We study energy-efficient operation of underground train networks by integrating power profile optimization with timetable design in a single mixed-integer optimization framework. The model minimizes traction energy by synchronizing braking and acceleration across trains sharing a power subnetwork to exploit regenerative energy and flexibly allocating running times to promote coasting. Unlike timetable-only approaches with fixed speed profiles per leg, we endogenize train velocity trajectories and couple them with departure and running-time decisions.
We discretize time, space, and velocity and develop two complementary formulations. The first constructs a multi-leg trajectory graph and computes shortest paths whose feasibility is governed by the chosen timetable; the joint problem is cast as a network design model and solved efficiently via a tailored Benders decomposition between timetable decisions and trajectory subproblems. The second is a logic-based binary formulation that encodes train states (position, speed, time) and their interactions through multipartite implication structures; specialized separation of constraints accelerates these binary components. Both formulations are embedded directly into the timetable model.
We further propose a framework that couples the discrete models with a continuous trajectory optimizer: the continuous layer supplies physics-consistent energy costs during model construction and certifies and refines discrete solutions, thereby reconciling discrete scheduling with accurate train and network physics.
A real-world case study with the public transport operator in Nuremberg VAG on both autonomous metro lines over 5-minute horizons demonstrates the practical impact of our approach, yielding approximately 16% energy reduction relative to an unoptimized timetable. The computational study compares the two formulations and quantifies the benefits of the proposed decomposition and cut-separation techniques.