Jam-absorption driving with a car-following model

https://doi.org/10.1016/j.physa.2015.03.036Get rights and content

Highlights

  • We introduce car-following behaviors to the jam-absorption driving (JAD).

  • Accelerations of cars and the stability of traffic flows in JAD are considered.

  • We numerically confirm the growth and decay of the perturbations caused by JAD.

  • The parameter regions for performing JAD without secondary jams are obtained.

Abstract

Jam-absorption driving (JAD) refers to the action performed by a single car to dynamically change its headway to remove a traffic jam. Because of its irregular motion, a car performing JAD perturbs other cars following it, and these perturbations may grow to become the so-called secondary traffic jams. A basic theory for JAD (Nishi et al. 2013) does not consider accelerations of cars or the stability of traffic flow. In this paper, by introducing car-following behaviors, we implement these elements in JAD. Numerous previous studies on the instability of traffic flow showed that even in a region whose density is below a critical density, perturbation may grow if its initial magnitude is large. According to these previous studies, we expect that the perturbations caused by JAD, if they are sufficiently small, do not grow to become secondary traffic jams. Using a microscopic car-following model, we numerically confirmed that the stability of a flow obeying the model depends on the magnitude of JAD perturbations. On the basis of this knowledge, numerical results indicate that parameter regions exist where JAD allows traffic jams to be removed without causing secondary traffic jams. Moreover, JAD is robust against a parameter of acceleration in the model, as well as the choice of car-following models.

Introduction

Traffic jams cause huge losses in modern society (e.g., 11 trillion yen in Japan in 2005  [1]). To mitigate traffic jams, several strategies have been investigated, such as ramp metering  [2], [3], variable speed limits  [4], [5], [6], and strategies that tune the behavior of individual cars (called “microscopic” behavior). An example of tuning microscopic behavior is adaptive cruise control (ACC), an onboard system that controls time headways and can improve the response to the car in front  [7]. Another microscopic strategy for reducing traffic jams was discussed by Beaty and involves the effect of a single car on the following cars with a large amount of headway. Beaty described his driving technique during passing through a number of stop-and-go waves  [8]: ‘I let a huge gap open up ahead of me, and timed things so I was arriving at the next “stop-wave” just as the last red brakelights were turning off ahead of me’. Beaty also remarked on the timing to approach a traffic jam  [9]: “But if you timed it right, the last of the jam would be gone just when you arrived at the actual jammed location. With the clot gone …won’t everyone take off at high speed? I sure would”. Beaty’s study is empirical and lacks theoretical support. Recently, a theoretical approach to explain “jam-absorption driving(JAD) by a single car was proposed  [10] (hereafter, this type of driving is called “JAD” and the car that performs JAD is called an “absorbing car”.) JAD is composed of two actions: “slow-in” and “fast-out”, as shown in Fig. 1. Slow-in consists of keeping a large headway to avoid being involved in a traffic jam. The traffic jam is removed because of this large headway. Fast-out is performed after slow-in and consists of following the car in front, which has exited from the traffic jam, with a short headway. The theory considers the so-called “memory effect” (or “frustration effect”)  [11], [12], which is the enlargement of the net time gap after being involved in a traffic jam and which causes the growth of traffic jams. Cars following the absorbing car avoid being involved in a traffic jam by JAD. Therefore, the memory effect is not expected to occur for these cars. Thus, JAD prevents the aggravation of the car-following behavior of the following cars. However, because of its irregular motion, JAD itself causes perturbations such as compression and expansion waves (Fig. 1). In the framework of this theory  [10], the compression and expansion waves should collide with each other and disappear to avoid the so-called “secondary traffic jams.” The theory indicates that a headway threshold exists that prevents secondary traffic jams.

The theory  [10] contains some weak points. The first is that it does not consider the accelerations of cars. The second is the lack of the stability of traffic flow, which determines whether secondary jams occur. The stability of traffic flow is closely related to the magnitude of perturbations. Below the critical density, there is a density region in which a perturbation grows if its amplitude exceeds the critical amplitude observed in macroscopic  [13], [14], microscopic car-following  [15] and cellular automaton  [16] traffic models. In addition, Bando et al. concluded from a microscopic car-following traffic model that density ranges exist where homogeneous flow and congested flow coexist  [17]. Moreover, Helbing and Moussaid analytically calculated the critical amplitudes of perturbations in a microscopic car-following traffic model  [18]. Based on these studies, we do not expect secondary jams to occur if the perturbations caused by the absorbing car are sufficiently small. According to these previous studies, we anticipate that in traffic flow composed of cars obeying a car-following model and maintaining extra-large gaps instead of minimal headway, traffic-flow stability depends on the magnitude of perturbations.

In this paper, we construct a framework of JAD that takes into account accelerations of cars and stability of traffic flow by introduction of car-following behaviors. First, we use a microscopic car-following model to numerically verify how the stability of traffic flow depends on the magnitude of perturbations. Next, we verify that a parameter region exists where the absorbing car avoids a traffic jam and prevents the growth of secondary jams. We also show that the validity of JAD is not influenced by the choice of a acceleration parameter value and a car-following model.

Section snippets

Model

Among the numerous car-following models proposed in the last several decades  [19], [20], [21], [22], [23], simple classical models are sufficient for confirming that the stability of traffic flow depends on the magnitude of the perturbations. For a simple model, we use the Helly model  [24], which is described by a set of linear differential equations vi̇(t)=k1[xi1(t)xi(t)Di(t)]+k2[vi1(t)vi(t)],Di(t)=d+Tdesvi(t), where xi(t) and vi(t) are the location and velocity of the car i at time t,

Settings

First, we describe the scenario used for numerical simulations without JAD: 1000 cars labeled i=1,,1000 from the head car to the tail car move in a uniform platoon in a single-lane road of infinite length, as shown in Fig. 2. Although we use a finite number of cars in the simulations, we assume that many cars run in columns in front of car 1 and behind car 1000. At time t=0, all the cars move at the same velocity vMAX and have the same gap D(vMAX)hbuf=(d+TdesvMAX)hbuf, which is greater than

Numerical simulations with JAD

Consider the situation in which the absorbing car performs JAD. The initial condition and the movement of the leading car are the same as in the scenario without JAD. Cars 2,,m1,m+1,,1000 obey the Helly model with the restriction 0vvMAX. Car m is the absorbing car and its movement is described as follows: At t=tS1, when the leading car starts to perturb traffic, car m starts to decelerate from vMAX to va with a constant acceleration αa. After reaching va, it maintains va for a period Ta.

Discussion

We developed a JAD with car-following behaviors to represent accelerations of cars and the stability of traffic flows. We investigated relationships between the growth of the perturbations caused by the leading car and an extra amount of initial gap. In addition, we formalized the relationships between the absorbing car’s low velocity va during JAD and the duration Ta during which the low velocity is maintained. We integrated the condition for which no secondary traffic jam is caused by JAD

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    Present address: Department of Mechanical and Aerospace Engineering, Graduate School of Engineering, Tottori University, 4-101 Minami, Koyama, Tottori 680-8552, Japan.

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