Optimal Driving Strategies for Two Successive Trains on Level Track With Safe Separation

When two trains travel along the same track in the same direction it is a common safety requirement that they should always be separated by at least two signals. If the signals are located at fixed positions that divide the track into sections, then safe separation means there must always be one clear section of track between the two trains. It is desirable to find optimal driving strategies that minimize total tractive energy consumption, but nevertheless allow the trains to reach their final destinations at the designated times, and remain safely separated everywhere. For each signal point there is a corresponding signal-point segment consisting of the two adjacent sections. The two trains can be safely separated on this segment, by specifying a signal-point segment clearance time and requiring that the leading train must leave before the specified time, and the following train must enter after the specified time. If feasible clearance times are specified for all signal points then the trains will be safely separated everywhere. Although the structure of the optimal strategies is well understood there is currently no efficient algorithm for calculating these strategies. In this paper we will show that on level track a Newton iteration can be used to determine the optimal strategies for each feasible set of signal-point clearance times. We also propose and justify a new greedy algorithm that can be used to find sets of optimal clearance times on level track. The theoretical results are illustrated with simple but realistic examples.