Abstract
Macroscopic traffic flow models are suited for large scale, network wide applications where the macro-characteristics of the flow are of prime interest. A clear understanding of the existing macro-level traffic flow models will help in modelling of varying traffic scenarios more accurately. Existing state-of-the-art reports on traffic flow models have not considered macro-level models exclusively. This paper gives a review of macroscopic modelling approaches used for traffic networks including recent research in the past decade. The modelling of the two main components of the network i.e. links and nodes are reviewed separately in two sections and solution procedures are discussed followed by a synthesis on the advantages and disadvantages of these models. This review should encourage efficient research in this area towards network level application of these models.
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References
Hoogendoorn, S.P., Bovy, P.H.L.: State-of-the-art of vehicular traffic flow modelling. In: Proc. Inst. Mech. Eng. I. 283–301 (2001)
Lighthill, M.J., Whitham, G.B.: On kinematic waves. II, A theory of traffic flow on long crowed roads. Proc. R. Soc. A 229, 281–345 (1955)
Richards, P.I.: Shockwaves on the highway. Oper. Res. 4(1), 42–51 (1956)
Transportation Research Board: Revised Monograph on Traffic Flow Theory: A State-of-the-Art- Report. Transportation Research Board, National Research Council, Washington, DC. http://www.tfhrc.gov/its/tft/tft.htm (1997). Accessed 14 March 2011
Munjal, P.K., Pipes, L.A.: Propagation of on-ramp density perturbations on unidirectional two and three lane freeways. Transp. Res. 5(4), 241–255 (1971)
Munjal, P.K., Hsu, Y.S., Lawrence, R.L.: Analysis and validation of lane drop effects on multilane freeways. Transp. Res. 5(4), 257–266 (1971)
Holland, E.N., Woods, A.W.: A continuum model for the dispersion of traffic on two-lane roads. Transp. Res. B 31(6), 473–485 (1997)
Greenberg, J.M., Klar, A., Rascle, M.: Congestion on multilane highways. SIAM J. Appl. Math. 63(3), 813–818 (2003)
Laval, J.A.: Some properties of a multi-lane extension of the kinematic wave model. Working paper, Intelligent Transportation Systems. University of California, Berkeley (2003)
Zhang, H.M., Jin, W.L.: A kinematic wave traffic flow model for mixed traffic. Transp. Res. Rec. 1802, 197–204 (2002)
Logghe, S., Immers, L.H.: Heterogeneous traffic flow modelling with the LWR model using passenger car equivalents. In: Proceedings of the 10th World Congress on ITS, Madrid. http://www.kuleuven.be/traffic/dwn/P2003E.pdf (2003). Accessed 13 March 2012
Chanut, S., Buisson, C.: A macroscopic model and its numerical solution for a two flow mixed traffic with different speeds and lengths. Transp. Res. Rec. 1852, 209–219 (2003)
Logghe, S.: Dynamic modelling of heterogeneous vehicular traffic. PhD Thesis, KU Leuven, Belgium (2003)
Wong, G.C.K., Wong, S.C.: A multi-class traffic flow model—an extension of LWR model with heterogeneous drivers. Transp. Res. A 36(9), 763–848 (2002)
Gavage, S.B., Colombo, R.M.: An n-populations model for traffic flow. Eur. J. Appl. Math. 14(5), 587–612 (2003)
Lebacque, J.P.: A two-phase bounded acceleration traffic flow model. Analytical solutions and applications. Transp. Res. Rec. 1852, 220–230 (2003)
Leclercq, L., Laval, J.A.: A multiclass car following rule based on the LWR model. In: Rolland C.A., Chevoir F., Gondrte P., Lassarre S., Lebaque J.P., Schreckenberg M. (eds.) Traffic and Granular Flow ’07, pp. 151–160. Springer, Heidelberg (2009)
Li, X., Lu, H.: An extension of LWR model considering slope and heterogeneous drivers. In: International Conference on Computer and Information Science, pp. 1236–1240. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5709505 (2010). Accessed 12 Nov 2011
Zhu, Z., Zhang, G., Wu, T.: Numerical analysis of freeway traffic flow dynamics under multi-class drivers. Transp. Res. Rec. 1852, 201–208 (2003)
Daganzo, C.F.: The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. B 28(4), 269–287 (1994)
Newell, G.F.: Comments on traffic dynamics. Transp. Res. B 23(5), 386–389 (1988)
Daganzo, C.F.: The cell transmission model, Part II: network traffic. Transp. Res. B 29(2), 79–93 (1995)
Nagel, K., Schreckenberg, M.: A cellular automation model for freeway traffic. J. Phys. I 2, 2221–2229 (1992)
Daganzo, C.F.: The lagged cell transmission model. In: Ceder A. (ed.) Transportation and Traffic Theory, Proceedings of the 14th International Symposium on Transportation and Traffic Theory, pp. 81–104. Pergamon, Amsterdam (1999)
Szeto, W.Y.: Enhanced lagged cell transmission model for dynamic traffic assignment. Transp. Res. Rec. 2085, 76–85 (2009)
Laval, J.A., Daganzo, C.: Working paper, Intelligent Transportation Systems. University of California, Berkeley (2004)
Hu, X., Wang, W., Sheng, H.: Urban Traffic flow prediction with variable cell transmission model. J. Transp. Sys. Eng. Inf. Technol. 10(4), 73–78 (2010)
Chen, X., Shi, Q., Li, L.: Location specific cell transmission model for freeway traffic. Tsinghua Sci. Technol. 15(4), 475–480 (2010)
Long, J., Gao, Z., Zhao, X., Lian, A., Orenstein, P.: Urban traffic jam simulation based on the cell transmission model. Netw. Spat. Econ. 11, 43–64 (2011)
Yperman, I.: The link transmission model for dynamic network loading. PhD Thesis, KU Leuven, Belgium (2007)
Nair, R., Mahmassani, H.S., Hooks, E.M.: A porous flow approach to modelling heterogeneous traffic in disordered system. Transp. Res. B (2011). doi:10.1016/j.trb.2011.05.009
Leveque, R.J.: Finite volume methods. In: Leveque, R.J. (ed.) Finite Volume Methods for Hyperbolic Problems, pp. 64–348. Cambridge University Press, Cambridge (2002)
Payne, H.J.: Models for freeway traffic and control. In: Bekey, G.A. (ed.) Mathematical Models of Public Systems, pp. 51–61. Simulation Council, La Jolla (1971)
Ross, P.: Traffic dynamics. Transp. Res. B 22(6), 421–435 (1988)
Kuhne, R.D.: Macroscopic freeway model for dense traffic—stop-start waves and incident detection. In: Volmuller J., Hamerslag R., Technische Universiteit Delft (eds.) Proceedings of the 9th International Symposium on Transportation and Traffic Theory, pp. 20–42. VNU Science Press, Utrecht (1984)
Kuhne, R.D.: Freeway control and incident detection using a stochastic continuum theory of traffic flow. In: Proceedings of the 1st International Conference on Applications of Advanced Technologies in Transportation Engineering, San Diego, CA, pp. 287–292 (1989)
Michalopoulos, P.G., Yi, P., Lyrintzis, A.S.: Continuum modelling of traffic dynamics on congested freeways. Transp. Res. B 27(4), 315–332 (1993)
Kotsialos, A., Papageorgiou, M.: Traffic flow modelling of large-scale motorway networks using the macroscopic modelling tool METANET. IEEE Trans. Intell. Transp. Syst. 3(4), 282–292 (2002)
Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)
Jin, W.L., Zhang, H.M.: The formation and structure of vehicle clusters in the Payne–Whitham traffic flow model. Transp. Res. B 37, 207–223 (2003)
Li, T.: Nonlinear dynamics of traffic jams. Physica D 207, 41–51 (2009)
Daganzo, C.F.: Requiem of second-order fluid approximations of traffic flow. Transp. Res. B 2(4), 277–286 (1995)
Liu, G., Lyrintzis, A.S., Michalopoulos, P.G.: Improved high-order model for freeway traffic flow. Transp. Res. Rec. 1644, 37–46 (1998)
Aw, A., Rascle, M.: Resurrection of “Second Order” models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000)
Moutari, S., Rascle, M.: A hybrid Lagrangian model based on the Aw–Rascle traffic flow model. SIAM J. Appl. Math. 68(2), 413–436 (2007)
Zhang, H.M.: A theory of non equilibrium traffic flow. Transp. Res. B 32(7), 485–498 (1998)
Zhang, H.M.: A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. B 36(3), 275–290 (2002)
Lebacque, J.P., Haj-Salem, H., Mammar, S.: Second order traffic flow modelling: supply–demand analysis of the inhomogeneous Riemann problem and of boundary conditions. In: Proceedings of the 16th Mini-EURO Conference and 10th Meet, pp. 108–115. EWGT. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.682 (2005). Accessed 22 Feb 2012
Lebacque, J.P., Mammar, S., Haj-Salem, H.: The Aw–Rascle and Zhang’s model: vacuum problems, existence and regularities of the solutions of the Riemann problem. Transp. Res. B 41, 710–721 (2007)
Helbing, D.: Gas-kinetic derivation of Navier–Stokes-like traffic equations. Phys. Rev. E 53(3), 2266–2381 (1996)
Prigogine, I., Andrews, F.C.: A Boltzmann-like approach of traffic flow. Oper. Res. 8(6), 789–797 (1960)
Paveri-Fontana, S.L.: On Boltzmann-like treatment of traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis. Transp. Res. 9, 225–235 (1973)
Hoogendoorn, S.P.: Multiclass continuum modelling of multiclass traffic flow. PhD Thesis, T99/5, TRAIL Thesis series, Delft University Press (1999)
Hoogendoorn, S.P., Bovy, P.H.L.: Modelling multiple user-class traffic flow. Transp. Res. B 34(2), 123–146 (2000)
Prigogine, I., Herman, R.: Kinetic Theory of Vehicular Traffic. American Elsevier, New York (1971)
Li, S.F., Zhang, P., Wong, S.C.: Conservation form of Helbing’s fluid dynamic traffic flow model. Appl. Math. Mech. (Engl. Ed.) 32(9), 1109–1118 (2011)
Helbing, D.: Derivation and empirical validation of a refined traffic flow model. Physica A 233(1–2), 253–282 (1996)
Helbing, D., Hennecke, A., Shvetsov, V., Trieber, M.: MASTER: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model. Transp. Res. B 35(2), 183–211 (2001)
Kerner, B.S., Rehborn, H.: Experimental properties of phase transitions in traffic flow. Phys. Rev. Lett. 49, 4030–4033 (1997)
Hoogendoorn, S.P., Bovy, P.H.L.: Multiclass macroscopic traffic flow modelling: a multi-lane generalization using gas-kinetic theory. In: Ceder A. (ed.) Transportation and Traffic Theory, Proceedings of the 14th International Symposium on Transportation and Traffic Theory, pp. 27–50. Pergamon, Amsterdam (1999)
Jiang, R., Wu, Q.S., Zhu, Z.J.: A new continuum model for traffic flow and numerical tests. Transp. Res. B 36, 405–419 (2002)
Helbing, D., Trieber, M.: Numerical simulation of macroscopic traffic equations. Comput. Sci. Eng. 1(5), 89–98 (1999)
Gazis, D.C., Herman, R., Rothery, R.W.: Nonlinear follow-the-leader models of traffic flow. Oper. Res. 9, 545–567 (1961)
Bando, M., Hasabe, K., Nakayama, A., Shibata, A., Sugiyama, Y.: Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035–1042 (1995)
Jiang, R., Wu, Q.S.: Extended speed gradient model for mixed traffic. Transp. Res. Rec. 1883, 78–84 (2004)
Tang, C.F., Jiang, R., Wu, Q.S., Wiwathanapataphee, B., Hu, Y.H.: Mixed traffic flow in anisotropic continuum model. Transp. Res. Rec. 1999, 13–22 (2007)
Del Castillo, J.M., Pintado, P., Benitez, F.G.: A formulation for the reaction time of traffic flow models. In: Daganzo C. (ed.) Transportation and Traffic Theory, Proceedings of the 12th International Symposium on Transportation and Traffic Theory, pp. 387–405. Elsevier, Amsterdam (1993)
Tang, C.F., Jiang, R., Wu, Q.S.: Extended speed gradient model for traffic flow on two-lane freeways. Chin. Phys. 16(6), 1570–1575 (2007)
Tang, T.Q., Huang, H.J., Zhao, S.G., Shang, H.Y.: A new dynamic model for heterogeneous traffic flow. Phys. Lett. A 373, 2461–2466 (2009)
Darbha, S., Rajagopal, K.R.: Aggregation of a class of nonlinear interconnected dynamical systems. Math. Probl. Eng. Theory Methods Appl. 7, 379–392 (2001)
Zhang, P., Wong, S.C., Shu, C.W.: A weighted essentially non-oscillatory numerical scheme for a multi-class traffic flow model on an inhomogeneous highway. J. Comput. Phys. 212(2), 739–756 (2006)
Ngoduy, D., Hoogendoorn, S.P., Van Zuylen, H.J.: Cross-comparison of numerical schemes for macroscopic traffic flow models. Transp. Res. Rec. 1876, 52–61 (2004)
Cremer, M., Papageorgiou, M.: Parameter identification for a traffic flow model. Automatica 17(6), 837–843 (1981)
Mohan, R., Ramadurai, G.: A Case for advanced traffic flow models in DTA. In: Presented at the 4th International Symposium on Dynamic Traffic Assignment. Martha Vineyard, Vineyard Haven (2012)
Papageorgiou, M.: Some remarks on macroscopic traffic flow modelling. Transp. Res. A 32(5), 323–329 (1997)
Daganzo, C.F., Lin, W.H., Del Castillo, J.M.: A simple physical principle for the simulation of freeways with special lanes and priority vehicles. Transp. Res. B 31(2), 103–125 (1997)
Helbing, D.: Derivation of non local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models. Eur. Phys. J. B 69, 539–548 (2009)
Bagnerini, P., Rascle, M.: A multi-class homogenized hyperbolic model of traffic flow. SIAM J. Math. Anal. 35(4), 949–973 (2003)
Colombo, R.M.: A 2 × 2 hyperbolic traffic flow model. Math. Comput. Model. 35, 683–688 (2002)
Tampere, C.M.J., van Arem, B., Hoogendoorn, S.P.: Gas Kinetic traffic flow modelling including continuous driver behavior models. Transp. Res. Rec. 1852, 231–238 (2003)
Jin, W.L., Zhang, H.M.: On the distribution schemes for determining flows through a merge. Transp. Res. B 37(6), 521–540 (2003)
Lebacque, J.P.: The Godunov scheme and what it means for first order traffic flow models. In: Lesort J.B. (ed.) Transportation and Traffic Theory, Proceedings of the 13th International Symposium on Transportation and Traffic Theory, pp. 647–677. Pergamon, Tarrytown, New York (1996)
Tampere, C.M.J., Corthout, R., Cattrysse, D., Immerse, L.H.: A generic class of first order node models for dynamic macroscopic simulation of traffic flows. Transp. Res. B 45, 289–309 (2011)
Ni, D., Leonard, J.D.: A simplified kinematic wave model at a merge bottleneck. Appl. Math. Model. 29(11), 1054–1072 (2005)
Lebacque, J.P.: Intersection modelling, application to macroscopic network traffic flow models and traffic management. In: Hoogerdorn SP., Luding S., Bovy P.H.L., Schreckenberg M., Wolf D.E. (eds.) Traffic and Granular Flow ’03, pp. 261–278. Springer, Heidelberg (2005)
Holden, H., Riserbo, N.H.: A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal. 26(4), 999–1017 (1995)
Herty, M., Klar, A.: Modelling, simulation and optimization of traffic flow networks. SIAM J. Sci. Comput 25(3), 1066–1087 (2003)
Coclite, G.M., Garavello, M., Piccoli, B.: Traffic flow on a road network. SIAM J. Math. Anal. 36(6), 1862–1886 (2005)
Durlin, T., Henn, V.: A delayed flow intersection model for dynamic traffic assignment. In: Proceedings of the 16th Mini-EURO Conference and 10th Meet, pp. 78–81. EWGT. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.77.2228 (2005). Accessed 14 March 2011
Lebacque, J.P., Koshyaran, M.M.: First order macroscopic traffic flow models: intersection modelling, network modelling. In: Mahmassani H.S. (ed.) Transportation and Traffic Theory, Proceedings of the 16th International Symposium on Transportation and Traffic Theory, pp. 365–386. Elsevier, Amsterdam (2005)
Garavello, M., Piccoli, B.: Traffic flow on a road network using the Aw–Rascle model. Commun. Partial Differ. Equ. 31(2), 243–275 (2006)
Herty, M., Rascle, M.: Coupling conditions for a class of “second-order” models for traffic flow. SIAM J. Math. Anal. 38(2), 595–616 (2006)
Herty, M., Moutari, S., Rascle, M.: Intersection modelling with a class of “second-order” models for vehicular traffic flow. In: Gavage, S.B., Serre, D. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 755–763. Springer, Heidelberg (2008)
Gentile, G., Meschini, L., Papola, N.: Fast heuristics for continuous dynamic shortest paths and all-or-nothing assignment. In: Proceedings of the 35th Annual Conference of the Italian Operation Research Society. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.157.2983 (2004). Accessed 5 May 2011
Gentile, G., Meschini, L., Papola, N.: Spillback congestion in dynamic traffic assignment: a macroscopic flow model with time varying bottlenecks. Transp. Res. B 41, 1114–1138 (2007)
Bliemer, M.J.: Dynamic queuing and spillback in an analytical multiclass network loading model. Transp. Res. Rec. 2029, 14–21 (2007)
Yuan, S.X., An, Y.S., Zhao, X., Fan, H.W.: Modelling and simulation of dynamic traffic flows at non-signalized T-intersections. In: Proceedings of the 2nd International Conference on Power Electronics and Intelligent Transportation Systems. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5406951 (2009). Accessed 16 May 2011
Tolba, C., Lefebvre, D., Thomas, P., El Moudni, A.: Continuous and timed Petri nets for the macroscopic and microscopic traffic flow modelling. Simul. Model. Pract. Theory 13, 407–436 (2005)
Liu, X., Dai, S.: Macroscopic models for interrupted traffic flow of signal controlled intersections. In: ISECS International Colloquium on Computing, Communication, Control and Management, pp. 78–81. http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5268043 (2009). Accessed 15 May 2011
Jin, W.L.: Continuous kinematic wave models of merging traffic flow. Transp. Res. B 44, 1084–1103 (2010)
Flotterod, G., Rohde, J.: Operational macroscopic modelling of complex urban road intersections. Transp. Res. B 45, 903–922 (2011)
Corthout, R., Flotterod, G., Viti, F., Tampere, C.M.J.: Non-unique flows in macroscopic first order intersection models. Transp. Res. B 46(3), 343–359 (2012)
Tuerprasert, K., Aswakul, C.: Multiclass cell transmission model for heterogeneous mobility in general topology of road network. J. Intell. Transp. Syst. Technol. Plan. Oper. 14(2), 68–82 (2010)
Lo, H.K., Szeto, W.Y.: A cell-based dynamic traffic assignment model: formulation and properties. Math. Comput. Model. 35(7–8), 849–865 (2002)
Acknowledgements
The authors thank the Ministry of Urban Development, Government of India, for sponsoring the Center of Excellence in Urban Transport at Indian Institute of Technology (IIT), Madras that enabled this research work. The second author also thanks the New Faculty Grant provided by IIT Madras that partially funded this research work. All findings and opinions in the paper are the authors and do not necessarily reflect the views of the funding agencies.
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Mohan, R., Ramadurai, G. State-of-the art of macroscopic traffic flow modelling. Int J Adv Eng Sci Appl Math 5, 158–176 (2013). https://doi.org/10.1007/s12572-013-0087-1
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DOI: https://doi.org/10.1007/s12572-013-0087-1