Lagrangian relaxation based speed trajectory optimization for multiple trains under virtual coupling with operational state transition

The utilization of virtual coupling in train operations offers significant potential to increase the railway transport capacity. Different from traditional signaling systems, virtual coupling allows trains to be coupled and decoupled during operation. As a result, train operations under virtual coupling differ fundamentally from those under conventional signaling systems. Therefore, traditional speed trajectory optimization methods used for regular operations are insufficient for the operation of virtually coupled trains. To optimize the speed trajectories under virtual coupling, a mixed integer nonlinear programming model is formulated. This model explicitly represents the different operational states of virtual coupling using binary variables, and there are time window constraints restricting the times of state transitions. The original model is converted into a linear model by a piecewise-linear approach. With the aim of enhancing computational efficiency, a Lagrangian relaxation method is adopted, where the dual problem can be solved efficiently because the safety distance constraints are relaxed. To provide evidence of the proposed method’s effectiveness in coupling and decoupling scenarios, numerical experiments are conducted based on a part of the Beijing-Shanghai high-speed railway. The results show that our model effectively optimizes the speed trajectories for virtually coupled trains while satisfying safety distance and time window constraints. In the coupling scenario with two trains, compared with using CPLEX directly, the speed trajectories obtained by LR have smaller energy consumption and larger train distance. The total objective function obtained by LR is 1.05 % lower. Moreover, in some cases with the coupling process where the direct method cannot find any feasible solution, LR can solve the problem within the same computational time limit, demonstrating that LR significantly improves computational efficiency in coupling scenarios.