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
Objectives of the Study Need for Study 1. 2. 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
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
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
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
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
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
Gap Data Extraction 8
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
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
Overview of critical gap results OVERVIEW OF CRITICAL GAP RESULTS DIFFERENT METHODS LOGIT 2.44 2.00 HARDER 3.11 2.82 RAFF 2.40 1.92 MLM 1.92 1.77 Modified MLM 1.62 1.64 CRRI 1.52 1.63 11
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
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 55.5 55.4 APPROCH (M) A1 A2 A3 A4 7.64 7.45 7.60 7.51 13
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 1.802 Ce = 1997.8 x e^(-0.0003 Vc) A 1997 B 0.000255 14
Entry CapacitY, Ce (Veh/hr) Developing Correction Factor for HCM Capacity Model Entry Capacity Vs Circulating Flow 3000 2500 HCM y = 1997.8e-3E-04x y = 2128e-6E-04x R² = 0.5546 2000 1500 HCM 1000 FIELD DATA 500 0 0 1000 2000 3000 4000 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
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
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) <1000 1000-2000 >2000 17
Correction factor from both methods METHOD 1: 2000 Entry Capacity Vs Circulating Flow y = 1799.2e-3E-04x y = 1459.1e-3E-04x 1800 Entry CapacitY, Ce (Veh/hr) Field Trendline y = 2128e-6E-04x 1600 y = 1036.4e-3E-04x 1400 1200 TRENDLINE Qc<1000 Qc Between 1000 to 2000 Qc>2000 Expon. (TRENDLINE) 1000 800 Expon. (Qc<1000) 600 Expon. (Qc Between 1000 to 2000) 400 200 Expon. (Qc>2000) 0 0 1000 2000 3000 Circulating Flow, Qc (Veh/hr) 4000 Expon. (HCM EQUATION) 18
Corrected entry capacity model (Method 1) Circulating flow Correction Factor <1000 1.11 1000-2000 1.37 >2000 1.93 Vc tc 500 900 1000 1200 1400 1800 2000 2200 2800 2900 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 Tf Tf CORRECTED Ce 1.802 1.802 1.802 1.802 1.802 1.802 1.802 1.802 1.802 1.802 2.00 2.00 2.00 2.47 2.47 2.47 2.47 3.48 3.48 3.48 1758 1587 1547 1470 1397 1261 1199 1139 977 953 CORRECTED FIELD Ce EQUATION 1584 1577 1431 1240 1394 1168 1074 1036 1021 919 922 723 876 641 591 569 507 397 494 374 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 1.11 1.37 1.93 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
Difference between Field and HCM equation Entry capacity ROUNDABOUT IN MIXED TRAFFIC CONDITIONS Method 2 700 Difference between Field and HCM equation Entry capacity Vs Circulating Flow 600 500 400 300 200 Difference between Field and HCM equation Entry capacity 100 0 0 500 1000 1500 2000 2500 3000 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
Entry CapacitY, Ce (Veh/hr) METHOD 2: 2000 1800 1600 1400 1200 1000 800 600 400 200 0 Field Trendline y = 2128e-6E-04x Entry Capacity Vs Circulating Flow trendline Corrected Equation HCM Equation Expon. (trendline) Expon. (HCM Equation) 0 500 1000 1500 2000 2500 3000 3500 Circulating Flow, Qc (Veh/hr) 21
Corrected entry capacity model (Method 2) Correction Factor (C.f)= 5 *(Vc0.5615) Vc 500 900 1000 1200 1400 1800 2000 2200 2700 2800 tc 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 1.82 tf 1.802 1.802 1.802 1.802 1.802 1.802 1.802 1.802 1.802 1.802 Vc 500 900 1000 1200 1400 1800 2000 2200 2700 2800 Ce 1758 1587 1547 1470 1397 1261 1199 1139 1002 977 Difference 184 258 275 305 334 386 411 434 489 499 Corrected Ce 1574 1329 1272 1165 1063 875 788 705 513 478 Field Trendline (Ce) 1577 1240 1168 1036 919 723 641 569 421 397 This method gives an uniform capacity equation with lesser variation from the field capacity equation 22
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) 56.3 57.05 78.1 APPROCH (M) A1 1.65 tf 1.6305 A 2208 B 0.000232 A3 A4 75.2 76.65 7.81 7.41 7.75 7.45 Value tc A2 23
Entry CapacitY, Ce (Veh/hr) COMPARISON OF FIELD EQUATION WITH CORRECTED EQUATION BY METHOD 2 (Data from RA-2) 2500 2000 Ce(trendline) 1500 y = 2108.4e-5E-05x 1000 y = 2293.7e-1E-05x 500 coorected hcm Expon. (Ce(trendline)) Expon. (HCM Equation) 0 0 500 1000 1500 2000 Circulating Flow, Qc (Veh/hr) METHOD 2 Qc 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 HCM Equation Ce(trendline) Correction CORRECTED Y= 2108*e^(0.0006 * Vc) Factor HCM HCM Equation 2077 123 2163 2285 2046 184 2095 2278 2016 233 2038 2270 1986 275 1988 2262 1956 313 1942 2254 1927 348 1899 2246 1898 381 1858 2238 1870 411 1820 2230 1842 440 1783 2222 1815 468 1748 2215 1788 495 1713 2207 2500 3000 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
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 1 444 1584 1 1080 444 1 828 192 1 144 900 2 540 1608 2 1224 516 2 852 108 2 168 1018 3 552 1452 3 1668 432 3 492 216 3 132 1000 4 612 2124 4 1932 456 4 948 192 4 168 924 5 540 2244 5 2136 384 5 900 113 5 144 1080 6 396 2328 6 2256 396 6 924 144 6 132 1044 7 600 2208 7 2088 324 7 588 144 7 204 1048 8 648 2172 8 2208 276 8 708 156 8 108 1124 9 672 2208 9 1248 240 9 720 123 9 192 900 10 420 1872 10 2052 372 10 924 192 10 168 1044 11 612 1896 11 2196 348 11 900 168 11 216 1200 12 516 2004 12 2112 384 12 1044 168 12 168 1178 13 528 1920 13 2256 336 13 1092 168 13 180 1080 25
Total Capacity CIRCULATING FLOW vs TIME Circulating Flow(Veh/hr) 2500 2000 ARM 3 Qe3=132 Qe2=1224 ARM 2 Qc=516 Qc= 108 ARM1 1000 ARM2 ARM3 500 ARM4 0 ARM 4 Qc= 1000 1500 0 5 10 15 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 ARM1 1452 1370 341 1029 - ARM2 240 1877 120 1757 672 ARM3 108 1942 76 1867 ARM4 900 1581 258 1323 1668 492 132 3321 720 192 3341 540 1224-144 3775 444 1080 828-3675 - 26 MAX 3775
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
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
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
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ANNEXURES 31
LOGIT METHOD 32
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.
HARDER METHOD 34
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
RAFF S METHOD 36
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
MAXIMUM LIKLIHOOD METHOD 38
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
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
HCM IIT ROORKEE METHOD 41
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
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