Traffic Flow Modelling

Population growth and economic growth come with an increase in traffic demand and—more often than not—increased levels of congestion and accompanying delays, pollution and decrease in safety. There are several strategies to reduce congestion, keep cities liveable, clean and safe and limit travel time increase. Examples are encouraging people to travel using modes of transport that put less strain on the transportation network, to encourage people to travel at different times or on different routes, to apply traffic management to use roads in a more efficient way or to expand the road network. For all these measures, it is important to know how traffic flow will actually look: where and when will there be congestion, what are the bottlenecks and where is the road capacity already sufficient? Traffic flow models support this assessment by describing and predicting traffic on roads. For example, they model the number of vehicles on the road and their speeds. Using the models, travel times and congestion can be predicted.

In the previous chapter, the main variables in traffic flow modelling were introduced. In this chapter, we discuss how they are related: obviously, high speeds seldom occur together with low headways, similarly, low densities create room for high speeds. Traffic flow models are based on the assumption that there is some relation between these variables. The relation between distance and velocity was first studied by Greenshields (The photographic method of studying traffic behavior. In: Proceedings of the 13th annual meeting of the highway research board (1934), pp 382–399) and called the fundamental relation (or fundamental diagram) later. Therefore, Greenshields is often regarded as the founder of traffic flow theory, and the fundamental diagram is the first model in the genealogical tree of traffic flow models (see Page 15).

The earliest family in the model tree incorporating dynamics are microscopic models. They are based on the assumption that drivers adjust their behaviour to that of the leading vehicle. Microscopic modelling has shown to be a fruitful line of thought, which is illustrated by the large part of the model tree taken up by this family (see the model tree on page 15. Microscopic models describe the longitudinal (car-following) and lateral (lane-changing) behaviour of individual vehicles. We focus on longitudinal behaviour.

Macroscopic traffic flow models forms arguably the largest family in the model tree, see page 15. They describe traffic flow as if it were a continuum flow and are often compared to, or derived in analogy with, continuum models for fluids. Individual vehicles are not modeled, however aggregated variables such as (average) density and (average) flow are used.

Numerical methods are used to approximate the solution of traffic flow models. This is needed because in most realistic cases it is impossible to solve the problems analytically. When a macroscopic model is applied, usually the space and time domains are divided into intervals: road segments (grid cells) and time steps. For each time step and at each grid cell the model equations are solved approximately using numerical methods. The result is the density in each grid cell, at each time step. Alternative methods are based on moving coordinates and will also be discussed. In this chapter, the focus is on the numerical methods themselves, with the main purpose that the reader should be able to apply the methods. After reading this chapter the reader will understand the basics of applying numerical methods to macroscopic traffic flow models. They can apply those methods to the models and can argue about the impact of choices such as a fixed vs. moving coordinate system and grid cell and time step size.

Mesoscopic traffic flow models were developed to fill the gap between the family of microscopic models that describe the behavior of individual vehicles and the family of macroscopic models that describe traffic as a continuum flow. Traditional mesoscopic models describe vehicle flow in aggregate terms such as in probability distributions. However, behavioral rules are defined for individual vehicles. The family includes headway distribution models, cluster models, gas-kinetic models and macroscopic models derived from them. Most recently, hybrid mesoscopic models have appeared as a new branch on the tree: they combine microscopic and macroscopic models.

In the previous chapters, four different families of traffic flow models were discussed. In this chapter we compare them, highlight their differences and discuss their applications.

This chapter provides solutions to a selection of the problems in this book.