## 4 Experimental Simulations and Results
### Results  
#### 4.2.1 Case 1: Algorithm Benchmarking on D1-D2 Section  
For this case, the section from D1 to D2, which spanned 1995 m with a planned running time of 150 s, was selected for analysis. The ESS-NSGA-III, NSGA-III, and NSGA-II algorithms were utilized to solve the multi-objective model, and their simulation results were compared. To assess the performance of the three algorithms, a variety of evaluation metrics were applied to the results, and they are summarized in Table 4. \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Name** & **Characteristic** \\ \hline \(m\) & Mass (kg) & \(4.7\times 10^{4}\) \\ \(\gamma\) & Rotating mass coefficient & 0.9 \\ \(v_{\text{max}}\) & Maximum operating speed (km/h) & 120 \\ \(F(v)\) & Maximum traction force (\(N\)) & \(-2.602\times 10^{-2}v^{3}+8.961\times v^{2}-1062v+5.22\times 10^{4}\) \\ \(B(v)\) & Maximum braking force (\(N\)) & \(\left(\left(4v+150\right)/\left(10v+15\right)+0.048\right)\times 3.443\times 10^{4}\) \\ \(\alpha_{1},\alpha_{2},\alpha_{3}\) & Basic operating resistance parameters & \(\alpha_{1}=2.461,\ \alpha_{2}=2.752\times 10^{-3}\), \(\alpha_{3}=6.776\times 10^{-4}\) \\ \(W_{i}(s)\) & Gradient resistance (\(N\)) & \(461.07\times i_{L}(s)\) \\ \(W_{R}(s)\) & Curve resistance (\(N\)) & \(2.76\times 10^{5}/R(s)\) \\ \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the railway catenary maintenance vehicle. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Population size (\(N_{p}\)) & 220 \\ Maximum number of iterations (\(l_{\text{max}}\)) & 50 \\ Cross-distribution parameter (\(q_{c}\)) & 30 \\ SBX probability (\(p_{c}\)) & 1.0 \\ PM variant parameters (\(\eta_{m}\)) & 20 \\ Polynomial variational probability (\(p_{m}\)) & 0.25 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the ESS-NSGA-III and NSGA-III algorithms. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Population size (\(N_{p}\)) & 220 \\ Maximum number of iterations (\(l_{\text{max}}\)) & 50 \\ Cross-distribution parameter (\(q_{c}\)) & 30 \\ SBX probability (\(p_{c}\)) & 1.0 \\ PM variant parameters (\(\eta_{m}\)) & 20 \\ Polynomial variational probability (\(p_{m}\)) & 0.25 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters of the NSGA-II algorithm. Among these metrics, spread [23] quantifies the diversity of the solution set by assessing the uniformity and range of its distribution in the objective space. The inverted generational distance (IGD) [24] serves as a comprehensive performance indicator, evaluating both convergence and diversity through the measurement of distances between the obtained solutions and the true Pareto front. Hypervolume (HV) [25], on the other hand, measures the volume of the objective space dominated by the solution set, providing insights into both convergence and diversity. For each generation, the non-dominated solution sets from the three algorithms were combined and underwent unified non-dominated sorting. This procedure generated a collection of Pareto-optimal solutions, which were dominated and then mapped in the objective space to establish the true Pareto front (PF). The PF was subsequently utilized to compute the performance metrics for algorithmic evaluation. Table 5 presents the percentage differences between the metrics for ESS-NSGA-III and those for NSGA-III and NSGA-II. Compared to NSGA-II, the ESS-NSGA-III algorithm achieved improvements of \(-12.89\%\), \(4.12\%\), and \(-5.79\%\) for the spread, HV, and IGD, respectively. These results indicate superior diversity and convergence characteristics in the four-dimensional optimization space. Furthermore, when compared to NSGA-III, ESS-NSGA-III exhibited improvements of \(-8.15\%\), \(0.77\%\), and \(-3.66\%\) for the spread, HV, and IGD, respectively, confirming the algorithm's ability to balance solution quality across conflicting objectives.