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2018 | OriginalPaper | Chapter

3. Traffic Flow Theory

Authors : Pushkin Kachroo, Kaan M. A. Özbay

Published in: Feedback Control Theory for Dynamic Traffic Assignment

Publisher: Springer International Publishing

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Abstract

This chapter presents the basic traffic flow theory which is used in the following chapters for control problem formulations. The theory develops the Lighthill–Whitham–Richards (LWR) model that uses the conservation law for traffic. Additionally, a density-dependent speed formula is used. There are many relationships available for this fundamental diagram, the chapter uses Greenshields’ formula for further analysis. Elementary partial differential equations (PDE) theory is also presented including the method of characteristics needed for the analysis of the traffic model. Shockwaves and weak solutions are discussed followed by a brief discussion of traffic measurements.

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previous chapter Traffic Assignment: A Survey of Mathematical Models and Techniques
next chapter Modeling and Problem Formulation
Footnotes
1
Implicit Function Theorem: Let F be a function of three variables of class \(C^1\) in an open set \(\mathcal {O}\) given by \(F(x,y,z)=C\). Then z can be solved in terms of x and y for (x, y, z) near the point \((x_{0},y_{0},z_{0})\) if \(F_z(x_{0},y_{0},z_{0})\ne 0\). Writing z as a function of x and y in the equation gives \(F(x,y,z(x,y))=C\). Differentiating with respect to x gives \(F_x+F_zz_x=0\) and differentiating with respect to y gives \(F_y+F_zz_y=0\). Therefore, we have \(z_x=-F_x/F_z\) and \(z_y=-F_y/F_z\). Hence, \(z(x,y)\approx z_0+z_xdx+z_ydy\).
 
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Metadata
Title
Traffic Flow Theory
Authors
Pushkin Kachroo
Kaan M. A. Özbay
Copyright Year
2018
Book
Feedback Control Theory for Dynamic Traffic Assignment

Print ISBN: 978-3-319-69229-6

Electronic ISBN: 978-3-319-69231-9

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-69231-9_3