A Many-Objective Marine Predators Algorithm for Solving Many-Objective Optimal Power Flow Problem

Abstract

Since the increases in electricity demand, environmental awareness, and power reliability requirements, solutions of single-objective optimal power flow (OPF) and multi-objective OPF (MOOPF) (two or three objectives) problems are inadequate for modern power system management and operation. Solutions to the many-objective OPF (more than three objectives) problems are necessary to meet modern power-system requirements, and an efficient optimization algorithm is needed to solve the problems. This paper presents a many-objective marine predators algorithm (MaMPA) for solving single-objective OPF (SOOPF), multi-objective OPF (MOOPF), and many-objective OPF (MaOPF) problems as this algorithm has been widely used to solve other different problems with many successes, except for MaOPF problems. The marine predators algorithm (MPA) itself cannot solve multi- or many-objective optimization problems, so the non-dominated sorting, crowding mechanism, and leader mechanism are applied to the MPA in this work. The considered objective functions include cost, emission, transmission loss, and voltage stability index (VSI), and the IEEE 30- and 118-bus systems are tested to evaluate the algorithm performance. The results of the SOOPF problem provided by MaMPA are found to be better than various algorithms in the literature where the provided cost of MaMPA is more than that of the compared algorithms for more than 1000 USD/h in the IEEE 118-bus system. The statistical results of MaMPA are investigated and express very high consistency with a very low standard deviation. The Pareto fronts and best-compromised solutions generated by MaMPA for MOOPF and MaOPF problems are compared with various algorithms based on the hypervolume indicator and show superiority over the compared algorithms, especially in the large system. The best-compromised solution of MaMPA for the MaOPF problem is found to be greater than the compared algorithms around 4.30 to 85.23% for the considered objectives in the IEEE 118-bus system.

1. Introduction

The optimal power flow (OPF) problem has played a major role in developing power system management and operation in the competitive power market [1,2]. The OPF problem is a large-scale, static, nonlinear optimization problem [3] aiming to optimize focused objective functions while maintaining several constraints. With the increase in fuel costs, mainly causing the higher generation costs, the fuel cost is frequently set as the objective function in thermal plants. Moreover, the development of many technologies in recent years causes a rise in power demand eventually resulting in higher generation costs and transmission line loss. So, transmission line loss becomes another important objective function to be minimized in modern power systems [4]. Moreover, the power generation in thermal plants emits emissions into the atmosphere causing an increase in pollution and global warming [5]. Environmental awareness has been increased, and emissions have been chosen as part of the objective function in the OPF problem [6,7,8]. In [6], a modified shuffle frog-leaping algorithm (MSFLA) was presented to solve the OPF problem considering economical and emission issues. The OPF problem was solved by hybrid particle-swarm optimization and a shuffle frog-leaping algorithm (PSO-SFLA) where the economic issues involving the prohibited zones, valve point effect and multi-fuel type of generation units, and emission problems were considered [7]. In [8], an improved particle-swarm optimization (IPSO) was introduced to solve the OPF problem considering cost, loss, voltage stability, and emission impacts as the objectives. In addition to the cost, loss, and environmental awareness, power blackouts often happen in many countries, such as North America and Europe [9], southern Sweden and eastern Denmark [10], and Italy [11,12], resulting in wide damages to power companies and people. One of the main causes of power blackouts is voltage instability consequently leading to voltage collapse generally occurring when the power demand significantly rises. Hence, voltage stability requires to be enhanced to improve power system reliability. One solution to enhance system voltage stability is to employ voltage stability indices (VSIs). VSIs are indicators applied to observe the proximity of power systems to voltage collapse. The values of VSIs are normally between 0 representing a no-load condition and 1 indicating a voltage collapse. System voltage stability can thus be enhanced if a VSI is used as the objective function to be minimized. From the aforementioned problems, four objectives including fuel cost, emission, transmission line loss, and VSI are considered as part of the objective functions in the OPF problem.

Solving the OPF problem is a challenging task that has attracted many researchers in the past decades. Various traditional optimization methods including quadratic programming [13], the interior point method [14], and nonlinear programming [15] have been introduced and applied to solve the OPF problem; however, with the deficient performance, these methods provided locally optimal solutions and consumed a large amount of effort and time. Metaheuristic algorithms are one of the methods which have been introduced to deal with these problems. In the beginning, most of the metaheuristic algorithms were proposed to solve single-objective optimization problems, including single-objective OPF (SOOPF) problems considering only one objective function at a time. Examples of these algorithms include evolutionary programming (EP) [16], ant colony optimization (ACO) [17], stochastic genetic algorithm (SGA) [18], grasshopper optimization algorithm (GOA) [19], and particle swarm optimization (PSO) [19]. With the requirement for power system improvements to satisfy electricity consumers in more than one aspect, the problem becomes multi-objective OPF (MOOPF) problems which consider two or three objective functions. Over the past few decades, to solve multi-objective problems, various methods consisting of indicator-based methods [20,21], decomposition-based methods [22,23,24], and Pareto-dominance-based methods [25,26] have been proposed. However, these algorithms consider all decision variables at a time resulting in performance deterioration when the number of decision variables increases. In the past few years, many metaheuristic algorithms have been introduced to overcome the problems and successfully solve MOOPF problems. Some of the algorithms are the slime mould algorithm (SMA) [27], fuzzy adaptive hybrid configuration oriented to a joint self-adaptive PSO and differential evolution algorithm (FAHSPSO-DE) [28], Harris hawks’ optimization (HHO) [29], hybrid firefly-Jaya algorithm [30], and modified pigeon-inspired optimization algorithm (MPIO) [31]. Recently, due to the continuous rise in electricity demand, environmental consciousness, and power security requirements, solving SOOPF and MOOPF problems are insufficient for modern power system management and operation. The problem then develops into many-objective OPF (MaOPF) problems in which more than three objective functions are simultaneously considered as part of the objective functions. MaOPF problems are more complicated than MOOPF problems, so a high-performance algorithm is needed to provide efficient solutions.

In many-objective optimization problems, the optimal tradeoffs between each conflict objective are called Pareto optimal solutions or Pareto fronts [32]. Various algorithms such as the improved competitive PSO (ICPSO) [33], weight-based ensemble machine learning algorithm (WBELA) [34], and kriging-assisted two-archive algorithm (KTA2) [35] have been introduced to solve many-objective optimization problems in different fields in the last few years. However, with the complication of the many-objective optimization problems including MaOPF problems, it is difficult to find high-quality Pareto optimal solutions providing great values in all objectives. So, only a few algorithms comprising the multi-objective evolutionary algorithm with many-stage dynamical resource allocation strategy (MOEA/D-MRA) [36], many-objective gradient-based optimizer (MaOGBO) [37], improved non-dominated sorting genetic algorithm III (I-NSGA-III) [38], and knee point-driven evolutionary algorithm (KnEA) [39] have been introduced to solve MaOPF problems.

Recently, several efficient metaheuristic algorithms have been proposed to successfully solve various optimization problems [40,41,42,43,44]. In [40], the African vultures’ optimization algorithm (AVOA) was proposed to solve benchmark functions and engineering problems. The Arithmetic optimization algorithm (AOA) [41] was introduced to solve welded beam design, tension/compression spring design, and some more mechanical problems. In [42], the artificial gorilla troops’ optimizer (GTO) was presented to solve benchmark functions and engineering problems. The artificial hummingbird algorithm (AHA) [43] was proposed to solve numerical test functions and challenging engineering design cases. In [44], the marine predators algorithm (MPA) was introduced to solve test functions, engineering benchmarks, and real-world engineering design problems. However, most of them have never been investigated in MaOPF problems. MPA is a well-proposed metaheuristic method inspired by predators in the ocean [44]. Many optimization problems in several areas, such as forecasting confirmed cases during the COVID-19 pandemic [45], feature selection problems [46], parameter estimation of photovoltaic models [47], and parameters’ identification of triple-diode photovoltaic models [48] have been efficiently solved by MPA. However, MPA has been rarely used to solve OPF problems [49,50]. Although the multi-objective optimization version of MPA has been proposed [51,52], MPA has never been used to solve MOOPF and MaOPF problems. So, this work proposes many-objective MPA (MaMPA) and evaluates the effectiveness of the MaMPA in solving SOOPF, MOOPF, and MaOPF problems. The MPA itself cannot solve multi- or many-objective optimization problems, so the non-dominated sorting, crowding mechanism, and leader mechanism are applied to MPA. Two test systems including IEEE 30-, and 118-bus systems are used in the simulation process. Four objectives, which are the following: fuel costs; emissions; transmission losses; and VSI are selected as the objective functions. The results from many other algorithms in the literature are used to compare the simulation results of the MaMPA, and the statistical results of MaMPA are investigated.

The main contributions of this work are listed below.

• The MaMPA is proposed to solve SOOPF, MOOPF, and MaOPF problems;
• The Pareto dominance concept is used to find Pareto fronts for MOOPF and MaOPF problems, the non-dominated sorting and crowding mechanism are used to organize the full repository, the leader mechanism is used to help calculate the equations of MPA, and the fuzzy decision method is applied to find the best-compromised solution for the MaOPF problems. The Pareto fronts are compared with those of PSO, PESA-II, SPEA2, and SSA based on the hypervolume indicator;
• The performance comparison of the proposed algorithm with other algorithms introduced in the literature is presented;
• The statistical results of the MaMPA are investigated.

The remainder of the paper is organized as follows. Section 2 introduces the MaOPF problem including objective functions and constraints. Section 3 presents the MPA formulations and implementation of MPA to solve the MaOPF problem. Section 4 provides the simulation results and comprehensive discussions. Finally, Section 5 concludes the work of the paper.

2. Many-Objective Optimal Power Flow (MaOPF) Problem

Since various issues including the economy, environment, and reliability are very important in modern power systems, the OPF problem turns into the MaOPF problem which is one of the many-objective optimization problems considering more than three objective functions. The many-objective optimization problem for minimization problem can be formulated as provided below:

$$\min f=\left\{f_1(x,u),f_2(x,u),\dots ,f_{Nobj}(x,u)\right\}$$

(1)

subject to:

$$g(x,u)=0$$

(2)

$$h(x,u)\le 0$$

(3)

where f is a vector of the objective functions to be minimized; g(x,u) and h(x,u) are the equality and inequality constraints, respectively; N_obj is the number of objective functions which is more than three for the many-objective problems; and x and u are vectors of state and control variables, respectively. For the OPF problem, the vector of state variables, x, includes slack-bus active power, load-bus voltages, generator-bus reactive powers, and complex power flows, and the vector of control variables comprises generator-bus active powers except for the slack bus, generator-bus voltages, transformer-tap ratios, and reactive powers of shunt VAR compensators.

Since many objective functions are considered at the same time where each objective conflicts with each other, the number of the obtained solutions which are a tradeoff between each objective is uncountable. These are the Pareto fronts which will be depicted later. In this section, the objective functions and constraints considered in this work are presented in the following subsections.

2.1. Objective Functions

This study considers four objective functions including fuel cost, emission, transmission loss, and VSI to be minimized to improve the power system operation performance.

2.1.1. Fuel Cost

To reduce the system operation cost, the fuel cost function is considered to be minimized which is formulated as given below:

$$f_1(x,u)={\displaystyle \sum _{i=1}^{N_g}\left(a_i{P}_{gi}^2+b_i{P}_{gi}+c_i\right)}$$

(4)

where f₁ is the fuel cost function; N_g is the number of generators; P_gi is the active power of the ith generating unit; and a_i, b_i, and c_i are the fuel cost coefficients.

2.1.2. Emissions

The amount of nitrogen oxides (NO_x) and sulfur oxides (SO_x) released into the atmosphere from the power generations can be expressed by the emission function which is mathematically calculated as follows:

$$f_2(x,u)={\displaystyle \sum _{i=1}^{N_g}\left({\gamma }_i{P}_{gi}^2+{\beta }_i{P}_{gi}+{\alpha }_i+{\xi }_i\exp ({\lambda }_i{P}_{gi})\right)}$$

(5)

where f₂ is the emission function; and γ_i, β_i, α_i, ζ_i, and λ_i are the emission coefficients.

2.1.3. Transmission Losses

Transmission loss reduction can reduce the amount of power generation resulting in a decrease in system costs. The transmission loss can be computed as expressed below:

$$f_3(x,u)={\displaystyle \sum _{k=1}^{N_{br}}{g}_k\left(V_i^2+V_j^2-2V_i{V}_j\cos ({\theta }_{ij})\right)}$$

(6)

where f₃ is the transmission loss function; N_br is the number of branches; g_k is the conductance of branch k; V_i and V_j are the voltage magnitude of buses i and j; and θ_ij is the voltage phase angle difference between two buses.

2.1.4. Voltage Stability Index

To enhance the system reliability, the voltage stability index (VSI) is improved to verify system stability and prevent the system from voltage collapse. The L-index is one of the voltage stability indices used to observe the voltage collapse proximity. The value of the L-index is between 0, indicating no-load condition, and 1, expressing voltage collapse. So, if the L-index value is minimized, the voltage stability of the system is enhanced. The L-index can be computed as the following equation:

$$L_j=\left|1-{\displaystyle \sum _{i=1}^{N_g}{F}_{ji}\frac{V_i}{V_j}}\right| , j\in {\alpha }_L$$

(7)

where α_L is the set of load buses; and the values of F_ji can be received from the matrix F_LG as follows:

$$F_{LG}=-{[Y_{LL}]}^{-1}[Y_{LG}]$$

(8)

To find the admittance matrix, consider a system consisting of N buses and N_g generators. The network equation between voltage and current is found as presented below:

$$\left[\begin{array}{c}{I}_L\\ I_G\end{array}\right]=\left[\begin{array}{cc}{Y}_{LL}& Y_{LG}\\ Y_{GL}& Y_{GG}\end{array}\right]\left[\begin{array}{c}{V}_L\\ V_G\end{array}\right]$$

(9)

where I_L, I_G, and V_L, V_G are the currents and voltages at the load buses and generator buses.

Thus, the maximum L-index value from all load buses is selected as the objective function to be minimized as the following equation:

$$f_4(x,u)=\max \left(L_j\right)$$

(10)

where f₄ is the maximum L-index value of all load buses; and L_j can be computed by (7).

2.2. Constraints

The constraints are classified into the equality and inequality constraints described as follows:

2.2.1. Equality Constraints

In this work, the active and reactive power balances presented in the equations below are the equality constraints:

$$P_{gi}-P_{di}=V_i{\displaystyle \sum _{j=1}^{N_b}{V}_j(G_{ij}\cos ({\theta }_{ij})+B_{ij}\sin ({\theta }_{ij}))}$$

(11)

$$Q_{gi}-Q_{di}=V_i{\displaystyle \sum _{j=1}^{N_b}{V}_j(G_{ij}\cos ({\theta }_{ij})-B_{ij}\sin ({\theta }_{ij}))}$$

(12)

where Q_gi is the reactive power of the ith generator; P_di and Q_di are active and reactive power demands at the ith bus; N_b is the number of buses; and G_ij and B_ij are the transfer conductance and susceptance, respectively, between buses i and j

2.2.2. Inequality Constraints

The inequality constraints are defined to ensure system security. The equations of the inequality constraints are provided as shown below:

$$P_{gi\min }\le P_{gi}\le P_{gi\max } ; i=1,2,\dots ,N_g$$

(13)

$$Q_{gi\min }\le Q_{gi}\le Q_{gi\max } ; i=1,2,\dots ,N_g$$

(14)

$$V_{gi\min }\le V_{gi}\le V_{gi\max } ; i=1,2,\dots ,N_g$$

(15)

$$V_{Li\min }\le V_{Li}\le V_{Li\max } ; i=1,2,\dots ,N_L$$

(16)

$$Q_{ci\min }\le Q_{ci}\le Q_{ci\max } ; i=1,2,\dots ,N_c$$

(17)

$$T_{i\min }\le T_i\le T_{i\max } ; i=1,2,\dots ,N_t$$

(18)

$$\left|S_{bri}\right|\le S_{bri\max } ; i=1,2,\dots ,N_{br}$$

(19)

where the subscripts min and max express the minimum and maximum values; V_g is the voltage of generators; Q_c is the shunt compensation capacitor; T is the transformer tap ratio; S_br is the branch complex power flow; and N_c, N_t, and N_L are the number of shunt compensation capacitors, transformer tap ratios, and load buses, respectively.

2.2.3. Constraint Handling

To satisfy the constraints of the state variables, the penalty function method is adopted to penalize the objective value when a state variable breaks the constraints. So, the penalized objective function is mathematically modeled as the provided equation:

$$\begin{array}{c}F(x,u)=f(x,u)+K_p{(P_{gslack}-P_{gslack}^{\lim })}^2+K_Q{\displaystyle \sum _{i=1}^{N_g}{(Q_{gi}-Q_{gi}^{\lim })}^2}\\ +K_V{\displaystyle \sum _{i=1}^{N_L}{(V_{Li}-V_{Li}^{\lim })}^2}+K_S{\displaystyle \sum _{i=1}^{N_{br}}{(S_{bri}-S_{bri}^{\lim })}^2}\end{array}$$

(20)

where f(x,u) is an objective function for one objective or a vector of the objective functions for more than one objective as described in Equation (1); F(x,u) is the penalized objective function for one objective or a penalized vector of the objective functions for more than one objective; K_P, K_Q, K_V, and K_S are penalty factors; and the superscript lim is the variable limits imposed as expressed below:

$$x^{\lim }=\left\{\begin{array}{c}{x}^{\max }\\ x\\ x^{\min }\end{array} \begin{array}{c}if\\ if\\ if\end{array}\right. \begin{array}{c}x>x^{\max }\\ x^{\min }\le x\le x^{\max }\\ x<x^{\min }\end{array}$$

(21)

where x^max and x^min are the maximum and minimum limits of the state variables.

3. Many-Objective Marine Predators Algorithm

The marine predators algorithm (MPA) is inspired by the foraging behavior of predators in the ocean, as well as the interaction between predator and prey [44]. The predators adopt the foraging tactics called Brownian and Lévy motions to improve their location and movement in searching for the prey. So, this section presents the Brownian and Lévy motions and their adaptation to the formulation of MPA. Then, the implementation of MPA for solving the MaOPF problem is introduced.

3.1. Brownian Motion

Brownian motion is a stochastic process whose step size is determined by a probability function specified by a normal distribution with zero mean ($\mu =0$) and unit variance (${\sigma }^2=1$). The motion’s Probably Density Function (PDF) at point x can be expressed below [53]:

$$f_B(x;\mu ,\sigma )=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{{(x-\mu )}^2}{2{\sigma }^2}\right)=\frac{1}{\sqrt{2\pi }}\exp \left(-\frac{x^2}{2}\right)$$

(22)

where x is a considered point.

3.2. Lévy Motion

Lévy motion or Lévy flight is one type of random walk. The random numbers generated based on the Lévy distribution can be found in the following equation [54]:

$$Levy(\alpha )=0.05\times \frac{x}{{\left|y\right|}^{1/\alpha }}$$

(23)

where Levy($\alpha$) is a random number based on Lévy distribution for an arbitrary value of index distribution ($\alpha$) ranging in 0.3 and 1.99; and x and y are both normal distribution variables that have standard deviations ${\alpha }_x$ and ${\alpha }_y$ expressed as follows:

$$x=Normal\left(0,{\alpha }_x^2\right)$$

(24)

$$y=Normal\left(0,{\alpha }_y^2\right)$$

(25)

where ${\alpha }_x$ can be computed as provided below:

$${\sigma }_x={\left[\frac{\Gamma (1+\alpha )\sin \left(\frac{\pi \alpha }{2}\right)}{\Gamma \left(\frac{(1+\alpha )}{2}\right)\alpha \cdot 2^{\frac{\left(\alpha -1\right)}{2}}}\right]}^{\frac{1}{\alpha }} ,{\sigma }_y=1 \mathrm{and} \alpha =1.5$$

(26)

where $\Gamma$ is Gamma function; and $\Gamma (x)=(x-1)!$.

3.3. Marine Predators Algorithm Formulation

The MPA adopts both Brownian and Lévy motions to balance the exploration and exploitation phases, so an efficient feasible solution can be obtained.

Initially, the solution is randomly uniformly generated within the limits of the provided equation:

$$X_0=X_{\min }+rand(X_{\max }-X_{\min })$$

(27)

where X₀ is the initially generated position; X_min and X_max are the minimum and maximum limits of the variables; and rand is a randomly uniformly generated vector between 0 and 1.

The objective function is then applied to each dispersed solution, and the greatest solution with the best objective value is adopted as the top predator in the optimization process. According to the survival of the fittest theorem, the top predator is used to build a matrix called Elite which is expressed as follows:

$$Elite={\left[\begin{array}{cccc}{X}_{1,1}^T& X_{1,2}^T& \cdots & X_{1,d}^T\\ X_{2,1}^T& X_{2,2}^T& \cdots & X_{2,d}^T\\ ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮\\ X_{n,1}^T& X_{n,2}^T& \cdots & X_{n,d}^T\end{array}\right]}_{n\times d}$$

(28)

where X^T is the top predator vector duplicated n times to build the Elite matrix; n is the number of searching populations; and d is the number of dimensions of the considered variables. In the searching process, predator and prey are both in the searching population. This is because while the predator is hunting for its prey, the prey is also hunting for its food. If the top predator is replaced by a superior predator, the Elite will be updated at the end of each iteration.

In each iteration, the positions of the predators are updated according to the Prey matrix which has the same dimension as Elite. The Prey is firstly initialized, and the best one is considered as the predator building up the Elite matrix. The Prey matrix can be written as the following equation:

$$Prey={\left[\begin{array}{cccc}{X}_{1,1}& X_{1,2}& \cdots & X_{1,d}\\ X_{2,1}& X_{2,2}& \cdots & X_{2,d}\\ ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮\\ X_{n,1}& X_{n,2}& \cdots & X_{n,d}\end{array}\right]}_{n\times d}$$

(29)

where X represents each prey in each dimension.

3.3.1. MPA Scenarios

In the optimization process, the MPA consists of three primary phases, each of which considers a particular velocity ratio while simulating the whole life cycle of a predator and prey. The particular iteration period is defined for each phase, and the movement of a predator and prey can be mathematically formulated as described below.

Phase A: In the first phase, a predator is moving faster than the prey or in a high ratio of velocity in the initial iterations, normally called the exploration phase. The prey begins to explore the search space by employing the Brownian approach to locate potential areas that may contain a feasible solution. So, the prey’s position is mathematically updated as the given equations:

If Iteration < $\frac{1}{3}$ maximum_iteration:

$$\begin{array}{c}stepsiz{e}_i={\overrightarrow{R}}_B\otimes \left(Elit{e}_i-{\overrightarrow{R}}_B\otimes Pre{y}_i\right) ,i=1,\dots ,n\\ Pre{y}_i=Pre{y}_i+P\cdot \overrightarrow{R}\otimes stepsiz{e}_i\end{array}$$

(30)

where ${\overrightarrow{R}}_B$ is a vector of numbers randomly generated based on the normal distribution presenting the Brownian motion; $\otimes$ is the entry-wise multiplication; P is a constant number set to 0.5; and $\overrightarrow{R}$ is a vector of numbers randomly uniformly generated between 0 and 1;

Phase B: This phase occurs in the middle of the process when predator and prey have the same movement or are in unit velocity ratio because both are searching for their prey. In this phase, the optimization process is changing from exploration to exploitation. So, half of the population (prey) is assigned for exploitation and the other half (predator) is imposed to explore the search space. To mathematically formulate this phase, the prey adopts the Lévy motion for exploration, and the predator applies the Brownian motion for exploitation as in the following equations:

If $\frac{1}{3}$ maximum_iteration < Iteration < $\frac{2}{3}$ maximum_iteration

For the first half of the population (prey):

$$\begin{array}{c}stepsiz{e}_i={\overrightarrow{R}}_L\otimes \left(Elit{e}_i-{\overrightarrow{R}}_L\otimes Pre{y}_i\right) ,i=1,\dots ,n/2\\ Pre{y}_i=Pre{y}_i+P\cdot \overrightarrow{R}\otimes stepsiz{e}_i\end{array}$$

(31)

where ${\overrightarrow{R}}_L$ in (31) is a vector of randomly generated numbers based on Lévy motion. The movement of prey is simulated by the multiplication of ${\overrightarrow{R}}_L$ and Prey. So, the step size added to the position of prey mimics the movement of prey in Lévy motion for exploitation.

For the other half of the population (predator):

$$\begin{array}{c}stepsiz{e}_i={\overrightarrow{R}}_B\otimes \left({\overrightarrow{R}}_B\otimes Elit{e}_i-Pre{y}_i\right) ,i=(n/2)+1,\dots ,n\\ Pre{y}_i=Elit{e}_i+P\cdot CF\otimes stepsiz{e}_i\end{array}$$

(32)

where the movement of the predator is simulated by the multiplication of ${\overrightarrow{R}}_B$ and Elite in Brownian motion; and the position of prey is updated by considering the predator movement in Brownian motion. CF is an adaptive parameter adopted to control the step size, and it can be calculated as follows:

$$CF={\left(1-\frac{Iter}{Max\_iter}\right)}^{\left(2\frac{Iter}{Max\_iter}\right)}$$

(33)

where Iter is the current iteration; and Max_iter is the maximum iteration number;

Phase C: The last phase happens when the predator is going faster than the prey in a low-velocity ratio. This situation occurs towards the end of the optimization process where the exploitation is mostly operated. The best motion for the predator in a low-velocity ratio (v = 0.1) is Lévy which is formulated as given below:

If Iteration > $\frac{2}{3}$ maximum_iteration

$$\begin{array}{c}stepsiz{e}_i={\overrightarrow{R}}_L\otimes \left({\overrightarrow{R}}_L\otimes Elit{e}_i-Pre{y}_i\right) ,i=1,\dots ,n\\ Pre{y}_i=Elit{e}_i+P\cdot CF\otimes stepsiz{e}_i\end{array}$$

(34)

by multiplying ${\overrightarrow{R}}_L$ and Elite, the motion of the predator is simulated in Lévy motion. The step size is then used to simulate the predator motion to help update the prey position by adding to the Elite position as in (34).

3.3.2. Effect of Eddy Formation and FADs

Environmental factors such as eddy formation, or the impact of Fish Aggregating Devices (FADs) are other factors that cause behavioral changes in marine predators. In [55], it was found that sharks spend more than 80% of their time around FADs, and the remaining 20% of their time will be spent making larger jumps in various dimensions, most likely in search of a different prey distribution. The local optima in the optimization process can be represented by the FADs when trapping in this area, and the larger jumps in various dimensions help avoid trapping in the local optima. So, the FADs’ effect can be formulated as follows in the provided equation:

$$Pre{y}_i=\left\{\begin{array}{c}Pre{y}_i+CF\left[{\overrightarrow{X}}_{\min }+\overrightarrow{R}\otimes \left({\overrightarrow{X}}_{\max }-{\overrightarrow{X}}_{\min }\right)\right]\otimes \overrightarrow{U} \mathrm{if} r\le FADs\\ Pre{y}_i+\left[FADs(1-r)+r\right]\left(Pre{y}_{r1}-Pre{y}_{r2}\right) \mathrm{if} r>FADs\end{array}\right.$$

(35)

where FADs = 0.2 represent the impact of FADs on the optimization process; U is a binary vector with arrays consisting of 0 and 1; where the binary vector is randomly generated between 0 and 1, and the array is changed to 0 if the array is less than FADs and 1 if the array is more than FADs; r is the uniformly randomly generated number between 0 and 1; X_min and X_max are the minimum and maximum limits; and r1 and r2 represent random indexes of the prey matrix.

3.3.3. Marine Memory Saving

One of the prominent points of MPA is that marine predators can well remember where they were successfully foraging which is simulated by marine memory saving. After updating the Prey and considering the FADs effect, this matrix is computed for fitness to update the Elite. Each solution in the current iteration is compared to its counterpart in the previous iteration, and if the current one is better, it replaces the previous one. With each iteration, this method increases the solution quality and represents predators returning to prey-rich areas after successful foraging.

3.4. Many-Objective Marine Predators Algorithm (MaMPA)

The MPA itself cannot solve multi- or many-objective optimization problems, but it can only solve the single-objective optimization problem. To solve the MaOPF problem, where four unrelated objective functions are considered in this work and the provided solutions are a trade-off between each objective, the Pareto dominance method is adopted. This method aims to find the trade-off which is a non-dominated solution and keep it in the repository traditionally called Pareto optimal fronts. The Pareto dominance method can be formulated as follows [8]:

$$\begin{array}{l}\forall i=\left\{1,2,\dots ,M\right\}, F_i(X_1)\le F_i(X_2)\\ \exists j=\left\{1,2,\dots ,M\right\}, F_j(X_1)<F_j(X_2)\end{array}$$

(36)

where the vector X₁ dominates X₂ when both conditions are satisfied; and M is the number of considered objective functions.

Since the number of non-dominated solutions is uncountable and the repository has a limited size, the non-dominated sorting and crowding mechanisms are also used in this work to organize the full repository to help the algorithm find the solutions with diversity in the limited size of the repository [56]. In addition, in the optimization process of the traditional MPA, the top predator (referred to as the best solution) is found in each iteration by simply comparing each prey from each iteration with the top predator from the previous iteration, and the top predator is also used to build the Elite matrix, as in (28). So, in this work, to find the top predator in the MOOPF and MaOPF problems where the number of the non-dominated solutions is uncountable, the leader mechanism is adopted using the roulette wheel and adaptive grid operators to provide the top predator in each iteration to be able to build the Elite matrix in (28).

With more than three dimensions of objectives in the MaOPF problems, the graph of Pareto fronts faces difficulty in being appropriately generated. Instead, a best-compromised solution from the optimal tradeoffs between all different objectives can be provided by using the fuzzy decision method [36] to help power the system operators to select the best solution from the various provided solutions. To apply the fuzzy decision method, fuzzy membership functions on each dimension of all non-dominated solutions are firstly computed as shown below:

$${\mu }_i^j=\left\{\begin{array}{c}1 , f_i^j\le f_{\min }^j\\ \frac{f_{\max }^j-f_i^j}{f_{\max }^j-f_{\min }^j} , f_{\min }^j\le f_i^j\le f_{\max }^j\\ 0 , f_i^j\ge f_{\max }^j\end{array}\right.$$

(37)

where $\mu$ is the fuzzy membership function, 1 ≤ i ≤ N_S, 1 ≤ j ≤ N_obj; N_S is the number of non-dominated solutions; and $f_{\max }^j$ and $f_{\min }^j$ are the maximum and minimum values of the objective j, respectively.

The summation of the fuzzy membership functions on all dimensions is then calculated and normalized by the following equation:

$${\mu }_i^j=\frac{{\displaystyle \sum _{i=1}^m{\mu }_i^j}}{{\displaystyle \sum _{j=1}^{NF}{\displaystyle \sum _{i=1}^m{\mu }_i^j}}}$$

(38)

where the non-dominated solution with the greatest summation value of the normalized function is referred to as the best-compromised solution.

The implementation of the proposed MaMPA for solving the MaOPF problem can be described in the following steps:

Step 1. Initialize the system data including costs and emission coefficients, active powers of generators, voltages of generator buses, transformer tap ratios, shunt compensation capacitors, maximum and minimum limits of the constraints, maximum iteration number, size of Pareto archive, and searching populations (Prey);

Step 2. Convert the constrained problem to an unconstrained one by (20);

Step 3. Calculate the fitness values of all initialized preys;

Step 4. Find the non-dominated prey by using the Pareto dominance concept as in (36) and keep them in the initial archive;

Step 5. Adopt a leader mechanism to find the top predator to construct the Elite matrix as shown in (28) and accomplish memory saving as described in Section 3.3.3;

Step 6. If the current iteration is less than one-third of the maximum iteration, update prey by (30). Then go to Step 10;

Step 7. If the current iteration is more than one-third of the maximum iteration and less than two-third of the maximum iteration, then go to Step 9;

Step 8. If the current iteration is more than two-thirds of the maximum iteration, update prey by (34). Then go to Step 10;

Step 9. If the first half of the populations is considered, update prey by (31), and if the other half of the populations is considered, update prey by (32);

Step 10. Move the components to be within their lower and upper limits;

Step 11. Evaluate the fitness values of all prey to detect the top predator to update the Elite. Then, accomplish memory saving;

Step 12. Apply the FADs effect on the population by using (35);

Step 13. Apply the Pareto dominance concept to find the non-dominated solutions by using (36) and keep them in the archive. If the archive is full, non-dominated sorting and crowding mechanism are adopted to handle the full archive;

Step 14. If the maximum iteration is reached, go to step 15; otherwise, go to step 4;

Step 15. Determine the best-compromised solution by (38). Then. the process is finished.

The flowchart of the MaMPA for solving MaOPF problems is presented in Figure 1.

4. Simulation Results and Discussions

In this section, MaMPA was used to solve SOOPF, MOOPF, and MaOPF problems. Two test systems including the IEEE 30- and 118-bus systems were used, and four objectives which are cost, emission, transmission loss, and VSI were chosen to be the objective functions in each system as in the 20 case studies shown in

Table 1. Case study presented in this work.

Case Study	Objectives	System
1	Cost	IEEE 30-bus
2	Emission	IEEE 30-bus
3	Loss	IEEE 30-bus
4	VSI	IEEE 30-bus
5	Cost	IEEE 118-bus
6	Emission	IEEE 118-bus
7	Loss	IEEE 118-bus
8	VSI	IEEE 118-bus
9	Cost, Emission	IEEE 30-bus
10	Cost, Loss	IEEE 30-bus
11	Cost, VSI	IEEE 30-bus
12	Emission, Loss	IEEE 30-bus
13	Cost, Emission, Loss	IEEE 30-bus
14	Cost, Emission, VSI	IEEE 30-bus
15	Cost, Emission	IEEE 118-bus
16	Cost, Loss	IEEE 118-bus
17	Cost, Emission, Loss	IEEE 118-bus
18	Cost, Emission, VSI	IEEE 118-bus
19	Cost, Emission, Loss, VSI	IEEE 30-bus
20	Cost, Emission, Loss, VSI	IEEE 118-bus

. The IEEE 30-bus system consists of 6 generators, 4 transformers, and 41 transmission lines, and the system active and reactive power demands were constantly 283.4 MW and 126.6 MVAR, respectively. The IEEE 118-bus system has 54 generators, 9 transformers, and 186 transmission lines, and the active and reactive power demands were considered fixed at 4242 MW and 1439 MVAR, respectively. The detailed data and system single-line diagrams of the IEEE 30- and 118-bus systems can be found in [57,58], respectively, and the emission coefficients for the IEEE 118-bus system are provided in [36]. The number of population and maximum iteration were both 100, for the IEEE 30-bus system. For the IEEE 118-bus system, the population and maximum iteration numbers were 100 and 500, respectively.

4.1. Single-Objective OPF Problem

The SOOPF problem considers only one objective to be optimized. Each objective including cost, emission, transmission loss, and VSI was individually selected to be the objective function. The results of the SOOPF problem solved by MaMPA for each considered system are presented in the following subsections.

4.1.1. IEEE 30-Bus System

In the IEEE 30-bus system, the results including the system state variables and objective values of solving SOOPF by MaMPA for cases 1–4 are provided in

Table A1. Optimal solutions provided by MaMPA in the IEEE 30-bus systems for cases 1–4.

	Case 1	Case 2	Case 3	Case 4
Pg1 (MW)	176.2376	64.1674	51.6974	79.6793
Pg2 (MW)	48.8430	67.6696	80.0000	79.9990
Pg5 (MW)	21.5184	50.0000	50.0000	49.9999
Pg8 (MW)	22.1248	35.0000	35.0000	34.9999
Pg11 (MW)	12.2011	30.0000	30.0000	29.9999
Pg13 (MW)	12.0000	40.0000	40.0000	13.1160
Vg1	1.0500	1.0500	1.0500	1.0500
Vg2	1.0381	1.0459	1.0477	1.0457
Vg5	1.0110	1.0275	1.0292	1.0467
Vg8	1.0194	1.0353	1.0366	1.0597
Vg11	1.1000	1.1000	1.1000	1.1000
Vg13	1.1000	1.0852	1.0850	1.0950
T6-9	0.9973	1.0136	1.0153	1.0378
T6-10	0.9000	0.9000	0.9000	0.9000
T4-12	1.0157	1.0097	1.0105	1.0336
T27-28	0.9403	0.9519	0.9529	0.9628
Qc10 (MVAR)	21.3750	14.6059	25.8997	29.9481
Qc24 (MVAR)	11.8515	30.0000	24.1261	29.9995
Cost (USD/h)	802.5449	945.0490	968.1335	921.5254
Emission (ton/h)	0.363619	0.204886	0.207294	0.224142
Loss (MW)	9.5249	3.4370	3.2974	4.3940
VSI	0.1388	0.1385	0.1384	0.1362

in Appendix A. The comparison results of MaMPA including real power generations, each objective value, and time with those of various methods for each considered objective are presented in

Table 2. Comparison results of case 1.

Algorithms	Pg1 (MW)	Pg2 (MW)	Pg5 (MW)	Pg8 (MW)	Pg11 (MW)	Pg13 (MW)	Cost (USD/h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
EP [16]	173.8480	49.9980	21.3860	22.6300	12.9280	12.0000	802.6200	0.357217	9.3900	-	51.4
EP-OPF [59]	175.0297	48.9522	21.4200	22.7020	12.9040	12.1035	803.5710	0.360125	9.7114	-	-
ACO [17]	181.9450	47.0010	21.4596	21.4460	13.2070	12.0134	802.5780	0.382000	9.8520	-	-
SGA [18]	179.3670	44.2400	24.6100	19.9000	10.7100	14.0900	803.6990	0.371129	9.5177	-	-
GOA [19]	-	48.0194	20.9145	20.2342	15.7260	13.5828	809.7410	-	10.0900	-	-
HHO [19]	-	47.4072	17.3766	17.8064	13.1780	17.1314	804.1407	-	7.9700	-	-
MVO [19]	-	51.1890	21.3110	21.1730	22.6990	16.5870	810.9011	-	7.6800	-	-
PSO [19]	-	31.9013	15.0000	10.0000	29.9591	12.0000	828.1315	-	8.3502	-	-
ABC [60]	-	-	-	-	-	-	800.6802	-	-	-	25.1
BBO [60]	-	-	-	-	-	-	800.9527	-	-	-	28.9
PSO [60]	-	-	-	-	-	-	800.7016	-	-	-	27.0
TLBO [60]	-	-	-	-	-	-	800.6735	-	-	-	26.6
AGTLBO [60]	177.1160	48.7445	21.3225	21.2933	11.9442	12.0017	800.4811	0.366200	9.0222	-	26.7
MaMPA	176.2376	48.8430	21.5184	22.1248	12.2011	12.0000	802.5449	0.363619	9.5249	0.1388	25.4

,

Table 3. Comparison results of case 2.

Algorithms	Pg1 (MW)	Pg2 (MW)	Pg5 (MW)	Pg8 (MW)	Pg11 (MW)	Pg13 (MW)	Cost (USD /h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
SFLA [6]	64.4840	71.3807	49.8573	35.0000	30.0000	39.9729	960.1911	0.206300	7.2949	-	-
MSFLA [6]	65.7798	68.2688	50.0000	34.9999	29.9982	39.9970	951.5106	0.205600	5.6437	-	-
GA [6]	78.2885	68.1602	46.7848	33.4909	30.0000	36.3713	936.6152	0.211700	9.6957	-	-
HMPSO-SFLA [7]	64.8148	68.0692	50.0000	34.9999	30.0000	40.0000	948.3052	0.205200	4.4839	-	-
IPSO [8]	67.0400	68.1400	50.0000	35.0000	30.0000	40.0000	954.2480	0.205800	5.3620	-	-
DSA [61]	64.0725	67.5711	50.0000	35.0000	30.0000	40.0000	944.40086	0.205826	3.2437	-	-
TLBO [62]	63.5221	68.7345	49.9931	34.9894	29.9824	39.9801	947.4392	0.205030	3.8016	-	-
MTLBO [62]	64.2924	67.6250	50.0000	35.0000	30.0000	40.0000	945.1965	0.204930	3.5174	-	-
GBICA [63]	64.3125	67.4938	50.0000	35.0000	29.9924	40.0000	944.6516	0.204900	3.3987	-	-
ACO [17]	64.3720	72.1604	49.5438	32.9099	28.6113	39.7390	945.5870	0.221000	3.9368	-	-
GSO [29]	60.9300	71.1200	50.0000	34.9300	30.0000	40.0000	951.1300	0.296000	3.5710	-	57.0
HHO [29]	60.9900	70.9800	50.0000	35.0000	30.0000	40.0000	950.9800	0.285000	3.5700	-	56.0
MF [29]	61.0200	70.9500	50.0000	35.0000	30.0000	40.0000	950.9300	0.295000	3.5720	-	61.0
SSA [29]	61.0200	70.9500	50.0000	35.0000	30.0000	40.0000	950.9300	0.295000	3.6000	-	62.0
WOA [29]	61.0800	70.8900	50.0000	35.0000	30.0000	40.0000	950.8200	0.295000	3.5710	-	62.0
PSO [60]	-	-	-	-	-	-	-	0.204840	-	-	27.5
BBO [60]	-	-	-	-	-	-	-	0.204900	-	-	29.1
ABC [60]	-	-	-	-	-	-	-	0.204840	-	-	26.2
TLBO [60]	-	-	-	-	-	-	-	0.204840	-	-	27.0
AGTLBO [60]	64.0640	67.5593	50.000	35.0000	30.0000	40.0000	944.3385	0.204800	3.2233	-	27.0
MaMPA	64.1674	67.6696	50.0000	35.0000	30.0000	40.0000	954.0490	0.204886	3.4370	0.1385	25.8

,

Table 4. Comparison results of case 3.

Algorithms	Pg1 (MW)	Pg2 (MW)	Pg5 (MW)	Pg8 (MW)	Pg11 (MW)	Pg13 (MW)	Cost (USD/h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
GWO [64]	51.8100	80.0000	50.0000	35.0000	30.0000	40.0000	968.3800	0.207310	3.4100	-	16.1
DE [64]	51.8200	79.9900	49.9900	35.0000	29.9800	40.0000	968.2300	0.207311	3.3800	-	16.7
MOHS [65]	66.2759	79.6413	46.8835	34.8880	29.1213	30.0558	928.5099	0.212890	3.5165	-	-
GOA [19]	-	68.6191	50.0000	27.5100	17.0215	24.6921	878.8137	-	5.8100	-	-
HHO [19]	-	43.1710	50.0000	29.9360	30.0000	28.8280	915.0934	-	4.5600	-	-
MVO [19]	-	25.8570	25.7890	21.2720	28.1110	12.9630	817.1171	-	9.7700	-	-
PSO [19]	-	80.0000	50.0000	10.0000	10.0000	40.0000	931.2200	-	10.3822	-	-
GSO [29]	52.1000	80.0000	50.0000	34.8000	30.0000	40.0000	968.2900	0.297000	3.5100	-	65.0
MF [29]	51.9000	80.0000	50.0000	35.0000	30.0000	40.0000	968.5600	0.296000	3.5000	-	62.0
SSA [29]	51.9000	80.0000	50.0000	35.0000	30.0000	40.0000	968.5600	0.296000	3.5000	-	61.0
WOA [29]	52.0300	79.9600	49.9700	34.9800	29.9800	39.9800	968.2100	0.296000	3.5000	-	61.0
PSO [60]	-	-	-	-	-	-	-	-	3.1709	-	27.7
BBO [60]	-	-	-	-	-	-	-	-	3.1725	-	29.5
ABC [60]	-	-	-	-	-	-	-	-	3.1314	-	26.8
TLBO [60]	-	-	-	-	-	-	-	-	3.1104	-	27.3
AGTLBO [60]	51.4932	79.9983	49.9997	34.9997	29.9999	39.9998	967.6336	0.207300	3.0906	-	27.5
MaMPA	51.6974	80.0000	50.0000	35.0000	30.0000	40.0000	968.1335	0.207294	3.2974	0.1384	26.6

and

Table 5. Comparison results of case 4.

Algorithms	Pg1 (MW)	Pg2 (MW)	Pg5 (MW)	Pg8 (MW)	Pg11 (MW)	Pg13 (MW)	Cost (USD/h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
SQP [66]	-	-	-	-	-	-	-	-	5.9930	0.1570	-
RGA [66]	-	-	-	-	-	-	-	-	5.0912	0.1386	25.7
CMAES [66]	-	-	-	-	-	-	-	-	5.1290	0.1382	27.1
MOPSO [66]	-	-	-	-	-	-	-	-	5.1090	0.1382	-
NSGA-II [66]	-	-	-	-	-	-	-	-	5.1280	0.1383	-
MNSGA-II [66]	-	-	-	-	-	-	-	-	5.1020	0.1382	-
ACDE [67]	-	-	-	-	-	-	-	-	-	0.1364	82.0
GWO [67]	-	-	-	-	-	-	-	-	-	0.1382	-
HHO [67]	-	-	-	-	-	-	-	-	-	0.1384	-
SKH [68]	-	-	-	-	-	-	-	-	-	0.1366	19.0
ECHT-DE [69]	-	-	-	-	-	-	-	-	-	0.1363	138.2
ARCBBO [70]	-	-	-	-	-	-	-	-	-	0.1369	-
MaMPA	79.6793	79.9990	49.9999	34.9999	29.9999	13.1160	921.5254	0.224142	4.3940	0.1362	27.1

where the objective value of the MaMPA when that objective was considered as the objective function are bold. The convergence curves of the MaMPA for cases 1–4 are presented in Figure 2.

It is observed from Table 2, Table 3, Table 4 and Table 5 that MaMPA could converge on the feasible solutions for all considered objective functions, cases 1–4, with slightly different objective values to the several algorithms in the literature. It also achieved better objective values than some algorithms such as GOA and MVO for cost, MF, SSA, and WOA for emission, MVO and PSO for loss, and SQP for VSI. The computational times of MaMPA are slightly faster or slower than various algorithms by a few seconds and much faster than some algorithms such as EP, GSO, HHO, MF, SSA, WOA, ACDE, and ECHT-DE. From Figure 2, it can be seen that MaMPA converged on the feasible solutions quickly within one-fourth of the maximum iteration for all considered objective functions. So, this can verify that MaMPA has a high quality for solving the SOOPF problem in the IEEE 30-bus system.

4.1.2. IEEE 118-Bus System

The performance of the MaMPA in solving SOOPF problems is verified by testing in the large system which is the IEEE 118-bus system. The optimal system state variables together with the optimal solutions when each objective is chosen to be the objective function can be seen in

Table A2. Optimal solutions provided by MaMPA when considering cost as the objective function in the IEEE 118-bus system (case 5).

Variables	Value	Variables	Value	Variables	Value	Variables	Value	Variables	Value
Pg1	4.6714	Pg65	340.2423	Vg1	1.0024	Vg65	1.0630	T8-5	1.0057
Pg4	1.0053	Pg66	344.0471	Vg4	1.0361	Vg66	1.0730	T26-25	1.0262
Pg6	11.7691	Pg69	433.9030	Vg6	1.0271	Vg69	1.0500	T30-17	1.0072
Pg8	0.5606	Pg70	0.0000	Vg8	1.0409	Vg70	1.0322	T38-37	1.0301
Pg10	392.6667	Pg72	0.0000	Vg10	1.0586	Vg72	1.0446	T63-59	1.0025
Pg12	84.6766	Pg73	0.0028	Vg12	1.0217	Vg73	1.0336	T64-61	0.9876
Pg15	0.0000	Pg74	24.9186	Vg15	1.0132	Vg74	1.0253	T65-66	0.9781
Pg18	0.0000	Pg76	40.1692	Vg18	1.0182	Vg76	1.0137	T68-69	1.0169
Pg19	17.7012	Pg77	3.2718	Vg19	1.0134	Vg77	1.0443	T81-80	0.9894
Pg24	2.1852	Pg80	410.8268	Vg24	1.0588	Vg80	1.0575	Qc5	23.7642
Pg25	191.0395	Pg85	10.0744	Vg25	1.0888	Vg85	1.0596	Qc34	25.3992
Pg26	279.1820	Pg87	3.8495	Vg26	1.0828	Vg87	1.0999	Qc37	8.5515
Pg27	15.1988	Pg89	472.9289	Vg27	1.0469	Vg89	1.0787	Qc44	29.9989
Pg31	7.0088	Pg90	2.0708	Vg31	1.0324	Vg90	1.0607	Qc45	29.9735
Pg32	7.8675	Pg91	4.7753	Vg32	1.0411	Vg91	1.0635	Qc46	29.9516
Pg34	3.2001	Pg92	1.2495	Vg34	1.0187	Vg92	1.0658	Qc48	0.0000
Pg36	12.0241	Pg99	0.0300	Vg36	1.0117	Vg99	1.0641	Qc74	29.8290
Pg40	55.0485	Pg100	212.6750	Vg40	1.0057	Vg100	1.0698	Qc79	29.9954
Pg42	77.1078	Pg103	140.0000	Vg42	1.0204	Vg103	1.0742	Qc82	0.7447
Pg46	19.9394	Pg104	5.3727	Vg46	1.0395	Vg104	1.0609	Qc83	22.2974
Pg49	189.9732	Pg105	14.7768	Vg49	1.0560	Vg105	1.0587	Qc105	29.8105
Pg54	49.1457	Pg107	15.7132	Vg54	1.0365	Vg107	1.0490	Qc107	30.0000
Pg55	23.0545	Pg110	17.3268	Vg55	1.0365	Vg110	1.0693	Qc110	3.7302
Pg56	42.7037	Pg111	37.4522	Vg56	1.0363	Vg111	1.0831	Cost (USD/h)	128,088.6942
Pg59	145.6645	Pg112	3.7043	Vg59	1.0564	Vg112	1.0581	Emis (ton/h)	6.8032
Pg61	141.5195	Pg113	3.6027	Vg61	1.0632	Vg113	1.0292	Loss (MW)	78.4546
Pg62	0.0000	Pg116	0.0000	Vg62	1.0603	Vg116	1.0952	VSI	0.0625

,

Table A3. Optimal solutions provided by MaMPA when considering emission as the objective function in the IEEE 118-bus system (case 6).

Variables	Value	Variables	Value	Variables	Value	Variables	Value	Variables	Value
Pg1	64.0673	Pg65	78.5766	Vg1	0.9877	Vg65	1.0576	T8-5	1.0380
Pg4	85.6764	Pg66	64.8584	Vg4	1.0084	Vg66	1.0298	T26-25	1.0998
Pg6	95.0024	Pg69	87.1242	Vg6	1.0050	Vg69	1.0302	T30-17	1.0263
Pg8	81.7911	Pg70	95.0795	Vg8	1.0347	Vg70	1.0527	T38-37	1.0227
Pg10	84.8757	Pg72	81.0996	Vg10	1.0422	Vg72	1.0824	T63-59	1.0219
Pg12	66.9270	Pg73	82.6974	Vg12	0.9983	Vg73	1.0634	T64-61	1.0429
Pg15	76.0727	Pg74	68.2487	Vg15	0.9989	Vg74	1.0358	T65-66	1.0084
Pg18	63.7309	Pg76	78.7266	Vg18	1.0052	Vg76	1.0136	T68-69	1.0131
Pg19	86.6316	Pg77	65.1822	Vg19	1.0037	Vg77	1.0216	T81-80	1.0179
Pg24	94.7514	Pg80	87.2192	Vg24	1.0541	Vg80	1.0234	Qc5	28.8729
Pg25	79.4694	Pg85	94.5080	Vg25	1.0361	Vg85	1.0431	Qc34	6.2554
Pg26	85.2969	Pg87	79.5313	Vg26	1.0725	Vg87	1.0635	Qc37	12.1667
Pg27	67.8921	Pg89	83.7218	Vg27	1.0281	Vg89	1.0402	Qc44	9.9278
Pg31	77.3676	Pg90	67.4321	Vg31	1.0254	Vg90	1.0318	Qc45	16.5402
Pg32	64.0451	Pg91	75.8764	Vg32	1.0270	Vg91	1.0433	Qc46	27.7579
Pg34	86.2332	Pg92	63.4069	Vg34	1.0099	Vg92	1.0373	Qc48	0.1537
Pg36	96.3716	Pg99	86.1364	Vg36	1.0073	Vg99	1.0548	Qc74	29.9983
Pg40	82.7879	Pg100	94.9840	Vg40	0.9998	Vg100	1.0501	Qc79	28.8732
Pg42	86.4091	Pg103	80.5549	Vg42	1.0056	Vg103	1.0609	Qc82	29.9996
Pg46	68.8862	Pg104	81.4427	Vg46	1.0198	Vg104	1.0535	Qc83	14.6246
Pg49	78.7307	Pg105	64.3667	Vg49	1.0162	Vg105	1.0507	Qc105	12.4426
Pg54	66.2104	Pg107	75.5434	Vg54	1.0086	Vg107	1.0527	Qc107	15.3549
Pg55	89.6142	Pg110	60.6496	Vg55	1.0104	Vg110	1.0719	Qc110	20.0449
Pg56	98.6324	Pg111	82.0147	Vg56	1.0091	Vg111	1.0902	Cost (USD/h)	175,654.5699
Pg59	84.5683	Pg112	91.1036	Vg59	1.0149	Vg112	1.0758	Emis (ton/h)	1.8268
Pg61	87.8397	Pg113	79.4047	Vg61	1.0228	Vg113	1.0126	Loss (MW)	1.8268
Pg62	67.9660	Pg116	85.6960	Vg62	1.0242	Vg116	1.0256	VSI	0.0638

,

Table A4. Optimal solutions provided by MaMPA when considering transmission loss as the objective function in the IEEE 118-bus system (case 7).

Variables	Value	Variables	Value	Variables	Value	Variables	Value	Variables	Value
Pg1	79.2962	Pg65	39.7697	Vg1	0.9986	Vg65	1.0445	T8-5	1.0250
Pg4	41.9684	Pg66	82.4272	Vg4	1.0113	Vg66	1.0178	T26-25	1.0022
Pg6	93.4360	Pg69	48.0714	Vg6	1.0134	Vg69	1.0178	T30-17	1.0095
Pg8	67.2358	Pg70	62.5577	Vg8	1.0317	Vg70	1.0172	T38-37	1.0156
Pg10	25.2477	Pg72	4.8808	Vg10	1.0318	Vg72	1.0184	T63-59	1.0240
Pg12	141.1926	Pg73	14.9595	Vg12	1.0085	Vg73	1.0184	T64-61	1.0198
Pg15	98.8427	Pg74	100.0000	Vg15	1.0127	Vg74	1.0166	T65-66	1.0116
Pg18	62.8468	Pg76	99.9963	Vg18	1.0138	Vg76	1.0046	T68-69	1.0776
Pg19	56.2988	Pg77	99.9994	Vg19	1.0130	Vg77	1.0152	T81-80	1.0069
Pg24	37.1550	Pg80	343.6913	Vg24	1.0200	Vg80	1.0214	Qc5	23.6248
Pg25	23.9125	Pg85	82.7307	Vg25	1.0194	Vg85	1.0191	Qc34	7.1127
Pg26	11.2993	Pg87	17.3755	Vg26	1.0317	Vg87	1.0195	Qc37	1.8497
Pg27	88.5769	Pg89	93.5984	Vg27	1.0180	Vg89	1.0183	Qc44	0.6465
Pg31	71.8922	Pg90	100.0000	Vg31	1.0179	Vg90	1.0117	Qc45	25.9688
Pg32	70.6395	Pg91	32.7792	Vg32	1.0175	Vg91	1.0165	Qc46	27.4017
Pg34	75.6676	Pg92	96.7686	Vg34	1.0163	Vg92	1.0155	Qc48	14.9718
Pg36	92.0627	Pg99	55.4364	Vg36	1.0142	Vg99	1.0207	Qc74	29.9529
Pg40	100.0000	Pg100	64.6881	Vg40	1.0145	Vg100	1.0185	Qc79	1.3349
Pg42	99.9950	Pg103	68.3646	Vg42	1.0167	Vg103	1.0233	Qc82	29.9963
Pg46	77.5342	Pg104	68.1967	Vg46	1.0211	Vg104	1.0214	Qc83	9.1679
Pg49	202.3555	Pg105	21.5187	Vg49	1.0203	Vg105	1.0189	Qc105	6.1990
Pg54	143.4796	Pg107	61.9437	Vg54	1.0183	Vg107	1.0217	Qc107	6.1352
Pg55	89.7263	Pg110	60.6194	Vg55	1.0188	Vg110	1.0223	Qc110	29.9901
Pg56	86.9104	Pg111	5.6696	Vg56	1.0180	Vg111	1.0229	Cost (USD/h)	163,295.7775
Pg59	240.2069	Pg112	52.2977	Vg59	1.0175	Vg112	1.0183	Emis (ton/h)	2.6964
Pg61	158.5317	Pg113	44.9178	Vg61	1.0210	Vg113	1.0188	Loss (MW)	11.1950
Pg62	64.8718	Pg116	29.0120	Vg62	1.0200	Vg116	1.0392	VSI	0.0665

and

Table A5. Optimal solutions provided by MaMPA when considering VSI as the objective function in the IEEE 118-bus system (case 8).

Variables	Value	Variables	Value	Variables	Value	Variables	Value	Variables	Value
Pg1	0.4486	Pg65	297.9908	Vg1	1.0055	Vg65	1.0521	T8-5	0.9401
Pg4	40.1660	Pg66	212.4324	Vg4	1.0517	Vg66	1.0808	T26-25	1.0892
Pg6	14.6342	Pg69	81.9709	Vg6	1.0342	Vg69	0.9790	T30-17	0.9219
Pg8	64.8623	Pg70	26.8173	Vg8	0.9504	Vg70	0.9916	T38-37	0.9505
Pg10	326.8203	Pg72	84.7585	Vg10	0.9575	Vg72	1.0816	T63-59	1.0071
Pg12	23.0027	Pg73	10.0040	Vg12	1.0292	Vg73	0.9656	T64-61	1.0281
Pg15	0.0113	Pg74	98.0428	Vg15	1.0415	Vg74	0.9919	T65-66	1.0160
Pg18	48.7747	Pg76	11.2165	Vg18	1.0449	Vg76	0.9779	T68-69	1.0609
Pg19	90.7062	Pg77	13.2903	Vg19	1.0444	Vg77	1.0367	T81-80	0.9331
Pg24	83.9210	Pg80	571.4437	Vg24	1.0136	Vg80	1.0868	Qc5	5.6328
Pg25	286.1231	Pg85	73.5674	Vg25	0.9891	Vg85	0.9870	Qc34	29.9999
Pg26	4.6937	Pg87	15.3395	Vg26	1.0412	Vg87	0.9746	Qc37	3.9437
Pg27	16.2361	Pg89	350.7334	Vg27	1.0633	Vg89	0.9649	Qc44	8.5375
Pg31	25.8398	Pg90	71.4154	Vg31	0.9513	Vg90	1.0027	Qc45	4.3897
Pg32	9.0424	Pg91	82.3990	Vg32	1.0157	Vg91	1.0573	Qc46	28.2219
Pg34	99.1197	Pg92	20.0460	Vg34	1.0627	Vg92	0.9906	Qc48	0.0181
Pg36	80.5757	Pg99	53.7099	Vg36	1.0615	Vg99	1.0900	Qc74	29.8006
Pg40	86.9357	Pg100	67.3055	Vg40	0.9844	Vg100	1.0273	Qc79	30.0000
Pg42	1.7163	Pg103	118.3309	Vg42	0.9966	Vg103	1.0366	Qc82	24.6194
Pg46	116.6165	Pg104	0.7734	Vg46	1.1000	Vg104	1.0242	Qc83	1.2281
Pg49	0.2618	Pg105	24.7107	Vg49	1.0560	Vg105	1.0235	Qc105	21.7857
Pg54	35.7633	Pg107	78.9323	Vg54	1.0226	Vg107	1.0169	Qc107	0.0053
Pg55	25.3053	Pg110	3.6567	Vg55	1.0168	Vg110	1.0371	Qc110	0.8670
Pg56	82.4442	Pg111	90.3582	Vg56	1.0199	Vg111	1.0662	Cost (USD/h)	150,972.3018
Pg59	106.5712	Pg112	39.8216	Vg59	1.0243	Vg112	1.0381	Emis (ton/h)	4.9794
Pg61	17.4251	Pg113	15.2741	Vg61	1.0447	Vg113	1.0664	Loss (MW)	104.4672
Pg62	97.8106	Pg116	42.7151	Vg62	1.0477	Vg116	0.9782	VSI	0.0596

in Appendix A. The optimal solutions consisting of the objective values and time generated by MaMPA for cost, loss, and VSI objectives are compared with those of several algorithms as expressed in

Table 6. Comparison results of case 5.

Algorithms	Cost (USD/h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
DE [64]	129,582.0000	-	79.41	-	89.5
GWO [64]	129,720.0000	-	79.58	-	101
GSA [71]	129,565.0000	-	76.19	-	-
HSA [72]	132,319.6000	-	-	-	-
FHSA [72]	132,138.3000	-	-	-	-
ICBO [73]	135,121.5704	-	62.73	0.0691	-
MSA [74]	129,640.7191	-	73.26	0.0615	-
PSOGSA [75]	129,733.5800	-	73.21	-	1280
SCA [76]	129,622.6500	-	76.50	-	-
MSCA [76]	129,620.2200	-	76.22	-	-
TLBO [77]	129,682.8440	-	-	-	1885
DSA [77]	129,691.6152	-	-	-	-
SSA [78]	129,675.0000	-	77.29	-	103.0
ISSA [79]	129,460.8351	-	-	0.0624	-
SP-DE [80]	135,055.7000	-	60.96	-	2400.0
Rao-3 [81]	129,220.6794	-	109.12	-	164.2
AMTPG-Jaya [69]	129,428.7030	-	76.05	-	94.3
PSO [60]	129,703.4800	-	-	-	1359.0
BBO [60]	129,734.0600	-	-	-	1595.0
ABC [60]	129,677.8300	-	-	-	1284.0
TLBO [60]	129,550.4900	-	-	-	741.8
AGTLBO [60]	129,543.5600	-	76.21	-	737.2
MaMPA	128,088.6942	6.8032	78.45	0.0625	171.6

,

Table 7. Comparison results of case 7.

Algorithms	Cost (USD/h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
SSA [78]	160,680.00	-	12.1930	-	103.0
BBO [71]	156,556,00	-	14.9200	-	-
GSA [71]	157,771.00	-	12.8800	-	-
SP-DE [80]	155,724.90	-	17.6946	-	2400.0
MaMPA	163,295.78	2.6964	11.1950	0.0665	172.4

and

Table 8. Comparison results of case 8.

Algorithms	Cost (USD/h)	Emis (ton/h)	Loss (MW)	VSI	Time (s)
NSGA-II [82]	-	-	-	0.0690	-
MNSGA-II [82]	-	-	-	0.0660	-
MaMPA	150,972.30	4.9794	104.4672	0.0596	202.8

while the reference comparison of solving SOOPF by considering the emission objective could not be found. The objective value of the MaMPA when that objective was considered as the objective function are bold as in the Tables. The convergence curves generated by MaMPA for each objective, cases 5–8, are shown in Figure 3.

In the IEEE 118-bus system, it is noticed that MaMPA provided significantly better feasible solutions for each objective than those of the various algorithms in the literature depicted in Table 6, Table 7 and Table 8. In particular, the cost value obtained by MaMPA was better than all compared algorithms for more than 1000 USD/h. The computational times of MaMPA are slower than some algorithms such as DE, GWO, SSA, Rao-3, and AMTPG-Jaya; however, they are faster than the other compared algorithms, especially PSOGSA, TLBO, SP-DE, PSO, BBO, and ABC. From the convergence curves of each objective generated by MaMPA in Figure 3, it is noticeable that MaMPA could quickly converge on the feasible solution within one-fifth of the maximum iteration. Thus, MaMPA has a high performance in solving SOOPF problems in this large system.

Hence, for the SOOPF problems, MaMPA could obtain superior solutions than various other algorithms for all considered objective functions in each test system, especially the large system, with a fast computational time. MaMPA could also quickly converge on the feasible solutions for all considered objectives in each test system.

4.1.3. Statistical Investigation

To further evaluate the performance of MaMPA in solving SOOPF problems, the statistical investigations including minimum, maximum, and average values and standard deviation of each objective value when that objective is an objective function are generated in both tested systems. The MaMPA was simulated for 30 independent runs for each test system.

The statistical investigations of MaMPA in the IEEE 30-bus system, which is a small system, are provided and compared with those of other algorithms in the literature as shown in

Table 9. Statistical comparison of MaMPA and other algorithms in the IEEE 30-bus system.

Case 1	Min	Max	Average	Std.
Algorithms
PSO [60]	800.7016	803.8826	801.4378	2.075
BBO [60]	800.9527	805.3919	802.3304	3.994
ABC [60]	800.6802	802.1506	800.9642	1.819
TLBO [60]	800.6735	801.1324	800.8705	0.542
AGTLBO [60]	800.4811	800.5587	800.5316	0.076
MaMPA	802.5449	802.5449	802.5449	1.65 × 10−8
Case 2	Min	Max	Average	Std.
Algorithms
PSO [60]	0.20484	0.20551	0.20496	0.009
BBO [60]	0.20490	0.20603	0.20532	0.010
ABC [60]	0.20484	0.20557	0.20505	0.009
TLBO [60]	0.20484	0.20495	0.20490	0.006
AGTLBO [60]	0.20482	0.20484	0.20483	0.000
MaMPA	0.204886	0.204886	0.204886	8.03 × 10−11
Case 3	Min	Max	Average	Std.
Algorithms
PSO [60]	3.1709	3.2235	3.1770	0.071
BBO [60]	3.1725	3.3178	3.2490	0.199
ABC [60]	3.1314	3.2074	3.1682	0.103
TLBO [60]	3.1104	3.1275	3.1133	0.062
AGTLBO [60]	3.0906	3.0920	3.0911	0.000
MaMPA	3.2974	3.2974	3.2974	9.5766 × 10−11
Case 4	Min	Max	Average	Std.
Algorithms
RGA [66]	0.13860	0.13956	0.13916	0.00033
CMAES [66]	0.13820	0.13836	0.13820	0.00019
ACDE [67]	0.13640	0.13680	0.13660	0.0001
SKH [68]	0.13660	0.13840	0.13720	0.0006
ECHT-DE [69]	0.13630	0.13720	0.13690	0.0002
ARCBBO [70]	0.13690	0.13870	0.13750	-
MaMPA	0.13625	0.13626	0.13625	7.9657 × 10−6

.

It can be noticed from Table 9 that the minimum, maximum, and average values of MaMPA for each objective are slightly different resulting in a very small standard deviation. When compared to other algorithms, MaMPA provided slightly worse objective values in terms of minimum, maximum, and average values while the standard deviations of MaMPA for all objectives are better than those of the compared algorithms. This means MaMPA has a great consistency, and system operators can certainly obtain a high-quality feasible solution in a single operation of the algorithm process.

In the IEEE 118-bus system, which is a large test system, the MaMPA was operated to provide the statistical results. The statistical comparison of MaMPA and other algorithms in the literature are presented in

Table 10. Statistical comparison of MaMPA and other algorithms in the IEEE 118-bus system.

Case 5	Min	Max	Average	Std.
Algorithms
SSA [78]	129,675.00	129,788.80	129,734.60	34.79
Rao-3 [81]	129,220.68	129,440.35	129,331.60	4.09
PSO [60]	129,703.48	131,573.7	129,867.01	610.70
BBO [60]	129,734.06	132,268.8	129,985.25	995.20
ABC [60]	129,677.83	130,896.6	129,732.46	446.80
TLBO [60]	129,550.49	129,691.0	129,607.63	72.14
AGTLBO [60]	129,545.56	129,562.3	129,552.92	8.53
MaMPA	128,088.69	128,669.94	128,370.72	141.68
Case 6	Min	Max	Average	Std.
Algorithms
MaMPA	1.8247	1.8439	1.8298	0.0048
Case 7	Min	Max	Average	Std.
Algorithms
SSA [78]	12.193	13.498	12.301	0.6600
MaMPA	11.1950	13.8413	12.3769	0.6550
Case 8	Min	Max	Average	Std.
Algorithms
MaMPA	0.05959	0.05967	0.05962	2.095 × 10−5

for the cost and loss objectives, cases 5 and 7, while the references for the statistical results of emission and VSI objectives, cases 6 and 8, could not be found.

It is observed from Table 10 that the minimum, maximum, and average values of each objective provided by MaMPA are slightly different from each other leading to low standard deviation values. Although the standard deviation when selecting cost as the objective function is higher than when choosing the other objectives as the objective functions, this is because the cost value for this large system is high compared to the values of the other objectives. When compared to other algorithms, for the cost objective, even if MaMPA has a higher standard deviation than some algorithms; the minimum, maximum, and average values are better than those of all compared algorithms. For the loss objective, MaMPA generated a better minimum loss and standard deviation values than those of SSA while the maximum and average values were slightly worse than those of SSA. Thus, MaMPA also has a high consistency in the large system, and high-quality feasible solutions can be generated in a few runs.

So, MaMPA is a high-consistency algorithm for solving SOOPF problems by providing small standard deviation values for each considered objective function in the IEEE 30- and 118-bus system. In the large IEEE 118-bus system, the minimum values of each objective generated by MaMPA are also better than the compared algorithms. The system operators can receive high-quality solutions within a few runs of the algorithm process.

4.2. Multi-Objective OPF Problem

In the MOOPF problems, two or three objective functions are simultaneously set as the objective functions. The MOOPF problems are solved by MaMPA in both IEEE 30- and 118-bus systems, and the results are compared with the compared algorithms based on the hypervolume indicator in the presented subsections.

4.2.1. IEEE 30-Bus System

Two and three objective functions are simultaneously chosen as part of the objective functions in the IEEE 30-bus system for the MOOPF problems. The two-dimensional Pareto fronts for some pairs of objectives, which are cases 9–12, are depicted in Figure 4. For the three objectives, Figure 5 presents the three-dimensional Pareto fronts for cases 13–14. The Pareto fronts for each considered case are compared with those of PSO [83], PESA-II [84], SPEA2 [26], and SSA [85] where the setting parameters of the compared algorithms can be found in the provided references.

To compare the Pareto fronts generated from all algorithms, hypervolume indicators for the fronts are calculated as presented in

Table 11. Hypervolume comparison of multi-objective OPF (MOOPF) problems in the IEEE 30-bus system.

Algorithms	Case 9	Case 10	Case 11	Case 12	Case 13	Case 14
PSO	0.07411	0.09921	0.00074	0.00132	0.05957	0.01160
PESA-II	0.07204	0.09274	0.00022	0.00065	0.05106	0.01010
SPEA2	0.07274	0.09764	0.00044	0.00059	0.05680	0.01073
SSA	0.07009	0.09347	0.00049	0.00098	0.05353	0.01014
MaMPA	0.07359	0.09780	0.00071	0.00131	0.05862	0.01156

where a higher value of the hypervolume indicates better generated Pareto fronts.

The results of MOOPF problems in the IEEE 30-bus system expressed in Figure 4 and Figure 5 show that MaMPA successfully generated Pareto fronts for all cases. Based on the hypervolume indicator values from Table 11, it is found that PSO provided the best Pareto fronts for all cases in this system, and MaMPA generated the second-best Pareto fronts for all cases. For cases 9–10, the Pareto fronts from all algorithms are very competitive while for cases 11–12, PSO and MaMPA obtained much greater Pareto fronts than those of PESA-II, SPEA2, and SSA. In cases 13 and 14, for the three objectives, PSO and MaMPA obtained the best and second-best Pareto fronts where the fronts from PSO and MaMPA have more diversity than those of PESA-II, SPEA2, and SSA, as is evident in Figure 5.

4.2.2. IEEE 118-Bus System

To guarantee the effectiveness of the MaMPA in the large system, this subsection investigates solving the MOOPF problems in the IEEE 118-bus system. By selecting two and three objectives as the objective functions, cases 15–18, the two- and three-dimensional Pareto fronts are presented in Figure 6 and Figure 7, respectively. For each case study, the generated Pareto fronts are compared with those of PSO [83] and SSA [85] while PESA-II and SPEA2 could not converge on the feasible solution within the set iterations. The setting parameters of the PSO and SSA can be found in the provided references.

After generating the Pareto fronts in this system, the hypervolume indicators for each case are computed to compare the generated fronts as presented in

Table 12. Hypervolume comparison of MOOPF problems in the IEEE 118-bus system.

Algorithms	Case 15	Case 16	Case 17	Case 18
PSO	0.04303	0.05891	0.01451	0.03351
SSA	0.00388	0.01504	0.00866	0.01153
MaMPA	0.04678	0.06317	0.04981	0.05855

where a higher hypervolume value means better Pareto fronts.

For the IEEE 118-bus system, MaMPA successfully provided Pareto fronts for each case study, as is evident in Figure 6 and Figure 7, while SSA generated a very narrow set of Pareto fronts as seen similar to a dot in Figure 6 and Figure 7 compared to those of MaMPA; PESA-II and SEPA2 could not find feasible solutions within the set maximum iteration. Based on the hypervolume indicator in Table 12, MaMPA generated significantly better Pareto fronts than those of PSO and SSA for all cases. The hypervolume values of MaMPA are better than those of PSO around 8.71, 7.23, 243.28, and 74.72% for cases 15–18, respectively, and of SSA around 1105.67, 320.01, 475.17, and 407.81% for cases 15–18, respectively. The percentage comparisons of the hypervolume values between MaMPA and SSA are very high because SSA provided a very narrow set of Pareto fronts as seen similar to a dot when compared to those of MaMPA.

Thus, for MOOPF problems, MaMPA has a high performance in providing better Pareto fronts than other methods in both tested systems. Although the Pareto fronts of MaMPA were slightly poorer than those of PSO for some cases in the IEEE 30-bus system, MaMPA could generate significantly better Pareto fronts than those of all compared algorithms in the IEEE 118-bus system which could be verified by the hypervolume values.

4.3. Many-Objective OPF Problem

For solving OPF problems comprising more than three objective functions, the problems become MaOPF problems. In this section, all four objective functions which are cost, emission, transmission loss, and VSI are simultaneously considered as the objective functions and solved by MaMPA in both tested systems. The fuzzy decision method is applied to provide the best-compromised solutions from the Pareto fronts for each test system since the Pareto fronts for more than three objectives are difficult to plot, and the hypervolume values are found to compare the Pareto fronts.

4.3.1. IEEE 30-Bus System

For the IEEE 30-bus system, when all four objectives are considered as the objective functions, case 19, the best-compromised solutions between each objective for the MaOPF problem of the MaMPA are compared with those of PSO, PESA-II, SPEA2, and SSA, as presented in

Table 13. Best compromised solutions for solving many-objective OPF (MaOPF) problems by using all considered objectives as the objective functions in the IEEE 30-bus system (case 19).

	Cost (USD/h)	Emission (ton/h)	Loss (MW)	VSI
PSO	863.0005	0.226071	4.9709	0.1424
PESA-II	863.8967	0.226163	5.0019	0.1394
SPEA2	861.6285	0.226465	5.0708	0.1390
SSA	857.4578	0.230565	5.1405	0.1404
MaMPA	862.1941	0.227876	4.9211	0.1388

. The calculated hypervolume values for each algorithm are depicted in

Table 14. Hypervolume comparison of MaOPF problems in the IEEE 30-bus system.

Algorithms	Case 19
PSO	0.03347
PESA-II	0.02849
SPEA2	0.03236
SSA	0.02876
MaMPA	0.03367

.

For solving the MaOPF problem in the IEEE 30-bus system, the best-compromised solution provided by MaMPA is competitive with the compared algorithms as in Table 13. The provided cost is the third best one, and the provided loss and VSI are the best values among the compared algorithms while the emission value is almost the worst one. However, based on the hypervolume indicator in Table 14, the hypervolume values of MaMPA are better than those of PSO, PESA-II, SPEA2, and SSA at around 0.59%, 18.18%, 4.05%, and 17.07%, respectively. So, MaMPA generated the best hypervolume value indicating that the Pareto fronts of MaMPA are the best for this case.

4.3.2. IEEE 118-Bus System

To ensure the effectiveness of the MaMPA in solving the MaOPF problem in the larger system, the IEEE 118-bus was tested. The best-compromised solution between the four objective functions of the MaMPA is compared with those of PSO and SSA as expressed in

Table 15. Best compromised solutions for solving the MaOPF problem by using all considered objectives as the objective functions in the IEEE 118-bus system (case 20).

	Cost (USD/h)	Emission (ton/h)	Loss (MW)	VSI
PSO	153,524.9888	4.2172	86.1397	0.4199
SSA	147,124.752	4.0393	56.3501	0.0648
MaMPA	140,797.6517	2.7576	44.8470	0.0620

and the hypervolume indicators are computed as in

Table 16. Hypervolume comparison of MaOPF problems in the IEEE 118-bus system.

Algorithms	Case 20
PSO	0.01378
SSA	0.02582
MaMPA	0.08339

while PESA-II and SPEA2 could not find a feasible solution within the given iterations.

In the IEEE 118-bus system, MaMPA generated highly efficient results in solving the MaOPF problem. MaMPA could provide the best-compromised solution which is remarkably superior to those of PSO and SSA in all considered objective values as in Table 15 while PESA-II and SPEA2 could not even converge on a feasible solution. The objective values of the best-compromised solution of MaMPA are better than those of PSO at around 8.29, 34.61, 47.94, and 85.23% for cost, emission, loss, and VSI objectives, respectively. They are also greater than those of SSA at about 4.30, 31.73, 20.41, and 4.38% for cost, emission, loss, and VSI objectives, respectively. From Table 16, the hypervolume values calculated by the Pareto fronts of MaMPA for this case are superior to those of PSO and SSA by about 505.15% and 222.97%, respectively. So, the hypervolume values also guarantee that MaMPA provided significantly better Pareto fronts than those of the compared algorithms for this case.

Hence, MaMPA has a high performance in solving MaOPF problems in the small tested system and in particular in the large tested system. The generated best-compromised solutions and Pareto fronts of MaMPA are competitive to those of the compared algorithms and much better than the compared algorithms when tested in the IEEE 30- and 118-bus systems, respectively. The hypervolume values could also verify the superior Pareto fronts generated by MaMPA to those of the compared algorithms.

5. Conclusions

In this paper, a MaMPA for solving the MaOPF problem is proposed. With the growth of electricity demand, environmental awareness, and power reliability requirements, more than three objective functions for the OPF problem, referred to as the MaOPF problem, are important to satisfy modern power system requirements. MPA is an efficient optimization algorithm and has been widely applied to efficiently solve different problems except for the MaOPF problem. The MPA itself cannot solve multi- or many-objective optimization problems, so the non-dominated sorting, crowding mechanism, and leader mechanism are applied to MPA in this work. Four objectives consisting of cost, emission, transmission loss, and VSI are selected to be the objective functions. The performance of the MaMPA is evaluated in the IEEE 30- and 118-bus systems. From the simulation results, MaMPA could find slightly different solutions and significantly better solutions in the IEEE 30- and 118-bus systems, respectively, for the SOOPF problems. The computational times of MaMPA are slower than some algorithms and much faster than various algorithms in the literature for both test systems. In the SOOPF problems, MaMPA also provided efficient statistical results with very small standard deviation values in both tested systems expressing high consistency of the generated results, and both standard deviation and minimum values for each objective provided by MaMPA are better than several algorithms in the literature. For the MOOPF problems, based on the hypervolume indicator, the two- and three-dimensional Pareto fronts generated by MaMPA are greater than those of PESA-II, SPEA2, and SSA while the fronts are slightly worse than those of PSO in the IEEE 30-bus system. In the IEEE 118-bus system, both two- and three-dimensional Pareto fronts provided by MaMPA are remarkably superior to those of PSO and SSA based on the hypervolume values while the fronts could not be obtained by PESA-II and SPEA2 within the given iterations. For the MaOPF problems, in the IEEE 30-bus system, the best-compromised solutions of MaMPA are very competitive showing two better objective values from four objectives than the compared algorithms while the Pareto fronts based on the hypervolume indicator of MaMPA are the best ones. In the IEEE 118-bus system, the best compromise solution of MaMPA presents absolute superiority over PSO and SSA in all objective values, and the Pareto fronts of MaMPA based on the hypervolume indicator are also significantly better than those of PSO and SSA while the other compared algorithms could not converge on the solution. Overall, MaMPA has a high performance in solving SOOPF, MOOPF, and MaOPF problems, especially in the large system, and with the high consistency of MaMPA, power system operators can generate high-quality solutions within a few runs. In future work, the self-adaptive method and two archive techniques can be applied to improve the MPA, so that the feasible solutions can be enhanced. In addition, more objective functions such as voltage deviation can be added to the objective functions to further improve the power system performance. The proposed MaMPA can also be adopted to solve other many-objective problems in different fields due to its high performance.