Minimum Energy Time Tables for Subway Operation - And Hamiltonian Feedback to Return to Schedule

Previous work by the authors studied energy optimal control of cam controlled subway trains going from one station to the next on a given schedule. An indirect competing Hamiltonian algorithm was created to handle the associated mixed integer programming problem. A direct approach was also created based on outer convexification, relaxation, and the Krein-Milman theorem. This paper extends the optimization, creating a new form of scheduling that produces a time table for the complete subway line that minimizes total energy consumption from start to end of the line, for a given total transit time. Example computations are given for optimizing the schedule across several stations in the New York subway system. A characteristic of the optimized time table is that the Hamiltonian for the transit from each station to the next must be the same value all over the line. In applications, trains can easily get behind the prescribed schedule, so one must have a method of returning to schedule. The authors previously presented feedback methods to modify the control in a nearly optimal manner en-route to the next station. This is also generalized here to develop feedback to return to schedule at a later station, or by the end of the line. An alternative that could be simpler to implement is also presented, to simply increase the Hamiltonian, which corresponds to requesting an increased average speed, and the motorman could use this until back on schedule.