Max-plus modeling of traffic on passenger railway lines with a junction: fundamental diagram and dynamic control

TL;DR

This thesis develops max-plus algebra-based mathematical models and control strategies for metro lines with junctions, enabling analysis of traffic regimes and dynamic control based on passenger demand.

Contribution

It extends existing linear metro line models to include junctions, proposing new train dynamics models and control laws using max-plus algebra.

Findings

Characterization of stationary traffic regimes

Derivation of traffic phase diagrams

Control laws adapting to passenger demand

Abstract

This thesis proposes mathematical traffic models and control laws for metro lines with one junction. The models are based on the ones developed for linear metro lines (without junction) in [12, 14]. The train dynamics are described with a discrete event traffic model. Two time constraints are considered. The first one imposes lower bounds on the train run and dwell times. The second one fixes a lower bound on the safe separation time between two trains. A model of the train dynamics on the junction is proposed, as well as control laws for the train run and dwell times, as a function of the passenger travel demand. Most of these models are written as linear systems in the max-plus algebra (polynomial matrix algebra), which permits the characterization of the stationary regime, and the derivation of traffic phase diagrams.