The Impact of Electric Vehicle Hosting Factors on Distribution Network Performance Using an Impedance-Based Heuristic Approach

Abstract

The fast adoption of electric vehicles (EVs) and the integration of renewable distributed generators (DGs) provide significant operational issues for radial distribution networks (RDNs), notably in terms of power losses, voltage variations, and system stability. This paper investigates the optimal placement and sizing of EV charging stations (EVCSs) and DGs under varying EV hosting factors (EV-HFs). An impedance matrix-based load flow method is developed, and a derived analytical formula for power loss calculation is proposed to improve computational efficiency. A weighted multi-objective function is developed to reduce active power losses and voltage variations while optimizing the voltage stability index and the yearly cost savings from energy loss. The optimization is performed using a deterministic heuristic procedure that incrementally adjusts the location and size of EVCSs and DGs until no further improvement in the fitness function is achieved. This stepwise approach provides fast convergence with low computational effort compared to population-based metaheuristics. The methodology is used on the IEEE 33-bus system under different loading conditions and EV-HFs. The results reveal that for 40% and 60% EV-HFs, active power losses decreased by about 57% compared with the basic case, while the minimum bus voltage improved from 0.9148 pu to 0.9654 pu and 0.9641 pu. The economic analysis demonstrates annual savings of up to USD 473,550, with a payback period between 7 and 8 years. These findings emphasize the need of integrated EVCS and DG planning in improving future distribution systems’ technical and economic performance.

1. Introduction

The growing utilization of electric vehicles (EVs) necessitates the integration of electric vehicle charging stations (EVCSs) into distribution networks. However, this integration introduces significant challenges, as it increases network loading, which in turn raises power losses and voltage drops along distribution branches, thereby reducing voltage levels. Previous studies have carefully explored the optimal location of EVCSs from an operational standpoint, considering factors such as waiting time, driving range, customer satisfaction, and market size [1]. Other studies have focused on decreasing power losses and voltage variations while including the idea of hosting factors (HFs), with specific emphasis on EV hosting factors [2,3]. This underscores the importance of further exploring the relationship between EV demand load and network performance.

A previous study aimed to reduce active and reactive power losses for various EV-HF values. The Quantum-Behaved Gaussian Mutational Dragonfly Algorithm (QGDA) was used for optimization [4], while Harmony Particle Swarm Optimization (PSO) improved voltage quality [5]. Additional research has explored the integration of EVCSs with other network components, such as distributed generators (DGs) and DSTATCOMs, to enhance efficiency and reliability [6,7,8]. Despite these efforts, the restricted treatment of EV-HFs prevents a thorough evaluation of such treatments. A prominent research trend is the cooperative optimization of power losses in networks that concurrently host both EVCSs and DGs [7,8]. Approaches have included hybrid artificial intelligence techniques combining Grey Wolf Optimizer with PSO, Modified Teaching–Learning-Based Optimization (TLBO), and Multi-Objective PSO (MOPSO). However, these studies frequently lack a thorough examination of the interplay between hosting factors and network performance, resulting in a large research gap.

Several studies have studied how to reduce energy losses and voltage variations using different optimization methodologies, such as Differential Evolution (DE), Grey Wolf Optimizer (GWO), adaptive Particle Swarm Optimization (PSO), and the Fuzzy Analytic Hierarchy Process (AHP) [9,10]. In ref. [11], the EV hosting factor (EV-HF) was supposed to be the whole demand load. Ref. [12] presents a detailed evaluation of ideal EVCS placement, taking into account different objectives, solution approaches, and geographic factors for enhancing charging infrastructure design in smart grids. In comparison, some studies have addressed uncertainties in EV-HFs using probabilistic modeling [13,14], while some studies have studied the use of renewable energy sources at charging stations to minimize grid demand [15]. This calls into doubt the applicability of standard EV-HF-based models in scenarios with significant renewable penetration. Optimization techniques such as GWO-PSO hybrids [13,15], Chicken Swarm Optimization (CSO), TLBO [16], and combined Lazy Greedy algorithms (Direct Gain and Effective Gain, LGEG) [17] have also been used to reduce power loss and voltage variations.

In addition, various mathematical and metaheuristic strategies have been employed for optimal EVCS placement with the aim of improving efficiency and reliability. Examples are Mixed Integer Nonlinear Programming (MINLP) [18], DE coupled with Harris Hawks Optimization (DE-HHO) [19], the Binary Atom Search method [20], and the BAT algorithm [21]. However, many of these methodologies frequently miss the effect of renewable distributed generation hosting factors (RDG-HFs), restricting their optimization reach. It is also noteworthy that the EV-HF was typically reported in discrete increments of 5%, 10%, 15%, or 20% [21]. While energy storage technologies are widely known for their capacity to reduce renewable unpredictability and supply–demand mismatches [22], the current study uses a strong optimization framework to maintain grid stability without depending on storage systems. The proposed method suggests that the optimal integration of EVCSs and PV production can significantly enhance grid performance in stochastic operating conditions. Its resilience ensures successful operation even without storage, giving a significant viewpoint for future grid integration difficulties. Future research might expand on this architecture by introducing storage systems to give greater optimization and flexibility. Meanwhile, several studies have identified differences in EV-HF distribution and their influence on network performance [23,24,25]. Also, TD3-CO-PI EMS offered a passenger-aware collaborative EMS to improve the operating efficiency of FCBs during hot weather, which improved hydrogen utilization and onboard energy system durability [26].

Recently, there has been a growing interest in the effect of wireless power transfer technology. Notably, various intelligent techniques have been highlighted in [27,28]. In [27], a new concept of the integrated inductive–capacitive core (IICC) structure, which uses the high conductivity characteristics of nanocrystalline materials and the distinctive laminated configuration of the nanocrystalline core, was used to build the resonant capacitor required in an LC resonant tank. For instance, in [28], a magnetic coupler made up of a flux pipe power supply rail and an H-type receiver is proposed to achieve the benefits of the magnetic couplers. Winding the rail coil unidirectionally in the traveling direction produces a reasonably uniform magnetic field in the traveling direction, which replaces the typical alternating magnetic field in the vertical direction.

Previous research’s shortcomings are mentioned below: Much of the previous research focused solely on the influence of RDGs, with little consideration for the significance of EV loads. While several studies have looked at the impact of EV loads, they usually disregard the hosting factor, instead portraying EV loads as a simple count of automobiles in the distribution network.

But in this manuscript, due to the different nature of loads on the RDNs, the consumed powers on the network are changed for each hour of the day; consequently, the EVs’ loadings on the network must be changed so the hosting factor of EVCSs is taken into account for this condition. The optimal selection of the placement and sizing of both EVCSs and DGs is determined at the maximum loading hour, and these placements are fixed for other hours, but the sizing of EVCSs and DGs is changed according to the loading condition at any other loading hour.

The contributions to the study are summarized as follows:

• Development of an impedance matrix-based load flow method using the DSB incidence matrix, providing a faster and computationally efficient alternative to traditional backward/forward sweep methods for RDN analysis.
• Development of a weighted multi-objective optimization model that balances power loss reduction, voltage deviation minimization, voltage stability enhancement, and economic advantages.
• Introduction of a deterministic heuristic optimization procedure for the placement and sizing of EVCSs and DGs, which incrementally improves the fitness function with low computational cost.
• Comprehensive case study on the IEEE 33-bus distribution system, showing that at 40% and 60% EV hosting factors, power losses can be reduced by ~57%, with significant improvements in voltage stability and annual economic savings.
• Techno-economic insights for system operators, including payback periods and lifetime profit analysis, demonstrating the practical feasibility of coordinated EVCS and DG planning.

2. Problem Formulation

The proposed problem formulation considers the hourly load variations and different EV-HF scenarios, with a focus on decreasing active and reactive power losses and improving the voltage stability index. The formulation integrates technical constraints, economic considerations, and a multi-objective optimization model, as described below.

2.1. Hosting Factor of EVCSs

The HF can be defined as the capability of a distribution network to accommodate additional EVCS load without violating operational constraints such as voltage limits, thermal ratings, or stability margins. It is expressed as the ratio between the allowable EV demand load power and the total system demand power as mentioned in Equation (1), which represents the simple form used to calculate it. HF is normally calculated using a set of technical factors that vary depending on the grid being analyzed.

$${\mathrm{P}}_{\mathrm{EVCS}}=HF\times \sum {\mathrm{P}}_{\mathrm{demand}}$$

(1)

Due to changing the capacities of the EVCSs, there are also varying effective ratings of DGs. Since EVCS ratings vary dynamically with the hourly load profile, the hosting factor directly influences the required DG capacities to maintain network performance. Given that the added loads actually increase the loading on the distribution branches, the thermal loading on some branches increases and the voltage drops on these branches also increase; consequently, the voltage at some buses is less than the base case. This only happens in the case of adding loads. However, the network is equipped with distributed generators that supply electric power to all distribution network loads so that the system performance is improved, which means that the branch currents are reduced and the minimum voltage on the network is increased; this means that the voltage profile on network buses is improved and the electrical losses on network can be minimized.

2.2. Economic Modeling of PV-Based DGs

PV generators are assumed to have energy storage batteries to make the system able to supply electric power to electrical distribution networks at any time and EV charging station demands are modeled as constant load power each hour based on the electric vehicle hosting factor (EV-HF) that is used for planning-stage analysis.

The cost of deploying PV-based DGs includes several components, such as solar modules, inverters, balance-of-system (BOS) costs, labor, overhead, and interconnection. The total cost of PV (PV Total_Cost) can be expressed as [29]

$$\mathit{PVTotal}\text{\_}\mathit{Cos}=\mathit{CPV}/W\times \mathit{PVsize}$$

(2)

where CPV/W is the total cost of PV per watt and PV_size is the size of the PV.

Annual cost saving of energy loss:

$$\mathrm{ACSEL} = K_E\times {\sum }_{i=1}^{24}\left(P_{losses of base}-P_{losses with DG}\right)\times 365$$

(3)

where $K_E$ = 0.06 USD/Kwhr.

2.3. System Constraints

Some technical requirements must be verified in RDNs, which are divided into three categories: power balancing, voltage restrictions, and DG actual power constraints. The RDG’s placement is an important factor in the RDN’s power equations.

2.3.1. Power Balance Constraint

The total power generated from the slack bus and DG units must balance the system load, EVCS demand, and network losses:

$$P_S+P_{DG}={\sum }_{i=1}^{N_b}{P}_{demand,i}+P_{EVCS}+P_{loss}$$

(4)

where $N_b$ is the bus number; $P_{S }$ is the active power generated at the slack bus bar; $P_{demand,i}$ is the total active power of the load at bus (i); $P_{EVCS}$ is the total active power of the EVCS; and $P_{loss}$ is the total active power loss.

2.3.2. Voltage Limit

Bus voltages must remain within specified limits to ensure reliable system operation:

$$0.95\le V_i\le 1.05 \mathrm{p}\mathrm{u}$$

(5)

The voltage at all buses must be in the acceptable range when connecting both EVCSs and DGs, which occurs by selecting the optimal sizing and placement of them. Given that the voltage drop on each distribution branch depends on the current flow in this branch, if this current is reduced, then the voltage drop decreases. But this current has two components: the real part that depends on the active power flow and the imaginary part that is related to reactive power flow. So if the active power flow is reduced using DGs, then the voltage drop reduces; consequently, the voltage level at all buses is enhanced and the electrical losses are minimized.

The DG must guarantee that the power injected at each optimized bus is within the minimum and maximum limits.

2.3.3. DG Capacity Limits

It should fall between the minimum ($P_{\mathrm{m}\mathrm{i}\mathrm{n} \_}^{DG}$) and maximum ($P_{\mathrm{m}\mathrm{a}\mathrm{x}}^{DG}$). The DG ratings are constrained to lie within minimum and maximum operating limits [25]

$$P_{\mathrm{m}\mathrm{i}\mathrm{n} \_}^{DG}\le P_{size}^{DG}\le P_{\mathrm{m}\mathrm{a}\mathrm{x}}^{DG}$$

(6)

2.4. Multi-Objective Optimization Model

The objective of this study is to find optimal locations and sizes of EVCSs and DGs such that system performance is enhanced under different EV-HFs. The optimization problem simultaneously (i) minimizes active power loss (P_loss); (ii) minimizes the voltage deviation index (VDI); (iii) maximizes the voltage stability index (VSI); and (iv) maximizes the annual cost saving of energy loss (ACSEL). These are combined into a weighted multi-objective function:

F_t = w₁ P_loss + w₂ VDI + w₃ (1/VSI) + w₄ (1/ACSEL)

(7)

where VSI (new operating condition) > VSI base case and where w₁, w₂, w₃, and w₄ are the weights assigned to each objective.

The values of them are as follows: w₁ = 0.45, w₂ = 0.25, w₃ = 0.1, and w₄ = 0.2. These values are used for normalization and sensitivity analysis for the impact of each one on the overall objective function.

The optimization model aims to determine the optimal sizing and locations for both EV charging stations and distributed generators. The optimal solution depends on minimizing the electrical power/energy losses and the voltage deviation index and increasing the voltage stability index for the electrical distribution network.

2.4.1. Power Loss

The unique technique focuses mostly on reducing $P_{loss}$ and voltage profile by properly allocating DGs and electric vehicle charging stations (EVCSs) inside the RDN to achieve the best performance. The overall real $P_{loss}$ can be expressed as

$$P_{loss}={\sum }_{h=1}^{24}{P}_{loss}\left(h\right) \mathrm{and} Q_{loss}={\sum }_{h=1}^{24}{Q}_{loss}(h)$$

(8)

$$P_{loss}\left(h\right)=real\left\{\right[I_{dem}\left(h\right) {]}^{*T} \left[Z_{bus}\right]\left[ I_{dem}\left(h\right)\right]\}$$

(9)

$$Q_{loss}\left(h\right)=imaj\left\{\right[I_{dem}\left(h\right) {]}^{*T} \left[Z_{bus}\right]\left[ I_{dem}\left(h\right)\right]\}$$

(10)

2.4.2. Voltage Deviation Index (VDI)

Voltage deviation is the difference between the nominal voltage (which equals 1 pu) and the actual bus voltage. The system’s voltage condition improves when the bus voltage approaches the nominal value. The VDI computation may be accomplished as [30]

$$VDI={\sum }_i^N{\left|1-V_i\right|}^2$$

(11)

where $V_i$ is the voltage at the ith node.

2.4.3. Voltage Stability Index (VSI)

This work presents a voltage stability index (VSI) for steady-state situations, with the goal of discovering nodes with increased susceptibility to voltage collapse. The index, abbreviated as VSI and stated by Equation (12) [31], is derived from power flow analysis and is used to measure voltage stability at each node. For an RDN to function properly, the VSI should be larger than zero. Nodes with lower VSI values require more compensation measures to ensure voltage stability. For Figure 1, the VSI can be calculated at bus q from Equation (12).

$$VSI={\left|V_p\right|}^4-4(Q_P{X}_k+P_P{R}_k){\left|V_p\right|}^2+4{\left(Q_P{X}_k+P_P{R}_k\right)}^2$$

(12)

where V_p is the bus voltage, R_k and X_k are line resistance and reactance, and P_P and Q_P represent effective active and reactive power, respectively.

3. Proposed Impedance-Based Heuristic Approach

This section presents the proposed methodology for the optimal allocation of electric vehicle charging stations (EVCSs) and distributed generators (DGs) in radial distribution networks (RDNs). The approach combines an impedance matrix-based load flow analysis with a deterministic heuristic optimization procedure. The methodology is designed to reduce power losses, reduce voltage deviations, and enhance the voltage stability index, while ensuring economic feasibility under varying electric vehicle hosting factors (EV-HFs).

3.1. Impedance Matrix-Based Load Flow Analysis

The RDN under study consists of a set of buses and radial distribution sections. Bus 1 is considered the slack bus, representing the main power source. The bus impedance matrix (Zbus) is derived from the distribution section-to-bus (DSB) incidence matrix. Generally, the bus voltage for any power network may be found from Equation (13).

$$[V_{bus}]=[Z_{bus}\left]\right[I_{bus}]$$

(13)

where [$V_{bus}$] and [$I_{bus}$] are column matrices of N bus and $Z_{bus}$ is the square matrix (N × N). But in the radial distribution network, the slack bus voltage is known. So, in this case,

$$[V_{bus}]=\left[V_1\right]+[Z_{bus}\left]\right[I_{bus}]$$

(14)

where [$V_{bus}$] and [$I_{bus}$] are column matrices of (N − 1) buses starting from bus 2 to bus N and [$Z_{bus}$] is the (N − 1) × (N − 1) matrix, which refers to the slack bus bar. Construction of the bus impedance matrix using the (DSB) matrix, which is (N − 1) × (N − 1) distribution section-to-bus-incidence matrix. This matrix can be obtained for any radial distribution network using the steps for the simplified network shown in Figure 2.

This distribution network has five DSs and six buses, and the method for constructing the DSB matrix is as follows.

• Step (1): Initialize the DSB matrix; DSB = zeros (5,5):

$$\mathrm{DSB}=\begin{array}{cc}bus& \begin{array}{ccccc}2& 3& 4& 5& 6\end{array}\\ \begin{array}{c}DS1\\ DS2\\ DS3\\ DS4\\ DS5\end{array}& \left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]& & & & \end{array}$$

(15)

• Step (2): for any distribution section (i), DSB(i, B) = 1 for section (i) supplying the load at bus B:

$$\mathrm{DSB}=\begin{array}{cc}bus& \begin{array}{ccccc}2& 3& 4& 5& 6\end{array}\\ \begin{array}{c}DS1\\ DS2\\ DS3\\ DS4\\ DS5\end{array}& \left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 0& 1& 1& 0& 0\\ 0& 0& 0& 1& 1\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1\end{array}\right]& & & & \end{array}$$

(16)

The bus impedance matrix of the distribution network is obtainable using the DSB matrix in the following equation:

$$\left[{\overline{Z}}_{bus}\right]=\left[\mathrm{D}\mathrm{S}\mathrm{B}{]}^T\right[{\overline{Z}}_s\left]\right[\mathrm{D}\mathrm{S}\mathrm{B}]$$

(17)

where ${[Z}_{bus}$] is the bus impedance matrix referring to the slack bus bar and [$Z_s$] is a diagonal matrix containing the impedance of ordered distribution sections, so

$$[Z_s]=\begin{array}{cc}bus& \begin{array}{ccccc}DS1& DS2& DS3& DS4& DS5\end{array}\\ \begin{array}{c}DS1\\ DS2\\ DS3\\ DS4\\ DS5\end{array}& \left[\begin{array}{ccccc}{\overline{Z}}_{s1}& 0& 0& 0& 0\\ 0& {\overline{Z}}_{s2}& 0& 0& 0\\ 0& 0& {\overline{Z}}_{s3}& 0& 0\\ 0& 0& 0& {\overline{Z}}_{s4}& 0\\ 0& 0& 0& 0& {\overline{Z}}_{s5}\end{array}\right]& & & & \end{array}$$

(18)

$$Z_{bus}=\begin{array}{cc}bus& \begin{array}{ccccc}2& 3& 4& 5& 6\end{array}\\ \begin{array}{c}2\\ 3\\ 4\\ 5\\ 6\end{array}& \left[\begin{array}{ccccc}{\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}\\ {\overline{Z}}_{s1}& {\overline{Z}}_{s1}+{\overline{Z}}_{s2}& {\overline{Z}}_{s1}+{\overline{Z}}_{s2}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}\\ {\overline{Z}}_{s1}& {\overline{Z}}_{s1}+{\overline{Z}}_{s2}& {\overline{Z}}_{s1}+{\overline{Z}}_{s2}+{\overline{Z}}_{s4}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}\\ {\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}+{\overline{Z}}_{s3}& {\overline{Z}}_{s1}+{\overline{Z}}_{s3}\\ {\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}& {\overline{Z}}_{s1}+{\overline{Z}}_{s3}& {\overline{Z}}_{s1}+{\overline{Z}}_{s3}+{\overline{Z}}_{s5}\end{array}\right]& & & & \end{array}$$

(19)

The mentioned detailed formula for calculating the bus impedance matrix is used to derive it, using the new DSB incidence matrix by directly applying Equation (17). Then the bus impedance matrix is obtained, and this matrix is constant during all operations, so it is constructed for the first iteration and used for the whole solution.

The load flow analysis using the bus impedance matrix is as follows:

$$\mathrm{But} \left[I_{bus}\right]=-\left[{\overline{I}}_{dem}\right]$$

(20)

$$\mathrm{Then} \left[{\overline{V}}_{bus}\right]=\left[{\overline{V}}_1\right]-\left[{\overline{Z}}_{bus}\right]\left[{\overline{I}}_{dem}\right]$$

(21)

[${\overline{V}}_{bus}$]=$\left[\begin{array}{c}{\overline{V}}_2\\ ⋮\\ {\overline{V}}_N\end{array}\right]$, [${\overline{V}}_1$] = $\left[\begin{array}{c}{\overline{V}}_1\\ {\overline{V}}_1\\ ⋮\\ {\overline{V}}_1\end{array}\right]$ (N − 1) column matrix

${\overline{I}}_{dem}= \left[\begin{array}{c}{\overline{I}}_{dem2}\\ {\overline{I}}_{dem3}\\ ⋮\\ {\overline{I}}_{demN}\end{array}\right]$(N − 1) column matrix

• Assume initially that all bus voltages start from bus number 2 to bus number N is equal to the slack bus voltage.

$${\overline{V}}_i^{\left(0\right)}={\overline{V}}_1, k=0 \left(\text{initialize the iteration counter}\right)$$

(22)

2.

Determine the demand load current at each bus.

$${\overline{I}}_{demi}^{\left(k\right)}={\displaystyle \frac{{\overline{S}}_{dem}^{*}}{{\overline{V}}_i^{k\ast }}}\forall i=2:N$$

(23)

3.

Then the new bus voltage can be calculated from Equation (24).

$$[{V_{bus}]}^{k+1}=\left[V_1\right]-[Z_{bus}]{{[I}_{dem}]}^K$$

(24)

4.

The voltage difference for this iteration is obtained from the Equation (25).

$$\left[\Delta V_{bus}^{K+1}\right]=\left|V_{bus}^{K+1}-V_{bus}^K\right|$$

(25)

5.

Compare the maximum voltage difference with an acceptable error (e = 0.00001 pu); if it is less than or equal to this acceptable error, then the final run of the load flow study is obtained. Otherwise, K = K + 1 and return to step 2.

6.

Then the distribution section currents can be obtained using the DSB matrix as follows:

$$[I_{DS}]=[\mathrm{DSB}\left]\right[I_{dem}]$$

(26)

The active and reactive power losses in Figure 3a are calculated as follows:

$$P_{loss}={\left|I_{s1}\right|}^2{R}_{s1}\mathrm{and} Q_{loss}={\left|I_{s1}\right|}^2{X}_{s1}$$

(27)

$$S_{loss}=P_{loss}+j{Q}_{loss}$$

(28)

where ${\overline{Z}}_{s1}=R_{s1}+j{X}_{s1}$

$${\overline{I}}_{s1}={\overline{I}}_{dem2}$$

(29)

$$S_{loss}={\left|I_{dem2}\right|}^2\ast {\overline{Z}}_{s1}$$

(30)

$$\left[Z_{bus}\right]=\left[Z_{s1}\right]$$

(31)

$${\overline{S}}_{loss}=[I_{dem} {]}^{*T} \left[Z_{bus}\right][ I_{dem}]$$

(32)

Also the total active and reactive power losses in Figure 3b can be computed by Equations (33) and (34), respectively.

$$P_{loss}={\left|I_{s1}\right|}^2{R}_{s1}+{\left|I_{s2}\right|}^2{R}_{s2}$$

(33)

$$Q_{loss}={\left|I_{s1}\right|}^2{X}_{s1}+{\left|I_{s2}\right|}^2{X}_{s2}$$

(34)

$$S_{loss}={\left|I_{s1}\right|}^2{\overline{Z}}_{s1}+{\left|I_{s2}\right|}^2{\overline{Z}}_{s2}$$

(35)

where ${\overline{Z}}_{s2}=R_{s2}+j{X}_{s2}$

$${\overline{I}}_{s2}={\overline{I}}_{dem3}, {\overline{I}}_{s1}={\overline{I}}_{dem2}+{\overline{I}}_{dem3}$$

(36)

where ${\overline{I}}_{dem2}=\left|I_{dem2}\right|\angle {\theta }_2$, ${\overline{I}}_{dem3}=\left|I_{dem3}\right|\angle {\theta }_3$, and ${\theta }_{23}={\theta }_2-{\theta }_3$

$$\mathrm{So}{\left|I_{s2}\right|}^2={\left|I_{dem3}\right|}^2,{\left|I_{s1}\right|}^2={\left|I_{dem2}\right|}^2+{\left|I_{dem3}\right|}^2+2\left|I_{dem2}\right|\left|I_{dem3}\right|\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{23}\right)$$

(37)

$$S_{loss}={\left|I_{dem2}\right|}^2{\overline{Z}}_{s1}+{\left|I_{dem2}\right|}^2\left({\overline{Z}}_{s1}+{\overline{Z}}_{s2}\right)+2\left|I_{l2}\right|\left|I_{l3}\right|{\overline{Z}}_{s1}\cos \left({\theta }_{23}\right)$$

(38)

Also the following relation applies:

$$\left[\begin{array}{cc}{\overline{I}}_{dem2}^{*}& {\overline{I}}_{dem3}^{*}\end{array}\right] \left[\begin{array}{cc}{\overline{Z}}_{s1}& {\overline{Z}}_{s1}\\ {\overline{Z}}_{s1}& {\overline{Z}}_{s1}+{\overline{Z}}_{s2}\end{array}\right] \left[\begin{array}{c}{\overline{I}}_{dem2 }\\ {\overline{I}}_{dem3}\end{array}\right]$$

(39)

$$=\left[\begin{array}{cc}{\overline{Z}}_{s1}{\overline{I}}_{dem2}^{*}& {\overline{Z}}_{s1}{\overline{I}}_{dem2}^{*}\\ {\overline{Z}}_{s1}{\overline{I}}_{dem3}^{*}& ({\overline{Z}}_{s1}+{\overline{Z}}_{s2}){\overline{I}}_{dem3}^{*}\end{array}\right] \left[\begin{array}{c}{\overline{I}}_{dem2}\\ {\overline{I}}_{dem3}\end{array}\right]$$

(40)

$$={\overline{Z}}_{s1} {\overline{I}}_{dem2} \left({\overline{I}}_{dem2}^{*}+{\overline{I}}_{dem3}^{*}\right)+{\overline{Z}}_{s1}{\overline{I}}_{dem3}\left({\overline{I}}_{dem2}^{*}+{\overline{I}}_{dem3}^{*}\right)+{\overline{Z}}_{s2}{\left|I_{dem3}\right|}^2$$

$$={\overline{Z}}_{s1}{\left|I_{dem2}\right|}^2+Z_{s1}{I}_{l3A}^{*}{\overline{I}}_{dem2}+{\overline{Z}}_{s1}{\left|I_{dem3}\right|}^2+{\overline{Z}}_{s1}{I}_{dem2}^{*}{\overline{I}}_{dem3}+{\overline{Z}}_{s2}{\left|I_{dem3}\right|}^2$$

$$={\overline{Z}}_{s1}{\left|I_{dem2}\right|}^2+\left({\overline{Z}}_{s1}+{\overline{Z}}_{s2}\right){\left|I_{dem3}\right|}^2+{\overline{Z}}_{s1}({\overline{I}}_{dem2}^{*}{\overline{I}}_{dem3}+{\overline{I}}_{dem2}{\overline{I}}_{dem3}^{*})$$

$$={\overline{Z}}_{s1}{\left|I_{dem2}\right|}^2+\left({\overline{Z}}_{s1}+{\overline{Z}}_{s2}\right){\left|I_{dem3}\right|}^2+2{\left|I_{dem2}\right|}^2{\left|I_{dem3}\right|}^2{\overline{Z}}_{s1}\cos ({\theta }_{23})$$

(41)

Then ${\overline{S}}_{loss}$ can be calculated for any RDN by Equation (42):

$${\overline{S}}_{loss}=[I_{dem} {]}^{*T} \left[Z_{bus}\right][ I_{dem}]$$

(42)

Also, here the detailed equations derive a new formula used for calculating electrical distribution network losses by applying Equation (42) instead of calculating them using the summation of each individual electrical loss in each distribution branch. So, by using Equation (42) above, the execution time for calculating the power losses can be reduced and the objectives can be calculated quickly, and the proposed method presents very accurate calculations with respect to radial distribution networks.

The novelty of Z_bus claimed in the manuscript is the direct method for the construction of it using the proposed DSB incidence matrix, i.e., the application of the DSB matrix to find all branch currents directly. The mentioned detailed equations that are used for calculating the bus impedance matrix are written to derive how Z_bus can be calculated using the new DSB incidence matrix by directly applying Equation (17), and the proposed derivation is used to obtain the electrical power losses using the Z_bus matrix, i.e., using one equation instead of calculating them using the summation of individual branch losses. So, the power flow solution using Z_bus is faster than other published methods.

The advantages of using the bus matrix method as a load flow solution method are the simplicity of constructing it using the DSB incidence matrix and also using the DSB matrix to calculate the distribution branch currents, and the electrical power losses are also determined very easily by one equation without any summations, so the execution time for applying the proposed technique is reduced compared with the exciting methods.

3.2. Heuristic Optimization Procedure at Maximum Loading Condition

The optimization problem is formulated to identify the ideal placement and sizing of EVCSs and DGs under the maximum network loading condition. The stepwise heuristic procedure is as follows:

7.

Identify the Maximum Loading Hour: From the daily load profile, determine the hour with the highest demand.

8.

Initial Placement of EVCSs:

○

Compute EVCS demand using hosting factors.

○

Assume two EVCSs with equal demand.

○

Place them on the first bus of the two most heavily loaded branches.

○

Run the load flow and compute the objective function F_t.

○

Search within these branches for placements that minimize F_t.

9.

Sizing of EVCSs:

○

Incrementally increase the size of one EVCS by ΔP and decrease the other by the same amount.

○

Recalculate F_t.

○

If improved, repeat until no further reduction in F_t is obtained.

10.

Placement and Sizing of DGs:

○

Locate DGs at the same buses as EVCSs.

○

Initialize their output as 35% of total demand (network load and EVCS demand).

○

Incrementally adjust DG sizes by ΔP.

○

Recalculate the total objective function F_t and update until the minimum is reached for each DG.

Steps for load flow applying the bus impedance matrix method, and as shown in Figure 4:

The presented optimization approach is compared with published methods for the 33-bus and 69-bus distribution networks. The comparison derives that the proposed approach is suitable for any distribution network and it is an effective approach compared to other methods. Knowing that the DGs are used to support the distribution network by supplying active power, it is assumed that they supply 35% of the total demand; this value is justified using sensitivity analysis on many loading conditions and is also applied on two electrical distribution networks. ΔP is assumed to be 1.0% of the total active power demand, and the convergence of the proposed approach depends on the value of total overall objective function. When transitioning from the current state condition of the network to another new state, if this objective function is minimized with the new allocation, then this operation is optimal compared to its previous one, but if the overall objective function is increased with the new allocation, then its previous operation is optimal and the process is stopped. This stopping criterion is illustrated in the flowcharts shown in Figure 5 and Figure 6.

At the end of this process, the optimal placement and sizing of both EVCSs and DGs under maximum loading conditions are obtained. Figure 5 and Figure 6 illustrates the flowchart of the optimization methodology at maximum loading.

3.3. Optimization Under Daily Loading Conditions

To extend the optimization beyond the maximum loading hour, the following steps are applied for each hour of the day:

• Perform load flow for the base case at every hour.
• Identify the maximum loading hour (h_max) and determine optimal EVCS and DG placement and sizing.
• Fix the locations of EVCSs and DGs obtained at h_max.
• For each hour h = 1, …, 24:

• Calculate EVCS demand using EV-HFs.
• Apply the heuristic procedure to optimize the sizing of EVCSs and DGs.
• If h = h_max, skip recalculation of placement.

5.

Repeat until h = 24, at which point the final daily optimal solution is obtained.

This approach ensures that the optimized locations from maximum loading are preserved, while sizing dynamically adapts to hourly variations in load and EV-HF. Figure 6 presents the flowchart of the daily optimization process.

The proposed impedance-based heuristic approach integrates a computationally efficient Zbus-based load flow with an iterative placement–sizing strategy for EVCSs and DGs. Unlike metaheuristic algorithms, this deterministic method achieves fast convergence with minimal computational effort, while ensuring improved network performance across different EV hosting factors and load conditions.

4. Simulation Results

The proposed impedance-based heuristic approach was implemented and verified on two distribution networks, which are the IEEE 33-bus and IEEE 69-bus radial distribution networks [32,33], which are shown in Figure 7 and Figure 8, respectively. The system was analyzed under different loading conditions and EV hosting factors (EV-HFs) to evaluate the effectiveness of coordinated EVCS and DG planning. Performance was evaluated in terms of active power loss reduction and voltage profile improvement, voltage stability index (VSI) improvement, and economic benefits.

4.1. 33-Bus Base Case Analysis

In the base case, the network was simulated without EVCSs or DGs.

Table 1. Results for the base case daily operation.

Hours	${\mathit{P}}_{\mathit{l}\mathit{o}\mathit{a}\mathit{d}}$(kW)	Losses (kW) (Base Case)	Reactive Power Loss (kVAR)	Minimum Voltage (Base Case) (pu)	VDI × 10−3	VSI
1	1297.1	21.8811	14.8497	0.9686	14	0.88
2	1276.1	21.4693	14.5608	0.9690	13.7	0.8817
3	1239.0	20.7606	14.065	0.9698	13.1	0.8846
4	1213.1	20.2802	13.73	0.9704	12.8	0.8867
5	1204.0	20.1142	13.6144	0.9706	12.6	0.8874
6	1697.5	51.1438	34.7552	0.9550	31.3	0.8317
7	2210.7	107.4919	73.381	0.9319	63.4	0.7542
8	2227.6	108.1548	73.8335	0.9318	64	0.7539
9	2866.1	148.6872	100.8082	0.9240	91	0.7288
10	2880.8	149.4229	101.314	0.9236	91.6	0.7277
11	2892.7	150.0219	101.726	0.9233	92.1	0.7269
12	2635.6	111.2300	75.3268	0.9322	69.2	0.7553
13	2550.4	105.8283	71.6977	0.9342	65.8	0.7616
14	2931.7	152.7907	103.5846	0.9224	93.9	0.7240
15	2982.9	156.6376	106.1622	0.9212	96.4	0.7202
16	3034.0	160.5628	108.7943	0.9200	98.9	0.7165
17	3148.6	167.5942	113.5962	0.9173	104.1	0.7081
18	3254.8	175.7117	119.0758	0.9148	109.5	0.704
19	3057.8	127.8623	86.7428	0.9238	81.2	0.7282
20	3073.7	122.1031	82.9468	0.9245	77.7	0.7305
21	3126.8	119.7738	81.5524	0.9242	76.3	0.7295
22	2993.2	103.1926	70.4735	0.9289	65.3	0.7445
23	1618.4	29.1630	20.0256	0.9615	19.4	0.8548
24	1591.1	28.4720	19.5301	0.9621	18.9	0.8570

shows the results of the daily hours for the base case study without EVCSs and DGs; these results include the total demand load power, the total active and reactive power losses, and the value of minimum voltage.

The results indicate high active power losses and significant voltage drops, with the minimum bus voltage falling below the acceptable limit of 0.95 pu during peak hours. The maximum loading on the RDN occurs at hour (18), which has active and reactive power losses of 175.7117 kW and 119.0758 kVAR, respectively, while the minimum voltage is 0.9148 pu. These results highlight the need for corrective measures to accommodate future EV penetration while maintaining system reliability.

4.1.1. Coordinated Integration of EVCSs and DGs

Introducing DGs alongside EVCSs provided substantial improvements in both technical and economic performance. The results for connecting EVCSs and DGs with a 40% hosting factor for EVCSs are illustrated in

Table 2. Results for 40% hosting factor for EVCS operation.

Hours	Total EVCS Consumed Power (kW)	DG (Size@Location) (kW)	EVCS (Size@Location) (kW)
1	518.84	957@11; 881.84@30	235 @11; 283.84@30
2	510.44	928@11; 861.44@30	233@11; 277.44@30
3	495.6	894@11; 840.6@30	230@11; 265.6@30
4	485.24	868@11; 830.24@30	230@11; 255.24@30
5	481.6	816@11; 765.6@30	221@11; 260.6@30
6	679	1130.64@11; 1248.36@30	378@11; 301@30
7	884.3	1673.3@11; 1502@30	388.3@11; 496@30
8	891.04	1676.6@11; 1501.44@30	389.6@11; 501.44@30
9	1146.44	1624@11; 1642.44@30	584@11; 562.44@30
10	1152.3	1630.5@11; 1649.8@30	587.5@11; 564.8@30
11	1157.04	1629.44@11; 1647.6@30	589.44@11; 567.60@30
12	1054.24	1537.43@11; 1606.69@30	562.43@11; 491.69@30
13	1020.16	1529.5@11; 1580.66@30	534.5@11; 485.66@30
14	1172.68	1618.2@11; 1644.48@30	603.2@11; 569.48@30
15	1193.16	1629.3@11; 1663.86@30	619.3@11; 573.86@33
16	1213.6	1630.82@11; 1675.78@30	637.82@11; 575.78@30
17	1295.44	1668.86@11; 1706.58@30	678.86@11; 616.58@30
18	1301.92	1662.3@11; 1708.62@30	690.3@11; 611.62@30
19	1223.12	1547.36@11; 1581.76@30	642.36 @11; 580.76@30
20	1229.48	1585.72@11; 1725.76@30	715.72 @11; 513.76@30
21	1250.72	1591.1@11; 1729.62@30	731.1@11; 519.62@30
22	1197.28	1549.8@11; 1719.48@30	714.8 @11; 482.48@30
23	647.36	1244.76@11; 1022.6@30	249.76@11; 397.6@30
24	636.44	1220@11; 1016.44@30	251@11; 385.44@30

@ means the busbar number.

, which includes the daily hour values of optimal EVCS and DG placement and sizing. The RDN performance under this condition is shown in

Table 3. RDN performance analysis results for a 40% hosting factor for EVCS operation.

Hours	Power Losses (kW)	% Decrease in Active Power Losses	Reactive Power Loss (kVAR)	% Decrease in Reactive Losses	Minimum Voltage (pu)	% Improvement in Voltage	VDI × 10−3	VSI
1	11.430	47.8	7.655	48.5	0.9807	1.25	5.2	0.9326
2	11.356	47.1	7.598	47.8	0.9808	1.22	5.2	0.9310
3	11.057	46.7	7.390	47.4	0.9812	1.18	5.1	0.9294
4	10.915	46.2	7.290	46.9	0.9813	1.12	5	0.9281
5	11.3385	43.6	7.565	44.4	0.9807	1.04	5.5	0.9251
6	24.324	52.44	16.648	52.1	0.9743	2.02	8.8	0.9400
7	48.598	54.8	33.595	54.2	0.9685	3.92	12	0.9588
8	48.895	54.8	33.787	54.2	0.9682	3.91	12.3	0.9586
9	64.028	56.9	43.933	56.4	0.9682	4.78	13	0.9563
10	64.286	56.9	44.121	56.5	0.9683	4.84	13	0.9566
11	64.5	57.1	44.25	56.5	0.9681	4.85	13.3	0.9563
12	47.948	56.9	32.825	56.4	0.973	4.38	9.9	0.9538
13	45.7297	56.8	31.313	56.3	0.9732	4.17	9.7	0.9544
14	65.61	57.1	44.964	56.6	0.9676	4.9	14	0.9550
15	67.127	57.1	46.014	56.7	0.9679	5.06	13.8	0.9550
16	68.7	57.2	47.08	56.7	0.9679	5.206	14	0.9544
17	71.338	57.4	48.831	57	0.9663	5.341	15.5	0.9541
18	74.545	57.5	50.987	57.2	0.9654	5.53	16.2	0.9533
19	55.121	56.9	37.339	56.9	0.9677	4.75	14.1	0.9484
20	52.201	57.2	35.628	57	0.9696	4.87	11.4	0.9502
21	51.422	57.1	35.066	57	0.9692	4.869	11.7	0.9497
22	44.992	56.4	30.669	56.5	0.9712	4.55	10.2	0.9490
23	14.527	50.2	9.892	50.6	0.9775	1.66	6.5	0.9403
24	14.2	50.1	9.655	50.6	0.9778	1.63	6.3	0.9400

, which includes the active and reactive power losses, and the minimum voltage on the distribution network; the percentages of improvement in performance are compared with the base case study. Similarly,

Table 4. Results for 60% hosting factor for EVCS operation.

Hours	Total EVCS Consumed Power (kW)	DG (Size@Location) (kW)	EVCS (Size@ Location) (kW)
1	778.26	1059.26@11; 1019@30	359.26@11; 419@30
2	765.66	1038.9@11; 926.76@30	358.9@11; 406.76@30
3	743.4	999.7@11; 963.7@30	359.7@11; 383.7@30
4	727.86	970.96@11; 946.9@30	360.96@11; 366.9@30
5	722.4	923.4@11; 889@30	343.4@11; 379@30
6	1018.5	1318.7@11; 1384.8@30	588.7@11; 429.8@30
7	1326.42	1853.24@11; 1743@30	593.24@11; 733@30
8	1336.56	1857.96@11; 1738.6@30	597.96@11; 738.6@30
9	1719.66	1900.66@11; 1912@30	892.66@11; 827@30
10	1728.48	1902.85@11; 1921.63@30	896.85@11; 831.63@30
11	1735.62	1910.62@11; 1915@30	900.62@11; 835@30
12	1581.36	1790.76@11; 1840.6@30	851.76@11; 729.6@30
13	1530.24	1770.24@11; 1810@30	811.24@11; 719@30
14	1759.02	1900.32@11; 1910.7@30	911.32@11; 847.7@30
15	1789.74	1930.74@11; 1930@30	941.74@11; 848@30
16	1820	1925@11; 1940@30	965@11; 855@30
17	1889.16	1965.16@11; 1960@30	1008.16@11; 881@30
18	1952.88	1990.88@11; 1986@30	1056.88@11; 896@30
19	1834.68	1860.68@11; 1857@30	980.68@11; 854@30
20	1844.22	1925.22@11; 1960@30	1087.22@11; 757@30
21	1876.08	1950.2@11; 1968.9@30	1120.2@11; 755.9@30
22	1795.92	1895.92@11; 1945@30	1079.92@11; 716@30
23	971.04	1364.04@11; 1212@30	389.04@11; 582@30
24	954.66	1337.86@11; 1201.8@30	387.86@11; 566.8@30

@ means the busbar number.

and

Table 5. RDN performance analysis results for a 60% hosting factor for EVCS operation.

Hours	Power Losses (kW)	% Decrease in Active Power Losses	Reactive Power Loss (kVAR)	% Decrease in Reactive Power Losses	Minimum Voltage (pu)	% Improvement in Voltage	VDI × 10−3	VSI
1	11.517	47.4	7.715	48.19	0.9805	1.23	5.3	0.9316
2	11.719	45.4	7.832	46.2	0.9803	1.17	5.6	0.9292
3	11.182	46.2	7.476	46.8	0.9808	1.13	5.2	0.9283
4	11.019	45.7	7.364	46.4	0.981	1.09	5.2	0.9268
5	11.389	43.4	7.601	44.2	0.9805	1.02	5.6	0.9242
6	24.4	52.3	16.711	52	0.974	1.98	8.9	0.9390
7	48.657	54.7	33.641	54.2	0.969	3.98	12.2	0.9585
8	48.975	54.7	33.84	54.2	0.9681	3.89	12.6	0.9581
9	64.047	56.9	43.934	56.4	0.9681	4.77	13.5	0.9549
10	64.314	56.9	44.125	56.4	0.9681	4.82	13.6	0.9548
11	64.533	56.9	44.251	56.5	0.9678	4.82	13.9	0.9549
12	47.978	56.9	32.816	56.4	0.9720	4.26	10.5	0.9519
13	45.779	56.7	31.320	56.3	0.9723	4.08	10.4	0.9526
14	65.682	57	44.965	56.6	0.9670	4.83	14.8	0.9536
15	67.156	57.1	45.997	56.7	0.9674	5.02	14.4	0.9539
16	68.758	57.2	47.054	56.7	0.9670	5.12	15.1	0.9525
17	71.454	57.4	48.854	56.9	0.9651	5.21	16.5	0.9523
18	74.678	57.5	51.024	57.1	0.9641	5.38	17.4	0.9513
19	55.231	56.8	37.391	56.9	0.9669	4.66	14.7	0.9472
20	52.308	57.2	35.656	57	0.9684	4.75	12.3	0.9485
21	51.521	56.9	35.107	61.9	0.9681	4.75	12.4	0.9483
22	45.072	56.3	30.693	56.4	0.9704	4.46	10.7	0.9479
23	14.593	49.9	9.937	50.4	0.9773	1.64	6.6	0.9401
24	14.266	49.9	9.700	50.3	0.9776	1.61	6.4	0.9398

display the related results at 60% hosting factor for EVCSs with optimal placement and sizing for both EVCSs and DGs.

At 40% EV-HF: Active power losses were reduced by approximately 57% compared with the base case. The minimum bus voltage improved from 0.9148 pu to 0.9654 pu. The VSI values increased, indicating enhanced network stability.

At 60% EV-HF: Losses were similarly reduced, though slightly less effective due to higher load demand. The minimum bus voltage improved to 0.9641 pu, remaining within acceptable operational limits. VSI improvements were sustained, demonstrating the robustness of the method under higher penetration.

These results confirm that the proposed methodology ensures reliable network performance even under heavy EV adoption.

Table 6. The impact of EV-HF on the objectives under different operating situations.

Operating Condition	Base Case	40% EV-HF	60% EV-HF
$E_{loss annual}$(kWh)	868.729 × 103	381.128 × 103	381.873 × 103
VSI at max. loading	0.704	0.9533	0.9513
VDI at max. loading	109.5 × 10−3	16.2 × 10−3	17.4 × 10−3
$V_{min}$ at max loading (pu)	0.9148	0.9654	0.9641
Costs of used DGs (USD)	-	3.405 × 106	3.976 × 106
Annual saving costs (USD)	-	468.85 × 103	473.55 × 103
Payback period (year)	-	7.3	8.4
Total profit for 25-year lifetime (USD).	-	8.3 × 106	7.86 × 106

presents the comparison between results obtained for the base case, the operation with a 40% EV-HF, and the operation with a 60% EV-HF with respect to the annual energy losses, VSI, VDI, Vmin, the total cost of DGs, the cost of saving energy, and the profit for the system assuming a lifetime of 25 years.

4.1.2. Voltage Profile Improvement

The optimized EVCS and DG allocation significantly enhanced the overall voltage profile of the IEEE 33-bus system. Voltage levels across all buses remained within the acceptable 0.95–1.05 pu band, eliminating the undervoltage problems observed in the base case and EV-only cases. The most pronounced improvements were observed at the weakest buses, where voltage recovery was nearly 5% compared with the base case. The following curve, shown in Figure 9, presents the voltage profile of both the light and maximum loading conditions for the base case study, which shows that the minimum voltage at maximum loading is 0.913 pu.

Figure 10 depicts the voltage on the network for the light and maximum loading conditions with an EVCS hosting factor of 40%; here, the minimum voltage is 0.9654 pu.

Figure 11 provides the voltage profile for the loading conditions at both hours 5 and 18 with an EVCS hosting factor of 60%: the minimum voltage is 0.9641 pu.

Figure 12 presents the voltage profile comparison at maximum loading condition for the base case, EVCS hosting factor of 40% operating condition, and EVCS hosting factor of 60% operating condition. Although the loading on the network is increased by 40% and 60%, respectively, the minimum voltage is increased to acceptable values compared to that occurring for base case operation.

Table 7. Comparison between published methods and the proposed approach at the heavy loading condition (hour 18) of the IEEE 33-bus distribution network.

Method	Operated with	Power Losses (kW)	Minimum Voltage (pu)	Simulation Time (Sec)	Total Number of Iterations
PSO [34]	EVCSs and DGs	77.4738	0.9562	NA	NA
CS [34]	EVCSs and DGs	77.2482	0.9563	NA	NA
AOA [34]	EVCSs and DGs	77.2475	0.9579	NA	NA
Proposed Approach	EVCSs and DG	74.545	0.9654	2.1	17
	EVCSs only	229.5840	0.8876	1.82	13
	DGs only	74.210	0.9692	1.82	13

presents the comparisons between published methods and the proposed approach for the 33-bus IEEE distribution network at the heavy loading condition operated at hour 18. The results show that the proposed approach is more effective than the published methods for the same operating condition.

4.2. 69-Bus Distribution Network Results

The application of the proposed approach on the IEEE 69-bus distribution network is presented with comparisons with some published methods at the same operating condition in

Table 8. Comparison between published methods and the proposed approach for the IEEE 69-bus distribution network at the same operating condition.

Method	Power Losses (kW)	Minimum Voltage (pu)	Simulation Time (Sec)	Total Number of Iterations
PSO [35]	83.2	0.984	NA	NA
ABC [35]	72	0.987	NA	NA
IWO [36]	13.64	0.9946	4.5	NA
WOA [37]	69.72	0.9780	NA	NA
SA [37]	16.26		NA	NA
Proposed Approach	8.2	0.9938	2.45	22

.

5. Conclusions

This research described an impedance-based heuristic technique for the optimum placement and size of electric vehicle charging stations (EVCSs) and distributed generators (DGs) in radial distribution networks with different electric vehicle hosting factors (EV-HFs). The proposed impedance matrix-based load flow and analytical loss formulation provided accurate and computationally efficient evaluation of network performance, enabling fast optimization across multiple operating conditions. Simulation results on the IEEE 33-bus system demonstrated that unmanaged EVCS integration leads to higher active power losses and voltage deviations, with minimum bus voltages dropping below acceptable limits. Coordinated EVCS and DG planning significantly improved network performance. At 40% and 60% EV-HFs, active power losses were reduced by nearly 57%, while the minimum bus voltage improved from 0.9148 pu (base case) to 0.9654 pu and 0.9641 pu, respectively. The voltage stability index (VSI) also increased, confirming improved system resilience. Economic analysis indicated strong financial feasibility. The integration of PV-based DGs provided annual cost savings of up to USD 473,550, with a payback period of 7–8 years and substantial long-term profit over the system’s lifetime. The methodology is practical for distribution network operators, as it combines technical benefits (loss minimization and voltage stability) with economic gains, while maintaining computational simplicity compared to population-based metaheuristics. The results of proposed approach for both the 33-bus and 69-bus IEEE electrical distribution networks are compared with the published methods, which demonstrates the effectiveness of the proposed approach.

Future work will concentrate on broadening the suggested framework to include uncertainties in EV demand and in renewable generation. The impact of variable operation costs such as maintenance costs for photovoltaic units and time-of-use electricity pricing should also be taken in calculation for economic calculation and payback time. Also, dynamic voltage stability analysis in high-penetration EV fast-charging scenarios and the incorporation of energy storage technologies to improve flexibility and resilience should be studied.