindmin≤PWindx≤PWindmax.Similarly, reactive power is also balanced at nodex as the following model:(8)QTGx+QComx−Qrqx=Ux∑y=1NnUyYxysinφx−φy−Xxycosφx−φy,where(9)QComxmin≤QComx≤QComxmax,withi=1,…,NCom.For the case that wind turbines are placed at nodex, the reactive power is also supplied by the turbine as the role of thermal generators. As a result, the reactive power balance is as follows:(10)QTGx+QWindx+QComx−Qrqx=Ux∑y=1NnUyYxysinφx−φy−Xxycosφx−φy,∈where QWindx is the reactive power generation of wind turbines at node x and is subject to the following constraint:(11)QWindmin≤QWindx≤QWindmax.


### 2.2.3. Other Inequality Constraints

These constraints are related to operating limits of electric components such as lines, loads, and transformers. Lines and loads are dependent on other operating parameters of other components like TUs, wind turbines, shunt capacitors, and transformers. However, operating values of lines and loads are very important for a stable operating status of networks. If the components are working beyond their allowable range, networks are working unstably, and fault can occur in the next phenomenon. Thus, the operating parameters of loads and lines must be satisfied as shown in the following models:(12)ULNmin≤ULNt≤ULNmax,witht=1,…,NLN,SBrq≤SBrmax,withq=1,…,NBr.In addition, transformers located at some nodes need to be tuned for supplying standard voltage within a working range. The voltage regulation is performed by setting tap of transformers satisfying the following constraint:(13)Tapmin≤Tapi≤Tapmax,withi=1,…,NT.


## 2.2.1. Physical Constraints regarding Thermal Generators

In the operating process of thermal generators, three main constraints need to be supervised strictly consisting of the limitation of real power output, the limitation of reactive power output, and the limitation of the voltage magnitude. The violation of any limitations as mentioned will cause damage and insecure status in whole system substantially. Thus, the following constraints should be satisfied all the time:(4)PTGimin≤PTGi≤PTGimax,withi=1,…,NTG,QTGimin≤QTGi≤QTGimax,withi=1,…,NTG,UTGimin≤UTGi≤UTGimax,withi=1,…,NTG.


## 2.2.2. The Power Balance Constraint

The power balance constraint is the relationship between source side and consumption side in which sources are TUs and renewable energies, and consumption side is comprised of loads and loss on lines. The balance status is established when the amount of power supplied by thermal generators equals to the amount of power required by load plus the loss.Active power equation at each nodex is formulated as follows:(5)PTGx−Prqx=Ux∑y=1NnUyYxycosφx−φy+Xxysinφx−φy.For the case that wind turbines supply electricity at nodex, the balance of the active power is as follows:(6)PTGx+PWindx−Prqx=Ux∑y=1NnUyYxycosφx−φy+Xxysinφx−φy,where PWindx is the power generation of wind turbines at node x and limited by the following constraint:(7)PWindmin≤PWindx≤PWindmax.Similarly, reactive power is also balanced at nodex as the following model:(8)QTGx+QComx−Qrqx=Ux∑y=1NnUyYxysinφx−φy−Xxycosφx−φy,where(9)QComxmin≤QComx≤QComxmax,withi=1,…,NCom.For the case that wind turbines are placed at nodex, the reactive power is also supplied by the turbine as the role of thermal generators. As a result, the reactive power balance is as follows:(10)QTGx+QWindx+QComx−Qrqx=Ux∑y=1NnUyYxysinφx−φy−Xxycosφx−φy,∈where QWindx is the reactive power generation of wind turbines at node x and is subject to the following constraint:(11)QWindmin≤QWindx≤QWindmax.


## 2.2.3. Other Inequality Constraints

These constraints are related to operating limits of electric components such as lines, loads, and transformers. Lines and loads are dependent on other operating parameters of other components like TUs, wind turbines, shunt capacitors, and transformers. However, operating values of lines and loads are very important for a stable operating status of networks. If the components are working beyond their allowable range, networks are working unstably, and fault can occur in the next phenomenon. Thus, the operating parameters of loads and lines must be satisfied as shown in the following models:(12)ULNmin≤ULNt≤ULNmax,witht=1,…,NLN,SBrq≤SBrmax,withq=1,…,NBr.In addition, transformers located at some nodes need to be tuned for supplying standard voltage within a working range. The voltage regulation is performed by setting tap of transformers satisfying the following constraint:(13)Tapmin≤Tapi≤Tapmax,withi=1,…,NT.


## 3. The Proposed Modified Equilibrium Algorithm (MEA)

### 3.1. Conventional Equilibrium Algorithm (CEA)

CEA was first introduced and applied in 2020 for solving a high number of benchmark functions. The method was superior to popular and well-known metaheuristic algorithms, but its feature is simple with one technique of newly updating solutions and one technique of keeping promising solutions between new and old solutions.The implementation of CEA for a general optimization problem is mathematically presented as follows.


#### 3.1.1. The Generation of Initial Population

CEA has a set ofN1 candidate solutions similar to other metaheuristic algorithms. The solution set needs to define the boundaries in advance, and then it must be produced in the second stage. The set of solution is represented by Z = [Zd], where d = 1, …, N1, and the fitness function of the solution set is represented by Fit = [Fitd], where d = 1, …, N1.To produce an initial solution set, control variables included in each solution and their boundaries must be, respectively, predetermined as follows:(14)Zd=zjd;j=1,…,N2;d=1,…N1,Zlow=zjlow;j=1,…,N2,Zup=zjup;j=1,…,N2,where N2 is the control variable number, zjd is the jth variable of the dth solution, Zlow and Zup are lower and upper bounds of all solutions, respectively, and zjlow and zjup are the minimum and maximum values of the jth control variable, respectively.The initial solutions are produced within their boundsZlow and Zup as follows:(15)Zd=Zlow+r1⋅Zup−Zlow;d=1,…,N1.


#### 3.1.2. New Update Technique for Variables

The matrix fitness fit is sorted to select the four best solutions with the lowest fitness values among the available set. The solution with the lowest fitness is set toZb1, while the second, third, and fourth best solutions with the second, third, and fourth lowest fitness functions are assigned to Zb2, Zb3, and Zb4. In addition, another solution, which is called the middle solution (Zmid) of the four best solutions, is also produced by(16)Zmid=Zb1+Zb2+Zb3+Zb44.The four best solutions and the middle solution are grouped into the solution setZ5b as follows:(17)Z5b=Zb1,Zb2,Zb3,Zb4,Zmid.As a result, the new solutionZdnew of the old solution Zd is determined as follows:(18)Zdnew=Z5brd+M⋅Zd−Z5brd+K×Mr21−M.In the above equation,Z5brd is a randomly chosen solution among five solutions of Z5b in equation (17), whereas M and K are calculated by(19)M=2signr3−0.51eA.Iter−1,(20)K=K0⋅Z5brd−r4⋅Zd,(21)A=1−IterN3Iter/N3,(22)K0=0,if&r5<0.5,r62,& otherwise.


#### 3.1.3. New Solution Correction

The new solutionZdnew is a set of new control variables zjd,,new that can be beyond the minimum and maximum values of control variables. It means zjd,,new may be either higher than zjup or smaller than zjlow. If one out of the two cases happens, each new variable zjd,,new must be redetermined as follows:(23)zjd,new=zjd,new,ifzjlow≤zjd,new≤zjup,zjlow,ifzjlow>zjd,new,zjup,ifzjd,new>zjup.After correcting the new solutions, the new fitness function is calculated and assigned to Fitdnew.


#### 3.1.4. Selection of Good Solutions

Currently, there are two solution sets, one old set and one new set. Therefore, it is important to retain higher quality solutions so that the retained solutions are equal toN1. This task is accomplished by using the following formula:(24)Zd=Zdnew,ifFitd>Fitdnew,Zdnew,ifFitd=Fitdnew,Zd,else,(25)Fitd=Fitdnew,ifFitd>Fitdnew,Fitdnew,ifFitd=Fitdnew,Fitd,else.


#### 3.1.5. Termination Condition

CEA will stop updating new control variables when the computation iteration reaches the maximum valueN3. In addition, the best solution and its fitness are also reported.


### 3.2. The Proposed MEA

The proposed MEA is a modified variant of CEA by using a new technique for updating new control variables. From equation (18), it sees that CEA only chooses search spaces around the four best solutions and the middle solution (i.e., Zb1,Zb2,Zb3,Zb4,Zmid) for updating decision variables whilst from search spaces nearby from the fifth best solution to the worst solution are skipped intentionally. In addition, the strategy has led to the success of CEA with better performance than other metaheuristics. However, CEA cannot reach a higher performance because it is coping with two shortcomings as follows:(1)
The first shortcoming is to pick up one out of five solutions in the setZ5b randomly. The search spaces may be repeated more than once and even too many times. Therefore, promising search spaces can be exploited ineffectively or skipped unfortunately.(2)
The second shortcoming is to use two update steps includingM×Zb1−Zd and K×M/r21−M, which are decreased when the computation iteration is increased. Especially, the steps become zero at final computation iterations. In fact, parameter A in equation (21) becomes 0 when the current iteration is equal to the maximum iteration N3. If we substitute A = 0 into equation (19), M becomes 0.Thus, the proposed MEA is reformed to eliminate the above drawbacks of CEA and reach better results as solving the OPF problem with the presence of wind energy. The two proposed formulas for updating new decision variables are as follows:(26)Zdnew1=Zd+MZb1−Zd+r7Zr1−Zr2,(27)Zdnew2=Zb1+MZd−Zb1+r8Zr3−Zr4.The two equations above are not applied simultaneously for the same old solutioni. Either equation (26) or equation (27) is used for the dth new solution. Zdnew1 in equation (26) is applied to update Zd if Zd has better fitness than the medium fitness of the population, i.e., Fitd < Fitmean. For the other case, i.e., Fitd ≥ Fitmean, Zdnew2 in equation (27) is determined.


## 3.1. Conventional Equilibrium Algorithm (CEA)

CEA was first introduced and applied in 2020 for solving a high number of benchmark functions. The method was superior to popular and well-known metaheuristic algorithms, but its feature is simple with one technique of newly updating solutions and one technique of keeping promising solutions between new and old solutions.The implementation of CEA for a general optimization problem is mathematically presented as follows.


### 3.1.1. The Generation of Initial Population

CEA has a set ofN1 candidate solutions similar to other metaheuristic algorithms. The solution set needs to define the boundaries in advance, and then it must be produced in the second stage. The set of solution is represented by Z = [Zd], where d = 1, …, N1, and the fitness function of the solution set is represented by Fit = [Fitd], where d = 1, …, N1.To produce an initial solution set, control variables included in each solution and their boundaries must be, respectively, predetermined as follows:(14)Zd=zjd;j=1,…,N2;d=1,…N1,Zlow=zjlow;j=1,…,N2,Zup=zjup;j=1,…,N2,where N2 is the control variable number, zjd is the jth variable of the dth solution, Zlow and Zup are lower and upper bounds of all solutions, respectively, and zjlow and zjup are the minimum and maximum values of the jth control variable, respectively.The initial solutions are produced within their boundsZlow and Zup as follows:(15)Zd=Zlow+r1⋅Zup−Zlow;d=1,…,N1.


### 3.1.2. New Update Technique for Variables

The matrix fitness fit is sorted to select the four best solutions with the lowest fitness values among the available set. The solution with the lowest fitness is set toZb1, while the second, third, and fourth best solutions with the second, third, and fourth lowest fitness functions are assigned to Zb2, Zb3, and Zb4. In addition, another solution, which is called the middle solution (Zmid) of the four best solutions, is also produced by(16)Zmid=Zb1+Zb2+Zb3+Zb44.The four best solutions and the middle solution are grouped into the solution setZ5b as follows:(17)Z5b=Zb1,Zb2,Zb3,Zb4,Zmid.As a result, the new solutionZdnew of the old solution Zd is determined as follows:(18)Zdnew=Z5brd+M⋅Zd−Z5brd+K×Mr21−M.In the above equation,Z5brd is a randomly chosen solution among five solutions of Z5b in equation (17), whereas M and K are calculated by(19)M=2signr3−0.51eA.Iter−1,(20)K=K0⋅Z5brd−r4⋅Zd,(21)A=1−IterN3Iter/N3,(22)K0=0,if&r5<0.5,r62,& otherwise.


### 3.1.3. New Solution Correction

The new solutionZdnew is a set of new control variables zjd,,new that can be beyond the minimum and maximum values of control variables. It means zjd,,new may be either higher than zjup or smaller than zjlow. If one out of the two cases happens, each new variable zjd,,new must be redetermined as follows:(23)zjd,new=zjd,new,ifzjlow≤zjd,new≤zjup,zjlow,ifzjlow>zjd,new,zjup,ifzjd,new>zjup.After correcting the new solutions, the new fitness function is calculated and assigned to Fitdnew.


### 3.1.4. Selection of Good Solutions

Currently, there are two solution sets, one old set and one new set. Therefore, it is important to retain higher quality solutions so that the retained solutions are equal toN1. This task is accomplished by using the following formula:(24)Zd=Zdnew,ifFitd>Fitdnew,Zdnew,ifFitd=Fitdnew,Zd,else,(25)Fitd=Fitdnew,ifFitd>Fitdnew,Fitdnew,ifFitd=Fitdnew,Fitd,else.


### 3.1.5. Termination Condition

CEA will stop updating new control variables when the computation iteration reaches the maximum valueN3. In addition, the best solution and its fitness are also reported.


## 3.1.1. The Generation of Initial Population

CEA has a set ofN1 candidate solutions similar to other metaheuristic algorithms. The solution set needs to define the boundaries in advance, and then it must be produced in the second stage. The set of solution is represented by Z = [Zd], where d = 1, …, N1, and the fitness function of the solution set is represented by Fit = [Fitd], where d = 1, …, N1.To produce an initial solution set, control variables included in each solution and their boundaries must be, respectively, predetermined as follows:(14)Zd=zjd;j=1,…,N2;d=1,…N1,Zlow=zjlow;j=1,…,N2,Zup=zjup;j=1,…,N2,where N2 is the control variable number, zjd is the jth variable of the dth solution, Zlow and Zup are lower and upper bounds of all solutions, respectively, and zjlow and zjup are the minimum and maximum values of the jth control variable, respectively.The initial solutions are produced within their boundsZlow and Zup as follows:(15)Zd=Zlow+r1⋅Zup−Zlow;d=1,…,N1.


## 3.1.2. New Update Technique for Variables

The matrix fitness fit is sorted to select the four best solutions with the lowest fitness values among the available set. The solution with the lowest fitness is set toZb1, while the second, third, and fourth best solutions with the second, third, and fourth lowest fitness functions are assigned to Zb2, Zb3, and Zb4. In addition, another solution, which is called the middle solution (Zmid) of the four best solutions, is also produced by(16)Zmid=Zb1+Zb2+Zb3+Zb44.The four best solutions and the middle solution are grouped into the solution setZ5b as follows:(17)Z5b=Zb1,Zb2,Zb3,Zb4,Zmid.As a result, the new solutionZdnew of the old solution Zd is determined as follows:(18)Zdnew=Z5brd+M⋅Zd−Z5brd+K×Mr21−M.In the above equation,Z5brd is a randomly chosen solution among five solutions of Z5b in equation (17), whereas M and K are calculated by(19)M=2signr3−0.51eA.Iter−1,(20)K=K0⋅Z5brd−r4⋅Zd,(21)A=1−IterN3Iter/N3,(22)K0=0,if&r5<0.5,r62,& otherwise.


## 3.1.3. New Solution Correction

The new solutionZdnew is a set of new control variables zjd,,new that can be beyond the minimum and maximum values of control variables. It means zjd,,new may be either higher than zjup or smaller than zjlow. If one out of the two cases happens, each new variable zjd,,new must be redetermined as follows:(23)zjd,new=zjd,new,ifzjlow≤zjd,new≤zjup,zjlow,ifzjlow>zjd,new,zjup,ifzjd,new>zjup.After correcting the new solutions, the new fitness function is calculated and assigned to Fitdnew.


## 3.1.4. Selection of Good Solutions

Currently, there are two solution sets, one old set and one new set. Therefore, it is important to retain higher quality solutions so that the retained solutions are equal toN1. This task is accomplished by using the following formula:(24)Zd=Zdnew,ifFitd>Fitdnew,Zdnew,ifFitd=Fitdnew,Zd,else,(25)Fitd=Fitdnew,ifFitd>Fitdnew,Fitdnew,ifFitd=Fitdnew,Fitd,else.


## 3.1.5. Termination Condition

CEA will stop updating new control variables when the computation iteration reaches the maximum valueN3. In addition, the best solution and its fitness are also reported.


## 3.2. The Proposed MEA

The proposed MEA is a modified variant of CEA by using a new technique for updating new control variables. From equation (18), it sees that CEA only chooses search spaces around the four best solutions and the middle solution (i.e., Zb1,Zb2,Zb3,Zb4,Zmid) for updating decision variables whilst from search spaces nearby from the fifth best solution to the worst solution are skipped intentionally. In addition, the strategy has led to the success of CEA with better performance than other metaheuristics. However, CEA cannot reach a higher performance because it is coping with two shortcomings as follows:(1)
The first shortcoming is to pick up one out of five solutions in the setZ5b randomly. The search spaces may be repeated more than once and even too many times. Therefore, promising search spaces can be exploited ineffectively or skipped unfortunately.(2)
The second shortcoming is to use two update steps includingM×Zb1−Zd and K×M/r21−M, which are decreased when the computation iteration is increased. Especially, the steps become zero at final computation iterations. In fact, parameter A in equation (21) becomes 0 when the current iteration is equal to the maximum iteration N3. If we substitute A = 0 into equation (19), M becomes 0.Thus, the proposed MEA is reformed to eliminate the above drawbacks of CEA and reach better results as solving the OPF problem with the presence of wind energy. The two proposed formulas for updating new decision variables are as follows:(26)Zdnew1=Zd+MZb1−Zd+r7Zr1−Zr2,(27)Zdnew2=Zb1+MZd−Zb1+r8Zr3−Zr4.The two equations above are not applied simultaneously for the same old solutioni. Either equation (26) or equation (27) is used for the dth new solution. Zdnew1 in equation (26) is applied to update Zd if Zd has better fitness than the medium fitness of the population, i.e., Fitd < Fitmean. For the other case, i.e., Fitd ≥ Fitmean, Zdnew2 in equation (27) is determined.


## 4. The Application of the Proposed MEA for OPF Problem

### 4.1. Generation of Initial Population

The problem of placing wind turbines in the transmission power network is successfully solved by using decision variables as follows: the active power generation and voltage of thermal generators (excluding power generation at slack node), generation of capacitors, tap of transformer, and position, active and reactive power of wind turbines. Hence,Zd is comprised of the following decision variables: PTGi (i = 2, …, NTG); UTGi (i = 1, …, NTG); QComi (i = 1, …, NCom); Tapi (i = 1, …, NT); PWindx (x = 1, …, NW); QWindx (x = 1, …, NW); and LWindx (x = 1, …, NW).The decision variables are initialized within their lower boundZlow and upper bound Zup as shown in Section 2.


### 4.2. The Calculation of Dependent Variables

Before running Mathpower program, control variables of wind turbines including active power, reactive power, and location are collected to calculate the new values of loads at the placement of the wind turbines. Then, the data of the load must be changed and then added in the input data of Mathpower program. Finally, other remaining decision variables are added to Mathpower program and running power flow for obtaining dependent variables includingPTG1; QTGi (i = 1, …, NTG); ULNt (t = 1, …, NLN); and SBrq (q = 1, …, NBr).


### 4.3. Solution Evaluation

The quality of solutionZd is evaluated by calculating the fitness function. Total cost and total active power loss are two single objectives, while the violations of dependent variables are converted into penalty values [66].


### 4.4. Implementation of MEA for the Problem

In order to reach the optimal solution for the OPF problem with the presence of wind turbines, the implementation of MEA is shown in the following steps and is summarized in Figure2.Step 1: select populationN1 and maximum iteration N2.Step 2: select and initialize decision variables for the population as shown in Section4.1.Step 3: collect variables of wind turbines and tune loads.Step 4: running Mathpower for obtaining dependent variables shown in Section4.2.Step 5: evaluate quality of obtained solutions as shown in Section4.3.Step 6: select the best solutionZb1 and set Iter = 1.Step 7: select four random solutionsZr1, Zr2, Zr3, and Zr4.Step 8: calculate the mean fitness of the whole population.Step 9: produce new solutions.If Fitd < Fitmean, apply equation (26) to produce new solution. Otherwise, apply equation (27) to produce new solutions.Step 10: correct new solutions using equation (23).Step 11: collect variables of wind turbines and tune loads.Step 12: running Mathpower for obtaining dependent variables shown in Section4.2.Step 13: evaluate quality of obtained solutions as shown in Section4.3.Step 14: select good solutions among old population and new population using equations (24) and (25).Step 15: select the best solutionZb1.Step 16: if Iter =N3, stop the search process and print the optimal solution. Otherwise, set Iter to Iter + 1 and back to Step 7.Figure 2
The flowchart of using MEA for solving the modified OPF problem.


## 4.1. Generation of Initial Population

The problem of placing wind turbines in the transmission power network is successfully solved by using decision variables as follows: the active power generation and voltage of thermal generators (excluding power generation at slack node), generation of capacitors, tap of transformer, and position, active and reactive power of wind turbines. Hence,Zd is comprised of the following decision variables: PTGi (i = 2, …, NTG); UTGi (i = 1, …, NTG); QComi (i = 1, …, NCom); Tapi (i = 1, …, NT); PWindx (x = 1, …, NW); QWindx (x = 1, …, NW); and LWindx (x = 1, …, NW).The decision variables are initialized within their lower boundZlow and upper bound Zup as shown in Section 2.


## 4.2. The Calculation of Dependent Variables

Before running Mathpower program, control variables of wind turbines including active power, reactive power, and location are collected to calculate the new values of loads at the placement of the wind turbines. Then, the data of the load must be changed and then added in the input data of Mathpower program. Finally, other remaining decision variables are added to Mathpower program and running power flow for obtaining dependent variables includingPTG1; QTGi (i = 1, …, NTG); ULNt (t = 1, …, NLN); and SBrq (q = 1, …, NBr).


## 4.3. Solution Evaluation

The quality of solutionZd is evaluated by calculating the fitness function. Total cost and total active power loss are two single objectives, while the violations of dependent variables are converted into penalty values [66].


## 4.4. Implementation of MEA for the Problem

In order to reach the optimal solution for the OPF problem with the presence of wind turbines, the implementation of MEA is shown in the following steps and is summarized in Figure2.Step 1: select populationN1 and maximum iteration N2.Step 2: select and initialize decision variables for the population as shown in Section4.1.Step 3: collect variables of wind turbines and tune loads.Step 4: running Mathpower for obtaining dependent variables shown in Section4.2.Step 5: evaluate quality of obtained solutions as shown in Section4.3.Step 6: select the best solutionZb1 and set Iter = 1.Step 7: select four random solutionsZr1, Zr2, Zr3, and Zr4.Step 8: calculate the mean fitness of the whole population.Step 9: produce new solutions.If Fitd < Fitmean, apply equation (26) to produce new solution. Otherwise, apply equation (27) to produce new solutions.Step 10: correct new solutions using equation (23).Step 11: collect variables of wind turbines and tune loads.Step 12: running Mathpower for obtaining dependent variables shown in Section4.2.Step 13: evaluate quality of obtained solutions as shown in Section4.3.Step 14: select good solutions among old population and new population using equations (24) and (25).Step 15: select the best solutionZb1.Step 16: if Iter =N3, stop the search process and print the optimal solution. Otherwise, set Iter to Iter + 1 and back to Step 7.Figure 2
The flowchart of using MEA for solving the modified OPF problem.


## 5. Numerical Results

In this section, MEA together with four other methods including FBI, HBO, MSGO, and CEA are applied for placing WPPs on the IEEE 30-node system with 6 thermal generators, 24 loads, 41 transmission lines, 4 transformers, and 9 shunt capacitors. The ssingle line diagram of the system is shown in Figure3 [67]. Different study cases are carried out as follows:Case 1: minimization of generation costCase 1.1: place one wind power plant at nodes 3 and 30, respectivelyCase 1.2: place one WPP at one unknown nodeCase 1.3: place two wind power plants at two nodesCase 2: minimization of power lossCase 2.1: place one wind power plant at nodes 3 and 30, respectivelyCase 2.2: place one WPP at one unknown nodeCase 2.3: place two wind power plants at two nodesFigure 3
The IEEE 30-node transmission network.The five methods are coded on Matlab 2016a and run on a personal computer with a processor of 2.0 GHz and 4.0 GB of RAM. For each study case, fifty independent runs are performed for each method, and the collected results are minimum, mean, maximum, and standard deviation values.


### 5.1. Electricity Generation Cost Reduction

#### 5.1.1. Case 1.1: Place One Wind Power Plant at Nodes 3 and 30, Respectively

In this study case, one power plant is, respectively, placed at node 3 and node 30 for comparison of the effectiveness of the placement position. As shown in [47], node 30 and node 3 are the most effective and ineffective locations for placing renewable energies. The found results from the five applied and JYA methods for the placement of WPPs at node 3 and node 30 are reported in Tables 2 and 3, respectively. Table 2 shows that the best cost of MEA is $764.33, while that of other methods is from $764.53 (CEA) to $769.963 (JYA). Exactly, MEA reaches better cost than others by from $0.2 to $5.633. Table 3 has the same features since MEA reaches the lowest cost of $762.53, while that from others is from $762.62 (CEA) to $768.039 (JYA). In addition, MEA can obtain less cost than others by from $0.09 to $5.509. The best cost indicates that MEA is the most powerful method among all the applied and JYA methods, while the standard deviation (STD) of MEA is the second lowest and it is only higher than that of HBO.Table 2
The results obtained by methods for placing one WPP at node 3.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum cost ($/h)764.69764.99766.85764.53764.33769.963Mean cost ($/h)767.23765.76782.38765.95765.94—Maximum cost ($/h)772.62766.71838.51768.83767.91—STD1.870.8715.781.051.01—N1103015303040N3100100100100100100Table 3
The results obtained by methods for placing one WPP at node 30.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum cost ($/h)763.56763.24765.74762.62762.53768.039Mean cost ($/h)765.91764.15780.02764.61764.36—Maximum cost ($/h)769.50765.82880.75766.62767.88—STD1.610.6822.191.281.23—N1103015303030N3100100100100100100For the effectiveness comparison between node 3 and node 30, it concludes that node 30 is more suitable to place WPPs. In fact, FBA, HBO, MSGO, CEA, MEA, and IAYA [47] can reach less cost for node 30. The cost of the methods is, respectively, $763.56, $763.24, $765.74, $762.62, $762.53, and $768.039 for node 30, but $764.69, $764.99, $766.85, $764.53, $764.33, and $769.963 for node 3.Figures4 and 5 show the best run of the applied methods for placing WPPs at node 3 and node 30, respectively. The curves see that MEA is much faster than the other methods even its solution at the 70th iteration is much better than that of others at the final iteration.Figure 4
The best runs of methods for the wind power plant placement at node 3.Figure 5
The best runs of methods for the wind power plant placement at node 30.


#### 5.1.2. Case 1.2: Place One WPP at One Unknown Node

In this section, the location of one WPP together with active and reactive power are determined instead of fixing the location at node 3 and node 30 similar to Case 1.1. Table4 indicates that MEA can reach better costs and STD than all methods. Table 5 summarizes the results of one WPP for Case 1.1 and Case 1.2. It is recalled that the power factor selection of wind power plant is from 0.85 to 1.0, while the active power is from 0 to 10 MW. For Case 1.2, HBO, CEA, and MEA can find the same location (node 30), while FBI and MSGO can find node 19 and node 26, respectively. Thus, the cost of FBI and MSGO is the worst, while others have better costs. The conclusion is very important to confirm that the renewable energy source placement location in the transmission power network has high impact on the effective operation. Figure 6 shows the best run of applied methods, and it also leads to the conclusion that MEA is the fastest among these methods.Table 4
The results obtained by five implemented methods for Case 1.2.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)763.96762.72765.22762.89762.52Mean cost ($/h)767.96764.32782.08764.59763.54Maximum cost ($/h)786.95766.45869.63767.68766.17STD4.450.9618.271.290.94N11030153030N3100100100100100Table 5
The optimal solutions obtained by methods for case 1.1 and case 1.2.
CaseMethodFBIHBOMSGOCEAMEAJYA [47]Case 1.1 (place WPP at node 3)Location of WPP333333Generation of WPP (MW)9.9910.009.9910.0010.009.1169Power factor of WPP0.970.890.880.850.890.85Minimum cost ($/h)764.69764.99766.85764.53764.33769.963Case 1.1 (place one WPP at node 30)Location of WPP303030303030Generation of WPP (MW)9.9810.0010.0010.0010.009.1478Power factor of WPP0.880.900.990.991.000.85Minimum cost ($/h)763.56763.24765.74762.62762.53768.039Case 1.2 (find the location of WPP)Location of WPP1930263030—Generation of WPP (MW)10.0010.0010.0010.0010.00—Power factor of WPP0.930.970.920.990.95—Minimum cost ($/h)763.96762.72765.22762.89762.52—Figure 6
The best run of applied methods for Case 1.2.Figure7 shows the fifty runs obtained by MEA. The black curve shows fifty sorted loss values, while blue bars show the location of the added WPP. In the figure, fifty runs are rearranged by sorting the cost from the smallest to the highest. Accordingly, the location of the runs is also reported. The locations indicate that node 30 is found many times, while other nodes such as 19 and 5 are also found, but the cost of nodes 19 and 5 is much higher than that of node 30.Figure 7
WPP location and cost of 50 rearranged runs obtained by MEA.


#### 5.1.3. Case 1.3: Place Two Wind Power Plants at Two Nodes

In this case, five methods are applied to minimize the cost for two subcases, Subcase 1.3.1 with two WPPs at node 3 and node 30 and Subcase 1.3.2 with unknown locations of two WPPs. The results for the two cases are reported in Tables6 and 7. MEA can reach the lowest cost for the two subcases, which is $728.15 for Subcase 1.3.1 and $726.77 for Subcase 1.3.2. It can be seen that the locations at nodes 30 and 3 are not as effective as locations at nodes 30 and 5. In addition to MEA, CEA also finds the same locations at nodes 30 and 5 for Subcase 1.3.2, and CEA reaches the second-best cost behind MEA. FBI, HBO, and MSGO cannot find the same nodes 30 and 5, and they suffer higher cost than CEA and MEA.Table 6
The results obtained by five implemented methods for Subcase 1.3.1.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)728.61728.40728.80728.39728.15Mean cost ($/h)731.27729.73736.72731.28728.74Maximum cost ($/h)738.69731.50762.72765.83730.55STD1.890.717.525.130.57N12060306060N3100100100100100Generation of WPP at node 30 (MW)9.9731109.95891010Generation of WPP at node 3 (MW)9.9693109.99999.985810Power factor of WPP at node 300.920.90040.98030.90480.8857Power factor of WPP at node 30.97820.93660.99920.98380.947Table 7
The results obtained by five implemented methods for Subcase 1.3.2.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)728.20727.41728.81726.84726.77Mean cost ($/h)730.67728.69731.75728.17728.04Maximum cost ($/h)733.77729.89739.08730.59730.35STD1.270.602.200.980.95N12060306060N3100100100100100Locations of WPPs30, 2430, 1924, 1930, 530, 5Generation of WPPs (MW)9.95, 9.962310, 9.999510, 9.998210, 1010, 10Power factor of WPPs0.9604, 0.93490.9605, 0.93940.9418, 0.91350.9342, 0.94580.9433, 0.8718Figure8 presents the cost and the locations of the two WPPs obtained by 50 runs. The black curve shows fifty values of loss sorted in the ascending order, while the blue and orange bars show the location of the first WPP and the second WPP. All the costs are rearranged from the lowest to the highest values. The figure indicates that the best cost and second-best cost are obtained by placing WPPs at nodes 30 and 5,while the next six best costs are obtained by placing WPPs at nodes 30 and 19. Other worse costs are found by placing the same nodes 30 and 5 or nodes 30 and 19. For a few cases, two WPPs are placed at nodes 30 and 24, but their cost is much higher. Clearly, node 30 is the most important, and node 5 is the next important location for supplying additional active and reactive power.Figure 8
WPP location and cost of 50 rearranged runs obtained by MEA for Subcase 1.3.2.Figures9 and 10 show the best run of applied methods for Subcases 1.3.1 and 1.3.2, respectively. Figure 9 shows a clear outstanding performance of MEA over other methods since the sixtieth iteration to the last iteration. The cost of MEA at the sixtieth iteration is smaller than that of other methods at the final iteration. Figure 10 also shows that MEA is much faster than FBI, HBO, and MSGO from the 30th iteration to the last iteration. The cost of MEA is always smaller than these methods from the 30th iteration to the last iteration. CEA shows a faster search than MEA from the first to the 80th iteration, but it is still worse than MEA from the 81st iteration to the last iteration. Obviously, MEA has a faster search than others.Figure 9
The best run obtained by five applied methods for Subcase 1.3.1.Figure 10
The best run obtained by five applied methods for Subcase 1.3.2.


### 5.2. Active Power Loss Reduction

#### 5.2.1. Case 2.1: Place One Wind Power Plant at Nodes 3 and 30, Respectively

In this section, one WPP is, respectively, located at nodes 30 and 3 for reducing power loss. Tables8 and 9 show the obtained results from 50 trial runs. The loss of MEA is the best for the cases of placing one WPP at node 3 and node 30. The best loss of MEA is 2.79 MW for the placement at node 3 and 2.35 MW for the placement at node 30, while those of others are from 2.8 MW to 3.339 MW for the placement at node 3 and from 2.37 MW to 2.67504 MW for the placement at node 30. On the other hand, all methods can have better loss when placing one WPP at node 30. Clearly, node 30 needs more supplied power than node 3. About the speed of search, Figures 11 and 12 indicate that MEA is much more effective than the other ones since its loss found at the 70th iteration is smaller than that of others at the final iteration.Table 8
The results obtained by methods as placing one WPP at node 3 for loss reduction.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum power loss (MW)2.912.872.912.802.793.3390Mean power loss (MW)3.353.084.013.113.10—Maximum power loss (MW)4.683.306.873.403.36—STD0.390.101.090.160.15—Generation of WPP (MW)9.15059.80768.93699.95599.99588.2827Power factor of WPP0.96010.99890.9740.916810.85Table 9
The results obtained by methods as placing one WPP at node 30 for loss reduction.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum power loss (MW)2.512.472.432.372.352.67504Mean power loss (MW)2.962.623.232.732.65—Maximum power loss (MW)4.502.855.693.182.97—STD0.430.090.890.170.14—N1103015303030N3100100100100100100Generation of WPP (MW)9.18219.82579.71119.99049.98539.95433Power factor of WPP0.97320.99470.99050.94550.99140.85Figure 11
The best runs of methods for minimizing loss as placing one WPP at node 3.Figure 12
The best runs of methods for minimizing loss as placing one WPP at node 30.


#### 5.2.2. Case 2.2. Place One WPP at One Unknown Node

In this section, five applied methods are implemented to find the location and power generation of one WPP. Table10 indicates that all methods have found the same location at node 30, but MEA is still the most effective method with the lowest loss even it is not much smaller than others. The loss of MEA is 2.39 MW, while that of others is from 2.45 MW to 2.7 MW. HBO is still the most stable method with the smallest STD. Figure 13 shows the best run of five methods. In the figure, MSGO has a premature convergence to a local optimum with very low quality, while other methods are searching optimal solutions. CEA seems to have a better search process than MEA from the 1st iteration to the 90th iteration, but then it must adopt a higher loss from the 91st iteration to the last iteration. Figure 14 presents the location and the loss of the proposed MEA for 50 runs. The rearranged losses from the lowest to the highest indicate that node 30 can reduce the loss at most, while other nodes such as 5, 7, 19, 24, and 26 are not suitable for reducing loss.Table 10
The results obtained by methods for placing one WPP at one unknown node.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.702.492.462.452.39Mean power loss (MW)3.172.683.542.822.74Maximum power loss (MW)4.723.186.343.293.43STD0.400.130.970.210.21N11030153030N3100100100100100Found position3030303030Generation of WPP (MW)8.89119.99989.90659.98559.9956Power factor of WPP0.98990.95030.99130.8950.9122Figure 13
The best runs of five applied methods for Subcase 2.2.Figure 14
WPP location and loss of 50 rearranged runs obtained by MEA for Subcase 2.2.


#### 5.2.3. Case 2.2. Place Two WPPs at Two Nodes

In this section, Subcase 2.2.1 is to place two WPPs at two predetermined nodes 3 and 30 and Subcase 2.2.2 is to place two WPPs at two random nodes. Tables11 and 12 show the results for the two studied subcases.Table 11
The results obtained by five methods for Subcase 2.2.1.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.372.432.322.282.26Mean power loss (MW)2.602.572.642.492.51Maximum power loss (MW)2.972.853.302.852.89STD0.140.090.250.130.13N12060306060N3100100100100100Location of WPPs3, 303, 303, 303, 303, 30Generation of WPPs (MW)9.6108, 9.81287.4719, 9.99366.8908, 9.96749.8404, 9.99999.9922, 9.9299Power factor of WPPs0.9814, 0.97290.9925, 0.93520.8752, 0.98410.9974, 0.95420.8597, 0.9973Table 12
The results obtained by five methods for Subcase 2.2.2.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.242.162.102.052.03Mean power loss (MW)2.612.372.462.272.31Maximum power loss (MW)3.932.644.242.702.77STD0.280.130.370.150.14N12060306060N3100100100100100Locations of WPPs30, 1919, 3030, 1930, 1924, 30Generation of WPPs (MW)9.83, 9.289.98, 9.979.70, 8.159.99, 9.989.95, 9.99Power factor of WPPs0.98, 0.961.00, 0.961.00, 0.920.94, 0.880.87, 0.99The two tables reveal that MEA can reach the lowest loss for both cases, 2.26 MW for Subcase 2.2.1 and 2.03 MW for Subcase 2.2.2. Clearly, placing WPP at the most effective node (node 30) and the least effective node (node 3) cannot lead to a very good solution of reducing total loss. While, the WPP placement at node 30 and node 24 can reduce the loss from 2.26 to 2.03 MW, which is about 0.23 MW and equivalent to 10.2%. When comparing to CEA, MSGO, HBO, and FBI, the proposed MEA can save 0.02, 0.06, 0.17, and 0.11 MW for Subcase 2.2.1 and 0.02, 0.07, 0.13, and 0.21 MW for Subcase 2.2.2. The mean loss of MEA is also smaller than that of MSGO, HBO, and FBI and only higher than that of CEA. The STD comparison is the same as the mean loss comparison. Figures15 and 16 show the search procedure of the best run obtained by five applied methods. Figure 15 indicates that MEA can find better parameters for wind power plants and other electrical components than other methods from the 75th iteration to the last iteration. Therefore, its loss is less than that of four remaining methods from the 75th to the last iteration. Figure 16 shows a better search procedure for MEA with less loss than other ones from the 55th iteration to the last iteration. The two figures have the same point that the loss of MEA at the 86th iteration is less than that of CEA at the final iteration. Compared to three other remaining methods, the loss of MEA at the 67th iteration for Subcase 2.2.1 and at the 56th iteration for Subcase 2.2.2 is less than that of these methods at the final iteration. Obviously, MEA is very strong for placing two WPPs in the IEEE 30-bus system.Figure 15
The best runs obtained by methods for Subcase 2.2.1.Figure 16
The best runs obtained by methods for Subcase 2.2.2.Figure17 shows the power loss and the location of the two WPPs for the fifty runs obtained by MEA. The black curve shows fifty sorted loss values, while the blue and orange bars show the location of the first WPP and the second WPP. The view on the bars and the curve sees that node 30 is always chosen, while the second location can be nodes 24, 19, 21, 5, and 4. The best loss and second-best loss are obtained at nodes 30 and 24, while other nodes reach much higher losses.Figure 17
Power losses and locations of two WPPs for 50 runs obtained by MEA.


### 5.3. Discussion on the Capability of MEA

In this paper, we considered the placement of WPPs on the IEEE 30-node system. The dimension of the system is not high, it is just medium. In fact, among IEEE standard transmission power systems such as IEEE 14-bus system, IEEE 30-bus system, IEEE 57-bus system, IEEE 118-bus system, etc. The considered system is not the largest system, and it has 6 thermal generators, 24 loads, 41 transmission lines, 4 transformers, and 9 shunt capacitors. With the number of power plants, lines, loads, transformers, and capacitors, the IEEE 30-bus system is approximately as large as an area power system in a province. By considering the placement of WPPs, the control variables of WPPs are location, active power, and reactive power. Therefore, there are six control variables regarding two placed WPPs, including two locations, two values of rated power, and two values of reactive power. In addition, other control variables regarding optimal power flow problem are 5 values of active power output for 6 THPs, 6 voltage values for THPs, 4 tap values for transformers, and 9 reactive power output values for shunt capacitors. On the other hand, the dependent variables are 1 value of the active power output for generator at slack node, 6 values of reactive power for THPs, 41 current values of lines, and 24 voltage values of loads. As a result, the total number of control variables for placing two WPPs in the IEEE 30-bus system is 30, and the total number of dependent variables is 72. In the conventional OPF problem, control variables have a high impact on the change of dependent variables, and updating the control variables causes the change of dependent variables. Furthermore, in the modified OPF problem, updating the location and size of WPPs also cause the change of control variables such as voltage and active power of THPs. Therefore, reaching optimal control parameters in the modified OPF problem becomes more difficult for metaheuristic algorithm. By experiment, MEA could solve the conventional OPF problem successfully for the IEEE 30-node system by setting 10 to population and 50 to iteration number and MEA could reach the most optimal solutions by setting 15 to population and 75 to iteration number. However, for the modified problem with the placement of two WPPs, the settings to reach the best performance for MEA were 60 for population and 100 for the iteration number. Clearly, the setting values were higher for the modified OPF problem. About the average simulation time for the study cases, Table13 summarizes the time from all methods for all study cases. Comparisons of the computation time indicate that MEA has the same computation time as FBI, HBO, MSGO, and CEA, but it has shorter time than JYA [47]. The average time for applied methods is about 30 seconds for the cases of placing one WPP and about 53 seconds for other cases of placing two WPPs, while the time is about 72 seconds for JYA for the cases of placing one WPP. The five algorithms approximately have the same average time because the setting of population and iteration number is the same. The reported time of the proposed method is not too long for a system with 30 nodes, and it seems that MEA can be capable for handling a real power system or a larger-scale power system. Therefore, we have tried to apply MEA for other larger scale systems with 57 or 118 nodes. For conventional OPF problem without the optimal placement of WPPs, MEA could solve the conventional OPF problem successfully. However, for the placement of WPPs in modified OPF problem for the IEEE 57-node system and the IEEE 118-node system, MEA could not succeed to reach valid solutions. Therefore, the highest shortcoming of the study is not to reach the successful application of MEA for placing WPPs on large-scale systems with 57 and 118 nodes.Table 13
Average computation time of each run obtained by methods for study cases.
MethodFBIHBOMSGOCEAMEAJYA [47]Case 1.1 (place one WPP at node 3)35.2130.6532.8228.1828.45∼72.4Case 1.1 (place one WPP at node 30)28.9326.0632.4527.0727.77∼72.4Case 1.2 (place one WPP at one unknown node)27.9331.9227.3227.6427.76—Subcase 1.3.1 (place two WPPs at nodes 3 and 30)51.0453.3653.5653.3852.57—Subcase 1.3.2 (place two WPPs at two unknown nodes)55.1854.0556.2754.1855.14—Case 2.1 (place one WPP at node 3)30.8227.1427.6531.1529.7∼72.4Case 2.1 (place one WPP at node 30)28.0228.728.3628.2028.04∼72.4Case 2.2 (place one WPP at one unknown node)28.2026.7528.2729.6227.40—Subcase 2.2.1 (place two WPPs at nodes 3 and 30)55.4156.4356.4356.8156.09—Subcase 2.2.2 (place one WPP at one unknown node)54.0454.1459.255.1956.12—It can be stated that CEA and MEA are powerful optimization tools for the IEEE 30-node system, but their capability on other large-scale systems or real systems is limited. The methods may need more effective improvement to overcome the mentioned limitation.


## 5.1. Electricity Generation Cost Reduction

### 5.1.1. Case 1.1: Place One Wind Power Plant at Nodes 3 and 30, Respectively

In this study case, one power plant is, respectively, placed at node 3 and node 30 for comparison of the effectiveness of the placement position. As shown in [47], node 30 and node 3 are the most effective and ineffective locations for placing renewable energies. The found results from the five applied and JYA methods for the placement of WPPs at node 3 and node 30 are reported in Tables 2 and 3, respectively. Table 2 shows that the best cost of MEA is $764.33, while that of other methods is from $764.53 (CEA) to $769.963 (JYA). Exactly, MEA reaches better cost than others by from $0.2 to $5.633. Table 3 has the same features since MEA reaches the lowest cost of $762.53, while that from others is from $762.62 (CEA) to $768.039 (JYA). In addition, MEA can obtain less cost than others by from $0.09 to $5.509. The best cost indicates that MEA is the most powerful method among all the applied and JYA methods, while the standard deviation (STD) of MEA is the second lowest and it is only higher than that of HBO.Table 2
The results obtained by methods for placing one WPP at node 3.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum cost ($/h)764.69764.99766.85764.53764.33769.963Mean cost ($/h)767.23765.76782.38765.95765.94—Maximum cost ($/h)772.62766.71838.51768.83767.91—STD1.870.8715.781.051.01—N1103015303040N3100100100100100100Table 3
The results obtained by methods for placing one WPP at node 30.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum cost ($/h)763.56763.24765.74762.62762.53768.039Mean cost ($/h)765.91764.15780.02764.61764.36—Maximum cost ($/h)769.50765.82880.75766.62767.88—STD1.610.6822.191.281.23—N1103015303030N3100100100100100100For the effectiveness comparison between node 3 and node 30, it concludes that node 30 is more suitable to place WPPs. In fact, FBA, HBO, MSGO, CEA, MEA, and IAYA [47] can reach less cost for node 30. The cost of the methods is, respectively, $763.56, $763.24, $765.74, $762.62, $762.53, and $768.039 for node 30, but $764.69, $764.99, $766.85, $764.53, $764.33, and $769.963 for node 3.Figures4 and 5 show the best run of the applied methods for placing WPPs at node 3 and node 30, respectively. The curves see that MEA is much faster than the other methods even its solution at the 70th iteration is much better than that of others at the final iteration.Figure 4
The best runs of methods for the wind power plant placement at node 3.Figure 5
The best runs of methods for the wind power plant placement at node 30.


### 5.1.2. Case 1.2: Place One WPP at One Unknown Node

In this section, the location of one WPP together with active and reactive power are determined instead of fixing the location at node 3 and node 30 similar to Case 1.1. Table4 indicates that MEA can reach better costs and STD than all methods. Table 5 summarizes the results of one WPP for Case 1.1 and Case 1.2. It is recalled that the power factor selection of wind power plant is from 0.85 to 1.0, while the active power is from 0 to 10 MW. For Case 1.2, HBO, CEA, and MEA can find the same location (node 30), while FBI and MSGO can find node 19 and node 26, respectively. Thus, the cost of FBI and MSGO is the worst, while others have better costs. The conclusion is very important to confirm that the renewable energy source placement location in the transmission power network has high impact on the effective operation. Figure 6 shows the best run of applied methods, and it also leads to the conclusion that MEA is the fastest among these methods.Table 4
The results obtained by five implemented methods for Case 1.2.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)763.96762.72765.22762.89762.52Mean cost ($/h)767.96764.32782.08764.59763.54Maximum cost ($/h)786.95766.45869.63767.68766.17STD4.450.9618.271.290.94N11030153030N3100100100100100Table 5
The optimal solutions obtained by methods for case 1.1 and case 1.2.
CaseMethodFBIHBOMSGOCEAMEAJYA [47]Case 1.1 (place WPP at node 3)Location of WPP333333Generation of WPP (MW)9.9910.009.9910.0010.009.1169Power factor of WPP0.970.890.880.850.890.85Minimum cost ($/h)764.69764.99766.85764.53764.33769.963Case 1.1 (place one WPP at node 30)Location of WPP303030303030Generation of WPP (MW)9.9810.0010.0010.0010.009.1478Power factor of WPP0.880.900.990.991.000.85Minimum cost ($/h)763.56763.24765.74762.62762.53768.039Case 1.2 (find the location of WPP)Location of WPP1930263030—Generation of WPP (MW)10.0010.0010.0010.0010.00—Power factor of WPP0.930.970.920.990.95—Minimum cost ($/h)763.96762.72765.22762.89762.52—Figure 6
The best run of applied methods for Case 1.2.Figure7 shows the fifty runs obtained by MEA. The black curve shows fifty sorted loss values, while blue bars show the location of the added WPP. In the figure, fifty runs are rearranged by sorting the cost from the smallest to the highest. Accordingly, the location of the runs is also reported. The locations indicate that node 30 is found many times, while other nodes such as 19 and 5 are also found, but the cost of nodes 19 and 5 is much higher than that of node 30.Figure 7
WPP location and cost of 50 rearranged runs obtained by MEA.


### 5.1.3. Case 1.3: Place Two Wind Power Plants at Two Nodes

In this case, five methods are applied to minimize the cost for two subcases, Subcase 1.3.1 with two WPPs at node 3 and node 30 and Subcase 1.3.2 with unknown locations of two WPPs. The results for the two cases are reported in Tables6 and 7. MEA can reach the lowest cost for the two subcases, which is $728.15 for Subcase 1.3.1 and $726.77 for Subcase 1.3.2. It can be seen that the locations at nodes 30 and 3 are not as effective as locations at nodes 30 and 5. In addition to MEA, CEA also finds the same locations at nodes 30 and 5 for Subcase 1.3.2, and CEA reaches the second-best cost behind MEA. FBI, HBO, and MSGO cannot find the same nodes 30 and 5, and they suffer higher cost than CEA and MEA.Table 6
The results obtained by five implemented methods for Subcase 1.3.1.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)728.61728.40728.80728.39728.15Mean cost ($/h)731.27729.73736.72731.28728.74Maximum cost ($/h)738.69731.50762.72765.83730.55STD1.890.717.525.130.57N12060306060N3100100100100100Generation of WPP at node 30 (MW)9.9731109.95891010Generation of WPP at node 3 (MW)9.9693109.99999.985810Power factor of WPP at node 300.920.90040.98030.90480.8857Power factor of WPP at node 30.97820.93660.99920.98380.947Table 7
The results obtained by five implemented methods for Subcase 1.3.2.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)728.20727.41728.81726.84726.77Mean cost ($/h)730.67728.69731.75728.17728.04Maximum cost ($/h)733.77729.89739.08730.59730.35STD1.270.602.200.980.95N12060306060N3100100100100100Locations of WPPs30, 2430, 1924, 1930, 530, 5Generation of WPPs (MW)9.95, 9.962310, 9.999510, 9.998210, 1010, 10Power factor of WPPs0.9604, 0.93490.9605, 0.93940.9418, 0.91350.9342, 0.94580.9433, 0.8718Figure8 presents the cost and the locations of the two WPPs obtained by 50 runs. The black curve shows fifty values of loss sorted in the ascending order, while the blue and orange bars show the location of the first WPP and the second WPP. All the costs are rearranged from the lowest to the highest values. The figure indicates that the best cost and second-best cost are obtained by placing WPPs at nodes 30 and 5,while the next six best costs are obtained by placing WPPs at nodes 30 and 19. Other worse costs are found by placing the same nodes 30 and 5 or nodes 30 and 19. For a few cases, two WPPs are placed at nodes 30 and 24, but their cost is much higher. Clearly, node 30 is the most important, and node 5 is the next important location for supplying additional active and reactive power.Figure 8
WPP location and cost of 50 rearranged runs obtained by MEA for Subcase 1.3.2.Figures9 and 10 show the best run of applied methods for Subcases 1.3.1 and 1.3.2, respectively. Figure 9 shows a clear outstanding performance of MEA over other methods since the sixtieth iteration to the last iteration. The cost of MEA at the sixtieth iteration is smaller than that of other methods at the final iteration. Figure 10 also shows that MEA is much faster than FBI, HBO, and MSGO from the 30th iteration to the last iteration. The cost of MEA is always smaller than these methods from the 30th iteration to the last iteration. CEA shows a faster search than MEA from the first to the 80th iteration, but it is still worse than MEA from the 81st iteration to the last iteration. Obviously, MEA has a faster search than others.Figure 9
The best run obtained by five applied methods for Subcase 1.3.1.Figure 10
The best run obtained by five applied methods for Subcase 1.3.2.


## 5.1.1. Case 1.1: Place One Wind Power Plant at Nodes 3 and 30, Respectively

In this study case, one power plant is, respectively, placed at node 3 and node 30 for comparison of the effectiveness of the placement position. As shown in [47], node 30 and node 3 are the most effective and ineffective locations for placing renewable energies. The found results from the five applied and JYA methods for the placement of WPPs at node 3 and node 30 are reported in Tables 2 and 3, respectively. Table 2 shows that the best cost of MEA is $764.33, while that of other methods is from $764.53 (CEA) to $769.963 (JYA). Exactly, MEA reaches better cost than others by from $0.2 to $5.633. Table 3 has the same features since MEA reaches the lowest cost of $762.53, while that from others is from $762.62 (CEA) to $768.039 (JYA). In addition, MEA can obtain less cost than others by from $0.09 to $5.509. The best cost indicates that MEA is the most powerful method among all the applied and JYA methods, while the standard deviation (STD) of MEA is the second lowest and it is only higher than that of HBO.Table 2
The results obtained by methods for placing one WPP at node 3.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum cost ($/h)764.69764.99766.85764.53764.33769.963Mean cost ($/h)767.23765.76782.38765.95765.94—Maximum cost ($/h)772.62766.71838.51768.83767.91—STD1.870.8715.781.051.01—N1103015303040N3100100100100100100Table 3
The results obtained by methods for placing one WPP at node 30.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum cost ($/h)763.56763.24765.74762.62762.53768.039Mean cost ($/h)765.91764.15780.02764.61764.36—Maximum cost ($/h)769.50765.82880.75766.62767.88—STD1.610.6822.191.281.23—N1103015303030N3100100100100100100For the effectiveness comparison between node 3 and node 30, it concludes that node 30 is more suitable to place WPPs. In fact, FBA, HBO, MSGO, CEA, MEA, and IAYA [47] can reach less cost for node 30. The cost of the methods is, respectively, $763.56, $763.24, $765.74, $762.62, $762.53, and $768.039 for node 30, but $764.69, $764.99, $766.85, $764.53, $764.33, and $769.963 for node 3.Figures4 and 5 show the best run of the applied methods for placing WPPs at node 3 and node 30, respectively. The curves see that MEA is much faster than the other methods even its solution at the 70th iteration is much better than that of others at the final iteration.Figure 4
The best runs of methods for the wind power plant placement at node 3.Figure 5
The best runs of methods for the wind power plant placement at node 30.


## 5.1.2. Case 1.2: Place One WPP at One Unknown Node

In this section, the location of one WPP together with active and reactive power are determined instead of fixing the location at node 3 and node 30 similar to Case 1.1. Table4 indicates that MEA can reach better costs and STD than all methods. Table 5 summarizes the results of one WPP for Case 1.1 and Case 1.2. It is recalled that the power factor selection of wind power plant is from 0.85 to 1.0, while the active power is from 0 to 10 MW. For Case 1.2, HBO, CEA, and MEA can find the same location (node 30), while FBI and MSGO can find node 19 and node 26, respectively. Thus, the cost of FBI and MSGO is the worst, while others have better costs. The conclusion is very important to confirm that the renewable energy source placement location in the transmission power network has high impact on the effective operation. Figure 6 shows the best run of applied methods, and it also leads to the conclusion that MEA is the fastest among these methods.Table 4
The results obtained by five implemented methods for Case 1.2.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)763.96762.72765.22762.89762.52Mean cost ($/h)767.96764.32782.08764.59763.54Maximum cost ($/h)786.95766.45869.63767.68766.17STD4.450.9618.271.290.94N11030153030N3100100100100100Table 5
The optimal solutions obtained by methods for case 1.1 and case 1.2.
CaseMethodFBIHBOMSGOCEAMEAJYA [47]Case 1.1 (place WPP at node 3)Location of WPP333333Generation of WPP (MW)9.9910.009.9910.0010.009.1169Power factor of WPP0.970.890.880.850.890.85Minimum cost ($/h)764.69764.99766.85764.53764.33769.963Case 1.1 (place one WPP at node 30)Location of WPP303030303030Generation of WPP (MW)9.9810.0010.0010.0010.009.1478Power factor of WPP0.880.900.990.991.000.85Minimum cost ($/h)763.56763.24765.74762.62762.53768.039Case 1.2 (find the location of WPP)Location of WPP1930263030—Generation of WPP (MW)10.0010.0010.0010.0010.00—Power factor of WPP0.930.970.920.990.95—Minimum cost ($/h)763.96762.72765.22762.89762.52—Figure 6
The best run of applied methods for Case 1.2.Figure7 shows the fifty runs obtained by MEA. The black curve shows fifty sorted loss values, while blue bars show the location of the added WPP. In the figure, fifty runs are rearranged by sorting the cost from the smallest to the highest. Accordingly, the location of the runs is also reported. The locations indicate that node 30 is found many times, while other nodes such as 19 and 5 are also found, but the cost of nodes 19 and 5 is much higher than that of node 30.Figure 7
WPP location and cost of 50 rearranged runs obtained by MEA.


## 5.1.3. Case 1.3: Place Two Wind Power Plants at Two Nodes

In this case, five methods are applied to minimize the cost for two subcases, Subcase 1.3.1 with two WPPs at node 3 and node 30 and Subcase 1.3.2 with unknown locations of two WPPs. The results for the two cases are reported in Tables6 and 7. MEA can reach the lowest cost for the two subcases, which is $728.15 for Subcase 1.3.1 and $726.77 for Subcase 1.3.2. It can be seen that the locations at nodes 30 and 3 are not as effective as locations at nodes 30 and 5. In addition to MEA, CEA also finds the same locations at nodes 30 and 5 for Subcase 1.3.2, and CEA reaches the second-best cost behind MEA. FBI, HBO, and MSGO cannot find the same nodes 30 and 5, and they suffer higher cost than CEA and MEA.Table 6
The results obtained by five implemented methods for Subcase 1.3.1.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)728.61728.40728.80728.39728.15Mean cost ($/h)731.27729.73736.72731.28728.74Maximum cost ($/h)738.69731.50762.72765.83730.55STD1.890.717.525.130.57N12060306060N3100100100100100Generation of WPP at node 30 (MW)9.9731109.95891010Generation of WPP at node 3 (MW)9.9693109.99999.985810Power factor of WPP at node 300.920.90040.98030.90480.8857Power factor of WPP at node 30.97820.93660.99920.98380.947Table 7
The results obtained by five implemented methods for Subcase 1.3.2.
MethodFBIHBOMSGOCEAMEAMinimum cost ($/h)728.20727.41728.81726.84726.77Mean cost ($/h)730.67728.69731.75728.17728.04Maximum cost ($/h)733.77729.89739.08730.59730.35STD1.270.602.200.980.95N12060306060N3100100100100100Locations of WPPs30, 2430, 1924, 1930, 530, 5Generation of WPPs (MW)9.95, 9.962310, 9.999510, 9.998210, 1010, 10Power factor of WPPs0.9604, 0.93490.9605, 0.93940.9418, 0.91350.9342, 0.94580.9433, 0.8718Figure8 presents the cost and the locations of the two WPPs obtained by 50 runs. The black curve shows fifty values of loss sorted in the ascending order, while the blue and orange bars show the location of the first WPP and the second WPP. All the costs are rearranged from the lowest to the highest values. The figure indicates that the best cost and second-best cost are obtained by placing WPPs at nodes 30 and 5,while the next six best costs are obtained by placing WPPs at nodes 30 and 19. Other worse costs are found by placing the same nodes 30 and 5 or nodes 30 and 19. For a few cases, two WPPs are placed at nodes 30 and 24, but their cost is much higher. Clearly, node 30 is the most important, and node 5 is the next important location for supplying additional active and reactive power.Figure 8
WPP location and cost of 50 rearranged runs obtained by MEA for Subcase 1.3.2.Figures9 and 10 show the best run of applied methods for Subcases 1.3.1 and 1.3.2, respectively. Figure 9 shows a clear outstanding performance of MEA over other methods since the sixtieth iteration to the last iteration. The cost of MEA at the sixtieth iteration is smaller than that of other methods at the final iteration. Figure 10 also shows that MEA is much faster than FBI, HBO, and MSGO from the 30th iteration to the last iteration. The cost of MEA is always smaller than these methods from the 30th iteration to the last iteration. CEA shows a faster search than MEA from the first to the 80th iteration, but it is still worse than MEA from the 81st iteration to the last iteration. Obviously, MEA has a faster search than others.Figure 9
The best run obtained by five applied methods for Subcase 1.3.1.Figure 10
The best run obtained by five applied methods for Subcase 1.3.2.


## 5.2. Active Power Loss Reduction

### 5.2.1. Case 2.1: Place One Wind Power Plant at Nodes 3 and 30, Respectively

In this section, one WPP is, respectively, located at nodes 30 and 3 for reducing power loss. Tables8 and 9 show the obtained results from 50 trial runs. The loss of MEA is the best for the cases of placing one WPP at node 3 and node 30. The best loss of MEA is 2.79 MW for the placement at node 3 and 2.35 MW for the placement at node 30, while those of others are from 2.8 MW to 3.339 MW for the placement at node 3 and from 2.37 MW to 2.67504 MW for the placement at node 30. On the other hand, all methods can have better loss when placing one WPP at node 30. Clearly, node 30 needs more supplied power than node 3. About the speed of search, Figures 11 and 12 indicate that MEA is much more effective than the other ones since its loss found at the 70th iteration is smaller than that of others at the final iteration.Table 8
The results obtained by methods as placing one WPP at node 3 for loss reduction.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum power loss (MW)2.912.872.912.802.793.3390Mean power loss (MW)3.353.084.013.113.10—Maximum power loss (MW)4.683.306.873.403.36—STD0.390.101.090.160.15—Generation of WPP (MW)9.15059.80768.93699.95599.99588.2827Power factor of WPP0.96010.99890.9740.916810.85Table 9
The results obtained by methods as placing one WPP at node 30 for loss reduction.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum power loss (MW)2.512.472.432.372.352.67504Mean power loss (MW)2.962.623.232.732.65—Maximum power loss (MW)4.502.855.693.182.97—STD0.430.090.890.170.14—N1103015303030N3100100100100100100Generation of WPP (MW)9.18219.82579.71119.99049.98539.95433Power factor of WPP0.97320.99470.99050.94550.99140.85Figure 11
The best runs of methods for minimizing loss as placing one WPP at node 3.Figure 12
The best runs of methods for minimizing loss as placing one WPP at node 30.


### 5.2.2. Case 2.2. Place One WPP at One Unknown Node

In this section, five applied methods are implemented to find the location and power generation of one WPP. Table10 indicates that all methods have found the same location at node 30, but MEA is still the most effective method with the lowest loss even it is not much smaller than others. The loss of MEA is 2.39 MW, while that of others is from 2.45 MW to 2.7 MW. HBO is still the most stable method with the smallest STD. Figure 13 shows the best run of five methods. In the figure, MSGO has a premature convergence to a local optimum with very low quality, while other methods are searching optimal solutions. CEA seems to have a better search process than MEA from the 1st iteration to the 90th iteration, but then it must adopt a higher loss from the 91st iteration to the last iteration. Figure 14 presents the location and the loss of the proposed MEA for 50 runs. The rearranged losses from the lowest to the highest indicate that node 30 can reduce the loss at most, while other nodes such as 5, 7, 19, 24, and 26 are not suitable for reducing loss.Table 10
The results obtained by methods for placing one WPP at one unknown node.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.702.492.462.452.39Mean power loss (MW)3.172.683.542.822.74Maximum power loss (MW)4.723.186.343.293.43STD0.400.130.970.210.21N11030153030N3100100100100100Found position3030303030Generation of WPP (MW)8.89119.99989.90659.98559.9956Power factor of WPP0.98990.95030.99130.8950.9122Figure 13
The best runs of five applied methods for Subcase 2.2.Figure 14
WPP location and loss of 50 rearranged runs obtained by MEA for Subcase 2.2.


### 5.2.3. Case 2.2. Place Two WPPs at Two Nodes

In this section, Subcase 2.2.1 is to place two WPPs at two predetermined nodes 3 and 30 and Subcase 2.2.2 is to place two WPPs at two random nodes. Tables11 and 12 show the results for the two studied subcases.Table 11
The results obtained by five methods for Subcase 2.2.1.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.372.432.322.282.26Mean power loss (MW)2.602.572.642.492.51Maximum power loss (MW)2.972.853.302.852.89STD0.140.090.250.130.13N12060306060N3100100100100100Location of WPPs3, 303, 303, 303, 303, 30Generation of WPPs (MW)9.6108, 9.81287.4719, 9.99366.8908, 9.96749.8404, 9.99999.9922, 9.9299Power factor of WPPs0.9814, 0.97290.9925, 0.93520.8752, 0.98410.9974, 0.95420.8597, 0.9973Table 12
The results obtained by five methods for Subcase 2.2.2.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.242.162.102.052.03Mean power loss (MW)2.612.372.462.272.31Maximum power loss (MW)3.932.644.242.702.77STD0.280.130.370.150.14N12060306060N3100100100100100Locations of WPPs30, 1919, 3030, 1930, 1924, 30Generation of WPPs (MW)9.83, 9.289.98, 9.979.70, 8.159.99, 9.989.95, 9.99Power factor of WPPs0.98, 0.961.00, 0.961.00, 0.920.94, 0.880.87, 0.99The two tables reveal that MEA can reach the lowest loss for both cases, 2.26 MW for Subcase 2.2.1 and 2.03 MW for Subcase 2.2.2. Clearly, placing WPP at the most effective node (node 30) and the least effective node (node 3) cannot lead to a very good solution of reducing total loss. While, the WPP placement at node 30 and node 24 can reduce the loss from 2.26 to 2.03 MW, which is about 0.23 MW and equivalent to 10.2%. When comparing to CEA, MSGO, HBO, and FBI, the proposed MEA can save 0.02, 0.06, 0.17, and 0.11 MW for Subcase 2.2.1 and 0.02, 0.07, 0.13, and 0.21 MW for Subcase 2.2.2. The mean loss of MEA is also smaller than that of MSGO, HBO, and FBI and only higher than that of CEA. The STD comparison is the same as the mean loss comparison. Figures15 and 16 show the search procedure of the best run obtained by five applied methods. Figure 15 indicates that MEA can find better parameters for wind power plants and other electrical components than other methods from the 75th iteration to the last iteration. Therefore, its loss is less than that of four remaining methods from the 75th to the last iteration. Figure 16 shows a better search procedure for MEA with less loss than other ones from the 55th iteration to the last iteration. The two figures have the same point that the loss of MEA at the 86th iteration is less than that of CEA at the final iteration. Compared to three other remaining methods, the loss of MEA at the 67th iteration for Subcase 2.2.1 and at the 56th iteration for Subcase 2.2.2 is less than that of these methods at the final iteration. Obviously, MEA is very strong for placing two WPPs in the IEEE 30-bus system.Figure 15
The best runs obtained by methods for Subcase 2.2.1.Figure 16
The best runs obtained by methods for Subcase 2.2.2.Figure17 shows the power loss and the location of the two WPPs for the fifty runs obtained by MEA. The black curve shows fifty sorted loss values, while the blue and orange bars show the location of the first WPP and the second WPP. The view on the bars and the curve sees that node 30 is always chosen, while the second location can be nodes 24, 19, 21, 5, and 4. The best loss and second-best loss are obtained at nodes 30 and 24, while other nodes reach much higher losses.Figure 17
Power losses and locations of two WPPs for 50 runs obtained by MEA.


## 5.2.1. Case 2.1: Place One Wind Power Plant at Nodes 3 and 30, Respectively

In this section, one WPP is, respectively, located at nodes 30 and 3 for reducing power loss. Tables8 and 9 show the obtained results from 50 trial runs. The loss of MEA is the best for the cases of placing one WPP at node 3 and node 30. The best loss of MEA is 2.79 MW for the placement at node 3 and 2.35 MW for the placement at node 30, while those of others are from 2.8 MW to 3.339 MW for the placement at node 3 and from 2.37 MW to 2.67504 MW for the placement at node 30. On the other hand, all methods can have better loss when placing one WPP at node 30. Clearly, node 30 needs more supplied power than node 3. About the speed of search, Figures 11 and 12 indicate that MEA is much more effective than the other ones since its loss found at the 70th iteration is smaller than that of others at the final iteration.Table 8
The results obtained by methods as placing one WPP at node 3 for loss reduction.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum power loss (MW)2.912.872.912.802.793.3390Mean power loss (MW)3.353.084.013.113.10—Maximum power loss (MW)4.683.306.873.403.36—STD0.390.101.090.160.15—Generation of WPP (MW)9.15059.80768.93699.95599.99588.2827Power factor of WPP0.96010.99890.9740.916810.85Table 9
The results obtained by methods as placing one WPP at node 30 for loss reduction.
MethodFBIHBOMSGOCEAMEAJYA [47]Minimum power loss (MW)2.512.472.432.372.352.67504Mean power loss (MW)2.962.623.232.732.65—Maximum power loss (MW)4.502.855.693.182.97—STD0.430.090.890.170.14—N1103015303030N3100100100100100100Generation of WPP (MW)9.18219.82579.71119.99049.98539.95433Power factor of WPP0.97320.99470.99050.94550.99140.85Figure 11
The best runs of methods for minimizing loss as placing one WPP at node 3.Figure 12
The best runs of methods for minimizing loss as placing one WPP at node 30.


## 5.2.2. Case 2.2. Place One WPP at One Unknown Node

In this section, five applied methods are implemented to find the location and power generation of one WPP. Table10 indicates that all methods have found the same location at node 30, but MEA is still the most effective method with the lowest loss even it is not much smaller than others. The loss of MEA is 2.39 MW, while that of others is from 2.45 MW to 2.7 MW. HBO is still the most stable method with the smallest STD. Figure 13 shows the best run of five methods. In the figure, MSGO has a premature convergence to a local optimum with very low quality, while other methods are searching optimal solutions. CEA seems to have a better search process than MEA from the 1st iteration to the 90th iteration, but then it must adopt a higher loss from the 91st iteration to the last iteration. Figure 14 presents the location and the loss of the proposed MEA for 50 runs. The rearranged losses from the lowest to the highest indicate that node 30 can reduce the loss at most, while other nodes such as 5, 7, 19, 24, and 26 are not suitable for reducing loss.Table 10
The results obtained by methods for placing one WPP at one unknown node.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.702.492.462.452.39Mean power loss (MW)3.172.683.542.822.74Maximum power loss (MW)4.723.186.343.293.43STD0.400.130.970.210.21N11030153030N3100100100100100Found position3030303030Generation of WPP (MW)8.89119.99989.90659.98559.9956Power factor of WPP0.98990.95030.99130.8950.9122Figure 13
The best runs of five applied methods for Subcase 2.2.Figure 14
WPP location and loss of 50 rearranged runs obtained by MEA for Subcase 2.2.


## 5.2.3. Case 2.2. Place Two WPPs at Two Nodes

In this section, Subcase 2.2.1 is to place two WPPs at two predetermined nodes 3 and 30 and Subcase 2.2.2 is to place two WPPs at two random nodes. Tables11 and 12 show the results for the two studied subcases.Table 11
The results obtained by five methods for Subcase 2.2.1.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.372.432.322.282.26Mean power loss (MW)2.602.572.642.492.51Maximum power loss (MW)2.972.853.302.852.89STD0.140.090.250.130.13N12060306060N3100100100100100Location of WPPs3, 303, 303, 303, 303, 30Generation of WPPs (MW)9.6108, 9.81287.4719, 9.99366.8908, 9.96749.8404, 9.99999.9922, 9.9299Power factor of WPPs0.9814, 0.97290.9925, 0.93520.8752, 0.98410.9974, 0.95420.8597, 0.9973Table 12
The results obtained by five methods for Subcase 2.2.2.
MethodFBIHBOMSGOCEAMEAMinimum power loss (MW)2.242.162.102.052.03Mean power loss (MW)2.612.372.462.272.31Maximum power loss (MW)3.932.644.242.702.77STD0.280.130.370.150.14N12060306060N3100100100100100Locations of WPPs30, 1919, 3030, 1930, 1924, 30Generation of WPPs (MW)9.83, 9.289.98, 9.979.70, 8.159.99, 9.989.95, 9.99Power factor of WPPs0.98, 0.961.00, 0.961.00, 0.920.94, 0.880.87, 0.99The two tables reveal that MEA can reach the lowest loss for both cases, 2.26 MW for Subcase 2.2.1 and 2.03 MW for Subcase 2.2.2. Clearly, placing WPP at the most effective node (node 30) and the least effective node (node 3) cannot lead to a very good solution of reducing total loss. While, the WPP placement at node 30 and node 24 can reduce the loss from 2.26 to 2.03 MW, which is about 0.23 MW and equivalent to 10.2%. When comparing to CEA, MSGO, HBO, and FBI, the proposed MEA can save 0.02, 0.06, 0.17, and 0.11 MW for Subcase 2.2.1 and 0.02, 0.07, 0.13, and 0.21 MW for Subcase 2.2.2. The mean loss of MEA is also smaller than that of MSGO, HBO, and FBI and only higher than that of CEA. The STD comparison is the same as the mean loss comparison. Figures15 and 16 show the search procedure of the best run obtained by five applied methods. Figure 15 indicates that MEA can find better parameters for wind power plants and other electrical components than other methods from the 75th iteration to the last iteration. Therefore, its loss is less than that of four remaining methods from the 75th to the last iteration. Figure 16 shows a better search procedure for MEA with less loss than other ones from the 55th iteration to the last iteration. The two figures have the same point that the loss of MEA at the 86th iteration is less than that of CEA at the final iteration. Compared to three other remaining methods, the loss of MEA at the 67th iteration for Subcase 2.2.1 and at the 56th iteration for Subcase 2.2.2 is less than that of these methods at the final iteration. Obviously, MEA is very strong for placing two WPPs in the IEEE 30-bus system.Figure 15
The best runs obtained by methods for Subcase 2.2.1.Figure 16
The best runs obtained by methods for Subcase 2.2.2.Figure17 shows the power loss and the location of the two WPPs for the fifty runs obtained by MEA. The black curve shows fifty sorted loss values, while the blue and orange bars show the location of the first WPP and the second WPP. The view on the bars and the curve sees that node 30 is always chosen, while the second location can be nodes 24, 19, 21, 5, and 4. The best loss and second-best loss are obtained at nodes 30 and 24, while other nodes reach much higher losses.Figure 17
Power losses and locations of two WPPs for 50 runs obtained by MEA.


## 5.3. Discussion on the Capability of MEA

In this paper, we considered the placement of WPPs on the IEEE 30-node system. The dimension of the system is not high, it is just medium. In fact, among IEEE standard transmission power systems such as IEEE 14-bus system, IEEE 30-bus system, IEEE 57-bus system, IEEE 118-bus system, etc. The considered system is not the largest system, and it has 6 thermal generators, 24 loads, 41 transmission lines, 4 transformers, and 9 shunt capacitors. With the number of power plants, lines, loads, transformers, and capacitors, the IEEE 30-bus system is approximately as large as an area power system in a province. By considering the placement of WPPs, the control variables of WPPs are location, active power, and reactive power. Therefore, there are six control variables regarding two placed WPPs, including two locations, two values of rated power, and two values of reactive power. In addition, other control variables regarding optimal power flow problem are 5 values of active power output for 6 THPs, 6 voltage values for THPs, 4 tap values for transformers, and 9 reactive power output values for shunt capacitors. On the other hand, the dependent variables are 1 value of the active power output for generator at slack node, 6 values of reactive power for THPs, 41 current values of lines, and 24 voltage values of loads. As a result, the total number of control variables for placing two WPPs in the IEEE 30-bus system is 30, and the total number of dependent variables is 72. In the conventional OPF problem, control variables have a high impact on the change of dependent variables, and updating the control variables causes the change of dependent variables. Furthermore, in the modified OPF problem, updating the location and size of WPPs also cause the change of control variables such as voltage and active power of THPs. Therefore, reaching optimal control parameters in the modified OPF problem becomes more difficult for metaheuristic algorithm. By experiment, MEA could solve the conventional OPF problem successfully for the IEEE 30-node system by setting 10 to population and 50 to iteration number and MEA could reach the most optimal solutions by setting 15 to population and 75 to iteration number. However, for the modified problem with the placement of two WPPs, the settings to reach the best performance for MEA were 60 for population and 100 for the iteration number. Clearly, the setting values were higher for the modified OPF problem. About the average simulation time for the study cases, Table13 summarizes the time from all methods for all study cases. Comparisons of the computation time indicate that MEA has the same computation time as FBI, HBO, MSGO, and CEA, but it has shorter time than JYA [47]. The average time for applied methods is about 30 seconds for the cases of placing one WPP and about 53 seconds for other cases of placing two WPPs, while the time is about 72 seconds for JYA for the cases of placing one WPP. The five algorithms approximately have the same average time because the setting of population and iteration number is the same. The reported time of the proposed method is not too long for a system with 30 nodes, and it seems that MEA can be capable for handling a real power system or a larger-scale power system. Therefore, we have tried to apply MEA for other larger scale systems with 57 or 118 nodes. For conventional OPF problem without the optimal placement of WPPs, MEA could solve the conventional OPF problem successfully. However, for the placement of WPPs in modified OPF problem for the IEEE 57-node system and the IEEE 118-node system, MEA could not succeed to reach valid solutions. Therefore, the highest shortcoming of the study is not to reach the successful application of MEA for placing WPPs on large-scale systems with 57 and 118 nodes.Table 13
Average computation time of each run obtained by methods for study cases.
MethodFBIHBOMSGOCEAMEAJYA [47]Case 1.1 (place one WPP at node 3)35.2130.6532.8228.1828.45∼72.4Case 1.1 (place one WPP at node 30)28.9326.0632.4527.0727.77∼72.4Case 1.2 (place one WPP at one unknown node)27.9331.9227.3227.6427.76—Subcase 1.3.1 (place two WPPs at nodes 3 and 30)51.0453.3653.5653.3852.57—Subcase 1.3.2 (place two WPPs at two unknown nodes)55.1854.0556.2754.1855.14—Case 2.1 (place one WPP at node 3)30.8227.1427.6531.1529.7∼72.4Case 2.1 (place one WPP at node 30)28.0228.728.3628.2028.04∼72.4Case 2.2 (place one WPP at one unknown node)28.2026.7528.2729.6227.40—Subcase 2.2.1 (place two WPPs at nodes 3 and 30)55.4156.4356.4356.8156.09—Subcase 2.2.2 (place one WPP at one unknown node)54.0454.1459.255.1956.12—It can be stated that CEA and MEA are powerful optimization tools for the IEEE 30-node system, but their capability on other large-scale systems or real systems is limited. The methods may need more effective improvement to overcome the mentioned limitation.


## 6. Conclusions

In this paper, a modified OPF (MOPF) problem with the placement of wind power plants in an IEEE 30-bus transmission power network was solved by implementing four conventional metaheuristic algorithms and the proposed MEA. Two single objectives taken into account were minimization of total generation cost and minimization of power loss. About the number of WPPs located in the system, two cases are, respectively, one WPP and two WPPs. About the locations of the WPPs, simple cases were to accept the result from the previous study [47]. Buses 30 and 3 were the most effective and ineffective locations. The results indicated that the placement of one WPP at bus 30 can reach smaller power loss and smaller fuel cost than at bus 3. For other complicated cases, the paper also investigated the effectiveness of locations by applying MEA and four other metaheuristic algorithms to determine the locations. As a result, placing one WPP at bus 30 has reached the smallest power loss and the smallest total fuel cost. For placing two WPPs, buses 30 and 3 could not result in the smallest fuel cost and the smallest power loss. Buses 30 and 5 were the best locations for the minimization of fuel cost, while buses 30 and 24 were the best locations for the minimization of power loss. Therefore, the main contribution of the study regarding the electrical field is to determine the best locations for the best power loss and the best total cost.For placing one WPP, fuel costs of MEA were the smallest and equal to $764.33 and $762.53 for locations at node 3 and node 30, whilst those of others were much higher and equal to $769.963 and $768.039, respectively. For placing two WPPs at two found locations, MEA has reached the cost of $726.77, but the worst cost of others was $728.81. The power losses of MEA were also reduced significantly as compared to others. For placing one WPP at node 3 and node 30, MEA has reached 2.79 and 2.35 MW, but those of others have been larger and equal to 3.339 and 2.67504 MW, respectively. For placing two WPPs at two found locations, the best loss of 2.03 MW was found by MEA and the worst loss of 2.24 MW was found by others. In summary, the proposed MEA could attain lesser cost than others from 0.28% to 0.73% and lesser power loss than others from 9.38% to 16.44%. Clearly, the improvement levels are significant. However, for other systems with larger scale, MEA could not succeed in determining the best location and size for WPPs. Thus, in the future work, we will find solutions to improve MEA for larger systems and real systems. In addition, we will also consider more renewable energy power plants, such as photovoltaic power plants and uncertainty characteristics of solar and wind speed. All considered complexities will form a real problem as a real power system, and contributions of optimization algorithms and renewable energies will be shown clearly.



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*Source: 1015367-2021-10-12.xml*