Single and Multi-Objective Optimal Power Flow Based on Hunger Games Search with Pareto Concept Optimization

Abstract

In this study, a new meta-heuristic optimization method inspired by the behavioral choices of animals and hunger-driven activities, called hunger games search (HGS), is suggested to solve and formulate the single- and multi-objective optimal power flow problem in power systems. The main aim of this study is to optimize the objective functions, which are total fuel cost of generator, active power losses in transmission lines, total emission issued by fossil-fueled thermal units, voltage deviation at PQ bus, and voltage stability index. The proposed HGS approach is optimal and easy, avoids stagnation in local optima, and can solve multi-constrained objectives. Various single-and multi-objective (conflicting) functions were proposed simultaneously to solve OPF problems. The proposed algorithm (HGS) was developed to solve the multi-objective function, called the multi-objective hunger game search (MOHGS), by incorporating the proposed optimization (HGS) with Pareto optimization. The fuzzy membership theory is the function responsible to extract the best compromise solution from non-dominated solutions. The crowding distance is the strategies carried out to determine and ordering the Pareto non-dominated set. Two standard tests (IEEE 30 bus and IEEE 57 bus systems) are the power systems that were applied to investigate the performance of the proposed approaches (HGS and MOHGS) for solving single and multiple objective functions with 25 studied cases using MATLAB software. The numerical results obtained by the proposed approaches (HGS and MOHGS) were compared to other optimization algorithms in the literature. The numerical results confirmed the efficiency and superiority of the proposed approaches by achieving an optimal solution and giving the faster convergence characteristics in single objective functions and extracting the best compromise solution and well-distributed Pareto front solutions in multi-objective functions.

1. Introduction

Power flow is one of the most important tools used to analyze and operate the power system by finding out active power losses in transmission lines and reactive power injection on lines and calculating the voltage at different buses. Optimal power flow (OPF) is a non-convex, nonlinear, and large-scale problem. The main aim of using OPF in power systems is to obtain the optimal objective functions by setting the control variables. These control variables involve real power output of generation units except the slack bus, the voltages magnitude at PV bus, regulating tap setting at transformers, and the source VAR compensator connected to transmission lines with satisfying the equality and inequality constraints. The objective functions that will be optimized are the total fuel cost of generation units, active power losses in transmission lines, total emission, voltage deviation, and the voltage stability index of system. The first presentation of the OPF problem formulation was presented by Dommel and Tinney 1968 [1].

Traditional and metaheuristic optimization methods are the types used to solve OPF problems in the power system. Linear programming, Newton methods, gradient based method, quadratic programming, and dynamic programming are the traditional methods that were proposed to solve OPF problems [2]. To overcome these weaknesses, many metaheuristic optimization methods were developed to solve OPF problems, such as differential evolution (DE) [3], genetic algorithm (GA) [4], modified artificial bee colony (MABC) [5], improved differential evolution (IDE) [6,7], adaptive constraint differential evolution (ACDE) [8], enhanced adaptive differential evolution with self-adaptive (EJADE-SP) [9], crisscross search based grey wolf optimizer (GS-GWO) [10], Harris hawks optimization [11], and fruit fly optimization (FFO) [12]. The previous metaheuristic optimization methods were applied to single-objective OPF problems.

Recently, multi-objective OPF problems (MOOPF) are solved by several optimization methods. These problems are described as nonlinear and large-scale. For example, improved differential evolution algorithm (IDEA) have been developed to solve multi-objective functions, called the multi-objective improved differential evolution algorithm (MOIDEA) [13]. The multi-objective slime mould algorithm (MOSMA) was proposed as a new optimization method to solve multi-objective optimal power flow (MOOPF) problems [14]. The multi-objective backtracking search algorithm (MOBSA) was suggested as a new multi-objective approach to solve and formulate MOOPF problems on power systems [15]. The multi-objective evolutionary algorithm (MOEA) was proposed as novel hybrid decomposition and local dominance to solve MOOPF problems with conflicting objectives [16]. The author in Ref. [17] suggested a new multi-objective method, called the multi-objective search group algorithm (MOSGA), which efficiently solves MOOPF problems in power systems.

In this work, the hunger games search algorithm was proposed to solve single objective OPF problems. In addition, the proposed algorithm HGS was developed to solve multi-objective optimal power flow (MOOPF) problems, namely, the multi-objective hunger games search (MOHGS). The proposed approach integrates the Pareto concept with the fuzzy membership approach to determine the nondominated solutions and extract the best compromise solution, respectively. The main aim of this approach is to optimize the objective function by satisfying the equality and inequality constraints. The objective functions that are minimized are total fuel cost, active power losses on transmission lines, total emission, voltage deviation at PQ busses, and the voltage stability index of whole system. In a multi-objective function. these objective functions that will be optimized may be conflicting simultaneously, which means when the first objective function is minimized, the second objective function will be maximized and vice versa. Therefore, the main goal in using the multi-objective optimal power flow (MOOPF) is to find out the compromise solution among multiple objective functions.

The hunger games search (HGS) is a stochastic optimization algorithm inspired by the behavior of social animal cooperation, which is proportional to their level of hunger, by Y. Yang et al. in 2021 [18]. The HGS is characterized by high convergence speed, powerful local exploitation, and global exploration. The main contributions of this paper can be summarized as follows:

• The hunger games search (HGS) and the developed approach, multi-objective hunger game search (MOOPF), are proposed to solve for single-and multi-objective optimal power flow (MOOPF) in power systems. Pareto concept (PC), fuzzy membership function (FMF), and crowding distance (CD) are the approaches used to find out non-dominated Pareto fronts (NDPF), extract the best compromise solution (BCS), and rank and reduce the Pareto repository, respectively.
• The IEEE 30-bus and IEEE 57-bus tests are the power systems applied to 25 studied cases by using single and multiple (Bi, Tri, Quad, and Quinta) objective functions. The numerical results obtained by the proposed approaches (HGS and MOHGS) will be compared with known optimization methods in the literature.

The rest of the paper will be arranged as follows: Section 2 introduces the formulation of OPF problems. Section 3 is focused on single- and multi-objective OPF frameworks. The simulation results of proposed approaches (HGS and MOHGS) are presented in Section 4. Finally, this paper will be finished with the conclusion.

2. The Formulation of the OPF Problem

The main aim of applied single- and multi-objective OPF in power systems is to optimize the objective functions of single and multiple objectives (Bi, Tri, Quad, and Quinta) through setting optimal control variables (active power output of generators except for active power output of slack bus, voltage magnitude of PV bus, Source VAR compensator, and tap setting regulating of transformers) with satisfying equality and inequality constraints, simultaneously. The mathematical model can be formulated as follows:

$$\begin{array}{ll}\mathrm{Optimize}& f\left(x,u\right)=f_1(x,u),f_2(x,u),\dots ,f_{N_{obj}}(x,u)\\ \mathrm{subjected}\mathrm{to}& \mathrm{g}\left(x,u\right)=0\\ & h\left(x,u\right)\le 0\end{array}$$

(1)

where $f\left(x,u\right),g\left(x,u\right),$ and $h\left(x,u\right)$ represent the objective function to be optimized, equality, and inequality constraints, respectively. $x$ and $u$ are the state and control variables, respectively.

2.1. The State and Variables

The mathematical expression of the state variable $x$ is:

$$x=\left[P_{G_1},\left|V_{L_1}\right|,\cdots ,\left|V_{L_{N_{PQ}}}\right|,Q_{G_1},\cdots ,Q_{G_{N_{PV}}},S_{l_1},\cdots ,S_{l_{N_l}}\right]$$

(2)

where $P_{G1}$ is the active power at slack bus; $V_L$ is the voltage magnitude at load bus (PQ buses); $Q_G$ is the reactive power at PV bus; and $S_l$ is the transmission line loading; $N_{PV}$, $N_{PQ}$, and $N_l$ denote numbers of generators, load buses, transmission lines, respectively. The mathematical expression of the state variable $u$ is:

$$u=[P_{G_2},\cdots ,P_{G_{N_G}},\left|V_{G_1}\right|,\cdots ,\left|V_{G_{N_G}}\right|,T_1,\cdots ,T_{N_T},Q_{C_1},\cdots ,Q_{C_{N_C}}]$$

(3)

where $P_G$ is the active power at the PV bus except the slack bus. $V_G$ is the voltage magnitude at PV buses. $T$ denotes the tap settings regulating transformers. $Q_c$ is the source VAR compensator. $N_G$, $N_T$, and $N_C$ are the numbers of generators, source VAR (shunt) compensators, and regulating transformers, respectively.

2.2. Objective Constraints

The constraints of OPF can be divided into equality and inequality constraints. The equality constraints represent the physical structure (the power balance equations) of the whole system.

$$\begin{gathered} P_G{}_i-P_D{}_i-\left|V_i\right|{\displaystyle \sum _{j=1}^{N_B}\left|V_j\right|\left(G_{ij}\cos ({\theta }_{ij})+B_{ij}\sin ({\theta }_{ij})\right)}=0 \forall i\in N \\ {Q}_G{}_i+Q_C{}_i-Q_D{}_i-\left|V_i\right|{\displaystyle \sum _{j=1}^{N_B}\left|V_j\right|\left(G_{ij}\sin ({\theta }_{ij})+B_{ij}\cos ({\theta }_{ij})\right)}=0 \forall i\in N \end{gathered}$$

(4)

where $N_B$ denotes the number of buses. $P_G$, $P_D$ are the active power output of generator and load demand, resepectively. $Q_G$, $Q_C$, and $Q_D$ are the reactive power output of generator, shunt VAR compensator, and load demand, respectively. $G_{ij}$ and $B_{ij}$ are the conductance and susceptance between bus $i$ and bus $j$, respectively. ${\theta }_{ij}$ is the voltage angle difference between bus $i$ and bus $j$.

The inequality constraints represent the operation limit of the equipment (generators, transformers, shunt compensators, and security). It can be presented as:

-

Generator constraints:

$$P_G{{}_i}^{min}\le P_G{}_i\le P_G{{}_i}^{max}\begin{array}{ccc}& & i=1,2,\dots ,N_G\end{array}$$

(5)

$$Q_G{{}_i}^{min}\le Q_G{}_i\le Q_G{{}_i}^{max}\begin{array}{ccc}& & i=1,2,\dots ,N_G\end{array}$$

(6)

$$V_G{{}_i}^{min}\le V_G{}_i\le V_G{{}_i}^{max}\begin{array}{ccc}& & i=1,2,\dots ,N_G\end{array}$$

(7)

where $P_G^{max}$, $Q_G^{max}$, and $V_G^{max}$ denote the maximum limit of active power, reactive power, and voltage magnitude of generators, respectively. $P_G^{min}$, $Q_G^{min}$, and $V_G^{min}$ are the minimum limit of active power, reactive power, and voltage magnitude of generators, respectively.

-

Transformer constraints:

$$T_j{}^{min}\le T_j\le T_j{}^{max}\begin{array}{ccc}& & j=1,2,\dots ,N_T\end{array}$$

(8)

where $T^{max}$ and $T^{min}$ represent the maximum and minimum limit of tap regulating at transformers, respectively. $N_T$ is the number of tap changers.

-

Shunt compensator constraints:

$$Q_C{{}_{{}_k}}^{min}\le Q_{C_k}\le Q_C{{}_{{}_k}}^{max}\begin{array}{ccc}& & k=1,2,\dots ,N_C\end{array}$$

(9)

where $Q^{max}$ and $Q^{min}$ denote the maximum and minimum limit of the shunt compensator. $N_C$ is the number of shunt capacitor sources.

-

Security constraints:

$$V_L{{}_i}^{min}\le V_L{}_i\le V_L{{}_i}^{max}\begin{array}{ccc}& & i=1,2,\dots ,N_L\end{array}$$

(10)

$$S_{L_m}\le S_{L_m}{}^{max}\begin{array}{ccc}& & m=1,2,\dots ,N_{nl}\end{array}$$

(11)

where $S_L^{max}$ is the maximum limit of MVA in the transmission line. $N_{nl}$ is the number of transmission lines.

2.3. Objective Functions

In this paper, the five most common objective functions were optimized to solve OPF problems, which are fuel cost, losses, emission, voltage deviation, and the voltage stability index.

-

Total Fuel Cost [$/h]

The mathematical formulation for the total fuel cost in the power system can be expressed as follows [19]:

$$F_C={\displaystyle \sum }_{i=1}^{N_G}{f}_i(P_{G_i})={\displaystyle \sum }_{i=1}^{N_G}\left(a_i{P}_{G_i}^2+b_i{P}_{G_i}+c_i\right)\begin{array}{cc}& \end{array}[\$/h]$$

(12)

$a_i,b_i,$ and $c_i$ denote fuel cost coefficients for generators.

-

Active power losses [MW]

The second objective function in this study is active power losses in transmission lines. It can be expressed as [19]:

$$F_{loss}={\displaystyle \sum }_{k=1}^{N_{nl}}{G}_i\left( V_i^2+V_j^2-2V_i{V}_j\cos {\delta }_{i,j}\right)\begin{array}{cc}& \end{array}[\mathrm{MW}]$$

(13)

where $F_{loss}$ is the total real losses in transmission lines. $G_i$ is the transfer conductance.

-

Total emission [ton/h]

This objective function aims to minimize the emission level in the atmosphere by reducing pollution gases, such as $N{O}_x$ and $S{O}_2$. The mathematical description of this objective function can be expressed as [7]:

$$F_{em}={\displaystyle \sum }_{i=1}^{N_G}{10}^{-2}\left({\alpha }_i+{\beta }_i{P}_{G_i}+{\gamma }_i{P}_{G_i}^2\right)+{\zeta }_i\exp \left({\lambda }_i{P}_{G_i}\right)\begin{array}{cc}& \end{array}[\mathrm{ton}/\mathrm{h}]$$

(14)

${\alpha }_i,{\beta }_i,{\gamma }_i,{\zeta }_i,$ and ${\lambda }_i$ denote fuel emission coefficients in generators.

-

Voltage deviation [p.u.]

The goal of this case is to optimize the voltage quality in the system by minimizing the voltage deviations at the PQ bus from 1.0 [p.u.] The mathematical expression of this objective is expressed as [20]:

$$F_{VD}={\displaystyle \sum }_{i=1}^{N_{PQ}}\left|V_i-1.0\right|\begin{array}{cc}& \end{array}[\mathrm{p}.\mathrm{u}.]$$

(15)

-

Voltage stability index

The last objective function is to enhance the voltage stability index in the whole system by minimizing the maximum value of the voltage stability indicator (L-index). The mathematical formulation of the L-index is defined as [20]:

$$L_j=\left|1-{\displaystyle \sum _{i=1}^{N_G}\left(F_{ji}\times \frac{V_i}{V_j}\right)}\right|\begin{array}{ccc}& & \end{array}j=1,2,\dots ,Nl$$

(16)

$$F_{ji}=-{\left[Y_1\right]}^{-1}\times \left[Y_2\right]$$

(17)

where $Y_1$ and $Y_2$ denote the sub-matrices of the admittance matrix $\left(Y_{bus}\right)$. It can describe this objective function as:

$$F_{VSI}=Max\left(L_j\right)\begin{array}{ccc}& & \end{array}j=1,2,\dots ,Nl$$

(18)

The above objective functions are related as a single objective function, and they can be obtained by the proposed HGS approach. In addition to the single objective function, the multi-objective function was proposed to solve the multi-objective optimal power flow (MOOPF). In this article, many objective functions were optimized (they may be conflicting) simultaneously while satisfying equality and inequality constraints. Generally, a multi-objective function can be expressed by the following equation:

$$f_m\left(x,u\right)={\displaystyle \sum _{m=1}^N{\omega }_m{f}_m(x,u)=}{\omega }_1{f}_1+{\omega }_2{f}_2+\dots +{\omega }_m{f}_m\begin{array}{cc}& m=1,2,\dots ,N\end{array}$$

(19)

where $f$ denotes the MOF with No. of functions, $f_m$ is individual OF, and $\omega$ is the weight coefficient at the ranges [0,1], $N$ denotes the No. of OF. The sum of weight coefficients is equal to 1. In this article, the weight coefficient of each OF is inverse the number of OF. If number of OF equal 2 (Bi objective function), the weight coefficient equal to 0.5, and so on. In other words, there is no preference for any OF over the others. Therefore, we do not have any prime significance for any OF.

3. Single- and Multi-Objective OPF Frameworks

3.1. Conventional Hunger Games Search

Hunger games search (HGS) is a new optimization technique inspired by the behavior of social animal cooperatives, which is proportional to their level of hunger. The processes that characterized this algorithm into two stages can be summarized as follows:

• Approach food

The mathematical model of this process is described as follows:

$$Y(t+1)=\{\begin{array}{c}Y(t)\cdot \left(1+rand(1)\right),r_1<l\\ V_1\cdot Y_b+S\cdot V_2\cdot \left|Y_b-Y(t)\right|,r_1>l,r_2>e\\ V_1\cdot Y_b-S\cdot V_2\cdot \left|Y_b-Y(t)\right|,r_1>l,r_2<e\end{array}$$

(20)

where $Y\left(t+1\right)$ is the position of an individual in the next iteration; $Y\left(t\right)$ is the position of an individual in the current iteration; $Y_b$ denotes the position of the individual chosen randomly in optimal individuals; rand (1), $r_1$, and $r_2$ are the random numbers within $\left[0,1\right];$ $V_1$ and $V_2$ represent the hunger weights; $S$ is the number within $\left[-a,a\right];$ $l$ is the parameter setting experiment value; The formula of $e$ is:

$$e=Sech\left(\left|f(i)-bf\right|\right)$$

(21)

where $f\left(i\right)$ denotes the fitness value; and $bf$ is the best fitness. $Sech$ is a hyperbolic function and can be described as:

$$\left(Sech\left(x\right)=\frac{2}{e^x+e^{-x}}\right)$$

(22)

$S$ can be formulated as:

$$\begin{array}{l}S=2\times A\times r-A\\ A=2\times \left(1-\frac{t}{Max\_iter}\right)\end{array}$$

(23)

where $r$ is a random number within $\left[0,1\right];$ and $Max\_iter$ is the maximum iteration.

2.

Hunger role

The characteristics of starvation can be expressed as a mathematical formulation as follows:

$$V_1(l)=\{\begin{array}{l}hun(i)\cdot \frac{N}{SHun}\times r_4\begin{array}{cc}& r_3<l\end{array}\\ 1\begin{array}{cc}\begin{array}{ccc}& & \begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}\end{array}& r_3>l\end{array}\end{array}$$

(24)

$$V_2(l)=\left(1-\exp \left(-\left|hun(i)-SHun\right|\right)\right)\times r_5\times 2$$

(25)

where $N$ is the number of populations, and $SHun$ is the sum of populations that feel hunger. $r_3$, $r_4$, and $r_5$ are random numbers within [0, 1]. $hun$ denotes the population’s hunger and can be formulated as:

$$\begin{array}{l}hun=\{\begin{array}{cc}0& Allfitness(i)==bf\\ hun(i)+h& Allfitness(i)!=bf\end{array}\\ h=\{\begin{array}{c}\begin{array}{cc}\begin{array}{cc}Lh\times (1+r),& \end{array}& Th<Lh\end{array}\\ \begin{array}{cc}\begin{array}{cc}Th& \begin{array}{ccc}& & \end{array}\end{array}& Th\ge Lh\end{array}\end{array}\\ Th=\frac{f(i)-bf}{wf-bf}\times r_6\times 2\times (ub-lb)\end{array}$$

(26)

where $AllFitness\left(i\right)$ and $f\left(i\right)$ are the fitness of each population; $h$ denotes the hunger sensation; $r_6$ is a random number within [0, 1]; $bf$ and $wf$ are the best and the worst fitness; $ub$ and $lb$ refer to the upper and lower bounds; and $Lh$ is the lower bound.

To describe the process of the proposed approach, HGS, to solve optimal power flow (OPF), a pseudo-code and flowchart are presented in Algorithm 1 and Figure 1.

Algorithm 1: Pseudo-code of HGS.
1.	Input all parameters $N,Max\_iter,l,D,SHun$
2.	Determine the location of population $X_i\left(i=1,2,\cdots ,N\right)$
3.	While $\left(T\le Max\_iter\right)$
4.	Calculate $f\left(i\right)$
5.	Update $bf,wf,Y_b,bl$
6.	Calculate the population’s hunger $hun$ by Equation (26)
7.	Calculate the hunger weight (1) $V_1$ by Equation (24)
8.	Calculate the hunger weight (2) $V_2$ by Equation (25)
9.	For each population
10.	Calculate $e$ by Equation (21)
11.	Update $S$ by Equation (22)
12.	Update location by Equation (20)
13.	End (For)
14.	$T=T+1$
15.	End (While)
16.	Return $bf,Y_b$

3.2. Multi-Objective Hunger Games Search (MOHGS)

A new approach was proposed to solve MOOPF problems (two or more objective functions) and optimize simultaneously, namely, the multi-objective hunger games search (MOHGS). Based on the number of objective functions, the Pareto concept (PC) is the proposed approach to find out the dominant and non-dominated solutions. The decision maker is responsible to determine the best compromise solution (BCS) from non-dominated solutions (NDS). Many approaches were suggested to determine the BCS, such as the fuzzy set theory, entropy criterions, and centroid concept [21]. The most popular approach used to find out the BCS is the fuzzy decision maker [21,22,23,24]. In this study, the fuzzy membership function (FMF) is the equation used to extract the BCS from NDS. Finally, the specific strategy that is employed to reduce and arrange the NDPF is the crowding distance (CD).

• Pareto concept (PC)

The Pareto concept (PC) is the method used to find the NDS. In general, a solution $x_1$ dominates solution $x_2$ when:

$$\begin{array}{l}\forall i\in \left\{1,2,\dots ,n\right\}:f_i\left(x_1\right)\le f_i\left(x_2\right)\\ \forall j\in \left\{1,2,\dots ,n\right\}:f_j\left(x_1\right)<f_j\left(x_2\right)\end{array}$$

(27)

2.

The best compromise solution (BCS)

The equation below is used to determine the BCS based on fuzzy membership function (FMF) (Figure 2).

$$u_i^k=\{\begin{array}{c}1\begin{array}{ccccc}& & & & \end{array}{F}_i\le F_i^{min}\\ \frac{F_i^{max}-F_i}{F_i^{max}-F_i^{min}}\begin{array}{cc}& \end{array}{F}_i^{min}<F_i<F_i^{max}\\ 0\begin{array}{ccccc}& & & & \end{array}{F}_i\ge F_i^{max}\end{array}$$

(28)

$$u^k=\frac{{\displaystyle \sum _{i=1}^{N_{obj}}{u}_i^k}}{{\displaystyle \sum _{k=1}^M{\displaystyle \sum _{i=1}^{N_{obj}}{u}_i^k}}}$$

(29)

where $F_i^{min}$ and $F_i^{max}$ represent the minimum and maximum values of NDS. $u_i^k$ denotes the normalized membership function of each NDS. $u^k$ is the BCS when having the maximum value [25].

3.

Crowding distance strategy

This strategy is employed to rank and reduce NDS in NDPF. The average of two neighboring NDS represents the value of crowding distance (CD). First, the fitness of for each population should be sorted in a descending order. Thereafter, the boundary solutions are determined as an infinite value for each objective function. For the remainder solutions, the corresponding diagonal length of the intermediate solutions is assigned. Figure 3 represents the CD of an individual solution as the diagonal length of the cuboid. It is expressed as follows:

$$c{d}_i=\prod _{n=1}^m\frac{\left|f_n\left(x_i+1\right)-f_n\left(x_i-1\right)\right|}{f_{n,min}},\begin{array}{ccc}& & i=1,2,\cdot \cdot \cdot \end{array},n_b$$

(30)

where $c{d}_i$ represents the crowding distance; $F_{n, min}$ denotes the minimum value of the nth objective function; $N_b$ is the No. of candidate solutions.

4.

Stages of multi-objective HGS

The phases of MOHGS can be summarized as follows:

Step 1: Initialize system data such as max iteration, No. of population, the sum of populations that feel hunger, No. of NDS, No. of control variables, the parameter setting experiment, etc.

Step 2: Initialize individual’s positions in the HGS.

Step 3: Calculate fitness values for each individual in the HGS.

Step 4: Sort NDS according to the fitness value and store it in the initial repository.

Step 5: Calculate the best and the worst fitness, the upper and lower bounds, and the position individual is chosen randomly.

Step 6: Calculate the hun by (25).

Step 7: Calculate $e,S,$ and the hunger weights ($V_{1}and{V}_2$) by (20), (22), (23), and (24).

Step 8: Update the position of the individual in the next iteration (Y(t+1)), based on (19).

Step 9: Calculate the fitness value of each HGS.

Step 10: Update the position of HGS.

Step 11: Sort the NDS of HGS and store it in the HGS repository.

Step 12: Combine the NDS in the initial and HGS repository with the NDS of the HGS, updated to find new NDS.

Step 13: The stopping criteria, if the number of non-dominated solutions is equal to 500, end. Otherwise, store the non-dominated solutions in the initial repository, return to step 3.

The processes used to solve MOOPF problems with the developed approach MOHGS can be expressed in Figure 4.

4. Simulation Results and Discussion

In this study, two bus power systems were tested, IEEE 30 and IEEE 57 bus, and twenty-five case studies were investigated to prove the viability and efficiency of the proposed approaches (HGS and MOHGS).

Table 1. The main characteristics of the systems.

System Characteristics	IEEE 30-Bus	IEEE 57-Bus
Buses	30	57
Branches	41	80
Generators	9 (Buses:1, 2, 5, 8, 11, and 13)	7 (Buses: 1, 2, 3, 6, 8, 9, 12)
Generator voltage limits	0.90–1.1 [p.u.]	0.90–1.1 [p.u.]
Load voltage limits	0.95–1.05 [p.u.]	0.94–1.06 [p.u.]
Limit of tap changer setting	0.90–1.1 [p.u.]	0.90–1.1 [p.u.]
Limit of VAR	0–5 [p.u.]	0–20 [p.u.]
Shunts	9 (Buses: 10, 12, 15, 17, 20, 21, 23, 24, and 29)	3 (Buses: 18, 25, 53)
Transformers	4 (Buses: 11, 12, 15, and 36)	17 (Buses: 19, 20, 31, 35, 36, 37, 41, 46, 54, 58, 59, 65, 66, 71, 73, 76, 80)
MW demand	283.4 [MW]	1250.8 [MW]
Control variables	24	33

presents the characteristics of these systems. The cases studied are summarized in

Table 2. The studied cases with different types.

System	Type of OF(s)	Case #	FC	Em	APL	VD	VSI
IEEE 30-bus	Single OF (s)	#1	√
		#2			√
		#3		√
		#4				√
		#5					√
	Bi-OF(s)	#6	√	√
		#7	√		√
		#8	√			√
		#9	√				√
		#10		√		√
		#11			√	√
		#12				√	√
	Triple-OF(s)	#13	√	√	√
		#14	√	√		√
		#15	√		√	√
		#16	√	√		√
		#17	√		√		√
		#18	√	√			√
		#19	√			√	√
	Quad-OF(s)	#20	√	√	√	√
		#21	√	√	√		√
	Quinta-OF(s)	#22	√	√	√	√	√
IEEE 57-bus	Single OF (s)	#23	√
		#24		√
		#25			√

. The simulation results were carried out on Intel Core (TM) i5-10500H 2.5GHz (12 CPUs) and 16384 (64 bit) MB RAM. The active power output of the generator without the slack bus, the voltage of the generator bus, tap setting regulating of transformers, and shunt capacitors VAR, which are connected on transmission lines, are the control variables that were set to obtain the optimal objective function.

4.1. IEEE 30-Bus Power System

In this article, the IEEE 30-bus system represents the first test system that was applied to validate the effectiveness and superiority of the proposed approaches (HGS and MOHGS), as shown in Figure 5. The control variable for the OPF problem for the IEEE 30-bus system is 24. The cost and emission coefficients of this system are illustrated in

Table 3. The cost and emission coefficients of generators for the IEEE 30 bus system.

Coefficient Generating Unit
	G1	G2	G5	G8	G11	G13
Fuel cost coefficient
a	0	0	00	0	0	0
b	2	1.75	1	3.25	3	3
c	0.00375	0.0175	0.0625	0.00834	0.025	0.025
Emission coefficient
α	4.091	2.543	4.258	5.326	4.258	6.131
β	−5.554	−6.047	−5.094	−3.55	−5.094	−5.555
γ	6.49	5.638	4.586	3.38	4.586	5.151
ζ	2.00 × 10−4	5.00 × 10−4	1.00 × 10−6	2.00 × 10−3	1.00 × 10−6	1.00 × 10−5
λ	2.857	3.33	8	2	8	6.67

. The maximum and minimum limits of voltage magnitude for generators and load bus are in the range [0.9–1.1] and [0.95–1.05], respectively.

4.1.1. Single-Objective OPF

Five objective functions were applied to demonstrate the efficiency and superiority of the proposed algorithm HGS. In a single OF, the objective functions that are optimized are total fuel cost [$/h], total emission [ton/h], active power losses [MW], voltage deviation [p.u.], and voltage stability index. The parameters chosen to solve a single OF are 1000 iterations and 250 population size.

Case #1: In this case, the rate of fuel cost minimization [$/h] is the objective function that was optimized by the HGS algorithm. The convergence plot of fuel cost [$/h] for the IEEE 30 bus system is depicted in Figure 6a. The total fuel costs are reduced from 901.6391 [$/h] (initial case) to 799.2202 [$/h] (best case) with a reduction loss of 11.36%, as listed in

Table 4. The best control variables obtained by HGS for cases #(1–5).

Item		Limit		Initial [26]	Case
		Max	Min		#1	#2	#3	#4	#5
[MW]	${\mathrm{P}}_1$	50	200	99.223	176.73	51.518	67.9326	99.3206	75.4018
	${\mathrm{P}}_2$	20	80	80	48.6285	79.9472	71.0846	56.2324	72.3195
	${\mathrm{P}}_5$	15	50	50	21.3556	49.9963	49.999	48.5423	49.6165
	${\mathrm{P}}_8$	10	35	20	20.9611	34.8738	34.999	34.7508	32.8099
	${\mathrm{P}}_{11}$	10	30	20	12.3146	29.9886	29.9995	19.9674	28.8962
	${\mathrm{P}}_{13}$	12	40	20	12.047	39.9871	32.7369	29.7117	27.9364
[p.u.]	${\mathrm{V}}_1$	0.95	1.1	1.05	1.10	1.09957	1.07979	1.00733	1.09893
	${\mathrm{V}}_2$	0.95	1.1	1.04	1.0879	1.09575	1.02963	0.99464	1.09953
	${\mathrm{V}}_5$	0.95	1.1	1.01	1.08115	1.07676	1.06095	1.06787	1.09397
	${\mathrm{V}}_8$	0.95	1.1	1.01	1.08895	1.09501	1.09796	1.05785	1.09956
	${\mathrm{V}}_{11}$	0.95	1.1	1.05	1.09935	1.09985	1.09540	1.05771	1.09950
	${\mathrm{V}}_{13}$	0.95	1.1	1.05	1.09998	1.09975	1.06955	0.99197	1.09772
[MVAr]	${\mathrm{Q}}_{\mathrm{c}10}$	0	5	0	4.40056	4.21049	4.05070	3.89253	4.96503
	${\mathrm{Q}}_{\mathrm{C}12}$	0	5	0	4.24071	4.19982	3.13443	1.68208	2.09009
	${\mathrm{Q}}_{\mathrm{c}15}$	0	5	0	4.47918	4.26252	3.70849	2.33112	4.61987
	${\mathrm{Q}}_{17}$	0	5	0	4.98546	4.31378	3.62262	3.14785	3.53632
	${\mathrm{Q}}_{\mathrm{c}20}$	0	5	0	4.62076	4.26326	4.13636	4.99605	4.95659
	${\mathrm{Q}}_{21}$	0	5	0	4.97743	4.27526	1.42028	4.06093	4.94313
	${\mathrm{Q}}_{\mathrm{c}23}$	0	5	0	4.41930	4.16337	2.22773	4.97509	4.66820
	${\mathrm{Q}}_{24}$	0	5	0	4.98744	4.23189	4.69879	4.99004	4.99429
	${\mathrm{Q}}_{29}$	0	5	0	3.78063	4.03580	3.42316	2.55515	4.99515
Tap Position	${\mathrm{T}}_{11}$	0.9	1.1	1.078	1.02609	1.01962	1.03789	0.97665	1.04588
	${\mathrm{T}}_{12}$	0.9	1.1	1.069	0.99766	1.04051	1.02467	0.98726	1.03403
	${\mathrm{T}}_{15}$	0.9	1.1	1.032	0.99149	0.95436	1.02072	0.99930	0.95605
	${\mathrm{T}}_{36}$	0.9	1.1	1.068	0.98371	0.99075	1.01205	0.96666	0.95025
Fuel cost [$/h]				901.639	799.220	966.837	933.703	889.278	920.183
Active power loss [MW]				5.6891	8.6424	2.9109	3.3515	5.1253	3.5803
Emission [ton/h]				0.2253	0.3673	0.2216	0.2174	0.2343	0.2204
Voltage deviation [p.u.]				1.1747	1.5671	1.7078	0.7971	0.1195	1.9699
Voltage stability index				0.1727	0.1199	0.1191	0.1314	0.1375	0.1119
Reduction rate				-	11.36%	48.83%	3.51%	89.83%	35.20%
Average resolution time [s/iter]				-	1.4834	1.631	1.613	1.575	1.609

The bold represents the best values of OF.

.

Case #2: The main aim of this case was to reduce the active power loss [MW] in the transmission lines using the HGS algorithm. The reduction rate in losses is 48.83% (reducing from 5.6891 [MW] to 2.9109 [MW]). The convergence speed of this case is illustrated in Figure 6b.

Case #3: Due to increasing environmental pollution, the new studies drew growing attention in emission. In this paper, this objective function deals with emission minimization. The variations of emission over iteration are described in Figure 6c. The best value of emission that can be achieved by the HGS algorithm is 0.2174 [ton/h]. The total emissions are reduced from 0.2253 [ton/h] to 0.2174 [ton/h] with a reduction rate equal to 3.51%, as illustrated in Table 4.

Case #4: To improve the voltage at PQ bus, the voltage deviation is the objective function that is considered. The sum voltage deviation reduced from 1.1747 to 0.1195 (reduction rate is 89.83%), as seen in Table 4. The convergence speed for this case is shown in Figure 6d.

Case #5: The fifth objective function is to enhance voltage stability of the whole system by minimizing the maximum value of the voltage stability indicator (L-max) using the HGS algorithm. The convergence characteristics of VSI using the proposed approach of HGS is depicted in Figure 6e. The reduction rate of this case is equal to 35.20% (compared between the initial case, which is 0.1727, and the optimal case using HGS, which is 0.1119) as given in Table 4.

The best control variables for cases (1–5) using the proposed algorithm HGS on the IEEE 30 bus system are depicted in Table 4. To prove the superiority and effectiveness of the performance of the HGS algorithm, the best results obtained by the HGS algorithm for fuel cost [$/h], active power losses [MW], emission [ton/h], voltage deviation [p.u.], and voltage stability index are compared with other recent optimization methods results reported in the literature, as seen in

Table 5. Comparison of the best results obtained by HGS with other algorithms for cases (1,2).

Case #1		Case #2
Method	FC [$/h]	Method	APL [MW]
Initial	901.6391	Initial	5.830
SCA [27]	800.1018	SSO [28]	3.8239
DSA [29]	800.3887	EM [30]	3.1775
JAYA [31]	800.479	GPU-PSO [32]	3.2601
MSA [33]	800.5099	EGA-DQLF [34]	3.2008
SP-DE [35]	800.4131	ASO [36]	3.1600
MGOA [37]	800.4744	EGA-EA [38]	3.2601
AMTPG-Jaya [39]	800.1946	GWO [10]	4.2905
TLBO [39]	800.4604	PSO [40]	5.1957
ABC [41]	800.6850	HPSO-DE [40]	5.1476
IABC [41]	800.4215	FAHSPSO-DE [40]	4.9989
SSO [28]	802.2580	IPSO [42]	5.0732
GPU-PSO [32]	800.53	HHO [43]	3.49
EGA [44]	802.06	SSA [43]	3.50
IGA [45]	800.805	WOA [43]	3.50
AGAPOP [46]	799.8441	MF [43]	3.50
ABC [47]	800.66	GWO [43]	3.51
PSOGSA [48]	800.49859	SMA [14]	2.9934
GWO [10]	802.7924	HGS	2.9109
ISSA [49]	800.4752
MFO [49]	800.7134
IHS [49]	800.5202
GA [49]	800.5272
SOS [50]	801.5733
SMA [14].	799.2557
HGS	799.2202

The bold line represents the best values of OF obtained by HGS.

and

Table 6. Comparison of the best results obtained by HGS with other algorithms for cases (3–5).

Case #3		Case #4		Case #5
Method	Em [ton/h]	Method	VD [p.u]	Method	VSI
Initial	0.3661	Initial	1.1747	Initial	0.1727
BSA [51]	0.2425	HFPSO [52]	0.1467	Jaya [31]	0.1243
SSO [28]	0.2315	EJADE-SP [9]	0.3752	AMTPG-Jaya [39]	0.1240
HHO [43]	0.2850	MABC [53]	0.1292	SSO [28]	0.1267
SSA [43]	0.2950	HGS	0.1195	NISSO [28]	0.12547
WOA [43]	0.2950			TLBO [39]	0.12444
MF [43]	0.2950			ARCBBO [54]	0.1369
GWO [43]	0.2960			ECHT-DE [55]	0.13632
SMA [14]	0.2175			SPEA [56]	0.1247
HGS	0.2174			DE [57]	0.1246
				SMA [14]	0.1136
				HGS	0.1119

The bold represents the best values of objective OF obtained by the HGS.

. The bold line represents the best values of OF obtained by HGS.

4.1.2. Bi-Objective OPF

In this subsection, two objective functions were optimized simultaneously to achieve the best compromise solution (BCS) of non-dominated solutions (NDS). The function used to find the BCS was the fuzzy membership function (FMF). The crowding distance (CD) was the method employed to rank and reduce NDS of the NDPF. The developed approach MOHGS was applied in power systems to demonstrate its performance to solve MOOPF problems. In this subsection, seven case studies were suggested to prove the efficiency and superiority of the MOHGS. The population size was 500 pollinators; the program was stopped when the number of NDS was equal to the number of NDS suggested, which was 500, or a number of iterations equal to 100 iterations. These cases can be summarized as follows:

Case #6: In the first case of Bi-OF, the total fuel cost and total emission were optimized simultaneously by using the proposed approach MOHGS. The BCS of fuel cost and emission were 827.735 [$/h] and 0.2587 [ton/h], respectively.

Case #7: The second case of this type, the total fuel cost and active power losses were considered as objective functions and optimized simultaneously. The BCS of fuel cost and losses were 826.842 [$/h] and 5.5946 [MW].

Case #8: The third case of bi-objectives functions that were optimized in this work were fuel cost and voltage deviation. The BCS of fuel cost and voltage deviation are 803.094 [$/h] and 0.1813 [p.u.]

Case #9: The fuel cost and voltage stability index are the objective functions that were minimized simultaneously in this case. The BCS was 801.491 [$/h] and 0.1213 of fuel cost and voltage stability index, respectively.

Case #10: The fifth case of Bi-OF was minimizing the total emission and voltage deviation simultaneously. The BCS of emission and voltage deviation were 0.2245 [ton/h] and 0.1508 [p.u.].

Case #11: The active power losses and voltage deviation represented the objective functions that were optimized simultaneously in this case. The BCS of losses and voltage deviation were 3.8636 [MW] and 0.2048 [p.u.].

Case #12: The voltage deviation and voltage stability index represented the last case of Bi objective OPF. The BCS of voltage deviation and voltage stability index were 0.3690 [p.u.] and 0.1280.

Figure 7a–g shows Pareto front solutions of the bi-objective functions obtained by the MOHGS on the IEEE 30-bus system. The best compromise solution of Bi OF for cases 6–12 is indicated by red diamonds, as shown in Figure 7.

4.1.3. Triple Objective OPF

In this type, three objective functions were considered simultaneously to obtain the BCS from NDS in the NDPF. Seven case studies are proposed; the population size is 500 pollinators. The stopping criteria of simulation running when the number of iterations equal to 100 iterations, or the number of non-dominated solutions equal to 500 pollinators. These cases can be summarized as follows:

Case #13: In the first case in this type, the fuel cost, emission, and losses combined were optimized simultaneously by using the proposed approach MOHGS to achieve the Pareto set solutions. The BCS of fuel cost, emission, and losses were 845.2988 [$/h], 0.2416 [ton/h], and 5.0562 [MW], respectively.

Case #14: In this case, the fuel cost, emission, and voltage deviation were considered objective functions and optimized simultaneously. The BCS of fuel cost, emission, and voltage deviation were 825.1766 [$/h], 0.2667 [ton/h], and 0.1953 [p.u.], respectively.

Case #15: The third case of triple-objective functions were minimization of fuel cost, losses, and voltage deviation. The BCS of fuel cost, losses, and voltage deviation were 817.1199 [$/h], 7.3484 [MW], and 0.1767 [p.u.], respectively.

Case #16: The active power losses, emission, and voltage deviation were the objective functions that were optimized simultaneously. The BCS was 3.8246 [MW], 0.2196 [ton/h], 0.2122 [p.u.] for losses, emission, and voltage deviation, respectively.

Case #17: The seventeenth case in this paper was minimization in fuel cost, losses, and voltage stability index, simultaneously. The BCS of fuel cost, losses, and voltage stability index were 834.4639 [$/h], 5.4740 [MW], 0.1198, respectively.

Case #18: In this case, the objective functions that were optimized included the fuel cost, emission, and voltage stability index. The BCS of fuel cost, losses, and voltage stability index were 835.0571 [$/h], 0.2504 [ton/h], 0.1192, respectively.

Case #19: Fuel cost, voltage deviation, and voltage stability index were the objective functions that were minimized simultaneously. The BCS of fuel cost, voltage deviation, and voltage stability index were 802.5842 [$/h], 0.4813 [p.u.], 0.1279, respectively.

The three-dimensional Pareto fronts set that were obtained by the developed approach MOHGS are illustrated in Figure 8a–g. The best compromise solutions of Tri OF for cases 13–19 are indicated by red diamonds, as shown in Figure 8.

4.1.4. Quad and Quinta Objective OPF

The last type of objective functions applied on the IEEE 30-bus system represents the Quad and Quinta objective functions. Two case studies on Quad objective functions and one case study on Quinta objective function are the suggested cases to solve MOOPF in this type. The stopping criteria are achieved when the No. of NDS is equal to 500 pollinators, or the No. of iterations reaches 500. It can be summarized as follows:

Case #20: Total fuel cost, losses, emission, and voltage deviation are the objective functions that were optimized in this case. The BCS obtained by MOHGS were 845.4721 [$/h], 5.7458 [MW], 0.2507 [ton/h], and 0.1400 [p.u.] for fuel cost, losses, emission, and voltage deviation., respectively.

Case #21: The objective functions that were optimized simultaneously were fuel cost, losses, emission, and voltage stability index. The BCS were 819.4061 [$/h], 7.4137 [MW], 0.2898 [ton/h], and 0.1389 for fuel cost, losses, emission, and voltage stability index, respectively.

Case #22: In this case, five objective functions were optimized simultaneously, which were fuel cost, losses, emission, voltage deviation, and voltage stability index. The best results of objective functions obtained by MOHGS were 818.7575 [$/h], 7.4471 [MW], 0.2912 [ton/h], 0.3272 [p.u.], and 0.1399 for fuel cost, losses, emission, voltage deviation, and voltage stability index, respectively.

Table 7. Better control variables obtained by HGS for cases (6−15).

Item		Case
		#6	#7	#8	#9	#10	#11	#12	#13	#14	#15
[MW]	${\mathrm{P}}_1$	120.99	125.59	177.232	176.954	87.646	62.266	145.554	107.558	127.609	141.273
	${\mathrm{P}}_2$	60.834	48.558	46.335	44.891	64.306	75.233	78.062	61.963	59.373	52.885
	${\mathrm{P}}_5$	28.719	27.875	21.539	18.819	47.005	48.636	24.003	32.594	26.056	29.708
	${\mathrm{P}}_8$	34.476	34.619	21.650	21.417	34.672	32.980	18.529	33.168	34.366	30.809
	${\mathrm{P}}_{11}$	21.773	26.675	13.122	14.813	27.983	29.414	10.806	29.236	27.365	22.131
	${\mathrm{P}}_{13}$	22.520	25.678	13.130	15.373	26.889	38.735	15.923	23.937	15.486	13.942
[p.u.]	${\mathrm{V}}_1$	1.092	1.097	1.051	1.095	1.003	1.031	1.053	1.097	1.072	1.068
	${\mathrm{V}}_2$	1.053	1.099	1.015	1.099	1.026	1.060	1.043	1.086	1.000	1.047
	${\mathrm{V}}_5$	1.093	1.063	1.053	1.099	1.076	1.002	0.972	1.082	1.068	1.040
	${\mathrm{V}}_8$	1.055	1.077	1.061	1.084	1.063	1.054	1.097	1.068	1.037	1.004
	${\mathrm{V}}_{11}$	1.095	1.099	1.037	1.085	0.986	1.040	1.083	1.099	1.055	1.037
	${\mathrm{V}}_{13}$	1.090	1.076	1.085	1.082	1.028	1.010	1.052	1.073	1.015	0.979
[MVAr]	${\mathrm{Q}}_{\mathrm{c}10}$	1.354	2.316	3.047	2.628	1.773	0.075	3.266	0.699	0.658	0.688
	${\mathrm{Q}}_{\mathrm{C}12}$	0.804	2.958	0.160	2.868	1.243	0.000	0.278	3.942	0.730	4.276
	${\mathrm{Q}}_{\mathrm{c}15}$	3.354	0.888	4.338	0.836	0.512	0.240	4.486	0.400	1.395	0.405
	${\mathrm{Q}}_{17}$	2.167	2.201	0.342	3.719	2.748	0.188	3.587	2.653	0.360	0.627
	${\mathrm{Q}}_{\mathrm{c}20}$	1.133	1.648	3.588	3.687	1.528	0.782	3.673	1.737	1.704	4.744
	${\mathrm{Q}}_{21}$	0.261	1.228	4.004	3.411	2.983	4.688	3.797	2.421	0.715	1.176
	${\mathrm{Q}}_{\mathrm{c}23}$	0.503	2.613	3.848	1.022	3.508	4.964	1.118	4.808	1.550	0.123
	${\mathrm{Q}}_{24}$	0.908	3.930	2.127	4.944	4.474	1.196	4.449	1.649	4.883	4.128
	${\mathrm{Q}}_{29}$	1.623	1.035	0.545	4.305	0.898	0.893	4.948	2.707	4.010	1.663
Tap Position	${\mathrm{T}}_{11}$	0.977	1.038	1.057	0.965	0.972	0.961	1.043	0.993	0.970	0.952
	${\mathrm{T}}_{12}$	0.962	0.983	0.971	0.974	0.950	0.963	1.098	0.972	0.978	0.971
	${\mathrm{T}}_{15}$	1.060	1.054	0.958	1.009	0.968	0.984	0.989	1.088	0.959	0.950
	${\mathrm{T}}_{36}$	1.000	0.997	0.955	0.953	0.954	0.951	0.951	0.972	0.973	0.954
GFC [$/h]		827.74	826.84	803.094	801.491	899.074	946.099	826.338	845.299	825.177	817.12
RPL [MW]		5.9178	5.5946	9.6092	8.8720	5.1007	3.8636	9.4757	0.2416	6.8612	7.3484
Em [ton/h]		0.2587	0.2619	0.3678	0.3668	0.2245	0.2208	0.3077	5.0562	0.2667	0.2868
VD [p.u.]		0.8944	0.9559	0.1813	1.3536	0.1508	0.2048	0.3690	1.0547	0.1953	0.1767
VSI		0.1373	0.1333	0.1410	0.1213	0.1413	0.1445	0.1280	0.1282	0.1392	0.1412
Average resolution time [s/iter]		2.723	4.7027	2.7262	2.6784	2.70	2.68	2.703	2.79	2.734	2.718

The bold represents the BCS of OF.

and

Table 8. The better control variables obtained by HGS for cases (16–22).

Item		Case
		#16	#17	#18	#19	#20	#21	#22
[MW]	${\mathrm{P}}_1$	67.936	121.170	113.261	179.444	119.182	142.724	143.244
	${\mathrm{P}}_2$	68.222	55.959	65.029	45.132	48.973	52.684	57.063
	${\mathrm{P}}_5$	49.162	34.921	29.736	19.931	40.045	33.242	32.352
	${\mathrm{P}}_8$	34.370	33.035	34.654	23.632	24.910	29.195	24.328
	${\mathrm{P}}_{11}$	29.153	27.184	22.918	11.624	18.664	16.913	20.928
	${\mathrm{P}}_{13}$	38.381	16.605	23.210	13.170	39.105	16.055	12.932
[p.u.]	${\mathrm{V}}_1$	1.015	1.083	1.099	1.086	1.006	1.052	1.074
	${\mathrm{V}}_2$	0.987	1.076	1.090	1.087	1.086	0.998	1.004
	${\mathrm{V}}_5$	1.032	1.099	1.079	1.026	1.088	1.005	1.026
	${\mathrm{V}}_8$	1.048	1.087	1.093	1.099	1.097	1.093	1.076
	${\mathrm{V}}_{11}$	1.094	1.083	1.087	0.980	0.996	1.096	1.013
	${\mathrm{V}}_{13}$	1.071	1.096	1.091	0.990	1.084	1.085	0.981
[MVAr]	${\mathrm{Q}}_{\mathrm{c}10}$	2.414	3.784	4.171	3.803	0.229	2.182	0.888
	${\mathrm{Q}}_{\mathrm{C}12}$	2.485	3.543	1.272	4.502	1.402	1.709	1.529
	${\mathrm{Q}}_{\mathrm{c}15}$	3.754	4.974	1.142	0.108	2.792	4.237	3.093
	${\mathrm{Q}}_{17}$	0.880	1.844	0.188	0.204	1.085	0.831	1.063
	${\mathrm{Q}}_{\mathrm{c}20}$	1.670	4.075	2.411	0.782	0.612	4.614	0.894
	${\mathrm{Q}}_{21}$	0.349	4.448	4.984	4.005	3.912	2.728	3.500
	${\mathrm{Q}}_{\mathrm{c}23}$	1.894	2.264	4.077	2.077	4.720	3.717	2.649
	${\mathrm{Q}}_{24}$	4.431	3.851	3.015	4.868	3.889	0.812	2.482
	${\mathrm{Q}}_{29}$	0.494	3.948	4.108	4.796	2.432	3.919	3.497
Tap Position	${\mathrm{T}}_{11}$	1.010	0.950	0.959	0.998	1.044	1.012	0.963
	${\mathrm{T}}_{12}$	0.981	1.026	0.980	0.979	0.976	1.075	1.024
	${\mathrm{T}}_{15}$	1.001	0.962	0.974	1.003	0.984	0.957	0.993
	${\mathrm{T}}_{36}$	0.963	0.954	0.953	0.952	1.028	1.013	0.976
GFC [$/h]		936.4713	834.4639	835.0571	802.5842	845.4721	819.4061	818.7575
RPL [MW]		3.8246	5.4740	5.4107	9.5336	5.7458	7.4137	7.4471
Em [ton/h]		0.2196	0.2559	0.2504	0.3746	0.2507	0.2898	0.2912
VD [p.u.]		0.2122	1.5161	1.6370	0.4813	0.1400	0.3514	0.3272
VSI		0.1444	0.1198	0.1192	0.1279	0.1456	0.1389	0.1399
Average resolution time [s/iter]		2.7456	2.718	2.7652	2.7479	2.7334	2.7668	2.8734

The bold represents the BCS of OF.

represent the better control variables obtained by MOHGS for Bi, Tri, Quad, and Quinta objective functions that are illustrated by cases (20–22).

The voltage profiles at each load bus are charted in Figure 9a. From Figure 9a, it can be concluded that the voltage magnitude in the load bus for cases (1, 2, 3, and 5) exceeds the maximum limit (1.05 [p.u.]) for some buses, while the voltage magnitude of PQ bus for case 4 (when the voltage deviation is considered as an objective function) have not exceeded the minimum and maximum limit (0.95–1.05) [p.u.]). In Bi objective functions, the voltage profiles of this type are illustrated in Figure 9b. The numerical results obtained by MOHGS illustrate that the voltage magnitude of PQ bus exceeds the maximum limit in cases when the voltage deviation is not considered an objective function (cases (1,2,4)). However, the voltages magnitude has not exceeded the minimum and maximum limit for the cases in which the voltage deviation is considered an objective function (cases 3,5,6,7)). Figure 9c illustrates that the voltage magnitude values of some buses in cases 8, 12, 13 (the cases where the voltage deviation is not considered an objective function) exceed the maximum [1.05 p.u.], while the voltage magnitude values in cases (9, 10, 11, 14) have not exceeded the minimum and maximum limit [0.95–1.05 p.u.]. The voltage profiles for types Quad and Quinta are illustrated in Figure 9d. Figure 9d illustrates that the voltage magnitude of the load bus for these cases has not violated the boundary limit [0.95–1.05] [p.u.].

Figure 10a–f shows the values of the reactive power output from the generators for all cases (1–22). It can be observed that the MVAr of some generators (1, 5, and 8) exceeds the minimum and maximum limit in cases (1, 2, 5, 10, 12, 19, and 20). While the remaining generators have not violated the constraints in all cases.

4.2. IEEE 57-Bus Power System

The second test system applied in the paper to validate the effectiveness and superiority of the proposed algorithm HGS is the IEEE 57 bus power system, as shown in Figure 11. The control variables in the OPF problem for the IEEE 57-bus system are 33. The cost and emission coefficients of IEEE 57 bus power system are illustrated in

Table 9. The cost and emission coefficients for generators for the IEEE 57-bus test system.

Coefficient Generating Unit
	G1	G2	G3	G6	G8	G9	G12
Fuel cost coefficient
a	0	0	0	0	0	0	0
b	2	1.75	3	2	1	1.75	3.25
c	0.00375	0.0175	0.025	0.00375	0.0625	0.0195	0.00834
Emission coefficient
α	4.091	2.543	6.131	3.491	4.258	2.754	5.326
β	−5.554	−6.047	−5.555	−5.754	−5.094	−5.847	−3.555
γ	6.49	5.638	5.151	6.39	4.586	5.238	3.38
ζ	2.0 × 10−4	5.0 × 10−4	1.0 × 10−5	3.0 × 10−4	1.0 −6	4.0 × 10−4	2.0 × 10−3
λ	2.857 × 10−1	3.33 × 10−1	6.67 × 10−1	2.66 × 10−1	8.0 × 10−1	2.88 × 10−1	2.0 × 10−1

. The optimal value control variables obtained by the proposed algorithm HGS are reported in

Table 10. Better control variables obtained by HGS for cases (23−25).

Item		Limit		Initial	Case
		Max	Min		#23	#24	#25
[MW]	$P_1$	0.0	576	478	147.65	180.33	191.87
	$P_2$	30.0	100	0	92.19	28.74	99.34
	$P_3$	40.0	140	40	46.14	124.68	138.93
	$P_6$	30.0	100	0	68.59	33.74	99.88
	$P_8$	100.0	550	450	463.81	390.93	282.28
	$P_9$	30.0	100	0	84.31	94.41	99.84
	$P_{12}$	100.0	410	310	363.11	409.80	353.99
[p.u.]	$V_1$	0.95	1.10	1.040	1.07	1.07	1.02
	$V_2$	0.95	1.10	1.010	1.07	1.08	1.02
	$V_3$	0.95	1.10	0.985	1.06	1.07	1.02
	$V_6$	0.95	1.10	0.980	1.06	1.06	1.02
	$V_8$	0.95	1.10	1.005	1.09	1.08	1.04
	$V_9$	0.95	1.10	0.980	1.08	1.06	1.04
	$V_{12}$	0.95	1.10	1.015	1.06	1.05	1.01
Tap Position	$T_{4-18}$	0.90	1.10	0.97	1.02	1.02	1.01
	$T_{4-18}$	0.90	1.10	0.978	1.02	1.05	1.04
	$T_{21-20}$	0.90	1.10	1.043	0.98	0.99	1.01
	$T_{24-25}$	0.90	1.10	1	1.03	1.04	0.99
	$T_{24-25}$	0.90	1.10	1	1.04	1.03	0.99
	$T_{24-26}$	0.90	1.10	1.043	1.06	1.00	1.01
	$T_{7-29}$	0.90	1.10	0.967	1.05	1.05	0.99
	$T_{34-32}$	0.90	1.10	0.975	0.98	0.97	0.98
	$T_{11-41}$	0.90	1.10	0.955	1.02	1.09	1.03
	$T_{15-45}$	0.90	1.10	0.955	1.00	1.05	1.01
	$T_{14-46}$	0.90	1.10	0.9	1.01	1.05	1.00
	$T_{10-51}$	0.90	1.10	0.93	1.03	1.09	0.97
	$T_{13-46}$	0.90	1.10	0.895	0.95	1.02	1.00
	$T_{11-43}$	0.90	1.10	0.958	1.03	1.02	1.08
	$T_{40-56}$	0.90	1.10	0.958	1.00	0.98	1.05
	$T_{39-57}$	0.90	1.10	0.98	0.99	1.00	1.02
	$T_{9-55}$	0.90	1.10	0.94	1.06	1.03	1.02
[MVAr]	$Q_{c18}$	0.00	20	10	12.58	26.08	11.25
	$Q_{25}$	0.00	20	5.9	22.16	21.56	10.78
	$Q_{53}$	0.00	20	6.3	12.78	10.89	18.02
GFC [$/h]				51353	41679.4	43740	45118
RPL [MW]				2.4129	15.24	11.829	15.6796
Em [ton/h]				27.868	1.3746	1.3048	0.9616
Reduction rate				-	18.84%	57.55%	60.15%
Average resolution time [s/iter]				-	12.257	12.225	12.24

The bold represents the better values of OF obtained by the proposed approach.

.

Single Objective OPF

The objective functions that are minimized are fuel cost [$/h], active power losses [MW], and emission [ton/h]. The parameters chosen in the proposed algorithm HGS to solve OPF problem are 100 iterations and 100 population sizes.

Case #23: Total fuel cost was minimized from 51,353 [$/h] (initial) to 41,679.4 [$/h] (the better), and the reduction rate was 18.84%, as illustrated in Table 10. The convergence plot of fuel cost [$/h] is depicted in Figure 12a.

Case #24: Active power losses are minimized from 27.868 [MW] (initial) to 11.829 [MW] (the better); the reduction rate is 57.55%. The convergence speed of this case is illustrated in Figure 12b.

Case #25: Figure 12c illustrates the variations of emission over iteration. The better value of emission is 0.962 [ton/h]. The reduction rate is 60.15%.

The better control variables for the IEEE 57 bus system that are obtained by the HGS algorithm are depicted in Table 10. The bold represents the better values of OF obtained by the proposed approach.

Table 11. Comparison of the numerical results obtained by HGS with other algorithms for cases (23−25).

Case #23		Case #24		Case #25
Method	FC [ton/h]	Method	RPL [p.u]	Method	Em [ton/h]
GOA [37]	41680	ABC [41]	12.6260	ABC [41]	1.2048
GA [37]	41685	MICA [58]	11.8826	IABC [41]	1.0484
TLBO [37]	41683	GA [59]	13.3983	Jaya [59]	1.1111
MO-DEA [23]	41683	PSO [59]	13.6673	GA [59]	1.189
ABC [47]	41694	HGS	11.83	PSO [59]	1.19
GSA [60]	41695			MTLBO [61]	1.0772
NPSO [62]	41699.52			MICA [58]	1.2246
KHA [63]	41709.3			GBBICA [64]	1.1724
EADDE [65]	41713.6			SKHA [66]	1.08
fuzzy GA [67]	41716.3			SSO [28]	1.7024
HGS	41679.4			NISSO [28]	1.03927
				HGS	0.9616

The bold represents the better values of OF obtained by the proposed HGS approach.

compares the numerical results obtained by HGS with other recent optimization methods for fuel cost [$/h], active power losses [MW], and emission [ton/h].

The voltage profiles at each load bus for the IEEE 57 bus test system are charted in Figure 13. Voltage limits [0.94–1.06 p.u.] were exceeded on some buses. The exceeding of admissible voltage deviation at load buses may be attributed to several reasons. One of the most important reasons is the fact that we have a single objective function for these cases (23–25), and the voltage deviation was not considered an objective function for these cases. The reactive power of generators (2, 6, and 9) has violated the limits, while for the reactive power output for the generators (1, 3, 8, and 12), the constraints were not violated. The reactive power output for the generators in cases (23–25) is shown in Figure 14.

4.3. Discussions

To demonstrate the efficiency and the performance of the proposed HGS algorithm and of the developed MOHGS approach, the proposed approaches were applied in power systems to solve single and multiple OPF. For solving multi-objective optimal power flow (MOOPF) problems, the authors have developed the proposed HGS algorithm considering a new approach, namely, the multi-objective hunger game search (MOHGS), by merging with Pareto concept optimization to obtain a better distribution of non-dominated solutions in the Pareto front set. In this article, the numerical results listed in Table 4 confirmed the ability of the proposed HGS algorithm to find good convergence and the better solution compared with state-of-the-art computational algorithms. Furthermore, the simulation results drawn in Figure 7 and Figure 8 confirmed the very good capability of the developed MOHGS approach to achieve the convergence characteristics and the well-distribution in the Pareto front set.

The single and multi-objective OPF problems faced many challenges, the most important of which were the accurate convergence to reach the global optimal solution and uniform distribution of the Pareto front set (high coverage). In other words, a balance should be found between coverage and convergence. Based on the free lunch theorem (NFL) [68], can one heuristic meta algorithm solve all optimization problems? This reason, in addition to convergence and coverage, represents the main reasons concerning the non-superiority of some algorithms over others. The proposed approaches provide more diversification in the search space and highly boost exploitation by founding very good quality solutions and a very good distribution as well. Furthermore, the proposed MOHGS approach can solve more complex problems with less computational effort. Briefly, the obtained results by HGS and MOHGS can provide most convenient solutions of Pareto front set with good coverage and convergence along one, two, and three objectives when applied to solve OPF problems.

5. Conclusions

In this study, a new optimization algorithm was tested, namely, the hunger games search (HGS), inspired by the behavior of social animal cooperation that is proportional to their level of hunger. This algorithm is proposed for solving the single objective optimal power flow problems in power systems to achieve the economical, technical, and environmental benefits. The HGS algorithm was developed to obtain better solutions for multiple conflicting objective functions simultaneously, namely, the multi-objective hunger games search (MOHGS). Various objective functions were considered, which are fuel cost [$/h], active power losses [MW], emission [ton/h], voltage deviation [p.u.], and voltage stability index. The optimization method that was used to determine the non-dominated solutions in the Pareto front set was the Pareto concept. The strategies that were used to extract BCS and rerank the non-dominated solutions were fuzzy set theory and crowding distance. To demonstrate the efficiency and capability of the proposed HGS algorithm and developed MOHGS approach, two power systems were used (IEEE 30-bus and IEEE 57-bus) for various single and multiple objective functions. The simulation results, illustrating the performance of the proposed approaches, are characterized by good convergence speed, well-distribution, and high efficiency. The comparison results with other well-known optimization methods confirmed the quality and robustness of the proposed approaches. Finally, the MOHGS is a reliable and robust optimization method to solve a multi-objective optimal power flow problem.