## 5 Computational Implementation and Results
### IEEE 33-Bus  
In Table 7 are listed the optimal solutions for the problem of the optimal placement and sizing of D-STATCOMs in electric distribution networks using the GAMS optimizers and the proposed optimization DCVSA approach. Please note that the solutions reported in Table 7 show that: (i) the best solution is reported by the proposed DCVSA with the selection of the nodes 14, 30 and 32 to locate the D-STATCOMs with sizes of 0.1599 MVAr, 0.3591 MVAr, and 0.1072 MVAr, respectively. These reduce the operating cost of the electric distribution grid by about US$14243 per year of operation, i.e., 12.63% with compared to the base case; and (ii) the GAMS solvers are stuck in local optimal solutions. The BONMIN solver reduces the total annual grid cost by about 9.13%, while the COUENNE achieves a reduction of only 4.57%. These problems in the solutions reported by the GAMS solvers are clearly attributable to the nonlinear non-convex behavior of the objective function and the power balance constraints, which makes it impossible to combine classical branch and bound method, which, with interior points, finds the global optimum. This is possible only in mixed-integer convex problems.  
To present the effect of the optimal location on the costs of the annual energy losses, we desegregate the investment and the operative costs associated with the installation of the D-STATCOMs reported in Table 7. In the Figure 3 are presented the desegregated costs for the IEEE 33-bus test system.  
The behavior of the annual investment and operating costs presented in Figure 3 shows that the proposed DCVSA decides to make an inversion in D-STATCOMs that duplicates the inversion made by the GAMS solvers. However, this behavior of the DCVSA leads to considerable reduction in the cost of annual energy losses and helps to improve the grid behavior significantly as compared with the GAMS solvers. In addition, if we observe the cost of annual energy losses of the BONMIM solver, these are about US $98.67 \(\times\) 10\({}^{3}\), which is itself higher than the summation of the costs reported with the proposed DCVSA which is about US $98.50 \(\times\) 10\({}^{3}\)--This result proves that the proposed approach is more suitable for solving complex mixed-integer non-convex optimization problems than the GAMS package as shown in [9]. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Discrete-continuous vortex search algorithm**} \\ \hline Population size:10 & Iterations’ number: 1000 \\ Population building: Gaussian Distribution & \\ \hline \multicolumn{2}{c}{**SAPF method**} \\ \hline Iterations’ number: 1000 & Convergence error: 1 \(\times\) 10\({}^{-10}\) \\ \hline \multicolumn{2}{c}{**Experimental tests in each test feeder**} \\ \hline Consecutive evaluations & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parametrization of the DCVSA. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Approach**} & \multicolumn{1}{c}{**Location and Size Node (MVAr)**} & \multicolumn{1}{c}{\(\boldsymbol{A_{\text{cost}}}\) **(US \$/Year)**} \\ \hline Caso & — & 112,740.90 \\ base & & 107,589.50 \\ COUENNE & & 102,447.29 \\ BONMIN & & 102,447.29 \\ \hline DCVSA & & 98,497.90 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal location and sizes of the D-STATCOMs in the IEEE 33-bus. To demonstrate the efficiency of the proposed DCVSA we report the first six solutions found by this approach in Table 8.  
Please note that the results in Table 8 demonstrate that: (i) after 100 consecutive evaluations, the proposed DCVSA is found to have a 36% efficiency at finding the same optimal solution, (ii) all the solutions in Table 8 are indeed better than the results achieved by the GAMS optimizers in reaching the final objective function value, and (iii) it is observed that the node 30 appears in all the solutions with sizes larger than 350 kvar, which implies that this is the most sensitive node for the placement of D-STATCOMs in the IEEE 33-bus test system.