## 4 Simulation Results and Discussions
### Single-Objective OPF Problem
#### 4.1.1 IEEE 30-Bus System
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Algorithms** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & **Cost** & **Emis** & **Loss** & **Loss** & **Time** \\  & & & & & **(MW)** & **(MW)** & **(USD/h)** & **(MM)** & **(MW)** & **(USD/h)** & **(MM)** & **(MW)** & **(s)** \\ \hline GWO [64] & 51.8100 & 80.0000 & 50.0000 & 35.0000 & 30.0000 & 40.0000 & 968.3800 & 0.207310 & 3.4100 & - & 16.1 \\ DE [64] & 51.8200 & 79.9900 & 49.9990 & 35.0000 & 29.9800 & 40.0000 & 968.2300 & 2.027311 & 3.3800 & - & 16.7 \\ MOHS [65] & 66.2759 & 79.6413 & 46.8835 & 34.8880 & 29.1213 & 30.0558 & 928.5099 & 0.212890 & 3.5165 & - & - \\ GOA [19] & - & 68.6191 & 50.0000 & 27.5100 & 17.0215 & 24.6921 & 878.8137 & - & 5.8100 & - & - \\ HHO [19] & - & 43.7120 & 50.0000 & 29.9360 & 30.0000 & 28.820 & 915.0934 & - & 4.5600 & - & - \\ MVO [19] & - & 28.5870 & 25.7890 & 21.2720 & 28.1110 & 29.638 & 817.171 & - & 9.7700 & - & - \\ PSO [19] & - & 80.0000 & 50.0000 & 10.0000 & 10.0000 & 40.0000 & 931.2200 & - & 10.3822 & - & - \\ GSO [29] & 52.1000 & 80.0000 & 50.0000 & 34.8000 & 30.0000 & 40.0000 & 968.2900 & 0.297000 & 3.5100 & - & 65.0 \\ MF [29] & 51.9000 & 80.0000 & 50.0000 & 35.0000 & 30.0000 & 40.0000 & 968.5600 & 0.296000 & 3.5000 & - & 61.0 \\ SSA [29] & 51.9000 & 80.0000 & 50.0000 & 35.0000 & 30.0000 & 40.0000 & 968.5600 & 0.296000 & 3.5000 & - & 61.0 \\ WOA [29] & 52.0300 & 79.9600 & 49.9700 & 34.9800 & 29.9800 & 39.9800 & 968.2100 & 0.296000 & 3.5000 & - & 61.0 \\ PSO [60] & - & - & - & - & - & - & - & - & 3.1709 & - & 27.7 \\ BBC [60] & - & - & - & - & - & - & - & - & - & 3.1725 & - & 29.5 \\ ABC [60] & - & - & - & - & - & - & - & - & - & 3.1314 & - & 26.8 \\ TLDO [60] & - & - & - & - & - & - & - & - & - & 3.1104 & - & 27.3 \\ AGTLBO [60] & 51.4932 & 79.9998 & 49.9997 & 34.9997 & 29.9999 & 39.9998 & 967.6336 & 0.207300 & 3.0906 & - & 27.5 \\ MaMPA & 51.6974 & 80.0000 & 50.0000 & 35.0000 & 30.0000 & 40.0000 & 968.1335 & 0.207294 & **3.2974** & 0.1384 & 26.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison results of case 3. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Algorithms** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & \(P_{gt}\) **(MW)** & **Cost** & **Emis** & **Loss** & **Cost** & **Time** \\  & & & & **(MW)** & **(MW)** & **(USD/h)** & **(MM)** & **(USD/h)** & **(MM)** & **(MW)** & **(s)** \\ \hline SQL [66] & - & - & - & - & - & - & - & - & 5.9930 & 0.1570 & - \\ RGA [66] & - & - & - & - & - & - & - & - & 5.0912 & 0.1386 & 25.7 \\ CMES [66] & - & - & - & - & - & - & - & - & 5.1290 & 0.1382 & 27.1 \\ MOPSO [66] & - & - & - & - & - & - & - & - & 5.1090 & 0.1382 & - \\ NSGA-1 [66] & - & - & - & - & - & - & - & - & 5.1280 & 0.1383 & - \\ MMSG-1 [66] & - & - & - & - & - & - & - & - & 5.1020 & 0.1382 & - \\ ACDE [67] & - & - & - & - & - & - & - & - & - & 0.1364 & 82.0 \\ GWO [67] & - & - & - & - & - & - & - & - & - & 0.1382 & - \\ HHO [67] & - & - & - & - & - & - & - & - & - & 0.1384 & - \\ SRH [68] & - & - & - & - & - & - & - & - & - & 0.1384 & - \\ ECHT-DE [69] &Figure 2: _Cont._  
It is observed from Tables 2-5 that MaMPA could converge on the feasible solutions for all considered objective functions, cases 1-4, with slightly different objective values to the several algorithms in the literature. It also achieved better objective values than some algorithms such as GOA and MVO for cost, MF, SSA, and WOA for emission, MVO and PSO for loss, and SQP for VSI. The computational times of MaMPA are slightly faster or slower than various algorithms by a few seconds and much faster than some algorithms  
Figure 2: Convergence curves of MaMPA for: (**a**) case 1; (**b**) case 2; (**c**) case 3; (**d**) case 4.  
such as EP, GSO, HHO, MF, SSA, WOA, ACDE, and ECHT-DE. From Figure 2, it can be seen that MaMPA converged on the feasible solutions quickly within one-fourth of the maximum iteration for all considered objective functions.