Energy-efficient train operation with steep track and speed limits: A novel Pontryagin's maximum principle-based approach for adjoint variable discontinuity cases

In this study, an energy-efficient speed trajectory planner is proposed for high-speed trains traveling on tracks with steep gradients and speed limits, especially for situations in which the speed limit has been reached, which causes adjoint variable discontinuity during calculation. New optimal switching rules at points where the speed limit is reached on steep tracks are derived by analysing the jump condition of state-constrained Pontryagin's maximum principle. Accordingly, a novel two-step algorithm for high-speed trains, including an optimal-cruise minimum-time algorithm and search-substitution algorithm, is designed to solve dynamic train models considering time-energy and space-energy conversions, respectively. Practical case studies demonstrates that the proposed method can save energy by approximately 3% and 10% in comparison to the approximate-optimal time-satisfied and minimum running time strategies, respectively. Moreover, the proposed method approximately consumes 0.98% and 1.62% of the computation time taken by discrete dynamic programming and reinforcement learning, respectively.