mesoscopic multiclass traffic flow modeling on multi-lane ... · pdf filemesoscopic multiclass...
Post on 16-Feb-2018
220 Views
Preview:
TRANSCRIPT
Mesoscopic multiclass traffic flow modelingon multi-lane sections
Guillaume Costeseque∗, Aurelien Duret
Inria Sophia-Antipolis Mediterranee& Universite de Lyon-IFSTTAR-ENTPE, LICIT
TRB Annual Meeting 2016, Washington DCJanuary 12, 2016
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 1 / 23
Motivations
Example: congested off-ramp
*
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 2 / 23
Motivations
Example: congested off-ramp
*
Requirements for modeling the upstream section:
1 multiclass2 non-FIFO
* [Richmond Bridge, c©Bay Area Council]G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 2 / 23
Motivations
Outline
1 Theoretical background
2 Mesoscopic formulation of multiclass multilane models
3 Numerical scheme
4 Conclusion and perspectives
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 3 / 23
Theoretical background
Outline
1 Theoretical background
2 Mesoscopic formulation of multiclass multilane models
3 Numerical scheme
4 Conclusion and perspectives
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 4 / 23
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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 5 / 23
Theoretical background Mesoscopic resolution
Mesoscopic resolution of the LWR model
Lagrangian-Space EulerianMeso Macron − x t − x
CLVariables
Pace p := 1v
Density k
Headway h := 1q= H(p) Flow q = Q(k)
Equation ∂np − ∂xH(p) = 0 ∂tk + ∂xQ(k) = 0
HJ
VariablePassing time T Label N
T (n, x) =
∫ x
−∞
p(n, ξ)dξ N(t, x) =
∫ +∞
x
k(t, ξ)dξ
Equation ∂nT − H (∂xT ) = 0 ∂tN − Q (−∂xN) = 0
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 6 / 23
Theoretical background Mesoscopic resolution
Mesoscopic: what for?
Strengths1 Consistent with micro and macro representations2 Large scale networks // spatial discontinuities OK3 Data assimilation (from Eulerian and Lagrangian sensors)
Weakness1 Single pipe2 Mono class3 No capacity drop at junctions
Developments1 Multilane and multiclass approach2 Relaxed FIFO assumption
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 7 / 23
Theoretical background Mesoscopic resolution
Mesoscopic: what for?
Strengths1 Consistent with micro and macro representations2 Large scale networks // spatial discontinuities OK3 Data assimilation (from Eulerian and Lagrangian sensors)
Weakness1 Single pipe2 Mono class3 No capacity drop at junctions
Developments1 Multilane and multiclass approach2 Relaxed FIFO assumption
−→ Moving bottleneck theory
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 7 / 23
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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 8 / 23
Mesoscopic formulation of multiclass multilane models
Outline
1 Theoretical background
2 Mesoscopic formulation of multiclass multilane models
3 Numerical scheme
4 Conclusion and perspectives
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 9 / 23
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
Flo
w (
veh/
s)
Fundamental Diagrams
RabbitsSlugs
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
Hea
dway
(s/
veh)
Fundamental Diagrams
RabbitsSlugs
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-dependentheadway-pace FD
H : (p, I ) 7→ H(p, I )
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 10 / 23
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
vBqD =
1
hB
R(vB , qD)
Introduce parameter δ ∈ [0, 1]
If δ = 0, strictlynon-FIFO
If 0 < δ < 1, reductionof the passing rate
If δ = 1, strictly FIFO
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 11 / 23
Mesoscopic formulation of multiclass multilane models Capacity drop
Time (s)40 60 80 100 120 140 160 180
Spa
ce (
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
Spa
ce (
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
Spa
ce (
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
Spa
ce (
m)
0
100
200
300
400
500
600
700
800
900
1000
Trajectories
T2(n, x)
T1(n, x)
δ = 0.6 δ = 1 (FIFO case)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 12 / 23
Mesoscopic formulation of multiclass multilane models Expression of the MCML model
System of coupled HJ PDEs
{
∂nT1 − H (∂xT1, I1) = 0, (rabbits)
∂nT2 − H (∂xT2, I2) = 0, (slugs)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 13 / 23
Mesoscopic formulation of multiclass multilane models Expression of the MCML model
System of coupled HJ PDEs
∂nT1 − H (∂xT1, I1) = 0, (rabbits)
∂nT2 − H (∂xT2, I2) = 0, (slugs)
H (∂xT1(n, ξ(n)), I1)− (1− δ)ξ (n∗2) ∂xT1(n, ξ(n))
≥N
N − 1H⊠
(
(1− δ)ξ (n∗2) , I1
)
, (2 → 1)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 13 / 23
Mesoscopic formulation of multiclass multilane models Expression of the MCML model
System of coupled HJ PDEs
∂nT1 − H (∂xT1, I1) = 0, (rabbits)
∂nT2 − H (∂xT2, I2) = 0, (slugs)
H (∂xT1(n, ξ(n)), I1)− (1− δ)ξ (n∗2) ∂xT1(n, ξ(n))
≥N
N − 1H⊠
(
(1− δ)ξ (n∗2) , I1
)
, (2 → 1)
whereH⊠(s, I ) = inf
p∈Dom(H(·,I )){H(p, I ) − sp}
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 13 / 23
Mesoscopic formulation of multiclass multilane models Expression of the MCML model
System of coupled HJ PDEs
∂nT1 − H (∂xT1, I1) = 0, (rabbits)
∂nT2 − H (∂xT2, I2) = 0, (slugs)
H (∂xT1(n, ξ(n)), I1)− (1− δ)ξ (n∗2) ∂xT1(n, ξ(n))
≥N
N − 1H⊠
(
(1− δ)ξ (n∗2) , I1
)
, (2 → 1)
whereH⊠(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 6= i
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 13 / 23
Mesoscopic formulation of multiclass multilane models Expression of the MCML model
System of coupled HJ PDEs
∂nT1 − H (∂xT1, I1) = 0, (rabbits)
∂nT2 − H (∂xT2, I2) = 0, (slugs)
H (∂xT1(n, ξ(n)), I1)− (1− δ)ξ (n∗2) ∂xT1(n, ξ(n))
≥N
N − 1H⊠
(
(1− δ)ξ (n∗2) , I1
)
, (2 → 1)
T2 (n, ξ(n)) ≥ T1(n∗
1, ξ(n)) + H (∂xT1(n∗
1, ξ(n)), I2) , (1 → 2)
whereH⊠(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 6= i
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 13 / 23
Numerical scheme
Outline
1 Theoretical background
2 Mesoscopic formulation of multiclass multilane models
3 Numerical scheme
4 Conclusion and perspectives
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 14 / 23
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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 15 / 23
Numerical scheme Lax-Hopf formulæ for the MCML model
Representation formulæ
∆n
n
x +∆x
x −∆x
n −∆n
x
1
κξT (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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 16 / 23
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− δhB
}
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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 17 / 23
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
1000S
pace
(m
)(alpha, delta) = (0.6, 0.8)
T2(n, x)
T1(n, x)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 18 / 23
Numerical scheme Simulation with a mixed traffic
50 100 150 200 250 300 350 400 450 500 550
Time (s)0
100
200
300
400
500
600
700
800
900
1000
Spa
ce (
m)
(alpha, delta) = (0.6, 0.8)
T1(n, x)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 19 / 23
Numerical scheme Simulation with a mixed traffic
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
Tra
vel t
ime
(s)
Class 1
20 40 60 80 100 120 140 160Vehicle label
40
60
80
100
120
140
160
180
Tra
vel t
ime
(s)
Class 2
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 20 / 23
Conclusion and perspectives
Outline
1 Theoretical background
2 Mesoscopic formulation of multiclass multilane models
3 Numerical scheme
4 Conclusion and perspectives
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 21 / 23
Conclusion and perspectives
A new event-based mesoscopic model for multi-class traffic flow onmulti-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(→ Aurelien’s presentation)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 22 / 23
Conclusion and perspectives
Thanks for your attention
Any question?
guillaume.costeseque@inria.fr
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 23 / 23
References
Some references I
Laval, J. A., Leclercq, L., 2013. The Hamilton–Jacobi partial differential equationand the three representations of traffic flow. Transportation Research Part B:Methodological 52, 17–30.
Leclercq, L., Becarie, C., 2012. A meso LWR model designed for networkapplications. In: Transportation Research Board 91th Annual Meeting. Vol. 118. p.238.
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 24 / 23
Complements
Mesoscopic resolution of the LWR model
Speed
Time
(i − 1)Space
x
Spacing
Headway
t
(i) Introduce
the pace p :=1
vthe headway h = H(p)
the passing time
T (n, x) :=
∫ x
−∞
p(n, ξ)dξ.
{
∂nT = h, (headway)
∂xT = p, (pace)
[Leclercq and Becarie(2012), Laval and Leclercq(2013)]
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 25 / 23
Complements
Lax-Hopf formula
1
u
H
p
1
κ
1
C
1
wκ
Assume
H(p) =
1
κp +
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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 26 / 23
Complements
Lax-Hopf formula
0
1
κ
x
n
n
x T (n, x)
T (n, 0)
T (0, x)
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 27 / 23
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
NQ∗
(
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∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 28 / 23
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
NQ∗
(
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) := supk∈Dom(Q)
{Q(k)− vk} .
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 28 / 23
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
vBqD =
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
C1
wκ
vB
R(vB , qD)
p
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 29 / 23
Complements
Mixed Neumann-Dirichlet boundary conditions
∂nTi (n, x0) = gi (n), on [n0,+∞),
∂nTi (n, x1) = gi (n), on [n0,+∞),
Ti (n0, x) = Gi(x), on [x0, x1],
for i ∈ {1, 2} . (5)
x
nn0
x1
x0
Downstream Supply
Upstream Demand
First trajectory
G. Costeseque∗ , A. Duret Meso Multiclass HJ model Washington DC, Jan. 12 2016 30 / 23
top related