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Chinese Physics B
Vol. 29, Issue 8, (2020)

A new car-following model with driver’s anticipation effect of traffic interruption probability

Guang-Han Peng^†

Author Affiliations

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

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DOI: 10.1088/1674-1056/ab9293 Cite this Article

Guang-Han Peng. A new car-following model with driver’s anticipation effect of traffic interruption probability[J]. Chinese Physics B, 2020, 29(8): Copy Citation Text

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Abstract

Traffic interruption phenomena frequently occur with the number of vehicles increasing. To investigate the effect of the traffic interruption probability on traffic flow, a new optimal velocity model is constructed by considering the driver anticipation term in the interruption case for car-following theory. Furthermore, the effect of driver anticipation in the interruption case is investigated via linear stability analysis. Also, the MKdV equation is obtained concerning the effect of driver anticipation in the interruption case. Moreover, numerical simulation states that the driver anticipation term in the interruption case contributes to the stability of traffic flow.

Keywords

interruption probability numerical simulation optimal velocity model traffic flow

\begin{array}{r} \begin{array}{lll} {\dot{x}}_{n} (t + τ) & = & p_{n + 1} V [Δ x_{n} + T_{1} (- {\dot{x}}_{n})] \\ + (1 - p_{n + 1}) V (Δ x_{n} + T_{2} Δ {\dot{x}}_{n}), \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} \frac{d v_{n} (t)}{d t} & = & a [V (Δ x_{n}) - v_{n} (t)] + λ_{1} V^{'} (Δ x_{n}) p_{n + 1} (- v_{n}) \\ + λ_{2} V^{'} (Δ x_{n}) (1 - p_{n + 1}) Δ v_{n} . \end{array} \end{array}

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\begin{array}{r} V (Δ x) = v_{max} / 2 [\tanh (Δ x - h_{c}) - \tanh h_{c}], \end{array}

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\begin{array}{r} \begin{array}{lll} Δ x_{n} (t + 2 τ) \\ = & Δ x_{n} (t + τ) + τ [V (Δ x_{n + 1} (t)) - V (Δ x_{n} (t))] \\ + λ_{1} V^{'} (Δ x_{n}) p_{0} [- Δ x_{n} (t + τ) + Δ x_{n} (t)] \\ + λ_{2} V^{'} (Δ x_{n}) (1 - p_{0}) \\ \times [Δ x_{n + 1} (t + τ) - Δ x_{n + 1} (t) - Δ x_{n} (t + τ) + Δ x_{n} (t)] . \end{array} \end{array}

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\begin{array}{r} x_{n}^{0} (t) = b n + V (b) t w i t h b = L / N . \end{array}

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\begin{array}{r} \begin{array}{lll} Δ y_{n} (t + 2 τ) \\ = & Δ y_{n} (t + τ) + τ V^{'} (b) [Δ y_{n + 1} (t) - Δ y_{n} (t)] \\ + λ_{1} V^{'} (b) p_{0} [- Δ y_{n} (t + τ) + Δ y_{n} (t)] \\ + λ_{2} V^{'} (b) (1 - p_{0}) \\ \times [Δ y_{n + 1} (t + τ) - Δ y_{n + 1} (t) - Δ y_{n} (t + τ) + Δ y_{n} (t)], \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} (e^{z t} - 1) [e^{z t} - λ_{2} (1 - p_{0}) V^{'} (b) (e^{i k} - 1) \\ + λ_{1} p_{0} V^{'} (b)] - τ V^{'} (b) (e^{i k} - 1) = 0. \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} z_{1} = \frac{V^{'} (b)}{1 + λ_{1} V^{'} (b) p_{0}}, \\ z_{2} = \frac{V^{'} (b) + 2 λ_{2} V^{'} (b) (1 - p_{0}) z_{1} - [3 + λ_{1} V^{'} (b) p_{0}] τ z_{1}^{2}}{2 [1 + λ_{1} V^{'} (b) p_{0}]} . \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} a & = & \frac{1}{τ} \\ = & \frac{[3 + λ_{1} V^{'} (b) p_{0}] V^{'} (b)}{{[1 + λ_{1} V^{'} (b) p_{0}]}^{2} + 2 λ_{2} V^{'} (b) (1 - p_{0}) [1 + λ_{1} V^{'} (b) p_{0}]} . \end{array} \end{array}

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\begin{array}{r} a > \frac{[3 + λ_{1} V^{'} (b) p_{0}] V^{'} (b)}{{[1 + λ_{1} V^{'} (b) p_{0}]}^{2} + 2 λ_{2} V^{'} (b) (1 - p_{0}) [1 + λ_{1} V^{'} (b) p_{0}]} . \end{array}

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\begin{array}{r} \begin{array}{lll} X = ε (n + b t), a n d T = ε^{3} t, 0 < ε \leq 1, \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} Δ x_{n} (t) = h_{c} + ε R (X, T) . \end{array} \end{array}

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\begin{array}{r} \begin{array}{lll} ε^{2} [K_{1} b - V^{'} (h_{c})] \partial_{X} R + ε^{3} G_{1} \partial_{X}^{2} R \\ + ε^{4} (K_{1} \partial_{T} R + G_{2} \partial_{X}^{3} R - G_{3} \partial_{X} R^{3}) \\ + ε^{5} {[3 b τ + λ_{1} V^{'} (h_{c}) p_{0} b τ - H] \partial_{T} \partial_{X} R \\ + G_{4} \partial_{X}^{4} R - G_{5} \partial_{X}^{2} R^{3}} = 0. \end{array} \end{array}

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\begin{array}{r} \begin{array}{l} ε^{4} (\partial_{T} R - W_{1} \partial_{X}^{3} R + W_{2} \partial_{X} R^{3}) \\ + ε^{5} (W_{3} \partial_{X}^{2} R + W_{4} \partial_{X}^{4} R + W_{5} \partial_{X}^{2} R^{3}) = 0. \end{array} \end{array}

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\begin{array}{r} T^{'} = W_{1} T, R = \sqrt{\frac{W_{1}}{W_{2}}} R^{'} . \end{array}

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\begin{array}{r} \partial_{T^{'}} R^{'} - \partial_{X}^{3} R^{'} + \partial_{X} {R^{'}}^{3} + ε M [R^{'}] = 0. \end{array}

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\begin{array}{r} M [R^{'}] = \sqrt{\frac{1}{W_{1}}} [W_{3} \partial_{X}^{2} R^{'} + \frac{W_{1} W_{5}}{W_{2}} \partial_{X}^{2} {R^{'}}^{3} + W_{4} \partial_{X}^{4} R^{'}] . \end{array}

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\begin{array}{r} {R^{'}}_{0} (X, T^{'}) = \sqrt{c} \tanh \sqrt{c / 2} (X - c T^{'}) . \end{array}

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\begin{array}{r} ({R^{'}}_{0}, M [R^{'}]) = \int_{- \infty}^{+ \infty} d X {R^{'}}_{0} (X, T^{'}) M [{R^{'}}_{0} (X, T^{'})] . \end{array}

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\begin{array}{r} c = 5 W_{2} W_{3} / (2 W_{2} W_{5} - 3 W_{1} W_{4}) . \end{array}

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\begin{array}{r} \begin{array}{lll} Δ x_{n} (t) & = & h_{c} + \sqrt{\frac{W_{1} c}{W_{2}} (\frac{τ}{τ_{c}} - 1)} \tanh \sqrt{\frac{c}{2} (\frac{τ}{τ_{c}} - 1)} \\ \times {j + [1 - c W_{1} (\frac{τ}{τ_{c}} - 1)] t} . \end{array} \end{array}

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\begin{array}{r} A = \sqrt{\frac{W_{1} c}{W_{2}} (\frac{τ}{τ_{c}} - 1)} . \end{array}

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\begin{array}{r} {\begin{cases} Δ x_{n} (0) = Δ x_{n} (1) = 4, n \neq N / 2, n = N / 2 + 1, \\ Δ x_{n} (0) = Δ x_{n} (1) = 4 + δ, n \neq N / 2, \\ Δ x_{n} (0) = Δ x_{n} (1) = 4 - δ, n = N / 2 + 1. \end{cases} \end{array}

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Guang-Han Peng. A new car-following model with driver’s anticipation effect of traffic interruption probability[J]. Chinese Physics B, 2020, 29(8):

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