Capacity Analysis of Interchange Ramp Terminals Final Report (1997) / Chapter Skim
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APPENDIX C Capacity Characteristics for Interchanges and Closely-Spaced Intersections
APPENDIX C Capacity Characteristics for Interchanges and Closely-Spaced Intersections
Pages 205-278
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From page 205... ...
Finally, the minimum discharge headway and saturation flow rate models are developed for the left-turn movements. The development of start-up lost time models for each of these movements is described in the next section.
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From page 206... ...
u~umate~y, the a~scnarg~ng stream converges to a relatively constant headway In He range of I.8 to 2.0 seconds per vehicle. This constant value represents the minimum discharge headway H of He queue; its Inverse represents the saturation flow rate s.
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From page 207... ...
= discharge time of the ~ queued vehicle (i = j-l to by, see, h, = headway of the vehicle in the id queue position' see; j = ``specified'' first queue position to discharge at the minimum discharge headway; and J= last queue position to discharge. The HCM (1J indicates Hat the saturation flow rate s can be computed from He minnnum discharge headway using He following equation: 3,600 s.
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From page 208... ...
As the results ot this research share this purpose, the models developed in this appendix are based on H5 rawer Man H lithe implications ofthis approach are discussed in the next section, however, subsequent to that discussion, all references to the m~nnnum discharge headway, saturation flow rate, or start-up lost the variables do not include the "5" subscript Equations C-l, C-2, and C-3 were used to compute the average discharge headway, saturation flow rate, and start-up lost time for the Trough movements included in the field studies.
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From page 209... ...
This effect can be seen in Figure C-l where the line representing average minimum discharge headway is "pulled" upward slightly by headway observations in queue positions five and six. The second factor affecting the amount of bias due to the use of headway s not at the m~nun~n value relates to the frequency of headway observation at each queue position.
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From page 210... ...
However, from the standpoint of statistically quantifying the effect of venous factors (e.g.' lane widths on minimum discharge headway or saturation flow rate, these differences can be very important. There are several implications that stem from He use of a biased estimate of the true magnum discharge headway.
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From page 211... ...
, found that the number of vehicles served per cycle had an effect on the minimum discharge headway. Specifically, he found that the headways observed for each queue position were lower when there were more vehicles queued behind that position.
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From page 212... ...
s where: Huh = through movement minimum discharge headway, sec/veh; us = speed at saturation flow, m/s; and I' = demand flow rate per lane (i.e., traffic pressure) , vpcpl.
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From page 213... ...
The effect of traffic pressure on discharge headway is shown in Figure C-3.
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From page 214... ...
Minimum Discharge Headway, sec/veh 2.2 ~ 2.0 + (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 100 obs)
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From page 215... ...
A A AA B BAA A A A AA B BAA A BA A A AN A A BCDDBBACA B BAA A A AD CAB CEEDBACAC GAB B ABA A B BA AB AACA ABB BB A A A BCBAA BA A CB A A 0 75 150 225 Distance to the Downstream Queue, m 300 375 Figure C-4. Effect of disfance-fo-queue on through movement minimum discharge headway.
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From page 216... ...
Nevertheless, the statistics in Table C-3 indicate that there is a statistically significant relationship between minimum discharge headway, traffic pressure, and distance to the back of queue. The root mean square error and number of observations can be used to estimate the minimum standard deviation (or precision)
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From page 217... ...
Model Statistics Value R2: ~ 0.04 | Root Mean Square Error 0.56 sec/veh Observations: | 7,704 Range of Model Variables Variable | Variable Name | Units ~Minimum H', Through movement min. discharge headway sec/veh 0.61 v,' Demand flow rate per lane (traffic pressure)
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From page 218... ...
Comparison of predicted arid measured" through movement minimum discharge headways. Several additional effects were also evaluated during the mode!
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From page 219... ...
Presumably, We error represented by this statistic is from random sources, however, there could also be some variation due to systematic effects. Knowledge oftypical values of the root mean square error for We dependent vanable can be a usefid gage to assess whether additional systematic error exists in the data.
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From page 220... ...
These ejects are shown in Figures C-6 and C-7 for the discharge headway and saturation flow rate, respectively.
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From page 221... ...
1 0 vpcpl 1 i 1 1 o 60 120 180 240 300 Distance to the Downstream Queue, m Figure C-7. Elect of distance-to-queue, Spillback occurrence, and tragic pressure on through movement saturation flow rate.
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From page 222... ...
The calibrated Trough movement minimum discharge headway model was converted into an equivalent saturation flow rate model. The form ofthis model was patterned after that used in Chapter 9 of the HCM (1)
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From page 223... ...
The trends shown are similar to those noted for the saturation flow rates shown in Figure C-7. Specifically, the adjustment factor (and saturation flow rate)
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From page 224... ...
An equivalent relationship, as it relates to minimum discharge headway, is: Hit = 1.73 ~ 2.60 R where: H.' = left-turn movement minimum discharge headway, sec/veh, and R = radius of curvature of the left-turn travel path (at center of palm, m.
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From page 225... ...
. Bonneson recommended Me following equation for predicting the minimum discharge headway of a left-turn movement as a function of radius and traffic pressure: H = 1.58 + - - 0.0121 v' -It R 0.245 where: v`' = demand flow rate per lane (i.e., traffic pressure)
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From page 226... ...
Figure C-9. Effect of traffic pressure on left-turn movement minimum discharge headway.
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From page 227... ...
The effects of these two factors were dissimilar in Me sense that drivers adopted shorter discharge headways dunng shorter phase durations or longer cycle lengths or both. A similar effect of phase duration on headway has been noted by Stokes et al 649.
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From page 228... ...
Minimum Discharge Headway, sec/veh (Legend: A = 1 obs, B = 2 obs, etc.) (Each data point represents the average of 75 obs)
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From page 229... ...
- Hit= left-turn movement minimum discharge headway. v,' = demand flow rate per lane (i.e., traffic pressure)
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From page 230... ...
As was noted In a previous examination of Table C-l, Me variability in minimum discharge headways among sites can often be largely explained by the bias due to queue position and frequency of observation. The ANOVA techniques descnbed previously were used to remove this bias arid, thereby, facilitate a closer examination of the elect of radius.
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From page 231... ...
. Nevertheless, He statistics in Table C-5 indicate a statistically significant relationship between minimum discharge headway, radius, traffic pressure, and g/C ratio.
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From page 232... ...
0.05 Root Mean Square Error: 0.44 sec/veh l Observations: | 4,153 Range of Model Variables Vanable Variable Name Units Minimum He Left-turn movement min. discharge headway sec/veh 0.83 Radius of curvature of travel paw meters 15 _ v,' Demand flow rate per lane (traffic pressure)
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From page 233... ...
The range in traffic pressure considered makes a difference of about 0.08 sec/veh in minimum discharge headway and about ~ 00 vphgp} in saturation flow rate. In contrast, the g/C ratio has almost twice Me effect as traffic pressure (i.e., a change of about 0.15 sec/veh and 160 vphgpl)
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From page 234... ...
Effect of traffic pressure, signal timing, and radius on left-turn movement saturation flow rate.
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From page 235... ...
In this regard, the saturation flow rate would be equal to the ideal rate when all factor effects are optimum for efficient traffic flow and the corresponding adjustment factors are equal to I.0. Based on this definition, it was determined that an infinite radius, a traffic pressure of 10.0 vpcpI, and a g/C ratio greater than 0.27 were representative of ideal conditions.
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From page 236... ...
Thus, factors that influence this speed (e.g., distance to queue, radius, etc.) also affect minimum discharge headway and saturation flow rate.
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From page 237... ...
Equations C-l, C-2, arid C-3, were used to compute the minimum discharge headway, saturation flow rate, and start-up lost time for the through movements studied. The relationship found between start-up lost time and saturation flow rate is shown in Figure C-17.
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From page 238... ...
Based on the trend in Me data shown in Figure C-17, the following model form was developed for Me start-up lost time model: Z = be ~ be so where: Is = start up lost time, see; and s' = saturation flow rate per lane under prevailing conditions; vphgpl.
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From page 239... ...
The data shown in Figure C-~8 confirm that We mode! is able to accurately predict the start up lost time over the range of saturation flow rates included In Me database.
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From page 240... ...
for Left-Turn Movements Effect of Saturation Flow Rate on Start-up Lost Time. Equations C-l, C-2, and C-3, were used to compute Me minimum discharge headway, saturation flow rate, and soup lost time for the left-turn movements studied.
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From page 241... ...
Specifically, startup lost time increases In a linear manner with increasing saturation flow rate. S=t-up lost tunes for We database range from about I.0 to 4.0 seconds for saturation flow rates ranging from 1,550 to 2,300 vphgpl.
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From page 242... ...
is able to accurately predict the start-up lost hme over the range of saturation flow rates included in Me database. Predicted Start-up Lost Time, sec.
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From page 243... ...
This relationship is shown in Figure C-22. As this figure indicates, the start-up lost times for saturation flow rates of 1,800 and 1,900 pcphgp!
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From page 244... ...
In contrast, the number of vehicles per cycle, minimum discharge headway, and clearance speed were found to be correlated wad green extension. Further examination of the first two of these ejects suggested that the number of vehicles per cycle relative to the maximum number that can be served per phase (i.e., volume-to-capacity ratio)
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From page 245... ...
In addition to X-rabo, clearance speed was also found to be correlated with green extension. Specifically, drivers used more of the yellow interval when they were traveling at higher speeds.
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From page 246... ...
These phases were observed at twelve interchange ramp terminals and at twelve intersection approaches. The green extension data used in this analysis represent observations made for both left-turn and through movements.
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From page 247... ...
Model Statistics Value R2 1 0.11 Root Mean Square Error: | 1.33seconds Observations: 9,044 Range of Model Variables Vanable | Vanable Name |Units| Minimum gY Effective Teen extension into the yellow interval see0.02 _ SL Approach speed limit 56 . G Green interval duration see10 H | Minimum disco ge headway | see T 1.7 v, demand flow rate in lane i vpcpl 2 Volume-to-capacity ratio in lane i na 0.08 , Calibrated Parameter Values Vanable Definition Value Std.
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From page 248... ...
As Figure C-25 illustrates, clearance lost time increases wad approach speed limit and decreases with increasing volume-to-capacity ratio. In general, it ranges hom ~ .0 to 3.0 seconds for typical speed limits arid uncongested conditiorls.
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From page 249... ...
The uneven allocation of traffic to the available traffic lanes is typically quantified In terms of the lane group Lane Utilization Factor U This utilization factor can be computed using the following equation, where the volumes used represent averages per signal cycle: v' N man U = ~(C-2 ,' ~ Hi where: U= lane utilization factor for He lane group; v',, = maximum demand flow rate in any of Nlanes, vpcpl; vi' = demand flow rate in lane i, i = I, 2, N
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From page 250... ...
When there are Tree arrivals per cycle, Equation C-27 predicts a lane utilization factor of 1.33 (= 2 veil * 2 lanes / 3 arrivals)
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From page 251... ...
= expected maximum lane flow rate based on the "most-even" distribution possible, vpcpl; v' = demand flow rate for the lane group, vpc. A key assumption made In Me development of Equation C-28, Mat the "even" and "uneven" arrival cases will occur with 50 percent probability (i.e., Me "~/2" factor shown)
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From page 252... ...
To demonstrate We predictive ability of Equation C-32, Equation C-3 ~ was used to compute the theoretic lane utilization factors for a range of flow rates arid traffic larches. These factors are plotted as data points in Figure C-26 along with the trend lines representing Equation C-32.
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From page 253... ...
= expected maximum demand flow rate in any of Nlarles, vpcpl; and flu = proportion of drivers that do not attempt to evenly distribute themselves. Lane Utilization Factor 2.2 2.0 1.8 1.6 1.4 0~Foints~om Eq.
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From page 254... ...
~ O at three through lanes) fall within the range of expected lane utilization factors shown In Figure C-27 when the volume exceeds about ~ O vpc.
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From page 255... ...
Equations C-34 and C-37 were combined to yield the generalized lane utilization model. This model is applicable to interchanges, adjacent intersections, arid over intersections where propositioning may occur.
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From page 256... ...
Table C-9. Calibrated lane utilization mode' Model Statistics | Value R2: 1 0.18 Root Mean Square Error: 0.1 1 Observations: 97 Range of Model Variables Variable Variable Definition | Units | Minimum U Lane utilization factor na 1.0 vim Maximum lane flow rate in any lane vpcpl 5.8 Member of lanes in the lane group na 2 v',` No.
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From page 257... ...
Comparison of predicted and measured lane utilization factors. C.4.3 Sensitivity Analysis The calibrated lane utilization mode} can be used to examine the relationship between lane utilization, lane group flow rate, and number-of-lanes.
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From page 258... ...
Next, the ability of the proposed models to predict capacity is demonstrated. Finally, a sensitivity analysis is conducted to illustrate the effect of saturation flow rate and lost time on capacity.
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From page 259... ...
can occur, see; s = saturation flow rate for the lane group under prevailing conditions, vphg; G = green signal interval, see; Y= yellowinterval,sec; RC = red clearance interval, see; Is = start-up lost time, see; le = clearance lost time, see; arid CP = clear period during cycle/phase when subject flow is unblocked (see Appendix D)
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From page 260... ...
The last two factors were developed for this research arid are specifically applicable to interchanges arid closely-spaced signalized intersections. A Bird adjustment factory was developed for this research that quantifies the effect of turn radius on He saturation flow rate of a left or right-tutn movement.
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From page 261... ...
The distar~ce-to-queue adjustment factor fD accounts for the adverse effect of downstream queues on the discharge rate of art upstream traffic movement. In genes, Me saturation flow rate is low for movements that have a downstream queue relatively near at the start of the phase, it is high for movements that are not faced with a downstream queue at the start of the phase.
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From page 262... ...
If spillback occurs during the phase, the saturation flow rate prior to the occurrence of the spillback is much lower than it would be if there were no spillback. Thus, the magnitude of the adjustment to We saturation flow rate is dependent on whether spillback occurs during Me subject phase.
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From page 263... ...
The effect of travel paw radius is tabulated In Table C-1 I It cart also be computed using We following equation: f = R 1 1 + 1.71 R where: fR = adjustment factor for Me radius of Me travel path, arid R = radius of curvature of Me left-turn travel paw (at center of path)
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From page 264... ...
Traffic Pressure Adjustment Factor. Saturation flow rates are generally found to be higher during peak traffic demand periods than during off-peak periods.
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From page 265... ...
More specifically, start-up lost time increases with saturation flow rate because it takes more time for the discharging queue to attain the higher speed associated with the higher saturation flow rate. The recommended start-up lost tunes corresponding to a range of saturation flow rates are provided in Table C-13.
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From page 266... ...
Clearance lost time can be computed as: ~ = Y ~ R - gy e c where: le = clearance lost time, see; Y= yellow~nterval,sec; RC = red clearance interval, see; and go= effective green extension into the yellow interval, sec.
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From page 267... ...
These values can also be computed using We following equation: "7 ~ U = 1 + 0.423 {N 1l +0.433N 2 al ) \ \ where: Ur = lane utilization factor for random lane-choice decisions; v' = demand flow rate for the lane group, vpc; and N= number of lanes In the lane group.
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From page 268... ...
~ Y' (C-54) where: Up = lane utilization factor for propositioning; vie= number of vehicles In the subject lane group that will be mining left at the downstream intersection, vpc; vies = number of vehicles in Me subject lane group Cat will be turning right at Me downstream Intersection, vpc; aIld Max(v 'fib V'dJ = luger of v a, "d v dr.
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From page 269... ...
; Sn= saturation flow rate for the subject lane under prevailing conditions assuming the "no spillback'' condition, vphgpl; C-65
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From page 270... ...
phase, m; so= saturation flow rate for the subject lane under prevailing conditions assuming the "w~thspiliback" condition, vphgpl; [, = average lane length occupied by a queued vehicle (see Equation C - 5) , m/vein; and Is = start-up lost fume, sec.
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From page 271... ...
Relationship between distance-to-queue and ideal signal onset. 100 90 80 70 60 50 40 30 10 lo Maximum Green Interval, see ~ ~spillb~ckfor~tual yr~errS dista, ,ct:-=mbir~ations- - ~ ~ ~ - ~ ~ ~ that fall below line.
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From page 272... ...
C.5.7 Predictive Ability of the Proposed Models As a verification of the calibrated saturation flow rate and start-up lost time models, the predicted and measured discharge times of several last-in-queue vehicles were compared. The predicted discharge time was computed using the following equation: T = J H + I (C-60)
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From page 273... ...
to predict He discharge time of the last queued vehicle observed dunng several hur~dred signal cycles. evaluation are shown In Figure C-32.
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From page 274... ...
The relative unpact of alternative combinations of saturation flow rate and start-up lost time on capacity are shown In Figure C-33. The capacity shown In this figure was computed using Equation C-39 with a 90-second cycle length and a green extension of 2.5 seconds.
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From page 275... ...
Start-up lost time was found to Increase with saturation flow rate. This increase is due to Me increased time it takes for the discharging queue to attain the higher speed associated win a higher saturation flow rate.
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From page 276... ...
Lane utilization factors based on Diver desire to preposition can vary widely, depending on the volume of traffic Mat is preposition~ng In the subject lane group. Capacity is dependent on the prevailing saturation flow rate, start-up lost dine, and clearance lost time.
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From page 277... ...
9. Special Report 209: Highway Capacity Manual.
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