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Kinetic Derivation of Aw–Rascle–Zhang-Type Traffic Models with Driver-Assist Vehicles

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Abstract

In this paper, we derive second order hydrodynamic traffic models from kinetic-controlled equations for driver-assist vehicles. At the vehicle level we take into account two main control strategies synthesising the action of adaptive cruise controls and cooperative adaptive cruise controls. The resulting macroscopic dynamics fulfil the anisotropy condition introduced in the celebrated Aw–Rascle–Zhang model. Unlike other models based on heuristic arguments, our approach unveils the main physical aspects behind frequently used hydrodynamic traffic models and justifies the structure of the resulting macroscopic equations incorporating driver-assist vehicles. Numerical insights show that the presence of driver-assist vehicles produces an aggregate homogenisation of the mean flow speed, which may also be steered towards a suitable desired speed in such a way that optimal flows and traffic stabilisation are reached.

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Notes

  1. This is consistent with the fact that the hydrodynamic regime appears naturally on a slow time scale, which is directly suggested by (18) through the rate \(\epsilon \).

References

  1. Albi, G., Choi, Y.-P., Fornasier, M., Kalise, D.: Mean field control hierarchy. Appl. Math. Optim. 76(1), 93–135 (2017)

    Article  MathSciNet  Google Scholar 

  2. Albi, G., Herty, M., Pareschi, L.: Kinetic description of optimal control problems and application to opinion consensus. Commun. Math. Sci. 13(6), 1407–1429 (2015)

    Article  MathSciNet  Google Scholar 

  3. Albi, G., Pareschi, L., Zanella, M.: Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. A 372, 2014 (2028)

    MATH  Google Scholar 

  4. Aw, A., Rascle, M.: Resurrection of “second order" models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000)

    Article  MathSciNet  Google Scholar 

  5. Banda, M.K., Herty, M.: Numerical discretization of stabilization problems with boundary controls for systems of hyperbolic conservation laws. Math. Control Relat. Fields 3(2), 121–142 (2013)

    Article  MathSciNet  Google Scholar 

  6. Bensoussan, A., Frehse, J., Yam, P.: Mean Field Games and Mean Field Type Control Theory. Springer Briefs in Mathematics, Springer, New York (2013)

    Book  Google Scholar 

  7. Borsche, R., Klar, A., Zanella, M.: Kinetic-controlled hydrodynamics for multilane traffic models. Phys. A. 587, 126486 (2022)

  8. Chiarello, F.A., Piccoli, B., Tosin, A.: Multiscale control of generic second order traffic models by driver-assist vehicles. Multiscale Model. Simul. 19(2), 589–611 (2021)

    Article  MathSciNet  Google Scholar 

  9. Chow, J.Y.J.: Informed Urban Transport Systems: Classic and Emerging Mobility Methods Towards Smart Cities. Elsevier, Amsterdam (2018)

    Google Scholar 

  10. Colombo, R.M., Herty, M., Mercier, M.: Control of the continuity equation with a non local flow. ESAIM Control Optim. Calc. Var. 17(2), 353–379 (2011)

    Article  MathSciNet  Google Scholar 

  11. Cristiani, E., Priuli, F.S., Tosin, A.: Modeling rationality to control self-organization of crowds: an environmental approach. SIAM J. Appl. Math. 75(2), 605–629 (2015)

    Article  MathSciNet  Google Scholar 

  12. Daganzo, C.F.: Requiem for second-order fluid approximations of traffic flow. Transport. Res. Part B 29(4), 277–286 (1995)

    Article  Google Scholar 

  13. Delis, A.I., Nikolos, I.K., Papageorgiou, M.: Macroscopic traffic flow modeling with adaptive cruise control: development and numerical solution. Comput. Math. Appl. 70(8), 1921–1947 (2015)

    Article  MathSciNet  Google Scholar 

  14. Delis, A.I., Nikolos, I.K., Papageorgiou, M.: A macroscopic multi-lane traffic flow model for ACC/CACC traffic dynamics. Transp. Res, Record (2018)

  15. Dimarco, G., Tosin, A.: The Aw-Rascle traffic model: Enskog-type kinetic derivation and generalisations. J. Stat. Phys. 178(1), 178–210 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  16. Düring, B., Toscani, G.: Hydrodynamics from kinetic models of conservative economies. Physica A 384(2), 493–506 (2007)

    Article  ADS  Google Scholar 

  17. Fornasier, M., Piccoli, B., Rossi, F.: Mean-field sparse optimal control. Philos. Trans. R. Soc. A 372(2028), 20130400 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  18. Garavello, M., Goatin, P., Liard, T., Piccoli, B.: A multiscale model for traffic regulation via autonomous vehicles. J. Differ. Equ. 269(7), 6088–6124 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  19. Gazis, D.C., Herman, R., Rothery, R.W.: Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9, 545–567 (1961)

    Article  MathSciNet  Google Scholar 

  20. Goatin, P., Göttlich, S., Kolb, O.: Speed limit and ramp meter control for traffic flow networks. Eng. Optim. 48(7), 1121–1144 (2016)

    Article  MathSciNet  Google Scholar 

  21. Gong,X., Piccoli,B., Visconti,G.: Mean-field limit of a hybrid system for multi-lane multi-class traffic. Preprint (arXiv:2007.14655), (2020)

  22. Greenberg, J.M.: Extensions and amplifications of a traffic model of Aw and Rascle. SIAM J. Appl. Math. 62(3), 729–745 (2001)

    Article  MathSciNet  Google Scholar 

  23. Gugat, M., Herty, M., Klar, A., Leugering, G.: Optimal control for traffic flow networks. J. Optim. Theory Appl. 126(3), 589–616 (2005)

    Article  MathSciNet  Google Scholar 

  24. Helbing, D.: Gas-kinetic derivation of Navier-Stokes-like traffic equations. Phys. Rev. E 53(3), 2366–2381 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  25. Helbing, D., Hennecke, A., Shvetsov, V., Treiber, M.: Micro- and macro-simulation of freeway traffic. Math. Comput. Model. 35(5–6), 517–547 (2002)

    Article  MathSciNet  Google Scholar 

  26. Herty, M., Kirchner, C., Klar, A.: Instantaneous control for traffic flow. Math. Methods Appl. Sci. 30(2), 153–169 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  27. Klar, A., Wegener, R.: Enskog-like kinetic models for vehicular traffic. J. Stat. Phys. 87(1–2), 91–114 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  28. Klar, A., Wegener, R.: Kinetic derivation of macroscopic anticipation models for vehicular traffic. SIAM J. Appl. Math. 60(5), 1749–1766 (2000)

    Article  MathSciNet  Google Scholar 

  29. Kolb, O., Göttlich, S., Goatin, P.: Capacity drop and traffic control for a second order traffic model. Netw. Heterog. Media 12(4), 663–681 (2017)

    Article  MathSciNet  Google Scholar 

  30. Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229, 317–345 (1955)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Ntousakis, I.A., Nikolos, I.K., Papageorgiou, M.: On microscopic modelling of adaptive cruise control systems. Transp. Res. Procedia 6, 111–127 (2015)

    Article  Google Scholar 

  32. Payne,H. J.: Models of freeway traffic and control. In: G. A. Bekey (ed.) Mathematical Models of Public Systems, volume 1 of Simulation Council Proc., pp 51–61. Simulation Councils (1971)

  33. Piccoli, B., Tosin, A.: Vehicular traffic: a review of continuum mathematical models. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, vol. 22, pp. 9727–9749. Springer, New York (2009)

    Chapter  Google Scholar 

  34. Piccoli,B., Tosin,A., Zanella,M.: Model-based assessment of the impact of driver-assist vehicles using kinetic theory. Z. Angew. Math. Phys. 71(5), 152/1–25 (2020)

  35. Prigogine, I., Andrews, F.C.: A Boltzmann-like approach for traffic flow. Oper. Res. 8(6), 789–797 (1960)

    Article  MathSciNet  Google Scholar 

  36. Prigogine, I., Herman, R.: Kinetic Theory of Vehicular Traffic. American Elsevier Publishing Co., New York (1971)

    MATH  Google Scholar 

  37. Rascle, M.: An improved macroscopic model of traffic flow: derivation and links with the Lighthill-Whitham model. Math. Comput. Model. 35(5–6), 581–590 (2002)

    Article  MathSciNet  Google Scholar 

  38. Schoettle,B., Sivak,M.: Potential impact of self-driving vehicles on household vehicle demand and usage. Technical Report UMTRI-2015-3, Transportation Research Institute, University of Michigan, February (2015)

  39. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Quarteroni, A. (ed.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 325–432. Springer, Berlin (1998)

    Chapter  Google Scholar 

  40. Stern, R.E., Cui, S., Delle Monache, M.L., Bhadani, R., Bunting, M., Churchill, M., Hamilton, N., Haulcy, R., Pohlmann, H., Wu, F., Piccoli, B., Seibold, B., Sprinkle, J., Work, D.B.: Dissipation of stop-and-go waves via control of autonomous vehicles: field experiments. Transp. Res. Part C 89, 205–221 (2018)

    Article  Google Scholar 

  41. Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)

    Article  MathSciNet  Google Scholar 

  42. Tosin, A., Zanella, M.: Control strategies for road risk mitigation in kinetic traffic modelling. IFAC-PapersOnLine 51(9), 67–72 (2018)

    Article  Google Scholar 

  43. Tosin, A., Zanella, M.: Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles. Multiscale Model. Simul. 17(2), 716–749 (2019)

    Article  MathSciNet  Google Scholar 

  44. Tosin, A., Zanella, M.: Uncertainty damping in kinetic traffic models by driver-assist controls. Math. Control Relat. Fields 11(3), 681–713 (2021)

    Article  MathSciNet  Google Scholar 

  45. Villani,C.: Contribution à l’étude mathématique des équations de Boltzmann et de Landau en théorie cinétique des gaz et des plasmas. PhD thesis, Paris 9, (1998)

  46. Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273–307 (1998)

    Article  MathSciNet  Google Scholar 

  47. Zhang, H.M.: A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. Part B 36(3), 275–290 (2002)

    Article  Google Scholar 

Download references

Acknowledgements

This research was partially supported by the Italian Ministry for Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018-2022) – Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino (CUP: E11G18000350001) and Department of Mathematics “F. Casorati”, University of Pavia, and through the PRIN 2017 project (No. 2017KKJP4X) “Innovative numerical methods for evolutionary partial differential equations and applications”. This work is also part of the activities of the Starting Grant “Attracting Excellent Professors” funded by “Compagnia di San Paolo” (Torino) and promoted by Politecnico di Torino. GD is member of GNCS (Gruppo Nazionale per il Calcolo Scientifico) of INdAM (Istituto Nazionale di Alta Matematica), Italy. AT and MZ are members of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM, Italy.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, University of Ferrara, Ferrara, Italy

    Giacomo Dimarco

  2. Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Turin, Italy

    Andrea Tosin

  3. Department of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Mattia Zanella

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  1. Giacomo Dimarco
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  2. Andrea Tosin
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  3. Mattia Zanella
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Correspondence to Andrea Tosin.

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Communicated by Francesco Ginelli.

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Dimarco, G., Tosin, A. & Zanella, M. Kinetic Derivation of Aw–Rascle–Zhang-Type Traffic Models with Driver-Assist Vehicles. J Stat Phys 186, 17 (2022). https://doi.org/10.1007/s10955-021-02862-7

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