Improved High-Order Model for Freeway Traffic Flow

Abstract

An improved high-order continuum model is developed based on hyperbolic conservation laws with relaxation, linearized stability analysis, and more realistic considerations of traffic flow. The improved high-order model allows smooth traveling wave solutions as well as contact shocks (different densities moving at the same speed), is able to describe the amplification of small disturbances on heavy traffic, and allows fluctuations of speed around the equilibrium values. Furthermore, unlike existing high-order models, it does not result in negative speeds at the tail of congested regions and disturbance propagation speeds greater than the traffic flow velocity because the improved model has a zero characteristic speed and a nonnegative characteristic speed that is equal to the traffic flow velocity. The relaxation time is a function of density and, in the equilibrium limit, the improved high-order model is consistent with the simple continuum model. The improved high-order model is compared with the simple continuum model. Exemplary test results suggest that the improved high-order model is intuitively correct. Comparison of numerical results with field data suggests that the improved high-order model yields lower error levels than the simple continuum model.

Get full access to this article

View all access options for this article.

References

Lighthill

M. H.

, and Whitham

G. B.

. On Kinematic Waves II: A Theory of Traffic Flow on Long Crowded Roads. Proc., Royal Society of London Series A, Vol. 229, 1955, pp. 317–345.

Payne

H. J.

Models of Freeway Traffic and Control. Mathematical Models of Public Systems. Simulation Counc. Proc. Ser., Vol. 1, 1971, pp. 51–61.

Payne

H. J.

FREFLO: A Macroscopic Simulation Model of Freeway Traffic. In Transportation Research Record 722, TRB, National Research Council, Washington, D.C., 1979, pp. 68–77.

Papageorgiou

A Hierarchical Control System for Freeway Traffic. Transportation Research, Part B, Vol. 17, 1983, pp. 251–261.

Papageorgiou

, Blosseville

J. M.

, and Hadj-Salem

. Macroscopic Modelling of Traffic Flow on the Boulevard Peripherique in Paris. Transportation Research, Part B, Vol. 23, 1989, pp. 29–47.

Michalopoulos

P. G.

, Yi

, Beskos

D. E.

, and Lyrintzis

A. S.

. Continuum Modeling of Traffic Dynamics. Proc., 2nd International Conference on Application of Advanced Technology in Transportation Engineering, Minneapolis, Minn., 1991, pp. 36–40.

Michalopoulos

P. G.

, Yi

, and Lyrintzis

A. S.

. Continuum Modeling of Traffic Dynamics for Congested Freeways. Transportation Research, Part B, Vol. 27, 1993, pp. 315–332.

Michalopoulos

P. G.

, Yi

, and Lyrintzis

A. S.

. Development of an Improved High Order Continuum Traffic Flow Model. In Transportation Research Record 1365, TRB, National Research Council, Washington, D.C., 1993, pp. 125–132.

Lyrintzis

A. S.

, Yi

, Michalopoulos

P. G.

, and Beskos

. Development and Field Testing of Advanced Continuum Traffic Flow Models for Congested Freeways. ASCE Journal of Transportation Engineering, Vol. 120, 1994, pp. 461–477.

10.

Phillips

W. F.

A Kinetic Model for Traffic Flow with Continuum Implications. Transportation Planning Technology, Vol. 5, No. 3, 1979, pp. 131–138.

11.

Kühne

Macroscopic Freeway Model for Dense Traffic—Stop-Start Waves and Incident Detection. 9th International Symposium on Transportation and Traffic Theory, VNU Science Press, 1984, pp. 21–42.

12.

Ross

Traffic Dynamics. Transportation Research, Part B, Vol. 22, 1988, pp. 421–435.

13.

Lyrintzis

A. S.

, Liu

G.-Q.

, and Michalopoulos

P. G.

. Development and Comparative Evaluation of High-Order Traffic Flow Models. In Transportation Research Record 1457, TRB, National Research Council, Washington, D.C., 1994, pp. 174–183.

14.

Castillo

J. M. D.

, Pintado

, and Bemtez

F. G.

. The Reaction Time of Drivers and the Stability of Traffic Flow. Transportation Research, Part B, Vol. 28, 1994, pp. 35–60.

15.

Daganzo

C. F.

Requiem for Second Order Fluid Approximations of Traffic Flow. Transportation Research, Vol. 29B, 1995, pp. 277–286.

16.

Liu

T.-P.

Hyperbolic Conservation Laws with Relaxation. Commu. Math. Phys., Vol. 108, 1987, pp. 153–175.

17.

Whitham

G. B.

Linear and Nonlinear Waves. John Wiley & Sons, New York, 1974.

18.

Rhee

H.-K.

, Aris

, and Amundson

N. R.

. First-Order Partial Differential Equations, Vol. 1. Prentice-Hall, Englewood Cliffs, N.J., 1989.

19.

Liu

G.-Q.

, Lyrintzis

A. S.

, and Michalopoulos

P. G.

. Numerical Simulation of Freeway Traffic Flow. ASCE Journal of Transportation Engineering, Vol. 123, 1997, pp. 503–513.

20.

Gerlough

D. L.

, and Huber

M. J.

. Special Report 165: Traffic Flow Theory. TRB, National Research Council, Washington, D.C., 1975.

21.

Liu

G.-Q.

Development, Implementation and Testing of High-Order Traffic Flow Models for Freeways. Ph.D. dissertation. Department of Civil Engineering, University of Minnesota, Minneapolis, 1997.

22.

Harten

, Lax

P. D.

, and van Leer

. On Upstream Differencing and Godunov-type Schemes for Hyperbolic Conservation Laws. SIAM Review, Vol. 25, No. 1, 1983, pp. 35–61.

23.

Michalopoulos

P. G.

, Kwon

, Lee

C.-F.

, Mahadevan

, and Kang

J.-G.

. Development of an Integrated Simulation Package for Freeway Design, Operations and Adaptive Traffic Management [Phase 1: Enhancement of the KRONOS Simulation Program]. Final Report, Center for Transportation Studies, University of Minnesota, Minneapolis, Minn., 1992.

24.

Press

, et al. Numerical Recipes in C. Cambridge University Press, Cambridge, 1994.

25.

Michalopoulos

P. G.

, and Lin

. A Freeway Simulation Program for Microcomputers. Proc., 1st National Conference on Microcomputers in Urban Transportation, ASCE, Calif., 1985, pp. 330–341.