Mixed Integer Linear Programming Based Speed Profile Optimization for Heavy-Haul Trains

Automatic heavy-haul train (HHT) operation technology has recently received considerable attention in the field of rail transportation. In this paper, a discrete-time-based mathematical formulation is proposed to address the speed profile optimization problem in order to ensure the safe, efficient, and economical operation of heavy-haul trains (HHTs). Due to the presence of long and steep downgrades (LSDs) on some heavy-haul lines, the brake forces of the HHT are typically jointly determined by air braking and electric braking. The time characteristics of the air braking, such as the command delay and the change process caused by the air pressure, are taken into account, and then formulas are presented to calculate the air brake force. In addition, the influence of the neutral section on the control of the electric braking is considered via space-based constraints. The resulting problem is a nonlinear optimal control problem. To achieve linearization, auxiliary 0-1 binary variables and the big-M approach are introduced to transform the nonlinear constraints regarding slope, curve, neutral section, air brake force, and air-filled time into linear constraints. Moreover, piecewise affine (PWA) functions are used to approximate the basic resistance of the HHT. Finally, a mixed integer linear programming (MILP) model is developed, which can be solved by CPLEX. The experiments are carried out using data from a heavy-haul railway line in China, and the results show that the proposed approach is effective and flexible.