The delayed-time effect of traffic flux on traffic stability for two-lane freeway

Journal Pre-proof The delayed-time effect of traffic flux on traffic stability for two-lane freeway HongZhuan Zhao, Dongxue Xia, ShuhongYang, Guanghan Peng

PII: DOI: Reference:

S0378-4371(19)31732-7 https://doi.org/10.1016/j.physa.2019.123066 PHYSA 123066

To appear in:

Physica A

Received date : 21 September 2019 Please cite this article as: H. Zhao, D. Xia, ShuhongYang et al., The delayed-time effect of traffic flux on traffic stability for two-lane freeway, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123066. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Highlights (for review)

Journal Pre-proof Research Highlights 1. A new lattice model is presented with the delayed-time effect for two-lane highway. 2. The neutral stability condition is obtained with the delayed-time effect on two lanes. 3. The early time effect is simulated to show the good results.

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4.The long-time effect is investigated to imply the positive effect of the delayed-time effect.

*Manuscript Click here to view linked References

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The delayed-time effect of traffic flux on traffic stability for two-lane freeway HongZhuan Zhaoa, Dongxue Xiab*, ShuhongYangb, Guanghan Pengc a

College of Architecture and Transportation Engineering, Guilin University Of Electronic Technology, Guilin, 541004

School of Computer Science and Communication Engineering, Guangxi University of Science and Technology, Liuzhou, 545006, China c

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College of Physical Science and Technology, Guangxi Normal University, Guilin 541004, China

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Abstract: The delayed-time effect may contribute to improving traffic flow. Then, a new lattice model in this paper is constructed with the consideration of the delayed-time effect of traffic flux for two-lane freeway. The linear stability condition can be acquired by applying linear theory. Moreover, numerical simulation shows that the delayed-time effect of traffic flux contributes to the stability of the traffic flow in the lattice model of two-lane traffic flow. Keywords: traffic flow, lattice hydrodynamic model, numerical simulation, nonlinear analysis

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PACS: 05.70.Fh, 05.70.Jk, 64.60.Fr 1. Introduction

With the rapid growth of vehicle ownership, traffic congestion has increasingly become one of the important bottlenecks in urban development. In order to improve the traffic environment, various types of traffic flow models [1-6] emerge as the times require. Especially in recent years, lattice model firstly proposed by Nagatani [7,8] has attracted more and more attention of scholars. Subsequently, some extended lattice models have been proposed by considering different traffic factors such as flow difference [9], optimal current difference [10], driver’s anticipation and interruption probability with passing [11,12], self-anticipative density and self-stabilization effect [14, 15], delayed-feedback flux. To investigate the lane changing behaviors, Nagatani [16] proposed a lattice model of two-lane traffic flow.

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Subsequently, some two-lane lattice models have been developed by considering various factors such as driver’s lane-changing aggressiveness [17], global average flux[18], average density difference [19], timid and aggressive behavior[20] and density difference [21]. What is more important, the history information of traffic flow has been

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investigated in previous lattice models on single lane [22-24]. Recently, Very recently, Redhu et al. [25] proposed a delayed-feedback control method in lattice model by considering the delayed-time flux of leading lattice on single lane. However, the delayed-time effect of traffic flux from the leading lattice’s historic flux has not been studied in case of lane change in the existing lattice models of two-lane traffic flow. Therefore, to explore the influence of the delayed-time effect of traffic flux on two lanes, a new two-lane lattice model will be proposed with the delayed-time

2. Modeling

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effect of traffic flux term in the following section.

The schematic diagram of two-lane traffic flow was described by Fig. 1 [16]. The lane changing rate is

  02V (  0 ) (  2, j 1  1, j )

 2, j 1  1, j . Here 

as

 2, j 1  1, j

. The lane changing rate is

  02V (  0 ) ( 1, j   2, j 1 )

as

is the rate constant coefficient with dimensionless. Therefore, Nagatani [16] put forward the

*Corresponding author e-mail: [email protected] (D.X. Xia)

Journal Pre-proof continuity equations as follows:

 t 1, j   0 ( 1, j v1, j  1, j 1v1, j 1 )    02V (  0 ) (  2, j 1  2 1, j   2, j 1 )

(1)

 t  2, j   0 (  2, j v2, j   2, j 1v2, j 1 )    02V (  0 ) ( 1, j 1  2  2, j  1, j 1 )

(2)

By adding Eqs. (1) and (2), we can obtain the continuity equation as below:

 t  j   0 (  j v j   j 1v j 1 )    02V (  0 ) (  j 1  2  j   j 1 )

 j v j  ( 1, j v1, j   2, j v2, j ) / 2 . In addition, the evolution equation [16] was including as

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where  j  ( 1, j   2, j ) / 2 ,

 t (  j v j )  a[  0V (  j 1 )   j v j ]

(4)

a  1  is driver’s sensitivity. V (  ) means the optimal velocity function [16] as below:

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where

(3)

V (  ) (vmax / 2)[tanh( 2 / 0   / 02  1 / c )  tanh( 1 / c )]

 c is the safety density. But Nagatani [16] did not consider the delayed-time effect of traffic flux. Accordingly,

we present a new evolution equation as below:

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where

(5)

 t (  j v j )  a[ 0V (  j 1 )  j v j ]  a[  j 1v j 1   j 1 (t   )v j 1 (t   )] where [  j 1v j 1   j 1 (t   )v j 1 (t   )] means the delayed-time effect of traffic flux.

(6)

 shows

the reaction

coefficient. Through eliminating the velocity in Eqs. (3) and (6), the density evolution is educed as below:

 j (t  2 )   j (t   )  02[V (  j 1 )  V (  j )]  [  j 1 (t   )  2  j 1   j 1 (t   )]   02V ( 0 ) [  j 1 (t   )   j 1  2  j (t   )   j 1 (t   )]  0

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3. Linear stability analysis

Assume a small deviation y j from the steady-state flow

(7)

 0 on site j as below:

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 j (t )  0  y j (t )

(8)

By linearizing Eq. (7), we get

y j (t  2 )  y j (t   )  02V ( 0 )( y j 1  y j )  [ y j 1 (t   )  2 y j 1  y j 1 (t   )]   02V ( 0 ) [ y j 1 (t   )  2 y j (t   )  y j 1 (t   )]  0 . Making y j  A exp( ikj  zt ) and expanding Eq. (9), we deduce

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Here V (  0 )  dV (  ) / d

  0

e2 z  e z  o2V ( 0 )(eik  1)  eik (e z  2  e z )   02V ( 0 ) e z (eik  2  eik )  0

Let

(9)

(10)

z  z1 (ik )  z2 (ik )2  . Therefore, we obtain z1    o2V (  0 )

(11)

Journal Pre-proof 1  2 3o2V ( 0 ) z2  [   o2V ( 0 )]o2V ( 0 ) 2 2

(12)

z2  0 . However, the uniform steady-state flow falls unstable when z2  0 . Hence

Then, traffic flow keeps stable as

we obtain the neutral stability condition as below:

1  2 (3  2 )  o2V ( 0 )

(13)

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 

 

1  2 (3  2 )  o2V ( 0 )

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Then, the stable conditions of the corresponding two lanes are obtained as

It is clear that the stable conditions returns to the stable condition of Nagatani’s model [16] when can plot the neutral stability lines under lane changing at

(14)

 =0. Therefore, we

  0.1 as shown in Fig. 2. From Fig. 2, we find that the

neutral stability lines fall down under lane changing with the increase of the reaction coefficient  . As a conclusion, the stable region has expanded with the reaction intensity



increasing under lane changing, which means that the

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delayed-time effect of traffic flux can improve traffic stability effectively on two lanes. 4. Numerical simulation

In this section, we mainly investigate the delayed-time effect under lane changing. Periodic boundary is used in simulation process. We chose initial parameters as: N=140,

c  0.25 , vmax  2 , a  2 , 0  0.25 ,   0.1 .

The initial lattice density Nos. 50 to 55 is taken as 0.5 and Nos. 56 to 60 is assumed as 0.2. And we show the evolution of density at different sites j =2; 25; 55 and 80 near the initial perturbation in following content. 4.1. Early time effect

0.4, respectively.

  0 in

  0 , 0.2, and

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The density evolution can be seen in Fig. 3 for sites Nos. 2, 25, 55 and 80 during t =1-300 s at

Fig.3(a) represents no delayed-time effect of traffic flux. Obviously, the kink-antikink

type of density wave and the oscillation occur on two lanes in Fig. 3(a). Furthermore, the fluctuation becomes smaller with the reaction coefficient  increasing from Fig. 3, which shows that the delayed-time effect of traffic flux can

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inhibit the traffic jams and quickly fall free flow. Moreover, the phase space plot (  (t ) -  (t - 1) ,  (t ) ) is described from Fig. 4 to Fig. 7 between t =1-500 s at sites-2, 25, 55 and 80 for

  0 and 0.2, respectively. The region of

dispersed points can reflect the density fluctuation of traffic flow. According to Figs. 4- 7, the space of phase plot shrinks when the delayed-time effect of traffic flux is considered. That is to say, the delayed-time effect of traffic flux improves the traffic congestion effectively even in early time stage.

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4.2. Long-time (steady-state) effect

Furthermore, the delayed-time effect of traffic flux will be investigated after a sufficiently long time namely at 10,000 s via simulation. Fig. 8 represents the density-time plot between time 10,000–10,300 atγ= 0.1 for

  0 , 0.2

and 0.4, respectively. It is obvious that the kink-antikink type of density waves occur under the initial perturbation for any position of lattices in view of Fig.8 (a) in two-lane system. But the amplitude of density waves shrinks as the reaction coefficient

 increases under lane changing from Fig.8(a) to Fig. 8(c). Especially, the amplitude of the

density wave retains constant after long time on two lanes since the steady condition is met from Fig.8(c). According to above simulations, it can be seen that the delayed-time effect of traffic flux strength the stability of the traffic flow. 6. Conclusion

Journal Pre-proof The delayed-time effect of traffic flux plays an important role on traffic stability on two-lane highway. To reveal the delayed-time effect, the linear stability condition is deduced in two-lane lattice model, which shows that the delayed-time effect can increase the stable region of space. Moreover, numerical simulation shows that the traffic stability is improved by considering the delayed-time effect for two-lane freeway under lane changing. And the traffic stability is continuously strengthened with the increase of the delayed-time effect in two-lane system, which implies that the delayed-time effect contributes to restrain traffic jam in two-lane lattice model. Acknowledgments and

61803113),

Guangxi

Natural

Science

Foundation,

China

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This work was supported by the National Natural Science Foundation of China (Grant Nos. 61963008, 61673168, (Grant

No.

2018GXNSFAA281274,

2018GXNSFAA138036, and 2018GXNSFAA050020), the Cooperative Education Program of Ministry of Education (201702117006), the Guangxi Key Laboratory of Trusted Software (kx201713), the Guangxi Science & Technology

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Base and Talent Special Project(GuikeAD18281015), the China Postdoctoral Science Foundation (2019M653313) and Doctor Scientific Research Startup Project Foundation of Guangxi Normal University, China (Grant no. 2018BQ007). References

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[3] H. Liu, D.H. Sun, M. Zhao, Nonlinear Dynamics 84 (2016) 881.

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[21] J.F. Tian, Z.Z. Yuan, B. Jia, M.H. Li, G.J. Jiang, Physica A 391 (2012) 4476. [22] J. Zhou, Z.K. Shi, H.L. Zhang, C.P. Wang, International Journal of Modern Physics C 28 (2017) 1750086. [23] H.Z. Zhao, G. Zhang,W.Y Li, T.L. Gu, D. Zhou, Physica A 503 (2018) 1204. [24] Y.C. He, G. Zhang, D. Chen, International Journal of Modern Physics B 33 (2019) 1950071. [25] P. Redhu, A.K. Gupta,. Communications in Nonlinear Science and Numerical Simulation 27 (2015) 263.

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Fig.1 The schematic model of traffic flow on a two-lane highway

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Fig. 2 Phase diagram in parameter space (  ; a) atγ= 0.1 for different





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Fig.3 Temporal density behavior of four sites between time 1–300 s at γ= 0.1 for different

Fig. 4 Scatter plot at site-2 between time t = 1–500 s atγ= 0.1 for (a)

 =0, and (b)  =0.2, respectively

 =0, and (b)  =0.2, respectively

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Fig. 5 Scatter plot at site-25 between time t = 1–500 s atγ= 0.1 for (a)

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 =0, and (b)  =0.2, respectively

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Fig. 6 Scatter plot at site-55 between time t = 1–500 s atγ= 0.1 for (a)

 =0, and (b)  =0.2, respectively

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Fig. 7 Scatter plot at site-80 between time t = 1–500 s atγ= 0.1 for (a)

Fig. 8 Density–time plot between time t = 10,000–10,300 atγ= 0.1for different values of

 , respectively