Macroscopic Traffic Flow Theory

Traffic speed, density and flow are the underlying variables of traffic analysis [92]. Traffic flow is defined as the number of vehicles, n, that pass some designated point on a highway during a time interval of length t. According to this definition, the traffic flow is given by

q = n

t, (3.1)

expressed in vehicles per time unit. It is, however, not only the number of vehicles that pass a point that are of interest, but also the amount of time that elapses between the arrival of successive vehicles at a specific point along a highway. This interarrival time of the vehicles is known as time headway, denoted by hti and is measured from a common point on each vehicle

(e.g. the front or rear bumper) as it passes a specific stationary point [161]. Headway may be related to flow by the relationship

q = Pnn i=1hti = _¯1 ht , (3.2) where ¯ ht= 1 n n X i=1 hti

represents the average time headway of n vehicles during a time interval of length t. Average speed may be defined in two ways, the first being the average speed at which vehicles travel when passing a specific stationary point, and the second based on the amount of time that vehicles require to traverse a set distance L. The first is known as the time mean speed, is denoted by ¯ utand is given by ¯ ut= 1 n n X i=1 ui, (3.3)

where ui represents the instantaneous speed of vehicle i when passing the designated point. The

second measure of average speed, known as the space mean speed, is given by

u = ₁ 1

n

Pn i=1L/t1i

where ¯ hs= 1 n n X i=1 hsi

is the average space headway between the n vehicles travelling along the specific stretch of road, again measured with respect to a common point on every vehicle [161]. Traffic density therefore provides an indication of how crowded the stretch of road under consideration is. It is, however, important to note that this definition of density does not take specific vehicle lengths, and as a result specific traffic composition, into account, as only the number of vehicles is considered. Based on these definitions, a simple identity may be formulated to showcase the basic relationship between speed, density and flow, called the fundamental relation of traffic flow theory [168], or the continuity equation [57]. This identity is given by

q = uρ, (3.6)

with typical units of flow, speed and density being vehicles per hour (veh/h), kilometres per hour (km/h), and vehicles per kilometre (veh/km), respectively. The significance of (3.6) is that it allows an analyst to estimate any of the three macroscopic variables, given the other two. This is especially useful when estimating density, which is often difficult to measure [91]. The Fundamental Diagrams

Greenshields [46] defined three basic traffic stream models, namely the Speed-Density Model, the Flow-Density Model, and the Speed-Flow Model, based on the fundamental relationship (3.6). These models give rise to the so-called fundamental diagrams of traffic flow theory, which provide a graphical representation of the statistical relationships between the macroscopic traffic flow variables of speed, flow and density, based on the premise that drivers act in a similar manner when faced with similar traffic conditions [57]. As a result, Maerivoet and de Moor [91] distinguished between three categories of traffic flow conditions, namely free-flow traffic, capacity-flow traffic and congested traffic.

Free-flow traffic occurs when vehicles are able to travel at their desired speeds, untroubled by queues or other slower moving vehicles. As a result, free-flow traffic typically prevails under light traffic flow conditions [91]. The desired, or free-flow, speeds depend on the vehicle, as well as the driver and road section characteristics, and the current weather conditions and traffic rules (e.g. speed limits) [57]. This desired, free flow speed, denoted by uf, is summarised by

the average speed of the vehicles travelling along the section of road under consideration. Due to the low traffic densities observed when free-flow traffic prevails, the space headway between the vehicles is typically large, and minor disturbances due to overtaking manoeuvres or sudden braking do not have a significant effect on the aggregated traffic flow, which may, as a result be considered to be stable [91].

As traffic density increases, so does traffic flow, due to the smaller space headways between individual vehicles. This trend continues until the flow along a lane reaches its maximum, known as capacity flow, denoted by qmax. This capacity flow depends not only on the current

traffic density, but also on the average speed along that specific lane. From (3.2) it is clear that capacity flow is reached at the point where the average time headway is at its minimum, which indicates tightly packed clusters of vehicles travelling at capacity-flow speed, which is typically lower than the free-flow speed [57]. These clusters of vehicles are, however, often unstable with the slightest braking action of one vehicle exhibiting a backward cascading effect, resulting in exaggerated braking by the following vehicles.

As the traffic density increases further, vehicles eventually start to slow down in order to avoid collisions caused by the decreased space and time headways. Due to these chain reactions of slowing vehicles, traffic flow starts to deteriorate, and the resulting, saturated traffic conditions are known as congested traffic [57]. Further increases in vehicle density will lead to so-called stop-and-go traffic, where vehicles often have to slow down significantly or even stop in order to avoid collisions. As traffic density further increases, the traffic becomes motionless, as the space headway between vehicles has reached a minimum bumper-to-bumper distance. In this state, the traffic conditions are referred to as jammed traffic. This maximum density, at which the traffic flow has deteriorated to such a point that the vehicles have become stationary, is known as the jam density, denoted by ρ_jam.

q0 Flow (q) qmax u0 uf Sp eed (u ) u0 uf ρ0 ρjam q0 Flo w (q ) qmax ρ0 ρjam Density (ρ) ρcrit ρcrit ucrit ucrit

Figure 3.1: The fundamental diagrams of macroscopic traffic flow theory, relating flow, speed and density (assuming a linear speed-density relationship). Adapted from Mannering and Kilareski [92].

The fundamental diagrams, employed to illustrate the relationships and traffic behaviour de- scribed above are shown in Figure 3.1. In this instance, a linear speed-density relationship is assumed. As may be seen from the fundamental diagram corresponding to Mannering and Ki-

Figure 3.1, maximum flow occurs at this critical speed. This diagram is not as intuitive to interpret as the others due to the fact that each speed value corresponds to two distinct flow values. The diagram is separated into two regions of flow by the horizontal line corresponding to the critical speed ucrit. The region above this line corresponds to free-flowing traffic. Points

along the curve in this region indicate that fewer vehicles pass a fixed point at higher speeds, with increased space headways separating them, while the point corresponding to the same flow on the curve below ucrit indicates that more vehicles travel past the fixed point, at a lower

speed, with decreased space headways separating them. Various empirical models have also been formulated in order to determine the fundamental diagrams for specific sections of road, based on nonlinear speed-density relationships.