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

Capacity drop and traffic control for a second order traffic model

Oliver Kolb 1, , Simone Göttlich 1, and  Paola Goatin 2,

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.

Keywords: Traffic flow, second order model, on-ramp coupling, numerical simulations, optimal control.
Mathematics Subject Classification: 90B20, 35L65.
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

M. Garavello and B. Piccoli, Traffic Flow on Networks, Springfield, MO: American Institute of Mathematical Sciences (AIMS), 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, PhD thesis, TU Darmstadt, 2011.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

M. Garavello and B. Piccoli, Traffic Flow on Networks, Springfield, MO: American Institute of Mathematical Sciences (AIMS), 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

O. Kolb, Simulation and Optimization of Gas and Water Supply Networks, PhD thesis, TU Darmstadt, 2011.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  Demand and supply functions on a fixed road for $\rho^{\max}=1$, $v^{\rm{ref}}=2$, $\gamma=2$ and the given values for $c$
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Figure 2.  1-to-1 junction
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Figure 3.  1-to-1 junction with on-ramp
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Figure 4.  On-ramp at origin
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Figure 5.  Outflow at a vertex
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Figure 6.  A single road
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Figure 7.  Density (top) and velocity (bottom) after 36 seconds for different choices of the relaxation parameter $\delta$ and for the LWR model
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Figure 8.  Two roads with an on-ramp in between
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Figure 9.  Actual outflow depending on the desired inflow at the on-ramp for the AR and the LWR model
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Figure 10.  Road network with an on-ramp at the node ''on-ramp''
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Figure 11.  Inflow profiles for the network in Figure 10
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Figure 12.  Queue at the origin ''in'' and the on-ramp with and without optimization
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Figure 13.  Flow at the origin ''in'' of the network (left) and at the node ''out'' (right) with and without optimization
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Figure 14.  Optimal control of $v_i^{\max}(t)$ on road2 (top left) and road3 (top right) and $u(t)$ at the on-ramp (bottom)
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Figure 15.  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
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Table 1.  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 $[\frac{\rm{cars}}{\rm{h}}]$
desiredactual
50050047.673.677.140004000
1000100047.673.677.145004500
15001500156.413.150.935544500
20001764160.211.050.635274500
25001764160.211.050.635274500
10001000148.017.851.636294500
500500137.223.852.837624000
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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 $[\frac{\rm{cars}}{\rm{h}}]$
desiredactual
50050047.673.677.140004000
1000100047.673.677.145004500
15001500156.413.150.935544500
20001764160.211.050.635274500
25001764160.211.050.635274500
10001000148.017.851.636294500
500500137.223.852.837624000
Table 2.  Properties of the roads in Figure 10
roadlength $[\text{km}]$$\rho^{\max}$ $[\frac{\text{cars}}{\text{km}}]$$v^{\rm{low}}$ $[\frac{\text{km}}{\text{h}}]$$v^{\rm{high}}$ $[\frac{\text{km}}{\text{h}}]$initial density $[\frac{\text{cars}}{\text{km}}]$
road1218010010050
road211805010050
road311805010050
road4218010010050
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roadlength $[\text{km}]$$\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 $[\frac{\text{cars}}{\text{km}}]$
road1218010010050
road211805010050
road311805010050
road4218010010050
Table 3.  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 control1871.71871.7834.9
ramp metering only1325.31325.3834.9
speed control only1122.8872.6834.9
both control types814.5818.4834.9
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AR, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$AR, $\frac{\partial p_i}{\partial v_i^{\max}}=0$LWR
no control1871.71871.7834.9
ramp metering only1325.31325.3834.9
speed control only1122.8872.6834.9
both control types814.5818.4834.9
Table 4.  Total travel time for different choices of the relaxation parameter $\delta$
$\delta$no controlopt. control, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}=0$
$5\cdot10^{-5}$2199.4868.8953.4
$5\cdot10^{-4}$2137.1860.1856.3
$5\cdot10^{-3}$1871.7814.5818.4
$5\cdot10^{-2}$731.4731.4731.4
$5\cdot10^{-1}$725.9725.9725.9
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$\delta$no controlopt. control, $\frac{\partial p_i}{\partial v_i^{\max}}\ne0$opt. control, $\frac{\partial p_i}{\partial v_i^{\max}}=0$
$5\cdot10^{-5}$2199.4868.8953.4
$5\cdot10^{-4}$2137.1860.1856.3
$5\cdot10^{-3}$1871.7814.5818.4
$5\cdot10^{-2}$731.4731.4731.4
$5\cdot10^{-1}$725.9725.9725.9
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