Traffic light control: A case study

Simone Göttlich; Ute Ziegler

doi:10.3934/dcdss.2014.7.483

Article Contents

2014, Volume 7, Issue 3: 483-501. Doi: 10.3934/dcdss.2014.7.483

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Traffic light control: A case study

Simone Göttlich^1, and
Ute Ziegler^2,

1.
Department of Mathematics, University of Mannheim, D-68131 Mannheim
2.
School of Business Informatics and Mathematics, University of Mannheim, D-68131 Mannheim

Received: May 2013

Revised: August 2013

Published: June 2014

Abstract / Introduction Related Papers Cited by

Abstract

This article is devoted to traffic flow networks including traffic lights at intersections. Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled as piecewise constant functions for red and green signals. The involved control problem is to find stop and go configurations depending on the current traffic volume. We propose a numerical solution strategy and present computational results.

Keywords:

Mathematics Subject Classification: Primary: 90B20, 35L50; Secondary: 90C06.

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References

[1]	G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks, Networks and Heterogeneous Media, 1 (2006), 57-84.doi: 10.3934/nhm.2006.1.57.
[2]	E. Brockfeld, R. Barlovic, A. Schadschneider and M. Schreckenberg, Optimizing traffic lights in a cellular automaton model for city traffic, Physical Review E, 64 (2001), 056132.doi: 10.1103/PhysRevE.64.056132.
[3]	Y. Chitour and B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 599-630.doi: 10.3934/dcdsb.2005.5.599.
[4]	C. Claudel and A. Bayen, Convex formulations of data assimilation problems for a class of hamilton-jacobi equations, SIAM Journal on Control and Optimization, 49 (2011), 383-402.doi: 10.1137/090778754.
[5]	G. M. Coclite, M. Garavello and B. Piccoli, Traffic flow on a road network, SIAM Journal on Mathematical Analysis, 36 (2005), 1862-1886.doi: 10.1137/S0036141004402683.
[6]	C. Daganzo, On the variational theory of traffic flow: well-posedness, duality and applications, Networks and Heterogeneous Media, 1 (2006), 601-619.doi: 10.3934/nhm.2006.1.601.
[7]	C. D'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation and Optimization of Supply Chains: A Continuous Approach, SIAM, Philadelphia, PA, 2010.doi: 10.1137/1.9780898717600.
[8]	C. D'Apice, R. Manzo and B. Piccoli, Packet flow on telecommunication networks, SIAM Journal on Mathematical Analysis, 38 (2006), 717-740.doi: 10.1137/050631628.
[9]	G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transportation Research Part B: Methodological, 45 (2011), 903-922 .
[10]	A. Fügenschuh, S. Göttlich, M. Herty, A. Klar and A. Martin, A discrete optimization approach to large scale supply networks based on partial differential equations, SIAM Journal on Scientific Computing, 30 (2008), 1490-1507.doi: 10.1137/060663799.
[11]	A. Fügenschuh, M. Herty, A. Klar and A. Martin, Combinatorial and continuous models for the optimization of traffic flows on networks, SIAM Journal on Optimization, 16 (2006), 1155-1176.doi: 10.1137/040605503.
[12]	S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogenous Media, 8 (2013), 685-705.
[13]	S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings on road networks, preprint, 2013.
[14]	S. Göttlich, S. Kühn and O. Kolb, Optimization for a special class of traffic flow models: combinatorial and continuous approaches, preprint, 2013.
[15]	M. Gugat, M. Herty, A. Klar and G. Leugering, Optimal control for traffic flow networks, Journal of Optimization Theory and Applications, 126 (2005), 589-616.doi: 10.1007/s10957-005-5499-z.
[16]	M. Herty and A. Klar, Modeling, simulation, and optimization of traffic flow networks, SIAM Journal on Scientific Computing, 25 (2003), 1066-1087.doi: 10.1137/S106482750241459X.
[17]	H. Holden and N. H. Risebro, A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis, 26 (1995), 999-1017.doi: 10.1137/S0036141093243289.
[18]	IBM ILOG CPLEX Optimization Studio, http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/.
[19]	S. Lämmer and D. Helbing, Self-control of traffic lights and vehicle flows in urban road networks, Journal of Statistical Mechanics: Theory and Experiment, (2008), P04019.
[20]	J. Lebacque and M. Khoshyaran, First order macroscopic traffic flow models for networks in the context of dynamic assignment, Transportation Planning, (2004), 119-140.doi: 10.1007/0-306-48220-7_8.
[21]	W. Lin and C. Wang, An enhanced 0-1 mixed-integer LP formulation for traffic signal control, IEEE Transactions on Intelligent Transportation Systems, 5 (2004), 238-245.doi: 10.1109/TITS.2004.838217.
[22]	P. Mazaré, A. Dehwah, C. Claudel and A. Bayen, Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model, Transportation Research Part B: Methodological, 45 (2011), 1727-1748.
[23]	L. Zhao, X. Peng, L. Li and Z. Li, A fast signal timing algorithm for individual oversaturated intersections, IEEE Transactions on Intelligent Transportation Systems, (2011), 1-4.doi: 10.1109/TITS.2010.2076808.
[24]	U. Ziegler, Mathematical Modelling, Simulation and Optimisation of Dynamic Transportation Networks with Applications in Production and Traffic, Ph.D Thesis RWTH Aachen University, 2013. Available from: http://darwin.bth.rwth-aachen.de/opus3/volltexte/2013/4452/.