Y. Wang, B. De Schutter, T.J.J. van den Boom, and B. Ning, "Optimal trajectory planning for trains under fixed and moving signaling systems using mixed integer linear programming," Control Engineering Practice, vol. 22, pp. 44-56, Jan. 2014.
The optimal trajectory planning problem for multiple trains under fixed block signaling systems and moving block signaling systems is considered. Two approaches are proposed to solve this optimal control problem for multiple trains: the greedy approach and the simultaneous approach. In each solution approach, the trajectory planning problem is transformed into a mixed integer linear programming (MILP) problem. In particular, the objective function considered is the energy consumption of trains and the nonlinear train model is approximated by a piece-wise affine model. The varying line resistance, variable speed restrictions, and maximum traction force, etc. are also included in the problem definition. In addition, the constraints caused by the leading train in a fixed or moving block signaling system are first discretized and then transformed into linear constraints using piecewise affine approximations resulting in an MILP problem. Simulation results comparing the greedy MILP approach with the simultaneous MILP approach show that the simultaneous MILP approach yields a better control performance but requires a higher computation time. Moreover, the performance of the proposed greedy and the proposed simultaneous MILP approach is also compared with that of the greedy and the simultaneous pseudospectral method, where the pseudospectral method is a state-of-the-art method for solving optimal control problems. The results show that the energy consumption and the end time violations of the greedy MILP approach are slightly larger than those of the greedy pseudospectral method, but the computation time is one to two orders of magnitude smaller. The same trend holds for the simultaneous MILP approach and the simultaneous pseudospectral method.