# Influence of Various Restrictions on Speed-Flow Models

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1 International Symposium on Fundamental Diagram - Greenshields 75 Influence of Various Restrictions on Speed-Flow Models Marian TRACZ, Stanisław GACA Krakow University of Technology, Poland

2 With less freedom of traffic, the average speed of traffic flow decreases. This can be seen in all models, which represent relationships between the speed of a traffic stream V, flow Q and its density k. A general form of this relationship can be expressed as follows: V = V ( X ) ΔV ( Q, Y ) 0 i i V ( X 0 i ) - is the free flow speed, which depends on a set of independent variables X i ΔV ( Q, Yi ) - is the adjustment expressing speed reduction

3 This model represents theoretical models - often used in traffic analyses and traffic control In these models potential impacts of posted or general speed limits are usually not taken into account Such simplification can have significant impact on results of optimization of traffic signal settings, on results of simulation, on road safety and environmental analyses Speed limits have also an important effect on initial speeds chosen to describe the relationship between traffic flow Q, speed V and density k

4 The paper presents traffic speed models developed on the basis of numerous measurements (76 sections, set of points). In studies of the free flow speed and speed reduction ΔV, the following issues were analyzed: impacts of the speed limits, traffic flow and its composition and time of a day on speedflow relationships verification of hypothesis that speed limits decrease influence of traffic flow on speed impacts of qualitative features of road, roadside and road surroudings on free flow speed (cross-section, road function, accessibility, intensity of development, roadside pedestrian traffic)

5 Speed analyses included sections of roads through built-up areas and streets in towns with numerous constraints of traffic freedom Essential factors constraining free traffic flow on these sections are: speed limits, effects of numerous turning maneuvers and merging from access points If one takes the model given in formula (1) as the point of departure, need to take into account the effects of choice of the data collection periods on the results of free flow speed analyses

6 Recorded variations of daily free flow speed data from one-hour intervals R 2 = 0,48 RURAL ROADS R2 R 2 = 0,87 Average speed Va [km/h] R 2 = 0,71 R1 ROADS THROUGH BUILT-UP AREAS T1 T2 STREETS 1/2 R 2 = 0,71 R 2 = 0, Hours

7 On sections of roads passing through small towns and on sections of streets in cities free flow speeds varied during daytime and increased during night-time. This can be explained by the fact that manual speed enforcement is less likely during the night. The dependence of speed on time of the day on road sections located in non-builtup areas has different characteristics than above. In this case reduction of speed was observed during night-time, which can be associated with worse visibility conditions.

8 The recorded variations of speed were the basis for selection of intervals for analysis of trends of relations between speed V and traffic volume Q. Selected period of analysis: 8 16 hr Figures 2 6 present results of speed measerements and obtained relations between speed and traffic flows on various road sections, in various periods before and after speed limit change. The speed on each lane was analyzed separately These relations were oriented on identification of impacts of speed limit change and influence of cross-section type

9 Average speed V a [km/h] Cross-section C2 - Curb lane - speed limit 60 km/h located in urban area, a two one-way lanes y = -0,0026x + 62,12 R 2 = 0, Traffic flow Q [veh/h/lane] Fig.2 Relationship between average speed and traffic flow

10 Average speed V a [km/h] Cross-section C2 - Curb lane - speed limit 50 km/h located in urban area, a two one-way lanes y = -4E-06x 2-0,0041x + 62,23 R 2 = 0, Traffic flow Q [veh/h/lane] before: 60 km/h Fig.3 Relationship between average speed and traffic flow after decreasing of general speed limit

11 Average speed V a [km/h] Cross-section C2 - Inner lane - speed limit 60 km/h y = -0,0044x + 69,13 R 2 = 0, Traffic flow Q [veh/h/lane] Fig.4 Relationship between average speed and traffic flow

12 Average speed V a [km/h] Cross-section C2 - Inner lane - speed limit 50 km/h y = -4E-06x 2-0,0026x + 67,79 R 2 = 0, Traffic flow Q [veh/h/lane] before: 60 km/h Fig.5 Relationship between average speed and traffic flow after decreasing of speed limit

13 Average speed V a [veh/h] Cross-section T1 - speed limit 50 km/h y = -3E-05x 2 + 0,0045x + 79,05 R 2 = 0, Traffic flow Q [veh/h/lane] before: 60 km/h section through a built-up area, a two lane (1x2) with paved shoulders Fig.6 Relationship between average speed and traffic flow after decrease of speed limit

14 Before and after analyses confirmed an impact of speed limit on character of speed-flow relation for large traffic flows Large spread of the average speed values at the same speed limit of 50 km/h, and also at other analysed limits of 60 and 90 km/h, shows significant impact of local factors, affecting speed choice in free-flow traffic what was the subject of next studies

15 Using data collected on the same roads, but in periods with different speed limits, determined were regression relationships between speed and lane traffic flow Q and share of heavy vehicles Hv (in %) Analyzed case Speed limit [km/h] Regression equation to estimate average speed on a lane Roads passing through built-up areas Data from day-time measurements Cross-section 60 V = 81, 94 0, Q 0,01 10,8 T V = 79, 61 0, Q 0,05 8,64 Cross-section 60 V = 77, 46 0, Q 0, 085 Hv 0,03 8,64 T2 50 V = 70, 46 0, 0202 Q + 0, 223 Hv 0,15 8,14 2 Cross-section 60 V = 6211, 0, 0264 Q + 0, Q + 0, 242 Hv 0,11 7,51 T V = 5712, 0, 0215 Q + 0, Q + 0, 454 Hv 0,17 11,1 R 2 S E

16 In analyzed ranges of traffic flows; - up to 1200 veh/h/lane in towns and - up to about 800 veh/h/lane in other cases, the influence of traffic flow on speed can be regarded as small. Traffic flow Q only to a small degree explains the registered variability of speed (R 2 = 0,01 0,17) If a parameter expressing the share of heavy vehicles in traffic stream is introduced to the regression equations, there is an increase in the value of R 2 and decrease in the value of S E ; however these changes are small

17 The influence of traffic flow level on average speed was larger for the speed limit of 50 km/h than for the limit of 60 km/h Large dispersion of speed values in the analyzed data sample makes it harder to estimate average speed as the function of traffic flow and share of heavy vehicles only The study confirmed previous assessments showing, that for free flow and partly constrained traffic flow, local conditions have considerable influence on average speed of traffic flow

18 In the first part general form of speed model was expressed as follows: V = V ( X ) ΔV ( Q, Y ) 0 i i V ( X 0 i ) ΔV ( Q, Yi ) - free flow speed - adjustment expressing speed reduction in relation to free flow speed What determines driver s choice of speed V 0 in free flow traffic on urban roads? The following factors were analyzed: type of cross-section, intensity of road surroundigs development, type of shoulders (or walkway) and accessibility

19 The significance of the effects of different variables was computed and as the result the derived model for estimating probability of the chances that average speed of cars in free flow traffic is lower or equal to critical value of i-th speed category: πi ln 1 π i = a i + 0, 284 0, 759 P P L , S + 0, 996 D D W 0, 267 S + 0, 261 D M G + 1, 257 π i the probability that the value of average speed is less or equal to critical value of i-th category, a i values of the coefficient for consecutive categorised intervals of average speed are as follows: a 1 = -5,255, a 2 = -3,603, a 3 = -1,951, a 4 = -0,6085 L length of a section passing through built-up area [km] S W, S P, S G types of shoulder: walkway, unpaved respectively D D, D M, D S symbols of dense, moderate or scattered development respectively D S

20 0,9 0,8 Intensive development, L = 1 km Probability [-] 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 walkway unpaved shoulder paved shoulder <50 <60 <70 <80 Average speed V a [km/h] The effect of the shoulder type on the probability that average speed of cars Va <Vi - estimate based on formula (6)

21 Can the free flow speed reduction V be determined only on the basis of traffic flow? Cross-section C1 10 ΔV [km/h] y = 2E-06x 2 + 0,0056x + 0,39 R 2 = 0, Cross-section C2 Traffic flow Q [veh/h/lane] y = 0,0061x R 2 = 0,04 ΔV [km/h] V < 0!!! Traffic flow Q [veh/h/lane]

22 On sections of dual-carriageway streets and on sections of passages through small towns/villages there were often cases when speed of traffic flows was larger than speeds of vehicles in free flow traffic Such an effect can occur in platoon traffic on sections with speed limits. Drivers who respect speed limits often adjust their speed to faster drivers. Such effect is usually disregarded in traffic models In case of traffic flows up to about 1000 veh/h the assumption that V is significantly influenced by the road cross-section was not confirmed The considerable dispersion of values in the analyzed tests with the same traffic flows indicates significant influence of random drivers' behaviors in free flow and partly constrained flow

23 In case of two-carriageway cross-sections and one-carriageway passages through towns/villages traffic flow is one of many factors which can affect V. If the share of heavy vehicles in traffic flow was taken into account, factor R 2 increased of 0,08 to 0,26; however, the obtained regression equations could not be qualified as useful in practice Only in case of one-carriageway streets the relation between V and Q and rate of heavy vehicles could be described with a simple regression equation with relatively high degree of determination R 2 = 0,40

24 Final Conclusions Even at general and posted speed limits, shape of traffic flow-speed relation - characteristic for the fundamental diagram - can be identified. However, characteristic for empirical results spread of recorded values, typical for randomness and impacts of local factors can be seen Legal speed limits influence free flow speed value V 0 and the value of V (speed reduction). The influence of traffic flow on decrease of average speed with the speed limit of 50 km/h was greater than for the limit of 60 km/h.

25 Speed limit decreases impact of heavy vehicles on real traffic flow speed Free flow speed varies in different periods of a 24-hour cycle. This variability depends on road location. On roads located in built-up areas there is a distinct increase in free flow speed at night-time. On roads located in non-built-up areas there is a decrease in free flow speed observed

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