Robust cooperative train trajectory optimization with stochastic delays under virtual coupling

Virtual coupling technology was recently proposed in railways, which separates trains by a relative braking distance (or even shorter distance) and moves trains synchronously to increase capacity at bottlenecks. This study proposes a real-time cooperative train trajectory planning algorithm for coordinating train movements under virtual coupling by considering stochastic initial delays. The algorithm uses mixed-integer programming models to estimate the delay propagation among trains, detect feasible coupled-running locations, and optimize the trajectories of the two trains such that they coordinate their speeds to achieve energy-efficient, punctual movements, as well as a safe coupled-running process. A robust optimization method is proposed to capture the stochastic delays as an uncertainty set, which is reformulated to its dual problem. Case studies of planning train trajectories for the classical virtual-coupling scenario suggest that (1) the coupled-running distance is greatly affected by the coordination of train timetables, delays, and safe separation constraints at switches; (2) the coordination of train movements for a coupled-running process imposes extra energy costs; and (3) the proposed method can detect feasible coupled-running locations and produce cooperative speed profiles in short computational times.