# Applications of Recent Metaheuristic Algorithms for Loss Reduction in Distribution Power Systems considering Maximum Penetration of Photovoltaic Units

Authors: Le Duy Luan Nguyen; Phuc Khai Nguyen; Viet Cuong Vo; Ngoc Dieu Vo; Thang Trung Nguyen; Tan Minh Phan
Journal: International Transactions on Electrical Energy Systems
 (2023)

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## Abstract

Radial distribution power systems (RDPSs) are essential in the power system to supply electricity to loads for developing industries, commerce, and services in countries. However, RDPSs have many distribution lines with high resistance, causing high active power loss. So, the core target of the study is to reduce the total active power loss, considering the highest penetration level of installed PVUs. Photovoltaic units (PVUs) are placed optimally by implementing the Coot optimization algorithm (COOA), the archimedes optimization algorithm (AOA), the transient search optimization algorithm (TSOA), the crystal structure algorithm (CrSA), the war strategy optimization algorithm (WSA), and the average and subtraction-based optimizer (ASBO). Two test RDPSs with 33 and 69 nodes are employed for four different study cases, including (1) the number of PVUs is three and the power factor is 1.0; (2) the number of PVUs is three, and the power factor is optimally determined; (3) the number of PVUs is determined optimally while the power factor is 1.0; and (4) the number of PVUs and the power factor of PVUs are optimally determined. For the first system, COOA can reach smaller power losses than others by 0.14% to 13.059% for Case 1, 0.5% to 77.89% for Case 2, 0.101% to 10.126% for Case 3, and 11.818% to 88.492% for Case 4. For the second system, COOA can reach a lower loss than others by 0.164% to 15.511% for Case 1, 0.774% to 78.6% for Case 2, 0.173% to 15.104% for Case 3, and 46.201% to 95.389% for Case 4. Especially in Case 4, the power loss is approximately equal to zero. The loss reduction is the first significant contribution of the study. In addition, the study also investigated the difference between the four cases for obtaining power loss reduction, the maximum penetration level of active and reactive power of PVUs, and the improvement of load voltage. As a result, Case 4 is superior to the three other cases. Thus, the number, peak power, power factor, and location of added PVUs in RDPSs should be simultaneously optimized, and one of the most effective optimization tools is COOA.


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## Body

## 1. Introduction

### 1.1. Motivations

Using nonrenewable fossil fuels such as oil, coal, and fracking gas causes the release of carbon dioxide and global warming [1, 2]. Thus, many countries have policies to encourage the development of renewable energy sources relying on natural processes such as wind, solar, hydropower, geothermal, and bioenergy due to many obtained benefits [3, 4]. The connection of renewable distributed generators, especially solar energy, to the distribution systems offers many significant advantages in terms of technology and economy, such as the high reliability of the system in providing power to loads and low distribution loss [5, 6]. Power loss in each distribution line, which can be obtained by calculating 3.R.I2 (where R is resistance and I is the operating current of the distribution line), can be significant due to the long distance between two nodes and low voltage of the system. On the other hand, the two reasons are also the consequence of a high voltage drop and a bad voltage profile, as shown in the formula ∆V=3.Z.I (where Z = R + jX and X is the reactance of the distribution line). To deal with the unexpected results, one of the most effective solutions is to reduce the current, and the target can be achieved by optimally placing PVUs in distribution systems [7, 8]. It was noted that the benefits of the placement mainly depended on the location and capacity of PVUs [9–11]. However, the total capacity of PVUs is also influenced by the number of PVUs. On the other hand, the potential of solar energy is very high, and PVUs can be employed to replace the supply from conventional power sources that are very expensive and highly influence the environment. So, the study focuses on the optimal site, size, and number of PVUs in distribution networks for reaching the highest power loss reduction and the best voltage profile by considering the highest penetration level of PVUs to eliminate the conventional power sources.


### 1.2. Literature Review

In recent years, the placement of PVUs in distribution networks has significantly developed, and the benefits from the free primary fuel have been extremely high. Based on the achievements, researchers also performed different combinations between PVUs and other devices and then investigated the advantages of the combinations. Based on the characteristics of the combination, the renewable energy integrated distribution system can be divided into five main applications. Although the applications of PVUs in power systems are not the same, the advantages obtained from the placement of PVUs in the systems are approximately the same, such as the reduction of power loss, the improvement of power system operation stability, the enhancement of voltage, and the reduction of costs regarding investment, operation, and maintenance. To investigate the impact of PVUs, researchers have mathematically formulated problems and then applied optimization algorithms to reach numerical results for evaluation and discussion. The details of the applications, applied algorithms, shortcomings, and advantages are expressed as follows:In the first application, the traditional RDPSs are combined only with PVUs. Many studies have been implemented to analyze the impact of RDPSs on the distribution system. The suitable installation of RDPSs was suggested to maximize the expected benefits. Do not use metaheuristic algorithms. The research studies [12–16] have implemented an analysis approach (AA) based on the configuration of considered RDPSs to calculate the current of branches and determine the suitable nodes to add PVUs. Then, the power loss optimization equation was solved to find the rated power of the added PVUs with pure active power generation. A general name for the approach in the studies is the distribution system configuration analysis approach. The approach had the advantage of eliminating the task of determining values for parameters of metaheuristics, but it suffered from the most serious disadvantage of being highly dependent on the implementation of distribution system configuration analysis. For small distribution systems with a small number of nodes and branches, the approach is very effective, but it is not the best choice for larger RDPSs with a higher number of nodes and branches. So, the approach is time-consuming for application in cases where the number of PVUs is high. In the study [14], an optimal power flow technique-based analysis approach (OPFAA) was proposed to reach higher effectiveness than the conventional approach its the significant limits. The results on two RDPSs have proved the real effectiveness of OPFAA since it could reach less loss than AA for different study cases. In the study [17], the analysis approach and conventional particle swarm optimization (PSO) were combined to develop a hybrid AA and PSO (AA-PSO) method. The method could reduce the impact of analysis on RDPS configuration and also estimate the task of determining the locations for metaheuristic algorithms. The AA-PSO method could reach a better loss than PSO, but the two approaches have used the same setting values for control parameters, which is an unfair setting for PSO. In fact, AA-PSO used a distribution system configuration analysis approach to determine location and then ran PSO to determine the rated power, while PSO had to finish the two tasks. Over the past ten years, metaheuristic algorithms were more successfully executed for dealing with power system optimization problems, and the optimization addition of PVUs in RDPSs was an example. In the study [18], five different improved particle swarm optimization (IPSO) methods were developed to reach the minimum loss for a 33-node distribution system and a 69-node distribution system. Conventional grey wolf optimization (GWO) [19] and another improved variant of GWO (IGWO) [20] successfully found optimal values of PVUs in distribution systems. GWO was superior to GSA and BA, while IGWO was as effective as the other methods. Specifically, the study [21–23] has applied moth flame optimization (MFO), the Lagrange multipliers (LM), and the manta ray foraging optimization (MRFO) for determining the best location for RDGs with the aim of minimizing total energy loss with consideration of operating constraints. By successfully connecting the RDPSs, the transmission loss is significantly reduced, and the voltage profile of the system is improved to the allowable limits. In addition, by considering many different aspects of the problem, multi-objective functions are also commonly used [24, 25]. In [24], the loss saving and the voltage drop of each voltage bus have been carefully considered as two components in the objective function. The siting and sizing of the RDPSs were suitably determined by using a genetic algorithm (GA). Similarly, in the study [25], the appropriate installation of RDGs is successfully found on distribution systems by applying biogeography-based optimization (BBO). Thanks to this proper installation, not only the power loss has been significantly reduced, but the bus voltage improvement and harmonic mitigations have also changed dramatically in the positive direction. In addition, other metaheuristic algorithms were also implemented for the optimal placement of PVUs in distribution systems, including the symbiotic organism search algorithm (SOSA) and improved SOSA (ISOSA) [1], the quasi-oppositional swine influenza algorithm (QOSIA) [2], the improved artificial bee colony algorithm (IABCA) [10], the OPFAA [14], the PSO and AA-PSO [17], teaching and learning algorithm (TLA), modified version of TLA (MTLA) [26], Krill herd algorithm (KHA) [27], Ant lion algorithm (ALA) [28], Grasshopper Optimization Algorithm (GOA) [29], Sin-Cos algorithm (SCA) [30], student psychology-based optimization (SPBO) [31], stochastic fractal search (SFS), chaotic stochastic fractal search (CSFS) [32], particle swarm optimization (PSO) [33], modified teaching learning based optimization (MTLBO) [33], Jaya algorithm (JAYA) [33], grey wolf optimizer (GWO) [33], improved raven roosting optimization (IRRO) [33], salp swarm algorithm (SSA) [34], Sunflower optimization (SFO) [34], coyote optimization algorithm (COA) [34], enhanced coyote optimization algorithm (ECOA) [34], sine cosine algorithm (SCA) [35], improve sine cosine algorithm ISCA [35], bat optimization algorithm (BA) [36], cuckoo search algorithm (CSA) [37], moth-flame optimization (MFO) [38], dragonfly algorithm (DA) [38], equilibrium optimizer (EO) [39], modified equilibrium optimizer (MEO) [39], adaptive equilibrium optimizer (AEO) [39], and improved equilibrium optimizer (IEO) [39]. Other applications of PVUs have a smaller number of research studies.The second application of PVUs is to combine PVUs with energy storage systems (ESSs) [40–42] The crowd search optimization (CSO) algorithm was applied to optimize the installation of PVUs and ESSs to cut power loss and enhance voltage stability in RDPSs [40]. In the study [41], a real power system in Algeria was supplied by conventional power plants PVUs, and ESSs whose capacities had to be determined within known locations. The artificial bee colony algorithm (ABC) has selected the location and size of PVUs and ESSs for a real distribution system, and it could save on energy costs by 24% and reduce loss costs by 20%.In the third application of PVUs, PVUs are integrated with wind power plants (WPs) and other power sources in the existing power grid to reduce the loss and cost of conventional power plants available in the gird. PVUs and WPs were installed in a small existing power system [43]. PVUs, WPs, and fuel cell units were combined to supply electricity to a load [44]. In [45], PVUs were combined with WPs, diesel generators, and batteries to be the main source for a predetermined load. In the fourth application of PVUs, the configuration of RDPSs was modified, and then PVUs, together with other power sources, were optimally placed for reducing loss and cost and improving the voltage profile of loads. In the study [46], PVUs and ESSs were placed in RDPSs after performing the configuration for the distribution systems. So, the optimal placement of PVUs and ESSs was implemented after determining the best configuration of the system. If the optimal placement of PVUs and ESSs was not implemented before the configuration, power loss would not be reduced effectively. The study [47] has applied the same direction of research as the study [46] by applying reconfiguration and optimizing the placement of PVUs and WPs instead of PVUs and ESSs.In the fourth application, PVUs are placed in microgrids with low voltage and a small demand for loads. The installed PVUs are used as the sole power source supplying full power to all loads, or they are a major power source together with the conventional power sources from thermal power plants. In addition, the energy storage system is also integrated with the added PVUs. In the study [48], the conventional distribution networks were transformed into resilient autonomous microgrids by the placement of PVUs. In the study [49], the reliability of microgrids with the integration of PVUs was evaluated, and different microgrid configurations were tried for the case of energy shortage. In the study [50], the microgrids were totally supplied by PVUs, and the benefits of the placement of PVUs in the grids were evaluated. The study [51] optimized the capacity configuration for a photovoltaic system and battery-basedmicro-grid. The location and capacity of PVUs were optimized for increasing the efficiency of the micro-grid, and the micro-grid was integrated into a conventional distribution network. In the study [16], PVUs with active and reactive generation in island microgrids were optimized in terms of location and capacity. Load demand and solar radiation were considered uncertain, and the uncertainty of generation was taken into account in the micro-grids.The fifth application of PVUs in the distribution system is also a hot topic in the current trend of developing electric vehicles and reducing the use of fossil fuel-based vehicles. PVUs are placed in distribution networks to generate electricity for existing loads and added electric vehicle charging stations (EVCSs). Grasshopper Optimization Algorithm (GOA) [52] and bear smell optimization algorithm (BSOA) [53] were proposed for optimizing the placement of PVUs, shunt capacitors (SCs), and EVCSs. The same strategy with two phases was applied in the two studies for achieving the reduction of load and enhancement of voltage. In the first phase, the location and capacity of PVUs and SCs were determined for improving the power factor of substations, minimizing power loss, and improving load voltage in distribution networks. In the second phase, the location of EVCS and the number of electric vehicles at each station were optimized with the purpose of reaching the highest number of electric vehicles at all stations. In the study [54], a multiobjective optimization problem simultaneously considering electric vehicle user loss, power loss of distribution, cost of charging stations, and load voltage was solved by using the nondominated sorting genetic algorithm II (NSGA-II). NSGA-II was in charge of finding the location and capacity of both PVUs and charging stations. The study [55] also solved a multiobjective optimization problem in distribution networks with the presence of PVUs and EVCSs; however, the study focused on both economic and technical issues such as power loss, voltage deviation, and voltage stability. Teaching-Learning-Based Optimization (TLBO) and Harries Hawk Optimization (HHO) were applied for two IEEE systems with 33 and 69 nodes, respectively. HHO was stated to be more effective than TLBO to the two systems. In the two studies [55, 56], PVUs were integrated into power networks where fast charging stations were located. The study supposed that a number of EVCSs was given and their capacity was respectively 5, 25, 50, and 100 MW. For each case of total capacity, the optimal site and size of PVUs were determined, and the power loss on all lines was calculated. The study placed PVUs in the network with 24 nodes and then used the Newton—Raphson method to calculate power loss and voltage of loads for evaluating the effectiveness and stability of the networks.In this paper, the maximum penetration of PVUs is considered in radial distribution power systems to calculate the highest possibility that PVUs can supply electricity to the system with and without the integrated generation of the power source in the base network. The main target of the paper is to use the highest capacity of PVUs in RDPSs and reach the most suitable power loss while satisfying operation conditions for other components, such as loads and branches. In the first study case, the main objective is to minimize the power losses on all distribution branches while the capacity of PVUs is not limited and the number of PVUs is given. In the second study case, the maximum penetration of PVUs is considered, and the number of PVUs is an optimal parameter that needs to be determined. In the two study cases, PVUs are divided into two different types: the first type has pure active power generation, and the second type has both active and reactive power generation. To reach the best solutions for the maximum penetration of PVUs in RDPSs with the smallest total power losses, we suggest using a novel metaheuristic algorithm named the Coot optimization algorithm (COOA), which was published in early 2021 [57]. The novel algorithm was compared to previous methods in terms of searching the global optimum and reaching the most stabilization. In addition, the novel algorithm was also implemented for eight other engineering design problems, and it was stated to be more effective than over thirty other algorithms. Recently, COOA has been applied for engineering optimization problems and it has obtained better results than other compared algorithms. In the study [58], the applications of COOA for improving the performance of photovoltaic arrays considering various operation and load cases were presented. COOA has successfully found optimal uncertain parameters for well-known photovoltaic arrays, and the obtained results have been equal to or better than those from previous studies using different metaheuristic algorithms. In the study [59], COOA was run to find optimal parameters for batteries based on lithium and ion. The major advantages of the found parameters were that they were highly accurate and highly efficient, with 99.25%, while those obtained by other algorithms suffered from poorer achievements. In the study [60], COOA was applied for the design of frequency-selective surfaces for 5G cellular communication networks. Obtained results proved that frequency-selective surfaces had high performance, such as maximization of realized gain, minimization of reflection coefficients, and the highest efficiency. In the study [61], COOA was applied to wireless sensor networks to overcome uneven distribution and low coverage. COOA was compared to six other metaheuristic algorithms and proved to be faster at convergent good solutions with a higher coverage rate. Thus, in this paper, we use COOA as an optimization tool to solve the placement of PVUs in RDPSs for reaching the highest capacity of PVUs and the lowest total power losses with different considered conditions. In this paper, COOA is implemented for two RDPSs with 33 and 69 nodes, respectively. In addition, we also apply other new algorithms, including the Archimedes optimization algorithm (AOA) [62], the transient search optimization algorithm (TSOA) [63], the crystal structure algorithm (CrSA) [64], the war strategy optimization algorithm (WSA) [65], and the average and subtraction-based optimizer (ASBO) [66] for the comparisons with COOA. The results, including power loss and capacity of PVUs, are compared to those from five implemented and other previous methods. The fact is that COOA is an effective optimization algorithm for the problem of determining the optimal parameters of PVUs in RDPSs. In summary, the contributions of the paper are as follows:(1)
Successfully apply six new algorithms, including COOA, AOA, TSOA, CrSA, WSA, and ASBO, to find the most suitable solutions for the optimal placement of PVUs in RDPSs.(2)
COOA can reach the smallest total power losses and the maximum penetration of PVUs for applied standard RDPSs. COOA is either as good as or much better than other applied ones.(3)
Prove the penetration-level impact of PVUs on loss reduction and the voltage profile of RDPSs.In addition to the section, the organization of other parts of the article is as follows: Section2 presents the objective and constraints of the problem. Section 3 describes the structure and shows mathematical formulations of COOA. Section 4 shows the implementation of COOA for PVUs determination in RDPSs. Section 5 shows results, comparisons, and discussions. Section 6 summarizes achievements and conclusions.


## 1.1. Motivations

Using nonrenewable fossil fuels such as oil, coal, and fracking gas causes the release of carbon dioxide and global warming [1, 2]. Thus, many countries have policies to encourage the development of renewable energy sources relying on natural processes such as wind, solar, hydropower, geothermal, and bioenergy due to many obtained benefits [3, 4]. The connection of renewable distributed generators, especially solar energy, to the distribution systems offers many significant advantages in terms of technology and economy, such as the high reliability of the system in providing power to loads and low distribution loss [5, 6]. Power loss in each distribution line, which can be obtained by calculating 3.R.I2 (where R is resistance and I is the operating current of the distribution line), can be significant due to the long distance between two nodes and low voltage of the system. On the other hand, the two reasons are also the consequence of a high voltage drop and a bad voltage profile, as shown in the formula ∆V=3.Z.I (where Z = R + jX and X is the reactance of the distribution line). To deal with the unexpected results, one of the most effective solutions is to reduce the current, and the target can be achieved by optimally placing PVUs in distribution systems [7, 8]. It was noted that the benefits of the placement mainly depended on the location and capacity of PVUs [9–11]. However, the total capacity of PVUs is also influenced by the number of PVUs. On the other hand, the potential of solar energy is very high, and PVUs can be employed to replace the supply from conventional power sources that are very expensive and highly influence the environment. So, the study focuses on the optimal site, size, and number of PVUs in distribution networks for reaching the highest power loss reduction and the best voltage profile by considering the highest penetration level of PVUs to eliminate the conventional power sources.


## 1.2. Literature Review

In recent years, the placement of PVUs in distribution networks has significantly developed, and the benefits from the free primary fuel have been extremely high. Based on the achievements, researchers also performed different combinations between PVUs and other devices and then investigated the advantages of the combinations. Based on the characteristics of the combination, the renewable energy integrated distribution system can be divided into five main applications. Although the applications of PVUs in power systems are not the same, the advantages obtained from the placement of PVUs in the systems are approximately the same, such as the reduction of power loss, the improvement of power system operation stability, the enhancement of voltage, and the reduction of costs regarding investment, operation, and maintenance. To investigate the impact of PVUs, researchers have mathematically formulated problems and then applied optimization algorithms to reach numerical results for evaluation and discussion. The details of the applications, applied algorithms, shortcomings, and advantages are expressed as follows:In the first application, the traditional RDPSs are combined only with PVUs. Many studies have been implemented to analyze the impact of RDPSs on the distribution system. The suitable installation of RDPSs was suggested to maximize the expected benefits. Do not use metaheuristic algorithms. The research studies [12–16] have implemented an analysis approach (AA) based on the configuration of considered RDPSs to calculate the current of branches and determine the suitable nodes to add PVUs. Then, the power loss optimization equation was solved to find the rated power of the added PVUs with pure active power generation. A general name for the approach in the studies is the distribution system configuration analysis approach. The approach had the advantage of eliminating the task of determining values for parameters of metaheuristics, but it suffered from the most serious disadvantage of being highly dependent on the implementation of distribution system configuration analysis. For small distribution systems with a small number of nodes and branches, the approach is very effective, but it is not the best choice for larger RDPSs with a higher number of nodes and branches. So, the approach is time-consuming for application in cases where the number of PVUs is high. In the study [14], an optimal power flow technique-based analysis approach (OPFAA) was proposed to reach higher effectiveness than the conventional approach its the significant limits. The results on two RDPSs have proved the real effectiveness of OPFAA since it could reach less loss than AA for different study cases. In the study [17], the analysis approach and conventional particle swarm optimization (PSO) were combined to develop a hybrid AA and PSO (AA-PSO) method. The method could reduce the impact of analysis on RDPS configuration and also estimate the task of determining the locations for metaheuristic algorithms. The AA-PSO method could reach a better loss than PSO, but the two approaches have used the same setting values for control parameters, which is an unfair setting for PSO. In fact, AA-PSO used a distribution system configuration analysis approach to determine location and then ran PSO to determine the rated power, while PSO had to finish the two tasks. Over the past ten years, metaheuristic algorithms were more successfully executed for dealing with power system optimization problems, and the optimization addition of PVUs in RDPSs was an example. In the study [18], five different improved particle swarm optimization (IPSO) methods were developed to reach the minimum loss for a 33-node distribution system and a 69-node distribution system. Conventional grey wolf optimization (GWO) [19] and another improved variant of GWO (IGWO) [20] successfully found optimal values of PVUs in distribution systems. GWO was superior to GSA and BA, while IGWO was as effective as the other methods. Specifically, the study [21–23] has applied moth flame optimization (MFO), the Lagrange multipliers (LM), and the manta ray foraging optimization (MRFO) for determining the best location for RDGs with the aim of minimizing total energy loss with consideration of operating constraints. By successfully connecting the RDPSs, the transmission loss is significantly reduced, and the voltage profile of the system is improved to the allowable limits. In addition, by considering many different aspects of the problem, multi-objective functions are also commonly used [24, 25]. In [24], the loss saving and the voltage drop of each voltage bus have been carefully considered as two components in the objective function. The siting and sizing of the RDPSs were suitably determined by using a genetic algorithm (GA). Similarly, in the study [25], the appropriate installation of RDGs is successfully found on distribution systems by applying biogeography-based optimization (BBO). Thanks to this proper installation, not only the power loss has been significantly reduced, but the bus voltage improvement and harmonic mitigations have also changed dramatically in the positive direction. In addition, other metaheuristic algorithms were also implemented for the optimal placement of PVUs in distribution systems, including the symbiotic organism search algorithm (SOSA) and improved SOSA (ISOSA) [1], the quasi-oppositional swine influenza algorithm (QOSIA) [2], the improved artificial bee colony algorithm (IABCA) [10], the OPFAA [14], the PSO and AA-PSO [17], teaching and learning algorithm (TLA), modified version of TLA (MTLA) [26], Krill herd algorithm (KHA) [27], Ant lion algorithm (ALA) [28], Grasshopper Optimization Algorithm (GOA) [29], Sin-Cos algorithm (SCA) [30], student psychology-based optimization (SPBO) [31], stochastic fractal search (SFS), chaotic stochastic fractal search (CSFS) [32], particle swarm optimization (PSO) [33], modified teaching learning based optimization (MTLBO) [33], Jaya algorithm (JAYA) [33], grey wolf optimizer (GWO) [33], improved raven roosting optimization (IRRO) [33], salp swarm algorithm (SSA) [34], Sunflower optimization (SFO) [34], coyote optimization algorithm (COA) [34], enhanced coyote optimization algorithm (ECOA) [34], sine cosine algorithm (SCA) [35], improve sine cosine algorithm ISCA [35], bat optimization algorithm (BA) [36], cuckoo search algorithm (CSA) [37], moth-flame optimization (MFO) [38], dragonfly algorithm (DA) [38], equilibrium optimizer (EO) [39], modified equilibrium optimizer (MEO) [39], adaptive equilibrium optimizer (AEO) [39], and improved equilibrium optimizer (IEO) [39]. Other applications of PVUs have a smaller number of research studies.The second application of PVUs is to combine PVUs with energy storage systems (ESSs) [40–42] The crowd search optimization (CSO) algorithm was applied to optimize the installation of PVUs and ESSs to cut power loss and enhance voltage stability in RDPSs [40]. In the study [41], a real power system in Algeria was supplied by conventional power plants PVUs, and ESSs whose capacities had to be determined within known locations. The artificial bee colony algorithm (ABC) has selected the location and size of PVUs and ESSs for a real distribution system, and it could save on energy costs by 24% and reduce loss costs by 20%.In the third application of PVUs, PVUs are integrated with wind power plants (WPs) and other power sources in the existing power grid to reduce the loss and cost of conventional power plants available in the gird. PVUs and WPs were installed in a small existing power system [43]. PVUs, WPs, and fuel cell units were combined to supply electricity to a load [44]. In [45], PVUs were combined with WPs, diesel generators, and batteries to be the main source for a predetermined load. In the fourth application of PVUs, the configuration of RDPSs was modified, and then PVUs, together with other power sources, were optimally placed for reducing loss and cost and improving the voltage profile of loads. In the study [46], PVUs and ESSs were placed in RDPSs after performing the configuration for the distribution systems. So, the optimal placement of PVUs and ESSs was implemented after determining the best configuration of the system. If the optimal placement of PVUs and ESSs was not implemented before the configuration, power loss would not be reduced effectively. The study [47] has applied the same direction of research as the study [46] by applying reconfiguration and optimizing the placement of PVUs and WPs instead of PVUs and ESSs.In the fourth application, PVUs are placed in microgrids with low voltage and a small demand for loads. The installed PVUs are used as the sole power source supplying full power to all loads, or they are a major power source together with the conventional power sources from thermal power plants. In addition, the energy storage system is also integrated with the added PVUs. In the study [48], the conventional distribution networks were transformed into resilient autonomous microgrids by the placement of PVUs. In the study [49], the reliability of microgrids with the integration of PVUs was evaluated, and different microgrid configurations were tried for the case of energy shortage. In the study [50], the microgrids were totally supplied by PVUs, and the benefits of the placement of PVUs in the grids were evaluated. The study [51] optimized the capacity configuration for a photovoltaic system and battery-basedmicro-grid. The location and capacity of PVUs were optimized for increasing the efficiency of the micro-grid, and the micro-grid was integrated into a conventional distribution network. In the study [16], PVUs with active and reactive generation in island microgrids were optimized in terms of location and capacity. Load demand and solar radiation were considered uncertain, and the uncertainty of generation was taken into account in the micro-grids.The fifth application of PVUs in the distribution system is also a hot topic in the current trend of developing electric vehicles and reducing the use of fossil fuel-based vehicles. PVUs are placed in distribution networks to generate electricity for existing loads and added electric vehicle charging stations (EVCSs). Grasshopper Optimization Algorithm (GOA) [52] and bear smell optimization algorithm (BSOA) [53] were proposed for optimizing the placement of PVUs, shunt capacitors (SCs), and EVCSs. The same strategy with two phases was applied in the two studies for achieving the reduction of load and enhancement of voltage. In the first phase, the location and capacity of PVUs and SCs were determined for improving the power factor of substations, minimizing power loss, and improving load voltage in distribution networks. In the second phase, the location of EVCS and the number of electric vehicles at each station were optimized with the purpose of reaching the highest number of electric vehicles at all stations. In the study [54], a multiobjective optimization problem simultaneously considering electric vehicle user loss, power loss of distribution, cost of charging stations, and load voltage was solved by using the nondominated sorting genetic algorithm II (NSGA-II). NSGA-II was in charge of finding the location and capacity of both PVUs and charging stations. The study [55] also solved a multiobjective optimization problem in distribution networks with the presence of PVUs and EVCSs; however, the study focused on both economic and technical issues such as power loss, voltage deviation, and voltage stability. Teaching-Learning-Based Optimization (TLBO) and Harries Hawk Optimization (HHO) were applied for two IEEE systems with 33 and 69 nodes, respectively. HHO was stated to be more effective than TLBO to the two systems. In the two studies [55, 56], PVUs were integrated into power networks where fast charging stations were located. The study supposed that a number of EVCSs was given and their capacity was respectively 5, 25, 50, and 100 MW. For each case of total capacity, the optimal site and size of PVUs were determined, and the power loss on all lines was calculated. The study placed PVUs in the network with 24 nodes and then used the Newton—Raphson method to calculate power loss and voltage of loads for evaluating the effectiveness and stability of the networks.In this paper, the maximum penetration of PVUs is considered in radial distribution power systems to calculate the highest possibility that PVUs can supply electricity to the system with and without the integrated generation of the power source in the base network. The main target of the paper is to use the highest capacity of PVUs in RDPSs and reach the most suitable power loss while satisfying operation conditions for other components, such as loads and branches. In the first study case, the main objective is to minimize the power losses on all distribution branches while the capacity of PVUs is not limited and the number of PVUs is given. In the second study case, the maximum penetration of PVUs is considered, and the number of PVUs is an optimal parameter that needs to be determined. In the two study cases, PVUs are divided into two different types: the first type has pure active power generation, and the second type has both active and reactive power generation. To reach the best solutions for the maximum penetration of PVUs in RDPSs with the smallest total power losses, we suggest using a novel metaheuristic algorithm named the Coot optimization algorithm (COOA), which was published in early 2021 [57]. The novel algorithm was compared to previous methods in terms of searching the global optimum and reaching the most stabilization. In addition, the novel algorithm was also implemented for eight other engineering design problems, and it was stated to be more effective than over thirty other algorithms. Recently, COOA has been applied for engineering optimization problems and it has obtained better results than other compared algorithms. In the study [58], the applications of COOA for improving the performance of photovoltaic arrays considering various operation and load cases were presented. COOA has successfully found optimal uncertain parameters for well-known photovoltaic arrays, and the obtained results have been equal to or better than those from previous studies using different metaheuristic algorithms. In the study [59], COOA was run to find optimal parameters for batteries based on lithium and ion. The major advantages of the found parameters were that they were highly accurate and highly efficient, with 99.25%, while those obtained by other algorithms suffered from poorer achievements. In the study [60], COOA was applied for the design of frequency-selective surfaces for 5G cellular communication networks. Obtained results proved that frequency-selective surfaces had high performance, such as maximization of realized gain, minimization of reflection coefficients, and the highest efficiency. In the study [61], COOA was applied to wireless sensor networks to overcome uneven distribution and low coverage. COOA was compared to six other metaheuristic algorithms and proved to be faster at convergent good solutions with a higher coverage rate. Thus, in this paper, we use COOA as an optimization tool to solve the placement of PVUs in RDPSs for reaching the highest capacity of PVUs and the lowest total power losses with different considered conditions. In this paper, COOA is implemented for two RDPSs with 33 and 69 nodes, respectively. In addition, we also apply other new algorithms, including the Archimedes optimization algorithm (AOA) [62], the transient search optimization algorithm (TSOA) [63], the crystal structure algorithm (CrSA) [64], the war strategy optimization algorithm (WSA) [65], and the average and subtraction-based optimizer (ASBO) [66] for the comparisons with COOA. The results, including power loss and capacity of PVUs, are compared to those from five implemented and other previous methods. The fact is that COOA is an effective optimization algorithm for the problem of determining the optimal parameters of PVUs in RDPSs. In summary, the contributions of the paper are as follows:(1)
Successfully apply six new algorithms, including COOA, AOA, TSOA, CrSA, WSA, and ASBO, to find the most suitable solutions for the optimal placement of PVUs in RDPSs.(2)
COOA can reach the smallest total power losses and the maximum penetration of PVUs for applied standard RDPSs. COOA is either as good as or much better than other applied ones.(3)
Prove the penetration-level impact of PVUs on loss reduction and the voltage profile of RDPSs.In addition to the section, the organization of other parts of the article is as follows: Section2 presents the objective and constraints of the problem. Section 3 describes the structure and shows mathematical formulations of COOA. Section 4 shows the implementation of COOA for PVUs determination in RDPSs. Section 5 shows results, comparisons, and discussions. Section 6 summarizes achievements and conclusions.


## 2. Problem Formulation

### 2.1. Description of the Considered Problem

Standard radial distribution systems are the connection between power sources and load through conductors, called distribution lines. Distribution lines have two major parameters, including resistance and reactance. Resistance and reactance, respectively, cause active power loss and reactive power loss on the distribution line; meanwhile, the two factors have a high impact on the voltage drop. The power losses and voltage drop can be obtained by using the following formulas:(1)PTLoss,k=3×Iop,k2×Rk,QTLoss,k=3×Iop,k2×Xk,VLoss,k=3×Iop,k×Rk×CosPFk+Xk×SinPFk,where PTLoss,k and QTLoss,k are active and reactive power losses on the kth distribution line; VTLoss,k is voltage drop on the kth distribution line; Iop,k is operating current of the kth distribution line; Rk and Xk are resistance and reactance of the kth distribution line; and PFk is power factor of the kth distribution line.The three equations above indicate that power loss and voltage drop are directly proportional to the operating current on thekth distribution line. If the line loads are high, the operating current will be high. As a result, power losses and voltage drops are high, unintentionally. To reduce the high power losses and the high voltage drop, the placement of PVUs is considered in the paper. PVUs are supposed to be pure active power generation sources or both active and reactive power generation sources. For the two cases, PVUs can supply a part of the demand or the full demand to loads, reducing the operating current on the kth distribution line. If the operating current of the kth distribution line after connecting PVUs at loads is Iop,kafter, it will certainly be smaller than Iop,k. Thus, active power loss, reactive power loss, and voltage drop become smaller, and PVUs are very useful in cutting power losses and improving voltage profiles.


### 2.2. Objective Function

Almost all previous studies have considered active power loss reduction the most important target when installing PVUs in RDPSs [39]. Because of the long length and high resistance of distribution branches, the active power loss is significant, and it should be reduced as much as possible. So, the main objective of the paper is as follows:(2)CuttingPTLoss=3×∑k=1NobrIop,k2×Rk,where PTLoss is the total power losses in all conductors; and Nobr is the number of distribution branches in the considered RDPSs.


### 2.3. Constraints

Constraints considered in the problem of installing PVUs in RDPSs are related to stable operation conditions of electric components of the girds, including conductors and loads. In addition, there are other constraints regarding PVUs, including installation position, rated power for each photovoltaic unit (PVU), and total power of all installed PVUs. The mentioned constraints are from electric components, while other constraints in connection with the power source, branches, and PVUs are the power balance equations. The details of the constraints are expressed as follows:


#### 2.3.1. Conductor Current Limits

Conductors are used to connect every two nodes, and they become distribution branches in distribution power grids [67]. The major parameter of the conductors is the maximum current, which must be equal to or higher than the operating current of the distribution branches. The difference between the grids with and without PVUs is the power sources and the current flowing on branches; however, the same constraint on the conductors for two study cases is as follows:(3)Ikmax≥Iop,k,k=1,…,Nobr.


#### 2.3.2. Load Voltage Limits

Electric loads need a supply of active and reactive power within an allowable voltage range. If the distribution branches supply power to loads beyond the operation range limit, the loads must stop working, and the power company must compensate for this disadvantage for customers. Thus, the voltage must be within the allowable operating range. According to the BS EN 50160 standard [68], the allowable limit of load voltage has to be between 0.9 and 1.1 pU. However, another study [69] has stated that the range is large and 0.9 pu is not good enough for loads. So, a better limit for load voltage should be from 0.95 pu to 1.05 pu. Therefore, it is essential to keep load voltage within an allowable range, as shown by the following inequality:(4)Vmin≤Vload,l≤Vmax,s=1,…,Nons.


#### 2.3.3. Constraints of PVUs

As for installing PVUs in RDPSs, some factors that need to be determined are the location of each PVU, the power limit of each PVU, and the total power of all installed PVUs in the RDPSs. Normally, the base grids have only one power source at node 1. So, all PVUs cannot be installed at the node, and the installation location must satisfy the following constraint [70]:(5)2≤LPVU,m≤Nons.In addition, the rated power of each PVU must follow the allowable limit below [71]:(6)PPVUmin≤PPVU,m≤PPVUmax;m=1,…,NoPVUs.As considering the maximum penetration of PVUs in RDPSs, the total power of all PVUs can be maximum and equal to the sum of all loads and losses as follows:(7)∑m=1NoPVUsPPVU,m≤∑l=1NonsPload,l+PTLoss.When power companies require the supply of reactive power from PVUs, the power factor of PVUs must be considered, and the reactive power of all PVUs is also taken into consideration. Limits of power factor and reactive power are as follows [72]:(8)PFmin≤PFPVU,m≤PFmax;m=1,…,NoPVUs,(9)∑m=1NoPVUsQPVU,m≤∑l=1NonsQload,l+∑i=1NobrQTLoss.In the equation (9), QPVU,m is the reactive power generation of the mth PVU and QTLoss is the total reactive power losses in all conductors. The two parameters QPVU,m and QTLoss can be determined as follows:(10)QPVU,m=tanarccosPFPVU,m.PPVU,m,QTLoss=3∑k=1NobrIop,k2.Xk.


#### 2.3.4. Power Balance Equations

Distribution power grids have two sides, the consumption side and the supply side, where the consumption side is comprised of demand for loads and losses. In contrast, the supply side is the generation of conventional power sources, and the added PVUs. The balance between the demand side and supply side is expressed by the two following equations [73]:(11)Pps+∑m=1NoPVUsPPVU,m−∑l=1NonsPload,l−PTLoss=0,Qps+∑m=1NoPVUsQPVU,m−∑l=1NonsQload,l−QTLoss=0.


## 2.1. Description of the Considered Problem

Standard radial distribution systems are the connection between power sources and load through conductors, called distribution lines. Distribution lines have two major parameters, including resistance and reactance. Resistance and reactance, respectively, cause active power loss and reactive power loss on the distribution line; meanwhile, the two factors have a high impact on the voltage drop. The power losses and voltage drop can be obtained by using the following formulas:(1)PTLoss,k=3×Iop,k2×Rk,QTLoss,k=3×Iop,k2×Xk,VLoss,k=3×Iop,k×Rk×CosPFk+Xk×SinPFk,where PTLoss,k and QTLoss,k are active and reactive power losses on the kth distribution line; VTLoss,k is voltage drop on the kth distribution line; Iop,k is operating current of the kth distribution line; Rk and Xk are resistance and reactance of the kth distribution line; and PFk is power factor of the kth distribution line.The three equations above indicate that power loss and voltage drop are directly proportional to the operating current on thekth distribution line. If the line loads are high, the operating current will be high. As a result, power losses and voltage drops are high, unintentionally. To reduce the high power losses and the high voltage drop, the placement of PVUs is considered in the paper. PVUs are supposed to be pure active power generation sources or both active and reactive power generation sources. For the two cases, PVUs can supply a part of the demand or the full demand to loads, reducing the operating current on the kth distribution line. If the operating current of the kth distribution line after connecting PVUs at loads is Iop,kafter, it will certainly be smaller than Iop,k. Thus, active power loss, reactive power loss, and voltage drop become smaller, and PVUs are very useful in cutting power losses and improving voltage profiles.


## 2.2. Objective Function

Almost all previous studies have considered active power loss reduction the most important target when installing PVUs in RDPSs [39]. Because of the long length and high resistance of distribution branches, the active power loss is significant, and it should be reduced as much as possible. So, the main objective of the paper is as follows:(2)CuttingPTLoss=3×∑k=1NobrIop,k2×Rk,where PTLoss is the total power losses in all conductors; and Nobr is the number of distribution branches in the considered RDPSs.


## 2.3. Constraints

Constraints considered in the problem of installing PVUs in RDPSs are related to stable operation conditions of electric components of the girds, including conductors and loads. In addition, there are other constraints regarding PVUs, including installation position, rated power for each photovoltaic unit (PVU), and total power of all installed PVUs. The mentioned constraints are from electric components, while other constraints in connection with the power source, branches, and PVUs are the power balance equations. The details of the constraints are expressed as follows:


### 2.3.1. Conductor Current Limits

Conductors are used to connect every two nodes, and they become distribution branches in distribution power grids [67]. The major parameter of the conductors is the maximum current, which must be equal to or higher than the operating current of the distribution branches. The difference between the grids with and without PVUs is the power sources and the current flowing on branches; however, the same constraint on the conductors for two study cases is as follows:(3)Ikmax≥Iop,k,k=1,…,Nobr.


### 2.3.2. Load Voltage Limits

Electric loads need a supply of active and reactive power within an allowable voltage range. If the distribution branches supply power to loads beyond the operation range limit, the loads must stop working, and the power company must compensate for this disadvantage for customers. Thus, the voltage must be within the allowable operating range. According to the BS EN 50160 standard [68], the allowable limit of load voltage has to be between 0.9 and 1.1 pU. However, another study [69] has stated that the range is large and 0.9 pu is not good enough for loads. So, a better limit for load voltage should be from 0.95 pu to 1.05 pu. Therefore, it is essential to keep load voltage within an allowable range, as shown by the following inequality:(4)Vmin≤Vload,l≤Vmax,s=1,…,Nons.


### 2.3.3. Constraints of PVUs

As for installing PVUs in RDPSs, some factors that need to be determined are the location of each PVU, the power limit of each PVU, and the total power of all installed PVUs in the RDPSs. Normally, the base grids have only one power source at node 1. So, all PVUs cannot be installed at the node, and the installation location must satisfy the following constraint [70]:(5)2≤LPVU,m≤Nons.In addition, the rated power of each PVU must follow the allowable limit below [71]:(6)PPVUmin≤PPVU,m≤PPVUmax;m=1,…,NoPVUs.As considering the maximum penetration of PVUs in RDPSs, the total power of all PVUs can be maximum and equal to the sum of all loads and losses as follows:(7)∑m=1NoPVUsPPVU,m≤∑l=1NonsPload,l+PTLoss.When power companies require the supply of reactive power from PVUs, the power factor of PVUs must be considered, and the reactive power of all PVUs is also taken into consideration. Limits of power factor and reactive power are as follows [72]:(8)PFmin≤PFPVU,m≤PFmax;m=1,…,NoPVUs,(9)∑m=1NoPVUsQPVU,m≤∑l=1NonsQload,l+∑i=1NobrQTLoss.In the equation (9), QPVU,m is the reactive power generation of the mth PVU and QTLoss is the total reactive power losses in all conductors. The two parameters QPVU,m and QTLoss can be determined as follows:(10)QPVU,m=tanarccosPFPVU,m.PPVU,m,QTLoss=3∑k=1NobrIop,k2.Xk.


### 2.3.4. Power Balance Equations

Distribution power grids have two sides, the consumption side and the supply side, where the consumption side is comprised of demand for loads and losses. In contrast, the supply side is the generation of conventional power sources, and the added PVUs. The balance between the demand side and supply side is expressed by the two following equations [73]:(11)Pps+∑m=1NoPVUsPPVU,m−∑l=1NonsPload,l−PTLoss=0,Qps+∑m=1NoPVUsQPVU,m−∑l=1NonsQload,l−QTLoss=0.


## 2.3.1. Conductor Current Limits

Conductors are used to connect every two nodes, and they become distribution branches in distribution power grids [67]. The major parameter of the conductors is the maximum current, which must be equal to or higher than the operating current of the distribution branches. The difference between the grids with and without PVUs is the power sources and the current flowing on branches; however, the same constraint on the conductors for two study cases is as follows:(3)Ikmax≥Iop,k,k=1,…,Nobr.


## 2.3.2. Load Voltage Limits

Electric loads need a supply of active and reactive power within an allowable voltage range. If the distribution branches supply power to loads beyond the operation range limit, the loads must stop working, and the power company must compensate for this disadvantage for customers. Thus, the voltage must be within the allowable operating range. According to the BS EN 50160 standard [68], the allowable limit of load voltage has to be between 0.9 and 1.1 pU. However, another study [69] has stated that the range is large and 0.9 pu is not good enough for loads. So, a better limit for load voltage should be from 0.95 pu to 1.05 pu. Therefore, it is essential to keep load voltage within an allowable range, as shown by the following inequality:(4)Vmin≤Vload,l≤Vmax,s=1,…,Nons.


## 2.3.3. Constraints of PVUs

As for installing PVUs in RDPSs, some factors that need to be determined are the location of each PVU, the power limit of each PVU, and the total power of all installed PVUs in the RDPSs. Normally, the base grids have only one power source at node 1. So, all PVUs cannot be installed at the node, and the installation location must satisfy the following constraint [70]:(5)2≤LPVU,m≤Nons.In addition, the rated power of each PVU must follow the allowable limit below [71]:(6)PPVUmin≤PPVU,m≤PPVUmax;m=1,…,NoPVUs.As considering the maximum penetration of PVUs in RDPSs, the total power of all PVUs can be maximum and equal to the sum of all loads and losses as follows:(7)∑m=1NoPVUsPPVU,m≤∑l=1NonsPload,l+PTLoss.When power companies require the supply of reactive power from PVUs, the power factor of PVUs must be considered, and the reactive power of all PVUs is also taken into consideration. Limits of power factor and reactive power are as follows [72]:(8)PFmin≤PFPVU,m≤PFmax;m=1,…,NoPVUs,(9)∑m=1NoPVUsQPVU,m≤∑l=1NonsQload,l+∑i=1NobrQTLoss.In the equation (9), QPVU,m is the reactive power generation of the mth PVU and QTLoss is the total reactive power losses in all conductors. The two parameters QPVU,m and QTLoss can be determined as follows:(10)QPVU,m=tanarccosPFPVU,m.PPVU,m,QTLoss=3∑k=1NobrIop,k2.Xk.


## 2.3.4. Power Balance Equations

Distribution power grids have two sides, the consumption side and the supply side, where the consumption side is comprised of demand for loads and losses. In contrast, the supply side is the generation of conventional power sources, and the added PVUs. The balance between the demand side and supply side is expressed by the two following equations [73]:(11)Pps+∑m=1NoPVUsPPVU,m−∑l=1NonsPload,l−PTLoss=0,Qps+∑m=1NoPVUsQPVU,m−∑l=1NonsQload,l−QTLoss=0.


## 3. Coot Optimization Algorithm

The main inspiration for forming the Coot optimization algorithm (COOA) came from the simulation of coots’ movement on the water surface toward the food supply area. The simulation consists of two main phases: the chaotic phase and the synchronous phase. In the first phase, coots in the flock are free to move around the search space. In the second phase, coots move in the same orientation, and they have to regulate their moving behavior by following leaders. Finally, the leaders also continuously regulate their direction in order to get the best way to optimal areas with abundant food sources and more comfortable living conditions. Similar to the other state-of-the-art metaheuristic algorithms, the COOA also requires a specific value of population at the beginning of the optimization process for the random generation of the initial population. Then, COOA implements the update for the whole population. The feature that distinguishes COOA from other metaheuristic algorithms is the update mechanism. Furthermore, the selection technique of COOA is also different from other methods. In the following sections, we present the main steps to execute the optimization process of COOA.


### 3.1. The Initialization

As mentioned already, the COOA needs the initial population that is obtained by the following equation:(12)Xi=rand1,D.Xmax−Xmin+Xmin;i=1,…,NoP.COOA has the same idea as the Coyote optimization algorithm (COA) [74]: to divide the population into members and leaders. In COA, the whole population is divided into some groups, and there is one leader for each group. On the contrary, COOA only separates the whole population into leaders and members. The quantity of leaders is LC, and the quantity of members is MQ. The two factors are determined by the following equation:(13)LC=a.NoP,(14)MQ=NoP−LC.In equation (13), a is a part of the population, and it can select within the range of [0; 1.0]. However, when setting the parameter to a high value close to 1, the leader quantity is high and the member quantity is low. This may lead to an inefficient search process. By experiment, it should be tried within the range [0; 0.2], and LC must be an integer value.After producing the initial coot swarm, the fitness function is determined to separate the swarm into a leader group with LC leaders and a member group with MQ members. The set ofX = [Xi] in equation (12) has the fitness set Fit = [Fiti]. Based on the rank of Fiti values, coots with better fitness values are gathered in the leader group, and other coots are put in the member group. Each member and its fitness function in the leader group are LPj and FitLj, where j = 1, …, LC. Similarly, each member and its fitness function in the member group are Xi and Fiti where i = 1, …, MQ.


### 3.2. New Solution Update

COOA produces new solutions corresponding to the updated positions of coots in two different groups for members and leaders. The positions of members are updated plentifully by using three different ways, while the positions of leaders are newly searched in two different ways. COOA uses several parameters as scaling factors in mathematical equations for updating new positions. These parameters are comprised ofW1 and W2. In addition, conditions for selecting the update method are also established by comparing random numbers RN1 and RN2 (which are randomized within 0 and 1) with RP1 and RP2. Among the parameters, RP1 and RP2 are set to 0.5 for all cases, while W1 and W2 have two different cases depending on the result of the comparison of RN1 with RP1 and RN2 with RP2. In the case of RN1 < RP1, W1 has a random number between 1.0 and 0.0. For other cases, i.e., RN1 ≥ RP1, W1 is a vector with the same size as D, and each term in the vector is randomized in the range between 1.0 and 0.0. The determination of W2 is performed similarly. The detail of the position update is expressed as follows:


#### 3.2.1. Solution Update for Members

Each coot will update their location for the next movement in this stage. There are three main methods to find the new location for each coot: the chaotic method, the middle method, and the leader method. The decision of selecting the most appropriate update method will be executed based on the comparison betweenRN1 (a random value generated in the interval [0, 1]) and the reference parameter RP1. The three applied methods are presented in detail as follows:(1) Chaotic Method. This update method is utilized if the two conditions RN1≤RP1 and i≠1 happen simultaneously. In the second condition, i is the order of the considered coots in the population. According to the chaotic method, each individual in the population is free to move around the search space. Therefore, their locations at a specific period are arbitrary, and they do not suffer any restriction except for the boundaries of search space. The determination for the new location is expressed as follows:(15)Xinew=Xi+H.W1.K−Xi.In the equation,Xinew is the new location of coot individual i and Xi is the previous location of coot individual. K and H are, respectively, stochastic and amplifying terms calculated by the following equation:(16)K=rand1,D.∗Xmax−Xmin+Xmin,H=1−It×1Itmax.(2) The Middle Method: the middle method will be utilized while the two conditions including RN1≤RP1 and i=1 are correct simultaneously. The considered coot will move to the central point of its own previous location and the previous location of another coot nearby. This behavior can be seen in real life when an arbitrary coot follows another coot but then it stops moving. The mathematical expression of this update method is as follows:(17)Xinew=0.5×Xi−1+Xi,where Xi−1 is the location of the (i−1) th Coot individual nearby the ith coot. In the viewpoint of executing optimization process, this mechanism is represented for self-update location belonging each solution toward the optimal point.(3) Leader Method. the leader method is utilized if the condition RN1>RP1 happens. As mentioned above, the movement of coots can refer to that of leaders. Thus, the leader method is also utilized in addition to the chaotic and middle methods. Therefore, Coot individuals must pay attention to the leader group. However, it is hard to follow the movement of all the leaders because each has a different location from the other remaining leaders. So, a leader with index SL in the leader group is selected by the following equation:(18)SL=1+Modi,LC.In equation (18), i is the coot index in the population. After selecting the SLth leader, the coot will update its new location based on the following equation:(19)Xinew=XSL+2.W1.cos2.r.π.XSL−Xi.In equation (19), XSL is the position of the SLth leader and r is a random number within −1 and 1.


#### 3.2.2. Solution Update for Leaders

On the journey to reach the optimal site, the leaders also need to update their own locations. The location update mechanism for leaders is mainly affected by the global best location (PGBest). This mechanism is clearly presented by using the two following equations:(20)LPjnew=J.W2.cos2.r.π.PGBest−LPj+PGBestifRN2<RP2,LPjnew=J.W2.cos2.r.π.PGBest−LPj−PGBestifRN2≥RP2.In the two equations above,LPjnew is the new location of the leader j. LPj is the previous location of the leader j where j = [1, …, LC]. PGBest is the best position found in the whole population at current iteration. J is the joint operator, determined as follows:(21)J=2−It×1Itmax.


### 3.3. Selection of Leaders and the Best Leader

COOA performs the site update for coot members first, and then the update of site for leaders is executed. After updating the position for each coot member, the evaluation for the new position is implemented, and the new position of the coot is compared with that of previous leaders. If the new position is more effective than the position of leaders, the coot is assigned to a new leader, and the old leader becomes the coot member. The first loop computation process continues to be executed until the last member (the MQth particle) is newly updated. Then the second loop computation process is carried out for leaders. Each leader updates the site, and then the new site is evaluated to compare with the best leader. If the new position has more potential than that of the best coot, they exchange roles. It means the leader becomes the best leader, and the best old leader becomes the leader. The second loop process is stopped when the last leader (the LCth leader) finishes the update of the site.


### 3.4. The Iterative Algorithm of COOA

The iterative algorithm of COOA for solving a general optimization problem is described in Figure1.Figure 1
Flowchart of applying COOA for an optimization problem.


## 3.1. The Initialization

As mentioned already, the COOA needs the initial population that is obtained by the following equation:(12)Xi=rand1,D.Xmax−Xmin+Xmin;i=1,…,NoP.COOA has the same idea as the Coyote optimization algorithm (COA) [74]: to divide the population into members and leaders. In COA, the whole population is divided into some groups, and there is one leader for each group. On the contrary, COOA only separates the whole population into leaders and members. The quantity of leaders is LC, and the quantity of members is MQ. The two factors are determined by the following equation:(13)LC=a.NoP,(14)MQ=NoP−LC.In equation (13), a is a part of the population, and it can select within the range of [0; 1.0]. However, when setting the parameter to a high value close to 1, the leader quantity is high and the member quantity is low. This may lead to an inefficient search process. By experiment, it should be tried within the range [0; 0.2], and LC must be an integer value.After producing the initial coot swarm, the fitness function is determined to separate the swarm into a leader group with LC leaders and a member group with MQ members. The set ofX = [Xi] in equation (12) has the fitness set Fit = [Fiti]. Based on the rank of Fiti values, coots with better fitness values are gathered in the leader group, and other coots are put in the member group. Each member and its fitness function in the leader group are LPj and FitLj, where j = 1, …, LC. Similarly, each member and its fitness function in the member group are Xi and Fiti where i = 1, …, MQ.


## 3.2. New Solution Update

COOA produces new solutions corresponding to the updated positions of coots in two different groups for members and leaders. The positions of members are updated plentifully by using three different ways, while the positions of leaders are newly searched in two different ways. COOA uses several parameters as scaling factors in mathematical equations for updating new positions. These parameters are comprised ofW1 and W2. In addition, conditions for selecting the update method are also established by comparing random numbers RN1 and RN2 (which are randomized within 0 and 1) with RP1 and RP2. Among the parameters, RP1 and RP2 are set to 0.5 for all cases, while W1 and W2 have two different cases depending on the result of the comparison of RN1 with RP1 and RN2 with RP2. In the case of RN1 < RP1, W1 has a random number between 1.0 and 0.0. For other cases, i.e., RN1 ≥ RP1, W1 is a vector with the same size as D, and each term in the vector is randomized in the range between 1.0 and 0.0. The determination of W2 is performed similarly. The detail of the position update is expressed as follows:


### 3.2.1. Solution Update for Members

Each coot will update their location for the next movement in this stage. There are three main methods to find the new location for each coot: the chaotic method, the middle method, and the leader method. The decision of selecting the most appropriate update method will be executed based on the comparison betweenRN1 (a random value generated in the interval [0, 1]) and the reference parameter RP1. The three applied methods are presented in detail as follows:(1) Chaotic Method. This update method is utilized if the two conditions RN1≤RP1 and i≠1 happen simultaneously. In the second condition, i is the order of the considered coots in the population. According to the chaotic method, each individual in the population is free to move around the search space. Therefore, their locations at a specific period are arbitrary, and they do not suffer any restriction except for the boundaries of search space. The determination for the new location is expressed as follows:(15)Xinew=Xi+H.W1.K−Xi.In the equation,Xinew is the new location of coot individual i and Xi is the previous location of coot individual. K and H are, respectively, stochastic and amplifying terms calculated by the following equation:(16)K=rand1,D.∗Xmax−Xmin+Xmin,H=1−It×1Itmax.(2) The Middle Method: the middle method will be utilized while the two conditions including RN1≤RP1 and i=1 are correct simultaneously. The considered coot will move to the central point of its own previous location and the previous location of another coot nearby. This behavior can be seen in real life when an arbitrary coot follows another coot but then it stops moving. The mathematical expression of this update method is as follows:(17)Xinew=0.5×Xi−1+Xi,where Xi−1 is the location of the (i−1) th Coot individual nearby the ith coot. In the viewpoint of executing optimization process, this mechanism is represented for self-update location belonging each solution toward the optimal point.(3) Leader Method. the leader method is utilized if the condition RN1>RP1 happens. As mentioned above, the movement of coots can refer to that of leaders. Thus, the leader method is also utilized in addition to the chaotic and middle methods. Therefore, Coot individuals must pay attention to the leader group. However, it is hard to follow the movement of all the leaders because each has a different location from the other remaining leaders. So, a leader with index SL in the leader group is selected by the following equation:(18)SL=1+Modi,LC.In equation (18), i is the coot index in the population. After selecting the SLth leader, the coot will update its new location based on the following equation:(19)Xinew=XSL+2.W1.cos2.r.π.XSL−Xi.In equation (19), XSL is the position of the SLth leader and r is a random number within −1 and 1.


### 3.2.2. Solution Update for Leaders

On the journey to reach the optimal site, the leaders also need to update their own locations. The location update mechanism for leaders is mainly affected by the global best location (PGBest). This mechanism is clearly presented by using the two following equations:(20)LPjnew=J.W2.cos2.r.π.PGBest−LPj+PGBestifRN2<RP2,LPjnew=J.W2.cos2.r.π.PGBest−LPj−PGBestifRN2≥RP2.In the two equations above,LPjnew is the new location of the leader j. LPj is the previous location of the leader j where j = [1, …, LC]. PGBest is the best position found in the whole population at current iteration. J is the joint operator, determined as follows:(21)J=2−It×1Itmax.


## 3.2.1. Solution Update for Members

Each coot will update their location for the next movement in this stage. There are three main methods to find the new location for each coot: the chaotic method, the middle method, and the leader method. The decision of selecting the most appropriate update method will be executed based on the comparison betweenRN1 (a random value generated in the interval [0, 1]) and the reference parameter RP1. The three applied methods are presented in detail as follows:(1) Chaotic Method. This update method is utilized if the two conditions RN1≤RP1 and i≠1 happen simultaneously. In the second condition, i is the order of the considered coots in the population. According to the chaotic method, each individual in the population is free to move around the search space. Therefore, their locations at a specific period are arbitrary, and they do not suffer any restriction except for the boundaries of search space. The determination for the new location is expressed as follows:(15)Xinew=Xi+H.W1.K−Xi.In the equation,Xinew is the new location of coot individual i and Xi is the previous location of coot individual. K and H are, respectively, stochastic and amplifying terms calculated by the following equation:(16)K=rand1,D.∗Xmax−Xmin+Xmin,H=1−It×1Itmax.(2) The Middle Method: the middle method will be utilized while the two conditions including RN1≤RP1 and i=1 are correct simultaneously. The considered coot will move to the central point of its own previous location and the previous location of another coot nearby. This behavior can be seen in real life when an arbitrary coot follows another coot but then it stops moving. The mathematical expression of this update method is as follows:(17)Xinew=0.5×Xi−1+Xi,where Xi−1 is the location of the (i−1) th Coot individual nearby the ith coot. In the viewpoint of executing optimization process, this mechanism is represented for self-update location belonging each solution toward the optimal point.(3) Leader Method. the leader method is utilized if the condition RN1>RP1 happens. As mentioned above, the movement of coots can refer to that of leaders. Thus, the leader method is also utilized in addition to the chaotic and middle methods. Therefore, Coot individuals must pay attention to the leader group. However, it is hard to follow the movement of all the leaders because each has a different location from the other remaining leaders. So, a leader with index SL in the leader group is selected by the following equation:(18)SL=1+Modi,LC.In equation (18), i is the coot index in the population. After selecting the SLth leader, the coot will update its new location based on the following equation:(19)Xinew=XSL+2.W1.cos2.r.π.XSL−Xi.In equation (19), XSL is the position of the SLth leader and r is a random number within −1 and 1.


## 3.2.2. Solution Update for Leaders

On the journey to reach the optimal site, the leaders also need to update their own locations. The location update mechanism for leaders is mainly affected by the global best location (PGBest). This mechanism is clearly presented by using the two following equations:(20)LPjnew=J.W2.cos2.r.π.PGBest−LPj+PGBestifRN2<RP2,LPjnew=J.W2.cos2.r.π.PGBest−LPj−PGBestifRN2≥RP2.In the two equations above,LPjnew is the new location of the leader j. LPj is the previous location of the leader j where j = [1, …, LC]. PGBest is the best position found in the whole population at current iteration. J is the joint operator, determined as follows:(21)J=2−It×1Itmax.


## 3.3. Selection of Leaders and the Best Leader

COOA performs the site update for coot members first, and then the update of site for leaders is executed. After updating the position for each coot member, the evaluation for the new position is implemented, and the new position of the coot is compared with that of previous leaders. If the new position is more effective than the position of leaders, the coot is assigned to a new leader, and the old leader becomes the coot member. The first loop computation process continues to be executed until the last member (the MQth particle) is newly updated. Then the second loop computation process is carried out for leaders. Each leader updates the site, and then the new site is evaluated to compare with the best leader. If the new position has more potential than that of the best coot, they exchange roles. It means the leader becomes the best leader, and the best old leader becomes the leader. The second loop process is stopped when the last leader (the LCth leader) finishes the update of the site.


## 3.4. The Iterative Algorithm of COOA

The iterative algorithm of COOA for solving a general optimization problem is described in Figure1.Figure 1
Flowchart of applying COOA for an optimization problem.


## 4. COOA Implementation of the Concerned Problem

### 4.1. Control Variable Selection

Optimal solutions of the installation of PVUs in RDPSs are comprised of the most suitable location, active power and power factor of all the installed PVUs for the lowest power loss and meeting all constraints exactly. So, each solutionXi must consider the site, size, and power factor as control variables. Each solution is represented by the following equation:(22)Xi=PPVU,m,i,PFPVU,m,i,LPVU,m,i;m=1,…,NoPVUs,where PPVU,m,i and PFPVU,m,i are active power and power factor of the mth PVU for the ith solution i; and LPVU,m,i is the node of placing the mth PVU for the ith solution.Each term in the solutionXi is always constrained by the following equation:(23)PPVUmin≤PPVU,m,i≤PPVUmax;m=1,…,NoPVUs,PFmin≤PFPVU,m,i≤PFmax;m=1,…,NoPVUs,2≤LPVU,m,i≤Nons;m=1,…,NoPVUs.


### 4.2. Control Variable Update and Correction

As shown in equation (22), the available control variables of the problem are PPVU,m,i,PFPVU,m,i and LPVU,m,i that should be newly updated in each computation iteration for improving solution quality. Section 3.2 is used to update the variables and newly obtained variables are represented as LPVU,m,inew, PPVU,m,inew, and PFPVU,m,inew. Unlike power generation PPVU,m,inew and power factor PFPVU,m,inew, location LPVU,m,inew must be an integer value, which is chosen from the lowest node (node 2) to the highest node (node Nons). So, after producing location LPVU,m,inew, it must be rounded to be an integer value first and it is then checked for correction as follows:(24)PPVU,m,inew=PPVU,m,inewifPPVUmin≤PPVU,m,inew≤PPVUmax,PPVUminifPPVUmin>PPVU,m,inew,PPVUmaxifPPVUmax<PPVU,m,inew.In the next step of correction, power generationPPVU,m,inew and power factor PFPVU,m,inew are checked for the following corrections:(25)PFPVU,m,inew=PFPVU,m,inewifPFPVUmin≤PFPVU,m,inew≤PFPVUmax,PFPVUminifPFPVUmin>PFPVU,m,inew,PFPVUmaxifPFPVUmax<PFPVU,m,inew,LPVU,m,inew=LPVU,m,inewifLPVU,m,inew∈2;Nons,NonsifNons<LPVU,m,inew,2if2>LPVU,m,inew.Currently, all the newly updated control variables are valid and can be used to find fitness-based solutions. The calculation of the fitness function is expressed in the following section.


### 4.3. Fitness Function

As all control variables are valid, they are added as input data for the forward-backward sweep technique [75]. The technique is then run for obtaining dependent variables including current Iop,k,i and voltage Vload,l,i. Current Iop,k,i is substituted into equation (2) for reaching the objective function, while both current Iop,k,i and voltage Vload,l,i are verified to calculate penalty values [25]. Finally, the fitness function of the solution i (Fiti) is determined by the following equation:(26)Fiti=PTLoss,i+β∑n=1Nobr∆Iop,k,i2+β∑l=1Nons∆Vload,l,i2,where β is constant chosen by experiment and set to 104 in the study; and ∆Iop,k,i and ∆Vload,l,i are current violation and voltage violation.


### 4.4. Application of COOA for the Considered Problem

The application of COOA for reaching optimal site, size, and power factor of PVUs by using COOA is presented in detail in Figure2.Figure 2
Flowchart of applying COOA for the studied problem.


## 4.1. Control Variable Selection

Optimal solutions of the installation of PVUs in RDPSs are comprised of the most suitable location, active power and power factor of all the installed PVUs for the lowest power loss and meeting all constraints exactly. So, each solutionXi must consider the site, size, and power factor as control variables. Each solution is represented by the following equation:(22)Xi=PPVU,m,i,PFPVU,m,i,LPVU,m,i;m=1,…,NoPVUs,where PPVU,m,i and PFPVU,m,i are active power and power factor of the mth PVU for the ith solution i; and LPVU,m,i is the node of placing the mth PVU for the ith solution.Each term in the solutionXi is always constrained by the following equation:(23)PPVUmin≤PPVU,m,i≤PPVUmax;m=1,…,NoPVUs,PFmin≤PFPVU,m,i≤PFmax;m=1,…,NoPVUs,2≤LPVU,m,i≤Nons;m=1,…,NoPVUs.


## 4.2. Control Variable Update and Correction

As shown in equation (22), the available control variables of the problem are PPVU,m,i,PFPVU,m,i and LPVU,m,i that should be newly updated in each computation iteration for improving solution quality. Section 3.2 is used to update the variables and newly obtained variables are represented as LPVU,m,inew, PPVU,m,inew, and PFPVU,m,inew. Unlike power generation PPVU,m,inew and power factor PFPVU,m,inew, location LPVU,m,inew must be an integer value, which is chosen from the lowest node (node 2) to the highest node (node Nons). So, after producing location LPVU,m,inew, it must be rounded to be an integer value first and it is then checked for correction as follows:(24)PPVU,m,inew=PPVU,m,inewifPPVUmin≤PPVU,m,inew≤PPVUmax,PPVUminifPPVUmin>PPVU,m,inew,PPVUmaxifPPVUmax<PPVU,m,inew.In the next step of correction, power generationPPVU,m,inew and power factor PFPVU,m,inew are checked for the following corrections:(25)PFPVU,m,inew=PFPVU,m,inewifPFPVUmin≤PFPVU,m,inew≤PFPVUmax,PFPVUminifPFPVUmin>PFPVU,m,inew,PFPVUmaxifPFPVUmax<PFPVU,m,inew,LPVU,m,inew=LPVU,m,inewifLPVU,m,inew∈2;Nons,NonsifNons<LPVU,m,inew,2if2>LPVU,m,inew.Currently, all the newly updated control variables are valid and can be used to find fitness-based solutions. The calculation of the fitness function is expressed in the following section.


## 4.3. Fitness Function

As all control variables are valid, they are added as input data for the forward-backward sweep technique [75]. The technique is then run for obtaining dependent variables including current Iop,k,i and voltage Vload,l,i. Current Iop,k,i is substituted into equation (2) for reaching the objective function, while both current Iop,k,i and voltage Vload,l,i are verified to calculate penalty values [25]. Finally, the fitness function of the solution i (Fiti) is determined by the following equation:(26)Fiti=PTLoss,i+β∑n=1Nobr∆Iop,k,i2+β∑l=1Nons∆Vload,l,i2,where β is constant chosen by experiment and set to 104 in the study; and ∆Iop,k,i and ∆Vload,l,i are current violation and voltage violation.


## 4.4. Application of COOA for the Considered Problem

The application of COOA for reaching optimal site, size, and power factor of PVUs by using COOA is presented in detail in Figure2.Figure 2
Flowchart of applying COOA for the studied problem.


## 5. Simulation Results

In this section, six applied algorithms, including COOA [57], AOA [62], TSOA [63], and CrSA [64], WSA [65], and ASBO [66], are tested on two RDPS with 33 and 69 for four study cases as follows:Case 1: The number of PVUs is predetermined to be three, and the power factor is supposed to be 1.0. Applied algorithms are in charge of finding the locations and active powers of the three PVUs.Case 2: The number of PVUs is predetermined to be three, but the power factor is not fixed at 1.0. Applied algorithms must find the optimal locations, active power, and power factor of three PVUs.Case 3: The number of PVUs is determined optimally while the power factor is fixed at 1.0. Applied algorithms are run to get the optimal number of PVUs, optimal locations, and active power of PVUs.Case 4: The number of PVUs and the power factor of PVUs are not predetermined. Applied algorithms are run to reach the optimal number of PVUs, location, active power, and power factor of PVUs.Case 1 and Case 2 above have the same characteristic of fixing the number of PVUs at 3, while Case 3 and Case 4 have the same characteristic of finding the optimal number of PVUs in each system. In addition, Case 1 and Case 3 have the same manner of fixing the power factor of each PVU at 1.0 (i.e., PVUs only produce active power), whereas Case 2 and Case 4 have the same feature of finding the optimal power factor for each PVU (i.e., PVUs produce both active power and reactive power). Among the four study cases, Case 4 is the most complicated for applied algorithms because the number of determined variables is the highest.The complete information on the two applied systems is reported in Table1. In the table, it is noted that the maximum penetration of installed PVUs can be found by calculating the sum of load demand and power loss of the base system without PVUs. For the 33-bus RDN, the maximum penetration of Case 1 and Case 3 is only about active power, and the maximum active power penetration is the sum of the active power demand of 3715 kW and the active power loss of 210.998 kW. Thus, the maximum penetration is obtained at 3926 kW. For Case 2 and Case 4 of the 33-bus RDN, the maximum penetration is about both active and reactive power, of which the maximum active power penetration is obtained as in Case 1 and Case 3. The maximum reactive power penetration is the sum of the reactive power demand of 2300 kVAR and the reactive power loss of 143.033 kVAR. So, the maximum reactive power penetration is equal to 2443.1 kVAR. Similarly, the maximum active and reactive power penetrations of the 69-bus RDN are 4025 kW and 2792.2 kVAR, respectively.Table 1
Data test networks and constraints.
Item33-bus RDN69-bus RDNLoad demand (MVA)(3715 +j2300) kVA(3800 +j2690) kVAPower loss (KVA)(210.998 +j143.033) kVA(225.001 +j102.165) kVALowest voltage (pu)0.904 pu0.909 puBase system voltage (kV)12.66 kV12.66 kVLimit of voltage (pu)0.9–1.050.9–1.05Limit of power factor0-10-1Maximum penetrationCases 1, 3: 3926 kWCases 1,3: 4025 kWCases 2, 4: (3926 +j2443.1) kVACases 2, 4: (4025 +j2792.2) kVAThe six applied algorithms are coded in the Matlab 2016a platform on a personal computer with 4 GB of RAM and a 2.0 GHz processor to reach the best active power loss of the systems in four cases. For each case of each system, each applied method is run 50 times to find the most optimal solutions and power loss for comparisons with previous methods. The settings for the population and iteration number for the applied methods are presented in Table2.Table 2
Setting of population and iteration number for applied methods.
MethodAOATSOCrSAWSAASBOCOOANoPItmaxNoPItmaxNoPItmaxNoPItmaxNoPItmaxNoPItmaxCase 1 of system 12015020150515020150715020150Case 2 of system 12025020250525020250725020250Case 3 of system 12050020500550020500750020500Case 4 of system 12012002012005120020120071200201200Case 1 of system 22020020200520020200720020200Case 2 of system 22030020300530020300730020300Case 3 of system 22050020500550020500750020500Case 4 of system 22012002012005120020120071200201200


### 5.1. Results Obtained for the IEEE 33-Node RDPS

In this section, the four applied methods are run for solving four study cases of the IEEE 33-node RDPS with the target of cutting active power loss and maximizing the penetration of PVUs. The one-line diagram of the system is presented in Figure3 and its whole data is taken from [76]. The best optimal solutions for fifty runs are saved and sorted from the best to the worst values in Figure 4. The figure indicates that approximately all loss values for COOA are below those of others in four cases. In addition, the number of runs with the same loss value as the best run obtained by COOA is high, while other methods cannot reach one best run with the same loss value as that from COOA. The total power of PVUs and the best power loss value from the best solutions of the compared methods are reported in Table 3 for Case 1 and Case 2, and in Table 4 for Case 3 and Case 4. In addition, the savings loss of COOA in% as compared to other methods is also calculated and presented to show the exact improvement level of COOA over these methods.Figure 3
The IEEE 33-node RDPS.Figure 4
Loss values of fifty runs were obtained by applying the method to four cases of system 1.Table 3
Comparative result for case 1 and case 2 of system 1.
MethodCase 1Case 2Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]2946.772.7870—3508.311.7410.000—ISOSA [1]2946.772.7870—3508.311.7410—QOSIA [2]2946.472.80.018—————AA [14]294772.78700.37357711.80.5—OPFAA [14]294772.7902.023522.211.740—AA-PSO [17]293072.790—348211.80.5—PSO [17]287072.890.141—349811.740—TLA [26]2741.975.543.64412.62————MTLA [26]3011.474.1011.77312.57————SPBO [31]2946.6772.79710.014—————SFS [32]294772.790—————CSFS [32]294772.790—————PSO [33]250483.7213.059—————MTLBO [33]275977.365.911—————JAYA [33]282676.665.052—————GWO [33]337873.430.876—————IRRO [33]294572.7870—————SSA [34]2270.4775.4711.504————SFO [34]2474.375.94.1011.577————COA [34]2302.2764.2281.328————ECOA [34]2459.974.62.431.284————SCA [35]2931.8672.830.059—3240.6315.3223.362—ISCA [35]2954.2672.790—3505.3511.760.162—BA [36]272073.40.835—3263.8912.354.931—CSA [37]290072.830.05910.25350012.354.93110.75EO [39]297974.02741.676—————MEO [39]290272.87950.127————AEO [39]272773.59881.103—————IEO [39]295372.78980.004—————AOA3616.777.7776.4163.162867.4353.12777.98.12TSOA2948.174.1781.8753.323447.6613.82915.0998.31CrSA2860.774.6322.4723.453381.817.99334.7478.33WSA2946.772.7890.0023.243478.1411.9611.8398.39ASBO3021.272.9380.2073.933494.7514.76620.4868.95COOA294772.78703.33507.0111.74108.28Table 4
Comparative result for cases 3 and 4 of system 1.
MethodCase 3Case 4Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]3860.363.56200.382—4490.12.00416.966—ISOSA [1]3775.963.38290.101—4364.41.88711.818—AOA2617.970.45310.1266.964511.9814.45988.49232.96TSOA3399.364.4651.7787.344424.015.88671.73034.09CrSA3562.564.9272.4777.404399.175.10967.43033.10WSA3301.165.1562.8196.913999.563.87757.08032.54ASBO3468.564.9212.4687.764475.225.91871.88235.95COOA3668.263.31907.674071.41.664033.34The applied COOA has the same loss as SOSA [1], ISOSA [1], AA [14], OPFAA [14], AA-PSO [17], SFS [32], CSFS [32], IRRO [33], and ISCA [35], for Case 1, and SOSA [1], ISOSA [1], OPFAA [14], and AA-PSO [17], for Case 2. COOA and these methods could reach a loss reduction of 65.5% forzz and 94.44% for Case 2, while that of others is smaller. As compared to five other applied algorithms, AOA, TSOA, CrSA, WSA, and ASBO, the applied COOA can reduce loss from 0.002% to 6.416% for Case 1, and from 1.839% to 77.900% for Case 2. As compared to other previously published algorithms, the loss of COOA is also smaller, from 0.004% to 13.059% for Case 1 and from 0.162% to 23.362% for Case 2. Clearly, COOA can improve the loss significantly.Regarding average computation time for each run, COOA and other applied algorithms have the same time, around 3 seconds for Case 1 and 8 seconds for Case 2, whereas that of others is either shorter or longer. It took methods in [34] under 2 seconds, methods in [26] about 13 seconds, and methods in [37], about 10 seconds. Simulation times can be different due to different settings for population and iteration, different computers, and different programming software. Other studies have neglected the information of the settings as well as computers and therefore COOA cannot be compared to these methods for the factors. However, the simulation time of around or under 10 seconds is not long for methods, and power loss is a major comparison factor in evaluating the performance of algorithms. In short, COOA is a really effective method for Case 1 and Case 2 of the IEEE 33-node system.Case 3 of the system becomes more challenging for applied methods since the number of PVUs as well as the number of variables have highly increased. COOA has reached less loss than all five other applied methods and two other compared methods in [1]. The loss is 63.319 kW for COOA, but that is from 63.3829 kW to 70.453 kW for others. By comparing the loss of the base system, COOA can reduce the loss up to 70%, while that of others is between 66.61% and 69.96%. As compared to others, COOA can reach a smaller loss from 0.101% to 10.126%. About the simulation time, it took COOA and other applied algorithms about 8 seconds to reach one candidate solution, while that time was not reported for methods in [1]. Clearly, for a more complicated case, COOA is significantly more robust than other compared methods. The view on the obtained solutions for COOA and the second-best method ISOSA indicates that the two methods have selected 9 PVUs, but the location and size of the nine are not the same. COOA has placed the nine PVUs at nodes 3, 7, 10, 16, 21, 24, 25, 30, and 32, while ISOSA has placed the PVUs at nodes 3, 6, 10, 16, 20, 24, 25, 30, and 32. The same locations for the two methods are nodes 3, 10, 16, 24, 25, and 32, while the three remaining locations are different. The maximum penetration of PVUs from COOA is 3668.2 kW, and that of ISOSA is 3775.9 kW. Clearly, placing three other PVUs at nodes 7, 21, and 30 obtained by COOA is more suitable than at nodes 6, 20, and 29 obtained by ISOSA. Also, COOA can determine more effective power for each PVU than ISOSA. Thus, COOA can reach less power loss while using a smaller maximum penetration of PVUs.Case 4 is more complicated than Case 3 by considering the power factor of PVUs as a variable. The loss of COOA is 1.664 kW, but that from others is from 1.887 kW to 14.459 kW. The values indicate COOA can reach less loss than the base system by 99.2%, while that of others is between 93.1% and 99.1% of the base system. COOA can reach less loss than these methods, from 11.818% to 88.492%. About the simulation time, it took COOA and other applied algorithms about 35 seconds to reach one candidate solution, while that time was not reported for methods in [1]. Clearly, Case 4 is more complicated than Case 3, and methods need a higher running time for finding optimal solutions. In the study case, COOA and the second-best method, ISOSA, have the same selection of nine PVUs but different locations and capacities for the PVUs. COOA placed nine PVUs at buses 6, 8, 14, 18, 21, 24, 25, 30, and 32, while ISOSA placed them at nodes 4, 10, 16, 21, 24, 25, 26, 30, and 32. The maximum penetration of PVUs from COOA is 4071.4 kVA, while that from ISOSA is 4364.4 kVA. Clearly, the maximum penetration of COOA is smaller than that of ISOSA, but COOA reaches lower loss than the method. The result indicates that COOA selected locations and sizes for PVUs more effectively than ISOSA.The four cases of System 1 indicate that COOA can reach the same power loss as other methods for simple cases but much smaller than other compared methods for complicated cases. So, COOA is a more effective performance method than the first system.The optimal solutions found by COOA for the system are reported in TablesS1 and S2 in the Supplementary Material.


### 5.2. Results Obtained for the IEEE 69-Node RDPS

In this section, the performance of COOA is investigated in the IEEE 69-node RDPS. The one-line diagram of the system is shown in Figure5, and its whole data is taken from [77]. The loss values of 50 trial runs are sorted from the best to the worst and then plotted in Figure 6. From the figure, we can see almost all the loss values of COOA are below those of others for four cases, and the four curves of COOA seem to be lined up with the same loss for all runs. In addition, COOA can find a high number of solutions with the same quality as the best solution, whereas the advantage cannot be seen in the curves of other algorithms. Thus, COOA is significantly more effective and stable than other compared methods.Figure 5
The IEEE 69-node RDPS.Figure 6
Loss values of fifty runs obtained by applied methods for system 2.The best loss from all compared methods is presented in Table5 (for Cases 1 and 2) and Table 6 (for Cases 3 and 4). Table 5 indicates the outstanding performance of COOA over almost all compared methods. For Case 1, COOA has the same power loss as three methods SOSA, ISOSA, and OPFAA but better loss than other remaining methods. For Case 2, COOA reaches the same loss as four methods SOSA, ISOSA, OPFAA, and ISCA [35] but better loss than other compared methods. As comparing the loss with the base system, COOA together with SOSA, ISOSA, and OPFAA can reduce the loss up to 69.14% for Case 1 and 98.1% for Case 2, while that of other methods is from 60.5% to 69.09% for Case 1, and from 91.14% to 98.09% for Case 2. COOA can reach less loss than others, from 0.164% to 16.685% for Case 1, and from 0.774% to 78.6% for Case 2. The total power of three PVUs from COOA is approximately the same as that of SOSA and ISOSA and different from that of other methods. The simulation time of COOA and other applied algorithms is about 10 seconds for Case 1 and 20 seconds for Case 2, while that from other studies is much different. It took SPBO in [31] about 172 seconds, methods in [34] about 4 seconds, methods in [14] about 6 seconds, and methods in [17] about 15 seconds for Case 1. It was noted that there was computation time reported in Case 2 for other studies and in Case 1 for the remaining studies. Generally speaking, the computation time is not much different among compared methods excluding SPBO in [31] with about 172 seconds, and searching performance can be evaluated by using the power loss as analyzed. Hence, COOA is a helpful algorithm for Cases 1 and 2 of the system.Table 5
Comparative results for cases 1 and 2 of the second system.
MethodCase 1Case 2Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]2626.269.42840.003—3120.54.26750.019—ISOSA [1]2626.269.42840.003—3120.54.26750.019—QOSIA [2]2785.2712.217—————AA [14]264269.620.2791.6232404.484.761—OPFAA [14]262669.430.0066.6631344.270.077—PSO [17]260069.540.164—32104.617.447—AA-PSO [17]256069.540.164—30704.30.774—TLA [26]3163.672.4064.11615.75————MTLA [26]2960.671.6253.0715.69————SPBO [31]2642.19869.450.035172.915————PSO [33]220083.3316.685—————MTLBO [33]275977.3610.256—————JAYA [33]311675.838.445—————GWO [33]3.37873.435.453—————IRRO [33]342771.142.409—————SSA [34]2065.2734.8964.244————SFO [34]2120.672.74.5034.475————COA [34]2068.572.54.243.646————ECOA [34]—71.83.3063.579————SCA [35]2283.671.773.266—4476.4710.3558.776—ISCA [35]2312.8471.663.117—3118.654.270.077—MFO [38]2647.5169.460.049—————DA [38]2524.371.12.354—————AOA2072.273.2025.15812.983602.3519.93978.619.06TSOA3445.872.694.4913.222962.29.16953.46319.47CrSA233871.2622.57610.42920.38.82451.64319.1WSA2622.869.42870.0049.823064.624.84411.91818.4ASBO2633.869.4330.0110.813037.434.7279.73820.56COOA262569.426010.463117.44.2667018.98Table 6
Comparative results for cases 3 and 4 of the second system.
MethodCase 3Case 4Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]3939.167.1550.173—4725.91.08246.201—ISOSA [1]3938.967.1550.173—4728.61.08246.201—AOA2535.478.96615.10430.034681.8812.624895.38976.38TSOA3736.769.2183.14831.563900.694.84587.98677.39CrSA3821.668.9842.81930.774608.135.347989.11579.56WSA3492.167.4780.65130.414614.593.49783.35473.81ASBO3254.567.9551.34831.753500.514.75187.74879.04COOA3403.267.039030.394622.500.5821068.45As shown in Table6, the power loss from COOA is less than all others for both Case 3 and Case 4. The loss of COOA is 67.039 kW for Case 3 and 0.5821 kW for Case 4, while that of others is from 67.155 to 78.966 kW for Case 3 and from 1.082 to 12.6248 kW for Case 4. The power loss of COOA can be less than in other cases, ranging from 0.173% to 15.104% for Case 3 and from 46.201% to 95.389% kW for Case 4. The computation time of COOA and other algorithms is about 30 seconds for Case 3 and about 75 seconds for Case 4. The power loss and simulation time indicate that COOA is superior to others in terms of the performance of solutions and search speed for the two study cases.For Cases 3 and 4, COOA uses 9 PVUs, while SOSA and ISOSA use 7 PVUs. For Case 3, the two compared methods, SOSA and ISOSA, have the same optimal solution that places PVUs at nodes 3, 8, 11, 18, 50, 61, and 64, while COOA places 9 PVUs at nodes 3, 8, 11, 15, 21, 33, 50, 61, and 64. COOA uses a total power of 3403.2 kW, while that of SOSA and ISOSA is much higher and equal to 3939.1 kW and 3938.9 kW. For Case 4, SOSA and ISOSA have the same locations as those in Case 3, while COOA has different locations from Case 3. These locations of COOA are 11, 12, 17, 22, 28, 49, 51, 61, and 64. The maximum penetration of COOA is smaller than that of SOSA and ISOSA. That is, respectively, 4725.9 and 4728.6 kVA for SOSA and ISOSA, but it is 4622.50 kVA for COOA. Clearly, the maximum penetration of PVUs determined by COOA is smaller than that of SOSA and ISOSA, but COOA can reach much lower losses than these methods. So, COOA is really much more effective than SOSA and ISOSA.Optimal solutions found by COOA for the system are reported in TablesS3 and S4 in Supplementary Material.


### 5.3. Impact of Penetration Level of PVUs on Power Loss Reduction and Voltage Improvement

In this section, the impact of the penetration level of PVUs in RDPs is analyzed to indicate the significance of PVUs for reducing power loss in conductors and increasing the voltage of loads. Figure7 shows four cases of System 1, in which brown bars show the active power of all installed PVUs, green bars show the reactive power of all installed PVUs, and black dots show the power loss of the system. Similarly, Figure 8 presents information on PVUs and power loss for four study cases in System 2. Case 1 and Case 2 limit the number of PVUs to 3, while Case 3 and Case 4 do not consider the limitation for the number of PVUs. On the other hand, Case 1 and Case 3 only consider active power generation for PVUs, while Case 2 and Case 4 consider both active and reactive power for PVUs.Figure 7
Power loss and power of PVUs for four cases of System 1.Figure 8
Power loss and power of PVUs for four cases of System 2.For System 1, the penetration power of the three PVUs was 2949.7 kW in Case 1, 2893.0 kW, and 1901.1 kVAR in Case 2. As a result, Case 2 was more effective than Case 1 in cutting power loss. The loss of Case 1 was 72.787 kW, while that of Case 2 was 11.741 kW. Case 3 and Case 4 did not constrain the number of PVUs, but Case 3 considered only active power generation and Case 4 considered both active and reactive power generation. The penetration power was 3668.2 kW for Case 3, and 3354.5 kW and 2148.4 kVAR for Case 4. The loss was 63.318 kW for Case 3 and 1.664 kW for Case 4. Clearly, Case 3 used a higher active power than Case 4, but it did not reduce loss as effectively as Case 4. It is recalled that the active power and reactive power of all loads in System 1 are 3715 kW and 2300 kVAR, and the maximum possible penetration of PVUs in System 1 shown in Table1 is 3926 kW and 2443.1 kVAR. Clearly, Case 3 had the highest penetration, but the loss was not the smallest.For System 2, the penetration power of PVUs and power loss approximate the same as in System 1. The loss in Case 2 was smaller than that of Case 1, and the loss in Case 4 was the smallest. The active power of three PVUs in Case 1 was equal to 2627.6 kW and smaller than 2547.8 kW in Case 2. On the contrary, Case 4 used higher active power generation than Case 3. It is recalled that the active power and reactive power of all loads shown in Table1 are 3800 kW and 2690 kVAR, and the maximum possible penetration of PVUs are 4025 kW and 2792.2 kVAR. Clearly, Case 4, with 3763.4 kW, had the highest penetration capacity of PVUs. This value was still much less than the maximum possible penetration (4025 kW) and slightly smaller than the load demand of 3800 kW. For Case 4, the sum of loss and load is 3800.582 kW, and the capacity of PVUs is 3763.4 kW. The power source at node 1 had to supply 37.1821 kW. Clearly, the penetration capacity of PVUs is very high and equal to 99.022%.Figures9 and 10 present voltage profiles of System 1 and System 2 for four study cases. The two figures have the same results regarding voltage improvement. After ranking the effectiveness of voltage improvement, Case 1, Case 2, Case 3, and Case 4 are, respectively, the worst, the third-best, the second-best, and the best. Approximately all nodes in Case 4 of Systems 1 and 2 have a voltage close to 1.0 pu. Case 3 is much worse than Case 4 in System 1, but Case 3 has approximately the same profile as Case 4 in System 2. In both systems, the voltage profile with the installation of PVUs is much better than that of base systems without PVUs. So, it can be concluded that the penetration of PVUs can improve the voltage of systems significantly, and the penetration with the highest active and reactive power generation has the best voltage profile.Figure 9
Comparison of voltage profiles for system 1.Figure 10
Comparison of voltage profiles for system 2.


## 5.1. Results Obtained for the IEEE 33-Node RDPS

In this section, the four applied methods are run for solving four study cases of the IEEE 33-node RDPS with the target of cutting active power loss and maximizing the penetration of PVUs. The one-line diagram of the system is presented in Figure3 and its whole data is taken from [76]. The best optimal solutions for fifty runs are saved and sorted from the best to the worst values in Figure 4. The figure indicates that approximately all loss values for COOA are below those of others in four cases. In addition, the number of runs with the same loss value as the best run obtained by COOA is high, while other methods cannot reach one best run with the same loss value as that from COOA. The total power of PVUs and the best power loss value from the best solutions of the compared methods are reported in Table 3 for Case 1 and Case 2, and in Table 4 for Case 3 and Case 4. In addition, the savings loss of COOA in% as compared to other methods is also calculated and presented to show the exact improvement level of COOA over these methods.Figure 3
The IEEE 33-node RDPS.Figure 4
Loss values of fifty runs were obtained by applying the method to four cases of system 1.Table 3
Comparative result for case 1 and case 2 of system 1.
MethodCase 1Case 2Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]2946.772.7870—3508.311.7410.000—ISOSA [1]2946.772.7870—3508.311.7410—QOSIA [2]2946.472.80.018—————AA [14]294772.78700.37357711.80.5—OPFAA [14]294772.7902.023522.211.740—AA-PSO [17]293072.790—348211.80.5—PSO [17]287072.890.141—349811.740—TLA [26]2741.975.543.64412.62————MTLA [26]3011.474.1011.77312.57————SPBO [31]2946.6772.79710.014—————SFS [32]294772.790—————CSFS [32]294772.790—————PSO [33]250483.7213.059—————MTLBO [33]275977.365.911—————JAYA [33]282676.665.052—————GWO [33]337873.430.876—————IRRO [33]294572.7870—————SSA [34]2270.4775.4711.504————SFO [34]2474.375.94.1011.577————COA [34]2302.2764.2281.328————ECOA [34]2459.974.62.431.284————SCA [35]2931.8672.830.059—3240.6315.3223.362—ISCA [35]2954.2672.790—3505.3511.760.162—BA [36]272073.40.835—3263.8912.354.931—CSA [37]290072.830.05910.25350012.354.93110.75EO [39]297974.02741.676—————MEO [39]290272.87950.127————AEO [39]272773.59881.103—————IEO [39]295372.78980.004—————AOA3616.777.7776.4163.162867.4353.12777.98.12TSOA2948.174.1781.8753.323447.6613.82915.0998.31CrSA2860.774.6322.4723.453381.817.99334.7478.33WSA2946.772.7890.0023.243478.1411.9611.8398.39ASBO3021.272.9380.2073.933494.7514.76620.4868.95COOA294772.78703.33507.0111.74108.28Table 4
Comparative result for cases 3 and 4 of system 1.
MethodCase 3Case 4Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]3860.363.56200.382—4490.12.00416.966—ISOSA [1]3775.963.38290.101—4364.41.88711.818—AOA2617.970.45310.1266.964511.9814.45988.49232.96TSOA3399.364.4651.7787.344424.015.88671.73034.09CrSA3562.564.9272.4777.404399.175.10967.43033.10WSA3301.165.1562.8196.913999.563.87757.08032.54ASBO3468.564.9212.4687.764475.225.91871.88235.95COOA3668.263.31907.674071.41.664033.34The applied COOA has the same loss as SOSA [1], ISOSA [1], AA [14], OPFAA [14], AA-PSO [17], SFS [32], CSFS [32], IRRO [33], and ISCA [35], for Case 1, and SOSA [1], ISOSA [1], OPFAA [14], and AA-PSO [17], for Case 2. COOA and these methods could reach a loss reduction of 65.5% forzz and 94.44% for Case 2, while that of others is smaller. As compared to five other applied algorithms, AOA, TSOA, CrSA, WSA, and ASBO, the applied COOA can reduce loss from 0.002% to 6.416% for Case 1, and from 1.839% to 77.900% for Case 2. As compared to other previously published algorithms, the loss of COOA is also smaller, from 0.004% to 13.059% for Case 1 and from 0.162% to 23.362% for Case 2. Clearly, COOA can improve the loss significantly.Regarding average computation time for each run, COOA and other applied algorithms have the same time, around 3 seconds for Case 1 and 8 seconds for Case 2, whereas that of others is either shorter or longer. It took methods in [34] under 2 seconds, methods in [26] about 13 seconds, and methods in [37], about 10 seconds. Simulation times can be different due to different settings for population and iteration, different computers, and different programming software. Other studies have neglected the information of the settings as well as computers and therefore COOA cannot be compared to these methods for the factors. However, the simulation time of around or under 10 seconds is not long for methods, and power loss is a major comparison factor in evaluating the performance of algorithms. In short, COOA is a really effective method for Case 1 and Case 2 of the IEEE 33-node system.Case 3 of the system becomes more challenging for applied methods since the number of PVUs as well as the number of variables have highly increased. COOA has reached less loss than all five other applied methods and two other compared methods in [1]. The loss is 63.319 kW for COOA, but that is from 63.3829 kW to 70.453 kW for others. By comparing the loss of the base system, COOA can reduce the loss up to 70%, while that of others is between 66.61% and 69.96%. As compared to others, COOA can reach a smaller loss from 0.101% to 10.126%. About the simulation time, it took COOA and other applied algorithms about 8 seconds to reach one candidate solution, while that time was not reported for methods in [1]. Clearly, for a more complicated case, COOA is significantly more robust than other compared methods. The view on the obtained solutions for COOA and the second-best method ISOSA indicates that the two methods have selected 9 PVUs, but the location and size of the nine are not the same. COOA has placed the nine PVUs at nodes 3, 7, 10, 16, 21, 24, 25, 30, and 32, while ISOSA has placed the PVUs at nodes 3, 6, 10, 16, 20, 24, 25, 30, and 32. The same locations for the two methods are nodes 3, 10, 16, 24, 25, and 32, while the three remaining locations are different. The maximum penetration of PVUs from COOA is 3668.2 kW, and that of ISOSA is 3775.9 kW. Clearly, placing three other PVUs at nodes 7, 21, and 30 obtained by COOA is more suitable than at nodes 6, 20, and 29 obtained by ISOSA. Also, COOA can determine more effective power for each PVU than ISOSA. Thus, COOA can reach less power loss while using a smaller maximum penetration of PVUs.Case 4 is more complicated than Case 3 by considering the power factor of PVUs as a variable. The loss of COOA is 1.664 kW, but that from others is from 1.887 kW to 14.459 kW. The values indicate COOA can reach less loss than the base system by 99.2%, while that of others is between 93.1% and 99.1% of the base system. COOA can reach less loss than these methods, from 11.818% to 88.492%. About the simulation time, it took COOA and other applied algorithms about 35 seconds to reach one candidate solution, while that time was not reported for methods in [1]. Clearly, Case 4 is more complicated than Case 3, and methods need a higher running time for finding optimal solutions. In the study case, COOA and the second-best method, ISOSA, have the same selection of nine PVUs but different locations and capacities for the PVUs. COOA placed nine PVUs at buses 6, 8, 14, 18, 21, 24, 25, 30, and 32, while ISOSA placed them at nodes 4, 10, 16, 21, 24, 25, 26, 30, and 32. The maximum penetration of PVUs from COOA is 4071.4 kVA, while that from ISOSA is 4364.4 kVA. Clearly, the maximum penetration of COOA is smaller than that of ISOSA, but COOA reaches lower loss than the method. The result indicates that COOA selected locations and sizes for PVUs more effectively than ISOSA.The four cases of System 1 indicate that COOA can reach the same power loss as other methods for simple cases but much smaller than other compared methods for complicated cases. So, COOA is a more effective performance method than the first system.The optimal solutions found by COOA for the system are reported in TablesS1 and S2 in the Supplementary Material.


## 5.2. Results Obtained for the IEEE 69-Node RDPS

In this section, the performance of COOA is investigated in the IEEE 69-node RDPS. The one-line diagram of the system is shown in Figure5, and its whole data is taken from [77]. The loss values of 50 trial runs are sorted from the best to the worst and then plotted in Figure 6. From the figure, we can see almost all the loss values of COOA are below those of others for four cases, and the four curves of COOA seem to be lined up with the same loss for all runs. In addition, COOA can find a high number of solutions with the same quality as the best solution, whereas the advantage cannot be seen in the curves of other algorithms. Thus, COOA is significantly more effective and stable than other compared methods.Figure 5
The IEEE 69-node RDPS.Figure 6
Loss values of fifty runs obtained by applied methods for system 2.The best loss from all compared methods is presented in Table5 (for Cases 1 and 2) and Table 6 (for Cases 3 and 4). Table 5 indicates the outstanding performance of COOA over almost all compared methods. For Case 1, COOA has the same power loss as three methods SOSA, ISOSA, and OPFAA but better loss than other remaining methods. For Case 2, COOA reaches the same loss as four methods SOSA, ISOSA, OPFAA, and ISCA [35] but better loss than other compared methods. As comparing the loss with the base system, COOA together with SOSA, ISOSA, and OPFAA can reduce the loss up to 69.14% for Case 1 and 98.1% for Case 2, while that of other methods is from 60.5% to 69.09% for Case 1, and from 91.14% to 98.09% for Case 2. COOA can reach less loss than others, from 0.164% to 16.685% for Case 1, and from 0.774% to 78.6% for Case 2. The total power of three PVUs from COOA is approximately the same as that of SOSA and ISOSA and different from that of other methods. The simulation time of COOA and other applied algorithms is about 10 seconds for Case 1 and 20 seconds for Case 2, while that from other studies is much different. It took SPBO in [31] about 172 seconds, methods in [34] about 4 seconds, methods in [14] about 6 seconds, and methods in [17] about 15 seconds for Case 1. It was noted that there was computation time reported in Case 2 for other studies and in Case 1 for the remaining studies. Generally speaking, the computation time is not much different among compared methods excluding SPBO in [31] with about 172 seconds, and searching performance can be evaluated by using the power loss as analyzed. Hence, COOA is a helpful algorithm for Cases 1 and 2 of the system.Table 5
Comparative results for cases 1 and 2 of the second system.
MethodCase 1Case 2Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]2626.269.42840.003—3120.54.26750.019—ISOSA [1]2626.269.42840.003—3120.54.26750.019—QOSIA [2]2785.2712.217—————AA [14]264269.620.2791.6232404.484.761—OPFAA [14]262669.430.0066.6631344.270.077—PSO [17]260069.540.164—32104.617.447—AA-PSO [17]256069.540.164—30704.30.774—TLA [26]3163.672.4064.11615.75————MTLA [26]2960.671.6253.0715.69————SPBO [31]2642.19869.450.035172.915————PSO [33]220083.3316.685—————MTLBO [33]275977.3610.256—————JAYA [33]311675.838.445—————GWO [33]3.37873.435.453—————IRRO [33]342771.142.409—————SSA [34]2065.2734.8964.244————SFO [34]2120.672.74.5034.475————COA [34]2068.572.54.243.646————ECOA [34]—71.83.3063.579————SCA [35]2283.671.773.266—4476.4710.3558.776—ISCA [35]2312.8471.663.117—3118.654.270.077—MFO [38]2647.5169.460.049—————DA [38]2524.371.12.354—————AOA2072.273.2025.15812.983602.3519.93978.619.06TSOA3445.872.694.4913.222962.29.16953.46319.47CrSA233871.2622.57610.42920.38.82451.64319.1WSA2622.869.42870.0049.823064.624.84411.91818.4ASBO2633.869.4330.0110.813037.434.7279.73820.56COOA262569.426010.463117.44.2667018.98Table 6
Comparative results for cases 3 and 4 of the second system.
MethodCase 3Case 4Total power of PVUs (kW)Power loss (kW)Loss reduction (%)CPU times (s)Total power of PVUs (kVA)Power loss (kW)Loss reduction (%)CPU times (s)SOSA [1]3939.167.1550.173—4725.91.08246.201—ISOSA [1]3938.967.1550.173—4728.61.08246.201—AOA2535.478.96615.10430.034681.8812.624895.38976.38TSOA3736.769.2183.14831.563900.694.84587.98677.39CrSA3821.668.9842.81930.774608.135.347989.11579.56WSA3492.167.4780.65130.414614.593.49783.35473.81ASBO3254.567.9551.34831.753500.514.75187.74879.04COOA3403.267.039030.394622.500.5821068.45As shown in Table6, the power loss from COOA is less than all others for both Case 3 and Case 4. The loss of COOA is 67.039 kW for Case 3 and 0.5821 kW for Case 4, while that of others is from 67.155 to 78.966 kW for Case 3 and from 1.082 to 12.6248 kW for Case 4. The power loss of COOA can be less than in other cases, ranging from 0.173% to 15.104% for Case 3 and from 46.201% to 95.389% kW for Case 4. The computation time of COOA and other algorithms is about 30 seconds for Case 3 and about 75 seconds for Case 4. The power loss and simulation time indicate that COOA is superior to others in terms of the performance of solutions and search speed for the two study cases.For Cases 3 and 4, COOA uses 9 PVUs, while SOSA and ISOSA use 7 PVUs. For Case 3, the two compared methods, SOSA and ISOSA, have the same optimal solution that places PVUs at nodes 3, 8, 11, 18, 50, 61, and 64, while COOA places 9 PVUs at nodes 3, 8, 11, 15, 21, 33, 50, 61, and 64. COOA uses a total power of 3403.2 kW, while that of SOSA and ISOSA is much higher and equal to 3939.1 kW and 3938.9 kW. For Case 4, SOSA and ISOSA have the same locations as those in Case 3, while COOA has different locations from Case 3. These locations of COOA are 11, 12, 17, 22, 28, 49, 51, 61, and 64. The maximum penetration of COOA is smaller than that of SOSA and ISOSA. That is, respectively, 4725.9 and 4728.6 kVA for SOSA and ISOSA, but it is 4622.50 kVA for COOA. Clearly, the maximum penetration of PVUs determined by COOA is smaller than that of SOSA and ISOSA, but COOA can reach much lower losses than these methods. So, COOA is really much more effective than SOSA and ISOSA.Optimal solutions found by COOA for the system are reported in TablesS3 and S4 in Supplementary Material.


## 5.3. Impact of Penetration Level of PVUs on Power Loss Reduction and Voltage Improvement

In this section, the impact of the penetration level of PVUs in RDPs is analyzed to indicate the significance of PVUs for reducing power loss in conductors and increasing the voltage of loads. Figure7 shows four cases of System 1, in which brown bars show the active power of all installed PVUs, green bars show the reactive power of all installed PVUs, and black dots show the power loss of the system. Similarly, Figure 8 presents information on PVUs and power loss for four study cases in System 2. Case 1 and Case 2 limit the number of PVUs to 3, while Case 3 and Case 4 do not consider the limitation for the number of PVUs. On the other hand, Case 1 and Case 3 only consider active power generation for PVUs, while Case 2 and Case 4 consider both active and reactive power for PVUs.Figure 7
Power loss and power of PVUs for four cases of System 1.Figure 8
Power loss and power of PVUs for four cases of System 2.For System 1, the penetration power of the three PVUs was 2949.7 kW in Case 1, 2893.0 kW, and 1901.1 kVAR in Case 2. As a result, Case 2 was more effective than Case 1 in cutting power loss. The loss of Case 1 was 72.787 kW, while that of Case 2 was 11.741 kW. Case 3 and Case 4 did not constrain the number of PVUs, but Case 3 considered only active power generation and Case 4 considered both active and reactive power generation. The penetration power was 3668.2 kW for Case 3, and 3354.5 kW and 2148.4 kVAR for Case 4. The loss was 63.318 kW for Case 3 and 1.664 kW for Case 4. Clearly, Case 3 used a higher active power than Case 4, but it did not reduce loss as effectively as Case 4. It is recalled that the active power and reactive power of all loads in System 1 are 3715 kW and 2300 kVAR, and the maximum possible penetration of PVUs in System 1 shown in Table1 is 3926 kW and 2443.1 kVAR. Clearly, Case 3 had the highest penetration, but the loss was not the smallest.For System 2, the penetration power of PVUs and power loss approximate the same as in System 1. The loss in Case 2 was smaller than that of Case 1, and the loss in Case 4 was the smallest. The active power of three PVUs in Case 1 was equal to 2627.6 kW and smaller than 2547.8 kW in Case 2. On the contrary, Case 4 used higher active power generation than Case 3. It is recalled that the active power and reactive power of all loads shown in Table1 are 3800 kW and 2690 kVAR, and the maximum possible penetration of PVUs are 4025 kW and 2792.2 kVAR. Clearly, Case 4, with 3763.4 kW, had the highest penetration capacity of PVUs. This value was still much less than the maximum possible penetration (4025 kW) and slightly smaller than the load demand of 3800 kW. For Case 4, the sum of loss and load is 3800.582 kW, and the capacity of PVUs is 3763.4 kW. The power source at node 1 had to supply 37.1821 kW. Clearly, the penetration capacity of PVUs is very high and equal to 99.022%.Figures9 and 10 present voltage profiles of System 1 and System 2 for four study cases. The two figures have the same results regarding voltage improvement. After ranking the effectiveness of voltage improvement, Case 1, Case 2, Case 3, and Case 4 are, respectively, the worst, the third-best, the second-best, and the best. Approximately all nodes in Case 4 of Systems 1 and 2 have a voltage close to 1.0 pu. Case 3 is much worse than Case 4 in System 1, but Case 3 has approximately the same profile as Case 4 in System 2. In both systems, the voltage profile with the installation of PVUs is much better than that of base systems without PVUs. So, it can be concluded that the penetration of PVUs can improve the voltage of systems significantly, and the penetration with the highest active and reactive power generation has the best voltage profile.Figure 9
Comparison of voltage profiles for system 1.Figure 10
Comparison of voltage profiles for system 2.


## 6. Conclusions

In this paper, six novel metaheuristic algorithms, including COOA, AOA, TSOA, CrSA, WSA, and ASBO, were employed for placing PVUs in RDPSs to reduce power loss while considering the maximum possible penetration of PVUs. Two standard IEEE 33-node and IEEE 69-node systems were employed for implementing four different study cases, which are summarized as follows:(1)
The number of PVUs was fixed at three for Cases 1 and 2, but not limited to Cases 3 and 4.(2)
The PVUs only produced active power for Case 1 and Case 3, but produced both active and reactive power for Case 2 and Case 4.From the result of the simulation, COOA had significant contributions in cutting power loss as follows:The loss reduction obtained by COOA for the four cases of placing PVUs is significant compared to other algorithms. However, it should be pointed out that the most effective study case is for the considered distribution systems. The comparison among the four study cases with different conditions can be summarized as follows:(1)
For the IEEE 33-node system, the highest penetration level was 2949.7 kW for Case 1, (2893.0 +j1901.1) kVA for Case 2, 3668.2 kW for Case 3, and (3354.5 + j2148.4) kVA for Case 4. The loss was 72.787 kW for Case 1, 11.741 kW for Case 2, 63.318 kW for Case 3, and 1.664 kW for Case 4.(2)
For the IEEE 69-node system, the highest penetration level was 2627.6 kW for Case 1, (2547.8 +j1796.1) kVA for Case 2, 3403.2 kW for Case 3, and (3763.4 + j2684.2) kVA for Case 4. The power losses in the four cases were 69.426 kW, 4.267 kW, 67.039 kW, and 0.5821 kW, respectively.(3)
For the two systems, the voltage of all nodes was approximately equal to 1.0 pu (the rated voltage) in Case 4, but that of other cases had a small or relatively high deviation from the expected value of 1.0 pu.The results indicate that the highest penetration level of PVUs could be reached in Case 4, which could reach a very high loss reduction as expected. So, it is suggested that the number of PVUs, their location, active power, and reactive power of each PVUs should be optimized by using metaheuristic algorithms.The paper had contributions in terms of the performance of COOA and the most suitable solution for using renewable energies in distribution systems. However, the paper also had limitations that should be improved in future work. Renewable energies are much more dependent on weather, and they are uncertain in producing and supplying power to loads. The consideration of uncertain factors such as wind speed and solar radiation is very important to calculating the benefits of the use of renewable energies in distribution systems. In addition, investment costs, operation costs, and the benefits of using renewable energies should become major objectives in future work. COOA or other existing metaheuristic algorithms can be selected for difficult future work.



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*Source: 9709608-2023-03-10.xml*