A new Cellular Automata Model including a decelerating damping effect to reproduce Kerner’s three-phase theory

Abstract

Most of the conventional traffic Cellular Automaton (CA) models based on the Nagel–Schreckenberg model (NaSch model) have two problems: an unrealistic deceleration dynamics when a vehicle agent collides with a preceding vehicle in a stopping event, and the problem with reproducing the synchronized flow in Kerner’s three-phase theory. In this paper, a revised stochastic Nishinari–Fukui–Schadschneider (S-NFS) model, belonging to the class of NaSch models, is presented. The proposed CA model, where a random braking effect is improved by considering the dependency on the velocity difference and heading distance with a preceding vehicle, is confirmed to overcome the two above-mentioned drawbacks.

Introduction

In traffic flow studies, it has been prompted to develop an appropriate model to reproduce the “real world” with reasonable accuracy and simplicity. The Cellular Automaton (CA) model in which vehicle agents are treated as discrete self-driven particles is one of the powerful approaches that allows flexibility and robustness [1], [2]. Through a field observation and numerical approaches with the CA model, basic physical mechanisms in real traffic flows are clarified. For example in relatively simple flow on real highways, it can be observed that the traffic phase transfers from a free flow to a congested state with increasing density; there exists a very unstable and irreversible phase (a metastable phase) between those two phases that can be reproduced by applying an appropriate model (e.g., Ref. [3]).

On the other hand, the agreement with empirical results by CA models still needs to be improved (e.g., Ref. [4]).

One of these reproducibility problems is the extent to which Kerner’s three-phase theory [5], [6] can be reproduced, namely, the reproducibility of the respective three phases: free flow (F), synchronized flow (S), and wide moving jam (J). In general, a flow field transfers from free flow to the congested phase through a metastable phase when the density increases, and the flux after the phase transition to congestion tends to decrease with increasing density. In the congested phase, there are two phases: a wide moving jam (J) in which a jam spreads upstream as a stop-and-go shock wave, and a synchronized flow (S) in which vehicle velocity and its flux are maintained at a certain level (at least, larger than that of the J phase). Note that, in this paper, we consider “continuous traffic flow with no significant stoppage” defined by Kerner’s theory [5], [6] as synchronized flow. So we do not use synchronized flow in a meaning of “synchronization of vehicle speeds across different lanes on a multilane road”. In real flow fields, the field observations confirm many complex phases, for example, some of these phases imply that two congested phases may coexist [5], [6]. Spontaneous phase transition from free flow to wide moving jam (F→J transition) is not observed in real traffic. A wide moving jam can emerge spontaneously only in synchronized flow (S→J transition). Therefore, a wide moving jam emerges spontaneously because of a sequence of F→S→J transitions. However, in most of the conventional CA models based on the Nagel–Schreckenberg (NaSch) model [7] synchronized flow is not reproduced very well. On the other hand, several previous authors have reported that synchronized flow can be reproduced by considering the appropriate deceleration process of each vehicle depending on the velocity difference or heading distance from the preceding car (e.g., Refs. [8], [9], [10], [11], [12], [13], [14]).

Another problem is that most CA models show unrealistic deceleration dynamics for each vehicle agent. If there is a relatively slower vehicle ahead, a real vehicle would gradually decelerate when approaching the preceding vehicle. However, most CA models based on the NaSch model only reproduce an unrealistically rapid deceleration where the focal vehicle stops by a collision like the preceding vehicle. Knospe et al. [15], [16] and Lee at al. [17] noticed this point. Following them, Lan et al. [18] reported that a realistic and smooth deceleration can be reproduced by introducing “piecewise-linear movement”.

Note that the above-mentioned two problems, namely, reproduction of the three phase theory and appropriate deceleration dynamics, are mutually related, because an unrealistically abrupt deceleration would cause a rapid growth of a stop-and-go wave, which rarely permits synchronized flow to emerge. If a model overestimates stopping probability, the excessive stop-and-go wave inevitably occurs. On the other hand, if a vehicle is allowed to gradually decelerate when approaching the tail-end of a jam, the stopping probability would be decreased. Thus, an occurrence of this excessive stop-and-go wave would be discouraged and synchronized flow could result.

In this paper, we establish a new model for random braking effects considering the velocity difference and distance to the preceding vehicle, which is able to prevent the occurrence of unrealistic deceleration events and also improves the reproducibility of synchronized flow.