# American Institute of Mathematical Sciences

June  2014, 7(3): 483-501. doi: 10.3934/dcdss.2014.7.483

## Traffic light control: A case study

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

Received  May 2013 Revised  August 2013 Published  January 2014

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.
Citation: Simone Göttlich, Ute Ziegler. Traffic light control: A case study. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 483-501. doi: 10.3934/dcdss.2014.7.483
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transportation Research Part B: Methodological, 45 (2011), 903-922 . Google Scholar [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.  Google Scholar [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.  Google Scholar [12] S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogenous Media, 8 (2013), 685-705. Google Scholar [13] S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings on road networks, preprint, 2013. Google Scholar [14] S. Göttlich, S. Kühn and O. Kolb, Optimization for a special class of traffic flow models: combinatorial and continuous approaches, preprint, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] , IBM ILOG CPLEX Optimization Studio,, , ().   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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/. Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] G. Flötteröd and J. Rohde, Operational macroscopic modeling of complex urban intersections, Transportation Research Part B: Methodological, 45 (2011), 903-922 . Google Scholar [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.  Google Scholar [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.  Google Scholar [12] S. Göttlich, M. Herty and U. Ziegler, Numerical discretization of Hamilton-Jacobi equations on networks, Networks and Heterogenous Media, 8 (2013), 685-705. Google Scholar [13] S. Göttlich, M. Herty and U. Ziegler, Modeling and optimizing traffic light settings on road networks, preprint, 2013. Google Scholar [14] S. Göttlich, S. Kühn and O. Kolb, Optimization for a special class of traffic flow models: combinatorial and continuous approaches, preprint, 2013. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] , IBM ILOG CPLEX Optimization Studio,, , ().   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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/. Google Scholar
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