Y. Wang, B. Ning, F. Cao, B. De Schutter, and T.J.J. van den Boom, "A survey on optimal trajectory planning for train operations," Proceedings of the 2011 IEEE International Conference on Intelligent Rail Transportation (ICIRT 2011)/IEEE International Conference on Service Operations and Logistics, and Informatics (SOLI 2011), Beijing, China, pp. 589-594, July 2011.
Because of the rising energy prices and environmental concerns, the calculation of energy-optimal reference trajectories for trains is significant for energy saving. On the other hand, with the development automatic train operation (ATO), the optimal trajectory planning is significant to the performance of train operation. In this paper, we present an integrated survey of this field. First, a nonlinear continuous-time train model and a continuous-space model of train operations are described, after which the optimal trajectory planning problem is formulated based on these two models. The various approaches in the literature to calculate the reference trajectory are reviewed and categorized into two groups: analytical solutions and numerical optimization. Finally, a short discussion of some open topics in the field of optimal trajectory planning for train operations are given.