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New approach for developing warrants of protected left-turn phase at signalized intersections
Transportation Research Part A 35 (2001) 561±574
www.elsevier.com/locate/tra
New approach for developing warrants of protected left-turn phase at signalized intersections A.F. Al-Kaisy a, J.A. Stewart b,* a
b
Department of Civil Engineering, QueenÕs University, Kingston, Ontario, Canada, K7L 3N6 Department of Civil Engineering, Royal Military College of Canada, Kingston, Ontario, Canada, K7K 5L0 Received 24 June 1999; received in revised form 10 December 1999; accepted 20 December 1999
Abstract The research embodied in this paper presents a new approach for the development of guidelines for the installation of a protected left-turn phase at signalized intersections when permissive-only left-turn operation is present. This approach is based on maintaining intersection trac operation at optimum eciency. Three analyses were presented and discussed and they involved the use of the new approach on some hypothetical basic scenarios at a four-legged intersection with single lane in each approach. The ®rst scenario involved exclusive left-turn lane operation while the other two scenarios involved shared-lane operation. Exhaustive signal optimization analyses were conducted using a signal optimization software package called ``Signal Expert''. Regression models were developed from optimization results that allow the analyst to make the decision on protected left-turn phase installation using the basic input data of signal timing design without the need to perform ®eld measurements. The regression results showed that the transition from permissive to protected/permissive left-turn operation, based on system optimization, is mainly a function of trac conditions and that this transition (interface) is predictable. The results also suggested that these warrants are of reasonable accuracy when compared with those in the current practice. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Signalized intersections; Delay; Protected left-turn warrants
1. General overview One of the most complicated issues in signal timing design of at-grade signalized intersections is to accommodate left turn movements on all approaches safely and eciently. In general,
*
Corresponding author. Tel.: 1-613-541-6000x6394; fax: 1-613-541-8336. E-mail address: [email protected] (J.A. Stewart).
0965-8564/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 5 - 8 5 6 4 ( 0 0 ) 0 0 0 0 9 - 4
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minimizing the number of change intervals in a signal phasing scheme increases the green time allocation for major trac movements and contributes, in most cases, to more ecient signal operations and higher intersection capacities. However, in some cases, left turn movements may create negative impacts on the eciency and safety of trac signal operations if not treated properly. At a typical four-legged signalized intersection operating under light trac conditions, the optimum signal timing plan normally involves two phases; one for each cross street. During each phase, the three trac movements, left, right and through, discharge at the end of the two opposing approaches with the right of way being assigned to through trac. Right turners proceed after yielding the right of way to pedestrians while left turners seek gaps in the oncoming trac and enter the desired approach after yielding to pedestrians crossing that approach. This phasing scheme is considered optimum under light trac conditions, as it yields the highest overall eciency and capacity provided an optimum green time allocation is in eect. As left turning volume and/or opposing through volume increase, a point is eventually reached when it is dicult for left turning trac to ®nd adequate gaps. Beyond this point, the left turners may have to wait for long periods before they can complete their maneuvers. Under these circumstances, a queue of left-turn vehicles typically builds up and it may extend upstream of the left-turn storage bay (when present) causing severe impacts on other trac movements on the same approach. Also, these trac conditions usually raise safety concerns, as drivers tend to accept higher risk in performing their turning maneuvers. The provision of a protected left-turn phase is considered the most eective way to solve this problem. It enables the intersection to handle greater left-turn volumes with less delay incurred by left turners and provides for safer trac operations. However, this protected phase, if not properly justi®ed, usually entails signi®cant negative consequences on intersection eciency and capacity. Worse, it could increase delay and safety hazards for left turners, such as the case of a protected left-turn under light trac conditions and no permissive left-turn maneuvers (Agent and Deen, 1979; Lin and Machemehl, 1983). The guidelines, now in practice, for the installation of protected left-turn phase will be brie¯y described in the following section. 2. Warrants and guidelines in current practice The literature review conducted for the purpose of this research con®rmed that, despite the numerous warrants and guidelines for protected left-turn phasing that are used by dierent agencies and jurisdictions in North America, it seems that none of these guidelines has achieved general acceptance. This observation agrees with some of the relevant studies in the literature (Lalani, 1986; Lin and Machemehl, 1983; Upchurch, 1986). In general, the guidelines for protected left-turn phase installation involve numerical warrants using three categories of criteria (McShane and Roess, 1990; Institute of transportation Engineers, 1992; Federal Highway Administration, 1983; Lin and Machemehl, 1983; Kell and Fullerton, 1991), namely; volume, delay, and accident experience. The following are the most common types of criteria within each of these categories. (a) Volume: Two common volume criteria have been used in the current practice; the absolute value of left-turn volume, and the cross product of left turn volume and opposing through
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volume. Prior to installation of a protected left-turn phase, the ability of an intersection to handle left-turn trac on a certain approach is based on the gap acceptance and its impact on permissive left-turn maneuvers. While the left-turn volume criterion does not account for this factor (gap acceptance) of permissive left-turn operation, the ``cross product'' criterion does. Using the ®rst criterion, a protected left-turn phase is warranted when the left-turn volume exceeds a certain threshold value regardless of other trac and pedestrian movements on the relevant approaches. The second criterion of ``cross product'' takes the following form: X Y P Constant where · X is Left-turn volume, · Y is Opposing through volume. Based on the well-accepted theories of gap acceptance process (Drew, 1968), the simple form above does not seem to capture the complexity involved in left-turn operations. In this context, Lin and Machemehl (1983) stated that ``it is hard to believe that complicated left-turn operations can be characterized by a single numerical value with reasonable accuracy, especially over a wide range of trac conditions.'' (b) Delay: Almost all warrants that fall within this category justify the installation of an exclusive left-turn phase based on some sort of indication of average delay incurred by left turners when permissive left-turn operation is present. None of these warrants accounts for trac operation at the intersection as a whole nor for how average delay incurred by left turners compare to that incurred by other trac movements such as cross street or opposing trac. (c) Accident Experience: Safety considerations have been among the most important warrants for installing protected left-turn phase and, in many cases, in eliminating permissive left-turn movements (Agent, 1987). These warrants specify some threshold values for left-turn related trac accidents on speci®c approach or opposing approaches during a one-or two-year period. Other criteria that are used as guidelines to install protected left-turn phase may involve leftturn capacity, intersection geometric features, system operation, and approach speed. The practice shows that these guidelines are also used in combination by many agencies. The foregoing description of these warrants con®rms the observation made by Upchurch (1986) that these guidelines are mostly based on habit and individual engineering judgement and preference rather than on strong, objectively based research. This is evident in the lack of uniform guidelines in practice and the great variety of criteria that are being used by dierent jurisdictions. Other important observations on the above warrants include: (a) The existing warrants for the installation of a protected left-turn phase address two main concerns, safety and eciency. These warrants involve some numerical threshold values that represent acceptable limits for the safety and/or eciency of left-turn operation and beyond which an exclusive left-turn phase is warranted. These acceptable limits are mainly based on judgement and they are subject to considerable variation among dierent jurisdictions. (b) Except for a capacity warrant, none of the above guidelines is established in the analytical procedures used in North American practice, i.e. the U.S. Highway Capacity Manual HCM (Transportation Research Board, 1997) or the Canadian Capacity Guide for Signalized Intersections CCGSI (Teply et al., 1995).
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(c) All the guidelines in practice overlook the overall trac operation at the intersection as they only examine the performance of left-turn trac operation. (d) The optimization of signal operations (based on some performance criterion) was not considered in developing any of the above warrants.
3. Research objective This research was initiated with two main objectives in mind. The ®rst objective was to examine a systematic approach for developing warrants for the addition of a protected left-turn phase at a signal setting with permissive-only left-turn operation. This approach is based on optimum system operation according to some performance criterion using the analysis and design procedures currently in practice. The second objective which is directly related to the ®rst one is to explore the rules and laws that determine when a protected/permissive left-turn plan, from system operation point of view, is superior to a permissive-only left-turn operation. In light of these objectives, safety considerations were considered beyond the scope of this research but will be considered in future research.
4. Rationale Under light trac conditions and low left-turn demands, it is expected that adding a protected left-turn phase to a signal timing scheme decreases the green time allocation for major trac movements and leads, in most cases, to a less ecient signal operations and lower intersection capacities. However, as trac demands on other approaches increase and/or the left-turn demand increases, a point will eventually be reached where adding a protected left-turn phase could improve the intersection overall eciency and capacity. This becomes evident if the severe impacts of left-turn queue spill backs on other trac movements are considered. The previous discussion implies that, at a speci®c signal setting, there must be a boundary in trac conditions where the system-optimum plan (according to some criterion) changes from permissive to protected/permissive left-turn operation. This boundary involves all the points in trac conditions where this change in optimum signal plan takes place. As such, it could be a line in a plane, a surface in 3-D space, or even a multidimensional interface depending on the number of trac variables that control this boundary. The research embodied in this paper is an attempt to investigate this change in optimum signal plans, i.e. the boundary in trac conditions beyond which the addition of a protected left-turn phase is justi®ed. Also, the association between this ``boundary'' and the key trac variables at a speci®c signal setting is investigated.
5. Methodology The proposed new approach to establish the protected left-turn warrants in this research will be described next in Section 5.1±5.4.
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5.1. Optimization criterion Theoretically, optimization criterion could involve any measure of performance that represent overall system operation, left-turn trac operation, or a combination of both. At the onset of this research, it was decided that optimum left-turn operation is not adequate to justify the installation of protected left-turn phase with all the potential impacts on other trac movements. System optimization was considered more reasonable for establishing the prospective warrants. Speci®cally, average overall delay per vehicle entering the intersection was deemed to be the most appropriate criterion that represents system operations. This was in part because motorists consider delay the most important and perceivable measure of performance of signal operations. Also, delay has historically been used as a measure of eectiveness at signalized intersections by the analysis and design procedures in practice (HCM and CCGSI). 5.2. Platform for optimization work: CCGSI procedures The analytical procedures of the CCGSI were the main analytical tool utilized in evaluating system operation. This design and analysis document is published by the Canadian District of the Institute of Transportation Engineers based on a considerable body of both Canadian and International Research. It is worth mentioning that the principles employed by the analytical procedures in the CCGSI and the U.S. HCM have identical theoretical foundations, and use the same delay equation (Teply et al., 1995; Transportation Research Board, 1997). Nevertheless, the application of these basic principles and the calibration of relationships may dier to re¯ect speci®c conditions in both countries (Teply et al., 1995). 5.3. Exhaustive optimization analyses using signal expert As it is crucial for this research to investigate the boundary between the system-optimum permissive plan and the system-optimum protected/permissive plan, sensitivity analyses to determine the optimum plan using trac variables that are believed to in¯uence this ``boundary'' were required. An enormous number of signal optimizations had to be done to generate the data that were necessary to establish these sensitivities. Practically, this extensive work would not have been possible without the aid of the signal optimization software package ``Signal Expert''. 5.3.1. Signal expert: a signal optimization software package This software package is able to simultaneously optimize the major signal-timing parameters of isolated intersections. These parameters include cycle length, green time split, and phase plan (Stewart, 1992; Stewart and Van Aerde, 1992; 1997). The ability to consider dierent phasing plans in the optimization process is a main feature of this software that is of critical importance to this research. Using Signal Expert, the analyst typically speci®es the valid ranges of cycle length and green time as well as the phase plans of interest. Also, Signal Expert allows the user to specify the cycle time increment step size and green time increment step size which determine the desired level of optimization accuracy. The software then exhaustively evaluates all combinations of cycle time, green split and phase plan. While this software incorporates the CCGSI procedures, links with the U.S. HCM procedures also exist within Signal Expert. Besides the optimum plan for each
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phasing scheme, Signal Expert output typically provides the global optimum plan with various statistics of interest on overall intersection operations. Each optimization run within Signal Expert normally involves a large number (in the order of tens of thousands) of applications of the CCGSI analytical procedures. 5.4. Multivariate linear regression technique The outcome of the sensitivity analyses is a series of tabulated data where the type of optimum signal plan (permissive or protected/permissive) is represented in a form of a binary variable and where the rows and columns represent dierent levels of two typical trac variables in a matrix form. An example of this tabulated data is presented in Fig. 1(a). Multivariate linear regression technique was deemed as the most appropriate analytical tool to establish the boundary between these two optimum signal plans. Also, it was decided that left-turn trac is the most appropriate trac variable to be used as a warrant for the protected/permissive signal plan and therefore it
Fig. 1. (a) Sample of optimization results where the type of left-turn operation is represented by a binary variable (1 permissive, 2 protected/permissive). (b) Data extracted from the sample table in (1(a)) with each pair of values representing a point on a line at the 4-D boundary between the two dierent signal plans.
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was considered as the dependent variable in the regression models. Other typical trac variables were used as predictors of the dependent variable (independent variables). The points at the ``boundary'' were extracted from these tables as shown in Fig. 1(b). In this particular example, each pair of values of two varied trac variables in this table along with the two other constant trac variables represent a point on the boundary (4-D interface). These sets of data were then reorganized in a ®nal tabulated form where the dependent variable appears in one column and the other predictor variables appear in several adjacent columns. The multivariate linear regression analysis is a robust statistical tool, which was used to establish the turning volume at the ``boundary'' in terms of other predictor trac variables. The end result is the production of a model that estimates the maximum left turning volume that can be accommodated in a permissive-phase plan (while maintaining optimum eciency) given the other trac volumes. If the forecast or actual left turn volume at the analyzed intersection exceeds this maximum value, a protected left turn phase would be warranted.
6. Experimental design In order to investigate the factors that determine when a protected/permissive left-turn plan is warranted in terms of operational eciency, the proposed approach was applied on several simple hypothetical con®gurations. Two geometric and lane striping con®gurations were investigated in this research, each representing a four-legged signalized intersection with single lane in each approach. The ®rst con®guration has an exclusive left-turn lane in the northbound direction, opposing through trac in the southbound direction, and through cross trac in eastbound and westbound directions. The second con®guration is similar to the ®rst one with a shared lane of left and through trac in the northbound direction. These hypothetical con®gurations and trac movements are shown in Fig. 2. The signal timing and optimization parameters that were input to Signal Expert are provided in Table 1. Three dierent analyses are presented in this paper. The ®rst analysis involves the con®guration presented in Fig. 2(a) with opposing trac and cross trac as predictor variables. A total of 532 optimization run using Signal Expert were utilized in developing the regression model of this analysis. It is important to mention here that a preliminary set of runs (392 run in this analysis) was necessary to explore the valid ranges of trac variables within which the change between the two plans takes place. This was also needed to have better sense of the optimum increment step size of each trac variable to be used in the ®nal set of runs. The second analysis involves the con®guration presented in Fig. 2(b) with shared-lane operation in the northbound direction. In this analysis, the cross trac in eastbound and westbound directions were held constant at 400 pcph each. The regression model developed in this analysis comprises left-turn trac as dependent variable and both adjacent through trac and opposing trac as predictor (independent) variables. The ®nal set of runs included a total of 343 optimization run that were used to develop the regression model. The same con®guration presented in Fig. 2(b) was utilized in the third and last analysis in this research. This time, the opposing trac in the southbound direction was held constant at 400 pcph while both cross trac and adjacent through trac were used as predictor variables. The
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Fig. 2. Con®gurations and trac movements at signalised intersections investigated in this study (a) with exclusive leftturn lane (b) with shared lane.
®nal set of runs involved a total of 441 optimization run in Signal Expert that was used to develop the regression model. Throughout these analyses, it was assumed that the intersection is isolated from the impact of any adjacent intersection and that there is no constraint on the queue size that could develop on any of the intersection approaches. Another important assumption is that all trac is uniform and is composed of passenger car units.
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Table 1 Signal timing and optimization parameters that were input to Signal Expert Signal timing input variable
Input value
Valid range for cycle length (s) Increment step size for cycle length (s) Valid range for green interval at phase i (s) Increment step size for green time (s) Amber interval (s) All-Red interval (s) Duration of congested period (min) Number of sneakers per cycle
50±200 2 10±100 1 3 2 15 2
7. Analysis of results The results of the three analyses will be discussed next in following sections. 7.1. Analysis one: exclusive left-turn lane This analysis investigated the superiority of protected/permissive left-turn operation over the permissive-only operation when the discharge from the subject approach during the green interval is mainly aected by gap acceptance dynamics. As mentioned earlier, the regression model was formulated so that left-turning trac at the ``boundary'' is predicted using two predictor variables, opposing trac and cross trac. Many transformations on the predictor variables were tried so as to reach the best ®t, i.e. highest coecient of determination and lowest standard error. The proposed model took the following form: Y 935:42
94:4 ln X1
009X2 ;
where · Y is Left-turn ¯ow rate above which an exclusive left-turn phase is warranted (pcph), · X1 is Cross trac ¯ow rate in each direction (pcph), · X2 is Opposing trac ¯ow rate (pcph). An examination of the regression output revealed some important observations. First, the F-test in the analysis of variance showed that the regression model is statistically signi®cant and this ruled out the possibility that all the regression coecients are 0 at 95% con®dence level. Also, the high F ratio (32.1) and the very small P values (all less than 0.0004) indicate that the predictor variables explained much of the variability of the dependent variable. In other words, the left turn trac at the ``boundary'' is signi®cantly dependent on the opposing and cross trac. However, the coecient of determination (R2 of 0.55) shows that about 45% of the variation in the left-turn trac is due to random ¯uctuation or due to factors other than these two predictor variables. This sounds somewhat strange especially if it is remembered that all possible trac movements are involved in this regression model. An examination of the factors that govern the signal operation at this particular setting revealed that another important factor was overlooked in the above
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regression model and this mostly explains the observation above. Speci®cally, it seems that the proportion of left-turn trac that sneaks during the amber intervals is signi®cant and yet it is independent from opposing and cross trac. As for the predictor variables, the t-tests show that all the coecients included in this equation are statistically signi®cant at 95% con®dence level. This implies that both opposing and cross trac have signi®cant in¯uence in determining the leftturn trac at the boundary between the two signal plans. The correlation matrix between the dependent and the predictor variables showed that that there is no co-linearity at all among the predictor variables and, therefore, the information these two predictor variables provide is not redundant. Also, the correlation between the dependent variable (left-turn trac) and predictor variables (cross and opposing trac) was shown to be high. This correlation behavior well satis®es the multiple linear regression assumptions. The other important observation which might contradict the general perception is that, cross trac showed much greater correlation with left-turn trac at the ``boundary'' than opposing trac did in a scenario where left-turn operation is mainly governed by the gap acceptance process. However, this observation sounds logical if one remembers that cross trac determines the green time split between the two phases while gap availability in opposing trac only aects the left-turn rate during the northbound/southbound phase. Despite the fact that the regression model was found to be highly signi®cant and the correlation matrix well satis®ed regression assumptions, it was observed that the assumptions of normality of error term and the constant variance of error term are not entirely satis®ed. This is shown in Figs. 3(a) and 3(b), respectively. It is interesting to note here that the residual shows increasing variance with increasing values of predicted Y (predicted left-turn trac). This is mainly attrib-
Fig. 3. (a) Standard residual frequency histogram for the ®rst analysis. (b) Scatterplot of residual versus predicted leftturn trac values for the ®rst analysis.
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uted to the proportion of left-turn trac accounted for by sneakers versus the rest of left-turn trac that make the maneuver during the green light indication. Logically, higher proportion of sneakers (constant number of left-turners multiplied by number of cycles) is associated with less variability in the error term (residual) and this explains the pattern of residual variance in Fig. 3(b). 7.2. Analysis two: shared lane/through and opposing trac as predictors This analysis investigates signal operation that is dramatically dierent from that in the previous analysis. The fact that the shared lane is the only lane in the northbound direction will cause through trac to be directly aected by the availability of gaps in the opposing trac. This is mainly due to the blockage eect of left turners waiting for a gap in the opposing trac when the green light is in eect in the northbound/southbound directions. More trac variables are involved in this scenario, as compared with the previous one, that could aect signal operations, namely, left-turn trac and adjacent through trac in the northbound direction, opposing trac in the southbound direction, and cross trac in the eastbound/westbound directions. In order to limit the amount of optimization work (for time constraints), this analysis was designed so that left-turn trac at the ``boundary'' is predicted using adjacent through and opposing trac only as predictor variables while holding cross trac constant at 400 pcph in each direction. The ®nal model that was shown to yield the best ®t took the following form: Y 704:32
95:6 ln X1
0:00000009X23 ;
where · Y is Left-turn ¯ow rate above which an exclusive left-turn phase is warranted (pcph). · X1 is Through trac ¯ow rate (pcph). · X2 is Opposing trac ¯ow rate (pcph). Despite the fact that the regression model is highly signi®cant as indicated by the F-test of the analysis of variance, it is easily observed that the coecient of the second predictor in the model is extremely small. This implies that this predictor variable does not explain a lot of the variation of the dependent variable. In fact, this observation could easily be con®rmed by examining the t-statistics of the model coecients. This examination showed clearly that the coecient of the second predictor is not signi®cant at 95% con®dence level. The coecient of determination (R2 0:62) shows that about 62% of the variance in left-turn trac is explained by the predictor variables. The standard error of estimate indicated that the absolute mean dierence between the observed (recorded from experiments) and predicted left-turn volumes is in the order of 48 pcph. The correlation matrix of the variables involved in this model showed no co-linearity, i.e. little correlation between predictor variables, which is essential to satisfy the multiple linear regression assumptions. Also, it showed that the correlation between the second predictor variable and the dependent variable was low and this agrees with expectations. The examination of the error term (residuals) of this model revealed reasonable closeness to the normal distribution and a roughly constant variance along the predicted left-turn ¯ow rates. To verify the observation regarding the signi®cance of the second predictor variable, the multiple linear regression analysis was repeated for this same scenario while omitting the opposing
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trac from the regression equation. The regression results showed that dropping opposing trac from the regression model had almost no impact on the predictive ability of the model. 7.3. Analysis three: shared lane/cross and through trac as predictors In light of the results of the previous analysis, it was logical to expect that some of the variation of left-turn trac might have been explained if cross trac were included in the regression model. As such, this analysis involves the same con®guration as that in the previous analysis but with dierent trac variables. The left-turn trac was predicted using cross trac and through trac as predictor variables. This was done while holding opposing trac constant at 400 pcph. In the previous analysis, opposing trac was shown to be insigni®cant in predicting left-turn trac at the ``boundary''. The regression model of the best ®t took the following form: Y 1060
324:34 lnX1 0:5
0:29X2 ;
where · Y is Left-turn ¯ow rate above which an exclusive left-turn phase is warranted (pcph). · X1 is Cross trac ¯ow rate in each direction (pcph). · X2 is Through trac ¯ow rate (pcph). An examination of the regression results revealed signi®cant improvement in the regression model as a result of the addition of the new predictor variable. The regression model was shown to be highly signi®cant as illustrated by the F-test F 76:19 and the P-values (all less than 10 11 ) in the analysis of variance. Also, the t statistics of the model coecients indicated that they are all signi®cant at 95% con®dence level, or in other words, the two predictor variables are signi®cant in predicting the left-turn trac at the ``boundary''. The coecient of determination R2 0:82 implies that 82% of the variability of the left-turn trac is explained by the predictor variables which is a signi®cant improvement from the previous model. Another indicator of the model improvement was the signi®cant decrease (28%) in the standard error. The examination of the model for regression assumptions through diagnostic tests also showed better compliance with those assumptions as compared with the previous model. The correlation matrix showed almost no correlation between predictor variables and high correlation between the predictor variables and the dependent variable. The distribution of residuals (error term) was shown to be closer to the normal curve than that in the previous analysis. Finally, the variance of the residuals along the range of predicted left-turn trac was shown to be reasonably constant. In conclusion, the model developed in this analysis is superior to that predicted in the previous analysis. The regression results and statistics indicate that the model could be used, with reasonable accuracy, to predict the value of left-turn trac at the boundary between permissive and protected/permissive plans when the system is operating at optimum eciency. 8. Summary of ®ndings and recommendations This paper presented a new approach for the development of warrants for the addition of protected left-turn phase at signalized intersections when permissive-only left-turn operation is
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present. The research project involved exhaustive optimization analyses of signal operation with respect to a speci®c objective criterion; the average delay per vehicle entering the intersection. Therefore, optimum overall system eciency was the basis on which the warrants in this research were developed. Three analyses, preliminary in nature, were presented and discussed. The most important ®ndings of this research can be summarized as follows: (a) The models developed by this research showed that the transition from permissive to protected/permissive left-turn operation, based on system optimization, is a function of a number of trac variables and not simply the left-turn and opposing through volume. This transition (boundary) was shown to be predictable particularly for the shared-lane operation scenario. The analyses also showed that the predictive ability of the proposed models could have been improved signi®cantly (particularly the model for exclusive left-turn lane) if sneakers during the amber interval were incorporated in the regression models. This is planned in the future work of this ongoing research. (b) This research revealed that engineering judgement is not sucient to determine the dierent factors aecting signal operations. For instance, none of the current warrants in practice involves cross trac as a factor in justifying the protected/permissive left-turn operation, yet the analyses presented in this paper con®rmed the important impact of this trac variable on signal operations. (c) This research indicated that contrary to most current warrants, the volume of opposing through trac may have little impact on when a protected left-turn phase is warranted. (d) In light of the state-of-practice in using the warrants for the addition of protected left-turn phase at signalized intersections, the use of models similar to those developed in this study is both reliable and advantageous. This is mainly due to the fact that, unlike most guidelines in practice, the proposed warrants are consistent with the well-accepted analytical procedures for the design and analysis of signalized intersections. Moreover, the application of these warrants requires no ®eld measurements as only the basic trac counts for signal timing design are needed. In light of the promising results of this initiative research, the authors recommend the use of this approach to develop a uniform set of warrants that cover most con®gurations and trac conditions that closely represent real life situations.
Acknowledgements The authors would like to express their gratitude to the Academic Research Program of the Department of National Defence of Canada for funding this research.
References Agent, K.R., 1987. Guidelines for the use of protected/permissive left-turn phasing. Institute transportation Engineers 57 (7), 37±42.
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Agent, K.R., Deen, R.C., 1979. Warrants for left-turn signal phasing. In: transportation research record 737, National Research Council, Washington, DC pp. 1±9. Drew, D.R., 1968. Trac Flow Theory and Control. McGraw-Hill Book Company, New York. Federal Highway Administration, 1983. Trac Control Devices Handbook, U.S. Department of Transportation, FHWA, Washington, DC. Institute of Transportation Engineers, 1992. Trac Engineering Handbook, 4th ed. Prentice Hall, Englewood Clis, New Jersey. Kell, J.H., Fullerton, I.J., 1991. Manual of trac signal design. Institute of Transportation Engineers, 2nd ed. Prentice Hall, Englewood Clis, New Jersey. Lalani et al, N., 1986. A Summary of the Use of Warrants for the Installation of Left-Turn Phasing at Signalized Intersections. ITE Journal 56 (4), 57±59. Lin, H., Machemehl, R.B., 1983. Developmental study of implementation guidelines for left-turn treatments. In: Transportation Research Record 905, National Research Council, Washington, DC pp. 96±105. McShane, W.R., Roess, R.P., 1990. Trac Engineering. Prentice Hall, Englewood Clis, New Jersey. Stewart, J.A., 1992. An automated approach to designing the timing parameters of isolated trac signals. Ph.D. Thesis, Queen's University, Kingston, Ontario, Canada. Stewart, J.A., Aerde, M., 1992. An expert assistant for selecting green splits, cycle times, and phasing schemes based on the canadian capacity guide procedures. Canadian Journal of Civil Engineering 19 (2), 245±251. Stewart, J.A., Aerde, M., 1997. Requirements for automating signal timing plan design. Canadian Journal of Civil Engineering 24 (1), 53±63. Teply, S. et al. 1995. Canadian capacity guide for signalized intersections. In: Institute of Transportation Engineers, 2nd ed. District 7, Canada. Transportation Research Board, 1997. Special Report 209, Highway Capacity Manual, National research Council, Washington, DC (update). Upchurch, J.E., 1986. Guidelines for selecting type of left-turn phasing. In: Transportation Research record 1069, National Research Council, Washington, DC pp. 30±38.
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New approach for developing warrants of protected left-turn phase at signalized intersections A.F. Al-Kaisy a, J.A. Stewart b,* a
b
Department of Civil Engineering, QueenÕs University, Kingston, Ontario, Canada, K7L 3N6 Department of Civil Engineering, Royal Military College of Canada, Kingston, Ontario, Canada, K7K 5L0 Received 24 June 1999; received in revised form 10 December 1999; accepted 20 December 1999
Abstract The research embodied in this paper presents a new approach for the development of guidelines for the installation of a protected left-turn phase at signalized intersections when permissive-only left-turn operation is present. This approach is based on maintaining intersection trac operation at optimum eciency. Three analyses were presented and discussed and they involved the use of the new approach on some hypothetical basic scenarios at a four-legged intersection with single lane in each approach. The ®rst scenario involved exclusive left-turn lane operation while the other two scenarios involved shared-lane operation. Exhaustive signal optimization analyses were conducted using a signal optimization software package called ``Signal Expert''. Regression models were developed from optimization results that allow the analyst to make the decision on protected left-turn phase installation using the basic input data of signal timing design without the need to perform ®eld measurements. The regression results showed that the transition from permissive to protected/permissive left-turn operation, based on system optimization, is mainly a function of trac conditions and that this transition (interface) is predictable. The results also suggested that these warrants are of reasonable accuracy when compared with those in the current practice. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Signalized intersections; Delay; Protected left-turn warrants
1. General overview One of the most complicated issues in signal timing design of at-grade signalized intersections is to accommodate left turn movements on all approaches safely and eciently. In general,
*
Corresponding author. Tel.: 1-613-541-6000x6394; fax: 1-613-541-8336. E-mail address: [email protected] (J.A. Stewart).
0965-8564/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 5 - 8 5 6 4 ( 0 0 ) 0 0 0 0 9 - 4
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minimizing the number of change intervals in a signal phasing scheme increases the green time allocation for major trac movements and contributes, in most cases, to more ecient signal operations and higher intersection capacities. However, in some cases, left turn movements may create negative impacts on the eciency and safety of trac signal operations if not treated properly. At a typical four-legged signalized intersection operating under light trac conditions, the optimum signal timing plan normally involves two phases; one for each cross street. During each phase, the three trac movements, left, right and through, discharge at the end of the two opposing approaches with the right of way being assigned to through trac. Right turners proceed after yielding the right of way to pedestrians while left turners seek gaps in the oncoming trac and enter the desired approach after yielding to pedestrians crossing that approach. This phasing scheme is considered optimum under light trac conditions, as it yields the highest overall eciency and capacity provided an optimum green time allocation is in eect. As left turning volume and/or opposing through volume increase, a point is eventually reached when it is dicult for left turning trac to ®nd adequate gaps. Beyond this point, the left turners may have to wait for long periods before they can complete their maneuvers. Under these circumstances, a queue of left-turn vehicles typically builds up and it may extend upstream of the left-turn storage bay (when present) causing severe impacts on other trac movements on the same approach. Also, these trac conditions usually raise safety concerns, as drivers tend to accept higher risk in performing their turning maneuvers. The provision of a protected left-turn phase is considered the most eective way to solve this problem. It enables the intersection to handle greater left-turn volumes with less delay incurred by left turners and provides for safer trac operations. However, this protected phase, if not properly justi®ed, usually entails signi®cant negative consequences on intersection eciency and capacity. Worse, it could increase delay and safety hazards for left turners, such as the case of a protected left-turn under light trac conditions and no permissive left-turn maneuvers (Agent and Deen, 1979; Lin and Machemehl, 1983). The guidelines, now in practice, for the installation of protected left-turn phase will be brie¯y described in the following section. 2. Warrants and guidelines in current practice The literature review conducted for the purpose of this research con®rmed that, despite the numerous warrants and guidelines for protected left-turn phasing that are used by dierent agencies and jurisdictions in North America, it seems that none of these guidelines has achieved general acceptance. This observation agrees with some of the relevant studies in the literature (Lalani, 1986; Lin and Machemehl, 1983; Upchurch, 1986). In general, the guidelines for protected left-turn phase installation involve numerical warrants using three categories of criteria (McShane and Roess, 1990; Institute of transportation Engineers, 1992; Federal Highway Administration, 1983; Lin and Machemehl, 1983; Kell and Fullerton, 1991), namely; volume, delay, and accident experience. The following are the most common types of criteria within each of these categories. (a) Volume: Two common volume criteria have been used in the current practice; the absolute value of left-turn volume, and the cross product of left turn volume and opposing through
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volume. Prior to installation of a protected left-turn phase, the ability of an intersection to handle left-turn trac on a certain approach is based on the gap acceptance and its impact on permissive left-turn maneuvers. While the left-turn volume criterion does not account for this factor (gap acceptance) of permissive left-turn operation, the ``cross product'' criterion does. Using the ®rst criterion, a protected left-turn phase is warranted when the left-turn volume exceeds a certain threshold value regardless of other trac and pedestrian movements on the relevant approaches. The second criterion of ``cross product'' takes the following form: X Y P Constant where · X is Left-turn volume, · Y is Opposing through volume. Based on the well-accepted theories of gap acceptance process (Drew, 1968), the simple form above does not seem to capture the complexity involved in left-turn operations. In this context, Lin and Machemehl (1983) stated that ``it is hard to believe that complicated left-turn operations can be characterized by a single numerical value with reasonable accuracy, especially over a wide range of trac conditions.'' (b) Delay: Almost all warrants that fall within this category justify the installation of an exclusive left-turn phase based on some sort of indication of average delay incurred by left turners when permissive left-turn operation is present. None of these warrants accounts for trac operation at the intersection as a whole nor for how average delay incurred by left turners compare to that incurred by other trac movements such as cross street or opposing trac. (c) Accident Experience: Safety considerations have been among the most important warrants for installing protected left-turn phase and, in many cases, in eliminating permissive left-turn movements (Agent, 1987). These warrants specify some threshold values for left-turn related trac accidents on speci®c approach or opposing approaches during a one-or two-year period. Other criteria that are used as guidelines to install protected left-turn phase may involve leftturn capacity, intersection geometric features, system operation, and approach speed. The practice shows that these guidelines are also used in combination by many agencies. The foregoing description of these warrants con®rms the observation made by Upchurch (1986) that these guidelines are mostly based on habit and individual engineering judgement and preference rather than on strong, objectively based research. This is evident in the lack of uniform guidelines in practice and the great variety of criteria that are being used by dierent jurisdictions. Other important observations on the above warrants include: (a) The existing warrants for the installation of a protected left-turn phase address two main concerns, safety and eciency. These warrants involve some numerical threshold values that represent acceptable limits for the safety and/or eciency of left-turn operation and beyond which an exclusive left-turn phase is warranted. These acceptable limits are mainly based on judgement and they are subject to considerable variation among dierent jurisdictions. (b) Except for a capacity warrant, none of the above guidelines is established in the analytical procedures used in North American practice, i.e. the U.S. Highway Capacity Manual HCM (Transportation Research Board, 1997) or the Canadian Capacity Guide for Signalized Intersections CCGSI (Teply et al., 1995).
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(c) All the guidelines in practice overlook the overall trac operation at the intersection as they only examine the performance of left-turn trac operation. (d) The optimization of signal operations (based on some performance criterion) was not considered in developing any of the above warrants.
3. Research objective This research was initiated with two main objectives in mind. The ®rst objective was to examine a systematic approach for developing warrants for the addition of a protected left-turn phase at a signal setting with permissive-only left-turn operation. This approach is based on optimum system operation according to some performance criterion using the analysis and design procedures currently in practice. The second objective which is directly related to the ®rst one is to explore the rules and laws that determine when a protected/permissive left-turn plan, from system operation point of view, is superior to a permissive-only left-turn operation. In light of these objectives, safety considerations were considered beyond the scope of this research but will be considered in future research.
4. Rationale Under light trac conditions and low left-turn demands, it is expected that adding a protected left-turn phase to a signal timing scheme decreases the green time allocation for major trac movements and leads, in most cases, to a less ecient signal operations and lower intersection capacities. However, as trac demands on other approaches increase and/or the left-turn demand increases, a point will eventually be reached where adding a protected left-turn phase could improve the intersection overall eciency and capacity. This becomes evident if the severe impacts of left-turn queue spill backs on other trac movements are considered. The previous discussion implies that, at a speci®c signal setting, there must be a boundary in trac conditions where the system-optimum plan (according to some criterion) changes from permissive to protected/permissive left-turn operation. This boundary involves all the points in trac conditions where this change in optimum signal plan takes place. As such, it could be a line in a plane, a surface in 3-D space, or even a multidimensional interface depending on the number of trac variables that control this boundary. The research embodied in this paper is an attempt to investigate this change in optimum signal plans, i.e. the boundary in trac conditions beyond which the addition of a protected left-turn phase is justi®ed. Also, the association between this ``boundary'' and the key trac variables at a speci®c signal setting is investigated.
5. Methodology The proposed new approach to establish the protected left-turn warrants in this research will be described next in Section 5.1±5.4.
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5.1. Optimization criterion Theoretically, optimization criterion could involve any measure of performance that represent overall system operation, left-turn trac operation, or a combination of both. At the onset of this research, it was decided that optimum left-turn operation is not adequate to justify the installation of protected left-turn phase with all the potential impacts on other trac movements. System optimization was considered more reasonable for establishing the prospective warrants. Speci®cally, average overall delay per vehicle entering the intersection was deemed to be the most appropriate criterion that represents system operations. This was in part because motorists consider delay the most important and perceivable measure of performance of signal operations. Also, delay has historically been used as a measure of eectiveness at signalized intersections by the analysis and design procedures in practice (HCM and CCGSI). 5.2. Platform for optimization work: CCGSI procedures The analytical procedures of the CCGSI were the main analytical tool utilized in evaluating system operation. This design and analysis document is published by the Canadian District of the Institute of Transportation Engineers based on a considerable body of both Canadian and International Research. It is worth mentioning that the principles employed by the analytical procedures in the CCGSI and the U.S. HCM have identical theoretical foundations, and use the same delay equation (Teply et al., 1995; Transportation Research Board, 1997). Nevertheless, the application of these basic principles and the calibration of relationships may dier to re¯ect speci®c conditions in both countries (Teply et al., 1995). 5.3. Exhaustive optimization analyses using signal expert As it is crucial for this research to investigate the boundary between the system-optimum permissive plan and the system-optimum protected/permissive plan, sensitivity analyses to determine the optimum plan using trac variables that are believed to in¯uence this ``boundary'' were required. An enormous number of signal optimizations had to be done to generate the data that were necessary to establish these sensitivities. Practically, this extensive work would not have been possible without the aid of the signal optimization software package ``Signal Expert''. 5.3.1. Signal expert: a signal optimization software package This software package is able to simultaneously optimize the major signal-timing parameters of isolated intersections. These parameters include cycle length, green time split, and phase plan (Stewart, 1992; Stewart and Van Aerde, 1992; 1997). The ability to consider dierent phasing plans in the optimization process is a main feature of this software that is of critical importance to this research. Using Signal Expert, the analyst typically speci®es the valid ranges of cycle length and green time as well as the phase plans of interest. Also, Signal Expert allows the user to specify the cycle time increment step size and green time increment step size which determine the desired level of optimization accuracy. The software then exhaustively evaluates all combinations of cycle time, green split and phase plan. While this software incorporates the CCGSI procedures, links with the U.S. HCM procedures also exist within Signal Expert. Besides the optimum plan for each
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phasing scheme, Signal Expert output typically provides the global optimum plan with various statistics of interest on overall intersection operations. Each optimization run within Signal Expert normally involves a large number (in the order of tens of thousands) of applications of the CCGSI analytical procedures. 5.4. Multivariate linear regression technique The outcome of the sensitivity analyses is a series of tabulated data where the type of optimum signal plan (permissive or protected/permissive) is represented in a form of a binary variable and where the rows and columns represent dierent levels of two typical trac variables in a matrix form. An example of this tabulated data is presented in Fig. 1(a). Multivariate linear regression technique was deemed as the most appropriate analytical tool to establish the boundary between these two optimum signal plans. Also, it was decided that left-turn trac is the most appropriate trac variable to be used as a warrant for the protected/permissive signal plan and therefore it
Fig. 1. (a) Sample of optimization results where the type of left-turn operation is represented by a binary variable (1 permissive, 2 protected/permissive). (b) Data extracted from the sample table in (1(a)) with each pair of values representing a point on a line at the 4-D boundary between the two dierent signal plans.
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was considered as the dependent variable in the regression models. Other typical trac variables were used as predictors of the dependent variable (independent variables). The points at the ``boundary'' were extracted from these tables as shown in Fig. 1(b). In this particular example, each pair of values of two varied trac variables in this table along with the two other constant trac variables represent a point on the boundary (4-D interface). These sets of data were then reorganized in a ®nal tabulated form where the dependent variable appears in one column and the other predictor variables appear in several adjacent columns. The multivariate linear regression analysis is a robust statistical tool, which was used to establish the turning volume at the ``boundary'' in terms of other predictor trac variables. The end result is the production of a model that estimates the maximum left turning volume that can be accommodated in a permissive-phase plan (while maintaining optimum eciency) given the other trac volumes. If the forecast or actual left turn volume at the analyzed intersection exceeds this maximum value, a protected left turn phase would be warranted.
6. Experimental design In order to investigate the factors that determine when a protected/permissive left-turn plan is warranted in terms of operational eciency, the proposed approach was applied on several simple hypothetical con®gurations. Two geometric and lane striping con®gurations were investigated in this research, each representing a four-legged signalized intersection with single lane in each approach. The ®rst con®guration has an exclusive left-turn lane in the northbound direction, opposing through trac in the southbound direction, and through cross trac in eastbound and westbound directions. The second con®guration is similar to the ®rst one with a shared lane of left and through trac in the northbound direction. These hypothetical con®gurations and trac movements are shown in Fig. 2. The signal timing and optimization parameters that were input to Signal Expert are provided in Table 1. Three dierent analyses are presented in this paper. The ®rst analysis involves the con®guration presented in Fig. 2(a) with opposing trac and cross trac as predictor variables. A total of 532 optimization run using Signal Expert were utilized in developing the regression model of this analysis. It is important to mention here that a preliminary set of runs (392 run in this analysis) was necessary to explore the valid ranges of trac variables within which the change between the two plans takes place. This was also needed to have better sense of the optimum increment step size of each trac variable to be used in the ®nal set of runs. The second analysis involves the con®guration presented in Fig. 2(b) with shared-lane operation in the northbound direction. In this analysis, the cross trac in eastbound and westbound directions were held constant at 400 pcph each. The regression model developed in this analysis comprises left-turn trac as dependent variable and both adjacent through trac and opposing trac as predictor (independent) variables. The ®nal set of runs included a total of 343 optimization run that were used to develop the regression model. The same con®guration presented in Fig. 2(b) was utilized in the third and last analysis in this research. This time, the opposing trac in the southbound direction was held constant at 400 pcph while both cross trac and adjacent through trac were used as predictor variables. The
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Fig. 2. Con®gurations and trac movements at signalised intersections investigated in this study (a) with exclusive leftturn lane (b) with shared lane.
®nal set of runs involved a total of 441 optimization run in Signal Expert that was used to develop the regression model. Throughout these analyses, it was assumed that the intersection is isolated from the impact of any adjacent intersection and that there is no constraint on the queue size that could develop on any of the intersection approaches. Another important assumption is that all trac is uniform and is composed of passenger car units.
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Table 1 Signal timing and optimization parameters that were input to Signal Expert Signal timing input variable
Input value
Valid range for cycle length (s) Increment step size for cycle length (s) Valid range for green interval at phase i (s) Increment step size for green time (s) Amber interval (s) All-Red interval (s) Duration of congested period (min) Number of sneakers per cycle
50±200 2 10±100 1 3 2 15 2
7. Analysis of results The results of the three analyses will be discussed next in following sections. 7.1. Analysis one: exclusive left-turn lane This analysis investigated the superiority of protected/permissive left-turn operation over the permissive-only operation when the discharge from the subject approach during the green interval is mainly aected by gap acceptance dynamics. As mentioned earlier, the regression model was formulated so that left-turning trac at the ``boundary'' is predicted using two predictor variables, opposing trac and cross trac. Many transformations on the predictor variables were tried so as to reach the best ®t, i.e. highest coecient of determination and lowest standard error. The proposed model took the following form: Y 935:42
94:4 ln X1
009X2 ;
where · Y is Left-turn ¯ow rate above which an exclusive left-turn phase is warranted (pcph), · X1 is Cross trac ¯ow rate in each direction (pcph), · X2 is Opposing trac ¯ow rate (pcph). An examination of the regression output revealed some important observations. First, the F-test in the analysis of variance showed that the regression model is statistically signi®cant and this ruled out the possibility that all the regression coecients are 0 at 95% con®dence level. Also, the high F ratio (32.1) and the very small P values (all less than 0.0004) indicate that the predictor variables explained much of the variability of the dependent variable. In other words, the left turn trac at the ``boundary'' is signi®cantly dependent on the opposing and cross trac. However, the coecient of determination (R2 of 0.55) shows that about 45% of the variation in the left-turn trac is due to random ¯uctuation or due to factors other than these two predictor variables. This sounds somewhat strange especially if it is remembered that all possible trac movements are involved in this regression model. An examination of the factors that govern the signal operation at this particular setting revealed that another important factor was overlooked in the above
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regression model and this mostly explains the observation above. Speci®cally, it seems that the proportion of left-turn trac that sneaks during the amber intervals is signi®cant and yet it is independent from opposing and cross trac. As for the predictor variables, the t-tests show that all the coecients included in this equation are statistically signi®cant at 95% con®dence level. This implies that both opposing and cross trac have signi®cant in¯uence in determining the leftturn trac at the boundary between the two signal plans. The correlation matrix between the dependent and the predictor variables showed that that there is no co-linearity at all among the predictor variables and, therefore, the information these two predictor variables provide is not redundant. Also, the correlation between the dependent variable (left-turn trac) and predictor variables (cross and opposing trac) was shown to be high. This correlation behavior well satis®es the multiple linear regression assumptions. The other important observation which might contradict the general perception is that, cross trac showed much greater correlation with left-turn trac at the ``boundary'' than opposing trac did in a scenario where left-turn operation is mainly governed by the gap acceptance process. However, this observation sounds logical if one remembers that cross trac determines the green time split between the two phases while gap availability in opposing trac only aects the left-turn rate during the northbound/southbound phase. Despite the fact that the regression model was found to be highly signi®cant and the correlation matrix well satis®ed regression assumptions, it was observed that the assumptions of normality of error term and the constant variance of error term are not entirely satis®ed. This is shown in Figs. 3(a) and 3(b), respectively. It is interesting to note here that the residual shows increasing variance with increasing values of predicted Y (predicted left-turn trac). This is mainly attrib-
Fig. 3. (a) Standard residual frequency histogram for the ®rst analysis. (b) Scatterplot of residual versus predicted leftturn trac values for the ®rst analysis.
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uted to the proportion of left-turn trac accounted for by sneakers versus the rest of left-turn trac that make the maneuver during the green light indication. Logically, higher proportion of sneakers (constant number of left-turners multiplied by number of cycles) is associated with less variability in the error term (residual) and this explains the pattern of residual variance in Fig. 3(b). 7.2. Analysis two: shared lane/through and opposing trac as predictors This analysis investigates signal operation that is dramatically dierent from that in the previous analysis. The fact that the shared lane is the only lane in the northbound direction will cause through trac to be directly aected by the availability of gaps in the opposing trac. This is mainly due to the blockage eect of left turners waiting for a gap in the opposing trac when the green light is in eect in the northbound/southbound directions. More trac variables are involved in this scenario, as compared with the previous one, that could aect signal operations, namely, left-turn trac and adjacent through trac in the northbound direction, opposing trac in the southbound direction, and cross trac in the eastbound/westbound directions. In order to limit the amount of optimization work (for time constraints), this analysis was designed so that left-turn trac at the ``boundary'' is predicted using adjacent through and opposing trac only as predictor variables while holding cross trac constant at 400 pcph in each direction. The ®nal model that was shown to yield the best ®t took the following form: Y 704:32
95:6 ln X1
0:00000009X23 ;
where · Y is Left-turn ¯ow rate above which an exclusive left-turn phase is warranted (pcph). · X1 is Through trac ¯ow rate (pcph). · X2 is Opposing trac ¯ow rate (pcph). Despite the fact that the regression model is highly signi®cant as indicated by the F-test of the analysis of variance, it is easily observed that the coecient of the second predictor in the model is extremely small. This implies that this predictor variable does not explain a lot of the variation of the dependent variable. In fact, this observation could easily be con®rmed by examining the t-statistics of the model coecients. This examination showed clearly that the coecient of the second predictor is not signi®cant at 95% con®dence level. The coecient of determination (R2 0:62) shows that about 62% of the variance in left-turn trac is explained by the predictor variables. The standard error of estimate indicated that the absolute mean dierence between the observed (recorded from experiments) and predicted left-turn volumes is in the order of 48 pcph. The correlation matrix of the variables involved in this model showed no co-linearity, i.e. little correlation between predictor variables, which is essential to satisfy the multiple linear regression assumptions. Also, it showed that the correlation between the second predictor variable and the dependent variable was low and this agrees with expectations. The examination of the error term (residuals) of this model revealed reasonable closeness to the normal distribution and a roughly constant variance along the predicted left-turn ¯ow rates. To verify the observation regarding the signi®cance of the second predictor variable, the multiple linear regression analysis was repeated for this same scenario while omitting the opposing
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trac from the regression equation. The regression results showed that dropping opposing trac from the regression model had almost no impact on the predictive ability of the model. 7.3. Analysis three: shared lane/cross and through trac as predictors In light of the results of the previous analysis, it was logical to expect that some of the variation of left-turn trac might have been explained if cross trac were included in the regression model. As such, this analysis involves the same con®guration as that in the previous analysis but with dierent trac variables. The left-turn trac was predicted using cross trac and through trac as predictor variables. This was done while holding opposing trac constant at 400 pcph. In the previous analysis, opposing trac was shown to be insigni®cant in predicting left-turn trac at the ``boundary''. The regression model of the best ®t took the following form: Y 1060
324:34 lnX1 0:5
0:29X2 ;
where · Y is Left-turn ¯ow rate above which an exclusive left-turn phase is warranted (pcph). · X1 is Cross trac ¯ow rate in each direction (pcph). · X2 is Through trac ¯ow rate (pcph). An examination of the regression results revealed signi®cant improvement in the regression model as a result of the addition of the new predictor variable. The regression model was shown to be highly signi®cant as illustrated by the F-test F 76:19 and the P-values (all less than 10 11 ) in the analysis of variance. Also, the t statistics of the model coecients indicated that they are all signi®cant at 95% con®dence level, or in other words, the two predictor variables are signi®cant in predicting the left-turn trac at the ``boundary''. The coecient of determination R2 0:82 implies that 82% of the variability of the left-turn trac is explained by the predictor variables which is a signi®cant improvement from the previous model. Another indicator of the model improvement was the signi®cant decrease (28%) in the standard error. The examination of the model for regression assumptions through diagnostic tests also showed better compliance with those assumptions as compared with the previous model. The correlation matrix showed almost no correlation between predictor variables and high correlation between the predictor variables and the dependent variable. The distribution of residuals (error term) was shown to be closer to the normal curve than that in the previous analysis. Finally, the variance of the residuals along the range of predicted left-turn trac was shown to be reasonably constant. In conclusion, the model developed in this analysis is superior to that predicted in the previous analysis. The regression results and statistics indicate that the model could be used, with reasonable accuracy, to predict the value of left-turn trac at the boundary between permissive and protected/permissive plans when the system is operating at optimum eciency. 8. Summary of ®ndings and recommendations This paper presented a new approach for the development of warrants for the addition of protected left-turn phase at signalized intersections when permissive-only left-turn operation is
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present. The research project involved exhaustive optimization analyses of signal operation with respect to a speci®c objective criterion; the average delay per vehicle entering the intersection. Therefore, optimum overall system eciency was the basis on which the warrants in this research were developed. Three analyses, preliminary in nature, were presented and discussed. The most important ®ndings of this research can be summarized as follows: (a) The models developed by this research showed that the transition from permissive to protected/permissive left-turn operation, based on system optimization, is a function of a number of trac variables and not simply the left-turn and opposing through volume. This transition (boundary) was shown to be predictable particularly for the shared-lane operation scenario. The analyses also showed that the predictive ability of the proposed models could have been improved signi®cantly (particularly the model for exclusive left-turn lane) if sneakers during the amber interval were incorporated in the regression models. This is planned in the future work of this ongoing research. (b) This research revealed that engineering judgement is not sucient to determine the dierent factors aecting signal operations. For instance, none of the current warrants in practice involves cross trac as a factor in justifying the protected/permissive left-turn operation, yet the analyses presented in this paper con®rmed the important impact of this trac variable on signal operations. (c) This research indicated that contrary to most current warrants, the volume of opposing through trac may have little impact on when a protected left-turn phase is warranted. (d) In light of the state-of-practice in using the warrants for the addition of protected left-turn phase at signalized intersections, the use of models similar to those developed in this study is both reliable and advantageous. This is mainly due to the fact that, unlike most guidelines in practice, the proposed warrants are consistent with the well-accepted analytical procedures for the design and analysis of signalized intersections. Moreover, the application of these warrants requires no ®eld measurements as only the basic trac counts for signal timing design are needed. In light of the promising results of this initiative research, the authors recommend the use of this approach to develop a uniform set of warrants that cover most con®gurations and trac conditions that closely represent real life situations.
Acknowledgements The authors would like to express their gratitude to the Academic Research Program of the Department of National Defence of Canada for funding this research.
References Agent, K.R., 1987. Guidelines for the use of protected/permissive left-turn phasing. Institute transportation Engineers 57 (7), 37±42.
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Agent, K.R., Deen, R.C., 1979. Warrants for left-turn signal phasing. In: transportation research record 737, National Research Council, Washington, DC pp. 1±9. Drew, D.R., 1968. Trac Flow Theory and Control. McGraw-Hill Book Company, New York. Federal Highway Administration, 1983. Trac Control Devices Handbook, U.S. Department of Transportation, FHWA, Washington, DC. Institute of Transportation Engineers, 1992. Trac Engineering Handbook, 4th ed. Prentice Hall, Englewood Clis, New Jersey. Kell, J.H., Fullerton, I.J., 1991. Manual of trac signal design. Institute of Transportation Engineers, 2nd ed. Prentice Hall, Englewood Clis, New Jersey. Lalani et al, N., 1986. A Summary of the Use of Warrants for the Installation of Left-Turn Phasing at Signalized Intersections. ITE Journal 56 (4), 57±59. Lin, H., Machemehl, R.B., 1983. Developmental study of implementation guidelines for left-turn treatments. In: Transportation Research Record 905, National Research Council, Washington, DC pp. 96±105. McShane, W.R., Roess, R.P., 1990. Trac Engineering. Prentice Hall, Englewood Clis, New Jersey. Stewart, J.A., 1992. An automated approach to designing the timing parameters of isolated trac signals. Ph.D. Thesis, Queen's University, Kingston, Ontario, Canada. Stewart, J.A., Aerde, M., 1992. An expert assistant for selecting green splits, cycle times, and phasing schemes based on the canadian capacity guide procedures. Canadian Journal of Civil Engineering 19 (2), 245±251. Stewart, J.A., Aerde, M., 1997. Requirements for automating signal timing plan design. Canadian Journal of Civil Engineering 24 (1), 53±63. Teply, S. et al. 1995. Canadian capacity guide for signalized intersections. In: Institute of Transportation Engineers, 2nd ed. District 7, Canada. Transportation Research Board, 1997. Special Report 209, Highway Capacity Manual, National research Council, Washington, DC (update). Upchurch, J.E., 1986. Guidelines for selecting type of left-turn phasing. In: Transportation Research record 1069, National Research Council, Washington, DC pp. 30±38.