## 6 Results and Discussions
### 85-Bus System
#### 6.2.2 Results with Reducing the Search Space by SBDMT  
The implementation of PODESCA on the OPSC problem is performed again with a reduced search space by the proposed SBDMT (reduced to 20 buses), with five locations also considered for SC placement. The comparison between the obtained search spaces by SBDMT, CLSF, and PLI is shown in Table 10, where it clearly demonstrates the difference between the three techniques. It also shows that the selected locations by PODESCA in the first case are all included in the search space.  
The obtained results by reimplementing PODESCA and utilizing the SBDMT's search space indicate that the same locations and sizes of SCs are selected.  
As a consequence, the results given in Tables 7-9 are attained again. However, those results are achieved with smaller SD (0.00725) comparing to its value without reducing the search space (0.01208), which means that the SBDMT increases PODESCA's consistency when solving the OPSC problem. Additionally, the convergence characteristics illustrated in Figure 5 show the difference between PODESCA's convergence rates without and with the SBDMT's search space. As depicted in this figure, the minimum total cost (91,018.47 $/year) is reached in both cases, but at the 51 st iteration without reducing the search space, and at the 23 rd iteration with the reduced search space. Those results validate the effectiveness as well as the robustness of each of the proposed PODESCA and SBDMT. Finally, the voltages along the 85-bus system considering the full-load level are depicted in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c c c c c c c} \hline \hline
**Technique** & \multicolumn{10}{c}{**The Reduced Search Space: Total: 20 Buses**} \\ \hline CLSF & 8 & 6 & 7 & 4 & 58 & 27 & 3 & 25 & 29 & 34 & 30 & 26 & 60 & 64 & 2 & 5 & 10 & 68 & 52 & 28 \\ PLI & 54 & 55 & 51 & 76 & 69 & 74 & 39 & 72 & 66 & 28 & 62 & 38 & 61 & 60 & 59 & 82 & 37 & 26 & 80 & 11 \\ SBDMT & 69 & 62 & 38 & 61 & 80 & 8 & 76 & 54 & 55 & 51 & 60 & 44 & 39 & 78 & 75 & 30 & 46 & 45 & 12 & 49 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The reduced search space by SBDMT comparing to CLSF and PLI for the 85-bus system.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Load level** & **Description** & **Without SCs** & **With SCs** & **Installed SCs (kVAr)** \\ \hline \multirow{8}{*}{100\%} & \(V_{min}\) (p.u.) & 0.8713 & 0.9207 & 5 locations \\  & VD (p.u.) & 0.8203 & 0.3121 & Total: 2000 \\  & APL (kV) & 315.714 & 148.25 & \\  & RPL (kVAr) & 198.370 & 92.520 & \\  & Ann. cost (\$/year) & 165,939.28 & 91,018.47 & \\  & Net sav. (\$/year) & - & 74,920.80 & \\ \hline \multirow{8}{*}{75\%} & \(V_{min}\) (p.u.) & 0.9068 & 0.9355 & 12 (350) \\  & VD (p.u.) & 0.4323 & 0.1888 & 30 (550) \\  & APL (kV) & 166.76 & 85.0098 & 60 (500) \\  & RPL (kVAr) & 104.83 & 52.4542 & Total: 1400 \\  & Ann. cost (\$/year) & 87,649.06 & 53,541 & \\  & Net sav. (\$/year) & - & 34,108.06 & \\ \hline \multirow{8}{*}{50\%} & \(V_{min}\) (p.u.) & 0.9397 & 0.9573 & 12 (250) \\  & VD (p.u.) & 0.1811 & 0.0834 & 30 (350) \\  & APL (kV) & 70.015 & 36.7358 & 60 (300) \\  & RPL (kVAr) & 44.034 & 22.6901 & Total: 900 \\  & Ann. cost (\$/year) & 36,800 & 25,668 \\  & Net sav. (\$/year) & - & 11,132 \\ \hline \multirow{2}{*}{Injected SCs (kVAr)} & Fixed (Location, kVAr) & \multicolumn{4}{c}{(12, 250), (30, 350), (60, 300)} \\  & Switched (Location, kVAr) & \multicolumn{4}{c}{(12, 200), (30, 200), (49, 350), (60, 200), (69, 250)} \\ \hline \hline \end{tabular}
\end{table}
Table 9: Optimal results of PODESCA for the 85-bus system considering different load levels.  
It can be clearly noted from Figure 6 that the voltages are well enhanced after adding the optimal SCs.