Speed-Flow & Flow-Delay Speed-Flow & Flow-Delay ModelsModels

Marwan AL-Azzawi

Project GoalsProject Goals

To develop mathematical functions to improve traffic assignment

To simulate the effects of congestion build-up and decline in road networks

To develop the functions to cover different traffic scenarios

BackgroundBackground

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.

Speed-Estimating ModelsSpeed-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 “desired speed” - 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”.

Speed-Flow relationships

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

S0

SF

SC

F C Flow (V)

Equation of S-F RelationshipEquation 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)

Flow-Delay CurvesFlow-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 Times tC

t0

C Flow (V)

Equation of F-D CurveEquation 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 critical importance.

Converting S-F into F-DConverting 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 CurvetC

tF

t0 F C Flow (V)

Incorporating Geometric LayoutsIncorporating 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).

ConclusionsConclusions

New S-F models should improve traffic assignment

New F-D curves help simulate affects of congestion

Further work on-going to develop model parameters for other road types