Analysis of Traffic Statics and Dynamics in Signalized Networks: A Poincaré Map Approach

An understanding of traffic statics and dynamics is critical for developing effective and efficient control strategies for a signalized road network with turning movements, especially under oversaturated conditions. In this study, we first describe traffic dynamics in a signalized double‐ring network with the link queue model, which is a space‐continuous approximation of the network kinematic wave model, and rewrite it as a switched affine system, assuming a triangular traffic flow fundamental diagram. Then we define periodic density evolution orbits as stationary states in the network and introduce a Poincaré map in densities, whose fixed points correspond to stationary states. With short cycle lengths and identical green times and retaining ratios in both rings, we are able to derive the closed form of the Poincaré map, from which we can analytically solve stationary states and study their stability properties; it is found that a stationary state can be asymptotically stable, Lyapunov stable, or unstable. By defining the network flow‐density relation in stationary states as the macroscopic fundamental diagram (MFD), we analytically derive an approximate closed‐form formula for MFD with green ratios and retaining ratios as parameters. We confirm that in stationary states the network flow rate may not be uniquely defined, and the network can reach a gridlock state at relatively low densities. We also analyze the convergence patterns to asymptotically stable gridlock states with different retaining ratios and initial densities. With more general signal settings and retaining ratios, we develop a secant method to numerically solve the fixed points of the Poincaré maps and plot the corresponding MFDs. Furthermore, numerical simulations are used to study traffic statics in a homogeneous signalized grid network; simulation results reveal a high level of similarity in traffic patterns between the signalized double‐ring and grid networks and validate analytical insights obtained from the former. This study provides a springboard for future analytical and numerical studies on traffic statics and dynamics in more general signalized road networks. The online appendix is available at https://doi.org/10.1287/trsc.2017.0740