# 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
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##### References:
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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