p.1 

OAR Box 1828 

Prepped by Ollie Stewart 

Document Number: 

298) IV-A-8 

Docket Number: 

A-83-27 

Printed 11/12/200811:12:59 AM Sheet 298 of 325 
P.2 

A-83-27 

IV-A-8 

DCN #85-231-372-16-78 

DETERMINATION OF MEAN S02 EMISSION LEVELS 

REQUIRED TO MEET A 1.2 LB S02/MILLI0N BTU 

EMISSION STANDARD FOR VARIOUS AVERAGING 

TIMES AND COMPLIANCE POLICIES 

Prepared for: 

Walter H. Stevenson 

U.S. Environmental Protection Agency 

Emission Standards and Engineering Division 

Durham, North Carolina 27711 

Prepared by: 

Gregory C. Giguere 

Suzanne C. Margerum 

Radian Corporation 

Progress Center 

3200 E. Chapel Hill Road/Nelson Highway 

Research Triangle Park, North Carolina 27709 

March 1985 
P.3 

1.0 INTRODUCTION 

The two most common methods used to control S0? emissions from 

coal-fired steam generating units are flue gas desulfurization (FGD) systems 

and direct combustion of low sulfur coal. Both of these methods demonstrate 

a natural variability with respect to S02 emissions discharged to the 

atmosphere. A knowledge of the variation or spread in S02 emissions with 

time and a statistical model which provides a close resemblance to the 

variability are required in order to understand the fluctuations in S02 

emission measurements. Due to the random fluctuation in emission levels 

both above and below the long-term average, the mean emission rate must be 

maintained at an emission level below the emission standard in order to 

assure compliance. 

Through analysis of the variability in emissions, a statistical 

procedure has been developed which allows the projection of long-term mean 

S 02 emission levels required to meet a 1.2 lb S02/million Btu emission 

limit. This technical note presents the required long-term mean S02 

emission rates, projected as a function of the averaging period (3-hour 

discrete, 24-hour discrete, 7-day rolling, 30-day rolling), enforcement 

policy (exceedance frequency: 1 per 10 years, 1 per 1 year, 1 percent of 

time), autocorrelation (daily values of 0.5, 0.7, 0.8), and relative 

standard deviation (daily values of 5 to 35 percent). 

Section 2.0 of this technical note briefly discusses the methodology 

used to determine long-term mean S02 emission levels. The projected mean 

emission levels to comply with a 1.2 lb S02/million Btu emission limit for 

various averaging periods and compliance policies are summarized in tabular 

form in Appendix A. Appendix B presents a more detailed description of the 

statistical approach to determining required mean emission levels. 

Appendix C lists the computer program used to project the required mean S02 

emission levels. 
P.4 

2.0 METHODOLOGY 

Sulfur dioxide (S02) emissions from any steam generating unit will vary 

with time. Steam generating units equipped with a flue gas desulfurization 

(FGD) system will exhibit a certain amount of variability in S02 emissions 

due to random fluctuations in operating parameters such as reagent quality, 

liquid and gas flow rates, and flue gas inlet S02 concentration (even 

ambient temperature). Similarly, there is a certain amount of natural 

variability in coal sulfur content due to the heterogeneous nature of coal, 

differences in mining practices, and differences in handling and preparation 

practices. There are several factors which describe the variability 

characteristics of S02 data and which influence the projected mean S02 

emission levels required for compliance: 

o Standard deviation and relative standard deviation; 

o Autocorrelation; 

o Emissions distribution (normal vs. lognormal); 

o Length of the averaging period and averaging method; and 

0 Compliance policy (exceedance rate) 

Each of these factors and its effect on the required mean emission level is 

briefly discussed below. The tables in Appendix A present required mean 

emission levels as a function of each of these factors. Appendix B contains 

complete details on the computation of the values presented in Appendix A. 

The standard deviation is an indicator of the spread of values that 

measures the variation of measurements from the average. The relative 

standard deviation (RSD) is defined as the ratio of the standard deviation 

to the average or mean and is typically used to describe emissions 

variability. The greater the relative standard deviation is, the greater 

the variability in observed S02 emission will be. Coals or S02 emissions 

data sets which have the same mean S02 level but have different S02 

variability levels (standard deviations or relative standard deviations) 

will have different levels of difficulty in complying with a 1.2 lb 
P.5 

S02/million Btu emission level. As increased variability is observed, a 

lower mean S02 emission level will be necessary to achieve compliance with 

an S02 emission standard (see Figure 1). 

Emission measurements taken at equal time intervals constitute a time 

series in which values are not necessarily independent. When data are 

collected in sequence, there can be a tendency for observations made close 

together in time to be more alike than those taken farther apart. A measure 

of this degree of association between observations in a time series is 

termed autocorrelation and can vary from -1.0 for inversely related 

observations to 1.0 for data which exhibits an extreme degree of linear 

association. S02 emissions values typically have a relatively high amount 

of autocorrelation (greater than 0.5), such that the value of each 

observation is expected to be similar to that of the previous observation. 

Overall, as the autocorrelation factor increases, a lower mean S02 emission 

level will be required for compliance, and this effect of autocorrelation on 

compliance determination is more important for longer averaging periods. 

However, autocorrelation is not as major a factor as RSD in determining S02 

variability and predicting long-term mean emission levels. A first order 

autoregressive time series model, abbreviated as AR(1), is used as part of 

the methodology to project the required long-term mean S02 emissions levels 

shown in Appendix A. This is the most basic autoregressive model and has 

been found in many cases to adequately fit emissions time series 

measurements and is, therefore, appropriate for this analysis. 

Observed S02 emissions data can be described by various emissions 

distributions. Emissions data have generally been found to be well 

represented by either the normal or lognormal distributions. The primary 

difference in the two distributions is that the lognormal (in which the 

natural logarithms of the data are normally distributed) is positively 

skewed. The required mean S02 emission levels necessary for compliance with 

a standard will differ depending on whether a lognormal or normal 

distribution is assumed, and the difference in the projected levels is 

primarily a function of the length of the averaging period and the RSD. For 

emissions data with lower RSD's, the required mean S02 levels are similar 

for the two distributions; however, in cases where high RSD's are expected, 
P.6 

the difference between projected levels is more pronounced (see Figures 2 

and 3) . Similarly, for short averaging periods the difference between the 

projected levels for lognormal and normal distributions can be ^ery 

pronounced, but as the length of the averaging period increases, the 

differences become negligible. Because the lognormal distribution is skewed 

to the right, its use results in the projection of a slightly lower mean 

being necessary for compliance than if the normal distribution had been 

assumed. 

The length of the averaging period also affects the required mean 

emission level necessary for compliance. Longer averaging periods dampen 

the effects of variability, resulting in required mean S02 emission rates 

closer to the actual emissions limit. The averaging period is defined as 

the period of time (hours or days) over which emission measurements are 

averaged in order to determine compliance with the S02 emission limit. 

A block average (discrete average) or a rolling average may be used to 

determine compliance. Discrete averages are computed on a separate, 

non-overlapping basis; rolling averages are calculated by adding the current 

value into the calculation of the average and deleting the oldest or least 

current value. With the use of discrete or block averages, the 

enforceability of an emissions standard is directly related to the averaging 

time. For example, a 24-hour average would be enforced for each discrete 

24-hour period and 365 such averages would be calculated per year. A 30-day 

average would be enforced for each discrete 30-day period and 12 such 

averages would be calculated per year. However, a 30-day average can be 

calculated and enforced on a daily basis through the use of a rolling 

average. Thus, rolling averaging periods allow for stricter enforcement • 

than discrete periods of the same length. For a given averaging period, the 

mean S02 level required for compliance is lower for a rolling average than 

for a discrete average since there is a greater potential for exceeding the 

emission limit when using a rolling average. 

The final factor affecting the required mean emission level is the 

compliance policy (allowed exceedance frequency). The allowed exceedance 

frequency is expressed as the number or percentage of exceedances assumed 
P.7 

over the specified time period for a particular emission limit, and can be 

thought of as an assumed violation rate. Of course, as the allowed 

exceedance frequency increases, the required mean S02 level also increases. 

The probability of violating the limit can be calculated for a specific 

averaging period and exceedance frequency; this probability is constant for 

each potential period of violation over the entire term of the averaging 

period. As an example, consider a 30-day rolling averaging period combined 

with an exceedance frequency of one per 10 years. There is a potential of 

having an exceedance or violation each day when using a 30-day rolling 

averaging period, since a new 30-day average emission level is computed each 

day. There are 3,650 days over a 10-year period; thus assuming one 

exceedance divided by the total number of potential exceedances (3,650) 

yields 0.00027. Therefore, the probability of violating the limit each day 

is a constant of 0.00027, and summing these daily probabilities over the 

3650 days results in a probability (or relative frequency) of one. The 

daily exceedance probability of 0.00027 translates to a standard normal 

distribution Z value of 3.46. 

The implication of the above example is that, in order to comply with 

the emission limit, the boiler should be run at a target level which is less 

than the given limit by an amount equal to 3.46 times the standard deviation 

of the 30-day rolling averages. 
P.8 

APPENDIX A 

This Appendix contains projected mean S02 emission levels required to 

meet a 1.2 lb. per million Btu emission limit based on different averaging 

time, compliance policies, and statistical assumptions. Each table 

corresponds to a distributional assumption (lognormal or normal) and an 

averaging period, both of which are specified in the title. To use the 

tables, select the daily autocorrelation (0.5, 0.7, or 0.8) desired; one of 

the three rows under the daily autocorrelation heading which corresponds to 

exceedance frequency (one per ten years, one per year, or one percent); and 

lastly, one of the seven columns which signify the daily relative standard 

deviation. The value in the body of the table is the mean S0? emission 

level) which the boiler operator must achieve in order to meet the 1.2 lb 

S 02 per million Btu emission limit given the particulars of emissions data 

variability and compliance practices. For example, in Table A-4, the 

required mean S0« emission level to comply with a 1.2 lb S02/million Btu 

limit based upon a 30-day rolling average period, a daily autocorrelation of 

0.7, an exceedance frequency of one per year, and a daily relative standard 

deviation of 20 percent, is 0.98 Ib/MM Btu. 

Lognormal Distribution Assumption Table 

3-Hour Discrete Averaging Period A-l 

24-Hour Discrete Averaging Period A-2 

7-Day Rolling Average Period A-3 

30-Day Rolling Average Period A-4 

Normal Distribution Assumption Table 

3-Hour Discrete Averaging Period A-5 

24-Hour Discrete Averaging Period A-6 

7-Day Rolling Average Period A-7 

30-Day Rolling Average Period A-8 

A-l 
P.9 

TABLE A-6. PROJECTED LONG TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 24-HOUR DISCRETE AVERAGING PERIOD 

AND A NORMAL DISTRIBUTION3,5 

Daily Relative Standard Deviation (Percent) 

Exceedance 

Frequency 5 10 15 20 25 30 35 

Daily Autocorrelation of 0.5 

One per 10 years 1.05 

One per year 1.07 

One percent 1.09 

Daily Autocorrelation of 0.7 

One per 10 years 1.04 

One per year 1.06 

One percent 1.08 

Daily Autocorrelation of 0.8 

One per 10 years 1.03 

One per year 1.06 

One percent 1.08 0.98 0.90 0.83 0.78 0.73 0.69 

aAssumes that 24-hour discrete averages are normally distributed. 

S02 emissions levels expressed in lb S02/million Btu heat input. 

0.93 

0.97 

1.00 

0.91 

0.96 

0.99 

0.90 

0.95 

0.84 

0.89 

0.93 

0.81 

0.87 

0.91 

0.80 

0.86 

0.76 

0.82 

0.86 

0.73 

0.80 

0.84 

0.73 

0.79 

0.69 

0.76 

0.81 

0.67 

0.73 

0.78 

0.67 

0.73 

0.64 

0.71 

0.76 

0.62 

0.68 

0.73 

0.62 

0.68 

0.59 

0.66 

0.71 

0.57 

0.63 

0.69 

0.57 

0.63 

A-2 
P.10 

TABLE A-7. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 7-DAY ROLLING AVERAGING PERIOD 

AND A NORMAL DISTRIBUTION3»

b 

Exceedance 

Frequency 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Dai 

5 

of 0.5 

1.09 

1.11 

1.13 

of 0.7 

1.07 

1.10 

1.11 

of 0.8 

1.06 

1.09 

1.10 

ly Relat 

10 

1.01 

1.04 

1.06 

0.97 

1.01 

1.04 

0.95 

0.99 

1.02 

ive Standard Deviation 

15 

0.93 

0.97 

1.01 

0.89 

0.94 

0.97 

0.86 

0.91 

0.95 

20 

0.87 

0.92 

0.95 

0.82 

0.87 

0.91 

0.79 

0.84 

0.89 

25 

0.81 

0.87 

0.91 

0.76 

0.82 

0.86 

0.72 

0.79 

0.83 

(Percent) 

30 

0.76 

0.82 

0.86 

0.71 

0.77 

0.82 

0.67 

0.73 

0.78 

35 

0.72 

0.78 

0.83 

0.66 

0.72 

0.77 

0.62 

0.69 

0.74 

aAssumes that 7-day rolling averages are normally distributed. 

S 02 emissions levels expressed in lb S02/million Btu heat input. 

A-3 
P.11 

TABLE A-l. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 3-HOUR DISCRETE AVERAGING PERIOD 

AND A LOGNORMAL DISTRIBUTION3,b 

Daily Relative Standard Deviation (Percent) 

Exceedance 

Frequency 5 10 15 20 25 30 35 

Daily Autocorrelation of 0.5 

One per 10 years 0.99 0.82 0.68 0.56 0.47 0.40 0.34 

One per year 1.02 0.86 0.74 0.63 0.54 0.47 0.41 

One percent 1.07 0.96 0.86 0.78 0.70 0.64 0.58 

Daily Autocorrelation of 0.7 

One per 10 years 0.99 0.81 0.67 0.56 0.47 0.39 0.33 

One per year 1.02 0.86 0.73 0.63 0.54 0.47 0.40 

One percent 1.07 0.96 0.86 0.77 0.70 0.64 0.58 

Daily Autocorrelation of 0.8 

One per 10 years 0.99 0.81 0.67 0.56 0.47 0.39 0.33 

One per year 1.01 0.86 0.73 0.63 0.54 0.46 0.40 

One percent 1.07 0.96 0.86 0.77 0.70 0.64 0.58 

aAssumes that 3-hour discrete averages are lognormally distributed. 

SOp emissions levels expressed in lb S02/million Btu heat input. 

A-4 
P.12 

TABLE A-5. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 3-HOUR DISCRETE AVERAGING PERIOD 

AND A NORMAL DISTRIBUTION3'

15 

Daily Relative Standard Deviation (Percent) 

Exceedance •• 

Frequency 5 10 15 20 25 30 35 

Daily Autocorrelation of 0.5 

One per 10 years 1.00 

One per year 1.03 

One percent 1.08 

Daily Autocorrelation of 0.7 

One per 10 years 1.00 

One per year 1.03 

One percent 1.08 

Daily Autocorrelation of 0.8 

One per 10 years 1.00 

One per year 1.03 

One percent 1.08 

3Assumes that 3-hour discrete averages are normally distributed. 

S 02 emissions levels expressed in lb S02/million Btu heat input. 

0.86 

0.90 

0.98 

0.86 

0.90 

0.98 

0.86 

0.90 

0.97 

0.76 

0.80 

0.89 

0.75 

0.80 

0.89 

0.75 

0.80 

0.89 

0.67 

0.72 

0.82 

0.67 

0.72 

0.82 

0.67 

0.72 

0.82 

0.61 

0.65 

0.76 

0.60 

0.65 

0.76 

0.60 

0.65 

0.76 

0.55 

0.60 

0.71 

0.55 

0.60 

0.71 

0.55 

0.60 

0.71 

0.51 

0.55 

0.67 

0.50 

0.55 

0.66 

0.50 

0.55 

0.66 

A-5 
P.13 

TABLE A-2. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 24-HOUR DISCRETE AVERAGING PERIOD 

AND A LOGNORMAL DISTRIBUTION3'

b 

Daily Relative Standard Deviation (Percent) 

Exceedance 

Frequency 5 10 15 20 25 30 35 

Daily Autocorrelation of 0.5 

One per 10 years 1.04 

One per year 1.07 

One percent 1.09 

Daily Autocorrelation of 0.7 

One per 10 years 1.03 

One per year 1.06 

One percent 1.08 

Daily Autocorrelation of 0.8 

One per 10 years 1.02 

One per year 1.05 

One percent 1.08 

3Assumes that 24-hour discrete averages are lognormally distributed, 

S02 emissions levels expressed in lb S02/million Btu heat input. 

0.90 

0.95 

0.99 

0.88 

0.93; 

0.97 

0.87 

0.93 

0.97 

0.78 

0.85 

0.90 

0.76 

0.83 

0.88 

0.74 

0.82 

0.87 

0.68 

0.77 

0.82 

0.65 

0.74 

0.80 

0.64 

0.73 

0.79 

0.60 

0.69 

0.76 

0.56 

0.66 

0.73 

0.55 

0.64 

0.72 

0.52 

0.62 

0.69 

0.49 

0.59 

0.66 

0.48 

0.58 

0.65 

0.46 

0.56 

0.64 

0.43 

0.53 

0.61 

0.41 

0.52 

0.60 

A-6 
P.14 

TABLE A-3. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 7-DAY ROLLING AVERAGING PERIOD 

AND A LOGNORMAL DISTRIBUTION3'

5 

Exceedance 

Frequency 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Dai 

5 

of 0.5 

1.09 

1.11 

1.13 

of 0.7 

1.07 

1.09 

1.11 

of 0.8 

1.05 

1.08 

1.10 

ly Relative Standard Deviation 

10 

0.99 

1.03 

1.06 

0.95 

1.00 

1.03 

0.93 

0.97 

1.01 

15 

0.90 

0.96 

0.99 

0.85 

0.91 

0.95 

0.82 

0.88 

0.93 

20 

0.82 

0.89 

0.93 

0.76 

0.83 

0.89 

0.72 

0.80 

0.85 

25 

0.75 

0.83 

0.88 

0.68 

0.76 

0.82 

0.64 

0.72 

0.64 

(Percent) 

30 

0.69 

0.77 

0.83 

0.61 

0.70 

0.77 

0.56 

0.66 

0.56 

35 

0.63 

0.72 

0.78 

0.55 

0.64 

0.72 

0.50 

0.60 

0.50 

3Assumes that 7-day rolling averages are lognormally distributed. 

S 02 emissions levels expressed in lb SOp/million Btu heat input. 

A-7 
P.15 

TABLE A-8. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 30-DAY ROLLING AVERAGING PERIOD 

AND A NORMAL DISTRIBUTION3'

5 

Exceedance 

Frequency 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Daily Autocorrelation 

One per 10 years 

One per year 

One percent 

Dai 

5 

of 0.5 

1.14 

1.15 

1.16 

of 0.7 

1.13 

1.14 

1.15 

of 0.8 

1.11 

1.13 

1.14 

ly Relat 

10 

1.09 

1.11 

1.13 

1.07 

1.09 

1.11 

1.04 

1.07 

1.09 

ive Standard Deviation 

15 

1.05 

1.07 

1.09 

1.01 

1.04 

1.06 

0.98 

1.01 

1.04 

20 

1.00 

1.04 

1.06 

0.96 

1.00 

1.03 

0.92 

0.96 

1.00 

25 

0.96 

1.00 

1.03 

0.91 

0.96 

0.99 

0.87 

0.92 

0.95 

(Percent) 

30 

0.93 

0.97 

1.10 

0.87 

0.92 

0.96 

0.82 

0.88 

0.92 

35 

0.89 

0.94 

0.98 

0.83 

0.89 

0.92 

0.78 

0.84 

0.88 

aAssumes that 30-day rolling averages are normally distributed. 

S 02 emissions levels expressed in lb S02/million Btu heat input. 

A-8 
P.16 

TABLE A-4. PROJECTED LONG-TERM MEAN S02 EMISSION LEVELS REQUIRED TO MEET 

1.2 LB LIMIT ASSUMING A 30-DAY ROLLING AVERAGING PERIOD 

AND A LOGNORMAL DISTRIBUTION3'

5 

Daily Relative Standard Deviation (Percent) 

Exceedance 

Frequency 5 10 15 20 25 30 35 

Daily Autocorrelation of 0.5 

One per 10 years 1.14 

One per year 1.15 

One percent 1.16 

Daily Autocorrelation of 0.7 

One per 10 years 1.13 

One per year 1.14 

One percent 1.15 

Daily Autocorrelation of 0.8 

One per 10 years 1.11 

One per year 1.13 

One percent 1.14 

aAssumes that 30-day rolling averages are lognormally distributed. 

S 02 emissions levels expressed in lb S02/million Btu heat input. 

1.09 

1.11 

1.12 

1.06 

1.08 

1.10 

1.03 

1.06 

1.08 

1.04 

1.07 

1.09 

0.99 

1.03 

1.06 

0.96 

1.00 

1.03 

0.99 

1.03 

1.05 

0.93 

0.98 

1.01 

0.89 

0.94 

0.98 

0.94 

0.99 

1.02 

0.88 

0.94 

0.97 

0.82 

0.89 

0.93 

0.90 

0.95 

0.99 

0.83 

0.89 

0.94 

0.77 

0.84 

0.89 

0.86 

0.92 

0.96 

0.78 

0.85 

0.90 

0.71 

0.79 

0.85 

A-9 
P.17 

is necessary to adjust for the covariance structure associated with a 

first-order time series process. The standard error is calculated as 

A-lo 
P.18 

APPENDIX B 

This Appendix is divided into two sections corresponding to the 

projected emission levels being calculated based on the assumption of a 

normal distribution or a lognormal distribution. 

Normal Distribution for Emission Averages 

Let X represent mean S02 emission for the given averaging period. 

Assume that X is normally distributed and let L be the mean of all X (y_) or 

the projected long term mean emission level (target level) required to meet 

a given emissions limit. Let C be the allowance for random variation in 

emissions or the difference between the target level and the emissions 

limit. Thus the emissions limit equals (L + C) . Using the same example 

discussed earlier in Section 2.0, the formula for computing C is derived as 

follows: 

P (X > L + C ) = 0.00027, where 

2 

X is normally distributed with mean p- and variance a - . 

P ( ^- ^ > — ) = 0.00027 

°x"

 CTx 

r 

P ( Z > — ) = 0.00027, where Z is a standard normal variate 

ax 

P ( Z > 3.46 ) = 0.00027 

— = 3.46 or C = 3.46 aa x

 x 

It is assumed that hourly S02 emissions follow an AR(1) time series 

process. Therefore in computing the standard deviation of the mean (o-), it 

B-1 
P.19 

is necessary to adjust for the covariance structure associated with a 

first-order time series process. The standard error is calculated as 

ax

 = (V

f

n M ) V 

where p is the lag 1 or first-order autocorrelation coefficient and a. is 

the hourly standard deviation. Since 

VVP I > 1//7T, 

the standard error has increased relative to the case where independence is 

assumed. The equation for f.(p) is derived using the theoretical 

autocorrelation function of <j> for an AR(1) process, where $ is the 

autocorrelation coefficient and k equals the lag. The covariance at lag k 

equals <p multiplied by the variance. After applying this result to the 

standard statistical formula for the variance of a sum, extensive algebraic 

manipulation leads to the following equation for f_(p): 

fn(p) = — n n(l 

1 1 + P 2p (1 - pn) 

n(l - P ) 

The final step in deriving the formula used to compute the target level 

tables is to develop an equation for a . 

(oh)Vyp) 

where h = hourly emission values 

n = number of hours in averaging period 

a, = k)\/Up) , 

where d = daily emission measurement 

B-2 
P.20 

ah = (RSD d nL)/-/ f

2 4 (p) , 

where RSD. = relative standard deviation on a daily basis. 

Recall that C = (Z ) a-, 

a x 

where Z is the critical value corresponding to the area (e.g., 0.00027 

above) for a given exceedance frequency and averaging period. 

C - (Za)(L)(RSDd)"[/fn(p)/f24(P) 

L + C Target level = L = - , for n > 25 

1 + (Za)(RSDd)-l/f

n(p)/f24( p) 

L + C 

1 + (Z HRSDJ VOe ) 

and L = M-£ , for n < 25 

where (L + C) equals the given emission limit of 1.2. 

Lognormal Distribution for Emission Averages 

This section will use the same notation as above. Assume that X is a 

lognormally distributed random variable. The value X such that 

a 

P (X < X ) = a 

a 

is related to the corresponding percentile Z of the standard normal 

distribution by the relation 

X = exp (u + Z a), 

a

 r a 

B-3 
P.21 

If X (mean emission level) is lognormally distributed, then Y = log(x) is 

normally distributed. 

log (L) = Uy + ay

2/2 

L = exp (Uy + ay

2/2) 

ay

2

 = log [1 + (RSD-)2] 

L + C = exp (py + Za ay) 

yy = log (L + C) - Za ay 

Thus given an emissions limit of (L + C) , solve for u . Then using 

substitution and the above equation for L, the target level is calculated. 

B-4 
P.22 

APPENDIX C 

Sample listings of the SAS programs used to predict the long-term mean 

S 02 emission levels required to meet a 1.2 lb SOp/million Btu emission 

standard are presented below. Both sample programs shown project the mean 

emission levels for averaging periods (3-hour discrete, 24-hour discrete, 

7-day rolling, and 30-day rolling) and three autocorrelation levels (0.5, 

0.7, and 0.8). Table C-1 presents a sample program assuming a lognormal 

distribution and RSD's of 25, 30, and 35 percent. Table C-2 presents a 

sample program assuming a normal distribution and RSD's of 10, 15, and 20 

percent. 

C-1 
P.23 

TABLE C-1. SAMPLE PROGRAM — LOGNORMAL DISTRIBTUION 

OOOIO /./ 

00020 // 

00030 //SYSIN DD * 

00040 HACRO OL 

00030 DATA ONEI 

00040 *t SET HOURLY AUTOCORR. (HACRO VALUE X) FOR GIVEN DAILY AUTOCORR.I 

00070 0«XI 

OOOtO NAV0«N» «* N IS HACRO VAR. FOR HOURS IN AVG PERIOD I 

000*0 FN»(<<l»0>/<l-0>-2*0t< l-D«*NAVO)/<NAVOt( !-0>«t2> >/HAVO>I 

00100 F24»<<<1*0)/<1-D)-2«D«( 1-D*«24 >/<21*< 1-D)*«3) )/2-l)» 

00110 IF N>24 THEN SR.FN-SQRT<FN/F2«>» 

00120 IF N<«24 THEN SR.FN»SQRT<FN>I 

00130 ttRSO.XX IS RSO OF HEAN OF X FOR GIVEN RSD OF XX AND AVERAGING PERIOD 

00140 OF LENOTH N HOURS. VARIATE X IS LOONORHALLY DISTRIBUTED! 

00130 RS0.23«SR.FN«0.23I RSD.30-3R.FNtO.30* RS0.33-SR.FNtO.331 

00140 <• S2.YXX IS THE VARIANCE OF Yt UHERE Y 13 THE LOQ QF THE MEAN 

00170 OF Xt Y IS NORNALLT DIST WITH VAR.S2-YXX ANO HSAN-H.f/* UHERE 

001tO XX IS THE GIVEN RSOI 

001*0 32.Y23-L0QU • RS0.23«*2>» S2.Y30«L0G( 1 • RSD.30**2)1 

00200 S2-Y33«L0Q<1 • RS0.33(t2)i 

00310 S.Y23«SQRT(S3.Y23>« S.Y30»S0RT<S2.Y30>» 9.Y33-S0RT(S2.YJ3)> 

00230 <t RSOXX IS THE VALUE OF TARGET LEVEL FOR RSO OF XX PERCEHT* 

00330 OATA EXC.PCT1* SET ONE! «t 1 PERCENT EXCEEDANCE FREQUENCY: 

00340 N.Y33«L00(ST> -2.324tS.Y23> 

00230 N_Y30«L00<ST> -2.324*3.Y30* 

00240 N.Y33>L00<ST) - 2.324*S.Y33I 

00270 RS023«EXP<H_Y23 + 0.3«S2.Y23>» 

002S0 RSD30>EXP(N.Y30 • 0.3*32.Y30>« 

00270 RS033«EXP<H_Y33 •• 0.3*S2-Y33>» 

00300 EXC.FREO'lt 

00310 OATA EXC.1YRI SET ONE* M t III 1 r* EXCEEDANCE FREQUENCY* 

00320 H.Y23"L00(ST)-Z1*3.Y23I 

00330 H.Y30*L0G(ST) - Z1CS.Y30* 

00340 H_Y33"LU0<ST) -Z1«S.Y331 

00330 R3023*EXP(N.Y23 * 0.3*S2-Y23>» 

00340 RSD30»EXP<H.Y30 * 0.3*32.Y30)i 

00370 RSD33*EXF(H.Y33 • 0.3<S2.Y33>* 

00380 EXC-FRE0»2I 

003*0 DATA EXC.IOYRI SET ONEI ** 1 IN 10 rp EXCEEDANCE FREQUENCY* 

00400 H_Y23»L00<ST> - :iO*S.Y23l 

00410 H.Y30«L00<ST> -Z10*S.Y30I 

00420 H_Y33"L00(ST> -Z10«3.Y33» 

00430 RSD23>EXF<N.Y23 > O.S«S2.Y23)i 

00440 RSD30*SXP(N.Y30 * 0.5*S2-Y30>* 

00430 RS033«EXP<N.Y33 • 0.3tS2-Y33)» 

00440 EXC.FREQ»3» 

00470 OATA AVOPERI SET EXC.tOYR EXC-1YR E^C.PCTl* 

004S0 R23>R0UN0(RSD23..01)* R30«ROUN0(RSD3O..01)IRSS'ROUNOCRSD33..01)1 

004*0 LABEL R23-'RSD - 23' 

00300 R30«'RSD • 30' 

00310 R33»'RSD • 33' I 

00320 PROC FORNATIVALUE EFO 1-QNE PERCENT 

00330 Z'.Ht PER YEAR 

00340 3-QME ?ER TEN YEARS* 

00330 PROC PRINT 0 LABEL* VAR R23 R30 R33I 

00340 tO EXC.FREO* 

00370 TITLE* 

00380 TITLES PROJECTEO S02 ENISSION LEVELS REQUIRED TO NEi" I.: Lb LIHIT* 

003*0 TITLE4 ASSUMING LOONORNAL DISTR1BUTIUN ?JR AVERftUE* 

00400 TITLES FOR DAILY AUTOCORRELATION OF Tl* 

00410 TITLE4 ANO T2 AVERACINQ PERIUDI 

C-2 
P.24 

TABLE C-1 (continued) 

00420 FORHAT EXC.FREO EFO. 

00430 X tt ENO OF HAIN HACRO DL) 

00440 •* SET HACRO VAR. VALUES FOR X.N.Zl.ZIOfTltT2 I 

00430 HACRO X 0.174*00*4 X M.«CF:0 N 720 X 

00440 HACRO Zl 2.78 X HACRO ZIO 3.44 X HACRO Tl 0.7T 

00470 HACRO T2 30 OAY ROLLING! 

00480 HACRO ST 1.2 X t* SET HACRO VAR OF STtFOR EHISSION LlHITU 

004*0 OL (S CALL HAIN HACRO TO PRINT 30 OAY AVO TABLE FrJ* 0 ..

, AUT3C0RR* 

00700 *t CHANGE HACROS N.T3 TO RUN 7 DAY AVG PERIOOI 

00710 HACRO N 148 X HACRO T3 7 OAY ROLL INOX DL 

00720 HACRO N 34 X HACRO T3 24 HOUR DISCRETE* OL 

00730 HACRO N 3 X HACRO Zl 3.40 X HACRO ZIO 3.*3 X 

00740 HACRO T3 3 HOUR OISCRETCX 

00730 DL 

00740 tt CHANGE HACRO VALUES TO RUN 0.3 AUTOCORRELATION! 

00770 HACRO X 0.93333747 X HACRO N 720 X 

00780 HACRO Zl 2.78 X HACRO ZIO 3.44 X HACRO Tl 0.3X 

00790 HACRO T2 30 OAY ROLLING! 

00800 OL 

00810 HACRO N 148 X HACRO T3 7 DAY ROLLINOX DL 

00830 MACRO N 34 X HACRO T2 24 HOUR DISCRETE DL 

00830 HACRO N 3 X HACRO Zl 3.40 X HACRO ZIO 3.*8 X 

00840 HACRO T2 3 HOUR OISCRETEX DL 

00830 *• CHANOE HACRO VALUES TO RUN 0.3 AUTOCORR. > 

00*40 MACRO X 0.98S73341 X HACRO N 720 X 

00870 MACRO Zl 2.78 X HACRO ZIO 3.44 X HACRO Tl 0.8X 

00880 HACRO T2 30 DAY ROLLINOX 

008*0 DL 

00*00 HACRO N 148 X HACRO T2 7 DAY ROLLINGX DL 

00*10 HACRO N 34 X HACRO T3 24 HOUR OISCRETEX DL 

00*30 HACRO N 3 X HACRO Zl 3.40 X HACRO ZIO 3.*3 X 

00*30 HACRO T2 3 HOUR OISCRETEX OL 

00*40 /« 

00930 // 

ENO OF DATA 

C-3 
P.25 

TABLE C-2. SAMPLE PROGRAM — NORMAL DISTRIBUTION 

OOOIO // 

00020 // 

00030 //SYSIN DO * 

00040 HACRO DOA 

00030 DATA ONE) 

00040 ** SET HOURLY AUTOCORR. (HACRO VALUE X) FOR GIVEN DAILY AUTOCORR.) 

00070 D'X! 

00080 F24-YI ** SET F UITH N OF 24 AVG FUNCTION FOR AUTOCORR. (HACRO Y>) 

000*0 N'NAVGI ** NAVO IS HACRO VALUE FOR HOURS IN AVG PERIOD* 

00100 FN>(((1+D)/(1-D>-2*D*<1-D«*N>/<N«(1-D>**2>>/N>* 

00110 IF N<23 THEN AR«SQRT(FN>) • • - ' 

00120 IF N>24 THEN AR«SQRT(FN/F24)I 

00130 t* RSOXX IS THE VALUE UF TARGET LEVEL FOR RSD OF XX PERCENT) 

00140 DATA EXC.PCT1I SET ONE* *t 1 PERCENT EXCEEDANCE FREOUENCY* 

00130 RS023>1.2/(1 • 2.324t.25tAR)) 

00140 RSD30*1.2/(1 • 2.324*.30*AR)» 

00170 RSD33"1.2/(1 + 2 . 324* . 33*AR)* 

00180 EXC.FRE0*1I 

00190 DATA EXC.1YR* SET ONE* It 1 IN I YR EXCEEDANCE FREQUENCY* 

00200 RSD23»l.2/(l t Z1«.23«AR>) 

00210 RSD30«1.2/<1 + Zlt.30«AR>! 

00330 RSD33»1.3/<1 + Zlt.33tAR)l 

00330 EXC.FREG-2) 

00240 DATA EXC.10YR* SET ONEI «* 1 IN 10 YR EXCEEDANCE FREQUENCY* 

00230 R9D23»1.2/(1 r Z10*.23*AR>) 

00240 RSD30»1.2/(1 • Z10*.30tAR)l 

00270 RSD33M.2/ U r Z10*.33*AR>* 

00280 EXC.FREQ'3! 

00290 DATA AVOPER* SET EXC.PCT1 EXC.1YR EXC.IOYRI J 

00300 R23»R0UND(RSD23..01) * R30*R0UND(RSD30t.01)I R3I-ROUND(RSD33. .01> > 

00310 LABEL R23»'RSD - J3' 

00320 R30»'RSD » 30' 

00330 R35*'RSD * 33 * 

00340 PROC FORHAT*VALUE EFO 1-ONE PERCENT 

00330 2-ONE PER YEAR 

00340 3»0NE PER TEN YEARS) 

00370 PROC PRINT 0 LABEL) VAR R23 R30 R3S) 

00380 ID EXC.FREO) 

00390 TITLE) 

00400 TITLES PROJECTED S02 EHISSION LEVELS REQUIRED TO rIEET 1.2 Lb LIMIT; 

00410 TITLE4 ASSURING NORHAL DISTRIBUTION l-'OR (V.'tR.IOF = 

00420 TITLES FOR DAILY AUTOCORRELATION OF Tl) 

00430 TITLE4 AND T2 AVERAGING PERIOD* 

00440 FORHAT EXC.FREO EFO. I 

00430 X *t END OF HAIN HACRO DOA* 

00440 ** SET HACRO VAR. VALUES FOR X>Y .NAVGtZl»ZtOfTliT2 ' 

00470 HACRO X 0.974V0094 X HACRO Y 0.93435319 X HACRO NA1.G 7ZQ ~ 

00480 HACRO Zl 2.78 X HACRO ZIO 3.14 X rlACRO Tl 0.7"? HACRO XZ 30 SAYX 

00490 DOA ** CALL HAIN HACRO TO PRINT 30 DAY AVG T.'.SLE .-OS 0.7 AUTOCORRi 

00300 ** CHANGE HACROS NAV0.T2 TO RUN 7 DAY AVG PERIOD* 

00310 HACRO NAVO 148 X HACRO T2 7 DAYX DOA 

00320 HACRO NAVG 34 X HACRO T2 24 HOURX DOA 

00330 HACRO NAVG 3 X HACRO Zl 3.40 X HACRO ZIO 3.73 X HACRO T2 3 HOURX 

00340 OOA 

00330 *« CHANGE HACRO VALUES TO RUN 0.3 AUTOCORRELATION* 

00340 HACRO X 0.93322747 X HACRO Y 0.70428974 X HACRO M.U'G 720 X 

00370 HACRO Zl 2.78 X HACRO ZIO 3.44 X HACRO Tl 0.5X MfilfiO T2 30 CAY* 

00380 DOA 

00390 HACRO NAVG 148 X HACRO T2 7 DAYX DOA 

00400 HACRO NAVG 24 7. HACRO T2 24 HOURX DOA 

00410 HACRO NAVG 3 X HACRO Zl 3.40 X HACRO ZIO 3.78 " 

00420 HACRO T2 3 HOURX DOA 

00430 ** CHANGE HACRO VALUES TO RUN 0.9 AUTOCORR. * 

00440 HACRO X 0.99373341 X HACRO Y 0.89444411 X HACRO NAVO 7J0 :: 

00430 HACRO Zl 2.78 X HACRO ZIO 3.44 X HACRO Tl 0.8X HACRO T2 30 DAYX 

00440 DOA 

00470 HACRO NAVG 148 X HACRO T2 7 DAYX OOA 

00480 HACRO NAVG 24 X HACRO T2 24 HOURX DOA 

00490 HACRO NAVG 3 X HACRO Zl 3.40 X HACRO ZIO 3.98 X 

00700 HACRO T2 3 HOURX DOA 

00710 /* 

00720 // 

ENO OF DATA 

J..-A.