What is the ideal underground metro vertical alignment for energy-efficient train control?

To further minimize the energy consumption in metro systems, the co-optimization of train control and underground metro vertical alignment (UMVA) has become a prominent research focus. However, a question still remains: what kind of ideal UMVA is most conducive to the energy-efficient train control (EETC). In this article, we formulate a coupled optimal control problem integrating both train control and UMVA. Six kinds optimal control structures of the coupled optimal control problem are derived based on the Pontryagin’s Maximum Principle (PMP). Algorithm to calculate the optimal ideal UMVA and corresponding control strategy of train is presented based on the analysis of the uniqueness of optimal control structures. At the same time, a mixed-integer quadratic programming (MIQP) model is developed to show that achieving the optimal tracking of the ideal UMVA ensures the energy-saving performance of the actual UMVA optimized under some simple but basic constraints. Based on the real data from the Wuxi Metro, it shows that the maximum energy-saving rate of the optimal train control under ideal UMVA reaches 42.05%, while decreasing to 27.08% when the real-world constraints on UMVA are considered.