Evaluation of Analytical Approximation Methods for the Macroscopic Fundamental Diagram

The Macroscopic Fundamental Diagram (MFD) describes the relation of average network flow, density and speed in urban networks. It can be estimated based on empirical or simulation data, or approximated analytically. Two main analytical approximation methods to derive the MFD for arterial roads and urban networks exist at the moment. These are the method of cuts (MoC) and related approaches, as well as the stochastic approximation (SA). This paper systematically evaluates these methods including their most recent advancements for the case of an urban arterial MFD. Both approaches are evaluated based on a traffic data set for a segment of an arterial in the city of Munich, Germany. This data set includes loop detector and signal data for a typical working day. It is found that the deterministic MoC finds a more accurate upper bound for the MFD for the studied case. The estimation error of the stochastic method is about three times higher than the one of the deterministic method. However, the SA outperforms the MoC in approximating the free-flow branch of the MFD. The analysis of the discrepancies between the empirical and the analytical MFDs includes an investigation of the measurement bias and an in-depth sensitivity study of signal control and public transport operation related input parameters. This study is conducted as a Monte-Carlo-Simulation based on a Latin Hypercube sampling. Interestingly, it is found that applying the MoC for a high number of feasible green-to-cycle ratios predicts the empirical MFD well. Overall, it is concluded that the availability of signal data can improve the analytical approximation of the MFD even for a highly inhomogeneous arterial.