Dynamic modeling of trip completion rate in urban areas with MFD representations

Abstract

A large part of the research utilizing the concept of Macroscopic Fundamental Diagram (MFD) relies heavily on an approximation of the trip completion rate derived for steady states. This model, referred to hereafter as the PL model, only requires the knowledge of the MFD, the average trip length and the current accumulation. An alternative approach, known as the trip-based model, allows to determine the trip completion rate exactly (provided that the road network is governed by an MFD) when the trip length distribution and inflow history are known. This work investigates the soundness of the PL approximation under time-varying inflow by comparing it to the more complex trip-based model. The trip length distribution is shown to be an important determinant of the accuracy of the PL model, not only via its mean but also via its coefficient of variation. The PL approximation is exact when trip length follows an exponential distribution, and relatively good when the coefficient of variation is close to 1. Other coefficients of variation lead to the emergence of hysteresis phenomena, whose properties depend on whether the coefficient of variation is smaller or greater than 1. A third type of model (the M model) is proposed to address the cases where the PL model does not provide sufficient accuracy. The M model has a dynamic behavior very similar to the one of the trip- based model, but it is described by an ordinary differential equation, thus being more suitable for control purposes. Despite their differences in accuracy, the PL and M models are found to perform equally well when integrated in a model predictive control framework.