Energy-efficient timetable and speed profile optimization with multi-phase speed limits: Theoretical analysis and application

•The coasting switching points on each section can be calculated precisely by using mathematical derivation.•The multi-phase-speed-limit sections are discretized into several segments to calculate the energy consumption.•A strictly convex quadratic programming is formulated by using Taylor Approximation.•It reduces the total energy consumption by 4.52% for one train finished one round trip. With ever-increasing energy consumption and associated costs, energy-efficient timetable and speed profile optimization approaches for metro systems have attracted much attention from both academic and industry communities. This paper develops an integrating timetable and speed profile optimization model with multi-phase speed limits to reduce the energy consumption for a metro line. First, the headway and cycle time for the next train are determined based on the passenger demand; and the optimal coasting-switching points on each section are theoretically analyzed with consideration of the multi-phase speed limits, headway and cycle time constraints. Second, the non-convex train scheduling problem is transformed to a strictly quadratic model by using the Taylor approximation. Finally, we apply the active set method (ASM) to find the approximate optimal solution and present numerical examples with real-world operational data from the Beijing Metro Yizhuang Line in China. The results shows that the developed approach can reduce the energy consumption by 4.52% for one train finished one round trip in comparison with the current timetable.