## 5 Results and Discussion
### Radial Configuration Results  
Table 5 shows the comparison between the SSA approach, the BONMIN and COUENNE solvers, and to the proposed GNDO approach that considers fixed reactive power injection throughout the day for the IEEE 33-bus system with a radial configuration. The numerical results presented in Table 5 allow stating that:  
* The GAMS solvers (COUENNE and BONMIN) are stuck in locally optimal solutions, with reductions of 11.22 and 16.12%, respectively, with respect to the benchmark case. This can be attributed to the fact that the MINLP structure of the model (1)-(8) causes exact solution methods based on Branch and Bound and combined with interior point methods to get stuck in local solutions without the ability to escape from them because of the non-convexity of the solution space. * The proposed GNDO approach improved the best solution reported in the literature, which was obtained with the SSA approach, i.e., by about USD 33.42 per year of operation. This improvement was reached since the GNDO detected that node 14 is a better location for one of the D-STATCOMs, instead of node 13 as per the SSA approach. * The total installed reactive power obtained with the GNDO approach was 869.84 kvar, while the solution found with the SSA approach installed 891.99 kvar. This implies that the proposed GNDO approach improved the objective function value by selecting a better set of nodes to place the D-STATCOMs, with the main advantage being that less investment is required for these devices. To demonstrate that the hourly variation of the reactive power injection with D-STATCOMs in distribution networks allows additional gains in the expected annual grid operating costs, Table 6 compares fixed and variable reactive power injection by fixing the nodes reached by the GNDO approach in the exact MINLP model (1)-(8) using the BONMIN solver as a solution method. The results in Table 6 show that an additional reduction of USD 1665.25 is obtained when variable reactive power injection is implemented with D-STATCOMs. That is to say, this is in comparison with the fixed injection scenario. This additional profit is reached when the total installed capabilities of the D-STATCOMs increase to 1034.67 kvar, which in turn implies an increase of 164.83 kvar with respect to the fixed injection scenario. With \begin{table}
\begin{tabular}{l c c|c c c|c c} \hline \hline
**Method** & \multicolumn{2}{c}{**Nodes**} & \multicolumn{2}{c}{**Sizes (Mvar)**} & \multicolumn{2}{c}{**Annual Costs (USD/year)**} & \multicolumn{1}{c}{**Red. (\%)**} \\ \hline Ben. Case & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & 130,613.90 & 0.00 \\ GNDO (Fixed) & \([14\) & \(25\) & \(30\) & \([0.23083\) & \(0.09996\) & \(0.53905]\) & 108,215.94 & 17.15 \\ \hline GNDO (Variable) & \([14\) & \(25\) & \(30\) & \([0.24092\) & \(0.10118\) & \(0.69257\) & 106,550.69 & 18.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Additional improvements reached when variable reactive power is used for daily compensation with D-STATCOMs for the radial grid configuration. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Method** & \multicolumn{2}{c}{**Nodes**} & \multicolumn{2}{c}{**Sizes (Mvar)**} & \multicolumn{2}{c}{**Annual Costs (USD/year)**} & \multicolumn{1}{c}{**Red. (\%)**} \\ \hline Ben. Case & \multicolumn{2}{c}{} & \multicolumn{2}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & 130,613.90 & 0.00 \\ COUENNE & \([5\) & \(6\) & \(11\) & \([0.0000\) & \(0.53600\) & \(0.27466\) & \(115,960,99\) & \(11.22\) \\ BONMIN & \([8\) & \(25\) & \(30\) & \([0.29796\) & \(0.09204\) & \(0.51265\) & \(109,560.85\) & \(16.12\) \\ SSA & \([13\) & \(25\) & \(30\) & \([0.25851\) & \(0.10547\) & \(0.52801\) & \(108,249.36\) & \(17.12\) \\ \hline GNDO & \([14\) & \(25\) & \(30\) & \([0.23083\) & \(0.09996\) & \(0.53905\) & \(108,215.94\) & \(17.15\) \\ \hline \hline \end{tabular}
\end{table}