Optimal driving strategies for trains on level track with bounds on energy consumption during specified intermediate time intervals

In order to manage electricity transmission and distribution it is now common practice for system operators to offer financial incentives that encourage large consumers to reduce energy usage during designated peak demand periods. For train operators on large rail networks it may be profitable—with selected individual journeys—to reduce energy usage during peak times and increase energy usage at other times rather than simply minimizing overall energy consumption. The desired reduction in energy usage on some intermediate time interval can often be achieved if the overall optimal driving speed is increased but the usual speedhold segment is interrupted by a segment of coast–speedhold–maximum acceleration with a lower optimal driving speed on the designated interval. In Howlett et al. (2023) it was shown that an interruption of this type is optimal if and only if the coast phase begins at the entry time to the designated interval and the acceleration phase begins at the exit time. In this paper we show that the above strategy is a suboptimal strategy and that the true optimal restricted strategy is one where the usual speedhold phase is interrupted by a segment of maximum acceleration–coast–speedhold–coast–maximum acceleration. The initial phase for this segment ends at the entry time to the designated interval and the final phase begins at the exit time. We will also extend the new strategy to find optimal restricted strategies for problems with energy consumption constraints on multiple predetermined intermediate time intervals.