## 5 Results and Discussion  
To validate the proposed solution methodology, the solution of the optimization model for locating and sizing D-STATCOMs in electrical distribution via the AESDC was implemented in MATLAB version R2021b on a computer with an Intel Core i7-10750H @2.6 GHz and 16.0 GB of DDR4 RAM at 2300 MHz on a 64-bit version of Microsoft Windows 10 Home. The validation of the proposed master-slave optimization model in conjunction with the improvement stage was carried out while considering the following facts:  
* Two configurations are considered for the IEEE 33-bus grid: a radial configuration with the original structure presented in Figure 2 and a meshed configuration with three tie-lines added to the radial topology, as proposed in [22].
* The GNDO approach, combined with the successive approximations power flow method, is used to obtain the initial nodal locations and sizes for the D-STATCOMs with fixed reactive power injection. Thus, the location of these devices is fixed in the exact MINLP model while aiming for additional gains for operating these reactive power compensators with variable reactive power throughout the day.
* For comparison, the MINLP model's exact solutions obtained with the BONMIN and COUENNE solvers and the recently developed Salp Swarm Algorithm (SSA) [22] are considered in order to confirm the effectiveness and robustness of the proposed optimization approach.  
Note that in order to ensure the effectiveness of the proposed master-slave optimization regarding the solution quality, 100 consecutive evaluations of the complete solution methodology were considered, with 1000 iterations for each of them. With these values, a statistical study of the GNDO performance was conducted considering mean, maximum, minimum, and standard deviation parameters. It is worth mentioning that in order to to make fair comparisons with the SSA approach presented by the authors of [22] as well as the implementation of the exact model in GAMS software with the BONMIN and COUNNE solvers, all results were obtained with our own implementations that were conducted during this research.  
It is worth mentioning that in order to ensure that all the numerical validations presented in this paper were correct, the following validation procedure was followed.  
* The solution of the multi-period optimal power flow problem for the benchmark case was implemented in MATLAB with our own scripts. However, to ensure the effectiveness of this approach, a comparative analysis with the DIgSILENT software was performed.
* The convergence analysis of the power flow problem for the proposed successive approximations power flow method was proven by the authors of [30], which im  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Paz** & **Value** & **Unit** & **Paz.** & **Value** & **Unit** \\ \hline C\({}_{\text{A}}\)wh & 0.1390 & USD\({}^{*}\)kWh & \(T\) & 365 & Days \\ A\({}_{\text{A}}\) & 0.50 & h & \(\alpha\) & 0.30 & USD/MVAr\({}^{3}\) \\ \(\beta\) & \(-\)305.10 & USD\({}^{*}\)MVAr\({}^{2}\) & \(\gamma\) & 127,380 & USD/MVAr \\ \(k_{1}\) & 6/2190 & 1/Days & \(k_{2}\) & 10 & Years \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parametric information to evaluate the objective function regarding the investment costs of the D-STATCOMs.  
plies that under the studied conditions this method always converges to the power flow solution.
* The solution of the MINLP model with he GNDO approach combined with the successive approximations method was also validated via GAMS and DlgSILENT.