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Transport behaviour at high densities
Transpn Res. Vol. 7, pp. 421-425.
Pergamon Press 1973.
SHORT TRANSPORT
Printed in Great Britain
COMMUNICATION
BEHAVIOUR
AT HIGH
DENSITIES
S. EL-HOSAINI Department of Civil Engineering, The City University, London, England (Received
15 January 1973)
INTRODUCTION MOST OF traffic behaviour theories were based on the statistical studies. Greenshields (1934) verified his theory statistically with the available data. Lighthill and Whitham (1955) based their theory on the application of fluid dynamics principles to various highway occurrences instead of using the statistical approach. Greenberg (1958) based his theory on the fluid dynamics analogy with satisfactory mathematical and statistical approaches and developed his study to obtain functional relations for the basis interactions between vehicles. At very high density, he found that the slope of the speed-density curve was decreasing gradually as traffic was approaching the jam density. This conclusion was not based on data below 7 m.p.h. Data was collected for this purpose at Edge Lane, Liverpool (1966) to cover the range below 7 m.p.h. A new traffic model based on the compressible gas dynamics was also developed. The model fitted the collected data satisfactorily and new results were obtained. THE
MODEL
Traffic was assumed to behave like a compressible gas under piston of gas dynamics may then be used for the high densities of traffic. The equations of gas dynamics used in this study are: (i) Equation
extracted
from the general p Y - *P Y--l
where
action.
The methods
energy equation + 8 22 = constant
P = pressure of gas P
=
Y
=
U
=
density of gas ratio between the specific volume under specific heat velocity of gas.
heat
(ii) Piston action
where c
= the sonic speed of the piston. 421
under
constant
pressure
and
the constant
Short
422
(iii) Adiabatic
communication
change for an ideal gas dP yP =-_ + P
Differentiate
(i) with respect to p and substitute du -=--. dp
For traffic use density k to replace density dv
(ii) and (iii) to get c2 PU
p and speed u to replace velocity --
dk =
c2 (1)
kv’
This is the basic equation to describe the behaviour density dependence of traffic speed. Speed-density-flow
u
of traffic. It has been solved to yield the
relationships
Equation (1) was solved to get the following three relationships: (i) By integration equation (1) and substitution of the boundary condition k = k, when v = 0
where c kj
of
= constant = the jam density
(ii) By substitution
of kh = kjhj = 5280 into equation h = hj . e(U*M2~2)
where h = headway (iii) By substitution
between
(2) (3)
vehicles
of 4 = kv into equation
(2) (4)
The normalized
flow and density: k
Assume a =kj and
Figure 1 shows the relationship between a and /3. This illustrates the traffic behaviour from k = 0 to k = k, where the maximum flow, qmax exists at a = 0.605. For qmax, dfllda = 0. Then, a = fl = em* kj = e*.k, q, = v,k, = e-*.v,ki where qm = VO = k, = Also q,,, =
qmax, 27at qmax k at qmax. ef .kjc, and c =
v,.
(5) (6)
423
Short communication
0
0
FIG. I. Theoretical
0.1
0.3
0.4
0.5
0.6
0.7
0.6
0.9
diagram of the normalized flow vs the normalized
This shows that c can be obtained a particular roadway. Different
0.2
if the optimum
speed, u, is known.
I.0
density.
c must be obtained
for
forms of the model:
v
=c
EXPERIMENTAL
(I
+2ln:)+
WORK
AND
DISCUSSION
An extensive volume-speed-density data of congested traffic were recorded using a purpose built automatic data collector (1966) and the Edge Lane data provided a basis for testing the model. In an attempt to fit the new model to this data, a semi-logarithmic graph was drawn between v2 and log,k as shown in Fig. 2. The model’s formula gave a coefficient of linear correlation, I = 0.98 as a “best-fit”. The relationship obtained was written in the following form v=c
(
2log.x
k, +
1
where c = 13.32 m.p.h. and k, = 150 v.p.m. This reveals that most drivers at very high densities, below 7 m.p.h., behave differently from Greenberg’s model, and individual drivers tend to decrease their speed with an increasing gradient to the (u-k) curve. Figure 3 shows the previous models of Greenberg and Greenshields together with the new model. While gradient stays constant in Greenshields’ model and decreases in Greenberg’s
Short communication
424 160
I
I
II
I
v2 =1774-355 160
II Lnk
R= 0.98 k, = 150
vpm mph
c = 1332
140
120
100
80 0 60 b 40
c
20
I
0
30
I
I
I
40 50
Density
"'i 200
k.
I
I
*
100
II
300 400 500
wm
FIG. 2. Statistical fitting of the model’s formula to the observed data.
30
I
I
I
I
-.-.--20
I
Greenshields model Greenberg model The new model
-
Density
FIG. 3. Fundamental
k,
wm
diagrams for the previous models and the new model.
Short communication
425
model, it increases in the new model as the density increases. Also, the rate of change of gradient (r.c.g.) between k = 120 and 140 v.p.m. was found -0*00125 in Greenberg’s model, while it is +0.017 in the new model. In a comparison between the basic equation of the new model dv/dk = -(?/kv), and the basic equation of Greenberg’s model dv/dk = -(c/k) or dvldk = -(cv/kv), it is found that the two basic equations are the same at one point only when v = c. The new model fitted satisfactorily the data collected at high densities for this purpose at Liverpool as shown in Fig. 3. This proved that the slope to the (v-k) curve increases gradually as the density approaches the jam density, particularly when speed falls under 7 m.p.h. Acknowledgement-The author wishes to acknowledge with thanks theassistanceandcooperation J. A. Proudlove for some of the concepts discussed in this paper.
of Professor
REFERENCES EL-HOSAINI, S. (1966). Measurements on congested traffic flows and theories of congestion. Ph.D. Thesis, University of Liverpool. GREENBERG,H. (1958). An analysis of traffic flow. Ops. Res. 7, 79-85. GREENSHIELDS, B. D. (1934). A study of traffic capacity. H. R. B. Proc. 14, 448477. LIGHTHILL,M. J. and WHITHAM,G. B. (1955). On kinematic waves II: A theory of traffic flow on long crowded roads. Proc. R. Sot. 229, 317-345.
Pergamon Press 1973.
SHORT TRANSPORT
Printed in Great Britain
COMMUNICATION
BEHAVIOUR
AT HIGH
DENSITIES
S. EL-HOSAINI Department of Civil Engineering, The City University, London, England (Received
15 January 1973)
INTRODUCTION MOST OF traffic behaviour theories were based on the statistical studies. Greenshields (1934) verified his theory statistically with the available data. Lighthill and Whitham (1955) based their theory on the application of fluid dynamics principles to various highway occurrences instead of using the statistical approach. Greenberg (1958) based his theory on the fluid dynamics analogy with satisfactory mathematical and statistical approaches and developed his study to obtain functional relations for the basis interactions between vehicles. At very high density, he found that the slope of the speed-density curve was decreasing gradually as traffic was approaching the jam density. This conclusion was not based on data below 7 m.p.h. Data was collected for this purpose at Edge Lane, Liverpool (1966) to cover the range below 7 m.p.h. A new traffic model based on the compressible gas dynamics was also developed. The model fitted the collected data satisfactorily and new results were obtained. THE
MODEL
Traffic was assumed to behave like a compressible gas under piston of gas dynamics may then be used for the high densities of traffic. The equations of gas dynamics used in this study are: (i) Equation
extracted
from the general p Y - *P Y--l
where
action.
The methods
energy equation + 8 22 = constant
P = pressure of gas P
=
Y
=
U
=
density of gas ratio between the specific volume under specific heat velocity of gas.
heat
(ii) Piston action
where c
= the sonic speed of the piston. 421
under
constant
pressure
and
the constant
Short
422
(iii) Adiabatic
communication
change for an ideal gas dP yP =-_ + P
Differentiate
(i) with respect to p and substitute du -=--. dp
For traffic use density k to replace density dv
(ii) and (iii) to get c2 PU
p and speed u to replace velocity --
dk =
c2 (1)
kv’
This is the basic equation to describe the behaviour density dependence of traffic speed. Speed-density-flow
u
of traffic. It has been solved to yield the
relationships
Equation (1) was solved to get the following three relationships: (i) By integration equation (1) and substitution of the boundary condition k = k, when v = 0
where c kj
of
= constant = the jam density
(ii) By substitution
of kh = kjhj = 5280 into equation h = hj . e(U*M2~2)
where h = headway (iii) By substitution
between
(2) (3)
vehicles
of 4 = kv into equation
(2) (4)
The normalized
flow and density: k
Assume a =kj and
Figure 1 shows the relationship between a and /3. This illustrates the traffic behaviour from k = 0 to k = k, where the maximum flow, qmax exists at a = 0.605. For qmax, dfllda = 0. Then, a = fl = em* kj = e*.k, q, = v,k, = e-*.v,ki where qm = VO = k, = Also q,,, =
qmax, 27at qmax k at qmax. ef .kjc, and c =
v,.
(5) (6)
423
Short communication
0
0
FIG. I. Theoretical
0.1
0.3
0.4
0.5
0.6
0.7
0.6
0.9
diagram of the normalized flow vs the normalized
This shows that c can be obtained a particular roadway. Different
0.2
if the optimum
speed, u, is known.
I.0
density.
c must be obtained
for
forms of the model:
v
=c
EXPERIMENTAL
(I
+2ln:)+
WORK
AND
DISCUSSION
An extensive volume-speed-density data of congested traffic were recorded using a purpose built automatic data collector (1966) and the Edge Lane data provided a basis for testing the model. In an attempt to fit the new model to this data, a semi-logarithmic graph was drawn between v2 and log,k as shown in Fig. 2. The model’s formula gave a coefficient of linear correlation, I = 0.98 as a “best-fit”. The relationship obtained was written in the following form v=c
(
2log.x
k, +
1
where c = 13.32 m.p.h. and k, = 150 v.p.m. This reveals that most drivers at very high densities, below 7 m.p.h., behave differently from Greenberg’s model, and individual drivers tend to decrease their speed with an increasing gradient to the (u-k) curve. Figure 3 shows the previous models of Greenberg and Greenshields together with the new model. While gradient stays constant in Greenshields’ model and decreases in Greenberg’s
Short communication
424 160
I
I
II
I
v2 =1774-355 160
II Lnk
R= 0.98 k, = 150
vpm mph
c = 1332
140
120
100
80 0 60 b 40
c
20
I
0
30
I
I
I
40 50
Density
"'i 200
k.
I
I
*
100
II
300 400 500
wm
FIG. 2. Statistical fitting of the model’s formula to the observed data.
30
I
I
I
I
-.-.--20
I
Greenshields model Greenberg model The new model
-
Density
FIG. 3. Fundamental
k,
wm
diagrams for the previous models and the new model.
Short communication
425
model, it increases in the new model as the density increases. Also, the rate of change of gradient (r.c.g.) between k = 120 and 140 v.p.m. was found -0*00125 in Greenberg’s model, while it is +0.017 in the new model. In a comparison between the basic equation of the new model dv/dk = -(?/kv), and the basic equation of Greenberg’s model dv/dk = -(c/k) or dvldk = -(cv/kv), it is found that the two basic equations are the same at one point only when v = c. The new model fitted satisfactorily the data collected at high densities for this purpose at Liverpool as shown in Fig. 3. This proved that the slope to the (v-k) curve increases gradually as the density approaches the jam density, particularly when speed falls under 7 m.p.h. Acknowledgement-The author wishes to acknowledge with thanks theassistanceandcooperation J. A. Proudlove for some of the concepts discussed in this paper.
of Professor
REFERENCES EL-HOSAINI, S. (1966). Measurements on congested traffic flows and theories of congestion. Ph.D. Thesis, University of Liverpool. GREENBERG,H. (1958). An analysis of traffic flow. Ops. Res. 7, 79-85. GREENSHIELDS, B. D. (1934). A study of traffic capacity. H. R. B. Proc. 14, 448477. LIGHTHILL,M. J. and WHITHAM,G. B. (1955). On kinematic waves II: A theory of traffic flow on long crowded roads. Proc. R. Sot. 229, 317-345.