Embed Size (px)
DESCRIPTION
Traffic Flow in Networks: Scaling Conjectures, Physical Evidence, and Control Applications. Carlos F. Daganzo U.C. Berkeley Center for Future Urban Transport www.its.berkeley.edu/volvocenter/. References. - PowerPoint PPT Presentation
Citation preview
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
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
0
300000
600000
900000
1200000
1500000
0 2000 4000 6000 8000 10000
Accumulation
Trav
el Produ
ction
Out
flow
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)
0
0.25
0.5
0.75
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 (d
imen
sion
less
)
Real World Experiment: The Detectors
0
15
30
45
0 20 40 60 80o u (%)
qu (v
hs/5
min
)
A1B1C1D1A2B2C2D2
0
10
20
30
40
0 20 40 60o u (%)
vu (k
m/h
r)
A1B1C1D1A2B2C2D2
(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.)
0.5
0.8
1.1
1.4
1.7
2
3:35 6:05 8:35 11:05 13:35 16:05 18:35 21:05 23:35time
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
0
10
20
30
40
0 4000 8000 12000n (vhs)
v T (k
m/h
r)
12/14/2001,3.30-13.30
12/14/2001,13.30-24.00
^
0
10
20
30
40
0 2000 4000 6000 8000 10000 12000
n (vhs)
v (k
m/h
r)
N' T < 25vhs (in 30 min)
N' T ≥ 25vhs (in 30 min)
%error < 1/√average N' T
%error < 2/√average N' T
^
^
0
1
2
3
4
5
3:35 6:05 8:35 11:05 13:35 16:05 18:35 21:05 23:35
time
P / D
(k
ms)
^^
(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)
n
0
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 120
0
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 1200
10000
20000
30000
40000
50000
60000
70000
0 20 40 60 80 100 120
Time
Trip
s En
ded
No Control With Control
TimeTr
ips
Ende
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)
0
5
10
15
20
25
30
0 2000 4000 6000 8000 10000
n (vhs)
v (k
m/h
r)
0
5
10
15
20
25
30
0 2000 4000 6000 8000 10000
n (vhs)
v (k
m/h
r)
