Final Report
THE EXEX METHOD: INCORPORATING
VARIABILITY IN SULFUR DIOXIDE EMISSIONS
INTO POWER PLANT IMPACT ASSESSMENT
Contract No. 68-01-3957
SAI No. 84-EF80-80R2
23 July 1980
PREPARED BY
SYSTEMS APPLICATIONS.!^.
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[40802]
Final Report
THE EXEX METHOD: INCORPORATING
VARIABILITY IN SULFUR DIOXIDE EMISSIONS
INTO POWER PLANT IMPACT ASSESSMENT
Contract No. 68-01-3957
SAI No. 84-EF80-80R2
23 July 1980
Prepared for
Office of Regional Programs
Office of Air Quality Planning and Standards
Environmental Protection Agency
Research Triangle Park, North Carolina 27711
Project Officer: B. J. Steigerwald
by
M. J. Hillyer
C. S. Burton
Systems Applications, Incorporated
950 Northgate Drive
San Rafael, California 94903
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DISCLAIMER
This report has been reviewed by the Office of Air Quality Planning and
Standards, U.S. Environmental Protection Agency, and approved for publica-
tion. Approval does not signify that the contents necessarily reflect the
views and policies of the U.S. Environmental Protection Agency, nor does
mention of trade names or commercial products constitute endorsement or recom-
mendation for use.
8
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EXECUTIVE SUMMARY
The principal topic addressed in this report is the assessment method-
ology used to estimate near-source ambient air quality effects of contaminant
emissions from coal-fired steam boilers. Attention is focussed specifically
on a technical description of a new and practical development in this method-
ology that explicitly accounts for both the variability inherent in sulfur
dioxide emissions from coal-fired boilers and in meteorological conditions
when assessing the (new or modified) source's attainment of ambient stan-
dards. Policy questions associated with this new development are not
discussed.
As currently practiced, air quality assessments entail the use of an
appropriate mathematical model to calculate ground-level contaminant concen-
trations and the number of times per year those concentrations equal or exceed
the applicable ambient air quality standard. If the computed second highest
concentration equals or exceeds an ambient standard more than twice per year,
emissions from the source are judged to be too great. Among the critical
factors affecting this judgement, particularly for isolated sources located in
flat terrain, is the selection of the meteorological conditions and contam-
inant emissions rates. If the emissions rate never varied, it would be pos-
sible to use such a model, employing a year's meteorological data to estimate
(within the uncertainties of the model and its input data) the number of times
an ambient standard is equalled or exceeded.
Named the ExEx Method, the new procedure, which utilizes the same
(appropriate) air quality dispersion model mentioned previously, recognizes
the problem associated with the use of a constant (usually maximum) emissions
rate for the assessment, and specifically accounts for the low probability
that maximum emissions and worst-case meteorology can occur simultaneously.
The technique does not presume that these two events cannot occur together,
but rather calculates the probability of their coincidence.
The technique by which emissions and meteorological fluctuations are com-
bined in the ExEx Method is a Monte Carlo simulation. In this technique,
random variations in emissions are simulated by first randomly selecting emis-
sions values from an appropriate probability distribution. Such emissions
values are then combined with air quality model estimates of atmospheric dis-
persion, and when the results of many samples are accumulated, an estimate is
ill
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obtained of the expected (average) number of times that an ambient standard is
equalled or exceeded in a year. The ExEx Method also provides estimates of
the probability that concentrations will exceed the standard two or more times
in a year. An important result of the ExEx Method is that the determinism
usually associated with air quality models in impact assessments is elimi-
nated, since it cannot be stated that the second highest concentration would
be, say, 400 ug S02/m3, but rather that the probability of occurrence of two
S02 concentrations greater than 400 ug/m3 in the course of a year would be,
say, 0.1 {10 percent, or once in ten years).
The majority of the report is devoted to a detailed description of the
formulation of the ExEx Method and to documentation of a computer code that is
applicable to isolated, coal-fired boilers located in flat terrain. Example
calculations of this initial version of the ExEx Method are presented for two
hypothetical 1000 MWe power plants—one with scrubbed and the other with
unscrubbed S02 emissions—using ground-level dispersion estimates obtained by
exercising the EPA CRSTER dispersion model. As currently implemented, the
ExEx Method utilizes several years of meteorological data and a distribution
of S02 emissions; the latter may be obtained from knowledge about the distri-
bution of {"as-burned") sulfur in the coal, stack monitors, or plant design
data. To date, practical experience in applying the current version of the
ExEx Method indicates costs that are comparable to, or less than, those
involved in the application of the EPA CRSTER model. Exercise of the ExEx
computer code involves use of the output of the CRSTER model for each year of
meteorological data, together with the distribution of S02 emissions. Judge-
ments about the acceptability of a source's S02 emissions are based on con-
sideration of:
> The_expected number of times an ambient standard is
exceeded.
> The probability of violating a standard (i.e., that the
standard is exceeded two or more times in a year).
Both measures of acceptability are provided for the user, since at this time
neither the suitability of a specific value of any measure has been estab-
lished, nor has any decision been made as to which measure is preferable.
The effort reported herein is believed to represent a first step toward
improving decisions associated with environmental protection and air quality
management by logically relating ambient and emissions standards in the impact
assessment, and by providing an analysis framework that is capable of
explicitly accounting for the principal elements of uncertainty in the assess-
ment decision. There are, however, many elements involved in an impact
assessment decision that have not been incorporated into the current version
of the ExEx procedure. Those worthy of further consideration are discussed in
the final chapter of the report and include: dispersion model uncertainties,
variations in background pollutant concentration, control equipment efficiency
variations, load variations, and the treatment of multiple sources.
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AUTHORS' NOTE
This report has been reviewed extensively, inside and outside the EPA,
for both technical content and clarity of exposition. In revising the report
prior to its publication in final form, we considered all the comments we
received. We would like to express our thanks to the reviewers; their careful
efforts have without doubt improved our rep'ort.
However, we have not responded to certain reviewers' comments, a situa-
tion that derives from one of three causes: first, we take exception to the
comment; second, we concur, but the comment covers work to be carried out in
the future; or third, we consider the subject of the comment outside the scope
of the report.
To clarify the third reason, we briefly discuss here what we consider to
be the scope of this report (what it is and is not intended to encompass). We
believe the ExEx methodology, whose development is described herein, is a step
in the direction of a more realistic assessment of the effects of S02 emis-
sions from point sources on their surroundings. In this report we describe
the ExEx method both in words (chapter 2) and by means of mathematical equa-
tions (chapter 3). We also describe some results for hypothetical sources
(chapter 4) and suggest possible directions for future work (chapter 5). In
an appendix we list a computer program we developed to implement the method.
What we do not discuss are those issues concerned with whether the method
is adopted, or the important details associated with its possible implementa-
tion. This report is premised on the view that such matters are properly
addressed by those responsible for policy within the EPA. In addition, we do
not deal with issues such as the adequacy of existing air quality dispersion
models or submodels. Issues associated with the inherent limitations of
existing models or model components—like the assumptions of steady-state con-
ditions and spatial homogeneity in Gaussian-based models, or suspected
deficiencies in currently used dispersion coefficients and plume rise
algorithms—are excluded. It is our view that the best, most acceptable (to
the EPA) air quality models can always be used. For the purpose of introduc-
ing, illustrating, and performing preliminary evaluations of the method, any
8H/21
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model would meet our needs. However, i^implemented, the model that is used
is an important factor to be considered.
Our overall purpose here is to develop and evaluate analysis tools whose
ultimate purpose is to improve decisions (by reducing uncertainties as to the
consequences of decisions) associated with environmental protection and air
quality management through ambient standards and emissions limits. The effort
reported herein represents a small step toward achieving this goal by provid-
ing an analytical framework that can explicitly account for major areas of
uncertainty in assessing near-source {< 100 km) S02 impacts of isolated coal-
fired boilers. Consideration of all sources of uncertainty remains to be
explicity incorporated into the ExEx methodology; thus it is a topic for
further work.
It may be of interest here to note that the ExEx method, as currently
implemented for an isolated source, could also use measured estimates
of atmospheric dispersal strengths; therefore, the method need not rely
on models.
BW21
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CONTENTS
DISCLAIMER i i
EXECUTIVE SUMMARY i i i
AUTHORS' NOTE .' v
LIST OF ILLUSTRATIONS ix
LIST OF TABLES x
LIST OF EXHIBITS *
ACKNOWLEDGMENTS xi
I INTRODUCTION l
II PICTORIAL OVERVIEW OF THE EXEX METHODOLOGY 5
III THEORETICAL MATTERS I4
A. Expected Exceedances and Violation Probability with
Constant Meteorology 15
B. Expected Exceedances and Violation Probability with
Variable Meteorology 1"
C. Formulation of the ExEx Method 26
D. Other Topics 33
1. The Probability Distribution of S02 Emissions 33
2. A Brief Consideration of Emissions Limits 33
3. The Multiple Source Problem 35
IV RESULTS 39
A. CRSTER Results 41
B. ExEx Results 43
C. Summary 63
V SUGGESTIONS FOR FURTHER WORK ••••• 65
> -.- i \ v i i
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APPENDIX
THE COMPUTER PROGRAM 68
REFERENCES 86
vlii
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ILLUSTRATIONS
1 Elements of the Simplified ExEx Method ....................... 6
2 Atmospheric Diffusion Model .................................. 8
3 Stochastic Point Source Emissions Treatment ...... . ........... 9
4 Simulation Model ............................................. H
5 Analysis of Exceedances and Violations from
ExEx Methodology ............................................. 12
6 Probability of a Violation of the Standard as
a Function of Daily Exceedance Probability,
Assuming Constant Meteorology ................................ 20
7 Expected Exceedances ....................... • ................. 49
8 Violation Probability ........................................ 55
9 Expected Exceedances Versus Violation Probability
Over Five Years at the Worst Site ............................ 62
10 Expected Exceedances at the Worst Site Versus Violation
Probability for the Complete Receptor Network Over
a Five-Year Period..' ......................................... 64
^-1 Flow Diagram for the ExEx Program ............................ 79
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TABLES
1 Applicable Federal Air Quality Standards for S02 2
2 Characteristics of Hypothetical Power Plants 40
3 CRSTER Results for the Hypothetical Power Plant
Simulations 42
4 Expected Exceedances at the Worst Site and Violation
Probability for the Scrubbed Plume 44
5 Expected Exceedances at the Worst Site and Violation
Probability for the Unscrubbed Plume 46
EXHIBITS
A-l ExEx Program Listing with Sample Input Data 71
A-2 ExEx Sample Output 80
8*1/21
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ACKNOWLEDGMENTS
Many people have contributed to the work described in this report.
Though we cannot list them all, we would like to acknowledge those individuals
whose participation merits special mention.
Bern Steigerwald, Tom Curran, and Walt Stevenson of OAQPS provided guid-
ance, ideas, and a great deal of perceptive advice during many discussions
over the course of the project.
Jerry Mersch and Russ Lee of OAQPS provided essential data for the proj-
ect, always promptly, and often at some inconvenience.
Bob Frost of SAI worked many long hours with the computing necessary to
carry out the work.
Hoff Stauffer and Ken Schweers of ICF, Incorporated, participated in many
helpful discussions.
Xi
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I INTRODUCTION
When fossil fuels are burned to generate electricity, an unavoidable con-
sequence is the introduction of contaminants into the atmosphere. The combus-
tion of fuels containing sulfur impurities results in the emission of sulfur
dioxide (S02) through the plant's stack. A principal difference between oil
or gas and coal combustion is the variability'of their S0£ emissions. Because
of the greater homogeneity of oil, SOg emissions from an oil-burning point
source are relatively constant for a particular fuel. Coal, on the other
hand, exhibits short-term variability in composition; the result is that S02
emissions from coal burning show stochastic variation from one time period to
another. In this report, we describe the formulation of a methodology to
assess the ground-level S02 impacts of coal burning sources, taking into
explicit account the statistical variability of the sulfur content of coal.
We also describe the coding and use of a computer program to implement the
methodology.
Variability in the sulfur content of coal derives from many causes. The
basic source of variability is within the coal itself as it is mined. Within
a mine, coal varies in its sulfur content from seam to seam and also within
individual seams. This variability is changed by handling between the mine
and the plant, and by subsequent treatment to prepare the coal for combus-
tion: movement from one stockpile to another, transportation, cleaning, and
crushing. The size of the combustion facility also has an effect on varia-
bility because of the averaging effect of large coal sample sizes. Finally,
the technology and performance characteristics of emissions control systems
installed at the plant depend on the coal composition as well as other fac-
tors. Thus, the control system affects the variability of S02 emissions from
the stack.
In assessing the impact of coal-fired facilities, current practice
involves the use of a mathematical simulation model of S02 emissions, trans-
port, and dispersion. In contrast to appropriate physical modeling
approaches, such models, in principle, enable calculations to be made of
ground-level concentrations, given plant emissions and meteorological data.
However, use of such a mathematical model introduces uncertainties into the
impact assessment as a result of the model's deficiencies. These should be
taken into account. In the work described in this report we have used the
EPA's single-source (CRSTER) model (EPA, 1977), which is a single-source
6-+/21
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Gaussian model. This model is recommended by the EPA for assessing S02
impacts from isolated point sources in flat terrain situations.
The EPA, as part of its legal responsibility to manage air quality, has
established National Ambient Air Quality Standards (NAAQS) and Prevention of
Significant Deterioration (PSD) increments.* Both the short term NAAQS and
PSD increments for S02 are defined in terms of concentration levels not to be
exceeded more than once per year. Sources are not permitted to cause or con-
tribute to the exceedance of these standards more than once per year. The
numerical values of these standards are shown in table 1.
TABLE 1. APPLICABLE FEDERAL AIR QUALITY
STANDARDS FOR S02
SO? Concentrations
(
Standard 3-Hour Average 24-Hour Average
PSD Class I 25 5
PSD Class II 512 91
PSD Class III 700 182
NAAQS 1300 365
If S02 emissions never varied and a perfect air quality model were
available, it would be possible to use that model, employing a year's meteoro-
logical data, to compute the second highest concentration at any point around
the plant and to judge whether this value exceeded the federal standard. The
problem with this approach in the context of stochastically varying emissions
is choosing an appropriate emissions rate to use in applying the model. The
second highest concentration computed using an average emissions rate requires
further adjustment if it is to represent the anticipated second highest con-
centration (since about 50 percent of the time emissions will be above the
average). On the other hand, if an upper-extreme emissions rate is used, it
can be argued that combining an unusually high emissions rate with unusually
poor meteorological conditions (the second worst in the year) is too stringent
to use for an assessment of attainment. What is evidently needed is a
methodology by which the stochastic variations in emissions rates are combined
with the day-to-day variations in meteorology to give a picture of the like-
* With the passage of the 1977 Clean Air Act Amendments, the PSO
increments originally established by the EPA were codified into law.
8W21
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lihood of violating the standard in a given year. Note that including the
uncertainty in emissions rates eliminates determinism in the model results, so
that we would not be able to state that the second highest concentration would
be, say, 400 yg S02An3, but rather that the probability of having two S02 con-
centrations greater than 400 ug/m3 in the course of a year would be, say, O.I
(or 10 percent). Stating the result in this way modifies the current practice
of judging acceptability, because an allowable probability of more than one
exceedance* of the standard must be specified (since this probability cannot
be reduced to zero). Thus, the method explicitly recognizes that, regardless
of the extent to which emissions are reduced, there always remains a finite,
however small, probability of exceeding the air quality standard (or
increment) during periods of unfavorable meteorological conditions.
i
The technique by which we combine the uncertainty in emissions rates with
the results of dispersion model computations is a Monte Carlo simulation.
This technique simulates the effects of random variations in a quantity by
randomly selecting values from an appropriate probability distribution and
accumulating results over many samples to obtain a picture of the form that a
long-term series of values should take.
The methodology we have developed utilizes a probability distribution of
3-hour or 24-hour average S02 emissions. This distribution can be obtained
from data collected on emissions estimates or coal sulfur content, or as esti-
mates based on plant design or operating characteristics. Meteorological
variability is treated by utilizing a multiyear (currently 5- to 10-year)
meteorological record of wind speed, wind direction, and atmospheric sta-
bility. Using these meteorological data and the EPA CRSTER dispersion model,
3-hour or 24-hour average estimates of atmospheric dispersion are calculated
*
This procedure would take into account both the meteorological and
emissions fluctuations. It should be noted that the fluctuations in
meteorology, as estimated by changes in ambient concentration levels for
a fixed emissions rate, are significantly greater than typical fluctua-
tions in emissions about the mean. Some may argue that, because of
this emissions fluctuations are unimportant. However, though emissions
fluctuations may be relatively unimportant in determining mean ground-
level concentrations over, say, a period of a year, such emissions
fluctuations will be very important in determining maximum ground-level
concentrations.
We recognize that the word "exceedance" is a coinage. However, we
believe that its use is sufficiently common in air quality data
analysis to convey clear meaning. Therefore, throughout this report
we have used "exceedance" to denote the observation or computation
of a ground-level concentration exceeding the level set in the
applicable standard, and "violation" to denote the occurrence of
two or more exceedances in a single year.
8W21
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for the entire meteorological record for a preselected array of critical
receptors, usually 180.
For each 3-hour or 24-hour average dispersion estimate in the meteoro-
logical record and the receptor array, an estimate of the appropriately aver-
aged SO^ emissions is randomly drawn from the emissions distribution and
combined with that dispersion estimate to produce a corresponding estimate of
the ground-level SOg concentration. This 3-hour or 24-hour average S02 con-
centration is compared with the applicable ambient standard, and a notation is
made if that concentration exceeds the standard. The same procedure is fol-
lowed to give sets of concentrations at each receptor for each time period
throughout the year. The number of exceedances of the standard, and whether a
standard violation (two or more exceedances) occurred at any receptor during
the year, is recorded. This simulation of a year's concentrations is then
repeated many (currently 1000) times to give an estimate of the long-run rate
of occurrence of standard exceedances and violations. This methodology thus
leads to estimates of both the expected number of exceedances of the appli-
cable standard at each receptor and the probability that a violation can be
expected to occur. In addition, we derive an estimate of the probability that
a violation will occur somewhere in the network of receptors in the vicinity
of the source. All of these estimates are provided for each meteorological
year and over all meteorological years.
In this chapter we have provided a brief introduction to our newly devel-
oped methodology. In chapters 2 and 3 we present more detailed descriptions
of the method and its theoretical development. Chapter 2 is pictorial in its
presentation; chapter 3 requires the reader to have some acquaintance with
elementary probability theory, though the main structure of the method can be
understood without such prior knowledge. Reference to texts such as Parzen
(1960) can supply any needed background. In chapter 4 we give details of
using the methodology to evaluate two hypothetical power plants. The two
plants are identical except that one emits a scrubbed plume and the other an
unscrubbed plume. Differences in impacts are, therefore, the result of dif-
ferences in exit temperature and velocity (and thus plume rise) rather than in
S02 emissions rates. In chapter 5 we list a series of concerns relevant to
the general problem of assessing exceedances and violation probabilities. We
point out those items that we have not covered in the present work and suggest
that they be considered for future study. The report concludes with an appen-
dix that gives a complete description (constituting a user's manual) for the
computer program that we developed to implement the methodology. The program
is written in standard FORTRAN and should be implementable on most computers.
8W21
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II PICTORIAL OVERVIEW OF THE EXEX METHODOLOGY
In the ExEx method a Monte Carlo simulation is the technique employed to
combine the variability in emissions rates with the results of dispersion
model computations. Stated simply, this methodology simulates the effects of
random variation in a quantity by randomly selecting values from an appropri-
ate probability distribution and accumulating results over many samples. An
estimate of how an actually realized long-term series of values should look is
thereby obtained.
All of the essential components of the ExEx methodology are depicted in
figure 1:
> A stochastic treatment of point source emissions
> A suitable atmospheric diffusion model
> A (Monte Carlo) simulation model.
Figure 1 shows the data and input requirements for each of these components,
as well as a summary of the results of the methodology and the use of these
results. The output of the simulation model is expressed in terms of exceed-
ances and violations:
> Expected (or average) number of exceedances of the ambient
standard(s) for each .receptor point specified in the dif-
fusion model for each meteorological year and for all
meteorological years.
> Probability of violating the ambient standard. Three
probabilities are provided: The first, provided for each
receptor and year, is the probability that two or more
exceedances will occur in that meteorological year. The
second is the average of the foregoing probabilities
(receptor violation probabilities) over all meteorological
years. The third is the probability that there will be
two or more exceedances at one or more receptors (in the
array of receptors) over the meteorological period. This
last quantity can be viewed as the probability that a vio-
8H/21
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Receptoi Array
Multi-yeat Meteoro-
logical Record
Fixed Operating Load
fixed iUnit) Emissions
Sfac* Paramet
ATMOSPHERIC
DIFFUSION
MODEL
X
STOCHASTIC
POINT SOURCE
EMISSIONS
TREATMENT
01
Coal Dalj
or
Design Data
Ambient
SlandarJtsI
SIMULATION
MODEL
\
RESULTS
Multi-year & single year for
• £ *pected number ot exceedances
o' ambient standard lor each
roc eplor
• Probability ot violating ambient
st.indard lor nach receptor and lor
around source
Specilicaiion ot
• AllowsblG number ot GMceedances. or
• Atlo*tabl9 violation probability
USE OF RESULTS
Specifications of emissions in
terms of:
• Annual average
• Not to be exceeded in any year
FIGURE 1. ELEMENTS OF THE SIMPLIFIED EXEX METHOD
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lation of the ambient standard will occur anywhere (or
somewhere) in the vicinity of the source.
These results can be used to determine allowable annual average emissions
and an emissions limit not to be exceeded in any year in the life of the
facility. Each of the principal elements of the methodology will be discussed
briefly.
Figure 2 illustrates the input and output of the atmospheric dispersion
model. Use of the model follows EPA-recommended procedure (EPA, 1977). It is
convenient to run the model with unit emissions (i.e., 1.0 Ib S02/MMBtu). It
is also convenient to view the output of the model as matrices (or tables)
consisting of pairs of 24-hour-average and 3-hpur-average normalized ground-
level concentrations for each year in the meteorological record. For five
years of meteorological data there would be 10 such matrices—five tables of
3-hour averages and five tables of 24-hour averages. Each matrix contains a
normalized concentration for each receptor and each time period. Thus, for a
24-hour time period, assuming the model is CRSTER, which utilizes 180 recep-
tors, there are 365 (or 366) x 180 entries in the matrix. For the 3-hour time
period, there are 2920 (or 2928) x 180 entries in the matrix.
Figure 3 depicts the input requirements and output options of the sto-
chastic treatment of source emissions. As noted in this figure, either stack
emissions, coal, or engineering design data may be used to generate a fre-
quency distribution of S02 emissions. This distribution can take the form of
either an analytical expression, currently assumed to be lognormal, or of a
table of values that specifies the number of occurrences for each S02 emis-
sions value. The averaging period for the S02 emissions data should match
that used in the dispersion model, i.e., 3-hour emissions data for 3-hour
ambient standards, and 24-hour emissions data for 24-hour ambient standards.
For the purposes of this report, we will assume that the 3-hour and 24-hour
S02 emissions distributions will be identical. For even larger sources and
coal sulfur variability that is not too large, it is logically conceivable
that 24-hour emissions could equal the annual average emissions. We are
unaware, however, of any data that support this possibility. In other situa-
tions, additional uncertainties can be introduced if the 3-hour and 24-hour
distributions are assumed to be identical, but at this time the magnitude of
such uncertainties is unknown.
A key feature of the methodology is the derivation, from the pollutant
emissions distribution and meteorological conditions, of the probability with
which the ambient standard will be exceeded in any averaging period. The cal-
culation proceeds as follows (for more detail see chapter 3): Suppose that,
for a particular period, the average ground-level concentration computed using
the CRSTER model is one-third the level of the standard. Then, because of the
linear relationship between emissions and concentrations included in CRSTER,
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Receotor Array
Multi-year Meteoro-
logical Record
Ei*ea Operating Load
in emissions
Slack Parameters
24-HOUR AVERAGE
GROUND-LEVEL NORMALIZED
CONCENTRA TIONS
ATMOSPHERIC
DIFFUSION
MODEL
1-HOUR AVERAGE
GROUND LEVEL NORMALIZED
CONCENTRATIONS
CO
Ctl -- (x/Q)
Normalized concentration
lor each receptor
tor each year.
I
FIGURE 2. ATMOSPHERIC DIFFUSION MODEL
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\
STACK EMISSIONS DATA
. S02 emissions rale
• Total heal input rate
• Averaging time and
sampling period
\
\
COAL DATA
. Percent sullur
. Mass ol sultur
. Heat value per unit mass
. Averaging lime and
sampling period
DETERMINE
FREQUENCY
DISTRIBUTION
OF
S02 EMISSIONS
\
ENGINEERING DESIGN DATA
\
ANALYTICAL
^EXPRESSIONS
2 a
Iba. S02 par mmBTU
_ -^ \
\
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It can be seen that, if the emissions rate used as input had been three times
higher, a concentration equal to the standard would have been computed. Thus,
for this averaging period, the probability of exceeding the standard is equal
to the probability of exceeding three times the emissions rate used for the
CRSTER calculations. This probability can be derived from the (known or
assumed) probability distribution of emissions rates. Use of the CRSTER model
makes this computation particularly easy because of the linear scaling of
emissions to concentrations, but, in principle, any model could be used so
long as there is a one-to-one correspondence of emissions and concentrations
(assuming this relationship is known).
Figure 4 illustrates how the results from both the dispersion model and
the stochastic SOg emissions treatment are used in the simulation model. The
example in figure 4 is based on comparison with a 24-hour-average ambient
standard. The upper left-hand portion of figure 4 shows the matrices of 24-
hour-average ground-level concentrations normalized to unit S02 emissions.
Each normalized concentration is designated by the symbol C^j, with the index
i denoting the day of the year and the index j denoting the receptor loca-
tion. The upper right-hand portion of figure 4 illustrates that 1000 vectors
are generated from the SQ^ emissions distributions, each vector containing 365
(or 366) randomly selected 24-hour-average S02 emissions. Each emissions
vector is designated 0^, with the index k specifying the particular sample.
For all receptors on each day in the meteorological year, the emissions value
corresponding to the appropriate day is used to scale the normalized concen-
trations, C^j. This results, for all receptors on that day, in a set of 24-
hour-average ground-level S02 concentrations, assuming that the randomly
selected S02 emissions occurred. The set consists of 365 (or 366) x 180 24-
hour S02 concentrations. This scaling process is repeated 1000 times for the
meteorological year and generates a three dimensional matrix consisting of
365 (or 366) x 180 x 1000 values. This three-dimensional matrix is depicted
in the lower portion of figure 4. For each year of meteorological data, one
such matrix (box) is generated. Thus, for a five year meteorological record
there are 5 such matrices, so that a total of 5 x 365 (or 366) x 180 x 1000
values are generated. Note that for the 3-hour ambient standard and a five
year record, 5 x 2920 (or 2928) x 180 x 1000 values are .generated. The pro-
gram implementing the methodology incorporates a screening technique to reduce
the number of operations so that only meteorological conditions that can
possibly lead to exceedances are considered. In a typical computer run, only
10 to 1000 values per meteorological year would be considered in the calcula-
tions.
The elements of these boxes (see figure 5) are used to provide estimates
of the expected number of exceedances of the ambient standard and of the
probability of violating the ambient standard. For each year, each receptor,
84/21 10
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y«»r
19xx
Ytar
1B7S
Year
1974
Year
1973
180 Receptor*
Normalized concentrations
lor each day, lor each
receptor and lor lint year;
1973
SCALE NORMALIZED
CONCENTRATIONS
1973 1973
C/y -Ofc = *//*
•fe-\
'•OK Ail «£C5PTOflS
WO ALL DA C3
MS PROCESS
ttPEATeO 10QQTIU£S
(^
v-
^S/BO RECSPTOFIS
\fOfi EACH YZAft
365 Of) 366 MHDOMi. Y
SFtfcreo f wssiotts.
2T
/ 366 OAAX AVfRA&E
( CQNCfHTRtriONS
\ F OK EACH ftSCSP TOfi
\
\
\
k
*»,».»
*2,1,1
*365,1,1
j
m
-------
FINAL YEAR
FIRST YEAR
T«0 RECEPTORS
Expected number ol etceedances lor last year —
180 RECEPTORS
Expected number ol exceedancet lor first year
Latt Year
180 RECEPTORS
Probability ol violating standard lor lint year
Expected number ol exceedancet
averaged over all yean
Probability of violating itandard '80 RECEPTORS
avenged over all year*
Probability ol violating ttandard
anywhere In the receptor array
Each trial is a simulation of 365
(or 366) days' concentration at
each receptor
Out of one trial for first year, a
sample 24-hour average concen-
tration at one receptor
For each year expected number of
exceedances and probability of
violation at each receptor
For all years expected number of
exceedance and violation proba-
bilities averaged over all years on
meteorological record
FIGURE 5. ANALYSIS OF EXCEEDANCES AND VIOLATIONS FROM EXEX METHODOLOGY
-------
and each trial, the expected number of exceedances is obtained. The expected
number of exceedances is the total number of exceedances of the ambient stan-
dard, after correcting for background,11 divided by 1000. The probability of
violating the ambient standard is also noted. For each receptor the proba-
bility of a violation is defined as the total number of occurrences of two or
more exceedances in a year for each trial, divided by the number of trials.
The expected number of exceedances for each receptor and the probability
of a violation for each receptor are also estimated for the entire meteoro-
logical period (record). This is done by averaging the expected number of
exceedances and violation probabilities over all years in the period. The
derivation of these average results is also shown in figure 5. In addition,
an estimate is made of the probability of violating an ambient standard any-
where in the array of receptors; this probability is defined as the total
number of trials (over all years in the meteorological period) for which there
are two or more exceedances at one or more receptors in the receptor array.
The method is applicable in principle to any point source of a nonreac-
tive pollutant (or any pollutant that can be treated as nonreactive).
Although the current version of the method utilizes the CRSTER model, this is
not a requirement for the application of the statistical methodology. Any
model that can be used to give dispersion estimates for each period in a
meteorological record could, in principle, be used to obtain the inputs to the
statistical model. In fact, if experimental dispersion estimates are avail-
able, these can also be used for input.
Availability of a computer program makes the application of the method-
ology practical. Current experience shows that costs of applications are com-
parable to, or less than, those of the CRSTER model.
The methodology can also be used to evaluate different regulatory sce-
narios. Currently, emissions limits are not uniform across states: Both
averaging time and allowed frequency of exceedance vary. The effect of dif-
ferent regulatory environments can be evaluated, provided meteorological and
emissions information is available at an appropriate level of temporal
averaging.
* A trial is defined as a simulation of 365 (or 366) days' concentrations
at each receptor.
* Here "background" is used to represent the ambient concentration that
would exist in the absence of the source under consideration.
13
-------
Ill THEORETICAL MATTERS
In this chapter we describe the theoretical development of a statistical
methodology for evaluating the impact of SC^-emitting point sources. This
method has become known as the ExEx, or expected exceedances, method. The
name is appropriate because, instead of deterministic calculation of predicted
ground-level S(>2 concentrations using a constant emissions rate, the method
employs a Monte Carlo technique to take explicit account of the stochastic
variation inherent in coal sulfur contents and the concomitant variation in
emissions rates.
The development of the ideas in this chapter is, as implied in the title,
theoretical. We believe that it is necessary to detail the development of the
method in the body of the report rather than in an appendix, since the
requirement that a Monte Carlo approach be used derives directly from the
theory. The reader who wishes to know how the method works, and not how it is
derived, could skim this chapter and rely on chapter 2 for a description of
the method (or skip directly to chapter 4).
To obtain the necessary meteorological input for application of the
method, we have used EPA's single-source (CRSTER) dispersion model {EPA,
1977). We point out here, however, that use of this particular model is in no
way required for the application of the statistical technique. Any method,
whether it be modeling or measurement, that will yield the required meteoro-
logical inputs can be used. Thus, the methodology described is flexible in
its accommodation of current and future dispersion models.
The development of the ExEx method is accomplished in three stages.
First, we develop expressions for calculating the number of exceedances of the
standard and the probability of a violation, assuming that meteorological con-
ditions remain constant throughout a year. This assumption enables a simple
development of the theory, illustrating many of the salient points of the ExEx
method. In the second section, we extend the development to include con-
sideration of meteorological variation. In the third section, we show the
64/21
-------
development of the ExEx method as it is implemented; particularly we show why
use of a Monte Carlo technique is necessary for the method.
In the fourth section we discuss briefly some other topics that were
studied in the course of our development of the ExEx method. These topics are
not part of the method's development, but they are important in the context of
point source impact assessment. First, we discuss the form of the probability
distribution of coal sulfur contents and sulfur dioxide emissions rates.
Next, we examine briefly the statistical consequences of some alternative
forms of emissions limits. This subject is relevant, since the evaluation of
the point source under consideration is obviously critically dependent on the
form of the selected emissions limit. The principles used for this analysis,
however, can be used to analyze any regulation once its form is stated.
Finally, we consider an extension of the main problem considered in this
report. Although our main concern in the report is evaluation of the impact
of single sources on their surroundings, in many situations several sources
contribute to pollutant concentrations at a given receptor point. We employ a
well-known statistical inequality to derive a lower bound to the probability
that emissions from multiple sources will not exceed a given value.
A. EXPECTED EXCEEOANCES AND VIOLATION PROBABILITY WITH CONSTANT
METEOROLOGY
The ExEx method is based on the computation of the numbers of exceedances
of the applicable S02 ambient air quality standard (referred to subsequently
in this report as "the standard"). The number of exceedances at a given
receptor point over a specified time period is not a fixed number. It varies
from day to day, month to month, and year to year, depending on, among other
factors, the meteorological conditions and the variation in the sulfur content
of the fuel (coal) as it is burned. For any specified time period, we could
not calculate the exact ground-level concentration that will be observed, even
if we had available a perfectly precise air quality simulation model and very
precise input data for that model. This uncertainty can be thought of as
arising from either the imperfect state of our knowledge of atmospheric pro-
cesses, or the inherently statistical nature of those processes. However, we
can state the number of exceedances that will occur, in probabilistic terms,
as an expectation.
The concept of expected value is well known in the field of probabil-
ity. Formally, the expected value of a random variable X is denoted by E(X)
and is given by
* Use of the Monte Carlo analysis is unnecessary if only results from
individual receptors are of interest; in this case the analytical
expressions developed in Section B are directly applicable.
'21 15
-------
E(x)
x Pr(X = x) = 2 *Px(x) if X is a discrete
all x all x random variable
/if x is a continuous
V ' random variable (i)
where Pr(X x) is the probability that X takes the value x, and PX(X)
or fx(t) ^s the Probability density function of the random variable X. If
many samples of the random variable are taken and the mean x is
computed, 7 tends to the expectation of random variable X as the number of
samples becomes large. That is,
= E(X) . (2)
Returning to our consideration of S02 distributions, for a specified
probability law describing the distribution of fuel (coal) sulfur contents, we
may infer that the distribution of St^ emissions has a corresponding distribu-
tion.* If meteorology were fixed and constant for every day, the ground-level
concentrations would also have a corresponding distribution.
Considering just one day, let us assume that the probability of observing
an S02 concentration above the standard is p. Then the formula for expecta-
tion given in eq. (1) can be used to compute the expected number of exceed-
ances for that day as follows: The random variable appropriate to this situa-
tion is a binomial (i.e., two-valued) variable that takes the value 1 if an
exceedance occurs, and 0 if no exceedance occurs. Thus the random variable
takes the value 1 with probability p and the value 0 with probability (1 - p),
and its expectation is given by
The correspondence between these two distributions can be quantified by
making assumptions about control technology effectiveness and relia-
bility. Given these assumptions and information about the distribution
of coal sulfur contents, the distribution of emissions can in principle
be derived.
16
-------
E(X) xPx(*) (3)
1 • p + 0 • (1-p)
or
E(X) = p (4)
Thus, the expected number of exceedances for the day under consideration Is
p.* If the probability of observing an exceedance were equal to p for all 365
days In a year (this could only occur if meteorology were fixed and constant),
the expected number of exceedances for the year' would be 365 p (or 366 p for a
leap year).
We will now estimate the probability that a violation of the standard
will occur during the year (i.e., the probability that more than the allowable
number of exceedances will occur). For S02, a violation is currently defined
as two or more exceedances during the year. For this definition of a viola-
tion, it is not sufficient to compare 365 p to 2, since 365 p is merely the
expected number of exceedances in a year. The actual number of exceedances
during a given year is a random variable (whose expectation is 365 p).
Thus, the occurrence of a violation will be evaluated as a probability.
Considering all 365 days in the year, and taking the probability of an exceed-
ance to be p on each day, we can estimate the probability of a violation as
Pr(violation) = Pr(2 exceedances) + Pr(3 exceedances)
+ ... + Pr(365 exceedances)
365
Pr(i exceedances)
The individual probabilities are given by the properties of the binomial
distribution. The following equation gives the probability of observing
i "successes" (exceedances) in n independent trials (days for a 24-hour stan
dard), assuming consecutive rather than running averages, with a constant
probability of success p at each trial:
* Note that p is less than 1, that is, the expected number of exceedances
for the one day is less than 1, though the actual number observed must
be either 0 or 1. The situation is analogous to die rolling, where the
expectation at each roll is 3.5, though this is not a realizable
outcome.
8W21 17
-------
p(1;n) - (J) p^l-p)"'1 • , (5)
where is the binomial coefficient and is equal to
i!(n-ijr
probability of observing a violation of the standard during the year, assuming
a constant of probability of exceedance p every day, is given by
365
Pr(violation) - £ (3J5) pVp)365"1 . (6)
i=2
This series has 364 terms and, though the terms generally can be assumed to
decay fairly rapidly, it is not worthwhile to evaluate eq. (6), since a simple
analytic expression is available. To derive this expression we first note
that the probability of a violation can be restated as
Pr(violation) = 1 - Pr(no violation)
* 1 - Pr(0 exceedances) - Pr(l exceedance) . (7)
From eq. (5), we have
'n \ 0
Pr(0 exceedances) = (_]p (1 - p)
- (1 - p)° (8)
and
Pr(l exceedance) = (J) P*U - p)""1 (9)
= np(l - p)""1 .
Combining eqq. (7), (8), and (9), we have
Pr(violation) = 1 - (1 - p)n - np(l - p)"'1
- 1 - (1 - p + np)(l - p)"'1 (io)
- 1 - [1 + (n - l)p](l - p)"-l
at/21
18
-------
For n - 365, eq. (10) becomes
Pr(violation) = 1 - (1 + 364 p)(l - p)364 . (11)
Figure 6 shows the probability of a violation given in eq. (11) as a function
of p. This figure reveals, for this simplified but instructive example, that
when the probability of exceeding the standard approaches 8/365 (i.e., 8
exceedances expected in one year), a violation of the standard becomes virtu-
ally certain. Note, however, that the analysis to this point does not repre-
sent a sufficiently realistic situation from which to draw conclusions about
the probability of violations, since we have assumed that the probability of
an exceedance is the same for every day during the year. In fact, of course,
this probability is not constant, but varies because of changes in meteoro-
logical conditions. This additional factor will be discussed in the following
section of this chapter.
There is one further point to make with the simple example illustrated in
figure 6. For this case with constant daily exceedance probability, the
probability corresponding to two expected exceedances in the year is 2/365, or
0.00548. This probability, when substituted in eq. (11), gives a probability
of violation of 0.595 (cf. figure 6). Thus, assuming an expected exceedance
of 2--which naive consideration might intuitively suggest to be virtually
certain to correspond to a violation—a probability of violation of only about
60 percent is computed. From this result we may infer further that, if the
daily probability of an exceedance is such that in the long run 2 exceedances
per year are expected, in 40 percent of those years no violation will occur.
B. EXPECTED EXCEEDANCES AND VIOLATION PROBABILITY WITH VARIABLE
METEOROLOGY
We now extend our analysis to the more realistic situation of variable
meteorology and a time-varying probability of exceeding and violating the
standard. First, we see that the expected number of exceedances for the whole
year is simply the sum of the expected exceedances for all time periods during
the year. That is, for a daily (i.e., 24-hour) time period,
365
Pi •
1=1
34/21 19
-------
0.002 0.004 0.006 0.008 0.010 0.012 0.014
Probability of an Exceedsnee on Any Day
0.016 0.018 0.020
FIGURE 6. PROBABILITY OF A VIOLATION OF THE STANDARD AS A
FUNCTION OF DAILY EXCEEDANCE PROBABILITY,
ASSUMING CONSTANT METEOROLOGY
-------
where E is the expected number of exceedances for the year and p^ is the
probability of an exceedance on day i.
To calculate the probability of a violation, we first note that eq. (7)
still applies, that is,
Pr(violation) = 1 - Pr(0 exceedances) - Pr(l exceedance)
As above, let pi be the probability of an exceedance on day i, so that 1 - p,
is the probability of not observing an exceedance on the same day. Then the
probability of observing zero exceedances throughout the year is the product
of the individual daily probabilities, assuming that the probability of an
exceedance on any day is statistically independent of the probabilities on any
other day:
Pr(0 exceedances) - (1 - Pj)(l -
P2)
• U -
(13)
365
Further, the probability that exactly one exceedance will be observed during
the year is the sum of the probabilities of observing an exceedance on day j
and observing no exceedances on any other day. This sum is given by
365
£>,
0=1
"365
TT
i=l
Pr(l exceedance) = / M p., /T ^1 - P^ • (^
\
Equation 14 can be written more conveniently, provided PJ * 1, so that
365
Z^j
r~n
365
TT »-',
provided pj * 1 for all j, or, referring back to eq. (13),
, (15a)
8W21
21
-------
365
Pr(l exceedance) =
Pr(0 exceedances) , (15b)
J-l
Thus, combining eqs. (7), (13), and (15), we have
365
Pr(violation) = l -
1=1
(1 - Pj) -
365
y Pj
7? X"PJ
"365
n <> - -,>
.i=l
365
365
J=l
(16a)
provided p,- *1 for all j.
Equation (16a) can be modified to make calculation more convenient if the
pi's are supplied in tabulated form. If there are n^, ng, .... n^ time peri-
ods with exceedance probabilities PJ, P£, ... p^, where
365
eq. (16a) can be rewritten
Pr(violation) = 1 - (1 - p.)
• (16b)
provided p^ * 1 for all j.
ewzi
22
-------
If some p^s are equal to 1, the equation takes the less convenient form
si
365
Pr(violation) = 1 -
(1 - p.)
which reduces to
365
p.
365
365
Pr(violation) =
365
TT"
1=1
(16c)
since the second term must be zero if any p-j = 1 (the probability of having no
exceedances during the year must be zero if there are some days when an
exceedance is certain).
These expressions for the probability of a violation (eqq. 16a, b, and
c), it must be emphasized, all apply only when it can be assumed that emis-
sions rates for any time period are statistically independent of those for any
other time period. This assumption is also included in the current implemen-
tation of the ExEx method, but other assumptions could be substituted.
Having derived expressions for the annual expected exceedances and viola-
tion probability in terms of the daily exceedance probabilities, we now derive
these probabilities from the results obtained from an atmospheric dispersion
model combined with a knowledge (known or assumed) of the probability distri-
bution of sulfur dioxide emissions or coal sulfur contents.
Atmospheric dispersion models are used to relate pollutant emissions to
ground-level concentrations. They take into account emissions rate, atmos-
pheric transport and dispersion, and, in some cases, other processes such as
chemical reactions and wet and dry deposition. For an inert pollutant (and,
in the present context, we consider S02 to be inert), which is emitted from an
elevated point source, the Gaussian dispersion model is commonly used. This
formulation assumes a continuous emissions source, a steady-state downwind
plume, and a Gaussian distribution of pollutant concentrations in both the
crosswind and vertical directions. The ground-level concentration is given by
23
-------
x =
WO 0 U
y z
exp
(17)
where the wind is advecting the plume at a speed u along the x-axis; plume
dispersal In the crosswind and vertical directions is parameterized by the
coefficients oy and o^, respectively. The pollutant emission from the source
is at a uniform rate Q and is assumed to be released at an effective stack
height H (which depends on the velocity and temperature of the emissions).
The type of dispersion model around which the ExEx method is designed
produces time-averaged concentrations at a number of points around the source
(called receptors), for an extended period (one year or more). This is
achieved by taking a formula such as eq. (17), substituting continuously
observed meteorological data, and averaging the computed concentrations for
the required time {typically, for SOg, 1 hour, 3 hours, or 24 hours). Thus,
concentrations at various receptors fluctuate according to the wind direction
and other meteorological factors. As noted earlier, the model we have used
exclusively in the present work is the single-source (CRSTER) model (EPA,
1977).
Using a dispersion model such as the single-source
vides a set of ground-level concentrations for an array
point source for a specified averaging period. However,
rate is assumed for this application;' thus the computed
simply values for atmospheric dispersion (averaged over
est) (x/Q)^jt* multiplied by the nominal emissions rate
lations. That is, the computed concentration in period i at receptor j is
(CRSTER) model pro-
of receptors around a
a constant emissions
concentrations are
the period of inter-
Q* used in the calcu-
««- (*)„ *
(18a)
or, alternatively, the dispersion in period 1 is given by
(18b)
In using these models for this application, it is convenient to exercise
them with a unit emissions rate, since concentrations computed by CRSTER
are linearly related to emissions rate [see eq. (17)].
Note that (x/q) here represents a variable rather than a mathematical
expression.
8W21
24
-------
The emissions rate corresponding to a concentration x is given by
. (19)
The particular emissions rate of interest in these calculations is that rate
which will produce an S02 concentration just equal to the standard, Xgf
Allowing for a background concentration B^ in period i, we obtain from
eq. (19) the emissions rate that will result in a computed concentration in
period i at receptor j which is just equal to
Q.
(20)
Combining eqq. (18) and (20) yields
where
Q-. = the S02 emissions rate in period i that will result
in a concentration just equal to the applicable
standard X at tne J"th receptor.
B- - the background concentration during time period i (in
1 all applications to date B^ has been assumed to be
constant both spatially and temporally, but this
assumption is not necessary to the development of our
analysis), •
x- • = the concentration computed by the dispersion model
1J for time period i at receptor j,
Q* = the nominal emissions rate used in the dispersion
calculations.
* Note that if B< is not constant, the transformation Ascribed by
eq (21) is noi necessarily monotonic, that is, xfcj > xti does not
necessarily imply that Qkj < Q^- (k * *)-
84/21 25
-------
The probability of exceeding the standard during,the time period i at receptor
j is just the probability that the emissions rate in that period is greater
than Q.JJ. This probability is obtained from the distribution of S02 emis-
sions rates, f(q), as
P.
f(q)dq . (22)
Expected exceedances for the year and the probability of a violation may then
be obtained by substituting the p-jj's calculated from eq. (22) into eqq. (12)
and (16), respectively. Carrying out this substitution yields the following
expression for the expected number of exceedances in a year for the averaging
period of interest:
365 365
f(q)dq . (23)
i^l i=l
C. FORMULATION OF THE EXEX METHOD
As outlined in the previous section, once the daily exceedance probabili-
ties are known at a specific receptor, the expected number of exceedances per
year and the probability of a standard violation for that (j-th) receptor may
be calculated. However knowing the individual probabilities of violations
occurring at 180 receptors separately does not seem to yield a reasonable
basis for evaluating the S02 impact of a power plant. For instance, is it
better to have a moderate probability of a violation at one receptor (say 0.4)
and zero probability everywhere else, or to have a small probability of viola-
tion (say 0.05) at 8 or 10 receptors and a 0.001 probability everywhere
else? What appears more appropriate is some simpler, overall measure of the
plant's impact. Specifically, a simpler measure is the probability that in
any year, or over several years, a violation that is attributable to the
facility will occur somewhere in the vicinity of the facility. This measure
seems to provide a satisfactory resolution of the problem in that attention is
shifted from individual receptors to the total area around the plant that
experiences S02 impacts as a result of the plant.
8«t/21 26
-------
We now turn to the calculation of the probability that a violation will
occur somewhere in the vicinity of the plant, given the 180 individual viola-
tion probabilities (derived from results of a Gaussian model such as
CRSTER). First we restrict the problem by noting that we can only calculate
an overall probability based on the specific receptor locations available,
that is, we can compute the probability of a violation somewhere in the net-
work of receptors. However, this restriction is only theoretical, since we
may assume without loss of generality that we can locate receptors at all
points of substantial interest.*
To develop an expression for the probability that a violation will occur
somewhere in the network, we first recall some results from probability theory
(see, for example, Parzen, 1960). If we denote" a series of events as E^,
Eg, ... and the probability of occurrence of these events as Pr(Ej),
Pr(E2) ... , the following result is obtained from the axiomatic definition of
the probability function: PrfEj u E2) Pr(E}) + Pr(E2) if E1 E2 = *. That
is, the probability of the union of two mutually exclusive events is the sum
of their probabilities. However, this definition is not useful in the present
case since the events of an exceedance at different receptors are not neces-
sarily mutually exclusive—obviously exceedances can occur at two receptors
simultaneously if they are located appropriately (i.e., sufficiently close to
one another). We therefore use the formula
E2) Pr(E!) + Pr(E2) = Pr(Ej n E2) , (24)
which applies without restriction to two events, Ej and E2, defined on the
same probability space. This expression can be extended to a large number of
events as follows:
Pr(E1UE2u...uEn)
i=l i < J
' *' * '
i < j < k
(25)
* The question of appropriately locating receptors when using dispersion
models is discussed in the ERA'S Guideline on Air Quality Models (EPA, 1978),
8W21 27
-------
where E^ is the complement of the event E^. The probability given in eq. (25)
can be evaluated if the events E^ are statistically independent:
n
Pr (E: ) . (26)
i=l
Combining eqq. (25) and (26), and recalling that
P(EC) 1 - P(E)
we have
P(E1U E£U ... U En) = 1 - Jj [1 - P(E.)] . (27)
Unfortunately, the events of interest for the present application are not
necessarily independent. Moreover, the degree of nonindependence of these
events is dependent on the characteristics of each individual problem
studied—in particular, the density of the receptor array and the meteor-
ology. This relationship can be illustrated by a simple set of examples.
Consider two receptors, each with a violation probability of 0.3. Appli-
cation of eq. (27) yields an estimate of 0.51 [= 1-{1-0.3)(1-0.3)] for the
probability that a violation will occur at one or the other, assuming statis-
tical independence (we will label this case A). Next, consider the receptors
to be colocated, so that when a violation occurs at one, a violation always
occurs at the other. (In probability terms, the conditional probability of a
violation at receptor 2, given a violation at receptor 1, Pr^lEj), is 1.0,
and vice versa.) Obviously, in this case Pr(EjU £3) ~ 0.3 (label this
case B). Conversely, suppose that the receptors are located such that viola-
tions never occur at both simultaneously, that is, Pr(E2|Ej) = 0. In this
case it is obvious that Pr(Ej u £3) 0.6 (case C). Thus, we obtain different
answers for the probability of a violation occurring at one or the other
receptor, depending on the nature of the dependence between the two receptors.
28
8«f/21
-------
These three examples can be clarified if the joint distribution of viola-
tions at the two receptors is considered. In case A (in which eq. [27] is
applicable), the occurrence of a violation at each receptor was assumed to be
statistically independent of events at the other. Thus, by definition, the
conditional probabilities are given by
and
Pr(E2|E1) = Pr(E2I
That is knowledge of whether a violation has occurred at one receptor yields
no information on the likelihood of a violation occurring at the other. This
is illustrated by the following tabulation of exceedance probabilities:
TABULATION OF VIOLATION PROBABILITIES
FOR CASE A: THE INDEPENDENT CASE
Violation
Receptor 2 No Violation
Total
Violation
0.09
0.21
0.3
Receptor 1
No Violation Total
0.21
0.49
0.3
0.7
0.7
29
8W21
-------
In this tabulation, the probability of the simultaneous occurrence of viola-
tions at receptors 1 and 2 is 0.09 (the product of the two marginal probabili-
ties, 0.3 x 0.3 - 0.09). Symbolically,
Pr(Ej n E2) -
Pr(E2)
which is required by the definition of statistical independence. The proba-
bility of a violation at receptor 1, given that a violation occurs at receptor
2, is 0.09/0.3 = 0.3. That is,
Pr(E, n EJ
(28)
From the tabulation, it can be seen that the probability of a violation at one
receptor ^r_ the other is 0.09 + 0.21 + 0.21 = 0.51, as before.
Turning to case 8 in which the receptors are colocated, the tabulation
now becomes:
TABULATION OF VIOLATION PROBABILITIES FOR
CASE B: CONDITIONAL PROBABILITY = 1
Receptor 2
Violation
No Violation
Total
Receptor 1
Violation
0.3
0.0
0.3
No Violation Total
0.0
0.7
0.3
0.7
0.7
8W21
30
-------
The tabulation shows that the probability of a violation at receptor 1, given
the occurrence of a violation at receptor 2, is 0.3/0.3 1.0, and the proba-
bility of a violation at one receptor or the other is 0.3 + 0.0 + 0.0 = 0.3.
Case C results in the tabulation:
TABULATION OF VIOLATION PROBABILITIES FOR
CASE C: CONDITIONAL PROBABILITY 0
Receptor 2
Violation
No Violation
Total
0.3
Receptor 1
Violation No Violation Total
0.0
0.3
0.3
0.4
0.3
0.7
0.7
Here the conditional probability Pr(E1|E2) 0.0/0.3 = 0.0, and the proba-
bility of a violation at one receptor or the other is 0.3 + 0.3 + 0.0 = 0.6.
To recapitulate these examples, in the first we considered the events at
each receptor to be statistically independent of one another. This is an
unrealistic assumption, since the receptors are dependent_via_ their relative
locations and the prevailing meteorological conditions. In the second and
third examples, dependencies were assumed to be known, and the desired proba-
bility of a violation at either or both receptors could be calculated. These
two examples, with conditional probabilities of 1.0 and 0.0, are special
cases, and, in general, the conditional probabilities can have any value
between 0.0 and 1.0. The desired probability of violation can be evaluated,
however, for any conditional probability, so long as it is known. For
instance, suppose that, as before, Pr^) = Pr(E2) - 0.3, and Pr{E1|E2) =
0.5. That is, given a violation at receptor 2, there is an even chance of a
violation also occurring at receptor 1. The tabulation for this case is:
8W21
31
-------
TABULATION OF VIOLATION PROBABILITIES:
CONDITIONAL PROBABILITY- 0.5
Receptor 1
Violation No Violation Total
Violation 0.15 0.15
Receptor 2 No Violation 0.15 0.55
0.3
0.7
Total 0.3 0.7
From the tabulation, it can be seen that in this case the probability of a
violation at receptor 1 or_ receptor 2 is 0.15 + 0.15 + 0.15 » 0.45.
An equation for the probability of a violation at one or both of two
receptors can be derived from eqq. (24) and (28):
U E2) PT{E!) + Pr(E2) - Pr(E1|E2) Pr(E2) . (29)
However, we cannot generalize eq. (29) to evaluate the probability of a viola-
tion somewhere in the receptor network because we do not know the conditional
probabilities. As stated earlier, these conditional probabilities depend on
relative receptor locations and meteorological conditions. Therefore, we use
the results of the CRSTER calculations to represent the day-to-day
dependencies of concentrations throughout the receptor array, and we use a
Monte Carlo technique to evaluate the effect of these dependencies on the
probability of a violation occurring somewhere in the vicinity of the source.
The Monte Carlo calculation is carried out by creating a large number
(currently 1000) of samples of a year's emissions by making random draws from
the probability distribution of emissions. (The appropriate form of the emis-
sions distribution is discussed elsewhere in this report.) The (randomly
selected) values for daily emissions are then combined with estimates of
atmospheric dispersion obtained from the single source (CRSTER) model to yield
ground-level concentrations at each receptor location. The number of exceed-
ances of the standard, and whether a violation of the standard (two or more
exceedances) occurred at any receptor during the year, are noted. This proce-
dure yields estimates of the numbers of exceedances and occurrences of viola-
tions at each receptor for the 1000 samples from the emissions distribu-
tions. Expected exceedances are obtained by summing over all samples and
dividing by 1000. Numbers of samples for which violations were recorded are
noted for each receptor individually and for the network as a whole.
-------
D. OTHER TOPICS
1. The Probability Distribution of S02 Emissions
As eq. (22) shows, calculation of the probability that the air quality
standard will be exceeded involves integrating a frequency (probability)
distribution of S02 emissions rates. This operation raises the following
question: What is an appropriate distribution to use? If specific informa-
tion for the source under study can be obtained, the best course is to use the
actual distribution. But if such specific information is not available, or if
a hypothetical case is being investigated, some emissions distribution must be
assumed.
Previous work (Rockwell, 1975; PEOCo, 1977; Foster, 1979) has centered on
determining distributions for sulfur contents of coal, which should be
directly related to S02 emissions distributions. Three distributions have
been studied: the normal, the lognormal, and the inverted gamma. In the
latest study, none of these distributions gave consistently better statistical
fits to data, but the lognormal was the closest of the three.
Consideration of the characteristics of coal burned at power plants leads
one to expect that, except in the rare case where a power plant utilizes coal
purchased from a single supplier (mine), no one standard distribution is
likely to adequately describe the emissions from a single plant. Purchases
will be made with the objective of meeting an applicable emissions limit.
Thus, purchases of relatively high sulfur coal from one source will be
balanced by purchases of low sulfur coal from another source. The resulting
distribution of emissions could very well be bimodal and not adequately
represented by any standard distribution.
In the work described in chapter 3, we have used lognormal emissions
distributions of emissions from two hypothetical power plants. This choice is
not based on our own data analysis or on any theoretical foundation; rather it
is one of convenience and custom. We believe that the results obtained are
reasonable and are applicable in a wider context.
2. A Brief Consideration of Emissions Limits
SOo emissions from point sources are regulated by imposition of emissions
limits. However, the effect on recorded emissions from a particular facility
of a prescribed emissions limit is dependent not only on the numerical value
of the limit, but also on the degrees of flexibility available to the regu-
lator with respect to emissions limits. These include the averaging time over
BW21 33
-------
which emissions are defined and the allowable frequency for exceeding the
limit. Since emissions are a stochastically varying quantity, no matter how
tightly the emissions from a plant are controlled, there will always remain a
finite probability of exceeding the imposed limits. This probability can be
made arbitrarily small if coal with a low enough sulfur content is burned, or
if scrubbing efficiency is extremely good. The probability of exceeding the
emissions limit can be made to approach zero by manipulation of the high emis-
sions end of the distribution (e.g., by "early warning" information systems),
or by rejecting noncompllance fuel. Our purpose in this section is to point
out how one can calculate emissions limit exceedance probabilities for
selected types of regulations. We use 24-hour averaging of emissions through-
out, but the same procedures apply to any averaging time, with appropriate
changes.
If the emissions limit is defined as a value not to be exceeded more than
once during a year, this requirement may be interpreted as implying that the
limit must only be exceeded with a probability of 1/365. Assuming the emis-
sions rates to be distributed lognormally, and following Larsen (1971), it can
be shown that this requirement further implies that the geometric mean (GM) of
the emissions distribution must be less than
m=-EL
(GSD)2-94
where EL = emissions limit and 6SD - the geometric standard deviation of the
SOg emissions distribution. Thus, the operator of this facility should buy
coal and operate a scrubber such that the geometric mean of the distribution
of the plant's S02 emissions Is less than m.
Another possible definition of the same emissions limit would be to
specify the mean of 24-hour values. In this case, the plant operator would
arrange coal purchases such that the long-term mean of the emissions distribu-
tion was equal to or less than the emissions limit. In actuality, such a
regulation might result in emissions that were slightly lower than in the
previous example, since facility operators might be inclined to institute a
margin of safety by purchasing coal that would result in mean emissions of,
say, 80 percent of the limit.
Another possible regulation would be to set the emissions limit as a 30-
day average and allow no more than two exceedances per month. If we let the
daily probability of an exceedance be constant and equal to p, and assume that
emissions for each day are independent of those for other days, we can write
(assuming a 30-day month):
8*1/21
34
-------
Pr(more than 2 exceedances)
= 1 Pr(0, 1, or 2 exceedances per month)
=1 (1 - p)30 . 30p(l - p)29 . (30) p2(1 . p)28
=!-(!- p)28[(l - p)2 + 30p(l - p) + 435p2]
1 . (1 . p)28(! _ 2p + P2 + 3Qp - 30p2 + 435p2)
= 1 - (1 - p)28(l + 28p.+ 406p2} . (30)
We can use eq. (30) to calculate the annual expected exceedances corresponding
to any desired probability of observing more than two exceedances per month.
For instance, if we set the probability of more than two exceedances at 1/13,
i.e., an expectation that two exceedances will occur in a given month just
less than once per year, we find that this probability corresponds to 12.4
expected exceedances per year. This result is obtained by setting the left
hand side of eq. (30) to 1/13 and solving for the daily probability of an
exceedance; this operation yields p 0.034. Thus the annual expected exceed-
ances are 365 x 0.034 = 12.4. Conversely, we calculate that if only one
exceedance per year is to be allowed, the corresponding probability of two
exceedances per month is only 0.00008. If running averages are used for the
regulation, the problem is more complicated because of the lack of indepen-
dence between successive exceedance probabilities.
This brief discussion is intended to indicate a need for careful examina-
tion of the consequences of any contemplated emissions limit regulatory stra-
tegy. Because of the variability inherent in a plant's emissions, the effect
of different forms of emissions limits is not always predictable without a
probabilistic analysis.
3. The Multiple Source Problem
One major problem in applying the methods described above to real situa-
tions is that of multiple sources. Whereas all of the development so far has
been restricted to consideration of a single source (or colocated multiple
sources; CRSTER is limited to a single plant, though that plant may have many
stacks, all considered to be colocated), many potential applications could
involve multiple sources. This extension of the method remains to be studied,
but we discuss here some simplified considerations that serve to describe the
nature of the problem.
Suppose n point sources exist, all emitting S02 at identical rates and
with identical frequency distributions. Suppose further that the emissions
35
-------
are Independently and lognormally distributed, with each plant having a mean
emissions rate u and variance a . We wish to calculate the probability that
the total of emissions from the plants is greater than Bnu, i.e., some factor
3 greater than the total mean emissions rate from all plants.
We first state Chebyshev's inequality:
Pr[|x - E(X)| < AS] >1 -±7 , (31)
where E(X) is the expected value of the random variable X, X is an arbitrary
constant, and S is the standard deviation of X. This inequality gives a lower
bound on the probability that the absolute value of the difference between a
random variable and its mean will exceed an arbitrary multiple of its standard
deviation. In this example, we put X = Jx-j, where the x^ are the emissions
from each individual plant, and S2 = n 0.95 . (33a)
yj >0
.1
Then note that if the distribution of £x. - ny is symmetrical, eq. (33a)
becomes i
- nut < (B- l)np >0.90 . (33b)
pl
Equation (33b) is equivalent to eq. (31) if
(B- l)nii= W)1/2 (34)
36
84/21
-------
and
1 - ^o- = 0.90
r
or
X -
Substituting X = vTo~ into eq. (34), we obtain
- (3 - l)ny *10 (na
or
(35)
/n
Thus, we can conclude that if there are 10 sources contributing to concentra-
tions at a particular location, and all 10 sources have identical emissions
distributions, there is at least a 95 percent probability that at any given
time the total emissions will be less than 1 + o/w. The term a/v is the coef-
ficient of variation of the individual emissions distributions (sometimes
called the relative standard deviation). Since typical values of this quan-
tity are about 0.2, we could state that there is at least a 95 percent proba-
bility that total emissions from all 10 plants will not be more than 20 per-
cent greater than the total mean emissions from the plants. This calculation
illustrates the result of the averaging effect of many plants, where if one
plant is emitting a large amount of S02, some other plant will be emitting
less to compensate. This considers only the variability of sulfur in the
fuel; the operating load is assumed constant in the above analysis.
A further refinement can be made by using the restricted form of Cheby-
shev's inequality (Cramer, [1946]; Mallows, [1956]):
Pr[|x - M{X)| < AS] >1 --^ . (36)
9X
37
-------
This form of the inequality holds only where an assumption of unimodality for
the distribution of X can be made, and the difference is measured from the
mode, M(X), instead of from the expected value. Working through the same cal-
culation as above, we find that A= 2.11 and
-1*^6) - (»>
Thus, for n = 10,
1 + 0
.67 (J) . (38)
Thus, by invoking an additional assumption of unimodality for the emissions
distribution, we can state that there is a 95 percent probability that the
total emissions from 10 plants will not be more than 13.4 percent greater than
the total mean emissions rate (assuming o/u 0.2).
38
8W21
-------
8W21
IV RESULTS
In this chapter we detail calculations made using the methodology devel-
oped in chapters 2 and 3. (The computer program is described in an appendix,
along with required input and sample outputs.) The calculations described in
this chapter were carried out for two hypothetical power plants. One of these
plants has a scrubber and the other does not; the presence of the scrubber
reduces the stack exit temperature and stack exit velocity. The characteris-
tics of each plant are shown in table 2. Note that the same emissions rate
was used in both sets of calculations; thus differences in computed X/Q values
are the result of different plume rise characteristics for the two plants.
Each plant is, in effect, utilizing fuels (coal) having different sulfur
contents. For the unscrubbed plant to have the same emissions rate as the
scrubbed plant, it must be burning higher sulfur coal.
Calculations of x/Q were made using the single source (CRSTER) model. By
setting Q - 1 lb/106 Btu for the model runs, the calculated concentrations of
X can be considered equal to x/Q. Concentrations for any other emissions
rate, Q', are then given by
Q' .
Q
calc
The CRSTER preprocessor runs were made using meteorological data from St.
Louis for the years 1973 through 1977 (Lee, 1979). These computed x/Q values
were then used as input to the ExEx method, and the exceedances and violations
of the selected standard concentrations were noted. Because of the linear
concentration scaling with emissions of CRSTER results, runs made with differ-
ent standard concentrations for comparison could also be thought of as runs at
the same standard concentration with coals having different mean sulfur val-
ues: For instance, a run using a standard concentration of, say, 100 gg/m3
and a coal producing mean emissions of 1.2 Ibs S02/106 Btu would produce
exactly the same results as a run with a standard of 200 ug/m3 and mean emis-
sions of 2.4 Ibs S02/106 Btu. That is, doubling all emissions rates results
in doubling all computed concentrations and, thus, it produces exactly the
same number of exceedances and violations of a standard set at double the
original. Because of this property, the graphs, shown in this chapter, of
probabilities of violation of a particular standard as a function of mean
sulfur content can easily be converted into graphs of probabilities of viola-
39
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TABLE 2. CHARACTERISTICS OF HYPOTHETICAL POWER PLANTS
Parameter
Size 1000 Mw (Two 500 Mw units)
Stack height 500 ft (typically good engineering practice)
S02 emissions Lognormally distributed
Stack exit temperature 175'F (scrubbed plant)
285"F (unscrubbed plant)
Stack exit velocity 13.14 m s"} (scrubbed plant)
15.42 m s"1 (unscrubbed plant)
Emissions rate 567 g/sec for each stack
corresponding to
1 Ib S02/MMBtu
Other Characteristics
St. Louis meteorology for period from 1973-1977
Flat terrain assumed
40
-------
tion for a given coal evaluated at various standard levels.
The cases studied were (for the scrubbed plume) emissions distributions
with GSD's of 1.1, 1.2, and 1.4, and (for the unscrubbed plume) emissions dis-
tributions with GSD's of 1.05, 1.15, and 1.25. The rationale for these
choices follows. The plant with the scrubbed plume was considered to be
representative of the emissions ceiling for new plants meeting the New Source
Performance Standard (NSPS) of 1.2 Ibs S02/MMBtu. Geometric standard devia-
tions of emissions distributions ranging from 1.1 to 2.3 have been calculated
for coal-fired power plants (ICF Incorporated, 1979); the great majority of
values fall in the range 1.1 to 1.4. Therefore, we chose this range of values
for our study of scrubbed plumes. The plant 'with the unscrubbed plume was
presumed to be representative of existing power plants with higher average
emissions not subject to NSPS emissions limitations. For a selected period in
the past, coal burned at the Labadie power plant had been found to have a mean
emissions rate of 4.8 Ibs S02/MMBtu and a GSD of 1.15 (Morrison, 1979). More-
over, the ICF work cited above had found GSD's of 1.04 - 1.18 for S02 emis-
sions distributions at the inlet to a scrubber. (Emissions at the inlet to a
scrubber should be representative of emissions from an unscrubbed plant; at
that point no S02 has been removed from the effluent gases.) Thus, 1.05 to
1.25 was chosen as a range covering the GSD of emissions distributions for an
unscrubbed plume.
The scrubbed power plant was studied using two concentration standards:
50 yg/m3 and 91 ug/m3. The latter is the PSD increment for Class II areas
(24-hour averaging time), and the former represents a case where only part
(4556) of this PSD increment is available for the plant's emissions. The
unscrubbed case was studied using concentration standards of 250 vg/mj and
325 wg/m3. These represent comparisons with the 24-hour NAAQS (365 vg/nr)
with constant background concentrations of 115 wg/m3 and 40 wg/nr, respec-
tively. As we pointed out above, however, by suitable rescaling the results
can be transformed to apply to any desired standard.
A. CRSTER RESULTS
Highest second high concentrations computed using CRSTER for each year's
meteorology are shown in table 3. These concentrations were calculated assum-
ing an S02 emissions rate of 1 Ib S02/MMBtu; to compute a concentration cor-
responding to any other rate, we merely multiple these numbers by that rate.
Thus, we can calculate that, ignoring other sources, in 1973 the scrubbed
plant would have violated the 24-hour average PSD Class II increment whenever
its emissions rate exceeded 1.95 Ibs S02/MMBtu (= 91/46.76), and that the
unscrubbed plume would have violated the 24-hour NAAQS whenever its emissions
exceeded 13.97 Ibs S02/MMBtu (365/26.13).
41
-------
TABLE 3. CRSTER RESULTS FOR THE HYPOTHETICAL
POWER PLANT SIMULATIONS
(Concentrations in wg/nr corresponding to
a unit emissions rate of 1 Ib S02/MMBtu)
Year of
Meteorology
1973
1974
1975
1976
1977
Highest Second High Concentration
In Receptor Network
Scrubbed Plume
46.76
39.85
59.01
45.24
39.19
Unscrubbed Plume
26.13
24.97
29.43
25.66
27.21
8%/21
42
-------
It may seem paradoxical that the scrubbed plume should yield higher com-
puted concentrations than the unscrubbed plume. However, since all CRSTER
computations were carried out using the same SC^ emissions rate (1 Ib/MMBtu),
the observed differences are all caused by different exit temperatures, and
thus different calculated plume rise (see table 2). The unscrubbed plume,
with a 110°F higher exit temperature, gives greater calculated plume rise, and
thus the plume is carried farther from the plant before its effect is observed
at ground level. Over this distance the plume has diffused more, and thus the
highest estimated concentrations are lower. Since an unscrubbed plant's emis-
sions can be 5 2/3 times those of a similarly sized scrubbed plant (assuming
85 percent scrubbing efficiency), the ratio between actual computed 1973 con-
centrations using real emissions rates can be 3.17 times higher for the
unscrubbed plant (= 5.67 x 26.13/46.76). Higher scrubber efficiencies would
result in even greater differences. (Of course, it is unlikely that scrubbed
and unscrubbed plants would burn coal with the same sulfur content.)
B. EXEX RESULTS
As outlined above, the CRSTER results computed using unit emissions were
used to represent five years of 24-hour-averaged dispersion x/Q values. The
yearly sequences were combined with randomly selected sequences of emissions
rates using the program described in appendix A. Expected exceedances and the
number of violations per 1000 sample emissions distributions were computed for
each of the 180 receptors in the CRSTER network. In addition, the number of
sample emissions distributions for which a violation was computed anywhere in
the receptor network was recorded, and this provided the probability of a
violation occurring anywhere around the plant. As discussed in chapters 2 and
3, this is intended to give a realistic assessment of the overall impact of
the power plant S02 emissions.
In carrying out the computations, we kept the sulfur emissions distribu-
tion constant and varied the standard against which computed concentrations
were compared. As discussed above, this is entirely equivalent to calcula-
tions carried out for a given standard with the appropriate mean sulfur diox-
ide emissions rates. We present a summary of the results of these computa-
tions in tables 4 and 5. In these tables, we show both the expected exceed-
ances at the worst site and the probability of a violation occurring somewhere
in the receptor network, averaged over the five-year meteorological record.
The data are plotted in figures 7 and 8, where the relationship of expected
exceedances at the worst receptor site and violation probabilities are presen-
ted as a function of emissions rate.
The mean emissions rates quoted are the arithmetic means of the distri-
butions and are given by exp(w + <£/2), where w = «n (geometric mean) and o
8 it/21
43
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BLE 4 EXPECTED EXCEEOANCES AT THE WORST SITE AND VIOLATION
PROBABILITY FOR THE SCRUBBED PLUME
(a) Calculations based on S02 emissions rate distribution
parameters: geometric mean—1.2 Ibs/MMBtu, geometric
standard deviation—1.1
Standard
(ng/rn3)
45
50
65
70
80
91
105
120
Expected
jxceedances
1.61
0.98
0.44
0.31
0.20
0.20
0.17
0.06
Violation
Probability
0.992
0.758
0.205
0.168
0.024
0.0
0.0
0.0
(b) Calculations based on S02 emissions rate distribution
parameters: geometric mean—1.2 Ibs/MMBtu, geometric
standard deviation—1.2
Standard
^i*}/m3)
50
60
70
80
91
100
120
140
Expected
Exceedances
1.11
0.64
0.34
0.20
0.18
0.16
0.08
0.02
Violation
Probability
0.912
0.383
0.195
0.078
0.011
0.002
0.0
0.0
44
-------
TABLE 4 (Concluded)
(c) Calculations based on S02 emissions rate distribution
parameters: geometric mean--1.2 Ibs/MMBtu, geometric
standard deviation—1.4
Standard Expected Violation
) Exceedances t Probability
55 1.11 0.963
70 0.49 0.448
80 0.25 0.225
91 0.18 0.096
105 0.12 0.024
120 0.09 0.006
140 0.05 0.001
160 0.03 0.0
45
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TABLE 5. EXPECTED EXCEEDANCES AT THE WORST SITE AND VIOLATION
PROBABILITY FOR THE UNSCRUBBED PLUME
(a) Calculations based on SOg emissions rate distribution
parameters: geometric mean—4.8 Ibs/MMBtu, geometric
standard deviation—1.05
Standard
(ug/m3)
100
112
125
135
150
175
200
225
Expected
Exceed ances
1.54
1.22
0.77
0.45
0.31
0.20
0.20
0.10
Violation
Probability
1.0
0.999
0.586
0.210
0.020
0.0
0.0
0.0
(b) Calculations based on S02 emissions rate distribution
parameters: geometric mean—4.8 Ibs/MMBtu, geometric
standard deviation—1.15
Standard
100
118
135
150
175
200
250
300
Expected
Exceedances
1.78
1.01
0.58
0.36
0.19
0.16
0.04
0.00
Violation
Probability
1.0
0.919
0.414
0.127
0.008
0.0
0.0
0.0
46
-------
TABLE 5 (Concluded)
(c) Calculations based on S02 emissions rate distribution
parameters: geometric mean—4.8 Ibs/MMBtu, geometric
standard deviation—1.25
Standard Expected Violation
(ug/m3) Exceedances Probability
100 1.70 1.0
c
125 0.94 0.886
137 0.66 0.608
150 0.45 0.318
200 0.14 0.008
225 0.10 0.001
250 0.06 0.0
300 0.02 0.0
47
-------
£n (geometric standard deviation). The arithmetic mean is a measure of long
term (for instance, annual) average emissions rates, and thus is a relative
measure of the total S02 discharged to the atmosphere by a facility.
A few comments on the computations are in order.
First, for a given concentration level, the receptor site with the high-
est expected exceedances is not necessarily the site with the highest viola-
tion probability. Second, if expected exceedances are plotted against viola-
tion probabilities for many examples, the points do not fall along a smooth
curve; rather, there is considerable scatter. Thus, there is no simple uni-
versal relationship between these two measures that holds for all cases. (See
figures 9 and 10 for examples of these plots.) These effects result from the
upper tails of the meteorology distributions having large discontinuities if
one or more days (periods) during the year have significantly worse dispersion
than all others. In an extreme case, one particularly bad day in an otherwise
"good" year (one with no more very bad dispersion values) can result in one
"guaranteed" exceedance every sample year but never in a violation, because
the next worst dispersion value cannot cause an exceedance even with the
highest possible emissions rate.
A further effect noted in the results is that when the five-year-average
violation probabilities are plotted against mean emissions rates (e.g.,
figure 8a), a smooth relationship is not always found; "knees" in the curves
for the scrubbed plume case were often observed at about 0.2 on the proba-
bility scale. The apparent reason for this effect is that, in the CRSTER
calculations for the scrubbed plume, one year (1975) was markedly worse than
the other four (see table 3), and thus the portion of the violation proba-
bility curves at low emissions rates is largely derived from 1975 results.
Figures 7 and 8 are self-explanatory, but some examples may clarify the
uses to which they may be put. Suppose we arbitrarily select an
("acceptable") violation probability of 0.1 for the Class II PSD increment of
91 ug/nr; that is, we accept a facility whose expected violation frequency
anywhere in the vicinity of the plant is once every 10 years. Then from
figure 8b, if a coal with an emissions rate 6SO of 1.2 is burned in a scrubbed
power plant, the mean emissions rate must be less than 1.45 Ibs S02/MMBtu in
order not to exceed this probability. Referring to figure 7b, this apparently
corresponds to expected exceedances of 0.28 at the worst site in the receptor
network, or to approximately one exceedance of the Class II PSD increment
every four years. Furthermore, using the results in table 2, and using the
mean emissions rate of 1.45, a highest second high concentration in the worst
year (1975) of 86 ng/m3 (= 1.45 x 59.01) would be computed using the CRSTER
model. The emissions rate expected to occur with a frequency of once per year
is 2.44 Ibs S02/WBtu - } exp [tn 1.45 - (* ^'2) 1 • 1.22-94 , which
8W21
48
-------
STANDARD: 50 yg/m
0.5
1.0 1.5 2.0 2.5
Mean Emissions Rate (Ibs S02/MHBtu)
(a) 6SD of S02 Emissions Distribution: 1.1—Scrubbed Plant
FIGURE 7. EXPECTED EXCEEDANCES
-------
s
1.8
1.6
1.4
vi 1 ?
41 i • *•
1 1.0
0.6
0.4
0.2
0
STANDARD:
50 vg/n
0.5
1.0 1.5 2.0 2.5
Mean Emissions Rate (Ibs S02/MMBtu)
3.0
(b) GSD of SO Emissions Distribution: 1.2—-Scrubbed Plant
FIGURE 7 (Continued)
-------
1.8
1.6
1.4
Ml 1 9
«i t • t
u
1 i.o
u
s.
0.8
0.6
0.4
0.2
STANDARD: 50 yg/n
0.5
1.0 1.5 2.0 2.5
Mean Emissions Rate (Ibs S07/MMBtu)
(c) GSD of S02 Emissions Distribution: 1.4—Scrubbed Plant
FIGURE 7 (Continued)
-------
U1
ro
1.8
1.6
1.4
1.2
1 i.o
I 0.8
u
,2 0.6
0.4
0.2
0
STANDARD: 250 yg/mj
6 8 10
Mean Emissions Rate (Ibs S02/MMBtu)
12
(d) 6SD of S02 Emissions Distribution: 1.05—Unscrubbed Plant
FIGURE 7 (Continued)
16
-------
tn
<*»
12
4 6 8 10
Mean Emissions Rate (Ibs SO^/MMBtu)
(e) GSD of S02 Emissions Distribution: 1.15—Unscrbbbed Plant
325 yg/nf
16
FIGURE 7 (Continued)
-------
2.2
2.0
1.8
1.6
jjj 1.4
1 '•«
•o
0)
0.8
0.6
0.4
0.2
0
X
UJ
STANDARD:
250 p(
4 6 8 10
Mean Emissions Rate (Ibs SO^/MHBtu)
12
325 Mg/md
16
(f) GSD of S02 Emissions Distribution: 1.25— Unscrubbed PUnt
FIGURE 7 (Concluded)
-------
(71
tn
1.0
91 ug/mj
0.5
1.0 ' 1.5 2.0 2.5
Mean Emissions Rate (Ibs S02/MHBtu)
(a) GSD of S02 Emissions Distribution: 1.1—Scrubbed Plant
3.0
FIGURE 8. VIOLATION PROBABILITY
-------
l.Or
01 '
01
0.5
1.0" 1.5 2.0 2.5
Mean Emissions Rate (Ibs S02/MMBtu)
(b) GSD of S02 Emissions Distribution: 1.2—Scrubbed Plant
FIGURE 8 (Continued)
-------
1.0
tn
1.0 1.5 2.0 2.5
Mean Emissions Rate (Ibs S02/MMBtu)
(c) 6SD of S02 Emissions Distribution: 1.4--Scrubbed Plant
FIGURE 8 (Continued)
3.0
-------
i.o
0.8
0.6
o
0.4
0.2
STANDARD:
I
4 6 8 10
Mean Emissions Rate (Ihs SO«/MMBtu)
12
14
(d) GSD of S02 Emissions Distribution: 1.05—Unscrubbed Plant
16
FIGURE 8 (Continued)
-------
1.0
STANDARD:
0.8
JO
•0
X)
o
Q.
O
0.6
tn
0.
6 8 10
Mean Emissions Rate (Ibs SO^/MHBtu)
12
14
16
(e) 6SD of S02 Emissions Distribution: 1.15— Unscrubbed Plant
FIGURE 0 (Continued)
-------
CTI
O
4 6 8 10
Mean Emissions Rate (Ibs SO^/MMBtu)
12
(f) 6SD of S02 Emissions Distribution: 1.25—Unscrubbed Plant
14
16
FIGURE 8 (Concluded)
-------
corresponds to a 1975 high second high of 144 ug/m3. That is, if the emis-
sions rate that occurs with probability 1/365 were to occur on the day with
the second worst dispersion during the year, the resulting highest ground-
level S02 concentration around the plant is computed as 144 ug/m3.
If a similar set of calculations is carried out for a 6SD of 1.4, the
mean emissions rate to ensure compliance with the 91 yg/m3 increment (at the
10 percent violation probability) is 1.27 Ibs S02/MMBtu. The expected number
of exceedances at the worst site is 0.18 (i.e., the increment can be expected
to be exceeded approximately once in six years). The mean expected highest
second high concentration is 75 yg/m3, and the highest second high concentra-
tion estimated with the extreme (1/365) emissions is 190 yg/m3.
Comparing the results for the 1.2 and 1.4 GSD values reveals that for the
17 percent increase in GSO (i.e., greater variance in emissions), a 12 percent
decrease in the annual average emissions rate is required to comply with the
91 yg/mj increment at the 10 percent violation probability level. In addi-
tion, this 17 percent higher GSD involves a 32 percent higher worst-case
(i.e., 1/365) emissions and worst-case highest second high 24-hour-average
ground-level S02 concentration (144 yg/m3 to 190 yg/m3).
Calculations such as these can be used to assess the effect of imposing
different emissions limits on a plant. We would emphasize, however, that the
results in this chapter apply only to the particular hypothetical plants and
terrain situation modeled, for the particular five years of meteorological
data. The results cannot be generalized to other circumstances, but they can
be used to determine the direction and order of magnitude of effects resulting
from changes in emissions distributions or other related matters. We believe
that the results given here will be useful in outlining the general relation-
ship between expected exceedances and violation probabilities. Figures 9 and
10 illustrate relationships for the expected number of exceedances at the
worst receptor (over a five vear period) and the average probability of vio-
lating the standard (91 yg/nr* PSD Class II increment) at that receptor, and
anywhere in the network, respectively. In figure 9 it is apparent that, for
these five years of St. Louis meteorology, these hypothetical power plants,
these ranges of emissions GSD values, and an expected number of exceedances of
1.0 at the worst site, the corresponding probability of violating the standard
is approximately 25 percent. Thus, over a long term period, we can expect
approximately one violation of 91 yg/m3 PSD Class II increment every four
years. The results of figure 9 also indicate that for a violation probability
of 10 percent at the worst receptor (i.e., one violation in 10 years at the
worst receptor), the corresponding expected number of exceedances is approxi-
* This worst-case condition is defined as the highest emissions rate and
poorest dispersion occurring on the same day.
61
-------
-------
mately 0.6 (the standard is expected to be exceeded approximately once every
two years).
The results in figure 10 indicate that, for an expected number of exceed-
ances of 1.0 at the worst receptor site, it is virtually certain that a
violation will be predicted by the model at at least one receptor. However,
for a predicted 10 percent probability of violations at one or more receptors
in the network, the expected number of exceedances at the worst receptor is
approximately 0.25, i.e., approximately once in every four years an exceedance
of the PSD Class II increment (91 yg/m3) can be expected at that receptor.
The probability of violating an ambient standard is sensitive to changes
in either the mean emissions rate or variability in emissions rates. Perusal
of figures 7 and 8 reveals interesting features of this sensitivity. It is
generally observed that:
> As the annual mean emissions rate varies by 25 percent for
emissions ranging from 0.75 to 5 Ibs S02/MMBtu, the proba-
bility of violating an ambient standard (from 50 to 91
wg/m3) increases from approximately 0.1 to 0.3, i.e., from
one violation in 10 years to one in approximately 3
years. For annual mean emissions rates between 5 and
10 Ibs S02/MMBtu, a 10 percent increase in the emissions
rate leads to a similar expected increase in the violation
probability (i.e., from 0.1 to 0.3).
> As the variability of emissions rates increases, the
probability of violating an ambient standard increases
somewhat for a range in variability from approximately 1.1
to 1.4 (as measured by the 6SO).
> The general shapes of the probability of violations vs.
annual emissions graphs for different GSO values were
similar, each showing an S-shaped curve between a proba-
bility of violation of zero and one, over a range of mean
emissions rates which depended on GSO.
C. SUMMARY
In this chapter we have detailed results obtained using a Monte Carlo
technique for simulating stochastically varying SOg emissions from two power
plants. The results presented enable an assessment of the relationships among
results from a single source Gaussian model, expected numbers of exceedances
of an air quality standard, and the probability of violating that standard
(I.e., observing two or more exceedances) 1n any given year.
8W21
63
-------
in
V
O
c
10
TJ
0)
01
u
X
LlJ
l»-
o
V
0)
1.8
1.6
1.4
0.8
0.6
0
0
Symbol
AA
6SD
1.1, 1.05
1.2, 1.15
Solid symbols represent
the unscrubbed plant.
i t 1
1 1
0.2 0.4 0.6 0.8 1.0
Expected Violation Probability
FIGURE 10. EXPECTED EXCEEDANCES AT THE WORST SITE VERSUS VIOLATION
PROBABILITY FOR THE COMPLETE RECEPTOR NETWORK OVER
A FIVE-YEAR PERIOD
64
-------
V SUGGESTIONS FOR FURTHER WORK
The work described in this report represents a new development in which
the variability inherent in sulfur dioxide emissions from coal-burning facili-
ties is explicitly accounted for in assessing their impact. This method
recognizes the problems with using a constant, maximum emissions rate for this
assessment and specifically accounts for the admittedly low probability that
maximum emissions and worst-case meteorology can occur simultaneously. At the
same time, the technique does not presume that these two events cannot occur
together, but rather assigns a realistic probability to their coincidence.
That we treated only the variability in SO^ emissions from isolated power
plants should not be misinterpreted; two important points must be remembered:
> Emissions of $02 are not the only, or major, cause of
variability in ground-level S02 concentrations.
> SOg emissions from other sources, and pollutants other
than SO, may merit such stochastic treatment.
We limited the scope of this report to a technical description of a relatively
simple version of the ExEx methodology and the presentation of example calcu-
lations for hypothetical power plant situations. This does not mean that we
are unaware of the policy implications associated with incorporating into
impact assessments factors that explicitly contribute to variability in
ground-level concentrations. The issues that can, should, and are being
raised are so numerous, and in some ways so fundamental, that we had to defer
their consideration because of their scope.
However, because of the attention that has been given to the potential
impact of implementing the ExEx methodology, we feel compelled to mention what
appears to be the greatest concern: The apparent increase in S02 emissions
over those allowed by current regulations and modeling (assessment) procedures
that could result, and the consequent increase in atmospheric loadings. We
wish to point out that both the direction (i.e., increase or decrease) and
magnitude of any changes in SOg emissions and atmospheric loadings are inex-
tricably linked to the policies and methods of implementation. As a conse-
quence, control of S02 emissions can be instituted, both in principle and in
practice, so as to increase or decrease $02 emissions or maintain them at the
65
-------
current rate. Having mentioned this issue briefly, we will proceed to address
additional technical topics that merit further investigation.
As noted previously, there are many other sources of uncertainty in the
evaluation of S02 impacts of coal burning facilities. Several such important
sources are discussed briefly:
> The atmospheric dispersion model used. Whether an impact
assessment is carried out as at present, using a constant
S02 emissions rate, or by the ExEx method with variable
emissions, the atmospheric dispersion models used yield
results that are uncertain to some extent because of
approximations inherent in their formulation or implemen-
tation, and uncertainties attributed to model inputs.
Such model uncertainties introduce a corresponding uncer-
tainty into impact assessments. It is recommended that
the magnitude of the effect should be investigated in fur-
ther studies.
> Background pollutant concentrations. In most real situa-
tions the background concentrations of a pollutant are not
constant at a given location, but fluctuate both from
statistically random atmospheric effects and because of
random influences of other nearby sources. These kinds of
effects could be readily incorporated into the ExEx
method.
> Scrubber or other control equipment efficiency. The
scrubber installed at a plant does not operate at constant
efficiency. Study results involving the evaluation of
variations in efficiency should be included in the model
to account for their effect in contributing to emissions
variability.
> Load variations. The emissions from a power plant depend
on the load. Load does not normally remain constant, but
rather fluctuates somewhat to correspond with demand.
Moreover, variations in load are, to a degree, autocorre-
lated, and they exhibit seasonal and diurnal trends. The
effects of these variations on S02 impacts should be
studied.
> Consecutive versus running average. In all of the work
described in this report, consecutive 3-hour or 24-hour
averages have been used, with the periods starting at
midnight on 1 January. However, compliance with the air
8*1/21
66
-------
quality standard is currently evaluated in terms of run-
ning averages—a new averaging period is started in every
hour of the year. Since the use of running averages can
result in higher maximum concentrations, the effect on
violation probabilities should be studied.
> Form of the S02 emissions distribution. As discussed in
chapter 3, no clear choice can be made as to a broadly
applicable form of emissions distribution. Additional
data are needed to establish suitable forms for the S02
emissions distribution. Incorporating different standard
distributions into the ExEx method could enable its sensi-
tivity to the form of S02 emissions distribution to be
evaluated; this would also enable the user to select an
alternative most suited to his application.
> Multiple sources. The current version of the ExEx is
applicable to isolated point sources only. Impact assess-
ments involving interacting sources in either urban or
rural environments are frequently required. Methods for
treating these situations should be examined.
> Random number generation. Implementation of a methodology
containing a machine-dependent process such as random
number generation can cause difficulties. Attention needs
to be given to procedures for ensuring consistency in
impact assessments carried out on different computers.
In addition to the technical issues listed above, there are other sources
of uncertainty in impact assessments, particularly those relating to the
formulation of regulations. For instance, the averaging period over which
emissions are defined, the number of exceedances allowed in a given period
(e.g., a month or a year) and the probability of violating any imposed stan-
dard are all matters that fall to the regulators to consider. The methods
described 1n this report can be used to investigate different types of regula-
tory alternatives and practices. Such studies are recommended with the
expanded capability of ExEx.
67
-------
APPENDIX
THE COMPUTER PROGRAM
68
-------
APPENDIX
THE COMPUTER PROGRAM
The computer program to implement the ExEx method is very simple: annual
emissions distributions are simulated, these distributions are combined with
dispersion values calculated by the Single Source (CRSTER) Model to give
ground-level SO? concentrations, these concentrations are compared against a
standard, and trie number of standard exceedances is counted. However, a few
subtleties must be noted to ensure that the program carries out the computa-
tions efficiently and accurately.
First, it is efficient to avoid storing even one year of concentration
outputs from CRSTER; for 3-hour averages in a leap year there are 366 x 8 x
180 - 527,040 individual values. Efficiency is achieved by storing only those
values that can result in exceedances when combined with possible emissions
values. This is done by computing a quantity called ZERO:
ZERQ = STANDARD CONCENTRATION .^
MEAN SO • (GSD)6
and eliminating those CRSTER concentrations that are smaller than ZERO. As
the formula Implies, ZERO is that x/Q value that results in a concentration
just equal to the standard value, when it 1s combined with an emissions rate
six standard deviations higher than the mean. Since such an emissions rate
occurs with probability ~ 10 , this procedure eliminates potential
exceedances with an upper bound of this probability. If all 527,040 3-hour-
average values were as large as ZERO, the total probability of missing an
exceedance by this procedure would be 527,040 x 10~8, or 0.00527, or around
0.5 percent. In actual fact, of course, the probability is much less; most
concentrations are several orders of magnitude less than the computed ZERO
and, thus, the actual probability of missing an exceedance is much less than
0.5 percent. However, the savings in computer time is spectacular; generally
we have observed that very few concentrations pass this screening procedure—
the number is usually less than a few hundred per year unless the standard
value is exceeded by a wide margin by the maximum concentration.
The second feature we point out is designed to make the program simulate
a lognormal distribution of emissions more accurately. The usual method for
21
-------
generating normally distributed random variables is to sum at least 12 uni-
formly distributed random variables. However, this method Is known to be
inaccurate in simulating the tails of the distribution outside the 30 limits
(Naylor, et al., 1966), or those values that occur with a frequency of 1/1000
or less. However, we are particularly interested in these extreme events,
which are the most likely to produce exceedances of the standard. Therefore,
we chose a method for generating normally distributed variables developed by
Box and Muller (1958), which is known to be more accurate in the distribution
tails. The uniform random number generator shown in the program, RANF(X), is
a congruential generator available as a system routine on the COC7600 com-
puter.
We present a listing of the program in exhibit A-l, with a sample set of
input data shown at the end. In our own version of the program, we store the
CRSTER output files in a series of disc files that are sequentially accessed
by the ExEx program, but this portion of the program can be adjusted to suit
the user. The CRSTER model, as supplied by the EPA, will generate files of
24-hour-average concentrations directly. To generate 3-hour-average concen-
trations, the user should insert an appropriate WRITE statement into CRSTER or
write a small program to average three 1-hour-average concentrations and write
these out to a file.
If the user wishes to study a particular power plant, and a lognormal
distribution of emissions is not appropriate, code can be substituted to
sample from any distribution, even one that is empirically derived. To do
this, we have set up a vector of length 10,000, with the statement
DIMENSION OSTRB (10000)
inserted after line 17. (In exhibit A-l, all code necessary to accommodate a
user-supplied distribution has been Inserted as comments and preceded by
C****.) This vector is filled out with the emissions values as follows. Sup-
pose the distribution to have 0.5 percent frequency of emissions of 2.1 Ibs
S02/MMBtu, 1.3 percent with 2.2 Ibs/NNBtu, 5.6 percent with 2.5 Ibs/MMBtu, and
so on. Then we Insert a DATA statement after line 30, as follows:
DATA DSTRB /SO * 2.1, 130 * 2.2, 560 * 2.5 . . . . /
The way in which ZERO is calculated is also changed. Instead of the standard
code at line 67, insert
ZERO = (STD * l.E-6) / DSTRB (10000)
Then, in place of lines 190 to 195, substitute the following code:
70
8«t/21
-------
DOCK. PROGRAM
C EXEX.PROGRAM
C*******************************************************
C
C PROGRAM TO IMPLEMENT THE EXEX METHOD
C
C WRITTEN BY M.J.HILLYER
C SYSTEMS APPLICATION INC
C 950 NORTHCATE DRIVE
C SAN RAFAEL
C CALIFORNIA 94903
C
c**********************#********************************
c
C CODE TO ENABLE SUBSTITUTION OF AN ARBITRARY DISTRIBUTION
C IS NOTED BY C****
C
C
DIMENSION NEXCD(180), NVIOL(186,9), CONCS(ISO)
DIMENSION IEXCDC180,9), EXPEXC(180,9), IVIOLC9)
DIMENSION NVIOL5(180), IEXCD5<180), EXPEX5C180)
DIMENSION tt(2928), INCR(9), CNCHRC10000), INDEX(10000)
DIMENSION MAX(9>
DIMENSION TITLEC20)
C**** DIMENSION DSTRBC10000)
C
C
LOGICAL LVIOL, THREE
C
C
DATA PI2 /6.2B3185308/
C**** DATA DSTRB S 50*2.1, 130*2.2, 560*2.5, .... '
C
C
C DEFINITIONS OF INPUT VARIABLES
C
C
C SMEAN — GEOMETRIC MEAN OF 802 EMISSIONS DISTRIBUTION
C GSD — GEOMETRIC STANDARD DEVIATION OF EMISSIONS DISTRIBUTION
C STD ~ CONCENTRATION AGAINST WHICH TO DEFINE EXCEEDANCES
C NSIM — NUMBER OF SAMPLES FOR MONTE CARLO PROCEDURE
C < GENERALLY SET TO 1000)
C ISTART— FIRST METEOROLOGICAL YEAR (E.G., 1973, ETC)
C NYRS — NUMBER OF YEARS OF METEOROLOGY AVAILABLE IN CRSTER OUTPUT
C IHCR — STARTING WITH 1 FOR THE FIRST MET. YEAR, THE INCREMENT
C DEFINING EACH SUBSEQUENT YEAR, EC, FOR FIVE SUCCESSIVE
C YEARS, ENTER 12345
C THREE — LOGICAL VARIABLE USED TO DEFINE THE AVERAGING PERIOD
C WHEN THREE IS 'TRUE', 3-HR AVERAGES ARE USED,
C WHEN THREE IS 'FALSE'. 24-HR AVERAGES ARE USED
G
G
G ***** READ DATA CARDS FOR RUN *****
C
G READ IN FREE FORMAT
G
G
EXHIBIT A-l. EXEX PROGRAM LISTING WITH SAMPLE INPUT DATA
71
-------
EXEX. PROGRAM
READ (5.1090) TITLE
READ (5,*) STD, SMEAN, GSD, NSIM, 1START, NYRS
READ (S,*) ( INCR(I),I=1.NYRS)
READ (5,*) THREE
C
C
C *** INITIALISE SOME VARIABLES FOR LATER USE ***
C
C
SSTD ALOG (CSD)
ZERO - (STD * l.E-6 ) ' ( SMEAR * EXP(6*SSTD) )
C**** ZERO = (STD * l.E-6 ) ' DSTR8(10000)
IF (THREE) GO TO 20
NOLEAP - 365
LEAPN =366
GO TO 23
20 ROLEAP = 2920
LEAPN = 2928
25 CONTINUE
DO 49 K = 1. 180
NVIOLSdD = 0
49 CONTINUE
DO 50 K = 1, 180
IEXCDS(K) = 0
50 CONTINUE
IVIOL5 - 0
C
C
C HERE STARTS THE MAIN LOOP, THROUGH THE MET YEARS
C
C THE VARIOUS LOOP INDICES ARE DEFINED AS FOLLOWS:
C
C L YEAR OF METEOROLOGICAL RECORD ( 1...5 OR 80 )
C K RECEPTOR NUMBER ( 1... 180 >
C I MONTE CARLO SAMPLE NUMBER ( 1... 1000 )
C J - INDEX OF 'SIGNIFICANT' CONCENTRATIONS (THESE
C ARE DEFINED BELOW) ( 1 )
C IPERD TIME PERIOD NUMBER ( 1...363,366,2920,OR 2928 )
C INDEX INDEX WHICH IS A COMPOSITE OF IPERD AND K, WHICH
C CAN BE DECODED TO RECOVER THE TIME PERIOD AND
C RECEPTOR NUMBER FOR A GIVEN CONCENTRATION
C
DO 600 L - l.NYRS
LL » L + 10
IYEAR = ISTART + INCRCL) - 1
NPERDS NOLEAP
C
C CHECK FOR LEAP YEAR
C
IF ( MOD( IYEAR,4).Ea.0 ) NPERDS - LEAPN
C
C INITIALISE IN-LOOP VARIABLES
C
DO 87 K = 1,180
NEXCD(K) = 0
87 CONTINUE
EXHIBIT A-l (Continued)
72
-------
EXEX.PROGRAM
DO 88 K = I,180
lEXGD(K.L) = O
88 CONTINUE
DO 89 K = 1, 180
NVIOL(K.L) = 0
89 CONTINUE
DO 90 K = 1,180
EXPEXC(K.L) =0.0
90 CONTINUE
LVIOL = .FALSE.
IVIOL(L) = 0
ICOUNT = 0
C
c
C READ THE CRSTER OUTPUT
C
C
C THE VARIABLE LL IS USED HERE TO DEFINE UNITS FROM
C WHICH THE CRSTER OUTPUT FOR THE VARIOUS YEARS IS
C READ. IF THE USER WISHES, ALL OF THE (POSSIBLY
C MULTIPLE) YEARS OUTPUTS COULD BE STORED ON THE
C SAME FILE AND READ WITH THE APPROPRIATE CODE.
C
C EACH 'READ' TAKES 180 VALUES FROM THE CRSTER OUTPUT
C FILE. THESE 180 VALUES ARE THE COMPUTED CONCENTRATIONS
C AT EACH RECEPTOR FOR ONE AVERAGING PERIOD.
C
C
DO 150 IPERD = l.NPERDS
READ(LL) CONCS
C
C
C CHECK FOR DISPERSION WHICH CAN NEVER RESULT IN AN EXCEEDANCE
C AND ELIMINATE THAT CONCENTRATION (DISPERSION CONDITION)
C FROM FURTHER CONSIDERATION. IF A CONCENTRATION IS
C GREATER THAN 'ZERO' ( CALCULATED ABOVE AS SIX STANDARD
C DEVIATIONS FROM THE MEAN), WRITE IT INTO AN ARRAY FOR
C USE IN LATER CALCULATIONS. FOR EVERY CONCENTRATION
C SAVED, GENERATE AN INDEX WHICH CAN LATER BE DECODED
C TO FIND OUT WHICH TIME PERIOD AND RECEPTOR NUMBER THE
C CONCENTRATION BELONGS TO.
C
C
ITEMP = 181 * (IPERD-1)
DO 140 K ^ 1,180
IF ( CONCSdO.LT.ZERO ) GO TO 140
ICOUNT « ICOUNT + 1
CNCHR( ICOUNT) = CONCS(K> * 1.E6
INDEX(ICOUNT) ITEMP + K
140 CONTINUE
C
C
MAXNO - ICOUNT
MAX(L) HAXNO
150 CONTINUE
C
EXHIBIT A-l (Continued)
73
-------
EXEX. PROGRAM
Q
C IF NO EXCEEDANCES, SKIP TO OUTPUT FOR THIS YEAR
C
C
IF ( MAXNO.EQ.O ) GO TO 55O
C
C
C EACH ITERATION OF THIS LOOP
C SIMULATES A YEAR'S OPERATION
C
C
DO 500 I l.NSIM
C
C GENERATE A SET OF RANDOM EMISSIONS RATES.
C THE METHOD OF BOX AND MULLER IS USED TO
C GENERATE NORMALLY DISTRIBUTED RANDOM VARIABLES.
C
C RANF(X) IS A RANDOM NUMBER GENERATOR
C WHICH GENERATES NUMBERS UNIFORMLY DISTRIBUTED
C ON THE INTERVAL (0,1)
C
DO 200 IPERD = l.NPERDS
Rl - RANF(X)
R2 = RANF(X)
RANDN = SORT (-2.*ALOG( Rl) > * COS (PI2 * R2)
O(IPERD) = SMEAN * ( EXP (SSTD * RANDN) )
200 CONTINUE
C
C
DO 200 IPERD = l.NPERDS
Rl = RANF(X)
I RAND = IFIX ( 10000*R1 ) + 1
Q( IPERD) = DSTRBCIRAND)
C*20O CONTINUE
C
C
C CALCULATE CONCENTRATION, CHECK FOR AN EXCEEDANCE AND,
C IF ONE IS FOUND. INCREMENT THE EXCEEDANCE COUNTER
C NEXCD FOR THE APPROPRIATE RECEPTOR.
C
C
DO 400 J l.MAXNO
IPERD = INDEX(J) / 181 + 1
CHI = CNCHR(J) * OA IPERD)
IF ( CHI.LT.STD ) CO TO 400
K = MOD ( INDEX(J).181 )
NEXCD(K) = NEXCD + 1
400 CONTINUE
C
C FOR EACH RECEPTOR SHOWING TWO OR MORE EXCEEDANCES,
C SET THE LOGICAL SWITCH WHICH INDICATES THAT A VIOLATIOH
C HAS BEEN DETECTED, AND INCREMENT THE VIOLATION COUNTER
C FOR THAT METEOROLOGICAL YEAR.
C
C
DO 420 K 1.180
EXHIBIT A-l (Continued)
74
-------
EXEX.PROGRAM
IF ( NEXCD(K) .LT.2 ) GO TO 420
NVIOL(K.L) = HVIOL(K.L) + 1
LVIOL = .TRUE.
420 CONTINUE
C
C
IF ( LVIOL ) IVIOL(L) IVIOL(L) + 1
LVIOL = .FALSE.
C
C
DO 450 K = 1,180
lEXCD(K.L) = IEXCD(K,L>, + NEXCD(K)
NEXCD(K) = 0
450 CONTINUE
C
500 CONTINUE
C
C
DO 520 K = 1, 180
EXPEXC(K.L) = FLOAT ( IEXCD(K,L) ) ' FLOAT (NSIM)
520 CONTINUE
C
C
C
C WRITE OUT RESULTS FOR YEAR BEING SIMULATED
C
C
550 KSTD - IFIX(STD)
WRITE (6,2000) TITLE
WRITE (6,2100) IYEAR, SMEAN, GSD, KSTD
WRITE (6,2200) (EXPEXC(K,L),K=1,180)
WRITE (6.230O) NSIM, (NVIOL(K.L),K=1,180)
WRITE (6.2400) IYEAR, NSIM, IVIOL(L)
600 CONTINUE
C
C
C ADD UP RESULTS FOR ALL YEARS
C
C
NSIM5 = NSIM * NYHS
DO 700 L = l.NYRS
DO 690 K * 1,180
NVIOL5CIO = NVIOL5(K) + NVIOL(K,L>
IEXCD5(K) IEXCD5(K) + lEXCD(K.L) ,M¥_.
EXPEX5(K) FLOAT ( IEXC05(K> ) / FLOAT (NSIM5)
690 CONTINUE
C
IVIOL5 = IVIOL5 + IVIOL(L)
700 CONTINUE
C
° PROB - FLOAT ( IVIOL5) / FLOAT (NSIM5J * 100.0
C
C
C WRITE OUT SUMMARY RESULTS
EXHIBIT A-l (Continued)
75
-------
EXEX.PROGRAM
C
C
C
C
C
C
C
C
C
C
C
C
C
C
WRITE (6,2000) TITLE
WRITE (6,2500) SMEAN, CSD, KSTD
WRITE (6,2200) ( EXPEX5( K),K=1,1CO)
WRITE (6,2300) NSIM5, ( NVIOL5,K=1,180)
WRITE (6,2600) NYRS, NSIM5, IVIOL5, PROB
WRITE (6,2700) ( L,MAX(L>,L=1,NYRS )
TERMINATE RUN
STOP
FORMAT STATEMENTS *****
1000 FORMAT(20A4>
2000 FORMAT( 111 1,20A4)
•>
"2100 FORMAT ( in-, I OX, 14, 12H METEOROLOGY,
8 4X,F5.2,lOn LD SO2/MMBTU COAL,
8 4X.60CSD = .F4.2.13X,14,
8 26H UC'M3 COMPARISON STANDARD)
C
2200 FORIL\T( HI-, 1OX.20HEXPECTED EXCEEDANCES,s/,
8 12X.6HRING 1,X.3(19X, I2(F6.2,2X),X),
8 12X.6IIRING 2,x,3(19X,I2(F6.2,2X).X),
8 l2X,6irRINC 3,X,3( 19X, 12(F6.2.2X),X) ,
8 12X.6HRING 4,X,3(19X, 12(F6.2,2X) ,X),
8 12X.6RR1NG 5,X,3(19X,12(F6.2,2X) ,X))
C
2300 FORMAT(1H0,10X.260NUMBER OF SAMPLES (OUT OF ,14,
8 51H) RESULTING IN VIOLATIONS (TWO OR MORE EXCEEDANCES),
Q 17H AT EACB RECEPTOR,XX,
8 12X.6URING 1,X,3<19X,12(16,2X),X),
8 12X.6RHING 2,X,3( 19X,12(I6.2X),X) ,
8 12X.6HTIINC 3,X,3( I9X, 12( 16,2X) ,/) ,
8 12X,6imiNG 4,X.3(19X, 12(16,2X> ,X),
8 12X.6DRING 5,X,3( 19X,12(I6.2X),X))
C
2400 FORMAT(IH-.xx, 12X.5IIUSINC, IS, 27Q METEOROLOGY, THE NUMBER OF,
8 21H SAMPLE YEARS (OUT OF, 15, 1H),X,12X,15HRESULTINC IN A
8 38HVIOLATION AT AT LEAST ONE RECEPTOR WAS, 15)
C
2500 FORMAT(ID-,390 RESULTS FOR ALL 5 METEOROLOGICAL YEARS,
8 4X.F5.2,1811 LB SO2XMMBTU COAL,
8 4X.6HCSD = .F4.2.13X.14.
8 26U UCXM3 COMPARISON STANDARD)
C
2600 FORMATCID-.xx,1OO USING ALL,12,220 YEARS OF METEOROLOGY,,
8 35H THE NUMBER OF SAMPLE YEARS (OUT OF,I5.1H).X,
EXHIBIT A-l (Continued)
76
-------
EXEX.PROGRAM
8 50H RESULTING IN A VIOLATION AT AT LEAST ONE RECEPTOR.
$ 4H WAS, 15, 111. ,/,28H THUS, THE PROBABILITY OF A ,
8 29UVIOLATION AT SOtlE RECEPTOR IS,F6. 1,411 PCT)
C
2700 FORMAT<1H1,3911 IN CASE YOU'RE INTERESTED, THE NUMBER ,
8 I7HOF CONCENTRATIONS,/,2BII LOOKED AT FOR EACH YEAR WAS,/,
@ 5(5H YR ,15,I10,/) )
END
#:!:*^^
^
EXAMPLE DATA INPUT
1000 MW UNSCRUBBED POWER PLANT - ST LOUIS HET DATA - 24 HOUR AVG CONG
118. 4.8 1. 15 2 1973 5
12345
0
*******^
EXHIBIT A-l (Concluded)
77
-------
DO 200 IPERD = 1, NPERDS
Rl = RANF(X)
IRAND IFIX (10000 * Rl) +1
Q(IPERD) = DSTRB (IRAND)
200 CONTINUE
This will result in an emissions rate that is randomly selected from the
entire distribution.
A flow diagram for the program is shown in figure A-l. As pointed out
above, the program is very simple and self-explanatory.
The input data are shown at the bottom of exhibit A-l. It consists of
four cards:
Card 1: Title (up to 80 characters).
Card 2: STD—the standard concentration for making concentration
comparisons.
SMEAN—the geometric mean of the emissions distribution (not
needed if an empirical distribution is being used—see
above).
GSD—Geometric standard deviation of emissions distribution (not
needed if an empirical distribution is being used).
NSIM—the number of sample years to use in the Monte Carlo
analysis—generally set to 1000.
ISTART—the first year of meteorological data (CRSTER output).
NYRS—the number of years of meteorological data (CRSTER output)
available.
Card 3: INCR--A set of increments that are used to increase ISTART to
equal the numbers of subsequent years. For instance, if
five successive years are available, INCR would be entered
as 1 2 3 4 5.
Card 4: THREE—A logical variable that indicates whether 3- or
24-hour average concentrations are being used.
A sample output is shown in exhibit A-2. Again, this material is self-
explanatory. We have run this program on a COC 7600 computer. However, we
believe it can be run on any computer with a standard FORTRAN compiler.
78
6W21
-------
FOR 1*1. NUMBER OF
SIMULATION SAMPLES
GENERATE SET OF RANDOM EMISSIONS
RA-E5 Qnm WHERE 1 IS SAMPLE NUMBER.
1 IS MET VEAR, AND m IS PERIOD NUM-
BER (£.G., 1 365)
2) CALCULATE GROUND LEVEL CONCENTRATIONS
'iklm
ft),
klm
CORRESPONDING TO SIGNIFICANT ( j) VALUES.
3) If ',k|m> «
COUNUR.
STD'
INCREMENT EICEEDANCE
AT END OF SIMULATION TEAR:
1) IF EXCEEOANCES > T AT ANT RECEPTOR,
INCREMENT VIOLATION COUNTER
?) IF ANf RECEPTOR HAS A VIOLATION. SET
VIOLATION INDICATOR FOR YEAR.
AFTER REQUIRED NUMBER OF SIMULATION SAMPLES, PRINT:
1) EXPECTED EXCEEOANCES AT EACH RECEPTOR
TOTAL NUMBEROf EXCEEDANCES
* NUMBER OF TEARS
2-, TOTAL VIOLATIONS AT EACH RECEPTOR
3} NUMBER OF YEARS WHEN VIOLATIONS MERE CALCU-
LATED ANYWHERE • NUMBER OF TEARS VIOLATION
INDICATOR HAS SET
PRINT OUTPUT SUMVM*
FIGURE A-1. FLOW DIAGRAM FOR EXEX PROGRAM
79
-------
iooo «5riJi«»ciiuB«eD~*oue» *LANT - ST LOUIS HET DATA - s« HOUR AVC CONC
1973 METEOROLOGY 4 80 LI 6O2/MHBTU COAL BSD - i IS
lie uo/n3 COMPARISON STANDARD
EIPECTED tlCCEDANCES
PING 1
RING 2
RING 3
RING 4
RING 9
NUn&EB OF
RINC 1
RING £
RING 3
RING 4
RING **
C 9t
0 OC
6 06
C 01
1 11
0 00
.O-OO
0 OO
0 00
0 00
0 00
O-OO
o.oo
0 00
0 00
SAMPLES
0
0
o
_.
o
243
0
0
ft
o
0
0
0
0
0
0
0 78
0 00
f> OC,
c i;
C 93
G O*
Q ftj .
C IS
O 03
0. OO
0 OC
O. OC ~-
0 00
0 00
0. 00
(OUT OF
o
0
0
1
191
0
0
ft
o
0
0
0
o
0
0
o oo
O 00
C OC
C 2i
0 00
c ::
6 46
o oo
0 34
o oo
0 00
.C.-OO-
0 OO
0 OO
0 00
1OOO)
0
0
0
16
o
0
1
O -
o
o
o
o
0
0
0
o oo
0 17
ft OO
o at.
C 0?
0 04
& oo
o oo
0 OO
c oo
0 OO
0 00 -
0 00
0 00
0. 00
RESULT.! MS
o
0
o
*
o
0
0
0
p
0
0
0
0
o
0
0
o oo
0 OO
o oo
0 51
0 IV
0 00
•O OO
0 00
0 00
o. oo
O 00
-O OO-
o oo
0 00
o oo
Cl Ol
0 OO
O C3--
0 IB
0 <»2
0 39
-O.-OO
0 OS
0 03
o.oo
O 00
- .0. oo.
0 00
O 00
0. OO
IN VIOLATIONS
o
0
0
71
3
o
o
. --O
o
0
o
0
o
0
0
o
0
o
- _
3
o
9
o
ft
o
o
0
o
0
0
0
0 OO
O 00
. 6 OS
O 96
O 61
O B4
•O--67 -
0 OO
o to
O. 00
0 OO
_ ~o OO-.
O. 00
o oo
0. 00
17UO OR
0
0
0
1
0
128
34
Q
o
o
0
o
0
0
0
0 OO
o oo
o oo
0 16
0 04
o os
A OO .
o oo
o oo
o oo
o.oo
. o.oo
0 OO
0 00
O 00
r.CRE EI
0
0
0
8
o
0
o
rt
o
0
o
o
o
0
0
0 OO
0 OO
o oo
0 IS
0 OO
o oo
. .. O.OO
0 OO
O O3
0 OO
0 00
o oo
o oo
o oo
0 OO
CtE DANCES!
o
o
0
_
o
o
0
o
ft
o
o
o
0
0
0
o
0 OO
0 00
o oo
0 13
0 00
o oo
O.-O9 — -
o. oo
o oo
0 OO
o oo
o oo
0 OO
0 00
o oo
AT EACH
0
0
0
4
o
0
o
Q
o
o
0
o
o
o
0
0 OO
0 OO
o oo
0 22
0 00
0 01
O.-O4— -
o oo
o oo
0 OO
0. OO
o oo
0 OO
0 00
0 OO
0 OO
0 03
0 13-
O 36
O O4
I 64
O..OO --
O O7
O 31
0 OO
0 OO
0.01
o. oo
0 00
O. 00
RECEPTOR
0
o
0
3
0
o
o
ft
0
0
0
o
o
0
o
o
o
1
2
o
636
o
n
13
0
o
o
0
0
o
USING 1973 METEQROLOQt. THE NUMBER OF SAMPLE YEARS (OUT OF 10OOJ
RESULTING IN A VIOLATION AT 'AT LEAST ONE RECEPTOR WAS 838
EXHIBIT A-2. EXEX SAMPLE OUTPUT
-------
1000 HU UNSCRUIBEO POWER PLANT - ST LOO 13 MET DATA - 24 HOUR AVO CONC
-- 1974-HCTgQ»OLOO¥
,0O-LI- BCaSMtlBTU-COAL. -«SD-> 1. 13 . — — -
118 UC/M3 COMPARISON STANDARD
EXPECTED EXCEEDANCES
RING S
0 82
0 Ol
._._. O 9B
RING 2
O. 03
O. 09
O 4O
RINO 3
0. OO
O. OO
RING 4
— O OO
O.OO
RINO 9
O. OO-
0.00
0. OO
0. O2
0.02
O. OO
0. 94
I 49
0. OO
O,O4-
•-O. OO
O OO
O-oo
o oo
O.OO
0. OO
0. 10
0. Ot
0 09
I 12
O.OO
1 27
o. oo
o oo
o. oo
o oo
O. OO-
*
o oo
o oo
O. O3
o. oo
- 0r-*7
o oo
o oo
o. oo
o oo
o oo
o. oo
0 OO
0. OO
O Ol
O 91
-O--44-
O OO
0 96
O OO
I O2
O 00
1 24
O 6O
0 96
O Ol
--O, OO-
O OO
0 OO
O OO
O OO
O- OO.
O OO
O 00
O. OO
O. 92
O. 2B
O 12
-O-&1—
O OO
O 00
0 00
O OO
O-OO.
O. OO
O 00
O. OCX
O. 63
0. 14
O Ol
-O.OO--
0 OO
O 00
O 00
O OO
-o-oo.
O OO
O. OO
O.OO
O. OO
O. 34
9 O*
O 04
O 00
O 00
O OO
- O. OO
O OO
O 00
O. OO
O 00
O 63
-O 00
o. to
O. 7O
. -O. 92
O. OO
O Ol
O 00
O 00
0 OO
O-OO-
0 OO
O OO
O. OO
O 00
0. 43
0-04- •
O OO
O. 37
O OO
- O.OO
O OO
O 00
O OO
O OO
-O. OO-
O. OO
O. 00
O OO
O OO
0 OS
-o-ov
32
IB
19
O- O3-
0 OS
0 OO
0 Ol
O 00
O OO
O. 39
0 OO
O OO
O OO
O O6
O 11
O 6B
O Ol
o oa
o oo
o oo
o oo
o oo
a oo
0. 00
o oo
0 OO
a>
NUHB6B OS.-S.'>f*P-U6S--tOm OS- 1OOO i- BESi-'LTAtlC- IM- UtOt-AT.l3N.S. -LIUO-OR SORE. EXCEEDAAICES l AT EACH RECEPTOR
RING 3
ft INC 4
O
0
IS
o
1
"Si
O
O
113
4OO
0
.0
0
0
0
o
0
o
0
0
0
0
0
0
0
0
o
0
0
a
o
o
o
o
a
i
o
27
4
O
0
136
0
SO
13
2
O
Q
O
0
0
0
0
a
0
0
38
0
109
3
O
O
Q
O
0
o
0
o
a
0
o
32 1.
o
0
3O
o
o
o
0
0
0
o
o
0
0
o
3
0 •
0
11
123
O
O
0
0
o
o
o
o
o
O
O
0
21
O
2
O
--O.-
0
0
o
0
0
0
0
0
a
5
3
77
O
USING 1474 METEOROLOGY. THE NUMBER OF SAMPLE YEARS (OUT QI-" lOOOl
RESULTING IN A VIOLATION AT AT LEAST ONE RECEPTOR MAS 997
EXHIBIT A-2 (Continued)
-------
tOOO Kta UNSCHOBIEO VOUCH W.ANT - ST lOUIS f*T DAT* - 34 HOUR AVO CONC
4 so LB soa/nngru COAL eso - i 13
118 UG/M3 COMPARISON STANDARD
EHFECTEO ElCEEDANCES
RING I
"INC 2
RIN8 3
RINO 4
RINO 5
0 03
0 OO
I. 21
1 24
1. 11
0 93
O 06
0 69
0 34
1. O4
0 37
1 OO
0. 13
2 63
O. 79
0 OO
I 78
O. 2O
0 94
O 67
O OO
1 3O
0 6O
O BO
O 03
O OO
1 13
O OO
O 66
n nn
O. 01
O. 29
O. 70
O 04
0. OS
0 OO
O 00
O 01
O. 01
O 33
O. Ol
O 96
O OS
O. OO
O. OO
O. 06
O. 41
O. 06
O 02
0. 71
0 Oi
O. 22
l.Ol
0 OO
O. 14
0. 77
0 01
O. 76
1 O*
0 09
O. 37
n nn
O 39
0. 44
k
0. OO
0 00
. . O..OO-
n nr
0 00
0. 00
0 00
0 OO
.-..&OO
. .O-OO
0 91
9 OO
0 OO
o oo
o oa
. 0-00.. .
o oo
O. 91
0 00
0 00
.- -O.OO.
. o-oa .
3 OO
0. OO
0 OO
0 00
3 oa
O O4
0 OO
0.00
0 OO
O-OO
1 «y>
o oo
a os
o oo
o oo
0,00
-0-00
o oo
0. OO
0 OO
o oo
0. OO .
. O-OO-.
o oo
o oo
0 OO
o oo
.. O-OO
o oo
0 00
o oo
0 OO
0-01
n QQ
0 OO
0 27
o oo
0 OO
o. 02
ft *YB
0. OO
O. 00
o oo
0 00
O-OO
0. OO
0 OO
o. ao
o oo
o oo
o. oo
o. oo
0 00
0. 00
o. oo
0. 00
0.00
0. OO
0 OO
0. OO
o. oo
o. oo
o. oo
o. oo
0 OO
0.00
0. OO
0 00
0 00
0. OO
0 OO
o. oo
0. OO
o oo
0 OO
0. 01
o. oo
0. OO
0. OO
0. OO
0.00
NUMBER OF SAMPLES (OUT OF 1000) RESULT4NC IN-VIOLATIONS -4TUO- OR- «OHe EICEEDANCESJ AT EACH RECEPTOR
00
ro
RINO t
a INC 2.
RING 3
RING 4
0
0
o
273
292
201
32
0
70
31
269
O
1~. _
0
0
0
0
— B1NC--3-.
0
a.
o
o
0
0
3
o
0
26
3
2
1
964
39
0
. -O.
0
0
0
0
O
0
0
0
64C
1
3
169
4
0
318
247
67
0
0
a
o
o
o
o
o
o
o
o
IBS
o
0
224
O
O
O
o
0
o
o
o
1
o
o
o
IB
O
o
o
0
0
o
o
13
o
28
O
0
103
0
0
6
0
1
0
o
0
16
0
0
2
--O
o
0
0
0
0
0
o
0
0
0
41
0
16
o
0
o
0
o
o
0
0
0
64
O
41
0
O
o
0
o
0
USING 1979 METEOROL3C
IN A
THE NUrDES
F SAMPLE YEARS (OUT OF 100O)
WAS 1OOO
EXHIBIT A-2 (Continued)
-------
iooo nw oNSCRijeaec POWER PLANT - ST LOUIS BET DATA - 24 HOVR AVO CONC
X976 .1ETEOROLOCY 4 8O LB S02/M13TU COAL SSD » 1 13
11B OCSH3 COnPAfilSON STANDARD
09
"INS %
RINC j
. —
9 ING 4
- .
RINC S
UU-.3E* Si
RINC I
Hl'-C- -•
RINC 3
. ~.. ._ .. .
RINC 4
HL.MC 5
O O2
i z:
O- 33
O OO
O 94
O O4
-O-OO
O OO
0 90
0 00
O OO
.. O-oa
a. oo
0 00
O OO
F iA.-..-.
3
272
O
0
179
0
'J
a
0
o
0
3
O
3
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0 59
0 00
o ss
0 04
0 00
0 66
..a. 2*
o oo
0 OO
0 O2
0 00
O.OO
o oo
O OC
o oo
ss -cur
;
0
3
0
0
3
0
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3
3
0
J
0
0
0 18
O OO
O. 06
0 00
O OO
1 66
O-OO.
0 00
O 49
o. oo
o oo
&. CO
0 00
0 00
0 CO
3f 1 z. '. ~.
9
0
T t
o
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^«*T
0
._. .. a
1
0
0
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0
0
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o oo
o oo
- _ - 3 Q^
0 23
o 01
O. 31
o_oo
o oo
O 02
0 00
o oo
. O 3C
0 3C
0 30
3 OC
, «*,. T :
0
o
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3
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2*
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-O
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3
0
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-0--00.
O 13
O 03
0- 32
a, oo
o oo
a oo
0 OO
o oo
a oa
0 30
O OO
O 00
.MS IN vliv.
O
a
0
2
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0
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- a
o
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0
3
^
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0
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0 00
•0. GO-..
0 58
O. OO
0 66
-a oa
0 OO
2 -i 1
o oo
0 OO
3 iO
0 CO
0 OO
o. oo
.-tlSSJS.
0
0
0
23
O
n
3
3
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3
3
3
0
0
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9 00
0 OO
. .O- O4-
0 20
0 00
0 1O
a oo
o oo
3 3O
0 00
1 00
- oo
0 OO
0 OO
o oo
. T-C OR «
0
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0
2
0
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0
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0
o
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0
0
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0 OO
o oo
0 69
O OI
o 10
o oo
0 OO
3 00
0 OO
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a oo
0 OO
0 00
0 OO
CRE Cl
o
o
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1000 1U UNSCRUBBE!) COUER PLANT - 9T LOUIS K*T DAT* - S4 MOUB AVO CONC
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« 3 PCT
EXHIBIT A-2 (Concluded)
-------
REFERENCES
Box, S.E.P., and M. E. Muller (1958), "A Note on the Generation of Normal
Deviates," Ann. Math. Stat., Vol. 29, p. 610.
Cramer, H. (1946), Mathematical MethodsofStatistics (Princeton
University Press, Princeton, New Jersey).
EPA (1977), "User's Manual for Single-Source (CRSTER) Model," EPA-450/2-
77-013, Environmental Protection Agency, Research Trianqle Park,
North Carolina.
EPA (1978), "Guideline on Air Quality Models," EPA-450/2-78-027, EPA,
Research Triangle Park, North Carolina.
Foster (1979), "A Statistical Study of Coal Sulfur Variability and Related
Factors," Draft, Foster Associates, Incorporated, Washington, D.C.
ICF, Incorporated (1979), "Summary of Information on Variability of Sulfur
Dioxide Emissions from Coal-fired Boilers," draft memorandum to
Office of Air Quality Standards, Environmental Protection Agency,
from ICF, Incorporated, Washington, D.C.
Larsen, R. I. (1971), "A Mathematical Model for Relating Air Quality
Measurements to Air Quality Standards," AP-89, Environmental
Protection Agency, Research Triangle Park, North Carolina.
Lee, R. S. (1979), private communication.
Mallons, C. L. (1956), "Generalizations of Chebyshev's Inequalities," jL_
Royal Statistical Soc., B, Vol. 18, p. 139.
Morrison, R. M. (1979), private communication.
PEDCo (1977), "Preliminary Evaluation of Sulfur Variability in Low Sulfur
Coals from Selected Mines," EPA-450/3-77-044, Environmental
Protection Agency, Research Trianqle Park, North Carolina.
8W21 86
-------
Naylor, T. H., et al. (1966), Computer Simulation Techniques (John Wiley &
Sons, New York, New York).
Parzen, E. (I960), Modern Probability Theory and Its Applications (John
Wiley & Sons, New York, New York).
Rockwell (1975), "Navajo Generating Station Sulfur Dioxide Field
Monitoring Program," Final Program Report, Volume I, Rockwell
International, Newbury Park, California.
/2
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