Chapter 34 Microscopic traffic flow modelig 34.1 Overview Macroscopic modelig looks at traffic flow from a global perspective, whereas microscopic modelig, as the term suggests, gives attetio to the details of traffic flow ad the iteractios takig place withi it. This chapter gives a overview of microscopic approach to modelig traffic ad the elaborates o the various cocepts associated with it. A microscopic model of traffic flow attempts to aalyze the flow of traffic by modelig driver-driver ad driver-road iteractios withi a traffic stream which respectively aalyzes the iteractio betwee a driver ad aother driver o road ad of a sigle driver o the differet features of a road. May studies ad researches were carried out o driver s behavior i differet situatios like a case whe he meets a static obstacle or whe he meets a dyamic obstacle. Several studies are made o modelig driver behavior i aother followig car ad such studies are ofte referred to as car followig theories of vehicular traffic. 34.2 Notatio Logitudial spacig of vehicles are of particular importace from the poits of view of safety, capacity ad level of service. The logitudial space occupied by a vehicle deped o the physical dimesios of the vehicles as well as the gaps betwee vehicles. For measurig this logitudial space, two microscopic measures are useddistace headway ad distace gap. Distace headway is defied as the distace from a selected poit (usually frot bumper) o the lead vehicle to the correspodig poit o the followig vehicles. Hece, it icludes the legth of the lead vehicle ad the gap legth betwee the lead ad the followig vehicles. Before goig i to the details, various otatios used i car-followig models are discussed here with the help of figure 34:1. The leader vehicle is deoted as ad the followig vehicle as ( + 1). Two characteristics at a istat t are of importace; locatio ad speed. Locatio ad speed of the lead vehicle at time istat t are represeted by x t ad v t respectively. Similarly, the locatio ad speed of the follower are deoted by x t +1 ad vt +1 respectively. The followig vehicle is assumed to accelerate at time t + T ad ot at t, where T is the iterval of time required for a driver to react to a chagig situatio. The gap betwee the leader ad the follower vehicle is therefore x t x t +1. Itroductio to Trasportatio Egieerig 34.1 Tom V. Mathew ad K V Krisha Rao
Directio of traffic v +1 + 1 Follower v Leader x x +1 x x +1 34.3 Car followig models Figure 34:1: Notatio for car followig model Car followig theories describe how oe vehicle follows aother vehicle i a uiterrupted flow. Various models were formulated to represet how a driver reacts to the chages i the relative positios of the vehicle ahead. Models like Pipes, Forbes, Geeral Motors ad Optimal velocity model are worth discussig. 34.3.1 Pipe s model The basic assumptio of this model is A good rule for followig aother vehicle at a safe distace is to allow yourself at least the legth of a car betwee your vehicle ad the vehicle ahead for every te miles per hour of speed at which you are travelig Accordig to Pipe s car-followig model, the miimum safe distace headway icreases liearly with speed. A disadvatage of this model is that at low speeds, the miimum headways proposed by the theory are cosiderably less tha the correspodig field measuremets. 34.3.2 Forbes model I this model, the reactio time eeded for the followig vehicle to perceive the eed to decelerate ad apply the brakes is cosidered. That is, the time gap betwee the rear of the leader ad the frot of the follower should always be equal to or greater tha the reactio time. Therefore, the miimum time headway is equal to the reactio time (miimum time gap) ad the time required for the lead vehicle to traverse a distace equivalet to its legth. A disadvatage of this model is that, similar to Pipe s model, there is a wide differece i the miimum distace headway at low ad high speeds. 34.3.3 Geeral Motors model The Geeral Motors model is the most popular of the car-followig theories because of the followig reasos: 1. Agreemet with field data; the simulatio models developed based o Geeral motors car followig models shows good correlatio to the field data. 2. Mathematical relatio to macroscopic model; Greeberg s logarithmic model for speed-desity relatioship ca be derived from Geeral motors car followig model. I car followig models, the motio of idividual vehicle is govered by a equatio, which is aalogous to the Newto s Laws of motio. I Newtoia mechaics, acceleratio ca be regarded as the respose of the particle Itroductio to Trasportatio Egieerig 34.2 Tom V. Mathew ad K V Krisha Rao
to stimulus it receives i the form of force which icludes both the exteral force as well as those arisig from the iteractio with all other particles i the system. This model is the widely used ad will be discussed i detail later. 34.3.4 Optimal velocity model The cocept of this model is that each driver tries to achieve a optimal velocity based o the distace to the precedig vehicle ad the speed differece betwee the vehicles. This was a alterative possibility explored recetly i car-followig models. The formulatio is based o the assumptio that the desired speed v desired depeds o the distace headway of the th vehicle. i.e.v t desired = v opt ( x t ) where v opt is the optimal velocity fuctio which is a fuctio of the istataeous distace headway x t. Therefore at is give by a t = [1/τ][V opt ( x t ) vt ] (34.1) where 1 τ is called as sesitivity coefficiet. I short, the drivig strategy of th vehicle is that, it tries to maitai a safe speed which itur depeds o the relative positio, rather tha relative speed. 34.4 Geeral motor s car followig model 34.4.1 Basic Philosophy The basic philosophy of car followig model is from Newtoia mechaics, where the acceleratio may be regarded as the respose of a matter to the stimulus it receives i the form of the force it receives from the iteractio with other particles i the system. Hece, the basic philosophy of car-followig theories ca be summarized by the followig equatio [Respose] α [Stimulus] (34.2) for the th vehicle (=1, 2,...). Each driver ca respod to the surroudig traffic coditios oly by acceleratig or deceleratig the vehicle. As metioed earlier, differet theories o car-followig have arise because of the differece i views regardig the ature of the stimulus. The stimulus may be composed of the speed of the vehicle, relative speeds, distace headway etc, ad hece, it is ot a sigle variable, but a fuctio ad ca be represeted as, a t = f sti (v, x, v ) (34.3) where f sti is the stimulus fuctio that depeds o the speed of the curret vehicle, relative positio ad speed with the frot vehicle. 34.4.2 Follow-the-leader model The car followig model proposed by Geeral motors is based o follow-the leader cocept. This is based o two assumptios; (a) higher the speed of the vehicle, higher will be the spacig betwee the vehicles ad (b) to avoid collisio, driver must maitai a safe distace with the vehicle ahead. Let x t +1 is the gap available for ( + 1)th vehicle, ad let x safe is the safe distace, v+1 t ad vt are the velocities, the gap required is give by, x t +1 = x safe + τv t +1 (34.4) Itroductio to Trasportatio Egieerig 34.3 Tom V. Mathew ad K V Krisha Rao
where τ is a sesitivity coefficiet. The above equatio ca be writte as x x t +1 = x safe + τv+1 t (34.5) Differetiatig the above equatio with respect to time, we get v t v t +1 = τ.a t +1 a t +1 = 1 τ [vt vt +1 ] Geeral Motors has proposed various forms of sesitivity coefficiet term resultig i five geeratios of models. The most geeral model has the form, [ a t αl,m (v t +1 = +1) m ] [v t (x t x t +1 )l v+1 t ] (34.6) where l is a distace headway expoet ad ca take values from +4 to -1, m is a speed expoet ad ca take values from -2 to +2, ad α is a sesitivity coefficiet. These parameters are to be calibrated usig field data. This equatio is the core of traffic simulatio models. I computer, implemetatio of the simulatio models, three thigs eed to be remembered: 1. A driver will react to the chage i speed of the frot vehicle after a time gap called the reactio time durig which the follower perceives the chage i speed ad react to it. 2. The vehicle positio, speed ad acceleratio will be updated at certai time itervals depedig o the accuracy required. Lower the time iterval, higher the accuracy. 3. Vehicle positio ad speed is govered by Newto s laws of motio, ad the acceleratio is govered by the car followig model. Therefore, the goverig equatios of a traffic flow ca be developed as below. Let T is the reactio time, ad t is the updatio time, the goverig equatios ca be writte as, v t = v t t x t = x t t a t +1 = [ + a t t t (34.7) + v t t α l,m (v t T (x t T +1 ) x t T +1 t + 1 2 at t t 2 (34.8) ][v t T v t T +1 ] (34.9) The equatio 34.7 is a simulatio versio of the Newto s simple law of motio v = u + at ad equatio 34.8 is the simulatio versio of the Newto s aother equatio s = ut + 1 2 at2. The acceleratio of the follower vehicle depeds upo the relative velocity of the leader ad the follower vehicle, sesitivity coefficiet ad the gap betwee the vehicles. Problem Let a leader vehicle is movig with zero acceleratio for two secods from time zero. The he accelerates by 1 m/s 2 for 2 secods, the decelerates by 1m/s 2 for 2 secods. The iitial speed is 16 m/s ad iitial locatio is 28 m from datum. A vehicle is followig this vehicle with iitial speed 16 m/s, ad positio zero. Simulate the behavior of the followig vehicle usig Geeral Motors Car followig model (acceleratio, speed ad positio) for 7.5 secods. Assume the parameters l=1, m=0, sesitivity coefficiet (α l,m ) = 13, reactio time as 1 secod ad sca iterval as 0.5 secods. Itroductio to Trasportatio Egieerig 34.4 Tom V. Mathew ad K V Krisha Rao
20 19 Leader Follower 18 Velocity 17 16 15 0 5 10 15 20 25 Time(secods) 30 Figure 34:2: Velocity vz Time Solutio The first colum shows the time i secods. Colum 2, 3, ad 4 shows the acceleratio, velocity ad distace of the leader vehicle. Colum 5,6, ad 7 shows the acceleratio, velocity ad distace of the follower vehicle. Colum 8 gives the differece i velocities betwee the leader ad follower vehicle deoted as dv. Colum 9 gives the differece i displacemet betwee the leader ad follower vehicle deoted as dx. Note that the values are assumed to be the state at the begiig of that time iterval. At time t=0, leader vehicle has a velocity of 16 m/s ad located at a distace of 28 m from a datum. The follower vehicle is also havig the same velocity of 16 m/s ad located at the datum. Sice the velocity is same for both, dv = 0. At time t = 0, the leader vehicle is havig acceleratio zero, ad hece has the same speed. The locatio of the leader vehicle ca be foud out from equatio as, x = 28+16 0.5 = 36 m. Similarly, the follower vehicle is ot acceleratig ad is maitaiig the same speed. The locatio of the follower vehicle is, x = 0+16 0.5 = 8 m. Therefore, dx = 36-8 =28m. These steps are repeated till t = 1.5 secods. At time t = 2 secods, leader vehicle accelerates at the rate of 1 m/s 2 ad cotiues to accelerate for 2 secods. After that it decelerates for a period of two secods. At t= 2.5 secods, velocity of leader vehicle chages to 16.5 m/s. Thus dv becomes 0.5 m/s at 2.5 secods. dx also chages sice the positio of leader chages. Sice the reactio time is 1 secod, the follower will react to the leader s chage i acceleratio at 2.0 secods oly after 3 secods. Therefore, at t=3.5 secods, the follower respods to the leaders chage i acceleratio give by equatio i.e., a = 13 0.5 28.23 = 0.23 m/s2. That is the curret acceleratio of the follower vehicle depeds o dv ad reactio time of 1 secod. The follower will chage the speed at the ext time iterval. i.e., at time t = 4 secods. The speed of the follower vehicle at t = 4 secods is give by equatio as v= 16+0.231 0.5 = 16.12 The locatio of the follower vehicle at t = 4 secods is give by equatio as x = 56+16 0.5+ 1 2 0.231 0.52 = 64.03 These steps are followed for all the cells of the table. The earliest car-followig models cosidered the differece i speeds betwee the leader ad the follower as the stimulus. It was assumed that every driver teds to move with the same speed as that of the correspodig leadig vehicle so that a t = 1 τ (vt+1 v t +1) (34.10) where τ is a parameter that sets the time scale of the model ad 1 τ ca be cosidered as a measure of the sesitivity of the driver. Accordig to such models, the drivig strategy is to follow the leader ad, therefore, such car-followig models are collectively referred to as the follow the leader model. Efforts to develop this stimulus fuctio led to five geeratios of car-followig models, ad the most geeral model is expressed Itroductio to Trasportatio Egieerig 34.5 Tom V. Mathew ad K V Krisha Rao
Table 34:1: Car-followig example t a(t) v(t) x(t) a(t) v(t) x(t) dv dx (1) (2) (3) (4) (5) (6) (7) (8) (9) t a(t) v(t) x(t) a(t) v(t) x(t) dv dx 0.00 0.00 16.00 28.00 0.00 16.00 0.00 0.00 28.00 0.50 0.00 16.00 36.00 0.00 16.00 8.00 0.00 28.00 1.00 0.00 16.00 44.00 0.00 16.00 16.00 0.00 28.00 1.50 0.00 16.00 52.00 0.00 16.00 24.00 0.00 28.00 2.00 1.00 16.00 60.00 0.00 16.00 32.00 0.00 28.00 2.50 1.00 16.50 68.13 0.00 16.00 40.00 0.50 28.13 3.00 1.00 17.00 76.50 0.00 16.00 48.00 1.00 28.50 3.50 1.00 17.50 85.13 0.23 16.00 56.00 1.50 29.13 4.00-1.00 18.00 94.00 0.46 16.12 64.03 1.88 29.97 4.50-1.00 17.50 102.88 0.67 16.34 72.14 1.16 30.73 5.00-1.00 17.00 111.50 0.82 16.68 80.40 0.32 31.10 5.50-1.00 16.50 119.88 0.49 17.09 88.84-0.59 31.03 6.00 0.00 16.00 128.00 0.13 17.33 97.45-1.33 30.55 6.50 0.00 16.00 136.00-0.25 17.40 106.13-1.40 29.87 7.00 0.00 16.00 144.00-0.57 17.28 114.80-1.28 29.20 7.50 0.00 16.00 152.00-0.61 16.99 123.36-0.99 28.64 8.00 0.00 16.00 160.00-0.57 16.69 131.78-0.69 28.22 8.50 0.00 16.00 168.00-0.45 16.40 140.06-0.40 27.94 9.00 0.00 16.00 176.00-0.32 16.18 148.20-0.18 27.80 9.50 0.00 16.00 184.00-0.19 16.02 156.25-0.02 27.75 10.00 0.00 16.00 192.00-0.08 15.93 164.24 0.07 27.76 10.50 0.00 16.00 200.00-0.01 15.88 172.19 0.12 27.81 11.00 0.00 16.00 208.00 0.03 15.88 180.13 0.12 27.87 11.50 0.00 16.00 216.00 0.05 15.90 188.08 0.10 27.92 12.00 0.00 16.00 224.00 0.06 15.92 196.03 0.08 27.97 12.50 0.00 16.00 232.00 0.05 15.95 204.00 0.05 28.00 13.00 0.00 16.00 240.00 0.04 15.98 211.98 0.02 28.02 13.50 0.00 16.00 248.00 0.02 15.99 219.98 0.01 28.02 14.00 0.00 16.00 256.00 0.01 16.00 227.98 0.00 28.02 14.50 0.00 16.00 264.00 0.00 16.01 235.98-0.01 28.02 15.00 0.00 16.00 272.00 0.00 16.01 243.98-0.01 28.02 15.50 0.00 16.00 280.00 0.00 16.01 251.99-0.01 28.01 16.00 0.00 16.00 288.00-0.01 16.01 260.00-0.01 28.00 16.50 0.00 16.00 296.00 0.00 16.01 268.00-0.01 28.00 17.00 0.00 16.00 304.00 0.00 16.00 276.00 0.00 28.00 17.50 0.00 16.00 312.00 0.00 16.00 284.00 0.00 28.00 18.00 0.00 16.00 320.00 0.00 16.00 292.00 0.00 28.00 18.50 0.00 16.00 328.00 0.00 16.00 300.00 0.00 28.00 19.00 0.00 16.00 336.00 0.00 16.00 308.00 0.00 28.00 19.50 0.00 16.00 344.00 0.00 16.00 316.00 0.00 28.00 20.00 0.00 16.00 352.00 0.00 16.00 324.00 0.00 28.00 20.50 0.00 16.00 360.00 0.00 16.00 332.00 0.00 28.00 Itroductio to Trasportatio Egieerig 34.6 Tom V. Mathew ad K V Krisha Rao
Acceleratio 1.5 1 0.5 0 Leader Follower 0.5 1 1.5 0 5 10 15 Time(secods) 20 25 30 Figure 34:3: Acceleratio vz Time mathematically as follows. a t+ T +1 ]m +1 = α l,m [v t T [x t T x+1 t T (vt T ]l v t T +1 ) (34.11) where l is a distace headway expoet ad ca take values from +4 to -1, m is a speed expoet ad ca take values from -2 to +2, ad α is a sesitivity coefficiet. These parameters are to be calibrated usig field data. 34.5 Simulatio Models Simulatio modelig is a icreasigly popular ad effective tool for aalyzig a wide variety of dyamical problems which are difficult to be studied by other meas. Usually, these processes are characterized by the iteractio of may system compoets or etities. 34.5.1 Applicatios of simulatio Traffic simulatios models ca meet a wide rage of requiremets: 1. Evaluatio of alterative treatmets 2. Testig ew desigs 3. As a elemet of the desig process 4. Embed i other tools 5. Traiig persoel 6. Safety Aalysis 34.5.2 Need for simulatio models Simulatio models are required i the followig coditios 1. Mathematical treatmet of a problem is ifeasible or iadequate due to its temporal or spatial scale Itroductio to Trasportatio Egieerig 34.7 Tom V. Mathew ad K V Krisha Rao
2. The accuracy or applicability of the results of a mathematical formulatio is doubtful, because of the assumptios uderlyig (e.g., a liear program) or a heuristic procedure (e.g., those i the Highway Capacity Maual) 3. The mathematical formulatio represets the dyamic traffic/cotrol eviromet as a simpler quasi steady state system. 4. There is a eed to view vehicle aimatio displays to gai a uderstadig of how the system is behavig 5. Traiig persoel 6. Cogested coditios persist over a sigificat time. 34.5.3 Classificatio of Simulatio Model Simulatio models are classified based o may factors like 1. Cotiuity (a) Cotiuous model (b) Discrete model 2. Level of detail (a) Macroscopic models (b) Mesoscopic models (c) Microscopic models 3. Based o Processes (a) Determiistic (b) Stochastic 34.6 Summary Microscopic traffic flow modelig focuses o the miute aspects of traffic stream like vehicle to vehicle iteractio ad idividual vehicle behavior. They help to aalyze very small chages i the traffic stream over time ad space. Car followig model is oe such model where i the stimulus-respose cocept is employed. Optimal models ad simulatio models were briefly discussed. 34.7 Problems 1. The miimum safe distace headway icreases liearly with speed. Which model follows this assumptio? (a) Forbe s model (b) Pipe s model (c) Geeral motor s model Itroductio to Trasportatio Egieerig 34.8 Tom V. Mathew ad K V Krisha Rao
(d) Optimal velocity model 2. The most popular of the car followig models is (a) Forbe s model (b) Pipe s model (c) Geeral motor s model (d) Optimal velocity model 34.8 Solutios 1. The miimum safe distace headway icreases liearly with speed. Which model follows this assumptio? (a) Forbe s model (b) Pipe s model (c) Geeral motor s model (d) Optimal velocity model 2. The most popular of the car followig models is (a) Forbe s model (b) Pipe s model (c) Geeral motor s model (d) Optimal velocity model Itroductio to Trasportatio Egieerig 34.9 Tom V. Mathew ad K V Krisha Rao