New formulations and solution approaches for train eco-driving problems

This paper investigates solution methods for train eco-driving problems that include the classic single-train eco-driving problem, the single-train eco-driving problem with intermediate time-window constraints, and the eco-driving problem for a fleet of trains under the green-wave policy. The latter two problems are particularly relevant in modern, busy railway networks. We start from proposing a relaxed continuous optimal control formulation for the classic single-train eco-driving problem and rigorously prove that the relaxation is exact. To solve this relaxed optimal control problem, we use the direct method by discretizing the independent variable in the problem and converting the problem to a nonlinear program, where the latter can be effectively solved to exact solutions. To further enhance the computational efficiency, we introduce valid inequalities for the nonlinear program. Numerical experiments are conducted to demonstrate the performance of our proposed method in terms of solution quality and computing time, which shows that our proposed method outperforms benchmark direct methods in solving the classic single-train eco-driving problem. Furthermore, we extend our proposed method to solve the other two aforementioned more complicated but practical eco-driving problems, and our proposed method can deliver exact solutions for the formulated nonlinear nonconvex programs within reasonable computing time.