This paper considers an optimal train control problem with two challenging, non-standard constraints: a speed constraint that is piecewise-constant with respect to the train's position, and control constraints that are non-smooth functions of the train's speed. We formulate this problem as an optimal switching control problem in which the mode switching times are decision variables to be optimized, and the track gradient and speed limit in each mode are constant. Then, using control parameterization and time-scaling techniques, we approximate the switching control problem by a finite-dimensional optimization problem, which is still subject to the challenging speed limit constraint (imposed continuously during each mode) and the non-smooth control constraints. We show that the speed constraint can be transformed into a finite number of point constraints. We also show that the non-smooth control constraints can be approximated by a sequence of conventional (smooth) inequality constraints. The resulting approximate problem can be viewed as a nonlinear programming problem and solved using gradient-based optimization algorithms, where the gradients of the cost and constraint functions are computed via the sensitivity method. A case study using data for a real subway line shows that the proposed method yields a realistic optimal control profile without the undesirable control fluctuations that can occur with the pseudospectral method.

Once a freight train is delayed on a busy railway under a quasi-moving-block signaling system, the speed of its following trains may fluctuate, without proper adjustment. Updating the reference speed profile of the delayed freight train is an urgent task to improve the train operation performance and rail utilization ratio. To this end, this article proposes a speed profile optimization method for following delayed trains that is based on the idea of model predictive control (MPC). The state change time of the block section ahead is predicted for each following train according to the speed profile of its preceding train, based on which the distance between the following train and each speed protection curve (DTC) is predicted. The DTC is taken as a key optimization index for the speed profile optimization, and the other optimization indices are train delay and energy consumption. A two-step hierarchical optimization algorithm is proposed in this article. In the upper level, each prediction step is defined as the traveling time of the preceding train in each block section ahead, and the train target cruising speed sequence is calculated, with the dimension decided by the prediction horizon, using particle swarm optimization. In the lower level, the optimal speed profile is calculated based on the rolling optimization algorithm of the train speed curve in the prediction horizon. The proposed algorithm repeats the optimization process with updated train information after each control horizon. Three simulations are presented, which consider a downhill scenario, steep uphill scenario, and temporary speed limit, respectively. Then, the MPC parameters are analyzed and the optimized speed profiles are compared with the other algorithm.

Maglev trains have the advantages of low noise, good climbing ability, and low environmental impact. Several maglev lines have been put into commercial operation in China in recent years, and the issue of energy saving in maglev trains has attracted many researchers' attention. Existing research on this topic mainly focuses on offline calculation of the optimal train control strategy and rarely considers the use of regenerative braking energy between trains. This paper proposes an online rolling optimization algorithm to obtain the energy-efficient control trajectory of maglev trains according to the actual running condition. In addition, the regenerative braking energy of nearby trains is also considered to adjust the maglev train's control stage to further reduce the net traction energy consumption. The pseudospectral method is used in each prediction horizon of the rolling optimization algorithm to solve the multi-phase optimal control problem. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.