Kinetic-Controlled Hydrodynamics for Traffic Models with Driver-Assist Vehicles
Abstract
We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.
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References
1.
S. M. Abuelenin and A. Y. Abul-Magd, Empirical study of traffic velocity distribution and its effect on VANETs connectivity, in Proceedings of the 2014 International Conference on Connected Vehicles and Expo (Vienna, Austria), IEEE, Washington, DC, 2014, pp. 391--395.
2.
J. P. Agnelli, F. Colasuonno, and D. Knopoff, A kinetic theory approach to the dynamics of crowd evacuation from bounded domains, Math. Models Methods Appl. Sci., 25 (2015), pp. 109--129.
3.
G. Albi, Y.-P. Choi, M. Fornasier, and D. Kalise, Mean field control hierarchy, Appl. Math. Optim., 76 (2017), pp. 93--135.
4.
G. Albi, L. Pareschi, and M. Zanella, Boltzmann-type control of opinion consensus through leaders, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20140138.
5.
A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916--938, https://doi.org/10.1137/S0036139997332099.
6.
M. K. Banda and M. Herty, Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws, Math. Control Relat. Fields, 3 (2013), pp. 121--142.
7.
D. Benedetto, E. Caglioti, F. Golse, and M. Pulvirenti, Hydrodynamic limits of a Vlasov-Fokker-Planck equation for granular media, Commun. Math. Sci., 2 (2004), pp. 121--136.
8.
A. Bensoussan, J. Frehse, and P. Yam, Mean Field Games and Mean Field Type Control Theory, SpringerBriefs Math., Springer, New York, 2013.
9.
D. S. Berry and D. M. Belmont, Distribution of vehicle speeds and travel times, in Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, The Regents of the University of California, 1951, pp. 589--602.
10.
A. V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation, Phys. Rev. E, 61 (2000), pp. 4576--4586.
11.
F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci. 22, World Scientific, River Edge, NJ, 1994, pp. 171--190.
12.
F. Bouchut, S. Jin, and X. Li, Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), pp. 135--158, https://doi.org/10.1137/S0036142901398040.
13.
J. A. Carrillo, Y.-P. Choi, and S. P. Perez, A review on attractive-repulsive hydrodynamics for consensus in collective behavior, in Active Particles, Vol. 1, N. Bellomo, P. Degond, and E. Tadmor, eds., Birkhäuser/Springer, Cham, 2017, pp. 259--298.
14.
C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Appl. Math. Sci. 106, Springer, New York, 1994.
15.
R. M. Colombo, M. Herty, and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 353--379.
16.
E. Cristiani, F. S. Priuli, and A. Tosin, Modeling rationality to control self-organization of crowds: An environmental approach, SIAM J. Appl. Math., 75 (2015), pp. 605--629, https://doi.org/10.1137/140962413.
17.
C. F. Daganzo, Requiem for second-order fluid approximation of traffic flow, Transportation Res., 29 (1995), pp. 277--286.
18.
A. I. Delis, I. K. Nikolos, and M. Papageorgiou, Macroscopic traffic flow modeling with adaptive cruise control: Development and numerical solution, Comput. Math. Appl., 70 (2015), pp. 1921--1947.
19.
A. I. Delis, I. K. Nikolos, and M. Papageorgiou, A macroscopic multi-lane traffic flow model for ACC/CACC traffic dynamics, Transp. Res. Record, 2672 (2018), pp. 178--192.
20.
M. Delitala and A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Models Methods Appl. Sci., 17 (2007), pp. 901--932.
21.
P. P. Dey, S. Chandra, and S. Gangopadhaya, Speed distribution curves under mixed traffic conditions, J. Transp. Eng., 132 (2006), pp. 475--481.
22.
B. Düring and G. Toscani, Hydrodynamics from kinetic models of conservative economies, Phys. A, 384 (2007), pp. 493--506.
23.
L. Fermo and A. Tosin, Fundamental diagrams for kinetic equations of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), pp. 449--462.
24.
A. Festa, A. Tosin, and M.-T. Wolfram, Kinetic description of collision avoidance in pedestrian crowds by sidestepping, Kinet. Relat. Models, 11 (2018), pp. 491--520.
25.
M. Fornasier, B. Piccoli, and F. Rossi, Mean-field sparse optimal control, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130400.
26.
P. Freguglia and A. Tosin, Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach, Commun. Math. Sci., 15 (2017), pp. 213--236.
27.
J. R. Frejo, I. Papamichail, M. Papageorgiou, and E. F. Camacho, Macroscopic modeling and control of reversible lanes on freeways, IEEE Trans. Intell. Transp. Syst., 17 (2016), pp. 948--959.
28.
D. C. Gazis, R. Herman, and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), pp. 545--567.
29.
M. Günther, A. Klar, T. Materne, and R. Wegener, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J. Appl. Math., 64 (2003), pp. 468--483, https://doi.org/10.1137/S0036139902404700.
30.
S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 3 (2008), pp. 415--435.
31.
Y. Han, A. Hegyi, Y. Yuan, S. Hoogendoorn, M. Papageorgiou, and C. Roncoli, Resolving freeway jam waves by discrete first-order model-based predictive control of variable speed limits, Transportation Res. Part C, 77 (2017), pp. 405--420.
32.
M. Herty, A. Klar, and L. Pareschi, General kinetic traffic models for vehicular traffic flows and Monte-Carlo methods, Comput. Methods Appl. Math., 5 (2005), pp. 155--169.
33.
M. Herty and L. Pareschi, Fokker-Planck asymptotics for traffic flow models, Kinet. Relat. Models, 3 (2010), pp. 165--179.
34.
M. Herty, A. Tosin, G. Visconti, and M. Zanella, Hybrid stochastic kinetic description of two-dimensional traffic dynamics, SIAM J. Appl. Math., 78 (2018), pp. 2737--2762, https://doi.org/10.1137/17M1155909.
35.
R. Illner, A. Klar, and T. Materne, Vlasov-Fokker-Plank models for multilane traffic flow, Commun. Math. Sci., 1 (2003), pp. 1--12.
36.
J. Jansson and F. Gustafsson, A framework and automotive application of collision avoidance decision making, Automatica J. IFAC, 44 (2008), pp. 2347--2351.
37.
B. S. Kerner, The Physics of Traffic, Understanding Complex Systems, Springer, Berlin, 2004.
38.
A. Klar and R. Wegener, Enskog-like kinetic models for vehicular traffic, J. Stat. Phys., 87 (1997), pp. 91--114.
39.
W. Leutzbach, Introduction to the Theory of Traffic Flow, Springer-Verlag, New York, 1988.
40.
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2004.
41.
A. K. Maurya, S. Das, S. Dey, and S. Nama, Study on speed and time-headway distributions on two-lane bidirectional road in heterogeneous traffic condition, Transp. Res. Proc., 17 (2016), pp. 428--437.
42.
G. Naldi, L. Pareschi, and G. Toscani, eds., Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Boston, MA, 2010.
43.
D. Ni, H. K. Hsieh, and T. Jiang, Modeling phase diagrams as stochastic processes with application in vehicular traffic flow, Appl. Math. Model., 53 (2018), pp. 106--117.
44.
I. A. Ntousakis, I. K. Nikolok, and M. Papageorgious, On microscopic modelling of adaptive cruise control systems, Transp. Res. Proc., 6 (2015), pp. 111--127.
45.
L. Pareschi and G. Russo, An introduction to Monte Carlo methods for the Boltzmann equation, ESAIM Proc., 10 (2001), pp. 35--75.
46.
L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, UK, 2013.
47.
S. L. Paveri-Fontana, On Boltzmann-like treatments for traffic flow: A critical review of the basic model and an alternative proposal for dilute traffic analysis, Transportation Res., 9 (1975), pp. 225--235.
48.
M. Peden, R. Scurfield, D. Sleet, D. Mohan, A. A. Hyder, E. Jarawan, and C. Mathers, World Report on Road Traffic Injury Prevention, Tech. report, World Health Organization, 2004.
49.
I. Prigogine and R. Herman, Kinetic Theory of Vehicular Traffic, Elsevier, New York, 1971.
50.
G. Puppo, M. Semplice, A. Tosin, and G. Visconti, Fundamental diagrams in traffic flow: The case of heterogeneous kinetic models, Commun. Math. Sci., 14 (2016), pp. 643--669.
51.
J.-M. Qiu and C.-W. Shu, Convergence of high order finite volume weighted essentially nonoscillatory scheme and discontinuous Galerkin method for nonconvex conservation laws, SIAM J. Sci. Comput., 31 (2008), pp. 584--607, https://doi.org/10.1137/070687487.
52.
P. Santi, G. Resta, M. Szell, S. Sobolevsky, S. H. Strogatz, and C. Ratti, Quantifying the benefits of vehicle pooling with shareability networks, Proc. Natl. Acad. Sci. USA, 111 (2014), pp. 13290--13294.
53.
C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), pp. 82--126, https://doi.org/10.1137/070679065.
54.
R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, R. Haulcy, H. Pohlmann, F. Wu, B. Piccoli, B. Seibold, J. Sprinkle, and D. B. Work, Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments, Transportation Res. Part C, 89 (2018), pp. 205--221.
55.
R. Tachet, P. Santi, S. Sobolevsky, L. I. Reyes-Castro, E. Frazzoli, D. Helbing, and C. Ratti, Revisiting street intersections using slot-based systems, PLoS ONE, 11 (2016), e0149607.
56.
G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), pp. 481--496.
57.
A. Tosin and M. Zanella, Control strategies for road risk mitigation in kinetic traffic modelling, IFAC-PapersOnLine, 51 (2018), pp. 67--72.
58.
M. Treiber, A. Kesting, and D. Helbing, Delays, inaccuracies and anticipation in microscopic traffic models, Phys. A, 306 (2006), pp. 71--88.
59.
C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), pp. 273--307.
60.
G. Visconti, M. Herty, G. Puppo, and A. Tosin, Multivalued fundamental diagrams of traffic flow in the kinetic Fokker--Planck limit, Multiscale Model. Simul., 15 (2017), pp. 1267--1293, https://doi.org/10.1137/16M1087035.
61.
World Health Organization, Global Status Report on Road Safety, Tech. report, World Health Organization, 2015.
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© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 30 July 2018
Accepted: 21 March 2019
Published online: 9 May 2019
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Ministero dell'Istruzione, dell'Università e della Ricerca https://doi.org/10.13039/501100003407 : PRIN 2017KKJP4X
Ministero dell'Istruzione, dell'Università e della Ricerca https://doi.org/10.13039/501100003407 : CUP: E11G18000350001
Politecnico di Torino https://doi.org/10.13039/100013000
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