A dynamical systems approach based on averaging to model the macroscopic flow of freeway traffic

Abstract The flow of traffic exhibits distinct characteristics under different conditions, reflecting the congestion during peak hours and relatively free motion during off-peak hours. This requires one to use different mathematical equations to describe the diverse traffic characteristics. Thus, the flow of traffic is best described by a hybrid system, namely different governing equations for the different regimes of response, and it is such a hybrid approach that is investigated in this paper. Existing models for the flow of traffic treat traffic as a continuum or employ techniques similar to those used in the kinetic theory of gases, neither of these approaches gainfully exploit the hybrid nature of the problem. Spurious two-way propagation of disturbances that are physically unacceptable are predicted by continuum models for the flow of traffic. The number of vehicles in a typical section of the highway does not justify its being modeled as a continuum. It is also important to recognize that the basic premises of kinetic theory are not appropriate for the flow of traffic (see [S. Darbha, K.R. Rajagopal, Limit of a collection of dynamical systems: an application to modeling the flow of traffic, Mathematical Models and Methods in Applied Sciences 12 (10) (2002) 1381–1399] for a rationale for the same). A model for the flow of traffic that does not treat traffic as a continuum or use notions from kinetic theory is developed here and corroborated with real-time data collected on US 183 in Austin, Texas. Predictions based on the hybrid system model seem to agree reasonably well with the data collected on US 183.

[1]  G. F. Newell.  Mathematical Models for Freely-Flowing Highway Traffic , 1955, Oper. Res..

[2]  M. Pulvirenti,et al.  Modelling in Applied Sciences: A Kinetic Theory Approach , 2004 .

[3]  I Okutani,et al.  Dynamic prediction of traffic volume through Kalman Filtering , 1984 .

[4]  Markos Papageorgiou,et al.  Applications of Automatic Control Concepts to Traffic Flow Modeling and Control , 1983 .

[5]  C. Daganzo.  THE CELL TRANSMISSION MODEL.. , 1994 .

[6]  Swaroop Darbha,et al.  LIMIT OF A COLLECTION OF DYNAMICAL SYSTEMS: AN APPLICATION TO MODELING THE FLOW OF TRAFFIC , 2002 .

[7]  Pravin Varaiya,et al.  Causes and cures of highway congestion , 2001 .

[8]  Harilaos N. Koutsopoulos,et al.  A microscopic traffic simulator for evaluation of dynamic traffic management systems , 1996 .

[9]  H. M. Zhang,et al.  A Car-Following Theory for Multiphase Vehicular Traffic Flow , 2003 .

[10]  Gordon F. Newell,et al.  A simplified car-following theory: a lower order model , 2002 .

[11]  L. A. Pipes.  An Operational Analysis of Traffic Dynamics , 1953 .

[12]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[13]  E. Montroll,et al.  Traffic Dynamics: Studies in Car Following , 1958 .

[14]  Kumbakonam R. Rajagopal,et al.  An Introduction to the Mechanics of Fluids , 1999 .

[15]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[16]  D. Gazis,et al.  Nonlinear Follow-the-Leader Models of Traffic Flow , 1961 .

[17]  H. M. Zhang.  A mathematical theory of traffic hysteresis , 1999 .

[18]  P. G. Gipps,et al.  A behavioural car-following model for computer simulation , 1981 .

[19]  Markos Papageorgiou,et al.  Real-time freeway traffic state estimation based on extended Kalman filter: a general approach , 2005 .

[20]  S Walker,et al.  CORSIM - CORRIDOR TRAFFIC SIMULATION MODEL , 1997 .

[21]  Louis A. Pipes,et al.  Car following models and the fundamental diagram of road traffic , 1967 .

[22]  Carlos F. Daganzo,et al.  A BEHAVIORAL THEORY OF MULTI-LANE TRAFFIC FLOW. PART I, LONG HOMOGENEOUS FREEWAY SECTIONS , 1999 .

[23]  Markos Papageorgiou,et al.  Parameter identification for a traffic flow model , 1979, Autom..

[24]  I. Prigogine,et al.  A Boltzmann-Like Approach for Traffic Flow , 1960 .

[25]  Swaroop Darbha,et al.  Intelligent Cruise Control Systems And Traffic Flow Stability , 1998 .

[26]  J. Hedrick,et al.  String stability of interconnected systems , 1996, IEEE Trans. Autom. Control..

[27]  G. F. Newell.  A simplified theory of kinematic waves in highway traffic, part I: General theory , 1993 .

[28]  H. Wehlan,et al.  Section-wise modeling of traffic flow and its application in traffic state estimation , 2001, ITSC 2001. 2001 IEEE Intelligent Transportation Systems. Proceedings (Cat. No.01TH8585).

[29]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[30]  Guanrong Chen,et al.  Kalman Filtering with Real-time Applications , 1987 .

[31]  R. B. Potts,et al.  Car-Following Theory of Steady-State Traffic Flow , 1959 .

[32]  B. Kerner,et al.  A microscopic model for phase transitions in traffic flow , 2002 .