Nonlinear Dynamics

, Volume 85, Issue 4, pp 2705–2717

# A new multi-anticipative car-following model with consideration of the desired following distance

• Jianzhong Chen
• Ronghui Liu
• Dong Ngoduy
• Zhongke Shi
Original Paper

## Abstract

We propose in this paper an extension of the multi-anticipative optimal velocity car-following model to consider explicitly the desired following distance. The model on the following vehicle’s acceleration is formulated as a linear function of the optimal velocity and the desired distance, with reaction-time delay in elements. The linear stability condition of the model is derived. The results demonstrate that the stability of traffic flow is improved by introducing the desired following distance, increasing the time gap in the desired following distance or decreasing the reaction-time delay. The simulation results show that by taking into account the desired following distance as well as the optimal velocity, the multi-anticipative model allows longer reaction-time delay in achieving stable traffic flows.

## Keywords

Multi-anticipative model Desired following distance Stability analysis Traffic flow

## Index

n

Index of vehicle

t

Time instant (s)

m

Number of preceding vehicles considered

## Model variables

$$x_{n}(t)$$

Position of vehicle n at time t (m)

$$\Delta x_{n+j,n}(t)$$

Headway $$x_{n+j}(t)-x_{n}(t)$$ (m) between the vehicle n and the leading vehicle $$n+j$$

$$v_{n}(t)$$

Velocity of vehicle n at time t (m/s)

$$\Delta x_{n}^\mathrm{des}(t)$$

Desired following distance (m)

## Model parameters

$$\alpha$$

Sensitivity coefficient of a driver to the difference between the optimal velocities and the actual velocity (1/s)

$$\beta$$

Sensitivity coefficient of a driver to the distance ($$1/\mathrm {s}^{2}$$)

$$t_\mathrm{d}$$

Reaction-time delay of drivers (s)

$$p_{j}$$

Weight of the optimal velocity function

$$q_{j}$$

Weight of the distance $$\Delta x_{n+j,n}(t-t_{d})/j$$

$$s_{0}$$

Stopping distance, including vehicle length (m)

T

Time gap (s)

a and b

Step function parameters for modelling $$\beta$$

$$s_\mathrm{c}$$

Critical distance (m) in the step function of $$\beta$$

$$V_{1}$$ and $$V_{2}$$

Parameters of the optimal velocity function (m/s)

$$C_{1}$$

Parameter of the optimal velocity function ($$\mathrm {m}^{-1}$$)

$$C_{2}$$

Parameter of the optimal velocity function

$$L_\mathrm{c}$$

Average length of vehicles (m)

## Notes

### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11102165), the Natural Science Basis Research Plan in Shaanxi Province of China (Grant No. 2015JM1013), the Fundamental Research Funds for the Central Universities (Grant No. 3102015ZY050), and the UK Rail Safety and Standards Board (Grant No. T1071). The first author would like to acknowledge the China Scholarship Council for sponsoring his visit to the University of Leeds.

## References

1. 1.
Brackstone, M., McDonald, M.: Car-following: a historical review. Transp. Res. F 2, 181–196 (1999)
2. 2.
Aghabayk, K., Sarvi, M., Young, W.: A state-of-the-art review of car-following models with particular considerations of heavy vehicles. Transp. Rev. 35, 82–105 (2015)
3. 3.
Bonsall, P., Liu, R., Young, W.: Modelling safety-related driving behaviour-impact of parameter values. Transp. Res. A 39, 425–444 (2005)Google Scholar
4. 4.
Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329, 199–329 (2000)
5. 5.
Wang, J., Liu, R., Montgomery, F.O.: A car following model for motorway traffic. Transp. Res. Rec. 1934, 33–42 (2005)
6. 6.
Gazis, D.C., Herman, R., Rothery, R.W.: Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9, 545–567 (1961)
7. 7.
Chandler, R.E., Herman, R., Montroll, E.W.: Traffic dynamics: studies in car following. Oper. Res. 6, 165–184 (1958)
8. 8.
Helly, W.: Simulation of bottlenecks in single lane traffic flow. In: Proceedings of the symposium on theory of traffic flow, pp. 207–238 (1959)Google Scholar
9. 9.
Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y.: Dynamics model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035–1042 (1995)
10. 10.
Helbing, D., Tilch, B.: Generalized force model of traffic dynamics. Phys. Rev. E 58, 133–138 (1998)
11. 11.
Jiang, R., Wu, Q.S., Zhu, Z.J.: Full velocity difference model for a car-following theory. Phys. Rev. E 64, 017101 (2001)
12. 12.
Tang, T.Q., Wang, Y.P., Yang, X.B., Wu, Y.H.: A new car-following model accounting for varying road condition. Nonlinear Dyn. 70, 1397–1405 (2012)
13. 13.
Tang, T.Q., Shi, W.F., Shang, H.Y., Wang, Y.P.: A new car-following model with consideration of inter-vehicle communication. Nonlinear Dyn. 76, 2017–2023 (2014)
14. 14.
Zheng, L.J., Tian, C., Sun, D.H., Liu, W.N.: A new car-following model with consideration of anticipation driving behavior. Nonlinear Dyn. 70, 1205–1211 (2012)
15. 15.
Kang, Y.R., Sun, D.H., Yang, S.H.: A new car-following model considering driver’s individual anticipation behavior. Nonlinear Dyn. 82, 1293–1302 (2015)
16. 16.
Nagatani, T.: Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction. Phys. Rev. E 60, 6395–6401 (1999)
17. 17.
Ge, H.X., Dai, S.Q., Dong, L.Y., Xue, Y.: Stabilization effect of traffic flow in an extended car-following model based on an intelligent transportation system application. Phys. Rev. E 70, 066134 (2004)
18. 18.
Nagatani, T., Nakanishi, K., Emmerich, H.: Phase transition in a difference equation model of traffic flow. J. Phys. A 31, 5431–5438 (1998)
19. 19.
Wilson, R.E., Berg, P., Hooper, S., Lunt, G.: Many-neighbour interaction and non-locality in traffic models. Eur. Phys. J. B 39, 397–408 (2004)
20. 20.
Chen, J.Z., Shi, Z.K., Hu, Y.M.: Stabilization analysis of a multiple look-ahead model with driver reaction delays. Int. J. Mod. Phys. C 23, 1250048 (2012)
21. 21.
Yu, L., Shi, Z.K., Zhou, B.C.: Kink-antikink density wave of an extended car-following model in a cooperative driving system. Commun. Nonlinear Sci. Numer. Simul. 13, 2167–2176 (2008)
22. 22.
Li, Z.P., Liu, Y.C.: Analysis of stability and density waves of traffic flow model in an ITS environment. Eur. Phys. J. B 53, 367–374 (2006)
23. 23.
Jin, Y.F., Xu, M., Gao, Z.Y.: KDV and Kink-antikink solitons in an extended car-following model. J. Comput. Nonlinear Dyn. 6, 011018 (2011)
24. 24.
Xie, D.F., Gao, Z.Y., Zhao, X.M.: Stabilization of traffic flow based on the multiple information of preceding cars. Commun. Comput. Phys. 3, 899–912 (2008)
25. 25.
Peng, G.H., Sun, D.H.: A dynamical model of car-following with the consideration of themultiple information of preceding cars. Phys. Lett. A 374, 1694–1698 (2010)
26. 26.
Ngoduy, D.: Linear stability of a generalized multi-anticipative car following model with time delays. Commun. Nonlinear Sci. Numer. Simul. 22, 420–426 (2015)
27. 27.
Treiber, M., Hennecke, A., Helbing, D.: Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E 62, 1805–1824 (2000)
28. 28.
Li, Y.F., Sun, D.H., Liu, W.N., Zhang, M., Zhao, M., Liao, X.Y., Tang, L.: Modeling and simulation for microscopic traffic flow based on multiple headway, velocity and acceleration difference. Nonlinear Dyn. 66, 15–28 (2011)
29. 29.
Lenz, H., Wagner, C.K., Sollacher, R.: Multi-anticipative car-following model. Eur. Phys. J. B 7, 331–335 (1999)
30. 30.
Treiber, M., Kesting, A., Helbing, D.: Delays, inaccuracies and anticipation in microscopic traffic models. Phys. A 360, 71–88 (2006)
31. 31.
Ossen, S., Hoogendoorn, S.P.: Multi-anticipation and heterogeneity in car-following: empirics and a first exploration of their implications. In: IEEE Intelligent Transportation Systems Conference, pp. 1615–1620 (2006)Google Scholar
32. 32.
Hoogendoorn, S., Ossen, S., Schreuder, M.: Empirics of multi-anticipative car-following behavior. Transp. Res. Rec. 1965, 112–120 (2006)
33. 33.
Farhi, N., Haj-Salem, H., Lebacque, J.P.: Multi-anticipative piecewise-linear car-following model. Transp. Res. Rec. 2315, 100–109 (2012)
34. 34.
Farhi, N.: Piecewise linear car-following modeling. Transp. Res. C 25, 100–112 (2012)
35. 35.
Hu, Y.M., Ma, T.S., Chen, J.Z.: An extended multi-anticipative delay model of traffic flow. Commun. Nonlinear Sci. Numer. Simul. 19, 3128–3135 (2014)
36. 36.
Treiber, M., Kesting, A.: Traffic flow dynamics: data, models and simulation. Springer, Berlin (2013)
37. 37.
Addison, P.S., Low, D.J.: A novel nonlinear car-following model. Chaos 8, 791–799 (1998)
38. 38.
Davis, L.C.: Modifications of the optimal velocity traffic model to include delay due to driver reaction time. Phys. A 319, 557–567 (2003)
39. 39.
Orosz, G., Wilson, R.E., Krauskopf, B.: Global bifurcation investigation of an optimal velocity traffic model with driver reaction time. Phys. Rev. E 70, 026207 (2004)
40. 40.
Orosz, G., Stépán, G.: Subcritical Hopf bifurcations in a car-following model with reaction-time delay. Proc. R. Soc. A 462, 2643–2670 (2006)
41. 41.
Sipahi, R., Niculescu, S.I.: Stability of car following with human memory effects and automatic headway compensation. Philos. Trans. R. Soc. A 368, 4563–4583 (2010)
42. 42.
Kesting, A., Treiber, M.: How reaction time, update time, and adaptation time influence the stability of traffic flow. Comput. Aided Civil Infrastruct. Eng. 23, 125–137 (2008)
43. 43.
Orosz, G., Moehlis, J., Bullo, F.: Robotic reactions: delay-induced patterns in autonomous vehicle systems. Phys. Rev. E 81, 025204(R) (2010)
44. 44.
Ngoduy, D., Tampere, C.M.J.: Macroscopic effects of reaction time on traffic flow characteristics. Phys. Scr. 80, 025802–025809 (2009)
45. 45.
Kang, Y.R., Sun, D.H.: Lattice hydrodynamic traffic flow model with explicit drivers’ physical delay. Nonlinear Dyn. 71, 531–537 (2013)
46. 46.
Ngoduy, D.: Analytical studies on the instabilities of heterogeneous intelligent traffic flow. Commun. Nonlinear Sci. Numer. Simul. 18, 2699–2706 (2013)
47. 47.
Ngoduy, D.: Generalized macroscopic traffic model with time delay. Nonlinear Dyn. 77, 289–296 (2014)
48. 48.
Chen, J.Z., Shi, Z.K., Yu, L., Peng, Z.Y.: Nonlinear analysis of a new extended lattice model with consideration of multi-anticipation and driver reaction delays. J. Comput. Nonlinear Dyn. 9, 031005 (2014)
49. 49.
Davoodi, N., Soheili, A.R., Hashemi, S.M.: A macro-model for traffic flow with consideration of driver’s reaction time and distance. Nonlinear Dyn. 82, 1–8 (2015)
50. 50.
Xing, J.: A parameter identification of a car following model. In: Steps Forward. Intelligent Transport Systems World Congress, pp. 1739–1745 (1995)Google Scholar
51. 51.
Van Winsum, W.: The human element in car following models. Transp. Res. F 2, 207–211 (1999)
52. 52.
Herman, R., Potts, R.B.: Single lane traffic theory and experiment. In: Proceedings of the Symposium on the Theory of Traffic Flow, pp. 120–146 (1961)Google Scholar
53. 53.
Chow, T.S.: Operational analysis of a traffic dynamics problem. Oper. Res. 6, 165–184 (1958)
54. 54.
Liu, R., Li, X.: Stability analysis of a multi-phase car-following model. Phys. A 392, 2660–2671 (2013)
55. 55.
Wilson, R.E., Ward, J.A.: Car-following models: fifty years of linear stability analysis-a mathematical perspective. Transp. Plan. Technol. 34, 3–18 (2011)
56. 56.
Treiber, M., Kanagaraj, V.: Comparing numerical integration schemes for time-continuous car-following models. Phys. A 419, 183–195 (2015)

## Authors and Affiliations

• Jianzhong Chen
• 1
• 2
Email author
• Ronghui Liu
• 2
• Dong Ngoduy
• 2
• Zhongke Shi
• 1
1. 1.College of AutomationNorthwestern Polytechnical UniversityXi’anChina
2. 2.Institute for Transport StudiesUniversity of LeedsLeedsUK