doi: 10.3934/nhm.2022006
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Stability analysis of microscopic models for traffic flow with lane changing

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

Received  October 2021 Revised  February 2022 Early access March 2022

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.

Citation: Matteo Piu, Gabriella Puppo. Stability analysis of microscopic models for traffic flow with lane changing. Networks and Heterogeneous Media, doi: 10.3934/nhm.2022006
References:
[1]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.

[2]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Structure stability of congestion in traffic dynamics, Japan J. Indust. Appl. Math., 11 (1994), 203-223.  doi: 10.1007/BF03167222.

[3]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical Review E, 51 (1995), 1035-1042. 

[4]

M. Brackstone and M. McDonald, Car-following: A historical review, Transportation Research Part F: Traffic Psychology and Behaviour, 2 (1999), 181-196.  doi: 10.1016/S1369-8478(00)00005-X.

[5]

M. Cassidy and J. Rudjanakanoknad, Increasing the capacity of an isolated merge by metering its on-ramp, Transportation Research Part B: Methodological, 39 (2005), 896-913.  doi: 10.1016/j.trb.2004.12.001.

[6]

R. E. ChandlerR. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Res., 6 (1958), 165-184.  doi: 10.1287/opre.6.2.165.

[7]

M. ErrampalliM. Okushima and T. Akiyama, Fuzzy logic based lane change model for microscopic traffic flow simulation, Journal of Advanced Computational Intelligence and Intelligent Informatics, 12 (2008), 172-181.  doi: 10.20965/jaciii.2008.p0172.

[8]

P. Goatin and E. Rossi, A multilane macroscopic traffic flow model for simple networks, SIAM J. Appl. Math., 79 (2019), 1967-1989.  doi: 10.1137/19M1254386.

[9]

X. GongB. Piccoli and G. Visconti, Mean-field of optimal control problems for hybrid model of multilane traffic, American Control Conference (ACC), 5 (2021), 1964-1969.  doi: 10.23919/ACC50511.2021.9482648.

[10]

R. Haberman, Mathematical Models, vol. 21 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998, Mechanical vibrations, population dynamics, and traffic flow, An introduction to applied mathematics, Reprint of the 1977 original.

[11]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141.  doi: 10.1103/RevModPhys.73.1067.

[12]

D. Helbing and M. Moussaïd., Analytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis of a simple traffic flow model, The European Physical Journal B - Condensed Matter and Complex Systems, 69 (2008), 571-581.  doi: 10.1140/epjb/e2009-00042-6.

[13]

D. Helbing and B. Tilch, Generalized force model of traffic dynamics, Phys. Rev. E, 58 (1998), 133-138.  doi: 10.1103/PhysRevE.58.133.

[14]

R. Herman and R. B. Potts, Single-lane traffic theory and experiment, In Theory of Traffic Flow, Elsevier, Amsterdam, (1961), 120–146.

[15]

M. HertyS. Moutari and G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.  doi: 10.1137/17M1151821.

[16]

N. Hodas and A. Jagota, Microscopic modeling of multi-lane highway traffic flow, American Journal of Physics, 71 (2003), 1247-1256.  doi: 10.1119/1.1613269.

[17]

R. Jiang, Q. Wu and Z. Zhu, Full velocity difference model for a car-following theory, Phys. Rev. E, 64 (2001), 017101/1–017101/4. doi: 10.1103/PhysRevE.64.017101.

[18]

E. Kallo, F. A and O. M. Lamberty S., Microscopic traffic data obtained from videos recorded on a german motorway, Mendeley Data, v1.

[19]

A. KestingM. Treiber and D. Helbing, General lane-changing model mobil for car-following models, Transportation Research Record, 1999 (2007), 86-94.  doi: 10.3141/1999-10.

[20]

A. Klar and R. Wegener, Hierarchy of models for multilane vehicular traffic i: Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001.  doi: 10.1137/S0036139997326946.

[21]

A. Klar and R. Wegener, Hierarchy of models for multilane vehicular traffic ii: Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011.  doi: 10.1137/S0036139997326958.

[22]

W. LvW.-G. Song and Z.-M. Fang, Three-lane changing behaviour simulation using a modified optimal velocity model, Physica A: Statistical Mechanics and its Applications, 390 (2011), 2303-2314.  doi: 10.1016/j.physa.2011.02.035.

[23]

W. LvW.-G. SongX.-D. Liu and J. Ma, A microscopic lane changing process model for multilane traffic, Phys. A, 392 (2013), 1142-1152.  doi: 10.1016/j.physa.2012.11.012.

[24]

G. PengX. CaiC. LiuB. Cao and M. Tuo, Optimal velocity difference model for a car-following theory, Physics Letters, Section A: General, Atomic and Solid State Physics, 375 (2011), 3973-3977.  doi: 10.1016/j.physleta.2011.09.037.

[25]

L. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281.  doi: 10.1063/1.1721265.

[26]

A. Reuschel, Vehicle movements in a platoon, Oesterreichisches Ingenieur-Archir, 4 (1950), 193-215. 

[27]

A. Reuschel, Vehicle movements in a platoon with uniform acceleration or deceleration of the lead vehicle, Zeitschrift des Oesterreichischen Ingenieur und Architekten-Vereines, 95 (1950), 50-62. 

[28]

J. Song and S. Karni, A second order traffic flow model with lane changing, J. Sci. Comput., 81 (2019), 1429-1445.  doi: 10.1007/s10915-019-01023-z.

[29]

T. Toledo, Driving behaviour: Models and challenges, Transport Reviews, 27 (2007), 65-84.  doi: 10.1080/01441640600823940.

[30]

Z. ZhengS. AhnD. Chen and J. Laval, Freeway traffic oscillations: Microscopic analysis of formations and propagations using wavelet transform, Transportation Research Part B: Methodological, 45 (2011), 1378-1388. 

[31]

Z. ZhengS. AhnD. Chen and J. Laval, The effects of lane-changing on the immediate follower: Anticipation, relaxation, and change in driver characteristics, Transportation Research Part C: Emerging Technologies, 26 (2013), 367-379.  doi: 10.1016/j.trc.2012.10.007.

show all references

References:
[1]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.

[2]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Structure stability of congestion in traffic dynamics, Japan J. Indust. Appl. Math., 11 (1994), 203-223.  doi: 10.1007/BF03167222.

[3]

M. BandoK. HasebeA. NakayamaA. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical Review E, 51 (1995), 1035-1042. 

[4]

M. Brackstone and M. McDonald, Car-following: A historical review, Transportation Research Part F: Traffic Psychology and Behaviour, 2 (1999), 181-196.  doi: 10.1016/S1369-8478(00)00005-X.

[5]

M. Cassidy and J. Rudjanakanoknad, Increasing the capacity of an isolated merge by metering its on-ramp, Transportation Research Part B: Methodological, 39 (2005), 896-913.  doi: 10.1016/j.trb.2004.12.001.

[6]

R. E. ChandlerR. Herman and E. W. Montroll, Traffic dynamics: Studies in car following, Operations Res., 6 (1958), 165-184.  doi: 10.1287/opre.6.2.165.

[7]

M. ErrampalliM. Okushima and T. Akiyama, Fuzzy logic based lane change model for microscopic traffic flow simulation, Journal of Advanced Computational Intelligence and Intelligent Informatics, 12 (2008), 172-181.  doi: 10.20965/jaciii.2008.p0172.

[8]

P. Goatin and E. Rossi, A multilane macroscopic traffic flow model for simple networks, SIAM J. Appl. Math., 79 (2019), 1967-1989.  doi: 10.1137/19M1254386.

[9]

X. GongB. Piccoli and G. Visconti, Mean-field of optimal control problems for hybrid model of multilane traffic, American Control Conference (ACC), 5 (2021), 1964-1969.  doi: 10.23919/ACC50511.2021.9482648.

[10]

R. Haberman, Mathematical Models, vol. 21 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998, Mechanical vibrations, population dynamics, and traffic flow, An introduction to applied mathematics, Reprint of the 1977 original.

[11]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Modern Phys., 73 (2001), 1067-1141.  doi: 10.1103/RevModPhys.73.1067.

[12]

D. Helbing and M. Moussaïd., Analytical calculation of critical perturbation amplitudes and critical densities by non-linear stability analysis of a simple traffic flow model, The European Physical Journal B - Condensed Matter and Complex Systems, 69 (2008), 571-581.  doi: 10.1140/epjb/e2009-00042-6.

[13]

D. Helbing and B. Tilch, Generalized force model of traffic dynamics, Phys. Rev. E, 58 (1998), 133-138.  doi: 10.1103/PhysRevE.58.133.

[14]

R. Herman and R. B. Potts, Single-lane traffic theory and experiment, In Theory of Traffic Flow, Elsevier, Amsterdam, (1961), 120–146.

[15]

M. HertyS. Moutari and G. Visconti, Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.  doi: 10.1137/17M1151821.

[16]

N. Hodas and A. Jagota, Microscopic modeling of multi-lane highway traffic flow, American Journal of Physics, 71 (2003), 1247-1256.  doi: 10.1119/1.1613269.

[17]

R. Jiang, Q. Wu and Z. Zhu, Full velocity difference model for a car-following theory, Phys. Rev. E, 64 (2001), 017101/1–017101/4. doi: 10.1103/PhysRevE.64.017101.

[18]

E. Kallo, F. A and O. M. Lamberty S., Microscopic traffic data obtained from videos recorded on a german motorway, Mendeley Data, v1.

[19]

A. KestingM. Treiber and D. Helbing, General lane-changing model mobil for car-following models, Transportation Research Record, 1999 (2007), 86-94.  doi: 10.3141/1999-10.

[20]

A. Klar and R. Wegener, Hierarchy of models for multilane vehicular traffic i: Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001.  doi: 10.1137/S0036139997326946.

[21]

A. Klar and R. Wegener, Hierarchy of models for multilane vehicular traffic ii: Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011.  doi: 10.1137/S0036139997326958.

[22]

W. LvW.-G. Song and Z.-M. Fang, Three-lane changing behaviour simulation using a modified optimal velocity model, Physica A: Statistical Mechanics and its Applications, 390 (2011), 2303-2314.  doi: 10.1016/j.physa.2011.02.035.

[23]

W. LvW.-G. SongX.-D. Liu and J. Ma, A microscopic lane changing process model for multilane traffic, Phys. A, 392 (2013), 1142-1152.  doi: 10.1016/j.physa.2012.11.012.

[24]

G. PengX. CaiC. LiuB. Cao and M. Tuo, Optimal velocity difference model for a car-following theory, Physics Letters, Section A: General, Atomic and Solid State Physics, 375 (2011), 3973-3977.  doi: 10.1016/j.physleta.2011.09.037.

[25]

L. Pipes, An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281.  doi: 10.1063/1.1721265.

[26]

A. Reuschel, Vehicle movements in a platoon, Oesterreichisches Ingenieur-Archir, 4 (1950), 193-215. 

[27]

A. Reuschel, Vehicle movements in a platoon with uniform acceleration or deceleration of the lead vehicle, Zeitschrift des Oesterreichischen Ingenieur und Architekten-Vereines, 95 (1950), 50-62. 

[28]

J. Song and S. Karni, A second order traffic flow model with lane changing, J. Sci. Comput., 81 (2019), 1429-1445.  doi: 10.1007/s10915-019-01023-z.

[29]

T. Toledo, Driving behaviour: Models and challenges, Transport Reviews, 27 (2007), 65-84.  doi: 10.1080/01441640600823940.

[30]

Z. ZhengS. AhnD. Chen and J. Laval, Freeway traffic oscillations: Microscopic analysis of formations and propagations using wavelet transform, Transportation Research Part B: Methodological, 45 (2011), 1378-1388. 

[31]

Z. ZhengS. AhnD. Chen and J. Laval, The effects of lane-changing on the immediate follower: Anticipation, relaxation, and change in driver characteristics, Transportation Research Part C: Emerging Technologies, 26 (2013), 367-379.  doi: 10.1016/j.trc.2012.10.007.

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 $
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 $
$ 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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