Nonlinear Dynamics

, Volume 67, Issue 3, pp 2255–2265 | Cite as

A new fundamental diagram theory with the individual difference of the driver’s perception ability

  • Tieqiao Tang
  • Chuanyao Li
  • Haijun Huang
  • Huayan Shang
Original Paper


Based on the driver’s individual difference of the driver’s perception ability, we in this paper develop a new fundamental diagram with the driver’s perceived error and speed deviation difference. The analytical and numerical results show that the speed-density and flow-density data are divided into three prominent regions. In the first region, the speed-density and flow-density data are scattered around the equilibrium speed-density and flow-density curves of the classical fundamental diagram theory, where the widths of these scattered data are very narrow and slightly increase with the real density (i.e., the scattered data appear as two thicker lines); the running speed is approximately equal to the free flow speed and the real flow approximately linearly increases with the real density. In the second region, the speed-density and flow-density data are scattered widely in a two-dimensional region, but the shapes of these widely scattered data are related to the properties of the driver’s perceived error and speed deviation difference. In the third region, the scattered speed-density and flow-density data appear but these scattered data will quickly degenerate into the equilibrium speed-density and flow-density curves with the increase of the real density. Finally, the numerical results illustrate that the new fundamental diagram theory also produces the F-line, U-line, and L-line. The shapes of the scattered data, F-line, U-line, and L-line are relevant to the properties of the driver’s perceived error and speed deviation difference. These results are qualitatively accordant with the real traffic, which shows that the new fundamental diagram theory can better describe some complex traffic phenomena in the real traffic system. In addition, the above results can help us to further explain why the widely scattered speed-density and flow-density data appear in the real traffic system and better understand the effects of the driver’s individual difference on traffic flow.


Fundamental diagram theory Driver’s perceived error Driver’s speed deviation difference Equilibrium flow 


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  1. 1.
    Chowdhury, D., Santen, L., Schreckenberg, A.: Statistics physics of vehicular traffic and some related systems. Phys. Rep. 329, 199–329 (2000) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Kerner, B.S.: Three-phase traffic theory and highway capacity. Physica A 333, 379–440 (2004) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Kerner, B.S., Rehborn, H.: Experimental features and characteristic of traffic jams. Phys. Rev. E 53, R1297–R1300 (1996) CrossRefGoogle Scholar
  4. 4.
    Kerner, B.S., Rehborn, H.: Experimental properties of complexity in traffic flow. Phys. Rev. E 53, R4275–R4278 (1996) CrossRefGoogle Scholar
  5. 5.
    Kerner, B.S., Rehborn, H.: Experimental properties of phase transitions in traffic flow. Phys. Rev. Lett. 79, 4030–4033 (1997) CrossRefGoogle Scholar
  6. 6.
    Kerner, B.S., Klenov, S.L., Konhäuster, P.: Asymptotic theory of traffic jams. Phys. Rev. E 56, 4200–4216 (1997) CrossRefGoogle Scholar
  7. 7.
    Kerner, B.S.: Experimental features of self-organization in traffic flow. Phys. Rev. Lett. 81, 3797–3800 (1998) CrossRefzbMATHGoogle Scholar
  8. 8.
    Kerner, B.S.: A theory of congested traffic flow. In: Proceedings of the 3rd International Symposium on Highway Capacity, Road Directorate, Denmark, vol. 2, pp. 621–642 (1998) Google Scholar
  9. 9.
    Kerner, B.S.: Congested traffic flow: observation and theory. Transp. Res. Rec. 1678, 160–167 (1999) CrossRefGoogle Scholar
  10. 10.
    Kerner, B.S.: Experimental features of the emergence of moving jams in free traffic flow. J. Physics A 33, L221–L228 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kerner, B.S.: Theory of breakdown phenomena at highway bottlenecks. Transp. Res. Rec. 1710, 136–144 (2000) CrossRefGoogle Scholar
  12. 12.
    Kerner, B.S.: Empirical features of congested patterns at highway bottlenecks. Transp. Res. Rec. 1802, 145–154 (2002) CrossRefGoogle Scholar
  13. 13.
    Kerner, B.S.: Empirical macroscopic features of spatio-temporal traffic pattern at highway bottlenecks. Phys. Rev. E 65, 046138 (2002) CrossRefGoogle Scholar
  14. 14.
    Kerner, B.S., Klenov, S.L.: A microscopic model for phase transition in traffic flow. J. Physics A 35, L31–L43 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kerner, B.S., Klenov, S.L., Wolf, D.E.: Cellular automata approach to three-phase traffic theory, J. Physics A 35, 9971–10013 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kerner, B.S., Klenov, S.L.: Microscopic theory of spatial-temporal congested traffic patterns at highway bottlenecks. Phys. Rev. E 68, 036130 (2003) CrossRefGoogle Scholar
  17. 17.
    Kerner, B.S.: The Physics of Traffic. Springer, Heidelberg (2004) Google Scholar
  18. 18.
    Kerner, B.S., Rehborn, H., Aleksic, M., Haug, A.: Recognition and tracking of spatial-temporal congested traffic patterns on freeways. Transp. Res. C 12, 369–400 (2004) CrossRefGoogle Scholar
  19. 19.
    Kerner, B.S.: Control of spatialtemporal congested traffic patterns at highway bottleneck. Physica A 355, 565–601 (2005) CrossRefGoogle Scholar
  20. 20.
    Kerner, B.S.: Microscopic three-phase traffic theory and its application for freeway traffic control. In: Proceeding of International Symposium on Transportation and Traffic Theory, pp. 181–203. Elsevier, Amsterdam (2005) CrossRefGoogle Scholar
  21. 21.
    Kerner, B.S., Klenov, S.L., Hiller, A., Rehborn, H.: Microscopic features of moving traffic jams. Phys. Rev. E 73, 046107 (2006) CrossRefGoogle Scholar
  22. 22.
    Kerner, B.S., Klenov, S.L.: Probabilistic breakdown phenomenon at on-ramp bottlenecks in three-phase traffic theory: congestion nucleation in spatially non-homogeneous traffic. Physica A 364, 473–492 (2006) CrossRefGoogle Scholar
  23. 23.
    Kerner, B.S.: On-ramp metering based on three-phase traffic theory. Traffic Eng. Control 48, 28–35 (2007) Google Scholar
  24. 24.
    Kerner, B.S.: Phase transition in traffic flow on multilane roads. Phys. Rev. E 80, 056101 (2009) CrossRefGoogle Scholar
  25. 25.
    Kerner, B.S.: Introduction to Modern Traffic Flow Theory and Control. Springer, Berlin (2009) CrossRefzbMATHGoogle Scholar
  26. 26.
    Gupta, A.K., Katiyar, V.K.: Phase transition of traffic states with on-ramp. Physica A 371, 674–682 (2006) CrossRefGoogle Scholar
  27. 27.
    Davis, L.C.: Driver choice compared to controlled diversion for a freeway double on-ramp in the framework of three-phase traffic theory. Physica A 387, 6395–6410 (2008) CrossRefGoogle Scholar
  28. 28.
    Gao, K., Jiang, R., Wang, B.H., Wu, Q.S.: Discontinuous transition from free flow to synchronized flow induced by short-range interaction between vehicles in three-phase traffic flow model. Physica A 388, 3233–3243 (2009) CrossRefGoogle Scholar
  29. 29.
    Tian, J.F., Jia, B., Li, X.G., Jiang, R., Zhao, X.M., Gao, Z.Y.: Synchronized traffic flow simulating with cellular automaton model. Physica A 388, 4827–4837 (2009) CrossRefGoogle Scholar
  30. 30.
    Huang, D.W.: How the on-ramp inflow causes bottleneck. Physica A 388, 63–70 (2009) CrossRefGoogle Scholar
  31. 31.
    Zhao, B.H., Hu, M.B., Jiang, R., Wu, Q.S.: A realistic cellular automaton model for synchronized traffic flow. Chin. Phys. Lett. 26, 118902 (2009) CrossRefGoogle Scholar
  32. 32.
    He, S., Guan, W., Song, L.: Explaining traffic patterns at on-ramp vicinity by a driver perception model in the framework of three-phase traffic theory. Physica A 389, 825–836 (2010) CrossRefGoogle Scholar
  33. 33.
    Lighthill, M.J., Whitham, G.B.: On kinematic waves II: a theory of traffic on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317–345 (1955) CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956) CrossRefMathSciNetGoogle Scholar
  35. 35.
    Payne, H.J.: Models of freeway traffic and control. In: Bekey, G.A. (ed.) Mathematical Models of Public System. Simulation Councils Proceedings Series, vol. 1, pp. 51–61 (1971) Google Scholar
  36. 36.
    Gazis, D.C., Herman, R.: The moving and “phantom” bottlenecks. Transp. Sci. 26, 223–229 (1992) CrossRefzbMATHGoogle Scholar
  37. 37.
    Kerner, B.S., Konhäuster, P.: Cluster effect in initial homogeneous traffic flow. Phys. Rev. E 48, R2335–R2338 (1993) CrossRefGoogle Scholar
  38. 38.
    Daganzo, C.F.: The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory. Transp. Res. B 28, 269–287 (1994) CrossRefGoogle Scholar
  39. 39.
    Bando, M., Hasebe, K., Nakayama, A., Shibata, A., Sugiyama, Y.: Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035–1042 (1995) CrossRefGoogle Scholar
  40. 40.
    Helbing, D.: Improved fluid-dynamic model for vehicular traffic. Phys. Rev. E 51, 3164–3169 (1995) CrossRefGoogle Scholar
  41. 41.
    Helbing, D.: Gas-kinematic derivation of Navier-Stokes-like traffic equations. Phys. Rev. E 53, 2366–2381 (1996) CrossRefMathSciNetGoogle Scholar
  42. 42.
    Wagner, C., Hoffmann, C., Sollacher, R., Wagenhuber, J., Schürmann, B.: Second-order continuum traffic flow model. Phys. Rev. E 54, 5073–5085 (1996) CrossRefGoogle Scholar
  43. 43.
    Daganzo, C.F.: A continuum theory of traffic dynamics for freeways with special lanes. Transp. Res. B 31, 83–102 (1997) CrossRefGoogle Scholar
  44. 44.
    Daganzo, C.F., Li, W.H., Castillo, J.M.: A simple physical principle for the simulation of freeways with special lanes and priority vehicles. Transp. Res. B 31, 103–125 (1997) CrossRefGoogle Scholar
  45. 45.
    Holland, E.N., Woods, A.W.: A continuum model for dispersion of traffic on two-lane roads. Transp. Res. B 31, 473–485 (1997) CrossRefGoogle Scholar
  46. 46.
    Zhang, H.M.: A theory of nonequilibrium traffic flow. Transp. Res. B 32, 485–498 (1998) CrossRefGoogle Scholar
  47. 47.
    Helbing, D., Treiber, M.: Numerical simulation of macroscopic traffic equations. Comput. Sci. Eng. 1, 89–99 (1999) CrossRefGoogle Scholar
  48. 48.
    Hebling, D., Hennecke, A., Treiber, M.: Phase diagram of traffic states in the presence of inhomogeneities. Phys. Rev. Lett. 82, 4360–4363 (1999) CrossRefGoogle Scholar
  49. 49.
    Shvetsov, V., Helbing, D.: Macroscopic dynamics of multilane traffic. Phys. Rev. E 59, 6328–6339 (1999) CrossRefGoogle Scholar
  50. 50.
    Treiber, M., Hennecke, A., Helbing, D.: Derivation, properties, and simulation of a gas-kinematic-based, non-local traffic model. Phys. Rev. E 59, 239–253 (1999) CrossRefGoogle Scholar
  51. 51.
    Aw, A., Rascle, M.: Resurrection of “second-order” models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    Treiber, M., Hennecke, A., Helbing, D.: Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E 62, 1805–1824 (2000) CrossRefGoogle Scholar
  53. 53.
    Nelson, P.: Synchronized traffic flow from a modified Lighthill-Whitham model. Phys. Rev. E 61, R6052–R6055 (2000) CrossRefGoogle Scholar
  54. 54.
    Helbing, D.: Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001) CrossRefGoogle Scholar
  55. 55.
    Helbing, D., Hennecke, A., Shvetsov, V., Treiber, M.: MASTER: macroscopic traffic simulation based on a gas-kinematic, non-local traffic model. Transp. Res. B 35, 183–211 (2001) CrossRefGoogle Scholar
  56. 56.
    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, 827–841 (2002) Google Scholar
  57. 57.
    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) CrossRefGoogle Scholar
  58. 58.
    Munoz, J.C., Daganzo, C.F.: Moving bottlenecks: a theory grounded on experimental observation. In: Proceeding of the 15th International Symposium of Traffic and Transportation Theory, pp. 441–462. Pergamon, Oxford (2002) Google Scholar
  59. 59.
    Aw, A., Klar, A., Materne, T., Rascle, M.: Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J. Appl. Math. 63, 259–278 (2002) CrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    Zhang, H.M.: A nonequilibrium traffic model devoid of gas-like behavior. Transp. Res. B 36, 275–290 (2002) CrossRefGoogle Scholar
  61. 61.
    Xue, Y., Dai, S.Q.: Continuum traffic model with the consideration of two delay scales. Phys. Rev. 68, 066123 (2003) Google Scholar
  62. 62.
    Helbing, D.: A section-based queueing-theoretical traffic model for congestion and travel time analysis in networks. J. Physics A 36, L593–L598 (2003) CrossRefzbMATHMathSciNetGoogle Scholar
  63. 63.
    Benzoni-Gavage, S., Colombo, R.: An n-populations model for traffic flow. Euro. J. Appl. Math. 14, 587–612 (2003) zbMATHMathSciNetGoogle Scholar
  64. 64.
    Tang, T.Q., Huang, H.J., Gao, Z.Y.: Stability of the car-following model on two lanes. Phys. Rev. E 72, 066124 (2005) CrossRefGoogle Scholar
  65. 65.
    Ossen, S.J., Hoogendoorn, S.P.: Car-following behavior analysis from microscopic trajectory data. Transp. Res. Rec. 1934, 13–21 (2005) CrossRefGoogle Scholar
  66. 66.
    Nagel, K., Nelson, P.: A critical comparison of the kinematic-wave model with observational data. In: Proceeding of the 15th International Symposium of Traffic and Transportation Theory, pp. 145–163. Elsevier, Amsterdam (2005) CrossRefGoogle Scholar
  67. 67.
    Treiber, M., Kesting, A., Helbing, D.: Delays, inaccuracies and anticipation in microscopic traffic models. Physica A 360, 71–88 (2006) CrossRefGoogle Scholar
  68. 68.
    Huang, H.J., Tang, T.Q., Gao, Z.Y.: Continuum modeling for two-lane traffic flow. Acta Mech. Sin. 22, 131–137 (2006) CrossRefzbMATHGoogle Scholar
  69. 69.
    Gupta, A.K., Katiyar, V.K.: A new multi-class continuum model for traffic flow. Transportmetrica 3, 73–85 (2007) CrossRefGoogle Scholar
  70. 70.
    Helbing, D., Tilch, B.: A power law for the duration of high-flow states and its interpretation from heterogeneous traffic flow perspective. Eur. Phys. J. B 68, 577–586 (2009) CrossRefzbMATHGoogle Scholar
  71. 71.
    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) CrossRefMathSciNetGoogle Scholar
  72. 72.
    Treiber, M., Kesting, A., Helbing, D.: Understanding widely scattered traffic flows, the capacity drop, and platoons as effects of variance-driven time gaps. Phys. Rev. E 74, 016123 (2006) CrossRefGoogle Scholar
  73. 73.
    Schönhof, M., Helbing, D.: Empirical features of congested traffic states and their implications for traffic modeling. Transp. Sci. 41, 135–166 (2007) CrossRefGoogle Scholar
  74. 74.
    Schönhof, M., Helbing, D.: Criticism of three-phase traffic theory. Transp. Res. B 43, 784–797 (2009) CrossRefGoogle Scholar
  75. 75.
    Treiber, M., Kesting, A., Helbing, D.: Three-phase traffic theory and two-phase models with fundamental diagram in the light of empirical stylized facts. Transp. Rev. B 44, 983–1000 (2010) CrossRefGoogle Scholar
  76. 76.
    Ngoduy, D.: Multiclass first order modelling of traffic networks using discontinuous flow-density relationships. Transportmetrica 6, 121–141 (2010) CrossRefGoogle Scholar
  77. 77.
    Bank, J.H.: Investigation of some characteristics of congested flow. Transp. Res. Rec. 1678, 128–134 (1999) CrossRefGoogle Scholar
  78. 78.
    Nishinari, K., Treiber, M., Helbing, D.: Interpreting the wide scattering of the synchronized traffic data by time gap statistics. Phys. Rev. E 68, 067101 (2003) CrossRefGoogle Scholar
  79. 79.
    Treiber, M., Helbing, D.: Macroscopic simulation of widely scattered synchronized traffic states. J. Physics A 32, L17–L23 (1999) CrossRefGoogle Scholar
  80. 80.
    Treiber, M., Helbing, D.: Memory effects in microscopic traffic models and wide scattering in flow-density data. Phys. Rev. E 68, 046119 (2003) CrossRefGoogle Scholar
  81. 81.
    Ngoduy, D.: Multiclass first-order traffic model using stochastic fundamental diagrams. Transportmetrica 7, 111–125 (2011) CrossRefGoogle Scholar
  82. 82.
    Yang, H.H., Peng, H.: Development of an errorable car-following driver model. Veh. Syst. Dyn. 48, 751–773 (2010) CrossRefGoogle Scholar
  83. 83.
    Li, J., Chen, Q.Y., Wang, Y., Ni, D.: Investigation of LWR model with flux function driven by random free flow speed. In: The 88th Transportation Research Board Annual Meeting, DVD-ROM, Washington, DC Google Scholar
  84. 84.
    Daganzo, C.F.: Requiem for second-order fluid approximations of traffic flow. Transp. Res. B 29, 277–286 (1995) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Tieqiao Tang
    • 1
    • 2
  • Chuanyao Li
    • 1
  • Haijun Huang
    • 2
  • Huayan Shang
    • 3
  1. 1.School of Transportation Science and EngineeringBeihang UniversityBeijingChina
  2. 2.School of Economics and ManagementBeihang UniversityBeijingChina
  3. 3.Information CollegeCapital University of Economics and BusinessBeijingChina

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