Traffic Modeling and Real-time Control for Metro Lines

We present in this article traffic flow and control models for the train dynamics in metro lines. The first model, written in the max-plus algebra, takes into account minimum running, dwell and safety time constraints, without any control of the train dwell times at platforms, and without consideration of the passenger travel demand. We show that the dynamics are stable and converge to stationary regimes with a unique asymptotic average growth rate. Moreover, the asymptotic average train time-headway, dwell time, as well as close-in time, are derived analytically, as functions of the number of running trains on the metro line. We then introduce, in a second model, the effect of the passenger demand on the train dwell times at platforms. We review that, if this effect is not well controlled, then the traffic is unstable. Finally, we propose a traffic control model which deals with this instability, by well controlling the effect of passenger arrivals on the train dwell times at platforms. We show that the dynamics are stable and converge to stationary regimes with a unique asymptotic average growth rate. We then calculate by numerical simulations the asymptotic average time-headway as a function of the number of running trains, compare the results with those of the max-plus algebra model, and derive the effect of the passenger travel demand on the frequency of the metro line, under the proposed control model.