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

When 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.