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7/29/2019 alazzawi_e2kpresentation1
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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
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