Traffic waves, known also as stop-and-go waves or phantom jams, appear naturally as traffic instabilities, also in confined environments as a ring-road. A multi-population traffic is studied on a ring-road, comprised of drivers with stable and unstable behavior. There exists a critical penetration rate of stable vehicles above which the system is stable, and under which the system is unstable. In the latter case, stop-and-go waves appear, provided enough cars are on the road. The critical penetration rate is explicitly computable, and, in reasonable situations, a small minority of aggressive drivers is enough to destabilize an otherwise very stable flow. This is a source of instability that a single population model would not be able to explain. Also, the multi-population system can be stable below the critical penetration rate if the number of cars is sufficiently small. Instability emerges as the number of cars increases, even if the traffic density remains the same (i.e., number of cars and road size increase similarly). This shows that small experiments could lead to deducing imprecise stability conditions.
Citation: Amaury Hayat, Benedetto Piccoli, Shengquan Xiang. Stability of multi-population traffic flows[J]. Networks and Heterogeneous Media, 2023, 18(2): 877-905. doi: 10.3934/nhm.2023038
Traffic waves, known also as stop-and-go waves or phantom jams, appear naturally as traffic instabilities, also in confined environments as a ring-road. A multi-population traffic is studied on a ring-road, comprised of drivers with stable and unstable behavior. There exists a critical penetration rate of stable vehicles above which the system is stable, and under which the system is unstable. In the latter case, stop-and-go waves appear, provided enough cars are on the road. The critical penetration rate is explicitly computable, and, in reasonable situations, a small minority of aggressive drivers is enough to destabilize an otherwise very stable flow. This is a source of instability that a single population model would not be able to explain. Also, the multi-population system can be stable below the critical penetration rate if the number of cars is sufficiently small. Instability emerges as the number of cars increases, even if the traffic density remains the same (i.e., number of cars and road size increase similarly). This shows that small experiments could lead to deducing imprecise stability conditions.
| [1] |
S. Albeaik, A. Bayen, M. T. Chiri, X. Gong, A. Hayat, N. Kardous, et al., Limitations and improvements of the intelligent driver model (IDM), SIAM J. Appl. Dyn. Syst., 21 (2021), 1862–1892. https://doi.org/10.1137/21M1406477 doi: 10.1137/21M1406477
|
| [2] |
M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E., 51 (1995), 1035. https://doi.org/10.1137/21M1406477 doi: 10.1137/21M1406477
|
| [3] | G. Bastin, J. M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, Cham: Birkhäuser, 2016. |
| [4] | S. Cui, B. Seibold, R. Stern, D. B. Work, Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations, 2017 IEEE Intelligent Vehicles Symposium (IV), (2017), 1336–1341. |
| [5] | M. L. Delle Monache, T. Liard, A. Rat, R. Stern, R. Bhadani, B. Seibold, et al., Feedback control algorithms for the dissipation of traffic waves with autonomous vehicles, Computational Intelligence and Optimization Methods for Control Engineering, (2019), 275–299. |
| [6] | F. Farokhi, K. H. Johansson, A study of truck platooning incentives using a congestion game, IEEE trans Intell Transp Syst, 16 (2015), 581–595. |
| [7] | M. R. Flynn, A. R. Kasimov, J. C. Nave, R. R. Rosales, B. Seibold, Self-sustained nonlinear waves in traffic flow, Phys. Rev. E, 79 (2009), 056113. |
| [8] |
D. C. Gazis, R. Herman, R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545–567. https://doi.org/10.1287/opre.9.4.545 doi: 10.1287/opre.9.4.545
|
| [9] | X. Gong, A. Keimer, On the well-posedness of the "bando-follow the leader" car following model and a "time-delayed version", [Preprint], (2022) [cited 2022 Feb 22]. Available from: https://www.researchgate.net/profile/Xiaoqian-Gong-2/publication/358443671.pdf |
| [10] |
M. Gugat, M. Herty, A. Klar, G. Leugering, Optimal control for traffic flow networks, J Optim Theory Appl, 126 (2005), 589–616. https://doi.org/10.1007/s10957-005-5499-z doi: 10.1007/s10957-005-5499-z
|
| [11] |
S. Hallé, B. Chaib-draa, A collaborative driving system based on multiagent modelling and simulations, Transp Res Part C Emerg Technol, 13 (2005), 320–345. https://doi.org/10.1016/j.trc.2005.07.004 doi: 10.1016/j.trc.2005.07.004
|
| [12] | A. Hayat, B. Piccoli, S. Truong, Dissipation of traffic jams using a single autonomous vehicle on a ring road, [Preprint], (2021) [cited 2022 Mar 14]. Available from: https://hal.science/hal-03354282 |
| [13] |
D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067–1141. https://doi.org/10.1103/RevModPhys.73.1067 doi: 10.1103/RevModPhys.73.1067
|
| [14] |
B. S. Kerner, P. Konhäuser, Structure and parameters of clusters in traffic flow, Phys. Rev. E, 50 (1994), 54–83. https://doi.org/10.1103/PhysRevE.50.54 doi: 10.1103/PhysRevE.50.54
|
| [15] |
M. Papageorgiou, H. Hadj-Salem, F. Middelham, Alinea local ramp metering: Summary of field results, Transp Res Rec, 1603 (1997), 90–98. https://doi.org/10.3141/1603-12 doi: 10.3141/1603-12
|
| [16] | H. Pohlmann, B. Seibold, Simple control options for an vehicle used to dissipate traffic waves, Technical Report, 2015. |
| [17] |
R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, et al., Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transp Res Part C Emerg Technol, 89 (2018), 205–221. https://doi.org/10.1007/s00115-018-0494-4 doi: 10.1007/s00115-018-0494-4
|
| [18] |
Y. Sugiyama, M. Fukui, M. Kikuchi, K. Hasebe, A. Nakayama, K. Nishinari, et al., Traffic jams without bottlenecks—experimental evidence for the physical mechanism of the formation of a jam, New J. Phys., 10 (2008), 033001. https://doi.org/10.1088/1367-2630/10/3/033001 doi: 10.1088/1367-2630/10/3/033001
|
| [19] | X. Sun, L. Muñoz, R. Horowitz, Highway traffic state estimation using improved mixture kalman filters for effective ramp metering control, 42nd IEEE International Conference on Decision and Control, 6 (2003), 6333–6338. |
| [20] |
A. Talebpour, H. S. Mahmassani, Influence of connected and autonomous vehicles on traffic flow stability and throughput, Transp Res Part C Emerg Technol, 71 (2016), 143–163. https://doi.org/10.1016/j.trc.2016.07.007 doi: 10.1016/j.trc.2016.07.007
|
| [21] |
M. Treiber, A. Hennecke, D. Helbing, Congested traffic states in empirical observations and microscopic simulations, Phys. Rev. E., 62 (2000), 1805. https://doi.org/10.1103/PhysRevE.62.1805 doi: 10.1103/PhysRevE.62.1805
|
| [22] | J. Wang, Y. Zheng, Q. Xu, J. Wang, K. Li, Controllability analysis and optimal control of mixed traffic flow with human-driven and autonomous vehicles, IEEE trans Intell Transp Syst, 22 (2020), 7445–7459. |
| [23] | M. Wang, W. Daamen, S. P. Hoogendoorn, B. van Arem, Cooperative car-following control: Distributed algorithm and impact on moving jam features, IEEE trans Intell Transp Syst, 17 (2015), 1459–1471. |
| [24] |
M. Wang, S. van Maarseveen, R. Happee, O. Tool, B. van Arem, Benefits and risks of truck platooning on freeway operations near entrance ramp, Transp Res Rec, 2673 (2019), 588–602. https://doi.org/10.1177/0361198119842821 doi: 10.1177/0361198119842821
|
| [25] | H. Yu, J. Auriol, M. Krstic, Simultaneous stabilization of traffic flow on two connected roads, 2020 American Control Conference (ACC), (2020), 3443–3448. |
| [26] |
H. Yu, M. Diagne, L. Zhang, M. Krstic, Bilateral boundary control of moving shockwave in lwr model of congested traffic, IEEE Trans. Automat. Contr., 66 (2020), 1429–1436. https://doi.org/10.1109/TAC.2020.2994031 doi: 10.1109/TAC.2020.2994031
|
| [27] |
H. Yu, M. Krstic, Traffic congestion control for aw–rascle–zhang model, Automatica, 100 (2019), 38–51. https://doi.org/10.1016/j.automatica.2018.10.040 doi: 10.1016/j.automatica.2018.10.040
|
| [28] | Y. Zheng, J. Wang, K. Li, Smoothing traffic flow via control of autonomous vehicles. IEEE Internet Things J., 7 (2020), 3882–3896. https://doi.org/10.1016/j.automatica.2018.10.040 |
Figures(2)
Amaury Hayat, Benedetto Piccoli, Shengquan Xiang. Stability of multi-population traffic flows[J]. Networks and Heterogeneous Media, 2023, 18(2): 877-905. doi: 10.3934/nhm.2023038
DownLoad: