## 5 Results and Discussion
### Results for the 33-Node AC MG Operating in Islanded Mode
#### 5.2.1 Statistical Analysis of the Objective Function \(FO_{1}\) (Operating Costs)  
When analyzing the islanded mode, it is not possible to evaluate a base case, as the system cannot operate under the generation and demand curves representing a typical day without violating the power limits imposed on the diesel generator. This highlights that, in this mode of operation, the smart management of the WTs installed in the MG is essential to prevent load disconnections. Upon evaluating the best solutions obtained by the analyzed methodologies with regard to \(FO_{1}\), it was observed that all of them achieved outstanding performance, reaching an average minimum cost of USD 12,352.5471 while fully complying with the operational constraints imposed on the MG in islanded mode. \begin{table}
\begin{tabular}{c|c c c|c c c} \hline \hline \multicolumn{6}{c}{Minimum Solution} & \multicolumn{4}{c}{Average Solution} \\ \hline Methodology & \(FO_{1}\) & \(FO_{2}\) & \(FO_{3}\) & \(FO_{1}\) & \(FO_{2}\) & \(FO_{3}\) \\ \hline MC & 15529.0698 & 1256.2840 & 14066.4777 & 15809.7272 & 1315.5942 & 14365.8213 \\ PGA & 11447.2650 & 545.5987 & 10265.1927 & 11454.6395 & 547.3763 & 10267.4865 \\ PSO & 11431.9460 & 577.7305 & 10259.5220 & 11553.4068 & 603.2329 & 10384.9829 \\ JAYA & 11386.6170 & 588.5812 & 10261.8013 & 11647.2806 & 682.1490 & 10511.3615 \\ GNDO & 12903.9259 & 811.2136 & 11589.6865 & 13547.2663 & 867.7252 & 12206.0300 \\ MVO & 11416.4589 & 618.8671 & 10262.3398 & 11465.3359 & 656.2798 & 10282.4269 \\ \hline \multicolumn{6}{c}{Standard deviation (\%)} & \multicolumn{4}{c}{Processing time (s)} \\ \hline Methodology & \(FO_{1}\) & \(FO_{2}\) & \(FO_{3}\) & \(FO_{1}\) & \(FO_{2}\) & \(FO_{3}\) \\ \hline MC & 0.6233 & 1.3077 & 0.6455 & 319.3222 & 318.3969 & 299.5148 \\ PGA & 0.0189 & 0.1703 & 0.0079 & 217.3473 & 216.7334 & 218.1240 \\ PSO & 0.9641 & 3.5881 & 1.2165 & 525.5258 & 502.6614 & 529.7702 \\ JAYA & 1.9563 & 6.8158 & 1.9286 & 196.3772 & 195.5680 & 200.7937 \\ GNDO & 1.4679 & 2.6803 & 1.8065 & 229.1272 & 219.8673 & 224.7762 \\ MVO & 0.3911 & 2.2502 & 0.2401 & 470.7246 & 536.7649 & 547.4273 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Solutions obtained for the 33-node MG operating in islanded mode. Figure 21: Apparent power dispatched by WTs for the 33-node AC MG during grid-connected operation. As with the analysis conducted for the MG in grid-connected mode, the best-performing metaheuristic technique was primarily identified based on its average solution and standard deviation since these indicators reflect both the effectiveness and consistency of each strategy. In this context, PGA achieved an average solution of USD 11454.6395, representing improvements of 27.5469%, 0.8549%, 1.6540%, 15.4469%, and 0.0933% over MC, PSO, JAYA, GNDO, and MVO. This behavior is clearly illustrated in Figure 22. It is important to note that, for the islanded operation mode, the performance of the PGA was directly compared against the other methodologies rather than the base case. This is because considering a base case for the islanded mode without implementing intelligent WT management would result in violations of the injection limits at the slack node during certain hours of the day. In Figure 23, a boxplot illustrates the 100 solutions obtained by each technique with regard to the operating costs of the MG in islanded mode. The plot clearly shows that the algorithm with the lowest standard deviation is PGA, with a value of 0.0189%. This result translates into improvements of 0.6044%, 0.9452%, 1.9374%, 1.4490%, and 0.3722% when compared with MC, PSO, JAYA, GNDO, and MVO. Figure 23: Boxplot of the solutions for \(FO_{1}\) in the 33-node AC MG operating in islanded mode. Figure 22: PGA’s operating cost reductions relative to other methodologies in the 33-node MG operating in islanded mode. When minimizing operating costs during islanded operation, JAYA reported the lowest processing time, completing the optimization in 196.3772 s. The proposed PGA followed close behind with 217.3473 s, which makes it the second-fastest strategy.