Analysis and validation of lane-drop effects on multi-lane freeways

Transpn Res. Vol. 5, pp. 257-266.

ANALYSIS

AND

Pergamon

Press 1971. Printed in Great Britain

VALIDATION MULTI-LANE

OF LANE-DROP FREEWAYS?

EFFECTS

ON

P. K. MUNJAL,YUAN-SHIHHsu and R. L. LAWRENCE System Development Corporation, Santa Monica, California, U.S.A. (Received

1 April

1971)

INTRODUCTION A LANE-DROPon a multi-lane freeway results in traffic density perturbations and usually causes congestion if the traffic is heavy. In this paper, a mathematical model that describes the perturbation phenomenon is first developed, and then the analysis is validated with aerial photographic data. The analysis is based on the equation of continuity of the hydrodynamic theory of traffic flow and a postulated cross-flow which assumes that the crossflow is proportional to the difference in density in the adjacent lanes, and it is in the direction of decreasing density. The results of the analysis indicate that dropping one lane affects the flow in all lanes upstream by the generation of four perturbation density waves. The exact perturbation along the distance from the lane-drop point depends on flow, concentration and boundary conditions. Two experimental cases are included in this paper. The first case is a combined lanedrop off-ramp, the second is a lane-drop following an on-ramp. In both cases, the analytical results compare favorably with the field data collected by aerial photography. PROPAGATION

OF DENSITY

WAVES

Gazis et al. (1962) analyzed the density oscillations with time between the various lanes of a multi-lane undirectional freeway, with emphasis on the question of stability. The lanechanging rule they assumed is a linear function of the densities. However, they neglected variation of the various densities with distance along the highway and considered the density oscillations as functions of time alone. Oliver and Lam (1967) assumed that lane-changing phenomena could be represented by a nonlinear function of densities. They performed statistical experiments on the Nimitz Freeway in California to determine the validity of their proposed lane-changing functions. Neither of the above papers considered the perturbation effects. This paper examines the disturbances in density caused by a lane-drop on a multi-lane freeway. The longitudinal waves of density and lane-changing phenomena are taken into account. The lane-changing function employed is the one proposed by Gazis et al. (1962), but variations of density with distance is now taken into account. The method of analysis is based on an extension of the kinematic wave method of Lighthill and Whitham (1955). This type of analysis has been applied to Freeway on-ramp studies by Munjal and Pipes (1969a, b) and to freeway off-ramp studies by Munjal (1971). t Prepared for the U.S. Department of Transportation, Federal Highway Administration. Office of Research, under Contract No. FH-1 l-7628, A Study of Analytical Models of Multi-Lane Traffic FlowStudy II. The opinions, findings and conclusions expressed in this publication are those of the authors, and not necessarily those of the Federal Highway Administration. 257

258

P. K. MUNJAL, YUAN-SHIH Hsu and R. L. LAWRENCE

In order to study the case of a uniform n-lane freeway, we assume that the speeddensity relationship for equilibrium conditions is known for every lane. A freeway is termed uniform if the speed-density relationship for equilibrium conditions is the same for each lane and the equilibrium density is the same across lanes. The present analysis is also applicable to nonuniform freeways. The equations of continuity used to describe the n-lane freeway are (D,+cD,)K+aAK=

0

(1)

where KT = [K,, K,, . . . . K,] and T denotes

the transpose,

and A is the n x n symmetric

matrix

1

-1

0

0

...

0

-1

2

-1

0

...

0

0

-1

2

-1

...

0

. ..

.. .

...

-1

2

-1

0

...

. ..

. ..

-1

1

A=

where 4, D, is the partial derivative operators with respect to t and x respectively; Ki, the density perturbation in lane i; c the wave speed; and a, a positive constant with dimension time-l. The wave speed c in equation (1) may take either positive or negative values. If the wave speed is positive, then the propagation due to the lane-drop perturbations would take place on the downstream side. However, if the wave speed is negative, the propagation due to the lane-drop perturbations would take place on the upstream side. The relationship between flow-concentration and wave speed c has been represented by Drake et al. (1967). This is shown in Fig. 1.

FIG. 1. Flow

concentration diagram.

The approximate flow-concentration diagram depicted in Fig. 1 consists lines. It can be shown readily that for densities greater than k,, the following

‘=

where q is the flow; q,,,, the maximum density; and U, the velocity.

I-zm,kj)

[ ‘-

01 k

i$

flow; km, the density at maximum

of two straight relations hold:

(2)

flow; kj, the jammed

Analysis and validation

of lane-drop

effects on multi-lane freeways

259

For densities lower than k, the relations are: 4= Fk

(4)

m

U= p

?n

= constant

(5)

The solution to equation (1) is given by K(x, t) = MB(x)M-‘4

(

r-f

(6)

1

where M is an orthogonal matrix such that MAM-’

= S

is a diagonal matrix with diagonal elements the eigenvalues of A. If we let the eigenvalues be A,, AZ,. . . . A,, then B%)

= Id,(x) &(x),

*a.,4d-41

where d*(x) = exp (- &y) and

&(r-;) =,(,, r-g The evaluation of A+ M and M-l can be found in many applied mathematics books (see, for example, Friedman, 1954). In the case n = 4, 1 M=

1 &2)-l

- 1

1

-1

1+J2

1

-1

l-42

1

M-l=&

1

-1

-(J(2)+

2

-2

-2

2-42

-J2

+J2

1) (7) I

2 -(2-J2)

I

and h,=O,

h,=2-r/2,

x,=2,

A4=2+J2

(9)

P. K. MUNJAL,YUAN-SHIHHsu and R. L. LAWRENCE

260

If we define

(10)

then

1d,u+d,p+d,

y+d4 S

(11)

where ur = J(2) - 1 and u, = J(2) + 1. Equation (11) is then the complete solution of wave propagation of the density perturbation on a four-lane freeway. Different types of perturbation are given by different boundary conditions. We will study each case in the following sections. DATA

COLLECTION

AND

REDUCTION

An aerial and ground-based system capable of recording the microscopic behavior of vehicles on a one-mile section of freeway was used as the data collection technique most suitable for the field study of freeway lane drops. A sequence of 70-mm color photographs were taken at known time intervals (one or two seconds) from a helicopter hovering about 4000 ft above the lane drop site. The camera selected for data collection was the Maurer 220 Pulsed Sequence camera, a lightweight camera designed for aerial reconnaissance applications. Magazines of 100 ft were used to produce about 7 min of film at 1-see or 14 min at 2-set frame intervals. Bell U7G-3B-1 and Bell Jet Ranger helicopters were used to collect data under minimum hovering conditions. The photographs obtained in this manner were then projected on a film reader, which permitted an accurate measurement of the coordinates of vehicles in the photographic image. The film reader selected for the data reduction system was the Benson-Lehner Model 29E Telereadex. The associated electronic accumulating readout and control unit was the Benson-Lehner Model 282E Telecordex. The output device is an IBM Model 523 Summary Card Punch. Details of the film reading techniques and the associated computer software to develop trajectories of vehicles relative to an actual ground-based coordinate system are in Goodwin and Barman (1970). Some of the important functions of these programs include: Correction of all photographic image points of interest for the optical distortion produced by the combined effect of the aerial camera lens and film magazine. Determination of the position and orientation of the aerial ordinates from camera and ground reference points. Transformation

of reference

points from film coordinates

Transformation

of auto coordinates

from film coordinates

camera

to ground to ground

in ground

coordinates. coordinates.

co-

261

Analysis and validation of lane-drop effects on multi-lane freeways Translation prespecified

of auto coordinate ground points.

points

into

distance

and lane position

Prediction of the position and speed of car in its previous speed of car in given frame. Matching computed

of auto position during processing

Smoothing

in previous frame of a given frame.

of the car trajectories

for improved

frame from the position

to the trajectory, estimate

relative

of position

Generation of data tape containing a car number, associated and corresponding times, speed and lane numbers of the car.

predicted

to a and

position

and speeds.

distances

along the road

The flow q, density (concentration) k and speed u are obtained from the aerial photographic data taken at an interval of 1 sec. These parameters are determined on a set of points along the freeway, say R,, R,, . . . . RK. For each point Rj that a car passes during the filmed period we will have a situation as follows : We observe the car at time t,, at a distance x0 and at time t,, + At at a distance x,. and R, is between them. At is the time interval of the is shown in the following figure. photograph taken, say 1 sec. The reiation&ip

Then the time that the vehicle passes 4 is tj = Rj-xoAt+to x1-x0

set

(12)

The velocity at Rj is (13) where c1 and o. are respectively the velocity at x1 and x,,. If the number of cars passing Rj in the photographed time period T is N,, then the flow quantity is &=$x3600

(14)

and the average velocity is

(1% and the density k~ = Qj/ ~j

(16)

These parameters were calculated for each lane on a multi-lane freeway and plotted lanewise at each of the Rj points. The plots of Figs. 3 and 6 are obtained from this technique.

P. K. MUNJAL, YUAN-SHIH Hsu and R. L. LAWRENCE

262

BOUNDARY

CONDITIONS

AND

VALIDATION

Different types of lane-drops are represented by different boundary conditions. Since aerial photographic data are available for the combination lane-drop off-ramp and the onramp followed by lane-drop, we shall concentrate on the validation of these two cases. 1. Combination lane-drop of-ramp. The geometry of the lane-drop site is shown in Fig. 2 with the concentration as a function of distance given in Fig. 3. c--TO

LOS ANGELES

TEsTr132

END TEST SITE

LANE

END LANE

BEGINNING

diagram

FT..Zr-91,5

FT.-T

_---t---------____ 6 FT. MEDIAN 1

__--_----_------__ 3

I

I

10 FT. SHOULDER TO SANTA ANA PLAN VIEW

FIG. 2. Geometric

design of lane-drop

site at Santa Ana freeway and Triggs Street.

40

lYz--I+

1

G_____9.

t -1100

I

I

-900

I

I

-700

I

,

-500

,

I

-300

-100

I

100

I

4

I

300

500 -

FIG. 3. Aerial photographic

,

,

700 DISTANCE feet

data for Santa Ana freeway and Triggs Street.

In general, the wave perturbation propagation is an exponentially decayed function of of the boundary conditions distance as we can see from equation (11). But the determination For experimental purposes, we use the boundary at the point of lane-drop is very involved. conditions from the photographic data and use the mathematical model to obtain the wave propagation. The result is shown in Fig. 4 for the case of lane-drop with off-ramp.

Analysis and validation

01 -1loD

I

1

I

-9w

of lane-drop

-700

-500

I

I

1

I

1

I

-300

-100

0

loo

300

500

DISTANCE feet

FIG.

2. On-ramp followed

263

effects on multi-lane freeways

1

700

-

4. Analytical result for Santa Ana freeway and Triggs Street.

by lane-drop

Another type of lane-drop is studied here. We have a four-lane freeway with an onramp merging to lane 1. After a distance of about 600 ft, lane 1 begins to merge with lane 2 with a merging distance of about 1000 ft. The detailed geometry of this lane-drop site is shown in Fig. 5 together with a concentration diagram from aerial photographic data shown in Fig. 6. -_--TO ON-RAMP

VENTURA

LANE DROP

DOWNSTREAM

LI”” s ,,‘,9SD

FT.+&3

FT.q-99,

END

,.._

u”‘Rop

I

LEDIAN (I1 FT.) LANES

FT.-

d

I

3 -i -

--

--_

-

---

--

--_

r

FIG. 5. Geometric

design of lane-drop

site at Ventura Freeway and Calabasas.

The present case study is more complicated than the previous one. We divided Fig. 5 into three sections. That is, up-stream on-ramp, down-stream on-ramp to end of lane-drop, down-stream end of lane-drop. Superposition technique is used here for which we apply the following procedure : (a) Assume flow is uniform for each lane at a far enough distance up-stream from onramp and in each section total flow does not vary with distance. This is because the concentration is low for each lane. The perturbation is then obtained.

264

P. K. MUNJAL, YUAN-SHIH

Hsu

and R. L. LAWRENCE

(b) At the end of the lane-drop, neglect the on-ramp effect and consider the flow as uniform up-stream with same value as in a. The boundary condition of concentration in lane 1 is taken to be zero. 40-

k

I

.3777

4177

-DISTANCE feet

FIG. 6.

Aerial

photographic

data for Ventura

Freeway

and Calabasas.

(c) Use the same boundary condition in b and assume uniform flow far down-stream to obtain propagation down-stream of end of lane-drop point. The result is shown in Fig. 7. In the above two cases, we can see that the mathematical model smoothes the perturbation curve. The fluctuations of the photographic data at some points result from the random driving habits of individual drivers or their stochastic attributes of traffic flow. Despite this, the analysis matches very well to photographic data. 40

0 500

1000

1500 2cm 2500 3000 LANE-DROP ON-RAMP ON-RAMP BEGIN BEGIN END

3500 4000 LANE-DROP END

4

DISTANCEfeet

FIG. 7.

Analytical

result for Ventura

Freeway

and Calabasas.

Analysis and validation

of lane-drop

effects on multi-lane freeways

26.5

3. Lane blocking Frequently a lane is blocked on a freeway due to an accident or other disturbance. This perturbation is more severe than previous cases. Since no field data are available at present, we can only present the analytical part with reasonable assumptions on the boundary conditions. A four-lane freeway is considered here with the same wave propagation equation given by equation (11). For light traffic flow, no flow reduction will be made if a lane is blocked. This can be justified by flow-concentration rule. Now we use this condition and assuming the perturbation densities have the following boundary conditions: lim&(x,r) z+Xz

= 0,

i = 1,2,3,4

(17)

Kl(O, t) = (kj - k,) H(t) I&(0, t) = (k’ -k,)

(18)

H(t),

i = 2,3,4

(19)

where ki is the jammed concentration, k, is the initial or equilibrium is the drscharge concentration. For no flow reduction

concentration

3k’ = 4k, must hold. From the above conditions, we get the solution in Fig. 8. If field data are available, some other boundary

and k’ (20)

from equation conditions

(11). The result is given

can be selected.

I

O.OC 0

I I

I 3

I 2 9

FIG.

1 4

I 5

I 6

UPSTREAM

8. Analytical results for lane blockage. CONCLUSION

The density perturbations in a multi-lane freeway have been studied on the basis of a hydrodynamic model. The model assumes that the cross-flow in the freeway is proportional to the difference in density of the adjacent lanes and that the equation of continuity applies. It is also assumed that the shock-wave speed of the highway is a constant. The results of the analysis indicate that regardless of the behaviour of the density in the lanes at x = 0, the effect of dropping the first lane, or for that matter any lane, produces

P. K. MUNJAL, YUAN-SHIH Hsu and R. L. LAWRENCE

266 waves

of perturbation

speed

c.

The

on the initial

density

perturbation or equilibrium

in all four

lanes

is an exponentially flow

of the highway decayed

concentration

and

function. boundary

that travel The

exact

condition

upstream

with

decay

depends

a

at the lane-drop

point. The

analysis

has been

validated

by the aerial

photographic

data

in two

different

cases.

would like to thank Dr. A. V. Gafarian, Dr. Louis Pipes and Mr. Browne for their help in the preparation of this paper.

Acknowledgement-We

Goodwin

REFERENCES DRAKE J. S., SCHOFERJ. L. and MAY A. D. (1967). A statistical analysis of speed density hypotheses. Highw. Res. Record No. 154, 53-87. FRIEDMANB. (1954). Principles and Techniques of Applied Mathematics. Wiley, New York. GAZIS D. C., HERMAN R. and WEISS G. H. (1962). Density oscillations between lanes of a multilane highway. Ops. Res. 10, 658-667. GOODWIN D. and BARMAN M. (1970). Vehicular trajectories semiautomatic aerial photographic data reduction system user’s manual. System Development Corp. Document TM(L)-4447/000/01, Santa

Monica, California,

1 March 1970.

LIGHTHILL M. J. and WHITHAM G. B. (1955). On kinematic waves II, a theory of traffic flow on long crowded roads. Proc. Roy. Sot., London, Series A, 229, No. 1178, 317-345.. MUNIAL P. K. (1970). A sirnole off-ram0 traffic model. Hiahw. Res. Record No. 334. 48-61. MUNJAL P. K. and PIPESL. A: (1969a). Propagation of on-ramp density perturbations on unidirectional two- and three-lane freeways. System Development Corp. Document TM-3858/007/00, Santa Monica,

California, MUNJAL P. K. multilane California,

March 1969. and PIPES L. A. (1969b). Propagation of on-ramp density waves on uniform unidirectional System Development Corp. Document TM-3838/016/00, Santa Monica, freeways. March 1970.

OLIVER R. M. and LAM T. (1967). Statistical experiments with a two-lane flow model. Science, pp. 17&180. American Elsevier, New York.

Vehicular

Trafic