STUDY OF CRITICAL GAP AND ITS EFFECT ON ENTRY CAPACITY...

STUDY OF CRITICAL GAP AND ITS EFFECT ONENTRY CAPACITY OF A ROUNDABOUT IN

MIXED TRAFFIC CONDITIONS

PRESENTED BY,Revathy Pradeep

School Of Planning And Architecture, New DelhiGUIDED BY ,

Dr. Sewa Ram and Prof. Dr. P.K SarkarAssociate Professor, Head of the Department

Dept. Of Transport Planning Dept. Of Transport Planning1

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

1. Presently we use Wardrop equation

for calculating weaving capacity of a

roundabout.

2. Whereas there is no exact method in

Indian scenario presently for

evaluating entry capacity of a

roundabout.

3. Hence a study of critical gap and its

relation to entry capacity in Indian

scenario need to be undertaken.

Objectives of the StudyNeed for Study

1. To critically review various

methods to evaluate critical

gap.

2. Derive a model relating

critical and entry capacity for

Mixed Traffic conditions.

2


ROUNDABOUT“ Roundabouts are intersections with a generally circular shape, characterizedby yield on entry and circulation around a central island” As per HCM 2010

FLOW PARAMETERS (Veh/Hr)Qe Entry flowQc Circulating flowa Component of entry, exit flowb Component of entry flow undergoing

weaving behaviorc Component of circulating flow exiting the

roundaboutd Component of circulating flow going for

right turning behavior and U turnQc+Qe Total flow in weaving section

p (b+c)/(a+b+c+d) 3


Entry Capacity of each arm of a Roundabout

Total Approach Capacity Of a roundabout

Weaving capacity of the roundabout

CAPACITY OF A ROUNDABOUTCAPACITY OF A ROUNDABOUT

The capacity of roundabouts depends on two major factors: Various Geometric components of the roundabout Circulating flow in the Roundabout

4

ENTRY CAPACITYENTRY CAPACITYThe capacity of a roundabout is directly influenced by flow patterns. Theintersection can be analyzed by both regression as well as analytical model.

Regression model: Analytical models:

Use of analytical models of preferred for our studysince this could help us study acceptability of existingcapacity models in the Indian scenario.For this study we will be using the GAP ACCEPTANCEMODEL


PARAMETERS IN GAP ACCEPTANCE MODELSPARAMETERS 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.

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.

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 andhomogeneous.

Follow up Time Follow up time is defined as the time span between two queued

vehicles entering the circulating stream in the same gap.

5


Data Collection Video graphic Survey was undertaken

SURVEY DURATION: 2Hours in morning andevening

Video camera should be focused oncomplete weaving section or entireroundabout

Data to be Extracted from the Video

Accepted and Rejected gap of Individual EntryVehicles

FIRST CIRCULATING VEHICLEFIRST CIRCULATING VEHICLE

GAP DATA EXTRACTION :GAP DATA EXTRACTION :

ENTRY VEHICLE at REFERENCELINEENTRY VEHICLE at REFERENCELINE

SECOND CIRCULATINGVEHICLESECOND CIRCULATINGVEHICLE

ENTRY VEHICLE AT EXIT LINEENTRY VEHICLE AT EXIT LINE

ENTRY VEHICLEENTRY VEHICLE REFERENCE LINEREFERENCE LINE

Marking entry and exit line Marking reference line

6


Gap Data ExtractionGap Data Extraction

Its exit time is noted down atthe end of the weaving

section

As an entry vehicle enters theweaving section the entry timeof its front and rare bumper is

noted down

After its entry the time of frontbumper of first circulation

vehicle crossing the dynamicreference line in weaving

section is noted down

The front and rare bumpertime of all circulating

vehicles are noted down

Till the entry vehicleaccepts a gap

Small carBig CarTwo WheelerThree WheelerLightCommercialVehicleTruckBusCYCLE

7


Gap Data ExtractionGap Data Extraction

8


Critical GapInput data for gap acceptanceInput 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 prioritystream which a minor street driver is ready to acceptfor crossing or entering the major stream conflictzone

INPUT PARAMETERS FOR GAPACCEPTANCE MODEL

9


Methods for evaluatingMethods for evaluating

Critical gap was Evaluatedusing the followingmethods:

1. Logit method

2. Harder method

3. Raff’s method

4. Maximumlikelihood method

5. Modified MLM (IITR)

given roundabout varying setsof Gaps data was taken and it is seen the modified MLM tends togive more or less reliable outputs that are close to the critical gapgiven by MLM method

Hence Modified MLM method shall be used for evaluating criticalgap of the roundabouts

Selection of a suitable methodSelection of a suitable method

10

METHOD INPUT DATA DRAWBACKS

LOGIT,HARDER,

Only Accepted Gap for aroundabout is collectedand taken as input data,

Calculates the probabilityof driver to accept the gapDoesn’t have a strongMathematical base

RAFF’s Accepted Gap andrejected gap for aroundabout is collectedand taken as input data

Doesn’t have a strongMathematical base

MLM Input Data is accepted andrejected gap of individualconsistent drivers

decreases the sample sizeconsiderably

ModifiedMLM (IIT R)

Input Data is accepted andrejected gap of individualconsistent drivers

Modified MLM method isfound to be the best toused


OVERVIEW OF CRITICAL GAP RESULTSDIFFERENT METHODS

LOGIT 2.44 2.00HARDER 3.11 2.82RAFF 2.40 1.92MLM 1.92 1.77Modified MLM 1.62 1.64CRRI 1.52 1.63

Overview of critical gap results

11


Tabulate Entry Flow andCirculating Flow from all

the arms of theroundabout

For a each value ofcirculating flow identify

the maximum Entry flowi.e. Entry Capacity

Plot Entry Capacity Vs.Circulating Flow

Fit an exponential trendline for Entry Capacity

Calculate Critical Gapand Average Follow-Up

Time

Input these values intoHCM Equation and

develop Capacity equationfor the roundabout

Compare HCM Equationwith Field Equation

Calculate correctionfactor to relate HCM

equation to FieldEquation

Validate Correctedequation with data setfrom other roundabout

Entry Capacity Analysis :MethodologyEntry Capacity Analysis :Methodology

12


ID TypeDIAMETER (M) APPROCH (M)

Minoraxis

Majoraxis dia(avg) A1 A2 A3 A4

Roundabout-1(RA1)Intersection of Nyaya Margand Satya Marg near Italy

Embassy

4 ARM 55.3 55.5 55.4 7.64 7.45 7.60 7.51

STUDY AREASTUDY AREA

13


HCM (2010) Entry Capacity ModelHCM (2010) Entry Capacity Model

cB*VeC = A * e

f

3600A =

tc ft 0.5*t

B =3600

Ce= Entry CapacityVc=Circulating Volumetf = follow-up time (s)tc = critical gap (s)

HCM Equation

Ce = 1997.8 x e^(-0.0003 Vc)

Parameters Value

tc 1.82

tf 1.802

A 1997

B 0.00025514

Where,

HCM Equation For RA1HCM Equation For RA1


Developing Correction Factor for HCM Capacity ModelDeveloping Correction Factor for HCM Capacity Model

HCMy = 1997.8e-3E-04x

y = 2128e-6E-04x

R² = 0.5546

0

500

1000

1500

2000

2500

3000

0 1000 2000 3000 4000

Entr

y Ca

paci

tY, C

e (V

eh/h

r)

Circulating Flow, Qc (Veh/hr)

Entry Capacity Vs Circulating Flow

HCM

FIELD DATA

Expon. (HCM )

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 1.82seconds

Also follow-up time is lower 15


CORRECTION FACTOR

METHOD 1Source: Roundabout Model CalibrationIssues and a Case StudyBy Rahmi Akcelik ,May 2005

• In this method the ratio of Entry capacity from HCMEquation to Entry capacity from field equation ismultiplied

• This the Difference in capacities is not constant thecirculating flow is divided into three bands. Andseparate correction factors are calculated for eachband

METHOD 2 • In this method the difference between entry capacityfrom field equation and HCM equation is plottedagainst Circulating Flow.

• A trend line is fit for this graph. An equation for thedifference in Entry capacity Values with respect to thecirculating volume Is developed.

• Based upon this equation a value is added as acorrection factor to the Entry capacity from HCNequation

16


Method 1

The maximum Capacity is obtainedunder very low circulation flowconditions. Hence There is a decrease incapacity with increase in circulating flowrate.

Hence to match the Observed Capacityvalue of Q1’ ,the follow-up Headway tobe specified (tf’) instead of estimatedvalue of (tf ) can be calculated by:

tf’= tf (Q1/Q1’)

Since difference is increasing with increase in Qc theCorrection factor can be applied to separate bands ofQc

Hence the Qc is divided into 3 separate band

Circulating flow (Qc)

<1000

1000-2000

>2000

Source: Roundabout Model Calibration Issues and a Case StudyBy Rahmi Akcelik ,May 2005

17


Correction factor from both methods

Field Trendliney = 2128e-6E-04x

y = 1799.2e-3E-04x

y = 1459.1e-3E-04x

y = 1036.4e-3E-04x

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 1000 2000 3000 4000

Entr

y Ca

paci

tY, C

e (V

eh/h

r)


Entry Capacity Vs Circulating Flow

TRENDLINE

Qc<1000

Qc Between 1000 to 2000

Qc>2000

Expon. (TRENDLINE)

Expon. (Qc<1000)

Expon. (Qc Between 1000 to2000)Expon. (Qc>2000)

Expon. (HCM EQUATION)

METHOD 1:METHOD 1:

18


Corrected entry capacity model (Method 1)

Ce= Entry CapacityVc=Circulating Volumetf = follow-up time (s),tc = critical gap (s)tf’ = Corrected Follow-up time (s)c = Correction Factor

Circulating flow Correction Factor

<1000 1.11

1000-2000 1.37

>2000 1.93

Vc tc Tf Tf CORRECTED Ce CORRECTEDCe

FIELDEQUATION

CORRECTIONFACTOR

500 1.82 1.802 2.00 1758 1584 15771.11900 1.82 1.802 2.00 1587 1431 1240

1000 1.82 1.802 2.00 1547 1394 11681200 1.82 1.802 2.47 1470 1074 1036

1.371400 1.82 1.802 2.47 1397 1021 9191800 1.82 1.802 2.47 1261 922 7232000 1.82 1.802 2.47 1199 876 6412200 1.82 1.802 3.48 1139 591 569

1.932800 1.82 1.802 3.48 977 507 3972900 1.82 1.802 3.48 953 494 374

INFERENCE:

This method doesn’t give auniform capacity equationthe values produced

This method of developingcorrected equation is notpreferable since in causesthe follow-up time to begreater than the criticalgap.

19

STUDY OF CRITICAL GAP AND ITS EFFECT ON ENTRY CAPACITY OF AROUNDABOUT IN MIXED TRAFFIC CONDITIONSMethod 2

0

100

200

300

400

500

600

700

0 500 1000 1500 2000 2500 3000

Diffe

renc

e be

twee

n Fi

eld

and

HCM

equ

atio

nEn

try

capa

city


Difference between Field and HCM equation Entry capacity Vs Circulating Flow

Difference between Field and HCMequation Entry capacity

The difference between the entry capacity from the field equation and the HCM equation isplotted 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 capacity from HCN

equation20


Field Trendliney = 2128e-6E-04x

0200400600800

100012001400160018002000

0 500 1000 1500 2000 2500 3000 3500

Entr

y Ca

paci

tY, C

e (V

eh/h

r)


Entry Capacity Vs Circulating Flowtrendline

Corrected Equation

HCM Equation

Expon. (trendline)

Expon. (HCM Equation)

METHOD 2:METHOD 2:

21


Corrected entry capacity model (Method 2)

Correction Factor (C.f)= 5 *(Vc0.5615)

This method gives an uniform capacity equation with lesser variation from thefield capacity equation

Vc tc tf Vc Ce Difference Corrected Ce Field Trendline (Ce)500 1.82 1.802 500 1758 184 1574 1577900 1.82 1.802 900 1587 258 1329 1240

1000 1.82 1.802 1000 1547 275 1272 11681200 1.82 1.802 1200 1470 305 1165 10361400 1.82 1.802 1400 1397 334 1063 9191800 1.82 1.802 1800 1261 386 875 7232000 1.82 1.802 2000 1199 411 788 6412200 1.82 1.802 2200 1139 434 705 5692700 1.82 1.802 2700 1002 489 513 4212800 1.82 1.802 2800 977 499 478 397

22


23

VALIDATIONVALIDATION

ID Type

DIAMETER (M) ICD (M) APPROCH (M)

Minoraxis

Majoraxis dia(avg) ICD 1 ICD 2 ICD

(avg) A1 A2 A3 A4

RA- 2(Intersection of Shanti Pathand Panchshel Marg near

US Embassy )

4 ARM 57.8 56.3 57.05 78.1 75.2 76.65 7.81 7.41 7.75 7.45

Parameters Value

tc 1.65

tf 1.6305

A 2208

B 0.000232


y = 2108.4e-5E-05x

y = 2293.7e-1E-05x

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500 3000

Entr

y Ca

paci

tY, C

e (V

eh/h

r)


COMPARISON OF FIELD EQUATION WITH CORRECTED EQUATION BY METHOD 2(Data from RA-2)

Ce(trendline)

HCM Equation

coorected hcm

Expon. (Ce(trendline))

Expon. (HCM Equation)

METHOD 2

QcCe(trendline)

Y= 2108*e^(0.0006 * Vc)Correction

FactorCORRECTED

HCM HCM Equation250 2077 123 2163 2285500 2046 184 2095 2278750 2016 233 2038 2270

1000 1986 275 1988 22621250 1956 313 1942 22541500 1927 348 1899 22461750 1898 381 1858 22382000 1870 411 1820 22302250 1842 440 1783 22222500 1815 468 1748 22152750 1788 495 1713 2207

The second model is consistent with thedata from other roundabout

The first method doesn’t work for alltype to roundabout as it appliescorrection to the average follow up timewhich is varying drastically in our case

The model from the second methodholds valid due to similarity in flowconditions of both the roundabouts

24


Total CapacityTabulate Entry and circulating Flowfrom all the four arms separately

Plot Circulating Flow Vs Time for allfour arms respectively

Identify the lowest value ofCirculating Flow

Calculate Entry Capacity for the armwith least Qc (Ce)

Note down entry flow in the otherthree arms at the same time

TOTAL CAPACITY =Ce+Qe1+Qe2+Qe3

Arm 1 Arm 2 ARM3 Arm4

Time Qe QC Time Qe QC Time Qe 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

ARM 1

Qc1=1608

Qe1=540

Qe2=1224

Qe3=132

Ce=1867Ce=1867

Qc=516Qc= 108Qc= 108

Qc= 1000

ARM

4

ARM 2

ARM 3

0

500

1000

1500

2000

2500

0 5 10 15

Circ

ulat

ing

Flow

(Veh

/hr)

Time

CIRCULATING FLOW vs TIME

ARM1

ARM2

ARM3

ARM4

Ce

Minimum Qc

HCMCe

CorrectionFactor

Corrected Ce ARM1 ARM2 ARM3 ARM4

TOTALCAPACITY

ARM1 1452 1370 341 1029 - 1668 492 132 3321

ARM2 240 1877 120 1757 672 - 720 192 3341

ARM3 108 1942 76 1867 540 1224 - 144 3775

ARM4 900 1581 258 1323 444 1080 828 - 3675

MAX 377526


Level of Service

An attempt has been made to identify the LOS of the roundabout basedupon out comings of the Ph.D. thesis undertaken by Dr. Sewa Ram.

The research made an effort to study the different roundabouts withdifferent 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 geometricand 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 outcomings of the Ph.D. thesis undertaken by Dr. Sewa Ram.

From the Research paper Roundabout 3 matches with the traffic and geometricconditions the our case study roundabout

The LOS of the Case study Roundabout comes out to be LOS C

28


Conclusion

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)

Correction Factor(C.f)= 5 *(Vc^0.5615)

29

ANNEXURES

31


LOGIT METHOD

32

STUDY OF CRITICAL GAP AND ITS EFFECT ON ENTRY CAPACITY OF AROUNDABOUT IN MIXED TRAFFIC CONDITIONSProcedure:

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 theseobservations 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 value of criticalgap.

33


HARDER METHOD

34


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 relativevalues:

N= number of all gaps of size , which are provided to entry vehiclesAi= number of accepted gaps of size iri= Ai/Ni

The ai values are corrected by a floating average procedure, where each is alsoweighted with the Ai values.

Finally the value of Critical gap (Tc) is found by calculating the probability of gap beingaccepted in a given interval

HARDER METHOD

Major Drawback of both Harder and Logit method is that they only take into accountthe 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 theydon’t have a strong mathematical background.

35


RAFF’S METHOD

36


37

RAFF’S METHOD

The earliest method for estimating critical gaps seems to be that by Raff. In thismethod the empirical distribution functions of accepted gaps Fa(t) and rejectedgaps Fr(t) Is taken.

When the sum of cumulative probabilities of accepted gaps and rejected gaps isequal 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 thenumber of accepted gaps smaller than critical gap.

Tc is that value of t at which the following functions intercept

where Fa (t) is the cumulative proportion of accepted gap; Fr (t) is thecumulative proportion of rejected gap; t is the headway of two continuedvehicles of circulating stream.

Raff’s method though considers both accepted and rejected gap of individual drivers it isnot backed any strong mathematical model.

1 − (t) and (t)


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 driverwaits to running into the roundabout.

The mean and variance of critical gap can be calculated by use of the maximumlikelihood function of probability theory

In this method it is assumed that critical gap follows lognormal distribution,

whereF(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 wasrejected;

Likelihood of critical gap is taken as the log of the above function.

[ − ]


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 andvariance of log of critical gap using the following equations.

Where, m is the mean of critical gaps 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 criticalgap causes errors in the outputs in cases were we have very low values of accepted andrejected gaps.

40

=ln ( + 1)

= ln( ) − 0.5and

= .


HCM IIT ROORKEE METHOD

41


n

c i i ci 1

Min Abs T R Abs A T

This method works on the assumption that the critical gap of individual drivers liesomewhere 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

This method gives results close to that critical gap values obtained by MaximumLikelihood Method.

The method is less tedious also removal of Use of Log function helps remove chances oferror during calculations.

42


MODIFIED MLM METHOD (By CRRI)

SQRT(0.5*[(Ai-Tc)^2+(Tc-Ri)^2]

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.

By the earlier equation there was a large variation in the values obtained by it.

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STUDY OF CRITICAL GAP AND ITS EFFECT ON ENTRY CAPACITY...

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