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Derivation of Non-local Macroscopic Traffic Equations and Consistent Traffic Pressures from Microscopic Car-Following Models

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Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 2062)

Abstract

This contribution compares several different approaches allowing one to derive macroscopic traffic equation directly from microscopic car-following models. While it is shown that some conventional approaches lead to theoretical problems, it is proposed to use an approach reminding of smoothed particle hydrodynamics to avoid gradient expansions. The derivation circumvents approximations and, therefore, demonstrates the large range of validity of macroscopic traffic equations, without the need of averaging over many vehicles. It also gives an expression for the “traffic pressure”, which generalizes previously used formulas. Furthermore, the method avoids theoretical inconsistencies of macroscopic traffic models, which have been criticized in the past by Daganzo and others.

Keywords

  • Traffic Pressure
  • Macroscopic Traffic Models
  • Gradient Expansion
  • Smoothed Particle Hydrodynamics
  • Theoretical Inconsistencies

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

First published in: The European Physical Journal B 69(4), 539–548, DOI: 10.1140/epjb/e2009-00192-5 (2009), © EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2009, reproduction with kind permission of The European Physical Journal (EPJ).

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Fig. 1

Notes

  1. 1.

    If another smoothing function is applied, the last term of (47) is replaced by a similar weighted mean value, as (41) reveals, but the essence stays the same. That is, the way of looking at the microscopic equations (i.e. the way of defining the density and velocity moments) potentially has some influence on the dynamics, but it is expected to be small.

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Acknowledgements

The author would like to thank for the inspiring discussions with the participants of the Workshop on “Multiscale Problems and Models in Traffic Flow” organized by Michel Rascle and Christian Schmeiser at the Wolfgang Pauli Institute in Vienna from May 5–9, 2008, with partial support by the CNRS.

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Helbing, D. (2013). Derivation of Non-local Macroscopic Traffic Equations and Consistent Traffic Pressures from Microscopic Car-Following Models. In: Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics(), vol 2062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32160-3_3

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