Efficient operation of underground railway systems is critical not only for maintaining punctual service but also for minimizing energy consumption, a key factor in reducing operational costs and environmental impact. To evaluate the energy consumption of the timetables, this paper delves into the development of mathematical models to accurately represent energy dynamics within the underground railway network.
We evaluate the total traction energy consumption in an underground railway network over a specified period, with the analysis discretized to a per-second basis.
At each second we evaluate the power flow in the transmission network, a direct current (DC) power grid with fixed powerstation voltages. Quadratic constraints arise when linking power, current, and voltage. To deal with the resulting computational complexity we compare two model formulations, one based on power flow and the other based on current flow. We demonstrate that the current flow model is easier to solve and develop a heuristic to further speed up the solution process.
We integrate the model addressing the power flows in the transmission network with the model that ensures the feasibility of the timetable. Central to our approach is the utilization of Benders row generation to tackle the complexity of the large integrated model. By decomposing the optimization problem into manageable subproblems, we enhance computational efficiency and scalability. To linearize the Benders subproblems we develop relaxations for the non-linear constraints and binary variables. We analyze the performance of the integrated timetabling power flow model on real world data provided by the VAG (Verkehrs-Aktiengesellschaft Nürnberg), the operator of public transport in Nuremberg, Germany. The simulated energy consumption deviates from the actual measurements by only around 1%. The calculated timetable increases energy efficiency by up to 0.8% compared to the previously used model. Further numerical studies demonstrate the effectiveness of the developed solving algorithms.