Optimal Operation for Medium-speed Maglev Trains

The maglev transportation system has obvious advantages in safety, reliability, and energy efficiency compared with other rail transit systems with the same carrying capacity. Research topics concerning optimal operations for maglev trains have attracted much attention in recent years. Since the basic resistance acting on the maglev train is more complex than that on the conventional train, it is more challenging to solve the time-optimal and energy-optimal problems, especially considering the operational and safety constraints, such as fluctuating track gradients, varying speed limits, and speed-dependent control bound constraints. This paper considers the time-optimal control problem for medium-speed maglev trains with operational and safety constraints. By expressing the basic resistance of the medium-speed maglev train as a piecewise-quadratic function of the train's speed, we formulate the time-optimal control problem as an optimal switching control problem with free terminal time and state-dependent switching conditions. To solve this problem, we first transform the optimal switching control problem with free terminal time into one with fixed terminal time. Then, we introduce a binary-valued function to govern the switching condition, and further replace the binary constraints with equivalent continuous constraints. The resulting optimal control problem is converted into a finite-dimensional optimization problem by applying the control parameterization and time-scaling transformation techniques. The speed limit and speed-dependent control bound constraints are then transformed into a set of point and integral constraints. Furthermore, the equivalent continuous constraints involving the binary-valued function are tackled using the exact penalty method. Finally, we obtain a nonlinear programming problem which can be solved using gradient-based optimization algorithms. A numerical example using data for a real line is given to demonstrate the effectiveness of the proposed approach.