Traffic Circle Design Based on Cellular Automata Simulation

Transcription

1 Traffc Crcle Desgn Based on Cellular Automata Smulaton Han Dong 1, a, X Gao 2, b and Huangjng Zhang 3, c 1 School of Computer Scence & Technology, North Chna Electrc Power Unversty, Baodng , Chna 2 School of Electrcal & Electronc Engneerng, North Chna Electrc Power Unversty, Baodng , Chna 3 School of Energy & Power Engneerng, North Chna Electrc Power Unversty, Baodng , Chna a @qq.com, b @qq.com, c @qq.com Abstract Nowadays, wth the development of socety and the progress of materal lfe, ownershp of vehcles s on the ncrease day by day. In order to solve the problem of traffc jam, we desgn some solutons of traffc crcles. For practcal purposes and unversalty, to begn wth, we choose conventonal non-controlled traffc crcle wth sngle lne to research. The whole model s dvded nto three submodels. Submodel 1 s a dynamc programmng model. Accordng to Wardrop Equaton, We get the relatonshp between traffc crcle s geometrc parameter and maxmum traffc volume. The submodel 2 s a traffc crcle model based on Cellular Automata. Wth the help of submodel 1, we desgn a traffc crcle wth optmal geometrcal parameter. We assume that the vehcle s arrval (.e., the headway) obey Posson dstrbuton wth mean λ, and we use the value of λ to represent the number of cars. The total delay tme s used to measure the total number of stoppng vehcles, whch s smlar to characterze the performance of the whole traffc crcle system. In order to solve the problem of submodel 2, we set up submodel 3. It s a Cellular Automata model, usng traffc lghts to control the ntersectons. We change the traffc crcle n submodel 2 nto an ntersecton wth the control of traffc lghts. Through the research above, we can draw up a concluson: It s sutable for a smooth road whle ntersecton wth sgnal lghts s sutable for a heavy traffc road. As for ths model, ts nnovaton s that we brng n Cellular Automaton to make smulaton. Submodel 3 s able to get more parameters by resettng the cycle of sgnal lght and smulatng agan. Therefore, our model s powerful and relable for varous types of dfferent traffc crcles' desgn. Keywords Traffc crcle, Wardrop Equaton, Cellular Automaton,Posson dstrbuton. 1. Restatement of the Problem Helpng desgn a traffc crcle s a problem whch can be splt nto two parts: desgnng ts geometry, and optmzng the roundabout traffc capacty. After some dscusson, we concluded that we should be able to use the Wardrop Equaton whch wdely apples to the roundabout traffc to analyze the geometry of the traffc crcle. Furthermore, we realzed that the traffc capacty s closely affected by how many seconds each lght should reman green, how many traffc lanes should be set and so on. And all of them can be artfcally controlled so that we can keep the roundabout capacty stable n the tme of day. In 219

2 addton, human factors are not consdered such as drvers emoton whch has reference value for the traffc crcle desgn. 2. Classfcaton of the Traffc Crcle There are varetes of traffc crcles, so clearng our target frst s of great mportance. To begn wth, a specfc type of the traffc crcle s needed, then we are able to analyze ts characterstcs. So t s avalable for us to get a reasonable result, whch can be used to other models. The detals are as follows. 2.1 Classfyng the crcle accordng to ts radus Regular crcle Ths s the most common type n the traffc desgn around the world. In the meanwhle, t s usually used to desgn an ntersecton wth a long mxed secton and mport channels wth the same wdth, whle the dameter of the ntersecton s above 25m. Small crcle The dameter of ths crcle s usually under 25m. And t s convenent for vehcles to broaden the channels when they get nto or get out of the crcle. Tny rotary The central area of the traffc crcle may be other shapes and the radus of the central crcle s less than 4m. Fg. 1 types of traffc crcle 2.2 Classfyng traffc crcles by the number of mport lanes Traffc crcle wth multple lanes Traffc crcle wth one lane 2.3 Classfyng traffc crcles by traffc lght control Traffc crcles wthout sgnal control Ths type s wdely appled to the vast majorty of natons, whch s the tradtonal management n traffc crcle. We dvde the traffc crcle nto two groups traffc crcles wthout specfc stop lnes and traffc crcle wth stop lnes. Traffc crcles wthout specfc stop lnes s also named non-controlled roundabout where vehcles blend wth the traffc flow n the roundabout freely. In ths case, drvers are able to decde to accelerate or slow down when arrvng at the ntersecton. However, vehcles enterng nto the roundabout should avod those movng on the roundabout when the traffc crcle sgn specfc stop lnes. Traffc crcles wth sgnal control It s a management that human adopts traffc sgnal to control vehcles when they enter nto the sland. When traffc s heavy, for ncreasng the number of passng vehcles n unt tme and decreasng the watng tme, we use traffc lght to control traffc flows. In order to smplfy the problem, frst of all, we choose the commonly used roundabout wth sngle lane whose traffc sgnals are not nstalled. On ths bass, two submodels can be establshed. 220

3 In submodel 1, to enlarge traffc capacty, we should fnd out the optmum geometry. Under the crcumstances, We adopt Wardrop Equaton to get the map of geometrc profle and best traffc capacty. Accordng to the nternatonal standard about the desgn of roundabouts, we set lmts to geometrc parameters and establsh a dynamc programmng model by usng Lngo software to get specfc desgnng values. In submodel 2, to verfy the theoretcal value mentoned above, we use a smulaton method based on Cellular Automata. In such a case, the entre sland s splt nto multple cells, and each of them has two knds of state--takng up by a car or not. Cars are only allowed to drve n the cellular one by one. After a perod of smulaton, the fnal state s avalable. By settng dfferent traffc flow, smulatng dfferent vehcles state, countng total delay tme n each smulaton process, we analyze the relatonshp between traffc flow and delay tme. At last, smulaton results can be used to drect the desgn of roundabout traffc lghts. 3. Assumptons (1). There s no suspended vehcle on the road. (2). The traffc crcle s located n the flat area. (3). Angles of gettng n or gettng out are n a proper range. (4). All of the vehcles enterng nto the crcle are the same type and pedestrans are prohbted. (5). It s average of the traffc flow n every entrance and ext, whch obeys the Posson dstrbuton. (6). All subjectve factors should be gnored such as drvers emoton so that the vehcles speed are only affected by traffc sgnals and vehcle s speed n front of the current one. 4. Modelng 4.1 Roundabout desgn model Based on the sland s bggest capacty, we adopt the famous Wardrop Equaton, whch s wdely appled to conventonal roundabout desgn. 221

4 max Q M e P 280W (1 )(1 ) w 3 W 1 l 6.1 W e / W 0.4 e e1 e2 s. t e1 / e P 1.0 W, e, e1, e2, P 0 Obvously, the mxed perod of maxmum capacty s assocated wth the desgn parameters. Through changng parameters, we can obtan the maxmum capacty of the roundabout. We adopt Lngo software to solve the problem. 4.2 Cellular Automata smulaton model We use ths model to smulate the moton of the car nsde or outsde the crcle. Based on past experence, cars arrvng at ntersectons obey Posson Dstrbuton. We change the probablty to fnd out when we need sgnal lght to control traffc flows enterng nto the crcle.model detals are as follows:frstly, we address the problem of optmzng a roundabout through separatng the roundabout and entrances connected wth t nto many equally spaced small squares and vehcles are only allowed to move n the small squares one by one. To model the roundabout more easly, we set some rules to control and renew cars condton. Rule 1: enterng the sland Vehcles enterng nto the road ntersectons obey Posson Dstrbuton: k e P( X k) k 0,1, 2 k! Rule 2: leavng the sland Supposng the number of exports s N around the roundabout. Otherwse, the possblty of leavng each ext s seemed to be the same. 1 P1 P2 PN N Rule 3: slowng down or stop If the speed of the current cell s greater than the sum of cell and the grd n front of t, then the car wll slow down or even stop. V t) V ( t) gap ( t) t ( 1 V ( t) gap ( ) 1 Rule 4: acceleratng If the speed of the current cell s slower than the front grd substractng one, then the cell choose to accelerate. V ( t) gap ( t) 1 V ( t) V ( t) 1 Based on all the rules above, we use Matlab to smulate. 222

5 4.3 Traffc sgnal lghts Takng flaws n model 2 nto account (when headway dstance s less than 4, the total delay tme of vehcles n traffc crcle s long. A dscusson of ths problem can be found n Secton 4.2), we are supposed to look for a new method to solve the problem. So the ntersecton traffc lght model s created. We set the traffc lght n the ntersecton, and then compare t wth traffc crcles, whch s hoped to make up for prevous defects. We also adopt Cellular Automata to smulate the vehcle's arrval and departure. Traffc crcle s replaced by ntersectons wth two ways and two lanes, whch s smlar to model 2. And graphs are shown as follows: Fg. 2 ntersecton traffc desgn We dvde the ntersecton nto many cellulars, whose length s the same as model 2 s. Then we set followng rules: Rule 1: arrvng rule Assume that vehcles arrval at ntersecton obey Posson dstrbuton. Rule 2:leavng rule There are four drectons vehcles can choose, ncludng gong straght, turnng left, turnng rght and turn around, and the probablty of four drectons are equal. Rule 3:speed rule When there s vehcle +1 n front of vehcle, the speed of vehcle and ts poston ( V 1( t), X 1( t)) at tme t need to be compared wth the speed and poston of vehcle +1 at tme t. Under ths crcumstance, our model can judge whether vehcles need to slow down. When the dstance s too small, vehcle choose to slow down or even stop. On the contrary, vehcle choose to accelerate. Rule 4:traffc lght rule If vehcle choose to go straght and ts next poston X 1( t) exceed stop lnes, then ths vehcle need to slow down or stop, whle vehcle choosng other drectons are out of traffc lghts control. In consderaton of regular arrval rules, the probablty of leavng from four exts are the same. Based on symmetry prncple, we can choose a par of ext and entrance at random. Detals about rule 3 and rule 4 are elaborated as follows: f X ( t) V ( t) X 1( t) X ( t) V ( t) X When state()=1 Stoplne V ( t 1) 0 V ( t 1) V ( t) 1 (rules for stoppng or slowng down) When state() X X Stoplne Stoplne 1 0

6 X ( t) V ( t) X Stoplne If X ( t) V ( t) X 1( t). Rule 5:poston at next tme V ( t 1) V ( t) 1 (rules for turnng) V ( t 1) V ( t) 1 (rules for acceleratng) V ( t 1) gap ( t) 1 (rules for slowng down) X ( t 1) X ( t) V ( t 1) We take all rules nto consderaton, and then use Cellular Automata to smulate. To begn wth, we set all vehcles on the ntal lane, at the same tme, they obey unformly random dstrbuton, whch means V ( 0) rand(1,2,3,4), 1,2, n. The method of smulatng tme s the same as model 2, and that means each step needs 2 seconds. We need to loop 1000 tmes. Each smulaton wll cost 2000 seconds. 4.4 Soluton to models the soluton of n Secton 2.2 We solve the dynamc programmng problem by usng Lngo software, and the results are as follows. At ths pont, the crcumference of the traffc crcle s as follows: 2 C 2 ( e2) m 2 the soluton of model 2 Accordng to the optmal parameters got n Secton 2.2, the crcumference of the traffc crcle s set to m, whch can be separated nto 48 cellular, of whch length s 3.9m. We set 10 cellular at each entrance and ther lengths are 39m. The total cycle number s 1000 tmes and each of t represents an actual tme about 2 seconds, whose total tme s 2000 seconds. By settng dfferent λ to smulate, we fnd the relatonshp between λ and delay tme. The results are as follows: It s clear to see that the relatonshp between delay tme and average space headway s nversely proportonal and the headway sze can be used to descrbe road congeston. The smaller the headway s, the heaver the traffc wll be.the lmted value of headway s zero, whch means roads get paralysed and delay tme s nfnte. When the space headway s greater than 5, there s no bg change on delay tme, and the number s small too, whch means roundabouts are capabale of dealng wth the traffc flow. So there s no need to use traffc sgnal to control t on condton that average space headway s more than 5. On the contrary, we can see a sgnfcant rsng trend of delay tme, so traffc lghts are needed to control 224

7 the traffc flow n the roundabout. To a certan degree, space headway represents a traffc flow n ths roundabout. So ths curve can be used as a gudance. Wth the help of t, we can desgn when to use traffc lghts to control traffc flow. Fg. 3 The delay tme of dfferent traffc flow the soluton of model 3 We set 4 lanes, whose length are equal to the length of 10 cellular. Length of each cellular s 3.9m.We regard the ntersecton of four access lanes as the crossroads wth traffc lghts. The traffc lghts perod T s set to be 4. The smulaton tme s 1000 tmes, whch cost 2000 seconds totally. The followng results are concluded by settng dfferent λ: Fg. 4 The smulaton results of the λ values correspondng to the total delay tme. 4.5 Evaluaton of results Comparng total delay tme of traffc crcle lanes wth ntersectons As we can see, when the value of λ s bgger than 5, the traffc crcle s n a good condton and the total delay tme s short too. As the value of λ ncrease, there wll be less decrease on total delay tme, whch matches the physcal truth very well. Owng to the small nfluence of sparse traffc, the total delay tme do not reduce sgnfcantly along wth the augment of λ. On the contrary, when the value of λ s smaller than 5, the stuaton of traffc crcle becomes worse. It s a sgnfcant phenomenon that the total delay tme ncrease sharply as the decrease value of λ, especally when the value of λ s less than 2. So, to replace the prevous desgn of traffc crcle, lookng forward to another method s of great mportance. That s the reason why we adopt model 3 to substtute the conventonal traffc crcle. 225

8 Fg. 5 the comparson of ntersecton and traffc crcle The smulaton result of ntersectons and traffc crcle are put n one graph, from t we are able to fnd out that when the value of the perod of traffc lght(t)s set to be 4 cellular, we can mprove congeston whose average space headway s less than 2. Nevertheless, we don t solve the problem when 2<λ<5. In order to fnd a method to deal ths matter, we can set dfferent T to smulate contnuously. To sum up, we can draw conclusons that ---when the quantty of the vehcle n the traffc crcle s small, traffc crcle can reduce total delay tme effcently. As the traffc goes heavy, the traffc crcle s fluency wll declne dramatcally, and that s the tme we need to choose traffc crcle to releve traffc pressure. That s to say, the method of traffc lghts desgn s sutable for the road where traffc s usually heavy, whle traffc crcle s sutable for the smooth traffc. Fg. 6 the result of changng traffc lght perod T Now, we can solve traffc jam ssue when λ<3 and we should chose the ntersecton sgnal lghts model n the case of T=6 and λ<3. 5. Testng the Model-Smulatons and Senstvty Analyss It s wdely beleved that traffc crcles are able to adjust themselves to releve the traffc pressure and the congeston level can be reduced by settng traffc sgnals. However, what we expected s really true? Wth the help of the orgnal experence, Wardrop Equaton are chosen to optmze our prevous desgn. On the bass, we adopt Cellular automata model to verfy whether the model s correct under dfferent condtons. Through analyzng the model, t s not reasonable to add traffc lghts to releve traffc pressure untl traffc flow n the crcle beng more than ts standard capacty. We set up a model from the aspect of qualtatve and quanttatve analyss to solve the problem of when sgnals and how many seconds each lght should reman green. We proved the subjectve conjecture n theory, whch also confrmed our model s valdty and practcablty. 226

9 In submodel 3, we set the perod of traffc lghts s 4. The problem s that, when 2<λ<5, we ddn t solve the problem of bad delayng tme. So, we wll change the perod n the next secton whle reman other condtons unchanged except perod. In ths background, we do another smulaton to fnd the optmal value. When we ncrease the perod to 6, the results are as follows. 6. Strengths and Weaknesses Our model s sutable for dfferent ctes. Through ths model, we can desgn traffc crcles wth optmal capacty. At the same tme, we apply Cellular automata smulaton to test whether ts capacty s best only to be gven the desgnng sze. Accordng to qualtatve and quanttatve analyss, we are able to analyze the relatonshp between delay tme and traffc flow. And then a better perod of traffc sgnal can be found, whch has certan gudng sgnfcance on urban crcle desgn. Nevertheless, there are stll some shortcomngs about our models. We gnore some objectve phenomenon, such as vehcles rregularty, pedestran s nfluence to traffc. In addton, to smplfy the model, we only smulate traffc crcles wth sngle lane based on all assumptons above. And the perod of red lght s the same as green lght s, whch may be dfferent from the real stuaton. 7. Techncal Summary In order to solve the ntersecton congeston problem, we come up wth two solutons and publsh them as follows. 7.1 Conventonal sngle non-control crcle Ths scheme s sutable for less traffc road especally when average headway tme (λ) s bgger than 5. Traffc crcle s geometrc parameter should be based on Wardrop Equaton as follows: e P 280W (1 )(1 ) Q W M 3 W 1 l Ideally, we can get an optmal traffc volume under the nternatonal standard lmt. And the optmal traffc volume s vehcles per hour. Other parameters are as follows. Each parameter correspondng to the poston below. 227

10 7.2 Lght control ntersecton Fg. 7 geometry parameters of traffc crcle Ths scheme s sutable for large traffc road especally when average headway tme (λ) s smaller than 3. Ths scheme s sutable for the two-way two-lane road. The cycle of traffc lghts (both red and green) s 6 second. We ll gve the schematc n detals. References Fg. 8 ntersecton traffc desgn [1] Fenfang Lu. The Study of two-lne traffc Flow Characterstcs Based on Cellular Automaton [D]. Central South Unversty, [2] YPeng Du. The Theory of Traffc Facltes Desgns and Study of Case. South Chna Unversty of Technology. Central South Unversty, [3] Bo Zhu. The Study of Mult-Agents Mxed-Traffc Smulaton Based on Cellular Automaton [D]. Shandong Unversty, [4] Halan Zhang. The Analyss of Roundabout s Capacty and the Measures of Improvement [D]. Chang an Unversty, [5] Xaobao Yang, Nng Zhang. Mathematcl Analyss of Effects of Lanes Number on Expressway Capacty [J]. Journal of Wuhan Unversty of Technology,