Schéma numérique basé sur une équation d'Hamilton-Jacobi : modélisation des intersections

Schéma numérique basé sur une équation
d’Hamilton-Jacobi: modélisation des intersections
Guillaume Costeseque
Joint work with R. Monneau & JP. Lebacque
Ecole des Ponts ParisTech, CERMICS
& IFSTTAR, GRETTIA
GdT ”Mathématiques pour la modélisation des Transports”
29 Novembre 2012 - Marne-la-Vallée
G. Costeseque (Université ParisEst) D5 Traffic flow models Marne-la-Vallée, Nov. 2012 1 / 40

Outline
1 Introduction
2 Proposed intersection model
3 Numerical simulations
4 Concluding remarks

Introduction
Outline
1 Introduction
Traﬃc ﬂow modelling
Link models
Overview of intersection modelling

Introduction Traffic flow modelling
Objectives
From Papageorgiou and Gentile keynote speeches,
Dynamic Traffic Assignment (DTA) models useful for:
Transport planning (off-line)
Traffic monitoring and control (real-time)
Dynamic Loading Network (DLN) model =
Link model (propagate flows on arcs)
+ Node model (manage flows at intersection)

Introduction Link models
LWR model
Introduced by Lighhill, Whitham (1955) and Richards (1956):
Flow dynamics analogy: Kinematic Wave (KW) theory
Conservation law for the vehicles densities:
ρt(x, t) + (Q(x, t))x = 0
Q(x, t) = ρ(x, t)V (x, t)
(1.1)
Assumption of equilibrium & Fundamental Diagram (FD) :
V (x, t) = Ve (ρ(x, t)) (1.2)
x x + ∆x
ρ(x, t)∆x
Q(x, t)∆t Q(x + ∆x, t)∆t
Speed V
ρcrit ρmax
Density ρ
Vmax
Vcrit

Introduction Overview of intersection modelling
Intersection modelling
Main congestion points on networks
Still an interesting research point
Jn+l
Jn+2
Jj
Jn+m
J1
Ji
Jn
Many approaches: point-wise VS spatial extended models, equilibrium
VS optimization models, with or without internal states etc.
[Lebacque and Khoshyaran, (2002, 2005, 2009)]
State-of-the-art review:
See [Tampère, Corthout, Catterysse, Immers (2011)]
and [Flötteröd and Rohde (2011)]

Proposed intersection model
Outline
1 Introduction
Hamilton-Jacobi equation
Numerical Scheme
Some mathematical results

Proposed intersection model Hamilton-Jacobi equation
HJ equation: settings
Point-wise intersection without internal state
No distinction on incoming or outgoing roads
No consideration on multiclasses or multilanes
J3
J1
J2
e1
e2
eN
JN
e3
uα := primitive of ρα solution of LWR equation on branch α...
...modulo γα the proportion of ﬂow sent or received by branch α
uα solution of:



uα
t + Hα(uα
x ) = 0 outside the node,
ut + max
α=1,...,N
H−
α (uα
x ) = 0 at the node.
(2.3)
Flow maximization at the node

Proposed intersection model Hamilton-Jacobi equation
HJ versus conservation laws
Conservation laws Hamilton-Jacobi
Variable ρα(x, t) uα(x, t)
Concentration in (x, t) Vehicle index in (x, t)
Equation ρα
t + (Qα(ρα))x = 0 uα
t + Hα (uα
x ) = 0
FD / Hamilt. Demand (∆) Decreasing part (H−
α )
Supply (Σ) Increasing part (H+
α )
Qmax
Density ρ
ρcrit ρmax
Flow Q
Qmax
Supply Σ
Density ρ
Demand ∆
Density ρ
ρcrit
ρcrit
ρmax
Qmax
H+
α (p)
pα
0 p
H−
α (p)

Proposed intersection model Numerical Scheme
Introduction to the scheme
Assumptions:
Conservation of vehicles
FIFO (First-In-First-Out) sequel
Maximization on the total though-flow
Coefficients γα fixed once for all

Proposed intersection model Numerical Scheme
Presentation of the NS
∆x and ∆t = space and time steps satisfying CFL condition
pα,n
i,± (resp. Wα,n
i ) discrete space (resp. time) derivative
Proposition (Numerical Scheme)



Wα,n
i = max (−H+
α (pα,n
i,− ), −H−
α (pα,n
i,+ )), for i ≥ 1, for all α


Un
0 := Uα,n
0 , for all α
Wα,n
0 = max
α=1,...,N
(−H−
α (pα,n
i,+ )),
for i = 0
(2.4)
Passing ﬂow = Minimum between Upstream Demand and
Downstream Supply [Lebacque (1996)]

Proposed intersection model Some mathematical results
Some results
From [Costeseque, Monneau, Lebacque (in progress)]:
First main result is Time and Space Gradient estimates
Second main result is the convergence of the numerical solution to
the HJ viscosity solution when (∆t, ∆x) → 0 (under suitable
assumptions)
From [Imbert, Monneau, Zidani (2011)]:
Existence and uniqueness of the viscosity solution for the HJ equation

Numerical simulations
Outline
1 Introduction
Example 1: Diverge
Example 2: Merge

Numerical simulations
Initial statements
We assume that the FD is bi-parabolic and we take:
Link flow capacity = free flow speed × critical density
Jam critical density = 20 veh/km/lane
Maximal density = 160 veh/km/lane
Focus on highway on or off-ramps modelling
Coefficients γα for incoming roads are capacity-proportional: see
[Cassidy and Ahn, (2005)], [Bar-Gera and Ahn, (2010)] and [Ni and
Leonard, (2005)]

Numerical simulations Example 1: Diverge
Diverge: Simulations
Representation of an oﬀ-ramp:
x > 0x < 0
x = 0
ρ1
ρ2
ρ3
Simplest case: one incoming road and two outgoing roads;
Single condition: u1(0, t) = u2(0, t) = u3(0, t) (continuity of the
index at the junction point)
Conﬁguration:
Branch Number of lanes Maximal speed γα
1 2 90 km/h 1
2 2 90 km/h 0.75
3 1 50 km/h 0.25

Diverge: Fundamental Diagrams
0 50 100 150 200 250 300 350
0
500
1000
1500
2000
2500
3000
3500
4000
Density (veh/km)
Flow(veh/h)
Fundamental diagrams per branch
1
2
3

Diverge: Initial conditions (t=0s)
−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 0s)
Position (m)
Density(veh/km)

Diverge: Numerical results (t=5s)
−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 5s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 5s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 5s)
Position (m)
Density(veh/km)

−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 10s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 10s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 10s)
Position (m)
Density(veh/km)

−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 20s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 20s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 20s)
Position (m)
Density(veh/km)

−200 −150 −100 −50 0
0
10
20
30
40
50
60
70
Road n° 1 (t= 30s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 2 (t= 30s)
Position (m)
Density(veh/km)
0 50 100 150 200
0
10
20
30
40
50
60
70
Road n° 3 (t= 30s)
Position (m)
Density(veh/km)

Diverge: Cumulative Vehicles Count
0 5 10 15 20 25
0
5
10
15
20
25
30
35
Time (s)
CumulativeNumberofVehicles
CVC on road n°1
Downstream station
Upstream station
−10 0 10 20 30
0
5
10
15
20
25
30
35
Time (s)
CVC on road n°2
Downstream station
Upstream station
−10 0 10 20 30
0
5
10
15
20
25
30
35
Time (s)
CVC on road n°3
Downstream station
Upstream station

Diverge: Trajectories
5.42868
6.514427.60015
8.68589
9.77163
10.8574
11.9431
13.0288
14.1146
15.2003
16.286
17.3718
18.4575
19.5433
20.629
21.7147
Trajectories on road n° 1
Time (s)
Position(m)
0 10 20 30
−200
−150
−100
−50
0
−1.47006−0.2934180.8832282.059873.236524.413165.589816.766467.94319.11975
10.2964
11.473
12.6497
13.8263
15.003
16.1796
17.356318.5329
Time (s)
Position(m)
0 10 20 30
0
50
100
150
200
−1.47006−0.2934180.8832282.059873.236524.413165.589816.766467.94319.1197510.2964
11.473
12.6497
13.8263
15.00316.1796
17.3563
18.5329
Time (s)
Position(m)
0 10 20 30
0
50
100
150
200

Numerical simulations Example 2: Merge
Merge: Simulations
Representation of an on-ramp:
x > 0x < 0
x = 0
ρ3
ρ2
ρ1
Simplest case: two incoming roads and one outgoing road
Conﬁguration:
Branch Number of lanes Maximal speed γα
1 3 90 km/h 0.8
2 1 70 km/h 0.2
3 3 90 km/h 1

Merge: Fundamental Diagrams
Branch Initial state Final state
1 50 veh/km 4900 veh/h 180 veh/km 4300 veh/h
0 50 100 150 200 250 300 350 400 450 500
0
1000
2000
3000
4000
5000
6000
Density (veh/km)
Flow(veh/h)
Fundamental diagrams per branch
1
2
3

Merge: Initial conditions (t=0s)
−200 −150 −100 −50 0
50
100
150
Road n° 1 (t= 0s)
Position (m)
Density(veh/km)
−200 −150 −100 −50 0
50
100
150
Road n° 2 (t= 0s)
Position (m)
Density(veh/km)
0 50 100 150 200
50
100
150
Road n° 3 (t= 0s)
Position (m)
Density(veh/km)

Merge: Numerical results (t=20s)
−200 −150 −100 −50 0
50
100
150
Road n° 1 (t= 20s)
Position (m)
Density(veh/km)
−200 −150 −100 −50 0
50
100
150
Road n° 2 (t= 20s)
Position (m)
Density(veh/km)
0 50 100 150 200
50
100
150
Road n° 3 (t= 20s)
Position (m)
Density(veh/km)

−200 −150 −100 −50 0
50
100
150
Road n° 1 (t= 50s)
Position (m)
Density(veh/km)
−200 −150 −100 −50 0
50
100
150
Road n° 2 (t= 50s)
Position (m)
Density(veh/km)
0 50 100 150 200
50
100
150
Road n° 3 (t= 50s)
Position (m)
Density(veh/km)

−200 −150 −100 −50 0
50
100
150
Road n° 1 (t= 100s)
Position (m)
Density(veh/km)
−200 −150 −100 −50 0
50
100
150
Road n° 2 (t= 100s)
Position (m)
Density(veh/km)
0 50 100 150 200
50
100
150
Road n° 3 (t= 100s)
Position (m)
Density(veh/km)

Merge: Flow distribution [Daganzo]
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
2000
2500
3000
Flow Q
1
(veh/h)
FlowQ
2
(veh/h)
Flows distribution at merge
Demand ∆
1
Demand ∆
2
Priority share γ
2
/ γ
1
Supply constraint Σ
3

Merge: Cumulative Vehicles Count
0 100 200 300 400
0
50
100
150
200
250
Time (s)
CVC on road n°1
Upstream station
Downstream station
0 100 200 300 400 500
0
50
100
150
200
250
Time (s)
CVC on road n°2
Upstream station
Downstream station
−100 0 100 200 300 400
0
50
100
150
200
250
Time (s)
CVC on road n°3
Upstream station
Downstream station

Merge: Trajectories
17.36824
34.73648
52.10472
69.47297
86.84121
104.2094
121.5777
138.9459
156.3142
173.6824
191.0507
208.4189
225.7871
243.1554
260.5236
277.8919
295.2601
312.6283
329.9966347.3648
Time (s)
Position(m)
0 50 100 150 200
−200
−150
−100
−50
0
18.30346
36.60692
54.91038
73.21384
91.5173
109.8208
128.1242146.4277164.7311183.0346201.3381219.6415237.945256.2484274.5519292.8553311.1588329.4623347.7657366.0692
Time (s)
Position(m)
0 50 100 150 200
−200
−150
−100
−50
0
9.497618
24.99524
40.49285
55.99047
71.48809
86.98571
102.4833
117.9809
133.4786
148.9762
164.4738
179.9714
195.469
210.9667
226.4643
241.9619
257.4595
272.9571
288.4547
303.9524
319.45
Time (s)
Position(m)
0 50 100 150 200
0
50
100
150
200

Concluding remarks
Outline
1 Introduction

Concluding remarks
Concluding remarks
Balance
Convergent numerical scheme
Special conﬁgurations (diverge / merge)
Use of the Cumulative Vehicles Count
Perspectives:
High order models (GSOM family)
Integrate internal node supplies (urban intersection)
Micro-Macro passage (conﬂicts modelling)

Concluding remarks
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr

Complements
Some references
G. Flötteröd and J. Rohde, Operational macroscopic modeling of
complex urban intersections, Transportation Research Part B:
Methodological 45(6), (2011), pp. 903-922.
C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach
to junction problems and application to traffic flows, Working paper,
Université Paris-Dauphine, Paris, (2011), 38 pages.
M.M. Khoshyaran and J.P. Lebacque, Internal state models for
intersections in macroscopic traffic flow models, TGF09, Shanghai
2009.
J.P. Lebacque and M.M. Koshyaran, First-order macroscopic traffic
flow models: intersection modeling, network modeling, 2005, pp.
365-386.
C. Tampère, R. Corthout, D. Cattrysse and L. Immers, A generic class
of first order node models for dynamic macroscopic simulations of
traffic flows, Transportation Research Part B, 45 (1) (2011), pp.
289-309.

Complements
References for conservation laws
Hyperbolic equation of conservation laws with discontinuous ﬂux:
ρt + (Q(x, ρ))x = 0 with Q(x, p) = 1{x<0}Q1(p) + 1{x≥0}Q2(p)
Uniqueness result for two branches:
See [Garavello, Natalini, Piccoli, Terracina (2007)]
and [Andreianov, Karlsen, Risebro (2010)]
Book of [Garavello, Piccoli (2006)] for conservation laws on networks:
Construction of a solution using the ”wave front tracking method”
No proof of the uniqueness of the solution on a general networks

Complements
Literature review
State-of-the-art review from [Tampère, Corthout, Catterysse, Immers
(2011)] and [Flötteröd and Rohde (2011)]: list of seven requirements:
General applicability for any n, m ≥ 1
Non-negativity of flows
Demand and Supply constraints
Vehicles conservation through the junction
Conservation of Turning Fractions (FIFO)
Maximization of Flows from an User perspective
Compliance to the invariance principle [Lebacque, Khoshyaran (2005)]
(Opt.) Internal node supplies: non-uniqueness! [Corthout et al., 2012]

Complements
HJ equation: settings
Point-wise intersection without internal state
No distinction on incoming or outgoing roads
No consideration on multiclasses or multilanes
J3
J1
J2
e1
e2
eN
JN
e3
Primitive of ρα solution of classical LWR equation on branch α:



uα(x, t) = uα(0, t) −
1
γα
x
0
ρα
(y, t)dy, for α ∈ {outgoing roads}
uα(x, t) = uα(0, t) +
1
γα
x
0
ρα
(y, t)dy, for α ∈ {incoming roads}
with γα the proportion of ﬂow sent or received by branch α

Complements
HJ formulation
Search of suitable continuous functions u, viscosity solutions of:



uα
t + Hα(uα
x ) = 0 on (0, T) × J∗
α,
ut + max
α=1,...,N
H−
α (uα
x ) = 0 on (0, T) × {0}.
(5.5)
submitted to an initial condition (globally Lipschitz continuous)
u(0, x) = u0(x), with x ∈ J. (5.6)
The Hamiltonians Hα are convex functions such that:
The function Hα : R → R is continuous and lim
|p|→+∞
Hα(p) = +∞;
There exists pα
0 ∈ R such that Hα is non-increasing on (−∞, pα
0 ] and
non-decreasing on [pα
0 , +∞).
We denote by H−
α and H+
α the corresponding functions.

Schéma numérique basé sur une équation d'Hamilton-Jacobi : modélisation des intersections

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