Development and Testing a New Ramp Weave Operational Analysis Method.

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XU, DEZHONG. Development and Testing a New Ramp Weave Operational Analysis Method. (Under the direction of Dr. Nagui Rouphail).

In the past several decades, the methodology of the Highway Capacity Manual (HCM)

weaving segments has been updated several times. In the latest version, HCM6, the method

predicts the speed of weaving and non-weaving traffic by using the predicted number of lane

changes. However, recent research has revealed that the speed and capacity prediction of HCM6

is not accurate and lacks sensitivity to important design parameters under certain conditions.

The goals of this study are to provide a simple method for ramp weave segments and to

ensure consistency with other freeway segment analyses in HCM6. It includes a new modeling

framework that maintains a continuum between the operation of the weaving segments to an

equivalent basic segment that has a similar volume, number of lanes, and free-flow speed. This

study focused on ramp weaves only. The proposed models were calibrated using data from seven

mostly local sites, including one site from NCHRP 3-75 and six additional sites with data

gathered from ground-based and drone videos. Several methods were tested for data reduction,

including image processing and manual extraction. The calibrated models were assessed using

several numerical sensitivity tests.

Based on the results of the tests, it is shown that the proposed approach is practical and

accurate. The proposed models are proven to have better goodness of fit to observed speeds than

HCM6, explaining about 75% of the variability in observed 5 min- mean speeds. The models’

sensitivities of speed and capacity to segment length, volume ratio, and demand are reasonable

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by Dezhong Xu

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the degree of Master of Science of Engineering

Civil Engineering

Raleigh, North Carolina 2019

APPROVED BY:

_______________________________ _______________________________ Nagui Rouphail Billy Williams

Committee Chair

_______________________________

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DEDICATION

First, I would like to express the deepest appreciation to my committee chair, Dr. Nagui

Rouphail, who gave me the opportunity and guided me through the main research project. I

would never be able to accomplish anything without his support.

I would also like to thank my parents, who have always supported me. Their care and trust

supported me in getting out of the depression and gaining the strength to face the challenges.

And I would like to offer my gratitude to my beloved, Dan Yu, who has inspired me with her

love, understanding, and company.

I am further grateful to thank all my friends for their support. I thank all professors in the Civil

Engineering Department for their excellent lectures and advises.

Finally, my gratefulness also goes to my cat Marra, who always stay by my side and

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BIOGRAPHY

Dezhong Xu received his B.S. degree in Civil Engineering at North Carolina State

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ACKNOWLEDGMENTS

The author wishes to sincerely thank Dr. Nagui Rouphail of North Carolina State

University. He is the first individual who developed the original framework. Without his advice

and patient teaching, the completion of this study would have been extremely difficult.

Special acknowledgment is also expressed to Dr. Behzad Aghdashi of North Carolina

State University and Dr. Lily Elefteriadou at the University of Florida as they contributed to all

phases of research, especially in model development, validation and review of the Highway

Capacity Manual method.

The author expresses his greatest regards to members of the Committee on Final

Examination, Dr. Nagui Rouphail, Dr. Behzad Aghdashi, and Dr. Billy William of North

Carolina State University, for their suggestions and comments. Additionally, the author wishes to

thank the Institute of Transportation and Research for providing the facilities during research.

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TABLE OF CONTENTS

LIST OF TABLES ... vi

LIST OF FIGURES ... viii

Chapter 1: Introduction ... 1

1.1 Research Objectives ... 1

1.2 Definition of Ramp Weave and Level of Service ... 2

1.3 Glossary of Terms ... 4

1.4 Thesis Organization ... 6

Chapter 2: Previous Works... 7

2.1 HCM Development History of Weaving Operational Analysis ... 7

2.2 Related Studies... 14

2.3 Works that Critically Analyzed HCM6 ... 15

Chapter 3: Methodology... 18

3.1 Scope of Work ... 18

3.2 Site Selection ... 18

3.3 Data Collection ... 20

3.4 Data Extraction ... 21

3.5 Data Filtering ... 22

3.6 Conceptual Model Formulation ... 24

3.7 Variables in SIW ... 29

Chapter 4: Model Result ... 45

4.1 Candidate Model Forms and Model Calibration ... 45

4.2 Procedure Implementation ... 69

4.3 Sensitivity Analysis to Ls ... 71

4.4 Sensitivity Analysis to VR ... 74

4.5 Sensitivity Analysis to Vrr ... 79

4.6 Limited Model Validation... 85

Chapter 5: Conclusions and Recommendations ... 89

5.1 Conclusions ... 90

5.2 Recommendations for future research ... 93

References ... 94

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LIST OF TABLES

Table 1.1 Level of Service Criteria of a Weaving Segment ... 3

Table 1.2 Terminology ... 4

Table 2.1 HCM1965 Relationship Between LOS and Quality of Flow on a Weaving Section . 9 Table 2.2 Quality of Flow and Maximum Lane Service Volumes in a Weaving Section ... 10

Table 2.3 Criteria for Unconstrained vs. Constrained Operation of Weaving Areas ... 11

Table 2.4 LOS Criteria for Freeway Weaving Sections in HCM1985 ... 12

Table 2.5 HCM1985 Limitations on Weaving Area ... 12

Table 2.6 LOS Criteria in HCM1997 ... 13

Table 3.1 Location of Study Sites ... 19

Table 3.2 Summary of Configuration and Data Points of the Site ... 23

Table 3.3 Candidate Variables that Explain the Effect of Flow and Configuration to SIW .... 30

Table 4.1 Parameter Calibration Result and Goodness of Fit for Different ‘a’ Value Models ... 47

Table 4.2 Model Calibration Results and Goodness of Fit Comparison Between Candidate Models and HCM6 ... 48

Table 4.3 Two-Tailed t-test Result of the Trendline of Each Model and HCM6 (Level of Significance = 0.05) ... 61

Table 4.4 Assumed OD volumes for Different VR ... 74

Table 4.5 Assumed OD volumes for Different Percentage of Ramp to Ramp Traffic ... 79

Table 4.6 Observed OD Volumes and SMS for each 5-min period ... 85

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LIST OF FIGURES

Figure 1.1 Flow Identification ... 2

Figure 1.2 Example of a Type A Weaving Segment ... 3

Figure 2.1 HCM1950 Traffic Volumes and Speed Relation Plot ... 7

Figure 2.2 Snapshot of HCM1965 Quality of Flow Curves and Relative Estimated Speeds ... 9

Figure 3.1 Image of the Drone ... 21

Figure 3.2 Graphical Illustration for Capacity and LOS Estimation ... 28

Figure 3.3 versus Normalized SMS Plot ... 32

Figure 3.4 Candidate 1 versus Normalized SMS Plot ... 33

Figure 3.7 Candidate 1 VR versus Normalized SMS Plot ... 34

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Figure 4.1 Observed SMS versus Model 1 Predicted SMS Plot... 54

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Figure 4.13 Observed SMS versus HCM6 Predicted SMS Plot ... 60

Figure 4.14 Model 1 Residual Plot ... 62

Figure 4.26 HCM6 Residual Plot... 68

Figure 4.27 Implementation Procedure Flow Chart ... 70

Figure 4.28 Capacity Sensitivity to Ls ... 72

Figure 4.29 Predicted Speed Sensitivity to Ls ... 73

Figure 4.30 Density Sensitivity to Ls ... 73

Figure 4.31 Capacity Sensitivity to VR (Ls = 250 ft) ... 76

Figure 4.32 Predicted Speed Sensitivity to VR (Ls = 250 ft) ... 76

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Figure 4.34 Capacity Sensitivity to VR (Ls = 2000 ft) ... 77

Figure 4.35 The result of Predicted Speed Sensitivity to VR Test (Ls = 2000 ft) ... 78

Figure 4.36 Density Sensitivity to VR Test (Ls = 2000 ft) ... 78

Figure 4.37 Capacity Sensitivity to %Vrr (Ls = 250ft) ... 81

Figure 4.38 Predicted Speed Sensitivity to %Vrr Demand (Ls = 250 ft) ... 82

Figure 4.39 Density Sensitivity to %Vrr Demand (Ls = 250 ft) ... 82

Figure 4.40 Capacity Sensitivity to %Vrr Demand (Ls = 2000 ft) ... 83

Figure 4.41 Predicted Speed Sensitivity to %Vrr Demand (Ls = 2000 ft) ... 83

Figure 4.42 Density Sensitivity to %Vrr Demand (Ls = 2000 ft) ... 84

Figure 4.43 Comparison between the Model 3 Predicted Speed and Observed Speed ... 86

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CHAPTER 1: INTRODUCTION

With decades of development, the HCM weaving segment analysis has been revised and

updated multiple times. The development direction of the HCM weaving segment analysis can

be summarized as developing simple and non-iterative models and framework that

accommodates different types of configuration and traffic operations. Some recent projects also

attempted to develop a new weaving methodology that maintains consistency in application with

other freeway segment methods, including basic freeway and ramp junctions. The current model

and analysis framework was developed in NCHRP 3-75 (Roess, et al., 2008) and later adopted in

the 2010 HCM (Transportation Research Board, 2010). Although it was updated more recently,

some researches pointed out that both speed and capacity predictions highly deviate from field

observations. This may be caused by the lack of sufficient ramp weave site data: NCHRP 3-75

only has one feasible ramp weave site. The discordant model form also generates discontinuities

in performance estimation across different types of freeway sections. The complex and lengthy

procedure of the operational analysis has also been questioned. Also, the current method requires

extensive field calibration and speed model validation. Specific problems are introduced later in

this paper.

1.1 Research Objectives

The objective of this research is to develop a simple, non-iterative and well-calibrated

speed and capacity models and framework for ramp weave segments. The models should also

improve the sensitivities to important traffic and configuration parameters such as volume ratio

(VR) and short segment length (Ls). Another objective is to keep the consistency of predictions

with other types of freeway segments. Even though the final calibrated models only address the

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model form to seamlessly transition among different types of freeway sections. The research

related information including the definition of ramp weave and level of service and all the inputs

variables used in the analysis are introduced next.

1.2 Definition of Ramp Weave and Level of Service

Weaving is the crossing of two or more uninterrupted traffic flows that have the same

direction in a limited segment length. In a simple weaving segment, traffic can be categorized

into four flows based on its origin and destination. In the four flows, freeway to freeway flow

and ramp to ramp flow are non-weaving flows. freeway to ramp flow and ramp to freeway flow

are weaving flows. Figure 1.1 below shows a summary of the flow identification.

Figure 1.1 - Flow Identification

The weaving segment has multiple configurations. Based on the number of lane change

required to finish the weaving maneuvers, the segment can be categorized into three different

types: Type A, Type B, and Type C (Transportation Research Board, National Research Council,

1985). Type A, also known as ramp weave, is the weaving segment configuration where vehicles

must make one lane change to complete weaving maneuvers. Figure 1.2 illustrates a typical Type

A weaving segment configuration.

Freeway Freeway

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Figure 1.2 - Example of a Type A Weaving Segment

The performance of the weaving segment can be explained by six levels of service

(LOS): A, B, C, D, E, and F. At LOS A, the traffic operates at a low density and at high average

space mean speed (SMS). In addition, the LOS F represents a traffic condition that has a high

density and low average space mean speed. A congestion or stop-and-go situation is very likely

to be observed in LOS F. The determination of the LOS is based on the density of the weaving

segment that is arithmetically related to the volume and overall traffic speed. Table 1.1 shows the

density criteria for different LOS in different types of segments. In addition, Equation (1), shows

the basic relationship among density, volume, and speed.

Table 1.1

Level of Service Criteria of a Weaving Segment

Level of Service

Maximum Density (pc/mi/ln) FREEWAY WEAVING

AREA MULTILANE AND C-D WEAVING AREAS

A 10 12

B 20 24

C 28 32

D 35 36

E <=43 <=40

F >43 >40

D = (v/N)

𝑆 (1)

Mainline

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1.3 Glossary of Terms

The input variables used in this analysis are illustrated and defined in Table 1.2. The

reader may find some unfamiliar variables in this table. These terms will be explained later in

this thesis.

Table 1.2 Terminology

Variable Meaning

𝑣 = freeway to freeway demand flow rate in the weaving segment (pc/hr);

𝑣 = freeway to ramp demand flow rate in the weaving segment (pc/hr);

𝑣 = ramp to freeway demand flow rate in the weaving segment (pc/hr);

𝑣 = ramp to ramp demand flow rate in the weaving segment (pc/hr);

𝑣 = on ramp demand flow rate (pc/hr), 𝑣 + 𝑣 ;

𝑣 = off ramp demand flow rate (pc/hr), 𝑣 + 𝑣 ;

𝑣 = weaving demand flow rate in the weaving segment (pc/hr), (𝑣 + 𝑣 )/ 𝑣;

𝑣 = non-weaving demand flow rate in the weaving segment (pc/hr), 𝑣 + 𝑣 ;

𝑣 = auxiliary lane usage flow rate (pc/hr), 𝑣 + 𝑣 ;

𝑣 = total demand flow rate in the weaving segment (pc/hr), 𝑣 + 𝑣 ;

𝑣 = per lane flow rate at capacity (pc/hr/ln);

𝑉𝑅 = volume ratio (decimal), 𝑣 / 𝑣 ;

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Table 1.2

Terminology （Continued)

Variable Meaning

𝑁 = number of lanes in the weaving segment (ln);

𝑆 = average speed of weaving vehicles within the weaving segment (mi/hr);

𝑆 = average speed of non-weaving vehicles within the weaving segment (mi/hr);

𝑆 = average speed of all vehicles within the weaving segment;

𝐹𝐹𝑆 = free-flow speed of the weaving segment (mi/hr);

𝑆 = space mean speed for the equivalent basic segment servicing the same total

demand flow rate (𝑣), with the same number of lanes (𝑁) and the same

free-flow speed (𝐹𝐹𝑆), (mi/hr);

𝑆𝐼𝑊 = speed impedance term due to weaving traffic turbulence (mi/hr);

𝐿 = short length of the weaving segment (ft), the distance between the end

points of barrier markings that prohibit or discourage lane changing;

𝑐 = weaving segment capacity per lane (pc/hr/ln);

𝑐 = basic freeway segment capacity per lane under equivalent ideal conditions

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1.4 Thesis Organization

The remainder of this thesis document is organized as follows. Chapter 2 introduces the

literature review that focuses on the historical development of HCM weaving operational

analysis as well as other models that were developed and recent works that critically analyzed

the HCM6. This is followed by Chapter 3 that explains the data preparation, conceptual model

formulation and candidate variables in the model. Chapter 4 describes the result of the model

calibration and preferred model forms. Procedure for model implementation of the operational

analysis framework and the results of selected models’ sensitivity analyses to segment and traffic

parameters are presented. Model validation tests are described next. Finally, Chapter 5 presents

the final recommended model, conclusions from the study and recommendations for future

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CHAPTER 2 : PREVIOUS WORKS 2.1 HCM Development History of Weaving Operational Analysis

The Highway Capacity Manual (HCM) was first introduced in 1950 (Bureau of Public

Roads, U.S. Department of Commerce, 1950). Until now, six major versions of HCMs, besides

the minor revised editions, were published. HCM1950 began the analysis of weaving segments

by using six data sites that were collected from the Pentagon Network and the San Francisco Bay

Bridge. Several findings were mentioned, including weaving vehicle behavior and the impact of

the speed to segment capacity. The data analysis results on traffic volumes and speed from the

six sites are presented as a plot in Figure 2.1. It should be noted that, in the plot, HCM1950

revealed the relationship between the minimum number of lanes and the traffic demand for the

first time.

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In 1965, Leisch and Normann developed a method based on the analysis result of

HCM1950 (Normann, 1957). Moreover, their method was added in HCM1965 (Highway

Research Board, National Research Council, 1965). Compared to HCM1950, HCM1965 defined

many concepts in weaving segments for the first time. For example, the HCM divided weaving

segments into 2 types: simple weaving sections and multiple weaving sections. Both types could

be further subdivided into one-sided or two-sided sections. The traffic flows in the weaving

segment were distinguished as weaving movements and non-weaving movements. The

measurement of the weaving segment length was stipulated. However, the most magnificent

concept in HCM1965 was the basic procedures and methodologies to design and evaluate the

weaving segments. The quality of flow was introduced as a measure of weaving section

operation. As Figure 2.2 shows below, the quality of flow had five designated classes from I to

V, which represent the congestion level from light to heavy. It should be noted that each curve in

the figure contained a number on it. The number, also known as k-factor, was presented as an

equivalency factor that expands the influence of the smaller weaving flow with a range from 1 to

3. The complete steps for measuring the weaving section performance were as follows: First, the

user locates a point based on segment length and weaving demand in the quality of the curve

plot. Then, by finding the nearest curve to the point, the class of the quality of flow and the

estimated speed can be identified. From Table 7.3 of HCM1965, which is shown in Table 2.1,

the known quality of flow can be converted to the LOS. The capacity of the segment was shown

in Table 7.2 of HCM1965, which is shown here in Table 2.2. However, the capacity was not

considered as a checker in evaluating the LOS for demand over the capacity condition. Even

though HCM1965 had a method for evaluating the segment performance, it was more focused on

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Figure 2.2 - Snapshot of HCM1965 Quality of Flow Curves and Relative Estimated Speeds

Table 2.1

HCM1965 Relationship Between LOS and Quality of Flow on a Weaving Section

QUALITY OF FLOW FREEWAYS AND MULTILANE RURAL

HIGHWAYS

LEVEL OF SERVICE

HIGHWAY PROPER

CONNECTING

COLLECTOR0DISTRIBUTOR ROADS AND OTHER

INTERCHANGE ROADWAYS

TWO-LANE RURAL HIGHWAYS

URBAN AND SUBURBAN ARTERIALS

A I-II II-III II III-IV

B II III II-III III-IV

C II-III III-IV III IV

D III-IV IV IV IV

E IV-V V V V

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Table 2.2

Quality of Flow and Maximum Lane Service Volumes in a Weaving Section

Quality of Flow Curve Max Lane SV Value (pcph)

I 2000

II 1900

III 1800

IV 1700

V 1600

From 1965 to 1985, several models were developed. Roess and McShane’s model

appeared in several forms, and its final form was introduced in Circular 212 (Transportation

Research Board, 1980). The model was iterative and intended to predict the average speed of

weaving and non-weaving vehicles. In addition, it introduced the required lanes change

categorized configuration and type of the operation into the analysis process. In 1984, Reilly

developed a model that utilized a density concept tied to weaving intensity to predict the average

speed for weaving and non-weaving traffic (W. Reilly, et al, 1984). HCM1985 merged the two

models above (Transportation Research Board, National Research Council, 1985). Reilly et al.’s

model was stratified to different configurations and types of operations. Equation (2) is the speed

equation from HCM1985. The equation implies that the traffic speed is related to the volume

ratio, traffic demand, number of lanes, and the length of the segment. The four constant

parameters (a, b, c, and d) in the equation were decided by the type of the segment and type of

operation. First, the speed was predicted by using unconstrained operation parameters. Then, by

comparing two variables, the number of lanes required for the weaving segment (Nw) and the

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unconstrained condition was justified. Table 2.3 shows the equation for calculating Nw and

Nw(max) in different types of configuration. The speed was predicted by using the parameters of

the constrained operation if it was proved that the traffic was under a constrained operation. The

predicted speed was then used in the determination of LOS of weaving and non-weaving traffic.

Table 2.4 shows the LOS criteria in HCM1985. The final segment’s LOS was the worst LOS

between the two. Interestingly, HCM1985 further provided a table of limitation for a weaving

segment, which is shown in

Table 2.5. The table had various limitation or maximum values indicated. However, the

limitation did not impact LOS determination. It only showed the accuracy of the LOS prediction.

𝑆 = 15 + 50

1 + 𝑎(1 + 𝑉𝑅 ) _𝑁𝑣 /𝐿 (2)

Table 2.3

Criteria for Unconstrained vs. Constrained Operation of Weaving Areas

Type of

Configuration No. of Lanes Required for Unconstrained Operation, NW

Max. No. of Weaving Lanes,

NW (max)

Type A 2.19 𝑁 𝑉𝑅 . 𝐿 . /𝑆 . 1.4

Type B 𝑁{0.085 + 0.703 𝑉𝑅 + . − 0.018 (𝑆 − 𝑆 )} 3.5

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Table 2.4

LOS Criteria for Freeway Weaving Sections in HCM1985

Level of Service Minimum Average Weaving Speed SW (MPH)

Minimum Average Non-Weaving Speed SNW (MPH)

A 55 60

B 50 54

C 45 48

D 40 42

E 35 35

F <35 <35

Table 2.5

HCM1985 Limitations on Weaving Area

Type of Configuration Weaving Capacity Maximum vW Maximum v/N Maximum Volume Ratio, VR Maximum Weaving Ratio, R Maximum Weaving Length, L

Type A 1,800 _pcph _pcphpl1,900

N VR 2 1.00 3 0.45 4 0.35 5 0.22

0.5 2,000 ft

Type B 3,000

pcph

1,900

pcphpl 0.80 0.5 2,500 ft

Type C 3,000 _pcph _pcphpl1,900 0.50 0.4 2,500 ft

The HCM1985 method was revised several times, but the model form was still used in

HCM2000. In 1997, HCM revised the table of limitation of weaving segments and the LOS

criteria (Transportation Research Board, National Research Council, 1997). HCM1997 used the

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Table 2.6, and the same criteria have been used until HCM6. Furthermore, the average

density was computed by using the total flow divided by the average space mean speed.

HCM2000 further revised the model by updating the constants for computation of the weaving

intensity factors and the coefficient in the equation of the number of lanes required for the

unconstrained condition (Transportation Research Board, 2000). In addition, HCM2000 updated

the limitation of the weaving segment and added pages of tables for capacity prediction. The

capacity was defined as any combination of flows that cause the density to reach LOS F

boundary condition which is 43 pc/ln/mi. Based on the configuration, the number of lanes, FFS,

segment length, and volume ratio, the user could find the rough estimated segment capacity.

However, the capacity prediction still did not impact the determination of the LOS.

Table 2.6

LOS Criteria in HCM1997

Level of Service

Maximum Density (pc/mi/ln)

Freeway Weaving Area Multilane and C-D Weaving Areas

A 10 12

B 20 24

C 28 32

D 35 36

E <= 43 <= 40

F > 43 >40

After HCM2000, the NCHRP 3-75 project was launched to develop a revised method for

weaving segments to improve the simplicity of model calibration as well as the consistency of

predictions with other types of freeway segments (Roess, et al., 2008). The recommended

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eliminate the need for determining the configuration type, Fazio recalibrated Reilly’s model by

adding lane change parameters (Fazio, 1985). HCM2010 adopted NCHRP 3-75’s approach, and

the same methodology form remained in HCM6 (Transportation Research Board, 2010). In

HCM2010, the speed of weaving and non-weaving was predicted based on the predicted lane

changes. Equation (3) and Equation (4) illustrate the weaving and non-weaving speed equation in

HCM2010. In addition, HCM changed the method for predicting the segment capacity. Two

capacity models were introduced: the density-based capacity model shown in Equation (5) and

the weaving demand capacity model shown in Equation (6). The final weaving segment capacity

was the smallest output of the two. Moreover, the predicted capacity became an important factor

for determining the final LOS. If the volume exceeded capacity, then the traffic was considered

to operate at LOS F.

𝑆 = 15 +𝐹𝐹𝑆 − 15

1 + 𝑊 , 𝑤ℎ𝑒𝑟𝑒:

𝑊 = 0.226(𝐿𝐶 𝐿 )

.

(3)

𝑆 = 𝐹𝐹𝑆 − (0.0072𝐿𝐶 ) − (0.0048𝑉

𝑁） (4)

𝑐 = 𝑐 − [438.2(1 + VR) . ] + (0.765𝐿 ) + (119.8𝑁 ) (5)

𝑐 =2400

𝑉𝑅 (𝑓𝑜𝑟 𝑁𝑤𝑙 = 2 𝑙𝑎𝑛𝑒𝑠)

𝑜𝑟 3500

𝑉𝑅 (𝑓𝑜𝑟 𝑁𝑤𝑙 = 3 𝑙𝑎𝑛𝑒𝑠)

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2.2 Related Studies

Various macroscopic and microscopic models have been developed in addition to the

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the merge, diverge, and freeway volume in the auxiliary lane and the freeway lane next to it

(Hess, 1963). But it was found that the operation characteristics are loosely tied to the word

description. In 1983, Leisch independently recalibrated his 1965 Leisch/Norman model. But, the

concept and form of the model did not change significantly.

The first microscopic model was developed by Moscowitz and Newman (Moskowitz &

Newman, July 1962). The model defined the lane-changing distribution between the auxiliary

lane and the freeway lane next to it. However, the model solely tied the lane-changing

distribution to the length of the segment. This model was then further calibrated in other studies

from 1988 to 1995 (M. Cassidy, et al, 1990; Cassidy & May, 1991; Windover & May, 1995;

Ostrom, et al, 1994). All the studies were funded by the California Department of Transportation

(CALTRANS) and the University of California at Berkley. Furthermore, the calibrated models

were all focused on lane changing in the right-most lane of the freeway and auxiliary lanes.

Those models were well-calibrated and provided far greater precision than the model by

Moscowitz and Newman. However, the workload to calibrate the model for different sites was

huge. In the early 2000’s, Lertworawanich and Elefteriadou introduced a methodology that uses

linear optimization and gap acceptance modeling to predict the weaving capacity (P.

Lertworawanich, 2003; Lertworawanich & Elefteriadou, 2001; Lertworawanich & Elefteriadou,

2003). The methodology was theoretically rational; however, the gap acceptance model in the

methodology was developed decades ago by Drew (Drew & D.R., 1967) and Raff and Hart (Raff

& Hart, 1950). Therefore, the model was not applicable to modern freeway flow characteristics.

2.3 Works that Critically Analyzed HCM6:

Even though the weaving segment operational analysis method in HCM6 was updated

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Several studies had found that the HCM6 density predictions deviate from field observation.

Field data collected from 93 sites in California showed that HCM6 overpredicted the density by

8% for balanced weaving segments and 24% for unbalanced weaving segments (Skabardonis &

Mauch, Jan 2015). Additional Bluetooth and video-recorded data revealed that the method

overpredicted the density by an average of 13.4%. The researchers did a follow-up study using

the data collected from Athens, Greece (Skabardonis, Papadimitriou, Halkias, & Kopelias,

2016). The follow-up study showed that HCM6 overestimated 17% density for situations where

the volume ratio (VR) was high.

The studies above also proved that HCM6 underestimates the capacity of weaving

segments, especially in cases where the volume ratio is high. The possible cause of the

underestimation is that HCM6 overemphasizes the impact of volume ratio or it uses the

underestimated basic freeway segment capacity. A study revealed that the observed basic

freeway capacity is significantly lower than the recommended number in the HCM (Kondyli,

George, Elefteriadou, & Bonyani, 2017). In addition, several studies questioned the assumption

of using a density of 43 pc/mi/ln to estimate the weaving segment capacity (Lertworawanich &

Elefteriadou, 2001; Lertworawanich & Elefteriadou, 2003; Lertworawanich & Elefteriadou,

2007). They found this density assumption is not rational and lacks data to validate it.

The HCM6 speed models have also been criticized. Zhou (Zhou, Rong, Wang, & Feng,

2015) found that, compared to the collected field data, HCM6 weaving speed prediction has an

error as high as 40%. In addition, his study also found that in some cases, the predicted weaving

speed is higher than the predicted non-weaving speed, which is counterintuitive. Another study

also found that the HCM6 speed estimation has low sensitivity to the weaving segment length

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when quadrupling the segment length, even with a high weaving intensity condition. The cause

of this problem was found that the non-weaving lane change model does not include the segment

length as a variable.

From the above-mentioned reviews, it is obvious that the HCM6 method needs further

improvement regarding the consistency of the capacity and the speed models to the basic

freeway segment, simplicity of the method, and the sensitivity of the models to the geometric

characteristics of the sites. These are also the motivations and objectives of this research. Several

sensitivity tests are added in the later chapter to illustrate the improvements of the new models

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CHAPTER 3 : METHODOLOGY 3.1 Scope of Work:

This research is intended to develop and test a new operational analysis for ramp weaves

only. Due to limitations in ramp weave sites in NCHRP 3-75, additional data are collected from

6 sites that locate in North Carolina. New data collection and extraction technologies such as

drone and image processing are involved and tested in this research.

After data are prepared, the speed model is designed to provide a simple and reliable

speed estimation for ramp weaves. The model contains fewer input variables and simpler

procedures than the current HCM6. Meanwhile, the model ensures consistency with the basic

freeway segment in estimating segment performance. For example, the model estimation is equal

to a basic freeway segment estimation at low volumes or when there is no weaving demand. The

designed model also has the potential to be calibrated to estimate merge and diverge junction

performance. The specific methodology of site selection, data collection, data extraction and

model formulation are introduced next.

3.2 Site Selection:

To develop and calibrate the ramp weave models, this study was intended to use the data

collected in NCHRP 3-75, which is the same dataset that was used to develop the HCM6

methodology. Even though NCHRP 3-75 collected data from 14 sites nationwide, only three

sites were Type-A weaving segments. Among the three segments, one was a collector

distribution (CD) road, and two were freeway weaving segments. However, of the two freeway

weaving segments, one site used the NGSIM data, which was collected at US 101 in California,

(31)

Six additional sites were surveyed to obtain enough data from ramp weaves. Those sites

were selected to keep variety in the length of the segment and traffic conditions in the dataset.

Moreover, due to a limited research budget, sites were selected only in North Carolina. The

specific locations of all sites are shown below in Table 3.1. The short length of the segments

varied from 268 to 2028 feet. The range of the number of lanes was from three to five. In the

dataset of the additional sites, the traffic condition varied from light to moderate, with a flow rate

that ranged from 418 to 1740 pc/hr/ln. The VR ranged from 0.08 to 0.54. The fraction of heavy

vehicles in most sites was from 2% to 4%, except for the I-95 site, which had 14% to 28% heavy

vehicles. The data were collected using both ground-based and drone-based videos.

Table 3.1

Location of Study Sites

Site Name Location Road Name (on-ramp) Road Name (off-ramp)

I-440 EB @ Ridge

Road Raleigh, NC Ridge Road Glenwood Avenue

I-40 EB @ Saunders Raleigh, NC S Saunders Street Hammond Road

I-40 WB @

Saunders Raleigh, NC Hammond Road S Saunders Street

Wade Avenue WB Raleigh, NC I-440 Blue Ridge Road

I-40 EB @ Cary

Town Blvd Raleigh, NC Cary Town Blvd I-440

I-95 SB @ Spring

Branch Dunn, NC E Cumberland Street Spring Branch Road

NCHRP 3-75

(32)

3.3 Data Collection:

At the I-440 Ridge Road site, the data were collected by two cameras that were mounted

on the bridge crossed above the middle of the segment. Each camera recorded one portion of the

directional traffic. The data was captured between 3:00 pm to 6:00 pm from Wednesday to

Friday. In total, nine hours of videos were collected. Excluding the period where traffic was

under a congested condition, approximately six hours of videos were used in the model

development.

The other sites’ data were collected by using a drone. The drone used in the study was the

DJI Inspire V-1 drone, which is shown in Figure 3.1. It recorded 4k resolution videos at an

elevation of 400 ft. above the segment. It recorded 10 to 15 minutes of video for each battery

cycle. It should be noted that the drone also required time to land, replace the battery, and take

off. Thus, the difference between the video length and the data collection period can be observed.

At the I-40 Saunders sites, the drone recorded the traffic in both directions from 4:00 to 6:00 pm

at the same time. A total of 89 minutes of videos were collected. The team captured one-hour

videos from 7:00 am to 8:30 am at the I-95 SB site. At the Wade Avenue site, 2 hours of footage

was collected from 8:00 am to 11:00 am. In addition, approximately 46 minutes of videos were

collected from 4:30 pm to 5:30 pm at the I-40 Cary Town Blvd site. In general, one hour of

(33)

Figure 3.1 - Image of the Drone

3.4 Data Extraction:

The data were aggregated into five-minute intervals to maintain sufficient sample sizes

and consistency with NCHRP 3-75 dataset. The data were reduced manually from videos. The

volumes were counted based on origin to destination (OD) at each five-minute interval. The

number of heavy vehicles was also counted in the OD. The timestamps were recorded when the

vehicle entered the on-ramp gore point and exited the off-ramp gore point. Therefore,

space-mean-speed (SMS) for the vehicle was calculated by dividing the gore to gore distance by

subtracted timestamps. Only a random sample of vehicles’ timestamps are recorded by OD to

extract speed information since the traffic volumes were large. The speed samples were then

weighted by the OD volumes to obtain the average space mean speed of the traffic in the

weaving segment in five-minute intervals. As to the configuration information, the length of the

segment was measured from on-ramp gore point to off-ramp gore point. The FFS for each site

was estimated by using the 85th percentile speed in the speed data that was downloaded from

(34)

calculated FFS. Thus, the prepared dataset contained the volume, number of heavy vehicles,

space mean speed information for each OD in five-minute intervals, length of the segment,

number of lanes, and the site’s FFS.

Besides the manual data reduction, some automatic image processing methods were also

tested. The tested video image processing tools were machine learning code that was developed

by the University of Florida and commercial service that was provided by a UK traffic video

analytics company named GoodVision (GoodVision Ltd., 2019). However, both tools were

proved that they are time consuming and require highly stable drone footage. Based on the

experience of implementing the tools, it suggests that the optimal weather for drone data

collection is cloudy and non-windy days for automatic image processing.

3.5 Data Filtering:

Since some of the videos were recorded during peak hours, parts of the prepared data

were found to occur under saturated conditions. Because the methodology of the research is

analyzing under-saturated traffic, data points that had a density higher than 43 mi/hr/ln or the

average space mean speed of all traffic lower than 40 mi/hr were excluded. The two data points

that followed the congested data point were considered as recovering from congestion and were

also excluded. After removing the oversaturated data points and other outliers, a total of 140,

five-minute data points (equivalent to 11 hours 40 minutes) remained in the dataset. Those data

points were used to calibrate the models. Table 3.2 shows a summary of the configuration and

data points for each site. After prepared the calibration dataset, the speed and capacity model

(35)

Table 3.2

Summary of Configuration and Data Points of the Site

Freeway Weaving Sites

Segment Short Length (ft)

Number of lanes

Range of Flow rate (pc/hr/ln)

VR Range

𝑉 𝐶 ∗

Number of 5-min Observations

I-440 @ Ridge

Road 268 4 1236 - 1665 0.16 - 0.28 0.52 - 0.7 66

NCHRP Sky03 360 3 926 - 1467 0.25 - 0.49 0.39 - 0.62 12

I-40 EB @

Saunders 976 5 1060 - 1360 0.23 - 0.3 0.44 - 0.56 13 I-40 WB

Saunders 1285 5 801 - 1047 0.17 - 0.27 0.33 - 0.43 13

I-95 SB @

Spring Branch 1234 3 418 - 604 0.08 - 0.37 0.17 - 0.25 11

Wade Avenue

WB 1135 3 716 - 1428 0.47 - 0.73 0.3 - 0.61 16

I-40 EB @ Cary Town

Blvd 2028 4 1518 - 1740 0.27 - 0.34 0.62 - 0.71 9

Total 140

(36)

3.6 Conceptual Model Formulation:

The two main motivations of this research are to simplify the current weave methodology

and ensure consistency between the freeway segment and the weaving segment. Conceptually,

with the same volumes and number of lanes and lengths, a weaving segment usually yields a

lower average space mean speed than a basic freeway segment. The difference between speeds is

caused by the turbulence of the weaving flows. In addition to the configuration, such as the

length of the segment, the turbulence is sensitive to the weaving demand. If the weaving segment

contains zero weaving traffic, then it is considered as operating as a basic freeway segment. In

other words, the predicted average speed of the weaving segment should equal to the speed

predicted from an equivalent basic freeway segment. Equation (7) illustrates the hypothesized

speed model, which presents the conceptual relationship between the weaving segment speed

(𝑆 ) and the equivalent basic segment speed (𝑆 ). The SIW is the speed impedance term that

caused by the weaving turbulence. The specific variables in SIW are introduced in the next

section.

𝑆 = 𝑆 − 𝑆𝐼𝑊 (7)

The difference of the speeds is normalized for analysis since the free flow speed (FFS) is

different across sites. Therefore, the transformation shown in Equation (8) was made. The X

variables in Equation (8) represent the variables included in the SIW, which contribute to the

weaving turbulence. The values (, , ,…) are the parameters of the model. To calibrate those parameters, the model is further transformed into a linearized form, as shown in Equation (9)

below. To find the optimal values for the parameters and validate the significance of each

variable, a linear regression analysis was made. Detailed results of the analysis are given in a

(37)

=  𝑋 𝑋 … . . 𝑋 . (8)

𝐿𝑛 ( ) = 𝐿𝑛 () + 𝐿𝑛 (𝑋 )+ 𝐿𝑛 (𝑋 ) + ⋯

+

𝐿𝑛 (𝑋 ) . ₍₉₎

Based on the analysis results, SIW is shown to have various explanatory variables, such

as volume ratio (VR) and the inverse of the weaving segment short length (Ls). Equation (10)

represents one of the model forms that has a good fit to the observed speed.

𝑆 = 𝑆 −  𝐹𝐹𝑆 (𝑉𝑅

𝐿 ) (10)

In this proposed model, the SIW does not include a term that presents the demand

over-capacity. Therefore, it was unknown whether the model could correctly present the demand

impact. In the initial model forms, all the SIW had a variable to present the demand impact to

the speed. However, after calibration, the variable in different models was found to be either

not significant or had a negative parameter, which is counterintuitive. It should be noted that the

term 𝑆 is calculated based on the demand flow rate. Therefore, the speed drop due to demand is

already addressed by 𝑆 . SIW only represents the additional effects of the turbulence caused by

weaving traffic. In this case, SIW thus is a fixed value when determining the volume at capacity

because the component of traffic (percentage of weaving traffic) does not change. Furthermore,

as explained below, this is important during estimating the capacity of the weaving segment.

Referring to HCM6, the density at capacity for the weaving segment is 43 pc/mi/lane

(Transportation Research Board, 2016). Therefore, the relationship between the speed at the

capacity per lane (𝑉 ) and the density can be expressed as follows:

(38)

The estimation of volume at capacity (𝑉) will also have two approaches because of the

𝑆 term that has two different equations for the two volume regimes. If 𝑉 is below the

breakpoint volume (BP), then 𝑆 is equal to the free flow speed (FFS). In this case, the capacity

can be calculated by Equation (12) as follows:

𝑉 = 43 ∗ (𝐹𝐹𝑆 − 𝑆𝐼𝑊) . (12)

In addition, since 𝑉 is unknown, it cannot be used to determine the volume regime. In

this case, 𝑉 ≤ 𝐵𝑃 condition can be transferred into the following equation:

𝑆𝐼𝑊 ≥ 𝐹𝐹𝑆 − . (13)

Now, SIW can be used to determine if 𝑉 is greater or less than BP. And when 𝑉 is

greater than the breakpoint volume, Equation (11) transforms into the following equation:

= (𝐹𝐹𝑆 −( / )( )

( ) ) − 𝑆𝐼𝑊 . (14)

In this case, 𝑉 appears on both sides of the equation. To calculate 𝑉, the equation is

further transformed into a quadratic equation by letting

𝑋 = 43 × (𝐹𝐹𝑆 − 𝑆𝐼𝑊)

and

𝑌 = 43 ×( ) ( ) .

Therefore, the quadratic equation is

𝑌𝑉 − 𝑉 (2𝑌𝐵𝑃 − 1) − (𝑋 − 𝑌𝐵𝑃 ) = 0 . (15)

By rearranging the equation above, 𝑉 can be expressed by

(39)

It should be noted that the capacity per lane calculated by Equation (12) and Equation

(16) is determined by the density at 43 pc/mi/ln. Referring to HCM6, the weaving segment

capacity is also controlled by the weaving demand. For ramp weaves, the weaving demand

should not exceed 2400 pc/hr. Therefore, HCM6 Equation 13-7 (shown in Equation (6)) is also

used to calculate the weaving segment capacity. The final weaving segment capacity is the lower

value between the capacity determined by the density and the capacity determined by the

weaving demand. Then, if the traffic is below the capacity (𝑉 < 𝑉 ), the segment average space

mean speed and density can simply be calculated by using the equations below:

𝑆 (𝑉) = 𝑆 (𝑉) −  𝐹𝐹𝑆 (𝑉𝑅 𝐿 )

and

(17)

𝐷 = . (18)

A graphical illustration can help to understand the above description. Figure 3.2

represents a speed and flow curve with an FFS of 60 miles per hour. The weaving density at the

capacity line (Dc= 43 pc/mi/ln) is represented by the dotted line. The SIW, in this case, was

assumed to be 10 miles per hour. Therefore, the capacity for this weaving segment is estimated

by using the line CDA, where the speed difference between 𝑆 and speed at capacity (line CD) is

equal to the SIW. Therefore, the capacity is 2,020 pc/hr/ln at point A and the speed at capacity is

47 miles per hour. Another scenario is when the traffic demand is 1,250 pc/hr/ln which is

presented by line EB in Figure 3.2, and the average space mean speed of the segment is proved

to be 50 miles per hour (point F) by using FFS (point E) minus the SIW. Therefore, the

(40)

point G, where SIW is zero (no weaving traffic), represents the highest possible capacity can be

estimated in this weaving segment.

Figure 3.2 - Graphical Illustration for Capacity and LOS Estimation 0

10 20 30 40 50 60 70

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

S

M

S

(

m

i/

hr

)

Volume (pc/hr/ln) Speed Volume Relationships

SIW E

V B Vc A

D C

CB

SIW _G

F

(41)

3.7 Variables in SIW:

This section describes the potential variables that can be involved in SIW. The variables

should present the effect of demand, weaving flows, and configuration since SIW illustrates

turbulence of the weaving traffic that causes the speed drop between the weaving segment and

the equivalent basic segment. The demand effect can be presented by and the effect of

weaving flows and configuration can be presented in various variables. Three hypothesizes can

be made to categorize and better explain those variables:

1. The turbulence of the weaving traffic is only related to the traffic movement. In other

words, the percentage of different combination of weaving flow in total traffic ( or

%𝑉 , 𝑤ℎ𝑒𝑟𝑒 𝑉 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑤𝑒𝑎𝑣𝑖𝑛𝑔 𝑓𝑙𝑜𝑤𝑠 ) is the

only impact factor for weaving turbulence. The configuration such as segment short

length (Ls) has no contribution.

2. The density of different weaving flows ( ) is the only contributor to SIW. The

percentage of weaving traffic does not impact the turbulence.

3. The traffic composition and configuration jointly contribute to the SIW (% ). The length

of the segment will enhance or dissolve the turbulence in this case.

Table 3.3 illustrates the different flow’s candidate variables that fit the different

hypotheses on candidate. It should be noted that besides the weaving flows, ramp to ramp is also

considered to be a flow that impacts the SIW. The reason for including this flow is that ramp to

ramp is usually a slow speed flow compared to the freeway to freeway flow, especially in short

segments. Therefore, it may contribute to the speed difference between the weaving segment and

(42)

lane are a combination of the ramp to ramp flow with freeway to ramp, ramp to freeway, and

weaving flow.

Table 3.3

Candidate Variables that may Explain the Effect of Flow and Configuration on SIW

Movement Candidate 1 Candidate 2 Candidate 3

Freeway to Ramp 𝑉

𝑉 𝑉 𝐿 𝑉 𝑉 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠)

Ramp to Freeway 𝑉

Ramp to Ramp 𝑉

Weaving Flow 𝑉𝑅 𝑉

𝐿

𝑉𝑅 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠)

On-ramp 𝑉

Off-ramp 𝑉

𝑉 𝑉 𝐿 𝑉 𝑉 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) Flow on

Auxiliary Lane *

(43)

The candidate variables in SIW were tested based on the trend analyses that plotted each

variable against the normalized space mean speed. Figure 3.3 below illustrates the demand

variable plot that was used in this trend analysis. Figure 3.4 to Figure 3.24 below show the plots

of candidate variables in hypotheses 1, 2 and 3.

Figure 3.3 shows that has a relative strong trend with the normalized space mean speed

though the variance of the speed trendline increase with . It implies that the demand impacts

the segment average space mean speed, but there are also other potential factors influence the

speed drop which causes the observed variance. Figure 3.5 shows that most candidate 1 variables

do not have strong or moderate trend with speed changes except and . Figure 3.12

clearly indicates that the length of the segment is impacting the significance of the weaving

turbulence. The variables including , , , and , clearly shows a

relationship with speed changes. Interestingly, the variable plot shows that the speed drops

significantly when the ramp to ramp over segment length ratio is high. It signifies that the ramp

to ramp flow is influencing the segment average speed especially for the short segment. At last,

based on the observation of Figure 3.19,

( ), ( ), ( ), and ( ) are

proved that they have relationship with segment average speed changes. And again, the

( ) variable plot shows a very strong contribution to the speed drops.

Overall, the variables that have moderate to strong trends with space mean speed changes

are , , , , ,

(44)

non-correlated variables are then combined with each other to form multiple candidate model

forms that are shown in the next section.

Figure 3.3 - versus Normalized SMS Plot 0.40

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.20 0.40 0.60 0.80

N

or

m

al

iz

ed

S

M

S

(

O

bs

er

ve

d

sp

ee

d

/ F

FS

)

V/Cb

(45)

Figure 3.4 - Candidate 1 versus Normalized SMS Plot

Figure 3.5 - Candidate 1 versus Normalized SMS Plot 0.40

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

N

or

m

al

iz

ed

S

M

S

Vrf/V

Vrf/V versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.10 0.20 0.30 0.40

N

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S

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Vfr/V

(46)

Figure 3.7 - Candidate 1 VR versus Normalized SMS Plot 0.40

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.02 0.04 0.06 0.08

N

or

m

al

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S

M

S

Vrr/V

Vrr/V versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.20 0.40 0.60 0.80

N

or

m

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S

M

S

VR

(47)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

N

or

m

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S

M

S

Von-ramp/V

Von-ramp/V versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.10 0.20 0.30 0.40

N

or

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S

M

S

Voff-ramp/V

(48)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.20 0.40 0.60 0.80

N

or

m

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S

M

S

Vaux/V

(49)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.50 1.00 1.50 2.00 2.50

N

or

m

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S

M

S

Vrf/Ls (pc/hr/ft) Vrf/Ls versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00

T

it

le

(50)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.20 0.40 0.60 0.80 1.00

N

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S

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S

Vrr/Ls (pc/hr/ft) Vrr/Ls versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00

N

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(51)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

N

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S

Von-ramp/Ls (pc/hr/ft) Von-ramp/Ls versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00

N

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(52)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

N

or

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S

M

S

Vaxu/Ls

(53)

Figure 3.18 - Candidate 3 ( ) versus Normalized SMS Plot

Figure 3.19 - Candidate 3 ( ) versus Normalized SMS Plot 0.40

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.50 1.00 1.50 2.00 2.50 3.00

N

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S

M

S

(Vrf/V)/Ls (ratio/mi) (Vrf/V)/Ls versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00

N

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(54)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.20 0.40 0.60 0.80 1.00 1.20

N

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S

(Vrr/V)/Ls (ratio/mi) (Vrr/V)/Ls versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 2.00 4.00 6.00 8.00

N

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(55)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 0.50 1.00 1.50 2.00 2.50 3.00

N

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S

(Von-ramp/V)/Ls (ratio/mi) (Von-ramp/V)/Ls versus Normalized SMS

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00

N

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(56)

0.50 0.60 0.70 0.80 0.90 1.00 1.10

0.00 2.00 4.00 6.00 8.00

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(57)

CHAPTER 4 : MODEL RESULTS 4.1 Candidate Model Forms and Model Calibration:

Twelve candidate model forms were created during the calibration, as listed below. It

should be noted that Model 5 contained a variable that has not been explained. The term

∗

is a transformation from the volume ratio. Because several prior works have shown

that on-ramp traffic has more contribution to the speed drop, it adds a weighting factor ‘a’ for the

ramp to freeway volume. Model forms 6 to 10 were motivated by the calibration results from

Model 1 to Model 4, because either had a negative parameter that is counterintuitive or proved

to be insignificant. Model 11 and 12 were the transformation from Model 8 and Model 9 which

separate the segment length as an independent variable.

Interestingly, these model forms can conceptually fit the ramp junctions as well. For

example, when Model 1 is used to predict the speed of a merging section, the variable will

become zero and the variable will present the ratio of merging demand over acceleration lane

length. But further studies are needed to validate this conceptual approach, this paper is focusing

on ramp weaves only.

Model 1: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗ ∗

Model 2: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗ ∗

Model 3: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗

( )

Model 4: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗

(58)

Model 5: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗ ∗

( )

Model 6: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗

Model 7: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗

Model 8: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗

( ) ∗ ( )

Model 9: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗

( ) ∗ ( )

Model 10: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗

( )

Model 11: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗ ∗ 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠)

Model 12: 𝑆 = 𝑆 − 𝛼 ∗ 𝐹𝐹𝑆 ∗ ∗ ∗ 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠)

All the model forms above were then transformed into linearized forms, was shown in

Equation (9). The parameters were calibrated using linear regression. However, due to the fact

that the Model 5 was still non-linear after transformation, a different approach was applied to it.

The ‘a’ value was tested within a range of values from 0.5 to 3.0. The linear regression analysis

was applied for each ‘a’ value assigned model. The optimal ‘a’ value was determined by

comparing the best goodness of fit among the models. The detailed model calibration result for

different ‘a’ value assigned to Model 5 is shown in Table 4.1. It should be noted that for the ‘a’

value from 0.5 to 1, the was not significant in the model, so there is no coefficient that can be

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the best adjusted R-squared value and the lowest sum of squared residual. Table 4.2 below shows

the parameter calibration result for all candidate models

Table 4.1

Parameter Calibration Result and Goodness of Fit for Different ‘a’ Value Models

Coefficient

a value α V

𝐶 VR SSR

Adjusted R-squared

0.5 0.0778 NA 0.707 2347.11 0.7308

0.6 0.0729 NA 0.740 2306.93 0.7354

0.7 0.0685 NA 0.769 2275.66 0.7390

0.8 0.0646 NA 0.795 2250.22 0.7419

0.9 0.0611 NA 0.818 2232.13 0.7440

1 0.0579 NA 0.838 2223.55 0.7450

1.1 0.0709 0.292 0.790 2215.87 0.7440

1.2 0.0698 0.327 0.798 2208.62 0.7448

1.3 0.0689 0.362 0.804 2205.55 0.7452

1.4 0.0681 0.395 0.810 2200.77 0.7458

1.5 0.0674 0.427 0.814 2201.21 0.7457

1.6 0.0667 0.458 0.818 2203.32 0.7455

1.7 0.0661 0.487 0.82 1 2205.91 0.7452

1.8 0.0656 0.515 0.823 2209.94 0.7447

1.9 0.0651 0.542 0.825 2214.89 0.7441

2 0.0646 0.568 0.826 2224.09 0.7431

2.1 0.0642 0.593 0.827 2230.67 0.7423

2.2 0.0638 0.617 0.827 2241.33 0.7411

2.3 0.0634 0.640 0.828 2248.52 0.7402

2.4 0.0631 0.661 0.827 2258.92 0.7390

2.5 0.0627 0.682 0.827 2270.09 0.7377

2.6 0.0624 0.702 0.827 2277.80 0.7368

2.7 0.0621 0.721 0.826 2289.25 0.7355

2.8 0.0618 0.740 0.825 2301.64 0.7341

2.9 0.0615 0.757 0.824 2312.62 0.7328

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Table 4.2

Model Calibration Results and Goodness of Fit Comparison Between Candidate Models and HCM6

Model Number

Parameters for 1 2 3 4 5 6 7 8 9 10 11 12 HCM6

𝛼 0.095 0.069 0.057 0.055 0.068 0.14 0.12 0.12 0.10 0.048 0.097 0.082 NA

𝑉

𝐶 -0.58 -0.83 NA NA 0.395 NA NA NA NA NA NA NA NA

𝑉

𝐿 0.52 NA NA NA NA NA NA NA NA NA NA NA NA

𝑉

𝐿 0.34 NA NA NA NA NA NA NA NA NA NA NA NA

𝑉

𝐿 NA 0.61 NA NA NA NA NA NA NA NA NA NA NA

𝑉

𝐿 NA 0.33 NA NA NA NA NA NA NA NA NA NA NA

𝑉𝑅

𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) NA NA 0.83 NA NA NA NA NA NA NA NA NA NA 𝑉

𝑉 ∗ 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) NA NA NA 0.83 NA NA NA NA NA NA NA NA NA 1.4 ∗ 𝑉 + 𝑉

𝑉 ∗ 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) NA NA NA NA 0.81 NA NA NA NA 0.898 NA NA NA 𝑉

𝐿 NA NA NA NA NA 0.51 NA NA NA NA NA NA NA

𝑉

𝐿 NA NA NA NA NA 0.25 NA NA NA NA NA NA NA

𝑉

𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) NA NA NA NA NA NA 0.58 NA NA NA NA NA NA 𝑉

𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) NA NA NA NA NA NA 0.20 NA NA NA NA NA NA 𝑉

𝑉

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Table 4.2

Model Calibration Results and Goodness of Fit Comparison Between Candidate Models and HCM6 (Continued)

Model Number

Parameters for 1 2 3 4 5 6 7 8 9 10 11 12 HCM6

𝑉 𝑉

𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠) NA NA NA NA NA NA NA 0.40 NA NA NA NA NA 𝑉

𝑉 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠)

NA NA NA NA NA NA NA NA 0.51 NA NA NA NA

𝑉 𝑉 𝐿 (𝑖𝑛 𝑚𝑖𝑙𝑒𝑠)

NA NA NA NA NA NA NA NA 0.37 NA NA NA NA

𝑉

𝑉 NA NA NA NA NA NA NA NA NA NA 0.45 NA NA

𝑉

𝑉 NA NA NA NA NA NA NA NA NA NA 0.35 NA NA

𝑉

𝑉 NA NA NA NA NA NA NA NA NA NA NA 0.51 NA

𝑉

𝑉 NA NA NA NA NA NA NA NA NA NA NA 0.30 NA

𝐿 NA NA NA NA NA NA NA NA NA NA -0.91 -0.94 NA

SSR 2242 1835 2224 2064 2201 2359 2126 1923 1663 2241 1893 1637 8727

SST 8847 8847 8847 8847 8847 8847 8847 8847 8847 8847 8847 8847 8847

Adjusted

R-squared 0.747 0.793 0.749 0.767 0.751 0.733 0.760 0.783 0.812 0.743 0.780 0.809 0.013

AICc 794.91 765.86 788.52 778.07 789.17 798.88 784.32 770.28 749.98 789.59 770.18 749.88 977.89

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By applying the results of the analysis to the model formula, the final models with

significant variables and calibrated parameters are listed below:

Model 1: 𝑆 = 𝑆 − 0.0952 ∗ 𝐹𝐹𝑆 ∗ . ∗ . ∗ .

Model 2: 𝑆 = 𝑆 − 0.0699 ∗ 𝐹𝐹𝑆 ∗ . ∗ . ∗ .

Model 3: 𝑆 = 𝑆 − 0.0579 ∗ 𝐹𝐹𝑆 ∗

( ) .

Model 4: 𝑆 = 𝑆 − 0.0555 ∗ 𝐹𝐹𝑆 ∗

∗ ( ) .

Model 5: 𝑆 = 𝑆 − 0.0681 ∗ 𝐹𝐹𝑆 ∗ . ∗ . ∗

∗ ( ) .

Model 6: 𝑆 = 𝑆 − 0.140 ∗ 𝐹𝐹𝑆 ∗ . ∗ .

Model 7: 𝑆 = 𝑆 − 0.124 ∗ 𝐹𝐹𝑆 ∗

( ) .

∗

( )

.

Model 8: 𝑆 = 𝑆 − 0.125 ∗ 𝐹𝐹𝑆 ∗

( )

. ∗

( )

.

Model 9: 𝑆 = 𝑆 − 0.109 ∗ 𝐹𝐹𝑆 ∗

( )

. ∗

( )

.

Model 10: 𝑆 = 𝑆 − 0.0481 ∗ 𝐹𝐹𝑆 ∗ . ∗

∗ ( ) .

Model 11: 𝑆 = 𝑆 − 0.0975 ∗ 𝐹𝐹𝑆 ∗ . ∗ . ∗ (𝐿 (𝑚𝑖𝑙𝑒𝑠)) .

Model 12: 𝑆 = 𝑆 − 0.0824 ∗ 𝐹𝐹𝑆 ∗ . ∗ . ∗ (𝐿 (𝑚𝑖𝑙𝑒𝑠)) .