0 ESTIMATING PARAMETERS FOR MODIFIED GREENSHIELD S MODEL AT FREEWAY SECTIONS FROM FIELD OBSERVATIONS Omor Sharif University of South Carolina Department of Civil and Environmental Engineering 00 Main Street Columbia, SC 0 Telephone: (0) -0 Fax: (0) -00 Email: omor.sharif@gmail.com Length of paper: Text = 0 words, Number of Figures = 0; Number of Tables = ; Total = 00 words Submission Date: December 0, 00 0 0
0 ABSTRACT To date various theoretical traffic stream models have been developed by researchers to describe traffic flow characteristics of real world. These models interpret the relationship among various traffic flow parameters such as speed, flow, density, occupancy etc. Greenshield s model, assuming linearly related speed and density, has been a popular choice for decades as the appropriate relationship and has enormous importance to traffic flow theory. In this project, we adopt Modified Greenshield s model to associate theoretical and field observations at three freeway sections of interstate 0 in North Carolina. Field measurements for a period of one month were obtained from online traffic flow databases. These dataset are then fitted to Modified Greenshield s model using least square regression analysis and yielded parameters (free flow speed, jam density etc) are recorded for each of the three freeway sections. To track the data reasonably well, we conclude that it is more appropriate to use a two regime model- a constant speed model for the free flow region and Modified Greenshield s model for near capacity and congested regions. Keywords: Traffic stream models, Greenshield s model, multi-regime models, macroscopic analysis, regression analysis. 0
Sharif 0 0 0. INTRODUCTION Various theoretical traffic stream models have been developed by researchers to describe traffic flow characteristics. These models interpret the relationship among various traffic flow parameters (speed, flow, density, occupancy, headway etc) and can be classified into two broad categories- single-regime models and multi-regime models. Single- regime models assume that a singular relationship holds true for the complete range of traffic flow. Multi-regime models, however, identify separate regimes (two or more) of flow conditions and propose independent relationships for each regime. Greenshield s model is the first of proposed single-regime models that was developed in and was based on field observation of speed and density measurements. The model requires estimation of two parameters, namely free flow speed and jam density for a roadway, to define the relationship between speed and density. Later a third new parameter was introduced into Greenshiled s model to improve upon some limitations of the model such as to minimize the disagreement between observed and model estimated values and to offer a more generalized approach. The revised model is known as Modified Greenshield s model (MGM). The parameters of MGM at three different freeway sections using best regression fit of real world traffic data are evaluated in this class project. However, based on inspection of field speed-density-flow distributions, we have found two regimes to be more appropriate to describe the relationship.. DATA EXTRACTION Field observations of various traffic flow parameters used in this project are obtained from two online database repsitories. These databases are www.inrix.com and www.traffic.com and can be accessed on-line with a qualified user account. A brief description of each of these sources are given below-. Source I- Inrix.com INRIX offers traffic information from various sources such as GPS-enabled vehicles and mobile devices, traditional road sensors and a variety of other sources. The provided data include realtime, historical and predictive traffic statistics on freeways, highways and secondary roadways and arterials. It archieves information on roadway location, direction, speed, travel time etc.. Source II- Traffic.com TRAFFIC provides traffic flow information including accidents and events on a specific route. Real time traffic flow data is obtained from fixed point sensors and GPS probes. The roadway locations where data can be extracted are identified with a unique station ID. The sensor readings include but not limited to direction of travel, lane, speed, volume, occupancy and vehicle class information. Both raw and aggreegated data are archieved and can extracted as reports. The location, time period and type of dataset that were made available for the class project are summarized in Table and for above two sources. In both cases, data was offered at three locations of Interstate 0 in state of North Carolina for eastbound and westbound direction. 0
Sharif 0 0 Table Dataset from Inrix.com Location Time Period Data available I-0 @ hours of weeks (Weekends excluded) Speed ( minute Davis Drive / Exit 0 0 Aug, 00 (Mon-Fri) average) I-0/I-0 @ 0 Aug, 00 (Mon-Fri) Hammond Road / Exit Aug, 00 (Mon-Fri) I-0 @ Harrison Avenue / Exit 0 Aug 0 Sept, 00 (Mon-Fri) Table Dataset from Traffic.com Location Time Period Data available I-0 @ hours of weeks (Weekends excluded) (All data are Davis Drive / Exit 0 0 Aug, 00 (Mon-Fri) collected at minute I-0/I-0 @ 0 Aug, 00 (Mon-Fri) intervals) Hammond Road / Exit Aug, 00 (Mon-Fri) Speed I-0 @ 0 Aug 0 Sept, 00 (Mon- Volume Harrison Avenue / Exit Fri) Occupancy Vehicle Class. DATA VISUALIZATION In this section several macroscopic traffic flow and speed characteristics are presented based on the field observations available. The visualization of these characteristics serves at least two purposes. Firstly, it offers a better understanding the various traffic characteristics at a roadway section. Secondly, by matching the obtained distribution at some location with established findings available in literature, it ensures that collected data is free from major errors. The dataset offers the possibility of investigating traffic characteristics at three freeway sections as noted in section. However, we provide an overview of the attributes only for I-0 @ Hammond Road / Exit to limit the contents of this report. Characteristics at other sections can be investigated in a similar manner. Also, speed-density-flow distributions are deferred until section, where we estimate the parameters for theoretical Modified Greenshield s Model using these distributions.. Macroscopic Flow Characteristics Several important macroscopic flow characteristics exist at a section but a few selected of them are presented here. We provide some temporal flow patterns that describe the changes in flow with time (daily and hourly variations). It should be noted that monthly and within the hour variation also exist but are not included here. Figure shows daily flow variation for different weekdays. For each day volume in converted to PCE and summed up for both directions and then averaged over four week data. The maximum average volume occurs on Friday. Figure and shows hourly variation of flow for different days for eastbound and westbound direction respectively. It is evident that the hourly variation deviates considerably on Friday compared to other days.
Sharif FIGURE Daily flow patterns for weekdays FIGURE Hourly flow patterns for different days in EB direction
Sharif FIGURE Hourly flow patterns for different days in WB direction Traffic flow rate also has directional distribution which also varies by hours and days. Figure shows a sample of this directional split of traffic for Thursday. Also, traffic has modal patterns and we can broadly classify the vehicles into two different modes namely passenger cars and non passenger cars. Figure shows hourly variation of modal flow averaged over the entire study period. FIGURE Directional Split on Thursday
Sharif 0 FIGURE Average hourly distribution of mode in the study period. Macroscopic Speed Characteristics Speed also varies with flow at different time period within a roadway section. Figure shows hourly speed distribution in westbound direction averaged over the entire study period. Figure shows speed class distribution in eastbound direction using weekday data of the complete study period. It is evident from the histogram that most speed values are above mph and below 0 mph. A few speed observations fall between mph and mph that represent congested conditions probably due to some incident.
Sharif FIGURE Hourly speed distributions with standard deviation in westbound direction 0 FIGURE Speed histogram from four week speed data (Eastbound direction). ESTIMATION OF PARAMETERS FOR MODIFIED GREENSHIELD S MODEL Greenshield s model is the first of proposed single-regime models and it was based on his field observation of speed and density. Supported by his field dataset, Greenshield concluded that speed varies linearly with density. The equation for his speed-density (u-k) model is shown below
Sharif u u ( f Where, k k j ) 0 0 0 u f = Free Flow Speed k j = Jam Density The model requires estimation of two parameters, namely free flow speed and jam density for a roadway. However, though an estimate of free flow speed is relatively easy, the estimate of jam density is not straightforward. The value of free flow speed usually falls between design speed of a roadway and posted speed limit. The value of jam density is difficult to measure in the field. Nonetheless, suggested values of to 0 vehicles per mile per lane can be used based on a stopped vehicle inhabits to ft of roadway. The use of this value is problematic since his model defines optimum density as half of jam density which is incompatible with the field observation of optimum density that generally fall between 0 to 0 vehicles per mile. To represent better actual field conditions and introducing a more generalized approach, an additional parameter was incorporated in the Greenshield s model. This revised model with third new parameter alpha is known as Modified Greenshield s model. Adjusting the value of α, a family of models can be constructed. The equation with the additional parameter α is shown below. u u ( f k k j ) Note, the modified model reduces to original model when α is set to one. As noted in section field observations are recorded at three different sections of a freeway. The freeway has lanes in both eastbound and westbound directions. Aggregated flow and speed data are available at each minute interval. The density can be easily computed using the fundamental traffic flow relationship shown below Flow( q ) Density( k ). Speed( u ) In calculating volume in PCE (Passenger car equivalent), non-passenger cars were converted to passenger cars using a factor of.. The aggregated minute flow rate for the entire approach is then converted into hourly rate per lane ie. pce/hr/lane. The speed is recorded in miles per hour and the density is expressed in pce/mile/lane. For the purpose of estimating the parameters for Modified Greenshield s Model, the plot of speed-density relationship were constructed using hours data available on a chosen day for each freeway section. To achieve a useful and effective speed-density relationship, a day and direction of travel was carefully selected such that data from both uncongested and congested periods are present and a meaningful relationship can be obtained for the entire range of density conditions. For Hammond Road we have used Friday, 0/0/00 data in eastbound direction.
Sharif 0 0 For Harrison Avenue we have used Thursday, 0//00 data in westbound direction. For Davis Drive we have used Friday 0/0/00 data in westbound direction. Once the plots are constructed there are two ways to estimate the parameters for the Modified Greenshield s model. The first approach is to determine the best regression fit to the plotted data and obtain the yielded parameters. The second approach is to select the parameter values by inspection from the plot of dataset and then use the parameters to define the model and appraise how well the model fit to the field observations by calculating mean deviations. For the class project, we adopted best regression fit using minimum least square error technique (first approach) to determine the model parameters. MS Excel standard solver add-in was used for regression analysis in this project. Based on inspection of field speed-density-flow distributions, we have found two regimes to be more appropriate to describe the relationships. This is because a discontinuity in field measured data exists at near capacity conditions. Regression analysis also verifies this finding since using a single modified Greenshield s model for the entire speed-density range yield large sum square errors. Consequently we adopted a constant speed model for the free flow regime and Modified Greenshield s Model for the congested flow regime. We define a break density k b that marks the transition between two regimes. The equations for these two regimes are shown below- u if k kb (Free Flow Regime) u u f k u f ( ) if b k j k k (Congested Flow Regime) Therefore we also need estimate the k b in addition to free flow speed (u f ), jam density (k j ) and α for the two regime model. The parameter k b was selected such that minimum sum square error is obtained through regression analysis. These estimated parameters are shown in Table. Table Estimated Parameters at Three Freeway Sections Parameter @ Hammond Rd @ Harrison Av @Davis Dr Free Flow Speed, U f (mph)... Jam Density, K j (pce/mile/lane) 0. 0 Additional Parameter, α...0 Break Point, K b (pce/mile/lane) Sum Square Errors 0..0 0. Figure -0 shows the three fundamental traffic flow diagrams for each of the three freeway sections with best regression fit superimposed on the datasets. The fit is based on the two regime model parameters shown in Table.
Sharif FIGURE Best fit model superimposed on dataset of Hammond Road
Sharif 0 FIGURE Best fit model superimposed on dataset of Harrison Avenue
Sharif FIGURE 0 Best fit model superimposed on dataset of Davis Drive
Sharif 0 0 0 0 It can be seen in Figure -0 that best fitted two regime model describe the field data reasonably well in most instances. The section at Davis Drive and Hammond Road has better fit compared to Harrison Avenue since they yield much lower sum square errors in comparison. During regression analysis the maximum possible jam density was set to 0 pce/mile/lane which is equal to bumper density (maximum possible physical density) assuming 0 ft long vehicles. The least square optimization assigns bumper density as jam density for Davis and Hammond section. For a density greater than 0 pce/mile/lane, all three sections start to reach capacity and become congested at higher density. There are relatively few observations available at capacity for the freeway sections, and the two regime model does not track the data well at capacity state.. CONCLUSION Parameters for traffic stream models are dependent on geometric and operational characteristics of a roadway as well as traffic characteristics. Also the field measurement procedures used to collect the data and the location of a roadway influence the nature of data. In most instances a single location cannot supply all the data (free-flow, near capacity and congested regions) required to describe the complete speed-flow relationship. This project used fixed station field measurements, obtained at three freeway locations, to fit into Modified Greenshield s model. However, we suggest that a two regime model be used, instead of single regime model. A constant speed model for free flow condition and Modified Greenshield s model for capacity and congested condition offers considerable improvement over a single modified Greenshield s model from the entire flow range as it tracks the real world data reasonably well. We have also provided an estimate of the break point density that marks the transition between two regimes.
Sharif REFERENCES [] A. May, Traffic flow fundamentals: Prentice Hall, 0. [] Gerlough, DL and Huber, MJ, Special Report : Traffic Flow Theory: A Monograph, TRB, National Research Council, Washington, DC,. [] Gartner, N. and Messer, C.J. and Rathi, A.K., Traffic flow theory: A state-of-the-art report, Transportation Research Board, Washington DC, 00. [] R. P. Roess, E. S. Prassas and W. R. McShane, Traffic Engineering, th Edition, Prentice Hall New Jersey, 00.