## 6 Results and Discussions
### 118-Bus System
#### 6.3.2 Results with Reducing the Search Space by SBDMT
In addition, the convergence characteristics of the proposed algorithm are compared without and with utilizing the SBDMT as illustrated in Figure 7, where it is obvious that the initial solutions in both cases are different due to the big difference in the search space's size, and the OF's values are reached with different convergence rates, where 103 iterations are required to obtain the optimal value (493,969.50 $/year) without reducing the search space. However, if the search space is reduced by the SBDMT, only 59 iterations are needed to reach the optimal value (492,225.66 $/year). Those results highly validate the effectiveness and robustness of both PODESCA and SBDMT for solving the OPSC problem for large-scale RDSs. Hence, since the best solutions are obtained by utilizing both PODESCA and SBDMT together for this system, the optimal results for different load levels are determined according to the reduced search space and listed in Table 16, where four fixed SCs with sizes of 950, 700, 300, and 900 kVAr are injected at buses 39, 72, 85, and 118, respectively. Whereas, the switched SCs are placed as 1150, 550, 600, 400, 850, 600, 1200, 1100 and 600 kVAr respectively at buses 34, 39, 48, 72, 74, 85, 91, 108 and 118. At last, the voltages along the 118-bus system considering the full-load level (obtained by the best solutions) are depicted in Figure 8, where a good improvement in voltages is achieved after adding the optimal SCs. Figure 7: Convergence characteristics of PODESCA for the 118-bus system. After testing the proposed methods on three different-scale RDSs, the following advantages can be pointed out:  
1. The rapid convergence towards the global optimal solutions is guaranteed by PODESCA due to its robust structure, which efficiently combines the mechanisms of DE, SCA, and QOT. By its unique hybridization, the search exploration and diversity are increased. It reliably keeps the best solutions in every generation and thus ensures their flexible flow to the optimal region inside the search space. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Load level** & **Description** & **Without SCs** & **With SCs** & \begin{tabular}{c} **Installed SCs** \\ **(kVar)** \\ \end{tabular} \\ \hline \multirow{6}{*}{100\%} & \(V_{\min}\) (p.u.) & 0.8688 & 0.9068 & 9 locations \\  & VD (p.u.) & 0.3576 & 0.1761 & Total: 9250 \\  & APL (kW) & 1297.95 & 837.89 & \\  & RPL (kVAr) & 978.54 & 630.72 & \\  & Ann. cost (§/year) & 682,202.52 & 492,225.66 & \\  & Net sav. (§/year) & - & 189,976.86 & \\ \hline \multirow{6}{*}{75\%} & \(V_{\min}\) (p.u.) & 0.9049 & 0.9288 & 39 (1500) \\  & VD (p.u.) & 0.1896 & 0.1067 & 72 (1100) \\  & APL (kW) & 697.24 & 479.34 & 85 (600) \\  & RPL (kVAr) & 527.16 & 362.01 & 91 (750) \\  & Ann. cost (§/year) & 36,6469.3 & 282.290 & 118 (1500) \\  & Net sav. (§/year) & - & 84,179.3 & Total: 5450 \\ \hline \multirow{6}{*}{50\%} & \(V_{\min}\) (p.u.) & 0.9385 & 0.9517 & 39 (950) \\  & VD (p.u.) & 0.0806 & 0.0504 & 72 (700) \\  & APL (kW) & 297.13 & 214.43 & 85 (300) \\  & RPL (kVAr) & 225.2 & 163.46 & 118 (900) \\  & Ann. cost (§/year) & 156,171.5 & 129,430 & Total: 2850 \\  & Net sav. (§/year) & - & 26,741.5 & \\ \hline \multirow{6}{*}{Injected SCs (kVAr)} & Fixed (Location, kVAr) & \
\begin{tabular}{c} (39, 950), (72, 700), (85, 300), (118, 900) \\ (34, 1150), (39, 550), (48, 600), (72, 400), (74, 850), (85, 600), (91, \\  \\  \\ \end{tabular} \\  & Switched (Location, kVAr) &
\begin{tabular}{c} (39, 950), (72, 700), (85, 300), (118, 900) \\ (34, 1150), (39, 550), (48, 600), (72, 400), (74, 850), (85, 600), (91, \\  \\  \\ \end{tabular} \\  & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \end{tabular}
\end{table} 
Table 16: Optimal results of PODESCA and SBDMT for the 118-bus system considering different load levels. Figure 8: Voltage profile of the 118-bus system considering full-load level using PODESCA and SBDMT. 2. The consistency and robustness of PODESCA are verified by indicating the small standard deviations and fast convergence rates. 3.