STUDY OF CRITICAL GAP AND ITS EFFECT ON ENTRY CAPACITY OF A ROUNDABOUT IN MIXED TRAFFIC CONDITIONS

Transcription

1 STUDY OF CRITICAL GAP AND ITS EFFECT ON ENTRY CAPACITY OF A ROUNDABOUT IN MIXED TRAFFIC CONDITIONS PRESENTED BY, Revathy Pradeep School Of Planning And Architecture, New Delhi GUIDED BY, Dr. Sewa Ram Associate Professor, Dept. Of Transport Planning and Prof. Dr. P.K Sarkar Head of the Department 1 Dept. Of Transport Planning

2 Objectives of the Study Need for Study Presently we use Wardrop equation 1. To critically review various for calculating weaving capacity of a methods to evaluate critical roundabout. gap. Whereas there is no exact method in Indian scenario evaluating entry presently capacity for of a 2. Derive a model critical and entry capacity for Mixed Traffic conditions. roundabout. 3. relating Hence a study of critical gap and its relation to entry capacity in Indian scenario need to be undertaken. 2

3 ROUNDABOUT Roundabouts are intersections with a generally circular shape, characterized by yield on entry and circulation around a central island As per HCM 2010 FLOW PARAMETERS (Veh/Hr) Qe Entry flow Qc Circulating flow a Component of entry, exit flow b Component of entry flow undergoing weaving behavior c Component of circulating flow exiting the roundabout d Component of circulating flow going for right turning behavior and U turn Qc+Qe Total flow in weaving section p (b+c)/(a+b+c+d) 3

4 CAPACITY OF A ROUNDABOUT Entry Capacity of each arm of a Roundabout Total Approach Capacity Of a roundabout Weaving capacity of the roundabout The capacity of roundabouts depends on two major factors: Various Geometric components of the roundabout Circulating flow in the Roundabout ENTRY CAPACITY The capacity of a roundabout is directly influenced by flow patterns. The intersection can be analyzed by both regression as well as analytical model. Regression model: Analytical models: Use of analytical models of preferred for our study since this could help us study acceptability of existing capacity models in the Indian scenario. For this study we will be using the GAP ACCEPTANCE MODEL 4

5 PARAMETERS IN GAP ACCEPTANCE MODELS Gap: A gap is defined as the time span between two consecutive circulating vehicles that create conflict with an entering vehicle. Critical Gap: Critical gap is defined as the minimum gap that all entering drivers of similar locations will accept, assuming all entering drivers are consistent and homogeneous. Follow up Time Follow up time is defined as the time span between two queued vehicles entering the circulating stream in the same gap. Headway: It is the time between two following vehicles and is measured from the first vehicle s front bumper to the following vehicle s front bumper. 5

6 Data Collection Video graphic Survey was undertaken SURVEY DURATION: 2Hours in morning and evening Data to be Extracted from the Video Accepted and Rejected gap of Individual Entry Vehicles Video camera should be focused on complete weaving section or entire roundabout GAP DATA EXTRACTION : Marking entry and exit line ENTRY VEHICLE Marking reference line REFERENCE LINE 6

7 Gap Data Extraction As an entry vehicle enters the weaving section the entry time of its front and rare bumper is noted down After its entry the time of front bumper of first circulation vehicle crossing the dynamic reference line in weaving section is noted down The front and rare bumper time of all circulating vehicles are noted down Till the entry vehicle accepts a gap Its exit time is noted down at the end of the weaving section 7

8 Gap Data Extraction 8

9 INPUT PARAMETERS FOR GAP ACCEPTANCE MODEL Critical Gap Input data for gap acceptance At an intersection we have: one major stream (priority movement) of volume qp one minor stream of volume qn tc = critical gap = minimum time gap in the priority stream which a minor street driver is ready to accept for crossing or entering the major stream conflict zone 9

10 Methods for evaluating Selection of a suitable method Critical gap was Evaluated using the following methods: METHOD INPUT DATA DRAWBACKS 1. Logit method 2. Harder method LOGIT, HARDER, Only Accepted Gap for a roundabout is collected and taken as input data, 3. Raff s method Calculates the probability of driver to accept the gap Doesn t have a strong Mathematical base 4. Maximum likelihood method RAFF s Doesn t have a strong Mathematical base 5. Modified MLM (IIT R) Accepted Gap and rejected gap for a roundabout is collected and taken as input data MLM Input Data is accepted and rejected gap of individual consistent drivers decreases the sample size considerably Modified MLM (IIT R) Input Data is accepted and rejected gap of individual consistent drivers Modified MLM method is found to be the best to used 10

11 Overview of critical gap results OVERVIEW OF CRITICAL GAP RESULTS DIFFERENT METHODS LOGIT HARDER RAFF MLM Modified MLM CRRI

12 Entry Capacity Analysis :Methodology Tabulate Entry Flow and Circulating Flow from all the arms of the roundabout For a each value of circulating flow identify the maximum Entry flow i.e. Entry Capacity Plot Entry Capacity Vs. Circulating Flow Input these values into HCM Equation and develop Capacity equation for the roundabout Calculate Critical Gap and Average Follow-Up Time Fit an exponential trend line for Entry Capacity Compare HCM Equation with Field Equation Calculate correction factor to relate HCM equation to Field Equation Validate Corrected equation with data set from other roundabout 12

13 STUDY AREA ID DIAMETER (M) Type Minor Major dia(avg) axis axis Roundabout-1(RA1) Intersection of Nyaya Marg 4 ARM 55.3 and Satya Marg near Italy Embassy APPROCH (M) A1 A2 A3 A

14 HCM (2010) Entry Capacity Model C e = A * e B*Vc Where, 3600 A = tf B= t c 0.5* t f 3600 Ce= Entry Capacity Vc=Circulating Volume tf = follow-up time (s) tc = critical gap (s) HCM Equation For RA1 Parameters Value tc 1.82 HCM Equation tf Ce = x e^( Vc) A 1997 B

15 Entry CapacitY, Ce (Veh/hr) Developing Correction Factor for HCM Capacity Model Entry Capacity Vs Circulating Flow HCM y = e-3E-04x y = 2128e-6E-04x R² = HCM 1000 FIELD DATA Expon. (HCM ) Circulating Flow, Qc (Veh/hr) The field equation is lower than HCM Equation. Capacity is overestimated Possible reasons could be lower value of Critical Gap. In U.S conditions critical gap comes out to be 2.8 seconds whereas for us the Critical Gap is 1.82 seconds Also follow-up time is lower 15

16 CORRECTION FACTOR METHOD 1 In this method the ratio of Entry capacity from HCM Equation to Entry capacity from field equation is multiplied This the Difference in capacities is not constant the circulating flow is divided into three bands. And separate correction factors are calculated for each band METHOD 2 In this method the difference between entry capacity from field equation and HCM equation is plotted against Circulating Flow. A trend line is fit for this graph. An equation for the difference in Entry capacity Values with respect to the circulating volume Is developed. Source: Roundabout Model Calibration Issues and a Case Study By Rahmi Akcelik,May 2005 Based upon this equation a value is added as a correction factor to the Entry capacity from HCN equation 16

17 Method 1 The maximum Capacity is obtained under very low circulation flow conditions. Hence There is a decrease in capacity with increase in circulating flow rate. Hence to match the Observed Capacity value of Q1,the follow-up Headway to be specified (tf ) instead of estimated value of (tf ) can be calculated by: tf = tf (Q1/Q1 ) Source: Roundabout Model Calibration Issues and a Case Study By Rahmi Akcelik,May 2005 Since difference is increasing with increase in Qc the Correction factor can be applied to separate bands of Qc Hence the Qc is divided into 3 separate band Circulating flow (Qc) < >

18 Correction factor from both methods METHOD 1: 2000 Entry Capacity Vs Circulating Flow y = e-3E-04x y = e-3E-04x 1800 Entry CapacitY, Ce (Veh/hr) Field Trendline y = 2128e-6E-04x 1600 y = e-3E-04x TRENDLINE Qc<1000 Qc Between 1000 to 2000 Qc>2000 Expon. (TRENDLINE) Expon. (Qc<1000) 600 Expon. (Qc Between 1000 to 2000) Expon. (Qc>2000) Circulating Flow, Qc (Veh/hr) 4000 Expon. (HCM EQUATION) 18

19 Corrected entry capacity model (Method 1) Circulating flow Correction Factor < > Vc tc Tf Tf CORRECTED Ce CORRECTED FIELD Ce EQUATION Ce= Entry Capacity Vc=Circulating Volume tf = follow-up time (s), tc = critical gap (s) tf = Corrected Follow-up time (s) c = Correction Factor INFERENCE: CORRECTION FACTOR This method doesn t give a uniform capacity equation the values produced This method of developing corrected equation is not preferable since in causes the follow-up time to be greater than the critical gap. 19

20 Difference between Field and HCM equation Entry capacity ROUNDABOUT IN MIXED TRAFFIC CONDITIONS Method Difference between Field and HCM equation Entry capacity Vs Circulating Flow Difference between Field and HCM equation Entry capacity Circulating Flow, Qc (Veh/hr) The difference between the entry capacity from the field equation and the HCM equation is plotted against the value of corresponding Circulating Flow It is seen that the difference is constantly increasing till Circulating Flow of 2500 veh/hr A trend line is fit for this graph. An equation for the difference in Entry capacity Values with respect to the circulating volume Is developed. Based upon this equation a value is added as a correction factor to the Entry 20capacity from HCN equation

21 Entry CapacitY, Ce (Veh/hr) METHOD 2: Field Trendline y = 2128e-6E-04x Entry Capacity Vs Circulating Flow trendline Corrected Equation HCM Equation Expon. (trendline) Expon. (HCM Equation) Circulating Flow, Qc (Veh/hr) 21

22 Corrected entry capacity model (Method 2) Correction Factor (C.f)= 5 *(Vc0.5615) Vc tc tf Vc Ce Difference Corrected Ce Field Trendline (Ce) This method gives an uniform capacity equation with lesser variation from the field capacity equation 22

23 VALIDATION DIAMETER (M) ID Type RA- 2 (Intersection of Shanti Path 4 ARM and Panchshel Marg near US Embassy ) Parameters Minor axis 57.8 ICD (M) Major ICD dia(avg) ICD 1 ICD 2 axis (avg) APPROCH (M) A tf A 2208 B A3 A Value tc A2 23

24 Entry CapacitY, Ce (Veh/hr) COMPARISON OF FIELD EQUATION WITH CORRECTED EQUATION BY METHOD 2 (Data from RA-2) Ce(trendline) 1500 y = e-5E-05x 1000 y = e-1E-05x 500 coorected hcm Expon. (Ce(trendline)) Expon. (HCM Equation) Circulating Flow, Qc (Veh/hr) METHOD 2 Qc HCM Equation Ce(trendline) Correction CORRECTED Y= 2108*e^( * Vc) Factor HCM HCM Equation The second model is consistent with the data from other roundabout The first method doesn t work for all type to roundabout as it applies correction to the average follow up time which is varying drastically in our case The model from the second method holds valid due to similarity in flow conditions of both the roundabouts 24

25 Total Capacity Tabulate Entry and circulating Flow from all the four arms separately Plot Circulating Flow Vs Time for all four arms respectively Identify the lowest value of Circulating Flow Calculate Entry Capacity for the arm with least Qc (Ce) Note down entry flow in the other three arms at the same time TOTAL CAPACITY = Ce+Qe1+Qe2+Qe3 Arm 1 Time Qe Arm 2 QC Time Qe ARM3 QC Time Qe Arm4 QC Time Qe QC

26 Total Capacity CIRCULATING FLOW vs TIME Circulating Flow(Veh/hr) ARM 3 Qe3=132 Qe2=1224 ARM 2 Qc=516 Qc= 108 ARM ARM2 ARM3 500 ARM4 0 ARM 4 Qc= Time Ce=1867 Ce Qc1=1608 Qe1=540 Minim um Qc ARM 1 Correct TOTAL HCM ion Correct CAPACI Ce Factor ed Ce ARM1 ARM2 ARM3 ARM4 TY ARM ARM ARM ARM MAX 3775

27 Level of Service An attempt has been made to identify the LOS of the roundabout based upon out comings of the Ph.D. thesis undertaken by Dr. Sewa Ram. The research made an effort to study the different roundabouts with different geometrics demonstrating varying traffic flow. From this study LOS off case study roundabouts was evaluated. One of the case study roundabout in the above study has similar geometric and traffic characteristics to the Roundabout-1 (Intersection of Nyaya Satya Marg). Hence LOS of this Roundabout is taken into consideration. 27

28 Level of Service An attempt has been made to identify the LOS of the roundabout based upon out comings of the Ph.D. thesis undertaken by Dr. Sewa Ram. From the Research paper Roundabout 3 matches with the traffic and geometric conditions the our case study roundabout The LOS of the Case study Roundabout comes out to be LOS C 28

29 Conclusion Correction Factor(C.f)= 5 *(Vc^0.5615) When the on field Entry capacity for circulating flow of 250Veh/hr is less than 2000veh/hr the above equation can used to calculate entry capacity. The total capacity comes out to be 3775Veh/hr for the case study roundabout Scope for further studies With a larger data base a Generic equation can be evaluated using the similar procedure. With a larger data set LOS can be evaluated for varying geometric conditions Effect of traffic composition of roundabout capacity Evaluation of capacity in PCU/hr (Static and dynamic) 29

30 30

31 ANNEXURES 31

32 LOGIT METHOD 32

33 Procedure: The time scale is divided into intervals of constant duration say t=0.5 For each vehicle queuing on the entry stream we have to observe all circulating stream gaps which are presented to the driver and, in addition, the accepted gap. From these observations we have to calculate the following frequencies and relative values: Ni= number of all gaps of size, which are provided to entry vehicles Ai= number of accepted gaps of size i Pa= Percentage of gaps accepted Calculate the probability of gaps being accepted in the given interval. ( ) Plot a graph of logarithm of the above obtained value versus the average gap size of the various interval. Develop a linear trend line for the points obtained. Critical gap will be at a point were Probality of acceptance will be 0.5 that is value of ( = 1 => ln (( ) )= 0 ) Hence on equating the equation of the trend line to zero we obtain the 33 value of critical gap.

34 HARDER METHOD 34

35 HARDER METHOD Procedure: The time scale is divided into intervals of constant duration say t=0.5 The centre of each interval i is denoted by ti For each vehicle queuing on the entry stream we have to observe all circulating stream gaps which are presented to the driver and, in addition, the accepted gap. From these observations we have to calculate the following frequencies and relative values: N= number of all gaps of size, which are provided to entry vehicles Ai= number of accepted gaps of size i ri= Ai/Ni The ai values are corrected by a floating average procedure, where each is also weighted with the Ai values. Finally the value of Critical gap (Tc) is found by calculating the probability of gap being accepted in a given interval Major Drawback of both Harder and Logit method is that they only take into account the accepted gaps of drivers they don t account for the gap rejection by the drivers. These methods only provide the probability of driver accepting a certain gap they don t have a strong mathematical background. 35

36 RAFF S METHOD 36

37 RAFF S METHOD The earliest method for estimating critical gaps seems to be that by Raff. In this method the empirical distribution functions of accepted gaps Fa(t) and rejected gaps Fr(t) Is taken. When the sum of cumulative probabilities of accepted gaps and rejected gaps is equal to 1 then a gap of length t is equal to critical gap tc. It means the number of rejected gaps larger than critical gap is equal to the number of accepted gaps smaller than critical gap. Tc is that value of t at which the following functions intercept 1 (t) and (t) where Fa (t) is the cumulative proportion of accepted gap; Fr (t) is the cumulative proportion of rejected gap; t is the headway of two continued vehicles of circulating stream. Raff s method though considers both accepted and rejected gap of individual drivers it is not backed any strong mathematical model. 37

38 MAXIMUM LIKLIHOOD METHOD 38

39 MAXIMUM LIKELIHOOD METHOD MLM uses two terms of gaps accepted gaps and maximum rejected gaps. The maximum rejected gap is the maximum value of all rejected gaps during a driver waits to running into the roundabout. The mean and variance of critical gap can be calculated by use of the maximum likelihood function of probability theory In this method it is assumed that critical gap follows lognormal distribution, [ ] Likelihood of critical gap is taken as the log of the above function. where F(ai) is the logarithm of the gap accepted by the ith driver; F(ri) is the logarithm of the maximum gap rejected by the ith driver, ri=0 if no gap was rejected; 39

40 Procedure The maximum gap rejected (ri) and Accepted Gap of individual drivers is recorded The likelihood of critical gap is calculated. The assume mean and variance of critical gap as 7.0 and 3.0. Now calculate mean and variance of log of critical gap using the following equations. =ln ( + 1) and = ln( ) 0.5 Where, m is the mean of critical gap s is the standard deviation Iterate the values of σ and µ. Substitute into the equation the calculate critical gap =. In MLM method the use of Log function multiple times in calculating likelihood of critical gap causes errors in the outputs in cases were we have very low values of accepted and rejected gaps. 40

41 HCM IIT ROORKEE METHOD 41

42 This method works on the assumption that the critical gap of individual drivers lie somewhere between the maximum rejected gap and accepted gap Ie. Ri+ x =Tc - (1) Ai - x = Tc - (2) When x->0 that gap obtained will be critical gap Hence if we minimize the following function we obtain critical gap n Min Abs Tc R i Abs Ai Tc i 1 This method gives results close to that critical gap values obtained by Maximum Likelihood Method. The method is less tedious also removal of Use of Log function helps remove chances of error during calculations. 42

43 MODIFIED MLM METHOD (By CRRI) By the earlier equation there was a large variation in the values obtained by it. A slight change in the equation helps reduce the huge variations in the calculated values. Here we use a slightly modified version of the earlier mentioned equation. SQRT(0.5*[(Ai-Tc)^2+(Tc-Ri)^2] 43