Modeling the dynamics of congestion in large urban networks using the macroscopic fundamental diagram: User equilibrium, system optimum, and pricing strategies

- Morning commute problem in urban networks with a well-defined MFD.
- A bathtub model of the network with time-dependent demand and state-variable capacity.
- Exact estimation and analytical approximations of the equilibrium condition.
- System optimum with minimum generalized cost by maximizing the flow in the network.
- Pricing users and taxing employers in the network.

The macroscopic fundamental diagram (MFD) is introduced in recent studies to present the relationship between the flow and the density of the network in large urban regions (neighborhoods). The MFD can be also rescaled to approximate network outflow as a function of the vehicular accumulation of the system in the morning commute problem. In this research, we develop a bathtub model (macro-scale traffic congestion model) by combining Vickrey's (1969) model of dynamic congestion with the MFD to formulate the user equilibrium over the peak as an ordinary differential equation (ODE). This problem can be solved numerically to estimate the exact solution of the morning commute problem. Alternatively, the morning commute problem can be solved analytically by approximating the solution of the ODE using a well-behaved function. Here, we present a quadratic and also a linear approximation of the equilibrium solution for a semi-quadratic MFD, considering that the declining part of the MFD is shown to be well estimated by a quadratic function. To optimize the system, we present pricing strategies for network users (dynamic tolling) and employers inside the region (dynamic taxing) that can minimize the generalized cost of the system by keeping the outflow maximized over the peak. Finally, we compare the exact and the approximate solutions of the problem, and also the proposed pricing strategies of the region in a numerical example.