Second-order macroscopic traffic models

Second-order macroscopic traffic flow models introduce a second dynamic equation compared to first-order models, i.e. the equation describing the dynamics of the mean speed of vehicles. Second-order models were introduced in the 70s as continuous models, the earliest one being the so-called Payne\u2013Whitham model. Some critiques arose on this class of models, focusing in particular on the dissimilarity between the flow of vehicles and the flow of molecules in fluids or gases. This criticism encouraged new developments of second-order models, leading to the model proposed by Aw and Rascle, and a similar model developed independently by Zhang. A discrete version of second-order models has been elaborated in the 90s, known as METANET. This discrete model, conceived both for freeway stretches and for networks, is very widespread in the engineering field and particularly suitable for prediction and control purposes

First-Order Macroscopic Traffic Models

Macroscopic traffic models describe the traffic behaviour at a high level of aggregation, i.e. the traffic dynamics is expressed through aggregate variables, such as traffic density, mean speed and flow. Macroscopic models rely on the analogy between the flow of vehicles and the flow of fluids or gases and are based on a limited number of equations that are relatively easy to handle. This chapter is devoted to describe a very relevant class of macroscopic models, i.e. first-order traffic flow models, which capture the dynamics of only one aggregate variable, namely, the traffic density. A very important first-order macroscopic model is the Lighthill\u2013Whitham\u2013Richards model, developed in the 50s, but still of interest nowadays both for theoretical analysis and practical applications. It is a continuous model, which describes the dynamics of the macroscopic variables through partial differential equations. The most famous discretised version of the Lighthill\u2013Whitham\u2013Richards model is the so-called Cell Transmission Model, developed in the 90s and very widespread in the communities of mathematicians and traffic engineers