The optimal solution to the energy-efficient train control in a multi-trains system–Part 2: the optimality and the uniqueness

AbstractWhen a train travels in a multi-trains system, the power flow of other trains and the track grades make up the spatial–temporal area (STA) for the train. Finding the optimal solution for the energy-efficient train control (EETC) problem in STA can help reduce the net energy consumption. This paper studies the analytic method to obtain the optimal solution. In Part 1, the algorithm for the problem was designed. The underlying structure of the algorithm is the connection between three optimal states through optimal feasible strategy. In Part 2, the optimality of the optimal feasible strategy is verified through a generalised local energy functional, and its uniqueness is proved based on the variational method. Additionally, we discuss the influence of external power on the optimal solution of the classical EETC problem. Case studies using data for a real freight railway line are given to illustrate our results.KEYWORDS: Optimal train controlenergy savingPontryagin’s Maximum Principlenet energy consumption AcknowledgmentsThis work was supported by the National Natural Science Foundation of China under Grant 62003283 and the National Key Research and Development Program of China under Grant 2021YFB2601500.Disclosure statementNo potential conflict of interest was reported by the author(s).Data availability statementThe data that support the findings of this study are available from the corresponding author upon reasonable request.Authors’ contributionsYu Rao: Conceptualization, Methodology, Software, Writing-original draft. Xiaoyun Feng: Methodology, Validation. Qingyuan Wang: Supervision, Visualization. Pengfei Sun: Conceptualization, Writing-review & editing.