Transcript Slide 1

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/
Luminy, October 2007
References
2
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
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).
approaches”
Definitions
Flow, q
= VKT / TL (veh/hr)
Density, k = VHT / TL (veh/km)
L
x
Speed, v = VKT / VHT (km/hr)
C-rate, f
= Completions / TL
(veh/km-hr)
T
(Daganzo, 1996)
Link Laws
• (q, k, v) related by FD
Flow, q
• q/f=d
qmax, Capacity
(Capacity; Max C-rate)
d, kms per
completion
C-rate, f
• Optimal density
fmax
k0
Max
completion
rate
Optimum
Density
Density, k
(Daganzo, 2007)
Composition: J Identical Links
Lj
Lj = L
dj = d
kj , qj , vj , fj
q (
 jVKT j )/(TLJ)

 j q j /J  q
k k ; f f
(Daganzo, 2007)
Conjectures
Network of identical links:
Jensen’s inequality: q ≤ Q(k)
Flow
Real Networks:
( ki , qi )
q
• An MFD exists
•
/
( kki , qqi )Network
Tripd completions
If vi ~
 constant:
q ~
 Q(k)
f ~
 Q(k) / d
flow 
~ Constant
f
C-rate
Density
(Daganzo, 2007)
San Francisco Simulation: No Control
1500000
1500000
1200000
Travel Production
Travel Production
Outflow
1200000
900000
900000
600000
600000
300000
300000
00
00
2000
2000
40004000
6000 6000
8000 8000
Vehicle
Accumulation
10000 10000
Accumulation
Accumulation
(Geroliminis & Daganzo, 2007a)
Real World Experiment:
Site Description
• Fixed sensors
500 ultrasonic detectors
– Occupancy and Counts per 5min
• Mobile sensors
140 taxis with GPS
– Time and position
– Other relevant data
(stops, hazard lights, blinkers etc)
10 km2
• Geometric data
Road maps
(detector locations, link lengths,
intersection control, etc.)
(Dec. 2001 data)
(Geroliminis & Daganzo, 2007b)
Real World Experiment:
The Demand
Occupancy by time-of-day
Flow by time-of-day
(Geroliminis & Daganzo, 2007b)
Real World Experiment:
The Detectors
0.75
qi /max {qi }
qi (dimensionless)
1
0.5
0.25
Detector #: 10-003D
Detector #: T07-005D
0
0
10
20
30
i (%)
oio(%)
40
50
60
70
Real World Experiment:
The Detectors
A1
30
30
B1 A1
C1 B1
C1
D1
D1
A2
A2
B2 B2
C2 C2
D2 D2
20
u
qvu (vhs/5min)
(km/hr)
45
40
15
10
00
00
20
20
u
u
40
o o(%)
(%)
40
60
60 80
(Geroliminis & Daganzo, 2007b)
Real World Experiment:
Taxis
Conjecture: Passenger carrying taxis use the same parts of the
network as cars
Then:
u  t   utaxi  t 
n  t   ntaxi  t    t 
 t  

(Geroliminis & Daganzo, 2007b)
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
• trip distance < 2 × “Euclidean distance”
and
(Geroliminis & Daganzo, 2007b)
Illustration of Filter Results
Taxi ID:1087
Date:12/14/2001
Direction:
A1→A2→A3→A4→A5→A6→A7→A8
Time
Position
17:11.30
17:22.00
17:26.00
17:48.00
19:00.30
19:34.30
19:40.00
19:57.00
A1
A2
A3
A4
A5
A6
A7
A8
A4
Trip
SEA
FULL
EMPTY
FULL
EMPTY
FULL
EMPTY
FULL
A8
A6
Area of
Analysis
A5
A7
A3
A2
1km
A1
(Geroliminis & Daganzo, 2007b)
Illustration of Filter Results (Cont.)
2
detectors
outbound / inbound
1.7
taxis
1.4
1.1
0.8
0.5
3:35
6:05
8:35
11:05
13:35
16:05
18:35
21:05
23:35
time
(Geroliminis & Daganzo, 2007b)
Real World Experiment:
Taxis
Conjecture: Passenger carrying taxis use the same parts of the
network as cars
Then:
u  t   utaxi  t 
n  t   ntaxi  t    t 
 t  

(Geroliminis & Daganzo, 2007b)
Real World Experiment:
Results
40
40
5
N' T < 25vhs
(in 30 min)
12/14/2001,
30
30
3.30-13.30
N' T ≥ 25vhs (in 30 min)
12/14/2001,
13.30-24.00
3
2
vT v^(km/hr)
(km/hr)
^ ^
P / D (kms)
4
%error < 1/√average N' T
20
20
%error < 2/√average N' T
1
10
10
0
3:35
6:05
0
0 0
0
8:35
11:05
13:35
16:05
18:35
21:05
23:35
time
2000
4000
4000
6000
^(vhs)
n^n
(vhs)
8000
8000
10000
12000
12000
(Geroliminis & Daganzo, 2007b)
Aggregate Dynamics
Given : inflow qin
qin
Output: e = G(n)
dn( t )
 qin ( t )  G( n( t ))
dt
n
e = G(n)
n
(Daganzo, 2007)
Finding: Effect of Control
With Control
70000
Trips Ended
60000
50000
40000
30000
20000
10000
0
0
20
40
60
80
100
120
80
100
120
Time
70000
60000
Trips Ended
No Control
50000
40000
30000
20000
10000
0
0
20
40
60
Time
Restrict vehicles from entering
(Geroliminis & Daganzo, 2007a)
Ring Road Simulation: No Control
(Daganzo, 1996)
Ring Road Simulation: Control
(Daganzo, 1996)
Ongoing Work: San Francisco
30
25
v (km/hr)
20
15
10
5
0
0
2000
4000
6000
8000
10000
n (vhs)
(Daganzo & Geroliminis , 2007)