The “Backward Looking” Effect in the Continuum Model Considering a New Backward Equilibrium Velocity Function

In this paper, a new continuum traffic model is developed considering the backward-looking effect through a new positive backward equilibrium speed function. As compared with the conventional full velocity difference model, the backward equilibrium velocity function, which is largely acceptably grounded from mathematical and physical perspectives, plays an important role in significantly enhancing the stability of the traffic flow field. A linear stability condition is derived to demonstrate the flow neutralization capacity of the model, whereas the Korteweg – de Vries – Burgers equation and the attendant analytical solution are deduced using nonlinear analysis to observe the traffic flow behavior near the neutral stability condition. A numerical simulation, used to determine the flow stability enhancement efficiency of the model, is also conducted to verify the theoretical results.

In 1955, a new dimension was added to the traffic flow dynamics by Lighthill, Whitham, and Richard with their so-called LWR model [29][30][31]. In this model, the fluid flow concept is implemented within the traffic flow system using a simple first-order continuity equation, with the model known as a macroscopic model. The dynamical equation of the LWR continuum model is as follows: where , , , and are the density, velocity, space, and time, respectively, of the traffic flow system.
However, the LWR model fails to accurately explain the non-equilibrium states of a real traffic flow field. In view of overcoming this limitation, Payne [32] suggested another high-order continuum model of traffic dynamics; the mathematical expression of which is as follows: where is the anticipation coefficient and is the relaxation time.
Meanwhile, in 1995, on the basis of the Lagrangian viewpoint, Bando et al. [33] proposed the optimal velocity (OV) model, a revolutionary traffic model, in which vehicles are regarded as granular dynamics of the flow field. Here, the vehicle's speed is adjusted by the OV function, which depends on the headway distance of the vehicle. This model has proven to be extremely convenient in terms of both theoretical analyses and numerical simulations since, given that all drivers have a unique sensitivity, the road is regarded as a single-lane system where overtaking is not permitted. The dynamical equation of this model is as follows: where is the unique sensitivity of all drivers, ( ) is the speed of the th vehicle at time , (∆ ( )) is the OV of the th vehicle, and ∆ ( ) is the headway distance of the th vehicle, as measured in terms of ∆ ( ) = +1 ( ) − ( ).
The OV model has certain limitations, with the acceleration and deacceleration of the model regarded as largely unrealistic. To address this inherent defect, Helbing and Tilch [34] developed the so-called generalized force model by introducing a negative speed difference to the OV model. Meanwhile, in 2001, Jiang et al. [35] demonstrated that, in addition to the negative speed difference, the positive velocity difference also has a great impact on increasing the flow stability. As such, the full velocity difference (FVD) model was proposed, with the attendant mathematical expression as follows: where ∆ ( ) is the velocity difference between the n+1th and the nth car, as quantified by ∆ ( ) = +1 ( ) − ( ), and is the sensitivity coefficient independent of .

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All of these conventional models fully rely on the preceding headway gap. However, in an actual traffic flow system, the speed characteristics of the follower vehicle would also come into play. In introducing this aspect, Nakayama et al. [36] proposed the BLOV model, an effective car-following model, in which a backward OV function was established alongside the forward optimal function. Although the BLOV car-following model is largely heralded from a real-life viewpoint, the model has several crucial drawbacks in terms of formulating the backward OV function, which is defined by a negative function that is regarded as unrealistic. To overcome this drawback, various robust car-following models have been proposed [37,38] with the incorporation of a new backward OV function as a positive function, which inevitably increases the stability of the flow field. Here, Zhaoze et al. [39] proposed a continuum traffic model based on the BLOV model, wherein the backward equilibrium speed is quantified by a negative function, which is regarded as unacceptable from a mathematical viewpoint. In the present study, a continuum traffic model is developed with consideration of a new backward equilibrium speed function. Here, the backward equilibrium speed is measured via a positive function depending on the local density, much like with the forward OV. The drawbacks of the existing models could be eliminated through the application of this function, with the stability of the flow field significantly enhanced.
The remainder of the paper is organized as follows. Section 2 describes the formulation of the proposed model, followed by a discussion on the neutral stability analysis in Section 3. Section 4 then presents the nonlinear analysis, whereas Section 5 outlines the numerical simulation and the model results. Finally, Section 6 concludes the paper with the main findings.

Model formulation
This continuum model is developed based on the concept of the backward-looking effect [37,38]. The dynamical equation of the FVD model with the backward-looking effect is as follows: where is the forward concentration of drivers and and are the forward and backward optimal velocities, respectively.
To convert the variables from the micro to the macro, the following transform is conducted: where ( , ) and ( , ) are the velocity and density, respectively, in the macroscopic system; ∆ is the distance between two adjacent vehicles; ℎ = 1 represents the average headway; and ( ) and ( ) are the equilibrium speed function for forward and backward, respectively.
Although considering the backward OV as a negative function of the forward OV [36,40,41], except for one or two outstanding works [37,38], a great number of backward-looking car-following models have been proposed. Similarly, in designing a continuum model for the backward-looking effect, the pioneering models regarded the backward equilibrium as a negative function of the forward equilibrium function, which is seen as unrealistic [39,42,43]. In our proposed model, we assume a new positive function for the backward equilibrium velocity, much like with the forward equilibrium speed. The functions of these equilibrium speeds can be defined as follows [44]: where is the maximum density of the flow field and and are the free-flow speed for the forward and backward equilibrium velocity functions, respectively. Generally, ≠ ; however, here, both cases were studied, that is, for = and for ≠ .
The term ( + ∆, ) can be expanded by the Taylor series as follows: By substituting the macroscopic variables into Eq. (5), we could obtain the following equation: Meanwhile, by combining Eq. (10) and Eq. (1), we could obtain the following equations in macro form: This presents the mathematical expression of the continuum traffic flow model incorporating the pressure effect of backward density.

Linear stability analysis
Linear stability analysis is conducted to examine the flow stability of the model. The vector form of the proposed model is as follows: where The eigenvalues of A are given by the following: Let us consider that 0 and 0 are the density and velocity at an initial time for a homogeneous traffic flow field, respectively. Here, the steady-state solution of the continuum model can be described as follows: Applying the small perturbation to the uniform flow system, we could obtain the following solution: where and are the number and frequency of the waves, respectively.
By substituting the perturbation solution Eq. (17) into Eq. (11), we could obtain the following equations by simplifying and neglecting the higher-order terms: ( + 0 )̂+ 0̂= 0, From Eq. (18), we could obtain a quadratic equation as follows : To ensure a stable flow, both roots of must have negative real parts. Thus, the traffic flow field will be stable under the following states: Therefore, the final neutral stability condition could be deduced from Eq. (20) in the following form: The imaginary part of is given as follows: For a small disturbance, the propagation critical speed of Eq. (21) is as follows: Based on the neutral stability condition described by Eq.  Fig. 1a and Fig.  1b. In fact, as Fig. 1 shows, it was confirmed that the flow stability of our model increased with the decrease in value and that the model performed better for = than for ≠ . It can thus be stated that the stability of the traffic flow field was substantially enhanced by the effect of the new backward equilibrium velocity function.

Nonlinear analysis
To discuss the nonlinear properties of the model, a nonlinear analysis is conducted. Here, to examine the system behavior near the neutral stability condition, a new coordinate system is introduced as follows [11]: By combining Eq. (11) with Eq. (24), we could obtain the following equations: where the traffic flow flux is the product of density and velocity, i.e., = . , with this equation deduced in terms of the following forms: By substituting Eq. (26) into Eq. (25), we could obtain the following: The flow flux can be expressed as follows: By equating the coefficients of and in Eq. (29), we could easily determine the parameters 1 and 2 as follows: Let us consider = ℎ +̂( , ) near the linear stability criteria. In expanding this through using the Taylor expansions and neglecting the higher-order terms of ̂, we could obtain the following form: Following this, we conducted the following variable transformations: After performing the transformation, the standard Korteweg-de Vries (KdV)-Burgers equation was obtained as follows: where 0 denotes an arbitrary constant.

Numerical simulations
Numerical simulations were conducted using the model alongside the finite difference method, with Eq. (11) discretized as follows: where = At the initial stage, Herrmann and Kerner [44] introduced the average density 0 as follows: where is the length of the road and 0 and ∆ 0 are respectively the initial density and density fluctuation near the equilibrium density.
In this numerical study, we assumed the following parameter settings:   Fig. 3b-d present diagrams for the proposed model. Here, various initial conditions were set for both case 1 and case 2, and the flow field was investigated in terms of a range of 100-10,000 s. Panels (b)-(d) correspond to = 0.95, 0.90, and 0.80, respectively, in both Fig. 2 and Fig. 3. The amplitude of the density fluctuation was significantly alleviated with the proposed model compared with that of the conventional FVD model, and the rate of the fluctuation was gradually mitigated with the decrease in value. Because of the introduction of the new backward equilibrium velocity function, the "honk" effect could be implemented, having a positive impact in terms of the driver having the capacity to adjust their velocity to the optimal level. In terms of these phenomena, the jamming area of the flow field was found to consistently disperse, with the flow domain shifting to a steady state within a short period of time. The proposed model performed better in case 1 than in case 2, and the subsequent investigations were thus conducted based on case 1 only. Fig. 4 shows the average velocity profile across the flow field at the initial state and the final state, with the investigation covering a range of 1,000-2,000 s for the initial state and a range of 9,000-10,000 s for the final state. The flow field gradually stabilized according to the time, which is a highly natural tendency, with the stability rate increasing with the decrease in value. As Fig. 4b-d shows, the vehicles enjoyed a largely steady-state environment at the final state, especially for p = 0.80, whereas an unstable situation emerged for the FVD model at the final state, as shown in Fig. 4a. Meanwhile, Fig. 5 shows the velocity profile of the entire domain on a single time step at 10 and 10,000 s. Here, the enhancement of the flow stability, which increased with the decrease in p value, exhibited a similar tendency as shown in Fig.  4, with Fig. 5a showing the results for the conventional FVD model and Fig. 5b-d those for our proposed model. Fig. 6 shows the hysteresis loop diagrams for velocity vs. density, with the flow field observed from 9,900 to 10,000 s. For the conventional FVD model, the trajectory loop exhibited a large amplitude, as shown in Fig 6a, with the vehicles encountering a high-level velocity instability. By contrast, as Fig. 6b-d shows, with our proposed model, the range of the hysteresis loops decreased gradually with the decrease in p value, indicating a smoother traffic flow compared with the conventional FVD model. As such, the numerical results demonstrated good agreement with the theoretical results.

Conclusion
In this study, we built a new continuum traffic flow model on the basis of the FVD model, with consideration of the follower honk effect through a new backward equilibrium speed function. The backward equilibrium speed function was improved with the introduction of a positive function, much like with the forward equilibrium function, rather than the negative function used in previous studies. For the effect of this new backward equilibrium speed function, every focal vehicle was alerted to consider their follower vehicle in addition to their preceding vehicle, and although the follower vehicle demanded more space using the horn effect to go forward, the focal vehicle attempted to adjust their velocity to the optimal value. These phenomena inevitably enhanced the stability of the flow field. The efficiency of the proposed model was considerably high compared with that of the conventional FVD model, which was confirmed via the neutral stability analysis. Meanwhile, the nonlinear analysis determined the flow behavior close to the critical point through the KdV-Burgers equation and the attendant wave solution. The numerical results verified the theoretical approach and demonstrated that the new backward equilibrium speed function had a significant impact on stabilizing the flow field.     The hysteresis loop diagrams of velocity vs density for the time steps 9900s to 10000s for (a) p=1.0, (b) p=0.95, (c) p=0.90 and (d) p=0.80