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Capacity drop and traffic control for a second order traffic model

The second author is supported by DFG grant GO 1920/4-1.
Abstract Full Text(HTML) Figure(15) / Table(4) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 90B20, 35L65.

    Citation:

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  • 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  • [1] A. AwA. KlarT. 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. BerthelinP. DegondM. 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. DelleMonacheB. 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 MonacheJ. ReillyS. SamaranayakeW. KricheneP. 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. FanM. 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. GoatinS. 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. HegyiB.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. HertyS. 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. JinQ.-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. LeclercqV.L. KnoopF. 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. PiccoliK. HanT.L. FrieszT. 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. ReillyS. SamaranayakeM.L. DelleMonacheW. KricheneP. 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. SiebelW. MauserS. 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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