Three-phase traffic theory is a theory of traffic flow developed by Boris Kerner between 1996 and 2002. It focuses mainly on the explanation of the physics of traffic breakdown and resulting congested traffic on highways. Kerner describes three phases of traffic, while the classical theories based on the fundamental diagram of traffic flow have two phases: free flow and congested traffic. Kerner's theory divides congested traffic into two distinct phases, synchronized flow and wide moving jam, bringing the total number of phases to three:
Free flow (F) Synchronized flow (S) Wide moving jam (J) The word "wide" is used even though it is the length of the traffic jam that is being referred to.
A phase is defined as a state in space and time.
Free flow (F) In free traffic flow, empirical data show a positive correlation between the flow rate (in vehicles per unit time) and vehicle density (in vehicles per unit distance). This relationship stops at the maximum free flow with a corresponding critical density . (See Figure 1.)
Congested traffic
Data show a weaker 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 vehicular traffic. He instead divides congestion into synchronized flow and wide moving jams.
In congested traffic, the vehicle speed is lower than the lowest vehicle speed encountered in free flow, i.e., the line with the slope of the minimal speed in free flow (dotted line in Figure 2) divides the empirical data on the flow-density plane into two regions: on the left side data points of free flow and on the right side data points corresponding to congested traffic.
Definitions [J] and [S] of the phases J and S in congested traffic
In Kerner's theory, the phases J and S in congested traffic are observed outcomes in universal spatial-temporal features of real traffic data. The phases J and S are defined through the definitions [J] and [S] as follows:
The "wide moving jam" phase [J]
A so-called "wide moving jam" moves upstream through any highway bottlenecks. While doing so, the mean velocity of the downstream front is maintained. This is the characteristic feature of the wide moving jam that defines the phase J.
The term wide moving jam is meant to reflect the characteristic feature of the jam to propagate through any other state of traffic flow and through any bottleneck while maintaining the velocity of the downstream jam front. The phrase moving jam reflects the jam propagation as a whole localized structure on a road. To distinguish wide moving jams from other moving jams, which do not characteristically maintain the mean velocity of the downstream jam front, Kerner used the term wide. The term wide reflects the fact that if a moving jam has a width (in the longitudinal road direction) considerably greater than the widths of the jam fronts, and if the vehicle speed inside the jam is zero, the jam always exhibits the characteristic feature of maintaining the velocity of the downstream jam front (see Sec. 7.6.5 of the bookB.S. Kerner, The Physics of Traffic, Springer, Berlin, New York 2004). Thus the term wide has nothing to do with the width across the jam, but actually refers to its length being considerably more than the transition zones at its head and tail. Historically, Kerner used the term wide from a qualitative analogy of a wide moving jam in traffic flow with wide autosolitons occurring in many systems of natural science (like gas plasma, electron-hole plasma in semiconductors, biological systems, and chemical reactions): Both the wide moving jam and a wide autosoliton exhibit some characteristic features, which do not depend on initial conditions at which these localized patterns have occurred.
The "synchronized flow" phase [S]
In "synchronized flow," the downstream front, where the vehicles accelerate to free flow, does not show this characteristic feature of the wide moving jam. Specifically, the downstream front of the synchronized flow is often fixed at a bottleneck.
The term "synchronized flow" is meant to reflect the following features of this traffic phase: (i) It is a continuous traffic flow with no significant stoppage, as often occurs inside a wide moving jam. The term "flow" reflects this feature. (ii) There is a tendency towards synchronization of vehicle speeds across different lanes on a multilane road in this flow. In addition, there is a tendency towards synchronization of vehicle speeds in each of the road lanes (bunching of vehicles) in synchronized flow. This is due to a relatively low probability of passing. The term "synchronized" reflects this speed synchronization effect.
Explanation of the traffic phase definitions based on measured traffic data
Measured data of averaged vehicle speeds (Figure 3 (a)) illustrate the phase definitions [J] and [S]. There are two spatial-temporal patterns of congested traffic with low vehicle speeds in Figure 3 (a). One pattern propagates upstream with an almost constant velocity of the downstream front, moving straight through the freeway bottleneck. According to the definition [J], this pattern of congestion belongs to the "wide moving jam" phase. In contrast, the downstream front of the other pattern is fixed at a bottleneck. According to the definition [S], this pattern belongs to the "synchronized flow" phase (Figure 3 (a) and (b)). Other empirical examples of the validation of the traffic phase definitions [J] and [S] can be found in the books and,B.S. Kerner, Introduction to Modern Traffic Flow Theory and Control: The Long Road to Three-Phase Traffic Theory, Springer, Berlin, New York 2009Boris S. Kerner, "Breakdown in Traffic Networks: Fundamentals of Transportation Science", Springer, Berlin, 2017 in the article as well as in an empirical study of floating car dataR.-P. Schäfer et al, "A study of TomTom's probe vehicle data with three-phase traffic theory". Traffic Engineering and Control, Vol 52, No 5, Pages 225–231, 2011 (floating car data is also called probe vehicle data).
Traffic phase definition based on empirical single-vehicle data
In Sec. 6.1 of the book has been shown that the traffic phase definitions [S] and [J] are the origin of most hypotheses of three-phase theory and related three-phase microscopic traffic flow models. The traffic phase definitions [J] and [S] are non-local macroscopic ones and they are applicable only after macroscopic data has been measured in space and time, i.e., in an "off-line" study. This is because for the definitive distinction of the phases J and S through the definitions [J] and [S] a study of the propagation of traffic congestion through a bottleneck is necessary. This is often considered as a drawback of the traffic phase definitions [S] and [J]. However, there are local microscopic criteria for the distinction between the phases J and S without a study of the propagation of congested traffic through a bottleneck. The microscopic criteria are as follows (see Sec. 2.6 in the book): If in single-vehicle (microscopic) data related to congested traffic the "flow-interruption interval", i.e., a time headway between two vehicles following each other is observed, which is much longer than the mean time delay in vehicle acceleration from a wide moving jam (the latter is about 1.3–2.1 s), then the related flow-interruption interval corresponds to the wide moving jam phase. After all wide moving jams have been found through this criterion in congested traffic, all remaining congested states are related to the synchronized flow phase.
Kerner's hypothesis about two-dimensional (2D) states of traffic flow
Steady states of synchronized flow
Homogeneous synchronized flow is a hypothetical state of synchronized flow of identical vehicles and drivers in which all vehicles move with the same time-independent speed and have the same space gaps (a space gap is the distance between one vehicle and the one behind it), i.e., this synchronized flow is homogeneous in time and space.
Kerner's hypothesis is that homogeneous synchronized flow can occur anywhere in a two-dimensional region (2D) of the flow-density plane (2D-region S in Figure 4(a)). The set of possible free flow states (F) overlaps in vehicle density with the set of possible states of homogeneous synchronized flow. The free flow states on a multi-lane road and states of homogeneous synchronized flow are separated by a gap in the flow rate and, therefore, by a gap in the speed at a given density: at each given density the synchronized flow speed is lower than the free flow speed.
In accordance with this hypothesis of Kerner's three-phase theory, at a given speed in synchronized flow, the driver can make an arbitrary choice as to the space gap to the preceding vehicle, within the range associated with the 2D region of homogeneous synchronized flow (Figure 4(b)): the driver accepts different space gaps at different times and does not use one unique gap.
The hypothesis of Kerner's three-phase traffic theory about the 2D region of steady states of synchronized flow is contrary to the hypothesis of earlier traffic flow theories involving the fundamental diagram of traffic flow, which supposes a one-dimensional relationship between vehicle density and flow rate.
Car following in three-phase traffic theory
In Kerner's three-phase theory, a vehicle accelerates when the space gap to the preceding vehicle is greater than a synchronization space gap , i.e., at (labelled by acceleration in Figure 5); the vehicle decelerates when the gap g is smaller than a safe space gap , i.e., at (labelled by deceleration in Figure 5).
If the gap is less than G, the driver tends to adapt his speed to the speed of the preceding vehicle without caring what the precise gap is, so long as this gap is not smaller than the safe space gap (labelled by speed adaptation in Figure 5). Thus the space gap in car following in the framework of Kerner's three-phase theory can be any space gap within the space gap range .
Autonomous driving in the framework of three-phase traffic theory
In the framework of the three-phase theory the hypothesis about 2D regions of states of synchronized flow has also been applied for the development of a model of autonomous driving vehicle (called also automated driving, self-driving or autonomous vehicle).
Traffic breakdown – a F → S phase transition
In measured data, congested traffic most often occurs in the vicinity of highway bottlenecks, e.g., on-ramps, off-ramps, or roadwork. A transition from free flow to congested traffic is known as traffic breakdown. In Kerner's three-phase traffic theory traffic breakdown is explained by a phase transition from free flow to synchronized flow (called as F →S phase transition). This explanation is supported by available measurements, because in measured traffic data after a traffic breakdown at a bottleneck the downstream front of the congested traffic is fixed at the bottleneck. Therefore, the resulting congested traffic after a traffic breakdown satisfies the definition [S] of the "synchronized flow" phase.
Empirical spontaneous and induced F → S transitions
Kerner notes using empirical data that synchronized flow can form in free flow spontaneously (spontaneous F →S phase transition) or can be externally induced (induced F → S phase transition).
A spontaneous F →S phase transition means that the breakdown occurs when there has previously been free flow at the bottleneck as well as both up- and downstream of the bottleneck. This implies that a spontaneous F → S phase transition occurs through the growth of an internal disturbance in free flow in a neighbourhood of a bottleneck.