An extended macroscopic model for traffic flow on curved road and its numerical simulation

  • Yu XueEmail author
  • Yicai Zhang
  • Deli Fan
  • Peng Zhang
  • Hong-di He
Original Paper


In this paper, we propose a full angular velocity difference model by introducing the angular velocity and displacement on curved road. The relation of transformation of microscopic model in form of angular variables into macroscopic one is deduced. Consequently, a corresponding continuum traffic flow model on curved road is derived. The critical condition for the steady traffic flow is obtained. Meanwhile, the stability condition of this continuum traffic model is compared with one of the microscopic model and lattice hydrodynamic traffic model on curved road. By nonlinear analysis, the KdV–Burgers equation which describes the density wave near the neutral stability line is derived. Furthermore, the numerical simulations are carried out to verify the validity of this macroscopic traffic. Results indicate the radius of curved road have a great impact on the formation of traffic jams. Local clustering effects in the state of instability of traffic flow give rise to the stop&go traffic jams. Numerical simulation also indicates the unstable region is shrunken under the increasing strength of the angular velocity difference.


Traffic flow Macroscopic model KdV–Burgers equation Curve road Radius 



The project supported by the National Natural Science Foundation of China (Grant Nos. 11262003 and 11302125) and the Natural Science Foundation of Guangxi, China (Grant No. 2018GXNSFAA138205).


  1. 1.
    Helbing, D.: Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Nagatani, T.: The physics of traffic jams. Rep. Progr. Phys. 65, 1331–1386 (2002)CrossRefGoogle Scholar
  3. 3.
    Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329, 199–329 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Helbing, D., Treiber, M.: Gas-kinetic-based traffic model explaining observed hysteretic phase transition. Phys. Rev. Lett. 81, 3042–3045 (1998)CrossRefGoogle Scholar
  5. 5.
    Newell, G.F.: Nonlinear effects in the dynamics of car following. Oper. Res. 9, 209–229 (1961)CrossRefzbMATHGoogle Scholar
  6. 6.
    Lighthill, M.J., Whitham, G.B.: On kinematic waves I. Flood movement in long rivers. Proc. R. Soc. Lond. A 229, 281–316 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Richards, P.I.: Shockwaves on the highway. Oper. Res. 4, 42–51 (1956)CrossRefGoogle Scholar
  8. 8.
    Whitham, G.B.: Exact solutions for a discrete system arising in traffic flow. Proc. R. Soc. Lond. 428, 49–69 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bando, M., Hasebe, K., Nakayama, A.: Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035–1042 (1995)CrossRefGoogle Scholar
  10. 10.
    Kai, N., Schreckenberg, M.: A cellular automaton model for freeway traffic. J. Phys. I(2), 2221–2229 (1992)Google Scholar
  11. 11.
    Paveri-Fontana, S.L.: On Boltzmann-like treatments for traffic flow: a critical review of the basic model and an alternative proposal for dilute traffic analysis. Transp. Res. 9, 225–235 (1975)CrossRefGoogle Scholar
  12. 12.
    Helbing, D.: Gas-kinetic derivation of Navier–Stokes-like traffic equations. Phys. Rev. E 53, 2366–2381 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Helbing, D.: Derivation and empirical validation of a refined traffic flow model. Physica A 233, 253–282 (1998)CrossRefGoogle Scholar
  14. 14.
    Lee, H.Y., Lee, H.W., Kim, D.: Origin of synchronized traffic flow on highways and its dynamic phase transitions. Phys. Rev. Lett. 81, 1130–1133 (1998)CrossRefGoogle Scholar
  15. 15.
    Kerner, B.S., Konhäuser, P.: Cluster effect in initially homogeneous traffic flow. Phys. Rev. E 48, R2335–R2338 (1993)CrossRefGoogle Scholar
  16. 16.
    Kerner, B.S., Konhäuser, P., Schilke, M.: Deterministic spontaneous appearance of traffic jams in slightly inhomogeneous traffic flow. Phys. Rev. E 51, 6243–6246 (1995)CrossRefGoogle Scholar
  17. 17.
    Kurtze, D.A., Hong, D.C.: Traffic jams, granular flow, and soliton selection. Phys. Rev. E 52, 218–221 (1995)CrossRefGoogle Scholar
  18. 18.
    Komatsu, T.S., Sasa, S.: Kink soliton characterizing traffic congestion. Phys. Rev. E 52, 5574–5582 (1995)CrossRefGoogle Scholar
  19. 19.
    Jiang, R., Wu, Q.S., Zhu, Z.J.: Full velocity difference model for a car-following theory. Phys. Rev. E 64, 017101(1)–017101(4) (2001)CrossRefGoogle Scholar
  20. 20.
    Payne, H.J.: Models of freeway traffic and control. Math. Methods Public Syst. 1, 51–61 (1971)Google Scholar
  21. 21.
    Daganzo, C.F.: Requiem for second-order fluid approximations of traffic flow. Transp. Res. Part B 29, 277–286 (1995)CrossRefGoogle Scholar
  22. 22.
    Aw, A., Rascle, M.: Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60, 916–938 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhang, H.M.: A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. Part B 36, 275–290 (2002)CrossRefGoogle Scholar
  24. 24.
    Pipes, L.A.: An operational analysis of traffic dynamics. J. Appl. Phys. 24, 274–281 (1953)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Jiang, R., Wu, Q.S., Zhu, Z.J.: A new continuum model for traffic flow and numerical tests. Transp. Res. Part B 36, 405–419 (2002)CrossRefGoogle Scholar
  26. 26.
    Xue, Y., Dai, S.Q.: Continuum traffic model with the consideration of two delay time scales. Phys. Rev. E 68, 066123(1)–066123(6) (2003)CrossRefGoogle Scholar
  27. 27.
    Berg, P., Mason, A., Woods, A.: Continuum approach to car-following models. Phys. Rev. E 61(2), 1056–1066 (2000)CrossRefGoogle Scholar
  28. 28.
    Cheng, R.J., Ge, H.X., Wang, J.F.: KdV–Burgers equation in a new continuum model based on full velocity difference model considering anticipation effect. Physica A 481, 52–59 (2017)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tang, T.Q., Huang, H.J., Xu, G.: A new macro model with consideration of the traffic interruption probability. Chin. Phys. B 387, 975–983 (2009)Google Scholar
  30. 30.
    Davoodi, N., Soheili, A.R., Hashemi, S.M.: A macro-model for traffic flow with consideration of driver’s reaction time and distance. Nonlinear Dyn. 83, 1621–1628 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Berg, P., Woods, A.W.: On-ramp simulations and solitary waves in a car-following model. Phys. Rev. E 64, 035602(1)–035602(4) (2001)CrossRefGoogle Scholar
  32. 32.
    Herrmann, M., Kerner, B.S.: Local cluster effect in difference traffic flow models. Physica A 255, 163–188 (1998)CrossRefGoogle Scholar
  33. 33.
    Helbing, D.: Derivation of non-local macroscopic traffic equations and consistent traffic pressures from microscopic car-following models. Eur. Phys. J. B 69(4), 539–548 (2009)CrossRefGoogle Scholar
  34. 34.
    Tang, T.Q., Huang, H.J., Shang, H.Y.: An extended macro traffic flow model accounting for the driver’s bounded rationality and numerical tests. Physica A 468, 322–333 (2017)CrossRefGoogle Scholar
  35. 35.
    Tang, T.Q., Li, P., Yang, X.B.: An extended macro model for traffic flow with consideration of multi static bottlenecks. Physica A 392, 3537–3545 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tang, T.Q., Huang, H.J., Shang, H.Y.: A new macro model for traffic flow with the consideration of the driver’s forecast effect. Phys. Lett. A 374, 1668–1672 (2010)CrossRefzbMATHGoogle Scholar
  37. 37.
    Peng, G.H., Song, W., Peng, Y.J., Wang, S.H.: A novel macro model of traffic flow with the consideration of anticipation optimal velocity. Physica A 398, 76–82 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sun, D.H., Peng, G.H., Fu, L.P., He, H.P.: A continuum traffic flow model with the consideration of coupling effect for two-lane freeways. Acta Mech. Sin. 27(2), 228–236 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zhai, Q.T., Ge, H.X., Cheng, R.J.: An extended continuum model considering optimal velocity change with memory and numerical tests. Physica A 490, 774–785 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Cheng, R.J., Ge, H.X., Wang, J.F.: An extended macro traffic flow model accounting for multiple optimal velocity functions with different probabilities. Phys. Lett. A 381, 2608–2620 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Liu, H.Q., Cheng, R.J., Zhu, K.Q., Ge, H.X.: The study for continuum model considering traffic jerk effect. Nonlinear Dyn. 83, 57–64 (2016)CrossRefGoogle Scholar
  42. 42.
    Gupta, A.K., Sharma, S.: Nonlinear analysis of traffic jams in an anisotropic continuum model. Chin. Phys. B 19, 110503 (2010)CrossRefGoogle Scholar
  43. 43.
    Gupta, A.K., Katiyar, V.K.: Analyses of shock waves and jams in traffic flow. J. Phys. A 38, 4069–4083 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Tang, T.Q., Huang, H.J., Shang, H.Y.: Influences of the driver’s bounded rationality on micro driving behavior, fuel consumption and emissions. Transp. Res. Part D Transp. Environ. 41, 423–432 (2015)CrossRefGoogle Scholar
  45. 45.
    Tang, T.Q., Li, C., Huang, H., Shang, H.: A new fundamental diagram theory with the individual difference of the drivers perception ability. Nonlinear Dyn. 67(3), 2255–2265 (2012)CrossRefGoogle Scholar
  46. 46.
    Ge, H.X., Zheng, P.J., Lo, S.M., Cheng, R.J.: TDGL equation in lattice hydrodynamic model considering driver’s physical delay. Nonlinear Dyn. 76, 441–445 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Ge, H.X., Cheng, R.J.: The “backward looking” effect in the lattice hydrodynamic model. Physica A 387, 6952–6958 (2008)CrossRefGoogle Scholar
  48. 48.
    Guo, Y., Xue, Y., Shi, Y., et al.: Mean-field velocity difference model considering the average effect of multi-vehicle interaction. Commun. Nonlinear Sci. Numer. Simul. 59, 553–564 (2018)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Zhang, Y.C., Xue, Y., Shi, Y., et al.: Congested traffic patterns of two-lane lattice hydrodynamic model with partial reduced lane. Physica A 502, 135–147 (2018)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhu, W.-X., Zhang, L.-D.: Friction coefficient and radius of curvature effects upon traffic flow on a curved road. Physica A 391, 4597–4605 (2012)CrossRefGoogle Scholar
  51. 51.
    Zheng, Y.M., Cheng, R.J., Ge, H.X.: The feedback control research on straight and curved road with car-following model. Phys. Lett. A 381, 2137–2143 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Kaur, R., Sharma, S.: Modeling and simulation of driver’s anticipation effect in a two lane system on curved road with slope. Physica A 499, 110–120 (2018)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Kaur, R., Sharma, S.: Analysis of driver’s characteristics on a curved road in a lattice model. Physica A 471, 59–67 (2017)CrossRefGoogle Scholar
  54. 54.
    Zhou, J., Shi, Z.K.: Lattice hydrodynamic model for traffic flow on curved road. Nonlinear Dyn. 83, 1217–1236 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Zhou, J., Shi, Z.K., Wang, C.P.: Lattice hydrodynamic model for two-lane traffic flow on curved road. Nonlinear Dyn. 85, 1423–1443 (2016)CrossRefzbMATHGoogle Scholar
  56. 56.
    Jin, Y.D., Zhou, J., Shi, Z.K., et al.: Lattice hydrodynamic model for traffic flow on curved road with passing. Nonlinear Dyn. 89, 107–124 (2017)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Gao, L.N., Zi, Y.Y., Yin, Y.H., Ma, W.X., Lü, X.: Backlund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 89, 2233–2240 (2017)CrossRefGoogle Scholar
  58. 58.
    Yin, Y.H., Ma, W.X., Liu, J.G., Lü, X.: Diversity of exact solutions to a (3 + 1)-dimensional nonlinear evolution equation and its reduction. Comput. Math. Appl. 76, 1275–1283 (2018)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Lü, X., Ma, W.X., Yu, J., et al.: Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 31, 40–46 (2016)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Lü, X., Ma, W.X., Yu, J., et al.: Envelope bright- and dark-soliton solutions for the Gerdjikov–Ivanov model. Nonlinear Dyn. 82, 1211–1220 (2015)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Lü, X.: Madelung fluid description on a generalized mixed nonlinear Schrödinger equation. Nonlinear Dyn. 81(1–2), 239–247 (2015)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Yu Xue
    • 1
    • 2
    Email author
  • Yicai Zhang
    • 1
  • Deli Fan
    • 1
  • Peng Zhang
    • 1
  • Hong-di He
    • 3
  1. 1.Institute of Physical Science and TechnologyGuangxi UniversityNanningPeople’s Republic of China
  2. 2.Key Laboratory for the Relativistic AstrophysicsNanningPeople’s Republic of China
  3. 3.Logistics Research Center and Shanghai Engineering Research Center of Shipping Logistics InformationShanghai Maritime UniversityShanghaiPeople’s Republic of China

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