A car following model for traffic flow simulation

Transcription

1 Inernaional Journal of Applied Mahemaical Sciences ISSN Volume 9, Number (206), pp. -9 Research India Publicaions hp:// A car following model for raffic flow simulaion Doudou GAYE, Roger Marcelin FAYE 2, and B. MAMPASSI 3 Absrac The raffic flow microscopic modeling is basically imporan for he developmen of specific ools for undersanding, simulaing and conrolling urban ransporaion sysems. Car-following models have been developed o describe he dynamical characerisics of he moving vehicles. In his paper, we presen a microscopic car following model based on he consideraion of he driving behavior on a single-lane road. Wih his model, we propose an approach which permis o ake ino accoun he phenomenon of anicipaion in driver behavior. A comparaive sudy wih he opimal velociy model is done. The proposed modeling approach is validaed by simulaion. The numerical simulaion shows ha he model can improve he represenaion of raffic flow. Keywords: Traffic modeling, Car following model, Microscopic model.. Inroducion The raffic flow modeling is oday an imporan opic o undersand perurbaions in urban raffic flow. There exiss many research for raffic flow modeling. Number of hem concern he sudy of microscopic modeling. Car Following models are among he mos suiable microscopic models o describe he movemen of each vehicle [3]. Some cars following models were invesigaed in [4, 6, ] o describe drivers behavior in raffic flow. These models are based in he fac ha drivers always reac o he simuli generaed by surrounding cars. I is shown ha such a reacion occurs wih a delay ime corresponding o he reacion ime of drivers, and according o some sensiiviy o simulaion. This behavior has been given in [5] and [6] and can be saed as: Reacion( + ) = Sensibili y Syimulus () However, previous experimenal sudies and models on car-following behavior have some imporan limiaions which make hem inconsisen wih real driving experiences [8]. Dakar Universiy, Senegal, gayedoudou@yahoo.fr 2 Dakar Universiy, Senegal, - roger.faye@ucad.edu.sn 3 Dakar Universiy, Senegal - mampassi@yahoo.fr

2 2 Doudou GAYE e al Since he model proposed by Pipes [7] was no improved unil Bando e al proposed an opimal velociy model []. Then many auhors have sudied car-following models using various heories. The proposed opimal velociy model by Bando and al. [] is one of he appropriae car-following models due o is pariculariy in describing numerous raffic flow properies such as insabiliy, changes in raffic congesion and formaion of sop-and-go waves [2]. In he opimal velociy model, acceleraion of a follower car is deermined according o a desired velociy, depending on he disance beween his car and is leader. This can be formulaed by he following: op V i x vi (2) x ai ( C Where V op i represens he desired velociy of he car i a ime, C is a sensiiviy consan, V i ( is he velociy of he car i a ime and where we have se: x x ( x ( ) i i wih x i ( and x i ( represening respecively he posiion of he leader car and is follower. The driving sraegy of he follower car is o mainain a safe velociy depending on he relaive disance. One can noiced ha he mos used form of he erm V op i x [] is given by: 0 x( x A op Vi x gx( x A x( x B (3) V max x B x( where g is an appropriae given funcion ha describes he raffic propery. However, he opimal velociy models have difficuly o avoid collisions in urgen braking cases. This is mainly due o he fac ha he phenomenon of anicipaion is no explicily aken ino accoun. Inspired by he opimal velociy model, some carfollowing models were developed o more realisically raffic. Many of hem can reproduce complex acual raffic phenomena, bu hey canno be used o sudy he influence of anicipaion in driving behavior [0]. Here afer, we presen a car following model based on he conrol mechanism of he disance beween a follower car and is leader in a closed loop feedback conrol sysem. Wih his model he phenomenon of anicipaion are explicily aken ino accoun by he inroducion of a differenial funcion in he driving process. The problem encounered by he opimal velociy model o avoid collisions in urgen braking cases is solved. 2. The proposed model Assume ha we have given a car named i which occupies a posiion x i a ime wih a velociy v i ( and acceleraion a i (. A he same ime, we consider he follower car i which occupies he posiion x i wih a velociy v i ( and acceleraion a i (. To represen he dynamical behavior of he follower a ime d, we assume ha he acceleraion of he follower is a funcion of he relaive velociy and

3 A car following model for raffic flow simulaion 3 he disance beween his follower and is leader. Then, we can wrie a d f ( v ( v (, x ( x ( ) (4) i ( i i i i where f is a given funcion which characerizes his dynamic. According o [9], in real raffic, a follower driver adjuss his velociy, by considering as well as he space headway and he speed difference wih his leader. The posiion difference plays an imporan role in he ransiion phase and he raffic congesion. So ha, he acion of he follower in erm of conrol of his car a ime d is proporional o he disance beween him and is leader. Thus his acion can be formulaed as k x ( x ( ) (5 ) c( i) ( i i where k is a reacion parameer. The phenomenon of acceleraion and deceleraion can be properly represened by anicipaing a slowdown or acceleraion zone [3]. Wih his anicipaion we can preven he oo fas advancing of cars. According o [0], a leader car can send is fuure velociy o he follower before changing is velociy, and hus he follower can obain in advance fuure running sae informaion of he leader o conrol his curren acceleraion for achieving he opimal sae [2]. Based on hese observaions we inroduce a posiion feedback conrol mechanism ino he sysem and we subsiue he reacion parameer k wih a Proporional-Differenial (PD) erm ino he model. This offers a new dynamic relaionship beween ( ) and he effecive acion e( i) ( ), represened by following uaion. c( i) ( ) (6) d e( i) ( c( i( T ) Td c( i) T d where T d is a response ime. This clearly shows ha for a given car, he velociy does no change insanly. For example, he ransiion from a zero velociy o a given ones is made progressively. For his velociy variaion, he acceleraion gradually changes from a maximum value o zero, where he velociy remains consan. This is relaed o he bounded naure of he acceleraion. From hese observaions, we assume ha he acceleraion are relaed o he effecive acion by he following relaion: d ( a ( ) (7) d a( i) ( c( i) i d d where is a consan parameer and a response ime. From Equaions 6 and 7 we finally deduce. Td. k. k a( i) ( vi ( vi ( T ) vi ( T ) xi ( T ) xi ( T ) (8) Acceleraion and deceleraion are no symmeric problems and he acceleraion response ime is greaer han he deceleraion response ime. Thus we define wo response imes and 2 such as 2. Hence, we have. Td. k. k a( i) ( vi ( vi ( T ) vi ( T ) xi ( T ) xi ( T ) (9 in he case of he deceleraing phase and )

4 4 Doudou GAYE e al a. Td. k (. k v ( T ) v ( T ) x ( T ) x ( ) (0) ( i) ( vi i i i i T in he case of he acceleraing phase. In his model, he acceleraion depends no only on he relaive disance and velociy bu also on he informaion speed of he follower a ime. Jiang e al. (200) and Xue (2002) found ha he relaive velociy plays an imporan role in he ransiion phase and he raffic congesion [3]. The uilibrium of he raffic occurs when followers and leaders have he same velociy c This gives vi ( vi ( V for all i and a i ( 0. Thus, a he uilibrium, he uaion 8 leads o he following uaion:. Td. k. k vi ( vi ( T ) vi ( T ) xi ( T ) xi ( T ) 0 ( ) From which, we deduce he following relaion: V S (2).k where S is he uilibrium disance beween cars a he uilibrium velociy V. To deermine he parameer k from uaion (2) we use he fundamenal diagram definiion which represens he characerisics of a road secion. This diagram provides he fundamenal uilibrium relaionship beween raffic densiy and velociy. The uilibrium condiion is given by he following uaion: S (3) where is he uilibrium densiy. Considering a road secion, using he definiion of he fundamenal diagram, we deduce he uilibrium relaionship linking he speed V o he densiy ha is V f ( ) (4) Figure : Example of uilibrium relaionship beween speed and densiy.

5 A car following model for raffic flow simulaion 5 The figure below gives an example of a linear relaionship beween speed and densiy. From he fundamenal diagram (Figure ) and uaion 3, we can deermine he uilibrium disance S ha mus separae a car i and is leader i, for a given uilibrium velociy V i S f ( V i ) i i (5) This uilibrium relaionship is used o deermine he parameer k from uaion 2. The calculaed maximum speed should no be greaer han ha allowed a he fixed free maximum densiy l max. To deermine S i, we shall use he fundamenal diagram of Figure. For insance, he above example leads o he definiion of he following relaionship: S min : he minimum uilibrium disance beween wo cars, max S min : he minimum uilibrium disance for which cars run a full l max velociy. Then, we mus have: k S min c ( k S max (6) 3. Numerical experimenaion To es our car following model approach, we have considered a road secion wih a given lengh and parameers. In paricular for all our numerical simulaion a leader vehicle is considered a he speed of 40km/h and locaed a he posiion x = 50m a he ime = 0. he behavior of many follower cars are simulaed. Figure 2 presens various car-followers posiions in a space-ime diagram while Figure 3 gives he velociy evoluion of hese car followers. From simulaion resuls, one can see ha velociies of car-followers increase when heir disance beween a leader is more and more greaer. Also, i is seen ha hese velociies end o be ual when car-followers posiions are more and more closed o a leader ones. This corresponds o he uilibrium siuaion. Figure 2: Car-followers posiions from a car leader driving a he speed 40 Km/h

6 6 Doudou GAYE e al Figure 3: Flow of car-followers speeds of a car leader driving a he speed 40 Km/h Figures 4 and 5 represen respecively, he posiion and velociy curves of cars in he case of several changes in velociy of he leader vehicle. Through hese resuls, we can observe he same behavior as before. So for differen variaions of he velociy of he leading car, we have he presence of differen uilibrium siuaions, separaed by acceleraion or deceleraion phases corresponding o ransiional phases observed in real raffic siuaions. Figure 4: Evoluion of cars under disruped raffic flow condiions

7 A car following model for raffic flow simulaion 7 Figure 5: Velociy variaion under disruped raffic flow condiions Figure 6 highlighs hese siuaions wih he represenaion of fundamenal diagrams. Real diagram are represened in red and he fundamenal diagram obained from simulaed resuls are represened in blue. On he lef, we have shown he speed depending on he densiy and on he righ we have represened he flow depending on he densiy. Figure 6: Fondamenal Diagrams

8 8 Doudou GAYE e al A comparaive sudy wih he Opimal Velociy Model (figures 7 and 8) shows ha problems relaed o he difficuly of he opimal velociy model o avoid collisions under cerain condiions are solved. Figure 7: Opimal Velociy Model vs Proposed Model: curves of posiion Figure 8: Opimal Velociy Model vs Proposed Model: curves of speed 4. Conclusion Microscopic raffic flow modeling is increasingly used by raffic managers. These are effecive ools for analyzing a wide variey of problems relaed o raffic condiions, bu hey can no be sudied by analyical mehods.

9 A car following model for raffic flow simulaion 9 Numerical resuls confirm ha his proposed approach can well describe car-followers models.from his approach his paper has conribued o he improvemen of some exising car-followers models. References [] Bando H., Hasebe K., Nakayama A., and Shibaa Y., A Dynamical Model of Traffic Congesion and Numerical Simulaion, Physics Review, Par E, Vol. 5, N. 2, Pages [2] Doniec A., Mandiau R., Espie S. and Piechowiak S., Comporemens anicipaifs dans les sysèmes muli-agens. Applicaion à la simulaion de rafic rouier, Revue d Inelligence Arificielle, Vol. 2 (2), Pages 83-22, [3] Gaye D., Faye R. and Mampassi B., A new approach for car following modeling, Inernaional Journal of Advanced Compuing, Engineering and Applicaion, Vol. 2, No. 5, Pages 0-08, 203. [4] Hoogendorn S. P. and Ossen S., Sae-of-he-ar of Vehicular Traffic Flow Modelling, Journal of Sysems and Conrol Engineering - Special Issue on Road Traffic Modelling and Conrol, Vol. 25(4), Pages , 200. [5] Kerner B. S. and Klenov S. L., Deerminisic approach o microscopic hreephase raffic heory, [6] Panwai S. and Dia H., Comparaive Evaluaion of Microscopic Car-Following Behavior, IEE ransacion on Inelligen Transporaion Sysems, Vol. 6, N.3, Pages 3-49, [7] Pipes L.A., An operaional analysis of raffic dynamics, Journal of Applied Physics, Vol. 24, Pages , 953. [8] Kim T., Lovell D. J. and Park Y., Limiaions of Previous Models on Car- Following Behavior and Research Needs, Transporaion Research Board, N , [9] Tian J. F., Jia B. and Li X. G., A New Car Following Model: Comprehensive Opimal Velociy Model, Communicaions in Theoreical Physics, Vol.55, Pages 9-26, 20. [0] Zhou T., Sun D., Kang Y., Li H. and Tian C., A new car-following model wih consideraion of he prevision driving behavior, Commun Nonlinear Sci Numer Simula, Vol. 9, Pages , 204. [] Weng Y. and Wu T., Car-following models of vehicular raffic, Journal for Zhejiang Universiy Science, China, Vol. 3, N. 4, Pages 42-47, [2] Yu S. and Shi Z., An improved car-following model considering headway changes wih memory, Physica A, Vol. 42, Pages -4, 205. [3] Zhu W. X. and Zhang L. D., A speed feedback conrol sraegy for carfollowing model, Physica A, Pages , 204

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