A discrete event traffic model explaining the traffic phases of the train dynamics on a linear metro line with demand-dependent control

Abstract

In this paper we present a mathematical model of the train dynamics in a linear metro line system with demanddependent run and dwell times. On every segment of the line, we consider two main constraints. The first constraint is on the travel time, which is the sum of run and dwell time. The second one is on the safe separation time, modeling the signaling system, so that only one train can occupy a segment at a time. The dwell and the run times are modeled dynamically, with two control laws. The one on the dwell time makes sure that all the passengers can debark from and embark into the train. The one on the run time ensures train time-headway regularity in the case where perturbations do not exceed a run time margin.We use a Max-plus algebra approach which allows to derive analytic formulas for the train time-headway and frequency depending on the number of trains and on the passenger demand. The analytic formulas, illustrated by 3D figures, permit to understand the phases of the train dynamics of a linear metro line being operated as a transport on demand system.

Real-time control of metro train dynamics with minimization of train time-headway variance

“…The first row presents the results for γ j = 0, i.e. without control where the dynamics (19) are a Max-plus linear system, as in [11]. It can easily be seen, that, depending on the initial train positions, we observe some areas with more trains on the line, and others with less, i.e.…”

“…Moreover, this irregularity persists over the simulation. In fact, it has been shown in [11] that the train time-headways along the line do not converge. They rather reach a periodic regime which makes their asymptotic average convergent.…”

“…We shortly review here the Max-plus linear train dynamics for a metro loop line, with demand-dependent dwell and run time controls [11]. The dwell and run time control laws are fixed as follows.…”

“…On the one side the Max-plus linear traffic model of section II-B from [11] guarantees stability and controls both dwell and run times to take into account the passenger travel demand. However, it does not harmonize train departure time intervals on the line.…”

“…In this paper, we refer to a Max-plus Theorem, see [1], [3], [9], having already been used in our preceding works. In the following, we present an enhanced version of the traffic model which we have proposed in [11]. It keeps its advantages, i.e.…”