Article Contents
Article Contents

# Stability analysis of microscopic models for traffic flow with lane changing

• *Corresponding author: Matteo Piu
• This paper investigates the mathematical modeling and the stability of multi-lane traffic in the microscopic scale, studying a model based on two interaction terms. To do this we propose simple lane changing conditions and we study the stability of the steady states starting from the model in the one-lane case and extending the results to the generic multi-lane case with the careful design of the lane changing rules. We compare the results with numerical tests, that confirm the predictions of the linear stability analysis and also show that the model is able to reproduce stop & go waves, a typical feature of congested traffic.

Mathematics Subject Classification: Primary: 76A30, 34D20, 70-10.

 Citation:

• Figure 1.  Vehicles in single-lane road

Figure 2.  Red: curve 11 in the $(\alpha_k, V'(h))$ polar coordinate plane. Black: critical curve of model 3

Figure 3.  $V(\cdot)$ function

Figure 4.  $V'(\cdot)$ function in blue, model 4 stability condition in red, model 3 stability condition in black

Figure 5.  On the left: all vehicles trajectories, on the right: velocity of vehicle 1

Figure 6.  On the left: all vehicles trajectories, on the right: velocity of vehicle 1

Figure 7.  On the left: all vehicles trajectories, on the right: velocity of vehicle 1

Figure 8.  On the left: all vehicles trajectories, on the right: velocity of vehicle 1

Figure 9.  On the left: all vehicles trajectories, on the right: velocity of vehicle 1

Figure 10.  On the left: all vehicles trajectories, on the right: velocity of vehicle 1

Figure 11.  Components of the vector $\mathcal{N}_n$, containing information on cell neighbours of the $n$-th vehicle

Figure 12.  Lane change from 1 to 2

Figure 13.  Lane change from 2 to 1

Figure 14.  Optimal velocity functions

Figure 15.  Top: vehicle trajectories in the two lanes. Bottom: number of vehicles versus time

Figure 16.  Top: vehicle trajectories in the two lanes. Bottom: number of vehicles versus time

Figure 17.  Top: vehicle trajectories in the two lanes. Bottom: number of vehicles versus time

Figure 18.  Top: vehicle trajectories in the two lanes. Bottom: number of vehicles versus time

Figure 19.  Desired velocity functions

Figure 20.  Test (a) - Top: vehicle trajectories in the three lanes. Bottom: number of vehicles versus time

Figure 21.  Test (b) - Top: vehicle trajectories in the three lanes. Bottom: number of vehicles versus time

Figure 22.  Top: vehicle trajectories in the three lanes. Bottom: number of vehicles versus time

Table 1.  Thresholds and perturbations

 from lane 1 to lane 2 from lane 2 to lane 1 perturbation $\varepsilon$ in lane 1 (slow lane) $\varepsilon<\dfrac{V_2(\bar{h}_2-d_s)-V_2(\bar{h}_2)}{\left(1+\frac{\gamma}{(\bar{h}_2-d_s)^2}\right) V_1'(\bar{h}_1)}<0$ $\varepsilon>d_s>0$ perturbation $\varepsilon$ in lane 2 (fast lane) $\varepsilon>d_s>0$ $\varepsilon<\dfrac{V_1(\bar{h}_1-d_s)-V_1(\bar{h}_1)}{\left(1+\frac{\gamma}{(\bar{h}_1-d_s)^2}\right) V_2'(\bar{h}_2)}<0$

Table 2.  Thresholds and perturbations

 $1\to 2$ & $3\to 2$ $2 \to 1$ $2 \to 3$ pert. $\varepsilon$ in lane 2 $\varepsilon>d_s>0$ $\varepsilon<\dfrac{V_1(\bar{h}_1-d_s)-V_1(\bar{h}_1)}{\left(1+\frac{\gamma}{(\bar{h}_1-d_s)^2}\right) V_2'(\bar{h}_2)}<0$ $\varepsilon<\dfrac{V_3(\bar{h}_3-d_s)-V_3(\bar{h}_3)}{\left(1+\frac{\gamma}{(\bar{h}_3-d_s)^2}\right) V_2'(\bar{h}_2)}<0$
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