Energy-efficient train control: From local convexity to global optimization and uniqueness

The optimal driving strategy for a train is essentially a power–speedhold–coast–brake strategy unless the track contains steep grades in which case the speedhold mode must be interrupted by phases of power for steep uphill sections and coast for steep downhill sections. The Energymiser®  device is used on freight and passenger trains in Australia and the United Kingdom to provide on-board advice for drivers about energy-efficient driving strategies. Energymiser®  uses a specialized numerical algorithm to find optimal switching points for each steep section of track. Although the algorithm finds a feasible strategy that satisfies the necessary optimality conditions there has been no direct proof that the corresponding switching points are uniquely defined. We use a comprehensive perturbation analysis to show that a key local energy functional is convex with a unique minimum and in so doing prove that the optimal switching points are uniquely defined for each steep section of track. Hence we also deduce that the global optimal strategy is unique. We present two examples using realistic parameter values.