Topics in the hydrodynamic theory of traffic flow

Transpn Res. Vol. 2, pp. 143-149. Pergamon Press t968. Printed in Great Britain

T O P I C S IN T H E H Y D R O D Y N A M I C

THEORY

OF TRAFFIC

FLOW

Louis A. PIPES University of California, Los Angeles, California, U.S.A.

(Received 11 August 1967)

1. I N T R O D U C T I O N The wave equation

THE HYDRODYNAMIC theory of traffic flow along long crowded highways replaces the individual vehicles involved in the actual flow with a continuous "traffic fluid" that has a certain concentration or density k ( x , t) (veh/mile) and a flow rate function q(x, t) (veh/hr), where x is a linear coordinate measured in the direction of the flow in miles and t is the time measured in hours. [See LighthiU and Whitham (1955), Richards (1956) and Pipes (1965).] If we make measurements of the flow rate q at two points a short distance Ax apart, the excess of q on the downstream side must equal the diminution of vehicles in the stretch of road between the two points. Therefore the law of conservation of vehicles gives the differential equation O x qAx = - Dt(kAx )

(1.1)

Dtk+ Dxq = 0

(1.2)

or

where Dl and D x are the partial derivatives with respect to t and x respectively. Equation (1.2) is recognized as the usual equation of continuity o f a compressible fluid. In order to determine the propagation of disturbances of flow density along a long crowded highway, it is necessary to assume that the flow rate q at any point depends explicitly only on the density of traffic k and on the position x along the highway, so that q = q(k,x)

(I.3)

If the road under consideration is homogeneous, the flow rate will not depend upon the position x along the highway, and (1.3) reduces to q = q(k) = G(k)

(1.4)

equation (1.4) may be regarded as the "equation of state" of the traffic fluid. If (1.4) is differentiated partially with respect to x, we obtain Dxq = DkqDxk

(1.5)

If we consider small variations of k about an average value, we can take Dkq = c

(1.6)

where c is a constant. In this case, we have (1.5) in the form Dxq = cD~k

143

(1.7)

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Louis A. PIPES

If (1.7) is now substituted into the equation of continuity (1.2), the result is = 0

Dtk+cDzk

(1.8)

Equation (1.8) is a form of the wave equation; its solution has the form k = ~(x-

(I.9)

ct)

where (I) is an a r b i t r a r y function. The solution (1.9) represents a wave of density k that travels in the direction of increasing x with a velocity c. A small perturbation in the concentration k is therefore propagated in the direction of traffic flow with a velocity c. 2. THE E Q U A T I O N OF STATE OF THE T R A F F I C F L U I D As a consequence of the definitions of the concentration k and the flow rate q, we have the important relation q = uk (2.1) where u is the speed of the traffic stream. As was shown in Section I, the velocity of perturbations of concentration along a homogeneous road is given by (2.2)

e = Dkq

I f we substitute (2.1) into (2.2), the result is c = Dk(uk ) = u+kD

ku

(2.3)

Experimentally [cf. Herman and Rothery (1967)], the quantity D k u is always found to be negative, therefore the term k D k u in (2.3) is negative and c is always less than the speed of the traffic stream u. It is convenient to define a parameter V by the equation V = - kD k u

(2.4)

In terms of this parameter (2.3) takes the form c = u- V

(2.5)

The quantity V is seen to be the speed of propagation of a disturbance back through a chain of vehicles measured from the frame of reference of the traffic stream u. By choosing special forms for the speed V in (2.4), we obtain various functional relations of the form q = G(k)

(2.6)

or different "equations of state" for the traffic fluid. Several special cases will now be considered. Special case 1 : V = Ak

As a first example of the use of (2.4) to obtain an equation of state, assume that the velocity V is proportional to the concentration k so that V = Ak

(2.7)

where A is a constant. I f (2.7) is substituted into (2.4), the result is V = Ak = - kDku

(2.8)

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145

If the differential equation (2.8) is integrated subject to the boundary conditions, u=0,

when

k=k s /

u=u,,,

when

k=0

u = u,,X 1 -

k/k s)

the result is

(2.9)

J (2.1 o )

This is the relation between the speed of the traffic stream u and the concentration k proposed by Greenshields (1934). u,,, is the maximum speed of the traffic stream on a free highway when the traffic density k is zero. k s is the so-called "jam concentration", this is the concentration at which traffic flow stops. Since q = k u , we obtain from (2.10) the relation q = ku,,,(1 - k / k s ) = G ( k ) (2.11) therefore the assumption (2.7) leads to the "equation of state" (2.11). In this case the waves of perturbation of density k travel with the speed e, where c = D k q = urn(1-2kikj)

(2.12)

This was the case discussed in detail by Riehards (1956). S p e c i a l c a s e 2: V = A k n+]

A generalization of case 1 will now be considered. Let it be assumed that the speed V is proportional to the (n + 1)st power of the concentration k. Equation (2.4) then takes the form V = A k n+l = - k D k u (2.13) where A is a constant. If we integrate the differential equation (2.13) and use the boundary conditions (2.9) the result is u = u,n [1 - ( k / k i ) '~+11 (2.14) The equation of state for this case is q = hum[1 - ( k / k ~ ) "+11 = G ( k )

(2.15)

the implications of the equation of state (2.15) with regard to a follow-the-leader model has been discussed by Pipes (1967). S p e c i a l c a s e 3 : V = Vo

It is interesting to study the case in which it is assumed that the velocity V has the c o n s t a n t value V0 [cf. Herman and Rothery (1967)]. For this case, equation (2.4) now takes

the form V=

Vo = - k D k u

(2.16)

If the differential equation (2.16) is integrated subject to the boundary condition, u=O

at

k=k i

(2.17)

the following expression for the speed of the traffic stream is obtained: u = V0 In ( k s / k )

(2.18)

q = k V o In ( k s / k ) = G ( k )

(2.19)

The equation of state is now given by

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Louis A. PIPEs

The m a x i m u m rate of flow given by (2.19) is qm, where (2.20)

qm = Vo k j / e

It can be shown that the constant velocity V~ is the speed of the traffic stream when the rate of flow q is a m a x i m u m . The velocity c of waves o f perturbation o f density is givcn by c = D k q = V o [ l n ( k j / k ) - 1]

(2.21)

Since m a x i m u m flow occurs when In ( k j / k ) = 1, we see that c = 0 when the rate of flow q is a m a x i m u m . The relation (2.19) was first obtained by Greenberg (1959) in an entirely different manner.

3. D I F F U S I O N P H E N O M E N A A n extension o f the simple wave theory of the hypothetical "traffic fluid" which is based on equations (1.2) and (1.4) takes into account a "diffusion effect" caused by the fact that each driver adjusts his speed to the concentration slightly ahead o f him. This behavior gives a dependence o f the flow on the concentration gradient k x = D x k as well as on the concentration. In order to take this effect into account we must replace the simple relation (1.4) by the equation q = q(k, kx) (3. l) where k x = D x k .

If (3.1) is differentiated partially with respect to x, the result is D z q = D k q D z k + Dk~ q D x k x

(3.2)

If we now let n = De~ q

(3.3)

D x q = c D~ k + n D ~ 2 k

(3.4)

c= Dxq,

then (3.2) m a y be written in the form

where Dx 2 is the second partial derivative with respect to x. We now substitute (3.4) into the continuity equation (1.2) to obtain Dtk+cDxk+nDx~k

= 0

(3.5)

Equation (3.5) is a partial differential equation o f the diffusion type. In order to determine the nature o f the waves propagated in accordance with equation (3.5) we assume that the traffic concentration k varies harmonically with the distance x along the highway with a wavelength ,~. In order to study how this type of concentration is propagated we assume a solution of the form k = k 0 exp (at) exp (ibx)

(3.6)

where b = 2rr/,~ is the wave number. The wave n u m b e r b is a real n u m b e r ; it mcasures the number o f waves in a distance o f 27r along x. I f (3.6) is substituted into (3.5) the result is a = nb ~ - icb

(3.7)

If this value o f a is now substituted into (3.6), the result is k = k 0 exp (nb 2 t) exp [ i b ( x - ct)]

(3.8)

The hydrodynamic theory of traffic flow

147

The solution (3.8) indicates that if we take into account the possible variation of the rate of flow with the concentration gradient k z, we obtain a wave of perturbation that travels along the x axis with a velocity c. I f the quantity n is positive, the disturbance grows exponentially with the time t. I f n is negative, the wave of perturbation is rapidly attenuated with the time. Disturbances having a short wavelength A are attenuated more rapidly than those of long wavelength. Since the rate of flow is greater when the concentration gradient k x is negative it appears that n = Dk~q is negative and the disturbance dies out exponentially with the time. This is a property of diffusion phenomena. The quantity n may be regarded as a sort of coefficient of diffusion for the hypothetical traffic fluid. 4. THE E F F E C T OF I N E R T I A Observations of actual traffic flows have indicated that the rate of flow increases when D t k is negative. This phenomenon leads to a sort of inertia effect. In order to study this effect, it is convenient to study the implications of assuming that the rate of flow q satisfies a relation of the form q = q(k, kt) (4.1) where k t = D t k . Differentiation of (4.1) leads to the relation, Dxq = cD x k +pDxt z k

(4.2)

where c = Dkq,

p = Dk, q

(4.3)

and Dxt ~ is the second-order mixed partial derivative with respect to x and t. I f (4.2) is substituted into the equation of continuity (1.2), we obtain Dtk+ cDxk+pDxtZk = 0

(4.4)

We again study the propagation of waves of concentration of wavelength ,~ by assuming a solution of 4.4) of the form k = k0ex p (at) exp (ibx),

(4.5)

where b = 2~-/~ is the wave number. If (4.5) is substituted into (4.4), the result is a + icb +paib = 0

(4.6)

a = - (pcb z + icb)/0

(4.7)

If we solve (4.6) for a, the result is where 0 = (1 +p2b2). I f the result (4.7) is substituted into (4.5), we obtain the following relation: k = k 0 exp ( - p c b 2 t/O) exp [ i b ( x - ct/O)]

(4.8)

I f p c is positive, we see from (4.8) that in this case the waves of concentration are attenuated exponentially with the time and travel in the x direction with a velocity v, where v = c/O,

0 = (1 +p2b2)

(4.9)

Since the waves have a wavelength ,~, we have b = 2rr/,L We note from this that the waves of shorter wavelength are attenuated more rapidly (provided p c is positive). The waves of shorter wavelength travel more slowly than the longer waves as may be seen from (4.9).

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Louis A. PIPES

5. T H E G E N E R A L CASE In the general case, it m a y be assumed that the rate o f flow q is a function o f the density k, the density gradient k x = D x k and the time rate o f concentration k t = D t k. We then have

q = q ( k , kx, k,)

(5.1)

I f we now c o m p u t e D x q f r o m (5.1) and substitute the result into the equation o f continuity (1.2), we obtain the equation

Dtk+cDzk+nDxZk+pDxt~k

= 0

(5.2)

where

c = Dkq,

n = Dk~ q,

p = Dk, q

(5.3)

If we desire to study the behavior o f waves o f concentration o f wavelength ,~, we try a solution o f (5.2) o f the form k = k 0 exp (at) exp (ibx), (5.4) where b = 2rr/,~. I f we substitute (5.4) into (5.2), we obtain

a + i c b - nb2 + ipab = 0

(5.5)

a = b ( n b - ie)/(l + ipb)

(5.6)

I f we solve (5.5) for a, we obtain I f we separate the real and imaginary parts o f a in (5.6), we find

a = b[b(n - p c ) - i(e + npb2)]/O

(5.7)

0 = (1 +p2b~)

(5.8)

where

Stability o f the flow In order for the flow governed by equation (5.2) to be stable, it is necessary that the amplitude o f the concentration k does not grow indefinitely large with the time t. I f we examine the solution (5.4), we see that stability is ensured provided that Re a ~<0

(5.9)

T h a t is, the real part o f a must be negative. We see by examining (5.7) that in order for the real part o f a to be negative we must have

pe>n

(5.10)

Therefore (5.10) is the condition for stability o f flow. If (5.10) is not satisfied, the concentration k will grow until the flow is stopped.

Behavior at m a x i m u m f l o w It is interesting to determine the nature o f the wave p h e n o m e n a when the flow is at its m a x i m u m value so that e = 0. I f we place e = 0 in equation (5.7), we obtain a = b 2 n / O - ib3pn/O

(5.11)

If this value o f a is substituted into the solution (5.4), the result is k = k o exp (b ~ nt/O) exp [ i b ( x - b2pnt)/O]

(5.12)

We see f r o m (5.13) that in order to ensure stability, the quantity n must be negative. If n is negative, the amplitude o f the wave o f concentration diminishes exponentially with the

The hydrodynamic theory of traffic flow

149

time. Waves of different wavelengths are p r o p a g a t e d with different velocities. The wave of c o n c e n t r a t i o n oscillates with the time with the frequency f given by f = b3pn/27rO

(5.13)

Since b = 2rr/A, the waves of shorter wavelength A oscillate more rapidly t h a n those o f long wavelength. T h e special case n = p c

It was shown above that the stability o f the flow is ensured provided that p c > n. It is interesting to study the n a t u r e of the wave p r o p a g a t i o n in the critical case for which p c = n. I f we place n = p c in e q u a t i o n (5.7), we o b t a i n a = - i(c + npb~)/O = - ibc

(5.14)

If we substitute this value of a into the solution (5.4), we obtain k = k 0 exp [ i b ( x - ct)]

(5.15)

The s o l u t i o n (5.15) shows that i n this case waves of c o n c e n t r a t i o n k travel i n the x direction with the velocity c regardless o f their wavelength. This is the same solution that we o b t a i n e d in Section 1 where we considered n = 0 a n d p = 0. REFERENCES GREENBERGH. (1959). An analysis of traffic flow. Ops Res. 7, 79-85. GREENSmELDSB. D. (1934). A study of traffic capacity. Proc. Highw. Res. Bd 14, 448-477. HERMANR. and ROTHERYR. (1967). Propagation of disturbances in vehicular platoons. Proceedings o f the Third lnternational Symposium on the Theory o f Traffic Flow, New York, 1965, pp. 14-25. Elsevier, New York. LIGHTmLL M. J. and WmTHAMG. B. (1955). On kinematic waves--II. A theory of traffic flow on long crowded roads. Proc. R. Soc. A 229, 317-345. PIPES L. A. (1965). Wave theories of traffic flow. J. Franklin Inst. 280, 23-40. PIPES L. A. (1967). Car following models and the fundamental diagram of road traffic. Transpn Res. 1, 21-29. RICHARDS P. I. (1956). Shock waves on the highway. Ops Res. 4, 42-51.