Mesoscopic multiclass traffic flow modeling on multi-lane sections

Mesoscopic multiclass traffic flow modeling
on multi-lane sections
Guillaume Costeseque∗, Aurélien Duret
Inria Sophia-Antipolis Méditerranée
& Université de Lyon-IFSTTAR-ENTPE, LICIT
TRB Annual Meeting 2016, Washington DC
January 12, 2016
G. Costeseque∗
, A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 1 / 23

Motivations
Example: congested oﬀ-ramp
*
G. Costeseque∗

Motivations
Example: congested oﬀ-ramp
*
Requirements for modeling the upstream section:
1 multiclass
2 non-FIFO
* [Richmond Bridge, c Bay Area Council]
G. Costeseque∗

Motivations
Outline
1 Theoretical background
2 Mesoscopic formulation of multiclass multilane models
3 Numerical scheme
4 Conclusion and perspectives
G. Costeseque∗

Theoretical background
Outline
3 Numerical scheme
G. Costeseque∗

Theoretical background Macroscopic models
Three representations of traﬃc ﬂow
Moskowitz’ surface
Flow
x
t
N
x
See also [Moskowitz(1959), Makigami et al(1971), Laval and Leclercq(2013)]
G. Costeseque∗

Theoretical background Mesoscopic resolution
Mesoscopic resolution of the LWR model
Lagrangian-Space Eulerian
Meso Macro
n − x t − x
CL
Variables
Pace p := 1
v Density k
Headway h := 1
q = H(p) Flow q = Q(k)
Equation ∂np − ∂x H(p) = 0 ∂tk + ∂x Q(k) = 0
HJ
Variable
Passing time T Label N
T(n, x) =
x
−∞
p(n, ξ)dξ N(t, x) =
+∞
x
k(t, ξ)dξ
Equation ∂nT − H (∂x T) = 0 ∂tN − Q (−∂x N) = 0
G. Costeseque∗

Mesoscopic: what for?
Strengths
1 Consistent with micro and macro representations
2 Large scale networks // spatial discontinuities OK
3 Data assimilation (from Eulerian and Lagrangian sensors)
Weakness
1 Single pipe
2 Mono class
3 No capacity drop at junctions
Developments
1 Multilane and multiclass approach
2 Relaxed FIFO assumption
G. Costeseque∗

Mesoscopic: what for?
Strengths
1 Consistent with micro and macro representations
2 Large scale networks // spatial discontinuities OK
3 Data assimilation (from Eulerian and Lagrangian sensors)
Weakness
1 Single pipe
2 Mono class
3 No capacity drop at junctions
Developments
1 Multilane and multiclass approach
2 Relaxed FIFO assumption
−→ Moving bottleneck theory
G. Costeseque∗

Theoretical background The moving bottleneck theory
Notations
(Eulerian)
vB
−w
−w
vB
k
Q
(N − 1) lanes N lanes
R(vB, qD)
(D) (U)
ξN(t)
x
(U) (D)
vB
kD (N − 1)κ Nκ
qD
NC
Q∗
(vB)
u
G. Costeseque∗

Mesoscopic formulation of multiclass multilane models
Outline
3 Numerical scheme
G. Costeseque∗

Mesoscopic formulation of multiclass multilane models
Settings
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Density (veh/m)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Flow(veh/s)
Fundamental Diagrams
Rabbits
Slugs
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Pace (s/m)
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
Headway(s/veh)
Fundamental Diagrams
Rabbits
Slugs
Stretch of road [x0, x1] (diverge at x1)
composed by N separate lanes
Two classes of users: “rabbits” (I = I1)
and “slugs” (I = I2).
Triangular class-dependent
headway-pace FD
H : (p, I) → H(p, I)
G. Costeseque∗

Mesoscopic formulation of multiclass multilane models Capacity drop
Capacity drop parameter
vB
k
Q
(D) (U)
kD (N − 1)κ Nκ
NC
(D)
(U)
δ = 1
δ = 0
vB
qD =
1
hB
R(vB, qD)
Introduce parameter δ ∈ [0, 1]
If δ = 0, strictly
non-FIFO
If 0 < δ < 1, reduction
of the passing rate
If δ = 1, strictly FIFO
G. Costeseque∗

Mesoscopic formulation of multiclass multilane models Capacity drop
Time (s)
40 60 80 100 120 140 160 180
Space(m)
0
100
200
300
400
500
600
700
800
900
1000
Trajectories
T2
(n, x)
T1
(n, x)
Time (s)
40 60 80 100 120 140 160 180
Space(m)
0
100
200
300
400
500
600
700
800
900
1000
Trajectories
T2
(n, x)
T1
(n, x)
δ = 0 δ = 0.4
Time (s)
40 60 80 100 120 140 160 180
Space(m)
0
100
200
300
400
500
600
700
800
900
1000
Trajectories
T2
(n, x)
T
1
(n, x)
Time (s)
40 60 80 100 120 140 160 180
Space(m) 0
100
200
300
400
500
600
700
800
900
1000
Trajectories
T2
(n, x)
T
1
(n, x)
δ = 0.6 δ = 1 (FIFO case)
G. Costeseque∗

Mesoscopic formulation of multiclass multilane models Expression of the MCML model
System of coupled HJ PDEs
∂nT1 − H (∂x T1, I1) = 0, (rabbits)
∂nT2 − H (∂x T2, I2) = 0, (slugs)
G. Costeseque∗




∂nT1 − H (∂x T1, I1) = 0, (rabbits)
∂nT2 − H (∂x T2, I2) = 0, (slugs)
H (∂x T1(n, ξ(n)), I1) − (1 − δ) ˙ξ (n∗
2) ∂x T1(n, ξ(n))
≥
N
N − 1
H⊠ (1 − δ) ˙ξ (n∗
2) , I1 , (2 → 1)
G. Costeseque∗




∂nT1 − H (∂x T1, I1) = 0, (rabbits)
∂nT2 − H (∂x T2, I2) = 0, (slugs)
H (∂x T1(n, ξ(n)), I1) − (1 − δ) ˙ξ (n∗
2) ∂x T1(n, ξ(n))
≥
N
N − 1
H⊠ (1 − δ) ˙ξ (n∗
2) , I1 , (2 → 1)
where
H⊠
(s, I) = inf
p∈Dom(H(·,I))
{H(p, I) − sp}
G. Costeseque∗




∂nT1 − H (∂x T1, I1) = 0, (rabbits)
∂nT2 − H (∂x T2, I2) = 0, (slugs)
H (∂x T1(n, ξ(n)), I1) − (1 − δ) ˙ξ (n∗
2) ∂x T1(n, ξ(n))
≥
N
N − 1
H⊠ (1 − δ) ˙ξ (n∗
2) , I1 , (2 → 1)
where
H⊠
(s, I) = inf
p∈Dom(H(·,I))
{H(p, I) − sp}
and n∗
i = the nearest leader from class i for vehicle n of class j = i
G. Costeseque∗




∂nT1 − H (∂x T1, I1) = 0, (rabbits)
∂nT2 − H (∂x T2, I2) = 0, (slugs)
H (∂x T1(n, ξ(n)), I1) − (1 − δ) ˙ξ (n∗
2) ∂x T1(n, ξ(n))
≥
N
N − 1
H⊠ (1 − δ) ˙ξ (n∗
2) , I1 , (2 → 1)
T2 (n, ξ(n)) ≥ T1(n∗
1, ξ(n)) + H (∂x T1(n∗
1, ξ(n)), I2) , (1 → 2)
where
H⊠
(s, I) = inf
p∈Dom(H(·,I))
{H(p, I) − sp}
and n∗
i = the nearest leader from class i for vehicle n of class j = i
G. Costeseque∗

Numerical scheme
Outline
3 Numerical scheme
G. Costeseque∗

Numerical scheme Lax-Hopf formula for the meso LWR model
Lax-Hopf formula & Dynamic Programming
Finite steps (∆n, ∆x)
∆n = κ ∆x.
n
x + ∆x
x − ∆x
n − ∆n
x
1
κ
x
n
∆x
∆n
T(n, x)
Solution reads:
T(n, x) = max
= free ﬂow
T(n, x − ∆x) +
∆x
u
,
T (n − ∆n, x + ∆x) +
∆x
w
= congested
.
See [Laval and Leclercq(2013)]
G. Costeseque∗

Numerical scheme Lax-Hopf formulæ for the MCML model
Representation formulæ
∆n
n
x + ∆x
x − ∆x
n − ∆n
x
1
κ
ξT (n)
Ti(n, x)
∆x
x
n
New supply constraint:
Ti (n, x) = max
= free ﬂow
Ti (n, x − ∆x) +
∆x
ui
,
Ti (n − ∆n, x + ∆x) +
∆x
w
= congested
,
coupling condition .
G. Costeseque∗

Numerical scheme Lax-Hopf formulæ for the MCML model
Representation formulæ
(Coupling conditions)



T1(n, x) = max T1(n, x − ∆x) +
∆x
u1
, T1(n − ∆n, x + ∆x) +
∆x
w
,
T2(n∗
2, x) +
1
1 − δ
hB
T2(n, x) = max T2(n, x − ∆x) +
∆x
u2
, T2(n − ∆n, x + ∆x) +
∆x
w
,
T1(n∗
1, x) + H
T1(n∗
1, x) − T1(n∗
1, x − ∆x)
∆x
, I2
(1)
G. Costeseque∗

Numerical scheme Simulation with a mixed traﬃc
Distribution per class: class 1 = 60% and class 2 = 40%
Capacity drop: δ = 0.8
50 100 150 200 250 300 350 400 450 500 550
Time (s)
0
100
200
300
400
500
600
700
800
900
1000Space(m) (alpha, delta) = (0.6, 0.8)
T2
(n, x)
T1
(n, x)
G. Costeseque∗

50 100 150 200 250 300 350 400 450 500 550
Time (s)
0
100
200
300
400
500
600
700
800
900
1000
Space(m)
(alpha, delta) = (0.6, 0.8)
T
1
(n, x)
G. Costeseque∗

Individual travel times
20 40 60 80 100 120 140 160 180 200 220 240
Vehicle label
40
60
80
100
120
140
160
180
Traveltime(s)
Class 1
20 40 60 80 100 120 140 160
Vehicle label
40
60
80
100
120
140
160
180
Traveltime(s)
Class 2
G. Costeseque∗

Conclusion and perspectives
Outline
3 Numerical scheme
G. Costeseque∗

A new event-based mesoscopic model for multi-class traffic flow on
multi-lane sections
Use of theory of moving bottlenecks
Among the perspectives:
Sensitivity analysis w.r.t. δ
Validation with real traffic data
Data assimilation for real-time applications
(→ Aurélien’s presentation)
G. Costeseque∗

Thanks for your attention
Any question?
guillaume.costeseque@inria.fr
G. Costeseque∗

References
Some references I
Laval, J. A., Leclercq, L., 2013. The Hamilton–Jacobi partial differential equation
and the three representations of traffic flow. Transportation Research Part B:
Methodological 52, 17–30.
Leclercq, L., Bécarie, C., 2012. A meso LWR model designed for network
applications. In: Transportation Research Board 91th Annual Meeting. Vol. 118. p.
238.
G. Costeseque∗

Complements
Mesoscopic resolution of the LWR model
Speed
Time
(i − 1)Space
x
Spacing
Headway
t
(i) Introduce
the pace p :=
1
v
the headway h = H(p)
the passing time
T(n, x) :=
x
−∞
p(n, ξ)dξ.
∂nT = h, (headway)
∂x T = p, (pace)
[Leclercq and B´ecarie(2012), Laval and Leclercq(2013)]
G. Costeseque∗

Complements
Lax-Hopf formula
1
u
H
p
1
κ
1
C
1
wκ
Assume
H(p) =



1
κ
p +
1
wκ
, if p ≥
1
u
,
+∞, otherwise,
Proposition (Representation formula (Lax-Hopf))
The solution under smooth boundary conditions is given by
T(n, x) = max T(n, 0) +
x
u
= free ﬂow
, T 0, x +
n
κ
+
n
wκ
= congested
. (2)
See [Laval and Leclercq(2013)]
G. Costeseque∗

Complements
Lax-Hopf formula
0
1
κ
x
n
n
x T(n, x)
T(n, 0)
T(0, x)
G. Costeseque∗

Complements
Math problem in Eulerian framework
Coupled ODE-PDE problem



∂tk + ∂x (Q(k)) = 0,
Q(k(t, ξN(t))) − ˙ξN(t)k(t, ξN (t)) ≤
N − 1
N
Q∗ ˙ξN(t) ,
˙ξN(t) = min {vb, V (k(t, ξN(t)+))} ,
(3)
with
k(0, x) = k0(x), on R,
ξN(0) = ξ0.
(4)
G. Costeseque∗

Complements
Math problem in Eulerian framework
Coupled ODE-PDE problem



∂tk + ∂x (Q(k)) = 0,
Q(k(t, ξN(t))) − ˙ξN(t)k(t, ξN (t)) ≤
N − 1
N
Q∗ ˙ξN(t) ,
˙ξN(t) = min {vb, V (k(t, ξN(t)+))} ,
(3)
with
k(0, x) = k0(x), on R,
ξN(0) = ξ0.
(4)
and Q∗ is the Legendre-Fenchel transform of Q
Q∗
(v) := sup
k∈Dom(Q)
{Q(k) − vk} .
G. Costeseque∗

Complements
Capacity drop parameter
ξN(t)
x
(U) (D)
vB
vB
k
Q
(D) (U)
kD (N − 1)κ Nκ
NC
u
(D)
(U)
δ = 1
δ = 0
vB
qD =
1
hB
R(vB, qD)
−w
−w
Case non−FIFO Case FIFO
(D)
(U)
(U) (D)
Flow
(MB)
(U)
pB =
1
vB
δ = 0 δ = 1
H
1
κ
1
u
hB =
1
qD
1
R(vB, qD)
vB
1
C
1
wκ
vB
R(vB, qD)
p
G. Costeseque∗

Complements
Mixed Neumann-Dirichlet boundary conditions



∂nTi (n, x0) = ˇgi (n), on [n0, +∞),
∂nTi (n, x1) = ˆgi (n), on [n0, +∞),
Ti (n0, x) = Gi (x), on [x0, x1],
for i ∈ {1, 2} . (5)
x
n
n0
x1
x0
Downstream Supply
Upstream Demand
First trajectory
G. Costeseque∗

Mesoscopic multiclass traffic flow modeling on multi-lane sections

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