A PDE-ODE model for traffic control with autonomous vehicles

Thibault Liard; Raphael Stern; Maria Laura Delle Monache; Thibault Liard; Raphael Stern; Maria Laura Delle Monache

doi:10.3934/nhm.2023051

Networks and Heterogeneous Media

2023, Volume 18, Issue 3: 1190-1206. doi: 10.3934/nhm.2023051

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A PDE-ODE model for traffic control with autonomous vehicles

1.
Univ. Limoges, CNRS, XLIM, UMR 7252, F-87000 Limoges, France
2.
Department of Civil, Environmental, and Geo- Engineering, University of Minnesota, 500 Pillsbury Dr SE, Minneapolis, MN 55406, USA
3.
Department of Civil and Environmental Engineering, University of California, Berkeley, Berkeley, CA, 94720, USA

Received: 12 November 2021 Revised: 10 May 2022 Accepted: 08 July 2022 Published: 07 April 2023

We consider a partial differential equation - ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model, the bulk flow of human drivers is represented by a scalar conservation law, while each autonomous vehicle is described by an ordinary differential equation. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. Next, we propose a control strategy for the speeds of the autonomous vehicles to minimize the average fuel consumption of the entire traffic flow. Existence of solutions for the optimal control problem is proved, and we numerically show that a reduction in average fuel consumption is possible with an AV acting as a moving bottleneck.
- scalar conservation laws,
- PDE-ODE systems,
- traffic control,
- traffic flow models,
- autonomous vehicles
Citation: Thibault Liard, Raphael Stern, Maria Laura Delle Monache. A PDE-ODE model for traffic control with autonomous vehicles[J]. Networks and Heterogeneous Media, 2023, 18(3): 1190-1206. doi: 10.3934/nhm.2023051

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Abstract

We consider a partial differential equation - ordinary differential equation system to describe the dynamics of traffic flow with autonomous vehicles. In the model, the bulk flow of human drivers is represented by a scalar conservation law, while each autonomous vehicle is described by an ordinary differential equation. The coupled PDE-ODE model is introduced, and existence of solutions for this model is shown, along with a proposed algorithm to construct approximate solutions. Next, we propose a control strategy for the speeds of the autonomous vehicles to minimize the average fuel consumption of the entire traffic flow. Existence of solutions for the optimal control problem is proved, and we numerically show that a reduction in average fuel consumption is possible with an AV acting as a moving bottleneck.

References

[1]	K. Ahn, H. Rakha, A. Trani, M. Van Aerde, Estimating vehicle fuel consumption and emissions based on instantaneous speed and acceleration levels, J Transp Eng, 128 (2002), 182–190. https://doi.org/10.1061/(ASCE)0733-947X(2002)128:2(182) doi: 10.1061/(ASCE)0733-947X(2002)128:2(182)
[2]	I. M. Berry, The effects of driving style and vehicle performance on the real-world fuel consumption of US light-duty vehicles, Doctoral Thesis of Massachusetts Institute of Technology, Cambridge, 2010.
[3]	C. Chalons, M. L. Delle Monache, P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled pde-ode problem, Interfaces Free. Boundaries, 19 (2017), 553–570. https://doi.org/10.4171/IFB/392 doi: 10.4171/IFB/392
[4]	M. Čičić, K. H. Johansson, Traffic regulation via individually controlled automated vehicles: a cell transmission model approach, 2018 21st International Conference on Intelligent Transportation Systems (ITSC), (2018), 766–771.
[5]	S. Cui, B. Seibold, R. Stern, D. B. Work, Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations, Proceedings of the IEEE Intelligent Vehicles Symposium, (2017), 1336–1341.
[6]	C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33–41. https://doi.org/10.1016/0022-247X(72)90114-X doi: 10.1016/0022-247X(72)90114-X
[7]	L. Davis, Effect of adaptive cruise control systems on traffic flow, Phys Rev E, 69 (2004), 066110. https://doi.org/10.1103/PhysRevE.69.066110 doi: 10.1103/PhysRevE.69.066110
[8]	M. L. Delle Monache, P. Goatin, A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, DISCRETE CONT DYN-S, 7 (2014), 435–447. https://doi.org/10.3934/dcdss.2014.7.435 doi: 10.3934/dcdss.2014.7.435
[9]	M. L. Delle Monache, P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J Differ Equ, 257 (2014), 4015–4029. https://doi.org/10.1016/j.jde.2014.07.014 doi: 10.1016/j.jde.2014.07.014
[10]	M. L. Delle Monache, P. Goatin, Stability estimates for scalar conservation laws with moving flux constraints, Netw. Heterog. Media, 12 (2017), 245–258. https://doi.org/10.3934/nhm.2017010 doi: 10.3934/nhm.2017010
[11]	M. L. Delle Monache, T. Liard, B. Piccoli, R. Stern, D. Work, Traffic reconstruction using autonomous vehicles, SIAM J Appl Math, 79 (2019), 1748–1767. https://doi.org/10.1137/18M1217000 doi: 10.1137/18M1217000
[12]	L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions, Boca Raton: CRC Press, 1991.
[13]	M. Garavello, P. Goatin, T. Liard, B. Piccoli, A multiscale model for traffic regulation via autonomous vehicles, J Differ Equ, 269 (2020), 6088–6124. https://doi.org/10.1016/j.jde.2020.04.031 doi: 10.1016/j.jde.2020.04.031
[14]	M. Guériau, R. Billot, N. E. El Faouzi, J. Monteil, F. Armetta, S. Hassas, How to assess the benefits of connected vehicles? A simulation framework for the design of cooperative traffic management strategies. Transp Res Part C Emerg Technol, 67 (2016), 266–279. https://doi.org/10.1016/j.trc.2016.01.020
[15]	K. Huang, X. Di, Q. Du, X. Chen, Stabilizing traffic via autonomous vehicles: A continuum mean field game approach, 2019 IEEE Intelligent Transportation Systems Conference (ITSC), (2019), 3269–3274.
[16]	F. Knorr, D. Baselt, M. Schreckenberg, M. Mauve, Reducing traffic jams via vanets, IEEE Transactions on Vehicular Technology, 61 (2012), 3490–3498. https://doi.org/10.1109/TVT.2012.2209690 doi: 10.1109/TVT.2012.2209690
[17]	P. D. Lax, Hyperbolic systems of conservation laws II, Commun. Pure Appl. Math., 10 (1957), 537–566. https://doi.org/10.1002/cpa.3160100406 doi: 10.1002/cpa.3160100406
[18]	T. Liard, B. Piccoli, On entropic solutions to conservation laws coupled with moving bottlenecks, Commun Math Sci, 19 (2021), 919–945. https://doi.org/10.4310/CMS.2021.v19.n4.a3 doi: 10.4310/CMS.2021.v19.n4.a3
[19]	T. Liard, B. Piccoli, Well-posedness for scalar conservation laws with moving flux constraints, SIAM J Appl Math, 79 (2019), 641–667. https://doi.org/10.1137/18M1172211 doi: 10.1137/18M1172211
[20]	M. J. Lighthill, G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. R. Soc. London A: Math. Phys. Sci., 229 (1955), 317–345. https://doi.org/10.1098/rspa.1955.0089 doi: 10.1098/rspa.1955.0089
[21]	G. Piacentini, P. Goatin, A. Ferrara, Traffic control via moving bottleneck of coordinated vehicles, IFAC-PapersOnLine, 51 (2018), 13–18. https://doi.org/10.1016/j.ifacol.2018.06.192 doi: 10.1016/j.ifacol.2018.06.192
[22]	R. A. Ramadan, B. Seibold, Traffic flow control and fuel consumption reduction via moving bottlenecks, arXiv: 1702.07995, [Preprint], (2017) [cited 2023 April 07 ]. Available from: https://doi.org/10.48550/arXiv.1702.07995
[23]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42–51. https://doi.org/10.1287/opre.4.1.42
[24]	W. Rudin, Principles of mathematical analysis, New York: McGraw-hill, 1976.
[25]	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.
[26]	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/10.1007/s00115-018-0494-4 doi: 10.1007/s00115-018-0494-4
[27]	E. Vinitsky, K. Parvate, A. Kreidieh, C. Wu, A. Bayen, Lagrangian control through deep-RL: Applications to bottleneck decongestion, 2018 21st International Conference on Intelligent Transportation Systems (ITSC), (2018), 759–765. https://doi.org/10.1016/j.trc.2016.07.007

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