Capacity drop and traffic control for a second order traffic model

Oliver Kolb; Simone Göttlich; Paola Goatin

doi:10.3934/nhm.2017027

Article Contents

2017, Volume 12, Issue 4: 663-681. Doi: 10.3934/nhm.2017027

This issue Previous Article The Lax-Oleinik semigroup on graphs Next Article Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces

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: October 31, 2016

Revised: January 31, 2017

Published: November 2017

The second author is supported by DFG grant GO 1920/4-1.

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Abstract

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:

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 $c$

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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 $[\frac{\rm{cars}}{\rm{h}}]$
desired	actual	$\rho_1$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 $[\frac{\rm{cars}}{\rm{h}}]$
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

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Table 2. 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

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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 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

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Table 4. 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

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