#### Abstract

This paper proposes a modified slime mold algorithm (MSMA) for finding optimal site and size of both photovoltaic generators (PGs) and shunt capacitors (SCs) in radial distribution power systems (RDPSs). The most suitable site together with the most effective size of PGs and SCs can reduce currents of conductors effectively, resulting in the significant reduction of active power loss on all conductors. MSMA and two other applied algorithms, slime mold algorithm (SMA) and equilibrium optimizer (EO), are implemented for three study cases of IEEE 33, 69, and 85 RDPSs including Case (1) place only PGs with both active and reactive power generations; Case (2) place both SCs and PGs with only active power generation; and Case (3) place both SCs and PGs with both active and reactive power generations. Comparisons with SMA and EO, MSMA can reach either the same loss or much smaller loss for study cases. Furthermore, MSMA also outperforms other previously published methods for the three study cases, and it can reach significantly smaller loss for the three cases by 1.5%, 30.55%, and 19.64% for system 1, 7.59%, 0.26%, and 0.32% for system 2, and 7.51%, 34.2%, and 22.65% for system 3. As comparing the loss among three study cases obtained by MSMA, Case 3 can reach less loss than Case 1 and Case 2 by 11.71% and 11.64% for system 1, 26.27% and 26.13% for system 2, and the same 26.6% for system 3. Thus, it is recommended that MSMA is the most effective algorithm and the placement of both SCs and PGs with active and reactive power generations is the most optimal.

#### 1. Introduction

##### 1.1. Motivation

A power system is comprised of three crucial parts including power plants, transmission systems, and distribution system [1]. Power plants are major power sources producing and transmitting electricity to transmission systems. Distribution systems are in charge of receiving the electricity from the transmission systems and distributing the received energy to loads. Load/electric customers are usually located in distribution systems, and the performance of the distribution systems has a great impact on the operation of loads. If the power quality and continuity of distribution systems are good, the loads will work stably and effectively [2]. However, as the economy is developing so fast, the demand of energy is also significantly increasing. Conventional power plants such as thermal power plants and hydropower plants cannot supply enough energy to loads. Furthermore, the power loss in the transmission and distribution system is very high due to the high distance from the plants to loads. Hence, the installation of renewable energies in distribution systems, especially the placement of the energy sources at loads, is a leading solution for reducing power loss and increasing the benefit of power companies and electric customers [3]. Derived from the advantages, photovoltaic generators (PGs), a popular renewable energy nowadays, are concerned to be installed in radial distribution power systems (RDPSs). Thanks to the high technology of inverters, PGs can provide loads with both active power and reactive power. However, the reactive power demand from the loads is higher than the capacity of PGs, while the distribution of reactive power from the conventional power source can cause a high loss. Therefore, shunt capacitors (SCs) are also a crucial electrical component that should be integrated in RDPSs. The simultaneous placement of PGs and SCs is very useful in reducing the use of power from the conventional power plants and current on distribution lines. As a result, electricity generation cost from the plants can be decreased significantly thanks to the active power generation from PGs and active power loss on all lines is also effectively reduced thanks to active power generation from PGs and reactive power generation from SCs. The disadvantages and benefits of the power system before and after placement of PGs and SCs are summarized in Figure 1.

##### 1.2. Literature Review

In recent years, a high number of studies have applied and proposed different algorithms for implementing the integration of PGs into RDPSs for cutting power loss, enhancing power quality, and reaching more economy benefits. Analytical algorithms were applied for large distribution systems [4] and a small system of a university [5]. These algorithms have focused on current of branches and voltage of nodes for determining location of PGs, while the calculation of size of PGs was solved by different iterative algorithms. Original metaheuristic algorithms have been applied such as SCA [6], CSA [7], WOA [8], and KHA [9]. These algorithms have been proved to be effective in loss reduction and voltage improvement when PGs were installed. Furthermore, results from them were much better than those from other algorithms. Modified metaheuristic algorithms were also applied to reach better results than original ones such as hybrid CSA-ABCA [10], MTLOA [11], MABCA [12], and MOALO [13]. In this regard, hybrid method [14] and HOA [15] were also effective methods. These methods were based on both analytical algorithms and other optimization tools in which analytical algorithms were used for selecting location and other optimization tools were used for finding size of PGs.

In addition to the presence of PGs in RDPSs, SCs, which cannot supply energy to loads but can reduce the current in DLs, have been concerned and installed in RDPSs for the purpose of decreasing loss and increasing stable voltage. Although SCs cannot replace PGs for active power supply, they can replace the conventional power sources for supplying reactive power to loads. Compared to PGs, the investment and maintenance costs of SCs are much chipper. Similar to the problem of PGs placement, the problem of effective reactive power compensation has been solved successfully by network configuration analysis methods, metaheuristic algorithms, and the combination of these analysis methods with these metaheuristic algorithms. The methods based on configuration analysis are MSM [16], TSM [17], and AA [18]. These methods used network configuration to find currents on lines and used the currents to determine suitable location for placing the electric component. The power of SCs was found by iterative algorithm so that the power loss reduction could not be improved better. On the contrary, metaheuristic algorithms such as PSO [19], FPA [20], MFPA [21], CSA [22], SFSO [23], TALA [24], and SMOA [25] could easily find more suitable solutions. However, conventional algorithms such as PSO [19] and FPA [20] could not reach the most effective solutions for large-scale systems as others. Due to the drawback, metaheuristic algorithms have been combined with other techniques based on network configuration analysis. For instances, LSF was combined with IHA in [26], with MHA in [27], and with IEAs in [28]. Search algorithm and refinement technique were suggested in [29]. In general, almost all the combined methods could reach smaller loss than metaheuristic algorithms. The high benefits of PGs and SCs have been proved and confirmed in previous studies. The loss could be reduced up to higher than 50%, and the voltage could be approximately equal to the rated value of 1.0 pu. Thus, the integration of PGs and SCs can bring higher benefits to RDPSs or power companies and customers.

The implementations of PGs and SCs placement have more difficulties than only PGs or only SCs placement implementation. Furthermore, if the PGs can produce both active and reactive power, the problem of both PGs and SCs placement also becomes more complicated. Each PG needs three parameters including placement site, active power generation, and reactive power generation, while SCs have two parameters including placement site and reactive power generation. Other studies also concerned the additional supply of active and reactive power, but they did not use solar radiations as primary fuel of electricity generators. They only called the placed generators distributed generators (DGs) in which there were three types of DGs including pure active power generation type, both active and reactive power generation types, and pure reactive power generations. The integration of different types of DGs at the same locations or different locations was investigated. In [30], SQP-BB was combined to determine the site and size of DGs, in which the site of each DG was found first and then the size was found. These installed DGs were supposed to be able to produce both active and reactive powers as a power plant in the power systems. The combined SQP-BB method was here compared with MAM and PSO by testing them on 33-node and 69-node systems. In [31], a system configuration analysis algorithm was proposed to find suitable placement locations for added DGs. And then, another proposed power flow method was applied to find the exact size of the determined locations. Different types of DGs have been installed for loss minimization, and several deterministic algorithms based on derivatives have been run for comparison. In [32], DGs were renewable energy generators that can produce both active and reactive powers. A repeated power flow (RPF) method was proposed to determine the site and capacity of each DG. The DGs’ site was chosen in the first step, and then, the size was computed by using RPF in the second step. In [33], a hybrid method based on WPSO-GSA was applied for 33-node and 85-node systems. These methods have been compared to other previous methods for the case of DGs producing only active power, and they were stated to be effective. However, they have not shown the highly effective performance for the case of DGs producing both active and reactive powers. The DGs and SCs placement integration in distribution systems was implemented by using different methods including system configuration analysis algorithm, metaheuristic algorithms, and hybrid metaheuristic algorithm. In [34], AA was compared to ABCA for 69-node and not compared to any methods for a 687-node system. It was concluded to be better than ABCA for a 69-node system and effective for placing both DGs and SCs for the 687-node system. In [35], BCACA was compared to binary PSO for 33-node and 69-node systems. In [36], BFA was compared with PSO and another analysis method for a 33-node system with three study cases including only DGs placement, only SCs placement, and both DGs and SCs placements. In [37], ABC-HS was formed to install DGs and SCs for loss reduction in [37]. The hybrid ABC-HS was compared with a LSF method and its original methods including ABCA and HSA for a 119-node system. In [38], IMDE was applied for 33-node and 69-node systems to install only DGs, only SCs, and both DGs and SCs. The comparisons indicated that only DGs placement was more effective than only SCs placement, and both SCs and DGs placement was the best. IMDE was compared to several metaheuristic algorithms, but it was not compared to conventional DE. In [39], WPSO was applied for a 33-node system with biobjective function including loss reduction and voltage improvement. WPSO was compared to other metaheuristic algorithms but not compared to conventional PSO. The study also stated that simultaneous placement of both DGs and SCs was more effective than separate placement of each type of DG. In [40], HGWA was applied for three systems with 33, 69, and 85 nodes. In [41], WPSO-GSA was reapplied for minimizing both active power and reactive power losses in distribution systems with 33 and 85 nodes. The two hybrid methods in [40, 41] have not been compared with their original methods because the original GWA, WPSO, and GSA have not been coded for reaching results. In [42], a HRA based on voltage evaluation was proposed to install only DGs, only SCs, and both DGs and SCs in 33, 69, and 119-node systems. The study has not proved the simultaneous placement of DGs and SCs was more effective than the separate installment of DGs or SCs. It has just shown the outstanding performance of HA compared to other existing metaheuristic algorithms and previously published analysis algorithms. In [43], a MSSA with the application of mutation technique was proposed for the application of benchmark functions and 33- and 69-node systems. The study has proved MSSA to be more effective than its conventional SSA and other metaheuristic algorithms.

On the contrary to other studies, AA [44] and HIC-GA [45] were proposed for installing DGs and SCs simultaneously where DGs were supposed to produce both active and reactive powers. The two methods have been applied for the same systems with 33 and 69 nodes and compared to other previous methods. Another hybrid PSO and GA method (PSO-GA) was also implemented for comparison with HIC-GA, while ICA and GA have not been run for comparison. In [14], different types of DGs were proposed and combined to inject active power and reactive power to grids. Pure active power generators, pure reactive power generators, and active and reactive power generators were considered DGs. The study has concluded that the combination of pure active power generators and pure reactive power generators was more effective than the use of active and reactive power generators. Different DGs at different locations could reduce power loss more effectively than different DGs at the same location. About the applied methods, the study [14] has proposed HMPSO based on AA and WPSO. For almost all study cases in 33- and 69-node systems, HMPSO could reduce the active power loss more effectively than MPSO. However, HMPSO was compared to few already published methods.

In general, almost all mentioned studies above have tried to contribute powerful optimization tools and/or contribute effective DGs placement methods. In addition, these studies have reached considerable loss reduction. However, there were shortcomings existing in the studies that need to be improved. The summary of shortcomings is expressed as follows:(1)Analytical methods suffer from the complexity of applications for different systems due to the use of circuit modeling. Different system configurations have different circuits, and there is not the same circuit modeling for different systems. Furthermore, the exactness of system analysis is the most leading factor for the success of DGs placement in distribution systems.(2)Proposed methods including modified and hybrid methods were not proved to be more effective than their original ones.(3)There was not the best integration of different DG types for reaching the highest loss announced.

##### 1.3. Objectives, Novelties, and Contributions

In this paper, unexpected issues in distribution systems including high active power loss, high electricity cost, and power failure risk as well as shortcomings of previous studies are solved satisfactorily. PGs and SCs are considered to be integrated in distribution system with different assumptions, and then, obtained results are judged to select the best solutions of the issues and shortcomings.

To solve the unexpected problems in distribution systems, the paper focuses on two specific objectives below:(1)Find the best method of generating active power and reactive power for loads in distribution systems. The study implements three study cases including Case (1) supply both active and reactive power generations by using PGs; Case (2) supply active power by using PGs and supply reactive power by using SCs; and Case (3) supply active and reactive power by using PGs and supply reactive power by using SCs. The three study cases have the same manner of producing both active power and reactive power to distribution systems, but the difference is generation locations. Active power and reactive power must be generated at the same nodes in Case 1, but active power and reactive power can be generated either at the same node or at different nodes. Case 3 is different from Case 1 and Case 2. In Case 3, PGs must produce active power and reactive power at the same locations, while SCs can produce reactive power at the same or different locations with PGs.(2)Minimize total active power loss on conductors in RDPSs. In all three study cases, location and power of PGs and SCs are optimized so that the total active power loss is minimum. The minimum loss value is used to determine the most optimal location and power of PGs and/or SCs for each study case.

As active power loss is calculated and electric component placement solutions are determined, the effectiveness of distribution systems can be evaluated exactly. The implementations to reach the expected results are also novelties as well as contributions of paper. The novelties can be summarized as follows:(1)Use three standard distribution systems and solve three study cases of PGs and SCs placement for each system. The novelty is also the first objective above, and it aims to find the best combination of electric components in distribution systems for cutting loss, reducing electricity generation cost from thermal power plants.(2)Propose a modified algorithm, called MSMA, and implement two other algorithms including EO and SMA for three study cases of each solved system. The novelty can solve the big problem of previous studies that used network configuration-based analysis algorithms as well as lowly effective metaheuristic algorithms.

As simulating the three study cases for the three systems by using the three algorithms, obtained results are very promising and these are also major contributions of the paper. These contributions are as follows:(1)Find disadvantages of the conventional SMA and propose solutions to eliminate the disadvantages for developing MSMA. The proposed MSMA is as good as or better than SMA, EO, and other previous for ten benchmark functions, but it is superior to these compared methods for approximately all study cases of the solved engineering problem.(2)Find the most effective solution of placing both PGs and SCs for reaching the lowest active power loss.(3)Reduce active power loss for three employed systems more effectively than previous studies.(4)Prove simultaneous use of PGs and SCs (where PGs are the active and reactive power generation sources) to be the most effective in active power loss reduction.

The proposed MSMA is developed by modifying the technique of generating new solutions from SMA. SMA was developed in 2020 for a set of benchmark functions and proved to be superior to many other metaheuristic algorithms [46]. The original technique in SMA is modified by using the four best solutions, and another new equation is applied for the modified technique. The ideal of using the four best solutions is taken from equilibrium optimizer (EO) [47], and EO is also applied for comparison with MSMA. In addition to SMA and EO, other four algorithms including TSA [48], JFA [49], NGOA [50], and BOA [51] are also run for ten benchmark functions for comparisons. These results indicate MSMA can be as good as or superior to these compared algorithms. Then, the real performance of MSMA continues to be investigated by solving three study cases of three different systems with 33, 69, and 85 nodes. As a result, it can conclude the proposed MSMA is an effective algorithm for the three systems.

#### 2. Problem Formulation

The significance of SCs and PGs can be evaluated via the active power loss reduction of RDPSs and the load node voltage improvement. However, the improvement of voltage is usually not regarded as the most important objective but the active power loss. The operation stability voltage limit has a range within a lower bound and an upper bound, while effective active power loss reduction can bring an extreme benefit. Although we consider the loss as a major objective and place PGs and SCs for loss minimization, the voltage is still improved sharply. Voltage is regarded as a constraint, and the load nodes must satisfy the voltage constraints. In addition, other constraints regarding distribution lines, PGs, SCs, and RDPSs are also taken into account. Objectives and constraints are presented in detail as follows.

##### 2.1. Objective Function

The placement of PGs and SCs at nodes in RDPS can provide loads with active power and reactive power. The supplies from the two electric components can decrease the current from the power source at the slack bus to other nodes and between each two nodes. As a result, the active power loss becomes smaller thanks to the reduction of current. However, the loss is also dependent on the resistance of the distribution lines. Therefore, the total active power loss must be considered as the objective function as follows:

##### 2.2. Considered Constraints

###### 2.2.1. System Constraints

The system considers active and reactive power balance constraints. The installed PGs and power source at node 1 can supply both active power and reactive power to loads, while the installed SCs can generate reactive power only. The original and added sources must be equal to the demand of load and losses online. These constraints are as follows [14]:

In Eq. (3), TRPL is the total reactive power loss obtained by [25]:

Parameters in Eq. (2) and Eq. (3) can be classified into four types including control variables, dependent variables, given data, and obtained values. *P*_{PG,i}, *Q*_{PG,I}, and *Q*_{SC,j} are control variables of the problem that need to be optimally determined by using metaheuristic algorithms. Active and reactive power demands ( and ) are given data of considered distribution systems. TAP_{L} and TRP_{L} are the two functions of current () obtained by using Eq. (1) and Eq. (4) where is a dependent variable. and are dependent variables that are obtained after reaching all others. However, the two dependent variables are not constrained and there are not penalty terms for the two variables. On the contrary, current () is constrained by the capacity of conductor and it is checked for calculating penalty term in fitness function.

Thermal limit of conductors: Conductors in RDPSs are constrained by the thermal limits when current flows inside the conductors. Each conductor connects two nodes together, and the operating current increases the thermal inside the conductor. To satisfy the thermal limit, the operating current must not be higher than the rated current of the conductor as shown in the following inequality [52]:

The current of the *m*th distribution line after the placement of PGs and SCs *I*_{PGSC,m} tends to be smaller than the current before the placement of PGs and SCs to reach the active power loss reduction. However, Eq. (5) is always applied for branch current to satisfy stable operation condition of distribution lines.

###### 2.2.2. Load Voltage Constraint

Loads in RDPSs can work stably if their voltage is not beyond a known allowable range. Thus, the load voltage must satisfy the following constraint [53]:

The voltage magnitude of the *k*th node *U*_{L,k} is a dependent variable of the problem that is obtained after considering the generation of PGs and SCs in RDPSs. If the voltage of at least one node is outside the allowable range, obtained solution of PGs and SCs placement is not valid and it should be rejected. So, the dependent variable is considered in fitness function by adding one penalty term.

###### 2.2.3. Generation Constraints of SCs and PGs

*Generation limit constraints*: When installing SCs and PGs, the generation limits of the two electrical components must satisfy the following constraints [54]:

Active power and reactive power generations of PGs ( and ) as well as reactive power generation of SCs () are control variables of the problem. So, the constraints above are not used to calculate penalty terms for the violation of the variables as other dependent variables such as and . However, these constraints are very important to redefine the generation of PGs and SCs when metaheuristic algorithms produce new generation values for each computation iteration.

*Placement site constraints*: In RDPS, the power source is located at node 1, while other nodes are the site of load. Thus, PGs and SCs can be installed at load nodes from node 2 to node *N*_{Ns} as long as the objective function is the best. The placement site of the installed PGs and SCs must meet the constraints below:

The sites of PGs and SCs are also control variables of the problem that need to be optimally determined by using metaheuristic algorithms. The sites of the installed components must be nodes in RDPSs. Thus, the constraint above supports the correction of wrong sites in case that metaheuristic algorithms select node 1 or nodes, which are higher than *N*_{Ns}. As a result, there are not penalty terms for the control variables in fitness function.

#### 3. The Proposed Method

##### 3.1. Conventional Slime Mold Algorithm

In early 2020, Shimin et al. have introduced SMA and shown SMA’s performance to readers by testing it on 30 benchmark mathematical functions with large search zones from −100 to 100. Similar to PSO, SMA is also comprised of four major steps including initial solution set generation, new solution generation, new solution correction, and high-quality solution selection. The detail of SMA is expressed as follows.

###### 3.1.1. Initial Solution Set Generation

It is supposed that we have a set of *N*_{So} solutions and each solution is represented by *S*_{n}*,* where *n* = 1, …, *N*_{So}. Each solution *S*_{n} in the current set (i.e., population) must be within its upper and lower bounds, and . In the first step, *S*_{n} is randomly produced bywhere and are the lowest and highest boundaries of the *n*th solution *S*_{n}. The three solutions are defined as follows:where and are the minimum and maximum values of the *h*th control variable and *H* is the number of control variables for each solution.

Each obtained *S*_{n} is evaluated by using fitness function . From the obtained fitness values, three main terms need to be determined including:(1)Fitness_{Best}: The lowest fitness value.(2)Fitness_{Worst}: The highest fitness value.(3)*S*_{Best}: The best solution with the lowest fitness value.

###### 3.1.2. New Solution Generation Technique

SMA has one newly produced generation for each iteration as PSO and DE. However, each update time of SMA can use two options, searching solutions nearby the so far best solution and searching around each current solution, while that of PSO and DE only searches around each current solution. The update technique of SMA is expressed bywhere is the *h*th variable of the new solution ; *x*_{h,Best} is the *h*th variable of the best solution; and *x*_{h,rnd1}, *x*_{h,rnd2} are the *h*th variable of the first and second randomly selected solutions.

In addition, other parameters including , and are also determined by

###### 3.1.3. Correction Technique

Correction technique is always applied in metaheuristic algorithms to fix the violation of control variables, which have been newly produced by using Eq. (15). Each new control variable is compared to its lower and upper limits, and then, it will be redefined as follows:

Finally, the new solutions are evaluated by calculating fitness function .

###### 3.1.4. Selection Technique

Selection technique is applied to retain good control variables and abandon bad control variables for the next generation in the next iteration. In addition, the fitness values are also compared to retain better ones and eliminate worse ones. The equations of the technique are formed as follows:

##### 3.2. Modified Slime Mold Algorithm

In MSMA, we propose to apply modifications to the new solution generation techniques with the intent to improve the performance of global and local search techniques that are not effective in the conventional SMA. The modifications are explained and carried out as follows.

###### 3.2.1. Modification of Global Search Technique

As shown in Eq. (15), the global search technique of SMA is carried out by using the first model, . However, is a function of current iteration and maximum iteration number in Eq. (16) and it tends to be close to zero as the current iteration is coming to the maximum iteration number *G*. To see the decrease of to zero, we have plotted upper bound and lower bound and in Figure 2. In the figure, we use *G* = 100. It is clear that is very close to zero at final iteration and it is zero at . So, the first model of Eq. (15) makes become very tiny and to be zero when moves to *G*. Therefore, the first model is not useful for the problem of determining the location and size of PGs and SCs. Locations can be from 2 to 33 for the IEEE 33-node system, to 69 for the IEEE 69-node system, and to 85 for the IEEE 85-node system. In addition, the size of PGs and SCs can be from 1 kW to 1000 kW or higher than 1000 kW. To overcome the shortcoming of SMA, MSMA uses another model as follows:

In the formula above, is the *h*th variable of one out of the top four best solutions in the current population. We take the ideal of using the top four best solutions from EO [47].

###### 3.2.2. Modification of Local Search Technique

SMA uses the second model in Eq. (15) for local search, finding a new solution around the best one. Search spaces around are explored by using a jumping step of in which and are two variables of two random solutions. The use of two random solutions for calculating the distance between the best solution and the new solution is not effective because there are some cases where two random solutions have a very large distance. However, for some cases, the distance between the two solutions is very small or zero if they are in the same local zone or global zone. To limit the unexpected cases, we use one out of the top four best solutions (i.e.,) and one random solution (i.e., ) with a required condition of . The condition can eliminate the case that the distance is zero or approximately zero (if is equal to 1 or approximately 1). As a result, the local search technique is modified as follows:

###### 3.2.3. Modification to Condition of Local and Global Search Techniques

As shown in Eq. (15), global search technique (i.e., ) is used if happens. If the condition is false, the local search technique (i.e., ) is applied. To clarify probability of global search and local search, we suppose that has a range value from 0 to 10. It is noted that is never less than because is the fitness of the best solution of the minimization problem and it is the lowest in the current population. For an engineering optimization problem, the deviation between fitness function values can be much higher than 10, but the result of is the same for other fitness deviations with higher values than 10. The explanation can be seen by observing Figure 3. Therefore, the probability that a random number *rnd*_{2} is higher than is very tiny and almost update cases use the local search technique in SMA. For balancing local and global search, we remove the condition of SMA and replace it with another condition, that is, *rnd*_{2} < 0.5. If the proposed condition is true, a local search is used and vice versa.

In summary, the three modifications carried out in MSMA are expressed by the formula below:

The whole search process of MSMA for a typical optimization problem is presented in the flowchart of Figure 4.

#### 4. The Application of MSMA for the Studied Problem

##### 4.1. Modified Slime Mold Algorithm

Metaheuristic algorithms are used to find the most suitable site and the most optimal power for installing SCs and PGs in RDPSs. Thus, the parameters of the electrical components are chosen to be the control variables. When reaching the parameters, they are added to data. Then, forward-backward sweep (FBS) method [55] is run to find other parameters such as the current of DLs, voltage drop, and load voltage. Among the three parameters, the current of DLs and load voltage are dependent variables. Thus, each solution *S*_{n} is represented by

And other dependent variables are represented by and .

##### 4.2. Assessment Function of Solutions

Each solution *S*_{n} is represented by control variables, but its quality is assessed by using both the control and dependent variables. The assessment function in Eq. (26) is comprised of the objective function and the penalty terms regarding the violation of dependent variables. The objective function is the total active power loss obtained by using Eq. (1), and the penalty terms are the violation of current and voltage [56].

##### 4.3. Control Variable Correction

As shown in Eq. (25), each solution has [(3 , *N*_{PGs}) + (2 *N*_{SCs})] control variables in which sites of PGs and SCs are discrete values but sizes of them are continuous values. In the flowchart of MSMA, these control variables are produced initially and then they are updated newly one time for each iteration. So, before running power flow for the modified radial distribution systems with the installation of PGs and SCs, the two types of control variables must be solved. Sites of PGs and SCs must be checked and corrected if they are not integer values. The sites are rounded up or down to become integer values, while the approximation process is not performed for sizes. At the last step, sites are checked for satisfying Eq. (27) and Eq. (28), while sizes of PGs and SCs are checked for satisfying Eq. (29), Eq. (30), and Eq. (31).

Added shunt capacitors and photovoltaic systems should be located at from node 2 to node *N*_{NS}. However, the approximation process of the sites can make them to be smaller than 2 or higher than *N*_{Ns}. Thus, the sites should be redefined by

Each photovoltaic system produces both active and reactive powers, while each shunt capacitor produces only reactive power. Thus, control variables and of photovoltaic systems together with of shunt capacitors are defined by

##### 4.4. The Iterative Algorithm

The iterative algorithm for applying MSMA for the studied problem is plotted in Figure 5.

#### 5. Numerical Simulation Results

In this section, the real performance of the proposed MSMA is investigated by comparing results from ten benchmark functions and three distribution systems with SMA, EO, and other previous algorithms. The proposed MSMA and other executed algorithms are run in the programming language of MATLAB 2019a platform and a computer with 2.2 GHz-8 GB of RAM.

##### 5.1. Performance of MSMA for Ten Benchmark Functions

In this part, ten benchmark functions are employed to run MSMA together with six other state-of-the-art algorithms including SMA [46], EO [47], TSA [48], JFA [49], NGOA [50], and BOA [51]. The ten benchmark functions were taken from [57] and are given in Table 1. For running these algorithms, population is set to 50 for NGOA and 100 for others, while the iteration number is selected to be 1,000 for all algorithms. Each algorithm is run for 100 independent trials, and summary of results is reported in Table 2.

For the first four functions f_{01}, f_{02}, f_{03}, and f_{04}, all applied methods could find very good results since minimum, mean, maximum, and standard deviation are equal to 0. The four functions have the same global optimum solution (zero solutions) and the same optimum value. From the function f_{05} to f_{10}, the applied algorithms have different results because the minimum value of function is no longer 0. Approximately all methods can find the same minimum, mean, and maximum values for function f_{05} excluding TSA with higher values of mean and maximum. The standard deviation of MSMA and NGOA is the same and smaller than that of others. So, MSMA is as effective as NGOA and more effective than others for function f_{05}. For f_{06}, TSA and SMA cannot reach the global optimum since their minimum values were, respectively, −209.997 and −209.999, while others could reach −210. The standard deviation of MSMA is only higher than that of BOA. Hence, MSMA is superior to others excluding the comparison with BOA for function f_{06.} For f_{07}, MSMA and others can find global optimum with value of −4.6935, whereas TSA fails. The mean and standard deviations of MSMA are smaller than those from others. So, MSMA is superior to other for the function f_{07}. For f_{08}, TSA and SMA are the worst since they reach higher minimum than others. Their minimums are −8.7221 and −9.5498, while that of others are the same and equal to −9.6602. However, the mean of MSMA is the lowest and it is very close to the optimum value, while that of others is much higher. So, MSMA is also more effective than others for function f_{08}. For functions f_{09} and f_{10}, all methods can find the same global optimum and have the same minimum, mean, and maximum values excluding TSA with higher mean for f_{09}. In addition, MSMA can reach the same standard deviation as others and smaller standard deviation than TSA and SMA. So, MSMA is superior to TSA and SMA for functions f_{09} and f_{10}. The comparisons of MSMA with other six applied algorithms for ten benchmark functions demonstrate that MSMA can be as good as these algorithms for several functions, but it is also more effective than others for remaining functions.

##### 5.2. Performance of MSMA for Three Distribution Systems

In this part, MSMA together with SMA and EO has implemented 50 runs for three standard systems with 33, 69, and 85 nodes by setting the population and iteration number as shown in Table 3. For each study system, three methods are run for the three following cases. Case 1: Install three PGs with active and reactive power generation. Case 2: Install three PGs and three SCs in which PGs only produce active power. Case 3: Install three PGs and three SCs in which PGs produce both active and reactive powers.

###### 5.2.1. Result Comparison for IEEE 33-Node System

In the section, MSA, EO, and the proposed MSMA are applied for three study cases of the IEEE 33-node RDPS to evaluating the performance of methods and the effectiveness of combined PGs. Figure 6 shows the one-line diagram of the system. Load demand, active power loss, and reactive power are 3715 kW and 2300 kVAR, 210.98 kW and 143.01 kVAR, respectively. Data of the system in detail are taken from [58].

The comparison of the results obtained by the methods for the three cases of system 1 is shown in Figure 7. In general, the methods in Case 3 have the lowest loss, while approximately all methods in Case 1 have the highest power loss excluding WPSO-GA [41] in Case 2. However, it is clearly identified that WPSO-GA was not implemented for Case 1. The proposed MSMA method can reach less loss than other methods or the same loss as other methods. For Case 1, the highest loss is 11.926 kW obtained by HIC-GA [45], while MSMA is one of the most powerful methods with 11.741 kW. For Case 2 and Case 3, the proposed MSMA is the best method with the smallest power loss, which is 11.731 kW for Case 2 and 10.366 kW for Case 3. Meanwhile, the worst method and the second-best method have the losses of 16.89 kW and 11.766 kW for Case 2, and 12.9 KW and 10.561 kW for Case 3.

The comparison of performance of EO, SMA, and the proposed MSMA can be observed via Figures 8–10. The loss values of 50 runs obtained by EO, SMA, and MSMA are sorted from the smallest to the highest values and plotted in Figure 8. The best and mean runs of the fifty runs are plotted in Figures 9 and 10. The three subfigures for the three cases in Figure 8 have the same point that all the loss values of MSMA are less than those of EO and SMA excluding some values of Case 1 at the 49th and 50th solutions. The best convergence characteristic curves of the three cases in Figure 9 indicate that MSMA can be faster than EO and SMA about twenty iterations. The fitness values of MSMA at the 80th iteration are less than those of EO and SMA at the last iteration for Case 1 and Case 3. For Case 2, the fitness value of MSMA at the 85th iteration is less than that of EO and SMA at the last iteration. The stability of MSMA is higher than that of EO and SMA as observed in Figure 10. The mean fitness values of MSMA are less than those of EO and SMA for all three study cases. Those of MSMA, EO, and SMA are 17.04, 18.15, and 18.73 kW for Case 1, 15.86, 19.48, and 17.26 kW for Case 2, and 14.08, 16.34, and 17.49 kW for Case 3, respectively. Furthermore, it needs to emphasize that the mean fitness values of MSMA at the previous iterations of the last iteration are also much less than those of EO and SMA at the last iteration. For instance, the mean fitness of MSMA is 18.13 kW at the 69th iteration for Case 1, 17.23 kW at the 65th iteration for Case 2, and 16.28 kW at the 54th iteration for Case 3.

Optimal solutions obtained by methods for the system are shown in Table S1, Table S2, and Table S3 in Supplementary Material. For Case 1, methods have chosen the same locations for PGs at nodes (13, 24, and 30) excluding EO with the selection at nodes (14, 24, and 30). However, the active power generation and power factor of PGs at the same nodes from these methods are not the same. For the first PG at node 13, the power generation of MSMA is 794.3 kW; meanwhile, the lowest and highest values are 752.8 kW from EO and 878 kW from HGWA. Similarly, the power generation of the second and the third PGs from MSMA is, respectively, 1069.7 and 1030 kW, while the lowest and highest power generations are 1061 kW from EO and 1186 kW from hybrid method for the second PG, and 1025 kW from SMA and 1454 kW from HGWA for the third PG. For Case 2, locations of PGs are nodes (13, 25, and 30) for WPSO-GSA, nodes (13, 24, and 30) for AA, EO, and SMA, and nodes (14, 24, and 30) for MSMA. Locations of SCs are nodes (13, 24, and 30) for AA, SMA, and MSMA, nodes (12, 29, and 30) for WPSO-GSA [41], and nodes (12, 24, and 30) for EO. Clearly, MSMA does not have the same locations of PGs as others, but it has the same locations of SCs as AA and SMA. On the other hand, MSMA and others have different values of active and reactive power generation. In Case 3, three applied algorithms place PGs at the same nodes (14, 24, and 30), while WPSO-GSA places PGs at nodes (12, 24, and 30). All methods have different locations of SCs. Those are nodes (6, 30, and 31) for WPSO-GSA, nodes (3, 8, and 26) for EO, nodes (6, 8, and 31) for SMA, and nodes (7, 9, and 32) for MSMA. About the size and power factor of PGs as well as the size of SCs, all methods have found different values. The analysis indicates that MSMA can find the same or different optimal parameters for PGs and SCs with other methods, but power loss, which is objective function, from MSMA is always the best among these compared methods.

###### 5.2.2. Result Comparison for IEEE 69-Node System

In this section, a standard system with 69 nodes is employed to simulate the three applied methods for three different cases of PGs and SCs placement. One-line diagram of the system is plotted in Figure 11. The system has a load demand of 3801.49 kW and 2694.6 kVAR, while its losses are 224.975 kW and 102.187 kVAR. The whole data of the system are taken from [38].

In the second system, the best losses of the applied methods are plotted in Figure 12. Similar to system 1, Case 1 of the system has the worst loss value and Case 3 has the best loss value. Case 1 has the highest loss from 4.26 to 4.61 kW, while those of Case 2 and Case 3 are from 4.252 to 4.2632 kW, and from 3.141 to 3.151 kW. The lowest loss values of the three cases are obtained by the proposed MSMA, while the higher values are by other compared methods.

Fifty power loss values arranged from the smallest to the highest are plotted in Figure 13. MSMA cannot reach all better values than EO and SMA for Cases 1 and 2 but for Case 3. Case 3 is the most complicated study case with the placement of PGs and SCs where PGs produce both active and reactive powers. Thus, the improvement of MSMA over EO and SMA is significant. Figure 14 and Figure 15 indicate that MSMA is much faster and more stable than EO and SMA. About the best run, the fitness values of MSMA at the 150th iteration for Case 2 and the 300th iteration for Case 3 are smaller than those of EO and SMA at the last iteration. About the mean run, the mean fitness values of MSMA are much smaller than those of EO and SMA at the last iteration. Moreover, the numbers in Figure 15 indicate that the mean fitness values of MSMA at the 178th, 169th, and 138th iteration for Case 1, Case 2, and Case 3 are also less than those of EO and SMA at the 200th, 200th, and 400th iteration.

Optimal solutions obtained by methods for the system are shown in Table S1, Table S2, and Table S3 in Supplementary Material. In Case 1, MSMA and other compared methods select the same nodes (11, 18, and 61) for PGs, while only hybrid method [41] selected nodes (18, 61, and 66). However, power generation and power factor from MSMA and these compared methods are totally different. In Case 2, EO and MSMA use the same nodes (11, 18, and 61), while AA and SMA use the same nodes (11, 17, and 61) for PGs. About SCs, SMA and MSMA chose the same nodes 11, 21, and 61, but the locations are nodes (11, 20, and 61) for AA and nodes (11, 19, and 61) for EO. All methods have different generations for PGs as well as SCs at the same nodes. In Case 3, the three applied algorithms select the same nodes (11, 18, and 61) for PGs, while EO and MSMA select the same nodes (8, 50, and 64) for SCs, which are different from SMA with nodes (8, 49, and 64). The three applied methods do have different values of active power generation and power factor for PGs, and reactive power generation for SCs. The analysis on PGs and SCs indicates that MSMA and other compared methods can find the same locations for PGs and SCs but only MSMA can find the best generations for PGs and SCs. As a result, MSMA is more effective than others in cutting active power loss. For the employed system, nodes 11, 18, and 61 are the most suitable locations for the placement of PGs for three study cases, while nodes 8, 50, and 64 are the most suitable locations for the placement of SCs.

###### 5.2.3. Result Comparison for IEEE 85-Node System

In the section, EO, SMA, and the proposed MSMA are implemented for three study cases of the largest system with 85 nodes shown in Figure 16. Active and reactive powers of load demand are 2570.28 kW and 2622.08 kVAR, while losses of the base system are, respectively, 224.32 kW and 141.00 kVAR. All data of the system are taken from [59].

The power losses of MSMA, EO, and SMA are compared with other previous methods as shown in Figure 17. For Case 1, the best losses of MSMA, EO, and SMA are the same and smaller than those of HGWA [40] by 1.242 kW, corresponding to 7.5% of HGWA. For Case 2, MSMA and EO have the same loss and less loss than SMA and WPSO-GA by 0.011 kW and 7.939 kW, which are equal to 0.07% and 34.16% of SMA and WPSO-GA. For Case 3, the three applied methods have the same loss and less loss than WPSO-GA by 3.289 kW, corresponding to 22.65% of WPSO-GA. It is noted that the best loss of Case 1 and Case 2 is the same. Observing from the obtained solutions of MSMA in Supplementary Material indicates that the locations of PGs (these PGs produce both active and reactive powers) for Case 1 are 9, 34, and 67, while three PGs and three SCs are located at 9, 34, and 67. Furthermore, active powers of PGs in Case 1 and Case 2 have the same values, while reactive powers of PGs in Case 1 and SCs in Case 2 have the same values. In fact, active power and power factor (PF) of three PGs in Case 1 are (971.6, 0.7011), (662.5, 0.7027), and (519.3, 0.7019). From the active power and power factor, the reactive power is obtained to be 987.84, 670.95, and 527.08 kVAR, while the active power of PGs and reactive power of SCs in Case 2 are 971.7, 662.4, and 519.3 kW and 988, 670.9, and 527 kVAR.

For the system, MSMA approximately has the same best loss as EO and SMA for the three study cases. As a result, the best convergence characteristic of the three algorithms is not much different and the outstanding performance of MSMA over EO and SMA is not reached for the system as shown in Figure 18. However, obtained 50 solutions and the mean convergence characteristic of these 50 runs shown in Figures 19 and 20 are good evidences to recognize the real performance of MSMA. In Figure 19, MSMA can reach many better solutions than EO for all three cases and SMA for Case 2 and Case 3 excluding Case 1. In Figure 20, mean loss of MSMA is much less than that of EO and SMA. Even the mean loss of MSMA at previous iterations can be significantly smaller than that of EO and SMA at the last iteration. For Case 1, mean loss of MSMA at the 419th iteration is 16.55 kW but that of EO and SMA at the 500th iteration is, respectively, 17.95 kW and 16.56 kW. For Case 2, mean loss of MSMA at the 299th iteration is 17.26 kW but that of EO and SMA at the 500th iteration is, respectively, 18.12 kW and 17.27 kW. For Case 3, mean loss of MSMA at the 369th iteration is 12.66 but that of EO and SMA at the 1000th iteration is, respectively, 13.28 kW and 12.67 kW. Thus, it can conclude that MSMA is much more stable and faster than EO and SMA for the system.

Optimal solutions obtained by methods for the system are shown in Table S1, Table S2, and Table S3 in Supplementary Material. For all three cases of the system, there is a coincidence that the three applied algorithms have the same locations for three PGs at nodes (9, 34, and 67). Also, the locations of three SCs from the three methods are the same in both Case 2 and Case 3. Case 2 has a special point of the locations of PGs and SCs since EO, SMA, and MSMA chose the same nodes 9, 34, and 67 for three PGs and three SCs. In addition, they also select approximately the same values for size and power factor of PGs and for size of SCs. The manner is the reason why the three algorithms have the same power loss for each study case. A previously applied algorithm, HGWA, selects different locations, different sizes, and different power factors from the three applied algorithms. Thus, HGWA must suffer from much higher power loss than the three applied algorithms. From the analysis on optimal solutions, it is suggested for the IEEE 85-node system that nodes 9, 34, and 67 are the most suitable locations for placing PGs, while 19, 52, and 80 are the most suitable locations for placing SCs.

##### 5.3. The Improvement of Voltage Profiles

The study only focuses on the active power loss reduction for radial distribution systems as shown in the objective function in Eq. (1). However, the reduction of using power from the conventional power source at slack node 1 leads to a very useful phenomenon of improving voltage profile for loads. Figures 21–23 show the voltage profiles of three systems for four cases including Case (1) base system without the installation of any PGs and SCs; Case (2) modified systems with active power generation using PGs and reactive power generation using SCs; and Case (3) modified systems with active and reactive power generations using PGs and reactive power generation using SCs. In general, the three figures have the same characteristic since base systems have the worst voltage profile and the deviation between base cases with other cases is very high, especially for the first and the third systems while three other cases have approximate voltage profiles. The worst voltage of base cases is about 0.9 pu for the first system, 0.91 pu for the second system, and 0.87 pu for the last system, while the worst voltage of three other cases is higher than 0.99 pu, which is slightly smaller than rated value of 1.0 pu. Clearly, modified systems with the additional supply of active power and/or reactive power can reach a significant voltage improvement.

The voltage profiles of three other cases have also a slight difference since voltage cures of Case 3 tend to be slightly better than Case 1 and Case 2. The difference can be seen clearly as observing from node 8 to node 19, node 26 and 27, and nodes 31, 32, and 33 in the first system. Similarly, Case 3 also has better voltage than Case 1 and Case 2 for nodes 52 and 66 for the second system, and nodes 21, 22, 23, 56, 57, 81, 82, 83, 84, and 85 for the last system. The outstanding voltage improvement of Case 3 indicates that the placement of both PGs and SCs (where PGs produce both active and reactive powers) is the most optimal for the purpose of voltage improvement.

##### 5.4. Discussion on the Performance of the Proposed MSMA

In this section, the real effectiveness of the proposed MSMA is analyzed in comparison with previous algorithms as applied for ten benchmark functions and three radial distribution systems with three cases for each system. As shown in Section 5.1, MSMA together with six other algorithms including SMA, EO, TSA, JFA, NGOA, and BOA was run for ten benchmark functions, and the results of these algorithms could be the same or different dependent on applied functions. For a clear view of possibility of reaching global optimums and a high stability, we summarize the evaluations regarding the possibility of reaching global optimums and the possibility of reaching a high stability in Table 4. In the table, we use Yes (Y) or No (N) to answer if the applied algorithm can reach global optimums and a high stability. The first letter is the answer for reaching global optimums, while the second letter is the answer for reaching a high stability. Over 100 runs, if methods could reach the best objective (the best objective is shown in Table 1 in Section 5.1) for at least one run, the first letter of the methods is Y. In other words, if the minimum in Table 2 is equal to the best objective in Table 1, the methods can reach the global optimum and it receives the answer Y for the first letter in Table 4. The high stability is evaluated if the average value in Table 2 and the best objective in Table 1 are the same or approximately the same. For this result, the evaluated method receives answer Y for the second letter and this method has a high stability. For the first four functions (*f*_{01}–*f*_{04}) and the last function (*f*_{10}), all methods receive the answer Y-Y. It means all methods could reach global optimums and the high stability for the five functions. However, for other remaining functions, all methods could not reach the achievement again. For function *f*_{05}, all methods could reach the global optimums, so the first letter is Y for all. But TSA could not reach the high stability since its mean and the best objective are, respectively, −49.8635 and 50. For function *f*_{06}, MSMA together with JFA, NGOA, and BOA could reach Y-Y, while the answers are N-N for SMA and TSA, and Y-N for EO. For the results, SMA and TSA could not reach the global optimums and its mean was much higher than the best objective. On the contrary, EO could reach the global optimums, but its mean was much higher than the best objective. For function *f*_{07}, MSMA together with NGOA and BOA could reach Y-Y; meanwhile, those are Y-N for SMA, EO, and JFA, and N-N for TSA. For function *f*_{09}, only TSA had to suffer from Y-N, while all methods could reach Y-Y. In summary, among seven applied algorithms, only MSMA and BOA could reach both global optimums and high stability for all ten functions. So, MSMA is more effective and/or more stable than others excluding the comparison with BOA.

Similarly, the possibility of reaching global optimums and the high stability of methods as applied for three cases of three RDPSs are also evaluated and reported in Table 5. In the table, the high stability of methods in previous studies could not be evaluated because these studies did not report the mean power loss for a number of runs. Methods reaching smaller or the same minimum power loss as that of MSMA can receive answer Y for global optimums, while others with higher power loss than MSMA must receive answer N. If MSMA found a higher power loss than at least one method, MSMA must receive answer N for the possibility of reaching global optimums. The first letter of MSMA is Y for all study cases, while others cannot reach the same results. Hybrid method [14], WPSO-GSA [41], and AA [44] could not reach the global optimums even for one study case. HGWA [40] could reach global optimums for Case 1 of systems 1 and 2 but for Case 1 of system 3. HIC-GA [45] could reach global optimums for Case 1 of system 2 but for Case 1 of system 1. The applied EO failed to reach global optimums and high stability for three cases of system 1 and system 2. For system 3, EO could reach global optimums as MSMA but its stability was low. SMA failed to reach global optimums for many cases excluding Case 1 of system 1, and Case 1 and Case 3 of system 3. The results indicate that MSMA is a really effective algorithm with higher possibility of reaching global optimums and higher stability than other compared methods.

In general, MSMA is a strong search tool for optimal placement of PGs and SCs in RDPSs with the purpose of reducing active power loss. The structure of MSMA is very simple, but its result is significantly effective. The major different characteristics between MSMA and others as well as between MSMA and SMA are the new solution generation mechanism. MSMA can take advantage of SMA, which is the local search around the so far best solution and the use of two scaling factors *α*_{2} and *α*_{3}. The characteristic of SMA is relatively effective in finding global optimums, which are nearby local optimums; however, SMA uses an ineffective global search strategy by using in which tends to be equal to zero as computation iteration is increased and equal to the maximum value. For three employed systems, optimal parameters of PGs and SCs (which are generation, power factor, and locations) are not zero and their real optimal values do not follow any rules to be reached. So, the use of cannot lead to success in exploiting global search. As comparing the global search mechanism of SMA and MSMA, that of MSMA is much more superior and that increase the possibility of reaching global optimums for MSMA, whereas that of SMA limits the chance of finding the global optimums. Furthermore, the use of top four solutions in both local and global search mechanisms of MSMA is also an outstanding advantage over SMA. The use can support MSMA to produce more suitable jumping steps between old and new solutions. As a result, MSMA is more stable than SMA thanks to the effective jumping steps. In fact, MSMA was more potential than SMA in finding global optimal solutions reflected via the best power loss and in reaching high stability reflected via mean of power loss. As overcoming the drawbacks of SMA, MSMA is also superior to other compared algorithms including EO, TSA, JFA, NGOA, hybrid method, HGWA, WPSO-GSA, AA, and HIC-GA.

#### 6. Conclusions

In this paper, three study cases of PGs and SCs placement have been implemented in RDPS for cutting active power loss. Only PGs with active and reactive power generations have been considered installed in RDPS in the first case, while both PGs and SCs have been installed in the second and third cases. However, PGs in the first and third cases could produce both active and reactive powers, but PGs in the second case could produce only active power. The three study cases have been simulated by applying two original metaheuristic algorithms (EO and SMA) and the proposed MSMA method. The obtained results compared among the studied cases, among the executed methods, and between MSMA with other previous methods can indicate the following conclusions:(1)The proposed MSMA was faster, more effective, and stable than EO and SMA for study cases of IEEE 33 and 69-node systems. For the last IEEE 85-node system, MSMA was more stable and faster than EO and SMA. MSMA could reach less loss than EO and SMA for the first two systems, but it reached the same loss as EO and SMA for the last system. For all study cases, the mean loss over 50 runs of MSMA was less than those of EO and SMA, and even the mean loss of MSMA before the last iteration was much less than that of EO and SMA at the last iteration.(2)About the three study cases, Case 1 had the highest loss while Case 3 had the smallest loss for all methods including the three executed methods and other previously published methods. The results indicated that the use of separated power sources for active power injection and reactive power injection was more effective than the combined injection of active and reactive power at the same locations.(3)MSMA was more effective than other previous methods over three study cases for three distribution systems. In addition, MSMA is also as good as or superior to other state-of-the-art algorithms for ten benchmark functions.

Thus, it is recommended that MSMA should be used as a powerful metaheuristic algorithm for finding the locations and size of PGs and SCs when installing the components in RDPS. Furthermore, the most optimal placement of these electrical components can be found to be the use of PGs and SCs, in which PGs produce both active and reactive power generations.

#### Abbreviation

DGs: | Distributed generators |

RDPSs: | Radial distribution power systems |

SMA: | Slime mold algorithm |

EO: | Equilibrium optimizer |

MSMA: | Modified slime mold algorithm |

PGs: | Photovoltaic generators |

SCs: | Shunt capacitors |

SCA: | Sine cosine algorithm |

CSA: | Cuckoo search algorithm |

WOA: | Whale optimization algorithm |

KHA: | Krill herd algorithm |

CSA-ABCA: | Cuckoo search algorithm and artificial bee colony algorithm |

MTLOA: | Modified teaching-learning optimization algorithm |

MABCA: | Modified artificial bee colony algorithm |

MALO: | Modified ant lion optimizer |

HOA: | Heuristic optimization algorithm |

DLs: | Distribution lines |

MSM: | Maximum savings method |

TSM: | Two-stage method |

AA: | Analytical approach |

PSO: | Particle swarm optimization |

FPA: | Flower pollination algorithm |

MFPA: | Modified flower pollination algorithm |

SFSO: | Stochastic fractal search optimization |

TALA: | Teaching and learning algorithm |

SMOA: | Slime mold optimization algorithm |

IHA: | Improved harmony algorithm |

LSF: | Loss sensitivity factor |

MHA: | Modified harmony algorithm |

IEAs: | Improved evolutionary algorithms |

PG: | Photovoltaic generator |

SQP-BB: | Sequential quadratic programming and branch-bound technique-based method |

MAM: | Modified analysis method |

WPSO-GSA: | Modified particle swarm optimization and gravitational search algorithm |

ABCA: | Artificial bee colony algorithm |

BCACA: | Binary collective animal behavior algorithm |

BFA: | Bacterial foraging algorithm |

ABC-HS: | Hybrid artificial bee colony algorithm and harmony search algorithm |

IMDE: | Modified differential evolution with intersect mutation technique |

WPSO: | Particle swarm optimization with weight factor |

GWA: | Grey wolf algorithm |

HGWA: | Hybrid grey wolf algorithm |

HRA: | Heuristic algorithm |

MSSA: | Modified Salp swarm algorithm |

SSA: | Salp swarm algorithm |

HIC-GA: | Hybrid imperialist competitive algorithm and genetic algorithm method |

ICA: | Imperialist competitive algorithm |

HMPSO: | Hybrid modified particle swarm optimization |

MPSO: | Modified particle swarm optimization |

TSA: | Tunicate swarm algorithm |

JFA: | Jellyfish algorithm |

NGOA: | Northern goshawk optimization algorithm |

BOA: | Bonobo optimization algorithm |

TAP_{L}: | Total active power loss |

I_{PGSC,m}: | Current of the mth distribution line after placement of PGs and SCs |

N_{DLs}: | Distribution line number |

R_{DL,m}, X_{DL,m}: | Resistance and reactance of the mth distribution line |

N_{PGs}, N_{SCs,}N_{Ns}: | Number of photovoltaic generators, shunt capacitors, and nodes |

P_{PG,i}, Q_{PG,i}: | Active and reactive power output of the ith photovoltaic generator |

P_{grid}, Q_{grid}: | Active and reactive power supplied by power source at node 1 |

Q_{SC,j}: | Reactive power output of the jth shunt capacitor |

P_{L,k}, Q_{L,k}: | Active and reactive power demand of load at the kth node |

: | Rated current of the conductor of the mth distribution line |

U_{L,k}: | Voltage of load at the kth node |

U^{Low}, U^{Up}: | Lower and upper bounds of load voltage |

,: | Lower bound and upper bound of active power for each photovoltaic generator |

,: | Lower bound and upper bound of reactive power for each photovoltaic generator |

,: | Lower bound and upper bound of reactive power for each shunt capacitor |

, G: | Current iteration and maximum iteration number |

rnd_{1}, rnd_{2}, rnd_{3}: | Random numbers within 0 and 1 |

, : | Sites of the jth shunt capacitor and the ith photovoltaic generator in the nth solution |

, : | Active and reactive power of the ith photovoltaic generator in the nth solution |

: | Reactive power of the jth shunt capacitor in the nth solution |

: | Current of the mth distribution line in the nth solution |

: | Voltage of load at node k in the nth solution |

δ: | Penalty factor |

#### Data Availability

Data of the employed systems were extracted from [38, 58, 59].

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work belongs to the Project of the Year 2022 funded by Ho Chi Minh City University of Technology and Education, Vietnam.

#### Supplementary Materials

Table S1, Table S2, and Table S3: optimal solutions for Case 1, Case 2, and Case 3 of the three test systems.* (Supplementary Materials)*