# American Institute of Mathematical Sciences

December 2017, 12(4): 663-681. doi: 10.3934/nhm.2017027

## Capacity drop and traffic control for a second order traffic model

 1 Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany 2 Inria Sophia Antipolis -Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 06902 Sophia Antipolis Cedex, France

Received  November 2016 Revised  February 2017 Published  October 2017

Fund Project: The second author is supported by DFG grant GO 1920/4-1

In this paper, we illustrate how second order traffic flow models, in our case the Aw-Rascle equations, can be used to reproduce empirical observations such as the capacity drop at merges and solve related optimal control problems. To this aim, we propose a model for on-ramp junctions and derive suitable coupling conditions. These are associated to the first order Godunov scheme to numerically study the well-known capacity drop effect, where the outflow of the system is significantly below the expected maximum. Control issues such as speed and ramp meter control are also addressed in a first-discretize-then-optimize framework.

Citation: Oliver Kolb, Simone Göttlich, Paola Goatin. Capacity drop and traffic control for a second order traffic model. Networks & Heterogeneous Media, 2017, 12 (4) : 663-681. doi: 10.3934/nhm.2017027
##### References:
 [1] A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278. doi: 10.1137/S0036139900380955. [2] A. Aw and M. Rascle, Resurrection of ''second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [3] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. [4] C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Communications in Mathematical Sciences, 5 (2007), 533-551. doi: 10.4310/CMS.2007.v5.n3.a2. [5] M.L. DelleMonache, B. Piccoli and F. Rossi, Traffic Regulation via Controlled Speed Limit, SIAM Journal on Control and Optimization, 55 (2017), 2936-2958. doi: 10.1137/16M1066038. [6] M.L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin and A.M. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), 22-39. doi: 10.1137/130908993. [7] S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Networks and Heterogeneous Media, 9 (2014), 239-268. doi: 10.3934/nhm.2014.9.239. [8] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Communications in Partial Differential Equations, 31 (2006), 243-275. doi: 10.1080/03605300500358053. [9] M. Garavello and B. Piccoli, Traffic Flow on Networks, Springfield, MO: American Institute of Mathematical Sciences (AIMS), 2006. [10] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Mathematical and Computer Modelling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016. [11] P. Goatin, S. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Engineering Optimization, 48 (2016), 1121-1144. doi: 10.1080/0305215X.2015.1097099. [12] J.M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM Journal on Applied Mathematics, 62 (2001), 729-745. doi: 10.1137/S0036139900378657. [13] B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Networks and Heterogeneous Media, 2 (2007), 227-253. doi: 10.3934/nhm.2007.2.227. [14] A. Hegyi, B.D. Schutter and H. Hellendoorn, Optimal coordination of variable speed limits to suppress shock waves, IEEE Transactions on Intelligent Transportation Systems, 6 (2005), 102-112. doi: 10.1109/CDC.2003.1273043. [15] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Networks and Heterogeneous Media, 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275. [16] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616. doi: 10.1137/05062617X. [17] W.-L. Jin, Q.-J. Gan and J.-P. Lebacque, A kinematic wave theory of capacity drop, Transportation Research Part B: Methodological, 81 (2015), 316-329. doi: 10.1016/j.trb.2015.07.020. [18] W. Jin and H. Zhang, On the distribution schemes for determining flows through a merge, Transportation Research Part B: Methodological, 37 (2003), 521-540. doi: 10.1016/S0191-2615(02)00026-7. [19] O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, PhD thesis, TU Darmstadt, 2011. [20] O. Kolb and J. Lang, Simulation and continuous optimization, in "Mathematical Optimization of Water Networks" (eds. A. Martin, K. Klamroth, J. Lang, G. Leugering, A. Morsi, M. Oberlack, M. Ostrowski and R. Rosen), Springer Basel, 162 (2012), 17-33. doi: 10.1007/978-3-0348-0436-3_2. [21] L. Leclercq, V.L. Knoop, F. Marczak and S.P. Hoogendoorn, Capacity drops at merges: New analytical investigations, Transportation Research Part C: Emerging Technologies, 62 (2016), 171-181. doi: 10.1109/ITSC.2014.6957839. [22] M.J. Lighthill and G.B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [23] C. Parzani and C. Buisson, Second-order model and capacity drop at merge, Transportation Research Record: Journal of the Transportation Research Board, 2315 (2012), 25-34. doi: 10.3141/2315-03. [24] B. Piccoli, K. Han, T.L. Friesz, T. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transportation Research Part C: Emerging Technologies, 52 (2015), 32-56. doi: 10.1016/j.trc.2014.12.013. [25] J. Reilly, S. Samaranayake, M.L. DelleMonache, W. Krichene, P. Goatin and A.M. Bayen, Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering, Journal of Optimization Theory and Applications, 167 (2015), 733-760. doi: 10.1007/s10957-015-0749-1. [26] F. Siebel, W. Mauser, S. Moutari and M. Rascle, Balanced vehicular traffic at a bottleneck, Mathematical and Computer Modelling, 49 (2009), 689-702. doi: 10.1016/j.mcm.2008.01.006. [27] P. Spellucci, A new technique for inconsistent QP problems in the SQP method, Mathematical Methods of Operations Research, 47 (1998), 355-400. doi: 10.1007/BF01198402. [28] P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Mathematical Programming, 82 (1998), 413-448. doi: 10.1007/BF01580078. [29] A. Srivastava and N. Geroliminis, Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model, Transportation Research Part C: Emerging Technologies, 30 (2013), 161-177. doi: 10.1016/j.trc.2013.02.006. [30] M. Treiber and A. Kesting, Traffic Flow Dynamics, Data, models and simulation, Translated by Treiber and Christian Thiemann, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.

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##### References:
 [1] A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278. doi: 10.1137/S0036139900380955. [2] A. Aw and M. Rascle, Resurrection of ''second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938. doi: 10.1137/S0036139997332099. [3] F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185-220. doi: 10.1007/s00205-007-0061-9. [4] C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Communications in Mathematical Sciences, 5 (2007), 533-551. doi: 10.4310/CMS.2007.v5.n3.a2. [5] M.L. DelleMonache, B. Piccoli and F. Rossi, Traffic Regulation via Controlled Speed Limit, SIAM Journal on Control and Optimization, 55 (2017), 2936-2958. doi: 10.1137/16M1066038. [6] M.L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin and A.M. Bayen, A PDE-ODE model for a junction with ramp buffer, SIAM Journal on Applied Mathematics, 74 (2014), 22-39. doi: 10.1137/130908993. [7] S. Fan, M. Herty and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Networks and Heterogeneous Media, 9 (2014), 239-268. doi: 10.3934/nhm.2014.9.239. [8] M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Communications in Partial Differential Equations, 31 (2006), 243-275. doi: 10.1080/03605300500358053. [9] M. Garavello and B. Piccoli, Traffic Flow on Networks, Springfield, MO: American Institute of Mathematical Sciences (AIMS), 2006. [10] P. Goatin, The Aw-Rascle vehicular traffic flow model with phase transitions, Mathematical and Computer Modelling, 44 (2006), 287-303. doi: 10.1016/j.mcm.2006.01.016. [11] P. Goatin, S. Göttlich and O. Kolb, Speed limit and ramp meter control for traffic flow networks, Engineering Optimization, 48 (2016), 1121-1144. doi: 10.1080/0305215X.2015.1097099. [12] J.M. Greenberg, Extensions and amplifications of a traffic model of Aw and Rascle, SIAM Journal on Applied Mathematics, 62 (2001), 729-745. doi: 10.1137/S0036139900378657. [13] B. Haut and G. Bastin, A second order model of road junctions in fluid models of traffic networks, Networks and Heterogeneous Media, 2 (2007), 227-253. doi: 10.3934/nhm.2007.2.227. [14] A. Hegyi, B.D. Schutter and H. Hellendoorn, Optimal coordination of variable speed limits to suppress shock waves, IEEE Transactions on Intelligent Transportation Systems, 6 (2005), 102-112. doi: 10.1109/CDC.2003.1273043. [15] M. Herty, S. Moutari and M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow, Networks and Heterogeneous Media, 1 (2006), 275-294. doi: 10.3934/nhm.2006.1.275. [16] M. Herty and M. Rascle, Coupling conditions for a class of second-order models for traffic flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616. doi: 10.1137/05062617X. [17] W.-L. Jin, Q.-J. Gan and J.-P. Lebacque, A kinematic wave theory of capacity drop, Transportation Research Part B: Methodological, 81 (2015), 316-329. doi: 10.1016/j.trb.2015.07.020. [18] W. Jin and H. Zhang, On the distribution schemes for determining flows through a merge, Transportation Research Part B: Methodological, 37 (2003), 521-540. doi: 10.1016/S0191-2615(02)00026-7. [19] O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, PhD thesis, TU Darmstadt, 2011. [20] O. Kolb and J. Lang, Simulation and continuous optimization, in "Mathematical Optimization of Water Networks" (eds. A. Martin, K. Klamroth, J. Lang, G. Leugering, A. Morsi, M. Oberlack, M. Ostrowski and R. Rosen), Springer Basel, 162 (2012), 17-33. doi: 10.1007/978-3-0348-0436-3_2. [21] L. Leclercq, V.L. Knoop, F. Marczak and S.P. Hoogendoorn, Capacity drops at merges: New analytical investigations, Transportation Research Part C: Emerging Technologies, 62 (2016), 171-181. doi: 10.1109/ITSC.2014.6957839. [22] M.J. Lighthill and G.B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089. [23] C. Parzani and C. Buisson, Second-order model and capacity drop at merge, Transportation Research Record: Journal of the Transportation Research Board, 2315 (2012), 25-34. doi: 10.3141/2315-03. [24] B. Piccoli, K. Han, T.L. Friesz, T. Yao and J. Tang, Second-order models and traffic data from mobile sensors, Transportation Research Part C: Emerging Technologies, 52 (2015), 32-56. doi: 10.1016/j.trc.2014.12.013. [25] J. Reilly, S. Samaranayake, M.L. DelleMonache, W. Krichene, P. Goatin and A.M. Bayen, Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering, Journal of Optimization Theory and Applications, 167 (2015), 733-760. doi: 10.1007/s10957-015-0749-1. [26] F. Siebel, W. Mauser, S. Moutari and M. Rascle, Balanced vehicular traffic at a bottleneck, Mathematical and Computer Modelling, 49 (2009), 689-702. doi: 10.1016/j.mcm.2008.01.006. [27] P. Spellucci, A new technique for inconsistent QP problems in the SQP method, Mathematical Methods of Operations Research, 47 (1998), 355-400. doi: 10.1007/BF01198402. [28] P. Spellucci, An SQP method for general nonlinear programs using only equality constrained subproblems, Mathematical Programming, 82 (1998), 413-448. doi: 10.1007/BF01580078. [29] A. Srivastava and N. Geroliminis, Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model, Transportation Research Part C: Emerging Technologies, 30 (2013), 161-177. doi: 10.1016/j.trc.2013.02.006. [30] M. Treiber and A. Kesting, Traffic Flow Dynamics, Data, models and simulation, Translated by Treiber and Christian Thiemann, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32460-4.
Demand and supply functions on a fixed road for $\rho^{\max}=1$, $v^{\rm{ref}}=2$, $\gamma=2$ and the given values for $c$
1-to-1 junction
1-to-1 junction with on-ramp
On-ramp at origin
Outflow at a vertex
Density (top) and velocity (bottom) after 36 seconds for different choices of the relaxation parameter $\delta$ and for the LWR model
Two roads with an on-ramp in between
Actual outflow depending on the desired inflow at the on-ramp for the AR and the LWR model
Road network with an on-ramp at the node ''on-ramp''
Inflow profiles for the network in Figure 10
Queue at the origin ''in'' and the on-ramp with and without optimization
Flow at the origin ''in'' of the network (left) and at the node ''out'' (right) with and without optimization
Optimal control of $v_i^{\max}(t)$ on road2 (top left) and road3 (top right) and $u(t)$ at the on-ramp (bottom)
Density (top) and velocity (bottom) behind the on-ramp in the uncontrolled case for different choices of the relaxation parameter $\delta$ and for the LWR model
Capacity drop effect
 inflow at on-ramp in $[\frac{\rm{cars}}{\rm{h}}]$} $\rho_1$in $[\frac{\rm{cars}}{\rm{km}}]$ $v_1$in $[\frac{\rm{km}}{\rm{h}}]$ $w_1$in $[\frac{\rm{km}}{\rm{h}}]$ outflow AR in $[\frac{\rm{cars}}{\rm{h}}]$ outflow LWR in $[\frac{\rm{cars}}{\rm{h}}]$ desired actual 500 500 47.6 73.6 77.1 4000 4000 1000 1000 47.6 73.6 77.1 4500 4500 1500 1500 156.4 13.1 50.9 3554 4500 2000 1764 160.2 11.0 50.6 3527 4500 2500 1764 160.2 11.0 50.6 3527 4500 1000 1000 148.0 17.8 51.6 3629 4500 500 500 137.2 23.8 52.8 3762 4000
 inflow at on-ramp in $[\frac{\rm{cars}}{\rm{h}}]$} $\rho_1$in $[\frac{\rm{cars}}{\rm{km}}]$ $v_1$in $[\frac{\rm{km}}{\rm{h}}]$ $w_1$in $[\frac{\rm{km}}{\rm{h}}]$ outflow AR in $[\frac{\rm{cars}}{\rm{h}}]$ outflow LWR in $[\frac{\rm{cars}}{\rm{h}}]$ desired actual 500 500 47.6 73.6 77.1 4000 4000 1000 1000 47.6 73.6 77.1 4500 4500 1500 1500 156.4 13.1 50.9 3554 4500 2000 1764 160.2 11.0 50.6 3527 4500 2500 1764 160.2 11.0 50.6 3527 4500 1000 1000 148.0 17.8 51.6 3629 4500 500 500 137.2 23.8 52.8 3762 4000
Properties of the roads in Figure 10
 road length $[\text{km}]$ $\rho^{\max}$ $[\frac{\text{cars}}{\text{km}}]$ $v^{\rm{low}}$ $[\frac{\text{km}}{\text{h}}]$ $v^{\rm{high}}$ $[\frac{\text{km}}{\text{h}}]$ initial density $[\frac{\text{cars}}{\text{km}}]$ road1 2 180 100 100 50 road2 1 180 50 100 50 road3 1 180 50 100 50 road4 2 180 100 100 50
 road length $[\text{km}]$ $\rho^{\max}$ $[\frac{\text{cars}}{\text{km}}]$ $v^{\rm{low}}$ $[\frac{\text{km}}{\text{h}}]$ $v^{\rm{high}}$ $[\frac{\text{km}}{\text{h}}]$ initial density $[\frac{\text{cars}}{\text{km}}]$ road1 2 180 100 100 50 road2 1 180 50 100 50 road3 1 180 50 100 50 road4 2 180 100 100 50
Optimization results for the network in Figure 10
 AR, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$ AR, $\frac{\partial p_i}{\partial v_i^{\max}}=0$ LWR no control 1871.7 1871.7 834.9 ramp metering only 1325.3 1325.3 834.9 speed control only 1122.8 872.6 834.9 both control types 814.5 818.4 834.9
 AR, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$ AR, $\frac{\partial p_i}{\partial v_i^{\max}}=0$ LWR no control 1871.7 1871.7 834.9 ramp metering only 1325.3 1325.3 834.9 speed control only 1122.8 872.6 834.9 both control types 814.5 818.4 834.9
Total travel time for different choices of the relaxation parameter $\delta$
 $\delta$ no control opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$ opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}=0$ $5\cdot10^{-5}$ 2199.4 868.8 953.4 $5\cdot10^{-4}$ 2137.1 860.1 856.3 $5\cdot10^{-3}$ 1871.7 814.5 818.4 $5\cdot10^{-2}$ 731.4 731.4 731.4 $5\cdot10^{-1}$ 725.9 725.9 725.9
 $\delta$ no control opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$ opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}=0$ $5\cdot10^{-5}$ 2199.4 868.8 953.4 $5\cdot10^{-4}$ 2137.1 860.1 856.3 $5\cdot10^{-3}$ 1871.7 814.5 818.4 $5\cdot10^{-2}$ 731.4 731.4 731.4 $5\cdot10^{-1}$ 725.9 725.9 725.9
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