MACROSCOPIC MODELING AND ANALYSIS OF URBAN VEHICULAR TRAFFIC
A macroscopic relation between the network-level average flow-rate and density, which is known as the macroscopic fundamental diagram (MFD), has been shown to exist in urban networks in stationary states. In the literature, however, most existing studies have considered the MFD as a phenomenon of urban networks, and few have tried to derive it analytically from signal settings, route choice behaviors, or demand patterns. Furthermore, it is still not clear about the definition or existence of stationary traffic states in urban networks and their stability properties. This dissertation research aims to fill this gap.
I start to study the stationary traffic states in a signalized double-ring network. A kinematic wave approach is used to formulate the traffic dynamics, and periodic traffic patterns are found using simulations and defined as stationary states. Furthermore, traffic dynamics are aggregated at the link level using the link queue model, and a Poincar𝑒́ map approach is introduced to analytically define and solve possible stationary states. Further results show that a stationary state can be Lyapunov stable, asymptotically stable, and unstable. Moreover, MFD is explicitly derived such that the network flow-rate is a function of the network density, signal settings, and route choice behaviors. Also the time for the network to be gridlocked is analytically derived.
Even with the link queue model, traffic dynamics are still difficult to solve due the discrete control at signalized junctions. Therefore, efforts are also devoted to deriving invariant continuous approximate models for a signalized road link and analyzing their properties under different capacity constraints, traffic conditions, traffic flow fundamental diagrams, signal settings, and traffic flow models. Analytical and simulation results show that the derived invariant continuous approximate model can fully capture the capacity constraints at the signalized junction and is a good approximation to the discrete signal control under different traffic conditions and traffic flow fundamental diagrams. Further analysis shows that non-invariant continuous approximate models cannot be used in the link transmission model since they can yield no solution to the traffic statics problem under certain traffic conditions.
For a signalized grid network, simulations with the link queue model confirm that important insights obtained for double-ring networks indeed apply to more general networks.