## 5 Computational Implementation and Results
### IEEE 69-Bus  
To present the effectiveness and robustness of the proposed DCVSA to solve the problem of the optimal siting and sizing of D-STATCOMs in the IEEE 69-bus test feeder, first six solutions reached by our proposed method are presented in Table 9. Please note that the base case for this test feeder has an annual cost of US $119,715.63.  
Regarding the solutions reported in Table 9 it is observed that: (i) in all the solutions the node 61 and 21 are recur, except in solution 2 where node 17 replaces node 21, which implies that these are the most suitable nodes to locate D-STATCOMs. On the basis of maximum reduction in the annual operating costs, node 61 with the D-STATCOM of the largest size appears in all the simulation cases; (ii) the solutions 3 to 6 are comparable because the differences among their annual costs are less than US $0.4100 per year of operation and the fact that in all these node 61 node features as the location for a D-STATCOM with a nominal rate of 0.5741 MVAr; and (iii) the proposed optimization approach for the IEEE 69-bus test system has an efficiency of the 49%, i.e., among 100 consecutive evaluations, 49 find global optimal solutions that cause the reduction of about 13.97% in the annual  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Sol.**} & \multirow{2}{*}{**Location and Size Node (MVAr)**} & \multicolumn{2}{c}{\(\mathbf{A_{\text{cost}}}\)} \\  & & **(US \$/Year)** & **Rep.** \\ \hline
1 & \{14(0.1599), 30(0.3591), 32(0.1072)\} & 98,497.90 & 36 \\
2 & \{11(0.0659), 14(0.1148), 30(0.4578)\} & 98,564.29 & 22 \\
3 & \{10(0.0642), 14(0.1175), 30(0.4574)\} & 98,565.03 & 10 \\
4 & \{11(0.0787), 15(0.1019), 30(0.4578)\} & 98,567.91 & 8 \\
5 & \{12(0.1110), 14(0.0666), 30(0.4591)\} & 98,569.08 & 2 \\
6 & \{12(0.0804), 15(0.0972), 30(0.4591)\} & 98,570.12 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Lists of solutions found by the proposed DCVSA in the IEEE 33-bus.  
Figure 3: Distribution of the operative and investment costs for the comparative and proposed approach.  
operative cost as compared to the base case when D-STATCOMs are installed at nodes 61, 61 and 64 with nominal rates of 0.0839 MVAr, 0.4601 MVAr, and 0.1139 MVAr, respectively.  
**Remark 5**.: _Comparisons with the GAMS solvers are not presented since the BONMIN and the COUENNE failed to solve the MINLP model for the IEEE 69-bus test feeder. This further demonstrates the efficiency and robustness of the DCVSA for addressing this complex optimization problem from the meta-heuristic optimization point of view._  
If we split the optimal solution yield by the proposed DCVSA in Table 9, then, the required inversion in D-STATCOMs is US $8372.08 against the reduction in the costs of the energy losses of US $25, 096.92 seen in the base case. This clearly compensates for the required inversions in the D-STATCOMs, because it leads to a reduction of 13.97% in the annual operating costs for the grid operator, i.e., US $16, 724.84 per year of operation.  
It is also important to mention the processing times the proposed approach takes. It is about 300.68 s to solve the optimization problem with excellent numerical results while the GAMS solvers diverge after 2187.96 s of exploration in the solution space.