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PAGE 1 1 COMPARISON OF FREEWAY CAPACITY ESTIMATION METHODS By VIPUL MODI A MASTER THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIE NCE UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Vipul Modi PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS First, I thank my advisor, Dr. Scott S Washburn who has patiently supported me throughout the course of my thesis with his encouragement, guidance and knowledge. I would also like to thank committee members Dr s Yafeng Yin and Dr. Lily Elefteriadou for their invaluable suggestions, support and feedback during the course of my thesis project I would also like to thank Se okjoo Lee TRC doctoral candidate for his assistance with data collection through the Central Data Warehouse, and my friend Lokesh from whom I learned programming that I used in my thesis project I would also take this opportunity to convey my wishes an d special thanks to my family members and my friends for their encouragement and support during my stay at University of Florida. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ ........... 4 LIST OF TABLES ................................ ................................ ................................ ...................... 7 LIST OF FIGURES ................................ ................................ ................................ .................... 8 LIST OF ABBREVIATIONS ................................ ................................ ................................ .... 10 ABSTRACT ................................ ................................ ................................ ............................. 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ ............. 13 Background ................................ ................................ ................................ ........................ 13 P roblem Statement ................................ ................................ ................................ ............. 13 Research Objective ................................ ................................ ................................ ............. 14 Organization of Report ................................ ................................ ................................ ....... 14 2 LITERATURE REVIEW ................................ ................................ ................................ ... 16 Introduction ................................ ................................ ................................ ........................ 16 Definition of Capacity ................................ ................................ ................................ ........ 16 Concept of Breakdown ................................ ................................ ................................ ....... 18 Methods to Estimate Capacity ................................ ................................ ............................ 22 Van Aerde Model ................................ ................................ ................................ ........ 22 Product Limit Method ................................ ................................ ................................ 25 Other Methods ................................ ................................ ................................ ............ 30 Factors that May Affect Capacity Values ................................ ................................ ........... 31 Free Flow Speed ................................ ................................ ................................ ......... 31 Lane width ................................ ................................ ................................ ........... 31 Lateral clearance ................................ ................................ ................................ .. 31 Number of lanes ................................ ................................ ................................ ... 32 Interchange density ................................ ................................ .............................. 34 Merge Diverge Areas on Freeways ................................ ................................ ............. 34 Summary ................................ ................................ ................................ ............................ 36 3 RESEARCH APPROACH ................................ ................................ ................................ 37 Introduction ................................ ................................ ................................ ........................ 37 Analysis Methods ................................ ................................ ................................ ............... 37 Capacity Estimation from Van Aerde Model ................................ ............................... 37 Capacity Estimation from Product Limit Method ................................ ......................... 38 Speed based breakdown identification ................................ ................................ .. 38 PAGE 6 6 Application of speed based threshold value method ................................ .............. 39 Capacity Estimated by Average of Maximum Flow Rates ................................ ........... 41 Data Collection ................................ ................................ ................................ .................. 43 Site Selection ................................ ................................ ................................ .............. 43 Data Source ................................ ................................ ................................ ................. 44 Data Processing ................................ ................................ ................................ .................. 46 Capacity Data Processor ................................ ................................ .............................. 46 Analysis period ................................ ................................ ................................ .... 46 Speed threshold value ................................ ................................ ........................... 47 Intervals preceding breakdown ................................ ................................ ............. 47 Data imputation ................................ ................................ ................................ .... 48 Downstream Breakdown Identifier ................................ ................................ .............. 48 4 DATA ANALYSIS AND RESULTS ................................ ................................ ................. 64 VAM Capacity Estimation ................................ ................................ ................................ 64 Stochastic Capacity Estimation ................................ ................................ .......................... 65 Applicability of PLM ................................ ................................ ................................ .. 65 Determination of Speed Threshold Values ................................ ................................ ... 66 Identification of Breakdown Ev ents ................................ ................................ ............. 67 PLM and Speed Flow Curves ................................ ................................ ...................... 68 Estimating Capacity Values ................................ ................................ ......................... 68 Average Maximum Flow Rate Capacity Estimation Method ................................ .............. 70 Comparisons and Results ................................ ................................ ................................ ... 73 5 SUMMARY AND CONCLUSIONS ................................ ................................ ................. 97 Conclusions ................................ ................................ ................................ ........................ 97 Advantages and Disadvantages of the Investigated Capacity Estimation Methods ....... 98 Stochastic estimation method ................................ ................................ ............... 98 Van Aerde model method ................................ ................................ ..................... 99 Average maximum flow rate method ................................ ................................ .... 99 Recommendations ................................ ................................ ................................ ............ 100 LIST OF REFERENCES ................................ ................................ ................................ ........ 101 BIOGRAPHICAL SKETCH ................................ ................................ ................................ ... 103 PAGE 7 7 LIST OF TABLES Table page 3 1 Data format for the data obtained from data source ................................ ........................ 62 3 2 Final selec ted sites and site description ................................ ................................ .......... 63 4 1 Capacity estimates and other parameters from VAM capacity estimation method .......... 88 4 2 Speed t hreshold values for upstream and downstream detectors ................................ ..... 89 4 3 Capacity estimates from stochastic capacity estimation method ................................ ..... 90 4 4 Capacity estimates from average of top 3% and top 5% highest flows ........................... 91 4 5 Threshold values for maximum flow rate in average flow estimation method ................ 92 4 6 Capacity estimates from average of flow rates above the flow rate that corresponds to 30 th & 35 th percentage of maximum flow rate ................................ ................................ 93 4 7 Average flow analysis x val ues for VAM capacity estimation method ........................... 94 4 8 Average flow analysis x values for stochastic capacity estimation method ..................... 95 4 9 Co mparison of capacity estimates from different analysis methods with HCM 2000 ...... 96 PAGE 8 8 LIST OF FIGURES Figure page 3 1 Acceptable Freeway Segment Configu rations ................................ ................................ 50 3 2 Snapshot of site T1 (Source: Google Earth) ................................ ................................ ... 51 3 3 Snapshot of site T2 (Source: Google Earth) ................................ ................................ ... 51 3 4 Snapshot of site T3 (Source: Google Earth) ................................ ................................ ... 52 3 5 Snapshot of site T4 (Source: Google Earth) ................................ ................................ ... 52 3 6 Snapshot of site T5 (Source: Google Earth) ................................ ................................ ... 53 3 7 Snapshot of site T6 (Source: Google Earth) ................................ ................................ ... 53 3 8 Snaps hot of site T7 (Source: Google Earth) ................................ ................................ ... 54 3 9 Snapshot of site T8 (Source: Google Earth) ................................ ................................ ... 54 3 10 Snapshot of site T9 (Source: Goo gle Earth) ................................ ................................ ... 55 3 11 Snapshot of site T10 (Source: Google Earth) ................................ ................................ 55 3 12 Snapshot of site T11 (Source: Google Earth) ................................ ................................ 56 3 13 Snapshot of site F1 (Source: Google Earth) ................................ ................................ ... 56 3 14 Snapshot of site F2 (Source: Google Earth) ................................ ................................ ... 57 3 15 Snapshot of site F3 (Source: Google Earth) ................................ ................................ ... 57 3 16 Snapshot of site F4 (Source: Google Earth) ................................ ................................ ... 58 3 17 Snapshot of site F5 (Source: Google Earth) ................................ ................................ ... 58 3 18 Snapshot of site F6 (Source: Google Earth) ................................ ................................ ... 59 3 19 Snapshot of site FV 1 (Source: Google Earth) ................................ ................................ 59 3 20 Capacity Data Processor utility program user interface ................................ .................. 60 3 21 Downstream Breakdown Identifier utility program user interface ................................ .. 61 4 1 Van Aerde Model fit to the speed flow data points for Site ID: T1 ................................ 76 4 2 Van Aerde Model f it to speed flow points for Site ID: T8 ................................ .............. 76 PAGE 9 9 4 3 Speed time series plot for site T4 ................................ ................................ ................... 77 4 4 Speed time series plot for site T7 ................................ ................................ ................... 77 4 5 Weibull curve fit with PLM curve, Site ID: 210391, NB North of Butler Blvd on I 95, District 2) ................................ ................................ ................................ ................ 78 4 6 Weibull curve fit with PLM curve Site ID: 610271, East of NW 57 Avenue on SR 826, District ................................ ................................ ................................ ................... 78 4 7 Speed Flow, Weibull and Van Aerde Model curves for Site ID T1 ................................ 79 4 8 Speed Flow, Weibull and Van Aerde Model curves for Site ID T2 ................................ 79 4 9 Speed Flow, Weibull and Van Aerde Model curves for Site ID T3 ................................ 80 4 10 Speed Flow, Weibull and Van Aerde Model curves for Site ID T4 ................................ 80 4 11 Speed Flow, Weibull and Van Aerde Model curves for Site ID T5 ................................ 81 4 12 Speed Flow, Weibull and Van Aerde Model curves for Site ID T6 ................................ 81 4 13 Speed Flow, Weibull and Van Aerde Model curves for Site ID T7 ................................ 82 4 14 Speed Flow, Weibull and Van Aerde Model curves for Site ID T8 ................................ 82 4 15 Speed Flow, Weibull and Van Aerde Model curves for Site ID T9 ................................ 83 4 16 Speed Flow, Weibull and Van Aerde Model curves for Site ID T10 ............................... 83 4 17 Speed Flow, Weibull and Van Aerde Model curves for Site ID T11 ............................... 84 4 18 Speed Flow, Weibull and Van Aerde Model curves for Site ID F1 ................................ 84 4 19 Speed Flow, Weibull and Van Aerde Model cu rves for Site ID F2 ................................ 85 4 20 Speed Flow, Weibull and Van Aerde Model curves for Site ID F3 ................................ 85 4 21 Speed Flow, Weibull and Van Ae rde Model curves for Site ID F4 ................................ 86 4 22 Speed Flow, Weibull and Van Aerde Model curves for Site ID F5 ................................ 86 4 23 Speed Flow, Weibul l and Van Aerde Model curves for Site ID F6 ................................ 87 4 24 Speed Flow, Weibull and Van Aerde Model curves for Site ID FV1 .............................. 87 PAGE 10 10 LIST OF ABBREVIATIONS HCM Highway Capacit y Manual FDOT Florida Department of Transportation VAM Van Aerde Model PLM Product Limit Method STEWARD Statewide Transportation Engineering Warehouse for Archived Regional Data U S United States CDP Capacity Data Processor DBI Downstream Breakdown Identi fier PAGE 11 11 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science COMPARISON OF CAPACITY ESTIMATION METHODS By Vipul Modi December 2009 C hair: Scott S Washburn Cochair: Yafeng Yin Major: Civil Engineering The capacity of a freeway segment is a critical factor f or the assessment of the traffic flow operations on freeway facilities. The Highway Capacity Manual HCM (2000) is considered to be one of the authoritative sources on capacity values for a variety of roadway types in the U.S. It provides a single set of capacity values for basic freeway segments as a function of free flow speed. These values are considered to be reasonably representat ive values for freeways located throughout the U.S., but it is recognized that lower or higher values may be more appropriate in any given location. However, the HCM does not provide any guidance on how its recommended values can be adjusted to reflect sig nificant differences in capacity due to local conditions nor how to directly measure or estimate capacity values. The objective of this research was to investigate various methods that can be used to arrive at an estimate of freeway capacity values, and t o recommend one of these methods to the FDOT for use in developing their own estimates of capacity for Florida freeways. Three methods were investigated: one that fits a mathematical function to plotted speed flow data points, from which the apex of the fu nction is taken as capacity; one that estimates a breakdown probability distribution based on flow rates preceding breakdown events, from which capacity can be taken to correspond to a certain percentile value of the breakdown probability distribution; and one that uses a simple average of a specified PAGE 12 12 percentage of the highest flow rates observed at a site. Based on the various advantages and disadvantages of each of the methods, the following w as concluded. T he method based on identifying breakdown events is most suitable for the determination of capacity at a site where a detailed operational analysis is desired. For example, at sites where different operational treatments (e.g., ramp metering) are going to be tried in an effort to improve operations and a n estimate of capacity that is as accurate as possible is desired. The method based on fitting a mathematical function to speed flow data is not as suitable as the previous method for detailed evaluations of operational treatments, but is still appropriate for the determination of general capacity estimates The average maximum flow rate capacity estimation method is most suitable for planning and preliminary engineering applications. PAGE 13 13 CHAPTER 1 INTRODUCTION Background The maximum number of vehicles that ca n be carried by a freeway lane is a critical factor for the planning, design, and analysis of freeway facilities Although definitions vary, the value used to represent the maximum number of vehicles that can be carried by a freeway lane is generally terme The Highway Capacity Manual HCM ( 2000) is considered to be one of the authoritative sources on capacity values for a variety of roadway types in the U.S. It provides a single set of capacity values for basic freeway segments as a function of free flow speed These values are considered to be reasonably representative values for freeways located throughout the U.S., but it is recognized that lower or higher values may be more appropriate in any given location. T he Florida Department of Transpor tation (FDOT), for one, believes that capacity values for Florida freeways might be lower than the values provided in the HCM. This belief is based on a preliminary basic analysis of freeway flow data Problem Statement W hile it is generally recognized tha t the capacity values provided in the HCM may not be perfectly applicable to all freeway locations, the HCM does not provide any guidance on how its recommended values can be adjusted to reflect significant differences in capacity due to local conditions. Although there are adjustments that can be made to the free flow speed, which in turn will affect the base capacity value, and also adjustments that can be made to the traffic demand, there is no mechanism for directly adjusting the base capacity values. F urthermore, the HCM does not provide a method that can be used for measuring or estimat ing capacity values. PAGE 14 14 Research Object i ve The objective of this research was to investigate various methods that can be used to arrive at an estimate of freeway capacity values and to recommend one of these methods to the FDOT for use in developing their own estimates of capacity for Florida freeway s The following tasks were performed in order to accomplish the desired results of this research objective: A detailed revi ew of previous research related to methods used to estimate the capacity of basic freeway segments as well as a review of the definitions of capacity that are used in these methods. From this review, the selection of one or more methods to test with Flori da freeway data. In addition to methods identified from the literature, a simple method is also developed which is easy to apply and will yield capacity estimates similar to those obtained through more mathematically and/or complex methods. A detailed surv ey of basic freeway segments across Florida to determine suitable sites for the collection of data to use with the selected method, or methods. Traffic data are obtained for the respective chosen sites across Florida and are prepared for the subsequent pro cess ing and analysis. Data were processed and analyzed according to the selected/developed capacity estimation methods. Different methods of estimating the capacity valu es are compared with each other as well as with the HCM value Organization of Report The remainder of this report is organized as follows. Chapter 2 discusses the previous studies on estimating the capacity values and the various methodologies implemented in order to estimate the capacity values for freeway segments. The chapter also discu sses the various factors which affects the capacity values on the freeway segment. Chapter 3 explains the selected methodologies used for estimating the capacity values, the description about the data and procedure applied for the site selection. Chapter 4 gives results of analyzed data for the selected PAGE 15 15 sites from all the estimation methods. Chapter 5 discusses the summary of the research study, describes the results and provides recommendations for future research on this study. PAGE 16 16 CHAPTER 2 LITERATURE REVIE W Introduction This chapter provides a review of the literature in several related areas. First, the definition s of capacity as discussed historically in the HCM and by other researchers are provided. Second, the concept of breakdown is discussed and vario us studies are presented which use the concept of breakdown to estimate the capacity on a freeway segment. Third, different capacity estimation methods as suggested and implemented by previous studies are discussed. Fourth, a discussion is provided on fact ors which may affect capacity values Fifth a brief summary on studies that discuss various capacity estimation methods is provided. Definition of Capacity The value used to represent the maximum number of vehicles that can be carried by a freeway lane is a variety of specific definitions of capacity have been offered by various sources/researchers. According to Agyemang Duah and Hall (1991) the capacity values of a freeway segment w ere d efined as the maximum 15 minut e flow values for two traffic state conditions namely: pre queue flows and queue discharge flows. In a study by Van Aerde (199 5 ), the capacity was defined as the apex value of the speed flow curves obtained by fitting speed flow data points to a mathematic al model on the basis of a simple car following model. With more research and studies, the definition of capacity was refined and was defined on the basis of the breakdown flows, which is discussed later in this chapter Brilon et al. (2005) defined capaci ty as the expected value of the Weibull distribution function; that is, the value at which there is a 50% probability of occurrence of a breakdown event. This definition of capacity accounted for the stochastic nature of the capacity values. PAGE 17 17 I n the most re cent edition of HCM (2000), the capacity o f a basic freeway segment is the maximum hourly rate at which persons or vehicles can be reasonably expected to traverse a point or a uniform section of a lane or roadway during a given time period, und er prevailing roadway, traffic and control conditions As the definition provided by the HCM constant and is considered to be stochastic in nature. Also, the f lows on different freeways are observed to vary under different conditions. Thus, a single value of the capacity value for a freeway facility does not reflect the real world observations and the capacity values are considered to be stochastic in nature. Ho wever, thi s edition and definition became the focus for determining the capacity as it is the most commonly accepted professional reference for traffic engineering analysis. The HCM (2000) provides the relationships between speed, flow and density which g ives capacity values for a freeway segments free flow speeds. According to the HCM (2000) and T RB Special Report 209 (1997), under ideal traffic and geometric conditions, freeways will operate with capacities of 2400 pc/h/ln These conditions and capacity are typically achieved on freeways with speeds of 70 mi/h or greater. As the free flow speed decreases, there is a slight decrease in the capacity values. For example, the capacity of a basic freeway segment with a free flow speed of 55 m i/ h is expected to be approximately 2250 pc /h/ln whereas for free flow speed of 70 m i/ h, the capacity is given as 2400 pc / h / l n These capacity values were arrived after observing the maximum flow values on various freeways across the U .S. Within a range of 55 mi/h to 70 mi/ h, the variation in the capacity values are observed but the HCM lacks in providing a methodology on how these capacity estimates are obtained. PAGE 18 18 These capacity estimates are based on the series of speed flow curves provided in HCM (2000) which gives relatio nships between speeds and flows for basic freeway segments for different free flow speed values. T hese speed flow curves are only provided for the un congested traffic state and the manual does not mention anything about post congested traffic states. Howe ver, it is observed that when the demand on the freeway exceeds the capacity, a transition takes place and the freeway system becomes congested. This state of transition from an un congested state to a congested state is defined as breakdown. It is thus ev ident that a breakdown phenomenon has a significant impact on the capacity value for a freeway segment. Although, the maximum flows observed on a freeway facility closely resemble the maximum pre breakdown flow values, the HCM does not discuss the breakdow n flows and the concept of breakdown in its editions. Thus, it becomes important to discuss the concept of breakdown from various studies and research performed in the past The following section introduces the concept of breakdown and explains the transit ion states from un congested to congested flows and vice versa. Concept of Breakdown A number of researchers have investigated the effects of transition from free flow conditions to congested conditions on freeways. With more r esearch on the transition sta tes, it was found that by studying the breakdown phenomenon, the determination of capacity values can be studied in a more comprehensive manner. It was also observed that the breakdown event does not necessarily occur at a maximum flow and breakdown can oc cur at flows lower or higher than those traditionally accepted as capacity. It was suggested then, that the breakdown is a better measure to associate specific flows with the capacity flows. With the concept of a breakdown state affecting the capacity valu es, the definition of capacity should also incorporate the probabilistic nature of the occurrence of the breakdown events. As a result, a new definition of the capacity value was developed as the volume below PAGE 19 19 which the facility conditions are acceptable an d above this volume, the facility condition becomes unacceptable. This transition between proper operation and non acceptable flow on a freeway occurs when the av erage travel speed is reduced from an acceptable speed level to a much lower value of congested conditions. These transitions usually involve a rather sudden speed reduction. The factors which affect the breakdown depend upon one region to another based up following section gives some of the brief descriptions and discussions on the past research related to breakdown. Elefteriadou et al. (1995) discussed the probabilistic nature of breakdown at freeway merge junctions. The authors analyzed the probabilistic aspect of ramp merge breakdown by examining and analyzing traffic data at ramp merge junctions for three sites using the NCHRP Project 3 37 data. It was observed that b reakdown was not the direct result of peak volumes. It was also observed that breakdown at ramp freeway junctions is a probabilistic variable and not deterministic. The authors concluded that capacity does not necessarily occur immediately before breakdown since it might include the cluster of vehicles from ramp which can cause the freeway segment. A study was conducted by Lorenz and Elefteriadou (2001) to estimate the capacity by the breakdown definition from congested flow and to measure the capacity through discharge flow measurements. The authors conducted an extensive analysis of speed and flow data collected at two freeway bottleneck locations in Toronto, Canad a, to investigate whether the probabilistic models previously developed replicated reality. The time series speed plots were examined at PAGE 20 20 tw o sites which concluded that a threshold speed at approximately 90 km/h existed between the non congested and congest ed regions. This threshold was used to define the breakdown but the cases where the average speeds dropped for a given interval was considered as a true breakdown. The period for this research was taken as 5 minutes or more. The authors observed the freque ncy of the breakdown events and it was concluded that with increase in flow rate, the probability of a breakdown event to occur increases. The authors also suggested that the capacity depends upon the probability of occurrence of a breakdown event. The res earchers also observed that the capacity depends upon the particular flow rate at which the facility breaks down and for a freeway facility the capacity depends upon the discharge flow following breakdown and the flow at which the breakdown occurs. However the research lacked the discussions on the effects of flows prior to the breakdown event. To study the pre breakdown flows, Elefteriadou and Lertworawanich (2003) examined freeway traffic data at two sites over a period of several days which focused on the non congested state to congested transition state. The authors observed the maximum pre breakdown flow, the breakdown flow and the flows following the breakdown. It was concluded by the authors that the numerical value of the maximum pre breakdown flow s was larger than the flows following the breakdown but the flows at which a breakdown event occurs is always less than the flows prior to the breakdown and after the breakdown. Zou and Levinson (2003) discussed that with timely traffic prediction the tra ffic control facilities can provide rapid and effective response. The authors used frequency domain tools in traffic flow studies instead of using time domain analysis methods. With the use of the frequency domain tools, one can detect the traffic breakdow n more effectively. It was found from the research that with the changing rate of the cross correlation between density dynamics and flow PAGE 21 21 rate, one can determine the transition from free flow phase to the congestion phase. The method as proposed by the aut hors suggests that an un returnable transition will occur only if the changing rate of the cross correlation exceeds a threshold. A new method was thus developed in order to detect the congested state on a freeway. But, due to complexity involved in this m ethod, the method to detect breakdown would be used as given by Elefteriadou and Lorenz (2001). Several authors, Minderhoud et al. (1997), Lorenz and Elefteriadou (2001), observed only traffic breakdowns at different flow rates to demonstrate the variabil ity of flows preceding a breakdown. But in order to have a better comprehensive theoretical concept, more systematic analysis should be performed. Brilon et al. (2005) discussed the concept of stochastic capacities which seems to be more realistic and mor e useful than the traditional use of single capacity values The author presented that with a transition of traffic flow from an un congested state to a congested state, the capacity value of a freeway segment is achieved. The paper examines the traffic fl ow patterns counted at 5 minute intervals over several months at different sites, which clearly showed that capacity can be well taken as Weibull distribution with a nearly constant shape parameter which represented the variance in the capacity values. Thi s was identified using the so called Product Limit Method, which is based on the statistics of lifetime data analysis. The investigations were an extension to the idea being proposed by Minderhoud et al. (1997). The author modified and extended the Product Limit Method to estimate the capacity values. As discussed in the paper, the stochastic analysis of capacity values was coherent with the real world results on German freeways. Overall, it was interpreted that the concept of randomness in capacity values is found to be more applicable as compared to other traffic engineering methodologies. PAGE 22 22 In an extended research to Brilon et al. (2005), Geistefeldt (2008) discussed the same approach of analyzing the capacity values by using the stochastic capacity concept The author compared the stochastic capacities for several freeways in Germany with the conventional capacity estimates. The conventional capacities from the speed flow diagram were estimated from the Van Aerde model. The capacity distribution function wa s used to find the breakdown probability that corresponds to the conventional capacity estimation. The author, after conducting the analysis for several freeways in Germany, concluded that these probability values could be suitably used for designing the f reeway segments on the basis of these obtained values. The author summarized that with the availability of five minute traffic data, 3% of the probability value on the capacity distribution function would fairly represent the freeway segment and with traff ic data in one hour data intervals, 40% of the probability value will represent the respective freeway segment. Methods to Estimate Capacity A lot of research has been done on the determination of freeway capacity and its variability Different studies hav e performed on the basis of different definitions of capacity values and various methods have been implemented to account for the stochastic nature of the capacity values. The following section summarizes more recent and well accepted methods of estimating freeway capacity values. Van Aerde Model Fitting a mathematical function to speed flow data plots to determine capacity dates back to Greenshields ( 1935). Since then, many researchers have tried using various mathematical functional forms to describ e the speed flow relationship. However, in many cases, the functional form either did not fit the data very well or was not necessarily consistent with traffic flow theory. A mathematical model developed by Van Aerde (1995) generally provides a good fit to PAGE 23 23 speed flow data and is consis tent with traffic flow theory. The mathematical model, as proposed by the author, is a continuous function and has the ability to explain all the traffic states in the fundamental speed flow diagram. The model is based on a simple c ar following model which is based on the minimum headway distance between consecutive cars. The headway is a combination of a constant term, a term which depends on the difference between the speed at any given time and the free flow speed and a term whic h depends on the speed at any given time. The model is given by : ( 2 1) Where: k = traffic density (veh/km) u = space mean speed (km/h) u f = free flow speed (km/h) c 1 c 2 c 3 as the parameters for the respective three terms These par ameters are calibrated by a non linear regression analysis as explained in Van Aerde and Rakha (1995). The procedure as proposed by the authors solves an optimization problem which calculates different parameters on the basis of a speed density relationshi p. The parameters calibrated from this method are used to determine the free flow speed, speed at capacity, capacity, and the jam density. From the optimization technique the model parameters are calculated from the following: PAGE 24 24 ( 2 2) ( 2 3) ( 2 4) ( 2 5) where: c 1 = fixed distance headway constant (km), c 2 = first variable distance headway constant (km 2 /h), c 3 = second variable distance hea dway constant (h), u f = free speed(km/h), u c = speed at capacity (km/h), q c = flow at capacity (veh/h), k j = jam density (veh/km), and m = is a constant used to solve f or the three headway constants (per hour) After the optimization problem is solved, the flow value at the apex of the mathematical model fitted to the speed flow data points is considered as the capacity estimate for the freeway segment. However, this model is only applied to the freeway segments which have considerable number of congested f low points in the speed flow diagram. With sufficient congested points, it is made sure that the data has flows in the range of the capacity level. Van Aerde and Rakha (1995) used the multivariate regression analysis procedure for performing the automated fitting of speed flow relationships for different roads based on loop PAGE 25 25 flow data on a variety of roadway types. Product Limit Method The Product Limit Method (PLM) for capacity es timation was proposed by Brilon et al. (2005) to estimate the capacity values by identifying the traffic flow breakdown events. The identification of these breakdown events are determined by application of a suitable algorithm but typically the breakdown e vents are identified by sudden drops in speed or sudden increase in occupancy values that are sustained for a certain period of time. This method determines the capacity based on the flows which causes the breakdown event. As a result these observed flows are found to be random in nature and accounts for the stochastic nature of the capacity values. To estimate the capacity, it is observed that these flows follow a particular mathematical distribution. These effects make the capacity distribution function a s follows: F c ( q ) = p ( c q ) ( 2 6) where, F c = capacity distribution function c = capacity q = traffic volume As discussed by Minderhoud et al. (1997) Brilon et al. (2005) modified the idea of defining the capacity distribution by proposing an analogy to the statistics of life time data analysis. The lifetime distribution function is given by: F ( t ) = 1 S ( t ) ( 2 7) PAGE 26 26 where, F ( t ) = distribution function of lifetime = p ( T t ) T = lifetime S ( t ) = survival function = p ( T > t ) The lifetime distribution functions are based on the life period of the experiment time but there might a possibility of certain lifetimes which exceeds the duration of the experiment. These are accounted as the censored data in the survival analysis function. In a similar manner, the traffic breakdown is regarde d as a failure event and it is analogous for estimating the capacity, c as the lifetime, T in the lifetime data analysis. The statistics of this lifetime analysis can be then used to estimate the parameters of the distribution function which includes the censored data. To estimate the survival function, Kaplan and Meier (1958) proposed the non parametric method parametric method is described as follows: j ( 2 8) where, ( t ) = estimated survi val function n j = number of individuals with a lifetime T t j d j = number of deaths t j Usually, in this function each observed lifetime is used as one t j value. For the above equation d j is always equal to 1. To estimate the distribution function for cap acity analysis, a similar analogy to Equation 2 9 is provided. The distribution function for the capacity analysis is given as: PAGE 27 27 ; i is set of {B,C2} ( 2 9) w here F c ( q ) = distribution function of capacity c q = traffic volume q i = traffic volume in interval i k i = number of intervals with a traffic volume of q q i d i = number of breakdowns at a volume of q i { B} = set of breakdown intervals (see below) Using this equation the traffic flow value, q, is kept into different categories or sets. These sets are obtained and named as set containing all the un congested flow values. The various sets used for this approach are defined as follows: B: The traffic pattern is un congested in time interval, i but the observed flow value just after the time interval causes a breakdown, i.e. the average speed value drops below the pre defined threshold value in the next interval i +1 F: The flow values for which the traffic flow is found to be un congested in interval i and in the following interval i + 1 This flow value in the interval i contains a censored value. This flow int erval reveals that the actual capacity in interval i is greater than the observed volume q i C1: The flow values in this interval are in congested state for interval i i.e. the average speed is below the threshold value. This flow interval, i, provides no information about the capacity value, so these intervals under the congested state are not considered for the data analysis. C2: The flow values in this interval, i are considered to be in un congested state, but the flow value causes a breakdown. Howev er, in contrast to classification B, traffic is congested at a downstream cross section during interval i or i 1 For this case, the breakdown at the observation point is supposed to be due to a spillback from downstream. As this flow interval also does not give information about the capacity values at the observation point, these flow values are also not considered for the data analysis. PAGE 28 28 After the flow datasets have been assigned with the respective set values, the distribution function F c ( q ) is plotted for the flow rates in B set. These set of values under B category are thus termed as B set flows and consequently the B set values. This distribution function is called as the PLM curves for the rest of the methodological procedure. The value of the distri bution function as estimated by the PLM will reach a value of 1 if the maximum observed volume is from the set B and for all the other volumes that are not in B set are assigned a value of 0. After the breakdown events are identified, the next step is to d efine the two elements which becomes the basis of using PLM. One of the elements is to select the data time intervals for which the analysis should be performed. This is decided on the basis of the availability of data. Another element is to identify the c riteria by which the breakdown events are identified. As a breakdown of traffic flow usually involves a significant speed reduction, breakdown events will be identified if there is drop in the average speed across the lanes for a certain period of time. Th e threshold speed is estimated using the method as described by Elefteriadou and Lorenz (2001). If the speed value drops in any interval, the traffic flow prior to that interval are considered as B set flows. While estimating the capacity functions, it is necessary to know more about the mathematical type of the distribution function F c ( x ) or the PLM curves To find the distribution, Brilon and Zurlinden ( 2003) suggested various plausible function types like Weibull, Normal and Gamma distribution. To estim ate the parameters of the distribution functions, a m axi mum likelihood technique is used. The likel ihood function is given by ( Lawless, 2003): ( 2 10) PAGE 29 29 where, f c ( q i ) = statistical density function of capacity c F c ( q i ) = cumulative distribution of capacity c n = number of intervals i = 1, if uncensored (breakdown classification B and C2) i = 0, elsewhere The likelihood function or its na tural logarithm has to be maximized to calibrate the parameters of the distribution functi on as per Lawless (2003). By comparing different types o f functions based on the value of the likelihood function, the Weibull distribution is estimated to be the function that fit the observations on all freeway sections and are in accordance with the PLM curves The expected Weibull distribution function will be : ( 2 11) Where, = shape parameter = scale parameter The Weibull distribution is thus checked if it fits well into the PLM estimation. The PLM can also be also used for traffic densities instead of volumes, q for capacity estimation. After the parameters, i.e. and are calibrated, the mean and the standard deviation of the function are calculated that are calculated by the following equations respectively: ( 2 12) ( 2 13) PAGE 30 30 where, and are the p arameters estimated by the distribution function is the Gamma distribution function is the mean or the expected value for the distribution function According to Brilon et al. (2005), the mean of the capacity distribution function or the Weibull curves gives the capacity estimates for the freeway segments. Other M ethods A study by Agyemang Duah and Hall (1991) examined data over 52 days during peak periods to investigate the possibility of a drop in capacity as a queue forms, and to recommend a numerical value for capacity. They compared the plot s of pre queue peak fl ows and queue discharge flows for 15 minute intervals in which the distribution of the plots was fairly similar. The authors observed the mean value of the 15 minute maximum flows and recomme nded 2,300 pc /h/ln as the capacity value of freeway segments for stable flow conditions and 2,200 pc /h/ln for post breakdown conditions Although the researchers recommended capacity values they did not discuss the variability in the capacity values obser ved. Doubts over this concept of capacit y as a constant value was raised by Ponzlet (1996) who demonstrated that capacities vary according to external conditions like dry or wet road surfaces, daylight or darkness, and prevailing purpose of the freeway, w hether it is used for long distance or metropolitan commuter traffic. To examine the variability of capacity values, Minderhoud et al ( 1997 ) recommended the use of the PLM to discuss the variability in the values of capacity. The method due to its sound t heoretical concepts was used to estimate the capacity distribution using the non congested flows. The author discussed that the effects of non congested flow rates for estimating the capacity values. It was observed that higher non congested flows had a si gnificant impact on the capacity values for a freeway facility. The study also d iscussed and compared other estimation PAGE 31 31 methods for the freeways and came to a conclusion that PLM can capture the variability in the capacity values. The study however, did not discuss about the flow rates following a congested state. Factors that M ay A ffect Capacity Values The following section discusses factors that have either been shown to affect, or has been hypothesized by some to affect the capacity values. Free Flow Spee d The speed flow and density flow relationships for a basic freeway segment are provided in HCM (2000) which vary according to the free flow speeds on the freeways. On the ba sis of these speed flow curves and level of service under certain conditions, the capacity is obtained from the curves. It is observed from the values provided by the HCM (2000) that with the decrease in the free flow speed, the capacity value for a freeway will decrease. However, this relationship is only provided for free flow speeds ranging from 50 mi/h to 70 mi/h. The research which led to the development of these curves found that a number of factors affect the free flow speed which indirectly affects the capacity of a freeway. These factors as discussed in the HCM (2000) are discus sed and listed next: Lane width W hen the average lane width across all the lanes is less than 12 ft, the base free flow speed is reduced. This implies that with decreasing lane widths, the free flow speed decreases and indirectly the capacity is also reduc ed. Lateral clearance When the right shoulder lateral clearance is less than 6 ft, the base free flow speed is reduced. Similarly, it implies that if the lateral clearance is reduced, the free flow speed and the capacity are also reduced. PAGE 32 32 Number of lanes The HCM (2000) considers the freeway segments with five or more lanes (in one direction) have the base conditions with respect to the number of lanes. The manual provides that with decrease in number of lanes, the base free flow speed decreases and so the capacity of a freeway segment. A detailed discussion on all the previous research is provided in the following section. However, it is interesting to note that this factor has been removed in the forthcoming 2010 HCM. A limited research has been performed in past for comparing the traffic characteristics on freeways w ith different numbers of lanes. Al Kaisy et al. (1999) developed a simulation approach for examining capacity and operational perfor mance at freeway off ramps areas but the into the effects of the number of lanes on a basic freeway segment. The authors investigated total upstream demand, off ramp demand, length of deceleration, off ramp free flow speed and number of lanes at mainline. The investigation gave an insight that th ere is a significant impact of number of lanes on capacity and operational performance. Th is study was based only on analyzing the freeway s at diverge or weaving area but it lacked analysis on the basic freeway segments. However, two studies have been con ducted in recent years to see the impact on capacity values if the number of lanes is increased on a freeway. The first study was conducted by Yang and Zhang (2005) which investigates the impact of the number of lanes on highways capacity. The authors give a better understanding of the relationship between highway capacity and its number of lanes upon the statistical analysis of the survey done on the freeways of Shanghai and Beijing with limited sites. The capacity values are estimated using the maximum su stained 15 minute rate of flow which can be accommodated by a uniform highway segment under prevailing and roadway conditions in the specified direction of interest. The statistical test is used to investigate the impact of the number of lanes on highway PAGE 33 33 c apacity. The variance analysis of single factor and t test are applied to test the inequality. It is found that the marginal decrease rate of average capacity per lane with increasing number of lanes is around 6.7%. They explored the possible explanations of the decrease which can be effect of increasing lane changing opportunities and cars interaction with increasing lanes on highways. The second study was conducted as in extension to the previous research by Yang et al. (2007), which investigates the impa ct of the number of lanes on freeway traffic characteristics. The research is being conducted and validated upon an extensive field survey of traffic flow in Beijing and Shanghai with more significant improvements observed on two lane (one way direction) f reeways, then on three lane freeways as compared to those on four lane freeways. They explored that in congested traffic conditions, at the same densities, both flow rate per lane and average speed decrease with increase in the number of lanes on uninterru pted freeway segments. However, in free flow conditions, the average speed increases with increasing number of lanes on freeways. The regional factor had a little impact on the differences among the flow density relationships on freeways with different num ber of lanes. The authors suggest that from the corresponding results observed, it can be interpreted whether building a wide road on a freeway is feasible or building a number of narrow roads around the freeway. These studies conclude that the average cap acity per lane on different lane freeways is not change a lot on different lane freeways. This change is observed due to lane changing activity and its ass ociated disturbance to traffic stream This observation may not be uniform on different lane freeways and hence, leads to differences in t he speed flow density relationships and effectively on the capacity values. PAGE 34 34 Interchange density The interchange densi ty of 0.5 interchanges per mile or a 2 mile interchange spacing is considered as the base conditions to calculate the free flow speeds. It is provided by the manual that with increasing interchanges per mile, the free flow speed decreases. It is implied th at the free flow speeds observed on shorter segments are higher as compared to free flow speeds observed on longer segments under the same conditions. As a result, the capacity for a longer segment is higher than for the short segment under the same condit ions. This factor has been changed to ramp density in the forthcoming 2010 HCM. Merge Diverge Areas on Freeways HCM (2000) does not indicate that merging or diverging maneuvers restrict the total capacity of the upstream or downstream basic freeway segment s. The influence of merging or diverging vehicles is primarily to add or subtract demand at the ramp freeway junction. Thus, it is believed that the capacity of a downstream basic freeway segment is not influenced by turbulence in a merge area. The capacit y will be the same as if the segment were a basic freeway segment. As on ramp vehicles enter the freeway at a merge area, the total number of ramp and approaching freeway vehicles that can be accommodated is thus defined as the capacity of the downstream b asic freeway segment. Similarly, the capacity of an upstream basic freeway segment is not influenced by the turbulence in a diverge area. The total capacity that may be handled by the diverging junction is limited either by the capacity of the approaching (upstream) basic freeway segment or by the capacity of the downstream basic freeway segment and the ramp itself. The basic approach to model merge and diverge areas in the HCM (2000) focuses on an influence area of 1500 ft including the acceleration or dec eleration lane and the two outside lanes which are in the merging or diverging area. The HCM recognizes that other freeway lanes may PAGE 35 35 be affected by merging or diverging operations and the impact of congestion in the vicinity of a ramp can extend beyond the 1500 ft influence area, but according to it, this defined area experiences most of the operational impacts across all levels of service. Thus, the operation of vehicles within the ramp influence area is the focus of the computational procedures in the HCM (2000). The HCM (2000) determines the capacity of a merge area by the capacity of the downstream freeway segment. Thus, the total flow arriving on the upstream freeway and the on ramp cannot exceed the basic freeway capacity of the departing downstream fr eeway segment. There is no evidence that the turbulence of the merge area causes the downstream freeway capacity to be less than that of a basic freeway segment. The freeway capacity per lane is always stated as an average across all lanes and the individu al lanes always carry proportionally less or more flow. In merge and diverge areas, through vehicles tend to move left to avoid turbulence, resulting in cases where inner lanes are very heavily loaded compared with lanes within the ramp influence area. Apa rt from HCM (2000), a study was conducted by Al Kaisy et al. (1999) to examine the capacity and operational performance at freeway diverge areas. The authors provided models to directly estimate the freeway capacity which were not mentioned in HCM (1997). A computer traffic simulation model INTEGRATION was used to explore the patterns of capacity and operational performance behavior at diverge areas under the impact of some key geometric and traffic variables. One of the control variables which were discuss ed in this research was number of lanes at mainline which was found to have a significant impact on the capacity and operational performance. The authors studied extensively the effect number of lanes at mainline on freeway capacity at ramp freeway diverge areas which is explained next. The research showed that for PAGE 36 36 freeways with more number of lanes, impact due to the spill back was not so drastic as the traffic had enough room to proceed through the diverge section through the left lanes on the freeways. I t was concluded that the number of lanes at mainline have an important impact on the capacity of the basic freeway segments when the diverge section is operated at the saturated traffic conditions, given that no queue spill back from the exit ramp is prese nt. It was also concluded that at a particular flow rate, the queue spill back has a significant impact on the th rough traffic and consequently on overall diverge section operation. This blockage effect was found to become less significant with the increas ing number of lanes. Summary To estimate the capacity for a basic freeway segment different studies are performed. The HCM (2000) gives fixed values for the capacity but does not consider the stochastic nature of the capacity values. However, different met hods have been identified that can be used to estimate the capacity values. First, the method that estimates a breakdown probability distribution based on flow rates preceding breakdown events and accounts for the stochastic nature of capacity values. Seco nd, the method that fits a mathematical function to plotted speed flow data points, from which the apex of the function is taken as capacity. PAGE 37 37 CHAPTER 3 RESEARCH APPROACH Introduction This chapter describes the approach taken to achieve the objectives of t his study. It provides a detailed discussion on the various analysis methods for capacity estimation, data collection, data reduction, and data processing. Analysis Methods As discussed in the literature review, capacity is considered as fixed value. Even under uniform traffic and roadway conditions, variability in maximum flow rate s (i.e., flow rate before breakdown) is observed. However, different methods of estimating the capacity values are provided from past studies. Thus, the focus of this study is to compare the capacity values as estimated by different methods. Three methods are used in this study to estimate the capacity of a freeway segment. The first method calculates the capacity value from Van Aerde Model, derived from the fundamental speed flo w diagram. The second method uses a combination of speed flow data plots and the Product Limit Method, as identified from the literature. The third method is a simple averaging scheme of high flow rates that can be applied easily to estimate the capacity v alues. Capacity Estimation from Van Aerde Model The first method used to estimate the capacity for a basic freeway segment is based on fitting a mathematical model to the speed flow data points. The relatively recent and widely accepted mathematical model proposed by Van Aerde (1995) is used, as described in the literature review chapter The regression analysis procedure that performs the fitting of the Van Aerde model is implemented in the software program SPD_CAL by Rakha (2007). The SPD_CAL program cal ibrates the four parameters free flow speed, speed at capacity, capacity PAGE 38 38 and jam density for a given data set The software uses a heuristic hill climbing technique to determine the optimum parameters The estimated capacity parameter is considered to b e the capacity estimate of the freeway segment which again, corresponds to the apex of the fitted function through the speed flow data plot. Capacity Estimation from Product Limit Method According to the PLM approach as per Brilon et al. (2005), one of th e key elements for the data analysis from PLM is to identify the breakdown events on the freeway segments. These events are typically identified by looking for sudden changes in traffic flow measurements or relationships such as speed, occupancy, and corr elation between volume and occupancy. In this study, it was decided to use average speed as the mechanism to identify traffic breakdown events. Speed based breakdown identification The breakdown events in this algorithm are identified using a speed thresho ld value. When the average speed drops below this threshold value for a specific period of time, a breakdown event is considered to have occurred. To find the speed threshold value for freeway segments, each study site is analyzed independently as per Elef teriadou and Lorenz (2001). According to the authors, for each of these study sites, the speed and vehicle count data are tabulated in one minute intervals for individual travel lanes and over all lanes. The vehicle count data are then expressed as equival ently hourly flow rates and the average speeds across all lanes are determined using the volume weighted average speed of all vehicles crossing the particular detector station. The speed and flow rate data are plotted in time series over each sample period with time on the x axis and the speed on the y axis. PAGE 39 39 To determine the breakdown condition or the transition state, the speed flow plots are plotted for a specific period of time for all the sites. It is recommended to examine the daily time series plot of speed rather than relying on scatter diagrams of many days of accumulated data. The first advantage in observing the daily traffic data is to obtain the relationship of speed with time, which the accumulated data cannot provide. The second advantage is th at inspection of the daily plots helps in identifying the points that represent transition between congested and uncongested flow so that one can observe a breakdown state. While analyzing the time series speed plot, a threshold or boundary value will exis t between the congested and uncongested regions. This threshold or boundary value of speed will be taken into account as an input for the PLM. This threshold value should be evaluated for all the study sites. In this study only the disturbances that will cause the average speed over all lanes to drop below the threshold value of speed for a certain period will be visualized to identify the breakdowns. A particular event will be considered as a true breakdown if and only if the average speed across all lane s drops below the threshold value for that particular time interval. Application of speed based threshold value method The speed threshold method is applied to each study site on the respective detector location for each day. On the basis of this method t he beginning of congestion times at study sites are identified and stored. If there is more than one event of congestion or breakdown event, the algorithm will run again and a second set of beginning of congestion times are stored separately in another dat abase. Once the beginning of the congestion periods or breakdown events is identified, the respective output for the particular breakdown is obtained. The output data are stored in a separate database which consists of the flow rates prior to the breakdown event with their respective B set values. These flow rates are determined on the basis of uncongested datasets as PAGE 40 40 described in the literature. This output which is referred as the uncongested flow datasets is developed according to the following steps: T he B set value of the flow rate just prior to the flow rate at which breakdown occurs is assigned a value of 1 and is referred as dataset {B}. All the other B set values of volumes are assigned a value of 0 and are referred as dataset {F}. All the flow rat es under the set {C2} are also obtained for the analysis and are eliminated from the flow rates in the set {F}. The final output from the above specifications becomes the main dataset for the remaining analysis. The output thus obtained is used for the pro babilistic modeling for developing the capacity distribution functions. The Weibull curves for all the study sites are developed from the maximum likelihood approach according to Elefteriadou et al. (2009). Brilon et al. (2005) refers to these Weibull curv es as the capacity distribution functions, F c ( q ) or the breakdown probability function. These functions give the probability of a breakdown event as a function of flow rate, on a freeway segment. The capacity distribution functions are first plotted for all the flow rates in B set. These plots are referred as PLM curves by Brilon et al. (2005) are determined according to Elefteriadou (2009). As per the methodology, the PLM curves fit the Weibull distribution and are referred as Weibull curves. For the dat a analysis, the fitting of the PLM curves to the Weibull distribution is checked by superimposing the two curves and the applicability of this assumption is validated. Both the PLM curves and the capacity distribution functions are then superimposed with t he speed flow curves as described in Geistefeldt (2008). The flow sets in the speed flow curves are plotted to their respective speed values. Only the flows from the {F} set or the uncongested flows are considered for these curves. After the plots are su perimposed with each other the capacity values are estimated from these graphs. Different approaches are performed to estimate the capacity values from the PAGE 41 41 superimposed graphs. One of the approaches as given by Brilon et al. (2005) is to calculate the exp ected mean s of all the Weibull distribution curve s obtained from analysis of all the study sites from Equation 2 12. This mean value is considered to be one of the capacity values for freeway segments. However, two different approach es are developed in thi s research study to estimate the capacity values. After the superimposition of these curves, a probability value is chosen from the capacity distribution function that corresponds to a flow rate, determined from the two approaches. On the basis of these sp ecific flow values, the range of the probability values are obtained for all the sites and a suitable probability value is chosen for the capacity estimation. This suitable probability value will be traced back from all the capacity distribution functions onto the speed flow curves to estimate the respective capacity for all the sites. The capacity values as estimated by the expected mean and by selecting a suitable probability value are compared with the capacity estimates from the VAM and the values as pr ovided by the HCM (2000). The following section discusses the applicability of the PLM on the basic freeway segments. Capacity E stimat ed by A verage of M aximum F low R ates Apart from the Van Aerde Model and the PLM, a simple methodology was developed to esti mate the capacity values. This method is independent of complex mathematical functions and breakdown events However it is dependent on the analysis period time over which this methodology is implemented. The average maximum flow rate analysis focus es only on the highest flows observed over a period of time at a particular detector location for given any segment Capacity values were estimated from the average maximum flow rate method according to two different approaches. For the application of these appro aches, the first step is to aggregate the flow rate data into five minute intervals. The second step is to convert the five minute flow rate data to hourly flow PAGE 42 42 rates The third step is to sort the data in from highest fl ow rates to lowest flow rates. The remaining steps are specific to each of the two approaches used for the average maximum flow rate analysis First, the average of top x % highest flows is taken such that the average value equals the capacity estimates as compared with Van Aerde Model and P LM estimation method s The consistency and the range of the x values are observed and a particular x value is chosen which would fairly give the capacity estimates as per Van Aerde Model and PLM. Second, the average of all highest flows is taken above a fl ow rate which is a certain x % of the highest flow observed for the respective site. Similarly, on the basis of the consistency and the range of x values in this approach, a particular x value is chosen which would give the capacity estimates as per Van Aer de Model and PLM. While selecting a particular x value for these approaches, the percentage error in the difference of the capacity estimates from average analysis with Van Aerde Model and PLM is least. It is also required to find a minimum analysis period for which these approaches are applicable. The identification of the minimum analysis period for this approach varies in several ways. These approaches are applied for any given dataset for a certain period of time. The initial testing analysis periods fo from two different approaches are obtained under each testing analysis period. These percent age values are compared with each other under a particular site for different analysis periods and the percentage difference is obtained. The consistency of the percentage values are observed over all the analysis period and on the basis of the minimum err or to the percentage values, the minimum analysis period on which these approaches can be performed is obtained. PAGE 43 43 The methodology and the steps as explained above for the Van Aerde Model, the PLM and the average maximum flow rate analysis are implemented o n the selected sites. The following section discusses data collection and data processing Data Collection This section describes the procedure used to identify all suitable field sites for capacity analysis, the sources from where the data were obtained f or these sites, the procedure to prepare the data, process the data and analyze the data using different methods. This section also explains the way to implement the methodology and to analyze the data on the selected sites. Site Selection Since the study focuses only on basic freeway segments, only freeway segments were finalized which are considered as basic freeway segments. For the purposes of this study, a basic freeway segment was considered to be a freeway segment that was at least 1500 ft in length, had a constant number of mainline lanes throughout this segment, whose operations were are not affected by the merge and diverge areas However, the freeway segments with unequal number of lanes before the on ramp and after the on ramps are also considere d as basic freeway segments provided the length of the freeway segment was at least 1500 ft. The freeway segments with consecutive on ramps were also considered for the study analysis. On the basis of these characteristics, three types of freeway segments were accepted as the basic freeway segments that are provided in Figure 3 1. Also, all the freeway segments with HOT or HOV lanes were excluded, and sites with significant vertical or horizontal geometry were excluded. The next step is to identify the desi red sites for the districts of Jacksonville, Ft. Lauderdale, Orlando, Miami and Tampa. An extensive study for all the freeway segments was performed to select a number of desired sites on the basis of some criteria. As the main objective for this study was to compare methods for freeway capacity estimation, it is necessary that the candidate sites PAGE 44 44 regularly experience high flow rates, and more preferably, that experience recurrent congestion due to high traffic demands. This was identified by calculating th e average per lane volume for most of the basic freeway segments in Florida. All the freeway segments which had reasonable high flows were considered for the data analysis for this research. However, the final sites were selected on the basis of the data a vailable. The candidate sites are selected on the basis of a review of maximum recorded flow rates on interstates across Florida. The data source used for these maximum flow rates was the Florida Data Highway DVD (2006) and Florida Traffic Information CD ( 2006) As the flow rates, available in the FDOT traffic DVD, are aggregated in hour intervals, this data aggregation level is not appropriate for the determination of capacity values. Ideally, the data at one or five minute of aggregation level are necessa ry for capacity estimation. Therefore, the data for the research study was obtained from the STEWARD (Statewide Transportation Engineering Warehouse for Archived Regional Data). Data Source As the scope of this study is limited Florida, it is required to o btain the data for all the potential sites. All the freeway segments across the state are initially considered for this study. The primary concern for data collection on these sites was the availability of useful and non erroneous data from the sources ava ilable. The availability of data was checked for all the freeways in Florida and within the scope of this research, five districts namely: District 2 (Jacksonville), District 4 (Ft. Lauderdale), District 5 (Orlando/Daytona), District 6 (Miami) and District 7 (Tampa) were chosen for the data analysis. The traffic data were obtained from STEWARD (Statewide Transportation Engineering Warehouse for Archived Regional Data), a website developed at University of Florida sponsored by Florida Department of Transport ation and its database server, for districts of Jacksonville, Ft. PAGE 45 45 Lauderdale, Orlando and Miami. The data are obtained for all the detector locations in these regional areas. All the detectors in these areas are found to be RTMS (Remote Traffic Microwave S ensor) detectors. The detectors at these locations are assumed to be properly functional. The traffic data including the volume, speed, occupancy, etc., are collected by these detectors for all the districts. A comprehensive list of other data parameters t hat should be collected at each detector location are also developed on the basis of the literature review. The list included elements that previous research found to affect capacity or factors that may affect capacity on a freeway. These factors are liste d below: Length of the basic freeway segments for all the sites Distance of the on ramps and off ramps from the basic segment Speed Limits on the freeways Terrain or grade of each freeway (%) Presence of any horizontal curvature on the site Volumes, speeds and occupancy by lane for a time period of 1 minute Percent heavy vehicles The data for all these districts are obtained from July 2007 to January 2009. However, the final data used and analyzed for each district is different depending on whether the dat a are reasonable to use. Apart from the traffic data, the detector configuration files are also obtained from the same source. The data available from STEWARD are then prepared in a compatible format for the data analysis and are converted into one minute data interval format for all the sites. The final datasets for the analysis ar e made available in one minute interval s Table 3 1 provides a typical format of the data available after data processing. Once the data are obtained, the final sites are selecte d for the analysis on the basis of the availability of the data. The final sites are selected on the basis of various points discussed in previous sections and are listed in Table 3 2 Figure s 3 2 to 3 19 provide the Google Earth images for all the selecte d PAGE 46 46 sites with their respective site ID. The next section describes the utility programs developed to process the collected data on the selected sites. Data Processing T o analyze the data for the PLM, two process ing programs were developed : the Capacity Data Processor and the Downstream Location Breakdown Identifier The details and various limitations to these processors are described in the following section: Capacity Data Processor The Capacity Data Processor (CDP) utility program is used to identify the breakdown events from a file of traffic data for a specific site The processor is developed based on several breakdown identification algorithms with a specific criterion through which the data can be analyzed to identify the breakdown events. The algorit hms contained in the CDP are explained in the section that describes the Product Limit Method O ne minute aggregated interval data are used with the CDP Some of the parameters, inputs and other elements which are used for the analysis from this processor are discussed next. Figure 3 20 provides a snapshot of the CDP The important elements used for the analysis using the Capacity Data Processor are the analysis time period for a given day, the algorithm method for identification of a breakdown event, the r espective threshold value, the number of intervals preceding the breakdown, the analysis method and the recovery time period for a breakdown event. These elements used for the data analysis are described next with the respective values used. Analysis perio d The analysis period for the data processor is user specific. It depends upon the user and the day of time, the user wants to apply the PLM. The ongoing research uses the data for each day from 5 AM to 10 PM. The reason to select such a period is to elimi nate the other non significant breakdown events where the traffic counts are very less. It is assumed that within this specific PAGE 47 47 time period, all the breakdown events which have traffic flows comparable to capacity values are considered. Speed threshold val ue The speed threshold value and the speed reduction time period are the key element s for the identification of breakdown events After analyzing the time series speed plots for the sites for a given period, the threshold value is determined. This value is considered as the threshold value and is used as an input for the capacity data analysis. The speed threshold value determines the congested flow intervals and the un congested flow intervals. From the speed threshold value, a breakdown is identified if th e speed drops below this threshold value for a period of five minutes. Thus, for a breakdown event to be identified, the average speed (volume weighted) must remain below the speed threshold value for a period of five minutes. Also, the breakdown event is recovered if the speed increases above this threshold value for a period of five minutes. Thus, for traffic flow to be considered to have recovered from a breakdown event (i.e., return to uncongested flow), the average speed (volume weighted) must remain a bove the speed threshold value for a period of five minutes. However, the number of intervals at each breakdown event is also an important measure of this methodology which is discussed next. Intervals preceding breakdown According to Brilon et al. (2005), all the time intervals preceding the breakdown from the start time of the data analysis are considered for the data analysis in PLM. If multiple breakdown events occur at the selected site for a particular day, the intervals after a breakdown event was re covered were considered till the next breakdown event is identified. These parameters are thus used as inputs to CDP to obtain the flow rates and the respective B set values for rest of the PLM analysis. PAGE 48 48 Data i mputation In some cases, the traffic flow data for a site has a missing entry for one or more time periods As the prepared data are available in one minute aggregations the data is read by the processor minute by minute. If a single one minute period of data is missing (i.e., non consecutive interva ls) in the input data, the CDP will impute that entry as the average of the next data entry and the previous data entry. In cases where consecutive time periods of data are missing, imputation does not take place and these time periods are ignored for pur poses of analysis. Downstream Breakdown Identifier Locations that experience congestion due to a downstream bottleneck or incident are n ot considered in the analysis. Thus, it is necessary to determine if the breakdown at a site is result of downstream con gestion that has propagated upstream. T he CDP identifies breakdown events regardless of where the congestion initiated. The Downstream Breakdown Identifier (DBI) utility program identifies breakdown events that resulted from downstream congestion This uti lity program requires only two inputs : 1) the PLM data file as obtained from CDP for the desired basic freeway segment location and 2) the results file from the capacity data processor for the detector at the downstream location. The PLM data at the basic freeway segment location is filtered on the basis of the following considerations: First, if the breakdown event at an downstream location occurs first as compared to that at the upstream detector location, the flow values at the upstream location are dis carded from the start time of the breakdown event at the downstream event till that event is recovered. Second, if the breakdown events at both locations occur at the same time, then the flow value preceding the breakdown event at the upstream location is discarded. PAGE 49 49 The output flow values, speed values and B set values from this processor are further used for the PLM analysis. The data is checked and the PLM and Weibull curves are determined to estimate the capacity value as pe r the methodology. Figure 3 21 provides a snapshot of the PLM data processor used for this analysis. PAGE 50 50 A) B) C) Figure 3 1. Acceptable freeway s egment c onfigurations A) Freeway segment with c onstant number of mainline lanes throughout the segment, B) Freeway segments with unequal number of lanes before the on ramp and after the on ramp with atleast 1500 ft in length and C) Freeway segment with consecutive on ramps with at least 1500ft in lengt h PAGE 51 51 Figure 3 2 Snapshot of site T1 (Source: Google Earth) Figure 3 3 Snapshot of site T2 (Source: Google Earth) PAGE 52 52 Figure 3 4 Snapshot of site T3 (Source: Google Earth) Figure 3 5 Snapshot of site T4 (Source: Google Earth) PAGE 53 53 Figure 3 6 Snapshot of site T5 (Source: Google Earth) Figure 3 7 Snapshot of site T6 (Source: Google Earth) PAGE 54 54 Figure 3 8 Snapshot of site T7 (Source: Google Earth) Figure 3 9 Snapshot of site T8 (Source: Google Earth) PAGE 55 55 Figure 3 10 Snapshot o f site T9 (Source: Google Earth) Figure 3 11 Snapshot of site T10 (Source: Google Earth) PAGE 56 56 Figure 3 12 Snapshot of site T11 (Source: Google Earth) Figure 3 13 Snapshot of site F1 (Source: Google Earth) PAGE 57 57 Figure 3 14 Snapshot of site F 2 (Source: Google Earth) Figure 3 15 Snapshot of site F3 (Source: Google Earth) PAGE 58 58 Figure 3 16 Snapshot of site F4 (Source: Google Earth) Figure 3 17 Snapshot of site F5 (Source: Google Earth) PAGE 59 59 Figure 3 18 Snapshot of site F6 (Source: Google Earth) Figure 3 19 Snapshot of site FV1 (Source: Google Earth) PAGE 60 60 Figure 3 20 Capacity Data Processor utility program user interface PAGE 61 61 Figure 3 2 1 Downstream Breakdown Identif i er utility program user interface PAGE 62 62 Table 3 1 Data format for the data obtained from data source Date Time CDWID FwySpd FwyVol FwyOcc SpdCV VolRatio SpdRatio 11/1/2007 0:00:00 210371 58.58 12 1.8 0.71 1.67 1.04 11/1/2007 0:01:00 210371 61.86 7 1.5 2.12 0 0 11/1/2007 0:02:00 210371 58.31 16 2.2 0.71 2.33 1.04 11/1/2007 0:03:00 210371 59 14 3.8 0 4 1.22 11/1/2007 0:04:00 210371 59.75 16 3.2 3.54 2.67 1.12 11/1/2007 0:05:00 210371 60.14 14 2.6 0 1.25 1.07 11/1/2007 0:06:00 210371 60.44 18 4.3 0.71 1.4 1.06 11/1/2007 0:07:00 210371 60.12 17 2.4 2.83 1.75 1.15 11/1/2007 0:08:00 210371 60.27 11 2 3.54 1.33 1.16 11/1/2007 0:09:00 210371 62.38 21 5.2 3.54 1.33 1.15 11/1/2007 0:10:00 210371 62.75 16 1.8 1.41 3.5 1.08 11/1/2007 0:11:00 210371 63.38 16 1.8 4.24 2.67 1.14 11/1/2007 0:12:00 210371 65.5 10 2 4.24 5 1.19 11/1/2007 0:13:00 210371 62.06 16 2.5 2.12 1.2 1.09 11/1/2007 0:14:00 210371 67.8 15 2 6.36 2.67 1.11 11/1/2007 0:15:00 210371 66.47 15 2 1.41 1.5 1.06 11/1/2007 0:16:00 210371 64.63 8 1.5 0.71 2 1.11 11/1/2007 0:17:00 210371 65.46 13 2 2.83 1.25 1.13 11/1/2007 0:18:00 210371 68.62 13 2.7 0 3 1.21 11/1/2007 0:19:00 210371 64.88 16 3.2 0.71 1.5 1.24 11/1/2007 0:20:00 210371 63.89 18 2.5 3.54 1.4 1.21 11/1/2007 0:21:00 210371 64.18 11 2 0 5 1.25 11/1/2007 0:22:00 210371 65.07 15 2.2 3.54 1.5 1. 35 11/1/2007 0:23:00 210371 61.68 22 3 7.07 2 1.23 11/1/2007 0:24:00 210371 63.57 14 1.8 1.41 2.33 1.09 11/1/2007 0:25:00 210371 65.38 13 4 2.12 6 1.1 11/1/2007 0:26:00 210371 63.74 19 2.7 0.71 2.67 1.04 11/1/2007 0:27:00 210371 62.21 14 2.2 2.12 1.5 1.07 11/1/2007 0:28:00 210371 61 6 1.8 0 0 0 11/1/2007 0:29:00 210371 62.55 11 2.8 0.71 1.33 1.1 11/1/2007 0:30:00 210371 63.56 16 4.6 0.71 2.67 1.04 PAGE 63 63 Table 3 2 Final selected sites and site description Site ID Site Description and Location Directio n Freewa y Speed 1 Data 2 Type 3 (mph) 3 Lanes T1 East of NW 57 Avenue EB SR 826 55 Aug08 Jan09 A T2 East of NW 57 Avenue WB SR 826 55 Aug08 Jan09 A T3 East of NW 67 Avenue EB SR 826 55 Aug08 Jan09 A T4 East of NW 47 Avenue EB SR 826 55 Aug08 Jan09 A T5 East of Wymore Rd WB I 4 55 Mar08 Nov08 A T6 East of SR 436 EB I 4 55 Mar08 Nov08 A T7 East of Wymore Rd EB I 4 55 Mar08 Nov08 A T8 Between Baymeadows and Butler Blvd NB I 95 70 Jul07 Feb08 A T9 North of Butler Blvd NB I 95 70 Jul07 Feb08 A T10 South of Spring Glen Road NB I 95 70 Jul07 Feb08 A T11 Between Old St. Augustine Rd and I 95 NB I 295 70 Jul0 8 Feb0 9 A 4 Lanes F1 South of NW 170 ST NB I 75 70 Aug08 Jan09 A F2 East of NW 27 Avenue NB SR 826 55 Aug08 Jan0 9 B F3 West of Central Florida Parkway EB I 4 70 Mar08 Nov08 A F4 At Kennedy Blvd WB I 4 55 Mar08 Nov08 C F5 North of Philips Highway NB I 95 70 Jul07 Feb08 B F6 North of I 295S NB I 95 70 Jul07 Feb08 B 5 Lanes FV1 At SW 26th Terrace/New River WB I 595 70 Aug08 Jan09 C 1 Speed l imit, 2 Data available for analysis, 3 Type of basic freeway s egment PAGE 64 64 CHAPTER 4 DATA ANALYSIS AND RE SULTS This chapter describes the application of the analysis methods and the respective results. The analysis methods that are used to estimate the capacity values are termed as VAM capacity estimation from the Van Aerde model, stochastic capacity estimation from the PLM, and average maximum flow rate estimation method. In the end, the capacity values from all the three methods are provided, followed by comparisons of capacity estimates from these three capacity estimation methods with the capacity values provided by HCM. VAM Capacity Estimation The VAM capacity estimation method is implemented through the Traffic Stream Calibration Software, SPD_CAL.exe, by Rakha (2007). As described in the methodology, it is an iterative heuristic procedure which calibrates the parameters for the Van Aerde model (1995). The parameters that are obtained from this program, on the basis of the traffic data input file, are the free flow speed, speed at capacity, capacity, and jam density. The inputs to this program are the flow rates in veh/h, the speeds in km/h and the density in veh/km. For all the selected sites across Florida, the input f iles with the flow rates, speeds and density were prepared. The data at the basic freeway segment detector location is used for creating these files. The data used for running this analysis method is similar to that used for the stochastic capacity estimat ion method. To avoid erroneous data entries in the datasets, all flow values more than 2700 veh/h/ln were eliminated. It should be noted that the minimum flow rate for this input file was considered to be 100 veh/h/ln, and the minimum speed was considered to be 12 mi/h, based on guidance from the software program. Data entries with flow rates less than 100 veh/h/ln were deleted from the datasets. The Van Aerde model curves fitted function to speed flow data points for two of the randomly selected study site s are provided in Figures 4 1 PAGE 65 65 and 4 2. The calibrated parameters obtained from the Traffic Stream Calibration program, for all sites, are shown in Table 4 1. Stochastic Capacity Estimation This section describes the various steps performed to implement the second analysis method, the stochastic capacity estimation method, to estimate the capacity values. First, the applicability of the PLM on basic freeway segments is described, followed by a description of the various steps performed for this analysis meth od. Applicability of PLM Brilon et al. (2005) used the PLM approach for a large number of sites on German freeways. However, the PLM was applied only at sites that had a bottleneck resulting from a lane drop. Brilon et al. (2005) did not apply the PLM at s ites where breakdown was the result of friction created at on ramp merges. The applicability of the PLM depends on whether a congestion event can occur at a basic freeway segment detector location Therefore, a small part of the large datasets for two site s was analyzed to check the applicability of the PLM For each site, breakdown events were observed at all the detector locations along the freeway segment. B reakdown events were identified for the detector station at the location of interest and the detec tor station immediately downstream of the location of interest for each day of the selected month It was observed that in most cases, the breakdown at the downstream of location of interest occurred first and in few cases, the breakdown at the location o f interest occurred first. For the cases where the breakdown occurred first at the location of interest, the incident data, obtained from Florida Highway Patrol, were checked and were found not to affect the breakdown at the location of interest. This smal l experiment shows that the breakdown can occur at the location of interest and not due to an incident or by the spillback of the queue from the downstream location. In contrast PAGE 66 66 to the PLM applied by Brilon et al. (2005), only those sites were considered t hat had a lane drop and the study sites at basic freeway segments were not considered. Therefore, it can be concluded that the PLM is applicable to basic freeway segments for estimation of capacity values. The description of the various steps for this anal ysis is provided next. These steps are also provided in Elefteriadou et al. (2009), with any significant deviations as implemented for this study denoted. Determination of Speed Threshold Values As the PLM is applicable to basic freeway segments, the first step in applying the stochastic capacity estimation method is to determine the speed threshold value. This value is one of the key inputs to the capacity data processor. The capacity of the freeway segments is estimated by identifying the breakdown events at the location of interest or at the basic freeway segments. These breakdown events are defined as breakdown events occurring at the location of interest, and not due to a breakdown at a downstream location. To identify these breakdowns at location of in terest, the breakdown events at both the location of interest and downstream location were identified. The speed threshold values were determined for both locations on the freeway segment. The speed threshold value for both locations is determined separate ly for each freeway segment. The threshold value is determined on the basis of a speed time series plot where speed and vehicle count are tabulated in 1 minute intervals for the selected site. The vehicle count is then expressed as an equivalent hourly flo w rate and the average speed across all lanes is determined using the volume weighted average speed of all vehicles crossing the particular detector station. The speed and flow rate data are plotted in time series over a specific period of time. After exte nsive examination of the available data, one month of data were used for the time series plots. For these plots, time is displayed on the x axis and speed on y axis. The speed threshold is PAGE 67 67 then determined visually, on the basis of speed drop for a period o f five minutes. If the average speed across the freeway section drops below this threshold value for a five minute of period, a breakdown event will occur. In Elefteriadou et al (2009), a breakdown event is identified at a detector location if the speed dr ops by 10 mi/h for ten minutes. The speed time series plot for two of the selected sites at its upstream locations are shown in Figure 4 3 and Figure 4 4 Table 4 2 summarizes the speed threshold values used for the analysis at the upstream and downstream locations. Identification of Breakdown Events After the speed threshold values for all the sites are determined, the next step is to load the prepared data into the CDP for identification of the breakdown events. The speed threshold values as obtained from the speed time series plots are used as an input to the CDP. All the breakdown events at the desired location and at the immediate downstream location are identified. These results are then loaded into the DBI, which filters out all the breakdown events a t the desired location that were due to a breakdown event at the downstream location. The next step is to check for erroneous data in the output file obtained from the DBI. All flow rates more than 2700 veh/h/ln are excluded from the data file to obtain an other dataset. In Elefteriadou (2009), all flow rates more than 3000 veh/h/ln are excluded. These erroneous data are usually a result of detector malfunctions and are considered as the outliers in the dataset. Also, all the flow rate values preceding a bre akdown event, or in the B set, are checked for any flow rates less than 1000 veh/h/ln are excluded from the datasets. At these flow rate values, the breakdown events are believed to have occurred due to an incident at the freeway segment. The final dataset obtained after excluding the erroneous data contained only the flow rates from the uncongested traffic flow regime. These datasets are used then for the remainder of the analysis. PAGE 68 68 PLM and Speed Flow Curves The PLM is then applied to the datasets obtained from the previous step. This method gives the PLM curves considered to be the capacity distribution function. According to Brilon et al. (2005), it was observed that the Weibull distribution provides the best fit of the PLM curves. Figure 4 3 and Figure 4 4 show that the Weibull distribution curves provide a good fit to the PLM curves, for two sites selected randomly from all 18 analysis sites. Again, the Weibull distribution is considered to be the capacity distribution function. The Weibull parameters are then estimated, for all sites, with the log likelihood estimation method. After the PLM and Weibull curves are determined, the speed flow data points are also plotted. It should be noted that only uncongested flows are used to plot the speed flow data poi nts. All three plots, i.e., PLM curves, Weibull distribution curves and speed flow data points are then superimposed with each other in a single graph as provided in Figure 4 5 and Figure 4 6 for all the sites as per Geistefeldt (2008). The left y axis rep resents the speed values in mi/h, the right y axis represents the breakdown probability for the Weibull distribution, and the x axis represents the flow rate in veh/h/ln. Estimating Capacity Values The next step in the data analysis is to superimpose only the speed flow data points and the Weibull curves. After the superimposition of these two plots, there is a need to find an appropriate value of the breakdown probability from the capacity distribution function, F ( q ) for all the selected sites. A probabili ty value from the distribution function is chosen which would reasonably represent the occurrence of a breakdown event and would best estimate the capacity for the selected freeway segment. The different approaches that can be used to estimate the capacity values from the breakdown probability function are described next. PAGE 69 69 First, a capacity value can be identified by visually selecting a point from the plot of speed flow data points that corresponds to the highest flow rate that is within a critical mass of data points (higher flow rates that are within a sparse area of the speed flow plot are not considered). However, the results from this method can be significantly biased for a couple of reasons: 1) the size of the data points and resolution of the plot ca n affect how dense an area of plotted points this approach is not recommended. Second, a probability value can be chosen from the capacity distribution fun ction that corresponds to the capacity estimates as determined from the VAM capacity estimation method. Third, for all the selected sites, an average can be taken of all the maximum flow rates that occur within ten minutes of breakdown. The average of all these maximum flow rates is calculated for the respective study sites to find a breakdown probability that corresponds to the average maximum flow rate. However, only flow rates greater than 1500 veh/h/ln were considered for this analysis. This flow rate i s termed as the maximum pre breakdown flow rate. The breakdown probability values from these two approaches, VAM capacity estimation and maximum pre breakdown flow rate, were determined and are represented by F c ( q ). Table 4 3 provides the F c ( q ) values for all the sites for each approach. The next step in the data analysis was to determine an appropriate breakdown probability value that would provide a reasonable estimate of the capacity for the analysis sites. To find this appropriate value, the average of all the F c ( q ) values under the respective approach was calculated. The range of the values for F c ( q ) under the VAM capacity estimate approach was found to be 0.7% to 10.67% and the respective average of all values was 3.98%. Similarly, the range of the val ues for F c ( q ) under the maximum pre breakdown flow rate approach was found to be 0.24% to 9.09%, and the respective average of all the values was 3.27%. Geistefeldt (2008) PAGE 70 70 used the average breakdown probability value under F c ( q ) to determines the capacity for a site. However, a value of 4% or 4 th percentile is used for this study to estimate the capacity values for all the selected sites. The flow values are then determined after the 5 th percentile value is traced back onto the speed flow data points. This flow is taken as the capacity estimate for the respective site from the stochastic capacity estimation method. Table 4 3 gives the F c ( q ) values and the capacity estimates for all the analysis sites, based on a 4 th percentile of the capacity distribution fu nction. Another approach to estimate the capacity is to determine the 50 th percentile F c ( q ) value as per Brilon et al. (2005). The capacity estimates for the 50 th percentile (i.e., mean value) of all the Weibull curves are also provided in Table 4 3. The g raphs for the capacity estimation from PLM for all analysis sites are shown in Figure 4 7 to Figure 4 2 4 with the speed flow data points plots for uncongested flow rates, speed flow data points for all the observed data points and the Weibull curves. The n ext section describes the third, and last, analysis method used for estimating capacity values. Average Maximum Flow Rate Capacity Estimation Method The average maximum flow rate capacity estimation is a simple method that was developed to estimate capacit y values without the complications of identifying breakdowns and/or estimating complex mathematical functions. Two variants of this approach were tested and are discussed below. The first step was to aggregate the flow rate data into five minute intervals. The datasets used for this analysis are for the detector station at the location of interest of that freeway segment. All the available data sets at the respective detector station location were considered for this analysis method. The second step was to convert the five minute flow rate data to hourly PAGE 71 71 flow rates (i.e., multiply by 12). The third step was to sort the data from highest flow rates to lowest flow rates. The remaining steps are specific to each of the two variants of this approach. For the app lication of the first variant, the average of the top 3% and the top 5% of the highest flow rates were taken. These average flow rates are taken as the capacity estimates for the freeway segments. These capacity values were compared with the capacity value s obtained from VAM capacity estimation and stochastic capacity estimation. Table 4 4 tabulates the values of the top 3% highest flows, the top 5% highest flows, the % error in difference of capacity estimates in comparison with VAM capacity estimates and PLM capacity estimates and the average number of breakdowns per day. On the basis of the error in difference in the respective capacity values for a particular site, it was observed that the average of the top 5% highest flow rates gives capacity estimates with less error as compared with VAM capacity estimates and PLM capacity estimates. In the second variant of this approach, the analysis performed is based on the maximum flow rate observed for the detector station at the location of interest for the free way segment. The aim of this approach is to calculate the average of all flow rates within a certain percentage of the maximum flow rate observed for the freeway segment of interest. The first step performed in this approach is to find flow rate below the maximum flow rate, such that the average of all flow rates between this maximum flow rate and lower flow rate will match the capacity estimates from the VAM capacity estimation method and stochastic capacity estimation method. The percent value of the maxi mum flow rate that equals the lower flow rate is then calculated. The percentage values from this analysis with the capacity estimates from the VAM capacity estimation method and the stochastic capacity estimation method are tabulated in Table 4 5. The PAGE 72 72 ave rage of these percentage values for the VAM capacity estimation method and the stochastic capacity estimation value were found to be 34.89% and 30.29%, respectively. The percentage values for all the selected sites were compared to each other and on the ba sis of the average of percentage values, the analysis was performed on the basis of 30% and 35% of the maximum flow rate observed for the freeway segment. The respective lower value that corresponds to 30% and 35% of the maximum flow rate was calculated. T he average of all flow rates that are within the 30% and 35% of the maximum flow rate is calculated and is taken as the capacity of the freeway facility. The percentage difference between these capacity estimates and those from the VAM capacity estimation method and stochastic capacity estimation method were calculated. Table 4 6 tabulates the lower flow rate that corresponds to the 30% and 35% of the maximum flow rate observed on the freeway segment, the capacity estimates on the basis of these two lower f low rate, the percentage difference of these capacity estimates with the capacity estimates from the VAM capacity estimation method and stocha stic capacity estimation method and the average number of breakdowns per day at each analysis site. The next step in this approach was to find the minimum time period for which this approach can be implemented. The time periods considered for this step were: three months, two months, one month, and two weeks of data for each analysis site. The start and end dates for each of the time periods were randomly selected from the available datasets, but generally avoiding major holiday periods. This analysis was also based on five minute flow rates converted to hourly flow rates. For each analysis site, and for each time peri od, the average of the top x % of the highest flow rates was calculated. The value of x was determined based on the capacity estimate obtained from the VAM capacity e stimation method for the site. In other words, a percentile value was chosen such that the average of these top x % flow rates matched with the PAGE 73 73 VAM estimated capacity value. The x value was also obtained relative to the stochastic capacity estimation method results The x values obtained were determined for all the time periods and were compared to each other. The percentage differences in the x values for different selected analysis periods were also calculated to check the consistency of datasets. Table 4 7 and Table 4 8 provides all the x values for each analysis period and the percentage diffe rence in x values as compared in different analysis periods. Comparisons and Results This section compares the capacity values provided by the HCM (2000) with the capacity estimates obtained from the analysis methods used in this study. The capacity values from the HCM (2000) are dependent on the free flow speed of the freeway segment, so the capacity value for a freeway segment with a free flow speed of 70 mi/h is given as 2400 pc/h/ln, and the capacity value for a free flow speed of 55 mi/h is given as 22 50 pc/h/ln. The capacity estimates from HCM (2000), VAM capacity estimation method, stochastic capacity estimation method and the average maximum flow rate analysis are presented in Table 4 9. It should be noted that the capacity values provided in the HCM (2000) are in passenger cars per hour per lane but for this research study, due to limited data on truck percentages flow rates were not adjusted for heavy vehicles and are estimated in vehicles per hour per lane. It was observed that the capacity value s provided by the HCM (2000) are higher than the capacity values estimated from the other thr ee analysis methods. From the stochastic analysis, the average of breakdown probabilities from the capacity distribution function that corresponds to VAM capacity estim ates and maximum flow rate within ten minutes of breakdown was found to be 3.98 % and 3.27%. The se averages closely resemble the average design breakdown pro bability provided by Geistefeld t (2008) which is 3%. Also, the range of breakdown probability values were observed to be between 1% and 10%, that were comparable to the range PAGE 74 74 of breakdown probability values, 0.6% to 5.7%, provided by Gesitefeld t (2008). Therefore, a 4 th percentile value of the capacity distribution functions which is the average o f the breakdown probabilities rounded up to the nearest integer percentage, is used to estimate the capacity values for the basic freeway segments. The capacity estimates at the 4 th percentile value were found to be lower in comparison with the values prov ided by HCM (2000). On the other hand, it is observed from the other definition of capacity i.e., the expected mean value or at the 50 th percentile of the capacity distribution function, the capacity estimates were higher as compared to capacity values giv en in HCM (2000). I n the maxi mum average flow rate analysis that from different averaging schemes, the capacity estimates were found to be lower than the capacity values given in HCM (2 000). In the first approach it was observed that the average of the to p 5% of highest flow rates were found to provide capacity estimates with less error over the average of top 3% highest flow rates when compared with capacity estimates from the stochastic capacity estimation method and capacity values given in HCM (2000). Similarly, in the second approach, it was observed that the capacity estimates from the average of highest flow rates above the flow rate that corresponds to the 35% of the maximum flow rate observed for a freeway facility, were found to have less error ov er the average of highest flow rates above the flow rate that corresponds to the 30% of the maximum flow rate when compared with capacity estimates from VAM capacity estimation method stocha stic capacity estimation method and the capacity values given in HCM (2000). The errors in the difference of capacity estimates between average maximum flow rate analysis and stochastic capacity estimates were found to be higher for sites with fewer number of breakdown events per day. It was observed that if the number of breakdown events per day is less than 0.5, the capacity estimates from the stochastic capacity analysis method were very high PAGE 75 75 as compared with capacity estimates from the average maximum flow rate analysis. Also, it was observed that capacity estimates from the stochastic analysis on one of the selected sites with 0.04 breakdown events per day was found to be high er than HCM (2000) capacity value Therefore, with fewer breakdown events per day the stochastic analysis and the average maximum flow rate ana lysis may not provide realistic capacity estimates. To find the minimum analysis period, over which the average maximum flow rate analysis is applicable, it was observed that the data remained consisten t of between analysis period s of two and three months The consistency of data was not maintained when comparisons were made for the other time durati ons The inconsistencies observed from this approach might be due to inclusion of a high flow rate period within the analysis period. Therefore, the consistenc y of the averaging schemes is maintained only for data sets of at least two months in duration. PAGE 76 76 Figure 4 1. Van Aerde Model fit to the speed flow data points for s ite ID: T1 Figure 4 2. Van Aerde Model fit to speed flow points for s ite ID: T8 PAGE 77 77 Figure 4 3 Speed time series plot for site T4 Figur e 4 4 Speed time series plot for site T7 PAGE 78 78 Figure 4 5 Weibull curve fit with PLM curve, Site ID: 210391, NB North of Butler Blvd on I 95, District 2 Figure 4 6 Weibull curve fit with PLM c urve Site ID: 610271, East of NW 57 Avenue on SR 826, District 6 PAGE 79 79 Figure 4 7 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T1 Figure 4 8 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T2 PAGE 80 80 Figure 4 9 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T3 Figure 4 10 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T4 PAGE 81 81 Figure 4 11 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T5 Figure 4 12 Speed Flow, Weibull and Van Aerde Mod el curves for s ite ID T6 PAGE 82 82 Figure 4 13 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T7 Figure 4 1 4 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T8 PAGE 83 83 Figur e 4 15 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T9 Figure 4 16 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T10 PAGE 84 84 Figure 4 17 Speed Flow, Weibull and Van Aerde Model curves for s ite ID T11 Figure 4 18 Speed Flow, Weibull and Van Aerde Model curves for S ite ID F1 PAGE 85 85 Figure 4 19 Speed Flow, Weibull and Van Aerde Model curves for s ite ID F2 Figure 4 20 Speed Flow, Weibull and Van Aerde Model curves for s ite ID F3 PAGE 86 86 Figure 4 21 Speed Flow, Weibull and Van Aerde Model curves for s ite ID F4 Fi gure 4 22 Speed Flow Weibull and Van Aerde Model curves for s ite ID F5 PAGE 87 87 Figure 4 2 3 Speed Flow, Weibull and Van Aerde Model curves for s ite ID F6 Figure 4 2 4 Speed Flow, Weibull and Van Aerde Model curves for s ite ID FV1 PAGE 88 88 Table 4 1. Capacity estimates and other p ar ameters from V AM capacity estimation method Site ID Lanes Capacity Free Flow Speed Speed at Capacity Jam Density (veh/h/ln) (mi/h) (mi/h) (veh/mi) T1 3 1693 60.10 44.50 256.47 T2 3 1790 59.23 40.40 229.28 T3 3 1632 61.59 36.67 273.05 T4 3 1746 73 .83 58.36 247.95 T5 3 2024 54.88 43.44 144.81 T6 3 2082 56.12 43.51 144.81 T7 3 2021 55.81 44.50 144.81 T8 3 1897 66.63 53.08 172.00 T9 3 1850 65.38 58.30 169.43 T10 3 1743 68.55 64.08 171.36 T11 3 2027 66.19 58.73 194.53 F1 4 1678 64.51 53.82 189. 70 F2 4 1729 63.21 56.43 193.40 F3 4 1692 55.44 41.89 228.64 F4 4 1905 59.11 45.56 174.90 F5 4 1851 68.86 62.27 144.81 F6 4 1976 69.17 54.94 186.00 FV1 5 1460 66.13 52.83 156.8 0 PAGE 89 89 Table 4 2. Speed t hreshold values for upstream and downstream detect ors Site ID Freeway Lanes Speed Threshold (mi/h) Upstream Downstream T1 SR 826 3 48 48 T2 SR 826 3 48 52 T3 SR 826 3 48 60 T4 SR 826 3 60 48 T5 I 4 3 44 44 T6 I 4 3 48 44 T7 I 4 3 48 48 T8 I 95 3 60 64 T9 I 95 3 60 56 T10 I 95 3 60 52 T11 I 295 3 60 60 F1 I 75 4 56 56 F2 SR 826 4 64 56 F3 I 4 4 48 48 F4 I 4 4 52 52 F5 I 95 4 56 60 F6 I 95 4 56 56 FV1 I 595 5 56 60 PAGE 90 90 Table 4 3 Capacity estimates from stochastic capacity estimation method 1 Capaci ty in veh/h/ln 2 Flow Rates below 1500 veh/h/ln were observed 3 Flow rate that corresponds to average of maximum flow rate within ten minutes of breakdown Site ID Dir. Lanes VAM Estimation Approach Pre Max Flow Rate 3 Capacity 1 Capacity 1 Breakdowns per day Capacity 1 F c (q) % Capacity 1 F c (q) % 50 th Percentile 4 th Percentile SR 826 (Mia) 3 1693 3.54 1646 2.64 2494 1789 1.43 SR 826 (Mia) 3 1790 4.43 1729 3.78 2534 1838 1.32 SR 826 (Mia) 3 1632 1.44 1656 1.42 2839 1978 0.55 SR 826 (Mia) 3 1746 3.09 1714 1.74 2757 1969 0.92 I 4 (Orl) 3 2024 7.37 2004 5.95 2434 1976 1.09 I 4 (Orl) 3 2082 5.28 1860 2.54 2980 2054 1.13 I 4 (Orl) 3 2021 10.18 1973 7.14 2485 1900 2.54 I 95 (Jax) 3 1897 1.98 1969 2.32 3 338 2224 0.23 I 95 (Jax) 3 1850 4.51 1959 7.14 2442 1890 1.08 I 95 (Jax) 3 1743 1.05 1852 1.29 3424 2290 0.52 I 295 (Jax) 3 2027 10.67 2006 9.09 2308 1920 0.80 I 75 (Mia) 4 1678 1.97 1665 0.80 2632 2015 0.12 SR 826 (Mia) 4 1729 1.95 2 2 3320 2096 0. 27 I 4 (Orl) 4 1692 3.27 1577 1.31 2775 1913 0.91 I 4 (Orl) 4 1905 7.12 1728 2.14 2431 1881 0.56 I 95 (Jax) 4 1851 1.36 1902 2.47 2217 1966 0.16 I 95 (Jax) 4 1976 0.79 1766 0.24 3720 2641 0.04 I 595 (Ft.L) 5 1460 1.58 1739 3.56 3036 1857 0.48 PAGE 91 91 Table 4 4. Capacity estimates from average of top 3% and top 5% highest flows Site ID Capacity 1 Capa city 2 Capacity 3 Capacity 4 %Change %Change %Change %Change Breakdowns per day A B C D A with C A with D B with C B with D SR 826 (Mia) 1824 1759 1693 1738 7.74 4.95 3.90 1.21 1.43 SR 826 (Mia) 1866 1787 1790 1787 4.22 4.42 0.19 0.00 1.32 SR 826 (Mia) 1743 1676 1632 1916 6.78 9.03 2.68 12.53 0.55 SR 826 (Mia) 1876 1812 1746 1912 7.48 1.88 3.81 5.23 0.92 I 4 (Orl) 2149 2096 2024 1940 6.19 10.77 3.57 8.04 1.09 I 4 (Orl) 2132 2053 2082 1988 2.40 7.24 1.39 3.27 1.13 I 4 (Orl) 2211 2146 2021 1855 9 .41 19.19 6.19 15.69 2.54 I 95 (Jax) 1978 1870 1897 2147 4.27 7.87 1.42 12.90 0.23 I 95 (Jax) 2119 2028 1850 1847 14.54 14.73 9.62 9.80 1.08 I 95 (Jax) 2108 2002 1743 2211 20.94 4.66 14.86 9.45 0.52 I 295 (Jax) 2087 2017 2027 1889 2.96 10.48 0.49 6.78 0.80 I 75 (Mia) 1656 1565 1678 1968 1.31 15.85 6.73 20.48 0.12 SR 826 (Mia) 1514 1443 1729 2014 12.43 24.83 16.54 28.35 0.27 I 4 (Orl) 1869 1777 1692 1852 10.43 0.92 5.00 4.05 0.91 I 4 (Orl) 1961 1908 1905 1840 2.95 6.58 0.16 3.70 0.56 I 95 (Jax) 1826 1730 1851 1946 1.35 6.17 6.54 11.10 0.16 I 95 (Jax) 1840 1731 1976 2563 6.88 28.21 12.40 32.46 0.04 I 595 (Ft.L) 1656 1530 1460 1779 13.42 6.91 4.79 14.00 0.48 1 Capacity estimates from average of top 3% highest flow rates ( veh /h/ln) 2 Capacity estimates from average of top 5% highest flow rates (veh/h/ln) 3 Capacity from VAM estimation method (veh/h/ln) 4 Capacity from stochastic capacity estimation method (veh/h/ln) PAGE 92 92 Table 4 5 Threshold values for maximum flow rate in average f low estimation method VAM capacity estimation method Stochastic capacity estimation method Site ID Max. Flow 1 Threshold Flow 1 % of Max. Flow Threshold Flow 1 % of Max. Flow T1 2688 1568 41.67 1660 38.24 T2 2684 1643 38.78 1680 37.41 T3 2696 1508 44.07 1720 36.20 T4 2696 1604 40.50 1816 32.64 T5 2528 1836 27.37 1752 30.69 T6 2696 1928 28.48 1900 29.52 T7 2700 1785 33.90 1608 40.45 T8 2352 1664 29.25 2192 6.80 T9 2552 1584 37.93 1624 36.36 T10 2502 1501 40.00 2202 12.00 T11 2424 1856 23.44 1612 3 3.49 F1 2670 1545 42.13 1836 31.24 F2 2679 1482 44.68 1773 33.82 F3 2691 1545 42.58 1698 36.90 F4 2211 1782 19.40 1734 21.57 F5 2169 1728 20.33 1899 12.45 F6 2307 1866 19.11 2 2 FV1 2690 1229 54.32 1534 42.97 1 Flow in veh/h/ln, 2 Threshold valu e not observed for capacity at 4 th percentile PAGE 93 93 Table 4 6 Capacity estimates from average of flow rates above the flow rate that corresponds to 30 th & 35 th percentage of maximum flow rate Site ID Max. Value Capacity 1 Capacity 2 Flow 3 Flow 4 Capacity 5 % Chang e % Change Capacity 6 % Change % Change Breakdowns per day A B C D E A with E A with F F B with E B with F T1 2688 1693 1738 1882 1747 2216 30.89 14.18 1933 23.87 8.05 1.43 T2 2684 1790 1787 1879 1745 2058 14.97 7.04 1916 11.97 4.24 1.32 T3 2696 1632 1916 1887 1752 2221 36.09 25.49 2048 12.29 3.54 0.55 T4 2696 1746 1912 1887 1752 2117 21.25 7.62 1879 7.52 4.57 0.92 T5 2528 2024 1940 1770 1643 1989 1.73 6.52 1892 0.66 4.25 1.09 T6 2696 2082 1988 1887 1752 2043 1.87 6.24 1952 0.54 4.97 1.13 T7 2700 2021 1855 1890 1755 2067 2.28 0.84 2004 8.79 5.47 2.54 T8 2352 1897 2147 1646 1529 1887 0.53 3.80 1825 15.15 17.94 0.23 T9 2552 1850 1847 1786 1659 2011 8.70 3.73 1919 6.40 1.53 1.08 T10 2502 1743 2211 1751 1626 1998 14.63 8.43 1890 12.75 17.47 0.52 T11 2424 2027 1889 1697 1576 1959 3.35 5.97 1906 2.03 0.73 0.80 F1 2670 1678 1968 1869 1736 2073 23.54 9.83 1843 2.88 8.54 0.12 F2 2679 1729 2014 1875 1741 2195 26.95 19.78 2071 4.72 1.19 0.27 F3 2691 1692 1852 1884 1749 2095 23.82 17. 79 1993 9.51 4.18 0.91 F4 2211 1905 1840 1548 1437 1761 7.56 12.28 1671 6.38 11.16 0.56 F5 2169 1851 1946 1518 1410 1743 5.83 8.75 1689 11.34 14.09 0.16 F6 2307 1976 2563 1615 1500 1815 8.15 11.99 1739 31.28 34.15 0.04 FV1 2690 1460 1779 18 83 1749 2254 54.38 46.78 2143 21.38 15.40 0.48 1 Capacity from VAM estimation method (veh/h/ln) 2 Capacity from stochastic capacity estimation method (veh/h/ln) 3 Threshold Value for 30% of maximum flow rate (veh/h/ln), 4 Threshold Value for 35% of maximum fl ow rate (veh/h/ln) 5 Average value of flow rates above flow rate that corresponds to 30% of max. flow rate (veh/h/ln) 6 Average value of flow rates above flow rate that corresponds to 35% of max. flow rate (veh/h/ln) PAGE 94 94 Table 4 7. Average flow analysis x valu es for VAM capacity estimation method Site ID 3 Months 2 Months % Change 1 Month % Change 2 Weeks % Change A (%) B (%) A to B C (%) B to C D (%) C to D T1 14.03 13.32 5.10 14.37 7.91 40.41 181.25 T2 10.53 10.79 2.47 11.85 9.91 15.89 34.02 T3 4.61 4. 33 6.11 4.07 5.92 4.28 5.16 T4 1.14 1.17 3.08 1.52 29.85 0.99 34.87 T5 1.98 2.22 11.93 2.28 2.61 1.81 20.72 T6 4.84 5.63 16.39 5.81 3.12 5.65 2.68 T7 1.18 1.08 7.97 0.82 24.17 1.29 57.52 T8 7.66 7.52 1.83 6.97 7.40 9.80 40.65 T9 9.03 8.45 6 .43 7.63 9.69 10.89 42.76 T10 2.03 1.94 4.42 3.60 86.03 7.40 105.34 T11 4.98 4.55 8.63 4.06 10.77 5.84 43.84 F1 2.17 2.28 4.83 2.58 13.16 3.47 34.75 F2 10.68 9.31 12.80 8.77 5.84 7.11 18.86 F3 2.58 3.92 51.76 4.39 12.13 3.58 18.44 F4 7.99 6.1 6 22.97 5.50 10.64 4.85 11.82 F5 5.94 4.55 23.40 3.40 25.27 0.90 73.53 F6 1.72 1.38 19.93 1.25 9.49 0.65 47.96 FV1 10.02 8.84 11.81 7.68 13.12 7.64 0.52 PAGE 95 95 Table 4 8. Average flow analysis x values for stochastic capacity estimation method Site ID 3 months 2 months % Change 1 month % Change Two Weeks % Change A (%) B (%) A to B C (%) B to C D (%) C to D T1 5.78 4.77 17.53 4.55 4.60 3.59 21.00 T2 4.60 3.43 25.48 2.62 23.56 0.77 70.68 T3 1.57 1.08 31.17 1.00 7.79 0.81 18.67 T4 2 .68 1.93 27.89 1.65 14.95 1.60 2.88 T5 9.54 9.24 3.17 8.63 6.62 12.14 40.70 T6 1.79 1.71 4.33 1.30 24.10 1.99 53.24 T7 14.68 13.78 6.18 13.09 4.94 18.03 37.69 T8 0.20 0.21 5.06 0.14 35.99 0.15 14.04 T9 9.10 9.40 3.33 10.31 9.67 9.71 5.81 T 10 0.57 0.56 1.16 0.34 39.24 0.36 5.13 T11 7.41 8.52 14.97 8.86 3.95 8.80 0.65 F1 0.31 0.47 53.00 0.64 36.24 0.26 59.43 F2 0.60 0.52 12.23 0.47 10.36 0.16 65.38 F3 0.11 0.02 84.18 0.04 101.54 0.08 124.00 F4 5.95 5.38 9.51 4.91 8.72 6.92 40.7 6 F5 0.39 0.46 17.91 0.54 17.22 0.42 21.45 F6 1 FV1 2 1,2 Flow rates that match the stochastic capacity estimates were not observed PAGE 96 96 Table 4 9 Comparison of capacity estimates from different analysis methods with HCM 2000 Si te ID Capacity 1 Capacity 2 %Change Capacity 3 %Change Capacity 4 %Change Capacity 5 % Change Breakdowns per day A B A with B C A with C D A with D E A with E SR 826 (Mia) T1 2250 1693 24.76 1738 29.46 1759 27.91 1933 14.09 1.43 SR 826 (Mia) T2 2250 1 790 20.44 1787 25.91 1787 25.91 1916 14.84 1.32 SR 826 (Mia) T3 2250 1632 27.47 1916 17.43 1676 34.25 2048 8.98 0.55 SR 826 (Mia) T4 2250 1746 22.4 1912 17.68 1812 24.17 1879 16.49 0.92 I 4 (Orl) T5 2250 2024 10.04 1940 15.98 2096 7.35 18 92 15.91 1.09 I 4 (Orl) T6 2250 2082 7.47 1988 13.18 2053 9.60 1952 13.24 1.13 I 4 (Orl) T7 2250 2021 10.18 1855 21.29 2146 4.85 2004 10.93 2.54 I 95 (Jax) T8 2400 1897 20.96 2147 11.78 1870 28.34 1825 23.96 0.23 I 95 (Jax) T9 2400 1850 2 2.92 1847 29.94 2028 18.34 1919 20.04 1.08 I 95 (Jax) T10 2400 1743 27.38 2211 8.55 2002 19.88 1890 21.25 0.52 I 295 (Jax) T11 2400 2027 15.54 1889 27.05 2017 18.99 1906 20.58 0.80 I 75 (Mia) F1 2400 1678 30.08 1968 21.95 1565 53.35 1843 23.21 0.12 SR 826 (Mia) F2 2250 1729 23.16 2014 11.72 1443 55.93 2071 7.96 0.27 I 4 (Orl) F3 2400 1692 29.5 1852 29.59 1777 35.06 1993 16.96 0.91 I 4 (Orl) F4 2250 1905 15.33 1840 22.28 1908 17.92 1671 25.73 0.56 I 95 (Jax) F5 2400 1851 22 .88 1946 23.33 1730 38.73 1689 29.63 0.16 I 95 (Jax) F6 2400 1976 17.67 2563 6.36 1731 38.65 1739 27.54 0.04 I 595 (Ft.L) FV1 2400 1460 39.17 1779 34.91 1530 56.86 2143 10.71 0.48 1 Capacity as provided by HCM 2000 (veh/h/ln) 2 Capacity as estim ated from VAM capacity estimation method (veh/h/ln) 3 Capacity as estimated from stochastic capacity estimation method (veh/h/ln) 4 Capacity as estimated from average of top 5% highest flow rates (veh/h/ln) 5 Capacity as estimated from average of flow rates between maximum flow rate and flow rate that corresponds to 35% of maximum flow rate of maximum flow rate (veh/h/ln) PAGE 97 97 CHAPTER 5 SUMMARY AND CONCLUSI ONS Th re e methods to estimate the capacity values of basic freeway segments for Florida freeways were inves tigated and compared to each other : the Van Aerde Model (VAM) stochastic capacity estimation based on the PLM and the average maximum flo w rate These methods were applied to 18 freeway sites across Florida with each site experiencing frequent congestio n. The STEWARD database server was used to obtain the traffic flow data used in the analysis. The Traffic Stream Calibration software (Rakha, 2007) was used to generate the VAM capacity estimat es The PLM approach, as proposed by Brilon et al. (2005), wa s used to generate the stochastic capacity estimates This approach also consisted of generating Weibull capacity distribution functions and comparing these curves to plotted speed flow data points to identify appropriate breakdown probability percentile v alues, and corresponding capacity values. T he average maximum flow rate method was developed as a simple alternative to the previous two more complicated methods for estimating capacity. This method consists simply of taking the average of a certain percen tage of the highest recorded flow rates. The minimum amount of data (from a time perspective) was also calculated for which this simple averaging method is applicable. Conclusions The capacity estimates from the VAM method the stochastic capacity method and the average maximum flow rate method were found to be lower than the capacity values given in the HCM (2000) for Florida freeways. From the stochastic capacity estimation method, it was observed that the average breakdown probabilit y from the Weibull c apacity distribution function that corresponds to the PAGE 98 98 VAM capac ity estimates and maximum flow rate within ten minutes of breakdown w as 3.98% and 3.27% respectively. The se percentile values or the average breakdown probabilit ies, compare closely with the average design breakdown probability provided by Geistefe l dt (2008) which was 3% for German freeways. Thus based on the results of this study and the Geistefeldt study, it appears that the use of a 4 th percentile value from the Weibull capacity distribut ion function will provide reasonable estimate s of freeway segment capacit ies It is also observed that the most reliable estimates of capacity from the stochastic capacity estimation method will be obtained when the analysis site has 0.5 or more breakdowns per day. Two variants of the averaging method were investigated. In the first variant, the average of the top 5% highest flow rates were taken as the capacity estimates for the freeway segments. In the second variant, the average of flow rates between max imum flow rate observed and flow that corresponds to 35% of the maximum flow rate were taken as the capacity estimates for the freeway segment. To obtain consistent estimates from this method, it was found that a minimum of two months data should be used Advantages and Disadvant age s of the Investigated Capacity Es t imation Methods Each of the three methods investigated in this study has its advantages and disadvantages for us e in capacity estimation, as described below. Stochastic estimation m ethod Advantag es : Method accounts for the stochastic nature of capacity Utilizes the concept of breakdown (without identifying breakdowns, one cannot be sure whether more traffic than the highest observed flow rates could be served) Capacity distribution function provid es flexibility in choosing capacity value (based on breakdown probability) that is appropriate to given application PAGE 99 99 Disadvantages : Very d ata processing intensive Very c omputationally intensive Determination of appropriate breakdown probability value is no t straightforward Van Aerde m odel m ethod Advantages : Does not require the identification of breakdowns Traffic flow theory basis (i.e., car following minimum headway rule) From a simplicit y perspective, a single capacity value is returned Is flexible in it s application to different types of freeways Disadvantages : Incorporation of congested data points increase accuracy, but capacity values are not tied directly to breakdown events Moderately data processing intensive Moderately computationally intensive Av erage m aximum f low r ate m ethod Advantages : Easy to understand E asy to apply Does not require the identification of breakdown events Disadvantages : No theoretical basis for capacity estimates Without incorporation of breakdown events or congested data point s, accuracy of estimated capacity values is unknown PAGE 100 100 In summary, t he stochastic capacity estimation method is most suitable for the determination of capacity at a site where a detailed operational analysis is desired. For example, at sites where different o perational treatments (e.g., ramp metering) are going to be tried in an effort to improve operations and an estimate of capacity that is as accurate as possible is desired. The VAM capacity estimation method is not as suitable as the stochastic estimation method for detailed evaluations of operational treatments, but is stil l appropriate for the determination of general capacity estimates T he average maximum flow rate capacity estimation method is most suitable for planning and preliminary engineering appl ications. Recommendations The focus of this study was on the comparison of different methods for capacity estimation, with the intention of choosing a method for a follow on study focused on determining specific capacity values for Florida freeways Given that the FDOT Systems Planning Office is looking to use these capacity estimates in its planning and preliminary engineering level of service analysis software, it is recommended that the follow on study apply the VAM method for the initial establishment o f these values. However, it is also recommended that the average maximum flow rate capacity estimation method be applied in this follow on study, for the purpose of identifying appropriate percentile values for this method (i n reference to the VAM method). For future updates to the initially established capacity values, the use of the average maximum flow rate method by itself should be sufficient This follow on study will also require more analysis sites. This will be possible as the Central Data Warehous e is expanded to include more Florida cities an d more sites within each city. To the extent possible, the follow on study should also investigate the effect of the following specific roadway and traffic factors on freeway segment capacity: number of lanes (as it relates to per lane capacity), merge/diverge activity, free f low speed, and truck percentag e. PAGE 101 101 LIST OF REFERENCES Al Kaisy, A., J.A. Stewart, and Van Aerde M. (1999). A simulation approach for examining capacity and operational performance at freew ay diverge areas. Canadian Journal of Civil Engineering, Vol. 26, pp. 760 770. Brilon, W. and Zurlinden H. (2003). Ueberlastungswahrscheinlichkeiten und Verkehrsleistung als Bemessungskriterium fuer Strassenverkehrsanlagen (Breakdown Probability and Traff ic Efficiency as Design Criteria for Freeways). Forschung Strassenbau and Strassenverkehrstechnik No. 870. Bonn. Brilon, W., Geistefeldt J. and Regler M. (2005). 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Traffic Stream Calibration Software, 7 July 20 09, < http://filebox.vt.edu/users/hrakha/Software.htm >. van Toorenburg, J. (1986). Praktijwaarden voor de capaiteit. Rijkswaterstaat dienst Verkeerskunde, Rotterdam. TRB Special Report 209, Highway Capacity Manual (1997). Transportation Research Board, Nat ional Research Council, Washington, DC. Van Aerde M. and Rakha H (1995). Multivariate Calibration of Single Regime Sp eed Flow Density Relationships. Vehicle Navigation and Information Conference (VNIS). IEEE, Piscataway NJ, USA 95CH35776, pp.334 341. Van Aerde, M. (199 5 ). Single regime speed flowdensity relationship for con gested and uncongested highways. Presented at the 74 th TRB Annual Conference, Washington, D C. Paper No. 950802. Yang, X and Zhang N. (2005). The marginal decrease of lane capacity wit h the number of lanes on highway. Proceedings of the Eastern Asia Society for Transportation Studies, Vol. 5, pp. 739 749, 2005. Yang, X., N. Zhang and Gao Z. (2007). Changes in traffic characteristics affected by number of lanes on freeways. Paper sub mitted at the 87 th meeting of the Transportation Research Board, Washington D.C. Zou, Xi and Levinson D. (2003), Detetcing the breakdown of traffic. Presented at Transportation Research Board Conference, January 12 16, Washington D.C. PAGE 103 BIOGRAPHICAL SKETC H Vipul Modi was born in the capital city of India, New Delhi. He completed his higher secondary school from Delhi Public School Farid abad and bachelors in Civil Engineering from Indian Institute of Technology (IIT) Bombay in April 2007. He joined Universi ty of Florida, Gainesville for his Masters in Transportation Engineering and graduated in December 2009. He has held many leadership positions in IIT Bombay and at the University of Florida. He held the position of President of ITE (Institute of Transporta tion Engineers) Student Chapter at University of Florida and has held the position of Treasurer of ITE Student Chapter. He has a passion to learn about different MRTS (Mass Rapid Transit Systems) and has knowledge of all the MRTS in different cities of the world. He has won many prizes at Inter collegiate festivals in the field of dramatics, choreography and sports. He likes to collect stamps, currencies, travel, photography, drive, listen to music and play all kinds of sports. 