Energy Consumption Optimal Control of the Train Movement

The paper is devoted to the solution of the energy minimization problem for a moving train. The train movement is governed by the system of the first order ordinary differential equations where the train speed and the distance along the track are the state variables. The provided locomotive power depends on the control function. The generated traction force is assumed to depend on the velocity of the train and on the control function. Each non-negative value of the control function determines a traction force control while negative values determine a braking force control. The cost functional is defined as the train energy. It is dependent on traction force, speed and control functions. The speed, distance and control functions are assumed bounded. Using the maximum principle and Lagrangian multipliers the system of equations constituting the necessary optimality conditions is formulated. Based on the analysis of the train movement the optimal trajectories in terms of train speed and associated optimal control functions are calculated. A new simplified method is used to calculate the set of the switching times implementing the optimal control function. Numerical examples are provided and discussed.