Lecture Objectives & Reading Material

CEE6984: Special Topics – Transportation Sustainability Traffic Flow Theory Basics and Longitudinal Vehicle Motion Modeling Hesham Rakha and Kyoungho Ahn Virginia Tech

Lecture Objectives & Reading Material 

Lecture Objectives: 

Model the motion of a single vehicle  



Model the motion of a vehicle considering its interaction with a lead vehicle  



Pipes car-following model Rakha-Pasumarthy-Adjerid (RPA) car-following model

Model the motion of a traffic stream 



Vehicle deceleration Vehicle acceleration

Flow continuity, hydrodynamic and traffic stream fundamental diagram

Reading material outlined in course syllabus

Transportation Sustainability

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1

Vehicle Deceleration Modeling Background 

Modeling:  



State-of-practice consider a vehicle kinematics model Better approach is to consider vehicle as a point mass and model all forces acting on the vehicle.

Delay: 

Occurs any time a vehicle travels at a speed less than the free-flow speed.

N æ u ç d = Dt å çç 1 - t uf è t =0ç

ö÷ ÷÷ ÷÷ ø

Where: d = Total delay for entire trip for a specific vehicle (s) ∆t = Analysis increment duration (s) N = Total number of increments along trip (unitless) (T/ ∆t) T = Total trip duration (s) ut = Vehicle speed at any instant “t” (m/s) uf = Facility free-speed (m/s)

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Vehicle Braking Vehicle Dynamics 

A vehicle braking while traveling uphill experiences a number of forces: 

Resultant force 



Friction resistance 





Wfcos

Weight resistance 



(W/g)a

Wsin

Aerodynamic resistance at CG Rolling resistance at pavement

Where: W = weight of vehicle (N) f = braking coefficient of friction (unitless) a = vehicle acceleration (m/s2) g = acceleration of gravity (m/s2) V0 = initial vehicle speed (m/s) Db = braking distance (m)  = angle of incline (degrees) G = grade (tan) (unitless) (W/g)a Ra Wf cos

W sin  CEE5604

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Vehicle Braking Aerodynamic Resistance Force 

Aerodynamic resistance originates from a number of sources:   



Turbulent flow of air around the vehicle body particularly the rear portion (85%) Friction of the air passing over the vehicle body (12%) Airflow through vehicle components (3%)

Aerodynamic resistance increases as air becomes more dense r v2 Ra =

Af V CD Ch



C1 H

2

C D C h Af

( 3.62 )

= C 1C D C h Af v 2

C h = 1 - 8.5 ´ 10-5 H

Vehicle frontal area (projected area of vehicle in direction of travel) (m2) Vehicle speed (km/h) Vehicle drag coefficient (unitless) Altitude coefficient (unitless) Air density at sea level and 15oC (1.2256 kg/m3) Constant (0.047284) Altitude (m)

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Vehicle Braking Aerodynamic Resistance Force 

The drag coefficient accounts for 3 aerodynamic sources: 

Measured from empirical data either from wind tunnel experiments or actual field tests 

Vehicle Type

Drag Coefficient

Automobile

0.25 – 0.55

Bus

0.50 – 0.70

Tractor-Trailer

0.60 – 1.30

Motorcycle Vehicle is allowed to decelerate from a known speed with other sources of resistance accounted for Vehicle

0.27 – 1.80 Drag Coefficient

1967 Chevrolet Corvette 0.50 Headlights

Windows Roof

CD

Change

1967 Volkswagen Beetle 0.46

Closed

Closed

Closed

0.363

0%

1977 Triumph TR77

Open

Closed

Closed

0.380

+5%

1977 Jaguar XJS

0.36

Closed

Open

Closed

0.381

+5%

1987 Acura Integra

0.34

Closed

Closed

Open

0.389

+7%

1987 Fort Taurus

0.32

Closed

Open

Open

0.447

+23%

1997 Lexus LS400

0.29

Open

Open

Open

0.464

+28%

1997 Infiniti Q45

0.29

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0.40

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Vehicle Braking Rolling Resistance Force 

Rolling resistance is the resistance generated from  



Sources of rolling resistance:   



Vehicle’s internal mechanical friction, and Pneumatic tires and their interaction with the roadway surface Deformation of tire on roadway surface (90%) Penetration of tire into the roadway surface (4%) Frictional motion due to tire slippage (fanning effect) (6%)

Three factors should be considered:   

Tire pressure, Roadway surface compression, and Tire deformation.

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Vehicle Braking Rolling Resistance Force 

é C ù Rr = frlW = êê r (c2 v + c3 ) úú éë 9.8066 ´ M ùû ë 1000 û

Rolling resistance: 







M

Asphalt

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Vehicle mass (kg),

Speed of vehicle (km/h), v Equal to coefficient of Rolling coefficient, and Cr rolling resistance (frl) c2, c3 Rolling resistance coefficients. multiplied by weight normal Tire Type c2 c3 to roadway surface Bias Ply 0.0438 6.100 Increases as pavement Radial 0.0328 4.575 surface deteriorates Pavement Type Pavement Condition Higher for asphalt versus Excellent concrete surface Concrete Good Poor Higher for “Bias Ply” Good versus “Radial” tires

Rakha

Cr 1.00 1.50 2.00 1.25

Fair

1.75

Poor

2.25 8

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Vehicle Braking Considering all Forces 



Taking moments around the contact points at the rear and front tires and assuming cos=1.0 Wf =

1é Wlr - h ( ma + Ra + W sin g ) ùû Lë

Wr =

1é Wl f + h ( ma + Ra + W sin g ) ùû Lë

Summing forces along the vehicle’s longitudinal axis Fb + frlW = -ma - Ra - W sin g 1 Wf = éëWlr + h ( Fb + frlW ) ùû L 1 Wr = éëWl f - h ( Fb + frlW ) ùû L



ma Ra

At maximum braking Fbf max = Wf 



=coefficient of road adhesion

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Rrlr Fbr

Fbf

Rrlf

W sin

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Vehicle Braking Considering all Forces Coefficient of Road Adhesion Pavement



Maximum

Sliding

Good, dry

1.00

0.80

Good, wet

0.90

0.60

Poor, dry

0.80

0.55

Poor, wet

0.60

0.30

Packed snow or ice

0.25

0.10

Fbf max  Fbr max 

W L

W L

l r

 h  f rl 

l f

 h  f rl 

To approach maximum braking forces (g): 

Vehicle systems must correctly distribute braking forces between the vehicle’s front and rear brakes 

Typically done by allocating the hydraulic pressures within the braking system in the ratio of Fbf max/Fbr max 



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Referred to as the Brake Force Ratio (BFRmax)

Percentage of Braking Force (PBF) on front and rear axle can be computed 100 100 l  h   f rl  PBFf  100  PBFr  BFR max  r 1  BFR 1  BFR max l f  h   f rl  max Rakha

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Vehicle Braking Considering all Forces 

Design of vehicle braking system is not an easy task:  Optimal brake-force proportioning changes with vehicle and road conditions  





Addition of vehicle cargo changes the weight distribution and the height of the center of gravity (h) Changes in road conditions change the road adhesion coefficient

If wheels lock:  Preferable to have front wheels lock first because rear wheel locking results in loss of steering control Anti-lock brakes serve two purposes:  Prevent the coefficient of road adhesion from dropping to slide values  Raise braking efficiency to 100% (b)

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Vehicle Braking Considering all Forces – Theoretical a = x

dv dx dv v = dx dt dx

ò a dx

v2

=

0 v2

ò v dv

x = -ò gb m v1

a =-

v1

x = -ò gb m v1 v2

But

Fb + å R gb m

v dv Fb + å R

x =

v dv hb mW + Ka v 2 + frlW  WG

Rakha

gb ( v12 - v22 )

2g ( hb m + frl  G ) Ignoring rolling resistance

x =

Assuming frl constant g W çæ h mW + Ka v12 + frlW  WG ÷ö x = b ln çç b ÷÷ 2Ka g çè hb mW + Ka v22 + frlW  WG ÷÷ø CEE5604

Ignoring aerodynamic resistance

gb ( v12 - v22 ) 2g ( hb m  G )

Where: b = Mass factor accounting for moments of inertia (1.04) (Wong, 1978) Ka = /2CDChAf v1 and v2 = Speed (m/s) 12

6

Vehicle Acceleration Modeling Vehicle Dynamics Approach 

The vehicle’s maximum acceleration can be computed using the resultant tractive force 

Vehicle dynamics model

Ff + Fr = Ma + Ra + Rrlf + Rrlr + Rg F = Ma + Ra + Rrl + Rg a=

F -R M

ma Ra

W sin  Fr

Rrlr

Ff

Rrlf

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Where: Ff, Fr = Front and rear axle tractive forces (N) F = Total tractive force (N) Ra = Aerodynamic resistance (N) Rrlf, Rrlr = Front and rear axle rolling resistance (N) Rrl = Total rolling resistance (N) Rg = Grade resistance (N) R = Total resistance force (N) M = Vehicle mass (kg) a = vehicle acceleration (m/s2)  = angle of incline (degrees) G = grade (tan) (unitless)

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Vehicle Acceleration Modeling Vehicle Dynamics Approach Fn (t ) - Rn (t ) t mn æ ö P Fn (t ) = min ççç 3600 fp bhd n , m 'n g m ÷÷÷ ÷ø çè un (t ) c r Rn (t ) = CdC h Af un (t )2 + mn g r 0 (cr 1un (t ) + cr 2 ) + mn gG (t ) 25.92 1000

u n (t +t ) = u n (t ) + 3.6

where Fn(t) is vehicle tractive force (N), Rn(t) is total resistance force (N), mn is vehicle mass (kg),fp is the driver throttle input [0,1], β is the gear reduction factor (unitless), ηd is the driveline efficiency (unitless), Pn is the vehicle power (kW), m’n is the mass of vehicle n on its tractive axle (kg), g is the gravitational acceleration (9.8067 m/s2), μ is the coefficient of friction (unitless), ρ is the air density at sea level (1.2256 kg/m3), Cd is the vehicle drag coefficient (unitless), Ch is the altitude correction factor (unitless), Af is the vehicle frontal area (m2), cr0 is the rolling resistance constant (unitless), cr1 is the rolling resistance constant (h/km), cr2 is the rolling resistance constant (unitless), and G(t) is the roadway grade at instant t (unitless).

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Car-following Modeling Overview 

Car-following behavior described as: 



Molecular approach to modeling traffic behavior

Automobile driving consists of three major factors:   

Driver (human) Car (machine) Environment Roadway  Surrounding traffic 



Modeling of vehicle motion: 

Can be described using a control-systems approach

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Car-following Modeling Control Systems 

Control systems can be classified as: 

Open-loop control system:  Not 



controlled while in motion

Unguided missile

Closed-loop or feedback control system:  Measures

the quantity to be controlled and compares to a desired value 



Person driving a car

Closed-loop systems constitute a unified theory independent of the particular application

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Car-following Modeling Feedback Control 

Variables:     

Reference input variable r(t) Controlled output variable c(t) Feedback signal b(t) Error e(t) Solving the control system can be achieved using a Laplace transformation of the time function 

R(s) = L[r(t)]

Input

Error

+

e(t)

r(t)

Output

g(t)

c(t)

Feedback b(t) Transportation Sustainability

h(t)

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Car-following Modeling Feedback Control 

System:  Driver attempts to maintain a desired speed  Errors in speeds are estimated by comparing the desired to actual speed  Control of speed is made by depressing the accelerator in proportion to the speed deficiency  This action provides more fuel to the car engine and the car speeds up  The speeding up of the car results in a vehicle speed that is displayed on the speedometer Input r(t)

Error e(t)

+

Driver g1(t)

Vehicle g2(t)

Output c(t)

Feedback

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Speedometer h(t) Rakha and Ahn

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Car-Following Models Overview  

Theory that describes how one vehicle follows another vehicle Car-following theories were developed primarily in the 1950s and 1960s Reuschel and Pipes were pioneers  Kometani and Sasaki  Forbes and Herman  Gazis  GM models  Wiedemann, Gipps, Fritzche, Van Aerde and Rakha, Treiber and Kesting, Rakha et al. 



Car-following theory continues to be developed to this day

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Car-Following Models Basic Notations Notations: n = Follower vehicle = Length of follower vehicle Ln xn = Position of follower vehicle = Speed of follower vehicle un an = Acceleration of follower vehicle

Notations: n-1 = Lead vehicle Ln-1 = Length of lead vehicle = Position of lead vehicle xn-1 un-1 = Speed of lead vehicle an-1 = Acceleration of lead vehicle

Ln

Ln-1

n

n-1

xn(t)

Xn-1(t) Xn-1(t)-xn(t)

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Car-Following Models Basic Definitions 

Distance headway (spacing): 

Distance from a selected point on the lead vehicle to the same point on the following vehicle 



sn(t) = Ln-1 + gn(t)  sn(t) = spacing of vehicle (n) at time t  Ln-1 = physical length of vehicle n-1  gn(t) = gap between vehicle “n-1” and “n” at time t

Time headway: 

Time headway between vehicles 

hn(t) = sn(t)/un(t)  hn(t) = time headway of vehicle n at time t  un(t) = speed of vehicle n at time t

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Car-Following Models Concept of Safe Headway 

Safe spacing includes:  Travel distance during time lag perception reaction time (s1) 



Length of lead vehicle plus safety buffer (s2) 



Assume constant speed

Safety buffer is spacing between rear bumper of lead and front bumper of follower when vehicles are stopped

Distance required to come to a complete stop (s3 & s4) n

s1

s2

n-1

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s3

s4

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Car-Following Models Concept of Safe Headway 

Safe spacing computed as: 



If both vehicles are traveling at equal speeds and deceleration rates are equal 



s1 + s2 + s3 – s4

s1 = Tun (t )

s3 = s4

Safe spacing:  

s = s1 + s2 = Tun(t) + sj Solving for the vehicle speed: un(t) = 1/T x (s – sj)  un(t) = 1/α x (s – sj)

s2 = s j s3 =



 Transportation Sustainability

s4 =

Where: α is the inverse of driver sensitivity factor

un2 (t ) 2a un2-1 (t -T ) 2a

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Car-Following Models Linear Car-Following – Pipes’ Model 

Characterizes the motion of a vehicle based on the rule: 



A safe distance is to allow yourself at least the length of a car between you and the vehicle ahead of you for every 10 mph of speed that you are traveling at

Model accounts for:   

Time lag: Speed at time “t” uses spacing at time “t-Δt” Minimum spacing when vehicles are fully stopped (sj) Driver sensitivity factor (α-1) (units of seconds-1)

sn (t - Dt ) = s j + aun (t ) Transportation Sustainability

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Car-Following Models Linear Car-Following – Pipes’ Model Model requires the calibration of 3 parameters to local conditions:





Free-flow speed, vehicle spacing when stopped, and driver sensitivity factor

140

Speed (km/h)

120 100 80

(

60

un (t ) = min l êéx n -1 (t - Dt ) - x n (t - Dt ) - s j úù , u f ë û

40

)

20 0 0

20

40

60

80

100

Distance headway (m) Transportation Sustainability

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Car-Following Models Linear Car-Following – Pipes’ Model 

Pipes’ model:   

Consistent with field data for speeds from 20 to 70 km/h Overestimates speeds for short headways Overestimates speeds for long headways 120

Speed (km/h)

100 80 Field results

60 40 20 0 0

20

40

60

80

100

Distance headway (m) Transportation Sustainability

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Van Aerde Model Overview  Model introduces curvature to the Pipes Model 

Excellent fit with field data

140

Speed (km/h)

120 100 80

Field results

60 40

sn (t - Dt ) = c1 +

20

c2

u f - un (t )

+ c3un (t )

0 0

20

40

60

80

100

Distance headway (m) Transportation Sustainability

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Van Aerde Model Calibration 

Calibration of model requires calibrating four macroscopic parameters: 

Free-flow speed, speed-at-capacity, capacity, and jam density 120

Speed (km/h)

100 80 60 40 Four-parameter Model Field Data

20 0 0

100

200

300

400

500

Headway (m) Transportation Sustainability

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RPA Car-following Model 

The INTEGRATION car-following model can be cast as: 

Minimum based on vehicle dynamics 



Aerodynamic, rolling, and grade resistance forces Engine tractive and maximum sustainable force between vehicle wheels and pavement

ì ü F (t ) - Rn (t ) ï ï ï ï un (t ) + 3.6 ⋅ n Dt, ï ï ï ï m ï ï ï ï un (t + Dt ) = min í 2 ý é ù é ù ïï -c1¢ + c3u f + sn (t ) - c1¢ - c3u f - sn (t ) - 4c3 sn (t )u f - c1¢u f - c2 ï ï ë û ë û ï ï ï ï ï ï 2 c ï ï î þ 3 Transportation Sustainability

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RPA Car-following Model Van Aerde Steady-state Model 

Van Aerde formulated as: sn (t ) = c1 + c3un (t + Dt ) +



c2 u f - un (t + Dt )

Van Aerde model parameters are calibrated as: æ u u u ç1 c = u - u ) ; c = çç 2u - u ) ; c = ( ( ççq ku ku ku f

1



2 j c

2

f

c

f

2

2 j c

f

c

f

3

è

c

j

ö÷ ÷÷ 2÷ ÷÷ c ø

Wave speed at jam density is computed as: w = - qcuf k j u f - qc

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RPA Car-following Model Collision Avoidance Constraints 2 æ u (t +t )2 - u n - 1 (t +t ) s n (t ) = sn (t ) + min ççç 0, n çè 25920 m fb hb g

u n (t +t ) =

ö÷ ÷÷ ø÷÷

u n -1(t +t )2 + 25920 m fb hb g ( s n (t ) - sn (t ) )

Where kj is jam density (veh/km) and un-1(t) is speed of vehicle n-1 at time t (km/h). This deceleration level is assumed to be equal to μfbηbg, where μ is the coefficient of roadway friction, fb is the driver brake pedal input [0,1], ηb is the brake efficiency [0,1], and g is the gravitational acceleration (9.8067 m/s2).

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RPA Car-Following Model

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Traffic Stream Parameters Overview 

Traffic stream parameters fall into two broad categories: 



Macroscopic parameters include: 



Macroscopic and microscopic parameters Flow (q), space-mean speed (u), and density (k)

Microscopic parameters include: 

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Vehicle headway (h), speed (u), and vehicle spacing (s)

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Traffic Flow Models 

Knowledge about the flow of traffic along a freeway is important to the traffic engineer in being able to assess traffic situations and make good decisions in a timely manner. There are three basic deterministic equations of traffic flow modeling:   



Equation of continuity (equation of mass-balance); Hydrodynamic relation; and Equation of traffic motion.

The traffic stream embodies these three concepts in its traffic flow characteristics, along an uninterrupted flow facility.

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Traffic Flow Continuity Equation 

The equation of continuity states that 

 

Vehicles accumulate on the freeway segment x if more vehicles enter the segment than leave it. All vehicles must be accounted for in the system. Mathematically, the equation of continuity (massbalance) is: dn = qin ( t ) - qout ( t ) dt

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Traffic Flow Continuity Equation 

Flow continuity equation:  The change in the number of vehicles on a length of roadway dx in an interval dt equals the difference between the number of vehicles entering the section at x and leaving the section at x+dx. ¶k ö÷ ¶q ö÷ æ æ k dx - çç k dt ÷dx = q dt - ççq + dx dt è è ¶t ø ¶x ø÷ ¶k ¶q + =0 ¶t ¶x dx (k,q) dt

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Hydrodynamic Relation 

The number of vehicles flowing along a freeway per unit time 



Equivalent to the number of vehicles crossing a line across the roadway per unit time.

The rate of traffic flow is: 



Product of the density of vehicles per unit length of freeway times the (space-mean) speed of the vehicles. The hydrodynamic relation of traffic flow is 

q=ku

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Traffic Stream Motion Equation 

The equation of motion: 



Traffic slows down over time as its density increases over space. For single lane traffic flow: 



“car-following” behavior

Empirical data: 

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A 3-D functional relationship can be established between the 3 variables

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19

Traffic Stream Motion Equation I-4 Data

Flow (veh/h/lane)

1500

1000

500

0 100

80

60

40

20

0

20

0

Speed (km/h)

80

60

40

Density (veh/km/lane)

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Traffic Stream Motion Equation Autobahn Data

2500

Flow (veh/h/lane)

2000 1500 1000 500 0 150

100

50

0

0

Speed (km/h)

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20

40

60

80

100

Density (veh/km/lane)

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40

20

Traffic Stream Motion Equation flow does not imply congestion

180 160 140 120 100 80 60 40 20 0

Speed (km/h)

Speed (km/h)

200 Low

0

500

1000

1500

2000

2500

3000

200 180 160 140 120 100 80 60 40 20 0

0

20

40

60

80

100

80

100

Density (veh/km)

3000

200 180 160 140 120 100 80 60 40 20 0

2500 Flow (veh/h)

Speed (km/h)

Flow (veh/h)

2000 1500 1000 500 0

0

100

200

300

400

500

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0

20

40

60

Density (veh/km)

Headway (m)

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Traffic Stream Motion Equation 

Four critical parameters: 







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Free-flow speed (uf):  Space-mean-speed of traffic when roadway is practically empty Speed-at-capacity (uc):  Space-mean-speed of traffic stream at maximum flow rate Capacity (qc):  Expected maximum sustainable flow rate over 15 minutes Jam density (kj):  Number of vehicles per unit distance when vehicles are completely stopped (queued)

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21

Traffic Stream Motion Equation Four traffic stream parameters define the shape of the speedflow-density relationship 180 160 140 120 100 80 60 40 20 0

Speed (km/h)

Speed (km/h)



0

500

1000

1500

2000

180 160 140 120 100 80 60 40 20 0

2500

0

20

40

60

80

100

80

100

Density (veh/km)

180 160 140 120 100 80 60 40 20 0

2500 2000 Flow (veh/h)

Speed (km/h)

Flow (veh/h)

1500 1000 500 0

0

100

200

300

400

0

500

20

40

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60

Density (veh/km)

Headway (m)

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Traffic Stream Motion Equation Greenshields Model u u  uf   f k  j

   k  uf 1   k      k j

  [1]  

Given that at capacity q/u=0 Setting the partial derivative of Eq. 3 to zero

uc 

Using the hydrodynamic relationship q=ku

 k q  kuf 1      k j

   

[2]

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  

[4]

Substituting Eq. 4 in Eq. 3

qc 

Substituting k for q/u in Equation 2

 u q  uk j 1     uf

uf 2 k j uf

[5]

4

Substituting Eq. 4 in Eq. 1 [3]

H. Rakha

kc 

kj

2

[6]

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Traffic Stream Motion Equation Van Aerde Model 

The Van Aerde functional form:  Is a single-regime model that combines the Greenshields and Pipes functional forms  Proposed by Van Aerde (1995) and Van Aerde & Rakha (1995)

k=

1 c1 + c3u +

c2 uf - u

Where: c1 = Fixed distance headway constant (km/veh), c2 = First variable distance headway constant (km2/h/veh), c3 = Second variable distance headway constant (h/veh), uf = Free-speed (km/h), u = Speed (km/h) kj = Jam density (veh/km).

CEE4604

45

Traffic Stream Motion Equation Van Aerde Model 

Calibration of model requires estimating four parameters:  

Free-flow speed, speed-at-capacity, capacity, and jam density Using these four parameters the constants c1, c2, and c3 are estimated

c1 =

CEE5604

u f (2uc - u f ) k j uc2

u f (u f - uc )

2

c2 =

2 j c

ku

Rakha

c3 =

uf 1 qc k j uc2

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Summary 

Car-following models: 



Describe the behavior of a vehicle following a lead vehicle

Require: Profile of lead vehicle  Boundary conditions for following vehicles  Time lag and car-following parameters 



RPA model considers a steady-state relationship, collision avoidance constraints and acceleration constraints

Transportation Sustainability

Rakha and Ahn

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