Evaluaion of a car-following model using sysems dynamics Arif Mehmood, rank Saccomanno and Bruce Hellinga Deparmen of Civil Engineering, Universiy of Waerloo Waerloo, Onario, Canada N2 3G1 Tel: 519 888 4567 Ex. 6596 Email: amehmood@uwaerloo.ca Absrac Models ha describe he processes by which drivers follow each oher in he raffic sream are generally referred o as car-following models. During he pas 50 years, driver behavior wihin he raffic sream has been sudied and models ha aemp o describe his behavior have been proposed. Car-following models have frequenly been developed for he purpose of incorporaion wih microscopic simulaion models. These models have hen been used o evaluae a wide range of poenial geomeric opions, operaional sraegies, and/or policies. In his paper we formulae a car-following model using he sysems dynamic SD approach. We compare he behavior of he proposed SD model o he GM model, a classic car-following model ha has been exensively described in he lieraure. These comparisons illusrae ha he proposed SD car-following model avoids several unrealisic characerisics of he GM model. The ulimae objecive is o use he proposed model o invesigae he mechanisms leading o rear-end crashes and o quanify he impac ha differen echnologies or policies e.g. driver vision enhancemen sysems may have on rear-end crash poenial. Keywords: Car-following, Driver behaviour, Sysems Dynamics, Microscopic raffic simulaion 1.0 Inroducion Rear-end crashes involving wo or more vehicles currenly represen abou one-fourh of all road crashes in Canada as well as in U.S. General Esimaes Sysems GES repored ha in 1998 rear-end crashes consiue approximaely 28% of all crashes in he U.S. In Onario rear-end crashes represen abou 23% of all crashes repored in any given year during 1993 o 1997 Onario Road Safey Annual repor, 1993-1997. While many injuries and faaliies are caused by rear-end crashes, rear-end crashes also cause approximaely 157 million vehicle-hours of delay annually in U.S., which is abou one-hird of all road crash-caused delays McGehee e al., 1992. Several facors may conribue o rear-end crashes, such as driver percepion/reacion ime, driver inaenion, high speed and following oo closely, and limiaions in visibiliy. According o Naional Safey Council Acciden acs, 1996 abou 90% of rear-end crashes resul from driver inaenion and/or following a lead vehicle oo closely. The sudy of driver behaviour
under car-following siuaions is essenial in developing advanced vehicle conrol and safey sysems ACSS. Such sysems seek o replicae human driving behaviour hrough parial conrol of he acceleraor, while removing poenial hazards ha may occur hrough driver mispercepion and acion. The developmen of effecive safey sysems for reducing rear-end crashes requires a horough undersanding of facors ha conribue o rear-end crashes. Driving an auomobile is a coninuous complex ask. I requires he driver o consanly scan he environmen and o respond properly in order o mainain conrol, avoid obsacles, and inerac safely wih oher vehicles. To build and inegrae he echnologies ha migh aver rear-end crashes, i is imporan o firs fully undersand driver behaviour in he car-following siuaions a a given locaion for a range of ransporaion condiions. Driver behaviour in he car-following siuaion has been sudied exensively since he 1950s. Over he years various models ha reflec he car-following behaviour of drivers for differen raffic assumpions and condiions have been developed see for example, he work by Chandler e al., 1958; orbes e al., 1958; orbes, 1963; Gazis e al., 1961; Herman and Pos, 1959; Herman and Rohery. 1969; May, 1967, Gipps, 1981; an Aerde, 1995; Hogema, 1998; and Zhang e al., 1998. By far he mos significan conribuion o he developmen of car-following heory was made by he General Moors GM researchers Chandler e al, Herman e al, Gazis. This paper reviews he exising car-following models from he heoreical and pracical sandpoin. These models have a number of shorcomings depending on heir formulaion, including failure o consider he longer view of he raffic sream i.e. several lead vehicles ineracing o effec he behaviour of a following vehicle, failure o consider a siuaion when relaive speed is zero a a small spacing, and failure o disinguish beween risk criical siuaions and oher siuaions for similar iniial speeds and spacing. The paper hen proposes a Sysem Dynamics SD model ha exends he feaures of hese exising car-following models by inroducing a concep of driver comfor zones effecing speed and spacing a differen poins of ime. The proposed SD model is evaluaed by comparing is resuls wih hose suggesed by he car-following models like he GM model for a given se of ransporaion condiions. Comparison beween he performance of he wo models shows ha he proposed SD model can overcome many shorcomings of he exising car-following models. 2.0 Exising car-following models Models ha describe he processes by which drivers follow each oher in he raffic sream are generally referred o as car-following models. Car-following models have been sudied exensively since as early as he 1950s. The earlies work focused on he principle ha vehicle separaion is governed by safey consideraions in which disance or ime headway beween vehicles are a funcion of speed. Pipes 1953 developed a car-following model on he assumpion ha drivers mainain a consan disance headway. His work was followed by orbes 1958 who assumed ha drivers mainain a consan ime headway. ollow-up research incorporaed facors such as spacing beween vehicles, speed differenial, and driver sensiiviy ino car-following models. These models are summarized in Table 1.
In Table 1, he car-following models by Chandler e al., 1958; Gazis e al., 1959, 1961; Edie, 1961; Newell, 1961; Herman and Rohery, 1963 and Bierley, 1963 assumed ha in car-following siuaion he ollowing ehicle driver observes only he vehicle immediaely ahead in deermining his or her speed. ox and ehman 1967, and Bexelius 1968 suggesed ha insead of following only he immediaely vehicle ahead, drivers in a car-following siuaion also observe he vehicles ahead of he lead vehicle. They incorporaed he effec of a second lead car in heir car-following models. Table 1: Car-following models Source: Corresponding Car-following Model Chandler e al. 1958 [ ] a = + α Gazis e al. 1959, 1961 = + a α Edie 1961 ] [ ] [ 2 a = + α Newell 1961 [ ] Gn a = + Herman and Rohery, 1963 ] [ ] [ ] [ a l m = + α Bierley 1963 [ ] [ ] a T + = + β α ox and ehman 1967 + = + 2 2 2 1 ] [ ] [ ] [ ] [ W W a T α Bexelius 1968 [ ] [ ] a T + = + β α Rockwell e al. 1968 [ ] a a β α + = + Where a +Ä = Acceleraion/deceleraion rae of ollowing ehicle driver a ime + Ä a = Acceleraion/deceleraion rae of ead ehicle driver a ime = ollowing ehicle speed a ime = ead ehicle speed a ime T = Targe 2 nd ead ehicle speed a ime = ollowing ehicle disance a ime = ead ehicle disance a ime = Simulaion ime sec Ä = Percepion-reacion ime sec or simulaion inerval G n = Empirical relaionship beween velociy and headway for acceleraion/deceleraion á, â, m, l, W 1, W 2 = Model parameers
In his paper, we will be using he GM model equaion 1 as a basis of comparison for carfollowing models in general because of he comprehensive field experimens conduced o validae is underlying assumpions May, 1990. a m [ ] + = α [ ] l 1 [ ] A common feaure of he above GM model is he assumpion ha driver behaviour can be represened as a simulus-response sysem. Sysem response is he driver's decision o accelerae or decelerae. The rae of acceleraion or deceleraion is a funcion of driver sensiiviy and he simulus. Simulus is assumed o be he difference in speed beween he ead and he ollowing ehicle. Driver sensiiviy is a funcion of he spacing beween he ead and he ollowing ehicle, he speed of he ollowing ehicle, and a calibraed coefficien. No wihsanding considerable work carried ou on car-following models following he iniial work by GM, he GM model srucure has remained as he basis of car-following behaviour. However, in order o calibrae and subsequenly evaluae he GM model many researchers have aemped o esimae he bes combinaion of parameers c, m, and l. Among hese he mos noable examples of his work are: May and Keller, 1967; Heyes and Ashworh, 1972; Ceder e al., 1976; Aron, 1988; Ozaki, 1993. A summary of opimal parameer combinaions o emerge is given in Table 1. Table 1: Mos reliable esimaes of parameer combinaions for he GM model Source C m l May and Keller 1967 1.33 x 10-3 0.8 2.8 Heyes and Ashworh 1972 0.8-0.8 1.2 Ceder e al. 1976 0.6 2.4 Aron 1988 2.45 / 2.67 / 2.46 0.655 / 0.26 / 0.14 0.676 / 0.5 / 0.18 Ozaki 1993 1.1 / 1.1 0.9 / -0.2 1 / 0.2 2.1 imiaions of exising GM car-following model Despie he dominance of he GM model and is varians in he research lieraure, his model exhibis several undesirable characerisics. Many of hese undesirable characerisics are common o car-following models in general and include Chakrobory and Kikuchi, 1999: 1. The response of he ollowing ehicle driver is based on only one simulus relaive speed. When he ead and ollowing vehicles are ravelling a he same speed, he acceleraion/deceleraion response dicaed by he GM model is zero, regardless of he curren spacing beween vehicles. Owing o he single simulus naure of he model, i fails o illusrae he behaviour of he ollowing ehicle driver a zero relaive speed. or example, consider he siuaion when he relaive speed beween wo successive vehicles is zero and he spacing is oo shor. or such a siuaion we expec he ollowing ehicle
driver will iniially decelerae o increase he spacing unil a comforable spacing is achieved o desirable and hen accelerae o reduce he speed difference beween he wo vehicles. This would occur if he ead ehicle is no ravelling a a speed ha exceeds desired speed of he ollowing ehicle driver. 2. The GM model assumes a symmerical behaviour for he ollowing ehicle driver. or example, consider wo cases: one wih a posiive relaive speed wih a cerain magniude and he oher wih a negaive relaive speed wih he same magniude, and all oher facors are idenical. In he ineres of safey, we expec he acceleraion in firs case o be lower han he deceleraion in absolue erms in second case. I has been observed ha drivers ac differenly depending on wheher spacing beween vehicles is increasing or decreasing more risk criical. Drivers "pay closer aenion o spacing decreases han o spacing increases" euzbach, 1988. 3. Anoher drawback of GM model is ha i assumes ha he ollowing ehicle driver observes only he ead ehicle in deermining his or her speed. In realiy, i has been observed ha drivers respond in relaion o he behaviour of several downsream vehicles, no jus he vehicle immediaely ahead ox and ehman 1967, Bexelius, 1968. In order o evaluae he properies of GM model, we have ranslaed i ino sock flow diagram in his paper. In a sock-flow diagram he logic of programming code is more readily demonsraed and visualized. This approach was adoped for he proposed model, which we are comparing o he GM model. igure 1 illusraes he basic feaures and assumpions underlying he sock-flow diagram for he car-following siuaion considered in his paper. The Targe, he ead and he ollowing vehicle are moving in he same lane wih he same iniial speed and iniial spacing. The speed limi of roadway is assumed 100 f/sec. The ead ehicle encouners some obsrucion refer o as Targe along is pah causing he driver o decelerae in order o avoid a crash wih he Targe. I has been assumed for his paper ha all vehicles ravel in he same lane and only speed changes are allowed. ollowing ehicle ead ehicle Targe obsrucion Iniial Spacing Iniial Spacing igure1: Car-following siuaion considered in his paper
3.0 Sock-flow diagram of GM model igure 2 illusraes he sock-flow diagram of GM model. The GM model equaions are given in Appendix-A o his paper. The GM model is considered ino four secors: 1 he Targe, 2 he ead ehicle, 3 he ollowing ehicle, and 4 he Spacing. Each secor performs cerain funcions and ineracs wih he oher secors hrough feedback links. The deails of each secor are described below. 3.1 Targe Secor In igure 2, he Targe Obsrucion Secor is an exogenous secor, whose funcion is o describe he Targe in erms of required speed, and spacing for he ead ehicle. The speed of he Targe is se exogenously by he user, while he speed and spacings of he ead and ollowing ehicles is deermined endogenously subjec o he GM model described in secion 2 of his paper. ollowing ehicle ead ehicle Targe Obsrucion Time o Adjsu Targe speed ollowing ehicle Speed Change in ollowing ehicle speed ead ehicle Speed Change in ead ehicle speed Targe Speed Change in Targe Speed Targe desired speed Spacing ead ehicle Disance ollowing ehicle Disance Spacing beween ead & ollowing Spacing beween Targe & ead Targe Disance igure 2: Sock-flow diagram of GM model In his secor he sock variable Targe Speed represens he speed of he Targe. The rae of change in he speed of he Targe is deermined by he speed of he Targe and desired speed of he Targe for an assumed inerval of ime over which his change akes place. Any change in he speed of he Targe will change is relaive posiion wih respec o he ead ehicle. The spacing beween he Targe and he ead ehicle is calculaed in he Spacing Secor.
3.2 ead and ollowing ehicle Secor The process describing he ead and he ollowing ehicle Secor is similar. The only difference beween he ead and ollowing ehicle Secor is ha he ead ehicle driver considers he Targe while adjusing his or her speed and he ollowing ehicle driver considers he ead ehicle while adjusing his or her speed. The acceleraion/deceleraion rae for boh he ead and he ollowing ehicle driver is deermined by he GM model equaion 1. The values of parameers in equaion 1 are aken from Chakrobory and Kikuchi 1999. These values are: c = 69, l = 2, m = 1. Chakrobory and Kikuchi esimaed he value of c for a given l and m by fiing he corresponding speed-densiy relaion o an observed speed-densiy daa se obained from he Queen Elizabeh Way freeway in Canada. 3.3 Spacing Secor As shown in igure 2, in his secor here are wo socks, one deermines he curren spacing beween he Targe and he ead ehicle, while he oher deermines he curren spacing beween he ead ehicle and he ollowing ehicle. The disance ravelled during he simulaion inerval by he Targe, he ead ehicle, and he ollowing ehicle is deermined by heir respecive speeds, acceleraion/deceleraion raes and he simulaion ime inerval. This disance is deermined by he equaion of moion given below. S = * d + 0.5 * a * d 2 2 Where S is he disance ravelled during he simulaion inerval d, 'a' is he acceleraion/deceleraion rae during d and is he beginning of d. The nex secion describes he sock-flow diagram of a revised car-following model as proposed in his paper. 4.0 A more realisic car-following model To alleviae he shorcomings of exising GM models discussed earlier in his paper, a more realisic car-following model is presened o replicae he behaviour of following vehicle drivers. Thus proposed model considers endogenously he speed, acceleraion/deceleraion of he ollowing vehicles, and spacing beween he ead and ollowing vehicles. Road geomery, pavemen condiions, and weaher condiions are conrolled exernally. The basic difference beween his model and he exising car-following models is ha exising car-following models consider each vehicle pair separaely whereas he revised model considers several vehicles a a ime. urhermore, he revised model is unique in car-following heory in ha i inroduces a concep of "desired safe spacing or operaing speed" ino he formulaion. The desired speed and spacing is subjecive sandard ha reflecs "feeling of safey" on he par of an individual driver faced wih a given raffic siuaion. The sock-flow diagram of proposed model is given in igure 3. In he proposed model only he ead and he ollowing ehicle secors are differen from he GM model while he Targe and
he Spacing secors boh are same as described for GM model. The ead and he ollowing ehicle secors of proposed model are described below. ollowing ehicle ead ehicle Targe Obsrucion Percepion Reacion Time of ollowing eh driver ollowing eh Req speed wr ead Speed limi ead eh Req speed Targe Speed Percepion Reacion ime of ead veh driver ollowing eh Speed Change in ollowing eh speed ead eh Speed Targe Speed Change in Targe Speed Change in ead ehicle speed ollowing veh Preferred headway Eff of spacing on oll veh required speed Eff of spacing on ead veh required speed ~ ~ Eff of T arge on oll eh speed ~ ead eh Preferred headway Targe desired speed Spacing beween ollowing vehicle and Targe Time o Adjus Targe Speed ollowing vehicle desired spacing ead vehicle desired spacing Spacing Change in ollowing eh speed Change in ead ehicle speed Change in Targe Speed ollowing vehicle disance Spacing beween ead & ollowing ead ehicle disance Spacing beween Targe & ead Targe disance Spacing beween ollowing vehicle and Targe igure 3: Sock-flow diagram of proposed model The process describing he ead ehicle secor is similar o he ollowing ehicle Secor as described in he nex paragraph. The only difference beween he ead and ollowing ehicle Secor is ha he driver of he ollowing ehicle considers boh he spacing beween he Targe and he ead ehicle, and he spacing beween he ead and he ollowing ehicle when adjusing his or her speed. While adjusing his or her speed, he ead ehicle driver considers only he spacing beween he Targe and iself. In he ollowing ehicle secor, he sock variable ollowing ehicle Speed represens he curren speed of he ollowing ehicle. The acceleraion/deceleraion rae of he ollowing
ehicle driver is assumed o depend on his required safe speed wih respec o he ead ehicle, his own curren speed, and his percepion reacion ime. We have assumed for his iniial applicaion ha he percepion reacion ime is 2.5 sec Olson, 1986. The required safe speed for he ollowing ehicle is esablished based on he curren speed of he ead ehicle, speed limi of roadway assumed 100 f/sec, desired and curren spacing of he ead and ollowing ehicles. The desired spacing of a driver a a paricular speed is he disance ha he or she considers is safe and aemps o mainain i. The desired spacing for he ollowing ehicle is assumed he produc of is curren speed and a consan preferred ime headway of he ollowing ehicle driver Winsum, 1999 and Aycin, 1997. The preferred ime headway is defined as a headway he driver wans o mainain under car-following siuaion. Winsim 1999 has repored ha here are subsanial differences in he value of preferred headway beween drivers. or example, drivers who are less skilled generally choose o drive a a larger ime headway. In his paper we have assumed he preferred ime headway of he ollowing ehicle driver is 1.5 sec. The curren spacing is he acual spacing beween vehicles calculaed in he spacing secor. The facors such as pavemen condiions, pavemen fricion, road geomery, and raffic condiions can affec he required safe speed for he ollowing or he ead ehicle. or our iniial analysis in his paper, we have assumed ideal condiions and hese oher relaionships have no been incorporaed ino he model. 5.0 Discussion of comparaive resuls beween GM and proposed model In his secion a comparison is made beween he GM model enumeraed earlier in his paper and he proposed model. Boh GM and proposed model are esed for assumed range of ransporaion condiions. In his paper we will discuss only four experimens. These experimens are discussed below. Experimen 1 In his experimen simulaion run sars for boh GM and proposed model where all variables are parameerised o reflec equilibrium in all socks. Iniially, we assume ha he speeds of he Targe, ead, and ollowing ehicles are he same a 80 f/sec. urhermore, he curren spacing beween he Targe and he ead ehicle, and beween he ead and ollowing ehicle is assumed iniially o be equal o heir desired spacing 120 f. We now assume ha he speed of he Targe is decreased 25% from an iniial value of 80 f/sec a ime = 10.0 sec o final value of 60 f/sec a ime = 20.0. This reducion in he speed of he Targe could be in response o he on-se of congesion or he presence of some debris along he pah of he Targe. igure 4 illusraes he speed profiles of he Targe, ead ehicle and ollowing ehicle over 40.0 sec of simulaion inerval obained using he GM and he proposed model PM based on he same adjused Targe speed. igure 5 illusraes he spacing beween he ead ehicle and he Targe, and he ead ehicle and he ollowing ehicle over his same simulaion inerval obained using he GM and he proposed model. As shown in igure 4 and 5, he reducion in he speed of he Targe resuls in a corresponding reducion in he curren spacing beween he Targe and he ead ehicle. The ead ehicle
driver perceives he reducion in he Targe speed and he spacing beween he Targe and ead ehicle, and reduces his speed accordingly o mach ha of he Targe. So also he reducion in he speed of he ead ehicle resuls in a corresponding reducion in he curren spacing beween he ead and he ollowing ehicle. The driver of he ollowing ehicle also reacs in a similar fashion o reduce his or her speed o mach ha of he ead ehicle. In igure 4, he Targe, ead, and ollowing ehicle speeds are adjused accordingly unil a new equilibrium speed profile is esablished. This occurs when he vehicle and Targe speeds are equal o each oher. In igure 5, he curren spacing beween he Targe and he ead ehicle is reduced as a resul of a reducion in he speed of he Targe. However, when he ead ehicle speed and Targe speed become equal, he curren spacing beween he Targe and he ead ehicle sabilizes a is new equilibrium value. Similarly he spacing beween he ead ehicle and he ollowing ehicle is reduced and unil i reaches a equilibrium where all speeds are equal. 85 80 Targe GM ollowing GM ollowing PM ead GM ead PM Speed f/sec 75 70 65 60 55 igure 4: 0 5 10 15 20 25 30 35 40 Time sec Speed profiles of he Targe, ead and ollowing vehicle obained using GM and proposed model PM in experimen 1. There are no dramaic difference in he resuls obained from he GM model and he proposed model for his experimen. Boh models reflec similar speed and spacing profiles. We noe ha for he proposed model he equilibrium spacing beween he Targe and he ead ehicle, and beween he ead and he ollowing ehicle is higher han suggesed by he GM car-following model. All vehicles are spaced a 90 f for he former and 80 f for he laer. The difference beween 90 f and 80 f spacing reflecs our assumpion in he proposed model concerning driver preference as assumed in his applicaion.
Spacing f 130 120 110 100 90 Spacing beween Targe and ead GM Spacing beween ead and ollowing GM Spacing beween Targe and ead PM Spacing beween ead and ollowing PM 80 70 0 5 10 15 20 25 30 35 40 Time sec igure 5: Spacing profiles obained using GM and proposed model PM in experimen 1 Experimen 2: In experimen 2, we assumed ha he Targe, he ead, and he ollowing vehicle all are moving a a consan speed of 80 f/sec wih a spacing of 20 f apar. igure 6 and 7 shows he speed and spacing profiles obained using he GM and PM model. As shown in igure 6, here is no variaion in he speed profiles obained using GM model. This is because GM model assumes ha for zero relaive speed he acceleraion/deceleraion response is zero regardless of he spacing beween vehicles. However, according o he proposed model, when spacing a a paricular speed is lower han desired spacing for ha speed, he driver deceleraes firs and hen acceleraes o adjus he spacing o mach wih his or her desired spacing. This is shown in igure 6, he ead and he ollowing vehicles deceleraes firs o increase he spacing and hen accelerae o mach heir spacing wih heir desired spacing. The variaions in spacing profiles obained using proposed model is shown in igure 7, while spacing profiles obained using GM model show no variaion because speed of he ead and he ollowing vehicle is unchanged. These resuls reflec a ype of "harmonic" relaionship in drivers behaviour as hey adjus heir speeds and spacing o an equilibrium posiion in response o he Targe posiion and speed. Drivers of he ead and he ollowing ehicles iniially reduce heir speeds dramaically o avoid a collision a he minimum spacing and hen increase heir speeds o a desired level, wih he corresponding desired spacing. or his experimen he desired spacing is 120 f beween he Targe and he ead ehicle, and beween he ead ehicle and he ollowing ehicle. This corresponds o desired spacing ha we assumed in he model for a speed of 80 f/sec. Under he GM car-following model he suggesed spacing is 20 f, which remains unchanged from our iniial assumpions.
Speed f/sec 85 80 75 70 65 60 55 50 45 40 igure 6: Targe GM ollowing GM ollowing PM ead GM ead PM 0 10 20 30 40 50 Time sec Speed profiles of he Targe, ead and ollowing vehicle obained using GM and proposed model PM in experimen 2. 130 Spacing f 110 90 70 50 30 10 Spacing beween Targe and ead GM Spacing beween ead and ollowing GM Spacing beween Targe and ead PM Spacing beween ead and ollowing PM 0 10 20 30 40 50 Time sec igure 7: Spacing profiles obained using GM and proposed model PM in experimen 2. Experimen 3 In experimen 3, we assumed ha he Targe, he ead, and he ollowing vehicle all are moving a a consan speed of 80 f/sec wih a spacing of 200 f apar. igure 8 and 9 shows he speed and spacing profiles obained using he GM and proposed model.
As shown in igure 8, here is no variaion in he speed profiles obained using GM model since relaive speed is zero. However, according o he proposed model, when spacing is large, he driver acceleraes firs and hen deceleraes o reduce spacing o mach wih his or her desired spacing. This is shown in igure 8, he ead and he ollowing vehicles acceleraes firs o decrease he spacing and hen accelerae o mach heir spacing wih heir desired spacing. The variaions in spacing profiles obained using proposed model is shown in igure 9, while spacing profiles obained using GM model show no variaion because speed of he ead and he ollowing vehicle is unchanged. Speed f/sec 110 100 90 80 70 60 50 Targe GM ollowing GM ollowing PM ead GM ead PM 40 igure 8: 0 10 20 30 40 50 Time sec Speed profiles of he Targe, ead and ollowing vehicle obained using GM and proposed model PM in experimen 3. The resuls of experimen 4 also reflec a ype of "harmonic" relaionship in drivers behaviour as hey adjus heir speeds and spacing o an equilibrium posiion in response o he Targe posiion and speed. Drivers of he ead and he ollowing ehicles iniially increase heir speeds a a given large spacing of 200 f and hen decrease heir speeds o a desired level, wih he corresponding desired spacing. In his experimen he desired spacing is also 120 f beween he Targe and he ead ehicle, and beween he ead ehicle and he ollowing ehicle. This corresponds o desired spacing ha we assumed in he model for a speed of 80 f/sec. Under he GM car-following model he suggesed spacing is 200 f, which remains unchanged from our iniial assumpions.
Spacing f 240 220 200 180 160 140 120 Spacing beween Targe and ead GM Spacing beween ead and ollowing GM Spacing beween Targe and ead PM Spacing beween ead and ollowing PM 100 0 10 20 30 40 50 Time sec igure 9: Spacing profiles obained using GM and proposed model PM in experimen 3. Experimen 4 In his experimen we assumed ha iniially he Targe, he ead, and he ollowing vehicle all are moving a a speed of 110 f/sec wih a spacing of 120 f apar. igure 10 and 11 shows he speed and spacing profiles obained using he GM and proposed model. As shown in igure 10 and 11, here is no variaion in he speed and spacing profiles obained using he GM model since relaive speed is zero. However, according o he proposed model, he ead and he ollowing ehicle driver adjus heir speeds and spacing o heir assumed desired levels. The desired speed in his experimen for boh he ead and he ollowing ehicle is equal o he assumed speed limi 100 f/sec while he corresponding desired spacing for boh he ead and he ollowing ehicle is 150 f. In igure 11, spacing beween he Targe and he ead ehicle is consanly increasing because speed of he Targe is higher han he speed of he ead ehcicle. The resuls of experimen 4 reflec our assumpion ha drivers will no follow heir Targe if he speed of he Targe is more han he speed limi. They will adjus heir speeds and spacing according heir desired levels. The GM car-following model suggess ha drivers always follow heir Targe even if he speed of he Targe is more han he speed limi.
Speed f/sec 115 110 105 100 95 90 85 Targe GM ollowing GM ollowing PM ead GM ead PM 80 igure 10: 0 10 20 30 40 50 Time sec Speed profiles of he Targe, ead and ollowing vehicle obained using GM and proposed model PM in experimen 4 Spacing f 620 520 420 320 220 120 Spacing beween Targe and ead GM Spacing beween ead and ollowing GM Spacing beween Targe and ead PM Spacing beween ead and ollowing PM 20 0 10 20 30 40 50 Time sec igure 11: Spacing profiles obained using GM and proposed model PM in experimen 4. 6.0 Conclusions In his paper we have discussed a number of exising car-following models and have idenified heir several shorcomings. We have presened a revised car-following model based on Sysem
Dynamics principles, which aemps o address many of hese shorcomings. The proposed model assumes ha drivers consider a longer view of he road ahead, raher han jus he behaviour of vehicle immediaely in fron. The model also akes ino accoun he driver's desired speed and spacing in relaion o increased risk of collisions. In his paper we compared he speed and spacing profiles of differen vehicles wih respec o he speed and posiion of he Targe. We conduced four experimens concerning speed and spacing of he ead and ollowing ehicle for differen iniial assumpions. or hese experimens he speed and posiion of a Targe ehicle is given over he simulaion period. These experimens sugges ha he proposed car-following model yields more realisic resuls han suggesed by he exising GM car-following model. In he proposed model drivers seek o mainain he speed and spacing ha is consisen wih heir undersanding of he risks involved for any raffic siuaion. Obviously he experimens conduced in his exercise have been simplified for he purpose of demonsraing he proposed model. In fuure we inend o carry ou a horough validaion of he proposed car-following model wih respec o observed micro-level vehicle racking daa.
Appendix-A: GM model equaions: Targe Obsrucion: Targe_Speed = Targe_Speed - d + Change_in_Targe_Speed * d INIT Targe_Speed = Targe_desired_speed INOWS: Change_in_Targe_Speed = Targe_desired_speed- Targe_Speed/Time_o_Adjsu_Targe_speed Targe_desired_speed = 80-sep20,10 Time_o_Adjsu_Targe_speed = 2.5 ead ehicle: ead_ehicle_speed = ead_ehicle_speed - d + Change_in_ead_ehicle_speed * d INIT ead_ehicle_speed = Targe_desired_speed INOWS: Change_in_ead_ehicle_speed = 69*ead_ehicle_Speed*Targe_Speed- ead_ehicle_speed/spacing_beween_targe_&_ead^2 ollowing ehicle: ollowing_ehicle_speed = ollowing_ehicle_speed - d + Change_in_ollowing_ehicle_speed * d INIT ollowing_ehicle_speed = Targe_desired_speed INOWS: Change_in_ollowing_ehicle_speed = 69*ollowing_ehicle_Speed*ead_ehicle_Speed- ollowing_ehicle_speed/spacing_beween_ead_&_ollowing^2 Spacing secor: Spacing_beween_ead_&_ollowing = Spacing_beween_ead_&_ollowing - d + ead_ehicle_disance - ollowing_ehicle_disance * d INIT Spacing_beween_ead_&_ollowing = 120 INOWS: ead_ehicle_disance = ead_ehicle_speed+0.5*change_in_ead_ehicle_speed*d OUTOWS: ollowing_ehicle_disance = ollowing_ehicle_speed+0.5*change_in_ollowing_ehicle_speed*d Spacing_beween_Targe_&_ead = Spacing_beween_Targe_&_ead - d + Targe_Disance - ead_ehicle_disance * d INIT Spacing_beween_Targe_&_ead = 120
INOWS: Targe_Disance = Targe_Speed+0.5*Change_in_Targe_Speed*d OUTOWS: ead_ehicle_disance = ead_ehicle_speed+0.5*change_in_ead_ehicle_speed*d
Appendix-B Proposed model equaions: Targe Obsrucion: Targe_Speed = Targe_Speed - d + Change_in_Targe_Speed * d INIT Targe_Speed = Targe_desired_speed INOWS: Change_in_Targe_Speed = Targe_desired_speed- Targe_Speed/Time_o_Adjus_Targe_Speed Targe_desired_speed = 80-sep20,10 Time_o_Adjus_Targe_Speed = 2.5 ollowing ehicle: ollowing_eh_speed = ollowing_eh_speed - d + Change_in_ollowing_eh_speed * d INIT ollowing_eh_speed = Targe_desired_speed INOWS: Change_in_ollowing_eh_speed = ollowing_eh_req_speed_wr_ead- ollowing_eh_speed/percepion_reacion_time_of_ollowing_eh_driver+targe_speed- ollowing_eh_speed/percepion_reacion_time_of_ollowing_eh_driver*eff_of_targe_ on_oll_eh_speed ollowing_vehicle_desired_spacing = ollowing_eh_speed*ollowing_veh_preferred_headway ollowing_veh_preferred_headway = 1.5 ollowing_eh_req_speed_wr_ead = ifead_eh_speed*eff_of_spacing_on_oll_veh_required_speed > Speed_limi hen Speed_limi else ead_eh_speed*eff_of_spacing_on_oll_veh_required_speed Percepion_Reacion_Time_of_ollowing_eh_driver = 2.5 Eff_of_spacing_on_oll_veh_required_speed = GRAPHSpacing_beween_ead_&_ollowing/ollowing_vehicle_desired_spacing 0.00, 0.00, 0.167, 0.26, 0.333, 0.47, 0.5, 0.64, 0.667, 0.78, 0.833, 0.9, 1, 1.00, 1.17, 1.09, 1.33, 1.17, 1.50, 1.23, 1.67, 1.28, 1.83, 1.31, 2.00, 1.33 Eff_of_Targe_on_oll_eh_speed = GRAPHSpacing_beween_ollowing_vehicle_and_Targe/ollowing_vehicle_desired_spacing 0.00, 1.00, 0.0833, 0.72, 0.167, 0.52, 0.25, 0.385, 0.333, 0.295, 0.417, 0.22, 0.5, 0.165, 0.583, 0.125, 0.667, 0.09, 0.75, 0.065, 0.833, 0.04, 0.917, 0.02, 1.00, 0.00
ead ehicle: ead_eh_speed = ead_eh_speed - d + Change_in_ead_ehicle_speed * d INIT ead_eh_speed = Targe_desired_speed INOWS: Change_in_ead_ehicle_speed = ead_eh_req_speed- ead_eh_speed/percepion_reacion_ime_of_ead_veh_driver ead_vehicle_desired_spacing = ead_eh_speed*ead_eh_preferred_headway ead_eh_preferred_headway = 1.5 ead_eh_req_speed = iftarge_speed*eff_of_spacing_on_ead_veh_required_speed>speed_limi hen Speed_limi else Targe_Speed*Eff_of_spacing_on_ead_veh_required_speed Percepion_Reacion_ime_of_ead_veh_driver = 2.5 Speed_limi = 100 Eff_of_spacing_on_ead_veh_required_speed = GRAPHSpacing_beween_Targe_&_ead/ead_vehicle_desired_spacing 0.00, 0.00, 0.167, 0.26, 0.333, 0.47, 0.5, 0.64, 0.667, 0.78, 0.833, 0.9, 1, 1.00, 1.17, 1.09, 1.33, 1.17, 1.50, 1.23, 1.67, 1.28, 1.83, 1.31, 2.00, 1.33 Spacing secor: Spacing_beween_ead_&_ollowing = Spacing_beween_ead_&_ollowing - d + ead_ehicle_disance - ollowing_vehicle_disance * d INIT Spacing_beween_ead_&_ollowing = ollowing_vehicle_desired_spacing INOWS: ead_ehicle_disance = ead_eh_speed+0.5*change_in_ead_ehicle_speed*d OUTOWS: ollowing_vehicle_disance = ollowing_eh_speed+0.5*change_in_ollowing_eh_speed*d Spacing_beween_Targe_&_ead = Spacing_beween_Targe_&_ead - d + Targe_disance - ead_ehicle_disance * d INIT Spacing_beween_Targe_&_ead = ead_vehicle_desired_spacing INOWS: Targe_disance = Targe_Speed+0.5*Change_in_Targe_Speed*d OUTOWS: ead_ehicle_disance = ead_eh_speed+0.5*change_in_ead_ehicle_speed*d Spacing_beween_ollowing_vehicle_and_Targe = Spacing_beween_ead_&_ollowing+Spacing_beween_Targe_&_ead
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