An Anisotropic Traffic Flow Model with Look-Ahead Effect for Mixed Autonomy Traffic

Abstract

In this paper we extend the Aw–Rascle–Zhang (ARZ) non-equilibrium traffic flow model to take into account the look-ahead capability of connected and autonomous vehicles (CAVs), and the mixed flow dynamics of human-driven and autonomous vehicles. The look-ahead effect of CAVs is captured by a non-local averaged density within a certain distance (the look-ahead distance). We show, using wave-perturbation analysis, that increased look-ahead distance loosens the stability criteria. Our numerical experiments, however, showed that a longer look-ahead distance does not necessarily lead to faster convergence to equilibrium states. We also examined the impact of spatial distributions and the market penetrations of CAVs and showed that increased market penetration helps to stabilize mixed traffic while the spatial distribution of CAVs has less effect on stability. The results revealed the potential to use CAVs to stabilize traffic and may provide qualitative insights into speed control in the mixed autonomy environment.

Introduction and Review of Related Work

Hydrodynamic Traffic Flow Models

Hydrodynamic traffic flow models, often given in the form of partial differential equations (PDEs) have been widely studied in the literature on traffic science. They have wide applications and are often used in traffic simulation, state estimation, and control design. The most well-known of them is the Lighthill–Whitham–Richards (LWR) model ( 1 , 2 ), which has the form

$${\rho }_t+{\left(\rho V\left(\rho \right)\right)}_x=0,$$

(1)

where $\rho$ is the density of traffic at location $x$ and time $t$ , and $V\left(\rho \right)$ is the equilibrium velocity as a function of density. The LWR model is essentially a scalar conservation law endowed with an equation of state that captures average driver behavior under stationary (or equilibrium) conditions. The LWR model is capable of modeling transitions from one stationary state to another, in the form of shock or acceleration waves. However, it lacks the ability to model some notable traffic flow phenomena, such as stop-and-go waves and traffic hysteresis. Various models, collectively known as higher-order traffic flow models, have been proposed to overcome the deficiencies of the LWR model. For example, analogous to shallow channel water flows, Payne and Whitham ( 3 , 4 ), respectively, introduced a momentum equation to capture speed evolution away from equilibrium, and proposed the first higher-order model:

$$\begin{array}{c}\{\begin{array}{c}{\rho }_t+{\left(\rho v\right)}_x=0,\hfill \\ \multicolumn{1}{c}{v_t+v{v}_x=\frac{V\left(\rho \right)-v}{\tau }-\frac{c_0^2}{\rho }{\rho }_x,}\end{array}\hfill \end{array}$$

(2)

where $\tau$ is a relaxation time constant, and $c_0<0$ is the “traffic sound speed.” But this model has two main drawbacks: it can produce negative travel speed (“wrong way travel”) and traffic information can travel faster than vehicles, which violates the anisotropic property of traffic flow, that is, vehicles cannot push other vehicles from behind to speed them up. To solve these problems, there are two models independently introduced in ( 5 , 6 ), and the inhomogeneous Aw–Rascle–Zhang (ARZ) model has the form:

$$\begin{array}{c}\{\begin{array}{c}{\rho }_t+{\left(\rho v\right)}_x=0,\hfill \\ \multicolumn{1}{c}{{(v+h\left(\rho \right))}_t+v{(v+h\left(\rho \right))}_x=\frac{V\left(\rho \right)-v}{\tau },}\end{array}\hfill \end{array}$$

(3)

where the constant $c_0$ in PW model is substituted by the convective derivative ( ${\partial }_t+v{\partial }_x$ ) of the pressure function $h\left(\rho \right)$ that accounts for the anticipation of drivers of downstream density changes.

The ARZ model has been widely used and studied since it was first proposed. Theoretical and numerical solutions of the ARZ model have been studied in, for example, ( 7 – 9 ). Others have extended the ARZ model: for example, Lebacque et al. ( 10 ) generalized the ARZ model to the generic second-order models (GSOM) where the pressure term is generalized to a non-linear velocity term. GSOM have then been used for data fitting ( 11 ) and extended with non-local densities ( 12 , 13 ).

Notably, first-order non-local models have been proposed and analyzed to incorporate look-ahead effects by defining vehicle velocity as a function of spatially averaged downstream density ( 14 , 15 ). However, as with the LWR model, they lack the dynamic velocity evolution and hyperbolic structure needed to capture non-equilibrium traffic dynamics and perform stability analysis, particularly in the context of mixed autonomy.

Stochastic first-order models ( 16 , 17 ), inspired by the noisy Burgers’ equation ( 18 ), have been proposed to capture scatter in the FD and certain non-equilibrium patterns by introducing variability and randomness. In contrast, second-order models, as continuum approximations of car-following dynamics, retain explicit velocity evolution needed to represent instabilities and wave propagation. These two approaches reflect different objectives: stochastic models focus on variability, while higher-order deterministic models are well suited to analyzing instability mechanisms and stability control in mixed traffic.

Multi-Class Hydrodynamic Traffic Flow Models

Real-world traffic has vehicles of different types and levels of performance, which can be categorized into vehicle classes. Each class of vehicle may interact with others in different ways and this can be captured by extending the aforementioned models to multi-class hydrodynamic traffic flow models. Starting with an extension of the LWR model, Wong and Wong ( 19 ) proposed a multi-class LWR model with heterogeneous drivers characterized by their choice of free-flow speeds. In particular, they gave an isotropic case where the speed of each class is a function of the total density. In a separate work, Zhang and Jin ( 20 ) proposed a multi-class LWR model considering critical density such that when traffic concentration reached a critical value, all the class of vehicles are mixed together and move as a group, and below the critical density the model is similar to Wong and Wong’s model. Ngoduy and Liu ( 21 ) proposed a generalized multi-class first-order simulation model based on an approximate Riemann solver, which is able to explain certain non-linear traffic phenomena on freeways. Logghe and Immers ( 22 ) proposed a new model where vehicle classes interact in a non-cooperative way, where slow vehicles act as moving bottlenecks for fast vehicles while fast vehicles have no influence on slow vehicles. Such relationships were presented in ( 23 ). Qian et al. ( 24 ) developed a macroscopic heterogeneous traffic flow model with pragmatic cross-class interaction rules.

There are also studies that have proposed non-equilibrium hydrodynamic models for mixed traffic flow, for example, ( 25 – 29 ). Specifically, Ngoduy et al. ( 25 ) proposed a multi-class gas-kinetic model where one class of vehicle is able to receive a warning message when there is downstream congestion. They further extended this in ( 26 , 27 ) to include cooperative adaptive cruise control (CACC). Mohan and Ramadurai ( 28 ) extend the ARZ model to a multi-class model using area occupancy (AO) which can capture the unique phenomena in lane-free traffic. Huang et al. ( 29 ) proposed a multi-class model where human-driven vehicles (HDVs) are modeled by the ARZ model and CAVs are modeled by a mean-field game. They also performed linear stability analysis for the mean-field game model.

CAVs as Agents for Traffic Stabilization

The traffic flow of HDVs can be unstable even without an external disturbance. For example, in Sugiyama et al. ( 30 ), a field experiment on a ring road with HDVs showed that stop-and-go waves can arise without the presence of bottlenecks when there is a sufficient number of vehicles on the road. A recent field experiment, on the other hand, showed that such stop-and-go waves can be eliminated with a single autonomous vehicle (AV) as a control agent to pace HDV traffic for the vehicles involved ( 31 ). Such improvements were also found in a larger field experiment of over 100 CAVs ( 32 ). This stabilization effect of an AV or CAV as a control agent has also been widely studied through traffic simulation using microscopic car-following models, for example, in ( 33 – 35 ). In these studies, it is shown that a single AV can stabilize multiple HDVs on a single-lane road by using its sensing capabilities and feedback control to adjust its speed.

The Main Contributions of This Paper

In this paper, we enhance the understanding on the look-ahead effect of CAVs in traffic flow modeling by extending the ARZ model with a non-local density parameter, which simulates the forward-looking capabilities of CAVs. This modification allows for a more realistic representation of how autonomous technologies might influence traffic flow dynamics.

We undertake a comprehensive theoretical stability analysis using wave-perturbation methods and demonstrate that the extended model for CAVs can achieve greater stability over longer look-ahead distances, offering a theoretical foundation for integrating CAVs into traffic systems.

Additionally, we further extend our model to a multi-class framework, accommodating both HDVs and CAVs. This extension is crucial to evaluate the stabilization effect of CAVs in various traffic conditions with the presence of HDVs. Through extensive simulations referencing the studies above, we evaluate how different configurations of look-ahead distances and vehicle distributions affect traffic flow stability.

The findings of this study contribute to ongoing discussions on traffic management in mixed autonomy environments. One of them suggests that moderate look-ahead distances might provide optimal stability conditions. Another notable finding is that with a relatively low penetration rate of CAVs, the mixed flow can be effectively stabilized, which is consistent with previous studies. Furthermore, evenly distributed CAVs achieve marginally better stabilization results compared with segregated distributions.

The remainder of this paper is organized as follows: We first introduce the modified ARZ model and interpret it as a model for CAVs. We then give a stability analysis of the model using wave perturbation. This is followed by a formulation of a multi-class extension of the modified ARZ model for mixed CAV-HDV traffic; the parameters of both models are analyzed via numerical experiments to test the stability of CAVs under different conditions. We end with conclusions and propose directions for future research.

An Extended ARZ Model with Look-Ahead Effects

We first extend the ARZ model to take into account the look-ahead capability of CAVs without explicitly modeling CAVs and HDVs as distinct classes. Here we assume that the CAVs are all equipped with range sensors and vehicle-to-vehicle communication to enable them to observe the density of a certain distance ahead, say $L_D$ . A visualized demonstration of this is given in Figure 1.

Figure 1.

A CAV’s front observation of the traffic density of a certain distance in front.

Instead of responding to the motion of the vehicle immediately in front, a CAV can take advantage of this look-ahead capability and adopt a speed that is based on the average traffic condition within this look-ahead distance, therefore reducing over- or under-reaction and smoothing its trajectory. This will in turn lead to greater stability in traffic. Following this argument, we modify the ARZ model with a new relaxation term that takes into account this look-ahead effect on traffic flow as follows:

$$\{\begin{array}{c}\begin{array}{c}{\rho }_t+{\left(\rho v\right)}_x=0,\hfill \\ \multicolumn{1}{c}{{(v+h\left(\rho \right))}_t+v{(v+h\left(\rho \right))}_x=\frac{v-V\left({\rho }^{*}\right)}{\tau },}\\ \multicolumn{1}{c}{{\rho }^{*}\left(L_D\right)=\frac{{\int }_x^{x+L_D}\rho (t,\xi )d\xi }{L_D},}\end{array}\hfill \end{array}$$

(4)

where the relaxation of $v$ is toward an equilibrium speed $V\left({\rho }^{*}\right)$ with ${\rho }^{*}$ as the average traffic density in the observation region $[x,x+L_D]$ . The observed average density is calculated by integration. Moreover, if $L_D$ goes to $0$ the model reduces to the original ARZ model. We can also generalize the average density with a weight function for biased observation:

• Remark 1. A more general weighted average density ${\rho }^{*}(L_D,w)$ with weight function $w\left(x\right)$ can be defined as

$${\rho }^{*}(L_D,w)=\frac{{\int }_x^{x+L_D}w\left(\xi \right)\rho (t,\xi )d\xi }{L_D},$$

(5)

where $w\left(x\right)$ satisfies ${\int }_x^{x+L_D}w\left(\xi \right)d\xi =1/L_D$ . With the weighted density, CAVs are considering vehicles in front with different sensitivities, similar to the microscopic multi-following model in Lenz et al. ( 36 ).

With the look-ahead (weighted) average density we are primarily focusing on the CACC logic in CAVs. There are many other complex dynamics and controls which can be implemented into the model Equation 4.

Additionally for periodic boundary conditions (i.e., traffic on a ring road), partial observation (look-ahead) is equivalent to full observation (look-ahead of the entire road) when $L_D=L$ , the length of the ring road.

For readers who are interested in the theoretical analysis such as solution existence, this model can be implicitly written as the non-local traffic model in Hamori and Tan ( 13 ) that has been shown to be well defined under certain constraints.

Stability Analysis of the Extended ARZ Model

In this section we will follow the classic wave-perturbation analysis approach ( 37 – 39 ) to analyze the stability of the extended ARZ model Equation 4.

For a given initial state, the steady state solution of the ARZ model is $(\rho ,v)=({\rho }_0,V\left({\rho }_0\right))$ for some $0<{\rho }_0<{\rho }_j$ where ${\rho }_j$ is the jam density. Now assume that the initial condition is perturbed by a small periodic disturbance:

$$\rho ={\rho }_0+\tilde{\rho };v=V\left({\rho }_0\right)+\tilde{\mathrm{V}},$$

(6)

where

$$\tilde{\rho }=R{e}^{\mathrm{ikx}+\sigma t};\tilde{v}=V{e}^{\mathrm{ikx}+\sigma t},$$

(7)

with $R,V$ has infinitesimal constant scales, and $k,\sigma$ are constants for the perturbation’s frequency and amplitude, respectively.

By neglecting second- or higher-order terms of $R$ and $V$ we can derive a linear system

$$\left[\begin{array}{c}\sigma +\mathrm{ik}\psi \mathrm{ik}{\rho }_0\hfill \\ \multicolumn{1}{c}{\sigma \varphi +\mathrm{ik}\psi \varphi -\frac{\zeta }{\tau }\sigma +\mathrm{ik}\psi +\frac{1}{\tau }}\end{array}\right]\left[\begin{array}{c}R\hfill \\ \multicolumn{1}{c}{V}\end{array}\right]=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$$

(8)

where $\psi =V\left({\rho }_0\right)>0$ , $\varphi =h^{\text{'}}\left(\rho \right)>0$ , $\zeta =(e^{\mathrm{ik}{L}_D}-1)V^{\text{'}}\left({\rho }_0\right)/\left(\mathrm{ik}{L}_D\right)$ with $\mathrm{Re}\left(\zeta \right)<0$ .

Following the calculation process in Ramadan et al. ( 39 ), we can deduce that traffic is stable when

$$h^{\text{'}}\left(\rho \right)+\frac{\left|\sin \left(k{L}_D\right)\right|}{k{L}_D}{V}^{\text{'}}\left(\rho \right)>0.$$

(9)

Since $\left|\sin x\right|/x$ is a decreasing function of $x$ for small $x$ , this equation implies that with a certain level of oscillation frequency, the stability criterion does not depend on $\tau$ and the range of stability can be increased with $L_D$ . Detailed calculation is shown at the end of this article.

• Remark 2. In particular, if $\rho (x+L)=\rho \left(x\right)$ for all $x\in \mathbb{R}$ , then for $L_D=L$ we have full observation of the road and the model will always be stable. As a consequence, we can derive ${\rho }^{*}={\rho }_0$ , which implies that all vehicles are relaxing toward equilibrium speed.

A Multi-Class Extension of the ARZ Model with Look-Ahead Effect

In this section, we propose a model for mixed autonomy traffic where HDVs and CAVs are modeled as distinctive classes. Similar to Huang et al. ( 29 ), in our model, the HDVs are reacting to the total density of traffic at its position. If we let ${\rho }^h$ denote density of HDVs and ${\rho }^c$ denote the density of CAVs, then the model has the form

$${\rho }_t^h+{\left({\rho }^h{v}^h\right)}_x=0,$$

(10a)

$${(v^h+h\left({\rho }^s\right))}_t+v^h{(v^h+h\left({\rho }^s\right))}_x=\frac{v^h-V\left({\rho }^s\right)}{\tau },$$

(10b)

$${\rho }_t^c+{\left({\rho }^c{v}^c\right)}_x=0,$$

(10c)

$${(v^c+h\left({\rho }^s\right))}_t+v^c{(v^c+h\left({\rho }^s\right))}_x=\frac{v^c-V\left({\rho }^{*}\right)}{\tau },$$

(10d)

$${\rho }^s={\rho }^h+{\rho }^c,$$

(10e)

where $v^h$ is the speed of HDVs and $v^c$ is the speed of CAVs. To effectively capture the look-ahead effect and vehicle interactions, we assume that CAVs share the same pressure function and relaxation constant as HDVs. Furthermore, to prevent excessive overtaking behavior, we assume that both CAVs and HDVs follow the same Fundamental diagram (FD). For such a mixed flow the stability can depend on the ratio and distribution of vehicles, and the control method of CAVs, which means that it is hard to obtain the stability condition analytically for the traffic flow model given in Equation 10. In this paper, we resort to numerical solutions of Equation 10 to explore the stability properties of this multi-class non-equilibrium model, which will be presented below.

• Remark 3. Practically, in mixed autonomy, CAVs might be capable of observing both the density and speed of surrounding HDVs to change their speed accordingly, which means the pressure term and relaxation term can be defined with consideration of the density of HDVs. We will consider such extensions in future work.

Numerical Solutions

To obtain numerical solutions for Equations 4 and 10, we adapted a forward scheme with an approximate Riemann solver in Ramadan et al. ( 39 ) that has low computation cost and preserves properties of finite volume methods. To calculate the average density, we use a Riemann sum to give an estimation of the integration term. Given $\Delta t$ , $\Delta x$ as time and space step size, $q=\rho (v+h\left(\rho \right))$ as a conserved flux variable, $i,n$ as space step variable and time step variable, and suppose that $L_D/\Delta t$ is a non-negative integer, then the update rule for approximate solutions of Equation 4 can be written as

$${\rho }_i^{n+1}={\rho }_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}({\left(F_{\rho }\right)}_{i+\frac{1}{2}}^n-{\left(F_{\rho }\right)}_{i-\frac{1}{2}}^n),$$

(11a)

$${\rho }_i^{*}=\frac{\sum _{j=1}^{\frac{L_D}{\Delta t}}{\rho }_j^{n+1}}{\frac{L_D}{\mathrm{\Delta }t}},$$

(11b)

$$\begin{array}{c}\multicolumn{1}{c}{q_i^{n+1}=q_i^n-\frac{\mathrm{\Delta }t}{\mathrm{\Delta }x}({\left(F_q\right)}_{i+\frac{1}{2}}^n-{\left(F_q\right)}_{i-\frac{1}{2}}^n)}\\ \multicolumn{1}{c}{-\frac{\mathrm{\Delta }t}{\tau }(V\left({\rho }_i^{*}\right)+h\left({\rho }_i^{n+1}\right)),}\end{array}$$

(11c)

where the update of the approximated flow $q_i^{n+1}$ is calculated after the update of the approximated density ${\rho }_i^{n+1}$ , and the update of the relaxation term adopts an implicit scheme to improve numerical stability. The average density ${\rho }_i^{*}$ is calculated by Riemann sum; the numerical fluxes ${\left(F_{\rho }\right)}_{i+\frac{1}{2}}^n,{\left(F_q\right)}_{i+\frac{1}{2}}^n$ are calculated by the Harten, Lax, and van Leer (HLL) approximate Riemann solver ( 40 ). The update rule for approximate solutions of Equation 10 can be similarly written by separately updating solutions of HDVs and CAVs using Equation 11.

For the model parameter values, we assumed that vehicles are on a ring road with length $L=1,000$  m, and set $\Delta t=0.05$ s, $\Delta x=5$ m, $\tau =3$ s, $h\left(\rho \right)=8*{((\rho -10)/(140-\rho ))}^{1/2}$ m/s, similar to Huang et al ( 29 ). The equilibrium speed model is a smooth function that combines the Greenshields model ( 41 ) and the Triangular FD model ( 42 ):

$$V\left(\rho \right)=\{\begin{array}{c}\begin{array}{c}{v}_f,\mathrm{if}\rho \le {\rho }_f;\hfill \\ \multicolumn{1}{c}{v_f(1-\frac{\rho -{\rho }_f}{{\rho }_j-{\rho }_f}),\mathrm{if}{\rho }_f\le \rho \le {\rho }_j;}\\ \multicolumn{1}{c}{0,\mathrm{if}\rho \ge {\rho }_j,}\end{array}\hfill \end{array}$$

(12)

where

${\rho }_f=10$ veh/km is the free-flow density,

${\rho }_j=140$ veh/km is the jam density, and

$v_f=20$ m/s is the free-flow speed.

The initial density is also a sinusoidal wave perturbation of equilibrium state similar to Huang et al. ( 29 ):

$${\rho }_0\left(x\right)=0.4*{\rho }_j+0.1*{\rho }_j*\sin (2\pi x/L),$$

(13)

where for mixed flow we substitute $\rho$ by ${\rho }^s$ . The initial velocity is then set as $v_0\left(x\right)=V\left({\rho }_0\left(x\right)\right)$ . In the following subsections, we will use two cases to evaluate the asymptotic stability of both models under different $L_D$ and vehicle mixes.

Investigation of the Look-Ahead Effect

In this scenario we evaluate the look-ahead distance $L_D$ on the convergence of the extended ARZ model. We will consider $L_D=15,100,1,000$ m and compare the model results with those from the ARZ model ( $L_D\to 0^{+}$ ). For $L_D=0^{+},100$ m we set the time duration as $T=600$ s and the others as $T=1,200$ s. The numerical results of the density and velocity evolution are shown in Figures 2 to 6.

Figure 2.

Density and velocity evolution of the ARZ model ( $L_D\to 0^{+}$ ), where the flow is not stable: (a) density evolution; and (b) velocity evolution.

Figure 3.

Density and velocity evolution of the modified model with $L_D=15$ m: (a) density evolution; and (b) velocity evolution.

Figure 4.

Density and velocity evolution of the modified model with $L_D=100$ m: (a) density evolution; and (b) velocity evolution.

Figure 5.

Density and velocity evolution of the modified model with $L_D=1,000$  m: (a) density evolution; and (b) velocity evolution.

Figure 6.

$L_{\infty }$ norm for density evolutions of different values of $L_D$ .

From the numerical results, we can observe that in all cases look-ahead helps to stabilize traffic, as the only unstable case is when $L_D\to 0^{+}$ . However, a longer look-ahead distance is not equivalent to faster convergence to an equilibrium state. With full observation, that is, $L_D=1,000$ m, or a shorter partial observation ( $L_D=15$ m) the convergence speed is much slower than with $L_D=100$ m. In the case of $L_D=15$ m, the look-ahead effect is not significant since this is no better than follow one-vehicle ahead. With the much longer $L_D$ , the redundant information from far away is also built into drivers’ responses, and therefore can be detrimental rather than beneficial to traffic stability when traffic conditions vary significantly over space. There seems to be a theoretical optimal look-ahead distance for achieving greater convergence and traffic stability, which may depend on parameter settings and even initial and boundary traffic conditions. We will explore this problem in our future work.

Investigation of Stability in Mixed Autonomy Traffic

In this scenario we investigate the potential for using CAVs to smooth and stabilize mixed traffic flow, considering two different spatial distributions of CAVs in the traffic mix.

Even Distribution

We first consider CAVs evenly distributed in the mixed traffic with penetration rates of 10%, 20%, and 40%. Based on the results of the single-class model, we choose the observation distance $L_D=100$ m for CAVs. For 10% and 20% penetration rates we set the time duration to be $T=1,200$ s and for 40% we set $T=600$ s. The numerical results of density and velocity evolution are shown in Figures 7 to 10.

• Remark 4. For the mixed plot, we plot the evolution of the total density and the HDVs’ velocity, since traffic flow of pure CAVs are already shown stable.

Figure 7.

Density and velocity evolution of the mixed flow model with 10% of CAVs evenly distributed: (a) density evolution; and (b) velocity evolution.

Figure 8.

Density and velocity evolution of the mixed flow model with 20% of CAVs evenly distributed: (a) density evolution; and (b) velocity evolution.

Figure 9.

Density and velocity evolution of the mixed flow model with 40% of CAVs evenly distributed: (a) density evolution; and (b) velocity evolution.

Figure 10.

$L_{\infty }$ norm for density evolutions of different rates of evenly distributed CAVs.

From these results, we can observe that, with 20% of the traffic being CAVs, the mixed flow stabilizes to smaller oscillations; at 40%, there is faster convergence to an equilibrium state; while at 10% the traffic fails to stabilize. Such results are consistent with those from a similar study ( 29 ).

Segregated Distribution

We now consider another type of distribution, that is, one in which CAVs and HDVs are segregated into two parts. With the same penetration rates, we let CAVs concentrate at around $x=500$ m and HDVs concentrate at the remaining locations. In detail, the initial density of CAVs is given as

$${\rho }^c\left(x\right)=\{\begin{array}{c}0.999{\rho }^s,\mathrm{if}\frac{1-r}{2}L<x<\frac{1+r}{2}L,\hfill \\ \multicolumn{1}{c}{0.001{\rho }^s,\mathrm{otherwise}.}\end{array},$$

(14)

where $r$ is the percentage of CAVs, and ${\rho }^h$ can be calculated by ${\rho }^h={\rho }^s-{\rho }^c$ . The small density is designed for numerical stability. We set the same simulation time as in the evenly distributed case. The numerical results of density and velocity evolution are shown in Figures 11 to 14.

Figure 11.

Density and velocity evolution of the mixed flow model with 10% of concentrated CAVs: (a) density evolution; and (b) velocity evolution.

Figure 12.

Density and velocity evolution of the mixed flow model with 20% of concentrated CAVs: (a) density evolution; and (b) velocity evolution.

Figure 13.

Density and velocity evolution of the mixed flow model with 40% of concentrated CAVs: (a) density evolution; and (b) velocity evolution.

Figure 14.

$L_{\infty }$ norm for density evolutions of different rate of concentrated CAVs.

These results show that the segregated distributions of mixed flow have similar asymptotic behaviors as even distributions. The main difference is that the initial waves have larger scales for segregated distributions where the HDVs are concentrated, since HDV traffic is less stable than CAV traffic. Overall, the convergence of mixed traffic is slower than the results obtained in Huang et al. ( 29 ), possibly because of large oscillations and inadequate information from HDVs. One possible improvement is to add predictive or feedback controls as previously investigated in car-following models for example, Zhou et al. ( 43 ), Jin and Meng ( 44 ).

Concluding Remarks

This paper makes extensions to a second-order non-equilibrium traffic flow model, that is, the ARZ model, to take into account the look-ahead capabilities of CAVs, in both single- and multi-class contexts. The look-ahead effect is captured by a modification of the relaxation term, which can be interpreted as a CAV’s attempt to adopt a target speed based on the average traffic conditions within its spatial observation range, similar to multi-following microscopic traffic models. The stability properties of both extended models are analyzed through wave-perturbation analysis, and the results show that a longer observation range yields a less restrictive stability condition. A numerical solution using forward schemes with approximate Riemann solvers is provided, and numerical experiments are carried out to examine the effects of various parameters and the spatial distribution of CAVS on both the stability of mixed autonomy traffic, and CAVs’ ability to stabilize mixed traffic flow. It is found that higher penetration rates of CAVs stabilize mixed traffic flow faster, which is consistent with similar studies in Huang et al. ( 29 ).

Our study reveals several new insights into mixed autonomy traffic. One interesting finding is that having more information on traffic conditions does not necessarily translate into better traffic control. In our particular setting, a moderate look-ahead distance of 100m .enables faster convergence to equilibrium than having the full observation of road conditions on the entire ring road. Another interesting finding is that the distribution of vehicles has little effect on the long-term stability of mixed traffic flow, but the initial oscillations for segregated distribution have larger amplitudes than in the even distribution case. However, these findings rely on the assumption that CAVs follow the same car-following dynamics as HDVs, with the uniformed look-ahead effect being the only advanced control mechanism considered. A promising direction for future research is to incorporate more realistic interaction models and explore the integration of advanced CAV control strategies, including the weighted look-ahead effect, feedback control, model predictive control (MPC), and reinforcement learning (RL), with data validations if possible. These insights can help CAV manufacturers design more effective control algorithms that could be of benefit to both parties in mixed autonomy traffic. In addition, traffic engineers will be better able to manage mixed autonomy flow by leveraging the sensing and control capabilities of CAVs.

Detailed Calculation of Stability Criteria

Here we outline the detailed derivation of Equation 9. The linearized system Equation 8 allows a nontrivial solution only if its determinant is zero, which is equivalent to

$$\sigma =-\mathrm{ik}\psi +\mathrm{ik}\frac{1}{2}{\rho }_0\varphi -\frac{1}{2\tau }(1+\mathrm{\Gamma }),$$

(15)

where $\mathrm{\Gamma }$ satisfies ${\mathrm{\Gamma }}^2=1-k^2{\tau }^2{\rho }_0^2{\varphi }^2-2\mathrm{ik}\tau {\rho }_0(\varphi +2\zeta )$ . The system is stable if $|\mathrm{Re}\left(\sigma \right)|<0$ , which is equivalent to $\left|\mathrm{Re}\left(\mathrm{\Gamma }\right)\right|<1$ . From the equation of $\mathrm{\Gamma }$ we can derive the expression of ${\left|\mathrm{Re}\left(\mathrm{\Gamma }\right)\right|}^2$ as

$${\left|\mathrm{Re}\left(\mathrm{\Gamma }\right)\right|}^2=\frac{1}{2}((1-{\beta }^2{k}^2+\omega )+\sqrt{{(1+{\beta }^2{k}^2-\omega )}^2+4({\gamma }^2-{\beta }^2)k^2+4\omega }),$$

(16)

where $\beta =\tau {\rho }_0\varphi$ , $\gamma =\tau {\rho }_0(\varphi +2\zeta )$ , and $\omega =\frac{1-\cos \left(k{L}_D\right)}{k{L}_D}{V}^{\text{'}}\left({\rho }_0\right)\beta$ is negligible for sufficiently small or large $k$ . Therefore, stability holds if $\left|\beta \right|>\left|\gamma \right|$ , or $\varphi +\left|\zeta \right|>0$ , which is equivalent to Equation 9.

• Remark 5. For intermediate $k$ , by solving $|4({\gamma }^2-{\beta }^2)k^2+4\omega |<0$ , the stability criterion becomes

$$h^{\text{'}}\left({\rho }_0\right)+\frac{\sin \left(k{L}_D\right)}{k{L}_D}{V}^{\text{'}}\left({\rho }_0\right)>\frac{\sin \left(k{L}_D\right)(\left(k{L}_D\right)-1)}{16\beta },$$

(17)

which implies Equation 9 is a sufficient stability condition when $k$ is intermediate.