Explanation of Observed Features of Self-Organization in Traffic Flow

4 Pages·1.1 MB·English

by  Martin Treiber

Explanation of observed features of self-organization in traffic flow
Martin Treiber and Dirk Helbing
II. Institute of Theoretical Physics, University of Stuttgart, Pfaffenwaldring 57/III, D-70550 Stuttgart, Germany
(February 1, 2008)
9 Basedonsimulationswiththe“intelligentdrivermodel”,amicroscopictrafficmodel,weexplainthe
9
recently discovered transition from free over “synchronized” traffic to stop-and-go patterns [B. S.
9 Kerner, Phys. Rev. Lett. 81, 3797 (1998)]. We obtain a nearly quantitative agreement with
1
empirical findings such as the “pinch effect”, the flow-density diagram, the emergence of stop-and-

n go waves from nonhomogeneous congested traffic, and thedimensions of their wavelength.
a
J

2
2
05.70.Fh,05.65.+b,47.54.+r,89.40.+k

]
h During the last years, theoretical and empirical inves- the gap sα(t) = [xα−1(t) xα(t) lα] and the velocity
ec tigations have identified different possible mechanisms difference∆vα(t)=[vα(t)−−vα−1(t−)]tothevehicle(α−1)
for a phase transition from free traffic to stop-and-go in front:
m
traffic on freeways. This includes deterministic [1–4]
at- and stochastic mechanisms [5–8] as well as effects of in- v˙α =a 1 vα δ s∗ 2 . (1)
t homogeneities [9–12]. In contrast, Kerner has recently " −(cid:18)v0(cid:19) −(cid:18)sα(cid:19) #
s described the detailled features of another transition
.
t to stop-and-go patterns [13] developing from “synchro- Accordingtothisformula,theaccelerationonafreeroad
a
δ
m nized” congested traffic [14,15] on German highways, (meaning sα ) is given by a[1 (vα/v0) ], where a
→ ∞ −
- which are compatible with empirical findings on Dutch is the maximum accelerationandv0 the desiredvelocity.
d highways [16]. Theexponentδ istypicallybetween1and5. Itallowsto
n Inthefollowing,weproposeaquantitativeexplanation describethattherealisticaccelerationbehaviorofdrivers
o
oftheseobservationsbasedonmicrosimulationswiththe liesbetweenaconstantaccelerationauptotheir desired
c
[ “intelligent driver model” (IDM). In particular, we will velocity v0 (δ ) and an exponential accelerationbe-
show the possible coexistence of different traffic states havior (δ =1)→. ∞
1 along the road behind an inhomogeneity of traffic flow. The braking term a(s∗/sα)2 depends Coulomb-like
v −
9 Itis,inupstreamdirection,associatedwiththesequence on the gap sα, as it is the case for the braking term
3 “homogeneouscongestedtraffic”[10,11](which,inamul- of the microscopic Wiedemann model [19]. Therefore,
2 tilane model, is related to the observed synchronization the acceleration term is negligible, if the gap sα drops
1 among lanes [9,17]) “inhomogeneous congested traf- considerably below the “effective desired distance” s∗.
0 →
fic” [11] (corresponding to the so-called “pinch region” With the relation
9
[13]) “stop-and-go traffic”, while we have free traffic
9 → ∗ vα∆vα
/ flow downstream of the inhomogeneity. s (vα,∆vα)=s0+max vαT + ,0 , (2)
t 2√ab
a Itwillturnoutthat,incontrasttopreviouslyreported (cid:18) (cid:19)
m traffic phenomena, this phenomenon relies on the exis-
it is constructed in a way that drivers keep a minimum
- tence ofasufficientlylargedensity regionofconvectively “jam distance” s to a standing vehicle, plus an addi-
d 0
stable traffic, in which traffic flow is unstable, but any
n tionalsafetydistancevαT,whereT isthesafetimehead-
perturbations are convectedawayin upstream direction.
o way in congested but moving traffic.
c Furthermore, one needs a traffic model in which the re- The nonequilibrium term proportional to ∆vα [2,8]
: sulting traffic flow inside of fully developed traffic jams
v reflects an “intelligent” braking strategy, according to
i is much lower (nearly zero) than in “synchronized” traf- which drivers restrict their deceleration to b in “nor-
X
fic. Inparticular,withoutsuitable modifications(seebe-
mal” situations (e.g., when approaching standing or
r low),thisisnotsatisfiedbythetrafficmodeldiscussedin
a slowervehiclesfromsufficientlylargedistances),butthey
Ref.[10]. Wealsopointoutthat,althoughtheIDMhasa
brake harder when the situation becomes more criti-
uniqueflow-densityrelationinequilibrium,itreproduces
cal, i.e., when the anticipated “kinematic deceleration”
the observed two-dimensional scattering of flow-density (∆v)2/(2sα), which is necessaryto avoida collisionwith
data at medium vehicle densities [14,13], even without
a uniformly moving leading vehicle (v˙α−1 = 0), exceeds
assuming a mixture of different vehicle types [18].
b. Notice that the acceleration a is typically lower than
The IDM is a continuous, deterministic model, in
thedesireddecelerationb,andthatbothaccelerationpa-
which the acceleration of a vehicle α of length lα at po- rametersdonotinfluencetheequilibriumflow-densityre-
sition xα(t) depends on its own velocity vα(t) as well as lation (“fundamental diagram”). Since it turns out that
1
neither multilane effects nor different types of vehicles solve or merge to a few “wide jams” (in which traffic
are relevant in the context of this study, we have as- comes to a standstill) with typical distances of 2 km up
sumed identical “driver-vehicle units” characterized by to5kmbetweenthem. Oncethejamshaveformed,they
the realistic parameters v = 120 km/h, δ = 4, a = 0.6 persist and propagate upstream at a constant propaga-
0
m/s2, b=0.9 m/s2, s =2 m, and T =1.5 s, apartfrom tion velocity without further changes of their shape. No
0
alocalizedchangeofv orT (seebelow). Forthe vehicle new clusters develop between the jams.
0
length we use l=5 m, but this value does not affect the To compare our simulation results directly with the
dynamics. empirical data published by Kerner [13], we investigated
We simulated an open freeway section of 20 kilometer thetemporalevolutionoftheaveragevelocityatsixsub-
length for time intervals up to 120 minutes, of which we sequent locations D1 through D6 that had the same dis-
display the most interesting parts only. In addition, we tances with respect to the inhomogeneity as in Ref. [13]
assumedan inhomogeneity of traffic flow that will be re- (see Fig. 2). The detector positions D1 through D4 are
sponsibleforthetransitionfromfreetocongestedtraffic, upstream of the inhomogeneity, D5 is directly at the in-
as described in Refs. [10,11]. However, as pointed out homogeneity, and D6 is downstream of it. In contrast
by Kerner [13], the self-organized patterns observed by to the simulation of Fig. 1, the capacity drop at the
him are not restricted to the vicinity of on-ramps. We inhomogeneity is so weak that free traffic is metastable
will confirm this by simulating different kinds of inho- in the overall system. At the inhomogeneity (D5), one
mogeneities (see Figs. 1 through 3) and comparing them observes homogeneous congested (“synchronized”) traf-
with the injection of vehicles at on-ramps (see Fig. 4). fic, at D4 one sees small oscillations that, around D3,
In Figure 1, we have assumed an inhomogeneity cor- developtostop-and-gowavesoflargeramplitude,andfi-
responding to a freewaysectionwhere people drivemore nally to jams (at D1 and D2). In the downstream direc-
carefully. This was modelled by setting the desired time tion, the congested traffic dissolves to free traffic (D6).
headwayfromT =1.5sto T =1.75sbetweenx=0km Apart from irregularities in the measured data due to
and x=0.3 km. In the simulations of Figs. 2 and 3, we fluctuations, the curves are in (semi-)quantitative agree-
havereducedthe desiredvelocityfromv =120km/hto ment with Kerner’s empirical findings [13].
0
v = 80 km/h in the same region. As initial conditions, We also plot one-minute data of the six “detectors”
0
we assumed homogeneous free traffic in equilibrium at in a flow-density diagram (Fig. 3), together with the
a flow of Q = 1670 vehicles/h (Figs. 1 through 3) or equilibrium flow-density relation Q (ρ) (“fundamental
0 e
1570 vehicles/h (Fig. 4). The actual initial conditions, diagram”). The lower boundary of the data points at
however, are only relevant for a short time interval. At mediumdensitiescorrespondstotheflow-densityrelation
the upstream boundary, we assume that vehicles enter belonging to the downstream front of a fully developed
thefreewayuniformlyatarateQ (t)=Q +∆Q(t)and jam. Inagreementwithobservations,thislineisthesame
in 0
drivewithavelocitycorrespondingtofreetrafficinequi- foralljams andcorrespondsto a unique propagationve-
librium. WhileinthesimulationofFig.1,thebreakdown locity of their downstream fronts. The twodimensional
of traffic flow is caused by exceeding the static freeway regionofpoints abovethis linerelatetocongestedtraffic
capacity at the inhomogeneity [11], in Figs. 2 and 3 it at D3, D4, and D5.
istriggeredbyatriangularlyshapedperturbation∆Q(t) To understand this scenario, we need some basic re-
of the inflow [10,11] between t = 10 min and t = 20 sults about the stability of homogeneous traffic with re-
min with a maximum of 200 vehicles per hour and lane specttolocalizedperturbations. Typically,therearefour
at t = 15 min. Furthermore, we used the “absorbing” “critical” densities ρci with ρc1 < ρc2 < ρc3 < ρc4 and
downstream boundary condition v˙α = 0. To minimize the following properties [20,21]: For low and very high
simulationtime,weintegratedEq. (1)withasimpleEu- densities (ρ < ρ or ρ > ρ ), traffic is stable with re-
c1 c4
ler scheme using a coarsetime discretizationof ∆t=0.4 specttoarbitraryperturbations,whileforρ <ρ<ρ ,
c2 c3
s, and translated the vehicles in each step according to it is linearly unstable. In the two density ranges in be-
xα(t+∆t)=xα(t)+vα∆t+1v˙α(∆t)2. However,smaller tween, homogeneous traffic is unstable only with respect
2
values of ∆t yielded nearly indistinguishable results. to perturbations exceeding a certain critical amplitude
Figure 1 gives a representative overview of the simu- ∆ρ (ρ) (“metastability”). Furthermore, there exists a
cr
lation result by means of a spatiotemporal density plot. range ρ < ρ < ρ with ρ > ρ , where traffic is
cv c3 cv c2
The resulting sequence of transitions is essentially the linearlyunstable,butconvectivelystable,i.e., allpertur-
same as observed [13]: After 10 minutes of free traffic, bations grow, but they are eventually convected away in
traffic breaks down near the inhomogeneity, resulting in upstreamdirection[22]. Actually, this rangecanbe very
homogeneouscongestedtraffic at this location, that per- large. For the IDM with the parameters used here, we
sists over a long time. Upstream of the inhomogeneity, have ρ =50 vehicles/km and ρ =100 vehicles/km.
cv c3
small oscillations develop that travel further upstream Let us assume that the inflow Q has a value larger
in
and grow to stop-and-go waves of relatively short wave- than Q , and that a phase transition from free to con-
c1
lengths (about 0.8 km). Finally, these waves either dis- gested traffic has occurred at the inhomogeneity. This
2
breakdown to congested traffic, which we identify with vectively stable. If “synchronized” traffic is linearly un-
“synchronized traffic”, here [10,11,17,18], was explained stable andfree traffic upstreamis metastable,upstream-
in [10]. Since it is connected with a drop of the effec- moving perturbations will grow and, when their ampli-
tive freeway capacity to the self-organized outflow Q˜ tudes became large enough, eventually form stop-and-go
out
from“synchronized”traffic(ST),theregionofcongested waves. To maintain this mechanism, the region of syn-
traffic will grow in upstream direction until the inflow chronized traffic behind the inhomogeneity must persist
Q (t) = Q (ρ (t)) to the freeway falls below the syn- (i.e. it must not dissolve to stop-and-go waves), which
in e in
chronized flow Q in the congested region behind the is only the case if it is convectively stable. The twodi-
ST
inhomogeneity. Without an on-ramp flow Q per mensionality of the congested branch of the flow-density
ramp
freeway lane, we have Q = Q˜ , otherwise it is diagram originates from the fact that the nonhomoge-
ST out
Q = (Q˜ Q ) [10]. In contrast to the empir- neous congested states are not in equilibrium.
ST out ramp
−
icalfindings explainedinRef.[10],the synchronizedflow Thephenomenonshouldbe widespreadsinceitistrig-
must be so high, here, that it is linearly unstable, which geredatrelativelysmallinhomogeneitieswhenevertraffic
implies QST > Qe(ρc3). Therefore, small perturbations flow Qin(t) exceeds a certainthresholdQc1. Note that it
will grow to larger oscillations, which propagate in up- can be also simulated with other traffic models like the
stream direction faster than the congested region grows. macroscopic gas-kinetic-basedmodel [21], if “frustration
Whenthe oscillationsreachthemetastableregionoffree effects” are additionally taken into account (see Fig. 4).
traffic upstream of the inhomogeneity, oscillations with The authors are grateful for financial support by the
an amplitude below the critical amplitude ∆ρ (ρ ) will BMBF (research project SANDY, grant No. 13N7092)
cr in
eventuallydisappear,theremainingoneswillcontinueto and by the DFG (Heisenberg scholarshipHe 2789/1-1).
growuntiltheyarefullydevelopedtrafficjams. Wepoint
outthatasequenceofsuchjamsissustained,becausethe
propagation velocity v and the outflow Q Q (ρ )
g out e c1
≈
from jams are characteristicconstants [23,24]. Finally, if
the inhomogeneity is such that the synchronized flow is
convectively stable, i.e., Q < Q = Q (ρ ), the per-
ST cv e cv [1] B. S. Kerner and P. Konh¨auser, Phys. Rev. E 50, 54
turbations cannotpropagate downstream(in contrastto
(1994).
small perturbations in free traffic). Hence, the front of
[2] D. Helbing, Verkehrsdynamik (Springer, Berlin, 1997).
dissolving traffic is smooth, then, and a small region of [3] C. Wagner et al.,Phys. Rev.E 54, 5073 (1996).
homogeneous congested traffic forms near the inhomo- [4] M. Bando, K. Hasebe, A. Nakayama, and Y. Sugiyama,
geneity [Figs. 1, 2(a), and 4]. Phys. Rev.E 51, 1035 (1995).
In Fig. 3, the flow-density relation of the downstream [5] K. Nagel and M. Schreckenberg, J. Phys. I (France) 2,
frontofafullydevelopedjam(wherethevelocityiszero) 2221 (1992).
correspondstoastraightline,theslopeofwhichisv [23]. [6] M. Schreckenberg, A. Schadschneider, K. Nagel, and N.
g
Notice that this line (which in Fig. 3 and in Ref. [13] is Ito, Phys. Rev.E 51, 2939 (1995).
[7] S.KraussandP.Wagner,Phys.Rev.E55,5597(1997).
labelled by “J”) lies considerably below the equilibrium
[8] D. E. Wolf, Physica A, in print (1999).
curveQ (ρ). Ontheotherhand,trafficflowintheregion
e [9] H.Y.Lee,H.W.Lee,and D.Kim, Phys.Rev.Lett. 81,
of oscillating congested traffic is nearly in equilibrium
1130 (1998).
as long as the oscillations are small. As the oscillations [10] D. Helbing and M. Treiber, Phys. Rev. Lett. 81, 3042
grow,thedatapointsgraduallyapproachthelineJ. This
(1998).
explainstheobservedtwodimensionalityofthecongested [11] D.Helbing,A.Hennecke,andM.Treiber,preprintcond-
partoftheflow-densitydiagram. Wepointoutthatmix- mat/9809324, submitted to Phys.Rev.Lett. (1998).
tures of different types of driver-vehicle units lead to to [12] D. Helbing and M. Treiber, Science 282, 2001 (1998).
further effects contributing to an even wider scattering [13] B. S.Kerner, Phys. Rev.Lett. 81, 3797 (1998).
of flow-density points in the congested regime [18]. [14] B. S. Kerner and H. Rehborn, Phys. Rev. E 53, R4275
In summary, we have shown that the emergence of (1996).
[15] B. S.KernerandH.Rehborn,Phys.Rev.Lett.79, 4030
stop-and-go waves out of synchronized traffic, their co-
(1997).
existence, and the twodimensional scattering of data in
[16] D. Helbing, Phys. Rev.E 55, R25 (1997).
the congested part of the flow-density diagram can be
[17] V. Shvetsov and D. Helbing, Macroscopic dynamics of
explained in the framework of “standard” traffic mod-
multi-lane traffic, Phys. Rev.E, submitted (1998).
els that have a unique equilibrium flow-density relation. [18] M. Treiber and D. Helbing, J. Phys. A: Math. Gen. 32,
The necessary conditions are a metastability of traffic L17 (1999).
flow,aflowinside oftrafficjams thatis muchlowerthan [19] R. Wiedemann, Simulation des Straßenverkehrsflusses.
in synchronizedcongestedtraffic, and a sufficiently large ”Schriftenreihe desIfV”,Vol.8(Institutfu¨rVerkehrswe-
densityregimeoflinearlyunstabletrafficflowthatiscon- sen, University of Karlsruhe, Germany,1974).
[20] B. S. Kerner, P. Konh¨auser, and M. Schilke, Phys. Lett.
3
A 215, 45 (1996). 2000
Flow at downstream jam fronts J
[21] M. Treiber, A. Hennecke, and D. Helbing, Phys. Rev. E
Fundamental diagram
59, 239 (1999). h) 1500
[22] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, s/
e
[23] B85.1S(.1K99e3r)n.er and H. Rehborn, Phys. Rev. E 53, R1297 vehicl 1000
(1996). w (
[24] B.S.Kerner,S.L.Klenov,andP.Konh¨auser,Phys.Rev. Flo 500
E 56, 4200 (1997).
0
0 50 100 150
ρ (vehicles/km) ρ (vehicles/km)
150
100 FIG. 3. Flow-density data (symbols) calculated from
50 themicroscopic simulation at thefourdetectorlocations dis-
played in Fig. 2, and in addition at x = −3.7 km (detector
0
-6 D2) and at x = 1.2 km (D6). Also shown is the equilibrium
-4 60 flow-densityrelation(solidline),andtheflow-densityrelation
at thedownstream fronts of fully developed jams (dashed).
40
-2
x (km) 20
0 t (min)
0
ρ (vehicles/km)
150
FIG. 1. Spatiotemporal density plot illustrating the
100
breakdown to “synchronized” traffic (smooth region of high
density) near an inhomogeneity, and showing stop-and-go 50
waves emanating from this region. Traffic flows in positive 0
-6
x-direction. The inhomogeneity corresponds to an increased -5
safe time headway T between x = 0 km and x = 0.3 km, -4 80
-3
reflecting more careful driving (see main text). Downstream -2 60
of the inhomogeneity, vehicles accelerate into free traffic. x (km) -1 0 40
1 t (min)
20
120 120
h) 90 (a) D5 (0.0 km) 90 (b) D4 (-0.5 km) FIG. 4. Spatiotemporal evolution of the traffic den-
km/ 60 60 sity according to the gas-kinetic-based traffic model [21].
V ( 30 30 The assumed inhomogeneity of traffic flow comes from an
0 0
20 40 60 80 100 20 40 60 80 100 on-ramp of length 200 m with an inflow of 220 vehicles
t (min) t (min) per hour and freeway lane. The inflow to the main road
is 1570 vehicles per hour and lane, and the breakdown of
120 120 traffic flow is triggered by a perturbation ∆Q(t) of the in-
h) 90 (c) D3 (-2.0 km) 90 flow with a flow peak of 125 vehicles per hour and lane (see
km/ 60 60 main text). The assumed model parameters are V0 = 120
V ( 30 30 (d) D1 (-5.2 km) km/h, T = 1.5 s, τ = 30 s, ρmax = 120 vehicles/km, and
0 0
20 40 60 80 100 20 40 60 80 100 γ = 1.2, while the parameters for the variance prefactor [21]
t (min) t (min) A(ρ) = A0 +∆A{tanh[(ρ−ρc)/∆ρ]+1} are A0 = 0.008,
∆A = 0.02, ρc = 0.27ρmax, and ∆ρ = 0.1ρmax. In or-
FIG.2. Temporal evolution of theaverage velocity deter- der to have a large region of linearly unstable but convec-
mined from theindividualvelocities of thevehicles that pass tively stable traffic, we introduced a “resignation effect”, i.e.
cross sections of the freeway during one-minute intervals at a density-dependent reduction of the desired velocity V0 to
the four “detector” positions D1, D3, D4 (upstream of the V0′(ρ)=V0−∆V/{1+exp[(ρ′c−ρ)/∆ρ′]} with ∆V =0.9V0,
′
inhomogeneity),andD5(attheinhomogeneity). Thenaming ρc=0.45ρmax, and ∆ρ=0.1ρmax.
of thedetectors and theirdistances with respect to theinho-
mogeneityarethesameasinRef.[13]. Theinhomogeneityis
realized byadropofthedesiredvelocity(seethemain text).
(a) Breakdown to homogeneous synchronized traffic around
t = 45 min. (b) Oscillating synchronized traffic (“pinch re-
gion”), (c) developing stop-and-go waves, and (d) resulting
traffic jams.
4