Optimal driving strategies for a fleet of trains on level track with prescribed intermediate signal times and safe separation.

We investigate analytic solutions to the problem of finding driving strategies which minimize total tractive energy consumption subject to constraints that ensure safe separation for a fleet of trains travelling on the same track in the same direction and minimize line occupancy time span for the fleet. It is known that a fleet of trains can be safely separated if intermediate signal times are prescribed by inequality constraints for each train at each signal location. The optimal driving strategy for each train is then uniquely defined by an optimal driving speed on each actively timed segment. The problem of finding optimal intermediate times has been solved for a fleet of two trains but for more than two trains the problem rapidly becomes intractable as the number of trains and signals increases. The main difficulty is in distinguishing between active and inactive constraints. This is exacerbated by the fact that an inactive constraint with one set of prescribed times may become active if the set of prescribed times changes. The curse of dimensionality means it is not feasible to check every different combination of designated active and inactive constraints and to optimize the corresponding prescribed times. Nevertheless we can formulate a substantial optimization problem with intermediate signal times prescribed by active equality constraints for all trains in the fleet. We find necessary and sufficient conditions on the optimal times and demonstrate viable solution algorithms to find the associated optimal speed profiles for each train. Compared to the schedule required for safe separation when each train uses the classic single-train optimal strategy our method provides a substantial reduction in the line occupancy time span with only a minimal increase in journey costs.