Transportaton Spatal deelopent of rban road networ traffc grdloc H.S. Q, Y.Y, Dan Ha Wang, *, Y.M. Be Receed: October 3, Resed: Febrary 4, Accepted: Jne 4 Abstract Abstract: Grdloc s an extree traffc state where ehcle cannot oe at all. Ths research stdes the deelopent of grdloc by theoretcal and nercal analyss. t s shown that the deelopent of grdloc can be dded nto seeral stages. The core of the deelopent s the eolton of congeston loop. A congeston loop s coprsed of a nber of consectely connected splloer lns. The eolton of a congeston loop always tends to be stable,.e. the state of all related lns tends to be dentcal.. Under the stable condton, traffc states of all lns are dentcal. A noel concept, rtal sgnal s proposed to descrbe the qee propagaton and splloer drng the stablzaton. Slaton reslts show that congeston propagates n an accelerated way. The preenton of the frst congeston loop s crcal. The acheed reslts hae potental se for ftre networ traffc control desgn and feld applcatons. Keywords: Vrtal sgnal; Sgnal cooperaton; Traffc congeston; Traffc control.. ntrodcton Grdloc s an extree traffc state nder. Under grdloc, no ehcle can oe forward. n rban networs, grdloc s often characterzed by soe consectely connected congeston lns. n ost cases t conssts of soe loops. The tgaton of the congeston of each ln wthn the grdloc depends on the downstrea ln wthn the sae grdloc. Hence at last traffc congeston of any ln cannot be cleared qcly. The foraton of grdloc s nflenced by aros factors, ncldng networ topology, sgnal paraeters, flow strctre etc. The deelopent of a grdloc noles the eolton of the traffc state of each ln wthn the grdloc. Ths t s ery portant to exane the ln traffc dynacs n the context of networ traffc flow fraewor. Wrght and Roberg (998) proded the frst nsght of ths topc. Ther research was slaton-based, wth an analytcal ln odel. n the ln odel, ahead flow and trnng flow were constant. Propagaton te of a downstrea obstrcton was the fncton of the flow rate. Accordng to the relatonshp between throgh flow qee propagaton te and trnng flow qee propagaton te, spatal strctre of the a was otlned. Ther slaton ethod was followed by any researchers, wth dfferent obectes. * orrespondng athor: wangdanha@z.ed.cn ollege of l Engneerng, Zheang Unersty, Hangzho, hna School of Transportaton Scence and Engneerng, Harbn nsttte of Technology, Harbn, hna Roberg and Abbess (998) appled a slaton odel to nestgate the dagnoss and treatent of traffc as. ong (8) explored the foraton of traffc a, focsng on the nflence of stop lne assgnent and channelzed secton length. Roberg-Orensten et al. (7) deeloped seeral alternate strateges for protectng networs fro grdloc and dsspatng traffc as once they had fored. The treatent focsed on the nstallaton of bans at specfc networ locatons. Dfferent fro aboe researches, Daganzo (7) arged that the odelng and control of the grdloc shold be establshed n an aggregate leel,.e. traffc dynacs shold be analyzed throgh aggregate paraeters. n hs research, the relatonshp between aggregate ehcle nber and syste otflow was assed to be nodal. f the ntal acclaton was on the decreasng part of ths relatonshp, then acclaton ncreased at a rate that tself grew wth acclaton,.e. grdloc too place. Howeer, the relatonshps between these aggregate leel traffc paraeters are not clear yet. Besdes, the two assptons,.e. slowly eolng deand and eenly dstrbted traffc state also shold be reexaned especally drng pea hor. Grdloc conssts of a seres of splloer lns. The nteracton between adacent ln qees plays an portant role n grdloc deelopent. Kerner () stded the grdloc foraton n a sgnal controlled ln. He showed that een nder sall nflow, when no traffc grdloc shold be expected, spontaneos traffc breadown wth sbseqent cty grdloc too place wth soe probablty. Dfferent fro Kerner s approach, we adopt a deternstc way whch descrbes the qee foraton and propagaton based on traffc wae theory. We focs or research on the nteracton between lns, nternatonal Jornal of l Engneerng, Vol. 3, No. 4A, Transacton A: l Engneerng, Deceber 5
rather than wthn the ln. We hope that ftre eprcal stdes can sole the proble: whch of the two approaches s ore assocated wth traffc grdloc n cty networ. There ay exst any solated grdlocs sltaneosly and the tal effect of sch grdlocs also contrbtes to the dynacs of networ traffc a. None of these lteratres aboe prode sefl nforaton abot ths topc. One reason aybe the dffclty n splloer odelng (hen, Sh et al. 9 ). The splloer eent s the ey for the deelopent of grdloc. t conses the capacty of pstrea ntersecton (K and assdy ). The balance between the deand and spply of pstrea lns s then enhanced. The resltng qee propagaton s accelerated. Ths paper realzes the splloer effect odelng wth the concept of speed of rtal sgnal, whch s deterned by traffc deand (pstrea arral) and spply (ln capacty, whch depends on sgnal splt). Another reason ay be the a eolton coplexty. The a eolton s nflenced by aros paraeters (Jn, Hang et al. ). Pre analytcal reslts can hardly be obtaned. rrent researches on one hand splfy the odeled syste (orthot, Flötteröd et al. ); on the other hand, slaton approach s appled. By the cobnaton of analytcal and slaton approaches, soe nterestng reslts abot grdloc deelopent are fond. All of these are realzed throgh the concept of rtal sgnal. The core dea was that qee eolton can be nterpreted as the propagaton of rtal red sgnal. When the qee occpes a specfc locaton, the rtal sgnal of ths locaton shows red, otherwse green sgnal lasts. Splloer can be odeled by the sae way. By applyng the dea to the networ leel, we show that the echans of the networ traffc a eolton presents a ore coplex natre. Secton lsts all related notatons. Secton 3 prodes the ln traffc odel, whch s based on traffc wae theory. Splloer odelng s also presented. Defnton and descrpton of relatonshps between congeston loops are gen n secton 4. Secton 5 deres the stablty condton both for sngle congeston loop and congeston loop grops. Based on the analyss of congeston loop, secton 6 otlnes the steps of the foraton of networ traffc a. onsderng the analytcal dffcltes, Secton 7 explores the grdloc deelopent when ln nflow s arable based on nercal slaton. Secton 8 concldes wth soe rears.. Notatons q - capacty of each lane, s short for axal ; - a densty; w - bacward wae speed; - free flow speed; f l - axal qee length wthn a cycle; ax G ( N, A) - the whole networ, N s node set and A s ln set; G - sb-graph coprsed by lns that belong to a congeston loop; G - copleentary set of G ; G - sb-graph coprsed by splloer lns; F G F - copleentary set of G F ; 3 {,,,...} - congeston loop expressed as a seres of consecte ntersectons; - the -th node at congeston loop ; - the -th ln at congeston loop ; - splt of the -th ln at congeston loop ; - pstrea node of node + + 3 {,,,...} - pstrea ln of ln ; ; - congeston loop expressed as a seres of lns; t R { R }- all rotes fro node node node ; - stable splt of congeston loop ; ; N - ntersectons nber n, N - lns nber n s - length of ln a; a s 3 - length of ln t 3 ; N N ; to s - rote length for t-th rote R t fro to node ; - sgnal cycle length; 3. n Traffc Model 3. rtcal condton for oersatraton qee onsder a one-lane ln controlled by sgnal wthot endogenos flow. The length s. nflow rate s q and otflow rate s q o. Red te and green te are r and g respectely. Asse that the fndaental dagra s of tranglar shape. Arrng flow s q and the correspondng densty s. Fg. Error! Reference sorce not fond. presents the foraton and dsperson of ehcle qee behnd the stop lne nder sgnal control. Green te s g and red te s r. At frst (the begnnng of red te), stoppng wae (lne OB n Fg. Error! Reference sorce not fond., whch represents the qee tal) propagates pstrea wth the elocty and a qee deelops. When the sgnal swtches to green, a startng wae (lne H.Sh. Q, Y.Y, Dan Ha Wang, Y.M. Be 389
AB n Fg. Error! Reference sorce not fond.) appears and propagates pstrea ether wth a greater speed (denoted as ). After te t, the startng wae catches p wth the stoppng wae and the qee has dspersed. A new wae (denoted by ts speed, ) coes nto beng. t taes the te t for the wae to propagate oer the stop lne. f the effecte green te g s eqal to or greater than t t, traffc state wll reprodce cycle by cycle. Otherwse, the wae propagaton profle wll ary fro cycle to cycle (see Fg. ). Drng the second cycle, stoppng wae frstly spreads wth speed whch aes the qee tal of ths cycle frther fro stop lne than the forer cycle. As te elapses, the qee tal becoes frther and frther. n traffc state s nstable. o l ax r g A t t t B x Stop lne r A B t Fg. Traffc wae propagaton nsatrated condton; oersatrated condton Based on the analyss aboe, soe forlas can be forlated: q q q q w; ; r ( r t) t t () r lax t (3) ax t (4) f () g t t s the crtcal condton for oersatraton. Ths eqaton deternes a specfc sgnal splt, whch s defned as the dson of cycle length by green te. The decrease of wll reslt n oersatraton. 3.. Oersatraton and splloer odelng When crtcal condton s not satsfed, ehcle qee wll grow as shown n Fg.. Each green sgnal shortens the qee length whle each red sgnal prolongs qee length. The length shortened or prolonged s proportonal to the te draton, becase wae speeds are fxed. Sppose h and h are qee length ncreent g r generated by green sgnal and red sgnal. Accordng to the geoetrc relatonshp n Fg. and (c), they can be dered as follows: h h r g r g Stoppng wae speed s the fncton of pstrea arral. Under nstable condton, qee eolton s deterned by ln sgnal paraeters and the nflow. l r g h r hr r g r h g l g l r g (c) h g Fg. Stoppng wae and startng wae wth nsffcent splt Vehcle qee eolton drectly reflects the traffc a deelopent. A noel concept, rtal sgnal s proposed to descrbe the qee dynacs. The core asspton sggests that there are (rtal) traffc sgnals eerywhere wthn the ln. For a specfc poston, f t has not been occped by ehcle qee, ths rtal sgnal shows green, otherwse red sgnal goerns the poston. The propagaton of ehcle qee s eqalent to the changes of rtal sgnal both n spatal and teporal denson. Fg. 3 presents the dea n a ln wthot ntal qee. Two locatons are chosen for descrpton. Drng the frst cycle, qee tal already spreads oer locaton, ths the rtal sgnal attached at locaton shows red. Bt rtal sgnal of locaton always dsplays green n the frst cycle. At the second cycle, the red te of the rtal sgnal at locaton ncreases, and the rtal red sgnal deelops at locaton ether. Ths the araton of rtal sgnal relates to the qee propagaton, whch depends on real t (5) (6) 39 nternatonal Jornal of l Engneerng, Vol. 3, No. 4A, Transacton A: l Engneerng, Deceber 5
sgnal paraeters and arrng flow. n realty, cycle nflow ay ary, whch leads to the generalzed forla: SVS ( ) () ( ) ( ) ( ) ( ) ( ) [ ] (3) where () s the stoppng wae speed at cycle. SVS() s the speed of rtal sgnal at cycle. Fg. 3 Method of rtal sgnal Fg. 4 presents the wae propagaton wthn a typcal cycle. Three ponts of the qee tal, naely x, y, z are labeled. For any locaton wthn x, rtal sgnal paraeters are dentcal to that at stop lne; for any locaton l wthn the nteral [x, y ], rtal red te s copted by: y l r r y x For any locaton beyond y, rtal red te s zero. Hence the three ponts represent the traffc state along the ln. They are defned as characterstc ponts, and can be recorded terately: x z y x hr z y h g The elocty of qee tal propagaton s called Speed of Vrtal Sgnal (SVS). SVS taes both stoppng wae and startng wae nto accont. Drng a cycle, the qee length ncreent: l h r h g (9) Therefore: r g l ( ) SVS SVS [ ] f ( q, ) (7) (8) () () t can be seen that SVS s a lnear fncton of sgnal splt. Qee tal eqaton of Eq. (8) s: x x SVS y y SVS z z SVS () Fg. 4 Vrtal Splt alclaton When the qee tal reaches pstrea ntersecton, splloer eent taes place. Een f a oeent gets RoW (rght of-way), t stll cannot traerse the ntersecton. Three types of sgnals at the ntersecton: rtal sgnal, dsplayed sgnal and operatonal sgnal. The ltate flow dynacs s deterned by the operatonal sgnal. The operatonal sgnal tself depends on the forer two types of sgnals. f both sgnals show green, the operatonal sgnal dsplays green, otherwse red goerns. n other words, operatonal sgnal s the reslt of the cooperaton between rtal sgnal and dsplayed sgnal. onsder two traffc sgnals. g, r, o s the green te, red te and offset for sgnal. For splcty we set o to zero. Slarly we hae g, r and o. s the cycle length. Accordng to the relatonshp between o, g and the relatonshp between o+g and, sx condtons are enerated, whch are shown n Fg. 5. The cooperaton of ltple sgnals,.e. ltple splloers, can be dered by the sae way. Snce the cooperaton of two sgnals s not green when ether of the s red, the oerall splt st be saller than both orgnal splts. Ths the cooperaton always decrease rtal splt. The decrease s called splt lost. Dfferent offset wll reslt n dfferent splt lost, whch then nflence the dynacs of traffc a. Traffc dynacs of splloer ln s ery sple. The traffc wae speed always eqals. And the traectory of each ehcle s slar to that of the front one, bt wth a teporal and spatal shft. Vrtal sgnal of all locatons are the sae wthn the cycle, whch eans that traffc state of the ln can be represented by a sngle splt. Ths featre s the start pont of followng research. H.Sh. Q, Y.Y, Dan Ha Wang, Y.M. Be 39
g o g g o g g o g g o g o g (c) g r g r g g g g ; r g g ; g o g g g o g o g (d) g g o r r o o g r o g g n{ g, o g } (e) g n{ g, o g }; r n{ g, o g }; n{ g, o g } o g Fg. 5 Vrtal sgnal cooperaton g g (f) g r g r g ondton ; ondton ; (c) ondton 3; (d) ondton 4;(e) ondton 5; (f) ondton 6; 4. Defnton of ongeston oop Drng pea hors, ehcle qee wll propagate arond the networ. Soe splloer lns connect consectely, whch aes p a loop. A congeston loop s defned as a seres of nterconnected lns wth the sae drecton,.e., all are clocwse or all are conterclocwse. All lns st be n splloer condton and hence perfor as the rtal sgnal on pstrea lns. 4.. Measrng congeston loop sze Two congeston loop easreents are establshed: M: ns nber of a congeston loop, MN N. A greater M eans there are ore splloer lns wthn a congeston loop. These lns ay perfor as the seeds for new congeston loops. M: Aerage rtal splt of all lns wthn the congeston loop, M ; fro ln traffc N dynacs we now that rtal splt can express the congeston degree of a splloer ln. Slarly M also easres the congeston leel of a loop. Fro the stablty analyss below, we can see that rtal splt of all lns wthn the sae congeston loop tend to be the sae. 4.. Relatonshp between congeston loops Generally there are for relatonshps between congeston loops n te and space: ln-sharng, nodesharng, xed and reprodcte. n-sharng congeston loops A ln ay belong to any congeston loops. These congeston loops are of ln-sharng relatonshp. Sppose congeston loop and congeston loop are ln-sharng congeston loops. The shared lns set s. Shared lns nber s N. When all lns are G G shared, these congeston loops are consdered to be dentcal st as n Fg. 6. Node-sharng congeston loops When two congeston loops share soe coon nodes, the relatonshp s of node-sharng type. n fact, ln-sharng congeston loops are node-sharng at the sae 39 nternatonal Jornal of l Engneerng, Vol. 3, No. 4A, Transacton A: l Engneerng, Deceber 5
te. They share the nodes that are drectly connected wth shared lns. Howeer, the opposte ay not be tre. Slar to the ln-sharng congeston loops, we hae shared nodes set and the nber of shared nodes N. Mxed type congeston loops Mxed type congeston loops, st as ts nae ples, are congeston loops that share soe nodes as well as soe lns. An exaple s shown n Fg. 6(c). ongeston loop and congeston loop hae for nodes and two lns n coon. Reprodcton The foraton of a congeston loop can trgger the deelopent of other congeston loops. ongeston loop prodced by exstng congeston loop s called chld congeston loop. The orgnal one s called parent congeston loop. t can be seen that the relatonshp between the parent congeston loop and chld congeston loop st be one of the three aboe relatonshps. Mxed type s an extenson of the ln-sharng and node-sharng relatonshp. Ths we only dscss the forer two relatonshps. Shared lns parent ongeston loop Shared nodes (c) chld ongeston loop Fg. 6 Types of loops relatonshp, : ln-sharng; : node-sharng(c):xed type; (d): reprodcte (d) 5. Stablty of ongeston oops 5.. Stablty of a sngle congeston loop lns, Sppose a congeston loop s coprsed by a seres of ln of ln,,,... Defne 3 + rtal splt of ln between rtal splt as the pstrea. Accordng to the ln traffc dynacs,,.e., s the cooperaton and that of soe other lns. Ths. Frtherore, Accordng to the loop shape, t can be conclded that:... N (4) Whle N, hence the necessary condton for the trth of Eq.Error! Reference sorce not fond. s: = = =...= (5) N Eq. (5) s the stablty prncple of a sngle congeston loop. The physcal eanng s that traffc states of all lns tend to conerge to the sae splt. Vrtal splt of all lns ay not be the sae at the begnnng. The congeston loop wll eole gradally. Snce each ln s nflenced drectly by ts downstrea ln, once there s splt dfference between any par of adacent lns,.e. traffc sate s dfferent, rtal splt wll propagate pstrea. Snce the wae speed nder splloer state, s constant, t wll tae the te sa for the rtal splt of ln B to propagate oer ln A. s A s the length of ln A. The propagaton speed of rtal splt wthn a congeston loop s ndependent on the splt tself bt only deterned by the physcal length of the congeston loop s. The stablty te then s s f the rtal splt reans nchanged drng the propagaton. f the rtal splt becoes saller drng the propagaton, the process wll H.Sh. Q, Y.Y, Dan Ha Wang, Y.M. Be 393
restart. The propagaton wll end ntl the rtal splts wthn the congeston loop are the sae. f there s an nner or oter dstrbance of rtal splt sch as an ncdent, a new stablzaton process wll begn and change the traffc state of all lns n a cyclcal way. The stable rtal splt s always saller than or eqal to the orgnal nal splt wthn the congeston loop. Under the worst condton where the rtal splt s zero, ehcles wthn the sae congeston loop cannot oe at all. Grdloc ltately fors. 5.. Stablty of a congeston loop grop ongeston loop grop s defned as a seres of nterconnected congeston loops. Each congeston loop st be n one of the three types of relatonshps referred to aboe wth other congeston loops. Wth respect to lnsharng congeston loops, snce the stablzaton process of a congeston loop noles all the lns, the stable rtal splt of all related congeston loops st be the sae,.e. nder stable condton, traffc states of all lns wthn a ln-sharng congeston loop grop are dentcal. Shared node Fg. 7 Node-sharng congeston loops ln and shared node As for node-sharng congeston loops, frst consder the coon node as shown n Fg. 7. Two congeston loops share the sae node. The lns n congeston loop s and. Slarly we hae and n congeston loop. De to the spatal relatonshp, we hae Eq. (6) and Eq. (7): (6) (7) Accordng to the stablty condton of a sngle congeston loop, the rtal splt of all lns wthn a congeston loop are eqal,.e.: ( a) ( b) Sbsttte the left sde n Eq. (6) sng Eq. (8): (8) (9) Sbsttte the left sde n Eq. (7) sng Eq. (8): () The necessary condton that garantees the trth of both Eq. (9) and Eq. () s: () Slarly: () Hence the stable rtal splt of both congeston loops st be the sae: (3) Slar reslts can be dered abot the xed type. n s, sngle congeston loop grop tend to be stable ltately. The stable traffc states for all congeston loops wthn the sae grop are dentcal. At the begnnng of the congeston loop grop foraton, the rtal splt of all lns ay be dfferent. The grop then wll eole gradally. As long as there s splt dfference between adacent lns, the stablzaton process wll last. Sppose there are two congeston loops, loop and loop n the sae grop. The rtal splt of ln (connected wth node ) s nal aong the grop whle other rtal splts are assed to be the sae. n (connected wth node ) wll then conerge to at least t wll tae for ln. The arsng qeston s how ch te to reach the stable condton. To answer ths, we constrct a sb-graph that conssts of the congeston loop grop G, =, { n } n. Snce the rtal splt always propagate pstrea, the spread rote of rtal splt st be n one of the rotes fro node to node n sb-graph G { n }. The spreadng speed of rtal sgnal s whch s fxed. Hence the te t wll tae for splt to decrease to s t n{ s } f nal rtal splt reans nchanged, where n{ s t } denotes the shortest path length fro node for node to node s greater than or eqal to. Actal stable te t n{ s } de to the fact that nal splt ay dnsh. Therefore for the whole grop, the stable te s greater than or eqal to the physcal length of axal shortest path fro all 394 nternatonal Jornal of l Engneerng, Vol. 3, No. 4A, Transacton A: l Engneerng, Deceber 5
nodes to propagaton sorce node dded by. 5.3. nteracton between congeston loop grops There ay sltaneosly exst ore than one congeston loop grop n the networ. When they are separated,.e. there s no splloer lns that connect the, the stablzaton processes of these congeston loop grops are solated. When soe splloer lns appear between these congeston loop grops and connect the, the saller rtal splt wll spread a these lns fro one grop to another. At last these congeston loop grops consttte a larger, new grop. The stable rtal splt of ths bgger grop s saller than any one of the orgnal splts. f the rtal splt changes (st be saller) drng the propagaton, the stablzaton process wold retrn bac to propagaton sorce wth the new splt, hence dsplays an oscllatory featre. f the ltate splt s zero, grdloc taes place, where no ehcle can oe forward. 6. Stages Of Networ Traffc ongeston Propagaton The rban traffc congeston can be dded nto seeral stages below: a) There s no oersatrated ln n the whole networ; b) Soe solated lns are oersatrated at frst, and the ehcle qees are growng gradally; c) Splloer taes place at soe nodes, whch leads to ore splloers. Howeer, there s no congeston loop yet. At ths stage, the propagaton of rtal splt s ndrectonal; d) Seeral splloer lns coprse soe congeston loops. At the sae te, these congeston loops conerge to the stable condton gradally, whch s accopaned by the worsenng of networ traffc state; e) Exstng congeston loops soon trgger ore loops to for, whch reslts n loop grops. At ths stage, traffc congeston ay not be controlled easly. When rtal splt of soe congeston loops or splloer lns becoes zero, grdloc taes place. arge scale traffc congeston coes nto beng. 7. Grdloc Slaton 7.. Slaton scenaro n realty traffc eents sch as splloer natrally decrease the otpt flow rate, whch seres as the deand of downstrea lns or nodes. Nonetheless, ncorporaton of sch sses nto the dedcton ay ntrodce analytcal dffcltes. The nercal stdy s carred ot to explore the dynac natre of grdloc wth the help of a dynac reson of the ln odel n secton. The slated networ s shown n Fg. 8. t s a grd networ wth the sze of 7*7. The lns are bdrectonal as shown n the fgre. There are 45 nodes and 36 lanes n s. The bondary nodes are OD nodes and nsde nodes are sgnal controlled. The phase settngs are shown ether. There are for phases: soth-north throgh phase, sothnorth left-trn phase, east-west throgh phase, east-west left-trn phase. The lost te for each phase s 3 seconds. Green te s assgned accordng to Webster s forla (Webster and obbe 966). Offset of the ntersectons are gen as follows: there are two offset ales, horzontal offset o and ertcal offset o. Offset of adacent nodes that s n horzontal relatonshp adopt horzontal offset. Offset of adacent nodes n ertcal relatonshp adopt ertcal offset. Becase the nflence of offset s not the aor concern, they both are set to zero drng the slaton. The gen flows are rotes flow. The rotes are the statc reasonable rotes defned n Sheff s lteratre (Sheff 985). There are 3 OD pars oerall that s lsted n Table. 684 reasonable rotes n s are fond ot. The flow of a ln s the s of related rotes flow. All rote flows are generated randoly. At the begnnng, ln flows are loaded drectly accordng to the rotes flow. And then we choose a ln, set the splt to a nor one (=. n the slaton). Hence oersatraton s ntrodced. n the slaton, the ntal oersatraton poston s set at left trn lane of the west approach at node 3. Then all lns eole accordng to the ln odel n the appendx. Here rerote effect of traffc a s neglected snce t relates to the drers ealaton of grdloc. We asse that ther rotes (reasonable rote between OD pars) are deterned once they enter the networ. Table OD pars and reasonable rotes Nber O ponts D ponts Nber of reasonable rotes O ponts D ponts Nber of reasonable rotes 45 7 45 7 44 5 44 5 3 43 43 3 4 4 5 4 4 5 5 4 7 4 5 7 6 4 7 4 6 7 3 33 5 33 3 5 6 6 7 9 5 9 7 5 34 7 34 7 Other paraeters are set as below:.6/h; f : 54/h; w : :33eh/; ( q can be dered fro the fndaental dagra, whch s abot eh/h.). H.Sh. Q, Y.Y, Dan Ha Wang, Y.M. Be 395
ongeston oop Nber grdloc reprodcng te(sec) 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 4 5 6 7 8 9 3 3 3 33 phase phase phase3 phase4 34 35 36 37 38 39 4 4 4 43 44 45 Phase settngs Fg. 8 Slaton scenaro 7.. Slaton reslts The slaton reslts are descrbed by the ndexes below: P: The nber of congeston loops, whch ares wth te; P: congeston loop sze at a specfc oent; P3: congeston loop reprodcng te; P4: congeston loop reprodcng rate,.e. the nber of congeston loops prodced per nt te; By defnton, we hae: p dp () t dt 4 (4) 7 6 5 4 3 6 4 ongeston oop reprodcng Te 8 6 ongeston oop reprodcng scale 5 6 7 8 9 3 5 5 5 3 35 Te(sec) Fg. 9 ongeston loops nber (P) eolton Fg. 9 presents the congeston loops nber cre. ongeston loops nber ncreases gradally wth te. The cre becoes sharper, whch ples that the congeston loop reprodce tself n an ncreasng speed. The horzontal dfference between neghborng ps of congeston loop nber can be nterpreted as congeston loop reprodcng te whle the ertcal dfference then can be seen as congeston loop reprodcton scale as shown n Fg. 9. Fg. shows the congeston loop reprodcng te needed for exstng congeston loop. t wll tae abot 5 seconds for the frst congeston loop to coe nto beng. Once t has fored, the reprodcton te sharply decreases, and the reprodcng process sees to accelerate to a relately stable speed. t ples that the control of the foraton of the frst congeston loop s crcal. f the control or preenton fals, the congeston loop wll qcly reprodce tself arond the networ. ongeston oop reprodcng te(sec) 5 5 3 4 te(sec) Fg. ongeston loop reprodcng te (P3) Fg. dsplays the reprodcton ndexes, naely congeston loop reprodcton scale and congeston loop reprodcton rate. The scale s easred by the ncreent of congeston loops nber. ongeston loop reprodcton rate s defned as the ncreent of loops nber per nt te. ongeston loop reprodcton scale presents an ncreasng trend. t s nderstandable becase the a propagaton wll natrally ncrease the reprodcng scale of a congeston loop. Snce the propagaton of ehcle qee s deterned by flow rate as well as sgnal paraeters, reprodcton rate whch s depend on the sae factors then ares n a chaotc profle whch s shown n Fg.. Fg. ges the loop sze cre drng slaton. The three sb fgres are axal sze, aerage sze and nal sze respectely. The axal loop sze s ncreasng all the te. Ths s not tre for aerage congeston loop sze becase saller congeston loops coe nto beng whch s coprsed by two lanes. Fg. (c) llstrates the nal congeston loop. At abot 4sec, two-lane congeston loops appear. Despte that 396 nternatonal Jornal of l Engneerng, Vol. 3, No. 4A, Transacton A: l Engneerng, Deceber 5
Mnal congeston loop sze(=nal M) Maxal congeston loop sze(=axal M) Aerage congeston loop sze(=aerage M) grdloc reprodcng scale grdloc reprodcng rate (grdlocs per second) the foraton of saller congeston loop ay decrease aerage congeston loop sze, the oerall trend of congeston loop sze stll rses whch s the natre reslt of congeston propagaton. ongeston oop reprodcng scale 5 5 ongeston oop reprodcng rate(per sec).5.5 3 4 te(sec) 3 4 te(sec) Fg. ongeston loop reprodcton ndex: congeston loop reprodcton scale ; congeston loop reprodcton rate (P4) 5 5 5 5 4 6 8 te(sec) 4 6 8 te(sec) 4 3 4 6 8 te(sec) (c) Fg. ongeston loop sze dynacs (P); axal congeston loop sze; aerage congeston loop sze; (c) nal congeston loop sze Fg. 3 presents one of the bggest congeston loop that s easred n ln nber at abot t=48.9s. t conssts of 8 lns that are dstrbted arond the networ. Dar gray lanes denote the lanes that are n splloer condton. Ble ones ean congeston loop lanes. We can see that not all the splloer lanes are congeston looped. Snce G eans the sb-graph coprsed by congeston loop lanes and G denotes that coprsed by splloer lanes, we F hae G G, whch represents the lanes that are n F splloer condton bt do not belong to any congeston loop. Fg. 4 ges the lanes proporton araton of these two sb-graphs: congeston loop sb-graph and splloer lane sb-graph. We can see that there are soe lanes whch are n splloer condton bt not congeston looped. Howeer, these lanes ay sere as the potental congeston loop seed. Fg. 4 presents the nflence of proporton dfference on congeston loop reprodcton. x- axs, whch s calclated by N N N, denotes the GF G G dfference between the two lane proportons. ongeston loop reprodcng te redces aganst the dfference. t s nderstandable becase these non-congeston-looped, splloer lanes wll probably consttte a new congeston loop. The ore they are, the qcer new congeston loop wll be reprodced. Fg. 5 ges an arbtrary chosen congeston loop. Ths congeston loop conssts of lanes. The stablzaton s presented by the axal, nal and aerage rtal splt of all the lanes n Fg. 5. At frst one lane s oersatrated. The congeston loop coes nto beng at H.Sh. Q, Y.Y, Dan Ha Wang, Y.M. Be 397
rtal Splt rtal Splt abot t=9 second. Howeer, at that oent, rtal splts of the related lanes wthn ths congeston loop are not dentcal. At abot t=38 second, the rtal splts of all lanes decrease to zero, whch eans that no car wthn ths congeston loop syste can oe ahead. Ths the stablzaton te s abot 38-9=9 seconds. Splloer lane Non splloer lane ongeston loop lane Non ongeston loop lane Fg. 3 The bggest congeston loop at t=48.9s Proporton of lanes.5.4.3.. congeston loop lanes splloer lanes ongeston loop reprodcng te 5 5 3 4 5 te(sec).5..5 N N N G G G F Fg. 4 Fracton of lanes and congeston loop ndex:fracton of lanes; P3 erss dfference of fracton.5.5 Maxal Vrtal Splt Mnal Vrtal Splt Aerage Vrtal Splt.5..5 congeston loop foraton oent satblty condton 5 3 35 4 Te(sec) 5 5 5 3 35 4 45 5 55 6 Te(sec) Fg. 5 A congeston loop and stablzaton process 398 nternatonal Jornal of l Engneerng, Vol. 3, No. 4A, Transacton A: l Engneerng, Deceber 5
8. onclson and Ftre Wor Traffc congeston of networ scale s ery coplex copared wth that wthn a ln. t not only noles road traffc dynacs analyss, bt also relates to the nteracton between adacent lns. Wth the help of the concept of rtal sgnal, we defne the congeston loop whose sze s deterned by the traffc state. Fro the stablty analyss, t can be seen that wth respect to a congeston loop grop nder stable nflow, there always exsts a stable condton where rtal splts for all lns are dentcal. Becase the rtal splt always dnshes, the stable process s accopaned by the deteroraton of traffc state. The slaton reslts sggest that once the orgnal congeston loop fors, the followng reprodcng rate soon ncreases sharply, whch ples the portance of the preenton of the frst congeston loop. Snce traffc a s characterzed by ehcle qee, and ehcle qee eolton tself s deterned by arral and ntersecton control easres as proposed n ths paper, ths control ethod based on the eolton featre can be establshed. Becase the congeston wold reassgn the rote flow, whch reslts n the araton of ln flow, and fnally changes the ln qee eolton. ncorporaton of these sses wll be or next wor, whch seres as the theoretc bass for networ a control. Acnowledgeent: Proect spported by the Natonal Natral Scence Fondaton of hna (Grant No. 548538, 53388, 578454); the Fndaental Research Fnds for the entral Unerstes (5QNA45). References [] hen J, Sh Z, H Y. A relaxaton schee for a ltclass lghthll-whtha-rchards traffc flow odel, Jornal of Zheang Unersty SENE A, 9, No., Vol., pp. 835-844. [] orthot R, Flötteröd G, Vt F, Tapère MJ., Nonnqe flows n acroscopc frst-order ntersecton odels, Transportaton Research Part B: Methodologcal,, No. 3, Vol. 46, pp. 343-359. [3] Daganzo F. Urban grdloc: Macroscopc odelng and tgaton approaches, Transportaton Research Part B: Methodologcal, 7, No., Vol. 4, pp. 49-6. [4] Jn S, Hang Z, Tao P, Wang D. ar-followng theory of steady-state traffc flow sng te-to-collson, Jornal of Zheang Unersty SENE A,, No. 8, Vol., pp. 645-654. [5] Kerner BS. Physcs of traffc grdloc n a cty, Phys Re E,, No. 4, Vol. 84, pp.. [6] K K, assdy MJ. A capacty-ncreasng echans n freeway traffc, Transportaton Research Part B: Methodologcal,, No. 9, Vol. 46, pp. 6-7. [7] ong J, Gao Z, Zhao X, an A, Orensten P. Urban Traffc Ja Slaton Based on the ell Transsson Model Networs and Spatal Econocs, 8, No., Vol., pp. 43-64. [8] Roberg-Orensten P, Abbess, Wrght. Traffc a slaton, Jornal of Maps, 7, No., pp. 7-. [9] Roberg P, Abbess R. Dagnoss and treatent of congeston n central rban areas, Eropean Jornal of Operatonal Research, 998, No., Vol. 4, pp. 8-3. [] Sheff Y. Urban transportaton networs: eqlbr analyss wth atheatcal prograng ethods, Prentce-Hall, nc, Englewood lffs, NJ 763, 985. [] Webster FV, obbe BM. Traffc Sgnals. Road Research Techncal, Paper No. 56, HMSO, ondon. U.K, 966. [] Wrght, Roberg P. The conceptal strctre of traffc as. Transp Polcy, 998, Vol. 5, pp. 3-35. H.Sh. Q, Y.Y, Dan Ha Wang, Y.M. Be 399