Energy-efficient train control: The two-train separation problem on level track

• We provide a new analytic solution to the two-train separation problem.
• We find optimal speed profiles for the leading train and the following train.
• We find optimal clearance times for each section of track.
• We provide a new necessary condition to test for optimality of the clearance times.
• We illustrate our findings with a simplified but realistic example.

When two trains travel in the same direction along the same track it is a common safety requirement that they must be separated by at least one signal. If the signals are located at fixed positions, they divide the track into separate sections and the safety requirement means that two trains cannot occupy the same section at the same time. Safe separation can be ensured by specifying supplementary section clearance times which define the latest allowed exit time for the leading train and the earliest possible entry time for the following train. The clearance times could initially be based on an existing timetable but we will show that adjusting these times can substantially decrease the total energy required by the trains.In this paper we find driving strategies that minimize total energy consumption and allow both trains to finish on time while adhering to the separation constraints imposed by the supplementary clearance times. We establish a new necessary condition to check whether a set of specified clearance times is optimal and discuss a heuristic procedure to find the optimal clearance times and the corresponding speed profiles. We illustrate our methods with a simplified but realistic case study.