## 5 Case Study
### Example Results
#### 5.2.3 Scenario 3: Uniting EE and EUE  
The proposed optimization model in this scenario combined the objective functions of EE and EUE, which belong to a multi-objective optimization. There are two main strategies for this problem: the direct solution path and the indirect solution path. On the one hand, the direct solution path can search for the optimal solution from a global scope, but the coding is too complicated. On the other hand, the indirect solution path is to assign weight coefficients to each objective function and weight the multi-objective model to a single one. In general, the weights are usually determined by the decision makers according to their subjective experience. This method takes full advantage of the knowledge of decision makers, but it is also subjective. However, this way is easier to get an optimal solution compared with the direct path. Furthermore, the proposed MAPSO is suitable for single objective optimization. Based on the above, we chose the indirect solution path to handle this issue, and how to change the multi-objective functions into a single one is presented below.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & \(\mathbf{E_{min}}\)**/(MW\(\cdot\)h)** & \(\mathbf{E_{mean}}\)**/(MW\(\cdot\)h)** & \(\mathbf{E_{max}}\)**/(MW\(\cdot\)h)** \\ \hline PSO & \(4.1618\times 10^{6}\) & \(4.5899\times 10^{6}\) & \(4.6000\times 10^{6}\) \\ MAPSO & \(4.1481\times 10^{6}\) & \(4.5871\times 10^{6}\) & \(4.6000\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of results with EUE objective.  
Figure 4: Comparison of convergence with EUE objective.  
The optimization direction and dimension of the objective function is in difference, so an appropriate treatment is necessary to solve this problem. Set \(\vartheta_{c}\) and \(\vartheta_{e}\). as weight coefficients of objective function \(C\) and \(E\), and the united objective function is \(\gamma\). The normalized and dimensionless process of the objective function values is expressed in Equation (19) [68].  
\[\gamma=min\bigg{\{}\vartheta_{c}\cdot\frac{C-C^{min}}{C^{max}-C^{min}}+ \vartheta_{e}\cdot\frac{E^{max}-E}{E^{max}-E^{min}}\bigg{\}} \tag{19}\]  
where \(\vartheta_{c}+\vartheta_{e}=1\).  
The optimization results of MAPSO and PSO are shown in Figure 5, along with the weight changes between 0.1 and 0.9 of objective function \(C\). We know that the cases of EE and EUE belong to the special situations when the weight coefficients of \(C\) are 1 and 0, which are not discussed here. It can be found from Figure 6 that PSO has great volatility when the weight coefficient of \(C\) is in the range of 0.2 and 0.5. When the weight coefficient is between 0.5 and 0.9, both PSO and MAPSO show a steady upward trend. Therefore, in conjunction with expert opinion, the average of 0.5 and 0.9 is taken as the weight coefficient of \(C\) here. Obviously, the weight coefficient of \(E\) is 0.3.  
Under these circumstances, the multi-objective optimization is carried out. The settings of the basic parameters and the laptop used are consistent with those in Section 5.2.1. The average solving time for computing PSO and MAPSO 20 times is 0.160904 s and 0.081907 s, respectively. Similarly,  
Figure 5: Convergence comparison of \(\gamma\) of PSO and MAPSO (a) and convergence diagram of PSO (b).  
Figure 6: \(\gamma\) of PSO and MAPSO under different \(\vartheta_{c}\) between 0.1–0.9.  
the convergence comparison of these two methods and the optimization results are shown in Figure 6 and Table 7.  
As shown in a, MAPSO runs about 40 times to achieve the optimal solution, and PSO operates about 50 times, which declines with a tiny step after the 30 th iteration as shown in Figure 5b. Compared with PSO, the minimum, mean and maximum values of \(\gamma\) with MAPSO are smaller, which is presented in Table 7. Additionally, the computing time of MAPSO is shorter than that of PSO. All of the results prove the effectiveness and feasibility of the proposed MAPSO.