# Set-Based Group Search Optimizer for Stochastic Many-Objective Optimal Power Flow
## 4 Case Study
### Simulation Results on the Different Proportion of REs  
In this part, a simulation study is conducted using a modified IEEE 57-bus and 118-bus system. Meanwhile, three types of renewable energy with various ratios are set in this research in order to examine the effects of such renewable energy with respect to the search performance of SetGSO: (1) 30% REs; (2) 40% REs; and (3) 50% REs. Tables 3 and 4 show the specific locations of wind farms and photovoltaic power generation for the above three schemes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**30\% REs**} & \multicolumn{2}{c}{**40\% REs**} & \multicolumn{2}{c}{**50\% REs**} \\ \hline \multirow{2}{*}{**Bus NO.**} & \multicolumn{2}{c}{**Type of Stochastic**} & \multirow{2}{*}{**Bus NO.**} & \multicolumn{2}{c}{**Type of Stochastic**} & \multirow{2}{*}{**Bus NO.**} & \multicolumn{2}{c}{**Type of Stochastic**} \\  & **Variables** & & & & & \\ \hline
15 & WT & 15 & WT & 15 & WT \\
18 & WT & 18 & WT & 18 & WT \\
19 & PV & 19 & PV & 19 & PV \\
55 & PV & 24 & WT & 24 & PV \\
56 & PV & 27 & PV & 27 & PV \\
74 & WT & 32 & WT & 32 & WT \\
76 & PV & 34 & WT & 34 & WT \\
77 & PV & 36 & WT & 36 & WT \\
112 & WT & 40 & WT & 40 & WT \\
113 & WT & 55 & PV & 55 & PV \\ - & - & 56 & PV & 56 & PV \\ - & - & 74 & - & 62 & WT \\ - & - & 76 & PV & 70 & WT \\ - & - & 77 & PV & 74 & WT \\ - & - & 92 & - & 76 & PV \\ - & - & 99 & PV & 77 & PV \\ - & - & 104 & - & 94 & WT \\ - & - & 112 & PV & 95 & PV \\ - & - & 113 & PV & 96 & PV \\ - & - & 116 & - & 92 & WT \\ - & - & - & - & 99 & PV \\ - & - & - & - & 104 & WT \\ - & - & - & - & 107 & WT \\ - & - & - & - & 110 & WT \\ - & - & - & - & 112 & PV \\ - & - & - & - & 113 & PV \\ - & - & - & - & 116 & WT \\ - & - & - & - & 84 & PV \\ - & - & - & - & 82 & PV \\ - & - & - & - & 83 & PV \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The uncertain renewable power position of WTs and PVs in the modified IEEE 118-Bus system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**30\% REs**} & \multicolumn{2}{c}{**40\% REs**} & \multicolumn{2}{c}{**50\% REs**} \\ \hline \multirow{2}{*}{**Bus NO.**} & \multicolumn{2}{c}{**Type of Stochastic**} & \multirow{2}{*}{**Bus NO.**} & \multicolumn{2}{c}{**Type of Stochastic**} & \multirow{2}{*}{**Bus NO.**} & \multicolumn{2}{c}{**Type of Stochastic**} \\  & **Variables** & & & & **Variables** & \\ \hline
21 & WT & 21 & WT & 21 & WT \\
22 & WT & 22 & WT & 22 & WT \\
48 & PV & 48 & PV & 48 & PV \\ - & - & 36 & WT & 36 & PV \\ - & - & 37 & PV & 37 & PV \\ - & - & - & - & 45 & WT \\ - & - & - & - & 46 & WT \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The uncertain renewable power position of WTs and PVs in the modified IEEE-57 Bus system. In the case of four set-based individuals, the novel stochastic many-objective optimization technique, SetGSO, and the conventional approach, SetGA, function autonomously. The outcomes are presented and compared in Tables 5 and 6. The metrics of mean Euclidean distance and hyper-volume indicate that the SetGSO approach has the potential to yield a superior Pareto front or Pareto optimum solution under similar circumstances. Furthermore, the SetGSO approach demonstrates superiority over the original SetGA method in terms of the average imprecision index, suggesting a higher level of solution stability. Furthermore, the results from Tables 5 and 6 also demonstrate that there was limited variability in the imprecision measure, while the hyper-volume measure displayed a significant rise with an increase in the number of wind and photovoltaic farms. This implies that the associated uncertainty increases significantly as the proportion of renewable energy sources in the system continues to rise. The quest for a stable optimal solution becomes challenging. Consequently, when balancing the convergence of the Pareto frontier solutions and addressing uncertainty, the algorithm tends to prioritize the pursuit of solutions with superior convergence rather than aiming solely for minimized imprecision.