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ARIZONA DEPARTMENT OF TRANSPORTATION
REPORT NUMBER: FHWA-AZ99-462
RHODES - ITMS CORRIDOR
CONTROL PROJECT
Final Report
Prepared by: Douglas Gettman, Larry Head, Pitu Mirchandani
Systems and Industrial Engineering Department
University of Arizona
Tucson, Arizona 85721
Prepared for:
Arizona Department of Transportation
206 South 17th Avenue
Phoenix, Arizona 85007
in cooperation with
U.S. Department of Transportation
Federal Highway Administration
The contents of this report reflect the views of the authors who
are responsible for the facts and the accuracy of the data
presented herein. The contents do not necessarily reflect the
official views or policies of the Arizona Department of
Transportation or the Federal Highways Administration. This
report does not constitute a standard, specification, or
regulation. Trade or manufacturer's names which may appear
herein are cited only because they are considered essential to
the objectives of the report. The U.S. Government and the State
of Arizona do not endorse products or manufacturers.
Technical Report Documentation Page
1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.
FHWA-AZ99-462
4. Title and Subtitle
RHODES-ITMS CORIDOR CONTROL PROJECT
5. Report Date
MAY1999
6. Performing Organization Code
7. Author 8. Performing Organization Report No.
Douglas Gettman, Larry Head; and Pitu Mirchandani
9. Performing Organization Name and Address
Systems and Industrial Engineering Department
The University of Arizona
Tucson, Arizona 85721
12. Sponsoring Agency Name and Address
10. Work Unit No.
11. Contract or Grant No.
.• SPR-PL-1(51)462
ARIZONA DEPARTMENT OF TRANSPORTATION
206 S.17TH AVENUE
13.Type of Report & Period Covered
Final Report 9/96 -5/99
PHOENIX, ARIZONA 85007 14. Sponsoring Agency Code
ADOT Project Manager: Stephen R. Owen, P.E.
15. Supplementary Notes
Prepared in cooperation with the U.S. Department of Transportation, Federal Highway Administration
16. Abstract
The RHODES-ITMS Coridor Control project addresses real-time control of ramp meters of a freeway segment, with
consideration of the trafic volumes entering and leaving the freeway from/to arterials, and the regulation of these volumes
via real-time seting of ramp metering rates.
Curent approaches to trafic-responsive control of ramp meters include (a) time-of day control, (b) locally responsive
strategies and (c) area-wide linear programming based approaches (curently implemented in parts of Europe). None of
these approaches are both real-time responsive to trafic conditions and consider the multiple objectives of minimizing
freeway travel times and decreasing congestion/queues at the interchanges.
A control system, MILOS (Multi-objective Integrated Large-scale Optimized ramp metering System), was developed that
determines ramp metering rates based on observed trafic on the freeway and its interchange arterials. MILOS temporally
and spatially decomposes the ramp-metering control problem into three hierarchical subproblems: (1) monitoring and
detection of trafic anomalies (to schedule optimizations at the lower levels of the control hierarchy), (2) optimization to
obtain area-wide coordinated metering rates, and (3) real-time regulation of metering rates to adjust for local conditions.
Simulation experiments were performed to evaluate the MILOS hierarchical system against (a) "no control" (i.e., when
no ramp metering is in efect) , (b) a locally trafic-responsive metering policy, and (c) an area-wide LP optimization
problem re-solved in 5-minute intervals. Three test scenarios were simulated (1) a short "burst" of heavy-volume flows to
all ramps, (2) a three-hour commuting peak, and (3) a three-hour commuting peak with a 30-minute incident occurring
somewhere in the middle of the coridor. The performance results indicate that MILOS is able to reduce freeway travel
time, increase freeway average speed, and improve recovery performance of the system when flow conditions become
congested due to an incident.
17. Key Words
Real-time trafic-adaptive control; Adaptive Rampmetering;
Freeway Trafic Control; Hierarchical
Control; Simulation; Optimization.
19. Security Classification 20. Security Classification
Unclassified Unclassified
18. Distribution Statement 23. Registrant's Seal
Document is available to the U.S.
Public through the National
Technical Information Service,
Springfield, Virginia, 22161
21. No. of Pages
208
22. Price
\\) ., , ... $1,:\(.NO,DER .:.ME!ERIC) CONVERSION FACTORS
APPROXIMATE CONVERSIONS TO SI UNITS APPROXIMATE CONVER SIONS FROM SI UNITS
Symbol When You Know Multiply By To Find Symbol . Symbol When You Know Multiply By To Find Symbol
LENGTH LENGTH
in inches 25.4 millimeters mm mm millimeters 0.039 inches in
ft feet 0.305 meters m m meters 3.28 feet ft
yd yards 0.914 meters m m meters 1.09 yards yd
ml miles 1.61 kilometers km km kilometers 0.621 miles mi
AREA AREA
in2 square inches 645.2 millimeters squared mm2 mm2 millimeters squared 0.0016 square inches in2
ft2 square feet 0.093 meters squared m 2 m2 meters squared 10.764 square feet ft2
yd2 square yards 0.836 meters squared m 2 . m2 meters squared 1.19 square yards yd2
ac acres 0.405 hectares ha ha hectares 2.47 acres ac
mi2 square miles 2.59 kilometers squared km2 km2 kilometers squared 0.386 square miles mi2
VOLUME VOLUME
ft oz fluid ounces 29.57 milliliters ml ml milliliters 0.034 fluid ounces fl oz
gal gallons 3.785 liters l l liters 0.264 gallons gal
ft3 cubic feet 0.028 meters cubed ma ma meters cubed 35.315 cubic feet ft3
yd3 cubic yards 0.765 meters cubed ma ma meters cubed 1.31 cubic yards yd3
NOTE: Volumes greater than 1000 l shall be shown in m3.
MASS MASS
oz ounces 28.35 grams g g grams 0.035 ounces oz
lb pounds 0.454 kilograms kg kg kilograms 2.205 pounds lb
T short tons (2000 lb) 0.907 megagrams Mg Mg megagrams 1 102 short tons (2000 lb) T
TE MPERATURE (exact) TEMPERATURE (exact)
Symbol When You Know Do The Following To Find Symbol Symbol When You Know Do The Following To Find Symbol
OF Fahrenheit ° F - 32 + 1.8 Celcius oc PC Celcius ° C X 1.8 + 32 Fahrenheit OF
temperature temperature temperature temperature
oc ·40 ·20 0 20 37 60 80 100 METER: a litle longer than a yard (about 1.1 yards)
I I I I 11 I I I LITER: a litle larger than a quart (about 1.06 quarts)
I I I I I GRAM: a litle more than the weight of a paper clip Of -40 0 32 80 98.6 160 212 MILLIMETER: diameter of a paper clip wire
water freezes body temperature . water bolls KILOMETER: somewhat further than 1/2 mile (about 0.6 mile)
•s1 is the symbol for the International System of Measurement
RHODES-ITMS Corridor Control Project
PREFACE
This report documents the work performed on the Corridor Control Subproject of the
RHODES-ITMS Project. This research effort was funded by the Arizona Department of
Transportation (ADOT) and the Maricopa Association of Govermnents (MAG). Essentially, the
scope of the Project was to develop a method to optimally control, in real time, the ramp meters
on a segment of a freeway. The control architecture used was based on extensions of the
hierarchical control concepts developed for the surface street network in the previous RHODES
Project funded by ADOT and the Pima Association of Governments.
The Corridor Control subproject was directed by the principal investigators, Pitu B.
Mirchandani and Larry Head, both of the Systems and Industrial Engineering Department at
the University of Arizona. This report is largely based on the dissertation written by Dr.
Douglas Gettman whose Ph.D. research was supported by the project.
In addition, Drs. Gettman, Head, and Mirchandani wish to acknowledge their appreciation to the
Project's Technical Advisory Committee (TAC) whose continual active participation,
technical input and support resulted in MILOS (the real-time ramp-metering system described in
this report) being very relevant to freeway ramp-metering control. The following individuals
served on the TAC at various times:
Jim Decker
Tim Wolfe
Dan Powell
Tom Parlante
Glenn Jonas
Jim Shea
Sarath Joshua
Paul Ward
Roy Turner
Piere Pretorius
Don Wiltshire
Alan Hansen
Tom Fowler
Jessie Yung
Steve Owen
Traffic Operations, City of Tempe
ADOT Technology Group
ADOT District 1
ADOT Traffic Engineering
ADOT Freeway Management
ADOT Traffic Engineering
Maricopa Association of Governments (previously at ATRC, ADOT)
Maricopa Association of Governments
Maricopa Association of Governments
Maricopa County Transportation and Development Agency
Maricopa County Transportation and Development Agency
Federal Highway Administration
Federal Highway Administration
Federal Highway Administration
RHODE-ITMS Project Manager,
Arizona Transportation Research Center (A TRC), ADOT
The contents of this report reflect the views of the authors who are responsible for the facts and
the accuracy of the data presented herein. The contents do not necessarily reflect the official
views of the Arizona Department of Transportation, Maricopa Association of Governments or
the Federal Highway Administration. This report does not constitute a standard, specification or
regulation.
RHODES-ITMS Corridor Control Project
EXECUTIVESUMARY
The RHODES-Integrated Traffic Management System (ITMS) Program addresses the
design and development of a real-time trafic adaptive control system for an integrated
system of freeways and arterials. The overall program was initiated in December 1993,
jointly funded by the Arizona Department of Transportation (ADOT) through the State
Planning and Research Program budget and the Maricopa Association of Governments
(MAG). The RHODES-ITMS program is overseen by ADOT's Ariz.:ona Transportation
· Research Center.
Subsequently, in September 1996, the RHODES-ITMS Corridor Control research project
was initiated which specifically addresses real-time control of ramp meters of a
freeway segment, with consideration of the trafic volumes entering and leaving the
freeway from/to arterials, and the regulation of these volumes via real-time setting of ramp
metering rates. This is the final report for the RHODES-ITMS Corridor Control Project.
RESEARCH CONCEPTS
Current approaches to controlling ramp meters to respond to varying traffic conditions,
those reviewed in this report, include (a) time-of day control, (b) locally responsive
strategies ( one such strategy is currently under consideration by ADOT), and ( c) areawide
linear programming (LP) based approaches ( currently implemented in parts of
Europe). None of these approaches are both fully responsive in real-time to prevailing
and predicted traffic conditions and consider the multiple objectives of minimizing
freeway travel times and decreasing congestion/queues at the interchange ramps and the.
coresponding arterial intersections.
In this research, a control system was developed, referred to as MILOS (Multi-objective
Integrated Large-scale Optimized ramp metering System), that determines ramp metering
rates based on observed and predicted trafic on the freeway and its interchange arterials.
MILOS has an hierarchical architecture to address the complexities of the real-time
freeway management problem, namely, (a) the dynamic and stochastic nature of state
changes, and (b) the existence of multiple objectives. MILOS temporally and spatially
decomposes the ramp-metering control problem into three hierarchical subproblems: (1)
monitoring and detection of traffic anomalies (to schedule optimizations at the lower
levels of the control hierarchy), (2) optimization to obtain area-wide coordinated metering
rates, and (3) real-time regulation of metering rates to adjust for local conditions.
The area-wide coordination problem at the second level of the hierarchical control system
is modeled as a "quadratic programming" (QP) optimization problem that considers the
impact of queue growth on the adjacent interchanges. A multi-criterion objective function
is used to trade-of between freeway travel times and congestion/queues at the
interchanges. The resulting nominal solution of the second-level area-wide optimization
problem is then provided to the third-level control function which locally adjusts these
nominal ramp-meter rates.
The third-level problem, referred to as predictive-cooperative real-time (PG-RT) rate
regulation problem, modifies, if necessary, the ramp metering rates based on local trafic
at each interchange. The PC-RT algorithm is based on a linear programming formulation
that uses a linearized approximation of a macroscopic :freeway flow model (in terms of
dynamic diference equations). The PC-RT algorithm pro-actively utilizes opportunities
to disperse queues or hold back additional vehicles when :freeway and ramp trafic
conditions are appropriate. The cost coeficients of the LP optimization objectives are
based on the multi-objectives trade-ofs considered in the second-level area-wide
coordination problem.
The optimization runs of the area-wide coordination problem and the PC-RT rate
regulation problem at each ramp are scheduled for execution by the highest-level rampdemand/:fr
eeway-flow monitoring system that is based on concepts :from "statistical
process controf' in production systems. Basically, this system functions as follows:
When the monitored conditions are within the expected variances in ramp demands and
:freeway flows, no optimization run is scheduled to obtain new ramp metering rates; when
the conditions are outside the expected variances then either the PC-RT algorithm (LP) is
run if the deviations are not too large, or the area wide QP is run when the deviations are
large, to obtain new ramp metering rates.
RESULTS
Simulation experiments were performed to evaluate the MILOS hierarchical system
against (a) "no control" (i.e., when no ramp metering is in efect) , (b) a locally trafficresponsive
metering policy currently under consideration by ADOT, and (c) an area-wide
LP optimization problem re-solved in 5-minute intervals. The simulation model was of a
small :freeway corridor in metropolitan Phoenix, Arizona (seven miles of State Route 202
with 7 of-ramps, 4 controllable on-ramps, and one freeway-freeway on-ramp
(hypothesized as controllable). Three test scenarios were simulated (1) a short "burst"
of heavy-volume flows to all ramps, (2) a three-hour comuting peak, and (3) a threehour
commuting peak with a 30-minute incident occuring somewhere in the middle of the
corridor.
The performance results indicate that MILOS is able to reduce freeway travel time,
increase :freeway average speed, and improve recovery performance of the system when
flow conditions become congested due to an incident. Specifically, when comparing with
the "no control" case, :freeway travel times were lowered by 8% - 36%, speeds were
increased by 3% - 18%, and recovery times were reduced by 6% - 25%. It also
performed better than the area-wide LP optimization that has been reported to perform
well in Europe. Locally responsive strategy performed well in light to moderate trafic
volumes, and, in fact, had lower :freeway travel times and faster speeds than MILOS;
however, for heavier volumes and incidents it had larger ramp-queues and longer recovery
times than MILOS.
1
AREAS OF FUTURE WORK
This research project identified several interesting future research, development and
deployment efforts. Development of (1) an algorithm to decompose a region into
subnetworks for MILOS control, (2) a model on route diversion and (3) methods to
estimate ramp demands and interchange turning probabilities are promising research areas.
Integration of MILOS with traffic-adaptive surface-street signal control and
incident/anomaly detection systems are developmental efforts that could make traffic
management even more real-time responsive. Finally, field testing and the deployment of
MILOS (and its future enhancements) should be an on-going effort towards the ITS goal
of implementing advanced trafic management systems that are safer, more efficient and
beneficial to the traveling public.
ACKNOWLEDGMENTS
The RHODES-ITMS Coridor Control Project was completed in January 1999. Project
oversight was provided by a Technical Advisory Committee (TAC) comprising of
representatives from key agencies. The project was administered by the Arizona
Transportation Research Center of ADOT. The following individuals served on the TAC
at various times:
Steve Owen
Tim Wolfe
Dan Powell
Tom Parlante
Glenn Jonas
Jim Shea
Alan Hansen
Tom Fowler
Jessie Yung
Sarath Joshua
Paul Ward
Roy Turner
Pierre Pretorius
Don Wiltshire
Jim Decker
RHODES-ITMS Project Manager,
Arizona Transportation Research Center (ATRC), ADOT
ADOT Technology Group
ADOT District 1
ADOT Trafic Engineering
ADOT Freeway Management
ADOT Trafic Engineering
Federal Highway Administration
Federal Highway Administration
Federal Highway Administration
MAG (previously at A TRC, ADOT)
MAG
MAG
Maricopa County Transportation and Development Agency
Maricopa County Transportation and Development Agency
Trafic Operations, City of Tempe
m
Table of Contents
CHAPTER 1: PROBLEM OVERVIEW ................................................................................................... 1
INTRODUCTION .................................................................................................................................... 1
THE FUNDAMENTAL FREEWAY MANAGEMENT PROBLEM ......................................................................... 2
ISSUES IN APPLICATION OF RAMP METERING AS A METHOD OF FREEWAY MANAGEMENT .......................... 4
THE MULTI-OBJECTIVE APPROACH TO FREEWAY SYSTEM MANAGEMENT ................................................. 6
RESEARCH METHODOLOGY ................................................................................................................... 7
SUMMARY OF THE FORTHCOMING CHAPTERS ......................................................................................... 8
CHAPTER 2: RAMP METERING LITERATURE REVIEW ••••••••••••••••••.....•...••.........•••.••••••••••••••••••••• 10
COSTS AND BENEFITS OF RAMP METERING ........................................................................................... 10
TYPES OF RAMP METERING ALGORITHMS ......................................................... .-. ................................. 11
TIME-OF-DAY METERING ALGORITHMS ................................................................................................ 11
LOCAL TRAFFIC-RESPONSIVE RAMP METERING ALGORITHMS ................................................................ 11
HYBRID RAMP METERING CONTROL ALGORITHMS ................................................................................ 12
INTEGRATED FREEWAY/SURFACE-STREET METERING ALGORITHMS ....................................................... 14
SUMMARY ......................................................................................................................................... 14
CHAPTER 3: HIERARCHICAL RAMP METERING CONTROL SYSTEM STRUCTURE ............ 16
INTRODUCTION .................................................................................................................................. 16
MULTI-LEVEL METHODS IN HIERARCHICAL CONTROL ............................................................................ 16
MULTI-LAYER HIERARCHICAL CONTROL SYSTEMS ................................................................................. 17
SET-POINT REGULATION METHODS ...................................................................................................... 18
THE MILOS HIERARCHICAL STRUCTURE ............................................................................................. 18
Modal decomposition of the MILOS hierarchy ............................................................................... 21
Subnetwork identification .............................................................................................................. 22
SPC-based anomaly detection and optimization scheduling layer ................................................... 24
Area-wide coordination layer .......... : ............................................................................................. 24
Predictive-cooperative real-time rate regulation layer ................................................................... 25
INTEGRATION OF MILOS WITH NECESSARY EXTERNAL SYSTEMS ......................................................... 25
SUMMARY ......................................................................................................................................... 26
CHAPTER 4: FREEWAY MACROSIMULATOR ................................................................................ 27
MODEL CONSTRUCTION ...................................................................................................................... 27
MODELING FLOW IN HEAVY CONGESTION ............................................................................................ 31
DYNAMIC MODELING OF RAMP QUEUES ............................................................................................... 32
SUMMARY OF MACROSCOPIC MODEL ................................................................................................... 33
STOCHASTIC EFFECTS AND DIVERSION BEHAVIOR ................................................................................. 35
SUMMARY ......................................................................................................................................... 35
CHAPTER 5: AREA-WIDE COORDINATION PROBLEM ........................•.................••••..•••...•......•. 37
INTRODUCTION .................................................................................................................................. 37
MATHEMATICAL DESCRIPTION OF THE AREA-WIDE COORDINATION PROBLEM ........................................ 37
Derivation of the objective junction ............................................................................................... 39
Consideration of queue storage limits ............................................................................................ 39
Development of a multi-criteria objective junction ......................................................................... 41
Setting costs according to interchange congestion level ................................................................. 42
Integration of surface-street flows in ramp demands ...................................................................... 43
Quadratic objective function summary ........................................................................................... 44
RESOLVING INFEASIBILITY .................................................................................................................. 45
AREA-WIDE COORDINATION PROBLEM SUMMARY ................................................................................ 46
OPERATION UNDER SEVERE CONGESTION ............................................................................................. 47
INTEGRATION WITH PREDICTIVE-COOPERATIVE REAL-TIME RATE REGULATION LAYER ........................... 48
PRELIMINARY EVALUATION OF AREA-WIDE COORDINATION PROBLEM ON SMALL EXAMPLE ................... 48
iv
Influence of /3 ............................................................................................................................... 50
Comparison of area-wide metering rate settings in macrosimulation .............................................. 51
Simulation test with extended queue dissipation ......................... : .................................................... 54
SUMMARY ......................................................................................................................................... 57
CHAPTER 6: PREDICTIVE-COOPERATIVE REAL-TIME RA TE REGULATION ALGORITHM ............ 58
INTRODUCTION .................................................................................................................................... 58
ADDING QUEUE MANAGEMENT TO STATE FEEDBACK CONTROL METHODS ............................................... 59
CENTRAL CONCEPT OF PC-RT RATE REGULATION ALGORITHM ............................................................... 59
Anticipated efects of PC-RT rate regulation algorithm .................................................................... 61
Basic function of the PC-RT rate regulation algorithm ..................................................................... 62
Reasonable and important assumptions ........................................................................................... 63
Linearization about an equilibrium state. ...................................................._ ,. ................................... 64
Elimination of the dynamic speed equation ...................................................................................... 66
Structure of the PC-RT objective function ........................................................................................ 69
Queue growth modeling .................................................................................................................. 70
Control variable modeling ............................................................................................................... 72
Derivation of the PC-RT cost coeficients from the QP solution ........................................................ 72
Computational procedure to obtain cost coeficients ........................................................................ 73
Modification to the linearization procedure for unstable conditions ................................................. 74
SUMMARY OF PC-RT MATHEMATICAL FORMULATION ........................................................................... 81
DIFFICULTY IN SOLVING THE MONOLITHIC PC-RT OPTIMIZATION PROBLEM ............................................ 82
Decomposition of full optimization problem into subproblems ......................................................... 83
SCENARIO PREDICTION ......................................................................................................................... 86
Scenario prediction example ........................................................................................................... 88
Construction of scenario rate tables ................................................................................................ 90
Infeasible PC-RT scenarios ............................................................................................................. 90
SUMMARY .......................................................................................................................................... 92
CHAPTER 7: SPC-BASED ANOMALY DETECTION ....................................................................... 94
INTRODUCTION .................................................................................................................................. 94
OVERVIEW OF STATISTICAL PROCESS CONTROL .................................................................................. 94
RELATIONSHIP OF SPC CONCEPTS TO FREEWAY CONTROL .................................................................... 97
JUSTIFICATION OF APPROXIMATELY-CONSTANT DEMAND ..................................................................... 98
SPC COMPUTATIONAL PROCEDURE ................................................................................................... 100
TRANSITION TO A NEW X LEVEL ....................................................................................................... 102
OTHER ISSUES IN SPC-BASED ANOMALY DETECTION ........................................................................... 104
SUMMARY ....................................................................................................................................... 105
CHAPTER 8: MILOS SOFTWARE IMPLEMENTATION ................................................................ 106
INTRODUCTION ................................................................................................................................. 106
INITIALIZATION MODULE ................................................................................................................... 108
SPC-BASED ANOMALY DETECTION MODULE ....................................................................................... 108
APPL YING NEW RATES ....................................................................................................................... 110
AREA-WIDE COORDINATION MODULE ................................................................................................. 110
PC-RT OPTIMIZATION MODULE .......................................................................................................... 111
EXAMPLE OF MILOS OPERATION ...................................................................................................... 113
SUMMARY ........................................................................................................................................ 115
CHAPTER 9: SIMULATION EXPERIMENTS .........................•......................................................•.. 117
INTRODUCTION ................................................................................................................................. 117
STRUCTURE OF THE SIMULATION EXPERIMENT .................................................................................... 117
CALIBRATION OF MACROSCOPIC MODEL TO SR202 CORSIM ournrr ................................................. 120
TEST CASE #l ................................................................................................................................... 127
Results for test case # 1 ................................................................................................................. 128
V
TEST CASE #2 ................................................................................................................................... 142
Results for test case #2 ................................................................................................................. 142
TEST CASE #3 ................................................................................................................................... 157
Results for test case # 3 ................................................................................................................. 158
SUMMARY ................ ··································· ...................................................................................... 172
CHAPTER 10: CONCLUSIONS ..•......•••••.••.••..•..............•..•.............•............•......•.....•.....••.•..••..•••••.•.... 174
GENERAL RESULTS ............................................................................................................................ 174
MILOS VERSUS NO CONTROL ............................................................................................................ 17 4
MILOS VERSUS THE LOCALLY TRAFFIC-RESPONSIVE METHOD ............................................................ 175
MILOS VERSUS LP METHOD ............................................................................................................. 176
SUMMARY ......... ························ ........................................................................................................ 177
DIRECTIONS FOR FURTHER RESEARCH .............................................................._ .. ................................ 181
APPENDIX A: SUPPORTING DATA ................................................................................................. 183
REFERENCES ...................................................................................................................................... 187
vi
List of Figures
Figure 1- 1. Empirical speed-volume measurements ....................................................... 2
Figure 1- 2. 15-minute flow time-series indicating congestion ........................................ 3
Figure 1- 3. 15-minute speed time-series indicating congestion ................. .' .................... 3
Figure 2- 1. Typical hierarchical control system structure ............................................. 13
Figure 3- 1. The pyramid structure of the MILOS hierarchy ......................................... 20
Figure 3- 2. Example freeway network ......................................................................... 22
Figure 3- 3. Initial decomposition of freeway network. ................................................. 23
Figure 3- 4. Alternative decomposition of freeway network .......................................... 23
Figure 4- I.· Typical shape of soft-limiter function ........................... .-.-............................ 33
Figure 5- 1. Ramp meter demand sources ..................................................................... 44
Figure 5- 2. Example problem ...................................................................................... 48
Figure 5- 3. Density evolution comparison ................................................................... 53
Figure 5- 4. Queue growth comparison ......................................................................... 54
Figure 5- 5. Density evolution comparisons for evaluation example 2 .......................... 55
Figure 5- 6. Queue growth comparisons, evaluation example 2 .................................... 56
Figure 6- 1. Prescribed maximum queue growth rate .................................................... 71
Figure 6- 2. Example of incorrect wave-speed model for congested section .................. 75
Figure 6- 3. Overcapacity segment results in upstream area-wide flow limitations ......... 77
Figure 6- 4. Alternative model for the over-capacity situation ....................................... 78
Figure 6- 5. Re-linearization for PC-RT and periodic solution of the QP ...................... 80
Figure 6- 6. Typical overlapping subsystem decomposition .......................................... 84
Figure 6- 7. Predicted trends for a given subproblem ..................................................... 88
Figure 7- 1. Typical SPC control chart .......................................................................... 95
Figure 7- 2. SPC limits and part specifications .............................................................. 96
Figure 7- 3. SPC chart showing sampled time-series .................................................... 96
Figure 7- 4. Detector time-series and underlying detection history ................................. 97
Figure 7- 5. Re-evaluation of "approximately constant" demand level.. ......................... 98
Figure 7- 6. Jumps between approximately-constant demand levels .............................. 99
Figure 8- 1. MILOS operational flow chart ................................................. : ............... 107
Figure 8- 2. SPC anomaly detection flow chart ........................................................... 109
Figure 8- 3. Area-wide coordination flow chart .......................................................... 111
Figure 8- 4. PC-RT optimization flow chart ............................................................... 113
Figure 8- 5. MILOS operational example ................................................................... 114
Figure 8- 6. MILOS operational example, continued .................................................. 115
Figure 9- 1. State Route 202 CORSIM link-node diagram .......................................... 121
Figure 9- 2. Comparison of density and speed measurements ..................................... 125
Figure 9- 3. SR202 comparisons, with stochastic input flows ..................................... 126
Figure 9- 4. Comparison of freeway travel time distributions ...................................... 130
Figure 9- 5. Comparison of queue time distributions .................................................. 130
Figure 9- 6. Comparison of average speed distributions .............................................. 131
Figure 9- 7. Comparison ofrecovery time distributions .............................................. 131
Figure 9- 8. Comparison of densities: no control ........................................................ 132
Figure 9- 9. Comparison of densities: TR w/QM ........................................................ 132
Figure 9- 10. Comparison of densities: LP, resolved each 5-minutes .......................... 133
Vll
Figure 9- 11. Comparison of densities: MILOS .......................................................... 133
Figure 9- 12. Comparison of queue growth: TR w/QM ............................................... 134
Figure 9- 13. Comparison of queue growth: LP, resolved each 5-minutes ................... 134
Figure 9- 14. Comparison of queue growth: MILOS ................................................... 135
Figure 9- 15. Comparison of metering rates: no control . ............................................. 136
Figure 9- 16. Comparison of metering rates: TR w/QM .............................................. 136
Figure 9- 17. Comparison of metering rates: LP, resolved each 5-minutes .................. 137
Figure 9- 18. Comparison of metering rates: MILOS .................................................. 137
Figure 9- 19. Total vehicles in system: no control.. ..................................................... 138
Figure 9- 20. Total vehicles in system: TR w/QM ...................................................... 139
Figure 9- 21. Total vehicles in system: LP, resolved each 5-minutes .......................... 140
Figure 9- 22. Total vehicles in system: MILOS .......................................................... 141
Figure 9- 23. Comparison of total travel time distributions ......................................... 144
Figure 9- 24. Comparison of queue time distributions ................................................. 144
Figure 9- 25. Comparison of average speed distributions ............................................ 145
Figure 9- 26. Comparison ofrecovery time distributions ............................................ 145
Figure 9- 27. Comparison of densities: no control.. ..................................................... 146
Figure 9- 28. Comparison of densities: TR w/QM ...................................................... 146
Figure 9- 29. Comparison of densities: LP, resolved each 5-minutes .......................... 147
Figure 9- 30. Comparison of densities: MILOS .......................................................... 147
Figure 9- 31. Comparison of queue growth: no control ............................................... 148
Figure 9- 32. Comparison of queue growth: TR w/QM ............................................... 148
Figure 9- 33. Comparison of queue growth: LP, resolved each 5-minutes ................... 149
Figure 9- 34. Comparison of queue growth: MILOS ................................................... 149
Figure 9- 35. Comparison of metering rates: no control . ................................ : ............ 150
Figure 9- 36. Comparison of metering rates: TR w/QM .............................................. 150
Figure 9- 37. Comparison of metering rates: LP, resolved each 5-minutes .................. 151
Figure 9- 38. Comparison of metering rates: MILOS .................................................. 151
Figure 9- 39. Total vehicles in system: no control.. ..................................................... 152
Figure 9- 40. Total vehicles in system: TR w/QM ...................................................... 153
Figure 9- 41. Total vehicles in system: LP, resolved each 5-minutes ........................... 154
Figure 9- 42. Total vehicles in system: MILOS .......................................................... 155
Figure 9- 43. SR202 model indicating incident location ............................................... 157
Figure 9- 44. Comparison of total travel time distributions ......................................... 160
Figure 9- 45. Comparison of queue time distributions ................................................. 160
Figure 9- 46. Comparison of average speed distributions ............................................ 161
Figure 9- 47. Comparison ofrecovery time distributions ............................................ 161
Figure 9- 48. Comparison of densities: no control... .................................................... 162
Figure 9- 49. Comparison of densities: TR w/QM ...................................................... 162
Figure 9- 50. Comparison of densities: LP, resolved each 5-minutes .......................... 163
Figure 9- 51. Comparison of densities: MILOS .......................................................... 163
Figure 9- 52. Comparison of queue growth: no control .. ............................................. 164
Figure 9- 53. Comparison of queue growth: TR w/QM ............................................... 164
Figure 9- 54. Comparison of queue growth: LP, resolved each 5-minutes ................... 165
Figure 9- 55. Comparison of queue growth: MILOS ................................................... 165
Figure 9- 56. Comparison of metering rates: no control .............................................. 166
Vlll
Figure 9- 57. Comparison of metering rates: TR w/QM .............................................. 166
Figure 9- 58. Comparison of metering rates: LP, resolved each 5-minutes .................. 167
Figure 9- 59. Comparison of metering rates: MILOS .................................................. 167
Figure 9- 60. Total vehicles in system: no control.. ..................................................... 168
Figure 9- 61. Total vehicles in system: TR w/QM ...................................................... 169
Figure 9- 62. Total vehicles in system: LP, resolved each 5-minutes ........................... 170
Figure 9- 63. Total vehicles in system: MILOS .......................................................... 171
Figure 10- 1. Comparison of typical metering rates, MILOS and LP .......................... 179
Figure 10- 2. Side-by-side comparison of metering rates for LP and MILOS .............. 180
IX
list of Tables
Table 1- 1. Characteristics of the proposed freeway control system ............................. 7
Table 4- 1. Parameters in macroscopic simulation equations ...................................... 29
Table 5- 1. Route-proportional matrix of example problem ........................................ 49
Table 5- 2. Ramp interchange data ............................................................................ 50
Table 5- 3. Comparison of metering rate coordination methods ................................. 50
Table 5- 4. Rate comparison for f3=1 ......................................................................... 51
Table 5- 5. Parameters for example problem .............................................................. 51
Table 5- 6. Initial conditions for simulation run ......................................................... 52
Table 5- 7. Preliminary method comparisons .................................. .- ......................... 54
Table 5- 8. Demand volumes for evaluation example 2 ............................................. 55
Table 5- 9. Initial conditions for evaluation example 2 .............................................. 55
Table 5- 10. Performance comparisons, evaluation example 2 ................................... 56
Table 6- 1. Ramp meter demand predictions .............................................................. 89
Table 6- 2. Upstream freeway demand predictions .................................................... 89
Table 6- 3. Rate table for next minute ........................................................................ 90
Table 6- 4. Rate table with infeasible optimization problems ..................................... 92
Table 9- 1. Trafic-responsive metering rates and thresholds ................................... 118
Table 9- 2. Route proportional matrices ................................................................... 123
Table 9- 3. Input volumes ........................................................................................ 123
Table 9- 4. Initial conditions and parameter values for State Route 202 ................... 124
Table 9- 5. State Route 202 macroscopic simulation parameters .............................. 126
Table 9- 6. Mean input rates, test case #1 ................................................................ 128
Table 9- 7. Performance results oftest case #1 ......................................................... 129
Table 9- 8. Average volume rates in each time segment, test case #2 ....................... 142
Table 9- 9. Performance comparisons, test case #2 .................................................. 143
Table 9- 10. Average volume rates in each time segment, test case #3 ...................... 157
Table 9- 11. Performance comparisons, test case #3 ................................................. 159
Table 9- 12. Comparison ofJ\1ILOS results with alternatives .................................... 173
Table 9- 13. Qualitative comparisons of J\1ILOS versus other algorithms ................. 173
X
Chapter 1 : Problem overview
Introduction
Trafic delay due to congestion on freeways and surface streets was approximated at 1.2
billion vehicle-hours in the United States in 1984 and projected to reach 6.9 billion
vehicle-hours by 2005 [Lindley, 1987]. User costs associated with traffic delay were
estimated at $100 billion in 1990 for the U.S. [Euler, 1990]. Total trips and commuter
miles are expected to grow significantly in most metropolitan areas as the trend towards
suburban sprawl continues. Construction of additional freeway lanes and wider surface
streets is certainly needed to respond to such societal needs. However, in many
situations, it is not possible to address capacity needs in dense urban areas with new.
construction. In these situations, capacity increases are possible only by adding modes of
travel (rail, subway, etc.) or reducing traveler delays with more efficient management of
the available system capacity.
Introduction of traffic management devices and systems (i.e. traffic signal systems, ramp
metering systems, lane channelization, HOV, etc.) has been shown to reduce delays and
increase capacity [FHW A, 1985]. Early estimates of the impacts of ITS technologies
range from 10% reduction in emission (relative to the projected increase in total vehiclemiles
traveled) to 20% savings in vehicle-delay and 30% reduction in stops [Mobility
2000, 1989]. These results are realized without significant spending on road widening
and adding miles of freeway. Benefit/cost ratios of 16: 1 and 22: 1 have been reported for
investment in ITS technologies in Los Angeles and Texas, respectively [Mobility 2000,
1989].
One of the most significant contributors to total vehicle-hours of delay is the daily
commute of travelers on the freeway from their homes to their place of business and viceversa.
On average, over 38% of total vehicle-hours of delay are recurrent, i.e. occurring
during the commuting hours, often referred to as the peak periods [Lindley, 1987].
Accidents and anomalous events, sometimes referred to as nonrecurrent congestionrelated
delay, account for the majority of the remaining delay factors. Thus, the largest
impacts on the reduction of total system delay for freeway operations, are realized by
efficiently managing the critical peak times and managing incident conditions efectively.
The fundamental freeway management problem
The central problem in freeway management can be described best by presenting the
fundamental diagram of freeway flow. Figure 1-1 illustrates speed versus volume on a
typical freeway segment in Phoenix, AZ [Technical Advisory Committee, 1997]. In this
figure, the upper concentration of points represents the uncongested flow "regime", where
traffic flows smoothly, i.e. without significant travel delays. The lower, less populated
collection of points represents 15-minute intervals when this freeway section was
congested, incurring trafic delays to travelers in and upstream of this section.
80
70
60
':2 50
40
Cl
Cl
30
20
10
0
0 50 10 150
Volume (vplph)
20
Figure 1-1. Empirical speed-volume measurements
250
Figures 1-2 and 1-3 depict the time-series of the same points in Figure 1-1. These figures
demonstrate that as the volume rises it becomes increasingly more precarious that the
speed will drop sharply and the system will "transition" to the congested flow regime. As
the system approaches closer and closer to the maximum flow rate, the transition can be
initiated given increasingly smaller shocks [Newell, 1993]. That is to say that the higher
the volume becomes, the more sensitive the system is to small anomalies in flow such as
anomalies induced by merging platoons of vehicles.
2
the undesirable congested flow regime. The dificulty is amplified because, as indicated
by Figures 1-2 and 1-3, capacity reductions usually occur when demand is greatest during
the peak morning and afternoon commuting times. Thus, without some form of
demand/capacity management, the sheer volume of demand for the freeway system
drives the network into congestion.
There are several basic technological forms of freeway management available to address
the freeway flow trade-of:
(a) Advising travelers to avoid certain sections and/or change their departure
times,
(b) Advising travelers (unilaterally) to maintain a certain speed, or
(c) Restricting access to the freeway system at certain locations.
Solution (a) describes passive advanced traveler information systems (ATIS) methods
which is outside the scope of this dissertation. Solution (b) has been investigated
[Karaaslan, et al., 1990; Smulders, 1993], but is likely to have compliance problems in
the U.S. Automated highway systems (AHS) eliminate this compliance problem, but are
relatively far from mass implementation due to regulatory concerns and cost issues
[Bender, 1991]. Solution (c) is generally referred to as ramp metering, and is the primary
topic of this research.
Issues in application of ramp metering as a method of freeway management
Ramp metering systems have existed since the early 1960's and have been used
efectively in many municipalities [Carlson, 1979; Marsden, 1981; Jacobson, 1989; HajSalem
et al., 1990; Hallenbeck and Nisbet, 1993; Wright, 1993] and in others with less
conclusive benefits [Lipp et al., 1992], but there is general consensus that ramp metering
systems can provide substantial benefits in throughput, travel-time, and congestion
reduction when applied appropriately. In fact, the efectiveness of on-ramp metering has
been substantial enough that a recent study proposed main-line metering as a tool for
congestion management [Haboian, 1997].
4
There does not exist a consensus, however, of what constitutes the most effective
metering rate(s), arising from the fact that the freeway management problem is difficult
to solve to optimality. Consider the following complicating factors:
(a) The state variables (i.e. volume, density, speed, ramp queues, etc.) change
dynamically over time;
(b) Although the behavior of individual travellers is somewhat deterministic in
that the drivers know where they are going, or at least have a trip purpose, the
behavior of trafic as a stream is stochastic and dificult to predict over long
time horizons;
(c) Flow anomalies (e.g. accidents, friction effects) occur at random,
unpredictable intervals;
( d) The ramp metering problem is a multi-dimensional one, with a large number
of state and control variables;
( e) The state variables are only partially observable at a limited number of fixed
locations where detectors are installed, and
(f) There are multiple stakeholders and, consequently, multiple objectives need to
be addressed in any freeway management policy.
The concerns of multiple stakeholders arise when you consider the fact that the freeway
system exists embedded inside and interacting with a larger network of surface-streets
and other modes of transportation. Previous research has not considered the complexity
associated with considering multiple stakeholders by:
(1) assuming that the freeway system exists in virtual isolation from the larger
surface-street network (e.g. usually assuming that ramp queuing capacity is
infinite), and
(2) choosing a single system-optimal optimization criterion that considers only the
efects of metering decisions on freeway conditions.
Freeway management policies developed or proposed to date have included, however,
considerations of dynamic state changes, stochasticity, multi-dimensionality,
unpredictability, and partial-observability in the freeway management problem.
5
The question remains, however, whether a system-optimal policy for the freeway control
problem alone is "system-optimal" for the entire transportation network. Here we
indicate the transportation network as the entire system of freeways and surface streets in
. a metropolitan area or municipality (or collection thereof). In fact, it is entirely plausible
that the freeway management policy "optimal" to the freeway conditions could be
counterproductive to the entire transportation network because the interactions between
the two surface street system and the freeway system are neglected.
This is especially true for ramp metering methods that (inevitably) create queues at the
freeway access points. These queues, if not suitably managed, can interfere with
operation of the surface-street system by extending into the adjacent interchange
(commonly known as spillback). Thus, the objectives at the ramp interface of the
surface-street manager and the freeway system manager conflict. The surface-street
manager would like to keep the ramp queue as short as possible and the freeway manager
would like to keep the queue as long as possible typically during congested conditions.
The multi-objective approach to freeway system management
The central issue addressed by this research is the consideration of the important
interaction between the surface-street system and the freeway system and their traffic
objectives in the development of a freeway access control (ramp metering) system. This
problem is addressed by using a multi-objective solution methodology. The trade-off
solutions produced by this solution method are defined by combining the two conflicting
objectives into a single, multi-criterion objective as opposed to other methods that
enumerate Pareto solutions [Haimes, et al., 1990]. In addition, because of the relative
size of the carrying capacity of the freeway with respect to the adjacent surface-street
system, the trade-of solution point is selected to maintain freeway performance that is at
least as good as management policies that do not consider the interactions of the two subsystems.
It will be shown in Chapter 9, via simulation, that acceptable freeway
performance similar to area-wide control methods that do not consider the effects on the
interchanges can be obtained by implementing a compromise solution while, at the same
time, providing queue management.
6
Research methodology
To mitigate the complicating factors of the multi-objective ramp queue management
issue, this research uses a structured approach based on previous work in freeway ramp
metering control systems, but utilizing new technologies where appropriate. Table 1-1
indicates the characteristic of the research methodology that addresses each of the
complicating factors.
Complicating factor Mitigating control system characteristic
Dynamic state changes Rolling-horizon optimization
Temporal-spatial decomposition
Stochasticity SPC-based anomaly detection
Temporal-spatial decomposition
Multi-dimensionality Temporal-spatial decomposition
Unpredictability Predictive scenario optimization
SPC-based anomaly detection
Partial-observability Predictive scenario optimization
Rolling-horizon optimization
Multiple objectives Multi-objective criterion functions
Cost coefficient trade-of weights
Table 1- 1. Characteristics of the proposed freeway control system
In brief, Table 1-1 identifies the characteristics of a hierarchical control system that
decomposes the large-scale freeway ramp metering into a series of optimization problems
of varying temporal and spatial resolution. The optimization problems are re-solved as
the parameters and conditions of the system change to continually adjust the control
strategy to the real-time behavior of the system. In addition, to mitigate the
unpredictability of the future system state, a predictive scenario-based optimization
scheme is implemented in real-time to prepare the local subsystem for the next short-term
stochastic disturbance.
7
Summary of the forthcoming chapters
The remainder of this document is structured as follows; Chapter 2 presents a brief
overview of previous work on the ramp metering problem. Chapter 3 outlines the
hierarchical structure of the research methodology and the temporal/spatial
decomposition of the control problem. Although hierarchical treatment of freeway
management is not new, the specific hierarchy proposed in this project is novel, in
particular the identification of subnetworks from a large-scale freeway system, and the
basis for interaction between the area-wide layer and the locally traffic-responsive layer
are new. Chapter 4 presents a popular and useful model of freeway trafic flow modified
slightly to more accurately represent the ramp-freeway interface under the presence of
congestion. Chapter 5 presents the area-wide coordination component of the hierarchical
control system that considers the impact of queue growth on the adjacent interchanges in
the optimization model. This optimization model is based on models available in the
literature but incorporates several additions: (1) a new multi-criterion objective function
and trade-off structure, (2) an alternative treatment of queue growth constraints, and (3)
modeling of demands from surface-street interchange flows.
Chapter 6 presents the locally traffic-reactive, predictive-cooperative real-time rate
regulation algorithm that provides additional capacity at the freeway/surface-street
interface. The basis for this optimization model is not new (i.e. linearization of the
nonlinear macroscopic flow model of Chapter 4 ), but the formulation of the scenariobased
linear-programming problem is new. The link to the solution of the area-wide
coordination problem of Chapter 5 using the dual information is entirely novel.
Chapter 7 presents the statistical process control concepts used to monitor system
operation and, in real-time, identify perturbations to the system states. This structure of
demand estimation and fluctuation identification in the context of freeway management
systems is an entirely new treatment of this modeling/estimation/optimization procedure.
Chapter 8 summarizes the hierarchical control system presented as components in
Chapters 5, 6, and 7 and presents the algorithmic operation of the system. Chapter 9
presents a simulation experiment that evaluates the hierarchical system against several
8
other ramp metering policies on a relatively small, but realistic, freeway management
problem in the metropolitan Phoenix, AZ area. Presentation of performance variance
information comparing metering methods has not been done before in freeway
management literature. Finally, Chapter 10 summarizes the results of the research.
9
Chapter 2: Ramp metering literature review
Costs and benefits of ramp metering
Ramp metering is the most widely used form of freeway control [Yagar, 1989]. Ramp
metering limits the rate at which vehicles enter the freeway system, thus potentially
reducing the possibility of bottlenecks, shock wave propagation, and congestion. A wide
range of benefits are available from the use of ramp metering [Arnold, 1987; Yagar,
1989; McShane and Roess, 1990]:
(1) minimizing the total travel time of freeway users
(2) efficient use of freeway capacity
(3) discouraging routes with high societal costs
(4) reducing the variance of corridor trip times
(5) decreasing local freeway congestion and shock waves resulting from merging
platoons
(6) decreasing the accident rate in freeway weaving sections.
Another study indicates that ramp meters can efficiently reduce system travel time,
although the savings are network dependent [Hellinga and Van Aerde, 1997]. Of course
there are disadvantages and adverse efects of ramp meters:
( 1) encouraging longer trip distances on diversion routes
(2) favoring through trafic over local trafic and short trips
(3) modifying the evolved "status quo" of unobstructed freeway entry
( 4) increasing the overall operating cost of the control system
(5) adversely afecting the surface street controller operation due to queue
spillback and diversion to oversaturated locations
An operational study in the Denver area showed no statistically-significant improvement
when a simple demand-capacity metering system was installed and evaluated [Lipp et al.,
1991]. Ramp metering advantages may also be strongly dependent on the existence of
good alternative routes, especially in the absence of effective queue management
10
strategies [Hellinga and Van Aerde, 1997]. Nevertheless, most large metropolitan areas
have some type of ramp metering installed or currently under installation, indicating that
practitioners have been convinced that the benefits (fiscal, social, temporal) of ramp
metering outweigh the costs of implementation, maintenance, and the adverse effects
mentioned above.
Types of :ramp metering algorithms
The vast array of ramp metering algorithms developed to date can be classified into one
of three general categories;
(I) fixed-time or time-of-day,
(2) trafic-responsive, and
(3) hybrids combining attributes of trafic-responsive and time-of-day algorithms.
Time-of-day metering algorithms
Time-of-day metering algorithms derive settings that apply during 10-30 minute intervals
based on historical origin-destination flow rates and demand volumes for an entire
commuting corridor or facility [Wattleworth and Berry, 1967; Messer, 1969; Yuan and
Kreer, 1971; Wang, 1972; Wang and May, 1973; Chen et al., 1974; USDOT, 1976;
Kahng et al., 1984].
The main drawback of fixed-time, time-of-day metering systems is the inability to handle
non-recurrent incidents, accidents, special events, and fluctuations in traffic flow that
may occur [Newman et al., 1970], since the actual demand may· not be close to the
demand used to derive the time-of-day metering rate. Recent studies have indicated that
although time-of-day and day-to-day patterns exist, the variability of the actual flows
from the historical average flows is significant enough to make some time-of-day settings
inefective [Rahka and Van Aerde, 1997].
Local traffic-responsive ramp metering algorithms
Trafic-responsive ramp metering algorithms measure variables such as speed, volume,
and occupancy on the freeway and apply metering rates that keep the local freeway
volume under capacity or at some desired set-point [Athans, 1969; Buhr et al., 1969;
11
Hardin, 1972; Estep, 1972; Pretty, 1972; HCM, 1985; Papageorgiou, 1989, 1991;
Middelham and Smulders, 1991; Nihan, 1991; Nihan and Berg, 1992; Davis, 1993;
Chang and Wu, 1994]. Other locally trafic-reactive ramp metering systems have been
. developed that merge vehicles into gaps in trafic [Drew et al., 1966; Wattleworth and
Courage, 1968; Brewer et al., 1969] but such systems have not been widely implemented.
Other types of traffic-reactive metering systems follow a pre-determined set of
relationships between metering rates and traffic variable measurements. Examples of
such systems are fuzzy and traditional rule-based expert systems· and neural networks
[Blumentritt et al., 1981; Rajan et al., 1986; Sasaki and Akiyama, 1987; Gray et al., 1990;
Stephanedes et al., 1992; Zhang et al., 1994; Zhang and Ritchie, 1995; Papageorgiou et
al., 1995].
The main drawbacks of using trafic-responsive ramp metering in a large-scale freeway
are:
(1) the absence of coordination between adjacent ramp meters, and
(2) the absence of consideration of the area-wide efects of local changes to the
metering rate.
Hybrid ramp metering control algorithms
Many hybrids and extensions of the basic trafic-responsive and time-of-day control
methods have been developed. These extensions address the generally recognized issue
that although day-to-day and time-of-day patterns exist, and can be exploited, their
realization on a specific day and time may be significantly diferent from the assumed
historical average pattern. Thus, hybrid ramp metering algorithms allow the system to
follow the underlying trends, but still react to temporal and/or spatial flow irregularities.
The most straightforward extension of time-of-day methods is the use of a rolling time
horizon and/or periodic re-optimization of the area-wide algorithm with new information
[Messer, 1969; Drew et al., 1969; May, 1979; Papageorgiou, 1980, 1983; Kahng et al,
1984; Chang and Wu, 1994; Asakura, 1995]. However, the issue has been raised of how
12
long such re-optimization intervals should be. Most studies used fixed update intervals of
5-15 minutes.
The complexity of the large, multi-variable ramp metering problem has also been
addressed by decomposition of the problem into smaller-scale descriptions of subsystems
which are each optimized independently [Isaksen and Payne, 1973; Looze et al., 1978;
Payne et al., 1979; Goldstein and Kumar, 1982; Papageorgiou, 1983; Kahng et al., 1984;
..
Payne et al., 1985]. Many hybrids offer a combination of the rolling-horizon extension
and spatial decomposition by establishing a hierarchical approach to the large-scale ramp
metering problem, similar to the organizational structure shown in Figure 2-3
[Papageorgiou, 1983].
Q) "' C:
0
0. "'
E
Q)
Adaptation
!
I cost coeficients, slowly-varying parameters
Optimization
set-points, reso urc e price, regulator parameters
'
Regulation
real-time meterin grates
'
Plant/System
Figure 2- 1. Typical hierarchical control system structure
These systems combine locally-optimized traffic-responsive control with guidance from
upper levels of the hierarchy regarding area-wide conditions, special events, and
incidents [Drew et al., 1969; May, 1979; Papageorgiou, 1984; Payne et al., 1985].
Hybrid metering algorithms based on hierarchical control structures can also support the
explicit consideration of optimization modes such as "normal flow", "congestion", and
13
"special-event" that are scheduled by the highest level(s) of the hierarchy [Papageorgiou,
1984; Pooran et al., 1994].
Integrated freeway/surface-street metering algorithms
Although there is a large body of work on freeway control algorithms, only exploratory
work has been done to produce ramp metering solutions that integrate information from
the surf ace-street system [Fan and Asmussen, 1990; Stephanedes and Chang, 1991, 1993;
Pooran et al., 1992, 1994; Han and Reiss, 1994]. Some work has recently been proposed
to develop freeway management solutions that derive both signal settings and metering
rates in a commuting corridor of surface streets and freeway [Cremer et al., 1990; Chang
et al., 1992; Papageorgiou, 1995; Zhang and Hobeika, 1997]. The failure to integrate the
two systems has been due to the technological barriers that have restricted application of
data-intensive ramp metering methods and the difficulty of modeling the two sub-systems
together for optimization purposes [Van Aerde et al., 1987]. As such, no results of field
implementation studies could be found in the literature.
However, as the technological barriers are being removed and real-time traffic
information is becoming readily available, a new focus on improving the system-wide
performance of the freeway and surface street network has emerged [Van Aerde and
Yagar, 1988]. Critical data such as origin-destination (and/or route-proportional)
matrices, time-varying demands, turning probabilities, and the like can be more reliably
estimated on-line as the Intelligent transportation systems (ITS) "infrastructure" of
communication networks and detection technology continues to be deployed.
Summary
For the past 35 years, much research has been done in the area of ramp metering control
systems. Even simple metering systems installed in the field have been shown to be
efective at improving freeway performance and having benefits that outweigh the
installation and recurrent operating costs. However, metering systems sometimes have
detrimental effects to the adjacent surface-streets when the ramp queue spills back into
the interchange. Methods to address the spillback problem at the interface between the
14
freeway system and surface-street system have only recently been established in the
research community and sparsely implemented in the field. Another drawback of ramp
metering algorithms based on local trafic is the lack of consideration for the system-wide
efects of the metering decisions and dis-proportionate queue growth rates [Benmohamed
and Meerkov, 1994]. The remainder of this document describes a hierarchical freeway
management system that builds on the successes of previous research in hybrid ramp
metering algorithms and adds consideration of the important problem of queue
management.
15
Chapter 3: Hierarchical ramp metering control system structure
Introduction
The ramp metering control system developed in this research is specifically designed to
address the complicating factors of the freeway management problem. Recall from
Chapter 1 that the freeway management problem is a difficult control and optimization
problem because of these factors. Previous work in freeway ra?1p metering control
systems has primarily focused on the complications caused by:
(1) dynamic state changes,
(2) stochasticity,
(3) multi-dimensionality, and
( 4) partial observability
without consideration of multiple objectives or the unpredictability of the future traffic
state.
The freeway control system developed in this research addresses all six of the
complicating factors by establishing a hierarchical system of layers that
(1) addresses embedded spatial and temporal descriptions of the ramp metering
control problem,
(2) considers concerns of both the freeway and surface-street systems in the
optimization problem(s),
(3) plans pro-active metering rates in real-time to respond to possible future
trafic states, and
( 4) re-schedules optimizations based on the stochastic fluctuations of the demand
processes.
Before detailing the characteristics of the hierarchical control model developed in this
research, we review some concepts and previous research in hierarchical optimization.
Multi-level methods in hierarchical control
The hierarchical approach to system control has substantial fundamental research
support, especially in the area of large-scale diferential equation systems [Mahmoud,
16
1977; Sandell et al., 1978; Wilson, 1979; Papageorgiou and Schmidt, 1980; Bemassou
and Titli, 1982; Papageorgiou, 1983]. Hierarchical control is particularly useful when the
system being controlled has an appreciably large set of state variables, and/or and
appreciably large set of control inputs. Large-scale control problems of this type are
primarily difficult because of appreciable computation time required to solve for the
"optimal" controls.
..
Such large-scale differential equation control problems are typically addressed by
decomposing the problem using a multi-level approach. The multi-level approach creates
a two-level optimization problem from a global optimization problem. Various methods
have been proposed to solve the two-level optimal control problem including interactionprediction
and interaction-balance procedures [Sandell et al, 1978; Wilson, 1979;
Papageorgiou and Mayr, 1982].
Multi-layer hierarchical control systems
A hierarchical control system can also describe a controller that solves the ramp metering
problem at several embedded layers of aggregation. Thus, the multi-layer hierarchical
approach typically indicates a structure where the targets, constraints, costs, and
parameters of a given layer are communicated from a higher-level layer and the given
layer communicates the targets, constraints, costs, and parameters to the lower-level
layer(s) in the hierarchy [Mahmoud, 1977]. Few general theories exist to describe the
efectiveness or expected performance of the multi-layer approach in system control
since the definition and structure of such "layers" are problem-dependent [Sandell et al,
1978]. This approach has been implemented to address the freeway control problem with
the layers being parameter estimation, incident detection, flow identification, and gapacceptance
ramp metering, respectively [Drew et al, 1969], although the gap-acceptance
metering method has not found widespread acceptance. A later extension by Messer
incorporated an LP-based area-wide coordination method at the "optimizing" layer of the
hierarchy [Messer, 1971]. Modeling of the freeway control problem with a hierarchical,
multi-layer approach has since persisted in the literature because of the natural way in
which it addresses the complicating factors of the problem.
17
Set-point regulation methods
The hierarchical approach also applies to the development of set-point regulation control
methods [Payne and Isaksen, 1973; Papageorgiou, 1983; Stephanedes and Chang, 1991].
A set-point regulation controller solves two separate optimization problems. One
optimization problem (or problems) is solved to obtain the set-point(s) of the system. A
second set of "optimization problems" are solved to obtain control laws that regulate the
system state, under the influence of external disturbances, to operate at the set-points.
A third layer (adaptation) resides above the upper-layer optimization problem to modify
the problem structure, parameters, and the like to the changing system conditions. The
set-point regulation control method has been successfully applied in many areas of
engineering such as chemical processing and aircraft control systems. Typically, because
of the natural structure (i.e. geographic size, multi-dimensionality) of the system being
controlled, the upper-layer control problem uses an aggregate model of the system to
reduce processing requirements. Then at the lower-layer, the control problem is
decoupled into independent subproblems that use a more detailed dynamic description of
a geographically smaller portion of the problem given the assumptions of eqn. 3-3. Thus,
because of the smaller size of the subproblems, more computational effort can be applied
to solve each subproblem in real-time.
The MILOS hierarchical structure
This research addresses the system-wide ramp metering control problem by using a
structured hierarchical framework hereafter referred to as the Multiobjective Integrated
Large-Scale Optimized ramp control System (MILOS). This framework is based on the
multi-layer approach to hierarchical process control using the set-point control method.
Although neither the multi-layer approach to hierarchical control nor the set-point control
method are contributions of this research, the hierarchical structure of the MILOS
framework includes the following contributions:
(a) consideration of multiple objectives in the optimization problem(s);
18
(b) integration of information about the current conditions of the adjacent
surface-street system;
(c) prediction of possible future system states in the development of pro-active
real-time metering rates; and
(d) computability in real-time.
These primary characteristics of MILOS are driven by the structure of the real-world
freeway/surface-street system. Thus, MILOS is composed of four hierarchically
embedded, interactive subsystems:
(1) locally reactive, predictive-cooperative real-time control,
(2) area-wide coordination,
(3) anomaly detection I optimization scheduling, and
( 4) subnetwork identification.
The structure of MILOS is a pyramid of modules that address smaller and smaller
geographic areas of the large-scale ramp metering problems as one progresses lower in
the hierarchy. The pyramid structure indicates that one optimization scheduler module
schedules the solution of several area-wide coordination problems that in tum schedule
the solution of several traffic-responive real-time metering problems. This pyramid
structure- is illustrated in Figure 3-1, motivated by the structure of the RHODES
hierarchical system for real-time surface-street traffic management and the RHODESITMS
system developed at the University of Arizona [Head et al., 1992; Head and
Mirchandani, 1993].
19
..
=
.,
w•
.
=-
't:l
'-<
.,
..
[l'J
.,
I\) =
n
.,
=-
0
0.
=-
..
'"I
n
·······················-···························.
: : MILOS SPC-based anomaly 1
detection •--+------,- recurring incidents, : planne events, historical g/c raitos,
Ii signal failures surface-street incidents
long-term ODs, flow pwfiles Optimization schedu ng
State and
parameter
estimation
Area-wide
current OD estinlates,I coordinator
Incident
detection
flows, TPs,
ramp queues
I I
..
real-time occupa:icy,
. ___ .,. __ , speed, volumes:
.
Predictive-cooperative
Real-time
Rate regulation
\
subnetwork definitions,
time-horizons,
trade-of parameters
------------.
Area-wide
coordinator
Area-wide
coordinator
slacks & dual variables
'
\
nominal metering rates,
nominal queues,
-----i---.
Predictive-cooperative
Real-time
Rate regulation
Predictiv
R
-···-·····--------------···············-··
volumes, TPs,
planned g/c ratios,,
spillback
Surface
street
control
system
real-time volumes, ----r----'
phase lengths, spillback,
emergency vehicles
Modal decomposition of the MILOS hierarchy
MILOS can be considered to operate in several modes; strategic, tactical, and operational.
At the highest level of the hierarchy, the strategic mode solves optimization problems
with time horizons on the order of hours, days, and weeks, as well as responding to
seasonal changes etc. The spatial influence of the strategic mode is the entire freeway
and interchange network. The objectives of the strategic mode are to:
(1) identify the "optimal" sub-network definitions that lower-level processors use
to solve de-coupled optimization problems,
(2) update parameters reflecting special events and long-term disturbances such
as work zones,
(3) update the slowly varying parameters in the system, and
(4) determine the optimization time horizons for the lower-level problems.
The strategic mode is fulfilled by the SPC anomaly detection module and the subnetwork
identifier module.
The tactical mode of the MILOS hierarchy solves optimization problems with time
horizons of hours and minutes, using the subnetwork definitions and parameters passed
from the strategic levels of the hierarchy. The spatial influence of a tactical-level module
or optimization problem is "several " ( e.g. 5-15) adjacent ramp meters and the associated
surface-street interchanges. The objectives of the tactical mode are to:
(1) plan coordinated metering rates for recurrent congestion,
(2) identify short-term flow fluctuations that require re-solution of the area-wide
and real-time optimization problems,
(3) react to changes in the relative congestion levels of the interchanges,
(4) balance queue growth rates in a given geographic sub-network, and
(5) respond to non-recurrent congestion generated by incidents.
The tactical mode is implemented by the SPC-based anomaly detection module and the
area-wide coordination modules.
At the lowest level of the hierarchy, the operational mode solves optimization problems
with time horizons of minutes, using the set-point metering rates and desired freeway
21
states provided by the tactical rnode modules. The spatial influence of the operational
level is a single ramp meter, a single interchange signal, and a small, relatively
predictable subsection of the freeway. By "relatively predictable" it is meant that
reasonable predictions for the next few minutes of flow can be made for this small
section using a mathematical model. The objectives of the operational mode are to:
(1) reduce ramp queue lengths when not detrimental to freeway conditions,
(2) plan metering rates pro-actively based on prediction of possible future states,
(3) react to short-term flow fluctuations that could cause freeway congestion, and
( 4) manage ramp queue spill back, if possible.
The operational mode is implemented by the predictive-cooperative real-time control
modules that each solve optimization problems local to a single ramp meter.
Subnetwork identification
The majority of this research is focused on the area-wide optimization (tactical level) and
traffic-responsive real-time control (operational level) algorithms. However, a role of the
strategic mode in the MILOS framework is to identify the problem boundaries for the
area-wide coordination and predictive-cooperative real-time control problems. Some
research has been done to develop a method to determine boundaries for surface-street
coordination problems [Moore and Jovanis, 1985], but little mention of such issues can
be found in freeway control literature. For example, consider the large freeway network
in Figure 3-2, where each node represents an interchange with a ramp meter (considering
the unidirectional case only).
Figure 3- 2. Example freeway network
22
Asmussen, 1990]. Thus, inter-agency cooperation would be necessary for crossboundary
coordination when a subnetwork crosses a boundary.
In this research , however, the subnetwork definitions at the area-wide coordination level
will be taken as given by the traffic management decision-makers. It should be a topic of
future work to develop the analytical subnetwork identification module of MILOS to
advise freeway system managers of modifications to the subnetwork decomposition
structure as current network conditions change.
SPC-based anomaly detection and optimization scheduling layer
The optimization procedures, at both the area-wide and real-time control layers, are
continually re-evaluated using a rolling-horizon approach. Rolling-horizon approaches to
traffic management have been proposed by many researchers in both surface-street and
freeway control [May, 1979; Gartner, 1983; Chang et al, 1992; Head, et al., 1992; Sen
and Head, 1997]. As the system evolves, the anomaly detector continually compares the
observed freeway flows and ramp demands to the expected flows and demands. When a
significant deviation from the expected state is detected, a new optimization run is
scheduled immediately. The SPC anomaly detection module is based on the concept of
control limits from the statistical process control (SPC) literature. This method is a
completely novel approach to the "integrated" demand estimation and optimization
scheduling problem and is discussed further in Chapter 7.
Area-wide coordination layer
The area-wide coordination layer provides the tactical decisionmaking of the MILOS
hierarchy. The area-wide coordination level allocates medium-term (i.e. 10-20 minute)
target or nominal ramp metering rates to maximize freeway throughput, balance ramp
queue growth rates, and minimize queue spillback into the adjacent surface-street
interchanges for a given subnetwork. The area-wide coordinator is based on a rollinghorizon
implementation of a multi-criteria quadratic programming optimization problem.
The area-wide coordinator interacts with the SPC anomaly detection module to identify
over-capacity congestion conditions and to modify the optimization constraints and
24
criteria appropriately during incident conditions. The area-wide coordinator is sensitive
to the needs of the adjacent surface streets by planning queue-growth rates according to
the relative congestion level of each interchange. Several aspects of this formulation of
the area-wide coordination problem are novel and discussed further in Chapter 5.
Predictive-cooperative real-time rate regulation layer
The predictive-cooperative real-time (PC-RT) rate regulation layer fulfills the
..
operational mode of the MILOS hierarchy. The PC-RT optimization problems are based
on a linearized description of the freeway state variables and are solved to minimize a
linear measure of additional travel-time savings. This additional travel-time savings is
above and beyond that due to the area-wide coordination solution by itself. At the
operational layer, the system model is more detailed than at higher layers of aggregation
[Papageorgiou, 1983; Payne et al., 1985; Fan and Asmussen, 1990]. Thus, linearization
allows the PC-RT rate regulation module to plan, in real-time, several pro-active
modifications to the nominal metering rates provided by the upper-layer area-wide
coordination module based on predicted scenarios of possible ramp and freeway flows in
the next few minutes. The scenario-based optimization structure of the PC-RT rate
regulation module is a new treatment of the real-time ramp metering problem and its
explicit connection to the solution of the area-wide coordination problem is completely
new. These issues are discussed further in Chapter 6.
Integration of MILOS with necessary external systems
MILOS, as shown in Figure 3-1, is primarily an optimization system. Parameter
estimation, especially turning-probability and route-proportional rate estimation, freeway
detector data collection/filtering, incident detection and surface-street performance data
all are taken as inputs to the MILOS hierarchy, and assumed to be "solved problems". Of
course, the successful implementation of an optimization routine such as MILOS is
highly dependent upon the reliability and accuracy of the external algorithms and
systems. In particular, MILOS requires real-time turning-probabilities, demand flows,
green splits, and queue lengths from the interchanges control system. Such information
25
requires the availability of a suficiently intelligent real-time signal controller and
communications network.
Summary
A Multiobjective Integrated Large-Scale Optimized ramp control System (MILOS) is
developed in this research. The framework is based on the multi-layer approach to
hierarchical process control using the set-point/regulation paradigm. The MILOS
framework is specifically structured to address the complicating characteristics of
dynamic state changes, stochasticity, multi-dimensionality, partial observability, the
existence of multiple objectives, and unpredictability that are inherent to the large-scale
freeway control problem. In addition, MILOS considers the efects of freeway control
decisions on the adjacent surface-street system at each level of the hierarchy. MILOS is
composed of four hierarchically embedded, interactive subsystems:
(1) area-wide coordination,
(2) predictive-cooperative real-time control,
(3) SPC-based anomaly detection and optimization scheduling, and
( 4) subnetwork identification,
based upon the decomposition of the large-scale control system into its strategic, tactical,
and operational processing modes. In the next chapter, a popular and useful macroscopic
flow model is discussed that is used (in Chapter 9) to evaluate the results of
implementing the MILOS systems in a simulated freeway environment.
26
Chapter 4: Freeway Macrosimulator
Model construction
A macroscopic freeway traffic simulator based on the enhanced FREFLO [Payne, 1971,
1979; Rathi et al., 1985] and META models [Papageorgiou, 1984; Cremer, 1989] is used
to evaluate and compare various ramp metering strategies developed in this research. The
FREFLO macroscopic traffic simulator is based on partial differential equation (PDE)
description of freeway traffic flow as a fluid of density p( x, t) and speed v( x, t) where x
indicates spatial variation and t indicates temporal variation of the fluid's density and speed
such that
J-v +-Jq =r-s Jx dt
Jv =v
Jv _ !_[v - v (p)+vJp ] Jt Jx T e Jx
Eqn. 4- 1
where r is the ramp meter input rate, sis the off-ramp (or end of freeway) output rate q(x, t)
is the flow rate, v.( p) is the equilibrium speed-density relationship, and v is the
aniticipation coeficient [Lighthill and Witham, 1955; Richards, 1956; Michalopolous et al.,
1986, 1991, 1993]. This set of PDEs describes the conservation of vehicle flow through
the freeway system and the dynamic relationship of speed and density. To evaluate p(x,t)
and v(x,t) for various input rates rand exit rates s at each point along the freeway, the
PDEs are discretized over space and time to obtain, using the simple Euler formula, the
diference equation description of the system
p/k + 1) = p/k) + :. (vlN)k)- V0ur)k)- sj (k) + rj (k))
J
Eqn. 4- 2
Eqn. 4- 3
27
½N)k) =a· Vj_1 (Pj-i (k.), vj-i (k)) + (1- a)· Vj(pj(k), vlk))
Vour)k) =a· Vj(pj(k ), vlk)) + (1 - a)· Vi+i (Pj+i (k), vj+I (k))
V.)pj(k), v)k)) = pj(k) · vj(k)
Eqn. 4- 4
where p/k) is the density, v/k) is the mean speed, and Vj(k) is the volume of vehicles in
freeway section} at time k. Additional terms are added in (4-3) that are not represented in
( 4-1) for the speed PDE. v.(r/k)) is an analytical speed-density characteristic such as
Eqn. 4- 5
The parameters v
1 Pmax, l, and m of (4-5), as well as the other parameters of (4-2), (4-3),
and (4-4), must be calibrated from field data and may vary over time and location. In
min[ .0. j]
particular, the time-interval T must be selected such that T < 1 to ensure that the
VJ
state updates are frequent enough that flows do not "skip" sections.
For simplicity, we assume that the parameters do not vary from location to location during
a given simulation. In addition, it is reasonable to assume that variations of the parameters
u
1, PMAX' l, and m of (4-5) are much slower than traffic flow dynamics and thus can be
assumed as constant over a simulation period. Full description of the derivation of the
remaining parameters in Table 4-1 can be found in [Payne, 1971; Cremer, 1989;
Papageorgiou, 1989].
28
Symbol Value
Lij length of section j (km)
T time interval duration (hr),
a E [ 0, 1], spatial discretization parameter
't time constant (km/hr), approximately the segment free-flow travel time
--
K constant (veh/km) to improve performance of eqn. 4-3 at low densities
V anticipation coefficient (veh/km2)
z on-ramp friction coeficient
f lane-drop friction coeficient
1 shaping parameter for speed-density characteristic
m shaping parameter for speed-density characteristic
Dr mean free-flow speed in section j (km/hr)
PMAX maximum density in a single traffic lane (veh/km)
0 if rj > 0 [Papageorgiou, 1989]. An additional term
Eqn. 4- 6
was added to (4-3) by Papageorgiou to represent the slowing efect from a lane-drop when
nj+J < nr Previous_ research has indicated that this addition more accurately reflects this
slowing efect than (4-3) without this term [Papageorgiou, 1989].
Equation 4-4 accounts for the spatial discretization of the flow model. The flow rate out of
section}, V0ur.fk), is expressed as a weighted sum of the flow rate p/k)*vfk) from section
j and section}+], Pj+lk)*vj+lk), such that a E [0,1] . Similarly, the flow rate into section
j VJN,lk) is expressed as a weighted sum of the flow rates from section}-], pjjk)*vjjk),
and}, pf k)*v/k), to smooth the behavior of the model. Equation 4-4 does not, however,
represent the general case where a segment} can haven feeder flows Vj,1N,lk), ... , ½.JN)k)
and/or m receiver links ½.our.lk), ... , Vj,our)k). Such cases are straightforward additions
to the single-source, single-receiver model by using weighted averages from the multiple
sources/sinks for computing upstream and downstream state variables, Pj+I' vj+J and pj_1,
vj-J . respectively [Papageorgiou, 1984].
30
Modeling flow in heavy congestion
(4-2), (4-3), and (4-4) have been shown to accurately reflect freeway traffic when
calibrated to a specific location during periods of moderate congestion [Papageorgiou,
1983, 1989; Cremer, 1989]. However, in situations of heavy congestion, the equations
must be modified to reflect the fact that vehicles cannot continue to flow from section to
section when a section is at the maximum density. The main reason for the continuing flow
from section to section even though the congestion is high is that the dynamic equation for
the section speed vfk) does not accurately represent the breakdown in speeds when a
section becomes congested. Previous researchers have addressed this by substituting the
equilibrium speed-density relationship v,(pfk)) for the dynamic speed-density relationship
during periods of high congestion [Rathi et al., 1985]. We take a similar approach here,
adding a threshold y to ( 4-4) such that if the density exceeds the threshold, the density at
the maximum capacity flow rate, the flow into or out of section j transitions to the
theoretical volume-density relationship. Hence, during periods of high congestion and
modify the structure of the density evolution equation while continuing to compute speeds
using (4-3).
Essentially, we add a threshold yto (4-2) such that the flow into or out of section j is equal
to zero during suficiently high congestion in the adjacent section. Hence,
½N . j (k) =a· ½-i(Pj-I (k), vj_1(k)) + (1- a)· ½(Pik), vik))
VouT)k) =a· ½(P/k), vik)) + (1-a)· ½+i(Pj+ 1 (k), vj+1 (k))
if p/k) y V.)p/k), v j (k)) = pj (k) · ve(P/k))
otherwise V.)pj(k), v/k)) = Pik) · v)k)
Eqn. 4- 7
Equation 4-7 is the modified form of (4-4). This condition helps to more accurately model
the stagnancy of flow when density becomes overcritical and speeds are very low.
Special conditions must be added for the implementation of ( 4-7) for sections at the
beginning and end of a freeway facility, so that if a congestion wave is passing upstream, it
does not stagnate in the first segment and limit the input flow rate. Thus, since we do not
have a measurement of po(k), we use pi (k-1 ). Given that the congestion wave is passing
upstream, the previous measurement of p1 estimates the current density of Po, which is not
available. This modification indicates that the source volume Vo(k) must be reduced as the
31
congestion wave passes out of the system before it can return to the nominal value.
Otherwise the density p i (k) in the first section will remain congested when the source
volume Vo(k) is reasonably large.
Dynamic modeling of ramp queues
In addition to the freeway state variables, it is also necessary to evaluate the queue lengths
q;(k) at each ramp such that
q;(k + 1) = q;(k) + r(d;(k)- rj (k))
q;(k) 0
where r;(k) is limited by
if q;(k) = 0
otherwise
Eqn. 4- 8
Eqn. 4- 9
where ri,MIN and r MAX are given minimum and maximum ramp metering rates, respectively.
Since r;(k) can be set higher than the demand rate when a queue is present (as high as the
saturation flow rate), we must restrict q;(k) to be non-negative since we cannot have
negative queues. At this level of modeling, we do not consider driver behavior in the
metering rate limitations. Thus, when a given metering rate is specified, (e.g. 456 veh/hr)
it is assumed that drivers can implement this rate precisely (e.g. not 445 veh/hr or 500
veh/hr). Recent results have indicated that this is usually not true in the real system,
especially at high metering rates when reaction time may be very close to the allocated
green-time of each metering signal [Banks, 1992; Decker, 1997].
To reflect the fact that vehicles are slowed, and possibly stopped, when entering and
exiting a freeway segment that is highly congested, we can add a soft-limiter
Eqn. 4- 10
to the on-ramp rate r;(k) and of-ramp rate s;(k). The shape of (4-10) is a sharp, but
continuous, transition about the critical point pc as illustrated in Figure 4-1.
32
0
density
Pmax
Figure 4- 1. Typical shape of soft-limiter function
The limiting effect is also applied to the off-ramp rates, since, undert;ongested conditions,
the of-ramp is also blocked after some critical density Pc is exceeded. Previous models
without such blockage terms would underestimate the clearance time required for
congestion to dissipate. Hence, (4-2) is further modified as
p)k + 1) = p)k) +: (½N)k)- VouT)k) + <;(pj (k))(r;(k)-s)k)))
J
and (4-8) as
qi (k + 1) = qlk) + r( di (k)- (p/k)) · rj (k))
qi (k)?: 0
Eqn. 4- 11
Eqn. 4- 12
to reflect the fact that a queue can develop at the ramp, even if the ramp metering rate is
higher than the demand rate, when freeway congestion blocks vehicles merging from the
ramp. Pc and PMAX must be carefully chosen (calibrated) in equation to reflect realistic
efects of queue-growth and restricted flow in the presence of freeway congestion.
Nevertheless, their addition to the macro-simulation model is to more accurately represent
congested conditions for comparison of various metering strategies in the evaluation
experiment of Chapter 9. Some preliminary evidence of the positive efects of these
additions are also shown in the benchmark test example of the area-wide coordination
problem in Chapter 5.
Summary of macroscopic model
The macroscopic description of freeway trafic and ramp queues used to evaluate ramp
metering strategies in this research project is a system of nonlinear diference equations
based on the fluid-flow model given by
p)k + 1) = p)k)+ :. (½N)k)-VouT)k) + ;(pj (k))(lj(k)-sj(k)))
J
33
v)k+l)=v)k)+ :(v,(pj (k))-v)k))+ :. v)k)(vj_1 (k)-v)k))
J
_ .!(nj+JPj+1(k)-njpj(k)
J-(r(nj+iroN)k)vj(k)
J 1: tJ.j njp)k)+TC tJ.j njp)k)+TC
½N)k) =a· "1-i(Pj-i (k), vj-i (k)) + (1- a)· ½(Pik), vik))
V0ur)k) =a· ½(Pik ), vj(k)) + (1- a)· "½+i (Pj+I (k), vj+I (k))
if p/k)y V.)pik),vj(k))=p/k)·v,(pj(k))
otherwise V.)pj(k), Vik))= p/k) · v/k)
q;(k + 1) = q;(k) + T( d;(k)- (Pik)) · rj(k))
q;(k) 0
if q;(k) = 0
otherwise
This model has the parameters [pc' cp, p crit' K, a, r, , l, m, p max' vf' 1, the specific
geometric and travel-behavior details [d1NJk), saFF,/k), L1P np b), and the control variables
r/k). The parameters must be calibrated precisely to the specific location characteristics
and driving population of the intended application area to obtain reasonable real-world flow
behavior from the model. Inappropriate choices for the model parameters can easily lead to
"unstable" model performance and inconclusive results. This model has been modified
slightly from previous instances of the model to reflect ; (a) the condition where queues are
built at the freeway on-ramps when the density in a section is too high to allow the current
on-ramp flow rate, and (b) the condition that of-ramp rates are also reduced when the
density becomes large because vehicles cannot physically move to the of-ramp to exit the
freeway system when the speed is near zero.
34
Stochastic efects and diversion behavior
Note that this macroscopic description of freeway flow is a detenninistic model. Empirical
data for real-world freeways indicates that the system does not evolve in a completely
detenninistic manner, but is highly afected by stochastic disturbances and flow
fluctuations. We address this issue by treating the input streams d/k) and the initial
upstream freeway input(s) Vo(k) as random variables, but leaving the evolution equations
deteninistic, as opposed to previous approaches based on adding acceleration noise to the
dynamic speed equations [Weits, 1988]. It is shown empirically in Chapter 9 that
considering the inputs d/k) and Vo(k) as random variables significantly improves the match
of the macroscopic model to a stochastic, microscopic model of a study area in Phoenix,
AZ. This microscopic simulation model, CORSIM, simulates travel of individual vehicles
in one-second increments and is well accepted for extensive simulation testing and
evaluation.
This macroscopic flow model also does not explicitly simulate diversion behavior or route
modification, but this can be "easily" added by modifying/updating the route-proportional
matrix (which detenines the of-ramp rates s/k) and demands d/k)) due to the current
conditions. Of course, as has been indicated in previous work, estimation and reestimation
of route-proportional matrices and diversion rates is very dificult and is a
subject of much research [Cremer and Keller, 1987; Madanat et al., 1995; Ashok and BenAkiva,
1993; Ding et al., 1996]. However, the ultimate success of a ramp metering control
system such as M1L0S in real-world freeway systems is highly dependent upon accurate
and reliable turning-proportions and/or route-proportional matrix estimation. This research
will assume that route-proportional matrices are given. In Chapter 7 we present a method
that could be used to detect changes to the route proportions and/or turning-probabilities,
but we do not further explore this possibility.
Summary
A macroscopic freeway traffic simulator based on the enhanced FREFLO and META
models was developed for evaluation of various ramp metering strategies developed in this
research project. The performance of the model was improved to represent highly
congested conditions, especially the simulation of ramp queues. A term was added to the
flow equations that represents the inability of vehicles to enter the freeway from the ramp
when the freeway is so congested that no merging maneuver can occur. Specific
simulation results showing the efects of the modeling enhancements are presented in
35
Chapter 9. The next three chapters develop the area-wide coordination, locally-reactive
real-time optimization, and SPC-based anomaly detection and optimization scheduling
layers of the MILOS hierarchical control structure.
36
Chapter 5: Area-wide Coordination Problem
Introduction
The area-wide coordination layer provides primarily the tactical decision-making of the
MILOS hierarchy by providing target ramp metering rates based on area-wide conditions
and aggregate traffic flows in each segment. The area-wide coordinator is based on two
rate coordination problem formulations from the literature
(1) Yuan and Kreer's queue-balancing problem [Yuan and KJ.:eer, 1971] and
(2) Wattleworth and Berry's throughput maximization problem [Wattleworth and
Berry, 1968].
These formulations are significantly modified in the model presented here. By using a
multi-criterion objective function, we combine the two conflicting objectives to address
both total system performance and user-specific performance benefits. The objective
function also includes
(a) consideration of the specific diferences in interchange congestion,
(b) physical capacity along the coridor, and
(c) agency/system-operator preference for incorporating queue-growth
considerations [Fan and Asmussen, 1990].
Mathematical description of the area-wide coordination problem
Consider a unidirectional freeway with N on-ramps and M of-ramps where the demand
(veh/hr) d; at each on-ramp i, i=l ... N is provided by either
( 1) the physical beginning of the freeway facility,
(2) a freeway-freeway connector, or
(3) a surface-street interchange.
We assume that demand d
0
= V
0 provided at the beginning of the freeway cannot be
controlled via ramp metering. The only freeway controls available are ramp metering rates
(veh/hr) r; i=2 ... N. By convention we assume that r1 = d0 , (i.e. the freeway input or the
first "ramp" is uncontrollable. Speed limits are assumed fixed in each freeway section, but
need not be equal everywhere. Speed advisories, such as those that could be provided by
variable message signs (VMS), are not considered in this algorithm.
The vehicular flows x
j in each freeway link j are determined by evaluating the routeproportional
flows from each on-ramp to each of-ramp, such that
37
j
,1_ .r. = x. \/1' L,/'-i,1, 1
i=l
where A;,j values have the special structure
Al.,]. =0 i<1'
'-Li'-i,1. ',.::;A.l,J.-I i>J.•
Eqn. 5- 1
Eqn. 5- 2
The matrix A = { A;) describes the proportion of the flow entering at ramp i that continues
through link j en route to its destination, it will be referred to as the route-proportional
matrix. The matrix A is assumed to be constant and known over the control period horizon,
T. In a steady-state input-output description of the freeway system such that
xj ( k) = xj \f k :s;:; T, it is assumed that all demand entering at ramp i bound for of-ramp j
will exit at of-ramp j during the time horizon T. Thus, we need only be concerned with
the physical limit of freeway capacity
j
L A;,/1 :;s CA \fj
i=l
Eqn. 5- 3
in each segment. The physical limit CAPj is derived for each segment from the volumedensity
curve specific to that segment. The volume-density curve can be empirically
derived (i.e. curve-fit) from observations or computed from the saturation flow rates,
number of lanes, merge area restrictions, and other factors as detailed in established
procedures [McShane and Roess, 1990]. Given the concerns noted in Chapter 1, it may be
advantageous to set a capacity CAPj for each link} in the optimization ·model that is slightly
less than the critical maximum volume to ensure stable flow. As will be shown in Chapter
9, it is dificult to maintain flows at the critical value CAPj without beginning a backwardtraveling
congestion wave, confirming the dificulties of the freeway management problem
as presented in Chapter 1.
An additional necessary set of constraints for the area-wide coordination problem is a
limitation on the minimum and maximum ramp metering rate, such that
'1,MIN :,'.; f; :,'.; '1,MAX i = I...N Eqn. 5- 4
38
where ,;,MAX = min(d;, sJ Here, s; is the saturation flow rate of the ramp i (a ramp could
have more than one lane) and ri,MIN is the slowest rate acceptable to drivers, such as two
vehicles per minute (120 veh/hr). The rate ri,MIN could be as low as zero if the ramp was
allowed to be and/or capable of being fully closed.
In this optimization formulation, metering/closure of a ramp only creates a queue at the
ramp and does not result in driver diversion. Diversion rates would be computed by an
external processor (not discussed in this research ) that updates the route-proportional
matrix and demands given the control decisions, e.g. [Cremer and Keller, 1987; Ashok and
Ben-Akiva, 1993; Madanat et al., 1995; Ding et al., 1997]. Chapter 7 discusses a system
identification procedure based on statistical process control which could be used to aid
diversion modeling by detecting changes to the flows xj in each link deviating from their
assumed nominal values xj .
Derivation of the objective function
Given the constraints detailed above, a popular objective function is to maximize the total
inputs to the freeway
max L1t·
reR
i=l
Eqn. 5- 5
This objective is derived from minimizing the total travel time in the freeway system, which
is a typical operational goal of freeway control [Wattleworth and Berry, 1968; Messer,
1971; May, 1979; Papageorgiou, 1983]. Using this objective, the current freeway
conditions p/k), and on-ramp demands d/k) must be continually monitored and compared
to the assumed steady-state values pj and d;. When the values of p/k) and d;(k) drift
outside of a reasonable upper or lower bound, the problem must be redefined and reoptimized
[Messer, 1971; May, 1979] as developed in Chapter 7.
Consideration of queue storage limits
The classical linear programming rate coordination problem is described by (5-3), (5-4),
and (5-5) [Wattleworth and Berry, 1968]. This formulation does not consider the
formation of ramp queues q;(k +I)= q;(k) + .6.T(d;(k)- ,;(k)) at each on-ramp due to the
39
application of metering rates lefis than the ofered demand. Thus, to reflect the physical
limitations of ramp queuing areas, an additional set of constraints must be added such that
Eqn. 5- 6
such that Qi is the physical limit on the number of vehicles that can be stored on the ramp
without causing spillback into the interchange (assuming some average vehicle length) and
T is the optimization time horizon. Q would be based on the length of the ramp storage
area and the average vehicle length. In operational practice, one ip.ay want to keep the
capacity limitation Q; slightly less than the physical limit of storage to provide an additional
cushion for unexpected surges in demand. The constraints (5-6) limit the rate at which the
queue is allowed to grow (based on a constant arrival rate) and fill to capacity during the
optimization horizon T. Vehicles queued at the ramp at the beginning of the optimization
period are included in the ofered demand di such that
d. = d +
qJ O) \;/ i 1 ext T Eqn. 5- 7
converting the queued vehicles q/0) into a flow rate (veh/hr) by assuming that all of the
vehicles queued "demand" to be discharged during the time horizon T.
Inclusion of constraints (5-6) would indicate that, in the absence of re-optimization during
the time horizon T, at t = t0 + T, several queues may be filled to capacity. This would
require, at least for a short time, r = ri.MAX (saturation flow rate) to clear the queue and to
create ramp capacity. This clearing at the maximum rate could have significant detrimental
efects to freeway conditions as the vehicles attempt to merge into traffic as a platoon. We
address the problem of alternately filling queues to capacity and dissipating them at the
saturation flow rate in two ways. First, we implement a rolling-horizon solution to the
area-wide coordination problem (5-3, 5-4, 5-5, 5-6) with frequent estimates of the current
constant demand di and queue lengths qi. Thus, as the unfilled queue storage capacity
begins to decrease as qi approaches Qi, the demand rate at the ramp increases and it becomes
more likely that queue dissipation will occur. Second, by modifying the nominal rate ri.N in
real-time such that ,i(k) = 1i.N + ,i(k) by solving a predictive-cooperative optimization
problem at each ramp for ,i(k), we can take advantage of the opportunities to dissipate
queues when
40
(a) the demand rate to the ramp meter is lower than expected and/or
(b) the freeway conditions are lighter than expected.
More details of the predictive-cooperative real-time optimization subproblems solved at
each ramp are provided in Chapter 6.
Development of a multi-criteria objective function
Inclusion of constraints (5-6) into the linear programming problem formulation provides
queue-growth management due to physical limitations of each ramp:. but does not control
queue-growth according to the prevailing congestion levels at each interchange. Such
constraints (5-6) also do not guarantee that equitable decisions will be made as to where to
hold vehicles in queues to provide freeway congestion relief. A quadratic optimization
criterion
Eqn. 5- 8
was proposed by Yuan and Kreer to address the need to balance ramp queues at each ramp,
such that q1
= q2
= ... = q11_1
= q1 (rather than hold many vehicles at some ramps and none
at others, such that q1 >> 0, q2 >> 0, q3
= ... = q11_1 = q11
= 0 [Yuan and Kreer, 1971]
which is a typical result of linear programming formulations such as (5-3), (5-4), and (5-5)
where the objective results in optimal solutions at the extrema of the feasible region). We
can thus use a combination of the two objectives (5-5) and (5-8) to obtain a compromise
solution that addresses both freeway throughput and ramp queue management.
Thus, we would like to simultaneously minimize freeway total travel time (by maximizing
on-ramp flow in steady-state) and balance ramp queues throughout the corridor. It would
be imprudent to simply add the cost functions (5-5) and (5-8) together because the units are
not the same (i.e. (veh/hr) and (veh/hr)2, respectively). As such, we use a simple
technique to combine objectives with difering units by dividing each objective by the
"ideal" cost and adding the dimensionless quantities. However, in (5-8) the optimal cost is
zero when r.* = d. i=l...N, and thus we cannot divide by the ideal cost solution to obtain a l l,
dimensionless objective for (5-8).
Thus, we modify objective (5-8) from minimizing the distance from the ideal point ri * =
di, i=l ... N, to maximizing the distance from the anti-ideal point. The anti-ideal point is
41
the (also usually infeasible) solution r;•= ri,MIN i=l ... N which creates the longest possible
queues at each ramp, and thus, the worst possible value for (5-8). Hence our single
objective now combines the distance for (5-5) from the ideal point r/ = d; , i=l ... N and the
distance for (5-8) from the_ anti-ideal point r;* = r;,MJN i=l ... N resulting in a compromise
objective function
max reR
N 2
l(d;-,J
1- i=I
N 2
I( di - 1t,MIN)
i=I
Eqn. 5- 9
Even though the terms from (5-5) and (5-8) are now in equivalent units, the relative
diference in the size of the two cost components for typical feasible choices of r; will still
influence in the importance attributed to each objective. We can provide decision-makers
with a preference between the two components by including a weighting factor /3 such that
(5-9) becomes
max
reR
Eqn. 5- 10
Setting f3 large will increase the importance of balancing ramp queues and setting /3 small
will decrease the importance on balancing queues and increase the importance of
maximizing freeway throughput.
Setting costs according to interchange congestion level
Although the objective (5-14) includes considerations for queue growth, the mechanism to
distinguish queue growth at one ramp over another is only the storage limitations in (5-6)
and the freeway conditions surrounding each ramp location. To reflect the current
congestion conditions at each interchange, we weight each of the components of objective
(5-10) with a weighting factor c; such that
± vm,i
m=I Cm i \-fl•
C '
V ; -
max(c;) Eqn. 5- 11
42
Eqn. 5- 13
where
r= N
(t,d,J
2 • Eqn. 5- 14
Le;( d; - \MJN)
i=l
Expanding the square and neglecting terms that do not contain the decision variables r; we
obtain the optimization problem
Eqn. 5- 15
subject to the constraints (5-3), (5-4), and (5-6) where c; is specified as in (5-11), d; is
specified as in (5-12), and all other parameters A, /3, T, Q;, r;,MiN• r;,MAx and q/0) are
specified from external data. (5-15) is a quadratic objective and (5-3), (5-4) and (5-6) are
linear constraints and thus the solution has a unique optimum when a feasible solution
exists.
Resolving infeasibility
It is possible, however, that the formulation posed in (5-3), (5-4), (5-6), and (5-15) does
not have a feasible solution. For example, an accident on a freeway link could reduce the
capacity considerably in that section, requiring many more vehicles to be metered at
upstream ramps than could be stored in the available ramp queues. In such a case,
constraints from (5-3), (5-4), or (5-6) must be relaxed to render the problem feasible.
Constraints (5-6) are the best candidate for relaxation, since we cannot increase the physical
carrying capacity of a freeway section in (5-3) or change the maximum (minimum) possible
metering rate, limited by the saturation flow rate (zero), in (5-4). Thus, we must opt to
allow spillback for a short time into the interchanges by increasing the queue storage
capacity in constraints (5-6) to accommodate the overflow in a system-equitable manner.
Let z; 0 i = l ... N be the extra capacity allocated at each ramp queue i to accommodate
the flow at that ramp. In the same way that the queue storage is balanced according to the
45
interchange congestion cost c; in a feasible problem, consider an allocation of the queue
overflow in a similar manner. Thus, the constraints (5-6) are modified such that
Eqn. 5- 16
A penalty term is added to the objective function (5-15) incorporating the cost of allowing
queue i to extend beyond its capacity Q over the time horizon T, such that the objective
function now becomes
max
reR J = L(l + 2/3rcA)r; - f3yc;r;2 - f32YC;Z/
i=l
Eqn. 5- 17
where /32 is an appropriately chosen scaling constant. In particular, /32 should be specified
large enough, say /32 = 100 /3, to induce Z; = 0 for all solutions that are feasible without
the inclusion of the additional capacity variables Z;, i=l ... N. Choosing a "small" value of
/32 can result in a solution where some ramps are specified to spill-back, and others are
allowed to flow unconstrained. This is much like the solution resulting from using an LP
method, but with the c; terms the congestion at each interchange is considered. It should be
noted here that the objective function (5-17) has no physical meaning with the introduction
f the penalty term /32 yc;z/ and is likely a negative quantity in the overcapacity situation.
However, each of the components of (5-17) is suitably derived to benefit both freeway and
surface-street system operation.
Area-wide coordination problem summary
The quadratic optimization problem is summarized as
subject to
i=l
46
Eqn. 5- 18
M
L vm,i
m=I Cm i \-ll' C -
'
; V
- max(c;)
i=l
d - d + (1 )d d q;(O) \-11· ; - PR,NB NB PL.SB - PR,SB SB + Pr,EB EB + T v
which can be solved with any constrained nonlinear programming or specialized quadratic
programming method, not detailed here. Note that the inclusion of the overcapacity
variables Z;, i=l...N ensures at least the feasible solution
r. = r. MIN'' z. = d.T- r. MINT- Q. i = l...N
l l, l l l, l Eqn. 5- 19
for reasonable (realistic) values of the freeway capacities CAPr In the evaluation results of
this chapter and Chapter 9, problem (5-18) is solved using the QP barrier algorithm of the
CPLEX math programming optimization software package [CPLEX, 1997]. For more
details of the iteration details of barrier optimization algorithms, see [Bazaraa et al., 1993].
Operation under severe congestion
If any
CA L 1\,/t,MIN
i=O
Eqn. 5- 20
then there is a severe limitation of capacity (an incident) in that section and even (5-18)
with the inclusion of the Z; variables will be infeasible. In this case, we can prescribe a
heuristic solution such that r; = ri,MIN for all ramps upstream of the congestion with
condition (5-20) and r; = ri.MAX downstream of the severely congested section.
In addition, the higher-layer processor(s) of the ramp metering control system should send
information to the surface-street controllers regarding the incident location (severe limitation
of capacity) and the prescribed emergency settings of ri,MIN· In the presence of ATIS, such
information could also be provided to travelers to increase the diversion efect away from
the congested segment. As the congestion clears, the anomaly detection module will detect
the favorable change to the state variables xj and re-run the area-wide optimization for a
new, feasible solution to the area-wide coordination problem (5-18).
47
from\to begin I 2 3 4 5
begin 1 0.95 0.9 0.85 0.8 0.75
1 I 0.95 0.9 0.85 0.8
2 1 0.95 0.9 0.85
3 I 0.95 0.9
4 I 0.95
5 1
Table 5- 1. Route-proportional matrix of example problem
MJLOS requires information about the interchange flows that comprise the ramp demand,
and needs performance information to derive the objective function cost coefficients. The
data in Table 5-2 includes the volumes and turning probabilities at each interchange, that
make up the demands to each ramp. Coupling these data with the green-time percentages
(GT%), the VIC ratios of each interchange are computed and the scaled cost coeficients ci
are computed from the VIC ratios. These data have been selected to produce values for the
demands at each ramp similar to the example problem of Papageorgiou [1983] and, at the
same time, to produce diferentiation in the congestion level at each interchange. As
indicated by the VIC ratio at each location, interchanges 3 and 4 are more congested than 1 ,
2, and 5. As a result, the QP coordination algorithm should store less vehicles on ramps 3
and 4 (relative to demand at that ramp and the storage capability of the ramp) than at ramps
1, 2, and 5.
The values chosen in Table 5-2 do not represent real locations, but they are intended to be
reasonable approximations of real behavior. The interchange parameters have also been
selected to illustrate the ability of the quadratic programming formulation to react to
diferences in the congestion level at adjacent ramps. The quadratic programming problem
(5-18) was solved using the data in Tables 5-1 and 5-2 with several values of /3, the tradeof
parameter between the two sub-objectives. The solutions, as listed in Table 5-6,
indicate that the LP method restricts on-ramp volume at ramps 3, 4, and 5 only, where the
freeway capacity is exceeded. The QP methods, since they take into account the queue
storage restriction, restrict on-ramp volume at all ramps. The last line of Table 5-3
indicates when the freeway would become over-capacity if no ramp metering was
implemented. Thus, in this example problem, if no ramp metering was implemented a
backward-traveling congestion wave would be started in the section containing ramp 3.
49
On-Ramp data begin 1 2 3 4 5
Northbd vol (veh/hr) 2200 1600 1800 1000 1100
Southbd vol (veh/hr) 2000 2000 1900 1000 900
Eastbd vol (veh/hr) 200 260 282 295 307
pnb,r 0.25 0.35 0.15 0.25 0.25
P,b,I 0.1 0.05 0.07 0.1 0.1
peb,t 0.05 0.07 0.05 0.15 0.05
Ramp demand d1 (veh/hr) 3000 684 610 355 355 342
Initial queue - 0 0 0 0 0
Time horizon (hrs) 0.5 0.5 0.5 0.5 0.5
Ramp storage (veh) - 40 30 40 40 50
GT%,EB - 0.18 U.:L) U,D 0.2 0.23
GT%,NBJSB - 0.55 0.6 0.7 0.55 0.6
GT%,sn1,n - 0.27 0.15 0.15 0.25 0.17
v/C ratio 1.67 1.55 1.74 1.24 1.15
scaled cost c1 0.95 0.89 1 0.71 0.66
Table 5- 2. Ramp interchange data
rampl ramp2 ramp3 ramp4 ramps
Demand 684 610 355 355 342
LP 684 610 320 275 275
QP,(i =l,(i 2 =100(i 642 555 295 275 253
QP, (i,;100,(i 2 =100(i . 644 550 275 275 271
QP, (i=0.01, (i2 = lOO(i 669 550 275 275 252
6-second cycle 600 600 355 355 342
freeway over capacity? NO NO YES YES YES
Table 5- 3. Comparison of metering rate coordination methods
Influence of /3
In this example, the diferences between the QP solutions for radically diferent values of /3
are negligible because of the queue-growth constraints (5-6). In addition, setting /32 =
10000/3 in each case prescribes a solution that allows no queues to spillback regardless of
their congestion level. It is possible with a lower setting for /32 (relative to the value of /3)
that spillback could occur at a ramp with a relatively uncongested interchange. As such,
care must be taken in setting the value for /32 when queue balancing is de-emphasized.
50
/3 has the most influence on the:resulting rate allocation when metering is required to avoid
freeway congestion near interchanges where the congestion is also considerable. In such a
situation, an LP approach that does not consider queue restrictions or interchange
congestion may apply restrictive metering at locations where the most adverse impact on the
surface-streets would result. The QP approach of (5-18) would enact metering upstream at
interchanges where the congestion was (possibly) lower (i.e. ci' < c; I i" < i) and resulting
ramp queues would have less efect on the surface-street congestion.
To further illustrate that the QP solution balances queues according to interchange
congestion, consider Table 5-4. Table 5-4 compares the rates for each ramp solved by the
QP method for /3=1, /32 =100/3. The queue storage size Q;, demand rated;, Ai,j' and cost
coeficient c; of each interchange all interact to produce the metering rates r;. Comparing the
rate at ramp 5 from the QP and LP in Table 5-3 provides some evidence that more vehicles
are held at uncongested ramps with the QP approach. Table 5-4 confirms this, since ramp
5 also has the largest ramp storage capacity Q5•
ramp 1 ramp2 Ramp3 ramp4 ramps
demand (veh/hr) 684 610 355 355 342
Metering rate (veh/hr) 642 555 295 275 253
Vehicle storage size 40 30 40 40 50
Cost coeficient 0.95 0.89 1 0.71 0.66
Percentage holdback 6.1 9.0 16.9 22.5 26.0
Maximum queue, 30-min horizon 21 28 30 40 45
Table 5- 4. Rate comparison for = 1
Comparison of area-wide metering rate settings in macrosimulation
We now compare the area-wide coordination optimization algorithm based on quadratic
programming (5-18) with the no-control and LP-control cases in the macrosimulation
environment of Chapter 4. For the example problem of Figure 5-2, we use parameters of
the macrosimulation similar to those used by Papageorgiou [1984] with the addition of the
cut-of level Pc which phases in the flow limitation in (4-11) and (4-12). These parameters
are listed in Table 5-5.
0.013 21.6 10 2 4 0. |