Stability analysis of microscopic models for traffic flow with lane changing

Matteo Piu; Gabriella Puppo

doi:10.3934/nhm.2022006

Article Contents

2022, Volume 17, Issue 4: 495-518. Doi: 10.3934/nhm.2022006

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Stability analysis of microscopic models for traffic flow with lane changing

Matteo Piu^1, , and
Gabriella Puppo^2,

1.
Sapienza - Università di Roma, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Via Scarpa 16, 00161 Roma, Italy
2.
Sapienza - Università di Roma, Dipartimento di Matematica, Piazzale Aldo Moro 5, 00185 Roma, Italy

^*Corresponding author: Matteo Piu
^*Corresponding author: Matteo Piu

Received: September 30, 2021

Revised: January 31, 2022

Early access: March 2022

Published: August 2022

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Abstract

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.

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

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Figure 1. Vehicles in single-lane road

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

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Figure 3. $ V(\cdot) $ function

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

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

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

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

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

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

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

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

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Figure 12. Lane change from 1 to 2

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Figure 13. Lane change from 2 to 1

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Figure 14. Optimal velocity functions

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

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

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

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

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Figure 19. Desired velocity functions

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

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

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

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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 $

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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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