1.225J
1.225J (ESD 205) Transportation Flow Systems
Lecture 3
Modeling Road Traffic Flow on a Link
Prof. Ismail Chabini and Prof. Amedeo Odoni
Lecture 3 Outline
‰ Time-Space Diagrams and Traffic Flow Variables
‰ Introduction to Link Performance Models
‰ Macroscopic Models and Fundamental Diagram
‰ Volume-Delay Function
‰ (Microscopic Models: Car-following Models
‰ Relationship between Macroscopic Models and Car-following Models)
‰ Summary
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Lecture 3, Page 2
•1
TimeTime-Space Diagram: Analysis at a Fixed Position
position
L
h1
x
h2
h3
h4
time
0
0
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T
Lecture 3, Page 3
Flows and Headways
‰ m(x): number of vehicles that passed in front of an observer at
position x during time interval [0,T]. (ex. m(x)=5)
‰ Flow rate: q( x) =
m( x )
T
‰ Headway hj(x): time separation between arrival time of vehicles i and
i+1
m( x)
h j ( x)
∑
j =1
=
h
(
x
)
‰ Average headway:
m( x )
‰ What is the relationship between q(x) and h (x)?
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Lecture 3, Page 4
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Flow Rate vs. Average Headway
‰ If T is large, T ≈
m( x )
∑ h ( x)
j =1
j
m( x)
1
T
=
≈
‰ Then,
q ( x ) m( x )
q( x) ≈
1
h ( x)
∑ h ( x)
j =1
j
m( x )
= h ( x)
This is intuitively correct.
‰ q(t) is also called volume in traffic flow system circles (i.e. 1.225)
‰ q(t) is also called frequency in scheduled systems circles (i.e. 1.224)
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Lecture 3, Page 5
TimeTime-Space Diagram: Analysis at Fixed Time
position
L
s1
s2
0
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t0
t
time
Lecture 3, Page 6
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Density and Average Spacing
‰ n(t): number of vehicles in a road stretch of length L at time t
‰ Density: k (t ) =
n (t )
L
‰ si(t): spacing between vehicle i and vehicle i+1
‰L≈
n (t )
∑ s (t )
i =1
‰
i
n(t )
1
L
=
≈
k (t ) n(t )
‰ k (t ) ≈
1
s (t )
∑ s (t )
i =1
i
n (t )
= s (t )
(Is this intuitive?)
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Lecture 3, Page 7
Performance Models of Traffic on a Road Link
‰ Link: a representation of a highway stretch, road from one
intersection to the next, etc.
‰ Example of measures of performance:
• Travel time
• Monetary or environmental cost
• Safety
‰ Main measure of performance: travel time
‰ 3 types of models:
• Macroscopic models: Fundamental diagram (valid in static
(stationary) conditions only. Long roads and long time periods)
• Microscopic models: Car-following models (no lane changes)
• Volume-delay functions
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Lecture 3, Page 8
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Macroscopic Flow Variables
‰ Three macroscopic flow variables of a link:
• Average density k (also denoted by ρ)
• Average flow q
• Average speed u (also denoted v)
‰ Relationships between variables:
• q = uk
• (k,q) curve: Fundamental diagram
• Fundamental diagram is a property of the road, the drivers and the
environment (icy, sunny, raining)
‰ 3 variables + 2 equations ⇒ only one variable can be an independent
variable (But one of the variables (k, u, q) can not be independent)
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Lecture 3, Page 9
Data Collected from Holland Tunnel (Eddie, 63)
Spe e d (km/hr)
Ave ra ge
Spa cing (m)
Conce ntra tion
(ve h/km)
Numbe r of
Ve hicle s
7.56
9.72
11.88
14.04
16.2
18.36
20.52
22.68
24.84
27
29.16
31.32
33.48
35.64
37.8
39.96
42.12
44.28
46.44
48.6
50.76
52.92
55.08
57.24
59.4
61.56
63.72
65.88
68.04
70.2
72.36
74.52
12.3
12.9
14.6
15.3
17.1
17.8
18.8
19.7
20.5
22.5
23.4
25.4
26.6
27.7
30
32.2
33.7
33.8
43.2
43
47.4
54.5
56.2
60.5
71.5
75.1
84.7
77.3
88.4
100.4
102.7
120.5
80.1
76.5
67.6
64.3
57.6
55.2
52.6
50
48
43.8
42
38.8
37
35.5
32.8
30.6
29.3
26.8
22.8
22.9
20.8
18.1
17.5
16.3
13.8
13.1
11.6
12.7
11.1
9.8
9.6
8.1
22
58
98
125
196
293
436
656
865
1062
1267
1328
1273
1169
1096
1248
1280
1162
1087
1252
1178
1218
1187
1135
837
569
478
291
231
169
55
56
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Lecture 3, Page 10
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(Density, Speed) Diagram for the Field Data
80
Speed (km/hr)
70
60
50
40
30
20
10
0
0
20
40
60
80
100
Density (veh/km)
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Lecture 3, Page 11
(Density, Speed) Diagram with a Fitted Curve
80
y = 0.0102x 2 - 1.7549x + 84.144
Speed (km/hr)
70
60
50
40
30
20
10
0
0
20
40
60
80
100
Density (veh/km)
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Lecture 3, Page 12
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(Density, Flow) Diagram from the Field Data
1400.0
Flow (veh/hr)
1200.0
1000.0
800.0
600.0
400.0
200.0
0.0
0
20
40
60
80
100
Density (veh/km)
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Lecture 3, Page 13
(Density, Flow) Diagram with a Fitted Curve
y = 0.0076x3 - 1.4481x 2 + 74.248x + 80.889
1400.0
Flow (veh/hr)
1200.0
1000.0
800.0
600.0
400.0
200.0
0.0
0
20
40
60
80
100
Density (veh/km)
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Lecture 3, Page 14
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(Flow, Speed) Diagram from the Field Data
80
Speed (km/hr)
70
60
50
40
30
20
10
0
0.0
500.0
1000.0
1500.0
Flow (veh/hr)
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Lecture 3, Page 15
(Spacing, Speed) Diagram from the Field Data
80
Speed (km/hr)
70
60
50
40
30
20
10
0
0
50
100
150
Spacing (m)
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Lecture 3, Page 16
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(Flow, Pace) Diagram from the Field Data
0.14
Pace (hr/km)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.0
500.0
1000.0
1500.0
Flow (veh/hr)
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Lecture 3, Page 17
Relationships between Flow Variables
(density, speed) diagram
(density, flow) diagram
q
u
umax
1
uc
qmax
Greenshiel d :
k
u = u max (1 −
)
k jam
3
3
3
1
2
kc
kjam
k
kc
2
k
kjam
• kjam: jam density (the highway stretch is like a parking lot!)
−1
• k jam = a car length
• q = uk
• qmax = q(kc) is the maximum flow, or link capacity
qmax
• uc = u ( k c ) =
kc
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Lecture 3, Page 18
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Fundamental Diagram
(density, speed) diagram
Fundamental diagram
q
u
umax
qmax
1
u=
3
3
uc
3
q
k
“stable”
kc
“unstable”
1
2
k
kjam
2
k
kjam
kc
• k ∈ [kc, kjam]: arise when flow is slower down stream due to lane
drops, a slow plowing-truck, etc
• kc is critical, since it marks the start of an “unstable” flow area
where additional input of cars decrease flow served by the highway
• (k, q) diagram is fundamental since it represents the three variable
as compared to the other diagrams
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Lecture 3, Page 19
Derived Diagrams
(flow, travel time) diagram
(flow, speed) diagram
u
t
true relationship
“unstable”
“stable”
classical
volume-delay
function
uc
“unstable”
“stable”
qmax
q
qmax
q
‰ In general, q cannot be used as a variable (why?)
‰ In the road network planning area:
• q is also called volume
• travel time is also called travel “delay”
• In the case of volume-delay functions, q is used as a variable
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Lecture 3, Page 20
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Examples of Classical VolumeVolume-Delay Functions
‰ Notation:
• q is the link flow
• t(q) is the link travel time
• c is the practical capacity
• α and β are calibration parameters
‰ Davidson’s function:
q
]
• t (q) = t (0) [1 + α
c−q
‰ US Bureau of Public Roads
q
• t (q ) = t (0)[1 + α ( ) β ]
c
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Lecture 3, Page 21
Observations on Classical VolumeVolume-Delay Functions
‰ Examples where the classical model may be acceptable:
• Delay at a signalized link
• q < qmax (mild congestion)
‰ What makes the classical model interesting?
• It is a function (There is only one value for a given q)
• Typical functions used are increasing with q, and
• their derivatives are also increasing (“it holds water” ⇔ it is
convex)
• The above are analytical properties that have been adopted to study
the properties of, and design solution algorithms for, network
traffic assignment models (Lectures 4-6)
• ⇒ An example of tradeoffs made between realism and computational tractability
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Lecture 3, Page 22
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Link Travel Time Models: CarCar-Following Models
‰ Notation:
Follower
Leader
n+1
n
0
1
k jam
ln+1
xn+1
Flow
‰ xn (t ) − xn +1 (t ) = spacing (space headway) = ln +1 (t ) +
x
L
xn
1
k jam
dxn (t )
= x&n (t )
‰ speed of vehicle n :
dt
dx&n (t ) d 2 xn (t )
=
= &x&n (t )
‰ acceleration (decceleration) of vehicle n :
dt
dt
‰ x& n (t ) − x& n +1 (t ) = l&n +1 (t )
‰ car-following regime: ln+1(t) is below a certain threshold
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Lecture 3, Page 23
Link Travel Time Models: CarCar-Following Models
0
Follower
Leader
n+1
n
xn+1
ln+1
1
k jam
Flow
xn
L
x
‰ Simple car-following model:
&x&n +1 (t + T ) = al&n +1 (t ) = a ( x& n (t ) − x&n +1 (t ))
T : reaction time (T ≈ 1.5 sec)
a : sensitivity factor (a ≈ 0.37 s −1 )
‰ Questions about this simple car-following model:
• Is it realistic?
• Does it have a relationship with macroscopic models?
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Lecture 3, Page 24
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From Microscopic Models To Macroscopic Models
‰ Simple car-following model: &x&n +1 (t ) = a ( x& n (t ) − x&n +1 (t )) (T = 0)
k
)
‰ Fundamental diagram: q = qmax (1 −
k jam
‰ Proof of “equivalency”
&x&n +1 ( y ) = a ( x& n ( y ) − x& n +1 ( y ))
&x&n +1 ( y ) dy = a ( x&n ( y ) − x&n +1 ( y ))dy = al&n +1 (t ) dy
t
∫ &x&
0
n +1
t
( y )dy = ∫ a l&n +1 (t )dy
0
un+1 (t ) − un +1 (0) = a (ln+1 (t ) − ln+1 (0))
un+1 (t ) = a ln+1 (t ) + un +1 (0) − aln+1 (0)
If ln+1 (t ) = 0, then un+1 (t ) = 0 ⇒ un+1 (0) − aln+1 (0) = 0
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Lecture 3, Page 25
From Microscopic Model to Macroscopic Model
un +1 (t ) = aln +1 (t ) = a (
1
1
−
)
k n +1 (t ) k jam
1
1
)
⇒ u = a( −
k k jam
k
1
1
⇒ q = uk = a ( −
) k = a (1 −
)
k k jam
k jam
If k = 0, then q = a
Since q = a ≥ a (1 −
⇒ q = qmax (1 −
k
k jam
k
k jam
), then a = qmax
)
‰ Note: if k → 0, then u → ∝. Does this make sense?
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Lecture 3, Page 26
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NonNon-linear CarCar-following Models
‰ &x&n +1 (t + T ) = a0
= a0
x& n (t ) − x& n +1 (t )
(xn (t ) − xn+1 (t ) )1.5
l& (t )
n +1
1.5


 ln +1 (t ) + 1 

k jam 

‰ If T = 0, the corresponding fundamental diagram is:
  k 0.5 
 
q = u max k 1 − 
  k jam  


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Lecture 3, Page 27
Flow Models Derived from CarCar-Following Models
&x&n +1 (t + T ) = a0 x&nm+1 (t + T )
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x&n (t ) − x& n +1 (t )
(xn (t ) − xn +1 (t ) )l
l
m
Flow vs. Density
0
0
1
0
1.5
0
  k
q = u max k 1 − 
  k jam

2
0

k 
q = u max  1 −


k
jam 

2
1

k
q = u max k exp  1 −

k jam

3
1
 1 k
q = u max k exp  − 
 2  k jam


k
q = q m 1 −

k jam





 k jam
q = u c k ln 
 k







0 .5












2




Lecture 3, Page 28
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Lecture 3 Summary
‰ Time-Space Diagrams and Traffic Flow Variables
‰ Introduction to Link Performance Models
‰ Macroscopic Models and Fundamental Diagram
‰ Volume-Delay Function
‰ Microscopic Models: Car-following Models
‰ Relationship between Macroscopic Models and Car-following
Models
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Lecture 3, Page 29
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