Multi-Objective Optimal Power Flow Problems Based on Slime Mould Algorithm

: Solving the optimal power ﬂow problems (OPF) is an important step in optimally dispatch-ing the generation with the considered objective functions. A single-objective function is inadequate for modern power systems, required high-performance generation, so the problem becomes multiobjective optimal power ﬂow (MOOPF). Although the MOOPF problem has been widely solved by many algorithms, new solutions are still required to obtain better performance of generation. Slime mould algorithm (SMA) is a recently proposed metaheuristic algorithm that has been applied to solve several optimization problems in different ﬁelds, except the MOOPF problem, while it outperforms various algorithms. Thus, this paper proposes solving MOOPF problems based on SMA considering cost, emission, and transmission line loss as part of the objective functions in a power system. The IEEE 30-, 57-, and 118-bus systems are used to investigate the performance of the SMA on solving MOOPF problems. The objective values generated by SMA are compared with those of other algorithms in the literature. The simulation results show that SMA provides better solutions than many other algorithms in the literature, and the Pareto fronts presenting multi-objective solutions can be efﬁciently obtained.


Introduction
In the competitive electricity market, optimal power flow (OPF) is one of the important tools to optimally dispatch generation with the considered objective function while satisfying system constraints [1][2][3]. The OPF aims to find the best feasible values of control variables providing the best objective value. Generally, the objective function considered by power companies is to minimize generation costs in order to achieve the highest profits of power dispatch. Minimizing fuel cost is a direct way to decrease the generation cost while minimizing transmission line loss is another way to reduce power generation resulting in generation cost reduction. However, with the higher power generation of thermal plants for satisfying present higher load demand, a large amount of emission is emitted that causes pollution in the air [4]. Hence, three objectives consisting of fuel cost, emission, and transmission line loss are considered as part of the objective function to be minimized in the OPF problems of this study.
The OPF is a nonlinear, nonconvex, large-scale, and static programming problem [5] which has attracted an effort from many researchers to apply various methods to solve the problem. In the past, some traditional techniques such as quadratic programming [6], interior point method [7], and nonlinear programming [8] were used to solve the OPF problem. However, these algorithms are usually trapped in the local optima, which returns a low-quality solution, and requires a large amount of computational time. To overcome these weaknesses, many other optimization methods have been introduced. One of the methods are metaheuristic algorithms which have recently been the most popular methods

1.
A method to solve the MOOPF problem in the power systems based on SMA is proposed. The Pareto dominance concept is applied to the SMA in order to store non-dominated Pareto fronts, and the crowding mechanism is adopted to deal with the Pareto repository.

2.
The SMA is adopted to solve single-and multi-objective optimal power flow problems in the IEEE 30-, 57-, and 118-bus systems.  3. The performance of SMA is investigated in terms of fuel cost, emission, and transmission loss improvement, and the results are compared with those of many other algorithms in the literature.
The rest of the paper is divided as follows. Problem formulation of the MOO including objective functions and constraints is presented in Section 2. Section 3 introduces the SMA and its mathematical model. In Section 4, simulation results, comparison results, and discussions are provided. The conclusion of this work is finally given in Section 5.

Problem Formulation
The MOO aims to find an optimal set of decision variables on solving more than one objective functions while satisfying a set of equality and inequality constraints. The obtained solutions from the MOO problem have a trade-off between each considered objective function; thus, the number of the solutions is uncountable. The set of these solutions is traditionally called Pareto optimal solutions or Pareto fronts. The MOO problem can be mathematically modeled as the given equation.
subject to g(x, u) = 0 (2) h(x, u) ≤ 0 where f is a vector of the considered objective function to be optimized, N ob is the number of objective functions, g(x,u) is the equality constraints, h(x,u) is the inequality constraints, x is a vector of state variables consisting of active power generation at the slack bus, voltage magnitudes at load buses, reactive power generations and apparent powers flowing in lines, and u is a vector of control variables including active power generations except at slack bus, voltage magnitudes of generators, transformer tap ratios, and shunt compensation capacitor reactive powers.

Objective Functions
In this work, three objective functions comprising fuel cost, emission, and transmission line loss are considered as the objective functions in the MOOPF problem. Each objective is explained as follows:

Fuel Cost
The total fuel cost of the generated real power of the interconnected units is generally considered as part of the objective functions to be minimized for thermal plants. The total fuel cost can be represented by the quadratic function and formulated as the provided equation.
where f TC is the total fuel cost function of generators ($/h), N g is the number of generators, a i , b i , c i are fuel cost coefficients of the ith generating unit, and P gi is the active power generation of the ith generating unit.

Emission
The total emission function is taken into account to be minimized to reduce pollutions released by plants in the atmosphere. The summation of various types of emissions such as SO x and NO x are computed to represent the total emission function. The total emission function is calculated by the presented equation.
where f TE is the total emission function of generators (ton/h), and γ i , β i , α i , ξ i , λ i are emission coefficients of the ith generating unit.

Transmission Line Loss
To further reduce generation cost and active power generation, transmission line loss is aimed to be minimized. The line loss function is computed as follows: where f TL is the total transmission line loss function (MW), N line is the number of transmission lines, g k is the conductance of the kth transmission line, V i is the voltage magnitude at the ith bus, V j is the voltage magnitude at the jth bus, and θ ij is the difference of voltage phase angle between buses i and j.

Constraints
A set of equality and inequality constraints is satisfied while optimizing the considered objective functions.

Equality Constraints
The equality constraints of the OPF problem are active and reactive power balance equations formulated as the following equations.
where Q gi is reactive power generations of the ith generating unit, P di and Q di are active and reactive power demands at the ith bus, respectively, N bus is the number of buses, and Gij and Bij are the transfer conductance and transfer susceptance between buses i and j, respectively.

Inequality Constraints
The inequality constraints of the OPF problem are set to satisfy system security as follows: P gimin ≤ P gi ≤ P gimax i = 1, 2, . . . , N g (9) Q gimin ≤ Q gi ≤ Q gimax i = 1, 2, . . . , N g (10) |S li | ≤ S limax (12) where P gimin and P gimax are the minimum and maximum active power generations of the ith generating unit, respectively, Q gimin and Q gimax are the minimum and maximum reactive power generations of the ith generating unit, respectively, V gimin and V gimax are the minimum and maximum voltage magnitudes of the ith generating unit, respectively, V gi is the voltage magnitude of the ith generating unit, S li is the apparent power flowing in branch i, S limax is the maximum apparent power flowing in branch i, V Limin and V Limax are the minimum and maximum load bus voltage magnitudes at the ith bus, respectively, V Li is the load bus voltage magnitude at the ith bus, Q cimin and Q cimax are the minimum and maximum shunt compensation capacitors installed at the ith bus, respectively, Q ci is the shunt compensation capacitor installed at the ith bus, T imin and T imax are the minimum and maximum transformer tap ratios at the ith bus, respectively, T i is the transformer tap ratio at the ith bus, N L is the number of load buses, N c is the number of shunt compensation capacitors, and N t is the number of transformers.

Constraint Handling
The inequality constraints of the state variables which are uncontrollable are merged into the penalized objective function to ensure that these variables are maintained within their limits. The objective function is then penalized as the following equation [26].
where J(x,u) is the penalized objective function, and K p , K Q , K V , and K s are the penalty factors for the real power generation limit violation at the slack bus, reactive power generation limit violation, load bus voltage violation, and the line apparent power flow violation, respectively. P lim gslack , V lim Li , Q lim gi , and S lim li are the limit values of the real power generation at the slack bus, reactive power generation, load bus voltage, and line apparent power flow, respectively, defined as the equations below.

Slime Mould Algorithm (SMA)
The SMA was recently proposed by Li, Shimin et al. [34] based on the behavior of slime mould in nature. Slime mould searches for food, encircles it, and releases enzymes to digest it. The slime mould characteristics can be explained through three main steps, consisting of approaching food, wrapping food, and oscillating, mathematically formulated as the following subsections.

Approach Food
To approach food, the slime mould follows the smell in the air which can be modeled as the given equation.
where X is the slime mould position, X b is the current position with the highest smell intensity (food position), X A and X B are two randomly chosen individuals from slime mould, t is the current iteration, r is a uniformly randomly generated value in the range of [0, 1], W is the adaptive weight of slime mould, vb is a uniformly generated value in the range of [−a, a], vc is a uniformly generated parameter in the range of [−b, b] where b is linearly decreased from 1 to 0 according to the iteration (b = 1 − t/MaxIter), and p is the probability index dynamically changed according to the following equation.
where i = 1, 2, . . . , n, n is the number of population, S(i) is the fitness value of X, and DF is the best fitness obtained so far. To find vb, the parameter a is computed by the presented equation.
where MaxIter represents the maximum iteration. The adaptive weight (W) to simulate the process of generating positive and negative feedbacks of the propagation wave of slime mould can be computed as follows: where bF is the best fitness provided in the current iteration, wF is the worst fitness provided in the current iteration, and SmellIndex is the sequence of the sorted fitness values.

Wrap Food
From the approaching food process, the slime mould changes the search process according to the concentration of food. When the food concentration is poor, the weight of the area is decreased, when the food concentration is satisfied, the weight of the area will be increased. However, to improve the exploration phase of the SMA, the location of slime mould can be updated by the following equation.
where ub and lb are the upper and lower limits of the decision variables, rand is the uniformly randomly generated value in the range of [0, 1], and z is set at 0.03 as it is the best value investigated in [34].

Oscillation
The parameters W, vb, and vc in the above equations are used to simulate the variations of slime mould when the process of food-finding is operated. W mathematically expresses the oscillation frequency of slime mould to improve the slime mould performance for selecting the best food source. When the slime mould discovers high-concentration food, Sustainability 2021, 13, 7448 7 of 21 they approach food faster, and when the food concentration is low, the slime mould more slowly approaches food.
From Equation (23), vb randomly oscillates in the range of [−a, a] and eventually tends to zero when the iteration is increased. In addition, vc randomly oscillates between [−1, 1] and gradually reaches zero with the increase of iteration. These two parameters simulate the behavior of slime mould to discover a better food source. Although a high-quality food source is discovered, slime mould still separates some of their parts to explore more areas in order to find a higher-quality food source and avoid trapping in the local optima.
For the MOOPF problems where the considered objective functions conflict with each other and the results are a trade-off between each objective, the traditional computations require a large amount of computational effort and time. The variation of the random search can overcome this drawback; thus, the SMA is used to solve the MOOPF problems. The Pareto dominance concept is applied to the SMA to keep the trade-off between the considered objectives that is the Pareto fonts in the repository [39]. The non-dominated sorting and the crowding mechanism are adopted to deal with the full repository [40]. The process of the SMA for solving MOOPF problems can be described in the Pseudo-code expressed in Algorithm 1 and the flowchart presented in Figure 1. Apply dominance concept to update the best fitness and worst fitness.
Calculate the W by (24) Update best position and best fitness Update a by (23) If rand < z Update X(t+1) by (26)-1 Update p by (22), vb, vc Move the components to be within its lower or upper bounds

Simulation Results and Discussion
The SMA was applied to solve both single-objective and multi-objective OPF problems in the IEEE 30-, 57-, and 118-bus systems to investigate its performance in terms of fuel cost, emission, and transmission loss reductions where 14 different cases shown in

Algorithm 1. Pseudo-code of SMA on solving MOOPF problems
Initialize the system data, the parameters NumPop, MaxIter, RepSize; Initialize the position of slime mould X i (i = 1, 2, . . . , n); Calculate the fitness values of the initialized slime mould; Determine non-dominated slime mould; Store the non-dominated slime mould in the initial repository; While t ≤ MaxIter Apply dominance concept to update bestFitness (bF); Apply dominance concept to update worstFitness (wF); Calculate the W by Equation (24); update bestFiness, bestPosition (X b ); update a by Equation (23); For each population If rand < z update position by Equation (26)

Simulation Results and Discussion
The SMA was applied to solve both single-objective and multi-objective OPF problems in the IEEE 30-, 57-, and 118-bus systems to investigate its performance in terms of fuel cost, emission, and transmission loss reductions where 14 different cases shown in Table 1 were evaluated. The simulation was operated in MATLAB. The data of the test systems and the results investigated by the SMA for solving single-and multi-objective OPF problems are presented in the following subsections.

IEEE 30-Bus System
The IEEE 30-bus system was applied to evaluate the performance of the SMA on solving the OPF problems. This system included six generators, four transformers, and 41 transmission lines. The system active and reactive power demands were 283.4 MW and 126.6 MVAR, respectively. The detailed data and the line diagram of the system can be found in [41]. For this system, the population number, maximum iteration number and size of the repository were all set at 100. Cases 1-7 including single-and multi-objective OPF problems were investigated in this system.

Case 4: Fuel Cost and Emission Minimizations
For the MOOPF problem, fuel cost, and emission were simultaneously considered as part of the objective functions to be minimized in case 4. The two-dimensional Pareto fronts provided by the SMA are compared with those of the high-performance well-known algorithm which is PSO obtained from [23] as presented in Figure 2.

. Case 4: Fuel Cost and Emission Minimizations
For the MOOPF problem, fuel cost, and emission were simultaneously considered as part of the objective functions to be minimized in case 4. The two-dimensional Pareto fronts provided by the SMA are compared with those of the high-performance wellknown algorithm which is PSO obtained from [23] as presented in Figure 2.

Case 5: Fuel Cost and Transmission Loss Minimizations
In case 5, the two objective functions selected for the MOOPF problem were fuel cost and transmission loss. Figure 3 shows the two-dimensional Pareto fronts obtained by the SMA when compared with those of the PSO.

Case 5: Fuel Cost and Transmission Loss Minimizations
In case 5, the two objective functions selected for the MOOPF problem were fuel cost and transmission loss. Figure 3 shows the two-dimensional Pareto fronts obtained by the SMA when compared with those of the PSO.

Case 7: Fuel Cost, Emission, and Transmission Loss Minimizations
For this case, all three objective functions including fuel cost, emission, and transmission loss were simultaneously considered as part of the objective functions in the MOOPF problem. The three-dimensional Pareto fronts generated by the SMA are shown in Figure 5.

Case 7: Fuel Cost, Emission, and Transmission Loss Minimizations
For this case, all three objective functions including fuel cost, emission, and transmission loss were simultaneously considered as part of the objective functions in the MOOPF problem. The three-dimensional Pareto fronts generated by the SMA are shown in Figure 5. For the IEEE 30-bus system, when solving the single-objective function, it can be seen that the SMA could provide slightly better values of each objective than many competitive algorithms such as GWO, WOA, HHO, and SSA and well-known algorithms such as PSO and GA. The computational times of the SMA on solving each objective function are found to be slightly slower than some algorithms and also slightly faster than some algorithms. Besides, for the MOOPF problems, the SMA could efficiently produce the two dimensional Pareto fronts which are slightly better than those of the PSO for the pair of costemission objectives, competitive to those of the PSO for the pair of cost-loss objective, and slightly worse than those of the PSO for the pair of emission-loss objectives. The threedimensional Pareto fronts could also be successfully generated with good diversity. It is worth mentioning that the system operators can judiciously select the fronts depending on the situation and aim. Thus, the SMA has a high performance in solving both singleand multi-objective OPF problems in the IEEE 30-bus system.

IEEE 57-Bus System
The IEEE 57-bus system was used to verify the performance of the SMA on solving the OPF problems in a larger system. It comprised seven generators, 15 transformers, and For the IEEE 30-bus system, when solving the single-objective function, it can be seen that the SMA could provide slightly better values of each objective than many competitive algorithms such as GWO, WOA, HHO, and SSA and well-known algorithms such as PSO and GA. The computational times of the SMA on solving each objective function are found to be slightly slower than some algorithms and also slightly faster than some algorithms. Besides, for the MOOPF problems, the SMA could efficiently produce the two dimensional Pareto fronts which are slightly better than those of the PSO for the pair of cost-emission objectives, competitive to those of the PSO for the pair of cost-loss objective, and slightly worse than those of the PSO for the pair of emission-loss objectives. The three-dimensional Pareto fronts could also be successfully generated with good diversity. It is worth mentioning that the system operators can judiciously select the fronts depending on the situation and aim. Thus, the SMA has a high performance in solving both singleand multi-objective OPF problems in the IEEE 30-bus system.

IEEE 57-Bus System
The IEEE 57-bus system was used to verify the performance of the SMA on solving the OPF problems in a larger system. It comprised seven generators, 15 transformers, and 80 transmission lines. The total active power demand was 1250.8 MW, and the total reactive power demand was 336.4 MVAR. The bus data and branch data of the system were presented in [48]. The population number was 100, the maximum iteration was 200, and the Pareto repository was 100. Single-and multi-objective OPF problems were solved in the IEEE 57-bus system as in cases 8-12. For the IEEE 57-bus system, the single-objective OPF problem considering each objective function was solved by the SMA as in cases 8-10. The simulation results consisting of the control variables, active power at the slack bus, fuel cost, emission, and line loss of the SMA for each case are shown in Table 6. The fuel cost was chosen as the objective function in this case. The fuel cost result obtained by the SMA is compared with those of various other algorithms in the literature such as PSO [23], dragonfly algorithm (DA) [23], GBICA [26], MGBICA [26], evolving ant direction differential evolution (EADDE) [49], Fuzzy-GA [50], particle swarm optimization with linearly decreasing inertia weight (LDI-PSO) [51], adaptive particle swarm optimization (APSO) [52], and new particle swarm optimization (NPSO) [53] as expressed in Table 7.

Case 9: Emission Minimization
In this case, the performance of the SMA was evaluated for solving the OPF problem when emission was considered to be minimized. The emission comparison results of the SMA with those of PSO [23], DA [23], GBICA [26], and MGBICA [26] are given in Table 8. Table 8. Comparison results of the SMA with other algorithms for case 9.

Case 10: Transmission Loss Minimization
This case simulated the OPF problem when transmission loss was an objective function. Table 9 shows the comparison results provided by the SMA with those of PSO [23] and DA [23]. Table 9. Comparison results of the SMA with other algorithms for case 10.

Case 11: Fuel Cost and Emission Minimizations
In this case, the MOOPF problem was investigated in the IEEE 57-bus system where fuel cost and emission were simultaneously considered as part of the objective functions. The two-dimensional Pareto fronts generated by the SMA are presented in Figure 6 when compared with those of PSO obtained from [23].

Case 11: Fuel Cost and Emission Minimizations
In this case, the MOOPF problem was investigated in the IEEE 57-bus system where fuel cost and emission were simultaneously considered as part of the objective functions. The two-dimensional Pareto fronts generated by the SMA are presented in Figure 6 when compared with those of PSO obtained from [23]. Figure 6. Two-dimensional Pareto fronts for case 11.

Case 12: Fuel Cost, Emission, and Transmission Loss Minimizations
The fuel cost, emission, and transmission loss were simultaneously minimized for the MOOPF problem in this case. The three-dimensional Pareto fronts provided by the SMA is displayed in Figure 7.

Case 12: Fuel Cost, Emission, and Transmission Loss Minimizations
The fuel cost, emission, and transmission loss were simultaneously minimized for the MOOPF problem in this case. The three-dimensional Pareto fronts provided by the SMA is displayed in Figure 7. In the larger system, which is the IEEE 57-bus system, the simulation results for each single-objective function provided by the SMA are found to be significantly better than those of many other algorithms in the literature. Moreover, the two-dimensional Pareto fronts generated by the SMA are significantly better than those of the PSO as evident in Figure 6, and the three-dimensional Pareto fronts could also be efficiently obtained with good diversity as in Figure 7. Hence, the SMA also has superiority over various other algorithms for the IEEE 57-bus system.

IEEE 118-Bus System
To verify the performance of the SMA in a large system, the MOOPF problem was solved in the IEEE 118-bus system. This system includes 54 generators, nine transformers, and 186 transmission lines. The active power demand was 4242 MW, and the reactive power demand was 1439 MVAR. The line diagram of this system, bus data, and branch data are presented in [54]. In the simulation, the population number and the maximum iteration were 100 and 500, respectively. The objective function was the fuel cost as in case 13 for the OPF problem, and the fuel cost and transmission loss were the objective func- In the larger system, which is the IEEE 57-bus system, the simulation results for each single-objective function provided by the SMA are found to be significantly better than those of many other algorithms in the literature. Moreover, the two-dimensional Pareto fronts generated by the SMA are significantly better than those of the PSO as evident in Figure 6, and the three-dimensional Pareto fronts could also be efficiently obtained with good diversity as in Figure 7. Hence, the SMA also has superiority over various other algorithms for the IEEE 57-bus system.

IEEE 118-Bus System
To verify the performance of the SMA in a large system, the MOOPF problem was solved in the IEEE 118-bus system. This system includes 54 generators, nine transformers, and 186 transmission lines. The active power demand was 4242 MW, and the reactive power demand was 1439 MVAR. The line diagram of this system, bus data, and branch data are presented in [54]. In the simulation, the population number and the maximum iteration were 100 and 500, respectively. The objective function was the fuel cost as in case 13 for the OPF problem, and the fuel cost and transmission loss were the objective functions for the MOOPF problem as in case 14.
The simulation results including control variables, active power generation at the slack bus (P g69 ), fuel cost, and transmission loss provided by the SMA when considering fuel cost as the objective function for this system are depicted in Table 10. This case selected the fuel cost as the objective function to be minimized. The fuel cost value provided by the SMA is compared with several algorithms including GWO [21], DE [21], DSA [55], TLBO [55], harmony search algorithm (HSA) [56], fuzzy harmony search algorithm (FHSA) [56], gravitational search algorithm (GSA) [57], hybrid PSO and GSA (PSOGSA) [58], improved colliding bodies optimization (ICBO) [59], moth swarm algorithm (MSA) [60], sine-cosine algorithm (SCA) [61], and modified SCA (MSCA) [61] as shown in Table 11. Table 11. Comparison results of the SMA with other algorithms for case 13.

Case 14: Fuel Cost and Loss Minimizations
For the IEEE 118-bus system, fuel cost and line loss were simultaneously chosen as the objective functions for the MOOPF in this case. The two-dimensional Pareto fronts generated by the SMA are presented in Figure 8. From the simulation results in the IEEE 118-bus system, which is the largest system in this study, it is found that the SMA could efficiently solve the single-objective OPF problem and provide considerably better fuel cost value than many other algorithms in the literature as evident in Table 11. Moreover, for solving the MOOPF problem, twodimensional Pareto fronts were successfully generated as evident in Figure 8 where the PSO could not converge to the solutions within the defined iterations.
Thus, it is found from all simulation results that the SMA has a high performance on solving the OPF problems including both single-and multi-objective OPF problems. The obtained results for all considered objectives are better than those of several other algorithms in the literature for all test systems where SMA could successfully provide highquality solutions in the large systems. The computational times of the SMA are also competitive to those of other algorithms. For the MOOPF problems, even though the considered objective functions including fuel cost, emission, and transmission loss conflict with each other, the SMA could provide the solutions which is a trade-off between each objective because of its high-quality random search property. The SMA could efficiently provide two-and three-dimensional Pareto fronts with good diversity in all test systems where the Pareto fronts are better than those of the PSO, which is an effective well-known algorithm. This is because the SMA has the process of producing positive and negative feedbacks of propagation wave to well balance the exploration and exploitation phases of the optimization process. From the simulation results in the IEEE 118-bus system, which is the largest system in this study, it is found that the SMA could efficiently solve the single-objective OPF problem and provide considerably better fuel cost value than many other algorithms in the literature as evident in Table 11. Moreover, for solving the MOOPF problem, two-dimensional Pareto fronts were successfully generated as evident in Figure 8 where the PSO could not converge to the solutions within the defined iterations.
Thus, it is found from all simulation results that the SMA has a high performance on solving the OPF problems including both single-and multi-objective OPF problems. The obtained results for all considered objectives are better than those of several other algorithms in the literature for all test systems where SMA could successfully provide high-quality solutions in the large systems. The computational times of the SMA are also competitive to those of other algorithms. For the MOOPF problems, even though the considered objective functions including fuel cost, emission, and transmission loss conflict with each other, the SMA could provide the solutions which is a trade-off between each objective because of its high-quality random search property. The SMA could efficiently provide two-and three-dimensional Pareto fronts with good diversity in all test systems where the Pareto fronts are better than those of the PSO, which is an effective well-known algorithm. This is because the SMA has the process of producing positive and negative feedbacks of propagation wave to well balance the exploration and exploitation phases of the optimization process.
In practical systems, the system operators can judiciously choose the suitable solution from the generated Pareto fronts by considering the situation and objective in order to achieve the high profit for the power generation or satisfy the power generation situation. For example, when the high demand is required, the operators should select the least transmission loss solution, so that the power generation is adequate for the required high demand. When the pollution situation deteriorates, the emission objective should be minimized to alleviate the situation. However, in the normal situation, the cost can be the best objective function to be minimized, so the power companies can dispatch the generation with the highest profit. It can be noticed from the simulation results that the SMA can sharply increase the profit for the power companies especially when the system is large. For example, in the IEEE 118-bus system, when fuel cost is considered as the objective function, the SMA can reduce the cost around 1685.4535 to 7225.0239 $/h or 40,450.8840 to 173,400.5736 $/day compared to other algorithms.

Conclusions
This paper presents a method to solve MOOPF problems based on SMA by considering fuel cost, emission, and transmission line loss as part of the objective functions to be minimized. The SMA is a recently proposed algorithm, which has been applied to successfully solve many optimization problems in several fields. However, the SMA has been rarely investigated on solving the single-objective OPF problem and never been evaluated on solving the MOOPF problem. So, this work has introduced a method to solve the MOOPF problems based on SMA and investigated the performance of the SMA on solving both single-objective and multi-objective OPF problems. The IEEE 30-, 57-, and 118-bus systems were employed to evaluate the performance of the SMA. The simulation results show that the SMA could successfully solve the single-objective OPF problem. The SMA provided slightly better solutions for the IEEE 30-bus system, moderately better solutions for the IEEE 57-bus system, and significantly better solutions for the IEEE 118-bus system in terms of the objective values than many other algorithms in the literature. For the MOOPF problems, the two-dimensional Pareto fronts generated by the SMA are better than those of the well-known high-performance algorithm which is the PSO algorithm. Especially in the IEEE 118-bus system, the SMA could efficiently obtain the Pareto fronts while PSO could not converge to the solutions within the imposed iteration. Moreover, the three-dimensional Pareto fronts could also be efficiently provided by the SMA where the operators should judiciously choose the appropriate solution depending on the objective and situation. Hence, the SMA is verified as a high-performance algorithm for solving both single-objective and multi-objective OPF problems, especially in large systems. In future work, the SMA can be applied to solve the OPF problems in practical systems where the system sizes are generally large, and many-objective OPF problems (more than three objective functions) can be solved based on SMA to further improve power system performance in several terms.