Fuzzy cellular model of signal controlled traffic stream

Transcription

1 Fuzzy cellular model of sgnal conrolled raffc sream Barłomej Płaczek Faculy of Transpor, Slesan Unversy of Technology, Krasńskego 8, Kaowce, Poland Absrac: Mcroscopc raffc models have recenly ganed consderable mporance as a mean of opmsng raffc conrol sraeges. Compuaonally effcen and suffcenly accurae mcroscopc raffc models have been developed based on he cellular auomaa heory. However, he real-me applcaon of he avalable cellular auomaa models n raffc conrol sysems s a dffcul ask due o her dscree and sochasc naure. Ths paper nroduces a novel mehod of raffc sreams modellng, whch combnes cellular auomaa and fuzzy calculus. The nroduced fuzzy cellular raffc model elmnaes man drawbacks of he cellular auomaa approach.e. necessy of mulple Mone Carlo smulaons and calbraon ssues. Expermenal resuls show ha he evoluon of a smulaed raffc sream n he proposed fuzzy cellular model s conssen wh ha observed for sochasc cellular auomaa. The comparson of boh mehods confrms ha he compuaonal cos of raffc smulaon s consderably lower for he proposed model. The model s suable for real-me applcaons n raffc conrol sysems. Keywords: road raffc modellng, fuzzy numbers, cellular auomaa, raffc sgnal conrol 1. Inroducon The developmen of adequae raffc models for applcaons n he road raffc conrol s a challengng ssue. Such models should presen a well-balanced rade-off beween accuracy and compuaonal complexy o enable on-lne processng of measuremen daa and raffc sae esmaon. The sae-of-he-ar raffc conrol mehods use macroscopc and mesoscopc models ha descrbe queues or groups of vehcles [1, 2, 3]. However, he ndvdual feaures relaed o vehcles are mporan from he raffc conrol pon of vew, as hey have a sgnfcan nfluence on he raffc performance. Modern sensng plaforms (e.g. vson-based monorng sysems [4, 5] and vehcular sensor neworks [6]) offer raffc daa concernng he parameers of parcular vehcles (poson, velocy, acceleraon, drecon, ec.). The vehcles can be used as sources of nformaon o deermne dealed raffc sream characerscs. Emergng echnologes n he road raffc monorng enable wreless communcaon beween sensng devces nsalled n vehcles (moble sensors) and he road envronmen for dynamc ransfers of measuremen daa [7]. These daa canno be fully ulsed for raffc conrol purposes when usng macroscopc or mesoscopc models. Mcro-smulaon as a mean of conrollng raffc sysems has recenly ganed he consderable mporance [8]. Compuaonally effcen and suffcenly accurae mcroscopc raffc models have been developed based on he cellular auomaa heory [9]. However, applcaon of he avalable cellular auomaa models n raffc conrol sysems s a dffcul ask due o her sochasc characer. Sochasc parameers n he cellular auomaa are necessary o represen model unceranes and o enable a model calbraon. Unforunaely, he raffc smulaon wh sochasc cellular auomaa requres he me consumng Mone Carlo mehod o be used. Long compuaonal me of he Mone Carlo smulaon s a crcal

2 dsadvanage for raffc conrol applcaons ha requre he resuls of smulaon o be obaned n a srcly lmed me-frames. In hs paper a new mcroscopc raffc model s nroduced, whch does no nvolve he Mone Carlo echnque and enables a realsc smulaon of sgnal conrolled raffc sreams. The model was formulaed as a hybrd sysem combnng a fuzzy calculus wh he cellular auomaa approach. The orgnal feaure dsngushng hs model from he oher cellular models s ha vehcle poson, s velocy and oher parameers are modelled by fuzzy numbers. Moreover, he rule of model ranson from one me sep o he nex s also based on fuzzy defnons of basc arhmec operaons. The applcaon of fuzzy calculus helps o deal wh mprecse raffc daa and o descrbe uncerany of he smulaon resuls. In fac, s mpossble o predc unambguously he effec of an applcaon of a specfc raffc conrol sraegy. Moreover, he curren raffc sae also canno be usually denfed precsely on he bass of he avalable measuremen daa. Therefore, he fuzzy numbers are used n a raffc model o descrbe he uncerany and precson of he smulaon npus and oupus. The model allows a sngle smulaon o ake no accoun many poenal scenaros (raffc sae evoluons) [10]. These facs along wh low compuaonal complexy make he model suable for real-me applcaons n raffc conrol sysems. The res of he paper s organsed as follows: Relaed works are revewed and analysed n Secon 2. Secon 3 addresses lmaons regardng he use of cellular auomaa for modellng he raffc a sgnalsed nersecons. Secon 4 nroduces he fuzzy cellular model of sgnalsed raffc sream and descrbes he raffc smulaon algorhm n deals. A comparson of smulaon resuls for he fuzzy cellular model and he Nagel-Schreckenberg sochasc cellular auomaa s presened n Secon 5. Fnally, conclusons are gven n Secon Relaed works Cellular auomaa have become a frequenly used ool for mcroscopc modellng of road raffc processes. Ther man advanages are hgh compuaonal effcency and fas performance when used n compuer smulaons. The cellular auomaa raffc models are dynamcal sysems wh dscree me, space and sae varables. Despe lmed accuracy on a mcroscopc scale, hey allow real raffc phenomena o be smulaed wh suffcen precson. A comprehensve and dealed revew of he cellular auomaa raffc models can be found n [11, 12]. Among many applcaons n he feld of road raffc modellng, he cellular auomaa were also used for smulaon and opmsaon of sgnal raffc conrol. In [13] a raffc smulaon ool for urban road neworks was proposed whch s based on Nagel- Schreckenberg (NaSch) sochasc cellular auomaa [14]. An nersecon model was consdered n hs work ncludng raffc regulaons (prory rules, sgns, and sgnalsaon). I was also suggesed ha for approprae seng of a deceleraon probably parameer he model yelds realsc me headways beween vehcles crossng sop lne a a sgnalsed nersecon. A modfed NaSch model for raffc flow conrolled by a raffc sgnal was proposed n [15]. Accordng o he nroduced modfcaon, he deceleraon probably for each vehcle s deermned as a funcon of free space n fron of he vehcle. The model was appled n order o smulae a sgnal conrolled raffc flow on a sngle-lane road. Several models of hs ype ha are based on he NaSch cellular auomaa can be found n he leraure (e.g.: model wh urnng-deceleraon rule [16], model wh ancpaon of change n raffc lghs [17]).

3 Schadschneder e al. [18] have presened a cellular auomaa model of vehcular raffc n sgnalsed urban neworks by combnng deas borrowed from Bham-Mddleon-Levne model of cy raffc [19] and he NaSch model of sngle lane raffc sream. The smlar model was adoped o calculae opmal parameers of raffc sgnal coordnaon plan ha maxmse a flow n a road nework [20]. In [21] a model of cy raffc was nroduced, whch s based on deermnsc elemenary cellular auomaa. Each cell n an elemenary cellular auomaon has only wo possble saes (0 and 1). Moreover, he sae of a cell depends only on he presen saes of s neares neghbours. The smplcy of hs model allows for he smulaon of large road neworks wh many nersecons. The emphass n hs approach was pu on he smplcy and scalably of he model raher han on realsm of he raffc smulaon. A cellular auomaa raffc model was also ulsed as an evaluaon ool n a genec algorhm for he raffc sgnals opmsaon [9]. The fness funcon n hs algorhm s evaluaed on he bass of raffc smulaon resuls. The opmsaon was performed for a road nework wh 20 sgnalsed nersecons. The resuls were compared for a sochasc and a deermnsc verson of he cellular auomaa model. I was observed ha he obaned populaon fness rankng s smlar for boh versons. However, he deermnsc cellular auomaa have enabled a remarkable speed-up of he genec algorhm execuon. The relaonshps beween parameers of he cellular auomaa models and sauraon flow raes a smulaed nersecon were analysed n [22]. The raffc models were nvesgaed n hs work o denfy he possbly of reproducng any desred value of he sauraon flow. Ths analyss was performed for boh he deermnsc and sochasc cellular auomaa. I was concluded ha he sochasc verson allows any value of he sauraon flow o be obaned by adjusng a deceleraon probably parameer. Varous arfcal nellgence echnques have been used n he feld of raffc modellng [23, 24]. In hs paper a cellular auomaa model of road raffc s combned wh fuzzy arhmec. Hybrd arfcal nellgence sysems ha combne he cellular auomaa and fuzzy ses are ypcally referred o as fuzzy cellular auomaa (FCA) [25]. FCA-based models have found many applcaons n he feld of complex sysems smulaon e.g. [26, 27]. A road raffc model of hs knd has been proposed n [28]. In such models, he local updae rule of classcal cellular auomaa s usually replaced by a fuzzy logc sysem conssng of fuzzy rules, fuzzfcaon, nference, and defuzzfcaon mechansms. A dfferen approach s used n hs paper: curren saes of he cells are deermned by fuzzy ses and a calculus wh fuzzy numbers s nvolved n he updae operaon. The nnovave feaures of he proposed mehodology are he elmnaon of nformaon loss caused by defuzzfcaon and he ncorporaon of uncerany n smulaon resuls. 3. Lmaons of cellular auomaa models Cellular auomaa models of road raffc descrbe veloces and posons of vehcles n dscree me seps. Poson x ndcaes a cell, whch s occuped by vehcle a me sep. Velocy v s expressed n cells per me sep and deermnes how many cells he vehcle advance a me sep. The dscree posons and veloces are updaed a each me sep accordng o he rule of he cellular auomaa. In order o compue velocy, he rule akes no accoun prevous velocy values, a maxmal velocy v max and number of free cells n fron of he vehcle a me sep (so-called gap) g. Realsc smulaon of he sgnalsed nersecon requres a raffc model, whch can be appropraely calbraed o reflec real values of he measured sauraon flow.e. he maxmum hourly vehcle flow rae, a whch he raffc s as dense as could reasonably be

4 expeced, passng an nersecon under prevalng roadway, raffc, and conrol condons. For cellular auomaa raffc models he calbraon s no a rval ask due o her dscree formulaon and lmed se of parameers Deermnsc cellular auomaa In case of deermnsc cellular auomaa models he sauraon flow rae can be evaluaed by he analyss of queue dscharge behavour. To hs end a raffc sream have o be modelled, whch consss of vehcles leavng a queue afer end of red sgnal. In such raffc sream a unform gap g exss beween vehcles ha reached he maxmal velocy v max. Every vehcle occupes one cell, hus he gap g corresponds o he raffc densy of 1/(g + 1) vehcles per cell. On hs bass he sauraon flow s can be calculaed n vehcles per me sep: vmax s. (1) g 1 Accordng o he above equaon, he sauraon flow n a deermnsc cellular auomaa model can be adjused n wo ways: by changng he maxmal velocy or changng he gap ha occurs beween vehcles movng wh maxmal velocy. The laer requres a modfcaon of he cellular auomaa rule. In pracce, s no possble o oban an arbrary sauraon flow rae because v max and g akes only neger values and a se of applcable rules ha reproduce he real-lfe raffc behavour s very lmed. In order o llusrae he ssue of deermnsc model calbraon, sauraon flow rae wll be analysed akng no accoun hree dfferen cellular auomaa rules. Fg. 1 compares he queue dscharge behavour for he hree consdered rules. In hs example, he maxmal velocy s wo cells per me sep. Numbers n Fg. 1 denoe veloces of vehcles and ndcae her posons (occuped cells), he symbol X represens a red sgnal for vehcles, whch s acve only a he frs me sep of he smulaon. The frs rule (R1) s smlar o he rule ha was proposed by Takayasu and Takayasu [29]. Accordng o hs model a sopped vehcle sars o move only f he gap n fron of g s wder han one cell: 0, v 1 0 g 1, v, (2) mn( v 1 1, g, vmax ), else, x v x 1. When applyng he above formulas, he resulng gap g beween vehcles n sauraed raffc sream s of 2 v max cells (Fg. 1 a). A smaller gap g s obaned for rule R2, whch skps he updae of a vehcle poson f he vehcle s sopped and s curren gap g s of one cell: v mn( v 1, g, ), (3) 1 vmax x, v 1 0 g 1, x x v, else. Usng rule R2 a sauraed raffc sream s formed wh wo gap wdhs: g = v max + 1 for vehcles wh even ndexes and g = 2 v max 1 for vehcles wh odd ndexes. Noe ha he average gap s 1.5 v max and boh above values are equal f v max = 2 (Fg. 1 b). The hrd analysed rule (R3) corresponds o he deermnsc case of he Nagel- Schreckenberg model [14] wh slowdown probably parameer p = 0: v mn( v 1, g, ), (4) 1 vmax x 1 x, v.

5 For hs rule he gap g has wdh of v max cells (Fg. 1 c). Gaps g and he resulng sauraon flow raes for all above rules are summarsed n Tab. 1. The sauraon flow raes s n vehcles per hour of green me were calculaed assumng ha v max = 2 and one me sep corresponds o one second. a) b) c) Fg. 1. Queue dscharge behavour for deermnsc cellular auomaa rules R1-R3 Tab. 1. Sauraon flow raes for deermnsc cellular auomaa rules R1-R3 Rule g [cells] s [vehs per me sep] s [vehs per hour] vmax R1 2v max 1440 v 1 R2 R3 1.5v v max max 2 max vmax 1.5vmax 1 vmax v 1 max The presened examples show ha s possble o oban only a lmed se of sauraon flow raes by manpulang parameers and rules of he deermnsc cellular auomaa models. Moreover, mnmal model modfcaons resul n sgnfcan changes of he sauraon flow. Thus, he deermnsc cellular auomaa models are no suffcen for he realsc raffc smulaon a sgnalsed nersecons Sochasc cellular auomaa Sochasc cellular auomaa models nclude some addonal probably parameers ha conrol he random aspecs of raffc smulaon. As was dscussed n [22], he probably

6 parameers have a drec nfluence on sauraon flow raes of he smulaed raffc sream. However, he modfcaon of he probably parameer for cellular auomaa model resuls no only n change of he average (expeced) value of he sauraon flow raes, bu also n change of her spread. Ths effec s llusraed n Fgs. 2 and 3 for he NaSch model. The NaSch model was used o smulae raffc a a sgnalsed nersecon. Durng he expermen he deceleraon probably parameer p was changed beween 0 and 0.8 wh ncremens of The smulaon of one hour perod was repeaed fve hundred mes for every value of he probably p. Sauraon flow rae was calculaed n each smulaon run. On hs bass a dsrbuon of he sauraon flow raes was deermned for every value of parameer p n he analysed range. The plo n Fg. 2 shows medans, 5-h, and 95-h percenles of he obaned sauraon flow dsrbuons. An example of he dsrbuon hsogram for p = 0.2 s presened n Fg. 6. The spread of he sauraon flow raes was evaluaed as a dfference beween 95-h and 5-h percenle (Fg. 3). Fg. 2. Sauraon flow rae vs. probably parameer p for NaSch model Fg. 3. Spread of sauraon flow values vs. probably parameer p for NaSch model

7 From he resuls n Fgs. 2 and 3 may be concluded ha an ncrease of probably p causes boh lower sauraon flow raes and hgher spread of s values. Thus, for he NaSch model s mpossble o ndependenly change he average value and he range of he sauraon flow raes. Anoher ssue s relaed o he dependency beween free-flow velocy v f and probably parameer: v f = v max p, whch s a well-known characersc of he NaSch cellular auomaa [11]. Due o hs dependency, any modfcaon of he probably parameer resuls also n he change of he free-flow raffc velocy. The aforemenoned muual dependences beween parameers n he sochasc cellular auomaa serously mpede he use of he probably parameer for raffc model calbraon. Applcaon of sochasc cellular auomaa for he evaluaon of raffc performance a a sgnalsed nersecon requres he Mone Carlo mehod o be used for esmaon of performance measures [9]. A number of raffc smulaon runs s necessary o esablsh a meanngful esmae. Therefore he applcably of he sochasc cellular auomaa s lmed due o he long compuaonal me of he Mone Carlo smulaon. Ths dsadvanage s crcal n raffc conrol applcaons, where he resuls have o be obaned faser han he real duraon me of he smulaed process. 4. Fuzzy cellular model A fuzzy cellular model of road raffc was developed o overcome he lmaons of cellular auomaa models ha were dscussed n prevous secon. The nroduced model combnes he man advanages of cellular auomaa models wh a possbly of realsc raffc smulaon a sgnalsed nersecons. The proposed mehod allows he raffc model o be calbraed n order o reflec real values and unceranes of measured sauraon flows. A raffc lane n he fuzzy cellular model s dvded no cells ha correspond o he road segmens of equal lengh. The raffc sae s descrbed n dscree me seps. These wo basc assumpons are conssen wh hose of he Nagel-Schreckenberg cellular auomaa model. Thus, he mehods presened n [11] are also applcable here for he deermnaon of cell lengh and vehcles properes. A novel feaure n hs approach s ha vehcle parameers are modelled usng ordered fuzzy numbers [30]. Moreover, he model ranson from one me sep () o he nex ( + 1) s based on fuzzy defnons of basc arhmec operaons. The road raffc sream s represened n he fuzzy cellular model as a se of vehcles. Each vehcle () s descrbed by s poson X (defned on he se of cells ndexes) and velocy V (n cells per me sep). Maxmal velocy V max s a parameer, whch s assgned o he raffc sream (a se of vehcles). In order o enable approprae modellng of sgnalsed nersecons, he sauraon flow S (n vehcles per hour of green me) was also aken no accoun as a parameer of he raffc sream. All he above quanes are expressed by rangular ordered fuzzy numbers Ordered fuzzy numbers The concep of ordered fuzzy numbers was nroduced n [30]. Accordng o he orgnal defnon, an ordered fuzzy number (A) s represened by an ordered par of connuous real funcons defned on he nerval [0; 1] (Fg. 4): A f, h,, :[0;1] R A A f A ha. (5) The correspondence beween he ordered fuzzy numbers and he classcal heory of convex fuzzy numbers was dscussed n [31]. I was shown ha f some specfc condons are sasfed by he par of funcons f A, h A hen can be ransformed no he membershp funcon µ(x), x R, whch represens a convex fuzzy number n he classcal sense.

8 The model of ordered fuzzy numbers provdes a que smple represenaon of nonprecse nformaon and also smple arhmec operaons. The man advanage of usng he ordered fuzzy numbers s he fac ha hs approach elmnaes several ssues relaed o he classcal fuzzy arhmec, whch s based on he so-called exenson prncple. In he classcal approach, boh he addon and subracon operaons ncrease fuzzness of he calculaed resul. In case of performng he sequences of operaons repeaedly, he applcaon of exenson prncple yelds resuls wh an overesmaed fuzzness (mprecson) ha have lle poenal o be useful. Ths fac was found o be a major obsacle mpedng he consrucon of a fuzzy verson of cellular auomaa model for he raffc smulaon. The ordered fuzzy numbers allow he mulple operaons o be performed whou an excessve ncrease of he fuzzness. I was assumed ha only rangular fuzzy numbers wll be used n he consrucon of a road raffc model, hus f A and h A wll be affne funcons. To be n agreemen wh he classcal denoaon of fuzzy ses (numbers), he ndependen varable of boh funcons wll be denoed by µ: (1) (2) (1) ( ) a ( a a ). (6) f A (3) (3) (2) h A ( ) a ( a a ). In he presened approach, he defnon of he ordered fuzzy number was modfed by (0) (4) nroducng an nerval I A [ a ; a ] o deermne range (codoman) of he funcons f A and h A : A f A, ha, I A, f A, ha :[0;1] I A. (7) Ths modfcaon was made o enable a concse descrpon of dependences beween fuzzy numbers represenng dfferen physcal quanes, whch values can vary n sgnfcanly dfferen ranges. For furher presenaon of he proposed model, wll be convenen o normalse he codoman of funcons f A and h A no he un nerval [0; 1]. Thus, he followng range-normalsed form of he ordered fuzzy number wll be used: A f, h, I, f, :[0;1] [0;1]. (8) A A A where: (0) (0) f A( ) a ha( ) a f A( ), ha( ). (9) I A I A are he range-normalsed counerpars of funcons f A and h A (Fg. 4). A h A Fg. 4. Trangular ordered fuzzy number and s range-normalsed counerpar

9 Herenafer, all he ordered fuzzy numbers are represened by 5-uples. The followng noaon s used o refer o her sandard and range-normalsed form respecvely: (0) (1) (2) (3) (4) (0) (1) (2) (3) (4) A ( a, a, a, a, a ), A ( a, a, a, a, a ). (10) A defnon of basc operaons on rangular ordered fuzzy numbers can be now formulaed usng he above represenaon: op( A, B) C c op( a, b ), m 0,...,4, (11) where A, B, C are ordered fuzzy numbers and op sands for a parcular operaon: addon, subracon, mulplcaon, mnmum or maxmum. Ths se of operaons s suffcen for consrucng a fuzzy raffc model based on he cellular auomaa approach Traffc smulaon algorhm In hs secon a raffc smulaon algorhm s nroduced. The mos mporan componens of hs algorhm are updae operaons ha enable he veloces and posons of vehcles o be compued a each me sep of he smulaon. The desgn of he updae operaons has a drec mpac on he resulng sauraon flow of he smulaed raffc sream. The proposed algorhm allows he raffc smulaon o be adjused n order o f a predeermned level of sauraon flow, whch s represened by a fuzzy number S. The proposed raffc smulaon algorhm (Algorhm 1) ulses a par of deermnsc rules (RL, RH) for updang he fuzzy cellular model. The rules RL and RH are used o (0) (4) compue posons x and x respecvely, where denoes curren me sep of he 1 1 smulaon. For he compuaon of x, (1) 1 x and (2) 1 x one of he avalable rules (RL or RH) s seleced a each me sep. The nroduced operaon of rules selecon allows he smulaed raffc sream o reach he assumed level of sauraon flow (S). (3) 1 Algorhm 1. Traffc smulaon wh fuzzy cellular model For = 1 o T do Updae raffc sgnals. For all vehcles ( = 1 o N) do Compue v 0) (0) and x usng rule RL. ( 1 For m = 1 o 3 do f x hen compue else compue Compue v 4) and ( v (4) 1 and 1 v and x usng rule RH, 1 x usng rule RL. x usng rule RH. As was dscussed n Secon 3.1, each deermnsc rule of cellular auomaa corresponds o a sngle value of he sauraon flow. The value of sauraon flow s deermned by wo parameers: maxmal velocy v max and gap g beween vehcles passng hrough an nersecon wh he maxmal velocy. We wll denoe he maxmal velocy for (0) (4) rules RL and RH as vmax and v max respecvely. Smlarly, he gaps for rules RL and RH wll be represened by g (0) and g (4). The correspondng sauraon flow values can be calculaed by usng Eq. (1). In he proposed algorhm he sauraon flow level s (0), acheved for rule RL, has o be lower han he sauraon flow level s (4) for he second rule (RH). The par of deermnsc rules RL, RH allows us o compue an nerval (0) (4) I [ x ; x ] represenng possble posons of -h vehcle ha correspond o he assumed X (0) (4) nerval of sauraon flow values I S [ s ; s ]. Upper char n Fg. 5 shows posons of sx

10 vehcles, deermned for he sauraon flow values from nerval I S. Black dos represen wo exreme confguraons of he raffc model ha are obaned by usng he wo dfferen (m) m deermnsc cellular auomaa rules. For each poson x, such ha x ( ) I X he followng equaly s sasfed: x vmax g. (12) I should be noed here ha he me ndces of varables x were omed for he sake of smplcy. On he bass of (12) we can deermne he maxmal velocy and gap for he vehcle poson x : (m) v max g (0) (4) (0) v x ( v v ), (13) max max max (0) (4) (0) g x ( g g ). (14) (m) Thus, he value of sauraon flow, whch corresponds o he poson x can be calculaed as: (0) (4) (0) vmax x ( vmax vmax ) s. (15) (0) (4) (0) g x ( g g ) 1 Ths leads o he non-lnear relaonshps beween posons x of parcular vehcles movng n a raffc sream and he sauraon flow s (Fg. 5). Fg. 5. Fuzzy cellular model of raffc sream: sauraon flow and vehcles posons The man am of he nroduced approach s o provde a road raffc model, whch can accep an ordered fuzzy number S as an npu parameer specfyng he level of he sauraon flow. The lower char n Fg. 5 shows vehcles posons ha were obaned by akng no accoun he assumed fuzzy value of he sauraon flow. These posons are represened by ordered fuzzy numbers X. The key nsgh s ha by ransformng he formula (15) we can deermne he expeced posons ha correspond wh he assumed level of sauraon flow: x (0) (0) s g vmax (4) (0) (4) (0) v v s g g 1 max max m. (16) The resulng value (m) s dencal for all vehcles n he raffc sream and for all me seps of he smulaon. I should be also noed ha Eq. (16) ogeher wh he known nervals I provdes complee descrpon of he fuzzy numbers X. Durng raffc smulaon he posons x have o be compued as neger cell ndces n accordance wh he deermnsc cellular auomaa rules. The symbol (m) n (16) was nroduced n order o dsngush he expeced posons from he curren posons X

11 deermned as a resul of he smulaon. Wh he above defnons, he am of achevng he predeermned sauraon flow level can be ranslaed no he followng requremen, whch needs o be sasfed n he raffc smulaon: x mn m. (17) The proposed raffc smulaon algorhm copes wh hs mnmsaon problem by selecng he cellular auomaa rule ha reduces he absolue dfference defned n (17). As was menoned a he begnnng of hs secon, he rule selecon s execued for all vehcles a each me sep of he smulaon and apples o he updae of posons x for m = 1, 2, 3. Exac defnon of he rule selecon operaon s presened n he form of f-hen saemen by he pseudo-code of Algorhm 1. Snce he fuzzy cellular model s desgned o be used for he smulaon of sgnalconrolled raffc sreams, has o ake no accoun drvers reacons o raffc sgnals. The nfluence of raffc sgnalsaon s smulaed by nroducng a se of cells n fron of whch he vehcles have o hal a red sgnals. Ths se wll be denoed by H. The hal cells n H correspond wh he locaons of sop lnes. The updae of a raffc sgnal nvolves he nseron of an approprae cell ndex x no he se H when he red sgnal has o be acvaed, and removal of he ndex x from H when he red sgnal has o be deacvaed. Thus, yellow me s consdered as a par of green phase Model mplemenaon The followng defnons provde a dealed descrpon of he fuzzy cellular model mplemenaon, whch s based on he deermnsc cellular auomaa rules R1 and R2 ha were dscussed n Secon 3.1. The rules denoed by RL and RH n he smulaon algorhm (Algorhm 1) are mplemened here as he rules R1 and R2 respecvely. The velocy of vehcle a me sep s compued usng he formula: mn( V 1, G, V A, (18) V 1 max ) where G s he fuzzy number of free cells n fron of a vehcle : L S G, mn( G,, G, ). (19) Gap L G represens he dsance beween -h vehcle and s lead vehcle ( 1): L G X X 1. (20) 1, If here s no lead vehcle n fron of he vehcle hen G s assumed o be equal o V max. The varable S G descrbes dsance of he -h vehcle o he neares red sgnal (.e. he hal cell x H): S g mn{ x x 1: x x x }. (21) H If here s no hal cell x sasfyng he condon n (21) hen L g. S g s assumed o be equal o The varables V, G, L G, S G, X, and V max are rangular ordered fuzzy numbers. Thus, he calculaons are performed accordng o he defnon (11). The subracon of 1 n (20) s handled by nerpreng he scalar value as an ordered fuzzy number, whch can be represened by 5-uple havng all he componens equal o one. A s a 5-uple of bnary values: 0, r, RL v 1 0 g 1, a (22) 1, else,

12 where m = 0,..., 5. Noe ha f a 0 hen v s compued by usng he rule RL (.e. R1). The mulplcaon by he bnary uple A n (18) s performed smlarly o he operaons on ordered fuzzy numbers, accordng o he defnon (11). Afer he deermnaon of veloces for all vehcles, her posons are updaed as follows: X 1 X V, B,, (23) where B s a 5-uple of bnary values: 0, r, RH v 1 0 g 1, b (24) 1, else. In he above formula, b akes value 0 when he rule RH (.e. R2) s used for he compuaon of x. 1 The varables r n equaons (22) and (24) are componens of anoher 5-uple (R ), whch s nroduced o descrbe he curren selecon of rules: RH, x (0) (4) r,, m 1,..., 3, r RL, r RH. (25) RL, else I should be noed here ha he above defnon of R s conssen wh he rule selecon operaon ha was nroduced for he fuzzy cellular model n Secon Comparson wh Nagel-Schreckenberg cellular auomaa model 5.1. Smulaon resuls Ths secon presens some resuls of raffc smulaons n a sgnalsed areral. The resuls of a smulaon performed wh he fuzzy cellular model are compared agans hose obaned by usng he Nagel-Schreckenberg (NaSch) sochasc cellular auomaa model. For he purpose of hs expermen he fuzzy cellular model was mplemened accordng o he defnons gven n Secon 4. The parameers of he fuzzy cellular model were adjused o mach he followng sengs of he sochasc NaSch model: me sep 1 s, cell lengh 7.5 m deceleraon probably p = 0.2 and maxmal velocy v max = 2 cells per me sep. Smlarly, for he fuzzy cellular model he maxmal velocy of vehcles was assumed as wo cells per me sep: v max 2 m. Thus, due o he applcaon of deermnsc rules R1 and R2, he resulng nerval of sauraon flow values for he fuzzy cellular model s I S = [1440, 1800] (n vehcles per hour of green me). The sauraon flow volume n he fuzzy cellular model s defned by he ordered fuzzy number S. Ths parameer was esmaed n order o reproduce he dsrbuon of sauraon flow raes observed n he NaSch model. Fg. 6 shows a hsogram of he sauraon flow values ha were obaned for he NaSch model durng 500 runs of a raffc smulaon a sgnalsed nersecon. The smulaon perod was 3600 seconds for each run. The expermenal daa presened n Fg. 6 were furher used for he deermnaon of he parameer S. The values of s (1), s (2) and s (3) were se respecvely as he 5-h percenle, medan and 95-h percenle of he sauraon flow raes dsrbuon. As a resul he followng fuzzy number was obaned, whch descrbes he sauraon flow n vehcles per hour of green me: S = (1440, 1503, 1575, 1638, 1800). The parameers (1) = 0.21, (2) = 0.43, (3) = 0.60 of he fuzzy cellular model were calculaed for he above level of sauraon flow accordng o Eq. (16).

13 Fg. 6. Hsogram of sauraon flow raes for NaSch model The cell lengh for he fuzzy cellular model was deermned on he bass of free-flow velocy comparson. For low raffc denses n NaSch model, vehcles move wh a freeflow velocy, whch can be calculaed n cells per me sep usng he formula: v f = v max p. In case of he fuzzy cellular model he free-flow velocy s equal o V max (noe ha V max s an ordered fuzzy number). The me sep for boh models corresponds o one second. Takng no accoun he parameers ha were deermned above, he cell lengh for fuzzy cellular model can be calculaed as 7.5 ( vmax p) / vmax = 6.75 m. Thus, n boh models he free-flow velocy s equal o 13.5 m/s (48.6 km/h). Smulaons of a raffc n sgnalsed one-way areral road were performed n order o compare he nroduced fuzzy cellular model and he NaSch cellular auomaa. Parameers of boh models were se as dscussed above. One-lane road was consdered n hs sudy because he analyss s focused on he dealed properes of he models ha are relaed o sngle lane raffc sreams. I was assumed ha hree sgnalsed nersecons are locaed a he modelled road. Toal lengh of he road s 3 km and he dsances beween nersecons are equal o 750 m. The nal condons of he raffc smulaons are deermned by a sngle queue lengh parameer (.e. number of vehcles wang n a queue a an nersecon). A he begnnng of each smulaon queues of equal lengh are formed for all hree nersecons. Addonally, he las vehcle s always nsered no he frs cell of he modelled road. Fg. 7 presens rajecores of he las vehcle n me-space dagrams. Black dashed and doed lnes show rajecores ha were deermned by usng he fuzzy cellular model (FCM). The grey colour ndcaes rajecores obaned for he NaSch model durng 500 runs of he raffc smulaon. Black horzonal bars correspond o red me nervals a he nersecons. Traffc sgnal mngs are smlar for all smulaed nersecons. The resuls presened n Fg. 7 a) were obaned for sgnal cycle me of 60 s and green phase of 30 s. For he second case (Fg. 7 b) he cycle me was 90 s and green phase was 45 s. Dependency beween he queue lengh parameer and he ravel me of he las vehcle s llusraed n Fg. 8. The ravel me s defned as me requred for he las vehcle o pass a sop lne a he hrd nersecon. The resuls of ravel me esmaon are compared for he wo analysed models. In case of he NaSch model applcaon, he percenles of ravel me dsrbuon are deermned on he bass of 500 smulaon runs for each queue lengh. Usng fuzzy cellular model, he ravel me s deermned n sngle smulaon run as an ordered fuzzy number accordng o he followng equaon: mn{ : x 333}, (26) l,

14 where l s an ndex of he las vehcle. The cell number 333 corresponds o locaon of he sop lne a hrd nersecon. a) b) Fg. 7. Trajecory of he las vehcle for wo sgnal cycle mes: a) 60 s b) 90 s Fg. 8. Travel me of he las vehcle vs. nal queue lengh Plos n Fg. 9 show number of vehcles on he modelled road (upsream of he hrd nersecon) for successve me seps of he raffc smulaon. Noe ha he vehcles are nsered no he modelled road only a he begnnng of smulaon. The number of vehcles decreases durng smulaon as subsequen vehcles pass over he sop lne of he hrd nersecon. The nal number of vehcles s deermned by he queue lengh parameer. The resuls presened n Fg. 9 a) and b) were obaned for he queue lengh of 30 and 70 vehcles respecvely. The resuls for NaSch model were esmaed afer 500 runs of he raffc smulaon. In he fuzzy cellular model, he number of vehcles a me sep s drecly calculaed for a sngle smulaon run as an ordered fuzzy number N : n { : x 333}, (27) where. denoes he cardnaly of a se.

15 a) b) Fg. 9. Number of vehcles on he modelled road for wo nal queue lenghs: a) 30 b) 70 The resuls shown n Fg. 8 and Fg. 9 were obaned for he raffc sgnal cycle of 60 s wh green phase of 30 s. The above sgnal mng parameers were used for all hree nersecons. As can be observed n he above resuls, he evoluon of he smulaed raffc sream s very smlar for boh models. Ths fac proves ha he proposed fuzzy cellular model can be appropraely calbraed o reproduce he raffc sream behavour for a gven dsrbuon of sauraon flows raes. The expermens have shown ha he accuracy of raffc smulaon s smlar for boh consdered models. However, he fuzzy cellular model avods he aforemenoned dsadvanages of he cellular auomaa (Secon 3). Frsly, he proposed model can be precsely calbraed by adjusng s parameers. Moreover, he uncerany of model parameers can be aken no accoun as he parameers are represened by fuzzy numbers. Secondly, he fuzzy cellular model does no need mulple smulaons because uses he fuzzy numbers o esmae he dsrbuons of raffc performance measures (ravel me, he number of vehcles n a gven regon, delays, queue lenghs, ec.) durng a sngle run of he raffc smulaon Compuaonal cos The mplemenaon of he NaSch model requres mulple raffc smulaon runs (see Algorhm 2). A each run, he smulaon resuls have o be sored. Afer K runs, he sored resuls are used o calculae dsrbuons of he raffc performance measures. The number of smulaon runs K has o be appropraely hgh n order o oban meanngful esmaes. For he expermens presened n hs secon he number of runs K was 500. The sochasc rule of NaSch cellular auomaa was decomposed no wo deermnsc rules, denoed by NSH and NSL. The NSH rule s conssen wh he R2 rule, whch was defned n Secon 3.1. I corresponds o he NaSch rule wh parameer p = 0. The NSL rule reflecs he operaon of he NaSch rule for p = 1. Thus, he velocy n he NSL rule s calculaed accordng o he followng formula: v, max( 0, mn( v 1 1, g, vmax ) 1). (28) The randomsaon sep of he NaSch model was mplemened n he smulaon algorhm by nroducng a selecon of he deermnsc rule (NSL or NSH). The selecon s

16 based on a random number [0;1), whch s drawn from a unform dsrbuon. Ths descrpon of he raffc smulaon algorhm enables s comparson wh he algorhm proposed for he fuzzy cellular model (Algorhm 1). Algorhm 2. Traffc smulaon wh he NaSch model For smulaon run 1 o K do For = 1 o T do Updae raffc sgnals. For all vehcles ( = 1 o N) do Generae random number. If p hen compue v and x 1 usng rule NSL, else compue v and x 1 usng rule NSH. Sore smulaon resuls. Le us assume ha he basc operaon n he raffc smulaon algorhm s he execuon of deermnsc cellular auomaa rule.e. he compuaon of he poson and velocy for a sngle vehcle. The number of basc operaons performed durng he raffc smulaon can be deermned for boh compared models by analysng he pseudo-code of Algorhm 1 and Algorhm 2. The raffc smulaon wh he NaSch model requres K T N basc operaons whereas durng he smulaon wh he fuzzy cellular model he basc operaon s execued 5 T N mes. I was assumed ha he number of vehcles N s consan n he analysed smulaon perod. The compuaonal cos of raffc smulaon s consderably reduced for he fuzzy cellular model because he number of smulaon runs K s always much greaer han 5 (usually amouns o several hundred runs). Moreover, he raffc smulaon wh he fuzzy cellular model does no need o sore paral resuls, hus requres less memory space han he smulaon wh he NaSch cellular auomaa. 6. Conclusons The fuzzy cellular model of sgnal conrolled raffc sream was proposed by combnng cellular auomaa wh fuzzy calculus. The presened approach benefs from advanages of he cellular auomaa models and elmnaes he man drawbacks ha have mpeded her applcaons n raffc conrol sysems. Parameers of he fuzzy cellular model enable a smple calbraon and allow he raffc smulaon o reflec predeermned sauraon flow raes. The fuzzy numbers are used n order o descrbe he uncerany and precson of he smulaon npus and oupus. Thus, he mprecse raffc daa can be ulsed n he proposed modellng approach for he esmaon of raffc performance [32]. The expermens repored n hs paper show ha he raffc smulaons wh he fuzzy cellular model are conssen wh hose performed by sochasc cellular auomaa. I was also demonsraed ha he applcaon of he nroduced model consderably reduces he compuaonal cos of raffc smulaon. These fndngs are of val mporance for real-me applcaons of mcroscopc models n he road raffc conrol.

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