## 5 Simulation Results and Discussion  
### Algorithm Performance Analysis  
The primary objective of this study is to solve the target function using the proposed EDMOA, ensuring both the normal operational load of high-speed railway trains and the stable operation of all system components. The algorithm optimizes the power and capacity of energy storage devices while minimizing the total cost and carbon emissions of the hybrid power system. For comparative validation, six benchmark algorithms--DMOA, GWO, Zebra Optimization Algorithm (ZOA) [48], Bird Swarm Algorithm (BSA) [49], Golden Jackal Optimization (GJO) [50], and Sand Cat Swarm Algorithm (SCSA) [51]--are selected to evaluate the rationality of the optimization results. Lower fitness values indicate superior algorithmic performance. The convergence curves of these algorithms for the target function are illustrated in the figures. All experiments are conducted under identical conditions with a population size of 100 and a maximum iteration count of 1000.  
The comparative analysis reveals the rationality of the optimization results obtained by EDMOA and validates the effectiveness of the algorithmic improvements proposed in this study. As illustrated in Figure 6, which shows the convergence curve of EDMOA being employed as the baseline for local magnification, EDMOA achieves a non-zero fitness value of 235,760, outperforming all benchmark algorithms. Specifically, the optimal fitness values are 237,104 for BSA, 236,195 for DMOA, 236,947 for GJO, 236,142 for GWO, 236,343 for SCSA, and 240,096 for ZOA. EDMOA demonstrates a performance advantage ranging from 435 points (compared to DMOA) to 4336 points (compared to ZOA). Furthermore, EDMOA exhibits enhanced capability to escape local optima and discover superior solutions during iterations. Runtime comparisons show that the original DMOA requires 2000.51 s for 1000 iterations, while the improved EDMOA completes the same task in 840.67 s, confirming significant computational efficiency gains. The iterative convergence curve also highlights EDMOA's phased optimization behavior, characterized by early exploration and later exploitation phases. To statistically validate these performance advantages, Wilcoxon signed-rank tests (\(\alpha\) = 0.5) were conducted on convergence trajectories across 30 independent runs. The tests confirmed improvements of EDMOA in both final solution quality (\(p=1.8\times 10^{-7}\)) and convergence speed (\(p=3.2\times 10^{-8}\)) are statistically significant. Collectively, the results demonstrate that EDMOA achieves the lowest fitness value in optimizing the target function for minimizing operational costs and carbon emissions while ensuring uninterrupted high-speed railway load demands.  
To systematically evaluate the necessity of components of EDMOA, we conducted an ablation study to assess their individual and combined contributions to algorithm performance. As shown in Table 2, the complete EDMOA implementation incorporating chaotic mapping, trust-region adaptation, and elite-guided search achieves the lowest fitness value of 235,760. The removal of the elite-guided mechanism results in a fitness increase to 235,949, indicating its critical role in enhancing solution quality. Furthermore, the absence of trust-region adaptation leads to a significant fitness degradation to 236,102, underscoring its importance in maintaining algorithm stability. These findings validate the necessity of all components and their synergistic effects in achieving optimal performance.