## 6 Conclusions  
The primary contribution of this study is the development of a multi-objective optimization algorithm for determining the optimal speed profiles of railway catenary maintenance vehicles. A multi-objective optimization model was first formulated for an operational scenario of railway catenary maintenance vehicles, integrating energy consumption, punctuality, comfort, and safety as optimization objectives. These objectives were included to ensure a sufficiently large solution space and to capture the complexities of real-world operations. Furthermore, the ESS-NSGA-III algorithm was introduced, incorporating the shift density distance and dynamic convergence distance to enhance the mating selection probability and environmental selection process, respectively.  
The proposed optimization model and algorithm were validated using an operational scenario of the Huoqiu-Caomiao section as a case study. In section D1-D2, the ESS-NSGA-III algorithm demonstrated improvements of \(-\)12.89%, 4.12%, and \(-\)5.79% for the spread, HV, and IGD metrics, respectively, compared to the NSGA-II algorithm. Similarly, improvements of \(-\)8.15%, 0.77%, and \(-\)3.66% were achieved when compared to the NSGA-III algorithm. Additionally, the Pareto front distribution confirmed that the ESS-NSGA-III algorithm exhibited superior convergence and diversity, reinforcing its effectiveness in solving complex multi-objective optimization problems.  
ESS-NSGA-III achieved a superior multi-objective optimization performance for section D1-D2, achieving the lowest energy consumption (1.3241 \(\times\) 104 J) compared to NSGA-III and NSGA-II. It also excelled in terms of punctuality (0.1 s), comfort (0.0197 m/s3), and safety (0.8201). These results indicate that ESS-NSGA-III is a robust and efficient algorithm for complex engineering optimization tasks.  
Additionally, the ESS-NSGA-III algorithm was applied to optimize sections D1-D9, resulting in the derivation of both punctual operation and delayed operation strategies. Compared to the minimum running time strategy, the punctual operation strategy reduced energy consumption, comfort, and safety indicators by 29.51%, 36.10%, and 14.93%, respectively. Similarly, the delayed operation strategy achieved reductions of 52.86%, 48.35%, and 27.38% in terms of the same indicators. These findings suggest that, within permissible operational limits, maximizing running time can significantly enhance the overall operational performance. The proposed method offers a valuable reference for drivers in real-world operations, enabling them to flexibly select control strategies based on specific requirements to improve operational quality.  
Conceptualization, B.Z., G.S., Y.W. and Y.Z.; methodology, B.Z. and G.S.; software, G.S; validation, G.S.; formal analysis, Y.W., C.Z. and X.W.; resources, B.Z. and C.Z.; data curation, X.W., Z.L. and X.L.; writing--original draft, B.Z. and G.S.; writing--review and editing, Y.W. and Y.Z.; visualization, B.Z., G.S. and Z.L.; supervision, B.Z. and G.S.; project administration, B.Z.; funding acquisition, B.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Joint Research Project of the Yangtze River Delta Science and Technology Innovation Community [Grant No. 2022 CSJGG1501], the Major Science and Technology Projects of Anhui Province [Grant No. 202203a05020008], the Key Research and Development Project of Anhui Province [Grant No. 202423d12050007], the 2023 Anhui Province Industrial Foundation Capability Enhancement Task List [Grant No. JZ2024 AFKJ0344], and the Key Project of Anhui Province Science and Technology [Grant No. 202423e12050001].  
Not applicable.  
Data availability Statement: Data are contained within the article.  
The authors declare no conflicts of interest.