Optimum placement of distributed generation resources, capacitors and charging stations with a developed competitive algorithm

Abstract

This study presents a novel approach for the optimal placement of distributed generation (DG) resources, electric vehicle (EV) charging stations, and shunt capacitors (SC) in power distribution systems. The primary objective is to improve power efficiency and voltage profiles while considering practical and nonlinear constraints. The proposed model combines competitive search optimization (CSO) with fuzzy and chaotic theory to develop an efficient and effective solution. The use of fuzzy theory in the model enables the identification of optimal locations for DG sources and SCs, leading to significant enhancements in power index, generation, power losses, and system voltage. Moreover, the proposed fuzzy method is employed to determine the best locations for EV charging stations, further optimizing the overall system performance. The theoretical analysis demonstrates substantial improvements in both accuracy and convergence speed, highlighting the robustness of the proposed approach. In addition, the utilization of chaos theory enhances the local search optimization process, making the proposed method more efficient in finding high-quality solutions. To validate the performance of the model, extensive simulations are conducted on a 69-bus distribution system and various test functions. The results consistently reveal the superiority of the proposed method compared to other conventional optimization techniques. The key contribution of this study lies in its development of a comprehensive and efficient approach for the optimal placement of DG, EV charging stations, and SCs in power distribution systems. The integration of CSO, fuzzy theory, and chaotic theory enables the simultaneous consideration of multiple objectives and constraints, resulting in enhanced power dissipation reduction and voltage profile improvement. The obtained results demonstrate the practical applicability and superiority of the proposed method, which can significantly benefit power system planners and operators in real-world scenarios.

1. Introduction

The electricity industry has witnessed significant changes in power generation systems, aiming to meet the growing demands and ensure reliable supply. One of the key advancements is the implementation of Distributed Generation (DG), a system that has greatly improved power supply reliability and prompted substantial investment [[1], [2], [3], [4], [5]]. The ever-increasing global population and economic development have escalated electricity demand. While traditional power plants could address these needs, their construction requires considerable time, space, and financial resources. To meet consumer demands efficiently and cost-effectively, governments and industry planners are now actively considering the development of distributed generation resources. However, the definition and scale of distributed generation capacity vary across different countries [[6], [7], [8], [9], [10]]. Integrating distributed power plants into the existing grid offers several benefits, including loss reduction, improved voltage profiles, and increased grid reliability. Inadequate placement of these power plants can lead to higher losses and elevated generation and transmission costs. Moreover, a crucial concern is the occurrence of islanding, where a fault in the grid can lead to the isolation of distributed generation plants, cutting off power supply to a limited number of consumers. To address these challenges, optimization methods are employed to determine the optimal installation locations and capacities of distributed generation resources, considering factors like loss reduction, voltage profile maintenance, and islanding scenarios [[11], [12], [13], [14]]. The placement of distributed generation requires careful planning and optimization to avoid economic and technical challenges for both parking lot investors and distribution system operators. A concept known as “vehicle-to-network" has been introduced, emphasizing the importance of connected vehicles in ancillary and regulatory services markets [15]. Although individual electrical devices alone have minimal impact on the power grid, recent studies have proposed aggregators to encourage vehicle owners to connect their vehicles to the grid, establishing a link between independent system operators and car owners. Such vehicle-to-network approaches have been studied in the context of frequency adjustment [5]. In the realm of electric vehicle studies, significant attention has been focused on optimizing charging schedules to achieve desired outcomes like losses and voltage profiles. Connected-vehicle technology plays a crucial role in these endeavors. Fig. 1 provides an overview of such systems and demonstrates how the components are interconnected.

Fig. 1.

A typical view of an intelligent system and how different components are connected.

Therefore, given the new needs of consumers, distribution networks planners should consider the technical aspects of the network, planning approaches, and network operation according to the needs of these consumers. In fact, the placement of parking spaces for electric vehicles in the distribution network leads to economic problems for the parking space seeker and to technical problems for the power system operator. The placement of these two main elements of the future distribution networks should allow a better use of the network facilities needed by the industry and the electricity applications in the digital community [16]. Placement of distributed generation -without technical planning and optimal placement leads to both economic problems for the parking lot investor and technical problems for the distribution network operator. The concept of vehicle-to-network was used to present a revenue and cost model to contribute to ancillary and regulatory services markets. In Ref. [17], the DG placement and EV parking for the radial network were studied. The distribution network, equipped with distributed generation units is considered as the units are installed in the network to interact with the electric vehicle parking to achieve the lowest losses in the distribution network. In this article, an attempt is made to upgrade the existing distribution grid with the presence of parkings and distributed generation resources in such a way that the reliability desired by the investor of an electric car parking is provided at the lowest cost. In addition, losses are minimized from the perspective of the grid operator. It is worth noting that the reliability from the investor's perspective is the reliability index introduced for the investor's decision and explained below. This index is calculated independently for each busbar, as it is independent of the load on other busbars in the distribution network. The proposed method is applied in Ref. [18] to present a method for optimal scheduling of CS and capacitors (CAP). To achieve a better balance between utilization and exploration, the quantum basis and Gaussian mutation methods are applied. In terms of reactive power return and congestion management, parking and capacitor allocation can be proposed.

In [19] the form of the distribution system is proposed. The novel logic technique aims to reduce power losses, by using the cooperation of stations. We wanted to compare this method with the current density method previously developed for individual charging stations. A control system and mathematical modelling are presented in Ref. [20]. The goal was to provide 800 V DC to novel electric vehicles while meeting the vehicles’ power requirements.

Optimum energy storage system (ESS) placement, sizing, and process are provided as indicated in Ref. [21]. Herein, the advantages and limitations of the proposed schemes are considered in terms of the variety of grid scenarios, the applied strategies and ESS. In Ref. [22], the development of a novel two-level optimization framework is presented with the aim of allocating the power-intensive wind generation plants and the battery ESS in terms of services. In this study, we tried to allocate two batteries ESS and one SC. In addition, the other participated in the distributed ancillary services. In Ref. [23], a novel method for determining the efficiency parameter in DG is presented, which takes into account, the economic and technical characteristics. The technical factors considered are the minimization of network load and power, while the economic characteristics include the optimal DG investment cost. A novel sensitivity index is presented to identify the locations of DG. In Ref. [24], a modified version of the particle swarm is applied. In Ref. [25], a fuzzy model for efficient positioning of VR and capacitors in DSs was proposed. In Ref. [26], a new model was proposed and SOS-NNA for ideal scheduling of DGs and CBs in RDNs considering optimization by various constraints. In Ref. [27], an arithmetic division was proposed which does not generate extra noise and thus increases the accuracy. In Ref. [28], the Salp algorithm was used at normal temperature under radial distribution conditions is used. In Ref. [29], TPA was proposed as a new optimization model for solving optimization problems. In Ref. [30], the optimization of the size for RES associated with DG units was proposed by GES in radial DN. The CPSO as an optimization model is used to reduce the energy loss, which is subject to various constraints. Various DG data were investigated to reduce the loss of DN. To the best of our knowledge, the acceptable efficiency cannot be generating by conventional multi-objective methods. Therefore, in this study, the fuzzy model is used to plan the distribution system, which is characterized by high performance. When the ideal placement of DGs and SCs is restored and the EV charging bar is improved, the power loss of the distribution system that satisfies the favorable characteristics is inflexible over common preparation models. The obligations of DG and power source capability at various systems by perfect capability are specified and distribution efficiency is required to trust power to casuals.

Placement of distributed generation -without technical planning and optimal placement leads to both economic problems for the parking lot investor and technical problems for the distribution s operator. The concept of vehicle-to-network was used to present a revenue and cost model to contribute to ancillary and regulatory services markets. However, in the studies conducted so far, there are no efficient models that consider different aspects such as electric vehicles, shunt capacitors, and the distributed generation resources simultaneously. In this study, the above cases were modeled first, and then, various issues, including charging, were considered to solve the real problem. The proposed model is a complex problem that cannot be adequately addressed by previous methods. To address this shortcoming of previous studies, this paper presents an algorithm based on particle competition and chaos theory. The proposed method is carried out in several steps. The multi-objective optimization technique for the competitive search optimization (CSO) algorithm is improved to achieve optimal performance. The optimal placement of EV is changed with the fuzzy CSO algorithm to improve the performance. Here, battery charging models are used to analyze the influence of EV on distribution system performance by load-flow models. The proposed work offers several novel contributions compared to the existing state-of-the-art literature in the field of optimal placement of distributed generation resources, electric vehicle charging stations, and shunt capacitors. The key points of novelty and contribution can be outlined as follows:

• i)Integrated Optimization Approach: Unlike many previous studies that focus on individual aspects of distributed generation, electric vehicles, or capacitors, this work presents an integrated optimization approach that simultaneously considers all three components. The model addresses the optimal placement problem in a holistic manner, taking into account the interactions and synergies between DG, EVs, and SCs, which results in more comprehensive and effective solutions.
• ii)Competitive Search Optimization: The proposed model employs competitive search optimization (CSO), a novel algorithmic approach that has shown promise in efficiently finding optimal solutions in complex optimization problems. The utilization of CSO allows for better exploration and exploitation of the solution space, leading to improved convergence speed and solution quality compared to traditional optimization methods.
• iii)Fuzzy and Chaotic Theory Integration: The combination of fuzzy theory and chaotic theory in the proposed model enhances the decision-making process and local search optimization. Fuzzy theory enables the model to handle uncertainty and imprecision in data and criteria, making it more robust in real-world scenarios. On the other hand, chaotic theory enhances the local search process, further improving the efficiency of finding high-quality solutions.
• iv)Multiple Objective Optimization: The study addresses the multi-objective nature of the optimal placement problem by considering practical and nonlinear constraints. The model aims to simultaneously minimize power dissipation and maximize voltage profile while accounting for various technical and operational constraints. This approach provides a more comprehensive and balanced optimization of the system, considering both economic and technical aspects.
• v)Comparative Analysis: In contrast to claiming uniqueness without justification, the proposed work provides a thorough comparative analysis with several optimization methods. By comparing the results with other conventional optimization techniques on a 69-bus distribution system and various test functions, the study establishes the superiority and advantages of the proposed method in terms of solution quality and efficiency.

In summary, the novelty of this work lies in its comprehensive and integrated optimization approach, combining competitive search optimization, fuzzy and chaotic theory, and multi-objective optimization to address the complex problem of optimal placement of distributed generation resources, electric vehicle charging stations, and shunt capacitors. The study presents a robust and efficient method that outperforms existing approaches and offers practical solutions for power system planners and operators in the context of the evolving energy landscape.

The content of the paper is presented in these sections as follows: In the second section, the modeling of the problem is presented with relevant mathematical principles. In the third section, the studied scheme is presented and the simulation results are discussed. Finally, in the fourth section, the conclusion is presented.

2. Problem formulation

Here, the developed CSO algorithm is used to determine the optimal size, allocate charging stations, SC and EV charging stations [31]. The placement problem is solved as follows: the CSO algorithm is used to obtain the DG optimum. Then, the distribution system is solved to find the optimal location of the charging stations and effectively control the loads of the electric vehicles. The five locations are analyzed for their optimal placement. Then, the CSO fuzzy algorithm is used to determine the optimal number of electric vehicles that have no impact on the distribution performance.

2.1. Modeling the battery load charge of an electric vehicle

In this study, it is predicted that due to economic and environmental factors, the use of electric vehicles will increase in the future, especially if they have the ability to connect to the power grid. The high market penetration of electric vehicles may have an impact on the grid; a large number of studies have been conducted to investigate the impact of electric vehicle charging on the grid. On the other hand, lithium-ion batteries have played a key role in improving vehicles, and extensive research has been conducted to develop these types of batteries [32,33]. Due to environmental concerns and stringent emission regulations, major automobile manufacturers have switched to electric and hybrid electric vehicles (HEVs) at great expense for research and development. However, providing the electricity required by these vehicles is a challenge for the development of the electrical devices, and rechargeable lithium-ion batteries are considered as the best option to overcome this challenge. In this study, among all energy storage systems, lithium-ion batteries are considered as capable energy sources and storage due to their energy density. The characteristic charging curve of an electric car battery for lithium-ion batteries is shown in Fig. 2. In this sense, Fig. 2 shows the charge-energy curve of the battery charge from complete discharge to complete charge and that of the state-of-charge (SOC).

Fig. 2.

Lithium-ion battery charge feature.

In this study, batteries in EV are ESS and have the ability to charge and discharge the chemical processes and exponential performances over time. Therefore, in this study, the characteristics of the transient mode of battery charging can be expressed by exponential equations [15]:

$${\mathrm{P}}_{\text{EV}}\left(\mathrm{t}\right)=\{\begin{array}{c}{\mathrm{P}}_{\text{EV}}^{\max }\left(1-{\mathrm{e}}^{\left(-\alpha \mathrm{t}/{\mathrm{t}}_{\mathrm{m}2}\right)}\right)0\le t\le {\mathrm{t}}_{\mathrm{m}2}\\ {\mathrm{P}}_{\text{EV}}^{\max }\left(\frac{{\mathrm{t}}_{\max }-\mathrm{t}}{{\mathrm{t}}_{\max }-{\mathrm{t}}_{\mathrm{m}2}}\right){\mathrm{t}}_{\mathrm{m}2}<t\le {\mathrm{t}}_{\max }\\ 0t>{\mathrm{t}}_{\max }\end{array}$$ (1)

Where ${\mathrm{P}}_{\text{EV}}\left(\mathrm{t}\right)$ is the immediate EV charging, and ${\mathrm{P}}_{\text{EV}}^{\max }$ is the highest charging load [15]:

$$\mathrm{\beta }{\mathrm{P}}_{\text{EV}}^{\max }={\mathrm{P}}_{\text{EV}}^{\max }\left(1-{\mathrm{e}}^{-\mathrm{\alpha }{\mathrm{t}}_{\mathrm{m}1}/{\mathrm{t}}_{\mathrm{m}2}}\right)$$ (2)

$$\mathrm{\alpha }=-\left(\frac{{\mathrm{t}}_{\mathrm{m}2}}{{\mathrm{t}}_{\mathrm{m}1}}\right)\ln \left(1-\mathrm{\beta }\right)$$ (3)

Where ${\mathrm{t}}_{\mathrm{m}1}$, ${\mathrm{t}}_{\mathrm{m}2}$ , and ${\mathrm{t}}_{\max }$ are 0.25, 4.5, and 5 h, respectively. The values of α and β are obtained from Fig. 2 β denotes the fraction for the upper limit charging demand and is 0.95, which corresponds to 95% of ${\mathrm{P}}_{\text{EV}}^{\max }$ at time ${\mathrm{t}}_{\mathrm{m}1}$. The value of α is obtained from Equation (3). If the batteries are charged from the initial charging mode ${\mathrm{P}}_{\text{EV}}^0$, the equation for the power charge is [15]:

$${\mathrm{P}}_{\text{EV}}\left(\mathrm{t}\right)={\mathrm{P}}_{\text{EV}}^{\max }\left(1-{\mathrm{e}}^{\left(\raisebox{1ex}{$-\alpha \mathrm{t}$}\!\left/ \!\raisebox{-1ex}{$\text{tfc}$}\right.\right)}\right)+{\mathrm{P}}_{\text{EV}}^0{\mathrm{e}}^{\left(\raisebox{1ex}{$-\alpha \mathrm{t}$}\!\left/ \!\raisebox{-1ex}{$\text{tfc}$}\right.\right)}\text{for}0<t<{\mathrm{t}}_{\text{fc}}$$ (4)

Where ${\mathrm{t}}_{\text{fc}}$ is the time required to fully charge the battery from the beginning of the charging position. In this study, the time a vehicle is connected to the grid for charging depends on a number of parameters; these include the start and end time of the charging process and the initial state of charge of the vehicle (SOC). In addition, the charging time is a function of the charging rate. The state of charge of the battery is equal to Ref. [15]:

$$\text{SOC}\left(\mathrm{t}+1\right)=\text{SOC}\left(\mathrm{t}\right)+{\mathrm{P}}_{\text{EV}}\left(\mathrm{t}\right)\Delta \left(\mathrm{t}\right)$$ (5)

In this study, to avoid overcharge hazards, the batteries must be shut down from the power after 100%. The battery is operated at lower power and therefore takes a long time to reach 100% battery charge [15,34]. In this study, considering the correction factor of 0.95, the charging load of electric vehicles is equal to Ref. [34]:

$${\text{PL}}_{\text{EV}}\left({\mathrm{N}}_{\mathrm{i}}\right)={\text{NV}}_{\mathrm{i}}\left({\mathrm{P}}_{\text{EV}}^{\max }\right)$$ (6)

$${\text{QL}}_{\text{EV}}\left({\mathrm{N}}_{\mathrm{i}}\right)={\text{NV}}_{\mathrm{i}}\left({\mathrm{P}}_{\text{EV}}^{\max }\right)\tan \left(\mathrm{\phi }\right)$$ (7)

The ${\text{NV}}_{\mathrm{i}}$ shows the number of EVs in ${\mathrm{N}}_{\mathrm{i}}\text{at}\text{the}$ ${\mathrm{i}}^{\text{th}}$ optimal node. ${\mathrm{N}}_{\mathrm{i}}$ ${\text{PL}}_{\text{EV}}$ and ${\mathrm{N}}_{\mathrm{i}}{\text{QL}}_{\text{EV}}$ are the actual and load of EV batteries in the node, respectively.

2.2. The competitive algorithm of the developed particles

Placement of distributed generation without technical planning and optimal placement leads to both economic problems for the parking lot investor and technical problems for the distribution network operator. The concept of vehicle-to-network was used to present a revenue and cost model to contribute to the ancillary and regulatory services markets. In this study, the CSO method is motivated by PSO, and the plan and development are unique. PSO particles update their speed and position with social learning and using p_best and g_best. In this sense, both indicate the best position of each solution on the respective path and the best overall position [35]. However, in the CSO design, the g_best and p_best were removed and a competition system between particle pairs was introduced. The CSO process is shown in Fig. 3.

Fig. 3.

Competitive mechanism in CSO algorithm.

In this study, Fig. 3 shows that two particles are selected from the ${\mathrm{P}}_{\mathrm{t}}$ population and then compete in each iteration. The winning and losing solutions are generated in the competitive process, and the values of the objective function of the losers are larger than those of the winners (in the minimization problem). The winners and losers are then placed in ${\mathrm{P}}_{\mathrm{t}+1}$ to generate a new population for the next iteration. In this study, ${\mathrm{P}}_{\mathrm{t}}$ represents the particles in the current iteration of t. Assume that the dimensions of the particles are equal to n, the position of the particles is determined by ${\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)$, and the velocity of the particles is determined by ${\mathrm{V}}_{\mathrm{i}}\left(\mathrm{t}\right).{\mathrm{P}}_{\mathrm{t}}$ swarm is divided into N/2 pairs and in each generation the competition is performed N/2 times. In the kth contest of the tth iteration, the loser updates its position and velocity by Ref. [35]:

$${\mathrm{V}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}+1\right)={\mathrm{R}}_1\left(\mathrm{k}.\mathrm{t}\right){\mathrm{V}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}\right)+{\mathrm{R}}_2\left(\mathrm{k}.\mathrm{t}\right)\left({\mathrm{X}}_{\mathrm{w}.\mathrm{k}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}\right)\right)+\mathrm{\phi }{\mathrm{R}}_3\left(\mathrm{k}.\mathrm{t}\right)\left({\mathrm{X}}_{\mathrm{k}}^{\prime }\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}\right)\right)$$ (8)

$${\mathrm{X}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}+1\right)={\mathrm{X}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}\right)+{\mathrm{K}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}+1\right)$$ (9)

Where ${\mathrm{X}}_{\mathrm{w}.\mathrm{k}}\left(\mathrm{t}\right)$ and ${\mathrm{X}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}\right)$ indicate the locations of the losers and winners, and the velocity is given by ${\mathrm{V}}_{\mathrm{l}.\mathrm{k}}\left(\mathrm{t}\right)$. ${\mathrm{X}}_{\mathrm{k}}^{\prime }\left(\mathrm{t}\right)$ is ${\mathrm{P}}_{\mathrm{t}}$ position, and $\mathrm{\phi }$ is the index just set in the method that controls the effects of ${\mathrm{X}}_{\mathrm{k}}^{\prime }\left(\mathrm{t}\right)$ in the method controls. In this procedure, all solutions that are repeated have a chance to competite and the winner goes directly to the ${\mathrm{P}}_{\mathrm{t}+1}$ generation. The loser goes to the ${\mathrm{P}}_{\mathrm{t}+1}$ swarm after updating the speed and position. The setting data of the algorithm are set to 1. In this study, compared to the PSO variables, the CSO significantly reduced the configuration overhead and improved the compatibility and efficiency of the algorithm.

Since the information is irregularly distributed in the search space, the use of a new operator such as chaos theory leads to much faster convergence and correctness of the final answer. The formulation of a dimensional problem based on the logistic method results from Ref. [36]:

$$\mathrm{X}\left(\mathrm{k}+1\right)=\text{AX}\left(\mathrm{k}\right)\left(1-\mathrm{X}\left(\mathrm{k}\right)\right)$$ (10)

Where, x(k) denotes the state in iteration k and A is the branching index. Fig. 4 shows that the dynamics of this function depends strongly on the index A. If one changes changing A, the periodic behavior becomes chaotic. If one sets A = 4, the function has a chaotic behavior.

Fig. 4.

Branching diagram of the logistic function for the change of the index A.

2.3. Optimum sizing and level determination of SCs and DGs

In this work, the improvement of the power, the minimization of the losses, and the improvement of the voltage profile of the system are considered. In this study, four fuzzy models have been introduced, which are presented below.

2.3.1. (S/S) power factor of the fuzzy (μ_PF)

DGs mainly operate with a power factor of 0.95 (PF). Therefore, this article aims to improve the power factor (S/S) of the substation to 0.95 [15]:

$$\text{PF}=\cos \left[\frac{{\mathrm{S}}_{\text{KW}}}{{\mathrm{S}}_{\text{KVA}}}\right]$$ (11)

Where ${\mathrm{S}}_{\text{KW}}$ and ${\mathrm{S}}_{\text{KVA}}$ show powers of substation [15]:

$${\mathrm{S}}_{\text{KW}}=\sum _{\mathrm{i}=1}^{\text{BN}}{\text{PL}}_{\mathrm{i}}+{\text{APL}}^{\text{DGSC}}-\sum _{\mathrm{j}=1}^{\text{NDG}}{\text{PDG}}_{\mathrm{j}}$$ (12)

Where ${\text{PDG}}_{\mathrm{i}}$ is the DG's capability, and NDG denotes numbers that start DG. Also, ${\text{PL}}_{\mathrm{i}}$ is the actual power load and BN is the bus-bar numbers. ${\text{APL}}^{\text{DGSC}}$ is the power dissipation of the power system by of SCs and DGs [15]:

$${\mathrm{S}}_{\text{KVAr}}=\sum _{\mathrm{i}=1}^{\text{BN}}{\text{QL}}_{\mathrm{i}}+{\text{QPI}}^{\text{DGSC}}-\sum _{\mathrm{j}=1}^{\text{NSC}}{\text{QC}}_{\mathrm{j}}-\sum _{\mathrm{k}=1}^{\text{NDG}}\left({\text{PDG}}_{\mathrm{K}}\times \tan \left({\mathrm{\phi }}_{\text{DG}}\right)\right)$$ (13)

Where ${\text{QL}}_{\mathrm{j}}$ denotes reactive load in the node, NSC denotes SC. ${\text{QPI}}^{\text{DGSC}}$ is equal to the reactive power losses from the power at DG and SC installation. ${\mathrm{\phi }}_{\text{DG}}$ refers to the factor angle of DG installations [15]:

$${\mathrm{S}}_{\text{KVA}}=\sqrt{{\left({\mathrm{S}}_{\text{KW}}\right)}^2+{\left({\mathrm{S}}_{\text{KVAr}}\right)}^2}$$ (14)

The fuzzy performance shown in Fig. 5 for the power factor S/S (PF) can be given as follows [15]:

$${\mathrm{\mu }}_{\text{PF}=}\{\begin{array}{c}0PF\le {\text{PF}}_{\min }\\ \frac{\left(\text{PF}-{\text{PF}}_{\min }\right)}{\left({\text{PF}}_{\mathrm{D}}-{\text{PF}}_{\min }\right)}{\text{PF}}_{\min }\le PF\le {\text{PF}}_{\mathrm{D}}\\ \begin{array}{l}\frac{\left({\text{PF}}_{\max }-\text{PF}\right)}{\left({\text{PF}}_{\max }-{\text{PF}}_{\mathrm{D}}\right)}{\text{PF}}_{\mathrm{D}}\le PF\le {\text{PF}}_{\max }\\ 1PF\ge {\text{PF}}_{\max }\end{array}\end{array}$$ (15)

Where ${\text{PF}}_{\mathrm{D}}$ is the value of the desired power factor. ${\text{PF}}_{\min }$ = 0.85, PFD = 0.95 and ${\text{PF}}_{\max }$ = 1.0 set:

Fig. 5.

Fuzzy membership function for S/S power factor.

2.3.2. Fuzzy permeability criterion of the distributed generation resources ${\mu }_{PDGI}$

DG penetration index (PDGI) was expressed as [15]:

$$\text{PDGI}=\frac{{\sum }_{\mathrm{i}=1}^{\text{NDG}}{\text{PDG}}_{\mathrm{i}}}{{\sum }_{\mathrm{j}=1}^{\text{BN}}{\text{PL}}_{\mathrm{i}}}$$ (16)

In this study, the PDGI is shown in Fig. 6. According to Fig. 6, the fuzzy DG penetration index is the same [15]:

$${\mathrm{\mu }}_{\text{LGDI}=}\{\begin{array}{l}0,\text{PGDI}\le \text{PGD}{\mathrm{I}}_{\min }\\ \frac{\left(\text{PGDI}-\text{PGD}{\mathrm{I}}_{\min }\right)}{\left(\text{PGD}{\mathrm{I}}_{\text{sp}}-\text{PGD}{\mathrm{I}}_{\min }\right)},\text{PGD}{\mathrm{I}}_{\min }\le \text{PGDI}\le \text{PGD}{\mathrm{I}}_{\text{sp}}\\ \frac{\left(\text{PGD}{\mathrm{I}}_{\max }-\text{PGDI}\right)}{\left(\text{PGD}{\mathrm{I}}_{\max }-\text{PGD}{\mathrm{I}}_{\text{sp}}\right)},\text{PGD}{\mathrm{I}}_{\text{sp}}\le \text{PGDI}\le \text{PGD}{\mathrm{I}}_{\max }\\ 0,\text{PGDI}>\text{PGD}{\mathrm{I}}_{\max }\end{array}$$ (17)

Where, ${\text{PGDI}}_{\text{sp}}$ represents the preferred access level of the DG. The target for the DGs is 50% of the actual load. ${\text{PGDI}}_{\min }$, ${\text{PGDI}}_{\text{sp}}$ and ${\text{PGDI}}_{\max }$ are 0.4, 0.5, and 0.6, respectively.

Fig. 6.

Fuzzy membership function for penetration of the distributed generation resources.

2.3.3. Fuzzy active power loss index ${\mathrm{\mu }}_{\text{APLI}}$

In this study, active power losses (APL) are obtained as follows [15]:

$$\text{Apl}=\sum _{\mathrm{i}=1}^{\text{BN}-1}\text{LP}\left(\mathrm{i}\right)$$ (18)

LP (i) denotes the actual power loss for the power system, one obtains [15]:

$$\text{LP}\left(\mathrm{i}\right)=\frac{\mathrm{R}\left(\mathrm{i}\right)\times \left({\mathrm{P}}^2\left(\mathrm{i}+1\right)+{\mathrm{Q}}^2\left(\mathrm{i}+1\right)\right)}{{\left|\mathrm{V}\left(\mathrm{i}+1\right)\right|}^2}$$ (19)

In this study, P(i+1) and Q(i+1) denote the injection of loads into the node ${\left(\mathrm{i}+1\right)}^{\text{th}}$, ${\mathrm{R}}_{\mathrm{i}}$ denotes the ith line, V(i+1) denotes the (i+1)^th voltage node. The APLI is expressed as [15]:

$$\text{APLI}=\frac{{\text{APL}}^{\text{DGSC}}}{{\text{APL}}^{\text{Base}}}$$ (20)

According to Fig. 7, the fuzzy (APLI) and the trapezoidal fuzzy are considered for the power losses ${\mathrm{\mu }}_{\text{APLI}}$ [15]:

$${\mathrm{\mu }}_{\text{APLI}=}\{\begin{array}{c}1forAPLI\le {\text{APLI}}_{\min }\\ \frac{\left({\text{APLI}}_{\max }-\text{APLI}\right)}{\left({\text{APLI}}_{\max }-{\text{APLI}}_{\min }\right)}for{\text{APLI}}_{\min }<APLI\le {\text{APLI}}_{\max }\\ 0forAPLI>{\text{APLI}}_{\max }\end{array}$$ (21)

In this study, the goal is to reduce the active power losses to the previously measured level. Therefore, in this task, ${\text{APLI}}_{\min }$ is chosen over the effective condition and ${\text{APLI}}_{\max }$ denotes one unit. The value of ${\text{APLI}}_{\min }$ can be chosen to achieve the preferred real low power.

Fig. 7.

Fuzzy membership function for active losses.

2.3.4. The fuzzified min and max voltage constrains

As shown in Fig. 8, the ${\mathrm{\mu }}_{\text{vi}}$ of the total node voltages is calculated as follows [15]:

$${\mathrm{\mu }}_{\text{Vi}}=\{\begin{array}{l}0,{\mathrm{V}}_{\mathrm{i}}\le {\mathrm{V}}_{\mathrm{L}1}\\ \frac{\left({\mathrm{V}}_{\mathrm{i}}-{\mathrm{V}}_{\mathrm{L}1}\right)}{\left({\mathrm{V}}_{\min }-{\mathrm{V}}_{\mathrm{L}1}\right)},{\mathrm{V}}_{\mathrm{L}1}<{\mathrm{V}}_{\mathrm{i}}<{\mathrm{V}}_{\min }\\ 1.0,{\mathrm{V}}_{\min }\le {\mathrm{V}}_{\mathrm{i}}\le {\mathrm{V}}_{\max }\\ \frac{\left({\mathrm{V}}_{\max }-{\mathrm{V}}_{\mathrm{i}}\right)}{\left({\mathrm{V}}_{\max }-{\mathrm{V}}_{\mathrm{L}2}\right)},{\mathrm{V}}_{\max }<{\mathrm{V}}_{\mathrm{i}}<{\mathrm{V}}_{\mathrm{L}2}\\ 0,{\mathrm{V}}_{\mathrm{i}}>{\mathrm{V}}_{\mathrm{L}2}\end{array}$$ (22)

Where ${\mathrm{V}}_{\mathrm{L}1}$ = 0.94, ${\mathrm{V}}_{\min }$ = 0.95, ${\mathrm{V}}_{\max }$ = 1.05, and ${\mathrm{V}}_{\mathrm{L}2}=1.06$ are considered. The fuzzy voltage constraint of DG is equal to Ref. [15]:

$${\mathrm{\mu }}_{\mathrm{V}}=\min \left({\mathrm{\mu }}_{{\mathrm{V}}_{\mathrm{i}}}\right)$$ (23)

Fig. 8.

Fuzzy membership function for voltage.

2.4. Fuzzy multiphase performance for the optimal sizing of SCs and DGs

In this study, the general objective fitness based on fuzzy membership coefficients is given by the following equation [15]:

$${\mathrm{J}}_{\mathrm{F}=}{\mathrm{W}}_1{\mathrm{\mu }}_{\text{PGDI}}+{\mathrm{W}}_2{\mathrm{\mu }}_{\text{PF}}+{\mathrm{W}}_3{\mathrm{\mu }}_{\text{APLI}}+{\mathrm{W}}_4{\mathrm{\mu }}_{\mathrm{V}}$$ (24)

Where, the coefficients of W parameters are united. Fuzzy multi-phase performance defined by the previous part was maximized by CSO in operational constraints. Currently, DG shows that the coefficient of the favorable nodes is limited to 25% for the power and also the injected power is limited to 25% for the power demand of the power system [15]:

$$0\le {\text{PDG}}_{\mathrm{i}}\le {\text{PDG}}_{\max }\text{fori}=1,2,3$$ (25)

$$0\le {\text{QC}}_{\mathrm{j}}\le {\text{QC}}^{\max }\text{forj}=1,2,3$$ (26)

${\text{PDG}}_{\mathrm{i}}$, ${\text{QC}}_{\mathrm{j}}$, DG and SC injection are in best positions.

2.5. Applying the CSO algorithm to solve the problem

In this study, this section describes how to find the SCs and DGs data optimally. Since the variables have different constraints, the modified distance calculation can be given [15]:

$$\text{NVT}=\text{NDG}+\text{NSC}+\text{NDGL}+\text{NSCL}$$ (27)

In this study, NDGL and NSCL denote DG and SC sites, respectively. In this sense, 12 variables are selected by sizing and placement. In this study, the 3 variables refer to the size of the DG and the next 3 variables refer to the size of the SC. The distance calculation ${\mathrm{D}}_{\text{ij}}$ for the variables DG and SC are given by Ref. [15]:

$${\mathrm{D}}_{\text{ij}}^{\mathrm{K}}\sqrt{\sum _{\mathrm{K}=1}^{\text{NDG}}{\left({\mathrm{X}}_{\mathrm{j}}^{\mathrm{K}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{K}}\right)}^2+1},\mathrm{k}=1,2,3$$ (28)

$${\mathrm{D}}_{\text{ij}}^{\mathrm{K}}\sqrt{\sum _{\mathrm{K}=\text{NDG}+1}^{\text{NDG}+\text{NSC}}{\left({\mathrm{X}}_{\mathrm{J}}^{\mathrm{K}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{K}}\right)}^2+1},\mathrm{k}=4,5,6$$ (29)

By above equation, the distance between different DG populations can be calculated. The following equations are used to determine the distances between the SC variables. Variables 7 to 9 were selected to optimize DG positions and variables 10 to 12 were selected to optimize SC positions [15]:

$${\mathrm{D}}_{\text{ij}}^{\mathrm{K}}=\sqrt{\sum _{\mathrm{K}=\text{NDG}+\text{NSC}+1}^{\text{NDG}+\text{NSC}+\text{NDGL}}{\left({\mathrm{X}}_{\mathrm{j}}^{\mathrm{K}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{K}}\right)}^2+1},\mathrm{k}=7,8,9$$ (30)

$${\mathrm{D}}_{\text{ij}}^{\mathrm{K}}=\sqrt{\sum _{\mathrm{K}=\text{NDG}+\text{NSC}+\text{NDGL}+1}^{\text{NVT}}{\left({\mathrm{X}}_{\mathrm{j}}^{\mathrm{K}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{K}}\right)}^2+1},\mathrm{k}=10,11,12$$ (31)

In this study, Equation (30) was used to find D_ij of DG, and Equation (31) can find the distance between population members at the optimal locations for SC variables.

2.6. Objective function formation

In this study, a multidimensional Fuzzy operation is used to determine the optimal number of locations and vehicles that have peak load for a simultaneous system. The EV power loss p and voltage profile are presented according to the descriptions. According to Fig. 9, the active power loss is [15]:

$$\text{LPEVI}=\frac{{\text{LPEV}}^{\text{EVDGSC}}}{{\text{LPEV}}^{\text{DGSC}}}$$ (32)

Fig. 9.

Real power loss performance with EV.

where ${\text{LPEV}}^{\text{EVDGSC}}$ denotes the active power loss due to EVs, SCs and DGs and ${\text{LPEV}}^{\text{DGSC}}$ denotes the active power loss with SCs and DGs. The fuzzy membership coefficient is equal to Ref. [15]:

$${\mathrm{\mu }}_{\text{Lpevi}}=\{\begin{array}{l}0,\text{LPEVI}\le {\text{LPEVI}}_{\min }\\ \frac{\left(\text{LPEVI}-{\text{LPEVI}}_{\min }\right)}{\left({\text{LPEVI}}_{\text{sp}}-{\text{LPEVI}}_{\min }\right)},{\text{LPEVI}}_{\min }\le \text{LPEVI}\le {\text{LPEVI}}_{\text{sp}}\\ \frac{\left({\text{LPEVI}}_{\max }-\text{LPEVI}\right)}{\left({\text{LPEVI}}_{\max }-{\text{LPEVI}}_{\text{sp}}\right)},{\text{LPEVI}}_{\text{sp}}\le \text{LPEVI}\le {\text{LPEVI}}_{\max }\\ 0,\text{LPEVI}>{\text{LPEVI}}_{\max }\end{array}$$ (33)

In this study, the power losses of the electric vehicle are increased so that the LPEVI will always be greater than one. In this sense, there is a 50% improvement in power losses at peak times. Moreover, ${\text{LPEVI}}_{\min }$, ${\text{LPEVI}}_{\text{sp}}$ and ${\text{LPEVI}}_{\max }$ are 1.0, 1.5 and 2.0, respectively. The μ_V shown in equation (22) is also valid for the optimal EV charging stations. The fuzzy fits of the sizing and location of EV are [15]:

$${\mathrm{J}}_{\text{FEV}}={\mathrm{W}}_{\mathrm{L}}{\mathrm{\mu }}_{\text{LPEVI}}+{\mathrm{W}}_{\mathrm{V}}{\mathrm{\mu }}_{\mathrm{V}}$$ (34)

Where ${\mathrm{W}}_{\mathrm{L}}$ and ${\mathrm{W}}_{\mathrm{V}}$ are weight coefficients and correspond to 0.9 and 1.2, respectively. The following equation (35) represents the limitation on the power demand of EV in relation to the maximum capacity of each line:

$$\left|p_L^{s.i.vn.ts.td}\right|\le p_{\mathit{max}}^{i.j}\times L_r^{i.j}\forall s\in S.i\in N_{cs}.ts\in TS.td\in TD$$ (35)

In the equation, $p_L^{s.i.vn.ts.td}$ denotes the power demand at the EV, $p_{\mathit{max}}^{i.j}$ represents the maximum capacity of the line, and $L_r^{i.j}$ signifies the gain factor associated with the line. This restriction ensures that the power demand at the EV does not exceed the maximum capacity of each line.

2.7. Wind energy

The energy output from wind power plants is directly correlated with the swiftness of the wind. To effectively model these power plants and address the associated uncertainties, it is imperative to incorporate a modeling framework for wind uncertainty. Wind speed exhibits variations across different seasons, a key feature that the presented model should duly consider. Additionally, the time-dependent nature of wind speed poses another important characteristic. For instance, if the wind speed is high in 1 h, there is a likelihood that it will remain elevated or transition to a lower speed in the subsequent hour. In the effort to model the uncertainty inherent in wind speed, probability functions are employed, particularly focusing on a distinct characteristic associated with high wind speeds. While these probability distributions proficiently capture the probabilities of different wind speeds, they do not inherently account for the temporal dynamics of wind speed fluctuations. To incorporate time dependence, the Markov chain method is utilized, categorizing wind speed values into states with specified ranges. Each time step assigns the wind speed to one of these states, with transition probabilities between states governed by a matrix, determined by the type index [10]:

$$P_{ij}=\frac{n_{ij}}{\sum _j{n}_{ij}}$$ (36)

To generate hourly wind speed time series, an initial step involves creating the cumulative transition probability matrix. Each row in this matrix is derived by summing the values of preceding rows in the transition probability matrix. Knowing the wind speed state at one-time step, the next state can be obtained by generating a random number using a uniform distribution and referencing the cumulative probability matrix. To capture seasonal variations, a Markov chain matrix is crafted for four distinct seasons. Assuming constant wind speed across wind power plants implies either a shared location or identical wind conditions for all capacities. Following wind speed modeling, electric power production is determined based on equation (37), with coefficients a, b, and c calculated from (38), (39), (40) [10]:

$$P_w=\{\begin{array}{l}0\\ P_r\left(a+bV+c{V}^2\right)\\ P_r\\ 0\end{array}\begin{array}{c}\begin{array}{cccc},& 0\le & V& <V\end{array}\\ \begin{array}{cccc},& V_{ci}\le & V& <V_r\end{array}\\ \begin{array}{cccc},& V_r\le & V& <V_{co}\end{array}\\ \begin{array}{cccc},& & V>& V_{co}\end{array}\end{array}\}$$ (37)

$$a=\frac{1}{{\left(V_{ci}-V_r\right)}^2}\left(V_{ci}\left(V_{ci}+V_r\right)-4V_{ci}{V}_r{\left(\frac{V_{ci}+V_r}{2V_r}\right)}^3\right)$$ (38)

$$b=\frac{1}{{\left(V_{ci}-V_r\right)}^2}\left(\begin{array}{cccc}4\left(V_{ci}+V_r\right)& {\left(\frac{V_{ci}+V_r}{2V_r}\right)}^3& -& \left(3V_{ci}+V_r\right)\end{array}\right)$$ (39)

$$c=\frac{1}{{\left(V_{ci}-V_r\right)}^2}\left(2-4{\left(\frac{V_{ci}+V_r}{2V_r}\right)}^3\right)$$ (40)

The minimum wind speed required for the turbine to initiate power generation is a crucial parameter, denoted in meters per second Vci (m/s). On the other hand, the upper limit of wind speed represents the threshold beyond which the turbine ceases to operate effectively, also measured in meters per second Vco (m/s). Vr denotes the rate of wind power.

3. Simulation results

In this study, the 69-bus system is used to propose model and algorithm. The structure of this system is shown in Fig. 10. The system in question has a size of 12.66 KV and its information can be found in Ref. [37]. Data are presented in Appendix A. First, the optimal dimension and proportion of SCs and DGs were determined and, then, three nodes were determined for the placement of DG plants. Optimal of SCs, DGs are optimal positions accessed with fuzzy CSO were shown in Table 1.

Fig. 10.

The under-study 69 bus system.

Table 1.

Optimal sizing of SCs and DGs with fuzzy CSO.

Node	DG	Node	SC
	Sizing (KW)		Sizing (kVAr)
22	761.38	69	105
63	451.47	2	675
64	688.26	52	675

3.1. Improving performance via SC and DG

The multi-objective fuzzy function was described in the previous sections. GA, PSO, and CSO methods are investigated for the optimal sizing of SCs and DGs. The fuzzy CSO model was compared with the fuzzy method GA [37], and the PSO method for obtaining the properties of CSO was considered in Table 2. In this study, moreover, it is able to achieve better substation factor and DG diffusion, GA and other fuzzy-based techniques.

Table 2.

69 node power system.

69 bus	Base Case	Conventional [31]	GA	PSO	CSO
S/S power (kW)	4027.19	2075.69	1938.24	1930.12	1928.12
S/S power (KvAr)	2796.77	1281.10	640.22	641.09	631.01
S/S Power	0.8214 lag	0.8510 lag	0.9500 lag	0.9500 lag	0.9500 lag
DG (kW)	…	1766.91	1901.10	1905.45	1901.01
power Loss (kW)	225.00	40.42	37.76	33.43	27.23
Voltage min (V(p.u.))	0.9092	0.9690	0.9731	0.9729	0.9783

A CSO voltage profile using other methods is presented in Fig. 11. Voltage profile is improved by combining DGs and SCs. The optimal voltage profile, i.e. voltage at each node modified beyond the preferred value of 0.95 p.u. Through the fuzzy PSO, GA, and CSO methods, for a 69-bus system, it is achieved that with the proposed technique, the voltage at each node is close to unity.

Fig. 11.

Comparison of voltage profiles for a 69-bus system.

The convergence comparison of different fitness exposed by the number of iterations in Fig. 12, one can see that CSO has much better convergence properties than PSO and GA.

Fig. 12.

Convergence evaluation of PSO and GA and CSO for 69-bus test system.

3.2. CSO sizing by optimal sizing

In this study, the CSO method has superior convergence and therefore, it was selected for the optimal measurement. In addition, EV charging was targeted in an integrated system with SCs and DGs. EV and peak distribution loads were determined. Based on the charging curve shown in Fig. 2, it is clear that the maximum charging demand for lithium-ion batteries at constant charge is 6.5 kW, as shown in Table 3. The optimal value for the locations and number of EVs by CSO are shown in Table 4, which illustrates the actual response load in terms of the EV charge for the 69-bus system.

Table 3.

Optimum sizing of 69-bus test system EV station.

Bus for EV	Number of EVs
47	45
36	21
66	36
42	55
19	33
Total No of EVs	190

Table 4.

Load consumed by EV battery in 69-bus system.

EV Load	Power Load (kW)	Power Load (KvAr)
Initial EV Load	1235.00	404.53
25% higher of EV	1560.00	511.87
50% higher of EV	1859.00	610.18

3.3. Distribution power system expansion factor study

The developed fuzzy CSO based on a two-stage system is shown that we have what to do. In the conventional simultaneous multi-objective approach, no parameter can be considered to achieve the preferred performance of the distribution power system, but it can be achieved by fuzzy functions. In general, the real power loss is 105.82 kW at full load and 41.02 kW in the proposed two-stage method [38]. Distribution power demand growth for 69-bus test system in listed in Table 5. In this study, Fig. 13 shows the effect of increasing demand in the distribution network while keeping the primary EV charging demand constant for the 69 busbar systems. From the figure, it can be seen that the optimal placement of SCs and DGs can minimize the losses. In this context, the active power is very high. In this study, the feeder load of the distribution power system is modified with SCs, DGs and the EV charging load is utilized. The loss minimization is many times better than the distribution grid.

Table 5.

Distribution power demand growth for 69-bus test system.

69 bus	Power Loss (kW) [38]	CSO approach	Minimum Voltage (p.u.) [38]	Two CSO approach
1.0	105.82	41.02	0.9375	0.9762
1.1	134.48	59.11	0.9279	0.9745
1.2	168.72	82.19	0.9182	0.9698
1.3	208.90	110.21	0.9083	0.9536
1.4	255.39	143.37	0.8984	0.9502
1.5	308.62	182.42	0.8877	0.9376

Fig. 13.

The result of the increase in the power distribution demand in the 69-bus test system.

3.4. EV demand factor analysis

In this study, the effect of electrical charging demand on the performance of a 69-bus distribution system, considering DG and SC is presented. Since the performance of the fuzzy CSO is better, SCs were proposed. Table 6 shows the effect of increasing EV load on the lowest node voltage under different load conditions. The values of SCs and DGs in many conditions are selected. According to Table 6, the voltage profiles can be increased up to 50% when the SCs and DGs of EV load are taking to account and the system performance can be considered as standard. Additionally, the effect of increasing the load of EVs due to energy losses is shown in Table 7. From Table 7, it can be seen that by increasing the load of electric vehicles, the losses increase. Moreover, through the DGs and SCs, the EVs are also built up under base conditions.

Table 6.

Effect of EV demand on voltage (p.u.).

69-bus	Base [15]	DG and SC	EV	25% higher EV	50% higher EV
0.4	0.9656	0.9898	0.9874	0.9852	0.9836
0.5	0.9567	0.9876	0.9865	0.9829	0.9818
0.6	0.9476	0.9864	0.9859	0.9810	0.9802
0.7	0.9383	0.9845	0.9843	0.9784	0.9765
0.8	0.9288	0.9832	0.9817	0.9776	0.9749
0.9	0.9191	0.9803	0.9786	0.9739	0.9735
1.0	0.9092	0.9798	0.9654	0.9728	0.9702

Table 7.

Effect of EV load on actual power loss (kW).

69-bus	Base [15]	DG and SC	EV	25% higher EV	50% higher EV
0.4	32.51	4.12	13.21	18.12	23.38
0.5	51.61	6.98	16.15	21.29	27.02
0.6	75.53	9.34	20.11	25.02	31.03
0.7	104.54	13.98	24.28	29.52	35.23
0.8	138.90	17.18	29.37	34.67	41.21
0.9	178.95	21.23	34.25	40.36	46.38
1.0	225.00	26.28	40.23	46.32	53.54

Table 8 shows the active power of an electric vehicle under different loading conditions. In this table, it is also clear that the support of in this table that by supporting the SCs and DGs limits the actual power supply below the interruption of the actual charging power of the EV load to full charge. It is possible to supply active power loading to the 69 bus-bar system. Table 9 shows the power over the EV charging at different demand conditions in the distribution system. The required reactive power of the interruption is higher than that assumed in all load conditions.

Table 8.

Effect of EV demand on the actual power (kW) required by the station.

69 bus	Base [15]	DG and SC	EV	25% higher EV	50% higher EV
0.4	1553.25	764.12	2008.11	2338.21	2642.09
0.5	1952.70	957.10	2201.87	2531.74	2836.25
0.6	2356.84	1150.25	2395.25	2725.32	3030.36
0.7	2766.07	1343.16	2589.84	2920.19	3225.27
0.8	3180.96	1537.14	2785.23	3115.21	3420.35
0.9	3600.92	1732.26	2979.45	3311.13	3616.29
1.0	4027.19	1928.13	3176.93	3507.26	3813.16

Table 9.

EV effects on power (KwAr) required via 69 bus-bar system.

69 bus	Base [15]	DG and SC	EV	25% higher EV	50% higher EV
0.4	1092.69	248.23	659.11	769.23	870.11
0.5	1370.85	311.19	723.28	833.02	934.28
0.6	1651.20	347.11	786.17	896.12	998.54
0.7	1933.83	437.28	851.25	961.17	1062.05
0.8	2218.88	501.35	915.56	1024.43	1127.23
0.9	2506.48	565.25	980.29	1089.89	1191.54
1.0	2796.77	630.14	1045.21	1155.12	1258.19

3.5. Transient response analysis

In this study, Fig. 14 shows the effects of EV charging on the bus distribution system voltage when the batteries switch from full discharge to full charge. The voltage profile depends on the loading positions of the system in the node and the EVs when charging is interrupted. It can be seen that by installing DG of shunt capacitors, the output voltage is maintained at a reasonable level.

Fig. 14.

Transient voltage at EV charging stations.

3.6. Impact of DG power fluctuations

In this study, the power of the load distribution is constantly changing. Accordingly, DGs are associated with uncertainties in renewable energy sources. The impact of these uncertainties on the system performance in charging EVs is important. Load variations are considered and 50% of DG is achievable, the rest we randomly combined into the DG mix. The distribution load changes and their effects are shown in Fig. 15(a–f) for the 69-bus test system. From the figure, it can be seen that the distribution system load change is more affected by the voltage and actual power losses. DG uncertainty causing variations consistent with the provided power-related changes, which means that the distribution system voltage profiles have little impact, compared to the actual power loss changes.

Fig. 15.

Distribution power system demand changes and DG randomness of the DGs, (a) load factor profile, (b) DG1 power, (c) DG2 power, (d) DG3 power, (e) minimum voltage value and (f) magnitude of loss, all based on battery charging time.

In this study, the purpose of integrating DG into the power grid is to provide a continuous charging voltage with a conventional load. The hybrid system, which consists of solar and wind power systems, depends on weather conditions, and it is important to consider the uncertainties in order to achieve an efficient charging load. The power generation through the solar DG system is not enough to meet the extreme demand; the charging system should be maintained by ESS and diesel sets.

3.7. Analysis of the proposed algorithm

To confirm the capabilities of the proposed model on standard test functions, a comparison was made between the standard harmonic search algorithm, particle swarm, and genetics. We have chosen a larger domain that has smooth parts that significantly complicate the search of the algorithm:

$$\begin{array}{l}\mathrm{f}\left(\mathrm{x}\right)=-{\sum _{\mathrm{i}=1}^{\mathrm{n}}\sin \left({\mathrm{x}}_{\mathrm{i}}\right)\left(\sin \left(\frac{\mathrm{i}\times {\mathrm{x}}_{\mathrm{i}}^2}{\mathrm{\pi }}\right)\right)}^{2\mathrm{m}}\\ \mathrm{m}=10,\mathrm{i}=1,...,\mathrm{n},-50\le {\mathrm{x}}_{\mathrm{i}}\le 50\end{array}$$ (41)

To have a better comparison, the best coefficients were taken from other literature and we considered the same initial population. The convergence of these algorithms is shown in Fig. 16.

Fig. 16.

Convergence of optimization algorithms.

Fig. 17 also shows how the distribution occurs after 30 different implementations with the proposed algorithms. The closeness of responses computed with the proposed algorithm shows its resilience and increased effectiveness. As we all know, the information flow between consumer and producer can have a direct impact on their performance. In future work, in addition to discussing capacitors and distributed generation sources, the presence of FACTs and load management devices will be investigated to determine their viability and increase grid reliability.

Fig. 17.

The best achieved value after 30 different performances.

4. Conclusion

In this study, the increasing use of electric vehicles poses new challenges for distribution system operators. Placement of DG resources and EV parking lots for the distribution network without engineering planning leads to economic problems of the parking lot investor. In this study, the simultaneous placement of distributed generation resources, shunt capacitors and electric vehicles parking in the distribution grid is investigated. The distribution network under study is improved by the presence of parking lots, distributed generation facilities and shunt capacitors. This improving provides the desired reliability and can bring the losses and voltage profiles to an optimal condition. The distribution system is used to find optimal locations from EV stations using the proposed fuzzy method. The theoretical work has shown that the accuracy and speed of convergence are greatly improved.

To solve the model, the competitive particle algorithm is used, which is combined with chaos and fuzzy theory to propose the optimal solution in the best way. The electric vehicles parking charge rate is considered as one of the parameters of the parking model. In this study, the parking charge rate affects the peak consumption of the parking lot and the consumption curve of the parking lot exit. The proposed algorithm based on different charging rates is implemented and then the results are studied. The results show that the determination of the charge rate contributes to the optimal placement of parking lots and distributed production resources–with a view of loss reduction. Investors can determine the coefficients based on the importance of the considered objectives in the investor's decision-making index.

They can present the candidate buses for parking lot installation and implement their goals at the site provided by the distribution system operator. In this study, the effects of EV load pickup in different loading positions were investigated and the obtained results show that the substation accepts the EV to high values for power supply. The voltage steps are reduced to an acceptable level by DGs and SCs even when the EV load increases. The impact of battery charging has a great impact on the voltages and when charging, the voltages are brought to an acceptable level by the support of DGs and SCs support. As we all know, the information flow between the consumer and the generator can have a direct impact on the performance of the two. In future work, in addition to discussing capacitors and distributed generation sources, the presence of FACTs and load management devices will be investigated to determine the viability and increase in grid reliability.

CRediT authorship contribution statement

Shifeng Wang: Data curation, Investigation, Methodology, Resources, Software. Zhixiang Li: Conceptualization, Data curation, Formal analysis, Investigation, Methodology. Mohammad Javad Golkar: Resources, Writing – original draft, Writing – review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. System data

Table A.

Data of some distribution networks.

Br. no. (jj)	IS(jj)	IR(jj)	(Ω)	(Ω)	Nominal load
					Receiving end node
					PLo kW)	QLo (kVAr)
1	1	2	0.0005	0.0012	0	0
2	2	3	0.0005	0.0012	0	0
3	3	4	0.0015	0.0036	0	0
4	4	5	0.0251	0.0294	0	0
5	5	6	0.366	0.1864	2.6	2.2
6	6	7	0.3811	0.1941	40.4	30
7	7	8	0.0922	0.047	75	54
8	8	9	0.0493	0.0251	30	22
9	9	10	0.819	0.2707	28	19
10	10	11	0.1872	0.0619	145	104
11	11	12	0.7114	0.2351	145	104
12	12	13	1.03	0.34	8	5
13	13	14	1.044	0.345	8	5.5
14	14	15	1.058	0.3496	0	0
15	15	16	0.1966	0.065	45.5	30
16	16	17	0.3744	0.1238	60	35
17	17	18	0.0047	0.0016	60	35
18	18	19	0.3276	0.1083	0	0
19	19	20	0.2106	0.069	1	0.6
20	20	21	0.3416	0.1129	114	81
21	21	22	0.014	0.0046	5	3.5
22	22	23	0.1591	0.0526	0	0
23	23	24	0.3463	0.1145	28	20
24	24	25	0.7488	0.2475	0	0
25	25	26	0.3089	0.1021	14	10
26	26	27	0.1732	0.0572	14	10
27	3	28	0.0044	0.0108	26	18.6
28	28	29	0.064	0.1565	26	18.6
29	29	30	0.3978	0.1315	0	0
30	30	31	0.0702	0.0232	0	0
31	31	32	0.351	0.116	0	0
32	32	33	0.839	0.2816	14	10
33	33	34	1.708	0.5646	9.5	14
34	34	35	1.474	0.4873	6	4
35	3	36	0.0044	0.0108	26	18.55
36	36	37	0.064	0.1565	26	18.55
37	37	38	0.1053	0.123	0	0
38	38	39	0.0304	0.0355	24	17
39	39	40	0.0018	0.0021	24	17
40	40	41	0.7283	0.8509	1.2	1
41	41	42	0.31	0.3623	0	0
42	42	43	0.041	0.0478	6	4.3
43	43	44	0.0092	0.0116	0	0
44	44	45	0.1089	0.1373	39.22	26.3
45	45	46	0.0009	0.0012	39.22	26.3
46	4	47	0.0034	0.0084	0	0
47	47	48	0.0851	0.2083	79	56.4
48	48	49	0.2898	0.7091	384.7	274.5
49	49	50	0.0822	0.2011	384.7	274.5
50	8	51	0.0928	0.0473	40.5	28.3
51	51	52	0.3319	0.1114	3.6	2.7
52	9	53	0.174	0.0886	4.35	3.5
53	53	54	0.203	0.1034	26.4	19
54	54	55	0.2842	0.1447	24	17.2
55	55	56	0.2813	0.1433	0	0
56	56	57	1.59	0.5337	0	0
57	57	58	0.7837	0.263	0	0
58	58	59	0.3042	0.1006	100	72
59	59	60	0.3861	0.1172	0	0
60	60	61	0.5075	0.2585	1244	888
61	61	62	0.0974	0.0496	32	23
62	62	63	0.145	0.0738	0	0
63	63	64	0.7105	0.3619	227	162
64	64	65	1.041	0.5302	59	42
65	11	66	0.2012	0.0611	18	13
66	66	67	0.0047	0.0014	18	13
67	12	68	0.7394	0.2444	28	20
68	68	69	0.0047	0.0016	28	20