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OCR for page 65
65
Chapter Six
CAPACITY
Vanous capacity models of TWSC intersections have been
documented In chapter two. Each mode! was developed
based on certain stated assumptions. To select the best
model, each must be tested against the collected field data.
Of Devious capacity models,Moate! ].l,Moa~e] 1.2, and
Mode! ].4 were selected as the candidate models as
outlined ~ chapter 3. The 1985 HEM uses Mode! 1.] and
the 1994 HCM uses Moated ].2.
A prunary objective of this research was to recommend a
model based on its fit relative to field conditions, which
were as close as possible to conditions assumed by the
Free recommended models, i.e. simple streams without
impedance effects or heavy horning flows that may require
nonu~tary weights to assess their contribution to the
conflicting volume. A secondary objective was to calibrate
these models by selecting the best Impedance methodology
and set of weights for calculating conflicting flow. To
some extent, these objectives are interrelated. Tests were
conducted first for the basic capacity models, which were
based on the twostream concept (see Figure 3~. No
hierarchy of traffic streams was considered. Next, tests
were conducted considering the existence of traffic streams
of different hierarchies. The Impedance effect of hither
rank steams on the lower rank stream, the weighting
factors used for calculating conflicting flow, the shared
lane situation on He minor stream approach, and the
platoon elect were considered during the model testing
Special considerations were given to conditions such as the
existence of a twoway lefcturn CrWLT) lane on the major
street, He existence of a raised median on the major streets
or the existence of a downstream bottleneck. Such
conditions, not assumed by theory, mean that He capacity
models cannot be applied directly In their current forms.
For the model tests, two measures of effectiveness were
defined: mean absolute error (MAE) and mean absolute
percent error (MAPE). Each was used to evaluate the
quality or "goodness of fit" of each model. Supplemental
parameters considered are parameters Dom He regression
analysis, such as the R2 value. The MAE and MAPE are
defined In Equations ~12 and ~13.
1 n
M4E=n~lCmicf'I
MAPS =
n `l I
~ ~ _ ~' I (113)
where n is He number of data points, cat is the mode!
capacity, veh/hr, and Of is He field capacitor, veh/hr.
A spreadsheet model was designed for conducting various
model testing tasks. The model was based on the 1994
HEM procedure for analyzing IWSC intersections. It
allowed the user to select venous geometric and traffic
input scenarios so Hat different tests could be conducted.
These input options included the critical gap and followup
 time, the platoon or bunching parameters, the impedance
method, and the weighting factors. For a complete
reference, see Appendix ~ of Working Paper 17 (NCHRP
346, 1995), Spreadsheet Model for Capacity and Delay
Model Testing at TWSC Intersections.
BASIC CAPACITY MODEL TESTING
The basic formulae for the recommended capacity models
were developed using the concept of two traffic steams.
Sites selected for~ng~ese basic models consisted only
of a major stream and a minor stream.
Figure 3 ~ through Figure 33 illustrate the results of the
simple tests of the three basic capacity models. The field
capacity was measured based on 5minute lateral data
using Equation 105. Site critical gap and followup time
were used for calculating the model capacity. Table 30 is
a summary of the statistical results of these models.
(~112)
The results indicated good correlations between the model
results and the field capacity measured using Equation
105. Model I.l gave the best results with an MAE of 46
veh/hr and an MAPE of 9.3 percent. The Model I.2 had an
almost identical result to Model 1.1, with an MAE of 46
veh/hr and an MAPE of 9.5 percent. Model 1.4 did not
give a better result than the other two models. Note that
the accuracy of the model result is based on 5minute
internal data or a ~ 12 veh/hr error range. By increasing
the internal length, the model forecast error would be
significantly reduced.
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66
The following conclusions were reached:
Models 1.1 and I.2 pronde identical results of
capacity forecasts, however, the Model 1.1 tends
to provide slightly better forecasting results than
Model 1.2.
Model 1.4, w~ththe or parameter for proportion of
tree flow vehicles, does not improve the capacity
forecasting results compared to the other two
models.
;vv~
800
2. 600
200
Figure 31. Model 1.1 Test
0 200 400 600
Fluid Capacity, whir
800 1000
owe
800
600
400
200
O
0 200 400 600
Fleld Capacitr, veh/hr
800 1~
Figure 32. Model 1.2 Test
mu
800
600
Too
200
o .
O 200 400 600
Add Capacity, vehkr
Figure 33. Model 1.4 Test
800 1~
Table 30. Regression and Statistics Results for Basic Capacity
Models
Constant
SId Err of Y Est
R Squared
No. of Observations
Degree of Freedom
X Coef.
St`1 Err of Coef.
MAE
MAPE
63.8
50.2
0.89
112
110
0.86
0.03
46
9.3%
90.0
48.8
O.g8
112
110
0.82
0.03
46
9.5%
WEIGHS OF OPPOSING FLOWS
5.5
S9.5
0.85
112
110
0.85
0.04
58
5.8%
The theoretical models all assume a very simple two
stream system for which the definition of conflicting flow
is unequivocal (see Figure 3~. The procedure included In
the 1994 HEM update for estimating capacity at TWSC
intersections requires the calculation of the total conflicting
flow for different minor steam movements (see Table I).
While calculating the total conflicting flow for the minor
streams, different proportions of each conflicting stream
are used. For example, if there is no exclusive right turn
lane, only half of the major street right tum volume is
counted as part of the total conflicting volume for the
minor stream movements. At a multilane site, say a two
lane by twolane street, only half of the total Trough
traffic from the left side is counted as part of the total
conflicting Bow for the minor street right turn movement,
and half of the total through traffic from the right side is
counted as part of the total conflicting Bow for the minor
street led turn movement At 4leg intersections, only half
of the through end right turn traffic on the opposing minor
approachis counted as pert office total conflicting flow for
the subject left turn movement. When the major street
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67
right turn movement is channeiized, the right turn volume
is not counted as the conflicting flow for the minor street
movements. For this situation, the weighting factor for the
major street right turn movement is zero.
The proportions used for calculating the total conflicting
flow are referred to as the weighting factors for each major
stream ~ the 1994 HEM procedure, the weighting factors
are either 0, 0.5, or I. The use of different weighting
factors is based on the real world observations that each
major stream has a different effect on We minor street
Diver. It was important, however, to verify that the use of
such weighting factors and Weir weights were reasonable
Trough We mode} testing process. It is also Important to
distinguish He difference between the impedance effect of
higher ranked flows and the contribution to conflicting
volume.
Because there were~insufficient data for testing the
weighting factors for He through and right turn movements
on the minor street opposing approach, this test
concentrated on the weighting factors for the major street
right turn movement, and the major street through
movement at multilane sites.
The opposing approach volumes make a relatively small
contobudon to the total conflicting flow when compared to
the Trough movements on the major street.
Major Street Right Turn Movement
Selection of the sites for testing the weighting factor for
the major street right turn movement was based on He
following criteria:
heavy major sheet right turn traffic
low major street left turn traffic
medium or high minor street traffic
It was Important to isolate other potential imp acts on the
capacity results while conduchng this test. For example,
the site should not have high major street left turn traffic
so that the impedance effect could be ignored. The minor
street traffic should not be so low as to make the field
measurement of the capacity difficult.
One issue needed to be cIanfied before Initiating He test:
that is, which critical gap and followup time should be
used? File measuring the critical gap and followup time
for each site, the major street right turn movement was
included to define the begin and end gap events. It was
treated In the same manner as the major through
movements. If the site specific critical gap and follo~up
time are used, should the weighting factor for the major
street nght turn movement equal one? If values other than
~ are used, say 0.5 from the HEM 1994, would this be He
same as discounting He effect of He major street right turn
movement?
A test was conducted to investigate this issue, using site
specific cntical gaps and followup times. The objective
was to find out what weighting factor for the major street
right ton movement would minimize the mode! forecasting
error. The optimum weighting factor for the major sheet
right turn movement was obtained for each site using
iterative methods. Similarly, the generalized critical gaps
and followup times obtained based on Table 28 and Table
29 were used and another set of optimum weighting
factors were obtained. Table 31 lists the two sets of
optimum weighting factors obtained from this test. It is
evident that using He sitespecific critical gaps and follow
up times always yields a weighting factor close to I, and
using~egeneraliz~ critical gaps and the followup times
always yields a weighting factor of less Han I.
The mode! testing results using bow the HEM default
weighting factor value and He optimum value (when
generalized critical gaps were used for major street right
turns are shown in Figure 34. The statistical results are
listed in Table 32. The field capacity is based on 5minute
intervals measured using Equation 105.
Figure 34 and the statistical results indicate no significant
difference between He mode! forecasting results when
using the HEM value of 0.5 and the optimum value of each
Site as the weighting factor for the major street right turn
movement.
The following conclusions can be drawn:
If the sitespecific cntical gap and followup time
are used, a weighting factor of ~ should be used
for He major street neat An movement. Using a
weighting factor other than ~ (e.g., 0.5) tends to
discount the effect.
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68
The generalized cntical gap and followup time
are based on the regression analysis while sewing
the percentage of major street night turn
movement to zero. ~ these gaps are used, the
effect of major sweet right turn movement should
be considered In the weighing factor.
.
Table 31. Optimum Weightng Factors for Major Street Right Turn Movement Using Different Critical Gap Values
She Specific
Generalized
When gen~ali~d critical gap and followup time
are used, the optimum weitht~g factors for the
major street right turn movement at each site are
close to 0.5, the same as Me value used in Me
1994 HEM.
to
o.g
1.0 1.0
0.S 0.4
1 ooo
600
200
o
400 600 800 1000
Fb d Capered veh~
o HCMOo0~rarn
Figure 34. Capacity Model Results Using HCM Weighting Factor
and the Optimum Weighting Factor for Major Street Right Turn
Movement
Table 32. Regression and Stabshcal Results for Capacity Model
Testing Using Different Weighting Factors for Me Major Skeet Right
Tum Movement
Constant
Std Err of Y Est
R Squared
No. of Observations
Degree of Freedom
X Coef.
Std E'TofCoef.
MAE
MAPE
80.9
66.9
0.S4
127
12S
0.81
0.07
S4
13.9
80.9
63.6
o.s6
127
12s
0.81
0.06
51
12.8
Major Street Through Movement at Multilane Sites
Selection of Me sites for testing Me weighting factor for
the major street Trough movement at multilane major
street sites considered Me following criteria:
· low major streetleR turn traffic or low impedance
medium or high minor street traffic
Two sets of weighting factors were tested: one is the HCM
value which is equal to I/n (where n is the number of
through lanes per direction) and the other is based on the
volume ratio c istributed on each lane. Sitespecif~c critical
gap and followup time values are used. As a result, He
weight~ngLactor for the major street nght turn movement,
whenever applicable, is set to I. Table 33 lists the
weighting factors calculated for each site based on the lane
volume ratios.
Table 33. Weighting Factors Calculated Based on Lane Volume
Ratio
swroos
SET101
NET208
NET214
NET21S
Nwr411
NWT412
0.S4
O.SS
0.42
0.46
0.46
0.45
lass
1 . _ __ .
Note: * implies data are not applicable
0.64
0.29
*
0.2S
*
*
*
Figure 35 shows the model results using both He HCM
weighting factor value and the factor calculated based on
He lane volume ratio. Field capacity was based on 5
minute interval data. No significant difference is observed
between the two weighting factor values used. It can be
seen that He model tends to overestimate He capacitor for
these sites. Note Hat other factors such as impedance were
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69
not taken into accost while Mung We mode} at this stage.
Some of the outliers can be exposed by the low minor
street volume during that interval, which makes Me field
capacity estimation difficult. Table 34 lists the regression
and statistical results of the test.
1000
it'
5 doo
200
o
0 200 loo 600
R  ~  ,
lo HCM Volt Rado
Boo 1000
Figure 35. Testing Weighing Factors for the Major Street Through
Movment at MultiLane Sites; 5minute Mutual Data
Table 34. Regression and Stabshcal Results for Capacity Model
Testing Using Different Weighting Factors for Me Major Street
Through Movement

, ~.2'
,.p~................
Constant
Std EN of Y Est
R Squared
No. of Observations
Degree of Freedom
X Coef.
Stir Err of Coef.
MAE
MAPE
202.6
122.9
0.47
14S
143
0.67
0.06
110
_ 29.6
222.7
114.8
O.SO
14S
143
0.66
0.06
114
32.0
It is concluded Dom He testing results that using He
weighting factors as included in the 1994 HEM for He
major street through movement can give satisfactory
results for most cases (i.e., the weighting factor for the
major street Trough movement can be obtained by fin,
where n is He number of through lanes). However, it can
also be stated that using lane proportions is just as valid,
since no significant difference was found. Using actual
lane proportions may provide better results In more severe
cases of lane imbalance at some multilane intersections.
IMPEDANCE EFFECI S
For He minor street left turn movement and through
movement, He conflicting major sweet left turn movement
has an additional effect besides being part of He
conflicting How. This additional effect has been referred to
as the impedance. Some previous studies have questioned
whether this impedance should also be included In the
capacitor estimation for He minor sweet left turn or Trough
movement.
Testing of the impedance elect was mainly focused on
whether or not considering the impedance improves the
capacityeshmationresult.Sitesw~highmajorstreetlefc
ton volume or high impedance were selected for this test.
Sitespecific cntical gaps and followup times were used,
so the weighting factors for the major street nght turn
movement were set to I. The mode! I.} capacity formula
was used. To account for impedance, the method of He
1994 HEM Update has been used (BnIon, Grossmann,
1988~.
Figure 36 illustrates the capacitor estimation results when
impedance was not included In the capacity calculation.
Figure 37 illustrates the mode} results when the impedance
calculated using the HCM method was used ~ He capacity
calculation. All He field capacities were measured using
Equation 105 based on 5minute intervals. The statistical
results are listed in Table 35.
The results show~atusing~eHCM impedance yields He
best capacitor estimates wig an MAE of 95 veh/hr and an
MAPE of 24.6 percent. If He impedance was not
considered, the results were not as good, wig an MAE of
~14 veh/br and an MAPE of 3 I.5 percent.
vw ~
~ Sw' ==
.~ 6^wn __
Q 4~w =
~O
2nwn ~
_
0 20 ~40^ 600
Field Capac ty, vehlhr
800 i^00
Figure 36. Model Testing Results When Iinpedance Is Not Included
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1000
600
400
200
o
400 600 800 1000
Field Capacity, veh/hr
Figure 37. Model Testing Results When Impedance is Included
Table 35. Regression and Statistical Results of Testing Me
Impedance Options
Constant
Std Err of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coeffic~ent(s)
Std Err of Coe£
MAE
MAPE
280.9
102.3
0.44
162
160
0.s4
O.OS
114
31.S%
216.9
112.8
0.45
162
160
0.60
0.05
9S
l . . 24.6% '
By examining Figure 37 and the mode} results where the
HEM impedance was used it was found that We mode}
usually ova; the capacity. About 70 percent of the
Ma points In Figure 37 are above the 45 degree line. This
led the project team to investigate other potential factors
Hat may reduce the mode} capacity. There are two possible
ways to reduce the mode} capacity estimation: one is to
increase the impedance effect, the other is to increase the
total conflicting volume. Because the HCM memos for He
impedance calculation is based on queuing theory and has
a consistent mathematical background, it was kept intact.
An increase In the toted canflichug flow can be achieved by
introducing hither weighting factors for He conflicting
streams. The first candidate stream was He major street
reacts Field observation has shown that He major street
left turn movement has a disproportionate effect on the
minor street capacity compared with other movements. It
not only reduces the number of usable gaps for He minor
sheet driver, but also impedes the minor street driver as a
reset of queuing and deceleration. Again, for each site He
optimum weighting factor for He major street left turn
movement was obtained by iteration. Sitespecific critical
gaps were used, and the weighting factor for the major
s~eetnght turn movement was set to I. Table 36 lists the
optimum weighting factors for the major street left turn
movement at each site.
As can be seen, He optimum weighting factor for the
major sheet left turn movement ranges between I.2 and
2.7 for the sites tested. The average weighing factor for
these sites is 2.~. The capacities were recalculated applying
these majorstreetlePc~rnweithting factors, and the result
is plowed In Figure 38. The regression and statistical
results are show In Table 37. Significant improvement of
the model forecasting results was achieved when higher
weighting factors were applied to the major street left turn
movement. The MAE was 85 veh/hr, and the MAPE was
20 percent. The number of data points above the 45 degree
line was reduced from 70 percent to 60 percent.
Table 36. Optimum Weighting Factors for the Major Street Left
Turn Movement
NET202
NET203
NET207
NET21 1
NET212
NET216
NET217
CET3 11
NWT402
NWT40S
Average
1.9
1.S
1.9
2.7
2.4
2.1
2.6
1.2
2.7
1.7
2.1
To sum~nanze the results of the Impedance testing, the
following can be concluded:
Consideration of the impedance effect In the
capacity estimation procedure is not double
counting That is' He major street left turn volume
should be included in both He total conflicting
flow and the impedance.
The method used In He HCM 1994 Update for
calculating the impedance is the most appropriate
method.
Besides the impedance effect of the major street
left tum movement, a higher weighting factor
should be used to account for the additional effect
of ibis movement resulting from deceleration time
approaching~e Intersection to make their tum. A
weighting factor of 2.0 is recommended for the
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major sweet left turn movement.
1000
6m
c' 400
is
200
O
o
400 600 800 1000
Field Capacity, vehlhr
Figure 38. Model Testing Results When Using the HCM Impedance
and the Optimum Major Street LeD Turn Weighting Factor
The regression end s~icaIresults are given In Table 37.
Table 37. Regression and Statistical Results, Major Street Len Turn
Weighting Factor
Constant
Std Err of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coefficient(s)
Std Err of Coe£
MAE
I MAPE
1S7.7
121.S
0.4S
162
160
0.65
0.06
8S
20.0%
VALIDATION TESTING
The recommended model and procedure were further
valida~us~ng~e sites collected In Phase ~ of the project
Figure 39 illustrates the results. Again, each data point
represents a 15minute lateral. Sites wad standard
geometry and unusual geometry are shown on the figure
wad different symbols.
800
~ 600
m
&400 1
to

0
200
O.
0 200 400 600 800
Field Capacity, veh/hr
· Unusual · Standard
Figure 39. Testing Capacity Model for Phase I Sites
The regression and statistical results for the standard sites
are shown In Table 38.
Table 38. Regression and Statistical Results, Phase I Data
Cons
Std E" of Y Est
R Squared
No. of Observations
Degrees of Freedom
X Coefficienys)
Std Err of Coef.
MAE
MAPE
3431
71.89
0.60
96
94
0.95
0.08
S9
22.9%
SUMMARY RESULTS AND CONCLUSIONS
There are some conditions for which the capacity mode}
may require additional subprocedures. So far, the
following situations have not been considered:
the existence of upstream signal
the existence of a twoway leftturnlane or a
mis~/shiped median on Me major street where a
twostage gap acceptance process was sometimes
observed
the existence of flared approach on Me minor
street that allows right turn sneakers
These special conditions usually result in specific
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72
operational phenomena which should be taken into
account. This is documented in detail in Chapter ~ of this
report.
If only sites with normal intersection geometry and traffic
flow characteristics are considered, the recommended
mode} and procedure usually give satisfactory results.
Figure 40 shows the capacitor estimation results for all
sites with normal intersection geometry and traffic flow
characteristics using the Mode} I.} with the recommended
procedure. Each point in the figure represents a 15minute
data Interval. Some points have a large variation between
field and mode} capacitor. This is mainly because the
critical gap and followup time used ~ the mode} are
significantly different Dom the site measured critical gap
and followup time. It was found that most of these sites
are located In the southeast sector where the intersection
locations are typicaDyr~al. However, these date points are
only a relatively sm41 portion of the whole data sample.
Table 39 is a summary of the capacity mode} testing
results for different scenarios. These scenarios include
using different critical gap and followup time
assumptions.
It should be pointed out that using the site specific gaps
yields the best result with an MAE of 66 veh/hr and an
MAPE of 17.7 percent. Using the recommended gaps
improved the result compared to using the 1994 HEM
values.
1000
>
.~
~ 500

~D
1
, c
al
O
~ O
_
a 9<
a.
O °

o

_~
oo ~
ale
`
e~
a O
3 
500
Field Capac ty,veh/hr
1000
Figure 40. Capacity Testing Results for Sites WE Nonnal
Geometry and Traffic Flow Characteristics
Table 39. Summary of Capacity Model Results Using Different
Critical Gap and FollowUp Time Values
16.89
lS7.16
O.SO
286
284
1.16
0.07
128
33.3%
1
.~. ~......... .............. ............. . . .......
.... , ......... ............. ................. .............. . ~
Constant 46.1S
St~iErr Y 96.61
R2 0.64
No. ON 286
Degree of Freedom 284
X CoeE 0.96
Std E=. Coef. 0.04
MAE 66
MAPE 17.7
56.64
118.02
0.S4
286
284
O.9S
O.OS
91
24.9%
l .
Some of the key findings throughout the capacity model
testing process are summanzed below.
.
.
.
.
Both Models 1.1 and 1.2 provide good capacity
estimates for TWSC intersections, as long as the
appropriate critical gap and followup time are
used. Mode} 1.4, which introduces platoon or
bunching factors to Mode} ~ .2, does not show any
improvement to the forecasting results compared
to the other models.
Using the site measured critical gap and followup
time gives the best capacity forecasting result.
Using the recommended critical gap and follow
up time does improve the forecasting result
compared with using the HEM critical gap and
followup time values.
Introducing impedance into the capacity
estimation procedure improves the capacity
forecasting results. The methodology included in
the 1994 HCM for calculating the impedance has
been shown to be appropriate.
The weighthng factors for calculation of opposing
volumes used in the 1994 HCM are reasonable for
the major street right turn movement and Trough
movement. However, applying a larger weighting
factor for the major street leR turn movement can
improve the mode} forecasting result. A value of
2.0 is recommended as the weighting factor for
He major street led turn movement.
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73
Various conditions have been identified where Me
cap Incite mode} and procedure may need
modification, e.g., upstream signal, TWLTE,
ra~sed/stnped median, to analyze the intersection
operations. Some ofthese situations may never be
completely resolved through analytical melons.
The only alternative and viable approach towards
solving these problems may be a simulation
technique. Chapter g addresses some of these
issues.
The following recommendations are made:
· The recommended capacity model: Model 1.1.
.
.
The recommended capacity estimation procedure:
1994 HEM.
The recommended parameter values for the
weighting factors for each traffic stream where
applicable:
1  through movement at single lane site;
1/n  Trough movement at multilane site when
calculating vp fro minor left or minor nght, n is
the number of through lanes;
0.5  major street right turn movement (0 if
channelized);
0.5  opposing through and right turn movements,
2.0  major street left turn movement.
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