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Speed-Flow & Flow-Delay

Models

Marwan AL-Azzawi


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Project Goals

To develop mathematical functions

to improve traffic assignment

To simulate the effects of congestionbuild-up and decline in road

networks

To develop the functions to coverdifferent traffic scenarios


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Background

In capacity restraint traffic assignment, a proper allocation

of speed-flow in highways, plays an important part in

estimating the effects of congestion on travel times and

consequently on route choice.

Speeds normally estimated as function of highway type

and traffic volumes, but in many instances the road

geometric design and its layout are omitted.

This raises a problem with regards to taking into account

the different designs and characteristics of different roads.


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Speed-Estimating Models Generally developed from large databases containing vehicle

speeds on road sections with different geometric characteristics,

and under different flow levels.

Multiple regression or multiple variant analysis used.

Example: S = DS 0.10B 0.28H 0.006V 0.027V* ....... (1)

DS = constant term (km/h) B = road bendiness (degrees/km)

H = road hilliness (m/km) V or V* = flow < or > 1200 (veh/h)

DS is desiredspeed - the average speed drivers would drive on a

straight and level road section with no traffic flow (road geometry is

the only thing restricting the speed of vehicles).

Desired and free-flow speed different - latter is speed under zero

traffic, regardless of road geometry. In fact, desired speed is only

a particular case of free-flow speed.


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Speed-Flow relationships

Speed(S) Figure 1: A typical speed-flow relationship

S0

SF

SC

F C Flow (V)


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Equation of S-F Relationship

S1(V) = A1 B1V V < F ........................ (2)

S2(V) = A2 B2V F < V < C ............ (3)

A1 = S0 B1 = (S0 SF) / F A2 = SF + {F(SF SC)/(C F)} B2 = (SF SC) / (C F)

S1(V) and S2(V) = speed (km/h)

V = flow per standard lane (veh/h)

F = flow at knee per standard lane (veh/h) C = flow at capacity per standard lane (veh/h)

S0 = free-flow speed (km/h)

SF = speed at knee (km/h)

SC = speed at capacity (km/h)


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Flow-Delay Curves

Exponential function appropriate to represent effects of congestion

on travel times.

At low traffic, an increase in flows would induce small increase in

delay. At flows close to capacity, the same increase would induce a much

greater increase in delays.

Time (t) Figure 2: Effects of Congestion on Travel TimestC

t0

C Flow (V)


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Equation of F-D Curve

t(V) = t0 + aVn V < C ........................ (4)

t(V) = travel time on link t0 = travel time on link at free flow

a = parameter (function of capacity C with power n)

n = power parameter input explicitly V = flow on link

Parameter n adjusts shape of curve according to link type. (e.g.

urban roads, rural roads, semi-rural, etc.)

Must apply appropriate values of n when modelling links of criticalimportance.


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Converting S-F into F-D

If time is t = L / S equations 2 and 3 could be written:

t1(V) = L / (A1 B1V) V < F .......................... (5)

t2(V) = L / (A2 B2V) F < V < C ............. (6)

These equations represent 2 hyperbolic (time-flow) curves of a

shape as shown in figure 3.

Use similar areas method to calculate equations. Tables 1 in

paper gives various examples of results.

Time (t) Figure 3: Conversion of Flow-Delay Curve

tC

tF

t0

F C Flow (V)


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Incorporating Geometric Layouts Example - consider rural all-purpose 4 lane road. If the speed

model is: S = DS aB bH cV - dV*

Let: So* = DS aB bH. Also, if only the region of low traffic flows

is taken (road geometry only affects speed at low traffic levels) then

d = 0

Hence equation is: S = S0* cV

Constant term S0* is geometry constrained free-flow speed, and

equation is geometry-adjusted speed-flow relationship. New

parameter n* from equation 9 (in paper) replacing S0 by S0*.

Example - DS = 108 km/h, B = 50 degrees/km, H = 20 m/km. Then

S0 = 108 0.10*0.5 0.28*20 = 97 km/h (i.e. the free-flow speed

S0 equal to 108 km/h is reduced by 11 km/h due to road geometry).


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Conclusions

New S-F models should improve

traffic assignment

New F-D curves help simulate affectsof congestion

Further work on-going to develop

model parameters for other roadtypes

Documents

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