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Analysis of the density wave in a new continuum model L.L. Lai, R.J. Cheng, Z.P. Li, H.X. Ge. Beijing. Shanghai. Los Angeles. Research Motivation. D. Chowdhury et al ： “ The aim of basic research in traffic science is to discover the fundamental laws governing traffic systems.”.
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Analysis of the density wave in a new continuum modelL.L. Lai, R.J. Cheng, Z.P. Li, H.X. Ge
Research Motivation D. Chowdhury et al： “The aim of basic research in traffic science is to discover the fundamental laws governing traffic systems.” There are much nonlinear research about density waves for the microscopic car following model and macroscopic lattice hydrodynamic model, while the corresponding investigation for hydrodynamic model is little comparatively.
Car Following Model • Bando et al. (1995) presented the Optimal Velocity Model with the consideration of relaxation process; • Helbing and Tilch (1998) developed the Generalized Force Model; • Jiang et al. (2001) proposed the Full Velocity Difference Model; • Ge et al. (2008) put forward the Two Velocity Difference Model.
Lighthill and Whitham and Richards (1995) proposed a simple continuum model (LWR Model); • Payne (1971) introduced ahigh-order continuum traffic flow modelincluding a dynamic equation; • A. Aw and M. Rascle (2000) introduced a second order model of traffic flow. Continuum Model
The micro–macro linkage • Jiang et al. put forward an anisotropic macro speed gradient continuum modelbased on the FVDM in 2002; • Zhang obtained another anisotropic continuum model from Pipes car following model in 2003; • Xue established another anisotropic hydrodynamic model by considering two different delay time scales in 2003.
The AD-CF Model In 2012, Zheng et al. proposed a new anticipation driving car-following (AD-CF) model where is the forecast time, representsthe estimation space headway in the next moment.
New Continuum Model Transfer the microscopic variables to the macroscopic ones: where is the relaxation time and represents the propagating time of a distance . ..
Apply the relation between the headway and the density given by Berg et al.:
The new improved continuum model: whereand .
Stability analysis 1. Characteristic Speed and are real numbers, so the new model is strictlyhyperbolic.
2. The stability condition Suppose that traffic flow is initially in a state differing infinitesimally from homogeneous flow: The neutral stability condition:
I. Introduce the coordinate II. The model is expressed by the flow and density. III. Expand the flow through balancing the equation, the parameters are determined. through IV. Near the stability condition, the small perturbations of density evolution can be described by the KdV equation. Nonlinear analysis
(b) (a) Fig.1. Temporal evolution of traffic density for: (a) =0.041veh/m, k=0.2s; (b) =0.046veh/m,k=0.2s. Simulation results
(b) (a) Fig.2. Temporal evolution of traffic density for: (a) =0.054veh/m, k=0.2s; (b) =0.073veh/m,k=0.2s. Simulation results
(b) (a) Simulation results Fig.3. Temporal evolution of traffic density for: (a) =0.073veh/m, k=0s; (b) =0.073veh/m, k=0.1s.
Summary • A new continuum model consideringthe effect of traffic anticipationis proposed based on the AD-CF model; • The stability condition and the KdV equation for the new model are derived; • The numerical simulations are given and the resultsshow that the effect of traffic anticipationcan stabilize the traffic flow effectively.
