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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 non-u~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 two-stream 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 two-way lefc-turn 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~lCmi-cf'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 follow-up - time, the platoon or bunching parameters, the impedance method, and the weighting factors. For a complete reference, see Appendix ~ of Working Paper 17 (NCHRP 3-46, 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 5-minute lateral data using Equation 105. Site critical gap and follow-up 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 5-minute 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 multi-lane site, say a two- lane by two-lane 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 4-leg 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 multi-lane 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 follow-up time should be used? File measuring the critical gap and follow-up 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 follow-up 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 follow-up 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 site-specific critical gaps and follow- up times always yields a weighting factor close to I, and using~egeneraliz~ critical gaps and the follow-up 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 5-minute 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 site-specific cntical gap and follow-up 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 follow-up 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 follow-up 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 Multi-lane Sites Selection of Me sites for testing Me weighting factor for the major street Trough movement at multi-lane 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. Site-specif~c critical gap and follow-up 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 Multi-Lane Sites; 5-minute 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 multi-lane 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. Site-specific cntical gaps and follow-up 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 5-minute 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 CoeŁ 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. Site-specific 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 CoeŁ 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 15-minute 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 sub-procedures. So far, the following situations have not been considered: the existence of upstream signal the existence of a two-way left-turn-lane or a mis~/shiped median on Me major street where a two-stage 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 15-minute data Interval. Some points have a large variation between field and mode} capacitor. This is mainly because the critical gap and follow-up time used ~ the mode} are significantly different Dom the site measured critical gap and follow-up 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 follow-up 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 Follow-Up 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 follow-up 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 follow-up 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 follow-up 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 multi-lane 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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Representative terms from entire chapter: