A Nonlinear Safety Equilibrium Spacing-Based Model Predictive Control for Virtually Coupled Train Set Over Gradient Terrains

The increasing demand for capacity in railway transportation has spawned the concept of virtual coupling (VC), which can further shorten the intertrain distance by applying communication technology. In this article, a centralized model predictive control (MPC) is developed for virtually coupled train set (VCTS) with a nonlinear safety equilibrium spacing policy over gradient rolling terrains. The tracking target of the controller in cruise phase is given based on the braking distance difference and speed difference of trains. A more precise braking distance model of trains explicitly considering the gradient is introduced for safety, based on which the state space applicable to the controller for VCTS is derived. To deal with the nonlinearities of the controller, a new optimization solving algorithm based on the continuation method and generalized minimum residual (GMRES) method is presented in this article. Finally, controller performances and empirical string stability of VCTS are verified by numerical simulations. It is proven that the proposed controller has a good effect for the safety and smoothness of the train operation on the railway line over gradient terrain through the test. The actual intertrain distance under the proposed controller is about 1 m closer to the equilibrium state than that under the same controller without considering the gradient at low speed (about 10 m/s) on the 0.3% downhill line, which verifies the safety of the controller. Besides, the influence of the parameters of the controller and optimization method on controller efficiency is analyzed, and for the latter, the simulation data show that the applied algorithm for the optimal control solver can save the calculation time more than seven times than the traditional algorithm.