/.1 I I I I J/ 1-7.' -.1. . ..I I I if/" J -:.", ," I ~ _0 0 0 0 .j .. ...-::~}~-:, I I , l I .:~ I X. I I , "--l_ I A model for traffic simulation by Hans-Thomas Fritzsche, Daimler-Benz AG The impairment to man and environment by a steadily increasing traffic volume requires concrete solutions now more urgently than ever before. This problem can, however, only be coped with from a scientific point of view, when founded knowledge exists of the very complex interconnections of the system called 'traffic'. Regarding the enormous costs of experimental investigations under- taken to influence traffic purposefully, the significance of theoretical considera- tions prior to these necessary field-tests can easily be seen. This paper will not only illustrate the development of such a model of traffic simulation, but also show first results thereof. Introduction. An attempt to descrihe the almostinexhaustible variety of road traffic in detail is wndemned to fail due to the utopica] volL1TIle of dat<l which would be ne<::ess<lry to do so_ The situ<ltion is very similar 10 that of thermodynamics. In traffic theory, abo macroscopic attempts are undertaken which - acwrding to hydrodynamics - describe theriowoftrafficasa<::ontinuuml.Obvious]v this kind of description al]ows only state- ments regarding global propet1ies such as stability, bifun.:ation. chaos, et<::., by using macroscopic variahles such as vehicle den- sity p, mean speed \-' and so forth. Using such models. whose eguations can be derived pre- dominant]y hy series expansion of a rnaster equation similar to the Boltzmann equation'. one can, forexamplc, understand the appear- ance of Stop-and-Go waves at sufficient traffi<::densitypJ. The necessity of an empirica] re]atlOn he- twem the mean speed v and the traffic den- sity pre. Ihe traffic flow q = p'\-' is, however. unsatisfadorv in this continuous-mechanical cOllsideration_ In traffic theory. the relation between anyone of three possihle pairs (JUtof v, If and p is called a fundamental diagram_ While such an application seems acceptable during stable states with moderate densities (p<30 cars,(lane-knl)-I). the extrapolation of the relation hetween me<ln speed and density up to the vehicle standstill density (Pm.x= 160 cars-(lanekmtl) isdoubtfu]*,sincc in these casesonecanhard]yspeakofareliabJecoITe- lation of the measured data (see Fig]). The non-existence of fundamental diagrams when equipping a vehicle with different electronic driving aids (e,g. Autonomous Intelligent Cruise Control (A ICC») so that the influence of such svstems on the traffic flow - and this, aher aiL should he the goal of traffic thcorv - cannot be studied. An attempt al a solution can be found in micmscopictrafficsimuIation,i,e.thecalcu- larion of trajectories of every single vehicle as a function of time I. OU! of the sum of all acceleration functions as a whole one can assume those macroscopic variables, which *ft i, ahoql1o:~tionable who:ther the fundamental diagram, as ealcutaled ulldeI' stalionary con- dilion" is valid in the case oftime-depemknt dynamicconsideralion<;_ May1994 characterise the tr<lffic, by integrating and suitahle averaging_ By considering vehicles individually one can, for example. predict the influence of speed limits and overtaking pro- hibitionsontheeapacityofaroad. One usable assumption would be that between the DYEs only nearest-neighbour interactions are relevant. i.e. for the aceelera- tion of the vehicle ion]y the constellation re its biding vehicle, J, is of importance (see Fig 2). For the wrfol!owillg. namely i. one presumes nO! only knowledge of its own po- sitionx" speed I',and acceleration h" htl!one ,lIso know~ the respective, momentarily valid data xi' vj' bj of Ihe Je"der j. The approxima- tion to pair interaction only results in the fact that the following vehicle i has no informa- tion of a possible change of acceleration of the leading vehide i in the future at its dis- posal. The DYE i presumes in its adjustment of its own driving behaviour a COI!5talll- lIcce!erv.lirlll ill lime of the DYEj. Fii!2.0ne-lollecar. jill/owinx sill/minn, DCE 'i'foliowIll,,/ewlnj" IT] IT] General]ytheproblemofascerl<liningtra- Jectories could be solved if one was ahle to find a dosed functionj,(x" .., Xi' ..,x,. VI' . Vi' ...V", b]...,b" ..,b" t) for the acceleration h, ofthe lth vehicle depending on the constel- ]ation of all II" vehicles and time t. The deci- sive point of view why such a function cannot be defined sensibly is4 that traffic cannot be seen <1Sa purely mechanica] system, but that it is more an interaction between physi<::s (equations of motion) and psychology (per- ception) since man is an important link of the vehide system as a whole and t]UlS cannot he ignored Mkroscopk single.]ane car-following model We now <lltemp! to develop a psy<::ho'physi- COlicar-following model which, at least. partially determines the functionsj,-'. Initially a single lane without oncoming traffic is considered. The characteristic propet1ies of the driver lper<::eplion. intention,..) and the vehicle (braking and accelerating per- formance) are implemented in a Driver- Vehicle-Element (DYE). FiJ? J. Me'i,mreJ tlQw-deflsllv relaririll./imda"ientaldilli!riml, Taken /rom tal'iern' and SI<'Ie/11'11Id'. lOOO ~ 1500 a . , ~ 1000 500 o The further consideration takes place in the phase spa<::e(difference of speed, dis- tance) of the fol]owing vehicle ire. the lead- ing vehicle j. As is known from physiology. the movement of an object can only he per- ceived when the reflcction on the retina has to exceed a cert"in minimum speed threshold, This threshold, on the one hand. individual while on the other hand it decreases with incrcasing perception time. As of now, however. one should always presume a time- constant parameter. Simple geometric con- siderations thus give a perceplio/l threshold for the realisation of a difference of speed ilv:=l-j- ~'i between the DYE i andj according to: !"J.VPiN/I'IP= :tkl'lNIPTI'(ilx - Auf :tj; t'J.x:=Xj-Xi The threshold kl'TNll'TI' for the minimum ve- locity with negative differences of velocity (PTN). i.e. decreasing distance, is smaller (ilv < 0 signals danger) than in the case of increasing distance (PTP). All describes the (gruss)stiUstanddist"ncej"an 'offset'. 10 go 100 317
Transcript
/.1
I I II
J/
1-7.' -.1..
..II Iif/" J -:.", ,"I
~ _0 0 0 0
.j..
...-::~}~-:,I I,
lI
.:~I
X.
I
I, "--l_I
Amodel for traffic simulation
by Hans-Thomas Fritzsche, Daimler-Benz AG
The impairment to man and environment by a steadily increasing traffic volumerequires concrete solutions now more urgently than ever before. This problemcan, however, only be coped with from a scientific point of view, when foundedknowledge exists of the very complex interconnections of the system called'traffic'. Regarding the enormous costs of experimental investigations under-taken to influence traffic purposefully, the significance of theoretical considera-tions prior to these necessary field-tests can easily be seen. This paper will notonly illustrate the development of such a model of traffic simulation, but alsoshow first results thereof.
Introduction. An attempt to descrihe thealmostinexhaustible variety of road traffic indetail is wndemned to fail due to the utopica]volL1TIleof dat<l which would be ne<::ess<lry todo so_ The situ<ltion is very similar 10 that ofthermodynamics. In traffic theory, abomacroscopic attempts are undertaken which
- acwrding to hydrodynamics - describetheriowoftrafficasa<::ontinuuml.Obvious]vthis kind of description al]ows only state-ments regarding global propet1ies such asstability, bifun.:ation. chaos, et<::., by usingmacroscopic variahles such as vehicle den-sity p, mean speed \-'and so forth. Using suchmodels. whose eguations can be derived pre-dominant]y hy series expansion of a rnasterequation similar to the Boltzmann equation'.one can, forexamplc, understand the appear-ance of Stop-and-Go waves at sufficienttraffi<::densitypJ.
The necessity of an empirica] re]atlOn he-twem the mean speed v and the traffic den-sity pre. Ihe traffic flow q = p'\-' is, however.unsatisfadorv in this continuous-mechanicalcOllsideration_ In traffic theory. the relationbetween anyone of three possihle pairs (JUtofv, If and p is called a fundamental diagram_While such an application seems acceptableduring stable states with moderate densities(p<30 cars,(lane-knl)-I). the extrapolation ofthe relation hetween me<ln speed and densityup to the vehicle standstill density (Pm.x= 160cars-(lanekmtl) isdoubtfu]*,sincc in thesecasesonecanhard]yspeakofareliabJecoITe-lation of the measured data (see Fig]). Thenon-existence of fundamental diagramswhen equipping a vehicle with differentelectronic driving aids (e,g. AutonomousIntelligent Cruise Control (A ICC») so that theinfluence of such svstems on the traffic flow- and this, aher aiL should he the goal oftraffic thcorv - cannot be studied.
An attempt al a solution can be found inmicmscopictrafficsimuIation,i,e.thecalcu-larion of trajectories of every single vehicleas a function of time I. OU! of the sum of allacceleration functions as a whole one canassume those macroscopic variables, which
*ft i, ahoql1o:~tionable who:ther the fundamentaldiagram, as ealcutaled ulldeI' stalionary con-dilion" is valid in the case oftime-depemkntdynamicconsideralion<;_
May1994
characterise the tr<lffic, by integrating andsuitahle averaging_ By considering vehicles
individually one can, for example. predict the
influence of speed limits and overtaking pro-hibitionsontheeapacityofaroad.
One usable assumption would be thatbetween the DYEs only nearest-neighbourinteractions are relevant. i.e. for the aceelera-tion of the vehicle ion]y the constellation re
its biding vehicle, J, is of importance (seeFig 2). For the wrfol!owillg. namely i. onepresumes nO! only knowledge of its own po-
data xi' vj' bj of Ihe Je"der j. The approxima-tion to pair interaction only results in the factthat the following vehicle i has no informa-tion of a possible change of acceleration ofthe leading vehide i in the future at its dis-posal. The DYE i presumes in its adjustmentof its own driving behaviour a COI!5talll-lIcce!erv.lirlll ill lime of the DYEj.
Fii!2.0ne-lollecar.jill/owinx sill/minn, DCE
'i'foliowIll,,/ewlnj"IT] IT]
General]ytheproblemofascerl<liningtra-Jectories could be solved if one was ahle tofind a dosed functionj,(x" .., Xi' ..,x,. VI'.
Vi' ...V", b]...,b" ..,b" t) for the accelerationh, ofthe lth vehicle depending on the constel-]ation of all II"vehicles and time t. The deci-sive point of view why such a function cannotbe defined sensibly is4 that traffic cannot beseen <1Sa purely mechanica] system, but thatit is more an interaction between physi<::s(equations of motion) and psychology (per-ception) since man is an important link of the
vehide system as a whole and t]UlS cannot heignored
Mkroscopk single.]anecar-following modelWe now <lltemp! to develop a psy<::ho'physi-COlicar-following model which, at least.partially determines the functionsj,-'. Initiallya single lane without oncoming traffic isconsidered. The characteristic propet1ies ofthe driver lper<::eplion. intention,..) and thevehicle (braking and accelerating per-formance) are implemented in a Driver-Vehicle-Element (DYE).
FiJ? J. Me'i,mreJ tlQw-deflsllvrelaririll./imda"ientaldilli!riml,Taken /rom tal'iern' andSI<'Ie/11'11Id'.
lOOO
~ 1500a.,~ 1000
500
o
The further consideration takes place inthe phase spa<::e(difference of speed, dis-tance) of the fol]owing vehicle ire. the lead-ing vehicle j. As is known from physiology.the movement of an object can only he per-ceived when the reflcction on the retina has toexceed a cert"in minimum speed threshold,This threshold, on the one hand. individualwhile on the other hand it decreases withincrcasing perception time. As of now,however. one should always presume a time-constant parameter. Simple geometric con-siderations thus give a perceplio/l thresholdfor the realisation of a difference of speedilv:=l-j- ~'ibetween the DYE i andj accordingto:
!"J.VPiN/I'IP= :tkl'lNIPTI'(ilx - Auf :tj;t'J.x:=Xj-Xi
The threshold kl'TNll'TI'for the minimum ve-locity with negative differences of velocity(PTN). i.e. decreasing distance, is smaller
(ilv < 0 signals danger) than in the case ofincreasing distance (PTP). All describes the(gruss)stiUstanddist"ncej"an 'offset'.
10 go 100
317
"'[
'00
-c 00
S-,
"00,
m
'"~
oc<
Be,lde, Ihl' Ihrc,huld, reg,lrding the dif-['erence (,fvelocilie, .i\'orb'Jlh [)VE I and}
there arc olhcr, which ,m~ inlruduced h' Je-,nihc the driving behaviourre Ihe di,tanlT.il bct\"een the vehide" The rull' ,tdted h,thel,lwthi!tthedi,lanl'elnmeUe,tobl'll1;!in-tained ,hou]d hc halflhl' ,rced me,\,ured in
I"m/h sh,)\.\.n on lhc specdnmclCf is 'Udl OJvariable, \.\.hich is rckrred tu ii, the desiredtlisl<ll/c(', ,\bbn.~vialcd here ;!sAn. Th]sAI> 1,prc,ul11l'l1 to dcpc"nd 1)11the sp<,ed I'"~of Ilw!'oliowingc;lrlasfulluws:
Th<' h<',tdway T/J i, ],X ,ecnlllh ;Il'l'ording tuIlwah(1vcIl1Cnli(Jnedruk
III the case lIloving iliin the typil',d di,-1;!J1L'e-keeping 11<'havioLiL dl-ivl'f' lhJ nOI f;dl
short ullwadway~ T, with m;!gnilude (15 ,ecGcncrally ,\ drivl'rwi]1 nOltend to I"eel' such a,hnndisl;mcelolhec;lrin front ofhill\ov<,r aInllg period 01 tllne, Thi, ,o-call<,d 'ri,,"\tlisl(llIr'I,',AN. I> l1<'ingdws<,n ,\" a fUllclionoIIJw']1<,<,d I',uftlw]<,adingl'arj:
TJwtypic,libeh;I\,iolirofadriv<,r\\h<,n'l']US-illg in' mal"e, tlw inlmductiun of ,\ furthl:r
di"tancc dqwnding Ihrc,hold llece"';lry a,lung a, the f,I~!<,r-m(wing DVE i (.il'< 01 hi!,- .adi,tar1l'<,.irlaf!!<,rlh,mitsde,ircdi1f).ilw]l] 11()ly<'ts<'<'th<,n<,c<,s,iIYh1l'hangcitsdri-
ving hehilviour;llthollgh illTWY regisler Ihatlhc biding Cdrj drives slov.'er (In]y I11lhecaseof-,teppingund<,rth<'i\l)thr<"lwldwillitbrak<,msLlchamannn,thalthedrivcrwili11<11bll below the ri,ky dislanrc/1N duringlhe br;lking procedure_ If 0111'is hral"iTlg Inumuch. i.e_ iflh<, Jislanl'<' during Ih<,hrabn~- -PWC<,dUf<'is a]v','ays tar abuve the ri,ky dis-lill1l'<' i\!<. the DVE i will Kcclcrdk whenn:cognising a pu,ilive difference uf "p<,<,d(slepl'mg o\<'r Ilw tl1l'<,sl1Old?T? J. H. how-
Fig.l.PI1",\."dia~I'(IIJI"'h"H'JlI('Ulllc('''rcl(,I/'''I'illgl),,ir:VIS_ f'll' am! AIr An. A\, AR. {khills ill 1"1'1
<,v<,r, Ih<, driver I dwvc righlllP 10 the ri,l"ydi,tanreAN.. ,I positive acceleraliun is onlythen chm<,n. wlwn a dislance considen~d toh<, "ale (al ,I re~JwrtivcJy LIJ-gl~r.p,)sitive dil'-f,'reIKe \11''!,c~l:d i'l\'> j,1'VII')' is reached_ ThisIhrt'"hold. inlhc' f'JII<Jv,ing de"c'rihed as \lill'tli'TwiCe, liS - dnalogoll" to the d<,slred Ji,-t;IIKeA{)
VIith a h<,atlway I, mth<, 1l1,lgnitLidc()f()tle,en)Jld,ls\\-rilknas:
Th<' pha"c "pilce i, subdivided by the,e I'ive
Ihre"hnld, PLY I'll' amJ .\/), it\. AN intu!'(1llrresp, fivcdre;l,(,eeFigJJ
-f)(/n,~('I-- Di,tance tu the l<,ading \1'-hid<'J ,mall<,rthall the ri"ky di,t,uKe,\/(, deceler,ltionn<,c<'-'S,!rV.i\'Si1N
.il'l'l,d and (i1N <j,r <;;II{)J- FolloHillg II: Areil len or I'll\-' ami
,\ho\'l' /11), no need j"Jr ;!l'lion (211' 'S.i1'I'I"Jand(i\f)<.irJ
- Free drillng Th<' aJ'<,as right ul PT;V;tnd ,lbo\<,/j{) r<,sp, right of PIP andabu\<,tlw "ilkt\ diq,ln(l:i\S, IIninllll-enL'eddrivlT\g [1.11- > j,\'I'IA) ,\1ld (A{) <.i\JlorlkV>j.I"'lr)and(;\S<.i111
-F()I")\,'il/,~ f: Rem,limler uf tJw ph,I"<''p;lcc:noneedloractiol1lij,]'/'I,'<.iv'Sj.1'I'II,J ;lnd (i\!( < j" < ,H))/ or [(.11' <':.i1"'II.J and {.il <ASi!
Not OIl]V for rl'aS(1IlS utdcm,lndforll1l;he;uJwavs
l)lle l!lll"t
The rl~;I'UIl i',)rdividlnl,.': Ihe I,Jlas<, "pace inlUIhe [lamed are,l, I~ th,lt bv intruducin!! tll(:s~. c
Ihre"IHdd" and ll1abnl:' U\C ntlhem (TCII/ICI[}he defined d-' inlcr,~cli"I1' olthe traj<'clury 01I)VElrespih]<'ddingL.ariwithlhe"ctllresh-
I~ \jI
~
i
j
1
. o ,
~8
olds_ El~h fJVE thLl~ cakulat<" from it' rre-,enl state Ih<,n<,xt poi1ll in lime. at which it
mus( choose ;lIlCW ;Icceleratiull Ra,eJonlhi, pnKedure slJch mudels af<' d<'~l'1'ibeJasbe])\1,.':n'elll-1I1,I1'/'("Ie. Thereby Ihc dC<:~tcra.liun i" alway, ,ldJu'-Ied immediately withinthe ;Ired' (1;lnger. free drivlI1g ur clu,ing inIl'\pectedeventsl \\'h<,n moving int(\olle0)lhe fullDwingarea, nochang<,ofaccelerationlake, p]al'<'_ B<"id<,~ the expcrted <:ventilhere are also ull<'\j1ed<'u ()ill'S: a fol1uwingcar i which i" ,dre;ldv br;lkiIH.' re,lch 10 ~. -l'h,lnge oj accl'kr.lti(111 ()f lhe 1eilding CdrJ('nh,\1ll'r,lreadi(11l!illle
A'alrcMlvmenlilHled,lhecIH)il'<,\,faecd-l't,\lion v.hiht cnlcring 1J110Ih<, area ofc]n\-illginl'go;tlhraking"Ji"uk<,nwilhthcaimofonl)ILlsltl1uchingtheri"kydi'lanc'eAli.\dwl1 thl'''J1l'cd, ufhn!h Icading and IDlluw-
wI; l'ilr, ar<, identical (i'lr= 01. P;lr;ltnet~r 11""'1(cIPP]'()"'{)'~ mS')i-,re"]1nllsihk forlhei~ad.<'Lju,\lc cunlwl over !h~ ,lcceleraling rest'br,\k.ingpclb],v.'herclhi,parameterd<,mamba dcrekr;ltiuli of at lea,l-h""II' \Vh<,nt'nteringlhe ;lrei! 01 Ire<, dri\ing al'cTkration take,p]al'<,withglwnva]ue".Thnefureth.:aci.:d-eratinn, uf Ilw DVE are p.lr!ially n)jl,1iI~t
fUIll'tion-' in limeSinre thl: maximulll C,IP;lhl<,deL'<,I<,ration
is reslriclcd (appro,,-_ 6 m,:) and all adJust-lIlenl(1l1lytak<,~placewl1l'nentninglhcilr~iI01 closing in. problem-' ari,e in the GlSesoflarge negillivediffer<,nces()f,p<,<,J,orinthrGise 01 strongly d<,cekrating leading \'chi-cI<,sj. Thl:!'[(1hkms;u-c:the 'spilce'ofafoJ-lowingvchiclcincl:dedtuhrakedowrttl1lherisky di,LII1CeARinl'r<,as<" noton]yrecipru-l'ally 10thI' (stiIlJpo-,siblc,capablcdiffcrcm:eufacl'<,]<,r,ltionandonlheolherh;!l1dre[,rliveto lhe dilTcrence ul ,peed exisling at th<,be.ginninglJI'thehrakingI'J'Oc<,s"Llllar<,d,Sincetl1<'J<,si]'<,ddl~tanct>i\f)-causinglhebrak-ingpn1l'css-depl:nlhuI11y lineally on Ihedillerence ill speed. n)lli,ions as a re,ultola'peed r<,dLlctlUn happ<,ning too ]al<' whilstdriving ,\1 high speeds <lrc illevil;lbk, Thisprohlem is ,olved hy inlwducing anotherIhre,hold_ call<,d "mkill); dislulIl'I', AB,
i'll" __,\','([h
:!.h",
+/Jjfor
((i'lr<()Jimd{.ioAIJJ)
in the 1"oIIU\\lJ1g manna: thel'losing-irt;\realS<,xl<'l1d<,dhythe,cP,lrtsoflhelr<,<,urivirtgand j'olluwill1! II areas, which Ji<,abov<,thedesircd di,t;lnce AI) al1d to the Idlll thehraking di,tanc<, AR (Fig 3), Thi-' lllodeJJingcaLise, an unsl<'ady change oftbe followingand Ir<,<, driving afe,IS. when Ih<, ]ead<,r jchan!!cs hi, aCCl:kr;lti"11. Rv this. a vehiclci,- .initi;1IlyJUCilled in Ilw fnl](1\\iing ;lr~iI willjump,o'de~ply' intnt)J<'cJo,ing-in,!rea-by an appn1l'ri,ltc change of acc<'l<,ralion of
Lane-changingThe dearlv diIT~r~!lt luncti()[wl a,se"mentoflh~ right anu lelt lane 01 a t\\/o-lane high-way -;IS <Itleast valid in EUl\1pe - 11L'ce",i-t;lte"dso adifferentiati(1!loflhetwu Kinds oflane-changing. Thus tlK: mutivation r~ th~'truc(Ureof,uchach,mgingoflancsfromth~right travelling lane onW th~ left ov~rtakinglan~ (right-hand ruk ofth~ road) i~ ohviou,lydiffn~nt to that of a chaneI' from the left toIherighthine.
\!..'ithout lo~ing oneself in d~(;lils or th~complex set of ru!cs when changing lane"mcrely the h;lsit:id~aof'Llchastructure is toh~explained. Sticking to the ideaoftlK: next-neighhour interactions of the microscopicsingle-lane car-fol1owing model. ;lg;lin onlypail interactions ;H'~ b~ing con~i,kred whench;mging lane~. Of the fuur interaction pair~existing on a two-hme higlm,'ay. . . (directand n~ighhollring leader and follower) threcarebeingeonsideJ\:dinthedeci~iveslnKture(Fig-+). When ov~rt;lking from th~ right to theleft lane, these are the DVEj. k and I, whenchanging back from left to right the dir~cIleader} is not heing consider~u Th~ d~ei~ivelogic i, hased on the location of the resl'ec-tive pha,e ~pace.
The desire of Ihe DVE i to chan!!e to thehcftiscausedbytheDVEjinlhatitwilihin-
del' DVE i predil'lahly Th()~e direl'l neigh-hoursk anJ Idrivmg in thelelt ];mecan, <lC-cordin!! to thcir con'l\:II.nion. en,1hle urprevenl Ih~ illl~!llied t:himge uf lane or evenletilhappenatadifferentlearlienjJointin
time. The condition olthe Cat driving aheadon the left lanek del'ides whether a ho:ncfitarises re to the' leader}. tl1c constdl;1tiun he-t\'/een DVE I and i must in anv cit,e be sofavourable. th<lt the left follower! will heh,mdic,1pped ;1t must ;IS sev~rely ,IS DVE ; itthat w~n: not to m~rwke.
On the other hand. the desire of changingh,!Ck from left tu right is rnotivateJhythetwodirect neighhuurs lollo\\'ing DVE i, namelym Uheoncjo.,tlingJ and !(theonejust havingovertaken). The pmsihility of ch,mging h;ICkis then giv~n. \\'h~n the potentially changingDVEi can follow (pn1hahly) the future leaderk on the right over a given period of limo:\\"ithout haviI1\'. to clwn!!e his an:eler<ltion,c c
Whenever;l <:himge 01' lane is n~ith~rpos-sible nur h~nefici,d, DVE i heh,lv~s in ref~r-~nc~ to his leaderj according tu the micro-scopic ~ingle-Iane-car-foll()wing Im1dcl
Thi, hrief description of the lane change~hould he sufficient 10 ijJustrat~ the nm-sciously modelled ;lsymmetry.
Boundan conditinnsIn traffic engineering one generally worKswith 'Iine,u sections'. i.e. vehicles are heingproducedr,mdomIY;llthebeginning()faslll1-ulation sel'liol1 and are taken out of the svs-tern at the end of such ,I section, The sectiol1u,ed tor measurements must be silll,11~dfarenough fmmhoth \~hielesoureeandsink toreduce their disturbing innu~nce on the traf-fic flow to;m 11l1;IVoidahleminimum.
SIKh prohlems are expelled immediatelyonee a dosedsvskm is used instead of suchan open 0111',For this reason tlwcol1cept offwr;Oific INllIIldllr\'nllldirilllll as Known fromphysics is adapted, The last UVEA! (H1lh~ re-spective lane is being assigned 10 the v~hi<:1e1(numhering with d~cr~,\sing eU-(1fJinatej asle;lder Figur;ltiv~ly one views a ~ystem o/;'I/v~hit:les un a dosed cirde without sourcesandsinKs(withoutconsideringthel;ltera] dy-namicsj,Thep<lramelers.i.e.llun1her()fvehi.cks N. number of lanes I( and cirt:urnkrenc~of the circle l!. delcrmil1e!hegJobal,ma(T()-scopic density p:
As compared to open systems, which de-mand as boundary conditions the in- ,md out-going tlow q = p . v, the circular track merelyd~mands the definition of the mean density p;th~ traffic t10w q adjusts, Obviously, the peri-odic boundary conditions are to he prderrediffundam~nta] findings arc predominant,
Besides the maeroseupil: parameters NandU one must g~nerally ddin~ a set of parame-ters for every single DYE, char;Kterising thevehide (max. hraking ability h,",,,, vehicle]engthAo. free acceleration, ...J and the driver(perception threshold kPTPIPIN-headway 1~,.,,). hl:an be seen, however. that the modeltends to a dlaotic h~haviour despitel:om-pletely identil:a] DYE du~ to the high d~greeof non-]inearities, Ana]ogous to the argu-m~ntation of th~ periodic houndary l:ondi-tions th~ simulation \'.'ith identical DYE does,however, have the bendit thilt it can be intcr-preted considerably more simply so that as ofnow the same mi(.:foscopic s~t of parametersis defined to he valid for all DYE.
ResultsWith this microscopic two-lilne car-follow-ing mode] different simulations have beencalculated. which wer~, by means of statis-til:s,post-prol:essed for the reason ofma(,To-scopic statements.
Here two examples are to b~ introducedwhich both hase on simulation runs withidentil:al DVE on <t two-lane circle. In thefirst example, the circumference U W.IS var-ied in view of l:alcul:l!ing fundamental dia-grams: in the second example, the effect orthe l:onsciously modelled asymmetry in lanechanging behaviour is illustrated.
Priorto this, one should mention that in itsstrid sens~ no validation'" of the modd hastak~nplacesofar. Theresultsofthetheoreti-C.l] t:a!culations do not merely show the h~-haviour l:Orresponding qualitativdy to rea]-ity, moreover they .lr~ quantitatively good,To ensure a correct parameter adapta _respel:tively physintl proceedings - an ~x-perimcnt adapt~d 10the model should he car-ried out (e.g. 'real' l:ar-following, circulartrack, driving simulator, ".). Up 10 now thevalidation is based on relative]y rough adapta
to measur~d data of different highwav sec-tions7. - -
*Especially the premise of identical DYE couldnot be kept valid any longer,
320
]n fig 5, th~ fundamental diagram is shown
- here the dep~ndenee of mean speed ontraffic volume for a homogeneous, two-lanecircle. ThenumherorvehiclesN(N=] ])waskept conslant. the externally defin~d densityp was varied hy means of differ~nt l:ircum-ferences U. The basic set ofdata,dderminedover 30 minutes r~spedive]y, was mademore dense by averaging re time (two-minuteintervals) and position (the respective l:ir-l:umfcrence U). CJear]v visible arc the char-acteristics which the fundamental diagramshows,namely:(I) Amhi!{uity: for each traffic !low q
(within the set of values) there are twodifferent levels of speed: in the nlse oflow-density p few, but quickly-movingDVE lead toth~ same vehicle !low If as is
the case of high-density p with lots ofDVEofs!ower speeds.
(ii) Meal! dt'l'illtion: The stability of fre~trattic, Le, of traffic with low density p,can he seenbvthesmall amount of meandata deviation within the upper branch ofsp~~d of the fundamental diagram. Onthe oth~r hand the instability of thebound re paniallybound traffil: ]eadstusUl:h a predominant mean deviation thatthis pan of the fundamental diagram nmhard]y be interpolated sensihly.
As a l:omparison with real measurcment (Fig6) shows, the model does not offer only qual-
itative]ygoodresults.Th~ following situation could be seen
\v'hen observing the dfed of asymmetry onthe structure oflane-ch.mging: on a two-lanel:ircle with t:ircumferene~ U == 1000 m 22identil:al DYE drive, wherehy as of co-ordi-n<lt~x = ]00 m the right respectively the leftlane is dosed over a lenglh of 200 m. Startand end of sUl:h a bottlen~l:k were simulatedby a DYE standing still; the inhomogenitywas not announced. The data of both 45-minute long simulations were th~n averagedover five-minute intervals, for the sake ofmore clarity the means of cad] variahle wereaveraged once more. Th~ position disp~rsalwas 25 m, esp~l:ially to enable more detailedstatements in the vil:initv of the bottleneck.
Figures 7a and 7b -show th~ position-depending densities, separated into the right
('*' symhols) and left Co' symbols) lan~, aswell as the meanofhoth (p,+ pt)l2 C+' sym-bo]s), In fronl of the hOlllcneck there areh.lrdly any qualitative difkr~nn~s: on hoth
the right and the left bottleneck c.lrsjam onthe closed lane and on th~otherhand the lane
reduction also leads to a strongly increaseddensity on the free lane du~ to DYE corningover from the closed lane, The relaxation\.1fdensityontheone-tral:kedsection(]OOm<x< 300 m) is analogous in both cases. Notice-able difkrenl:es occur aftcr the hOllleneck:while the re-openingofthe lane on the righl(as driFin!{ lane) leads to a quick adaptationof the density of both left and right lane, thisrelaxation takes nmsiderably more timewhen re-opening the !tjt lane (as overlakingbne). The explanation of this phenomenonlieswithintheasymmetryofthelane-l:hang_ing. Through the botlleneck on the right theDYE are forl:ed onto the overtaking lane.After the hottlcncl:k the right]ane is hardlyol:l:upied sn that the DYE can change hackimm~diately: the vehieles distribute them-selves very quickly over both lan~s. This isdifferent in thecaseofa left bottleneck: sincethe desire to l:hange tn the left only takesplal:e when a hindrance through thc leadingcar can be predicted, the DYE, however, ae.l:elerateafterthebonlenel:kmoreorless;lsawhole, lane change seldom takes p]al:e after aleft bottleneck. Thus. an even distribution ofthe DYE onto both lanes only takes placerelativelv far behind the bott]eneck.
Using th~ l:ourse of mean velocities v _notationasahovewiththedifkrenl:e,thatthe'+' symbols arises from the weighted meanp,Vr+ PI I',of hoth right and Idt lane - anp,+ piimportant difference belween the left andright hotlleneck can he s~en (figs 7c and 7d).The ]evel of speeds - esp~l:ially in the areaof lane reduction - is reduced less stronglyin the case otelosing the right lane as in thecase uf dosing the left lane. The reason forthis again lies in the asymmetry of lane-changing: a hindrance on the lane causes thedesiretoovert<tke'timely',whil:hcausesanmneased hannonisation of the traffic t1ow.On the other hand, a DYE on the ov~rtakinglane is not deliberately motivated by its Ie,HJ-ing car to change lane in the case of ahindrance, whereby more frequently DYEonthe overtaking lane cometn aSlandstill andthey then lower the level of sp~ed whenchanging over to theoth~rlanerespectively.
"1000 m. The hOlllelleck i.' moddl"d hy dosing 0111'lillie re5l'ecli"eIy III
IJ(l.I'iliollX = 100 m ()\'l'r a /nlSIIt of2()() ,,'- The datil are iH'cragu/ 01'('1'ililen'a/.\- Of/l'IIRI" '!f25 m alldperio,[of.J5/11inwe,l,(a)illld(h):Dcmiryp(.r): '*' symhol5, riglil/"''''Pr 'o'IVlJlhols, Iefl/ollepl: '+'sYl1lho/5.l1Iewldc//51r1'(p,+p,)12(e) amlld). Velocily \'Ix), NOIaliOlI aI iil (a)al1'/ IhJ- The mewl \'docily
Both charactcristio::s.lhc lov..-cr (reducing)intlucno::e on the mean speed bcfon: andwithin the bu(tleneck as well ,tS the 4uil"kerrelax<\tiunufdensitvbehindthebottleneck incaseofaclosingtherightlaneasopposedtoahindrance on the lett lane, can clear!y be seenwhen observing thc respedive position de-pending tlow q (figs 7e and 70. The explana-tion of these phenomena by the asymmetrywithin lane changing behaviour is strength-ened by their intluence on thc frcquenl"Y oflane changing (Figs 7g and 7h).
ConclusiunsThe o::umparison of the case-studies betweenright and left botlleneo::ks has dearly shownthe effect of ass em me try on "me-changing_furthermore and especially, however. firstcono::lusions of the drawbacks and benefits ofthe <lddition of another Jane either on the right(as driving lane) or on the Icft (as ovenakinglaneJcou!dbe dr<lwn.
jfonc is interested in maximum safetv. i_e,in low c<lrdcnsity re an even distribution ofcarsonbothlanes.Figs7ato7fclearlyfavourthe idea of adding another lane on the right.
Gl:nerally this exampJe of both boule-neo::ks shuws the potential of the microscopictraffic simul,ttion as opposl:d to the macro-scopic procedure,s, which in turn have thebl:ncfitofshortercaJculatingtimesandlargeraesthetics_ Neverthdess the macroscopictraffic theory does after al! depend on the factthat the microscopic simulation places ,It itsdisposal fundamemal diagrams, especially inlhl:l"aseofclcctronicdrivingaid,_
ACKt,,'()WLEDGMENTSTilisopporlllllilV i,\'lakl'lI to <'Xl're:Js(IIIc.lCl'pli(JI!IIIgmlilllde10 Dr..lng. P. KOJllwellsn. ,,'/wr!id/!(il01l/Y/Jrol'idc!>"l!l'l'ol"lwillthi.>1IIIfailillgpo,I'ili\'l'at-Ii/wi" /() di,I'C!l.Hi/J/l hw aiio ,,'ilh !!IallY concep-liOlw/ ('Nllrilml;,ms_ TOKl'Iha ,,-i/!rProf Dr R. S,Kemcr hI' {'II'w"cd <I,I'cjell/ific pel'spec/i\'c 4/wr(i('/I/l'ory_
Thefimll1cjtl/.",pport {~f1h" HAl FT is grll!ejiil/y(I('kl'ow/n/ger!.
REFERENCES'L\GHTHILL,M. J, ~nd G. W. WITH.-\M_On kinc-
11'1I1UWS,W. E. Kinetic modeJ fur tralIi<: nuwU.S_ Depal1meIH of Transportation. Wa<,h-inglon,OC.J,)77.
'KOIINE.R. Traffic 110w'on maJorruads, Phys. BI"470),1991,201-204(in(jcrman)_4HLRMAN,R. and R. W. ROTHERY.Car.following
and stcady-state now. Pmc" 2nd /iu, SVlllp_Tllcory of Row I Traf(ic Fit"" (t96.h OECD.l'aris.IlJ65_
'WIEDEMA!\N,R_ Sinllltalion of road traffic nowReports or the Institule for Tran'ipol1 andCommunication Universily of Karhwhe,is~ue8, 1974 (in German),
