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

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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
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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.
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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
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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.
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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.
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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.
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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-
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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,

-------
                                                        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

-------
  \
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
_ -^ \
\


-------
        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

-------
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

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     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

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      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

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       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

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             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

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      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

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           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
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        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
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                                                30

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        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:
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                                             31

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                  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.


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        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

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        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

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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

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       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

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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

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       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
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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

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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

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                      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

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             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

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                    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

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                   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

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          £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

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          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

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         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

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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

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                        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

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        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

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      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
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RING 2



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           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
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                     	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
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                     77
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                               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
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                                   RIN8 3
                                   RINO 4
                                   RINO 5
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                                  NUMBER OF SAMPLES (OUT OF 1000) RESULT4NC IN-VIOLATIONS -4TUO- OR- «OHe EICEEDANCESJ AT  EACH RECEPTOR
00
ro
                                   RINO  t
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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
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                                      USINS  1976 METECROLOCr.  THE NUMBER Of  SAMPLE (EARS 1OUT OF 100OI
                                      «E"iJL'!v; fi A  l\r.l c.';i,<, a- AT ^Fi" ' f.£ •JF:E°'~P wa5  »••»
                                                                    EXHIBIT A-2  (Continued)

-------
                   1000 1U UNSCRUBBE!) COUER PLANT - 9T LOUIS K*T DAT* - S4 MOUB AVO CONC
                            1477 nETEORJLOCY
                                                4 BO LB SC2/nnBTU COAL
                                                                         eso •  i  is
                                                                                                118 UG/H3 COMPARISON STANDARD
00
EtPECTEO 6«CEfcOANCES ' "" ~~ ~" 	 	
RINC 1
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NU-.fitH C.F
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                                                                 EXHIBIT  A-2  (Concluded)

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                                           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,
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           Foster (1979), "A Statistical Study of Coal  Sulfur Variability and Related
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           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

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       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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