Analyzing the impact of automated vehicles on uncertainty and stability of the mixed traffic flow

Transportation Research Part C 112 (2020) 203–219

Contents lists available at ScienceDirect

Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Analyzing the impact of automated vehicles on uncertainty and stability of the mixed traffic flow

T

Fangfang Zhenga, Can Liua, Xiaobo Liua, , Saif Eddin Jabarib, Liang Lua ⁎

a

School of Transportation and Logistics, National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Southwest Jiaotong University, Chengdu 611756, PR China Division of Engineering, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi P.O. Box 129188, United Arab Emirates

b

ARTICLE INFO

ABSTRACT

Keywords: Mixed traffic flow Stochastic Lagrangian model Uncertainty Stability

This paper proposes a stochastic model for mixed traffic flow with human-driven vehicles (HVs) and automated vehicles (AVs). The model is formulated in Lagrangian coordinates considering the heterogeneous behavior of human drivers. We further derive a first and second order approximation of the stochastic model describing the mean and the covariance dynamics, respectively, under different combinations of HVs and AVs in the traffic stream (e.g., randomly distributed in the stream, at the front of the stream, in the middle of the stream and in the rear of the stream). The proposed model allows us to explicitly investigate the interaction between AVs and HVs considering the uncertainty of human driving behavior. Six performance metrics are proposed to measure the impact of AVs on the uncertainty of HVs’ behavior, as well as on the stability of the system. The numerical experiment results show that AVs have significant impact on the uncertainty and stability of the mixed traffic flow system. Larger AV penetration rates can reduce the uncertainty inherent in HV behavior and improve the stability of the mixed flow substantially. Whereas AVs’ reaction time only has subtle impact on the uncertainty of the mixed stream; as well as the position of AVs in the traffic stream has marginal influence in terms of reducing uncertainty and improving stability.

1. Introduction One important feature of human-driven traffic flow is traffic instability, which can be frequently observed on highways where small disturbances are amplified along a traffic stream, leading to stop-and-go traffic jams called ‘phantom jams’ (Gazis and Herman, 1992). It has significant adverse impacts on traffic safety, efficiency and sustainability (Treiber and Kesting, 2013). Common triggers of traffic oscillation (or stop-and-go traffic) include lane changes near merges and diverges (Ahn and Cassidy, 2007; Bertini and Leal, 2005; Laval, 2006; Laval and Daganzo, 2006; Zheng et al., 2011), lane drops (Bertini and Leal, 2005) or changes of road geometrics (Jin and Zhang, 2005). However, traffic instability is not always associated with these triggers. Some empirical studies revealed that the stochastic behavior of drivers might cause unstable traffic flow dominated by traffic waves in certain flow regimes (Ahn and Cassidy, 2007; Laval, 2006; Laval and Daganzo, 2006). In the literature, car following models are widely used to investigate the mechanisms behind the propagation of traffic oscillations. The majority of these models are deterministic with elegant stability properties, which fail to capture the uncertainty in human driving behavior. A number of stochastic car-following models have been developed by adding time-varying random noise (e.g.,

⁎

Corresponding author. E-mail address: [email protected] (X. Liu).

https://doi.org/10.1016/j.trc.2020.01.017 Received 4 June 2019; Received in revised form 17 January 2020; Accepted 23 January 2020 0968-090X/ © 2020 Elsevier Ltd. All rights reserved.

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

white noise) to deterministic models (Laval and Leclercq, 2008; Laval et al., 2014; Treiber and Kesting, 2018; Treiber et al., 2006). Though they can better capture the properties of traffic oscillation caused by human driving behavior, this type of stochastic models can produce negative traffic variables (e.g., vehicle speed). To address this, (Jabari and Liu, 2012) and (Jabari and Liu, 2013) developed stochastic queuing techniques with random time headways that result in a stochastic Eulerian model, Jabari, et al. (2018) modeled a stochastic Lagrangian counterpart, Jabari et al. (2019) further developed a random field that preserves the non-negativity in state estimates, and Ngoduy et al. (2019) developed a continuous stochastic car following model by applying Langevin equations and an extended Cox-Ingersoll-Ross (CIR) stochastic process to limit the negative trajectories of the stochastic vehicle speeds. They further applied the model to multi-lane freeways where the stochastic optimal velocity model (SOVM) is combined with lane changing estimation based on a deep learning approach (Lee et al., 2019). In general, there still remain issues related to the physical accuracy of the sample paths of existing stochastic traffic models. The main issue is that stochasticity is introduced in the time dimension and the implicit assumption of independent increments, which results in sample paths that are prone to aggressive oscillations in the time dimension. The impact of stochastic human driving behavior on traffic instability has been investigated both theoretically and empirically. If traffic flow is stable, perturbations of the leading vehicle are dampened out (or not amplified) when propagating along the vehicle string. Li et al. (2019) proposed a frequency-domain stability analysis method which is capable of quantifying speed variations of a vehicle platoon analytically. Ngoduy et al. (2019) derived stochastic linear stability conditions which can theoretically capture the effect of the random acceleration on traffic instability. An experimental study conducted with a 51-car-platoon by Jiang et al. (2018) indicated that the traffic speed is a better indicator of traffic (in)stability rather than the vehicle spacing. They further proposed a mechanism of traffic instability which can be described as the competition between stochastic factors and speed adaption effects. With the development of connected and automated vehicle technologies, attention has shifted from conventional human-driven vehicle (HV) traffic to fully automated vehicle (AV) traffic. The research on AV traffic mainly focuses on AV trajectory planning (Guanetti et al., 2018; Li and Li, 2019; Lu et al., 2019; Ma et al., 2016; Yu et al., 2018; Zhou et al., 2017), optimization and control (Feng et al., 2018; Guo et al., 2019; Yu et al., 2018) considering fully controllable and deterministic behavior of AVs. However, it is widely expected that the mixed traffic, HVs and AVs, will be prevalent in the next 20–30 years before full AV’s era. In this case, vehicle control and stability become more complicated due to the uncertainty of human driving behavior. Therefore, how to accurately model the interaction between HVs and AVs, in particular considering the stochastic behavior of human drivers is crucial for efficient and reliable estimation, control and optimization of the mixed traffic flow. Zhu and Zhang (2018) investigated the stability of the mixed traffic flow of human-driven vehicles and autonomous vehicles. A Bando’s model and a modified Bando’s model are used to describe the human driver behavior and the autonomous car flow, respectively. Talebpour and Mahmassani (2016) proposed a framework to investigate the impact of connected and autonomous vehicles (CAVs) on the stream’s stability and throughput of the traffic system and their results show that CAVs can help improve string stability in the system. In particular, autonomous vehicles are effective in preventing shockwave propagation. From the cyber physical perspective, a deterministic model for mixed traffic has been developed by Jin et al. (2020) considering the driver’s reaction delay and the information of multiple vehicles ahead obtained by CAVs. The stability of the system was investigated under different CAV penetration rates, the driver’s reaction delay and the number of vehicles from which CAVs can obtain information. The stability of the mixed traffic flow can be significantly improved by proper control of CAVs as has been both theoretically and empirically investigated in the literature. For instance, Ghiasi et al. (2019) developed a control strategy of AVs to harmonize speed and stabilize mixed traffic flow. A field experiment study (Stern et al., 2018) demonstrates that a single autonomous vehicle can dampen the stop-and-go waves via a set of simple control strategies with calibrated parameters. The one question that remained for their experiment is whether their results are sufficiently representative because their qualitative and quantitative results are based on one single experiment for each control strategy. Most of previous work assumes deterministic behavior of human-driven vehicles and automated vehicles when they investigate problems related to the mixed traffic flow. Though recent work done by Chen et al. (2019) addresses the impact of heterogeneous vehicle behavior including acceleration rates, desired speeds and car following behavior on the traffic dynamics and throughput of the mixed traffic. Ghiasi et al. (2019) proposed an analytical stochastic model to estimate highway capacity for mixed traffic considering the CAV penetration rate, the CAV platooning intensity and mixed traffic headway settings. Still research on traffic (in) stability for the mixed traffic, as well as the influence of AVs on uncertainty and stability of the mixed traffic is rather limited. In this study, we adopt a stochastic version of Newell’s nonlinear speed-spacing relation as proposed in (Zheng et al. 2018) to model HVs’ dynamics explicitly considering the uncertainty in human driving behavior. We formulate the traffic dynamics in Lagrangian coordinates which move with vehicles. By doing so, the coupling between macroscopic models and microscopic models becomes simpler, facilitating the investigation of individual vehicle dynamics with reduced nonlinearity compared with those in traditional Eulerian coordinates (Laval and Leclercq, 2013; Leclercq et al., 2007; Tchrakian and Basu, 2012; van Wageningen-Kessels et al., 2013). We further derive first and second-order approximations of the stochastic Lagrangian model analytically for the mixed traffic flow. The second order (e.g., spacing variance) allows us to explicitly investigate the interaction between AVs and HVs to assess the impact of AVs (e.g., AV penetration rate, position of AVs in a stream) on the uncertainty of the mixed traffic flow as well as on traffic stability. The paper is organized as follows. Section 2 derives a stochastic Lagrangian model for mixed traffic flow considering the heterogeneous behavior of human drivers and deterministic behavior of AVs. We further derive the mean dynamics and the covariance dynamics of the proposed model considering different combinations of HVs and AVs in the traffic stream. In Section 3, we propose six performance metrics including the mean spacing variance, the mean speed variance, the variance of mean speed, three metrics of percentage improvement to measure the impact of AVs on the uncertainty and the stability of the mixed traffic system. Section 4 conducts numerical experiments and investigates the effects of different AV penetration rate, AVs’ position in the traffic stream, and 204

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

AVs’ reaction time on the uncertainty and stability of the system. Section 5 concludes the paper and provides some discussion. 2. A stochastic model for the mixed traffic For the mixed traffic flow, heterogeneous behavior of HVs can be captured by using driver specific speed-spacing relations, i.e., parametric uncertainty as developed analytically (and validated experimentally) in Jabari et al. (2014). We assume homogeneous and deterministic behavior of AVs which is deemed as a reasonable assumption commonly used in AV modeling and control (Ge and Orosz, 2018; Ye and Yamamoto, 2018). A stochastic Lagrangian model for human-driven traffic developed by (Zheng et al., 2018) is included for the sake of completeness. Based on this model, we derive a stochastic model incorporating the characteristic of autonomous vehicles (AVs) in the mixed traffic condition. 2.1. Stochastic model without AVs We consider a system with N + 1 human-driven vehicles (HVs), where vehicle n = 0 is the platoon leader, n = 1 is the immediate follower, and n = N is the trailing vehicle in the system. We denote the position and speed of vehicle n at time t [0, T ] by x n (t ) and vn (t ) , respectively, where T < is the time horizon. The spacing between vehicles n and n 1 is given by

sn (t )

xn

1 (t )

(1)

x n (t )

and the position dynamics for vehicle n are given by: t

x n (t ) = xn (0) +

0

vn ( ) d ,

(2)

where x n (0) is the initial position of vehicle n; vn ( ) represents the speed of vehicle n at time instant . To capture heterogeneity in the human driver population, we employ driver-specific speed-spacing relations. We adopt Newell’s non-linear car following model (Castillo and Benítez, 1995; Newell, 1961) with speed-spacing relation cnHV

Vn (s ) = vnHV ,f

vnHV ,f e

(s dnHV )

vnHV ,f

(3)

,

HV HV where the driver-specific parameters (vnHV , f ; dn ; cn ) respectively represent human driver n’s desired (free-flow) speed, the minimum safety distance, and the inverse of the reaction time of driver n when their speed is restricted by the trajectory of their leader. By utilizing the speed-spacing relation, we write t

x n (t ) = xn (0) +

0

Vn (sn ( )) d .

(4)

The spacing dynamics are derived by substituting Eq. (4) into Eq. (1) given by

sn (t ) = sn (0) +

t 0

(vn

1 (sn 1 (

))

vn (sn ( ))) d .

(5)

To introduce stochasticity, we let the parameters be random variables. This is interpreted as uncertainty about the driver characteristics. We write the (stochastic) parameters as functions of , where is the random space, and assume the parameters of one driver are independent of the parameters of other drivers, but the parameters themselves are not necessarily independent for each driver. As such, we have the stochastic parameter vector ( ) (vfHV , d HV , c HV )( ) with joint distribution function F and the HV HV parameter tuple for each driver n, n = (vnHV , f , dn , cn ) , is drawn independently from this common distribution: n ∼F . The stochastic speed-spacing relation is written as

V (s, ) = vfHV ( )

vfHV ( ) e

c HV ( ) (s d HV ( )) vfHV ( )

(6)

.

The stochastic dynamic model evolves according to

sn (t , ) = sn (0, ) +

t 0

(Vn

1 (sn 1 (

, ))

Vn (sn ( , ))) d

(7)

and the stochastic spacing dynamics can be simulated using the following recursion

sn (t + t , ) = sn (t , ) + t (Vn

1 (sn 1 (t ,

))

Vn (sn (t , )))

(8)

We assume the same boundedness properties in Jabari et al. (2014) and Zheng et al. (2018) to ensure physically reasonable sample path characteristics. Specifically, we assume that a finite constant c HV , max < exists, which binds cnHV ( ) from above with HV , max probability 1. This is utilized to select a time discretization, t = n/ c , which respects the Courant-Friedrichs-Lewy (CFL) condition. 2.2. Stochastic model with AVs Now we assume N + 1 mixed human-driven and autonomous vehicles travel along a single lane road without lane changes and 205

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

Fig. 1. Four sequence combinations of HVs and AVs in the traffic stream.

overtaking. Let ‘HV’ denote the human-driven vehicle and ‘AV’ denote the autonomous vehicle, for a certain type of vehicle i, i {HV , AV } , the speed-spacing relation is given by:

V (s , )

Vi( ) =

(s )

if

,

i = HV

, elseif i = AV

,

(9)

where V i is the speed of vehicle type i (i {HV , AV }) ; V (s, ) is the stochastic speed-spacing relation of HVs given by Eq. (6). We apply Newell’s (stationary) speed-spacing relation to describe the car following behavior of AVs as well; therefore (s ) is the deterministic speed-spacing relation for AVs given by:

(s ) =

vfAV

c AV (s d AV ) AV v fAV vf e ,

(10)

and where respectively represent the AV’s desired speed, minimum safety distance and the inverse of the reaction time when stationary. The stochastic dynamic model in the mixed traffic flow condition evolves according to: vfAV ,

d AV

c AV

t

sn (t , ) = sn (0, ) +

0

(V i n

1 (sn 1 (

, ), )

V i n (sn ( , )), ) d ,

(11)

) is the speed of vehicle n which belongs to vehicle type i (i {HV , AV }) . Eq. (11) can be further classified in accordance where with the four leader/follower combinations of HVs and AVs in the traffic stream shown in Fig. 1. Case k = 1: The leading vehicle n-1 is a HV and the following vehicle n is a HV (as shown in Fig. 1(a)). In this case, the dynamic model of Eq. (11) represents the stochastic dynamics of human-driven vehicles only as given by (7). Case k = 2: The leading vehicle n-1 is a HV and the following vehicle n is an AV (as shown in Fig. 1(b)). In this case, the stochastic dynamic model (10) becomes: Vni (

t

sn (t , ) = sn (0, ) +

0

(Vn

1 (sn 1 (

, ), )

n (sn (

, ))) d ,

(12)

where the stochastic speed-spacing relation of the leading HV, Vn 1 (s ( , ), ) , is given by Eq. (6); and the deterministic relation of the following AV n (s ( , )) is given by Eq. (10). Case k = 3: The leading vehicle n-1 is an AV and the following vehicle n is a HV (as shown in Fig. 1(c)). In this case, the stochastic dynamic model (10) becomes: t

sn (t , ) = sn (0, ) +

0

(

n 1 (sn 1 (

, ))

Vn (sn ( , ), )) d

(13)

Case k = 4: The leading vehicle n-1 is an AV and the following vehicle n is an AV (as shown in Fig. 1(d)). In this case, the dynamic model of equation (10) evolves according to:

sn (t , ) = sn (0) +

t 0

(

n 1 (sn 1 (

, ))

n (sn (

, ))) d .

(14)

With Eqs. (7) and (12)–(14), we can simulate the stochastic dynamics for mixed traffic. One significant advantage of this model is that the heterogeneity of human drivers is considered explicitly. Hence, we can further investigate the impact of AVs on traffic instability (i.e., in terms of speed variations) in mixed traffic. Not only can the average impact be quantified but its variability due to the uncertainty of human driver behavior can also be quantified with the proposed model. 206

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

2.3. Mean dynamics and variability Eqs. (7) and (12)–(14) demonstrate the non-linear characteristic of the stochastic dynamic model for the mixed traffic condition. Applying this model to analyze uncertainty and instability of the mixed traffic requires a process of sampling. In this section, we derive a surrogate stochastic model so that the above stochastic Lagrangian model can be approximated by mean dynamics and covariance dynamics explicitly considering different characteristics of human-driven vehicles and autonomous vehicles. By doing so, the uncertainty of the mixed traffic system and the impact of AVs on the uncertainty can be quantified analytically, i.e., using the covariance dynamics which reflect the stochastic behavior of the system. 2.3.1. Mean dynamics Due to the stochasticity in the speed-spacing relation of HVs, V (s, ) , and the resulting stochasticity in the spacings, simply taking expectations will not deliver the desired results. Therefore, we apply the ensemble-averaging to determine the mean dynamics as the limiting process. Let M denote the ensemble size and m = 1, ..,M the index of the stochastic process in the ensemble. The ensemble averaged spacing is given by

snM (t , ) =

1 M

M

snm (t , ),

(15)

m =1 th

) represents the m spacing process for n = 1, …, N. For vehicle n = 1, …, N, the ensemble-averaged process where evolves according to snm (t ,

snM (t , ) = sn (0) +

1 M

M

t 0

m=1

(V i, m (snM 1 ( , ), ·)

where i and j represent the vehicle type, i , j

s M (t , ) = s (0) +

M

1 M

t 0

m =1

V j, m (snM ( , ), ·)) d ,

{snM (

, )}

(16)

{HV , AV } . To determine the mean dynamics, we first write Eq. (16) in vector form

D k V m (s M ( , ), ·) d ,

(17)

, ) ( , where DV m (s M ( )) [v0 ( ) ( ), ) V i, m (sNM 1 ( ), ) V j, m (sNM ( ), )]T with V i, m ( ) the mth speed process for vehicle type i; and k represents the cases defined in the previous section with the leading vehicle and the following vehicle of {HV HV , HV AV , AV HV , AV AV } . Next, we write the mean dynamic process of spacing as [s1M

sM (

s¯ (t ) = s (0) +

)]T ,

) sNM ( , V i, m (s1M

t 0

D k V¯ (¯s ( )) d ,

(18)

where s¯ ( ) is defined as s¯ ( ) = [¯s1 ( ) s¯N ( large numbers, we have that

1 M

M

V m (s , ) m=1

)]T

and V¯ (¯s ( ))

[V¯ (¯s1 ( )) V¯ (¯sN ( ))]T is the mean speed-spacing relation. By the law of

V¯ (s ) almost surely

M

(19)

Convergence results: According to the statement as proved by Zheng et al. (2018), the ensemble-averaged process converges to the mean dynamics as follows

lim ||s M (·, )

M

s¯ (·)||t = 0 for all t

[0, T ]

(20)

The results above can be generalized to any vehicle size scaling of n . In essence, we have assumed n = 1 thus far. The ensemble averaged process and the limit (mean) spacing process can be generalized to

s M (t , ) = s (0) +

1 M n

M

t

m=1

0

D k V m (s M ( , ), ·) d

(21)

and

s¯ (t ) = s (0) +

1 n

t 0

D k V¯ (¯s ( )) d .

(22)

2.3.2. Covariance dynamics The mean dynamics in Eq. (22) can be considered as a first-order approximation of the stochastic Lagrangian model for mixed traffic flow. In this section, we derive the covariance dynamics which constitutes a second moment of the stochastic system. First of all, we consider the amplified deviation process M (t ,

)

M (s M (t , )

(23)

s ¯(t )), 207

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

where M is a scaling parameter ensuring the covariance matrix pertaining to proved in Zheng et al. (2018). By expanding Eq. (23), we derive M (t ,

) =

t 0

1 M n

=

t 0

M m=1

1 M n

t 0 t 0

)= +

as

M

1 n M 1 n M

t 0

D k (V m (s M ( , ), )

1 M n

M

M

V¯ (¯s ( ))) d

0

M m=1 M m =1

D k (V m (¯s ( ), )

D k V m (¯s ( ), ) +

n M

(24)

D k V m (¯s ( ), ) d to the right-hand side of Eq. (24), we obtain

m=1 D k (V m (s M (

M

1

D k V¯ (¯s ( )) d

m =1

V m (¯s ( ), )) d

, ))

V¯ (¯s ( ))) d

(25) 1 M

Since we have by definition, Eq. (23), that s M ( , ) = s¯ ( ) + t

(t , ) is the same as that pertaining to s (t , ) as

m=1

t 1 0 M n

Adding and subtracting M (t ,

D k V m (s M ( , ), ) d -

M

m =1

1 M

M(

M(

, ) , the first term on the right side of Eq. (25) can be written

V m (¯s ( ), ) d

, )

(26)

According to Zheng et al. (2018), Eq. (26) converges to

1 n

t 0

D k diag [ V¯ (¯s ( ))] ( , ) d ,

(27)

( ) V¯ (¯s (t )) ( ) V¯ (¯s d ds

where V¯ (¯s (t )) =

T

d ds

and diag [s] is a diagonal matrix with diagonal elements given by the vector s. We N (t )) further define H (¯s (t )) diag [ V¯ (¯s (t ))] to simplify notation. The limit process of the second term on the right side of Eq. (25) can be derived by applying the central limit theorem and the continuous mapping theorem given by

1 n

t 0

Dk

1

1/2 (¯ s(

)) dW ( , ),

(28)

where ( ) is the covariance matrix with the diagonal elements given by Var (V (¯sn (t ), )) = E(V (¯sn (t ), ) is a N-dimensional Wiener process. Hence, the limiting deviation process ( , ) can be derived as

(t , ) =

(0) +

1 n

t 0

D k (H (¯s ( )) ( , ) d +

1/2 (¯ s(

)) dW ( , ))

V¯ (¯sn

(t )))2

, and W ( ,

)

(29)

And the differential form is given by

d (t , ) =

1 k D (H (¯s (t )) (t , ) dt + n

1/2 (¯ s (t )) dW

(t , ))

(30)

The limit deviation process ( , ) is a Gaussian process with covariance process given by P ( ) = E ( , ) ( , )T . The evolution of the covariance can be calculated by taking the derivative of P ( ) resulting in the following matrix differential equation describing the evolution of the covariance dynamics of the mix traffic flow

dP (t ) 1 k 1 T dt k = D H (¯s (t )) P (t ) + P (t ) H (¯s (t ))(D k )T + D (¯sn (t ))(D k )T , dt n n n2

(31)

where (s ) = . Due to different position combinations of HVs and AVs in the traffic stream, we have four cases of Eq. (31) for each leader/ follower pair: Case 1: The leading vehicle n-1 is a HV and the following vehicle n is a HV as well: In this case, we obtain the covariance dynamics for the human-driven traffic only as 1/2 (s ),

1/2 (s )

dPn (t ) 1 1 T dt = DH (¯sn (t )) Pn (t ) + Pn (t ) H (¯sn (t )) DT + D (¯sn (t )) DT , dt n n n2 where DH (¯sn (t )) =

( ) V¯ (¯s d ds

n 1 (t ))

(32)

( ) V¯ (¯s (t )), which is determined by the mean speed-spacing relation of the leading vehicle d ds

n

D (¯sn (t )) = Var (V (¯sn 1 (t ), )) Var (V (¯sn (t ), ))=E(V (¯sn 1 (t ), ) V¯ (¯sn 1 (t ))) 2 and the following vehicle; and E(V (¯sn (t ), ) V¯ (¯sn (t )))2 , which is determined by the stochastic speed-spacing relation of the leading vehicle and the following vehicle. Case 2: The leading vehicle n-1 is a HV and the following vehicle n is an AV: In this case, the evolution of the covariance dynamics becomes 208

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

dPn (t ) 1 1 T = DV H (¯sn (t )) Pn (t ) + Pn (t ) H (¯sn (t )) DV dt n n

( ) V¯ (¯s

where DV H (¯sn (t )) =

( )

T

+

T,

(¯sn (t )) DV

(33)

¯ (¯sn (t )))2 . E( (¯sn (t )) 2 ¯ (¯sn (t ))) = 0 . Thus, Considering the deterministic speed-spacing relation of AVs, we can derive E( (¯sn (t )) DV (¯sn (t )) = E(V (¯sn 1 (t ), ) V¯ (¯sn 1 (t ))) 2 , which is determined by the stochastic speed-spacing relation of the leading HV n-1. Eq. (33) can be further written as d ds

d ds

n 1 (t ))

(¯sn (t )) ; and DV

dt DV n2

dPn (t ) 1 1 T = DV H (¯sn (t )) Pn (t ) + Pn (t ) H (¯sn (t )) DV dt n n

T

(¯sn (t )) = E(V (¯sn

+

1 (t ),

)

V¯ (¯sn

1

(t ))) 2

dt (¯sn - 1 (t )), n2

(34)

where (¯sn 1 (t )) = E(V (¯sn 1 (t ), ) V¯ (¯sn 1 (t )))2 is the speed variance of the leading HV n-1. Case 3: The leading vehicle n-1 is an AV and the following vehicle n is a HV: In this case, we can derive the covariance dynamics as

dPn (t ) 1 = D dt n where D

sn (t )) Pn (t ) V H (¯

+

( )

1 T Pn (t ) H (¯sn (t )) D n

V

T

+

dt D n2

V

(¯sn (t )) D

V

T,

(35)

( )

d d ¯ ¯ (¯sn 1 (t ))) 2 E(V (¯sn (t )) V¯ (¯sn (t ))) 2 = = ds (¯sn 1 (t )) V (¯sn (t )) , and D V (¯sn (t )) = E( (¯sn 1 (t )) ds 2 ¯ E (V (¯sn (t )) V (¯sn (t ))) , which is determined by the stochastic speed-spacing relation of the following HV n. Therefore, Eq. (35) can be simplified as

sn (t )) V H (¯

dPn (t ) 1 D = dt n

sn (t )) Pn (t ) V H (¯

+

1 T Pn (t ) H (¯sn (t )) D n

V

T

+

dt (¯sn (t )) n2

(36)

Case 4: The leading vehicle n-1 is an AV and the following vehicle n is an AV as well: In this case, the differential equation describing the covariance dynamics can be written as

dPn (t ) 1 1 T = D H (¯sn (t )) Pn (t ) + Pn (t ) H (¯sn (t )) D dt n n

( )

( )

(¯sn 1 (t )) where D H (¯sn (t )) = ¯ (¯sn D (¯sn (t )) = E( (¯sn 1 (t )) derive Eq. (36) can be further simplified as d ds

d ds

T

+

dt D n2

(¯sn (t )) D

T,

(37)

(¯sn (t )) ; Due to the assumption of deterministic speed-spacing relation of AVs, we can 1 (t )))

2

E( (¯sn (t ))

dPn (t ) 1 1 T = D H (¯sn (t )) Pn (t ) + Pn (t ) H (¯sn (t )) D dt n n

¯ (¯sn (t )))2 = E( (¯sn

1 (t ))

(¯sn

1 (t )))

2

E( (¯sn (t ))

T.

(¯sn (t ))) 2 = 0 .

(38)

Eqs. (32), (34), (36) and (38) provide the analytical formulation of spacing variance dynamics for different combinations of HVs and AVs in the mixed traffic stream. 3. Metrics of uncertainty and stability for the mixed traffic flow 3.1. Metrics of uncertainty In mixed traffic flow, due to different properties of HVs and AVs, in particular the heterogeneity of HVs, the traffic dynamics are stochastic in that the spacing between two adjacent vehicles (position) is not a deterministic value but rather follows a certain distribution as illustrated in Fig. 2. The covariance dynamics can by analytically calculated using Eqs. (32), (34), (36) and (38) as derived in Section 2. It needs to be emphasized that the covariance dynamics reflect the uncertainty in HVs’ driving behavior which is considered as the source of uncertainty in the mixed traffic system. Thus, we define two metrics for the uncertainty of the mixed traffic flow, namely the mean spacing variance and the mean speed variance over the mixed traffic stream and over the time period considered. For each time instant t, we compute the mean spacing variance and the mean speed variance as 2 spacing (t )

=

1 N

N 2 sn

(t )

(39)

n=1

and 2 speed (t )

=

1 N

N 2 vn

(t )

(40)

n= 1

The mean spacing variance and the mean speed variance over all vehicles and over the time interval are given by spacing

=

1 NT

T

N 2 sn

(t )

(41)

t=1 n=1

and 209

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

n=0 S1(t)

n=1

......

x

n=i-1

Si(t)

si

n=i n=N-1

SN(t)

......

n=N

t

t

Fig. 2. Uncertainty of spacing dynamics in the mixed traffic stream.

speed

1 NT

=

T

N 2 vn

(t ),

(42)

t=1 n=1

(t ) is the spacing variance at time t given by Eqs. (32), (34), (36) and (38), depending on the combinations of HVs and AVs where in the traffic stream and sn is the spacing between vehicle n and its preceding vehicle n-1; v2n (t ) is the speed variance derived from the ensemble averaged process. Since we have the ensemble averaged speed-spacing relation which converges to the mean speed-spacing relation given by Eq. (19). Then, for a sufficiently large M, V¯ (s ) can be very well approximated by 2 sn

1 M

V¯ (s )

M

V m (s )

(43)

m =1

The speed variance for vehicle n is given by 2 vn

(t ) =

M

1 M

(vnm (s (t ), )

v¯n (s (t )))2 .

(44)

m=1

It is noteworthy that we assume that AVs are homogeneous and the speed spacing relation is deterministic. Therefore, the uncertainty metrics provide the possibility to investigate the impact of AVs (e.g., penetration rates, combination of AVs and HVs) on the system uncertainty (such as suppression of stochastic behavior of HVs). 3.2. Metrics of instability As argued in Jiang et al. (2018), traffic speed might be a better indicator of traffic instability rather than vehicle spacing, because vehicles can have different spacing under the same speed. The speed variance has been used to quantify traffic oscillation by Stern et al. (2018) in their empirical study as well. Therefore, in order to quantify the instability of mixed traffic, we define the instability metrics as the variance of the mean speed over all vehicles and over the time. For each time interval t, the variance of the mean speed is computed as 2 s peed (t )

=

1 N

N

(v¯n (t ) n=1

1 N

N

v¯n (t ))2 ,

(45)

n

where v¯n (t ) is the mean speed derived from the ensemble averaged process given by Eq. (43). Over a total number of time intervals T, the variance of the mean speed are calculated as speed

=

1 NT

T

N

(v¯n (t ) t=1 n=1

1 N

N

v¯n (t )) 2 .

(46)

n=1

3.3. Metrics of performance improvement We define four metrics to quantify the impact of AVs (AVs’ penetration rate and AVs’ position in the traffic stream) on the 210

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

uncertainty of HVs’ behavior and the stability of the system. These metrics measure the reduction of uncertainty as the percentage difference between the average of spacing variance with a certain AVs’ penetration rate ‘ ’ and that with no AVs given by

Ispacing =

0 spacing spacing 0 spacing

× 100%

(47)

and 0 speed

Ispeed =

speed 0 speed

× 100%,

(48)

where spacing, speed represent the mean spacing variance and the mean speed variance, respectively, with penetration rate ‘ ’ given by Eqs. (41) and (42); 0spacing and 0speed represent the mean spacing variance and the mean speed variance without AVs, respectively; ( {5%, 10%, 15%, 20%, 25%, 30%, 35%, 40%, 45%, 50%}) is the penetration rate of AVs. Similarly, the metric used to measure the improvement in stability is given by

Pspeed =

0 speed

speed 0 speed

× 100% ,

(49)

where is the variance of the mean speed without AVs; and AVs given by Eq. (46). 0 speed

speed

is the variance of the mean speed with penetration rate ‘α’ of

4. Numerical experiments 4.1. Scenario settings and parameters Numerical experiments are conducted to visualize and analyze the mixed traffic flow properties using our proposed stochastic model. Fig. 3 shows the test scenario setting where there is a single-lane straight highway section with 100 vehicles (mixed of HVs and AVs) driving from the upstream to the downstream (eastbound). We set the start of the section as x = 0 and increases downstream. The bottleneck location is set to the location at x = 4.5 km. The total length of the simulated section is 12 km. Two cases are considered: Case 1: The first vehicle in the mixed traffic stream drives with the free flow speed v = 100 km/hr, and the initial spacing between vehicles is a random variable with (µ, 2) = (35m, 25m2) . Case 2: The first vehicle in the mixed traffic stream drives with the speed v = 60 km/hr, and the initial spacing between vehicles is a random variable with (µ, 2) = (25m, 9m2) . Fig. 4 illustrates different capacities for upstream and downstream of the bottleneck, and the corresponding speed spacing relation. In this experiment, we consider different traffic settings upstream and downstream of the bottleneck. For the traffic settings upstream of the bottleneck, the parameters of HVs, vf ( , t ), d ( , t )andc ( , t ) , are independent Beta random variables with supports max HV HV HV HV [v fmin , up, v f , up ] = [90, 110] km / hr , [d up, min , dup, max ] = [6, 9] meters and [cup, min, cup, max ] = [1950, 4250] veh / hr ; and the parameters of AVs are AV AV deterministic with vfAV , up = 100 hr , cup = 4400veh / hr , dup = 5m . To model different capacities for downstream of the bottleneck, we

km

AV HV HV set different jam densities in the car following model, where [ddown , min , ddown, max ] = [10, 15] meters for HVs and ddown = 8.4m for AVs. In order to assess the impact of AVs on the performance of the mixed traffic flow in terms of uncertainty and stability, we consider two elements

- Penetration rate: we set penetration rate of AVs to 5%, 10%, 15%, 20%, 25%, 30%, 35%, 40%, 45% and 50% and see how the penetration rate of AVs impacts the uncertainty of HVs and the stability of the traffic stream differently. - Combination of HVs and AVs in the traffic stream: We consider four scenarios where AVs are randomly distributed in the stream (scenario 1: AV-RD), AV platoon is at the front of the stream (scenario 2: AV-FS), AV platoon is in the middle of the stream (scenario 3: AV-MS) and AV platoon is in the rear of the stream (scenario 4: AV-RS).

HV

X=0

L1=4.5km

Bottleneck

X=4.5km driving direction Fig. 3. Test scenario setting. 211

AV

L2 = 7.5km

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

Fig. 4. Relation between traffic variables: (a) Fundamental diagram; (b) Speed-spacing relation.

(a) 0% AVs

(b) 10% AVs

(c) 30% AVs

(d) 50% AVs

Fig. 5. A single path of density dynamics with different penetration rates for scenario 1 where AVs are randomly distributed in the traffic stream (AV-RD) (Case 1).

212

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

(a) 0% AVs

(b) 10% AVs

(c) 30% AVs

(d) 50% AVs

Fig. 6. A single path of density dynamics with different penetration rates for scenario 1 where AVs are randomly distributed in the traffic stream (AV-RD) (Case 2).

4.2. Results 4.2.1. Impact of AVs on traffic dynamics In our experiment settings, vehicles first travel at the free flow speed along the highway section until a bottleneck appears. The reduced capacity downstream of the bottleneck results in traffic congestion which propagates upstream the bottleneck. Figs. 5 and 6 illustrate the density dynamics of simulated sample paths. They are generated with the proposed stochastic model for the mixed traffic flow with different AV penetration rates (0%, 10%, 30% and 50%) for case 1 and case 2, respectively. In Fig. 5, AVs are randomly distributed in the mixed traffic flow, corresponding to scenario 1 (AV-RD) of case 1 where the first vehicle drives with their desired speed. With the increase of the AV penetration rate, the congestion dynamics in terms of congestion duration is significantly reduced. Similar results can be observed in Fig. 6, where the first vehicle drives with the speed of 60 km/h corresponding to the initial condition of higher density. 4.2.2. Impact of AVs on the uncertainty of the mixed traffic In order to investigate the influence of AVs on the suppression of HV’s stochastic behavior, we apply the metrics defined in Section 3.1 (Eqs. (39)–(42), (47) and (48)) to quantitatively analyze the impact. Fig. 7 illustrates the mean speed variance dynamics with different penetration rates (from 0% to 50% AVs) under four scenarios. At the beginning of the simulation, vehicles travel with their desired speed and the traffic is in the free flow condition (stage 1). At the bottleneck a queue forms due to the reduction of the capacity and propagates upstream of the bottleneck (stage 2). The traffic flow propagates with the reduced capacity downstream of the bottleneck (stage 3). Three observations can be made from Fig. 7: Firstly, the uncertainty of HVs’ behavior in the free flow condition is much higher than that is in the congested condition. Secondly, the increase of AVs in the traffic stream can help reduce the stochastic behavior of HVs both in free flow conditions and congested conditions. Thirdly, the position of AVs of in the mixed traffic stream does not bring significant difference on damping stochastic behavior of HVs, though slightly better results can be 213

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

(b) Scenario 2: AV-FS

(a) Scenario 1: AV-RD

(d) Scenario 4: AV-RS

(c) Scenario 3: AV-MS

Fig. 7. Mean speed variance dynamics of HVs with different AV penetration rates under four scenarios for Case 1.

observed for scenario 2 (AV-FS) and scenario 3 (AV-MS) in the congested condition. Fig. 8 shows the mean speed variance dynamics for case 2. Similarly, we can observe that the uncertainty of HVs is much lower in congested conditions where vehicles are fully restricted. When traffic is less congested (e.g., at the beginning of the simulation) or in the free flow (e.g., downstream of the bottleneck), HV drivers have more freedom to choose their own desired speed. In this case, influence of heterogeneous behavior becomes significant with much higher mean speed variance indicating higher uncertainty. Fig. 9 shows the mean spacing variance (HV-HV and AV-HV) of four scenarios with different AVs’ penetration rates for case 1 and case 2. For case 1 as illustrated in Fig. 9(a), the increase of AVs’ penetration rate does not show clear impact on the uncertainty of the system for scenario 1 where AVs are randomly distributed in the traffic stream. For scenarios 2, 3 and 4, it can be clearly observed that with the increase of AVs’ penetration rate, the mean spacing variance decreases. Similarly, for case 2 as shown in Fig. 9(b), the mean spacing variance decreases with the increase of AVs’ penetration rate for all scenarios. Fig. 10 illustrates the mean speed variance (HVs only) of four scenarios for both cases. More consistent results can be concluded for all four scenarios: the increase of the AVs’ penetration rate can help reduce the mean speed variance (uncertainty of HVs’ behavior). Tables 1 and 2 indicate the percentage reduction of the mean speed variance under different AVs’ penetration rate. When the penetration rate increases from 5%

(a) Scenario 1: AV-RD

(b) Scenario 2: AV-FS

(c) Scenario 3: AV-MS

(d) Scenario 4: AV-RS

Fig. 8. Mean speed variance dynamics of HVs with different AV penetration rates under four scenarios for Case 2. 214

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

Fig. 9. Mean spacing variance with different AV penetration rates under four scenarios (AV-RD, AV-FS, AV-MS, AV-RS): (a) Case 1; (b) Case 2.

Fig. 10. Mean speed variance with different AV penetration rates under four scenarios (AV-RD, AV-FS, AV-MS, AV-RS): (a) Case 1; (b) Case 2. Table 1 Percentage reduction of the mean speed variance Ispeed (%) for Case 1. α (%) Scenario

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

AV-RD AV-FS AV-MS AV-RS

4.69 3.84 4.72 3.64

8.51 7.20 8.81 9.19

14.18 12.26 14.55 12.99

18.21 16.46 19.24 17.86

23.80 22.20 25.38 22.85

28.38 26.17 30.21 27.61

33.50 31.55 35.31 31.97

37.60 36.66 39.78 36.89

43.24 42.36 45.38 42.81

48.54 47.06 50.30 47.37

Table 2 Percentage reduction of the mean speed variance Ispeed (%) for Case 2. α (%) Scenario

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

AV-RD AV-FS AV-MS AV-RS

4.84 5.85 5.53 5.01

9.63 10.03 9.55 8.60

13.39 15.17 14.15 13.06

17.97 20.24 19.01 18.16

23.48 25.71 24.00 22.53

28.14 29.85 28.02 26.59

32.43 34.36 32.93 30.93

38.21 40.05 37.88 36.52

42.11 44.57 42.60 40.67

47.56 49.95 47.79 46.01

to 50%, the uncertainty of HVs in terms of the average of speed variance can be reduced by nearly 50% (ranging from 46.01% to 50.30%) for all scenarios. The impact of different combinations of HVs and AVs in the traffic stream on uncertainty reduction is marginal. The percentage reduction follows the similar pattern for all scenarios as well, in which scenario 3 (AV-MS) performs slightly better than the others for case 1; while scenario 2 (AV-FS) performs best among all scenarios for case 2. 215

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

Fig. 11. Variance of mean speed with different AV penetration rates (from 0% to 50% with 5% increment) under four scenarios (AV-RD, AV-FS, AVMS, AV-RS): (a) Case 1; (b) Case 2.

4.2.3. Impact of AVs on the stability of the mixed traffic flow In this simulation example, the impact of AVs on the stability of the traffic stream is assessed based on the metric defined in Sections 3.2 and 3.3. Fig. 11 illustrates the variance of mean speed under four scenarios with different penetration rates for case 1 (a) and case 2 (b). The results are consistent for the variance of mean speed which decreases when the AVs’ penetration rate increases from 5% to 50% for all scenarios both for case 1 (Fig. 11(a)) and case 2 (Fig. 11(b)). Detailed quantitative results can be found in Figs. 12 and 13 and Tables 2 and 3. The stability of the mixed traffic flow (in terms of the percentage reduction of the variance of mean speed) improves by up to 19% as the penetration rate increases from 5% to 50%. In this simulation experiment, we can observe that different combination of HVs and AVs in the traffic stream has marginal impact on improvement of the stability. Scenario 3 (AVMS) performs slightly better than the others for case 1, and scenario 1(AV-RD) performs best for case 2 (see Table 4). 4.2.4. Sensitivity analysis of AVs’ reaction time on uncertainty and stability In order to investigate how different AVs’ reaction times affect uncertainty and stability of mixed traffic flow, we consider different reaction times of AVs ranging from 0.2 s to 0.8 s. The settings in this numerical example is similar to that of Case 1 in Section 4.2.2, where the first vehicle in the mixed traffic stream drives with free flow speed of 100 km/h with 20% AV penetration rate. Fig. 14(a) shows the influence of different AVs’ reaction times on the uncertainty of HVs. It can be clearly seen that the AVs’ reaction time has marginal impact on the uncertainty of the mixed traffic flow, where the difference of the mean speed variance among four scenarios with different reaction times is subtle. Fig. 14(b) illustrates the variance of mean speed under four scenarios with different reaction times. With the increase of the reaction time from 0.2 s to 0.3 s, the stability increases (the variance of mean speed decreases) substantially for scenarios 2, 3 and 4. The effect is particular obvious for scenario 4 where the AV platoon is in the rear of the mixed traffic stream. In this scenario, the behavior of the AV platoon does not influence that of HVs. The lower reaction time of AVs could produce higher speeds according to Newell’s speed-spacing relation. Thus, the speed difference between HVs and AVs becomes more significant leading to the larger speed variance among all vehicles in the mixed stream. When the reaction time increases, the impact of the reaction time becomes less significant. As for scenario 1 where AVs are randomly distributed in the traffic stream, different reaction time has subtle influence on stability of the mixed traffic. One possible explanation is that the behavior of HVs are more evenly affected by that of AVs leading to lower speed variance.

Fig. 12. Percentage reduction of the variance of mean speed as a function of AV penetration rates under different scenarios for Case 1: (a) Scenario 1: AV-RD; (b) Scenario 2: AV-FS; (c) Scenario 3: AV-MS; (d) Scenario 4: AV-RS. 216

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

Fig. 13. Percentage reduction of the variance of mean speed as a function of AV penetration rates under different scenarios for Case 2: (a) Scenario 1: AV-RD; (b) Scenario 2: AV-FS; (c) Scenario 3: AV-MS; (d) Scenario 4: AV-RS. Table 3 Percentage reduction of the variance of mean speed Pspeed (%) for Case 1. α (%) Scenario

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

AV-RD AV-FS AV-MS AV-RS

2.34 1.27 3.01 0.92

4.19 2.08 4.52 1.25

4.17 2.22 6.26 1.63

7.58 3.45 7.43 2.85

6.39 5.62 9.15 4.16

9.17 6.69 11.24 5.49

10.24 8.72 12.77 7.06

12.96 10.46 14.83 10.10

14.29 12.68 16.93 12.98

15.74 15.63 18.00 13.28

Table 4 Percentage reduction of the variance of mean speed Pspeed (%) for Case 2. α (%) Scenario

5%

10%

15%

20%

25%

30%

35%

40%

45%

50%

AV-RD AV-FS AV-MS AV-RS

2.46 0.85 2.54 0.58

3.04 1.79 3.75 0.90

4.98 1.86 4.73 1.15

7.40 3.49 5.16 1.38

8.80 4.17 7.37 2.86

11.17 6.55 9.95 4.88

12.84 7.55 11.67 7.44

15.54 10.64 12.60 8.28

16.69 11.57 15.04 11.75

18.96 13.95 15.36 12.40

Fig. 14. The impact of AVs’ reaction time on performance of the mixed traffic under four scenarios with 20% AV penetration rate: (a) mean speed variance (uncertainty); (b) variance of mean speed (instability).

217

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

5. Conclusions and discussion This paper aims at developing a stochastic Lagrangian model to reveal the impact of AVs on the stochastic behavior and stability of mixed traffic flow with HVs and AVs. We use a stochastic version of the Newell-Franklin speed-spacing relation for HVs by considering parametric uncertainties, e.g., by treating free flow speed, minimum safety distance and the slope of the speed-spacing relation when stationary as random variables, in the same sense as Zheng et al. (2018). The behavior of AVs is assumed to be homogeneous and the deterministic Newell-Franklin speed-spacing relation is adopted. We further derive the second order approximation of the stochastic model as the mean dynamics and the covariance dynamics considering different combination of HVs and AVs in the traffic stream. Six performance metrics, namely mean spacing variance spacing , mean speed variance speed , variance of mean speed speed , percentage reduction of uncertainty (Ispacing , Ispeed ) and percentage improvement of stability (Pspeed ), based on the derived mean dynamics and the covariance dynamics are proposed to measure the stochastic behavior of HVs and the stability of the mixed traffic system. Numerical experiments are conducted to investigate the impact of AVs with various penetration rates and different positions of AVs in the traffic stream on traffic dynamics, uncertainty and stability of the mixed traffic flow. The simulation results show that the increase of the AV penetration rate from 5% to 50% can significantly help reduce the uncertainty and improve the stability of the mixed traffic system significantly both in the free flow and the congested condition. The position of AVs in the stream does not give significant impact difference on the suppression of uncertainty and the improvement of stability. The sensitivity analysis of AVs’ reaction time shows marginal impact on the uncertainty of the mixed traffic stream. Among the proposed metrics, the mean spacing variance and the variance of mean spacing could not generate clear impact measurement on uncertainty and stability of the mixed traffic system. Whereas the metrics of mean speed variance and the variance of the mean speed provides more consistent results which is in accordance with the results in Jiang et al. (2018). The primary purpose of this study is to establish analytical formulations describing the variance of the mixed traffic system, thus to provide physical insights into impact of AVs on the stochastic behavior of HVs and the stability of the system. Some simplifications are made regarding the parameters of HVs in the stochastic model, e.g., the parameters (the free flow speed, the minimum safety distance and the inverse of the reaction time) are assumed to follow the Beta distribution which can be calibrated once real-world trajectory data becomes available. The behavior of AVs is assumed to be deterministic in the proposed model though the sensitivity analysis of AVs’ reaction time is provided in the numerical example. The findings of this study reveal great potential of AVs to improve the stability and reduce the uncertainty of the mixed traffic flow. Intuitively, as a next step we would like to extend the model to include the heterogeneity and stochasticity of AVs; and furthermore, to develop control strategies for AVs to improve the stability of the mixed traffic flow, in particular taking the stochastic behavior of HVs into account. CRediT authorship contribution statement Fangfang Zheng: Conceptualization, Methodology, Writing - original draft. Can Liu: Investigation, Visualization. Xiaobo Liu: Supervision, Writing - review & editing. Saif Eddin Jabari: Writing - review & editing. Liang Lu: Software, Validation. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This research is funded by National Science Foundation of China under project codes 61673321 and 71671147, Sichuan Provincial Department of Science and Technology under project code 2019JDTD0002, and Didi Chuxing Company. References Ahn, S., Cassidy, M.J., 2007. Freeway traffic oscillations and vehicle lane-change maneuvers. Transportation & Traffic Theory Papers Selected for Presentation at Isttt. Bertini, R.L., Leal, M.T., 2005. Empirical study of traffic features at a freeway lane drop. J. Transp. Eng. 131 (6), 397–407. Castillo, J.M.D., Benítez, F.G., 1995. On the functional form of the speed-density relationship—I: General theory. Transportation Res. Part B: Methodological 29 (5), 373–389. Chen, D., Srivastava, A., Ahn, S., Li, T., 2019. Traffic dynamics under speed disturbance in mixed traffic with automated and non-automated vehicles. Transportation Res. Part C: Emerging Technol. Feng, Y., Yu, C., Liu, H.X., 2018. Spatiotemporal intersection control in a connected and automated vehicle environment. Transportation Res. Part C: Emerging Technol. 89, 364–383. Gazis, D.C., Herman, R., 1992. The moving and ‘‘phantom” bottlenecks. Transportation Sci. 26 (3), 223–229. Ge, J.I., Orosz, G., 2018. Connected cruise control among human-driven vehicles: Experiment-based parameter estimation and optimal control design. Transportation Res. Part C: Emerging Technol. 95, 445–459. Ghiasi, A., Li, X., Ma, J., 2019. A mixed traffic speed harmonization model with connected autonomous vehicles. Transportation Res. Part C: Emerging Technol. 104, 210–233. Guanetti, J., Kim, Y., Borrelli, F., 2018. Control of connected and automated vehicles: State of the art and future challenges. Ann. Rev. Control 45, 18–40. Guo, Y., Ma, J., Xiong, C., Li, X., Zhou, F., Hao, W., 2019. Joint optimization of vehicle trajectories and intersection controllers with connected automated vehicles: Combined dynamic programming and shooting heuristic approach. Transportation Res. Part C: Emerging Technol. 98, 54–72.

218

Transportation Research Part C 112 (2020) 203–219

F. Zheng, et al.

Jabari, S., Dilip, D., Lin, D., Thonnam Thodi, B., 2019. Learning traffic flow dynamics using random fields. IEEE Access 7, 130566–130577. Jabari, S., Liu, H., 2012. A stochastic model of traffic flow: Theoretical foundations. Transportation Res. Part B: Methodological 46 (1), 156–174. Jabari, S., Liu, H., 2013. A stochastic model of traffic flow: Gaussian approximation and estimation. Transportation Res. Part B: Methodological 47 (1), 15–41. Jabari, S., Zheng, F., Liu, H., Filipovska, M., 2018. Stochastic Lagrangian Modeling of Traffic Dynamics. D.C., Transportation Research Board of the National Academies, Washington, pp. 18–04170. Jabari, S., Zheng, J., Liu, H., 2014. A probabilistic stationary speed-density relation based on Newell’s simplified car-following model. Transportation Res. Part B: Methodological 68, 205–223. Jiang, R., Jin, C.-J., Zhang, H.M., Huang, Y.-X., Tian, J.-F., Wang, W., Hu, M.-B., Wang, H., Jia, B., 2018. Experimental and empirical investigations of traffic flow instability. Transportation Res. Part C: Emerging Technol. 94, 83–98. Jin, S., Sun, D.-H., Zhao, M., Li, Y., Chen, J., 2020.. Modeling and stability analysis of mixed traffic with conventional and connected automated vehicles from cyber physical perspective. Phys. A: Stat. Mech. Appl 124217. Jin, W.L., Zhang, Y., 2005. Paramics simulation of periodic oscillations caused by network geometry. Transportation Res. Rec. J. Transportation Res. Board 1934 (1), 188–196. Laval, J., 2006. Stochastic processes of moving bottlenecks: approximate formulas for highway capacity. Transportation Res. Rec. J. Transportation Res. Board 1988 (1), 86–91. Laval, J.A., Daganzo, C.F., 2006. Lane-changing in traffic streams. Transportation Res. Part B: Methodological 40 (3), 251–264. Laval, J.A., Leclercq, L., 2013. The Hamilton-Jacobi partial differential equation and the three representations of traffic flow. Transportation Res. Part B: Methodological 52, 17–30. Laval, J.A., Leclercq, L., 2008. Microscopic modeling of the relaxation phenomenon using a macroscopic lane-changing model. Transportation Res. Part B: Methodological 42 (6), 511–522. Laval, J.A., Toth, C.S., Zhou, Y., 2014. A parsimonious model for the formation of oscillations in car-following models. Transportation Res. Part B: Methodological 70, 228–238. Leclercq, L., Laval, J., Chevallier, E., 2007. The Lagrangian coordinates and what it means for first order traffic flow models. In: 17th International Symposium on Transportation and Traffic Theory (ISTTT17), pp. 735–753. Lee, S., Ngoduy, D., Keyvan-Ekbatani, M., 2019. Integrated deep learning and stochastic car-following model for traffic dynamics on multi-lane freeways. Transportation Res. Part C: Emerging Technol. 106, 360–377. Li, L., Li, X., 2019. Parsimonious trajectory design of connected automated traffic. Transportation Res. Part B: Methodological 119, 1–21. Li, X., Wang, Y., Tian, J., Jiang, R., 2019. Stability analysis of stochastic linear car-following models. Transportation Sci. forthcoming. Lu, G., Nie, Y., Liu, X., Li, D., 2019. Trajectory-based traffic management inside an autonomous vehicle zone. Transportation Res. Part B: Methodological 120, 76–98. Ma, J., Li, X., Zhou, F., Hu, J., Park, B.B., 2016. Parsimonious shooting heuristic for trajectory design of connected automated traffic part II: Computational issues and optimization. Transportation Res. Part B: Methodological. Newell, G.F., 1961. Nonlinear effects in the dynamics of car following. Oper. Res. 9 (2), 209–229. Ngoduy, D., Lee, S., Treiber, M., Keyvan-Ekbatani, M., Vu, H.L., 2019. Langevin method for a continuous stochastic car-following model and its stability conditions. Transportation Res. Part C: Emerging Technol. 105, 599–610. Stern, R.E., Cui, S., Delle Monache, M.L., Bhadani, R., Bunting, M., Churchill, M., Hamilton, N., Haulcy, R.m., Pohlmann, H., Wu, F., Piccoli, B., Seibold, B., Sprinkle, J., Work, D.B., 2018. Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments. Transportation Res. Part C: Emerging Technol. 89, 205–221. Talebpour, A., Mahmassani, H.S., 2016. Influence of connected and autonomous vehicles on traffic flow stability and throughput. Transportation Res. Part C: Emerging Technol. 71, 143–163. Tchrakian, T., Basu, B., 2012. Modeling non-equilibrium traffic dynamics in a Lagrangian framework. Nonlinear Dyn. 67 (3), 1957–1968. Treiber, M., Kesting, A., 2018. The Intelligent Driver Model with stochasticity – New insights into traffic flow oscillations. Transportation Res. Part B: Methodological 117, 613–623. Treiber, M., Kesting, A., 2013. Microscopic calibration and validation of car-following models – a systematic approach. Procedia - Social Behav. Sci. 80, 922–939. Treiber, M., Kesting, A., Helbing, D., 2006. Delays, inaccuracies and anticipation in microscopic traffic models. Phys. A: Stat. Mech. Appl. 360 (1), 71–88. van Wageningen-Kessels, F., Yuan, Y., Hoogendoorn, S.P., van Lint, H., Vuik, K., 2013. Discontinuities in the Lagrangian formulation of the kinematic wave model. Transportation Res. Part C: Emerging Technol. 34, 148–161. Ye, L., Yamamoto, T., 2018. Modeling connected and autonomous vehicles in heterogeneous traffic flow. Phys. A: Stat. Mech. Appl. 490, 269–277. Yu, C., Feng, Y., Liu, H.X., Ma, W., Yang, X., 2018. Integrated optimization of traffic signals and vehicle trajectories at isolated urban intersections. Transportation Res. Part B: Methodological 112, 89–112. Zheng, F., Jabari, S.E., Liu, H.X., Lin, D., 2018. Traffic state estimation using stochastic Lagrangian dynamics. Transportation Res. Part B: Methodological 115, 143–165. Zheng, Z., Ahn, S., Chen, D., Laval, J., 2011. Applications of wavelet transform for analysis of freeway traffic: Bottlenecks, transient traffic, and traffic oscillations. Transportation Res. Part B: Methodological 45 (2), 372–384. Zhou, F., Li, X., Ma, J., 2017. Parsimonious shooting heuristic for trajectory design of connected automated traffic part I: Theoretical analysis with generalized time geography. Transportation Res. Part B: Methodological 95, 394–420. Zhu, W.-X., Zhang, H.M., 2018. Analysis of mixed traffic flow with human-driving and autonomous cars based on car-following model. Phys. A: Stat. Mech. Appl. 496, 274–285.

219