# Speed-Flow & Flow-Delay Models - PowerPoint PPT Presentation

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Speed-Flow & Flow-Delay Models. Marwan AL-Azzawi. Project 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. Background.

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Speed-Flow & Flow-Delay Models

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## Speed-Flow & Flow-Delay Models

Marwan AL-Azzawi

### Project 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

### 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.

### 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 “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”.

### Equation of S-F Relationship

• S1(V) = A1 – B1VV < F........................(2)

• S2(V) = A2 – B2VF < V < C............(3)

• A1 = S0B1 = (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 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.

### Equation of F-D Curve

• t(V) = t0 + aVnV < C........................(4)

• t(V) = travel time on linkt0 = travel time on link at free flow

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

• n = power parameter input explicitlyV = flow on link

• Must apply appropriate values of n when modelling links of critical importance.

### 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.

### 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).

### Conclusions

• 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