## 4 Simulation Results and Discussions
### Single-Objective OPF Problem
#### 4.1.3 Statistical Investigation  
To further evaluate the performance of MaMPA in solving SOOPF problems, the statistical investigations including minimum, maximum, and average values and standard deviation of each objective value when that objective is an objective function are generated in both tested systems. The MaMPA was simulated for 30 independent runs for each test system. The statistical investigations of MaMPA in the IEEE 30-bus system, which is a small system, are provided and compared with those of other algorithms in the literature as shown in Table 9.  
It can be noticed from Table 9 that the minimum, maximum, and average values of MaMPA for each objective are slightly different resulting in a very small standard deviation. When compared to other algorithms, MaMPA provided slightly worse objective values in terms of minimum, maximum, and average values while the standard deviations of MaMPA for all objectives are better than those of the compared algorithms. This means \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case 1** & **Min** & **Max** & **Average** & **Std.** \\ \hline PSO [60] & 800.7016 & 803.8826 & 801.4378 & 2.075 \\ BBO [60] & 800.9527 & 805.3919 & 802.3304 & 3.994 \\ ABC [60] & 800.6802 & 802.1506 & 800.9642 & 1.819 \\ TLBO [60] & 800.6735 & 801.1324 & 800.8705 & 0.542 \\ AGTLBO [60] & 800.4811 & 800.5587 & 800.5316 & 0.076 \\ MaMPA & 802.5449 & 802.5449 & 802.5449 & \(1.65\times 10^{-8}\) \\ \hline
**Case 2** & **Min** & **Max** & **Average** & **Std.** \\ \hline PSO [60] & 0.20484 & 0.20551 & 0.20496 & 0.009 \\ BBO [60] & 0.20490 & 0.20603 & 0.20532 & 0.010 \\ ABC [60] & 0.20484 & 0.20557 & 0.20505 & 0.009 \\ TLBO [60] & 0.20484 & 0.20495 & 0.20490 & 0.006 \\ AGTLBO [60] & 0.20482 & 0.20484 & 0.20483 & 0.000 \\ MaMPA & 0.204886 & 0.204886 & 0.204886 & \(8.03\times 10^{-11}\) \\ \hline
**Case 3** & **Min** & **Max** & **Average** & **Std.** \\ \hline PSO [60] & 3.1709 & 3.2235 & 3.1770 & 0.071 \\ BBO [60] & 3.1725 & 3.3178 & 3.2490 & 0.199 \\ ABC [60] & 3.1314 & 3.2074 & 3.1682 & 0.103 \\ TLBO [60] & 3.1104 & 3.1275 & 3.1133 & 0.062 \\ AGTLBO [60] & 3.0906 & 3.0920 & 3.0911 & 0.000 \\ MaMPA & 3.2974 & 3.2974 & 3.2974 & \(9.5766\times 10^{-11}\) \\ \hline
**Case 4** & **Min** & **Max** & **Average** & **Std.** \\ \hline RGA [66] & 0.13860 & 0.13956 & 0.13916 & 0.00033 \\ CMAES [66] & 0.13820 & 0.13836 & 0.13820 & 0.00019 \\ ACDE [67] & 0.13640 & 0.13680 & 0.13660 & 0.0001 \\ SKH [68] & 0.13660 & 0.13840 & 0.13720 & 0.0006 \\ ECHT-DE [69] & 0.13630 & 0.13720 & 0.13690 & 0.0002 \\ ARCBBO [70] & 0.13690 & 0.13870 & 0.13750 & - \\ MaMPA & 0.13625 & 0.13626 & 0.13625 & \(7.9657\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Statistical comparison of MaMPA and other algorithms in the IEEE 30-bus system. MaMPA has a great consistency, and system operators can certainly obtain a high-quality feasible solution in a single operation of the algorithm process. In the IEEE 118-bus system, which is a large test system, the MaMPA was operated to provide the statistical results. The statistical comparison of MaMPA and other algorithms in the literature are presented in Table 10 for the cost and loss objectives, cases 5 and 7, while the references for the statistical results of emission and VSI objectives, cases 6 and 8, could not be found. It is observed from Table 10 that the minimum, maximum, and average values of each objective provided by MaMPA are slightly different from each other leading to low standard deviation values. Although the standard deviation when selecting cost as the objective function is higher than when choosing the other objectives as the objective functions, this is because the cost value for this large system is high compared to the values of the other objectives. When compared to other algorithms, for the cost objective, even if MaMPA has a higher standard deviation than some algorithms; the minimum, maximum, and average values are better than those of all compared algorithms.