A Slime Mould Algorithm Programming for Solving Single and Multi-Objective Optimal Power Flow Problems with Pareto Front Approach: A Case Study of the Iraqi Super Grid High Voltage

Abstract

Optimal power flow (OPF) represents one of the most important issues in the electrical power system for energy management, planning, and operation via finding optimal control variables with satisfying the equality and inequality constraints. Several optimization methods have been proposed to solve OPF problems, but there is still a need to achieve optimum performance. A Slime Mould Algorithm (SMA) is one of the new stochastic optimization methods inspired by the behaviour of the oscillation mode of slime mould in nature. The proposed algorithm is characterized as easy, simple, efficient, avoiding stagnation in the local optima and moving toward the optimal solution. Different frameworks have been applied to achieve single and conflicting multi-objective functions simultaneously (Bi, Tri, Quad, and Quinta objective functions) for solving OPF problems. These objective functions are total fuel cost of generation units, real power loss on transmission lines, total emission issued by fossil-fuelled thermal units, voltage deviation at load bus, and voltage stability index of the whole system. The proposed algorithm SMA has been developed by incorporating it with Pareto concept optimization to generate a new approach, named the Multi-Objective Slime Mould Algorithm (MOSMS), to solve multi-objective optimal power flow (MOOPF) problems. Fuzzy set theory and crowding distance are the proposed strategies to obtain the best compromise solution and rank and reduce a set of non-dominated solutions, respectively. To investigate the performance of the proposed algorithm, two standard IEEE test systems (IEEE 30 bus IEEE 57 bus systems) and a practical system (Iraqi Super Grid High Voltage 400 kV) were tested with 29 case studies based on MATLAB software. The optimal results obtained by the proposed approach (SMA) were compared with other algorithms mentioned in the literature. These results confirm the ability of SMA to provide better solutions to achieve the optimal control variables.

1. Introduction

Power flow (PF), also known as load flow, is one of fundamental issues in electrical power systems. The main idea of power flow analysis is to find out the reactive power output in transmission lines, the voltage at buses, and total losses in the whole system at operation conditions. In recent decades, optimal power flow (OPF) has been given extensive interest by researchers because it is one of the most important tools used in power management systems to achieve the reliable operation and planning of electrical power systems [1]. To optimize objective functions in the power system, OPF needs to set the control variables while respecting equality and inequality constraints because OPF is a non-convex, nonlinear, and large-scale problem. The active power output of the generation units without the slack bus, the voltages at PV buses, reactive power compensators, and tap transformers settings are the control variables that are tuned. The generation fuel cost (GFC), real power loss (RPL) in the transmission lines, emission (Em), voltage deviation (VD), and voltage stability index (VSI) in the whole system are the objective functions that will be optimized. OPF was first presented by Carpentier in 1962 [2].

Two types of optimization methods that have been proposed to solve OPF problems are classical and intelligent optimization methods. Several classical methods have been applied, such as linear and nonlinear programming, interior point method, the Newton method, quadratic programming, and mixed-integer programming [3,4]. Although these techniques provide an optimal solution, their drawbacks cannot reach a local minimum if it is assumed that the initial point is not close to the solution. Further, the quality of solutions is inversely proportional to a number of control variables. In addition, due to increasing the number of non-linear constraints, the problems are more complex. The second type of optimization algorithms is intelligent optimization techniques, such as grey wolf optimizer (GWO) [5], hunger games search (HGS) [6], Harris hawks optimization [7], Nomadic People Optimizer [8], and the honey badger algorithm (HBA) [9].

A Slime Mould Algorithm (SMA) is one of new optimization algorithms that has been proposed to solve the OPF problem in the power system. SMA is a meta-heuristic algorithm inspired by the diffusion and foraging conduct of slime mould proposed in 2020 by S. Li et al. [10]. SMA has many features, such as:

(i)

The mathematical model used in this algorithm is unique. It uses adaptive weights, which are allowed to produce positive and negative feedback in the simulation process for propagation wave.

(ii)

The form path of connection food is optimal using a bio-oscillator.

(iii)

The ability and propensity for exploration and exploitation is excellent.

Several articles have solved single-objective OPF problems using intelligent optimization techniques, such as differential evolution (DE) [11], modified artificial bee colony (MABC) [12], improved differential evolution (IDE) [13,14], Harris hawks optimization [15], and the moth swarm algorithm (MSA) [16]. On the other hand, several approaches have been proposed to solve multi-objective optimization (MOO), such as a weighted sum [17], the penalty function method [18], ε-constant [19], the non-dominated sorting genetic algorithm-based approach [20], and the strength Pareto evolutionary algorithm [21]. The most popular method used to solve multi-objective optimization (MOO) problems is Pareto optimization (PO) [22]. One of the main features of this method is a comparison of conflicting objective functions (OFs) to choose favourable solutions [23]. The fuzzy membership approach is the approach taken in the decision-making process to select the best compromise solution in the Pareto front computations.

Multi-objective optimal power flow (MOOPF) is most important in power systems operation and planning because of its ability to find the best compromise solution for more than one objective function simultaneously [24]. The Pareto concept is incorporated with many optimization methods to arrange the non-dominated solutions and set the generation probability for individuals. Many optimization algorithms have been proposed to solve MOOPF in electrical power system, such as the Multi-Objective Improved Differential Evolution Algorithm (MOIDEA) [25], multi-objective backtracking search algorithm (MOBSA) [26], Jaya Optimization [27], Multi-Objective Manta Ray Foraging Optimizer (MOMRFO) [28], Multi-Objective Ant Lion Optimizer (MOALO) [29], and Harris Hawks Optimization (HHO) [30].

In this paper, a newly proposed algorithm (proposed in 2021), named the Slime Mould Algorithm (SMA), is suggested to solve a single-objective function on three systems: IEEE 30 bus and IEEE 57 bus test systems, and one practical system (Iraqi Super Grid High voltage 400 kV). In addition, the proposed algorithm SMA was developed by incorporating it with Pareto concept optimization to generate a new approach, named the Multi-Objective Slime Mould Algorithm (MOSMS), to solve multi-objective (Bi, Tri, Quad, and Quinta) optimal power flow problems. The approach used to extract best compromise solution is fuzzy set theory. Generation fuel cost (GFC), emission (Em), real power losses (RPL), voltage deviation (VD), and voltage stability index (VSI) are the objective functions that will be optimized. It can be summarized the main contribution as follows:

• The Slime Mould Algorithm was developed to solve single- and multi-objective optimal power flow to achieve the economic, environmental, and technical benefits of power systems.
• The Pareto concept is the approach taken to rank store non-dominated Pareto fronts, crowding distance is the mechanism to reduce the Pareto repository, and fuzzy set theory is the theory applied to extract the best compromise solution.
• Two standard IEEE test systems (IEEE 30 bus IEEE 57 bus systems) and a one practical network (Iraqi Super Grid High Voltage 400 kV) were applied with 29 case studies for single- and multi-objective (Bi, Triple, Quad, and Quinta) functions.
• The optimal results obtained by the proposed algorithm were compared with other recent optimization methods in the literature.

The remainder of this paper can be summarized as follows: Section 2 present the OPF problem formulation, including the general OPF formulation, objective functions, and operational constraints. Section 3 is the mathematical model of Slime Mould Algorithm (SMA). Section 4 introduce the strategy taken in multi-objective solutions. Section 5 discussed the numerical results for 29 cases and compared them with other recent optimization methods. Finally, the conclusions are presented in Section 6.

2. OPF Problem Formulation

In power systems, the objective functions can be optimized by set control variables as optimally with satisfied the equality and inequality constraints. The mathematical model of OPF problems can be described by the following:

$$\begin{array}{lll}\mathrm{Optimize}& f\left(x,u\right) & \\ \mathrm{subjected} \mathrm{to}& {\mathrm{g}}_i\left(x,u\right)=0 & i=1,2,\dots ,\mathrm{m}\\ & h_i\left(x,u\right)\le 0& i=1,2,\dots ,\mathrm{p}\end{array}$$

(1)

These vectors can be symbolized as:

$$x=\left[P_{G_1},\left|{\mathrm{V}}_{L_1}\right|,\cdots ,\left|V_{L_{PQ}}\right|,Q_{G_1}\cdots Q_{G_{PV}}\right]$$

(2)

$$u=\left[P_{G_2},\cdots ,P_{G_{PV}},\left|V_{G_1}\right|,\cdots ,\left|V_{G_{PV}}\right|,\right.\left.T_1,\cdots ,T_{NT},Q_{C_1},\cdots ,Q_{C_{NC}}\right]$$

(3)

2.1. Objective Functions

The objective functions will be optimized are generation fuel cost (GFC), real power losses (RPL), emission (Em), voltage deviation (VD), and voltage stability index (VSI).

• Generation Fuel Cost (GFC) (USD/h)

The mathematical formula that has been described to GFC is [31]:

$$F_{GFC}={\displaystyle \sum }_{i=1}^{N_G}\left(a_i{P}_{G_i}^2+b_i{P}_{G_i}+c_i\right)\begin{array}{cc}& \end{array}(USD/h)$$

(4)

2.

Real Power Losses (RPL) (MW)

The mathematical formula that has been described to GFC is [31]:

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

(5)

3.

Emission (Em) (ton/h)

Emission (Em) can be expressed of the mathematical description of greenhouse gases emissions, such as $NOx$ and $SOx$, as follows:

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

(6)

4.

Voltage Deviation (VD) (p.u.)

The voltage level at buses is a very important factor to achieve stability and economic benefits by keeping the voltages of each bus close to the reference voltage. The formula that expressed of voltage deviation is given by [32]:

$$F_{VD}={\displaystyle \sum _{i=1}^{N_{PQ}}\left|V_i-V_{ref}\right|}\begin{array}{cc}& \end{array}(p.u.)$$

(7)

Here, $V_{ref}$ denotes the rated voltage magnitude, which the value is 1.0 (p.u.).

5.

Voltage Stability Index (VSI)

The maximum value of the voltage stability indicator (L-index) will be minimized by enhancing the voltage stability of the whole system. The following equation represents the mathematical formula of VSI [32]:

$$F_{VSI}=Max\left(L_j\right)$$

(8)

$$L_j=\left|1-{\displaystyle \sum _{i=1}^{N_G}\left(F_{ji}\times \frac{V_i}{V_j}\right)}\right|$$

(9)

$$F_{ji}=-{\left[Y_{LL}\right]}^{-1}\times \left[Y_{LG}\right]$$

(10)

2.2. Constraints

In OPF, equality constraints (active and reactive powers) represent the physical structure of the whole system. It can be expressed as [33]:

$$P_g{}_i-P_d{}_i=\left|V_i\right|{\displaystyle \sum _{j=1}^N\left|V_j\right|\left(g_{ij}\cos {\delta }_{ij}+b_{ij}\sin {\delta }_{ij}\right)}\begin{array}{cc}& \end{array}\begin{array}{cc}& \end{array}\forall i\in N$$

(11)

$$Q_g{}_i+Q_c{}_i-Q_d{}_i=\left|V_i\right|{\displaystyle \sum _{j=1}^N\left|V_j\right|\left(g_{ij}\sin {\delta }_{ij}-b_{ij}\cos {\delta }_{ij}\right)}\begin{array}{cc}& \end{array}\forall i\in N$$

(12)

The other constraints are inequality constraints (generator, transformer, shunt compensator, and security). These constraints represent the operation limit to achieve the stable operation of the system. It can be described as follows [33]:

1.

Generator Constraints:

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

(13)

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

(14)

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

(15)

2.

Shunt Compensator Constraints:

$$Q_c{{}_{{}_k}}^{\min }\le Q_c{}_{{}_k}\le Q_c{{}_{{}_k}}^{\max }\begin{array}{ccc}& & k=1,2,\dots ,N_c\end{array}$$

(16)

3.

Transformer Constraints:

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

(17)

4.

Security Constraints:

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

(18)

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

(19)

3. The Mathematical Model of the Slime Mould Algorithm (SMA)

The Slime Mould Algorithm (SMA) is a new optimization algorithm inspired by the diffusion and behaviour conduct of slime mould in nature and proposed by S. Li et al. in 2021 [10]. The processes of SMA by approaching food, wrapping food, and oscillating can be summarized as follows:

3.1. Approach Food

The approaching behaviour can be expressed as mathematical formulae as follows:

$$Y(t+1)=\left\{\begin{array}{c}{Y}_b(t)\cdot ub\cdot \left(V\cdot Y_A(t)-Y_B(t)\right), r<p\\ uc\cdot Y(t), r\ge p\hfill \end{array}\right.$$

(20)

The formula of $p$ is as follows [10]:

$$p=\tanh \left(\left|fit(i)-df\right|\right)$$

(21)

where 𝑖 ∈ 1,2, …, 𝑛. It can be expressed as follows:

$$a=\mathrm{arctanh}\left(-\left(\frac{t}{t\_\max }\right)+1\right)$$

(22)

The formula of V is as follows:

$$W(\mathrm{SmIndex}(i))=\left\{\begin{array}{c}1+r\cdot \log \left(\frac{bf-R(i)}{bf-wf}+1\right),\begin{array}{cc}& condition\end{array}\\ 1-r\cdot \log \left(\frac{bf-R(i)}{bf-wf}+1\right),\begin{array}{cc}\begin{array}{cc}& \end{array}& other\end{array}\hfill \end{array}\right.$$

(23)

$$\mathrm{SmIndex}=\mathrm{sort}\left(\mathrm{R}\right)$$

(24)

3.2. Wrap Food

To update the slime mould location, the mathematical formula can be expressed as:

$$Y=\left\{\begin{array}{c}rand\cdot \left(ub-lb\right)+lb,\begin{array}{cc}\begin{array}{cc}& \end{array}& rand<z\end{array}\hfill \\ Y_b(t)+ub\cdot \left(V\cdot Y_A(t)-Y_B(t)\right),\begin{array}{cc}& r<p\end{array}\hfill \\ uc\cdot Y(t),\begin{array}{cc}\begin{array}{cccc}\begin{array}{cccc}& & & \end{array}& & & \end{array}& r\ge p\end{array}\hfill \end{array}\right.$$

(25)

3.3. Oscillation

The variations of slime mould will be simulated to find the food by the parameters $V, ub$, and $uc$. To select the best food source, the slime mould should be improved by the oscillation frequency, which is mathematically expressed by $V$. The velocity of slime mould to discover food depends on the concentration of food. If the concentration of food is high, then the velocity of slime mould will be faster, but the velocity of slime mould will be slow if the concentration of food is low. $ub$ oscillates in the interval $\left[-a, a\right]$ as randomly and decreased to zero when the iteration is increased. In addition, $uc$ oscillates in the range $\left[-1,1\right]$ and gradually decreases to zero when the iteration increases. The flowchart and Pseudo-code of SMA are expressed in Figure 1 and Algorithm 1.

Algorithm 1. Pseudo-code of the Slime Mould Algorithm (SMA)
1.	Select the values of parameters $pop\_size, Max\_t$
2.	Determine the positions of slime mode $Y_i \left(\mathrm{i} =1, 2,\dots , \mathrm{N}\right)$
3.	While $\left(t \le Max\_t\right)$
4.	Calculate $R\left(i\right)$ of all slime mode
5.	Update $bf,Y_b$
6.	Calculate $W$ Equation (23)
7.	For $each search portion$
8.	Update $p, ub, uc$
9.	Update positions by Equation (25)
10.	End (For)
11.	$t=t+1$
12.	End (While)
13.	Return $bf, Y_b$

4. Multi-Objective Slime Mould Algorithm (MOSMA)

The main purpose of using multi-objective optimization is to optimize two or more objectives simultaneously (mostly conflicting and non-commensurable objectives) in power systems. In MOSMA, the concept used for classification the dominated and non-dominated solutions, based on objective functions, is Pareto dominance. The fuzzy decision-maker is the strategy taken to extract the best compromise solution.

4.1. Pareto Optimization Approach

The Pareto optimization approach can solve the problems related to single-objective optimization directly. Therefore, it is much more difficult to determine a suitable solution for problems related to multi-objective optimization. One of the popular solutions to solve multi-objective optimization problems is simplified into single-objective optimization by determining the different weights of each objective and summing these objectives. Due to the conflicting objective functions, it is not easy to determine the optimal solution to multi-objective optimization problems. In other words, if a solution is not superior to other solutions based on the objective function, the solution is not dominated, then these solutions are called the non-dominated solutions. The Pareto dominates solutions achieved when:

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

(26)

where $F_i\left(X_1\right)$ and $F_i\left(X_2\right)$ are the $ith$ objective function values of solutions $X_1$ and $X_2$. Based on Equation (26), the solution $X_1$ should be dominated solution $X_2$ to satisfy Pareto front non-dominated solutions. In Pareto concept optimization, the fitness of objective function will be compared with each other for all solutions. The dominated solutions are achieved when the fitness of objective function for a solution is higher than the fitness of objective function of any other solutions. On the contrary, the obtained solutions called dominated solutions. The main aim of Pareto concept optimization is to obtain the set of non-dominated solutions of multi-objective optimization problems. Therefore, the solutions obtained by Pareto optimization do not represent the optimal solution of each objective function. The Pareto fronts non-dominated (PFND) solutions represent a set of non-dominated solutions that are plotted as curve in the solutions space.

The set of solutions that could not dominate each other are called Pareto optimal solutions. These sets will be continuously updated and stored to solve multi-objective problems.

4.2. Best Compromise Solution

It necessary to unify the values of objective function in a similar range because there are different ranges. The incompatibility with the sets can be indicated as value 0, while the full compatibility indicated as value 0, as shown in Figure 2. Each objective function has a membership function as follows:

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

(27)

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

(28)

The maximum value of $u^k$ represents the best compromise solution [27].

4.3. Phases of MOSMA

The stages of MOSMA can be summarized as follows:

Step 1: Initialize the main parameters, such as the no. of population, no. of control variables, no. of non-dominated solutions, max iterations, etc…

Step 2: Initialize the population of slime mould.

Step 3: Calculate the fitness function of each individual of the initial population.

Step 4: Sort the initial population according on the fitness function of each individual and save the non-dominated solutions into initial repository.

Step 5: Calculate the best and the worst compromise solution of the initial population according to Equation (28).

Step 6: Calculate the weight of slime mould of non- dominant solutions by (24).

Step 7: Calculate the parameter $a$ by (22).

Step 8: Update the position of slime mould according to (25).

Step 9: Update the position of the slime mould to be within lower or upper bounds.

Step 10: Calculate the fitness value of each slime mould position.

Step 11: Sort the non-dominated solutions of slime mould position and store them in the slime mould repository.

Step 12: Combined the non-dominated solutions in the initial repository with the non-dominated solutions in the slime mould repository to find new non-dominated solutions.

Step 13: Verify the stopping criteria and check if the new non-dominated solutions are equal to or more than the number of non-dominated that have been suggested. If this is the case, then the program will end. Otherwise, store the new non-dominated solutions as in the initial repository and return to Step 5.

The process of the multi-objective slime mould algorithm (MOSMA) to solve multi-objective optimal power flow can be described in the flowchart in Figure 3

4.4. Crowding Distance

A crowding distance is a strategy used to reduce the non-dominate solutions by calculating the average distance for two neighbouring solutions. First, the fitness value of the objective function must be sorted in ascending order depending on the nearest neighbours. Then, the fitness values of the boundary solutions are evaluated as the infinite distance value. The fitness value of the intermediate solutions is equal to the distance of the corresponding diagonal length. Figure 4 represents the diagonal length of the cuboid to calculate the crowding distance. It is expressed as follows:

$$C{D}_i=\prod _{n=1}^m\frac{\left|F_n\left(X_i+1\right)-F_n\left(X_i-1\right)\right|}{F_{n,\min }},\begin{array}{ccc}& & i=1,2,\cdot \cdot \cdot \end{array},N_b$$

(29)

where $C{D}_i$ denotes the crowding distance, $F_{n, min}$ represents the minimum value of nth objective function, and $N_b$ is the number of candidate solutions.

5. Simulation Results

To demonstrate the effectiveness and performance of SMA to solve OPF problems, two standard systems (IEEE-30 bus system and IEEE 57-bus test system) and one practical system (Iraqi Super Grid High Voltage ISGHV 400 kV) were used to test with 29 cases for various objective functions. The characteristics of these systems are presented in

Table 1. The main characteristics of the systems applied in this study.

System Characteristics	IEEE-30	IEEE 57	ISGHV 400 kV
Buses	30	57	28
Branches	41	80	44
Generators	9 (Buses:1, 2, 5, 8, 11 and 13)	7 (Buses: 1–2–3–6–8–9–12)	14 (Buses: 1–14)
Generator voltage limits	0.9–1.1 [p.u.]	0.9–1.1 [p.u.]	0.9–1.1 [p.u.]
Load voltage limits	0.95–1.05 [p.u.]	0.95–1.05 [p.u.]	0.95–1.05 [p.u.]
Limit of tap changer setting	0.9–1.1 [p.u.]	0.9–1.1 [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]	5994 [MW]
Control variables	24	33	27

.

Table 2. Various case studies.

Type of System	Type of OF(s)	Case #	FC	Em	RPL	VD	VSI
IEEE 30-bus	Single OF(s)	Case #1	√
		Case #2			√
		Case #3		√
		Case #4				√
		Case #5					√
	Bi-OF(s)	Case #6	√	√
		Case #7	√		√
		Case #8	√		√
		Case #9	√				√
		Case #10		√		√
		Case #11			√	√
		Case #12				√	√
	Triple-OF(s)	Case #13	√	√	√
		Case #14	√	√		√
		Case #15	√		√	√
		Case #16	√		√		√
		Case #17	√	√			√
		Case #18	√			√	√
		Case #19		√	√	√
	Quad-OF(s)	Case #20	√	√	√	√
		Case #21	√	√	√		√
	Quinta-OF(s)	Case #22	√	√	√	√	√
IEEE 57-bus	Single OF(s)	Case #23	√
		Case #24			√
		Case #25		√
ISGHV 400 kV (28 bus)	Single OF(s)	Case #26	√
		Case #27			√
		Case #28				√
		Case #29					√

describes the studies that have been applied.

5.1. IEEE 30-Bus Power System

The data of the IEEE 30 bus system are given in [34]. The main characteristics of IEEE 30-bus power system are given in Table 1. The coefficients of cost and emission of generators are given in

Table A1. The coefficients of cost and emission of generators for the IEEE 30-bus test 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

. Figure A1 represents the single-line diagram of the IEEE 30 bus test system.

5.1.1. Single-Objective OPF on IEEE 30-Bus Power System

Five objective functions were optimized to solve OPF problem— the generational fuel cost (GFC), real power loss (RPL), emission (Em), voltage deviation (VD), and voltage stability index (VSI)—by setting the parameters of the control variables (active power output of generators except for the slack bus, the voltage of PV bus, tap ratio of transformers, and shunt VAR compensator). In total, 1000 iterations and 250 population sizes were the values chosen in SMA to solve OPF problem. The finest settings of optimal control variables to find the optimal objective function for five cases are reported in

Table 3. Optimal control variables obtained by SMA for Cases 1–5.

Item		Limit		Initial [35]	Case 1	Case 2	Case 3	Case 4	Case 5
		Max	Min
$P_g$ [MW]	${\mathrm{P}}_1$	50	200	99.223	176.9638	52.66583	67.23830	122.9787	153.8864
	${\mathrm{P}}_2$	20	80	80	48.54495	79.60434	71.62589	45.29825	23.85954
	${\mathrm{P}}_5$	15	50	50	21.21892	49.92812	49.99828	47.87340	36.47932
	${\mathrm{P}}_8$	10	35	20	21.38800	34.60913	34.97310	29.73724	34.96892
	${\mathrm{P}}_{11}$	10	30	20	11.92001	29.92682	29.99790	21.98221	20.91042
	${\mathrm{P}}_{13}$	12	40	20	12.02793	39.65335	33.16002	21.74010	19.63984
$V_g$ [p.u.]	${\mathrm{V}}_1$	0.95	1.1	1.05	1.09999	1.09997	1.07207	0.99759	1.09925
	${\mathrm{V}}_2$	0.95	1.1	1.04	1.09786	1.09672	1.09211	0.98059	1.09803
	${\mathrm{V}}_5$	0.95	1.1	1.01	1.08201	1.09541	1.03596	1.06050	1.08448
	$V_8$	0.95	1.1	1.01	1.08952	1.08871	1.04344	1.04797	1.08762
	${\mathrm{V}}_{11}$	0.95	1.1	1.05	1.09966	1.09989	1.08511	1.09352	1.09901
	${\mathrm{V}}_{13}$	0.95	1.1	1.05	1.09959	1.09955	1.04750	1.06366	1.09924
Shunt Element [MVAr]	${\mathrm{Q}}_{\mathrm{c}10}$	0	5	0	3.47456	4.90651	0.01712	4.99757	4.17957
	${\mathrm{Q}}_{\mathrm{c}12}$	0	5	0	3.78440	4.71029	0.14186	0.03554	4.07682
	${\mathrm{Q}}_{\mathrm{c}15}$	0	5	0	2.76611	3.61236	0.02957	1.18163	4.17146
	${\mathrm{Q}}_{17}$	0	5	0	2.99716	0.01457	0.00000	0.72789	4.83945
	${\mathrm{Q}}_{\mathrm{c}20}$	0	5	0	4.57258	4.58251	2.02777	4.97550	2.86524
	${\mathrm{Q}}_{21}$	0	5	0	4.98153	4.04716	0.00000	4.98735	3.04483
	${\mathrm{Q}}_{\mathrm{c}23}$	0	5	0	2.50862	0.01196	0.00457	4.95938	4.94036
	${\mathrm{Q}}_{24}$	0	5	0	4.99707	4.28293	3.02812	4.96797	4.95660
	${\mathrm{Q}}_{29}$	0	5	0	2.10969	3.95118	0.63863	3.30036	4.99499
Tap Position	${\mathrm{T}}_{11}$	0.9	1.1	1.078	1.03080	1.04738	1.05597	1.01913	1.05077
	${\mathrm{T}}_{12}$	0.9	1.1	1.069	1.03860	1.03586	1.00876	1.04096	1.01569
	${\mathrm{T}}_{15}$	0.9	1.1	1.032	0.95241	0.99520	1.01355	0.95167	1.04096
	${\mathrm{T}}_{36}$	0.9	1.1	1.068	0.98279	1.00199	1.00406	0.97008	0.95084
GFC [USD/h]				901.639	799.2557	964.5746	936.1166	868.0514	834.0165
RPL [MW]				5.6891	8.6691	2.9934	3.5935	6.2099	6.3446
Em [ton/h]				0.2253	0.3681	0.2213	0.2175	0.2569	0.31
VD [p.u.]				1.1747	1.4192	1.4677	0.4754	0.1097	1.7545
VSI				0.1727	0.1237	0.1229	0.1483	0.1371	0.1136
Reduction rate				-	11.36%	47.38%	3.46%	90.66%	34.23%

The optimal values of the objective functions are given in bold.

.

• Case #1: To the optimal operating point, the generation fuel cost (GFC) used to operate generation units should be minimized by setting the control variables. Therefore, the first objective function of this study is to achieve generation fuel cost (GFC) minimization using the Slime Mould Algorithm (SMA). Table 3 illustrated that generation fuel cost has (GFC) been minimized to 799.2557 (USD/h). The remainder results of this case are 8.6691 (MW), 0.3681 (ton/h), 1.4192 (p.u.), and 0.1237 of RPL, Em, VD, and VSI.
• Case #2: The second objective function is to minimize the real power losses (RPL) in the transmission lines. The real power losses (RPL) will be reduced from 5.6891 (MW) at the initial case to 2.9934 (MW) at the optimal case with a reduction rate equal to 47.38%. The values of GFC, Em, VD, and VSI of this case are equal to 964.5746 (USD/h), 0.2213 (ton/h), 1.4677 (p.u.), and 0.1229, respectively.
• Case #3: Recently, emission (Em) studies have received growing attention due to environmental pollution and global warming. In this case, reducing emissions is the primary aim of the proposed SMA. The best result obtained by SMA to reduce emission is 0.2175 (ton/h). The reduction rate between the optimal case (0.2175(ton/h)) and the initial case (0.2253 (ton/h)) is 3.46%.
• Case #4: The fourth objective function in this paper is voltage deviation (VD) minimization. The aim of this minimizing is to improve voltage profiles at each load bus. The reduction rate of this case is 90.66% (compared between the initial case, which is 1.1747 (p.u.), and the optimal case, which is 0.1097 (p.u.)). The rest values of this case are 868.0514 (USD/h), 6.2099 (MW), 0.2569 (ton/h), and 0.1371 of GFC, RPL, Em, and VSI, respectively.
• Case #5: To achieve more stability of whole power system, the voltage stability indicator (${\mathrm{L}}_{\max }$) was minimized using the SMA. In this case, the VSI was minimized from 0.1727 at initial case to 0.1136 at the optimal case with a reduction rate equal to 34.23%. The remainder results of GFC, RPL, Em, and VD are 834.0165 (USD/h), 6.3446 (MW), 0.31 (ton/h), and 1.7545 (p.u.), respectively, as given in Table 3.

To demonstrate the effectiveness and superiority of SMA, the results of the objective function for five cases abstained by proposed approach SMA were compared with other meta-heuristics algorithm results reported in the literature.

Table 4. Comparison of the results obtained by the SMA with other methods for Cases (1 and 2).

Case 1		Case 2
Method	GFC [USD/h]	Method	RPL (MW)
Initial	901.6391	Initial	5.830
MSA [16]	800.5099	GWO [30]	3.51
SCA [36]	800.1018	SSA [30]	3.50
SSO [37]	802.2580	WOA [30]	3.50
DSA [38]	800.3887	MF [30]	3.50
JAYA [39]	800.479	HHO [30]	3.49
GPU-PSO [40]	800.53	SSO [37]	3.8239
SP-DE [41]	800.4131	EM [42]	3.1775
MGOA [43]	800.4744	EGA-DQLF [44]	3.2008
TLBO [45]	800.4604	ASO [46]	3.1600
AMTPG-Jaya [45]	800.1946	EGA-EA [47]	3.2601
GWO [48]	802.7924	GWO [48]	4.2905
ABC [49]	800.6850	PSO [50]	5.1957
IABC [49]	800.4215	HPSO-DE [50]	5.1476
EGA [51]	802.06	FAHSPSO-DE [50]	4.9989
IGA [52]	800.805	IPSO [53]	5.0732
AGAPOP [54]	799.8441	SMA	2.9934
ABC [55]	800.66
PSOGSA [56]	800.49859
GA [57]	800.5272
IHS [57]	800.5202
MFO [57]	800.7134
ISSA [57]	800.4752
SOS [58]	801.5733
SMA	799.2557

The values obtained by the proposed algorithm are given in bold.

and

Table 5. Comparison of the results obtained by the SMA with other methods for Cases (3, 4, and 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
GWO [30]	0.2960	HFPSO [59]	0.1467	SSO [37]	0.1267
SSA [30]	0.2950	EJADE-SP [60]	0.3752	NISSO [37]	0.12547
WOA [30]	0.2950	MABC [61]	0.1292	Jaya [39]	0.1243
MF [30]	0.2950	SMA	0.1097	TLBO [45]	0.12444
HHO [30]	0.2850			AMTPG-Jaya [45]	0.1240
SSO [37]	0.2315			ARCBBO [62]	0.1369
BSA [63]	0.2425			ECHT-DE [64]	0.13632
SMA	0.2175			SPEA [65]	0.1247
				DE [66]	0.1246
				SMA	0.1136

The values obtained by the proposed algorithm are given in bold.

prove the effectiveness of performance SMA over the other recent algorithms. Figure 5a–e illustrates the convergence speed of single objective functions using the SMA optimization method to solve the OPF problem in the IEEE 30 bus test system. These figures prove the superiority and efficiency of the proposed approach by providing a good characteristics rate.

5.1.2. Multiple-Objective OPF on IEEE 30-Bus Power System

To demonstrate the performance of the SMA on multi-objective optimization problems, MOSMA was applied to solve MOOPF problems. The number of populations is 500, and the stopping criteria of simulation running when no. of non-dominated solutions is equal to 500 or the no. of iterations equal to 500.

First, seven bi-objective functions (Cases 6–12) were studied and presented as follows:

• Case #6: In this case, MOSMA was applied to optimize the GFC and Em simultaneously. The best compromise solutions of GFC and Em are (832.8647 (USD/h), 0.2514 (ton/h)).
• Case #7: The GFC and RPL were optimized simultaneously. The best compromise solutions of GFC and RPL are (840.960 (USD/h), 4.8762 (MW)).
• Case #8: In Case 8, the two objectives function that were minimized and considered simultaneously are GFC and VD. The best compromise solution of this case is 802.0533 ($/h) and 0.3267 (p.u.) of GFC and VD, respectively.
• Case #9: The minimum GFC and VSI were simultaneously considered. The best compromise solutions obtained by MOSMA in this case are 800.1309 (USD/h) and 0.1172.
• Case #10: This case shows the minimization of Em and VD simultaneously. The proposed MOSMA provided the best compromise solutions, which are 0.2184 (ton/h) and 0.2074 (p.u.).
• Case #11: The RPL and VD were optimized simultaneously. The best compromise values obtained by the proposed MOSMA of RPL, and VD are 3.1922 (MW) and 0.5417 (p.u.), respectively.
• Case #12: The last case of the bi-objective functions type is the minimization of VD and VSI simultaneously. The proposed MOSMA provided the best compromise values for VD and VSI, which are 0.3102 (p.u.) and 0.1284, respectively.

The Pareto front, according to non-dominated solutions obtained by the proposed MOSMA of bi-objective functions, is shown in Figure 6a–g.

The Triple-objective functions are considered and presented as follows:

• Case #13: The GFC, RPL, and Em were presented as objective functions to optimize simultaneously. The best compromise of GFC, RPL, and Em obtained by the proposed MOSMA are 867.5282 (USD/h), 4.3416 (MW) and 0.2300 (ton/h), respectively.
• Case #14: In this case, the minimization of GFC, Em, and VD have been optimized simultaneously as objective functions. The best compromising GFC, Em, and VD obtained by the proposed approach are 841.554 (USD/h), 0.2531 (ton/h) and 0.2214 (p.u.), respectively.
• Case #15: In Case 15, the three objectives function that were minimized considered simultaneously are GFC, RPL, and VD. The best compromise values of this case are 844.6107 (USD/h), 6.0058 (MW), and 0.2279 (p.u.) of GFC, RPL, and VD, respectively.
• Case #16: The minimization of GFC, RPL, and VSI were optimized simultaneously. The best compromising of GFC, RPL, and VSI obtained by the proposed MOSMA are 841.4057 (USD/h), 5.0766 (MW) and 0.1176, respectively.
• Case #17: This case shows the minimization of GFC, Em, and VSI simultaneously. The proposed MOSMA provided the best compromise values, which are 850.7178 (USD/h), 0.2476 (ton/h), and 0.1158.
• Case #18: The GFC, VD, and VSI are optimized simultaneously. The best compromise values obtained by proposed MOSMA of GFC, VD, and VSI are 804.4035 (USD/h), 0.5409 (p.u.), and 0.1291 respectively.
• Case #19: The last case of triple-objective functions type is the minimization of Em, RPL, and VD simultaneously. The proposed MOSMA provided the best compromise values for Em, RPL, and VD are 0.2183 (ton/h), 3.9925 (MW) and 0.2414 (p.u.), respectively.

The Pareto front, according to non-dominated solutions obtained by the proposed MOSMA of triple-objective functions, is shown in Figure 7a–g. The red diamonds indicate the best compromise solution of the triple objective functions for Cases 13–19.

Finally, the Quad and Quinta-objective functions are considered and presented as follows:

• Case #20: In this case, the GFC, Em, RPL, and VD were optimized simultaneously. The best compromise solutions obtained by the developed framework are 832.3665 (USD/h), 0.2675 (ton/h), 6.4495 (MW) and 0.2189 (p.u.), of GFC, Em, RPL, and VD, respectively.
• Case #21: The GFC, Em, RPL, and VSI were optimized simultaneously. The best compromising of GFC, Em, RPL, and VSI obtained by the proposed MOSMA are 847.723 (USD/h), 0.2466 (ton/h), 5.1423 (MW) and 0.1183, respectively.
• Case #22: The last case of the multiple-objective functions type is the minimization of GFC, Em, RPL, VD, and VSI simultaneously. The proposed MOSMA provided the best compromise values for GFC, Em, RPL, VD, and VSI, which are 824.7751 (USD/h), 0.2753 (ton/h), 6.3599 (MW), 0.5111 (p.u.) and 0.1290, respectively.

Table 6. Optimal control variables obtained by MOSMA for Cases 6–13.

Item		Case 6	Case 7	Case 8	Case 9	Case 10	Case 11	Case 12	Case 13
$P_g$ [MW]	${\mathrm{P}}_1$	126.521	139.156	177.458	177.570	70.974	58.480	170.349	92.703
	${\mathrm{P}}_2$	57.394	48.924	46.455	50.162	70.791	77.896	70.719	68.717
	${\mathrm{P}}_5$	26.635	28.495	20.722	19.629	49.648	48.772	17.612	37.093
	${\mathrm{P}}_8$	34.242	34.955	21.449	18.869	34.749	33.597	19.081	34.532
	${\mathrm{P}}_{11}$	24.051	21.583	12.920	14.099	29.757	29.299	10.539	29.626
	${\mathrm{P}}_{13}$	20.555	16.447	14.204	12.017	32.509	39.018	14.182	25.070
$V_g$ [p.u.]	${\mathrm{V}}_1$	1.098	1.100	1.045	1.099	0.988	1.042	0.958	1.097
	${\mathrm{V}}_2$	1.098	1.100	1.045	1.099	1.011	1.048	1.097	1.097
	${\mathrm{V}}_5$	1.098	1.100	1.045	1.099	1.079	1.017	0.959	1.097
	${\mathrm{V}}_8$	1.098	1.100	1.045	1.099	1.056	1.048	1.097	1.097
	${\mathrm{V}}_{11}$	1.098	1.100	1.045	1.099	1.098	1.095	1.035	1.097
	${\mathrm{V}}_{13}$	1.098	1.100	1.045	1.099	1.098	1.095	1.090	1.097
Shunt Element [MVAr]	${\mathrm{Q}}_{\mathrm{c}10}$	0.001	1.014	0.279	0.047	0.001	0.001	0.486	1.097
	${\mathrm{Q}}_{\mathrm{c}12}$	0.055	0.316	1.045	1.812	0.057	0.000	0.765	1.097
	${\mathrm{Q}}_{\mathrm{c}15}$	0.000	0.326	0.440	0.116	0.004	0.005	0.179	0.124
	${\mathrm{Q}}_{17}$	0.485	0.469	0.015	0.229	0.000	0.000	1.097	0.976
	${\mathrm{Q}}_{\mathrm{c}20}$	1.675	4.991	2.378	4.926	0.000	0.000	4.915	2.925
	${\mathrm{Q}}_{21}$	0.070	4.991	2.846	4.975	0.144	0.000	4.915	2.095
	${\mathrm{Q}}_{\mathrm{c}23}$	0.325	4.991	2.916	4.972	0.000	0.007	4.915	3.283
	${\mathrm{Q}}_{24}$	0.003	4.991	1.608	4.972	0.001	1.029	4.915	2.774
	${\mathrm{Q}}_{29}$	0.043	2.394	0.775	4.972	1.335	0.000	4.915	0.822
Tap Position	${\mathrm{T}}_{11}$	1.023	0.999	1.007	1.005	1.014	1.022	1.063	1.027
	${\mathrm{T}}_{12}$	0.967	0.968	0.977	0.968	0.976	0.961	1.028	1.041
	${\mathrm{T}}_{15}$	1.006	0.989	0.962	1.022	0.965	1.006	0.971	0.952
	${\mathrm{T}}_{36}$	0.987	0.992	0.952	0.952	0.952	0.954	0.951	0.980
GFC [USD/h]		832.8647	840.960	802.0533	800.1309	935.1719	965.5084	866.5512	867.5282
RPL [MW]		5.4534	4.8762	9.6647	8.9461	5.0218	3.1922	15.0962	4.3416
Em [ton/h]		0.2514	0.2465	0.3826	0.3698	0.2184	0.2214	0.2713	0.2300
VD [p.u.]		0.9403	1.5133	0.3267	1.6453	0.2074	0.5417	0.3102	1.3322
VSI		0.1329	0.1243	0.1447	0.1172	0.1452	0.1370	0.1284	0.1288

The best compromise solutions for objective functions are given in bold.

and

Table 7. Optimal control variables obtained by MOSMA for Cases (14–22).

Item		Case 14	Case 15	Case 16	Case 17	Case 18	Case 19	Case 20	Case 21	Case 22
$P_g$ [MW]	${\mathrm{P}}_1$	113.137	118.406	117.802	103.918	170.859	71.869	129.158	111.560	135.012
	${\mathrm{P}}_2$	72.911	57.143	46.295	79.270	47.625	72.514	56.560	61.157	43.921
	${\mathrm{P}}_5$	28.035	36.667	34.245	21.055	25.984	49.962	36.327	34.110	29.727
	${\mathrm{P}}_8$	30.461	25.476	34.859	34.696	16.724	34.973	25.233	22.732	30.329
	${\mathrm{P}}_{11}$	24.979	22.381	29.887	29.757	17.284	29.979	22.187	29.610	24.467
	${\mathrm{P}}_{13}$	20.555	29.333	25.389	20.181	13.752	28.095	20.384	29.372	26.304
$V_g$ [p.u.]	${\mathrm{V}}_1$	1.023	1.043	1.099	1.098	1.094	1.040	1.041	1.097	1.078
	${\mathrm{V}}_2$	0.985	1.043	1.099	1.098	1.094	1.027	1.041	1.097	1.073
	${\mathrm{V}}_5$	1.003	1.043	1.099	1.098	1.094	0.997	1.041	1.097	1.052
	${\mathrm{V}}_8$	1.074	1.043	1.099	1.098	1.094	1.009	1.041	1.097	1.064
	${\mathrm{V}}_{11}$	1.074	1.043	1.095	1.098	1.094	1.100	1.041	1.097	1.016
	${\mathrm{V}}_{13}$	1.061	1.043	1.099	1.098	0.968	1.036	1.041	1.097	1.039
Shunt Element [MVAr]	${\mathrm{Q}}_{\mathrm{c}10}$	1.385	1.043	0.558	0.024	0.154	0.006	0.220	0.464	3.800
	${\mathrm{Q}}_{\mathrm{c}12}$	0.294	1.043	1.034	0.316	0.127	0.043	1.041	0.965	1.808
	${\mathrm{Q}}_{\mathrm{c}15}$	4.090	1.043	1.081	1.098	0.349	0.009	0.557	0.069	3.370
	${\mathrm{Q}}_{17}$	1.014	1.043	0.246	0.875	0.465	0.001	0.859	0.000	4.664
	${\mathrm{Q}}_{\mathrm{c}20}$	1.994	3.095	4.972	4.939	1.710	3.970	1.570	0.877	3.839
	${\mathrm{Q}}_{21}$	0.010	3.095	4.227	4.939	0.083	0.792	2.404	4.903	4.180
	${\mathrm{Q}}_{\mathrm{c}23}$	0.053	3.095	4.972	4.939	1.400	0.076	3.047	4.903	1.270
	${\mathrm{Q}}_{24}$	2.928	3.095	4.972	4.939	4.289	0.755	2.089	4.575	1.575
	${\mathrm{Q}}_{29}$	0.210	3.095	4.919	4.939	4.341	3.209	3.047	4.903	4.640
Tap Position	${\mathrm{T}}_{11}$	0.967	0.963	0.991	1.037	0.963	0.988	1.036	1.000	1.021
	${\mathrm{T}}_{12}$	1.049	0.985	1.029	0.990	1.097	0.971	0.979	0.957	0.990
	${\mathrm{T}}_{15}$	0.981	1.050	1.024	0.999	0.958	0.995	0.961	0.959	0.993
	${\mathrm{T}}_{36}$	0.953	0.953	0.963	0.951	0.952	0.961	0.965	0.954	0.956
GFC [USD/h]		841.554	844.611	841.4057	850.7178	804.4035	928.292	832.3665	847.723	824.7751
RPL [MW]		6.6773	6.0058	5.0766	5.4772	8.8206	3.9925	6.4495	5.1423	6.3599
Em [ton/h]		0.2531	0.2545	0.2502	0.2476	0.3495	0.2183	0.2675	0.2466	0.2753
VD [p.u.]		0.2214	0.2279	1.6214	1.7186	0.5409	0.2414	0.2189	1.6979	0.5111
VSI		0.1436	0.1370	0.1176	0.1158	0.1291	0.1440	0.1408	0.1183	0.1290

The best compromise solutions for objective functions are given in bold.

present the results of optimal control variables and objective functions obtained by the proposed MOSMA for Cases 6–22.

Figure 8 shows the voltage profiles of all buses for Cases 1–22. Figure 8a shows the voltage profiles of all buses for Cases 1–5. Figure 8a proves that the proposed algorithm is effective only in Casess 3 and 4, but the results extracted from Cases 1, 2, and 5 are infeasible solutions because many load buses have voltage values exceeding the maximum limit for load buses, 1.05 (p.u.). Figure 8b shows the voltage profiles of the bi-objective function for Cases 6–12. Figure 8b proves that the proposed algorithm is effective only in cases in which the voltage deviation is considered an objective function, namely in Cases 8, 10, 11, and 12. However, the results obtained from cases that the voltage deviation is not considered an objective function for Cases 6, 7, and 9 are infeasible solutions because many load buses have voltage values exceeding the maximum limit for load buses, 1.05 (p.u.). Figure 8c shows the voltage profiles of the triple-objective function for Cases 13–19. The proposed approach was effective for Cases 14, 15, 18, and 19. The cases that the voltage load bus exceeded the maximum limit for load buses, 1.05 (p.u.), are 13, 16, and 17. Figure 8d shows the voltage profiles of Quad and Quinta objective function for Cases 20–22. The results obtained by Case 21 are infeasible solutions because many load buses have voltage values exceeding the maximum limit for load buses, which is 1.05 (p.u.). The results obtained from cases (20,22) are effective because the value of the voltage load bus is within the range [0.95–1.05] (p.u.).

Figure 9 illustrates the reactive power output of generation units. From this figure, it can be observed that generator 1 exceeded the limits [ −20 to 200 MVAr] in Cases 4, 5, 6, 10, 12, 16, and 17, while in the remaining cases, the constraints were not violated. The generators (2 and 8) violated the constraints in Case 12 only. The remaining generators (5, 11, and 13) satisfied all constraints in this study.

5.2. IEEE 57-Bus Power System

In this subsection, the IEEE 57- bus power system was applied to validate of performance of proposed MOSMA. The total generation capacity of this system is 1975.9 MW [34]. The main characteristics of IEEE 57- bus power system is given in Table 1. Figure A2 represents the single-line diagram of the IEEE 57 bus system. The coefficients of the cost and emission of generators are given in

Table A2. The coefficients of cost and emission of 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

.

Single-Objective OPF on IEEE 57-Bus Power System

To demonstrate the superiority and performance of the proposed method on IEEE 57- bus power system, three single objective functions (Cases 23–25) were considered. The number of iterations and population sizes are 1000 and 250, respectively.

Table 8. Optimal control variables obtained by the SMA algorithm for Cases 23-25 on the IEEE 57-bus system.

Item		Max	Min	Initial	Case 23	Case 24	Case 25
[MW]	$P_1$	0.0	576	478	142.26	188.27	188.26
	$P_2$	30.0	100	0	89.97	99.58	22.33
	$P_3$	40.0	140	40	45.18	139.66	139.83
	$P_6$	30.0	100	0	70.04	99.56	99.80
	$P_8$	100.0	550	450	463.46	276.00	300.14
	$P_9$	30.0	100	0	94.12	99.95	99.78
	$P_{12}$	100.0	410	310	359.91	362.06	409.74
[p.u.]	$V_1$	0.95	1.10	1.040	1.08	1.07	1.10
	$V_2$	0.95	1.10	1.010	1.08	1.08	1.10
	$V_3$	0.95	1.10	0.985	1.08	1.06	1.10
	$V_6$	0.95	1.10	0.980	1.09	1.05	1.10
	$V_8$	0.95	1.10	1.005	1.10	1.07	1.10
	$V_9$	0.95	1.10	0.980	1.09	1.08	1.09
	$V_{12}$	0.95	1.10	1.015	1.08	1.07	1.09
Tap Position	$T_{4-18}$	0.90	1.10	0.97	0.97	0.99	1.07
	$T_{4-18}$	0.90	1.10	0.978	1.00	1.02	0.98
	$T_{21-20}$	0.90	1.10	1.043	0.95	1.06	1.04
	$T_{24-25}$	0.90	1.10	1	0.97	1.04	1.01
	$T_{24-25}$	0.90	1.10	1	0.95	1.04	0.97
	$T_{24-26}$	0.90	1.10	1.043	0.97	1.05	1.02
	$T_{7-29}$	0.90	1.10	0.967	0.98	1.03	0.98
	$T_{34-32}$	0.90	1.10	0.975	0.98	0.97	1.01
	$T_{11-41}$	0.90	1.10	0.955	0.99	1.08	1.00
	$T_{15-45}$	0.90	1.10	0.955	1.00	1.05	0.96
	$T_{14-46}$	0.90	1.10	0.9	1.00	1.07	0.96
	$T_{10-51}$	0.90	1.10	0.93	1.01	1.03	0.97
	$T_{13-46}$	0.90	1.10	0.895	0.97	1.00	0.95
	$T_{11-43}$	0.90	1.10	0.958	0.96	1.04	0.98
	$T_{40-56}$	0.90	1.10	0.958	0.97	0.98	1.02
	$T_{39-57}$	0.90	1.10	0.98	0.96	1.03	1.04
	$T_{9-55}$	0.90	1.10	0.94	0.99	1.06	0.98
[MVAr]	$Q_{\mathrm{c}18}$	0.00	20.0	10	12.11	0.00	20.00
	$Q_{25}$	0.00	20.0	5.9	5.04	11.87	14.57
	$Q_{53}$	0.00	20.0	6.3	15.29	19.68	12.81
FC [USD/h]				51353	41633.61	45157	44911
Em [ton/h]				2.4129	1.3624	0.9595	1.1126
RPL [MW]				27.868	14.3018	14.7671	9.2874
VD [p.u.]				1.1264	2.4684	2.5298	3.8873
VSI					0.2561	0.2890	0.2145
Reduction rate				-	18.93%	60.24%	66.67%

The optimal values of objective function are given in bold.

represents the optimal setting of control variables and optimal results of objective functions obtained by proposed method. The convergence speed of proposed method is illustrated in Figure 10a–c.

• Case #23: The GFC is the objective function that has been considered. The minimum of GFC obtained by the proposed method is 41633.61 (USD/h). The remainder results of this case are 14.3018 (MW), 1.3624 (ton/h), 2.4684 (p.u.), and 0.2561 of RPL, Em, VD, and VSI, respectively. The reduction rate between the optimal case and the initial case is 18.93%.
• Case #24: The minimization of Em is the objective function of this case. The optimal result obtained from this case is 0.9595 (ton/h) using the proposed approach. The reduction rate of emission reaches 60.24% when compared to the initial case (2.4129 (ton/h)) and the optimal case (0.9595 (ton/h)). The rest values of GFC, RPL, VD, and VSI are 45157 (USD/h), 14.7671 (MW), 2.5298 (p.u.), and 0.2890, respectively.
• Case #25: The main objective function of this case is the reduction of RPL. The best result obtained by the proposed approach is 9.2874 (MW) with a reduction rate equal to 66.67%. The remainder values of GFC, Em, VD, and VSI are 44911 (USD/h), 1.1126 (ton/h), 3.8873 (p.u.), and 0.2145, respectively.

To prove the superiority and performance of the SMA on the IEEE 57-bus system, the optimal results of the objective function obtained by the proposed approach SMA were compared with the optimal results obtained by other recent optimization methods reported in the literature. Table 9 proved the effectiveness of proposed algorithm over the other recent algorithms. Figure 9a–c shows the convergence rate of single-objective functions using the SMA optimization method to solve the OPF problem in the IEEE 57-bus test system. These figures prove the superiority and efficiency of the proposed approach by providing a good characteristics rate.

5.3. Iraqi Super Grid High Voltage 400 kV (ISGHV 400 kV)

The Iraqi Super Grid High Voltage 400 kV (ISGHV 400 kV) has been chosen as practical system to prove the ability and applicability of the proposed SMA as shown in Figure A3 [67]. The main characteristics of ISGHV is given in Table 1. The bus number 01 (MUSP) represents the swing bus, and the total load demand is 5994 MW.

Table A3. The cost coefficients of generators for the ISGHV network.

Gen	a	b	c	Gen	a	b	c
1	275	0.35	0.0012	8	0	0	0
2	0	0	0	9	250	0.5	0.02
3	200	3.5	0.04	10	300	2.2	0.003
4	2581	2.155	0.05	11	200	0.652	0.002
5	1698	11.91	0.03	12	159	0.561	0.002
6	154	7.05	0.0136	13	120	0.8	0.0025
7	200	0.64	0.0017	14	685	3.1	0.0158

presents the cost coefficients of the ISGHV network. The nodes and lines data of the ISGHV (400 kV) [68] are detailed in

Table A4. The node’s data of ISGHV (400 kV).

Bus No.	Type	Bus Name	Voltage		Load		Generation		Q Injection
			Mag [p.u.]	Angle Deg.	MW	MVAr	MW	MVAr
1	Slack	MUSP	1.04	0	206	56	159.4	2347.4	0
2	PU	MMDH	1.015	9.525	0	0	690	−92.4	0
3	PU	GNENW	1.01	7.695	150	75	250	−120.3	0
4	PU	BAJP	1.02	6.74	125	93	406	−203.8	−155
5	PU	BAJG	1.02	6.792	0	0	591	−91	0
6	PU	KRK4	1.017	5.691	130	10	240	−194	−100
7	PU	QDSG	1.01	−0.773	0	0	735	115.6	0
8	PU	HDTH	1.02	2.645	200	50	203	−198.7	0
9	PU	MUSG	1.02	0.243	0	0	369	−2082.4	0
10	PU	KUTP	1.03	−0.654	0	0	478	−70	0
11	PU	GKHRP	1.025	1.618	0	0	600	−295.4	0
12	PU	NSRP	1.02	−2.117	423	101	775	−230.3	−100
13	PU	HRTHP	1.01	−9.663	155	72	332	69.9	−50
14	PU	KAZG	1.0096	−9.48	200	101	208	24.6	−50
15	PQ	MSL4	1.0119	7.252	650	302	0	0	−50
16	PQ	BGS4	1.0246	−0.671	0	0	0	0	0
17	PQ	BGW4	1.0084	−0.563	576	302	0	0	−150
18	PQ	BGE4	1.0075	−1.678	849	295	0	0	−50
19	PQ	BGN4	1.0086	−1.053	413	149	0	0	−50
20	PQ	AMN4	1.0183	−1.248	127	56	0	0	0
21	PQ	BGC4	1.0099	−0.671	50	182	0	0	0
22	PQ	DYL4	1.0029	−2.029	84	22	0	0	−50
23	PQ	KUT4	1.0249	−4.277	260	108	0	0	−100
24	PQ	QIM4	1.0132	1.222	109	40	0	0	−50
25	PQ	BAB4	1.033	0.041	308	185	0	0	0
26	PQ	KDS4	1.0316	−0.695	213	152	0	0	−50
27	PQ	AMR4	0.9988	−10.44	311	161	0	0	−100
28	PQ	BSR4	1.0052	−10.26	455	145	0	0	0

and

Table A5. The line data of ISGHV (400 kV).

Line	Bus		Line	R [p.u.]	X [p.u.]	Charging [p.u.]
	From	To
L1	15	2	2	0.00144	0.01177	0.36439
L2	15	3	1	0.001777	0.016154	0.478634
L3	15	4	1	0.0042	0.03437	1.06426
L4	15	6	1	0.004984	0.04531	1.34251
L5	3	4	1	0.003294	0.02994	0.887224
L6	4	5	1	0.00002	0.0002	0.00584
L7	4	17	2	0.00483	0.04393	1.30165
L8	4	8	1	0.00345	0.03132	0.92808
L9	5	6	1	0.0018	0.01635	0.48447
L10	6	18	1	0.00496	0.04511	1.333667
L11	17	19	1	0.00093	0.00847	0.25099
L12	17	21	1	0.000607	0.005516	0.163436
L13	17	8	1	0.005049	0.045901	1.360021
L14	16	20	2	0.00082	0.00749	0.22181
L15	16	21	1	0.000953	0.00866	0.25682
L16	16	1	1	0.00122	0.01015	0.31897
L17	16	9	1	0.001094	0.009106	0.286176
L18	16	26	1	0.00308	0.02795	0.82827
L19	18	19	1	0.00029	0.00262	0.07763
L20	18	20	1	0.00043	0.00394	0.11674
L21	18	22	1	0.00087	0.00788	0.23348
L22	19	7	2	0.00015	0.00138	0.04086
L23	20	10	1	0.002427	0.022064	0.653744
L24	23	10	1	0.001734	0.01576	0.46696
L25	23	12	1	0.00432	0.03928	1.1639
L26	23	27	1	0.00479	0.04354	1.28998
L27	8	24	1	0.00292	0.02391	0.74035
L28	1	9	1	0.000125	0.001043	0.032791
L29	1	25	2	0.00081	0.00673	0.21165
L30	25	11	1	0.000898	0.00736	0.227
L31	25	26	1	0.00233	0.01935	0.60812
L32	11	26	1	0.002267	0.01857	0.5752
L33	26	12	1	0.00383	0.03485	1.03256
L34	12	14	1	0.00439	0.03993	1.18316
L35	27	13	1	0.0029	0.0264	0.78216
L36	13	14	2	0.00118	0.01076	0.3187
L37	13	28	1	0.000672	0.006107	0.180947
L38	14	28	1	0.000563	0.005122	0.151762
L39	15	2	2	0.00144	0.01177	0.36439

respectively.

Single-Objective OPF on ISGHV 400 kV

• Case #26: The objective function in this case is the minimization of GFC of the ISGHV network. The GFC was reduced from 39565 (USD/h) in the initial case to 20740 (USD/h) in the optimal case, with a reduction rate equal to 47.58%. The GFC, VD, and VSI are equal to 45.2254 (MW,) 0.4678 (p.u.) and 0.0826, respectively, as tabulated in Table 10.
• Case #27: In this case, the objective function that was optimized is RPL by means of SMA. The optimal result of RPL was reduced to 18.6087 (MW) compared with the initial case, which was equal to 42.3834 (MW) The reduction rate of this case is 56.09%. The rest results of GFC, VD, and VSI are equal to 36784 (USD/h), 0.7725 (p.u.) and 0.0815, respectively, as tabulated in Table 10.
• Case #28: The third objective function of this subsection is to improve the voltage profiles by minimizing the voltage deviation (VD) at the load bus from 1.0 (p.u.) The VD was minimized from 0.2013 (p.u.) in the initial case to 0.0625 (p.u.) in the optimal case. The reduction rate of VD is 68.95%.

The remainder results of GFC, RPL, and VSI are equal to 49843 (USD/h), 54.8430 (MW), and 0.0917, respectively, as tabulated in Table 10.

• Case #29: The last case of this article is the voltage stabilization reinforcement by minimization of the maximum voltage stability index (L-index) of the system load buses. The VSI is minimized to 0.0749 in the optimal case compared with 0.0886 in the initial case. The reduction rate of VSI is 15.46%. The GFC, RPL, and VD are equal to 45500 (USD/h), 19.9760 (MW), and 1.2955 (p.u.), respectively, as tabulated in Table 10.

Figure 11a–d illustrates the convergence rate single-objective functions using the SMA optimization method to solve the OPF problem in the ISGHV 400 kV network. The optimal control variables obtained by the proposed method provide the optimal objective functions tabulated in Table 10.

5.4. Performance Comparison

This subsection presents the performance and efficiency of the proposed approach (SMA and MOSMA) to solve single- and multi-objective optimal power flow problems. The standards and all their variants were evaluated to solve real-world problems. The SMA and MOSMA were carried out on all cases to achieve the best solutions for single- and multi-objective problems. The researchers faced two main challenges to solve single- and multi-OPF problems: the speed convergence toward the global optimum (single and multi OF) and the good distribution of the Pareto front (Multi OF). In other words, the balance between convergence and coverage should be found to determine the effectiveness of the algorithm. For example, the results obtained by the proposed algorithm from Cases 1–5 were compared with other recent algorithms as shown in Table 4, Table 5 and Table 9. These results confirmed the efficiency and superiority of the proposed algorithm. It is worth mentioning that none of the meta-heuristics algorithms can be superior to all optimization algorithms in solving all optimization problems, according to the no free lunch theorem (NFL) [69]. This is the main reason leading to no superior algorithm on all sides (coverage and convergence). This is very clear when applying the proposed approach to multi-objective functions. Therefore, it is difficult to compare the proposed approach MOSMA with other methods in terms of the results.

Table 9. Comparison of the results obtained by SMA with other methods from Cases 23-25 on the IEEE 57-bus system.

Case 23		Case 24		Case 25
Method	FC [USD/h]	Method	Em [ton/h]	Method	RPL [MW]
Initial	51353	Initial	2.4129	Initial	27.8683
TLBO [43]	41683	PSO [27]	1.19	PSO [27]	13.6673
GA [43]	41685	GA [27]	1.189	GA [27]	13.3983
GOA [43]	41680	Jaya [27]	1.1111	ABC [49]	12.6260
ABC [55]	41694	SSO [37]	1.7024	MICA [70]	11.8826
MO-DEA [71]	41683	NISSO [37]	1.03927	SMA	9.2874
GSA [72]	41695	ABC [49]	1.2048
NPSO [73]	41699.52	IABC [49]	1.0484
KHA [74]	41709.3	MICA [70]	1.2246
EADDE [75]	41713.6	MTLBO [76]	1.0772
fuzzy GA [77]	41716.3	GBBICA [78]	1.1724
SMA	41633.61	SKHA [79]	1.08
		SMA	0.9595

The optimal values of objective function are given in bold.

Table 9. Comparison of the results obtained by SMA with other methods from Cases 23-25 on the IEEE 57-bus system.

Case 23		Case 24		Case 25
Method	FC [USD/h]	Method	Em [ton/h]	Method	RPL [MW]
Initial	51353	Initial	2.4129	Initial	27.8683
TLBO [43]	41683	PSO [27]	1.19	PSO [27]	13.6673
GA [43]	41685	GA [27]	1.189	GA [27]	13.3983
GOA [43]	41680	Jaya [27]	1.1111	ABC [49]	12.6260
ABC [55]	41694	SSO [37]	1.7024	MICA [70]	11.8826
MO-DEA [71]	41683	NISSO [37]	1.03927	SMA	9.2874
GSA [72]	41695	ABC [49]	1.2048
NPSO [73]	41699.52	IABC [49]	1.0484
KHA [74]	41709.3	MICA [70]	1.2246
EADDE [75]	41713.6	MTLBO [76]	1.0772
fuzzy GA [77]	41716.3	GBBICA [78]	1.1724
SMA	41633.61	SKHA [79]	1.08
		SMA	0.9595

The optimal values of objective function are given in bold.

Based on the above, the simulation results obtained by the SMA and MOSMA for both single- and multi-objective functions have a high performance and provide high-quality solutions to solve OPF problems. The computational times of proposed approach (SMA and MOSMA) are competitive compared to other recent algorithms. In the multi-objective function and based on high-quality random search property of MOSMA, the objective functions (despite conflict with each other) provide the trade-off solutions among of each objective function. In Pareto fronts, the MOSMA provides good convergence, high efficiency, and a good distribution of two and three dimensions.

Table 10. Optimal control variables obtained by the SMA for four cases on the ISGHV network.

Item		Max	Min	Initial	Case 26	Case 27	Case 28	Case 29
[MW]	$P_1$	150	1200	159.383	1096.691	813.1153	1158.4575	888.0881
	$P_2$	130	988	690	987.9720	325.8517	447.2378	351.6373
	$P_3$	250	750	250	250.0141	384.4645	434.8705	284.2415
	$P_4$	120	1320	406	120.0362	395.7200	563.6893	305.5577
	$P_5$	120	636	591	120.0380	170.0734	310.7864	614.2338
	$P_6$	50	260	240	50.2201	73.2614	127.6456	65.0107
	$P_7$	180	910	735	732.5920	896.2568	449.9110	879.2067
	$P_8$	60	660	203	659.9335	427.6555	281.8446	166.4127
	$P_9$	50	500	369	67.3979	152.3144	216.3835	155.9652
	$P_{10}$	250	1320	478	250.0869	559.2488	645.6229	416.5426
	$P_{11}$	250	1250	600	583.3062	288.9408	619.7411	406.4446
	$P_{12}$	210	840	775	635.0647	837.6637	442.9369	813.4249
	$P_{13}$	100	440	332	435.5222	439.0323	225.7120	425.6579
	$P_{14}$	50	250	208	50.3508	248.9596	123.9482	241.5635
[p.u.]	$V_1$	0.95	1.1	1.04	1.0394	1.0592	1.0055	1.0937
	$V_2$	0.95	1.1	1.02	1.0305	1.0609	1.0055	1.0937
	$V_3$	0.95	1.1	1.01	1.0089	1.0586	1.0055	1.0937
	$V_4$	0.95	1.1	1.02	1.0182	1.0604	1.0055	1.0937
	$V_5$	0.95	1.1	1.02	1.0180	1.0604	1.0055	1.0937
	$V_6$	0.95	1.1	1.02	1.0281	1.0562	1.0055	1.0937
	$V_7$	0.95	1.1	1.01	1.0305	1.0567	1.0055	1.0937
	$V_8$	0.95	1.1	1.02	1.0286	1.0614	1.0055	1.0937
	$V_9$	0.95	1.1	1.02	1.0398	1.0582	1.0055	1.0937
	$V_{10}$	0.95	1.1	1.03	1.0507	1.0652	1.0055	1.0937
	$V_{11}$	0.95	1.1	1.03	1.0530	1.0594	1.0055	1.0937
	$V_{12}$	0.95	1.1	1.02	1.0647	1.0682	1.0055	1.0937
	$V_{13}$	0.95	1.1	1.01	1.0448	1.0459	1.0055	1.0937
	$V_{14}$	0.95	1.1	1.01	1.0448	1.0497	1.0055	1.0937
FC [USD/h]				39565	20740	36784	49843	45500
RPL [MW]				42.3834	45.2254	18.6087	54.8430	19.9760
VD [p.u.]				0.2013	0.4678	0.7725	0.0625	1.2955
VSI				0.0886	0.0826	0.0815	0.0917	0.0749
Reduction rate				-	47.58%	56.09%	68.95%	15.46%

The optimal values of the objective function are given in bold.

Table 10. Optimal control variables obtained by the SMA for four cases on the ISGHV network.

Item		Max	Min	Initial	Case 26	Case 27	Case 28	Case 29
[MW]	$P_1$	150	1200	159.383	1096.691	813.1153	1158.4575	888.0881
	$P_2$	130	988	690	987.9720	325.8517	447.2378	351.6373
	$P_3$	250	750	250	250.0141	384.4645	434.8705	284.2415
	$P_4$	120	1320	406	120.0362	395.7200	563.6893	305.5577
	$P_5$	120	636	591	120.0380	170.0734	310.7864	614.2338
	$P_6$	50	260	240	50.2201	73.2614	127.6456	65.0107
	$P_7$	180	910	735	732.5920	896.2568	449.9110	879.2067
	$P_8$	60	660	203	659.9335	427.6555	281.8446	166.4127
	$P_9$	50	500	369	67.3979	152.3144	216.3835	155.9652
	$P_{10}$	250	1320	478	250.0869	559.2488	645.6229	416.5426
	$P_{11}$	250	1250	600	583.3062	288.9408	619.7411	406.4446
	$P_{12}$	210	840	775	635.0647	837.6637	442.9369	813.4249
	$P_{13}$	100	440	332	435.5222	439.0323	225.7120	425.6579
	$P_{14}$	50	250	208	50.3508	248.9596	123.9482	241.5635
[p.u.]	$V_1$	0.95	1.1	1.04	1.0394	1.0592	1.0055	1.0937
	$V_2$	0.95	1.1	1.02	1.0305	1.0609	1.0055	1.0937
	$V_3$	0.95	1.1	1.01	1.0089	1.0586	1.0055	1.0937
	$V_4$	0.95	1.1	1.02	1.0182	1.0604	1.0055	1.0937
	$V_5$	0.95	1.1	1.02	1.0180	1.0604	1.0055	1.0937
	$V_6$	0.95	1.1	1.02	1.0281	1.0562	1.0055	1.0937
	$V_7$	0.95	1.1	1.01	1.0305	1.0567	1.0055	1.0937
	$V_8$	0.95	1.1	1.02	1.0286	1.0614	1.0055	1.0937
	$V_9$	0.95	1.1	1.02	1.0398	1.0582	1.0055	1.0937
	$V_{10}$	0.95	1.1	1.03	1.0507	1.0652	1.0055	1.0937
	$V_{11}$	0.95	1.1	1.03	1.0530	1.0594	1.0055	1.0937
	$V_{12}$	0.95	1.1	1.02	1.0647	1.0682	1.0055	1.0937
	$V_{13}$	0.95	1.1	1.01	1.0448	1.0459	1.0055	1.0937
	$V_{14}$	0.95	1.1	1.01	1.0448	1.0497	1.0055	1.0937
FC [USD/h]				39565	20740	36784	49843	45500
RPL [MW]				42.3834	45.2254	18.6087	54.8430	19.9760
VD [p.u.]				0.2013	0.4678	0.7725	0.0625	1.2955
VSI				0.0886	0.0826	0.0815	0.0917	0.0749
Reduction rate				-	47.58%	56.09%	68.95%	15.46%

The optimal values of the objective function are given in bold.

6. Conclusions

In this paper, a new meta-heuristic optimization algorithm inspired by the diffusion and foraging conduct of slime mould, called the Slime Mould Algorithm, was proposed to solve single- and multi-objective OPF problems. The objective functions that were considered are the generation fuel cost (GFC), real power losses (RPL) in the transmission lines, total emission (Em) issued by fossil-fuelled generation units, voltage deviation (VD) at buses, and voltage stability index (VSI) of whole system. The Pareto concept is the approach proposed to solve multi-objective OPF problems by determining the set of non-dominated solutions (Pareto front). The theory used to extract the best compromise solution is fuzzy set theory. In multiple-objective functions, MOSMA was developed to find optimal solution for two to five conflicting objective functions simultaneously. To validate the MOSMA performance, three different power systems were applied, two standard IEEE test systems (IEEE 30-bus and IEEE 57- bus power systems) and one practical system (Iraqi Super Grid High Voltage 400 kV), with 29 case studies of single- and multi-objectives functions. The simulations results confirmed that the convergence speed of SMA is impressive. To demonstrate the robustness and superiority of SMA, the optimal results of objective function was compared with other recent meta-heuristics optimization methods. The SMA provides a favourable performance, competitive optimizer, and better convergence speed to solve OPF problems in the power system.