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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
×
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Suggested Citation:"6. Model Development, Calibration, and Validation." National Academies of Sciences, Engineering, and Medicine. 2023. Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance. Washington, DC: The National Academies Press. doi: 10.17226/27264.
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Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

6. MODEL DEVELOPMENT, CALIBRATION, AND VALIDATION This chapter describes the methodology used for the development of RTOR volume and capacity estimation models for different right-turn lane configurations. In addition, this chapter describes the validation of these models by testing with a subset of the data that was excluded from the model-development process and set aside specifically for validation. Certain options in the model-development process, such as whether to produce separate or combined models for interchange and intersection locations, are also explored in this chapter. The chapter concludes by presenting recommended models considering the probable availability of data to practitioners and ease of use. 6.1 Modeling of Right-Turn-on-Red Volume The first approach to modeling of RTOR explored in this study was the development of models of RTOR volume. Such models would be easy to integrate into the existing HCM, given that the current methodology asks the user to provide an RTOR measurement that is subtracted from the total right-turn volume for subsequent analysis. For this study, several different types of statistical models were developed to estimate the RTOR volume as a function of conflicting volumes, signal timing, and other data. The models were developed for different lane configurations. This section describes the development of these models. 6.1.1 Model Selection For developing regression-based RTOR volume estimation models, flow rates were calculated using 5-minute counts of vehicles departing an intersection, as described in the previous chapter. Count variables may have many observations with values of zero and are often positively skewed. Such skewed variables tend to violate the assumptions of ordinary least squares (OLS) regression. Log-transformations are often used to satisfy the assumptions of OLS regression. However, in data with observations that have a value of zero, log-transformation is not recommended (O’Hara and Kotze 2010). Moreover, linear regression of count variables can yield impossible results, such as negative flow rates. A generalized linear model (GLM) expands the idea of OLS regression so that the response variable is linearly related to the explanatory variables via a specified link function. In the case of linear regression, the link function is identity. A commonly used GLM for predicting count variables is Poisson regression, which uses log as the link function and has the following form: log( μ) = β0 + β1 X 1 + β2 X 2 +  + βk X k Equation 50 Here, μ is the expected number of events, β0 , β1 , β2 ,  , βk are the regression coefficients, and X 0 , X 1 , X 2 ,  , X k are the explanatory variables. The coefficients are estimated using maximum likelihood estimation. Poisson regression assumes that the responses follow a Poisson distribution for each level of X, i.e., the mean and variance of the response variable are equal. 77

For skewed data, the assumption of mean and variance being equal does not hold true. Negative binomial and quasi-Poisson regression are able to accommodate this problem. These models do not require an assumption of the variance being equal to the mean. Negative binomial regression considers variance as a quadratic function of the mean, whereas quasi-Poisson regression considers variance to vary linearly with the mean (Ver Hoef and Boveng 2007). Another issue that often occurs with count-based models is the presence of excess zeros in the data. The models discussed so far may not fit the data very well when there are many zeros. Zero-inflated and zero hurdle models are two popular models that can address such problems. In a zero-inflated model, it is assumed that observations of zero have two different origins: structural and sampling. Structural zeros are modeled using a binomial probability model, and the remaining values are modeled using a Poisson or negative binomial model. Hurdle models provide additional flexibility by modeling all the zeros separately from the non-zeros. In hurdle models, whether a count variable has a zero or positive value is governed by a binomial probability model. The conditional distribution of the positive values is governed by a zero- truncated Poisson or binomial model (Feng 2021). Another model that may be suitable for predicting RTOR volume is the negative binomial mixed model. This model accounts for the variation that is explained by the independent variables of interest (fixed effects) as well as the variation that is not explained by those variables (random effects). A negative binomial mixed model considers the sample correlation by incorporating random effects into the commonly used fixed effects negative binomial model (Zhang et al. 2017). As an alternative to developing a regression model for RTOR volume, another possibility is to model the proportion of RTOR. Proportion data are bounded by the interval [0,1]. A suitable GLM to model such data where the outcome is binomial is logistic regression. Such models compute the proportion as the ratio of the number of target events to the total number of trials (Chen et al. 2017). 6.1.2 Model-Development Process Different categories of regression models, as explained in the previous section, were tested to compare their predictive accuracies. Under each category, three types of regression models were fitted, as shown in Figure 35. Model 1 was fitted with the best set of predictors obtained using “glmulti” package in R (Calcagno and de Mazancourt 2010). This package generates all possible model formulas, fits them, and returns the models that have the best fit. The best set of predictors was found after examining the correlations and the signs of the slopes of the independent variables. Because some of the variables used in Model 1 may not always be available to practitioners, a second model (Model 2) was developed after considering a subset of the independent variables that would be more readily available. Finally, an even simpler model (Model 3) using only a single explanatory variable was developed for each category of regression model. The reason for including Model 3 was the use of such models in previous studies, as discussed in Chapters 2 and 3. Such models may provide reasonable “back of the envelope” estimations and have value for practitioners. 78

Figure 35. Regression models considered for analysis We first developed models for intersections with RTOR occurring from an exclusive right-turn lane. Data from five intersections were set aside for model validation, and the remaining data were used for model development. Descriptive statistics of the variables are shown in Table 14 and a correlation matrix is shown in Figure 36. The descriptive statistics are provided to give a sense of the range of variation in the dataset. The maximum observed RTOR flow rate was 672 veh/h and the mean was 79.6 veh/h across all of the 5-minute observations. The correlation matrix shows that some independent variables are highly correlated with each other. Because highly correlated variables may contribute similar information to the model, we avoided including highly correlated variables together within the same model. 79

Table 14. Descriptive statistics Variable N Max Mean Std Dev Right-turn-on-red flow rate (veh/h) 4,809 672 79.6 75.4 Red duration (%) 4,079 1.0 0.6 0.2 Conflicting thru flow rate during red (veh/h/ln) 4,809 1,788 307.7 266.6 Opposing left flow rate during red (veh/h/ln) 4,809 1,056 57.3 73.8 Shadowed left flow rate during red (veh/h/ln) 4,809 474 51.9 57.1 U-turn flow rate during red (veh/h/ln) 4,809 108 2.0 7.7 Conflicting ped flow rate during red (ped/h) 4,809 204 2.9 11.8 Parallel ped flow rate during red (ped/h) 4,809 156 0.9 5.7 Right-turn flow rate during green (veh/h/ln) 4,809 1,116 96.7 110.3 Conflicting thru flow rate during green (veh/h/ln) 4,809 12 0.002 0.2 Opposing left flow rate during green (veh/h/ln) 4,809 240 7.3 22.4 Shadowed left flow rate during green (veh/h/ln) 4,809 468 20.0 54.6 U-turn flow rate during green (veh/h/ln) 4,809 48 0.3 2.4 Conflicting ped flow rate during green (veh/h/ln) 4,809 48 0.1 1.2 Parallel ped flow rate during green (ped/h) 4,809 168 2.1 9.0 Total right-turn flow rate (veh/h/ln) 4,809 1,176 176.3 147.5 Right-turn-on-red percent (%) 4,809 100.0 45.2 29.9 Total conflicting thru flow rate (veh/h/ln) 4,809 1,788 307.7 266.6 Total opposing left-turn flow rate (veh/h/ln) 4,809 1,056 64.7 76.0 Total shadowed left-turn flow rate (veh/h/ln) 4,809 474 71.9 65.9 Total parallel pedestrian flow rate (ped/h) 4,809 168 3.0 10.6 Total conflicting pedestrian flow rate (ped/h) 4,809 204 3.0 12.0 80

Figure 36. Correlation matrix We started our analysis by fitting the Poisson regression model, which assumes that the mean and variance are equal. The Akaike information criterion (AIC) value was the highest for Model 1 and lowest for Model 3. Next, we determined whether there was overdispersion in the data using statistical tests. The test results are shown in Table 15 which indicate that overdispersion does exist. To account for this overdispersion, the negative binomial and quasi-Poisson regression models were fitted. 81

Table 15. Statistical tests for overdispersion Method Evidence Residual deviance/residual df > 1.5 Chi-square test p-value <0.001 Standardized Pearson Residuals versus predicted means The distribution of the RTOR flow rate, shown in Figure 37, clearly indicates the presence of excess zeros in the dataset. The Vuong test was performed, which has the null hypothesis that excess zeros are not a result of a separate process (Vuong 1989). The p-value from the Vuong test was significant at a 95% confidence level, indicating that the data are incompatible with the null model, thus confirming the presence of excess zeros. The zero-inflated negative binomial (ZINB) model and negative binomial hurdle model were fitted to account for the excess number of zeros. These models have two components: one for zeros and another for non-zero observations. For the non-zero component, variables for the models were selected similarly to the methods described earlier. The red-to-green ratio and total right-turn flow rate were used for the model component pertaining to zeros because these variables are likely to be responsible for the structural zeros. For Model 3, only the total right-turn flow rate was used for the zero-model component. Figure 37. Distribution of RTOR flow rate 82

In the models considered so far, the average effect of the independent variables was considered, where the average was taken over all of the states where data were collected. Since there are multiple study locations in some states (most notably North Carolina, Florida, and Oregon), there is a chance of violating the assumption of independence. This can be resolved by adding a random effect for state that will result in a different intercept for each jurisdiction in the dataset. For this purpose, we fitted negative binomial mixed models with random effects for state. The random effects for Model 1 are shown in Figure 38. This diagram shows the deviation from the mean RTOR flow rate, along with the standard error, for the data from each state. The values lie between 0.5 and 2, meaning that at most there was a difference of 2 veh/h/ln between the results from any particular state and the mean RTOR flow rate. Similar results were obtained for the other two models. The results demonstrate that it is not necessary to consider geographic location in the model development. Figure 38. Model 1 random effects Finally, models for RTOR proportion were developed using logistic regression. These models can be represented using the following equation: exp [ β0 + β1 X 1 + β2 X 2 +  + βk X k ] y= Equation 51 1 + exp [ β0 + β1 X 1 + β2 X 2 +  + βk X k ] Here, y is the expected RTOR proportion, β0 , β1 , β2 ,  , βk are the regression coefficients, and X 0 , X 1 , X 2 ,  , X k are the explanatory variables. After fitting the models, the expected RTOR flow rates can be obtained by multiplying y and the total right-turn flow rate. 83

6.1.3 Model Refinement 6.1.3.1. Handling of Interchanges by Separate Model or Indicator Variable As discussed in Chapter 2, in some pervious research (Hawley and Bruggeman 2009), differences were found in RTOR volumes observed at interchanges compared to non-interchange locations. At interchanges, drivers transition from driving on a freeway to a surface street, turning movements tend to dominate at the ramp approaches, and dual right turns are not uncommon on ramp approaches. Two options for handling interchanges were considered: generating a separate model for interchange ramps, or developing a combined model using an indicator variable for interchange ramps. Some additional analysis was undertaken to determine which option would be better. Using the ZINB regression for Model 1, negative binomial for Model 2, and logistic regression for Model 3, four different models were assessed and compared for each scenario: • Model A used data from intersections only. • Model B used data from interchanges only. • Model C used data from both intersections and interchanges, without any categorical variable for the junction type. • Model D used data from both intersections and interchanges, with a categorical variable for the junction type. The results for exclusive, shared, and dual right-turn lane scenarios are shown in Table 16, Table 17, and Table 18, respectively. The results show that model C performed better in most cases for exclusive and shared right-turn lane scenarios. However, for dual right-turn lanes, model D performed better. Table 16. RMSE of models for exclusive right-turn lane scenario Model form Model type Model A Model B Model C Model D Model 1 Zero-inflated negative binomial 51.3 57.9 51.0 52.3 Model 2 Negative binomial 52.3 64.9 52.5 55.4 Model 3 Logistic regression 55.9 55.7 55.6 57.9 Table 17. RMSE of models for shared right-turn lane scenario Model form Model type Model A Model B Model C Model D Model 1 Zero-inflated negative binomial 45.9 66.7 44.6 51.1 Model 2 Negative binomial 51.8 53.9 46.1 49.0 Model 3 Logistic regression 43.5 44.4 46.6 48.4 84

Table 18. RMSE of models for dual right-turn lane scenario Model form Model type Model A Model B Model C Model D Model 1 Zero-inflated negative binomial 50.2 51.0 35.9 36.1 Model 2 Negative binomial 49.1 32.8 32.8 30.6 Model 3 Logistic regression 32.8 42.8 28.9 26.5 By combining larger amounts of data together in a single model, the combined models C and D are generally able to yield lower RMSE than models A and B. In particular, model B has the smallest amount of data available and tends to exhibit higher RMSE than the others. It is somewhat surprising that the addition of an indicator variable for interchange does not improve RMSE (i.e., the RMSE of model D tends to be higher than that of model C, except for some dual right-turn scenarios). The results suggest that differences in the performance of RTOR are not likely to result simply from the subject approach occurring at an interchange. From these results, it appears to be appropriate to proceed with a single model for both interchange and non- interchange locations, with the use of an indicator variable for interchanges in the case of dual right-turn lanes. 6.1.3.2 Selection of Mathematical Forms for Recommended Models Five intersections for exclusive and shared right-turn lanes and two intersections for dual right- turn lanes were set aside for model validation. The RMSE was calculated for each form of model to compare their performance for different lane configurations. The results are shown in Table 19, Table 19, Table 20 for the exclusive, shared, and dual right-turn lane scenarios respectively. The RMSE values for Model 1 show that zero-inflated and zero hurdle models performed comparatively better than the other models for each scenario. For Model 2 and Model 3, negative binomial and logistic regression performed better for the dual right-turn lane scenario. Overall, Model 1 with the best set of predictors performed better than Model 2 and Model 3 for the three different lane configurations. Table 19. RMSE of models for single exclusive right-turn lane scenario Model type Model 1 Model 2 Model 3 Poisson 60.7 69.3 87.4 Negative binomial 61.8 54.1 87.4 Quasi-Poisson 60.7 69.3 87.4 Zero-inflated negative binomial 50.9 55.0 79.4 Negative binomial hurdle 50.9 55.0 79.6 Logistic regression 67.9 83.1 54.5 Mixed effect model 78.0 56.5 80.6 85

Table 20. RMSE of models for shared right-turn lane scenario Model type Model 1 Model 2 Model 3 Poisson 41.4 38.5 45.5 Negative binomial 42.7 39.9 45.5 Quasi-Poisson 41.4 38.5 45.5 Zero-inflated negative binomial 41.7 39.1 43.5 Negative binomial hurdle 41.7 39.1 43.5 Logistic regression 41.7 38.6 38.0 Mixed effect model 45.1 42.7 46.7 Table 21. RMSE of models for dual right-turn lane scenario Model type Model 1 Model 2 Model 3 Poisson 43.1 69.1 36.7 Negative binomial 47.6 60.6 45.9 Quasi-Poisson 43.1 69.1 36.7 Zero-inflated negative binomial 45.6 59.1 44.3 Negative binomial hurdle 45.6 59.1 44.3 Logistic regression 88.0 89.8 83.7 Mixed effect model 45.0 26.8 23.9 Another consideration was the development of models that included useful correlations with the input data. Our objective was to develop models that would reduce the predicted RTOR flow rate as conflicting flow rate increased and that would increase the predicted RTOR flow rate with variables that would logically tend to increase it (e.g., the total right-turn flow rate, duration of red as expressed by red-to-cycle ratio, and the shadowed left-turn flow rate). For a model to be useful, it was desired that variables would be included to address the impacts of the conflicting through and left-turn volumes and of the shadowed left turn. Finally, the independent variables needed to be significant. Based on the RMSE calculations and other variable criteria, we decided to use ZINB for Model 1, negative binomial regression for Model 2, and logistic regression for Model 3. However, a difficulty arose in the application of ZINB to the dual right-turn lane scenario. Although a model could be estimated and the RMSE values could be calculated, the algorithm for estimating the ZINB model did not converge, and it was not possible to assess the significance of the variables included in the model. The hurdle model did converge but did not produce useful results meeting all of the criteria mentioned earlier. In particular, there was a tendency for models run on the dual right-turn lane data to predict that RTOR volume would increase with opposing left-turn volume and decrease with shadowed left-turn volume, which would not produce sensible results in applications of RTOR volume estimation. The issue seemed to be related to the relatively small amount of data available for estimation of the dual right-turn lane models. To address this issue, we combined the data for the single and dual right-turn lanes into a single model and included an indicator variable for the presence of dual lanes. This type of model is presented for Model 1 for dual right-turn lanes, although this model is also able to cover the single exclusive right-turn lane scenario. 86

6.1.3.3 Development of Additional Models After initial model development, based on panel feedback and internal discussion, several additional models were developed with particular use cases in mind. One goal was to develop a model that did not include any signal timing data. For this purpose, several models were tested in which only the other volumes present at the intersection were used as independent variables, and any independent variables reflecting the signal timing, such as the proportion of red time, were excluded. After exclusion of these variables, however, in most of the resulting models the coefficients for conflicting volumes did not yield sensible results. It is anticipated that an increase in the conflicting volume would be associated with a decrease in the RTOR volume. However, in some of these trial models the opposite correlation was observed, and after exclusion of these instances, the remaining models did not have enough remaining independent variables to be useful. Signal timing information is needed to estimate the LOS. To address the case where an RTOR estimate is needed but no signal timing data are available, a method for estimating the signal timing was implemented in the spreadsheet accompanying the practitioner guide and in the HCM Computational Engine. The method is described in the practitioner guide. The method selects a cycle length based on the principle of seeking a target degree of intersection saturation for the given demand volumes and balances splits by equalizing volume-to-capacity ratios. Reflecting on the better performance of the ZINB models (Model 1) and the selection of variables for Model 2, a variation on Model 1 was developed that replaced movement flow rates on red with total flow rates, which are more likely to be available in practice. This new set of models was called Model 1B, and the previous Model 1 was renamed Model 1A. The performance of Model 1B was similar to 1A in terms of overall goodness of fit for the exclusive and shared right-turn lane scenarios. For the dual right-turn lane scenario, attempts to develop an equivalent Model 1B using similar sets of variables did not produce a useful model with correlations to conflicting volumes that would yield sensible results. The inclusion of opposing and shadowed left turns yielded coefficients that had the opposite effect (i.e., the RTOR volume would increase with greater opposing left-turn flow rate and decrease with greater shadowed left- turn flow rate), yet exclusion of these variables would cause still other variables to no longer produce sensible results. 6.1.4 Recommended Models of Right-Turn-on-Red Volume Among the seven model types used in our analysis, the final recommended models for different scenarios were selected with consideration of both the performance and complexity of the models. ZINB regression was selected as the appropriate model type for use in models that make use of as many variables as possible from the dataset available for this study. For models based only on data likely to be available to practitioners (e.g., flow rates and the red-to-green ratio of the subject approach), negative binomial regression was considered to be the most suitable model because of its performance and simpler equation form. The model can be represented by an 87

equation similar to the Poisson regression model. Finally, logistic regression for RTOR proportion using only the red-to-green ratio was selected as the simplest model as an alternative for potential use when conflicting vehicle volume data might not be available. From the results of the previous section, it was determined that a categorical variable for an interchange location is appropriate for the dual right-turn lane scenario. The final models are presented in Table 22, Table 23, Table 24 for the single right-turn lane, shared through and right-turn lane, and dual right-turn lane scenarios, respectively. Table 22. Recommended models for single exclusive right-turn lane scenario Model 1A Model 1B ZINB (All Variables) ZINB (Easier-to-Attain Variables) (a) Zero-Inflated Models Count Zero-Inflated Count Zero-Inflated Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Flow Rate (veh/h/ln) Constant 2.923 *** 1.167 *** 2.793 *** 1.167 *** Red-to-cycle ratio 1.389 *** -5.020 *** 1.486 *** -5.020 *** Conflicting thru flow rate during red (veh/h/ln) –1.290×10–4 *** Shadowed left-turn flow rate during red 2.489×10–3 *** (veh/h/ln) Total conflicting thru flow rate (veh/h/ln) –2.069×10–4 *** Total opposing left-turn flow rate (veh/h/ln) –3.069×10–4 *** Total shadowed left-turn flow rate (veh/h/ln) 6.990×10–4 *** Total right-turn flow rate (veh/h/ln) 3.360×10–3 *** –0.01037 *** 3.558×10–3 *** –0.01037 *** Conflicting pedestrian flow rate during red –2.517×10–3 *** (ped/h) Total conflicting pedestrian flow rate (ped/h) –2.233×10–3 *** Presence of parallel pedestrian crosswalk –0.06377 *** One receiving lane –0.1024 *** –0.05420 Shadowed left turn is present 0.1291 *** Observations 4,390 4,390 Log likelihood –21,932 –22,070 Model 2 Model 3 (b) Other Models Negative Binomial Logistic Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Proportion Constant 2.497 *** –2.321 *** Red-to-cycle ratio 1.743 *** 3.470 *** Total conflicting thru flow rate (veh/h/ln) –2.025×10–4 *** Total opposing left-turn flow rate (veh/h/ln) –4.152×10–4 *** Total shadowed left-turn flow rate (veh/h/ln) 9.084×10–4 *** Total right-turn flow rate (veh/h/ln) 3.869×10–3 *** Total conflicting pedestrian flow rate (ped/h) –2.302×10–3 ** Observations 4,390 4,390 Log likelihood –22,955.02 –91,018.58 Akaike Information Criterion 45,924.04 182,041.10 Note: * p < 0.1; ** p < 0.05; *** p < 0.01. 88

Table 23. Recommended models for shared through and right-turn lane scenario Model 1A Model 1B ZINB (All Variables) ZINB (Easier-to-Attain Variables) (a) Zero-Inflated Models Count Zero-Inflated Count Zero-Inflated Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Flow Rate (veh/h/ln) Constant 2.670 *** 1.458 *** 2.678 *** 1.459 *** Red-to-cycle ratio 1.438 *** –2.734 *** 1.262 *** –2.728 *** Conflicting thru flow rate during red (veh/h/ln) –2.870×10–4 *** Opposing left-turn flow rate during red –9.837×10–4 *** (veh/h/ln) U-turn flow rate during red (veh/h/ln) –2.733×10–3 * Conflicting pedestrian flow rate during red –1.939×10–3 *** (ped/h) Total conflicting thru flow rate (veh/h/ln) –1.941×10–4 ** Total opposing left-turn flow rate (veh/h/ln) –9.304×10–4 *** Total shadowed left-turn flow rate (veh/h/ln) 1.523×10–3 *** Total right-turn flow rate (veh/h/ln) 3.692×10–3 *** –0.01406 *** 3.607×10–3 *** –0.01223 *** Total conflicting pedestrian flow rate (ped/h) –2.088×10–3 *** Presence of conflicting bicycle lane –0.1871 *** One receiving lane –0.2827 *** –0.04132 *** Observations 1,533 1,533 Log likelihood –7,004 –7,002 Model 2 Model 3 (b) Other Models Negative Binomial Logistic Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Proportion Constant 2.013 *** –2.462 *** Red-to-cycle ratio 1.725 *** 2.844 *** Total opposing left-turn flow rate (veh/h/ln) –1.180×10–3 *** Total right-turn flow rate (veh/h/ln) 4.441×10–3 *** Total conflicting pedestrian flow rate (ped/h) –1.200 ×10–3 * Observations 1,533 1,533 Log likelihood –7,510 –37,865 Akaike Information Criterion 15,031 75,733 Note: * p < 0.1; ** p < 0.05; *** p < 0.01. 89

Table 24. Recommended models for dual right-turn lane scenario Model 1A Model 1B (a) Zero-Inflated Models (including single ZINB (all variables) ZINB (Easier-to-attain variables) exclusive lane data) Count Zero-Inflated Count Zero-Inflated Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Flow Rate (veh/h/ln) Constant 2.390 *** 1.245 2.351 *** 1.245 Presence of more than one right-turn lane –0.2293 *** –0.2079 *** Intersection is interchange ramp 0.1343 *** 0.1410 *** Red-to-cycle ratio 1.334 *** –5.160 1.467 *** –5.160 Opposing left-turn flow rate during red (veh/h/ln) –2.461×10–4 * Parallel pedestrian flow rate during red (ped/h) –2.428×10–3 * Conflicting pedestrian flow rate during red (ped/h) –2.224×10–3 *** Total conflicting thru flow rate (veh/h/ln) –2.235×10–4 *** Total opposing left-turn flow rate (veh/h/ln) –3.373×10–4 *** Total shadowed left-turn flow rate (veh/h/ln) 3.348×10–5 Total right-turn flow rate (veh/h/ln) 5.260×10–3 *** –0.02175 5.281×10–3 *** –0.02168 Total conflicting pedestrian flow rate (ped/h) –2.670×10–3 *** Presence of parallel pedestrian crosswalk: Yes –0.04242 * Observations 5746 5746 Log likelihood –22410 –22390 (b) Other Models (Using only dual right-turn Model 2 Model 3 lane data) Negative Binomial Logistic Dependent Variable RTOR Flow Rate (veh/h/ln) RTOR Proportion Constant 1.530 *** –2.293 *** Intersection is interchange ramp 0.4177 *** 0.4159 *** Red-to-cycle ratio 2.470 *** 2.851 *** Total opposing left-turn flow rate (veh/h/ln) –2.539×10–3 *** Total right-turn flow rate (veh/h/ln) 3.582×10–3 *** Total conflicting pedestrian flow rate (ped/h) –1.736×10–3 * Observations 609 609 Log likelihood –2,942 –9003 AIC 5,896 18,012 Note: * p < 0.1; ** p < 0.05; *** p < 0.01. 6.1.5 Final Model Validation As mentioned earlier, a subset of data was set aside for validation of the models, a process that was undertaken continuously during model development with the goal of arriving at a set of final models that would be both useful, well-fitted to the data, and able to provide reasonable results. The performance of the final models is presented here with application to this validation dataset. Figure 39 presents the results for all 12 models. Each model is represented by a scatterplot showing the predicted versus the actual performance. The labels show the RMSE calculated on the same dataset in parentheses. 90

Single, Model 1A (51.00) Shared, Model 1A (49.27) Dual*, Model 1A (55.89) 900 300 600 800 250 500 Predicted (veh/h/ln) Predicted (veh/h/ln) Predicted (veh/h/ln) 700 600 200 400 500 150 300 400 300 100 200 200 50 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 0 100 200 300 Actual (veh/h/ln) Actual (veh/h/ln) Actual (veh/h/ln) Single, Model 1B (51.86) Shared, Model 1B (53.10) Dual*, Model 1B (52.17) 700 300 500 600 250 Predicted (veh/h/ln) Predicted (veh/h/ln) Predicted (veh/h/ln) 400 500 200 400 300 150 300 200 100 200 50 100 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 0 100 200 300 Actual (veh/h/ln) Actual (veh/h/ln) Actual (veh/h/ln) Single, Model 2 (52.48) Shared, Model 2 (48.11) Dual, Model 2 (57.71) 800 300 350 700 300 250 Predicted (veh/h/ln) Predicted (veh/h/ln) Predicted (veh/h/ln) 600 250 200 500 200 400 150 150 300 100 200 100 50 50 100 0 0 0 0 100 200 300 400 500 600 0 100 200 300 400 0 50 100 150 Actual (veh/h/ln) Actual (veh/h/ln) Actual (veh/h/ln) Single, Model 3 (20.04) Shared, Model 3 (25.62) Dual, Model 3 (19.34) 80 70 60 70 60 50 60 50 Predicted (%) Predicted (%) 40 Predicted (%) 50 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 25 50 75 100 0 25 50 75 100 0 25 50 75 100 Actual (%) Actual (%) Actual (%) Figure 39. Final model validation results. *Dual Model 1A/1B show data from both single and dual right-turn lane validation datasets. 91

The RTOR flow rate models (1A, 1B, and 2) generally exhibit correlation between the predicted and actual flow rates. The dataset for the single right-turn lanes had the most data, and the predicted values tend to correlate well with the actual values in the validation dataset. The shared through and right-turn lane models had somewhat less data and tend to exhibit some correlation, although the predicted models tend to overestimate the RTOR flow rate when the actual values are closer to the low end (around 100 veh/h/ln or less). The dual right-turn lane models have one group of data points that exhibit good correlation, but there are some outlier points where the predicted RTOR flow rate is higher than the actual flow rate. The RTOR proportion model (Model 3) is intended to offer a simple prediction model using only the red-to-cycle ratio and, in the case of dual right-turn lanes, an interchange indicator variable. Since the model is greatly simplified, it is not surprising that there is a high degree of spread in the data. The predicted values do tend to correlate with the actual values. However, because the red-to-cycle ratio rarely reaches very low values, the predicted RTOR proportions rarely extend to the lower ranges and are never less than about 0.15 for any of the models, while some actual values did have RTOR proportions in that range, including some zeros. 6.2 Modeling of Right-Turn-on-Red Capacity Two different approaches were used to model RTOR capacity for different lane configurations. The first approach is based on the gap-acceptance theory for two-way stop-controlled intersections as used in previous studies. In the second approach, simulation was used to derive the basic form of equations for RTOR capacity under different scenarios. 6.2.1 Basic Model Structure To begin, it is necessary to first agree upon a terminology for the various movements and signal intervals about which the analysis is concerned. Chapter 4 presented a framework for this analysis that identified the intervals affecting RTOR at different intersections; this is summarized below in brief. For a definition of the terms for vehicle and pedestrian movements, refer to Figure 14 in Chapter 4, which presents a view of an example intersection, and Figure 25, which defines the intervals in terms of the overall signal phase configuration. A typical eight-phase configuration is assumed for the following analysis. The characteristics of the intervals during which RTOR can take place are as follows: • During RTOR interval 1, the shadowed left-turn movement on the intersecting street is in service, and the right-turn movement is usually unobstructed. If there are a significant number of U-turns (V3), this may affect the situation. It is possible that pedestrians (P2) may be served during this interval, as with two-stage pedestrian crossings, but this type of operation is not common in the United States. • During RTOR interval 2, the conflicting through movement is in service, and the subject right-turning traffic must yield to the conflicting through vehicles (V1) and pedestrians (P1) on the crossing street. 92

• During RTOR interval 3, the opposing left-turn movement is in service, and the subject right- turning traffic must yield to the opposing left-turn movement (V2). Considering the above three intervals as separate opportunities for RTOR to take place, the total RTOR capacity cRTOR can be expressed as cRTOR = c1 + c2 + c3 Equation 52 where c1 , c2 , and c3 represent the capacity for intervals 1, 2, and 3, respectively, as shown in Figure 25. The mathematical forms of the capacity equations for each interval must be determined next. These may vary by interval because of the situation that occurs in each interval and by the type of lane configuration on the subject right turn (exclusive, dual, or shared). Thus, there may be up to nine different forms: • Exclusive – Interval 1; Exclusive – Interval 2; Exclusive – Interval 3 • Dual – Interval 1; Dual – Interval 2; Dual – Interval 3 • Shared – Interval 1; Shared – Interval 2; Shared – Interval 3 A synthesis of previous studies was presented in Chapter 2, and an exploration of model outputs was presented in Chapter 4. Much of the previous research suggested the use of a gap-acceptance model similar to that used in the two-way stop control methodology. This type of model does make sense to employ, given that the RTOR maneuver is very similar. The main difference is that RTOR is limited to taking place during a specific amount of time, determined in part by the duration of each interval as well as the amount of conflicting traffic during each interval. It is assumed that the conflicting traffic tends to follow a typical pattern of queuing whereby a queue accumulates during red, this queue is released during the initial portion of green at the saturation flow rate, and this is followed by a lower flow rate representative of the overall demand for the movement. In this case, there is a potential for gaps during the latter subinterval with a lower flow rate. Therefore, we can divide each interval into two subintervals: a gapless subinterval and a gap subinterval. During some intervals (e.g., the service of the shadowed left turn), there is no conflicting traffic, so the entire interval can be considered to be the gap subinterval. A single expression for capacity based on a gap-acceptance model could therefore be used for all three intervals. This expression is adjusted according to the duration of the gap subinterval and may take on different forms according to the lane configuration. Thus, we can reduce the number of expressions from 9 to 3. In addition, there potential for a vehicle or two to be served during phase transitions, similar to “sneakers” on permitted left-turn phases who finish the turn as the permitted interval ends and the next phase begins. In the following analysis, the focus is on the development of an expression for capacity related to the gap subintervals, and a separate term for “sneakers” is not explicitly 93

included; however, such a term could be added and would likely contribute a constant value to the overall capacity. Two different models were developed for capacity. Model 1 was developed using different expressions for capacity for different lane configuration types found in the literature review. Model 2 was developed using a gap-acceptance analysis assisted by microsimulation to establish the forms of the equations and to arrive at a unified model form for RTOR capacity for all lane configuration types, with adjustments to handle specific lane configurations. 6.2.2 Model Using Equation Forms from the Literature From the literature review presented in Chapter 2 and the comparison of previous models presented in Chapter 4, several previous models were selected to construct models of capacity for different lane configurations. As a starting point, we begin by using the two-way stop control capacity modified by the signal timing, as proposed by Liu (1995), for intervals 2 and 3:  Vt  exp  − c c  c{2,3} = Vc  3600  g − g s Equation 53  Vt  C 1 − exp  − c f   3600  where: Vc = total conflicting flow rate (veh/h) tc = critical gap (s) t f = follow-up time (s) g = effective green time for the conflicting traffic (s) g s = portion of effective green used for clearance of conflicting queue (s) C = average cycle length (s) The rightmost term in this formula limits the capacity to a fraction of the time that the conflicting phase is active and no longer producing traffic at saturation flow, which would be very unlikely to have any gaps. During that interval, the capacity is a function of the distribution of gaps, which is a function of the arrival flow rate. This model assumes that the arrivals are randomly distributed. A similar formula would be used for RTOR during both the conflicting through ( c2 ) or the conflicting left-turn ( c3 ) intervals. For interval 1, the RTOR capacity when the shadowed left turns receive an exclusive green display can be expressed as a function of the signal interval of the shadowed left turn and the follow-up time (Creasey et al. 2011): 94

g SHLT 3600 c1 = Equation 54 C tf Here, g SHLT is the effective green duration for the shadowed left-turn phase and t f is the follow- up time for the subject right-turn movement, both expressed in seconds. The above equations are applicable when RTOR occurs from an exclusive right-turn lane. For a shared right-turn lane, the capacity during each interval needs to be multiplied by the probability of RTOR occurrence from a shared lane, which is given as follows (Creasey et al. 2011): 1 (1 − p ) 3600 PRTOR = Equation 55 Vs p C where: Vs = total shared lane volume (veh/h) p = ratio of through vehicles to the total volume in the shared lane For dual right-turn lanes, Chen et al. (2012) proposed using two regimes where RTOR maneuvers can take place. Regime A can be defined as the occurrence of RTOR under acceptable gaps in the conflicting through or left-turn traffic, while regime B represents RTOR movements during the shadowed left turn. For regime A, capacity models consider three possible gap acceptance patterns. The capacity of the left-side right-turn lane during regime A is   qtc2   qt f1   exp  −  1 − exp  −  cleft  S  q2 =⋅  3600  + q1 ⋅ q2 exp  − q (tc2 + t f2 )  ⋅  3600    qt q 3600   2  1 − exp  − f2    qt f2    1 − exp  −     3600    3600   Equation 56  qtc1   exp  −   q 2  3600   + ⋅1 q  qt f1   1 − exp  −   3600   where: q1 = conflicting volume in the rightmost lane (veh/h) q2 = conflicting volume in the left lane (veh/h) 95

q = total conflicting volume (veh/h) = q1 + q2 tc1 = RTOR critical gap, gap closed by vehicles in the rightmost lane (s) tc2 = RTOR critical gap, gap closed by vehicles in the left lane (s) t f1 = RTOR follow-up time, gap closed by vehicles in the rightmost lane (s) t f2 = RTOR follow-up time, gap closed by vehicles in the left lane (s) S = cycle split for regime A (%) Vehicles departing from the curb lane usually turn into the rightmost lane of the right-hand cross street. The capacity of a curb lane during regime A is expressed as follows:   qtc1   qt f2   exp  −  1 − exp  −    3600  q1 ⋅ q2  q (tc1 + t f1 )   3600  ccurb S  q1 =⋅ + exp  − ⋅  qt f1  q  3600    qt f1   2  1 − exp  −   3600   1 − exp  −       3600   Equation 57  qtc2   exp −   q22  3600   + ⋅ q  qt f2   1 − exp  −   3600   During regime B, there is no conflicting traffic, and hence Equation 54 can be used for estimating the capacity during this regime. In summary, the equations for Model 1 are as shown in Table 25. Table 25. Equations comprising RTOR capacity Model 1. Single Exclusive Dual Shared Equation 54 (applied for Equation 54 multiplied by Interval 1 (c1) Equation 54 each lane) Equation 55 Equation 53 multiplied by Interval 2 (c2) Equation 53 Equation 56 + Equation 57 Equation 55 Equation 53 multiplied by Interval 3 (c3) Equation 53 Equation 56 + Equation 57 Equation 55 6.2.3 Use of Simulation Results to Determine New Model Forms To develop a new mathematical model of RTOR capacity, simulation networks with different lane configurations were developed in microsimulation to establish a dataset in which the intersections operate at capacity. This enabled the impact of one variable on the capacity to be identified independently. Although the simulation model is only a rough representation of real- 96

world operation and subject to errors in calibration, it is assumed that the trends in operation will be similar and that the resulting model forms can be applied to analysis of real-world data. VISSIM was used for simulation modeling. A series of idealized intersections was created to cover the situations of exclusive, dual, and shared right-turn lanes. A sufficiently high right-turn flow rate was introduced to ensure that the movement would operate at capacity. Gaps between vehicles departing from the intersection were calculated using raw data from VISSIM vehicle travel time sections. Unsaturated green times were calculated using the departure headways of vehicles on conflicting movements. Equations for the RTOR capacity during the unsaturated green period were obtained by using a curve fitting procedure with the R software package. Additional details are provided in the following discussion for each of the three lane configurations. The overall approach taken here is similar to that of a previous study by members of the research team (Emtenan and Day 2021). 6.2.3.1 Single Exclusive Right-Turn Lane A simulation model of an intersection with two approaches, one with an exclusive right-turn lane and the other with a conflicting through lane, was developed in VISSIM to investigate the single exclusive right-turn lane scenario. A volume of 1,500 veh/h was generated on the input link for the subject right turn. Fixed-time traffic control was used with cycle lengths of 180 sec and four different green durations ranging from 40 to 160 sec. A long cycle length was used to ensure that saturation flow would be sustained. Conflicting volumes were varied from 50 to 950 veh/h to establish scenarios with a wide range of available gaps for the RTOR movement. Priority rules and conflict markers were carefully positioned to examine the impact of different gap acceptance behaviors on RTOR capacity. The minimum gap time for priority rules was varied from 3 to 8 seconds. As discussed earlier, the RTOR interval is divided into a gapless subinterval and a gap subinterval. To determine the length of the gap subinterval, the first step is to determine the queue clearance time of the conflicting traffic. For this, we can employ the HCM equation for conflicting queue service time (TRB 2016), which is q C (1 − P ) g s = 3600 Equation 58 s q  CP  − 3600 3600  g  where: gs = queue service time on the conflicting approach (s) g = effective green time on the conflicting approach (s) C = cycle length (s) q = arrival flow rate on the conflicting approach (veh/h/ln) 97

P = proportion of vehicles arriving during green on the conflicting approach s = adjusted saturation flow rate on the conflicting approach (veh/h/ln) With the simulation data, we calculated g s from the departure headway between the RTOR vehicles. For the remaining duration of the RTOR interval, the capacity was calculated for different conflicting flow rates and priority rule minimum gap times. First, the number of RTOR vehicles served by each gap was calculated. The critical gap was determined from the cumulative number of rejected and accepted gaps. It was found to be nearly equal to the VISSIM priority rule minimum gap time. Therefore, we used the value of the priority rule minimum gap time as the critical gap value for our analysis. The RTOR saturation flow rate sRTOR can be written as a function of the follow-up time tf: 3600 sRTOR = Equation 59 tf This would be the capacity if there were no conflicting volume. As the conflicting flow rate increases, the RTOR capacity decreases. The rate of decay of the RTOR capacity depends on the critical gap. From the simulation data, we found that the RTOR capacity follows a similar pattern to an exponential decay curve, as shown in Figure 40. Therefore, an equation similar to an exponential decay equation was used to model RTOR capacity c′ during the unsaturated green period of the conflicting traffic: =c′ sRTOR exp [ − λ ⋅ VC ] Equation 60 Here, λ is the rate of decrease of RTOR capacity with increasing conflicting flow rate Vc. It can be expressed as a function of critical gap tc. Using simulation data, the following relation was obtained: tc − 1 λ= Equation 61 500 98

Figure 40. RTOR capacity during the unsaturated green period of the conflicting traffic for exclusive right-turn lane Finally, the RTOR capacity for interval i can be obtained after multiplying c′ by the ratio of the unsaturated portion of green and the cycle length: g − gs ci = c′ Equation 62 C Assuming that the same dynamics occur with a protected opposing left-turn movement, the above expressions can be used for any of the three intervals 1, 2, or 3. In the case of interval 1 (the shadowed left turn), the queue service time g s would become zero, and Equation 62 would simplify to the potential capacity (Equation 60) multiplied by the g C ratio of the conflicting movement. 6.2.3.2 Dual Right-Turn Lane A similar procedure was followed for determining the capacity of RTOR from dual right-turn lanes. An intersection with dual right-turn lanes and dual conflicting through lanes was modeled in VISSIM. Green durations, priority rule minimum gap times, and conflicting volumes were varied, similar to the single right-turn lane model. Forms of equations for the RTOR capacity of each lane were found that were similar to those obtained for the single right-turn lane. 99

The fitted model for the RTOR capacity during the unsaturated green period of the conflicting traffic is shown in Figure 41 for the inner and outer right-turn lanes. If cL and cR are the capacities of RTOR from the left-side right-turn lane and the curb lane, respectively, then the total RTOR capacity is as follows: c cL + cR = Equation 63 For both cL and cR , Equation 62 is used, but the parameters governing its operation depend in part on the behavior of the conflicting movement and the selection of gap acceptance parameters. Each lane may have different gap acceptance behavior, so the calculations are done independently for each lane. For the shadowed left-turn interval, Equation 62 is used with g s = 0. 100

Figure 41. RTOR capacity during the unsaturated green period of the conflicting traffic for dual right-turn lanes: (a) left-side right-turn lane and (b) curb lane. 101

6.2.3.3 Shared Through and Right-Turn Lane For the shared through and right-turn lane scenario, the proportion of vehicles turning right from the shared lane was varied in addition to green durations, priority rule minimum gap times, and conflicting volumes. As described earlier, for a single exclusive right-turn lane, the RTOR saturation flow rate sRTOR was found to be a function of the follow-up time tf (Equation 59). However, in a shared lane, the actual RTOR flow rate is limited by the proportion of through vehicles using the shared lane, p. Therefore, the RTOR flow rate in the absence of conflicting traffic is adjusted by a factor fT : 3600 vRTOR =fT ⋅ sRTOR =fT ⋅ Equation 64 tf From an analysis of the simulation data, the following expression was obtained for this adjustment factor: fT = 0.01 ⋅ e 4.3 p Equation 65 The following expression gives the potential RTOR capacity: =c′ vRTOR exp [ − λ ⋅ CV ] Equation 66 Using the simulation data for curve fitting, the following equation form was obtained for λ: 4 ⋅ p + 0.3 ⋅ tc − 1 λ= Equation 67 1000 The following expression was obtained for the RTOR capacity: g − gs c = c′ Equation 68 C Similar to the analysis of the exclusive right-turn lane, Equation 68 is used for all three intervals. In the case of interval 1 (shadowed left turn), g s becomes zero and the equation becomes the potential capacity (Equation 66) multiplied by the g/C ratio. Figure 42 shows the fitted model for the RTOR capacity during the unsaturated green period of the conflicting traffic for different conflicting volumes, critical gaps, and right-turning vehicle proportions. 102

Figure 42. RTOR capacity during the unsaturated green period of the conflicting traffic for shared through and right-turn lane 6.2.3.4 Overall Forms of Capacity Model 2 In summary, Model 2 was developed with the assistance of data taken from microsimulation, with different equations considered for the three types of lane configurations as needed. The application of the equations to the various intervals and scenarios is explained in Table 26. Table 26. Equations comprising RTOR capacity Model 2. Single Exclusive Dual Shared Equation 62, gs = 0 (applied to Interval 1 (c1) Equation 62, gs = 0 each lane separately) Equation 68, gs = 0 Equation 62 (applied to each Interval 2 (c2) Equation 62 lane separately) Equation 68 Equation 62 (applied to each Interval 3 (c3) Equation 62 lane separately) Equation 68 6.2.4 Adjustment Factors for Pedestrians Because pedestrians may influence the RTOR movement at an intersection, appropriate adjustment factors should be applied to the models to obtain a more accurate prediction of RTOR capacity. The adjustment factors can be determined using a procedure similar to the HCM-based procedure for permitted left-turn movements (TRB 2016). According to this procedure, pedestrian flow rate during pedestrian service time is determined first: 103

C v pedg v ped = ≤ 5000 Equation 69 g ped where: v pedg = pedestrian flow rate during the pedestrian service time (ped/h) v ped = pedestrian flow rate in the subject crossing (both directions) (ped/h) C = cycle length (s) g ped = pedestrian service time (s) The average pedestrian occupancy OCCpedg can be obtained from v pedg OCC pedg = ; when v pedg ≤ 1000 2000 Equation 70 v pedg 0.4 + OCC pedg = ≤ 0.9; when v pedg > 1000 10000 RTOR vehicles complete their maneuver based on the availability of gaps in the conflicting traffic after the queue clears. The occupancy of the relevant RTOR-pedestrian conflict zone is a function of the probability of accepted gap availability and pedestrian occupancy. It can be computed from g ped − g q v0 ( OCC pedu ) e −5 OCCr = 3600 Equation 71 g p − gq where: OCCr = relevant conflict zone occupancy g p = effective green time for the conflicting through movement v0 = opposing demand flow rate (veh/h) Finally, the pedestrian adjustment factor f p can be calculated using the time the conflict zone is unoccupied: f p = 1 − OCCr Equation 72 104

6.2.5 Model Validation Two locations for each lane configuration were used for model validation. These locations were chosen because they had some of the highest amounts of RTOR and observation of video showed that the RTOR movement operated close to capacity much of the time at these sites. Some site characteristics and flow rates for these locations are presented in Table 27. The pedestrian volumes in these locations were very low. Therefore, for model validation it was assumed that pedestrians did not interfere with the RTOR movement. Table 27. Site characteristics and descriptive statistics of validation sites Total Total opposing Red RTOR flow conflicting left-turn Opposing duration rate thru flow rate flow rate RTOR Conflicting left-turn (sec) (veh/h/ln) (veh/h/ln) (veh/h/ln) Location lane thru lanes lanes Mean Max Mean Max Mean Max Mean Max NE 112th Ave & NE 18th St Exclusive 2 2 151 171 200 284 197 258 4 8 N Hoagland Blvd & US 192 Exclusive 1 1 400 414 264 380 1,395 1,540 178 236 I-10 & S Acadian Thruway Shared 0 1 626 649 392 464 0 0 391 464 Naperville Wheaton Rd &Ogden Ave Shared 2 1 696 696 272 312 1,037 1,136 42 56 Civic Dr & Ygnacio Valley Rd Dual 3 2 586 622 48 70 1,577 1,800 122 152 I-680 NB Off- Ramp & Ygnacio Valley Rd Dual 1 1 508 559 358 424 68 124 2 8 HCM-recommended default values of critical gap and follow-up time were used for model validation. The videos were revisited to reconfirm the average cycle lengths and green durations at the selected locations. For shared through and right-turn lane scenarios, a right-turn proportion of 0.5 for the two selected sites was found to be appropriate based on observations of the video. However, any proportion may be used in the final models. Two types of models were tested as described earlier. Model 1 uses a gap-acceptance theory-based model (Table 25), while Model 2 uses a mathematical model derived from analysis of simulation data in this study (Table 26). To test the models, we applied these equations using the data for the sites listed in Table 27 and compared the resulting capacity with the actual flow rates observed in the field, which were the highest reported flow rates we observed in our dataset. The chart for each model in Figure 43 shows observed RTOR capacity on the horizontal axis and capacity predicted by the model on the vertical axis. The reference line represents where the observed and predicted capacities are equal. The charts show that for the exclusive and dual right- 105

turn lane scenarios, both models exhibited performance with points relatively close to the reference line. For the shared through and right-turn lane scenario, the models sometimes underestimated the RTOR capacity. One possible reason for this is that the RTOR models developed in this study do not directly take into consideration lane utilization, whereas in the real world it is possible that fewer through vehicles may select the right-turn lane. This was true of both models in the literature as well as the new model forms developed during this research and suggests a potential model refinement. Figure 43. Predicted versus observed RTOR capacity for different lane configurations: (a) Model 1 and (b) Model 2 The performance of the models was compared by calculation of RMSE. The results are shown in Table 28. Overall, the models for the single exclusive and dual right-turn lane scenarios performed reasonably well, while the models for the shared through and right-turn lane scenario had considerably more error. Model 2 had lower error for the single exclusive right-turn lane scenario, both models had very similar error values for the shared through and right-turn lane scenario, and Model 1 had lower error for the dual right-turn lane scenario. From this, we may conclude that Model 2 yields the best capacity estimate in the case of the single exclusive right-turn lane scenario while Model 1 is better in the case of the dual right-turn lane scenario. 106

Table 28. RMSE of RTOR capacity models Lane configuration Model 1 Model 2 Single Exclusive 62.6 50.6 Shared 125.8 125.4 Dual 76.5 83.6 6.2.6 Use of the Capacity Models The capacity models presented here are able to provide estimates of the capacities of the right-turn movements that are available when RTOR is permitted at an intersection. These capacities are provided for the three intervals during which RTOR is possible (illustrated by Figure 25). One of the most important reasons for estimating capacity would be to determine the delay at an intersection. The HCM delay equation includes an expression for the v/c ratio, and the reader might initially be given to think that the additional capacity for a right-turn movement could be added to the existing value of capacity that is provided by the right-turn movement during the effective green. However, modification of the delay expression cannot be done this easily. The existing delay equation is based on an analysis of random arrivals at the intersection served by a single green interval, but RTOR introduces multiple new intervals in which the right-turning traffic may be served. Figure 44 shows queue accumulation polygons for potential delay models: • No modeling of RTOR (Figure 44a) • Adjustment of right-turn volume (Figure 44b) • Adjustment of right-turn capacity (Figure 44c) • With RTOR intervals, where the right-turn flow rate is greater than the RTOR flow rate (Figure 44d) • With RTOR intervals, where the right-turn flow rate is less than the RTOR flow rate (Figure 44e) The area of the polygon determines the uniform delay. It is not immediately clear whether the current delay equation, the kernel of which is the term for delay for uniformly arriving vehicles at a signal with two intervals (red and green), can be directly adapted to a signal with multiple intervals, each of which has its own dynamics. In addition, the adjustments of the delay equation for reduced volume (Figure 44b) or greater capacity (Figure 44c) are not able to produce the same polygon shape because there are fewer points by definition. However, it seems more feasible to obtain reasonable results by adjusting the volume (Figure 44b), which can control the peak of the distribution, as opposed to increasing the capacity (Figure 44c), which is not able to counter any overestimation of delay that occurs before the start of green. 107

(a) Queued Vehicles Time Effective Red Effective Green (b) Queued Vehicles Time Interval 1 Interval 2 Interval 3 Effective Green (c) Queued Vehicles Time Interval 1 Interval 2 Interval 3 Effective Green (d) Queued Vehicles Time Interval 1 Interval 2 Interval 3 Effective Green (e) Queued Vehicles Time Interval 1 Interval 2 Interval 3 Effective Green Figure 44. Queue accumulation polygons: (a) no RTOR, (b) adjustment of RTOR volume, (c) adjustment of RTOR capacity, (d) with RTOR where right-turn arrival flow rate is greater than RTOR flow rate, (e) with RTOR where right-turn arrival flow rate is less than RTOR flow rate. 108

Interval 1, the shadowed left-turn phase, would see some queue accumulation if the arrival rate exceeds the RTOR flow rate (Figure 44d); otherwise, little queue accumulation should occur (Figure 44e). Intervals 2 and 3 (conflicting through and opposing left) each have periods where RTOR is possible, leading to a decreased rate of queue accumulation when the arrival rate exceeds the RTOR flow rate (Figure 44b) or a decrease in the number of vehicles queued if the RTOR flow rate exceeds the arrival rate (Figure 44c). In all cases for this example, the overall size of the queue polygon is smaller than that where no RTOR flow rate occurs (Figure 44a). Additional analysis would be needed to compare the delay expressions resulting from the original HCM delay equation with modified forms of the equations and determine the best way to implement new expressions of capacity. 6.3 Conclusion This chapter presented the outcomes of model development and validation for models estimating RTOR volume and capacity. First, appropriate models to predict RTOR volume from vehicle counts were described in detail. The chapter then discussed the procedures for developing RTOR volume models for different lane configurations. The models were then validated, and an investigation of separate model requirements for the type of junction was performed. A set of recommended models was presented that consider data availability and model complexity. The second section of the chapter focused on RTOR capacity modeling. Two different approaches were used to model RTOR capacity. One approach was based on the gap-acceptance theory, whereas the other approach used simulation data to derive the basic form of equations for predicting RTOR capacity. The models presented here could be used to augment the signalized intersection methodology of the HCM, which in its current form is known to produce inaccurate estimates of delay and LOS for right-turn movements by largely ignoring the additional capacity that may result from the RTOR maneuver. In the absence of field data, the analyst is currently asked to assume an RTOR flow rate of zero. An improved approach would be to use a predicted RTOR flow rate from the recommended estimation models to adjust the total right-turn flow rate. Since RTOR contributes to the total right-turn capacity, an alternative approach would be to adjust the total right-turn capacity by estimating the capacity contributed by the RTOR intervals as estimated by the models described in this chapter. These approaches would result in a more accurate estimation of delay and LOS at signalized intersections compared to the current HCM methodology. Another potential application of the RTOR volume models can be to improve the current pedestrian methodology of the HCM. The RTOR flow rates to be used as an input to the method for determining performance measures for pedestrian LOS can be estimated using the models developed in this study. 109

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Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance Get This Book
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 Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance
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The determination of the level-of-service (LOS) at signalized intersections is an important activity for decision-making in the allocation of resources for managing public roads, estimating the impact of new developments, and designing signal timing plans. There is a need to develop models of right-turn-on-red (RTOR) volume to permit users of the Highway Capacity Manual methodology to estimate the RTOR rather than rely on collection of field data, which often does not include RTOR as a separate quantity.

NCHRP Web-Only Document 368: Right-Turn-on-Red Operation at Signalized Intersections with Single and Dual Right-Turn Lanes: Evaluating Performance, from TRB's National Cooperative Highway Research Program, addresses these needs through the development of models for RTOR volume prediction and the development of improved guidance for whether to allow RTOR.

The document is supplemental to NCHRP Research Report 1068: Right-Turn-on-Red Site Considerations and Capacity Analysis: Practitioner's Guide.

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