A cooperative scheduling model for subway trains

For subway systems, the energy put into accelerating the trains can be reconverted into electric energy by using the motors as generators during braking phase. Generally speaking, except a small part is used for on-board purposes, most of the recovery energy is transmitted backwards along the conversion chain and fed back into the catenary. However, since the low catenary voltage DC systems, the transmission losses are very high. In order to improve the utilization of recovery energy, this paper proposes an optimization approach to cooperate the acceleration and brake times of successive trains such that the recovery energy from the braking train can be directly used by the accelerating train. First, we formulate a quadratic programming model to optimize the cooperative degree with trip time constraint and time window constraints. Furthermore, we solve the optimal solution by using the Kuhn-Tucker conditions. Finally, we present a numerical example based on the operation data from Beijing Yizhuang subway line of China, which illustrates that the proposed model can improve the cooperative degree by 9.08%.