Abstract
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.
Highlights
The minimization of the energy consumption by a train moving from one station to the other is a central issue of the railway transport both from the environmental and economic perspective
The train movement between two stations has to be completed in a given time and to satisfy the infrastructure and traffic conditions
In general for zero slope tracks, the optimal strategy consists in power, speed-hold, coast, brake phases [3, 4, 13, 20, 22]
Summary
The minimization of the energy consumption by a train moving from one station to the other is a central issue of the railway transport both from the environmental and economic perspective. In general for zero slope tracks, the optimal strategy consists in power, speed-hold, coast, brake phases [3, 4, 13, 20, 22]. In this paper we consider the energy minimization problem for train moving on the different tracks. The movement of the train is governed by an ordinary differential equations where the velocity and the distance traveled along the track are the state functions. The optimal control function is calculated using the analytical solutions to state and adjoint equations. The simplified algorithm is used to calculate the optimal switching times and optimal velocities profiles. These features make the proposed approach to solve this train optimal control problem different from the approaches already developed in literature [4, 5, 10, 19, 20].
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