Energy-efficient multi-curve optimization in urban rail transit: Stability enhancement under operational uncertainties and curve adjustments

The multi-curve optimization problem involves selecting train speed curves for nominal timetables and configuring candidate curves embedded in the Automatic Train Operation (ATO) system for train rescheduling. In practice, train speed curves planned under nominal conditions are frequently disrupted by uncertainties such as delays and fluctuations in passenger flow, which may require rescheduling, where the actual speed curves can only be selected from the candidate train speed curves. This rescheduling process leads to deviations between rescheduled (actual) and nominal energy performance. Existing research has not fully addressed the impact of rescheduling on energy consumption from a planning perspective, a critical gap for improving the efficiency of energy-efficient timetables under uncertainty. To fill this gap, we define the stability of energy-efficient train timetables as a quantifiable metric, assessing deviations in terms of both energy reduction and delay control. To minimize actual energy consumption, this study incorporates stability-based constraints into a two-stage stochastic programming model, combining an energy-efficient scheduling stage with a bi-level programming stage for speed curve rescheduling, which introduces nonlinear complexities. Two logic-based Benders decomposition algorithms, including a novel multi-scenario dynamic programming method, solve the model. Using actual data from the Beijing Yizhuang Line, we conducted two sets of numerical experiments to validate the performance of the model and algorithms. Compared to a benchmark two-stage model without optimizing the candidate train speed curves, our approach achieves average stability improvements of 2.74% for in-sample tests and 2.40% for out-of-sample tests, with gains surpassing 4.00% under more challenging delay scenarios, alongside reductions in energy consumption.