## 5 Conclusions  
In this paper, a MaMPA for solving the MaOPF problem is proposed. With the growth of electricity demand, environmental awareness, and power reliability requirements, more than three objective functions for the OPF problem, referred to as the MaOPF problem, are important to satisfy modern power system requirements. MPA is an efficient optimization algorithm and has been widely applied to efficiently solve different problems except for the MaOPF problem. The MPA itself cannot solve multi- or many-objective optimization problems, so the non-dominated sorting, crowding mechanism, and leader mechanism are applied to MPA in this work. Four objectives consisting of cost, emission, transmission loss, and VSI are selected to be the objective functions. The performance of the MaMPA is evaluated in the IEEE 30- and 118-bus systems. From the simulation results, MaMPA could find slightly different solutions and significantly better solutions in the IEEE 30- and 118-bus systems, respectively, for the SOOPF problems. The computational times of MaMPA are slower than some algorithms and much faster than various algorithms in the literature for both test systems. In the SOOPF problems, MaMPA also provided efficient statistical results with very small standard deviation values in both tested systems expressing high consistency of the generated results, and both standard deviation and minimum values for each objective provided by MaMPA are better than several algorithms in the literature. For the MOOPF problems, based on the hypervolume indicator, the two- and three-dimensional Pareto fronts generated by MaMPA are greater than those of PESA-II, SPEA2, and SSA while the fronts are slightly worse than those of PSO in the IEEE 30-bus system. In the IEEE 118-bus system, both two- and three-dimensional Pareto fronts provided by MaMPA are remarkably superior to those of PSO and SSA based on the hypervolume values while the fronts could not be obtained by PESA-II and SPEA2 within the given iterations. For the MaOPF problems, in the IEEE 30-bus system, the best-compromised solutions of MaMPA are very competitive showing two better objective values from four objectives than the compared algorithms while the Pareto fronts based on the hypervolume indicator of MaMPA are the best ones. In the IEEE 118-bus system, the \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithms** & **Case 20** \\ \hline PSO & 0.01378 \\ SSA & 0.02582 \\ MaMPA & **0.08339** \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Hypervolume comparison of MaOPF problems in the IEEE 118-bus system. best compromise solution of MaMPA presents absolute superiority over PSO and SSA in all objective values, and the Pareto fronts of MaMPA based on the hypervolume indicator are also significantly better than those of PSO and SSA while the other compared algorithms could not converge on the solution. Overall, MaMPA has a high performance in solving SOOPF, MOOPF, and MaOPF problems, especially in the large system, and with the high consistency of MaMPA, power system operators can generate high-quality solutions within a few runs. In future work, the self-adaptive method and two archive techniques can be applied to improve the MPA, so that the feasible solutions can be enhanced. In addition, more objective functions such as voltage deviation can be added to the objective functions to further improve the power system performance. The proposed MaMPA can also be adopted to solve other many-objective problems in different fields due to its high performance. Conceptualization, S.K. ; methodology, S.K. ; software, S.K. ; validation, S.K., A.S. and S.P. ; formal analysis, S.K. ; investigation, S.K. ; resources, S.K. ; data curation, S.K. ; writing--original draft preparation, S.K. ; writing--review and editing, S.K., A.S. and S.P. ; visualization, S.K. ; supervision, S.K., A.S. and S.P. ; project administration, S.K., A.S. and S.P. ; funding acquisition, S.K. All authors have read and agreed to the published version of the manuscript.