
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00026509/00001
Material Information
 Title:
 An economical alternative for control of ground level concentrations of sulfur dioxide from electric power plants using the method of production costing
 Creator:
 Hilson, David Wayne, 1944
 Publication Date:
 1976
 Language:
 English
Subjects
 Subjects / Keywords:
 Air quality ( jstor )
Coal ( jstor ) Emission control systems ( jstor ) Environmental policy ( jstor ) Plumes ( jstor ) Pollutant emissions ( jstor ) Sulfur ( jstor ) Sulfur dioxide ( jstor ) Unit costs ( jstor ) Wind velocity ( jstor )
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright David Wayne Hilson. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 3154502 ( OCLC )
0025651541 ( ALEPH )

Downloads 
This item has the following downloads:

Full Text 
AN ECONOMICAL ALTERNATIVE FOR CONTROL OF GROUND LEVEL CONCENTRATIONS OF SULFUR DIOXIDE FROM ELECTRIC POWER PLANTS USING THE METHOD OF PRODUCTION COSTING
By
DAVID WAYNE HILON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976
To my wife, Carol, and sons, Jamie and Michael, for their patience and understanding and above all their love and cooperation.
ACKNOWLEDGMENTS
I would like to express my genuine appreciation and gratitude to the members of my supervisory committee: Dr. Robert L. Sullivan, Chairman, Dr. Olle I. Elgerd, Cochairman, and Dr. Paul Urone. I would especially like to thank Dr. Robert L. Sullivan for his patience, encouragement and scholarly conduct. In particular, I would like to acknowledge that his suggestion to use the new production costing method in this work proved to be invaluable. I remember well my initial skepticism and later enthusiasm.
I would also like to thank Dr. Lynn D. Russell for his support and patience, which was far beyond what could have been expected.
I want to express my appreciation to the members of the Power
Research Staff of the Tennessee Valley Authority for their cooperation in providing relevant information on their Sulfur Dioxide Emission Limitation program.
Finally, a special thanks to Mrs. Beth Beville for her editorial assistance and for the typing of this dissertation.
iii
TABLE OF CONTENTS
ACKNOWLEDGMENTS. ........ ............... .iii
ABSTRACT.. ..... . . . . . . vi
CHAPTER 1: INTRODUCTION . . . . . 1
1.1 Historical Background of Clean
Air Legislation ... .................. 2
1.2 Statement of Purpose and
Chapter Outline . . . . 6
CHAPTER 2: ALTERNATIVES FOR CONTROL OF SULFUR DIOXIDE
EMISSIONS. . . . . ... .. 8
2.1 Continuous Emission Control ........ .. 8
2.1.1 Low Sulfur Fuel. ........... 9
2.1.2 Fuel Oil Desulfurization ... ... 11
2.1.3 Coal Desulfurization ........... 11
2.1.4 Coal Gasification and Liquefaction . 13 2.1.5 Flue Gas Desulfurization . . 14
2.1.6 Fluidizedbed Combustion ........ 15
2.2 Noncontinuous Emission Control . ... 17
2.2.1 Fuelswitching ..... ........ 18
2.2.2 Loadswitching . . . 19
2.2.3 Tall Stacks . . . 19
2.3 Arguments for a National Mix of
Alternatives ............... .. 22
CHAPTER 3: STEADY STATE MATHEMATICAL DISPERSION MODEL ..... 28
3.1 Gradient Transport Model. ............ 29
3.2 Steady State Gradient Transport Model. ... 34 3.3 Steady State Statistical Model . . 44 3.4 Long Term Dispersion Model . . 51
3.5 Plume Rise, Wind Speed at Stack Height,
and Emission Formulas . .. . 68
iv
TABLE OF CONTENTS (continued)
CHAPTER 4: THE METHOD OF PRODUCTION COSTING . ..... 79
4.1 The Load Duration Curve ............. 81
4.2 The Effective Load Duration Curve ........ 99
4.3 The Effective Load Duration Curve for
Segments of Units ......... . 109
4.4 Estimating Short Term Maximum Concentrations .. .111 CHAPTER 5: DISCUSSION OF THE DIGITAL COMPUTER PROGRAM ...... .122
5.1 The Digital Computer Program ........ .. 122
5.2 Discussion of the Application of the Computer
Program to a Realistic Size System ....... .131
5.3 Conclusion and Future Work. ......... ...137
APPENDIX A . . . . . ... .138
REFERENCES . . . . . ... .146
BIOGRAPHICAL SKETCH ................... .... .151
v
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
AN ECONOMICAL ALTERNATIVE FOR CONTROL OF GROUND LEVEL
CONCENTRATIONS OF SULFUR DIOXIDE FROM ELECTRIC POWER
PLANTS USING THE METHOD OF PRODUCTION COSTING
By
David Wayne Hilson
August, 1976
Chairman: Dr. Robert L. Sullivan Cochairman: Dr. Olle I. Elgerd
Major Department: Electrical Engineering
An alternative approach for reducing ground level concentrations
of sulfur dioxide emitted from electric power plants is presented. The approach involves the commitment of generating units using an extended version of the method of production costing. It is shown that this method, in addition to providing a strategy for meeting the Federal Ambient Air Quality Standards, can be used as a planning model for fuel purchases, plant siting, and stack height determinations. It is easily adapted to any size electric utility system.
The historical background leading to the promulgation of emission rate and air quality standards is first discussed. The advantages and disadvantages of several of the more prominent existing methods are also discussed. Arguments for a national mix of alternative strategies are presented, as well as some viewpoints by prominent persons on the issue of continuous or noncontinuous emission controls.
This alternative approach involves the use of a steady state, long
term (a month or more) mathematical dispersion model. The dispersion model is derived in a novel manner from the steady state gradient transport
vi
equation and will be shown to be the Gaussian dispersion model used extensively for predicting pollutant concentrations from point sources (stacks). The statistical approach to the Gaussian dispersion model is discussed and the fundamental relationship connecting the two approaches is shown to be only a special case of a more general connection discovered and presented here.
The theory of production costing is extended to include monotonically increasing or decreasing nonlinear functions of the electrical output of the generator. This makes it possible to use the method of production costing to predict long term estimates of the sulfur dioxide emissions as a function of the commitment order of the various fossil and nonfossil electric generating units. Larsen's technique for relating maximum concentrations to average concentrations is incorporated into the approach in order to extend its usefulness.
A digital computer technique is proposed to determine the optimal
commitment order of the generating units, mix of fuels, and sulfur contents. A widely used model for predicting average concentrations near the source is shown to be in error and is corrected here. A simple numerical technique for mathematically constructing the "effective" load duration curve is presented and shown to involve only convolving of the unitsnever deconvolving. This technique avoids many of the computational problems present in previous models. Finally, the application of the computer program to a realistic size electric utility system is presented and discussed.
V i
CHAPTER 1
INTRODUCTION
The emissions of various forms of sulfur compounds into the atmosphere have been a concern of man for some time now, although it has only been during the last few decades that it has received so much public attention. Hesketh (1972) reports
As early as 1300, a royal decree was issued in London
prohibiting the use of lowgrade coal for heating because it
created excessive smoke and soot. The only known case of capital
punishment because of an air pollution violation occurred
in the 13th century when a Londoner violated this order.
Sulfur in fuels burns to sulfur dioxide. In 1600 sulfur
dioxide was the first chemical to be specifically recognized
as an air pollutant. However, it was not until about 1940
when air pollution, as such, became important. (p. 1)
Kellog et al. (1972) state
1) Man is now contributing about one half as much as
nature to the total atmospheric burden of sulfur compounds,
but by A.D. 2000 he will be contributing about as much,
and in the Northern Hemisphere alone he will be more than
matching nature.
2) In industrialized regions he is overwhelming natural
processes, and the removal processes are slow enough
(several days, at least) so that the increased concentration
is marked for hundreds to thousands of kilometers downwind. (p. 595)
Stoker and Seager (1972) have determined that approximately one half of the total atmospheric burden of sulfur compounds results from the combustion of fossil fuels (primarily coal and oil) in the production of electric power. Thus, while recognizing that there are other pollutants, as well as other sources, we will confine ourselves in this
1
2
work to the consideration of the emissions of sulfur dioxide from electric power generating plants. (See Chapter 3 for a discussion of the emissions of sulfur dioxide from the combustion of coal or oil containing elemental sulfur or sulfur compounds.)
1.1 Historical Background of Clean Air Legislation
In 1963 Congress passed the first Clean Air Act, which was amended several times during the next few years. The 1967 Clean Air Amendment required that the Department of Health, Education and Welfare (HEW) set air quality criteria. The first such criteria were published in February, 1969 for sulfur dioxide and particulates. The criteria contained data, available at that time, on the adverse health effects of pollutants and control techniques for controlling the pollutants, but did not contain standards.
On December 2, 1970 the Federal Environmental Protection Agency (EPA) was established by an act of Congress. Among its many functions is the supervision of air pollution control, which formerly came under the auspices of the Department of HEW through the National Air Pollution Control Administration (NAPCA). Some weeks later the 91st Congress passed the Clean Air Amendments of 1970. On December 31, 1970 the President signed them into law as Public Law 91604. These amendments are often referred to as the Clean Air Act of 1970.
One of the many provisions in the act calls for the EPA Administrator to establish Federal Air Quality Standards, and on April 30, 1971 these standards were issued and published in the Federal Register (see EPA (1971a)). These standards included both primary and secondary standards as called for in the Act. Section 109 (b) (1) specified that primary ambient standards should be selected which, allowing for adequate margins
3
of safety, would protect public health. Section 109 (b) (2) specified secondary ambient standards to protect public welfare from any known or anticipated adverse effects from pollutants. Table 1 summarizes these standards. The secondary standards for the annual arithmetic mean and 24 hour average were adopted but subsequently dropped.
Air quality standards are not enforceable, in that the air cannot be told to be clean with any assurance that the air will somehow do so. Therefore, emission standards are required which are enforceable. The Clean Air Act of 1970 provides for emission limitations to be set by the individual states through State Implementation Plans (EPA (1971b)). These plans are to set emission limits on all manmade sources such that the ambient air quality standards are met within three years of the approval by the EPA of each state implementation plan, with possible extension to mid1977 through state initiative. By mid1972 the states had submitted their implementation plans.
By this time it had become obvious to those in the EPA and in the utility industry that there simply was not enough low sulfur fuel or other control technologies (see Chapter 2) available to meet and maintain the national ambient air quality standards as called for in the 1970 Act. In the Fall of 1972, the EPA initiated the "clean fuels policy". This policy was designed to encourage states to delay or relax their sulfur dioxide emissions standards in areas where primary ambient standards were not being exceeded, and to give priority for low sulfur fuel to plants which were threatening the primary ambient standards. In addition, this policy asked that the states promulgate revised implementation plans that did not contain emission limitations more stringent than required to meet ambient standards (see EPA (1974)). According to the EPA (1975)
4
Table 1. National Primary and Secondary Ambient Air Quality Standards.
Primary Standards in Secondary Standards in
Micrograms per Cubic Meter Micrograms per Cubic Meter
Annual 80 24 hour 365
3 hour  1300
"This program has resulted in 42 million tons of coal being made legally acceptable through revisions in the sulfur regulations of State Implementation Plans." (p. 16)
In the Fall of 1973, the muchpublicized Arab oil embargo further complicated the fuel supply and pressure began to increase from the electric utility industry to amend the Clean Air Amendments of 1970 once again. On March 13, 1974, Rep. Nelsen (RMinn) introduced a bill (HR 13464) to amend the Clean Air Act. This bill proposed 13 amendments be adopted by Congress. The EPA opposed two of the amendments but all 13 were transmitted on behalf of the Administration by the EPA Administrator with a letter explaining the differing points of view on the two amendments.+ (We will mention these differing points of view in a discussion of alternatives in Chapter 2.)
One significant change that would occur if these amendments were
adopted, would be the extension of the compliance date for meeting ambient air quality standards to 1985. Anticipating that Congress would hold hearings during 1975 or 1976 on these proposed amendments, and that the electric utility industry would be asked to respond at the hearings, the electric utility industry formed the Clean Air Coordinating Committee (CACC) in early 1975. This Committee, which consists of representatives from the principal utility industry trade associations and both public and investorowned electric utilities, has been responsible for coordinating the major efforts of the utility industry to respond to Congress or the regulatory agencies for information regarding the views of the utility
tLetter dated March 22, 1974 from Russell E. Train, EPA Administrator, to President Richard M. Nixon, in forwarding the proposed bill, "The Clean Air Act Amendments of 1974."
6
industry on the proposed amendments (see CACC (1975)). It should be noted that as of this writing (June, 1976) the Congressional hearings have not been held, but are tentatively scheduled for Spring 1977 (after the Presidential election).
We complete this section with a quote from the CACC (1975) that summarizes their feelings (and they represent the electric utility industry) on the state of existing air pollution regulations:
The field of clean air regulation is in a state of flux.
Congress is currently reexamining the Clean Air Act and the Environmental Protection Agency's implementation of
that Act to date. EPA is also reassessing its own policies
under the Act. The Federal Energy Administration and
EPA are about to implement the Energy Supply and Environmental
Coordination Act (ESECA). The Federal Power Commission
is charged with certain responsibilities with regard to the
adequacy and costs of national electricity supply. The Treasury Department, the Department of Commerce and the
Department of the Interior have broad responsibilities
with regard to control of mounting inflation, alleviation of growing unemployment and wiser development and use of national resources. The Energy Resources Council has ongoing responsibility within the Executive Branch for
coordinating national energy policy with other federal
policies, such as protection of the human environment, the
economic wellbeing of our citizens, national defense,
the balance of trade and foreign policy. (p. 1)
1.2 Statement of Purpose and Chapter Outline
It is the purpose of this dissertation to present an economical
alternative for control of ground level concentrations of sulfur dioxide from electric power plants using the method of production costing. In view of the ambiguous status of clear air regulations discussed above, it is not surprising that there are numerous alternative approaches to the control of emissions of sulfur dioxide and/or ground level concentrations of sulfur dioxide from electric power plants. We intend to present a novel approach that is an outgrowth and extension of the work of Sullivan and Hilson (1975).
7
Since there are so many approaches to the control of sulfur dioxide from electric power plants, we will devote Chapter 2 to a discussion of the relevant features of the more prominent alternative approaches in existence today. In Chapter 3 we will present a novel solution to the steady state gradient transport equation that will lead to the classical Gaussian dispersion model. In Chapter 4 we will present the method of production costing and extend its theory to include certain generalized functions of the electrical output of an electric generator. Also, in Chapter 4, we will present a statistical method due to Larsen (1969) that extends the usefulness of the approach taken in this work. Finally, in Chapter 5, we will apply the model developed in Chapters 3 and 4 to a realistic size electric utility system. This will involve a computer program which is also discussed in Chapter 5. A statement on notation ends this chapter.
The notation used in this dissertation is discussed as it is first encountered. This is appropriate since it is not necessary to introduce any nomenclature that is not standard mathematical notation. Two items worthy of mention are 1) vectors are always underlined; and 2) a bar over a letter indicates average value, e.g., u is read u bar and is the mathematical symbol for average or expected value of u.
CHAPTER 2
ALTERNATIVES FOR CONTROL OF SULFUR DIOXIDE EMISSIONS
In this chapter we present a brief discussion of some of the
relevant features of the more prominent approaches to the control of sulfur dioxide emissions from electric power plants. These control approaches can be conveniently divided into two broad categories: continuous emission controls and noncontinuous emission controls as is often done in the literature (see PEDCo (1975) and EPA (1975)).
Continuous emission controls can further be divided into precombustion processes and postcombustion processes. Likewise, noncontinuous emission controls can be divided into fuelswitching or loadswitching or a combination of both. In Section 2.1 we will discuss briefly the continuous emission control techniques. Then, in Section 2.2 we will discuss the noncontinuous emission control approaches. Finally, in Section 2.3 we will present arguments for an interim (up to 1985) mix of both continuous and noncontinuous emission controls of sulfur dioxide from electric power plants, and present the method proposed in this work.
2.1 Continuous Emission Control
Continuous emission control techniques limit atmospheric emissions of pollutants (sulfur dioxide) to some fixed or permanent levels, usually expressed in pounds of pollutant emitted per million BTU of heat input. These levels are to be consistent with achieving State Implementation Plan emission limitations and with maintaining ambient air quality standards
9
under the worst case meteorological conditions. The continuous emission control approaches can be separated into precombustion and postcombustion methods. We will consider the precombustion processes first. We will discuss both the use of naturally occurring low sulfur fuels and the processing of high sulfur fuels before combustion.
2.1.1 Low Sulfur Fuel
Naturally occurring low sulfur fuels used for electric power generation include low sulfur coal, low sulfur oil, and natural gas. Natural gas is in short supply in the U.S. and has been given the lowest supply priority for use as a fuel by electric power plants by the Federal Power Commission (PEDCo (1975)). The 1973 Arab oil embargo followed by dramatic increases in crude oil prices, has produced concern about dependence on foreign oil. This development has meant that except for some power plants already burning oil along the east and west coasts, low sulfur oil is of limited usefulness as a clean fuel. Therefore, we only consider low sulfur coal as a viable source of low sulfur fuel.
There are large reserves of low sulfur coal in the Northern Great
Plains, as well as in the East. The reserves in the Northern Great Plains are located far from the major eastern markets and pose transportation problems and added costs. The low sulfur coal reserves in the East are subject to strip mining legislation since much of it is located on steep slopes (EPA (1975)).
The use of low sulfur coal has some environmental problems. For
example, most electrostatic precipitators in use today to control particulate emissions depend on the presence of sulfur to decrease the resistivity of the particles. The removal of this sulfur by burning low sulfur coal, therefore, decreases the effectiveness of the precipitators,
10
sometimes by a factor of ten (Oglesby and Nichols (1970)). The use of coal as a combustible fuel also creates fly ash disposal problems and dust emissions from coalhandling activities.
There are derating problems associated with the conversion to low sulfur coal. Oilfired plants could only be converted if they were designed originally to burn coal and subsequently converted to oil, or if they were designed for dual oil/coal firing (PEDCo (1975)). In addition, even some units burning high sulfur coal would experience deratings in the range of 10 to 30 percent because of limited pulverizing capability (Perry (1974)). For new plants, of course, these deratings do not occur since the boiler would be designed for low sulfur coal.
The sulfur content limitation for new plants is defined by the New
Source Performance Standardst (see EPA (1971c)). For all other facilities the EPA (1975) states
The emission limitation is determined through consideration
of the following factors: (1) dispersion characteristics
of the source and the surrounding area, (2) background
concentration of the pollutant [sulfur dioxide], (3) total
emissions rate of all sources in the area, and (4) expected rate of growth in emissions. The applicable sulfur content
limitation is based on allowable emission rates. If the determination of the allowable emission rate is correct
and the supply of coal with the necessary sulfur content is
assumed, this alternative is better than 99 percent
effective in assuring that air quality goals are achieved. (p. 21)
In addition to the increased cost due to capital conversion of
existing plants, and the increased effective cost due to plant deratings, there is an increased fuel cost for the low sulfur coal. The fuel cost increase is generally more acceptable to electric utilities, since these costs can be passed on to the consumer through the now familiar "fuel
This limitation is 1.2 pounds of sulfur dioxide per million BTU heat input.
adjustment clause." Whereas capital costs can only be recovered through rate base changes which necessitate rate hearings.
In summary, it is seen that low sulfur coal is a viable option for meeting emission limitations and ambient air quality standards. Its principal constraints are supply and technical problems associated with using them in certain types of boilers (PEDCo (1975)).
2.1.2 Fuel Oil Desulfurization
Fuel oil desulfurization is only one of several approaches we will
discuss that involve preprocessing of the fuel before combustion. Hence, it is a precombustion process. The removal of sulfur from crude oil has been demonstrated at many refineries, e.g., Phillips, Humble, Exxon, Chevron, Gulf, Standard, and others, all have installations that are in operation today.
The desulfurization process is capital intensive, requiring on the order of 50 to 70 million dollars for installation. In addition, there are significant operating costs, and the process requires energy that amounts from 5 to 10 percent of the energy contained in the oil itself (Nelson (1973)). The principal drawback to the use of low sulfur oil as an alternative to meeting environmental constraints is its supply. The shortage of capital and unpredictable costs of crude oil have severely limited plans for new installations (EPA (1974)). For this reason it is probably not a viable option, except as mentioned before, at those plants on the east and west coast that are already using imported crude oil.
2.1.3 Coal Desulfurization
This process involves physical coal cleaning (sometimes called coal washing) and chemical coal cleaning (sometimes called solvent refining).
12
This latter cleaning process is relatively new, and in fact, the availability is forecast to be in the late 1980's (EPA (1975)). For this reason we will only discuss the physical coal cleaning process.
This process generally involves three steps. The first step is to
separate the pulverized coal by size. The second step involves "cleaning" in a wet medium where the difference in specific gravity between clean coal and impure coal is used to separate them. Finally, the fine cleaning involves the use of a froth flotation in which the separation is achieved by floating the fine coal away from the coal containing sulfur and other minerals (Bodle and Vyas (1974)).
The limiting factor in this coal washing process is that only the pyritic sulfur (mineral compounds containing sulfur) is removed. The organic sulfur (bound in the coal molecule) is not affected. Even so, the EPA (1975) estimates that about 14 percent of the U.S. coal reserves can be cleaned to conform to the New Source Performance Standards. In addition, the EPA (1975) estimates another 55 percent could be cleaned to emit less than 4.0 pounds of sulfur dioxide per million BTU heat input. (The Tennessee Valley Authority (TVA) has a plant that has been assigned an emission limit of 5.2 pounds while the American Electric Power System (AEP) has a plant with an emission limit of 6.0 poundstt per million BTU heat input.)
Coal used in the preparation of coke is presently cleaned and so the technology is available. Also energy requirements and costs are modest (PEDCo (1975)). A significant drawback is that only about 14 percent
tPersonal communication with E. David Daugherty, TVA, Chattanooga, TN. 1Personal communication with John C. Hoebel. AEP, New York, N.Y.
13
of the U.S. coal reserves could be cleaned to meet the New Source
Performance Standards (1.2 pounds), and so coal cleaning is at best
an interim approach or the first stage of some other approach. Also,
10 to 30 percent of the coal cleaned by weight is refuse and must be
disposed of in landfills which involves environmental problems. It is,
nevertheless, a viable option for the near future.
2.1.4 Coal Gasification and Liquefaction
Many processes for converting coal to a (low sulfur) liquid or gas
are in existence or under development today. Yet, this approach is best
considered as a long term approach and not of much value to the electric
utility industry in the immediate future. This will be established by
citing two recent reports. The report by PEDCo (1975) states
Highsulfur coals are also converted to clean fuel by
gasification and liquefaction. The systems in use today,
however, are generally inefficient and expensive; many
have considerable potential for adverse environmental
impact. Problems of low efficiency probably can be
solved, principally in operation of combinedcycle systems now under intensive development. Very high
costs of installation and operation, however, coupled with shortages of the required manpower and materials, could preclude competitive commercialization of these
processes between now and 1990. (pp. xiixiii)
Further, the EPA (1975) reports
While coal gasification [3nd liquefaction] processes are
currently in commercial use, the widescale commercial
application of coal conversion is dependent upon
improvements in technology, as well as easing of constraints on the construction and expansion of a synthetic
fuels industry. Technological advances that increase
gasifier efficiencies, ameliorate environmental impacts of the process, and decrease costs must occur. On the
construction side, it is estimated that a fullscale
plant would require 3 to 5 years and 1.5 million
manhours to construct. This is a substantial commitment
of manpower. Additional manpower with special training
is required to operate the plant. Therefore it is
concluded that coal conversion processes will have little
impact [on electric power systems] between now and 1985. (p. 23)
14
As has been demonstrated by the above references, the conversion of coal to a liquid or gas for widespread use by the electric utility industry is not expected before about 1985 to 1990. Thus it is not a viable approach for the near future. (As is explained in Section 2.3, the alternative approach presented in this work is a viable option for the interim period from now to about 1985.)
2.1.5 Flue Gas Desulfurization
This is the first of two techniques we will discuss that involve the removal of sulfur after the combustion process. Hence, flue gas desulfurization (FGD) is a postcombustion continuous control approach. This is the approach the EPA most strongly favors (see Section 2.3). At the present time there are about a dozen different FGD processes which have been demonstrated in fullscale commercial practice (Princiotta (1972)). Only limestone scrubbing has been successful for extended periods of time on coalfired electric power plants (EPRI (1975)).
The limestone scrubbing process is a nonregenerable process, in
that it produces a waste sludge that must be disposed of in an environmentally approved mannerusually by ponding or landfills. For disposal by ponding, a 100 megawatt coalfired unit would require 0.5 acre at a depth of 50 feet each year the FGD unit is in operation (PEDCo (1975)).
PEDCo (1975) reports
The disposal of scrubber sludges entails potential
pollution of land and water. Surface waters such as
rivers, streams, lakes, and ponds can be contaminated
by leaching and percolation of sludge liquor into
ground water through soil and sludge storage areas.
Large areas of land could deteriorate from storage of
considerable amounts of sludge materials that typically
contain 50 to 75 percent water. This land could be made useless by nonsettling characteristics of the
sludge. (pp. 7778)
15
In Japan scrubber sludges are oxidized to form fiber gypsum used in the production of wallboard; and if the economic incentives were present, the same might be possible in the U.S. (EPRI (1975)).
Regenerable processes produce a marketable byproduct, usually
sulfur or sulfuric acid. The most promising of these are the magnesium oxide (MagOx), the catalytic oxidation (CatOx), and the sodium solution scrubbing (WellmanLord) processes.
The operations of the FGD system will require from three to six percent of a power plant's total (gross) energy input (PEDCo (1975)). Early FGD processes were unreliable because of chemical and mechanical problems; however, more recently installed systems are achieving better than 90 percent availability (EPA (1975)). In addition to sludge disposal there is the possibility of increased local concentrations of sulfur dioxide near the plant brought about by decreased buoyancy of the plume. This can be overcome by reheating the stack gas before it is emitted.
The principal objection by most electric utility planners has been the immense capital costs associated with the installation of a system they do not feel is a proven technology (see Section 2.3). Nevertheless, the EPA (1974) feels "...[FGD] represents the most practical medium or long term solution to the sulfur oxides problem for a large number of coalfired power plants." (p. 9) In this work we consider FGD systems as a viable approach to controlling sulfur dioxide emissions.
2.1.6 Fluidizedbed Combustion
This approach, which entails a combustion modification, is one of the most promising approaches for using fossil fuels in the production of electric energy. It offers the very real possibility of greater thermal efficiencies than conventional boilers. The thermal efficiency of a
16
conventional.boiler with FGD (scrubbing) is about 37 percent, and no substantial improvement is expected. For pressurized fluidizedbed boilers, thermal efficiencies of 38 percent for firstgeneration units and up to 47 percent for secondgeneration units are predicted (PEDCo (1975)).
In addition, fluidizedbed boilers should be able to burn many types of fuels and combinations of fuels. For example, all ranks of coal, blends of oil, gas, rejects from coalcleaning plants, and even municipal refuse are possible fuels.
The use of a sorbent, such as limestone, reacts with the sulfur dioxide formed during combustion, and if the sorbent is continuously withdrawn (oncethrough sorbent) the sulfur is effectively removed. Also proposed are systems that use separate vessels to extract the sulfur from the sorbent and then reuse the sorbent (sorbentregeneration). This latter process eliminates the disposal problem of the oncethrough sorbent. The coal fed to a fluidizedbed combustion process does not have to be as fine as that fed to a pulverizedcoal boiler so particulates (fly ash) will be more coarse and, therefore, more easily collected.
Cost estimates by the EPA (1975) indicate that pressurized fluidizedbed systems should have capital cost savings of 15 to 20 percent and operating cost savings of 10 to 15 percent when compared to conventional boilers with a FGD process. For later generation pressurized systems the operating cost savings may be 25 percent. In addition, space requirements are comparable to existing plants.
The EPA (1974) has a program underway to demonstrate a pilotplant
pressurized system by 1982. It is hoped that by the late 1980's widespread use of these systems will be possible. Thus, as promising an approach as
it is, the fluidizedbed combustion alternative is not considered in this work as a viable option for the near term (up to 1985). In the next section we will discuss the noncontinuous emission control techniques.
2.2 Noncontinuous Emission Control
The control approaches discussed in this section have been referred to as: intermittent control systems (ICS), supplementary control systems (SCS), dynamic emission controls (DEC), and various other names that imply noncontinuous control. A noncontinuous emission control system has been defined by the EPA (1973) as a "...system whereby the rate of emissions from a source is curtailed when meteorological conditions conducive to high groundlevel pollutant concentrations exist or are anticipated." (p. 25697) In effect, the control is accomplished by reducing emission levels selectively rather than continuously, based on pollutant dispersion characteristics to assure that ambient air quality standards are not exceeded.
The Tennessee Valley Authority (TVA) calls its noncontinuous emission control system the sulfur dioxide emission limitation program (SDEL). SDEL is the most comprehensive SCS in operation today (PEDCo (1975)). It is controversial but workable, as evidenced by the successful operation at their Paradise power plant in Drakesboro, Ky. since 1969 (Montgomery et al. (1975)). The SDEL program has been installed at nine TVA power plants, which have been divided into class 1 and class 2 type programs. The class
1 programs are less complex and operate in an openloop frame; whereas, the class 2 programs are operated in a closedloop mode. The feedback is via a telemetering system that transmits realtime data from sulfur dioxide monitors located strategically throughout the geographical area affected by the sulfur dioxide emissions. A typical class 2 program
18
involves (see Montgomery et al. (1975))
1) a sitespecific pollutant dispersion model 2) 12 sulfur dioxide ambient monitors (field)
3) sulfur dioxide emission monitors (stack)
4) at least one instrumented meteorological tower
5) a rawinsonde system
6) several minicomputers
7) a telecommunication system
8) an environmental data station to be operated by a
staff of five
Both the class 1 and class 2 programs are supported by the TVA meteorological forecast center in Muscle Shoals, Ala.
The noncontinuous methods for reducing emissions involve fuelswitching, loadswitching, a combination of the two, and tall stacks. Tall stacks are obviously a continuous method once they are built. However, since they do not reduce emissions, and have been proposed along with SCS, they are generally thought of as a noncontinuous emission technique.
2.2.1 Fuelswitching
This approach involves switching to a low sulfur fuel when meteorological conditions are conducive to high ground level concentrations of sulfur dioxide. This necessitates the storage of alternate fuels and the ability to effect the switch. All of the necessary components for fuelswitching are currently available (EPA (1975)). Some limitations involve the ability to switch rapidly enough to avoid violating the ambient standards, increased particulate emissions, and insignificant decreases in the total annual emissions.
19
Some positive points are that this approach becomes effective as soon as the ability to switch fuels is installed; and it represents an efficient use of the scarce low sulfur fuels.
2.2.2 Loadswitching
This approach is similar to fuelswitching except that when conditions are such that an emission reduction is required, the electrical load on the particular plant is switched to another plant in the interconnected electrical transmission system. Again, the components are commercially available. This system along with tall stacks is the primary basis for the SDEL program. It should be pointed out that loadswitching has long been a practice of the electric utility industry for economically loading of generating units and, also, for recovering load that has been dropped due to forced outages of generating units.
Similar to the fuelswitching approach, drawbacks to this approach include questions as to the reliability of predicting when control is needed. Total emissions are not reduced and, in fact may increase, since the alternative plant may be using a "dirtier" fuel or may be more heavily loaded.
2.2.3 Tall Stacks
Any process that regularly generates a sizeable plume must use a stack (chimney). Two criteria involving "good engineering practice" have evolved:
1) The stack should be high enough to allow the plume to escape
from the wakes created in the lee of local buildings and
topographic features.
20
2) The exit gas velocity should be high enough to allow the
plume to escape from the wake of the stack itself.
The first criterion is usually met by building the stack 21/2 times the height of the nearest building or topographic structure, The second criterion is usually met with stack gas exit velocities of 11/2 times the maximum expected normal wind velocities at the top of the stack. Thus a stack exit velocity of 40 meters per sec (m/sec) would be required for a wind speed of 60 miles per hour (mph) (PEDCo (1975)).
The EPA (1974) has stated
EPA's "Tall Stack Policy" encourages the use of stacks
conforming to good engineering practice, which is a function
of the individual facility configuration and local terrain features. In general, this policy results in stack heights
sufficiently tall to minimize ground level effects caused by
aerodynamic wakes, eddies, and downwash, and those caused
by high winds during neutral atmospheric stability conditions.
In some cases good engineering practice requires a
relatively tall stack to overcome adverse terrain features.
However, use of excessively tall stacks in an attempt
to avoid reducing emissions merely results in dispersion of sulfur dioxide, sulfates, and acid aerosols over wide
areas. Their use as a substitute for permanent [continuous] emission controls, in addition to a harmful effect on health
and welfare, would be inconsistent with the ...[Clean Air
Act]. (pp. 1415)
The use of tall stacks has frequently been misunderstood. For example, in the July, 1975 EPRI Progress Report, a feature article by Yeager (1975) is entitled "Stacks vs. Scrubbers." Stacks are a necessary part of any emission control system, continuous or noncontinuous. We will view, therefore, the use of a tall stack as a method to be used in conjunction with (not in lieu of) some other method of control.
It should be mentioned that under present Federal regulations
(EPA (1971c)) noncontinuous emission controls are applicable only to power plants for which construction was begun prior to the issuance of the
21
New Source Performance Standards on August 17, 1971. With a doubling rate of ten years, about one half of the power plants in 1981 will still be eligible to use noncontinuous emission controls. However, this approach at best must be considered as an interim (up to 1985) method.
Two of the drawbacks common to all noncontinuous emission control strategies to date are
1) It is difficult to enforce a noncontinuous emission control
system in a multisource environment, since one source cannot
control the emissions of another source.
2) Insignificant decreases in total emissions are effected. Thus,
increased levels of suspended sulfates and "acidrain" problems
are possible.
Two of the advantages common to noncontinuous strategies are
1) Noncontinuous controls are relatively inexpensive when compared
to continuous strategies.
2) The control becomes effective as soon as the program begins
operation, and lead times for initiation of the control are
significantly less than for most continuous controls.
We have discussed some general aspects of noncontinuous emission
controls, and some of the specific requirements of the SDEL program which is being used by TVA (see Montgomery et al. (1975)). Other models which have been proposed are Sullivan (1972), Sullivan and Hackett (1973), Gent and Lamont (1971), and Lamont et al. (1975). All of these models suffer from the same kinds of limitations mentioned above. In the next section we will discuss a method proposed in this dissertation that is actually a hybrid of continuous and noncontinuous emission controls, which incorporates some of the better features of both control approaches.
22
2.3 Arguments for a National Mix of Alternatives
In Section 2.2 we discussed the noncontinuous emission control
approaches. All of the models mentioned were short term models. That is, their primary control effort is directed toward the 3 hour and 24 hour ambient air quality standards. Because these models only reduce emissions during adverse meteorological conditions, their principal drawback is that they do not significantly reduce anissions over the long term. A second limitation, as mentioned before, is the difficulty of enforcement in a multisource environment.
The model we are proposing in this work avoids the principal objection of noncontinuous emission controls, in that it does significantly reduce long term emissions of sulfur dioxide. In this sense it is a continuous approach. Since it also involves loadswitching, fuelswitching, and tall stacks, it has elements of a noncontinuous approach. Hence, this approach is best described as a hybrid approach. We prefer to call it an economical alternative approach to all the other methods; keeping in mind it is only meant to be an interim approach valid until about 1985.
Our model uses a dispersion model (Chapter 3) to predict long term air quality, based on long term emissions determined, a priori, through a probabilistic production costing mechanism (Chapter 4). That is, we simulate a power system, use a dispersion model to predict air quality for some future period (usually a month), and through the method of production costing, select the optimal mix of loadswitching and fuelswitching. Tall stacks are not a necessity, but are desirable as discussed in Section 2.2. Before applying the method of production costing, we must extend its theory to include certain other generalized functions of the electrical output of electric generators. This is done in Chapter 4.
23
The use of the extended production costing method allows us to select the order of committing units based on minimal incremental additions of concentrations of sulfur dioxide at ground level. In effect, this is loadswitching; but since it is done beforehand, it avoids most of the drawbacks of loadswitching discussed in the previous section.
Since with this model we specify the fuels and their maximum sulfur contents beforehand, we can consider fuel supply constraints, boiler types, and fuel costs in determining the optimal mix of fuels (fuelswitching). As is discussed in Chapter 5, the model lowers the sulfur contents of the fuels, where necessary, to operate the power system in an environmentally acceptable manner. When environmental constraints are met, the model operates in an economical (fuel costs minimized) mode.
Since the model determines the optimal mix of fuels (actually sulfur
contents in a specified type of fuel) beforehand, it is useful as a planning technique for fuel purchasing. In addition, effects on the economical or environmental costs to the power system can be studied relative to fuel changes, plant siting (location), stack heights, weather changes, and many other variables.
Before developing the model, we want to present some arguments by prominent persons in favor of a mix of alternative approaches on a national scale rather than selecting just one or two approaches.
In a letter to Secretary Morton, Chauncey Starr, President of ERPI states
The primary standards can be met by flue gas
desulfurization devices such as "scrubbers", or alternatively, by an intermittent control system
tLetter dated March 26, 1975 from Dr. Chauncey Starr, President of the Electric Power Research Institute (EPRI) to the Honorable Rogers CB. Morton, Secretary of the interior.
24
involving a mixture of fuel, operating schedules, and
tall stacks, all adjusted to meet the daytoday
conditions of the region. Scrubbers are roughly ten times more costly than intermittent control systems.
As noted in the attached memorandum, scrubbers do have the advantage of removing sulfur from the flue gas and
thus minimizing uncertain, longterm, lowlevel but
cumulative effects of sulfur dispersal on the regional
environment. For this reason, EPA has preferred
scrubbers. In either casescrubbers or intermittent
controlsthe criteria for public health can be met.
In the memorandum referred to above by Dr. Starr, entitled,
"Issues and Conclusions on the Use of Intermittent Control," it is stated
in the conclusion
Finally, an effective SOx control strategy must
recognize that nearterm achievement and maintenance of existing SO2 ambient air quality standards depends
on understanding the following:
a.) Source impact on ground level ambient SO02 concentration is not directly related to mass emission.
This encouraging circumstance provides the basis
for allocating the limited quantity of low
sulfur fuel and control technology available to the specific sources where it is most needed to
achieve and maintain the standards.
b.) Intermittent control must be applied to the
majority of large utility and industrial sources
not only because it is the most cost effective
approach but because it is the only method which can be made available in sufficient
quantity in the nearterm.
c.) Extensive imposition of stack gas desulfurization
for controlling sulfates is unjustified not
only because the standard has not been
established, but because the current stateoftheart is technically ineffective for this
control function. (pp. 1011)
William Lalor, Senior Vice President of Southern Services, Inc. in
reference to Russell Train, the EPA Administrator, says
Statement of William G. Lalcr, Jr., Senior Vice President, Southern Services, Inc., before the U.S. House of Representatives, Committee on Interstate and Foreign Commerce, April 2, 1974.
25
Mr. Train seems to be saying that intermittent control systems are satisfactory insofar as sulfur oxides are concerned but that sulfur oxides are not the problem.
Therefore, we should spend a great deal of money to control
sulfur oxides on the grounds that sulfates at some level may be a problem. This makes no sense to me.
I feelthat Mr. Train has set up a sulfate straw man
because the current uniform sulfur oxide emission
standard cannot be defended. (p. 3)
William Donham Crawford, President of the Edison Electric Institute,
in reference to a meeting with Frank Zarb, Federal Energy Administration
Administrator, states
Mr. Zarb stated that the Administration did not wish
to repeat last spring's situation where the Administration
had recommended several amendments to the Clean Air Act,
but EPA refused to support two of them (intermittent
control systems (ICS), and "no significant deterioration").
He noted that the proposals had languished before Congress
ever since without hearings being scheduled. He
thought better results could be obtained from Congress
if the Administration presented a unified front. He
said tentative agreement had been reached with
Mr. Train, and that an amendment would be submitted
in January along the following lines: Scrubbers would
be designated as the best or ultimate technology
and permanent emission reduction systems should be
the goal; however, it would be recognized that more
development is needed, that time extensions should be allowed, that research and development should be
continued, that ICS should be authorized in the interim,
and that flexibility should be permitted. He stated
that FEA and EPA would testify in support of the
amendment, and that the industry should be prepared to
present its views fully at Congressional hearings.
The meeting above was held in late December, 1974. In January, 1975
the Clean Air Coordinating Committee (CACC) was formed (see Chapter 1).
George C. Freemen, Jr., Special Counsel for CACC statestt
tLetter dated December 26, 1974 from W. Donham Crawford, President of Edison Electric institute, to Aubrey Wagner, Chairman of the Board of Directors of the Tennessee Valley Authority.
ttLetter dated April 30, 1975 to the Honorable Frank G. Zarb, Administrator,
Federal Energy Administration, from George C. Freemen, Jr., Special
Counsel for the Electric Utility Industry, CACC.
26
We believe that in the interim, while we will find
the long term answers and solutions, the government
should adopt a policy that requires the electric utility industry to assure attainment of present federal air quality standards, which effectively
protect public health and welfare, but at the same time permits each plant to do so through
whatever mix of technology, fuels and operating
strategies will do the job. This would minimize
consumer costs and contribute to this nation's
energy selfsufficiency.
Finally, Yeager (1975), Program Manager, EPRI states
The issue of tall stacks versus scrubbers is
fundamentally an administrativ.e controversy and not
a technical one. This is the main reason for the delay in resolving it. In reality, a technically feasible national strategy for achieving the established sulfur dioxide (S02) ambient air quality standards
over the next decade must employ both stacks and
scrubbers. Such a strategy is not only the least
costly to the nation but the only one consistent with
available supplies of domestic fuel and control
technology. (p. 2)
In the same article Yeager (1975) states
In summary, a truly feasible national SO2 control
strategy must consider the roles of both continuous
and intermittent [noncontinuous] controls and use
established techniques to apply them selectively. Not
only will such a strategy effect significant cost
savings to the nation but it is the only one that can
be used to allocate limited clean fuels and control technology over the next decade without sacrificing the present national ambient S02 standards. (p. 16)
As mentioned above (Crawford meeting with Zarb) the EPA, which had
opposed noncontinuous emission controls was willing to change its
position. We will close this chapter with statements of policy from
the EPA (1974) and EPA (1975), which remain in effect today (June, 1976).
EPA (1974) states
...temporary use of intermittent controls in
carefully selected circumstances will facilitate
more rapid attainment of the current primary sulfur
dioxide standards without the necessity for power
27
plant shutdown, will allow the continued use of the
nation's high sulfur coal reserves while control
technology, which will make it environmentally
acceptable is being installed, and will also allow
time to increase the availability of low sulfur fuels.
(p. ii)
Further in the same report, the EPA (1974) states
Accordingly, intermittent control is currently
considered an acceptable control measure only in cases where constant emission reduction measures
are unavailable, and only until such measures become
available. Under this philosophy, authorized
intermittent control systems are referred to as
"Supplementary Control Systems," meaning that
they are intended to supplement available constant
emission controls. (p. 14)
Finally in summarizing the report, the EPA (1974) states
For a variety of reasons, it will be necessary
to apply intermittent controls or tall stacks temporarily on some plants to minimize public health
impacts from sulfur dioxide until adequate emission
control measures can be applied. In a few cases,
interim controls may be required until the early 1980's.
The use of intermittent controls will be minimized
and will be discontinued as soon as possible.
Intermittent controls will not be sanctioned for long term use where constant control measures are
available. (pp. 2728)
Then in 1975 in its Report to Congress on Control of Sulfur Oxides,
required by the Energy Supply and Environmental Coordination Act of
1974.(ESECA), the EPA stated their latest position (EPA (1975)):
EPA believes, however, that a number of isolated
power plants can use supplementary control systems,
which require additional emission reduction on an intermittent basis, as an interim strategy for a
number of years without increasing the risk to
public health. (p. 36)
We will now put politics aside. In the next chapter we begin the
theoretical and technical development of the alternative model we
have proposed.
CHAPTER 3
STEADY STATE MATHEMATICAL DISPERSION MODEL
In this chapter the dispersion model used throughout this dissertation is developed. While it would be desirable to have a dispersion model capable of predicting accurate ground level concentrations of sulfur dioxide emitted from electric power plants, it is not necessary that the estimates be exact. This is because decisions about the commitment order of the generating units (Chapter 4) are based on comparing alternative strategies (dispatching orders) and selecting that strategy whose incremental increase in ground level concentrations throughout a geographical area is the smallest. It is for this reason that the mathematical dispersion model developed in this chapter need not be exactindeed, the physics of the earth's atmosphere are so complex that it is unlikely that an exact model will ever be found. If the dispersion model contains a bias and that bias is consistent, then the bias will disappear through the comparison process. Thus, we are concerned with the effect on concentrations of one strategy relative to another.
A gas such as sulfur dioxide emitted high into the earth's atmosphere from the stack (chimney) of an electric power plant will undergo
As is commonly done throughout the electric power industry, we will use the words dispatching and committin of generating units interchangeably.
2P
29
considerable dispersion before reaching ground level (Turner (1970)), This dispersion is an extremely complex, physical mechanism that involves large scale, as well as small scale turbulent eddies that mix, dilute, shear, and diffuse the gas with the atmosphere. Because of the complexity of turbulent dispersion in the atmosphere, no one to date has proposed a mathematical model capable of describing all the motions that actually take place (Tesche et al. (1976)).
There are two general approaches to developing an atmospheric
dispersion model. The two approaches are the gradient transport model credited to Fick (1855) and the statistical model formulated by Taylor (1921) and applied by Lin (1960). Both approaches have had wide acceptance by researchers and workers. Both have some limitations due mainly to certain simplifying assumptions in the interest of practicality. Both, as will be shown, lead to a Gaussian dispersion function but with different second moments.
In this dissertation the gradient transport method is chosen as a starting point for developing the dispersion model. We shall arrive at the classical result in a simple but novel manner. In addition, we shall present and discuss the statistical approach and show it is identical to our derived result under certain conditions. Finally, we shall present the plume rise formulas used with the dispersion model and the formula relating gross emissions of sulfur dioxide to electrical power out of the generator. Subsequently in Chapter 4 the emission formula will be modified to account for incremental emissions due to incremental changes in the electrical power produced by a generator.
30
3.1 Gradient Transport Model
The gradient transport model is chosen because it can be derived from basic laws of physics. What happens to a pollutant after it is emitted into the atmosphere is, indeed, a complex question. The large scale winds move it downwind from its source, while small scale winds (turbulent eddies) diffuse and mix it as it is moved downwind. The atmospheric temperature, pressure, and humidity influence the movement as well as chemical reactions of the pollutant with other agents present. Sunlight, clouds, and precipitation cause reactions and may remove some of the pollutant from the atmosphere or convert it to another possibly more dangerous agent. For example, the conversion of sulfur dioxide to acidsulfates is considered by the Environmental Protection Agency (EPA) to pose a serious health problem.
A precise model of dispersion would have to include all of the above
processes, plus other weather variables, irregular terrain, variable sources, and types of deposition (removal from the atmosphere). While it may seem a hopeless task at this point to develop a meaningful model, it is not. This is because we are interested in concentrations averaged over a period of time. In this respect we are fortunate to have the laws of statistics in our favor. Although the individual molecules seem to move in a completely random fashion, when viewed macroscopically a definite plume pattern is evident. If one recalls viewing a visible plume, it is seen to flow away from the stack in the direction of the transporting winds (large scale), while it spreads out and mixes due to the turbulence of the eddy wind (small scale). This plume pattern that emerges is due to the laws of statistics and to physical laws such as the conservation of mass, momentum, and energy. The atmosphere must obey the conservation of mass for both the pollutant and the air. In equation form (sometimes called the equation of continuity)
31
this is written as
g+ V J = 0 (3.11)
dt
where X is the concentration of the pollutant in micrograms per cubic meter (vg/m3) and J is the flux of the pollutant in micrograms per square meter per second (ug/m2sec).
Just as ideal gases are known to diffuse from regions of higher
concentrations to regions of lower concentrations, the gradient transport theory assumes that the pollutant flux will be proportional to and in the direction of maximum decrease in concentrations per unit distance. The proportionality constant is called the diffusivity constant and has the units square meters per second (m2/sec). Stated in mathematical terms, the above becomes, for one dimension
J = K aX (3.12)
x x Sx
where Kx is the proportionality constant (diffusivity constant) described above for the xdirection. Equation (3.12) has analogies in heat conduction, neutron flows, and molecular motions. It is credited to Fick (1855) and is often termed Fickian diffusion. In three dimensions (cartesian) equation (3.12) becomes
J = (K x a + K a + Kz ) (3.13)
xy y ay z z
where (a a a ) are the cartesian system unit vectors in the x, y, and
x y z
z directions,respectively. For an isotropic system (K = Ky = Kz = K)
tAny consistent set of units may be chosen. Micrograms per cubic meter has been chosen by the U.S. Government and will be used throughout this dissertation.
32
equation (3.13) becomes simply J = KV X (3.14)
Substitution of equation (3.13) into (3.11) yields
( + (K ) + (KX ) (3.15)
x Ox Ux y y ay Bz z Bz
Since the concentration of the pollutant may change in time as well as space (due to wind transport), the left hand side of equation (3.15) may be expanded as
d = dX+ dx + xdY +dx dz
t 3t x dt 3y dt az dt
The concentration at a particular point in space (x, y, z) is seen to change with time (ax/at), as well as with the movement of the pollutant with velocities dx/dt, dy/dt, and dz/dt. The velocity dx/dt is the velocity with which the pollutant is being transported in the xdirection, and hence, is the mean wind speed u in the xdirection. That is
 dx
dt
Similarly we define
v d=
dt
the mean wind speed in the ydirection and
dz
W=dt
the mean wind speed in the zdirection. Defining u = dx/dt is an approximation because dx/dt is, in fact, the instantaneous xvelocity
33
of the pollutant which would include the small scale eddy turbulence, as well as the large scale mean wind transport. Again, we rely on the law of averaged to render the instantaneous eddy effects insignificant to the mean wind effects, since in the large they tend to cancel (Slade (1968)). Their effects are not completely ignored, since their combined effects are included on the right hand side of equation (3.15).
One of the principal limitations of all dispersion models is the inability to describe adequately the wind. We know the wind changes speed and direction in time and space, but to model it accurately is a problem of the first magnitude. The simplest model consists of describing the wind by three independent components averaged over time. If u represents the wind speed, it may be written as
u = ua + v a + wa
x y z
where u, v, and w are the components of mean wind speed described above. We choose to use this rather inaccurate model for the wind because we are interested in concentrations representative of, first, steady state, and second, long term (a month to a year). As a result, the model for the wind is adequate, and in fact, is the one used by most workers even for short term concentrations.
Below, equation (3.15) is rewritten with two additional terms, R and S (Slade (1968))
X+ Dx + y + w )) + (K X + ( (Kz )+R+S
t x y z ax ax y y By z z Dz
(3.16)
The presence of R in equation (3.16) is to include the rate of production (if any) of the pollutant due to cha nical reactions,. S is the source of
34
emissions (positive) and may include deposition (negative) at the position (x, y, z) in the atmosphere.
Equation (3.16) is very general and many simplifications are needed to solve it. It is the basis for all gradient transport studies. It was first investigated by Schmidt (1925) and Richardson (1926). Because of the K terms in the equation, gradient transport theory is often termed Ktheory. In the next section we will attempt to solve equation (3.16) with appropriate boundary conditions.
3.2 Steady State Gradient Transport Model
It is instructive to consider each term of equation (3.16). Since in this work we are concerned with the steady state solution to equation (3.16), we immediately set the first term on the left, DX/9t = 0. The next three terms on the left are the.transport terms due to large scale mean winds (sometimes called advection terms). If one recalls viewing a visible plume, the pattern that evolves is one of a plume streaming off in one predominant direction. We will take that downwind direction to be the xdirection (ax). It is assumed then, that the horizontal (a y) mean wind speed, v, and the vertical (az) mean wind speed, w, are small compared to the downwind mean wind speed, u. Thus, the mathematical model for the steady state wind is further simplified to
U= ua x (3.21) and the direction is taken to coincide with the observable downwind direction.
Inspection of equation (3.16) indicates there are two dispersion effects taking place in the xdirection. The first, on the lefthand side, is the advection or transport term due to the mean wind speed, u,
35
and the second, on the righthand side, is the diffusion term (Kxterm) due to the turbulent eddies. In the downwind direction, the advection term far outweighs the diffusion term under normal circumstances. We assume this to be true for our steady state model. Mathematically, this is equivalent to
X >> X
u X x (Kx ) (3.22)
In the horizontal and vertical directions, just the opposite is true. The diffusion terms outweigh the advection terms significantly. Thus
v << (K ) (3.23)
ay ~y y 3y
and
w << a (K .) (3.24)
3z ;z z az
Equations (3.23) and (3.24) are consistent with the assumption leading to equation (3.21).
The R term in equation (3.16) represents the chemical reactions that may take place to generate or remove the pollutant. In more complicated studies where many pollutants are being modeled, this term serves to couple many equations of the form of equation (3.16), thereby greatly complicating the problem. In this work we are modeling only sulfur dioxide. We will take R to be zero. As regards the removal of sulfur dioxide, this amounts to slight overestimates, which means being conservative in our decisionmaking process. The generation of sulfur dioxide, whether chemical or otherwise, will be included in the S (source) term.
36
The final term in equation (3.16) is the source term, S. This
term represents the production and deposition of sulfur dioxide at any point (x, y, z), in the atmosphere. As explained above, the deposition is ignored, which produces conservative estimates. Electric power plant stacks are the only sources of sulfur dioxide considered here. This is because we want to evaluate and compare the effect each plant has on adding sulfur dioxide at ground level throughout some appropriate geographical area. Background contributions, such as area sources of sulfur dioxide can be included afterwards as a percentage of the total concentrations.
Given a system of n stacks, there would be sources at the following points in space
(x1, Y1, Zl), (x2' Y2' 2)'...(xn Y ,n' Zn)
where (xi, Yi', zi) is the location of the ith stack (i = 1,2,...,n). Everywhere else S would be zero. The simplest method of handling these multiple sources is to model each separately and then superimpose the concentrations due to each source. That superposition is valid follows from the fact that pollutants regardless of their source (location) must follow the same basic dispersion laws.
The use of the superposition principle means equation (3.16) need only be solved once for a given location of the cartesian coordinate system. The solution for another source at a different location simply requires a coordinate system translation and a new source strength.
For a single source located at the origin (0,0,0), and all of the above simplifications, equation (3.16) reduces to
37
(Ky 2) + (K ) + S(Q, 0, 0, 0) (3.25)
ax 3y y Sy Sz z @z
where S(Q,0,0,0) is the source of strength, Q, (pg/sec) located at the origin. Instead of including the source in the dispersion equation, as it is in equation (3.25), it may be included in a boundary condition. The appropriate boundary condition requires that the concentration at any point in space moving (at speed u) across an infinite zyplane must equal the source strength, Q. Mathematically this is equivalent to
J X dydz = Q (3.26)
and is another statement of the conservation of mass. There are two other boundary conditions that are necessary. The concentrations must approach zero at very large distances from the source, and, the concentration must be infinite at the source.
The remaining assumption, before actually solving equation (3.25), involves the two eddy diffusivity constants, K and K We will assume y z
K to be independent of y and K to be independent of z. This is not y z
tantamount to assuming an isotropic atmosphere, as is frequently done. In fact, as will be shown later, Ky and K are both functions of the
y z
downwind spatial coordinate, x. Equation (3.25) and the three boundary conditions are rewritten below with the above assumptions incorporated
2 2
u = K 2 + K iX (3.27)
ax y 2 z 2
tPhysically, the yzplane would have to terminate at the earth's surface. Mathematically, it is convenient to ignore the earth's surface in equation (3.26), and to include its effect through boundary reflections. This is discussed in Section 3.4.
38
u X dydz = Q (3.27a)
lim X = 0 (3.27b)
r +
lim X = (3.27c)
rO 0
where r = (x2 + y2 + z2 ) 1/2. Equation (3.27) and its associated boundary conditions will now be solved.
There are many approaches to the solution of equation (3.27). One of the more familiar yet complicated methods is through the use of Green's function (see Lamb et al. (1974)). The following solution is presented here because it is thought to be novel. This approach to the solution was arrived at independently, and was not discovered during the extensive literature search undertaken prior to writing this dissertation.
The approach is to assume a product solution to equation (3.27), apply boundary condition (3.27a) to evaluate arbitrary constants, and finally, verify that the result satisfies boundary conditions (3.27b) and (3.27c). The form of the product solution is
X(x,y,z) = C.F(y,x).G( zx)
where C is an arbitrary constant. Substituting the assumed product solution into equation (3.27) and rearranging yields
1 3F d F 1 @G 2 G
Sax Ky 2 x Kz ] (3.28)
y az
where it is understood that F is a function of y and x only and G is a function of z and x only. Consider the following argument as applied to
39
equation (3.28). For a fixed x, a change in y, and hence F, would in general change the left hand side of the equality but not the right hand side. The opposite is true for a fixed x, and a change in z, and hence G. A change on the right hand side should occur, but no change on the left hand side. For equation (3.28) to be a true equality it must be satisfied for all values of x, y, and z and not for just arbitrary values. The inescapable conclusion is that both sides of equation (3.28) must be a constant, such that coordinate changes leave the value unchanged. This constant is referred to in mathematical literature as a separation constant.
It is expedient at this point to consider this separation constant zero, rather than to carry it through the manipulations only to have it lumped in with the other constants in the evaluation of the boundary condition (3.27a). This effectively separates (3.28) into the following two equations
2 a2F aF
y 2 ax (3.29) and
2 32G aG
2 2 2G G (3.210)
2 2
where 2 K /u and B2 = Kz/U. Notice that the two equations are identical in form so that only one of them has to be solved.
To solve equation (3.29), again assume a product solution of the form
F(y,x) = exp(py + qx) (3.211)
tIt is a product since exp(a)exp(b) = exp(a+b).
40
where exp = 2.71828..., the Naperian exponential. Substituting equation (3.211) into (3.29) yields
2 p2F(y,x) = qF(y,x) or
q = 22
Let p2 = s, so that q = s and p =js. Then
F(y,x) = exp(jsy a2s2x)
is a solution to equation (3.29). A particular value of s yields a particular solution. The total solution is the linear combination of all possible solutions. The total solution then, is
Co
F(y,x) = I exp(jys a2xs2)ds
CO
which evaluates as follows:
Co
F(y,x) = I exp(a2xs2)[cos(ys)j sin(ys)]ds
00 CO
F(y,x) = I exp(ac2xs2)cos(ys)dsj I exp(a2xs2)sin(ys)ds
CO
F(y,x) = 20 exp(a2's2)cos(ys)ds (3.212)
0
The last step follows since the first integral is an even function while the second integral is an odd function. From a table of common integrals, equation (3.212) becomes
41
F(y,x) 1 / ()2 exp(y 2/4 2x)
Since the partial differential equation for G(z,x) equation
(3.210), is identical in form to equation (3.29), we have for G(z,x)
G(z,x) = ( )/2 exp(z2/482x)
Recall that the form of the solution was assumed to be
X = CF(y,x)G(z,x) thus
2 2
X = C(1 ) exp [ 42x a B x 4a x 4 x which upon substitution for a and a becomes
2 2
Cu ()exp [ x uz 1 (3.213)
JEK x 4K x 4Kzxyz
Equation (3.213) can be placed in the standard form of a Gaussian function if we let
2
K =  (3.214) and
K = Z (3.214a)
z 2x
Then
X 02 exp[1Y_2 1 z 2] (3.215)
yz 2 2 z
42
The arbitrary constant C can now be determined from the boundary condition (3.27a), which is repeated here for convenience
uX dydz = Q (3.27a) Substitution of equation (3.215) into boundary condition (3.27a) and
rearranging yields
u2rrnC L exp[ (z) 2] exp[ )]dy dz
z z y y
The integral in the brackets evaluates to v2E (from the properties of a Gaussian function). This leaves
3/2 1 1 z
(2T) 3/2 Ciu a exp[ !( ) ]dz Q
0
and again, the integral evaluates to v2, and we have
4x2Cu = Q
or
c = Q/
4T2
Substitution of this constant into equation (3.215) yields
2 2
Q/u exp[ 12 ( X(x,y,z) = 2T n e ) 2 (3.216) y z y z
43
Equation (3.216) is the steady state solution to the dispersion
equation for a single source of strength Q at the origin. It is actually the product of a Gaussian function of y and a Gaussian function of z. At first glance it does not appear to be a function of x. However, inspection of equations (3.214) and (3.214a) reveal that oy and az are both functions of x. The rationale for substituting the a's (standard deviations) for the K's (eddy diffusivity constants) is so the Gaussian functions appear in standard form and so a comparison with the result from the statistical approach (Section 3.3) will be possible.
For completeness, we must verify that boundary conditions (3.27b) and (3.27c) are also met. This is easiest to do by inspecting equation (3.213). The presence of the exponential makes it obvious that X approaches zero as r approaches infinity, and so boundary condition (3.27b) is satisfied. To verify that X approaches infinity as r approaches zero (boundary condition (3.27c)), notice that the presence of the squared distance in the exponential numerator means the exponential approaches unity as r approaches zero. Then, the x in the denominator outside the exponential is the governing quantity, and it obviously approaches infinity as r and hence, x approaches zero.
Equation (3.216) is rewritten below to emphasize the product of two Gaussian functions
2 2
X(x,y,z) = 1 exp[ f(z) ] 1 exp[ ) ] (3.217) z y
Equation (3.217) shows that the concentrations are proportional to the source strength, Q, and inversely proportional to the mean wind speed, u in the downwind direction, a... Also the concentrations are Gaussian dis
44
tributed in the vertical, az, and horizontal, a directions.
The solution obtained above, equation (3.217), can be verified under more general conditions than were used in the derivation. The practice used here was to obtain the solution by appropriate mathematical manipulations without justifying each steo. In the end the solution is justified on its own merits; that is, it satisfies the partial differential equation (3.27), and boundary conditions (3.27a), (3.27b), and (3.27c). In the next section we present the statistical approach to dispersion modeling. We will see that it, too, leads to a product solution of two Gaussian functions.
3.3 Steady State Statistical Model
The statistical approach to dispersion modeling of pollutants emitted into the atmosphere will, as we will see, lead to a Gaussian formulation much the same as the gradient transport theory formulation. We will present the condition under which the two lead to an identical result. The statistical approach has its beginnings in a study of turbulence of the atmosphere by Taylor (1921). Unlike the gradient transport theory, this approach does not begin by considering the physical laws of materials in the atmosphere. Instead, probability theory is utilized to evaluate the likelihood of the material being modeled traversing from source to a particular point in the atmospheric space. The concentration of the material at that point is then proportional to its probability of arriving there.
The following analogy is described by Slade (1968) and is included
here in a summary version to emphasize some of the underlying assumptions of the statistical approach. A classroom experiment is performed by an instructor and his students in which the instructor tosses a coin and
45
passes it to the student to his right or left in the middle of the front row, depending on whether the coin comes up heads or tails. That student, in turn, continues the experiment by tossing the coin and passing it over his right or left shoulder to a student in the second row, depending on the outcome of the toss. Similarly, the students continue tossing the coin and passing it back until it reaches the back row. If this process is repeated with a large number of coins, a trend begins to appear with most of the coins going to students near the middle of the last row and fewest to those near the end of the last row.
There are several features of this experiment that have parallels in statistical atmospheric diffusion. First, the probabilistic nature of the travel from source to final position (point in atmosphere) as illustrated by the mechanism of tossing the coin. Second, conservation of
coins (mass) must not be violated, i.e., all coins must be accounted for. Third, deposition may occur if someone puts coins in his pocket. Fourth, a distribution of coins (concentrations) is observed to occur with increasing regularity as the nuriber of coins injected (emitted) into the experiment (atmosphere) increases.
The experiment above has been mathematically formalized by
Chandrasekhar (1943), who showed that the resulting distribution is Bernoulli's distribution. For large number of coins (pollutants emitted), the central limit theorem requires that the distribution approach the Gaussian function. For "spreading" in the horizontal and vertical direction and transport by the mean wind, u, in the downwind direction
(ax), this becomes
2 2
X 1 exp[ ) ] 1 exp[ ,.) ] (3.31)
oz y
46
where, Q, the source strength and, u, are as a result of the conservation of mass. The standard deviations, oz and y, are statistical properties necessary to represent this dispersion. How they are determined is discussed shortly. Notice that equation (3.31) is identical to equation (3.217). Equation (3.31) is only presented here for discussion and comparison with the solution to the gradient transport model. Its derivation can be found in Gifford (1955).
Although equations (3.31) and (3.217) appear identical, there is
a subtle difference in the two. Equation (3.31) is derived with standard deviations ('s) which are statistical in nature. Equation (3.217) is derived with eddy diffusivity constants (see equation (3.213)). The connection between the two is through equations (3.214) and (3.214a) repeated here for convenience
2
K =  (3.214)
y 2x
and
2
K Uz (3.214a)
z 2x
Many researchers assume the eddy diffusivity constants are independent of position as their name implies (see Lamb et al. (1974), Slade (1968), Tesche et al. (1976)). This assumption leads to the following incorrect result, which they take to be the fundamental relationship connecting the gradient transport theory to the statistical dispersion theory
d 2
dt (2Ti =2Ki i = y or z (3.32)
47
Equation (3.32) is valid only if the assumption of the K's being independent of position is valid. We will show that assumption leads to a relationship between the a's and the xcoordinate that is not substantiated in field tests. We will develop the argument for the Kz and oz relationship (3.214a) but the same also holds true for K and ay,
y
equation (3.214).
If Kz is not a function of x then
az2 = a2x (3.33)
where a2 is a proportionality constant. This leads to
u a2x a2
z 2x 2
or Kz is proportional to the mean wind speed, u. Intuitively, we feel that the turbulence in the zdirection should not be a function only of the mean wind speed in the xdirection since they are perpendicular. Even more convincing that equation (3.32) is not true, in general, is that field tests conducted by Pasquill (1961) have shown that ,z follows a power law equation of the form
az = axb (3.34)
or
z2 a2x2b (3.35)
which is in direct opposition to equation (3.33) except whenever b=1/2. The parameters a and b are related to atmospheric stability to be dis
48
cussed next. Pasquill (1962) has reported values of b ranging from 0.4 to 2.1 depending on the stability class of the atmosphere.
Assuming equation (3.35) to be valid instead of equation (3.33), since it has been experimentally verified, then Kz must be a function of the xcoordinate, since from equation (3.214a)
Kz = 2 X (3.36)
2
thus
2xK
d 2 d 2xKz
dt z dt
u
d 2 2Kz dx 2x dKz dt z dt dt u u
d (a2) = 2K + 2x dz (3.37)
d z z dx
where the last step follows since, u = dx/dt. Equation (3.37) is seen to contain an extra term when compared with equation (3.32). Substitution of equation (3.36) into equation (3.37) yields
d(z2) = 2a2 x2bI + 2x a2u (2bl)x2b2
dt z 2
2 22d (z2) 2a 2b1 au 2b2
( x + 2x (2b1)x
d 2 __+z (2b1)
dt z x x
2
d 2 uz2b udt z x
49
d (az2) = 4bKz (3.38)
Again we have two equations, (3.38) and (3.32) that contradict each other unless b = 1/2, which is simply not experimentally verified.
As mentioned earlier, the gradient transport theory and the statistical approach lead to identical results, providing an appropriate relationship connecting the eddy diffusivity constants and the standard deviations is used. We have shown that by choosing the eddy diffusivity constants as functions of the xcoordinate (see equation (3.36)), and adopting Pasquill's (1961) power law form for the standard deviations
0z = axb (3.39)
and
y = cxd (3.39a)
leads to an appropriate connection in that the fundamental connection between the two theories, equations (3.214) and (3.214a), is satisfied.
We choose the power law form for the standard deviation terms, az and ay, since experimental field tests have shown it is an appropriate form. The parameters a, b, c, and d are functions of the stability of the atmosphere. Pasquill (1961) has categorized the atmosphere into six classes, A through F, where A is the most unstable (most turbulent) and F is the most stable (least turbulent). His original categorization was based on wind speed, cloudiness, and sunshine intensity. More recent models (Roberts et a!. (1970), Busse and Zimmerman (1973), and Air Quality Display Model (1969)) have retained the same categories, (AF),
50
but use the vertical temperature gradient as a means of selecting the appropriate stability class. The stability of the atmosphere is best defined as the capability for enhancing or resisting vertical motion. While the stability is obviously a function of many meteorological parameters (particularly wind gustiness and vertical temperature gradients) today's models, almost universally, consider only vertical temperature gradients (lapse rates).
Since we will make extensive use of the neutral stability class
(D), we take this opportunity to describe it. If a volume of air is forced upward, it will encounter lower pressures and expand and cool. Theoretically, if no heat is exchanged between the volume of air and the surrounding atmosphere, the volume of air will cool at about the rate of one degree centigrade per one hundred meters vertical (10C/100 M). This lapse rate is termed the dry adiabatic lapse rate and is characteristic of neutral stability, which means there is no tendency for the volume of
air to gain or lose bouyancy.
If the lapse rate is greater than the dry adiabatic lapse rate
(cooling faster than one degree centigrade per one hundred meters), the atmospheric condition is termed unstable (superadiabatic), and this class tends to enhance vertical motion in the atmosphere and therefore disperses more. If the lapse rate is less than the dry adiabatic rate (subadiabatic), the atmosphere is slightly stable, and a volume of air forced upwards becomes more dense and will, as a result, experience a downward force returing it to its original height. Stable conditions do not favor dispersion, and this condition is commonly called an "inversion". Neutral
conditions are most representative of long term or average conditions, since they are associated with overcast skies and moderate wind speeds (Turner (1970)).
51
In this work we chose to use the gradient transport theory or
Ktheory to derive the dispersion model, since it could be related to basic physical laws. It should be pointed out that there are those who are critical of Ktheory (see Calder (1965)). A Russian worker, Monin, (1959), refers to Ktheory as a "semiempirical" theory of diffusion. Actually, it is, since Fick's law is empirical (but then so is Newton's law).
The most compelling reason for acceptance of the Gaussian formulation as valid is that a widely different approach, the statistical approach, leads to the same formulation. The acceptance of the Gaussian formulation is so widespread among researchers that all the major dispersion models in use today incorporate the Gaussian distribution.
Finally, we are able to avoid the argument over which of the two
approaches to the Gaussian formulation is most valid, since we are actually using a hybrid of gradient transport theory and statistical theory. Our dispersion model is based on the solution of a simplified diffusion equation and our model incorporates the statistical properties of the atmosphere through the use of standard deviations related to the xcoordinate.
In the next section our steady state dispersion model is further
modified to be representative of long term (monthly or seasonal) concentrations. In addition boundary reflections are considered and incorporated into our mathematical dispersion model.
3.4 Long Term Dispersion Model
Up to this point in our development of an appropriate dispersion
model, we have assumed steady state, i.e., variations with time are set equal to zero. The length of tinie for which our mathematical model
52
(equation (3.217)) is valid varies from a few minutes to an hour or so, depending on the length of time the mean wind speed and direction, stability parameters, and emissions are constant. One of the main contributions of this research is through the use of stochastic production costing as a basis for the comparison of the effects of one dispatching strategy relative to another strategy on ground level concentrations of sulfur dioxide. In order for that comparison to be meaningful, the time frames of the dispersion model and the production costing model must be the same.
As will be shown in Chapter 4, the appropriate time scale for the production costing model is a month, a season, or even a year. Thus, we now undertake to modify our steady state dispersion model to a long term dispersion model, where it is now understood that long term implies monthly, seasonal, or annual estimates of the concentrations of sulfur dioxide.
Before developing the long term model for dispersion, it is convenient to modify the model through a coordinate translation. Equation (3.217) was derived, in Section 3.2, as if the source of the emissions were at the origin (0,0,0) of the cartesian coordinate system. Actually, for electric power plants (the only sources considered here), the emissions are from a tall stack (see Figure 1). Inspection of Figure 1 reveals that the plume actually disperses symmetrically about an "effective" stack height, H, rather than the actual stack height, Ha. The effective stack height is defined as the height at which the centerline of the plume becomes essentially horizontal (dotted line in Figure 1). Formulas for determining the effective stack height are discussed in Section 3.5. It simplifies matters to place the origin of the coordinate system at the base of the stack. The source, then, is at position (O,O,H). Equation (3.217) with this coordinate system translation becomes
53
x
H
Figure 1. Coordinate System Showing the Plume Dispersing
About an Effective Height in the Vertical and
Horizontal Directions.
54
exp [2 0 2 20
Xxp M]  exp [()2] (3.41)
u J z2o z V2 y2s y
Throughout this dissertation, we are most concerned with ground
level concentrations of sulfur dioxide, and we take this opportunity to consider only ground level (z=O) concentrations. Equation (3.41) becomes
= 1 exp 1 H 2 1 exp [ )2] (3.42)
u v/ 2 z 2 2 ay
z y
Notice that equation (3.42) is no longer a function of z. We shall present empirical formulas in Section 3.5 for determining H and u, so that they become deterministic inputs. Thus, the only variables in equation (3.42) are x, y, and Q. Remember that y and a are functions y z
of x. Equation (3.42) is rewritten below
X(x,y,Q) = g(H) g(y) (3.43) where
g(H) = 1 exp [ ~() 2] (3.43a)
/21ra z z
and
g(y) 1 exp [ (y )2] (3.43b)
Implicit in equation (3.43) is that the xdirection is the downwind direction. Obviously, for long periods of time this direction will change many times. A particularly convenient way to represent these changes in wind direction and also wind speed is to use historical wind rose data representative of the time period for which concentration estimates are desired. Usually, 16 wind directions are assumed (16 major compass points). The wind rose, then, gives for each wind direction the frequency of
55
occurrence and the mean wind speed. For a good first order approximation, the neutral stability class D may be assumed (Turner (1970)).
Since 16 wind directions have been assumed, it is convenient to think of the stack as located at the center of a circle of radius, x, and the circle to consist of 16 sectors with each sector centered along one of the 16 major compass points. Consider, for example, the sector shown in Figure 2.
The wind rose will give information about the frequency of the wind and its speed, for example, in the direction of east. Thus, to obtain a meaningful estimate of the concentration as a function of x in the eastern direction, it is necessary to consider the average concentrations over the crosswind (horizontal) distance from y= d/2 to y=d/2. We will define this crosswind average concentration later, but for now we distinguish it from the concentration function in equation (3.43) by placing a bar over it. Thus, the crosswind average concentration function is Y(x,Q). This crosswind averaging will, in effect, remove the explicit dependence of the concentrations on the ycoordinate and will make the concentrations functions of the xcoordinate (the radial distance from the stack), the emissions, Q, and the frequency, f, with which the wind blows toward that sector. So, for a given value of x and a particular wind rose, 16 concentration estimates are obtained (one for each direction).
The average concentration estimate for a particular direction as a function of the radial distance, x, is the product of the crosswind average concentration, X, and the percentage of time the wind blows in that direction (the frequency, f, in that direction). Thus
V(x:Q,f) = f 9(x,Q) (3.44)
y=d/2
e  East Y= d/2
r x
Figure 2. Eastern Sector Showing Downwind and
Crosswind Directions.
57
It is important to remember that the concentration function, Ax, (equation (3.43)) is a function of only two random variables, Q and y, for a given value of x. All other parameters are deterministic. Then, by definition
x(xQ) = jf x(x,y,Q)p(Q,y) dQdy (3.45)
where p(Q,y) is the joint probability density function of Q and y. For obvious physical reasons, the emissions, Q, are independent (statistically) of the ycoordinate and so
p(Q,y) = p(Q)p(y)
and, then
X(xQ.) = (H~ yJQ Qg(y)p(Q)p(y)dQdy (3.46)
which can be written as
X(x,Q) = (H) g(y)p(y) Qp(Q)dQ dy (3.47)
u y Qp(y
Note that the expression in the brackets in equation (3.47) is, by definition, the average or mean value of emissions. We shall label it Q. Then, equation (3.47) becomes
X(x) 9=_ g(H) g(y)p(y)dy (3.48)
u y
where the functional dependence on Q has been dropped, since Q is deterministic. How Q is computed is discussed in Section 3.5.
It is reasonable to assume that a concentration at a particular distance, y, from the x axis within the sector is as likely as a
concentration at any other value of y within the sector for the same value of the xcoordinate. That is, within the sector and a particular radial distance, x, from the stack, the concentrations should be weighted equally. Then
p(y) = c
where c is a constant. But application of the definition of a probability density function requires
d
d c dy = 1
2
or
c = I/d
This yields for X(x)
d
(x) = 2LI 2 1 exp [ Y()2]dy
u d 2 2 cy
U 2 y Consider the change of variables
S= Y/Oy
then dS= d, since a is not a function of y. So
ay y
X(x) = I1 exp (1 s2)dS
d 2
d d S1 V2 where
S= d/2Oy
From Figure 2, d is given by
d =xe
59
(In fact, d=2 tan()x = .398x, whereas d= 6x .393x or about a one percent error). So
x(x) 16 Q g(H).G(S) where
G(S1) 1 exp ( S2)dS
42 7T 51 2 and
S2Trx x
1 162Cry 16y
For the assumed neutral stability class D (Pasquill (1961))
a = cxd = 0.32 x0.78 so
S1 = 0.614 x0"22
Refer to Table 2 for several values of x, S1, and G(S1) evaluated from standard mathematical tables. For distances from the stack of one kilometer or more, the function G(S1) is for all practical purposes equal to unity. Then
X(x) = 16Qg(H)
27x
u
or
((x)= 16 1exp [ 1( ) 2]
Finally, from equation (3.44), we have
16 f 1 H 2
(x,f) 3/2 exp [ 2( )H 2] (3.49)
(2r) uya zz
60
Table 2. Integrated Gaussian Function for Several Values of Downwind Distance.
Downwind Limit of Integrated Distance Integration, Gaussian in Meters, S Function,
x 1 G(S1)
100 1.69 .925 500 2.41 .982 1000 2.81 .994 5000 4.00 .999 >5000 >4.00 1.000
61
Equation (3.49) is very similar to one developed by Calder (1971), who used polar rather than rectangular coordinates.
Equation (3.49), when modified to include boundary reflections,
will be the long term dispersion model used throughout the remainder of this dissertation. In Chapter 5 the results of Chapters 3 and 4 are applied to the study of a realistic size electric power system with multiple generating units. Also, a computer program developed during this research is presented and discussed. For distances closer to the source than one kilometer, equation (3.49) must be modified. This is done also in Chapter 5.
Equation (3.49) is plotted in Figure 3 as a function of the radial distance, x. It is seen to begin at zero concentration, increase to a maximum, and then fall off rapidly to zero again.
As mentioned in Chapter 2, the EPA has set a radius of liability of 25 miles (40.2 km) for each point source. It is instructive to determine from equation (3.49), at what radial distance, x, the maximum concentration occurs as a function of effective stack height for a given set of meteorological conditions and emissions. Equation (3.49) is rewritten below as
k 1 I H 2
X = k (]exp [ 2 axx
where k is a constant containing the fixed emissions and meteorological parameters. If X is differentiated with respect to x and the result set equal to zero, the maximum is found to occur at
1 1
xmax b j Hb (3.410)
a (b+l)
62
S
4J
w 500
U
o
400
E
r
300
S.
200
c,
0
,
O
5 10 15 20 25
Downwind Distance in Kilometers
Figure 3. Plot of Concentrations Vs. Downwind
Distance from the Source.
63
Figure 4 shows a plot of equation (3.410) as a function of H from one hundred meters to one kilometer. The values of a and b are 0.22 and
0.78 respectively (Pasquill (1961)). Figure 4 reveals that the maximum occurs farther downwind for increasingly higher effective stack heights. Even for the unrealistically high one kilometer effective stack height, the maximum occurs well within the 25mile liability radius set by the EPA. One must keep in mind that while higher effective stack heights produce maximums farther downwind, these maximums are decreasing significantly with higher effective stacks. This is shown in Figure 5.
Our long term dispersion model, equation (3.49), is valid only if the effective stack height is far enough away from the earth's surface that its presence can be ignored. Physically this can never be the case, and we must include the effect of the earth's surface in the model. The most effective way to include this physical barrier is to assume total reflection of the pollutant at the earth's surface. While some deposition at the ground is known to occur, we are being conservative in our estimates by ignoring it. The most expedient method for handling these reflections, mathematically, is through the concept of a virtual source or image source located symmetrically with respect to the ground plane, to the actual source. This technique is widely used in heatconduction theory.
The translation of the source from origin (0,0,0) to effective stack height (O,O,H) caused the vertical dispersion term to change from z to zH. For a (virtual) source located a distance underground of H meters, the vertical dispersion due to this second source should disperse about H, or appear in the exponential as a z+H term. Thus, the vertical dispersion term becomes
(vertical dispersion) exp [ zH 2+exp 1 z+2 S_2 z 2 zz
64
25 Mile Radius
35 S28
14
,
w
,:)
E
200 400 600 800 1000
Effective Stack Height in Meters
Figure 4. Plot of Downwind Distance to Maximum Concentration
vs. Effective Stack Height of Source.
65
S.
500
s=
c
* 400
o
5300
0
2 400
E 100
.9E
200 400 600 800 1000
Effective Stack Height in Meters
Figure 5. Plot of Maximum Concentration vs. Effective
Stack Height.
C
2 0
.9
x o
Stack Height.
66
Reflections of the pollutant may also occur off a stable layer of air above the stack (inversion) in much the same way. If the height of the inversion (sometimes called mixing height) is Hm, then we can include its effect by assuming a virtual source at H + (H H) = 2 H H.
m m m
Then the vertical dispersion exponents have the following three terms: zH (due to the actual source), z+H (due to ground level reflections), and z2Hm+H (due to inversion layer reflections). There can also be reflections off the inversion layer after reflections off the ground. This leads to a term, z2H1H. Likewise, there can be reflections off the ground after reflections off the inversion layer. This leads to a term, z+2HmH. Consider the case where reflections occur, first off the ground, second off the inversion, and third off the ground. This leads to a term, z+2H +H. The symmetry is complete now, since all possible combinations
m
of signs in front of Hm and H have been shown. Additional multiple reflections are theoretically possible and each new reflection changes only the numerical coefficient on H This is because of the additional distance traversed from ground to inversion, H m. Consideration of all possible reflection combinations leads to the following infinite set of terms
zH, I [z(2nHm+H)], I [z(2nHmH)]
n= m n=l
z+H, I [z+(2nHm+H)], I [z+(2nH H)]
n=l n= m
It is important to remember that these terms actually appear in exponentials, and we have a sum of exponentials rather than the sum of the terms above.
Since we are primarily concerned with ground level concentrations, we let z=O and after some grouping of like terms, we have
67
S 2p 2] 2nH H 2nH +H gR(H) = 2 exp[ 2( 1 2 exp[ 2( )2]+exp[ 2( m )2 z n=l z z
Equation (3.43a) is now changed to include reflections and is written
g(H) gR(H) (3.411)
Our long term dispersion model including reflections is written below in its entirety for future reference
X(x,f) 3/2 fQ exp[ ( 2] +
(2) u z z
Sexp 2nH H 2nH +H 2
Sexp[ ( m ) + exp[ 1( ) ] (3.412)
n=l z z
In the computer program discussed in Chapter 6, it is seldom
necessary to include terms beyond n=3, since the large exponents cause the exponentials to approach zero very rapidly. An internal check in the program compares values of the infinite summation terms to the first term (n=O term) beginning with n=l. Values that do not contribute more than 0.1 percent are discarded.
Inversion layers aloft are not typical of long term meteorological conditions. The most useful long term dispersion model, then, includes ground level reflections but not inversion layer reflections. This can be accomplished in equation (3.412) by setting H = m, which causes the infinite summation to vanish, leaving
(x,f) = 2.03f exp[ .(,) 2 (3.413)
ZXU 2
Equation (3.413) is the one we will use most frequently in modeling the pollutant dispersion from electric power plants.
68
In the next section the plume rise formulas are presented. A formula for calculating u is presented, and the techniques for calculating gross emissions is developed.
3.5 Plume Rise, Wind Speed at Stack Height, and Emissions Formulas
We begin this section with a discussion of the many plume rise
formulas in use today. Inspection of Figure 1 shows that there are two distinct regions of the plume dispersion. In the first region, the plume rises vertically from the stack, due to its stack exit velocity, to some higher position termed the effective stack height. From there the prevailing mean wind carries the plume downwind while it disperses. It is important to remember that even though the plume enters the atmosphere at (O,O,Ha), the actual stack height, it is treated mathematically as if it entered the atmosphere at (O,O,H), the effective stack height with zero vertical velocity.
The early attempts to estimate the effective stack of a buoyant plume were primarily empirical formulas based on observation. The point at which the plume centerline becomes essentially horizontal is highly subjective and explains why there have been so many different plume rise formulas developed over the years. Tesche et al. (1976) quotes Briggs (1969) as saying, "There were over 30 documented formulas as of 1969, and he [Briggs] estimated that two more would be added each year."
Disappointed by the vast differences (sometimes as much as three to one) in estimates of plume rise, researchers began attempting to develop plume rise models using rigorous physical laws. Two general approaches developed and it was encouraging, at first, that both led to the same "twothirds power law." The first approach, proposed by M orton et al. (1956), uses entrainment theory, while the second approach, Csanady (1965),
69
uses gradienttransfer closure arguments. We will outline the entrainment theory, pointing out the assumptions that affect its application to our dispersion model.
The approach begins by equating the buoyant force of the plume to the time rate of change of the vertical momentum of the plume
2 AT d 2
irr gp = 7 (Tr pw) (3.5I)
where
r = radius of plume (m)
g = acceleration of gravity (m/sec2) p = density of plume material ( g/m3)
T = ambient temperature (OK)
AT = TsT = temperature excess between plume and
surroundings (oK)
w = vertical plume speed (m/sec)
The plume is assumed to be adiabatic which leads to
2
QH = 7r uAT = constant (3.52) where
QH = heat emission (m3 K/sec)
u = mean wind speed throughout the plume in the downwind
direction (m/sec)
For equation (3.52) to be strictly valid, requires that
1
AT a
T
since u is assumed constant.
70
Using equation (3.52), we can replace the lefthand side of
equation (3.51) with pgQH/Tu. The righthand side can be rewritten as
d 2 d 2w dz
dz
Ft (Trr pw) = d (Frpw) dt
Recognizing that dt = w, we have
dt
w d (Tr2pw) = pw d (r2w)
which becomes
d 2 2 dw 2 dr
w dz (r pw) = upwr T + upw dz Equation (3.51) can now be rewritten as
2 dw 2 dr2 gQH(3.53)
wr w  + ,w dz Tu (3.53)
dz dz Tu
Briggs (1971) assumes that the plume radius increases linearly with height during the ascent to effective stack height, i.e.
r = yz
where y is a constant determined experimentally. Now equation (3.53) becomes
2 2 dw 2 2 gQH
Try z w + y w 2z T (3.54)
Assume w=azn, where a and n are constants, as the solution to equation (3.54). This yields
2 2 n nl 2 2 2n QH
7Ty z az naz + 2Try a z 2 = Tu
or
2 2 2n+l 2 2 2n+l gQH
na y z + 2ayz (3.55) Tu
71
Because of the adiabatic assumption, the righthand side of equation (3.55) is a constant (independent of z). Thus, we must have
2n+l = 0
or
1
Then
1 2 Try2 22 2 H
~aw + 2a2vy =Tu
or
2gQH
a = [ 2T ]1/2
a2 I
3Txy Tu SO
2gQH 1/2 1/2
r= 22 ] z = Az (3.56)
3Try Tu
where A is recognized as a constant.
Since by definition, the downwind speed is
dx u dt
and the vertical wind speed of the plume is
dz
dt
their ratio is
dz = w = w(z) (3.57)dx
u u
since by equation (3.56) w is a function of z. Rearranging equation (3.57) and integrating, we have
72
 z1/2 dz = dx
A 
1 z3/2 x
A 3/2
U
or
z (3Ax 2/3 9 A2)1/3 (x)2/3
2t 4 u
Substituting equation (3.56) and grouping terms, we have
3gQH 1/3 (3.58)
2[X ]2/3 (3.58)
2ry Tu u
Equation (3.58) shows that the height, z, the plume rises above the actual stack height is proportional to the downwind distance, x, raised to the twothirds power; hence, it is often called Briggs' twothirds power law.
Equation (3.58) is often written in terms of the buoyancy flux, F, which is defined as
2 AT
F = gvr AT (3.59) where v is the velocity of the plume material, and the other terms are as previously defined. Note for the initial buoyancy flux, v=w (the stack exit velocity), while for the buoyancy flux at or near the effective stack height, v=u (the mean downwind speed).
Using equation (3.59) we can rewrite equation (3.58) as 1/3
3uF 1 2/3
z ] x (3.510)
2y v u
It should be pointed out that while equation (3.510) is valid in form for all x, it is of use to us only near the stack (v=w) or near the effective stack height (v=u), since v is unknown elsewhere. This is not a limitation,
73
actually, since for our dispersion model we are only interested in z at or near the effective stack height.
Baker and Jacobs (1971) have suggested that the plume has essentially reached its effective height at a downwind distance of 10 times the actual stack height. With this modification and v=u, equation (3.510) becomes
z = (3F 1/3 1 (10H )2/3 (3.511)
2 a
2y u
where
2 AT
F = gr u T
Remembering that the adiabatic assumption required constant heat emission,
1
QH' and therefore AT ac we note that the buoyancy flux, F, is also
t T
a constant. Since it is easier in practice to determine the parameters at stack height, we chose the initial buoyancy flux rather than the final. This means
F = gr2 AT (3.512) where it is understood that r is now the inside radius of the actual stack, and w is the stack gas exit velocity. Tesche et al. (1976) report that the entrainment constant, y, is approximately 0.66 for a neutral atmosphere.
To avoid confusion with the zcoordinate, in the future we will define the increase above the actual stack as Ah. Thus equations (3.511) and (3.512) become
Ah = 15.0 T) (Ha)2/3 (3.513)
u
This is equivalent to assuming conservation of momentum.
74
and the effective stack height is, then, given by
H = Ha + Ah (3.514)
While there are literally dozens of plume rise formulas in the
literature, we have chosen equation (3.513) for our model, since it is partly based on physical laws. In addition, a second theoretical approach, the gradienttransfer closure (Csanady (1965)), leads to the same form. It should be remembered that the model is empirical due to the entrainment constant relationship (r = yz). The assumption of adiabatic conditions is also a limitation, although not a serious one for neutral atmospheric conditions (Liu (1975a)).
In the interest of being objective, we should point out that
Thomas et al. (1970) have studied several plume rise models in field tests including the one presented here, and report that it has a slight tendency to underpredict the rise for light winds and overpredict for strong winds. They report, however, that "the use of the 2/3 power law formula is considered preferable, provided information for the meteorological parameters is available." We again emphasize that since our decisions will be based on comparisons, a good consistent estimate of plume rise is satisfactory for our purposes.
Recently, there has evolved a new approach to modeling the plume rise phenomenon. This approach involves numerical solutions to equation (3.16), the generalized dispersion equation, near the stack, (see Shir (1970) and Liu (1975b)). This approach requires extensive computer computations and to date has not been shown to be a significant improvement over the twothirds law model we chose. This approach shows much promise, however, and may in the future replace all the empirical models. It is considered
75
too early in its developmental stage to be of use to us now, but undoubtedly, this will change in the future.
The mean wind speed, u, is used in both the long term dispersion model and the plume rise model. Wind speed is known to increase with height, i.e., u=u(z). Ideally, the mean wind speed for the dispersion model should be representative of the wind about the plume centerline (effective stack height), while the mean wind speed for the plume rise model should be representative of the wind throughout the vertical region from the actual stack height to the effective stack height. To avoid this complication, most models, including the one used here, use the mean wind speed at stack height, Ha. This allows for sampling the wind at one height and eliminates the errors that would arise from inaccurate determinations of the effective stack height.
For the plume rise model, the entrainment coefficient, y, is
determined empirically from field tests where u is selected at the actual stack height. Likewise, for the dispersion model the stability parameters, a and b, are determined empirically with u selected at the actual stack height. Since u at 10 meters off the ground is the most common method for reporting mean wind speeds, we need a formula to relate the mean wind speed at stack height to the mean wind speed at 10 meters.
Busse and Zimmerman (1973) suggest a power law relation for the neutral atmosphere of the form
H
u = ul0 (a)1 (3.515) where u10 is the mean wind speed at 10 meters, and the other parameters are as previously defined.
76
Finally, we complete this section by presenting the formula for determining gross emissions of sulfur dioxide from the stack. Several definitions will be presented first.
Heatinput can be defined as the rate of heat, in BTU's/hrt necessary to produce a given amount of electrical power out of the generator. Several of the emissions standards set by the U. S. Government are in terms of pounds per million BTUtt heatinput (see Chapter 1). In mathematical terms, the heatinput, HI, is
HI P Hr
where
P = electrical power generated in megawatts, (MW)
H = heat rate in BTU's per 103 watthour (BTU/KWH)
As an example, a 500 MW output for a unit having a 9,000 BTU/KWH heat rate requires
6 BTU)
H = (500x10 6w)(9000
I KWH
HI = 4.5x109 BTU/H
or four and onehalf billion BTU's per hour.
If this were a coalfired unit, the coal consumption, Cc, would be
H
C c Hv
where Hv is the heating value of the coal in BTU's per pound. For this
tWe regret this sudden shift from the 'MKS system of units to the British
system of units, but the use of BTU's is so widespread in the electric
power industry that the use of Joules or Wattseconds would be distracting.
ttEven the U.S. Government is inconsistent, since it uses the MKS system
for concentrations (ig/m
77
unit using a coal having 12,000 BTU's per pound, the coal consumption would be
C 4.5x109 BTU/H
c 12000 BTU/1b
Cc 3.75x105 lb/H
or 375,000 pounds per hour of coal.
If this coal contained Ps percent sulfur then the sulfur consumption Sc would be
C. p
S = cs c 100
where ps is the sulfur content of the coal in percent. For this unit, if ps = three percent, then the sulfur consumption is
Sc = (3.75x105 lb/H) 3 100
Sc = 1.125x104 lb/H
or 11,250 pounds per hour of sulfur.
During the combustion process, the sulfur is oxidized to sulfur dioxide (SO2). The atomic weight of sulfur is 16 while the two oxygen atoms together have an atomic weight of 16. Thus the weight of sulfur dioxide is twice that of elemental sulfur. This means that for each pound of sulfur consumed in the furnace, there will be two pounds of sulfur dioxide produced and emitted from the stack. This assumes complete (100 percent) oxidation.t Thus, for this unit, the sulfur dioxide produced, Q, is given by
tThe EPA allows certain furnaces 95 percent oxidation, or a weight factor of 1.9 rather than 2.0. Our choice of 2.0 is in keeping with a conservative estimate.
78
Q = 2 Sc
Q = 2 1.125x104 lb/H
Q = 2.250xl04 lb/H
or 22,500 pounds per hour of sulfur dioxide produced.
It is desirable to have a formula for calculating gross emissions of sulfur dioxide, Q, directly. This is shown below:
20P Hr p
20P Hr s (3.516) where all of the parameters are as defined above. Substitution of P=500, Hr=9,000, Ps=3, and Hr=12,000, verifies the result as 22,500 pounds per hour of sulfur dioxide produced.
Since the U.S. Government has selected micrograms per cubic meter for reporting concentration, equation (3.516) is modified to the following form
(2.53x106)P Hr P
(253x Hr s (3.517)
where Q is now given by micrograms per second. As is shown in Chapter 6, it is necessary to have both equations (3.516) and (3.517). Ground level concentrations are determined through the dispersion model using equation (3.517), after it is modified in Chapter 4 to reflect average (long term) emissions. Emission standards, on the other hand, are computed on the basis of pounds of sulfur dioxide per million BTU heatinput, which requires the use of equation (3.516).
CHAPTER 4
THE METHOD OF PRODUCTION COSTING
In this chapter we present the method of production costing as it is used by today's electric utility companies for estimating the long term fuel costs of the supply of electrical energy. We will extend the theory to include other functions of the output of the generating units. In particular, we will be interested in the application of this extension to the estimation of long term emissions of sulfur dioxide.
The production costing method, then, is applied to the problem of determining the best estimates of long term emissions of sulfur dioxide from each source (generating unit), and these estimates are used as inputs to the mathematical dispersion model presented in Chapter 3. Through the use of the production costing method and the dispersion model, decisions can be made based on the order of dispatching generating units that minimize the incremental deterioration of ground level air quality throughout some appropriate geographical region.
In the past the electric utilities have experienced exponential
growth rates due to the everincreasing demand for electric energy. A doubling rate of ten years has not been uncommon, and some areas, for example Florida, have experienced doubling rates closer to seven years. Faced with these short lead times, utility system planners have relied heavily on larger and larger capacity generators. Along with these increased capacities has naturally been an increased complexity in design and operation of these large units. As manufacturers of electric generators
79
80
pushed the limits of their technologies, a new phenomenon began to emergethat of an increasing unavailability of these units due to forced outages brought about by malfunctions in boilers or auxiliary parts. It became obvious during the early sixties that a technique for incorporating these forced outage rates into the various system planning models had to be developed.
In 1967, French researchers (Baleriaux et al. (1967)) introduced a probabilistic simulation technique, that included forced outage rates for estimating the fuel costs of operating various electric generating units within an electric utility system. Hydroelectric and pumpedstorage projects were assigned fictitious fuel costs based on operating costs for purposes of comparison. Booth (1972) incorporated a dynamic programming algorithm with the method introduced by Baleriaux et al. (1967) and introduced his generation expansion program as applied to the State Electricity Commission of Victoria, Australia.
In addition toconsideration of the forced outage rates of the
generating units, the production costing model as applied by Booth (1972) could consider all types of generating units, i.e., nuclear, fossil, hydroelectric, gas turbine, combined cycle, and pumpedstorage. This has gained the production costing method increasing acceptance within the electric utility industry in several areas of application. Sager et al. (1972) have also used the method for evaluating generation production costs. Sullivan (1974) has used the method for generation reserve planning, while Sullivan and Hilson (1975) have used it in estimating ground level concentrations of sulfur dioxide from nearby electric generating plants.
In Section 4.1 we will extend the mathematical theory of production costing to include any monotonically increasing or decreasing function of
81
the output, P, of a generator. Since in Chapter 3 we were able to write the emission rate of sulfur dioxide (equation (3.517)) as a function of the output, P, we can make use of this extension of the theory to predict long term emission rates from individual generators. In order to derive this extension, it will be necessary to take a detailed look at the mathematics of production costing.
4.1 The Load Duration Curve
At the heart of production costing are three important concepts:
I. Load duration curves 2. Forced outage rates
3. Order of dispatching generating units
In this section we will discuss the use of a load duration curve in estimating the long term output of a generator. We will derive the necessary mathematics to extend its use to any monotonically increasing or decreasing function of the output.
One of the major contributions of this research is the extension of the production costing method to include these generalized functions of the output of individual generators. The only restriction is that these functions be monotonically increasing or decreasing functions of the output, P. Simply stated, this means there must be a one to one relationship between the function of P and the output, P. Thus, for a given value of P there must be a unique value of the function of P and vice versa. Emission rates, fuel costs, and fuel consumption are functions of the output, P, that satisfy this restriction. In practice, then, this is a restriction only in the mathematical sense.
Figure 6 shows a typical load duration curve. Load duration curves are normally assembled from historical data. The abscissa represents
82
P
T/2 T
* Time in Hours Figure 6. Typical Load Duration Curve.
the number of hours that the system load, PL' equaled or exceeded the ordinate value of load. Thus, if T were the total time period under consideration, then for T/2 hours the system load equaled or exceeded PL,T/2 megawatts (see Figure 6). If the values along the abscissa are divided by the time period, T, the abscissa values become a fraction of time. In this case we can say that one half the time the system load equaled or exceeded PL,T/2 megawatts, or the system load equaled or exceeded PL,T/2 with a probability of 0.5. Note that by considering the abscissa values as probabilities, we can use historical data as representative of future data for the purpose of predicting future loads, providing that the load duration curve for the time period, T, is a valid representation of the future period (see Jenkins and Joy (1974)).
Generally, the shape of the load duration curves for different periods of time are the same. The actual values of the loads may be different. Most electric utilities have a fair degree of expertise in load forecasting, and in this work, we assume that the load duration curves are representative of the particular future time period under consideration.
In the development of probabilistic simulation models, it has been
more convenient to reverse the roles of the ordinate and abscissa of load duration curves. Figure 7 shows this "inverted" load duration curve. The interpretation, now, is that the ordinate value is the forecasted probability that the abscissa value (load) is equaled or exceeded during the time period for which the curve is valid. From this point on when we speak of a load duration curve, it should be understood that we mean an inverted load duration curve.
tTheoretically, one should only include the number of hours PL exceeded the ordinate value of load. In practice it is convenient to include also the number of hours PL equaled the ordinate value of load. The distinction is important only for discontinuous load duration curves which are not used here (Sullivan (1976)).
84
1.0
0.5
f I
P P P
L,min L,T/2 L,max
System Load in Megawatts
Figure 7. Typical "Inverted"' Load Duration Curve.
85
Since we will be deriving the long term estimate of functions of the generator output, P, from fundamental mathematics, it is necessary to develop the load duration curve from more basic probability theory.
Figure 8 shows a typical probability density function of system loads, fL(PL)t, where the "L" indicates system load. The cumulative probability distribution function, F'L(P L ), is defined as (see Figure 9a)
F'L(PL) JL(PL)dPL
The backwards cumulative distribution function, FL(PL), can be defined as (see Figure 9b)
FL(PL) = 1 F'L(PL) (4.11) Equation (4.11) can be rewritten as
FL L L L)dPL LL ( L)dPL (4.12) where the first integral on the right hand side of equation (4.12) is, by definition of the probability density function, equal to one. Then, equation (4.12) may be written as
FL(L) = f L (PL)dPL (4.13)
Inspection of Figure 9b reveals that it is identical to the (inverted) load duration curve of Figure 7. Equation (4.13), then, is the mathematical description of a load duration curve. In Chapter 3 we used p(.) for the probability density function. From now on we use f(.) for the probability density function to avoid confusion with the generator outputs, P, and the system load, PL'
86
P. P
L,min PL,max System Load in Megawatts Figure 8. Typical Probability Density Function of
System Loads.
*
oe o
L,min L,max L,min L,max
System Load in Megawatts System Load in Megawatts
(a) Cumulative (b) Backward Cumulative
Figure 9. Cumulative and Backward Cumulative Distribution Functions of Systems Loads.
88
In order to illustrate the technique for determining the expected
output of a generator from the load duration curve, consider the following argument. Refer to Figure 10. Suppose that the system generators are scheduled such that generator 1 is loaded until its maximum before generator 2 is loaded, and then generator 3 follows after generator 2 is completely loaded, and so forth.
Suppose also that the maximum output of generator 1 is Pl
which is less than PL,min' the system minimum forecasted load. Then generating unit 1 must be loaded to its maximum for the entire time period, since from the load duration curve (Figure 10), FL(PL)=1.0, or the probability that the system load exceeds the capability of unit 1 is equal to unity.
In this case the output of unit 1 is deterministic rather than
random. The probability density function of a deterministic value is a diracdelta function. Thus
f(P1) = 6(P1Pl,max) (4.14) where 6() is the notation for the diracdelta function. Then the expected value of the output of unit 1 is by definition
a 0 l,max
1 P1 f(P1)dP = P1 f(P)dP1 (4.15)
m JO
where the last step follows as a practical consideration. Substitution or equation (4.14) into (4.15) and evaluating yields
1,max
P =0 P1 6(P1P1 )dP1
or
1 1,max
89
1.0
o
1,max L,mi n L,max
System Load in Megawatts
Figure 10. Load Duration Curve.
90
where the last step follows from the sampling theorem for diracdelta functions.
It is no surprise that the expected value of unit 1 turned out to be its maximum capability. The fact that unit 1 was dispatched first, and its maximum capability was less than the minimum system load, determined that it be fully loaded for the entire period. Notice that the area under the load duration curve in Figure 10 from 0 to Pl,max is P1,max' since the ordinate value is unity.
Carrying this illustration one step further, suppose that it has been determined that generating unit i will be loaded only after the first iI units have been dispatched. How this is actually determined will be discussed in Chapter 5. This is the same as saying that unit i will not be loaded until the system load reaches PT megawatts, where PT is the sum of the outputs of the first iI units. In addition, as the system load grows from PT to PT+Pi,max unit i will be loaded from 0 to Pi,max (see Figure 11).
In effect, the probability of dispatching unit i is FL(PT), the probability that the system load equals or exceeds PT. Likewise, the probability that unit i is fully loaded is FL(PT + P i,max). Similarly, the probability of unit i being called on to supply Pa megawatts (0 < Pa < P.imax ) is the probability that the system load if P +P or greater, which is precisely FL(PT+Pa). Since Pa is an intermediate output between no load and full load for unit i, we know the probability of any output for unit i between no load and full load. The average value of the output of unit i, then, could be determined by multiplying each possible output by the probability it is required and summing all these products. In the limit, this average value for the output of unit i is simply the area under the load duration curve from PI to P +Pimax iTi 3.max
91
SU
n
i
P P +P.
T T 1,max
System Load in Megawatts
Figure 11. Load Duration Curve Showing Commitment
of Unit i.
92
The important point in the above discussion is that by deciding
beforehand not to allow unit i to be loaded until the system load grew beyond PT' and to load unit i to its maximum as the system load grew to PT +Pi,max we have forced unit i to assume the backward cumulative distribution function or load duration curve of the system loads.
In the above discussion, it was shown in an heuristic manner that
the expected value of unit i was given by the area under the load duration curve where unit i was placed (see Figure 11). This will be proven in the following discussion.
In order to make the discussion more general, consider that it is required to find the expected value of some function (not necessarily linear) of the output of a particular generating unit. The only restriction on the function will be that it be a monotonically increasing or decreasing function of the output, as discussed earlier. In addition, suppose it is necessary to know the expected value of the function over an interval (segment) from a to b. Let the function be given by
Q = Q(P)
where P is the output of the generating unit.
By definition, the expected value of this function over the interval a to b is
= Q f(Q)dQ (4.16)
a
where f(Q) is the probability density function of Q. Since Q is a
tThe derivation that follows is for some particular generating unit, e.g., the ith unit. However, to keep the notation from becoming too cumbersome, we will not use the subscript i, but it should be understood.
93
function of P, equation (4.16) can be rewritten as
Q= Q(P)f(P)dP (4.17)
where f(P) is the probability density function of the output, P, of the generator. By definition, f(P) can also be written as
f() = dF'(P) dF(P)
= dP dP
where F'(P) is the cumulative distribution function and F(P) is the backward cumulative distribution function of P (see equation (4.11)).
Equation (4.17) can, then, be written as
b
= Q(P)dF(P) (4.18)
a
Note that in the formulation of equation (4.18), the expected value of Q is a RiemannStijlets integral, since it is integration with respect to a function F(P), instead of a variable. Now suppose that the unit is forced to follow the load duration curve from PT to PT+ba megawatts. That is, the unit is dispatched from a to b megawatts, only as the system load grows from PT to PT+ba megawatts. Then for the interval, PT PL P T+ba, for the system load, there is a corresponding interval, a 5 P b, for the generating unit. These corresponding limits require that
P = PLP T+a (4.19) and
dP = dPL
Now since the generating unit follows the load duration curves from PT to PT+ba, its backward cumulative distribution function F(P), may

Full Text 
PAGE 1
AN ECONOMICAL ALTERNATIVE FOR CONTROL OF GROUND LEVEL CONCENTRATIONS OF SULFUR DIOXIDE FROM ELECTRIC POWER PLANTS USING THE METHOD OF PRODUCTION COSTING By DAVID WAYNE HILSON A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REOUIRFMFNTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA wr^m^wrr mHaiiii*. me^ffummtimtimmtitt ^ & a HtimmsAreM
PAGE 2
d To my wife, Carol, and sons, Jamie and Michael, for their patience and understanding and above all their love and cooperation. "**' "^TT I *u.'ipiiii"LiSfliÂ£iBp Trurrnr^nfyptifti Tim II
PAGE 3
ACKNOWLEDGMENTS I would like to express my genuine appreciation and gratitude to the members of my supervisory committee: Dr. Robert L. Sullivan, Chairman, Dr. Qlle I. Elgerd, Cochairman, and Dr. Paul Urone. I would especially like to thank Dr. Robert I. Sullivan for his patience, encouragement and scholarly conduct. In particular, I would like to acknowledge that his suggestion to use the new production costing method in this work proved to be invaluable. I remember well my initial skepticism and later enthusiasm. I would also like to thank Dr. Lynn D. Russell for his support and patience, which was far beyond what could have been expected. I want to express my appreciation to the members of the Power Research Staff of the Tennessee Valley Authority for their cooperation in providing relevant information on their Sulfur Dioxide Emission Limitation program. Finally, a special thanks to Mrs. Beth Seville for her editorial assistance and for the typing of this dissertation. m ^ ^* > ^;j '* '* *^^* f*y ilt'^?*M?wii# *t ii if i  f taii i i^ia** vtfiifliiiwi
PAGE 4
TABLE OF CONTENTS ACKNOWLEDGMENTS iii ABSTRACT vi CHAPTER 1: INTRODUCTION .... 1 "1.1 Historical Background of Clean Air Legislation 2 1.2 Statement of Purpose and Chapter Outline 6 CHAPTER 2: ALTERNATIVES FOR CONTROL OF SULFUR DIOXIDE EMISSIONS 8 2.1 Continuous Emission Control 8 2.1.1 Low Sulfur Fuel 9 2.1.2 Fuel Oil Desulfurization 11 2.1.3 Coal Desulfurization 11 2.1.4 Coal Gasification and Liquefaction .... 13 2.1.5 Flue Gas Desulfurization 14 2.1.6 Fluidizedbed Combustion 15 2.2 Noncontinuous Emission Control 17 2.2.1 Fuelswitching 18 2.2.2 Loadswitching 19 2.2.3 Tall Stacks 19 2.3 Arguments for a National Mix of Alternatives 22 CHAPTER 3: STEADY STATE MATHEMATICAL DISPERSION MODEL 28 3.1 Gradient Transport Model. ..... 29 3.2 Steady State Gradient Transport Model ..".... 34 3.3 Steady State Statistical Model 44 3.4 Long Term Dispersion Model 51 3.5 Plume Rise, Wind Speed at Stack Height, and Emission Formulas 68 TV
PAGE 5
TABLE OF CONTEiJTS (continued) CHAPTER 4: THE METHOD OF PRODUCTION. COSTING 79 4.1 The Load Duration Curve 81 4.2 The Effective Load Duration Curve 99 4.3 The Effective Load Duration Curve for Segments of Units .109 4.4 Estimating Short Term Maximum Concentrations .111 CHAPTER 5: DISCUSSION OF THE DIGITAL COMPUTER PROGRAM 122 5.1 The Digital Computer Program 122 5.2 Discussion of the Application of the Computer Program to a Realistic Size System 131 5.3 Conclusion and Future Work 137 APPENDIX A 138 REFERENCES 146 BIOGRAPHICAL SKETCH 151  ~ ' .^^ W' < t 'i W'Tg *?f i > T Ty *[^ghf"it** iiiiliwi II Â— Minni I iiii I iiiiiiiiiiim m i
PAGE 6
Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN ECONOMICAL ALTERNATIVE FOR CONTROL OF GROUND LEVEL CONCENTRATIONS OF SULFUR DIOXIDE FROM ELECTRIC POWER PLANTS USING THE METHOD OF PRODUCTION COSTING By David Wayne Hi 1 son August, 1976 Chairman: Dr. Robert L. Sullivan Cochairman: Dr. 01 le I. Elgerd Major Department: Electrical Engineering An alternative approach for reducing ground level concentrations of sulfur dioxide emitted from electric power plants is presented. The approach involves the commitment of generating units using an extended version of the method of production costing. It is shown that this method, in addition to providing a strategy for meeting the Federal Ambient Air Quality Standards, can be used as a planning model for fuel purchases, plant siting, and stack height determinations. It is easily adapted to any size electric utility system. The historical background leading to the promulgation of emission rate and air quality standards is first discussed. The advantages and disadvantages of several of the more prominent existing methods are also discussed. Arguments for a national mix of alternative strategies are presented, as well as some viewpoints by prominent persons on the issue of continuous or noncontinuous emission controls. This alternative approach involves the use of a steady state, long term (a month or more) mathematical dispersion model. The dispersion model is derived in a novel manner from the steady state gradient transport Vl
PAGE 7
equation and will be shown to be the Gaussian dispersion model used extensively for predicting pollutant concentrations from point sources (stacks). The statistical approach to the Gaussian dispersion model is discussed and the fundamental relationship connecting the two approaches is shown to be only a special case of a more general connection discovered and presented here. The theory of production costing is extended to include monotonically increasing or decreasing nonlinear functions of the electrical output of the generator. This makes it possible to use the method of production costing to predict long term estimates of the sulfur dioxide emissions as a function of the commitment order of the various fossil and nonfossil electric generating units. Larsen's technique for relating maximum concentrations to average concentrations is incorporated into the approach in order to extend its usefulness. A digital computer technique is proposed to determine the optimal commitment order of the generating units, mix of fuels, and sulfur contents. A widely used model for predicting average concentrations near the source is shown to be in error and is corrected here. A simple numerical technique for mathematically constructing the "effective" load duration curve is presented and shown to involve only convolving of the unitsnever deconvolving. This technique avoids many of the computational problems present in previous models. Finally, the application of the computer program to a realistic size electric utility system is presented and discussed. "^'Â•>Â•*.,'*MÂ• Â•Â• Â— Â— .Â—.Â— ^
PAGE 8
CHAPTER 1 INTRODUCTION The emissions of various forms of sulfur compounds into the atmosphere have been a concern of man for some time now, although it has only been during the last few decades that it has received so much public attention. Hesketh (1972) reports As early as 1300, a royal decree was issued in London prohibiting the use of lowgrade coal for heating because it created excessive smoke and soot. The only known case of capital punishment because of an air pollution violation occurred in the 13th century when a Londoner violated this order. Sulfur in fuels burns to sulfur dioxide. In 1600 sulfur dioxide was the first chemical to be specifically recognized as an air pollutant. However, it was not until about 1940 when air pollution, as such, became important, (p. 1) Kellog et al (1972) state 1) Man is now contributing about one half as much as nature to the total atmospheric burden of sulfur compounds, but by A.D. 2000 he will be contributing about as much, and in the Northern Hemisphere alone he will be more than matching nature. 2) In industrialized regions he is overwhelming natural processes, and the removal processes are slow enough (several days, at least) so that the increased concentration is marked for hundreds to thousands of kilometers downwind, (p. 595) Stoker and Seager (1972) have determined that approximately one half of the total atmospheric burden of sulfur compounds results from the combustion of fossil fuels (primarily coal and oil) in the production of electric power. Thus, while recognizing that there are other pollutants, as well as other sources, we will confine ourselves in this "^"Â— Â— ^^"**'^"^*^ ^ 'Â• ''
PAGE 9
work to the consideration of the emissions of sulfur dioxide from electric power generating plants. (See Chapter 3 for a discussion of the emissions of sulfur dioxide from the combustion of coal or oil containing elemental sulfur or sulfur compounds.) 1 1 Historical Background of Cle an Air Legislation In 1963 Congress passed the first Clean Air Act, which was amended several times during the next few years. The 1967 Clean Air Amendment required that the Department of Health, Education and Welfare (HEW) set air quality criteria. The first such criteria were published in February, 1969 for sulfur dioxide and particulates. The criteria contained data, available at that time, on the adverse health effects of pollutants and control techniques for controlling the pollutants, but did not contain standards. On December 2, 1970 the Federal Environmental Protection Agency (EPA) was established by an act of Congress. Among its many functions is the supervision of air pollution control, which formerly came under the auspices of the Department of HEW through the National Air Pollution Control Administration (NAPCA). Some weeks later the 91st Congress passed the Clean Air Amendments of 1970. On December 31, 1970 the President signed them Into law as Public Law 91604. These amendments are often referred to as the Clean Air Act of 1970. One of the many provisions in the act calls for the EPA Administrator to establish Federal Air Quality Standards, and on April 30, 1971 these standards were issued and published in the Federal Register (see EPA (1971a)). These standards included both primary and secondary standards as called for in the Act. Section 109 (b) (1) specified that primary ambient standards should be selected which, allowing for adequate margins
PAGE 10
of safety, would protect public health. Section 109 (b) (2) specified secondary ambient standards to protect public welfare from any known or anticipated adverse effects from pollutants. Table 1 suirmarizes these standards. The secondary standards for the annual arithmetic mean and 24 hour average were adopted but subsequently dropped. Air quality standards are not enforceable, in that the air cannot be told to be clean with any assurance that the air will somehow do so. Therefore, emission standards are required which are enforceable. The Clean Air Act of 1970 provides for emission limitations to be set by the individual states through State Implementation Plans (EPA (1971b)). These plans are to set emission limits on all manmade sources such that the ambient air quality standards are met within three years of the approval by the EPA of each state implementation plan, with possible extension to mid1977 through state initiative. By mid1972 the states had submitted their implementation plans. By this time it had become obvious to those in the EPA and in the utility industry that there simply was not enough low sulfur fuel or other control technologies (see Chapter 2) available to meet and maintain the national ambient air quality standards as called for in the 1970 Act. In the Fall of 1972, the EPA initiated the "clean fuels policy". This policy was designed to encourage states to delay or relax their sulfur dioxide emissions standards in areas where primary ambient standards were not being exceeded, and to give priority for low sulfur fuel to plants which were threatening the primary ambient standards. In addition, this policy asked that the states promulgate revised implementation plans that did not contain emission limitations more stringent than required to meet ambient standards (see EPA (1974)). According to the EPA (1975) u i I >^ j  ^y i WJU! aiii wMj t I i Na g M^n w TB g M iB i iHilidt ^r^^ Â— > '^
PAGE 11
Table 1. National Primary and Secondary Ambient Air Quality Standards. Primary Standards in Micrograms per Cubic Meter Secondary Standards in Micrograms per Cubic Meter Annual 24 hour 3 hour 80 365 1300 if ffWB ^ gwi a t ? in J I II II I ^ g '" iP i g' t JiMi i r iw c ii g. iiwtti .mxn i a MtoM
PAGE 12
"This program has resulted in 42 million tons of coal being made legally acceptable through revisions in the sulfur regulations of State Implementation Plans." (p. 15) In the Fall of 1973, the muchpublicized Arab oil embargo further complicated the fuel supply and pressure began to increase from the electric utility industry to amend the Clean Air Amendments of 1970 once again. On March 13, 1974, Rep. Nelsen (RMinn) introduced a bill (HR 13464) to amend the Clean Air Act. This bill proposed 13 amendments be adopted by Congress. The EPA opposed two of the amendments but all 13 were transmitted on behalf of the Administration by the EPA Administrator with a letter explaining the differing points of view on the two amendments, (We will mention these differing points of view in a discussion of alternatives in Chapter 2.) One significant change that would occur if these amendments were adopted, would be the extension of the compliance date for meeting ambient air quality standards to 1985. Anticipating that Congress would hold hearings during 1975 or 1976 on these proposed amendments, and that the electric utility industry would be asked to respond at the hearings, the electric utility industry formed the Clean Air Coordinating Committee (CACC) in early 1975. This Committee, which consists of representatives from the principal utility industry trade associations and both public and investorowned electric utilities, has been responsible for coordinating the major efforts of the utility industry to respond to Congress or the regulatory agencies for information regarding the views of the utility 'Letter dated March 22, 1974 from Russell E. Train, EPA Administrator. to President Richard M. Nixon, in forwarding the proposed bill, "The Clean Air Act Amendments of 1974." Ill I i y" "nri t i ^
PAGE 13
Industry on the proposed amendments (see CACC (1975)). It should be noted that as of this writing (June, 1976) the Congressional hearings have not been held, but are tentatively scheduled for Spring 1977 (after the Presidential election). We complete this section with a quote from the CACC (1975) that sunmarizes their feelings (and they represent the electric utility Industry) on the state of existing air pollution regulations: The field of clean air regulation is in a state of flux. Congress is currently reexamining the Clean Air Act and the Environmental Protection Agency's implementation of that Act to date. EPA is also reassessing its own policies under the Act. The Federal Energy Administration and EPA are about to implement the Energy Supply and Environmental Coordination Act (ESECA). The Federal Power Commission is charged with certain responsibilities with regard to the adequacy and costs of national electricity supply. The Treasury Department, the Department of Commerce and the Department of the Interior have broad responsibilities with regard to control of mounting inflation, alleviation of growing unemployment and wiser development and use of national resources. The Energy Resources Council has ongoing responsibility within the Executive Branch for coordinating national energy policy with other federal policies, such as protection of the human environment, the economic wellbeing of our citizens, national defense, the balance of trade and foreign policy, (p. 1) 1 .2 Statement of Purpose and Chapter Outline It is the purpose of this dissertation to present an economical alternative for control of ground level concentrations of sulfur dioxide from electric power plants using the method of production costing. In view of the ambiguous status of clear air regulations discussed above, it is not surprising that there are numerous alternative approaches to the control of emissions of sulfur dioxide and/or ground level concentrations of sulfur dioxide from electric power plants. We intend to present a novel approach that is an outgrowth and extension of the work of Sullivan and Hi 1 son (1975). himiH0 fiHfth*f>i
PAGE 14
Since there are so many approaches to the control of sulfur dioxide from electric power plants, we will devote Chapter 2 to a discussion of the relevant features of the inore prominent alternative approaches in existence today. In Chapter 3 we will present a novel solution to the steady state gradient transport equation that will lead to the classical Gaussian dispersion model. In Chapter 4 we will present the method of production costing and extend its theory to include certain generalized functions of the electrical output of an electric generator. Also, in Chapter 4, we will present a statistical method due to Larsen (1959) that extends the usefulness of the approach taken in this work. Finally, in Chapter 5, we will apply the model developed in Chapters 3 and 4 to a realistic size electric utility system. This will involve a computer program which is also discussed in Chapter 5. A statement on notation ends this chapter. The notation used in this dissertation is discussed as it is first encountered. This is appropriate since it is not necessary to introduce any nomenclature that is not standard mathematical notation. Two items worthy of mention are 1) vectors are always underlined; and 2) a bar over a letter indicates average value, e.g., u is read u bar and is the mathematical symbol for average or expected value of u. iii n i i i n iii III w Mn.L L IT i<"^^ i ; i r iii >
PAGE 15
CHAPTER 2 ALTERNATIVES FOR CONTROL OF SULFUR DIOXIDE EMISSIONS In this chapter we present a brief discussion of some of the relevant features of the more prominent approaches to the control of sulfur dioxide emissions from electric power plants. These control approaches can be conveniently divided into two broad categories: continuous emission controls and noncontinuous emission controls as is often done in the literature (see PEDCo (1975) and EPA (1975)). Continuous emission controls can further be divided into precombustion processes and postcombustion processes. Likewise, noncontinuous emission controls can be divided into fuelswitching or loadswitching or a combination of both. In Section 2.1 we will discuss briefly the continuous emission control techniques. Then, in Section 2.2 we will discuss the noncontinuous emission control approaches. Finally, in Section 2.3 we will present arguments for an interim (up to 1985) mix of both continuous and noncontinuous emission controls of sulfur dioxide from electric power plants, and present the method proposed in this work. 2 1 Continuous Emi ssion Control Continuous emission control techniques limit atmospheric emissions of pollutants (sulfur dioxide) to some fixed or permanent levels, usually expressed in pounds of pollutant emitted per million BTU of heat input. These levels are to be consistent with achieving State Implementation Plan emission limitations and with maintaining ambient air quality standards '" MHWM F aia .THg^g^ ^ln ai Wia i n f U SMMw aaaHB *Bi. i iT iffT ^
PAGE 16
under the worst case meteorological conditions. The continuous emission control approaches can be separated into precombustion and postcombustion methods. We will consider the precombustion processes first. We will discuss both the use of naturally occurring low sulfur fuels and the processing of high sulfur fuels before combustion. 2.1.1 Low Sulfur Fuel Naturally occurring low sulfur fuels used for electric power generation include low sulfur coal, low sulfur oil, and natural gas. Natural gas is in short supply in the U.S. and has been given the lowest supply priority for use as a fuel by electric power plants by the Federal Power Commission (PEDCo (1975)). The 1973 Arab oil embargo followed by dramatic increases in crude oil prices, has produced concern about dependence on foreign oil. This development has meant that except for some power plants already burning oil along the east and west coasts, low sulfur oil is of limited usefulness as a clean fuel. Therefore, we only consider low sulfur coal as a viable source of low sulfur fuel. There are large reserves of low sulfur coal in the Northern Great Plains, as well as in the East. The reserves in the Northern Great Plains are located far from the major eastern markets and pose transportation problems and added costs. The low sulfur coal reserves in the East are subject to strip mining legislation since much of it is located on steep slopes (EPA (1975)). The use of low sulfur coal has some environmental problems. For example, most electrostatic precipitators in use today to control particulate emissions depend on the presence of sulfur to decrease the resistivity of the particles. The removal of this sulfur by burning low sulfur coal 5 therefore, decreases the effectiveness of the precipitators.
PAGE 17
10 sometimes by a factor of ten {Oglesby and Nichols (1970)). The use of coal as a combustible fuel also creates fly ash disposal problems and dust emissions from coalhandling activities. There are derating problems associated with the conversion to low sulfur coal. Oilfired plants could only be converted if they were designed originally to burn coal and subsequently converted to oil, or if they were designed for dual oil/coal firing (PEDCo (1975)). In addition, even some units burning high sulfur coal would experience deratings in the range of 10 to 30 percent because of limited pulverizing capability (Perry (1974)). For new plants, of course, these deratings do not occur since the boiler would be designed for low sulfur coal. The sulfur content limitation for new plants is defined by the New Source Performance Standards"^ (see EPA (1971c)). For aJM other facilities the EPA (1975) states The emission limitation is determined through consideration of the following factors: (1) dispersion characteristics of the source and the surrounding area, (2) background concentration of the pollutant [sulfur dioxide], (3) total emissions rate of all sources in the area, and (4) expected rate of growth in emissions. The applicable sulfur content limitation is based on allowable emission rates. If the determination of the allowable emission rate is correct and the supply of coal with the necessary sulfur content is assumed, this alternative is better than 99 percent effective in assuring that air quality goals are achieved, (p. 21) In addition to the increased cost due to capital conversion of existing plants, and the increased effective cost due to plant deratings, there is an increased fuel cost for the low sulfur coal. The fuel cost increase is generally more acceptable to electric utilities, since these costs can be passed on to the consumer through the now familiar "fuel This limitation is 1.2 pounds of sulfur dioxide per million BTU heat input. J 1 W I > HHllj WmiWTTnMiMiMi M W
PAGE 18
n adjustment clause." Whereas capital costs can only be recovered through rate base changes which necessitate rate hearings. In summary, it is seen that low sulfur coal is a viable option for meeting emission limitations and ambient air quality standards. Its principal constraints are supply and technical problems associated with using them in certain types of boilers (PEDCo (1975)). 2.1.2 Fuel Oil Desulfurization Fuel oil desulfurization is only one of several approaches we will discuss that involve preprocessing of the fuel before combustion. Hence, it is a precombustion process. The removal of sulfur from crude oil has been demonstrated at many refineries, e.g., Phillips, Humble, Exxon, Chevron, Gulf, Standard, and others, all have installations that are in operation today. The desulfurization process is capital intensive, requiring on the order of 50 to 70 million dollars for installation. In addition, there are significant operating costs, and the process requires energy that amounts from 5 to 10 percent of the energy contained in the oil itself (Nelson (1973)). The principal drawback to the use of low sulfur oil as an alternative to meeting environmental constraints is its supply. The shortage of capital and unpredictable costs of crude oil have severely limited plans for new installations (EPA (1974)). For this reason it is probably not a viable option, except as mentioned before, at those plants on the east and west coast that are already using imported crude oil. 2.1.3 Coal Desulfurization This process involves physica l coal cleaning (sometimes called coal washing) and ch emical coal cle aning (sometimes called solvent refining). l n^ ^ ^ t i ttM^ ^ ov^ i
PAGE 19
12 This latter cleaning process is relatively new, and in fact, the availability is forecast to be in the late 1980's (EPA (1975)). For this reason we v/ill only discuss the physical coal cleaning process. This process generally involves three steps. The first step is to separate the pulverized coal by size. The second step involves "cleaning" in a wet medium where the difference in specific gravity between clean coal and impure coal is used to separate them. Finally, the fine cleaning involves the use of a froth flotation in which the separation is achieved by floating the fine coal away from the coal containing sulfur and other minerals (Bodle and Vyas (1974)). The limiting factor in this coal washing process is that only the pyritic sulfur (mineral compounds containing sulfur) is removed. The organic sulfur (bound in the coal molecule) is not affected. Even so, the EPA (1975) estimates that about 14 percent of the U.S. coal reserves can be cleaned to conform to the New Source Performance Standards. In addition, the EPA (1975) estimates another 55 percent could be cleaned to emit less than 4.0 pounds of sulfur dioxide per million BTU heat input. (The Tennessee Valley Authority (TVA) has a plant that has been assigned an emission limit of 5.2 pounds"^, while the American Electric Power System (AEP) has a plant with an emission limit of 6.0 pounds^''" per million BTU heat input.) Coal used in the preparation of coke is presently cleaned and so the technology is available. Also energy requirements and costs are modest (PEDCo (1975)). A significant drawback is that only about 14 percent Personal communication with E. David Daugherty, iVA, Chattanooga, TN, Personal commumcation with John C. Hoebel AEP, Mew York, M.Y. *W iftf i^ > t t0 m
PAGE 20
13 of the U.S. coal reserves could be cleaned to meet the Mew Source Performance Standards {1.2 pounds), and so coal cleaning is at best an interim approach or the first stage of some other approach. Also, 10 to 30 percent of the coal cleaned by weight is refuse and must be disposed of in landfills which involves environmental problems. It is, nevertheless, a viable option for the near future. 2.1.4 Coal Gasification and Liquefaction Many processes for converting coal to a (low sulfur) liquid or gas are in existence or under development today. Yet, this approach is best considered as a long term approach and not of much value to the electric utility industry in the immediate future. This will be established by citing two recent reports. The report by PEDCo (1975) states Highsulfur coals are also converted to clean fuel by gasification and liquefaction. The systems in use today, however, are generally inefficient and expensive; many have considerable potential for adverse environmental impact. Problems of low efficiency probably can be solved, principally in operation of combinedcycle systems now under intensive development, yery high costs of installation and operation, however, coupled with shortages of the required manpower and materials, could preclude competitive commercialization of these processes between now and 1990. (pp. xiixiii) Further, the EPA (1975) reports While coal gasification [and liquefaction] processes are currently in commercial use, the widescale commercial application of coal conversion is dependent upon improvements in technology, as well as easing of constraints on the construction and expansion of a synthetic fuels industry. Technological advances that increase gasifier efficiencies, ameliorate environinental impacts of the process, and decrease costs must occur. On the construction side, it is estimated that a fullscale plant would require 3 to 5 years and 1.5 million manhours to construct. This is a substantial commitment of manpower. Additional manpower with special training is required to operate the plant. Therefore it is concluded that coal conversion processes will have little impact [on electric powersystems] between now and 1985. (p. 23) y" T ^ >^wat M rUiitif^iw mirt
PAGE 21
14 As has been demonstrated by the above references, the conversion of coal to a liquid or gas for widespread use by the electric utility industry is not expected before about 1985 to 1990. Thus it is not a viable approach for the near future. (As is explained in Section 2.3, the alternative approach presented in this work is a viable option for the interim period from now to about 1985) 2.1.5 Flue Gas Desulfurization This is the first of two techniques we will discuss that involve the removal of sulfur after the combustion process. Hence, flue gas desulfurization (FGD) is a postcombustion continuous control approach. This is the approach the EPA most strongly favors (see Section 2.3). At the present time there are about a dozen different FGD processes which have been demonstrated in fullscale commercial practice (Princiotta (1972)) Only limestone scrubbing has been successful for extended periods of time on coalfired electric power plants (EPRI (1975)). The limestone scrubbing process is a nonregenerable process, in that it produces a waste sludge that must be disposed of in an environmentally approved mannerusuany by ponding or landfills. For disposal by ponding, a 100 megawatt coalfired unit would require 0.5 acre at a depth of 50 feet each year the FGD unit is in operation (PEDCo (1975)). PEDCo (1975) reports The disposal of scrubber sludges entails potential pollution of land and water. Surface waters such as rivers, streams, lakes, and ponds can be contaminated by leaching and percolation of sludge liquor into ground water through soil and sludge storage areas. Large areas of land could deteriorate from storage of considerable amounts of sludge materials that typically contain 50 to 75 percent water. This land could be made useless by nonsettling characteristics of the sludge, (pp. 7778)
PAGE 22
15 In Japan scrubber sludges are oxidized to form fiber gypsum used in the production of vvall board; and if the economic incentives were present, the same might be possible in the U.S. (EPRI (1975)). Regenerable processes produce a marketable byproduct, usually sulfur or sulfuric acid. The most promising of these are the magnesium oxide (MagOx), the catalytic oxidation (CatOx), and the sodium solution scrubbing (WellmanLord) processes. The operations of the FGD system will require from three to six percent of a power plant's total (gross) energy input (PEDCo (1975)). Early FGD processes were unreliable because of chemical and mechanical problems; however, more recently installed systems are achieving better than 90 percent availability (EPA (1975)). In addition to sludge disposal there is the possibility of increased local concentrations of sulfur dioxide near the plant brought about by decreased buoyancy of the plume. This can be overcome by reheating the stack gas before it is emitted. The principal objection by most electric utility planners has been the immense capital costs associated with the installation of a system they do not feel is a proven technology (see Section 2.3), Nevertheless, the EPA (1974) feels "...[FGD] represents the most practical medium or long term solution to the sulfur oxides problem for a large number of coalfired power plants." (p. 9) In this work we consider FGD systems as a viable approach to controlling sulfur dioxide emissions. 2.1.6 Fluidizedbed Combustion This approach, which entails a combustion modification, is one of the most py^omising approaches for using fossil fuels in the production of electric energy. It offers the very real possibility of greater thermal efficiencies than conventional boilers. The thermal efficiency of a ffWIT Slff f r y 'twni ilj i VZ i> yr.? Mlfem>iti*'fcan>aiM<:.^^!wm>*iM
PAGE 23
16 conventional, boiler with FGD (scrubbing) is about 37 percent, and no substantial improvement is expected. For pressurized fluidizedbed boilers, thermal efficiencies of 38 percent for firstgeneration units and up to 47 percent for secondgeneration units are predicted (PEDCo (1975)). In addition, fluidizedbed boilers should be able to burn many types of fuels and combinations of fuels. For example, all ranks of coal, blends of oil, gas, rejects from coalcleaning plants, and even municipal refuse are possible fuels. The use of a sorbent, such as limestone, reacts with the sulfur dioxide formed during combustion, and if the sorbent is continuously withdrawn (oncethrough sorbent) the sulfur is effectively removed. Also proposed are systems that use separate vessels to extract the sulfur from the sorbent and then reuse the sorbent (sorbentregeneration) This latter process eliminates the disposal problem of the oncethrough sorbent. The coal fed to a fluidizedbed combustion process does not have to be as fine as that fed to a pulverizedcoal boiler so particulates (fly ash) will be more coarse and, therefore, more easily collected. Cost estimates by the EPA (1975) indicate that pressurized fluidizedbed systems should have capital cost savings of 15 to 20 percent and operating cost savings of 10 to 15 percent when compared to conventional boilers with a FGD process. For later generation pressurized systems the operating cost savings may be 25 percent. In addition, space requirements are comparable to existing plants. The EPA (1974) has a program underway to demonstrate a pilotplant pressurized system by 1982. It is hoped that by the late 1980's widespread use of these systems will be possible. Thus, as promising an approach as ^ H '^' ig ^i II I LuMi i Miinti w inri m mr > >i
PAGE 24
17 it is, the fluidizedbed combustion alternative is not considered in this work as a viable option for the near term (up to 1985). In the next section vie will discuss the noncontinuous emission control techniques. 2.2 Noncontinuous Emission Control The control approaches discussed in this section have been referred to as: intermittent control systems (ICS), supplementary control systems (SCS), dynamic emission controls (DEC), and various other names that imply noncontinuous control. A noncontinuous emission control system has been defined by the EPA (1973) as a "...system whereby the rate of emissions from a source is curtailed when meteorological conditions conducive to high groundlevel pollutant concentrations exist or are anticipated." (p. 25697) In effect, the control is accomplished by reducing emission levels selectively rather than continuously, based on pollutant dispersion characteristics to assure that ambient air quality standards are not exceeded. The Tennessee Valley Authority (TVA) calls its noncontinuous emission control system the sulfur dioxide emission limitation program (SDEL). SDEL is the most comprehensive SCS in operation today (PEDCo (1975)). It is controversial but workable, as evidenced by the successful operation at their Paradise power plant in Drakesboro, Ky. since 1969 (Montgomery et al (1975)). The SDEL program has been installed at nine TVA power plants, which have been divided Into class 1 and class 2 type programs. The class 1 programs are less complex and operate in an openloop frame; whereas, the class 2 programs are operated in a closedloop mode. The feedback is via a telemetering system that transmits realtime data from sulfur dioxide monitors located strategically throughout the geographical area affected by the sulfur dioxide emissions. A typical class 2 program m irrillJ H ll irri Wlf l rwil i' Â— i lli liaiiiriwia nimtti^Tii i =.. iw^amimmtam m n n ii rw
PAGE 25
18 involves (see Montgomery et al (1975)) 1) a sitespecific pollutant dispersion model 2) 12 sulfur dioxide ambient monitors (field) 3) sulfur dioxide emission monitors (stack) 4) at least one instrumented meteorological tower 5) a rawinsonde system 6) several minicomputers 7) a telecommunication system 8) an environmental data station to be operated by a staff of five Both the class 1 and class 2 programs are supported by the TVA meteorological forecast center in Muscle Shoals, Ala. The noncontinuous methods for reducing emissions involve fuelswitching, loadswitching, a combination of the two, and tall stacks. Tall stacks are obviously a continuous method once they are built. However, since they do not reduce emissions, and have been proposed along with SCS, they are generally thought of as a noncontinuous emission technique. 2,2,1 Fuel switching This approach involves switching to a low sulfur fuel when meteorological conditions are conducive to high ground level concentrations of sulfur dioxide. This necessitates the storage of alternate fuels and the ability to effect the switch. All of the necessary components for fuel switching are currently available (EPA (1975)). Some limitations involve the ability to switch rapidly enough to avoid violating the ambient standards, increased particulate emissions, and insignificant decreases in the total annual emissions.
PAGE 26
19 Some positive points are that this approach becomes effective as soon as the ability to switch fuels is installed; and it represents an efficient use of the scarce low sulfur fuels. 2.2.2 Loadswitching This approach is similar to fuelswitching except that when conditions are such that an emission reduction is required, the electrical load on the particular plant is switched to another plant in the interconnected electrical transmission system. Again, the components are commercially available. This system along with tall stacks is the primary basis for the SDEL program. It should be pointed out that loadswitching has long been a practice of the electric utility industry for economically loading of generating units and, also, for recovering load that has been dropped due to forced outages of generating units. Similar to the fuelswitching approach, drawbacks to this approach include questions as to the reliability of predicting when control is needed. Total emissions are not reduced and, in fact may increase, since the alternative plant may be using a "dirtier" fuel or may be more heavily loaded. 2.2.3 Tall Stac ks Any process that regularly generates a sizeable plume must use a stack (chimney). Two criteria involving "good engineering practice" have evolved: 1) The stack should be high enough to allow the plume to escape from the wakes created in the lee of local buildings and topographic features.
PAGE 27
20 2) The exit gas velocity should be high enough to allow the plume to escape from the wake of the stack itself. The first criterion is usually met by building the stack 21/2 times the height of the nearest building or topographic structure, The second criterion is usually met with stack gas exit velocities of 11/2 times the maximum expected normal wind velocities at the top of the stack. Thus a stack exit velocity of 40 meters per sec (m/sec) would be required for a wind speed of 60 miles per hour (mph) (PEDCo (1975)), The EPA (1974) has stated era's "Tall Stack Policy" encourages the use of stacks conforming to good engineering practice, which is a function of the individual facility configuration and local terrain features. In general, this policy results in stack heights sufficiently tall to minimize ground level effects caused by aerodynamic wakes, eddies, and downwash, and those caused by high winds during neutral atmospheric stability conditions. In some cases good engineering practice requires a relatively tall stack to overcome adverse terrain features. However, use of excessively tall stacks in an attempt to avoid reducing emissions merely results in dispersion of sulfur dioxide, sulfates, and acid aerosols over wide areas. Their use as a substitute for permanent [continuous] emission controls, in addition to a harmful effect on health and welfare, would be inconsistent with the ...[Clean Air Act], (pp. 1415) The use of tall stacks has frequently been misunderstood. For example, in the July, 1975 EPRI Progress Report, a feature article by Yeager (1975) is entitled "Stacks vs. Scrubbers." Stacks are a necessary part of any emission control system, continuous or noncontinuous. We will view, therefore, the use of a tall stack as a method to be used in conjunction with (not in lieu of) some other method of control. It should be mentioned that under present Federal regulations (EPA (1971c)) noncontinuous emission controls are applicable only to power plants for which construction was begun prior to the Issuance of the
PAGE 28
21 New Source Performance Standards on August 17, 1971. With a doubling rate of ten years, about one half of the power plants in 1981 will still be eligible to use noncontinuous emission controls. However, this approach at best must be considered as an interim (up to 1985) method. Two of the drawbacks common to all noncontinuous emission control strategies to date are 1) It is difficult to enforce a noncontinuous emission control system in a multi source environment, since one source cannot control the emissions of another source. 2) Insignificant decreases in total emissions are effected. Thus, increased levels of suspended sulfates and "acidrain" problems are possible. Two of the advantages common to noncontinuous strategies are 1) Noncontinuous controls are relatively inexpensive when compared to continuous strategies. 2) The control becomes effective as soon as the program begins operation, and lead times for initiation of the control are significantly less than for most continuous controls. We have discussed some general aspects of noncontinuous emission controls, and some of the specific requirements of the SDEL program which is being used by TVA (see Montgomery et al (1975)). Other models which have been proposed are Sullivan (1972), Sullivan and Hackett (1973), Gent and Lamont (1971), and Lamont et al (1975). All of these models suffer from the same kinds of limitations mentioned above. In the next section we will discuss a method proposed in this dissertation that is actually a hybrid of continuous and noncontinuous emission controls, which Incorporates some of the better features of both control approaches. ''"' *'M '*'^>g*i*^^/^ja Tiji iwi Wirj a' t) iii & M Â—i in M j j iwiacj
PAGE 29
2 3 Arguments for a National Mix of Alternatives In Section 2.2 we discussed the noncontinuous emission control approaches. All of the models mentioned were short term models. That is, their primary control effort is directed toward the 3 hour and 24 hour ambient air quality standards. Because these models only reduce emissions during adverse meteorological conditions, their principal drawback is that they do not significantly reduce anissions over the long term. A second limitation, as mentioned before, is the difficulty of enforcement in a multisource environment. The model we are proposing in this work avoids the principal objection of noncontinuous emission controls, in that it does significantly reduce long term emissions of sulfur dioxide. In this sense it is a continuous approach. Since it also involves loadswitching, fuelswitching, and tall stacks, it has elements of a noncontinuous approach. Hence, this approach is best described as a hybrid approach. We prefer to call it an economical alternative approach to all the other methods; keeping in mind it is only meant to be an interim approach valid until about 1985. Our model uses a dispersion model (Chapter 3) to predict long term air quality, based on long term emissions determined, a priori, through a probabilistic production costing mechanism (Chapter 4). That is, we simulate a power system, use a dispersion model to predict air quality for some future period (usually a month), and through the method of production costing, select the optimal mix of loadswitching and fuelswitching. Tall stacks are not a necessity, but are desirable as discussed in Section 2.2. Before applying the method of production costing, we must extend its theory to include certain other generalized functions of the electrical output of electric generators. This is done in Chapter 4.
PAGE 30
23 The use of the extended production costing method allows us to select the order of committing units based on minimal incremental additions of concentrations of sulfur dioxide at ground level. In effect, this is loadswitching; but since it is done beforehand, it avoids most of the drawbacks of loadswitching discussed in the previous section. Since with this model we specify the fuels and their maximum sulfur contents beforehand, we can consider fuel supply constraints, boiler types, and fuel costs in determining the optimal mix of fuels (fuelswitching). As is discussed in Chapter 5, the model lowers the sulfur contents of the fuels, where necessary, to operate the power system in an environmentally acceptable manner. When environmental constraints are met, the model operates in an economical (fuel costs minimized) mode. Since the model determines the optimal mix of fuels (actually sulfur contents in a specified type of fuel) beforehand, it is useful as a planning technique for fuel purchasing. In addition, effects on the economical or environmental costs to the power system can be studied relative to fuel changes, plant siting (location), stack heights, weather changes, and many other variables. Before developing the model, we want to present some arguments by prominent persons in favor of a mix of alternative approaches on a national scale rather than selecting just one or two approaches. In a letter to Secretary Morton, Chauncey Starr, President of ERPI + states The primary standards can be met by flue gas desulfurization devices such as "scrubbers", or alternatively, by an intermittent control system t Letter dated March 26, 1975 from Dr. Chauncey Starr, President of the Electric Power Research Institute (EPRI) to the Honorable Rogers C,B. Morton, Secretary of the Interior. yWytW Wf ligi'f *? T *^.^ g '> WP I ?'* i uWtf firtgHrii^ih 1**m, mMtmm m tV^ m mSa< .
PAGE 31
24 involving a mixture of fuel, operating schedules, and tall stacks, all adjusted to meet the daytoday conditions of the region. Scrubbers are roughly ten times more costly than intermittent control systems. As noted in the attached memorandum, scrubbers do have the advantage of removing sulfur from the flue gas and thus minimizing uncertain, longterm, lov/level but cumulative effects of sulfur dispersal on the regional environment. For this reason, EPA has preferred scrubbers. In either caseÂ— scrubbers or intermittent controlsthe criteria for public health can be met. In the memorandum referred to above by Dr. Starr, entitled, "Issues and Conclusions on the Use of Intermittent Control," it is stated in the conclusion Finally, an effective SOx control strategy must recognize that nearterm achievement and maintenance of existing SO2 ambient air quality standards depends on understanding the following: a.) Source impact on ground level ambient SO7 concentration is not directly related to mass emission. This encouraging circumstance provides the basis for allocating the limited quantity of low sulfur fuel and control technology available to the specific sources where it is most needed to achieve and maintain the standards. b.) Intermittent control must be applied to the majority of large utility and industrial sources not only because it is the most cost effective approach but because it is the only method which can be made available in sufficient quantity in the nearterm. c.) Extensive imposition of stack gas desulfurization for controlling sulfates is unjustified not only because the standard has not been established, but because the current stateoftheart is technically ineffective for this control function, (pp. 1011) William Lalor, Senior Vice President of Southern Services, Inc. in reference to Russell Train, the EPA Administrator, says + 'Statement of William G. Lalor, Jr., Senior Vice President, Southern Services, Inc., before the U.S. House of Representatives, Committee on Interstate and Foreign Commerce, April 2, 1974.
PAGE 32
25 Mr. Train seems to be saying that intermittent control systems are satisfactory insofar as sulfur oxides are concerned but that sulfur oxides are not the problem. Therefore, we should spend a great deal of money to control sulfur oxides on the grounds that sulfates at some level may be a problem. This makes no sense to me. I feel that Mr. Train has set up a sulfate straw man because the current uniform sulfur oxide emission standard cannot be defended, (p. 3) William Donham Crawford, President of the Edison Electric Institute, in reference to a meeting with Frank Zarb, Federal Energy Administration f Administrator, states Mr. Zarb stated that the Administration did not wish to repeat last spring's situation where the Administration had recommended several amendments to the Clean Air Act, but EPA refused to support two of them (intermittent control systems (ICS), and "no significant deterioration"). He noted that the proposals had languished before Congress ever since without hearings being scheduled. He thought better results could be obtained from Congress if the Administration presented a unified front. He said tentative agreement had been reached with Mr. Train, and that an amendment would be submitted in January along the following lines: Scrubbers would be designated as the best or ultimate technology and permanent emission reduction systems should be the goal; however, it would be recognized that more development is needed, that time extensions should be allowed, that research and development should be continued, that ICS should be authorized in the interim, and that flexibility should be permitted. He stated that FEA and EPA would testify in support of the amendment, and that the industry should be prepared to present its views fully at Congressional hearings. The meeting above was held in late December, 1974. In January, 1975 the Clean Air Coordinating Committee (CACC) was forFned (see Chapter 1). +4George C. Freemen, Jr., Special Counsel for CACC states fLetter dated December 25, 1974 from W. Donham Crawford, President of Edison Electric Institute, to Aubrey Wagner, Chairman of the Board of Directors of the Tennessee Valley Authority. Letter dated April 30, 1975 to the Honorable Frank G. Zarb, Administrator, Federal Energy Administration, from George C. Freemen, Jr., Special Counsel for the Electric Utility Industry, CACC. iiniyrhi iiiiiiTilri iini m Tinrmi r i i n Â— i t\arm 'r fciiiw 'lfTi'f>T rcr r mtr ,t
PAGE 33
26 Ue believe that in the interim, while we will find the long term answers and solutions, the government should adopt a policy that requires the electric utility industry to assure attainment of present federal air quality standards, which effectively protect public health and welfare, but at the same time permits each plant to do so through whatever mix of technology, fuels and operating strategies will do the job. This would minimize consumer costs and contribute to this nation's energy selfsufficiency. Finally, Yeager (1975), Program Manager, EPRI states The issue of tall stacks versus scrubbers is fundamentally an admini strati v.e controversy and not a technical one. This is the main reason for the delay in resolving it. In reality, a technically feasible national strategy for achieving the established sulfur dioxide (SO2) ambient air quality standards over the next decade must employ both stacks and scrubbers. Such a strategy is not only the least costly to the nation but the only one consistent with available supplies of domestic fuel and control technology, (p. 2) In the same article Yeager (1975) states In summary, a truly feasible national SOn control strategy must consider the roles of both continuous and intermittent [noncontinuous] controls and use established techniques to apply them selectively. Not only will such a strategy effect significant cost savings to the nation but it is the only one that can be used to allocate limited clean fuels and control technology over the next decade without sacrificing the present national ambient SO2 standards, (p. 16) As mentioned above (Crawford meeting with Zarb) the EPA, which had opposed noncontinuous emission controls was willing to change its position. We will close this chapter with statements of policy from the EPA (1974) and EPA (1975), which remain in effect today (June, 1976) EPA (1974) states ...temporary use of intermittent controls in carefully selected circumstances will facilitate more rapid attainment of the current primary sulfur dioxide standards without the necessity for power
PAGE 34
27 plant shutdown, will allow the continued usa of the nation's high sulfur coal reserves while control technology, which will make it environmentally acceptable is being installed, and will also allow time to increase the availability of low sulfur fuels. (p. ii) Further in the same report, the EPA (1974) states Accordingly, intermittent control is currently considered an acceptable control measure only in cases where constant emission reduction measures are unavailable, and only until such measures become available. Under this philosophy, authorized intermittent control systems are referred to as "Supplementary Control Systems," meaning that they are intended to supplement available constant emission controls, (p. 14) Finally in summarizing the report, the EPA (1974) states For a variety of reasons, it will be necessary to apply intermittent controls or tall stacks temporarily on some plants to minimize public health impacts from sulfur dioxide until adequate emission control measures can be applied. In a few cases, interim controls may be required until the early 1980's. The use of intermittent controls will be minimized and will be discontinued as soon as possible. Intermittent controls will not be sanctioned for long term use where constant control measures are available, (pp. 2728) Then in 1975 in its Report to Congress on Control of Sulfur Oxides required by the Energy Supply and Environmental Coordination Act of 1974. (ESECA), the EPA stated their latest position (EPA (1975)): EPA believes, hovjever, that a number of isolated power plants can use supplementary control systems, which require additional emission reduction on an intermittent basis, as an interim strategy for a number of years without increasing the risk to public health, (p. 36) We will now put politics aside. In the next chapter we begin the theoretical and technical development of the alternative model we have proposed.
PAGE 35
CHAPTER 3 STEADY STATE MATHEMATICAL DISPERSION MODEL In this chapter the dispersion model used throughout this dissertation is developed. While it would be desirable to have a dispersion model capable of predicting accurate ground level concentrations of sulfur dioxide emitted from electric power plants, it is not necessary that the estimates be exact. This is because decisions about the commitment order of the generating units (Chapter 4) are based on comparing alternative strategies (dispatching orders) and selecting that strategy whose incremental increase in ground level concentrations throughout a geographical area is the smallest. It is for this reason that the mathematical dispersion model developed in this chapter need not be exactindeed, the physics of the earth's atmosphere are so complex that it is unlikely that an exact model will ever be found. If the dispersion model contains a bias and that bias is consistent, then the bias will disappear through the comparison process. Thus, we are concerned with the effect on concentrations of one strategy relative to another. A gas such as sulfur dioxide emitted high into the earth's atmosphere from the stack (chimney) of an electric power plant will undergo As is commonly done throughout the electric power industry, we will use the words dispatching and committ ing of generating units interchangeably.
PAGE 36
29 considerable dispersion before reaching ground level (Turner (1970)). This dispersion is an extremely complex, physical mechanism that involves large scale, as well as small scale turbulent eddies that mix, dilute, shear, and diffuse the gas with the atmosphere. Because of the complexity of turbulent dispersion in the atmosphere, no one to date has proposed a mathematical model capable of describing all the motions that actually take place (Tesche et al (1976)). There are two general approaches to developing an atmospheric dispersion model. The two approaches are the gradient transport model credited to Pick (1855) and the statistical model formulated by Taylor (1921) and applied by Lin (1960). Both approaches have had wide acceptance by researchers and workers. Both have some limitations due mainly to certain simplifying assumptions in the interest of practicality. Both, as will be shown, lead to a Gaussian dispersion function but with different second moments. In this dissertation the gradient transport method is chosen as a starting point for developing the dispersion model. We shall arrive at the classical result in a simple but novel manner. In addition, we shall present and discuss the statistical approach and show it is identical to our derived result under certain conditions. Finally, we shall present the plume rise formulas used with the dispersion model and the formula relating gross emissions of sulfur dioxide to electrical power out of the generator. Subsequently in Chapter 4 the emission formula will be modified to account for incremental emissions due to incremental changes in the electrical power produced by a generator. Â•~r" Â— 11" li ~"i TiiiiMinrTi mill I iinrjii iiT'i Ti I hi nr im
PAGE 37
30 3 Â• "! Gradi ent Transport Model The gradient transport model is chosen because it can be derived from basic laws of physics. What happens to a pollutant after it is emitted into the atmosphere is, indeed, a complex question. The large scale winds move it downwind from its source, while small scale winds (turbulent eddies) diffuse and mix it as it is moved downwind. The atmospheric temperature, pressure, and humidity influence the movement as well as chemical reactions of the pollutant with other agents present. Sunlight, clouds, and precipitation cause reactions and may remove some of the pollutant from the atmosphere or convert it to another possibly more dangerous agent. For example, the conversion of sulfur dioxide to acidsulfates is considered by the Environmental Protection Agency (EPA) to pose a serious health problem. A precise model of dispersion would have to include all of the above processes, plus other weather variables, irregular terrain, variable sources, and types of deposition (removal from the atmosphere). While it may seem a hopeless task at this point to develop a meaningful model, it is not. This is because we are interested in concentrations averaged over a period of time. In this respect we are fortunate to have the laws of statistics in our favor. Although the individual molecules seem to move in a completely random fashion, when viewed macroscopically a definite plume pattern is evident. If one recalls viewing a visible plume, it is seen to flow away from the stack in the direction of the transporting winds (large scale), while it spreads out and mixes due to the turbulence of the eddy wind (small scale). This plume pattern that emerges is due to the laws of statistics and to physical laws such as the conservation of mass, momentum, and energy. The atmosphere must obey the conservation of mass for both the pollutant and the air. In equation form (sometimes called the equation of continuity) Â•g" ^Tn > i wM r i ft< ai iii^pi J^ 'lli^ ^^ ^f I ^ P1 iil .^ wi n T w .<^^. ^^r*v lp^i^^
PAGE 38
31 this is written as ^$+ V Â• J = (3.11) dt where x is the concentration of the pollutant in micrograms per cubic meter (ug/m ) and J is the flux of the pollutant in micrograms per square meter per second (ug/m sec). Just as ideal gases are known to diffuse from regions of higher concentrations to regions of lower concentrations, the gradient transport theory assumes that the pollutant flux will be proportional to and in the direction of maximum decrease in concentrations per unit distance. The proportionality constant is called the diffusivity constant and has the units square meters per second (m /sec). Stated in mathematical terms, the above becomes, for one dimension J = K 1^ (3.12) X X 9x where K is the proportionality constant (diffusivity constant) described A above for the xdirection. Equation (3.12) has analogies in heat conduction, neutron flows, and molecular motions. It is credited to Pick (1855) and is often termed Fickian diffusion. In three dimensions (cartesian) equation (3.12) becomes J = (^xl^ix^^%^^zlfiz' (3.13) where (a^ a a^ ) are the cartesian system unit vectors in the x, y, and X Â— y z z directions respectively. For an isotropic system (K = K = K = K) "^Any consistent set of units may be chosen. Micrograms per cubic meter has been chosen by the U.S. Government, and will be used throughout this dissertation. ,il(ff^=*^M*NM*YW>*^l,nH' T''ll'Jt>"i^ ^'* ^^ii ffr TTrt^'=Wr i ^'^lAiw<'y^g>*?^t**#lr ^*^o*f r^ if i ^ V^ i*^ '^f<*'^
PAGE 39
32 equation (3.13) becomes simply J = KV Â• X (3.14) Substitution of equation (3.13) into (3.11) yields It 3x ^ X 8x ^ 9y ^ y dy dz ^ z dz Ulb) Since the concentration of the pollutant may change in time as well as space (due to wind transport), the left hand side of equation (3.15) may be expanded as eft 9t 3x dt 8y dt 3z dt The concentration at a particular point in space (x, y, z) is seen to change with time [dx/^t], as well as with the movement of the pollutant with velocities dx/dt, dy/dt, and dz/dt. The velocity dx/dt is the velocity with which the pollutant is being transported in the xdirection, and hence, is the mean wind speed u" in the xdirection. That is dx U = rr dt Similarly v>fe define ^ dt the mean wind speed in the ydirection and dz '^ = dt the mean wind speed in the z~direction. Defining U = dx/dt is an approximation because dx/dt is, in fact, the instantaneous xvelocity '7>Ma)rsi2irM. Â•^tnn H'n*!i.a'i
PAGE 40
33 of the pollutant which would include the small scale eddy turbulence, as well as the large scale mean wind transport. Again, we rely on the law of averaged to render the instantaneous eddy effects insignificant to the mean wind effects, since in the large they tend to cancel (Slade (1968)). Their effects are not completely ignored, since their combined effects are included on the right hand side of equation (3.15). One of the principal limitations of &}]_ dispersion models is the inability to describe adequately the wind. We know the wind changes speed and direction in time and space, but to model it accurately is a problem of the first magnitude. The simplest model consists of describing the wind by three independent components averaged over time. If u represents the wind speed, it may be written as u_ = U a + 7 a + w a, X ^' Â— z where u, v, and w are the components of mean wind speed described above. We choose to use this rather inaccurate model for the wind because we are interested in concentrations representative of, first, steady state, and second, long term (a month to a year). As a result, the model for the wind is adequate, and in fact, is the one used by most workers even for short term concentrations. Below, equation (3.15) is rewritten with two additional terms, R and S (Slade (1968)) 3t 3x "^ ^ Sy "^ '' 3z 3x ^'^x 3x^ 8y ^\ 9y ^ ^ ^^2 dz^^^^^ (3.16) The presence of R in equation (3.16) is to include the rate of production (if any) of the pollutant ciue to chemical reactions, S is the source of ii fciariiw ii iTiai
PAGE 41
34 emissions (positive) and may include deposition (negative) at the position (x, y, z) in the atmosphere. Equation (3.16) is very general and many simplifications are needed to solve it. It is the basis for all gradient transport studies. It was first investigated by Schmidt (1925) and Richardson (1925). Because of the K terms in the equation, gradient transport theory is often termed Ktheory. In the next section we will attempt to solve equation (3.16) with appropriate boundary conditions. 3.2 Steady State Gradient Transport Model It is instructive to consider each term of equation (3.16). Since in this work we are concerned with the steady state solution to equation (3.16), we immediately set the first term on the left, 9x/9t = 0. The next three terms on the left are the transport terms due to large scale mean winds (sometimes called advection terms). If one recalls viewing a visible plume, the pattern that evolves is one of a plume streaming off in one predominant direction. We will take that downwind direction to be the xdirection (a^). It is assumed then, that the horizontal (a^ ) mean wind speed, v, and the vertical (a_ ) mean wind speed, w, are small compared to the downwind mean wind speed, u. Thus, the mathematical model for the steady state wind is further simplified to u=ua^ (3.21) and the direction is taken to coincide with the observable downwind direction. Inspection of equation (3,16) indicates there are two dispersion effects taking place in the xdirection. The first, on the lefthand side, is the advection or transport term due to the mean wind speed, u, "O.""LtatTll ..Uui Tr T l ~ Â•a gJ lIfW MrfÂ§ XtlUll^t.MSt,^,m^^,^lr,,zS
PAGE 42
35 and the second, on the righthand side, is the diffusion term (K^terni) due to the turbulent eddies. In the downwind direction, the advection term far outweighs the diffusion term under normal circumstances. We assume this to be true for our steady state model. Mathematically, this is equivalent to "If^^lri^li' (322) In the horizontal and vertical directions, just the opposite is true. The diffusion terms outweigh the advection terms significantly. Thus 7^Â— (K ^) f3 23^ 3y 3y ^ y dy' ^^'^ ^' and ^!^^(^l^) (3.24) 92 9z z 3z Equations (3.23) and (3.24) are consistent with the assumption leading to equation (3.21). The R term in equation (3.16) represents the chemical reactions that may take place to generate or remove the pollutant. In more complicated studies where many pollutants are being modeled, this term serves to couple many equations of the form of equation (3.16), thereby greatly complicating the problem. In this work we are modeling only sulfur dioxide. We will take R to be zero. As regards the removal of sulfur dioxide, this amounts to slight overestimates, which means being conservative in our decisionmaking process. The generation of sulfur dioxide, whether chemical or otherwise, will be included in the S (source) term. Il'>r***r "DfT ^^ KlB' Â— Mn' i < w J i t M j onM^^a^fl.^wM W iWJ fn
PAGE 43
36 The final term in equation (3.16) is the source term, S. This term represents the production and deposition of sulfur dioxide at any point (x, y, z), in the atmosphere. As explained above, the deposition is ignored, which produces conservative estimates. Electric power plant stacks are the only sources of sulfur dioxide considered here. This is because we want to evaluate and compare the effect each plant has on adding sulfur dioxide at ground level throughout some appropriate geographical area. Background contributions, such as area sources of sulfur dioxide can be included afterwards as a percentage of the total concentrations. Given a system of n stacks, there would be sources at the following points in space (xj, y^, z^), (X2, y2, Z2),...,(x^, y^, z^) where (x., y., 2.) is the location of the ith stack (i = l,2,...,n). Everywhere else S would be zero. The simplest method of handling these multiple sources is to model each separately and then superimpose the concentrations due to each source. That superposition is valid follows from the fact that pollutants regardless of their source (location) must follow the same basic dispersion laws. The use of the superposition principle means equation (3.16) need only be solved once for a given location of the cartesian coordinate system. The solution for another source at a different location simply requires a coordinate system translation and a new source strength. For a single source located at the origin (0,0,0), and all of the above simplifications, equation (3.16) reduces to ^Â•*w*w<*0TiKaf^ ^f?awiMl T M' *r raaw .*' iif/i#miiti tr sim fi^ tiiUS
PAGE 44
37 u^ 3x ar^l^'^'^^if^^^^Q'Q'^'^) (3.25) where S(Q, 0,0,0) is the source of strength, Q. (yg/sec) located at the origin. Instead of including the source in the dispersion equation, as it is in equation (3,25), it may be included in a boundary condition. The appropriate boundary condition requires that the concentration at any point in space moving (at speed u") across an infinite"^ zyplane must equal the source strength, Q. Mathematically this is equivalent to 00 00 ux dydz = Q Â• (3.26) ooJ oo and is another statement of the conservation of mass. There are two other boundary conditions that are necessary. The concentrations must approach zero at very large distances from the source, and, the concentration must be infinite at the source. The remaining assumption, before actually solving equation (3.25), involves the two eddy diffusivity constants, K and K We will assume K^ to be independent of y and K to be independent of z. This is not tantamount to assuming an isotropic atmosphere, as is frequently done. In fact, as will be shown later, K and K are both functions of the downwind spatial coordinate, x. Equation (3.25) and the three boundary conditions are rewritten below with the above assumptions incorporated "JV4*^4 (3.27) ay dZ t Physically, the yzplane would have to terminate at the earth's surface Mathematically, it is convenient to ignore the earth's surface in equation (3.25), and to include its effect through boundarv reflections. This is discussed in Section 3.4. i>ii^a^qwÂ—Banaiaw J^.,tfc*cr i^.s*
PAGE 45
38 u X dydz = Q (3.27a) lim X = (3.27b) r ^ CO lim X = (3.27c) r ^ 2 2 2 1/2 where r = (x + y + z ) Equation (3.27) and its associated boundary conditions will now be solved. There are many approaches to the solution of equation (3.27). One of the more familiar yet complicated methods is through the use of Green's function (see Lamb et al (1974)). The following solution is presented here because it is thought to be novel. This approach to the solution was arrived at independently, and was not discovered during the extensive literature search undertaken prior to writing this dissertation. The approach is to assume a product solution to equation (3.27), apply boundary condition (3.27a) to evaluate arbitrary constants, and finally, verify that the result satisfies boundary conditions (3.27b) and (3.27c). The form of the product solution is X(x,y,2) = CF(y,x).G(z,x) where C is an arbitrary constant. Substituting the assumed product solution into equation (3.27) and rearranging yields JjT7 9E_K ^1IriT^G ,, S^G, MOON oj dZ where it is understood that F is a function of y and x only and G 1s a function of z and x only. Consider tne follov/ing argument as applied to ^?Pi'jfr'*"lft'*'''y* '" ^ 'f T '' \f !^if v irn Mgfm\ ,~^y^iMi^tJ' iisMir'i^ &
PAGE 46
39 equation (3.28). For a fixed x, a change in y, and hence F, would in general change the left hand side of the equality but not the right hand side. The opposite is true for a fixed x, and a change in z, and hence G. A change on the right hand side should occur, but no change on the left hand side. For equation (3.28) to be a true equality it must be satisfied for all values of x, y, and z and not for just arbitrary values. The inescapable conclusion is that both sides of equation (3.28) must be a constant, such that coordinate changes leave the value unchanged. This constant is referred to in mathematical literature as a separation constant. It is expedient at this point to consider this separation constant zero, rather than to carry it through the manipulations only to have it lumped in with the other constants in the evaluation of the boundary condition (3.27a). This effectively separates (3.28) into the following two equations 2 9^F 9F a 9y 2 dx (3.29) and 2 B 23^ (3.210) dZ 2 Â— 2 Â— where a = K /u and 6 = K^/u. Notice that the two equations are identical in form so that only one of them has to be solved. To solve equation (3.29), again assume a product solution of the t form F(y,x) = exp(py + qx) (3.211) t It is a product since exp(a)exp(b) = exp(a+bj wi7Â™rtaiftt. i T i 'T? ii n III ii rrlii rin 'i Mi MM y Â• ifciriiiriTirr
PAGE 47
40 where exp = 2.71828..., the Naperian exponential. Substituting equation (3.211) Into (3.29) yields or 2 2 a p F(y,x) = qF(y,x) 2 2 q = a p 2 2 2 2 Let p = s so that q = a s and p = js. Then 2 2. F(y,x) = exp(:!"jsy a s x) is a solution to equation (3.29). A particular value of s yields a particular solution. The total solution is the linear combination of all possible solutions. The total solution then, is F(y,x) = exp(*jys a xs )ds which evaluates as follows: F(y,x) = F(y,x) = f 2 2 exp(a xs )[cos(ys)*j sin(ys)]ds 00 2 2 r 7 9 exp(a xs )cos(ys)dsij exp(a xs )sin(ys)ds F(y,x) = f 2 2 2 exp(a xs )cos(ys)ds (3.212) The last step follows since the first integral is an even function while the second integral is an odd function. From a table of comn^.on integrals, equation (3.212) becomes fr &'m^s^Himiiif^^^}mnm'i^u\*^
PAGE 48
I 41 F{y,x) = ^ () exp(y /4a x) Since the partial differential equation for G(z,x) equation (3.210), is identical in form to equation (3.29), we have for G(z,x) / \ l/'n'\l/2 I 2, .,,2, G(z,x) = g () exp(z /46 x) Recall that the form of the solution was assumed to be X = CF(y,x)G(z,x) thus 2 2 ^ ^ 4a2x 4a2x which upon substitution for a and 3 becomes = % (>P [f^ ^1 (3.213) Equation (3.213) can be placed in the standard form of a Gaussian function if we let ua2 K = ^ (3.214) y 2x and K, >^ (3,214a) Then X = Â§^ exp[1(^)2 \if)h (3.215) y z y z
PAGE 49
42 The arbitrary constant C can now be determined from the boundary condition (3.27a), which is repeated here for convenience u X dydz = Q (3.27a) Substitution of equation (3.215) into boundary condition (3.27a) and rearranging yields u2TrC ^ exp[l(^) ] f 1 exp[h^) ^^y dz = Q y y The integral in the brackets evaluates to v^ (from the properties of a Gaussian function). This leaves (2ir} Cu ^ exp[i(?) ]dz = Q and again, the integral evaluates to /Zii, and we have 47r^Cu = Q or C = 5Al 2 Substitution of this constant into equation (3.215) yields ""Â•Â•^i = 2% pc(^i' hip (3.216) n r'=Tfrrrii^rT>i^iifcVi^ivf7a t Â•Â•d
PAGE 50
43 Equation (3.215) is the steady state solution to the dispersion equation for a single source of strength Q at the origin. It is actually the product of a Gaussian function of y and a G=>ussian function of z. At first glance it does not appear to be a function of x. However, inspection of equations (3.214) and (3.214a) reveal that a and a are both functions of x. The rationale for substituting the a's (standard deviations) for the K's (eddy diffusivity constants) is so the Gaussian functions appear in standard form and so a comparison with the result from the statistical approach (section 3.3) will be possible. For completeness, we must verify that boundary conditions (3.27b) and (3.27c) are also met. This is easiest to do by inspecting equation (3.213). The presence of the exponential makes it obvious that X approaches zero as r approaches infinity, and so boundary condition (3.27b) is satisfied. To verify that X approaches infinity as r approaches zero (boundary condition (3.27c)), notice that the presence of the: squared distance in the exponential numerator means the exponential approaches unity as r approaches zero. Then, the x in the denominator outside the exponential is the governing quantity, and it obviously approaches infinity as r and hence, x approaches zero. Equation (3.215) is rewritten below to emphasize the product of two Gaussian functions 2 2 X(x,y,z) =^ Â— ^Â— exp[l(f) ] JÂ— exp[^;f) ] (3.217) u v^ a^ ^ "^z /2Tr a ^ ^y Equation (3.217) shows that the concentrations are proportional to the source strength, Q, and inversely proportional to the mean wind speed, il in the downwind direction, a^ Also the concentrations are Gaussian dis^:^'rm^ n mi ^u. x '^m^^Sl^m^f^%,fyiÂ€ffAf!fm>^v^^3& '#^iHri^>'*ira^\nA^2,>S%^^iia^_ '^'^tfwirAktw^
PAGE 51
44 tributed 1n the vertical, a,, and horizontal, a directions. The solution obtained above, equation (3.217), can be verified under more general conditions than were used in the derivation. The practice used here v/as to obtain the solution by appropriate mathematical manipulations without justifying each steo. In the end the solution is justified on its own merits; that is, it satisfies the partial differential equation (3,27), and boundary conditions (3.27a), (3.27b), and (3.27c). In the next section we present the statistical approach to dispersion modeling. We will see that it, too, leads to a product solution of two Gaussian functions. 3 3 Steady State Statistical Model The statistical approach to dispersion modeling of pollutants emitted into the atmosphere will, as we will see, lead to a Gaussian formulation much the same as the gradient transport theory formulation. We will present the condition under which the two lead to an identical result. The statistical approach has its beginnings in a study of turbulence of the atmosphere by Taylor (1921). Unlike the gradient transport theory, this approach does not begin by considering the physical laws of materials in the atmosphere. Instead, probability theory is utilized to evaluate the likelihood of the material being modeled traversing from source to a particular point in the atmospheric space. The concentration of the material at that point is then proportional to its probability of arriving there. The following analogy is described by Slada (1968) and is included here in a summary version to emphasize some of the underlying assumptions of the statistical approach. A classroom experiment is performed by an instructor and his students in which the instructor tosses a coin and *>l*t i^'>0rtW^'^*tVi'i^.*ifciuiih:unilri3.^
PAGE 52
45 passes it to the student to his right or left in the middle of the front row, depending on whether the coin comes up tieads or tails. That student, in turn, continues the experiment by tossing the coin and passing it over his right or left shoulder to a student in the second row, depending on the outcome of the toss. Similarly, the students continue tossing the coin and passing it back until it reaches the back row. If this process is repeated with a large number of coins, a trend begins to appear with most of the coins going to students near the middle of the last row and fewest to those near the end of the last row. There are several features of this experiment that have parallels in statistical atmospheric diffusion. First, the probabilistic nature of the travel from source to final position (point in atmosphere) as illustrated by the mechanism of tossing the coin. Second, conservation of coins (mass) must not be violated, i.e., all coins must be accounted for. Third, deposition may occur if someone puts coins in his pocket. Fourth, a distribution of coins (concentrations) is observed to occur with increasing regularity as the number of coins injected (emitted) into the experiment (atmosphere) increases. The experiment above has been mathematically formalized by Chandrasekhar (1943), who showed that the resulting distribution is Bernoulli's distribution. For large number of coins (pollutants emitted), the central limit theorem requires that the distribution approach the Gaussian function. For "spreading" in the horizontal and vertical direction and transport by the mean wind, u", in the downwind direction (a), this becomes X' ~Â— exp[l(^)^ ^_exp[. l ^T PiHrVit" i W d" i irftiifc^l.^jgotMMi ^ i t J^ y. i^ina^=ajn
PAGE 53
46 where, Q, the source strength and, u, are as a result of the conservation of mass. The standard deviations, ct^ and a are statistical properties necessary to represent this dispersion. How they are determined is discussed shortly. Notice that equation (3.31) is identical to equation (3.217). Equation (3.31) is only presented here for discussion and comparison with the solution to the gradient transport model. Its derivation can be found in Gifford (1955). Although equations (3.31) and (3.217) appear identical, there is a subtle difference in the two. Equation (3.31) is derived with standard deviations (a's) which are statistical in nature. Equation (3.217) is derived with eddy diffusivity constants (see equation (3.213)). The connection between the two is through equations (3.214) and (3.214a) repeated here for convenience and S' 2x %' h' z 2x (3.214) (3.214a) Many researchers assume the eddy diffusivity constants are independent of position as their name implies (see Lamb et al (1974), Slade (1968), Tesche et_al^ (1976)). This assumption leads to the following incorrect result, which they take to be the fundamental relationship connecting the gradient transport theory to the statistical dispersion theory dt ^^""^ 2K. i = y or 2 (3.32)
PAGE 54
47 Equation (3.32) is valid only if the assumption of the K's being independent of position is valid. We will show that assumption leads to a relationship between the a's and the xcoordinate that is not substantiated in field tests. We will develop the argument for the K and a relationship (3.214a) but the same also holds true for K and a y y equation (3.214). If K is not a function of x then ^2^ = a^x (3.33) 2 where a is a proportionality constant. This leads to 2 2 \f u a X a 77 or K^ is proportional to the mean wind speed, IF. Intuitively, we feel that the turbulence in the zdirection should not be a function only of the mean wind speed in the xdirection since they are perpendicular. Even more convincing that equation (3.32) is not true, in general, is that field tests conducted by Pasquill (1961) have shown that a follows a power law equation of the form 0^ = d^x (3.34) or o^^ ~ aV^ (3.35) f which is in direct opposition to equation (3.33) except whenever b=l/2. The parameters a and b are related to atmospheric stability to be disr>;MW7giaiLai'*JitP C ^,4 tfaca^lt^^yy< iF*Jrttt.fyu;t f .:5>^
PAGE 55
48 ciissed next. Pasquill (1962) has reported values of b ranging from 0.4 to 2.1 depending on the stability class of the atmosphere. Assuming equation (3.35) to be valid instead of equation (3.33), since it has been experimentally verified, then K must be a function of the xcoordinate, since from equation (3.214a) y a^u 2bl S^T"^ (3.36) thus u d 2, 2K J dK d ,_ 2. .Â„ Â„ ^\ dt (^z ) = ^\ + 2x ^ (3.3.7) where the last step follows since, u" = dx/dt. Equation (3.37) is seen to contain an extra term when compared with equation (3.32). Substitution of equation (3.36) into equation (3.37) yields ^(,^2)^2A^2b1^2,^A(2,_^)^2b2 d /_ 2, 2 2 ua^ ua (a/) =f+ ^(2bl) dt ^"2 Â• x X 2 d 2. ^^ dt('^z (a/) = 2b z .^i^Vi,F=mci frWP*'(> TlWWfy*Sr^a' afc,***4Srtaiawifi
PAGE 56
49 i K^) 4bK, (3.38) Again we have two equations, (3.38) and (3.32) that contradict each other unless b = 1/2, which is simply not experimentally verified. As mentioned earlier, the gradient transport theory and the statistical approach lead to identical results, providing an appropriate relationship connecting the eddy diffusivity constants and the standard deviations is used. We have shown that by choosing the eddy diffusivity constants as functions of the xcoordinate (see equation (3.36)), and adopting Pasquill's (1961) power law form for the standard deviations ^Z = ax'' (3.39) and ^y ^ ex (3.39a) leads to an appropriate connection in that the fundamental connection between the two theories, equations (3.214) and (3.214a), is satisfied. We choose the power law form for the standard deviation terms, a z and a^, since experimental field tests have shown it is an appropriate form. The parameters a, b, c, and d are functions of the stability of the atmosphere. Pasquill (1961) has categorized the atmosphere into six classes, A through F, where A is the most unstable (most turbulent) and F is the most stable (least turbulent). His original categorization was based on wind speed, cloudiness, and sunshine intensity. More recent models (Roberts et al (1970), Busse and Zimmerman (1973), and Air Quality Display Model (1969)) have retained the same categories, (A~F), r Â— .iii.vr>">^ltim 'W>)r ^o
PAGE 57
5& but use the vertical temperature gradient as a means of selecting the appropriate stability class. The stability of the atmosphere is best defined as the capability for enhancing or resisting vertical motion. While the stability is obviously a function of many meteorological paraimeters (particularly wind gustiness and vertical temperature gradients) today's models, almost universally, consider only vertical temperature gradients (lapse rates). Since we will make extensive use of the neutral stability class (D), we take this opportunity to describe it. If a volume of air is forced upward, it will encounter lower pressures and expand and cool. Theoretically, if no heat is exchanged between the volume of air and the surrounding atmosphere, the volume of air will cool at about the rate of one degree centigrade per one hundred meters vertical (TC/lOO M), This lapse rate is termed the dry adiabatic lapse rate and is characteristic of neutral stability, which means there is no tendency for the volume of air to gain or lose bouyancy. If the lapse rate is greater than the dry adiabatic lapse rate (cooling faster than one degree centigrade per one hundred meters), the atmospheric condition is termed unstable (superadiabatic) and this class tends to enhance vertical motion in the atmosphere and therefore disperses more. If the lapse rate is less than the dry adiabatic rate (subadiabatic), the atmosphere is slightly stable, and a volume of air forced upwards becomes more dense and will, as a result, experience a downward force returing it to its original height. Stable conditions do not favor dispersion, and this condition is commonly called an "inversion". Neutral conditions are most iepresentative of long term or average conditions, since they are associated with overcast skies and moderate wind speeds (Turner (1970)). ..'tfTW#<f^iM l i^ rtV "l ^I^ ^S a T? i*?yt**k
PAGE 58
St In this work we chose to use the gradient transport theory or Ktheory to derive the dispersion model, since it could be related to basic physical laws. It should be pointed out that there are those who are critical of Ktheory (see Calder (1965)). A Russian worker, Monin, (1959), refers to Ktheory as a "semiempirical" theory of diffusion. Actually, it is, since Pick's law is empirical (but then so is Newton's law). The most compelling reason for acceptance of the Gaussian formulation as valid is that a widely different approach, the statistical approach, leads to the same formulation. The acceptance of the Gaussian formulation is so widespread among researchers that all the major dispersion models in use today incorporate the Gaussian distribution. Finally, we are able to avoid the argument over which of the two approaches to the Gaussian formulation is most valid, since we are actually using a hybrid of gradient transport theory and statistical theory. Our dispersion model is based on the solution of a simplified diffusion equation and our model incorporates the statistical properties of the atmosphere through the use of standard deviations related to the xcoordinate. In the next section our steady state dispersion model is further modified to be representative of long term (monthly or seasonal) concentrations. In addition boundary reflections are considered and incorporated into our mathematical dispersion model. 3.4 Long Term Dispersion Model Up to this point in our development of an appropriate dispersion model, we have assumed steady state, i.e., variations with time are set equal to zero. The length of time for which our mathematical model 1 *r**wgtfwgBMtt(Twiriaaa
PAGE 59
52 (equation (3.217)) is valid varies from a few minutes to an hour or so, depending on the length of time the mean wind speed and direction, stability parameters, and emissions are constant. One of the main contributions of this research is through the use of stochastic production costing as a basis for the comparison of the effects of one dispatching strategy relative to another strategy on ground level concentrations of sulfur dioxide. In order for that comparison to be meaningful, the time frames of the dispersion model and the production costing model must be the same. As will be shown in Chapter 4, the appropriate time scale for the production costing model is a month, a season, or even a year. Thus, we now undertake to modify our steady state dispersion model to a long term dispersion model, where it is now understood that long term implies monthly, seasonal, or annual estimates of the concentrations of sulfur dioxide. Before developing the long term model for dispersion, it is convenient to modify the model through a coordinate translation. Equation (3.217) was derived, in Section 3.2, as if the source of the emissions were at the origin (0,0,0) of the cartesian coordinate system. Actually, for electric power plants (the only sources considered here), the emissions are from a tall stack (see Figure 1). Inspection of Figure 1 reveals that the plume actually disperses symmetrically about an "effective" stack height, H, rather than the actual stack height, H The effective stack height is defined as the height at which the centerline of the plume becomes essentially horizontal (dotted line in Figure 1). Fonnulas for determining the effective stack height are discussed in Section 3.5. It simplifies matters to place the origin of the coordinate system at the base of the stack. The source, then, is at position (0,0,H). Equation (3.217) with this coordinate system translation becomes 'frii^ ' i> a i ^r rpwi i fc50 i ii^MM wdh
PAGE 60
53 Figure 1. Coordinate System Showing the Plume Dispersing About an Effective Height in the Vertical and Horizontal Directions.
PAGE 61
54 X = ^ Z^exp [l(.^i)2] L_ exp [i(J^)2] (3.4.1 ) u /27TO2 "^z /2Tra^ ^ ''y Throughout this dissertation, we are most concerned with ground level concentrations of sulfur dioxide, and we take this opportunity to consider only ground level (z=0) concentrations. Equation (3.41) becomes X = ^ ^ exp [i(i^)2] 1exp [\{f)h (3.42) z y "^ Notice that equation (3.42) is no longer a function of z. VJe shall present empirical, formulas in Section 3.5 for determining H and IT, so that they become deterministic inputs. Thus, the only variables in equation (3.42) are x, y, and Q. Remember that a and a are functions of X. Equation (3.42) is rewritten bel ow = 1 where X(x,y,q) = ^ Â• g(H) Â• g(y) (3.43) u g(H) = ^ exp [kil)2] (3.4.33) /27ra^ z and g(y) = =Â— exp [\{^)h (3.43b) y ^ Implicit in equation (3.43) is that the xdirection is the downwind direction. Obviously, for long periods of time this direction will change many times. A particularly convenient way to represent these changes in wind direction and also wind speed is to use historical wind rose data representative of the time period for which concentration estimates are desired. Usually, 15 wind directions are assumed (16 major compass points). The wind rose, then, gives for each wind direction the frequency of *""Â• Â— ^firl Â• Â— i Tm i T i ii i t Tnr' i i itriinl r mt fl mi u
PAGE 62
55 occurrence and the mean wind speed. For a good first order approximation, the neutral stability class D may be assumed (Turner (1970)). Since 16 wind directions have been assumed, it is convenient to think of the stack as located at the center of a circle of radius, x, and the circle to consist of 16 sectors with each sector centered along one of the 16 major compass points. Consider, for example, the sector shown in Figure 2. The wind rose will give information about the frequency of the wind and its speed, for example, in the direction of east. Thus, to obtain a meaningful estimate of the concentration as a function of x in the eastern direction, it is necessary to consider the average concentrations over the crosswind (horizontal) distance from y= d/2 to y=d/2. We will define this crosswind average concentration later, but for now we distinguish it from the concentration function in equation (3.43) by placing a bar over it. Thus, the crosswind average concentration function is xi>^,Q). This crosswind averaging will, in effect, remove the explicit dependence of the concentrations on the ycoordinate and will make the concentrations functions of the xcoordinate (the radial distance from the stack), the emissions, Q, and the frequency, f, with which the wind blows toward that sector. So, for a given value of x and a particular wind rose, 15 concentration estimates are obtained (one for each direction). The average concentration estimate for a particular direction as a function of the radial distance, x, is the product of the crosswind average concentration, J, and the percentage of time the wind blows in that direction (the frequency, f, in that direction). Thus x(x,Q,f) f x(x,Q) (3.44) ***>*'g^g""^Â— ''^r yi Â— '^ i 1tf*rWhaii^.t^uli
PAGE 63
30 y=d/2 East y=d/2 h i Figure 2. Eastern Sector Showing Downwind and Crosswind Directions. kf ~ C'a<^,lV^^fi:::.?taL.^ ;iPt^V. Icj^iM't.
PAGE 64
57 It is important to remember that the concentration function, x, (equation (3.43)) is a function of only two random variables, Q and y, for a given value of x. All other parameters are deterministic. Then, by definition x(x,Q) = x(x,y,Q)p(Q,y) dQdy (3.45) where p(Q,y) is the joint probability density function of Q and y. For obvious physical reasons, the emissions, Q, are independent (statistically) of the ycoordinate and so p(Q>.y) = p(Q)p(y) and, then X(X:,Q) g(H) Qg(y)p(Q)p(y)dQdy y^ (3.46) which can be written as x(x,Q) = ^ 1 g(y)p(y) f Qp(Q)dO dy U J Y JQ (3.47) Note that the expression in the brackets in equation (3.47) is, by definition, the average or mean value of emissions. We shall label it Q. Then, equation (3.47) becomes X(x) = ^g(H) g(y)p(y)dy u Â•'y (3.48) where the functional dependence on Q has been dropped, since Q is deterministic. How Q is computed is discussed in Section 3.5. It is reasonable to assume that a concentration at a particular distance, y, from the x axis within the sector is as likely as a it:^yfc6^f.^biTlaiin K' 'W Wfc ffq?>ifMr*iK*^tj a B*i
PAGE 65
58 concentration at any other value of y within the sector for the same value of the xcoordinate. That is, within the sector and a particular radial distance, x, from the stack, the concentrations should be weighted equally. Then p(y) = c where c is a constant. But application of the definition of a probability density function requires d 2 d 2 c dy = 1 or c = 1/d This yields for x(x) ( \ Q g H) u ^^ 2 y Consider the change of variables 2 _J r I/y^2,^ d ^rexp [j{^) ]dy /2TTa.. ^y S= y/o^ a. then dS = ^^, since a, is not a function of y. So y J x(x)=?^f'^L d 1 exp (i s^)dS where Si = d/2ay From Figure 2, d is given by d = xe na r.. ^M^i>*4i .TT'*Bj^if,>LW'w^'^*ips'i*rf
PAGE 66
59 (In fact, d=2 tan(T^)x = .398x, whereas d^y. .393x or about a one lb 10 percent error) So ^M'^^mGis^) where g 1 ^' /2tF and 1 r2, G(S^) ^ exp (^ S'')dS h n 2ttx X 1 162a 16a y y For the assumed neutral stability class D (Pasquill (1961)) a^ Vcx^ = 0.32 x^S so S^ = 0.614 x^2 Refer to Table 2 for several values of x, S,, and G(S,) evaluated from standard mathematical tables. For distances from the stack of one kilometer or more, the function G(S, ) is for all practical purposes equal to unity. Then x(x) = 2^g(H) u or {2nr^ u >^^2 2 a^ Finally, from equation (3.44), we have X(x.f) = Â— 16 _fSL g^p [_ l(iL)2j (3_4_9j (2.)^'^^ uxa^ 2a^ ..i.m i jnr >, ^ .^t t^MS i i tift i rn f ffyvFT i yr o tfiifi: ^ ., ilTt:^^^ Â— Â•Â•^.i** fj fc?,'ia
PAGE 67
60 Table 2. Integrated Gaussian Function for Several Values of Downwind Distance. Downwind Limit of Integrated Distance Integration, Gaussian in Meters, S. Function, X G(S^) 100 1.69 .925 500 2.41 .982 1000 2.81 .994 5000 4.00 .999 >5000 >4.00 1.000 twtkrt^ f ii>^i STi wu. ii. il rt Wr'*'i''*^l''1wi^ /^ f i ifPJ Tfyrwii a ge Ai *^ ^ .''t fj *i V ii^t^\\mf\\ni i^M''iJtirt*tmti,imnMw
PAGE 68
61 Equation (3.49) is yery similar to one developed by Calder (1971), who used jDolar rather than rectangular coordinates. Equation (3.49), when modified to include boundary reflections, will be the long term dispersion model used throughout the remainder of this dissertation. In Chapter 5 the results of Chapters 3 and 4 are applied to the study of a realistic size electric power system with multiple generating units. Also, a computer program developed during this research is presented and discussed. For distances closer to the source than one kilometer, equation (3.49) must be modified. This is done also in Chapter 5 Equation (3.49) is plotted in Figure 3 as a function of the radial distance, x. It is seen to begin at zero concentration, increase to a maximum, and then fall off rapidly to zero again. As mentioned in Chapter 2, the EPA has set a radius of liability of 25 miles (40.2 km) for each point source. It is instructive to determine from equation (3.49), at what radial distance, x, the maximum concentration occurs as a function of effective stack height for a given set of meteorological conditions and emissions. Equation (3.49) is rewritten below as X=k (^)exp[1(^)2] A where k is a constant containing the fixed emissions and meteorological parameters. If x is differentiated with respect to x and the result set equal to zero, the maximum is found to occur at 1 1 '^ ^7^^ <^^^> <"^frV'iS.**^e.niP'T%J*'IVli>rfi1^(;*ittHliM
PAGE 69
62 5 10 15 20 25 Downwind Distance in Kilometers Figure 3. Plot of Concentrations Vs. Downwind Distance from the Source. Â•.'Â•P^VfrfTJP**^ ,Uf.^ l 'i ^M^i^i^^ III BMiil II 111 II ir" Â— Vi'llnrtiMr ir iiiiiV'if II
PAGE 70
63 Figure 4 shows a plot of equation (3.410) as a function of H from one hundred meters to one kilometer. The values of a and b are 0.22 and 0.78 respectively (Pasquill (1961)). Figure 4 reveals that the maximum occurs farther downwind for increasingly higher effective stack heights. Even for the unreal istically high one kilometer effective stack height, the maximum occurs well within the 25mile liability radius set by the EPA. One must keep in mind that while higher effective stack heights produce maximums farther downwind, these maximums are decreasing significantly with higher effective stacks. This is shown in Figure 5. Our long term dispersion model, equation (3.49), is valid only if the effective stack height is far enough away from the earth's surface that its presence can be ignored. Physically this can never be the case, and we must include the effect of the earth's surface in the model. The most effective way to include this physical barrier is to assume total reflection of the pollutant at the earth's surface. While some deposition at the ground is known to occur, we are being conservative in our estimates by ignoring it. The most expedient method for handling these reflections, mathematically, is through the concept of a virtual source or image source located symmetrically with respect to the ground plane, to the actual source. This technique is widely used in heatconduction theory. The translation of the source from origin (0,0,0) to effective stack height (0,03H) caused the vertical dispersion term to change from z to zH. For a (virtual) source located a distance underground of H meters, the vertical dispersion due to this second source should disperse about H, or appear in the exponential as a z+H term. Thus, the vertical dispersion term becomes (vertical dispersion) = Â— ^Â— exp [h^h + exp [i(^^] /27Ta ^ "^z "^ "^z Utrmml v lAitimUikVtmsi
PAGE 71
54 25 Mile Radius 35 SOl E o o c to o 3 o a 28 21 14 7 200 400 600 800 1000 Effective Stack Height in Meters Figure 4. Plot of Downwind Distance to Maximum Concentration vs. Effective Stack Height of Source. noMiaHoe^Mt^f^ rfopw 4. Ue* ar^:iSSt^w%:srT,^:Â—:
PAGE 72
0) o 65 J2 o 500 0) o su 400 = 300 t/1 E O Â•rÂ— M !T3 S+J C O) u c o o E E i~" X 200 100 200 400 600 SOO 1000 Effective Stack Height in Meters Figure 5. Plot of Maximum Concentration vs. Effective Stack Height. "^'^l^'rf f^fWf ^^flW^iti"*Â— T^*T** Â— ril*^y^'WlW^si.H:^" 'IwM fatf f Â— ws^? "f y ^^m'tr^ik (JlMr^r iJMil ilKÂ£.=^=ic<*.,^r> r MitmS^v^rjstisnim
PAGE 73
66 Reflections of the pollutant may also occLir off a stable layer of air above the stack (inversion) in much the same way. If the height of the inversion (sometimes called mixing height) is H then we can include its effect by assuming a virtual source at !i + (H H) = 2 H H, mm m Then the vertical dispersion exponents have the following three terms: zH (due to the actual source), z+H (due to ground level reflections), and Z2H +H (due to inversion layer reflections). There can also be reflections off the inversion layer after reflections off the ground. This leads to a term, z2H H. Likewise, there can be reflections off the ground after reflections off the inversion layer. This leads to a term, z+2Hj^H. Consider the case where reflections occur, first off the ground, second off the inversion, and third off the ground. This leads to a term, z+211 +H. The symmetry is complete now, since all possible combinations of signs in front of H and H have been shov/n. Additional multiple reflections are theoretically possible and each new reflection changes only the numerical coefficient on H This is because of the additional m distance traversed from ground to inversion, H Consideration of all m possible reflection combinations leads to the following infinite set of terms zH, I [z(2nH +H)], I Cz(2nH H)] n=l ^ n=l ^ z+H, I [z+(2nH +H)], I [z+(2nH H)] n=l ^ n=l ^ It is important to remember that these terms actually appear in exponentials, and we have a sum of exponentials rather than the sum of the terms above. Since we are primarily concerned with ground level concentrations, we let z=0 and after some grouping of like terms, we have ==s5ni5^?^i>^3ia.'^ *>*'Â—'Â• wi L i ii i ii ,i iMaiHri j i<.,_
PAGE 74
57 Equation (3.4s3a) is now changed to include reflections and is written g(H) =Â— ^gR(H) (3.411) Our long term dispersion model including reflections is written below in its entirety for future reference ^^;p7?V^ exp[2(5^)]. 2nH^H 2nH +H I exp[i(^L)2] exp[i(JL_)2] (3.412) n1 z z In the computer program discussed in Chapter 6, it is seldom necessary to include terms beyond n=3, since the large exponents cause the exponentials to approach zero very rapidly. An internal check in the program compares values of the infinite summation terms to the first term (n=0 term) beginning with n=l Values that do not contribute more than 0.1 percent are discarded. Inversion layers aloft are not typical of long term meteorological conditions. The most useful long term dispersion model, then, includes ground level reflections but not inversion layer reflections. This can be accomplished in equation (3.412) by setting H^^ = which causes the infinite summation to vanish, leaving v^Y f\ 2.03fQ r "!/ H ^2. ^' ^ "^ ^^p!^" ?V^ ^ (3.413) Z z Equation (3.413) is the one we will use most frequently in modeling the pollutant dispersion from electric power plants. TTi ^ ..p ? j ff^ *y^ inrn iT rTT wl*. ti ra i a fc gii.#>it i
PAGE 75
68 In the next section the plume rise formulas are presented. A formula for calculating u is presented, and the techniques for calculating gross emissions is developed. 3.5 Plume Rise, Wind Speed at Stack Height, and Emissions Formulas We begin this section with a discussion of the many plume rise formulas in use today. Inspection of Figure 1 shows that there are two distinct regions of the plume dispersion. In the first region, the plume rises vertically from the stack, due to its stack exit velocity, to some higher position termed the effective stack height. From there the prevailing mean wind carries the plume downwind while it disperses. It is important to remember that even though the plume enters the atmosphere at (0,0,H^), the actual stack height, it is treated mathematically as if it entered the atmosphere at (0,0,H), the effective stack height with zero vertical velocity. The early attempts to estimate the effective stack of a buoyant plume were primarily empirical formulas based on observation. The point at which the plume centerline becomes essentially horizontal is highly subjective and explains why there have been so many different plume rise formulas developed over the years. Tesche et al (1976) quotes Briggs (1969) as saying, "There were over 30 documented formulas as of 1969, and he [Briggs] estimated that two more would be added each year," Disappointed by the vast differences (sometimes as much as three to one) in estimates of plume rise, researchers began attempting to develop plume rise models using rigorous physical laws. Two general approaches developed and it was encouraging, at first, that both led to the same "twothirds power law." The first approach, proposed by Morton et al (1956), uses entrainment theory, rfhile the second approach. Csanady (1955), Tinir'TTTr" n TTnr' i nT'TTl i ii fTn i ~TW]if iiMW'Mrii T^ i M i i iiiii rii i aMnT i i iÂ— i fn'fc i iiimiiiiiMi H imh in i ini n i M nini r ni" wi
PAGE 76
69 uses gradienttransfer closure arguments. We will outline the entrainment theory, pointing out the assumptions that affect its application to our dispersion model The approach begins by equating the buoyant force of the plume to the time rate of change of the vertical momentum of the plume Trr^gp Y ^ (Tir^pw) (3.51) where r = radius of plume (m) 2 g = acceleration of gravity (m/sec ) 3 p = density of plume material (ug/m ) T = ambient temperature (K) AT = T T = temperature excess between plume and surroundings (K) w = vertical plume speed (m/sec) The plume is assumed to be adiabatic which leads to 2Â— Qu = irr uAT = constant (3.52) where 3 Q = heat emission (m K/sec) n u = mean wind speed throughout tha plume in the downwind direction (m/sec) For equation (3.52) to be strictly valid, requires that AT a Â— p T since u is assumed constant. *'>^/u^'^*^rit'^*1rr)' ^ r w ji ; >ril*1iiri'ir.^^rili^1Â— .tbi Â— ^ril Â— T iT rir^amV^^ii*n'^i
PAGE 77
70 Using equation (3.52), we can replace the lefthand side of equation (3.51) with pgQu/Tu. The righthand side can be rewritten as dz Recognizing that ^ = w, we have d / 2 \ d / 2 \ w ^ (rrr pw) = 'fipw ^ (r w) which becomes I" ^ M M n i i^Hfa > ? Â• <^^ ^ ^^^^ J i^Ai >Ty ^m^^?^ >  7Wg Ki>Jl ri i f iigj am i> ^ i inCJ i ftW*B ;: ^ n I i H L ^tW^^^Trt>>i:i# ^iM fgwiW Vg'ai^Mrvie" ^f ^ ^A 'I.i:^ill WlMttT=^i=Â£5SSi^^f:i^,::*J*rt<*:^*#M,
PAGE 78
71 Because of the adiabatic assumption, the righthand side of equation (3.55) is a constant (independent of z). Thus, we must have 2n+l = or Then or n.i 1 2 2 ^ Â„ 2 2 9Ql 5a Try + 2a tty = Â— z: ^ Tu Stty Tu so 2gQ 1/2 _,,Â„ Stty Tu where A is recognized as a constant. Since by definition, the downwind speed is Â— dx Â•^ = dt and the vertical wind speed of the plume is '^ dt their ratio is iz = w = w^zl (3.57) u u since by equation (3.56) w is a function of z. Rearranging equation (3.57) and integrating, we have T l Â• I
PAGE 79
72 ^ u A 3/2or z= (3AX)2/3^ (a2)1/3(^)2/3 2ir ^ u Substituting equation (356) and grouping terms, we have 3gQn 1/3 ^ ,,^ z = [^] [^]"'^ (3.58) 2Try Tu u Equation (3.58) shows that the height, z, the plume rises above the actual stack height is proportional to the downwind distance, x, raised to the twothirds power; hence, it is often called Briggs' twothirds power law. Equation (3.58) is often written in terms of the buoyancy flux, F, which is defined as F = gvr^ ^(3.59) where v is the velocity of the plume material, and the other terms are as previously defined. Note for the initial buoyancy flux, v=w (the stack exit velocity), while for the buoyancy flux at or near the effective stack height, v=u (the mean downwind speed). Using equation (3.59) we can rewrite equation (3.58) as 2y v u It should be pointed out that while equation (3.510) is valid in form for all x, it is of use to us only near the stack (v=w) or near the effective stack height (v=u), since v is unknown elsewhere. This is net a limitation. 1 "T'TtTl JTi .^yiÂ— iiii ^aÂ— iTTÂ— n f mffcfri'ru^lr I'ri'i'iTi 'i" fii
PAGE 80
73 actually, since for our dispersion model we are only interested in z at or near the effective stack height. Baker and Jacobs (1971) have suggested that the plume has essentially reached its effective height at a downwind distance of 10 times the actual stack height. With this modification and v^u", equation (3.510) becomes ,. (.3^)1/3 i Â„ 2/3 (3^^_Â„j o a Zy u where c 2AT F = gr u ^ Remembering that the adiabatic assumption required constant heat emission, Q^, and therefore M a ^ we note that the buoyancy flux, F, is also t "^ a constant. Since it is easier in practice to determine the parameters at stack height, we chose the initial buoyancy flux rather than the final. This means F = gr w 4j. (3.512) where it is understood that r is now the inside radius of the actual stack, and w is the stack gas exit velocity. Tesche et al (1976) report that the entrainment constant, y, is approximately 0.66 for a neutral atmosphere. To avoid confusion with the zcoordinate, in the future we will define the increase above the actual stack as Ah. Thus equations (3.511) and (3.512) become Ah'^z.i!^(M:^)V3,Â„,2/3 ^ u 'This is equivalent to assuming conservation of momentum. I' til w y t ** Â— l "* r? 1 1 p >iWw*., > 1 1 ^.MJ i i ify i .ws
PAGE 81
74 and the effective stack height is, then, given by H = Hg + Ah (3.514) While there are literally dozens of plume rise formulas in the literature, we have chosen equation (3.513) for our model, since it is partly based on physical laws. In addition, a second theoretical approach, the gradienttransfer closure (Csanady (1965)), leads to the same form. It should be remembered that the model is empirical due to the entrainment constant relationship (r = yz) The assumption of adiabatic conditions is also a limitation, although not a serious one for neutral atmospheric conditions (Liu (1975a)). In the interest of being objective, we should point out that Thomas et al (1970) have studied several plume rise models in field tests including the one presented here, and report that it has a slight tendency to underpredict the rise for light winds and overpredict for strong winds. They report, however, that "the use of the 2/3 power law formula is considered preferable, provided information for the meteorological parameters is available." We again emphasize that since our decisions will be based on comparisons, a good consistent estimate of plume rise is satisfactory for our purposes. Recently, there has evolved a new approach to modeling the plume rise phenomenon. This approach involves numerical solutions to equation (3.16), the generalized dispersion equation, near the stack, (see Shir (1970) and Liu (1975b)). This approach requires extensive computer computations and to date has not been shown to be a significant improvement over the twothirds law model we chose. This approach shows much promise, however, and may in the future replace all the empirical models. It is considered
PAGE 82
75 too early in its developmental stage to be of use to us now, but undoubtedly, this will change in the future. The mean wind speed, u, is used in both the long term dispersion model and the plume rise model. Wind speed is known to increase with height, i.e., u=u(z). Ideally, the mean wind speed for the dispersion model should be representative of the wind about the plume center! ine (effective stack height), while the mean wind speed for the plume rise model should be representative of the wind throughout the vertical region from the actual stack height to the effective stack height. To avoid this complication, most models, including the one used here, use the mean wind speed at stack height, H^. This allows for sampling the wind at one height and eliminates the errors that would arise from inaccurate determinations of the effective stack height. For the plume rise model, the entrainment coefficient, y, is determined empirically from field tests where u" is selected at the actual stack height. Likewise, for the dispersion model the stability parameters, a and b, are determined empirically with u" selected at the actual stack height. Since u at 10 meters off the ground is the most common method for reporting mean wind speeds, we need a formula to relate the mean wind speed at stack height to the mean wind speed at 10 meters. Busse and Zimmerman (1973) suggest a power law relation for the neutral atmosphere of the form H "^= "^lO ^TU)^'''^ (3.515) where ujq is the mean wind speed at 10 meters, and the other parameters are as previously defined. ftk'^CiiiV'SE.trMVSM'., ', ^ iuaA^c^lRi^KkJ\l.^BÂ£j^>ni:
PAGE 83
75 Finally, we complete this section by presenting the formula for determining gross emissions of sulfur dioxide from the stack. Several definitions will be presented first. t Heatinput can be defined as the rate of heat, in BTU's/hr necessary to produce a given amount of electrical power out of the generator. Several of the emissions standards set by the U. S. Government are in terms of pounds per million BTU heatinput (see Chapter 1). In mathematical terms, the heatinput, Hj, is H, = P H^ where P = electrical power generated in megawatts, (MW) H = heat rate in BTU's per 10^ watthour (BTU/KWH) As an example, a 500 MW output for a unit having a 9,000 BTU/ heat rate requires Hj = ( 500x1 0^w)( 9000 ^) Hj = 4.5x10^ BTU/H or four and onehalf billion BTU's per hour. If this were a coalfired unit, the coal consumption, C would be "l where H is the heating value of the coal in BTU's per pound. For this We regret this sudden shift from the Mi(S system of units to the British system of units, but the use of BTU's is so widespread in the electric power industry that the use of Joules or Wattseconds would be distracting. +4" Even the U.S. Government is inconsistent, since it uses the MKS system 3 for concentrations (ug/i^ ) P ig nH WW g Â— iJH **f^ .:y t T"^ f^ c a)=Wiiii4< H tnmt i Mr i i iM iri i r ii immii iim iiBiiimii i mh i h I im nin ii i ., v;; aM M LMi ^ eg a fai;rf]iÂ£paftftjcfcsiaKgr.*t*w^
PAGE 84
11 unit using a coal having 12,000 BTU's per pound, the coal consumption would be P Â„ 4.5x10^ BTU/H c 12000 BTU/lb C^ = 3.75x10^ Ib/H or 375,000 pounds per hour of coal. If this coal contained p percent sulfur then the sulfur consumption S would be C Â• p S = ^ 'c 100 where p^ is the sulfur content of the coal in percent. For this unit, if p = three percent, then the sulfur consumption is Â•3! (3.75x10^ Ib/H) Â• 3 ^C 100 S = 1.125x10^ Ib/H c or 11,250 pounds per hour of sulfur. During the combustion process, the sulfur is oxidized to sulfur dioxide (SO2). The atomic weight of sulfur is 16 while the two oxygen atoms together have an atomic weight of 16. Thus the weight of sulfur dioxide is twice that of elemental sulfur. This means that for each pound of sulfur consumed in the furnace, there will be two pounds of sulfur dioxide produced and emitted from the stack. This assumes complete (100 percent) oxidation. Thus, for this unit, the sulfur dioxide produced, Q, is given by t The EPA allows certain furnaces 95 percent oxidation, or a weight factor of 1.9 rather than 2.0. Our choice of 2.0 is in keeping with a conservative estimate. ira iW iM i a^iii I 'miT i i ii T t Tr i a Â— I. m ri7i 1 v ii icju>jsc.g
PAGE 85
78 Q 2 S^ Q = 2 Â• 1.125x10^ Ib/H Q = 2.250x10^ Ib/H or 22,500 pounds per hour of sulfur dioxide produced. It is desirable to have a formula for calculating gross emissions of sulfur dioxide, Q, directly. This is shown below: 20P Â• H Â• p Q = s (3.516) where all of the parameters are as defined above. Substitution of P=500, H^=9,000, p^=3, and H^=l 2,000, verifies the result as 22,500 pounds per hour of sulfur dioxide produced. Since the U.S. Government has selected micrograms per cubic meter for reporting concentration, equation (3.516) is modified to the following form (2.53xlO^)P Â• H Â• p Q = H, (3.517) where Q is now given by micrograms per second. As is shown in Chapter 6, it is necessary to have both equations (3.516) and (3.517). Ground level concentrations are determined through the dispersion model using equation (3.517), after it is modified in Chapter 4 to reflect average (long term) emissions. Emission standards, on the other hand, are computed on the basis of pounds of sulfur dioxide per million BTU heatinput, which requires the use of equation (3.516). "Fl 't I T" ** T I I'l Wn'P' 1 1 n T ^H^m bi i iii Hit tiT i ga i m i iir m lMi g''^fk* rtinfl Ti^.ar .JiniMirTrt Â•jtfar i TTfc i B J i^^ n' a\ i^'rAAriSr.^Tim^mtTm.
PAGE 86
CHAPTER 4 THE METHOD OF PRODUCTION COSTING In this chapter we present the method of production costing as it is used by today's electric utility companies for estimating the long term fuel costs of the supply of electrical energy. We will extend the theory to include other functions of the output of the generating units. In particular, we will be interested in the application of this extension to the estimation of long term emissions of sulfur dioxide. The production costing method, then, is applied to the problem of determining the best estimates of long term emissions of sulfur dioxide from each source (generating unit), and these estimates are used as inputs to the mathematical dispersion model presented in Chapter 3. Through the use of the production costing method and the dispersion model, decisions can be made based on the order of dispatching generating units that minimize the incremental deterioration of ground level air quality throughout some appropriate geographical region. In the past the electric utilities have experienced exponential growth rates due to the everincreasing demand for electric energy. A doubling rate of ten years has not been uncomm.on, and some areas, for example Florida, have experienced doubling rates closer to seven years. Faced with these short lead times, utility system planners have relied heavily on larger and larger capacity generators. Along with these increased capacities has naturally been an increased complexity in design and operation of these large units. As manufacturers of electric generators 79 T t'\ \i!M m III Â—iiilir miWa lit ir'i lir T w ">^T' fli'iiW['l'iri?i'ii7f i l i'i r'" i ni liilT'rTiiiilM'i"Â— rfin'i r l ar dim i
PAGE 87
80 pushed the limits of their technologies, a new phenomenon began to emergethat of an increasing unavailability of these units due to forced outages brought about by malfunctions in boilers or auxiliary parts. It became obvious during the early sixties that a technique for incorporating these forced outage rates into the various system planning models had to be developed. In 1967, French researchers (Baleriaux et al (1967)) introduced a probabilistic simulation technique, that included forced outage rates for estimating the fuel costs of operating various electric generating units within an electric utility system. Hydroelectric and pumpedstorage projects were assigned fictitious fuel costs based on operating costs for purposes of comparison. Booth (1972) incorporated a dynamic programming algorithm with the method introduced by Baleriaux et al (1957) and introduced his generation expansion program as applied to the State Electricity Commission of Victoria, Australia. In addition to consideration of the forced outage rates of the generating units, the production costing model as applied by Booth (1972) could consider all types of generating units, i.e., nuclear, fossil, hydroelectric, gas turbine, combined cycle, and pumpedstorage. This has gained the production costing method increasing acceptance within the electric utility industry in several areas of application. Sager et al (1972) have also used the method for evaluating generation production costs, Sullivan (1974) has used the method for generation reserve planning, while Sullivan and Hilson (1975) have used it in estimating ground level concentrations of sulfur dioxide from nearby electric generating plants. In Section 4.1 we will extend the mathematical theory of production costing to include any monotonically increasing or decreasing function of wWifwip.i
PAGE 88
81 the output, P, of a generator. Since in Chapter 3 we were able to v/rite the emission rate of sulfur dioxide (equation {3.517)) as a function of the output, P, we can make use of this extension of the theory to predict long term emission rates from individual generators. In order to derive this extension, it will be necessary to take a detailed look at the mathematics of production costing. 4.1 The Load Duration Curve At the heart of production costing are three important concepts: 1. Load duration curves 2. Forced outage rates 3. Order of dispatching generating units In this section we will discuss the use of a load duration curve in estimating the long term output of a generator. We will derive the necessary mathematics to extend its use to any monotonically increasing or decreasing function of the output. One of the major contributions of this research is the extension of the production costing method to include these generalized functions of the output of individual generators. The only restriction is that these functions be monotonically increasing or decreasing functions of the output, P. Simply stated, this means there must be a one to one relationship between the function of P and the output, P. Thus, for a given value of P there must be a unique value of the function of P and vice versa. Emission rates, fuel costs, and fuel consumption are functions of the output, P, that satisfy this restriction. In practice, then, this is a restriction only in the mathematical sense. Figure 6 shows a typical load duration curve. Load duration curves are normally assembled front historical data. The abscissa represents ll ll fci iiiii i iir^Tir i^ rt iihi I ft i rflliitt ntlt ff i i iÂ— in r^liin 7 i [ ^firrr^ "ninf i tnt i^iirTrf^f^"!' .7 ';'.. "' 'r'~'';',',>7''.^f 7t&v^<^.^Ki.u^i^'aa>(Â£aier>>fi*.M^r.i
PAGE 89
82 L,max T/2 Tirrje in Hours Figure 6. Typical Load Duration Curve. MP IffCT ff'i' ^ ii^iiw rtM i i m t ^i i iM >T^yiac
PAGE 90
83 the number of hours that the system load, P. equaled''' or exceeded the ordinate value of load. Thus, if T were the total time period under consideration, then for T/2 hours the system load equaled or exceeded P p megawatts (see Figure 6). If the values along the abscissa are divided by the time period, T, the abscissa values become a fraction of time. In this case we can say that one half the time the system load equaled or exceeded P _,Â„ megawatts, or the system load equaled or L 1 / ^ exceeded P, p,Â„ with a probability of 0.5. Note that by considering the abscissa values as probabilities, we can use historical data as representative of future data for the purpose of predicting future loads, providing that the load duration curve for the time period, T, is a valid representation of the future period (see Jenkins and Joy (1974)). Generally, the shape of the load duration curves for different periods of time are the same. The actual values of the loads may be different. Most electric utilities have a fair degree of expertise in load forecasting, and in this work, we assume that the load duration curves are representative of the particular future time period under consideration. In the development of probabilistic simulation models, it has been more convenient to reverse the roles of the ordinate and abscissa of load duration curves. Figure 7 shows this "inverted" load duration curve. The interpretation, now, is that the ordinate value is the forecasted probability that the abscissa value (load) is equaled or exceeded during the time period for which the curve is valid. From this point on when we speak of a load duration curve, it should be understood that we mean an inverted load duration curve. ''"Theoretically, one should only include the number of hours Pl exceeded the ordinate value of load. In practice it is convenient to include also the number of hours Pl equaled the ordinate value of load. The distinction is important only for discontinuous load duration curves which are not used here (Sullivan (1975)). I rtinrYi Â— yj "tr i l i^ V i n i niMnmii i i '^i a f i v n rr i^ TiMi T fii m imm '^Ti'T^ti r'Tr^m m^,' in
PAGE 91
84 Â•^ 1.0 I 0.5 o sp p ^L.min L,T/2 L,max System Load in Megawatts Figure 7. Typical "Inverted'' Load Duration Curve, w iii itii i i ig riTTi iiiii PMp/ i M f.?yi^wggHf^=ww iff ?^ig w. iiWii ^grW h'i^ fP f t a i yT. r ii f I C i g Â• fttli f'i' m m f ii y mir nt >m> frf^f ^ '= v*'4':i^fÂ— *Â•*=rt''tlH^* ^J ^ >~ ^^
PAGE 92
85 Since we will be deriving the long tern estimate of functions of the generator output, P, from fundamental mathematics, it is necessary to develop the load duration curve from more basic probability theory. Figure 8 shows a typical probability density function of system loads, fi_(Pi_) > where the "L" indicates system load. The cumulative probability distribution function, F'i_^^L^' ^^ defined as (see Figure 9a) F'l(Pl) ^ fP. ^\i\U\ The backwards cumulative distribution function, F (P ), can be defined as (see Figure 9b) Equation (4.11) can be rewritten as rP, (4.11) Fl(P,) f, (P, )dP ^f, (P, )dP, (4.12) where the first integral on the right hand side of equation (4.12) is, by definition of the probability density function, equal to one. Then, equation (4.12) may be written as f.iPO fL(PL)dPL (4.13) Inspection of Figure 9b reveals that it is identical to the (inverted) load duration curve of Figure 7. Equation (4.13), then, is the mathematical description of a load duration curve. In Chapter 3 we used p() for the probability density function. From now on we use f() for the probability density function to avoid confusion with the generator outputs, P, and the system load, P, s'mjlSaifc^^^Â—.. ^i^.lJ^^ .l^:..^?.^r^wi;.j^.J' Â— ._
PAGE 93
86 o SL,min L,max System Load in Megawatts Figure 8. Typical Probability Density Function of System Loads. ifflB'TI'Ni Â•>Â•"<
PAGE 94
87 yC:iLLLqBqoJd X re to E +J Â—1 tJ re 3 re CD tu Ol c > Â•rÂ— t> a re re 1^Â— o 3 Â— : E 3 c o Â•^E E CU T3 +j SE _1 CO re O >) s Â•rÂ— oo J^ t> o 3 re XI CQ Â•1Â— ^Â—^ +> jd 00 Backward Cumulative Di ystems Loads. X T3 OO re CO E E 4J re MA +J o _J re OJ S > O Ol re r2: 3 CJ c E E Â•r3 3 C_3 U_ re o M 01 >i 00 Ol > re 3 3 cn 0) 5cn A:mLq?qoJ.d irfflfrl[f<)ifr^'Vl'i'^irfiMrTSaB*r'i V
PAGE 95
88 In order to illustrate the technique for determining the expected output of a generator from the load duration curve, consider the following argument. Refer to Figure 10. Suppose that the system generators are scheduled such that generator 1 is loaded until its maximum before generator 2 is loaded, and then generator 3 follows after generator 2 is completely loaded, and so forth. Suppose also that the maximum output of generator 1 is P, 1 ,max which is less than P. the system minimum forecasted load. Then generating unit 1 must be loaded to its maximum for the entire time period, since from the load duration curve (Figure 10), F, (P, )=1.0, or the probability that the system load exceeds the capability of unit 1 is equal to unity. In this case the output of unit 1 is deterministic rather than random. The probability density function of a deterministic value is a diracdelta function. Thus where 6() is the notation for the diracdelta function. Then the expected value of the output of unit 1 is by definition P A r "" (1 ,max P = P,f(PjdP = I P^flPjdP (4.15) where the last step follows as a practical consideration. Substitution or equation (4.14) into (4.15) and evaluating yields P, ^1 = 1 ,max P, 6(PtPt )dPT Â„ 1 M l,max' 1 u or P = P 1 l.max ?'^ F *l>ffr^*? Til" Til i T ii W i 'a 'Bw ii wii ii f li f i l liTT'iifiin^'iiiM^Tiinirirr'f itr Â• ln^'~l*^ 'Â— ^"^'^^ '" "Â— "" f^>ip:Â£n>aisra^.>ivirrs>:.t;w^n.^=.'hif^i>3<iu&nN3^
PAGE 96
89 ^ en O SQ. P P 1 ,max L ,ini n L.max System Load in Megawetts Figure 10. Load Duration CiirvG i linBUB BII nP" H rwriM > i f  ^ ^ ffl.  TlBJi iIW l  ll liaii*Wffir i
PAGE 97
90 where the last step follows from the sampling theorem for diracdelta functions. It is no surprise that the expected value of unit 1 turned out to be its maximum capability. The fact that unit 1 was dispatched first, and its maximum capability was less than the minimum system load, determined that it be fully loaded for the entire period. Notice that the area under the load duration curve in Figure 10 from to P, ^^Â„ I jmax is P, since the ordinate value is unity. 1 ,max ^ Carrying this illustration one step further, suppose that it has been determined that generating unit i will be loaded only after the first i1 units have been dispatched. How this is actually determined will be discussed in Chapter 5. This is the same as saying that unit i will not be loaded until the system load reaches Pj megawatts, where Py is the sum of the outputs of the first i1 units. In addition, as the system load grows from Pr to Py+P,Â„.Â„> unit i will be loaded from to P. ^,Â„ ^ 111 jmax 1 ,max (see Figure 11 ) In effect, the probability of dispatching unit i is F. (P,), the probability that the system load equals or exceeds P,. Likewise, the probability that unit i is fully loaded is F, (P_ + P. ). Similarly, ^ ^ J L T 1 ,max the probability of unit i being called on to supply P megawatts (0 < P, < p. ) is the probability that the system load if P^+P or a 1 5 max i a greater, which is precisely F, (P^+P ), Since P is an intermediate L i a a output between no load and full load for unit i, we know the probability of any output for unit i between no load and full load. The average value of the output of unit i, then, could be determined by multiplying each possible output by the probability it is required and summing all these products. In the limit, this average value for the output of unit i is simply the area under the load duration curve frorii P.. to P^+P. '^ I T 1 ,max nnri rrT i i "' n~'^ y i ii ii ii~nTTirTi 'iir iiirrn i wiiini i iff f iTj f M '~i la ~MrriTT riir';*f n i i sa fr iin i> i Â— i it i iri"inaB"T
PAGE 98
91 ^ o Q. T T 1 ,max System Load in Megav/atts Figure 11. Load Duration Curve Showing Commitment of Unit i
PAGE 99
G9 The important point in the above discussion is that by deciding beforehand not to allow unit i to be loaded until the system load grew beyond P^, and to load unit i to its maximum as the system load grew to '^T'*"'^i max' we have forced unit i to assume the backward cumulative distribution function or load duration curve of the system loads. In the above discussion, it was shov>/n in an heuristic manner that the expected value of unit i was given by the area under the load duration curve where unit i was placed (see Figure 11). This will be proven in the following discussion. In order to make the discussion more general, consider that it is required to find the expected value of some function (not necessarily linear) of the output of a particular generating unit. The only restriction on the function will be that it be a monotonically increasing or decreasing function of the output, as discussed earlier. In addition, suppose it is necessary to know the expected value of the function over an interval (segment) from a to b. Let the function be given by Q = Q(P)"^ where P is the output of the generating unit. By definition, the expected value of this function over the interval a to b is f^ Q = Q f(Q)dQ (4.16) 'a where f(Q) is the probability density function of Q. Since Q is a The derivation that follows is for some particular generating unit, e.g., the ith unit. However, to keep the notation from becoming too cumbersome, we will not use the subscript i, but it should be understood. fii^MtV HpHtraa n^ i tfwm ^ i ff H > w > ii tarfiwi^BMitiffca5Ca'a;aF itfinii wrnri
PAGE 100
93 function of P, equation (4.16) can be rewritten as rb Q(P)f(P)dP (4.17) where f(P) is the probability density function of the output, P, of the generator. By definition, f(P) can also be v^ritten as tfp) dPiPl ::dFiP) '\^l rl[) HP dP dp where F'(P) is the cumulative distribution function and F(P) is the backward cumulative distribution function of P (see equation (4.11)). Equation (4.17) can, then, be written as fb Q = Q(P)dF(P) (4.18) a Note that in the formulation of equation (4.18), the expected value of Q is a RiemannStijlets integral, since it is integration with respect to a function F(P), instead of a variable. Now suppose that the unit is forced to follow the load duration curve from P_ to P.+ba megawatts. That is, the unit is dispatched from a to b megawatts, only as the system load grows from Pj. to Pj.+ba megawatts. Then for the interval, Pj Pj Pj+ba, for the system load, there is a corresponding interval, a P ^ b, for the generating unit. These corresponding limits require that P = Pj_P^+a (4.19) and dP = dP^ Now since the generating unit follows the load duration curves from Pjto Pj+ba, its backward cumulative distribution function F(P), may
PAGE 101
94 be expressed in terms of the load duration curve as F(P) = [y(PLPj) u(Pj_P^b+a)]F_(P^) where y() is the step function and is related to the diracdelta function by (4.110) dP y(Pc) = 6(Pc) where c is a constant. Then by the equation above, we have for equation (4.110) dF(P) [5(PLPy)6(PLPT^^^)^ FL(PL)dPL + [li(PLPT)V'(PLPTt>+a)] dF^(P^) (4.111) Substitution of equation (4.111) into (4.18) and changing the limits to reflect the change of variables (equation (4.19) )yields Q = ! Q(PLPT+a) [6(P^_P^)6(PLPyb+a)] F_(PL)dP^_ P,+ba P,+ba Q(P^P.^+a)[u(P^P^)u(PLP^b+a)] dFjP^) T The first integral can be simplified by the sampling theorem, and the second integral can be simplified by noting that the step functions are identical to the limits of integration. Hence, they are unnecessary. This leaves Q = Q(a)F^(P^)+Q(b)F^(P.,.+ba) P +ba Q(PLP^+a)dF^_(P^) (4.112) Then by use of the partial integration formula, the integral in equation (4.112) becomes
PAGE 102
95 P,+ba r I Q(P^_P^+a)dF^(P^) = Q(b)FjP^+ba)+Q(a)Fj_(P^) P[.+ba + f F^(PL)dQ(P^P^+a) (4.113) Substitution of equation (4.113) into (4.112) and simplifying yields Q = P^+ba FL(PL)dQ(PLPT+^) (4.114) The restriction that Q and P have a one to one relationship satisfies the requirement of the partial integration formula that Q be of bounded variation on the interval a to b. Assume for the moment that the function Q is the output P of the generator. That is Q(P) = P so Then Q = P dQ. = dP and by equation (4.19) dP = dP and so equation (4.114) becomes Q = P P^+ba r T Fj_(PL)tiPL which by definition is the area under the load duration curve from Pj to Pp+ba, as earlier hypothesized. tjiy>c;ic:*.s;=;
PAGE 103
95 In Chapter 3 the emission rate of sulfur dioxide (in yg/sec) was shown to be (equation (3.517)) Q = (ZSSxIO^^)? Â• H^ Â• p^ For realistic size generating units, the heat rate, H is a function of the generator output, P. Thus the emission rate should be written as Q = k Â• P Â• H^(P) (4.115) where k is independent of P and is given by (2.53xl0^)p^ H. Differentiating equation (4.115) with respect to P yields f ^k^[PH,(P)] (4.116) The derivative on the right side of equation (4.115) is, by definition, the incremental heat rate, IHR() of the generating unit (in BTU/KWH). Returning to equation (4.114) and using equation (4.19), we have P..+ba F^(Pj_)dQ(P) which by equation (4.116) becomes Q P^+ba F^(P_)k IHR(?)dP Since P=P^Py+a, and dP=dP we finally have P,.+ba Q = k P^ IHR(PP^+a)!^(F^)dP, (4.117)
PAGE 104
97 Equation (4.117) is the expected value cf the emission rate of sulfur dioxide from a generator dispatched over the interval a to b, corresponding to the system's load duration curve from Pj to P^+ba. Note that for the limits of integration in equation (4.117) the incremental heat rate of the generator assumes values from a to b as would be expected, k'hen Q is multiplied by the time period, T, (in seconds) for which the load duration curve is valid, the result is the expected emissions of sulfur dioxide (in micrograms) the generating unit would produce over the interval a to b. This concept is used in Chapter 5 to compare the effect various generating units have on ground level concentrations of sulfur dioxide, as each additional segment of energy under the load duration curve is dispatched. The general approach taken in deriving equation (4.117) makes it easy to compute the expected value of other functions of P. For example, suppose one desires the expected cost per hour of operating unit over some range (a to b) of its output, P. Let Q(P) be replaced by C(P), the cost per hour. Next suppose this unit is placed under the load duration curve from P^ to P,+ba, then from equation (4.114) C = P +ba F^_(PjdC(PLPy+a) The cost per hour of operating the unit, considering only the fuel costs, is given by C(P) = fj, Â• P Â• H^(P) (4.118) where f is the fuel cost in dollars per 1000 BTU's (S/lO'^ BTU) and the other parameters are as previously defined. ai>>^ fTlMBl II i"" i'rTii'iT'i i'""
PAGE 105
98 Differentiation of equation (4.118) yields ^ ^ c ^ [PH^(P)] = f^IHR(P) (4.119) Again, using the change of variables, P=P, Pj+a, we have C = f P +ba IHR(P^P^+a)F_(P^)dP^ (4.120) Note equations (4.120) and (4.117) differ only by a constant. We can relate them as Q = (k/f^) C (4.121) In Chapter 5 we use equation (4.120) to compute the costs of dispatching the units over various segments, and then use equation (4.121) to compute the emissions. Note for a nuclear unit or a hydroelectric unit k=0 (since p.=0) and the emissions are zero. For this reason, we cannot s compute all the costs from the emissions, which is why we did not write equation (4,121) as C = (f^/k) Q At the beginning of this section we listed three important concepts that are fundamental to the production costing method. The first item, the load duration curve, has been discussed in this section. The second item, the forced outage rates, is considered in the next section. The final item, the order of dispatching the generating units, is taken up in Chapter 5. All of the results of this section are valid only if the generating unit is available all of the time, i.e., a forced outage rate of zero.
PAGE 106
Obviously this can never be the case, and our results must be modified to include nonzero forced outage rates. This is done in the next section. 4 2 The Effective Load Duration Curve In this section we need to extend the results of the previous section to include nonzero forced outage rates of the generating units. The follovn'ng discussion is similar to the one given by Jenkins and Joy (1974), except they considered constant heat rate curves. We will not make this restriction and so our results will be more general. Refer to Figure 12, where the units have been placed under the load duration curve in the numerical sequence, 1,2,3,4 Equation (4.120) gives the cost per hour of operating a unit over an interval a to b, under the load duration curve from P^ to P_+ba. Initially, we will be interested in the first unit to be dispatched, which from Figure 12 is unit 1. We are also interested, in this section, only in the complete loading of a unit. In the next section we again consider segmental dispatching. In this case, a equals zero and b equals P, the 1 ,max 4maximum output of unit 1. The cost per hour of unit 1 is then given by P +P Ci =fci jp ihRi(PlPt)^(Pl)^Pl (^21) T where Py is the total amount of pov/er dispatched ahead of unit 1, which in this case is zero. As mentioned before, equation (4.21) is valid only for the period of time that unit 1 is available for supplying power. If unit 1 is 'In Section 4.1 we were able to derive the results without the necessity of unit number subscripts. In this section, however, since we are considering several units we must use unit number subscripts, e.g., Pis the output of the ith unit.
PAGE 107
100 ^ o u U U n n n i i 1 t t t 1 2 3 4+ 4u n i N \ ^ \ ^ \ + \ U n i U U t n n \ 44\! System Load in Megawatts Figure 12. Load Duration Curve Showing Unit Commitments.
PAGE 108
101 unavailable (forced out of service) then all the remaining units in Figure 12 must slide to the left under the load duration curve as shown in Figure 13. During this time unit 1 cannot generate any power, since it is out of service. Let p. be the availability of unit i and q. be the unavailability (forced outage rate) of unit i. Then since the unit can only be in one of two situations (states) we have p. + q. = 1 where p. and q. are expressed as fractions less than one. The expected cost of unit 1 is less than the value in equation (4.21) by a factor p, or ^1 = Pi ^cl Pt+Pi ,max IHR^(P^_P^)F^(P^)dPL (4.22) ^T Notice that since units 2, 3, and 4, etc., are dispatched after unit 1, whether they are available or not does not affect the position of unit 1, and thus does not affect the cost given in equation (4.22). Inspection of Figures 12 and 13 reveals a different situation for unit 2. Its cost is seen to depend not only on its availability but also on the availability of unit 1. This is because during the fraction of the time period, T, that unit 1 is unavailable, q, unit 2 is shifted to the left and hence occupies a different position under the load duration curve. The cost for unit 2, then, will consist of two terms. The first term will depend on the availability of unit 1, while the second term will depend on the unavailability of unit 1. Both terms depend on the availability of unit 2, since its cost is zero while it is unavailable. The expression for the first term (Figure 12) becomes
PAGE 109
102 Its o su U U n n n i i i t t t 2 3 4 u i i + n i X \ t \ 5 \ \ U n i t U n i t 7 i System Load in Megawatts Figure 13. Load Duration Curve Showing Unit Commitments if Unit 1 is Unavailable. Â•stSHTTk ^*atai Â•Â• fr7gi'fni';'iif1'T^Vi'fn'iiYii'TTrr^ '=*r].^ ^m
PAGE 110
lo: Cg (first term) P]P2^"c2 V^2,niax (4.23) where P is now the total amount of power dispatched ahead of unit 2 and is P, Notice that equation (4.23) is similar in form to 1 ,max equation (4.22), with the additional parameter, p, For the second term, in determining the expected cost of unit 2; unit 1 must be unavailable. In this case the cost is given by (see Figure 13) CÂ„ (second term) = qip2f o 2, max IHR2(PL)F^(P^)dP^ (4.24) We can rewrite equation (4.24) by changing the limits of integration, such that CÂ„ (second term) = qip2f o P +P 1 ,max 2, max IHR(PP, )F, (P P, )dP, (4.25) 1 ,max 2 L 1 ,max L L 1 ,max L Since Pj is the total amount of power dispatched ahead of the unit under consideration, and we are now considering unit 2, Pj. must equal P, Then equation (4.25) becomes .max T 2, max ^2 (^^^"^ term) q^p^f^^  IHR2(PlPt)^(Pl' 1 ^max^^^L (4.25) where, in the interest of conformity with later results, we have left the load duration term in equation (4.26) in terms of P, ^ 1 ,max The total expected cost of unit 2 is the sum of the first and second terms, and becomes T 2, max C, Â• ,,f,, J IHR^IP^P^) [lfL^='fL {'27) iiiTMTi V'iri*^r^rrr'i"r'r^'n'i'^'T'i''J>"'i17?i*'iMtir'i~fi i tfÂ— iriiiMlr>i^ Â— att i~^iii i
PAGE 111
104 Comparing equation (4.22) and (4.27) reveals that the term in the brackets of equation (4.27) has replaced the load duration curve term in equation (4.22). For this reason it is often called the "effective" load duration curve. We can define the effective load duration curve as EFl(Pl)^=P,F,(PlH,,F^(P,P,_Â„J (4.28) where EF, (P, ) is the effective load duration curve after one unit (the **1 first unit)has been dispatched. Substitution of equation (4.28) into (4.27) yields T 2, max ^2 P2^c2 I IHR2(PlPt)EF_(P^_), dP^ (4.29) pj n where it should be remembered that P, is the total amount of power dispatched before unit 2. In this case P^ equals P, Since Pj includes the total power dispatched ahead of unit 2, whether available or not, it is often termed the effective or equivalent load. Simply stated, the equivalent load is the actual load plus the additional load on the remaining units due to the unavailability of units already dispatched (often called capacity on outage). Our formulation is unique, in that it is unnecessary to make this arbitrary definition of the new random variable, capacity on outage. We are able to avoid this additional complication through the change of limits of integration from equation (4.24) to (4.25). Figure 14 shows a plot of F, (P, ), F, (P, P. ^_), and EF, (P, ), Notice that the effective L L L L I ,max L >1 load duration curve makes the system load look larger. This is a direct result of the inclusion of the forced outage rates, q., of the generating units.
PAGE 112
105 ^ o s1 ,max System Load in Megawatts Figure 14. Load Duration Curve [F, (P, )], Shifted Load Duration Cuive [F, (P, Pn )], and Effective Load Duration Curve [EF (P. )]
PAGE 113
105 We can show this mathematicany by the following argument. Let P, = A, then we want to show 1 ,max p^F, (P_)+q^F^(P^A) ^ F^_(P^) (4.210) The following steps are selfexplanatory. q^FjP^A) ^ (lp)Fj_(P^) F^P^A) > F_(P^) (4.211) Inequality (4.211) is always true, and therefore so is inequality (4.210), since the backward cumulative distribution, F. {Â•), never increases. Thus, a shift in load backwards by A can never decrease. Notice that in Figure 14, the shift to the left in load by A (equal to Pi m=v) T" effect shifts the F, (P, ) curve to the right as is shown by t )mdX L L ''(P, Pi )Â• L L 1 .max Inspection of Figure 13 reveals that the expected cost of unit 2 is independent of whether or not the units to the right (units 3,4,...) are available. Unit 3, however, will depend on the availability of both units 1 and 2, since the forced outage of either one or both shifts unit 3 to the left under the load duration curve. Thus the cost of unit 3 will have four terms representing the four possible combinations of units 1 and 2 being available or not available for producing energy. These ternis are similar to those in determining the expected cost of unit 2. When both units 1 and 2 are available, C^ vn"ll depend on PiP2Fl(Pl)
PAGE 114
107 If unit 1 is unavailable, C will depend on qiL^VPl,max) and if unit 2 is unavailable but unit 1 is available, C", will depend on Finally, if both units 1 and 2 are unavailable, Co will depend on ^lL(V^l,max''2,max) Similar to before, we will define the effective load duration curve after two units are dispatched as E\(PL)2PlP2^L(PL)^qiP2^(PLPl,max) ^ Pil( VP2,max)^^1^2f^L^PLPl ,.axP2,.ax) Then the expected cost for unit 3 becomes T 3, max C, = p.f 3 c3 IHR3(PlP^)EFl(P^) dP, T (4.212) where Pjis the total amount of power dispatched ahead of unit 3 (to the left of unit 3 under the load duration curve), and in this case is equal to P, +P^ 1 ,max 2, max Using arguments similar to those used in developing the expected costs for units 1, 2, and 3, we can write the expected cost for the ith unit to be loaded as P^+P. T 1 ,max i ^ Pi^ci Jp ihr,(PlPt)ef,_(Pl) dP, (4.213) ^.=iiiUa/!i,si3SiÂ£Â£Siu.t .'^^Â•t^
PAGE 115
108 where Pj is the sum of all the power already dispatched, or is i1 T i^ J. max and EF, (P, ). is the effective load duration curve after the dispatching ""ji~l of i1 units, and is given by EF, (P, ) = (P.P.P^ ... P, JF (P, ) LLj_ i l. O 11 Li_ ^'i^'>2''3 l'il'\<''L''l,n,ax' + ... + {^^^^^^ ... qi.i)FL(PLPi,, axPiUmax) ^^^^^^ where the p's and q's in the parentheses above are the 2^ ~ possible combinations of the i1 units being available or unavailable. The effective load duration curve, equation (4.214), is often termed a convolution equation. This is because equation (4.214) can be derived from basic statistical mathematics, involving the integral of the product of two density functions, which is a convolution equation. Then one speaks of "convolving in" the units with the load duration curve and "deconvolving out" the units when they are removed (unavailable). We have not taken this approach, since it requires detailed mathematics that tend to obscure the results. In the approach we have taken, it is never necessary to deconvolve any units. Our approach includes, at each step, the appropriate units and thus avoids deconvolution with its wellrecognized problems. ^llliM)uM*^h in" ~r TTTi 1"Â— n Â— T^'V ,'ii'^7'7 . r ^.u .Â•!. Â•Â— ^r 'ii^ti^m a tmi ^M f j^isixai ^a^y Wii i ar^ ul ~ l en iat^sa Â— =jia^;=^wr'c. iB a iMii^r^illwf=ay^^.apT i<;~'> tfij
PAGE 116
109 In the next section, we will extend the results of this section to include dispatching segments of units rather than the entire unit. This makes the approach amenable to the comparison process to be discussed in Chapter 5. 4.3 The Effective Load Duration Curve for Segments of Units In the previous section, the forced outage rate of the ith unit was given by q., and was described as the fraction of the time period, T, that the ith unit was expected to be unavailable for supplying energy. In this section we want to consider dispatching segments of a unit as opposed to dispatching a unit's full capacity. We will not consider partial outages of generating units. If a unit is unavailable for service then a11_ its segments from no load to full load are unavailable. Likewise, if a unit is available then all its segments are available. In Section 4.2, we found that in determining the effective load duration curve, it was only necessary to consider those units already dispatched (to the left of the unit about to be dispatched under the load duration curve). Those units to be dispatched afterwards did not affect the position under the load duration curve of the unit about to be dispatched. The same interpretation can be applied here with a slight modification. That is, in forming the effective load duration curve, any unit that has had at least one segment already dispatched can affect the position of the segment about to be dispatched and, therefore must be included. Any unit that has not had any segments dispatched will be to the right of the segment about to be dispatched and cannot affect its position. Thus, it need not be included in forming the effective load duration curve. n "III mi^^ I
PAGE 117
no Suppose vie are about to dispatch the second segment of unit i. Any units that have had at least one segment already dispatched must be included in an equation of the form of equation (4.214). Unit i has already had one segment dispatched. Should it be included in the effective load duration curve? The answer lies in the answer to the following question. Does the availability or unavailability of the first segment of unit i affect the position under the load duration curve of the second segment? While on first thought it seems it does, the answer is no. This is because if the first segment is unavailable, then so is the second segment. Remember that partial outages are not being considered. Thus, if we are about to dispatch the second segment, both it and the first segment must be available, and as a result, unit i must not be included in the formation of the effective load duration curve. Most of the useful production costing models to date (see Booth (1972), Jenkins and Joy (1974), and Sullivan (1974)) form the effective load duration curve one step at a time as each new segment is dispatched. As each segment is about to be dispatched, that unit must be "deconvolved out" of the effective load duration curve. This requires an iterative procedure that takes many computations and may involve accumulating errors of the roundoff type. We are able to avoid this problem by forming, at each step, the appropriate effective load duration curve considering only those units that have had at least one segment dispatched, with the single exception of the unit about to be dispatched. Thus, for the ith unit about to be dispatched over its segment a to b: (0 a < b P. Â„,Â„) 1 ,maX we have for its expected cost P^+b C = p,f^. f IHR.(P, F^)ER (P, ), dP, (4.31) egClmVStaP^ Â— Â— .^ Â— Â— ; ^Â•Â•a^>.J.!?.'.Â— ?.r a!su:. iii.i[,Â— .^t.r^, ia>f.aj/JBKi2iKtru^ ^J TM t Jii Â•' T' tm'' 'Â— M^nrf fiM^iiltrr U^n^iiFift^V Â£! iinlln>ikc%lir
PAGE 118
Ill L^ L' + (q^P2P3 ... Pi_,)FL(PLPi) MP^q2P3 ... Pi.,)FL(PLP2) + ... + (q^qgqa ... qil)f'L^\"P^Â•Â•Â•^Â•l) (4.32) The term Pj(j=l ,2, ,il ) in equation (4.32) is the amount of power of the jth unit already dispatched. Mote that equation (4.32) is M^ry similar to equation (4.214), except P. replaces P. Also Pt is now J J jmax 1 given by i1 Pj = .1 Pj Â• (4.33) J=l ^ Equations (4.31), (4.32), and (4.33) form the basic equations used in our production costing model to determine expected values of fuel costs and emission rates of various generating units when dispatched over a particular segment of their capacity, In Chapter 5 we will apply this production costing model along with the dispersion model developed in Chapter 3 to study the effects on ground level concentrations of sulfur dioxide our dispatching scheme can have. 4.4 Estimating Short Term Maximum Concentrations In Chapter 3 we developed the mathematical model for estimating long term ambient concentrations of sulfur dioxide at ground level due to emissions from electric power plants. Earlier in this chapter, we where EFj (P, ) is now given by j i1 I
PAGE 119
112 presented the method of production costing and extended the theory to include certain generalized functions of the output of electric generators. In particular, we were interested in long term estimates of fuel costs and emissions of sulfur dioxide as a function of the position of a segment of a generating unit under the load duration curve. Since the load duration curve was representative of some time period, T, that is generally a month or longer, these estimates or expected values were considered to be long term estimates. Before discussing the application of the models of Chapters 3 and 4 to a realistic size electric utility system (Chapter 5), we want to present and discuss, in this section, a method for predicting maximum short term concentrations (on the order of hours) from the estimates of long term average concentrations of sulfur dioxide. The method was developed by Larsen (1969). We will make use of his method to extend the usefulness of our model in predicting both long term and short term estimates of ground level concentrations of sulfur dioxide. In Chapter 1 we noted that the U.S. Government through the EPA has issued both long term standards (annual averages) and short term standards (3 hour and 24 hour maximum concentrations). Although we will be primarily concerned with the long term averages, this method allows us also to report the predicted short term maximum concentrations. Thus, we could elect to control on either the short term or long term concentrations (Chapter 5). In this section we will develop those aspects of Larsen's method that we find of use in our model. The complete development and detailed derivation is given by Larsen (1971). Since Larsen's method assumes the lognormal distribution, the reader is referred to Appendix A for a discussion of the mathematics of the lognormal distribution as well as seme of its properties. ii n i^ iiiB'H'M Â— I li'ii riM ifn ili^iB I'l Â— ^ Â•..j,^ i^iiB  r trm'a t n ^MBJiltun'i if.l Pi i 'ir i^MiiinHiii^^~iinrrhli(liiriliii7TilHT~*
PAGE 120
13 Larsen (1971) in reference to his method states, "The two dominant features of this new method are derived from observations that indicate (1) air pollutant concentrations are approximately lognormally distributed for all pollutants in all cities for all averaging times; and (2) median concentration is proportional to averaging time to an exponent." (p. 1) The observations referred to above were conducted by recording continuous air pollutant concentration data for seven pollutants (including sulfur dioxide) in six cities over a threeyear period (see Larsen et a1 (1967)). During this threeyear period, Larsen et al (1967) performed a study which showed that concentrations of various pollutants follow a lognormal distribution. If one plots the logarithm of the concentrations. In x> versus the number of standard deviations, z, from the median, then a straight line results in the form In X oz+u where a is the standard deviation and is the mean of the normal distribution after the transformation (A1) from Appendix A. Thus, a is seen to be the slope of the line, and y is the intercept. Using equations (A7) and (A11), this becomes lnx=2lns+lnm 3 y or X = "IgSg^ (4.41) where m is the geometric mean, and s is the geometric standard deviation (see Appendix A). Equation (4.41) is valid only when all quantities are related to the same averaging time. Thus for 24 hour concentrations, we must have 24 hour values for m and s '^'hen we have g g ^24 ^ ''g24 '^'g24'^ ^4.42) ':>cniWiat4ii;u^an^9'Savis7=''rtf''ni'?dfl7llliÂ£i;aUEti t irrniim liTniiii <ir>Mwirr^^^'fii^Mri ^'^^^^'~^: ....^. ^ .i^sniikJt i n il i 1 1 ^im r i ii ii ibih h i hhmmtthii "nnar^
PAGE 121
114 Since we will be interested in various short term averaging times, it will be necessary to convert m and s from one averaging time to another. Two characteristics on which the method is based are stated here: 1) Median concentrations are proportional to averaging time to an exponent. 2) For the longest time period, t the arithmetic mean equals the median. Since, as is shown in Appendix A for lognormal distributions, the median equals the geometric mean, we can express characteristic one, above, as m. = kt^ (4.43) Taking logarithms of both sides of equation (4.43) we have In m = In k + P In t (4.44) g Equation (4.44) would plot as a straight line on loglog paper. Choosing three points on that line representing the total averaging time, t ., averaging time t and t, we can write three equations. These are In m tot = In k + P In t^ ^ (4.45) g tot In m = In k + P In t (4.45a) ga a In m = In k + P In t^^ (4.45b) Using characteristic two the first of these three may be written as In m = In k + P In t^^^ (4.45) where m is the mean concentration for the total time period. Using equation (4.45) and (4.45a) we can write ln(m/mg^) = P In (t^^^/t^) or cSBTt r m BW g=iaHn'aigk*'gRja . .. Â— ..Â— .. Â— <=jj,j:.^ ^^.~,r.^ ^^^,^^^j^,. .^ ^ .r=^ Â—. lh ^i^ twrr. Mflifiriii i v^, m 'iii T r r 'T'h' 7 Â• r Â— i i i r n t Â— r 1 1 ir r v nn i ff r "*n nt T~r~' r'r
PAGE 122
IIS ln(rn/m ) '"''Hot' a^ Similarly, choosing equation (4.45b) we can write ln(m/m ) P = ln(t /t ) ("^^8) '"^Hot^ b^ Using equation (A12) which is repeated here for convenience, I we have i 12[ E[x] = exp(y+ a^) (At12) Then since the mean of the concentrations is also the expected value, we have m = E[x] also recall that u = In m g and a = In s g Taking logarithms and using the above substitutions, we can write 1 2 In m = In m + 2 (In s ) (4.49) Using equation (4.49) and equating equations (4.47) and (4.48) yields (in Sg,)' ^"'Wb' I I' f or finally V = (^gb^" (4.410) i where gywffltWIPfrSII^^Ttf^tt*?^^.^ ^M ili ^BJ Ul ff?a 'li' IW ft r i^iet'ZTWÂ£Kf>^g^Ma ,^y* ifjwy>.
PAGE 123
116 Equations (4.410) and (4. 411) relate the geometric standard deviation, s_., for averaging time, t to the geom.etric standard deviation, s 9 a gb for averaging time, t^^. Also t is the total averaging tine period (usually a year). For clarity, we will present an example involving all these quantities at the end of this section. One can also develop a relationship between the geometric means, similar to equations (4.410) and (4.411). However, it is not necessary to do so, since equation (4.49) relates the geometric mean to the geometric standard deviation. Thus, we need only change the time base for s and then relate m to it, through equation (4.49). That is, rewriting 9 equation (4.49), we have % = Â—7177 Â— ^zr (^Â•412) ^^ exp[2(ln Sgg) ] The use of equation (4.41) for predicting short term concentrations depends on z, the number of standard deviations from the median that the concentration is located. This distance, in turn, depends on the relative frequency, f^, with which the particular concentration is expected to occur. For example, the highest one hour concentration in one year (8,760 hours) would occur with a frequency of For ranking the order of highest concentrations, Pearson and Hartley (1962) have suggested the following formula ^, ^ (4.41.3) itiviiNli!ii>;^^n na'af J g Eiii ; ii>i *< .^^vJ:^^gu^^.=^
PAGE 124
117 where f is the frequency (as a decimal), n is the number of samples, and r is the rank, e.g., r=l is the highest, r=2 is the second highest, etc. For our purposes equation (5.213) becomes t f = ^(r.4) (4.414) ^ ^tot where t and t (in hours) are as described earlier, a tot The number of standard deviations from the mean can then be written as lf^ z 1 2 exp (2" 2 ) "^s or 0.5f = I exp (1 s^) ds (4.415) Â•'o We have solved equation (4.415) for z, for various values of t (which relate to f by equation (4.414)) with t.^ equal to 8,760 hours (1 year). Table 3 shows the z values for the first and second highest concentrations for several values of t (in hours). The solution of a equation (4.415) is incorporated into the computer program discussed in Chapter 5, so that highest concentrations for any averaging time of interest may be predicted. \'ie will finish this chapter with two examples that illustrate the use of the pertinent equations of this chapter, as they are used in the computer algorithm of Chapter 5. Example (1 ) Suppose we are given 3 m = 45 (ug/m annual ) and iTi~t~"~y'i Â— rir"iiiii'ri'tiiiTTr i m rr "i ifr ~>i'i"t ii t i n'l 'H'^iriiiifr iiTrini Â— i in '~Ti'^ ir iii mi i r Â— i T""!' rn ir n mTtr^'rc Â•to=n"**^i**'^Mi>'i'^i,iiBirt>>*rir*i "Tir*^ *^!^
PAGE 125
118 Table 3, Number of Standard Deviations from the Median. t in hours z1 1st highest 1 3.81 2 3.64 3 3.53 4 3.46 z2, 2nd highest 3.57 3.42 3.26 3.18 '' V. m em 'tt ii i w i' ^" > f h ^t p fc*> i ^ti iiraiii jf^tt rj^msr ^ n Mi iaw> ffi ft if i ir ^ s> iw/ M. i *q w*affiitf*rriMii^wii4ifi
PAGE 126
19 Sg24 = 2.21 and we want to determine the highest and second highest threehour concentrations. Since the arithmetic mean, m, is an annual value, the total time period, t is 8,760 hours. Note that the arithmetic mean is the output of our long term dispersion model (Chapter 3), The first step is to find s^ by use of equations (4,410) and (4.411). This is done as follows s = (s )" where n r ln (8,760/3) ,1/2 Mn (8,760/24)J '^^ Then s = (2.21)^^^ = 2.51 g3 m and from equation (4.412) m = T Â— ^ ^ = 29.41 9^ exp[^(ln 2.51)^] To determine z we use Table 3 and find z (highest) = zl = 3.53 and z (second highest) = z2 = 3.26 Then from equation (4.41) X3 (highest) = mg3 {s^^f^ X3 (highest) 29.41 (2.51)^^^ or MMilMite fi i^ J wpiia m, I* i gB7 iKPir^*^wiiii[!PS1f'w>:;!T=' 'JtJ i r ii'^rs lwli.^itfT^
PAGE 127
120 X3 (highest) 752.7 (yg/m^) 3 .which compares favorably with the 1,300 (yg/m ) threehour maximum standard not to be exceeded more than once per year. For the second highest we have X3 (second highest) = 29.41 (2.51)^^^ q Xj (second highest) = 594.6 (^g/m ) Example. (2) As mentioned above. Table 3 is based on a total averaging time of one year. To illustrate its use for other total averaging times, suppose the arithmetic mean in Example (1) is for a threemonth period (2,190 hours). Further, suppose tliat we desire estimates of the highest and second highest onehour concentrations. Then we have p in (2,190/1) 1/2 Hn (2,190/24)^ "" 'Â•^' Then Sg,={Sg2,)"= (2.21)^2' S T ~ 2 82 and 45 m = ^ K= 26.33 ^ exp ij (In 2.82)^] Notice that the ratio of one hour to 2,190 hours is the same as four hours to 8,760 hours (one year). Than using Table 3 for four hours, we have z (highest) = zl = 3.46 z (second highest) = z2 = 3.18 riuin.tMiTTtt*rt
PAGE 128
121 rhus X3 (highest) = 26.33 (2.82) <3 7 0,^3.46 Xo (highest) = 945.3 ( g/m^) while X3 (second highest) = 26.33 (2.82)^^^ X3 (second highest) = 708.2 ( g/rr?) This completes this chapter. In the next chapter we put the models of Chapters 3 and 4 together into a computer program to simulate a realistic size electric power system.
PAGE 129
CHAPTER 5 DISCUSSION OF THE DIGITAL COMPUTER PROGRAM In this the final chapter, we will discuss the computer program we have developed for selecting the optimal strategy for committing electric generating units so as to meet the Federal Ambient Air Quality Standards. As pointed out in the earlier chapters, this approach involves: 1) estimating the ground level concentrations of sulfur dioxide from electric power plants through a steady state long term dispersion model (Chapter 3), 2) selecting an optimal unit commitment strategy through the extended method of production costing (Chapter 4), and 3) estimating the highest and second highest concentrations via Larsen's technique (Chapter 4). The actual technique for selecting the optimal commitment strategy has been purposely delayed until this chapter. In the next section this technique is discussed, along with the other salient features of the computer program. 5.1 The Digital Computer Program The most expedient method for discussing the computer program used in this work is to follow the sequence of steps executed by the computer and outlined in Figure 15. The flow diagram (Figure 15) begins with the input data and ends with the output results. Both of these are discussed thoroughly in Section 5.2. For these reasons the discussion here begins j with the second block in the flow diagramthe computation of the environmental coefficients. 122
PAGE 130
1?3 SET CONTROL TO AIR QUALITY NO IS CONTROL SET TO AIR QUALITY YES COMPUTE NEW SULFUR CONTENT^ SEARCH FOR BEST CASE MÂ£. ML INPUT TIIE DATA COMPUTE ENVIRONMENTAL COEFFICIENTS jskL SET CONTROL TO ECONOMICS ^ ia/SELECT UNIT COMMITMENTS CHECK FOR AIR QUALITY VIOLATIONS IS CON NONE ROL SE1 TO ECONOMICS YES OUTPUT THE RESULTS Figure 1 5. Flow Diagram of Computer Program
PAGE 131
124 The environmental coefficients are also called influence coefficients and sometimes called "x over Q" coefficients (Turner (1970)). These definitions come from equations (3.412) and (3.413). If x is divided by Q the resulting equation on the right is only a function of the meteorological parameters and the effective stack height. This is shown below for equation (3.413) X = 2^ exp [1 (f )2] (5.11) Q a. Xu ^ ^z The term on the right of equation (5.11) is independent of the emissions, Q. The concentration, due to a particular generating unit, can be determined whenever both the environmental coefficient and the emissions of the particular unit are known, thus separating the dispersion calculations from the production costing calculations. The dispersion calculations need only be made once for a given set of meteorological conditions. As soon as the input data is complete, the program computes an environmental coefficient for each grid (see Section 5.2) due to each generating unit and stores these results for future use. Next, the control is set to economics, and the program selects the unit commitments. This is done by committing the unit segments to serve the load (Chapter 4) in an order that minimizes the total system fuel costs. That is, at each step each unit in a group (Section 5.2) is a candidate for supplying the next increment (segment) of energy. Equation ( 4.31) is used in the program to compute the fuel costs of each candidate (unit), and the one whose fuel cost is the minimum is selected and placed in that position under the effective load duration curve. This process is continued until all generating units have been committed.
PAGE 132
Obviously, as the last segment of a unit is committed, it is no longer a candidate for any further positions. Since the criterion for selecting a unit's segment for position under the effective load duration curve is based on fuel costs, the control is said to be based on economics. The order in which the units are conmitted, as well as the individual segment fuel costs, are stored for future reference. The individual segment emissions can then be determined from equation (4.121), which is repeated here for convenience Q = (KfJC (4.121) These emissions are used, along with the environmental coefficients, in the next block of the flow diagram to check for air quality violations. A violation is considered to have occurred if any grid has a concentration level is excess of 60 percent of the annual primary standard of 80 micrograms per cubic meter. The selection of 60 percent is in keeping with a conservative strategy; and the actual percentage level could easily be changed to any other level desired. If no violations have occurred, the program outputs the results and terminates. If at least one violation occurs, the control is set to air quality as is shown in the flow diagram; and the program again selects unit commitments. However, in this case, the selection is based on air quality in the following manner. For each candidate for a given position under the effective load duration curve, the program computes the resulting emissions of sulfur dioxide. Next using the environmental coefficients, the program computes and sums the ground level concentrations of sulfur dioxide that would occur throughout all the grids (geographical area) for each candidate occupying that position under the effective load duration curve. That
PAGE 133
1 i J* unit's segment which produces the minimum total ground level concentrations is selected. This process is continued until all the units have been committed, thus defining the desired commitment order and associated segment fuel costs for future use. Since the individual emissions can easily be recovered through the use of equation (4.121), it is unnecessary to store them. The selection process is based on minimizing total concentrations of sulfur dioxide; hence, the control is said to be based on air quality. Next, the program checks again for air quality violations. If there are none, the control of the program is routed to the block, "search for best case." If there are violations, the control is routed to the block, "compute new sulfur contents." We will discuss both these blocks beginning with the latter block first. If there is at least one violation, there is no way to avoid air quality violations by reordering the unit commitments, since all units have been committed in an order that minimizes the total concentrations. Thus, if the energy is to be produced, the emissions, and hence the sulfur contents must be lowered. There are various schemes for deciding which units should have their sulfur contents lowered; for example, a Lagrange multiplier formulation, that minimizes increased fuel costs due to more expensive low sulfur fuels, subject to the inequality that the grid concentrations all be less than or equal to the 60 percent standard. While this method is preferred in theory, it is of little practical use here. This is because in practice only those units that do not suffer significant deratings (Chapter 2) are actually candidates for lower sulfur fuels. The decision as to which units must burn lower sulfur fuels is likely to be made on the basis of avoiding combustion problems rather than on increased fuel costs.
PAGE 134
"97 For this reason, we have selected the procedure, favored by the EPA (1975), of a direct rollback of emissions on all sources capable of burning lower sulfur fuels. As discussed in the next section, the user can specify whether a generating unit is capable of lower sulfur contents or not, as well as a minimum low sulfur content. The program assumes a three percent increase in fuel prices for each one quarter percent reduction in sulfur content. This amounts to a 16 percent increase in fuel costs for a full percent reduction in sulfur content (PEDCo (1975)). In the program the procedure is to select the grid having the highest concentration (most serious violation) and to roll back the emissions of the preselected units to bring this grid into conformance with the 60 percent standard. Next, the new sulfur contents are lowered further to the nearest one quarter percent. For example, 3.26 percent is lowered to 3.25 percent; while 3.24 percent is lowered to 3.00 percent sulfur content. This is in keeping with a conservative approach and also to avoid specifying unrealistic numbers, such as 3.17 percent. Since it is possible that some grid violations are caused by units not selected for lower sulfur contents, the program checks to insure that ari_ violations have been brought into conformance. If some violations are still found to be present, the procedure outlined in the paragraph above is repeated until all violations have disappeared. Also, when the program control is first transferred to "compute new sulfur contents," a check is made to see if the violations can be removed by placing all selected units at their specified minimum low sulfur content. (If not specified, a 0.5 percent minimum is assumed.) If violations still occur at this point, the program terminates and informs the user that it is not possible to meet the 60 percent standard under the conditions he has specified.
PAGE 135
128 Notice in the flow diagram that after computing the low sulfur coni tents, the control is transferred to the block "set control to economics."  r This is done since we are assured that no violations will occur with all : units committed on the basis of air quality, or else the program would have i terminated. Since the lower sulfur contents have been lowered to the nearest quarter of a percent, it is possible that no violations will occur with units committed on the basis of economics. This is checked; and if none are found to occur, the program outputs the results, which includes the commitment order based on economics and the new lower sulfur contents. If at least one violation is found to occur, the program transfers control to the block, "search for best case," which is discussed next. Notice that the flow diagram is drawn so that the program reselects the order based on air quality. This is because changing the sulfur contents may change the commitment order. Actually, this repeated selection is performed while in the block "compute new sulfur contents," to insure that there are no violations occuring based on both the new lower sulfur contents and the new commitment order. The final block to be discussed in the flow diagram in this section is the block, "search for best case." This block is necessary since when control is transferred to this block, two important facts are known: 1) Violations will occur if units are committed on the basis of economics. 2) Violations will not occur if units are committed on the basis of air quality. As discussed in the next section, the units are placed in groups, and each group of units is either committed on the basis of economics or air quality. If there are three Â•jroups, there are eight (2") cases
PAGE 136
129 (possibilities) to be considered. The first case (all on economics) ; i has been considered, and the eighth case (all on air quality) has also t 1 been considered. Since the program has stored the conmitment orders of [ both these cases, it is possible to consider the other six cases by [ I "swapping" in and out the commitment order of the units within groups. I i This is done in the program. ; All eight possible combinations of the three groups being committed I on economics or air quality are considered and air quality violations are checked. For those cases that do not have any air quality violations, the fuel costs are computed and the "best case" is selected as the one with the minimum fuel cost. If all the cases are found to contain air quality violations, then the case selected is the one in which all groups are committed based on air quality (case 8 for this example), and control passes to the block, "output the results." Before discussing the input and output blocks in the next section, we need to consider the dispersion model near the generating source. As mentioned in Section 3.4, it must be modified for distances closer than one kilometer to the source. We find it more convenient to make this modification here, since it goes well with the discussion of the computer program. In the program, sources are placed at the center of a square grid of dimensions It by It. For example, if r equals 2500 (m) then the grid is 5000 (m) on each side. We will consider the grid to be roughly a circle of radius r, for the purposes of obtaining the average concentration throughout the grid due to the source in the grid. This source square average concentration will be given by 16 1 ri^x(x,f.) Xsc ^ ) ^ ~ f~dx (5.12) ^i = l "^ JO 1
PAGE 137
130 where x(x,f^) is given by equation (3,412) or (3.413), r is one half the grid linear dimension, and f. is the frequency with which the wind blov/s tov/ard the ith sector. Note that the f. in equation (3.412) or (3.413) identifies the particular term as representing the ith sector. Equation (5.112) can be rewritten as X_ =M X(x)dx il f ) ss r jQ .^^ 1 or 1 Tsince the summation equals one, and X(x,f.) X(x) = ~f7^ (5.14) For the case of no inversion layer aloft equation (5.14) becomes X(x) =%^ exp [^(^)2] (5.15) Notice that in equation (5.12) it was only necessary to integrate over the xcoordinate. The integration over the ycoordinate was done in Chapter 3. Also each f. is treated as the probability of concentrations appearing in the ith sector due to the source in the source square. Hanna (1972) derives the source square long term concentration formula as ^ .i,2i, r^ ss _.Z (f?) JÂ„ '<(^'<''< (516) Ttr Â•'0 where the details can be found in Henna (1972). The ur tenri is due to the area of the area of the circle of radius r. Notice that the units of equation (3.16) are yg/m whicli appears incorrect. Perhaps the discrepancy Â£.^Â£;b?e.^&iii.~i.
PAGE 138
1 "?7 'An unpublished report available from the author. is due to summing over the area and then dividing by the area, when the crosswind distance has been summed and divided already in the averaging process of Chapter 3. We believe it is only necessary to sum and divide j by the linear distance r as we have done in equation (5.13). Notice [ 3 also, the units in equation (5.13) are correct (yg/m ). I 5 2 Discussi on of the Application of the Computer Program to a Realistic Size System I i The digital computer program discussed in this chapter has been \ written as a source program in the BASIC language. It is an interactive program in the sense that the program prompts the user for the input data [ and asks the user to make decisions about whether certain less relevant j output should be printed. The program has been operating on a Hewlett i Packard Access System 2000 timeshared minicomputer, but should be compatible with any digital computer supporting the BASIC language with i file structures. For the following discussion, the reader is referred to the report. User's Guide to ENVECO" '", where a detailed description of the use of the program is given. The program begins execution by reading the input data from a file (BASIC formatted file). The user is then given the opportunity to change any input data. Any changes to the input data are automatically recorded in the file. In this manner each successive use of the program has the user's most recent configuration. The details of constructing the user's initial configuration (also called base case) are given in the User's Guide to EMVECQ The program has been written with the flexibility to
PAGE 139
132 return to the user's base case at any time. This feature allows the user a simple procedure for undoing configuration changes in the input data  he mav no longer desire. 1 i As mentioned above as soon as the program reads in the input data, [ the user is given the opportunity to make changes in the input data. The [ procedure for doing this is discussed next. The input data are divided [ into two categories, system data and generating unit data. We will discuss [ the system data first. I I The system data are further divided into six sections. The first section is the load duration curve data. These data represent 100 ordinate  values equally spaced along the abscissa from zero to the maximum expected I i system load, P The program asks the user if he wishes to see or I L,maX 1 change any of these data. If so, he can do so; if not the program continues j to the next section, the wind rose frequencies. Beginning with this second section, each remaining section of system data are displayed at the user's terminal and the user is asked for any changes. All questions asked by the program should be answered yes or no. The program is written such that any answer other than "no" is automatically interpreted as "yes". This means that in response to a question on whether certain data should be printed, a mistaken entry (other than no) would default to yes and print out the data. Also in the input stream, a mistaken enter (other than no) leaves the user with the opportunity to enter the new data rather than have the program continue executing with the wrong (old) data. In the second section of system data, the program prints the 16 wind rose frequencies from northnortheajt to north. The user should realize that a change in cne frequency may chcnge other frequencies. The
PAGE 140
133, program automatically sums the 16 frequencies and divides each one by the sum to insure that the sum of the 16 wind rose frequencies is equal to one. In the third section of system data, the program inquires as to whether or not there is a stable layer aloft. If not, the mixing depth is considered infinite (see Chapter 3) and equation (3.413) is selected. If yes, equation (3.412) is selected, and the program prompts the user to enter the height of the stable layer in meters. In the fourth section of system data, the program prints the time period in hours for which the simulation is to be representative, the grid size in meters, and the number of grid rows and columns. The geographical area is considered to consist of rows and columns of square grids. The maximum number of 12 rows and 12 columns could be changed in the dimension statement. Each source (generating unit) is considered to reside at the center of its grid; and each grid concentration is computed for the center of the grid and considered representative throughout that grid. A maximum of 16 generating units can be accommodated. After viewing these data, the user may change any or all of them (or none of them). In the fifth section of the system data, the program prints the 10 meter wind speed in meters per second, the ambient temperature in degrees Kelvin, the number of groups of units, and the number of segments the units have to be committed (Chapter 4), A maximum of four groups of units is possible and each group can contain up to four generating units. The placing of units into a group to be com.mitted before any units are committed in the next group, is a tradeoff between accuracy and computation time. Inspection of equation (4.32) shows that the effective load duration curve becomes unwieldy for more than about four units
PAGE 141
134 In the sixth and final section of the system data, the program prints the number of generating units in each group. As in all the other sections, the user can elect to change these numbers. This completes the system data. The second category of input data, the generating unit data, is divided into three sections for each generating unit. This category begins with the program asking the user if he wishes to change any of the generating unit data. If the answer is no, then the computations begin (see Section 5.1). If the answer is yes, the program asks for the group number and unit number of the generating unit, and displays the current data for that unit in three sections. In the first section of generating unit data, the program prints the maximum output in megawatts, the grid row and column (location), and the availability (decimal) of the particular generating unit, and then asks for any changes. In the second section of generating unit data, the program prints the incremental heat rate coefficients, the fuel type, the fuel cost in dollars per million BTU, and the sulfur content in percent. (Note the incremental heat rate curve for each unit is stored as a quadratic function of the form 2 a+ P+yP .) After printing this data, the program asks for any changes. In the third and final section of generating unit data, the program prints the stack height in meters, the stack radius at the top in meters, the stack gas exit velocity in meters psr second, the fuel heating value in BTU's per pound, and th minimum sulfur content in percent. The program next asks for any changes. This completes the generating unit data for the particular unit the user specified. The progra.Ti aqain asks the user it he wishes to change any of the generating unit data. If so, the process described above is repeated.
PAGE 142
i35 This second category of input (generatiny unit data) is continued until the user's response indicates he no longer wishes to modify any generating unit data. The input data are summarized in Table 4, v/here the dotted lines separate the sections described above. The ability to change or modify any of the input data shown in Table 4 makes this program a \'ery powerful tool for evaluating operating strategies and for performing planning studies and simulations. The reader is referred to the U ser's Guide to ENVECO for a detailed description of the output. We will only mention here some of the more important aspects of the output from the program. We will end this section with a partial list of some of the information the program can provide: 1) The results of the effective stack height computations (Chapter 3) for those units burning coal or oil. 2) The basis for which each group of units was committed, i.e., on economics or air quality. 3) The commitment order of the generating units within each group. 4) Whether or not the block (see Figure 15) "compute new sulfur contents," was invoked, and if so, the new schedule of sulfur contents. 5) The generating unit fuel costs, emissions, and emission rates. 6) The total group and system fuel costs and emissions. 7) The predicted grid concentrations of sulfur dioxide. 8) The 24 hour lst and 2nd highest predicted concentrations. 9) The 3 hour lst and 2nd highest predicted concentrations. 10) The additional fuel costs (if any) for committing some of the units on the basis of air quality.
PAGE 143
136 Category 1 System Data Table 4. Input Data Summary Category 2 Generating Unit Data LDC array Wind rose frequencies Height of stable Layer aloft Time period Grid size Number of rows Number of columns 10m wind speed Ambient temperature Number of groups Number of segments Number of units in group 1 Number of units in group 2 Number of units in group 3 Number of units in group 4 Maximum output Grid row Grid column Unit availability IHR coefficients: Alpha Beta Gamma Fuel type Fuel cost Sulfur Content Stack height Stack radius Stack gas velocity Stack gas temperature Fuel heating value Minimum sulfur content
PAGE 144
137 5 3 Con clusion and Future W ork We have presented a new approach for reducing ground level concentrations of sulfur dioxide due to electric power plants. Since the approach involves time frames on the order of a month to a year, it is not an intermittent control scheme, and therefore, also reduces emissions of sulfur dioxide. It is an approach to be used as a supplement to and not in lieu of, constant emission controls. We have concluded that this approach, as an interim {up to 1985) control technique, should be accepted among other i approaches in keeping with the best interests of our nation. In this respect f we will conserve valuable low sulfur fuels, make them available to sources  more likely to violate the primary standards, and meet the primary and  secondary standards in a reasonable and acceptable manner; until such time I as widespread use of constant emission controls are possible, both from an engineering and socioeconomic standpoint. A novel solution to the gradient transport dispersion equation has been presented and shown to lead to the familiar and widely used Gaussian dispersion model. Hanna's (1972) equation for predicting average concentrations near the source has been corrected in this work; and a more fundamental relationship connecting the gradient transport theory to the statistical theory of dispersion has been presented. The extension of the theory of production costing to include certain nonlinear functions presented in this work has enriched the method. Finally, in addition to the usefulness of this approach in evaluating operating strategies, it will likely find many uses as a planning technique by system planners. In view of this, suggestions for future work include the extension of the theory of production costing to include any nonlinear Â• i function and useful techniques for considering partial outages of generating units.
PAGE 145
APPENDIX A Since, as is mentioned in Chapter 4, concentrations are approximately lognormally distributed, we must have an understanding of some of the properties of the lognormal distribution that are used in Section 4.4. In the interest of completeness, these properties are discussed in this Appendix. Also, while the lognormal distribution is not uncommon, the reader is probably not as familiar with its properties as he is with the properties of, say, the normal distribution. The lognormal distribution has had applications in many areas of engineering (see Aitchison and Brown (1957)). One use of the lognormal distribution has been for problems whose variate is limited to the positive real axis. A function that is normally distributed after the transformation y = In X (A1) where In is the natural logarithm, is said to be lognormally distributed. Its probability density function, f(y), isgiven as F(y) ^ 7^ exp [y i^fl t Its cumulative distribution function will be F(y) = ry f(y)dy Using the transformation (A1), the cumulative distribution becomes t In Chapter 4 the cumulative distribution function, F'()> had a nrime over it to distinguish it from the backward cumulative distribution function, F(). In this appendix the prime is unnecessary and so we drop it. 138
PAGE 146
139 F(ln x) = f "^ ^ exp [\ (ln^^Ji.)2jdx (A2) JO X/2TO "^ Equation (A~2) can be rev/ritten as /Â•In X F(ln x) = I f(x)dx ^0 where F(x) = 1^ exp [i(IlLx_:iJi)2] (As) X/2TTa Equation (A3), then, is the probability density function for a lognormal distribution. It is plotted in Figure 15 for i.i=0 and a=l Equation (A3) is seen to be characterized by two parameters, y and a. We need estimators of these parameters to be able to describe a lognormal distribution. We will use the maximum likelihood estimator. For the estimator of y we have L(y) = f(x,X2,...,x^;y) (A4) where L(*) is the likelihood function, x, ,X2,. ,Xj^ are sample values from a sample size n, and f (x, ,X2 ,x ;y) is the joint density function of the n samples and the parameter to be estimated, y. The maximum likelihood estimator assumes statistical independence and so equation (A4) becomes L(y) = f(x^;y)f(x2;y)...f(x^;y) where 1 r 1 ."'^ 'i^,2. f(xr'Vi) = Â— izexp [^ ( Â— ) 1 ..,^; ..p u2 ^ '^^" 1 n T ^ In x.y L(y) = (~~r) n (L exp [\ (Â— M^]) V li'O 1=1 1 mJa.c,^^SÂ£.
PAGE 147
140 To determine the maximum likelihood, it is necessary to take the derivative of L with respect to y and set the result to zero. However, the same value of u that maximizes L will maximize In L. The use of the natural logarithm greatly simplifies the computation. Thus n 1 n In x.y 2 In L(y) = n In (/2^a) I In x ^ 1 ( ^) i=l ^ ^ i=l Then, taking the derivative and setting it equal to zero, we have , n In x.y Â£ [in L(y)] = 1 ; (^) = from which n I In x. = ny i=l or 1 Y In x. (A5) u = y In X. n i^l Notice that the maximum likelihood estimator of y is seen to be analogous to the sample mean, except that the logarithm of the sample is used. If exponentials of both sides of equation (A5) are taken, we have 1 n exp(y) = exp {jI In x.) i=l ^ 1 exp(y) = exp [ln( n x.)] i=l n exp(y) = [ n x.]^/" (A5) i=l The right hand side of equation (A6) is, by definition, the geometric mean, m thus m = exp(y) or y = In m (A7) 9 Thus, equation (A3), the lognormal probability density function, becomes
PAGE 148
HI T , In x.ln m ^ x/2TTa Similarly for the estimator of o, the likelihood function is n , 1 In x.ln m ^ L(a) = {^f n f exp[l( V^)2] ^^a i1 X. ^ u Taking the logarithm yields n n In x.ln m ^ In L(a) = n ln(/2^a) J (In x )i I ( '^f i=l ^ "^ i=l "^ and differentiating and setting the result to zero, we have n In x.ln m ^ ^[1nL(a)] ^I ( '^ ^)^ = or n ,ln x.ln m .2 1/2 a = [ I ^ ^^^ 2_) ] (A9) i = l Similar to the relationship between the geometric mean and y m = exp(u) 9 we define the geometric standard deviation as sÂ„ exp(a) (A10) 9 where we have chosen s rather than a to be consistent with Larsen (1959) g g Equations (A9) and (A10) together relate the standard geometric deviation to the samples. Note the similarity of equation (A9) to the sample standard deviation of a normal function. Similar to equation (Ar7) we rewrite (A10) as a = In s (A11) g
PAGE 149
142 and In m. (A7) where equation (A7) is repeated here for convenience. The need for the geometric mean and standard deviation becomes clearer in Section 4.4. To complete this appendix v/e need to determine the mean, median, and mode of the lognormal distribution. Recall that for the normal distribution, they are all three the same and equal to y. The mean, m, or expected value, E[x], is m=E[x] = E[x] xf (x)dx 1 1 Â„ r 1 /lnxiJ\2, X exp [p( ^ ] dx Consider the change of variable y = In x Then 1 E[x] = E[exp{y)] = exp(y) 33exp l2^ ^ J co /Zrra [l(^)^]dy which can be rewritten after grouping of some terms as E[x] = Â— [" exp [X. (y22(y+a''^)y4y2)]dy 2a Now, if we complete the square of the terms in parentheses, and let a = \x^Q we get Â•03 E[x] = expdif ^f^) ^I exp[k^^)^]dy 2Tro ^ o i. c
PAGE 150
!43 or 1 2 E[x] = exp (y+ 2a ) (A12) since the term in brackets evaluates to one. The median, by definition, is the 50th percentile position. That is, the abscissa value for which the cumulative distribution function equals 1/2. From equation (A12) this becomes In X 1 ^ exp [i (ll^J=Ii)2]dx 2tjo So we want to find the abscissa value, In x for which the right hand side evaluates to 1/2. Again, we use the logarithm transformation so and y = In' x y. = In X Â•'o /o y2?( 1 r l/yy>2, Â— exp [ji^"") ] no 2' o dy which is in the form of a normal distribution. For a normal distribution, we know the median is y. Thus Yq = M from which Xq = expfp) lie now corns to one of the most important properties of a Icgnormally distributed function, because from equation (A.7) p = in m
PAGE 151
144 or g r'^/ and therefore the median equals the geometric mean, lie will express this as an equation since we will need to refer back to it. Thus Median = x m (A.13) 9 Finally, the mode is the maximum value of the probability density function (our function is unimodal ) Differentiating equation (A3) and setting it equal to zero yields VZna or _T ( In x)p ^ Q a 2 (In x)y = a In X = ua 2 X = exp(ya ) (A14) Equation (A14) is the equation for the mode of a lognormal distribution, Figure 16 shoves a lognormal density function for y^O and 0=1, for which mode = exp (1 ) = .368 median = exp(O) =1.00 and mean = exp (1/2) =^ 1 .65 and these values are shov;n in the figure.
PAGE 152
145 X3 Q O sD'^o'^^ median "^^a" Concentrations In Micrograms per Cubic Meter Figure 16. Probability Density Function for a Lognormal Distribution.
PAGE 153
REFERENCES Air Quality Display Model [1969] TRW SysteiTi Group, Available from National Technical Inforaation Service, NTIS No. PB189194, Washington, D.C. J. Aitchison and J.A.C. Brown [1957] Lognormal Distribution Cambridge University Press, Cambridge, England. P.J. Baker and B.E.A. Jacobs [1971] "Pulsed Emission Chimney," Civil Eng. Publ Works Rev. pp. 199200. H. Baleriaux, E. Jamoulle, and Fr. Linard de Guertechin [1967] "Simulation de 1 'exploitation d'un pare de machines, thermiques de production d'electricite' couple a des \ stations de pompage," Review E (edition SRBE) Vol. 5, [ No. 7, pp. 324. I W.K. Bodle and K.C. Vyas \ [1974] "Clean Fuels from CcalSNG Special Report," Oil and Gas Journal Vol. 72, August 26, 1974. R.R. Booth [1972] "Power System Simulation Model Based on Probability Analysis," IEEE Trans, on Power Apparatus and Systems Vol. PAS91, No. 1, pp. 6269. G.A. Briggs [1969] "Plume Rise," Atomic Energy Commission Critical Review Seri es Washington, D.C. [1971] "Plume Rise: A Recent Critical Review," Nuclear Safet y, Vol. 12. pp. 1524. A.D. Busse and J.R. Zimmerman [1973] U sers Guide for the CI imatological Dispersion Model U~S. Environmental Protection^Agency, EPAR473024, Washington, D.C. K.L. Calder [1965] "On the Equation of Atmospheric Diffusion," Quart. J. Roy. Mete orol. Soc Vol. 91, No. 390, pp. 514517. [1971] "A CI imatological Model for Multiple Source Urban Air Pollution," P roc. 2r.(i M eeti n g of t h e Expert P anel on Ai r Pel 1 uti on Model i no", NATA Comnii ttee on the' Changes of Modern Society, Paris, France. 146 I ma ^xiattwrv^a I>1
PAGE 154
147 S. Chandrasekhar [1943] "Stochastic Problems in Physics and Astronomy," Rev. Mod. Phys. Vol. 15, pp. 189. Clean Air Coordinating Committee [1975] Preliminary Report of the Clean Air Coordinating Committee, Presented before U.S. Senate Committee on Public Works, April, 1975. G.T. Csanady [1965] "The Buoyant Motion Within a Hot Gas Plume in a Horizontal Wind," J. Fluid Mech. Vol. 22, pp. 225239. Electric Power Research Institute [1975] Status of Stack Gas Technology for 50? Control Project 209, Palo Alto, California. Environmental Protection Agency [1971a] "National Primary and Secondary Air Quality Standards," Federal Register Vol. 35, No. 84, p. 8186. [1971b] "Requirements for Preparation, Adoption, and Submittal of Implementation Plans," Federa l Register Vol. 36, No. 158, p. 15486. [1971c] "Standards of Performanc2 for New Stationary Sources," F ederal Register Vol. 36, No. 159, Part II, p. 28476. [1973] "Use of Supplementary Control Systems and Implementation of Secondary Standards," Federal Register Vol. 38, Mo. 178, pp. 2569725703. [1974] Natio nal Strategy for C ontrol of Sulfur Oxides from Electric Power Plants U.S. EPA," July 10, 1974. Washington, D.C, [1975] Report to Congress on Control of Sulfur Oxides U.S. EPA, Report No. EPA450/175001, Washington, D.C. A. Pick [1855] "Uber Diffusion," Ann. Physik Chem. Vol. 2, No. 94, pp. 5986. M.R. Gent and J.W. Lamont [1971] "MinimumEmission Dispatch," IEEE Tra nsactions Vol. PA590, No. 5, pp. 26502660. F.A. Gifford, Jr. [1955] "A SimultaneousEulerian Turbulence Experiment," Monthly Weather Rev. Vol. 83, No. 12, pp. 293301. S.R. Hanna [1972] "A Computer Model for Estimating Longterm Ground Level Concentrations from Area and Point Sources," Atmospheric Turbulence Dispersion Laboratory, Oak Ridge, Tennessee. H.E. Hesketh [1972] Understanding and Contro l ling Air Po llutio n, Ann Arbor Sciences Pub,, Ann Artor, Mich. iiiii'ifiiriTirT' ^i ti 1*1 I II 111 j fci niiiiB MB I I ii p 1 1 Bin III 1 1 a I w I III I II I III I III n I nil I" iM iitn^r "irfi rti mii 'ii i ii ii i~r' i'
PAGE 155
148 R.T. Jenkins and D.S. Joy [1974] "Wein Automatic System Planning Package (WASP)An Electric Utility Optimal Generation Expansion Planning Computer Code,' Report ORML4945, Oak Ridge National Laboratory, Oak Ridge, Tennessee. W.W. Kellog, R.D. Cadle, E.R. Allen, A.L. Lazrus, and E.A. Martel [1972] "The Sulfur Cycle," Science Vol. 175, No. 4022, pp. 587596. D.V. Lamb, F.I. Badgley, and A.T. Rosano, Jr. [ 1 974 J A Critical Review of iMathematical Diffusion Modeling Techni ques for Pre dictin g Air Quality with Re la tion to M otor Vehicle Transportation Departments of Atmospheric ^iences and Civil Engineering, Univ. of Washington, Seattle, Washington. J.W. Lament, K. Sim, and E.P. Hamilton [1975] "A Multi Area Environmental Dispatching Algorithm," 9th PICA Conf New Orleans, Louisiana. R.I. Larsen, C.E. Zimmer, D.A. Lynn, and K.G. Blemel [1967] "Analyzing Air Pollutant Concentration and Dosage Data," J. Air Poll. Contr. Assoc Vol. 17, pp. 8593. R.I. Larsen [1969] "A New Mathematical Model of Air Pollutant Concentration Averaging Time and Frequency," J. Air Poll. Contr. Assoc Vol. 19, No. 1, pp. 2430. [1971] A Mathematical Model for Relating Air Quality Measurements to Air Quality Standards U.S. EPA Publication, No. AP89, Washington, D.C. C.C. Lin [1960] "On a Theory of Dispersion by Continuous Movements," Proc. Natl. Acad. Sci Vol. 46, pp. 566570. M.K. Liu [1975a] "A Simple Reactive Plume Model," Report E17512, Systems Applications, Inc., San Rafael, California. [1975b] "Numerical Modeling of Buoyant Plumes in the Vicinity of the Stack," Report E17528, System Applications, Inc., San Rafael, California. A.S. Monin [1959] "Smoke Propagation in the Surface Layer of the Atmosphere," Advances in Geophysic s, Vol. 6, pp. 331344. T.L. Montgomery, J.W. Frey, and W.B. Norris [1975] "Intermittent Control System," Environmental Science and Technolog y, Vol. 9, pp. 528533.. iarMkMffl>L^Wf75
PAGE 156
149B.F^. Morton, G. Taylor, and J.S. Turner [1956] "Turbulent Gravitational Convection from Maintained and Instantaneous Sources," Proc. Roy. Soc. (London) Vol. 234, pp. 123. W.L. Nelson [1973] "Process Costimating Series," O il and Gas Journal Vol. Hi pp. 2739. Â• S. Oglesby and G.B. Nichols [1970] A Manual of Electrostatic Precipitati on T echnology Southern Research Institute, 3irminghaiTU~7il a Â• PEDCo Environmental Specialists, Inc. [1975] Assessm ent of Alternative Strategies for the Attainment and Maintenance o f Na tional Am bient Air Quality Standards for Sulfur Oxides, Prepared for U.S. Environmental Protection Agency, Contract No. 68021375, Task Order No. 17. F. Pasquill [1951] "The Estimation of the Dispersion of Windborne Material," Me teorol. Mag ., Vol. 90, No. 1063, pp. 3349. [1962] "Some Observed Properties of MediumScale Diffusion in the Atmosphere," Quart. J. Roy. Meteorol. Soc Vol. 88, No. 375, pp. 7079. E.S. Pearson and H.O. Hartley [1962] Biometlcha T a ble s for S tatist ici ans Vol. 1 Cambridge University Press', New York, p. "175. H. Perry [1974] "Coal Conversion Technology," Chem. Eng. Vol. 81, pp. 88102. F.T. Princiotta [1972] "Control of Sulfur Oxide Pollution from Power Plants," Presented at the American Public Power Association Engineering and Operations Workshop Anaheim, Calif. L.F. Richardson [1926] "Atmospheric Diffusion Shown on a DistantNeighbour Graph," Proc. Ro y. Soc. (London) Ser. A, Vol. 110, pp. 709737. J.J. Roberts, E.J. Croke, A.S. Kennedy, J.E. Norco, and L.A. Conley [1970] A Mu lti pleSour ce Urban Atmospheric Dispersion Model Argonne National Laboratory, ANL/ESCC007. M.A. Sager, R.J. Ringlee, and A.J. Wood [1972] "A New Generation Production Cost Program to Recognize Forced Outages," IEEE T rans, on Power A pparatus and Systems, Vol. PAS9l,'No. 5, "ppT" 21l4'24. W. Schmidt [1925] "Uer Massenaustausch in Freier Luft and Verwandte Ersche'inungen," P robleme de r Kosm ichen Physik Vol, 7, pp. 4753, in I in Â• ii 'il ii l i iiiw
PAGE 157
150 '' C.C. Shir [1970] "A Pilot Study in Numerical Techniques for Predicing Air Pollutant Distribution Downwind from a Line Stack," AtTOS. Env. Vol. 4, pp. 387407. D.H. Slade [1968] Meteorology and A to mic Energy Air Resources Laboratory, U.S. Atomic Energy Commission, Washington, D.C. H.S. Stoker and S.L. Seager [1972] Air and Water P ollution Scott Foresman and Co., Glenview, Illinois, pp. 5355. R.L. Sullivan [1972] "Minimum Pollution Dispatching," Presented at IEEE PES Summer P ower Meeting Paper C724687, San Francisco, California. [1974] "A Comprehensive Generation Reserve Planning Technique," IEEE Conf. C744297, Anaheim, California. [1975] Power System "Planning McGrawHill International, Nov. 1976. R.L. Sullivan and D.F. Hackett [1973] "Air Quality Control Using a Minimum Pollution Dispatching Al gor i thm Eny ironm ental Science and Technology Vol. 7, p. 1019". R.L. Sullivan and D.W. Hilson [1975] "Computer Aided Ambient Air Quality Assessment for Generation System Planning," 9th PICA Conf New Orleans, Louisiana. G.I. Taylor [1921] "Diffusion by Continuous Movements," Proc. London Math Soc Vol. 20, pp. 196202. T.W. Tesche, G.Z. Whitten, M.A. Yocke, and M. Liu [1976] Theoretical, Numerical, and Physical Techniques for Characterizing Power PlantTTumes ETictric Power Research Institute, Project No. EC144. F.W. Thomas, S.B. Carpenter, and W.C. Colbaugh [1970] "Plume Rise Estimates for Electric Generating Stations," Journal of the Ai r Pollution Contro l Ass ociat ion," Vol. 20, No. 3'. D.B. Turner [1970] Workbook of At m ospheric Dispersion Est imates, U.S. Department of Health, Education :.nd Welfare, Available from National Technical Information Service, NTIS No. PB1 91482. K.E. Yeager [1975] "Stacks /i. Scrubbers," Resear ch Progress R eport FF3, Fo'jsi] Fiifil a'ld PJva;,{.ed System:Division, Palo Alto, Califo.^nia, M *(*ifT*Wt(llllTi.1ll,"
PAGE 158
BIOGRAPHICAL SKETCH David Wayne Hilson was born in Dothan, Alabama on November 21, 1944. He graduated from Miami Edison Senior [ligh School in June, 1962. He received the degrees of Bachelor and Master of Science in Electrical Engineering from Cornell University in June, 1956, and June, 1967, respectively. After graduating he worked for several years as an application engineer in the electric power systems group for the Leeds and Northrup Company. He entered the Graduate School of the University of Florida in January, 1971. He has been an assistant professor of Electrical Engineering at the University of Tennessee at Chattanooga, Chattanooga, Tennessee, since September, 1974. During the past two years, he has also served as a consultant to the Power Research Staff of the Tennessee Valley Authority in environmental evaluation and planning. David Wayne Hilson is married to the former Carol Ann Hendricks, and they have two sons, James Wayne and Michael Keith. He is a member of the Order of Engineer, Eta Kappa Nu, and the Institute of Electrical and Electronics Engineers, and the Power Engineering Society. 1 En 'Jf lkJBB
PAGE 159
I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / Robert L. Sullivan, Chairman Associate Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 01 le I. ElgerC Cochairman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Paul Urone Professor of Environmental Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partia" fulfillment of the requirements for the degree of Doctov of Philosophy, August, 1976 Â„^eap. College of Engineering Dean, Graduate School
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EFAXXEFDU_POSY31 INGEST_TIME 20141215T20:32:35Z PACKAGE AA00026509_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES

