Intervenant - date
Macroscopic Fundamental Diagrams: Estimation methods
Ludovic Leclercq, Université de Lyon, IFSTTAR, ENTPE, COSYS-LICITETH, SeminarApril, 29th - 2015
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The LICIT Laboratory
• A joined research unit belonging to IFSTTAR and ENTPE– IFSTTAR: French National Research Center on Transportation and Network Management
– ENTPE: Civil Engineering school with a Transportation Program
• IFSTTAR and ENTPE are members of Université de Lyon• The LICIT has two research teams dedicated to road transportation system modeling and control :– MOMI (data driven approaches) – N.E. El Faouzi (head of the lab)
– AMMET (model driven approaches) – L. Leclercq
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MFD definition
Flow
Density
FD (local level)
Signals
MFD (network level)
Density or accumulation
FD + Network structure (topology / signal timings) + Route choices = MFD
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Outline
• Short recap of existing estimation methods for the Macroscopic Fundamental Diagram (MFD)
• Cross-comparison of the different methods for two typical networks (Leclercq, Chiabaut et al, part B, 2014)– Analytical VS. Edie methods– Edie VS. loops methods– Loops VS. Probes methods
• Using MFD for simulation purpose (Leclercq, Parzani et al, ISTTT21, 2015)
• Conclusion
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Cross-comparaison of different methods
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Studied Networks
I−1 O−2
1 2 3 4 5 6 7 8 9 10
200 m
nodes with traffic signals
(a) − SC1
I/O−1
I/O−7
I/O−13
I/O−19
I/O−2
I/O−8
I/O−14
I/O−20
I/O−3
I/O−9
I/O−15
I/O−21
I/O−4
I/O−10
I/O−16
I/O−22
I/O−5
I/O−11
I/O−17
I/O−23
I/O−6
I/O−12
I/O−18
I/O−24
1
7
13
19
25
31
2
8
14
20
26
32
3
9
15
21
27
33
4
10
16
22
28
34
5
11
17
23
29
35
6
12
18
24
30
36
200 m
(b) − SC2
I−1 O−2
1 2 3 4 5 6 7 8 9 10
200 m
nodes with traffic signals
(a) − SC1
I/O−1
I/O−7
I/O−13
I/O−19
I/O−2
I/O−8
I/O−14
I/O−20
I/O−3
I/O−9
I/O−15
I/O−21
I/O−4
I/O−10
I/O−16
I/O−22
I/O−5
I/O−11
I/O−17
I/O−23
I/O−6
I/O−12
I/O−18
I/O−24
1
7
13
19
25
31
2
8
14
20
26
32
3
9
15
21
27
33
4
10
16
22
28
34
5
11
17
23
29
35
6
12
18
24
30
36
200 m
(b) − SC2
A Urban corridor
A square grid network
We mainly focus on homogeneous loadings(the methodology is presented in the paper)
Numerical simulations are provided by a LWR mesoscopic simulator (Leclercq and Becarie, 2012)
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Different estimation methods
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Analytical Method (1) – Cuts for a corridor
t
x
hyperlink
Flow
Density or accumulation
(Daganzo and Geroliminis, 2008)
Copy
Copy
Moving observer (mean speed V)V
Minimum overtaking flow R(V)R(V)
Cut: Q=minV(KV+R(V))
V
R(V)
MFD
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Analytical Method (2) - The sufficient graph
The cost R(V) for different V can be calculated using a sufficient graph defined by three kinds of edges
A
Vj>0
B
(c)
t
x
edge (a) edge (b) edge (c)
Vj
Cut
NFD
Rj
(a)
K (or N)
QMoving observer path
Vj>0
Vj<0
urba
n co
rrido
r
green phase red phase
(b)
t
x
link i
(Leclercq and Geroliminis, 2013)
The estimation is tight only if the network is homegenously loaded
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Numerical Methods (1) - Edie’s definitions
I 1 I 2
1 2 3 4 5 6 7 8 9 10
200 m
nodes with traffic signals
(c)
I/O−1
I/O−7
I/O−13
I/O−19
I/O−2
I/O−8
I/O−14
I/O−20
I/O−3
I/O−9
I/O−15
I/O−21
I/O−4
I/O−10
I/O−16
I/O−22
I/O−5
I/O−11
I/O−17
I/O−23
I/O−6
I/O−12
I/O−18
I/O−24
1
7
13
19
25
31
2
8
14
20
26
32
3
9
15
21
27
33
4
10
16
22
28
34
5
11
17
23
29
35
6
12
18
24
30
36
200 m
(d)
(a)
Virtual loop detector
0
li
t t+∆t
link i
tk(x)
tk(0)
tk−nκ(l
i−x)(li)
veh fx
x+∆xS’w
(b)
x
t
Δt
Link i
For one link:
Veh k
∑ ∑τ= ∆ ∆ = ∆ ∆q
d
x t k x t
' and
'i
kk
i
kk
d’k∑ ∑τ= ∆ ∆ = ∆ ∆q
d
x t k x t
' and
'i
kk
i
kk
τ’k
For the whole network:
∑ ∑= = =Q l Q l K l K l V Q K/ / /i ii
i ii
∑ ∑= = =Q l Q l K l K l V Q K/ / /i ii
i ii
Δx
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I 1 I 2
1 2 3 4 5 6 7 8 9 10
200 m
nodes with traffic signals
(c)
I/O−1
I/O−7
I/O−13
I/O−19
I/O−2
I/O−8
I/O−14
I/O−20
I/O−3
I/O−9
I/O−15
I/O−21
I/O−4
I/O−10
I/O−16
I/O−22
I/O−5
I/O−11
I/O−17
I/O−23
I/O−6
I/O−12
I/O−18
I/O−24
1
7
13
19
25
31
2
8
14
20
26
32
3
9
15
21
27
33
4
10
16
22
28
34
5
11
17
23
29
35
6
12
18
24
30
36
200 m
(d)
Virtual loop detector
0
li
t t+∆t
link i
tk(x)
tk(0)
tk−nκ(l
i−x)(li)
veh fx
x+∆xS’w
(b)
Numerical Methods (2) – Loop detectors
For one link:Δt
Link i
x
t
Loop
∑ ∑τ= ∆ ∆ = ∆ ∆q
d
x t k x t
' and
'i
kk
i
kk
∑ ∑τ= ∆ ∆ = ∆ ∆q
d
x t k x t
' and
'i
kk
i
kkΔx
For the whole network:
∑ ∑= = =Q l Q l K l K l V Q K/ / /i ii
i ii
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Numerical Methods (3) – Probe vehicles
Veh k∑
∑τ= ∈
∈
Vd "
"k
k K
kk K
Probe vehicles provide a direct estimate for the mean network speed V:
Distance traveled
Travel time related to the Δt period
Mean flow Q should be determined by another way (loops)
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Cross-comparison of the different methods
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Analytical VS. Edie methods (1) – SC1
Unsurprisingly, cuts match with the simulation results
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(c) − SC2
K [veh/m]
Q [veh/s]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(d) − SC2
K [veh/m]
Q [veh/s]Upper enveloproute weightingtypical routeE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
(a) − SC1
K [veh/m]
Q [veh/s]
A methodE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(b) − SC2
K [veh/m]
Q [veh/s]
I/O−1&6I/O−2&5I/O−3
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Analytical VS. Edie methods (2) – SC2
I−1 O−2
1 2 3 4 5 6 7 8 9 10
200 m
nodes with traffic signals
(a) − SC1
I/O−1
I/O−7
I/O−13
I/O−19
I/O−2
I/O−8
I/O−14
I/O−20
I/O−3
I/O−9
I/O−15
I/O−21
I/O−4
I/O−10
I/O−16
I/O−22
I/O−5
I/O−11
I/O−17
I/O−23
I/O−6
I/O−12
I/O−18
I/O−24
1
7
13
19
25
31
2
8
14
20
26
32
3
9
15
21
27
33
4
10
16
22
28
34
5
11
17
23
29
35
6
12
18
24
30
36
200 m
(b) − SC2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(c) − SC2
K [veh/m]
Q [veh/s]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(d) − SC2
K [veh/m]
Q [veh/s]Upper enveloproute weightingtypical routeE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
(a) − SC1
K [veh/m]
Q [veh/s]
A methodE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(b) − SC2
K [veh/m]
Q [veh/s]
I/O−1&6I/O−2&5I/O−3
Analytical method(one turn routes)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(c) − SC2
K [veh/m]
Q [veh/s]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(d) − SC2
K [veh/m]
Q [veh/s]Upper enveloproute weightingtypical routeE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
(a) − SC1
K [veh/m]
Q [veh/s]
A methodE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(b) − SC2
K [veh/m]
Q [veh/s]
I/O−1&6I/O−2&5I/O−3
Analytical method(two turns routes)
The analytical method is only applicable for routes not for the global network
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Analytical VS. Edie methods (3) – SC2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(c) − SC2
K [veh/m]
Q [veh/s]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(d) − SC2
K [veh/m]
Q [veh/s]Upper enveloproute weightingtypical routeE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.1
0.2
0.3
0.4
0.5
(a) − SC1
K [veh/m]
Q [veh/s]
A methodE method
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.050.1
0.150.2
0.250.3
0.350.4
(b) − SC2
K [veh/m]
Q [veh/s]
I/O−1&6I/O−2&5I/O−3
Non homogeneousloadings
The global capacity is reduced in simulation because only an integer number of vehicle can pass the green at each signal cycleThe typical route method appears to provide the closest results compared to simulations
Different ways to estimate the global MFD
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Loops VS. Edie methods (1)
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(d) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(a) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(c) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
30 m
100 m
170 m
Loops are not able to capture the spatial dynamics within links
Intervenant - date
t
x
κ kc k0
0A B
C
(a)
t
x
0 kc k0
κ
x
xq
k’0 linkt1(x) t2(x)
(b)
A correction method for loops observations
K qu
qw
qu
ql t
x xq2
( )i C
0 0 0 0
0
κκ
τκ
= + − − +
The mean spatial density can be derived from the observations at x
xt k x q
uqw
qu
( )( )C
0
0 0τ
κ
( )=
−
− −
q0
li
tc
xk(x)
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Loops (corrected) VS. Edie methods – SC1
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(d) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(a) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(c) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(d) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(a) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(c) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
Urban corridor
Intervenant - date
Loops (corrected) VS. Edie methods – SC2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(d) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(a) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(c) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(d) − SC2
Heterogeneous loadings K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(a) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5(c) − SC1
K [veh/m]
Q [veh/s]
x=170 mx=100 mx=30 mx→ U(200)Edie
Grid network
The correction method provides few improvements in congestion for such a
network
Intervenant - date
Probes VS Edie methods (1)
0 0.05 0.1 0.150
0.1
0.2
0.3
0.4
0.5(a) − SC1
K [veh/m]
Q [veh/s]
ε= 2%ε= 5%ε= 10%ε= 20%Edie
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
0.05
0.1
0.15
0.2
0.25
0.3
0.35
(b) − SC2
K [veh/m]
Q [veh/s]
ε= 2%ε= 5%ε= 10%ε= 20%Edie
Low penetration rates provide accurate estimation for the mean speed
Loops are still needed to capture the mean flow
Intervenant - date
Conclusion (1)
• The Edie method:– always provides the exact estimation of the MFD but requires all the trajectories
• The analytical method:– the analytical method leads to a tight estimation for a homogeneously loaded urban corridor
– For a grid network, the main question is how to scale up the route MFDs to a global MFD.
– The typical route method provides a appealing balance between simplicity and accuracy
Intervenant - date
Conclusion (2)
• The loops method:– Not relevant for estimating MFD except if loops are numerous and uniformly distributed
– A correction method can be proposed to reduce the discrepancies due to local observations
• The probes method:– Probes provide a accurate estimation of the network mean speed even for low penetration rate (<20%)
– Loops are still needed for estimating the mean flow
Intervenant - date
Conclusion (3)
• In this paper we only focus on stationary situations
• The question of filtering stationary states at a network level is very challenging and needs to be investigated
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Using MFD for simulation purpose
Intervenant - date
The macroscopic traffic variables
• the number of vehicles (accumulation), n;;• the travel production, P;;• the mean speed, V;;• the mean travel distance, L;;• the outflow, Q=P/L;;
q
qqq
q
qqq
q
q
Same networks with same productionsTwo outflows due to different route distributions(and different mean travel distance)
Case 1 Case 2
Intervenant - date
Mean travel distance vs OD matrix
reservoirsignal
o1,1 o1,2 o1,3 o1,4
d2,1 d2,2 d2,3 d2,4
d1,1
d1,2
O1
D2
D1250 m
100 m
(a)
0 50 100 150 200 250 300200
300
400
500
600
700τ=0%τ=10%τ=20%
τ=30%τ=40%τ=50%
τ=60%τ=70%τ=80%
τ=90%τ=100%
n
L [m] (b)
0 50 100 150 200 250 300200
300
400
500
600
700
n
L1 [m] (O1→D1) (c)
0 50 100 150 200 250 300200
300
400
500
600
700
n
L2 [m] (O1→D2) (d)
Intervenant - date
Outflow estimations
0 0.5 1 1.5 20
0.5
1
1.5
210%
−10%
20%
−20%
Effective outflow [veh/s]
Estim
ated
out
flow
[veh
/s]
(a)
(m1) − see Tab.1(m2)
0 0.5 1 1.5 20
0.5
1
1.5
210%
−10%
20%
−20%
Effective outflow [veh/s]
(b)
(m1)(m3)(m4)
0 0.5 1 1.5 20
0.5
1
1.5
210%
−10%
20%
−20%
Effective outflow [veh/s]
Estim
ated
out
flow
[veh
/s]
(c)
(m1)(m5)(m6)
0 0.5 1 1.5 20
0.5
1
1.5
210%
−10%
20%
−20%
Effective outflow [veh/s]
(d)
(m1)(m7)(m8)
Method label
Route lengths
(L1 & L2)
OD Matrix
(τ)
Traffic conditions (n) Calculation method
(m1) L=E(L)
(m2) L=αE(L1)+(1-α)E(L2)
(m3) L=E(L|τ)
(m4) L=αE(L1|τ)+(1-α)E(L2|τ)
(m5) L=a n+b
(m6) L=α(a1n+b1)+(1-α) (a2n+b2)
(m7) L=a|τ n+b|τ
(m8) L=α(a1|τ n+b1|τ)+(1-α) (a2|τ n+b2|τ)
Intervenant - date
Conclusion
Intervenant - date
General conclusion
• MFD is a very promising tool to represent traffic dynamics at large scale
• Lots of researches still to be done:– To properly estimation MFD accounting for heterogeneous network loadings and the influence of OD matrix
– To properly use this concept for simulation purpose
MAGnUM
Intervenant - date
Thank you for your attention
COSYS/LICIT – Ifsttar/Bron and ENTPE25, avenue François Mitterand69675 Bron Cedex - [email protected]
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