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.
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References
Payne HJ (1971) Models of freeway traffic and control. Math Model Public Syst 28:51–61
Whitham GB (1974) Linear and nonlinear waves. Wiley, New York
Daganzo CF (1995) Requiem for second-order fluid approximations of traffic flow. Transp Res Part B 29:277–286
Aw A, Rascle M (2000) Resurrection of “second order” models of traffic flow. SIAM J Appl Math 60:916–938
Zhang HM (2002) A non-equilibrium traffic model devoid of gas-like behavior. Transp Res Part B 36:275–290
Garavello M, Piccoli B (2016) Traffic flow on networks. American Institute of Mathematical Sciences
Garavello M, Han K, Piccoli B (2006) Models for vehicular traffic on networks. American Institute of Mathematical Sciences
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 69:549–562
Greenberg JM (2001) Extensions and amplifications of a traffic model of Aw and Rascle. SIAM J Appl Math 62:729–745
Rascle M (2002) An improved macroscopic model of traffic flow: derivation and links with the Lighthill-Whitham model. Math Comput Model 35:581–590
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 41:710–721
Garavello M, Piccoli B (2006) Traffic flow on a road network using the Aw-Rascle model. Commun Partial Differ Equ 31:243–275
Herty M, Rascle M (2006) Coupling conditions for a class of second-order models for traffic flow. SIAM J Math Anal 38:595–616
Herty M, Moutari S, Rascle M (2006) Optimization criteria for modelling intersections of vehicular traffic flow. Netw Heterog Media 1:275–294
Kerner B (1998) Experimental features of self-organization in traffic flow. Phys Rev Lett 81:3797–3800
Colombo R (2003) Hyperbolic phase transitions in traffic flow. SIAM J Appl Math 63:708–721
Blandin S, Work D, Goatin P, Piccoli B, Bayen A (2011) A general phase transition model for vehicular traffic. SIAM J Appl Math 71:107–127
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 52:31–55
Goatin P (2006) The Aw-Rascle vehicular traffic flow model with phase transitions. Math Comput Model 44:287–303
Colombo RM, Goatin P, Piccoli B (2010) Road networks with phase transitions. J Hyperbolic Differ Equ 7:85–106
Colombo RM, Garavello M (2014) Phase transition model for traffic at a junction. J Math Sci 196:30–36
Papageorgiou M, Blosseville J-M, Hadj-Salem H (1989) Macroscopic modelling of traffic flow on the Boulevard Périphérique in Paris. Transp Res Part B 23:29–47
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 24:345–359
Messmer A, Papageorgiou M (1990) METANET: a macroscopic simulation program for motorway networks. Traffic Eng Control 31:466–470
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 3:282–292
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 383–388
Papageorgiou M, Kotsialos A (2002) Freeway ramp metering: an overview. IEEE Trans Intell Transp Syst 3:271–281
Bellemans T, De Schutter B, De Moor B (2006) Model predictive control for ramp metering of motorway traffic: a case study. Control Eng Pract 14:757–767
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 13:185–209
Hegyi A, De Schutter B, Hellendoorn J (2005) Optimal coordination of variable speed limits to suppress shock waves. IEEE Trans Intell Transp Syst 6:102–112
Cremer M (1979) Der Verkehrsfluss auf Schnellstrassen (Traffic flow on freeways), Fachberichte Messen 3, Steuern, Regeln. Springer, Berlin
Carlson RC, Papamichail I, Papageorgiou M, Messmer A (2010) Optimal mainstream traffic flow control of large-scale motorway networks. Transp Res Part C 18:193–212
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 12:1261–1276
Tang TQ, Huang HJ, Zhao SG, Shang HY (2009) A new dynamic model for heterogeneous traffic flow. Phys Lett A 373:2461–2466
Mohan R, Ramadurai G (2017) Heterogeneous traffic flow modelling using second-order macroscopic continuum model. Phys Lett A 381:115–123
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 25–30
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 8781–8785
Logghe S, Immers LH (2008) Multi-class kinematic wave theory of traffic flow. Transp Res Part B 42:523–541
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 1780–1785
Pasquale C, Sacone S, Siri S (2014) Two-class emission traffic control for freeway systems. In: Proceedings of the 19th IFAC world congress, pp 936–941
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 55:85–99
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 80:384–408
Special report 209 (1994) Highway capacity manual, 3rd edn. Transportation Research Board, Washington DC
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 36:725–742
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Ferrara, A., Sacone, S., Siri, S. (2018). Second-Order Macroscopic Traffic Models. In: Freeway Traffic Modelling and Control. Advances in Industrial Control. Springer, Cham. https://doi.org/10.1007/978-3-319-75961-6_4
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DOI: https://doi.org/10.1007/978-3-319-75961-6_4
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