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Second-Order Macroscopic Traffic Models

  • Antonella Ferrara
  • Simona Sacone
  • Silvia Siri
Chapter
Part of the Advances in Industrial Control book series (AIC)

Abstract

Second-order macroscopic traffic flow models introduce a second dynamic equation compared to first-order models, i.e. the equation describing the dynamics of the mean speed of vehicles. Second-order models were introduced in the 70s as continuous models, the earliest one being the so-called Payne–Whitham model. Some critiques arose on this class of models, focusing in particular on the dissimilarity between the flow of vehicles and the flow of molecules in fluids or gases. This criticism encouraged new developments of second-order models, leading to the model proposed by Aw and Rascle, and a similar model developed independently by Zhang. A discrete version of second-order models has been elaborated in the 90s, known as METANET. This discrete model, conceived both for freeway stretches and for networks, is very widespread in the engineering field and particularly suitable for prediction and control purposes.

References

  1. 1.
    Payne HJ (1971) Models of freeway traffic and control. Math Model Public Syst 28:51–61Google Scholar
  2. 2.
    Whitham GB (1974) Linear and nonlinear waves. Wiley, New YorkzbMATHGoogle Scholar
  3. 3.
    Daganzo CF (1995) Requiem for second-order fluid approximations of traffic flow. Transp Res Part B 29:277–286CrossRefGoogle Scholar
  4. 4.
    Aw A, Rascle M (2000) Resurrection of “second order” models of traffic flow. SIAM J Appl Math 60:916–938MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhang HM (2002) A non-equilibrium traffic model devoid of gas-like behavior. Transp Res Part B 36:275–290CrossRefGoogle Scholar
  6. 6.
    Garavello M, Piccoli B (2016) Traffic flow on networks. American Institute of Mathematical SciencesGoogle Scholar
  7. 7.
    Garavello M, Han K, Piccoli B (2006) Models for vehicular traffic on networks. American Institute of Mathematical SciencesGoogle Scholar
  8. 8.
    Helbing D, Johansson AF (2009) On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models. Eur Phys J 69:549–562Google Scholar
  9. 9.
    Greenberg JM (2001) Extensions and amplifications of a traffic model of Aw and Rascle. SIAM J Appl Math 62:729–745MathSciNetCrossRefGoogle Scholar
  10. 10.
    Rascle M (2002) An improved macroscopic model of traffic flow: derivation and links with the Lighthill-Whitham model. Math Comput Model 35:581–590MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lebacque J-P, Mammar S, Haj-Salem H (2007) The Aw-Rascle and Zhang’s model: vacuum problems, existence and regularity of the solutions of the Riemann problem. Transp Res Part B 41:710–721CrossRefGoogle Scholar
  12. 12.
    Garavello M, Piccoli B (2006) Traffic flow on a road network using the Aw-Rascle model. Commun Partial Differ Equ 31:243–275MathSciNetCrossRefGoogle Scholar
  13. 13.
    Herty M, Rascle M (2006) Coupling conditions for a class of second-order models for traffic flow. SIAM J Math Anal 38:595–616MathSciNetCrossRefGoogle Scholar
  14. 14.
    Herty M, Moutari S, Rascle M (2006) Optimization criteria for modelling intersections of vehicular traffic flow. Netw Heterog Media 1:275–294MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kerner B (1998) Experimental features of self-organization in traffic flow. Phys Rev Lett 81:3797–3800CrossRefGoogle Scholar
  16. 16.
    Colombo R (2003) Hyperbolic phase transitions in traffic flow. SIAM J Appl Math 63:708–721MathSciNetCrossRefGoogle Scholar
  17. 17.
    Blandin S, Work D, Goatin P, Piccoli B, Bayen A (2011) A general phase transition model for vehicular traffic. SIAM J Appl Math 71:107–127MathSciNetCrossRefGoogle Scholar
  18. 18.
    Blandin S, Argote J, Bayen AM, Work DB (2013) Phase transition model of non-stationary traffic flow: definition. properties and solution method. Transp Res Part B 52:31–55CrossRefGoogle Scholar
  19. 19.
    Goatin P (2006) The Aw-Rascle vehicular traffic flow model with phase transitions. Math Comput Model 44:287–303MathSciNetCrossRefGoogle Scholar
  20. 20.
    Colombo RM, Goatin P, Piccoli B (2010) Road networks with phase transitions. J Hyperbolic Differ Equ 7:85–106MathSciNetCrossRefGoogle Scholar
  21. 21.
    Colombo RM, Garavello M (2014) Phase transition model for traffic at a junction. J Math Sci 196:30–36MathSciNetCrossRefGoogle Scholar
  22. 22.
    Papageorgiou M, Blosseville J-M, Hadj-Salem H (1989) Macroscopic modelling of traffic flow on the Boulevard Périphérique in Paris. Transp Res Part B 23:29–47CrossRefGoogle Scholar
  23. 23.
    Papageorgiou M (1990) Modelling and real-time control of traffic flow on the Southern part of Boulevard Périphérique in Paris: part I: modelling. Transp Res Part A 24:345–359CrossRefGoogle Scholar
  24. 24.
    Messmer A, Papageorgiou M (1990) METANET: a macroscopic simulation program for motorway networks. Traffic Eng Control 31:466–470Google Scholar
  25. 25.
    Kotsialos A, Papageorgiou M, Diakaki C, Pavlis Y, Middelham F (2002) Traffic flow modeling of large-scale motorway networks using the macroscopic modeling tool METANET. IEEE Trans Intell Transp Syst 3:282–292CrossRefGoogle Scholar
  26. 26.
    Cremer M, May AD (1986) An extended traffic flow model for inner urban freeways. In: Preprints of 5th IFAC/IFIP/IFORS International conference on control in transportation systems, pp 383–388Google Scholar
  27. 27.
    Papageorgiou M, Kotsialos A (2002) Freeway ramp metering: an overview. IEEE Trans Intell Transp Syst 3:271–281CrossRefGoogle Scholar
  28. 28.
    Bellemans T, De Schutter B, De Moor B (2006) Model predictive control for ramp metering of motorway traffic: a case study. Control Eng Pract 14:757–767CrossRefGoogle Scholar
  29. 29.
    Hegyi A, De Schutter B, Hellendoorn H (2005) Model predictive control for optimal coordination of ramp metering and variable speed limits. Transp Res Part C 13:185–209CrossRefGoogle Scholar
  30. 30.
    Hegyi A, De Schutter B, Hellendoorn J (2005) Optimal coordination of variable speed limits to suppress shock waves. IEEE Trans Intell Transp Syst 6:102–112CrossRefGoogle Scholar
  31. 31.
    Cremer M (1979) Der Verkehrsfluss auf Schnellstrassen (Traffic flow on freeways), Fachberichte Messen 3, Steuern, Regeln. Springer, BerlinCrossRefGoogle Scholar
  32. 32.
    Carlson RC, Papamichail I, Papageorgiou M, Messmer A (2010) Optimal mainstream traffic flow control of large-scale motorway networks. Transp Res Part C 18:193–212CrossRefGoogle Scholar
  33. 33.
    Carlson RC, Papamichail I, Papageorgiou M (2011) Local feedback-based mainstream traffic flow control on motorways using variable speed limits. IEEE Trans Intell Transp Syst 12:1261–1276CrossRefGoogle Scholar
  34. 34.
    Tang TQ, Huang HJ, Zhao SG, Shang HY (2009) A new dynamic model for heterogeneous traffic flow. Phys Lett A 373:2461–2466CrossRefGoogle Scholar
  35. 35.
    Mohan R, Ramadurai G (2017) Heterogeneous traffic flow modelling using second-order macroscopic continuum model. Phys Lett A 381:115–123CrossRefGoogle Scholar
  36. 36.
    Deo P, De Schutter B, Hegyi A (2009) Model predictive control for multi-class traffic flows. In: Proceedings of the 12th IFAC symposium on transportation systems, pp 25–30Google Scholar
  37. 37.
    Liu S, De Schutter B, Hellendoorn H (2014) Model predictive traffic control based on a new multi-class METANET model. In: Proceedings of the 19th IFAC world congress, pp 8781–8785CrossRefGoogle Scholar
  38. 38.
    Logghe S, Immers LH (2008) Multi-class kinematic wave theory of traffic flow. Transp Res Part B 42:523–541CrossRefGoogle Scholar
  39. 39.
    Caligaris C, Sacone S, Siri S (2007) Optimal ramp metering and variable speed signs for multiclass freeway traffic. In: Proceedings of the European control conference, pp 1780–1785Google Scholar
  40. 40.
    Pasquale C, Sacone S, Siri S (2014) Two-class emission traffic control for freeway systems. In: Proceedings of the 19th IFAC world congress, pp 936–941CrossRefGoogle Scholar
  41. 41.
    Pasquale C, Papamichail I, Roncoli C, Sacone S, Siri S, Papageorgiou M (2015) Two-class freeway traffic regulation to reduce congestion and emissions via nonlinear optimal control. Transp Res Part C 55:85–99CrossRefGoogle Scholar
  42. 42.
    Pasquale C, Sacone S, Siri S, De Schutter B (2017) A multi-class model-based control scheme for reducing congestion and emissions in freeway networks by combining ramp metering and route guidance. Transp Res Part C 80:384–408CrossRefGoogle Scholar
  43. 43.
    Special report 209 (1994) Highway capacity manual, 3rd edn. Transportation Research Board, Washington DCGoogle Scholar
  44. 44.
    Al-Kaisy AF, Hall FL, Reisman ES (2002) Developing passenger car equivalents for heavy vehicles on freeways during queue discharge flow. Transp Res Part A 36:725–742Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical, Computer and Biomedical EngineeringUniversity of PaviaPaviaItaly
  2. 2.Department of Informatics, Bioengineering, Robotics and Systems EngineeringUniversity of GenoaGenoaItaly

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