Optimal Integration of New Technologies and Energy Sources into Radial Distribution Systems Using Fuzzy African Vulture Algorithm

Abstract

In the contemporary global context, excessive fossil fuel consumption remains a critical issue, particularly within the transportation sector. Electric vehicles offer a promising alternative due to their durability and reduced greenhouse gas emissions. However, their rapid adoption has introduced significant challenges, including increased network power losses, deteriorating voltage profiles, and declining substation power factors. This study proposes an approach that integrates fuzzy objective optimization with African Vulture Optimization (AVO) to determine the optimal sitting and sizing of distributed generations (DG), shunt capacitors (SC), and electric vehicle charging stations (EVCS) within radial distribution systems (RDS). The proposed methodology is evaluated on the standard IEEE-69 bus RDS. A detailed comparative analysis between the proposed simultaneous optimization approach for DGs, SCs, and EVCSs and with the traditional two-staged method is presented. The findings indicate that the proposed strategy not only matches but surpasses the performance of existing strategies for the reduction of power losses and enhancement of bus voltage profiles. Key findings include a significant reduction in active and reactive power line loss, with losses minimized by 85.90% and 82.15%, respectively. In addition, an improvement in the minimum bus voltage to 0.98 p.u. is also achieved. Thereafter, the proposed issue is solved in different loading scenarios to present the effectiveness of the approach under different operating conditions. This research effectively demonstrates the complexities introduced by EVCS integration and addresses the issue with simultaneous optimal sitting and sizing of DGs, SCs, and EVCSs that significantly enhance the sustainability and efficiency of RDS.

1. Introduction

1.1. Motivation

In a period characterized by swift technical progress and increasing worldwide apprehensions regarding environmental sustainability, the energy sector is experiencing a substantial transition. At the forefront of this shift are electric vehicles (EVs), which are increasingly replacing conventional gasoline-powered automobiles. This transformation is primarily motivated by the necessity to diminish carbon dioxide emissions and confront the impending exhaustion of fossil fuel reserves. However, the widespread adoption of EVs poses complex challenges to electrical distribution systems, particularly in terms of their impact on energy demand and grid stability. The increased deployment of EVs is reshaping energy consumption patterns and introducing a range of technical challenges for existing distribution infrastructures. These challenges include elevated peak load demands, amplified feeder currents, and the consequent degradation of voltage profiles within the network. The situation becomes even more complex with the contemporaneous integration of distributed generation (DG) sources into the grid, where poorly planned DG placements can exacerbate power losses and cause undesirable voltage fluctuations.

Building on insights from previous research, it is evident that the interaction between EVs, DGs, and SCs in radial distribution systems (RDS) requires careful attention. These elements are interdependent, and any misalignment can lead to suboptimal performance, including increased power losses and compromised voltage stability [1]. The existing literature emphasizes the importance of an integrated approach to energy management within distribution networks. Such an approach is crucial for improving energy efficiency, minimizing losses, and ensuring reliable, high-quality power delivery to consumers.

In response to these challenges, this research aims to contribute to the ongoing discussion on energy management in RDS. Specifically, we focus on the optimal sitting of DGs, SCs, and electric vehicle charging stations (EVCSs) in RDS. To tackle this intricate challenge, we propose a novel fuzzy multi-objective approach combined with the African Vulture Optimization Algorithm (AVOA). This methodology seeks to balance multiple objectives, including improving network efficiency, reducing power losses, improving voltage profiles, and facilitating the smooth incorporation of EVs.

Our motivation for this research stems from a strong commitment to addressing contemporary challenges in energy management. This work was carried out to advance the field of distribution system optimization by providing practical solutions that can help create a greener, more efficient, and robust energy system. By leveraging the potential of EVs, DGs, and SCs, this work aspires to contribute to the development of modern, environmentally conscious electrical RDS.

1.2. Literature Review

The optimal sitting and sizing (OSS) of DGs in RDS is a crucial aspect of modern energy system design due to its role in boosting operational efficiency, reliability, and sustainability. The optimal integration of DGs in RDS is carried out considering various aspects, such as power loss minimization, voltage profile improvement, and cost-effective operation. Various strategies and algorithms, including classical approaches and advanced metaheuristics, have been proposed in the past to address these challenges [2,3]. The authors in ref. [4] proposed a comprehensive framework that integrates the analytic hierarchy process with particle swarm optimization (PSO) to optimize DG placement. This approach successfully balances technical, economic, and environmental objectives, with its practical applicability demonstrated through a case study involving a 51-bus RDS. In relation to distribution network expansion, an extensive model that considers multiple factors, including costs, power procurement, energy dissipation, DG investments, operational expenses, and environmental impacts was studied in ref. [5]. This study also addresses uncertainties in DG integration using probabilistic methods, employing probability distributions to account for potential variability. This research presents an enhanced harmony search algorithm, which is compared with a genetic algorithm and the standard PSO, demonstrating its effectiveness in managing the complex objectives associated with network expansion. Another significant contribution is made by the authors in ref. [6], who present a multi-objective opposition-based chaotic differential evolution technique designed to optimize DG placement. This method focuses on minimizing power losses, optimizing economic outcomes, and mitigating voltage deviations on IEEE standard 33-bus and 69-bus RDS. In ref. [7], the authors introduced the improved Salp swarm algorithm (ISSA) for the OSS of DGs in RDS, enhancing system performance by reducing power loss and costs, and improving voltage profiles. The ISSA improves on the traditional Salp swarm algorithm with updated equations for better exploration that can avoid premature convergence, which were tested on Iraqi-36, IEEE-33, and IEEE-69 bus RDS. The ISSA outperforms traditional methods, achieving significant cost savings, power loss reduction, and system stability improvements, proving its superiority over genetic algorithms and other conventional approaches.

The shift towards renewable energy generation is increasingly important in DG integration due to its environmental advantages, as well as technical and economic benefits. In ref. [8], the authors proposed a novel method for the OSS of photovoltaic DG units in RDS. This approach employs a loss sensitivity factor and a unique variant of the bat algorithm, significantly curtailing power losses and enhancing voltage profiles. The method outperforms other optimization techniques in tests piloted on IEEE 33-bus and 69-bus RDS. Similarly, ref. [9] introduced an improved PSO (IPSO) algorithm aimed at optimizing the sitting, sizing, and power factor of biogas-fueled DGs in RDS. The IPSO method demonstrates superior performance compared to standard PSO and other recent algorithms, effectively reducing active power loss and voltage deviation while improving the voltage stability of the standard IEEE 33-bus RDS. Additionally, the study presented in ref. [10] outlines a multi-criterion, multi-objective approach for the OSS of DG units, focusing on technical, economic, and environmental indices. By utilizing a combination of sensitivity analysis and the analytic hierarchy process (AHP) for decision-making, this method is verified in a standard IEEE 33-bus RDS. The findings demonstrate that AHP-based scenarios produce superior results compared to non-prioritized base instances, particularly when utilizing a hybrid genetic algorithm (GA) and PSO optimization method.

Shunt capacitors are vital in electrical power distribution systems and are primarily used to enhance the power factor and voltage stability by compensating for the reactive power demand of the loads. Previous studies emphasize their contribution to promoting energy efficiency, minimizing losses, and improving system reliability, especially in RDS. The OSS of shunt capacitors (SCs) has been found to mitigate power losses, improve voltage regulation, and boost overall system performance in various past applications. Moreover, the simultaneous OSS of SCs and DGs in RDS are observed to yield more favorable and advantageous results in comparison to their individual allocation. Recent research has placed significant emphasis on developing methods to efficiently integrate these components, aiming to improve system efficiency, reduce power losses, and enhance voltage profiles in RDS. In ref. [11], various methodologies and algorithms were proposed to determine the OSS of DGs and SCs, with the goal of enhancing overall system functionality. Building on this, ref. [12] introduced a hybrid harmony search algorithm (HSA) that addresses the placement of DGs and SCs in RDS. Similarly, ref. [13] presented the intersect mutation differential evolution (IMDE) algorithm, which focuses on the OSS of DGs and SCs simultaneously. The IMDE method effectively minimizes power loss and operational costs, demonstrating superior performance when compared to other optimization techniques. Addressing the complexities introduced by nonlinear loads and inverter-based DGs, ref. [14] proposed the use of the biogeography-based optimization algorithm for simultaneously finding the optimal position and size of DGs and SCs. In another advancement, ref. [15] presented a hybrid optimization method that combines analytical and metaheuristic techniques utilizing the Salp swarm algorithm for optimizing DG and SC placements in RDS. This method exhibits improved power loss reduction compared to existing approaches on standard IEEE 33-bus, 69-bus, and 94-bus Portuguese RDS. Further contributions are made in ref. [16], which proposes an OSS strategy for renewable energy distributed generation (RE-DG) and SCs. The study employs the artificial bee colony algorithm and demonstrates its effectiveness in public hospital distribution systems. The results show a significant reduction in power losses, improved voltage profiles, and economic benefits through the optimized integration of RE-DG units and SCs. The authors in ref. [17] introduced an optimization method using the energy valley optimizer for the strategic placement of DG units and SCs, minimizing energy loss, improving voltage stability, and promoting sustainability. Tested on the ALG-AB-Hassi-Sida 157-bus network, it outperformed existing algorithms, demonstrating superior technical and economic efficiencies while significantly reducing energy costs over 24 years.

In an era marked by escalating concerns over the finite supply of fossil fuels and the urgent imperative to combat environmental issues linked to greenhouse gas emissions, the shift from conventional vehicles to electric vehicles (EVs) represents a critical milestone on the path to a more sustainable future. This transformative shift has catalyzed extensive research into the electrification of transportation systems, with a particular emphasis on optimizing the sitting of electric vehicle charging stations (EVCSs). These charging stations are foundational to the broader adoption of EVs, mitigating the challenges associated with range limitations and promoting the seamless integration of electric mobility into our daily lives. As the EV market continues to surge, the need for a meticulously planned network of EVCSs has become increasingly apparent. The research in ref. [18] delved into the intricate landscape of optimal EVCS placement, encompassing a comprehensive exploration of diverse methodologies, objective functions, constraints, and strategies to provide valuable insights that can aid in the efficient deployment of charging infrastructure. The growing demand for EVs necessitates strategically positioned EVCS and renewable energy sources (RESs) to address grid stress and enhance sustainability. The authors in ref. [19] presented a review for optimizing the allocation of RES-based DG units and EVCSs in active distribution networks, focusing on multi-criteria decision analysis methods to balance technical, economic, social, and environmental factors. In ref. [20], the authors addressed the stochastic nature of renewable energy sources (RES) and EV load demand introducing a novel method for optimal RES and EVCS siting and sizing, considering various objectives such as power loss reduction and cost minimization using a hybrid GA–PSO approach. The work in ref. [21] focused on optimizing the sitting and sizing of EVCSs in Allahabad, India, addressing the challenges caused by the city’s growing population and the need for alternative transportation. It utilizes a hybrid algorithm combining the features of GA and PSO approaches. The GA-based improved PSO demonstrates superior performance in enhancing the RDS’s voltage profile compared to the GA and PSO individually. Real-time charging navigation for EVs is explored in ref. [22], which proposes a comprehensive optimization model for the joint optimal planning of DGs and EVCSs using a mixed integer second-order cone programming for efficient problem-solving. The challenges posed by integrating fast EVCSs were addressed in ref. [23] on an IEEE-25 unbalanced RDS system using a transient search optimization algorithm. The study demonstrates how DG mitigates the negative impacts of EVCSs by decreasing the power loss and enhancing the voltage profile of the system. The authors in ref. [24] tackled the issues generated in RDS due to the inclusion of EVCSs by optimally integrating renewable energy-based DGs. This study formulates and resolves a multi-objective optimization issue utilizing both PSO and cuckoo search algorithm techniques on IEEE 33-bus radial and meshed distribution systems. The work in ref. [25] explored the placement of DGs with EVs in distribution systems to improve key performance metrics such as power loss, voltage deviation, and short circuit current in a 16-bus system. By using hybrid GA–Monte Carlo simulation optimization, the study optimally pairs DGs and EVs for different load models. The research focuses on enhancing system efficiency and reducing environmental impact, offering valuable insights for future planning in smart grid applications.

Recently, researchers have been focused on the simultaneous OSS of DGs, SCs, and EVCSs in RDS. These approaches provide a holistic solution for enhancing the power quality, minimizing losses, and alleviating voltage levels. In ref. [26], a two-stage optimization framework was introduced, integrating a fuzzy objective strategy with the grasshopper optimization algorithm (GOA) to optimize the sitting and sizing of DGs, SCs, and EVCSs in RDS. The initial stage employs fuzzy GOA to enhance the substation power factor, reduce real power line losses, and increase the voltage profile. Meanwhile, in the second stage, fuzzy GOA optimizes the number and placement of EVCSs, considering pre-allocated DGs and SCs. As research in this field evolves, these optimization techniques will become increasingly essential in the development of smart grids and sustainable power distribution networks.

1.3. Main Contributions

• The African Vulture Optimization Algorithm (AVOA), a newly developed meta-heuristic approach, has demonstrated remarkable effectiveness in tackling a diverse array of engineering and non-engineering problems [27]. However, a review of the literature reveals that AVOA’s potential has not yet been explored for solving the OSS issues of DGs, SCs, and EVCSs. This gap in the research serves as the motivation for employing AVOA to tackle the identified problem. To elevate the system performance in this work, the objective functions are modified using the fuzzy logic approach and, thereafter, the studied AVOA is utilized to obtain the optimal results. The primary contributions of this proposed study are indicated below:

• The AVO algorithm is utilized for the first time for the OSS of DGs, SCs, and EVCSs in RDS, considering various objectives.

• The adopted problem is solved in two ways: (i) a studied two-stage optimal placement approach with EVCS placed after DG and SC deployment and (ii) the proposed simultaneous OSS of DGs, SCs, and EVCSs.

• The performance of both approaches is compared to demonstrate the differences in their effectiveness in solving the adopted issue.

• The proposed method is evaluated under different load scenarios to showcase its adaptability for different conditions.

2. Problem Formulation

This study involves fuzzifying multiple objective functions and subsequently applying the selected techniques to achieve the optimal outcome. The specifics of the objective functions are elucidated in the subsequent subsections.

2.1. Objective Function Formulation

Fuzzy logic has the advantage of allowing the user to select a value of interest from a wide range of possibilities [27]. In this part, fuzzy membership functions are used for finding the distribution system’s performance parameters that converge towards the predefined desired values.

The fuzzy multi-objective function ($J_F$) expressed in terms of considered individual fuzzy objective functions is shown in Equation (1).

(1)$Maximize J_F={\mu }_{PF}+{\mu }_{PDGI}+{\mu }_{APLI}+{\mu }_{VT}+{\mu }_{PLIEV}$

The various fuzzy objective functions are:

• (a). ${\mu }_{PF}$ —Fuzzified substation (S/S) power factor.

• (b). ${\mu }_{PDGI}$—Fuzzified DG penetration.

• (c). ${\mu }_{APLI}$—Fuzzified active power line loss.

• (d). ${\mu }_{VT}$—Minimum of all fuzzy membership values for all discrete node voltages.

• (e). ${\mu }_{PLIEV}$—Fuzzified EVs power loss.

2.2. Structuring of Objective Function

The fuzzified functions allow for the simultaneous consideration of conflicting objectives. This is achieved through fuzzy membership functions, which quantify the degree of satisfaction for each objective. The fuzzy membership functions are used to obtain the distribution system’s performance parameters that converge towards the predefined desired values.

2.2.1. Fuzzification of Substation (S/S) Power Factor (μ_PF)

Equation (2) may be used to determine the fuzzy membership function for the substation (S/S) power factor (PF) illustrated in Figure 1, where $P{F}_D$ is the projected power factor level. In this equation, the values of parameters are taken as PF_min = 0.85, $P{F}_D$ = 0.95, and PF_max = 1.0. Equation (2) presents the triangular membership function for the substation power factor, emphasizing its role in achieving a lagging power factor of 0.95 to ensure efficient grid operation.

(2)${\mu }_{PF}=\left\{\begin{array}{c}0 \mathit{for}PF\le P{F}_{min}\hfill \\ {\displaystyle \frac{\left(PF-P{F}_{min}\right)}{\left(P{F}_D-P{F}_{min}\right)}}\mathit{for} P{F}_{min}\le PF\le P{F}_D\hfill \\ {\displaystyle \frac{\left(P{F}_{max}-PF\right)}{\left(P{F}_{max}-P{F}_D\right)}}\mathit{for} P{F}_D\le PF\le P{F}_{max}\hfill \end{array}\right.$

2.2.2. Fuzzification of DG Penetration (μ_PDGI)

The right-angle triangle function shown in Figure 2 is used to represent the DG’s penetration limit, which may be determined using Equations (3) and (4). The DG penetration indicator (PDGI) is determined by dividing the total installed DG capacity by the total active power load.

(3)$PDGI={\displaystyle \frac{{\sum }_{i=1}^{NDG}PD{G}_i}{{\sum }_{j=1}^{BN}P{L}_i}}$

(4)${\mu }_{PDGI}=\left\{\begin{array}{c}0 \mathit{for} PDGI\le PDG{I}_{min}\hfill \\ {\displaystyle \frac{\left(PDGI-PDG{I}_{min}\right)}{\left(PDG{I}_{SP}-PDG{I}_{min}\right)}}{ \mathit{for} \mathit{PDGI}}_{min}\le PDGI\le PDG{I}_{SP}\hfill \\ {\displaystyle \frac{\left(PDG{I}_{max}-PDGI\right)}{\left(PDG{I}_{max}-PDG{I}_{SP}\right)}}{ \mathit{for} \mathit{PDGI}}_{SP}\le PDGI\le PDG{I}_{max}\hfill \\ 0 \mathit{for} \mathit{PDGI} >PDG{I}_{max}\hfill \end{array}\right.$

Here, the fuzzified design parameters are denoted by $PDG{I}_{max}$ and $PDG{I}_{min}$. $PDG{I}_{SP}$ is designated as a fraction of the total real power load chosen to represent the desired level of DG penetration for the requisite total capacity of DG units [28]. The values of $PDG{I}_{min}, PDG{I}_{SP},$ and $PDG{I}_{max}$ are chosen as 0.4, 0.5, and 0.6, respectively.

2.2.3. Fuzzification of Active Power Line Loss (${\mathsf{\mu }}_{\mathrm{APLI}})$

The active power loss index (APLI) is determined by the ratio of APL with DGs and SCs to the base case scenario, as articulated in Equation (5).

(5)$APLI={\displaystyle \frac{AP{L}^{DGSC}}{AP{L}^{BC}}}$

where $AP{L}^{DGSC}$ and $AP{L}^{BC}$ are the active power losses in the presence of DGs and SCs and with the base case, respectively.

The trapezoidal membership function, depicted in Figure 3, is engaged to represent the objective of APL reduction within the fuzzy domain. This study aims to reduce APL to a level beneath a certain threshold. Accordingly, $APL{I}_{min}$ is determined based on utility requirements, while $APL{I}_{max}$ is set to unity. The fuzzy active power line loss index is articulated mathematically using the following Equation (6).

(6)${\mu }_{APLI}=\{\begin{array}{cc}1 & \mathrm{for} APLI\le APL{I}_{min}\\ {\displaystyle \frac{\left(\mathrm{A}PL{I}_{max}-APLI\right)}{\left(APL{I}_{max}-\mathrm{A}PL{I}_{min}\right)}} & \mathrm{for} APL{I}_{min}\le APLI\le APL{I}_{max}\\ 0& \mathrm{for} APLI>APL{I}_{max}\end{array}$

The fuzzy value ${\mu }_{APLI}$ contributes to the overall objective function by penalizing higher power losses, guiding the optimization algorithm to identify configurations that minimize losses.

2.2.4. Fuzzification of Bus Voltage Profile (${\mathsf{\mu }}_{{\mathrm{V}}_{\mathrm{i}}})$

The bus voltages in the RDS are denoted by the fuzzy membership function (illustrated in Figure 4), which is calculated as per Equation (7).

(7)${\mu }_{V_i}=\left\{\begin{array}{c} 0 \mathrm{for}{V}_i\le V_{L1}\hfill \\ {\displaystyle \frac{\left(V_i-V_{L1}\right)}{\left(V_{min}-V_{L1}\right) }}for V_{L1}<V_i<V_{min}\hfill \\ 1.0 \mathrm{for}{V}_{min}\le V_i\le V_{max}\hfill \\ {\displaystyle \frac{\left(V_{max}-V_i\right)}{\left(V_{max}-V_{L2}\right)}} for V_{max}<V_i<V_{L2}\hfill \\ 0 \mathrm{for}{V}_i>V_{L2}\hfill \end{array}\right.$

The distribution network’s fuzzified voltage profile performance index (${\mu }_{V_i}$) is defined as the minimum of all fuzzy membership values for all discrete node voltages and is denoted by Equation (8). The ${\mu }_{V_i}$ quantifies the satisfaction level of the voltage at any given node based on the predefined acceptable range (i.e., 0.95 p.u.–1.05 p.u.).

(8)${\mu }_{VT}=\min ({\mu }_{V_i})$

2.2.5. Formation of Fuzzy Objective Function for Optimum Placement of EVs (${\mathsf{\mu }}_{\mathrm{PLIEV}})$

A fuzzy multi-objective function, incorporating the objectives of fuzzy real power loss and fuzzy voltage profile, is utilized to decide the optimal number and location of EVs while accounting for the distribution system’s peak load. The power loss index with EVs can be expressed using Equation (9). Referring to Figure 5, the fuzzy active power loss index is represented by Equation (10). The active power loss index with EVs (PLIEV) is characterized as:

(9)$\begin{array}{c}PLIEV={\displaystyle \frac{PLE{V}^{EVDGSC}}{PLE{V}^{DGSC}}}\end{array}$

In Equation (8), $PLE{V}^{EVDGSC}$ represents the real power loss in RDS with EVs, DGs, and SCs, and $PLE{V}^{DGSC}$ denotes the real power loss with DGs and SCs only.

(10)${\mu }_{PLIEV}=\left\{\begin{array}{c}0 \mathrm{for} PLIEV\le PLIE{V}_{min}\hfill \\ {\displaystyle \frac{\left(PLIEV-PLIE{V}_{min}\right)}{\left(PLIE{V}_{SP}-PLIE{V}_{min}\right)}} for PLIE{V}_{min}\le PLIEV\le PLIE{V}_{SP}\hfill \\ {\displaystyle \frac{\left(PLIE{V}_{max}-PLIEV\right)}{\left(PLIE{V}_{max}-PLIE{V}_{SP}\right)}}{ \mathrm{for} \mathrm{PLIEV}}_{SP}\le PLIEV\le PLIE{V}_{max}\hfill \\ 0 \mathrm{for} \mathrm{PLIEV} >PLIE{V}_{max}\hfill \end{array}\right.$

The power loss increases with EV load, and hence, the PLIEV will always be greater than one. The values of PLIEV_min, PLIEV_SP, and PLIEV_max are 1.0, 1.5, and 2.0, respectively.

2.3. Network Constraints

In each iteration of the optimization technique, to obtain the solution for the OSS of considered multi-units in RDSs, several constraints must be satisfied. The constraints considered in this work are listed below.

2.3.1. Equality Constraints

The equality constraints for power balance are presented below:

(11)${\sum }_{i=1}^{DG}{P}_{\mathrm{dg}i}+P_g-APL=P_d$

(12)${\sum }_{i=1}^{DG}{Q}_{\mathrm{dg}i}+{\sum }_{i=1}^{SC}{Q}_{ci}+Q_g-RPL=Q_d$

where $P_{\mathrm{dg}i}$ and $Q_{\mathrm{dg}i}$ are the real and reactive powers generated, respectively, by the DG placed at bus i. $P_g$ and $Q_g$ are the generated real and reactive powers of the network, respectively. $Q_{ci}$ is the reactive power injected at bus i. $P_d$ and $Q_d$ are the total real and reactive power demands, respectively.

2.3.2. Inequality Constraints

Bus voltage constraints: the bus voltages must be kept within the minimum ($V_{\min }$) and the maximum $(V_{max}$) bounds of voltage at each bus and are expressed below.

(13)$V_{min}\le V_i\le V_{max}$

DG size constraint: the active power infused by $D{G}_i$ at the ith bus must be within the minimum limit ($D{G}_{min}$) and the maximum limit ($D{G}_{max}$).

(14)$D{G}_{min}\le D{G}_i\le D{G}_{max}$

SC size constraint: similarly, the size constraint of SC for reactive power compensation may be expressed by Equation (15):

(15)$Q{C}_{min}\le Q{C}_j\le Q{C}_{max}$

where $Q{C}_{min}$ and $Q{C}_{max}$ represent the minimum and the maximum permissible kVAr limits for shunt capacitors.

Power factor constraint: the power factor must remain within the specified ranges, as presented by Equation (16):

(16)$P{F}_{min}\le P{F}_j\le P{F}_{max}$

where $P{F}_{min}$ and $P{F}_{max}$ represent the minimum and maximum PF limits of the installed DGs.

3. Application of AVOA to Solve OSS Problem of DGs, SCs, and EVCSs in RDS

In comparison to analytical approaches, heuristics or meta-heuristics can offer near-optimal solutions in a shorter amount of time for complex non-linear situations. The studied AVOA is one such approach investigated in this work, which is briefly discussed in the subsequent subsection.

3.1. African Vulture Optimization Algorithm

AVOA is a newly established metaheuristic optimization technique inspired by the hunting and foraging actions of African vultures [27]. These vultures, known for their scavenging and opportunistic hunting strategies, exhibit complex social and biological behaviors that make them highly efficient at locating food sources over large areas. By modeling these behaviors, AVOA has been designed to decipher composite optimization problems, particularly in cases where traditional algorithms may struggle to find the global optimal value.

Optimization algorithms like AVOA are essential in fields such as engineering, energy systems, and data science, where solving multi-objective, non-linear, or constrained problems is critical. AVOA joins a class of bio-inspired algorithms such as PSO and GA but distinguishes itself through its focus on the specific dynamics of vulture foraging strategies, particularly their coordinated search patterns and ability to exploit resources efficiently.

3.1.1. Inspiration from Vulture Behavior

African vultures exhibit unique foraging behaviors that are essential to the functionality of AVOA. These birds are known for their strong vision, which allows them to spot carcasses from miles away, as well as their ability to coordinate within a group to optimize their foraging efforts. Key aspects of vulture behavior that inspired the algorithm include:

3.1.2. Group Coordination

Vultures often forage in groups, which increases the chances of locating a food source. Once a carcass is spotted by one vulture, others in the vicinity are quick to join. This collaborative foraging behavior is modeled in AVOA as the vultures communicate and share information about potential solutions (in optimization, this is analogous to global and local searching).

3.1.3. Efficient Exploration and Exploitation Capabilities

Vultures balance exploration (searching for food over large areas) and exploitation (feeding on a discovered food source). This balance is critical in optimization problems, where a good algorithm must explore the solution space thoroughly while exploiting the best solutions found so far.

3.1.4. High Mobility and Sensing Capabilities

African vultures cover vast distances using soaring flight, enabling them to explore large regions with minimal energy. In AVOA, this behavior translates into the ability of virtual vultures to explore a large search space efficiently without becoming stuck in local optima.

3.2. Mechanism of the African Vulture Optimization Algorithm

AVOA emulates the foraging and social behaviors of vultures to efficiently investigate and utilize a solution space. The algorithm is typically initialized with a population of vultures (candidate solutions). Each vulture searches for food (an optimal solution) based on its position in the search space, the best positions found by other vultures, and the proximity to food sources. Different steps involved in AVOA include the following.

3.2.1. Identifying the Best Vulture in Each Group

Upon establishing the starting population, the fitness of each solution is assessed. The highest-performing solution is identified as the primary vulture in the first group, and the second-best solution assumes leadership in the second group. Utilizing Equation (17), the residual solutions are attracted to these optimal solutions, directed by the probabilities ${\phi }_i$ calculated according to Equation (18). The probabilities, determined by parameters ${\theta }_1$ and ${\theta }_2$, are meticulously calibrated before the initiation of the search procedure to guarantee that their sum equals one.

(17)$Bes{t}_{Vul}\left(i\right)=\left\{\begin{array}{c}B{V}_1 if{\phi }_i={\theta }_1\\ B{V}_2 if{\phi }_i={\theta }_2\end{array}\right.$

(18)${\phi }_i={\displaystyle \frac{F_i}{{{\displaystyle \sum }}_{i=1}^n{F}_i}}$

where $Bes{t}_{Vul}\left(i\right)$ is the vulture chosen in the present iteration to advance towards the optimal solution.

3.2.2. The Starvation Rate of Vultures

The hunger rates of vultures markedly influence their foraging behavior, as well-nourished vultures can traverse longer distances. To incorporate this dynamic into the algorithm, Equation (19) models the fluctuation of hunger levels (Hun) over time.

(19)$Hun=(2{\partial }_1+1)\times \rho \times (1-{\displaystyle \frac{{\mathrm{iteration}}_i}{{\mathrm{iteration}}_{\max }}})+(\mathrm{c} \times ({\sin }^{w }({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{{\mathrm{iteration}}_i}{{\mathrm{iteration}}_{\max }}})+\cos ({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{{\mathrm{iteration}}_i}{{\mathrm{iteration}}_{\max }}})-1\left)\right)$

where $\rho$ represents a parameter in the range of [−1, 1] that changes with each iteration, c is a randomly chosen integer within the range of [−2, 2], and ${\partial }_1$ represents a variable having a value ranging from 0 to 1.

3.2.3. Exploration Phase

The exploration phase leverages the vultures’ keen vision and meticulous search strategies. This phase employs two distinct tactics, determined by the parameter, as defined in Equation (20). The vultures’ exploratory behavior is guided by Equations (21) and (22), which delineate their quest for feeding while considering their present levels of satisfaction.

(20)$\phi \left(i+1\right)={\{}_{Equation\left(22\right) \mathrm{if} {\phi }_1 \ge {\partial }_{\phi 1}}^{Equation\left(21\right) \mathrm{if} {\phi }_1 \ge {\partial }_{\phi 1}}$

(21)$\phi \left(i+1\right)={\mathrm{Best}}_{\mathrm{Vul}}\left(i\right)-\left|H\times Bes{t}_{Vul}\left(i\right)-\phi \left(i\right)\right|\times Hun$

(22)$\phi \left(i+1\right)={\mathrm{Best}}_{\mathrm{Vul}}\left(i\right)-Hun+{\partial }_2\left(ub-lb\right){\partial }_3+lb$

where $\phi \left(i+1\right)$ denotes the vulture’s position vector in the succeeding iteration, and H signifies the location to which vultures randomly migrate to safeguard the food from competitors. $\phi \left(i\right)$ is the current location vector of the vulture, and ${\partial }_{\phi 1}$ represents a stochastic value within the range of 0 to 1. ∂₂ denotes a stochastic value inside the interval [0, 1]. ub and lb signify the upper and lower limits of the variables, respectively. ∂3 is utilized to augment the unpredictability of the coefficient.

3.2.4. Exploitation Phase

The exploitation phase, aimed at focusing on promising areas within the search space, occurs in two distinct stages driven by the parameters $\phi 2$ and $\phi 3$. The initial stage of exploitation involves either rotating flight or siege-fight strategies, selected according to parameter $\phi 2$. The strategy to be applied in this first stage is determined by Equation (23). The fight for food is represented by Equation (24), in which vultures vie for accessible food resources. The rotational flight patterns of vultures are depicted in Equation (25), modeling their movement during the rotational flight phase.

(23)$\phi \left(i+1\right)={\{}_{Equation\left(25\right) \mathrm{if} \phi 2 \ge {\partial }_{\phi 2}}^{Equation\left(24\right) \mathrm{if} \phi 2 \ge {\partial }_{\phi 2}}$

(24)$\phi \left(i+1\right)=\left|H\times Bes{t}_{Vul}\left(i\right)-\phi \left(i\right)\right| \left(Hun+{\partial }_4\right)-\left(Bes{t}_{Vul}\left(i\right)-\phi \left(i\right)\right)$

(25)$$\begin{gathered} \phi \left(i+1\right)=Bes{t}_{Vul}\left(i\right)-\left(Bes{t}_{Vul}\left(i\right){\displaystyle \frac{{\partial }_5\times \phi \left(i\right)}{2\pi }}\mathit{cos}(\phi \left(i\right)\right)+ \\ Bes{t}_{Vul}\left(i\right){\displaystyle \frac{{\partial }_6\times \phi \left(i\right)}{2\pi }}\mathit{sin}(\phi \left(i\right))) \end{gathered}$$

The parameters $\phi 2$ and $\phi 3$ are utilized to determine the techniques accessible in the initial and subsequent phases, respectively. The ${\partial }_4$ parameter denotes a random variable ranging from 0 to 1, utilized to augment the random coefficient. Both ${\partial }_5$ and ${\partial }_6$ variables are stochastic numbers inside the interval [0, 1]. The second phase of exploitation entails vultures either congregating around food sources or participating in fierce competition, as dictated by the parameter $\phi 3$, as seen in Equation (26). The vulture movement in the second phase is characterized by Equations (27) and (28), which consider variables such as proximity to food sources and levy flight behavior.

(26)$\phi \left(i+1\right)={\{}_{Equation\left(28\right) \mathrm{if} \phi 3 \ge {\partial }_{\phi 3}}^{Equation\left(27\right) \mathrm{if} \phi 3 \ge {\partial }_{\phi 3}}$

(27)$\phi \left(i+1\right)={\displaystyle \frac{1}{2}}\left(B{V}_1\left(i\right)-{\displaystyle \frac{B{V}_1\left(i\right)\phi \left(i\right)}{B{V}_1\left(i\right)-\phi {(i)}^2}}Hun+B{V}_2\left(i\right)-{\displaystyle \frac{B{V}_2\left(i\right)\phi \left(i\right)}{B{V}_2\left(i\right)-\phi {(i)}^2}}Hun\right)$

(28)$\phi \left(i+1\right)=Bes{t}_{Vul}\left(i\right)-\left|Bes{t}_{Vul}\left(i\right)-\phi \left(i\right))\right|\times Hun\times \mathrm{Levy}\left(d\right)$

where $B{V}_1\left(i\right)$ and $B{V}_2\left(i\right)$ denote the optimal vultures from the first and second groups in the present iteration, respectively; $V_{LF}$ represents the levy flight patterns calculable via Equation (29); d signifies the dimensions of the problem; $\alpha$ and $\beta$ are parameters constrained within the range [0, 1]; and $\mu$ is a constant set to a default value of 1.5.

(29)$V_{LF}\left(x\right)=0.01{\displaystyle \frac{\alpha \times {\left\{\frac{\Gamma \left(1+\mu \right)\times \sin (\pi \mu /2)}{\Gamma \left(1+\mu 2\right)\times \mu \times 2\times \left(\left(\mu -1\right)/2\right)}\right\}}^{1/\mu }}{{\left|\beta \right|}^{\frac{1}{\mu }}}}$

In ref. [27], the effectiveness of the AVO algorithm is rigorously validated using a comprehensive set of benchmark test functions. The results are further supported by an in-depth statistical analysis, highlighting the algorithm’s robustness, optimization accuracy, and reliability across diverse optimization scenarios. The parameter settings of the optimization AVOA are shown in Table 1.

The flow of the proposed methodology is presented in Figure 6, whereas Figure 7 represents the flow chart of the AVO algorithm.

4. Results and Discussions

This study evaluates the efficacy of AVOA in addressing the OSS of DGs, SCs, and EVCSs by utilizing the standard IEEE 69-bus RDS. The nominal voltage of the 69-bus RDS is considered 12.66 kV and its apparent base power is 10 MVA during the calculation. The total base load demand of the system is (3802.1 + j2694.6) kVA. The details on the studied 69-bus RDS may be found in ref. [29]. In the proposed work, the maximum limit for SC sizing at a single location is taken as 600 kVAr with a step size of 100 kVAr. The DG’s maximum limit at a single location is considered 8 MW. The bus voltage constraint is considered within the limit of 0.9 p.u. to 1.1 p.u. For all of the algorithms under study, the population size of the algorithm and the maximum number of iterations are set at 50 and 100, respectively. The proposed solution to the identified problem is implemented using MATLAB^R 2021 software. The system used for running different algorithms consists of an Intel Core i5 processor of the 12th generation with 16 GB RAM. This work considers a maximum of three DG units and the same number of SC units for optimal placements at a single location. Similarly, it is assumed that a maximum of five bus locations may be used for optimal EVCS deployment with a maximum charging capacity of 50 EVs per location. The results are evaluated considering the three cases as mentioned below:

• (a). Case 1: Two-stage multi-objective AVOA-based method for optimal sitting and sizing of DGs, SCs, and EVCSs in IEEE 69-bus RDS. The optimal sitting of DGs and SCs is performed in the first stage and followed by the optimal placement of EVCS in the second stage.

• (b). Case 2: Simultaneous sitting of DGs, SCs, and EVCSs with multi-objective AVOA approach in IEEE 69-bus RDS.

• (c). Case 3: Simultaneous OSS of DGs, SCs, and EVCSs in IEEE 69-bus RDS with varying loading conditions

For each of the mentioned case studies, different optimization methods have been employed, and their respective outcomes are compared to evaluate their efficacies. In the first stage of Case 1, the OSS of DGs and SCs are identified using both fuzzy AVOA and fuzzy AOA. Thereafter, these results are compared against existing approaches found in the literature, including fuzzy GA [26], fuzzy PSO [26], and fuzzy GOA [26]. However, due to limited data availability, the results in the second stage of Case 1 are compared solely between fuzzy AVOA and fuzzy AOA. Similarly, Case 2 also utilizes fuzzy AVOA and fuzzy AOA for simultaneous placement of DGs, SCs, and EVCSs. Their outcomes are evaluated for comparative performance analysis. Here, the word fuzzy appended ahead of each algorithm refers to their application in evaluating fuzzified objectives. In Case 3, the simultaneous OSS of DGs, SCs, and EVCSs in IEEE 69-bus RDS is extended with varying loading conditions utilizing fuzzy AVOA and fuzzy AOA.

4.1. Case 1: Two-Stage OSS of DGs, SCs, and EVCS in IEEE 69-Bus RDS

The OSS of DGs and SCs in an IEEE 69-bus RDS is investigated in this case utilizing a two-stage multi-objective AVOA. The fuzzy AVOA-based method outcomes are compared to fuzzy AOA, fuzzy GA, fuzzy GOA, and fuzzy PSO in the first stage. The comparison of the obtained results is made in Table 2. The fuzzy AVOA obtains optimal DG placements at bus numbers 2, 61, and 17, with a DG capacity of 1225 kW, 379 kW, and 14 kW, respectively. The optimal reactive power compensations offered by DGs are 593 kVAr, 184 kVAr, and 7 kVAr at bus 61, 17, and 2, respectively. The algorithm finds the best placement of SCs at buses 2, 63, and 60, with optimal sizes of 600, 400, and 200 kVAr, respectively. The proposed fuzzy AVOA-based OSS of DGs and SCs significantly reduced the APL from 224.96 kW (base case) to 19.32 kW. This performance surpasses other techniques, where the next best method, fuzzy AOA, achieved an APL of 25.40 kW. In terms of percentage, the reduction in APL was found to be 91.41%, 88.71%, 83.18%, 85.11%, and 87.85%, respectively, under fuzzy AVOA (the proposed), fuzzy AOA (studied), fuzzy GA [26], fuzzy PSO [26], and fuzzy GOA [26]. The fuzzy AVOA also minimized the total RPL from 102.15 kVAr to 13.74 kVAr, whereas fuzzy AOA resulted in RPL reduction of up to 15.29 kVAr. All the studied and compared methods improved the substation power factor from 0.82 (Base Case) to 0.95 p.u. lagging. Additionally, the minimum bus voltage was upraised from 0.91 p.u. to 0.98 p.u. for fuzzy AVOA, outperforming other algorithms such as fuzzy GA and fuzzy PSO, which achieved the minimum voltages of 0.9734 p.u. and 0.9752 p.u., respectively. The comparison of the convergence profile and the enhancement in bus voltage profile may be seen in Figure 8. From Table 2 and Figure 8, it may be observed that the fuzzy AVOA performs better than its other counterparts for stage 1 of Case 1.

In the second stage, EVCSs are optimally placed considering the optimal location and capacity of DGs and SCs to be the same, as found in stage 1. The details of the obtained result for stage 2 of Case 1 may be observed in Table 3 and Figure 9. The proposed fuzzy AVOA optimally places EVCSs at buses 2, 19, 65, 69, and 7, with a total number of EVs of 10, 10, 49, 11, and 49, respectively. The real power demand from EVs at the abovementioned locations amounted to 65 kW, 319 kW, 72 kW, and 319 kW, respectively. After integrating EVCSs, the APL slightly increased to 36.58 kW for fuzzy AVOA, which is a significant improvement compared to the base case value but slightly higher than the result in stage 1 due to the additional power demands imposed due to the EV charging. The fuzzy AOA yielded an APL of 46.06 kW, which is more than the fuzzy AVOA value. The fuzzy AVOA method reduced the RPL to 21.22 kVAr, still far superior to the base case of 102.15 kVAr but slightly higher than in stage 1. Similarly, the fuzzy AOA achieved a RPL of 23.48 kVAr. As in stage 1, the substation power factor in stage 2 remained at 0.95. The minimum bus voltage slightly decreased to 0.97 p.u. in comparison to stage 1, which still remains a significant improvement over the base case of 0.91 p.u.

The results demonstrate the efficacy of the proposed two-stage fuzzy AVOA approach in diminishing power losses, enhancing voltage profiles, and improving power factor in the adopted RDS. In stage 1, the proposed method successfully identifies optimal locations and sizing of DGs and SCs, demonstrating its superiority over conventional techniques. The reduction in APL and RPL is particularly noteworthy, as it directly impacts energy efficiency and cost savings. Stage 2 introduces the placement of EVCSs after the DG and SC installations, reflecting the growing integration of EV loads in modern power grids. Although the inclusion of EVs leads to a slight increase in power losses compared to stage 1, the overall performance remains significantly better than the base case. Thus, it may be concluded that the strategic placement of EVCSs ensures the minimal impact of EV loads on the network’s operational parameters, maintaining high power factor and voltage stability.

4.2. Case 2: Simultaneous OSS of DGs, SCs, and EVCSs in IEEE 69-Bus RDS

This study investigated the simultaneous OSS of DGs, SCs, and EVCSs utilizing the fuzzy AVOA and fuzzy AOA approaches. The obtained results are presented in a comparative manner in Table 4. The optimal DG placements under fuzzy AVOA are obtained at buses 55, 62, and 37, with sizes of 1133 kW, 1142 kW, and 10 kW, respectively. Similarly, the approach finds the best placement of SCs at buses 24, 59, and 53, with optimal sizes of 400 kVAr, 300 kVAr, and 100 kVAr, respectively. The optimal locations for EVCSs are found at bus locations 2, 49, 56, 61, and 37, with real power demands ranging from 65 kW to 299 kW, whereas the reactive power demands ranged from 31 kVAr to 145 kVAr. Fuzzy AOA obtained the optimal location for DGs: bus numbers 37, 55, and 62. The optimal size of DGs in these locations in terms of active power compensation were found to be 10 kW, 1133 kW, and 1142 kW, respectively. The respective reactive power compensations are 5 kVAr, 549 kVAr, and 553 kVAr. Similarly, the reactive power compensation by SCs are 400 kVAr, 100 kVAr, and 300 kVAr at bus numbers 24, 53, and 59, respectively. It is observed that the fuzzy AVOA-based approach for the simultaneous OSS of considered units results in the reduction of APL from 224.96 kW to 31.73 kW, which is an 85.90% reduction as compared to the base case. It also outperformed the fuzzy AOA-based obtained simultaneous OSS of DGs, SCs, and EVCS (mentioned in Table 4), which reduced the APL to 36.88 kW (i.e., an 83.61% reduction with respect to the base case). The RPL is also reduced from 102.15 kVAr to 18.23 kVAr under fuzzy AVOA. The approach again surpasses fuzzy AOA, which has achieved a reduction in RPL up to 21.36 kVAr. In the case of fuzzy AVOA, the active power drawn from the substation is decreased from 4026.85 kW to 2452.12 kW, which is a bit higher than 2263.77 kW obtained under fuzzy AOA. Similarly, under fuzzy AVOA, the reactive power drawn from the substation decreased from 2796.25 kVAr to 805.66 kVAr, which is again a bit higher compared to fuzzy AOA’s 743.41 kVAr. This is due to the fact that the aggregate optimal capacity of DGs and SCs found by utilizing fuzzy AOA is more than fuzzy AVOA, which results in higher compensation offered by them, thereby relieving the burden in the substation. Both methods improved the substation power factor to 0.95, but fuzzy AVOA enhanced the minimum bus voltage to 0.98 p.u. as compared to 0.96 p.u. obtained under fuzzy AOA. The convergence and the comparative bus voltage profiles of the studied algorithms are showcased in Figure 10. In comparison to the results obtained in Case 1, the fuzzy AVOA-based simultaneous OSS of the considered unit in Case 2 further reduces the APL by 2.16%. Further, the RPL is also further reduced by 2.92% in comparison to the approach used in Case 1. Similarly, the minimum bus voltage achieved in this case is 0.98 p.u., which is a little better than that of Case 1.

The simultaneous OSS of DGs, SCs, and EVCSs in the IEEE 69-bus RDS using the fuzzy AVOA-based approach demonstrates its superiority over traditional methods such as fuzzy AOA. The significant reduction in APL and RPL, combined with improved voltage profiles and power factor, underscores the efficacy of this approach. The results show that the proposed fuzzy AVOA outperforms fuzzy AOA in reducing both active and reactive power losses. This improvement can be attributed to the algorithm’s enhanced search capabilities in simultaneously solving the OSS issues of DGs, SCs, and EVCSs. The significant reduction in APL and RPL compared to the base case are key indicators of the improved efficiency achieved with the proposed method. Ensuring consistent voltage profiles is a critical issue in power distribution networks. Fuzzy AVOA effectively boosts the minimum bus voltage from 0.91 p.u. to 0.98 p.u., thus ensuring improved voltage stability across the network. This improvement is crucial for avoiding issues like voltage sags, which can affect sensitive equipment and overall network reliability. This indicates the algorithm’s effectiveness in reducing the overall power demand on the substation, which can result in long-term financial savings and improved operational performance.

4.3. Case 3: Simultaneous OSS of DGs, SCs, and EVCSs in IEEE 69-Bus RDS with Varying Loading Conditions

In real-world power distribution systems, fixed loading scenarios are not a viable approach due to the dynamic nature of load demands, which vary with time. A system optimized for a specific load condition may not perform effectively under different loading scenarios, leading to technical inefficiencies. For instance, a system optimized for light load conditions may face issues such as higher line losses, off-limit bus voltages, and reverse power flows during high-demand periods. To address these challenges, this study analyzes a simultaneous optimization approach for the placement of DGs, SCs, and EVCSs in the radial distribution system, considering variable loading scenarios. The load factors (LF) were set at 1.0 (full load), 0.75 (medium load), and 0.5 (light load). The obtained results are presented in a comparative manner in Table 5. The results and discussions are presented below.

• (a). Full Load Condition (Load Factor = 1.0)—In the full load scenario, where the system operates at maximum demand, the base case showed the highest power losses, with an APL of 224.96 kW and an RPL of 102.15 kVAr. The fuzzy AOA method reduced these values significantly to 36.88 kW (APL) and 21.36 kVAr (RPL), representing improvements of 83.6% and 79.1%, respectively. However, the proposed fuzzy AVOA method achieved even better results, reducing APL to 31.73 kW and RPL to 18.23 kVAr—additional reductions of 13.96% and 14.67% compared to fuzzy AOA. Voltage stability also saw considerable improvement: the minimum bus voltage increased from 0.91 p.u. (base case) to 0.96 p.u. (fuzzy AOA) and further to 0.98 p.u. (fuzzy AVOA). The substation power factor improved from 0.82 in the base case to 0.95 for both fuzzy AOA and fuzzy AVOA, reducing system stress and enhancing efficiency. The proposed method’s superior optimization of DG (buses 55, 62, and 37) and SC placement (buses 24, 59, and 53) minimized losses while maintaining system reliability under high-demand conditions.

• (b). Medium Load Condition (Load Factor = 0.75)—At a medium load, the system operates at approximately 75% of its maximum demand. Power losses decreased naturally due to lower demand but the optimization methods showed distinct performance differences. The base case APL was 121.01 kW, which fuzzy AOA reduced to 18.59 kW—a reduction of 84.6%. The proposed fuzzy AVOA method further decreased the APL to 15.21 kW, achieving an additional 18.2% improvement over fuzzy AOA. Similarly, RPL decreased from 55.09 kVAr (base case) to 9.53 kVAr (fuzzy AOA) and further to 9.97 kVAr (fuzzy AVOA). The minimum bus voltage improved from 0.93 p.u. (base case) to 0.99 p.u. in both optimization methods, highlighting the effectiveness of these approaches in voltage regulation. The proposed method excelled in resource allocation, strategically placing DGs at buses 62, 69, and 15 and SCs at buses 67, 15, and 2 to minimize losses while maintaining system balance. EV integration was also optimized, with EVs located at buses 4, 7, 21, 67, and 34, ensuring effective load distribution and improved system stability compared to the fuzzy AOA.

• (c). Light Load Condition (Load Factor = 0.5)—In the light load scenario, the system operates at its lowest demand, where overvoltage issues and reduced power flow can create challenges. The base case showed an APL of 51.59 kW and a RPL of 23.55 kVAr. Fuzzy AOA reduced these values to 8.82 kW (APL) and 5.25 kVAr (RPL), achieving reductions of 82.9% and 77.7%, respectively. However, the proposed fuzzy AVOA method achieved superior results, with APL reduced to 7.09 kW (an additional 19.64% improvement over fuzzy AOA) and RPL reduced to 4.09 kVAr (22.1% improvement over fuzzy AOA). The minimum bus voltage was consistently maintained at 0.99 p.u. in both fuzzy AOA and fuzzy AVOA, ensuring voltage stability. The proposed method’s DG and SC optimization (DGs at buses 62, 69, and 36; SCs at buses 2, 58, and 36) mitigated overvoltage issues effectively, demonstrating better adaptability to light load conditions. Additionally, EV placement and management were more efficient in the fuzzy AVOA method, with EVs distributed across buses 2, 29, 56, 68, and 38, balancing the load and maintaining system reliability.

Across all loading scenarios, the fuzzy AVOA method consistently outperformed the base case and fuzzy AOA in reducing power losses and improving voltage stability. Compared to fuzzy AOA, the proposed method achieved additional reductions in APL (13–20%) and RPL (14–22%) while maintaining a high substation power factor of 0.95. Voltage stability was also enhanced, with the minimum bus voltage consistently maintained at 0.98–0.99 p.u. in the proposed method, compared to 0.96–0.99 p.u. in fuzzy AOA and 0.91–0.96 p.u. in the base case. The optimized placement and sizing of DGs and SCs in the fuzzy AVOA method provided better system balance and efficiency, particularly in handling variable loading conditions. EV integration was managed more effectively, with improved load distribution and prevention of overloading or overvoltage issues. The convergence and the comparative bus voltage profiles of the studied algorithms are showcased in Figure 11 and Figure 12. The proposed fuzzy AVOA method’s superior performance demonstrates its dynamic adaptability and scalability for modern distribution systems. By minimizing power losses, improving voltage regulation, and enhancing substation efficiency, the method ensures technical benefits. These results validate the fuzzy AVOA method as a robust and reliable solution for optimizing radial distribution systems under diverse loading scenarios.

5. Conclusions

This study demonstrates the effectiveness of the proposed fuzzy AVOA method for the simultaneous OSS of DGs, SCs, and EVCSs in a standard IEEE 69-bus RDS. The superiority of the proposed fuzzy AVOA-based approach for the simultaneous OSS of considered units is established over a studied two-stage OSS of the units found using fuzzy AVOA. In stage 1 of Case 1, firstly, the optimal sitting of DGs and SCs achieved a drastic saving in power losses, with a remarkable enhancement in network voltage stability and power factor. In Case 1, the APL was reduced from 224.96 kW to 19.32 kW (91.41% reduction) and the reactive power losses (RPL) dropped from 102.15 kVAr to 13.74 kVAr. The substation power factor improved from 0.82 to 0.95, and the minimum bus voltage increased from 0.91 p.u. to 0.98 p.u. In stage 2, EVCSs are optimally integrated after the deployment of DGs and SCs, which showed a slight increase in losses due to the additional EV load but still maintained substantial improvements over the base case. After EVCS integration, the APL rose slightly to 36.58 kW, maintaining a reduction of 83.73% compared to the base case.

In contrast, in Case 2, where sitting and sizing of DGs, SCs, and EVCSs are optimized simultaneously using the proposed fuzzy AVOA, the APL was reduced to 31.73 kW (85.90% reduction) and the RPL decreased to 18.23 kVAr (82.15% reduction). The minimum bus voltage was enhanced to 0.98 p.u., outperforming Case 1 by achieving 2.16% lower APL and 2.92% lower RPL. The results obtained demonstrate more enhancement compared to Case 1 regarding decreases in power losses, as well as improvements in voltage stability and power factor. The detailed analysis of the results presented in the two case studies using different approaches for the deployment of the considered units confirms that the simultaneous OSS of the considered units in Case 2 is preferable over the approach used in Case 1. In Case 3 (Variable Load Conditions), under full load, APL decreased from 224.96 kW (base case) to 31.73 kW (fuzzy AVOA), and RPL decreased from 102.15 kVAr to 18.23 kVAr. Voltage stability improved consistently, with minimum bus voltage maintained at 0.98–0.99 p.u., and the substation power factor was enhanced to 0.95. Compared to fuzzy AOA, fuzzy AVOA achieved additional reductions in APL (13–20%) and RPL (14–22%) across all scenarios. The method provides technical benefits, proving robust and scalable for modern distribution. Overall, fuzzy AVOA effectively reduces power losses, improves voltage stability, and enhances power factor, demonstrating its robust capability for real-world RDS optimization, especially in integrating DGs, SCs, and EVCSs.

Author Contributions

Conceptualization, G.S.; methodology, G.S. and S.S. (Sumeet Sahay); software, S.S. (Sumeet Sahay) and S.R.B.; validation, G.S., S.R.B. and A.V.J.; formal analysis, S.S. (Sumeet Sahay), D.K.G. and S.S. (Sarita Samal); investigation, S.S. (Sumeet Sahay), S.R.B. and D.K.G.; resources, A.V.J. and D.K.G.; data curation, S.S. (Sumeet Sahay) and S.S. (Sarita Samal); writing—original draft preparation, S.S. (Sumeet Sahay) and S.R.B.; writing—review and editing, A.V.J., D.K.G., A.-G.M. and N.B.; visualization, D.K.G., A.-G.M. and N.B.; supervision, G.S., A.-G.M. and N.B.; project administration, A.V.J., A.-G.M., and N.B.; funding acquisition, A.-G.M. and N.B. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data supporting this research will be made available on request from the corresponding authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. S/S power factor.

Figure 2. DG penetration limit.

Figure 3. Active power line loss index.

Figure 4. Node voltage fuzzy membership.

Figure 5. Membership function for active power loss with EVs.

Figure 6. Design and implementation of the proposed technique.

Figure 7. Flow chart of AVO algorithm.

Figure 8. Performance results for IEEE 69-bus RDS: (a) voltage profile, (b) convergence curve (Case 1, stage 1).

Figure 9. Performance results for IEEE 69-bus RDS: (a) voltage profile, (b) convergence curve (Case 1, stage 2).

Figure 10. Performance results for IEEE 69-bus RDS: (a) voltage profile, (b) convergence curve (Case 2).

Figure 11. Voltage profile performance results for IEEE 69-bus RDS for variable loading conditions: (a) LF = 1, (b) LF = 0.75, (c) LF = 0.50 (Case 3).

Figure 12. Convergence curve for IEEE 69-bus RDS for variable loading conditions: (a) LF = 1, (b) LF = 0.75, (c) LF = 0.50 (Case 3).

Table 1

Parameter settings of AVOA.

Algorithm	Parameters	Value
AVOA	θ1	0.8
	θ2	0.2
	w	2.5
	φ1	0.6
	φ2	0.4
	φ3	0.6

Table 2

Summary of results of Case 1, stage 1 for IEEE 69-bus RDS.

Parameters	Base Case	Fuzzy AOA(Studied)	Fuzzy AVOA(Proposed)	Fuzzy GA[26]	Fuzzy PSO[26]	Fuzzy GOA[26]
Total APL (kW)	224.96	25.40	19.32	37.83	33.49	27.34
Total RPL (kVAr)	102.15	15.29	13.74	NA	NA	NA
DG location	-	5, 69, 61	2, 61, 17	NA	NA	22, 63, 64
DG size (kW)	-	1267, 366, 1267	14, 1225, 379	NA	NA	761.38, 451.47, 688.26
DG size (kVAr)	-	614, 177, 614	7, 593, 184	NA	NA	NA
SC location	-	10, 69, 63	2, 63, 60	NA	NA	69, 2, 59
SC size (kVAr)	-	600, 200, 200	600, 400, 200	NA	NA	105, 675, 675
S/S * active power (kW)	4026.85	927.29	2203.21	1938.93	1930.11	1928.42
S/S reactive power (kVAr)	2796.25	304.86	724.21	640.22	641.08	631.08
S/S power factor	0.82	0.95	0.95	0.9500 lag	0.9500 lag	0.9500 lag
Min. bus voltage (in p.u.)	0.91	0.98	0.98	0.9734	0.9752	0.9761

* S/S stands for system.

Table 3

Summary of results of Case 1, stage 2 for IEEE 69-bus RDS.

Parameters	Base Case	Fuzzy AOA(Studied)	Fuzzy AVOA(Proposed)
Total APL (kW)	224.96	46.06	36.58
Total RPL(kVAr)	102.15	23.48	21.22
DG location	-	5, 69, 61	2, 61, 17
DG size (kW)	-	1267, 366, 1267	14, 1225, 379
DG size (kVAr)	-	614, 177, 614	7, 593, 184
SC location	-	10, 69, 63	2, 63, 60
SC size (kVAr)	-	600, 200, 200	600, 400, 200
EV location	-	2, 16, 6, 27, 65	2, 19, 65, 69, 7
Number of EV	-	37, 16, 11, 17, 50	10, 10, 49, 11, 49
Real power EV (kW)	-	241, 104, 72, 111, 325	65, 65, 319, 72, 319
Reactive power EV (kVAr)	-	116, 50, 35, 54, 157	31, 31, 154, 35, 154
S/S * active power (kW)	4026.85	1004.95	1634.47
S/S reactive power (kVAr)	2796.25	340.66	547.88
S/S power factor	0.82	0.95	0.95
Min. bus voltage (in p.u.)	0.91	0.97	0.97

* S/S stands for system.

Table 4

Summary of results of Case 2 for IEEE 69-bus RDS.

Parameters	Base Case	Fuzzy AOA(Studied)	Fuzzy AVOA(Proposed)
Total APL (kW)	224.96	36.88	31.73
Total RPL (kVAr)	102.15	21.36	18.23
DG location	-	5, 61, 65	55, 62, 37
DG size (kW)	-	1004, 1034, 382	1133, 1142, 10
DG size (kVAr)	-	486, 501, 185	549, 553, 5
SC location	-	3, 38, 28	24, 59, 53
SC size (kVAr)	-	200, 500, 100	400, 300, 100
EV location	-	22, 28, 31, 50, 4	2, 49, 56, 61, 37
Number of EV	-	15, 34, 36, 14, 31	46, 10, 46, 10, 27
Real power EV (kW)	-	98, 221, 234, 91, 202	299, 65, 299, 65, 176
Reactive power EV (kVAr)	-	47, 107, 113, 44, 98	145, 31, 145, 31, 85
S/S * active power (kW)	4026.85	2263.77	2452.12
S/S reactive power (kVAr)	2796.25	743.41	805.66
S/S power factor	0.82	0.95	0.95
Min. bus voltage (in p.u.)	0.91	0.96	0.98

* S/S stands for system.

Table 5

Summary of results of Case 3 for IEEE 69-bus RDS with variable loading conditions.

Parameters	Base Case	Fuzzy AOA(Studied)	Fuzzy AVOA(Proposed)	Base Case	Fuzzy AOA(Studied)	Fuzzy AVOA(Proposed)	Base Case	Fuzzy AOA(Studied)	Fuzzy AVOA(Proposed)
	Load Factor = 1			Load Factor = 0.75			Load Factor = 0.50
Total APL (kW)	224.96	36.88	31.73	121.01	18.59	15.21	51.59	8.82	7.09
Total RPL (kVAr)	102.15	21.36	18.23	55.09	9.53	9.97	23.55	5.25	4.09
DG location	-	5, 61, 65	55, 62, 37	-	9, 62, 15	62, 69 15	-	67, 61, 49	62, 69, 36
DG size (kW)	-	1004, 1034, 382	1133, 1142, 10	-	22, 939, 939	950, 12, 360	-	634, 582, 135	634, 563, 10
DG size (kVAr)	-	486, 501, 185	549, 553, 5	-	11, 455, 455	460, 6, 174	-	307, 282, 65	307, 273, 5
SC location	-	3, 38, 28	24, 59, 53	-	15, 45, 39	67, 15, 2	-	14, 2, 29	2, 58, 36
SC size (kVAr)	-	200, 500, 100	400, 300, 100	-	200, 300, 200	300, 200, 100	-	200, 500, 600	400, 200, 600
EV location	-	22, 28, 31, 50, 4	2, 49, 56, 61, 37	-	13, 37, 32, 24, 5	4, 7, 21, 67, 34	-	40, 48, 54, 67, 2	2, 29, 56, 68, 38
Number of EV	-	15, 34, 36, 14, 31	46, 10, 46, 10, 27	-	10, 10, 10, 10, 10	40, 50, 10, 10, 10	-	50, 45, 48, 10, 42	29, 36, 10, 10, 10
Real power EV (kW)	-	98, 221, 234, 91, 202	299, 65, 299, 65, 176	-	65, 65, 65, 65, 65	260, 325, 65, 65, 65	-	325, 292, 312, 65, 273	189, 234, 65, 65, 65
Reactive power EV (kVAr)	-	47, 107, 113, 44, 98	145, 31, 145, 31, 85	-	31, 31, 31, 31, 31	126, 157, 31, 31, 31	-	157, 142, 151, 31, 132	91, 113, 31, 31, 31
S/S * active power (kW)	4026.85	2263.77	2452.12	2972.43	1295.00	2324	1952.54	1826.62	1318.89
S/S reactive power (kVAr)	2796.25	743.41	805.66	2075.66	409.89	790	1370.59	601.84	533.26
S/S power factor	0.82	0.95	0.95	0.82	0.95	0.95	0.82	0.95	0.95
Min. bus voltage (in p.u.)	0.91	0.96	0.98	0.93	0.99	0.98	0.96	0.99	0.99

* S/S stands for system.

References

1. Colbu, S.C.; Babcila, D.M.; Nita, A.P.; Popescu, D.; Olteanu, S.C. A comparative study of support vector machine and risk deterministic approach on a photovoltaic system. UPB Sci. Bull. Ser. C; 2024; 86, pp. 67-80.

2. Kumar, A.; Verma, R.; Choudhary, N.K.; Singh, N. Optimal placement and sizing of distributed generation in power distribution system: A comprehensive review. Energy Sources, Part A: Recovery. Util. Environ. Eff.; 2023; 45, pp. 7160-7185.

3. Kumar, M.; Soomro, A.M.; Uddin, W.; Kumar, L. Optimal multi-objective placement and sizing of distributed generation in distribution system: A comprehensive review. Energies; 2022; 15, 7850. [DOI: https://dx.doi.org/10.3390/en15217850]

4. Tanwar, S.S.; Khatod, D.K. Techno-economic and environmental approach for optimal placement and sizing of renewable DGs in distribution system. Energy; 2017; 127, pp. 52-67. [DOI: https://dx.doi.org/10.1016/j.energy.2017.02.172]

5. Rastgou, A.; Moshtagh, J.; Bahramara, S. Improved harmony search algorithm for electrical distribution network expansion planning in the presence of distributed generators. Energy; 2018; 151, pp. 178-202. [DOI: https://dx.doi.org/10.1016/j.energy.2018.03.030]

6. Kumar, S.; Mandal, K.K.; Chakraborty, N. Optimal DG placement by multi-objective opposition based chaotic differential evolution for techno-economic analysis. Appl. Soft Comput.; 2019; 78, pp. 70-83. [DOI: https://dx.doi.org/10.1016/j.asoc.2019.02.013]

7. Neda, O.M. Optimal amalgamation of DG units in radial distribution system for techno-economic study by improved SSA: Practical case study. Electr. Power Syst. Res.; 2025; 241, 111365. [DOI: https://dx.doi.org/10.1016/j.epsr.2024.111365]

8. Khenissi, I.; Guesmi, T.; Alshammari, B.M.; Alqunun, K.; Almalaq, A.; Alturki, M.; Neji, R. A hybrid chaotic bat algorithm for optimal placement and sizing of dg units in radial distribution networks. Energy Rep.; 2024; 12, pp. 1723-1741. [DOI: https://dx.doi.org/10.1016/j.egyr.2024.07.042]

9. Alao, M.A.; Popoola, O.M.; Ayodele, T.R. An improved particle swarm optimisation algorithm for optimum placement and sizing of biogas-fuelled distributed generators for seasonal loads in a radial distribution network. Energy Rep.; 2024; 12, pp. 1531-1550. [DOI: https://dx.doi.org/10.1016/j.egyr.2024.07.049]

10. Tanwar, S.S.; Prajapat, G.P.; Rathod, R.; Bansal, S.K.; Sisodia, S. AHP based technical, economic and environmental impact analysis with optimal DG placement in radial distribution network. Int. J. Appl. Power Eng. (IJAPE); 2024; 13, [DOI: https://dx.doi.org/10.11591/ijape.v13.i3.pp530-538]

11. Leghari, Z.H.; Kumar, M.; Shaikh, P.H.; Kumar, L.; Tran, Q.T. A critical review of optimization strategies for simultaneous integration of distributed generation and capacitor banks in power distribution networks. Energies; 2022; 15, 8258. [DOI: https://dx.doi.org/10.3390/en15218258]

12. Muthukumar, K.; Jayalalitha, S. Optimal placement and sizing of distributed generators and shunt capacitors for power loss minimization in radial distribution networks using hybrid heuristic search optimization technique. Int. J. Electr. Power Energy Syst.; 2016; 78, pp. 299-319. [DOI: https://dx.doi.org/10.1016/j.ijepes.2015.11.019]

13. Khodabakhshian, A.; Andishgar, M.H. Simultaneous placement and sizing of DGs and shunt capacitors in distribution systems by using IMDE algorithm. Int. J. Electr. Power Energy Syst.; 2016; 82, pp. 599-607. [DOI: https://dx.doi.org/10.1016/j.ijepes.2016.04.002]

14. Ghaffarzadeh, N.; Sadeghi, H. A new efficient BBO based method for simultaneous placement of inverter-based DG units and capacitors considering harmonic limits. Int. J. Electr. Power Energy Syst.; 2016; 80, pp. 37-45. [DOI: https://dx.doi.org/10.1016/j.ijepes.2016.01.030]

15. Selim, A.; Kamel, S.; Mohamed, A.A.; Yu, J. Placement and sizing of distributed generations and shunt capacitors in radial distribution systems using hybrid optimization technique. Electr. Power Compon. Syst.; 2023; 51, pp. 421-437. [DOI: https://dx.doi.org/10.1080/15325008.2023.2173827]

16. Anuar MS, S.; Muhtazaruddin, M.N.; Rahman MA, A.; Amran, M.E. Optimal RE-DG and Capacitor Placement for Cost-Benefit Maximization in Malaysia Distribution System. Proceedings of the International Conference on Robotics, Vision, Signal Processing and Power Applications; Penang, Malaysia, 5–6 April 2021; Springer Nature: Singapore, 2021; pp. 85-91.

17. Fettah, K.; Guia, T.; Salhi, A.; Mouassa, S.; Bosisio, A.; Shirvani, R. Optimal Allocation of Capacitor Banks and Distributed Generation: A Comparison of Recently Developed Metaheuristic Optimization Techniques on the Real Distribution Networks of ALG-AB-Hassi Sida, Algeria. Sustainability; 2024; 16, 4419. [DOI: https://dx.doi.org/10.3390/su16114419]

18. Ahmad, F.; Iqbal, A.; Ashraf, I.; Marzband, M. Optimal location of electric vehicle charging station and its impact on distribution network: A review. Energy Rep.; 2022; 8, pp. 2314-2333. [DOI: https://dx.doi.org/10.1016/j.egyr.2022.01.180]

19. Aljumah, A.; Darwish, A.; Csala, D.; Twigg, P. A Review on the Allocation of Sustainable Distributed Generators with Electric Vehicle Charging Stations. Sustainability; 2024; 16, 6353. [DOI: https://dx.doi.org/10.3390/su16156353]

20. Mozafar, M.R.; Moradi, M.H.; Amini, M.H. A simultaneous approach for optimal allocation of renewable energy sources and electric vehicle charging stations in smart grids based on improved GA-PSO algorithm. Sustain. Cities Soc.; 2017; 32, pp. 627-637. [DOI: https://dx.doi.org/10.1016/j.scs.2017.05.007]

21. Awasthi, A.; Venkitusamy, K.; Padmanaban, S.; Selvamuthukumaran, R.; Blaabjerg, F.; Singh, A.K. Optimal planning of electric vehicle charging station at the distribution system using hybrid optimization algorithm. Energy; 2017; 133, pp. 70-78. [DOI: https://dx.doi.org/10.1016/j.energy.2017.05.094]

22. Luo, L.; Gu, W.; Wu, Z.; Zhou, S. Joint planning of distributed generation and electric vehicle charging stations considering real-time charging navigation. Appl. Energy; 2019; 242, pp. 1274-1284. [DOI: https://dx.doi.org/10.1016/j.apenergy.2019.03.162]

23. Bhadoriya, J.S.; Gupta, A.R.; Kumar, A.; Ray, R.; Maini, S. Enhancement of the distribution network in the presence of EV charging stations augmented by distributed generation. Electr. Eng.; 2023; 105, pp. 3703-3717. [DOI: https://dx.doi.org/10.1007/s00202-023-01901-8]

24. Guindi, M.; Kamel, R.M. Optimal location and sizing of renewable distributed generations and electric vehicle charging stations. Renew. Energy; 2024; 235, 121272. [DOI: https://dx.doi.org/10.1016/j.renene.2024.121272]

25. Dubey, P.K.; Singh, B.; Singh, D. Integration of Distributed Generations and electric vehicles in the distribution system. Eng. Appl. Artif. Intell.; 2024; 137, 109036. [DOI: https://dx.doi.org/10.1016/j.engappai.2024.109036]

26. Gampa, S.R.; Jasthi, K.; Goli, P.; Das, D.; Bansal, R.C. Grasshopper optimization algorithm based two stage fuzzy multiobjective approach for optimum sizing and placement of distributed generations, shunt capacitors and electric vehicle charging stations. J. Energy Storage; 2020; 27, 101117. [DOI: https://dx.doi.org/10.1016/j.est.2019.101117]

27. Abdollahzadeh, B.; Gharehchopogh, F.S.; Mirjalili, S. African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems. Comput. Ind. Eng.; 2021; 158, 107408. [DOI: https://dx.doi.org/10.1016/j.cie.2021.107408]

28. Goli, P.; Makkena, S.; Gampa, S.R.; Das, D. Fuzzy ant colony optimization technique for predefined performance of distribution systems considering DGs and shunt capacitors. Proceedings of the 2019 North American Power Symposium (NAPS); Wichita, KS, USA, 13–15 October 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 1-6.

29. Satyanarayana, S.; Ramana, T.; Sivanagaraju, S.; Rao, G.K. An efficient load flow solution for radial distribution network including voltage dependent load models. Electr. Power Compon. Syst.; 2007; 35, pp. 539-551. [DOI: https://dx.doi.org/10.1080/15325000601078179]