Nonlinear Dynamics

, Volume 85, Issue 4, pp 2705–2717 | Cite as

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

  • Jianzhong ChenEmail author
  • Ronghui Liu
  • Dong Ngoduy
  • Zhongke Shi
Original Paper


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.


Multi-anticipative model Desired following distance Stability analysis Traffic flow 

List of symbols



Index of vehicle


Time instant (s)


Number of preceding vehicles considered

Model variables


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 \(n+j\)


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 (\(1/\mathrm {s}^{2}\))


Reaction-time delay of drivers (s)


Weight of the optimal velocity function


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


Stopping distance, including vehicle length (m)


Time gap (s)

a and b

Step function parameters for modelling \(\beta \)


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

\(V_{1}\) and \(V_{2}\)

Parameters of the optimal velocity function (m/s)


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


Parameter of the optimal velocity function


Average length of vehicles (m)



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.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

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

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