## 5 Numerical Results
### Investigation of the Proposed Method's Performance on the IEEE 30-Bus System  
In the section, the results obtained by the proposed ICOA for the standard IEEE 30-bus system are compared to those from other implemented methods consisting of WCA, SSA, SFOA, COA, COA1, and COA2, and other methods available in previous studies. As mentioned in Section 5.1 above, the best optimal solution with the lowest fitness function is the first comparison factor for concluding the effectiveness of methods. Secondly, \(N_{\text{sage}}\) and \(Ti\) are the second factor for concluding the search speed of the proposed method. Finally, the average fitness of all runs is compared to other existing methods, and 50 runs of all implemented methods are plotted in the same figure for comparison of search stability. All the values regarding the three factors are reported in Tables 5-7 for TPL, TVD, and \(L\) index objectives. The minimum TPL in Table 5 shows that the proposed ICOA method can find the best solution for the case of continuous variables and discrete variables. The power loss of ICOA is the lowest value with 4.5128 MW for the case of continuous variables and 4.5138 MW for the case of discrete variables, while the best power loss in other studies is much higher. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \(N_{\text{sage}}\) & 9 & 6 & 4 & 3 \\  & \(N_{\text{pack}}\) & 2 & 3 & 4 & 5 \\ \hline \multirow{4}{*}{**ICOA**} & Min. (MW) & 0.1250 & 0.1246 & 0.1242 & 0.1249 \\  & Avg. (MW) & 0.1262 & 0.1273 & 0.1264 & 0.1273 \\  & Max. (MW) & 0.1279 & 0.1494 & 0.1425 & 0.1391 \\  & Std. dev. (MW) & 0.0008 & 0.0039 & 0.0025 & 0.0032 \\  & CPU time (s) & 7.3 & 6.9 & 6.9 & 6.8 \\  & SR (\%) & 96 & 96 & 94 & 98 \\ \hline \multirow{4}{*}{**COA**} & Min. (MW) & 0.1316 & 0.1307 & 0.1252 & 0.1284 \\  & Avg. (MW) & 0.1392 & 0.1324 & 0.1267 & 0.1325 \\ \cline{1-1}  & Max. (MW) & 0.1453 & 0.1585 & 0.1308 & 0.1532 \\ \cline{1-1}  & Std. dev. (MW) & 0.0082 & 0.0093 & 0.0011 & 0.0123 \\ \cline{1-1}  & CPU time (s) & 9.7 & 9.5 & 9.4 & 9.2 \\ \cline{1-1}  & SR (\%) & 76 & 68 & 70 & 72 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results found by the proposed ICOA approach for \(L\) index objective of the IEEE 30-bus system. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Min.**} & \multirow{2}{*}{**Avg.**} & \multicolumn{2}{c}{**Average**} & \multirow{2}{*}{**\(N_{\text{sage}}\)**} & \multirow{2}{*}{**\(Ti\) (pu)**} & \multirow{2}{*}{**CV**} & \multirow{2}{*}{**DV**} & \multirow{2}{*}{**SR (\%)**} \\  & & & & & & & \\ \hline ICOA & 4.5128 & 4.7037 & 6.36 & 2000 & 1 & X & 100 \\ ICOA & 4.5138 & 4.870 & 6.78 & 2000 & - & X & 100 \\ WCA & 4.58218 & 4.76097 & 8.7 & 2000 & 1.37 & X & 100 \\ SSA & 4.7175 & 4.9522 & 6.3 & 2000 & 0.99 & X & 89 \\ SFOA & 4.7364 & 5.1177 & 6.9 & 2000 & 1.08 & X & 87 \\ COA & 4.5777 & 4.9205 & 8.2 & 2000 & 1.29 & X & 92 \\ COA1 & 4.534 & 4.9128 & 7.2 & 2000 & 1.13 & X & 100 \\ COA2 & 4.5328 & 4.9196 & 7.8 & 2000 & 1.23 & X & 100 \\ SGA [14] & 4.5692 & - & - & 18,000 & - & - & - \\ PSO-AWI [19] & 4.8458 & 4.8761 & 8.762 & 4000 & 1.44 & X & X & - \\ PSO-AAC [19] & 4.8449 & 4.8702 & 8.68 & 4000 & 1.42 & X & - \\ SPSO-AAC [19] & 4.5262 & 4.5564 & 9.092 & 4000 & 1.49 & X & - \\ PSO-CF [19] & 4.5258 & 4.5711 & 8.48 & 4000 & 1.39 & X & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Result comparison of TPL for the IEEE 30-bus system. For the case of continuous variables, the study [23] employed three methods in which the best one, HPSO-TS, found the most promising solution with 4.5213 MW. The study [19] implemented eight improved PSO approaches, and the best one determined the best solution with 4.52560 MW. The study [50] employed three methods in which the best one reported 4.5943 MW for power loss. Clearly, the proposed method found the best solution with 4.5128 MW, while the second-best method, GSA [29], reported 4.51431 MW, and the worst method, TS [23], reported 4.9203 MW.