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ISSN 1055- 1425
March 2010
This work was performed as part of the California PATH Program of the
University of California, in cooperation with the State of California Business,
Transportation, and Housing Agency, Department of Transportation, and the
United States 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 State of California. This
report does not constitute a standard, specification, or regulation.
Final Report for Task Order 6323
CALIFORNIA PATH PROGRAM
INSTITUTE OF TRANSPORTATION STUDIES
UNIVERSITY OF CALIFORNIA, BERKELEY
Development of an
Adaptive Corridor Traffic Control Model
UCB- ITS- PRR- 2010- 13
California PATH Research Report
Will Recker, Xing Zheng, Lianyu Chu
CALIFORNIA PARTNERS FOR ADVANCED TRANSIT AND HIGHWAYS
FINAL REPORT
Caltrans RTA 65A0208 ( PATH T. O. 6323)
Optimal Control for Corridor Networks:
A Mathematical Modeling Approach
Prepared by:
Will Recker, Xing Zheng
Institute of Transportation Studies
University of California, Irvine
Irvine, CA 92697
Lianyu Chu
California Center for Innovative Transportation
University of California, Berkeley
Berkeley, CA 94720
November, 2009
STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION
TECHNICAL REPORT DOCUMENTATION PAGE
TR0003 ( REV. 10/ 98)
1. REPORT NUMBER
CA10- 0759
2. GOVERNMENT ASSOCIATION NUMBER
3. RECIPIENT’S CATALOG NUMBER
5. REPORT DATE
November 2009
4. TITLE AND SUBTITLE
Optimal Control for Corridor Networks: A Mathematical Modeling
Approach
6. PERFORMING ORGANIZATION CODE
7. AUTHOR( S)
Will Recker, Lianyu Chu, Xing Zheng
8. PERFORMING ORGANIZATION REPORT NO.
UCB- ITS- PRR- 2010- 13
10. WORK UNIT NUMBER
193
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Institute of Transportation Studies
University of California, Irvine
Irvine, CA 92697- 3600
11. CONTRACT OR GRANT NUMBER
65A0208
13. TYPE OF REPORT AND PERIOD COVERED
Final Report
12. SPONSORING AGENCY AND ADDRESS
California Department of Transportation
Division of Research and Innovation, MS- 83
1227 O Street; Sacramento CA 95814
14. SPONSORING AGENCY CODE
15. SUPPLEMENTAL NOTES
None
16. ABSTRACT
This research develops and tests, via microscopic simulation, a real- time adaptive control system for corridor
management in the form of three real- time adaptive control strategies: intersection control, ramp control and an
integrated control that combines both intersection and ramp control. The development of these strategies is
based on a mathematical representation that describes the behavior of traffic flow in corridor networks and
actuated controller operation. Only those parameters commonly found in modern actuated controllers ( e. g., Type
170 and 2070 controllers) are considered in the formulation of the optimal control problem. As a result, the
proposed strategies easily could be implemented with minimal adaptation of existing field devices and the
software that controls their operation. Microscopic simulation was employed to test and evaluate the
performance of the proposed strategies in a calibrated network. Simulation results indicate that the proposed
strategies are able to increase overall system performance and also the local performance on ramps and
intersections. Prior to testing the complete model, separate tests were conducted to evaluate the intersection
control model on: 1) an isolated intersection, and 2) a network of intersections along an arterial. The complete
model was then tested and evaluated on the Alton Parkway/ I- 405 corridor network in Irvine, California. In testing
the optimal control model, we simulated a variety of conditions on the freeway and arterial subsystems that
cover the range of demand from peak to non- peak, incident to non- incident, conditions. The results of these
experiments were evaluated against full- actuated operation and found to offer improved performance.
17. KEY WORDS
Adaptive Traffic Control, Corridor Management,
Mathematical Modeling, Optimal Control
18. DISTRIBUTION STATEMENT
No restrictions. This document is available to the
public through the National Technical Information
Service, Springfield, VA 22161
19. SECURITY CLASSIFICATION ( of this report)
None
20. NUMBER OF PAGES
59
21. PRICE
N/ A
Reproduction of completed page authorized
DISCLAIMER STATEMENT
This document is disseminated in the interest of information exchange. The contents of this
report reflect the views of the authors who are responsible for the facts and accuracy of the data
presented herein. The contents do not necessarily reflect the official views or policies of the State
of California or the Federal Highway Administration. This publication does not constitute a
standard, specification or regulation. This report does not constitute an endorsement by the
Department of any product described herein.
For individuals with sensory disabilities, this document is available in Braille, large print,
audiocassette, or compact disk. To obtain a copy of this document in one of these alternate
formats, please contact: the Division of Research and Innovation, MS- 83, California Department
of Transportation, P. O. Box 942873, Sacramento, CA 94273- 0001.
1
Table of Contents
List of Figures ............................................................................................................................... . 2
List of Tables ............................................................................................................................... .. 3
Executive Summary ....................................................................................................................... 4
1. Introduction ............................................................................................................................ 6
2. Methodological Approach ..................................................................................................... 8
2.1 Intersection control module ............................................................................................. 8
2.2 Ramp meter control model ............................................................................................... 9
2.3 Freeway model ............................................................................................................... 11
2.4 Optimal corridor control formulation ............................................................................ 17
2.5 Path to deployment ........................................................................................................ 17
2.6 Testing and evaluating the proposed control models ..................................................... 18
3. Theoretical Development of the Intersection Control Model .......................................... 19
3.1 Conceptualization .......................................................................................................... 19
3.2 Determining vehicle arrival flow rate j
i
...................................................................... 22
3.3 Determining vehicle departure number ( j)
i N G and spillover spill j
i Q ............................ 25
3.4 Determining future vehicle arrival flow rate j 1
i
......................................................... 28
3.5 Determining optimal maximum green 1
max
j
i G ................................................................. 29
3.6 Determining optimal phase split j 1
i G ........................................................................... 30
3.7 Determining optimal minimum green 1
min
j
i G .................................................................. 33
3.8 Determining optimal passage setting j 1
i
.................................................................... 33
4. Theoretical Development of Ramp Control Model ........................................................... 35
5. Consideration of Freeway Delay......................................................................................... 44
6. Development of Integrated Control Model........................................................................ 45
7. Simulation Evaluation ......................................................................................................... 50
7.1 Simulation model setup.................................................................................................. 50
7.2 Evaluation of intersection control model ....................................................................... 52
7.3 Evaluation of ramp control model ................................................................................. 55
7.4 Evaluation of combined intersection/ ramp control model ............................................. 57
8. Concluding Remarks ............................................................................................................... 58
References ............................................................................................................................... ..... 60
2
List of Figures
Figure 1. Typical Ramp Metering Configuration ........................................................................... 9
Figure 2. Typical Open- loop Ramp Metering Control ................................................................. 10
Figure 3. Typical Closed- loop Ramp Metering Control ............................................................... 10
Figure 4. Expected Speed- Density Relationship .......................................................................... 12
Figure 5. Field Speed- Density Data .............................................................................................. 13
Figure 6. Single Lane Speed- Density Field Data Correspondence with Model ........................... 14
Figure 7. Single Lane Flow- Density Field Data Correspondence with Model ............................. 14
Figure 8. Single Lane Flow- Speed Correspondence with Model ................................................. 15
Figure 9. All Lanes Speed- Density Correspondence with Model ................................................ 15
Figure 10. All Lanes Flow- Density Correspondence with Model ................................................ 16
Figure 11. All Lanes Flow- Speed Correspondence with Model ................................................... 16
Figure 12. Irvine Triangle Network .............................................................................................. 18
Figure 13. Dual- ring Controller Phasing Diagram ....................................................................... 19
Figure 14. Dual- ring Controller Stages ......................................................................................... 19
Figure 15. Phase State ................................................................................................................... 20
Figure 16. Pattern of Arrivals/ Departures ..................................................................................... 21
Figure 17. Approach Volumes ...................................................................................................... 28
Figure 18. Circular Dependency ................................................................................................... 31
Figure 19. Test Network ............................................................................................................... 51
Figure 20. Study Intersection ........................................................................................................ 53
Figure 21. Flow Profile for Each Phase ........................................................................................ 53
Figure 22. Study Onramp .............................................................................................................. 55
Figure 23. Flow Profile for Onramp and Freeway Section ........................................................... 56
Figure 24. Overall System Performance Comparision of the Three Control Models .................. 58
3
List of Tables
Table 1. Parameters for the Study Intersection ............................................................................. 53
Table 2. Performance of the Intersection Control ......................................................................... 54
Table 3. Performance of the Network Control .............................................................................. 55
Table 4. Performance of the Ramp and Mainline Sections .......................................................... 56
Table 5. Performance of the Network ........................................................................................... 57
Table 6. Performance of the Network ........................................................................................... 57
Abstract
This research develops and tests, via microscopic simulation, a real- time adaptive control
system for corridor management in the form of three real- time adaptive control strategies:
intersection control, ramp control and an integrated control that combines both
intersection and ramp control.
The development of these strategies is based on a mathematical representation that
describes the behavior of traffic flow in corridor networks and actuated controller
operation. Only those parameters commonly found in modern actuated controllers ( e. g.,
Type 170 and 2070 controllers) are considered in the formulation of the optimal control
problem. As a result, the proposed strategies easily could be implemented with minimal
adaptation of existing field devices and the software that controls their operation.
Microscopic simulation was employed to test and evaluate the performance of the
proposed strategies in a calibrated network. Simulation results indicate that the proposed
strategies are able to increase overall system performance and also the local performance
on ramps and intersections. Prior to testing the complete model, separate tests were
conducted to evaluate the intersection control model on: 1) an isolated intersection, and
2) a network of intersections along an arterial. The complete model was then tested and
evaluated on the Alton Parkway/ I- 405 corridor network in Irvine, California.
In testing the optimal control model, we simulated a variety of conditions on the freeway
and arterial subsystems that cover the range of demand from peak to non- peak, incident
to non- incident, conditions. The results of these experiments were evaluated against full-actuated
operation and found to offer improved performance.
Key Words: Adaptive Traffic Control, Corridor Management, Mathematical Modeling,
Optimal Control
4
Executive Summary
This project developed and tested, via microscopic simulation, a real- time adaptive control
system for corridor management. Although the focus of the development is on signal controllers
designed for operation on arterial street networks, the formulation of the adaptive control
strategy explicitly includes interaction with freeway ramp control devices, which are also
designed to react adaptively to both the onramp flow, as determined by the operation of adjacent
intersection signal controllers, and the traffic state on the mainline freeway. The resulting control
strategy is based on a mathematical representation that describes the behavior of real- life
processes ( traffic flow in corridor networks and actuated controller operation). In formulating the
optimal control problem, we have restricted our attention to control of only those parameters
commonly found in modern actuated controllers ( e. g., Type 170 and 2070 controllers). By doing
this, we hope to ensure that the procedures developed herein can be implemented with minimal
adaptation of existing field devices and the software that controls their operation.
In the methodological approach taken, we assume that the traffic arrival pattern can be
represented as a queue with Poisson arrivals, and from queuing theory we first develop estimates
of both the effective green time ( equal to actual displayed green interval), and the vehicle arrival
flow, departure number and spillovers for the expired signal phase based on the known controller
parameter settings. Similarly, we estimate upstream contributions to the target intersections from
known parameters at the upstream intersections and readouts from the corresponding signal
displays. Dynamic turning fractions at the target intersection, which cannot be known a priori,
are estimated based on a moving average model.
Maximum green settings provide constraints for the decision of optimal phase splits, which are
determined by solving a non- linear optimization problem with the objective to be minimizing
total intersection control delay per cycle. The expression used for delay is a generalization of the
well- known Webster formulation. These optimized phase splits are used to determine optimal
phase minimum green and passage settings.
The outputs of the adaptive control model for intersection signalization are the product of a
stochastic optimal control problem that returns dynamic values for the three parameters of
actuated controllers— phase minimum green parameter ( subject to its absolute minimum based
on such other conditions as pedestrian waiting time and start- up lost time), phase passage
parameter and phase maximum green parameter— that control its responsiveness to stochastic
fluctuations in traffic conditions ( other parameters, e. g., yellow interval, clearance interval, phase
sequencing, are determined principally in regard to safety and geometric considerations);
contrasted to current controller operation, in which these parameters are static/ preset, in our
formulation they are dynamically set in response to estimates of demand.
Three real- time adaptive control strategies: an intersection control, ramp control and an
integrated control that combines both intersection and ramp control are proposed. Microscopic
simulation was employed to test and evaluate the performance of the proposed strategies in a
calibrated network. Prior to testing the complete model, separate tests were conducted to
evaluate the intersection control model on: 1) an isolated intersection, and 2) a network of
intersections along an arterial. The complete model was then tested and evaluated on the Alton
Parkway/ I- 405 corridor network in Irvine, California. In testing the optimal control model, we
5
simulated a variety of conditions on the freeway and arterial subsystems that cover the range of
demand from peak to non- peak, incident to non- incident, conditions. The results of these
experiments were evaluated against full- actuated operation and found to offer improved
performance.
Simulation results indicate that the proposed strategies are able to increase overall system
performance and also the local performance on ramps and intersections.
6
1. Introduction
The objective of this project is to develop and test, via microscopic simulation, a real- time
adaptive control system for corridor management. Although the focus of the development is on
signal controllers designed for operation on arterial street networks, the formulation of the
adaptive control strategy explicitly includes interaction with freeway ramp control devices,
which are also designed to react adaptively to both the onramp flow, as determined by the
operation of adjacent intersection signal controllers, and the traffic state on the mainline freeway.
The proposed control strategy is based on a mathematical representation that describes the
behavior of real- life processes ( traffic flow in corridor networks and actuated controller
operation). In formulating the optimal control problem, we have restricted our attention to
control of only those parameters commonly found in modern actuated controllers ( e. g., Type 170
and 2070 controllers). By doing this, we hope to ensure that the procedures developed herein can
be implemented with minimal adaptation of existing field devices and the software that controls
their operation.
A typical advantage of an adaptive signal controller is that, in the case of intersection control, the
cycle length, phase splits, and even the phase sequence, may vary from cycle to cycle, in a
manner that satisfies the demands of the current traffic pattern. To some extent, actuated
controllers are themselves “ adaptive” in the sense that they vary these same outcomes, but do so
subject to a set of predefined, fixed, parameters that do not “ adapt” to current conditions. For the
functionality of truly adaptive controllers, a set of on- line optimized phasing and timing
parameters are needed.
Existing adaptive controls, such as SCOOT ( Robertson and Bretherton, 1991), make incremental
adjustments to the current signal plan for the next cycle, in response to the changing traffic
demands. In another real- time network control, SCATS ( Lowrie, 1992; Sims, 1979), the local-level
intersection controller decides its timing parameters on the basis of the degree of saturation,
and then incrementally adjusts to varying traffic conditions. The major drawback of these
systems is that they are not proactive and therefore, cannot accommodate significant transients
effectively. RHODESTM, a real- time traffic- adaptive signal control system developed at the
University of Arizona, uses a traffic flow arrivals algorithm – PREDICT ( Head, 1995) – to
improve effectiveness when calculating online phase timings. In the PREDICT algorithm,
detector information on approaches of every upstream intersection, together with the traffic state
( arrival and queues), and control plan for the upstream signals are used to predict future traffic
volume. It assumes that all surrounding upstream intersections have fixed- time signalized
planning, an assumption that is violated in virtually every modern system.
In none of these previous systems do the embedded traffic flow prediction models fully utilize
available detector information and control features. Consequently, their applicability is confined
only to particular factors, and thus restricted in achieving comprehensively good performance.
For any signalized intersection, at least three kinds of information— vehicle actuated detector
information, signal timing plan and current signal phase information— can be exploited to infer a
relatively rich body of information that can be used in adapting the operation of the signal
controller to current, or expected, conditions. Here, we develop a traffic flow prediction model
7
based on the actuated phase control strategy and other features, such as phase minimum green
parameter, phase passage parameter and phase maximum green parameter, together with related
detector information gleaned from actuated- signalized upstream intersections to estimate the
future arrivals at downstream intersections. To better utilize all available information, our traffic
flow prediction model is divided into an approach volume prediction and the corresponding
turning proportion estimation. Based on the time of actuation in the upstream detector of
neighboring intersections, together with current signal state and control tactics of the neighboring
intersections, the arrival pattern of vehicles is predicted. Then by using the exit/ entry passage
detector cycle/ phase counts in the neighboring intersections, the turning percentage for each
movement is estimated. As a result, the model can utilize instantaneous information that is
currently available but not used, and thus assist fine- tuning intersection performance without any
additional hardware investment.
The development and adoption of adaptive control procedures for signalized intersections have
been hampered by two fundamental impediments to their successful implementation— those that
are theoretically sound invariably have been specified in terms of parameters and control options
that simply are not within the lexicon of control devices and typically involve complex mixed-integer-
programming formulations that do not lend themselves to real- time solution, and those
that do manipulate parameters employed in modern actuated control devices are based on highly
simplified approximations and simplifications to both control response and traffic measurement.
Consistent modeling of traffic signal operations inevitably includes some sort of conditional
piece- wise functions in the mathematical representation. For example, such a representation is
the basis of the dispersion- and- store model where the inflow to a link is dispersed and is
subsequently stored at its end if the signal at the adjacent intersection is “ Red,” or the similar
store- and- forward model where the inflow is assumed to travel at a constant travel time, a
general relationship of the corresponding outflow discharge would be described by a function
that is conditional on the signal indication and the prevailing traffic conditions. Specifically, the
outflow is equal to zero if the signal is “ Red”, and equal to the minimum of the flow rate of the
stored vehicles and the saturation flow rate if “ Green”. Within the context of a mathematical
programming problem this function is represented by some sort of constraint( s).
Typically, this task has been approached either by considering specific aspects of the process
behavior that narrow the applicability of the model and restrict the insight of the findings, or via
its questionable manipulation in the solution procedure of the corresponding problem. For
example, when designing optimal signal control strategies for surface street networks based on
the store- and- forward model, Singh and Tamura ( 1974), D'Ans and Gazis ( 1976), and
Papageorgiou ( 1995) assumed that oversaturated conditions prevail. The control variables are the
green per cycle ratios given a cycle of fixed duration, so that the outflow discharge is calculated
as the product of the saturation flow rate and the green per cycle ratio. In their formulations,
traffic signal operation is not explicitly modeled, and the oversaturation assumption restricts the
applicability of the control strategy that of a single- ring, 2- phase, fixed cycle controller. As
another example, Chang et al. ( 1994) develop signal control strategies for mixed surface
street/ freeway networks by manipulating the outflow discharge function based on the values of
the current state and the previously determined control variable, with the solution algorithm
assigning the minimum of the two arguments to the link outflow. In other cases, the conditional
8
piece- wise function is expressed in the form of minimum or maximum operators; see, e. g.,
Stephanedes and Chang ( 1993), and Ziliaskopoulos ( 2000).
Despite the theoretical consistency of optimal control formulations based on such piece- wise
functions, the impracticality of their solution in real- time and their general inconsistency with the
operation of existing control devices ( e. g., by specifying control transition commands that cannot
be understood by existing controller logic) have rendered their practical implementation virtually
impossible. In the approach taken herein, we avoid this pitfall by formulating the optimal control
problem for a signalized intersection in terms of parameters ( phase minimum green parameters,
phase passage parameters and phase maximum green parameters) featured in any modern
actuated controller, based on a theoretically consistent model of stochastic traffic flow.
2. Methodological Approach
2.1 Intersection control module
In the approach taken here, we assume that the traffic arrival pattern can be represented as a
queue with Poisson arrivals, and from queuing theory ( e. g., Cox and Smith, 1961) first develop
estimates of both the effective green time ( equal to actual displayed green interval), and the
vehicle arrival flow, departure number and spillovers for the expired signal phase based on the
known controller parameter settings. Similarly, we estimate upstream contributions to the target
intersections from known parameters at the upstream intersections ( a total of four) and readouts
from the corresponding signal displays; depending on the expected travel time from the
contributing intersection, these values may be drawn from a completed cycle or from an ongoing
cycle of operation that commenced just prior to the forecast period for the target intersection.
Dynamic turning fractions at the target intersection, which cannot be known a priori, are
estimated based on a moving average model.
Based on maximum cycle length restrictions, we set phase maximum green parameters based on
Webster’s functions, accounting for any spillover from previous cycles of operation. These
maximum green settings provide constraints for the decision of optimal phase splits, which are
determined by solving a non- linear optimization problem with the objective to be minimizing
total intersection control delay per cycle. The expression for delay is given by Darroch ( 1964),
which is a generalization of the well- known Webster formulation. These optimized phase splits
are used to determine optimal phase minimum green and passage settings. All these timing
parameters will be used for the upcoming control cycle as well as provide signal timing data for
further optimizations.
As specified, the outputs of the adaptive control model for intersection signalization are the
product of a stochastic optimal control problem that returns dynamic values for the three
parameters of actuated controllers— phase minimum green parameter ( subject to its absolute
minimum based on such other conditions as pedestrian waiting time and start- up lost time),
phase passage parameter and phase maximum green parameter— that control its responsiveness
to stochastic fluctuations in traffic conditions ( other parameters, e. g., yellow interval, clearance
interval, phase sequencing, are determined principally in regard to safety and geometric
9
considerations); contrasted to current controller operation, in which these parameters are
static/ preset, in our formulation they are dynamically set in response to estimates of demand.
2.2 Ramp meter control model
Based on the procedures described in Section 4, below, real- time approach volumes ( demand) at
the entry ramps downstream of the intersections that feed the ramps are estimated. Owing to the
proximity of the intersections to the respective ramp meters, the arrival pattern at the point of
metering will be determined using platoon dispersion principles. The departure pattern will be
determined as an output of the ramp control model, which will have as its control parameter the
instantaneous metering headway, subject to certain installation parameters ( e. g., queue override
headway, merge queue override headway), and to controller operation protocol.
Caltrans Type 170 metering controllers comprise a number of control elements based on
inductive loop detector data inputs. A typical freeway configuration is shown in Figure 1.
Under current deployment, the headway component of the signal controller uses input from: 1)
the upstream detector station, 2) the downstream detector station, 3) the excessive queue detector,
and 4) the merge detector station. Basically, the upstream and downstream detectors are used to
calculate an appropriate metering headway based on conditions on the mainline freeway, while
the queue and merge detectors are used to override the calculated headway based on conditions
on the ramp. The actual ramp signal sequencing is determined by input from the demand
( Checkin) detector and the passage ( Checkout) detector.
A typical open- loop control operation is shown in Figure 2 below, in which the objective of the
control is to keep the total demand downstream at a value that does not exceed the capacity
downstream:
Ramp Meter
US
US
US
Upstream System Detectors
DS
DS
DS
Downstream System Detectors
M
Merge Detector
CO
Checkout Detector
CI
Checkin Detector
Q
Queue Detector
qutp () qdtown ()
qRt A ()
qRtD ()
Figure 1. Typical Ramp Metering Configuration
10
1, ( 1) 0
0, ( 1) 0
i
i
DDeelalayy
( i 1)
Count
1, ( ) 0
0, ( ) 0
e i
e i
e( i) JR JR J( i)
u( i 1) O( i 1)
1
J( i)
Occupancy
JR JR
1, ( 1) 0
0, ( 1) 0
i
i
DDeelalayy
( i 1)
Count 1, ( ) 0
0, ( ) 0
e i
e i
e( i) JR JR J( i)
u( i 1) O( i 1)
1
J( i)
Occupancy
JR JR
Input
Signal
Input
Sig nal
Upstream System Detector
Downstream System Detector
0
3600
T
( ) c down q
up q
( ) c down up e q) q
q 3600 ( ) 1
Override
Figure 2. Typical Open- loop Ramp Metering Control
In this controller, the reference input is the downstream capacity, ( ) c down q , and the metering
headway is computed as the headway corresponding to a ramp flow rate that would lead to the
total downstream demand being less than or equal to capacity. Note that in this open loop design,
the downstream detectors are not used; only the upstream flow rate ( which is external to the
control system) is utilized.
An example of a simple closed- loop control system is shown in Figure 3 in which the objective
is to maintain the downstream speed at a certain prescribed level, REF x .
1, ( 1) 0
0, ( 1) 0
i
i
DDeelalayy
( i 1)
Count
1, ( ) 0
0, ( ) 0
e i
e i
e( i) JR JR J( i)
u( i 1) O( i 1)
1
J( i)
Occupancy
JR JR
1, ( 1) 0
0, ( 1) 0
i
i
DDeelalayy
( i 1)
Count 1, ( ) 0
0, ( ) 0
e i
e i
e( i) JR JR J( i)
u( i 1) O( i 1)
1
J( i)
Occupancy
JR JR
Input
Signal
Input
Signal
Upstream System Detector
Downstream System Detector
REF x
REF down e x x
Override
0
3600
T
down q
5280
100( ) D L L
down k ( ) 1
down x
P K
( ) P REF down K x x
Figure 3. Typical Closed- loop Ramp Metering Control
11
In this controller, the count and occupancy from the downstream detector stations are used to
compute an estimate of the downstream speed, which is then compared to the input reference
speed; a proportional control is then used to calculate the ramp metering headway. In this
application the upstream system detectors are not used.
In neither of these typical installations is the system- wide performance an explicit consideration
in the setting of parameters under which ramp meter controllers operate. In the work presented
here, we formulate the ramp control element of our real- time adaptive control corridor model
under assumptions of stochastic queuing, with demand input determined from the output of the
associated intersection discharge model, and with output determined in accordance with
minimizing the delay to the combined corridor system, comprised of: intersection delay, ramp
delay, and freeway delay.
2.3 Freeway model
It is well- known that there is an inherent relationship among the speed ( and, correspondingly,
delay), flow, and density of traffic on a freeway ( often referred to as the “ fundamental diagram
of traffic flow”). Less well- known is the exact form of this relationship. For analytical
formulations, such as ours, it is nonetheless necessary to impose a mathematically tractable
relationship. Although a number of such relationships have been proposed, based on their
mathematical simplicity ( see, e. g., Greenshield’s linear model), few of these are consistent with
observed data; most of these models predict a gradual decrease in speed ( linear, in the case of
Greenshield’s model) as traffic density increases. In fact, our experience suggests that speed
remains relatively constant until we reach a point where there are sufficient numbers of vehicles
to cause interference in the traffic stream, resulting in the need or desire among drivers to change
lanes, accelerate and brake. At this point, we know that things can quickly deteriorate to “ stop-and-
go” conditions, i. e., congestion, with a precipitous drop in speed. That is, what we expect to
see in the way of a relationship between speed and density is something like that shown in Figure
4 below.
12
" Expected" Speed - Density Relationship
0
10
20
30
40
50
60
70
80
0 20 40 60 80 100 120 140 160
Density ( veh/ mi/ lane)
Speed ( mph)
f S
c k j k
Figure 4. Expected Speed- Density Relationship
This figure depicts a speed– density relationship in which speed remains relatively constant at a
value equal to the free- flow speed until we reach capacity, and then speed decreases somewhat
unstably from that point to stop- and- go conditions. The corresponding flow– density picture
suggests that an underlying theoretical model of the form shown in Figure 5 below ( in red)
would give results that closely approximate conditions observed in the field. Such a theoretical
model would have the attractive feature of being ( piecewise) linear ( but not smooth, i. e., not
having continuous derivatives). Unlike the Greenshield formulation, the linearity here would be
in the flow– density relationship, rather than in the speed– density relationship.
13
Field Data: All Lanes Flow - Density Relationship
0
2000
4000
6000
8000
10000
12000
0 100 200 300 400 500 600 700
Density ( veh/ mi)
Flow ( veh/ hr)
Stable Flow Unstable Flow
Figure 5. Field Speed- Density Data
Such a model was first proposed by Gordon Newell ( of UC Berkeley). Known as the
“ triangular” flow – density relationship, it has the mathematical form:
;
1 ;
f c
c
c j c
j c
S k k k
q q k k k k k
k k
( 2.1)
Sinceq k x x q k, the equations above imply the following speed – density relationship
for the “ triangular” flow – density relationship:
;
1 ;
1
f c
f j
j c
j
c
S k k
x kS kk k k k
k
( 2.2)
How closely does the “ triangular” flow model replicate field conditions? Below, in Figures 6- 11,
we superimpose the model results for 80 mph, 2,300 veh/ hr/ lane f c S q and
211 veh/ mi/ lane. j k ( Ordinarily, we would use a formal statistical analysis, such as “ least
squares regression” to find the best fit, but here we simply pick some values that seem to fit the
data.)
14
Field Data: Lane 2 Speed - Density Relationship
0
20
40
60
80
100
120
0 50 100 150 200 250
Density ( veh/ mi/ lane)
Speed ( mph)
" Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane
Figure 6. Single Lane Speed- Density Field Data Correspondence with Model
Field Data: Lane 2 Flow - Density Relationship
0
500
1000
1500
2000
2500
3000
3500
4000
0 50 100 150 200 250
Density ( veh/ mi/ lane)
Flow ( veh/ hr)
" Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane
Figure 7. Single Lane Flow- Density Field Data Correspondence with Model
15
Field Data: Lane 2 Flow - Speed Relationship
0
500
1000
1500
2000
2500
3000
3500
4000
0 20 40 60 80 100 120
Speed ( mph)
Flow ( veh/ hr)
" Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane
Figure 8. Single Lane Flow- Speed Correspondence with Model
Field Data: All Lanes Speed - Density Relationship
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700 800 900
Density ( veh/ mi)
Speed ( mph)
" Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane
Figure 9. All Lanes Speed- Density Correspondence with Model
16
Field Data: All Lanes Flow - Density Relationship
0
2000
4000
6000
8000
10000
12000
0 100 200 300 400 500 600 700 800 900
Density ( veh/ mi)
Flow ( veh/ hr)
" Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane
Figure 10. All Lanes Flow- Density Correspondence with Model
Field Data: All Lanes Flow - Speed Relationship
0
2000
4000
6000
8000
10000
12000
0 20 40 60 80 100 120
Speed ( mph)
Flow ( veh/ hr)
" Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane
Figure 11. All Lanes Flow- Speed Correspondence with Model
The typical goal for efficient operations is to design a ramp control strategy that processes the
maximum number of vehicles, while maintaining uncongested, or “ high- speed,” conditions. In
work conducted herein, we first “ fit” the triangular flow model to loop data for each section of
the freeway in our corridor. Then, using the calibrated speed- flow- density models, we specify
17
freeway delay in terms of the mainline volumes ( determined from loop stations at the entry
boundary to the corridor) and the controlled discharge from the entry ramps within the corridor.
2.4 Optimal corridor control formulation
The procedures outlined in Sections 2.1, 2.2, and 2.3 above specify the total delay components in
the corridor network— intersection delay, ramp delay, and freeway delay— in terms of a set of
control variables ( gap settings, maximum green settings, minimum green settings, and ramp
meter headway settings) that can be dynamically adjusted in response to detector inputs and
known controller responses. Nominally, these adjustments would be guided by achieving some
system optimal condition, e. g., minimization of total system delay, and achieved through solving
the accompanying nonlinear optimization problem. For practical application, it is important to
recognize that, in most cases, the arterial and freeway/ ramp subsystems reside under different
jurisdictional control. ( For example, in the corridor used as the test network, the
arterial/ intersection components are under control of the City of Irvine ( COI), while the
freeway/ ramp components are under the control of Caltrans District 12.) We thus specify the
system objective as a multi- objective minimization function— minimization of freeway/ ramp
delay and minimization of arterial signal delay— and develop solutions for optimal control that
specify the efficient frontier; i. e., the set of non- dominated control options. In this way, we not
only preserve the autonomy of the individual operating agencies, but also are able to present a set
of global solutions that translate directly into the recommended set of options for use in
CARTESIUS applications.
2.5 Path to deployment
The ultimate goal of this project is to set the stage for deploying a prototype of the optimal
corridor control system in a real- world setting for evaluation and testing. It is primarily because
of this overriding goal that we have specified the adaptive control procedures solely in terms of
those parameters common to existing signal control devices ( e. g., Type 170, Type 2070, and
NEMA controllers), and utilize only those data provided by inductance loop detectors. As a
result, upon successful completion of the adaptive control protocol, its deployment in the field is
restricted only by the ability to communicate parameter value updates to the field devices at
regular intervals.
To facilitate deployment, our development work is conducted on a corridor network for which
we have at least limited authority to conduct tests involving closed- loop control. On the arterial,
we have installed a system of Type 2070 controllers at all signalized intersections that operate
independently from the local COI system. Work is currently underway to place management of
these controllers under CTNet, the latest version of which supports serial and TCP/ IP
communications; a secondary system based on state- of- the- art Siemens ACTRA Central Traffic
Control System with custom- designed Input Acquisition Software is in place as a backup, should
the CTNet configuration prove problematic. Software has been developed, and laboratory tested,
that permits real- time adaptive control of Caltrans District 12 ramp meters in the study area. We
have established real- time communication with these control devices and also receive real- time
raw data streams from loop detectors within the study area.
18
The scope of the current effort includes the development of the corridor adaptive control model
and its testing and evaluation in a simulation environment. Prior to testing the complete model,
separate tests were conducted to evaluate the intersection control model on: 1) an isolated
intersection, and 2) a network of intersections along an arterial. The complete model is then
tested and evaluated on the Alton Parkway/ I- 405 corridor network. Although actual deployment
is beyond the scope of the current effort, pending the results of the evaluation of the simulated
network, it is envisioned that the adaptive control system can be incorporated as a service within
the CARTESIUS deployment under CTNet ( in separate, complementary PATH/ Caltrans
projects).
2.6 Testing and evaluating the proposed control models
In order to test and evaluate the proposed control models, the optimal control formulation has
been developed as an API in Paramics. The test network has been drawn for a subsection of the
so- called “ Irvine Triangle” Paramics network ( Figure 12) that has been extensively coded and
calibrated as part of the Caltrans ATMS Testbed program.
Figure 12. Irvine Triangle Network
In testing the optimal control model, we simulate a variety of conditions on the freeway and
arterial subsystems that cover the range of demand from peak to non- peak, incident to non-incident,
conditions. The results of these experiments are evaluated against full- actuated
operation ( these models have already been coded as API functions within Paramics). In the first
phase of the evaluation, we are interested only in the performance of the arterial subsystem,
rather than in the combined performance of the freeway- arterial system— this latter aspect of the
study is addressed in subsequent testing.
19
3. Theoretical Development of the Intersection Control Model
3.1 Conceptualization
Consider the dual- ring actuated controller shown in Figure13 below:
Barrier
1 2 3 4
5 6 7 8
2
4
6
8
1
3
5
7
Figure 13. Dual- ring Controller Phasing Diagram
Depending on the values of controller parameters and the traffic arrival pattern, at most six
distinct “ stages” will be realized; e. g.,
Barrier
3 4
7 8
1 2
5 6
Figure 14. Dual- ring Controller Stages
For the i th phase, designate the phase split for the j th cycle by j
i G , and duration of red phase by
j
i R . Then, the complete breakdown of any particular cycle j for phase i can be represented as
follows:
20
Figure 15. Phase State
In Figure 15:
1
2
Cycle length of cycle
Start- up lost time associated with phase during cycle
Clearance lost time associated with phase during cycle
Effective red time associated with phas
j
i
j
i
j
i
j
ei
C j
l i j
l i j
R
e during cycle
Effective green time associated j with phase during cycle
ei
i j
G i j
Here we assume that the lost times 1i l and 2i l , and thus the total lost time i L , for phase i are
constant through all cycles, and the effective green, j
ei G , is equal to the actual displayed green.
Designating the mean arrival flow rate for phase i during cycle j by j
i
and the constant mean
saturation flow rate for phase i through all cycles by i S , the pattern of arrivals/ departures for any
particular phase i is as shown below:
21
Gap- out case 1 Max- out case 1
Gap- out case 2 Max- out case 2
Gap- out case 3 Max- out case 3
Figure 16. Pattern of Arrivals/ Departures
In Figure 16:
min
max
Minimum green time associated with phase during cycle
Maximum green time associated with phase during cycle
Queue service time associated with phase during cycle
j
i
j
i
j
qi
G i j
G i j
G i j
22
As can be seen, depending on the phase termination mode ( either gap- out or max- out) and the
values of queue service time, there are six cases that describe distinct arrival/ departure patterns.
3.2 Determining vehicle arrival flow rate j
i
The gap- out and max- out situations are considered separately to determine j
i
. In gap- out
control, the green phase terminates when the vehicle gap ( headway) larger than the unit
extension ( gap setting) occurs. Let j
i
denote the gap setting for phase i during cycle j, and j
i Z
the waiting time for the occurrence of the first vehicle gap of at least j
i
. Based on Poisson
arrival process, the associated headway distribution is given by ( ) j exp( j )
i i t t. Denote by
( t) the probability density of the delay in waiting for a gap of at least j
i
. The probability that
the first gap is j
i
is then
0 0
0
j ( )
i H t t dt
( 3.1)
where H( ) is the Heaviside function.
Then, ( t) is given by
0 1 ( t) ( t) ( t) ( 3.2)
where ( t) is the Dirac delta function, and 1 ( t) is the contribution due to having to wait for at
least one vehicle to pass before a gap of at least j
i materializes. The probability that an arbitrary
gap is at least j
i
is given by
0
j ( )
i H t t dt
( 3.3)
Then
1 ( ) j( )
i t Z t ( 3.4)
where j( )
i Z t dt is the probability that a vehicle arrives during the time interval ( t, t dt) and no
gap of at least j
i
has been detected up to that point.
Then
23
( ) 0 ( ) j( )
i t t Zt ( 3.5)
Let
0 0 ( ) ( ) 1
( ) ( ) 1
j
i
j
i
t t Ht
t t Ht
( 3.6)
Then, 0 ( t) dt is the probability that the first gap is in the time interval ( t, t dt) and it is not a
gap of at least j
i
, and ( t) dt is the probability that a succeeding gap is in the time interval
( t, t dt) and is not a gap of at least j
i
. If a vehicle passes at time t and no gap of at least j
i
has materialized, it is either the first vehicle to do so, or the last such event occurred at some time
t and the succeeding gap was not a gap of at least j
i
. These two possibilities are captured
by
0
0
( ) ( ) ( ) ( )
t
j j
i i Z t t Z t d ( 3.7)
Denote by
*
0
f ( s) estf( t) dt
( 3.8)
the Laplace transform of f ( t) . Then, the transformation of the convolution integral above is
*
* 0
*
( ) ( )
1 ()
j
i
Z s s
s
( 3.9)
But,
*
* * 0
0 0 *
( ) ( ) ( )
1 ()
j
i
s Z s s
s
( 3.10)
or,
*
( ) ( ) j j
i i
j j
i i
j
i
j
i
s
s e s
s e
( 3.11)
Let j n
i Z denote the n th moment of j
i Z . Then
24
*
0 0
( ) ( 1) ( )
j n n n n
i n
s
Z t tdt d s
ds
( 3.12)
Then, the expected wait time j
i E Z for the first gap of at least j
i
duration is given by
exp j j 1
j i i j
i j i
i
E Z
( 3.13)
Therefore, the phase split can be expressed by
min min
exp j j 1 exp j j 1
j j i i j j j i i
i i i j i i i i j
i i
G L G L G
( 3.14)
In Eq. ( 3.14), all variables except j
i
are known signal timing parameters obtained from the
expired phase, and thus the vehicle arrival flow rate j
i
can be determined by solving the
nonlinear inverse function 1( j)
i F , i. e.,
j 1( j)
i i F ( 3.15)
where
min
exp 1
0
j j
j j j i i
i i i i j
i
F G L G
In max- out- controlled termination of green, arriving vehicles keep actuating the extension
detector until maximum green limit is reached. Therefore, it is safe to presume that the number
of vehicles arriving from the end of minimum green to the end of phase green is greater than the
minimum vehicle arrivals sufficient to invoke max- out conditions, i. e.,
min
min ( )
j j
j j j i i i
i i i i j
i
G L G G L G
or,
j 1
i j
i
( 3.16)
Specifically, we assume here that j
i
is approximately equal to the mean of 1/ j
i
and i S , i. e.,
j ( 1 )/ 2
i j i
i
S
( 3.17)
25
Also, j
i
can be determined with the known parameters j
i
and i S .
3.3 Determining vehicle departure number ( j)
i N G and spillover spill j
i Q
The number of vehicle departures and any spillover depend on the value of queue service time,
j
qi G . Let j ( 0)
i Q denote the queue remaining at the end of the prior green phase and j
i Q denote
the number of arrivals associated with phase i during the effective red j j
i i R L. Then j
qi G is the
convolution of j ( 0) j
i i Q Q busy periods in a queue with Poisson arrivals with rate j
i
and
constant service time 1/ i i s S . Then, from queuing theory ( e. g., Cox and Smith, 1961, p 55),
0
| 0
1
j j
j j j i i i
qi i i j
i i
Q Q s
E G Q Q
s
( 3.18)
Under the assumption of Poisson arrivals, j
i Q is Poisson distributed with mean j j
i i i R L ,
0
1
j j j j
j i i i i i
qi j
i i
Q R L s
E G
s
or,
0
j j j j
j i i i i
qi j
i i
Q R L
E G
S
( 3.19)
Referring to the arrival/ departure pattern as shown in Figure 16 above, the value of j
qi G may lie
in one of three different ranges:
Case 1:
min j j
qi i G G
or,
min
0
j j j j
i i i i j
j i
i i
Q R L
G
S
or,
26
min
min
0
j j
i i i j
j j i
i i i
SG Q
R L G
( 3.20)
Case 2:
min j j j
i qi i i G G G L
or,
min
0
j j j j
j i i i i j
i j i i
i i
Q R L
G GL
S
or,
min
min
0 ( ) 0
j j j j
i i i j i i i i
j j i j j
i i i i i
SG Q S G L Q
R L G R G
( 3.21)
Case 3:
j j
qi i i G G L
or,
0
j j j j
i i i i j
j i i
i i
Q R L
G L
S
or,
( ) 0
j j
j i i i i
i j j
i i
S G L Q
R G
( 3.22)
Note that in max- out- controlled termination, max j j
i i i G L G .
As can be seen, three consecutive, non- overlapped numerical intervals regarding to the value of
j
i
are illustrated by inequalities ( 3.20), ( 3.21) and ( 3.22), which are also expressed in terms of
known timing parameters. Based on the j
i
determined by Eq. ( 3.15) or Eq. ( 3.17), only one
inequality ( i. e., only one case) is “ true” for the gap- out or max- out situation. Hence, the vehicle
departure number during the phase split, ( j )
i N G , and spillover spill j
i Q can be determined by the
following equations that correspond to the true case.
27
Case 1 or 2:
( j) j j j j
i i qi i i i qi N G SG G L G
or,
( j) j j j j
i i i qi i i i N G S G G L
or,
0
( )
j j j
j j i i i i j j
i i i j i i i
i i
Q R L
N G S G L
S
or,
( j) j 0 j j j
i i i i i N G Q G R ( 3.23)
And,
spill j j ( 0) j j j ( j )
i i i i i i Q Q G R NG
or,
spill j 0
i Q ( 3.24)
Case 3:
( j) j
i i i i N G S G L ( 3.25)
And,
spill j j ( 0) j j j ( j )
i i i i i i Q Q G R NG
Or,
spill j j ( 0) j j j j
i i i i i i i i Q Q G R S G L ( 3.26)
For Eq. ( 3.23) to Eq. ( 3.26), ( j)
i N G and spill j
i Q can be determined with such known parameters
obtained from the expired phase as j ( 0)
i Q , j
i G , j
i R and j
i
.
28
3.4 Determining future vehicle arrival flow rate j 1
i
k m
2
4
6
8
1
3
5
7
3
j
m q
6
j
m q
16 3 6
j j j
k m m q q q
Figure 17. Approach Volumes
at intersection m during cycle j .
( j) ( j)/ j
im im m E q N G C ( 3.27)
where
1 2 3 4
Cj Gj Gj Gj Gj or 5 6 7 8
Cj Gj Gj Gj Gj
Therefore,
( j ) ( j) ( j ) ; 1636,2527,3858,4714
rsk tm um E q E q E q rstu ( 3.28)
where intersection m is on the respective upstream approach to the phase [ 1,6], [ 2,5], [ 3,8], and
[ 4,7] movements at intersection k.
And, the turning fractions for cycle j can be expressed as
; 16,25,38, 47,61,52,83,74
j
j ik
ik j j
ik rk
TF ir
( 3.29)
where j
ik TF denotes the percent of traffic on the approach contributing to phase i that is assigned
to phase i during cycle j.
Next, forecast j 1
ik TF by some form of “ moving average” model; e. g.,
1
1 ; ; 1 o
o o
j j
j n
ik n ik j j j j n
n j j n j j
TF TF
( 3.30)
29
Then, compute j 1, 1, ,8
ik i as
1 1
1 1
, 16, 25,38, 47
, 16, 25,38, 47
j j j
ik ik irk
j j j
rk rk irk
TF q ir
TF q ir
3.5 Determining optimal maximum green 1
max
j
i G
Here, we determine maximum green using Webster’s formulas, i. e.,
1 1 1
max j j 3600 / j ( 0)
i i i i D C Q S ( 3.31)
where
max C Maximum allowable cycle length , say 120 seconds
j1( 0) spill j
i i Q Q ( 3.32)
The term 1
max 3600 / j ( 0)
i C Q represents an approximation of the equivalent apparent arrival
rate ( per hour) due to incorporating the leftover queue from the previous cycle ( i. e., spill j
i Q )
present at onset of green. And, the critical path through each ring is determined by
Left Side Conditions: Critical Path = * *
* * 1 1 1 1
12,56
j j j j
i m im i m
i m D D Max D D
Right Side Conditions: Critical Path = * *
* * 1 1 1 1
34,78
j j j j
r n r n r n
r n D D Max D D
( 3.33)
Employing Webster’s optimal distribution of green, we obtain for the left and right portions of
the ring, Web j 1
left G and Web j 1
right G , respectively,
* *
* * * *
* * * *
* *
* * * *
* * * *
1 1
1
1 1 1 1 max
1 1
1
1 1 1 1 max
j j
Web j i m
left j j j j i m r n
i m r n
j j
Web j r n
right j j j j i m r n
i m r n
D D
G C L L L L
D D D D
D D
G C L L L L
D D D D
( 3.34)
Using the same philosophy, we calculate for the critical movements:
30
*
* *
*
* *
*
* *
*
* *
1
1 1
1 1
1
1 1
1 1
1
1 1
1 1
1
1 1
1 1
j
Web j i Web j
i j j left
i m
j
Web j m Web j
m j j left
i m
j
Web j r Web j
r j j right
r n
j
Web j n Web j
n j j right
r n
D
G G
D D
D
G G
D D
D
G G
D D
D
G G
D D
( 3.35)
And, for the non- critical movements:
1
1 1 * *
1 1
1
1 1 * *
1 1
1
1 1 * *
1 1
1
1
1 1
; 12,56;
; 12,56;
; 34,78;
j
Web j i Web j
i j j left
i m
j
Web j m Web j
m j j left
i m
j
Web j r Web j
r j j right
r n
j
Web j n Web j
n j j right
r n
G D G im im i m
D D
G D G im im i m
D D
G D G r n r n r n
D D
G D G
D D
1; r n 34,78; r n r* n*
( 3.36)
Then, set maximum greens according to Webster’s optimal phase splits
1 1
max j Web j ; 1, ,8
i i G G i
3.6 Determining optimal phase split j 1
i G
A nonlinear optimization problem is formulated to determine optimal phase splits, with the
objective to be minimizing total intersection control delay during the upcoming cycle. The
optimization of phase splits is also called " Critical Intersection Control ( CIC)" and it is
considered a first generation UTCS control strategy; our formulation of this strategy explicitly
incorporates stochastic factors incorporated with optimal control. The delay expression is given
by Darroch ( 1964), which is a generalization of the well- known Webster formulation,
1
1 12 1 1
1 1 1
( ) 2 ( 0) 1 1 ; 1,2, ,8
2( 1 ) 1
j
j i j j j
i j i i j i i j
i i i i i
E W R R Q s i
s s
31
or,
1
1 12 1 1
1 1 1
( ) 2 ( 0) 1 1 ; 1,2, ,8
2( )
j
j i i j j j
i j i i j i j
i i i i i i
E W S R R Q i
S SS
( 3.37)
where j 1
i W is the waiting time per cycle. The optimization problems can be expressed by
8
1
1
j
i
i
Min E W
( 3.38)
Based on the circular dependency relationship in dual- ring structure as shown in Figure 18, the
term j 1
i R can be expressed as
1
1 2 3 4
1 1
2 3 4 1
1 1
3 4 1 2
1 1 1 1
4 1 2 3
1
5 6 7 8
1 1
6 7 8 5
1 1 1
7 8 5 6
1 1 1 1
8 5 6 7
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
( 3.39)
Figure 18. Circular Dependency
32
We note that Eq. ( 3.39) does not contain terms 1
4
G j and 1
8
G j , and thus these two variables
cannot be expressed in the optimization problem. Here, a rolling horizon scheme is applied by
substituting 1
1
R j with 2
1
R j , and 1
5
R j with 2
5
R j , then we have
2 1 1 1
1 2 3 4
1 1
2 3 4 1
1 1
3 4 1 2
1 1 1 1
4 1 2 3
2 1 1 1
5 6 7 8
1 1
6 7 8 5
1 1 1
7 8 5 6
1 1 1 1
8 5 6 7
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
( 3.40)
Then, the optimization problem becomes
1 2
2,3,4,6,7,8 1,5
j j
i r
i r
Min E W E W
( 3.41)
In the expression for j 2
r E W , i. e.,
2
2 22 2 2
2 2 2
( ) 2 ( 0) 1 1 ; 1,5
2( )
j
j r r j j j
r j r r j r j
r r r r r r
E W S R R Q r
S SS
we assume j 2( 0) 0
r Q , and j 2
r
can be estimated by some form of “ moving average” model,
e. g.,
1 1
2
1 1
1 1
; ; 1 o
o o
j j
j n
r n r j j j j n
n j j n j j
( 3.42)
In addition, two constraints are considered in formulating the optimization problem:
( 1) Barrier condition. According to the concept of dual- ring control, the timing period in ring A
should be equal to the timing period in ring B on either side of the barrier, i. e.,
1 1 1 1
1 2 5 6
1 1 1 1
3 4 7 8
j j j j
j j j j
G G G G
G G G G
( 3.43)
( 2) Equilibrium condition. The phase green is expected to be large enough to service all the
vehicles that arrive during the effective red and effective green ( plus initial queue) in order
33
to avoid oversaturation delay, i. e., to terminate the phase by gap- out control and invoke no
vehicle spillover ( refer to gap- out Case 1 and 2); therefore,
1 1
max
1 1
j j
i i i
j j
qi i i
G G L
G G L
( 3.44)
where
1 1 1 1
1
1
0
j j j j
j i i i i
qi j
i i
Q R L
G
S
Therefore, the complete optimization problem is expressed by
1 2
2,3,4,6,7,8 1,5
j j
i r
i r
Min E W E W
subject to
1 1 1 1
1 2 5 6
1 1 1 1
3 4 7 8
1 1 1
max
j j j j
j j j j
j j j
qi i i i i
G G G G
G G G G
G L G G L
3.7 Determining optimal minimum green 1
min
j
i G
Minimum green is set equal to queue service time if queue service time is less than the pre-determined
( i. e., traditional) minimum green, 0
mini G , otherwise, set it equal to the pre- determined
minimum green, i. e.,
1 1 1 0
min min
1 0 1 0
min min min
if
if
j j j
i qi qi i
j j
i i qi i
G G G G
G G G G
or,
1 1 0
min min j min j ,
i qi i G G G ( 3.45)
3.8 Determining optimal passage setting j 1
i
Recall that the optimized phase is expected to be terminated by gap- out control; therefore, the
phase split can be expressed by Eq. ( 3.14), i. e.,
34
1 1
1 1
min 1
exp j j 1
j j i i
i i i j
i
G L G
Then,
1 1 1
1 min
1
ln 1 j j j
j i i i i
i j
i
G G L
( 3.46)
Note here that, the natural logarithm in Eq. ( 3.46) requires 1 1 1
min 1 j j j
i i i i G G L be
greater than zero. According to inequality ( 3.44), i. e.,
j1 j1
qi i i G G L
we have
1 1 1 1
min min
j j j j
qi i i i i G G G L G ( 3.47)
And, according to Eq. ( 3.45), we have
1 1 1 1 1 0
min min
1 0 1 1 1 0
min min min
if
if
j j j j j
qi qi i i i qi i
j j j j
qi i i i i qi i
G G G L G G G
G G G L G G G
or,
1 1 1 0
min min
1 1 1 0
min min
0 if
0 if
j j j
i i i qi i
j j j
i i i qi i
G L G G G
G L G G G
Then,
1 1 1 1 0
min min
1 1 1 1 0
min min
1 1 if
1 1 if
j j j j
i i i i qi i
j j j j
i i i i qi i
G L G G G
G L G G G
( 3.48)
and the requirement imposed by the natural logarithm is satisfied. Substituting inequality ( 3.48)
into Eq. ( 3.46), we have
1 1 0
min
1 1 0
min
0 if
0 if
j j
i qi i
j j
i qi i
G G
G G
( 3.49)
35
4. Theoretical Development of Ramp Control Model
Assume Poisson arrivals at a ramp; i. e.,
( )
!
i t
i
t e
P t
i
( 4.1)
Where the mean arrival rate V and variance 2 are given by
V 2 t ( 4.2)
The corresponding headway distribution is given by
Pr( h t) 1 e t ( 4.3)
Note that if the arrivals are formed by a sum of Poisson arrivals,
( ) ; ; 1, ,
!
mk
k k
i k k
P t m e m t k n
i
( 4.4)
then,
1
( ) ; ;
!
M i n
i k k k
k
P t e M M m m t
i
( 4.5)
Note: The following derivation parallels that of Hokstad ( 1979).
Consider a stationary ramp queue. Let X denote the queue waiting time ( not including the
“ service time” once the vehicle arrives at the ramp meter stop line). Let Y denote the “ service
time,” which is simply the metering headway, or the inverse of the current ramp metering rate,
R M ; initially, we assume that Pr( Y ) F( ) .
Assume that the ramp has a finite storage capacity, R C ( expressed in ft.). Then, it is assumed
that, once the queue length reaches this limit, any further ramp- bound vehicles will be diverted.
This condition is specified by
vehicle will join the ramp queue
vehicle will be diverted and not join the ramp queue
K
X Y
K
( 4.6)
where
R ; Average length of a vehicle
V
V
K C Y L
L
( 4.7)
36
Therefore, X K .
Following Takacs ( 1955), let ( t) denote the waiting time at the ramp at time instant t. Denote
0 ( 0) with distribution function *
0 0 Pr( x) W ( x) . Let Pr( ( t) x) W* ( t, x).
Consider W*( t t, x). The event ( t t) x can occur in the following mutually exclusive,
and exhaustive, ways:
1. During the interval ( t, t t) no event occurs; the probability of this outcome is
1 t o( t). Then , for this outcome, Pr ( t) x t W* ( t, x t) .
2. During the interval ( t, t t) one event occurs— the probability of this outcome is
t o( t)— and the waiting time ( t) y, where 0 y x . For this, we must have
Y x y, which occurs with probability:
*
0
( ) (, )
x
u F x u dW t u ( 4.8)
Then , for this outcome,
*
0
*
0
Pr ( ) ( ) ( ) ( , )
( ) (, ) ( )
x
u
x
u
t x t t o t Fx udW tu
t F x u dW t u o t
( 4.9)
3. During the interval ( t, t t) more than one event occurs; the probability of this outcome is
o( t).
Then,
* * *
0
( , ) 1 (, ) ( ) (, ) ( )
x
u W t t x t W t x t t F x u dW t u o t ( 4.10)
But,
*
W*( t, x t) W*( t, x) W ( t, x) t o( t)
x
( 4.11)
So,
37
*
* * *
0
*
* * *
0
( , ) 1 (, ) ( , ) ( ) (, ) ( )
( , ) ( , ) ( , ) ( ) ( , ) ( )
x
u
x
u
W t t x t W t x W t x t t F x u dW t u o t
x
W t x W t x t t W t x t F x u dW t u o t
x
( 4.12)
Or,
* * *
* *
0
( , ) (, ) (, ) ( , ) ( ) ( , ) ( )
x
u
W t t x W t x W t x W t x F x u dW t u o t
t x
( 4.13)
Taking the limit as t 0 ,
* *
* *
0
( , ) ( , ) ( , ) ( ) ( , )
x
u
W t x W t x W t x F x u dW t u
t x
( 4.14)
Consider stationary solutions, i. e., solutions satisfying
*
W ( t, x) 0 W*( t, x) W( x)
t
( 4.15)
where we define
( ) lim *( , ) Pr( )
Pr( 0) ( 0)
t
W x W t x X x
Q X W
( 4.16)
We note that, from the condition X K , W( K) 1. Then, w( x) dW( x) dx exists for all x 0 ,
and is defined by the integral- differential equation
0
( ) ( ) ( ) ( ) ( )
w x dW x W x x F x u dW u
x
( 4.17)
The probability that the waiting time will be between u and u u is simply dW( u) times the
probability that the vehicle will join the queue; this latter probability is simply the probability
that the service time Y is K u, or F( K u). ( Recall W( K) 1.) Then,
0
( ) ( ) ( )
x
W x F K u dW u ( 4.18)
and
38
0 0
0
( ) ( ) ( ) ( ) ( )
( ) ( ) (); 0
x x
x
w x F K u dW u F x u dW u
F K u F x u dW u x K
( 4.19)
We assume that the “ service times” are independent with cumulative distribution function ( cdf)
F( y) Pr( Y y). In the case where the “ service times” can be assumed to be equal to the
metering headway, 1
R b M ,
1,
( ) Pr( ) ( )
0,
y b
F y Y y H b
y b
( 4.20)
where H( ) is the Heaviside step function.
Let n be an integer satisfying
nb K ( n 1) b ; n 1,2,
( Note: we exclude the case n 0 since it corresponds to the case in which no vehicle is allowed
to enter the system; we also treat the case n 1 separately.) Assume n 2 . Divide the interval
0, K into n 2 subintervals; i. e.,
1
1
: ( 1) ; 0,1,2, , 2
: ( 1)
:
:
k
n
n
n
I x kb x k b k n
I x n b x K b
I xK b x nb
I xnb x K
( 4.21)
Let
( ) ( ) ; , 0,1, , 1
( ) ( ) ; , 0,1, , 1
k k
k k
W x W x x I k n
w x w x x I k n
( 4.22)
Then
0
0 0
( ) ( ) ( ) ( ) ; 0
( ) ( ) ( ) ; 0
x
x x
w x F K u F x u dW u x K
dW u F x u dW u x K
( 4.23)
0 0 w ( x) W ( x) ( 4.24a)
39
2
0 1 1
0 ( 1)
2
0 1 1
0 ( 1)
1 1
( 1) ( 1)
1 1
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( 1) (
b b kb x
k k k
b k b kb
b b x b
k
b k b
kb x x b
k k k
k b kb k b
k k k
w x dW u dW u dW u dW u
dW u dW u dW u
dW u dW u dW u
W kb W k b W x
1 1
1 1 1
1
) ( ) ( ) ( 1)
( ) ( ) ( ) ( ) ; Note: ( ) ( )
( ) ( ) ; 1,2, , 1
k k k
k k k k k k
k k
W kb W x b W k b
W kb W x W kb W x b W kb W kb
W x W x b k n
( 4.24b)
1 2 ( ) ( ) ( ) n n n w x W K b W x b ( 4.24c)
1 1 1 ( ) ( ) ( ) n n n w x W K b W x b ( 4.24d)
0
0
0 0
0
0
( ) ( ) ( ) ( ) ; 0
1 0
( ) ( ) ( ) ( ) ; 0
( ) ; 0
( )
x
x x
x
w x F K u F x u dW u x b
F K u dW u F x u dW u x b
dW u x b
W x
0
0 0 0
w ( x) dW ( x) W ( x) W( x) ce x
dx
But,
0
0 W( 0) Q ce c Q
So,
0 W ( x) Qe x ( 4.25a)
1
1 1 0
( )
1
( ) ( ) ( ) ( )
( ) x b
w x dW x W x W x b
dx
W x Qe
40
1 ( )
1
1
( ) ( )
( ) 1 ( )
x b
x b
dW x W x Qe
dx
W x Qe e x b
2
2 2 1
( )
2
( ) ( ) ( ) ( )
( ) x b 1 b( 2 )
w x dW x W x W x b
dx
W x Qe e x b
2 ( )
2
2
2 2
2
( ) ( ) 1 ( 2 )
( ) 1 ( ) ( 2 )
2
x b b
x b b
dW x W x Qe e x b
dx
W x Qe e x b e x b
In general,
0
( ) ( ) ( ) ; 0,1, , 1
!
k j
x jb j
k
j
W x Qe e x jb k n
j
( 4.25b)
And, from ( 4.24c) and ( 4.24d),
1 2
n( ) ( ) ( )
n n
dW x W K b W x b
dx
1 2
1
0 0
( ) ( ) ( ) ( ) ( 1 )
n n
n n k k
k k
W x x K BW x W K n k b W x n k b
( 4.25c)
1
1 1
n ( ) ( ) ( )
n n
dW x W K b W x b
dx
1
1 1
0
() ( ) ( ) ( ) ( )
n
n n k k
k
W x Q x K bW K b W K n kb W x n kb
( 4.25d)
Observe that, from ( 4.25d),
1 1 ( ) ( ) n n W K Q bW K b ( 4.26)
And, from the condition X K , W( K) 1. So,
1 ( ) 1 n Q bW K b ( 4.27)
Evaluating ( 4.25b) for k n 1, we get
41
1
1
0
1
( )
1
0
( ) ( ) ( )
!
( ) ( ) ( 1)
!
n j
x jb j
n
j
n j
K b j b j
n
j
W x Qe e x jb
j
W K b Qe e K j b
j
( 4.28)
Substituting ( 4.28) into ( 4.27)
1
( )
0
( ) ( 1) 1
!
n j
K b j b j
j
Q b Qe e K j b
j
From which,
1
( )
0
1
1
( )
0
1 ( ) ( 1) 1
!
1 ( ) ( 1)
!
n j
K b j b j
j
n j
K b j b j
j
Q be e K j b
j
Q be e K j b
j
( 4.29)
Recall, from ( 4.16), i. e.,
( ) lim *( , ) Pr( )
Pr( 0) ( 0)
t
W x W t x X x
Q X W
( 4.16)
So, the cumulative distribution function for X, the queue waiting time, under the conditions of
Poisson arrivals with mean arrival rate , and metering headway 1
R b M , and finite storage
capacity, R V C K L b, is given by:
1
1
( )
0
Pr( 0) ( 0) 1 ( ) ( 1)
!
n j
K b j b j
j
x W be e K j b
j
0 Pr( X x) W( x) Qe x ; 0 x b
0
Pr( ) ( ) ( ) ( ) ; 1,2, , 1; ( 1)
!
k j
x jb j
k
j
X x W x Qe e x jb k n kb x k b
j
1 2
1
0 0
Pr( ) ( ) ( ) ( ) ( ) ( 1 ) ;
n n
n n k k
k k
X x W x x K bW x W K n kb W x n kb K b x nb
42
1 1
1
0
Pr( ) ( ) ( ) ( )
( ) ( ) ;
n n
n
k k
k
X x W x Q x K bW K b
W K n k b W x n k b nb x K
Observe that the probability that a random arrival J joins the system is given by
1 Pr( ) Pr Arbitrary arrival enters ramp system ( ) n J WK b ( 4.30)
Or, from ( 4.27),
Pr( J ) 1 Q
b
( 4.31)
Let M denote the number of vehicles queued on the ramp. Then,
Pr( ) Pr( ) ( ) ; 0,1, , 1 m M m X mb Wmb m n ( 4.32a)
1 2
1
0 0
Pr( ) ( ) ( 1) ( ) ( ) ( 1)
n n
n n k k
k k
M n W nb n b K W K b W K n kb W k b
( 4.32b)
The mean queue length is simply
0
( ) Pr( )
n
m
E M M m
( 4.33)
Or,
43
0
0
1
0
1 2 1
1
0 0 0
1 2
1
0 0
( ) 1 Pr( )
1 Pr( )
1 Pr( ) ( )
1 ( 1) ( ) ( ) ( 1) ( )
1 ( 1) ( ) ( ) ( 1) (
n
m
n
m
n
m
m
n n n
n k k m
k k m
n n
n k k m
k k
E M M m
n M m
n M n Wmb
n n b K W K b W K n k b W k b W mb
n n bKW Kb WK nkb W k b W
1
0
1 2 1
1 0
0 0 1
1 2 2
1 1 0
0 0 0
)
1 ( 1) ( ) ( ) ( 1) ( ) ( 0)
1 ( 1) ( ) ( ) ( 1) ( 1) ( 0)
n
m
n n n
n k k m
k k m
n n n
n k k k
k k k
mb
n n bKW Kb WK nkb W k b Wmb W
n n bKW Kb WK nkb W k b W k b W
But,
1 ( 1) ( 1) ; 0,1, , 2 k k W k b W k b k n
So,
1
1 0
0
( ) 1 ( 1) ( ) ( ) ( 0)
n
n k
k
E M n n b K W K b W K n k b W
But, 0 W( 0) Q and, from ( 4.17), 1 ( ) 1 n Q bW K b . So,
1
1 1
0
( ) 1 ( 1) ( ) ( ) 1 ( )
n
n k n
k
EM n n b KW K b W K n kb bW K b
Or,
1
1
0
( ) ( ) ( )
n
n k
k
EM n K nbW K b W K n kb
( 4.34)
Using Little’s formula, see e. g., Kleinrock ( 1975), the expected number of vehicles on the ramp,
( ) q E M , is given by
( ) ( ) ( 1 ) q E M E M Q ( 4.35)
The arrival rate of those vehicles that actually enter the ramp is given by
44
R Pr( J ) ( 4.36)
But, from ( 4.31), i. e.,
Pr( J ) 1 Q
b
( 4.31)
So,
1
R
Q
b
( 4.37)
Little’s formula gives
( ) ( ) q R EM EX ( 4.38)
Then, the mean waiting time can be computed as
1
1
0
( ) ( ) ( ) ( 1 ) ( ) 1
( 1 ) ( 1 )
( ) ( )
1
( 1 )
q
R
n
n k
k
E X E M E M Q b E M b
Q Q
n K nbW K b W K n kb
b
Q
( 4.39)
where
0 W ( x) Qe x
0
( ) ( ) ( ) ; 0,1, , 1
!
k j
x jb j
k
j
W x Qe e x jb k n
j
1
1
( )
0
1 ( ) ( 1)
!
n j
K b j b j
j
Q be e K j b
j
5. Consideration of Freeway Delay
As discussed in Section 2, the flow– density picture presented by actual field data suggests an
underlying theoretical model of the form first proposed by Gordon Newell ( of UC Berkeley),
known as the “ triangular” flow – density relationship, it has the mathematical form:
45
;
1 ;
f c
c
c j c
j c
S k k k
q q k k k k k
k k
( 2.1)
;
1 ;
1
f c
f j
j c
j
c
S k k
x kS kk k k k
k
( 2.2)
Here, we adopt this model to represent the freeway component of the corridor system. A feature
of this representation is that the freeway speed remains relatively constant for densities below the
critical density, c k ; thus, there is no appreciable freeway delay for values c k k . We note also
that
; c
f
k q k k
S
( 5.1)
Downstream of a ramp entry point, provided that densities are restricted to be c k ,
u R ;
c
f
k q M k k
S
( 5.2)
where u q is the mainline flow rate immediately upstream of the ramp. Or,
R ;
u c
f
k k M k k
S
( 5.3)
Enforcing such conditions will result in ( approximately) zero delay to the freeway. In the
optimal control formulation based on minimizing total delay ( as well as any combination of
component delays), this condition places the following constraint on the solution:
R
u c
f
k M k
S
( 5.4)
6. Development of Integrated Control Model
The integrated control of the combined intersection and ramp system can be formulated as a
nonlinear, multi- objective, programming problem. Consider some intersection k that provides
46
access to freeway entry ramp R . Let R
ik denote the proportion of traffic associated with NEMA
phase i at intersection k that contribute flow to ramp R ; specifically, R 0
ik for phases that do
not feed the ramp, R 1
ik for phases that exclusively feed the ramp, and 0 R 1
ik for phases in
which it is optional to feed the ramp. Then, during any particular cycle of operation of length C ,
the ramp arrival rate, , is determined by
NEMA
R j
ik ik
i
q
( 6.1)
where j
ik q are determined via the procedure outlined in Section 3.4 above; i. e., from ( 3.27)
( j) ( j)/ j
ik ik k E q N G C ( 3.27)
where
4
1
j j
k ik
i
C G
Then, from ( 4.39) above and noting that 1
R b M , the mean waiting time can be computed as
1
1
1 1 1
1
0 1
( ) ( ) ( ) ( 1 ) ( ) 1
( 1 ) ( 1 )
( ) ( )
1
( 1 )
q
R
R
n
R n R k R
k
R
E X E M E M Q b E M M
Q Q
n K nM W K M W K n kM
M
Q
( 6.2)
where
0 W ( x) Qe x
1 1
0
( ) ( ) ( ) ; 0,1, , 1
!
R
k j
x j M j
k R
j
W x Qe e x jM k n
j
1 1
1
1
1( ) 1
0
1 ( ) ( 1)
!
R R
K M n j j M j
R R
j
Q Me e K j M
j
And, from ( 4.37) and ( 4.38) above, i. e.,
1 1 R R
Q Q M
b
( 4.37)
47
E( Mq) R E( X) ( 4.38)
the total expected delay due to ramp metering is
2
( ) ( )
( )
R q
R
D EM EX
E X
( 6.3)
Substituting ( 6.2) and ( 4.37),
1 2
1 1 1
1
0 1
( ) ( )
1 1
( 1 )
n
R n R k R
k
R R
n K nM W K M W K n kM
D Q M
Q
( 6.4)
So, the problem of minimizing the ramp delay can be stated as:
1 2
1 1 1
1
0 1
,
( ) ( )
1 1
R ( 1 )
n
R n R k R
k
M R R
n K nM W K M W K n kM
MinD Q M
Q
( 6.5)
subject to:
NEMA
R j
ik ik
i
q
( 6.5a)
0 W ( x) Qe x ( 6.5b)
1
0
( ) ( ) ( ) ; 0,1, , 1
!
k j
x jb j
k R
j
W x Qe e x jM k n
j
( 6.5c)
1 1
1
1
1( ) 1
0
1 ( ) ( 1)
!
R R
K M n j j M j
R R
j
Q Me e K j M
j
( 6.5d)
R
u c
f
k M k
S
( 6.5e)
Min Max
R R R M M M ( 6.5f)
, 0 R M ( 6.5g)
Recall that the problem of minimizing signal delay, S D , is given by ( 3.41) above, i. e.,
48
1 2
2,3,4,6,7,8 1,5
j j
i r
i r
Min E W E W
subject to
1
1 12 1 1
1 1 1
( ) 2 ( 0) 1 1
2( )
j
j i i j j j
i j i i j i j
i i i i i i
E W S R R Q
S SS
2
2 22 2 2
2 2 2
( ) 2 ( 0) 1 1
2( )
j
j r r j j j
r j r r j r j
r r r r r r
E W S R R Q
S SS
2 1 1 1
1 2 3 4
1 1
2 3 4 1
1 1
3 4 1 2
1 1 1 1
4 1 2 3
2 1 1 1
5 6 7 8
1 1
6 7 8 5
1 1 1
7 8 5 6
1 1 1 1
8 5 6 7
1 1 1 1
1 2 5 6
1
3
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
G G G G
G G
1 1 1
4 7 8
1 1 1
max
j j j
j j j
qi i i i i
G G
G L G G L
1 1 1 1
1
1
0
j j j j
j i i i i
qi j
i i
Q R L
G
S
In a multi- objective formulation, the ramp and signal delays ( for intersections feeding freeway
entry ramps) form a two- element set:
1 2
2,3,4,6,7,8 1,5
1 1 1 1 2
, 1 1( ) 0 ( ) 1 1
( 1 )
j j
i r
i r
n
Qn K nMR Wn K MR k Wk K n kMRMR
Q
E W E W
D ( 6.6)
and the multi- objective problem can be stated as:
1 2
2,3,4,6,7,8 1,5
1 1 1 1 2
, 1 1( ) 0 ( ) 1 1
( 1 )
j j
i r
i r
n
Min Q n K nMR Wn K MR k Wk K n kMR MR
Q
E W E W
D
( 6.7)
49
subject to:
1
1 12 1 1
1 1 1
( ) 2 ( 0) 1 1
2( )
j
j i i j j j
i j i i j i j
i i i i i i
E W S R R Q
S SS
( 6.8)
2
2 22 2 2
2 2 2
( ) 2 ( 0) 1 1
2( )
j
j r r j j j
r j r r j r j
r r r r r r
E W S R R Q
S SS
( 6.8b)
2 1 1 1
1 2 3 4
1 1
2 3 4 1
1 1
3 4 1 2
1 1 1 1
4 1 2 3
2 1 1 1
5 6 7 8
1 1
6 7 8 5
1 1 1
7 8 5 6
1 1 1 1
8 5 6 7
1 1 1 1
1 2 5 6
1
3
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j j j j
j
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
R G G G
G G G G
G G
1 1 1
4 7 8
1 1 1
max
j j j
j j j
qi i i i i
G G
G L G G L
( 6.8c)
1 1 1 1
1
1
0
j j j j
j i i i i
qi j
i i
Q R L
G
S
( 6.8d)
NEMA
R j
ik ik
i
q
( 6.8e)
0
W ˆ ( x) Qe x ( 6.8f)
1
0
ˆ ( ) ( ) ( ) ; 0,1, , 1
!
k j
x jb j
k R
j
W x Qe e x jM k n
j
( 6.8g)
1 1
1
1
1( ) 1
0
1 ( ) ( 1)
!
R R
K M n j j M j
R R
j
Q Me e K j M
j
( 6.8h)
R
u c
f
k M k
S
( 6.8i)
Min Max
R R R M M M ( 6.8j)
For the special case in which we wish to minimize total delay, T D , which is simply the sum of
the ramp and signal delay ( for intersections feeding freeway entry ramps), i. e.,
50
DT DS DR ( 6.9)
the problem can be stated as:
1 2
2,3,4,6,7,8 1,5
1 1 1 1 2
1 1( ) 0 ( ) 11
( 1 )
j j
i r
i r
n
Min Q n K nMR Wn K MR k Wk K n kMR MR
Q
E W E W
( 6.10)
subject to the conditions imposed by ( 6.8).
7. Simulation Evaluation
7.1 Simulation model setup
The proposed control strategies are tested and evaluated using a scalable, high- performance
microscopic simulation package, Paramics ( Cameron, G. D. B. and Duncan, G. I. B., 1996).
Paramics has been widely used in the testing of various algorithms and evaluation of various
Intelligent Transportation System ( ITS) strategies because of its powerful Application
Programming Interfaces ( API), through which users can access the core models to customize and
extend many features of the underlying simulation model, without having to deal with the
underlying proprietary source codes. The proposed adaptive control model is implemented as a
Paramics plug- in through API programming. It is noted that, although the theoretical models for
adaptive control are developed under the assumption of Poisson arrivals ( in order to obtain
tractable mathematical results), in the simulation the arrival patterns are determined by the
microsimulation and are, in general, not Poisson ( particularly for peak flow conditions). As a
result, the models themselves may represent only a crude approximation to actual conditions; it
can be expected that, relaxing the assumption of Poisson arrivals ( to the extent that such is
possible) would produce improved results.
51
Figure 19. Test Network
The study network is shown as in Figure 19, which is so- called the “ Irvine Triangle” located in
southern California. A previous study calibrated this network in Paramics for the morning peak
period from 6 to 10 AM ( Chu, L. et al., 2004). This network includes a 6- mile section of freeway
I- 405, a 3- mile section of freeway I- 5, a 3- mile section of freeway SR- 133 and several adjacent
surface streets, including two streets parallel to I- 405 ( i. e. Alton Parkway and Barranca Parkway),
one street parallel to I- 5 ( Irvine Center Drive), and three crossing streets to I- 405 ( i. e. Culver
Drive, Jeffery Road, and Sand Canyon Avenue). A total of thirty- eight signals under free- mode
actuated control are included in this network.
Three traffic demand scenarios are set up to test the proposed control models:
1. Existing demand scenario: this scenario corresponds to the existing traffic condition for
the morning peak period; demands are obtained from the calibrated simulation model
directly ( Chu, et al 2004);
2. Medium demand scenario: demands are equivalent to 75% of the existing demand
scenario;
3. Low demand scenario: demands are equivalent to 50% of the existing demand scenario.
Simulations are performed for a 4 ½ - hour period for each scenario under the baseline control and
the adaptive control, respectively. The baseline control corresponds to the free- mode actuated
intersection control and the traffic- responsive ramp metering control in the existing network. The
first 30 minutes are considered as the warm- up period for vehicles to fill in the network, and only
52
the last four hours of simulation are analyzed. Five simulation runs are conducted per scenario in
order to generate statistically meaningful results. The mean value of simulation results are used
for analysis.
7.2 Evaluation of intersection control model
In the adaptive intersection control model, the maximum allowable cycle length, Cmax, is set
equal to 100 seconds for each signal. The total lost time, L, is 4 seconds for each actuated phase.
The saturation flow rate, S, is equal to 1900 veh/ hr/ lane for each through movement phase, and
1800 veh/ hr/ lane for each left- turn movement phase. And, to avoid some potential problems in
the simulation network, those optimized control parameters that can take on unreasonably small
values are further adjusted based on the following rules:
1. If the minimum green time is extremely short ( e. g., < 4 sec), it is set to be 4 seconds.
2. If the maximum green time is shorter than the minimum green time, it is set equal to the
minimum green.
3. If the unit extension is not greater than 1/ S, which may cause “ early gap- out” right after
the minimum green, it is set equal to 1/ S + 0.1 seconds.
Two groups of performance measures are used for the model evaluation:
1. For isolated intersections: Vehicle Spillover ( VSO), Maximum Queue Length ( MQL)
and Vehicle Travel Delay ( VTD).
2. Overall system performance: Average Travel Time ( ATT), Average Vehicle Speed
( AVS), Vehicle Mileage Traveled ( VMT) and Vehicle Hours Traveled ( VHT).
As an example, a T- intersection is selected to show the performance of the proposed control
model at individual intersections. This intersection corresponds to the junction of Irvine Center
Drive and the off ramp from Southbound I- 405, as shown in Figure 20. Phases 2 and 6 are
assigned to the through movements and operated as min- recall phases, while phase 4 is assigned
to the left- turn movement with no recall function. The extension detectors ( 6' ×6') for through
phases are placed 300 ft upstream from the stop line, and the call and extension detectors ( 5' ×50')
for left- turn phase are placed right behind the stop line. The baseline control parameters for this
signal are shown in table 1.
53
Figure 20. Study Intersection
Table 1. Parameters for the Study Intersection
Phase 2 4 6
Min Green ( sec) 8 5 8
Max Green ( sec) 40 24 40
Unit Extension ( sec) 5.0 2.0 5.0
Yellow and Red ( sec) 4.0 4.0 4.0
Here, we present only the simulation results from scenario 1 to demonstrate the impact of the
proposed control model at this T- intersection. Figure 21 shows the arrival flow profiles for the
three phases during the simulation period. Phases 2 and 4 experience two “ peak” periods—
around 8am and 9am, respectively— and phase 6 experiences a relatively steady and low level of
flow. The profiles under both baseline and adaptive control for each signal phase are very similar
due to the use of the same demands for simulation.
Figure 21. Flow Profile for Each Phase
54
Table 2 lists the performance measurements resulting from the simulation results for each phase.
It can be seen that the vehicle spillover in phase 2 has been decreased by 9, and in phase 4 has
been decreased by 8. A possible reason is phase 6 has relatively low flow rate and thus no
spillover occurs in this phase. Some reduction in maximum queue length has been achieved with
the biggest improvement being 23.5 feet in phase 2. No improvement has been gained in the
maximum queue length for phase 4, but the travel time for this phase has been reduced by 128.8
vehicle seconds. The travel time is also reduced for phases 2 and 6. The overall results show
some improvement for the entire intersection in each measure of performance.
Table 2. Performance of the Intersection Control
VSO
( number)
MQL
( feet)
VTD
( second)
Phase 2
Baseline 29 77.1 683.4
Adaptive 20 53.6 599.3
Improvement 9 23.5 84.1
Phase 4
Baseline 19 60.1 378.4
Adaptive 11 60.1 249.6
Improvement 8 0 128.8
Phase 6
Baseline 0 23.0 379.9
Adaptive 0 19.7 362.2
Improvement 0 3.3 17.77
Overall
Baseline 48 160.2 1441.7
Adaptive 31 133.4 1211.1
Improvement 17 26.8 230.6
The performance of the entire network for all three scenarios is shown in Table 3. It is found that
the network under adaptive control performs better than the baseline free- mode actuated
control— drivers spend less time in the network and travel more distance with improved traveling
speed. It is also found that the performance in scenario 1 is better than that in the other two
scenarios, and scenario 3 has gained the least improvement. One possible reason underlying this
result is that the extremely low- level traffic flow may behave freely in the network without being
affected by the change of control strategies. On the other hand, it can be concluded that, although
the vehicle arrival pattern is assumed to be a Poisson process in the model formulation, the
performance of the signalized network can also be improved by the proposed adaptive control.
55
Table 3. Performance of the Network Control
ATT
( second)
AVS
( mile/ hr)
VMT
( mile)
VHT
( hour)
Scenario 1
Baseline 344.3 43.9 760920.0 17367.6
Adaptive 331.0 45.9 762491.2 16725.5
Improvement (%) 3.86 4.56 0.21 3.70
Scenario 2
Baseline 257.0 59.1 575585.6 9788.2
Adaptive 255.2 59.5 576046.1 9671.8
Improvement (%) 0.86 0.68 0.08 1.19
Scenario 3
Baseline 249.5 60.9 382327.4 6284.3
Adaptive 248.4 61.1 382649.9 6263.4
Improvement (%) 0.48 0.33 0.01 0.33
7.3 Evaluation of ramp control model
In the adaptive ramp control model, the maximum allowable metering rate, rmax, is set equal to
900 veh/ hr/ lane ( i. e., 15 veh/ min/ lane), and minimum allowable metering rate, rmin, is set equal to
240 veh/ hr/ lane ( i. e., 4 veh/ min/ lane). And, the queue/ merge overide operation applies as needed.
Two groups of measure of performance are used for the model evaluation:
1. For isolated ramps: Ramp Vehicle Travel Delay ( RVTD), ( Freeway) Mainline Vehicle
Travel Delay ( MVTD), and Total Vehicle Travel Delay ( TVTD). Note that only the freeway
section into which the ramp merges is considered here.
2. Overall system performance: ATT, AVS, VMT and VHT.
As an example, the ramp that corresponds to the onramp from Southbound Jeffery Road to
Northbound I- 405 ( Figure 22) is selected to demonstrate the performance of the proposed control
model at individual ramps. The onramp has two vehicle travel lanes merging into one that
connects to the four- lane mainline section. The system detectors downstream of the ramp are
placed to collect those data ( e. g., occupancy and flow) that can be used as input to the ramp
control model.
Figure 22. Study Onramp
56
Figure 23. Flow Profile for Onramp and Freeway Section
Figure 23 shows the flow profile for scenario 1 under both baseline and adaptive control for the
onramp and freeway, respectively. The profiles are plotted with smoothed lines based on the
flow data measured at 15- minute intervals.
Table 4 lists the simulation results with the delay measurements. It can be seen that the vehicle
travel delay on ramp is reduced by 976.0 seconds, which means about 70% improvement has
been achieved for the ramp. The vehicle travel delay on freeway mainline is also reduced by 52.5
seconds and totally, the vehicle travel delay has been reduced by 1028.5 seconds.
Table 4. Performance of the Ramp and Mainline Sections
RVTD
( second)
MVTD
( second)
TVTD
( second)
Baseline 1392.3 645.7 2038.0
Adaptive 416.3 593.2 1009.5
Improvement 976.0 52.5 1028.5
Improvement (%) 70.1 8.1 50.0
The performance measures for the entire network for each of the three scenarios are shown in
Table 5. It is found that the network under adaptive control performs better than the baseline
traffic responsive metering control— drivers spend less time in the network and travel greater
distance with improved traveling speed. Similar to the intersection results, the performance in
scenario 1 is better than that in the other two scenarios, and scenario 3 has gained the least
improvement.
.
57
Table 5. Performance of the Network
ATT
( second)
AVS
( mile/ hr)
VMT
( mile)
VHT
( hour)
Scenario 1
Baseline 344.3 43.9 760920.0 17367.6
Adaptive 311.2 48.7 765988.7 15745.9
Improvement (%) 9.61 10.93 0.67 9.34
Scenario 2
Baseline 257.0 59.1 575585.6 9788.2
Adaptive 256.1 59.3 577270.3 9731.4
Improvement (%) 0.35 0.34 0.29 0.58
Scenario 3
Baseline 249.5 60.9 382327.4 6284.3
Adaptive 248.8 61.1 382900.2 6255.1
Improvement (%) 0.28 0.33 0.15 0.46
7.4 Evaluation of combined intersection/ ramp control model
Here, the combined intersection/ ramp control is evaluated using the overall system performance
measures, ATT, AVS, VMT and VHT. Table 6 lists the simulation results for each of the three
scenarios. Again, the network under adaptive intersection control and ramp control performs
better than the baseline actuated intersection control and traffic- responsive metering control—
drivers spend less time in the network and travel more distance with improved traveling speed.
And, the performance in scenario 1 is better than that in the other two scenarios, and scenario 3
has gained the least improvement.
Table 6. Performance of the Network
ATT
( second)
AVS
( mile/ hr)
VMT
( mile)
VHT
( hour)
Scenario 1
Baseline 344.3 43.9 760920.0 17367.6
Adaptive 305.5 49.6 766071.2 15442.5
Improvement (%) 11.27 12.98 0.68 11.08
Scenario 2
Baseline 257.0 59.1 575585.6 9788.2
Adaptive 255.5 59.4 577585.6 9653.9
Improvement (%) 0.58 0.51 0.35 1.37
Scenario 3
Baseline 249.5 60.9 382327.4 6284.3
Adaptive 248.3 61.2 383196.6 6252.1
Improvement (%) 0.48 0.49 0.23 0.51
Figure 24 compares the three control models using overall system performance measures. It is
found that the combined intersection/ ramp control model performs the best in terms of ATT,
AVS, VMT, and VHT. The improvement on VMT is very minor since the simulation period for
each scenario starts from a free flow condition and ends with another free flow condition.
58
Figure 24. Overall System Performance Comparison of the Three Control Models
8. Concluding Remarks
This project introduces three real- time adaptive control strategies, including an intersection
control, ramp control and an integrated control that combines both intersection and ramp control.
The development of these strategies is based on a mathematical representation that describes the
behavior of real- life processes ( traffic flow in corridor networks and actuated controller
operation). Only those parameters commonly found in modern actuated controllers ( e. g., Type
170 and 2070 controllers) are considered in the formulation of the optimal control problem. As a
result, the proposed strategies could be easily implemented with minimal adaptation of existing
field devices and the software that controls their operation.
Microscopic simulation was employed to test and evaluate the performance of the proposed
strategies in a calibrated network. Simulation results indicate that the proposed strategies are able
to increase overall system performance and also the local performance on ramps and
intersections. Prior to testing the complete model, separate tests were conducted to evaluate the
intersection control model on: 1) an isolated intersection, and 2) a network of intersections along
an arterial. The complete model was then tested and evaluated on the Alton Parkway/ I- 405
corridor network in Irvine, California.
In testing the optimal control model, we simulated a variety of conditions on the freeway and
arterial subsystems that cover the range of demand from peak to non- peak, incident to non-incident,
conditions. The results of these experiments were evaluated against full- actuated
operation and found to offer improved performance.
59
The scope of the current effort includes the development of the corridor adaptive control model
and its testing and evaluation only in a simulation environment. Although actual deployment is
beyond the scope of the current effort, we believe that the results of the evaluation of the
simulated network warrant further investigation of incorporation of the adaptive control system
as a service within the planned CARTESIUS deployment under CTNet ( in separate,
complementary PATH/ Caltrans projects).
Such research deployment could relatively easily be conducted on the Alton Parkway/ I- 405
corridor network for which we have at least limited authority to conduct tests involving closed-loop
control. On the arterial, we have installed a system of Type 2070 controllers at all
signalized intersections that operate independently from the local City of Irvine system. We have
established real- time communication with these control devices and also receive real- time raw
data streams from loop detectors within the study area. In addition, software has been developed,
and laboratory tested, that permits real- time adaptive control of the Caltrans District 12 ramp
meters in this corridor.
Future efforts will be made to improve this model by: ( 1) seeking a more sophisticated algorithm
that models the actual traffic flow pattern in signalized network; ( 2) further developing this
model in order for the application in coordinated control systems; ( 3) comparing this model with
other adaptive control strategies; and ( 4) incorporating access/ egress choice model in the
integrated corridor control.
60
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Click tabs to swap between content that is broken into logical sections.
Description
| Rating | |
| Title | Development of an adaptive corridor traffic control model |
| Subject | TE228.A1 P36 no. 2010-13; Transportation corridors--California--Irvine--Management--Mathematical models.; Adaptive control systems--Mathematical models.; Electronic traffic controls--Mathematical models. |
| Description | Performed in cooperation with the California Dept. of Transportation and the Federal Highway Administration.; "March 2010."; Includes bibliographical references (p. 58-59). |
| Creator | Recker, Wilfred W. |
| Publisher | California PATH Program, Institute of Transportation Studies, University of California at Berkeley |
| Contributors | Zheng, Xing.; Chu, Lianyu.; California. Dept. of Transportation.; University of California, Berkeley. Institute of Transportation Studies.; Partners for Advanced Transit and Highways (Calif.) |
| Type | Text |
| Language | eng |
| Relation | Available online.; http://www.path.berkeley.edu/PATH/Publications/PDF/PRR/2010/PRR-2010-13.pdf; http://worldcat.org/oclc/643116260/viewonline |
| Date-Issued | [2010] |
| Format-Extent | 59 p. : charts, maps ; 28 cm. |
| Relation-Is Part Of | California PATH research report, UCB-ITS-PRR-2010-13; California PATH research report ; UCB-ITS-PRR-2010-13. |
| Transcript | ISSN 1055- 1425 March 2010 This work was performed as part of the California PATH Program of the University of California, in cooperation with the State of California Business, Transportation, and Housing Agency, Department of Transportation, and the United States 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 State of California. This report does not constitute a standard, specification, or regulation. Final Report for Task Order 6323 CALIFORNIA PATH PROGRAM INSTITUTE OF TRANSPORTATION STUDIES UNIVERSITY OF CALIFORNIA, BERKELEY Development of an Adaptive Corridor Traffic Control Model UCB- ITS- PRR- 2010- 13 California PATH Research Report Will Recker, Xing Zheng, Lianyu Chu CALIFORNIA PARTNERS FOR ADVANCED TRANSIT AND HIGHWAYS FINAL REPORT Caltrans RTA 65A0208 ( PATH T. O. 6323) Optimal Control for Corridor Networks: A Mathematical Modeling Approach Prepared by: Will Recker, Xing Zheng Institute of Transportation Studies University of California, Irvine Irvine, CA 92697 Lianyu Chu California Center for Innovative Transportation University of California, Berkeley Berkeley, CA 94720 November, 2009 STATE OF CALIFORNIA DEPARTMENT OF TRANSPORTATION TECHNICAL REPORT DOCUMENTATION PAGE TR0003 ( REV. 10/ 98) 1. REPORT NUMBER CA10- 0759 2. GOVERNMENT ASSOCIATION NUMBER 3. RECIPIENT’S CATALOG NUMBER 5. REPORT DATE November 2009 4. TITLE AND SUBTITLE Optimal Control for Corridor Networks: A Mathematical Modeling Approach 6. PERFORMING ORGANIZATION CODE 7. AUTHOR( S) Will Recker, Lianyu Chu, Xing Zheng 8. PERFORMING ORGANIZATION REPORT NO. UCB- ITS- PRR- 2010- 13 10. WORK UNIT NUMBER 193 9. PERFORMING ORGANIZATION NAME AND ADDRESS Institute of Transportation Studies University of California, Irvine Irvine, CA 92697- 3600 11. CONTRACT OR GRANT NUMBER 65A0208 13. TYPE OF REPORT AND PERIOD COVERED Final Report 12. SPONSORING AGENCY AND ADDRESS California Department of Transportation Division of Research and Innovation, MS- 83 1227 O Street; Sacramento CA 95814 14. SPONSORING AGENCY CODE 15. SUPPLEMENTAL NOTES None 16. ABSTRACT This research develops and tests, via microscopic simulation, a real- time adaptive control system for corridor management in the form of three real- time adaptive control strategies: intersection control, ramp control and an integrated control that combines both intersection and ramp control. The development of these strategies is based on a mathematical representation that describes the behavior of traffic flow in corridor networks and actuated controller operation. Only those parameters commonly found in modern actuated controllers ( e. g., Type 170 and 2070 controllers) are considered in the formulation of the optimal control problem. As a result, the proposed strategies easily could be implemented with minimal adaptation of existing field devices and the software that controls their operation. Microscopic simulation was employed to test and evaluate the performance of the proposed strategies in a calibrated network. Simulation results indicate that the proposed strategies are able to increase overall system performance and also the local performance on ramps and intersections. Prior to testing the complete model, separate tests were conducted to evaluate the intersection control model on: 1) an isolated intersection, and 2) a network of intersections along an arterial. The complete model was then tested and evaluated on the Alton Parkway/ I- 405 corridor network in Irvine, California. In testing the optimal control model, we simulated a variety of conditions on the freeway and arterial subsystems that cover the range of demand from peak to non- peak, incident to non- incident, conditions. The results of these experiments were evaluated against full- actuated operation and found to offer improved performance. 17. KEY WORDS Adaptive Traffic Control, Corridor Management, Mathematical Modeling, Optimal Control 18. DISTRIBUTION STATEMENT No restrictions. This document is available to the public through the National Technical Information Service, Springfield, VA 22161 19. SECURITY CLASSIFICATION ( of this report) None 20. NUMBER OF PAGES 59 21. PRICE N/ A Reproduction of completed page authorized DISCLAIMER STATEMENT This document is disseminated in the interest of information exchange. The contents of this report reflect the views of the authors who are responsible for the facts and accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California or the Federal Highway Administration. This publication does not constitute a standard, specification or regulation. This report does not constitute an endorsement by the Department of any product described herein. For individuals with sensory disabilities, this document is available in Braille, large print, audiocassette, or compact disk. To obtain a copy of this document in one of these alternate formats, please contact: the Division of Research and Innovation, MS- 83, California Department of Transportation, P. O. Box 942873, Sacramento, CA 94273- 0001. 1 Table of Contents List of Figures ............................................................................................................................... . 2 List of Tables ............................................................................................................................... .. 3 Executive Summary ....................................................................................................................... 4 1. Introduction ............................................................................................................................ 6 2. Methodological Approach ..................................................................................................... 8 2.1 Intersection control module ............................................................................................. 8 2.2 Ramp meter control model ............................................................................................... 9 2.3 Freeway model ............................................................................................................... 11 2.4 Optimal corridor control formulation ............................................................................ 17 2.5 Path to deployment ........................................................................................................ 17 2.6 Testing and evaluating the proposed control models ..................................................... 18 3. Theoretical Development of the Intersection Control Model .......................................... 19 3.1 Conceptualization .......................................................................................................... 19 3.2 Determining vehicle arrival flow rate j i ...................................................................... 22 3.3 Determining vehicle departure number ( j) i N G and spillover spill j i Q ............................ 25 3.4 Determining future vehicle arrival flow rate j 1 i ......................................................... 28 3.5 Determining optimal maximum green 1 max j i G ................................................................. 29 3.6 Determining optimal phase split j 1 i G ........................................................................... 30 3.7 Determining optimal minimum green 1 min j i G .................................................................. 33 3.8 Determining optimal passage setting j 1 i .................................................................... 33 4. Theoretical Development of Ramp Control Model ........................................................... 35 5. Consideration of Freeway Delay......................................................................................... 44 6. Development of Integrated Control Model........................................................................ 45 7. Simulation Evaluation ......................................................................................................... 50 7.1 Simulation model setup.................................................................................................. 50 7.2 Evaluation of intersection control model ....................................................................... 52 7.3 Evaluation of ramp control model ................................................................................. 55 7.4 Evaluation of combined intersection/ ramp control model ............................................. 57 8. Concluding Remarks ............................................................................................................... 58 References ............................................................................................................................... ..... 60 2 List of Figures Figure 1. Typical Ramp Metering Configuration ........................................................................... 9 Figure 2. Typical Open- loop Ramp Metering Control ................................................................. 10 Figure 3. Typical Closed- loop Ramp Metering Control ............................................................... 10 Figure 4. Expected Speed- Density Relationship .......................................................................... 12 Figure 5. Field Speed- Density Data .............................................................................................. 13 Figure 6. Single Lane Speed- Density Field Data Correspondence with Model ........................... 14 Figure 7. Single Lane Flow- Density Field Data Correspondence with Model ............................. 14 Figure 8. Single Lane Flow- Speed Correspondence with Model ................................................. 15 Figure 9. All Lanes Speed- Density Correspondence with Model ................................................ 15 Figure 10. All Lanes Flow- Density Correspondence with Model ................................................ 16 Figure 11. All Lanes Flow- Speed Correspondence with Model ................................................... 16 Figure 12. Irvine Triangle Network .............................................................................................. 18 Figure 13. Dual- ring Controller Phasing Diagram ....................................................................... 19 Figure 14. Dual- ring Controller Stages ......................................................................................... 19 Figure 15. Phase State ................................................................................................................... 20 Figure 16. Pattern of Arrivals/ Departures ..................................................................................... 21 Figure 17. Approach Volumes ...................................................................................................... 28 Figure 18. Circular Dependency ................................................................................................... 31 Figure 19. Test Network ............................................................................................................... 51 Figure 20. Study Intersection ........................................................................................................ 53 Figure 21. Flow Profile for Each Phase ........................................................................................ 53 Figure 22. Study Onramp .............................................................................................................. 55 Figure 23. Flow Profile for Onramp and Freeway Section ........................................................... 56 Figure 24. Overall System Performance Comparision of the Three Control Models .................. 58 3 List of Tables Table 1. Parameters for the Study Intersection ............................................................................. 53 Table 2. Performance of the Intersection Control ......................................................................... 54 Table 3. Performance of the Network Control .............................................................................. 55 Table 4. Performance of the Ramp and Mainline Sections .......................................................... 56 Table 5. Performance of the Network ........................................................................................... 57 Table 6. Performance of the Network ........................................................................................... 57 Abstract This research develops and tests, via microscopic simulation, a real- time adaptive control system for corridor management in the form of three real- time adaptive control strategies: intersection control, ramp control and an integrated control that combines both intersection and ramp control. The development of these strategies is based on a mathematical representation that describes the behavior of traffic flow in corridor networks and actuated controller operation. Only those parameters commonly found in modern actuated controllers ( e. g., Type 170 and 2070 controllers) are considered in the formulation of the optimal control problem. As a result, the proposed strategies easily could be implemented with minimal adaptation of existing field devices and the software that controls their operation. Microscopic simulation was employed to test and evaluate the performance of the proposed strategies in a calibrated network. Simulation results indicate that the proposed strategies are able to increase overall system performance and also the local performance on ramps and intersections. Prior to testing the complete model, separate tests were conducted to evaluate the intersection control model on: 1) an isolated intersection, and 2) a network of intersections along an arterial. The complete model was then tested and evaluated on the Alton Parkway/ I- 405 corridor network in Irvine, California. In testing the optimal control model, we simulated a variety of conditions on the freeway and arterial subsystems that cover the range of demand from peak to non- peak, incident to non- incident, conditions. The results of these experiments were evaluated against full-actuated operation and found to offer improved performance. Key Words: Adaptive Traffic Control, Corridor Management, Mathematical Modeling, Optimal Control 4 Executive Summary This project developed and tested, via microscopic simulation, a real- time adaptive control system for corridor management. Although the focus of the development is on signal controllers designed for operation on arterial street networks, the formulation of the adaptive control strategy explicitly includes interaction with freeway ramp control devices, which are also designed to react adaptively to both the onramp flow, as determined by the operation of adjacent intersection signal controllers, and the traffic state on the mainline freeway. The resulting control strategy is based on a mathematical representation that describes the behavior of real- life processes ( traffic flow in corridor networks and actuated controller operation). In formulating the optimal control problem, we have restricted our attention to control of only those parameters commonly found in modern actuated controllers ( e. g., Type 170 and 2070 controllers). By doing this, we hope to ensure that the procedures developed herein can be implemented with minimal adaptation of existing field devices and the software that controls their operation. In the methodological approach taken, we assume that the traffic arrival pattern can be represented as a queue with Poisson arrivals, and from queuing theory we first develop estimates of both the effective green time ( equal to actual displayed green interval), and the vehicle arrival flow, departure number and spillovers for the expired signal phase based on the known controller parameter settings. Similarly, we estimate upstream contributions to the target intersections from known parameters at the upstream intersections and readouts from the corresponding signal displays. Dynamic turning fractions at the target intersection, which cannot be known a priori, are estimated based on a moving average model. Maximum green settings provide constraints for the decision of optimal phase splits, which are determined by solving a non- linear optimization problem with the objective to be minimizing total intersection control delay per cycle. The expression used for delay is a generalization of the well- known Webster formulation. These optimized phase splits are used to determine optimal phase minimum green and passage settings. The outputs of the adaptive control model for intersection signalization are the product of a stochastic optimal control problem that returns dynamic values for the three parameters of actuated controllers— phase minimum green parameter ( subject to its absolute minimum based on such other conditions as pedestrian waiting time and start- up lost time), phase passage parameter and phase maximum green parameter— that control its responsiveness to stochastic fluctuations in traffic conditions ( other parameters, e. g., yellow interval, clearance interval, phase sequencing, are determined principally in regard to safety and geometric considerations); contrasted to current controller operation, in which these parameters are static/ preset, in our formulation they are dynamically set in response to estimates of demand. Three real- time adaptive control strategies: an intersection control, ramp control and an integrated control that combines both intersection and ramp control are proposed. Microscopic simulation was employed to test and evaluate the performance of the proposed strategies in a calibrated network. Prior to testing the complete model, separate tests were conducted to evaluate the intersection control model on: 1) an isolated intersection, and 2) a network of intersections along an arterial. The complete model was then tested and evaluated on the Alton Parkway/ I- 405 corridor network in Irvine, California. In testing the optimal control model, we 5 simulated a variety of conditions on the freeway and arterial subsystems that cover the range of demand from peak to non- peak, incident to non- incident, conditions. The results of these experiments were evaluated against full- actuated operation and found to offer improved performance. Simulation results indicate that the proposed strategies are able to increase overall system performance and also the local performance on ramps and intersections. 6 1. Introduction The objective of this project is to develop and test, via microscopic simulation, a real- time adaptive control system for corridor management. Although the focus of the development is on signal controllers designed for operation on arterial street networks, the formulation of the adaptive control strategy explicitly includes interaction with freeway ramp control devices, which are also designed to react adaptively to both the onramp flow, as determined by the operation of adjacent intersection signal controllers, and the traffic state on the mainline freeway. The proposed control strategy is based on a mathematical representation that describes the behavior of real- life processes ( traffic flow in corridor networks and actuated controller operation). In formulating the optimal control problem, we have restricted our attention to control of only those parameters commonly found in modern actuated controllers ( e. g., Type 170 and 2070 controllers). By doing this, we hope to ensure that the procedures developed herein can be implemented with minimal adaptation of existing field devices and the software that controls their operation. A typical advantage of an adaptive signal controller is that, in the case of intersection control, the cycle length, phase splits, and even the phase sequence, may vary from cycle to cycle, in a manner that satisfies the demands of the current traffic pattern. To some extent, actuated controllers are themselves “ adaptive” in the sense that they vary these same outcomes, but do so subject to a set of predefined, fixed, parameters that do not “ adapt” to current conditions. For the functionality of truly adaptive controllers, a set of on- line optimized phasing and timing parameters are needed. Existing adaptive controls, such as SCOOT ( Robertson and Bretherton, 1991), make incremental adjustments to the current signal plan for the next cycle, in response to the changing traffic demands. In another real- time network control, SCATS ( Lowrie, 1992; Sims, 1979), the local-level intersection controller decides its timing parameters on the basis of the degree of saturation, and then incrementally adjusts to varying traffic conditions. The major drawback of these systems is that they are not proactive and therefore, cannot accommodate significant transients effectively. RHODESTM, a real- time traffic- adaptive signal control system developed at the University of Arizona, uses a traffic flow arrivals algorithm – PREDICT ( Head, 1995) – to improve effectiveness when calculating online phase timings. In the PREDICT algorithm, detector information on approaches of every upstream intersection, together with the traffic state ( arrival and queues), and control plan for the upstream signals are used to predict future traffic volume. It assumes that all surrounding upstream intersections have fixed- time signalized planning, an assumption that is violated in virtually every modern system. In none of these previous systems do the embedded traffic flow prediction models fully utilize available detector information and control features. Consequently, their applicability is confined only to particular factors, and thus restricted in achieving comprehensively good performance. For any signalized intersection, at least three kinds of information— vehicle actuated detector information, signal timing plan and current signal phase information— can be exploited to infer a relatively rich body of information that can be used in adapting the operation of the signal controller to current, or expected, conditions. Here, we develop a traffic flow prediction model 7 based on the actuated phase control strategy and other features, such as phase minimum green parameter, phase passage parameter and phase maximum green parameter, together with related detector information gleaned from actuated- signalized upstream intersections to estimate the future arrivals at downstream intersections. To better utilize all available information, our traffic flow prediction model is divided into an approach volume prediction and the corresponding turning proportion estimation. Based on the time of actuation in the upstream detector of neighboring intersections, together with current signal state and control tactics of the neighboring intersections, the arrival pattern of vehicles is predicted. Then by using the exit/ entry passage detector cycle/ phase counts in the neighboring intersections, the turning percentage for each movement is estimated. As a result, the model can utilize instantaneous information that is currently available but not used, and thus assist fine- tuning intersection performance without any additional hardware investment. The development and adoption of adaptive control procedures for signalized intersections have been hampered by two fundamental impediments to their successful implementation— those that are theoretically sound invariably have been specified in terms of parameters and control options that simply are not within the lexicon of control devices and typically involve complex mixed-integer- programming formulations that do not lend themselves to real- time solution, and those that do manipulate parameters employed in modern actuated control devices are based on highly simplified approximations and simplifications to both control response and traffic measurement. Consistent modeling of traffic signal operations inevitably includes some sort of conditional piece- wise functions in the mathematical representation. For example, such a representation is the basis of the dispersion- and- store model where the inflow to a link is dispersed and is subsequently stored at its end if the signal at the adjacent intersection is “ Red,” or the similar store- and- forward model where the inflow is assumed to travel at a constant travel time, a general relationship of the corresponding outflow discharge would be described by a function that is conditional on the signal indication and the prevailing traffic conditions. Specifically, the outflow is equal to zero if the signal is “ Red”, and equal to the minimum of the flow rate of the stored vehicles and the saturation flow rate if “ Green”. Within the context of a mathematical programming problem this function is represented by some sort of constraint( s). Typically, this task has been approached either by considering specific aspects of the process behavior that narrow the applicability of the model and restrict the insight of the findings, or via its questionable manipulation in the solution procedure of the corresponding problem. For example, when designing optimal signal control strategies for surface street networks based on the store- and- forward model, Singh and Tamura ( 1974), D'Ans and Gazis ( 1976), and Papageorgiou ( 1995) assumed that oversaturated conditions prevail. The control variables are the green per cycle ratios given a cycle of fixed duration, so that the outflow discharge is calculated as the product of the saturation flow rate and the green per cycle ratio. In their formulations, traffic signal operation is not explicitly modeled, and the oversaturation assumption restricts the applicability of the control strategy that of a single- ring, 2- phase, fixed cycle controller. As another example, Chang et al. ( 1994) develop signal control strategies for mixed surface street/ freeway networks by manipulating the outflow discharge function based on the values of the current state and the previously determined control variable, with the solution algorithm assigning the minimum of the two arguments to the link outflow. In other cases, the conditional 8 piece- wise function is expressed in the form of minimum or maximum operators; see, e. g., Stephanedes and Chang ( 1993), and Ziliaskopoulos ( 2000). Despite the theoretical consistency of optimal control formulations based on such piece- wise functions, the impracticality of their solution in real- time and their general inconsistency with the operation of existing control devices ( e. g., by specifying control transition commands that cannot be understood by existing controller logic) have rendered their practical implementation virtually impossible. In the approach taken herein, we avoid this pitfall by formulating the optimal control problem for a signalized intersection in terms of parameters ( phase minimum green parameters, phase passage parameters and phase maximum green parameters) featured in any modern actuated controller, based on a theoretically consistent model of stochastic traffic flow. 2. Methodological Approach 2.1 Intersection control module In the approach taken here, we assume that the traffic arrival pattern can be represented as a queue with Poisson arrivals, and from queuing theory ( e. g., Cox and Smith, 1961) first develop estimates of both the effective green time ( equal to actual displayed green interval), and the vehicle arrival flow, departure number and spillovers for the expired signal phase based on the known controller parameter settings. Similarly, we estimate upstream contributions to the target intersections from known parameters at the upstream intersections ( a total of four) and readouts from the corresponding signal displays; depending on the expected travel time from the contributing intersection, these values may be drawn from a completed cycle or from an ongoing cycle of operation that commenced just prior to the forecast period for the target intersection. Dynamic turning fractions at the target intersection, which cannot be known a priori, are estimated based on a moving average model. Based on maximum cycle length restrictions, we set phase maximum green parameters based on Webster’s functions, accounting for any spillover from previous cycles of operation. These maximum green settings provide constraints for the decision of optimal phase splits, which are determined by solving a non- linear optimization problem with the objective to be minimizing total intersection control delay per cycle. The expression for delay is given by Darroch ( 1964), which is a generalization of the well- known Webster formulation. These optimized phase splits are used to determine optimal phase minimum green and passage settings. All these timing parameters will be used for the upcoming control cycle as well as provide signal timing data for further optimizations. As specified, the outputs of the adaptive control model for intersection signalization are the product of a stochastic optimal control problem that returns dynamic values for the three parameters of actuated controllers— phase minimum green parameter ( subject to its absolute minimum based on such other conditions as pedestrian waiting time and start- up lost time), phase passage parameter and phase maximum green parameter— that control its responsiveness to stochastic fluctuations in traffic conditions ( other parameters, e. g., yellow interval, clearance interval, phase sequencing, are determined principally in regard to safety and geometric 9 considerations); contrasted to current controller operation, in which these parameters are static/ preset, in our formulation they are dynamically set in response to estimates of demand. 2.2 Ramp meter control model Based on the procedures described in Section 4, below, real- time approach volumes ( demand) at the entry ramps downstream of the intersections that feed the ramps are estimated. Owing to the proximity of the intersections to the respective ramp meters, the arrival pattern at the point of metering will be determined using platoon dispersion principles. The departure pattern will be determined as an output of the ramp control model, which will have as its control parameter the instantaneous metering headway, subject to certain installation parameters ( e. g., queue override headway, merge queue override headway), and to controller operation protocol. Caltrans Type 170 metering controllers comprise a number of control elements based on inductive loop detector data inputs. A typical freeway configuration is shown in Figure 1. Under current deployment, the headway component of the signal controller uses input from: 1) the upstream detector station, 2) the downstream detector station, 3) the excessive queue detector, and 4) the merge detector station. Basically, the upstream and downstream detectors are used to calculate an appropriate metering headway based on conditions on the mainline freeway, while the queue and merge detectors are used to override the calculated headway based on conditions on the ramp. The actual ramp signal sequencing is determined by input from the demand ( Checkin) detector and the passage ( Checkout) detector. A typical open- loop control operation is shown in Figure 2 below, in which the objective of the control is to keep the total demand downstream at a value that does not exceed the capacity downstream: Ramp Meter US US US Upstream System Detectors DS DS DS Downstream System Detectors M Merge Detector CO Checkout Detector CI Checkin Detector Q Queue Detector qutp () qdtown () qRt A () qRtD () Figure 1. Typical Ramp Metering Configuration 10 1, ( 1) 0 0, ( 1) 0 i i DDeelalayy ( i 1) Count 1, ( ) 0 0, ( ) 0 e i e i e( i) JR JR J( i) u( i 1) O( i 1) 1 J( i) Occupancy JR JR 1, ( 1) 0 0, ( 1) 0 i i DDeelalayy ( i 1) Count 1, ( ) 0 0, ( ) 0 e i e i e( i) JR JR J( i) u( i 1) O( i 1) 1 J( i) Occupancy JR JR Input Signal Input Sig nal Upstream System Detector Downstream System Detector 0 3600 T ( ) c down q up q ( ) c down up e q) q q 3600 ( ) 1 Override Figure 2. Typical Open- loop Ramp Metering Control In this controller, the reference input is the downstream capacity, ( ) c down q , and the metering headway is computed as the headway corresponding to a ramp flow rate that would lead to the total downstream demand being less than or equal to capacity. Note that in this open loop design, the downstream detectors are not used; only the upstream flow rate ( which is external to the control system) is utilized. An example of a simple closed- loop control system is shown in Figure 3 in which the objective is to maintain the downstream speed at a certain prescribed level, REF x . 1, ( 1) 0 0, ( 1) 0 i i DDeelalayy ( i 1) Count 1, ( ) 0 0, ( ) 0 e i e i e( i) JR JR J( i) u( i 1) O( i 1) 1 J( i) Occupancy JR JR 1, ( 1) 0 0, ( 1) 0 i i DDeelalayy ( i 1) Count 1, ( ) 0 0, ( ) 0 e i e i e( i) JR JR J( i) u( i 1) O( i 1) 1 J( i) Occupancy JR JR Input Signal Input Signal Upstream System Detector Downstream System Detector REF x REF down e x x Override 0 3600 T down q 5280 100( ) D L L down k ( ) 1 down x P K ( ) P REF down K x x Figure 3. Typical Closed- loop Ramp Metering Control 11 In this controller, the count and occupancy from the downstream detector stations are used to compute an estimate of the downstream speed, which is then compared to the input reference speed; a proportional control is then used to calculate the ramp metering headway. In this application the upstream system detectors are not used. In neither of these typical installations is the system- wide performance an explicit consideration in the setting of parameters under which ramp meter controllers operate. In the work presented here, we formulate the ramp control element of our real- time adaptive control corridor model under assumptions of stochastic queuing, with demand input determined from the output of the associated intersection discharge model, and with output determined in accordance with minimizing the delay to the combined corridor system, comprised of: intersection delay, ramp delay, and freeway delay. 2.3 Freeway model It is well- known that there is an inherent relationship among the speed ( and, correspondingly, delay), flow, and density of traffic on a freeway ( often referred to as the “ fundamental diagram of traffic flow”). Less well- known is the exact form of this relationship. For analytical formulations, such as ours, it is nonetheless necessary to impose a mathematically tractable relationship. Although a number of such relationships have been proposed, based on their mathematical simplicity ( see, e. g., Greenshield’s linear model), few of these are consistent with observed data; most of these models predict a gradual decrease in speed ( linear, in the case of Greenshield’s model) as traffic density increases. In fact, our experience suggests that speed remains relatively constant until we reach a point where there are sufficient numbers of vehicles to cause interference in the traffic stream, resulting in the need or desire among drivers to change lanes, accelerate and brake. At this point, we know that things can quickly deteriorate to “ stop-and- go” conditions, i. e., congestion, with a precipitous drop in speed. That is, what we expect to see in the way of a relationship between speed and density is something like that shown in Figure 4 below. 12 " Expected" Speed - Density Relationship 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 120 140 160 Density ( veh/ mi/ lane) Speed ( mph) f S c k j k Figure 4. Expected Speed- Density Relationship This figure depicts a speed– density relationship in which speed remains relatively constant at a value equal to the free- flow speed until we reach capacity, and then speed decreases somewhat unstably from that point to stop- and- go conditions. The corresponding flow– density picture suggests that an underlying theoretical model of the form shown in Figure 5 below ( in red) would give results that closely approximate conditions observed in the field. Such a theoretical model would have the attractive feature of being ( piecewise) linear ( but not smooth, i. e., not having continuous derivatives). Unlike the Greenshield formulation, the linearity here would be in the flow– density relationship, rather than in the speed– density relationship. 13 Field Data: All Lanes Flow - Density Relationship 0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 700 Density ( veh/ mi) Flow ( veh/ hr) Stable Flow Unstable Flow Figure 5. Field Speed- Density Data Such a model was first proposed by Gordon Newell ( of UC Berkeley). Known as the “ triangular” flow – density relationship, it has the mathematical form: ; 1 ; f c c c j c j c S k k k q q k k k k k k k ( 2.1) Sinceq k x x q k, the equations above imply the following speed – density relationship for the “ triangular” flow – density relationship: ; 1 ; 1 f c f j j c j c S k k x kS kk k k k k ( 2.2) How closely does the “ triangular” flow model replicate field conditions? Below, in Figures 6- 11, we superimpose the model results for 80 mph, 2,300 veh/ hr/ lane f c S q and 211 veh/ mi/ lane. j k ( Ordinarily, we would use a formal statistical analysis, such as “ least squares regression” to find the best fit, but here we simply pick some values that seem to fit the data.) 14 Field Data: Lane 2 Speed - Density Relationship 0 20 40 60 80 100 120 0 50 100 150 200 250 Density ( veh/ mi/ lane) Speed ( mph) " Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane Figure 6. Single Lane Speed- Density Field Data Correspondence with Model Field Data: Lane 2 Flow - Density Relationship 0 500 1000 1500 2000 2500 3000 3500 4000 0 50 100 150 200 250 Density ( veh/ mi/ lane) Flow ( veh/ hr) " Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane Figure 7. Single Lane Flow- Density Field Data Correspondence with Model 15 Field Data: Lane 2 Flow - Speed Relationship 0 500 1000 1500 2000 2500 3000 3500 4000 0 20 40 60 80 100 120 Speed ( mph) Flow ( veh/ hr) " Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane Figure 8. Single Lane Flow- Speed Correspondence with Model Field Data: All Lanes Speed - Density Relationship 0 20 40 60 80 100 120 0 100 200 300 400 500 600 700 800 900 Density ( veh/ mi) Speed ( mph) " Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane Figure 9. All Lanes Speed- Density Correspondence with Model 16 Field Data: All Lanes Flow - Density Relationship 0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 700 800 900 Density ( veh/ mi) Flow ( veh/ hr) " Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane Figure 10. All Lanes Flow- Density Correspondence with Model Field Data: All Lanes Flow - Speed Relationship 0 2000 4000 6000 8000 10000 12000 0 20 40 60 80 100 120 Speed ( mph) Flow ( veh/ hr) " Triangular Model" with S f = 80 mph, k j = 211 veh/ mi/ lane Figure 11. All Lanes Flow- Speed Correspondence with Model The typical goal for efficient operations is to design a ramp control strategy that processes the maximum number of vehicles, while maintaining uncongested, or “ high- speed,” conditions. In work conducted herein, we first “ fit” the triangular flow model to loop data for each section of the freeway in our corridor. Then, using the calibrated speed- flow- density models, we specify 17 freeway delay in terms of the mainline volumes ( determined from loop stations at the entry boundary to the corridor) and the controlled discharge from the entry ramps within the corridor. 2.4 Optimal corridor control formulation The procedures outlined in Sections 2.1, 2.2, and 2.3 above specify the total delay components in the corridor network— intersection delay, ramp delay, and freeway delay— in terms of a set of control variables ( gap settings, maximum green settings, minimum green settings, and ramp meter headway settings) that can be dynamically adjusted in response to detector inputs and known controller responses. Nominally, these adjustments would be guided by achieving some system optimal condition, e. g., minimization of total system delay, and achieved through solving the accompanying nonlinear optimization problem. For practical application, it is important to recognize that, in most cases, the arterial and freeway/ ramp subsystems reside under different jurisdictional control. ( For example, in the corridor used as the test network, the arterial/ intersection components are under control of the City of Irvine ( COI), while the freeway/ ramp components are under the control of Caltrans District 12.) We thus specify the system objective as a multi- objective minimization function— minimization of freeway/ ramp delay and minimization of arterial signal delay— and develop solutions for optimal control that specify the efficient frontier; i. e., the set of non- dominated control options. In this way, we not only preserve the autonomy of the individual operating agencies, but also are able to present a set of global solutions that translate directly into the recommended set of options for use in CARTESIUS applications. 2.5 Path to deployment The ultimate goal of this project is to set the stage for deploying a prototype of the optimal corridor control system in a real- world setting for evaluation and testing. It is primarily because of this overriding goal that we have specified the adaptive control procedures solely in terms of those parameters common to existing signal control devices ( e. g., Type 170, Type 2070, and NEMA controllers), and utilize only those data provided by inductance loop detectors. As a result, upon successful completion of the adaptive control protocol, its deployment in the field is restricted only by the ability to communicate parameter value updates to the field devices at regular intervals. To facilitate deployment, our development work is conducted on a corridor network for which we have at least limited authority to conduct tests involving closed- loop control. On the arterial, we have installed a system of Type 2070 controllers at all signalized intersections that operate independently from the local COI system. Work is currently underway to place management of these controllers under CTNet, the latest version of which supports serial and TCP/ IP communications; a secondary system based on state- of- the- art Siemens ACTRA Central Traffic Control System with custom- designed Input Acquisition Software is in place as a backup, should the CTNet configuration prove problematic. Software has been developed, and laboratory tested, that permits real- time adaptive control of Caltrans District 12 ramp meters in the study area. We have established real- time communication with these control devices and also receive real- time raw data streams from loop detectors within the study area. 18 The scope of the current effort includes the development of the corridor adaptive control model and its testing and evaluation in a simulation environment. Prior to testing the complete model, separate tests were conducted to evaluate the intersection control model on: 1) an isolated intersection, and 2) a network of intersections along an arterial. The complete model is then tested and evaluated on the Alton Parkway/ I- 405 corridor network. Although actual deployment is beyond the scope of the current effort, pending the results of the evaluation of the simulated network, it is envisioned that the adaptive control system can be incorporated as a service within the CARTESIUS deployment under CTNet ( in separate, complementary PATH/ Caltrans projects). 2.6 Testing and evaluating the proposed control models In order to test and evaluate the proposed control models, the optimal control formulation has been developed as an API in Paramics. The test network has been drawn for a subsection of the so- called “ Irvine Triangle” Paramics network ( Figure 12) that has been extensively coded and calibrated as part of the Caltrans ATMS Testbed program. Figure 12. Irvine Triangle Network In testing the optimal control model, we simulate a variety of conditions on the freeway and arterial subsystems that cover the range of demand from peak to non- peak, incident to non-incident, conditions. The results of these experiments are evaluated against full- actuated operation ( these models have already been coded as API functions within Paramics). In the first phase of the evaluation, we are interested only in the performance of the arterial subsystem, rather than in the combined performance of the freeway- arterial system— this latter aspect of the study is addressed in subsequent testing. 19 3. Theoretical Development of the Intersection Control Model 3.1 Conceptualization Consider the dual- ring actuated controller shown in Figure13 below: Barrier 1 2 3 4 5 6 7 8 2 4 6 8 1 3 5 7 Figure 13. Dual- ring Controller Phasing Diagram Depending on the values of controller parameters and the traffic arrival pattern, at most six distinct “ stages” will be realized; e. g., Barrier 3 4 7 8 1 2 5 6 Figure 14. Dual- ring Controller Stages For the i th phase, designate the phase split for the j th cycle by j i G , and duration of red phase by j i R . Then, the complete breakdown of any particular cycle j for phase i can be represented as follows: 20 Figure 15. Phase State In Figure 15: 1 2 Cycle length of cycle Start- up lost time associated with phase during cycle Clearance lost time associated with phase during cycle Effective red time associated with phas j i j i j i j ei C j l i j l i j R e during cycle Effective green time associated j with phase during cycle ei i j G i j Here we assume that the lost times 1i l and 2i l , and thus the total lost time i L , for phase i are constant through all cycles, and the effective green, j ei G , is equal to the actual displayed green. Designating the mean arrival flow rate for phase i during cycle j by j i and the constant mean saturation flow rate for phase i through all cycles by i S , the pattern of arrivals/ departures for any particular phase i is as shown below: 21 Gap- out case 1 Max- out case 1 Gap- out case 2 Max- out case 2 Gap- out case 3 Max- out case 3 Figure 16. Pattern of Arrivals/ Departures In Figure 16: min max Minimum green time associated with phase during cycle Maximum green time associated with phase during cycle Queue service time associated with phase during cycle j i j i j qi G i j G i j G i j 22 As can be seen, depending on the phase termination mode ( either gap- out or max- out) and the values of queue service time, there are six cases that describe distinct arrival/ departure patterns. 3.2 Determining vehicle arrival flow rate j i The gap- out and max- out situations are considered separately to determine j i . In gap- out control, the green phase terminates when the vehicle gap ( headway) larger than the unit extension ( gap setting) occurs. Let j i denote the gap setting for phase i during cycle j, and j i Z the waiting time for the occurrence of the first vehicle gap of at least j i . Based on Poisson arrival process, the associated headway distribution is given by ( ) j exp( j ) i i t t. Denote by ( t) the probability density of the delay in waiting for a gap of at least j i . The probability that the first gap is j i is then 0 0 0 j ( ) i H t t dt ( 3.1) where H( ) is the Heaviside function. Then, ( t) is given by 0 1 ( t) ( t) ( t) ( 3.2) where ( t) is the Dirac delta function, and 1 ( t) is the contribution due to having to wait for at least one vehicle to pass before a gap of at least j i materializes. The probability that an arbitrary gap is at least j i is given by 0 j ( ) i H t t dt ( 3.3) Then 1 ( ) j( ) i t Z t ( 3.4) where j( ) i Z t dt is the probability that a vehicle arrives during the time interval ( t, t dt) and no gap of at least j i has been detected up to that point. Then 23 ( ) 0 ( ) j( ) i t t Zt ( 3.5) Let 0 0 ( ) ( ) 1 ( ) ( ) 1 j i j i t t Ht t t Ht ( 3.6) Then, 0 ( t) dt is the probability that the first gap is in the time interval ( t, t dt) and it is not a gap of at least j i , and ( t) dt is the probability that a succeeding gap is in the time interval ( t, t dt) and is not a gap of at least j i . If a vehicle passes at time t and no gap of at least j i has materialized, it is either the first vehicle to do so, or the last such event occurred at some time t and the succeeding gap was not a gap of at least j i . These two possibilities are captured by 0 0 ( ) ( ) ( ) ( ) t j j i i Z t t Z t d ( 3.7) Denote by * 0 f ( s) estf( t) dt ( 3.8) the Laplace transform of f ( t) . Then, the transformation of the convolution integral above is * * 0 * ( ) ( ) 1 () j i Z s s s ( 3.9) But, * * * 0 0 0 * ( ) ( ) ( ) 1 () j i s Z s s s ( 3.10) or, * ( ) ( ) j j i i j j i i j i j i s s e s s e ( 3.11) Let j n i Z denote the n th moment of j i Z . Then 24 * 0 0 ( ) ( 1) ( ) j n n n n i n s Z t tdt d s ds ( 3.12) Then, the expected wait time j i E Z for the first gap of at least j i duration is given by exp j j 1 j i i j i j i i E Z ( 3.13) Therefore, the phase split can be expressed by min min exp j j 1 exp j j 1 j j i i j j j i i i i i j i i i i j i i G L G L G ( 3.14) In Eq. ( 3.14), all variables except j i are known signal timing parameters obtained from the expired phase, and thus the vehicle arrival flow rate j i can be determined by solving the nonlinear inverse function 1( j) i F , i. e., j 1( j) i i F ( 3.15) where min exp 1 0 j j j j j i i i i i i j i F G L G In max- out- controlled termination of green, arriving vehicles keep actuating the extension detector until maximum green limit is reached. Therefore, it is safe to presume that the number of vehicles arriving from the end of minimum green to the end of phase green is greater than the minimum vehicle arrivals sufficient to invoke max- out conditions, i. e., min min ( ) j j j j j i i i i i i i j i G L G G L G or, j 1 i j i ( 3.16) Specifically, we assume here that j i is approximately equal to the mean of 1/ j i and i S , i. e., j ( 1 )/ 2 i j i i S ( 3.17) 25 Also, j i can be determined with the known parameters j i and i S . 3.3 Determining vehicle departure number ( j) i N G and spillover spill j i Q The number of vehicle departures and any spillover depend on the value of queue service time, j qi G . Let j ( 0) i Q denote the queue remaining at the end of the prior green phase and j i Q denote the number of arrivals associated with phase i during the effective red j j i i R L. Then j qi G is the convolution of j ( 0) j i i Q Q busy periods in a queue with Poisson arrivals with rate j i and constant service time 1/ i i s S . Then, from queuing theory ( e. g., Cox and Smith, 1961, p 55), 0 0 1 j j j j j i i i qi i i j i i Q Q s E G Q Q s ( 3.18) Under the assumption of Poisson arrivals, j i Q is Poisson distributed with mean j j i i i R L , 0 1 j j j j j i i i i i qi j i i Q R L s E G s or, 0 j j j j j i i i i qi j i i Q R L E G S ( 3.19) Referring to the arrival/ departure pattern as shown in Figure 16 above, the value of j qi G may lie in one of three different ranges: Case 1: min j j qi i G G or, min 0 j j j j i i i i j j i i i Q R L G S or, 26 min min 0 j j i i i j j j i i i i SG Q R L G ( 3.20) Case 2: min j j j i qi i i G G G L or, min 0 j j j j j i i i i j i j i i i i Q R L G GL S or, min min 0 ( ) 0 j j j j i i i j i i i i j j i j j i i i i i SG Q S G L Q R L G R G ( 3.21) Case 3: j j qi i i G G L or, 0 j j j j i i i i j j i i i i Q R L G L S or, ( ) 0 j j j i i i i i j j i i S G L Q R G ( 3.22) Note that in max- out- controlled termination, max j j i i i G L G . As can be seen, three consecutive, non- overlapped numerical intervals regarding to the value of j i are illustrated by inequalities ( 3.20), ( 3.21) and ( 3.22), which are also expressed in terms of known timing parameters. Based on the j i determined by Eq. ( 3.15) or Eq. ( 3.17), only one inequality ( i. e., only one case) is “ true” for the gap- out or max- out situation. Hence, the vehicle departure number during the phase split, ( j ) i N G , and spillover spill j i Q can be determined by the following equations that correspond to the true case. 27 Case 1 or 2: ( j) j j j j i i qi i i i qi N G SG G L G or, ( j) j j j j i i i qi i i i N G S G G L or, 0 ( ) j j j j j i i i i j j i i i j i i i i i Q R L N G S G L S or, ( j) j 0 j j j i i i i i N G Q G R ( 3.23) And, spill j j ( 0) j j j ( j ) i i i i i i Q Q G R NG or, spill j 0 i Q ( 3.24) Case 3: ( j) j i i i i N G S G L ( 3.25) And, spill j j ( 0) j j j ( j ) i i i i i i Q Q G R NG Or, spill j j ( 0) j j j j i i i i i i i i Q Q G R S G L ( 3.26) For Eq. ( 3.23) to Eq. ( 3.26), ( j) i N G and spill j i Q can be determined with such known parameters obtained from the expired phase as j ( 0) i Q , j i G , j i R and j i . 28 3.4 Determining future vehicle arrival flow rate j 1 i k m 2 4 6 8 1 3 5 7 3 j m q 6 j m q 16 3 6 j j j k m m q q q Figure 17. Approach Volumes at intersection m during cycle j . ( j) ( j)/ j im im m E q N G C ( 3.27) where 1 2 3 4 Cj Gj Gj Gj Gj or 5 6 7 8 Cj Gj Gj Gj Gj Therefore, ( j ) ( j) ( j ) ; 1636,2527,3858,4714 rsk tm um E q E q E q rstu ( 3.28) where intersection m is on the respective upstream approach to the phase [ 1,6], [ 2,5], [ 3,8], and [ 4,7] movements at intersection k. And, the turning fractions for cycle j can be expressed as ; 16,25,38, 47,61,52,83,74 j j ik ik j j ik rk TF ir ( 3.29) where j ik TF denotes the percent of traffic on the approach contributing to phase i that is assigned to phase i during cycle j. Next, forecast j 1 ik TF by some form of “ moving average” model; e. g., 1 1 ; ; 1 o o o j j j n ik n ik j j j j n n j j n j j TF TF ( 3.30) 29 Then, compute j 1, 1, ,8 ik i as 1 1 1 1 , 16, 25,38, 47 , 16, 25,38, 47 j j j ik ik irk j j j rk rk irk TF q ir TF q ir 3.5 Determining optimal maximum green 1 max j i G Here, we determine maximum green using Webster’s formulas, i. e., 1 1 1 max j j 3600 / j ( 0) i i i i D C Q S ( 3.31) where max C Maximum allowable cycle length , say 120 seconds j1( 0) spill j i i Q Q ( 3.32) The term 1 max 3600 / j ( 0) i C Q represents an approximation of the equivalent apparent arrival rate ( per hour) due to incorporating the leftover queue from the previous cycle ( i. e., spill j i Q ) present at onset of green. And, the critical path through each ring is determined by Left Side Conditions: Critical Path = * * * * 1 1 1 1 12,56 j j j j i m im i m i m D D Max D D Right Side Conditions: Critical Path = * * * * 1 1 1 1 34,78 j j j j r n r n r n r n D D Max D D ( 3.33) Employing Webster’s optimal distribution of green, we obtain for the left and right portions of the ring, Web j 1 left G and Web j 1 right G , respectively, * * * * * * * * * * * * * * * * * * * * 1 1 1 1 1 1 1 max 1 1 1 1 1 1 1 max j j Web j i m left j j j j i m r n i m r n j j Web j r n right j j j j i m r n i m r n D D G C L L L L D D D D D D G C L L L L D D D D ( 3.34) Using the same philosophy, we calculate for the critical movements: 30 * * * * * * * * * * * * 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j Web j i Web j i j j left i m j Web j m Web j m j j left i m j Web j r Web j r j j right r n j Web j n Web j n j j right r n D G G D D D G G D D D G G D D D G G D D ( 3.35) And, for the non- critical movements: 1 1 1 * * 1 1 1 1 1 * * 1 1 1 1 1 * * 1 1 1 1 1 1 ; 12,56; ; 12,56; ; 34,78; j Web j i Web j i j j left i m j Web j m Web j m j j left i m j Web j r Web j r j j right r n j Web j n Web j n j j right r n G D G im im i m D D G D G im im i m D D G D G r n r n r n D D G D G D D 1; r n 34,78; r n r* n* ( 3.36) Then, set maximum greens according to Webster’s optimal phase splits 1 1 max j Web j ; 1, ,8 i i G G i 3.6 Determining optimal phase split j 1 i G A nonlinear optimization problem is formulated to determine optimal phase splits, with the objective to be minimizing total intersection control delay during the upcoming cycle. The optimization of phase splits is also called " Critical Intersection Control ( CIC)" and it is considered a first generation UTCS control strategy; our formulation of this strategy explicitly incorporates stochastic factors incorporated with optimal control. The delay expression is given by Darroch ( 1964), which is a generalization of the well- known Webster formulation, 1 1 12 1 1 1 1 1 ( ) 2 ( 0) 1 1 ; 1,2, ,8 2( 1 ) 1 j j i j j j i j i i j i i j i i i i i E W R R Q s i s s 31 or, 1 1 12 1 1 1 1 1 ( ) 2 ( 0) 1 1 ; 1,2, ,8 2( ) j j i i j j j i j i i j i j i i i i i i E W S R R Q i S SS ( 3.37) where j 1 i W is the waiting time per cycle. The optimization problems can be expressed by 8 1 1 j i i Min E W ( 3.38) Based on the circular dependency relationship in dual- ring structure as shown in Figure 18, the term j 1 i R can be expressed as 1 1 2 3 4 1 1 2 3 4 1 1 1 3 4 1 2 1 1 1 1 4 1 2 3 1 5 6 7 8 1 1 6 7 8 5 1 1 1 7 8 5 6 1 1 1 1 8 5 6 7 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j R G G G R G G G R G G G R G G G R G G G R G G G R G G G R G G G ( 3.39) Figure 18. Circular Dependency 32 We note that Eq. ( 3.39) does not contain terms 1 4 G j and 1 8 G j , and thus these two variables cannot be expressed in the optimization problem. Here, a rolling horizon scheme is applied by substituting 1 1 R j with 2 1 R j , and 1 5 R j with 2 5 R j , then we have 2 1 1 1 1 2 3 4 1 1 2 3 4 1 1 1 3 4 1 2 1 1 1 1 4 1 2 3 2 1 1 1 5 6 7 8 1 1 6 7 8 5 1 1 1 7 8 5 6 1 1 1 1 8 5 6 7 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j R G G G R G G G R G G G R G G G R G G G R G G G R G G G R G G G ( 3.40) Then, the optimization problem becomes 1 2 2,3,4,6,7,8 1,5 j j i r i r Min E W E W ( 3.41) In the expression for j 2 r E W , i. e., 2 2 22 2 2 2 2 2 ( ) 2 ( 0) 1 1 ; 1,5 2( ) j j r r j j j r j r r j r j r r r r r r E W S R R Q r S SS we assume j 2( 0) 0 r Q , and j 2 r can be estimated by some form of “ moving average” model, e. g., 1 1 2 1 1 1 1 ; ; 1 o o o j j j n r n r j j j j n n j j n j j ( 3.42) In addition, two constraints are considered in formulating the optimization problem: ( 1) Barrier condition. According to the concept of dual- ring control, the timing period in ring A should be equal to the timing period in ring B on either side of the barrier, i. e., 1 1 1 1 1 2 5 6 1 1 1 1 3 4 7 8 j j j j j j j j G G G G G G G G ( 3.43) ( 2) Equilibrium condition. The phase green is expected to be large enough to service all the vehicles that arrive during the effective red and effective green ( plus initial queue) in order 33 to avoid oversaturation delay, i. e., to terminate the phase by gap- out control and invoke no vehicle spillover ( refer to gap- out Case 1 and 2); therefore, 1 1 max 1 1 j j i i i j j qi i i G G L G G L ( 3.44) where 1 1 1 1 1 1 0 j j j j j i i i i qi j i i Q R L G S Therefore, the complete optimization problem is expressed by 1 2 2,3,4,6,7,8 1,5 j j i r i r Min E W E W subject to 1 1 1 1 1 2 5 6 1 1 1 1 3 4 7 8 1 1 1 max j j j j j j j j j j j qi i i i i G G G G G G G G G L G G L 3.7 Determining optimal minimum green 1 min j i G Minimum green is set equal to queue service time if queue service time is less than the pre-determined ( i. e., traditional) minimum green, 0 mini G , otherwise, set it equal to the pre- determined minimum green, i. e., 1 1 1 0 min min 1 0 1 0 min min min if if j j j i qi qi i j j i i qi i G G G G G G G G or, 1 1 0 min min j min j , i qi i G G G ( 3.45) 3.8 Determining optimal passage setting j 1 i Recall that the optimized phase is expected to be terminated by gap- out control; therefore, the phase split can be expressed by Eq. ( 3.14), i. e., 34 1 1 1 1 min 1 exp j j 1 j j i i i i i j i G L G Then, 1 1 1 1 min 1 ln 1 j j j j i i i i i j i G G L ( 3.46) Note here that, the natural logarithm in Eq. ( 3.46) requires 1 1 1 min 1 j j j i i i i G G L be greater than zero. According to inequality ( 3.44), i. e., j1 j1 qi i i G G L we have 1 1 1 1 min min j j j j qi i i i i G G G L G ( 3.47) And, according to Eq. ( 3.45), we have 1 1 1 1 1 0 min min 1 0 1 1 1 0 min min min if if j j j j j qi qi i i i qi i j j j j qi i i i i qi i G G G L G G G G G G L G G G or, 1 1 1 0 min min 1 1 1 0 min min 0 if 0 if j j j i i i qi i j j j i i i qi i G L G G G G L G G G Then, 1 1 1 1 0 min min 1 1 1 1 0 min min 1 1 if 1 1 if j j j j i i i i qi i j j j j i i i i qi i G L G G G G L G G G ( 3.48) and the requirement imposed by the natural logarithm is satisfied. Substituting inequality ( 3.48) into Eq. ( 3.46), we have 1 1 0 min 1 1 0 min 0 if 0 if j j i qi i j j i qi i G G G G ( 3.49) 35 4. Theoretical Development of Ramp Control Model Assume Poisson arrivals at a ramp; i. e., ( ) ! i t i t e P t i ( 4.1) Where the mean arrival rate V and variance 2 are given by V 2 t ( 4.2) The corresponding headway distribution is given by Pr( h t) 1 e t ( 4.3) Note that if the arrivals are formed by a sum of Poisson arrivals, ( ) ; ; 1, , ! mk k k i k k P t m e m t k n i ( 4.4) then, 1 ( ) ; ; ! M i n i k k k k P t e M M m m t i ( 4.5) Note: The following derivation parallels that of Hokstad ( 1979). Consider a stationary ramp queue. Let X denote the queue waiting time ( not including the “ service time” once the vehicle arrives at the ramp meter stop line). Let Y denote the “ service time,” which is simply the metering headway, or the inverse of the current ramp metering rate, R M ; initially, we assume that Pr( Y ) F( ) . Assume that the ramp has a finite storage capacity, R C ( expressed in ft.). Then, it is assumed that, once the queue length reaches this limit, any further ramp- bound vehicles will be diverted. This condition is specified by vehicle will join the ramp queue vehicle will be diverted and not join the ramp queue K X Y K ( 4.6) where R ; Average length of a vehicle V V K C Y L L ( 4.7) 36 Therefore, X K . Following Takacs ( 1955), let ( t) denote the waiting time at the ramp at time instant t. Denote 0 ( 0) with distribution function * 0 0 Pr( x) W ( x) . Let Pr( ( t) x) W* ( t, x). Consider W*( t t, x). The event ( t t) x can occur in the following mutually exclusive, and exhaustive, ways: 1. During the interval ( t, t t) no event occurs; the probability of this outcome is 1 t o( t). Then , for this outcome, Pr ( t) x t W* ( t, x t) . 2. During the interval ( t, t t) one event occurs— the probability of this outcome is t o( t)— and the waiting time ( t) y, where 0 y x . For this, we must have Y x y, which occurs with probability: * 0 ( ) (, ) x u F x u dW t u ( 4.8) Then , for this outcome, * 0 * 0 Pr ( ) ( ) ( ) ( , ) ( ) (, ) ( ) x u x u t x t t o t Fx udW tu t F x u dW t u o t ( 4.9) 3. During the interval ( t, t t) more than one event occurs; the probability of this outcome is o( t). Then, * * * 0 ( , ) 1 (, ) ( ) (, ) ( ) x u W t t x t W t x t t F x u dW t u o t ( 4.10) But, * W*( t, x t) W*( t, x) W ( t, x) t o( t) x ( 4.11) So, 37 * * * * 0 * * * * 0 ( , ) 1 (, ) ( , ) ( ) (, ) ( ) ( , ) ( , ) ( , ) ( ) ( , ) ( ) x u x u W t t x t W t x W t x t t F x u dW t u o t x W t x W t x t t W t x t F x u dW t u o t x ( 4.12) Or, * * * * * 0 ( , ) (, ) (, ) ( , ) ( ) ( , ) ( ) x u W t t x W t x W t x W t x F x u dW t u o t t x ( 4.13) Taking the limit as t 0 , * * * * 0 ( , ) ( , ) ( , ) ( ) ( , ) x u W t x W t x W t x F x u dW t u t x ( 4.14) Consider stationary solutions, i. e., solutions satisfying * W ( t, x) 0 W*( t, x) W( x) t ( 4.15) where we define ( ) lim *( , ) Pr( ) Pr( 0) ( 0) t W x W t x X x Q X W ( 4.16) We note that, from the condition X K , W( K) 1. Then, w( x) dW( x) dx exists for all x 0 , and is defined by the integral- differential equation 0 ( ) ( ) ( ) ( ) ( ) w x dW x W x x F x u dW u x ( 4.17) The probability that the waiting time will be between u and u u is simply dW( u) times the probability that the vehicle will join the queue; this latter probability is simply the probability that the service time Y is K u, or F( K u). ( Recall W( K) 1.) Then, 0 ( ) ( ) ( ) x W x F K u dW u ( 4.18) and 38 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) (); 0 x x x w x F K u dW u F x u dW u F K u F x u dW u x K ( 4.19) We assume that the “ service times” are independent with cumulative distribution function ( cdf) F( y) Pr( Y y). In the case where the “ service times” can be assumed to be equal to the metering headway, 1 R b M , 1, ( ) Pr( ) ( ) 0, y b F y Y y H b y b ( 4.20) where H( ) is the Heaviside step function. Let n be an integer satisfying nb K ( n 1) b ; n 1,2, ( Note: we exclude the case n 0 since it corresponds to the case in which no vehicle is allowed to enter the system; we also treat the case n 1 separately.) Assume n 2 . Divide the interval 0, K into n 2 subintervals; i. e., 1 1 : ( 1) ; 0,1,2, , 2 : ( 1) : : k n n n I x kb x k b k n I x n b x K b I xK b x nb I xnb x K ( 4.21) Let ( ) ( ) ; , 0,1, , 1 ( ) ( ) ; , 0,1, , 1 k k k k W x W x x I k n w x w x x I k n ( 4.22) Then 0 0 0 ( ) ( ) ( ) ( ) ; 0 ( ) ( ) ( ) ; 0 x x x w x F K u F x u dW u x K dW u F x u dW u x K ( 4.23) 0 0 w ( x) W ( x) ( 4.24a) 39 2 0 1 1 0 ( 1) 2 0 1 1 0 ( 1) 1 1 ( 1) ( 1) 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1) ( b b kb x k k k b k b kb b b x b k b k b kb x x b k k k k b kb k b k k k w x dW u dW u dW u dW u dW u dW u dW u dW u dW u dW u W kb W k b W x 1 1 1 1 1 1 ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ; Note: ( ) ( ) ( ) ( ) ; 1,2, , 1 k k k k k k k k k k k W kb W x b W k b W kb W x W kb W x b W kb W kb W x W x b k n ( 4.24b) 1 2 ( ) ( ) ( ) n n n w x W K b W x b ( 4.24c) 1 1 1 ( ) ( ) ( ) n n n w x W K b W x b ( 4.24d) 0 0 0 0 0 0 ( ) ( ) ( ) ( ) ; 0 1 0 ( ) ( ) ( ) ( ) ; 0 ( ) ; 0 ( ) x x x x w x F K u F x u dW u x b F K u dW u F x u dW u x b dW u x b W x 0 0 0 0 w ( x) dW ( x) W ( x) W( x) ce x dx But, 0 0 W( 0) Q ce c Q So, 0 W ( x) Qe x ( 4.25a) 1 1 1 0 ( ) 1 ( ) ( ) ( ) ( ) ( ) x b w x dW x W x W x b dx W x Qe 40 1 ( ) 1 1 ( ) ( ) ( ) 1 ( ) x b x b dW x W x Qe dx W x Qe e x b 2 2 2 1 ( ) 2 ( ) ( ) ( ) ( ) ( ) x b 1 b( 2 ) w x dW x W x W x b dx W x Qe e x b 2 ( ) 2 2 2 2 2 ( ) ( ) 1 ( 2 ) ( ) 1 ( ) ( 2 ) 2 x b b x b b dW x W x Qe e x b dx W x Qe e x b e x b In general, 0 ( ) ( ) ( ) ; 0,1, , 1 ! k j x jb j k j W x Qe e x jb k n j ( 4.25b) And, from ( 4.24c) and ( 4.24d), 1 2 n( ) ( ) ( ) n n dW x W K b W x b dx 1 2 1 0 0 ( ) ( ) ( ) ( ) ( 1 ) n n n n k k k k W x x K BW x W K n k b W x n k b ( 4.25c) 1 1 1 n ( ) ( ) ( ) n n dW x W K b W x b dx 1 1 1 0 () ( ) ( ) ( ) ( ) n n n k k k W x Q x K bW K b W K n kb W x n kb ( 4.25d) Observe that, from ( 4.25d), 1 1 ( ) ( ) n n W K Q bW K b ( 4.26) And, from the condition X K , W( K) 1. So, 1 ( ) 1 n Q bW K b ( 4.27) Evaluating ( 4.25b) for k n 1, we get 41 1 1 0 1 ( ) 1 0 ( ) ( ) ( ) ! ( ) ( ) ( 1) ! n j x jb j n j n j K b j b j n j W x Qe e x jb j W K b Qe e K j b j ( 4.28) Substituting ( 4.28) into ( 4.27) 1 ( ) 0 ( ) ( 1) 1 ! n j K b j b j j Q b Qe e K j b j From which, 1 ( ) 0 1 1 ( ) 0 1 ( ) ( 1) 1 ! 1 ( ) ( 1) ! n j K b j b j j n j K b j b j j Q be e K j b j Q be e K j b j ( 4.29) Recall, from ( 4.16), i. e., ( ) lim *( , ) Pr( ) Pr( 0) ( 0) t W x W t x X x Q X W ( 4.16) So, the cumulative distribution function for X, the queue waiting time, under the conditions of Poisson arrivals with mean arrival rate , and metering headway 1 R b M , and finite storage capacity, R V C K L b, is given by: 1 1 ( ) 0 Pr( 0) ( 0) 1 ( ) ( 1) ! n j K b j b j j x W be e K j b j 0 Pr( X x) W( x) Qe x ; 0 x b 0 Pr( ) ( ) ( ) ( ) ; 1,2, , 1; ( 1) ! k j x jb j k j X x W x Qe e x jb k n kb x k b j 1 2 1 0 0 Pr( ) ( ) ( ) ( ) ( ) ( 1 ) ; n n n n k k k k X x W x x K bW x W K n kb W x n kb K b x nb 42 1 1 1 0 Pr( ) ( ) ( ) ( ) ( ) ( ) ; n n n k k k X x W x Q x K bW K b W K n k b W x n k b nb x K Observe that the probability that a random arrival J joins the system is given by 1 Pr( ) Pr Arbitrary arrival enters ramp system ( ) n J WK b ( 4.30) Or, from ( 4.27), Pr( J ) 1 Q b ( 4.31) Let M denote the number of vehicles queued on the ramp. Then, Pr( ) Pr( ) ( ) ; 0,1, , 1 m M m X mb Wmb m n ( 4.32a) 1 2 1 0 0 Pr( ) ( ) ( 1) ( ) ( ) ( 1) n n n n k k k k M n W nb n b K W K b W K n kb W k b ( 4.32b) The mean queue length is simply 0 ( ) Pr( ) n m E M M m ( 4.33) Or, 43 0 0 1 0 1 2 1 1 0 0 0 1 2 1 0 0 ( ) 1 Pr( ) 1 Pr( ) 1 Pr( ) ( ) 1 ( 1) ( ) ( ) ( 1) ( ) 1 ( 1) ( ) ( ) ( 1) ( n m n m n m m n n n n k k m k k m n n n k k m k k E M M m n M m n M n Wmb n n b K W K b W K n k b W k b W mb n n bKW Kb WK nkb W k b W 1 0 1 2 1 1 0 0 0 1 1 2 2 1 1 0 0 0 0 ) 1 ( 1) ( ) ( ) ( 1) ( ) ( 0) 1 ( 1) ( ) ( ) ( 1) ( 1) ( 0) n m n n n n k k m k k m n n n n k k k k k k mb n n bKW Kb WK nkb W k b Wmb W n n bKW Kb WK nkb W k b W k b W But, 1 ( 1) ( 1) ; 0,1, , 2 k k W k b W k b k n So, 1 1 0 0 ( ) 1 ( 1) ( ) ( ) ( 0) n n k k E M n n b K W K b W K n k b W But, 0 W( 0) Q and, from ( 4.17), 1 ( ) 1 n Q bW K b . So, 1 1 1 0 ( ) 1 ( 1) ( ) ( ) 1 ( ) n n k n k EM n n b KW K b W K n kb bW K b Or, 1 1 0 ( ) ( ) ( ) n n k k EM n K nbW K b W K n kb ( 4.34) Using Little’s formula, see e. g., Kleinrock ( 1975), the expected number of vehicles on the ramp, ( ) q E M , is given by ( ) ( ) ( 1 ) q E M E M Q ( 4.35) The arrival rate of those vehicles that actually enter the ramp is given by 44 R Pr( J ) ( 4.36) But, from ( 4.31), i. e., Pr( J ) 1 Q b ( 4.31) So, 1 R Q b ( 4.37) Little’s formula gives ( ) ( ) q R EM EX ( 4.38) Then, the mean waiting time can be computed as 1 1 0 ( ) ( ) ( ) ( 1 ) ( ) 1 ( 1 ) ( 1 ) ( ) ( ) 1 ( 1 ) q R n n k k E X E M E M Q b E M b Q Q n K nbW K b W K n kb b Q ( 4.39) where 0 W ( x) Qe x 0 ( ) ( ) ( ) ; 0,1, , 1 ! k j x jb j k j W x Qe e x jb k n j 1 1 ( ) 0 1 ( ) ( 1) ! n j K b j b j j Q be e K j b j 5. Consideration of Freeway Delay As discussed in Section 2, the flow– density picture presented by actual field data suggests an underlying theoretical model of the form first proposed by Gordon Newell ( of UC Berkeley), known as the “ triangular” flow – density relationship, it has the mathematical form: 45 ; 1 ; f c c c j c j c S k k k q q k k k k k k k ( 2.1) ; 1 ; 1 f c f j j c j c S k k x kS kk k k k k ( 2.2) Here, we adopt this model to represent the freeway component of the corridor system. A feature of this representation is that the freeway speed remains relatively constant for densities below the critical density, c k ; thus, there is no appreciable freeway delay for values c k k . We note also that ; c f k q k k S ( 5.1) Downstream of a ramp entry point, provided that densities are restricted to be c k , u R ; c f k q M k k S ( 5.2) where u q is the mainline flow rate immediately upstream of the ramp. Or, R ; u c f k k M k k S ( 5.3) Enforcing such conditions will result in ( approximately) zero delay to the freeway. In the optimal control formulation based on minimizing total delay ( as well as any combination of component delays), this condition places the following constraint on the solution: R u c f k M k S ( 5.4) 6. Development of Integrated Control Model The integrated control of the combined intersection and ramp system can be formulated as a nonlinear, multi- objective, programming problem. Consider some intersection k that provides 46 access to freeway entry ramp R . Let R ik denote the proportion of traffic associated with NEMA phase i at intersection k that contribute flow to ramp R ; specifically, R 0 ik for phases that do not feed the ramp, R 1 ik for phases that exclusively feed the ramp, and 0 R 1 ik for phases in which it is optional to feed the ramp. Then, during any particular cycle of operation of length C , the ramp arrival rate, , is determined by NEMA R j ik ik i q ( 6.1) where j ik q are determined via the procedure outlined in Section 3.4 above; i. e., from ( 3.27) ( j) ( j)/ j ik ik k E q N G C ( 3.27) where 4 1 j j k ik i C G Then, from ( 4.39) above and noting that 1 R b M , the mean waiting time can be computed as 1 1 1 1 1 1 0 1 ( ) ( ) ( ) ( 1 ) ( ) 1 ( 1 ) ( 1 ) ( ) ( ) 1 ( 1 ) q R R n R n R k R k R E X E M E M Q b E M M Q Q n K nM W K M W K n kM M Q ( 6.2) where 0 W ( x) Qe x 1 1 0 ( ) ( ) ( ) ; 0,1, , 1 ! R k j x j M j k R j W x Qe e x jM k n j 1 1 1 1 1( ) 1 0 1 ( ) ( 1) ! R R K M n j j M j R R j Q Me e K j M j And, from ( 4.37) and ( 4.38) above, i. e., 1 1 R R Q Q M b ( 4.37) 47 E( Mq) R E( X) ( 4.38) the total expected delay due to ramp metering is 2 ( ) ( ) ( ) R q R D EM EX E X ( 6.3) Substituting ( 6.2) and ( 4.37), 1 2 1 1 1 1 0 1 ( ) ( ) 1 1 ( 1 ) n R n R k R k R R n K nM W K M W K n kM D Q M Q ( 6.4) So, the problem of minimizing the ramp delay can be stated as: 1 2 1 1 1 1 0 1 , ( ) ( ) 1 1 R ( 1 ) n R n R k R k M R R n K nM W K M W K n kM MinD Q M Q ( 6.5) subject to: NEMA R j ik ik i q ( 6.5a) 0 W ( x) Qe x ( 6.5b) 1 0 ( ) ( ) ( ) ; 0,1, , 1 ! k j x jb j k R j W x Qe e x jM k n j ( 6.5c) 1 1 1 1 1( ) 1 0 1 ( ) ( 1) ! R R K M n j j M j R R j Q Me e K j M j ( 6.5d) R u c f k M k S ( 6.5e) Min Max R R R M M M ( 6.5f) , 0 R M ( 6.5g) Recall that the problem of minimizing signal delay, S D , is given by ( 3.41) above, i. e., 48 1 2 2,3,4,6,7,8 1,5 j j i r i r Min E W E W subject to 1 1 12 1 1 1 1 1 ( ) 2 ( 0) 1 1 2( ) j j i i j j j i j i i j i j i i i i i i E W S R R Q S SS 2 2 22 2 2 2 2 2 ( ) 2 ( 0) 1 1 2( ) j j r r j j j r j r r j r j r r r r r r E W S R R Q S SS 2 1 1 1 1 2 3 4 1 1 2 3 4 1 1 1 3 4 1 2 1 1 1 1 4 1 2 3 2 1 1 1 5 6 7 8 1 1 6 7 8 5 1 1 1 7 8 5 6 1 1 1 1 8 5 6 7 1 1 1 1 1 2 5 6 1 3 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j R G G G R G G G R G G G R G G G R G G G R G G G R G G G R G G G G G G G G G 1 1 1 4 7 8 1 1 1 max j j j j j j qi i i i i G G G L G G L 1 1 1 1 1 1 0 j j j j j i i i i qi j i i Q R L G S In a multi- objective formulation, the ramp and signal delays ( for intersections feeding freeway entry ramps) form a two- element set: 1 2 2,3,4,6,7,8 1,5 1 1 1 1 2 , 1 1( ) 0 ( ) 1 1 ( 1 ) j j i r i r n Qn K nMR Wn K MR k Wk K n kMRMR Q E W E W D ( 6.6) and the multi- objective problem can be stated as: 1 2 2,3,4,6,7,8 1,5 1 1 1 1 2 , 1 1( ) 0 ( ) 1 1 ( 1 ) j j i r i r n Min Q n K nMR Wn K MR k Wk K n kMR MR Q E W E W D ( 6.7) 49 subject to: 1 1 12 1 1 1 1 1 ( ) 2 ( 0) 1 1 2( ) j j i i j j j i j i i j i j i i i i i i E W S R R Q S SS ( 6.8) 2 2 22 2 2 2 2 2 ( ) 2 ( 0) 1 1 2( ) j j r r j j j r j r r j r j r r r r r r E W S R R Q S SS ( 6.8b) 2 1 1 1 1 2 3 4 1 1 2 3 4 1 1 1 3 4 1 2 1 1 1 1 4 1 2 3 2 1 1 1 5 6 7 8 1 1 6 7 8 5 1 1 1 7 8 5 6 1 1 1 1 8 5 6 7 1 1 1 1 1 2 5 6 1 3 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j R G G G R G G G R G G G R G G G R G G G R G G G R G G G R G G G G G G G G G 1 1 1 4 7 8 1 1 1 max j j j j j j qi i i i i G G G L G G L ( 6.8c) 1 1 1 1 1 1 0 j j j j j i i i i qi j i i Q R L G S ( 6.8d) NEMA R j ik ik i q ( 6.8e) 0 W ˆ ( x) Qe x ( 6.8f) 1 0 ˆ ( ) ( ) ( ) ; 0,1, , 1 ! k j x jb j k R j W x Qe e x jM k n j ( 6.8g) 1 1 1 1 1( ) 1 0 1 ( ) ( 1) ! R R K M n j j M j R R j Q Me e K j M j ( 6.8h) R u c f k M k S ( 6.8i) Min Max R R R M M M ( 6.8j) For the special case in which we wish to minimize total delay, T D , which is simply the sum of the ramp and signal delay ( for intersections feeding freeway entry ramps), i. e., 50 DT DS DR ( 6.9) the problem can be stated as: 1 2 2,3,4,6,7,8 1,5 1 1 1 1 2 1 1( ) 0 ( ) 11 ( 1 ) j j i r i r n Min Q n K nMR Wn K MR k Wk K n kMR MR Q E W E W ( 6.10) subject to the conditions imposed by ( 6.8). 7. Simulation Evaluation 7.1 Simulation model setup The proposed control strategies are tested and evaluated using a scalable, high- performance microscopic simulation package, Paramics ( Cameron, G. D. B. and Duncan, G. I. B., 1996). Paramics has been widely used in the testing of various algorithms and evaluation of various Intelligent Transportation System ( ITS) strategies because of its powerful Application Programming Interfaces ( API), through which users can access the core models to customize and extend many features of the underlying simulation model, without having to deal with the underlying proprietary source codes. The proposed adaptive control model is implemented as a Paramics plug- in through API programming. It is noted that, although the theoretical models for adaptive control are developed under the assumption of Poisson arrivals ( in order to obtain tractable mathematical results), in the simulation the arrival patterns are determined by the microsimulation and are, in general, not Poisson ( particularly for peak flow conditions). As a result, the models themselves may represent only a crude approximation to actual conditions; it can be expected that, relaxing the assumption of Poisson arrivals ( to the extent that such is possible) would produce improved results. 51 Figure 19. Test Network The study network is shown as in Figure 19, which is so- called the “ Irvine Triangle” located in southern California. A previous study calibrated this network in Paramics for the morning peak period from 6 to 10 AM ( Chu, L. et al., 2004). This network includes a 6- mile section of freeway I- 405, a 3- mile section of freeway I- 5, a 3- mile section of freeway SR- 133 and several adjacent surface streets, including two streets parallel to I- 405 ( i. e. Alton Parkway and Barranca Parkway), one street parallel to I- 5 ( Irvine Center Drive), and three crossing streets to I- 405 ( i. e. Culver Drive, Jeffery Road, and Sand Canyon Avenue). A total of thirty- eight signals under free- mode actuated control are included in this network. Three traffic demand scenarios are set up to test the proposed control models: 1. Existing demand scenario: this scenario corresponds to the existing traffic condition for the morning peak period; demands are obtained from the calibrated simulation model directly ( Chu, et al 2004); 2. Medium demand scenario: demands are equivalent to 75% of the existing demand scenario; 3. Low demand scenario: demands are equivalent to 50% of the existing demand scenario. Simulations are performed for a 4 ½ - hour period for each scenario under the baseline control and the adaptive control, respectively. The baseline control corresponds to the free- mode actuated intersection control and the traffic- responsive ramp metering control in the existing network. The first 30 minutes are considered as the warm- up period for vehicles to fill in the network, and only 52 the last four hours of simulation are analyzed. Five simulation runs are conducted per scenario in order to generate statistically meaningful results. The mean value of simulation results are used for analysis. 7.2 Evaluation of intersection control model In the adaptive intersection control model, the maximum allowable cycle length, Cmax, is set equal to 100 seconds for each signal. The total lost time, L, is 4 seconds for each actuated phase. The saturation flow rate, S, is equal to 1900 veh/ hr/ lane for each through movement phase, and 1800 veh/ hr/ lane for each left- turn movement phase. And, to avoid some potential problems in the simulation network, those optimized control parameters that can take on unreasonably small values are further adjusted based on the following rules: 1. If the minimum green time is extremely short ( e. g., < 4 sec), it is set to be 4 seconds. 2. If the maximum green time is shorter than the minimum green time, it is set equal to the minimum green. 3. If the unit extension is not greater than 1/ S, which may cause “ early gap- out” right after the minimum green, it is set equal to 1/ S + 0.1 seconds. Two groups of performance measures are used for the model evaluation: 1. For isolated intersections: Vehicle Spillover ( VSO), Maximum Queue Length ( MQL) and Vehicle Travel Delay ( VTD). 2. Overall system performance: Average Travel Time ( ATT), Average Vehicle Speed ( AVS), Vehicle Mileage Traveled ( VMT) and Vehicle Hours Traveled ( VHT). As an example, a T- intersection is selected to show the performance of the proposed control model at individual intersections. This intersection corresponds to the junction of Irvine Center Drive and the off ramp from Southbound I- 405, as shown in Figure 20. Phases 2 and 6 are assigned to the through movements and operated as min- recall phases, while phase 4 is assigned to the left- turn movement with no recall function. The extension detectors ( 6' ×6') for through phases are placed 300 ft upstream from the stop line, and the call and extension detectors ( 5' ×50') for left- turn phase are placed right behind the stop line. The baseline control parameters for this signal are shown in table 1. 53 Figure 20. Study Intersection Table 1. Parameters for the Study Intersection Phase 2 4 6 Min Green ( sec) 8 5 8 Max Green ( sec) 40 24 40 Unit Extension ( sec) 5.0 2.0 5.0 Yellow and Red ( sec) 4.0 4.0 4.0 Here, we present only the simulation results from scenario 1 to demonstrate the impact of the proposed control model at this T- intersection. Figure 21 shows the arrival flow profiles for the three phases during the simulation period. Phases 2 and 4 experience two “ peak” periods— around 8am and 9am, respectively— and phase 6 experiences a relatively steady and low level of flow. The profiles under both baseline and adaptive control for each signal phase are very similar due to the use of the same demands for simulation. Figure 21. Flow Profile for Each Phase 54 Table 2 lists the performance measurements resulting from the simulation results for each phase. It can be seen that the vehicle spillover in phase 2 has been decreased by 9, and in phase 4 has been decreased by 8. A possible reason is phase 6 has relatively low flow rate and thus no spillover occurs in this phase. Some reduction in maximum queue length has been achieved with the biggest improvement being 23.5 feet in phase 2. No improvement has been gained in the maximum queue length for phase 4, but the travel time for this phase has been reduced by 128.8 vehicle seconds. The travel time is also reduced for phases 2 and 6. The overall results show some improvement for the entire intersection in each measure of performance. Table 2. Performance of the Intersection Control VSO ( number) MQL ( feet) VTD ( second) Phase 2 Baseline 29 77.1 683.4 Adaptive 20 53.6 599.3 Improvement 9 23.5 84.1 Phase 4 Baseline 19 60.1 378.4 Adaptive 11 60.1 249.6 Improvement 8 0 128.8 Phase 6 Baseline 0 23.0 379.9 Adaptive 0 19.7 362.2 Improvement 0 3.3 17.77 Overall Baseline 48 160.2 1441.7 Adaptive 31 133.4 1211.1 Improvement 17 26.8 230.6 The performance of the entire network for all three scenarios is shown in Table 3. It is found that the network under adaptive control performs better than the baseline free- mode actuated control— drivers spend less time in the network and travel more distance with improved traveling speed. It is also found that the performance in scenario 1 is better than that in the other two scenarios, and scenario 3 has gained the least improvement. One possible reason underlying this result is that the extremely low- level traffic flow may behave freely in the network without being affected by the change of control strategies. On the other hand, it can be concluded that, although the vehicle arrival pattern is assumed to be a Poisson process in the model formulation, the performance of the signalized network can also be improved by the proposed adaptive control. 55 Table 3. Performance of the Network Control ATT ( second) AVS ( mile/ hr) VMT ( mile) VHT ( hour) Scenario 1 Baseline 344.3 43.9 760920.0 17367.6 Adaptive 331.0 45.9 762491.2 16725.5 Improvement (%) 3.86 4.56 0.21 3.70 Scenario 2 Baseline 257.0 59.1 575585.6 9788.2 Adaptive 255.2 59.5 576046.1 9671.8 Improvement (%) 0.86 0.68 0.08 1.19 Scenario 3 Baseline 249.5 60.9 382327.4 6284.3 Adaptive 248.4 61.1 382649.9 6263.4 Improvement (%) 0.48 0.33 0.01 0.33 7.3 Evaluation of ramp control model In the adaptive ramp control model, the maximum allowable metering rate, rmax, is set equal to 900 veh/ hr/ lane ( i. e., 15 veh/ min/ lane), and minimum allowable metering rate, rmin, is set equal to 240 veh/ hr/ lane ( i. e., 4 veh/ min/ lane). And, the queue/ merge overide operation applies as needed. Two groups of measure of performance are used for the model evaluation: 1. For isolated ramps: Ramp Vehicle Travel Delay ( RVTD), ( Freeway) Mainline Vehicle Travel Delay ( MVTD), and Total Vehicle Travel Delay ( TVTD). Note that only the freeway section into which the ramp merges is considered here. 2. Overall system performance: ATT, AVS, VMT and VHT. As an example, the ramp that corresponds to the onramp from Southbound Jeffery Road to Northbound I- 405 ( Figure 22) is selected to demonstrate the performance of the proposed control model at individual ramps. The onramp has two vehicle travel lanes merging into one that connects to the four- lane mainline section. The system detectors downstream of the ramp are placed to collect those data ( e. g., occupancy and flow) that can be used as input to the ramp control model. Figure 22. Study Onramp 56 Figure 23. Flow Profile for Onramp and Freeway Section Figure 23 shows the flow profile for scenario 1 under both baseline and adaptive control for the onramp and freeway, respectively. The profiles are plotted with smoothed lines based on the flow data measured at 15- minute intervals. Table 4 lists the simulation results with the delay measurements. It can be seen that the vehicle travel delay on ramp is reduced by 976.0 seconds, which means about 70% improvement has been achieved for the ramp. The vehicle travel delay on freeway mainline is also reduced by 52.5 seconds and totally, the vehicle travel delay has been reduced by 1028.5 seconds. Table 4. Performance of the Ramp and Mainline Sections RVTD ( second) MVTD ( second) TVTD ( second) Baseline 1392.3 645.7 2038.0 Adaptive 416.3 593.2 1009.5 Improvement 976.0 52.5 1028.5 Improvement (%) 70.1 8.1 50.0 The performance measures for the entire network for each of the three scenarios are shown in Table 5. It is found that the network under adaptive control performs better than the baseline traffic responsive metering control— drivers spend less time in the network and travel greater distance with improved traveling speed. Similar to the intersection results, the performance in scenario 1 is better than that in the other two scenarios, and scenario 3 has gained the least improvement. . 57 Table 5. Performance of the Network ATT ( second) AVS ( mile/ hr) VMT ( mile) VHT ( hour) Scenario 1 Baseline 344.3 43.9 760920.0 17367.6 Adaptive 311.2 48.7 765988.7 15745.9 Improvement (%) 9.61 10.93 0.67 9.34 Scenario 2 Baseline 257.0 59.1 575585.6 9788.2 Adaptive 256.1 59.3 577270.3 9731.4 Improvement (%) 0.35 0.34 0.29 0.58 Scenario 3 Baseline 249.5 60.9 382327.4 6284.3 Adaptive 248.8 61.1 382900.2 6255.1 Improvement (%) 0.28 0.33 0.15 0.46 7.4 Evaluation of combined intersection/ ramp control model Here, the combined intersection/ ramp control is evaluated using the overall system performance measures, ATT, AVS, VMT and VHT. Table 6 lists the simulation results for each of the three scenarios. Again, the network under adaptive intersection control and ramp control performs better than the baseline actuated intersection control and traffic- responsive metering control— drivers spend less time in the network and travel more distance with improved traveling speed. And, the performance in scenario 1 is better than that in the other two scenarios, and scenario 3 has gained the least improvement. Table 6. Performance of the Network ATT ( second) AVS ( mile/ hr) VMT ( mile) VHT ( hour) Scenario 1 Baseline 344.3 43.9 760920.0 17367.6 Adaptive 305.5 49.6 766071.2 15442.5 Improvement (%) 11.27 12.98 0.68 11.08 Scenario 2 Baseline 257.0 59.1 575585.6 9788.2 Adaptive 255.5 59.4 577585.6 9653.9 Improvement (%) 0.58 0.51 0.35 1.37 Scenario 3 Baseline 249.5 60.9 382327.4 6284.3 Adaptive 248.3 61.2 383196.6 6252.1 Improvement (%) 0.48 0.49 0.23 0.51 Figure 24 compares the three control models using overall system performance measures. It is found that the combined intersection/ ramp control model performs the best in terms of ATT, AVS, VMT, and VHT. The improvement on VMT is very minor since the simulation period for each scenario starts from a free flow condition and ends with another free flow condition. 58 Figure 24. Overall System Performance Comparison of the Three Control Models 8. Concluding Remarks This project introduces three real- time adaptive control strategies, including an intersection control, ramp control and an integrated control that combines both intersection and ramp control. The development of these strategies is based on a mathematical representation that describes the behavior of real- life processes ( traffic flow in corridor networks and actuated controller operation). Only those parameters commonly found in modern actuated controllers ( e. g., Type 170 and 2070 controllers) are considered in the formulation of the optimal control problem. As a result, the proposed strategies could be easily implemented with minimal adaptation of existing field devices and the software that controls their operation. Microscopic simulation was employed to test and evaluate the performance of the proposed strategies in a calibrated network. Simulation results indicate that the proposed strategies are able to increase overall system performance and also the local performance on ramps and intersections. Prior to testing the complete model, separate tests were conducted to evaluate the intersection control model on: 1) an isolated intersection, and 2) a network of intersections along an arterial. The complete model was then tested and evaluated on the Alton Parkway/ I- 405 corridor network in Irvine, California. In testing the optimal control model, we simulated a variety of conditions on the freeway and arterial subsystems that cover the range of demand from peak to non- peak, incident to non-incident, conditions. The results of these experiments were evaluated against full- actuated operation and found to offer improved performance. 59 The scope of the current effort includes the development of the corridor adaptive control model and its testing and evaluation only in a simulation environment. Although actual deployment is beyond the scope of the current effort, we believe that the results of the evaluation of the simulated network warrant further investigation of incorporation of the adaptive control system as a service within the planned CARTESIUS deployment under CTNet ( in separate, complementary PATH/ Caltrans projects). Such research deployment could relatively easily be conducted on the Alton Parkway/ I- 405 corridor network for which we have at least limited authority to conduct tests involving closed-loop control. On the arterial, we have installed a system of Type 2070 controllers at all signalized intersections that operate independently from the local City of Irvine system. We have established real- time communication with these control devices and also receive real- time raw data streams from loop detectors within the study area. In addition, software has been developed, and laboratory tested, that permits real- time adaptive control of the Caltrans District 12 ramp meters in this corridor. Future efforts will be made to improve this model by: ( 1) seeking a more sophisticated algorithm that models the actual traffic flow pattern in signalized network; ( 2) further developing this model in order for the application in coordinated control systems; ( 3) comparing this model with other adaptive control strategies; and ( 4) incorporating access/ egress choice model in the integrated corridor control. 60 References Cameron, Gordon D. B. and Duncan, Gordon I. D. ( 1996). PARAMICS- Parallel Microscopic Simulation of Road Traffic. The Journal of Supercomputing, Vol. 10, No. 1, 25- 53. Chang, G- L., Wu, J., and Lieu, H. ( 1994) Real- time incident- responsive corridor control: a successive linear programming approach. Proceedings of the 4th Annual Meeting of IVHS America, Atlanta, GA, Vol. 2, pp. 907- 918. 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( 1992) “ SCATS: A Traffic Responsive Method of Controlling Urban Traffic Control”, Roads and Traffic Authority. Papageorgiou, M. ( 1995) An Integrated Control Approach for Traffic Corridors. Transportation Research- C3, No. 1, pp. 19- 30. Robertson, D. I. and Bretherton, R. D. ( 1991) “ Optimizing Networks of Traffic Signals in Real Time – the SCOOT Method”, IEEE Transaction on Vehicular Technology, Vol. 34, pp. 11- 15. Sims, A. G., ( 1979), “ The Sydney Coordinated Adaptive Traffic System.”, Proceedings of the Engineering Foundation Conference on Research Priories in Computer Control of Urban Traffic Systems, pp. 12- 27. Singh, M. G. and Tamura, H. ( 1974) Modelling and hierarchical optimization for oversaturated urban road traffic networks. International Journal of Control, Vol. 20, No. 6, pp. 913- 934. 61 Stephanedes, Y. J., and Chang, K- K. ( 1993) Optimal Control of Freeway Corridors. Journal of Transportation Engineering, Vol. 119, No. 4, pp. 504- 514. Takacs, L. ( 1955). Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hungary. 6, pp 101- 129. Ziliaskopoulos, A. K. ( 2000) A linear programming model for the single destination system optimum dynamic traffic assignment problem. Transportation Science, Vol. 34, No. 1, pp. 37- 49. |
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