Multi-Objective Optimal Integration of Distributed Generators into Distribution Networks Incorporated with Plug-In Electric Vehicles Using Walrus Optimization Algorithm

Abstract

The increasing adoption of plug-in electric vehicles (PEVs) leads to negative impacts on distribution network efficiency due to the extra load added to the system. To overcome this problem, this manuscript aims to optimally integrate distributed generators (DGs) in radial distribution networks (RDNs), while including uncoordinated charging of PEVs added to the basic daily load curve with different load models. The main objectives are minimizing the network’s daily energy losses, improving the daily voltage profile, and enhancing voltage stability considering various constraints like power balance, buses’ voltages, and line flow. These objectives are combined using weighting factors to formulate a weighted sum multi-objective function (MOF). A very recent metaheuristic approach, namely the Walrus optimization algorithm (WO), is addressed to identify the DGs’ best locations and sizes that achieve the lowest value of MOF, without violating different constraints. The proposed optimization model along with a repetitive backward/forward load flow (BFLF) method are simulated using MATLAB 2016a software. The WO-based optimization model is applied to IEEE 33-bus, 69-bus, and a real system in El-Shourok City-district number 8 (ShC-D8), Egypt. The simulation results show that the proposed optimization method significantly enhanced the performance of RDNs incorporated with PEVs in all aspects. Moreover, the proposed WO approach proved its superiority and efficiency in getting high-quality solutions for DGs’ locations and ratings, compared to other programmed algorithms.

1. Introduction

The growing concerns over carbon dioxide emissions and global warming have led to a trend towards zero-emission PEVs, which are projected to play a vital role in minimizing pollutants in the transportation sector [1]. In addition, efficient management of PEVs and their charging schedules can potentially reduce petrol prices [2]. However, unlike traditional vehicles that rely on internal combustion motors, PEVs are integrated into the power grid through charging piles, which are used to refuel by recharging their batteries. Hence, PEVs are considered as an extra load for distribution networks. Moreover, uncoordinated charging of PEVs and improper integration of charging locations may lead to negative impacts on distribution networks, such as overloading of transformers and transmission lines, higher energy losses, a rise in peak demand, power quality issues, poor voltage profiles, and voltage stability concerns [3]. A detailed review of the PEV market, charging strategies, and the impacts of charging demand on utilities is presented in [4]. A probabilistic model is given in [5] to evaluate the electric vehicle (EV) load, in which power flow analysis was performed for one week to investigate the different technical impacts of EVs on a practical distribution network in Saudi Arabia. The results expected were that, for every 10% (1 million) increase in the deployment rate of EVs, the maximum demand and power losses would increase by 2% on average. The impacts of EV load on voltage stability and overloading have been investigated in [6] under various scenarios. The voltage regulation issue is addressed in [7], using hybrid electric vehicles (HEVs) under varying load profiles. According to [7], HEVs are widely divided into two main categories: (1) ICE-based HEVs, in which the internal combustion engine (ICE) represents the main energy source; and (2) fuel-cell-based hybrid electric vehicles (FCHEV), in which a fuel cell is the main energy source. HEVs usually have auxiliary rechargeable sources, such as batteries, ultra-capacitors, and flywheel energy storage system (FESS). These auxiliary sources assist in the fulfilment of energy demand.

Many approaches have been utilized to optimally allocate DGs in distribution networks for various objectives, such as power loss reduction, improving voltage profile, and voltage stability concerns [8,9,10,11,12,13]. The literature shows that the radial distribution network suffers from a significant percentage of power losses, which can reach 13% as pointed out in [14]. With small sizes of DG units integrated into RDN, their generated active and/or reactive power may significantly affect active and reactive power flow through the branches of the network, resulting in a high influence on the network’s technical performance [15]. However, improper selection of DG location and/or its rating may have negative impacts, leading to higher power loss and a significant reduction in bus voltage. Therefore, finding near-optimal locations and sizes of DGs in RDN is a critical challenge for researchers. In the literature, the problem of DG allocation has been formulated as an optimization problem solved using various approaches, including analytical [16,17,18], deterministic [19,20], and stochastic-based metaheuristic techniques. However, it has been found that both analytical and deterministic methods may be effective, reliable, and require less computation time only in the case of small-scale systems with a limited number of DGs.

As a result, stochastic optimization approaches, especially metaheuristic techniques, have become more popular for solving such non-linear complex problems with multiple DGs of various types, via minimization of single or multi-objectives, and dealing with different equality and inequality constraints [21]. The use of an enhanced artificial ecosystem-based optimizer (EAEO) was proposed in [22] to optimally allocate DGs in three distribution networks with different sizes, including 33-bus, 69-bus, and 119-bus RDNs. The optimization problem was formulated based on single or multi-objective fitness functions to reduce power losses, improve voltage profile, and enhance the stability of the systems. The manta ray foraging optimization approach (MRFO) was implemented in [23] to solve the DG allocation problem and to minimize the total power losses in 33-bus, 69-bus, and 85-bus RDNs. Techno-economic objectives were handled in [24] using a multi-objective whale optimization algorithm (MOWOA) to find the best locations and ratings of DGs. Power loss minimization and voltage deviation reduction were taken as the technical aims, whereas the economic aims included the cost of energy loss reduction as well as DG cost, including capital, operation, and maintenance costs. The chimp optimization algorithm (COA) was proposed in [25] to solve the problem of DG allocation and sizing in the large 119-bus RDN. The objectives to be minimized included economic costs and the reduction of energy not-supplied index (ENSI).

A hybrid optimization approach was proposed in [26], combining the strength points of the salp swarm algorithm (SSA) and particle swarm optimization (PSO), to reduce active and reactive power losses and to improve the voltage profile in the 69-bus RDN, while considering time-varying and voltage-dependent load demands. The black widow optimization approach (BWO) is proposed in [27] as a population-based optimizer to optimize power flow for cost effective operation of microgrids. A novel decentralized stochastic recursive gradient (DSRG) approach is proposed in [28] to solve the optimal power flow problem in multi-area power systems based on local information in a fully decentralized manner. An improved beluga whale optimization algorithm (IBWO) was implemented in [29] for optimal planning of DG locations and sizes, considering DG uncertainties and the demand response in distribution networks. The IBWO was applied to minimize power losses, reduce operating costs, and improve the voltage profile. More stochastic metaheuristic-based approaches dealing with DG allocation problems are given in [21,30], which consider various objectives and constraints. In general, the stochastic-based optimization techniques have some difficulties in convergence speed and may get trapped in local optimums. In addition, some of these approaches are complicated in structure and need multiple parameters to be tuned. Hence, the continuous seeking for metaheuristic optimization algorithms characterized by simple structure, less tunable parameters, and improved convergence rate becomes a vital motivational key for researchers to get near-optimal results.

WO is classified as a swarm-based metaheuristic optimization algorithm inspired by walruses’ social behaviors in migrating, roosting, breeding, and foraging [31]. Like dolphins and bats, walruses depend more on sound localization, instead of vision, to communicate with their peers, forage, and exchange information about feeding locations. WO emulates social behaviors and combines the unique features of walruses by modeling populations’ updating procedures depending on danger and safety signals. By these two signals, WO makes a combination of the exploration phase (Migration of the herd) and exploitation phase (Reproduction of the herd) and hence balances the trade-off between early convergence and avoiding, trapped into local optima. Moreover, walruses may also receive a danger signal from their guards when foraging underwater. Therefore, walruses will escape from predation to another safe position underwater. WO emulates this behavior, mainly during the late iteration, leading to some kind of disruption in the population which helps the optimizer to enhance global exploration in the search space during the last iterations. In addition, WO employs a simple construction and requires fewer user-defined parameters. All these merits and features encourage the application of WO in this manuscript to solve DG allocation problems.

For effective planning of a distribution network incorporating DGs, it is important to include the load demand from PEVs within the network. From the literature, it has been observed that some of the recent DG allocation studies have considered EV demand in their analysis using various metaheuristic approaches [32,33,34,35,36]. However, few papers have investigated the mitigation of PEV impacts in distribution networks under different behaviors of charging via DG optimal allocation, considering various load models with a time-varying nature. In this manuscript, two charging scenarios are considered, the off-peak charging scenario (OPC) and the on-peak charging scenario (PC), based on the charging behavior of PEV owners as presented in [37]. The total charging power profiles for both charging scenarios are obtained based on the total number of PEVs, the charging profile for each type of PEV, and the charging time probability distribution given in [37]. The obtained charging demand is integrated into the residential buses in the distribution network and added to the basic domestic load. Then, a detailed study is performed to investigate the impacts of both charging behaviors on RDNs with a time-varying, voltage-dependent load model for 24 h. Finally, unity and non-unity power factor DGs are optimally integrated into the RDNs under study to mitigate the negative impacts of PEV charging demand and improve system performance. The main objectives of this study are the minimization of energy losses, daily voltage profile improvement, and the enhancement of daily voltage stability of the network incorporated with PEVs via the introduction of DGs with optimal locations, sizes, and power factors, using the recent approach of a walrus optimizer. Therefore, the main contributions in this paper are as follows:

• It is the first time to apply the recently developed WO approach to determine the near-optimal locations and ratings of DGs in RDNs incorporated with PEVs;
• Two standard 33-bus and 69-bus RDNs beside one real distribution system of ShC-D8 in Egypt are analyzed, considering a time-varying voltage-dependent load model;
• Two types of DGs, with unity and non-unity power factors, are used in this study while adapting WO so that the first decision variables must be integer numbers representing DGs’ locations and the others represent DGs’ ratings and power factors;
• Constraints like power balance, buses’ voltages, and line flow are taken into consideration in the optimization model to present a more realistic problem;
• The efficacy and superiority of the proposed WO are verified by comparing it to other approaches.

2. Distribution Network Modelling Framework

2.1. DG Modelling

For planning studies in distribution networks, DGs may be modeled either as a PQ model or as a PV model [38]. In this manuscript, the PQ model is utilized, in which a DG is modeled as a load with negative active and reactive power. Hence, the equivalent load demand at any bus integrated with a DG unit is expressed as follows:

$$P_{d,eq}=P_d-P_{DG}$$

(1)

$$Q_{d,eq}=Q_d-Q_{DG}$$

(2)

where $P_d$ and $Q_d$ are the active and reactive load demands at the bus; $P_{DG}$ and $Q_{DG}$ represent the DG’s active and reactive power supplied to the bus; and $P_{d,eq}$ and $Q_{d,eq}$ denote the equivalent active and reactive load demand at the bus with DG penetration. $Q_{DG}$ is calculated based on $P_{DG}$, the type, and the value of the DG power factor (${PF}_{DG}$), as illustrated in Equation (3).

$$Q_{DG}=\left\{\begin{array}{c}+P_{DG}\tan \left({cos}^{-1}\left({PF}_{DG}\right)\right) for \mathrm{l}\mathrm{a}\mathrm{g}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{g} {PF}_{DG} (\mathrm{D}\mathrm{G} \mathrm{i}\mathrm{n}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{s} \mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}) \\ -P_{DG}\tan \left({cos}^{-1}\left({PF}_{DG}\right)\right) for \mathrm{l}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} {PF}_{DG} (\mathrm{D}\mathrm{G} \mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{r}\mathrm{b}\mathrm{s} \mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} \mathrm{p}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r})\end{array}\right.$$

(3)

2.2. Load Modelling

In the current manuscript, three distinct time-varying voltage-dependent load models (i.e., industrial, commercial, and residential) are considered. The practical active and reactive powers at each load bus are modeled by employing the following polynomial expressions:

$$P_{d,t}\left(i\right)=\left(P_{d,t}^0\left(i\right)+P_{EV,t}\left(i\right)\right)\times V_{i,t}^{\alpha }$$

(4)

$$Q_{d,t}\left(i\right)=Q_{d,t}^0\left(i\right)\times V_{i,t}^{\beta }$$

(5)

where $P_{d,t}$ and $Q_{d,t}$ are the actual active and reactive load power at bus $i$ during time interval $t$; $P_{d,t}^0$ and $Q_{d,t}^0$ are the initial active and reactive load power at bus $i$ during time interval $t$; and $P_{EV,t}$ is the active power load of PEVs at bus $i$ during the $t$-hour interval. The voltage coefficients $\alpha$ and $\beta$ of various load models are given in

Table 1. Load category and voltage coefficients.

Load Model	$\mathit{\alpha }$	$\mathit{\beta }$
Industrial Load (IL)	0.18	6
Residential Load (RL)	0.92	4.04
Commercial Load (CL)	1.51	3.40

[39].

2.3. Load Flow Model

Compared to a transmission network, a distribution network involves buses and branches arranged in a radial and weakly connected structure. The slack bus represents the main source of power. Figure 1 depicts a schematic representation of a distribution network with two buses: sending bus m and receiving bus n. In our study, the backward/forward load flow (BFLF) method is more appropriate to perform load flow analysis than other conventional algorithms, due to the radial nature and high R/X ratio of distribution networks [40].

Based on Kirchhoff’s laws, BFLF updates branches’ currents and the voltages at each bus in an iterative manner using backward and forward sweeps, respectively. After many iterations, the complex voltage at each bus of the network is obtained. Both active and reactive power fed into the receiving bus $n$ from the sending bus $m$ are calculated using Equations (6) and (7), respectively.

$$P_{mn}={\displaystyle \frac{V_m{V}_n}{R_{mn}^2+X_{mn}^2}}\cos ({\theta }_{mn}-{\delta }_m+{\delta }_n)-{\displaystyle \frac{V_n^2}{R_{mn}^2+X_{mn}^2}}\cos \left({\theta }_{mn}\right)$$

(6)

$$Q_{mn}={\displaystyle \frac{V_m{V}_n}{R_{mn}^2+X_{mn}^2}}\sin ({\theta }_{mn}-{\delta }_m+{\delta }_n)-{\displaystyle \frac{V_n^2}{R_{mn}^2+X_{mn}^2}}\sin ({\theta }_{mn})$$

(7)

where $V_m$ and $V_n$ are the voltage magnitudes at bus $m$ and bus $n$, respectively; ${\delta }_m$ and ${\delta }_n$ are the voltage angles at bus $m$ and bus $n$, respectively; and $R_{mn}$, $X_{mn}$, and ${\theta }_{mn}$ are, respectively, the resistance, reactance, and impedance angle of branch $m$, $n$. Equations (8) and (9) are used respectively to calculate the active power loss ($P_{mn}^{loss}$) and reactive power loss ($Q_{mn}^{loss}$) in branch $m$, $n$.

$$P_{mn}^{loss}=I_{mn}^2{R}_{mn}={\displaystyle \frac{P_{mn}^2+Q_{mn}^2}{V_n^2}} · R_{mn}$$

(8)

$$Q_{mn}^{loss}=I_{mn}^2{X}_{mn}={\displaystyle \frac{P_{mn}^2+Q_{mn}^2}{V_n^2}} · X_{mn}$$

(9)

where $I_{mn}$ is the magnitude of current passing through branch $m$, $n$.

2.4. Objective Function

Technical benefit enhancement is the main concern in this manuscript. Therefore, a weighted sum multi-objective function (MOF) is formulated, as shown in Equation (10), which involves reducing power loss, enhancing voltage stability, and reducing bus voltage deviation.

$$minimize\left(MOF\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\left(w_1\times LI\right)+\left(w_2\times (1/VSI)\right)+(w_3\times VDI)\right\}$$

(10)

where $w_1$, $w_2$, and $w_3$ are the weighting factors; $PLI$—power loss index; $VSI$—voltage stability index; and $VDI$—voltage deviation index. The weighting factors are defined by users with values ranging from 0 to 1, and should always add up to one. All indices are defined as follows.

2.4.1. Power Loss Index (PLI)

The network power loss during time interval $t$ is expressed, as shown in Equation (11), based on Equation (8).

$$P_t^{loss}=\sum _{m=1}^{N_b-1}\sum _{n=2}^{N_b}{\displaystyle \frac{P_{mn,t}^2+Q_{mn,t}^2}{V_{n,t}^2}}·R_{mn}$$

(11)

where $N_b$ is the total number of buses in the network; and $P_{mn,t}$ and $Q_{mn,t}$ are, respectively, the active and reactive power fed into the receiving bus $n$ from the sending bus $m$. PLI is defined in Equation (12), which represents the ratio of the network’s daily real power loss with DG placement $(P_{daily, DG}^{loss})$ to the network’s daily real power loss without DG $(P_{daily}^{loss})$.

$$PLI={\displaystyle \frac{P_{daily, DG}^{loss}}{P_{daily}^{loss}}}={\displaystyle \frac{\sum _{t=1}^{24}{P}_{t, DG}^{loss}}{\sum _{t=1}^{24}{P}_t^{loss}}}$$

(12)

where $P_{t, DG}^{loss}$ and $P_t^{loss}$ are, respectively, the real power loss during the $t$-hour interval for the network with and without DG placement, and are calculated based on Equation (11).

2.4.2. Voltage Stability Index (VSI)

The random increase in load demand, especially with the higher penetration level of EV charging stations, may lead to a voltage collapse phenomenon in distribution networks. A stability index (SI), presented in [41], indicates the extent to which the bus voltage is steady and unlikely to collapse. In this study, it is required to maximize the SI at each receiving bus, calculated as illustrated in Equation (13).

$${SI}_{n,t}=V_{m,t}^4-4\left(P_{mn,t}{R}_{mn}+Q_{mn,t}{X}_{mn}\right)V_{m,t}^2-4{(P_{mn,t}{X}_{mn}-Q_{mn,t}{R}_{mn})}^2$$

(13)

The minimum value of the SI among all receiving buses is selected at each time interval $t$ during the day and then aggregated to be maximized with DG placement, as illustrated in Equation (14).

$$VSI={\displaystyle \frac{{VSI}^{DG}}{{VSI}^{WODG}}}={\displaystyle \frac{\sum _{t=1}^{24}min\left({SI}_{n,t}^{DG}\right)}{\sum _{t=1}^{24}min\left({SI}_{n,t}^{WODG}\right)}} n=\mathrm{2,3},\dots .N_b$$

(14)

where ${SI}_{n,t}^{DG}$ and ${SI}_{n,t}^{WODG}$ are, respectively, SI values for each receiving bus $n$ during the $t$-hour interval with and without DG placement, and are calculated based on Equation (13).

2.4.3. Voltage Deviation Index (VDI)

Improving the daily voltage profile is one of the distribution network operator’s main responsibilities. In addition, it represents a big concern for all consumers regarding meters and appliances. Hence, voltage profile can be enhanced by minimizing the VDI, as illustrated in Equation (15).

$$VDI={\displaystyle \frac{{VDI}^{DG}}{{VDI}^{WODG}}}={\displaystyle \frac{\sum _{t=1}^{24}max\left|1-V_{n,t}^{DG}\right|}{\sum _{t=1}^{24}max\left|1-V_{n,t}^{WODG}\right|}} n=\mathrm{2,3},\dots ..,N_b$$

(15)

where ${VDI}^{DG}$ and ${VDI}^{WODG}$ are the values of VDI with and without DG, respectively; and $V_{n,t}^{DG}$ and $V_{n,t}^{WODG}$ are, respectively, voltage magnitudes at each receiving bus n during the $t$-hour interval with and without DG placement. The total voltage deviation (TVD) is also determined in this study to measure the daily total amount of deviation in voltage magnitude for all load buses, as shown in Equation (16).

$$TVD={\sum }_{t=1}^{24}{\sum }_{n=2}^{N_b}\left|1-V_{n,t}\right| n=\mathrm{2,3},\dots ..,N_b$$

(16)

2.5. Constraints

2.5.1. Power Balance Constraints

These kinds of constraints are associated with load flow procedures and can be presented, as shown in Equations (17) and (18), for active and reactive power balance, respectively.

$$P_{slack,t}+P_{DG,t}^T=P_{d,t}^T+P_t^{loss}$$

(17)

$$Q_{slack,t}+Q_{DG,t}^T=Q_{d,t}^T+Q_t^{loss}$$

(18)

where $P_{slack,t}$ and $Q_{slack,t}$ are the active and reactive powers supplied at the slack bus during the $t$-hour interval; $P_{DG,t}^T$ and $Q_{DG,t}^T$ are, respectively, the total active and reactive power from DGs during the $t$-hour interval; $P_{d,t}^T$ is the total active power demand involving main load and PEVs’ charging demand during time interval $t$; and $Q_{d,t}^T$ is the total reactive power of the network main load at time $t$.

2.5.2. Bus Voltage Constraints

It is essential to keep the voltage magnitude of all buses within the acceptable boundaries after allocating DGs in the distribution network as follows:

$$V^{min}\le V_i\le V^{max} i=\mathrm{1,2},\dots ,N_b$$

(19)

where $V^{min}$ and $V^{max}$ are the minimum and maximum acceptable limits of voltage magnitude.

2.5.3. Sizing Limits of DGs

The optimal size of each DG is designed considering predefined power limits as follows:

$$P_{DG}^{min}\le P_{DG}^k\le P_{DG}^{max} k=\mathrm{1,2},\dots ,N_{DG}$$

(20)

$${PF}_{DG}^{min}\le {PF}_{DG}^k\le {PF}_{DG}^{max} k=\mathrm{1,2},\dots ,N_{DG}$$

(21)

where $P_{DG}^k$ is the rated power supplied from the $k^{th}$ DG unit; $P_{DG}^{min}$ and $P_{DG}^{max}$ are the minimum and maximum predefined limits for DG active power; ${PF}_{DG}^k$ is the power factor of the $k^{th}$ DG unit; ${PF}_{DG}^{min}$ and ${PF}_{DG}^{max}$ are the allowable limits for the DG power factor; and $N_{DG}$ is the total number of DGs.

2.5.4. Line Flow Constraints

The power transmitted through each line in the network should not exceed the maximum allowable capacity of the line as follows:

$$S_l\le S_l^{max} l=\mathrm{1,2},\dots ,N_{br}$$

(22)

where $S_l$ represents the power flow in branch $l$; $S_l^{max}$ denotes the maximum allowable power capacity of $l^{th}$ branch; and $N_{br}$ is the total number of branches in the network.

2.5.5. Treatment of Constraints

It is worth mentioning here that power balance equality constraints are attained within load flow calculations. DG sizing limits are self-constrained by the floor and ceiling bounds of the proposed WO approach. Other inequality constraints (bus voltage and line flow) are handled using the following penalty factors:

$$P_v={\gamma }_v.\left[\sum _{i=1}^{N_b}max \left(0,V_i-V^{max}\right)+\sum _{i=1}^{N_b}\mathrm{m}\mathrm{a}\mathrm{x}(0,V^{min}-V_i)\right]$$

(23)

$$P_{Lf}={\gamma }_{Lf}.\left[\sum _{l=1}^{N_{br}}max \left(0,S_l-S_l^{max}\right)\right]$$

(24)

where $P_v$ and $P_{Lf}$ represent the bus voltage and line flow penalty factors, respectively. ${\gamma }_v$ and ${\gamma }_{Lf}$ are the weights of penalty factors that have large positive values. These penalty factors are associated with each violated inequality constraint and added to the objective function to avoid the infeasible solution area.

3. The WO Approach

3.1. General Overview

WO is classified as a swarm-based metaheuristic optimization algorithm inspired by walruses’ social behaviors in migrating, roosting, breeding, and foraging [31]. The walrus is one of the biggest marine mammals, aside from whales. The temperate waters in the arctic or adjacent regions are home to walruses. Walruses are sociable animals that are typically seen in groups of dozens, hundreds, and sometimes even thousands. The walrus has two distinct features that assist it in various activities for survival and self-defense. The first feature is its sharp sensitivity of touch, resulting from hundreds of sensitive whiskers surrounding the upper lip. A pair of well-developed upper canine teeth is the second feature, which is considered a most unique characteristic used for self-defense and searching in the sand and mud for food. Like dolphins and bats, walruses depend more on sound localization instead of vision to communicate with their peers, forage, and exchange information about feeding locations.

When the mating season starts, Walruses mark their territories on the sand. The most powerful males tend to have the best desirable locations, with spaces varying depending on the number of females attracted. Owing to the strong social behavior of walruses, they use a collective defense technique to protect themselves against predators such as killer whales. The walrus’s experience with long-term survival keeps it vigilant. Vigilantes in walrus herds are considered to be the determining factor that influences the herd’s direction. WO emulates social behaviors and combines the unique features of walruses by modeling populations’ updating procedures depending on danger and safety signals. By these two signals, WO makes a combination of exploration phase (migration of herd) and exploitation phase (reproduction of herd) and hence balances the trade-off between early convergence and avoiding being trapped in local optima. Also, the behavior of the WO population mimics the roosting behavior of a walrus herd, which is different in males, females, and juveniles, while updating their positions. The detailed mathematical modeling of WO is further explained in the next subsection.

3.2. Mathematical Model of WO

The optimization procedures of the proposed WO are performed in four phases: initialization, production of danger and safety signals, migration (exploration), and reproduction (exploitation) [31].

3.2.1. Phase (1): Initialization

The modeling of the WO begins with a set of initial random bounded solutions ${(S}_i)$ that serve as the initial population matrix (S).

$$S=\left[\begin{array}{c}{S}_1\\ S_2\\ \begin{array}{c}⋮\\ S_i\\ \begin{array}{c}⋮\\ S_N\end{array}\end{array}\end{array}\right]={\left[\begin{array}{cc}\begin{array}{ccc}{s}_1^1& s_1^2& \dots \\ s_2^1& s_2^2& \dots \\ ⋮& ⋮& ⋮\end{array}& \begin{array}{ccc}{s}_1^j& \dots & s_1^{dim}\\ s_2^j& \dots & s_2^{dim}\\ ⋮& ⋮& ⋮\end{array}\\ \begin{array}{ccc}{s}_i^1& s_i^2& \dots \\ ⋮& ⋮& ⋮\\ s_N^1& s_N^2& \dots \end{array}& \begin{array}{ccc}{s}_i^j& \dots & s_i^{dim}\\ ⋮& ⋮& ⋮\\ s_N^j& \dots & s_N^{dim}\end{array}\end{array}\right]}_{N\times dim}, i=\mathrm{1,2},\dots ,N j=\mathrm{1,2},\dots , dim$$

(25)

$$s_i^j=s_{i,min}^j+rand\times (s_{i,max}^j-s_{i,min}^j)$$

(26)

where $N$ is the population size; $dim$ represents the dimension of decision variables in each solution vector ${(S}_i)$; $s_i^j$ denotes the $j^{th}$ decision variable in the $i^{th}$ solution; $s_{i,min}^j$ and $s_{i,max}^j$ are the lower and upper boundaries of $s_i^j$; and $rand$ is a random number inside the range [0, 1]. Each candidate solution has its own fitness value $(F_i)$, arranged in the following vector:

$$Fitness={\left[\begin{array}{ccc}{F}_1& F_2& \begin{array}{ccc}\dots & F_i& \begin{array}{cc}\dots & F_n\end{array}\end{array}\end{array}\right]}_{1\times N}^T$$

(27)

Ninety percent of the walrus herd (WO population) is composed of adults, equally distributed between males and females. The remaining 10% of walruses are considered to be juveniles. All candidate solutions update their positions continuously in the search space via iterations which follow the strategies considered in the next phases.

3.2.2. Phase (2): Giving Danger and Safety Signals

In a walrus herd, one or two walruses serve on patrol as guards. They determine and influence the herd’s direction by launching danger alerts instantly if any unexpected circumstances are found. When these danger signals are very high, migration to other more suitable areas for survival is the best choice for the walrus herd. In contrast, safety signals released by the vigilantes mean roosting and foraging instead of migrating. In WO, the following definitions are formulated to define these two kinds of signals, which play a vital role in balancing between exploration and exploitation [31]:

$$Danger\text{\_}signal=A\times R$$

(28)

$$A=2\times \alpha$$

(29)

$$\alpha =1-k/it\text{\_}max$$

(30)

$$R=2\times r_1-1$$

(31)

$$Safety\text{\_}signal=r_2$$

(32)

where $A$ and $R$ represent danger coefficients; $\alpha$ is a descending factor that decreases from 1 to 0 via iterations, $k$ and $it\text{\_}max$ are, respectively, the iteration count and the maximum predefined number of iterations; and $r_1$ and $r_2$ are random numbers inside the range [0, 1].

3.2.3. Phase (3): Migration (Exploration)

When danger signals are very high, exceeding a certain predefined level, the walrus herd will have to migrate to another, more suitable area for survival. In WO, this process is more likely to occur during the early stage of the algorithm, providing an opportunity to maximize global search. Hence, the position of each solution in the WO population will be updated as follows [31]:

$$S_i^{k+1}=S_i^k+Migration\text{\_}step$$

(33)

$$Migration\text{\_}step={(S}_m^k-S_u^k)\times \beta \times r_3^2$$

(34)

$$\beta =1-1/\left(1+exp\left(-10\times {\displaystyle \frac{k-0.5\times it\text{\_}max}{it\text{\_}max}}\right)\right)$$

(35)

where $S_i^{k+1}$ and $S_i^k$ are, respectively, the new and current position of the $i^{th}$ solution vector; $Migration\text{\_}step$ denotes the step length of walrus movement; $S_m^k$ and $S_u^k$ are two randomly selected solution vectors from the population at iteration $k$, representing vigilantes in the herd; $\beta$ represents the migration step control coefficient varying smoothly with iterations; and $r_3$ is a random number inside the range [0, 1].

3.2.4. Phase (4): Reproduction (Exploitation)

Contrary to the migration phase, the walrus herd tends to reproduce in its current area instead of migrating, when dangers are at a low level. Based on the level of safety signal, the reproductive stage may be divided into two main activities: roosting onshore and foraging underwater. In WO, these activities are more likely to occur late in the algorithm, providing an opportunity to maximize local search. The mathematical models representing both activities are as follows:

Roosting Behavior

In this behavior, the three main categories of WO population—males, females, and juveniles—in the walrus herd update their positions in three different manners. Halton sequence distribution is adopted to randomly redistribute the solution vectors related to male walruses. The new position of the female walrus is directed by the male walrus $({Male}_i^k)$ and the walrus in the lead $\left(S_{best}^k\right)$. Throughout the algorithm’s progression, the female walrus is more influenced by the leader than the mate.

$${Female}_i^{k+1}={Female}_i^k+\alpha \times \left({Male}_i^k-{Female}_i^k\right)+(1-\alpha )\times \left(S_{best}^k-{Female}_i^k\right)$$

(36)

where ${Female}_i^{k+1}$ and ${Female}_i^k$ are the new and current positions of the $i^{th}$ female walrus, respectively; and ${Male}_i^k$ is the current position of $i^{th}$ male walrus. Juvenile walruses are the most vulnerable to predators, like polar bears. Therefore, juvenile walruses have to change their position to avoid predation.

$${Juvenile}_i^{k+1}=\left(O-{Juvenile}_i^k\right)\times P$$

(37)

$$O=S_{best}^k+{Juvenile}_i^k\times LF$$

(38)

where ${Juvenile}_i^{k+1}$ and ${Juvenile}_i^k$ are the new and current positions of the $i^{th}$ juvenile walrus, respectively; $P$ represents the distress factor of the juvenile walrus, having random value inside the range [0, 1]; $O$ denotes a safer position; and $LF$ is a random vector depending on the Lévy function.

$$Lévy\left(a\right)=0.05\times (g/{\left|h\right|}^{1/a})$$

(39)

where $g$ and $h$ represent two variables having a normal distribution, $g N\left(0,{\sigma }_g^2\right)$, $h N\left(0,{\sigma }_h^2\right)$.

$${\sigma }_g={\left[{\displaystyle \frac{\Gamma \left(1+\epsilon \right)sin\left(\pi \epsilon /2\right)}{\Gamma \left(0.5+0.5\epsilon \right)\epsilon \times 2^{\frac{\epsilon -1}{2}}}}\right]}^{{\displaystyle \frac{1}{\epsilon }}}, {\sigma }_h=1, \epsilon =1.5$$

(40)

where ${\sigma }_g$ and ${\sigma }_h$ are the standard deviations.

Foraging Behavior

Fleeing and gathering activities are the two main kinds of foraging underwater. Considering fleeing activity, walruses may also receive a danger signal from their guards when foraging underwater. Therefore, walruses will escape from predation to another safe position underwater. WO emulates this behavior mainly during late iterations. This leads to some kind of disruption in the population, which helps the optimizer enhance global exploration in the search space throughout the late iterations.

$$S_i^{k+1}=S_i^k\times R-\left|S_{best}^k-S_i^k\right|\times r_4^2$$

(41)

where $\left|S_{best}^k-S_i^k\right|$ is the distance between the leader walrus and the current walrus; and $r_4$ is a random number inside the range [0, 1]. During gathering activity, walruses update their positions to identify sea regions with higher availability of food by sharing information and cooperating with other walruses in the herd. WO models this behavior to achieve improvement in local exploitation, and to obtain the best possible solutions in a specific area within the search space.

$$S_i^{k+1}=\left(S_1+S_2\right)/2$$

(42)

$$S_1=S_{best}^k-a_1\times b_1\times \left|S_{best}^k-S_i^k\right|$$

(43)

$$S_2=S_{second}^k-a_2\times b_2\times \left|S_{second}^k-S_i^k\right|$$

(44)

$$a=\beta \times r_5-\beta$$

(45)

$$b=\mathrm{t}\mathrm{a}\mathrm{n}(\theta )$$

(46)

where $S_1$ and $S_2$ are two weighting vectors influencing the gathering activities of the herd; $S_{second}^k$ denotes the current position of the second walrus; $\left|S_{second}^k-S_i^k\right|$ represents the distance between the second walrus and the current walrus; $a$ and $b$ are the coefficients of gathering; $\theta$ has a value from 0 to $\pi$; and $r_5$ is a random number inside the range [0, 1].

3.3. Procedures of WO

Figure 2 depicts a flowchart that summarizes different procedures of the WO approach. As indicated in the Figure, the absolute value of the danger signal determines the transition, either to the migration phase (global exploration) or to the exploitation phase. In addition, the safety signal plays a vital role during exploitation, which selects between roosting and foraging behaviors. Moreover, the two main activities of foraging behavior are directed by danger signals.

3.4. Application of WO Approach on the DG Allocation Problem

This section presents a method for how to apply the proposed WO technique to determine the best locations and sizes of DGs integrated into radial distribution networks in the presence of a PEV charging load. Figure 3 and Figure 4 show the different sections of each solution in the WO population related to the unity power factor DG and non-unity power factor DG, respectively. As indicated in the Figures, WO is conveniently adapted so that the first section of decision variables in each solution is assigned to integer numbers representing DG locations. The rated values of DG active power (in MW) are memorized in the second section, while the third section is added only for the non-unity power factor DGs and represents the values of the DG power factors. At the beginning, the parameters of WO are defined, such as the population size and maximum number of iterations. Also, the configuration of the network under study, branches, and bus data are described, including the charging demand of PEVs. The initial population is formulated, which includes a number of candidate solutions, and each solution is arranged in the form illustrated in Figure 3 and Figure 4. The fitness value of each candidate solution is evaluated using the load flow model and the formulated MOF presented in Section 2. Penalty factors are associated with each violated inequality constraint and added to the MOF to avoid infeasible solutions. The procedures for updating the population are performed as presented in Figure 2 to get better solutions. When reaching the maximum number of iterations, the updating process is terminated and the best sizes, locations, and power factors of DGs are obtained.

4. Results and Discussions

The application of the proposed WO approach to the problem of DG allocation is applied on three radial distribution networks, an IEEE 33-bus, IEEE 69-bus, and a real system, ShC-D8 in Egypt, while incorporating PEV charging loads. The configurations of the 33-bus and 69-bus systems are illustrated in Figure 5 and Figure 6, respectively. The buses of both systems are classified and grouped as given in [42] and presented in

Table 2. Classification of buses for the 33-bus and 69-bus networks.

Network	Bus Type	Bus Numbers
33-bus network	Residential buses	2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18
	Commercial buses	19,20,21,22,23,24,25
	Industrial buses	26,27,28,29,30,31,32,33
69-bus network	Residential buses	2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,
	Commercial buses	47,48,49,50,51,52,66,67,68,69
	Industrial buses	28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46

. The data of all branches and buses for both systems is given in [43] with base values of 100 MVA and 12.66 kV. The power demands in the bus data are considered the peak demands, i.e., 1 p.u. The typical hourly load pattern [42] for both active and reactive power demands of different load models during the day is shown in Figure 7 with a 1 p.u. peak.

The main objective is to determine the optimal locations and ratings of DGs so that they mitigate the negative technical impacts of PEVs’ charging load on the system. The configuration of the network under study, branches’ data, and load bus data are encoded using MATLAB (R2016a) on a Dell laptop with specifications: Intel^® Core i5 CPU 1.6 GHz, 16 GB RAM Dell. In addition, the proposed WO procedures, the load flow model using the BFLF method, and the required fitness function are encoded and implemented in MATLAB to solve the optimization problem. Two types of conventional DGs are used in this study: unity power factor DGs (supplying only active power) and non-unity power factor DGs (supplying both active and reactive power). The adopted parameters of the WO approach for the studied networks and the required inequality constraints are presented in

Table 3. The adopted parameters of the proposed WO approach and the constraints.

Parameter	33-Bus	69-Bus and Shc-D8
$\mathrm{Population} \mathrm{size} \left(N\right)$	20	30
$\mathrm{Maximum} \mathrm{iteration} \left(it\text{\_}max\right)$	150	200
$\left(w_1,w_2,w_3\right)$	(0.5, 0.25, 0.25)	(0.5, 0.25, 0.25)
Voltage boundaries	$0.9 \mathrm{p}.\mathrm{u}.\le V_i\le 1.05 \mathrm{p}.\mathrm{u}.$
$P_{DG}^{min}\le P_{DG}^k\le P_{DG}^{max}$	$0\le P_{DG}^k\le 5 MW$
${PF}_{DG}^{min}\le {PF}_{DG}^k\le {PF}_{DG}^{max}$	$0.65\le {PF}_{DG}^k\le 1$

.

To investigate the effect of PEV charging load on the system and to mitigate the expected negative influences on the technical performance of the system using DGs, the following case studies are framed.

• Case 0: Without PEVs and DG units;
• Case 1: With PEVs under the PC scenario and without DG units;
• Case 2: With PEVs under the OPC scenario and without DG units;
• Case 3: Optimal allocation of unity power factor DGs in the presence of PEVs under a PC scenario;
• Case 4: Optimal allocation of non-unity power factor DGs in the presence of PEVs under a PC scenario.

4.1. PEV Charging Demand

In this study, two charging scenarios are considered based on the charging behavior of PEV owners [37]. The first is the PC scenario, in which all PEVs’ charging requests are given during peak hours of load demand. The second OPC scenario represents the EV owners who preferred charging during off-peak time due to lower electricity prices, especially during the early morning or at midnight hours. The total charging power profiles for both charging scenarios are obtained based on the total number of PEVs, the charging profile for each type of PEV, and the charging time probability distribution given in [37].

Table 4. Battery capacity of different types of PEVs (kWh).

Vehicle Type	PEV30	PEV40	PEV60
Compact sedan	7.8	10.4	15.6
Mid-size sedan	9	12	18
Mid-size SUV	11.4	15.2	22.8
Full-size SUV	13.8	18.4	27.6

shows the battery capacity of various types of PEVs with different all-electric ranges (AERs). In this work, four types of PEVs are considered with AER (PEV60). The maximum charging power level is considered as 6 kW, based on the Oak Ridge National Laboratory charging (ORNL) standard of 240 V AC/30 A [44]. The grid energy ${(E}_g)$ required to charge the PEV’s battery is calculated for each PEV type using Equation (47).

$$E_g=\left({\displaystyle \frac{{SOC}_d-{SOC}_i}{{\eta }_{ch}}}\right)\times E_c$$

(47)

where ${SOC}_d$ denotes the desired state of charge (SOC) of the battery; ${SOC}_i$ is the initial SOC of the battery when the PEV is plugged into the grid for charging; ${\eta }_{ch}$ represents charging efficiency; and $E_c$ denotes the energy capacity of the PEV’s battery.

Once $E_g$ is calculated, based on the ORNL charging standard, the charging profile for each type of PEV is obtained. Then, the charging power profile for a certain number of PEVs is determined based on the charging time probability distribution shown in Figure 8. Finally, this charging demand may be integrated into the distribution network at any bus and added to the basic load. It is assumed that 50 PEVs are integrated at each residential bus, where 45% of this number are compact sedans with 15.6 kWh batteries, 25% are midsize sedans with 18 kWh batteries, and the remaining PEVs are equally divided between midsize SUVs and full-size SUVs equipped with 22.8 kWh and 27.6 kWh batteries, respectively. The values of ${SOC}_i$, ${SOC}_d$, and ${\eta }_{ch}$ are assumed to be 20%, 80%, and 88%, respectively. All PEVs are charged only on the residential buses of the system. The charging load demand is determined for each charging scenario and integrated at each residential bus.

4.2. IEEE 33-Bus System

The investigation of the proposed method is carried out on the radial distribution network of the 33-bus. The initial 24-h load flow study is performed in case 0 without PEV and DG penetration. Both the voltage profile and stability profile for case 0 are shown in Figure 9 and Figure 10, respectively. It is noted that the least values for voltage magnitude and stability index for case 0 are, respectively, 0.917 p.u. and 0.707 p.u., and both occur at bus 18 during the 17th-hour interval. Figure 11 gives the hourly variation of both total active and reactive power losses. The highest levels of losses occur at the 17th-hour interval, with values reaching 187.84 kW and 124.84 kVAr for active and reactive power losses, respectively, as indicated in Figure 11. The impacts of PEV charging load demand on the network under study are examined for PC and OPC scenarios in cases 1 and 2, respectively. Figure 12 shows the charging demand profile penetrated into the system due to PEVs. As noted from Figure 12, the load of PEV charging during the PC scenario reaches the maximum level at the 16th-hour interval, with a value reaching 2045.68 kW. In contrast, the peak values of PEV charging load occur during early morning or midnight hours, considering the OPC scenario. The maximum level of PEV charging load during the OPC scenario occurs at the 2nd-hour interval after midnight, with a value reaching 2045.68 kW. The results obtained from a comparative analysis between case 0, case 1, and case 2 are summarized in

Table 5. Comparison of performance between case 0, case 1, and case 2 for 33-bus system.

Parameter	Case 0	Case 1	Case 2
$\mathrm{Daily} \mathrm{energy} \mathrm{losses} E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$	2305.8	3513.8	3014.5
$TVD (\mathrm{p}.\mathrm{u}.)$	26.2939	30.9496	30.9186
$\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$	0.917 (18th bus, 17th hour)	0.8586 (18th bus, 16th hour)	0.8986 (18th bus, 24th hour)
$\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$	0.707 (18th bus, 17th hour)	0.5435 (18th bus, 16th hour)	0.6519 (18th bus, 24th hour)
$\mathrm{Daily} \mathrm{substation} \mathrm{energy} (\mathrm{k}\mathrm{W}\mathrm{h})$	63,728.12	74,821.87	74,816.87
$\mathrm{Maximum} \mathrm{substation} \mathrm{power} (\mathrm{k}\mathrm{W})$	4077.6 (17th hour)	5644.5 (17th hour)	4077.6 (17th hour)

.

From Table 5, it is observed that the highest level of substation power occurs in case 1, reaching a value of 5644.5 kW during the 17th-hour interval. Meanwhile, it is the same in cases 0 and 2, with a level of 4077.6 kW at the 17th-hour interval. In addition, the daily energy losses in case 0 are determined to be 2305.8 kWh but increased to 3014.5 kWh in case 2 and then again to 3513.8 kWh in case 1. Despite there being a noticeable difference between the values of energy loss in cases 1 and 2, the daily energy supplied from the substation is almost the same. This is because of the existence of voltage-dependent load models drawing more energy in case 2, which has a more improved voltage profile than case 1. The hourly voltage profile of the system for case 1 is shown in Figure 13. It is observed that the minimum voltage occurs at bus 18 during the 16th-hour interval, with a value equal to 0.8586 p.u. Also, Figure 13 shows that, during peak hours (14th:18th) in case 1, a number of residential buses (from bus 9 to bus 18) have a voltage magnitude below 0.9 p.u. that could be a serious technical issue regarding power quality at these buses.

The hourly variations in minimum voltage magnitude and minimum SI are shown in Figure 14 and Figure 15. As noted from the Figures as well as Table 5, the minimum values of voltage magnitude and SI in case 1 reach their lowest levels (0.8586 p.u. and 0.5435 p.u., respectively) at the 18th bus, and during peak hours, specifically at the 16th hour. Hence, the PC scenario has a significant negative influence on consumers’ appliances during peak hours, which represent the period with the highest demand for electricity. In contrast, OPC scenario (case 2) has its lowest levels of voltage magnitude (0.8986 p.u.) and SI (0.6519 p.u.) also at the 18th bus, but at midnight (24th hour). Hence, the OPC scenario has fewer negative impacts on consumers compared with the PC scenario. Figure 16 depicts the hourly changes in total active power losses. There is a sharp increase in active power loss in case 1 between the 12th and 16th time intervals, reaching its maximum value of 410.7 kW at the 16th hour before slightly decreasing to its least level of 27 kW at the 5th hour. In addition, there is a slight difference in maximum active power losses between case 0 and case 2 compared to case 1, with values reaching 187.84 kW at the 17th hour for case 0 and 198 kW at the 24th hour for case 2.

Overall, energy losses are increased in both charging scenarios because of: (1) PEVs in this study representing an extra load for distribution networks, without providing vehicle-to-grid (V2G) service; and (2) uncoordinated charging of PEVs and improper integration of charging locations. Various strategies can be applied to mitigate the effects of PEVs on distribution networks. For instance, optimal scheduling of PEVs’ charging/discharging profiles and smart charging strategies play a vital role in reducing energy losses. In addition, choosing the optimal locations for parking lots leads to minimizing the negative impacts of PEVs on the grid. Moreover, EVs with smart V2G and grid-to-vehicle (G2V) services could participate in an energy management system to stock excessive energy during light loading, supply stored energy in EV batteries during power shortage intervals, and use optimal scheduling to consume network loads smoothly. DGs are also one of the effective strategies that can be utilized to mitigate the negative impacts of PEVs on distribution networks. Due to the higher negative technical impacts of the PC scenario on the system than the OPC scenario, DGs are considered in our study to be optimally allocated with a PEV charging load under PC only.

Two types of conventional DGs are studied: unity power factor DGs, used in case 3, and non-unity power factor, used in case 4. The proposed WO approach is applied for cases 3 and 4, and the findings are compared with other programmed optimization techniques, including Particle Swarm Optimizer (PSO), Water Cycle Algorithm (WCA), Backtracking Search Algorithm (BSA), and Bitterling Fish Optimizer (BFO). To provide a fair comparison, all the studied techniques as well as the proposed WO approach have the same population size and the same maximum number of iterations. In addition, each decision variable has the same lower and upper limits for all considered approaches. The number of DGs used in this study is limited to four DGs because it is found that the value of MOF is slightly reduced as the number of DGs increases in the system, and there would not be a significant improvement with introducing more than four DGs into the network.

The obtained results of case 3 are organized in

Table 6. Optimal results of case 3 after installing DGs with different number in the 33-bus system.

DG Number	Item	PSO (Studied)	BSA (Studied)	WCA (Studied)	BFO (Studied)	WOA (Proposed)
2	(Bus, DG size (MW))	(14, 0.8905)	(14, 0.8766)	(14, 0.8905)	(14, 0.8766)	(14, 0.8410)
		(32, 0.8041)	(31, 0.8471)	(32, 0.8041)	(31, 0.8471)	(30, 0.9549)
	$E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$	1734.61	1723.21	1734.61	1723.21	1703.19
	$TVD \left(\mathrm{p}.\mathrm{u}.\right)$	13.93	13.85	13.93	13.85	13.76
	$\%$$\mathrm{Reduction} \mathrm{in} E_{loss}$	50.63%	50.96%	50.63%	50.96%	51.53%
	$MOF$	0.5778	0.5754	0.5778	0.5754	0.5744
	$\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$	0.9153	0.9151	0.9153	0.9151	0.9147
	$\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$	0.7017	0.7013	0.7017	0.7013	0.7
	Convergence time (s)	106.58	101.68	104.30	125.82	108.48
3	(Bus, DG size (MW))	(8, 0.7661)	(14, 0.6602)	(8, 0.7394)	(14, 0.8133)	(14, 0.8317)
		(14, 0.6217)	(6, 1.0632)	(14, 0.6217)	(25, 0.7242)	(24, 0.7868)
		(32, 0.6449)	(32, 0.5248)	(31, 0.6821)	(30, 0.8781)	(31, 0.8015)
	$E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$	1595.22	1568.63	1591.95	1557.84	1555.2
	$TVD \left(\mathrm{p}.\mathrm{u}.\right)$	13.355	13.1915	13.32	13.2477	13.2375
	$\%$$\mathrm{Reduction} \mathrm{in} E_{loss}$	54.6%	55.36%	54.69%	55.66%	55.74%
	$MOF$	0.5569	0.5528	0.5565	0.5519	0.5491
	$\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$	0.9143	0.9141	0.9143	0.9148	0.915
	$\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$	0.6988	0.6981	0.6987	0.7004	0.7009
	Convergence time (s)	115.84	109.80	111.64	135.42	117.28
4	(Bus, DG size (MW))	(8, 0.7567)	(6, 0.7375)	(8, 0.6228)	(6, 1.0441)	(6, 0.7659)
		(14, 0.6006)	(14, 0.6996)	(13, 0.6679)	(14, 0.6898)	(14, 0.6994)
		(25, 0.6123)	(25, 0.5281)	(25, 0.5567)	(25, 0.4929)	(24, 0.6535)
		(30, 0.8391)	(30, 0.6713)	(31, 0.6756)	(32, 0.4515)	(31, 0.6045)
	$E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$	1536.27	1475.08	1483.86	1467.56	1462.05
	$TVD \left(\mathrm{p}.\mathrm{u}.\right)$	12.4946	12.8768	12.9023	12.8717	12.8194
	$\%$$\mathrm{Reduction} \mathrm{in} E_{loss}$	56.28%	58.02%	57.77%	58.23%	58.39%
	$MOF$	0.5431	0.539	0.5401	0.5371	0.5349
	$\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$	0.9174	0.9154	0.9136	0.916	0.9154
	$\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$	0.7084	0.702	0.6967	0.704	0.702
	Convergence time (s)	120.00	116.62	114.86	140.68	122.28

with various numbers of unity power factor DGs. Regarding the results shown in Table 6, it is noted that the proposed WO approach outperformed other studied optimization techniques, especially with a higher number of decision variables (i.e., using three DGs and four DGs). The least value of MOF and the best percentage reduction in energy losses are achieved via the WO algorithm using four DGs, with values equal to 0.5349 and 58.39%, respectively. The variation of MOF with iteration number for three DGs’ penetration of unity power factor is shown in Figure 17; the proposed WO response is the best while searching for best fitness (least value) compared to other studied approaches. In addition, it is observed from Figure 17 that the WO algorithm avoids premature convergence to the local minimum, thanks to effective balancing between global exploration and local exploitation.

After installing the DGs, it is crucial to assess the network’s voltage profile. Figure 18 shows the hourly voltage profile of the 33-bus system with four unity power factor DGs. Compared to the voltage profile of case 1 (see Figure 13), a clear improvement is achieved using the proposed methodology. Also, the minimum voltage magnitude is improved from 0.8586 p.u. to 0.9154 p.u. at bus 18 during the 16th hour. Moreover, it is noted that the voltage profile exceeds the substation voltage (1 p.u.) without violating the maximum voltage limit (1.05 p.u.) during off-peak hours, and that is because of light loading and the occurrence of reverse power flow.

In case 4, the optimal results are obtained and arranged in

Table 7. Optimal results of case 4 after installing different numbers of DGs in the 33-bus system.

DG Number	Item	PSO (Studied)	BSA (Studied)	WCA (Studied)	BFO (Studied)	WOA (Proposed)
2	$(\mathrm{Bus}, \mathrm{DG} \mathrm{size} (\mathrm{MW})/{\mathrm{D}\mathrm{G}}_{\mathrm{P}\mathrm{F}})$    $E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$ TVD$(\mathrm{p}.\mathrm{u}.)$ % Reduction in Eloss $MOF$  $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u})$ Convergence time (s)	(13, 0.678/0.938) (30, 0.903/0.721) 1094.09 12.639 68.86% 0.4787 0.9179 0.7105 122.46	(13, 0.807/0.934) (30, 0.935/0.732) 1070.10 12.604 69.55% 0.4744 0.9258 0.7347 118.45	(14, 0.814/0.923) (30, 0.892/0.687) 1091.63 12.735 68.93% 0.4771 0.9316 0.7531 124.24	(14, 0.741/0.952) (30, 0.943/0.750) 1067.85 12.606 69.61% 0.47263 0.9244 0.7303 134.61	(14, 0.798/0.964) (30, 0.945/0.750) 1063.95 12.607 69.72% 0.47204 0.9266 0.7371 125.63
3	$(\mathrm{Bus}, \mathrm{DG} \mathrm{size} (\mathrm{MW})/{\mathrm{D}\mathrm{G}}_{\mathrm{P}\mathrm{F}})$      $E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$ TVD$(\mathrm{p}.\mathrm{u}.)$ % Reduction in Eloss $MOF$ $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u})$ Convergence time (s)	(14, 0.647/0.953) (25, 0.551/0.790) (30, 0.864/0.676) 927.9 12.219 73.59% 0.4486 0.9211 0.72 138.90	(13, 0.818/0.937) (25, 0.618/0.797) (30, 0.895/0.775) 905.29 12.374 74.24% 0.4464 0.9275 0.74 135.88	(13, 0.765/0.951) (25, 0.400/0.666) (30, 0.936/0.770) 915.64 12.258 73.94% 0.4477 0.9235 0.727 141.56	(14, 0.775/0.964) (25, 0.608/0.908) (30, 0.903/0.738) 884.12 12.284 74.84% 0.44107 0.927 0.739 151.46	(14, 0.768/0.964) (24, 0.763/0.909) (30, 0.889/0.736) 870.23 12.276 75.23% 0.43947 0.927 0.739 142.23
4	$(\mathrm{Bus}, \mathrm{DG} \mathrm{size} (\mathrm{MW})/{\mathrm{D}\mathrm{G}}_{\mathrm{P}\mathrm{F}})$        $E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$ TVD$(\mathrm{p}.\mathrm{u}.)$ % Reduction in Eloss $MOF$ $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u})$ Convergence time (s)	(8, 0.495/0.797) (13, 0.633/0.976) (25, 0.586/0.913) (31, 0.731/0.790) 843.85 12.31 75.98% 0.43761 0.9244 0.7303 156.89	(8, 0.390/0.893) (14, 0.602/0.953) (24, 0.832/0.971) (30, 0.743/0.701) 805.78 12.3 77.07% 0.431 0.9258 0.7347 154.55	(8, 0.639/0.906) (14, 0.584/0.971) (24, 0.701/0.901) (31, 0.622/0.717) 817.19 12.33 76.74% 0.43232 0.9263 0.7362 162.43	(6, 0.813/0.883) (14, 0.639/0.97) (25, 0.535/0.904) (31, 0.556/0.705) 806.78 12.29 77.04% 0.43062 0.9258 0.7346 169.88	(6, 0.703/0.936) (14, 0.649/0.968) (24, 0.659/0.904) (30, 0.652/0.666) 786.15 12.35 77.63% 0.42798 0.9268 0.7379 158.20

when installing a single and multiple non-unity power factor DG. The proposed WO approach proved its superiority to achieve the best results compared to other studied algorithms. In addition, it is found that DGs with optimal power factor give better results compared with unity power factor DGs (refer to Table 6 and Table 7). Installing four DGs with optimal locations, ratings, and power factors, obtained via the WO approach, gives the least value of energy losses, reaching 786.15 kWh with a percentage reduction of 77.63%.

Figure 19 shows the variation of MOF with iteration number for three DGs’ penetration of non-unity power factor; the response of WO is also the best compared to other studied approaches. The hourly voltage profile of the 33-bus system with four non-unity power factor DGs is presented in Figure 20. It is clear that case 4 gives better improvement in voltage profile than case 3 (see Figure 18 and Figure 20). Regarding the results in Table 6 and Table 7, the minimum voltage after installing four DGs, based on the WO approach, is enhanced to 0.9268 p.u. at bus 18 during the 16th hour in case 4, compared to 0.9154 p.u. in case 3.

The variations of total active power loss and substation power during the day for cases 3 and 4, after installing four DGs based on the WO approach, compared to case 1, are shown in Figure 21 and Figure 22. As shown in Figure 21, the maximum level of active power loss reached 410.7 kW at the 16th hour in case 1, but decreased severely to 132.07 kW in case 3, and then again slightly to 100.63 kW in case 4. Figure 22 reveals that the maximum values of substation power in cases 3 and 4 are respectively decreased to values of 2808.04 kW and 2866.31 kW, compared to 5644.5 kW in case 1 at the 17th hour. Also, it is noted that substation power is reversed during off-peak hours due to light loading and surplus generation of DGs.

Moreover, the performance of the proposed WO approach is investigated via calculating statistical measures like minimum, maximum, mean, median, variance, and standard deviation. The WO algorithm is executed 20 times for the 33-bus system using four DGs in case 4, and its performance is compared against other programmed approaches, including BFO, WCA, BSA, and PSO. The statistical analysis verified the robustness and efficacy of WO and its ability to get either the final best value of objective function or a very close result to it every time with the least values of variance and standard deviation, as shown in

Table 8. Statistical comparison of various optimizers applied on 33-bus system using four DGs in case 4.

Vehicle Type	PSO	BSA	WCA	BFO	WO
Minimum	0.4376	0.431	0.4323	0.4306	0.42798
Maximum	0.4493	0.4448	0.4454	0.4385	0.4361
Mean	0.4419	0.4380	0.4397	0.4357	0.4294
Median	0.4395	0.4386	0.4399	0.4359	0.4285
Variance	1.969 × 10−5	1.697 × 10−5	1.782 × 10−5	5.197 × 10−6	4.979 × 10−6
Standard Deviation	4.437 × 10−3	4.119 × 10−3	4.222 × 10−3	2.28 × 10−3	2.231 × 10−3

. The minimum evaluated MOF obtained by various optimization techniques throughout 20 independent runs is illustrated in Figure 23. Only case 4 is considered for this investigation using four DGs with non-unity power factors penetrated into the 33-bus system. As noted from Figure 23, the proposed WO algorithm outperformed the BFO, BSA, WCA, and PSO in providing highly consistent results.

4.3. IEEE 69-Bus System

In addition, the proposed WO and other studied optimization approaches are investigated on the large-scale 69-bus radial distribution network. The hourly bus voltage profile and stability profile for case 0 are shown in Figure 24 and Figure 25, respectively. It is noted that the lowest levels of voltage magnitude and stability index for case 0 occur at bus 65 during the 17th-hour interval, with values equal to 0.917 p.u. and 0.707 p.u., respectively. PEV charging load demand is integrated into the 69-bus network at each residential bus to examine its impacts considering PC and OPC scenarios in cases 1 and 2, respectively.

Table 9. Comparison of performance between case 0, case 1, and case 2 for 69-bus system.

Parameter	Case 0	Case 1	Case 2
$E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$	2911.95	4172	3760.33
$TVD (\mathrm{p}.\mathrm{u}.)$	30.58	38.61	38.57
$\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$	0.9092 (65th bus, 17th hour)	0.8874 (27th bus, 16th hour)	0.9092 (65th bus, 17th hour)
$\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$	0.6834 (65th bus, 17th hour)	0.6202 (27th bus, 16th hour)	0.6834 (65th bus, 17th hour)
$\mathrm{Daily} \mathrm{substation} \mathrm{energy} (\mathrm{k}\mathrm{W}\mathrm{h})$	69,202.95	86,899.58	86,855.38
$\mathrm{Maximum} \mathrm{substation} \mathrm{power} (\mathrm{k}\mathrm{W})$	4327 (17th hour)	7391 (16th hour)	5183 (1st hour)

summarizes a comparative analysis between case 0, case 1, and case 2. It is noted that the highest energy losses occur in case 1, reaching a value equal to 4172 kWh. In addition, case 1 scored the highest level of substation power, reaching a value of 7391 kW during the 16th-hour interval. Moreover, case 1 gives the highest negative influences on the 69-bus network regarding voltage deviation.

Figure 26 shows the hourly bus voltage profile for case 1 with the PC scenario. It is observed that the minimum voltage occurs at bus 27 during the 16th-hour interval, with a value equal to 0.8874 p.u. (see Table 9). Also, Figure 26 shows that, during peak hours (15th, 16th, and 17th), in case 1, a number of residential buses (from bus 15 to bus 27) and industrial buses (from bus 61 to bus 65) have voltage magnitudes below 0.9 p.u. that could pose a serious technical issue regarding power quality at these buses. Hence, DGs are integrated into the 69-bus network with a PC scenario to reduce network losses, enhance voltage stability, and improve the bus voltage profile. The proposed WO is implemented to optimally integrate four DGs, either with unity power factor (case 3) or with non-unity power factor (case 4).

The optimal results are obtained and arranged in

Table 10. Optimal results of case 3 and 4 after installing four DGs in the 69-bus system.

Case	Item	PSO (Studied)	BSA (Studied)	WCA (Studied)	BFO (Studied)	WOA (Proposed)
3	(Bus, DG size (MW))       $E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$ TVD (p.u.) % Reduction in $E_{loss}$ MOF $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$ Convergence time (s)	(11, 0.3464) (21, 0.2869) (20, 0.3172) (61, 1.5408) 1660 16.664 60.21% 0.50509 0.9289 0.7447 425.98	(11, 0.8332) (20, 0.4303) (61, 0.8686) (64, 0.4948) 1626.9 16.595 61% 0.50252 0.9254 0.7333 375.65	(12, 0.5325) (22, 0.4398) (57, 0.1327) (61, 1.3877) 1625.7 16.6 61.03% 0.50303 0.9266 0.7371 380.46	(11, 0.8332) (20, 0.4303) (61, 0.8686) (64, 0.4948) 1626.9 16.595 61% 0.50252 0.9254 0.7333 482.72	(12, 0.5496) (21, 0.4467) (61, 1.0959) (64, 0.3616) 1622.3 16.63 61.11% 0.50055 0.9268 0.7378 450.44
4	$(\mathrm{Bus}, \mathrm{DG} \mathrm{size} (\mathrm{MW})/{\mathrm{D}\mathrm{G}}_{\mathrm{P}\mathrm{F}}$         $E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$ TVD (p.u.) % Reduction in $E_{loss}$ MOF $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$ Convergence time (s)	(12, 0.719/0.969) (24, 0.369/0.931) (58, 0.131/0.997) (61, 1.323/0.799) 804.95 16.81 80.71% 0.39627 0.9338 0.7603 455.82	(11, 0.607/0.929) (20, 0.483/0.970) (27, 0.035/0.810) (61, 1.424/0.825) 789.61 16.81 81.07% 0.39422 0.9338 0.7604 395.23	(12, 0.719/0.969) (24, 0.369/0.931) (58, 0.131/0.997) (61, 1.323/0.799) 804.95 16.81 80.71% 0.39627 0.9338 0.7603 408.00	(11, 0.56/0.928) (17, 0.339/0.967) (24, 0.216/0.956) (61, 1.425/0.824) 783 16.85 81.23% 0.39349 0.9344 0.7623 505.56	(11, 0.61/0.928) (21, 0.502/0.964) (61, 1.147/0.819) (64, 0.247/0.841) 779.62 16.746 81.3% 0.39256 0.9335 0.7594 482.00

, which proved the superiority of the proposed WO approach to achieve better results in large-scale radial networks, compared to other studied algorithms. Installing four DGs with optimal locations, ratings, and power factors, based on the WO approach, will reduce the daily energy losses from 4172 kWh to 779.62 kWh, with a percentage reduction equal to 81.3%. Figure 27 shows the variation of MOF with iteration number for four DGs’ penetration of non-unity power factor The response of WO is best compared to other studied approaches. The hourly voltage profiles of the 69-bus system with four unity and non-unity power factor DGs are presented in Figure 28 and Figure 29, respectively. It is clear that case 4 provides a better improvement in voltage profile than case 3 (see Figure 28 and Figure 29).

4.4. ShC-D8 System in Egypt

Moreover, the proposed WO and other studied optimization approaches are applied on the ShC-D8 real system in Egypt. This system is supplied from a 66–22 kV substation feeding 45 load buses via three feeders; the configuration of such a system is shown in Figure 30. The data of all branches and buses is given in [45,46], with base values of 100 MVA and 22 kV. All buses in the system are considered residential buses with time-varying demand, as given in [46], representing the Egyptian daily load curve, in order to achieve more realistic findings.

The hourly bus voltage profile and stability profile for case 0 are shown in Figure 31 and Figure 32, respectively. It is noted that the lowest levels of voltage magnitude and stability index for case 0 occur at bus 18 during the 22nd-hour interval, with values equal to 0.9446 p.u. and 0.7962 p.u., respectively. To investigate the impacts of uncoordinated charging of electric vehicles on the system, extra charging load demand of PEVs is integrated into the system at each residential bus, considering PC and OPC scenarios in cases 1 and 2, respectively.

Table 11. Comparison of performance between case 0, case 1, and case 2 for ShC-D8 system.

Parameter	Case 0	Case 1	Case 2
$E_{loss} (\mathrm{k}\mathrm{W}\mathrm{h})$	2508.2	5479.3	4984.7
$TVD (\mathrm{p}.\mathrm{u}.)$	19.01	26.75	26.04
$\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$	0.9446 (18th bus, 22nd hour)	0.9042 (18th bus, 16th hour)	0.9226 (18th bus, 2nd hour)
$\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$	0.7962 (18th bus, 22nd hour)	0.6683 (18th bus, 16th hour)	0.7246 (18th bus, 2nd hour)
$\mathrm{Daily} \mathrm{substation} \mathrm{energy} (\mathrm{k}\mathrm{W}\mathrm{h})$	96,774	125,397.4	125,624.85
$\mathrm{Maximum} \mathrm{substation} \mathrm{power} (\mathrm{k}\mathrm{W})$	6658.4 (22nd hour)	9767.5 (16th hour)	7858.5 (2nd hour)

presents a comparative examination of cases 0, 1, and 2. It is noted that case 1 causes the worst system performance regarding energy losses, voltage deviation, and voltage stability. In addition, case 1 scored the highest level of substation power, reaching a value of 9767.5 kW during the 16th-hour interval. The hourly bus voltage profile for case 1 with the PC scenario is shown in Figure 33. It is observed that the minimum voltage occurs at bus 18 during the 16th-hour interval, with a value equal to 0.9042 p.u. (see Table 11). DGs are integrated into this real system to reduce the negative impacts resulting from PEV penetration, considering the worst charging scenario (i.e., case 1). The proposed WO and other programmed optimization algorithms are implemented to optimally integrate four DGs, either with unity power factor (case 3) or with non-unity power factor (case 4).

Referring to the optimal results listed in

Table 12. Optimal results of case 3 and 4 after installing four DGs in ShC-D8 system.

Case	Item	PSO (Studied)	BSA (Studied)	WCA (Studied)	BFO (Studied)	WO (Proposed)
3	$(\mathrm{Bus}, \mathrm{DG} \mathrm{size} (\mathrm{MW}))$ Eloss (kWh) TVD (p.u.) % Reduction in $E_{loss}$ $MOF$ $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$ Convergence time (s)	(16, 1.6496) (26, 0.8889) (31, 0.8623) (43, 1.3353) 2458.07 13.006 55.14% 0.5784 0.9381 0.7743 275.55	(9, 0.6425) (14, 1.4122) (30, 1.4404) (44, 1.4008) 2416.18 12.903 55.9% 0.5776 0.9348 0.7637 248.22	(15, 1.6342) (30, 1.4684) (41, 0.8595) (46, 0.72) 2448.53 13.023 55.31% 0.5795 0.9352 0.7649 267.42	(7, 1.1041) (16, 1.1766) (29, 1.5388) (43, 1.3424) 2394.45 12.934 56.3% 0.5742 0.9358 0.7671 305.22	(10, 1.1922) (17, 0.8611) (29, 1.5441) (43, 1.3415) 2376.03 12.875 56.64% 0.5691 0.9371 0.7711 284.21
4	$(\mathrm{Bus}, \mathrm{DG} \mathrm{size} (\mathrm{MW})/{\mathrm{D}\mathrm{G}}_{\mathrm{P}\mathrm{F}})$ Eloss (kWh) TVD (p.u.) % Reduction in $E_{loss}$ $MOF$ $\mathrm{Minimum} \mathrm{voltage} (\mathrm{p}.\mathrm{u}.)$ $\mathrm{Minimum} SI (\mathrm{p}.\mathrm{u}.)$ Convergence time (s)	(14, 1.457/0.768) (25, 0.900/0.770) (31, 0.869/0.827) (43, 1.255/0.806) 1184.43 12.321 78.38% 0.4578 0.9431 0.8205 309.92	(8, 1.068/0.776) (16, 0.879/0.761) (29, 1.430/0.806) (44, 1.262/0.805) 1142.48 12.356 79.14% 0.4513 0.9453 0.8217 262.66	(9, 1.428/0.896) (16, 0.599/0.561) (30, 1.306/0.757) (43, 1.125/0.756) 1154.92 12.472 78.92% 0.4531 0.9427 0.8175 289.12	(10, 1.230/0.770) (17, 0.593/0.770) (29, 1.427/0.806) (42, 1.346/0.804) 1134.85 12.329 79.29% 0.4496 0.9465 0.8253 342.64	(9, 1.068/0.779) (16, 0.819/0.758) (29, 1.431/0.806) (43, 1.263/0.805) 1109.51 12.302 79.75% 0.4485 0.9481 0.8289 321.26

, it is noted that installing non-unity PF DGs gives the best results compared to unity PF DGs. Installing four DGs with optimal locations, ratings, and power factors, based on the WO approach, will reduce daily energy losses from 5479.3 kWh to 1109.51 kWh, with a percentage reduction equal to 79.75%.

The hourly voltage profiles of the ShC-D8 system with four DGs penetration, considering cases 3 and 4, are shown in Figure 34 and Figure 35, respectively. By comparing the two figures, it is clear that case 4 provides the best improvement in voltage profile. The least values of voltage magnitude occur at bus 18 during the 16th-hour interval for both cases 3 and 4, with values equal to 0.9371 p.u. and 0.9481 p.u., respectively.

5. Conclusions

This paper presents an application of the recently developed WO approach to determine near-optimal ratings and locations of DGs in RDNs incorporated with PEVs. The main objective is to minimize the network’s daily energy losses, improve the daily voltage profile, and enhance voltage stability, considering various constraints involving power balance, buses’ voltages, and line flow. The PEV load demand is added to the basic daily load curve of the network based on two charging models, PC and OPC. The daily basic load is modeled as a time-varying, voltage-dependent load consisting of residential, industrial, and commercial demands. Two types of DG units are investigated in this study, unity power factor DGs (supplying only active power) and non-unity power factor (supplying both active and reactive power).

The proposed optimization model integrates both WO and MOF, along with a repetitive BFLF load flow method, and the simulations are performed using MATLAB 2016a software. It is found that the PC scenario results in the highest negative impacts on RDNs compared to the OPC scenario. Hence, DGs are considered in this study to be optimally allocated with PEV charging load under PC only. From the obtained results and detailed analysis, it is concluded that the performance of RDNs (in terms of energy loss minimization, voltage profile improvement, and voltage stability enhancement) in the presence of extra PEV loads is improved with near-optimal DG allocation and sizing. The proposed WO algorithm shows an efficient capability to solve such kinds of optimization problems. Moreover, the simulation results verify the superiority and efficacy of the proposed WO approach in comparison with other programmed algorithms of Particle Swarm Optimizer (PSO), Water Cycle Algorithm (WCA), Backtracking Search Algorithm (BSA), and Bitterling Fish Optimizer (BFO) in terms of convergence curve and solution quality.

Regarding the findings of the 33-bus system, the proposed WO approach succeeded in achieving the lowest energy losses of 786.15 kWh in case 4, with the installation of four DGs with non-unity power factor on buses 6, 14, 24, and 30, and with active power ratings of 0.703 MW, 0.649 MW, 0.659 MW, and 0.652 MW, respectively. In addition, WO gives the least value of MOF of 0.42798 in case 4 compared to PSO, WCA, BSA, and BFO with values equal to 0.43761, 0.43232, 0.431, and 0.43062, respectively. Regarding the 69-bus network, the highest reduction in energy losses is achieved by applying the proposed WO approach in case 4, with a percentage of 81.3% with respect to case 1. The proposed WO gives the lowest value of energy losses of 779.62 kWh in case 4 using four DGs, compared to PSO, WCA, BSA, and BFO with values equal to 804.95, 804.95, 789.61, and 783, respectively. Regarding the ShC-D8 system in Egypt, installing four DGs with optimal locations, ratings, and power factors based on the WO approach provides the highest reduction in daily energy losses, from 5479.3 kWh to 1109.51 kWh, with a percentage reduction of 79.75%. The BFO algorithm comes in second place with a value equal to 79.29%. Additionally, the best value of minimum voltage level (0.9481 p.u. at bus 18) is achieved via the WO approach by installing four DGs with non-unity power factors on buses 9, 16, 29, and 43, with active power ratings of 1.068, 0.819, 1.431, and 1.263, respectively.