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Macroscopic Traffic VariablesTraffic flow theory (TFT)
Zhengbing He, Ph.D.,http://zhengbing.weebly.com
School of traffic and transportation, Beijing Jiaotong University
October 30, 2015
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Measurement intervals
Three types of intervals:
S1-such as aerial photograph
S2-such as loop detectors
S3-Edie’s generalizeddefinitions.When trajectory data areavailable, they are the mostappropriate.
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Remote traffic monitor sensor (RTMS)
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Video surveillance
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Drone???
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Macroscopic Variables
Three main macroscopic variables:
Density (veh/km)
Flow (veh/h)
Speed (km/h)
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Density (veh/km)
Definition: vehicle number per distance unit at a time slice.
S1: k(x , t0, L) = nL — from the definition.
S3: k(x , t,A) = nL = ndt
Ldt ≈ total time spent by all vehicles in AArea A
we write as k(A) = t(A)|A|
S2: k(x0, t,T ) =
∑m
dxvi
Tdx =
∑m
1vi
T
— from total time spent by all vehicles in AArea A in S3
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Flow (rate) (veh/h)
Definition: vehicle number passing a cross-section per time unit.We call the maximum possible flow as capacity. (The capacity ofa highway lies between 1800 ∼ 2400 veh/h per lane.)
S2: q(x0, t,T ) = mT — from the definition.
S3: q(x , t,A) = mdxTdx ≈ Total distance covered by vehicles in A
Area A
we write as q(A) = d(A)|A|
S1: q(x , t0, L) =∑
n vidtdtL =
∑n viL
— from Total distance covered by vehicles in AArea A in S3
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
(Mean) speed (km/h)
Definition: the mean speed v as the quotient of the flow rate andthe density; i.e., v = q/k
Two types:
space-mean speed: v = q/k
time-mean speed: vt = 1m
∑m vi
Note: time-mean speed does not comply with v = q/k (thefundamental diagram)
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
(Mean) speed (km/h)
S1: using the flow and density equations in S1
v(x , t0, L) =∑
n viL /n
L= 1n
∑n vi
Note: space-mean speed=time-mean speed. We thus definethe space-mean speed for a location interval (S1) as: 1
n
∑n vi
S2: using the flow and density equations in S2
v(x0, t,T ) = mT /
∑m
1vi
T = m∑m
1vi
S3: the measurement interval A no longer appears:v(x , t,A) = q(x ,t,A)
k(x ,t,A) ≈ Total distance covered by vehicles in ATotal time spent by vehicles in A
we write as v(A) = d(A)t(A)
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Distinguishing space-mean and time-mean speed
Example problem: in a two-lane road, vehicles travel at 60km/hon the left lane and at 120km/h on the right lane. Assume that1200 veh/h pass on both lanes. What are the density, the flow, the(space-mean) speed and the time-mean speed on this road?
The solution is:
q = 1200 ∗ 2 = 2400veh/h
k = 1200/60 + 1200/120 = 30veh/km
v = q/k = 80km/h
vt = (120 + 60)/2 = 90km/h
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Relative occupancy (no unit)
The occupancy o of a vehicle is easy to obtain in S2; i.e., loopdetectors in practice. The relative occupancy b in S2 is givenby:
b(x0, t,T ) = 1T
∑m oi – Recall the definition of occupancy
If assume all vehicles have the same length l , we can get:b(x , t0, L) = lk(x , t0, L) – How to derive it
Note: this formula is not appropriate in practical situationsbecause inhomogeneity of the real traffic flow. If want to finddensity by using detectors, it is relatively better to usek = q
v .
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Homogeneous and stationary traffic
Homogeneous traffic: the vehicles are identical;
Stationary traffic: the traffic state (described by themacroscopic variables) does not change over time
Figure: Examples of non-stationary traffic
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Summary about Edie’s definitions
Edie’s definitions are important because it allows one to comparethe behavior of two traffic streams even if they have been observedin different ways, such as the comparison of A and B.
One should try to avoid the non-stationary area, such as C, becauseone value of the macroscopic variableonly corresponds to one traffic state.
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Summary about macroscopic traffic variables
The discrete nature of traffic requires time intervals of at least halfa minute if we want to achieve meaningful macroscopicinformation. When the time intervals exceed a duration of fiveminutes, certain dynamic characteristics are lost.
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables
Thank you!
Zhengbing He, Ph.D., http://zhengbing.weebly.com Macroscopic Traffic Variables