Integrating Power Profile Optimization with Timetabling for Underground Train Networks

We study energy-efficient operation of underground train networks, where energy from regenerative braking is usable only if another train in the same electrically isolated subnetwork accelerates simultaneously. Timetabling models for this setting typically fix one velocity profile per leg and running time, which limits the matching of braking and accelerating phases. We drop this assumption and optimize the power profile of every leg jointly with departure and running times in a single mixed-integer program. After discretizing time, distance, and velocity, we develop two formulations together with algorithms that make them tractable at practical instance sizes. The first builds a multi-leg trajectory graph whose arcs are enabled or blocked by the timetable decisions, turning the problem into network design. We solve it by a tailored Benders decomposition with Pareto-optimal cuts. The second formulation replaces the explicit graph by binary state variables and links leg-level and network-level energy through multipartite implications, whose polytope we exploit for separation. A computational study on real-world data from the underground train network in Nuremberg compares both formulations and quantifies the effects of decomposition and separation. Within the discretized model, optimized solutions save about 16% energy over an unoptimized reference and 7% over a timetable optimized without power profiles.

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