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A two-dimensional data-driven model for traffic flow on highways

  • * Corresponding author: Giuseppe Visconti

    * Corresponding author: Giuseppe Visconti
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  • Based on experimental traffic data obtained from German and US highways, we propose a novel two-dimensional first-order macroscopic traffic flow model. The goal is to reproduce a detailed description of traffic dynamics for the real road geometry. In our approach both the dynamics along the road and across the lanes is continuous. The closure relations, being necessary to complete the hydrodynamics equation, are obtained by regression on fundamental diagram data. Comparison with prediction of one-dimensional models shows the improvement in performance of the novel model.

    Mathematics Subject Classification: Primary: 90B20, 35L65; Secondary: 35Q91, 91B74.


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  • Figure 1.  Experimental diagrams from the A3 German highway using $20$ minutes of recorded video. Top row: flux-density (left) and speed-density (right) diagrams in $x$-direction. Bottom row: flux-density (left) and speed-density (right) diagrams in $y$-direction

    Figure 2.  Two-dimensional trajectories extrapolated from the German data-set. In red we show the trajectories of vehicles crossing a lane while traveling

    Figure 3.  Fundamental diagrams in $x$ direction. Left: macroscopic data computed each $0.2$ seconds and aggregated over $60$ seconds. Right: macroscopic data computed each $1$ second and aggregated over $30$ seconds

    Figure 4.  Data-fitting of the experimental diagrams. Top row: functions approximating the flux-density (left) and speed-density (right) diagrams in $x$-direction. Bottom row: functions approximating flux-density (left) and speed-density (right) diagrams in $y$-direction

    Figure 5.  Log-log graph of the convergence test on the Gaussian initial datum (left panel) and on the double sine wave (right panel)

    Figure 6.  Top-left: initial density profile at time $T_{\text{init}} = 407.4$ seconds. Top-right: simulated density profile after $0.5$ seconds. Bottom-left: real density profile at final time. Bottom-right: difference between the simulated and the real density profiles

    Figure 7.  Top-left: initial density profile at time $T_{\text{init}} = 870.9$ seconds. Top-right: simulated density profile after $0.5$ seconds. Bottom-left: real density profile at final time. Bottom-right: difference between the simulated and the real density profiles

    Figure 8.  Left: $15$ seconds of simulation between $400-415$ seconds, showing the $1$-norm error each $0.5$ seconds (the top panel). Right: $15$ seconds of simulation between $863-878$ seconds, showing the $1$-norm error each $0.5$ seconds (the top panel). The mid and the bottom panels show the variation of density in the time intervals and on the whole recorded time period, respectively

    Figure 9.  Error plots comparing the predictive accuracy of the 1D model (2) (red data) and of the 2D model (3) (blue data). Each panel refers to different initial density profiles computed by using the kernel density estimation approach. On the $x$-axis we show the percentage of the traveling time with respect to the total time to cover the road section at the maximum speed

    Figure 10.  Error plots comparing the predictive prediction of traveling time of the 1D model (2) (red data) and of the 2D model (3) (blue data). On the $x$-axis we show the percentage of the traveling time with respect to the total time to cover the road section at the maximum speed

    Figure 11.  Relative error between $\boldsymbol{E}_{\boldsymbol{\alpha}^{\boldsymbol{y}}, \boldsymbol{p}^{\boldsymbol{y}}}$ and $E_{\alpha^y_{\text{opt}}, p^y_{\text{opt}}}(T_{\text{fin}})$ for each pair of possible parameters in the vectors $\boldsymbol{\alpha}^{\boldsymbol{y}}$ and $\boldsymbol{p}^{\boldsymbol{y}}$

    Figure 12.  Experimental diagrams in $y$-direction resulting from the US101 highway and using $15$ minutes of recorded data (07:50 - 08:05 a.m.). The macroscopic quantities are obtained by aggregating each $100$ meter sections and every $1$ second. Left panel: flux-density diagram. Right: speed-density diagram

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