School of Engineering
Assignment Title Greenshield’s and Greenberg’s Model
Name
: Waseem Akram
Lecture
: Miss Thillai
Subject
: Traffic Engineering
Matric Number : 092-90-4968 Program
: BCE
Date
: 10 August 2010
Marks:
Greenshield’s Model (1935) Introduction: To figure out the exact relationship between the traffic parameters, a great deal of research has been done over the past several decades. The results of these researches yielded many mathematical models. Some important models among them will be discussed. Speed-Flow Model: Macroscopic stream models represent how the behavior of one parameter of traffic flow changes with respect to another. Most important among them is the relation between speed and density. The first and most simple relation between them is proposed by Greenshield. Greenshield assumed a linear speeddensity relationship as illustrated in figure to derive the model. The equation for this relationship is shown below.
Relation between speed and density •
Description
–
v = mean speed
–
k = density
–
vf = free flow speed
–
kj = jam density
–
When density becomes zero, speed approaches free flow speed
•
Relation between speed and flow
•
Relation between flow and density
Boundary conditions
•
–
Maximum flow
–
Density corresponding to max. flow
–
Speed corresponding to max. flow
Model parameters –
Jam density
–
Free flow speed
•
Density corresponding to max. flow –
We have
–
Differentiating
•
Maximum flow
•
Speed corresponding to max. flow
•
Calibration
–
Determination of model parameters o
Free flow speed (vf)
o
Jam density (kj)
where x is density and y denotes speed
Example –
Calibrate Greenshields model using the data give in the table
–
Find the maximum flow
–
Find the density corresponding to a speed of 30 km/hr
No
K
v
1
171
5
2
129
15
3
20
40
4
70
25
Solution Denoting y = v and x = k, The solution is tabulated as shown below. x(k) y(v)
(
) (
) (
)(
) (
)
171
5
73.5
-16.3
-1198.1
5402.3
129
15
31.5
-6.3
-198.5
992.3
20
40
-77.5
18.7
-1449.3
6006.3
70
25
-27.5
3.7
-101.8
756.3
390
85
-2947.7
13157.2
= 97.5,
,b= equation will be,
= -0.2
= 21.3. From
= 21.3 + 0.2
97.5 = 40.8 So the linear regression
Here
= 40.8 and
= 0.2 This implies,
= 204 veh/km The basic parameters of
Greenshield's model are free flow speed and jam density and they are obtained as 40.8 kmph and 204 veh/km respectively. To find maximum flow, i.e.,
= 2080.8 veh/hr
Density corresponding to the speed 30 km/hr can be found out by substituting = 40.8 - 0.2
k Therefore, k =
. i.e, 30
= 54 veh/km Answer
Greenberg’s Model (1959) In order to use this model for any traffic stream, one should get the boundary values, especially free ow speed (vf ) and jam density (kj ). This has to be obtained by field survey and this is called calibration process. Although it is difficult to determine exact free flow speed and jam density directly from the field, approximate values can be obtained from a number of speed and density observations and then fitting a linear equation between them. Let the linear equation be y = a+bx such that y is density k and x denotes the speed v. For example the Greenberg’s model
u2 u2 S ut L xo 2a f 2 a l can be used in general when a f is the following vehicle deceleration, al is the lead vehicle deceleration, u is the speed,, L is the length of vehicle and x o is the safety margin between vehicles. By letting al = , af = 6.0; al = 6.0, af = 6.0; al = 6.0, af = 2.0 and AASHTO stopping with brick wall the following results can be obtained Actually it can be shown that the Greenberg model and the car following model can be shown to result in s similar relationship
u u0 ln(
kj k
)
Or u
1 k e u0 kj Using uo=23 and kj = 185 the volume at various speed can be computed (Value from “Traffic Flow Fundamentals “ by Adolf May) speed 0 10 20 30 40 50 60
concentration 185 119.7699975 77.53974221 50.19964721 32.49952228 21.04036597 13.62164638
Spacing 28.54054 44.0845 68.09411 105.18 162.4639 250.9462 387.6183
Volume 0 440.845 1361.882 3155.401 6498.557 12547.31 23257.1
Example of traffic changes from Lane A to Lane B: When flow conditions change from one state to another there is a wave that develops both in front of the change and in back of the change. In front of the traffic slows and volume increases. Upstream the traffic arrives and is slowed to the speed of the moving platoon of traffic. Changes of this type occur due to traffic incidents such as collisions and /or slow moving traffic entering the traffic stream. This is best understood by looking at the changes as they occur on a typical diagram of volume versus concentration. Assume for example that traffic instantaneously changes from state B to state A where state B is up stream and has a volume of 1200 vph and 40 mph and state A is 2000 vph at 20 mph. The resulting shock wave (the difference between the speed of the forward moving traffic and the backward moving platoon of traffic is defined by the equation
AB
q q A qB 2000 1200 800 11.4mph k k A k B 100 30 70
This is the speed of the forward moving shock wave after the change of state. The backward moving wave is from state B to a jammed concentration state with 0 speed . The speed of this backward moving wave is
BC
qC qB 0 1200 1200 7.1mph kC k B 200 30 170
Greenberg's logarithmic model:
Greenberg assumed a logarithmic relation between speed and density. He proposed,
Greenberg's logarithmic model
This model has gained very good popularity because this model can be derived analytically. (This derivation is beyond the scope of this notes). However, main drawbacks of this model is that as density tends to zero, speed tends to infinity. This shows the inability of the model to predict the speeds at lower densities
Comparison of the various flow models is shown below: Macoscopic flow l= m= uf= kf=
2.2 1 50 250
kact 0 10 25 50 100 150 200 250
Equation u 50 48.94939 46.84521 42.7522 33.34894 22.91359 11.7459 0
Composite q u 0 #DIV/0! 489.4939 80.4719 1171.13 57.56463 2137.61 40.23595 3334.894 22.90727 3437.038 12.77064 2349.18 5.578589 0 0
Greenberg q u #DIV/0! 804.719 1439.116 2011.797 2290.727 1915.596 1115.718 0
50 48 45 40 30 20 10 0
Greenshild q u 0 50 480 48.03947 1125 45.24187 2000 40.93654 3000 33.516 3000 27.44058 2000 22.46645 0 18.39397
Underwood q u 0 50 480.3947 49.84026 1131.047 49.00993 2046.827 46.15582 3351.6 36.30745 4116.087 24.33761 4493.29 13.90187 4598.493 6.766764
Northwest q 0 498.4026 1225.248 2307.791 3630.745 3650.642 2780.373 1691.691
Greenshield k u u f (1 ) kj
70
4000
non lin
50
Green
40
Greenberg
non lin
5000
60
Greenberg
3000 Greenshiel d
Greenshield 30
k u u f ln( j ) k Underwood
Underwood
20
NW
2000 1000
Underwoo d
10
0
0 0
2000
4000
0
6000
100
200
300
NW
k u u f exp( ) kj 90
Nothwestern k u u f exp(0.5( ) 2 ) kj
80
Composite
60
u
(1 m )
u
(1 m ) f
k (1 ( )(l 1) ) kj
70 non lin Greenberg Greenshild Under NW
50
l 1
40
0 m 1
30 20 10 0 0
50
100
150
200
250
300
Conclusions and recommendations To apply this model details of the road network of the area are needed. The more detailed the information on the network the more accurate the results will be. However, the more complex the computation will be. It is important to note that using a more detailed road network means more time is required to set up the model. The model set up needs to make assumptions concerning how many vehicles depart from each node, so if a detailed network is used it could be time consuming to set up the model in terms of assigning vehicles to each node:
It is relatively quick to set up such a model for small areas;
It can be simply modified;
Large areas can be modelled using relatively simple networks;
A number of different evacuation routes and locations of safe havens can be tested quickly.
The current status of mathematical models for speed-flow concentration relationships is in a state of flux. The models that dominated for nearly 30 years are incompatible with the data currently being obtained, but no replacement models have yet been developed
References i.
http://www.civil.iitb.ac.in/tvm/1100_LnTse/119_lntse/plain/plain.html
ii.
http://ops.fhwa.dot.gov/publications/fhwahop06006/chapter_4.htm
iii.
http://www.mikrotik.com/testdocs/ros/2.9/ip/traffic-flow.pdf
iv.
http://www.trafficengland.com/motorwayflow.aspx
v.
http://www.webs1.uidaho.edu/niatt_labmanual/Chapters/trafficflowtheory/Introduction/index .htm
vi.
http://www.trafficflowseo.com/
vii.
http://www.tft.pdx.edu/
viii.
http://searchenterprisewan.techtarget.com/tip/0,289483,sid200_gci1173559,00.html
