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Research Papers

A Novel Lattice Model on a Gradient Road With the Consideration of Relative Current

[+] Author and Article Information
Jin-Liang Cao

College of Automation,
Northwestern Polytechnical University,
Xi'an, Shaanxi 710072, China
e-mail: cjl1210@163.com

Zhong-Ke Shi

College of Automation,
Northwestern Polytechnical University,
Xi'an, Shaanxi 710072, China

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 6, 2014; final manuscript received January 28, 2015; published online April 16, 2015. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 10(6), 061018 (Nov 01, 2015) (10 pages) Paper No: CND-14-1239; doi: 10.1115/1.4029701 History: Received October 06, 2014; Revised January 28, 2015; Online April 16, 2015

In this paper, a novel lattice model on a single-lane gradient road is proposed with the consideration of relative current. The stability condition is obtained by using linear stability theory. It is shown that the stability of traffic flow on the gradient road varies with the slope and the sensitivity of response to the relative current: when the slope is constant, the stable region increases with the increasing of the sensitivity of response to the relative current; when the sensitivity of response to the relative current is constant, the stable region increases with the increasing of the slope in uphill and decreases with the increasing of the slope in downhill. A series of numerical simulations show a good agreement with the analytical result and show that the sensitivity of response to the relative current is better than the slope in stabilizing traffic flow and suppressing traffic congestion. By using nonlinear analysis, the Burgers, Korteweg–de Vries (KdV), and modified Korteweg–de Vries (mKdV) equations are derived to describe the triangular shock waves, soliton waves, and kink–antikink waves in the stable, metastable, and unstable region, respectively, which can explain the phase transitions from free traffic to stop-and-go traffic, and finally to congested traffic. One conclusion is drawn that the traffic congestion on the gradient road can be suppressed efficiently by introducing the relative velocity.

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Figures

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Fig. 1

Illustration of vehicles running on the gradient road: uphill and downhill

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Fig. 2

The maximal flux with different θ, “+,” and “–” represent uphill and downhill, respectively

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Fig. 3

Phase diagram in the parameter space (ρ,a) for model A and model B, as λ = 0, θ = 0. (a) Model A and (b) model B.

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Fig. 4

Phase diagram in the parameter space (ρ,a) for model B with different θ, as λ = 0.1. (a) Uphill and (b) downhill.

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Fig. 5

Phase diagram in the parameter space (ρ,a) for model B with different λ, as θ = 0

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Fig. 6

Space–time evolution of density waves after t = 10,000 s with different λ, as θ = 3 deg. (a) λ = 0.0, (b) λ = 0.1, (c) λ = 0.2, and (d) λ = 0.3.

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Fig. 7

Space–time evolution of density waves after t = 10,000 s with different θ, as λ = 0.1. (a) θ = −6 deg, (b) θ = −3 deg, (c) θ = 0 deg, and (d) θ = 6 deg.

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Fig. 8

Snapshot of density waves at t = 10,150 s with different λ and θ. (a) θ = 3 deg and (b) λ = 0.1.

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Fig. 9

Limit cycles in the density–velocity plane with different λ and θ. (a) θ = 0 deg and (b) λ = 0.0.

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