SIMULATION MODELLING OF PERMITTED LEFT-TURN SATURATION FLOW RATE BASED ON OPPOSING THROUGH-FLOW DEGREE OF SATURATION

This paper presents the modelling of the saturation flow rate of the permitted left-turn in an exclusive lane. In the proposed model, the total permitted left-turn saturation flow rate is determined as a sum of saturation flow rates during the effective green time and the intergreen period. Primarily, the permitted left-turn saturation flow rate during the effective green time is modelled based on the opposing through-flow degree of saturation and the number of opposing through-flow lanes. The relation be-tween the permitted left-turn saturation flow during the effective green time and these variables was examined using data from the simulation experiments in VISSIM. To our knowledge, this is the first study of the permitted left-turn saturation flow modelling based on the opposing through-flow degree of saturation instead of the opposing through-flow rate and signal-timing parameters. The proposed model was validated based on data collected at seven intersections with a permitted left-turn served in an exclusive lane. The permitted left-turn saturation flow rate could be accurately determined based on the opposing through-flow degree of saturation and the number of opposing lanes according to the RMSE of 58.4 pcu/h.


INTRODUCTION
Saturation flow rate is a crucial input parameter for signal-timing optimisation, due to its effect on capacity and control delay estimation.The saturation flow rate is an hourly rate at which previ-ously queued vehicles can traverse an intersection approach, assuming that green signal is available at all times and no lost times are experienced [1].
The saturation flow rate, besides various geometric and traffic conditions, depends on a movement type.Hence, the left-turn lane saturation flow is different from the through lane due to the lower turning speed and movement radius [2].Also, the left-turn saturation flow rate is affected by the left-turn mode whereby left turns can be served as permitted, protected or permitted-protected.The permitted mode requires a left-turn vehicle to yield to both opposing vehicles and pedestrians, and to wait for an available acceptable gap in the opposing flow.So, permitted left turns, either served in shared lanes or exclusive lanes, seriously affect intersection operations [3].Hence, estimating the left-turn saturation flow rate when it operates in the permitted mode is a complex problem.
Therefore, the discharge characteristics of the permitted left turns require considering several components of the permitted left-turn saturation flow rate (PLTSF).Wu in [4] suggests that there are three components of left-turn departures.The first component, named jumpers, consists of vehicles that turn in front of the first opposing vehicle.The second component refers to vehicles that turn during the green time, while the third component, named sneakers, consists of vehicles that pass an intersection during the intergreen period.
The PLTSF model, presented in this paper, considers the second and third PLTSF components, while the second component is examined during the effective green time (PLTSFe).It is important to develop the PLTSFe model that includes all variables affecting left turns.However, using signal-timing parameters in the PLTSFe estimation is complicated, considering that they are the output from signal timing optimisation.This paper proposes PLTSFe modelling based on the opposing through-flow degree of saturation since it unites the effects of the opposing through-flow rate and signal-timing parameters.Hence, those variables are indirectly incorporated.Including the opposing flow degree of saturation, instead of separate variables, simplifies the PLTSFe estimation.The number of opposing flow lanes should not be neglected, considering its established effect on the PLTSFe.So, this paper hypothesises that the PLTSFe depends on the opposing through-flow degree of saturation and the number of opposing through-flow lanes.Also, this paper proposes that the number of sneakers per cycle, influencing the third component (PLTSFs), could be estimated based on the waiting space length if field data are not available.

LITERATURE REVIEW
In this section, PLTSF models proposed by the Highway Capacity Manual (HCM 2016), the Australian Road Research Broad (ARRB), and the Canadian Capacity Guide (CCG 3rd edition) are presented in more detail.These relevant models will be used for comparison with the model proposed in this paper.The majority of authors have developed the PLTSF model considering second and third components, so the PLTSF is estimated as a sum of the PLTSF during the green time and the PLTSF during the intergreen period (PLTSFs) [1,2,[4][5][6].Existing models do not consider the first component because the number of jumpers is usually small [4].On the other hand, some studies have proposed models for the PLTSF estimation regardless of the mentioned PLTSF components [7][8][9].
The second and most important component, the PLTSF during the green time, is usually examined in two ways: during the effective green time [6] or during the unsaturated green time [1,2,4,5].Thereby, the effective green time refers to the effective green time for the opposing traffic flow [6], while the unsaturated green time represents the portion of green time after the clearance of the opposing queue [4].The second approach requires defining the method for the unsaturated green time estimation, which additionally complicates the PLTSF model development and application.
The second PLTSF component depends on the opposing through-flow rate, the number of opposing lanes, and signal-timing parameters as variables that affect left turns during the (effective or unsaturated) green time.Still, some models included only the opposing through-flow rate [1], while others combined the opposing through-flow rate and the number of opposing lanes [2,4].On the other hand, Akçelik in [5] incorporated the opposing through-flow rate and signal-timing parameters in the model.Only the Canadian Capacity Guide model includes all listed variables [6].
The third PLTSF component is a result of sneakers during the intergreen period (PLTSFs).This PLTSF component can be of great importance, especially if the opposing flow is saturated during the entire green time since there are no available gaps to turn left during the green time.The PLTSFs depends on the number of sneakers per cycle and the number of cycles per hour.The number of sneakers per cycle is defined in several ways.In papers [1,2,5], the number of sneakers is given as a fixed, experimentally determined value.In [4], the number of sneakers is a function of the opposing through-flow rate, while in [6], it is a function of the intersection waiting space length.It is expected that the number of sneakers varies among intersections depending on the intersection geometry, so the approach with fixed predefined values could not be reliable.

METHODOLOGY
Based on the literature review, it is concluded that the PLTSF model should consider second and third PLTSF component.In this study, the PLTSF is determined as a sum of the PLTSF during the effective green time (PLTSFe) and the PLTSF during the intergreen period (PLTSFs), as given by Equation 1 where S L s is permitted left-turn saturation flow rate during the intergreen period [pcu/h] and other variables are previously defined in pcu/h.The PLTSFs depends on the number of cycles per hour, i.e. on the cycle length and the number of sneakers per cycle.On an hourly basis, the maximum number of sneakers represents the permitted left-turn capacity during the intergreen periods.Considering that the capacity is equal to the saturation flow rate multiplied by the effective green time ratio [1], the PLTSFs is calculated using where: All of the presented PLTSF models consider the second and third PLTSF components.However, in the CCG 3 model, the second component is examined during the effective green time, while in the HCM 2016 and ARRB models, this component is examined during the unsaturated green time.The methodology applied in the modelling of the second PLTSF component differs among the studies.Namely, the CCG 3 model is developed based on field data.This model includes all variables affecting left turns as the opposing through-flow rate, the number of opposing lanes, and signal-timing parameters.The HCM 2016 and ARRB models are based on the gap-acceptance theory, and these models include only the opposing through-flow rate as a model variable.However, the ARRB model includes signal timing parameters in the total PLTSF estimation.
The suggested number of sneakers varies between the presented models.The ARRB and HCM 2016 models define one fixed value, while the CCG 3 model defines the range of values depending on the waiting space length.It is important to note that the ARRB model recommends using the defined number of sneakers if the field data are not available [5]. .

Experimental system design
In this paper, the PLTSFe was analysed and modelled using the microsimulation model PTV VISSIM.For the research purpose, VISSIM was set to represent local conditions according to the research presented in [13].
In VISSIM, the through-lane saturation flow rate depends on the parameters of the Wiedemann 74 model, which is given by Equations 5 and 6.Values of the parameters were adopted to represent local conditions based on the through-lane saturation flow of 1,850 pcu/h [13].So, bx_mult and bx_add were 3.48 m and 2.48 m, respectively.Adopted values of parameters led to the left-turn lane saturation flow rate of 1,650 pcu/h.This saturation flow rate is valid in local conditions [14], but it is also close to values cited in [5,6,15]. where: which is normally distributed around 0.5 with a standard deviation of 0.15.Since the PLTSFe depends on the availability of critical headways in the opposing flow, it was necessary to modify this parameter in VISSIM.Different values of the critical headway were defined depending on the number of opposing through lanes.When the left-turn was opposed by one-lane flow, the 5-second critical headway was used, based on the research conducted in local conditions [13].The accepted value is close to values cited in the literature as 5 ѕ [5,16], 4.95 [4], 5.1 s [17], 5.7 [8], 4.5 ѕ [1] and 6 s [18].When the left-turn was opposed by the two-lane flow, the critical headway of 6 seconds was accepted according to [16], due to the lack of appropriate research in local conditions.
To represent urban driver behaviour, the desired speed distribution was defined in the range from 35 to 58 km/h.The desired speed for the left-turn was It is important to determine the number of sneakers precisely since the higher number of sneakers demonstrates that more left-turn vehicles can pass through the intersection [2].Considering assumption that the number of sneakers varies among intersections depending on the intersection geometry, this paper recommends measuring the number of sneakers in the field or estimating it by dividing the waiting space length by the average passenger car length, as given by Equation 3. Thereby, the intersection waiting space represents a space between the stop line and the point where left-turn vehicles stop and wait for an acceptable gap.The main objective of this paper is to model the PLTSFe based on the number of opposing lanes and the opposing through-flow degree of saturation, which is a function of the opposing throughflow rate and signal-timing parameters, as given by Equation 4x where: x O -opposing through-flow degree of saturation, S O -opposing through-flow saturation flow rate [pcu/h] and other variables are previously defined.
Measuring the PLTSFe in the field is difficult due to a limited sample of research sites and numerous combinations of variables affecting left turns.However, the permitted left-turn saturation flow has already been examined using simulation [2,4,7,10,11].Simulation is a widely used technique that allows imitating the operations of various types of real-world facilities or processes by using computers [12].So, simulation is an alternate, easier way to simulate various traffic and geometric conditions and measure PLTSFe.
The following two sections describe the experimental design and procedure used in this research.The next section describes the PLTSFe model development based on data collected by simulation experiments.The last section presents the methodology for the PLTSF model testing and comparison.green time.So, the sneakers were excluded given the purpose of modelling only the PLTSFe using the simulation.
Traffic composition was set to passenger cars to exclude the effect of heavy vehicles on the PLTSFe.The left-turn flow rate was set to provide constant saturated conditions during the simulation.

Experimental procedure
The investigation of the PLTSFe dependence on the opposing through-flow degree of saturation requires various values of the opposing flow degree of saturation during simulation.Different values of the opposing degree of saturation, representing different traffic conditions in the opposing flow, were achieved by changing the following variables: -The opposing through-flow rate: from 0 to 1,900 pcu/h/lane, with increment step 100 pcu/h.-Signal-timing parameters: defined to provide the effective green time ratio in the range 0.1-0.9 with increment step 0.1 (the fixed cycle length of 100 s).
Combining the preceding values of variables resulted in 531 experimental scenarios in total.Namely, there were 180 scenarios for one opposing lane (20 opposing through-flow rates and 9 effective green time ratios) and 351 scenarios for two opposing lanes (39 opposing through-flow rates and 9 effective green time ratios).For each of 531 scenarios, ten simulation runs were executed using different random number seeds.A warm-up period of 600 s was followed by a one-hour simulation experiment (3,600 s).Collected data were averaged from ten simulation runs for each scenario.

PLTSFe model development
The average opposing through-flow rate was used for calculating the opposing through-flow degree of saturation, applying Equation 4. Due to the constant left-turn demand, the average left-turn flow represents a left-turn capacity during the effective green time.According to [1], the capacity is used for the PLTSFe calculation for each of 531 scenarios, applying Equation 7S C L L e e m = (7) where C L e is permitted left-turn capacity during the effective green time [pcu/h] and other variables are previously defined. in the range from 20 to 25 km/h, with a maximum deceleration of 2 m/ѕ 2 .Applied left-turn desired speed is close to value cited in the literature of 22 feet/s ≈ 24 km/h [4].
The PLTSFe examination requires the experimental system to meet the following prerequisites: -Permitted left-turn mode, -Exclusive left-turn lane, -One or two opposing through lanes, -A 3.

Data collection for PLTSF model testing and comparison
The final step of the defined methodology refers to the collecting of data for model testing and comparison with prominent widely used models, such as the ARRB model, the CCG 3 model and the HCM 2016 model.These PLTSF models, as the proposed model, consider the second and third PLTSF components.
The testing and comparison were conducted regarding real data collected in local conditions.Seven four-leg intersections in Belgrade were selected based on requirements defined during the experimental system design (permitted left-turn mode, exclusive left-turn lane, constant left-turn demand, one or two opposing through lanes, mostly a passenger car flow, closely a 3.5-meter lane width, approximately a flat approach grades, no parking, no public transport, no pedestrians or a negligible number of pedestrians and sunny weather).Video camera was installed at intersections to record left-turn flow, opposing through-flow and signal-timing parameters for one hour.The waiting space length was measured in the field for each intersection.Collected data, given in Table 2 for each intersection, represent the input data for model testing and comparison.
The opposing through-flow (Q O ), the total leftturn capacity (C L ), the maximum number of sneakers (n L s ) and signal-timing parameters (c, g e ) were obtained manually from the recordings and given in Table 2 for each intersection.The opposing throughflow represents the total flow rate on the approach, not per lane.The total permitted left-turn capacity is determined as a sum of left-turn vehicles during Hence, the database consists of the PLTSFe and the opposing through-flow degree of saturation for each simulation experiment.The PLTSFe and the corresponding values of the opposing flow degree of saturation are plotted in Figure 2. The figure shows that the PLTSFe decreases with the increase of the degree of saturation.However, data points are concentrated along two curves depending on the number of opposing lanes.Results of the two-sample t-test (p=0.0005<0.05)showed that the PLTSFe statistically differs when left turns are opposed by the one-lane flow and by the two-lane flow.
Based on the previous result, the relationship between the PLTSFe and the opposing flow degree of saturation is modelled separately depending on the number of opposing lanes.The model for the PLTS-Fe estimation is given by Equation 8, with R 2 of 0.99 and 0.97 when the left-turn is opposed by one-lane and two-lane flow, respectively.
where all variables are as previously defined.
The values of R 2 indicate that 99% and 97% of the variance of the PLTSFe is explained with the opposing flow degree of saturation, when the leftturn is opposed by one-lane and two-lane flow, respectively.
The results confirm the hypothesis that the PLTS-Fe depends on the opposing through-flow degree of saturation and the number of opposing through-flow lanes.Hence, the PLTSFe can be estimated based on the opposing flow degree of saturation instead of the opposing flow rate and signal-timing parameters.by the effective green time ratio, so the observed PLTSF was calculated by applying Equation 2, but using the total permitted left-turn capacity in Table 2 instead of the capacity during the intergreen period.It is important to emphasise that the permitted left-turn capacity at intersection 6 consists only of the capacity during the intergreen period due to the constant saturation in the opposing flow during the green time.Other data in Table 2 were used for the PLTSF estimation using the proposed model and aforementioned relevant models.It is important to note that the number of sneakers was adopted as referred to in each model.In the CCG 3 model, the number of sneakers was estimated using the waiting space length.Given that the waiting space is longer than 9 m at each intersection, the accepted number the effective green time and sneakers during the intergreen period.The maximum number of sneakers represents the maximum number of left-turn vehicles that pass the intersection during the intergreen period.
The total opposing through-flow and signal-timing parameters were used to calculate the effective green time ratio (λ), the opposing through-flow degree of saturation (x O ) and the unsaturated green time (g u ), which are given in Table 2.The opposing through-flow degree of saturation was calculated using Equation 4, while the unsaturated green time was estimated according to [5].
Data in Table 2 were further used for the PLTSF estimation (Table 3).As explained earlier, the capacity is equal to the saturation flow rate multiplied   flow is saturated during the entire green time, and the PLTSF is entirely the result of sneakers during the intergreen periods.The CCG 3 model considers all variables affecting left turns but underestimates the effect of the number of opposing lanes.On the other hand, the HCM 2016 model overestimates observed PLTSF at each intersection, regardless of the number of opposing lanes.This model overestimates the observed PLTSF with the observed-to-modelled ratio from 0.53 to 0.92.Among the considered models, the HCM 2016 model is the only one that does not include signal-timing parameters in the PLTSF model, which leads to the PLTSF overestimation.
Also, the RMSE is calculated considering intersections with one or two opposing lanes and considering all intersections (Table 4).The RMSE confirms that all analysed models, except the HCM model, determine the PLTSF close to observed values for one-opposing-lane intersections.At these intersections, the ARRB and the proposed model determine the PLTSF with the lowest RMSE of 51.5 pcu/h and 68.1 pcu/h, respectively.Although all models overestimate observed PLTSF values when the left-turn is opposed by the two-lane flow, the proposed model overestimates with the lowest RMSE of 17.9 pcu/h.In summary, the proposed model estimates the PLTSF in local conditions more precisely than other models even though the ARRB and CCG 3 models estimate PLTSF more precisely for some one-opposing-lane intersections.Considering all intersections, the proposed model, on average, determines the PLTSF with the lowest RMSE of 58.4 pcu/h (Table 4).According to the results, the proposed model is more precise for two-opposing-lanes intersections than for one-opposing-lane intersections.However, a reliable conclusion needs further research due to the inclusion of only two intersections with two opposing lanes and similar values of the degree of saturation at these intersections.
The finding that the proposed model determines the PLTSF with low deviations is supported by the result of the paired two-sample t-test.The result in-of sneakers was 3 pcu/cycle.On the other hand, in the HCM 2016 model, the number of sneakers is the fixed value of 2 pcu/cycle.The number of sneakers in the ARRB model was measured in the field and given in Table 2, while in the proposed model, the number of sneakers was determined using Equation 3, whereby the waiting space length was divided by the average passenger car length of 5 m.

RESULTS AND DISCUSSION
The model testing and comparison is based on the ratio of the observed PLTSF to the modelled PLTSF and the root mean square error (RMSE).Furthermore, the proposed model is validated based on the results of the paired two-sample t-test.
The observed-to-modelled ratio is determined and given in Table 3 for each intersection and each model.
Results show that the proposed, CCG 3 and ARRB models slightly underestimate or overestimate PLTSF values when left-turn is opposed by the one-lane flow.The proposed model determines that the PLTSF is the most precise at intersections 1, 3 and 5, while the ARRB or the CCG 3 model is more precise at intersections 2 and 4. Hence, the proposed model determines the PLTSF more accurately at lower and higher values of the opposing flow degree of saturation than at median values.Despite different approaches in the PLTSFe estimation (effective or unsaturated green time), those three models estimate the PLTSF close to observed values at one-opposing-lane intersections.
However, all models overestimate observed PLTSF values when the left-turn is opposed by the two-lane flow (intersections 5 and 6).The proposed model overestimates observed values at these intersections the least, with the observed-to-modeled ratio of 0.94 and 0.99.The ARRB model overestimates the PLTSF at two-opposing-lanes intersections since it does not consider the number of opposing lanes and overestimates the unsaturated green time.The unsaturated green time overestimation is evident at intersection 6, where the opposing The proposed model was evaluated using data collected at seven intersections, and the presented results lead to the following conclusions.
The paper hypothesis that PLTSFe depends on the opposing through-flow degree of saturation and the number of opposing lanes is confirmed.Firstly, the results of the two-sample t-test confirmed that the PLTSFe statistically differs when left turns are opposed by the one-and the two-lane flow.This result supports the earlier finding that PLTSF depends on the number of opposing lanes [2,4,6,15].Secondly, the dependency of the PLTSFe on the opposing flow degree of saturation is confirmed with R 2 of 0.99 and 0.97 for the opposing one-lane and twolane flow, respectively.
The RMSE of 58.4 pcu/h indicates that the PLTSF could be accurately determined based on the opposing through-flow degree of saturation and the number of opposing lanes.According to model comparison, the proposed model is the most precise for local conditions.The proposed model is applicable in signal timing optimisation regardless of the number of opposing lanes, unlike other analysed models.Also, the proposed model allows PLTSF estimation in the signal timing optimisation based on the desired opposing flow degree of saturation instead of the signal-timing parameters, which are the output from this procedure.
dicates that the PLTSF determined by the proposed model is not significantly different from the observed PLTSF due to the p-value of 0.6426>0.05.
There is a brief review of the suggested number of sneakers (Table 5).The assumptions that the number of sneakers varies among intersections and the predefined values could not be suitable for every intersection were confirmed.Values suggested by the HCM 2016 and ARRB models statistically differ from the observed values with p of 0.0010 and 0.0002, respectively.Hence, the recommendation in the ARRB model to use the observed values instead of suggested is justified.It was also confirmed by previous results of model comparison.The CCG 3 model defines the number of sneakers depending on the intersection waiting space, and the number of sneakers statistically differs at α=0.05 with p of 0.0465.Still, this model estimates PLTSF close to observed values based on previous results.However, this method should be more sensitive to values of waiting space length.The number of sneakers determined by the proposed method does not statistically differ from the values measured in the field, based on the paired two-sample t-test (p=0.1458>0.05).

CONCLUSIONS
Accurate estimation of the saturation flow rate is essential, considering that it is an input variable in the signal timing optimisation and the capacity and delay estimation.The modelling of the PLTSF is an intricate procedure, so researchers applied different approaches (simulation, field study, gap-acceptance theory) to model the PLTSF considering different affecting variables.In this study, the PLTSF is determined as a sum of the PLTSF during the effective green time (PLTSFe) and the PLTSF during the intergreen period (PLTSFs).To our knowledge, this is the first time that the PLTSFe is modelled based
ws -waiting space length [m], l pc -average passenger car length [m].

Figure 1 -
Figure 1 -Intersection layout and signal phasing sequence used in experiments

Figure 2 -
Figure 2 -The relationship between the PLTSFe, the degree of saturation and the number of opposing lanes

Table 1 -
Review of the relevant PLTSF models

Table 2 -
Collected data for model testing and comparison

Table 3 -
Estimated permitted left-turn saturation flow rates and comparison

Table 4 -
Model comparison