Presentation at Indam workshop, Rome (March 8, 2017)

1. Queue length estimation on urban corridors
Guillaume Costeseque
with Edward S. Canepa (KAUST) and Chris G. Claudel (UT, Austin)
Inria Sophia-Antipolis M´editerran´ee
VIII Workshop on the Mathematical Foundations of Traffic
March 08, 2017
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 1 / 25

2. Motivation
Traffic control strategies
[Source: TRI Old Dominion University website]
Main control schemes:
Highways
Variable speed limits
Ramp metering
Dynamic lane management
Arterial streets
Adaptative traffic signal timings
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 2 / 25

3. Motivation
Traffic control strategies
[Source: TRI Old Dominion University website]
Main control schemes:
Highways
Variable speed limits
Ramp metering
Dynamic lane management
Arterial streets
Adaptative traffic signal timings
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 2 / 25

4. Motivation
Why introducing bounded acceleration?
Traffic light: What scalar conservation laws theory teaches us
∂tk + ∂x Q(k) = 0,
Q(k) = min {vf k , w (k − κ)}
k
(A)
(B)
(C)
(A) (A)
(A)
(A)
(B)
(C)
x
t
vf
w
Q
0 κ
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5. Motivation
Why introducing bounded acceleration?
Car trajectories (Assuming no Italian taxi drivers...)
t
x
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25

6. Motivation
Why introducing bounded acceleration?
Bounded acceleration phase [Lebacque, 2003, Leclercq, 2007]
vf
w
Q
0 κ
(A)
(B)
(C)
k
t(C)
x
(B)
(A) (A)
(A)
(A)
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25

7. Motivation
Why introducing bounded acceleration?
Car trajectories with bounded acceleration phase
t
x
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25

8. Motivation
Outline
1 Introduction
2 Optimization problem
3 Model and data constraints
4 Application to Lankershim Bvd, LA
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9. Introduction
Outline
1 Introduction
2 Optimization problem
3 Model and data constraints
4 Application to Lankershim Bvd, LA
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10. Introduction Quick review of queue length estimation methods
Queue length estimation at signalized intersections:
[data-driven] input-output techniques
(-) Need good estimate of the initial queue length
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 6 / 25

11. Introduction Quick review of queue length estimation methods
Queue length estimation at signalized intersections:
[data-driven] input-output techniques
(-) Need good estimate of the initial queue length
[data-driven] statistical/probabilistic approaches
(-) Strongly depend on realistic vehicles arrival patterns
VS sparsely available GPS data
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 6 / 25

12. Introduction Quick review of queue length estimation methods
Queue length estimation at signalized intersections:
[data-driven] input-output techniques
(-) Need good estimate of the initial queue length
[data-driven] statistical/probabilistic approaches
(-) Strongly depend on realistic vehicles arrival patterns
VS sparsely available GPS data
[model based] “shockwaves-based” approach
(-) Previous works do not account for bounded acceleration
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 6 / 25

13. Introduction Our approach
Our focus
“Shockwaves-based” approach:
optimization-based framework [Anderson et al., 2013]
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14. Introduction Our approach
Our focus
“Shockwaves-based” approach:
optimization-based framework [Anderson et al., 2013]
+ explicit solutions for the macroscopic traffic flow models
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 7 / 25

15. Introduction Our approach
Our focus
“Shockwaves-based” approach:
optimization-based framework [Anderson et al., 2013]
+ explicit solutions for the macroscopic traffic flow models
Basic assumptions:
triangular fundamental diagram (FD)
Q(k) = min {vf k , w(k − κ)}
piecewise affine conditions
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 7 / 25

16. Introduction LWR and LWR-BA models
LWR model [Lighthill and Whitham, 1955, Richards, 1956]: scalar
conservation law
∂tk + ∂x Q(k) = 0, on (0, +∞) × R, (1)
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 8 / 25

17. Introduction LWR and LWR-BA models
LWR model [Lighthill and Whitham, 1955, Richards, 1956]: scalar
conservation law
∂tk + ∂x Q(k) = 0, on (0, +∞) × R, (1)
LWR model with bounded acceleration
[Lebacque, 2002, Lebacque, 2003, Leclercq, 2002, Leclercq, 2007]
⎧
⎪⎨
⎪⎩
∂tk + ∂x Q(k) = 0, if v = Ve (k) ,
∂tk + ∂x (kv) = 0
∂tv + v∂xv = a
if v < Ve (k) ,
(2)
a is the maximal acceleration rate
Ve : k → Ve(k) equilibrium speed such that Q(k) = kVe(k)
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 8 / 25

18. Introduction Hamilton-Jacobi setting
Consider the Moskowitz function
M(t, x) =
+∞
x
k(t, y)dy (3)
such that
∂x M = −k and ∂tM = kv
Then the LWR with bounded acceleration can be recast as
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
∂tM − Q (−∂x M) = 0, if v = Ve (−∂xM) ,
∂tM + v∂x M = 0,
∂tv + v∂x v = a,
if v < Ve (−∂xM)
(4)
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19. Introduction Hamilton-Jacobi setting
Explicit solutions
Viability theory + Lax-Hopf formula
[Claudel and Bayen, 2010a, Claudel and Bayen, 2010b]
=⇒ explicit solutions
LWR model
LWR model with bounded acceleration
[Mazar´e et al., 2011] [Qiu et al., 2013]
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20. Optimization problem
Outline
1 Introduction
2 Optimization problem
3 Model and data constraints
4 Application to Lankershim Bvd, LA
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21. Optimization problem Initial and boundary conditions
Piecewise affine conditions
c
(l)
intern
t
c
(i)
ini
c
(j)
down
c
(j)
up
xn
x0
x
t0 tmax
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22. Optimization problem Initial and boundary conditions
Piecewise affine conditions
Initial conditions
c
(i)
ini (x) =
−ki x + bi , if x ∈ [xi , xi+1],
+∞, else,
Upstream boundary conditions
c
(j)
up (t) =
qj t + dj , if t ∈ [tj , tj+1],
+∞, else,
Downstream boundary conditions
c
(j)
down(t) =
pj t + bj , if t ∈ [tj , tj+1],
+∞, else,
Internal boundary condition
c
(l)
intern(t, x) =
M(l) + q
(l)
intern(t − t
(l)
min), if (t, x) ∈ D(l),
+∞, else
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23. Optimization problem Setting of the MILP
Decision variable
y := . . . , ki , . . .
initial densities
, . . . , qj , . . .
upstream flows
, . . . , pj , . . .
downstream flows
, . . . , M(l)
, q
(l)
intern, . . .
internal conditions
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24. Optimization problem Setting of the MILP
Decision variable
y := . . . , ki , . . .
initial densities
, . . . , qj , . . .
upstream flows
, . . . , pj , . . .
downstream flows
, . . . , M(l)
, q
(l)
intern, . . .
internal conditions
Optimization problem as a Mixed Integer Linear Programming (MILP)
Maximize g(y)
subject to
Amodely ≤ bmodel, (model constraints),
Cdatay ≤ ddata, (data constraints).
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 14 / 25

25. Optimization problem Setting of the MILP
Decision variable
y := . . . , ki , . . .
initial densities
, . . . , qj , . . .
upstream flows
, . . . , pj , . . .
downstream flows
, . . . , M(l)
, q
(l)
intern, . . .
internal conditions
Optimization problem as a Mixed Integer Linear Programming (MILP)
Maximize g(y)
subject to
Amodely ≤ bmodel, (model constraints),
Cdatay ≤ ddata, (data constraints).
Objective function: maximize the downstream outflows
g(y) = (0Rn , 0Rm , 1Rm , 0Ro ×Ro ) · yT
=
m−1
j=0
pj
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 14 / 25

26. Optimization problem Queue estimation
Algorithm
1 Compute the optimal solution to the MILP
y∗
:= . . . , k∗
i , . . .
initial densities
, . . . , q∗
j , . . .
upstream flows
, . . . , p∗
j , . . .
downstream flows
, . . . , M(l)
∗
, q
(l)
intern
∗
, . . .
internal conditions
= argmaxy g(y)
2 Compute the traffic states M and k = −∂x M thanks to the explicit
solutions [Qiu et al., 2013]
3 Deduce queue lengths by computing for any time step the extremal
points of
Qε(t) := (α, β)
ξ ≤ α < β ≤ χ,
|k(t, z) − κ| ≤ ε, ∀z ∈ [α, β]
where ε > 0 is a prescribed sensitivity parameter
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 15 / 25

27. Model and data constraints
Outline
1 Introduction
2 Optimization problem
3 Model and data constraints
4 Application to Lankershim Bvd, LA
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28. Model and data constraints Model constraints
Compatibility conditions
Proposition (Compatibility conditions [Claudel and Bayen, 2011])
Consider a family of value conditions cj and define their minimum
c(t, x) := min
j∈J
cj (t, x).
Then, the solution M of the LWR-BA PDE verifies
M(t, x) = c(t, x), for any (t, x) ∈ Dom (c) ,
if and only if
Mci
(t, x) ≥ cj (t, x), for all i, j ∈ J, and (t, x) ∈ Dom(cj ).
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 17 / 25

29. Model and data constraints Model constraints
xn
x0
x
tmax
t
xi
xi+1
w
w
w
vf
c
(i)
ini
t0
(i)
(ii)
(iii)
(iv)
xn
x0
x
t0 tmax
t
tj tj+1
c
(j)
up
vf
(iv)
vf
(iii)
(v)
vf
(i)
(ii)
w
Check
Mc
(i)
ini
≥ c
(j)
up
and
Mc
(j)
up
≥ c
(i)
ini
only for crossing points of
domains of influence
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30. Model and data constraints Data constraints
Data constraints
Assume that the data constraints are linear w.r.t. the decision variable y
Cdatay ≤ ddata.
1 Downstream outflow constraint (red light)
pj = 0, ∀ j s.t. Ωred ∩ [tj , tj+1] ̸= ∅,
2 [Loops] Upstream flow data qmeas with errors emeas
flow
(1 − emeas
flow )qmeas
(t) ≤ qj ≤ (1 + emeas
flow )qmeas
(t), ∀ t ∈ [tj , tj+1]
3 [GPS] Travel times data dmeas
travel with errors emeas
time
M (tmeas
exit − dmeas
travel − emeas
time , ξ) ≤ M(tmeas
exit , χ) ≤ M (tmeas
exit − dmeas
travel + emeas
time , ξ) .
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 19 / 25

31. Application to Lankershim Bvd, LA
Outline
1 Introduction
2 Optimization problem
3 Model and data constraints
4 Application to Lankershim Bvd, LA
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32. Application to Lankershim Bvd, LA
NGSIM dataset (2006)
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33. Application to Lankershim Bvd, LA
NGSIM dataset (2006)
monitored section = 5 blocks and 4 signalized intersections
individual trajectories for each vehicle (+2,400) over 30 min
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34. Queue Estimation on Networks
24
Link 1

35. Queue Estimation on Networks
25
Link 2

36. End of the talk
Thanks for your attention
Any question?
guillaume.costeseque@inria.fr
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37. References
Some references I
Anderson, L. A., Canepa, E. S., Horowitz, R., Claudel, C. G., and Bayen, A. M.
(2013).
Optimization-based queue estimation on an arterial traffic link with measurement
uncertainties.
Transportation Research Board 93rd Annual Meeting. Paper 14-4570.
Claudel, C. G. and Bayen, A. M. (2010a).
Lax–Hopf based incorporation of internal boundary conditions into
Hamilton–Jacobi equation. Part I: Theory.
Automatic Control, IEEE Transactions on, 55(5):1142–1157.
Claudel, C. G. and Bayen, A. M. (2010b).
Lax–Hopf based incorporation of internal boundary conditions into
Hamilton–Jacobi equation. Part II: Computational methods.
Automatic Control, IEEE Transactions on, 55(5):1158–1174.
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38. References
Some references II
Claudel, C. G. and Bayen, A. M. (2011).
Convex formulations of data assimilation problems for a class of Hamilton–Jacobi
equations.
SIAM Journal on Control and Optimization, 49(2):383–402.
Lebacque, J.-P. (2002).
A two phase extension of the LWR model based on the boundedness of traffic
acceleration.
In Transportation and Traffic Theory in the 21st Century. Proceedings of the 15th
International Symposium on Transportation and Traffic Theory.
Lebacque, J.-P. (2003).
Two-phase bounded-acceleration traffic flow model: analytical solutions and
applications.
Transportation Research Record: Journal of the Transportation Research Board,
1852(1):220–230.
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 27 / 25

39. References
Some references III
Leclercq, L. (2002).
Mod´elisation dynamique du trafic et applications `a l’estimation du bruit routier.
PhD thesis, Villeurbanne, INSA.
Leclercq, L. (2007).
Bounded acceleration close to fixed and moving bottlenecks.
Transportation Research Part B: Methodological, 41(3):309–319.
Lighthill, M. J. and Whitham, G. B. (1955).
On kinematic waves II. A theory of traffic flow on long crowded roads.
Proceedings of the Royal Society of London. Series A. Mathematical and Physical
Sciences, 229(1178):317–345.
Mazar´e, P.-E., Dehwah, A. H., Claudel, C. G., and Bayen, A. M. (2011).
Analytical and grid-free solutions to the Lighthill–Whitham–Richards traffic flow
model.
Transportation Research Part B: Methodological, 45(10):1727–1748.
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 28 / 25

40. References
Some references IV
Qiu, S., Abdelaziz, M., Abdellatif, F., and Claudel, C. G. (2013).
Exact and grid-free solutions to the Lighthill–Whitham–Richards traffic flow model
with bounded acceleration for a class of fundamental diagrams.
Transportation Research Part B: Methodological, 55:282–306.
Richards, P. I. (1956).
Shock waves on the highway.
Operations research, 4(1):42–51.
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 29 / 25

41. Appendices
Outline
5 References
6 Appendices
Initial condition: free-flow case
Initial condition: congested case
Upstream condition: free-flow case
Upstream condition: congested case
Downstream condition: free-flow case
Downstream condition: congested case
Junction setting
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42. Appendices Initial condition: free-flow case
xn
x0
x
tmax
t
vf
vf
w
xi+1
(iii)
t0
c
(i)
ini
xi
(i)
(ii)
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43. Appendices Initial condition: congested case
xn
x0
x
tmax
t
xi
xi+1
w
w
w
vf
c
(i)
ini
t0
(i)
(ii)
(iii)
(iv)
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44. Appendices Upstream condition: free-flow case
xn
x0
x
t0 tmax
t
tj tj+1
vf
vf
c
(j)
up
(iii)
(ii)
(i)
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45. Appendices Upstream condition: congested case
xn
x0
x
t0 tmax
t
tj tj+1
c
(j)
up
vf
(iv)
vf
(iii)
(v)
vf
(i)
(ii)
w
G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 34 / 25

46. Appendices Downstream condition: free-flow case
xn
x0
x
t0 tmax
t
tj+1tj
c
(j)
down
w w
w
(i)
(ii)
(iii)
(iv)
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47. Appendices Downstream condition: congested case
vf
(v)
w
xn
x0
x
t
w
vf
w
tmaxt0
c
(l)
intern
t
(l)
maxt
(l)
min
x
(l)
min
vf
(vi)
(ii)
(iv)
(i)
(iii)
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48. Appendices Junction setting
frampin
fout
fin
frampout
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