## 4 Simulation Results
### HIC-GWA Robustness Analysis  
Robustness analysis, which is a non-empirical form of confirmation, is an indispensable procedure in studying complex phenomena. A sensitivity analysis for case studies 1 and 11 has been performed to evaluate the robustness of the considered metaheuristic. Each parameter of the HIC-GWA has been perturbed by changing the values up and down. Likewise, optimization parameters values have been changed also to check the global effect of parameter's variations on the solution of the OPF problem. The equivalent Pareto solutions are illustrated in Table 18. The deviation ratio between normal and perturbed solutions is calculated using the following equation:  
\[\textit{Deviation}\ (\%)=\frac{\textit{Normal}\ \textit{Solution}-\textit{perturbed}\ \textit{Solution}}{\textit{Normal}\ \textit{Solution}}\times 100, \tag{62}\]  
Small deviations affirm the robustness of the HIC-GWA to variation of parameters in solving OPF problems.  
To confirm the robustness of the HIC-GWA, we compare best and worst fuel cost averages to recently published OPF optimization methods in Table 19. The proposed HIC-GWA has consistently better solutions over 30 trial runs.  
Table 20 shows the convergence speed of the HIC-GWA compared to recently published optimization methods. With 14.34 (s), HIC-GWA is second fastest to MFO by one hundredth of a second.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Time (s)** \\ \hline MICA-TLA & 30.74 \\ LTLBO & 22.78 \\ HMTSO-SFLA & 19.06 \\ MPSO & 16.05 \\ MDE & 15.63 \\ MSA & 14.91 \\ FPA & 14.79 \\ HIC-GWA & 14.34 \\ \hline \hline \end{tabular}
\end{table}
Table 20: Case 2 simulation time.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multicolumn{2}{c}{**30 Bus Power System**} & \multicolumn{2}{c}{**118 Bus Power System**} \\ \cline{2-5}  & **Cost (S/h)** & **Deviation (\%)** & **Parameters** & **Cost (S/h)** & **Deviation (\%)** \\ \hline Normal Solution & 798.20 & 0.0 & Normal Solution & 129,633.70 & 0.0 \\ \(N_{\textit{Century}}=15+5\) & 797.38 & +0.1017 & \(N_{\textit{Century}}=200+30\) & 129,631.93 & +0.00137 \\ \(N_{\textit{Century}}=15-5\) & 799.07 & \(-0.1102\) & \(N_{\textit{Century}}=200-30\) & 129,636.79 & \(-0.00228\) \\ \(N_{\textit{Sup}}=5+2\) & 797.33 & +0.1082 & \(N_{\textit{Sup}}=40+10\) & 129,632.44 & +0.00098 \\ \(N_{\textit{Sup}}=5-2\) & 797.00 & +0.1491 & \(N_{\textit{Sup}}=40-10\) & 129,630.66 & +0.00235 \\ \(N_{\textit{Sup}}=5+2\) & 797.12 & +0.1341 & \(N_{\textit{Sup}}=10+3\) & 129,631.27 & +0.00149 \\ \(N_{\textit{Sup}}=5-2\) & 798.98 & \(-0.0984\) & \(N_{\textit{Sup}}=10-3\) & 129,634.92 & \(-0.00094\) \\ All (up) & 797.06 & +0.1420 & All (up) & 129,630.94 & +0.00213 \\ All (Down) & 799.08 & \(-0.1110\) & All (Down) & 129,645.84 & \(-0.00936\) \\ \hline \hline \end{tabular}
\end{table}
Table 18: Sensitivity analysis for IEEE 30 bus and 118 bus power systems.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Best Cost (S/h)** & **Worst Cost (S/h)** & **Average Cost (S/h)** \\ \hline MSA & 646.84 & 648.03 & 646.86 \\ MPSO & 646.73 & 656.23 & 649.86 \\ MDE & 650.28 & 653.40 & 651.26 \\ MFO & 649.27 & 650.62 & 649.89 \\ FPA & 651.38 & 654.33 & 652.96 \\ LTLBO & 647.43 & 647.86 & 647.47 \\ ABC & 649.09 & 659.77 & 654.08 \\ GABC & 647.03 & 647.12 & 647.08 \\ HIC-GWA & 645.85 & 647.03 & 645.87 \\ \hline \hline \end{tabular}
\end{table}
Table 19: Comparisons of the results obtained for case 2.