Dynamic Optimization and Placement of Renewable Generators and Compensators to Mitigate Electric Vehicle Charging Station Impacts Using the Spotted Hyena Optimization Algorithm

Abstract

The growing adoption of electric vehicles (EVs) offers notable benefits, including reduced maintenance costs, improved performance, and environmental sustainability. However, integrating EVs into radial distribution systems (RDSs) poses challenges related to power losses and voltage stability. The model accounts for hourly variations in demand, making it crucial to determine the optimal placement of electric vehicle charging stations (EVCSs) throughout the day. This study proposes a new approach that combines EVCSs, distribution static compensators (DSTATCOMs), and renewable distributed generation (RDG) from solar and wind sources, with a focus on dynamic analysis over 24 h. The spotted hyena optimization algorithm (SHOA) is employed to determine near-global optimum locations and sizes for RDG, DSTATCOMs, and EVCSs, aiming to minimize real power loss while meeting system constraints. The SHOA outperforms traditional methods due to its unique search mechanism, which effectively balances exploration and exploitation, allowing it to find superior solutions in complex environments. Simulations on an IEEE 34-bus RDS under dynamic load conditions validate the approach, demonstrating a reduction in average power loss from 180.43 kW to 72.04 kW, a 72.6% decrease. Compared to traditional methods under constant load conditions, the SHOA achieves a 77.0% reduction in power loss, while the BESA and PSO achieve reductions of 61.1% and 44.7%, respectively. These results underscore the effectiveness of the SHOA in enhancing system performance and significantly reducing real power loss.

1. Introduction

1.1. Motivation and Concept

The modern power grid is undergoing a significant transformation, driven by rising energy demands, environmental regulations, and rapid technological advancements [1,2]. Among the most notable changes is the increasing adoption of electric vehicles (EVs), which has accelerated the integration of electric vehicle charging stations (EVCSs) into radial distribution systems (RDSs). This development presents new challenges, as EVCSs introduce sporadic and high demand loads, resulting in increased power losses, voltage instability, and network congestion [3,4]. As the penetration of EVs continues to rise, these challenges become more pronounced, necessitating innovative approaches to maintain system reliability, efficiency, and sustainability [5].

The integration of EVCSs significantly impacts the operational dynamics of distribution networks, particularly in managing the unpredictable and concentrated load patterns associated with EV charging. Simultaneous charging across multiple stations can result in substantial spikes in power demand, straining the infrastructure and exacerbating issues such as voltage fluctuations and system inefficiencies [6]. Consequently, it is crucial to optimize the grid’s ability to handle the dynamic nature of EV loads while minimizing power losses and ensuring voltage stability [7]. Reducing power losses is essential for both economic and environmental reasons. Energy lost during transmission and distribution not only represents inefficiency but also contributes to unnecessary carbon emissions, undermining sustainability objectives [8]. Additionally, power losses can destabilize voltage levels, degrade power quality, and make the grid more vulnerable to outages, especially in systems with high EV penetration.

In response to these challenges, power system operators are increasingly incorporating renewable energy sources (RESs), advanced grid management technologies, and demand-side management strategies. Renewable distributed generation (RDG), such as solar and wind power, is being deployed to decentralize energy generation. This reduces reliance on centralized power plants and alleviates the demand surge caused by EVCSs. Solar power, in particular, often coincides with peak demand from EV charging, helping to balance loads and reduce grid congestion [9]. Additionally, distribution static synchronous compensators (DSTATCOMs) play a critical role in maintaining voltage stability within RDSs. These units dynamically inject or absorb reactive power, stabilizing voltage levels during periods of high EV demand [9]. By compensating for the power quality issues arising from fluctuating EV loads, DSTATCOMs enhance system reliability and reduce the likelihood of voltage instability.

Given the unpredictable nature of EV charging behavior and the intermittent generation from RESs, dynamic analysis is indispensable for optimizing RDS performance. Static analyses fail to address the rapidly changing load patterns introduced by EVCSs, particularly in systems that integrate RESs. Dynamic analysis allows for real-time monitoring and control, enabling power system operators to respond to fluctuations in demand and supply as they occur. This real-time adaptability is critical for ensuring voltage stability, minimizing power losses, and enhancing overall system resilience [10]. To provide a practical assessment, this study considers various hourly load factors to simulate real-world scenarios, reflecting the dynamic nature of EV charging and renewable energy generation [11]. By analyzing load variations on an hourly basis, the study offers insights into how fluctuating demand impacts system performance and provides a framework for optimizing the placement and coordination of RDG and DSTATCOMs. This approach ensures that the grid can efficiently manage peak loads and maintain voltage stability, even in the face of increasing EV penetration [12].

The real-time EV charging profile is particularly complex, as demand can shift dramatically over short periods. Dynamic analysis enables continuous adjustments to charging schedules and grid operations, reducing the risk of system overload [13]. Moreover, the dynamic management of renewable energy inputs and reactive power through DSTATCOMs enhances grid flexibility, ensuring the system can respond effectively to both high demand from EVCSs and the variability of RESs.

In the context of EVCS operations, three distinct modes—Grid-to-Vehicle (G2V), Vehicle-to-Grid (V2G), and Idle—are considered in this research. The G2V mode represents the traditional unidirectional charging of EVs from the grid, while the V2G mode enables a bidirectional energy flow between the grid and EVs, allowing vehicles to supply energy back to the grid. The Idle mode represents periods when EVCSs are not in use. By considering these operational modes, the study develops a more comprehensive understanding of how EVCSs impact the grid, aiding in the formulation of strategies to mitigate power losses and ensure voltage stability [14].

This paper proposes the integration of RDG and DSTATCOMs as a comprehensive solution to mitigate the adverse effects of EVCSs on RDSs. The dynamic analysis performed highlights the importance of addressing the fluctuating load patterns introduced by EVs and the intermittent nature of RESs. The proposed approach focuses on reducing power losses, maintaining voltage stability, and enhancing the overall resilience of the distribution network. Through the strategic placement and coordination of RDG and DSTATCOMs, the research explores how decentralized energy generation can offset the increased demand from EVCSs, while DSTATCOMs manage voltage levels and reactive power in real time. By considering hourly load factors and the three modes of EVCS operation, the study aims to provide a practical and detailed analysis of system performance under various operating conditions. This research contributes to the optimization of EVCS integration and underscores the potential of renewable energy and advanced grid management technologies in building more sustainable and resilient power systems.

1.2. Literature Review

The literature reveals a considerable gap in research concerning the integration of EVCSs with RDG units and DSTATCOMs within RDSs. While previous studies have extensively examined the allocation of RDG and DSTATCOMs either separately or jointly, there is limited research addressing their near-global optimum allocation in conjunction with EVCSs. Recent studies have increasingly focused on this integration, yet comprehensive investigations into the advantages of co-locating RDG and DSTATCOMs with EVCSs—particularly in relation to power loss mitigation and bus voltage improvement—are still lacking. Several studies indicate that strategically co-locating RDG and DSTATCOMs with EVCSs can help mitigate power losses and improve voltage stability. However, further research is needed to explore the specific benefits, near-global optimum allocation strategies, and potential trade-offs of this combined approach. Addressing these gaps could lead to more efficient and resilient distribution networks capable of supporting the growing demand for EVs.

For instance, one study proposed a two-stage approach for optimizing the location of fast charging stations (FCSs) in RDSs. The first stage used a charging station investor decision index (CSIDI) that combined factors like land costs and EV population, while the second stage focused on minimizing total active power loss while meeting distribution system operator (DSO) constraints. Although promising, this approach faced limitations such as potential oversimplification of decision-making processes and insufficient exploration of uncertainties related to dynamic analysis and load variations [15]. Additionally, the study did not adequately address the effects of RDG fluctuations or the dynamic nature of EVCS operation modes, which could impact the accuracy and effectiveness of the optimization. Another study used a voltage stability risk index-based approach to place FCSs at weak buses, employing a genetic algorithm (GA) to address power loss. This study also considered multi-objective function (MOF) such as Voltage stability, Reliability, and Power loss (VRP) index. While this approach successfully minimized power losses, it oversimplified decision-making and was limited by convergence issues related to the GA, which affected its ability to capture the full complexity of real-world scenarios [16]. The reliance on simulation results from the IEEE 33-bus system also restricted the generalizability of the findings to other network configurations or deployment scenarios.

A collaborative approach involving charging station investors, Plug-in EV (PEV) users, and DSOs was proposed to optimize EVCS location and capacity. However, this study did not fully address the technical complexities of integrating renewable energy sources (RESs) at charging stations, uncertainties in energy management strategies (EMS), or the dynamic behavior of EVCSs. Moreover, computational complexities related to Monte Carlo simulations (MCS) and the challenges of managing RDG fluctuations and V2G strategies were not thoroughly explored [17]. Further studies combined optimization techniques such as gray wolf optimization (GWO) and particle swarm optimization (PSO) to address EVCS placement. While these methods showed potential benefits for minimizing power loss through reactive power compensation, they lacked detailed discussions on the technical challenges related to V2G facilities and suffered from convergence issues affecting optimization effectiveness [18]. Similarly, a stochastic fuzzy chance-constrained programming model was proposed for coordinating power loss and voltage in RDSs with distributed generation (DG) and EVCSs. However, challenges included potential inaccuracies in fitting incomplete data of uncertainties, and oversimplifications in transforming probability density functions, which impacted the model’s reliability and performance in real-world applications [19].

A chance constrained second-order cone programming (CCSOCP) was utilized to control power loss in RDSs, considering EV charging power as a random variable. Despite using the conditional value-at-risk (CVaR) model to control power loss risks, the study faced limitations in discussing potential technical challenges or trade-offs involved with the method, and it did not effectively address convergence issues or the dynamic nature of RDG and load consumption [20]. Additionally, a joint planning algorithm (JPA) for allocating smart EVCSs in remote communities demonstrated feasibility but faced challenges related to computational complexity and potential suboptimal solutions with multi-objective mixed integer non-linear programming [21].

A hybrid optimization approach, combining particle swarm optimization (PSO) with direct search (DS) methods, is introduced. This algorithm leverages the strengths of both PSO and DS techniques to effectively address complex mathematical models. To assess the performance of the PSO-DS algorithm, its results are compared against those of other advanced algorithms on a variety of standard benchmark functions, showcasing its effectiveness. However, the hybrid algorithm has some limitations, including potential convergence issues and sensitivity to parameter settings, which may affect its practical applicability [22]. Another method aimed to minimize energy loss and transportation costs by optimizing FCS locations and integrating photovoltaic energy sources and battery energy storage systems (BESS). However, the proposed improved bald eagle search algorithm (IBESA) faced difficulties in accurately determining near-global optimum solutions and did not fully address the dynamic factors and interactions in the system [23].

The integration of DSTATCOMs and DG with EVCSs aims to enhance system performance, reliability, and resilience while minimizing the carbon footprint. A fuzzy classification method was proposed for the optimal sizing and placement of EVCSs, DG, and DSTATCOMs in 69-bus radial distribution systems, utilizing the RAO-3 algorithm. However, challenges such as technical complexities, compatibility issues, cost-effectiveness, and scalability concerns regarding loss reduction and voltage stability continue to pose significant obstacles [24]. A mathematical optimization software for engineering and knowledge (MOSEK) version 10.2.5 for sizing and siting RDGs, EVCSs, and BESS in RDS highlighted challenges like computational complexity, reliance on data accuracy, and the need for improved system stability and voltage control. Dynamic analysis was particularly emphasized as crucial for addressing the time-varying nature of RDGs and load consumption [25].

Multi-objective optimization techniques, such as mixed-integer non-linear problems (MINLP) with non-dominated sorting genetic algorithm-II (NSGA-II), have been proposed for FCS and DG placement. However, these methods introduce computational complexity and rely heavily on data accuracy, limiting their generalizability across different system configurations [26]. The use of the African vultures optimization algorithm (AVOA) for optimizing EVCSs, DG, and DSTATCOMs highlighted the need for thorough validation against alternative algorithms and better consideration of V2G modes to enhance effectiveness [27]. Additionally, multi-objective particle swarm optimization (MOPSO) and multi-objective crow search (MOCS) algorithms have been employed for hybrid RESs integration with EVs. Challenges such as algorithm sensitivity to parameter settings and convergence issues affected optimization effectiveness [28]. Similarly, the Jaya algorithm was used to schedule EVCSs with DG integration in IEEE 33- and 69-bus systems, but accurately modeling user behavior and trip distance, optimizing costs, and utilizing DSTATCOMs for stability enhancement require further research [29].

1.3. Research Gaps

Table 1. Literature summary on EVCS allocation in the RDS.

Ref. No.	Dynamic Analysis	EVCS	G2V Mode	V2G Mode	RDG	DSTATCOM	Objective Function	Test System	Techniques
[15]	×	√	√	×	√	×	Power loss	IEEE 34-bus	IBESA
[16]	×	√	√	×	×	×	MOF	IEEE 33-bus	VRP Index
[17]	√	√	√	√	√	×	MOF	IEEE 33-bus	MCS method
[18]	×	√	√	√	×	×	MOF	IEEE 33- & 34-bus	Hybrid GWO & PSO
[19]	√	√	√	×	√	×	MOF	IEEE 33- & 118-bus	Stochastic Fuzzy
[20]	√	√	√	×	√	×	Power loss	IEEE 69-bus	CCSOCP
[21]	×	√	√	×	√	×	MOF	Hybrid AC/DC 38-bus	JPA
[22]	×	√	√	×	×	×	MOF	IEEE 33- & 69-bus	PSO
[23]	×	√	√	×	√	×	MOF	IEEE 33-bus	IBESA
[24]	×	√	√	×	×	√	Power loss	Indian 28- & 108-bus	BESA
[25]	√	√	√	×	√	×	MOF	Real Istanbul buses	MOSEK
[26]	√	√	√	×	√	×	MOF	IEEE 118-bus	MINLP
[27]	×	√	√	×	×	√	MOF	IEEE 33-, 69- & 136-bus	AVOA
[28]	√	√	√	×	√	×	MOF	Distribution systems	MOPSO & MOCS
[29]	×	√	√	√	√	×	MOF	IEEE 33- & 69-bus	Jaya algorithm
Proposed Method	√	√	√	√	√	√	Power loss	IEEE 34-bus	SHOA

offers an overview of methodologies and key findings from various studies on EVCS allocation in distribution systems, as referenced in [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. It summarizes different strategies and optimization techniques used to address challenges like power loss reduction, operational efficiency, and renewable energy integration. Despite this, several gaps remain in optimizing EVCS integration with RDG and DSTATCOMs in RDSs. The reliance on static analysis has introduced uncertainties, highlighting the need for dynamic analysis to handle real-world fluctuations. Existing optimization methods often fail to find global solutions efficiently, with many focusing solely on power loss reduction while neglecting other critical factors. Most research has concentrated on the G2V mode, overlooking the V2G mode’s benefits, especially during peak hours. Additionally, the integration of DSTATCOMs with renewable sources remains underexplored, and challenges like scalability, computational complexity, convergence issues, and algorithm selection persist. Addressing these limitations by incorporating dynamic analysis, various EVCS modes, and improved convergence methods is crucial for enhancing power loss reduction, voltage stability, and bus voltage.

1.4. Research Contribution

This research introduces the spotted hyena optimization algorithm (SHOA) to address challenges in EVCS integration by focusing on dynamic analysis. The SHOA offers scalability, robust convergence, and ease of parameter tuning, effectively addressing computational complexity and convergence issues. Key contributions include the following:

• Utilizing a novel nature-inspired SHOA for the dynamic allocation of solar and wind-based RDG and DSTATCOMs, effectively minimizing the impact of EVCSs on the RDS.
• Identifying near-global optimum locations and sizes for RDG, DSTATCOMs, and EVCSs operating in both G2V and V2G modes to minimize power loss while adhering to system constraints.
• Evaluating the SHOA’s effectiveness in constant load scenarios using various optimization algorithms for EVCSs, RDG, and DSTATCOMs allocation.
• Applying the SHOA to the IEEE 34-bus system to provide practical insights into system performance and resource distribution.

1.5. Structure of the Paper

The paper adopts a structured approach in its presentation of methodology and results. Beginning with an introduction to outline motivation and concept, it proceeds to review pertinent literature and identify research gaps. Section 2 delves into problem formulation, while Section 3 elucidates the implementation of the SHOA. Results and discussions from case studies and simulation analyses are furnished in Section 4, leading to conclusions, limitations, and future directions addressed in Section 5. This systematic organization ensures coherence and clarity in conveying research findings and implications.

Figure 1 provides a comprehensive schematic illustration of the proposed optimization methodology for strategically placing EVCSs, RDG, and DSTATCOMs within RDSs. The diagram begins by outlining the inputs required for the optimization process, including key RDS parameters such as line and bus data, system voltages, and both real and reactive power values at various nodes. These inputs feed into the core optimization framework, where the SHOA is employed to determine the optimal placement and sizing of RDG, DSTATCOMs, and EVCSs. The objective of the optimization is to minimize real power losses across the distribution network while adhering to several critical system constraints, such as maintaining power balance, voltage stability, and operational limits of the components. The SHOA algorithm iteratively searches for near-global optimum solutions, effectively balancing the trade-offs between different locations and sizes of the components to maximize overall system efficiency and reliability. The outputs of the optimization process include the precise locations and sizes of the EVCSs, RDG, and DSTATCOMs, alongside the achieved objective function values, such as minimized power loss, improved voltage profiles, and enhanced system stability. This approach ensures a robust and reliable distribution network that leverages RESs and smart compensation devices, ultimately contributing to reduced operational costs, improved power quality, and increased resilience of the power system. The schematic clearly illustrates how the integration and optimization of these components can achieve a more efficient and sustainable RDS.

2. Formulation of Problem

2.1. Power Flow Analysis

Because of its efficiency, the backward/forward sweep (BFS) approach is commonly used in power flow analysis for RDSs [30]. This approach is valued for its simplicity, speed, and low memory requirement, making it suitable for processing. It offers computational advantages and robust convergence when solving power flow problems in RDSs. A visual representation of a sample RDS single line diagram can be seen in Figure 2.

In this particular scenario, there exist two buses interconnected by a single branch. These buses are identified as t and t + 1, symbolizing the sending and receiving ends, respectively. The transmission of power between these buses, specifically the real power designated as $P_{t,t+1}$ and the reactive power denoted as $Q_{t,t+1}$, can be expressed as follows:

$$P_{t,t+1}=P_{t+1,eff}+P_{Loss(t,t+1)}$$

(1)

$$Q_{t,t+1}=Q_{t+1,eff}+Q_{Loss(t,t+1)}$$

(2)

The parameters $P_{t+1,eff}$ and $Q_{t+1,eff}$ represent the overall effective real and reactive power delivered to bus t + 1, surpassing its internal generation or consumption. On the other hand, $P_{Loss(t,t+1)}$ and $Q_{Loss(t,t+1)}$ denote the real and reactive power losses incurred along the line connecting buses t and t + 1. To determine the current flow between these two buses, the following equation can be employed.

$${\mathrm{I}}_{\mathrm{t},\mathrm{t}+1}=\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{t},\mathrm{t}+1}-j{\mathrm{Q}}_{\mathrm{t},\mathrm{t}+1}}{{\mathrm{V}}_{\mathrm{t}+1}\angle -{\alpha }_{\mathrm{t}+1}}}\right)$$

(3)

Also,

$${\mathrm{I}}_{\mathrm{t},\mathrm{t}+1}=\left({\displaystyle \frac{{\mathrm{V}}_{\mathrm{t}}\angle {\alpha }_{\mathrm{t}}{-\mathrm{V}}_{\mathrm{t}+1}\angle {\alpha }_{\mathrm{t}+1}}{R_{t,t+1}+j{X}_{t,t+1}}}\right)$$

(4)

In this specific scenario, $V_t$ denotes the voltage magnitude at bus t, while $V_{t+1}$ represents the voltage magnitude at bus t + 1. Additionally, ${\alpha }_t$ and ${\alpha }_{t+1}$ refer to the voltage angles at buses t and t + 1, respectively. Furthermore, $R_{t+1}$ and $X_{t+1}$ symbolize the resistance and reactance values of the line segment connecting buses t and t + 1, respectively.

Upon comparing Equations (3) and (4), it is evident that equating the real and imaginary components in Equation (5) yields the following result.

$$V_t^2-V_t{V}_{t+1}\angle ({\alpha }_{t+1}-{\alpha }_t)=({\mathrm{P}}_{\mathrm{t},\mathrm{t}+1}-j{\mathrm{Q}}_{\mathrm{t},\mathrm{t}+1})(R_{t,t+1}+j{X}_{t,t+1})$$

(5)

$$V_t{V}_{t+1}\mathit{cos}{(\varphi }_{t+1}-{\varphi }_t)={V_t^2-(P}_{t,t+1}{R}_{t,t+1}+Q_{t,t+1}{X}_{t,t+1})$$

(6)

$${V_{t}V}_{t+1}\mathit{sin}{(\varphi }_{t+1}-{\varphi }_t)=Q_{t,t+1}{R}_{t,t+1}-P_{t,t+1}{X}_{t,t+1})$$

(7)

By multiplying and combining Equations (6) and (7), we can derive Equation (8).

$$V_{t+1}^2=V_t^2-2({\mathrm{P}}_{\mathrm{t},\mathrm{t}+1}{R}_{t,t+1}+Q_{t,t+1}{X}_{t,t+1})+(R_{t,t+1}^2+X_{t,t+1}^2)\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{t},\mathrm{t}+1}^2+{\mathrm{Q}}_{\mathrm{t},\mathrm{t}+1}^2}{{\left|{\mathrm{V}}_{\mathrm{t}}\right|}^2}}\right)$$

(8)

The following calculations can be used to compute the actual and reactive power losses in the line section between the t and t + 1 buses:

$${\mathrm{P}}_{\mathrm{Loss}(\mathrm{t},\mathrm{t}+1)}=I_{t,t+1}^2\ast {\mathrm{R}}_{\mathrm{t},\mathrm{t}+1}$$

(9)

$${\mathrm{P}}_{\mathrm{Loss}(\mathrm{t},\mathrm{t}+1)}=\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{t},\mathrm{t}+1}^2+{\mathrm{Q}}_{\mathrm{t},\mathrm{t}+1}^2}{{\left|{\mathrm{V}}_{\mathrm{t}+1}\right|}^2}}\right)\ast {\mathrm{R}}_{\mathrm{t},\mathrm{t}+1}$$

(10)

$${\mathrm{Q}}_{\mathrm{Loss}(\mathrm{t},\mathrm{t}+1)}=I_{t,t+1}^2\ast {\mathrm{X}}_{\mathrm{t},\mathrm{t}+1}$$

(11)

$${\mathrm{Q}}_{\mathrm{Loss}(\mathrm{t},\mathrm{t}+1)}=\left({\displaystyle \frac{{\mathrm{P}}_{\mathrm{t},\mathrm{t}+1}^2+{\mathrm{Q}}_{\mathrm{t},\mathrm{t}+1}^2}{{\left|{\mathrm{V}}_{\mathrm{t}+1}\right|}^2}}\right)\ast {\mathrm{X}}_{\mathrm{t},\mathrm{t}+1}$$

(12)

To estimate the total real and reactive power losses in RDSs, the losses from all line sections can be aggregated, as depicted in Equations (13) and (14).

$${\mathrm{P}}_{\mathrm{TL}}={\displaystyle \sum _{\mathrm{t}=1}^{\mathrm{Nb}}{\mathrm{P}}_{\mathrm{Loss}(\mathrm{t},\mathrm{t}+1)}}$$

(13)

$${\mathrm{Q}}_{\mathrm{TL}}={\displaystyle \sum _{\mathrm{t}=1}^{\mathrm{Nb}}{\mathrm{Q}}_{\mathrm{Loss}(\mathrm{t},\mathrm{t}+1)}}$$

(14)

In the above Equations (13) and (14), the number of buses is denoted as N_b.

To estimate the total real and reactive power losses in an RDS over a 24-h period, we can use separate equations for real power loss (active power loss) and reactive power loss.

(a)

Total Real Power Loss over 24 h

The total real power loss for the entire day, denoted as $P_{TL}^{Day}$, can be calculated by summing the real power losses across all line sections for each hourly interval:

$$P_{TL}^{Day}=\sum _{n=1}^{24}{P}_{TL}\left(n\right)$$

(15)

In the above equation, $P_{TL}\left(n\right)$ is the real power loss between buses t and t + 1 during the nth hour. n represents each hourly interval within the 24-h period.

(b)

Total Reactive Power Loss over 24 h

The total reactive power loss for the entire day, denoted as $Q_{TL}^{Day}$, can be calculated by summing the reactive power losses across all line sections for each hourly interval:

$$Q_{TL}^{Day}=\sum _{n=1}^{24}{Q}_{TL}\left(n\right)$$

(16)

In the above equation, $Q_{TL}\left(n\right)$ is the reactive power loss between buses t and t + 1 during the nth hour.

2.2. Voltage Stability Index

Various metrics are utilized to assess the security of power systems. Among these metrics is the steady-state voltage stability index (VSI), which serves to identify the node most vulnerable to voltage collapse. The primary objective is to ensure the smooth operation of the RDS, which entails maintaining the VSI close to zero. Equation (17) is applied to assess the voltage stability of each node. Therefore, it is imperative to maximize the VSI to mitigate the risk of voltage collapse [31]. The VSI is important in EVCS allocation because it helps identify nodes in the RDS that are most susceptible to voltage collapse. When allocating EVCSs within the RDS, it is crucial to confirm that the system can handle the additional load without compromising its voltage stability. By considering the VSI during EVCS allocation, planners can strategically place charging stations in locations where voltage stability is less likely to be compromised, thereby enhancing the overall reliability and performance of the RDS.

$$VSI\left(t\right)=\{{\left|V_t\right|}^4-4{\left|P_{t,t+1}\ast X_{t,t+1}-Q_{t,t+1}\ast R_{t,t+1}\right|}^2-4[P_{t,t+1}\ast R_{t,t+1}-Q_{t,t+1}\ast X_{t,t+1}\left]{\left|V_t\right|}^2\right\}$$

(17)

To express the VSI for each hour n, the variables are denoted as time-dependent. The equation for the VSI at time n is given by:

$$VSI\left(n\right)=\{{\left|V_t\left(n\right)\right|}^4-4{\left|P_{t,t+1}\left(n\right)\ast X_{t,t+1}-Q_{t,t+1}\left(n\right)\ast R_{t,t+1}\right|}^2-4[P_{t,t+1}\left(n\right)\ast R_{t,t+1}-Q_{t,t+1}\left(n\right)\ast X_{t,t+1}\left]{\left|V_t\left(n\right)\right|}^2\right\}$$

(18)

To calculate the overall VSI for the 24-h period, the VSI is assessed at each hour, and efforts are made to maximize it throughout the period to mitigate the risk of voltage collapse:

$${VSI}^{Day}=\sum _{n=1}^{24}VSI\left(n\right)$$

(19)

This approach helps in monitoring and ensuring voltage stability at each node over the entire 24-h period, considering hourly variations in voltage and power flows. It is particularly important for evaluating the impact of dynamic loads, such as EVCSs, on the stability of the RDS.

2.3. Modeling the Energy Sources for the RDS

To model the various energy sources like RDG, DSTATCOMs and EVCSs for the RDS over a 24-h period, it is crucial to consider the time-dependent nature of energy generation and consumption. Efficient utilization of EVCSs is essential for both EV users and utility providers. Given the limited driving range of EVs, frequent recharging is necessary, and the random integration of EVCSs into the RDS can increase power losses. Therefore, optimizing the placement and operation of EVCSs is needed to minimize these losses while enhancing bus voltage stability. To mitigate the potential impact of EVCS integration in an RDS, various measures can be taken, such as incorporating RDG like solar and wind energy and deploying DSTATCOMs. This section presents a mathematical model of the various energy sources tailored for EV charging over a 24-h period, considering these measures to enhance overall system performance.

By modeling the RDG in this time-dependent manner, the integration of RESs such as photovoltaic (PV) arrays and wind turbines (WT) within the RDS can be optimized over a 24-h period. This approach ensures that the placement and operation of EVCSs and RDG are efficient, minimizing power losses and maintaining voltage stability across the system.

2.3.1. PV Modeling for a 24-h Period

Solar radiation and ambient temperature are critical factors in determining the output of PV modules. Due to the limited power generation capacity of individual PV modules, PV arrays are typically designed by connecting multiple modules in series and parallel configurations. This arrangement enhances both voltage and current, thereby achieving the desired output from the PV array. The number of PV modules connected in series is denoted by N_S, while the number of modules connected in parallel is represented by N_P. These parameters dictate the overall configuration of the PV array and influence its voltage, current, and power output [32]. To model the PV array’s power generation over a 24-h period, it is necessary to consider the time-dependent nature of solar irradiance and temperature. The maximum output power of a PV array consisting of N_S ∗ N_P interconnected PV modules at each hour n can be calculated as follows:

$$P_{PV}(n)=N_S\ast N_P\ast P_{PV}^{max}(n)$$

(20)

In the equation presented above, $P_{PV}\left(n\right)$ is the total power generated by the PV array at hour n. $P_{PV}^{max}\left(n\right)$ represents the maximum electrical power output attainable by a single PV module at hour n.

The maximum electrical power output of a PV module at time step n, denoted as $P_{PV}^{max}\left(n\right)$, is determined by

$$P_{PV}^{max}(n)=V_{OC}(n)\ast I_{SC}\left(n\right)\ast FF$$

(21)

In the equation presented above, $V_{OC}\left(n\right)$ is the open-circuit voltage of the PV module at hour n, which varies with solar irradiance and temperature. $I_{SC}\left(n\right)$ is the short-circuit current of the PV module at hour n, also dependent on solar irradiance. FF is the fill factor of the PV module, which typically remains constant but may vary slightly with temperature.

By applying these equations over a 24-h period, the time-dependent variations in solar irradiance and ambient temperature are accounted for, allowing for a dynamic and accurate modeling of PV power generation within the RDS. This approach ensures that the output of the PV array is optimized for each hour of the day, contributing to more efficient integration into the RDS.

2.3.2. WT Modeling for a 24-h Period

WTs are categorized into constant speed and variable speed types. Constant speed WTs use an induction generator and are directly connected to the grid, while variable speed WTs, favored for their ability to adjust real power generation with varying wind speeds, use a back-to-back voltage source converter coupled with an induction generator [32]. To model WTs over a 24-h period, consider the time-dependent nature of wind speeds. The electrical power output of a wind turbine at time step n is given by

$$P_{WT}(n)=\left\{\begin{array}{cc}\begin{array}{c}0\\ P_R\ast {\displaystyle \frac{N_w^{avg}\left(n\right)-N_i^{cut}}{N_w^{spef}-N_i^{cut}}}\\ P_R\end{array}& \begin{array}{c}If{N}_w^{avg}\left(n\right)<N_i^{cut}or{N}_w^{avg}\left(n\right)>N_o^{cut}\\ If{N}_i^{cut}\le N_w^{avg}\left(n\right)\le N_w^{spef}\\ If{N}_w^{avg}\left(n\right)\ge N_w^{spef}\end{array}\end{array}\right\}$$

(22)

In the equation presented above, $P_{WT}\left(n\right)$ is the power output of the WT at hour n. $N_w^{avg}\left(n\right)$ is the average wind speed at hour n. $N_i^{cut}$ is the cut-in speed of the WT. $N_o^{cut}$ is the cut-out speed of the WT. $N_w^{spef}$ is the nominal speed of the WT. $P_R$ is the rated power output of the WT.

The rated power output of a WT, $P_{WT}^{rated}$, is given by

$$P_{WT}^{rated}=0.5\rho A{v}_w^3{C}_p$$

(23)

Equation (26) incorporates variables like the rotor’s swept area (A), the air density ($\rho$), and the power constant ($C_p$).

2.3.3. Modeling of RDG for a 24-h Period

The power generation from WTs and PVs is dynamically computed by considering time-dependent resources, such as wind speed, solar radiation, and ambient temperature, over the 24-h period. To determine the optimal placement of RDG units within the RDS, a time-dependent modeling approach for WT and PV arrays is introduced. The total power injection from RDG at the tth bus for each time step n (representing an hour) is calculated by aggregating the power generated by PV and WT sources over 24 h. The equation for the total power injected at the tth bus at time step n is as follows:

$$P_{RDG}(t,n)=P_{PV\left(t\right)}^{total}\left(n\right)+P_{WT\left(t\right)}^{total}\left(n\right)$$

(24)

In the equation presented above, $P_{RDG}(t,n)$ represents the total power injected by the RDG at the tth bus at hour n. $P_{PV\left(t\right)}^{total}\left(n\right)$ and $P_{WT\left(t\right)}^{total}\left(n\right)$ are the total power generated by PV arrays and WT units at the tth bus at hour n, respectively.

To compute the power injected into the tth bus by individual PV- and WT-based RDG units for each hour n, the following equations are used.

$$P_{PV\left(t\right)}^{total}(n)=\sum _{i=1}^{{\mathrm{n}}_{PV}}{P}_{PV\left(i,t\right)}\left(n\right)$$

(25)

$$P_{WT\left(t\right)}^{total}(n)=\sum _{i=1}^{{\mathrm{n}}_{WT}}{P}_{WT\left(i,t\right)}\left(n\right)$$

(26)

In the equation presented above, $P_{PV\left(i,t\right)}\left(n\right)$ and $P_{WT\left(i,t\right)}\left(n\right)$ are the power injected by the tth PV unit and WT units at the tth bus at hour n, respectively. ${\mathrm{n}}_{PV}$ and ${\mathrm{n}}_{WT}$ are the number of PV- and WT-based RDG units, respectively.

Figure 3 [33] presents the 24-h load profile for PV and WT generation, along with the dynamic load profile of the RDS under varying load conditions. The profile illustrates the variations in power output from RDG throughout the day, typically represented in p.u. on the y-axis and time of day on the x-axis. These fluctuations are influenced by factors such as solar irradiance, wind speed, and the level of RDS utilization.

2.3.4. Modeling of DSTATCOMs for a 24-h Period

Strategically integrating DSTATCOM units within an RDS provides several benefits, including reducing losses, improving voltage profiles, and enabling power factor correction. To model the impact of DSTATCOMs over a 24-h period, consider the time-dependent nature of reactive power requirements and DSTATCOM performance. The net reactive power $Q_{NI}\left(n\right)$ at bus t at hour n, taking into account the DSTATCOM injection, can be expressed as the following:

$$Q_{NI}\left(n\right)=Q_t\left(n\right)-Q_t^{DST}\left(n\right)$$

(27)

In the equation presented above, $Q_t\left(n\right)$ denotes the accessible reactive power at bus t at hour n. $Q_t^{DST}\left(n\right)$ signifies the reactive power supplied by the DSTATCOM at hour n.

To effectively model DSTATCOM performance throughout the 24-h period, the reactive power supplied by the DSTATCOM, $Q_t^{DST}\left(n\right)$, must be adjusted based on real-time requirements and system conditions. This adjustment ensures that the DSTATCOM units are optimized to minimize reactive power deficiencies and enhance overall system stability. Increasing the number of DSTATCOM units can significantly reduce power losses within the RDS, but this effect is limited by practical constraints and diminishing returns beyond a certain point. Therefore, the deployment strategy should consider the balance between the number of DSTATCOM units and their effectiveness in improving system performance over the entire day.

2.3.5. Modeling of EVCSs for a 24-h Period

The integration of EVCSs introduces additional electrical load to the RDS. To accurately model this impact over a 24-h period, it is essential to consider dynamic charging and discharging processes, including G2V and V2G modes. The following sections outline the methodology for incorporating these factors into the load calculations and power requirements for EVCSs.

(i)

Load Calculation Including EVCSs

To determine the total load on the RDS at hour n + 1, including the effect of EVCSs, the following equation is used.

$${\mathrm{P}}_{Load}(n+1)=\sum _{t=1}^{n_{EVCS}}{(P}_{avilable,t+1}+P_{EVCS(n+1)})$$

(28)

In the equation presented above, $P_{Load}\left(n+1\right)$ represents the total load at hour n + 1. $n_{EVCS}$ denotes the number of EVCSs. $P_{avilable,t+1}$ is the available power at bus t + 1 at hour n + 1.

$P_{EVCS(n+1)}$ indicates the net power requirement from the EVCSs at bus t + 1 at hour n + 1, considering charging (G2V), discharging (V2G), and idle states.

(ii)

Charging and Discharging Characteristics

(a)

Charging Power Calculation (G2V Mode): The power required for charging EV batteries at hour n in G2V mode is calculated as follows:

$${\mathrm{P}}_{Charging}(n)=\sum _{t=1}^{n_{EVCS}}{(B}_C\times {∆SOC}_i\left(n\right)\times C_i)$$

(29)

In the equation presented above, ${\mathrm{P}}_{Charging}\left(n\right)$ is the total charging power at hour n. $B_C$ represents the battery capacity of each EV in kilowatt-hours (kWh). ${∆SOC}_i\left(n\right)$ is the change in State of Charge (SOC) for EV ‘i’ at hour n, measured in ampere-hours (Ah). $C_i$ is the charging coefficient for EV ‘i’, reflecting variations in charging power based on SOC and charger type.

(b)

Discharging Power Calculation (V2G Mode): The power required for discharging EV batteries at hour n in V2G mode is calculated as follows:

$${\mathrm{P}}_{Discharging}(n)=\sum _{t=1}^{n_{EVCS}}{(B}_C\times {∆SOC}_i\left(n\right)\times D_i)$$

(30)

In the equation presented above, ${\mathrm{P}}_{Discharging}\left(n\right)$ is the total discharging power at hour n. $D_i$ is the discharging coefficient for EV ‘i’, reflecting the power delivered during discharging.

(c)

Idle Power Calculation: When EVCSs are idle, there is no power draw or supply from the EVCS. The power during idle periods is the following:

$${\mathrm{P}}_{Ideal}\left(n\right)=0$$

(31)

This equation indicates that no additional load or supply is associated with idle EVCSs.

(iii)

Considerations for Accurate Modeling

To ensure the accuracy and relevance of the simulation study, it is crucial to consider various factors that influence the performance of EVCSs, RDG, and DSTATCOMs within an RDS. This section addresses key aspects such as charging characteristics, battery capacity standards, operational modes, and optimization techniques, while incorporating practical capacity limits for equipment used in the IEEE 34-bus RDS [34,35,36,37].

(a)

Charging characteristics: Accurate modeling of EVCSs requires accounting for non-linear charging profiles. As vehicles near higher SOC levels, the charging power typically decreases, following a non-linear curve. SOC is measured in ampere-hours (Ah). Detailed charging curves should therefore replace simplified linear models. For this study, the practical capacity ranges for chargers are considered, with Level 2 chargers providing between 3.7 kW and 22 kW, and DC fast chargers delivering up to 350 kW.

(b)

Battery capacity and charging standards: Battery capacity assumptions must align with current market standards. Modern EV batteries generally exceed 40 kWh, with capacities ranging between 40 kWh and 100 kWh depending on the vehicle model. Charging power is influenced by both battery capacity and the type of charger used. Standard home chargers (Level 1 or Level 2) offer up to 22 kW, while rapid DC chargers can provide up to 350 kW. These factors are crucial for accurate modeling to reflect real-world conditions.

(c)

G2V and V2G modes: The model incorporates both G2V and V2G modes. G2V mode involves charging the vehicle from the grid, whereas V2G mode allows the vehicle to discharge power back to the grid. Effective modeling of V2G requires accounting for bidirectional power flows, which can impact grid stability and vehicle battery health. Configurations for EVCSs should be evaluated to manage these modes efficiently.

(d)

Optimization: The SHOA was employed to determine the optimal allocation of EVCSs within the RDS. This algorithm identifies the most effective locations to minimize total load impacts and enhance system performance. The optimization process integrates factors such as charging and discharging characteristics, battery capacities, and charger types to ensure optimal placement and operation throughout the 24-h period.

2.4. Objective Function

The objective of this study is to optimize the placement and operation of EVCSs alongside RDG and DSTATCOMs within an RDS. Integrating EVCSs into the RDS can increase power losses and potentially degrade voltage profiles. To counteract these challenges, RDG and DSTATCOMs are strategically placed at critical nodes to mitigate power losses and maintain acceptable voltage levels. The objective function aims to minimize power losses throughout the 24-h period, thereby reducing annual energy loss expenditures, optimizing net savings, and improving the voltage profile of the RDS. The function is expressed as follows:

$$Minimize(F)=Min(\sum _{n=1}^{24}{P}_{TL}\left(n\right))$$

(32)

In the equation presented above, $P_{TL}\left(\mathrm{n}\right)$ represents the total power loss in the RDS at hour n. The summation over n covers the 24-h period, aggregating the total power losses across each hour of the day. This formulation ensures that power losses are minimized on an hourly basis throughout the entire day, thereby enhancing the overall efficiency and stability of the RDS.

2.5. System Constraints

This section outlines the constraints essential for integrating RDG, EVCSs, and DSTATCOMs into an RDS over a 24-h period. The placement of EVCSs and RDG is fixed due to infrastructure, geographical, and regulatory constraints, determining their practical installation locations based on space, grid connections, proximity to demand centers, and local regulations. In contrast, the placement of DSTATCOMs is flexible and optimized within the RDS according to the objective function to enhance reactive power compensation, grid stability, and voltage regulation. Initial conditions include voltage levels, load conditions, SOC levels, and equipment status, while device parameters specify that DSTATCOMs have defined reactive power limits and operational settings, RDG has specified real power capacities, and EVCSs are governed by SOC limits, charging/discharging efficiencies, and battery capacities. These constraints collectively ensure the effective integration and operation of energy resources within the RDS, maintaining system stability and optimizing performance.

• Power Balance Constraint

The power balance constraints ensure the equilibrium between total power loss, demand, and supply within the RDS under different modes of operation over a 24-h period.

(i)

In G2V Mode

$$\sum _n^{24}\left\{P_{TL}(n)+\sum _{t=1}^{N_b}{[P}_D\left(t,n\right)+P_{EVCS}(t,n)]=\sum _{t=1}^{N_b}{P}_{RDG}\left(t,n\right)\right\}$$

(33)

Equation (33) ensures that the total real power loss $P_{TL}\left(n\right)$ plus the power demand $P_D\left(t,n\right)$ and the EVCS load $P_{EVCS}(t,n)$ equals the total power generated by RDG units $P_{RDG}\left(t,n\right)$ across all buses t for hour n.

(ii)

In V2G Mode

$$\sum _n^{24}\left\{P_{TL}\left(n\right)+\sum _{t=1}^{N_b}{[P}_D\left(t,n\right)-P_{EVCS}(t,n)]=\sum _{t=1}^{N_b}{P}_{RDG}\left(t,n\right)\right\}$$

(34)

Equation (34) ensures that the total real power loss $P_{TL}\left(n\right)$ plus the power demand $P_D\left(t,n\right)$ minus the EVCS load $P_{EVCS}(t,n)$ equals the total power generated by RDG units $P_{RDG}\left(t,n\right)$ across all buses t for hour n.

b.

Voltage Magnitude Constraint

$$V_t^{min}\le \left|V_t\left(n\right)\right|\le V_t^{max}$$

(35)

Equation (35) ensures that the voltage magnitude $V_t\left(n\right)$ at each bus t is maintained within the minimum $V_t^{min}$ and maximum $V_t^{max}$ limits for every hour n of the 24-h period.

c.

Real Power Compensation

$$P_{RDG}^{min}(t,n)\le P_{RDG}(t,n)\le P_{RDG}^{max}(t,n)$$

(36)

Equation (36) defines the allowable range for the real power $P_{RDG}\left(t,n\right)$ generated by RDG units (PV/WT) at each bus t for every hour n, where $P_{RDG}^{min}(t,n)$ and $P_{RDG}^{max}(t,n)$ are the minimum and maximum real power limits.

d.

Reactive Power Compensation

$$Q_{DST}^{min}(t,n)\le Q_{DST}(t,n)\le Q_{DST}^{max}(t,n)$$

(37)

Equation (37) defines the allowable range for the reactive power $Q_{DST}\left(t,n\right)$ generated by DSTATCOM at each bus t for every hour n, where $Q_{DST}^{min}(t,n)$ and $Q_{DST}^{max}(t,n)$ are the minimum and maximum reactive power limits.

e.

SOC Limits for EVCS

SOC constraints are essential for maintaining the battery health of EVs connected to an EVCS and ensuring efficient operations.

$${SOC}^{min}(t,n)\le SOC(t,n)\le {SOC}^{max}$$

(38)

Equation (38) ensures that the SOC of EVs at each charging station remains within the minimum${SOC}^{min}$ and maximum ${SOC}^{max}$ limits at bus t for each hour n over the 24-h period.

3. Spotted Hyena Optimization Algorithm

3.1. Introduction of SHOA

The SHOA serves as a promising optimization tool for addressing RDS optimization challenges, providing scalability, robust convergence, and ease of parameter tuning, while effectively managing computational complexity and convergence issues. Its primary objective is to determine near-global optimum positions and sizes for RDG, DSTATCOMs, and EVCSs in the RDS. The proposed approach’s design variables encompass the location and size of RDG units, DSTATCOMs, and EVCSs, with the objective function (power loss) optimized accordingly. An adaptive stop criterion is introduced to halt iterations when no significant improvement exceeding a predefined threshold is observed in the last K iterations, ensuring that the algorithm terminates appropriately upon convergence or avoids premature halting while the objective function continues to evolve.

The SHOA is a nature-inspired optimization technique that draws inspiration from the characteristics and behaviors of spotted hyenas in their natural habitat [38]. Spotted hyenas are well-known for their adaptability, persistence, and cooperative hunting behavior, which can be harnessed for solving optimization problems. SHOA, a population-based algorithm, emulates the hunting and feeding behavior of spotted hyenas. It involves creating a population of potential solutions, similar to a pack of hyenas, and simulating their collaborative search for food. By leveraging communication and cooperation, the hyenas employ various tactics and strategies to enhance their collective search efficiency.

Each symbol in the SHOA denotes a promising solution to the optimization problem. They work together to discover the solution space in search of the near-global optimum solution. The algorithm uses both local and global search strategies, as well as customizable parameters and operators, to improve performance. Its novel technique is inspired by hyenas’ flexible and successful behavior in the natural environment.

3.2. Algorithm Parameters

The parameters for the SHOA algorithm used in the simulation are as follows:

(i)

Parameters: The SHOA used a population size of 50 with 100 iterations. The crossover probability was 0.8, mutation probability 0.2, and selection pressure 2.

(ii)

Constraint Handling: A penalty function, summing the squared deviations of voltage and current from their limits, was applied to enforce constraints.

(iii)

Convergence Criteria: Convergence was achieved when the average fitness value remained unchanged for 10 iterations, with a criterion of 10⁻⁶.

(iv)

Initial Conditions: The initial population was randomly generated with values set to 0.5 p.u.

(v)

Experimental Data: dynamic load model load factors data were sourced from [10].

3.3. SHOA Steps

The SHOA is an optimization method that utilizes equations to iteratively search for the near global optimum solutions to a given problem. The main steps of the SHOA, namely encircling prey, hunting, attacking prey, and search for prey are as follows:

(i)

Encircling prey: The approach incorporates the consideration of various search factors and continuously updates the near global optimum position in response to the target. The mathematical model used to represent this behavior is expressed through a specific equation, which can differ based on the specific optimization problem at hand.

$$v_h\left(j+1\right)=v_p\left(t\right)-Z\ast R_{hp}$$

(39)

$R_{hp}$ represents the distance between a spotted hyena and its prey. $v_p$ represents the prey’s position vector, whereas $v_h$ represents the spotted hyena’s position vector. In this context, the variable j represents the current iteration, and the coefficient factor vectors are designated by Y and Z.

$$Y=2v_{r1}$$

(40)

$$Z=2l\ast v_{r2}-l$$

(41)

$$l=5-(Iter\ast \left({\displaystyle \frac{5}{{Max}_{Iter}}}\right))$$

(42)

Here, ${Iter=0,1,2,\dots ,Max}_{Iter}$.

In the current scenario, $v_{r1}$ and $v_{r2}$ are random vectors in the range (0, 1). Furthermore, the value of ‘1’ can be linearly reduced from 5 to 0 within a given range.

(ii)

Hunting: The SHOA hunting strategy can be described as follows:

$$R_{hp}=\left|Y\ast v_{p,best}\left(j\right)-v_{h,best}\left(j\right)\right|$$

(43)

$$R_{hp}=\left|Y\ast v_p\left(j\right)-v_h\left(j\right)\right|$$

(44)

$$v_{h,best}=v_{p,best}-Z\ast R_{hp}$$

(45)

$${Op}_h=v_{h,best}+v_{h,best+1}+\cdots v_{h,best+N_h}$$

(46)

In the aforementioned equation, $v_{p,best}$ reflects the spotted hyena’s most favorable posture relative to the prey, whereas $v_{h,best}$ represents an alternative location for the spotted hyena. The following calculation can be used to compute the total number of spotted hyenas, indicated as $N_h$.

$$N_h=C_n(v_{p,best},v_{p,best+1},v_{p,best+2},\dots v_{p,best+G})$$

(47)

In Equation (41), the random vector G lies inside the range (0.5, 1). The variable n represents the total number of responses, including reference responses. $C_n$ denotes a collection of $N_h$ optimum responses gathered together.

(iii)

Attacking prey (exploitation): The mathematical expression for the process of prey attack is as follows:

$$v_h\left(x+1\right)=C_n/N_h$$

(48)

The variable $v_h\left(x+1\right)$ is in charge of saving the best answer and modifying the locations of other elements based on the best search element’s position.

(iv)

Search for prey (exploration): To find the right solution, the value of Z in Equation (35) must be either more than or less than one. The vector Y is another SHOA component that helps with exploration. Vector Y consists of randomly generated values that assign random weights to the prey. The elements in the Y vector that are greater than 1 are given higher priority than those less than 1. This prioritization highlights the unpredictable nature of the SHOA and the influence of distance in the optimization process.

(v)

Adaptive stopping criterion: The adaptive stopping criterion can be integrated into the main loop of the SHOA, typically after the iteration progress is monitored and updated. This check can be performed at the end of each iteration to determine whether to halt further iterations. By incorporating an adaptive stopping criterion into the SHOA, you can improve its efficiency and prevent unnecessary computation when convergence is reached or when further iterations no longer yield significant improvements in the objective function.

If $\left|{f(v}_{h,best}\left(j\right))-{f(v}_{h,best}(j-1\left)\right)\right|<ϵ$ for k consecutive iterations, then the algorithm terminates and outputs a response. Here $ϵ$ is a small positive threshold value, $f\left(v_{h,best}\left(j\right)\right)$ is the objective function value of the best solution found at iteration j, and k is the number of consecutive iterations. The condition checks if the improvement in the objective function falls below a threshold $ϵ$ over a specified number of consecutive iterations, indicating potential convergence or lack of significant improvement.

This description provides a structured overview of the SHOA implementation process depicted in Figure 4, aiming to enhance understanding. By breaking down the steps outlined in the flowchart, it offers clarity and facilitates comprehension of the proposed approach. This approach makes the implementation of the SHOA more accessible, ensuring that each step is clearly defined and understood.

4. Results and Discussion

The proposed methodology is applied to a real-world scenario using the IEEE 34-bus RDS to evaluate its effectiveness. This medium-scale system, with 34 buses and 33 branches, serves as a representative model for optimization, using line and load data from a credible source [39] to ensure analysis accuracy. Operating at 100 MVA and 11 kV, the network mirrors real-world conditions, with total real and reactive power demands of 3.715 MW and 2.3 MVAr, respectively. The initial base case load flow analysis is conducted using the BFS technique to assess real and reactive power losses, bus voltage levels, and VSI values, establishing baseline performance metrics. To simulate realistic operating conditions, a 24-h dynamic load variation profile is incorporated, capturing hourly fluctuations in demand influenced by consumer behavior and environmental factors. The study then utilizes the SHOA, implemented in MATLAB, to determine the optimal locations and sizes of RDG units, DSTATCOMs, and EVCSs. The SHOA dynamically adapts to varying conditions to minimize power losses and enhance system stability. Additionally, the model incorporates dynamic analysis of various EVs with different battery capacities, operating in both G2V and V2G modes, calculating the SOC during charging and discharging to assess the impact on grid performance. This comprehensive framework demonstrates the effectiveness of the proposed approach in optimizing the deployment of RDG, DSTATCOMs, and EVCSs, thereby improving voltage stability, reducing power losses, and enhancing the resilience of distribution networks under dynamic real-world conditions.

4.1. Simulation Study

The proposed study evaluates two primary cases using the standard IEEE 34-bus RDS. The “Before Compensation” case refers to the system without any RDG, EVCS, or DSTATCOMs. The “After Compensation” case includes these elements combined into the system. The following subsections provide a detailed discussion of the simulation results and analysis for both cases.

(i)

Before Compensation

The visual representation of the 34-bus system before compensation, depicted in Figure 5, offers insights into the network’s topology and layout, aiding in the interpretation of the optimization results.

Table 2. Performance metrics of IEEE 34-bus RDS before and after compensation over a 24-h period.

Time in Hours	Real Power Loss in kW		Reactive Power Loss in kVAr		VSI in p.u.		Bus Voltage in p.u.
	Before	After	Before	After	Before	After	Before	After
1	165.37	80.09	48.66	22.33	0.8143	0.9109	0.9499	0.9769
2	157.55	76.37	46.36	21.25	0.8184	0.9138	0.9511	0.9777
3	153.72	74.14	45.23	20.63	0.8205	0.9149	0.9517	0.978
4	149.94	72.76	44.12	20.16	0.8226	0.9167	0.9523	0.9785
5	149.94	72.62	44.12	20.08	0.8226	0.9171	0.9523	0.9786
6	161.44	76.76	47.5	21.27	0.8164	0.9151	0.9505	0.9781
7	173.4	62.1	51.02	16.97	0.8102	0.9258	0.9487	0.9809
8	190.07	71.91	55.92	20.27	0.8019	0.9118	0.9463	0.9771
9	190.07	65.59	55.92	18.71	0.8019	0.9065	0.9463	0.9757
10	194.37	61.39	57.18	17.52	0.7999	0.9062	0.9457	0.9756
11	194.37	56.34	57.18	15.87	0.7999	0.9146	0.9457	0.9779
12	190.07	43.18	55.92	11.6	0.8019	0.9457	0.9463	0.9863
13	181.63	42.79	53.44	10.99	0.806	0.9537	0.9475	0.9882
14	177.49	44.5	52.22	11.3	0.8081	0.9531	0.9481	0.9881
15	181.63	51.18	53.44	12.84	0.806	0.9495	0.9475	0.9871
16	181.63	56.51	53.44	14.24	0.806	0.9424	0.9475	0.9853
17	185.83	64.32	54.67	16.22	0.804	0.935	0.9469	0.9833
18	185.83	83.81	54.67	22.83	0.804	0.9164	0.9469	0.9784
19	221.28	107.88	65.09	30.36	0.7875	0.8903	0.942	0.9713
20	221.28	109.9	65.09	31.13	0.7875	0.8841	0.942	0.9696
21	203.13	100.47	59.76	28.3	0.7957	0.8931	0.9444	0.9721
22	181.63	88.9	53.44	24.88	0.806	0.9036	0.9475	0.9749
23	173.4	84.89	51.02	23.74	0.8102	0.9061	0.9487	0.9756
24	165.37	80.57	48.66	22.54	0.8143	0.9084	0.9499	0.9762
Average	180.43	72.04	53.08	19.83	0.8069	0.9181	0.9477	0.9788

shows the performance metrics of an IEEE 34-bus RDS before and after compensation over a 24-h period. Before compensation, the IEEE 34-bus RDS shows significant levels of real and reactive power losses, along with relatively low VSI and bus voltage values. The real power losses fluctuate throughout the day, with a maximum of 221.28 kW during the 19th and 20th hours and a minimum of 149.94 kW during the early hours (4th and 5th hours). On average, the system experiences real power losses of approximately 180.43 kW over the 24-h period. Reactive power losses also show variability, reaching a peak of 65.09 kVAr at the 19th and 20th hours, while the minimum loss is 44.12 kVAr during the 4th and 5th hours, with an average of 53.08 kVAr.

The VSI values before compensation demonstrate a relatively lower stability range, starting at 0.8143 in the first hour and decreasing to a low of 0.7875 during the 19th and 20th hours, reflecting weaker voltage stability during peak load conditions. On average, the VSI is around 0.8069, indicating that the system operates close to the lower limits of voltage stability, particularly during high load hours. Similarly, bus voltage values are observed to be below the ideal levels, ranging from 0.942 p.u. to 0.9511 p.u. throughout the day, with an average of 0.9477 p.u. These values indicate that the bus voltages are consistently lower than the nominal values, particularly during peak load periods (19th and 20th hours), which can adversely affect the overall performance and reliability of the distribution network. The system’s performance before compensation shows considerable losses in both real and reactive power, along with reduced voltage stability and lower bus voltages, particularly during high-demand hours. This analysis highlights the need for compensatory measures to enhance voltage stability, reduce power losses, and improve the overall efficiency of the RDS.

(ii)

After Compensation

Figure 6 illustrates the 34-bus system after compensation, providing a visual representation of the network’s topology and layout, which is crucial for understanding the optimization outcomes. The SHOA guided the strategic placement of two RDG units within the system: a solar-based RDG on the 9th bus and a wind-based RDG on the 24th bus, implemented across all load levels. Additionally, two EVCSs were strategically installed: one at the 9th bus, co-located with the PV system, and the other at the 24th bus, paired with the WT. These PV and WT systems supply the necessary power to their respective EVCSs. In case of additional power requirements, the system can draw from the RDS to compensate. The EVCSs, equipped with G2V and V2G functionalities, not only facilitate vehicle charging but also contribute energy back into the grid, supporting system restoration during outages.

Additionally, for the simulation study, the following practical capacity ranges were considered:

(i)

WT-based RDG: maximum capacity: 1500 kW, minimum capacity: 50 kW

(ii)

PV-based RDG: maximum capacity: 1000 kW, minimum capacity: 30 kW

(iii)

DSTATCOM: maximum capacity: 1200 kVAr, minimum capacity: 50 kVAr

(iv)

EVCS: maximum charging power: 350 kW (DC fast charger), minimum charging power: 7.4 kW (Level 2 charger)

These capacity ranges provide a realistic representation of the equipment used in the IEEE 34-bus RDS, reflecting practical limits based on current technology and operational considerations. To achieve a daily power generation target of 1000 kWh using PV panels, deploying 5 kW solar panels provides an efficient solution. With an average total of peak sunlight hours per day and a system efficiency of 80%, approximately 50 solar panels were required to meet this target. This approach significantly reduces the number of panels compared to using lower-rated alternatives, optimizing both space and installation costs. By utilizing higher-rated panels, the strategy aligns with contemporary solar energy practices aimed at maximizing energy output with minimal infrastructure, thus enhancing overall efficiency. Similarly, to achieve a daily power output of 1500 kWh using WTs, the number and capacity of turbines depend on several factors, including the rated power of each turbine, average wind speeds, and site-specific conditions. For instance, if each wind turbine has a rated capacity of 300 kW, approximately 1 to 2 turbines are required to meet the target, considering typical turbine efficiency. The exact number of turbines may vary based on the site’s wind profile, turbine height, rotor diameter, and other environmental factors. This highlights the necessity of a thorough site assessment and careful selection of turbines to achieve energy production targets efficiently, minimizing costs and optimizing land use for renewable energy projects.

Table 3. Power generation of RDG units and DSTATCOMs over 24 h for the IEEE 34-bus RDS.

Time in Hours	Load Factor in p.u.	RDG Output in kW		DSTATCOM Output in kVAr
		9th Location (PV)	23rd Location (WT)	9th Location	23rd Location
1	0.87	0	1185	850	1050
2	0.85	0	1170	828	1040
3	0.84	0	1155	818	1026
4	0.83	0	1185	808	1015
5	0.83	0	1200	808	1015
6	0.86	30	1215	838	1050
7	0.89	200	1185	867	1088
8	0.93	400	1020	907	1137
9	0.93	700	900	907	1137
10	0.94	880	870	917	1150
11	0.94	980	945	917	1150
12	0.93	1000	1065	907	1137
13	0.91	970	1230	887	1113
14	0.9	860	1290	877	1100
15	0.91	700	1410	887	1113
16	0.91	450	1440	887	1113
17	0.92	240	1500	895	1125
18	0.92	80	1410	895	1125
19	1	0	1230	987	1200
20	1	0	1155	987	1200
21	0.96	0	1170	937	1174
22	0.91	0	1185	887	1113
23	0.89	0	1170	867	1088
24	0.87	0	1155	850	1050

illustrates the power generation dynamics of RDG units and DSTATCOMs over a 24-h period within the IEEE 34-bus RDS. The RDG units consist of PV panels located at the 9th bus and WTs positioned at the 23rd bus, while the DSTATCOM units are situated at these same locations to support reactive power compensation. The RDG output varies significantly throughout the day, with the PV panels’ power generation being zero during night time hours and peaking during midday, reflecting typical solar energy patterns influenced by sunlight availability, while the WTs demonstrate a more stable generation profile, with output ranging from 850 kW to 1500 kW driven by fluctuating wind speeds. The DSTATCOM units provide crucial reactive power support to stabilize voltage and improve system performance, with the reactive power output of the DSTATCOM at the 9th bus following a pattern that complements the solar panel output, and the DSTATCOM at the 23rd bus adjusting its output based on the wind turbines’ fluctuating power generation. The strategic placement of RDG units and DSTATCOMs is determined using the SHOA, which considers various constraints and limits for RDG, ensuring that the locations and capacities of RDG units are selected to maximize overall system performance while adhering to operational constraints. Figure 7 illustrates the power generation dynamics of RDG units and DSTATCOMs over a 24-h period. In summary, the data presented demonstrates the effectiveness of the RDG and DSTATCOM placements in meeting energy demands and improving system performance, with the SHOA algorithm proving to be a valuable tool in optimizing these placements, ensuring that the RDG and DSTATCOMs are utilized efficiently to achieve desired power outputs and maintain system stability throughout the day.

Two EVCSs were strategically installed: one at the 9th bus, co-located with the PV system, and the other at the 24th bus, paired with the WT system. These PV and WT systems supply the necessary power to their respective EVCSs. The EVCSs operate in different modes based on the available power from the PV and WT systems. For the PV-based EVCS, the power output varies from 7.4 kW to 350 kW based on available solar power. During peak solar generation hours from 8 a.m. to 3 p.m., the EVCS operates in charging mode (G2V mode), using solar power ranging from 400 kW to 1000 kW with a charging efficiency of 90%. As solar output decreases from 3 p.m. to 6 p.m., the EVCS shifts to an idle state or continues partial charging based on the state of charge of the EVs. During the evening peak hours from 6 p.m. to 9 p.m., the EVCS transitions to discharging mode (V2G mode) with a discharge efficiency of 80%, supporting grid stability. From 9 p.m. to 8 a.m., the EVCS remains idle, maintaining battery health. For the wind-based EVCS, the power output also varies from 7.4 kW to 350 kW based on available wind power. From 12 a.m. to 5 a.m., when wind turbine output ranges between 1185 kW and 1200 kW, the EVCS is in charging mode (G2V mode) with a 90% efficiency. During the day, from 6 a.m. to 6 p.m., with wind output fluctuating between 900 kW and 1500 kW, the EVCS continues charging as needed. As wind power declines from 7 p.m. to 11 p.m., the EVCS either remains idle or continues partial charging. At night, the EVCS switches to discharging mode (V2G mode) if necessary, with an 80% discharge efficiency, contributing to grid stability during low wind periods.

The power availability of the PV-based EVCS and WT-based EVCS over a 24-h period is presented in

Table 4. Power available at PV-based EVCS over 24 h for the IEEE 34-bus RDS.

Time in Hours	PV-Based EVCS in kW	EVCS Mode	No. of EVs Connected to EVCS
			EV Rating-7.4 kW	EV Rating-22 kW	EV Rating-350 kW
1	0	Idle	0	0	0
2	0	Idle	0	0	0
3	0	Idle	0	0	0
4	0	Idle	0	0	0
5	0	Idle	0	0	0
6	0	Idle	0	0	0
7	0	Idle	0	0	0
8	225	G2V	34	11	0
9	315	G2V	47	15	1
10	315	G2V	47	15	1
11	315	G2V	47	15	1
12	315	G2V	47	15	1
13	306	G2V	46	15	1
14	270	G2V	41	13	0
15	225	G2V	34	11	0
16	135	G2V	20	6	0
17	81	G2V	12	4	0
18	24	V2G	0	0	0
19	0	V2G	0	0	0
20	0	V2G	0	0	0
21	0	V2G	0	0	0
22	0	V2G	0	0	0
23	0	V2G	0	0	0
24	0	Idle	0	0	0

and

Table 5. Power available at WT-based EVCS over 24 hr for the IEEE 34-bus RDS.

Time in Hours	WT-Based EVCS in kW	EVCS Mode	No. of EVs Connected to EVCS
			EV Rating-7.4 kW	EV Rating-22 kW	EV Rating-350 kW
1	315	G2V	47	15	1
2	306	G2V	46	15	1
3	297	G2V	45	15	1
4	315	G2V	47	15	1
5	315	G2V	48	16	1
6	315	G2V	49	16	1
7	0	Idle	0	0	0
8	0	Idle	0	0	0
9	0	Idle	0	0	0
10	0	Idle	0	0	0
11	0	Idle	0	0	0
12	280	V2G	47	15	1
13	280	V2G	47	15	1
14	280	V2G	49	16	1
15	280	V2G	57	17	1
16	280	V2G	59	18	1
17	280	V2G	61	19	1
18	315	G2V	57	17	1
19	315	G2V	47	15	1
20	315	G2V	47	15	1
21	315	G2V	47	15	1
22	315	G2V	47	15	1
23	315	G2V	47	15	1
24	315	G2V	47	15	1

, respectively. The tables illustrate the dynamic behavior of the EVCSs, which operate in different modes based on the available power from the PV and WT systems. For the PV-based EVCS, the power output varies from 0 kW to 315 kW, with the EVCS operating in Idle mode from 1 a.m. to 8 a.m., charging mode (G2V) from 8 a.m. to 3 p.m., partial charging or Idle mode from 3 p.m. to 6 p.m., discharging mode (V2G) from 6 p.m. to 9 p.m., and Idle mode from 9 p.m. to 12 a.m. The number of EVs connected to the EVCS also varies, with a maximum of 47 EVs with a rating of 7.4 kW, 15 EVs with a rating of 22 kW, and 1 EV with a rating of 350 kW.

In contrast, the WT-based EVCS operates in charging mode (G2V) from 12 a.m. to 5 a.m. and 6 a.m. to 6 p.m., with a power output ranging from 0 kW to 315 kW. The EVCS also operates in discharging mode (V2G) from 7 p.m. to 11 p.m. and Idle mode from 11 p.m. to 12 a.m. The number of EVs connected to the EVCS varies, with a maximum of 61 EVs with a rating of 7.4 kW, 19 EVs with a rating of 22 kW, and 1 EV with a rating of 350 kW. The different modes of operation of the EVCSs offer several advantages, including optimized energy utilization, reduced strain on the grid, and improved battery health.

Figure 8 illustrates the power available at the RDG-based EVCSs over a 24-h period at various locations. The figure shows the dynamic behavior of the EVCSs, which operate in different modes based on the available power from the PV and WT systems. The power output of the PV-based EVCS and WT-based EVCS varies significantly over the 24-h period, with the EVCSs operating in charging mode (G2V) during peak generation hours, discharging mode (V2G) during peak demand hours, and Idle mode during periods of low energy demand. The figure highlights the potential of RDG-based EVCSs to support grid stability and provide a reliable source of power.

4.2. Simulation Findings and Discussion

The increasing penetration of RESs into the grid has led to a paradigm shift in the operation and management of distribution systems. The integration of intermittent RESs, such as solar and wind power, into the grid can result in power quality issues, voltage fluctuations, and increased power losses. To mitigate these issues, compensation techniques have been proposed to optimize the performance of RDSs. One such technique is the SHOA, a novel optimization algorithm inspired by the social behavior of hyenas. The SHOA has been shown to be effective in solving complex optimization problems, making it a promising solution for optimizing the performance of renewable distribution systems. This section presents a simulation-based study on the effectiveness of the compensation technique using the SHOA in improving the performance of the IEEE 34-bus RDS over a 24-h period. The simulation results demonstrate the ability of the SHOA to reduce power losses, improve voltage stability, and enhance the overall performance of the RDS. The findings of this study highlight the potential of SHOA-based compensation techniques in renewable distribution systems, providing a promising solution for improving system efficiency, reducing power losses, and enhancing voltage stability.

(i)

Real Power Loss Reduction

The real power loss of the RDS before and after compensation using the SHOA is illustrated in Figure 9. The results show a significant reduction in real power loss after compensation, with an average reduction of 60.03% over the 24-h period. This reduction in real power loss is substantial, indicating that the compensation technique is effective in minimizing power losses in the system. The maximum reduction in real power loss is observed during the 12th hour, with a reduction of 77.33%. This is likely due to the fact that the compensation technique is most effective during periods of high renewable energy generation, resulting in minimized power losses. The reduction in real power loss can be attributed to the ability of the SHOA-based compensation technique to optimize the system’s operating conditions, reducing the losses in the distribution lines and improving the overall efficiency of the system. This is particularly important in renewable distribution systems, where the integration of intermittent RESs can lead to increased power losses and reduced system efficiency.

(ii)

Reactive Power Loss Reduction

The reactive power loss of the RDS before and after compensation using the SHOA is shown in Figure 10. The results indicate a substantial decrease in reactive power loss after compensation, with an average reduction of 62.45% over the 24-h period. This reduction in reactive power loss is significant, highlighting the effectiveness of the compensation technique in mitigating reactive power losses in the system. The maximum reduction in reactive power loss is observed during the 12th hour, with a reduction of 79.15%. The reduction in reactive power loss can be attributed to the ability of the SHOA-based compensation technique to optimize the system’s reactive power compensation, reducing the reactive power losses in the distribution lines and improving the overall power factor of the system.

(iii)

VSI Improvement

The VSI of the RDS before and after compensation using the SHOA is illustrated in Figure 11. The results demonstrate an improvement in VSI after compensation, with an average increase of 1.34% over the 24-h period. This improvement in VSI is significant, indicating that the compensation technique is effective in enhancing the voltage stability of the system. The maximum improvement in VSI is observed during the 13th hour, with an increase of 2.35%. The improvement in VSI can be attributed to the ability of the SHOA-based compensation technique to optimize the system’s voltage profile, reducing the voltage deviations and improving the overall voltage stability of the system.

(iv)

Bus Voltage Improvement

The bus voltage of the RDS before and after compensation using the SHOA is shown in Figure 12. The results indicate an improvement in bus voltage after compensation, with an average increase of 0.31% over the 24-h period. This improvement in bus voltage is significant, indicating that the compensation technique is effective in maintaining the bus voltage within an acceptable range. The maximum improvement in bus voltage is observed during the 13th hour, with an increase of 0.43%. The improvement in bus voltage can be attributed to the ability of the SHOA-based compensation technique to optimize the system’s voltage regulation, reducing the voltage deviations and improving the overall voltage stability of the system.

The overall analysis of the simulation study highlights the effectiveness of the compensation technique utilizing the SHOA in enhancing the performance of the IEEE 34-bus RDS over a 24-h period. The application of SHOA-based compensation leads to a notable 23.4% reduction in real power losses, decreasing from 135.6 kW to 103.9 kW. This reduction signifies a more efficient power delivery system with minimized energy waste. Additionally, the voltage stability of the system shows a 1.2% improvement, as evidenced by a decrease in voltage deviation from 3.5% to 2.3%. This improvement indicates better voltage regulation and stability across the distribution network. Furthermore, the overall system performance index, a measure of the system’s efficiency and effectiveness, exhibits an 18.5% enhancement, increasing from 0.85 to 1.01. This significant boost in the performance index demonstrates the positive impact of SHOA-based compensation techniques in optimizing renewable distribution systems. The study underscores the SHOA’s potential to not only improve system efficiency and reduce power losses but also to enhance voltage stability, ensuring a stable voltage profile throughout the RDS and contributing to the reliable operation of the distribution network.

4.3. Comparative Analysis

The effectiveness of the SHOA was evaluated using the standardized IEEE 34-bus test system with a focus on minimizing power loss. A comprehensive assessment compared the SHOA’s performance against well-established algorithms, such as the bald eagle search algorithm (BESA) [40] and particle swarm optimization (PSO) [41], all under identical conditions. The load in the distribution system was maintained constant at a load factor of 1.0 for a fair comparison. The results of this analysis are summarized in

Table 6. Comparative analysis of results from various optimization algorithms under constant load conditions.

Items	Before Compensation	After Compensation
		SHOA	BESA [40]	PSO [41]
Location & Capacity of EVCS (G2V) in kW	Not Available	287 (9)	328 (13)	339 (6)
Location & Capacity of EVCS (V2G) in kW	Not Available	220 (23)	310 (27)	208 (30)
Location & Capacity of PV in kW	Not Available	980 (9)	859 (13)	790 (6)
Location & Capacity of WT in kW	Not Available	1020 (23)	989 (27)	880 (30)
Location & Capacity of DSTATCOM in kVAr	Not Available	890 (9) 1000 (23)	730 (13) 870 (27)	630 (6) 780 (30)
${\mathrm{P}}_{\mathrm{T}\mathrm{L}}$ in kW	221.29	50.91	86.14	122.53
${\mathrm{Q}}_{\mathrm{T}\mathrm{L}}$ in kVAr	65.09	14.38	22.22	31.92
${\mathrm{V}\mathrm{S}\mathrm{I}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ in p.u.	0.7875	0.9228	0.8960	0.8348
${\mathrm{V}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ in p.u.	0.9420	0.9801	0.9729	0.9558
Convergence Time (s)	--	8.86	10.52	14.06

and illustrated through Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. The comparative analysis reveals that the SHOA outperforms both the BESA and PSO in several key performance metrics. Notably, the SHOA achieved the lowest real power loss (86.14 kW) and reactive power loss (22.22 kVAr) compared to the BESA and PSO, demonstrating superior efficiency in power loss reduction. Additionally, the SHOA’s ability to enhance voltage stability is evident from the highest VSI_min value (0.8960) and the lowest voltage deviation (0.9729 p.u.), which surpasses the results obtained using the BESA and PSO. These improvements are achieved with a convergence time of just 10.52 s, faster than PSO (14.06 s) and comparable to the BESA (8.86 s).

Figure 13 demonstrates that the SHOA achieves a notable reduction in real power loss, lowering it to 86.14 kW, a substantial decrease compared to the losses incurred using the BESA (122.53 kW) and PSO (121.84 kW). This significant reduction in real power loss highlights the SHOA’s superior performance in optimizing power flow and minimizing energy dissipation within the IEEE 34-bus RDS. Effective management of real power losses is crucial for improving the overall energy efficiency of the distribution system, as it directly impacts the amount of energy that can be utilized for end use rather than being wasted as losses. The enhanced capability of the SHOA to minimize these losses results in lower operational costs and contributes to the system’s economic efficiency. By reducing real power loss, the SHOA not only improves the operational performance of the distribution system but also supports sustainability goals by decreasing energy wastage. This reduction translates into cost savings and improved reliability, making the SHOA a more advantageous choice compared to the BESA and PSO for optimizing real power loss in RDSs.

As depicted in Figure 14, the SHOA excels in managing reactive power losses, achieving a reduction to 22.22 kVAr. This represents a significant improvement com-pared to BESA’s 31.92 kVAr and PSO’s 30.50 kVAr. Effective management of reactive power is crucial for maintaining voltage stability and optimizing overall system per-formance. The SHOA’s ability to minimize reactive power loss demonstrates its effi-ciency in balancing the reactive power demands within the system, thereby enhancing power quality. By reducing reactive power losses, the SHOA contributes to improved voltage regulation and system efficiency, minimizing the potential for voltage drops and ensuring more stable and reliable operation of the distribution network. This effi-cient reactive power management underscores the SHOA’s advantage in optimizing power distribution and enhancing overall system performance.

Figure 15 shows that the SHOA achieves the highest VSI of 0.8960, surpassing BESA’s 0.8348 and PSO’s 0.8552. The VSI is a critical measure of a power system’s abil-ity to maintain stable voltage levels despite fluctuations in load conditions. The SHOA’s superior VSI performance indicates its effectiveness in enhancing voltage sta-bility across the distribution network. By achieving a higher VSI, the SHOA reduces the risk of voltage collapses and instabilities, which can otherwise lead to power outages or system failures. This capability ensures a more reliable power supply, as the system can better handle varying load demands without compromising voltage stability. The improved VSI performance highlights the SHOA’s strength in optimizing voltage reg-ulation, contributing to a more stable and resilient power distribution network.

Figure 16 illustrates that the SHOA maintains the highest minimum bus voltage of 0.9729 p.u., which is notably higher than BESA’s 0.9558 p.u. and PSO’s 0.9682 p.u. This superior performance in voltage maintenance reflects the SHOA’s effectiveness in op-timizing the distribution network’s voltage profile. By achieving a higher minimum bus voltage, the SHOA ensures that voltage levels across the network remain more stable and within the desired operational range. This capability is critical because maintaining higher bus voltages reduces the risk of voltage sags and potential voltage collapse, which can adversely affect the performance and reliability of the power system. Higher bus voltages contribute to enhanced system efficiency by minimizing power losses associated with low voltage conditions and improving the overall quality of power delivered to consumers. This stability helps in preventing voltage-related is-sues such as equipment malfunctions and power quality disturbances, thereby ensur-ing a more reliable and efficient distribution system. The ability to maintain higher bus voltages underscores the SHOA’s effectiveness in managing and optimizing voltage levels, leading to improved system reliability and performance.

The efficacy of an algorithm in attaining the optimal solution is significantly influenced by its reliability in convergence. Figure 17 illustrates that the SHOA converges to the optimal objective value within just 12 iterations, showcasing its high convergence rate. This rapid convergence rate, coupled with its stability and effectiveness in exploring near-global solutions, underscores the SHOA’s superiority over the BESA and PSO. The SHOA’s performance is marked by a swift, precise convergence process that consistently yields optimal solutions, making it a robust and efficient choice for optimizing renewable distribution systems. This analysis highlights the substantial advantages of the SHOA in enhancing system performance, reducing power losses, and maintaining voltage stability.

In summary, the comparative analysis demonstrates the SHOA’s distinct advantages over the BESA and PSO. The SHOA not only reduces power losses and reactive power losses more effectively but also enhances voltage stability and maintains higher bus voltages. Its rapid convergence further underscores its practical value in optimizing renewable distribution systems. The results validate the SHOA as a powerful tool for improving system efficiency and performance, making it a preferable choice for addressing complex optimization challenges in power systems.

5. Conclusions

The integration of EVCSs into RDSs introduced both opportunities and challenges, particularly in terms of power loss and voltage stability. This study presented a novel optimization method utilizing the SHOA to enhance the placement of PV- and WT-based RDG and DSTATCOMs, alongside EVCSs. The key contributions of this work were substantiated through quantitative results that demonstrated substantial improvements in system performance. Over a 24-h period, the SHOA achieved a 60.1% reduction in real power loss, decreasing from 180.43 kW to 72.04 kW. It also resulted in a 62.7% reduction in reactive power loss, from 53.08 kVAr to 19.83 kVAr. Additionally, the VSI improved by 13.8%, and bus voltage increased by 3.3% following compensation. Compared to other algorithms, the SHOA provided a 77.0% reduction in power loss, outperforming BESA, which achieved a 61.1% reduction, and PSO, which achieved a 44.7% reduction. In terms of reactive power loss reduction, the SHOA performed 35.2% better than the BESA and 54.9% better than PSO. Furthermore, the SHOA demonstrated convergence times that were 15.8% and 37.1% faster than those of the BESA and PSO, respectively. These results underscore the effectiveness of the SHOA in minimizing power loss, enhancing voltage stability, and offering efficient convergence. The practical benefits and superior performance of this optimization approach highlight its potential for significantly improving RDS operations and addressing the challenges associated with EVCS integration.