Unformatted text preview: CE 480 Transportation CE 480 Transportation Engineering Traffic Flow Fundamentals Fundamentals
Fundamentals Primary Elements of Traffic Streams
– Flow
– Density Gap/headway – Speed Used in planning, designing and evaluating traffic engineering measures on a highway system Applications
Applications Traffic flow theory is used for: – Design of left turn storage lengths in left turn lanes
– Average delay at intersections, freeway ramp merging areas
– Simulation Mathematical algorithms used to study complex interrelationships of a traffic stream
– Estimate impacts of: Travel time and delay Air pollution Fuel consumption Traffic Flow (Q)
Traffic Flow (Q) Flow (Q): hourly rate at which vehicles pass a point on a street/highway
Q = n × 3600/T where:
Q = hourly flow in vehicles per hour (vph) n = number of vehicles passing a point on the roadway in ‘T’ seconds Density (K)
Density (K) Also referred to as concentration
Number of vehicles traveling over a unit length of highway in an instant in time Unit length is usually 1 mile
∴ Unit of density is typically vehicles per mile (vpm) Speed (V)
Speed (V) Distance traveled by a vehicle during a unit of time
Expressed in mph or ft/sec
Two types of speed
– Time mean speed (Vt)
– Space mean speed (Vs) SMS is always used in traffic flow theory applications SMS weights slower moving vehicles more heavily
– Based on the amount of time they occupy the highway
TMS is always greater than SMS Time Mean Speed (TMS)
Time Mean Speed (TMS)
Vt
Average speed of all vehicles passing a point on a highway over a specified time period (point measure) Σ (d/ti) = i
TMS = Σ (vi)
i
n Where, n TMS in ft/sec or miles/hr
d = distance traveled (ft or mile) ti = travel time of ith vehicle (sec or hour) n = number of travel times observed
Vi = speed of ith vehicle (fps or mph) Space Mean Speed (SMS)
Space Mean Speed (SMS) Vs
Average speed of all vehicles occupying a given section of highway over some specified time period describes a length d
n
of a highway
Σ (ti)/n Σ (1/vi) SMS =
where, SMS in ft/sec or miles/hr
d = distance traveled (ft or mile)
ti = travel time of ith vehicle (sec or hour) Time Headway (h)
Time Headway (h) Difference between the time the front of a vehicle arrives at a point on the highway and the time the front of the next vehicle arrives at that same point
Usually expressed in seconds
Relationship between headway and flow h = 1/Q = 3600/Q 1 2
Time headway in seconds Space Headway (s)
Space Headway (s) Distance between the front of a vehicle and the front of a following vehicle
Usually expressed in feet Relationship with density (K)
1 2 s = 1 mile/K = 5280/K Space headway in feet Time and Space Headway
Time and Space Headway 2 Gap 1 Space headway (spacing) in feet
Time headway in seconds Fundamental Equation of a Vehicular Traffic Stream
QKV Relationship
QKV Relationship
Flow (veh/hr) = Density (veh/mile) x Speed (miles/hr) Q = K x V Where, Q= Flow K= Density V= Speed (also denoted by ‘u’ or ‘s’) Traffic Stream Model SPEEDDENSITY Speed (mph)
Behavior nonlinear Vf Vf = free flow speed
Kj = jam density Kj Density (veh/mile) Traffic Stream Model FLOWDENSITY
Flow (veh/h)
dq/dk=0 @ max flow Qm Qm = maximum flow
Km = optimum density 0 Km Kj Density (veh/mile) Basic Traffic Stream Models SPEEDFLOW
Speed (km/h) Vf Qm = maximum flow
Vm = optimum speed Vm 0 Qm Flow (veh/h) QKV Curves
QKV Curves
Flow (veh/hr) qmax Speed (miles/hr) Speed (miles/hr) Density (veh/mile) Density (veh/mile) Flow (veh/hr) Y = mX + b
Slope of VK line is (Vf/Kj) Vf
V1
Vm
V2 I Area under the speed
density curve is flow Optimum Flow II Kj Qmax Q1 = Q1
Flow Speed QKV Relationship
QKV Relationship K1 Km
Density K2 Qmax is considered to be the ‘capacity’ of a roadway K1V1 K2V2 Typical ranges of jam density are: 180 vpmpl < Kj < 225 vpmpl
(vpmpl) = veh per mile per lane Free Flow Speed
Free Flow Speed Varies based on:
– Type of facility – Time of day Some constrictions include: – Presence of police officers, speed capabilities of vehicle, friction, road curvature, etc. – Avg. free flow speeds on highway ≈ 75 to 80 MPH
– On residential streets free flow speed ≈ 35 MPH Constrictions may include more curves, presence of children, etc. Greenshield’s Model
Greenshield’s Model Macroscopic approach
Linear relationship between speed and density Vs = Vf – (Vf/Kj)K Derived formulas:
Km = Kj/2 Vm = Vf/2
Qmax = KjVf/4 Where: Vs = Space mean speed, also referred to as V
Vf = free flow speed
Kj = jam density
Km = density at which max flow occurs
Qmax = maximum flow Example 1: Greenshield’s Example 1: Greenshield’s Model Given:
Find: –
–
– Qm
Vm
Kj V = 54.5 – 0.24 K – havg at Qmax
– savg at Qmax
– space gap at Qmax
At Q = 2,500 VPH, what are the values for K and V? Example 2: Greenshield’s Example 2: Greenshield’s Model Given: V and K are linearly related
– V1 = 45 MPH, K1 = 60 VPM
– V2 = 30 MPH, K2 = 105 VPM Compute:
–
–
–
–
– K and V at Q = 1500 VPH
Vf
Qmax
V1
havg at Qmax
savg at Qmax
V2
K at V = 65 mph
Speed Y = mx + b
(60,45) (105, 30) K1 K2 Kj Greenburg’s Model
Greenburg’s Model
Basic equation V = Vm ln(Kj/K) Example 3: Greenburg’s Example 3: Greenburg’s Model Given: V = 17.2 ln(228/K)
Compute: Qm, Km, Vm and Kj Since Q = KV, Q = K[17.2 ln (228/K)] = 17.2[K ln(228/K)]
dQ/dK = 17.2[K(1/K) + (ln 228 – lnK)]
dQ/dK = 0 ⇒Km
17.2 [1 + (ln 228 – ln Km)] = 0
[1 + (ln 228 – ln Km)] = 0
Ln Km =ln 228 – 1 = ln 228 – ln e
Ln Km = ln (228/e)
∴ K = (228/e) = 84 VPM Proof of Greenburg’s Model Example 3: Greenburg’s Model Compute Vm at K = Km, V = Vm by definition
∴ Vm = 17.2 ln (228/Km) = 17.2 ln [228/(228/e)] = 17.2 ln e V = 17.2 MPH
m OR Vm = 17.2 ln (228/84) = 17.2 MPH Example 3: Greenburg’s Model Compute Qmax Qmax = VmKm = 17.2 MPH x 84 VPM = 1,445 VPH Compute Kj
At Kj, V = 0 V = 17.2 ln (228/Kj) = 0
ln (228/Kj) = 0 e0 = 228/Kj = 1
K = 228
j ...
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 Fall '09
 Nevils
 V2 Institute for the Unstable Media, Qmax

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