† Corresponding author. E-mail:
Project supported by the National Key R&D Program of China (Grant No. 2018YFB1601000).
Existing traffic flow models give little consideration on vehicle sizes. We introduce the solid angle into car-following theory, taking the driver’s perception of the leading vehicle’s size into account. The solid angle and its change rate are applied as inputs to the novel model. A nonlinear stability analysis is performed to analyze the asymmetry of the model and the size effect of the leading vehicle, and the modified Korteweg–de Vries equation is derived. The solid angle model can explain complex traffic characteristics and provide an important basis for modeling nonlinear traffic phenomena.
Nowadays, the traffic congestion problems our society is facing have been the focus undeniably today, influencing people’s daily life and schedule. In order to solve these problems, many scholars focus on the traffic flow theory which is a kind of core theory for traffic management. The definition of traffic flow is a system used to describe the interacting vehicles of which the feature is self-driven and many-particle.[1] Car-following theory, which can explain the complex phenomena of traffic flow such as critical phase transitions,[2–15] is indispensable in microscopic traffic modeling.
The theory of car following has been developed for a long time. Since 1950, numerous researchers have come up with different forms of car-following models successively. From the perspective of statistical physics, vehicles are considered as self-driven interacting particles. The physicists build microscopic car following models by introducing some essential factors which can describe the basic properties of the actual traffic flow.
A classic idea has been widely accepted that the car-following behavior is a complex process which can be regarded as a response to a stimulus. As a result, many researchers have proposed different kinds of car-following models based on the above idea of stimulus-response framework. Of all car-following models, the kind of models concerning with optimal velocity is very classical in which the acceleration is expressed as a response to the stimulus containing different kinds of factors. The models listed below are more representative.
Bando[2] proposed an optimal velocity (OV) model in the 1990s to solve the problem of the excessive acceleration in Newell model
The parameters of the OV model were calibrated by Helbing and Tilch.[20] In addition, they pointed out the existing problems and proposed a generalized force (GF) model
Considering the effect of positive speed difference, Jiang et al.[21] came up with a more comprehensive model called full velocity difference (FVD) model
Recently, lots of researchers proposed extended car-following models based on the former research results, which is of great significance for development of car-following theory. Sun et al.[22] proposed an extended car-following model by considering drivers memory and average speed of preceding vehicles with control strategy, they used different methods to analyze the model. Wang et al.[23] came up with a novel two-lane lattice hydrodynamic model considering lane changing behavior and passing behavior on curved roads. It is a breakthrough in traditional car following models. Besides, they[24] considered the influence of other factors, the empirical lane changing rate and the self-stabilization effect, and put forward an extended lattice hydrodynamic model.
Nevertheless, car following is a behavior that is highly constrained in the driver’s decision. Apparently, the drivers are unable to estimate the distance and velocity exactly. They usually use their sensory ability to judge the behavior of the leading car and respond accordingly. Considering the realistic situation, Jin et al.[18] proposed a new model using the visual angle information as stimulus from the perspective of statistical physics based on the stimulus-response framework. The expression is as follow:
In the past research, it is assumed crucially in the current car-following model that all of vehicles are modeled as particles, and the type and size of vehicles are neglected. Nevertheless, the drivers of the following vehicles will be influenced enormously by the size of the leading vehicles. In addition to relative velocity and distance with proceeding vehicle, it can be found that the leading vehicle types could have a great impact on the car following behavior. Due to the larger size and the worse braking capacity of trucks, the headway is larger when a driver follows a truck rather than a car,[16] so we can conclude from the above phenomena that the driving behavior is impacted by the type of leading vehicle.[17] Thus, it is significant to model the effects of vehicle size on car following behavior. To our knowledge, the effects of vehicle size on the car-following behavior have not been taken into consideration in most of the models.[18] We consider that it is of importance to put forward a solid angle model with considering the effect of vehicle size and to explore the characteristics of the novel model.
In this paper, we propose a dynamical model of the car following behavior using the solid angle and its change rate as variables for the first time so that we replace the original variables, such as the distance and velocity, by introducing the concept of solid angle into the previous classic car-following model. In Section
Most of the previous models were built if the assumption is founded that the drivers have accurate perception of the distance, velocity and acceleration of themselves and leading vehicles. However, there is no psychological evidence to support these assumptions. Previously we have proposed a visual angle model (VAM) to consider drivers’ visual perception behavior.[18] In this model we introduced the width of the leading vehicles only, which fails to completely describe the effect of the leading vehicles’ size.[19] Therefore, this paper introduces the concept of solid angle to establish the stability of the traffic flow influenced by the vehicle size.
In geometry, a solid angle is the angle of an object to a specific point in three-dimensional space, and it is an analogy of the planar angle. The solid angle describes the scale of the size of the object measured by the observer standing at a certain point. The apex of the solid angle is the point from which the object is viewed, and the object is said to subtend its solid angle from that point. In a car-following scenario, the apex of the solid angle is the following car driver, and the given object is the leading car.
A unit sphere is constructed with the observation point as its center. The projected area of any object projected onto the unit sphere is the solid angle of the object relative to the observation point. Therefore, the solid angle is an area on the unit sphere, which is similar to the planar angle being an arc length on the unit circle. A solid angle in steradians can be expressed by the following equation:
Considering the sizes of leading vehicles and the solid angle of drivers, we replace the distance and speed in the original optimal velocity model by the solid angle and its change rate. To detect the vehicle movement, the solid angle is regarded as a stimulus, which depends on the leading vehicle size and the distance between the front bumper of the follower and the rear bumper of the leader, as shown in Fig.
In this paper, we substitute the space headway and the velocity difference with the solid angle and its change rate, respectively. Then, we propose a novel model called the solid angle model, or SAM for short. The model can be formulated as
The acceleration describing the solid angle stimulus and the following driver’s response is expressed in Eq. (
Considering the stability of a steady traffic flow, it is assumed that all vehicles are dimensionally identical, moving at a uniform optimal velocity and distance headway. Therefore, the location of the n-th vehicle is
![](cpb_29_6_060504/cpb_29_6_060504_ieqn1.gif)
In order to explore the characteristics of SAM’s stability, a small disturbance is added to the steady state
Substituting Eq. (
![]() | Fig. 2. Comparison of headway-sensitivity relationship diagrams with different car-following models. |
The sizes of leading vehicles also influence the stability regions of SAM, as shown in Fig.
Figure
When the steady traffic flow is affected by a random disturbance, the change of the traffic flow state near the critical point will be transmitted to the upstream in the form of density wave. In this section, we obtain a nonlinear wave equation using the long wavelength mode to depict the propagation behavior so that we achieve the purpose of analyzing the slowly varying behavior at long waves.
Plugging Eq. (
Under both conditions of αc = (1 + ε2)α and b = V′, the second-order and third-order terms of ε can be eliminated near the critical point (hc, αc). The simplified result is shown as follow:
The coexisting phase represented by the kink–antikink soliton solution is made up by the headways of the low-density free flow phase and high-density congested phase. The solution of the mKdV equation is an accepted expression for describing the state of traffic flow near the critical point, which is achieved by analyzing the nonlinear stability.
The neutral and coexisting curves with different lambdas are shown in Fig.
Similar to the neutral curves, the coexisting curves also exhibit asymmetric characteristics. In the congested state, the stability region is smaller than that in the uncongested state because the drivers will pay more attention to changes in the size of the leading vehicles, and then make a more timely response. The results illustrate that using solid angle under the consideration of vehicle size can enhance the linear and nonlinear stability of traffic flow.
As shown in Fig.
In order to further explore the properties of SAM, numerical results are presented to explore the characteristics of phase transition. In the simulations, the n th vehicle’s velocity and position are calculated using the following equations:
Kink-antikink density waves, which are an important feature of traffic congestion in the unstable region, are shown in Fig.
The asymmetric characteristics of the SAM can be easily observed in Fig.
Spatiotemporal vehicle trajectory diagrams also clearly exhibit the effects of vehicle’s size, as shown in Fig.
The main contribution of this paper is to propose the influence of the solid angle on the car following theory. This allows us to take the size of leading vehicles into account, which is an issue hard to investigate in previous studies. The novel SAM is an improvement of the OV model. Stability analysis results indicate that the neutral and coexisting stability curves are asymmetric, and such that the steady and metastable regions are enlarged as the leading vehicle’s size increases. The SAM’s validity is also be verified through the corresponding numerical simulations, which show that the size of the leading vehicle strongly influences and enhances the stability of traffic.
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