mfd leclercq 20150429 -...

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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


(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


(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


(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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I−1 O−2

1 2 3 4 5 6 7 8 9 10

200 m


(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]


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]


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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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]


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


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

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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


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


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


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


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

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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


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


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


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


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

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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

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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

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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

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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

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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

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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)

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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%


(b)

(m1)(m3)(m4)

0 0.5 1 1.5 20

0.5

1

1.5

210%

−10%

20%

−20%


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%


(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

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mfd leclercq 20150429 -...

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