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ABSTRACT

THE ADJUSTMENT OF ELECTRIC POWER GENERATION
TO ALTERNATIVE $02 STANDARDS: A CASE STUDY
IN ENVIRONMENTAL PROTECTION REGULATION

BY

James Michael Falvey

The object of this study was to compare the costs of generation under
two forms of environmental protection standards. The first form is the

ambient standard and restricts hourly average ground level concentration

 

of SO2 to one part per million or less. The second form is the emission
standard, and in this case, is assumed to be a uniform restriction on the
sulphur content of the fuel itself. The allowable sulphur content was set
so that hourly average ground level concentrations would not exceed one ppm.
A case study approach was utilized using the generating system of
Consumers Power Company as the case in point. Atmospheric dispersion esti-
mation equations were selected after a review of the dispersion literature.
The distribution functions of the relevant atmospheric variables were esti-
mated using past weather data for the mid-Michigan area or the opinion of
the state Climatologist where datum was lacking. In addition to the atmos-
pheric variability, temporal variation in the demand for power was considered.
Against this background of natural and man-made variation, the emission

standard has to consider worst conditions (i.e., worst atmospheric condi-

tions and highest system output) as determining the allowable sulphur

James Michael Falvey

content of the fuel. The ambient standard, however, allows ongoing fuel
adjustment to match varying weather and load conditions; it also allows
exploitation of differences instation design. With the price of fuel
inversely related to sulphur content, it was expected that the ambient
standard would result in substantially lower fuel costs.

Under the uniform emission standard, simulated fuel costs were 37
percent above fuel costs under the ambient standard. This saving under
the ambient standard results from the inclusion of both natural and man-
made variability. In spite of these savings the conclusions of this study
point out the problems of enforcing an ambient standard, and lend some
support for the predominance of emission standards seen in the real world.

In an attempt to capture strengths of each of these standards, a
station specific fuel restriction was simulated. This third form of regu-
lation, in addition to capturing the enforcement ease characteristic of
emission standards, can also exploit the design differences between sta-
tions. This third alternative was encouraging, costing only 12 percent

more than the ambient standard.

THE ADJUSTMENT OF ELECTRIC POWER GENERATION
TO ALTERNATIVE 802 STANDARDS: A CASE STUDY
IN ENVIRONMENTAL PROTECTION REGULATION

BY

James Michael Falvey

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1973

DEDICATION

To Wallace and Clair Murphy -

from whom I have learned much.

ii

ACKNOWLEDGMENTS

The author wishes to thank Professor Bruce T. Allen for many helpful
comments and an abundance of encouragement. Thanks also go to John Reynolds
of Consumers Power Company and Norton Strommen, formerly the State Climatol-
ogist, National Weather Service, U. S. Department of Commerce, for their

help in acquisition and interpretation of data.

iii

II.

III.

TABLE OF CONTENTS
Page
IntrOdUCtiOn O O O O O O O O O O O O O I O O O O O O I O O O O 1

1.1 The Purpose of the Study. . . . . . . . . . . . . . . . . l

1.2 Environmental Regulation. . . . . . . . . . . . . . . . . 2
1.2.1 Pollution: A Working Definition and

Some Implications. . . . . . . . . . . . . . . . . 2

1.2.2 The Two 802 Regulations to be Evaluated. . . . . . 3

1.2.3 Evaluation . . . . . . . . . . . . . . . . . . . . 4

1.3 The Organization of the Study . . . . . . . . . . . . . . 5

1.3.1 Chapter II: Theory. . . . . . . . . . . . . . . . 5

1.3.2 Chapter III: Meteorology and Dispersion . . . . . 6

1.3.3 Chapter IV; The Generating System . . . . . . . . 6

1.3.4 Chapter V: Analysis . . . . . . . . . . . . . . . 6

1.3.5 Chapter VI: Results and Conclusions . . . . . . . 6

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Description of the Functional Parts of th

Generating System . . . . . . . . . . . . . . . . . . . . 7
2.3 Profit Maximization for the Firm. . . . . . . . . . . . . 8

2.3.1 The Power Constraint . . . . . . . . . . . . . . . 9

2.3.2 The 802 Constraint . . . . . . . . . . . . . . . . 10

2.3.3 Cost Minimization with 802 Constraint. . . . . . . 11
2.4 Fuel Selection. . . . . . . . . . . . . . . . . . . . . . 12
2.4.1 Conceptualization of the Fuel Selection Problem. . 12
A Procedure to Choose Fuel Sets . . . . . . . . . . . . . 14
2.5.1 The Optimum SOZ Allocation Between Units . . . . . 15
2.5.2 The Optimum Allocation of Station Load to

Individual Units . . . . . . . . . . . . . . . . . 19
2.5.3 The Allocation of System Load to Individual
Stations . . . . . . . . . . . . . . . . . . . . . 29

2.6 Consideration of Stochastic Factors . . . . . . . . . . . 29

2.6.1 Meteorology. . . . . . . . . . . . . . . . . . . . 29

2.6.2 Load Pattern . . . . . . . . . . . . . . . . . . . 30

2.6.3 Treatment of Stochastic Factors. . . . . . . . . . 30
2.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 31

2.5

Meteorology and Dispersion . . . . . . . . . . . . . . . . . . 32
3 O l IntrOduction. . O O O O O O O O O O O O O O O O O O O O O 32

iv

IV.

VI.

3.3

3.5

Meteorology . . . . . . . . . . . . . . . . . . . . .
3.2.1 Introduction . . . . . . . . . . . . . . . . .
3.2.2 Stability. . . . . . . . . . . . . . . . . . .
Calculation of Dispersion . . . . . . . . . . . . . .
3.3.1 Introduction . . . . . . . . . . . . . . . . .
3.3.2 Calculation of Effective Stack Height. . . . .
3.3.3 Calculation of Dispersion Incorporating
Effective Stack Height . . . . . . . . . . . .
3.3.4 Concentration Versus Time. . . . . . . . . . .
Summary of Meteorology and Dispersion Estimation. . .
3.4.1 The State of the Art . . . . . . . . . . . . .
3.4.2 Use of the Dispersion Model - A New Direction.
3.4.3 The Use of the Dispersion Model - Conceptual
Significance . . . . . . . . . . . . . . . . .
Meteorological Parameter Values Used. . . . . . . . .
3.5.1 Hourly Mean Wind Velocity. . . . . . . . . . .
3.5.2 Ambient Temperature. . . . . . . . . . . . . .
3.5.3 Atmospheric Stability. . . . . . . . . . . . .
3.5.4 Meteorological Assumptions . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . .

Generating systelI‘. O O O O O O O O O O O O O O O O O 0

Introduction. . . . . . . . . . . . . . . . . . . . .
System Load . . . . . . . . . . . . . . . . . . . . .
Thermal Efficiency. . . . . . . . . . . . . . . . . .
Dispersion Characteristics of Generating Units. . . .
Summary . . . . . . . . . . . . . . . . . . . . . . .

Analysj-S O O O O O O O O O O O O O O O O O O O O O O O O O

5.1
5.2
5 3

5.4

Introduction. . . . . . . . . . . . . . . . . . . . .
A Restatement of the Problem. . . . . . . . . . . . .
The Assumptions . . . . . . . . . . . . . . . . . . .
The Generating System Assumptions. . . . . . .
Generating Station Assumptions . . . . . . . .
Generating Unit Assumptions. . . . . . . . . .
Fuel and Fuel Use Assumptions. . . . . . . . .
Meteorological Assumptions . . . . . . . . . .
alysis. . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . .
The Generating Station Output Expansion Path .
Allocation of System Load Between Stations . .

t-3

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Results and Conclusions. . . . . . . . . . . . . . . . . .

6.1
6.2

Introduction. . . . . . . . . . . . . . . . . . . . .
Specific Results. . . . . . . . . . . . . . . . . . .
6.2.1 Generation Without SO2 Regulation. . . . . . .
6.2 2 Generation Subject to MAGLC Regulation . . . .
6.2.3 Generation Subject to USWF Regulation. . . . .
6 2 4 A Third Alternative Regulation . . . . . . . .

33
33
33
35
35
35

39
47
48
48
49

49
50
50
50
51
51
52

53

53
53
54
56
57

66

66
66
67
67
67
68
68
68
69
69
69
78

‘ 86

86
86
87
87
88
88

6.3

6.4

APPENDIX I

General Results and Comparisons . . . . .
6.3.1 Comparison of Results. . . . . . .
6.3.2 Causes of Cost Differences Between

Regulations. . . . . . . . . . . .
6.3.3 Summary of Comparative Results . .
Conclusions . . . . . . . . . . . . . . .
6.4.1 Introduction . . . . . . . . . . .

Alternative

6.4.2 Implications for Spatial Dispersion. . . . . .
6.4.3 The Evaluation of the Alternative Regulations.

vi

89
89

91
93
94
94
95
97

99

LIST OF TABLES

Table IV-l. Generating Unit Characteristics.

Table VI-l. Results of Alternative 802 Regulation on Selected Items.

vii

Figure
Figure

Figure

Figure

Figure
Figure
Figure

Figure

Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure

Figure

Figure

Figure

11-1.

11-2.

11-3.

11-4 0

11-50

11-6.

III-l.

III-2 .

III-3.

III-4o

IV-lo

Iv-Z o

IV-3.

LIST OF FIGURES

Four fuel price-SO relationship.

2

Two intersecting linear production functions.

Isoquant map of a three unit generating station with super-

imposed isocost and isoSO2 lines.

Expansion paths traced out from the isocost line and the

isoSO2 line.

constraint line SS.

Isoquant map and SO2

Constrained least cost expansion path.
Effective stack height.

Distribution of ground level concentrations of SO released

from an elevated source. 2
Horizontal standard deviations of a plume.
Vertical standard deviations of a plume.

Input output relation for Campbell unit 2.

Input output relation for standard units.

Comparison of input output relationships for all units.

Least cost and least SO

2 expansion path.

Actual output expansion about the constraint point.
Actual output expansion to station capacity.

Graphic determination of the maximum possible error in
optimum load allocation.

Piecewise linear approximation of the discontinuous non-

linear constrained output expansion path.

Fuel price-sulphur characteristics.

viii

CHAPTER I

INTRODUCTION

1.1 The Purpose of the Study

This is a study to compare the estimated costs of two different envi-
ronmental quality protection schemes. The costs estimated will be only
the fuel costs faced by the firm being regulated.

The two specific methods of regulation investigated are both assumed
to have the same objective: the control of environmental damage caused

by ground level SO concentrations resulting from fossil-fuel-fired steam

2
electric power generation.

The first regulation is set in terms of a maximum allowable $02 con-
centration at ground level. The second regulation sets a maximum allow—
able sulphur per BTU standard on the fuel burned. The firm in question
is Consumers Power Company, whose generating capacity includes 24 conven-
tional fossil fuel units in six principal plants -- it will be these units
to which the two forms of regulation will be applied in order to compare
relative costs.

Although this study is of a specific type of environmental regulation
imposed on a specific firm, it is really no more than a single case study
in the area of regulation in general. Although a large amount of effort

will be expended in understanding and describing the underlying natural

phenomena, the technical aspects of power generation, and the behavior of

I
b I

"V

the firm, the object of the study is the evaluation of regulation. Eval-
uation can only be carried out after a general understanding of the under-

lying features unique to the particular example used as a case study.

1.2 Environmental Regulation
The assumption was made above that the point of environmental protec-
tion policy is to control environmental damage. The additional question
of how much damage is optimum is not considered; only the question of dam-
age protection versus regulation is addressed here.
1.2.1 Pollution: A Working Definition and Some Implications
Definition: An energy or material input to the environment
can be defined as a pollutant if it, in some sense, negatively
affects human use of that environment. The effect may be either
direct or indirect, local or global, reversible or irreversible,
and the use affected may be something as ill defined as a pleasant

view. The point is the same; the value of the environment has
been damaged.

 

The distinction to be made in this definition is the separation be-
tween stimuli and effects.' It is the effects that are of concern; stimuli
are of interest only because of effects. Unfortunately, from the point of
view of the regulator, environmental regulation is often more easily en-
forceable at the point of introduction of the stimuli into the ecosphere.
This is especially true of production processes in which the effluent is
introduced through a pipe or smoke stack of fixed location.

If damage is a single valued function of effluent flow alone, then
effects can be accurately controlled by controlling the flow of effluent.
If, however, there are many other variables that influence effects in addi-
tion to the flow rate of effluent, control of the effluent alone may be a

rather crude approach,especially if not polluting is costly.

These are all questions pertinent to the choice of damage protection

standard. The literature notes two basic classes of standards. One is

the ambient standard, and defines regulation in terms of the receiving
body; ground level concentration regulation used in this study is essen-
tially an ambient standard. The second general regulation type is the
effluent standard, which defines the permissable behavior in units of
effluent output, or effluent causing input. The sulphur per BTU standard
to be evaluated in this study is such a standard. The controversy over
the two types of standards has been both confused and bitter.

1.2.2 The Two 80 Regulations to be Evaluated

2

The first scheme of SO2 regulation considered is a standard defined
in terms of concentrations at the point of maximum ground level concen-
tration, irrespective of the location of that point. Standards of this
type are usually specified in terms of an allowable mean concentration
level over some stipulated time span.2 The time interval in this study
will be arbitrarily set at one hour. Hence the first regulation consi-
dered will impose the restriction on the generating company that the maxi-
mum one hour mean concentration of 802 at ground level shall not exceed
some specified intensity.

The second form of regulation considered will be a restriction on the
chemical composition of the fuel burned: the firm is constrained to burn

only those fuels which demonstrate a sulphur/BTU ratio less than or equal

to some specified figure.

 

lSee Engdahl [1973] for a brief history of the problem and a summary
of current regulation in the 0.5., or Schorr [1973] for a statement of the
current position of the E.P.A., which favors movement toward ambient stan-
dards, and Senator Edmund Muskie's attack on that position.

2For example, the secondary standards set by the E.P.A. specified
three separate means for three time intervals:

0.02 ppm maximum annual arithmetic mean concentration

0.1 ppm maximum 24-hour mean concentration

0.5 ppm maximum 3-hour mean concentration.

1.2.3 Evaluation

In the above description of the two SO2 regulations, numerical values
for both the concentration level and the sulphur/BTU ratio have intention-
ally been left out. In selecting values to use, one is immediately faced
with the question of what is the optimum damage level. As mentioned pre-
viously, no attempt will be made in this study to address that question.
The approach, rather, will be to focus on the respective values chosen for
each standard with the object being that the numbers used should result in
equal environmental damage under either standard, irrespective of the dam—
age level itself. Comparison of costs to the firm will be meaningless
unless damages are equated under both standards.

The problem of equating damages still has to be solved. Unfortunately
802 damage is a complex phenomenon -- it depends on ground level concentra-
tions over extended time periods as well as relatively short term fluctua-
tions. It will be assumed here that the effects of short-term fluctuations
are dominant, and that the one-hour mean concentration level is an adequate
descriptive statistic. The implication of this assumption is that to equate
damages under the two standards, both the standard set in terms of maximum
allowable hourly mean concentration and the standard set in terms of a
maximum allowable sulphur/BTU ratio, should be set so that the maximum
hourly mean concentration level will be the same under either standard.

The difference in fuel costs under the two standards will be due to
the amount of variability in the load on the generating system and the
degree of variability of meterological conditions (meteorology will have
a strong influence on dispersion of the stack gas). The standard set in

terms of ground level concentrations allows the generating company to vary

both fuel burned and the location of load generation, and thus should result

«nflw

in lower costs of generation. The question is: how great will the differ-
ence in costs be?

The philosophical question underlying all of this pertains to the
object of regulation: What is the justification for regulation? The only
tenable reason for environmental regulation is that which has been assumed
above: environmental regulation exists to control environmental damage.

If this is accepted, then regulation should be on effects rather than on

behavior. If ground level concentration of 80 is the best available proxy

2
for damage, then SO2 regulations should be specified in terms of that proxy.
In contrast to the concentration standard, the restriction on the
sulphur/BTU ratio of the fuel is a behavioral restriction. It takes no

account of other adjustments the firm could make to decrease damage.
On the surface, the sulphur/BTU regulation appears clearly inferior
to the first standard. However, it has two strong advantages: it is

simple and it is easily enforced. For these two reasons it is a real

alternative, especially from the viewpoint of the regulator.

1.3 The Organization of the Study
The remainder of the thesis is in five chapters. Most of the material
in those chapters will be concerned with developing either input informa-

tion for, or the implications of, the SO constraint set in terms of ground

2
level concentration. This is due to the complexity of operationalizing
that regulation.
1.3.1 Chapter II: Theory

The theory chapter will examine the optimization problem faced by

the firm restricted to keeping SO concentrations below some specified

2

level.

1.3.2 Chapter III: Meteorology and Dispersion

This chapter will describe the effect of meteorology on the dispersion
of stack gases. The relationship between ground level concentrations of
802 and total 802 leaving the stack will be estimated.
1.3.3 Chapter IV: The Generating System

This chapter will describe the fossil-fuel-fired segment of Consumer's
Power Company. Production functions will be estimated for each of the 24
units. Parameters necessary in using the dispersion equations of the pre-
vious chapter will be developed. The load pattern faced by the system will
be estimated.
1.3.4 Chapter V: Analysis

The analysis chapter will take the material developed in Chapters II,
III and IV and estimate the costs of generation under the two sets of
regulation.
1.3.5 Chapter VI: Results and Conclusions

The final chapter will discuss the relative costs estimated, particu-

larly in light of the narrowness of the analysis. General conclusions

pertaining to environmental regulation will be drawn.

CHAPTER II

THEORY

2.1 Introduction

This chapter attempts to provide a conceptual framework within which
the costs of power generation, subject to a maximum allowable ground level
80 concentration standard, can be estimated. The previous chapter postu-

2

lated an alternative standard set in terms of equivalent SO released per

2
BTU input. Consideration of the alternative standard will be postponed

until Chapter V.

2.2 Description of the Functional Parts of the Generating System

The generating system of an electric power company consists of all
the generating u£i£§_the firm has available to meet the power requirements
of the system. In a system consisting of m thermal generating units, the
jth unit (j = 1,2,...,m) displays a unique relationship between megawatt
hour output, Kj, and BTU input such that Kj = fj(BTUj) is monotonically
increasing up to its capacity, Rj' If transmission losses are assumed
to be zero, then the load on the system, KT, is equal to the sum of the

m

individual outputs, or KT = '§1K.. Although the typical generating system
consists of a mixture of hydio, nuclear and fossil fuel generating units,
the concern here is exclusively with conventional fossil-fuel-fired units.

In most generating systems, several fossil fuel units are usually

located at the same location in order to obtain economy in the use of

common subsidiary equipment such as transportation and transmission facil-
ities. This group of contiguous units is referred to as a generating
station. Typically the system will consist of several generating stations

scattered over its operating area.

2.3 Profit Maximization for the Firm

It is assumed that the electric utility, although regulated, attempts
to maximize profits -- regulation being merely a constraint on the ways
in which maximizing behavior can be displayed. This behavioral assumption
is generally applied to all firms, but the background conditions against
which the electric power company attempts to maximize profits are not
typical of most firms. The two main features that distinguish electric
utilities are: (l) the price of its output is fixed by a regulatory commis-
sion in the short run, and (2) it is legally bound to meet the demand for
its output.

Since the firm cannot control the demand for its output and since
the price of its output is fixed, profit maximizing behavior can only be
directed toward the cost term of the profit equation. Assuming labor and
capital to be fixed, profit maximization reduces to fuel cost minimization.
Assuming a generating system made up of m generating units, with each unit
free to operate on a fuel mixture3 of n potential fuels, the objective
function can be written as
m n
2

(l)

min 2 = Z Ci'pi'

j=1 i=1 3 3
. . .th

where cij denotes the number of units per time of the 1 fuel burned at

the jth generating unit and pij denotes the corresponding price per unit

of that fuel.

 

3That is, the units will operate with no loss in thermal efficiency,
and the mixing process will be costless.

2.3.1 The Power Constraint

In Section 2.2 reference was made to the fact that electric utilities
are legally bound to meet the demand for power. This means that the firm
is constrained to generate power equal to consumer demand, or in equation
form

m
X K. = K , (2)

where Kj is the megawatt (MW) output of the jth unit and KT is the total
demand on the system. This equation ignores transmission losses which are
assumed to be zero throughout.

At the individual unit level there is an additional constraint on
output -- each unit is constrained to a range of output between zero and

capacity, R.. Thus the system power constraint is more accurately written

(K <R.)=K . (3)

n r13 I4

As was mentioned previously, each generating units displays a mono-

tonically increasing relationship between BTU input and MW output (the

 

point where :BTU :_0 might in fact be used as a definition of capacity).

Thus there are m production functions, Kj = fj(BTUj), which can be con-
verted into MW fuel relationships, K. = f.(£ci.bi.), where again Ci' de-

th 1 k 3 3 th 3
notes units per time of the i fuel burned in the j generating unit
and bij is the corresponding BTU content of the fuel. Thus the power
constraint can be rewritten

m n
E (f.( X Ci'bi') < R.) = K . (4)
One additional feature peculiar to electric production warrants

mention here: unlike material producing firms, electric utilities have

10

no cheap way of stockpiling output.4 There is no easy way to dampen the
oscillations of demand for electric energy -- production must follow demand
through time, and the temporal variation is wide. Thus the innocuous
appearing KT may present serious problems in the analysis.

2.3.2 The $02 Constraint

In this chapter the SO constraint level, S, is assumed to be a single

2

uniform maximum allowable parts—per-million (PPm) concentration of 802 at

ground level. It is further assumed that there is independence between

stations with respect to SO concentrations, but that within any station

2

the contribution to SO2 concentrations of each unit sums arithmetically
. 5
to the total local concentration.
Ground level concentrations of SO2 depend on two things: (1) the

total amount of SO2 in the stack gas and (2) the degree of dispersion to

which the $02 is subjected. The total SO2 depends simply on the sulphur
inputs to the furnace, or the sulphur content of the fuel, times the amount
of fuel burned. Dispersion is much more complex. It depends upon the
release conditions of the stack gas (stack height, exit temperature and
exit velocity), and the condition of the atmosphere (meteorological state)
into which the gas is released.

For already installed units, stack height is fixed, and although exit
velocity and temperature are not constant, their relationship to the rate
of BTU utilization is fixed by the design of the unit. The meteorological

state is a random variable whose distribution would depend upon the geo-

graphic location of the generating system.

 

4Pumped storage units are an exception. A pumped storage unit utilizes

unused capacity of base load units during off peak hours to pump water to an
elevated reservoir. When needed, this water is released to drive a generator.

5 . . .
The second assumption becomes less valid as the number of units, stacks,
and other design parameters increase.

ll

. . .th
some measure of the degree of disperSion for the j

Letting Dj

unit, i.e., total 80 out divided by the resulting

2
ground level concentration,
and M = meterological state,
then dispersion can be expressed as a function of the rate of BTU consumption

and the meteorological state in the following form:

J g] i 13 13 '

where the function g. incorporates the stack height and stack and furnace
J
O O I O t . O I
deSign characteristics of the 3 h unit. To express the contribution of the

.th . . . . .
3 unit to station 802 concentration levels, Dj must be combined With a

. .th .
measure of the total 802 emitted by the 3 unit.

. . 6
Let sij = SO2 content of the ith fuel burned at the 3th unit,

 

Sj = parts per million of $02 at ground level contributed by
the jth unit,
then
n
2(C. .s )
i=1 1] 13

The assumption of independence between stations but additivity within

stations makes the SO2 constraint a station constraint. For a station of q

units, the SO2 constraint can be written as

n
2 (c
i:

 

(7)

2.3.3 Cost Minimization with SO2 Constraints

The effect, on the firm, of the SO regulation is that it adds one

2

additional constraint (equation 7) for each station in the generating system.

 

n

. . .th .
Thus 2 Ci'sij = total SO2 output per unit time from the 3 unit.
i=1

6

12

Thus for a system of t generating stations where station 1 consists of
units 1 to a, station 2 consists of units (a+1) to b,...,station t con-

sists of units (v+l) to m, the constrained optimization problem can be

written
min 2 = I 2(cijpij) (l)
l 3
subject to
§(f.(2}cijbij) _<_ R3.) = KT (4)
j i
and subject to
2c..s.. £c..s,. Zc..s..
a i ij 1] b i ij ij m i ij 1]
X _—D—_- _<_S, X -—D——— .<_S’°"' X ——D—' _<_S. (7)
j=l j j=a+1 j j=v+1 j

2.4 Fuel Selection
It has been assumed throughout that capital (particularly the basic
operating piece, the generating unit?) is fixed. This implies that the

shape of each of the m production functions, Kj = f.(£ci bij), and the
J i

j
m dispersion functions, Dj = gj( 2(cijbij),M), are fixed. In addition,
i
the SO2 constraint level, S, and the load characteristics, KT (more accu-

rately the distribution of KT), are determined exogenously. In addition,
the variable M is stochastic. This leaves the firm able to affect optimi-
zation mainly through fuel selection.
2.4.1 Conceptualization of the Fuel Selection Problem

The firm is viewed as having to choose t sets of fuels, one set at
each of its t stations. Each of the fuel sets is chosen from n possible
fuel types, each with known price, BTU content and sulphur content. It

is assumed that there is some degree of irreversibility involved in the

 

7 . . . .
The term "generating unit" includes everything from the furnace to
and including the generator.

13

choice of fuels —- once contracted for, the firm can only change the fuel
set at considerable expense. The reason for this assumption is that most
fuel contracts are for an extended time period.

The size of the fuel set (i.e., the number of fuels included in the
set) cannot, in theory, be determined a priori. There are, however, cost
factors that should limit the size of the maintained fuel set. One factor
is the cost of maintaining fuel stocks. It seems intuitively clear that
the physical storage space alone will limit the number of fuels. Another
factor is due to scale economies in both purchase and transportation of
fuels which would dictate against using small amounts of each of a large
number of fuels. This suggests that an additional problem exists in deter-
mining the price per unit of any particular fuel, since price will depend
not only on the fuel itself, but also on the quantity used. If fuels are
considered 1,2,...,n at a time, the price of a particular fuel would
change as the size of fuel set changed, due to changes in the amount of
that particular fuel that was used. For our purposes, the price of a
particular fuel can be considered constant, so long as the number of
fuels in the fuel set isconstant.

Thus far the choice of fuels has been approached at the station level.
Whatever fuel set that is chosen for the station is available to each of
the units there. However, any two generating units at the same station
may vary with respect to operating characteristics to the extent that the
same fuel may have different price characteristics due to differing amounts
of fuel treatment necessitated by differing furnace designs. Thus the
fuel choice for the station must ultimately be based on the individual

units. That group of t fuel sets which simultaneously meets the system

14

power constraint and the t 50 constraints and yields minimum costs is

2

the optimum.

2.5 A Procedure to Choose Fuel Sets

The optimum fuel set at any single station depends on the fuel sets
chosen at the remaining (t-l) stations.8 Thus a single station cannot be
looked at in isolation.

The approach used here will be to compare all the possible fuel set
combinations for the t stations. For this comparison estimates of KT and
M will be used. If the least cost use of each of the combinations can be
determined, the problem is solved -- we merely pick that combination which
yields minimum system fuel costs.

Given n fuel types at t stations the number of combinations to be
considered could be astronomical. ‘The number of fuel sets, CS, to be

considered at a single station would be

:3

n!

s r=l (n-r)!(r)! ’

D
II
S:

 

where r is the number of fuels considered at a time. Given CS fuel sets
at each of the t stations, the number of combinations to be considered

for the system, Cs' would be

t
= C .
CS ( S)
Unless both n and t are numerically small, the problem appears to be over-

powering.

 

8The optimum fuel set at any one station depends on its share of the
total system load, KT. Its share is KT minus output of the (t-l) remaining

stations. The load carried by the (t-l) stations will depend upon com-
parative costs of generation, or ultimately on the (t-l) fuel sets they
are burning.

15

The numerical value of t is equal to the number of generating stations
in the system. A quick survey of power systems in the U.S. shows that
few systems have more than ten stations and most have between three and
six stations. The value of n is potentially larger, but in most cases
some fuels can be discarded off-hand as being uneconomic. It also may be
possible to find station fuel combinations that dominate all other fuel
sets at that station. These possibilities depend on the data and cannot
be evaluated a priori. At this point it is assumed that the number of
combinations is manageable, and the problem of valuation of the different
fuel sets is considered.

Any comparison of fuel sets must make certain that the figures being
compared represent minimum cost usage of those fuel sets. In order that
a group of fuel sets be used in an efficient manner, there are three
necessary conditions:

(1) The station 802 constraint, 5, must be optimally allocated
between the units at each generating station.

(2) The production level at the station must be optimally allocated
between the generating units at that station.

(3) The demand load on the system, KT, must be optimally allocated
between stations.

If these conditions are met, the fuel sets are being used in an optimal
manner.
2.5.1 The Optimum SO2 Allocation Between Units

Assume that an arbitrarily chosen fuel set is considered at a station

consisting of two generating units. The fuel set can be visualized in two

dimensional space as follows.

P/BTU

 

 

SOZ/BTU

Figure II-l. Four Fuel Price-SO2 Relationship

In the diagram we are looking at the price, SO relationship for an

2
assumed fuel set of four fuels. Two separate relationships are shown
to symbolize use in two different units, with unit 2 demanding more fuel
treatment than unit 1. It can be shown that the relevant price-SO2 rela-
tionship lines will always be convex. Any fuel lying northeast of a line
between any two other fuels can be thrown out as uneconomic since any
point on the line can be reached by mixing. The negative slope implies
that the cost minimizing firm will always use the highest sulphur fuel
possible. The question for the firm to answer is: How should the sulphur
constraint, S, be split between the operating units?

Assume two units are operating at given outputs, ii and E5. Assume

further that sulphur output of each unit can be varied at will, so long

as (S1 + $2) = 8.9 How should the mix of fuels be set to yield the least

cost solution?

Fixed values of K and K imply that the input of BTU's to units 1

1 2

and 2 are constant. It also implies that the values of D1 and D2 are fixed

for any specific meteorological state (see Section 2.3.2 and equation (5)).

 

9Although the regulation allows S + 52 :_S, the least cost solution
demands 8 + S = S. This is implied y the inverse relationship between

SOz/BTU and Price/BTU.

17

Since the output of each unit is fixed, and meteorological conditions
are assumed constant, the only variable in the SO2 constraint equation is

the sulphur content (SOZ/BTU) of the fuel mix used. Rewriting the 802

constraint (equation 7) as

(SOZ/BTU)ZBTU2

l
+ =5. (8)
D
D1 2

(SOZ/BTU)lBTU

 

 

and realizing that AS + A82 = 0 we have

1

A(SOz/BTU)lBTU1 A(SOZ/BTU)ZBTU2
D + D = 0 . (9)
1 2

 

 

Rewriting equation (9) yields the required relationship between changes
at the two units as

A(SOz/BTU)l = (DI/Dz)(BTUz/BTU1)A(SOZ/BTU)2. (10)

The change in total cost associated with any such change in SOZ/BTU will
depend on the slope of the SOz/BTU, Price/BTU relationship of the particu-
lar fuel set considered.lO Let 2 denote fuel costs:

2 = (P/B'I'U)1 ° BTU + (P/BTU)2 ° BTU . (11)

1 2

The change in fuel costs due to altering the SO outputs of the two units

2

can be written as

A2 = A(SOZ/BTU)1[A(P/BTU)l/A(SOZ/BTU)1]BTU

1
(12)
- A(SOz/BTU)2[A(P/BTU)2/A(SOZ/BTU)2]BTU2.
Substituting (10) into (12) and setting A2 = 0 yields
[(01/02) (BTUz/BTUl)A (SOZ/BTU) 2] [A(P/BTU)1/ MSOz/BTU) 1] (BTUI)
= A(302/BTU)2[A(P/BTU)2/A(302/BTU)2](BT02), (13)

which reduces to

 

10Although the fuel set is fixed and individual fuel prices are

fixed, changes in both price and SO2 can be affected by changing the fuel

mixture.

18

 

 

A(P/BTU)2
D—l _ 13(502/13'm)2 (14
D2 " A(P/BTU)1 ° )

 

A(SOz/BTU)l

Thus the optimal allocation of S to the two units occurs where the contri-
butions of the two units equal S and the ratio of the dispersion values is
inversely equated to the slopes of the price—SO2 lines.

The strength of the optimizing rule is that it is independent of out-
put levels, and so holds for all levels of output.

This result must hold for any two units at the same station. Thus
for optimum allocation of the SO2 constraint to each of the units located

at a station consisting of q units,

 

 

q

E S = S and

, 1
i=1

A(P/BTU).
1_

Di A(SOz/BTU).
—= for all i #j (i = 1,2,...,q;
D. A P BTU . .

J (/ )1 3:1'2'000Iq)°

 

A(SOZ/BTU)i

In the case under consideration each of the price-SO relationships

2
are piecewise linear relationships, discontinuous at each of the pure
fuel points, with constant slopes existing between the pure fuel points.
Thus, in general, equality between dispersion ratios and slopes will not
be attainable. These considerations do not change the results except
that instead of equality between dispersion ratios and slope ratios, we
have to come as close to equality as the fuel lines and dispersion func-
tions permit. In general there will be an optimum where, with q units,
there will be at least q-l of the units operating on the points of dis-

continuity (i.e., on a single fuel rather than a mixture), and at most,

1 unit mixing fuels.

19

2.5.2 The Optimum Allocation of Station Load to Individual Units
The problem of allocating any arbitrarily chosen level of station

output, K to the individual units at that station is complicated by the

Sr
SO2 constraint. This is a result of the BTU input itself being an argu-
ment in both the production function and the dispersion function -- a change
in output for any particular unit necessitates a change in BTU input, which
in turn implies a change in dispersion.

Starting again with a simple case, assume that the station consists
of but two units, with combined output fixed and equal to KS. For fixed
meteorological conditions, the outputs of units 1 and 2, K1 and K2 respec-

tively, imply known unique values of D1 and D2. The relative values of

D1 and D2 imply unique fuel prices for the two units, so long as equations
[8] and [14] are satisfied, and the price/BTU, SOZ/BTU relationship of the

fuel is known. Total fuel costs are simply

P P
z _ BTU1(BTU)1 + BTU2‘BTU)2 '

The problem lies in evaluating changes in fuel cost due to changing the
allocation of KS to the two units. Both BTU levels and prices will change
as a result of the new allocation.

Assuming that the production functions, the dispersion functions, and

the fuel price-SO relationships are all continuously differentiable, the

2

change in cost due to changing the output of unit 1 can be expressed as

dBTU dP dP dBTU

 

 

dz 1 l l 1
———-= . p +-——— - BTU + ———-- (15)
dKl dKl 1 dKl 1 dKl dKl

The last term can be ignored since it is the product of two deriva-
tives, leaving the usual result of the rule for differentiation of the~

product of two functions.

20

The cost of the change in output at unit 2 will be identical in form;
only the subscripts will change. The change in total costs will be the

difference between the two or:

 

 

 

 

[ dz dz] [(iBTU:L P dPl BTU ] [dBTU P dP2 B U 1 (l6)
—— _ _—_ o + __ o - o + —— T .
dKl dK2 dK1 l dKl 1 dK2 2 dK2 2
dBTU.
Both the sign and the magnitude of Pi' BTUi, and dK 1 can be deter-
dP. i
mined. The problem lies in evaluating dii'
i

The change in price per unit of fuel depends upon: 1) the effect of
changes in Ki on Di (AKi implies ABTUi implies ADi), 2) the effect of

equal and opposite 1 changes in K1 and K2 on the ratio of their respec-
D

tive dispersion values (610, and 3) the relationship between SOZ/BTU and
price/BTU of the fuel set -- in particular, the rate of change of the slope
of that relationship.

In order to get the combined effects, the fuel mix has to be reopti-
mized, i.e., equations (8) and (14) have to be satisfied for each load
allocation. Unless there exist special properties of the production

functions, dispersion functions or the fuel set, neither the sign nor the
dpi
magnitude of dE— can be evaluated before reoptimizing via (8) and (14).
i
Unfortunately such a procedure would be exceedingly tedious.
dP.
An alternative approach would be to temporarily assume that gig-was
i
small enough to have no effect on the change in costs. Thus starting at

any arbitrarily chosen value of K1 and K2, with equations (8) and (14)

dBTU

dK
1

 

determining P and P at that point, a comparison of (

° P
1 2 1) and

 

11Since the point of focus is the change in allocation of K between

K and K , any increase in K1 must be offset by an equal decrease in K2,
and vice versa.

21

 

(23%2 ° P2) can be used12 to determine the direction of movement of the
2
change in allocation of Ks to K1 and K2.
Let A = (dBTU ° P )/(§§EE-° P ). If A > 1, the direction of movement
dK1 l dK 2

is towards increased output in unit 2 and equally decreased output in unit

1. Using P and P as constants, output can be changed (while recalcula-

 

 

1 2
. BTU . .
ting new values for dBTU and d ) until either A = l or K = 0.
dKl dK2 1

If the point is reached were A = 1, then a new set of values for P1
and P2 have to be determined by applying (8) and (14). With the new values
of P1 and P2, A should then be recalculated.

If the change in allocation leads instead to the result K = 0, a

1

different procedure is necessary. When Kl = 0, then K must be equal to

2

KS. It is entirely possible that even though A > 1, the direction of
movement may be wrong. If the two production functions cross, and Ks is

less than that level of output at which they cross, the ratio of deriva-

tives may be misleading. Figure II—2 is instructive.

BTU Input

 

I
1 1_
V I

K MW Out ut
KS C P

Figure 11-2. Two Intersecting Linear Production Functions

 

12Throughout this section is is assumed that dZBTU/dK? is either §_0
or 3_0 for all i = 1,2,...,q units. This implies that there can be, at
most, one point of intersection of production functions for any two units.
The assumption will be checked against the Production functions estimated
in Chapter IV.

22

The diagram shows a simple case of linear production functions inter-

secting at output KC. At any level of station output less than KC (Kl +

dBT dBTU
-—Jl'<-——- . Unless P > P by an amount greater than
dKl dK2 2 l

the divergence in derivatives, strict adherence to the value of the ratio

= <
K2 KS KC),

A would lead to K1 = KS, K2 = O, which would not be the least cost alloca-

tion. This result is due to the production functions intersecting the
BTU axis at points where BTU > 0. To correct for this error, whenever

KS < KC and the A ratio drives either K1 or K2 to zero, costs at that

point must be compared to costs of reversing the allocation (i.e., if

Kl = 0, K2 = KS then calculate costs for Kl = KS, K2 = 0). So long as

K
S

|v

Kc, this problem does not exist.

Due to the problems of determining changes in prices caused by changes
in power allocation, the above analysis has been concerned heavily with com-
parative thermal efficiencies of units followed by after-the-fact determi-
nation of prices. It appears that, since the emphasis is so heavily on
the production function, the best procedure would be to choose the initial

allocation of KS to Kl' K with complete disregard for prices, i.e., based

2
on thermal efficiencies. Thus, instead of starting with arbitrary values

of K , K start at the point where QEIE-= dBTU.
1 dKl dK2

exist, expand output of the unit with the smaller derivative until its

 

2, If equality does not

output is equal to K In the case of KS < KC (Figure II-l). Slope has no

S.
meaning as mentioned earlier, and comparative total BTU inputs of (K1 =

KC, K2 = 0) versus (Kl = 0, K

point is chosen based on thermal efficiency, then prices can be determined

2 = KC) must be calculated. Once the initial

using equations (8) and (14).

Equation (8) sets the sum of the SO contributions equal to the regu-

2

lation and equation (14) splits the regulated amount optimally between the

23

units. When equations (8) and (14) are satisfied, the resulting fuel

prices, P and P are the optimum fuel mix prices. These prices will,

1 2'
in general, necessitate a change in the allocation of power between units
from the original allocation based on thermal efficiency. The new power
allocation is the start of the next iteration.

It may be instructive to look at a simplified expansion path in some
detail. Assume that we consider a station consisting of three generating
units, each with a linear production function, and each choosing from the
same twg_fuels. Assume further that the firing order of the three units
is predetermined, that the dispersion efficiencies are identical and that
output is not an argument in the dispersion functions. In Chapter V these
assumptions will be relaxed.

In Figure II-3, isoquants are drawn in two-fuel space. They are
straight lines running at a 450 angle to the axes, because the factor
inputs are perfect substitutes in production. It is assumed in the dia-
gram that the three generating units are of 300, 200 and 100 MW capacity,
with production functions of the form MWi = ai + Bi(BTU), (i = 1,2,3).
Although the quadrant is dense with isoquants, those drawn in are of
particular interest. The isoquants labelled 0, 300 and 500 are "thick" --
this thickness is the graphical representation of the absolute size of

a a and a respectively.

1' 2 3

In Figure II-4, two separate expansion paths are derived directly
from Figure 11-3. The price line PHPL would yield an expansion path of
corner solutions along the high sulphur coal axis. Given the relative
prices depicted, the least cost expansion path is derived in Figure II-4.

The least SO2 expansion path is likewise derived from Figure 11-3,

where the line SHSL shows the relative sulphur contents of the two coals.

24

Figure II-3. Isoquant map of a three unit generating station with super-

imposed isocost and isoSO2 lines.

Figure II-4. Expansion paths traced out from the isocost line and the

isoSO2 line.

High Sulphur BTU's

 

25

 

 

Low Sulphur BTU's

Least 802 Expansion Path

I Least Cost Expansion Path
l
' l
' I

 

Fuel Costs

|
! 1 I
300 500 600
MW Output

 

26

As in the least cost expansion path, the least 802 expansion path is
derived from corner solutions except in this case, they occur along the
low sulphur coal axis.

In Figure 11-6, LSLS and LCLC are the least 502 and least cost paths
taken from Figure II-4. What is new in Figure 11—6 is the constrained
least cost expansion path. The derivation of this comes from Figure II-5.

Given the assumption that the dispersion of $02 from each of the three
units is identical and independent of output, then for any particular mete-
orological state, ground level concentration is a linear function of the
combined sulphur inputs to the three furnaces. The slope of the line in
our two-fuel space is the ratio of sulphur per BTU of the two fuels, and
the position of the line is determined by the regulation level itself and
the meteorological state. As drawn in Figure 11-5, the constraint line SS
intersects the high sulphur axis at the point of intersection of isoquant
350 with that axis. For all output levels less than 350, the constraint
is irrelevant, and the least cost expansion path would proceed along the
high sulphur coal axis. Beyond 350, the coal mix will be determined by
following the SS constraint line. As can be seen, not only will increased
output be produced completely with low sulphur coal, but, low sulphur will
be substituted for high sulphur over part of the previous 350 units of
output, i.e., the absolute amount of high sulphur coal burned will de-
crease. This substitution is referred to as "back mixing" in Chapter V.
It is because of this phenomenon that the constrained least cost expan-
sion curve shows costs increasing faster than the LS curve beyond 350
units of output in Figure 11-6.

Without the simplifying assumptions, the effect of the SO constraint

2

on the allocation of station load between units is unpredictable. However,

27

Figure 11-5. Isoquant map and $02 constraint line SS.

Figure II-6. Constrained least cost expansion path.

High Sulphur BTU’s
(D

 

 

 

28

 

 

 

Low Sulphur BT U ’5

L5

Constrained Least Cost Expansion Path

 
     

 

13 HS
0
U I
- l

3 I
ILL l

HsI l I I

I l l J
3 35° 500 600

MW. Output

29

a few observations can be made. First, the relationship between dispersion
functions and production functions is crucial. If dispersion efficiency is
directly related to thermal efficiency, the effect of the S02 constraint
will be to strengthen the motivation for using the thermally most efficient
unit mix. Second, the shape of the dispersion function is important. It
can safely be assumed that the first derivative of dispersion with respect
to BTU input is non-negative. The sign of the second derivative is at this
point unknown. If the sign is positive, there will be a tendency for the
$02 constraint to lead to generating the load with fewer units. If nega-
tive, the tendency will be to use more units. The shape of production
functions will also be crucial. The next two chapters will answer some
of these questions.
2.5.3 The Allocation of System Load to Individual Stations

The problem of system load allocation is dependent entirely on the
results of sections 2.5.1 and 2.5.2. Once the least cost expansion path
of output is determined for each station, the problem of allocating system
load between stations is solved by simply comparing individual station

output expansion paths -- equating marginal station generating costs will

yield the optimum solution.

2.6 Consideration of Stochastic Factors
2.6.1 Meteorology

All references to meteorology made thus far have alluded to the fact
that the meteorological "state" heavily influences the dispersion of stack
gas. This state can be conceptualized as a multidimensional vector with
the values assumed by each component determining the state. The values

assumed by each component are beyond the control of the firm and display

30

a wide range of variability. A listing of components would include such
things as temperature, wind speed, and an atmospheric stability parameter.
The next chapter will deal with these components in depth, but for now,

all that is necessary is to recognize that the meteorological state is
multidimensional with the value of each component being a random variable.
The probability distribution of the individual components is inconsequential;
what is necessary is an estimate of their joint distribution.

2.6.2 Load Pattern

The load demanded of the generating system varies greatly over time.
There is a daily, a weekly, and an annual cycle involved in this variation.
In contrast to the joint distribution of meteorological effects, the load
distribution function should be easily estimated from past history.

2.6.3 Treatment of Stochastic Factors

Once the probability distribution estimates have been made for both
the system load and the meteorological state, each of the distributions
can be broken into intervals, with the number of intervals being deter-
mined by the degree of accuracy desired.

Assume that the load pattern probability distribution and the meteo-
rological probability distribution are broken into m and n intervals respec-
tively. Using the midpoint of the interval as the value of each interval
and assigning probability to each value based on the area under the curve
over each interval will yield m and n discrete values with attached
probabilities.

The analysis of sections 2.5.1, 2.5.2, and 2.5.3 will yield solutions
to the problem of minimizing fuel costs for known values of the system load

(KT) and the meteorological state (M). Having discretized the two probability

31

functions, the problem now is to minimize fuel costs for m separate values
of KT and n separate values of M.

The approach that will be used will be to take each of the m values
of KT, one at a time, and produce cost estimates for each of the n meteoro-
logical states, weighted by the probability of that state occurring. Like-
wise the cost figure will be weighted for each of the m values of KT by the
probability of that value of KT occurring. The total costs of generation

will be fuel costs summed over both KT and M.

2.7 Results

All of section 2.5 (2.5.1-2.5.3) and 2.6 (2.6.1-2.6.3) deal with
solving the problem of estimating fuel costs of electric power generation,
or more accurately, minimum costs of power generation. It should be borne

in mind that so far we have only estimated minimum costs for a single fuel

 

set. Returning to section 2.5 on pages 14 and 15, it is clear that the whole

procedure of 2.5.1-2.6.3 has to be repeated numerous times (CS times) in

order to choose the optimum fuel set. That fuel set which gives minimum

 

fuel costs is the optimum, and the associated cost is the cost of power
generation for our system of given parameters, subject to the hypothesized

SO2 constraint.

2.8 Conclusions

The conversion of this chapter into a practical vehicle for estimating
dollar costs depends heavily on many factors that have thus far been left
indefinite. Two of these factors are the specified form and parameter
values of the individual unit production and dispersion functions. The
next two chapters will deal with these problems. In Chapter V the results
of Chapters III and IV will be combined with the material of this chapter

towards a computational solution procedure.

CHAPTER III

METEOROLOGY AND DISPERSION

3.1 Introduction

The purpose of this chapter is to clarify and describe the relation-
ship between the generation of electric power and the resulting ground
level concentrations of 802. The actual concentration level experienced
will be a result of two distinct conditions: 1) the amount of SO emitted

2
and 2) the degree of atmospheric dispersion to which that quantity of 802
is subjected.

The amount of SO2 put out in conjunction with power generation is
relatively simple to quantify. Given the thermal efficiency of a gener-
ating unit and the level of output that unit is operating at, a known
level of BTU input is implied. If,in addition,the sulphur/BTU ratio of
the fuel burned is known, one simply multiplies total BTU input times
twice13 the sulphur/BTU ratio to get the total 802 output of the unit.
The dispersion conditions are more complex.

The dispersion of $02 in the atmosphere is determined by both the

release conditions of the 502 and the condition of the atmosphere into
which it is released.

The release conditions can, for each unit, be adequately described

in terms of four parameters: 1) the point of release in the vertical

 

13On a mass basis, sulphur has a molecular weight twice that of

oxygen. Thus upon combustion, the mass of SO2 will be twice the original
mass of sulphur.
32

33

dimension (stack height), 2) the velocity of the stack gas at exit, 3) the
temperature of stack gas and 4) the diameter of the stack. Parameters 2,
3 and 4 determine what is referred to as bouyancy flux. Of these four
parameters, 1 and 4 are fixed and 2 and 3, although variable, vary in a
known direct relationship with output of that unit.

The state of the atmosphere into which the stack gas is released is
crucial in effecting ground level concentration intensities. Ambient tem—
perature, wind velocity and directional variability in both the horizontal
and vertical plane, and the vertical temperature profile all strongly in-
fluence dispersion. They represent stochastic processes which are beyond
the control of the firm. In view of their impact on dispersion, these
meteorological phenomena will be treated first and in more depth than the

release conditions.

3.2 Meteorology
3.2.1 Introduction
The essential relation between meteorology and atmospheric
dispersion involves the wind in the broadest sense of the term.
Wind fluctuation over a very wide range of time and space scales
accomplish dispersion and strongly influence other processes asso-
ciated with it [ASME, 19681.14
Wind, as defined in lay language, plus the condition of the atmosphere
to either suppress or enhance vertical motion are the main determinates of
atmospheric dispersion. This tendency to resist or suppress vertical motion
is referred to as the stability of theatmosphere. The degree of stability
is crucial to dispersion estimates.
3.2.2 Stability

If a small volume of air is forced upwards in the atmosphere, it will,

in theory, encounter lower pressure, expand and cool. Assuming no exchange

 

14The bulk of the material in this chapter is from this publication.

34

of heat between the small volume of air and its atmospheric environment,
the rate at which the small volume cools due to expansion is defined as

the dry adiabatic lapse rate, and is equal to -1 degree centigrade per 100

 

meters of ascent. Although there is always some exchange of heat in the
atmosphere, the concept is an important one.

The stability of the atmosphere is determined by a comparison of the
actual vertical temperature profile with the dry adiabatic lapse rate. The

actual temperature stratification is known as the environmental lapse rate.

 

If the environmental lapse rate is greater (absolutely, e.g., -2°/100 m)
than the dry adiabatic lapse rate, the volume of gas released upward will
tend to become less dense than the surrounding atmosphere and upward motion
will be enhanced. This is referred to as an unstable atmosphere. Alter-
natively, a stable atmosphere exists when the environmental lapse rate is
less than the dry adiabatic lapse rate, and a neutral atmosphere is one in
which the two lapse rates are nearly identical.

Another measure of atmospheric stability is known as the potential

temperature. The potential temperature 63 is defined as the temperature

 

that would be assumed by a parcel of dry air brought adiabatically from
its initial elevation in the atmosphere to a standard pressure of 1000 mb.
A potential temperature that increases with elevation denotes a stable
atmosphere; if it decreases with elevation the atmosphere is unstable.

The atmospheric condition receiving the most attention from pollution
authorities is the phenomenon referred to as a thermal inversion. An in-
version exists when the vertical temperature profile is reversed, resulting

in increased temperature readings as the elevation increases. When an
inversion occurs, vertical motion is severely resisted. However, much of

the concern with low level inversions is only pertinent to pollutants

35

released at ground level. Since an inversion is simply an extremely
stable situation, pollutants released at ground level will stay at ground
level, where the damage potential is highest. In the case of power plant
emissions, gases are released at an elevation provided by the stack height
itself. With vertical motion resisted in either direction, the 802 will

tend to stay above ground level, and so yield extremely low ground level

readings, except in areas of high topographic relief.

3.3 Calculation of Dispersion
3.3.1 Introduction

In calculating the dispersion of a gas from an elevated point source,
the dispersion process is separated into two sequential parts. The first

step is to calculate the effective stack height, which includes the rise

 

of the plume above the actual stack height. This establishes a theoret-
ical origin of the plume. The second step is to calculate dispersion
using the theoretical origin as the point source of the effluent.
3.3.2 Calculation of Effective Stack Height

The theoretic estimation of stack gas dispersion demands consider-
ation of not only the actual height of the stack from which the gas is
emitted, but also consideration of the rise of the gas above stack height.
This rise above stack height is due to the initial velocity of the gas
emerging from the stack (the range of velocity for the generating units
under consideration is from about 13 ft/sec. to nearly 120 ft/sec.), and
a buoyancy effect due to exit gas temperature which is well above ambient
(again, for the units considered, the range is from approximately 230°F
to 330°F). The actual stack height plus the resulting rise of the center
line of the plume is defined as the effective stack height. The following

diagram is instructive.

36

Figure III-1. Effective stack height.

 

 

 

 

 

 

 

 

38

The theoretic origin, as shown in Figure III-l, is used for the com-

putation of dispersion. The A.S.M.E.'s "Recommended Guide for the Predic-

tion of Airborne Effluents" suggests using two different equations for the

calcu

(Ah)

lation of the rise of plumes.

For plumes rising in a stable atmosphere the rise above stack height

is estimated by

F .33
Ah - 2(559 .

For neutral and unstable conditions the equation recommended is

w

6

Terms in the above equations are

 

E = mean wind speed at actual stack height (m/sec)
D 2 Ts-T
F = buoyancy flux = gV (—) a where
s 2 T
a
. . 2
g = acceleration due to graVity (m/sec )
VS = vertical efflux velocity (m/sec)
D = stack diameter (m)

T = stack gas temperature (OK)
T = ambient temperature at stack height (OK)
G = stability parameter = 3—-A§-,

60 A2

9 = potential temperature at stack height (OK)

%g-= lapse rate of potential temperature (0K)

The authors suggest extreme caution should be used when interpreting

results for low speed wind conditions, and state that they are unreliable

when

the wind speed at stack height is less than 7 m/sec.

39

3.3.3 Calculation of Dispersion Incorporating Effective Stack Height

The two atmospheric flow features of greatest importance to the dis-
persion of stack gases are the wind speed and the turbulence characteristics
the stack gas is subjected to. The wind speed acts to disperse the gas by
providing separation between gas molecules. The turbulence acts to spread
the gas out in the plane perpendicular to wind direction.

The ASME manual advises the use of a simple form of the basic Gaussian
distribution equation:

2 2 2 2
-{h /2oz + y /20y}

X(x o) = no 3 E e
IYI Y z
. . 3 3
where X = concentration (units/m , e.g., gm/m ) of pollutant at

(x,y,z)

(x,y,z) where x is the distance directly downwind, y is distance in the

cross wind direction and z is the distance vertically.

Q = pollutant release rate (units/sec., e.g., gm/sec.)
oy,oz = crosswind and vertical plume standard deviations (m)
U = mean wind speed at stack height (m/sec.)

h = effective stack height = hS + Ah where

hS = actual stack height.

Since we are interested in ground level concentrations only, 2 has
been set equal to zero. By setting y = 0, we will obtain concentrations
directly down wind.

-{h2/20:}

X — - e
(x,o,o) no 0 u
y z

This equation is more suitable, since the highest values will always be
found directly downwind (y = 0).
Figure III-2 shows the crosswind distribution of ground level

centrations as a gradually broadening set of bell-shaped curves. The

Figure III-2.

40

Distribution of Ground level concentrations of SO
released from an elevated source.

2

sou ac E
)/

 

 

(0,0,0)

41

42

quantities CY and oz are the crosswind and vertical standard deviations

of the dispersing plume and are functionally related to the distance down-
wind. One means of estimating 0y and Oz often used where instrumentation
is lacking is to use a power law relationship of the form

0 = bxq
where the coefficient b and the exponent q vary according to meteorological
conditions. The graphs and values for b and q are shown below. They are
intended for estimating hourly mean concentrations, and are acceptable for

our purposes (see Figures III-3 and III-4).

The equation for the maximum ground level concentration is

X =_22_o_z
max em-ih2 CY
with the point of maximum concentration occurring at the distance where
oz = h/VFE .

Two additional ground level concentration equations are put forward

in the ASME manual:

1) Inversion fumigations:

X =9:—4-g.
1F hoyu

The reader is cautioned on the use of this equation with the statement
that the answers derived may be off by as much as a factor of 5.

This equation attempts to estimate concentrations that occur during
the breakup of low-level temperature inversions. This phenomenon is usu-
ally of short duration, ranging from a few minutes to an hour. Unless
frequently experienced in Michigan, it can be ignored.

2) Multiple Sources

In many cases, and particularly with respect to generating units,

ground level concentration levels depend upon the contribution of not a

Figure III-3.

43

Horizontal standard deviations of a plume.
[Sourcez ASME, 1968]

44

(Hourly Mean Value)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10,000 I lllllll I 11111" I Ill/Ill-
E /
1,000 1,- I:
_' a
I .I
d
(m) _ q
I— d
100 _ -
(_ .1
: I
Power Law Equations _
for o
.91
10 Very Unstable 0.40x 86—":
E7 Unstable 0. 36x' 78 :
P Neutral 0.32x°71 -
: Stable 0.3lx' 1
l 11 l‘llllll I l llJll l lcllll
100 1,000 10,000 100,000

x (m)

45

Figure III-4. Vertical standard deviations of a plume.
[Sourcez ASME, 1968]

(Hourly Mean Values)

 

 

 

 

 

 

 

 

 

10'000: I l IIrWI F I IriIII I II n
"" /
_. ‘/ )2
"’ / I ""
L. / / -
‘//’ /
"' / an
r ‘/ 4
/ I
1 000 / / #/
P ‘
(m) _, [’2’ _
I- / _-
r— ’ ..
P / .-
100 __ 41/ _
: ”—p-n-———:
I— / r’ "
I—n ’ _1
Power Law Equations ‘-
For 0
z .91
10 /, Very Unstable 0.40x 86__1:
Unstable 0.33x' ‘
h— .78 —
—- Neutral 0.22x ‘
-' .71 -
__ Stable 0.3lx ~
III- --I
1 l lllllll 1 (1111“ l 111”“
100 1,000 10,000 100,000

x (m)

47

single stack, but several stacks (i.e., the generating station). The ASME
manual suggests the following equation but cautions that knowledge of the
problem is extremely limited

0.8
X =
max(N) Xmax(l)N '

where xmax(1) is the maximum concentration determined for a single stack
and N is the number of stacks.

This estimate of concentration seems intuitively untenable for our
purposes, unless the stacks are the same height, have the same conditions
of release (velocity, temperature) and the corresponding generating units
are operating at the same levels of output.

Referring back to Figure III-2, in the case of multiple stacks we have
as many concentration surfaces as we have stacks -- the ground level con-
centration at any point in the x,y plane is the vertical sum of the heights
of the individual concentration surfaces overlying that point. The point
of global maximum has to be found, and its height determined.

This assumes that one stack does not affect the dispersion of the
other stacks. This is, in fact, probably not true -- multiple stacks seem
to enhance the dispersion of each individually. Thomas et a1. [1963] pre-
sent data which indicate this.

Given the lack of information on interactions between units, the
assumption made in Chapter II (Section 2.3.2) will be retained: maximum
ground level concentration is the simple arithmetic sum of the individual
unit maxima. This assumption will yield estimates of the global maximum
that will be biased toward overstatement of concentration levels.

3.3.4 Concentration Versus Time
All the equations presented above represent estimates of one hour mean

concentrations. Short-term peaks may be substantially higher, but since the

48

concern here is only with one hour mean concentration levels (i.e., the
regulation is written in terms of hourly mean concentrations), transient

peaks will be ignored.

3.4 Summary of Meteorology and Dispersion Estimation
3.4.1 The State of the Art

It should be evident from Section 3.3 that the equations put forth
are, at best, rough approximations. There are many reasons for this fact.

Very little comprehensive empirical work has been done on stack gas
dispersion to date. A large-scale study initiated by the Tennessee Valley
Authority in 1963 probably represents the most complete experimental work
done so far, but even there, the design of much of the work is not appli-
cable to this study.

The paucity of empirical work with results that can be generalized
is due heavily to the complexity of the phenomenon itself. The dispersion
process, between exit of the stack gas and resulting ground level concen-
tration, takes time. Over this time interval, meteorological conditions
may be changing. The rise of the plume is difficult to accurately docu-
ment. The instrumentation to simultaneously measure ground level concen-
trations and temperature and wind velocities at many different heights is
extensive. The effects of unique local terrain may be crucial.

The ASME approach which is adopted in this study is derived theoreti-
cally from the physics of the problem, starting with the mechanics and
thermodynamics inherent in the dispersion of a large volume of gas released
into the atmosphere with known initial temperature and velocity. The model
assumes a level uncomplicated terrain, an assumption which is not badly

violated in most of Michigan's Lower Peninsula.

49

3.4.2 Use of the Dispersion Model - A New Direction

Historically, dispersion models have been used by electric utilities
to determine adequate stack height, and as such they are usually employed
in the planning stage, prior to construction of the unit. Given the firm's
criterion of allowable concentrations of $02, and given the sulphur content
of the fuel to be used, the engineer uses the dispersion estimate to deter-
mine how high the stack should be built so as not to exceed the allowable
concentration with some degree of confidence.

The use of the dispersion equation in this paper is fundamentally
different. In this study, already installed units are considered, with
given fixed stack heights. The equation will be used here to determine
allowable sulphur content of the fuel. In addition to changing the depen-
dent variable, there is another dimension of difference. Whereas dispersion
equations were developed to estimate worst probable cases (i.e., generating
unit running at full capacity, meteorological conditions such as to give
highest concentration levels), the equations used here will be employed
for a range of both load and meteorological conditions.

3.4.3 The Use of the Dispersion Model - Conceptual Significance

In spite of the misgivings voiced on the accuracy of results derived

from the dispersion model, the attempt to predict dispersion is important.

First, it seems obvious that even in the case where SO regulation is

2
phrased in terms of an SOZ/BTU ratio, some implicit idea of the relation
between output at the stack and ground level concentration must be envoked,
even if never explicitly stated. The explicit statement used here is at
least open to criticism and debate.

The dispersion model is more significant from an ecological point of

view. Ecologists put a great deal of emphasis on the concept of "environ-

50

mental capacity." This is the natural capacity of the environment to
tolerate and process stimuli without being damaged by that stimuli. The
dispersing ability of the atmosphere is very close to this concept. There-
fore the dispersion model can be viewed as an attempt to estimate this
capacity. As would be expected, this capacity to disperse would vary from
area to area, and conclusions reached for the Michigan region may not hold
for other regions. With this in mind we now turn to the necessary assump-
tions on the probability distribution of meteorological conditions (the

local determinants of atmospheric dispersion) for the Michigan area.

3.5 Meteorological Parameter Values Used
3.5.1 Hourly Mean Wind Velocity

Four hourly mean velocity ranges of equal probability have been esti-
mated from wind rose data gathered over a five-year period at Capital City
Airport, Lansing, Michigan. This data was gathered at a 52' elevation above
ground level and has been multiplied by 2 to approximate wind velocity at
stack height. It is assumed that the four range means are representative
of wind distributions for all five coal fired station locations. The esti-

mates are as follows:

 

 

l. ui(meters/sec.) P(ui)
1 5.36 .25
2 10.73 .25
3 14.30 .25
4 19.67 .25

3.5.2 Ambient Temperature
As in the case of wind velocity, ambient temperature has been esti-

mated from data gathered at Capital City Airport. The data used are average

51

daily temperature readings for each day of the year over the 30-year
period 1931-1960. The range of temperatures was broken into four segments
of equal probability with the average value in each range selected to

represent that range.

1 TAi( K) P(TAi)
1 269 .25
2 277 .25
3 287 .25
4 294 .25

3.5.3 Atmospheric Stability

Unlike temperature and wind velocity, there is no good historical
documentation of stability conditions for the mid-Michigan area. For this
reason estimates have been made which rely heavily on informal conversations
with the state Climatologist, The necessary information is listed below.

The values of Uz/Uy have been taken directly from Figures III-3 and III-4.

 

 

Classification (oz/0y) (AG/A2) Probability
Stable 0.1935 1.3°c./100m. .40
Neutral 0.6875 n.a. .30
Unstable 0.9167 n.a. .25
Very unstable 1.0000 n.a. .05

3.5.4 Meteorological Assumptions
In using the above estimates, two assumptions will be made for analytic
simplicity. The first assumption is that the three meteorological variables,

5, TA' and oz/oy are independently distributed. Thus the joint probability

is merely the product of the three individual probabilities. The second

assumption is that at any instant, all five coal burning stations experience

52

identical meteorological conditions. These two assumptions are simplifying
assumptions made about phenomena for which the data necessary to estimate

the true distributions is totally lacking.

3.6 Summary

In Chapter II, section 2.3.2, the dependence of ground level concen-
trations on bothlhe amount of SO2 in the stack gas, and the degree of
dispersion which the stack gas is subjected to, was emphasized. It was
further noted that that dispersion itself depends upon release conditions
(determined by the design of the machine, i.e., within the realm of the
firm's control) and the state of the atmosphere into which the $02 is
released (beyond the firm's control). This chapter attempts to describe
the interrelation between release conditions and the receiving atmosphere.
The description provided in the equation used to estimate ground level
concentrations identifies those variables in the natural world for which
input numbers are needed. The last sections of this chapter specify those
chosen and the assumptions under which they will be used.

This chapter has essentially dealt with the natural world half of the

phenomena. It remains for the next chapter to describe the man-made side

of the dichotomy.

CHAPTER IV

THE GENERATING SYSTEM

4.1 Introduction

The fossil-fuel-fired steam-electric segment of Consumers Power
Company's generating capacity consists of 24 units located at six princi-
ple generating stations. These units range in size from 35 MW to 385 MW
with the combined capacity of the units being 2,816 MW. The dates of
installation of the 24 units range from 1939 to 1966. This capacity made
up more than 80 percent of total company capacity as of 1971. The re-
mainder consisted of 71 MW of nuclear, 135 MW of hydro and 406 MW of gas
turbine peaking units. In addition to company owned capacity, Consumers
Power Company can, as a member of the Michigan Power Pool, draw on outside
capacity. The exchange of power via the pool will be essentially ignored
in this study, as will be spinning reserve requirements and both scheduled

and non-scheduled downtime for maintenance of units.

4.2 System Load

The demand for power faced by the generating company is largely beyond
its control. As noted in Chapter II, both the absolute amount of power
demanded and the fluctuations in demand heavily influence the cost of
power generation.

In order to estimate the distribution function of the annual system

load, actual load figures for the year 1971 will be used. However, these

53

54

figures represent total system load, whereas the concern in this study is
completely with the conventional steam generating segment of that system.
To adjust the system figures, the 71 MW of nuclear generation and 136 MW

of hydro generation will be assumed to be base load capacity and subtracted
out of the total load figures. The figures below represent the adjusted
data broken into ten segments of equal probability, with the mean of each

interval being used to represent the entire segment.

SYSTEM LOAD (KS)

 

 

IE. P(Ks)
1324 MW .1
1589 MW .1
1728 MW .1
1880 MW .1
2089 MW .1
2377 MW .1
2631 MW .1
2768 MW .1
2886 MW .1
3063 MW .1

4.3 Thermal Efficiency

As noted in the introduction to this chapter, there is a wide range
in both size and date of installation of the 24 conventional units. These
variations are paralleled by an equally wide range if underlying design
differences, and thus, as should be expected, thermal efficiencies vary
widely.

Although no two units, no matter how similar in design, can be ex-
pected to exhibit an identical range of thermal efficiencies, units can
be grouped together without much loss of accuracy in estimating production

functions. Consumers Power Company has provided data on the 24 conven-

tional units broken into five groups. The criteria of inclusion in one

55

of the five groups is based on steam temperature and pressure; this grouping
is sufficient for the needs of this study.

In addition to the five groups provided, a sixth group has been created
by combining three pairs of units at the Weadock station. This simplifi-
cation is justified since the units within any pair are identical and both
are served by the same stack. Although the two units at the Campbell sta-
tion are also served by a single stack, the design of the two units is dif-
ferent, and so the two units are treated individually.

The entire gas turbine capacity has been thrown into one group for
simplicity. Although individual heat rates range from roughly 12,000BTU/KWH
to 15,000 BTU/KWH, all are arbitrarily treated as a single, 406 MW unit with
constant returns of 13,500 BTU/KWH.

Although the BTU-MWH relationship for the standard thermal units was
thought to be non-linear a priori, a visual check of the points portrayed
in two dimensional space suggested that a linear approximation of the MW‘
BTU relationship would be sufficient. A linear regression equation of
the form Yi = a + BXi (with Yi = BTU x 106 input and Xi = MW output) was
used. Although the differences between the estimated value of BTU input
and the actual BTU value (Yi - §i) vary systematically over the range of
output, which would suggest a curvilinear relationship, the size of the
maximum difference never exceeded three percent. Table IV-l at the end
of this chapter shows the estimated production function equations (column 5).

Graphs of two of the equations relative to the data points are shown
at the end of this chapter. Figure IV-l is for the 385 MW number 2 unit
at the Campbell station; Figure IV‘2 is for the seven standard units which
range from 35-66 MW capacity. (It should be noted that the axes in the

two figures are not the same.)

56

A few observations are relevant at this point. First, along with
increases in steam pressure and temperature have come improved incremental
thermal efficiency. Thus, as temperature and pressure increase, the esti-
mated curve will become less steep. This phenomenon is entirely consistent
with a priori knowledge. Second, as pressure and temperature increase the
value assumed by the constant term, 0, increases. This at first appears
somewhat puzzling -- why should a more efficient technology result in a
larger constant term? Part of the explanation lies in the fact that for
the units considered, higher pressures and temperatures are found in the
larger units. The size of the constant term is probably directly related
to the size of the unit. One would expect the BTU input necessary to keep
the unit spinning to be positively related to the size of the unit. Third,
the inverse relation between the size of a, the constant term, and the size
of 8, the slope, implies that the production functions for any two units
of different pressure and temperature values will intersect at some posi-
tive output level. This possibility was mentioned in section 2.5.2 of

Chapter II, and is shown in Figure IV-3 at the end of this chapter.

4.4 Dispersion Characteristics of Generating Units

The observation was made in Chapter III that the dispersion of stack
gases depends upon atmospheric conditions and release conditions, with re-
lease conditions determined by the design of the unit and the rate at which
the unit is operating. It is those design characteristics of the unit
which affect dispersion that are of concern here.

The design characteristics affecting dispersion are those that contri-
bute to plume rise, or in the language of Chapter III, effective stack

height. They are denoted in the dispersion estimation equations as hS

57

(actual stack height), D (stack diameter), VS (exit velocity), and Ts
(exit temperature). The values of hS and D are shown in columns 6 and 7
of Table IV-l. The estimated relationships of v; and Ts to thermal input
to the furnace are shown in columns 8 and 9 respectively.

It should be noted in Table IV-l that some of the units at the Weadock
station have been combined and/or renumbered; unit 1 as shown in the table
is made up of two identical 35 MW units; unit 2 is two identical 50 MW units
and unit 3 consists of two identical 66 MW units. Units 4 and 5 are unal-
tered except for a change in identification number (true identification is
number 7 and 8 respectively). The reason for the three pairs of units is
that each of the pairs is served by a single stack. At the Campbell sta-
tion, units 1 and 2 are also served by a single stack. In this case the
design of the two units is quite different, and so each is treated indivi-
dually when considering the thermal to electrical energy conversion. Con-
sideration of dispersion characteristics treats the two units as one, and
assumes that unit 2 (being thermally more efficient) will always be fired
first and run up to capacity before unit 1 is fired up. Exit velocity and
temperature become a function of the sum of the BTU inputs to the two

furnaces.

4.5 Summary

This chapter deals with the man-made half of the output and dispersion
dichotomy. Whereas Chapter III dealt with the atmosphere as the nonpassive

receptacle into which the SO is dumped, this chapter deals with the condi-

2

tions under which the SO2 is released. It provides a spatial orientation

of the generating units, and through the production functions and the system

load figures, provides the background information necessary to determine

58

both release conditions and amounts. One further piece of information is
lacking before the analysis can be undertaken and that is the fuel set with
which the firm is operating. This information will be provided in the

next chapter.

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.uu«uaauouuouunu was: unaumuocwo

.HI>H candb

 

 

 

L 4.: __~l'-_'.1v_

Figure IV-l.

60

Input output relation for Campbell unit 2.

61

on o A .>>.$_
com can con ca. 64 o

:oou

\ if cog

.OON—

\

Indw niaw

:ooVu

L.ooun

 

 

 

Figure IV-2.

62

Input output relation for standard units.

63

GOP

cm

00

:5 B 332.60 .x.
ov

 

I
d

L

.ooqm

:ooQo

+ooo&

.ooogi

A

 

Kigaedea 7. x emu IeaH

r.
‘1
-MA

 

 

Figure IV-3.

64

Comparison of input output relationships for all
units.

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CHAPTER V

ANALYSIS

5.1 Introduction

The purpose of this chapter is to combine the material of Chapters II,
III and IV so that the object of the study, as stated in Chapter I, can be
accomplished.

In Chapter II a general conceptual approach to determine the costs

of power generation subject to SO restrictions was developed. The treat-

2
ment of the problem was intentionally kept as broad as possible. In this
chapter the scope will be narrowed considerably in order to yield a trac-

table analysis of the problem. New assumptions will be introduced to—

gether with some that have already been stated in previous chapters.

5.2 A Restatement of the Problem

The generating company, Consumers Power Company, is assumed to attempt
to minimize the total costs of power generation subject to the following
three constraints:

1) Capital equipment (the basic generating units) is fixed.

2) The load, KT, on the system is exogenously determined and the
firm is legally bound to meet that load.

3) The firm is faced with one of two possible regulations aimed at

controlling damage from SO2 emissions. One regulation is set in terms of

66

67

a maximum allowable SOz/BTU ratio on the fuel burned. The other regulation
is a maximum allowable ground level concentration of 802.

The problem for analysis is the difference between these two standards;
specifically, the difference in costs of generation to the generating com-
pany is to be estimated.

Before getting into the analysis, a number of assumptions, some of

which have been stated elsewhere, will be stated here.

5.3 The Assumptions

The assumptions to be stated fall into five groups, depending upon
the level at which they enter the analysis. The natural separation is
preserved in the listing which follows.
5.3.1 The Generating System Assumptions

1) The system is assumed to operate in isolation from other genera-
ting systems, i.e., there is no cooperative power pooling arrangement to
provide additional generating capacity.

2) The firm has complete knowledge of the time path of system load
over the relevant future.

3) There is no spinning reserve requirement maintained by the firm.

4) Transmission losses are zero over the entire franchise area.
5.3.2 Generating Station Assumptions

1) There is complete independence between any two generating stations
at any single station has no

with respect to SO i.e., the output of SO

2' 2

effect on the concentration of $0 at any other station location.

2

2) Within any station, the maximum ground level concentrations of $02

is the sum of the maximum ground level concentrations that would result from

the individual units.

68

5.3.3 Generating Unit Assumptions

1) For each steam unit, the relationship between BTU input and MW
output is of the form BTU = a + 8(MW).

2) All units will operate without failure over the relevant time
period, i.e., zero down time.

3) The gas turbine units are not treated individually, but are lumped
together as if there was a single gas turbine whose production function was
BTU = 8(MW).

5.3.4 Fuel and Fuel Use Assumptions

1) For the coal types considered, thermal efficiency of the unit is
not affected by the choice of coa1,whether a single type or a mixture is
used: the BTU-MW relationship is independent of fuel choice.

2) Coal mixing can be accomplished instantaneously and costlessly.

3) In this analysis, two coals and natural gas are the only three

. . . . l
fuels conSidered. The relevant characteristics are listed below. 5

 

 

 

Fuel type Cost BTU (as received) Sulphur content
Low sulphur coal $8.00/ton 12,500/1b 1.1%
High sulphur coal $5.30/ton 11,800/1b 3.3%
Natural gas $ .50/l,000ft3 1,000/ft3 0%

5.3.5 Meteorological Assumptions

1) In Chapter III, three meteorological parameters were found to
have significant influence on dispersion. Each of the three, wind speed,
ambient temperature and atmospheric stability,are seen as random variables,

15The two coals listed are respectively the lowest sulphur coal and

the lowest priced coal from a 1971 listing supplied by Consumers Power
Company. The gas data is only approximate. See Appendix I for a com-
plete list of fuels and the reason for choosing these two coals.

 

69

and for the purpose of this analysis these three random variables are
assumed to be independent of each other.
2) At any point in time, the meteorological conditions are assumed

to be identical at every one of the six generating stations.

5.4 The Analysis
5.4.1 Introduction

In searching for an algorithm to yield a numerical solution to the
optimization problem set up in Chapter II, an approach less precise than
the iterative procedure suggested there has been selected. The lack of
precision can be justified by the relative simplicity of the following
approach.
5.4.2 The Generating Station Output Expansion Path

To begin with, suppose one looks at an output expansion path for a
hypothetical generating station made up of four coal burning units in
Figure V—l. The least cost output expansion path would use only high
sulphur (HS) coal and utilize units in descending order of thermal effi-
ciency. It is due to this ordering that the slope of each successive
generation cost segment is steeper than the previous one. If the no load
input (a in the production function) is ignored, HSHs is the least cost
expansion path for that station, and will be the path followed so long as
the SO constraint is not operative.

2
A second path, LSLS, has been drawn which represents the minimum SO2
output expansion path. Because the order of thermal efficiency and disper-
sion efficiency are assumed here to be the same, the minimum SO2 expansion

path is merely a projection of the minimum cost expansion path. Each point

on the LsLs path is merely (price LS/price HS) times the equivalent output

 

 

Figure V-l.

Least cost and least SO

70

2

expansion paths.

 

71

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72

point on the HSHS curve. The derivation of Figure V-l is similar to that
of Figure II-4 in Chapter II.

It is obvious a cost minimizing firm would expand output along the
HS path if that alternative was open. Therefore the HS line is the relevant
output expansion path up to the point where the $02 constraint is reached.
The output level at which the constraint is reached will depend on the
meteorological state (64 different meteorological states will be consi-
dered). Once the 802 constraint is reached, further increases in power
output will necessitate coal mixing to satisfy the constraint. The mixing
process and the resulting generation costs are worth looking at in detail
at this point. The simplifying assumptions used in Figures II-S and II-6
will be dropped here.

Let us call the point of maximum allowable output, using 100 percent
HS coal, Kj (j = l,...,5 to denote station number). What will happen to
costs as power output is increased beyond Kj? Equation (14) in Chapter II
implies that the unit with the lowest dispersion capability should use the
low sulphur coal. Therefore the expanded output should be produced from
low sulphur coal. This is depicted in Figure V-2 by the counter-clockwise rota-
tion of the cost line AB to a new position AC. There is, in addition, a second
cost increase due to "back mixing" of fuel. In spite of the fact that low
sulphur coal is used in moving to the right from point A, the total output
of $02 from the stack increases. This will necessitate using low sulphur
coal on "previous" units of output, i.e., once the point A is reached,
movement to the right of that point, even though utilizing low sulphur
coal will dictate corresponding mixing to the left of that point also.

Thus the change in costs will be greater than the ratio of prices of the

two coals, since each expansion of output would necessitate mixing on

.37

73

 

Figure V-2. Actual output expansion about the constraint point.

 

 

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75

previous units of output. Thus the costs of the increase in output beyond
Kj might be visualized as a parallel leftward shift of line segment AC; for
each expansion of output a new position of AC is determined and the corre-
sponding fuel cost is read off the newly positioned AC line. This is, of
course, wrong. The output expansion path for all output less than Kj is
the original HSHS line. The expansion path at output greater than Kj can
be determined by shifting AC to the left, but the actual path must be a
rotation of AC counter-clockwise about the point A. This rotation is indi-
cated by the line segment AD.

Complicating the rotation of line segment AB through AC to AD is the
effect of increased output on the dispersion of stack gas. This was assumed
away in Figures II-5 and II—6. From the dispersion equations of Chapter III
it is obvious that the relationship is nonlinear. It can be shown that
under those meteorological conditions which result in high concentration

levels, the increase in dispersion resulting from increased output is

slight. Incorporating this effect into the graph will yield the curve AB.

The exact shape of the curve is of little consequence here. The important
point is that slope is decreasing throughout. This will have important

implications for the analysis.

Up to this point we have been focusing on relatively small increases
in output. It is now necessary to move out further along the output expan-
sion path. This is depicted in Figure V-3.

Moving to the right from A, the slope of AB' should be decreasing as
pointed out above. At 8' unit 2 has reached capacity and unit 3 is brought

on line. Over both curve segments AB' and B"C', increased SO outputs will

2
be compensated for by back mixing on unit 2. At some point C', back mixing

on unit 2 is no longer possible; unit 2 is operating on 100 percent LS coal

and increased SO2 output to the right of C' can only be compensated for by

,2?

76

Figure V-3. Actual output expansion to station capacity.

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78

lowering the $02
The discontinuity at point C' depicts the effect of the difference in dis-
persion efficiencies between unit 1 and unit 2.

At point D', station generating capacity is reached. Units 2 and 3
are burning 100 percent Ls coal and unit 1 is mixing fuel. This agrees
with one of the findings in Chapter II: at most, 1 out of q units at the
station would operate on a fuel mix, the remaining q-l will use pure fuels
(either 100 percent HS or 100 percent LS).

Under sufficiently less favorable meteorological conditions the path-
way HSD' would intersect the LSLS pathway. Given only those two fuels,
output would have to be curtailed at that point, since increased output
even with all units burning 100 percent LS would exceed the constraint
level. Atmospheric dispersive capacity would have been surpassed before
station generating capacity was reached.

5.4.3 Allocation of System Load Between Stations

In the simplest case meteorological conditions are sufficiently favor-
able to allow all coal units to operate on 100 percent HS coal. With the
knowledge of fuel prices and the requisite production function information,
average cost can be calculated for each unit, assuming the unit operates
at full capacity. The procedure used here will be to bring units on line
in order of their average costs. The emphasis on comparing average costs
rather than marginal costs is necessitated by the characteristics of the
production functions. It will be recalled from Chapter IV that the esti-
mated relationship between BTU input and MW output was Y = a + Bx, and
further, an inverse relation between the size of a and the size of 8 was
noted there (section 4.3) for the units being considered. If one were to

start at zero output and build up output incrementally, a would dominate,

output of unit 1, i.e., back mixing is initiated at unit 1.

'17

79

and units would be added in order of a size, from smallest to largest.
This could lead to the least efficient load allocation. The average cost
comparison will circumvent this problem; it could, in fact, be interpreted
as a marginal cost approach with the size of the output change restricted
to capacity size increments.

In general, if units are brought on line in order, with each unit run
up to capacity before another unit is added, system load will be met with
the last unit added running at less than full capacity. However, the order
in which that unit was brought on line was based on average costs at full
capacity. A new average cost figure must be calculated at the particular
partial load, and compared to the cost of generating that output on the
remaining units. The unit with the lowest cost will be chosen. This will
not necessarily lead to the precise optimum, but given the range of units
considered it will not only be close, but the maximum possible error can
be determined.

In Figure V-4 we assume that the first n units have been fired in order
of average costs, and that the sum of their capacity outputs has left us
Ko MW short of system demand. Assume two unused units remain in the system.
The above procedure would first look at the average cost OA, then come back
along AA to point A'; calculate the slope of OA' and compare it to the re-
spective average costs, OB', for unit B. The costs are obviously lower for
unit B to produce K.o output than for unit A to produce KO.

This is not necessarily the optimum solution. Under the developed
firing order of units, the point 0 in Figure V-4 represents the combined
output of the most efficient n units. It is truly an optimum allocation
of units to produce Ehg£_level of output. Further, given that we start

with that allocation of output 0, unit B becomes the most efficient way

12.9- a .

80

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optimum load allocation.

 

 

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82

to produce an additional KO of output. However, this does pgt_imply

that the n units producing output 0 and unit B producing output‘KO is the
optimum way to produce the system demand 0 + KO. However, the furthest
our solution could be from the true optimum would be represented by the
distance B'C. This would only occur if the nth unit added had capacity
equal to (KA-KO) and an average cost at capacity equal to the average cost
of unit A running at capacity.

In the more complex case, at least one of the five coal burning sta-
tions reaches the SO2 constraint before system load is met. In this event
the procedure used in the simple case can be utilized except that instead
of dealing with linear cost curves, the cost curves will, in general, be
nonlinear beyond the point where fuel mixing begins. Again a diagram is
helpful.

Figure V-5 is essentially copied from Figure V—3 with the end points
of each curve connected by a straight line. The slope of the connecting
lines yields the average cost of output over the entire curve segment. So
long as the whole curve segment is taken, the straight line is an adequate
representation. This leads to an approach almost identical to the allo-
cation procedure to deal with the simpler case above. Instead of dealing
with the average slope of a linear function plus a constant, here the con-
cern is with the average slope of a non-linear function. Analytically
there is little difference. Each curve segment is treated as if it were
an individual unit. The only real difference is that at any of the five
COal burning stations the order of consideration of curve segments within
theIstation is fixed; C'D' cannot be considered until B'C' has been util-
12633. The way the optimum solution will be approximated is the same as in

the: case without operative SO2 constraints. The average cost figures,

83

Figure V-5. Piecewise linear approximation of the discontinuous
nonlinear constrained output expansion path.

 

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85

calculated between end points, are used. Output will expand along the path
of lowest average cost until system demand is reached. At this point, costs
for the last unit (which will generally be operating between end points)
must be recalculated, and compared to the cost of generation on other units,
at that particular output level. Again, as in Figure V-4, the maximum error
will lie between the original average cost line considered and the lowest

average cost at actual output.

CHAPTER VI

RESULTS AND CONCLUSIONS

6.1 Introduction

The object of this study, as has been repeatedly stated, is to compare
alternative forms of SO2 regulation. The two forms of regulation are:
l) a regulation defined in terms of a maximum allowable ground level con-
centration of SO2 (henceforth abbreviated as MAGLC) measured in parts per
million (ppm), and 2) a uniform systemrwide fuel regulation (USWF) speci-
fied in terms of a maximum allowable sulphur content per BTU, measured in
grams sulphur per million BTU (S/MBTU).

In light of the lack of solid information on the full relationship
between $02 concentrations and damage, it has been assumed that there is
a one-to-one correspondence between damage and the maximum ground level
concentration experienced. Additional dimensions such as frequency, dura-
tion, or the amplitude of fluctuations have been assumed to be inconse-
quential. Armed with these assumptions, one can set the two regulations
on an equal damage basis merely by setting the S/MBTU ratio in regulation
2 at a level such that resulting ground level concentrations under worst
conditions will equal the maximum allowable ppm under regulation 1.
6.2 Specific Results

Before examination of the cost figures resulting from the simulation

of the alternative regulations, some base line information is helpful in

establishing a point of reference.

86

87

6.2.1 Generation Without $02 Regulation
As a starting point a calculation was made assuming that the company

was not restricted by any form of SO regulation. The company was, however,

2
restricted to a choice between the two coals selected in Chapter V for
its five coal burning stations. Without SO2 regulation the cheaper of
the two coals ($.225 per MBTU) is the obvious choice. The choice of
generating units is likewise simplified to a comparison of thermal effi-
ciencies only.

The resulting expected annual cost of fuel is (to the nearest $1000)
$44,271,000. This figure includes $4,711,000 worth of gas used to fire
the gas turbine units and the gas burning conventional steam units at
the Morrow station. The difference of $39,560,000 represents expenditure
for coal. This figure represents an input of 175,822,222 MBTU and a re-
sulting maximum system-wide ground level concentration of 3.014 ppm 502
which would occur at the Weadock station. Probability of occurrence would
be .00125, or slightly less than lJ.hours per year.

6.2.2 Generation Subject to MAGLC Regulation

For analysis of MAGLC regulation costs, the weather conditions of
Chapter III and the load conditions of Chapter IV were used. The simu-
lation was run with the firm restricted to some combination of the two
coals selected in Chapter V. The constraint level was set at 1 ppm maxi-
mum hourly average.

Total fuel costs were calculated to be $45,538,000, of which $4,711,000
was again the expected expenditure on natural gas. The resulting $40,827,000
coal cost is 3.2 percent above the unrestricted case. All five coal burning

stations were able to meet the constraint, with only Weadock, and to a

lesser extent Cobb, coming close to not being able to meet the constraint.

88

Under worst weather conditions at Weadock (U = 14.3 m., TA = 2940K,
02/0y = 1.000) a coal mix of nearly 98 percent low sulphur coal had to
be used. At the other side of the spectrum, the Campbell station, using
931x high sulphur coal,did not come close to the constraint (.664 ppm
under worst conditions). The ability of Campbell, and to a lesser extent,
Karn, to operate on high sulphur coal combined with mixing being necessary 3
at the remaining stations only 25 percent of the time is the reason that
MAGLC costs are so close to the costs of unrestricted generation.
6.2.3 Generation Subject to USWF Regulation

Under this regulation a coal mix was determined which would, under
worst conditions, result in concentration no greater than 1 ppm. Since
the fuel regulation is uniform over all coal-burning stations, it is
determined by the station whose characteristics yield the highest concen-
trations. Weadock is that station, under meterorological conditions
listed in section 6.2.2 above. The necessary coal mix is one which has
a sulphur content of 421 grams/MBTU. This mix is slightly lower in sul-
phur (i.e. more expensive) than would occur under identical meteorological
and load conditions, but with MAGLC regulation. MAGLC would allow dif-
ferent mixes in different units (Equation [14] of Chapter II) where USWF
is uniform not only across stations but also across all units. Cost per
MBTU under USWF regulation is 5.3176, and this cost is the same at all
five coal-burning stations. As should be anticipated, total expected coal
costs are high: $55,843,000.
6.2.4 A Third Alternative Regulation

Although the above two regulations were the only ones originally
considered for study, the wide divergence in cost between the two prompted

consideration of a third alternative, which it was hoped, would avoid

89

the enforcement costs of MAGLC regulation while retaining some of its
potential economy. This third alternative is, like USWF regulation,
based on sulphur content of the fuel (easy enforcement), but the content
is allowed to differ at each station. Each station must burn, at all
times, a fuel of sufficiently low sulphur content so that under worst
conditions, ground level concentrations will not exceed 1 ppm. This
is a station-by-station fuel regulation (SSF). Being based on worst
condition criteria, it will not allow the firm to take advantage of
either favorable meteorological conditions or low load situations.
However it does, at least, allow the company to take advantage of the
superior dispersive capability of some stations -- Campbell does not
have to pay according to the relatively poor dispersive capability of
Weadock.

Under SSF, coal costs for the five coal-burning stations are a sur-

prisingly low $45,918,000.

6.3 General Results and Comparisons
6.3.1 Comparison of Results

Table VI-l below shows a summary of the values assumed by selected
variables under the three cases of regulation and the initial no regula-
tion case. The emphasis so far has been completely on the total coal
costs. Table VI-l shows more completely the tradeoffs between fuel costs,
ground level concentrations, and use rates of high and low sulphur coal
as effected by alternative regulation. The total MWH figures for each
station are of interest because they show the effect of each regulation
on the allocation of output between stations. Campbell and Karn are

unaffected because their units are both thermally and dispersively most

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efficient. Whiting versus Weadock and Cobb, on the other hand, show
shifts. This is because Whiting is more efficient dispersively, but
less efficient thermally.
6.3.2 Causes of Cost Differences Between Alternative Regulations

The three regulations looked at in this study give three distinctly
different cost figures in spite of the fact that each regulation is based

on ground level concentrations of SO below 1 ppm. A brief outline of

2
the reasons for the differences is presented below.

The differences in costs are a result of differences in flexibility
allowed the firm in adjusting to certain background variability. This
background variability is made up of meteorological variability, varia-
bility of unit design and their combination into stations, variability
in the demand on the system and variability of price and sulphur content
of the fuel itself.

1) Fuel Variation

In all four cases simulated (including the no regulation case) the
necessary choice was between some combination of the same two fuels. The
mixture possibility line in $/MBTU, sulphur/MBTU space has the equation
Y = -.0001092X + .36357 for 399 :_X :_1269. If there existed a greater
range of variation between the two fuel prices, total fuel costs of gen-
eration would likewise show a greater range. Suppose, for example, the
fuel mix line had the same equation, but that X was bounded on the upper
end by 2000 (i.e., Y = -.0001092X + .36357 for 399 :_X :_2000). This would
result in a new high sulphur coal cost of l4.5¢/MBTU or approximately 65%
of the old price. Cost of no regulation generation would be reduced by

this 65% figure. Costs under MAGLC restrictions, although not by the

 

92

full 65%, would be greatly reduced. SSF would show reduced costs because
Campbell could handle the higher sulphur content coal but USWF regulation
would remain unchanged.

2) Meteorological Variation

 

Meteorological variation, to a large extent, determines the variability
in the allowable amount of 802 that can be emitted. In addition to the I 1
background variation in meteorological conditions, there are two additional I
sources of variation that have been ruled out by assumption. The first
is the possible difference between the probability distribution of meteoro—
logical variables at different locations. The second is even with iden-
tical distribution functions, the timing of conditions would be different
at different locations, i.e. at any instant, identical conditions cannot
be expected to exist across all stations.

The background variability is introduced by the probability distri-
bution assumed for the meteorological variables. Under MAGLC regulation,
the firm is allowed to take advantage of the variability; under both SSF
and USWF regulation there is no way for the firm to exploit favorable
meteorology.

3) Generating Unit and Station Design Variation

This variability is, in one way or another, "designed" into the gen-
erating system. Between units there are differences in thermal efficien-
cies and dispersion characteristics that are fixed by the underlying design
of the unit. The way in which these units are grouped together to form
stations displays a degree of variety. The three forms of regulation
simulated respond to the variation in different degrees.

Under USWF regulation, the firm is saddled with having to meet not

only worst weather conditions, but also worst station design conditions.

 

93

It is unable to capitalize on the dispersive capability of the Campbell
-station. With SSF regulation, station dispersion differences are cap-
tured, but variation between the dispersive capability of units is not.
This is because all units at a particular station would burn the exact
same mix of coals. If the regulation is a MAGLC constraint, both differ-
ences between stations and between units can be captured. Although this
difference between units is not an obvious one it does exist. At the
Cobb station, for example, worst conditions are E = 10.73 m/sec, Ta =
2940K, oz/oy = 1.000 and station output at full capacity. This calls

for a single station wide fuel of 495.6 gm sulphur per MBTU and a total
hourly station coal cost of $1646. Under MAGLC, units 1—4 would be run
on 100 percent low sulphur coal (399 gm per MBTU) and unit 5 would operate
with mix yielding 911 gm. per MBTU. Total hourly cost would be $1622

or $24 cheaper.

4) Demand Variation

 

It has been mentioned repeatedly that the demand for electricity
varies widely. To meet the changing demand the power output of the
individual stations must shift accordingly. For any given fuel mix

this implies that the total SO output will fluctuate. At low levels

2
of output the potential for utilizing a higher sulphur coal mix may exist.
Under MAGLC, the firm is allowed to take advantage of this variation.
Since both SSF and USWF are worst condition regulations (in this case,
highest output) this dimension of variability is lost.
6.3.3 Summary of Comparative Results

What emerges from the above discussion is that MAGLC regulation leaves

the firm flexible enough to take advantage of all four sources of varia-

tion: variation in fuels, in weather, in system design and in output

94

demand levels. At the other extreme USWF regulation allows for none of
the man-made or natural variations. The resulting cost difference is
not surprising. The compromise regulation, SSF, although leaving the
firm unable to react to meteorological or demand variation, does allow
limited response to design variation. The surprising result is that
under SSF regulation, costs are so 12!, This can be explained by the
fact that nearly 55 percent of the total generation comes from Campbell

(unaffected by SSF) and Karn (fuel price raised approximately 1¢ per MBTU).

6.4 Conclusions
6.4.1 Introduction

At the very outset of this study a distinction was drawn between
what the two types of regulation represented. One is a behavioral regu-
lation ("thou shalt not burn fuel containing more than x grams of sulphur
per MBTU"); the other is a regulation on effects ("burn what you please,
but do not exceed an hourly average concentration of 1 ppm").

Any environmental regulation can only really be rationalized for
its control of effects; damage control is the end, behavioral regulation
may or may not be a legitimate means to that end. It can only really be
judged on its effectiveness in accomplishing the desired ends. Before
the overall effectiveness of the three alternatives is evaluated, a sep-
arate, but related, means-end question will be addressed. The question
is on the legitimacy of spatial dispersion of production facilities as
a general damage control strategy. Although not a part of this study,

the question is prompted by results of this work.

95

6.4.2 Implications for Spatial Dispersion

"All generalizations are false...including the
generalization that all generalizations are false!"

One of the general statements that has been made in the past per-

taining to production facilities and environmental effects is the

following: As the size of production units that release harmful efflu-
ents increases, environmental damage will increase. And further, not
only will the damage increase monotonically with size, but damage will, i
at least beyond some point, begin to increase faster than size. From
this implicitly assumed generality it is often argued that smaller pro-
duction units scattered more uniformly about the countryside would result
if the full costs (i.e., private costs plus external costs) of production
facilities were used. From this position it is but a short step to the
position that concentration in itself is bad, and dispersion becomes an
end in and of itself. Such a confusion of means and ends, although
logically sloppy does little harm unless the underlying implicit assump-
tion is wrong. Then the dispersion of production facilities becomes
counter productive, and because it is taken as an end in itself, it does
not receive the close scrutiny it would have had it been properly recog-
nized as merely a means to an end.

The results of this study show no clear relationship between size
and damage (as approximated by ground level concentrations of 502). Size
alone does not give sufficient information. This in itself does not

devastate the assumed generality. However, the single most conspicuous

 

l6Quote is from a lecture in a political science course taught by

Professor Frederick Schuman in the early 1960's; original source unknown.

96

piece of information coming out of the results is that the Campbell sta—
tion with the largest MW capacity also has by far the lowest ground level
concentrations of 802. This result does, in fact, emasculate the state-

ment as a generality.

 

What are the reasons behind this curious result (i.e., largest sta-
tion, lowest concentration)? The answer consists of essentially three
parts: 1) The two units at the Campbell station are thermally very effi-
cient; 2) stack height is the largest in the system (123 m); 3) both
units are served by a single stack. All three warrant a closer look,
especially as related to size and concentration.

1) Thermal Efficiency

 

The reason that thermal efficiency will effect SO2 concentration
is that there is a clear inverse relationship between thermal effi-

ciency and SO output, other things being equal. The relationship

2
between thermal efficiency and size is less clear -- the larger
units in the system are more efficient; they are also newer and
thus utilize an improved technology. Unless the technology is
size specific, there may be no relationship between size and effi-

ciency.

2) Stack Height

 

The height of a stack is important in the dispersion of stack
gas -- it has particular influence under restrictive weather con-
ditions. There is no necessary technological connection between
stack height and generating unit capacity. However, there is an
economic connection: increased stack height is expensive, and

cannot be justified for small units.

97

3) Single Stack Utilization
The use of a single stack by both generating units greatly
increases the dispersive capability of both. This results from an
increase in plume rise due to a greater buoyancy flux as a conse-
quence of the larger volume of exit gas at higher temperature. It
is this phenomenon that does the greatest violence to the generality:
combining station 802 output into a single stack is the extreme
boundary of spatial concentration, yet it results in greater dis-
persion and therefore less effect.
6.4.3 The Evaluation of the Alternative Regulations
A comparison of the simulated regulatory schemes are documented
in Table VI-l and their differences are analyzed in sections 6.3 through
6.3.3. Such comparisons are necessarily incomplete because they do not
include the full costs faced by the firm, nor do they evaluate the effec-
tiveness of the regulation itself.
1) Cost Discrepancies
There are two types of bias in the cost figures. The first
type effects the total costs of each regulatory scheme but does
not effect the comparative figures. This uniform bias is a result
of understatement of all coal costs due to the omission of trans-
portation costs and an understatement of generation costs as a
result of simplifying assumptions with respect to reserve require-
ments and maintenance of units. It will not affect conclusions
based on differences in costs.
The second bias introduced is that injected by the assumption

of costless fuel mixing. This cost would be borne only in the MAGLC

98

case, and thus makes ground level regulation appear cheaper than it
is relative to the other forms of regulation. This point will be
returned to again in the discussion of regulation itself.

2) Regulation: Effects, Effectiveness and Enforcement

 

From the introduction of this study the question of behavior
versus effects has been a recurring theme. This study has shown that
the potential savings to the firm from effects regulation (MAGLC) is
great when compared to USWF regulation. However, this is not a fair
comparison. The cost of detection and enforcement of a 1 ppm ground
level concentration may be staggering, and in fact, when added to
fuel mixing costs, may reverse the ordering of the two alternatives.

The third alternative, SSF regulation, is an attempt to bridge
the gap between the two extremes: it captures some of the variation
handled so well by ground level regulation, while at the same time
taking advantage of the ease of regulation embraced in the station
wide regulation.

The real question is the cost of regulation. If regulation
costs were the same for each of the three alternatives, there would
be no question of which form to take: regulation should be based
on the criterion of ground level concentration. This would not
imply that the firm mix fuels; it might act as if regulated by SSF
and use a single fuel at each station. But this decision would be
for the firm to make, and would depend upon the relative cost of
the two responses. This is not a statement of the firm's rights,

but merely a statement of efficiency.

 

APPENDIX

APPENDIX I

The list of fuels supplied by Consumers Power Company included six

coals. For our purposes the relevant characteristics are as follows:

 

 

Dollar Cost Sulphur
Item Source Type Per Million BTU's Per Million BTU's
1 District 8 Contract $.265 727
2 District 8 Spot .320 399
3 Ohio Contract .237 1194
4 Ohio Spot .248 1226
5 Midwestern Contract .225 1269
6 Midwestern Spot .288 1204

For this particular case study, we wish to select those two coals
which will yield the minimum fuel cost while meeting the SO2 constraint.
The results of the actual analysis (as was discussed in Chapter VI)
show that under worst weather conditions the Weadock generating station
would have to be able to attain a fuel mixture of less than 421 gm. sul-
phur per 106 BTU's. This implies that of the two coals chosen for the
system, one must be coal number 2 above. The question then is which of
the remaining five coals should be chosen as the second coal.

Figure A-l is helpful. The six points represent the six coals located
in dollar-sulphur space. It is obvious that coals 3, 4 and 6 can be ex-
cluded at once: a combination of 5 and 2 can be utilized which would be

both cheaper and of lower sulphur content. The only non-trivial choice

is between 1 and 5. Should the two fuels be 2 and l or 2 and 5?

99

SJ}?

100

Figure A-l. Fuel price-sulphur characteristics.

101

mb.—b :o_=_E too ...:_n__:m mEEo

 

 

 

com. com . coo _ can
flow.
0
nl.
m7 / . m...
m./ s
u d
m
.3 haw:
m / / : mu.
4 / / a
/ l
n.
o. s

 

.om.

 

102

If we use the usual convention of denoting line segments by the numbers
or letters located at their end points, then it can be seen that by mixing
coal 1 and coal 2 any combination of cost and sulphur content can be
reached along the line 1,2. Similarly line 2,5 represents the possible
combinations of coal 2 mixed with coal 5. If we now picture the sulphur
axis as the allowable amount of sulphur, it is obvious that in the allow-
able range of approximately 400 to 725 grams sulphur per 106 BTU, the com-
bination of coal 2 and coal 1 is the cheapest. However if the allowable
output exceeds 725 grams, there can be no further decrease in cost per
106 BTU with coals 2 and l -- cost remains at $.265. At point A the two
combinations are the same. Beyond an allowable output of roughly 900
grams sulphur, combination 2,5 is cheaper. The choice between combinations
thus depends upon the frequency distribution of the allowable output.

That in turn depends on the frequency distribution of the relevant meteoro-
logical conditions. As the computations of the actual analysiswill show, the
allowable output exceeds 1269 grams substantially more than half the time.
Therefore the fuel combination chosen for the five coal burning stations

is 2 and 5.

103

BIBLIOGRAPHY
The American Society of Mechanical Engineers, "Recommended Guide for the
Prediction of Airborne Effluents," New York, 1968. T”

Engdahl, Richard B., "A Critical Review of Regulations for the Control of
Sulfur Oxide Emissions," Journal of the Air Pollution Control Asso-
ciation, May 1973, pp. 365-375.

 

"Environmental Protection Agency Sets National Air Quality Standards,"
Journal of the Air Pollution Control Association, June 1970, pp. 352-
353.

Schorr, Burt, "EPA Plans Revised Sulphur Dioxide Curbs That Could Help
Some Utilities, Smelters," The Wall Street Journal, Thursday, June 28,
1973, p. 8.

 

Thomas, F. W., Carpenter, S. B. and Gartrell, F. E., "Stacks - How High?"
Journal of Air Pollution Control, May 1963, pp. 198-204.

 

104

GENERAL REFERENCES

Skrotzki, Bernhardt G. A., and Vopat, William A., Power Station Engineering

 

and Economy, New York: McGraw Hill, 1960.

 

Stern, Arthur C. (editor), Air Pollution, Volumes 1-3, New York: Academic
Press, 1968.

 

n.

’n

 

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