## 5 Controller Optimization Procedure
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Controller** & **Algorithm** & **ITAE** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(N\) & \(U_{sh}\) & \(O_{sh}\) & \(T_{s}\) \\  & & & & & & & (mHzHz) & (mHz) & (s) \\ \hline \multirow{4}{*}{PIDF} & ALO & 0.2867 & \(-\)0.8400 & 0.1743 & \(-\)1.8700 & 0.0010 & \(-\)59.761 & 6.6323 & 14.5059 \\  & GOA & 0.2146 & \(-\)1.2721 & 0.2020 & 1.1019 & 0.5233 & \(-\)59.761 & 3.8077 & 12.3677 \\  & GWO & 0.2867 & \(-\)0.8372 & 0.1741 & \(-\)0.0677 & 0.0390 & \(-\)59.761 & 6.8512 & 14.5088 \\  & HHO & 0.2228 & \(-\)1.2141 & 0.1981 & 0.9253 & 1.0212 & \(-\)59.761 & 2.4283 & 13.8521 \\  & JSO & 0.1683 & \(-\)1.5236 & 0.2116 & 2.6151 & 0.4902 & \(-\)59.761 & 0.0000 & 12.4524 \\  & WOA & 0.2081 & \(-\)1.0997 & 0.1849 & 1.8759 & 0.5992 & \(-\)59.761 & 0.0000 & 13.1046 \\ \hline \multirow{4}{*}{**Controller**} & **Algorithm** & **ITAE** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(\lambda\) & \(\mu\) & \(U_{sh}\) & \(O_{sh}\) & \(T_{s}\) \\  & & & & & & & (mHzHz) & (mHz) & (s) \\ \hline \multirow{4}{*}{FOPID} & ALO & 0.2095 & \(-\)1.6244 & 0.1564 & 0.8213 & 1.0842 & 0.1311 & 3.1125 & 12.1313 \\  & GOA & 0.1818 & \(-\)0.6785 & 0.0437 & 0.1998 & 1.5684 & 0.6036 & \(-\)60.171 & 0.0000 & 17.3533 \\ \cline{1-1}  & GWO & 0.2157 & \(-\)0.7583 & 0.0280 & 0.0980 & 1.7795 & 0.0645 & \(-\)60.171 & 1.5037 & 18.1233 \\ \cline{1-1}  & HHO & 0.2418 & \(-\)0.9179 & 0.0295 & 0.5486 & 1.6904 & 0.0001 & \(-\)60.171 & 0.0000 & 18.2348 \\ \cline{1-1}  & JSO & 0.1479 & \(-\)2.1209 & 0.0817 & 1.6633 & 1.3457 & 0.1281 & \(-\)60.171 & 0.0000 & 14.3062 \\ \cline{1-1}  & WOA & 0.2156 & \(-\)3.0000 & 0.0540 & 2.6208 & 1.4169 & 0.0001 & \(-\)60.171 & 0.0000 & 16.1266 \\ \hline \multirow{4}{*}{**Controller**} & **Algorithm** & **ITAE** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(N\) & \(b\) & \(c\) & \(U_{sh}\) & \(O_{sh}\) & \(T_{s}\) \\  & & & & & & & (mHz) & (mHz) & (s) \\ \hline \multirow{4}{*}{2 DOF-PIDF} & ALO & 0.3194 & \(-\)0.0004 & 0.3316 & \(-\)0.0173 & 0.2535 & 2.1355 & 0.6743 & \(-\)50.361 & 0.0000 & 15.0948 \\  & GOA & 0.2509 & 0.0250 & 0.9357 & 0.0694 & 1.4305 & 1.0224 & 1.0163 & \(-\)40.875 & 22.854 & 15.61 For the FOPID controller, the results indicated that (in descending order) the HHO (0.2418) algorithm yielded the highest ITAE value, followed by the GWO (0.2157), WOA (0.2156), ALO (0.2095), GOA (0.1818), and JSO (0.1479). Interestingly, it was observed that all algorithms produced undershoots of \(-\)69.171 mHz, with no significant differences among them. However, the overshoots were relatively zero, except in the cases of the ALO (3.1125 mHz) and GWO (1.5037 mHz). In terms of settling time, the HHO (18.2348 s) and GWO (18.1233 s) exhibited the longest settling times. Convergence curves for FOPID were plotted using different meta-heuristic optimization techniques to investigate their performance behavior. The results showed that the HHO, GWO, WOA, and ALO converged prematurely and failed to reduce the objective function effectively. In contrast, the JSO and GOA produced smoother convergence curves, resulting in lower objective function values. For the last controller (2 DOF-PIDF) considered for the frequency load problem in the shipboard MG, the ITAE value for the different algorithms (in descending order) was obtained to be the ALO (0.3194), HHO (0.2748), GOA (0.2509), WOA (0.2370), GWO (0.0476), and JSO (0.0208). Furthermore, it was observed that the WOA (38.175 mHz), GOA (22.854 mHz), and HHO (10.853 mHz) exhibited the largest overshoots, while the GOA  
Figure 7: Optimization procedure results: **(a)** frequency variation with PIDF controller optimized with 6 algorithms **(left)** and corresponding convergence curves **(right)**, **(b)** frequency variation with FOPID controller optimized with 6 algorithms **(left)** and corresponding convergence curves **(right)**, **(c)** frequency variation with 2 DOF-PIDF controller optimized with 6 algorithms **(left)** and corresponding convergence curves **(right)**.