## 4 Case Analysis and Discussion
### Results Analysis  
Based on the above simulation parameters, the mathematical model of the stand-alone wind-photovoltaic-battery system with a hybrid electrolyzer was optimized by each single objective function. To demonstrate the performance of the hybrid electrolyzer, the optimization results of AWE, PEME, and AWE + PEME were compared with the proposed method. The optimization results for each objective function and the configuration outcomes under the three electrolyzer applications are illustrated in Table 2 and Figure 4, respectively. Figure 3: The power curve from the new energy source and load under a day. Table 2 and Figure 4 demonstrate that when the economy of hydrogen production \(J_{1}\) was taken as the optimization objective, the rated power of AWE and PEME was 40.27 MW and 33.23 MW in the hybrid electrolyzer, respectively. The objective function value for \(J_{1}\) was the smallest at 442,790.62, and the objective function value for \(J_{2}\) was 1216, less than the 1259.66 from a 69.59 MW single AWE electrolyzer. Similarly, when the utilization rate of hydrogen production \(J_{2}\) was a single optimization objective, the planning results showed that the hybrid electrolyzer operated at a single rated power of 62.15 MW for the PEME. Here, the objective function value for \(J_{2}\) was the smallest at 905.29, and the objective function value for \(J_{1}\) was 478,127.59, smaller than the 486,189.99 from a single AWE electrolyzer with the same rated power. This demonstrates that the hybrid electrolyzer has clear advantages over a single AWE electrolyzer. Conversely, when compared with a single PEME electrolyzer under the objective of \(J_{2}\), the optimization results indicate that the rated power of AWE is zero, suggesting that PEME is significantly more efficient in terms of the hydrogen production utilization rate. When focusing on the economy of hydrogen production \(J_{1}\), the hybrid electrolyzer outperformed a single PEME electrolyzer due to the substantial cost advantage of AWE applications. Figure 5 displays the implementation process of the proposed method and compares the results with other methods, which showed a significant contradiction between the two optimization objective functions. Specifically, as one optimization object's value decreased, the other's value increased. The Pareto front shown in Figure 5 was divided into three distinct regions: AB, BC, and CD. Each region represents notable differences in slope. In the AB region, while \(f_{1}(\mathbf{X})/\min f_{1}(\mathbf{X})\)  increased slowly, \(f_{2}(\mathbf{X})/\min f_{2}(\mathbf{X})\)  decreased rapidly. This means that by relaxing the optimization requirements for objective function \(J_{1}\), a greater benefit can be achieved for optimization objective \(J_{2}\), which is worthwhile. After moving into the BC region, the decrease in \(f_{2}(\mathbf{X})/\min f_{2}(\mathbf{X})\)  became more gradual, indicating that the cost of sacrificing optimization objective \(J_{1}\) to improve \(J_{2}\) was increasing. In the CD region, after incurring significant losses on optimization objective \(J_{1}\), the resulting benefit gained on optimization objective \(J_{2}\) was minimal, which is quite unreasonable. Furthermore, observing the changes in slope along the Pareto front, point B showed the most dramatic variation. Therefore, it is justifiable that point B should be recommended as the optimal solution. Notably, compared with the other methods, point B was successfully \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Objective** & \multicolumn{2}{c}{**AWE**} & \multicolumn{2}{c}{**PEME**} & \multicolumn{2}{c}{**AWE + PEME**} \\ \cline{2-7}
**Function** & \(\mathbf{J_{1}}\) & \(\mathbf{J_{2}}\) & \(\mathbf{J_{1}}\) & \(\mathbf{J_{2}}\) & \(\mathbf{J_{1}}\) & \(\mathbf{J_{2}}\) \\ \hline min(\(J_{1}\)) & 450,244.91 & 1259.66 & 451,513.58 & 1033.22 & 442,790.62 & 1216.00 \\ min(\(J_{2}\)) & 486,189.99 & 1101.03 & 478,127.59 & 905.29 & 478,127.59 & 905.29 \\ \hline \hline \end{tabular}
\end{table}