Luminy, October 2007

Traffic Flow in Networks:Scaling Conjectures, Physical Evidence,

and Control Applications

Carlos F. DaganzoU.C. Berkeley Center for Future Urban Transport

www.its.berkeley.edu/volvocenter/

References

1. Daganzo, C.F. (1996) “The nature of freeway gridlock and how to prevent it" in Transportation and Traffic Theory, Proc. 13th Int. Symp. Trans. Traffic Theory (J.B. Lesort, ed) pp. 629 646, Pergamon Elsevier, Tarrytown, N.Y.

2. Daganzo, C.F. (2007) “Urban gridlock: macroscopic modeling and mitigation approaches” Transportation Research B 41, 49-62; “corrigendum” Transportation Research B 41, 379.

3. Daganzo, C.F. and Geroliminis, N. (2007) “How to predict the macroscopic fundamental diagram of urban traffic” Working paper, Volvo Center of Excellence on Future Urban Transport, Univ. of California, Berkeley, CA (submitted).

4. Geroliminis N., Daganzo C.F. (2007a) “Macroscopic modeling of traffic in cities” 86th Annual Meeting Transportation Research Board, Washington D.C.

5. Geroliminis, N. and Daganzo, C.F. (2007b) “Existence of urban-scale macroscopic fundamental diagrams: some experimental findings” Working paper, Volvo Center of Excellence on Future Urban Transport, Univ. of California, Berkeley, CA (submitted).

2

T

x L

Definitions

Flow, q = VKT / TL (veh/hr)

Density, k = VHT / TL (veh/km)

Speed, v = VKT / VHT (km/hr)

C-rate, f = Completions / TL (veh/km-hr)

(Daganzo, 1996)

Link Laws

k0

Optimum Density

Density, kfmax

Max completion rate

C-rate, f

Flow, q

qmax, Capacity

d, kms per completion

(Daganzo, 2007)

• (q, k, v) related by FD

• q / f = d

• Optimal density (Capacity; Max C-rate)

Composition: J Identical Links

Lj Lj = L dj = d

kj , qj , vj , fj

f f ; k k

(Daganzo, 2007)

q/Jq)/(TLJ)VKT(qj jj j

Network of identical links: Jensen’s inequality: q ≤ Q(k)

If vi ~ constant:

q ~ Q(k)

f ~ Q(k) / d

Density

q( ki , qi )

d ( ki , qi )k q

f

Flow

C-rate

(Daganzo, 2007)

Conjectures

Real Networks:

• An MFD exists• Trip completions / Network flow ~ Constant

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Vehicle Accumulation

San Francisco Simulation: No Control

(Geroliminis & Daganzo, 2007a)

• Fixed sensors500 ultrasonic detectors

– Occupancy and Counts per 5min

• Mobile sensors140 taxis with GPS

– Time and position– Other relevant data

(stops, hazard lights, blinkers etc)

• Geometric dataRoad maps(detector locations, link lengths, intersection control, etc.)

(Dec. 2001 data)

10 km2

(Geroliminis & Daganzo, 2007b)

Real World Experiment: Site Description

Real World Experiment: The Demand

Occupancy by time-of-day Flow by time-of-day

(Geroliminis & Daganzo, 2007b)

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0.25

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1

0 10 20 30 40 50 60 70o i (%)

q i/m

ax {q

i}

Detector #: 10-003D Detector #: T07-005D

Real World Experiment: The Detectors

oi (%)

q i (

dim

ensi

onle

ss)

Real World Experiment: The Detectors

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0 20 40 60 80o u (%)

qu (

vhs/

5min

)

A1B1C1D1A2B2C2D2

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0 20 40 60o u (%)

vu (

km

/hr)

A1

B1

C1

D1

A2

B2

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D2

(Geroliminis & Daganzo, 2007b)

Real World Experiment:Taxis

Conjecture: Passenger carrying taxis use the same parts of the network as cars

(Geroliminis & Daganzo, 2007b)

Then:

taxi

taxi

u t u t

n t n t t

t t

Filters to determine full vs. empty taxis

A stop is a passenger move, if:• hazard lights are ON or• parking brake is used or• left blinker is ON and taxi stops > 45 sec or • speed < 3 km/hr for >60sec

A trip is valid if:• trip duration > 5 min and length > 1.5 km and • trip distance < 2 × “Euclidean distance”

(Geroliminis & Daganzo, 2007b)

A1

A3 A2

Taxi ID:1087 Date:12/14/2001

Direction:

A1→A2→A3→A4→A5→A6→A7→A8

Time Position Trip17:11.30 A1

17:22.00 A2

17:26.00 A3

17:48.00 A4

19:00.30 A5

19:34.30 A6

19:40.00 A7

19:57.00 A8

A5

A4

A8

A6

A7

1km

SEA

Area of Analysis

FULLEMPTYFULL

EMPTYFULL

EMPTYFULL

Illustration of Filter Results

(Geroliminis & Daganzo, 2007b)

Illustration of Filter Results (Cont.)

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time

outb

ound

/ in

boun

d

detectors

taxis

(Geroliminis & Daganzo, 2007b)

Real World Experiment:Taxis

Conjecture: Passenger carrying taxis use the same parts of the network as cars

(Geroliminis & Daganzo, 2007b)

Then:

taxi

taxi

u t u t

n t n t t

t t

Real World Experiment: Results

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v T (k

m/h

r)

12/14/2001,3.30-13.30

12/14/2001,13.30-24.00

^

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N' T ≥ 25vhs (in 30 min)

%error < 1/√average N' T

%error < 2/√average N' T

^

^

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time

P /

D

(km

s)^

^

(Geroliminis & Daganzo, 2007b)

Aggregate Dynamics

Given : inflow qin

Output: e = G(n)

e = G(n)

qin

n ))t(n(G)t(qdt

)t(dnin

(Daganzo, 2007)

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Time

Tri

ps

En

de

d

No Control With Control

Time

Tri

ps

En

de

d

Restrict vehicles from entering

Finding: Effect of Control

(Geroliminis & Daganzo, 2007a)

Ring Road Simulation: No Control

(Daganzo, 1996)

Ring Road Simulation: Control

(Daganzo, 1996)

Ongoing Work: San Francisco

(Daganzo & Geroliminis , 2007)

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