Stability analysis and the fundamental diagram for mixed connected automated and human-driven vehicles

Physica A 533 (2019) 121931

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Stability analysis and the fundamental diagram for mixed connected automated and human-driven vehicles Zhihong Yao a,b,c,e , Rong Hu a,b , Yi Wang a,b , Yangsheng Jiang a,b , Bin Ran c , ∗ Yanru Chen d , a

School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Southwest Jiaotong University, Chengdu, Sichuan 610031, China c TOPS Laboratory, Department of Civil and Environmental Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI 53706, USA d School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, Sichuan 610031, China e Chongqing Key Laboratory of Traffic and Transportation, Chongqing Jiaotong University, Chongqing 400074, China b

highlights • The different car-following models are applied to capture the driver behavior of CAVs and HDVs. • The stability of mixed traffic flow under different penetration rates of CAVs was analyzed. • The fundamental diagram under different penetration rates was derived and the sensibility of parameters on the fundamental diagram was analyzed.

• The simulation experiment was designed to verify the effectiveness of the proposed model.

article

info

Article history: Received 22 March 2019 Received in revised form 10 June 2019 Available online 11 July 2019 Keywords: Mixed traffic flow Fundamental diagram Car-following Connected automated vehicles Stability Penetration rates

a b s t r a c t With the development of connected automated vehicles (CAVs), traffic flow on the road is generally mixed with human-driven vehicles (HDVs) and CAVs. In this paper, the full velocity difference (FVD) model and cooperative adaptive cruise control (CACC) model validated by PATH laboratory of University of California, Berkeley, are used to describe the car-following (CF) driving behavior of HDVs and CAVs, respectively. Firstly, an analytical method for the stability of mixed traffic flow is developed and the stability conditions under different penetration rates of CAVs are obtained. Then, the fundamental diagram model of mixed traffic flow under different penetration rates is derived and the influence factors of the fundamental diagram are analyzed. Finally, a simulation experiment is designed to verify the effectiveness of the proposed model. The simulation results show that there are some differences between the simulation data and the theoretical results under different penetration rates. The simulation data are fluctuating, but they are both on the theoretical curve. The overall trend is consistent with the theoretical results, which proves the validity of the fundamental diagram analysis results. © 2019 Elsevier B.V. All rights reserved.

∗ Corresponding author. E-mail address: [email protected] (Y. Chen). https://doi.org/10.1016/j.physa.2019.121931 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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Z. Yao, R. Hu, Y. Wang et al. / Physica A 533 (2019) 121931

1. Introduction With the development of connected automated vehicles technology, the information can exchange between vehicles (V2V) and between vehicles and infrastructure (V2I) through wireless communication by installing onboard units (OBUs) and roadside units (RSUs). Based on this technology, drivers and vehicles can make safe and reliable decisions about acceleration choice and executing lane-changing maneuvers. Therefore, CAVs technology can greatly improve traffic safety and efficiency [1–3]. However, the penetration of Level 4 CAVs in the urban road is only 24.8% in 2045 from the latest research [4]. This indicates that traffic flow will present the characteristics of mixed with CAVs and HDVs. Therefore, it will be of great theoretical significance and practical value to study the characteristics of mixed traffic flow with CAVs and HDVs. Recently, there has been considerable work done for CAVs, to improve traffic efficiency and safety, including driving behavior [5,6], control optimization [7–9], capacity [10,11], traffic flow stability [12–21] and so on. There is a lot of existing literature [12–21] which studied the stability and fundamental diagram on a mixed traffic flow consisting of HDVs and CAVs. To the best of our knowledge, these studies can be divided into two categories. Firstly, most current works [12– 17,22,23] used the same car-following model to describe the driving behavior of CAVs and HDVs. In Ref. [12–14,22,23], the intelligent driver model (IDM) [24] with different parameters was used to capture CAVs and HDVs. Ngoduy [12] proposed a linear stability analysis to find the stability threshold of heterogeneous traffic flow. The analytical results showed how CAVs percentages affect the stability of multi-class traffic flow. Talebpour and Mahmassani [13] presented a framework to study the influence of CAVs on traffic flow stability and throughput. The analysis showed that CAVs can improve string stability and throughput. Sun et al. [22] studied major methods for analyzing local and string stability for three types of car-following models in mixed with CAVs and HDVs. The simulation results showed that the time delay and connectivity both significantly affect the string stability. Xie et al. [23] developed a generic car-following framework to derive a linear stability condition for the heterogeneous traffic. The simulation results showed that connected vehicles (CVs) can obviously enhance the stability of traffic flow and improve traffic efficiency in particular when traffic is in congestion. Monteil et al. [14] applied L2 linear control theory to optimize string stability of the mixed traffic flow. The results showed that the optimization strategy systematically leads to increased traffic flow stability. These studies all based on IDM [24], in addition to IDM, the optimal velocity model (OVM) [25] and two-state safe-speed model (TSM) [26] are also used in some researches [15–17]. Zhu and Zhang [15] studied the fundamental diagrams and density waves of the mixed traffic flow by using the OVM car-following model. The numerical results indicated that the density–volume curve had a critical point. Before the critical point, the bigger proportion of autonomous vehicles (AVs) may lead to a greater volume for mixed traffic flow. Monteil et al. [16] considered the impact of CAVs on the stability of traffic flow, derived traffic-statedependent conditions for the sign of the solitary wave amplitude. The results provided a new insight supporting the future implementation of CAVs systems. Ye et al. [17] presented a proposed methodology for modeling CAVs in heterogeneous traffic flow and investigated the impact of setting dedicated lanes for CAVs on traffic flow throughput. Secondly, some studies [18–21] used different models to capture the driving behavior of CAVs and HDVs. Yuan et al. [18] investigated traffic flow characteristics in a mixture traffic system consisting of adaptive cruise control (ACC) vehicles and HDVs based on a hybrid modeling approach. The hybrid modeling approach used the modified comfortable driving (MCD) model and the constant time headway (CTH) model to capture the driving behavior of ACC and HDVs. The results showed that the hybrid modeling approach outperformed both the cellular automaton (CA) model and the car-following model, and was able to capture the three phases (i.e., free flow, jammed flow, and synchronized traffic) in traffic flow. Qin et al. [19,20] presented a stability analysis method for mixed CAV flow based on the OVM and electronic throttle angle (ETA) feedback [27]. The stability chart indicated that the proposed method can reduce traffic emissions and fuel consumption. Ye et al. [21] proposed a heterogeneous traffic flow model to study the possible impact of CAVs on the traffic flow. The simulation results showed that the road capacity increases with an increase in the CAVs penetration rate within the heterogeneous flow. Table 1 summarizes the recent studies [12–21] on stability analysis and a fundamental diagram model for mixed CAVs and HDVs, with three major limitations. First, most current works [12–17,22,23] used the same car-following model to describe the driving behavior of CAVs and HDVs. Considering CAVs and HDVs have different driving characteristics, it is very important to choose different models to capture the driving behavior of CAVs and HDVs. Second, some studies [18–21] adopted different models, but they did not simultaneously analyze the stability and fundamental diagram. Finally, in many studies [12,14–23], sensitivity analysis only considers a few factors, such as penetration rates and headway. Therefore, to overcome these limitations, the full velocity difference (FVD) car-following model [28,29] and the PATH laboratorycalibrated CACC model [30,31] are selected to capture the car-following behavior of HDVs and CAVs, respectively. Then, the stability conditions and the fundamental diagram analytical model are proposed. Finally, the influence factors of the capacity of mixed traffic flow are discussed, and simulation experiments are designed to verify the validity and rationality of the proposed model. Therefore, the key contributions of this work are summarized as follows: (1) To consider the difference between the driving behavior of HDVs and CAVs, two car-following models (FDV and CACC) are used to describe HDVs and CAVs, respectively. (2) To analysis the linear stability of the mixed traffic flow and some factors (e.g., speed, penetration rate, and desired time headway) which affects the stability.

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Table 1 Summary of modeling method, stability, fundamental diagram, and sensitivity analysis for mixed CAVs and HDVs environment. Publication

Modeling method

Stability

Method

HDVs

CAVs

Yuan et al. [18]

CA/CF

MCD

CTH

Ngoduy [12] Monteil et al. [16] Talebpour et al. [13]

CF CF CF

IDM OVM/ IDM IDM

IDM OVM/ IDM IDM

✓ ✓ ✓

Sun et al. [22] Qin et al. [19] Xie et al. [23] Ye et al. [21]

CF CF CF CA/CF

IDM OVM IDM TSM

IDM ETA IDM ACC

✓ ✓ ✓

Zhu et al. [15]

CF

OVM

OVM

✓

Qin et al. [20] Ye et al. [17]

CF CA

IDM TSM

ACC/CACC TSM

✓

Monteil et al. [14] This paper

CF CF

IDM FVD

IDM CACC

✓ ✓

Fundamental diagram

Sensitivity analysis

✓

• • • • • • • • • • • • • • • • •

Penetration rate Desired time headway Penetration rate Penetration rate Penetration rate Desired time headway Minimum safe gap distance Number of vehicles Penetration rate Penetration rate Penetration rate Desired time headway Penetration rate Headway Penetration rate Penetration rate Desired time headway

• • • •

Penetration rate Free speed Desired time headway Minimum safe gap distance

✓ ✓ ✓

✓

✓

(3) To propose a new analytical fundamental diagram model of the mixed traffic flow and analysis the sensitivity parameters, such as penetration rate, free speed, minimum safe gap distance, and desired time headway. (4) To design a simulation experiment to verify the validity and rationality of the proposed model. The remainder of this paper is structured as follows. Section 2 presents car-following models for CAVs and HDVs. The stability analysis for mixed CAVs and HDVs is studied in Section 3. Section 4 proposes the analytical model and sensitivity analysis of the fundamental diagram for mixed CAVs and HDVs. Simulation experiment and discussions are presented in Section 5. Finally, conclusions and recommendations are delivered in Section 6. 2. Car-following models As a kind of microscopic traffic behavior, the car-following model is also a bridge to connect micro and macro traffic flow. The existing car-following models mainly focus on human-driven vehicles (HDVs) and connected automated vehicles (CAVs). Therefore, there are many models for HDVs [32,33], which can be divided into four categories: stimulationresponse, safety distance, psychophysiology, and artificial intelligence. The car-following model of CAVs is mainly an intelligent driver model (IDM) [24] and a CACC model calibrated by the PATH laboratory of the University of California at Berkeley [30,31]. This study selects the full velocity difference (FVD) car-following model in the stimulus–response model [28,29] and the PATH laboratory-calibrated CACC model [30,31] to describe the driving behavior of HDVs and CAVs. 2.1. Human-driven vehicles The FVD car-following model [28] can better describe the car-following behavior of HDVs than OVM [25,34]. Therefore, the FVD car-following model is selected to describe the following behavior of HDVs. In addition, to reflect the influence of headway on the speed difference, Wang et al. [35] improved the FVD model [28] which had a better calibration effect than OVM [25,34], as shown in Eq. (1).

v˙ =

dv dt

= κ [V (∆x) − v ] +

λ ∆v ∆x − L

(1)

where κ and λ are the sensitivity coefficients, ∆x is the space headway between the current vehicle and the preceding vehicle, v is the speed of the current vehicle, v˙ is the derivative form of the speed v , ∆v is the speed difference between the current vehicle and the preceding vehicle, L is the vehicle length, and V (∆x) is the optimized speed function as follows: V (∆x) = vf

)] α 1 − exp − (∆x − L − s0 ) vf

[

(

where vf is the free flow velocity, α is the sensitive parameter, and s0 is the minimum safe distance.

(2)

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Z. Yao, R. Hu, Y. Wang et al. / Physica A 533 (2019) 121931 Table 2 The value of desired time headway. Desired time headway tc (s)

Driver acceptance ratio

0.6 0.7 0.9 1.1

57% 24% 7% 12%

Substituting Eq. (2) into Eq. (1), the desired acceleration calculation equation of the improved FVD can be obtained, as shown in Eq. (3).

)] } { [ ( λ α ∆v v˙ = κ vf 1 − exp − (∆x − L − s0 ) − v + vf ∆x − L

(3)

The value of the parameters in Eq. (3) are taken from Wang et al. [35], and are vf = 33.0 m· s−1 , s0 = 2.46 m,

κ = 0.629 s−1 , λ = 4.10 s−1 , α = 1.26 s−1 , and L = 5.0 m. 2.2. Connected automated vehicles

The IDM [24] and the CACC car-following model calibrated by the PATH laboratory [30,31] were used to describe the car-following behavior of CAVs. Compared with IDM, the CACC model is based on real vehicle trajectory data and more capable of capturing the following characteristics of CAVs. Therefore, the CACC car-following model is used to describe the car-following behavior of CAVs, as shown in Eq. (4).

{

v = vp + kp e + kd e˙ e = ∆x − s0 − L − tc v

(4)

where vp is the speed of the previous control time, e is the error between the actual vehicle spacing and the desired vehicle spacing, e˙ is the derivative form of e, s0 is the minimum safe distance, tc is the desired headway, kp and kd are the control parameters. After deriving the velocity in Eq. (4), the desired acceleration calculation equation of the CACC car-following model can be obtained, as shown in Eq. (5).

v˙ =

kp (∆x − L − s0 ) − kp tc v + kd ∆v kd tc + ∆t

(5)

The CACC car-following model was calibrated by the PATH laboratory of the University of California at Berkeley [31] and the optimal parameters of the CACC model are ∆t = 0.01 s, kp = 0.45 s−1 , and kd = 0.25. In addition, according to Ref. [36], the acceptance ratios of drivers with different expected time headway are different, as shown in Table 2. 3. Stability analysis 3.1. Definition of stability There is a multitude of studies on the stability of the car-following models [13,20,37–40]. According to the Ref. [13], the control equations of all car-following models can be expressed by the following equation: x˙ = v

(6)

v˙ = f (v, ∆v, ∆x)

(7)

According to the linear stability of the car-following model [13,20], Ref. [13] unified the control equation of the car-following model into the acceleration control equation, and gave the unstable condition of the control equation. 1

(8) (fv )2 − f∆v fv − f∆x < 0 2 If Eq. (8) is satisfied, that is F < 0, it means that the traffic flow is unstable. Among them, fv , f∆v , f∆x are the partial differential of the control equation for speed, speed difference, and headway distance, which can be calculated by Eq. (9). F =

⏐ ⎧ ∂ f (v, ∆v, ∆x) ⏐⏐ ⎪ ⎪ f = v ⎪ ⏐ ⎪ ∂v ⎪ (⏐V (∆x∗ ),0,∆x∗ ) ⎪ ⎨ ∂ f (v, ∆v, ∆x) ⏐⏐ f∆v = ⏐ ∂ ∆v ⎪ ⎪ ⏐ (V (∆x∗ ),0,∆x∗ ) ⎪ ⎪ ∂ f (v, ∆v, ∆x) ⏐⏐ ⎪ ⎪ ⎩f∆x = ⏐ ∆x (V (∆x∗ ),0,∆x∗ )

(9)

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Fig. 1. Linear stability curve for HDVs traffic flow.

where ∆x∗ and V (∆x∗ ) are the space headway and its corresponding speed in the stable traffic flow [41,42]. 3.2. Stability analysis for homogeneous traffic flow 3.2.1. Human-driven traffic flow The Eq. (3) of HDVs is used to differentiate the speed, speed difference, and space headway. Then, the fvH , f∆Hv , and f∆Hx of HDVs can be obtained, as shown in the Eq. (10).

⎧ H f = −κ ⎪ ⎪ ⎨v λα ( ) f∆Hv = α s0( − vf ln 1) − v/vf ⎪ ⎪ ⎩ H f∆x = κα 1 − v/vf

(10)

According to the condition of the linear instability, the linear instability equation for HDVs traffic flow can be calculated, as shown in Eq. (11). FH =

1( 2

fvH

)2

− f∆Hv fvH − f∆Hx =

1 2

(−κ)2 +

( ) λακ ( ) − κα 1 − v/vf α s0 − vf ln 1 − v/vf

(11)

The Eq. (11) indicates that the instability discriminant value FH is determined by the speed. Taking the velocity v as the independent variable and the stability discriminant value FH as the ordinate, the curve of the linear stability discriminant value of HDVs traffic flow can be obtained, as shown in Fig. 1. Fig. 1 shows that the traffic flow has two stable and one unstable speed interval. When the speed is in [0, 2.97] and [21.08, 33], the linear stability discriminant value FH is greater than or equal to 0. This indicates that the traffic flow is in a stable state. On the contrary, when the speed is in (2.97, 21.08), the discriminant value FH is less than 0, which means the traffic flow is unstable in this speed internal. 3.2.2. Connected automated traffic flow In the same way, the Eq. (5) of CAVs can be differentiated in terms of speed, speed difference, and headway distance. Then, the fvC , f∆Cv and f∆Cx of CAVs traffic flow can be obtained, such as Eq. (12).

⎧ kp tc ⎪ C ⎪ ⎪f v = − ⎪ k t + ∆t ⎪ d c ⎪ ⎨ kd C f∆v = ⎪ kd tc + ∆t ⎪ ⎪ ⎪ kp ⎪ C ⎪ ⎩f∆x = kd tc + ∆t

(12)

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According to the linear instability condition of the platoon, the equation for determining the linear stability of CAVs traffic flow can be calculated, as shown in Eq. (13). FC =

1( 2

fvC

(

)2

− f∆Cv fvC − f∆Cx =

kp tc kp tc + 2kd

) −

2 (kd tc + ∆t )2

kp kd tc + ∆t

=

0.45 0.25tc + 0.01

>0

(13)

From Eq. (13), the linear stability discriminant value FC is directly proportional to kp and also inversely proportional to kd , tc and ∆t. In addition, the value is always greater than 0 because tc > 0. Therefore, the CAVs traffic flow is stable under different speed. 3.3. Stability analysis for heterogeneous traffic flow In accordance with the mixed traffic flow instability judgment condition given in Ref. [43,44], which is shown in Eq. (14).

∑ [ 1 ( )2 n n

2

fv

]

⎞2

⎛

− f∆nv fvn − f∆nx ⎝

∏

f∆mx ⎠ < 0

(14)

m̸ =n

where n is the index number of a vehicle, f n is the control equation for the nth vehicle, fvn , f∆nv , and f∆nx are the partial differentials of the control equation f n of the nth vehicle to the speed, speed difference, and space headway, respectively. According to Ref. [44], the mixed traffic flow instability determination equation is consistent with the platoon linear stability discrimination, which is only related to the proportion of vehicles, but not to the relative position of vehicles. Therefore, assuming the mixed flow contains N vehicles and the penetration rate of CAVs is p. In this study, the HDVs and CAVs are denoted by H and C, respectively. Then, the linear stability equation of the mixed flow is shown in Eq. (15).

[

]

[( )N (1−p)−1 ( )Np ]2 1 ( H )2 N (1 − p) fv − f∆Hv fvH − f∆Hx × f∆Hx f∆Cx 2 [ + Np

]

[( )N (1−p) ( )Np−1 ]2 1 ( C )2 f∆Cx <0 fv − f∆Cv fvC − f∆Cx × f∆Hx 2

(15)

Further, the Eq. (15) can be expressed as Eq. (16).

(1 − p)

[ ( ) 1 H 2

] [ ( ) ] 2 − f∆Hv fvH − f∆Hx p 12 fvC − f∆Cv fvC − f∆Cx + <0 ( H )2 ( C )2

fv

2

f∆x

(16)

f∆x

In this study, SH and SC can be represented by Eq. (17).

⎧ ⎪ ⎪ ⎪ ⎪ ⎨SH = ⎪ ⎪ ⎪ ⎪ ⎩SC =

1 2

( H )2

1 2

( C )2

fv

fv

− f∆Hv fvH − f∆Hx FH = ( )2 ( H )2 H f∆x

C

f∆x

C

− f∆ v fv − ( C )2 f∆x

f∆Cx

= (

(17)

FC

f∆Cx

)2

Then, the Eq. (16) can be expressed as: S = (1 − p) SH + pSC < 0

(18)

By analyzing the Eqs.(15)–(18), the linear stability discriminant value S under the mixed traffic flow is related to the vehicle speed v , the penetration rate p, and the desired time headway tc of the CAVs. Therefore, the stability discrimination values under different speed, penetration rate, and desired time headway are obtained, as shown in Fig. 2. Fig. 2 shows that with the increase of the desired time headway (tc ), the penetration rate in stable state decreases gradually. In particular, when the desired time headway is 1.1 s, the penetration rate of CAVs in the stable mixed traffic flow is less than 47%. In addition, the steady speed interval gradually decreases with the increase in the penetration rate of CAVs. By further analyzing the unstable region of the mixed traffic flow, a mixed traffic flow stable region and a stable value heat map can be obtained, as shown in Fig. 3. From Figs. 2 and 3(a), the speed interval of the unstable region of the mixed traffic flow gradually decreases, as the penetration rate of CAVs increases. Specifically, when the desired time headway is 1.1 s and the penetration rate is greater than 46.7%, the mixed traffic flow will not be affected by the speed and remain in a stable state. Meanwhile, Fig. 3(b) indicates that the region surrounded by the black curve and the x-axis is an unstable region, that is, the mixed traffic flow is unstable under any corresponding velocity and penetration rate in this region; on the contrary, the mixed traffic flow outside this region is in a stable state. In addition, Fig. 3(b) shows that the size of the stable region decreases with the increase in the desired time headway (tc ). This indicates that the design of a larger desired time headway (tc ) of CAVs is conducive to the stability of mixed traffic flow.

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Fig. 2. Stability curve for the mixed traffic flow.

4. Fundamental diagram 4.1. Analytical model When the traffic flow is stable, the speed difference and speed differential value between the vehicles are both zero. Therefore, the space headway in this state is a stable space headway. If the speed difference ∆v and the speed differential v˙ in the Eqs.(3) and (5) are zero, the space headway in the steady state of the mixed traffic flow can be obtained, as shown in the Eq. (19).

{

h∗H = ∆x∗ = L + s0 −

vf α

(

ln 1 −

v vf

)

h∗C = ∆x∗ = L + s0 + v tc

(19)

where h∗H and h∗C indicate the space headway under the steady state of the HDVs and CAVs, respectively, and ∆x∗ is the stable space headway. If there are N vehicles in the mixed traffic flow, the penetration rate of CAVs is p, the average space headway in the stable mixed traffic flow is the linear average of the space headway of the HDVs and CAVs, as shown in Eq. (20). h∗ =

N (1 − p) h∗H + Nph∗C N

= (1 − p) h∗H + ph∗C

where h∗ is the average space headway in the mixed traffic flow steady state.

(20)

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Fig. 3. Stability v.s. velocity and penetration for mixed traffic flow.

In accordance with the relationship between traffic flow density and average space headway, the density of mixed traffic flow can be obtained: 1000 k= (21) h∗ where k is the density of the mixed traffic flow in a steady state, the unit is veh/km. Substituting Eqs.(19) and (20) into Eq. (21), the relationship between the density and the velocity can be calculated.

{[ k = 1000 ×

v tc +

( )] ( )}−1 vf v vf v ln 1 − p + L + s0 − ln 1 − α vf α vf

(22)

Similarly, according to the macroscopic traffic flow relationship of volume, density, and velocity, the relationship between volume and velocity can be obtained.

{[ ( )] ( )}−1 vf v vf v q = 3600 × v · v tc + ln 1 − p + L + s0 − ln 1 − α vf α vf

(23)

where q is the volume in a steady state of the mixed traffic flow. Therefore, according to Eqs.(22) and (23), the relationship between volume, density, and velocity of the mixed traffic flow under different penetration rate can be calculated, as shown in Fig. 4.

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Fig. 3. (continued).

Fig. 4 reveals that as the penetration rate of CAVs increases, the capacity of the mixed traffic flow gradually increases. This is consistent with reality, indicating that CAVs can effectively improve the capacity of existing traffic systems. In addition, Fig. 4(a) presents that when the penetration rate of CAVs is 100%, the minimum density is 36.69 veh/km, and the volume is the largest. Moreover, the capacity of the mixed traffic flow does not increase infinitely with the increase in the penetration rate of CAVs, which is related to the free flow speed, the minimum safety distance, and the desired time headway. The impact of these factors on the fundamental diagram will be specifically analyzed below. 4.2. Sensitivity analysis 4.2.1. Free speed The fundamental diagram of the mixed traffic flow under different free flow velocities is analyzed by fixing the minimum safety distance s0 = 2 m and the desired time headway tc = 0.6 s. Four free flow speeds of the mixed flow (HDVs and CAVs) vf = 60, 80, 100, and 120 km/h are tested in this section. The results are shown in Fig. 5. Fig. 5 shows that as the free flow velocity increases, the capacity of the mixed traffic flow gradually increases. When the free-flow speeds are 60, 80, 100, and 120 km/h, the minimum density of CAVs are 58.27, 48.8, 41.97, and 36.82 veh/km, respectively, and the volumes are 3392, 3816, 4122, and 4352 veh/h at p = 100%. Therefore, this indicates that CAVs can greatly improve traffic capacity with high velocity.

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Fig. 4. The relationship between volume, density, and velocity of the mixed traffic flow under different penetration. (t c = 0.6 s)

4.2.2. Minimum safe gap distance The fundamental diagram of mixed traffic flow under different minimum safety distance is analyzed by fixing free-flow velocity vf = 33 m/s and the desired time headway tc = 0.6 s. Four minimum safety distances of the mixed flow (HDVs and CAVs) s0 = 1.5, 2.0, 2.5, and 3.0 m are tested in this section. The results are shown in Fig. 6. Fig. 6 presents that the capacity of the mixed traffic flow gradually decreases with the increase in the minimum safety distance. When the minimum safety spacing are 1.5, 2.0, 2.5, and 3.0 m, the minimum density of CAVs are 38.91, 38.17, 37.45, and 36.76 veh/km, respectively, and the corresponding volumes are 4482, 4397, 4315, and 4235 veh/h at p = 100%. Therefore, contrary to the conclusion of the free flow velocity, the reduction of the minimum safe spacing is advantageous for the improvement of the mixed traffic flow capacity. 4.2.3. Desired time headway The fundamental diagram of mixed traffic flow under different desired time headways is analyzed by fixing the freeflow velocity vf = 33 m/s and the minimum safe spacing s0 = 2 m. The desired time headway of CAVs is set to tc = 0.6, 0.7, 0.9, and 1.1 s. The fundamental diagram of the mixed traffic flow under different desired time headways is shown in Fig. 7. From Fig. 7, the capacity of the mixed traffic flow is gradually reduced with the increase in the desired time headways. In addition, compared with the free-flow velocity and the safe headway distance, the influence of the desired time headway on the mixed traffic flow capacity is greater. When the safety time headway are 0.6, 0.7, 0.9, and 1.1 s, the minimum density of CAVs are 37.51, 33.49, 27.58, and 23.44 veh/km, respectively, and the corresponding volumes are

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Fig. 5. Influence of the free velocity on the fundamental diagram of the mixed traffic flow.

4321, 3858, 3177, and 2700 veh/h at p = 100%. Therefore, this indicates that the design of the smaller desired time headway is advantageous for the improvement of the mixed traffic flow capacity. Fig. 7 shows that when the intersection occurs (e.g., tc = 0.9, 1.1 s), the greater the penetration rate of CAVs but the smaller capacity of the mixed traffic flow. That is, reflected on the intersection point (the density is 70.55 veh/km when tc = 0.9 s) which is consistent with the critical points in the literature [15]. When the density is greater than the density of the intersection point, as the density increases, the capacity of the mixed traffic flow with a high penetration rate of CAVs reduces faster. This is because a pure CAVs has a lower capacity than a pure HDVs under such circumstance. Moreover, Fig. 7 shows that as the desired time headway increases, the intersection point of the curves will move forward, that is, the density corresponding to the intersection point will become smaller. Therefore, the desired time headway is one of the important parameters reflecting the driving behavior of CAVs, further reducing the desired time headway and improving the traffic capacity of the mixed traffic flow to a large extent. 5. Numerical analysis 5.1. Simulation setup In order to prove the validity of the fundamental diagram model of the mixed traffic flow, the simulation experiment is modeled in microscopic simulation software, named Vissim [45]. Referring to the Ref. [13], we design the simulated road section, as shown in Fig. 8. The simulated road section is a hypothetical one-lane [13,36] with an on-ramp located in the middle of the section. A gap-acceptance based lane-changing model [46] is selected for merging maneuvers. Specifically,

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Fig. 6. Influence of the minimum safe gap distance on the fundamental diagram of the mixed traffic flow.

this lane-changing model consists of three steps: (1) consider a lane change, (2) choose left or right lane, and (3) search for an acceptable gap to change lanes. The section is 6 km long with a speed limit of 120 km/h and the input volume of the main road varies from 1000 to 5000 veh/h. The car-following models and merging model are realized in MATLAB and embed into Vissim by Vissim COM technology [47] to capture the driving behavior of different vehicles. All the experiments are performed in a laptop computer with an Intel 2.5 GHz CPU and 8 GB memory. The CAVs are generated with a new generated random number between 0 and 1 when it less than the penetration rate. In addition, detectors are placed in the road section with an interval of 500 m, which collect traffic volume and density every 10 min. The simulation duration is 2 h, including 30 min of simulated warm-up time, 30 min of dissipation time, and 1 h of data acquisition time. The simulation accuracy is 0.1 s, and the same scene is simulated 3 times with different random seeds. In the simulation experiment, the minimum safety distance s0 and the desired time headway tc are set as 2 m and 0.6 s, respectively. The fundamental diagram of the mixed traffic flow is obtained under different CAVs penetration rates. Specifically, the penetration rates p is set to 0, 20%, 40%, 60%, 80%, and 100%. The simulation results are shown in Fig. 9. The red curve and the blue curve are the theoretical curves with 100% and 0% penetration rates, respectively. The black curve and points are the theoretical curve and the density–volume data of the simulation results with a given penetration rate. 5.2. Results and discussions Fig. 9 reveals that there is some difference between the simulation data and the actual curve. This indicates that the traffic flow in the simulation environment is difficult to reach a stable state. As a result, the obtained simulation results

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Fig. 7. Influence of the desired time headways on the fundamental diagram of the mixed traffic flow.

Fig. 8.

Diagram of the simulation road segment..

are different from the theoretical results. However, the dispersion degree of simulation data points decreases with the increase of CAVs penetration rates. This shows that CAV can improve the stability of mixed traffic flow. In addition, Fig. 9 shows that the simulation data fluctuates, but they are both on the two sides of the theoretical curve. The overall trend is consistent with the theoretical results, which proves the correctness of the analysis results of the fundamental diagram in the mixed traffic flow. Meanwhile, the corresponding maximum volume rate of the scatter plot also gradually increases

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Fig. 9. Result of the simulation experiment.

with the increase in the penetration rate of CAVs from Fig. 9. Therefore, this is also consistent with the analysis results of the analytical model, which proves the validity of the analytical model.

6. Conclusions and recommendations

The primary objective of this study is to research the stability and fundamental diagram model of the mixed traffic flow in CAVs environment. The FVD model and the CACC model of the PATH laboratory respectively described the following behavior of HDVs and CAVs. Based on this, the stability criterion and fundamental diagram model under the mixed traffic flow conditions were derived. Through the stability, sensitivity and numerical simulation analysis, the following conclusions can be drawn:

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(1) On one hand, when the penetration rate of CAVs is less than 47%, the mixed traffic flow has an unstable speed interval which gradually decreases with the increase in the penetration rate of CAVs. On the other hand, when the penetration rate of CAVs is greater than or equal to 47%, the mixed traffic flow is in a stable state. (2) The traffic capacity of the road segment gradually increases with the increase in the penetration rates, which indicates that CAVs can effectively improve the capacity of existing roads. (3) The free flow velocity, the minimum safety distance, and the desired time headway have an impact on the traffic capacity. The greater the free flow velocity is, the greater the corresponding traffic capacity will be; the smaller the minimum safety distance and the desired time headway are, the larger the capacity is. In addition, the desired time headway is an important parameter to reflect the driving behavior of CAVs; additionally, further reducing desired time headway can improve the traffic capacity of the mixed traffic flow to a large extent. (4) From the numerical simulation analysis, the simulation results are consistent with the analytical results, which proves the rationality and effectiveness of the fundamental diagram analytical model of the mixed traffic flow. The effects of time delays [48] (e.g. human delay and vehicle response times) are not considered in this work, which leads to the excessive capacity calculated by the proposed model. Therefore, these factors will be taken into account in the next study to further analyze the stability and fundamental diagram of the mixed flow. In addition, the different traffic phases [18] are not distinguished in current work. In the next step, it will obtain more interesting results to analyze the different traffic phases. Acknowledgments The paper received research funding support from the Open Fund Project of Chongqing Key Laboratory of Traffic and Transportation, China (2018TE01), the National Science Foundation of China (5157765, 61703064, 71771190), Chengdu Science and Technology Bureau, China Project (No. 2017-RK00-00362-ZF), Chongqing Research Program of Basic Research and Frontier Technology, China (cstc2017jcyjAX0473), Southwest Jiaotong University Graduate Academic Training and Promotion Program, China (2019KCJS46), and Education Department of Hunan Province, China (No. 16B008). The authors are grateful to the three anonymous reviewers for sharing their research insights and providing helpful comments to improve the quality of the paper. References [1] L.C. Davis, Nonlinear dynamics of autonomous vehicles with limits on acceleration, Physica A 405 (2014) 128–139, http://dx.doi.org/10.1016/j. physa.2014.03.014. [2] L. Ye, T. 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