Three phase traffic theory

The three phase traffic theory is an alternative traffic theory developed by Boris Kerner. It is mainly concerned with the physics of traffic congestion at freeways. Kerner describes three phases, opposed to the classical theories based on the fundamental diagram of traffic flow, which considers only two phases: "free flow" and "congestion". The congestion phase is divided into two distinct phases by Kerner, which results in the following three phases:

# Free flow (F)
# Synchronised flow (S)
# Wide moving jams (J)A "phase" is described as a "state" in "time" and "space".

Free Flow (F)

In free flow conditions, drivers can choose their own speed. Empirical data show a positive relationship between the flow $q$ (expressed in vehicles per time unit) and the density $k$ (expressed in vehicles per distance unit). This relationship is cut off at a maximum point of free flow, $q\_\{max\}$ with a corresponding critical density $k\_\{crit\}$.
The accompanying graphs contain fictitious data points; however, many studies show a similar pattern.

Congestion (S and J)

The fundamental diagram shows a much less strong relationship between flow and density in congested conditions. Therefore, Kerner argues that the fundamental diagram, as used in classical traffic theory, cannot adequately describe the complex dynamics of traffic under congested conditions. He instead divides congestion into "synchronised flow" and "wide moving jams".

When the number of vehicles on a road, i.e. the density, becomes too high, the state of the traffic is said to be metastable. This means that when small perturbations occur, the state is still stable (synchronised flow); however when larger perturbations occur, the traffic becomes unstable and moving jams will emerge (wide moving jams). Another interpretation of the metastability is the following: when the flow rate out of a moving jam $q\_\{out\}$ is lower than the maximum flow $q\_\{max\}$, the speed of the downstream front is higher than the speed of the upstream front. The downstream front will "catch up" with the upstream front, and the moving jam will disappear.

The metastability is one of the differences of Kerner's theory with classical traffic flow theories. In classical theory, a fundamental diagram is used, where it is hypothesized that when the density exceeds $k\_\{crit\}$ the traffic becomes "unstable" and moving jams occur spontaneously.

Wide moving jams (J)

The wide moving jam phase is a jam that moves "upstream" and is "wide". A moving jam is said to be wide if the width of the moving jam (in the direction of the road) considerably exceeds the width of the jam fronts. Inside the wide moving jam, the mean speed of the vehicles is equal to the mean speed of the vehicles at the "downstream front" of the jam. At the downstream front, vehicles speed up to free flow again; at the upstream front, vehicles come from free flow and have to reduce speed. On average, a wide moving jam maintains the mean speed of the downstream front, even if the jam propagates through other traffic states or bottlenecks. The traffic flow (the number of vehicles per time unit) in a wide moving jam is reduced strongly.

Kerner's empirical results show that some characteristics of the wide moving jam are independent of traffic demand and bottleneck characteristics of where and when the jam emerged. These characteristic parameters can depend on weather conditions, road conditions, et cetera. The speed of the downstream front of the wide moving jam (in the upstream direction) is a typical parameter, as well as the flow downstream of the downstream front (free flow conditions). This means that multiple wide moving jams will have the same parameters under the same conditions. These parameters are therefore up to a certain amount "predictable".

Kerner finds that the flow rate downstream of a wide moving jam is lower than the maximum possible flow rate in free flow (upstream of the wide moving jam). In other words, after people drive out of a wide moving jam, the traffic flow is typically lower than before they drove into the wide moving jam. This is explained by the driver behaviour: upstream of the upstream front of the wide moving jam, the speeds are nearly homogeneous. In these homogeneous conditions, drivers accept a considerably lower time gap, leading to a higher flow. However, when drivers accelerate out of a moving jam from the standstill, the speeds are not homogeneous, leading to larger time gaps between the cars and a lower flow.

As soon as a wide moving jam propagates through an upstream bottleneck, it is called a "foreign wide moving jam". A foreign wide moving jam can influence considerably on other moving jams. For example, the growth of narrow moving jam that is close to the downstream front of the foreign moving jam can be suppressed.

Synchronised flow (S)

In contrast to wide moving jams, the speeds of the vehicles within the synchronised flow phase can vary quite a lot. The downstream front of the synchronised flow is often fixed at a location, usually a bottleneck somewhere on the road. The traffic flow in this phase can still remain close to free flow, although the speeds of the vehicles are reduced.

Another difference between synchronised flow and wide moving jams is that synchronised flow usually has a fixed downstream front. A Synchronised Flow Pattern (SP) with a fixed downstream front and a fixed upstream front is called a Localised Synchronised Flow (LSP). It is however possible that the SP propagates upstream. When only the upstream front moves upstream, the SP is called a Widening Synchronised Flow Pattern (WSP) – the downstream front stays at the effectual bottleneck and the width of the SP grows. It is even possible that both the upstream and downstream front move upstream - the downstream front is then not fixed at the effectual bottleneck; this is called a Moving Synchronised Flow Pattern (MSP). The difference between these SPs and wide moving jams however is, that when a WSP or MSP reaches an upstream bottleneck, a "catch-effect" occurs. The SP is "caught" at the upstream bottleneck and stops moving upstream. A wide moving jam is never caught and just keeps on moving upstream.

Opposed to the wide moving jams, the synchronised flow, even when it is moving (MSP), does not have characteristic parameters. For example, the speed of the downstream front of an MSP can vary over a wide range during pattern propagation and can be different for different MSPs.

Expanded Congestion Patterns

Expanded Congestion Patterns (EP) are complex congestion patterns that occur when two or more bottlenecks are closely located to each other. As was stated before, when an SP propagates upstream (a WSP or MSP), the SP is caught at an upstream bottleneck. However, when the upstream bottleneck is close to the effectual bottleneck downstream, the SPs are not caught. The pattern that then occurs is called an EP. EPs can consist of SP only, but usually wide moving jams begin to emerge in synchronised flow. The EP then consist of both SP and wide moving jams.

F → S transition

The emergence of synchronised flow is called the "breakdown phenomenon", or the "F → S transition". In such a transition, the average speed of the vehicles in the synchronised flow phase is reduced, although the traffic flow can still remain more or less constant. It has been found that the breakdown phenomenon has a probabilistic nature: the higher the flow rate, the more likely a breakdown.

Kerner finds, using empirical data, that synchronised flow can emerge "spontaneously" or "induced" in free flow conditions. Spontaneous emergence of synchronised flow means that, due to a high traffic demand at a bottleneck, a phase transition occurs. Induced breakdown is caused by external disturbances in traffic flow. This is usually related to the propagation of synchronised flow or a wide moving jam, that originate from a downstream bottleneck.

Synchronised flow usually only emerges at bottlenecks. A bottleneck where Synchronised Flow occurs on a regular basis is called an "effectual bottleneck".

S → J transition

Wide moving jams do not emerge spontaneously, but can only emerge in synchronised flow conditions. Therefore, the emergence of wide moving jams can be summarized as a F → S → J transition. Firstly, synchronised flow emerges out of free flow. Then, the synchronised flow compresses (densities become higher, speeds decrease). This self-compression is called "the pinch effect". Kerner finds that the higher the density in synchronised flow are, the higher the frequency of moving jam occurrence is. In the pinch regions of synchronised flow, "narrow moving jams" emerge and grow. When these narrow moving jams grow, wide moving jams will emerge. These wide moving jams propagate upstream and will continue to propagate, even if they pass through synchronised flow regions or bottlenecks.

Criticism of the theory

The theory has been criticized for two primary reasons. First, the theory is almost completely based on measurements on the Bundesautobahn 5 in Germany. It may be that this road has this pattern, but other roads in other countries have other characteristics. Future research must show the validity of the theory on other roads in other countries around the world. Second, it is not clear how the data was interpolated. Kerner uses fixed point measurements (loop detectors), but draws his conclusions on vehicle trajectories, which span the whole length of the road under investigation. These trajectories can only be measured directly if floating car data is used, but as said, only loop detector measurements are used. How the data in between was gathered or interpolated, is not clear.

References

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See also

* Fundamental Diagram
* Traffic flow
* Traffic wave
* Traffic congestion