A Novel Stochastic Optimizer Solving Optimal Reactive Power Dispatch Problem Considering Renewable Energy Resources

Abstract

Optimal Reactive Power Dispatch (ORPD is thought of as a noncontinuous, nonlinear global optimization problem. Within the system’s constraints, the ORPD manages to accomplish the reactive power flow. Due to its more intricate linkage of variables, the reactive power issue is more challenging to resolve than the optimum power flow issue. With the existence of renewable energy resources (RERs), solving the ORPD problem to attain the most stable and secure system condition has become a more challenging task. The goal of this article is to solve the objective function of ORPD combined with RERs using a metaheuristic novel optimizer named the African Vultures Optimization Algorithm abbreviated by (AVOA), where the formulation of the ORPD issue including minimization of three single objective functions as follows, voltage deviation, system operating cost, and real power loss, is introduced and also transmission power loss minimization is embraced with the simultaneous incorporation of the optimal renewable energy resources (RERs). Where the ORPD problem complexity grows exponentially with a mixture of continuous and discrete control variables, two distinct continuous and discrete types of optimization variables are considered, and the proposed single objective functions that meet different operating constraints are then transformed into a coefficient multi-objective ORPD problem and elucidated using the weighted sum approach. To validate the suggested algorithm’s effectiveness in addressing the ORPD issue, it is evaluated on three standard IEEE networks: the IEEE-30 bus small-scale network, the IEEE-57 bus medium-scale network, and the IEEE-118 bus large-scale network using different scenarios and the outcomes are compared to these other popular optimization techniques. The findings show that the suggested AVOA algorithm provides an efficient and sturdy high-quality solution for tackling ORPD situations and vastly enhances the overall system performance of power at all scales.

1. Introduction

The perfect and optimal solution to one of the most critical problems in the planning and operation of electric power systems known as (ORPD) takes great attention and continuous efforts and has piqued the interest of researchers due to its critical role in modern energy management systems, where ORPD is an optimum reactive power distribution in the power system that minimizes a predetermined objective function while accommodating a variety of constraints [1,2].

To accomplish these objectives, the optimal adaptation of specified control variables such as transformer tapping settings, generator buses’ voltages, and reactive power resource distribution, such as the VAR shunt compensator, is needed. Once the problem of ORPD is resolved, the equality constraints and the inequality constraints are kept inside their permissible operational bounds [3,4].

Renewable energy resources, such as wind energy, solar energy, and microturbine generators have helped lower system power losses while also enhancing power system security and dependability during the previous decade. Furthermore, RESs have a significant influence on the energy market since they must specify the system’s required reactive power to achieve the specified objective functions [5].

The optimization of reactive power issues deals mainly with an objective to minimalize the absolute value of total voltage deviation (TVD) leading to enhancement in the voltage profile, the reduction of reactive power cost, and reduction of the electric network active power losses (Ploss), consequently, reducing the total operating cost. Therefore, (TVD), (Cost), and (Ploss) minimization have thus been selected as the study’s objective functions. In recent decades, several approaches with excellent convergence characteristics, such as (DP) (IPM), (LP), and (QP) have been developed and employed to address the ORPD issue. These approaches can alternatively be classified as traditional optimization methods or mathematical optimization methods. However, these approaches have certain limits and obstacles when dealing with discrete variables and non-differentiable objective functions in ORPD. One of the most negative aspects of these approaches is that they necessitate many iterations of numbers and extensive calculations, so taking a lot of time to provide answers without rounding and assumptions leading to municipal perfect solutions. As a result, these strategies were unable to cope with non-linear and complicated situations such as the ORPD issue. As an outcome, more modern optimization methods have been continuously developed and, as a consequence, there are a large variety of optimization solvers used in various power system applications, such as the ORPD issue, leading to improvements in the downsides of traditional techniques for determining ORPD solutions [5,6,7]. Recently, metaheuristic scheme exploitation and development have proven successful results in resolving optimum RPD issues [2].

Table 1 shows what has been accessed and searched recently on the topic of ORPD problems, where many optimization algorithms are being used for solving the ORPD issues, which include various single objective functions such as minimizing voltage deviation, decreasing active power loss, voltage stability index improvement, minimizing operating cost, minimizing fuel cost, L-index, emission, and combining many objectives as a multi-objective function [8].

Table 1. Optimization Methods for ORPD Problem Solution.

Research References	Year	Applied Methodology for ORPD Solution	Validation Test System							Optimization Objectives
			Benchmark Functions	IEEE 14—Bus System	IEEE 30—Bus System	IEEE 39—Bus System	IEEE 57—Bus System	IEEE 118—Bus System	IEEE 300—Bus System	Active Power Losses Minimization	Voltage Deviation Minimization	Voltage Stability Index Improvement	Cost Minimization	Emission Minimization	Uncertainty Impact
[1]	2019	(A-CSOS)			√					√	√
[2]	2020	(FO-DPSO)			√		√			√	√
[3]	2019	Hybrid fuzzy-JAYA optimization algorithm		√	√			√		√	√
[4]	2019	(EP)			√					√
[5]	2021	(SMA)			√			√		√	√	√	√		√
[6]	2020	(CBA)		√		√	√	√	√	√	√	√
[7]	2020	(GWO) algorithm			√					√	√	√	√
[9]	2019	(IRCGA)	√		√		√	√		√	√	√
[10]	2019	(MSCA)			√					√	√	√
[11]	2019	(IALO)			√		√	√		√	√	√
[12]	2020	(ISSO)			√			√		√	√	√
[13]	2020	(PCA-RCGA)			√					√					√
[14]	2020	(LAPO)			√					√	√				√
[15]	2020	(FPSOGSA)			√		√			√	√
[16]	2020	(FPSOGSA)			√		√			√	√
[17]	2020	(FPSOGSA)			√					√	√	√			√
[18]	2020	(HPSOFA)		√	√					√	√
[19]	2020	JAYA algorithm		√	√		√	√		√					√
[20]	2020	(MPFA)					√	√		√
[21]	2020	(WWO) algorithm			√					√
[22]	2020	JAYA algorithm		√	√					√	√
[23]	2020	(MPA)			√					√	√				√
[24]	2021	(MOPSO) algorithm		√	√					√	√				√
[25]	2021	(IHBO)			√		√	√		√	√	√
[26]	2021	(MVMO) algorithm			√		√			√
[27]	2021	JAYA algorithm			√		√			√	√				√
[28]	2021	Modified JAYA algorithm			√		√			√	√				√
[29]	2021	JAYA algorithm		√	√		√	√		√
[30]	2021	Rao-3 Algorithm			√		√	√		√	√				√
[31]	2021	(ICOA)			√		√	√		√	√	√
[32]	2021	A Modified Inertia Weight of Particle Swarm Optimization (PSO)		√	√					√
[33]	2021	(IDE) algorithm	√	√	√					√
[34]	2022	(CTFWO) algorithm			√		√			√	√
[35]	2022	(GWFA)		√	√	√				√	√
[36]	2022	(FA)		√						√	√				√
[37]	2022	(IMPA)			√					√	√	√			√
[38]	2022	Coot Algorithm (CA)					√			√	√			√
[39]	2022	(HPSOBA)			√		√			√	√
[40]	2022	(SCA)		√	√		√			√
[41]	2022	(AMOAIA)			√					√	√	√

Based on the recent research papers listed in Table 1, which is concerned with and addresses the subject of ORPD research, some of the details on which the research choices were based and the most important points of motivation for research in these selected aspects were listed, based on the identification of the most common study aspects as percentages of the total research papers in the table, estimated at (40 research papers).

From the side of the study relating to the validation test system, there are (12) research papers, with a percentage of (30%) based on the IEEE 14—Bus System; there are (36) research papers with a percentage of (90%) based on the study of the IEEE 30—Bus System, (2) research papers with a percentage of (5%) based on the IEEE 39—Bus System, (19) research papers with a percentage of (47.5%) based on the IEEE 57—System, (12) research papers with a percentage of (30%) based on the IEEE 118—Bus System, and there is only (1) research Paper with a percentage of (2.5%) based on the IEEE 300—Bus System. Therefore, the aspects of the study were chosen according to the three highest aspects according to the IEEE (30, 57, and 118) bus validation test systems.

The study according to the optimization objective functions was identified and selected with the same approach, there are (40) research papers with a percentage of (100%) based on active power loss minimization, (30) research papers with a percentage of (75%) based on voltage deviation minimization, (12) research papers with a percentage of (30%) based on voltage stability index improvement, (2) research papers with a percentage of (5%) based on cost minimization, and there is only (1) research paper with a percentage (2.5%) based on emission minimization. Other than specifying the cost minimization, the aspects of the study were chosen according to the highest aspects of the study including active power loss minimization and voltage deviation. Thus, three aspects of the study were selected as single- and multi-optimization objective functions.

The suggested AVOA algorithm is implemented on the IEEE standard testing power systems of the IEEE (30, 57, and 118) bus in this article to showcase its efficiency in resolving the ORPD problem. The analyzed objective functions in this research are the minimization of active power loss, the system’s overall operating cost minimization, and total voltage deviation minimization. According to the simulation findings, the AVOA algorithm is more capable, and all results confirm that it is more convenient than the compared suggested metaheuristic algorithms.

In summary, the following are the primary contributions of this research:

• The African Vultures Optimization Algorithm (AVOA) is proposed for the first time to solve a real-world optimization problem such as the ORPD problem.
• In the AVOA, vulture populations are designed with a Levy mutation, which can help to avoid falling into the local minima.
• The suggested AVOA is used to divide the optimization problem into three distinct objective functions (voltage deviation, operating cost, and power loss) as single- and multi-optimization objective functions considering continuous and discrete types of optimization control variables.
• The standard IEEE (30-bus, 57-bus, and 118-bus) testing systems are employed in this research to analyze the impact and demonstrate the consistency of the suggested AVOA algorithm to reach the near-optimal solutions to the self-posed problem and its application scalability on a challenging testing process from the metaheuristic literature.
• The effect of RES insertion on solving the ORPD is analyzed by two methods. The first method is to find the optimal solutions in two steps. The first method is finding the optimal allocation of RES for standard IEEE then determining the optimal solution to the ORPD challenge for the modified IEEE system after inserting the RES in the first step. The second method to find the optimal solution in one step by finding the optimal RES allocation and the rest of the optimal control variables, which means the control variables are increased (problem dimensions increased).
• Finally, IEEE test systems incorporated with the RERs are presented to investigate the dominance of the analyzed AVOA algorithm among other metaheuristic methods for finding the best objective functions solution with fast convergence characteristics.

The remaining parts of this paper are structured as follows. Part 2 describes the ORPD issue mathematical formulations, Part 3 discusses the suggested AVOA approach, Part 4 describes the discussions of all simulation results and covers all proposed case studies. Finally, Part 5 gives the paper’s conclusion.

2. ORPD Problem Formulation

The ORPD issue attempts to optimize an objective function by making a good change to power system control variables while complying with various constraints. Overall, the problem of optimization can be expressed as follows [42,43,44,45,46,47]:

$$\begin{gathered} Minimize F\left(x,u\right) \\ \mathrm{Subject} \mathrm{to}:\{\begin{array}{c}{A}_j\left(x,u\right)=0, j=1,\dots ,c\\ B_j\left(x,u\right)\le 0, j=1,\dots ,d\end{array} \end{gathered}$$

where the objective function is $F\left(x,u\right)$, $x$ signifies the dependent variables vector (state variables), $u$ signifies the independent variables vector (control variables), and $A_j$ and $B_{j }$ are equality and inequality constraints, respectively. The $c$ and $d$ variables signify the number of equality constraints and the number of inequality constraints, respectively.

$$x=\left[P_{G1},V_{L1 }\dots V_{L,NPQ },Q_{G,1}..Q_{G,NG} ,S_{TL,1 }..S_{TL,NTL}\right]$$

(1)

where the power of slack bus is signified by$P_{G1}$, the voltage of the load bus is signified by$V_{L }$, the reactive power output of the generator is signified by$Q_G$, the transmission line’s apparent power flow is signified by $S_{TL }$, $NTL$ reflects the transmission lines number, $NG$ reflects the generation buses number, and $NPQ$ reflects the load buses number.

$$u=\left[P_{G,2}..P_{G,NG},V_{G,1 }..V_{G,NG },Q_{C,1}..Q_{C,NC} ,T_1..T_{NT}\right]$$

(2)

where the generator’s active power output is signified by $P_G$, the generation bus voltage is signified by $V_{G }$, the shunt compensator’s injected reactive power is signified by $Q_C$, $NC$ signifies the shunt compensator bank units’ number, and $NT$ signifies the transformer’s number; $T$ signifies the transformer’s tapping settings.

2.1. Objective Functions

In this work, there are three objective functions: voltage deviation, operating cost, and active power loss, which are formulated as a single- and multi- objective function.

2.1.1. Single Objective Functions

The proposed objective functions can be performed as follows:

Voltage Deviation Minimization

Voltage instability has become an essential issue that must be addressed for the modern power system to operate securely. The objective function, demonstrated as (VD), is used to assess power system voltage instability by reducing the sum of voltage deviations at each load bus from a specified voltage. Therefore, the minimization of voltage deviations (VD) is the first objective function, which is expressed as follows:

$$\begin{array}{c}{F}_1=\min \left(VD\right)\\ F_1={\displaystyle {\displaystyle \sum }_{i=1}^{NPQ}}\left|V_i-V_{ref}\right|\end{array}$$

(3)

where the voltage reference is $V_{ref}$ and commonly equal to one.

Cost Minimization

The second objective function seeks to reduce total operating cost. The ORPD problem mostly focuses on this OF. For each generator, it can be written as the polynomial quadratic function shown below:

$$\begin{array}{c} F_2=\min \left(Cost\right)\\ F_2={\displaystyle {\displaystyle \sum }_{i=1}^{NG}}{F}_i\left(Q_{Gi}\right)={\displaystyle {\displaystyle \sum }_{i=1}^{NPV}}\left(a_i+b_i{Q}_{Gi}+c_i{Q}^2{}_{Gi}\right) \frac{\$}{h}\end{array}$$

(4)

where the $i$th generator’s fuel cost is $F_i$, and the $i$th generator’s cost coefficients are $a_i$, $b_i$, and $c_i$.

Real Power Loss Minimization

In terms of economics, power suppliers invariably create the better utilization of active power and available transmission capacity. As a result, minimizing real power loss on transmission lines becomes a priority for the power department. The minimization of transmission active loss ($Ploss$) is the third objective function, which can be expressed as follows:

$$\begin{array}{c}{F}_3=min \left(Ploss\right)\\ F_3={\displaystyle {\displaystyle \sum }_{i=1}^{NTL}}{G}_{ij}\left(V^2{}_i+V^2{}_j-2 V_i{V}_j\mathit{cos}{\delta }_{ij}\right) MW\end{array}$$

(5)

where $G_{ij}$ is the conductance of transmission, and ${\delta }_{ij}$ is a phase difference of voltages.

2.1.2. Multi-Objective Functions

The main goal of solving multi-objective issues is to optimize numerous independent objective functions at the same time. The multi-objective problem is defined as follows:

$$Min F\left(x,u\right)=\left[F_1\left(x,u\right),F_2\left(x,u\right),\dots ,F_i\left(x,u\right)\right]$$

(6)

where $i$ is the number of objective functions; the Pareto optimization approach or weight factors as follows can be used to solve multi-objective functions:

$$\begin{array}{c}Min F_4={\displaystyle {\displaystyle \sum }_{i=1}^3}{w}_i F_i\left(x,u\right)\\ F\left(x,u\right)=w_1{F}_1+w_2{F}_2+w_3{F}_3=w_1{\displaystyle {\displaystyle \sum }_{i=1}^{NPQ}}\left|V_i-V_{ref}\right|+w_2{\displaystyle {\displaystyle \sum }_{i=1}^{NPV}}\left(a_i+b_i{Q}_{Gi}+c_i{Q}^2{}_{Gi}\right)\\ +w_3{\displaystyle {\displaystyle \sum }_{i=1}^{NTL}}{G}_{ij}\left(V^2{}_i+V^2{}_j-2 V_i{V}_j\cos {\delta }_{ij}\right)\end{array}$$

(7)

where $w_1$, $w_2$, and $w_3$ are weight factors chosen depending on the relative importance of one objective to others. Typically, the weight factor values are chosen as follows:

$${\displaystyle \sum }_{i=1}^n{w}_n=1$$

(8)

2.2. System Constraints

The system already has several constraints, which are listed as follows:

2.2.1. Equality Constraints

The equality of two constraints containing active and reactive power balance must be fulfilled for power system operating conditions and are stated as follows.

$$P_{Gi}=P_{Di}+\left|V_i\right|{\displaystyle \sum }_{j=1}^{NB}\left|V_j\right|\left(G_{ij}\cos {\delta }_{ij}+B_{ij}sin{\delta }_{ij}\right)$$

(9)

$$Q_{Gi}=Q_{Di}+\left|V_i\right|{\displaystyle \sum }_{j=1}^{NB}\left|V_j\right|\left(G_{ij}\sin {\delta }_{ij}-B_{ij}cos{\delta }_{ij}\right)$$

(10)

where $P_{Gi}$ is the generated active power at bus $i$ and $Q_{Gi}$ is the generated reactive power at bus $i$. The active demand load at bus $i$ and reactive demand load at bus $i$ are signified by $P_{Di}$ and $Q_{Di}$, respectively. The conductance and susceptibility between bus $i$ and bus $j$are depicted by$G_{ij}$ and $B_{ij}$, respectively.

2.2.2. Inequality Constraints

The inequality constraints, which reflect the system components’ operating bounds, include the following:

-

Generator Constraints

Under any condition, the generator should be operating within its lower and upper bounds. The maximum and minimum boundaries of output active power, output reactive power, and bus voltages are given below:

$$P_{Gi}{}^{min}\le P_{Gi}\le P_{Gi}{}^{max} i=1,2,\dots ,NG$$

(11)

$$V_{Gi}{}^{min}\le V_{Gi}\le V_{Gi}{}^{max} i=1,2,\dots ,NG$$

(12)

$$Q_{Gi}{}^{min}\le Q_{Gi}\le Q_{Gi}{}^{max} i=1,2,\dots ,NG$$

(13)

-

Transformer Constraints

The boundaries of maximum and minimum transformer tapping settings are given below as follows:

$$T_i{}^{min}\le T_i\le T_i{}^{max} i=1,2,\dots ,NT$$

(14)

-

Shunt Capacitor Bank Constraints

Shunt capacitor banks’ VAR compensation capacity must be limited to the lower and upper bounds shown below as follows:

$$Q_{Ci}{}^{min}\le Q_{Ci}\le Q_{Ci}{}^{max} i=1,2,\dots ,NC$$

(15)

-

Load Voltage Constraints

The magnitude of load bus voltage must be kept within a tolerable limit, as follows:

$$V_{Li}{}^{min}\le V_{Li}\le V_{Li}{}^{max} i=1,2,\dots ,NPQ$$

(16)

-

Security Constraints

To avoid overload, each transmission line’s apparent power flow should be restricted to its allowed limits, which is shown below:

$$S_{Li}\le S_{Li}{}^{max} i=1,2,\dots ,NTL$$

(17)

3. African Vultures Optimization Algorithm

A new nature-inspired metaheuristic algorithm was introduced in [48] and Figure 1 displays the flowchart and steps for the proposed AVOA by modeling and replicating African vulture life patterns and foraging activities. The African vulture agents have N as the population in the metaheuristic algorithms. Its size is determined by the issue to be solved, and each vulture’s position space is described in d dimensions. Vulture populations are divided into three categories: the best solution is the first-best vulture, the second solution is the second-best vulture, and the remaining vultures are allocated to the third group. The AVOA solution’s fitness value can reflect the upsides and downsides of vultures. As a result, two of the finest solutions are deemed the strongest and best vultures, with the remaining vultures attempting to approach the best. In general, all vultures in the AVOA want to be near the best while avoiding the worst. The AVOA approach may be divided into five stages in the foraging stage.

Figure 1. Flowchart for AVOA proposed optimization algorithm.

-

Stage 1: Population Classification

In this phase, the fitness of all solutions is calculated, and the best solution is identified as the best and first vulture, the second solution is identified as the second-best vulture using Equation (18), and the other vultures are allocated to the third group.

$$R_i=\{\begin{array}{c}Best Vultur{e}_{1 }if p_i=L_1\\ Best Vultur{e}_2 if p_i=L_2\end{array}$$

(18)

where $Best Vultur{e}_{1 }$represents the best vulture, $Best Vultur{e}_2$ denotes the second-best one, $L_1$ and $L_2$are two random values in the range of (0, 1) and their total is 1. Equation (19) is used to determine $p_i$, which is accomplished using the roulette-wheel technique.

$$p_i=\frac{F_i}{{{\displaystyle \sum }}_{i=1}^n{F}_i}$$

(19)

where the fitness of the first and second two groups of vultures is represented by $F_i$, and $n$ is the total number of vultures in both groups.

-

Stage 2: Vulture Famine Rate

If the vultures are not famished, they have enough stamina to fly longer distances in search of food, but if they are, they lack the energy to sustain their long-distance flight. As a result, the hungry vultures will become hostile. Vulture exploration and exploitation stages may thus be built around this tendency. The $F_i$, a hunger level, of the $i\mathrm{th}$vulture at the $t\mathrm{th}$ iteration, is calculated using Equation (20), and it is used as a signal of the vulture’s transition from exploration to exploitation.

$$F=\left(2\times ran{d}_1+1\right)\times z\times \left(1-\frac{iteration{i}_i}{maxiterations}\right)+t$$

(20)

where $F$ denotes that the vultures have had their fill, $ran{d}_1$ is a variable with a random value between 0 and 1, $z$ is a variable with a random value between (−1,1) that varies with each iteration, and t is determined by Equation (21).

$$t=h\times \left(\sin \omega \left(\frac{\pi }{2}\times \frac{iteration{i}_i}{maxiterations}\right)+\cos \omega \left(\frac{\pi }{2}\times \frac{iteration{i}_i}{maxiterations}\right)-1\right)$$

(21)

where the parameter $\omega$, which is provided in advance, determines the vulture’s likelihood of executing the exploitation step. Furthermore, $iteration{i}_i$is the current iteration number, $maxiterations$ is the total number of iterations, and $h$ is a random integer between −2 and 2.

According to the equation, $F$ decreases steadily as the number of iterations grows (20). When the value of$\left|F\right|$ is greater than one, the vultures start the exploration stage and look for fresh food in various areas. Otherwise, vultures enter the exploitation stage, hunting for better food in the immediate surroundings.

-

Stage 3: Reconnaissance Phase

Vultures have excellent vision in the wild, helping them to seek food and recognize dead animals fast. Vultures, on the other hand, may have difficulty finding food since they spend a long time assessing their surroundings before flying huge distances in search of food. The AVOA’s vultures can check several locations randomly by two separate strategies based on the value of ${\mathrm{P}}_1$ in the range of (0, 1).

During the exploration phase, a random number ($rand{p}_1$) between 0 and 1 is utilized to determine one of the techniques. If the value of the $rand{p}_1\le P_1$, parameter Equation (21) is used. Otherwise, Equation (22) is used.

$$P\left(i+1\right)=R\left(i\right)-D\left(i\right)\times F_i$$

(22)

$$P\left(i+1\right)=R\left(i\right)-F_i+ran{d}_2\times \left(\left(ub-lb\right)\times ran{d}_3+lb\right)$$

(23)

where $R\left(i\right)$ is one of the best vultures chosen in the current iteration using Equation (18), $F_i$ is the rate of vulture satiation calculated in the current iteration using Equation (20), $ran{d}_2$ is a random number between 0 and 1, and $lb$ and $ub$ are the variables’ lower and upper bounds, respectively. $ran{d}_3$is used to give a high random coefficient at the search environment scale to boost the diversity and search for diverse search space areas.

$D\left(i\right)$, which reflects the distance between the vulture and the current optimum, is calculated by Equation (24).

$$D\left(i\right)=\left|X\times R\left(i\right)-P\left(i\right)\right|$$

(24)

where $P\left(i\right)$ is the $i\mathrm{th}$ vulture’s location and $X$ is a random integer between 0 and 2.

-

Stage 4: Utilization Phase (First Stage)

The AVOA’s efficiency stage is investigated at this step. If $\left|F_i\right|$ value is less than one, the AVOA begins the first step of exploitation. The parameter $P_2$ in the range (0, 1) is used to determine which method is used. At the start of this phase, $P_2$ generates a random number between 0 and 1. If this $rand{p}_2$is more than or equal to the parameter$P_{2 },$the siege-fight method is implemented slowly. Otherwise, rotational flying is employed by Equation (25).

$$P\left(i+1\right)=\{\begin{array}{ll}D\left(i\right)\times \left(F_i+rand{p}_4\right)-d\left(t\right) & if P_2\ge rand{p}_2\\ R\left(i\right)-\left(S_1+S_2\right) & if P_2<rand{p}_2\end{array}$$

(25)

$S_1$ and $S_2$ are determined using Equations (26) and (27), respectively, as respects:

$$S_1=R\left(i\right)\times \left(\frac{ran{d}_5\times P\left(i\right)}{2\pi }\right)\times \cos \left(P\left(i\right)\right)$$

(26)

$$S_2=R\left(i\right)\times \left(\frac{ran{d}_6\times P\left(i\right)}{2\pi }\right)\times \sin \left(P\left(i\right)\right)$$

(27)

where $ran{d}_5$ and $ran{d}_6$ are random numbers between 0 and 1, respectively.

-

Stage 5: Utilization Phase (Second Stage)

If$\left|F_i\right|$ is less than 0.5, this stage of the algorithm is executed.$ran{d}_3$ is created in the range (0,1) at the commencement of this phase. If the parameter$P_3$is greater than or equal to$ran{d}_3$, the goal is to attract a variety of vultures to the food supply, resulting in competing behavior. As a result, utilizing the equation, the vulture’s location may be updated (28).

$$P\left(i+1\right)=\frac{A_1+A_2}{2}$$

(28)

Equations (29) and (30) are used to compute $A_1$ and $A_2$, respectively.

$$A_1=BestVultur{e}_1\left(i\right)-\frac{BestVultur{e}_1\left(i\right)\times P\left(i\right)}{BestVultur{e}_1\left(i\right)-{\left(P\left(i\right)\right)}^2}\times F_i$$

(29)

$$A_2=BestVultur{e}_2\left(i\right)-\frac{BestVultur{e}_2\left(i\right)\times P\left(i\right)}{BestVultur{e}_2\left(i\right)-{\left(P\left(i\right)\right)}^2}\times F_i$$

(30)

Similarly, as the AVOA reaches its second stage, the vultures would congregate around the best vulture to scavenge the leftover food. As a result, using an equation, the vultures’ location may be updated (31).

$$P\left(i+1\right)=R\left(i\right)-\left|d\left(t\right)\right|\times F_i\times Levy\left(d\right)$$

(31)

where $d$ is the problem dimensions.

The AVOA’s efficacy was boosted by adopting Lévy flight ($LF$) patterns generated from Equation (32).

$$LF\left(x\right)=0.001\times \frac{u\times \sigma }{{\left|\upsilon \right|}^{\frac{1}{\rho }}}$$

(32)

$$\sigma ={\left(\frac{\mathsf{\Gamma }\left(1+\beta \right)\times \sin \left(\frac{\pi \beta }{2}\right)}{\mathsf{\Gamma }\left(1+\beta 2\right)\times \beta \times 2\times \left(\frac{\beta -1}{2}\right)}\right)}^{\frac{1}{\rho }}$$

(33)

where, $\upsilon$ and $u$ are random numbers between 0 and 1, respectively, and $\beta$ is a constant number of 1.5.

Figure 1 and Table 2 show the AVOA algorithm flowchart and parameters.

Table 2. The AVOA algorithm parameters.

Algorithm	Parameters	Value
AVOA	$L_1$	0.8
	$L_1$	0.2
	$w$	2.5
	$P_1$	0.6
	$P_2$	0.4
	$P_3$	0.6

4. Simulation Results and Discussion

To demonstrate the usefulness of the proposed AVOA, it was utilized to fix the typical IEEE (30, 57, and 118) bus test systems with the goal of reducing actual power loss, total voltage variations, and total operational cost as a single- and multi-objective function considering RES with a continuous and discrete control variable handling. Table 3 lists ten cases that were used in this study. All simulation studies have been run on MATLAB 2022a, a 2.8 GHz Intel Pentium i7 PC with 16 GB of RAM. After 200 iterations for all test systems, the numerical optimum values for the five methods were determined. Furthermore, the simulation studies were acquired after 50 independent runs for each of the test situations. The five algorithms were executed on a total population of 30 particles. The control variables handling in our codification cases with two distinct types of optimization variables considering continuous and discrete variables require a special initialization. The continuous variables are initialized as follows, e.g., $P_{Gi}=$random[$P_{Gi}{}^{min}$,$P_{Gi}{}^{max}$] and $V_{Gi}=$random[$V_{Gi}{}^{min}$,$V_{Gi}{}^{max}$]. However, tap changers and the shunt capacitor bank compensators in the case of discrete variables are rounded off to their closest decimal values.

Table 3. Case studies for different objective functions of ORPD problems.

Case No	Objective Function
1	Minimization of voltage-level deviation in IEEE-30 bus.
2	Minimization of operational cost in IEEE-30 bus.
3	Minimization of transmission power loss in IEEE-30 bus.
4	Minimization of transmission power loss incorporating optimal renewable energy sources in IEEE-30 bus.
5	Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with continuous control variable settings in IEEE-30 bus.
6	Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with discrete control variable settings in IEEE-30 bus.
7	Minimization of transmission power loss in IEEE-57 bus.
8	Minimization of transmission power loss incorporating optimal renewable energy sources in IEEE-57 bus.
9	Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with continuous control variable settings in IEEE-118 bus.
10	Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with discrete control variable settings in IEEE-118 bus.

This procedure is accomplished by incorporating the rounding operator into each step of the AVOA algorithm. Mathematically, the rounding can be written as follows: round (random$[T_i{}^{min}$,$T_i{}^{max}]$, step size), and round (random$[Q_{ci}{}^{min}$,$Q_{Ci}{}^{max}]$, step size). After the initialization phase, the solution vector is updated by changing the previous solution to find a better new solution. Table 4 contains a description of the proposed test systems, including several control variables. Table 5 shows the settings for the control variable limits for each test system [49].

Table 4. The description of proposed test systems.

Parameters	30-Bus	57-Bus	118-Bus
Control variable numbers	19	25	77
Generator numbers	6	7	45
Transformer Tap numbers	4	15	9
Q-shunt numbers	9	3	14

Table 5. Control variable limits for various IEEE cases (pu).

Parameters	Variables	Min-Value	Max-Value	Step Size
30-Bus	$V_G$	0.95	1.1	Continuous
	$T_i$	0.9	1.1	0.01 (20-level)
	$Q_C$	0	5MVAR	0.05 (100-step)
57-Bus	$V_G$	0.9	1.1	Continuous
	$T_i$	0.9	1.1	Continuous
	$Q_C$	0	30MVAR	Continuous
118-Bus	$V_G$	0.94	1.06	Continuous
	$T_i$	0.9	1.1	0.01 (20-level)
	$Q_C$	0	30MVAR	0.3 (100-step)

4.1. IEEE 30-Bus Test System

Figure 2 depicts the IEEE 30-bus test system with 19 control variables including 6 generators, 4 tap changing transformer settings, 9 shunt VAR compensators, 41 branches (37 lines and 4 tap changing transformers), and the total real and reactive power demands are 238.4 MW and 126.2 MVAR, respectively. The exact details of lines and buses for the IEEE- 30 bus system are described in [16,34]. Furthermore, in this research, the system is restricted as follows: the magnitude range of all buses’ voltages is 0.95 pu to 1.1 pu. The tap-changing transformers, on the other hand, range in power from 0.9 pu to 1.1 pu. Likewise, the shunt VAR compensator limits are assumed to be in the range of 0 to 5 MVAR.

Figure 2. Standard IEEE 30 bus test system.

4.1.1. Case 1: Minimization of Voltage-Level Deviation in IEEE 30 Bus

In this case, the AVOA algorithm is being used to improve the function of minimizing voltage deviation. The findings mentioned in Table 6 point out that the voltage deviation index is decreased with all proposed algorithms within the accepted range. The AVOA achieves the best and lowest voltage deviation (0.103 pu) and enhances the voltage profile at major buses (around 1 pu) as displayed in Figure 3. In addition, Figure 4 shows how the AVOA achieves a pattern of decreasing system voltage divergence characteristics that outperforms comparable algorithms. In Table 6, the comparison of solutions obtained using these five methods, the AVOA algorithm greatly outperforms other population-based optimization strategies.

Table 6. Optimal IEEE-30 bus test system control variables for curtailment VD.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	1.023328	1.049871	1.008875	1.011852	1.037673
$V_{G2}$ (pu)	1.008159	1.036918	1.003797	1.008226	1.040367
$V_{G5}$ (pu)	1.000557	1.010459	1.018456	1.003649	1.025815
$V_{G8}$ (pu)	0.994860	0.985178	0.999638	1.006250	0.994378
$V_{G11}$ (pu)	1.082926	0.975178	1.087355	1.013930	0.992829
$V_{G13}$ (pu)	1.008800	1.092579	1.005192	0.997207	1.001126
$T_{11}$ (6-9)	1.047965	1.051309	1.090717	0.973667	0.997475
$T_{12}$ (6-10)	0.921887	0.927919	0.919338	0.973680	0.900000
$T_{15}$ (4-12)	0.984708	1.012708	0.974276	0.973665	0.939858
$T_{36}$ (28-27)	0.937218	0.926008	0.965334	0.973737	0.939329
$Q_{\mathrm{C}10}$ (MVAR)	5.000000	4.398053	2.184212	5.000000	2.368492
$Q_{\mathrm{C}12}$ (MVAR)	5.000000	0.429754	4.994771	5.000000	1.280472
$Q_{\mathrm{C}15}$ (MVAR)	5.000000	3.127819	3.855124	5.000000	1.006742
$Q_{\mathrm{C}17}$ (MVAR)	0.000000	3.826915	4.686445	5.000000	1.237033
$Q_{\mathrm{C}20}$ (MVAR)	4.456893	4.590043	4.986836	5.000000	5.000000
$Q_{\mathrm{C}21}$ (MVAR)	5.000000	4.029262	4.267873	5.000000	4.746975
$Q_{\mathrm{C}23}$ (MVAR)	5.000000	2.063050	4.507320	5.000000	3.478525
$Q_{\mathrm{C}24}$ (MVAR)	0.000000	1.468192	4.999888	5.000000	4.288032
$Q_{\mathrm{C}29}$ (MVAR)	0.000000	3.338243	2.533782	5.000000	0.000000
Power Losses (MW)	5.502730	5.789450	5.794690	5.627020	5.815130
Voltage Deviation (pu)	0.141010	0.293620	0.103090	0.126710	0.124770

Figure 3. The AVOA voltage profile with the other compared methods for case 1.

Figure 4. The AVOA convergence characteristics with the other compared methods for case 1.

According to Table 7, all proposed algorithms succeeded in minimizing the VD objective function considering operational constraints and achieving real and reactive power balance. Also, the performance evaluation of the proposed algorithms based on VD and the “number of runs is 50” with statistical analysis is illustrated in Table 8. The AVOA algorithm’s narrow range between minimum and maximum values presents a minimum stander deviation value (0.02875 pu). Additionally, it achieves a suitable mean value of computational time (18.85260 s) among other compared algorithms.

Table 7. Balance of active and reactive power for proposed algorithms with case 1.

Method	Balance of Active Power			Balance of Reactive Power
	Load (MW)	Generation (MW)	Loss (MW)	Load (MVAR)	Generation (MVAR)	Shunt VAR Compensation (MVAR)	Shunt Admittance and Branch Charging (MVAR)	Loss (MVAR)
PDO	283.400	288.903	5.503	126.200	92.603	29.457	30.486	26.346
RSO	283.400	289.189	5.789	126.200	93.894	27.271	32.816	27.782
AVOA	283.400	289.195	5.795	126.200	88.998	37.016	28.309	28.123
ROA	283.400	289.027	5.627	126.200	72.498	45.000	33.133	24.431
SPO	283.400	289.215	5.815	126.200	96.332	23.406	31.325	24.863

Table 8. Statistical analysis for proposed algorithms with case 1.

Method	Minimum Value	Maximum Value	Mean Value	Standard Deviation	Mean Value of Computational Time
PDO	0.14101	225.70993	23.85192	60.80698	16.83349 (s)
RSO	0.29362	1624.08688	194.90665	325.54255	42.39668 (s)
AVOA	0.10309	0.2324700	0.150460	0.028750	18.85260 (s)
ROA	0.12671	36.849600	5.888400	12.66745	100.80976 (s)
SPO	0.12477	201.57414	6.740770	29.61497	119.25339 (s)

4.1.2. Case 2: Minimization of Operational Cost in IEEE 30 Bus

When using the AVOA algorithm without considering the DG’s involvement, the target function is thought to be fuel cost minimization in this case. The fuel cost reduction convergence graph is shown in Figure 5 and the AVOA algorithm gets the optimal solution according to a remarkable and quick convergence rate and gets the lowest numbers with 100 iterations. Table 9 shows the appropriate control variables changes and the optimal cost minimization values; the findings show how superior the AVOA algorithm is against other heuristic optimization techniques and also show a considerable reduction in fuel costs to 81.268916 USD/h and the average computing time for a single loop is 0.08. Also, the majority of obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 6. These outcomes demonstrate how the AVOA method performs well in terms of fuel cost minimization.

Figure 5. The AVOA convergence characteristics with the other compared methods for case 2.

Table 9. Optimal IEEE 30-bus test system control variables for lessening operational cost.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	1.077223	1.010074	1.078933	1.009823	1.096953
$V_{G2}$ (pu)	1.062954	0.970207	1.040035	1.009823	1.093400
$V_{G5}$ (pu)	1.034902	0.950000	0.997439	1.009860	1.100000
$V_{G8}$ (pu)	0.987255	0.950000	0.950000	1.009823	1.036836
$V_{G11}$ (pu)	0.966547	0.950000	0.988204	1.010017	1.100000
$V_{G13}$ (pu)	1.058877	0.950000	1.013312	1.009823	1.100000
$T_{11}$ (6-9)	1.021251	1.004746	0.932451	0.972293	0.928126
$T_{12}$ (6-10)	0.958663	0.900000	0.930201	0.972293	0.964485
$T_{15}$ (4-12)	0.921516	0.900000	0.906104	0.947242	1.087568
$T_{36}$ (28-27)	0.973659	0.900000	0.954467	0.947285	0.943896
$Q_{\mathrm{C}10}$ (MVAR)	5.000000	2.754954	0.000000	4.718844	5.000000
$Q_{\mathrm{C}12}$ (MVAR)	5.000000	2.543246	0.324557	4.718844	4.977415
$Q_{\mathrm{C}15}$ (MVAR)	0.000000	2.222816	0.118031	4.720256	0.000000
$Q_{\mathrm{C}17}$ (MVAR)	0.000000	3.212513	4.210257	4.718844	5.000000
$Q_{\mathrm{C}20}$ (MVAR)	5.000000	2.973234	4.581713	4.720256	0.318032
$Q_{\mathrm{C}21}$ (MVAR)	5.000000	0.000000	0.000000	4.718844	1.081768
$Q_{\mathrm{C}23}$ (MVAR)	5.000000	2.625242	0.000000	4.720256	5.000000
$Q_{\mathrm{C}24}$ (MVAR)	0.000000	0.504981	0.574697	4.720256	1.921813
$Q_{\mathrm{C}29}$ (MVAR)	5.000000	0.000000	5.000000	4.718844	3.431373
Cost (USD/h)	82.703420	351.212637	81.268916	150.071347	103.635994
Power Losses (MW)	6.597461	6.946435	7.964927	5.743704	5.777907
Voltage Deviation (pu)	0.299451	1.198932	0.383136	0.243850	1.518099

Figure 6. The AVOA voltage profile with the other compared methods for case 2.

All proposed algorithms succeeded in minimizing the operational cost without considering DGs and achieving real and reactive power balance as shown in Table 10. Also, statistical analysis is illustrated in Table 11; the performance evaluation of the proposed algorithms with a number of runs is 50. The AVOA algorithm’s narrow range between minimum and maximum values presents a minimum stander deviation value (22.99958 pu). Additionally, it achieves a minimum mean value of computational time (15.85748 s) among other compared algorithms.

Table 10. Balance of active and reactive power for proposed algorithms with case 2.

Method	Balance of Active Power			Balance of Reactive Power
	Load (MW)	Generation (MW)	Loss (MW)	Load (MVAR)	Generation (MVAR)	Shunt VAR Compensation (MVAR)	Shunt Admittance and Branch Charging (MVAR)	Loss (MVAR)
PDO	283.400	288.841	5.441	126.200	84.227	24.035	32.995	33.013
RSO	283.400	288.521	5.121	126.200	88.280	24.321	26.979	35.262
AVOA	283.400	288.793	5.393	126.200	86.205	23.382	31.356	32.021
ROA	283.400	288.820	5.420	126.200	82.678	24.221	34.025	33.718
SPO	283.400	288.674	5.274	126.200	84.765	23.216	32.154	32.497

Table 11. Statistical analysis for proposed algorithms with case 2.

Method	Minimum Value	Maximum Value	Mean Value	Standard Deviation	Mean Value of Computational Time
PDO	1006.71192	1517.74404	1162.93396	158.66803	23.09068 (s)
RSO	1181.05238	3126.24346	1474.70360	329.13510	21.79979 (s)
AVOA	972.25120	1078.00374	992.75867	22.999580	15.85748 (s)
ROA	1015.93296	1215.87610	1109.35461	64.874550	98.75274 (s)
SPO	991.15484	1648.76614	1143.98243	172.34129	78.69903 (s)

4.1.3. Case 3: Minimization of Transmission Power Losses in IEEE 30 Bus

In this case, the target function is thought to be real power loss saving. The AVOA algorithm was employed to accomplish the best solution and establish the perfect sets of the control variable, which diminishes system losses. Table 12 shows how the AVOA algorithm outperforms these prior methods; when the AVOA algorithm is used without taking DG into account, real power losses are dramatically lowered to 4.54647 MW. In Figure 7 real power losses using the AVOA method are steeply converged and fully achieve the perfect solution with 120 iterations, indicating a remarkable and quick convergence rate. Also, the majority of obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 8. The outcomes demonstrate how the AVOA method performs well in terms of real power loss minimization compared to that found using published population-based optimization techniques.

Table 12. Optimal IEEE 30-bus test system control variables for decline power losses.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	1.073600	1.091200	1.100000	1.100000	1.100000
$V_{G2}$ (pu)	1.063800	1.085800	1.094403	1.095900	1.095600
$V_{G5}$ (pu)	1.039600	1.073700	1.074998	1.072400	1.075000
$V_{G8}$ (pu)	1.041100	1.038700	1.076819	1.079600	1.077400
$V_{G11}$ (pu)	1.098600	1.043100	1.099993	1.100000	1.100000
$V_{G13}$ (pu)	1.094100	1.095400	1.100000	1.100000	1.099000
$T_{11}$ (6-9)	0.993800	0.962000	1.042528	1.025200	0.978500
$T_{12}$ (6-10)	0.966300	0.968800	0.900024	1.040200	0.988700
$T_{15}$ (4-12)	0.983100	1.036400	0.980079	1.028200	0.995900
$T_{36}$ (28-27)	0.954100	0.988300	0.966956	1.002600	0.976800
$Q_{\mathrm{C}10}$ (MVAR)	5.000000	3.840300	4.659089	5.000000	4.073100
$Q_{\mathrm{C}12}$ (MVAR)	5.000000	2.919700	3.784615	5.000000	0.440700
$Q_{\mathrm{C}15}$ (MVAR)	0.000000	1.070700	4.998465	5.000000	4.977900
$Q_{\mathrm{C}17}$ (MVAR)	5.000000	0.760800	4.976225	5.000000	5.000000
$Q_{\mathrm{C}20}$ (MVAR)	0.000000	0.920200	4.843913	5.000000	5.000000
$Q_{\mathrm{C}21}$ (MVAR)	5.000000	3.404000	4.877789	5.000000	3.665900
$Q_{\mathrm{C}23}$ (MVAR)	4.062800	4.981000	4.678076	5.000000	4.254900
$Q_{\mathrm{C}24}$ (MVAR)	5.000000	3.509100	4.983888	5.000000	4.322500
$Q_{\mathrm{C}29}$ (MVAR)	0.000000	3.637300	2.469794	5.000000	2.833000
Power Losses (MW)	4.839510	5.132850	4.546470	4.627150	4.573080
Voltage Deviation (pu)	1.266310	1.024060	1.988920	1.586570	1.825720

Figure 7. The AVOA convergence characteristics with the other compared methods for case 3.

Figure 8. The AVOA voltage profile with the other compared methods for case 3.

All proposed algorithms succeeded in saving real power losses and achieving real and reactive power balance as shown in Table 13. Also, statistical analysis is illustrated in Table 14. The performance evaluation of the proposed algorithms with a number of runs is 50. The AVOA algorithm’s narrow range between minimum and maximum values presents a minimum stander deviation value (0.042 pu). Additionally, it achieves a minimum mean value of computational time (16.00185 s) among other compared algorithms.

Table 13. Balance of active and reactive power for proposed algorithms with case 3.

Method	Balance of Active Power			Balance of Reactive Power
	Load (MW)	Generation (MW)	Loss (MW)	Load (MVAR)	Generation (MVAR)	Shunt VAR Compensation (MVAR)	Shunt Admittance and Branch Charging (MVAR)	Loss (MVAR)
PDO	283.400	288.240	4.840	126.200	82.622	29.063	36.897	22.382
RSO	283.400	288.533	5.133	126.200	87.795	25.043	36.383	23.021
AVOA	283.400	287.946	4.546	126.200	70.324	40.272	36.000	20.395
ROA	283.400	288.027	4.627	126.200	63.608	45.000	38.654	21.062
SPO	283.400	287.973	4.573	126.200	73.003	34.568	38.824	20.194

Table 14. Statistical analysis for proposed algorithms with case 3.

Method	Minimum Value	Maximum Value	Mean Value	Standard Deviation	Mean Value of Computational Time
PDO	4.840	16,002.689	1504.755	3986.054	46.108410 (s)
RSO	5.133	17,6611.603	13,757.799	29,157.813	17.582580 (s)
AVOA	4.546	4.708000	4.587000	0.042000	16.001850 (s)
ROA	4.627	3531.414	519.6950	1172.257	108.32703 (s)
SPO	4.573	22,390.582	1351.229	5029.313	79.124780 (s)

4.1.4. Case 4: Minimization of Transmission Power Loss for Standard IEEE 30 Bus Incorporating RES

The reduction of real power losses by implementing the AVOA method when accommodating the DG at node 27 is the target function in this case, even more significantly lowering the power losses, reaching 2.67438 MW as stated in Table 15. In Figure 9, real power losses using the AVOA method are steeply converged and fully achieve the perfect solution with 100 iterations, indicating a remarkable and fast convergence rate. Also, the majority of obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 10. The results demonstrate the AVOA algorithm’s advantage over other heuristic approaches for tackling this sort of issue, and they support the effectiveness of the suggested DG placement strategy.

Table 15. Optimal IEEE 30-bus test system control variables for decline in power losses with incorporation of RES.

Control Variables		PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)		1.100000	1.099455	1.100000	1.097868	1.100000
$V_{G2}$ (pu)		1.099989	1.099851	1.097978	1.096221	1.094982
$V_{G5}$ (pu)		1.099587	1.024629	1.081564	1.085981	1.073478
$V_{G8}$ (pu)		1.093189	1.068854	1.088944	1.082906	1.077110
$V_{G11}$ (pu)		1.085935	1.019485	1.099938	1.095299	1.100000
$V_{G13}$ (pu)		1.012753	1.033458	1.100000	1.071204	1.100000
$T_{11}$ (6-9)		1.099679	1.077608	0.988583	1.042668	1.013376
$T_{12}$ (6-10)		1.099712	0.918801	0.958993	1.074469	1.028824
$T_{15}$ (4-12)		1.099075	1.016538	0.988412	1.043277	1.010851
$T_{36}$ (28-27)		1.090130	1.063849	0.984554	1.078806	0.984372
$Q_{\mathrm{C}10}$ (MVAR)		5.000000	2.367058	3.455445	2.682247	0.275218
$Q_{\mathrm{C}12}$ (MVAR)		5.000000	4.952253	4.950191	1.501447	4.670871
$Q_{\mathrm{C}15}$ (MVAR)		5.000000	2.472453	4.388203	3.121859	4.671834
$Q_{\mathrm{C}17}$ (MVAR)		5.000000	2.635213	4.167731	3.102620	3.951102
$Q_{\mathrm{C}20}$ (MVAR)		2.938907	4.859103	2.978210	1.318443	4.908250
$Q_{\mathrm{C}21}$ (MVAR)		5.000000	0.260195	5.000000	0.704469	1.595587
$Q_{\mathrm{C}23}$ (MVAR)		5.000000	2.101977	1.768431	3.208966	2.680394
$Q_{\mathrm{C}24}$ (MVAR)		5.000000	1.679141	4.453273	0.569255	4.916036
$Q_{\mathrm{C}29}$ (MVAR)		3.705237	0.630078	3.422900	2.628873	5.000000
RES Location and Size	Location	21.00000	25.00000	28.00000	21.00000	21.00000
	MW	46.35348	48.73839	46.07358	44.31754	46.19743
	MVAR	0.000000	6.568155	11.33584	11.81921	16.39360
Power Losses (MW)		3.035540	4.980920	2.674380	3.040910	2.797390
Voltage Deviation (pu)		0.691760	0.648760	2.064220	1.013430	1.897910

Figure 9. The AVOA convergence characteristics with the other compared methods for case 4.

Figure 10. The AVOA voltage profile with the other compared methods for case 4.

All proposed algorithms succeeded in saving the real power losses and achieving real and reactive power balance considering DGs as shown in Table 16. Also, statistical analysis is illustrated in Table 17. The performance evaluation of the proposed algorithms with a number of runs is 50. The AVOA algorithm’s narrow range between minimum and maximum values presents a minimum stander deviation value (0.637 pu). Additionally, it achieves a minimum mean value of computational time (20.66890 s) among other compared algorithms.

Table 16. Balance of active and reactive power for proposed algorithms with case 4.

Method	Balance of Active Power				Balance of Reactive Power
	Load (MW)	Generation (MW)	RES (MW)	Loss (MW)	Load (MVAR)	Generation (MVAR)	RES (MVAR)	Shunt VAR Compensation (MVAR)	Shunt Admittance and Branch Charging (MVAR)	Loss (MVAR)
PDO	283.4	240.082	46.353	3.036	126.2	58.434	0.0000	41.644	47.542	21.419
RSO	283.4	239.643	48.738	4.981	126.2	80.243	6.5681	21.957	37.258	19.827
AVOA	283.4	240.001	46.073	2.674	126.2	54.796	11.335	34.584	38.962	13.478
ROA	283.4	242.123	44.317	3.041	126.2	69.091	11.819	18.838	41.999	15.547
SPO	283.4	240.000	46.197	2.797	126.2	49.402	16.393	32.669	38.566	10.831

Table 17. Statistical analysis for proposed algorithms with case 4.

Method	Minimum Value	Maximum Value	Mean Value	Standard Deviation	Mean Value of Computational Time
PDO	3.03600	23451.0380	4069.32700	6986.33300	23.117360 (s)
RSO	4.98092	5587.77200	3635.62384	1459.32771	18.682660 (s)
AVOA	2.67400	5.44200000	3.30300000	0.63700000	20.668900 (s)
ROA	3.04100	707.269000	29.5230000	106.320000	188.90750 (s)
SPO	2.79700	13072.7640	587.630000	2454.28500	83.338790 (s)

4.1.5. Case 5: Minimization of Multi-Objective Functions (Voltage-Level Deviation, Operational Cost, and Transmission Power Loss) with Continuous Control Variable Settings in IEEE-30 Bus

The applicability of the AVOA algorithm for solving a combinatorial complex ORPD optimization problem involving nonlinear muti-objective functions with a nonlinear restricted constraint is presented in this case. The AVOA algorithm is proposed for solving a multi-objective function minimization problem based on a weighted sum approach with continuous control variable settings, three single distinct objective functions (voltage-level deviation, operational cost, and transmission power loss) are transformed into a coefficient multi-objective ORPD problem tested on the IEEE-30 bus small-scale power system, and its results are illustrated in Table 18 and show that the AVOA method outperforms these prior methods with a minimum multi-objective value of 103.336931. In Figure 11, the AVOA method is converged and accomplishes the best solution against other compared techniques. All obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 12.

Table 18. Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with continuous control variable settings in IEEE-30 bus.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	1.034042	1.013040	1.096031	1.015558	1.099988
$V_{G2}$ (pu)	1.000626	1.002160	1.070756	1.011386	1.088032
$V_{G5}$ (pu)	0.950586	0.957289	1.028974	1.003144	1.067478
$V_{G8}$ (pu)	0.954355	0.970801	0.982896	1.010000	1.023205
$V_{G11}$ (pu)	1.030482	0.992876	0.997983	1.001603	1.093794
$V_{G13}$ (pu)	1.080495	1.074294	0.994995	1.021175	0.981983
$T_{11}$ (6-9)	0.972125	0.946234	0.985301	0.970950	1.004438
$T_{12}$ (6-10)	0.979401	0.962047	0.901766	0.996247	0.959636
$T_{15}$ (4-12)	0.900002	0.935633	0.954967	0.989841	1.038180
$T_{36}$ (28-27)	0.934839	0.910310	0.966368	0.973880	1.016444
$Q_{\mathrm{C}10}$ (MVAR)	0.000000	4.296961	0.373418	4.238483	2.887637
$Q_{\mathrm{C}12}$ (MVAR)	0.000000	4.868191	4.655968	4.323851	0.000000
$Q_{\mathrm{C}15}$ (MVAR)	0.000000	0.495375	1.635248	4.105074	1.799884
$Q_{\mathrm{C}17}$ (MVAR)	0.000000	3.882412	4.359865	4.211507	1.983525
$Q_{\mathrm{C}20}$ (MVAR)	5.000000	0.369930	1.755742	3.917401	4.822479
$Q_{\mathrm{C}21}$ (MVAR)	2.731975	1.833711	2.966848	4.251728	4.552650
$Q_{\mathrm{C}23}$ (MVAR)	0.000000	2.368489	4.552541	4.323851	4.999714
$Q_{\mathrm{C}24}$ (MVAR)	5.000000	0.057049	4.277262	4.077535	3.441456
$Q_{\mathrm{C}29}$ (MVAR)	4.884375	4.504252	3.904529	4.121847	4.479210
Voltage Deviation (pu)	0.396291	0.394974	0.178213	0.147804	0.435536
Cost (USD/h)	90.02432	88.97314	50.13732	80.29645	57.96144
Power Losses (MW)	6.620585	5.952060	6.891535	5.554982	5.844435
Multi-objective function	153.919925	157.098928	103.336931	138.203123	136.375579

Figure 11. The AVOA convergence characteristics with the other compared methods for case 5.

Figure 12. The AVOA voltage profile with the other compared methods for case 5.

4.1.6. Case 6: Minimization of Multi-Objective Functions (Voltage-Level Deviation, Operational Cost, and Transmission Power Loss) with Discrete Control Variable Settings in IEEE-30 Bus

When dealing with a mixture of continuous and discrete control variables, an additional case is introduced with these optimization algorithms because it significantly increases the complexity of the ORPD problem. When the discrete variables number increases, the complexity increases exponentially. Therefore, two distinct types of optimization control variables are considered in this case as follows; generator set points as a continuous control variables type, and transformer tap settings with generator reactive power compensations as a discrete control variable type.

In this case, with the IEEE 30 bus test small-scale power system, generators’ voltage set points are typically in the (0.95, 1.1) pu range, transformer taps vary within the (0.9, 1.1) pu range with a step of 0.01 discretized into 20 levels as (0.9 0.91 0.92...until, 1.1), and a 5 MVAR maximum capacity for all capacitor banks with a step of 0.05 MVAR. Also, to show the ability of the proposed AVOA algorithm for solving a multi-objective function dealing with these continuous and discrete control variables, all results are illustrated in Table 19 and prove that the AVOA algorithm outperforms these prior methods with a minimum multi-objective value of 234.854930. In Figure 13, the AVOA method is converged and accomplishes the best solution against other compared techniques. Also, the majority of obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 14.

Table 19. Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with discrete control variable settings in IEEE-30 bus.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	0.950000	0.950000	1.013608	1.082459	1.100000
$V_{G2}$ (pu)	0.950000	0.950000	1.013608	1.082459	1.100000
$V_{G5}$ (pu)	0.950000	0.950000	1.013608	1.082459	1.100000
$V_{G8}$ (pu)	0.950000	0.950000	1.013608	1.082459	1.100000
$V_{G11}$ (pu)	0.950000	0.950000	1.013608	1.082459	1.100000
$V_{G13}$ (pu)	0.950000	0.950000	1.013608	1.082459	1.100000
$T_{11}$ (6-9)	0.900000	0.900000	1.010000	1.030000	1.100000
$T_{12}$ (6-10)	0.900000	0.900000	1.010000	1.030000	1.100000
$T_{15}$ (4-12)	0.900000	0.900000	1.010000	1.030000	1.100000
$T_{36}$ (28-27)	1.100000	0.900000	1.010000	1.030000	1.100000
$Q_{\mathrm{C}10}$ (MVAR)	0.000000	0.000000	4.600000	2.700000	5.000000
$Q_{\mathrm{C}12}$ (MVAR)	0.000000	0.000000	4.600000	4.400000	4.850000
$Q_{\mathrm{C}15}$ (MVAR)	0.000000	0.000000	4.600000	2.900000	5.000000
$Q_{\mathrm{C}17}$ (MVAR)	2.200000	0.000000	4.600000	5.000000	0.000000
$Q_{\mathrm{C}20}$ (MVAR)	0.000000	0.000000	4.600000	3.250000	5.000000
$Q_{\mathrm{C}21}$ (MVAR)	0.000000	0.000000	4.600000	0.000000	1.300000
$Q_{\mathrm{C}23}$ (MVAR)	0.200000	0.000000	4.600000	5.000000	5.000000
$Q_{\mathrm{C}24}$ (MVAR)	0.000000	0.000000	4.600000	0.900000	0.700000
$Q_{\mathrm{C}29}$ (MVAR)	0.000000	1.250000	4.600000	1.700000	5.000000
Voltage Deviation (pu)	2.136450	1.340780	0.322600	0.978970	0.914060
Cost (USD/h)	132.4474	133.1194	90.48282	78.08285	96.59046
Power Losses (MW)	7.937040	7.276330	5.648940	5.034010	5.117670
Multi-objective function	9036.805590	1952.706490	234.854930	330.749040	450.021300

Figure 13. The AVOA convergence characteristics with the other compared methods for case 6.

Figure 14. The AVOA voltage profile with the other compared methods for case 6.

4.2. Standard IEEE-57 Bus Test System

This work considers a medium-scale IEEE 57-test system portrayed in Figure 15, to assess the scalability of the suggested methods. This system has a total generation capacity of 1975.9 MW and is made up of 80 transmission lines, containing 25 control variables including 7 generators, 3 shunt VAR compensators, and 15 transformers [6,16,34]. Furthermore, the system is restricted as follows: the voltage magnitude range for all buses is 0.9 pu to 1.1 pu. The tap-changing transformers, on the other hand, range in power from 0.9 pu to 1.1 pu. Likewise, the shunt VAR compensator limits are assumed to be between 0 and 30 MVAR.

Figure 15. Standard IEEE 57 bus.

4.2.1. Case 7: Minimization of Transmission Power Loss in IEEE 57 Bus

In this case, the target function is thought to be real power loss saving. The AVOA algorithm was employed to accomplish the best solution and establish the perfect sets of the control variable, which diminishes system losses, the findings of which are given in Table 20, which shows how the AVOA algorithm outperforms these prior methods. When the AVOA algorithm is used without taking DG into account, real power losses are dramatically lowered to 27.51680 MW. In Figure 16, real power losses using the AVOA method are steeply converged and fully achieve the perfect solution with 160 iterations, exhibiting the AVOA method’s remarkable and quick convergence rate. Also, the majority of obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 17. The outcomes demonstrate how the AVOA method performs well in terms of real power loss minimization compared to that found using published population-based optimization techniques.

Table 20. Optimal IEEE 57-bus test system control variables for reducing power losses.

Control Variables	PDO	RSO	AVOA	ROA	SPO
${\mathit{V}}_{\mathbf{1}}$ (pu)	1.014757	1.079382	1.080374	1.017385	1.076876
${\mathit{V}}_{\mathbf{2}}$ (pu)	0.978060	1.067958	1.066919	1.017385	1.070644
${\mathit{V}}_{\mathbf{3}}$ (pu)	0.953511	1.063965	1.034293	1.017385	1.073361
${\mathit{V}}_{\mathbf{6}}$ (pu)	0.978504	1.058594	0.990386	1.017385	1.075436
${\mathit{V}}_{\mathbf{8}}$ (pu)	1.007692	1.068161	1.020260	1.017385	1.080711
${\mathit{V}}_{\mathbf{9}}$ (pu)	0.968363	1.044105	1.020613	0.995944	1.061063
${\mathit{V}}_{\mathbf{12}}$ (pu)	0.989555	1.059060	1.036204	1.017385	1.076054
${\mathit{T}}_{\mathbf{19}}$ (4-18)	0.913049	1.003605	0.943555	1.017385	1.021544
${\mathit{T}}_{\mathbf{20}}$ (4-18)	0.900000	0.984822	1.085978	1.017385	1.076384
${\mathit{T}}_{\mathbf{31}}$ (21-20)	0.974656	1.068002	0.948072	0.971602	1.097143
${\mathit{T}}_{\mathbf{35}}$ (24-25)	0.908226	1.040848	0.940854	1.017385	1.090065
${\mathit{T}}_{\mathbf{36}}$ (24-25)	0.900001	1.069429	0.938419	1.017385	1.039761
${\mathit{T}}_{\mathbf{37}}$ (24-26)	1.001831	1.060441	0.946427	1.017385	1.094090
${\mathit{T}}_{\mathbf{41}}$ (7-29)	0.949960	1.065368	0.910877	0.929103	0.994970
${\mathit{T}}_{\mathbf{46}}$ (34-32)	0.969328	1.045765	1.043665	1.017385	1.072752
${\mathit{T}}_{\mathbf{54}}$ (11-41)	0.950228	0.961513	1.001356	0.971472	0.997184
${\mathit{T}}_{\mathbf{58}}$ (15-45)	0.900430	1.029488	0.979318	1.017385	1.008150
${\mathit{T}}_{\mathbf{59}}$ (14-46)	0.906230	1.022964	1.068994	0.997602	0.965880
${\mathit{T}}_{\mathbf{65}}$ (10-51)	0.947442	1.053327	1.067980	1.017385	1.051045
${\mathit{T}}_{\mathbf{66}}$ (13-49)	0.900006	1.048173	1.021711	1.017385	1.049060
${\mathit{T}}_{\mathbf{71}}$ (11-43)	0.911247	1.020260	1.040798	0.916027	1.099736
${\mathit{T}}_{\mathbf{73}}$ (40-56)	0.939646	1.024146	0.993105	1.005665	0.990529
${\mathit{T}}_{\mathbf{76}}$ (39-57)	0.963492	0.990351	0.954516	1.010802	1.020521
${\mathit{T}}_{\mathbf{80}}$ (9-55)	0.948951	1.099189	0.963165	0.958351	1.001199
${\mathit{Q}}_{\mathbf{18}}$ (MVAR)	0.000000	8.482640	1.053096	21.99263	13.08836
${\mathit{Q}}_{\mathbf{25}}$ (MVAR)	0.000000	26.36727	21.99351	21.99263	19.31112
${\mathit{Q}}_{\mathbf{53}}$ (MVAR)	28.42860	27.78365	28.737910	21.992630	20.723070
Power Losses (MW)	50.57143	32.60550	27.51680	47.24557	39.26724
Voltage Deviation(pu)	1.280390	1.742980	2.730820	1.692840	1.897380

Figure 16. The AVOA convergence characteristics with the other compared methods for case 7.

Figure 17. The AVOA voltage profile with the other compared methods for case 7.

In this case, all proposed algorithms succeeded in saving real power losses, but only AVOA was excellent against all other proposed algorithms and succeeded in achieving real and reactive power balance as shown in Table 21.

Table 21. Balance of active and reactive power for proposed algorithms with case 7.

Method	Balance of Active Power			Balance of Reactive Power
	Load (MW)	Generation (MW)	Loss (MW)	Load (MVAR)	Generation (MVAR)	Shunt VAR Compensation (MVAR)	Shunt Admittance and Branch Charging (MVAR)	Loss (MVAR)
PDO	1250.800	1279.878	50.571	336.400	326.620	28.429	143.501	183.683
RSO	1250.800	1276.460	32.605	336.400	257.251	62.634	155.551	146.074
AVOA	1250.800	1278.323	27.517	336.400	281.251	51.785	163.840	160.487
ROA	1250.800	1278.710	47.246	336.400	274.348	65.978	131.320	154.731
SPO	1250.800	1276.611	39.267	336.400	263.156	53.123	157.923	151.276

4.2.2. Case 8: Minimization of Transmission Power Loss for Standard IEEE 57 Bus Incorporating RES

Implementing the AVOA method when accommodating the DG at node 14 even more significantly lowered power losses, reaching 25.61081 MW as stated in Table 22. These results demonstrate the AVOA algorithm’s advantage over other heuristic approaches for tackling this sort of issue, and they support the effectiveness of the suggested DG placement strategy. In Figure 18, real power losses using the AVOA method are steeply converged and fully achieve the optimal solution at 25 rounds, exhibiting the AVOA algorithm’s quick convergence rate. Also, the majority of obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 19. The results demonstrate the AVOA algorithm’s advantage over other heuristic approaches for tackling this sort of issue, and they support the effectiveness of the suggested DG placement strategy.

Table 22. Optimal IEEE 57-bus test system control variables for reducing power losses with incorporation of RES.

Control Variables		PDO	RSO	AVOA	ROA	SPO
${\mathit{V}}_{\mathbf{1}}$ (pu)		1.077386	0.900000	0.993069	0.983857	1.100000
${\mathit{V}}_{\mathbf{2}}$ (pu)		1.065985	0.900000	0.988758	0.983857	1.097155
${\mathit{V}}_{\mathbf{3}}$ (pu)		1.060514	0.900000	1.016191	0.983857	1.098401
${\mathit{V}}_{\mathbf{6}}$ (pu)		1.055375	0.900000	1.027202	0.983857	1.100000
${\mathit{V}}_{\mathbf{8}}$ (pu)		1.081048	0.900000	1.033914	0.983857	1.100000
${\mathit{V}}_{\mathbf{9}}$ (pu)		1.053142	0.900000	1.013254	0.983857	1.076958
${\mathit{V}}_{\mathbf{12}}$ (pu)		1.057999	0.900000	1.020582	0.983857	1.100000
${\mathit{T}}_{\mathbf{19}}$ (4-18)		1.051213	0.900000	0.974094	0.983857	1.100000
${\mathit{T}}_{\mathbf{20}}$ (4-18)		1.051761	0.900000	1.010956	0.983857	1.070695
${\mathit{T}}_{\mathbf{31}}$ (21-20)		1.012635	0.900000	1.010942	0.983857	1.099511
${\mathit{T}}_{\mathbf{35}}$ (24-25)		1.058798	0.900000	1.010118	0.983857	1.100000
${\mathit{T}}_{\mathbf{36}}$ (24-25)		1.024535	0.900000	0.998917	0.983857	1.100000
${\mathit{T}}_{\mathbf{37}}$ (24-26)		0.992774	0.900000	1.029908	0.983857	1.100000
${\mathit{T}}_{\mathbf{41}}$ (7-29)		1.018937	0.900000	1.051328	0.983857	1.072223
${\mathit{T}}_{\mathbf{46}}$ (34-32)		1.033953	0.900000	1.003320	0.983857	1.064969
${\mathit{T}}_{\mathbf{54}}$ (11-41)		0.9933	0.900000	0.987018	0.983857	1.100000
${\mathit{T}}_{\mathbf{58}}$ (15-45)		1.085949	0.900000	0.996392	0.983857	1.065345
${\mathit{T}}_{\mathbf{59}}$ (14-46)		1.083144	0.900000	1.000851	0.983857	1.100000
${\mathit{T}}_{\mathbf{65}}$ (10-51)		0.983436	0.900000	0.971795	0.983857	1.100000
${\mathit{T}}_{\mathbf{66}}$ (13-49)		1.077092	0.900000	1.029415	0.983857	1.100000
${\mathit{T}}_{\mathbf{71}}$ (11-43)		0.999101	0.900000	0.997948	0.983857	0.981359
${\mathit{T}}_{\mathbf{73}}$ (40-56)		0.949339	0.900000	1.019042	0.983857	1.100000
${\mathit{T}}_{\mathbf{76}}$ (39-57)		0.999857	0.900000	0.976653	0.983857	0.932947
${\mathit{T}}_{\mathbf{80}}$ (9-55)		1.010469	0.900000	1.015313	0.983857	1.099985
${\mathit{Q}}_{\mathbf{18}}$ (MVAR)		0.000000	0.000000	28.94143	6.976593	29.99944
${\mathit{Q}}_{\mathbf{25}}$ (MVAR)		20.40472	0.000000	13.72425	7.742438	30.00000
${\mathit{Q}}_{\mathbf{53}}$ (MVAR)		22.54991	4.195559	20.54241	3.631028	15.04518
RES Location and Size	Location	22.00000	14.00000	14.00000	22.00000	29.00000
	MW	202.6786	0.000000	220.1538	46.53750	0.000000
	MVAR	0.000000	134.2083	131.2234	74.03436	32.52395
Power Losses (MW)		29.64456	36.34125	25.61081	39.88670	28.32156
Voltage Deviation (pu)		1.746140	3.979550	1.370980	1.337450	2.328250

Figure 18. The AVOA convergence characteristics with the other compared methods for case 8.

Figure 19. The AVOA voltage profile with the other compared methods for case 8.

4.3. Standard IEEE-118 Bus Test System

This work considers a large-scale IEEE 118-bus test system to evaluate the scalability of the proposed methods, which has 77 control variables including 186 transmission lines, 54 generators, 14 shunt VAR compensators, 9 tap changing transformers, and a total system demand of 4242 MW. Furthermore, in this research, the IEEE 118-bus system is restricted as follows: the voltage magnitude range for all generator buses is 0.94 pu to 1.06 pu. The tap-changing transformers, on the other hand, range in power from 0.9 pu to 1.1 pu. Likewise, the shunt VAR compensator limits are assumed to be between 0 and 30 MVAR and its details are described in [49,50,51].

4.3.1. Case 9: Minimization of Multi-Objective Functions (Voltage-Level Deviation, Operational Cost, and Transmission Power Loss) with Continuous Control Variable Settings in IEEE-118 Bus

As explained in case 5, the applicability of the proposed AVOA algorithm for solving a multi-objective ORPD problem is presented and tested on the IEEE-118 bus large-scale power system and its results are illustrated in Table 23 and show that the AVOA algorithm outperforms these prior methods with the minimum multi-objective value of 265394.15. In Figure 20, the AVOA method is converged and fully accomplishes the best solution against other compared techniques. All obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 21.

Table 23. Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with continuous control variable settings in IEEE-118 bus.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	0.940000	0.940000	0.940000	1.023631	1.060000
$V_{G4}$ (pu)	0.940000	0.940000	0.940000	1.000612	1.060000
$V_{G6}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G8}$ (pu)	1.009497	0.940000	0.940000	0.986338	1.060000
$V_{G10}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G12}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G15}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G18}$ (pu)	0.940000	0.940000	0.940000	1.048302	1.060000
$V_{G19}$ (pu)	0.967036	0.940000	0.940000	0.974983	1.060000
$V_{G24}$ (pu)	1.059021	0.940000	0.940000	0.941251	1.060000
$V_{G25}$ (pu)	1.051391	0.940000	0.940000	0.941251	1.060000
$V_{G26}$ (pu)	0.940000	0.940000	0.940000	1.009417	1.060000
$V_{G27}$ (pu)	0.940000	0.940000	0.940000	1.024970	1.060000
$V_{G31}$ (pu)	0.940000	0.940000	0.940000	1.046760	1.060000
$V_{G32}$ (pu)	0.968184	0.940000	0.940000	1.018317	1.060000
$V_{G34}$ (pu)	0.940000	0.940000	0.940000	1.046706	1.060000
$V_{G36}$ (pu)	0.960118	0.940000	0.940000	0.941251	1.060000
$V_{G40}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G42}$ (pu)	0.940000	0.940000	0.940000	1.038727	1.060000
$V_{G46}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G49}$ (pu)	0.966030	0.940000	0.940000	0.941251	1.060000
$V_{G54}$ (pu)	0.944383	0.940000	0.940000	1.000019	1.060000
$V_{G55}$ (pu)	0.940000	0.940000	0.940000	0.957834	1.060000
$V_{G56}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G59}$ (pu)	0.967256	0.940000	0.940000	0.941251	1.060000
$V_{G61}$ (pu)	0.946649	0.940000	0.940000	0.941251	1.060000
$V_{G62}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G65}$ (pu)	0.940000	0.940000	0.940000	1.049611	1.060000
$V_{G66}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G69}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G70}$ (pu)	1.032090	0.940000	0.940000	1.047609	1.060000
$V_{G72}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G73}$ (pu)	0.940000	0.940000	0.940000	0.983141	1.060000
$V_{G74}$ (pu)	0.959571	0.940000	0.940000	0.941251	1.060000
$V_{G76}$ (pu)	1.005403	0.940000	0.940000	0.941251	1.060000
$V_{G77}$ (pu)	0.961494	0.940000	0.940000	0.958203	1.060000
$V_{G80}$ (pu)	0.956331	0.940000	0.940000	0.941251	1.060000
$V_{G85}$ (pu)	1.018228	0.940000	0.940000	0.957806	1.060000
$V_{G87}$ (pu)	0.973018	0.940000	0.940000	0.941251	1.060000
$V_{G89}$ (pu)	0.960041	0.940000	0.940000	0.941251	1.060000
$V_{G90}$ (pu)	1.036898	0.940000	0.940000	0.941251	1.060000
$V_{G91}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G92}$ (pu)	0.940167	0.940000	0.940000	0.980899	1.060000
$V_{G99}$ (pu)	0.940000	0.940000	0.940000	0.949593	1.060000
$V_{G100}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G103}$ (pu)	0.944658	0.940000	0.940000	1.049690	1.060000
$V_{G104}$ (pu)	1.014380	0.940000	0.940000	0.998008	1.060000
$V_{G105}$ (pu)	0.947759	0.940000	0.940000	0.941251	1.060000
$V_{G107}$ (pu)	0.940000	0.940000	0.940000	0.979954	1.060000
$V_{G110}$ (pu)	0.940000	0.940000	0.940000	0.941251	1.060000
$V_{G111}$ (pu)	0.940000	0.940000	0.940000	1.047427	1.060000
$V_{G112}$ (pu)	0.942125	0.940000	0.940000	1.024985	1.060000
$V_{G113}$ (pu)	1.038235	0.940000	0.940000	0.941251	1.060000
$V_{G116}$ (pu)	0.940502	0.940000	0.940000	0.941251	1.060000
$T$(5–8)	1.097681	0.900000	0.900000	1.026183	1.100000
$T$(25–26)	0.900000	0.900000	0.900000	0.901198	1.100000
$T$(17–30)	0.900000	0.900000	0.900000	0.901198	1.100000
$T$(37–38)	0.964377	0.900000	0.900000	0.977837	1.100000
$T$(59–63)	0.983106	0.900000	0.900000	0.901198	1.100000
$T$(61–64)	0.900000	0.900000	0.900000	1.034980	1.100000
$T$(65–66)	0.918860	0.900000	0.900000	0.901198	1.100000
$T$(68–69)	1.097386	0.900000	0.900000	0.980136	1.100000
$T$(80–81)	1.081179	0.900000	0.900000	0.911889	1.100000
$Q_{\mathrm{C}34}$ (MVAR)	30.000000	0.300615	16.558994	0.000000	30.000000
$Q_{\mathrm{C}44}$ (MVAR)	30.000000	0.000000	19.977174	1.034545	30.000000
$Q_{\mathrm{C}45}$ (MVAR)	30.000000	0.000000	0.000000	0.000000	30.000000
$Q_{\mathrm{C}46}$ (MVAR)	30.000000	0.000000	14.754032	0.000000	30.000000
$Q_{\mathrm{C}48}$ (MVAR)	0.000000	0.000000	0.362552	0.000000	30.000000
$Q_{\mathrm{C}74}$ (MVAR)	0.000000	0.000000	18.454059	10.947679	30.000000
$Q_{\mathrm{C}79}$ (MVAR)	0.000000	0.051377	6.558655	22.525961	30.000000
$Q_{\mathrm{C}82}$ (MVAR)	30.000000	0.000000	17.561728	0.000000	3.566820
$Q_{\mathrm{C}83}$ (MVAR)	0.000000	0.000000	24.318348	0.000000	30.000000
$Q_{\mathrm{C}105}$ (MVAR)	0.000000	0.000000	23.581277	18.371447	30.000000
$Q_{\mathrm{C}107}$ (MVAR)	27.017612	0.000000	12.489382	0.000000	30.000000
$Q_{\mathrm{C}110}$ (MVAR)	0.000000	0.059911	11.410370	0.000000	30.000000
Voltage Deviation (pu)	3.323260	4.722720	4.619960	3.144940	3.793920
Cost (USD/h)	21,150.22192	14,385.27743	14,385.27743	55,189.38505	24,022.84895
Power Losses (MW)	213.86978	187.78444	187.62808	287.49896	149.84819
Multi-objective function	363,863.9088	269,452.7983	265,394.1500	4,314,259.2273	578,631.4217

Figure 20. The AVOA convergence characteristics with the other compared methods for case 9.

Figure 21. The AVOA voltage profile with the other compared methods for case 9.

4.3.2. Case 10: Minimization of Multi-Objective Functions (Voltage-Level Deviation, Operational Cost, and Transmission Power Loss) with Discrete Control Variable Settings in IEEE-118 Bus

For a small-scale IEEE-30 network presented previously in case 5, the problem is solved as multi-objective functions considering continuous control variables and the findings show a minimization of the objective function with a small value more than that obtained with discrete control variables presented in case 6. Therefore, on the large-scale network, the same cases are studied to prove the validity of the results and the ability of the AVOA optimizer for handling and solving the ORPD optimization issues. In this case, with the IEEE-118 bus test large-scale power system, generators’ voltage set points are typically in the (0.95, 1.1) pu range, transformer taps vary within the (0.9,1.1) pu range with steps of 0.01 discretized into 20 levels as (0.9 0.91 0.92...until, 1.1), and a 300 MVAR maximum capacity for all capacitor banks with steps of 0.3 MVAR. As explained in case 6, to prove the ability of the proposed AVOA algorithm for solving a multi-objective function dealing with these continuous and discrete control variables, all results are illustrated in Table 24 and prove that the AVOA algorithm outperforms these prior methods with a minimum multi-objective value of 269,219.7522. In Figure 22, the AVOA method is steeply converged and fully accomplishes the best solution against other compared techniques. All obtained solutions employing proposed heuristic optimization techniques are particularly viable, owing to the voltage magnitude accepted at all system load buses, as shown in Figure 23.

Table 24. Minimization of multi-objective functions (voltage-level deviation, operational cost, and transmission power loss) with discrete control variable settings in IEEE-118 bus.

Control Variables	PDO	RSO	AVOA	ROA	SPO
$V_{G1}$ (pu)	1.057845	0.940000	0.940000	1.060000	1.060000
$V_{G4}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G6}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G8}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G10}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G12}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.059344
$V_{G15}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G18}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G19}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G24}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G25}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G26}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G27}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G31}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G32}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G34}$ (pu)	1.020603	0.940000	0.940000	1.060000	1.060000
$V_{G36}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.045414
$V_{G40}$ (pu)	0.987362	0.940000	0.940000	1.060000	1.060000
$V_{G42}$ (pu)	1.012975	0.940000	0.940000	1.060000	1.060000
$V_{G46}$ (pu)	1.058959	0.940000	0.940000	1.060000	1.060000
$V_{G49}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G54}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.047553
$V_{G55}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G56}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G59}$ (pu)	0.942984	0.940000	0.940000	1.060000	1.060000
$V_{G61}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G62}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G65}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G66}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G69}$ (pu)	0.940804	0.940000	0.940000	1.060000	1.060000
$V_{G70}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G72}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G73}$ (pu)	1.052856	0.940000	0.940000	1.060000	1.060000
$V_{G74}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G76}$ (pu)	1.060000	0.940000	0.940000	1.060000	1.060000
$V_{G77}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G80}$ (pu)	0.940338	0.940000	0.940000	1.060000	1.060000
$V_{G85}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G87}$ (pu)	0.940051	0.940000	0.940000	1.060000	1.060000
$V_{G89}$ (pu)	0.973574	0.940000	0.940000	1.060000	1.060000
$V_{G90}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G91}$ (pu)	1.000984	0.940000	0.940000	1.060000	1.060000
$V_{G92}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.047490
$V_{G99}$ (pu)	0.943547	0.940000	0.940000	1.060000	1.060000
$V_{G100}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G103}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G104}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G105}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G107}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G110}$ (pu)	1.045764	0.940000	0.940000	1.060000	1.060000
$V_{G111}$ (pu)	0.943852	0.940000	0.940000	1.060000	1.060000
$V_{G112}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$V_{G113}$ (pu)	1.058386	0.940000	0.940000	1.060000	1.060000
$V_{G116}$ (pu)	0.940000	0.940000	0.940000	1.060000	1.060000
$T$(5-8)	1.100000	0.900000	0.900000	1.090000	1.100000
$T$(25-26)	0.900000	0.900000	0.900000	1.090000	1.100000
$T$(17-30)	0.900000	0.900000	0.900000	1.090000	1.100000
$T$(37-38)	0.900000	0.900000	0.900000	1.090000	1.050000
$T$(59-63)	0.900000	0.900000	0.900000	1.090000	1.100000
$T$(61-64)	0.900000	0.900000	0.900000	1.090000	1.100000
$T$(65-66)	0.900000	0.900000	0.900000	1.090000	1.100000
$T$(68-69)	0.960000	0.900000	0.900000	1.090000	1.100000
$T$(80-81)	0.900000	0.900000	0.900000	1.090000	1.100000
$Q_{\mathrm{C}34}$ (MVAR)	0.000000	0.000000	0.000000	30.000000	25.800000
$Q_{\mathrm{C}44}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	28.200000
$Q_{\mathrm{C}45}$ (MVAR)	0.000000	4.800000	4.800000	30.000000	16.500000
$Q_{\mathrm{C}46}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	30.000000
$Q_{\mathrm{C}48}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	1.200000
$Q_{\mathrm{C}74}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	5.100000
$Q_{\mathrm{C}79}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	30.000000
$Q_{\mathrm{C}82}$ (MVAR)	0.000000	1.800000	1.800000	30.000000	30.000000
$Q_{\mathrm{C}83}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	28.200000
$Q_{\mathrm{C}105}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	30.000000
$Q_{\mathrm{C}107}$ (MVAR)	0.000000	0.000000	0.000000	30.000000	30.000000
$Q_{\mathrm{C}110}$ (MVAR)	30.000000	0.000000	0.000000	30.000000	15.000000
Voltage Deviation (pu)	3.763680	4.712360	4.712360	3.799160	3.689060
Cost (USD/h)	28,881.62974	23,950.97836	13,386.31841	16,049.54453	15,183.90299
Power Losses (MW)	254.090000	187.734780	187.734780	148.856000	149.640070
Multi-objective function	892,353.9000	269,219.7522	269,219.7522	486,760.0000	571,536.9707

Figure 22. The AVOA convergence characteristics with the other compared methods for case 10.

Figure 23. The AVOA voltage profile with the other compared methods for case 10.

5. Conclusions

In this research, an efficient optimization optimizer named AVOA is developed for handling ORPD optimization issues. Furthermore, ten approaches for addressing a single- and multi-objective ORPD issue were proposed. The suggested methods were evaluated and confirmed on a variety of standard testing systems including IEEE (30, 57, and 118) bus systems with and without RES, and considering two distinct discrete and continuous types of optimization variables. The results proved that the optimal allocation of renewable energy sources simultaneously with the optimal power flow gives better results than that if it happens separately. This was achieved by adding the location and size of the distributed generation as a control variable and increasing the dimension of the ORPD problem. The results validate AVOA’s excellent performance and efficacy in tackling ORPD optimization challenges, when solving a multi-objective ORPD optimization problem considering continuous control variables, and all the findings show a minimization of the objective function with a small value more than that obtained with discrete control variables and prove the validity of the results and the ability of the AVOA optimizer to handle and solve ORPD optimization issues. Additionally, the best-achieved results from the IEEE-standard systems prove that the suggested approach overcomes the challenges associated with this test system. The AVOA’s results were also compared to other newly developed meta-heuristic approaches. The simulated findings demonstrate that the AVOA surpasses other compared strategies for handling ORPD when it comes to robustness and efficacy. In upcoming work, the suggested AVOA can be modified or mixed with other metaheuristic algorithms for addressing other complex optimization problems in dissimilar fields, for instance optimally distributed generation allocation seeing the vagueness of RES, optimal hybrid RES planning and design, the parameter estimation of fuel cells, and photovoltaic models.