Solvability Conditions of Power-Flow Equation for DC Traction Systems Considering Maximum Power Consumption and Injection of Electric Vehicles

DC traction systems, known for their advantages in large passenger capacity and low energy consumption, have been widely used in transportation services. However, with the increasing adoption of electric vehicles, DC traction systems with bidirectional power characteristics may operate under excessively stressed conditions, which could lead to system instability. In this paper, we analyze the solvability condition of bidirectional power-flow equations. First, we establish the bidirectional powerflow equation of a DC traction system. Then, the set of feasible power is proven to be a convex set. Due to the convexity of the feasible power set, and by applying the separating hyperplane theorem, we derive the first necessary and sufficient solvability condition based on a linear matrix inequality. This condition allows us to determine critical boundaries, such as critical substation voltage, critical power consumption and injection power consumption. Finally, the second analytical condition is obtained by applying Brouwers fixed-point theorem, enabling the determination of the maximum power consumption and injection. Compared to existing results, our sufficient condition is less conservative. Simulation results verify the correctness of the proposed solvability condition, providing valuable guidelines for developing a reliable DC traction system.