Optimal Allocation of Biomass Distributed Generators Using Modified Hunger Games Search to Reduce CO₂ Emissions

Abstract

Biomass is a renewable energy source because it is contained in organic material such as plants. This paper introduces a modified hunger games search for solving global optimization and biomass distributed generator problems. The hunger search algorithm is a very recent optimization algorithm. Despite its merits, it still needs some modifications. The proposed approach includes a new binary τ-based crossover strategy with satisfaction fulfillment step mechanisms. This new algorithm is designed to improve the original hunger games search algorithm by addressing some of its shortcomings, specifically, in solving problems related to global optimization such as finding the best possible solutions for biomass distributed generators. To assess the power of the new approach, its performance was evaluated on the IEEE CEC’2020 test suite against five recent and competitive algorithms. This comparison process included applying the Wilcoxon sign rank and Friedman statistical tests. Reducing the system losses and enhancing the network’s voltage profile are two main issues in the stability of radial distribution networks. Optimal allocation of biomass distributed generators in radial distribution networks can not only improve their stability but also guarantee good service to the customers. Consequently, this research work suggests an effective strategy based on the proposed approach to produce the optimal positions, sizes, and power factors of the biomass distributed generators in the network. Accordingly, the target is to mitigate the network’s active power loss such that the power flow and the bus voltage have to be maintained at their standard limits. Three distribution networks were considered for validating the superiority of the new proposed algorithm. These networks are the IEEE 33-bus, IEEE 69-bus, and IEEE 119-bus. The obtained results were compared with the gravitational search algorithm, whale optimization algorithm, grey wolf optimizer, Runge Kutta method, and the original hunger search algorithm. The new approach outperformed the other considered approaches in obtaining the optimal parameters, which mitigated the power loss to 11.6300, 5.2291, and 145.489 kW, with loss reduction of 94.49%, 97.68%, and 88.79% for the three networks, respectively.

1. Introduction

Nowadays, everybody who lives on Earth is a witness to the vast and terrible changes in the world’s climate. Directly or indirectly, these changes have negative impacts on all living organisms, not only human beings. Therefore, energy sources that produce carbon dioxide (CO₂), which is the main cause of this problem, have to be used as little as possible [1]. From another perspective, securing different renewable and green sources of energy has become a topic of extreme importance for every single country. This can clearly be concluded following the start of the Russia–Ukraine conflict. Thus, this new type of energy source has two main features. First, it will act as a substitution for fossil fuels and, second, it will be an environmentally friendly source. In the same context, the United Nations conducts a conference every year to discuss the climate change dilemma. The last conference meeting was in Sharm El Sheikh, Egypt, on 6–18 November 2022 (https://cop27.eg/, accessed on 15 November 2022), which aimed to unite the world to tackle these changes. This conference’s goal was to accelerate global climate actions through emissions reduction based on the agreements and commitments of the previous meetings at COP26, which was held in Glasgow, UK, 2021. The principal goal of the conferences is still to decrease the world’s global warming temperature by 1.5 °C. This motivates researchers to work in the same direction by developing new sources of energy that satisfy the two main features mentioned before. Biomass is a renewable energy sources as it is contained in organic material such as plants [2]. It is safe because it is carbon-free. Moreover, it is economical, and hence secures a source of profits for the producers. Biomass gasification is a way to produce fuel gas from these materials by undertaking some chemical operations. Consequently, biomass distributed generators (BDGs) are a more promising form of technology for connection with utilities than traditional distributed generators. However, the success of using other sources of energy, such as biomass-based fuel, will definitely help in reducing the massive use of CO₂-based fuels, and hence reduce CO₂ emissions.

Fortunately, there is a very close connection between biomass gasification and the reduction in CO₂ emissions. Intuitively, increasing the use of biomass fuels will definitely decrease the use of fossil fuels, and hence decrease CO₂ emissions. Korberg et al. identified the suitability of biomass gasification to employ biomass resources in an efficient way [3]. They studied and analyzed the hourly energy in the renewable energy system for Denmark and Europe with the goal of reaching a net-zero energy system by the year 2050. Their results indicated that the use of biofuels such as biomass gasification will reduce the energy system costs, and hence the fuel’s costs in comparison to those of CO₂-based fuels.

Sun et al. conducted an experiment using the chemical looping deoxygenated gasification (CLDG) process to demonstrate the role of deoxygenation on syngas refinement and CO₂ utilization [4]. They concluded that the value of syngas yields is increased by 16.37% and the CO₂ concentration is decreased by 44.16%. Habibollahzade et al. applied tri-objective grey wolf optimization to determine the most suitable feedstock for high cold gas and exergy efficiencies and low CO₂ emissions of the gasification system [5]. Their goal was to maximize the cold gas and exergy efficiencies and minimize the CO₂ emissions. González and Sandoval conducted a review study to investigate the restrictions that govern the sustainability of the gasification process [6]. They found that there are three crucial factors that affect the process of small-scale power generators efficiencies. These factors are land availability, policies development, and syngas effects. Caglar et al. studied the use of three simultaneous gasifying agents, namely, air, H₂O, and CO₂, in the gasification process [7]. The performance was indicated by the product yields, cold gas efficiency, exergy efficiency, CO₂ emissions, and heat requirement of the gasifier. The results proved that using CO₂ as a gasification agent provides many benefits for CO₂ recycling and biomass gasification. Pandey et al. studied the influences of varying the concentrations of CO₂ and oxygen in the gasifying agent on the performance of the gasifier [8]. They concluded that involving the CO₂ in the gasification process helps the reaction of converting the CO₂ into CO.

The optimization operation can be applied to achieve the best values of the controlling parameters that produce either the minimum or the maximum value of the cost (objective) function [9]. Furthermore, optimization has seen remarkable advances in various scientific fields [10,11]. In optimization problems, the formulation of a single-objective function is represented mathematically according to [11]:

$$\mathrm{minimize}\mathrm{or}\mathrm{maximize}\mathrm{f}\left(\overrightarrow{X}\right)\mathrm{subject}\mathrm{to}:L{B}_j\hspace{1em}{X}_j\hspace{1em}U{B}_j,\mathrm{j}=1,2,\dots ,\mathrm{D}.$$

(1)

where $\mathrm{f}\left(\overrightarrow{X}\right)$ represents the cost function, $\overrightarrow{X}$ denotes the decision variables’ vector with dimension D, and every variable $X_j$ is bounded by its lower $L{B}_j$ and upper $U{B}_j$ limits of the inputs’ search area.

In population-based searching approaches, the algorithm begins by generating a set of randomly proposed solutions. Evolutionary algorithms (EAs) are a type of population-based algorithm in which the candidate solutions are evolved through three processes to guide the solutions towards new and promising search areas within the search boundaries. These processes are the selection, mutation, and crossover methods. Recently, the hunger games search (HGS) has increased in popularity in many fields and applications of engineering problems. In [12], the authors developed an enhanced variant of the hunger games search called the IHGS algorithm; the authors employed the IHGS to obtain the values that optimize the performance of a buck converter system. In [13], the authors combined the hunger games search and the whale optimization algorithms to develop a hybrid optimizer. This new optimizer was applied to solve global optimization applications, where the merits of the two algorithms were merged together to produce a new searching strategy. In [14], the authors combined the HGS with a multi-strategy (MS) framework to solve the global optimization and feature selection process. In the machine learning area, the HGS was utilized as a searching algorithm to obtain the internal parameters’ values that optimize the random vector functional link (RVFL) [15]. In [16], a hybrid optimizer that combines the HGS algorithm with the differential evolution (DE) algorithm was introduced to solve the HGS issues such as the local minima and premature convergence.

Many researchers have dealt with the biomass DG (BDG) problem by considering it as a constrained optimization function. Some of these approaches can be reported as follows: In [17], the authors considered the distribution network enhancement and the environmental emissions as the targets while integrating the BDGs into the radial distribution networks (RDNs). Moreover, the power loss mitigation, the energy sales excess, and the annual costs were considered in the problem. Two distribution networks were analyzed in that work, which are IEEE 33-bus and 141-bus networks. The authors in [18] integrated the BDGs in an RDN via identifying the capacities and locations of photovoltaic (PV), wind, and BDGs for an introduced Q−PQV bus pair. Additionally, capacitor banks were installed in the network using a probabilistic approach [19]. Furthermore, the solar irradiance and random wind behaviors were represented by the beta and Weibull distributions. To install wind, solar, and BDGs in an RDN, the authors in [20] represented the active power loss, branch current capacity, environmental impact mitigation index, voltage deviation, and economic index as a multi-objective optimization problem. Moreover, the problem was solved by a particle swarm optimizer (PSO). In [21], the authors obtained the optimal parameters, such as the sites and sizes of the biomass-based DGs in distribution networks, using symbiotic organisms’ search. Moreover, the technical and economic aspects related to the BDGs were considered. Their analysis was applied to the two networks of the IEEE 33-bus and IEEE 69-bus. The authors in [22,23] analyzed the IEEE 13-bus network to conduct the power flow in a radial network with BDGs by combining the shuffled frog leaping optimizer and the probabilistic three-phase power flow. The authors in [24] employed triple PSO and an ordinal optimizer to determine the best operating parameters of the network with integrated PV, WT, and BDG that minimize the total cost, which was considered as the objective function. Fathy [25] presented a new approach of the artificial hummingbird algorithm (AHA) to allocate the biomass DGs in a radial network such that the power loss and voltage deviation are minimized. Moreover, a multi-objective AHA was introduced to achieve both targets. In [26], a capuchin search algorithm (CapSA) was used to solve the problem of integrating biomass DGs in a distribution network via identifying their sizes, locations, and power factors. The power loss reduction was the target in that work. A chaotic student psychology-based optimizer was introduced in [27] to determine the optimal sites and sizes of PV-battery, wind-battery, and biomass systems in radial networks. Raut et al. [28] employed an improved equilibrium optimizer to reconfigure a dynamic network with PV, wind, and biomass DGs. Moreover, a multi-objective problem was solved to minimize the power loss and maximize the profit. Khasanov et al. [29] presented an improved artificial ecosystem optimizer to solve the problem of integrating DGs in a distribution network such that the active power loss was mitigated.

In the same context, this work introduces a newly proposed method called the modified HGS (mHGS) based on the binary τ-Based Crossover (τBX) to enrich the children’s solutions with valuable information from the candidate parent solutions; consequently, the generated children inherit the shared information from the parent’s characteristics.

This strategy employs the fact that better offspring will inherit the features of better parents. From this postulation, a new solution will be created by exchanging information between the solutions and their better parents to hopefully generate better offspring. The performance of mHGS is applied to deal with the complex objective function in the IEEE CEC’2020 test suite, as well as a real application, such as integrating the BDGs into electric distribution networks. Three RDNs of IEEE 33-bus, IEEE 69-bus and IEEE 119-bus were analyzed to mitigate the active power loss via identifying the optimal positions, sizes, and power factors of the installed units. In comparison to several other metaheuristic algorithms, including the original hunger games search (HGS), the resulting values revealed that the new mHGS is outstanding.

The contributions of the paper can be outlined as follows:

• We proposed an efficient method called mHGS;
• The proposed mHGS introduces the mechanisms of the crossover strategy to boost algorithm diversity;
• The proposed mHGS was evaluated on the CEC’2020 test suite and compared with other competitors;
• mHGS was applied to three RDNs of the IEEE 33-bus, IEEE 69-bus, and IEEE 119-bus;
• The superiority of the proposed mHGS technique is demonstrated.

The work presented in this paper is organized as follows: Section 2 briefly introduces the standard hunger games search (HGS). Section 3 introduces the mathematical equations and the associated pseudo-code needed to fully understand the proposed mHGS method. Additionally, Section 4 discusses the resulting experimental values obtained by the new mHGS applied to the CEC’2020 test suite. Section 5 presents the second experimental results on three RDNs of the IEEE 33-bus, IEEE 69-bus, and IEEE 119-bus. Finally, Section 6 concludes the presented work and introduces some future prospects.

2. Hunger Games Search (HGS)

The HGS algorithm is a recent optimizer that was developed by Yang et al. [30]. It simulates the hunger-related activities of creatures and their behavioral choices. The authors utilized the approach of hunger to design an adaptive weight, and the consequences of the hunger are employed at each step of the search procedure to establish an efficient search mechanism.

2.1. Approach Food

In the HGS algorithm, the individual’s communication and foraging behavior can be represented as follows [30]:

$$\overrightarrow{X\left(t+1\right)}=\left\{\begin{array}{cc}\overrightarrow{X\left(t\right)}\cdot \left(1+\mathrm{randn}\left(1\right)\right),& r_1<l\hfill \\ \overrightarrow{W_1}\cdot \overrightarrow{X_b}+\overrightarrow{R}\cdot \overrightarrow{W_2}\cdot \left|\overrightarrow{X_b}-\overrightarrow{X\left(t\right)}\right|,& r_1>l,r_2>E\hfill \\ \overrightarrow{W_1}\cdot \overrightarrow{X_b}-\overrightarrow{R}\cdot \overrightarrow{W_2}\cdot \left|\overrightarrow{X_b}-\overrightarrow{X\left(t\right)}\right|,& r_1>l,r_2<E\hfill \end{array}\right.$$

(2)

where $\overrightarrow{R}$ is in the range of [−𝑎, 𝑎], $r_1$, $r_2$ and $\mathrm{randn}\left(1\right)$ are normally distributed random generators having limits are in the range of [0, 1], 𝑡 refers to the index of the current step, $\overrightarrow{W_1}$ and $\overrightarrow{W_2}$ denote the weight values of hunger, $\overrightarrow{X_b}$ indicates the location of the best solution in the current step, $\overrightarrow{X\left(t\right)}$ denotes the current solution’s position, and $l$ is a constant parameter.

The parameter 𝐸 is evaluated as follows [30]:

$$E=\sec \mathrm{h}\left(\left|F\left(i\right)-BF\right|\right)$$

(3)

where $i\in 1,2,\dots ,n,F\left(i\right)$ refers to the objective function value of the i^th solution, $BF$ is the resulting best objective value in the current step, and $\sec \mathrm{h}$ is a hyperbolic function which takes the formula $\left(\sec \mathrm{h}\left(x\right)=\frac{2}{e^x+e^{-x}}\right)$.

The parameter $\overrightarrow{R}$ can be written according to [30]:

$$\overrightarrow{R}=2\times a\times \mathrm{rand}-a$$

(4)

$$a=2\times \left(1-\frac{t}{Max\_FEs}\right)$$

(5)

where $\mathrm{rand}$ is a normally distributed randomly generated number limited by the range of [0, 1], and $Max\_FEs$ denotes the uppermost number of function evaluations.

2.2. Hunger Role

The hunger features of individuals in the search space are interpreted mathematically by weights for each search individual. The formula of $\overrightarrow{W_1}$ in is shown in Equation (6) [30]:

$$\overrightarrow{W_1\left(l\right)}=\left\{\begin{array}{c}\mathrm{hungry}\left(i\right)\cdot \frac{N}{\mathrm{SHungry}}\times r_4,r_3<l\\ \\ 1\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{r}_3>l\end{array}\right.$$

(6)

The parameter $\overrightarrow{W_2}$ is formulated in Equation (7) [30]:

$$\overrightarrow{W_2\left(l\right)}=\left(1-\exp \left(-\mathrm{hungry}\left(i\right)-\mathrm{SHungry})\right)\right)\times r_5\times 2$$

(7)

where $\mathrm{hungry}$ denotes the hunger feeling of the solution; $N$ is the proposed number of solutions; and $\mathrm{SHungry}$ is the sum of hungry features of all current solutions, that is sum(hungry); $r_3$, $r_4$ and $r_5$ are normally distributed randomly generated number and their limits are in the range of [0, 1].

The parameter $\mathrm{hungry}\left(i\right)$ is formulated mathematically as follows [30]:

$$\mathrm{hungry}\left(i\right)=\left\{\begin{array}{cc}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \mathrm{F}\left(i\right)==BF\\ \mathrm{hungry}\left(i\right)+H,\hspace{1em}\hspace{1em}& \mathrm{F}\left(i\right)!=BF\end{array}\right.$$

(8)

The formula for 𝐻 can be seen as follows [30]:

$$TH=\frac{F\left(i\right)-BF}{WF-BF}\times r_6\times 2\times \left(UB-LB\right)$$

(9)

$$\mathrm{H}=\left\{\begin{array}{cc}LH\times \left(1+r\right),& TH<LH\\ TH,& TH\ge LH\end{array}\right.$$

(10)

where $r_6$ is a normally distributed randomly generated number and its limit is in the range of [0, 1]; WF represents the current worst objective value; $LB$ and $UB$ are the lower and upper boundaries of the considered parameters search area, respectively. The hunger sensation 𝐻 is saturated by the value of the lower bound.

3. The Proposed mHGS Optimizer

In the following, the authors introduce an explanation of the proposed mHGS optimization mechanism. The shortcomings of the conventional HGS algorithm are also introduced, followed by the optimization process of the proposed mHGS.

3.1. Shortcomings of the Original HGS

The original HGS optimizer suffers from fast convergence in an early stage. In the latter iterations, the HGS algorithm maintains a constant convergence without new evolution in the fitness value. This constant progress indicates that the algorithm becomes stuck in a local minima region instead of walking towards the global regions. Consequently, the HGS algorithm suffers premature convergence, and is subject to local minimum issues, due to the fast convergence.

The optimization steps of the mHGS are formulated in Algorithm 1, which illustrates the main steps of the mHGS algorithm. Similar to other metaheuristics, the introduced algorithm is initialized with a randomly generated population of size N, and some of the steps are inherited from the original HGS algorithm, i.e., the solutions are evaluated to obtain their objective values. To define the best and worst objective solutions, the hunger weights are calculated to adjust the solutions according to the food need. The algorithm diversity is boosted, i.e., it integrates with two solutions that are selected randomly. Solution A is selected from the third section with the high fitness solutions, while solution B is from the whole population. This randomly distributed selection is applied to avoid the local minima. Further, a novel crossover strategy called $binary\tau -BasedCrossover\left(\tau BX\right)$ is introduced and employed to boost the algorithm diversity. Finally, the satisfaction fulfillment step is employed to strengthen the solutions with low objective values. The searching process is continued until a stopping criterion is met to declare the end of the optimization process. For more clarification, detailed illustrations of the proposed algorithm’s parameters are presented in the following subsections. In addition, the flowchart of the new mHGS is provided in Figure 1.

$$\begin{array}{c}\hfill \overrightarrow{X\left(t+1\right)}=\left\{\begin{array}{}\mathrm{(11a)}& \overrightarrow{X_b}+vc·\left(X\left(A\right)-X\left(B\right)\right),\hfill & \mathrm{rand}<VC2\mathrm{(11b)}& \overrightarrow{X_b}+vb·\left(S_Y-\overrightarrow{W_2}·S_Z\right),\hfill & r_2>VC2\mathrm{(11c)}& X\left(A\right)-\overrightarrow{W_2}\cdot \left(S_Y-S_Z\right),\hfill & r_2\le VC2\end{array}\right.\end{array}$$

where $S_Y$ and $S_Z$ are two solutions generated by the crossover strategy presented in Equation (16); A and B are two positions selected randomly over the population size; and $\overrightarrow{W_2}$ is the hungry weight of each solution, computed in the original HGS in Equation (7).

$$vb=2·\mathrm{shrink}·\mathrm{r}-\mathrm{shrink}$$

(12)

where $vc$ is in the range of [$-VC2$, $VC2$], $\mathrm{shrink}=2·\mathrm{VC}2.$, and the $VC2$ operator is tuned according to the optimization progress as follows:

$$VC2=1-\frac{FE}{Max\_FEs}$$

(13)

where $FE$ and $Max\_FEs$ are the current function evaluation number and the maximum number of function evaluations, respectively.

$$VC1=\sec \mathrm{h}\left(\left|F\left(i\right)-BF\right|\right)$$

(14)

3.2. The Satisfaction Fulfillment Step

This satisfaction fulfillment process is applied to the worst solutions whose probability of hungry weight $VC1$ is greater than 0.5.

$$\overrightarrow{X\left(t+1\right)}=\overrightarrow{X\left(t+1\right)}+\mathrm{rand}·\left(\overrightarrow{X\left(t\right)}-\overrightarrow{X\left(t+1\right)}\right)ifVC1<0.5$$

(15)

where $\overrightarrow{X\left(t+1\right)}$ represents the new ith solution, $\overrightarrow{X\left(t\right)}$ is the current ith solution, and $\mathrm{rand}$ is a uniform number generated randomly through the range [0, 1].

3.3. The Binary τ-Based Crossover (τBX)

The crossover’s main objective is to enrich the children’s solutions with valuable information from the candidate parent’s solutions. Consequently, the generated children inherit the shared information from the parent’s characteristics. This helps a new solution to be generated by exchanging the information between the current solutions and the better parents’ solutions to inherent and generate promising offspring.

In binary crossover, two parents are selected for crossover and offspring, which are generated based on the similarity vector of the parents. This type of crossover simulates the binary crossover observed in nature such as the simulated binary crossover (SBC). In this study, a novel crossover mechanism is proposed called the binary τ-Based Crossover (τBX). This is explained mathematically as follows:

$$\mathrm{Crossover}=\left\{\begin{array}{c}\acute{X_1}=\tau ·X_1+\left(\tau -1\right)·\left(X_1+X_2\right)+\left(\tau -1\right)·\left(X_1-X_2\right)\\ \\ \acute{X_2}=\tau ·X_2+\left(\tau -1\right)·\left(X_2+X_1\right)+\left(\tau -1\right)·\left(X_2-X_1\right)\end{array}\right.$$

(16)

where $\tau$ is the crossover control parameter, explained as follows:

$$\tau =e^{-rand\left(dim\right)}$$

(17)

where $rand\left(dim\right)$ is a vector of normally distributed random generators that is limited in the range of [0, 1], of the same $X_{1}and{X}_2$ size. The $X_{1}and{X}_2$ are the parent solutions, and $\acute{X_1}and\acute{X_2}$ are the two generated solutions after the crossover.

Algorithm 1: Pseudo-code of the mHGS.

Input: 𝑁, 𝑀𝑎𝑥_FEs, 𝑙, 𝐷, 𝑆𝐻𝑢𝑛𝑔𝑟𝑦 parameters.

Output: Return 𝐵𝐹, 𝑋𝑏.

Initialization: Generate the initial positions of all solutions 𝑋𝑖 (𝑖 = 1,2, …, 𝑁).

While (FE ≤ 𝑀𝑎𝑥_FEs)

Compute the cost function values of all proposed solution

𝑈𝑝𝑑𝑎𝑡𝑒 𝐵𝐹, 𝑊𝐹, 𝑋𝑏, 𝐵𝐼

Compute the 𝐻𝑢𝑛𝑔𝑟𝑦 by Equation (8)

Compute the 𝑊1 by Equation (6)

For 𝑒𝑎𝑐𝒉 𝐼𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠

Update vb, VC1 in Equations (12) and (14).

𝑈𝑝𝑑𝑎𝑡𝑒 𝑅 by Equation (4)

Apply the satisfaction fulfillment step in Equation (15)

End 𝐅𝐨𝐫

𝑡 = 𝑡 + 1

End While

Return 𝐵𝐹, 𝑋𝑏

4. Experimental Series 1: Assessing mHGS in Solving Global Optimization Problems

The proposed mHGS efficiency was examined on the benchmark function test suite suggested in the IEEE-CEC’2020. Moreover, a comparative study was conducted between the simulation findings of the new mHGS and those resulting from several different classes of existing search algorithms, including the gravitational search algorithm (GSA) [31], whale optimization algorithm (WOA) [32], grey wolf optimizer (GWO) [33], Runge Kutta method (RUN) [34], and hunger search algorithm (HGS).

4.1. IEEE-CEC’2020 Test Suite

The IEEE-CEC’2020 test suite [35] was used to evaluate the performance of the proposed mHGS algorithms and the other competitors. Ten test functions, associated with the optimal values Fi*, were included in the CEC’2020 benchmark functions, including unimodal, multimodal, hybrid, and composition functions, as reported in

Table 1. CEC’2020 test suite description.

Category	No.	Functions	Fi*
Uni-modal function	F1	Shifted and Rotated Bent Cigar Function	100
Multi-modal functions	F2	Shifted and Rotated Schwefel’s Function	1100
	F3	Shifted and Rotated Lunacek bi-Rastrigin Function	700
	F4	Expanded Rosenbrock’s plus Griewangk’s Function	1900
Hybrid Functions	F5	Hybrid Function 1 (N = 3)	1700
	F6	Hybrid Function 2 (N = 4)	1600
	F7	Hybrid Function 3 (N = 5)	2100
Composition Functions	F8	Composition Function 1 (N = 3)	2200
	F9	Composition Function 2 (N = 4)	2400
	F10	Composition Function 3 (N = 5)	2500

. Moreover, the search range of [−100, 100]^D was the same search range defined for all test functions.

4.2. Parameters Setting

To establish a stable and fair assessment environment, the evaluation of the mHGS algorithm and the other comparative approaches was conducted for 30 experimental runs. Moreover, forty-five thousand function evaluations (FEs) were applied to all cases. Prior to conducting the searching process, some parameters were required to be initialized or set. The parameters of every algorithm, including the proposed mHGS, were defined according to

Table 2. Parameters’ values for the considered optimizers.

Optimizer	Values
Common Parameters	N = 30 (Number of proposed solutions) MAX_FEs = 45,000 (Number of uppermost function evaluations) Dim = 10 (Number of controlling variables) Runs = 30 (Number of experimental runs)
GSA	Alfa = 10, G0 = 244.
WOA	alpha = 1
GWO	A = 2.7778, C1 = 2.7778
RUN	g = 1, mu = 0.9
HGS	shrink = 3.2

. In engineering applications, identifying the appropriate set of parameter values that produces the optimal performance for every test method is a critical issue. Therefore, the Taguchi method for robust design parameters was applied. This method uses the Orthogonal Array (OA) and mean analysis to study the influence of these parameters’ variations on each algorithm according to the statistical measures of the experimental results. A fractional factorial matrix of numbers is structured in the OA so that each row denotes the level of factors in each run and each column denotes a particular variable that may be altered at each run.

4.3. Statistical Analysis

To assess and compare the performances of the considered algorithms, a statistical analysis was conducted based on the outcomes of the CEC’2020 cost functions. This analysis includes the calculation of mean (AV) and standard deviation (SD) values of the resulting optimal solutions obtained at every experimental run.

Table 3. The AV and SD of the 30-experimental-run study of the considered optimizers applied to the CEC’2020 cost functions when the dimension is 10.

F	Measure	GSA	WOA	GWO	RUN	HGS	mHGS
F1	AV SD	9.75 × 102 1.12 × 103	4.80 × 105 9.2 × 105	3.13 × 107 1.10 × 108	3.65 × 103 2.22 × 103	1.07 × 105 2.56 × 105	7.21 × 102 7.80 × 102
F2	AV SD	1.38 × 103 1.98 × 102	2.06 × 103 2.68 × 102	1.52 × 103 2.38 × 102	1.59 × 103 2.23 × 102	1.85 × 103 2.57 × 102	1.28 × 103 1.25 × 102
F3	AV SD	7.16 × 102 3.22	7.81 × 102 2.61 × 101	7.27 × 102 9.96	7.51 × 102 1.60 × 101	7.52 × 102 1.73 × 101	7.16 × 102 2.56
F4	AV SD	1.90 × 103 2.80 × 10−1	1.91 × 103 4.80	1.90 × 103 8.23 × 10−1	1.90 × 103 7.63 × 10−1	1.90 × 103 2.13	1.90 × 103 1.22
F5	AV SD	2.29 × 105 1.97 × 105	8.44 × 104 1.15 × 105	8.88 × 103 5.84 × 103	2.37 × 103 4.41 × 102	2.22 × 105 1.82 × 105	2.33 × 103 3.76 × 102
F6	AV SD	1.64 × 103 1.89 × 101	1.60 × 103 3.02 × 10−1	1.60 × 103 2.52 × 10−1	1.60 × 103 3.39 × 10−1	1.60 × 103 3.70	1.60 × 103 2.15 × 10−1
F7	AV SD	1.12 × 104 4.97 × 103	2.71 × 104 2.43 × 104	7.17 × 103 6.11 × 103	2.59 × 103 8.38 × 102	1.60 × 104 1.14 × 104	2.41 × 103 2.53 × 102
F8	AV SD	2.30 × 103 1.31 × 10−1	2.32 × 103 9.67	2.31 × 103 7.60	2.31 × 103 4.65	2.53 × 103 4.79 × 102	2.30 × 103 1.17
F9	AV SD	2.66 × 103 1.08 × 102	2.74 × 103 7.70 × 101	2.74 × 103 5.38 × 101	2.75 × 103 1.08 × 101	2.77 × 103 1.13 × 101	2.65 × 103 1.23 × 102
F10	AV SD	2.94 × 103 1.04 × 101	2.94 × 103 2.08 × 101	2.94 × 103 1.41 × 101	2.93 × 103 2.23 × 101	2.94 × 103 2.99 × 101	2.93 × 103 2.33 × 101
Friedman mean rank		3.1 3	4.9 5	3.5 4	3 2	5.2 5	1.3 1

shows the calculated mean and SD values obtained from each algorithm for the examined cost functions with a dimension of 10 parameters (Dim = 10). The boldface numbers in the table denote the best results.

The calculated average (AV) and its associated standard deviation SD values are illustrated in Table 3 for the 30-experimental-run study of the considered optimizers applied to the CEC’2020 benchmarks at Dim = 10. In the table, the resulting best (optimal) values are illustrated in boldface. As shown in Table 3, the new approach (mHGS) obtained the best result for the unimodal F1 cost function followed by the well-known GSA and RUN algorithms. Furthermore, the new mHGS performed well in the case of multi-modal F2, F3, and F4 cost functions. However, in the case of hybrid cost functions, F5, F6, and F7, the mHGS optimizer achieved a superior performance compared to the other competitor approaches. Alternatively, for the composite cost functions F8 to F10, the mHGS approach achieved a better result compared to the competitors, and was similar to the recent RUN on F10. Referring to the conducted Friedman test, the new mHGS performance was better than that of the GSA, WOA, GWO, RUN, and HGS. In a specific manner and based on the Friedman test, the mHGS was ranked first among all of the algorithms.

4.4. Converging Analysis

One of the essential analyses in the optimization field is convergence analysis. Therefore, the behaviors of the algorithms in reaching the best solutions were plotted, as shown in Figure 2. Each plot gives information about the speed and accuracy of its corresponding algorithm. By investigating the plots in Figure 2, the new mHGS optimizer shows a convergence curve that reaches the near-optimal solutions faster than most of the other algorithms. Accordingly, this proposed algorithm can be considered viable, especially for a problem that requires fast processing time such as the online-optimization applications.

4.5. Boxplot Behavior Analysis

The boxplot graph is a powerful tool that analyzes and explains graphically how the empirical data are distributed and where the data distributions are interpreted into quartiles [36]. The minimum and maximum edges of the boxplot whiskers refer to the highest and lowest obtained data values. Furthermore, the lower and upper ends of each rectangle represent both the lower and upper quartiles, respectively. Further, the boxplots illustrate the data distribution and its degree of agreement, noting that the narrower the boxplot the higher the symmetry between the obtained data. Figure 3 illustrates the distributed boxplots’ graphical representation of the results obtained for the ten CEC’20 cost functions, F1 to F10, when the problem’s dimension is 10. It demonstrates that, for most of the cost functions, the mHGS achieves narrower boxplot distributions that have minimum values relative to the other optimizers. Moreover, the plots in Figure 3 demonstrate that the new mHGS optimizer is consistent and is able to effectively locate the promising search areas.

4.6. Wilcoxon Rank Test Analysis

Another test was conducted to exhibit the degree of significance for the resulting values, namely, the Wilcoxon rank-sum test. This test was applied to explore the difference between the new mHGS and the other competing optimizers. Details about the Wilcoxon test can be found in [37]. Despite MAs’ searching mechanism being naturally stochastic, their performances are still precise and acceptable in many applications. Accordingly, this test was used for the proposed mHGS algorithm outcomes, and proved that the algorithm’s findings are not only stable but also not stochastic.

Table 4. The comparison and p-value results based on the Wilcoxon test for the ten CEC’2020 cost functions.

Competing Optimizer	Criterion	Better	Similar	Worse	p-Value
MHGS vs. GSA	Average	7	3	-	0.020
MHGS vs. WOA	Average	10	-	-	0.002
MHGS vs. GWO	Average	8	2	-	0.002
MHGS vs. RUN	Average	7	3	-	0.004
MHGS vs. HGS	Average	8	2	-	0.002

presents the results of the Wilcoxon rank-sum test for the new mHGS and its five competitors, one by one, when applied to the ten cost functions, F1-F10. The criterion of comparison is based on the average value. The “Better”, “Similar”, and “Worse” labels denote the three expected cases of comparison when the mHGS is better, similar, or worse than the other method, respectively. To rely on the results shown in Table 4, they must be significant. Therefore, the significance test, known as “p-value”, is applied to decide whether the null hypothesis should be rejected or not. Basically, the significance value should be less than 5% to reject the null hypothesis. From the table, all the p-values (see the last column in Table 4) are less than 5%, which implies that the resulting values of Wilcoxon’s rank-sum test are reliable and significant.

5. Experimental Series 2: Applying mHGS to Allocate Biomass Distributed Generators

5.1. Problem Formulation

If P_j and Q_j represent the active and reactive power needed by the load, respectively, the power loss in this transmission line is defined by the following relation [25]:

$$P_{ij}^{loss}={\left|I_{ij}\right|}^2{R}_{ij}=\frac{\left(P_j{}^2+Q_j{}^2\right)}{{\left|V_i\right|}^2}$$

(18)

Therefore, in the case of a distribution network, the total active power loss is expressed as follows [26]:

$$P^{loss}={\displaystyle \sum }_{m=2}^{n_d}{\displaystyle \sum }_{n=1}^{n_d}\frac{\left(P_n{}^2+Q_n{}^2\right)}{{\left|V_m\right|}^2}{R}_{mn}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}m\ne n,\forall m,n\in n_d$$

(19)

where $n_d$ denotes the bus number, the active and reactive demands at the n^th bus are denoted by $P_n$ and $Q_n$, respectively, and $V_m$ is the voltage at the m^th bus.

Equation (20) generalizes the active power loss expressed in Equation (19) using the following relation [26]:

$$P^{loss}={\displaystyle \sum }_{j=1}^{n_g}{\displaystyle \sum }_{i=1}^{n_g}{b}_{ij}{P}_i{P}_j+{\displaystyle \sum }_{i=1}^{n_g}{b}_{i0}{P}_i+b_{00}$$

(20)

where the number of generators is represented by $n_g$, the generated power from the i^th unit is represented by $P_i$, and the output power from the j^th unit is represented by $P_j$.

The mathematical representation for the objective function is as follows [26]:

$$Min\hspace{1em}\hspace{1em}{P}^{loss}=P_g^{T}B{P}_g+P_g^T{B}_0+B_{00}$$

(21)

where the vector of generating-units’ power is denoted by $P_g^T$, and the loss coefficients’ matrices are denoted by $B$, $B_0$, and $B_{00}$.

5.2. Results and Discussion

The suggested strategy was applied to three different RDNs, IEEE 33-bus, IEEE 69-bus, and IEEE 119-bus. The suggested mHGS was performed with a 25-particle population size, a 2500-iteration optimization process, and 10 independent runs. Additionally, the results were compared with GSA, WOA, GWO, RUN, and original the HGS, in addition to the stochastic fractal search algorithm (SFSA) [38]. Moreover, Friedman and ANOVA statistical tests were used to prove the superiority of the suggested mHGS.

5.2.1. Objective: IEEE 33-Bus Network

The IEEE 33-bus comprises 33 nodes and 32 branches, and the nominal voltage and base MVA of the network are 12.66 kV and 100 MVA, respectively. The architecture of this network is shown in Figure 4. The active and reactive demands are 3.715 MW and 2.3 MVAr, respectively. The system data are validated in [36] and the active and reactive losses are found to be 210.99 kW and 143.13 kVAr, respectively. The considered cost function, which mitigates the total active power loss, is achieved via integrating the three biomass-based DGs into the network. The new mHGS is utilized to optimize the IEEE 33-bus compared to other optimizers, and the optimal obtained results are given in

Table 5. The optimal results obtained via different optimizers applied to the IEEE 33-bus.

Algorithm	Optimal Place	Optimal Capacity (kW/kVar)	pf	PLoss (kW)	V.D (pu)	VSI−1 (pu)	VSI (pu)	Loss Reduction
GSA	28	1043.18/735.64	0.817	37.7552	0.0141	1.1813	0.8465	82.11%
	24	1029.15/772.82	0.799
	7	655.412/499.32	0.795
WOA	25	358.162/227.21	0.844	19.4532	0.0009	1.0413	0.9604	90.78%
	12	1072.48/571.59	0.882
	30	1001.74/815.77	0.775
GWO	14	665.332/381.27	0.868	12.7826	0.0012	1.0419	0.9598	93.94%
	30	1077.27/999.16	0.733
	24	835.076/526.49	0.846
RUN	30	1039.47/1039.1	0.707	12.3773	0.0008	1.0386	0.9628	94.13%
	15	721.328/315.05	0.916
	24	1048.76/542.92	0.888
HGS	13	847.683/146.75	0.985	22.1601	0.0022	1.0574	0.9457	89.49%
	32	778.009/1023.3	0.605
	24	967.489/892.82	0.735
SFSA [38]	13	792.7/374.7	0.904	11.762	0.000619	1.0319	0.9691	36.38%
	24	1030.0/523.2	0.892
	30	1042.2/1015.2	0.716
mHGS	14	747.608/347.58	0.907	11.6300	0.0006	1.0343	0.9668	94.49%
	30	1050.82/1019.4	0.718
	24	1081.29/519.76	0.901

. The proposed mHGS succeeded in achieving the best power loss at a value of 11.6300 kW by installing three BDGs with sizes of 747.608, 1050.82, and 1081.29 kW with optimal power factors of 0.907, 0.718, and 0.901 on buses 14, 30, and 24, respectively. This integration mitigated the voltage deviation to 0.0006 pu. The voltage stability index (VSI) and its inverse (VSI⁻¹) in this case were 0.9668 and 1.0343 pu, respectively. These results reveal that the network with the installed BDGs via the proposed mHGS is stable. By comparison, the conventional HGS achieved active power loss of 22.1601 kW via installing 847.683, 778.009, and 967.489 kW BDGs with 0.985, 0.605, and 0.735 power factors on buses 13, 32, and 24 respectively. Moreover, SFSA [38] achieved a power loss of 11.762 kW, mitigating it by 36.38%, via installing BDGs with active power of 792.7, 1030.0, and 1042.2 kW and reactive power of 374.7, 523.2, and 1015.2 kVAr on buses 13, 14, and 30 respectively. The worst power loss was 37.7552 kW, obtained via GSA. Regarding the loss reduction, the proposed mHGS achieved the best mitigation of 94.49%, whereas the worst one was HGS with 82.11%. The statistical parameters of best, worst, mean, median, variance, and SD for all considered optimizers were calculated and are tabulated in

Table 6. Statistical parameters of different optimizers applied to the IEEE 33-bus.

	GSA	WOA	GWO	RUN	HGS	mHGS
Best	37.755	19.453	12.783	12.377	22.160	11.630
Worst	68.894	39.685	23.662	47.931	39.059	28.954
Mean	50.815	29.070	18.571	21.198	28.588	15.819
Median	50.507	28.014	19.528	19.451	28.798	13.593
Variance	132.16	31.741	8.6936	95.751	23.017	31.310
Stand. Dev.	11.496	5.6339	2.9485	9.7852	4.7976	5.5956
Consumed time (s)	398.92	276.91	347.031	407.725	121.582	125.232

. The results reveal that the new optimizer is superior to the competitors in terms of best, worst, mean, and median, whereas for variance and Stand. Dev. it was not ranked first. However, the most important issue is that it achieved the best fitness value. Additionally, the computation time for one run was executed and is tabulated in Table 6; it is clear that the HGS and mHGS achieved the lowest computational times of 121.582 and 125.232 s, respectively.

Figure 5 shows the voltage profile obtained through all the considered optimizers for the IEEE 33-bus network with and without BDGs. It is clear that the suggested mHGS helped in enhancing the network’s voltage profile compared to the other approaches. The variations in the fitness value throughout the optimization processes resulting from the considered approaches are illustrated in Figure 6. The curves confirm the superiority of the proposed mHGS in achieving the best performance compared to the others. The suggested mHGS confirmed its competence and superiority in integrating the BDGs into the IEEE 33-bus network.

5.2.2. Objective: IEEE 69-Bus Network

The analysis was extended to the IEEE 69-bus network, which includes the 69-node and 68-branch configuration, as given in Figure 7. In this network, the nominal voltage and the MVA are 12.66 kV and 100 MVA, respectively. The total demands are 3.8 MW and 2.69 MVAr, while the total active and reactive power losses of the original network are 225 kW and 102.16 kVAr, respectively. Moreover, the voltage deviation of the original network is 0.099337 pu. The proposed mHGS, with the other algorithms, was applied to install three BDGs for mitigating the network active power loss, and the resulting values are listed in

Table 7. The resulting optimal values through different optimizers applied to the IEEE 69-bus.

Algorithm	Optimal Place	Optimal Capacity (kW/kVar)	pf	PLoss (kW)	V.D (pu)	VSI−1 (pu)	VSI (pu)	Loss Reduction
GSA	23	508.37/443.33	0.754	22.6155	0.0027	1.0809	0.9252	89.95%
	8	499.89/1594.9	0.299
	61	1233.1/782.03	0.844
WOA	61	1485.9/1279.3	0.758	11.8816	0.0014	1.0399	0.9616	94.72%
	12	950.96/556.66	0.863
	38	1423.2/270.93	0.982
GWO	22	332.56/235.82	0.816	5.2618	0.0005	1.0355	0.9658	97.66%
	61	1500.0/1185.4	0.785
	11	588.49/389.49	0.834
RUN	11	600.63/367.64	0.853	5.232	0.0005	1.0342	0.9669	97.67%
	20	356.18/237.78	0.832
	61	1500.0/1195.9	0.782
HGS	62	1500.0/1039.4	0.822	8.9086	0.0017	1.0601	0.9433	96.04%
	12	516.02/388.29	0.799
	25	180.24/186.39	0.695
mHGS	11	588.69/328.48	0.873	4.2291	0.0005	1.0341	0.9670	97.68%
	18	369.57/258.79	0.819
	61	1499.8/1201.5	0.780

. The suggested mHGS outperformed the other considered approaches employed to install BDGs in the IEEE 69-bus network, achieving the best fitness value of 5.2291 kW with a loss reduction of 97.68%. To accomplish this goal, three BDGs have to be installed on buses 11, 18, and 61 in the network with the following features: capacities of 588.69, 369.57, and 1499.8 kW and power factors of 0.873, 0.819, and 0.78, respectively. In this case, the voltage deviation, VSI, and its inverse are 0.0005, 0.9670, and 1.0341 pu, respectively. Based on these results, and after installing the BDGs, the stability of the network was satisfied via the suggested approach. However, the conventional HGS achieved a power-loss value of 8.9086 kW with a reduction of 96.04%, which is less than the value obtained by the proposed algorithm. The worst case was obtained by the GSA optimizer, as it achieved a power loss of 22.6155 kW with only a reduction of 89.95% via installing 508.37, 499.89, and 1233.1 kW BDGs with 0.754, 0.299, and 0.844 power factors on buses 23, 8, and 61, respectively. Figure 8 illustrates the resulting voltage profiles via all considered optimizers of the considered IEEE 69-bus network with and without installing the BDGs. The curves show a great improvement in the network after installing the BDGs using the new approach. Moreover, Figure 9 shows the convergence curves of the fitness function for all considered algorithms applied to the investigated network. From the plotted curves, the superiority and competence can clearly be inferred for the suggested mHGS performance compared to the others.

5.2.3. IEEE 119-Bus Network

Validating the new mHGS on a large-scale distribution network is essential to assess its capability. This can be performed via conducting the analysis on the IEEE 119-bus network as an example. The topology of a such network is given in Figure 10. This network contains 119-node and 115-tie-line architecture with a rated voltage and a base MVA of 11 kV and 100 MVA, respectively. The details of the network, including the branch and bus data, can be found in [39]. Moreover, the network’s active and reactive load power values are 22.709.7 kW and 17.041.1 kVAr, respectively. Furthermore, the overall active and reactive power losses of the original network are 1298.091 kW and 978.736 kVAr, respectively. The network has a voltage stability index of 0.5697 pu and a voltage deviation of 0.357641 pu. In this network, it is assumed that the five BDGs with maximum limits of 5 MW are integrated. The suggested approach was applied to the IEEE 1119-bus network, and the obtained results are tabulated in

Table 8. The optimal results obtained via different optimizers applied to the IEEE 119-bus.

Algorithm	Optimal Place	Optimal Capacity (kW/kVar)	Pf	PLoss (kW)	V.D (pu)	VSI−1 (pu)	VSI (pu)	Loss Reduction
GSA	88	665.85/1267.8	0.465	665.4651	0.5451	1.9581	0.5107	48.74%
	101	2043.5/1074.8	0.885
	112	2190.0/4982.0	0.402
	32	2596.1/2834.6	0.675
	69	1971.1/2317.2	0.648
WOA	101	2110.7/914.71	0.918	341.251	0.1974	1.3635	0.7334	73.71%
	112	4577.3/2809.9	0.852
	75	3347.4/1874.1	0.873
	61	1507.2/1463.1	0.718
	66	1789.4/3361.3	0.469
GWO	68	709.59/244.57	0.945	207.5368	0.0246	1.1671	0.8568	84.01%
	77	1805.1/1364.5	0.798
	100	2401.7/1582.1	0.835
	43	1121.3/1750.8	0.539
	114	3689.4/2690.9	0.808
RUN	100	2098.3/1062.9	0.892	152.6763	0.0075	1.1076	0.9029	88.24%
	73	2352.7/1999.2	0.762
	113	3929.9/2932.9	0.801
	82	2235.5/1670.3	0.801
	44	1431.1/1199.4	0.766
HGS	31	269.10/422.89	0.537	348.3066	0.0095	1.0792	0.9266	73.17%
	93	2824.0/1988.7	0.818
	74	1678.5/2597.2	0.543
	47	2072.1/153.26	0.997
	114	2940.5/3876.2	0.604
mHGS	43	1554.5/1380.9	0.748	145.489	0.0159	1.1425	0.8753	88.79%
	94	1544.9/860.64	0.874
	114	3302.2/2846.33	0.757
	75	2641.2/1426.38	0.879
	81	3209.0/3288.00	0.698

in comparison with the other considered approaches. The new mHGS suggested installing five BDGs with capacities of 1554.5, 1544.9, 3302.2, 2641.2, and 3209.0 kW with power factors of 0.748, 0.874, 0.757, 0.879, and 0.698 on buses 43, 94, 114, 75, and 81, respectively. This integration reduced the network’s active power loss to 145.489 kW with a power reduction of 88.79%. Moreover, the voltage deviation, VSI, and its inverse in this case were 0.0159, 0.8753, and 1.1425 pu, respectively. The results reveal that the network is stable after installing the BDGs based on locations and sizes obtained via the proposed mHGS. The RUN algorithm was ranked second, with an active power loss of 152.6763 kW and a reduction of 88.24%. The worst approach was GSA, which achieved power loss of 665.4651 kW. Figure 11 shows the voltage profiles of the network with and without installing the BDGs through all the considered optimizers. Figure 12 shows the convergence curves of the fitness function for all considered algorithms applied to the investigated network. From the plotted curves, the superiority and competence can also be inferred for the suggested mHGS performance compared to the others.

Based on the obtained results mentioned above, it can be concluded that the new mHGS approach is competent and reliable when installing the biomass-based DGs in the radial distribution network. This was confirmed by applying the suggested approach to three different networks.

6. Conclusions

This paper presents a new modified variant of hunger games search (HGS) called mHGS for solving global optimization and biomass distributed generator (BDG) problems. The proposed mHGS approach includes a new binary τ-Based Crossover (τBX) strategy with the satisfaction fulfillment step mechanisms. This strategy proposed solutions to overcome the drawbacks of the classical hunger games search in dealing with global optimization problems, such as the BDG problems. The new mHGS performance was evaluated through ten IEEE CEC’2020 benchmarks against five competitive algorithms. The results of the proposed mHGS were compared to the findings of the other optimizers based on the statistical values of the Wilcoxon sign rank and Friedman tests. Then, the best distribution of the BDGs in a radial distribution network (RDN) was determined by mHGS to enhance the stabilization of the network. Three distribution networks of IEEE 33-bus, IEEE 69-bus, and IEEE 119-bus were considered to validate the superiority of the proposed mHGS. The resulting performance was compared with those resulting from the GSA, WOA, GWO, RUN, and original HGS optimizers. The new mHGS is capable of reducing the power loss to 11.6300, 5.2291, and 145.489 kW, with loss reduction of 94.49%, 97.68%, and 88.79% for the IEEE 33-bus, 69-bus, and 119-bus networks, respectively. These results showed that the new approach outperformed the other considered approaches. Therefore, the success of using other sources of energy, such as biomass-based fuel, will definitely help in reducing the massive use of CO₂-based fuels, and hence reduce the CO₂ emissions. A multi-objective version of mHGS is recommended in future work to mitigate both power loss and voltage deviation of distribution networks.