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STUDY OF FREEWAY MERGE BOTTLENECKS AND DEVELOPMENT OF NEW
LOCAL TRAFFIC RESPONSIVE RAMP NIETERING STRATEGY

By

Heung-Un Oh

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

2000

ABSTRACT

STUDY OF FREEWAY MERGE BOTTLENECKS AND DEVELOPMENT OF NEW
LOCAL TRAFFIC RESPONSIVE RANIP NfETERING STRATEGY

By

Heung-Un Oh

Most local traffic responsive ramp metering strategies are based on the concept that the
bottleneck and the worst section at the ramp merge are both located downstream of the
on-ramp. The worst section typically refers to the section that experiences the highest
volume, regardless of speed and density. Consequently, ramp metering typically is based
on the measurement downstream of the on-ramp for congestion prevention and clearance
purposes. However, the concept is still controversial, since several bottleneck studies
have reported different observations with respect to the location of the worst section.

In this study, the hypothesis is stated and tested that, under congested conditions,
while the bottleneck section of the on-ramp is located downstream of the ramp, the worst
section, in terms of density and speed, is upstream of the on-ramp.

Through the study of traffic stream models and field data observations, it is
concluded that the hypothesis is acceptable. Furthermore, no change indicative of a
breakdown was observed downstream of an on-ramp prior to congestion. It therefore is
concluded that, as the breakdown occurs first upstream, modification of existing local
traffic responsive control strategies which reflect the findings of the study is expected to

result in better management of traffic at ramp merge areas.

This suggests the introduction of new ramp metering strategies for upstream
monitoring, as a basis for control. The new ramp metering strategy introduced in this
study is MALINEA, which is a modified version of ALINEA, an existing ramp metering
strategy.

After the study of parameters and associated theories, MALINEA was constructed
as a mathematical and macroscopic flow model and was verified using field data.
MALINEA was tested as a feedback control and ramp metering model using simulation.
Comparisons of MOEs, obtained from simulation runs of MALINEA and ALINEA under
similar conditions, showed that MALINEA demonstrated a superior performance.
Therefore, MALINEA was introduced as a new ramp metering strategy with good

potential.

Dedicated to My Family

iv

ACKNOWLEDGMENTS

From the beginning of my graduate education to the conclusion of my research I
continuously have needed encouragement, guidance, criticism and advice. I received this
in abundant measure from Dr. Virginia P. Sisiopiku, for whose wise and willing counsel I
always shall be gratefiJl.

My dissertation has required my own devotion, of course, but also that of an
experienced advisor. I am thankful for the time, effort and fine intellect of Dr. William C.
Taylor who served so ably and cordially in this capacity. I thank also Drs. Thomas L.
Maleck and Raoul D. LePage whose advisory committee time and wisdom helped form
both the purpose and perspective of this dissertation.

I hardly can express sufficiently my deep appreciation of my wife Eun-Mi, and my
sons Jong-Taek and Min-Tack, who have sustained me both spiritually and physically

throughout this endeavor.

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................... ix
LIST OF FIGURES ........................................................................................................ xi
CHAPTER 1
INTRODUCTION ........................................................................................................... 1
1.1 Existing Ramp Control Strategies .......................................................................... 1
1.2 Bottleneck Concept and New Hypothesis for Worst Section .................................. 2
1.3 Desired New Ramp Metering Algorithm ................................................................ 4
1.4 ALINEA Model ..................................................................................................... 5
1.5 Advantages of ALINEA ......................................................................................... 6
1.6 Potential ALINEA Model Modification ................................................................. 7
1.7 Research Objectives ............................................................................................... 8
CHAPTER 2
DATA DESCRIPTION AND RAMP VOLUME APPROXIMATION .......................... 10
2.1 Data Description .................................................................................................. 10
2.2 Ramp Volume Approximation ............................................................................. 12
CHAPTER 3
WORST SECTION ANALYSIS AND RESULTS ........................................................ 14
3.1 Traffic Stream Models ......................................................................................... 14
3.2 Empirical Results ................................................................................................. 19
3.3 Breakdown Procedure .......................................................................................... 27
3.4 Summary ............................................................................................................. 28
CHAPTER 4
PARAMETER CHARACTERISTICS .......................................................................... 31
4.1 Typical Trend of A-and K-Curves at Ramp Merge Bottleneck .............................. 32
4.1.1 A-Curve Trend Study ................................................................................. 32
4.1.2 K-Curve Trend Study ................................................................................. 42
4.2 Traffic Characteristic of A (Occupancy Relationship)-Curves .............................. 43
4.2.1 Linear Assumption of A-Curve ................................................................... 43
4.2.2 Fluctuation under Congestion ..................................................................... 52
4.2.3 Location Specific A-Curve ......................................................................... 55
4.3 Time Lag Effect .................................................................................................. 62
CHAPTER 5
MODIFIED MODEL CONSTRUCTION AND VERIFICATION ................................ 68
5.1 Upstream Occupancy Estimation by Ramp Volume ............................................. 69
5.1.1 Model Construction ..................................................................................... 69
5.1.2 Model Verification ...................................................................................... 73
5.2 Upstream Occupancy Estimation by Downstream Occupancy .............................. 76

vi

5.2.] Model Construction ..................................................................................... 76

5.2.2 Model Verification ...................................................................................... 76
CHAPTER 6
APPLICATION OF MALINEA .................................................................................... 81
6.1 Flow Chart of MALINEA Ramp Metering ........................................................... 81
6.2 K-Parameter Application ...................................................................................... 83
6.2.1 Description .................................................................................................. 83
6.2.2 K-Parameter Calculation ............................................................................. 83
6.3 A-Parameter Application ...................................................................................... 84
6.3.1 A-Parameter Calculation ................... . .......................................................... 84
6.3.2 Sensitivity of A ........................................................................................... 89
6.4 n-Selection ........................................................................................................... 90
6.4.] n-Calculation ............................................................................................... 90
6.4.2 n-Application During Operation .................................................................. 90
6.5 Applicability under Uncongested Condition ......................................................... 92
6.5.1 Applicability under ALINEA ...................................................................... 92
6.5.2 Applicability under MALINEA ................................................................... 93
6.6 Detector Location ................................................................................................ 94
6.7 Desirable Upstream Occupancy ........................................................................... 98
6.7.1 Desirable Occupancy Based on Traffic Characteristics ................................ 98
6.7.2 Desirable Occupancy Based on Feedback Control Theory ........................... 99
CHAPTER 7
OPERATIONAL TEST USING SIMULATION ......................................................... 100
7.1 Introduction of Operational Test ........................................................................ 100
7.2 Simulation Run Plan .......................................................................................... 100
7.2.1 Definition .................................................................................................. 101
7.2.2 Monte Carlo Simulation in Ramp Metering Studies ................................... 101
7.2.3 Flow Charts ............................................................................................... 103
7.2.4 Variable Description ................................................................................. 103
7.2.5 Monte Carlo Sampling .............................................................................. 108
7.2.6 Data Description ....................................................................................... 112
7.2.7 MOEs ....................................................................................................... 112
7.3 Results and Discussion ...................................................................................... 1 15
7.3.1 MALINEA and ALINEA Comparison ...................................................... 1 15
7.3.2 “Wrong” A-effect ...................................................................................... 119
7.3.3 “Wrong” n-effect ...................................................................................... 120
7.4 Summary ........................................................................................................... 120
CHAPTER 8
SUMMARY AND CONCLUSIONS ........................................................................... 125
8.1 Summary ........................................................................................................... 125
8.2 Conclusions ....................................................................................................... 129
8.3 Limitations and Recommendations .................................................................... 130
8.3.1 Effect of Detailed Geometric Conditions ................................................... 130

vii

8.3.2 Speed Based Model ................................................................................... 130

8.3.3 FRESIM Issues ......................................................................................... 131
8.3.4 Effect on Parallel Corridor ........................................................................ 131
BIBLIOGRAPHY ....................................................................................................... 133

viii

LIST OF TABLES

Table 2.1 Geometric Characteristics of Major Detectors ................................................ 11
Table 3.1 Detector Data between 6:30 and 6:50 AM at Detectors 63 and 64 .................. 29
Table 4.1 Detector Data between 5:59 and 6:14 at Detectors 100 and 101 ..................... 36
Table 4.2 Examples of Linear Regression Results .......................................................... 50
Table 4.3 Sample Description for A-Curve Slope Predictor Study .................................. 60
Table 4.4 Coefficient Description for A under All Variables .......................................... 61
Table 4.5 Coefficient Description for A under Variables of LA and DD ......................... 61
Table 5.1 SE between Estimated and Observed Occupancies for MALINEA ................. 76
Table 5.2 SE between Estimated and Observed Occupancies for Upstream Occupancy

Estimation Model .......................................................................................................... 77
Table 6.1 Standard Error by MLM ................................................................................. 87
Table 6.2 Standard Error by SS ..................................................................................... 88
Table 6.3 Sensitivity of A-Values Based on MLM and SE Results ................................. 89
Table 7 .1 Contribution of Each Variable to Deviation of 0,. ........................................ 106
Table 7 .2 Calculation of Chi-Square Test Value for Mainline Volume ......................... 109
Table 7 .3 Calculation of Chi-Square Test Value for Ramp Discharge Volume ............. 109
Table 7.4 Basic Condition of Simulation Test .............................................................. 112
Table 7.5 MOE Comparison (case: MALINEA vs. ALINEA) ..................................... 118
Table 7.6 MOE Comparison (case: A = 0.84 vs. 1.24) ................................................. 122
Table 7.7 MOE Comparison (case: A = 0.84 vs. 0.44) ................................................. 122
Table 7.8 MOE Comparison (case: n =3 vs. 1) ............................................................. 123
Table 7.9 MOE Comparison (case: n =3 vs. 2) ............................................................. 123

ix

Table 7.10 MOE Comparison (case: n =3 vs. 4) ........................................................... 124

Table 7.11 MOE Comparison (case: n =3 vs. 5) ........................................................... 124

LIST OF FIGURES

Figure 2.1 Data Description at Ramp Merge .................................................................. 12
Figure 3.1 Importance of Location (May, p288) ............................................................ 15
Figure 3.2 Comparison of Traffic Characteristics Upstream and Downstream of On-
Ramp ................................................................................................................... 18
Figure 3.3 Schematic of Study Site ................................................................................ 19
Figure 3.4 Comparison of % Occupancy under Congestion at Detectors 62, 63, 64, and 65
(2/ 18/97) .............................................................................................................. 21
Figure 3.5 Relationship between Upstream and Downstream Volume at Ramp Merge
(2/18/97, Detectors 63, 64) ................................................................................... 22
Figure 3.6 Relationship between Upstream and Downstream Speed at Ramp Merge
(2/18/97, Detectors 63, 64) ................................................................................... 23
Figure 3.7 Relationship between Upstream and Downstream Occupancy at Ramp Merge
(2/18/97, Detectors 63, 64) ................................................................................... 26
Figure 4.1 A-Curve Shape between Detectors 100 and 101 (5/13 PM) ........................... 33
Figure 4.2 Volume-Occupancy at Detector 101 (5/13 PM) ............................................ 34
Figure 4.3 A-Curve Trend between Detectors 100 and 101 (1/15 AM) ........................... 38
Figure 4.4 Volume-Occupancy at Detector 64 (1/15 AM) .............................................. 39
Figure 4.5 A-Curve Trend between Detectors 92 and 93 (7/29 PM) ............................... 40
Figure 4.6 Volume-Occupancy at Detector 93 (7/29 PM) ............................................. 41
Figure 4.7 General A-Curve Shape ................................................................................ 42
Figure 4.8 Approximation of A-Curves Based on Athol’s and Greenshields' .................. 49
Figure 4.9 Change of Upstream Occupancy by Ramp Volume (2/ 10 AM, Detectors 63-
64) ....................................................................................................................... 54
Figure 4.10 A-Curve Trend (February (AM), Detectors 63-64) ..................................... 56
Figure 4.11 A-Curve Trend (February (AM), Detectors 72-73) ..................................... 57

xi

Figure 4.12 A's Comparison by Time Lag before Congestion ......................................... 63

Figure 4.13 A's Comparison by Time Lag under Congestion .......................................... 64
Figure 4.14 A's Comparison by Time Lag under Congestion at 63-64 ............................ 65
Figure 5.1 Ai /A0 Comparison under Congestion at 63-64 ............................................. 72
Figure 5.2 Comparison of Estimated vs. Observed 0.. under Congestion (Estimation by
Ramp Volume, 2/14 AM, Detectors 72-73) .......................................................... 74
Figure 5.3 Sequential Comparison of Observed and Estimated 0.. under Congestion
(Estimation by Ramp Volume, 2/14 AM, Detectors 72-73) .................................. 75
Figure 5.4 Comparison of Estimated vs. Observed 0.. under Congestion (Estimation by
0.1, 2/14 AM, Detectors 72-73) ............................................................................. 79
Figure 5.5 Sequential Comparison of Observed and Estimated 0.. under Congestion
(Estimation by 0.1, 2/14 AM, Detectors 72-73) ..................................................... 80
Figure 6.1 Flow Chart of MALINEA Ramp Metering .................................................... 82

Figure 6.2 Correlation Trend between Ramp Volume Change and Upstream Occupancy
under Congested Conditions(Prior 20-mins Based, 2/20 AM, Detectors 63 -64).... 91

Figure 6.3 Comparison of Estimated and Observed 0.. during Congestion Dissipation

(Estimation by Ramp Volume, 2/ 10 AM, Detectors 72-73) .................................. 95
Figure 6.4 Relationship between Desired and Real Occupancy ...................................... 97
Figure 7.1 MALINEA Simulation Procedure ............................................................... 104
Figure 7 .2 ALINEA Simulation Procedure .................................................................. 105
Figure 7.3 Distribution for Mainline Volume ............................................................... 110
Figure 7.4 MALINEA Operation Example .................................................................. 1 16
Figure 7.5 ALINEA Operation Example ...................................................................... 117

xii

Chapter 1

INTRODUCTION

1.1 Existing Ramp Control Strategies

Ramp metering is a positive freeway control strategy, the purpose of which is to prevent
or reduce congestion. Existing ramp metering has been studied and implemented in two
distinctive ways, i.e. local ramp control and system ramp control (Chang, Messer, and
Urbanik 1994; US. Department of Transportation 1985).

Local ramp control is the simplest ramp metering strategy that uses isolated or local
pre-timed control. Ramp volumes are established based on traffic characteristics at a
given location, which are then fine tuned through local field observations. Local ramp
control operates on either a pre-timed or traffic responsive basis. Pre-timed ramp
metering plans are typically established from the observed mainline volume. Local traffic
responsive control is regarded as an advanced type of control. During traffic responsive
system control, real time measurements of trafiic variables are taken and used to decide
the real time ramp metering rate for each individual ramp.

System ramp control refers to “coordinated ramp control” or “integrated ramp
control”. The algorithm for metering each ramp in the control system is based on the
demand-capacity considerations for the whole system rather than on the demand-capacity
constraints at each individual ramp. Like a local ramp control, a system ramp control
strategy‘ can be operated on a pre—timed or traffic responsive basis. In pre-timed system
control, each ramp-metering rate is determined in accordance with available capacity

constraints at the other ramps. These metering rates are computed from historical data

during each control interval. Under traffic responsive system control, real time
measurements are taken of trafiic variables (usually, volume, occupancy, and/or speed).
On the basis of these measurements, both an independent and an integrated metering rate
are calculated for each entrance ramp (on-ramp). Of these two metering rates, the one that
is the more restrictive is selected for use during the next successive control interval.

Local trafiic responsive ramp metering control is known to be usefiJl both for
congestion prevention and clearance. As far as congestion prevention is concerned, ramp
metering may assist a freeway section to be managed at the optimal state without
breakdown. Clearance purpose-ramp metering is also important for traffic management
since expected and/or unexpected conditions can make freeway traffic movement worse
at any time. Examples of such conditions include excessive flow rates, incidents, weather
changes, and diversion rate changes.

Several algorithms have been developed for traffic responsive local control.
ALINEA, RIJKWATERSTAAT, and W&J are representative algorithms for ramp
metering and their effectiveness has been practically confirmed (Haj-Salem, Graham, and
Middleham 1991). ALINEA is known as the best traffic responsive local control strategy
in France. It uses the occupancy rate downstream of the on-rarnp to calculate the on-ramp
volume. RIJKWATERSTAAT, on the other hand, uses upstream and downstream
volume to determine the on-ramp volume to optimize the downstream section. Finally,
W&J uses the past speed-flow relationship and current demand surplus to optimize the

bottleneck.

1.2 Bottleneck Concept and New Hypothesis for Worst Section

The guidelines for selection of ramp control systems indicate that existing ramp-metering
strategies are based on the assumption that a bottleneck section is the freeway section
immediately downstream of the ramp (Blumentritt et a1. 1981). The concept results from

(C

the definition of bottleneck as . a section of roadway to which more demand is
delivered than can be processed” (McShane and Roess 1990). The definition implies that
the bottleneck is decided only on the basis of traffic flow, without consideration of
density and speed conditions. So, the question can be raised whether the worst section
(in terms of flow rate) corresponds to the section that demonstrates the worst density
and/or the worst speed conditions. Proper identification of the worst section is important
for control purposes because the traffic characteristic of the worst section may influence
the measures of effectiveness (MOEs) of the facility.

McShane and Roess (1990) studied two cases of bottleneck conditions. In the first
case, the demand flow rate is lower than the bottleneck capacity. Under such conditions,
the bottleneck endures lower speed and higher density considering the traffic
characteristic curve of uncongested traffic flow. In the second case, the demand flow rate
is higher than the bottleneck capacity. In this case, the bottleneck results in higher density
and lower speed at the immediate upstream section because the queuing may occur
upstream of the bottleneck.

Introduction of the above theory to the ramp merge area is not trivial, due to the
special conditions native at the ramp merge. These include the presence of two traffic

demands (ramp volume and mainline volume), discontinuous geometry (acceleration lane

and taper), and potential differences in vehicle interactions as a result of the merge.

To assist in better understanding the actual traffic behavior at ramp merge areas a
hypothesis is introduced and tested in this paper as follows:

Hypothesis: Under congestion, while the bottleneck section of ramp merge is

downstream of the on-ramp, the worst section in terms of density and speed, is

upstream of the on—ramp.

Acceptance of the hypothesis implies that the upstream section of ramp merge may have
been overlooked during freeway congestion management. Consequently, reduction of the
throughput of a metered freeway section is likely, since congestion may still exist during
ramp metering, and downstream capacity may be reduced. The downstream capacity
reduction was explained by Hall and Agyemang-Duah (1991), who found an observable
drop of 5 to 6 percent capacity due to queue discharge, compared to pre—queue capacity
conditions.

The proposed hypothesis can be partly supported by earlier research studies, which
described empirically the shockwave or alternative queue formation processes at the
ramp merge. Elefteriadou, Roess, and McShane (1995) observed that queues are
developed downstream of the merge section, but only after the formation of queues
upstream. Banks (1990) indicated that the normal point of queue formation is about 1500
ft upstream of the merge and that shock-waves are observed in the vicinity of the on-

ramp.

1.3 Desired New Ramp Metering Algorithm
If the hypothesis is acceptable, congestion control based on the measurements upstream
of the on-ramp should be considered as a new ramp metering strategy. The attribute

desirable for the new ramp control strategy is not limited to the congestion control at the

upstream of ramp merge. As in any other existing ramp control strategy, continuous and
rapid response should be guaranteed, since the purpose of control is not only prevention
but also clearance.

In this respect, modification of existing control algorithms may be an appropriate
approach if consistent parameters can be found between existing and modified ramp
control strategies.

The ALINEA model is one of the widely used local ramp controls, of which
practical effectiveness has been well known and its main attributes are occupancy based
control and feedback control. These can be regarded as strengths, in that occupancy is a
good indicator of congestion, and in that feedback is the repeating error-minimizing
control algorithm.

Therefore, a desirable new ramp metering strategy will be introduced here, based
on ALINEA model principles. The new model will be called MALINEA because it
represents a “Modified ALINEA”. MALINEA and ALINEA models are mathematically

linked using parameters, characteristics of which are from a traffic stream model.

1.4 ALINEA Model
ALINEA was introduced by Papageorgiou, Haj-Salem, and Blosseville (1991). Its
closed—loop feedback control law is used to update the metering rate periodically based

on the following algorithm:

Q’(t+ 1) = Q71) + Kl0d(t+1) - 0d(t)l (H)

where

Q’ (t+1): metering rate to be applied in the next time period;

Q’(t): metering rate applied in the last time period;

0d(t+ 1) : desired occupancy at the downstream detector for the next time period;
0d(t): measured occupancy at the downstream detector during the last time period;
and

K : a parameter obtained from downstream characteristics.

ALINEA’s downstream occupancy based control strategy is explained by Haj-Salem and

Papageorgiou (1995) as:

“Because the main aim of ALINEA ramp metering is to maintain capacity flow

downstream of the merge area, the control strategy for each controllable on—ramp should

be based on the downstream measurements”.

1.5 Advantages of ALIN EA

Papageorgiou, Haj-Salem, and Blosseville (1991) described advantages of the ALINEA

model for ramp metering as follows:

Simpler than other known algorithms,

Requires a minimal amount of real time measurements (detectors),

Easily is adjustable to particular traffic condition because only one parameter is to be
adjusted in a prescribed way,

In real life experiments has proven to be more efficient in preventing congestion and
preserving capacity flow, compared to other known algorithms,

Can be embedded in a coordinated on—ramp control system,

Can be modified easily in case of changing operational requirements,

o Is highly robust with respect to inaccuracies and different kinds of disturbances, and

. Is theoretically supported by automatic control theory.

Additionally, its efficacy has been qualified through field studies. In France, the
effectiveness of ALINEA was compared to the effectiveness of other traffic responsive or
fixed time controls. ALINEA was found to perform better than the other traffic
responsive strategies in terms of the amount of time spent by an average vehicle in
congestion and their total amount of travel time spent in the system (Haj-Salem, Graham,
and Middleham 1991). Moreover, it was proven that improvement in total time spent
achieved by ALINEA under recurrent and non-recurrent congestion is considerable

compared to no control (Haj-Salem and Papageorgiou 1995).

1.6 Potential ALIN EA Model Modification
If the hypothesis presented proves acceptable, modification of ALINEA is desirable for
the upstream ramp control strategy. Given that there is an occupancy relationship
between upstream and downstream of the on—ramp, the ALINEA model can be revised
based on this relationship.

If a parameter exists which explains the occupancy relationship between upstream

and downstream conditions, the equation can be expressed as:

0,.(z) = A * 04(1) (1.2)

where
04(t): measured occupancy at the downstream detector during the last time period;

0..(t): measured occupancy at the upstream detector for the last time period; and

A : parameter.

When equation (1.2) is combined with equation (1.1), a modified equation can be

introduced as equation (1 .3).

Q'(t+1)= Q'(t) + K[0..(t+1) - 0..(t)] /A (1.3)

where
0..(t+ 1) : desired occupancy at the upstream detector for the next time period,
K : parameter, and

all other parameters as defined earlier.

The variables 0..(t), 0.. 1+1), and new parameter A are replaced or added to ALINEA.
Therefore, the calculation of the metering rate is not a function of the downstream
occupancy, but the upstream occupancy of the on-ramp in equation (1.3). Equation (1.3)
is a preliminary model because the quantification of each variable or parameter has not
been performed and operational characteristics are not yet clarified. The following

chapters will investigate those issues in detail.

1.7 Research Objectives

The first objective of this research is to identify the relationship between traffic
characteristics upstream and downstream of an on-ramp under congested conditions. For
this purpose, a hypothesis - that the location of the worst section in terms of density and

speed at the ramp merge under congestion is upstream of the on-ramp - is proposed and

tested. Traffic stream models and field data are employed as methodologies for testing
the hypothesis, both theoretically and practically. The breakdown process is reviewed as
a supplementary study to detail the tested results. Based on the assumption that the
hypothesis proves to be acceptable, the modified ALINEA (MALINEA) algorithm is
discussed as a new ramp metering strategy to handle the upstream congestion at a ramp
merge.

The second objective of this research is to introduce MALINEA as a plausible
alternative. For this purpose, detailed studies are performed including:

o Parametric studies in terms of traffic characteristics and conditions,

Model generalization, construction, and verification,

Parameter and variable selection for effective application, and

Feedback operation study using simulation.

Chapter 2

DATA DESCRIPTION AND RAMP VOLUME APPROXIMATION

2.1 Data Description

To achieve the purposes of the study, serial speeds, occupancies and volumes both at the
upstream and downstream of the on-ramp are needed. These data will allow testing the
hypothesis that the location of the worst section, in terms of density and speed at the
ramp merge under congestion, is the upstream of the on-ramp. These data will also be
used to identify parameter characteristics because each data point reflects the traffic
characteristics and conditions at a point in time. By adding ramp volume, these data will
be used to verify new models and to identify the simulation conditions to test the
feedback operation.

Data were obtained from the Michigan Intelligent Transportation System (MITS)
which covers the freeways within the Detroit Metropolitan Area. NflTS covers 285
detectors on the entire Ford Freeway (I-94) within the City of Detroit (22.9 km), all of the
Lodge (US-10) Freeway within the city of Detroit (19.6 km), part of I-75 (37 km of the
Fisher Freeway east of Jefi‘ries), and 5.5 km of the Chrysler Freeway (I-75) south of the
Ford Freeway.

Data collection was done from January to August in 1997. A large portion of data
were not appropriate for the purpose of this study because the data did not show the
expected traffic patterns on the freeway. For example, the data from May to August
frequently showed one- or two-lane blockage patterns of traffic, which seemed to have

originated from a construction zone. Several data sets from January to February indicated

10

considerably lower capacity that was confirmed to be due to snowy weather. An unstable
data pattern, which appeared to be a result of a hardware problem, was also found.
Additionally, some data showed just the queue backup or the free flow state, which were
not appropriate for a bottleneck study, in that there was not a temporal capacity state
during a day. Therefore, data fi'om a limited numbers of days and detectors were used for

the study. Table 2.1 shows the geometric characteristics of major detectors used in this

 

 

 

study.
Table 2.1 Geometric Characteristics of Major Detectors
Detectors Freeway Crossing No. of Lanes UD DD Bottleneck

Line Street Mainline Ramp (m) JmL
100-101 1-94 WB 30th 3 1 6 505 No
118-119 1-94 EB Livemois 3 1 13 531 No
193-194 US-lO NB Grand Blvd 3 l 113 126 Yes
199-200 US-lO NB Chicago 3 1 46 316 Yes
201-202 US-lO NB Webb 3 1 219 227 No
55-56 1-94 WB Moross 3 l 3 438 No
63-64 I-94 WB Charmers 3 l 29 507 Yes
68-69 I-94 WB Conner 3 l 3 314 Yes
72-73 1-94 WB Gratiot 3 1 2 454 Yes
92-93 I-94 WB Trumbell 3 l 80 394 Yes
69-71 1-94 WB French 3 l 534 505 No
126-127 1-94 EB Grand River 3 1 157 268 No

 

 

 

 

 

 

 

 

 

 

Pairs of measurements, which were obtained at locations upstream and downstream
of the on-ramp respectively, are often used as a basic unit in this study. Figure 2.1
describes the available data at the ramp merge, which include speed, volume, and
occupancy at each detector. The physical distances between the upstream or downstream
detectors to the ramp nose were measured for parametric analysis. Data obtained were

formatted on a daily basis and grouped into two twelve-hour periods (one for PM and one

11

for AM conditions). Data were aggregated on a minute-by-minute basis resulting in 720
time slices within each data set.

During the study, the volume-occupancy relationship curve showed a unique value
of the optimal occupancy, which ranged from 10 to 15%. This range is relatively lower
than the range of 15-30% that has been proposed by existing researchers. This partly may
be explained by the difference from location specific characteristics such as average

vehicle length and detector zone length.

 

 

 

 

UD DD
< x —>
13 Cl

Upstream detector Downstream detector
S..(t) = Speed (mph) SJ!) = Speed (mph)
Q..(t) = Volume Qdfl) = Volume

(veh/min) (veh/min)

0..(t) = Occupancy (%) 0..(t) = Occupancy (%)

Figure 2.1 Data Description at Ramp Merge

2.2 Ramp Volume Approximation

As ramp volumes were available only at limited locations, calculated ramp volumes were
frequently used for the study. This calculation is based on the concept that part of the
downstream volume flows in from the upstream. As the travel time of vehicles from the
upstream to the downstream detector location is known, the ramp volume can be
extracted from the difference between downstream and upstream measurements. The

result was plausible because data showed that the travel time between two-measurement

12

points is as small as less than 2 minutes and the speed is relatively constant during that
period (speed change by time slice was as gradual as less than 9% at 95% confidence

limit). The mathematical calculation steps are expressed as follows:

Time mean speed, S = (Sd+S..)/2 (2.1)

Travel time, T (min) = (UD+DD)/S (2.2)

where
UD: Distance between ramp nose and upstream detector,
DD: Distance between ramp nose and downstream detector,
Sd: Speed measured at downstream detector, and

5.: Speed measured at upstream detector.
The upstream volume portion Qdupfl) among the downstream measurement Q(t) at the
time slice 1 can be calculated as a firnction of the past upstream volume and travel time T

considering the weight of the past time slice. That is, where INT(7) represents the integer

value of T,

Qdupfl) = Qu(l-INT(T))[T— 1NT(T)]+Qu(l-1NT(7)— 1)[1— (T- 1NT(T))] (23)

Therefore, the ramp volume portion Q’ (t) at the time slice 1 is

Q71) = Qdfl) - Qdupfl) (2-4)

13

Chapter 3

WORST SECTION ANALYSIS AND RESULTS

In the following paragraphs, the worst section at ramp merge locations is defined as the
section (upstream or downstream of the on-ramp) that demonstrates higher density or
lower speed under congested conditions. Theoretical and empirical studies are performed
to test the hypothesis that, under congestion, the worst section with respect to density and
speed is the one upstream of the on-ramp. Results from the traffic stream model analysis,

the empirical analysis and the breakdown procedure are discussed below.

3.1 Traffic Stream Models

Traffic flow theory principles can be applied to the study of the worst section at ramp
merge areas. Toward this direction, the research work of McShane and Roess (1990), as
well as May’s location study (1990) offers valuable guidance as they both dealt with the
study of traffic data characteristics along bottleneck sections based on traffic stream
models. The review below covers the simplest bottleneck cases to stress the fact that the
field location significantly affects the resulting speed-flow-density relationship.

Figure 3.1 is extracted from May's work (1990, p288), and presents a typical
bottleneck with normal conditions in section A and a lane reduction in section B. The
hollow circles represent the possible resulting data points in sections A and B. Based on
the diagram, it is evident that the lane closure results in capacity conditions in section B.
Clearly, location limits the range of traffic characteristics such as flow rate, speed, and

density.

14

 

 

[ Section A

 

 

 

SA

 

 

 

 

 

 

 

 

 

 

QA
—-—>
kA
o
T 00
o
S A

————->

k1

 

 

Section B

 

 

 

SB

 

 

 

qr;

 

 

 

k I
B

 

SB

 

 

 

k3

Figure 3.1 Importance of Location (May, p288)

By transferring this concept to the ramp merge area, the interpretation of traffic
characteristics at ramp merge may become easier and more detailed. For clear
interpretation, however, it is necessary to consider the diversity of traffic characteristics
under congested and uncongested states.

Figure 3.2 shows the speed-volume and the density-volume relationships at a ramp
merge bottleneck location. For simplification purposes, the transition effect due to
geometry and traffic characteristics between downstream and upstream of the on-ramp
are not considered, while the capacity of the upstream is assumed to be identical to that of
the downstream. The congested and uncongested states are studied separately. The

volume-density (occupancy)-speed characteristics are denoted as follows:

Uncongested state
(k... kd): a pair of occupancies that are observable simultaneously before congestion
where: k. represents an occupancy at the upstream of an on—ramp.
kd represents an occupancy at the downstream of an on-ramp.
(q.., q..): a pair of volumes that are observable simultaneously before congestion
where: q.. represents a volume at the upstream of an on-ramp.
qd represents a volume at the downstream of an on-ramp.
(s.., 3.1): a pair of speeds that are observable simultaneously before congestion
where: s.. represents a speed at the upstream of an on-ramp.
3.. represents a speed at the downstream of an on-ramp.
C ongested state

(km kdc): a pair of occupancies that are observable simultaneously under congestion

16

where: k... represents an occupancy at the upstream of an on-ramp.
kdc represents an occupancy at the downstream of an on-ramp.
(q... , qdc): a pair of volumes that are observable simultaneously under congestion
where: q... represents a volume at the upstream of an on-ramp.
q... represents a volume at the downstream of an on-ramp.
(5..., 3...): a pair of speeds that are observable simultaneously under congestion
where: 3... represents a speed at the upstream of an on—ramp.

sdc represents a speed at the downstream of an on-ramp.

Figure 3.2 (a) shows density and volume relationships under uncongested and congested
conditions. The demand downstream of the ramp is the summation of the on-ramp and
the mainline demand. Thus, upstream volume (i.e. the mainline volume), is always less
than, or equal to, downstream volume.

Under uncongested conditions, as q.. S qd, the traffic characteristic curve in 2 (a)
results in k,. s k. for all possible values. In the congested state, the demand exceeds the
maximum flow rate. Under such conditions as q... s qdc, the traffic characteristics curve in
2 (a) shows that the upstream density is higher or equal to that of the downstream (kg
1:...) for all possible values.

Figure 3.2 (b) presents the speed-volume relationship under uncongested and
congested states. In the uncongested state, q.. s q.., and s.. 2 sd for all possible values,
Under congested conditions, q... s qdc, and s... s 5...; for all possible values.

In summary, under congested conditions and based on the traffic stream model, the

upstream volume is lower than the downstream volume (q... S (1.1.), the upstream density

17

Density or Occupancy

 

 

 

 

 

\ Congestion
kdc ‘>.
Before
/ Congestion
kd /

 

 

 

 

 

 

 

 

 

Volume

 

q“ qd quc qdc

(6!)

Speed

s. \\

\ Before
3., Congestion

Congestion

 

 

 

 

 

 

 

 

 

 

Sdc

 

 

 

 

 

Volume

 

q no qdc (In qd

(b)

Figure 3.2 Comparison of Traffic Characteristics Upstream and Downstream of On-
Ramp

l8

(occupancy) is greater than the downstream density (k... 2 kdc), and the upstream speed is
lower than the downstream speed (sacs 3...). Based on these results, the worst section is
defined as the section upstream of the ramp merge that demonstrates higher densities and
lower speeds. Therefore, the traffic stream model confirms the hypothesis proposed

above, regarding the location of the worst section at merge areas under congestion.

3.2 Empirical Results
The following paragraphs describe the procedure used and results obtained from an
empirical study performed in order to determine the location of the worst section relative

to the merge point. A merge location on I-94 is selected as the test bed.

488m ‘ 29m 507m 5 554m

 

/\
/\
(

 

 

 

 

 

E] C]

T

62 63 64 65 ‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D Detector position

Figure 3.3 Schematic of Study Site

19

The test location was within the Detroit Metropolitan Area and experienced recurrent
congestion. The merge absorbs the on-ramp inflow from Chalmers Road onto I-94
westbound. There are 3 freeway lanes on the mainline. A system of four permanent
detectors allowed observation of traffic characteristics just upstream of the merge point
(detector 63), firrther upstream (detector 62), at the downstream of merge (detector 64)
and further downstream (detector 65). A sketch of the study site is shown in Figure 3.3.
The horizontal alignment was straight, with ramp to mainline volume ratios of 5 to 20%.
Volume measurements took place on February 18, 1997 from 00:01 AM to 12:00 PM. No
entrance or exit ramps exist between detectors 62 and 65, other than the study on-ramp on
Chalmers Road.

Prior to performing the ramp merge study concerning detectors 63 and 64, one
should ensure that the bottleneck due to ramp merge is the only bottleneck during
congestion between zones 62 and 65. Otherwise, a far downstream or far upstream
bottleneck may bias the study results and affect the study conclusions.

Figure 3.4 compares percent occupancy values at detector locations 62, 63, 64, and
65 based on readings obtained over 50 minutes of congested conditions (i.e. from the start
of congestion at 6:40 AM until its end at 7:30 AM). The highest occupancies are
observed at the detector located upstream of the merge location (detector 63). Based on
these data, under congested conditions the worst section (bottleneck) is located around
detector 63.

Figure 3.5 presents a plot of the relationship between the upstream and downstream
volume based on 12 hour, minute-by-minute volume data (720 readings). The volume
represents the total volume of the 3 lanes in a minute. The downstream volume (veh/min)

is always greater than the upstream volume (veh/min), regardless of congestion condition

20

occ (%)

   

Time (min) L t'
ocaron

Figure 3.4 Comparison of % Occupancy under Congestion at

Detectors 62, 63, 64 and 65 (2/18/97)

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(before, during, or after congestion).

The maximum downstream volume value observed is 117 veh/min (7,020 veh/h).
Downstream volume occasionally exceeds 6,900 veh/h, the 3-lane freeway capacity at
ramp merge reported in the HCM (1997). Upstream volume reached a value of 102
veh/min (6,120 veh/h) which is below the capacity value of 6,900 veh/h. This observation
is identical to that of Hall and Agyemang-Duah (1991), who indicated that, immediately
upstream of an on-ramp, capacity is not reached under unstable flow conditions.

Figure 3.6 shows the relationship between speeds collected upstream and
downstream of the study ramp merge location. Under uncongested conditions (speeds
over 65 mph), the upstream speed is approximately equal to the downstream speed. This
result is different from the previous theoretical consideration based on traffic stream
models. However, a large number of empirical research studies dealing with speed-flow
relationships confirm this behavior. Banks (1989), Hall and Hall (1990), Chin and May
(1991), Hall and Agyemang-Duah (1991) and Ringert and Urbanik (1993) support the
idea that speed remains nearly constant under uncongested flow.

The relationship between speeds observed during congestion (speeds below 50
mph) upstream and downstream of the study ramp merge is relatively clear. Downstream
speed values obtained under congested conditions are always greater than the
corresponding upstream values. Downstream speed values range from 26 to 53 mph
while corresponding upstream speeds range from 22 to 43 mph. The variation observed
under congestion can be explained by the speed oscillation effect that was described by
Mika, Kreer, and Yuan (1969) and Kosh et a1. (1983).

Figure 3.6 also shows clearly the congestion development and recovery states. At

the start of congestion, rapid deterioration of traffic conditions can be observed as the

24

average upstream speeds drop from 63 to 41 mph in just five minutes (63 to 53 mph for
the downstream). During the congestion recovery state, speeds rise continuously from 42
to 80 mph and non-congested conditions exist thereafter.

Figure 3.7 shows the relationship between upstream and downstream percent
occupancy at the ramp merge. Several days of observations at the site confirmed that the
optimal occupancy ranges from 10 to 13%. These values are lower than those suggested
in the literature. An earlier observation by Mika et a1. (1969) at a nearby location
indicated that the optimal occupancy is about 15%. Optimal occupancy value is used in
order to distinguish between the congested and uncongested state. Assuming an optimal
occupancy value of 12%, the upstream occupancy looks typically greater than the
downstream occupancy under congested conditions. Large oscillation results in the large
scatter shown in Figure 3.7 during congestion. In the uncongested state, the upstream
occupancy is lower than the downstream occupancy. The concentrated data trend implies
a high correlation between the downstream and upstream occupancy values.

The plot in Figure 3.7 clearly defines the congested and uncongested states and the
transition between the two. Percent occupancy values under congested conditions range
from approximately 12 to 20% at the upstream of the ramp merge. Under uncongested
conditions they range from 1 to 6%. The transition from uncongested to congested
conditions occurs when occupancies start rising consistently and rapidly from 6% to
12%. The mechanism is reversed at the end of congestion, when a consistent and rapid
drop of occupancies from 17% to 6% occurs.

The results from the empirical study strongly support the hypothesis that the worst
section at ramp merge locations in terms of speed and occupancy, is located upstream of

the on-ramp.

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Moreover, the results are in general agreement with the traffic stream model, with the
exception of the upstream and downstream speed relationship under uncongested

condition.

3.3 Breakdown Procedure

The breakdown procedure at the ramp merge has been discussed for a long time. The
discussion has been developed over two opposite arguments. The first supports the notion
that the breakdown occurs downstream of the on-ramp first and then propagates
upstream. The other argument suggests that the phenomenon occurs vice versa.

Newman (1963) described the initiation of queuing at a merge location and
suggested that queuing develops first downstream of an on-ramp as drivers increase their
gaps following the merge section. Based on Newman's study, the HCM (1997) suggests
that the on-ramp influence area ranges from the nose point to 1,500 ft downstream of it.

Banks’ observation (1990) of the flow process in the vicinity of a high volume
fixed bottleneck on a metered freeway led to exactly the opposite conclusion. He
observed the queue formation using detector data and videotapes. Based on his study, the
typical point of queue formation was about 1,500 ft upstream of the merge. Elefteriadou
et al. (1995) added that queues are observed downstream of a merge section, but after the
formation of queues upstream. They described the sequence of events before breakdown
as being (a) entrance of a large cluster to the freeway system, (b) subsequent small speed
drop at the beginning of the bottleneck, (c) spreading of the speed drop upstream, and (d)
creation of the breakdown.

The worst section studies discussed above compared upstream and downstream

traffic characteristics and confirmed that lower upstream speed or larger upstream percent

27

occupancy represents congested conditions. This idea can be used to trace the procedure
of breakdown or queue beginning, if detailed time serial measurements of traffic
characteristics are available.

As an example, Table 3.1 shows detailed values of traffic characteristics upstream
and downstream of the ramp merge location at the study site. The maximum downstream
flow rate (117 veh/min) is sustained from 6:38 to 6:40 AM. This time period corresponds
to the start of transition from uncongested to congested conditions. During this period,
while the occupancy downstream remains constant (11%), the upstream occupancy starts
increasing, showing the first signs of breakdown occurrence. Also, the speed drop
upstream occurs earlier (at 6:39 AM) and is more rapid compared to downstream (e. g.
from 6:38 to 6:42 AM, 12 mph speed reduction is observed upstream versus a 5 mph
speed reduction downstream).

In summary, it is evident that the breakdown symptom appears upstream, while
maximum flow is sustained downstream of the on-ramp. This leads to the conclusion
that, even if the downstream is managed at capacity, the breakdown still can occur
upstream of the on-ramp. In other words, congestion can occur upstream without any
advanced change in downstream traffic characteristics. This inference closely fits the

results of the traffic stream model study in Figure 3.2.

3.4 Summary

This chapter focused on the determination of the worst section of the bottleneck at the
ramp merge location. The study was motivated by the fact that most local traffic
responsive ramp metering strategies implemented today aim at optimization of the

downstream of the on-ramp although the approach with this concept may generate

28

Table 3.1 Detector Data between 6:30 and 6:50 AM at Detectors 63 and 64

 

 

 

 

 

 

 

 

Upstream detector 63 Downstream detector 64
Time Volume Occupancy Speed Volume Occupancy Speed
(veh/minl 1%) (mph) (veil/min) fiL (mph)
6:30 98 8 78 110 9 76
6:31 100 8 76 113 10 74
6:32 104 9 73 1 15 10 72
6:33 101 9 69 116 10 71
6:34 101 10 64 115 11 69
6:35 99 10 63 116 ll 68
6:36 98 10 63 114 ll 69
6:37 100 10 63 115 11 68
6238* 103 10 64 117 ll 68
6:39* 104 11 63 117 ll 68
6240* 102 11 60 117 11 65
6:41 103 12 57 114 11 64
6:42 101 13 52 112 ll 63
6:43 100 14 48 111 12 59
6:44 99 14 47 108 13 56
6:45 94 15 43 105 12 54
6:46 93 16 41 107 13 54
6:47 95 16 41 101 14 48
6:48 93 16 39 99 15 45
6:49 86 17 35 99 14 46
6:50 86 17 35 99 14 47

 

 

 

 

 

 

 

* : transition from uncongested to congested state

29

 

unexpected congestion at the upstream of the on-ramp.

First, a hypothesis was set, based on which, under congestion, the bottleneck

section at a ramp merge occurs downstream of the on-ramp. However, the worst section,

in terms of density and speed, is located upstream of the on-ramp. Then the hypothesis

was tested and confirmed, both theoretically and empirically. Finally, the breakdown

procedure at an on-ramp was studied. No breakdown indicative change was observed

downstream of the on-ramp during congestion development.

Several conclusions can be drawn fi'om the work presented above.
The volume-density (occupancy)-speed relationship in the traffic stream model is
based on the concept that the location limits the range of flow rate, speed, and density
values. This concept is developed for the general bottleneck location but also applies
to bottlenecks at the ramp merge. Therefore, typical relationships between traffic
characteristics upstream and downstream of on-ramps can be identified using this
concept.
Under congestion, the worst section at the ramp merge is upstream of the on-ramp,
even though the bottleneck develops on the downstream side.
Breakdown first occurs upstream of the ramp merge. Therefore, even when ramp
metering is used to manage the downstream flow at capacity, the breakdown occurred
upstream of the on-ramp can cause upstream congestion.
Existing local traffic responsive ramp metering strategies are typically based on the
downstream measurements. Modification of these strategies, to account for worst
section conditions upstream of the on-ramp, may have a great potential toward
improvement of ramp metering and reduction of delays in the vicinity of metered on-

ramp locations.

30

Chapter 4

PARAMETER CHARACTERISTICS

Chapter 1 introduced the principles of ALINEA and “Modified ALINEA.” Chapter 3
showed that the hypothesis proved acceptable, thus “Modified ALINEA” will be firrther
explored as a potential strategy to improve the upstream condition of the ramp merge.
Before using the MALINEA model presented as equation (1.3), its characteristics should
be firlly studied for effective model construction and application.

One of the simplest ways toward characterizing parameters A and K involving the
model formulation is to study the corresponding A and K curves. Parameter A represents
the slope of the A-curve, where the A-curve is defined as the occupancy relationship trend
between upstream and downstream sections of the on-ramp. Parameter K represents the
slope of the volume-occupancy curve.

In section 4.1, the typical shape of A and K-curve trends at the merge bottleneck is
explained using traffic flow characteristic. Then, in section 4.2, the characteristics of the
A-curve trend are discussed in more detail in terms of linearity, the cause of scattered data
under congestion, and location specific characteristics. These characteristics play an
important role in the construction of the MALINEA model and application of the A-
parameter, which is explained in Chapters 5 and 6. Section 4.3 handles the time. The time
lag has been considered as a very influential parameter in feedback control theory. The

result in this chapter will be used in the construction of the MALINEA in Chapter 5.

31

4.1 Typical Trend of A and K -Curves at Ramp Merge Bottleneck
The A-parameter is obtained from the occupancy values observed simultaneously
upstream and downstream of the on-ramp. Several trends of curves may express this
relationship. As the occupancy is the only variable involved in the construction of an A-
curve, the trend of the A-curve is expected to reflect the traffic condition of the ramp
merge including back-up from downstream, congestion dissipation from upstream, merge
bottleneck, and incident presence.

On the other hand, the K-parameter results from the volume-occupancy relationship
curve (K-curve). Therefore, as a traffic characteristic curve, the shape of the K-curve

trend obviously reflects the traffic condition of the ramp merge, also.

4.1.1 A-Curve Trend Study

Case 1: Backup from Far Downstream

Figure 4.1 shows an A-curve trend obtained at detectors 100 and 101. The pair at each
point represents upstream and downstream occupancies for a given time slice. For
example, the pair (4, 5) represents downstream occupancy 4% and upstream occupancy
5%. Accordingly, the passage of serial pairs may reflect a change of traffic condition.
Two simple passages are observed within downstream occupancy (04) range of 10 to
17%. One of the two passages represents congestion propagation from downstream. In
this case, the passage starts at the beginning of congestion and the downstream
occupancy (0..) is considerably greater than the upstream occupancy (0..) within the
range. Figure 4.2, which shows the volume-occupancy curve at downstream detector 101,
also supports this interpretation. Note that the maximum flow rate is not reached, while a

sudden jump into congestion state occurs. The maximum flow rate at a nearby

32

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location with the similar geometric condition has been observed as approximately 120
veh/min for 3 lanes as shown on the curve in Figure 4.4. Existing research explained “no
maximum flow rate” in detail. May (1990), McShane and Roess (1990), Drake, Schofer,
and May (1967), Hall and Ageyamang-Duah (1991) explained a lower maximum flow
rate or a lower occupancy in the queue from a bottleneck. They reached the conclusion
that the bottleneck or ramp volume takes up a portion of the capacity, the queue forms at
the bottleneck and then reduced flows are possible within the queue. In this respect, as
detector 101 is at the downstream of an on-ramp, another source of bottleneck far
downstream must have brought out the lower maximum flow rate at detector 10].

Outside the occupancy value range of 10 to 17%, an approximately linear trend is
found. This infers that after congestion is fully developed, one A-curve trend can be

approximated as a curve with consistent slope even under the congestion caused by queue

backup.

Case 2: Congestion Dissipation from Upstream

The other passage in 0.. range of 10 to 17% in Figure 4.1 represents congestion
dissipation from an upstream volume reduction. Table 4.1 shows detailed values of traffic
characteristics upstream and downstream of the ramp merge location at the study site.
The congestion at the upstream detector ends at 6:00 PM and at the downstream detector
at 6:08 PM. A 12% occupancy is assumed as the optimal occupancy. Speed recovery is
faster at the upstream of the on-ramp than that at the downstream. Existing research about
backward recovery at the bottleneck explains this phenomenon. If there were backward
recovery, there should be an observation of the approximated capacity flow at

downstream detector 101.

35

Hurdle and Datta (1983) suggested that capacity should be defined as queue
discharge flow, Hall and Ageyamang-Duah (1991) suggested duality of capacity, which
means that both queue discharge flow and pre-queue flow can be regarded as capacity,
and the former is less than the latter by as much as 5-6%. Assuming that the pre-queue
maximum flow in this study is 120 veh/min, the queue discharge flow rate 113 to 115
veh/min should have been observed if the queue dissipation is from the downstream. As
shown in Figure 4.2, the maximum flow at the downstream detector observed is only 85

veh/min.

Table 4.1 Detector Data between 5:59 and 6:14 at Detectors 100 and 101

 

 

 

 

Upstream detector 100 Downstream detector 101
Time(PM) Volume Occupancy Speed Volume Occupancy Speed
(veh/min) (%) (mph) (veh/ min) (%) (mph)
5 :59 67 13 41 75 18 28
6:00 69 12 47 74 16 30
6:01 70 1 1 52 74 16 30
6:02 69 10 54 73 15 31
6:03 70 9 58 74 15 32
6:04 64 8 . 61 70 15 31
6:05 65 8 63 66 14 31
6:06 63 7 64 65 14 3 1
6:07 62 7 65 65 13 33
6:08 63 6 69 65 12 36
6:09 63 6 69 65 l l 41
6: 10 61 6 70 63 10 46
6:1 1 61 6 74 63 9 52
6: 12 60 6 74 62 8 57
6:13 61 5 75 62 8 59
6: 14 63 6 75 65 8 60

 

 

 

 

 

 

 

 

36

Case 3: Merge Bottleneck
Figure 4.3 shows an A-curve trend under the merge bottleneck without backup from
downstream. The slope seems to be consistent for the entire range of values; two
different regimes seem to exist in terms of the scattered pattern. These will be discussed
in detail in the next section.

Figure 4.4 shows the volume-occupancy relationship at the downstream of the on-
ramp. It indicates that the on-ramp is a bottleneck on the freeway line because the
maximum flow rate of 120 veh/min can not be reached in the case of upstream discharge

or downstream queue backup.

Case 4: Incident Case

Tracing detailed values of traffic characteristics upstream and downstream of the ramp
merge location can provide a means for identifying an incident in the A-curve trend.
Figure 4.5 shows an A-curve trend of the incident case. Data was collected at the
detectors 92 and 93 on US-10 on July 29, 1997. At point a, the downstream occupancy is
10% while the upstream occupancy is 11%. At point b, which represents the occupancy
coordinate l-min later, the occupancy at the upstream increases to 20% while the
downstream shows a far from congested condition of which occupancy is 7%. Figure 4.6
also shows the volume-occupancy relationship at the downstream detector. It is shown

that the curve development is very erratic compared to the merge bottleneck case.

Summary
A summary is offered next, based on the results from case studies of A-curve trends for

various traffic conditions.

37

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For the merge case, there exists an approximately linear A-curve trend both under
congested and uncongested conditions. For the queue backup case from far downstream
or queue dissipation from upstream, the A-curve trend is downward from that of the ramp
bottleneck case. This is plotted as oval or in Figure 4.7. The incident case is distinguished
by the upward A-curve trend from that of the bottleneck case. This is plotted as oval ,B in

Figure 4.7

(%)

 

 

 

04(°/o)

Oval or represents the case of queue backup from far downstream or queue dissipation from upstream
Oval [3 represents the case of incident

Figure 4.7 General A-Curve Shape

4.1.2 K-Curve Trend Study

The K-curve trend has been well known because it is a part of the fundamental traffic
stream model at the downstream of the ramp merge. Moreover, the ALINEA model has
been using its slope as a constant parameter. Figures 4.2, 4.4, and 4.6 show the K-curve

trend case by case. All figures indicate that there are rapid occupancy drop-downs from

42

congested to uncongested condition when the volume sustained was about 80 veh/min. It
was explained previously that these drop-downs are associated with mainline volume
reduction, as these have been observed consistently at the end of all congested cases.
Rapid occupancy jump-up is shown in Figure 4.2. As explained previously, this is related
to queue backup from downstream. Therefore, the K-curve trend at the downstream of
ramp merge is characterized as approximately linear at merge bottleneck as in the
ALINEA model.

The study of detailed data was used in order to examine how A and K-curve trends
are generated and what traffic conditions are associated with the curve trend. Then A and
K-curve trends at the merge bottleneck were simplified. As a result, the bottleneck
location can be identified which will be used frequently during this study. Detectors 193-
194, 68-69, 199-200, 92-93, 63-64, and 72-73 were selected for fiirther analysis as merge

bottlenecks frequently had been observed there.

4.2 Traffic Characteristic of A (Occupancy Relationship)—Curves

4.2.1 Linear Assumption of A-Curve

The A-parameter plays the role of the linkage between ALINEA and MALINEA.
Therefore, the A—parameter is important to characterize MALINEA as an effective model.
If the A-parameter value depends on the occupancy level, then the application of
MALINEA would be more complicated and need more precise consideration. As the A-
parameter is expressed by the slope of the A-curve, linearity of the A-curve deserves to be
studied. Characteristics of the A-curve are studied based on three different approaches.
Two widely known traffic stream models, namely, Greenshields’ (1935) and Athol’s

models (1965) were considered first and the field data analysis was added.

43

A-Curve Based on Greenshields’ Study

Greenshields’ model is regarded as one of the basic traffic stream flow models, from
which many new theories and high ordered models have been developed. Its main
characteristic is the quadratic shape in describing density (occupancy)-volume
relationship. In this model, the A-curve is expressed by upstream and downstream
occupancies. Therefore, its slope is induced through the density-volume equation as

follows. Greenshields’ model is expressed as:

Sf 2
= k — —k .
q Sf k, (4 1)
given q: volume (veh/h),
sf : free-flow speed (mph or km/h),

k, : jam density (veh/mile, veh/km, or %), and

k : density (veh/mile , veh/km, or %),

Therefore, density is

 

/ 461
k = 0.5k -(1i 1- ) (4.2)
1 kjsf

where
“+” under congestion, and

“ ’3

under uncongested state

44

Accordingly, the density upstream and downstream of the on-ramp can be described

k, =O.5kj(1: /1— 4‘1“ ) (4.3)
kjfif
k -05k 1+ 1 4q 44

respectively as:

 

 

where,
kn = upstream density,
kd = downstream density,
qu = upstream volume, and

qd = downstream volume.

As the ratio of the densities is,

 

4
(1i 1___fl_)
ku kjff
—= (4.5)
kd 4qd
(1i 1— )
kij

the slope of volume-density curve is

dk l 4 —-
“ =——(1——qi) 2 (4.6)
dqu sf kjsf

 

45

l
dkd _ 1 1 4qd -‘2‘
dqd sf [6ij

 

(4.7)

From equations (4.6) and (4.7), the slope of a curve for uncongested state is

flu— : 5111.. ————ka/ _ 4q" (4 8)
dkd dCId [€ij - 4qu

and, the slope of a curve for congested state is

 

 

 

 

dkuc _ dquc .kasf — ‘1qu (4 9)

dkdc — dqdc [€ij —4un

The curve slope can be inferred from equations (4.8) and (4.9). Under congestion at the
downstream side, if qd equals k, Uf/4 then the slope has no value based on the equations.
When qu or quc gets close to kjsf/4, the slope goes to 0 because qu < qd and quc < qdc are
always accomplished as described in Chapter 3. On the other hand, when q“, qd, quc or qdc
goes to O, as is in the case of heavily congested or free flow state, the slope goes to 1.
Therefore, the slope of the A-curve can be equal to l at both sides of the A-curve and
become zero near the center of the curve, that is, at the beginning of downstream
congestion. For this reason, the curve slope can be symmetric over the beginning point of

congestion. All inferences above are when dqu / dqd and dquc / dqdc are managed at

unity.

46

Equations (4.8) and (4.9) show that there is a point of no value on the A-curve. This is
explained by equation (4.5) in detail. As qu _< qd and qua 5614: are established, equation
(4.5) infers that the k, /kd _< 1 under uncongested condition and kW/kdcz 1 under congested

condition. This characteristic is reflected as jumping-up of the curve at a threshold of the

congestion beginning point.

A-Curve Based on Athol’s Study
Athol (1963) suggested a linear relationship between volume and occupancy (density). In
this section, his suggestion is quantified using mathematical equations. Athol’s traffic

stream model may be expressed as:

q=(b+7:—)k (4.11)

qc = bkc +6 (4.12)

where
k0: optimal density,
k: density under uncongested condition,
kc: density under congested condition,
q: volume under congested condition,
qc: volume under uncongested condition, and

b, c : constants.

The occupancy relation could be expressed as,

47

——=— (4.13)

(4. 14)

Differentiation of equation (4.13) and (4.14) give the slope of the volume-density curve

that is expressed as:

 

£1 = 31511 (4 15)
dkd 61%
dkuc : dquc (4 l6)
dkdc dqdc

Equations (4.15) and (4.16) imply that the slope of the A-curve is constant when
dqu / dqd and dquc / dqdc are managed as a unity. This holds regardless of the value b
and c and implies that the same slope is applicable for both congested and uncongested
conditions.

A discrepancy at the threshold is considered to be true in this case as is in
Greenshields’. This is because the two models have an identical trend resembling the U
shape. Equations (4.13) and (4.14) can explain the discrepancy. As downstream volume
is always greater than upstream volume, equation (4.13) stays always less than 1 and
equation (4.14) greater than 1. As a result, a jump at the congestion starting point can be

observed as is in Greenshields’.

48

The symmetric slope and the discrepancy over the beginning of congestion make
the curve approximation under Athol’s model as shown in Figure 4.8. The figure also
shows the characteristic of the symmetric A-curve, the slope and the discrepancy of the

A-curve approximation based on Greenshields’.

 

 

 

 

 

 

 

Upstream density
(occupancy)
Upstream
congestion
point
Athol’s
I xx” _ Greenshields’
T Downstream density
(occupancy)
Downstream

congestion point

Figure 4.8 Approximation of A-Curves Based on Athol’s and Greeshields’

A-Curve Based on Field Data Study
As previously shown in Figure 4.3, the field data shows controversial results. First of all,
the discrepancy at the threshold appears evident as is in both Greenshields’ and Athol’s.

Secondly, the symmetric slope, which is positively supported by Greenshields’ and

Athol’s, was not consistent with the field data.

49

Thirdly, while the non-linear (Greenshields’) effect appears small, an assumption of a
linear relationship between 0“ (%) and 0d (%) values yields R—square value (RSQ) of
0.996 under the uncongested state. Results for more locations are shown in Table 4.2.
The RSQ under congestion is less than that of the uncongested condition due to large

fluctuations and biased clustering of data.

Table 4.2 Examples of Linear Regression Results

 

 

 

 

 

 

 

 

 

 

 

Zones Time Uncongested state Congested state
Reg. function RSQ Reg. function RSQ
199-200 4/16 AM 0.9196x 0.983 1.064x — 0.381 0.956
193-194 4/16 PM 1.159x 0.970 1.040): + 2.392 0.952
68-69 5/13 AM 1.001x 0.998 1.359x — 7.045 0.879
92-93 8/12 PM 0.829x 0.922 1.625x — 8.066 0.625
63-64 2/19 AM 0.842x 0.986 0.806x + 4.878 0.552
72-73 2/21 AM 0.987x 0.994 1.198x-0.799l 0.974

Discussion

The characteristics of the upstream and downstream occupancy relationship have been
studied based on Greenshields’ model, Athol’s model and the field data. The discrepancy
at the threshold was confirmed by all three studies. The discrepancy was very explicit for
the occupancy relationship.

Linearity of the A-curve is still controversial because it is supported by just Athol’s
theory and field data in this study. However, the valuable result based on Athol’s theory,
that linearity of the volume-occupancy curve leads to linearity of the A-curve, is
confirmed.

Linearity of the volume-occupancy curve was suggested in much of the existing

research. Indirect support of the linear volume-occupancy curve under congestion is

50

found in shockwave propagation studies where the inverse of the slope in the volume-
occupancy curve represents the shockwave speed. There are several studies of linear
propagation of shockwaves. Existing research has studied the constant value for the
propagation speed. Iwasaki (1981) studied the linear characteristic of propagation speed.
He suggested the propagation speed of 17-24 km/h. Kosh et al. (1983) suggested the
speed of 18 to 20 km/h. Mika, Kreer, and Yuan (1969) suggested the speed of 25 km/h.

More importantly, a variety of field observations have shown the linear trend of
volume-occupancy curves. Drake, Schofer, and May (1967), Cedar and May (1976),
May and Keller (1967), and Bell (1987) observed linearity of the volume-occupancy
relationship in their studies.

In MALINEA, the non-linear characteristic of the A-curve may not be meaningful
considering its feedback control based system. The A-parameter is regarded as a system
control parameter in MALINEA feedback control, the role of which is to produce control
input reflecting the current status. Thaler and Pastel (1962) stated that when there is only
a mild degree of non-linearity in the control parameter, the effects of such nonlinearity
may not be observable except by careful measurement. Moreover, there are quite
convenient linear approximation methods such as tangent-square approximation and least
square approximation. These concepts are similar to Papageorgiou’s description (1990)
based upon which linear feedback control is applicable to nonlinear systems that are
linearized around a desired steady state.

Following this reasoning, it may be concluded that the A-parameter that is obtained

from a linear A-curve can be regarded as a constant value.

51

4.2.2 Fluctuation under Congestion
As discussed in the previous section, the fluctuation under congestion leads to lower
RSQ values of the A-curve when linear models are fitted to the models obtained under the
uncongested state. This section explains the cause of fluctuation, which provides the
basic concept of model construction and A-parameter selection.

The fluctuation under congestion on the basic freeway section has been studied for
a long time. Mika, Kreer, and Yuan (1969) indicated that traffic flow is categorized into
two distinct modes (a) steady flow, where the usual linear and parabolic relations
between density, flow rate, and speed apply; and (b) an oscillatory mode, in which speed
and density exhibit out-of-phase periodicities when plotted as a function of time. It was
added that the system switches from one mode of flow to the other when operation is near
capacity.

Giving more detail, Kosh et al. (1983) observed the fluctuation based on its causes.
They pointed out that traffic flow under the congested condition has some specific
characteristics such as the following:
. Oscillations in traffic flow are not uniform over the congested road sections.
. There are small amplitudes immediately upstream from the bottlenecks. As they

propagate upstream, the ripples grow larger, and the wave numbers fewer.
. The oscillations of the two neighboring lanes become more synchronous as they
propagate toward upstream.

Based on Kosh’s study, Iwasaki (1991) studied shockwaves from the oscillatory
characteristic of traffic flow in Japan. His results indicated that there exists an accordion-
like action at the upstream of bottleneck under congestion and explained the fluctuation

by shockwaves and the effect of the bottleneck.

52

Fluctuation at the ramp merge may not be different from that of any other
bottleneck on a basic freeway section. A unique attribute is the existence of the ramp
volume as a possible cause of the fluctuation. Elefteriadou, Roess, and McShane (1995)
studied the effect of ramp flows on breakdown at the freeway ramp merge. They
suggested that the breakdown at the ramp freeway junction is a probabilistic variable
depending on the ramp volume characteristic.

Based on the above research, it can be inferred that the fluctuation at the ramp
merge under congestion may be caused by the ramp volume. The fluctuation effect is
known to be quantifiable by several variables such as occupancy or velocity oscillation
by time. Figure 4.9 shows the relationship between occupancy change and ramp volume.
The occupancy change by minute is used as a measurement of fluctuation and the
volume is used as an indicator of the cause. In the figure, the more scattered data is
observed with the higher ramp volume. As in an existing research about breakdown at
basic freeway section and ramp merge (Elefteriadou, Roess, and McShane 1995), the
fluctuation of the ramp merge appears to follow the probabilistic behavior of the ramp
volume. The probabilistic behavior denotes that the change of the ramp volume is not
necessarily incurring fluctuation in the upstream occupancy. But the ramp volume may
be one of the main factors to determine the change of occupancy fluctuation under
congestion.

Therefore, it may be concluded that the fluctuation and low RSQ in the A-curve
construction at the ramp merge are inevitable characteristics under congestion. More
importantly, MALINEA’s control theory, that the upstream occupancy is changed by the

on-ramp volume control, is confirmed by this observation.

53

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4.2.3 Location Specific A-Curve

Traffic characteristics such as speed, volume, and density (occupancy) can vary
considerably due to geometric, surface, weather condition, etc. As the A-curve slope is
the function of traffic characteristics, it could be dependent on geometric or site-specific
conditions. Therefore, the relationship between geometric and site-specific characteristics
of the A-curve slope is studied in this section.

The effect of site characteristics on the A-curve can be explained easily by an
example that compares two data sets from two distinct major merge bottlenecks. Figures
4.10 and 4.11 show trends at two ramp merges, at detector locations 63-64 and 72-73
respectively. Normal weekday morning data during February 1997 were accumulated for
this purpose. It may be concluded that although the data were accumulated over several
days, specific trends of the A-curve exist for each location under both uncongested and
congested states.

Detector location can be an influential variable on the A-curve slope. Kosh et al.
(1983) observed the occupancy oscillation upstream of the bottlenecks. It was described
that as the oscillation propagated upstream, the ripple grows larger, and the waves are
reduced in number. Following this theory, the upstream occupancy of the on—ramp may
vary, depending on the distance to the bottleneck cause. On the other hand, Newman
(1963) described varying occupancies at the downstream of the on-ramp during
congestion. He observed that drivers increase their gaps following the merge. Likewise,
this implies that the downstream occupancy of the on-ramp may vary depending on the
distance to the cause of bottleneck.

Based on the above observations, two studies were performed to show the

variability of the A-curve slope as the detector location moves closer to the ramp merge.

55

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The first study was focused on the occupancy distribution from far upstream to far
downstream of the on-ramp. This was to identify how the occupancy changes before and
after the ramp merge section. The second one was performed partly based on the first
result and was the predictor study for the A-curve slope. The predictor study for the A-
curve slope also considers geometric characteristics and other parameters with a potential

to influence the slope.

Occupancy Distribution Study
Occupancy distributions from the far upstream to the far downstream of the on—ramp on
serial detector zones on the I-94 freeway line were studied. The observation was done in
the morning of February 18, 1997 at detectors 62, 63, 64, and 65. Figure 3.3 described
the distances between detectors and detectors to the ramp nose.

Occupancy distributions from far upstream to far downstream of the on-ramp under
a congested condition is shown in Figure 3.4. The highest occupancy values appear at the
location immediately upstream of the on-ramp with the slightly less occupancy values
observed at the far upstream. Immediately downstream of the on-ramp, occupancies drop
down considerably and, at the far downstream, occupancies look slightly increased. The
reason for the increased occupancy at the far downstream detector can not be clarified
because many variables such as queue back-up from the next ramp, or any incident, could
affect the result.

On the other hand, relatively high occupancy at the far upstream may be explained
by the ramp merge effect or queue back-up effect because no ramp exists between the far
upstream and the upstream detector. Data obtained during the other collection periods

(February 1997) showed consistent results for upstream and far upstream occupancies.

58

The average occupancies at 65, 64, 63, and 62 zones under congestion were 13.8, 13.4,
15.5, 14.1 respectively. From this comparison, it is clear that the detector location before

or after the ramp merge affects the occupancy measurements.

Predictors of A-Curve Slope

As the A-curve slope (A-parameter) is a function of upstream and downstream
occupancies and the detector location affects the occupancy measurement, the detector
location must affect the A-curve slope. In addition, several other possible variables can be
included for this study. If a freeway section is geometrically straight, a geometric
component which distinguishes a ramp merge from others could be length of acceleration
lane. Ramp volume could be a variable influencing the A-curve slope because the ramp
volume may affect the fundamental traffic flow curve at the downstream of the on-ramp
and accordingly affect the A-curve slope.

Therefore, detector locations, ramp volumes and acceleration lane lengths are tested
as possible predictors of the A-curve slope. For this purpose, 30 samples were obtained
from 15 ramp merge locations. Table 4.3 shows a description of each sample, data for
which were collected under uncongested conditions. For congested conditions, the value
was inconsistent for all variables. This may be attributed to the fluctuation effect or to the
biased occupancy distribution during a day. However, based on the characteristics of the
symmetric slope of the A-curve, the value of uncongested condition can provide useful
understanding of the characteristics of both congested and uncongested conditions.

Tables 4.4 and 4.5 show the results of multivariate linear regression for the sample.
The results show that the distance from the ramp nose to the downstream detector zone

(DD) and the length of acceleration lane (LA) are influential factors.

59

Table 4.3 Sample Description for A-Curve Slope Predictor Study

 

 

 

 

 

 

 

 

 

Detector LA (m) UD (m) DD (m) RVOL (%) A
118-119 (7/28 AM) 202 13 531 0.0609 0.8266
118-119 (3/14 AM) 202 13 531 0.0310 0.7722
125-126 (7/28 AM) 277 158 269 0.1500 0.965
199-200 (4/15 AM) 202 46 316 0.0830 0.9214
199-200 (4/16 AM) 202 46 316 0.1044 0.9072
201-202 (4/16 AM) 303 219 227 0.2304 0.9327
201-202 (4/17 AM) 303 219 227 0.2790 0.9485
193-194 (4/22 AM) 167 113 126 0.1360 1.1365
193-194 (4/16 AM) 167 113 126 0.1357 1.1587

98-101 (7/28 PM) 202 80 394 0.006 1.0493
92-93 (8/11 AM) 202 80 394 0.1357 0.8754
92-93 (8/20 AM) 227 3 314 0.006 0.8495
68-69 (5/13 AM) 227 3 314 0.0412 1.0068
68-69 (5/12 AM) 202 373 304 0 0.9734
69-71 (3/18 AM) 202 373 304 0 0.9225
69-71 (3/19 PM) 177 3 438 0.1404 0.8117
55-56 (8/18 AM) 177 3 438 0.1408 0.75
55-56 (2/17 AM) 202 373 304 0.1793 0.8106
69-71 (6/16 AM) 177 534 505 0.106 0.9797

99-101 (6/26 AM) 177 534 505 0.0690 0.8745

99-101 (5/15 AM) 277 447 417 0 0.8734

113-114 (7/30 AM) 277 447 417 0.1375 1.0721
112-114 (3/11 PM) 177 6 505 0.0901 0.8455
100-101 (5/16 AM) 177 6 505 0 0.8961
100-101 (5/12 AM) 252 516 507 0 0.8952
62-64 (2/21 AM) 252 516 507 0.31 0.8333
63-64 (1/15 AM) 252 29 507 0.2288 0.809
63-64 (2/21 AM) 252 29 507 0.2369 0.8024
72-73 (1/21 AM) 202 2 454 0.1034 0.9867
72-73 (1/22 AM) 202 2 454 0.1022 0.9862

 

LA: length of acceleration lane,
UD: distance from ramp nose to upstream detector,
DD: distance from ramp nose to downstream detector , and
RVOL: ramp volume portion in downstream volume.

60

 

Table 4.4 Coefficient Description for A under All Variables

 

 

 

 

 

 

 

 

 

Variable B (Factor) Std. Error T Sig.
(Constant) 1.264 .075 16.854 0.000
LA -4.511E-O4 0.000 -l.364 0.185
RVOL -.108 0.144 -0.749 0.461
UD 9.444E-06 .000 0.166 0.869
DD -6.443E-04 .000 -7.048 0.000
R 0.822
R square 0.676
MSE 3.372E-3

 

 

Table 4.5 Coefficient Description for A under Variables of LA and DD

 

 

 

 

 

 

 

 

 

Variable B (Factor) Std. Error T Sig.
(Constant) 1.279 .070 18.396 0.000
LA -5.81E-04 0.000 -2.238 0.034
DD -6.37E-04 0.000 -7.218 0.000
R 0.818
R square 0.669
MSE 3.193E-3

 

 

61

 

 

Testing the hypothesis that each parameter is equal to 0, which means the parameter is

not an influential value, shows that parameter DD has a p-value of 0.000 while other

factors such as UD, the acceleration length (LA), and the ramp volume portion in the

downstream volume of the on-ramp (RVOL), have 0.869, 0.185 and 0.461 respectively.

The RSQs obtained when DD, UD, LA, and RVOL were considered independently were

0.608, 0.010, 0.030, and 0.006 respectively. These results indicate the following:

. DD and LA are influential factors in determining the A-parameter.

. DD is the most significant factor in terms of statistical significance and individual
RSQ.

. As DD increases, the A-parameter decreases.

4.3 Time Lag Effect

Time lag, n is defined as the response time between upstream and downstream occupancy
measurements, or between the upstream occupancy measurement and the ramp volume
discharge. The ALINEA model does not consider the time lag between the ramp volume
discharge and the downstream response. Instead, the requirement of measurement
location suggested that congestion originating from excessive on-ramp volumes is visible
in the measurements (Papageorgiou 1991). Following ALINEA’s assumption, only the
response time of occupancy measurements between upstream and downstream detector
zones is considered as a time lag in this section. The time lag as the response time
between upstream occupancy and ramp volume discharge will be considered again in
Chapter 6. To decide the optimal time lag, the RSQ between the occupancy
measurements are used. Figures 4.12, 4.13, 4.14 compare RSQs obtained from model

fitting between 0.10) and 0,,(t) data sets at ramp merges.

62

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The comparison is based on different locations or days. After investigating RSQ trends,
the representative time lag for each day can be found.

Figure 4.12 shows that best results are obtained under zero time lag under
uncongested condition. This result is very reasonable considering that the travel time
between the two detector zones is relatively short. For example, given the distance
between two zones ranging from 317 to 544 m and an average speed of 112 km/h (70
mph), the travel time calculated ranges from 10 to 20 sec, which is less than 1 min.

Figure 4.13 shows the time lag between Od(t) and 0,,(t) data sets for different
locations and data collection days under congestion. The highest RSQ values appear with
the time lag of 1 to 2 mins, indicating that time lag exists under congestion.

Figure 4.14 show the time lag at detector locations 63 and 64 for different days of
data collection. It is shown that the time lag is not uniform at a specific location for
different days. The optimal time lag for a given location typically ranged from 1 to 2
mins based on RSQ comparison.

A unique characteristic of RSQ trends at the ramp merge under congestion is
observed. For all the RSQ trends, a relatively high correlation has been sustained for a
while, even before and after the highest RSQ is reached. This can be explained by the
queue propagation procedure and the instant downstream occupancy increase due to
inflow from the on-ramp.

The queue propagation has been observed to initiate at the shoulder lane and
expand to other lanes, while longitudinally, it propagates upstream first and downstream
later. Therefore, the queue propagation effect could be reflected on the first minute’s
observation after volume discharge. The procedure has been explained by Elefteriadou et

al. (1995) and Blumentritt (1981).

66

The instant increase of downstream occupancy can be explained by the short travel
time between the ramp discharge and the downstream measurement points even under
heavy congestion. For example, in the case when vehicle speed is as low as 20 mph, the
vehicle can travel over 500 m in a minute, under congestion. Thus, relatively high RSQs
are supposed to be obtained as soon as the ramp discharge begins, although the highest
one should be obtained at the optimal time lag.

The time lag under congestion can be reflected in MALINEA in two ways. First,
the time lag can be considered as a type of MALINEA parameter and secondly, it can be

reflected in the control function. These are explained more in Chapter 5.

67

Chapter 5

MODIF [ED MODEL CONSTRUCTION AND VERIFICATION

In Chapter 1, the possibility of ALINEA modification was suggested as a countermeasure
to the limitation of existing local traffic responsive ramp metering. Detailed studies of the
parameters involved were performed in Chapter 4. It was shown that the A-parameter
could be derived from an A-curve that represents the occupancy relationship between
upstream and downstream occupancy measurements. A new parameter n (time lag)
proved to be desirable to explain the best relationship between upstream and downstream
occupancies of the on-ramp. Thus, in this chapter, the modified model that considers both
A and n shall be considered. Two models will be constructed based on the results of
previous chapters. The first is the upstream occupancy estimation (prediction) model
using ramp volumes. This model is a mathematical expression of MALINEA. The other
is the upstream occupancy estimation (prediction) model using downstream occupancies.
The upstream occupancy estimation (prediction) under congestion is important in
traffic operations because the upstream proved to be the worst section of ramp merge in
terms of speed and occupancy (Chapter 3). As the throughput of the freeway mainline is
determined based on the worst section condition, the accuracy of upstream occupancy

estimate is highly correlated to the quality of travel time or delay estimation at the ramp

merge.

68

 

5.1 Upstream Occupancy Estimation by Ramp Volume

The upstream occupancy at the ramp merge can be estimated from ramp volume. The
upstream occupancy estimation model using ramp discharge volumes easily can be
derived from the basic MALINEA model by adding the time lag n. This function
provides upstream occupancy estimation as a function of the ramp volume given the

parameters A, K, and n.

5.1.1 Model Construction

The basic MALINEA model is given in equation (1 .3) as:

0u(t+1) = (Q'(t+1) - Q71) )A/K + 021(1) (51)

If time lag n can be added to equation (5.1), equation (5.2) can be introduced. Parameters
A p and KP are expressed as the variable with the subscription p because they may not
always represent the optimal time lag n. This may happen when there is difference

between expected and observed optimal time lags.

0.1mm!) = (Q’(t+ 1) — Q70 )A/ p + 0.(t+n) (5.2)

The calculation of implemented ramp volume (Q’ (H 1)) needs each value of the variable
to be provided. The desirable upstream occupancy 0u(t+n+1) and parameters Ap and KP
generally are given before implementation; and the current ramp volume at the time t,
Q71), is obtained during implementation. However, 0u(t+n) is still an unobtainable

variable because it is the upstream occupancy at the next time period.

69

Through the following procedure, 0,,(t+n) will be estimated in terms of obtainable
variables and the general equation for MALINEA will be introduced. Equation (5.2) can

be rewritten with initial parameters A0 and K0 as:

0u(t+n+ 1) = (Q’ (1+ 1) — Q’(t) )AMKO + 0,.(t+n). (5.3)

If n>0, 0u(t+n) estimation is needed, then equation (5.2) can be rewritten as:

Est’d 0..(t+n) = (Q71) — Q’(t-1) ) Al/K, + 0..(z+n-1). (5.4)

When n is 1, 0u(t+n-1) equals 0,,(1). If n>1, Ou(t+n-1) estimation is needed, equation

(5.2) can be rewritten as:

Est’d 0..(t+n-1) = (Q’(t-1)— Q’(t-2) )Az/Kz + 0..(t+n-2). (5.5)

When n is 2, 0u(t+n-2) equals 0,.(t) and no more estimation is needed. If n>2, 0u(t+n-1)

estimation is needed, then equation (5.2) can be rewritten as:

Est’d 0,.(t+n-2) = (Q’(t-2) — Q’(t-3) )A3/K3 + 0..(t+n-3). (5.6)

When n is 3, 0u(t+n-3) equals 0,.(1) and no more estimation is needed. Likewise, the

relationship could be extended to the case of n >3, 4, 5, Therefore, equations (5.3),

(5.4), (5.5), and (5.6) can be combined and a general equation can be summarized as:

70

0u(t+n+1) = (Q’(t+ 1) — Q70) Ar/Ko + (Q?!) -— Qfl-I) ) Az/Kz
+(Q’(t-1) —Q’(t-2)) A 2/K2 +. . .+(Q’(t-n+ 1) -— Q’ (t-n)) A n/Kn +0.0). (5.7)

Equation (5.7) is an identical expression to the general linear discrete system in the
feedback theory that is characterized by the time discrete application of the system
parameter, which is An/Kn in equation (5.7) (Cadzow 1973). This form of expression is
somewhat complex. However, if the characteristics of the parameter such as A1) and Kp
during the discrete procedure are clarified or simplified, a simpler expression is possible.

KP is the parameter used in ALINEA and is acquired from the required
measurement location so that congestion originating from excessive on—ramp volumes is
visible in the measurements. As this implies no time lag, KP is supposed to be constant,
regardless of the discrete time interval.

With respect to A p, the experimental data at the ramp merge can provide the basis
for the simplification. The data used in the time lag study in Chapter 4 can be used again
for this purpose. For comparison, the A-parameter at each time discrete (A p) is divided by
the A-parameter at the optimal time discrete (A). If this ratio is consistent over time, it
can be assumed that A p is supposed to be constant regardless of the discrete time interval.
Figure 5.1 shows serial ratios of the A-parameters (A/Ao). The ratio Ap/Ao is shown to
range from approximately 0.9 to 1 in the first four time slices. This range is relatively
small, considering the possible range of the ratio in the time discrete procedure in
feedback theory is 0 to 1. The small range has been observed across other locations
studied. For example, at detectors 92-93 and 193-194, it ranged from approximately 0.9
to 1.0; at detectors 72-73, 193-194, and 199-200, it ranged from approximately 0.85 to

1.0.

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72

If it is assumed that the A-parameter changes during the time discrete procedure are
negligible, and the K-parameter is constant, equation (5.7) can be simplified into a new

estimation or prediction equation as:

Est’d 0,. t+n+1)= [Q’(t+ 1) - Q’(t-n)]A/K +0u(t) (5.8)
01'

Q'(t+ 1) = [0u(t+n+1) — 0u(t)]K/A + Q’(t-n). (5.9)

5.1.2 Model Verification

The upstream occupancy estimation using equation (5.8) is performed using field data for
verification of the new model. The standard error (SE) is used as the indicator of
efficiency, as was true in previous chapters. Table 5.1 shows SE for 720 observations on
the specific day ranges from 0.760 to 1.129%. These values are small considering that the
occupancy ranges from 10 to 30% under congestion. Figure 5.2 shows the plot of the
estimated versus the observed 011(1) under congestion. For each 0,.(1), the SE seems to be
approximately uniform over the whole range of 0.0). This can be a strength of the new
model in terms that it is applicable to the whole congestion period. Figure 5.3 visually
shows the sequential comparison between estimated and observed 0,10).

Therefore, equation (5.8), which is a mathematical expression of MALINEA, is
verified as a good ramp merge analysis model that estimates (predicts) the upstream
occupancy, given the past time period occupancy and the ramp discharge rate. The
MALINEA model, as a ramp control strategy combined with feedback, will be discussed

in Chapter 7.

73

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75

Table 5.1 SE between Estimated and Observed Occupancies for MALINEA

 

 

 

Detectors Time Lday) Standard error (f/o)
63—64 2/14 AM 0.7803
72-73 2/ 14 AM 0.7602

199-200 4/16 PM 1.1294
193-194 4/16 PM 1.0383
72-73 2/20 AM 0.9793
63-64 2/20 AM 0.9550
72-73 2/ 10 AM 1.0499
63-64 2/10 AM 0.9041

 

 

 

 

5.2 Upstream Occupancy Estimation by Downstream Occupancy

The upstream occupancy can be estimated by the downstream occupancy. The fimction
for this purpose is easily derived from the MALINEA model. This function provides the
upstream occupancy estimation or prediction by the downstream occupancy, given the
parameter A and the time lag n. As the estimation is based on the macroscopic variables,
if the verification is added, this estimation model can be used as a macroscopic ramp

merge analysis model.
5.2.1 Model Construction
The model can be derived from the MALINEA model or directly from the occupancy

relationship between upstream and downstream of the on-ramp. Because the variable K is

the slope of the upper branch of the volume occupancy curve,

K -—- [U(t+1) — (1(1) 1/ [0,, (1+1) —— 0d (1)] (5.10)

If this function is input into MALINEA, the upstream occupancy estimation function is

derived as:

 

 

Est’d 0u(t+1+n) = [0d (t+1) — 0d (1)]A + 0.. (t+n) (5.11)

5.2.2 Model Verification

The model estimates are compared with real data. As shown in Figure 5.4, the
distribution of SE is approximately normal and the variance of the error is almost
constant over the data range. This can be strength of the model in terms that it is
applicable to the whole congestion period (This is a common attribute in ramp volume
based and downstream occupancy based models). SEs between estimated and observed
upstream occupancy values for other data sets are shown in Table 5.2. SE values range
from 0.499 to 0.891%. These values are low compared to the usual occupancy range of

12-30%.

Table 5.2 SE between Estimated and Observed Occupancies for Upstream
Occupancy Estimation Model

 

 

 

Detector Time (day) Standard error (%)
72-73 1/22 AM 0.6349
63-64 1/22 AM 0.8755
63-64 1/15 AM 0.8917
72-73 5 1/17 AM 0.4999
72-73 1/13 AM 0.7448
72-73 2/10 AM 0.7651
63-64 2/10 AM 0.7777

199-200 4/ 16 PM 0.8190

193-194 4/16 PM 0.7582

 

 

 

 

 

Figure 5.5 provides the sequential comparison between observed and estimated 0,.(t)
under congested conditions for a 152 minute data collection period at detector zone 73-

74. From the comparison between Tables 5.1 and 5.2, it may be concluded that the latter

77

model is more efficient than the former. This may be explained by the fact that equation
(5.11) has one more parameter, that is K, than equation (5.8) assuming that addition of

parameters increases the uncertainty of the equation.

78

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80

Chapter 6

APPLICATION OF MALIN EA

Issues associated with the application of variables and parameters included in MALINEA
are presented in this chapter. Issues considered include how to set the control flow of
MALINEA and how to obtain parameters such as K, A, and n. Then model applicability
under uncongested conditions, location of detectors and selection of the desirable

occupancy are also discussed as they may be important toward model implementation.

6.1 Flow Chart of MALINEA Ramp Metering

Figure 6.1 presents the MALINEA model flow chart, which describes the logic flow of
the MALINEA ramp metering. The ramp metering procedure of MALINEA is similar to
that of ALINEA. As discussed previously, MALINEA needs several variables such as K,
A, n, and the desired upstream volume secured before application. Using current or past
upstream occupancies and ramp volume, MALINEA determines the ramp metering rate
for the next time step. Practically, the geometry and the preliminary decision confine the
implemented ramp metering rate (Blumentritt et al. 1981). Therefore, after calculation of
the metering rate, calibration may follow and then the ramp rate is implemented. As
MALINEA itself is a feedback control algorithm, it may decide automatically and
continuously whether the control should continue or not. Given continuation, all steps
above are to be repeated again. Desired upstream volumes can be changed at every

iteration following operator’s decisions.

81

INPUTK,A,n

 

 

 

 

 

 

SET Target 0.,

 

 

 

 

t=t+l

 

 

 

 

 

 

 

 

0,, (t+n+1) = 0,,

 

 

 

 

 

 

INPUT 0,,(1), Q’(t-n), Q'(t-n+1) ..... Q’ (t)

 

 

 

 

 

Q' (”U =[0u(f+n+1)- 0n (WK/A“ Q'fl-n)

 

 

Yes

 

O

 

OrderQ’(t+1)

 

 

 

Yes

 

9+

v
Q’ n+1) =M

F

 

 

 

 

 

*M is the maximum ramp volume

Figure 6.1 Flow Chart of MALINEA Ramp Metering

82

6.2 K-Parameter Application

6.2.1 Description

Haj-Salem, Graham, and Middleham (1991) described K as a regulation parameter. They

suggested that a value of K= 7O veh/h gives good results in real life experiments and that

K is the only parameter to be adjusted in the implementation phase of ALINEA. With

respect to the theoretical characteristics of the K-parameter, they described that:

. ALINEA model is insensitive for a wide range of K—values.

. Increasing K leads to higher reaction time of the regulator and shorter regulation time.
The opposite effects were observed when reducing the K—value.

. For extremely high values of K, the regulator may have an oscillatory, unstable

behavior.

6.2.2 K-Parameter Calculation
K-values can be derived either from the ALINEA model and/or be obtained from the

volume-occupancy curve. Based on the ALINEA fiinction,

K =AQ at ramp / A061 (6.1)

The above equation reflects a characteristic of the ALINEA algorithm based on which K
is a function of the ramp volume change.

On the other hand, as Papageorgiou, Haj-Salem, and Blosseville (1991) described,
the K-value can be considered as the slope of the volume-occupancy curve at the location
of occupancy collection. In this case, the K—value is positive on one side of the curve, and

is represented as:

83

K =AQ at mainline downstream of on-ramp / A061 (6.2)

From the comparison of equations (6.1) and (6.2), it can be concluded that the ALINEA
model is a traffic flow analysis model which can be applied only to the case that AQ at
the mainline, upstream of the on-ramp, equals 0. Therefore, the feedback law is needed in
ALINEA for error correction between equations (6.1) and (6.2).

As K is a slope of the tangent of the fimdamental volume-occupancy diagram at the
location of occupancy collection, equation (6.2) is adequate for the ALINEA application.
Thus, AQ should be the total volume over the number of lanes during a time slice. As
discussed previously, the K—value may differ from location to location. Thus, calibration

of the K-value is needed for each location, based on observed data.

6.3 A-Parameter Application

The role of the A-parameter in the MALINEA model is the exact opposite of that of K.
That is, as A——) 00, the feedback effect and the occupancy change effect disappear. If A—>
0, extremely high Q’(t+ 1) results and, hence, too much fluctuation may be brought about.
Therefore, extremely large or extremely small values of A are not desirable as is the case

of the K-parameter.

6.3.1 A-Parameter Calculation
Three calculation methods for A-parameters are compared in this section. These include
a. 0.; -0u regression curve, b. maximum likelihood method, and c. symmetric slope

assumption.

84

04-0.. Regression Curve

The simplest way to obtain the A-parameter values is by plotting the 0d -0u regression
curve and employing the linear slope of the curve. The slope under uncongested
conditions is very easy to obtain and shows high correlation between upstream and
downstream occupancies. Under congested conditions, as discussed in Chapter 4,
obtaining a reliable curve is not that easy. This is because most of the data are clustered
near the beginning point of congestion with large deviation. The slope obtained varies

considerably day by day.

Maximum Likelihood Method
The Maximum Likelihood Method (MLM) is one of the methods recommended for a
linear approximation in feedback control theory (Thaler and Pastel 1962). The MLM
calculation fianction can be derived through couples of mathematical steps in the
following (Quandt 1958).

When y,- and yigsr represent observed and estimated variables respectively, the
density fiinction of the normally and independently distributed error term (yi- WEST) with

mean zero and standard deviation afor a total number of observations, 1 is:

1

27:0

 

1
eXPI- EU: — yiESTH (63)

where

85

t
2
201' ’J’iEST)
0' = 1:] t (6.4)

 

The likelihood of a sample oft observations is therefore,

1

(fi)’ apt—3:726,- —y,-EST>2 1 (6.5)

i=1

 

Taking the logarithm of the likelihood function L on equation (6.3),

L = —t[logx/27r +logc7 + 0.5] (6.6)

Ify,- = 0,11) and yiEST = [04(1) — 04(i-1)]A+0u(i-1),

 

t . . . . 2
L = _t[log\/§;+O'Slogz(A[0d(’)‘Odo—1)]—[0u(l)-0u(l—1)]) ]

i=1 1

— 0.510gt + 0.5] (6.7)

 

Then, it is desirable to find the value of A which maximizes the above function. If

function L is differentiated with respect to A and the derivative is set equal to zero, the A-

value is calculated as in equation (6.8).

86

1
2m. (1') - 0d (i — 1))(0. (i) — 0.. (i — 1»
A = 1:1 (6.8)

t
2mm — Odo —- 1»2

i=1

 

This equation aims at a simplified MLM fiinction in the calculation of the A-value. Table
6.1 shows the results of the upstream occupancy estimation based on MALINEA, where

A-values are calculated by the MLM procedure.

Table 6.1 Standard Error by MLM

 

 

 

 

 

 

Detector zones Time A by SE of 0.,
MLM estimate (occupancy, %)
63-64 1/15 AM 0.8408 0.8917
63-64 1/21 AM 0.8417 0.7281
63-64 1/22 AM 0.7875 0.8702
63-64 1/24 AM 0.8790 0.5668
63-64 2/10 AM 0.6135 0.7776
63-64 2/14 AM 0.7912 0.6553
72-73 1/13 AM 0.9602 0.7431
72-73 1/17 AM 0.6914 0.4735
72-73 1/21 AM 0.9452 0.6402
72-73 1/22 AM 0.9612 0.6414
72-73 1/24 AM 0.8373 0.5994
72-73 2/10 AM 0.8747 0.7651
72-73 2/ 14 AM 0.8659 0.7547
199-200 4/16 PM 0.8161 0.1458
193-194 4/16 PM 0.8026 0.1947

 

 

The standard error of occupancy ranges from 0.61 to 0.96%. These values are small,
given that the congestion begins at occupancy of 12 to 15% and the occupancy during

congestion is sustained at higher than those values.

87

An important advantage of the use of the MLM application for the A-parameter
calculation is that MLM makes the A-parameter traceable because the A-value can be

obtained in each time slice during implementation.

Symmetric Slope Approach
Since the characteristic of symmetric slope (SS) is applicable to Greenshields’ and
Athol’s curves, the A-value calculation under an uncongested state can be used as an

indirect calculation of the A-value under congestion.

Table 6.2 Standard Error by SS

 

 

 

 

 

 

 

Detector zones Time A by SE of 0.,
SS estimate Qccupancy, %)
63-64 1/15 AM 0.8401 0.8917
63-64 1/21 AM 1.037 0.7436
63-64 1/22 AM 0.8377 0.8755
63-64 1/24 AM 0.9164 0.5698
63-64 2/10 AM 0.9942 0.8530
63-64 2/14 AM 0.8222 0.6599
72-73 1/13 AM 1.0204 0.7448
72-73 1/17 AM 0.9904 0.4992
72-73 1/21 AM 0.9867 0.6407
72—73 1/22 AM 0.9862 0.6421
72-73 1/24 AM 0.9326 0.6029
72-73 2/10 AM 0.9949 0.7726
72-73 2/14 AM 0.9611 0.7634
199-200 4/16 PM 0.9196 0.1507
193-194 4/16 PM 1.073 0.2180

 

 

Table 6.2 shows the standard error obtained under the SS assumption. The results
obtained using SS and MLM are comparable in terms of standard errors of upstream
occupancies. It can be seen that A-values from the SS approach are generally greater than

those from MLM. Moreover, A-values fi'om SS give somewhat higher standard errors

88

than those from MLM. Therefore, it can be argued that MLM is the better way of

calculating the A-value.

6.3.2 Sensitivity of A
The differences between results from MLM and SS can be used for a sensitivity study.
Table 6.3 shows the sensitivity of A-values that are based on calculation results in Tables

6.1 and 6.2.

 

Table 6.3 Sensitivity of A-Values Based on MLM and SE Results

 

 

 

 

 

 

 

Detector zones Time AA ASE Sensitivity
(A by MLM-A (SE by MLM- ASE/AA
by SS) SE by SSL

63-64 1/15 AM -0.0007 0 0.0000
63-64 1/21 AM 0.1953 0.0155 0.0794
63-64 1/22 AM 0.0502 0.0053 0.1056
63-64 1/24 AM 0.0374 0.003 0.0802
63-64 2/10 AM 0.3807 0.0754 0.1981
63-64 2/14 AM 0.031 0.0046 0.1484
72-73 1/13 AM 0.0602 0.0017 0.0282
72-73 1/17 AM 0.299 0.0257 0.0860
72-73 1/21 AM 0.0415 0.0005 0.0120
72-73 1/22 AM 0.025 0.0007 0.0280
72-73 1/24 AM 0.0953 0.0035 0.0367
72-73 2/10 AM 0.1202 0.0075 0.0624
72-73 2/14 AM 0.0952 0.0092 0.0966
199-200 4/16 PM 0.1035 0.0049 0.0473
193-194 4/16 PM 0.2704 0.0233 0.0862

 

 

ASE/AA is used to explain the sensitivity of the MALINEA model for varying A-
parameters. It is shown that the value of ASE/AA ranges from 0 to 0.198. This short range
confirms that the error of the A-parameter is widely acceptable. This result positively

accords to the K-value characteristic of ALINEA. That is, because the A-value is the

89

denominator of K, a wide range of A-values can be applicable to MALINEA similarly to

the range of the K—value that can be applicable to the ALINEA model.

 

6.4 n-Selection

6.4.1 n-Calculation

The parameter n is regarded as the time lag between the 0.1 and 0,. in Chapter 4. n
explains the existence of the time lag and the parameter behaviors under a time discrete

system. However, it can not reflect the correlation between the ramp discharge rate and

 

the upstream occupancy change when the distance from the nose to the downstream
detector or the metering location is too large. For that reason, it is desirable to calculate n
based on the correlation of the on—ramp volume change and the upstream occupancy

change. From the MALINEA model, equation (6.9) can be derived.

[0,. (t+n+1) - 0,. (t+n)]/ [Q’(t+ 1) — Q71) ]= A/K (6.9)

As A/K is assumed to be constant, it is possible to consider the correlation between the
numerator and the denominator for a given pair of observations. For R” = coefficient of

correlation between (0., (t+n+1) - 0,, (t+n)) and (Q’(t+ I) — 070) during a time interval,

Optimaln={n|R,,=Max (Rn, n=1,2,3..)} (6.10)

6.4.2 n-Application During Operation
Figure 6.2 shows the correlation series between (0.. (t+n+1) - 0,. (t+n)) and (Q'(t+1) —

Q (1)) by the time lag, n. 20-minute based coefficients of correlation are serially plotted

90

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91

for a 120 minute data set. Zero time lag is observed dominantly over the entire congestion
period inferring that the time lag can be approximately constant during the whole
congestion period. At the first time and the last time slice (which denotes the beginning
and the end of congestion), the time lag effect is negligible. The result suggests that the
time lag is a traceable variable during operation. It may be concluded that this result

makes traffic responsive change of the time lag possible during implementation.

6.5 Applicability under Uncongested Condition

As ramp metering is implemented both for congestion prevention and clearance purposes,
the implementation period is not limited to the congested state and can be extended to
periods before or afier congestion. For that reason, a good ramp metering strategy should
be applicable to any range of traffic conditions without concerns about malfunction.
Therefore, it is reasonable to raise the question of whether MALINEA can be used for

both congested and uncongested states without changing parameter values and variables.

6.5.1 Applicability under ALINEA

Based on equation (6.11), Haj-Salem and Papageorgiou (1995) argued that the ALINEA
model is applicable to both congested and uncongested conditions. This reasoning was
from the fact that the measured occupancy Odfl) at any time slice, 1 in the uncongested
states is lower than the desired occupancy 0d(t+1). That is, in this case, the occupancy
feedback term of the ALINEA model becomes positive and the ordered on-ramp volume
Q’ (t+ 1) is increased as compared to its last volume Q’(t). This routine always works
under the uncongested state given that 0d(z‘) is less than 0d(t+1) even in the case that

occupancies fluctuate between the optimum and free flow states.

92

Q'(1+1)=Q'(t)+K[0d(t+1) - 0d(’)] (6.11)

A limitation exists for ALINEA in the case that the 04(t+1) is fixed at a value over the
optimal occupancy (capacity state). In this case, [Od(t+ 1) - Od(t)] may be positive under
Od(t) on the range from 0d(1+1) to the optimal occupancy regardless of the traffic flow

state. This makes Q’ (1+ 1) continuously greater than Q’(t) even under the congested state.

6.5.2 Applicability under MALIN EA

Q’(t+ 1) = [Ou(t+n+1) — 0u(t)]K/A + Q'(t-n) (6.12)

All above statements about ALINEA are applicable to MALINEA because the structure
of MALINEA is very similar to that of the ALINEA model as shown in equation (6.12).
In MALINEA, 10A always has a positive value. Therefore, if the target occupancy 0.,
(t+n+1) is fixed to the optimal occupancy (capacity state), 0,, (t) is under uncongested
state always less than 0,, (t+n+1) and Q’ (1+1) is always greater than Q’(t-n). This
indicates that after several iterations of control, the discharged ramp volume would reach
the ramp demand or the ramp capacity that is the maximum volume possible under the
given conditions.

When the desired occupancy is fixed at a value higher than the optimal occupancy
(occupancy at capacity), the same limitation which exists in the ALINEA model is
inevitable in MALINEA because Q’(t+ 1) may be consistently greater than Q'(t-n) while

0.,(1) ranges from Ou(t+n+ 1) to the optimal occupancy.

93

To avoid the limitation described above, two alternatives are possible. The first is
to use the optimal occupancy as a desirable occupancy over the whole congestion period.
The second alternative is to change the desirable occupancy to the optimal occupancy at a
specific point of occupancy during congestion. Since this alternative may be easily
adopted by simply revising the control algorithm, this is recommendable in field
operation.

Figure 6.3 shows a plot of the 0u estimation based on the ramp volume in
MALINEA. The first time slice is congested with an occupancy value of 22% and the last
time slice is uncongested with an occupancy value of 6%. The same A, K, n values are
applied to both the uncongested and congested states. Although there were no changes in
parameters during the transition to the uncongested state, it is confirmed that MALINEA
worked consistently, both under congested and uncongested conditions. It was found that
there is a smaller stande error of occupancy in the uncongested state. More specifically,
for occupancy values over 12%, the standard error was 1.05% while for occupancy

values less than 12%, the standard error was 0.447%.

6.6 Detector Location

Proper detector positioning depends on the overall control strategies being used (US.
Department of Transportation 1985). In ALINEA, it is recommended that the
downstream measurement location be placed at the position where congestion originating
from excessive on—ramp volumes is visible in the measurements (Papageorgiou et al.
1991). Additionally, a location 400 m downstream of the on-ramp nose was found to

work adequately in an implementation test (Keen, Schofield, and Hay 1986).

94

 

rm“

 

IlHl 5.! in

3.2» 288.8
.52 SR 68:25 083— .3 83883.0.an :28235 838080

82:: aQ cutomnc 0:: 38:53“ .8 85:88.5 m6 2:3,.—
Eé we:
_.0F F0 E K F0 5 3‘ 5 _.N I.

_ _ _ _ _ _ _ _ H L

 

 

0..

 

 

0..

 

 

10N

0N

 

 

 

00

 

 

 

umammno l uEmEzww l

 

 

(%) "o

95

The location of downstream measurements in MALINEA may not be different from
those in ALINEA.

The final decision on the physical location of the downstream detector should
consider the effect of the detector on the MALINEA model. As discussed in Chapter 4, A
and n in MALINEA to some extent are correlated with the distance from the ramp nose to
the detector location. Therefore, a change in the downstream detector location can lead to
a change in the value of such parameters.

The upstream detector location can be a point of concern in MALINEA. As
discussed in Chapter 3, the traffic characteristic at the ramp merge differs according to
the on-ramp location. Considering this fact, the upstream detector should be upstream of
the on-ramp. The physical beginning point of the upstream point of the on-ramp can be
the ramp nose, because the ramp nose can be regarded as the beginning point of traffic
merging.

Another important issue for consideration is how far from the nose point the
upstream detector can be placed. As far as the excessive demand continues on the
freeway mainline, the detection of upstream congestion may be possible far upstream
from the nose point. The only problem that occurs when the detector is located far
upstream is the excessive time lag observable in detection and queue dissipation. This is
because the queue formation or dissipation has its own speed. This idea originates from
shockwave theory and is supported widely in practice. Urbanik (1986), Hall and
Agyemang-Duah (1991), Mika et al. (1969), Kosh et al. (1983), and Iwassaki (1981)
observed the propagation speed under congestion. The speeds range from 10 to 25 km/h

(166 to 416 m/min) depending on the observation case. Therefore, for the time lag in

96

 

Occupancy

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Real 0.,
Desired . .
0" Congestion
0d >— ———————————————————— - — — — ———————
(Optimum) Before
Congestion
Volume
qu qd(opfimM)
(3)
Occupancy
04 Congestion
(Optimum _________________________ ._ .......
Desired Real 0., Before
0“ k‘ Congestion
Volume
qu qa (optimum
(b)

Figure 6.4 Relationship Between Desired and Real Occupancy

97

 

detection or dissipation of congestion to be reduced, the upstream detector should be

located at a point as close as possible to the ramp nose.

6.7 Desirable Upstream Occupancy

6.7.1 Desirable Occupancy Based on Traffic Characteristics

Selection of the desired occupancy 0., 1+ 1) can be another interesting issue. In ALINEA,
no requirement exists to set the desired value. Haj-Salem (1995) and Papageorgiou, Haj-
Salem, and Blosseville (1991) mentioned that the optimal occupancy is the typical value.

In MALINEA, a unique characteristic exists for congestion clearance or prevention,
that is, the desired occupancy can be lower than the optimal occupancy at the upstream
without any concern about throughput reduction at the downstream. Traffic flow
characteristics studied in Chapter 3 indicate that the upstream occupancy is always higher
than the downstream occupancy under congestion and vice versa under an uncongested
condition. Therefore, when the downstream occupancy is managed optimally, there exist
two distinguishable occupancy levels. The first is when desired occupancy is higher than
optimal occupancy. As shown in Figure 6.4 (a), the real occupancy in this case may
fluctuate around the desired occupancy with no transition to the uncongested condition,
even though the downstream occupancy is managed constantly at the optimal. The
occupancy fluctuation that may be probable in real field operation is expressed as an
arrow (Real 0., in Figures (a) and (b)).
Then, it is desirable to reduce the desired upstream occupancy to the optimal value

or I ess, to secure both the optimal occupancy at the downstream and more throughputs at
the upstream. As indicated in Chapter 3, the optimal occupancy at the upstream may not

be an existing value because of the location limitation. Although the occupancy at the

98

 

 

upstream is less than optimal, the occupancy at the downstream can be managed at the
optimal, as has been discussed above. Therefore, as shown in Figure 6.4 (b), the
occupancy at the upstream can be managed at the uncongested state, and the optimal
throughput at the downstream can still be achieved.

The extent to which occupancy at the upstream is lower than optimal occupancy
can be determined easily through the occupancy relationship value, A. For example, if the
A—value and the optimal occupancy are determined to be 0.85 and 10% respectively,
using the past field data, the desired upstream occupancy can be 8.5%. This value could

be more conservatively adjusted considering upstream occupancy fluctuations.

6.7.2 Desirable Occupancy Based on Feedback Control Theory

Feedback characteristics could limit the desired occupancy. When congestion is severe,
the large difference between current and desired occupancies may delay the response
time.

Biernson (1988) indicated that the stability of the feedback could not be achieved
with a large value of the feedback transfer function. Based on his theory, a practical limit
on an acceptable feedback design should be Max lGibl < 1.5 (where the feedback transfer
function, Gib, equals the ratio of the measured output value to the desired value).
Therefore, the excessively large measured output value that could occur in the case of a
heavy demand is not desirable for feedback control. A step-wise change of the desirable

occupancy to the Optimal can be a countermeasure to achieve continuous stability.

99

 

ll

Chapter 7

OPERATIONAL TEST USlN G SHVIULATION

7.1 Introduction of Operational Test

In Chapter 5, verification of MALINEA as a mathematical ramp merge analysis function
was performed. This proved MALINEA deserved the efficacy in upstream occupancy
estimation. However, as the estimation was only based on acquired data without
feedback, the operational effectiveness of MALINEA is still in question. The feedback
reflects the interactive operation between ramp volumes and occupancies at the ramp
merge. Therefore, the feedback operation test of the MALINEA model can be the
operational test of its ramp metering. The operational test may not be possible without
real life implementation of the ramp control strategy. In most cases, for practical and
economic reason, simulation is used to test a new ramp metering strategy.

In the present chapter, Monte Carlo simulation is employed to test MALINEA as a
control strategy when the feedback operation works during the simulation process.
Subsequently, this makes it possible to compare MALINEA and ALINEA operations and
to confirm the characteristics of MALINEA parameters such as A and n. As a result,
simulation tests are expected to increase the understanding of MALINEA’s performance

when feedback control is given.

7.2 Simulation Run Plan
Before the simulation experiment, simulation run plans should be defined for a clear

understanding of the results. First, the simulation definition and Monte Carlo technique

100

 

 

examples in the existing ramp metering research are introduced. Then, flow charts,
variable description, and data description are provided for understanding the conditions

for the simulation experiment.

7.2.1 Definition

Simulation may be defined as a technique that imitates the operation of a real world
system. A simulation model is usually based on a set of assumptions about the operation
of the system, expressed as a mathematical or logical relationship between the objects of
interest in the system. Different from the exact mathematical solutions available with
most analytical models, the simulation process involves executing or running the model
through time, usually on a computer, to generate representative samples of the measures
of performance. In this respect, simulation may be seen as a sampling experiment on the
real system, with the result being sample points.

A static simulation model is a representation of a system at a particular point in
time. A classic static simulation method is called Monte Carlo Simulation. On the other
hand, a dynamic simulation is a representation of a system as it evolves over time. The
procedure generating random variables fiom the probability distributions or random
variate generation is called as Monte Carlo Sampling. The principle of sampling from the

distribution is based on the frequency interpretation of real world data (Winston 1987).

7.2.2 Monte Carlo Simulation in Ramp Metering Studies
Monte Carlo (MC) simulation can be a very useful tool for simulation of ramp metering,
because the boundary conditions or probability distributions of traffic volume, capacity,

speed, occupancy at the mainline or at the ramp can be obtained or calculated easily from

101

 

field data. Another advantage of the MC technique is that the probabilistic nature of its
output can be used to evaluate specific parameters or variables. For this purpose,
Gafarian and Halati (1986) used the MC technique to evaluate the efficacy of the ratio of
NETSIM parameters of speed, delay, etc. using confidence intervals. For coordinated or
isolated ramp metering, there are several cases of MC technique usage. Nihan and Davis
(1989) studied the efficiency of estimators of intersection splitting probabilities using MC
simulation. The random variables were the departure of traffic and the propagation of
traffic from each on-ramp to each off-ramp. Bieberitz (1994) used the MC technique for
optimizing the ramp metering rate in a predetermined section of a freeway mainline.

MC simulation has several advantages in a MALINEA operational test. First of all,
it gives the opportunity for a simplified test as the MC simulation uses mathematical
variables without any traffic characteristic analysis. More importantly, the concept of
MALINEA and ALINEA fits the MC simulation well. Mathematical expressions of
MALINEA and ALINEA, are established on the assumption that the mainline volume is
constant and that only the change of ramp volume leads to the occupancy change
doWnstream of the on-ramp, as explained in section 6.2. Therefore, the feedback concept
is needed to handle the uncertainty of the mainline volume during the MALINEA and
ALINEA. In the MC simulation, the uncertainty of the mainline volume can be easily
quantified using the past volume distribution and feedback is also accomplished using
iterative characteristics of simulation. Consequently, the MC simulation can simulate the

uncertainty and the feedback operation in MALINEA and ALINEA.

102

 

7.2.3 Flow Charts

Flow charts of the MALINEA and ALINEA simulation procedure are presented in
Figures 7.1 and 7.2 respectively. Each flow chart consists of two major sub-structures,
i.e., the ramp metering rate calculation and the random value calculation. The ramp-
metering rate is calculated using MALINEA or ALINEA without any random variable
consideration, as supposed to be done within a ramp metering controller. As a result, the
next step ramp metering rate Q’(t+ 1) is calculated in each iteration.

The second sub-structure uses the randomness of MALINEA or ALINEA
variables. Mainline volume change Q and current ramp merge occupancies such as
Od(t+1) and 0,,(t+1) are determined using their randomness within this structure.
Through these two substructures, the randomness of traffic conditions at the ramp merge
is supposed to create an occupancy difference between the calculated controller output
and the current field data. The difference fi'om these two substructures is the error of

operation and shall be handled by the feedback operation.

7.2.4 Variable Description

The variables involved in MALINEA and ALINEA algorithms should be defined as a
random or fixed variables considering their probabilistic characteristic or role in the
algorithm during simulation. MALINEA and ALINEA variables such as K, A, n, ramp
volume, and mainline occupancy are considered as main variables. During simulation,
occupancy is expressed in the form of the mainline volume. Ramp volume is expressed as
the error of the ramp discharge for easy quantification of uncertainty. The error of the
ramp volume discharge is the difference of the ordered and the implemented ramp

volume.

103

 

 

 

 

INPUT K, A, n, Var(Q’), Var(Q)

l

SET Target 0.,

 

 

 

 

 

 

 

 

 

t=t+1

 

 

 

Yes

 

l

0:4 (t+n+1):0u P'-

L
INPUT 0.. (t). Q’ (t-n). Q’ (t—n+1) Q (t). 0. (t)

I

Plan’d Q’ (t+l)=[0,, (t+n+1)—0.,(t) jK/A +Q’ (t-n)

 

 

 

 

 

 

 

 

CH -_. u _-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Yes
Q’ (t+1) =0
Q’(t+1)>24 1, Yes
Q’ (t+1) =24
1\
Generation of Random Q’ (t+1), Q
l

 

 

Q’ (t+1),Q’ (t-n+1) = Random

0.. (1+1) = [Q'(t-n+1)-IQ'(t-n)+Q/A/K + OJ!)

0,, (t+1) = [Q’(t+1) l—Q’ (t)+Q]/K + 0,, (t)
OUTPUT 0,, (HI; Q’ (HI), 0,, (HI)

 

 

 

 

 

 

 

 

 

 

 

 

 

‘N.

Figure 7.1 MALIN EA Simulation Procedure

104

INPUT K, Var(Q’ ), Var(Q’)

 

 

 

 

 

 

 

 

 

 

 

SET Target 0,,
l

0d (”U = 0.1
l

 

 

 

INPUT 0.11), Q7141). Q7141”) Q’ (0.04“)

l

Plan’d Q71”) = [04(t+1) - 04 (01K + Q' (t)

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

Yes
Q’ (t+1)<0
t=t+1
No ' t+1 =0
/\ Q ( )
Q’ (t+1)>24 Yes
Tl
No I! Q’ (t+1) =24
F
Generation of Random Q’ (t+1), Q
1
Yes Q’ (t+1), Q’ (t-n+1)=Random

 

 

 

l
0:: t+1) : [g(t'n+1) —' Qr(t'n)+Q]A/K + 0a (t)

 

 

 

 

 

0.! (1+1) = [Q’(t+1)— Qr(’)+Q]/K + 04(1)
1 l
OUTPUT 0,, (t+1), Q’ (t+1), Od (t+1)

 

 

 

 

 

    

 

Continue ?

 

No

Figure 7.2 ALINEA Simulation Procedure

105

The K could be a random variable because the slope of the volume-occupancy curve is
not fixed and can be changed by momentary or daily conditions of the road. However, the
randomness can be ignored when its effect is small. Table 7.1 is obtained from random
collection of available data and shows that the effect of the randomness of K is not large
relative to other variables. This is supported by the fact that ALINEA has been known to

be insensitive for a wide range of K-values as documented in the literature.

Table 7.1 Contribution of Each Variable to Deviation of 0,,

 

 

 

 

 

 

Variable Time Location STD 0., Change per
of Of of STD
observation Observation variable For occupancy
of 15%
K 2/14 72-73 0.15 0.73
1/15 63-64 0.07 0.35
2/20 72-73 0.21 0.998
A 2/14 72-73 0.11 1.38
1/15 63-64 0.03 0.54
4 2/20 72-73 0.10 1.807
Mainline 2/14 72-7 3 8 .04 2.19
volume 1/15 63-64 3.3 0.89
2/20 72-73 3.1 1.25
Error of ramp COV 0.20 3 for 15% 3
volume

 

 

 

 

 

 

The A-parameter also shows the attributes of randomness resulting from the volume-
occupancy curve characteristic. Table 7.1 shows that the contribution of randomness in A
is higher than that in K. However, MALINEA has been described in previous chapters as

being insensitive to the A-parameter as well as to the K-parameter.

106

 

 

For the purpose of testing the simulation, the two variables above are chosen as
constant. This assumption helps to eliminate a type of error during simulation, which may
come from its dependence on the mainline flow rate or the ramp volume discharge error.

Parameter n is assumed as a fixed value during simulation. This is in accordance
with the observation that n was almost fixed during a day’s congestion as shown in
Chapter 6.

Mainline volume is regarded as a random variable during the simulation. Table 7.1
shows that 0,, is sensitive to the change of mainline volume. In real life operations, the
mainline volume possesses the characteristic of a random variable. This can be inferred
from the fact that the headway can be regarded as a random variable. May ( 1990), Kell
(1962), Grecco and Sword (1968), Daou (1964), Wohl and Martin (1967), Dawson and
Chimini (1968) supported the notion that headways on a freeway mainline are randomly
distributed.

Although the discharge volume is a fixed variable during ramp metering, an error
in the implemented discharge rate over the ordered rate can exist. Papageorgiou, Haj-
Salem, and Blosseville (1991) observed this error and indicated that it is completely
independent of the ALINEA algorithm. In this study, the variation of the error is
expressed by a Coefficient of Variation (COV). Five sets of data analyzed for 40 minutes
under pre-timed ramp metering in the Detroit Area show COVs of the discharged ramp
volume range from 0.2 to 0.23. Based on this, a COV of 0.2 is selected for use for the
simulation study.

ALINEA and MALINEA algorithms can produce a ramp volume less than zero or
an infinite value based on their equations. During simulation, the ramp flow discharge

rate is constrained with a maximum value of 24 vphpl and a minimum value of zero. This

107

 

is based on the maximum ramp capacity of an urban entrance is 1,450 vphpl (AASHTO
1994)

0,, (t+n+1) in MALINEA and 0d (t+1) in ALINEA are the desirable values that
may be assigned by the controller or traffic operator. For this reason, these variables are

assumed to be fixed during the simulation.

7.2.5 Monte Carlo Sampling

The type of the random simulator is an issue for the MC simulation. As the mainline
volume and the ramp discharge error are suggested as random variables, their test of
fitness to a specific distribution is required for MC sampling.

For identification of the mainline volume distribution, the mainline volume is
quantified as a ratio of Q(t+1)/Q(t). This change is intended to help the long period of
volumes considered in a distribution. Supposed that Q(t+ 1)/Q(t) is a random variable, the
hypothesis that Q(t+ 1)/Q(t) follows a normal distribution can be tested for goodness of fit
by a chi-square test. A distribution of Q(t+1)/Q(t), which is obtained at detectors 63-64
on 1/ 15 1997, is visually presented in Figure 7 .3. It approximated a normal distribution
with mean value 0999 and standard deviation 0.04. The calculated test value is 18.75 as
Table 7.2 shows. It is found that x200) quantile at significant level 0.025 is 20.48.
Therefore, the hypothesis that Q(t+1)/Q(t) follows a normal distribution can not be
rejected at level 0.025.

To test the normality of the ramp discharge error, a set of pre-timed ramp discharge
volumes are employed from I-696. As shown in Table 7.3, the test value is 7.61, whereas
x2(4) quantile at significant level 0.025 is 11.14. Therefore, the hypothesis that the ramp

discharge error follows a normal distribution can not be rejected at significance level

108

 

Table 7.2 Calculation of Chi-Square Test Value for Mainline Volume

 

 

 

 

 

 

 

 

 

 

Bin F requency* (9‘) Frequency (1) (f-f‘)"2/f
1 0.902 2 1 l
2 0.922 8 3 8.3333
3 0.942 7 7 0
4 0.962 14 16 0.25
5 0.982 17 26 3.1153
6 1.002 37 29 2.2068
7 1.022 23 30 1.6333
8 1.042 26 22 0.7272
9 1.062 12 14 0.2857
10 1.082 6 5 0.2
11 1.102 2 2 0
12 1.122 2 l 1
Sum 156 156 18.7516

 

 

Table 7.3 Calculation of Chi-Square Test Value for Ramp Discharge Volume

 

 

 

 

 

 

 

 

Bin Frequency“ (1") Frequency (f) (f-f")"2/f
1 3 l 1 0
2 4 7 3 5.3333
3 5 9 7 0.5714
4 6 11 12 0.0833
5 7 6 8 0.5
6 8 5 8 1.125
sum 39 39 7.6130

 

 

109

’W' ‘.

Frequency

 

30

 

20«

10*

Std. Dev = .04
Mean = .999
N = 157.00

 

 

 

Figure 7.3 Distribution for Mainline Volume

110

0.025. Accordingly, the distribution of the mainline flow rate and the ramp discharge
error are assumed to follow the normal distributions.

A simple technique for generating a random observation from a normal distribution

 

is obtained by applying the central limit theorem. Hillier and Lieberman (1990) provided

a simple technique for this purpose. Because a random number has a uniform distribution

between 0 and 1, it has mean 0.5 and standard deviation 1/ m . Therefore, this theorem

implies that the sum of z uniform random numbers has approximately a normal

 

distribution with mean 2/ 12 and standard deviation V 2 / 1 2 . If r1, r1, r2... r2 are a

sample of a uniform random numbers, then,

2

x= 0' Zn+(,u-§ a ). (7.1)

V2/ 12 i=1

 

 

Equation 7.1 is a random variable observation from an approximately normal distribution
with mean ,u and standard deviation (1 Hillier and Lieberman (1990) suggested that this
approximation is an excellent one except in the tails of the distribution, even with small
value of z. The values of z from 5 to 10 may be adequate; 2 =12 is a convenient value,

because it eliminates the square root from the preceding expression. Therefore,

12
x a air,- + (p— 60) (7.2)

i=1

111

7.2.6 Data Description
The ramp merge at detector location 63 and 64 are selected for data analysis because it
shows recurrent congestion during the morning peak hour. Table 7.4 summarizes the

basic conditions of the simulation test based on collected data or assumed parameter

 

 

 

 

 

 

 

values.
Table 7.4 Basic Condition of Simulation Test
Description Mean or Calculation equation STD
A 0.8401 0.028
K 3.1105 0.073
Length of congestion (min) 31 -
Main line demand (veh / min) 102 3.300
n min (Assumed) 3 -
Initial 0,, (%) 18 -
Initial 0,, (%) 20 -
Desired occupancy (%) 16 -
No. of simulation run 500
Vol (veh/min) -3.1105*0,- + 142.41 -
Speed (fi/min) 15237.2*exp(-0.0921* 0, ) -
Influenced downstream length (m (fi)) 457 (1,500)
Influenced upstream length (m (11)) 457 (1,500) -
7.2.7 MOEs

The results from the simulation runs are compared based on several measures of

effectiveness (MOEs). Selected MOEs are:

. Occupancy: among the traffic characteristics, speed and occupancy are known to be
highly correlated with each other (Athol 1965). Therefore, occupancy is a good
indicator of the traffic condition. Although a ramp metering strategy is targeting a

desired occupancy, the resulting occupancy could be different from the ordered

112

 

 

 

occupancy. This can lead to late clearance of congestion and ineffectiveness of ramp
metering.

. Recovery speed: recovery speed here is defined as the time it takes to reach the
ordered occupancy for the first time. Feedback control does not guarantee the
immediate response to the desired occupancy. This brings out the warming-up time

and lengthens congestion periods more than expected. The late response increases the

 

total travel time as the congestion period is lengthened.

. COV (coefficient of variation): the statistical meaning of coefficient of variation

 

represents the standard deviation divided by the mean. COV is very useful in order to
quantify the scatter of a random variable (Harr 1987). The scatter of traffic
characteristics during congestion is viewed as a measure of ineffectiveness. Large
scatter may indicate the ineffectiveness of ramp metering because overestimation or
underestimation of the current system may follow.

. Ramp discharge rate: Ramp discharge rate can be a measure of effectiveness for the

 

ramp volume. However, the efficiency of this MOE may be not linked to that of the
total system because a more restricted discharge rate under congestion generally leads
to higher effectiveness in the system.
Total travel time and total travel distance are the most generally accepted MOEs in
freeway traffic operation. Sisiopiku (1998) provided the mathematical concept of these
MOEs. Based on the concept, the detailed equations that consider l-min based time slices
can be expressed as follows:

. Total non-delay travel time (TNT?)

113

d,uk Li X
TNTT(veh— hrs): 22(5 QU)

1 SU

(7.3)

where

L,- = length of section i (d for downstream of on-ramp, u for upstream of on-ramp)

 

(miles), l

Qij = number of vehicle at j time slice (j =1, 2. . . , k) (veh/min), and

 

V“ ...._......
.1“

Sij = speed on section i at j time slice (mph).

. Accumulated mainline delay (MD)

(Q5-"d -Q,-,~)
60

 

MD(veh— hrs)— Z , if Q?” —Q,-j)20 (7.4)

j: l

where

 

Q?” = mainline demand at j time slice (veh/min), and

other parameters as defined above.

. Accumulated ramp delay (RD)

(de -Qj) .

RD(veh— hrs): Z 60 ,1f(Q;d-Q;)2o (7.5)

j=1

114

where

de = ramp demand at j time slice (veh/min), and

Q; = metering rate at j time slice (veh/min).

. Total travel time (T77)

TTT(veh—hrs)= 7NTT+1VID+RD

. Total Travel Distance (TTD)

d,u k

TTD(miles) = ZZ(L,. x Q”)
i 1'

7.3 Results and Discussion

7.3.1 MALIN EA and ALINEA Comparison

Figures 7 .4 and 7.5 offer a visual comparison between MALINEA and ALINEA results.
First of all, it is shown how the two feedback models may work operationally during real
life implementation. The feedback effect appears to be reflected on each figure because
the fluctuated occupancy is converging on the desired occupancy 16%. In addition, the

statistical comparison of simulation results is shown in Table 7.5. Both results are

interpreted as follows:

- The average ordered ramp volume values, 0,; and 0,, in MALINEA are lower than

those in ALINEA. This is explained by the fact that 0,, > 0,, at the ramp merge under

congestion, thus, MALINEA results in greater restriction.

115

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Recovery time up to the desired 0,, is shorter during MALINEA compared to
ALINEA. This can be explained by the fact that MALINEA has a feedback system

based on the upstream measurement while ALINEA has it based on the downstream

measurement.

In terms of COV, MALINEA is more effective in upstream side and ALINEA is

more effective in downstream side. This effect originates from the difference of

measurement location.

Except for the accumulated ramp delay, MALINEA performs better than ALINEA
with respect to delays. One interesting result is that MALINEA gives higher benefits
in downstream non-delay travel time. However, this is expected because as the

downstream occupancy decreases, the downstream throughput increases based on the

fundamental traffic stream model.

Table 7.5 MOE Comparison (case: MALINEA vs. ALINEA)

 

 

 

MLM (A=0.84) ALINEA P value based on
paired t-test
Avg U, (veh/min) 2.3348 10.1038 0.0000
COV of U, 1.7058 1.1682 0.0000
Recovery time up to 8.7333 17.1667 0.0000
Target 0,, (min)
Avg 0,,(%) 13.9533 16.6345 0.0000
COV of 0,, 0.1774 0.1757 0.4479
Avg 0,,(%) 17.8196 21.5140 0.0000
COV of 0,, 0.1133 0.2481 0.0000
TNTT at downstream 18.1503 21.5497 0.0000
T NTT at upstream 22.8766 27.0901 0.0000
MD 7.6791 13.5015 0.0000
RD 11.8519 7.5295 0.0000
TTD 1650.3646 1471.830] 0.0000
TTT 60.5578 69.6710 0.0000
Sum of TNTT 41.0267 48.6399 0.0000

 

 

 

 

 

118

 

if"!

 

The large difference between the total travel distances (T TDs) in Table 7.5 may not be
understandable in terms that TTD should be identical in the demand-capacity analysis
(May 1990). However, it should be noted that, based on equation 7.7, the throughput
increase can leads to the TTD increase although this may occur only in the case of a short

operational period.

7.3.2 “Wrong” A-effect

Tables 7.6 and 7.7 show MOE comparisons based on two different A-values. A=0.84 is

obtained from MLM and A=1.24 and 0.44 are chosen as “wrong” A-values to

demonstrate the effect of the use of a high and a low A-value. The results are summarized
as follows:

. The ramp volume using the “correct” A is almost always equal to or less than that
using the “wrong” A. This means the “correct” A controls ramp volume more strictly
than the “wrong” A.

o The “correct” A gives equal or less recovery time than the “wrong” A.

. COVs of 0,, and 0,, using the “correct” A is almost always equal to or less than those
using the “wrong” A.

0 Except for the accumulated ramp delay (RD), all MOEs about delay show that the
“correct” A gives the best result. Especially in TTT and (short termed) TTD, the
differences are significant.

In several MOEs, the difference is statistically significant. However, the difference of

MOEs between the results are very small so that the results confirm the previously

mentioned statement that the sensitivity to variations of A is very low.

119

 

 

7.3.3 “Wrong” n-effect
Tables 7.8, 7.9, 7.10, and 7.11 show the effect of a “wrong” n value selection on MOE’s.
The case n =3 is the one for the simulation design and n =1, 2, 4 and 5 are selected as

examples of the “wrong” It case. The results are summarized as follows:

 

The ramp volume using the “correct” 11 is almost always equal to or less than that
using the “wrong” n.
. 0,, with the “correct” n is greater or almost always equal than that with the “wrong” n

while the 0,, difference range is as small as 0.06 to 0.2%.

 

. Recovery time with the “correct” It is equal or less than that with the “wrong” n.

. All COVs show that the “correct” n gives more stable traffic results than the “wrong”
It.

. The control with the “correct” n value leads to better results in delay and travel time
except for accumulated ramp delay. Especially in TTT and (short termed) TTD, the
differences are significant.

Based on the study, when the “wrong” It value is selected, MALINEA still works with

small difference in MOEs. As the difference of MOEs between the results are small, it is

supported that the “wrong” n effect is not considerable.

7.4 Summary

Base on the simulation testing, the following conclusions can be reached.

. MALINEA algorithms works as a ramp metering strategy during the feedback control
operation.

. The difference between MALINEA and ALINEA results is very significant in most

MOEs.

120

Except for the accumulated ramp delay, MALINEA performs better than ALINEA
with respect to delays.

MALINEA is insensitive to the A-value. However, the value from MLM gives the
best effectiveness in most MOEs including total travel time spent (777) and (short
termed) total travel distance (TT D).

MALINEA is insensitive to the “wrong” n while the “correct” n gives the best
effectiveness in most MOEs including total travel time spent (77?) and (short termed)

total travel distance (TTD).

121

 

 

Table 7.6 MOE Comparison (case: A = 0.84 vs. 1.24)

 

 

 

 

MLM (A=0.84) Large A (A=1.24) P value based on
paired t-test
Avg U, (veh/min) 2.3348 2.3370 0.4927
COV of U, 1.7058 1.4825 0.0000
Recovery time up to 8.7333 9.6833 0.1071
Target 0,, (min)
Avg 0,,(%) 13.9533 13.6528 0.0652
COV of 0,, 0.1774 0.1776 0.4857
Avg 0,,(%) 17.8196 17.827755 0.2391
COV of 0,, 0.1133 0.1152 0.0890
TNTTat downstream 18.1503 18.6465 0.0188
TNTT at upstream 22.8766 22.9482 0.2090
MD 7 .6791 7.7685 0.2063
RD 11.8519 11.7597 0.0409
TTD 1650.3646 1636.257 0.0238
TTT 60.5578 61.1231 0.0912
Sum of TNTT 41.0267 41.6662 0.0113

 

 

 

Table 7.7 MOE Comparison (case: A = 0.84 vs. 0.44)

 

 

 

 

 

 

MLM (A=0.84) Large A (A=0.44) P value based on
paired t-test
Avg U, (veh/min) 2.3348 2.8317 0.0000
COV of U, 1.7058 2.1219 0.0000
Recovery time up to 8.7333 8.7666 0.4818
Target 0,, (min)
Avg 0,,(%) 13.9533 13.9778 0.4502
COV of 0,, 0.1774 0.1923 0.0006
Avg 0,,(%) 17.8196 17.9980 0.0016
COV of 0,, 0.1133 0.1140 0.2995
TNTI’at downstream 18.1503 18.5587 0.0388
TNTT at upstream 22.8766 23.1075 0.0013
MD 7.6791 7.9731 0.0010
RD 11.8519 11.5059 0.0000
TTD 1650.3646 1635.417 0.01312
TIT 60.5578 61.1452 0.0664
Sum of TNTT 41.0267 41.6662 0.0113

 

122

 

 

 

 

 

Table 7.8 MOE Comparison (case: n =3 vs. 1)

 

 

 

 

 

 

n=3 (Correct) n=l (Wrong) P value based on
paired t-test
Avg U, (veh/min) 2.3348 3.0213 0.0000
COV of U, 1.7058 1.6701 0.1620
Recovery time up to 8.7333 9.3333 0.2514
Target 0,, (min)
Avg 0,,(%) 13.9533 13.6672 0.0799
COV of 0,, 0.1774 0.1907 0.0227
Avg 0,,(%) 17.8196 17.8711 0.1949
COV of 0,, 0.1133 0.1248 0.0000
TNTT at downstream 18.1503 18.5323 0.0434
TNTI’ at upstream 22.8766 23.0430 0.0192
MD 7.6791 7.8687 0.0280
RD 11.8519 11.6871 0.0007
TTD 1650,3646 1637,6481 0.0278
777' 60.5578 61.1312 0.0690
Sum of TNYT 41.0267 41.5753 0.0227

 

Table 7.9 MOE Comparison (case: n =3 vs. 2)

 

 

 

 

 

 

n=3 (Correct) n=2 (Wrong) P value based on
paired t-test
Avg U, (veh/min) 2.3348 3.3771 0.000
COV of U, 1.7058 1.7877 0.010
Recovery time up to 8.7333 9.1000 0.3117
Target 0,, (min)
Avg 0,(%) 13.9533 14.046 0.3245
COV of 0,, 0.1774 0.1947 0.0038
Avg 0,,(%) 17.8196 18.0294 0.0004
COV of 0,, 0.1133 0.1363 0.0000
TNYT at downstream 18.1503 18.5922 0.0246
TNTT at upstream 22.8766 23.1535 0.0000
MD 7.6791 7.9657 0.0019
RD 11.8519 11.3924 0.0000
TTD 1650.3646 1634.29 0.0077
TTT 60.5578 61.1040 0.0802
Sum of TNTI" 41.0267 41.7458 0.0043

 

123

 

 

 

 

 

Table 7.10 MOE Comparison (case: n =3 vs. 4)

 

 

 

 

 

 

n=3 (Correct) n=4 (Wrong) P value based on
paired t-test
Avg U, (veh/min) 2.3348 3.0054 0.0000
COV of U, 1.7058 1.7374 0.1801
Recovery time up to 8.7333 9.1667 0.2642
Target 0., (min)
Avg 0,,(%) 13.9533 14.1667 0.1394
COV of 0,, 0.1774 0.1874 0.0590
Avg 0,,(%) 17.8196 18.0561 0.0000
COV of 0,, 0.1133 0.12645 0.0000
TNTT at downstream 18.1503 18.6069 0.0228
T NTT at upstream 22.8766 23.2194 0.0000
MD 7.6791 8.0711 0.0000
RD 11.8519 11.4673 0.0000
TTD 1650.3646 1632.696 0.004
TTT 60.5578 61.3648 0.0214
Sum of TNTT 41.0267 41.8263 0.0022

 

Table 7.11 MOE Comparison (case: n =3 vs. 5)

 

 

 

 

 

 

n=3 (Correct) =5 (Wrong) P value based on
paired t-test
Avg U, (veh/min) 2.3348 3.0122 0.0000
COV of U, 1.7058 1.8002 0.0003
Recovery time up to 8.7333 9.2833 0.2558
Target 0., (min)
Avg 0,,(%) 13.9533 13.9593 0.4883
COV of 0,, 0.1774 0.2000 0.0004
Avg 0,,(%) 17.8196 18.1120 0.0000
COV of0,, 0.1133 0.1354 0.0001
TNTT at downstream 18.1503 18.5961 0.0243
TNT!" at upstream 22.8766 23.3138 0.0000
MD 7.6791 8.1722 0.0000
RD 11.8519 11.2680 0.0000
TTD 1650.3646 1630.509 0.0013
TTT 60.5578 61.3501 0.0178
Sum of TNTT 41.0267 41.9098 0.0006

 

124

 

 

Chapter 8

SUMNIARY AND CONCLUSIONS

8.1 Summary
This study began with the review of existing ramp-metering strategies, which are based
on the assumption that the bottleneck section is the freeway section immediately
downstream of the on—ramp. The concept results from the definition of bottleneck as a
section of roadway in which more demand is delivered than can be processed. The
definition implies that the bottleneck is decided only on the basis of traffic flow, without
consideration of density and speed conditions. Thus, a question was raised whether the
worst section (in terms of flow rate) corresponds to the section that demonstrates the
worst density and/or the worst speed conditions. The answer to this question is important
for proper ramp metering.

Theoretical and empirical studies were performed to test the hypothesis that, under
congestion, the worst section with respect to density and speed is the one located
upstream of the on-ramp. Results from the traffic stream model analysis, the empirical
analysis and the breakdown procedure were as follows:

. The volume-density (occupancy)-speed relationship in the traffic stream model is
based on the concept that the location limits the range of flow rate, speed, and density
values. This concept is developed for the general bottleneck location, but also applies
to bottlenecks at the ramp merge. Therefore, typical relationships between traffic
characteristics at the upstream and downstream of on—ramps can be identified using

this concept.

125

 

 

L1 .

 

. Under congestion, the worst section at the ramp merge is upstream of the on-ramp,
even though the bottleneck develops on the downstream side.

. Breakdown first occurs upstream of the ramp merge. Therefore, even when ramp
metering is used to manage the downstream flow at capacity, the breakdown
upstream of the on-ramp can cause upstream congestion development.

. Existing local traffic-responsive ramp metering strategies are typically based on
downstream measurements. Modification of these strategies, to account for worst
section conditions upstream of the on-ramp, may have great potential for the
improvement of ramp metering, and for reduction of delays in the vicinity of metered
on-ramp locations.

Therefore, a desirable new ramp metering strategy was introduced with fundamental

principles similar to the popular ramp metering algorithm ALINEA. The new model was

called MALINEA because it represents “Modified ALINEA.” MALINEA and ALINEA
models were mathematically linked using parameters, with characteristics from traffic
stream models.

Detailed studies of the MALINEA parameters were performed. The parameters
included K (the slope of occupancy-volume curve), A (the slope of the relationship
between the upstream and downstream occupancies), and n (the time lag between the
upstream and downstream occupancies). It was found that there exists a consistent trend
of the A-curve based on traffic conditions. For the merge case, there was approximately a
linear A-curve trend both under congested and uncongested traffic conditions. For the
queue backup case from far downstream, or the queue dissipation from upstream, the A-
curve trend was lower than that of the ramp bottleneck case. The K-curve also had a

consistent trend, as had been discussed in the ALINEA study.

126

 

 

The characteristic of the A-curve was studied based on Greenshields’ theory,
Athol’s theory and the field data. The discrepancy at the threshold of the A-curve was
explained by all three studies. The linearity of A-curve was controversial because it was
supported only by Athol’s theory and field data. A valuable result based on Athol’s
theory is that linearity of the volume-occupancy curve leads to linearity of the A-curve,
while linearity of the volume-occupancy curve has been confirmed in many previous
studies. Literature research shows that mild non-linearity does not matter in feedback
control theory. Following this reasoning, the A-parameter was suggested as a constant,
the value of which was obtained from the linear characteristic of the A-curve.

The A-parameter was found to have location specific characteristics. Several results
were briefly discussed. The distance between the ramp nose and the downstream detector
location (DD), and the length of acceleration lane (LA), were influential factors in
determining the A—parameter. DD was the most significant factor in terms of its statistical
significance and RSQ. It was found that the A-value decreases as DD increases.

Based on the above results, models were constructed and verified. The A-parameter
proved that it could be derived from the A-curve which represents the occupancy
relationship between upstream and downstream occupancy measurements. A new
parameter, n, proved desirable for obtaining the best relationship of the A-parameter.
Thus, the modified model that considered both A and n was introduced. Two models were
constructed based on the parameter characteristics. The first was the upstream occupancy
estimation (prediction) model using ramp volume, which was also a mathematical
expression of MALINEA. The other was the upstream occupancy estimation (prediction)

model using downstream occupancy. The verification process showed that standard

127

 

 

errors were relatively small in both models and that the estimated and observed values
were highly correlated.

For application of MALINEA, various issues such as how to obtain the proper
value for each parameter, and how to set each variable, were studied. For the calculation
of the A-parameter, the results from the symmetric slope (SS) and the maximum
likelihood method (MLM) were compared in terms of standard errors of upstream
occupancy. It was found that MLM was the best way to calculate the A-value.

Although there were no changes of the factors during transition to an uncongested
state, it was found that MALINEA works consistently under both congested and
uncongested conditions, and that there is a smaller standard error of occupancy under the
uncongested state than under the congested state.

Detector location was studied. Location of the downstream measurement in
MALINEA was assumed not different from that in ALINEA. It was suggested that the
upstream detector is located at a point as close as possible to the ramp nose point. An
additional suggestion was to reduce the desired upstream occupancy to the optimal or
less. This was explained by the fact that when the upstream occupancy is less than
optimal, the downstream occupancy can be managed at the optimal.

Feedback characteristics could limit the desired occupancy. It was suggested that
when congestion is severe, the fixed desired occupancy might delay the response time, in
that the stability of feedback could not be achieved with the excessive value of feedback
transfer fimction. Therefore, the excessively large measured output value, which could
occur in the case of heavy demand, was not desirable for MALINEA.

Simulation was employed to consider the feedback effect during ramp metering.

This step was possible because a mathematical function of MALINEA was established

128

 

 

and its verification was followed with field data. MALINEA was compared to ALINEA

and its parameter characteristics were discussed based on feedback control and ramp

metering. Following are the results from the simulation.

MALINEA algorithms work as ramp metering strategies during the feedback control
operation.

The difference between MALINEA and ALINEA results is very significant in most
MOEs.

Except for the accumulated ramp delay, MALINEA performs better than ALINEA
with respect to delays.

MALINEA is insensitive to the A-value. However, the value from MLM gives the
best effectiveness in most MOEs, including total travel time spent (T77) and (short
termed) total travel distance (TT D).

MALINEA is insensitive to the “wrong” It while the “correct” n gives the best
effectiveness in most MOEs, including total travel time spent (T77) and (short

termed) total travel distance (TT D).

8.2 Conclusions

The relationship between traffic characteristics upstream and downstream of the on-ramp

under congested conditions was investigated based on traffic flow theory and field data. It

was concluded that the upstream is almost always the worst section at the ramp merge, in

terms of speed and occupancy. This suggests the introduction of new ramp metering

strategies for upstream monitoring, as a basis for control. The new ramp metering

strategy introduced in this study is MALINEA, which is a modified version of ALINEA,

an existing ramp metering strategy. After the study of parameters and associated theories,

129

 

 

 

MALINEA was constructed as a mathematical and macroscopic flow model and was
verified using field data. MALINEA was tested as a feedback control and ramp metering
model using simulation. Comparisons of MOEs obtained from simulation runs of
MALINEA and ALINEA under similar conditions, showed that MALINEA
demonstrated superior performance. Therefore, MALINEA was introduced as a new

ramp metering strategy with good potential.

8.3 Limitations and Recommendations

8.3.1 Effect of Detailed Geometric Conditions

The concept that the section upstream of the on-ramp is the worst one at the ramp merge,
in terms of speed and occupancy, was generalized. However, the effects from detailed
geometric characteristics such as high curvature, short acceleration lane, high grade,
weaving conditions, etcetera were not considered in this study, although these factors are
influential in capacity and then inevitably affect the definition of the worst section. The

study of these factors is highly recommended.

8.3.2 Speed Based Model

As a desirable ramp metering strategy, only the occupancy based model MALINEA was
considered in this study. Considering that speed is highly correlated to occupancy or
density, the speed-based model is expected to provide results similar to those of the
occupancy based model. However, as existing researches indicate, effectiveness might be

different from model to model, and thus a speed based model is worthy of study.

130

8.3.3 FRESIM Issues

More simulations or field tests are desirable for testing the model. FRESIM, a popular
FHWA supported simulation software was once considered for testing MALINEA as
another option. This was because FRESIM is known to have the ability to simulate traffic
responsive ramp metering such as ALINEA. However, the current FRESIM version was
not appropriate to simulate MALINEA for two reasons. First, the concept that the
upstream is the worst section at the ramp merge, in terms of speed, was not a usual case
in FRESIM; that is, the worst section at the ramp merge was not clearly expressed in
FRESIM. Second, the volume-occupancy-speed curve from car-following theory
assumed in FRESIM, hardly fits the field data. Therefore, traffic volumes under
congestion were highly overestimated and, accordingly, mainline delays (especially,
accumulated mainline delays) were underestimated. FRESIM’s inaccuracy in predicting
ramp merge was also confirmed by Road Management & Engineering Journal (1997).
Moreover, Dijker et a1. (1998) showed the difference of measured and car-following
modeled macroscopic characteristics. Therefore, an advanced FRESIM model version is
desirable to test MALINEA. This may make testing MALINEA simple under various

conditions, as well as help accurate explanation of traffic behavior at ramp merge.

8.3.4 Effect on Parallel Corridor

The simulation result showed that MALINEA increases accumulated ramp delay while
decreasing all other types of delays. This leads to an inference that MALINEA’s benefit
is limited to mainline traffic. However, this may not be true in field operations. As the
ramp metering increases the throughput of the on-ramp, diversion to a parallel corridor

decreases and traffic condition on the parallel corridor can be improved. Haj-Salem and

131

 

 

 

Papageorgiou (1995) insisted that this effect be clear under both recurrent and non-
recurrent congestion during ALINEA field tests.

Because reliable testing may require numerous field tests on a network basis, long
observation periods and accurate calibration, the question still remains concerning the
positive effects of MALINEA on parallel corridors. Therefore, additional field tests are
desirable for more accurately quantifying the effect of the MALINEA metering strategy

on parallel corridors.

132

 

 

 

 

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133

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