Final report of ITS Center project: Signal timing algorithm
For the Center for ITS Implementation Research
A U.S. DOT University Transportation Center
“Stochastic Traffic
Signal Timing Optimization”
Principal Investigators:
Anil Kamarajugadda
Department of Civil
Engineering
Center for Transportation
Studies
University of Virginia
Dr. Byungkyu “Brian”
Park
Department of Civil
Engineering
Center for Transportation
Studies
University of Virginia
August 2003
Disclaimer
The contents of this report reflect the views of the authors, who are responsible
for the facts and the accuracy of the information presented herein. This document
is disseminated under the sponsorship of the Department of Transportation, University
Transportation Centers Program, in the interest of information exchange. The U.S.
Government assumes no liability for the contents or use thereof.
Research Report No. UVACTS-15-0-44
August 2003
Stochastic Traffic Signal Timing Optimization
By:
Anil Kamarajugadda
Dr. Byungkyu “Brian” Park
A Research
Project Report
For the
Center for ITS Implementation Research (ITS)
A U.S. DOT University Transportation Center
Anil Kamarajugadda
Department of Civil Engineering
Email: adk3w@virginia.edu
Dr. Byungkyu “Brian” Park
Department of Civil Engineering
Email: bpark@virginia.edu
Center for Transportation Studies at
the University of Virginia produces outstanding transportation professionals,
innovative research results and provides important public service. The Center
for Transportation Studies is committed to academic excellence, multi-disciplinary
research and to developing state-of-the-art facilities. Through a partnership
with the Virginia Department of Transportation’s (VDOT) Research Council (VTRC),
CTS faculty hold joint appointments, VTRC research scientists teach specialized
courses, and graduate student work is supported through a Graduate Research Assistantship
Program. CTS receives substantial financial support from two federal University
Transportation Center Grants: the Mid-Atlantic Universities Transportation Center
(MAUTC), and through the National ITS Implementation Research Center (ITS Center).
Other related research activities of the faculty include funding through FHWA,
NSF, US Department of Transportation, VDOT, other governmental agencies and private
companies.
Disclaimer: The contents of this report
reflect the views of the authors, who are responsible for the facts and the accuracy
of the information presented herein. This document is disseminated under the sponsorship of the Department
of Transportation, University Transportation Centers Program, in the interest
of information exchange. The U.S. Government
assumes no liability for the contents or use thereof.
| 1. Report No. |
2. Government Accession No. |
3. Recipient’s Catalog No. |
| UVACTS-15-0-44 |
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| 4. Title and Subtitle |
5. Report Date |
| Stochastic Traffic Signal Timing Optimization |
August 2003 |
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6. Performing Organization Code |
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| 7. Author(s) Anil Kamarajugadda and Byungkyu “Brian” Park |
8. Performing Organization Report No. |
| |
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| 9. Performing Organization and Address |
10. Work Unit No. (TRAIS) |
| Center for Transportation Studies |
|
| University of Virginia |
11. Contract or Grant No. |
| PO Box 400742 Charlottesville, VA 22904-7472 |
|
| 12. Sponsoring Agencies' Name and Address |
13. Type of Report and Period Covered |
| Office of University Programs,
Research and Special Programs Administration US Department of Transportation 400 Seventh Street, SW Washington DC 20590-0001 |
|
Final Report |
|
|
|
14. Sponsoring Agency Code |
| |
|
|
| 15. Supplementary
Notes |
|
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| 16. Abstract |
| Signalized intersections are a critical element of an
urban road transportation system and maintaining these control systems at their
optimal performance for different demand conditions has been the primary concern
of the traffic engineers. Currently, average
control delay is used as a performance measure of a signalized intersection. The
control delay is estimated using the delay equation provided by the Highway Capacity
Manual (HCM). The HCM delay equation is a function of multiple input parameters
arising from geometry, traffic and signal conditions. Variables like volume, green
time and saturation flow rate that influence delay computations are stochastic
variables, which follow their characteristic distribution. This implies that delay
has to be estimated as a distribution as against the point estimate, the average
delay. Various simulation programs
and optimization techniques have evolved that aid the traffic engineer in the
optimization process. None of the optimization programs consider the day-to-day
stochastic variability in the delay during the optimization process. The purpose
of this research is to estimate variability in delay at signalized intersections
and incorporate the variability in the optimization process. An analytical methodology to compute the variance of
delay for an isolated intersection and arterial intersections is developed. First,
delay variance is computed for an isolated intersection using expectation function
method for undersaturated conditions and integration method for oversaturated
conditions. The variance computation for an isolated intersection is expanded
to arterial intersections using the integration method and the analytically approximated
platoon dispersion model. The delay variance estimates are then utilized in the
optimization of intersections. A genetic algorithm approach is used in the optimization
process using either average delay or the 95th percentile delay as an objective
function. The results of the optimization, especially for isolated intersections,
have shown considerable improvement over SYNCHRO, a signal optimization program,
when evaluated using microscopic simulation programs SIMTRAFFIC and CORSIM. However,
the results of arterial optimization did not show any significant improvement
over the SYNCHRO.
|
| 17 Key Words |
18. Distribution Statement |
| Stochastic Optimization, Traffic Signal, Simulation,
Variability, HCM, Genetic Algorithms, Level of Service. |
No restrictions. This document is available
to the public. |
| 19. Security Classif. (of this report) |
20. Security Classif. (of this page) |
21. No. of Pages |
22. Price |
| Unclassified |
Unclassified |
130 |
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| | | | | | | |
ABSTRACT
Signalized
intersections are a critical element of an urban road transportation system and
maintaining these control systems at their optimal performance for different demand
conditions has been the primary concern of the traffic engineers. Currently, average control delay is used as
a performance measure of a signalized intersection. The control delay is estimated
using the delay equation provided by the Highway Capacity Manual (HCM). The HCM
delay equation is a function of multiple input parameters arising from
geometry, traffic and signal conditions. Variables like volume, green time and
saturation flow rate that influence delay computations are stochastic variables,
which follow their characteristic distribution. This implies that delay has to
be estimated as a distribution as against the point estimate, the average delay.
Various simulation programs and optimization
techniques have evolved that aid the traffic engineer in the optimization process.
None of the optimization programs consider the day-to-day stochastic variability
in the delay during the optimization process. The purpose of this research is
to estimate variability in delay at signalized intersections and incorporate the
variability in the optimization process.
An
analytical methodology to compute the variance of delay for an isolated intersection
and arterial intersections is developed. First, delay variance is computed for
an isolated intersection using expectation function method for undersaturated
conditions and integration method for oversaturated conditions. The variance computation
for an isolated intersection is expanded to arterial intersections using the integration
method and the analytically approximated platoon dispersion model.
The
delay variance estimates are then utilized in the optimization of intersections.
A genetic algorithm approach is used in the optimization process using either
average delay or the 95th percentile delay as an objective function. The results
of the optimization, especially for isolated intersections, have shown considerable
improvement over SYNCHRO, a signal optimization program, when evaluated using
microscopic simulation programs SIMTRAFFIC and CORSIM. However, the results of
arterial optimization did not show any significant improvement over the SYNCHRO.
TABLE OF CONTENTS
Approval Sheet……………………………………………………………………………………………….ii
Abstract……………………………………………………………………………………………………...iv
Table of Contents……………………………………………………………………………………………vi
Table of Figures……………………………………………………………………………………………viii
Table of tables……………………………………………………………………………………………….ix
CHAPTER 1 INTRODUCTION......................................................................................................................................
1
1.1 Background
…………………………………………………………………………1
1.1.1 Delay Estimation………………………………………………………………………….2
1.1.2 Optimization………………………………………………………………………………4
1.2 Objectives
…………………………………………………………………………4
1.3 Scope and Structure
………………………………………………………………………4
CHAPTER 2
LITERATURE REVIEW ………………………………………………………………..6
2.1 Delay Background ………………………………………………………………..6
2.2 Platoon
Dispersion …………………………………………………………...….10
2.3 Uncertainty
Analysis ………………………………………………………………12
2.4 Optimization
………………………………………………………………13
2.4.1 TRANSYT-7F
………………………………………………………………14
2.4.2 SYNCHRO
………………………………………………………………14
2.4.3 Genetic Algorithms
……………………………………………………...……….15
CHAPTER 3 METHODOLOGY……………….. 19
3.1 Highway Capacity
Manual Delay Equation for Isolated intersection. 19
3.1.1 Stochastic
Variables Identified……...…………………………………………………………..19
3.2 Inputs. 21
3.3 Overview of
the Methodology. 23
3.4 Undersaturated
isolated intersection. 25
3.4.1 Assumptions……………………………………………………...………………………25
3.4.2
Expectation function method …………………………………………………………….26
3.4.2.1
Expectation Functions………………………………………………………………26
3.4.3 Taylor Series Expansion…………………………………………………………………32
3.5.4 Calculation of HCM delay
variability…………………………………………………...33
3.5 Oversaturated
Intersections. 37
3.6 Arterial Intersections. 40
3.6.1 HCM
Delay Equation for an arterial intersection……………………………………….42
3.6.2 Platoon Dispersion Model……………………………………………………………….43
3.6.2.1 Upstream Discharge Pattern…………………………………………………...44
3.6.2.2 Simplification of the Platoon Dispersion Model……………………………….47
3.6.3 Estimation of the Arrival Pattern/ The Progression Factor……………………………..50
3.6.3.1 Combining Through and Left Volumes at downstream
intersection…………...53
3.6.4 Delay Variance Computations…………………………………………………………...54
3.7 LOS Computations………………………………………………..……………...………55
3.8 Optimization.……………………………………………………………………………..57
3.8.1 Optimization
Procedure………………………………………………………………….57
3.8.1.1 Isolated Intersection……………………………………………………………58
3.8.1.2 Arterial Intersection……………………………………………………………62
3.9 Evaluation of the optimization result 63
3.9.1 SIMTRAFFIC
Evaluation………………………………………………………………..63
3.9.2 CORSIM
Evaluation……………………………………………………………………..64
Chapter 4 DELAY VARIABILITY FOR UNDERSATURATED
INTERSECTIONS...............................
65
4.1 Example Delay Variance Estimation for
an Undersaturated Intersection………………….65
4.2 Evaluation of the Expectation Function
Method……………………………………………68
4.3 Evaluation of the Variance of Delay
Under Different Demand Conditions………………...700
4.4 Evaluation of the Mean and Variance
of Delay for Different Distributions………………..71
4.5
Evaluation of the Variance of Delay Under Different Demand Variance Conditions……...73
CHAPTER
5 DELAY ESTIMATION FOR OVERSATURATED
INTERSECTIONS….……...…...74
5.1
Single Approach…………………………………………………………………………………..…..74
5.2
Evaluation of the Integration Method………………………………………………………..…….76
5.3
Evaluation of the Average and Variance of Delay under Different Demand
Conditions….. 78
5.4
Intersection…………………………………………………………………………………………….79
5.4.1 Example…………………………………………………………………………………..81
CHAPTER 6 ARTERIAL
INTERSECTION………………………………………………………………………..84
6.1
Estimation of Arrival Pattern / Progression Factor…………………………………..….…..….84
6.2
Delay Variance Computations Using the Progression Factor………………………….……...87
CHAPTER 7 OPTIMIZATION……………….…………………………………………………………….……….89
7.1
Scenario I: Moderate Traffic……………….............................................................................90
7.1.1 Setting……………………………………………………………………………………90
7.1.2 Timing
Plan Development………………………………………………………………..92
7.1.3 Evaluation………………………………………………………………………………..93
7.1.3.1 Experimenatl
Design…………………………………………………………...93
7.1.3.2 SYNCHRO
Evaluation…………………………………………………………93
7.1.3.2.1
Discussion..…………………………………………………………...95
7.1.3.3 CORSIM
Evaluation……………………………………..……………………..95
7.1.3.3.1
Evaluation Criteria……………………..………………………….....96
7.1.3.3.2
Discussion………………………………………………………….....95
7.2 Scenario
II: Heavy Traffic…………………..……………………………………………………100
7.2.1 Setting……………………………………..……………………………………………100
7.2.2 Timing
Plan Development……………………………………………….……………..101
7.2.2.1 Average
Delay……………………………………………………………...…101
7.2.2.2 95th
Percentile Delay………………………………………………………….102
7.2.2.3 SYNCHRO….…………………………………………………………………103
7.2.3 SIMTRAFFIC
Evaluation and Discussion……………………………………….……..104
7.2.4 CORSIM
Evaluation……………………………………………………………………106
7.2.4.1 Discussion……………………………………………………………………..107
7.3 Scenario
III: Arterial Intersection………………………………………………………….…….106
7.3.1 Setting…………………………………………………………………………………..110
7.3.2 Timing
Plan Development……………………………………………………….……..111
7.3.3 Evaluation
and Discussion…………………………………………………………..…111
CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS...…………………………………...115
8.1 Conclusions………………………………………………………………………………………115
8.2 Recommendations…………………………………………………………………………….…..117
REFERENCES……………………………………………………………………………………………119
LIST OF FIGURES
Figure 1. Flowchart of intersection
signal timing optimization. 24
Figure 2 Flow Chart of the HCM delay
variance computation for undersaturated intersection.
36
Figure 3. Flow Chart of the HCM delay
variance computation for oversaturated intersections.
38
Figure 4. Discharge pattern at an upstream
intersection considering only through volumes.
45
Figure 5. Discharge
pattern at the upstream intersection considering through and left vehicles…………………………………………………………………………………….………………..45
Figure 6. Figure
depicting the Platooned arrivals at downstream..………………………………………..46
Figure 7. Depicting the platoon dispersion
process. 51
Figure 8.
LOS ranges with delay distribution…...………………………………………………………….56
Figure 9.
NEMA Phase notation……………………………………………………………………………58
Figure 10.
Comparison between approximate equation and HCM delay equation n=5…………………...67
Figure 11.
Expectation Function Method versus Monte Carlo Simulation……………................................69
Figure 12.
Confidence intervals on delay for different degrees of saturation…………...............................70
Figure 13. Expected delay under selected
input distributions. ………72
Figure 14. Standard deviation of delay
for different distributions of the input volume.
73
Figure 15. Expectation Function Method
versus Monte Carlo Simulation.
77
Figure 16. Average delay compared with
HCM delay for over saturated conditions.
78
Figure 17. Standard Deviation of delay
with degree of saturation. 79
Figure 18. Example layout of an isolated
intersection. 81
Figure 19. Example situation for arterial
intersections. 84
Figure 20.
Layout of the hypothetical intersection…….………………………………................................91
Figure 21. Comparison of SYNCHRO and GA (Average) optimized
timing plan using SYNCHRO – HCM delays............………………………………………………………………………………………....94
Figure 22. Comparison of SYNCHRO and
GA (Average) optimized timing plan using SYNCHRO – Percentile delays 95
Figure 23. SYNCHRO Timing Plan compared
with the GA Average optimized timing plans in CORSIM...
98
Figure 24. SYNCHRO Timing Plan Compared
with the GA 95th Percentile optimized timing plans in CORSIM 99
Figure 25. GA convergence using the
average delay for optimization. 102
Figure 26. Convergence of GA algorithm
to the optimal solution.
103
Figure 27. Comparison between SYNCHRO
and GA.. 105
Figure 28. Comparison between 95th
percentile and average delay optimization.
106
Figure 29. SYNCHRO Timing Plan compared with the GA Average
optimized timing plans in CORSIM……………………………………………………………………………………………..109
Figure 30. SYNCHRO Timing Plan Compared with the GA 95th
Percentile optimized timing plans in CORSIM……………………………………………………………………………………………..109
Figure 31. XY scatter plot of the delay
values from GA optimized timing plan and SYNCHRO optimized timing plan 114
Figure 32. XY scatter plot of the delay
values from GA optimized timing plan and SYNCHRO optimized timing plan 114
LIST OF TABLES
Table 1. HCM level of service criteria……………………………………………………………………...55
Table 2. Delay Confidence Interval with
respect to Coefficient of Variance.
73
Table 3. Example demand conditions for
an isolated intersection. 81
Table 4. Example Problem delay mean
and variance. 82
Table 5. Demand conditions for Scenario
I 91
Table 6. Comparison of GA and SYNCHRO
green times. 92
Table 7. Comparisons of SYNCHRO and
GA timing plans, T-Test result 98
Table 8. Example inputs for Optimization
of an isolated intersection.
100
Table 9. Various timing plans for scenario
II 104
Table 10.
Comparisons of SYNCHRO and GA timing plans, T-Test result…..…………………………... 108
Table 11. Input demand conditions for
the hypothetical intersection. 110
Table 12. Result of the optimization
for the hypothetical arterial 111
Table 13. Paired T-test on the delay
estimates from GA optimized timing plan and SYNCHRO optimized timing plan. 114
CHAPTER 1 INTRODUCTION
1.1 Background
Transportation systems are an integral part
of a modern day society designed to provide efficient and economical movement
between the component parts of the system and offer maximum possible mobility
to all elements of our society. A competitive, growing economy requires
a transportation system that can move people, goods, and services quickly and
effectively. Road transportation is a
critical link between all the other modes of transportation and proper functioning
of road transportation, both by itself and as a part of a larger interconnected
system, ensures a better performance of the transportation system as a whole.
Signalized intersections, as a critical element of an urban road transportation
system, regulate the flow of vehicles through urban areas. Traffic flows through
signalized intersections are filtered by the signal system (stopping of vehicles
during red time) causing vehicular delays. Vehicular delay at signalized intersections
increases the total travel time through an urban road network, resulting in a
reduction in the speed, reliability, and cost-effectiveness of the transportation
system. Increase in delay results in the degradation of the environment through
increases in air and sound pollution. Thus, delay can be perceived as an obstacle
that has a detrimental effect on the economy. It
has been the traffic engineers’ endeavor to quantify delay and optimize the signal
system to perform at a minimum delay.
1.1.1
Delay Estimation
Delay
estimate measures reflect the driver discomfort, frustration, fuel consumption
and lost travel time. Numerous equations
have been developed for the estimation of delay. In the U.S., delay is estimated
using the Highway Capacity Manual (HCM). The HCM equation is a function
of multiple input parameters arising from geometry, traffic and signal conditions.
The HCM procedure for a signalized intersection uses average demand flow rate
and saturation flow rate in order to estimate volume to capacity ratio and the
corresponding performance measure, delay. The Level of Service (LOS) is then determined
from a predefined range of average control delay values.
In practice, for gathering inputs
for the evaluation of delay in the field, most efforts are given to the estimation
of capacity, while traffic volumes are collected just for a day or two, with the
exception of locations with existing surveillance systems. The delays are then
calculated based on the average demand observed from the data collection process.
Conversely, the demand volumes are subject to stochastic variability and usually
follow a Normal distribution for daily variability and Poisson distribution for
cyclic variability. Further, variables like green time and saturation flow rate
(in addition to the volume) that influence delay computations are stochastic variables
which follow their characteristic distribution. In addition, the average control
delay at signalized intersection in the real world might vary depending on traffic
conditions including different arrival distributions, percentage of trucks, and
driver characteristics. Considering that delay is governed by a number of stochastic
variables, it is imperative that delay also be considered a stochastic variable.
This implies that delay has to be defined through a distribution as against the
point estimate, the average delay.
This implies that an average delay
value obtained from the data collection may not reflect the actual performance
of the intersection. In other words, an average delay value of say 35 seconds
per vehicle (LOS D) obtained on the data collection day has no significance if
the 95th percentile confidence interval of delay varies from 20 to
50 seconds per vehicle (LOS B to LOS D). Further, LOS is not a good performance measure
when the delay value lies borderline of two LOS categories. For example, an average
delay of 34.9 seconds is considered as LOS C, while 35.0 is LOS D.
Computation of the variability of delay often requires information on the
demand variability like variance. With the use of advanced vehicle detection and
communication technology, traffic count data from signalized intersections are
extensively archived in places such as the Smart Travel Laboratory (STL) at the
University of Virginia. The data in the STL is provided from Northern Virginia
Smart Travel Signal System. The Management Information System for Transportation
(MIST) at Northern Virginia controls over 1,000 signalized intersections, and
system detectors report vehicle counts, speed, and occupancy every 15 minutes.
Thus, vehicle demand variations can be easily captured and analyzed. Then, the
delay variance could be estimated through sampling processes like Monte Carlo
Simulations and Latin Hypercube Design. However, sampling techniques provide inconsistent
results from different runs and requires a fairly large sample size to get close
to the analytical values. Thus an analytical methodology that overcomes the drawbacks
of the sampling procedures is desirable.
1.1.2
Optimization
Various simulation programs and optimization
techniques have evolved that aid the traffic engineer in the optimization process.
Delay and its derivative are used as the objective function in most optimization
software. For example, SYNCHRO optimizes based on the percentile delay while TRANSYT-7F
optimizes based on factor that involves the average delay called the disutility
index. However, delay is a stochastic variable and optimizing a signalized intersection
for an average value fails the system performance for extreme demand conditions. Thus, a methodology that optimizes the intersection considering
stochastic variability would be very useful.
1.2
Objectives
The
objectives of this analytical research are to
a) Develop
an analytical methodology that estimates the variability of HCM delay equation
for both undersaturated and oversaturated conditions, and
b) Optimize
the signalized intersections considering stochastic variability, and evaluate
its performance using microscopic simulation programs like SIMTRAFFIC and CORSIM.
1.3
Scope and Structure
An analytical methodology to compute the variance
of delay for isolated and arterial intersections is developed. First, delay variance
is computed for an isolated intersection using expectation functions for undersaturated
conditions and numerical integration for oversaturated conditions. The variance
computation for an isolated intersection is expanded to an arterial intersection
using the platoon dispersion model. The HCM delay equation is utilized in the
computations and is assumed to be valid. However, the same methodology could be
used with any analytical delay computation equation. The stochastic variability
in delay is studied for the variability in the demand volumes only and the effect
of variable saturation flow rates and green times is not studied.
The delay variance estimates are
then utilized in the optimization of intersections.
A genetic algorithm approach is used in the
optimization process. Average delay, 95th percentile delay, and the
total delay are some of the delay derivatives used in the optimization process.
In addition to optimizing an isolated intersection, arterial network optimization
is also presented. In the optimization of arterials, only two intersections are
considered in the present report. However, the methodology could be expanded to
a full fledged arterial.
This report commences with a literature review. Chapter 2 reviews the delay
computations, uncertainty quantification, the platoon dispersion model and the
genetic algorithm technique. Chapter 3 outlines the methodology in detail with
sufficient examples. Chapters 4 - 7 present the results and comparisons with simulation
models. Finally, the report concludes with conclusions and recommendations.
CHAPTER 2 LITERATURE REVIEW
This
chapter presents the relevant literature that has been reviewed as a part of this
research. This chapter commences with a discussion on the delay equations that
have been developed beginning with the early 20th century. The platoon
dispersion model, which is the link between isolated and arterial intersections,
is reviewed next. This is followed by an overview of the uncertainty analysis
where the pros and cons of the sampling procedures are presented. Finally, the
optimization procedure utilized by various simulation programs is discussed followed
by the Genetic Algorithm procedure.
2.1
Delay Background
Signalized
intersections were developed in England in the early 20th century.
With the introduction of these controls to maneuver conflicting streams of vehicular
and passenger traffic, researchers have concentrated on estimating delays due
to these controls and in developing the optimum signal timings to minimize delay
especially for pre-timed signals. Webster’s equation is one of the fore most delay
equations developed in 1958 assuming practical distributions like Poisson (random)
arrivals with uniform discharge headways [1]. Webster introduced three terms to
the delay equation as shown below.
(1)
where,
d is the average delay per vehicle,
c is the cycle time,
l is the ratio of the effective
green to the cycle length,
q is the flow rate,
s is the saturation flow rate, and
x is the degree of saturation.
The first term in Equation (1) represents the delay when traffic is considered
to arrive at a uniform rate. The second term is a correction to consider the random
nature of the arrivals. The third term is the empirical correction term introduced
to give a closer fit to the simulated delay values. Furthermore, Webster used
differential calculus techniques on the developed delay estimate to compute the
cycle length for the minimum average delay.
Akcelik further developed the delay equation by utilizing the coordinate
transformation technique to obtain a time-dependent equation that is applicable
to signalized intersections [2]. A generalized delay equation of the form shown
in Equation (2) was developed that embraces the Australian and Canadian delay
formulas as well.
(2)
where,
d = average overall delay
c = signal cycle time in seconds
u = g/C (ratio of effective green to
the cycle length)
x = degree of saturation
T = flow period in hrs
m, n = calibration parameters
x0 = degree of saturation below which the second term of the
delay formula is zero
In the U. S., the Highway Capacity Manual (HCM) delay equation is utilized
in delay computations. The HCM 2000 propounds that delay be computed using the
following equation [3]:
(3)
Where,
C = cycle length in seconds,
g = effective green time in seconds,
X = degree of saturation (v/c),
v = demand volume in vehicles/hour
T = duration of analysis period hours,
k = incremental delay factor, 0.5 for pre-timed signals,
i = upstream filtering/metering adjustment factor, 1 for isolated intersection,
and
c = capacity in vehicles per hour.
Engelbrecht et al. have validated the HCM delay equation for oversaturated
conditions and for different period of analysis [4]. Delays estimated by the HCM
2000 delay model were observed to be in close agreement with the delay estimates
from TRAF-NETSIM simulations.
In addition to computing point estimates like the average delay, attempts
have been made to quantify the variability in delay due to random nature of the
arrivals. Olsweski developed numerical methods to calculate average delay
and estimate the distribution of the average cyclic delay. The methodology is
based on sequential calculation of queue length probabilities with different arrival
processes and was not applicable to practical situations [5].
Fu et al. modeled analytical equations to compute the variance of delay
based on a simulation study [6]. The model for variance had to be calibrated extensively
and the calibration depended on the delay definition used in the simulation program. Further, a signalized intersection optimization based
on variance minimization was conducted and the results were similar to that obtained
from average delay minimization.
The arrival process observed downstream of a traffic signal is expected
to differ from that observed upstream of the signal due to platooning of the vehicles.
Thus, the delay equation developed for an isolated intersection has been modified
to be applicable to arterial intersections. A progression factor was introduced
for estimating delays for intersections in an arterial to account for this process.
Rouphail developed a set of progression factors that adjusts delays at coordinated
intersections using time-space diagrams and flow counts [7]. Levinson, also
attempted to compute the signal delay for platooned arrivals for two extreme conditions
[8]:
a) When
the first vehicle in the platoon arrives during a green interval and is unimpeded
and
b) When
the first vehicle in the platoon arrives during the red period and is impeded
by queued vehicles.
Olszewski computed delay for a pretimed signal
when the arrival rate is non-uniform by utilizing the step arrival rate model
[9]. A significant finding of his research
was that the progression effects the uniform delay term and not the overflow delay
term in the HCM delay equation. Teply presented a practical system to evaluate
the signal coordination at a series of intersections and studied the quality of
signal progression based on the time-space charts developed from surveys and simulation
[10].
2.2
Platoon Dispersion
“The
on-off nature of traffic signal tends to create bunches or “platoons” of vehicles.
The platoons of vehicles disperse as they travel away from the lights due to the
different speeds of the individual vehicles” [11]. Thus the arrival pattern at
an intersection downstream from another signal is different from an isolated intersection.
Robertson developed the platoon dispersion model for the Road Research Laboratory
in United Kingdom in 1969. The dispersion model was developed based on the observations
made at four sites in West London at approximately 300, 600 and 1000 ft downstream
of the stop bar. The predicated flow rate at any time step is expressed as a linear
combination of the original platoon flow rate in the corresponding time step (with
a lag of t) and the flow rate of the predicted platoon in the step immediately
preceding it. Equation (24) presents the recursive model developed by Robertson.
A smoothing factor ‘F’ is used in the model to best fit the actual and calculated
platoon shapes and is inversely proportional to the travel time on the link. The
arrivals at the downstream intersection are estimated depending on the discharge
patterns from upstream intersection. The smoothing factor is found to be site
specific and depends on the road width, gradient, parking, opposing flow level,
etc [11].
Rouphail developed a closed form solution for the recursive model developed
by Robertson, and studied the effect of platoon dispersion on signal coordination
and delay estimation. Flow rates in the predicted platoon measured at the kth
interval of the jth simulated cycle are expressed in terms of the demand
and capacity rates at the source intersection in addition to signal-control and
travel-time parameters [12].
Successful implementation of the dispersion model needs parameters like
the dispersion factor a and travel-time factor b (Refer to Equation (24))
to be calibrated according to the conditions of the arterial. McCoy et al. attempted
to calibrate the platoon dispersion model for passenger cars under low friction
traffic flow conditions and suggested appropriate values for a and
b to
0.21 and 0.97 for two-way-two-lane street and 0.15 and 0.91 on a four lane divided
highway [13]. McTrans suggests that the degree of platoon dispersion
on internal links can be calibrated for local conditions by using the platoon
dispersion factor (PDF). High platoon dispersion factors indicate heavy friction, (i.e., urban central
business districts (CBD) areas having significant amounts of parking, turning,
pedestrians, and narrow lane widths), which conspire to reduce platoon intensities. Low platoon dispersion factors indicate low friction, (i.e., ideal suburban
high- type arterial street conditions) that allows increased platoon intensities [14]. A
value of 0.35 for a was found to be suitable for U.S. conditions.
2.3
Uncertainty Analysis
Uncertainty
is inherent in any system or model and it has been the engineers’ endeavor to
reduce uncertainty to the minimum wherever possible and to quantify the uncertainty
in the system. Sampling techniques have been widely used in transportation engineering
to quantify variability [15]. Sampling techniques involve the running of the model
for a selected set of inputs based on their probability distributions to generate
the probability distribution of the output. These sampling procedures lack accuracy
because every simulation run produces different results and a large sample size
might be necessary for the convergence to the true solution. Some of the widely
used sampling methods for uncertainty analysis are Monte Carlo Simulation and
Latin Hypercube Sampling.
Monte Carlo simulation, a simple random sampling (SRS) procedure, is the
most widely used sampling method for computer experiments because they are quick
and easy to implement for high dimension problems. Many of the initial studies
in computer experiments investigated the distribution of the response given "random"
inputs. However, with increasing complexity of the problems, an improved design
strategy called the Latin Hypercube Design is utilized..
A Latin Hypercube Design improves the distribution of input variables in
the design of a sample and the design with the best distribution of points is
selected. Ideally, the Latin Hypercube design generates a minimal number of input
combinations that are spread as evenly as possible in the experimental space.
A Latin Hypercube design with the design points more uniformly spaced can be chosen
by measuring the variability of the number of design points in a randomly located
sub region of the experimental design space. To ensure that each of the input
variables Xk has all portions of its distribution represented by
the input values, the range of Xk is divided into N strata of equal
marginal probability 1/N and one sample is picked from each stratum [16].
Other probabilistic methods like expectation functions have evolved. Expectation
functions can be used to quantify uncertainty contributed by uncertain input parameters
and expectation functions overcome the drawbacks of the sampling procedures. The
moments of the output variable about the mean are estimated based on the distribution
of the input parameters. Exact knowledge of these moments is used in identifying
the distribution of the output random variable. Tyagi et al. discusses various
distributions for these input parameters and developed generic expectation functions
[17].
Very little literature was found on the uncertainty analysis of delay under
variable demand conditions. Olszewski attempted to develop the probability distribution
of delay while Fu et al. calibrated a model that computes the variance of delay
assuming that the arrivals follow a Poisson distribution [6, 18]. The developed
calibrated model was compared to the theoretically developed Markov Chain model
and was found to comply well within a range of degree of saturation values.
2.4
Optimization
One
of the primary objectives of this research is to optimize signalized intersections
using stochastic variability in delay. Delay is a stochastic variable and its
variability has to be accounted for in the optimization process. Simulation and
optimization programs like SYNCHRO and TRANSYT-7F do not consider the variability
in delay for their optimization.
2.4.1
TRANSYT-7F
TRANSYT is macroscopic optimization
and simulation tool originally developed in the United Kingdom by the Transport
and Road Research Laboratory (TRRL) [11]. It is a model that considers platoon
dispersion for its computations. TRANSYT-7F is a U.S. version of TRANSYT
developed by the University of Florida. TRANSYT-7F uses a delay derivative (Disutility
Index) as the objective function during the optimization process [14]. The
delay definition used is the entire amount of time spent while not traveling at
the prevailing cruise speed. TRANSYT-7F measures
this by periodically counting the number of vehicles queued at a signal and integrates
this series of counts over time. Uniform and residual delays are computed based
on the area under the uniform queue profile (queue.out from Spyglass). Incremental
delay is computed by using the Highway Capacity Manual equation, where certain
input parameters (e.g., capacity) are obtained directly from TRANSYT-7F simulation,
and other input parameters (e.g. duration of the analysis) are obtained directly
from the input data file. TRANSYT-7F uses genetic algorithms or the hill climbing method for optimizing
cycle length, splits, phase sequences and offsets.
2.4.2
SYNCHRO
SYNCHRO, developed by Trafficware Inc., is a software package
that can model and optimize traffic signal timings. SYNCHRO minimizes a parameter
called percentile delay in its optimization. The Percentile Delay is the weighted
average of a delay corresponding to the 10th, 30th, 50th, 70th and 90th percentile
volumes. SYNCHRO accommodates for progression by calculating the progression factor
(PF) used in the delay equation using the ratio of uniform delay calculated by
SYNCHRO with coordination and uniform delay calculated by SYNCHRO assuming random
arrivals. Furthermore, SYNCHRO uses quasi-exhaustive search in offset optimization
[19].
2.4.3
Genetic Algorithms
Genetic
Algorithms are search algorithms based on the mechanics of natural selection and
evolution. John Holland, his colleagues and his two students at the University
of Michigan developed these algorithms [20].
A genetic algorithm process starts with a random set of individuals called
the population. The individuals in a population are represented in the form of
binary strings. These strings are then acted upon by operators, which produce
a different population every generation, and then this cycle is repeated until
certain termination criteria are met. A simple genetic algorithm is composed of
three operators:
§
Reproduction
§
Crossover
§
Mutation
The reproduction is a process in which individuals are selected based upon
their fitness value or the objective function. This operator is an artificial
version of natural selection, the survival of the fittest. The reproduction operator
is implemented in algorithmic form in a number of ways. Roulette wheel selection
and tournament selection are some of them. After reproduction, a crossover operator
involving two steps is operated. Firstly, members of the newly reproduced strings
in the mating pool are mated at random. Secondly, each pair of strings undergoes
crossover as follows: an integer position k along the string is selected uniformly
at random between 1 and the string length minus one. Two new strings are created
by swapping all characters, between positions k+1 and the string length. A mutation
is an operator to move the function from local maxima and minima. A simple mutation
involves generating a random number for every digit in the binary string and if
the number is less than a predefined mutation probability (usually 0.05) the digit
in the binary string is flipped.
These genetic algorithms have been gaining significance in its applications
for transportation signal system optimization. Foy et al. have used GA to develop
a demand responsive (adaptive) optimization technique to control traffic signals
(Traffic GA) [21]. A street network with four intersections shaped in a square
configuration and each intersection being connected to two other intersections
by perpendicular roadways was simulated. A two phase signal control was assumed
for all the four intersections. Nine decision variables were involved in the optimization
process, the total green for all the phases and two variables (for all the four
intersections) one for the phase sequence and the other for the proportion of
the green time allocated the phases. These nine variables were coded into a 24
bit string. In Traffic GA, the inverse of the total average wait time was used
as an objective function. A simulation model was developed by them to be used
as an evaluator. Each GA run consisted of 50 individuals and the program was run
for 60 times for every set of conditions. After 60 generations, the member with
a minimum fitness value was chosen as the best solution for the given set of conditions.
Traffic GA was run repeatedly while using the newest traffic data and new signal
timing plans better suited to the present conditions are displayed. Balanced conditions
of green phase times and a reasonable cycle lengths were obtained as a function
of the traffic demand. The traffic GA results and the theory of convergence indicate
that GAs may be able to solve more difficult problems than traditional control
strategies and search methods.
Park et al. [22] have developed
a procedure that optimizes all the traffic control parameters (i.e., cycle length,
green split, offset, and phase sequence) for oversaturated and undersaturated
conditions. The procedure utilizes genetic algorithm-based program to optimize
the four parameters simultaneously as well as model queue blocking effects. Delay
multiplied by -1 is utilized as an evaluation function for the optimization process.
The genetic algorithm-based signal optimization program was implemented at two
closely-spaced signalized intersections within 100 meters of each other. The GA
optimizer utilized up to 250 generations with a population size of 10 per generation,
a crossover probability of 0.4, and a mutation probability of 0.03. An elitist
method was used for the GA selection process. The results of genetic algorithm
optimizer indicate that the GA optimizer searches more frequently for a good cycle
length range. It was found that the proposed GA-based program provides acceptable
solutions within reasonable amount of time [22].
The above study was extended
to deal with three different optimization strategies and was tested under different
intersection spacing: 100, 200, and 300 meters [23]. Three types of objective
functions are considered namely, throughput maximization, average delay minimization,
and modified average delay minimization with a penalty function. An arbitrary
arterial system consisting of four intersections was selected in order to test
the GA-based program. Of the three objective functions, the delay minimization
strategy is observed to be applicable to both undersaturated and oversaturated
conditions. The GA-based program and TRANSYT-7F timing plans were compared. The
GA-based program yielded less queue time than that of TRANSYT-7F on the basis
of multiple CORSIM runs [23].
CHAPTER 3 METHODOLOGY
As
mentioned earlier, the primary objectives of this research are to develop an analytical
methodology that estimates the variability of HCM delay equation for both undersaturated
and oversaturated conditions, and further, to optimize the signalized intersections
considering stochastic variability in the demand volumes. The methodology for
the realization of these objectives will be presented in this chapter. Furthermore,
the results of the optimization are compared with that of SYNCHRO and the timing
plans from both the optimization processes are evaluated using microscopic simulation
programs SIMTRAFFIC and CORSIM. The results will be presented in the chapter 7.
The following is an overview of
the methodology chapter. Firstly, the HCM delay equation for a signalized intersection
is presented and its stochastic variables are identified. Then, the methodology
developed for the delay variability estimation of an isolated intersection is
outlined. Undersaturated and oversaturated intersections are dealt differently
and the methodology for each condition is presented separately. The delay variability
estimation for an arterial intersection is presented next. Finally, the genetic
algorithm based optimization procedure is presented along with a brief overview
of the C++ program developed for the optimization. The results of the methodology
will be presented in the later chapters.
The
Highway Capacity Manual (HCM) delay equation has been presented in the literature
review. The HCM delay equation (Equation (3)) has a number of input variables
some of which are subject to stochastic variability. The
following section presents in detail the stochastic input variables to the HCM
delay equation.
3.1.1
Stochastic Variables Identified
Of
the multitude of input variables presented in Equation (3), effective green time,
saturation flow rate, and degree of saturation are identified to be stochastic
variables. The reasons for identifying only three variables are presented here.
The effective green time is computed using the following equation
(4)
Where,
g
= effective green time,
G = displayed green time,
Y = yellow time,
Rc = red clearance time, and
L = lost time.
The lost time in Equation (4) changes from driver to driver and hence induces
stochastic variability into the effective green time. However, the magnitude of
the variability in lost times is negligible compared to the variability of the
volumes and it is difficult to quantify the variability in the lost times that
driver’s experience, as it would involve collecting data on human perception.
Likewise, saturation flow rate also has stochastic variability arising from the
presence of heavy vehicles and the temporal variations in saturation flow estimation.
Nevertheless, as the variability of effective green time and saturation flow rate
are relatively small, for this exploratory research the variations in traffic
demand are only considered.
The variability in the degree of saturation arises from the random nature
of the traffic arrivals at an intersection. The degree of saturation is nothing
but the ratio of the volume to the capacity. The assumptions of constant green
time and saturation flow rate fixes the capacity which is a function of green
time and saturation flow rate. This implies that the degree of saturation (X),
which is governed by the volume (V) and capacity, will vary according to the volume
only. The variability in the traffic demand (V) is inherent to any traffic system
due to the random nature of the arrivals at an intersection and this induces stochastic
variability into the degree of saturation. The variability of the arrivals could
be identified from one cycle to the other and/or from one day to another. Since
the degree of saturation (X) is equal to demand volume (V) divided by a constant
capacity (c), the mean of X will be the average demand divided by a constant capacity
and the standard deviation of X will be the standard deviation of demand divided
by the capacity. Further, the distribution of X will be the same as that of V
with the mean and standard deviation scaled down by a factor (capacity) as mentioned
above. Hence, given the mean, variance and distribution of the demand volumes,
the stochastic properties of X can be computed and these values of mean and variance
are further used for delay variance computations.
The proposed methodology computes the delay
variance at a signalized intersection. The variables required for delay variance
computations are mentioned here. Firstly, HCM delay for an intersection
is computed by taking the weighted average of the delays from all the lane groups
with their corresponding average volumes. Similarly, the delay variance is also
computed for all the lane groups separately and then aggregated. Therefore, the
data required for the delay variance computations are required for each lane group
separately.
As mentioned earlier, computation
of the variability of delay requires the mean, variance and distribution of all
the stochastic variables. As volume is the only variable that is being
considered for stochastic variability, information on the average volume, variance
and distribution is needed for every lane group. In addition to stochastic details, other information
like the signal timing plan, saturation flow rates and other variables that are
essential to HCM delay computation are necessary. The following is a detailed
list of all the variables required for an isolated intersection computation
·
Average volume, variance and distribution for every lane
group
·
Green times for every lane group
·
Cycle length for the intersection
·
Saturation flow rate by lane groups (left turn and through
volumes may have different saturation flow rates)
·
Duration of analysis period (T)
·
Incremental delay factor, upstream filtering/metering
adjustment factor.
For an arterial intersection delay variance computation, the following additional
inputs are required:
·
Cruise time on the arterial link
·
Platoon dispersion factor (a) and
the empirical factor (b)
These additional factors are used to calibrate the platoon dispersion model
and are explained in section 3.6.2.
The mean and variance of volumes could be obtained from data collection.
A signal control system like the Management Information System for Transportation
(MIST) of Northern Virginia can provide volume information. The distributions
of the volumes are usually assumed to be Poisson for cycle-by-cycle variability
and Normal, Uniform or any other feasible distribution for day-to-day variability.
However, the distributions mentioned are valid only for an isolated intersection.
For an intersection in an arterial, the arrivals are screened through an upstream
intersection and therefore, the distribution of arrivals at a downstream intersection
is different from the standard distributions and has to be estimated. The platoon
dispersion model is used in the estimation process and the parameters required
to calibrate the model are presented in section 2.2.
The following flow chart (Figure 1) is an overview of the methodology involved
in the optimization procedure. Two modules are marked as the delay variance computation
module and the optimization module. The delay variance computation module involves
inputting the data and computing the delay variance for different demand conditions.
The optimization module involves the signal timing development process using the
genetic algorithms. The different aspects involved in the flowchart will be presented
in the later sections of the methodology chapter.
Figure
1. Flowchart
of intersection signal timing optimization
The
following section presents the methodology involved in the delay variance computations
for an undersaturated intersection. The methodology for an isolated undersaturated
intersection involves the expectation function method. First, the assumptions are highlighted followed
by the expectation function method. Then, the Taylor series expansion used to
simplify the HCM delay equation is explained. Finally, the procedure for HCM delay
variance computation is presented.
The
following assumptions are made in developing the methodology.
§
Intersection is operating under undersaturated conditions.
§
Saturation flow rate and effective green times are constant.
The first assumption is valid as has been observed in the literature review.
The second assumption implies that after considering the variability in the volumes,
the 99.99th percentile upper confidence limit on the degree of saturation
has to be less than one. Otherwise, the methodology for an oversaturated intersection
has to be applied. The validity of the third assumption is discussed earlier in
this chapter under the inputs section 3.2.
The
expectation method is an analytic procedure that overcomes the shortcomings of
sampling procedures. This method involves expectation functions. In this methodology,
each stochastic input variable is considered a random variable following a distribution
with known mean and variance. Since the output (from the function) is dependent
on the input variables, the output is also a random variable whose higher order
moments are to be calculated based on the variation of the input variables. The
expectation method can be used to calculate the first and higher order moments
of an output variable that is a function of several independent random variables
in multiplicative, additive and combined forms. The following section explains
the expectation functions utilized in the methodology.
3.4.2.1
Expectation Functions
Expectation functions provide the expectation
values of the powers of the variable given the mean, variance, and distribution
of the variable. That is, given the mean (µ), variance (s2), and the distribution of a random variable
X, expectation functions provide the expectation of Xn as a function
of 
and s2 (i.e., E(Xn) = f(
,σ2)).
The nature of the function depends on the distribution of X.
Tyagi et al. developed generic expectation function equations, which are
functions of the mean and coefficient of variance (COV) of an input random variable
[17]. The input random variable can have a Uniform, Triangular, Lognormal, Gamma,
Exponential, or Normal distribution. With a prior knowledge of the mean, variance
and the distribution of the input random variable the expectation values of the
higher powers of the variables can be computed. For example, consider an input
variable (X) that follows a distribution with a mean value of mX
and a coefficient of variance of CVX (i.e., the ratio of Standard deviation
to mean). The expectation values of powers of X (depending on the distribution)
are presented below for Normal, Uniform, Log-normal and Gamma distributions [17].
For
Normal distribution,
When r is even
(5a)
When r is odd (5b)
Where,
r is the power of the variable for which the expectation value is being
computed, and
z is the unit normal variable.
For
Uniform Distribution,
(6)
For Lognormal distribution,
(7)
For
Gamma distribution,
(8)
Where,
G is
a gamma function.
Poisson is a count distribution. A cycle-by-cycle
distribution for volumes implies that the volume counts from different cycles
follow a Poisson distribution. Therefore, the expectation values are generated
for these cyclic volume counts and then translated to hourly volumes.
The generalized expression for Poisson distribution
is as shown in equation below
Where,
l is the mean value of the
variable x.
Poisson distribution is a
discrete distribution. Therefore, the generalized expression for computing the
expectation values could be written as
.
For n=1,
.
This equation implies that the mean of a Poisson distribution
is l! which
is true.
For n=2,
. x2 in the
Right Hand Side (RHS) of this equation can be expressed as 
and simplified as follows:
The above equation implies
that the variance of Poisson distribution is l2+l-l2
(i.e., Var(x) = E(x2)-E(x)2)
= l which is the basic property of a Poisson
distribution. Similarly, computing the expectation values for up to n=6 the following
equations are obtained. The expectation values for Poisson
distribution are as shown below
(9)
Where,
l is the mean of the random variable, and
U
is a random variable representing the cyclic counts.
The expectation values in
Equation (9) are applied onto the counts and, these count expectation values have
to be translated to that of the hourly volumes and then the degree of saturation
(X). The count (U), vehicles per cycle, could be related
to the degree of saturation X using the following process.
The
U in the above equation represents the volume counts for a time period equal to
a cycle length (i.e., C secs). Volume is the vehicle count per hour, therefore
the counts per cycle have to be converted to the counts per hour. This implies
that if the vehicles are counted for every C seconds (where C is the cycle length),
the volume will turn out to be (3600/C) times the count.
The following equations are
developed:
(10)
Where,
U
is the count for a period of C secs,
V
is the arrival volume, and
c
is the capacity.
Using
Equation (10), the following expression for the degree of saturation (X) is obtained.
The expectation values of U and X can also be related as shown.
(11)
Using
Equation (11) and the expectation values for counts Equation (9), the expectation
values for the degree of saturation (X) are obtained. The expectation values of
X obtained using the above equations are as presented in Equation (12):
(12)
Using these expectation values in a simplified delay equation, an approximated polynomial obtained from the Taylor series expansion,
the expectation values for delay and its higher powers are obtained. The variance
of delay from day-to-day and cycle-to-cycle is computed. The reasons for
simplifying the delay equation and the process involved are discussed in the following
section.
3.4.3
Taylor series expansion
The
HCM delay equation could be conceptualized as a function of demand volume if the
other stochastic variables are considered to be constant. However, it is noted
that since expectation values have been developed for power functions or additive
and multiplicative terms of these power functions, this method cannot be applied
directly on the HCM delay equation. The HCM delay equation has to be transformed
to an equation involving only additive and multiplicative terms of the powers
of X. This can be realized by approximating the delay equation to a polynomial
as a function of X. Taylor series expansion on the delay equation is used for
the transformation.
Given that the delay equation is a function of a single stochastic variable
X, the delay equation is approximated as a univariate polynomial of X using Taylor
series expansion.
The
generalized Taylor Series expansion of any function F(x) is:
(13)
Where,
F(X) = the function being approximated,
and
X0 = the point about
which the equation is expanded
The Xo in the above equation is the point where the function
is expanded. The approximated function F(X) yields values that are very close
to the true values around the point Xo and is exact at the point Xo.
Since traffic demand volume is a random variable with a particular mean and variance,
traffic demand volumes vary around the mean volume in a pattern depending on the
distribution of volume. When Taylor Series expansion (Equation (13)) is used to
approximate the HCM delay equation, it is logical to utilize a value of X0
equal to the mean volume. This will result in the best approximations. It is noted
that different mean values yield different approximation equations.
In Equation (13), the F(X) is replaced with D(X) implying that delay is
a function of X only. This would yield an approximate equation of the generalized
form
(14)
Where,
D(X) is the delay function,
X is the degree of saturation,
αj are the constants,
and
n is the number of terms or the order
of the polynomial.
The n value is determined on the basis of how well the approximation replicates
the HCM delay curve within reasonable percentile confidence intervals for X. Usually,
for low degrees of saturation, a value of 3 or 4 should suffice. An example of
the comparison between the delay equation and the approximation is shown in chapter
4.
3.4.4
Calculation of HCM
delay variability
This section presents half of the variance computations
module depicted in Figure 1. The methodology involved in the delay variance
computations for an undersaturated intersection, which is the right portion of
the variance computation module, is presented here. Figure 2 depicts the delay variance computation
in detail through a flow chart.
The steps I and II in Figure 2have
already been explained in the sections 3.4.2 and 3.4.3. Step III involves computing
the expectation values for the powers of X which has been explained in section
3.4.2. The expectation values for Xr are computed depending
upon the probabilistic distribution, mean and variance of the arrival flow as
elicited earlier from Equations (5)-(8), (9) and (12). Step IV involves the computation
of the expectation values for delay. The expectation values for the delay (D)
and for D2 are calculated from the expectation values of
Xr as shown below. Using Equation (14) and the expectation values generated
from the Equations (5)-(8), (9) and (12), the expectation of delay is calculated
as in Equation (15).
(15)
Similarly, the delay equation is squared, the expectation
of D2 is computed, and from these values, the variance of delay is
calculated as
(16)
Equation (16) is the step V portion of the flow chart where the delay variance
is obtained. Furthermore, if the distribution of delay and its percentile values
are known, the confidence intervals of delay can be computed. Using the standard
deviation of delay (σ) from Equation (16) and mean value of delay (m) from Equation (15), the confidence interval is computed
as follows.
C.I.
= m ± σ
× (percentile value) (17)
The percentile values are calculated from
statistical tables depending upon the distribution and the percentile. For example,
in case of Normal distribution, the 95th percentile value is 1.96, while 99.99
percentile uses the value of 3. The day-to-day variability computations are the
simplest as the expectation values can be computed directly from the Equations
(5)-(8). The cycle-to-cycle variability computations involve the expectation values
from the Poisson distribution Equation (12).
Figure
2. Flow Chart of the HCM delay variance computation for undersaturated intersection
For an undersaturated intersection, by definition the volumes are less than
the capacity implying that the degree of saturation (X) is less than one. Considering
the stochastic variability in the volumes, if the 99.99th percentile
confidence intervals of X are found to be less then 1, the expectation methodology
could be used.
Conversely, by the definition of an oversaturated intersection, the volumes
are very high at an intersection such that they do exceed the capacity. Furthermore,
with the inherent variability in the volumes, the degree of saturation encompasses
the regions of X≤ 1 and X >1 and delay changes its equation at the value
of X=1 and hence expectation methodology provides results that might not be accurate.
Therefore, to compute the variance of delay for an oversaturated intersection,
an integration technique is utilized. The flow chart presented in Figure 3 depicts
the process of delay variability estimation for oversaturated conditions. A detailed
methodology is also explained below.
Figure
3. Flow Chart of the HCM delay variance computation for oversaturated intersections
The first step in delay variance computations involves gathering the relevant
data and computing the average and variance of the degree of saturation which
will be used for further calculations. The next step involves computing the expectation
values for delay and delay squared using integration as is explained below.
The expectation value for any function f(x) can be computed from the following
integral:
.
(19)
Where,
f(x) = the function for which
expectation values are needed, and
P(x) = probability distribution
of x.
The
above integration is computed within the limits over which x varies and in the
present case; x is the degree of saturation (X) and is assumed to be in the range
[0, 3] as the degree of saturation is not usually expected to exceed 3.
For the present case, the function f(x) is of the form
shown in Equation (18)
