## 4 Case and Discussion
### Discussion  
Firstly, this section discusses the influence of the step parameter \(\Delta c\) of the SSO method on the search stability and speed. Table 4 shows the calculation time and optimization target value of the SSO method under the step parameter \(\Delta c\), and Figure 7 shows the change process of the corresponding curve. Table 4 and Figure 7 show that (1) with the decrease in step parameter \(\Delta c\), the solution time of the SSO method increases; especially when \(\Delta c\leq 0.03\), the solution time almost doubles. However, when the step parameter \(\Delta c\) is large, \(\Delta c\in[0.08,2.5]\), the solution time of the SSO method fluctuates in a small range between 250\(-\)350 s. (2) With the decrease in step parameter \(\Delta c\), the fluctuation ranges of \(J_{1}/\min(J_{1})\) and \(J_{2}/\min(J_{2})\) are 1.51-1.60 and 2.37-2.40, respectively. This means that the accuracy of the SSO method is insensitive to the step parameter \(\Delta c\). Therefore, in the SSO method, a larger step size can be selected for calculation by selecting the step parameter \(\Delta c\). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\Delta c\) & **0.02** & **0.03** & **0.05** & **0.08** & **0.1** & **0.15** & **0.18** & **0.2** & **0.25** \\ \hline Search time & 688.67 & 442.67 & 358.85 & 325.88 & 280.44 & 309.58 & 316.85 & 330.09 & 263.99 \\ \(J_{1}/\mid\min(J_{1})\mid\) & 1.52 & 1.51 & 1.51 & 1.52 & 1.51 & 1.53 & 1.54 & 1.60 & 1.52 \\ \(J_{2}/\mid\min(J_{2})\mid\) & 2.40 & 2.40 & 2.40 & 2.40 & 2.40 & 2.40 & 2.38 & 2.37 & 2.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimization results under different parameters \(\Delta c\) using the SSO method. Then, to verify the applicability of the SSO method, this section establishes two optimization objectives \(J_{3}\) and \(J_{4}\) from the aspects of node voltage and new energy station in the DES, which are defined as shown in Expressions (19) and (20). At time \(t\), the \(i\)-th node voltage and the power have been defined as \(V_{i}(t)\) and \(P_{\text{New},i}(t)\), respectively. The duration of the typical day and the number of grid nodes have been set to \(T\) and \(N_{\text{b}}\). The voltage reference value is \(V_{\text{base}}\). \[\text{min}\quad J_{3}=\max\limits_{t=1}^{T}\biggl{(}\max\limits_{i=1}^{N_{ \text{b}}}\left(\text{abs}\bigl{(}V_{i}^{2}(t)/V_{\text{base}}^{2}-1\bigr{)} \bigr{)}\right) \tag{19}\]  
\[\text{min}\quad J_{4}=-\sum\limits_{t=1}^{T}\sum\limits_{i=1}^{N_{\text{b}}} \left(P_{\text{New},i}(t)\right) \tag{20}\]  
Table 5 represents the comparison results under the dual objectives of \((J_{1},J_{3})\) with single-objective optimization, methods 1, 2 and 3 and the SSO method, respectively. Figure 8 shows the optimization results under four different methods and the stepper search process of the SSO method. In Table 5, comparing the results of single-objective optimization, methods 1 and 2 and the SSO method can effectively balance the contradiction between the two objectives, but method 3 is not. Method 3 focuses more on the objective function \(J_{1}\), resulting in an excessive loss of \(J_{3}\). Obviously, it is not good. Compared to the other methods, using the best results of each objective function as a baseline, methods 1 and 2 decrease by 0.4879 and 0.5170 in the \(J_{1}/\min(J_{1})|\) direction and improve by 0.5731 and 0.5448 in the \(J_{3}/\min(J_{3})|\) direction. Then, the SSO method reduces 0.5031 in the \(J_{1}/\min(J_{1})\) direction and enhances 0.5581 in the \(J_{3}/\min(J_{3})|\) direction. Apparently, the SSO method is more balanced. In addition, the solving time of the proposed method is superior to method 3 and reduces the solution time by about 30 times, but it is inferior to the other methods. This phenomenon and its causes are consistent with Table 3.