**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.

In all the models considered here, the metro line is discretized (in space) in a number of segments (blocks). In the model of Chapter 3, the train run, dwell, and safe separation times are lower-bounded on every segment. The train dynamics out of the junction are modeled as in the case of linear metro lines [12, 14]. One of the main contributions of Chapter 3 is the model of the train dynamics on the junction. The train dynamics model for the entire line (with the junction) is shown to be linear in the max-plus algebra and is shown to reach a stationary regime. The asymptotic average growth rate of the train dynamics, interpreted as the asymptotic average train time-headway, is derived analytically. It is given as a function of the train run, dwell, and safe separation times, and of the total number of trains, as well as the diﬀerence between the number of trains on the two branches. This derivation permits to obtain phase diagrams of the train dynamics, called here fundamental diagrams, as in road traﬃc. Eight traﬃc phases of the train dynamics are derived analytically and interpreted in terms of traﬃc. Moreover, based on the closed-form solutions of the traﬃc phases, macroscopic control laws are proposed for the traﬃc on a line with a junction.

Chapter 4 proposes an extension of the model of Chapter 3. In the ﬁrst section, the model of the train dynamics on a linear metro line [12, 14] is extended with the run and dwell times as functions of the passenger travel demand. It is shown that the train dynamics remain linear in the max-plus algebra. The stationary regime is characterized, and the asymptotic average growth rate of the train dynamics (asymptotic average train time-headway) is derived analytically, as a function of the parameters of the line (lower bounds on the run, dwell, and safe separation times), and of the total number of trains, as well as the passenger travel demand. It is suggested that margins on the train run times can be used to extend train dwell times at platforms in case of train delays, which improves the robustness of the train dynamics. In the second section of Chapter 4, this extension is applied to a line with a junction, combining the models of Chapter 3 and Chapter 4 (section 1). Similarly, the traﬃc phases of the train dynamics are derived analytically, giving the asymptotic average train time-headway as a function of the parameters of the line and of the total number of trains, the diﬀerence between the number of trains on the two branches, as well as the passenger travel demand.

Finally, Chapter 5 proposes three simulation cases illustrated on metro line 13 of Paris (with one junction). The ﬁrst case illustrates the macroscopic control on the number of trains depending on the passenger travel demand volume. The second case shows the macroscopic control of the number of trains on the branches in case of a perturbation on the train travel times. The third case gives a simulation of the demand-dependent train dynamics with perturbed initial time-headways and shows how an additional dwell time control harmonizes the train time-headways.

**Keywords:**

Traﬃc ﬂow theory, discrete event systems, physics of traﬃc, railway traﬃc modeling, traﬃc control.