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 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

Equation of S-F Relationship S 1 (V) = A 1 – B 1 VV < F (2) S 2 (V) = A 2 – B 2 VF < V < C (3) A 1 = S 0 B 1 = (S 0 – S F ) / F A 2 = S F + {F(S F – S C )/(C – F)}B 2 = (S F – S C ) / (C – F) –S 1 (V) and S 2 (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) –S 0 = free-flow speed (km/h) –S F = speed at knee (km/h) –S C = 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) = t 0 + aV n V < C (4) –t(V) = travel time on linkt 0 = travel time on link at free flow –a = parameter (function of capacity C with power n) –n = power parameter input explicitlyV = 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-D If time is t = L / S equations 2 and 3 could be written: –t 1 (V) = L / (A 1 – B 1 V)V < F (5) –t 2 (V) = L / (A 2 – B 2 V)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:S o * = 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 = S 0 * – cV Constant term S 0 * is geometry constrained free-flow speed, and equation is geometry-adjusted speed-flow relationship. New parameter n* from equation 9 (in paper) replacing S 0 by S 0 *. Example - DS = 108 km/h, B = 50 degrees/km, H = 20 m/km. Then S 0 = 108 – 0.10*0.5 – 0.28*20= 97 km/h (i.e. the free-flow speed S 0 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