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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

[21]

L. Leclercq, V.L. Knoop, F. Marczak, 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, 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, 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, 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, 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, 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, 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.

show all references

References:
[1]

A. Aw, A. Klar, T. Materne, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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.

[21]

L. Leclercq, V.L. Knoop, F. Marczak, 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, 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, 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, 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, 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, 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, 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.

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 $c$
Figure 2.  1-to-1 junction
Figure 3.  1-to-1 junction with on-ramp
Figure 4.  On-ramp at origin
Figure 5.  Outflow at a vertex
Figure 6.  A single road
Figure 7.  Density (top) and velocity (bottom) after 36 seconds for different choices of the relaxation parameter $\delta$ and for the LWR model
Figure 8.  Two roads with an on-ramp in between
Figure 9.  Actual outflow depending on the desired inflow at the on-ramp for the AR and the LWR model
Figure 10.  Road network with an on-ramp at the node ''on-ramp''
Figure 11.  Inflow profiles for the network in Figure 10
Figure 12.  Queue at the origin ''in'' and the on-ramp with and without optimization
Figure 13.  Flow at the origin ''in'' of the network (left) and at the node ''out'' (right) with and without optimization
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)
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
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
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}}]$$v^{\rm{high}}$ $[\frac{\text{km}}{\text{h}}]$initial density $[\frac{\text{cars}}{\text{km}}]$
road1218010010050
road211805010050
road311805010050
road4218010010050
roadlength $[\text{km}]$$\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
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
$\delta$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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