CALIBRATION OF THE PLATOON DISPERSION MODEL BY CONSIDERING THE IMPACT OF THE PERCENTAGE OF BUSES AT SIGNALIZED INTERSECTIONS

Transcription

1 CALIBRATION OF THE PLATOON DISPERSION MODEL BY CONSIDERING THE IMPACT OF THE PERCENTAGE OF BUSES AT SIGNALIZED INTERSECTIONS By Youan Wang, Graduate Research Assistant MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, School of Traffic and Transportation, Beijing Jiaotong University Beijing, P.R. China Tel: -, Fax: -, Xumei Chen, Ph.D., Professor (Corresponding Author) MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, School of Traffic and Transportation, Beijing Jiaotong University Beijing, P.R. China Tel: -, Fax: -, Lei Yu, Ph.D., P.E. Professor of Texas Southern University Yangtze River Scholar of Beijing Jiaotong University College of Science, Engineering and Technology, Texas Southern University Cleburne Avenue, Houston, Texas Tel: --, Fax: --, Yi Qi, Ph.D., Associate Professor Department of Transportation Studies, Texas Southern University Cleburne Avenue, Houston, Texas Tel: --, Fax: --, Submitted for Presentation at the th Transportation Research Board Annual Meeting and Publication in the Transportation Research Record Washington D.C., January, Date of Submission: July, Word Count: (Text) + * (Tables) + * (Figures) =

2 Wang, Chen, Yu, and Qi. ABSTRACT Despite the wide use of Robertson s platoon dispersion model, only a limited number of studies have calibrated this model under different traffic conditions at signalized intersections. The research in this paper calibrated the platoon dispersion model on the basis of field data collected from a signalized intersection in Beijing, China, in which the impact of the percentage of buses was considered in the calibration. First, a video of the platoon dispersion was recorded at this intersection. The link travel time of the vehicles in the platoon was extracted from the video. Then, the parameters of the platoon dispersion model were estimated on the basis of the average and standard deviations of the fleet link travel times. It was found that the derived parameters varied, with the observed percentages of buses ranging from to % or to %, which shows the impact of the percentage of buses on the platoon dispersion under specific conditions. Thus, regression models were developed to reflect such impact on the parameters of the platoon dispersion model. To evaluate the effectiveness of the calibration method for the platoon dispersion model, the downstream flow profiles derived from the calibrated model were compared to the field observed downstream flow profiles within the platoon dispersion process at the intersection studied. Finally, the influence of the time step on the calibrated platoon dispersion model was also analyzed. The results showed that the calibrated model in this study had a high accuracy. The calibrated platoon dispersion model can be applied to represent the process of platoon dispersion at signalized intersections where the impact is affected by the percentage of buses. It can contribute to signal timing optimization of the intersections. Keywords: Platoon dispersion model; Parameter calibration; Percentage of buses; Fitting model; Validation analysis

3 Wang, Chen, Yu, and Qi. INTRODUCTION To develop an effective traffic-signal control strategy, the ability to accurately measure or predict vehicle arrival patterns while approaching traffic signals is critical. As a key element in traffic optimization and simulation models, the widely used Robertson s platoon dispersion model contributes to better predictions of arrival pattern platoons at intersections. Robertson s platoon dispersion models attempt to simulate the dispersion of a traffic stream as a platoon travels along a roadway by estimating vehicle arrivals at downstream locations (). However, a successful application of the model requires an appropriate calibration of its parameters, especially considering different traffic conditions at signalized intersections. With the development of public transportation in the big cities of China, relatively high volumes of buses use intersections, and this is especially the case in Beijing. Compared with passenger cars, buses differ in terms of operation mode and vehicle characteristics. Because of such differences, platoon dispersion is influenced by the percentage of buses running through intersections. The process of the platoon dispersion varies with changes in the percentage of buses in transit. Therefore, the platoon dispersion model should be calibrated by considering the impact of the percentage of buses traveling through these signalized intersections. The calibrated model could be used to develop effective traffic-signal timing plans because it can predict the arrival flow at the downstream. In this study, on the basis of data collected from a signalized intersection in Beijing, China, the platoon dispersion model was calibrated by considering the percentage of buses traveling the intersection. Analysis focuses on the correlation between the parameters of the platoon dispersion model and the percentage of buses. To assess the effectiveness of the calibrated model, the downstream flow profiles derived from the calibrated model were compared to the field-observed downstream flow profiles within the platoon dispersion process at the intersection studied. In addition, the influence of the step on the calibrated platoon dispersion model is also analyzed. OVERVIEW OF EXISTING STUDIES As the most widely used platoon dispersion model, the Robertson s model has been used in various traffic optimization and simulation models, including TRANSYT (), TRANSYT-F (), and INTEGRATION (). The expression of this model () is shown as follows: ' ' qt= F qt T+ ( F) qt t () F= () + a β Ta ' where q t is the estimated arrival flow (veh/h) at the downstream during a time interval t; q t is the departure flow at the upstream during a time interval t; T is the time-lag factor, which is found as β Ta ; t is the modeling time-step duration; α is the platoon dispersion factor; β is the travel-time factor; F is the smoothing factor, a coefficient that indicates the extent of platoon dispersion; and T a is the average link travel time. Many studies have focused on the calibration of the parameters α and β when the Robertson s platoon dispersion model is applied. Different values of α and β dictate the flow of the arrival patterns of vehicles approaching an intersection, which thus significantly affect the optimal timings of the signal. Therefore, the appropriate calibration of α and β is truly important to the effective signal control and intersection operations. Existing studies have already shown

4 Wang, Chen, Yu, and Qi. that the values of α and β provided by the TRANSYT-F Users Guide () where β is a fixed value of., and α is one of three values,.,., or. are not appropriate for many application scenarios, especially in Beijing, where the percentage of buses is relatively high. Numerous studies have been conducted since, in which α and β were calibrated on the basis of field studies, and the objective was to minimize the sum of the squared errors between the field-observed and estimated downstream flow profiles when considering different scenarios (). Moreover, some researchers have studied the dynamic relationship between parameters, which can reflect dynamic traffic conditions. Yu and Van Aerde () proposed an alternate technique for calibrating these parameters within the platoon dispersion model. This technique was formed on the basis of the statistical analysis of link travel times rather than the more traditional goodness-of-fit tests between observed and projected vehicle-arrival distributions. They also established three equations to calibrate α, β, and the smoothing factor F on the basis of the average link travel time and its standard deviation. Yu () found that the technique was especially suited for applications in advanced traffic-management networks in which the required link travel time data could be obtained on a real-time basis. Yu also found that the calibration results showed that platoon dispersion parameters varied for different standard deviations of link travel times, even on the same street, and therefore, the platoon dispersion parameters must be calibrated on a site-specific basis. Instead of using travel times at -sec interval during the calibration process, Rakha and Farzaneh () studied the impact of the time step on the calibration and presented numerical examples that provided different time steps. They also studied the changes of α and β under conditions in which that six loop detectors were placed on the roadway. The first was located immediately downstream of the signalized intersection within m of the intersection, and the other five were located downstream of the signalized intersection at -m intervals. They found that the values of parameters changed with different data detected from different loop detectors (). Besides analyzing the impact of time step and road length on the calibration of the parameters, Bie et al. () addressed the impact of the number of lanes on the platoon dispersion of traffic flows in low-friction conditions, which are defined as no parking on the road, divided lanes, turning provisions, and suburban high-type arterial. The platoon dispersion factor was recalibrated by using the data on road segments with different numbers of lanes. Although some existing studies have focused on the platoon dispersion model by considering different traffic conditions such as the traffic friction, road length, and number of lanes, when calibrating the platoon dispersion model, few researchers have considered the impact of the percentage of buses. Meanwhile, because of the different operating modes of buses and the high volume of buses at signalized intersections in Beijing, it is important to consider the impact of buses in the calibration. For this paper, a comprehensive field study was conducted, and field data were used to calibrate the parameters α, β, and F. The relationship between the platoon dispersion parameters and the percentage of buses under low-friction conditions is identified. DATA COLLECTION AND MODEL CALIBRATION Field Data Collection To calibrate the Robertson s platoon dispersion model, a field site was selected, as shown in Figure. This field site was the signalized intersection of Zhongguancun Street and Zhongguancun South Road, in Beijing. Zhongguancun Street is an arterial road. Zhongguancun South Road is a collector road, which serves to move traffic from local streets to arterial roads. There is no on-street parking, bus stop, and bus lane within this intersection. The intersection is

5 Wang, Chen, Yu, and Qi. not left-turn prohibited. Vehicle queuing at a signalized intersection takes place when vehicles cannot depart from the intersection during the red phase for the approach. Consequently, a queue of vehicles is formed. The queuing is widespread on Zhongguancun Street during peak hours. This signalized intersection was chosen primarily for its high traffic and high bus volume, with bus lines on this road. Most of these bus lines serve the trips from the northern area to the center of Beijing. There are about buses per hour in one direction. Also, three successive pedestrian overpasses run upstream and downstream of Zhongguancun Street, where camcorders recorded the platoon dispersion process. The camcorders were placed in three locations, Sites A-C (Figure ). Site A captured the last tail of the platoon. Site B captured the head of the platoon. Site C captured the vehicles arriving at the downstream. The three camcorders were programmed to record at the same time. Intersection of Zhongguancun Street and Zhongguancun South Road Haidian East st Street Zhongguancun Street Zhongguancun South Road FIGURE Intersection of Zhongguancun Street and Zhongguancun South Road.

6 Wang, Chen, Yu, and Qi. Greenbelt Zhongguancun Street Haidian dongyi Street Zhongguancun Street Pedestrian overpass Camcorder Camera range Stop line m Stop line A B C Zhongguan cun South Road m FIGURE Layout of the field study site on Zhongguancun Street in Beijing. Data Processing When the platoon dispersion at the study site was recorded, all the cameras were set to the same start time. According to the license plate and the shape of each vehicle from the same platoon, each arrival time and departure time could be extracted from the videos recorded at Sites A and B. The vehicle arrival times at the downstream pedestrian overpasses could be extracted from the video recorded at Site C. All these data were extracted by manual observation. On the basis of each vehicle s departure time at the upstream and the arrival time at the downstream, the link travel time of each vehicle in the same platoon could be derived. The standard deviations of the link travel times could be calculated accordingly. When the standard deviations of link travel times were less, it meant that the vehicles in the platoon were much closer to each other and could proceed through the intersections smoothly. Thus, intersection signal timings could be designed for better passing efficiency. Model Calibration As provided earlier, Yu and Van Aerde () proposed a technique for calibrating platoon-dispersion parameters on the basis of link travel time. Three equations were established to calibrate α, β, and the smoothing factor F on the basis of average link travel time and its standard deviation. This technique can take into account the impact of different traffic conditions, such as the percentage of buses at a signalized intersection, on the platoon dispersion, which, thus, is used in this paper. The equations () applied for calibrating α, β, and the smoothing factor F can be expressed as follows: + σ () a =, + + σ T a

7 Wang, Chen, Yu, and Qi. β = + α, () + σ () F=, σ where α is the platoon dispersion factor; σ is the standard deviation of link travel times of vehicles in each platoon; T a is the average link travel time; β is the travel-time factor; and F is the smoothing factor. The vehicle departure time at the upstream and the arrival time at the downstream of a platoon could be obtained from the field data from the intersection of Zhongguancun Street and Zhongguancun South Road. Travel times of platoons, including all types of vehicles departing from the surveyed intersection, were obtained from the videos recorded on the pedestrian overpasses. Among them, the data of platoons were used for model calibration, and the other platoons were used for model validation. The link travel time of the majority of vehicles was between and sec, and the number of these vehicles accounted for % of the total. A program written in MATLAB calculated α, β, and F, according Equations () to (). Furthermore, the percentage of buses in each platoon was extracted from the field data, which ranged from. to.%. To analyze the impact of the percentage of buses on the platoon dispersion parameters, a regression analysis was conducted. The relationship between the percentage of buses and the platoon dispersion factor α is shown in Figure. Percentage of buses vs. α. Value of α y =.x -.x +.x +. R² =..%.%.%.%.%.% Percentage of buses FIGURE Relationship between the percentage of buses and the platoon dispersion factor α. The regression relationship between the percentage of buses and platoon dispersion factor α was derived, as shown in Equation (): α =. x. x +. x +., () where x is the percentage of buses. In Figure, it can be observed that α increases with an increase in the percentage of buses ranging from to % and to %. Contrary to the TRANSYT-F Users Guide, α is not the fixed value. for low-friction conditions. This finding implies that, with an increase in the

8 Wang, Chen, Yu, and Qi. percentage of buses ranging from to % or to % at the signalized intersection, the speed of the platoon dispersion decreases. The relationship between the percentage of buses and the travel time factor β is shown in Figure. Equation () shows the corresponding mathematical relationship. β =. x +. x. x +. () Value of β Percentage of buses vs. β.. y = -.x +.x -.x +. R² =...%.%.%.%.%.% Percentage of buses FIGURE Relationship between the percentage of buses and the travel time factor β. As shown in Figure, the travel time factor β decreases with an increase in the percentage of buses from to % or to %, as opposed to the fixed value. suggested by the TRANSYT-F Users Guide. This finding indicates that, with an increase in the percentage of buses from to % or to % at the signalized intersection, vehicles move more slowly and require more time to arrive at the downstream. It should be noted that not only do the buses move slowly, they also affect the speed of nearby cars. Thus, the travel time for all vehicles to arrive at the downstream increases, especially when the bus is the first vehicle in the platoon. The smoothing factor F is also influenced by the percentage of buses, as shown in Figure. The curve shows that the smoothing factor F decreases with an increase in the percentage of buses from to % or to %. It means when the percentage of buses from to % or to % is higher, the speed of platoon dispersion decreases. Therefore, the buses in the platoon affect platoon dispersion under specific conditions. The mathematical relationship is shown by Equation (). F. x. x. x. = + + ()

9 Wang, Chen, Yu, and Qi.. Percentage of buses vs. F.. Value of F.... y = -.x +.x -.x +. R² =...%.%.%.%.%.% Percentage of buses FIGURE Relationship between the percentage of buses and the smoothing factor F. Figures - show that the percentage of buses and the three parameters of the platoon dispersion model indicate a good correlation with R-squares of.,., and.. Therefore, the impact of the percentage of buses on platoon dispersion can be well determined through the established relationships of Equations () (). Besides, it also can be observed that the values of α, β, and F do not appear to be affected by the percentage of buses between %. This finding suggests that the percentage of buses between % has little impact on platoon dispersion. Therefore, the percentage of buses could be divided into three groups that range from to %, to %, and to %. This finding can be used in the model for signal timing optimization, and the impact of buses should be considered when the percentage of buses falls within the range of % or %. MODEL VALIDATION To demonstrate the validity of the calibrated model, a comparison between the platoon dispersion derived from the calibrated model and the field observations was conducted. On the basis of the regression Equations ()-(), three different percentages of buses for three platoons are used as examples. (It should be noted that neither these three percentages of buses nor the data of the corresponding platoons were used in the earlier regression analysis.) The three sets of platoon dispersion parameters were then calibrated and the results are given in Table. TABLE Calibration Results of Parameters of the Platoon Dispersion Model Scenarios Buses (%) α β F Furthermore, the downstream flow profiles can be derived from the calibrated parameters. The observed downstream flow profiles were obtained from the field data. When the time step of the platoon dispersion model was set to sec, the flow of downstream profiles could be

10 Wang, Chen, Yu, and Qi. calculated through the model in Equation (). The flow of field downstream profiles could be derived from the observation. These two kinds of flow are compared in Figure. The horizontal axis refers to the travel times of each vehicle in the platoon. (a) Percentage of buses=.% Observation Calibrated model Flow(veh/s) Time(s) (b) Percentage of buses=% Observation Calibrated model Flow(veh/s) Time(s) (c) Percentage of buses=.% Observation Calibrated model Flow(veh/s) Time(s) FIGURE Observed and estimated downstream flow profiles of Site C on Zhongguancun Street.

11 Wang, Chen, Yu, and Qi. From Figure, it can be seen that the trends of two kinds of flow are quite similar. The differences between the two kinds of flows are calculated. Since the vehicles from the access road impose interrupted effects on the platoons of the Zhongguancun Street, some flow observations exceed the values derived from the calibrated model. This explains why part of ratios in Figure is higher than one vehicle per second. The differences among the three scenes, none of which are greater than one vehicle, are %, %, %, respectively. Therefore, the predicted data closely match the field data. In addition, a quantitative measure used by Hanna (), Song (), and Wu () the normalized mean square error (NMSE) as shown in Equation (), is used in this paper to evaluate the quality of the calibrated parameters. When the value of the NMSE is close to, the model selection and fitting are better and data prediction is more accurate. where ( q ) oi qpi n NSME=, n q *q i= o p q oi is the observed flow at time i; q pi is the estimated flow at time i; () q o is the average of the observed flow; and q p is the average of the estimated flow. In an accurate model, the NMSE should be close to. Song () proposed that NMSE <. is the acceptable limit. The values of NMSE of the three scenarios in Table are.,., and., respectively. The results show that for different bus percentage scenarios, the estimated flow profiles on the basis of calibrated platoon-dispersion parameters all fall within the acceptable range of accuracy. TIME STEPS FOR THE PLATOON DISPERSION MODEL According to a study by Rakha and Farzaneh (), the time step influenced the accuracy of the platoon dispersion model. To validate the calibrated platoon dispersion model further, the impact of the time step is analyzed. A comparison of the observed versus the estimated downstream flow profiles under the time steps for - sec was conducted. When the percentage of buses is.%, α is., β is., and F is.. The downstream flow profiles estimated from the calibrated platoon dispersion model and the flow profiles with the -sec time step are shown in Figure (a). The results for the other time steps are shown in Figure.

12 Wang, Chen, Yu, and Qi. (a) Time step=s (b) Time step=s Observation Calibrated model Observation Calibrated model Flow(veh/s) Time(s) Flow(veh/s) Time(s) (c) Time step=s (d) Time step=s Observation Calibrated model Observation Calibrated model Flow(veh/s) Time(s) Time(s) FIGURE Observed and estimated downstream flow profiles for different time steps. Results show that the platoon dispersion process varies for different time steps in the platoon dispersion model. When the time steps are - sec, the NMSE values for the model are.,.,.,., and., respectively. It is interesting to observe that when the time step increases from to sec, the accuracy of the model improves. However, when the time step increases from to sec, the accuracy of the model diminishes. Thus, the time step of the platoon dispersion model indeed affects accuracy. The analysis demonstrates that the widely used -sec time step might not be the best for estimating downstream flow profiles, so the value of the time step varies along with different situations. CONCLUSIONS The calibration of the Robertson s platoon dispersion model remains an open question in the literature. This study has addressed the issue of using data collected from the field to calibrate the Robertson s platoon dispersion model under varying percentages of buses at the intersection that was studied. The data recording Sites A-C are from the Zhongguancun Street intersection, a well-divided arterial with typical low-friction traffic flow conditions. Videos of the platoon dispersion were recorded at the intersection, from which the arrival and departure times were extracted. The parameters of the platoon dispersion model were calibrated and the impact of the percentage of buses on the parameters was studied. Regression equations in which the platoon dispersion parameters varied as the function of the percentage of buses at the intersection were established. To assess the quality of the calibrated platoon dispersion model, the estimated downstream flow profiles from the calibrated model were compared to the observed downstream flow profiles. The influence of the time step on the calibrated platoon dispersion model was also Flow(veh/s)

13 Wang, Chen, Yu, and Qi. analyzed. It has shown in this study that the calibrated platoon dispersion model works in Beijing. The methodology proposed in this paper can be applied in other signalized intersection where the impacts of the percentage of buses or other factors need to be considered. The following conclusions can be summarized from this study. First, the number of buses at the signalized intersections indeed has a significant impact on the platoon dispersion factor. The relationship between the percentage of buses and parameters of the platoon dispersion model can be established as a polynomial formula. As the percentage of buses increases in from to % or to %, the value of α increases, and the values of β and F decrease. Second, the calibrated values of α, β, and F differ significantly from the default values suggested by the TRANSYT Users Guide. The calibrated values of α are much smaller than the default value.. The values of β are larger than the default value.. This difference can be attributed partially to the influence of buses at the intersection. Finally, the implementation of the proposed calibration approach is expected to improve the accuracy of the platoon dispersion model, which is important to the design of traffic-signal control plans. It was also verified that the time step indeed has an impact on the quality of the platoon dispersion model. The -sec time step was found to be the best time step in this study. Because this paper focuses on field data collected from Beijing, the study has some limitations. The impact of other traffic conditions, the relationship between the parameters of the platoon dispersion model and the traffic conditions, as well as the application of the platoon dispersion model in the signal timing plan, can be considered. Therefore, further studies should be extended to other factors, including traffic friction and road length. To capture the features of traffic conditions, the dynamic relationship between the parameters of the platoon dispersion model and the traffic conditions can be analyzed further. Besides, on the basis of the improved platoon dispersion model, future study can focus on optimizing the signal timing plan, thus reducing delays at intersections. ACKNOWLEDGEMENT The authors acknowledge that this paper was prepared based on NSFC # and the Program for New Century Excellent Talents in University (NCET--). This research is partially supported by the National Science Foundation (NSF) under Grant #. REFERENCES. Rakha, H. and M. Farzaneh. Issues and Solutions to Macroscopic Traffic Dispersion Modeling. Journal of Transportation Engineering, Vol., No.,, pp. -.. Maher, M. A Comparison of the Use of the Cell Transmission and Platoon Dispersion Models in TRANSYT. Transportation Planning and Technology, Vol., No.., pp. -.. McTrans Center, University of Florida. TRANSYT-F Users Guide. Gainesville, USA,.. Farzaneh, M. Modeling Traffic Dispersion. Doctoral thesis, Virginia Polytechnic Institute and State University, Blacksburg, America,.. Robertson, D. I. TRANSYT - A Traffic Network Study Tool, Report LR, Washington, DC, USA: Transport and Road Research Library,.. Farzaneh, M. and H. Rakha. Procedures for Calibrating TRANSYT Platoon Dispersion

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