Macroscopic Fundamental Diagram: Alternative Theoretical Analysis and Implications for Traffic Control

This paper presents the theory and analysis related to the aggregated macroscopic fundamental diagram and presents specific implications for traffic control. The paper presents the aggregation results for the three fundamental variables, traffic density, speed, and traffic flow, and the relationship among the aggregated versions of these for the Greenshields' model and a piecewise affine model. The development of the algebraic relationships is followed by stochastic analysis to obtain aggregation results in the limiting sense. The dynamics of the aggregated variables are studied, and the idea of dynamic region for aggregation in terms of dynamic reachability is utilized. We also provide an analysis of the error bounds that can be utilized during perimeter control design using MFDs. Finally, the implications of this new analysis are studied in terms of traffic control for an aggregated region, followed by traffic control simulations. Two separate control problems are formulated and studied, which include a) MFD-based control strategy on freeways and b) MFD-based modeling and control of urban sub-networks. Control methodologies used for the two problems include conservation law-based direct control design and feedback linearization control, respectively.