Train Trajectory Optimization of Opposite Trains on Single-Track Railway Lines

This paper studies the train trajectory optimization problem of two opposite trains on single-track railway lines. A two-train trajectory optimization problem is formulated using the multiple-phase optimal control model, where the timetable as well as train performance parameters, track gradients, curves, speed limits and energy-saving requirements are taken into account. Multiple-tracks at stations are reflected in the model, so that trains may use different tracks at stations. A meeting constraint is developed to avoid head-on conflicts between opposite trains on open tracks. The trajectories of the two opposite trains are optimized simultaneously by a pseudospectral method with minimizing energy consumption as the objective function. In addition, an optimization method is developed for energy-efficient time-distance paths for two opposite trains based on the twotrain trajectory optimization model. The method is applied in case studies of two opposite trains running on a Dutch singletrack railway corridor. The results show that our method is able to produce optimal trajectories and time-distance paths for both trains within a short time.