## 6 Results and Discussions
### 69-Bus System
#### 6.1.2 Results with Reducing the Search Space by SBDMT  
PODESCA is applied again on the OPSC problem but in this case with a reduced search space by the proposed SBDMT. By applying this technique, the search space is significantly reduced, where only 15 buses are considered for selecting the optimal locations of SCs. For further validation of the proposed SBDMT, the obtained search space is compared to that from CLSF and PLI as illustrated in Table 6.  
It can be easily seen that the three techniques give different locations. It is also clear that the selected locations by PODESCA in the first case are included in the search space. By utilizing the SBDMT's search space, PODESCA's results are obtained. It is observed that the same optimal locations and sizes are selected by the proposed algorithm. Accordingly, the same results are attained as presented in Tables 3-5.  
Nevertheless, the SD in this case is smaller than that obtained without using the SBDMT, where it reached the value of 0.00443. Those results prove that SBDMT effectively reduces the search space while keeping the most critical and suitable buses for SC placement.  
This technique also helps to increase PODESCA's consistency by reducing the SD. In addition to that, the convergence characteristics of the proposed algorithm are plotted and compared without and with utilizing the SBDMT as depicted in Figure 3. This figure clearly demonstrates that the same OF's value (85,913.48 $/year) is achieved in both cases but with different convergence rates, where 30 iterations are needed to reach the optimal value without reducing the search space. On the other hand, only 13 iterations are needed to get to the optimal value if the search space is reduced by the  
\begin{table}
\begin{tabular}{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: 15 Buses**} \\ \hline CLSF & 57 & 58 & 7 & 6 & 61 & 60 & 10 & 59 & 55 & 56 & 49 & 12 & 54 & 13 & 14 \\ PLI & 61 & 64 & 59 & 65 & 21 & 12 & 11 & 62 & 17 & 18 & 16 & 8 & 24 & 68 & 69 \\ SBDMT & 61 & 49 & 64 & 65 & 62 & 59 & 63 & 60 & 58 & 66 & 57 & 21 & 17 & 20 & 18 \\ \hline \hline \end{tabular}
\end{table}
Table 6: The reduced search space by SBDMT comparing to combined loss sensitivity factor (CLSF) and power loss index (PLI) for the 69-bus system.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Load level** & **Description** & **Without SCs** & **With SCs** & **Installed SCs (kVAr)** \\ \hline \multirow{6}{*}{100\%} & \(V_{min}\) (p.u.) & 0.9092 & 0.9302 & 20 (250) \\  & VD (p.u.) & 0.0993 & 0.0622 & 61 (1150) \\  & APL (kW) & 225.02 & 147.78 & Total: 1400 \\  & RPL (kVAr) & 102.10 & 68.90 & \\  & Ann. cost ($/year) & 118,270.5 & 85,913.48 & \\  & Net sav. ($/year) & - & 32,356.50 & \\ \hline \multirow{6}{*}{75\%} & \(V_{min}\) (p.u.) & 0.9335 & 0.9476 & 61 (900) \\  & VD (p.u.) & 0.0536 & 0.0365 & Total: 900 \\  & APL (kW) & 121.04 & 83.065 & \\  & RPL (kVAr) & 55.065 & 38.643 & \\  & Ann. cost ($/year) & 63,618.62 & 48,779 & \\  & Net sav. ($/year) & - & 14,839.62 & \\ \hline \multirow{6}{*}{50\%} & \(V_{min}\) (p.u.) & 0.9567 & 0.9650 & 61 (550) \\  & VD (p.u.) & 0.0229 & 0.0162 & Total: 550 \\  & APL (kW) & 51.610 & 36.057 & \\  & RPL (kVAr) & 23.537 & 16.824 & \\  & Ann. cost ($/year) & 27,126.32 & 22,321.64 & \\  & Net sav. ($/year) & - & 4804.68 & \\ \hline \multirow{2}{*}{Injected SCs (kVAr)} & Fixed (Location, kVAr) & \multicolumn{4}{c}{(61, 550)} \\  & Switched (Location, kVAr) & \multicolumn{4}{c}{(20, 250), (61, 600)} \\ \hline \hline \end{tabular}
\end{table}
Table 5: Optimal results of PODESCA for the 69-bus system considering different load levels.  
proposed SBDMT. Those results verify the effectiveness and robustness of both proposed PODESCA and SBDMT for solving the OPSC problem.  
Finally, Figure 4 shows the voltages along the 69-bus system considering the full-load level, where it can be obviously remarked that a good enhancement in voltages is achieved after adding the optimal SCs.