macroscopic traffic variables

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

http://zhengbing.weebly.com


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.



Remote traffic monitor sensor (RTMS)



Video surveillance



Drone???



Macroscopic Variables

Three main macroscopic variables:

Density (veh/km)

Flow (veh/h)

Speed (km/h)



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



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



(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)



(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)



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



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 .



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



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.



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.



Thank you!



macroscopic traffic variables

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