Cooperative optimal train operation algorithm for utilizing regenerative braking energy

The efficient use of regenerative braking energy between trains can significantly reduce the net traction energy consumption in urban rail transit systems, and has attracted considerable attention in recent years. An effective way of using regenerative braking energy is to adjust the train's coasting or cruising phase to an acceleration phase to match the braking phase of a nearby train within the same substation. This paper formulates the train driving phase adjustment problem as a cooperative optimal control problem, where a novel energy constraint is introduced to ensure efficient use of regenerative braking energy during the adjusted phase with optimal traction effort. By applying the control parameterisation method and elaborate handling of the energy constraint and the non-smooth control bound constraint, we transform the formulated problem into a nonlinear programming problem. We then use the sequential quadratic programming algorithm to solve the resulting problem, with the gradients of the cost and constraint functions computed using the sensitivity method. Numerical examples based on real-world data for a subway line show the effectiveness of the proposed method and its advantages over a genetic algorithm in terms of energy savings and computational efficiency.