Y. Wang, B. De Schutter, T.J.J. van den Boom, and B. Ning, "Optimal trajectory planning for trains - A pseudospectral method and a mixed integer linear programming approach," Transportation Research Part C, vol. 29, pp. 97-114, Apr. 2013.
The optimal trajectory planning problem for train operations under constraints and fixed arrival time is considered. The varying line resistance, variable speed restrictions, and varying maximum traction force are included in the problem definition. The objective function is a trade-off between the energy consumption and the riding comfort. Two approaches are proposed to solve this optimal control problem. First, we propose to use the pseudospectral method, a state-of-the-art method for optimal control problems, which has not used for train optimal control before. In the pseudospectral method, the optimal trajectory planning problem is recast into a multiple-phase optimal control problem, which is then transformed into a nonlinear programming problem. However, the calculation time for the pseudospectral method is too long for the real-time application in an automatic train operation system. To shorten the computation time, the optimal trajectory planning problem is reformulated as a mixed-integer linear programming (MILP) problem by approximating the nonlinear terms in the problem by piecewise affine functions. The MILP problem can be solved efficiently by existing solvers that guarantee to return the global optimum for the proposed MILP problem. Simulation results comparing the pseudospectral method, the new MILP approach, and a discrete dynamic programming approach show that the pseudospectral method has the best control performance, but that if the required computation time is also take into consideration, the MILP approach yields the best overall performance. More specifically, for the given case study the control performance of the pseudospectral approach is about 10% better than that of the MILP approach, and the computation time of the MILP approach is two to three orders of magnitude smaller than that of the pseudospectral method and the discrete dynamic programming approach.