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High-fidelity macroscopic traffic equations
PHYSICA ELSEVIER
Physica A 219 (1995) 391-407
High-fidelity macroscopic traffic equations Dirk Helbing 11. Institute of Theoretical Physics, University of Stuttgart, 70550 Stuttgart, Germany Received 11 May 1995
Abstract The Euler-like traffic equations that can be derived for quasi-homogeneous traffic situations are corrected for non-equilibrium effects. This yields additional transport terms describing a flux of velocity variance. Other modifications resulting in a bulk viscosity term arise from the adaptive behavior of drivers with respect to changes of the traffic situation. Finally, corrections due to the space requirements of vehicles as well as due to finite reaction- and braking-times are introduced. These are responsible for a van der Waals-like relation for 'traffic pressure' and modified relations for the kinetic coefficients.
1. Introduction In a previous paper [ 1 ] an Euler-like macroscopic traffic model was derived from a gas-kinetic traffic equation which was constructed from elementary laws of individual driver behavior. This model already overcomes some shortcomings of former macroscopic traffic models which are mostly of a more or less phenomenological nature or based on lax analogies with the equations of ordinary fluids [2,3]. Nevertheless, due to the presupposed approximation of local equilibrium, the Euler-like traffic equations are still not perfect. They only apply to traffic situations with moderate deviations from equilibrium which implies small gradients and slow temporal changes of vehicle density, average velocity, and velocity variance. Therefore a higher-order approximation of the solution of the gas-kinetic traffic equation will be derived in Section 2. The applied method is analogous to the derivation of Navier-Stokes equations for ordinary fluids from the Boltzmann equation [ 4 - 7 ] . It bases on a Chapman-Enskog expansion [8,9] which is known from kinetic gas theory and yields Euler(-like) equations in the zeroth-order approximation of local equilibrium, whereas Navier-Stokes(-like) equations are found in first-order approximation. However, due to additional terms in the gas-kinetic traffic equation compared with the 0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD10378-437 1 (95)00175-1
D. Helbing/Physica A 219 (1995) 391-407
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Boltzmann equation [ 10] the corresponding mathematical procedure is more complicated than for ordinary gases. Nevertheless it is still possible to derive correction terms of the Euler-like traffic equations. These describe a flux of velocity variance which is related with a finite skewness of the velocity distribution in non-equilibrium situations. In comparison with the conventional Navier-Stokes equations the resulting Navier-Stokes-like traffic equations include an additional covariance equation due to the fact that drivers try to get ahead with a certain individual desired velocity. Moverover, they contain additional terms arising from the acceleration and interaction of vehicles. However, because of the one-dimensionality of the investigated traffic equations no shear viscosity term occurs. In the remaining sections we will introduce some corrections and modifications of the Navier-Stokes-like traffic equations. These will take into account traffic-specific characteristics arising from the circumstance that driver-vehicle units are active systems. Up to now they have been neglected by most macroscopic traffic models: (i) Drivers show an adaptive behavior with respect to changes of the traffic conditions. According to Section 3 this leads to an additional bulk viscosity term. (ii) The distance between succeeding vehicles is often of the order of magnitude of the space required by themselves so that vehicles cannot be idealized as point-like objects. The space needed by each vehicle (= vehicle length plus safe distance) can be described by a van der Waals-like relation for 'traffic pressure' (cf. Section 4). (iii) Drivers of vehicles have a finite reaction time to changes of traffic flow conditions. This causes modifications of the transport terms (cf. Eqs. (44) and (77)). The resulting macroscopic traffic model includes the most important effects of drivervehicle behavior and can therefore be denoted as a 'high-fidelity' traffic model.
2. Derivation of the Navier-Stokes-like traffic equations In Ref. [ 1] a gas-kinetic traffic equation has been developed for the temporal evolution of the phase-space density ,~(r, v, vo, t) of vehicles with desired velocity vo and actual velocity v at place r. For an uni-directional multi-lane freeway without entrances and exits it reads
O~
o-7+
O(~v)
+
O(~_2_)
P(r'v't)[p~(vo;r,t)_Po(vo;r,t) ]
T
(la)
O0
= ¢ l - p ) f aw f dwolv-wl
(r,v, wo,
vo, t)
(lb)
U V
- ( l - p ) f dw f dwolw-vlb(r,w, wo,t)~(r,v, vo, t), 0
where
(lc)
D. Helbing/PhysicaA 219 (1995) 391-407 ~(r,v,t)
:=f dvop(r,v, vo, t)
393 (2)
is a reduced phase-space density. The second term of (la) describes a temporal change of/3(r, v, v0, t) due to vehicle motion with velocity v and is called a convection term. The third term delineates an adjustment of the actual velocity v to the desired velocity v0 within the relaxation time ~', i.e. it reflects the drivers' acceleration behavior. The fourth term arises from an adaptation of the actual distribution of desired velocities Po(vo; r, t) to the reasonable distribution of desired velocities P~ (v0; r, t) which is determined by speed limits and traffic conditions (gradient of the road; tog, rain, snow, or ice on the road). T ~ l s is about the reaction time. The terms ( l b ) and ( l c ) describe vehicular pair interactions due to which one vehicle must decelerate to the velocity of a slower one. p denotes the probability that a slower car can be immediately overtaken. (For a detailed discussion of Eq. ( l ) cf. Refs. [ 11,1] .) Integrating Eq. ( l ) with respect to vo yields the following reduced equation:
O~ O(vD) 0 ( ~'o(v;r,t)-v) Ot + ---~r + -~v ~ ( r, v, t ) r a
O0
= (1 - p ) ~ ( r , v , t ) / d w ( w -
v)~(r,w, t),
(3)
, /
o where we have introduced
f/o( v; r, t ) := [ dvo voP(-:~-' v--~°-'t), . j ~tr, v,t)
(4)
Eq. (3) allows the derivation of dynamic equations for macroscopic quantities like the spatial density
p(r.t)::/,.l,.o,,r,...o,,,:l,.,,....,,
(5)
of vehicles per lane, the average velocity
, ~(r,v,t) V(r,t) =-- (v):= f avv --~,}~
(6)
of vehicles, and their velocity variance
O(r,t) =- ( [ v - V(r,t) ] 2) : = . / dv [v - V(r,t) ] 2~(r'v't) p(r,t) = @2)_ [V(r,t)]2.
(7)
These equations read
Op O(pV) Ot Or _ SV + vSV _ Ot Or
(8)
--+--=0,
1 l a'P + _[Ve(p, V,O) _ V] ' p Or T
(9)
D. Helbing/Physica A 219 (1995)391-407
394
O0 O0 --+V-at ar
279 OV p ar
1 Off 2 -- +-[Oe(p,V,O,J,C) p Or r
--0].
(10)
Here we have used the abbreviations
Ve(p, V,O) := V0 - r ( p , V,O) [ 1 - p ( p , V,O) ] P ( r , t),
(11)
where
Vo(r,t) : = / d v f d v o v o
#(r'v'v°'t) p(r,t)
(12)
is the average desired velocity, and
Oe(p,V,O,J,C) :=C(r,t) - ½r(P,V,O)[l - p ( p , V , O ) ] f f ( r , t ) .
(13)
Moreover, we have introduced the so-called traffic pressure [ 12-14]
l / /
P ( r , t ) .- - p(r, t)
dv(v-V)~(r,v,t)
dw(v-w)p(r,w,t)
#
= ]dv(v
- V)2~(r,v,t) = p ( r , t ) O ( r , t )
(14)
,
J
the flux of velocity variance
1/
f l ( r , t ) .- p(r,t~
dr(v-
V)2~(r,v,t)
/
dw(v-
w)~(r,w,t)
= f dv (v - V)3/5(r, v, t) = p(r, t)F(r, t)
(15)
(which corresponds to the 'heat flow' in conventional fluid-dynamics), and the covari-
ance C(r,t)
:=Jdvofdv(v-v)(oo-v°)P(r'v'v°'t)p(r,t) = f d v ( v - V ) [ V o ( v ) - v o j , ~ pp( r( r, v, t, )t ) "
(16)
In order to close the system of F.qs. (8), (9), and (10), the functions C(r,t) and i f ( r , t) must be appropriately determined in dependence of p(r, t), V(r, t), and O(r, t). In Ref. [ 1] this has been done via the zeroth-order approximation of local equilibrium
~ ( r , v , t ) ~ ~(o)(r,v,t) .-
p(r,t)
x/2crO(r, t)
exp{-[v - V(r,t)]2/20(r,t)}.
(17)
This implies
f f ( r , t ) ~ fl¢o)(r,t) =0, and leads to the Euler-like traffic equations.
(18)
D. Helbing/Physica A 219 (1995) 391-407
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However, approximation (17) is only valid if the velocity distribution
/5(r,v,t) P(v;r,t) .- - p(r,t)
(19)
relaxes very rapidly towards the local equilibrium distribution
P(o)(v;r,t):= /5(o)(r,v,t) _ 1 exp{-[v-V(r,t)]2/20(r,t)}. p(r,t) x/2qrO(r,t)
(20)
This is only the case for small deviations from stationary and spatially homogeneous traffic. Otherwise the finite adaptation time 7, the relaxation process lasts will lead to corrections. Therefore we will now derive an approximate dynamic equation for the deviation
3/5(r,v,t) := ~(r,v,t) -/5(o)(r,v,t)
(21)
from the local equilibrium distribution /5(o)(r,v, t). Utilizing that the correction term 8/5(r, v, t) will normally be small compared to p(o)[p(r, t), V(r, t), O(r, t)] we have
8/5(r,v,t) < < / 5 ( o ) [ p ( r , t ) , V ( r , t ) , 8 ( r , t ) ] ,
(22)
and get
a/5
-
a {_f/o(v)-v)
kp
;
/
0/5(0) + vC~p(o) + 0 ( /5(0) Vo(v) - v ) • -
at
(23)
-W-r
(For a detailled discussion of this approximation cf. [4,5,7].) Now, introducing the abbreviation d
0
dt
at + v"~r'
0 (24)
we can write
0/5(o_____Z ) + vO/5(o) = d/5(o____ 2 _ 0/5(0) dp + 0/5(0______2) dV + 0/5(o______2) d0 Ot Or dt Op dt OV dt O0 dt -
/5(o___2)dp /5(o)~ dV /5(o) ( ( S v ) 2 p dt + - - o - o v - ~ + - ~ ---~-
1) dO d---t'
(25)
with
8v := v - V.
(26)
Relations for dp/dt, dV/dt, and dO/dt can be obtained from the macroscopic traffic equations (8), (9), and (10) via d
0
d t - Ot +
V0_
O
Or +6v~r"
With (14), (18), and (13) we find
(27)
D. Helbing/Physica A 219 (1995) 391-407
396
dp = Svap _ p a V , --~ Or Or
=Bray
dt
Or
(28a)
1 a(pO___2
p
Or
+
l[Ve(p,V,O) _ V], 7"
d_O0=SvO0 _ 2 0 0V
dt
Or
-~r + 21.[C(r, t) - O].
(28b) (28c)
For the interaction term we apply a linear approximation in tS~(r, v, t) which is justified by relation ( 2 2 ) . The result is
f dw(w-v)k(r,w,t) (1 - p ) ~ ( o ) ( r , v , t ) p ( V - v) - f dwL(v,w;r,t) 8~(r,w,t),
(1 - p ) ~ ( r , v , t )
(29a)
where we have introduced a linear operator L with the components
L ( v , w ; r , t ) := (1 - p ) p ( r , t ) { [ v -
V ( r , t ) ] 6 ( v - w) + P ( o ) ( v ; r , t ) ( v - w)}. (29b)
Here, tS(v - w ) denotes Dirac's delta function. The linear operator L possesses an infinite number o f eigenvalues 1/7-u (cf. [7,15-18] ). The relevant eigenvalue is the smallest one since it characterizes temporal changes that take place on the time scale we are interested in. It is o f the order o f the average interaction rate per vehicle [5,4,6]
1 1-P/dvfdwlw_vl~(r,w,t)~(r,v,t) To "- p(r, t)
U e~J(1-p)p(r,t) /dv f dw[14)-l~[P(o)(w~r,t)P(o)(D;r,t) = (1 -
(29c)
The other eigenvalues are somewhat larger [7,15-18] (i.e. r;, < r0 f o r / z 4: 0) and they describe fast fluctuations which can be adiabatically eliminated [ 19]. As a consequence, we can make the so-called 'relaxation time approximation' l
f d w L ( v , w ; r , t ) 6 ~ ( r , w , t ) ~ 6~(r,v,t)/7-o.
(29d)
Inserting ( 2 3 ) , ( 2 5 ) , (28) and (29) into (3) we finally obtain l To illustrate this. assume that we are confronted with a linear differential equation dx/dt = --Lx with a linear operator (or matrix) L, This equation has the general solution x(t) = ~-]~t`at,x , e x p ( - t / r , ) where the eigenvalues l / r , are the solutions of the characteristic equation det(L - 1/~',) = 0 and the eigenvectors x , satisfy the equation Lx, = x , / r , . If 1/~-, >> I/~'o for g 4:0 we have x(t) ~, aoxo exp(-t/r0) after a very short time t > 3max{~', : /z 4. O} (adiabatic approximation). This implies - L x ( t ) = dx(t)/dt
--x(t)/~'O.
D. Helbing/PhysicaA 219 (1995) 391-407
397
6P(r'v't)~-7°{Pp---~) (6vOp-P~r)\ Or +-~
6V Or
+ --~
0
+~
p Or 1
6V-~r
r [((By) 3
= -~(o)~'o [\ ~-~
+ l(c-o)
Or + r Ve- V )
+(1 3div~ O0
')
+ f°(°) - v °P(°-----A) + 1 (°%(v)) Ov
1
-2-6 / - g + Tg (v~ - v)a~
((~)2
~'/5(o)
Or + -(C(o)r - O)
-~k, ~
1 +(1-p)pSv
]
.
(30)
The explicit form of the function
I/o(v) := ao + al 6v + a z ( 6 v ) 2 + . . . + an(6v) n
(31)
(where n is an arbitrary integer) must now be determined by means of the consistency
conditions
/ I
dv [~(o)(r,v,t) +8~(r,v,t)l - p(r,t),
(32)
do v [/5(o)(r, v, t) + 6/5(r, v, t) ] - p(r, t) g(r, t),
(33)
f dv I v - V(r,t)12[~(o)(r,v,t) +6~(r,v,t)l "p(r,t)O(r,t),
(34)
since the relations for p(r, t), V(r, t), and O(r, t) were not approximated in contrast to the relation for ff(r,t). In order to additionally guarantee compatibility with the equilibrium relation
go(v) = v0 + ,~v
(35)
(cf. Ref. [ 1 ] ), we try to find a solution of the form
go(v) = ao + a16v,
(36)
which implies n = 1. Inserting this into (30) we get 8/5(r, v, t) ~ -/5(o)ro +I(c-O)
202
20 / -~r + - ~ ( Ve - V) 6v
( (6v)201)
D. Helbing/PhysicaA 219 (1995)391-407
398 -~
ao+al~V-(V+~v)
( 8__~)
q"
-
1
+ - (al -- 1) + (1 T
-p)p6v
]
. (37)
On the other hand, the consistency conditions (32) to (34) require that 8t5(r, v, t) has the form
6/5(r,v,t) L_ y ( r , t ) ( 3 3 v 6 \01/2 where y ( r , t) is the
(8v)3"~
03/2 /] p(r,t)P(o)(v;r,t),
(38)
skewness of the velocity distribution
P(v; r, t) =/5(0) (r, v, t) + 8/5(r, v, t) p(r,t)
(39)
From (37) and (38) follows
ao = Ve + T( 1 - p)pO = Vo,
al = C/O,
(40)
so that
f'o(v) = Vo + (C/O) 3v.
(41)
Since in equilibrium we have O = C (cf. [ 1 ] ), this relation indeed fulfils the compatibility condition (35). Moreover we find
C(r,t) = / d r [ v :/dvl
V(r,t)][Vo(v) - Vo]P(v;r,t) .log(8v)2~P(v;r,t) =O-~,
(42)
which guarantees consistency. Summarizing our considerations, for non-equilibrium situations we have found the correction term (~/5(r, v, t) ~ -/5(0)70
(By) 202
3 8v~
80 -2-0 J Or"
(43)
This is obviously a consequence of the finite adaptation time To which is characterized by the interaction-free time ro causing a delayed adjustment o f / 5 ( r , v, t) to the local equilibrium/5(o) (r, v, t). In order to take into account the effects of finite reaction time and braking time we must add a time period r t > 0 to the interaction-free time To. Hence, ~'0 must be replaced by the corrected adaptation time 7", := r0 + 7-'. With
(44)
D. Helbing/Physica A 219 (1995) 391-407
P(v;r,t)~
1-r.\
~
~-~
]~r
399
P(o)(v;r,t)
=[1 3/(6t)(3& (&)3 x
1
~/2¢rO(r, t)
e x p { - [ v - V(r, t)]2/20(r, t)},
(45)
we can now calculate a corrected relation for J ( r , t). We find
,.7(r, t) = --KOO/Or,
(46)
K := 3pr.O,
(47)
where K is called a kinetic coefficient. According to (46), in non-equilibrium (nonhomogeneous) traffic situations an additional 'transport term' appears which describes a flux of velocity variance. This is a consequence of the finite skewness J 3/ = ~
K =
00
DO3/2 Or
3r. 00
-
~
(48)
Or
of the velocity distribution. Finally, we must find an appropriate relation for the covariance C(r, t). Luckily one can derive from the gas-kinetic traffic equation ( 1 ) a dynamic equation for C(r, t) since relation (41) implies
/ d v f dvo(av)2avo#(r,v, vo, t) = / dv(Bv)2[Vo(v) - Vo]#(r,v,t) C
=f dv(&)3C~(r,v,t) = ab
(49)
(Bvo := vo - Vo). The covariance equation (which is derived in the appendix) reads
Ot
Or
Or + -(O0-
p Or C) -
p Or +
(1 - p ) p C v / O ,
(50)
T
where
Oo(r,t)
:=fdvf dv°[v°-v°(r't)lz~(r'v'v°'t)p(r,t)
(51)
is the variance of desired velocities. Summarizing our results, we have found the following macroscopic traffic equations:
Op
a(pV)
+ - 0t Or
=0,
(52)
OV+vaV _ l o? +![v~(p,v,o) -Vl, Ot
Or
p Or
r
(53)
D. Helbing/PhysicaA 219 (1995) 391-407
400
O0 vOO - ~ 4- ~r =
2198V l a (KaO~ 00 p --@rr4- p -@rr\. Or f 4- 2-( C -- O ) 4- (1-- P ) K-O-r
(54)
and
ec 0--7+
vaC _coy Or =
ar
eVo par
l e (;oo) + p ~r \ Or J
1
+-[Ce(p,V,,O,C) - C ] 4- (1 - - p ) f a O
7r
(55)
with the kinetic coefficient
[ := K(C/O) = 3pr.C,
Ce(p,V,o,c)
:= o0 - ( 2 / V ~ ) T ( 1
(56) - p)pCv/-&
(57)
These equations are the Navier-Stokes-like traffic equations. 2 Compared with the Navier-Stokes equations for ordinary fluids they possess the additional terms ( V e - V ) / r and 2(Oe - O ) / r which are due to acceleration and interaction processes. Due to the one-dimensionality of the considered traffic equations, the velocity equation (53) does not include a shear viscosity term. The variance equation (54) is related to the equation of heat conduction. However, O does not have the interpretation of 'heat' but only of velocity variance, here. Finally, the Navier-Stokes-like traffic equations include the additional covariance equation (55) arising from the tendency of drivers to get ahead with a certain desired velocity v0.
3. Adaptive driver behavior and bulk viscosity The term - ( l/p)@79/Or of velocity equation (53) describes an anticipation effect reflecting that drivers accelerate when the 'traffic pressure' P = pO lessens, i.e. when the density p or the variance O decreases. However, drivers additionally react to spatial changes of average velocity. This effect can be modelled by the modified pressure relation
P ( p , V,O) := pO - rlaV/ar,
(58)
which gives velocity equation (53) a similar form like variance equation (54) and covariance equation (55). In order to present reasons for relation (58) let us assume that drivers switch between two driving modes m E { 1,2} depending on the traffic situation. Let m = 1 characterize a brisk, m = 2 describe a careful driving mode. Then, we can split the density p(r, t) into partial densities pro(r, t) that delineate drivers who are in state m:
pl(r,t) + p2(r,t) = p(r,t).
(59)
2 Often the "Navier-Stokesequation' denotes the velocityequation only. However,since it is easier to name the whole set of connected equations (containing the respectivedensity equation, velocityequation, etc.) by one single term, the notation of Ref. [7] has been applied here.
D. Helbing/PhysicaA 219 (1995) 391-407
401
Both densities are governed by a continuity equation, but we have transitions between the two driving modes with a rate R(pl, V) so that
Opl 3 O---t- = - O--r(Pl V) - R(pl, V),
(60a)
0p2 3 0--t- = - 3 r (p2V) + R(p - P2, V).
(60b)
Adding both equations we see that the original continuity equation (52) is still valid. Now, defining the substantial time derivative D Dt
--
0 Ot
:=--+V--
0 Or
(61)
we can rewrite (60a) and obtain
Dpl Dt -
?V Pl-~r
R(pI,V).
(62)
D / D t describes temporal changes in a coordinate system that moves with velocity V. Assuming that Pl relaxes rapidly we can apply the adiabatic approximation [ 19] D p l / D t ,~ O,
(63)
which is valid on the slow time-scale of the macroscopic changes of traffic flow. This leads to
R(pl, V) ~ -plOV/Or.
(64)
Relation (63) implies that the density Pl of briskly behaving drivers is approximately constant in the moving coordinate system whereas the density p2 = p - pl of carefully behaving drivers varies with the traffic situation: ?V Dp2 ~ Dt -P~r "
(65)
P2 increases when the average velocity spatially decreases (3V/Or < 0) since this may indicate a critical traffic situation. According to relations (60), (64) incessant transitions between the two driving modes take place as long as traffic flow is spatially non-homogeneous (i.e. OV/ar 4: 0). This leads to corrections of the pressure relation. Expanding P with respect to the variable R which characterizes the disequilibrium between the two driving modes we find [20]
79(p'O'R) = 79(p'0'0) - O~R R=opl-~OV r +....
(66)
With the equilibrium relation T'(p, O, 0) = pO and 3T' 7] :=Pl ~ R=O'
(67)
402
D. Helbing/Physica A 219 (1995) 391-407
we finally obtain the desired result 79(p, O, R) -- 79(p, V,O) = pO - flaY~Or.
(68)
The corresponding additional term O/Or(rlOV/Or) in velocity equation (53) causes a spatial smoothing of sudden velocity changes and thus reduces the risk of accidents. (This result is comparable to the ones obtained for microsimulations including changes between different driving strategies, cf. Ref. [21].)
4. Modifications due to finite space requirements We will now introduce some corrections that are due to the fact that vehicles are no point-like objects but need, on average, a space of s ( V ) = l + VT
(69)
each. Here, l is about the average vehicle length whereas V/" corresponds to the safe distance each driver should keep to the next vehicle ahead. T is again the reaction time. Consequently, if A N ( r , t) := p(r, t) Ar means the number of vehicles that are at a place between r - A r / 2 and r + Ar/2, the effective density is AN(r,t) p(r,t) AN(r,t)s[V(r,t) ] = 1 - p(r,t)s[V(r,t) ] "
p(r,t) = Ar-
(70)
Since A N ( r , t ) s ( V ) is the space which is occupied by A N ( r , t) vehicles, the effective density is the number AN(r, t) of vehicles per effective free space A r - A N ( r , t ) s ( V ) . The reduction of available space by the vehicles leads to an increase of their interaction rate. Therefore, we have oo
-~-
:=(1 - p ) tr
/ / dw
d w o l v - w l ~ f r , v, wo, t ) p f r , w, vo, t)
i! L,
- (1-p)
f dw/dwolw-vJO(r,w,
w o , t ) ~ ( r , v , vo, t),
(71)
0
with ~(r,v, vo,t) :=
~ ( r , v , vo, t) 1 -p(r,t)s[V(r,t)]"
(72)
Consequently, we obtain the corrected relation l
,
-- :=(1--
v/-6
(73)
In addition, we must replace 79 and 3" by 79i . -
79 1 -- p s ( V )
and
j i :=
,Y 1 - ps(V) '
(74)
D. Helbing/Physica A 219 (1995) 391-407
403
respectively [22]. For the kinetic coefficients r/, K, and ( we obtain the corrected relations 7 ! :=
Kt
~
:=
1 - ps(V)
1 - ps(V)'
(,:=
(
l - ps(V)
K
= 3Or,C.
= 3or, O,
(75)
The corrected formula
pO
OO = l -
(76)
ps(V)
for the equilibrium pressure corresponds to the pressure relation of van der Waals for a 'real gas'. According to (76), the traffic pressure diverges for p ---, Pmax := l/l which causes a deceleration of vehicles. The corrected kinetic coefficients r / ( p , V , O ) , K~(p,V,O), and (~(p, gO, C) also diverge for p ~ Pmax [22]. We find for example
K~°~Z;~ 3Or'O =
3pr~O 1 -
(77)
ps(V)'
so that the divergence of Kt is a consequence of the finite reaction- and braking-time r t. This divergence causes a homogenization of traffic flow since the second spatial derivatives rlO2V/ar2, Ka20/ar 2, and sra20/0r 2 produce a spatial smoothing of average velocity V, variance O, and covariance C respectively. It is the divergence of 'traffic pressure' and kinetic coefficients for p --~ Pmax that prevents the spatial density p from exceeding the maximum density Pmax [23].
5. S u m m a r y and outlook This paper continued the project of deriving macroscopic traffic equations from elementary laws of driver-vehicle behavior. On the way to the construction of a 'highfidelity' model we started with calculating a corrected solution of the gas-kinetic traffic equation developed in a previous paper. The procedure applied the method of Chapman and Enskog which utilized the Euler-like traffic equations that resulted in zeroth-order approximation. In this way a correction term for the flux of velocity variance has been found which has the meaning of a transport tenn. It originates from the finite skewness of the velocity distribution in non-equilibrium situations which is a consequence of the finite adaptation time for approaching the local equilibrium distribution. We have also included in our model additional effects which arise from the circumstance that driver-vehicle units are active systems. The adaptive behavior with regard to the respective traffic conditions, for example, are connected with transitions between a brisk and a careful driving mode. This leads to an additional bulk viscosity term which causes a spatial smoothing of sudden velocity changes and, by this, reduces the risk of
404
D. Helbing/Physica A 219 (1995) 391--407
accidents. A shear viscosity term does not occur in the macroscopic traffic equations due to the one-dimensionality of the considered traffic equations. Moreover, we have introduced modifications that take into account the finite reactionand braking-time as well as the finite space requirements of vehicles (= vehicle length plus safe distance). This resulted in a van der Waals-like relation for 'traffic pressure' and a divergence of the kinetic coefficients when maximum density is approached. These corrections guarantee that the maximum density cannot be exceeded (which could happen in former traffic models). Present research activities focus on the simulation of traffic flow on the basis of the derived 'high-fidelity' model. In addition, the proposed traffic model can be extended to a model which explicitly treats the interactions between the different lanes of a uni-directional multi-lane freeway originating from overtaking and lane changing [24].
Appendix A. Calculation of the covariance equation In the following we will use the abbreviation
- fk, xlP(r,v, vo, t) av l a v o v ~voj ~ . j ptr, t)
(vk(vo) l) :=
(A.1)
Multiplying the gas-kinetic traffic equation ( 1 ) by vvo and integrating with respect to v and vo we obtain the equation
~ (p(vvo) ) + ~ (p{do0)) a
+fdv°fdvvv°a~-( av _ 'v°-u 7" a(pWo) -
a(pC)
0-----7- + - - ~ _
_
+ 2 0 (pVC)or
+
(A.2a)
a(pV2Vo) ar
a(pVoO)
+ --Or
-/dvo/dvvo~V°
- ~p ( W ~ ' - Wo)
-v T
-t-
o
(p((Sv)2t~VO)) V;-Vo o_~
-pV
(A.2b)
_ 8(pVVo) + a(p(7__._~)+ O(pV2Vo) + 20(PVC) + 0(Vo79) Ot Ot Or Or Or __ P [ (Vo)2 + Oo - V o V - C ] 7
V / - Vo T
- pV o (X3
u u
o
(A.2c)
D. Helbing/PhysicaA 219(1995)391--407 x ~3(r, w, wo, t)~(r,
v, vo, t)
4O5 (A.2d)
p ) f d v f dwvf,'o(w)(w-v)~(r,v,t)~(r,w,t)
=(1-
w~>v
/
(A.2e)
w
(A.2f) w<~/;
6w<6v
x/5(r,
V + 6w, t)~(r, V + 6v, t)
=-(l-p)p
2
77 dx
-oo
x ~1
(A.2g)
(
dyy 2 Vo+~9 -~ ,] h(x,y)
0
exp(-x2/40)
~ e1x p ( - y 2 / 4 O )
=_(, _p)p2 (VoO+__~Cv/_O+ 2Cv/~+ ,2
(A,2h) (A.2i) (A,2j)
where
Vd(r, t)
:=
/ dvo voPd(vo;r, t).
(A.3)
In order to come from (A.2a) to (A.2b) we applied partial integration and used the relation
(v2vo)=((V+t~v)2(Vo+&vo)) =V2Vo+2VC+VoO+{(6v)26vo}.
(A.4)
(A.2c) follows with (49) and ((vo) 2 -
roy) =
((Vo + & o ) 2) - ((Vo + a v o ) i v + & ) )
= [ ( v o) 2 + 0o1 - ( ~ b v + C ) .
(A.5)
From (A.2d) to (A.2e) we applied definition (4) and made the approximation
/dwf(w)~(r,w,t) ~ 0
dwf(w)~(r,w,t),
(A.6)
--oo
which is very well justified as long as 3v/O < V and only invalid if V ~ 0 or c ~ 0. In order to arrive at (A.2f) we exchanged the variables v *-+ w in the first term of (A.2e) which corresponds to a renaming of variables. Whereas the transformation of
D. Helbing/Physica A 219 (1995) 391-407
406
variables v ~ V+6v, w ~ V + 6 w applied in (A.2g) is trivial, in (A.2h) we made the substitutions
t3v+6w=v+w-2V--~
x,
&v-Sw=v-w-~
y.
(A.7)
Moreover, we inserted expression (45) into /5(r, V + 8w, t), ~(r, V + By, t), and we introduced the abbreviation 1-g
0]-/2
2"X
31/
O312
2"xy2
2"X3
2'2(X2_ y2) 2 V ~ + 3 • 2303/2 q- 2303/2 - + 240
1 +
2
2" [ ~ x - y 1 - -~ /z 2
x
- -
,
{ y [ 3 x + y
h(x,y):=
2'2(X6 __ y6) 32 • 2803
2'2(X4
__
y4)
3. 2502
T2(x4y2 _ x2y4) 3 ' 2803
(A.8)
(A.2i) follows by integration with respect to x and y with the relations oo
dz z 2k+l e x p [ - z 2 / ( 4 0 ) ] = 0,
(A.9)
--(20
faz O0
z2kexp[-z2/(40)]=l
• 3 . . . ( 2 k - 1 ) ~/-~(20) '+1/2,
(A.10)
0 oo
dz z 2k+l e x p [ - z 2 / ( 4 0 ) ] = ½k!(40) k+l = 2 . 4 . . . 2 k . ( 2 0 ) k+l.
(A.I 1)
0 Approximation (A.2j) is valid for small 2' which is normally guaranteed since y = 0 in equilibrium. From (A.2) one obtains
ac
P-~ =
_ c o p _ VVoO_p
Ot
ov
, OVo _ r o y @
at - pVo-~ - pv--~
-2VC~r-2pC~r-2pV~r
-VOO~-'poVOOr
+P-[(vo)Z + O o - V o v - c ]
V / - Vo + pV o
7"
-(1-p)p2[VoO+(-~+2)Cv/O
av
,2aVo
or - 2pW°-~r - p v -~r ~
OrO ( ' f - ~ )
T
] .
(A.12)
In order to simplify Eq. (A.12) one must insert relations (52) and (53) for Op/Ot and aV/at respectively. For aVo/Ot one can derive the equation OVo + VC~l,~ =
1 0 ( p C ) + V~ - Vo
Ot
p
Or
Or
T
(A.13)
D. Helbing/PhysicaA 219 (1995)391-407
407
This easily follows via e q u a t i o n
~ o ( r , vo, t) +
[V(vo)~o(r, vo, t) ]
p(r,t) T [Pd(vo;r,t) -Po(vo;r,t)] =0, (A.14)
where /5o(r, vo,
t) := f
~ ' ( v o ; r , t ) :=
f
dop(r, v, vo,t),
(A.15)
~(r,v, vo, t) dvv ~o(r, vo, t)
(A.15)
Eq. ( A . 1 4 ) can be o b t a i n e d from ( l ) by integration with respect to v.
References 111 D. Helbing, Physica A 219 (1995) 375, this issue. 121 P. Nelson, Transport Theory and Statistical Physics 24 (1995) 383. [31 D. Helbing, to be published (1995). 14 ] J. Jfickle, Einftihrung in die Transporttheorie (Vieweg, Braunschweig, Germany, 1978). [51 A. Rieckers and H. Stumpf, Thermodynamik, Vol. 2 (Vieweg, Braunschweig, Germany, 1977). 16] K. Huang, Statistical Mechanics, 2nd edition (Wiley, New York, 1987). 171 R.L. Liboff, Kinetic Theory (Prentice-Hall, London, 1990). 18] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition (Cambridge University Press, New York, 1970). [ 9 ] D. Enskog, The Kinetic Theory of Phenomena in Fairly Rare Gases (Dissertation, Uppsala, 1917). [ 10] L. Bohzmann, Lectures on Gas Theory (University of California, Berkeley, 1964). I 11 ] S.L. Paveri-Fontana, Transportation Research 9 (1975) 225. [121 1. Prigogine, in: Theory of Traffic Flow, ed. R. Herman (Elsevier, Amsterdam, 1961). 113] W.F. Phillips, Kinetic Model for Traffic Flow (Report No. DOT/RSPD/DPB/50-77/17, National Technical Information Service, Springfield, Virginia 22161, 1977). 1141 W.E Phillips, Transportation Planning and Technology 5 (1979) 131. [151 G.E. Uhlenbeck and G.W. Ford, Lectures in Statistical Mechanics (American Math. Soc., Providence, R1, 1963). [161 C.W. Wang-Chang and G.E. Uhlenbeck, in: Studies in Statistical Mechanics, eds. J. deBoer and G.E. Uhlenbeck (North-HoUand, Amsterdam, 1970). [17] H. Grad, Phys. Fluids 6 (1963) 147. [ 18] H. Grad, in: Rarefied Gas Dynamics Symposium, Vol. 1., ed. J. Laurmann (Academic Press, New York, 1963). [ 19] H. Haken, Advanced Synergetics (Springer, Berlin, 1983). 120 ] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987). [21] S. Migowsky, T. Wanschura and P. Ruj~in,Z. Phys. B 95 (1994) 407. [22] Y.L. Klimontovich, Statistical Physics (Harwood Academic, Chur, Swizzerland, 1982). 123] D. Helbing, Phys. Rev. E 51 (1995) 3164. [ 24] D. Helbing, Modeling multi-lane traffic flow with queuing effects, submitted to Transportation Research B (1995).
Physica A 219 (1995) 391-407
High-fidelity macroscopic traffic equations Dirk Helbing 11. Institute of Theoretical Physics, University of Stuttgart, 70550 Stuttgart, Germany Received 11 May 1995
Abstract The Euler-like traffic equations that can be derived for quasi-homogeneous traffic situations are corrected for non-equilibrium effects. This yields additional transport terms describing a flux of velocity variance. Other modifications resulting in a bulk viscosity term arise from the adaptive behavior of drivers with respect to changes of the traffic situation. Finally, corrections due to the space requirements of vehicles as well as due to finite reaction- and braking-times are introduced. These are responsible for a van der Waals-like relation for 'traffic pressure' and modified relations for the kinetic coefficients.
1. Introduction In a previous paper [ 1 ] an Euler-like macroscopic traffic model was derived from a gas-kinetic traffic equation which was constructed from elementary laws of individual driver behavior. This model already overcomes some shortcomings of former macroscopic traffic models which are mostly of a more or less phenomenological nature or based on lax analogies with the equations of ordinary fluids [2,3]. Nevertheless, due to the presupposed approximation of local equilibrium, the Euler-like traffic equations are still not perfect. They only apply to traffic situations with moderate deviations from equilibrium which implies small gradients and slow temporal changes of vehicle density, average velocity, and velocity variance. Therefore a higher-order approximation of the solution of the gas-kinetic traffic equation will be derived in Section 2. The applied method is analogous to the derivation of Navier-Stokes equations for ordinary fluids from the Boltzmann equation [ 4 - 7 ] . It bases on a Chapman-Enskog expansion [8,9] which is known from kinetic gas theory and yields Euler(-like) equations in the zeroth-order approximation of local equilibrium, whereas Navier-Stokes(-like) equations are found in first-order approximation. However, due to additional terms in the gas-kinetic traffic equation compared with the 0378-4371/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSD10378-437 1 (95)00175-1
D. Helbing/Physica A 219 (1995) 391-407
392
Boltzmann equation [ 10] the corresponding mathematical procedure is more complicated than for ordinary gases. Nevertheless it is still possible to derive correction terms of the Euler-like traffic equations. These describe a flux of velocity variance which is related with a finite skewness of the velocity distribution in non-equilibrium situations. In comparison with the conventional Navier-Stokes equations the resulting Navier-Stokes-like traffic equations include an additional covariance equation due to the fact that drivers try to get ahead with a certain individual desired velocity. Moverover, they contain additional terms arising from the acceleration and interaction of vehicles. However, because of the one-dimensionality of the investigated traffic equations no shear viscosity term occurs. In the remaining sections we will introduce some corrections and modifications of the Navier-Stokes-like traffic equations. These will take into account traffic-specific characteristics arising from the circumstance that driver-vehicle units are active systems. Up to now they have been neglected by most macroscopic traffic models: (i) Drivers show an adaptive behavior with respect to changes of the traffic conditions. According to Section 3 this leads to an additional bulk viscosity term. (ii) The distance between succeeding vehicles is often of the order of magnitude of the space required by themselves so that vehicles cannot be idealized as point-like objects. The space needed by each vehicle (= vehicle length plus safe distance) can be described by a van der Waals-like relation for 'traffic pressure' (cf. Section 4). (iii) Drivers of vehicles have a finite reaction time to changes of traffic flow conditions. This causes modifications of the transport terms (cf. Eqs. (44) and (77)). The resulting macroscopic traffic model includes the most important effects of drivervehicle behavior and can therefore be denoted as a 'high-fidelity' traffic model.
2. Derivation of the Navier-Stokes-like traffic equations In Ref. [ 1] a gas-kinetic traffic equation has been developed for the temporal evolution of the phase-space density ,~(r, v, vo, t) of vehicles with desired velocity vo and actual velocity v at place r. For an uni-directional multi-lane freeway without entrances and exits it reads
O~
o-7+
O(~v)
+
O(~_2_)
P(r'v't)[p~(vo;r,t)_Po(vo;r,t) ]
T
(la)
O0
= ¢ l - p ) f aw f dwolv-wl
(r,v, wo,
vo, t)
(lb)
U V
- ( l - p ) f dw f dwolw-vlb(r,w, wo,t)~(r,v, vo, t), 0
where
(lc)
D. Helbing/PhysicaA 219 (1995) 391-407 ~(r,v,t)
:=f dvop(r,v, vo, t)
393 (2)
is a reduced phase-space density. The second term of (la) describes a temporal change of/3(r, v, v0, t) due to vehicle motion with velocity v and is called a convection term. The third term delineates an adjustment of the actual velocity v to the desired velocity v0 within the relaxation time ~', i.e. it reflects the drivers' acceleration behavior. The fourth term arises from an adaptation of the actual distribution of desired velocities Po(vo; r, t) to the reasonable distribution of desired velocities P~ (v0; r, t) which is determined by speed limits and traffic conditions (gradient of the road; tog, rain, snow, or ice on the road). T ~ l s is about the reaction time. The terms ( l b ) and ( l c ) describe vehicular pair interactions due to which one vehicle must decelerate to the velocity of a slower one. p denotes the probability that a slower car can be immediately overtaken. (For a detailed discussion of Eq. ( l ) cf. Refs. [ 11,1] .) Integrating Eq. ( l ) with respect to vo yields the following reduced equation:
O~ O(vD) 0 ( ~'o(v;r,t)-v) Ot + ---~r + -~v ~ ( r, v, t ) r a
O0
= (1 - p ) ~ ( r , v , t ) / d w ( w -
v)~(r,w, t),
(3)
, /
o where we have introduced
f/o( v; r, t ) := [ dvo voP(-:~-' v--~°-'t), . j ~tr, v,t)
(4)
Eq. (3) allows the derivation of dynamic equations for macroscopic quantities like the spatial density
p(r.t)::/,.l,.o,,r,...o,,,:l,.,,....,,
(5)
of vehicles per lane, the average velocity
, ~(r,v,t) V(r,t) =-- (v):= f avv --~,}~
(6)
of vehicles, and their velocity variance
O(r,t) =- ( [ v - V(r,t) ] 2) : = . / dv [v - V(r,t) ] 2~(r'v't) p(r,t) = @2)_ [V(r,t)]2.
(7)
These equations read
Op O(pV) Ot Or _ SV + vSV _ Ot Or
(8)
--+--=0,
1 l a'P + _[Ve(p, V,O) _ V] ' p Or T
(9)
D. Helbing/Physica A 219 (1995)391-407
394
O0 O0 --+V-at ar
279 OV p ar
1 Off 2 -- +-[Oe(p,V,O,J,C) p Or r
--0].
(10)
Here we have used the abbreviations
Ve(p, V,O) := V0 - r ( p , V,O) [ 1 - p ( p , V,O) ] P ( r , t),
(11)
where
Vo(r,t) : = / d v f d v o v o
#(r'v'v°'t) p(r,t)
(12)
is the average desired velocity, and
Oe(p,V,O,J,C) :=C(r,t) - ½r(P,V,O)[l - p ( p , V , O ) ] f f ( r , t ) .
(13)
Moreover, we have introduced the so-called traffic pressure [ 12-14]
l / /
P ( r , t ) .- - p(r, t)
dv(v-V)~(r,v,t)
dw(v-w)p(r,w,t)
#
= ]dv(v
- V)2~(r,v,t) = p ( r , t ) O ( r , t )
(14)
,
J
the flux of velocity variance
1/
f l ( r , t ) .- p(r,t~
dr(v-
V)2~(r,v,t)
/
dw(v-
w)~(r,w,t)
= f dv (v - V)3/5(r, v, t) = p(r, t)F(r, t)
(15)
(which corresponds to the 'heat flow' in conventional fluid-dynamics), and the covari-
ance C(r,t)
:=Jdvofdv(v-v)(oo-v°)P(r'v'v°'t)p(r,t) = f d v ( v - V ) [ V o ( v ) - v o j , ~ pp( r( r, v, t, )t ) "
(16)
In order to close the system of F.qs. (8), (9), and (10), the functions C(r,t) and i f ( r , t) must be appropriately determined in dependence of p(r, t), V(r, t), and O(r, t). In Ref. [ 1] this has been done via the zeroth-order approximation of local equilibrium
~ ( r , v , t ) ~ ~(o)(r,v,t) .-
p(r,t)
x/2crO(r, t)
exp{-[v - V(r,t)]2/20(r,t)}.
(17)
This implies
f f ( r , t ) ~ fl¢o)(r,t) =0, and leads to the Euler-like traffic equations.
(18)
D. Helbing/Physica A 219 (1995) 391-407
395
However, approximation (17) is only valid if the velocity distribution
/5(r,v,t) P(v;r,t) .- - p(r,t)
(19)
relaxes very rapidly towards the local equilibrium distribution
P(o)(v;r,t):= /5(o)(r,v,t) _ 1 exp{-[v-V(r,t)]2/20(r,t)}. p(r,t) x/2qrO(r,t)
(20)
This is only the case for small deviations from stationary and spatially homogeneous traffic. Otherwise the finite adaptation time 7, the relaxation process lasts will lead to corrections. Therefore we will now derive an approximate dynamic equation for the deviation
3/5(r,v,t) := ~(r,v,t) -/5(o)(r,v,t)
(21)
from the local equilibrium distribution /5(o)(r,v, t). Utilizing that the correction term 8/5(r, v, t) will normally be small compared to p(o)[p(r, t), V(r, t), O(r, t)] we have
8/5(r,v,t) < < / 5 ( o ) [ p ( r , t ) , V ( r , t ) , 8 ( r , t ) ] ,
(22)
and get
a/5
-
a {_f/o(v)-v)
kp
;
/
0/5(0) + vC~p(o) + 0 ( /5(0) Vo(v) - v ) • -
at
(23)
-W-r
(For a detailled discussion of this approximation cf. [4,5,7].) Now, introducing the abbreviation d
0
dt
at + v"~r'
0 (24)
we can write
0/5(o_____Z ) + vO/5(o) = d/5(o____ 2 _ 0/5(0) dp + 0/5(0______2) dV + 0/5(o______2) d0 Ot Or dt Op dt OV dt O0 dt -
/5(o___2)dp /5(o)~ dV /5(o) ( ( S v ) 2 p dt + - - o - o v - ~ + - ~ ---~-
1) dO d---t'
(25)
with
8v := v - V.
(26)
Relations for dp/dt, dV/dt, and dO/dt can be obtained from the macroscopic traffic equations (8), (9), and (10) via d
0
d t - Ot +
V0_
O
Or +6v~r"
With (14), (18), and (13) we find
(27)
D. Helbing/Physica A 219 (1995) 391-407
396
dp = Svap _ p a V , --~ Or Or
=Bray
dt
Or
(28a)
1 a(pO___2
p
Or
+
l[Ve(p,V,O) _ V], 7"
d_O0=SvO0 _ 2 0 0V
dt
Or
-~r + 21.[C(r, t) - O].
(28b) (28c)
For the interaction term we apply a linear approximation in tS~(r, v, t) which is justified by relation ( 2 2 ) . The result is
f dw(w-v)k(r,w,t) (1 - p ) ~ ( o ) ( r , v , t ) p ( V - v) - f dwL(v,w;r,t) 8~(r,w,t),
(1 - p ) ~ ( r , v , t )
(29a)
where we have introduced a linear operator L with the components
L ( v , w ; r , t ) := (1 - p ) p ( r , t ) { [ v -
V ( r , t ) ] 6 ( v - w) + P ( o ) ( v ; r , t ) ( v - w)}. (29b)
Here, tS(v - w ) denotes Dirac's delta function. The linear operator L possesses an infinite number o f eigenvalues 1/7-u (cf. [7,15-18] ). The relevant eigenvalue is the smallest one since it characterizes temporal changes that take place on the time scale we are interested in. It is o f the order o f the average interaction rate per vehicle [5,4,6]
1 1-P/dvfdwlw_vl~(r,w,t)~(r,v,t) To "- p(r, t)
U e~J(1-p)p(r,t) /dv f dw[14)-l~[P(o)(w~r,t)P(o)(D;r,t) = (1 -
(29c)
The other eigenvalues are somewhat larger [7,15-18] (i.e. r;, < r0 f o r / z 4: 0) and they describe fast fluctuations which can be adiabatically eliminated [ 19]. As a consequence, we can make the so-called 'relaxation time approximation' l
f d w L ( v , w ; r , t ) 6 ~ ( r , w , t ) ~ 6~(r,v,t)/7-o.
(29d)
Inserting ( 2 3 ) , ( 2 5 ) , (28) and (29) into (3) we finally obtain l To illustrate this. assume that we are confronted with a linear differential equation dx/dt = --Lx with a linear operator (or matrix) L, This equation has the general solution x(t) = ~-]~t`at,x , e x p ( - t / r , ) where the eigenvalues l / r , are the solutions of the characteristic equation det(L - 1/~',) = 0 and the eigenvectors x , satisfy the equation Lx, = x , / r , . If 1/~-, >> I/~'o for g 4:0 we have x(t) ~, aoxo exp(-t/r0) after a very short time t > 3max{~', : /z 4. O} (adiabatic approximation). This implies - L x ( t ) = dx(t)/dt
--x(t)/~'O.
D. Helbing/PhysicaA 219 (1995) 391-407
397
6P(r'v't)~-7°{Pp---~) (6vOp-P~r)\ Or +-~
6V Or
+ --~
0
+~
p Or 1
6V-~r
r [((By) 3
= -~(o)~'o [\ ~-~
+ l(c-o)
Or + r Ve- V )
+(1 3div~ O0
')
+ f°(°) - v °P(°-----A) + 1 (°%(v)) Ov
1
-2-6 / - g + Tg (v~ - v)a~
((~)2
~'/5(o)
Or + -(C(o)r - O)
-~k, ~
1 +(1-p)pSv
]
.
(30)
The explicit form of the function
I/o(v) := ao + al 6v + a z ( 6 v ) 2 + . . . + an(6v) n
(31)
(where n is an arbitrary integer) must now be determined by means of the consistency
conditions
/ I
dv [~(o)(r,v,t) +8~(r,v,t)l - p(r,t),
(32)
do v [/5(o)(r, v, t) + 6/5(r, v, t) ] - p(r, t) g(r, t),
(33)
f dv I v - V(r,t)12[~(o)(r,v,t) +6~(r,v,t)l "p(r,t)O(r,t),
(34)
since the relations for p(r, t), V(r, t), and O(r, t) were not approximated in contrast to the relation for ff(r,t). In order to additionally guarantee compatibility with the equilibrium relation
go(v) = v0 + ,~v
(35)
(cf. Ref. [ 1 ] ), we try to find a solution of the form
go(v) = ao + a16v,
(36)
which implies n = 1. Inserting this into (30) we get 8/5(r, v, t) ~ -/5(o)ro +I(c-O)
202
20 / -~r + - ~ ( Ve - V) 6v
( (6v)201)
D. Helbing/PhysicaA 219 (1995)391-407
398 -~
ao+al~V-(V+~v)
( 8__~)
q"
-
1
+ - (al -- 1) + (1 T
-p)p6v
]
. (37)
On the other hand, the consistency conditions (32) to (34) require that 8t5(r, v, t) has the form
6/5(r,v,t) L_ y ( r , t ) ( 3 3 v 6 \01/2 where y ( r , t) is the
(8v)3"~
03/2 /] p(r,t)P(o)(v;r,t),
(38)
skewness of the velocity distribution
P(v; r, t) =/5(0) (r, v, t) + 8/5(r, v, t) p(r,t)
(39)
From (37) and (38) follows
ao = Ve + T( 1 - p)pO = Vo,
al = C/O,
(40)
so that
f'o(v) = Vo + (C/O) 3v.
(41)
Since in equilibrium we have O = C (cf. [ 1 ] ), this relation indeed fulfils the compatibility condition (35). Moreover we find
C(r,t) = / d r [ v :/dvl
V(r,t)][Vo(v) - Vo]P(v;r,t) .log(8v)2~P(v;r,t) =O-~,
(42)
which guarantees consistency. Summarizing our considerations, for non-equilibrium situations we have found the correction term (~/5(r, v, t) ~ -/5(0)70
(By) 202
3 8v~
80 -2-0 J Or"
(43)
This is obviously a consequence of the finite adaptation time To which is characterized by the interaction-free time ro causing a delayed adjustment o f / 5 ( r , v, t) to the local equilibrium/5(o) (r, v, t). In order to take into account the effects of finite reaction time and braking time we must add a time period r t > 0 to the interaction-free time To. Hence, ~'0 must be replaced by the corrected adaptation time 7", := r0 + 7-'. With
(44)
D. Helbing/Physica A 219 (1995) 391-407
P(v;r,t)~
1-r.\
~
~-~
]~r
399
P(o)(v;r,t)
=[1 3/(6t)(3& (&)3 x
1
~/2¢rO(r, t)
e x p { - [ v - V(r, t)]2/20(r, t)},
(45)
we can now calculate a corrected relation for J ( r , t). We find
,.7(r, t) = --KOO/Or,
(46)
K := 3pr.O,
(47)
where K is called a kinetic coefficient. According to (46), in non-equilibrium (nonhomogeneous) traffic situations an additional 'transport term' appears which describes a flux of velocity variance. This is a consequence of the finite skewness J 3/ = ~
K =
00
DO3/2 Or
3r. 00
-
~
(48)
Or
of the velocity distribution. Finally, we must find an appropriate relation for the covariance C(r, t). Luckily one can derive from the gas-kinetic traffic equation ( 1 ) a dynamic equation for C(r, t) since relation (41) implies
/ d v f dvo(av)2avo#(r,v, vo, t) = / dv(Bv)2[Vo(v) - Vo]#(r,v,t) C
=f dv(&)3C~(r,v,t) = ab
(49)
(Bvo := vo - Vo). The covariance equation (which is derived in the appendix) reads
Ot
Or
Or + -(O0-
p Or C) -
p Or +
(1 - p ) p C v / O ,
(50)
T
where
Oo(r,t)
:=fdvf dv°[v°-v°(r't)lz~(r'v'v°'t)p(r,t)
(51)
is the variance of desired velocities. Summarizing our results, we have found the following macroscopic traffic equations:
Op
a(pV)
+ - 0t Or
=0,
(52)
OV+vaV _ l o? +![v~(p,v,o) -Vl, Ot
Or
p Or
r
(53)
D. Helbing/PhysicaA 219 (1995) 391-407
400
O0 vOO - ~ 4- ~r =
2198V l a (KaO~ 00 p --@rr4- p -@rr\. Or f 4- 2-( C -- O ) 4- (1-- P ) K-O-r
(54)
and
ec 0--7+
vaC _coy Or =
ar
eVo par
l e (;oo) + p ~r \ Or J
1
+-[Ce(p,V,,O,C) - C ] 4- (1 - - p ) f a O
7r
(55)
with the kinetic coefficient
[ := K(C/O) = 3pr.C,
Ce(p,V,o,c)
:= o0 - ( 2 / V ~ ) T ( 1
(56) - p)pCv/-&
(57)
These equations are the Navier-Stokes-like traffic equations. 2 Compared with the Navier-Stokes equations for ordinary fluids they possess the additional terms ( V e - V ) / r and 2(Oe - O ) / r which are due to acceleration and interaction processes. Due to the one-dimensionality of the considered traffic equations, the velocity equation (53) does not include a shear viscosity term. The variance equation (54) is related to the equation of heat conduction. However, O does not have the interpretation of 'heat' but only of velocity variance, here. Finally, the Navier-Stokes-like traffic equations include the additional covariance equation (55) arising from the tendency of drivers to get ahead with a certain desired velocity v0.
3. Adaptive driver behavior and bulk viscosity The term - ( l/p)@79/Or of velocity equation (53) describes an anticipation effect reflecting that drivers accelerate when the 'traffic pressure' P = pO lessens, i.e. when the density p or the variance O decreases. However, drivers additionally react to spatial changes of average velocity. This effect can be modelled by the modified pressure relation
P ( p , V,O) := pO - rlaV/ar,
(58)
which gives velocity equation (53) a similar form like variance equation (54) and covariance equation (55). In order to present reasons for relation (58) let us assume that drivers switch between two driving modes m E { 1,2} depending on the traffic situation. Let m = 1 characterize a brisk, m = 2 describe a careful driving mode. Then, we can split the density p(r, t) into partial densities pro(r, t) that delineate drivers who are in state m:
pl(r,t) + p2(r,t) = p(r,t).
(59)
2 Often the "Navier-Stokesequation' denotes the velocityequation only. However,since it is easier to name the whole set of connected equations (containing the respectivedensity equation, velocityequation, etc.) by one single term, the notation of Ref. [7] has been applied here.
D. Helbing/PhysicaA 219 (1995) 391-407
401
Both densities are governed by a continuity equation, but we have transitions between the two driving modes with a rate R(pl, V) so that
Opl 3 O---t- = - O--r(Pl V) - R(pl, V),
(60a)
0p2 3 0--t- = - 3 r (p2V) + R(p - P2, V).
(60b)
Adding both equations we see that the original continuity equation (52) is still valid. Now, defining the substantial time derivative D Dt
--
0 Ot
:=--+V--
0 Or
(61)
we can rewrite (60a) and obtain
Dpl Dt -
?V Pl-~r
R(pI,V).
(62)
D / D t describes temporal changes in a coordinate system that moves with velocity V. Assuming that Pl relaxes rapidly we can apply the adiabatic approximation [ 19] D p l / D t ,~ O,
(63)
which is valid on the slow time-scale of the macroscopic changes of traffic flow. This leads to
R(pl, V) ~ -plOV/Or.
(64)
Relation (63) implies that the density Pl of briskly behaving drivers is approximately constant in the moving coordinate system whereas the density p2 = p - pl of carefully behaving drivers varies with the traffic situation: ?V Dp2 ~ Dt -P~r "
(65)
P2 increases when the average velocity spatially decreases (3V/Or < 0) since this may indicate a critical traffic situation. According to relations (60), (64) incessant transitions between the two driving modes take place as long as traffic flow is spatially non-homogeneous (i.e. OV/ar 4: 0). This leads to corrections of the pressure relation. Expanding P with respect to the variable R which characterizes the disequilibrium between the two driving modes we find [20]
79(p'O'R) = 79(p'0'0) - O~R R=opl-~OV r +....
(66)
With the equilibrium relation T'(p, O, 0) = pO and 3T' 7] :=Pl ~ R=O'
(67)
402
D. Helbing/Physica A 219 (1995) 391-407
we finally obtain the desired result 79(p, O, R) -- 79(p, V,O) = pO - flaY~Or.
(68)
The corresponding additional term O/Or(rlOV/Or) in velocity equation (53) causes a spatial smoothing of sudden velocity changes and thus reduces the risk of accidents. (This result is comparable to the ones obtained for microsimulations including changes between different driving strategies, cf. Ref. [21].)
4. Modifications due to finite space requirements We will now introduce some corrections that are due to the fact that vehicles are no point-like objects but need, on average, a space of s ( V ) = l + VT
(69)
each. Here, l is about the average vehicle length whereas V/" corresponds to the safe distance each driver should keep to the next vehicle ahead. T is again the reaction time. Consequently, if A N ( r , t) := p(r, t) Ar means the number of vehicles that are at a place between r - A r / 2 and r + Ar/2, the effective density is AN(r,t) p(r,t) AN(r,t)s[V(r,t) ] = 1 - p(r,t)s[V(r,t) ] "
p(r,t) = Ar-
(70)
Since A N ( r , t ) s ( V ) is the space which is occupied by A N ( r , t) vehicles, the effective density is the number AN(r, t) of vehicles per effective free space A r - A N ( r , t ) s ( V ) . The reduction of available space by the vehicles leads to an increase of their interaction rate. Therefore, we have oo
-~-
:=(1 - p ) tr
/ / dw
d w o l v - w l ~ f r , v, wo, t ) p f r , w, vo, t)
i! L,
- (1-p)
f dw/dwolw-vJO(r,w,
w o , t ) ~ ( r , v , vo, t),
(71)
0
with ~(r,v, vo,t) :=
~ ( r , v , vo, t) 1 -p(r,t)s[V(r,t)]"
(72)
Consequently, we obtain the corrected relation l
,
-- :=(1--
v/-6
(73)
In addition, we must replace 79 and 3" by 79i . -
79 1 -- p s ( V )
and
j i :=
,Y 1 - ps(V) '
(74)
D. Helbing/Physica A 219 (1995) 391-407
403
respectively [22]. For the kinetic coefficients r/, K, and ( we obtain the corrected relations 7 ! :=
Kt
~
:=
1 - ps(V)
1 - ps(V)'
(,:=
(
l - ps(V)
K
= 3Or,C.
= 3or, O,
(75)
The corrected formula
pO
OO = l -
(76)
ps(V)
for the equilibrium pressure corresponds to the pressure relation of van der Waals for a 'real gas'. According to (76), the traffic pressure diverges for p ---, Pmax := l/l which causes a deceleration of vehicles. The corrected kinetic coefficients r / ( p , V , O ) , K~(p,V,O), and (~(p, gO, C) also diverge for p ~ Pmax [22]. We find for example
K~°~Z;~ 3Or'O =
3pr~O 1 -
(77)
ps(V)'
so that the divergence of Kt is a consequence of the finite reaction- and braking-time r t. This divergence causes a homogenization of traffic flow since the second spatial derivatives rlO2V/ar2, Ka20/ar 2, and sra20/0r 2 produce a spatial smoothing of average velocity V, variance O, and covariance C respectively. It is the divergence of 'traffic pressure' and kinetic coefficients for p --~ Pmax that prevents the spatial density p from exceeding the maximum density Pmax [23].
5. S u m m a r y and outlook This paper continued the project of deriving macroscopic traffic equations from elementary laws of driver-vehicle behavior. On the way to the construction of a 'highfidelity' model we started with calculating a corrected solution of the gas-kinetic traffic equation developed in a previous paper. The procedure applied the method of Chapman and Enskog which utilized the Euler-like traffic equations that resulted in zeroth-order approximation. In this way a correction term for the flux of velocity variance has been found which has the meaning of a transport tenn. It originates from the finite skewness of the velocity distribution in non-equilibrium situations which is a consequence of the finite adaptation time for approaching the local equilibrium distribution. We have also included in our model additional effects which arise from the circumstance that driver-vehicle units are active systems. The adaptive behavior with regard to the respective traffic conditions, for example, are connected with transitions between a brisk and a careful driving mode. This leads to an additional bulk viscosity term which causes a spatial smoothing of sudden velocity changes and, by this, reduces the risk of
404
D. Helbing/Physica A 219 (1995) 391--407
accidents. A shear viscosity term does not occur in the macroscopic traffic equations due to the one-dimensionality of the considered traffic equations. Moreover, we have introduced modifications that take into account the finite reactionand braking-time as well as the finite space requirements of vehicles (= vehicle length plus safe distance). This resulted in a van der Waals-like relation for 'traffic pressure' and a divergence of the kinetic coefficients when maximum density is approached. These corrections guarantee that the maximum density cannot be exceeded (which could happen in former traffic models). Present research activities focus on the simulation of traffic flow on the basis of the derived 'high-fidelity' model. In addition, the proposed traffic model can be extended to a model which explicitly treats the interactions between the different lanes of a uni-directional multi-lane freeway originating from overtaking and lane changing [24].
Appendix A. Calculation of the covariance equation In the following we will use the abbreviation
- fk, xlP(r,v, vo, t) av l a v o v ~voj ~ . j ptr, t)
(vk(vo) l) :=
(A.1)
Multiplying the gas-kinetic traffic equation ( 1 ) by vvo and integrating with respect to v and vo we obtain the equation
~ (p(vvo) ) + ~ (p{do0)) a
+fdv°fdvvv°a~-( av _ 'v°-u 7" a(pWo) -
a(pC)
0-----7- + - - ~ _
_
+ 2 0 (pVC)or
+
(A.2a)
a(pV2Vo) ar
a(pVoO)
+ --Or
-/dvo/dvvo~V°
- ~p ( W ~ ' - Wo)
-v T
-t-
o
(p((Sv)2t~VO)) V;-Vo o_~
-pV
(A.2b)
_ 8(pVVo) + a(p(7__._~)+ O(pV2Vo) + 20(PVC) + 0(Vo79) Ot Ot Or Or Or __ P [ (Vo)2 + Oo - V o V - C ] 7
V / - Vo T
- pV o (X3
u u
o
(A.2c)
D. Helbing/PhysicaA 219(1995)391--407 x ~3(r, w, wo, t)~(r,
v, vo, t)
4O5 (A.2d)
p ) f d v f dwvf,'o(w)(w-v)~(r,v,t)~(r,w,t)
=(1-
w~>v
/
(A.2e)
w
(A.2f) w<~/;
6w<6v
x/5(r,
V + 6w, t)~(r, V + 6v, t)
=-(l-p)p
2
77 dx
-oo
x ~1
(A.2g)
(
dyy 2 Vo+~9 -~ ,] h(x,y)
0
exp(-x2/40)
~ e1x p ( - y 2 / 4 O )
=_(, _p)p2 (VoO+__~Cv/_O+ 2Cv/~+ ,2
(A,2h) (A.2i) (A,2j)
where
Vd(r, t)
:=
/ dvo voPd(vo;r, t).
(A.3)
In order to come from (A.2a) to (A.2b) we applied partial integration and used the relation
(v2vo)=((V+t~v)2(Vo+&vo)) =V2Vo+2VC+VoO+{(6v)26vo}.
(A.4)
(A.2c) follows with (49) and ((vo) 2 -
roy) =
((Vo + & o ) 2) - ((Vo + a v o ) i v + & ) )
= [ ( v o) 2 + 0o1 - ( ~ b v + C ) .
(A.5)
From (A.2d) to (A.2e) we applied definition (4) and made the approximation
/dwf(w)~(r,w,t) ~ 0
dwf(w)~(r,w,t),
(A.6)
--oo
which is very well justified as long as 3v/O < V and only invalid if V ~ 0 or c ~ 0. In order to arrive at (A.2f) we exchanged the variables v *-+ w in the first term of (A.2e) which corresponds to a renaming of variables. Whereas the transformation of
D. Helbing/Physica A 219 (1995) 391-407
406
variables v ~ V+6v, w ~ V + 6 w applied in (A.2g) is trivial, in (A.2h) we made the substitutions
t3v+6w=v+w-2V--~
x,
&v-Sw=v-w-~
y.
(A.7)
Moreover, we inserted expression (45) into /5(r, V + 8w, t), ~(r, V + By, t), and we introduced the abbreviation 1-g
0]-/2
2"X
31/
O312
2"xy2
2"X3
2'2(X2_ y2) 2 V ~ + 3 • 2303/2 q- 2303/2 - + 240
1 +
2
2" [ ~ x - y 1 - -~ /z 2
x
- -
,
{ y [ 3 x + y
h(x,y):=
2'2(X6 __ y6) 32 • 2803
2'2(X4
__
y4)
3. 2502
T2(x4y2 _ x2y4) 3 ' 2803
(A.8)
(A.2i) follows by integration with respect to x and y with the relations oo
dz z 2k+l e x p [ - z 2 / ( 4 0 ) ] = 0,
(A.9)
--(20
faz O0
z2kexp[-z2/(40)]=l
• 3 . . . ( 2 k - 1 ) ~/-~(20) '+1/2,
(A.10)
0 oo
dz z 2k+l e x p [ - z 2 / ( 4 0 ) ] = ½k!(40) k+l = 2 . 4 . . . 2 k . ( 2 0 ) k+l.
(A.I 1)
0 Approximation (A.2j) is valid for small 2' which is normally guaranteed since y = 0 in equilibrium. From (A.2) one obtains
ac
P-~ =
_ c o p _ VVoO_p
Ot
ov
, OVo _ r o y @
at - pVo-~ - pv--~
-2VC~r-2pC~r-2pV~r
-VOO~-'poVOOr
+P-[(vo)Z + O o - V o v - c ]
V / - Vo + pV o
7"
-(1-p)p2[VoO+(-~+2)Cv/O
av
,2aVo
or - 2pW°-~r - p v -~r ~
OrO ( ' f - ~ )
T
] .
(A.12)
In order to simplify Eq. (A.12) one must insert relations (52) and (53) for Op/Ot and aV/at respectively. For aVo/Ot one can derive the equation OVo + VC~l,~ =
1 0 ( p C ) + V~ - Vo
Ot
p
Or
Or
T
(A.13)
D. Helbing/PhysicaA 219 (1995)391-407
407
This easily follows via e q u a t i o n
~ o ( r , vo, t) +
[V(vo)~o(r, vo, t) ]
p(r,t) T [Pd(vo;r,t) -Po(vo;r,t)] =0, (A.14)
where /5o(r, vo,
t) := f
~ ' ( v o ; r , t ) :=
f
dop(r, v, vo,t),
(A.15)
~(r,v, vo, t) dvv ~o(r, vo, t)
(A.15)
Eq. ( A . 1 4 ) can be o b t a i n e d from ( l ) by integration with respect to v.
References 111 D. Helbing, Physica A 219 (1995) 375, this issue. 121 P. Nelson, Transport Theory and Statistical Physics 24 (1995) 383. [31 D. Helbing, to be published (1995). 14 ] J. Jfickle, Einftihrung in die Transporttheorie (Vieweg, Braunschweig, Germany, 1978). [51 A. Rieckers and H. Stumpf, Thermodynamik, Vol. 2 (Vieweg, Braunschweig, Germany, 1977). 16] K. Huang, Statistical Mechanics, 2nd edition (Wiley, New York, 1987). 171 R.L. Liboff, Kinetic Theory (Prentice-Hall, London, 1990). 18] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition (Cambridge University Press, New York, 1970). [ 9 ] D. Enskog, The Kinetic Theory of Phenomena in Fairly Rare Gases (Dissertation, Uppsala, 1917). [ 10] L. Bohzmann, Lectures on Gas Theory (University of California, Berkeley, 1964). I 11 ] S.L. Paveri-Fontana, Transportation Research 9 (1975) 225. [121 1. Prigogine, in: Theory of Traffic Flow, ed. R. Herman (Elsevier, Amsterdam, 1961). 113] W.F. Phillips, Kinetic Model for Traffic Flow (Report No. DOT/RSPD/DPB/50-77/17, National Technical Information Service, Springfield, Virginia 22161, 1977). 1141 W.E Phillips, Transportation Planning and Technology 5 (1979) 131. [151 G.E. Uhlenbeck and G.W. Ford, Lectures in Statistical Mechanics (American Math. Soc., Providence, R1, 1963). [161 C.W. Wang-Chang and G.E. Uhlenbeck, in: Studies in Statistical Mechanics, eds. J. deBoer and G.E. Uhlenbeck (North-HoUand, Amsterdam, 1970). [17] H. Grad, Phys. Fluids 6 (1963) 147. [ 18] H. Grad, in: Rarefied Gas Dynamics Symposium, Vol. 1., ed. J. Laurmann (Academic Press, New York, 1963). [ 19] H. Haken, Advanced Synergetics (Springer, Berlin, 1983). 120 ] J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987). [21] S. Migowsky, T. Wanschura and P. Ruj~in,Z. Phys. B 95 (1994) 407. [22] Y.L. Klimontovich, Statistical Physics (Harwood Academic, Chur, Swizzerland, 1982). 123] D. Helbing, Phys. Rev. E 51 (1995) 3164. [ 24] D. Helbing, Modeling multi-lane traffic flow with queuing effects, submitted to Transportation Research B (1995).