1'
United States
Environmental Protection
Agency
Atmospheric Sciences Research ~^~ -\I
Laboratory * tw -^
Research Triangle Park NC 27711 '/ F, \N
Research and Development
EPA/600/S3-86/036 Sept. 1986
&EPA Project Summary
Applications of Decision
Theory Techniques in Air
Pollution Modeling
Robert G. Lamb and Saroj K. Hati
This study applies methods of opera-
tions research to two basic areas of air
pollution modeling: (1) the generation
of wind fields for use in models of re-
gional scale transport, diffusion and
chemistry and (2) the application of
models in studies of optimal pollution
control strategies. The work is illus-
trated in the context of a hypothetical
problem in which optimal sites and
emissions control plans are sought for
two new power plants. The study ad-
dresses all aspects of the problem start-
ing with stochastic specification of the
wind fields in the region of interest,
proceeding to the development of sim-
ple models that relate pollutant emis-
sions to both short- and long-term aver-
aged concentration, and concluding
with the incorporation of game theory
concepts into mathematical methods
of finding optimum solutions to multi-
objective problems. Five objectives are
considered in this study: minimization
of plant operating cost and minimiza-
tion of given short-period maximum
and long-period averaged concentra-
tions of each of two pollutants. The op-
timization procedure attempts to fulfill
these objectives jointly while comply-
ing with specified constraints on the
overall system.
This study emphasizes concepts and
technique development rather than the
solution of actual problems.
This Project Summary was devel-
oped by ERA's Atmospheric Sciences
Research Laboratory, Research Triangle
Park, NC, to announce key findings of
the research project that is fully docu-
mented in a separate report of the same
title (see Project Report ordering infor-
mation at back).
Introduction
This study applies methods of opera-
tions research to two aspects of air pol-
lution modeling: (1) the generation of
wind fields for use in models of regional
scale transport, diffusion and chemistry
and (2) the application of diffusion mod-
els in studies of optimal pollution con-
trol strategies, particularly the siting of
new industries. The first application
represents an attempt to implement an
idea developed by one of the authors
(R. G. Lamb) in an earlier study of the
formulation of wind fields for use in re-
gional scale air pollution models. It was
shown in that study that the physical
laws of fluid motion and a set of discrete
meteorological observations do not
uniquely define the wind field. Rather
they delineate an infinite set of func-
tions any one of which is a possible de-
scription of the atmospheric flow that
existed during the period the observa-
tions were made. This inherent uncer-
tainty should be accounted for explicitly
in air pollution model applications so
that decision makers can use model out-
puts properly. This study develops a
technique that affects this type of model
enhancement.
Since models are used extensively to
assess the effectiveness of proposed
solutions to air quality problems, it
would be advantageous to have a
means by which a computer could work
with models to find optimal solutions to
given problems. Moreover, in the area
of policy analysis, cost/benefit ratios
-------
must often be estimated for a myriad of
hypothetical policy options. For exam-
ple, what is the cost of the additional
emissions controls needed on a given
source to achieve compliance with a
given secondary standard; and how
does this cost compare with that of the
controls that one would find necessary
were the cost and the degree of viola-
tion of the secondary standard treated
as objectives to be minimized jointly?
Answering questions of this kind has
not been practical heretofore because
techniques have not been available for
applying air pollution models to multi-
criteria optimization problems. This
study uses concepts of game theory to
develop a technique of this kind.
Ensembles of Wind Fields
If the horizontal wind (u,v) in a shal-
low layer of the atmosphere is repre-
sented at any instant t, by a complex
Fourier series containing 2(2M+1)2 am-
plitudes, compatibility with the princi-
ple of mass conservation reduces the
number of independent amplitudes to
4M(M+1). In other words, the mathe-
matical description of the entire wind
field (u,v) in a given region at time t1; is
represented by a single point in a
4M(M+1) dimensional phase space in
which each coordinate axis is associ-
ated with one of the independent
Fourier amplitudes. If at time t1( one has
J observations of the wind, say (uj, Vj),
j = 1, . . J, made at arbitrary locations
(*j, Vj) j = 1, . . .J, these define a hyper-
plane of dimension 4M(M+1)-2J in
the phase space. Each point on the hy-
perplane is a possible description of the
flow at time tj.
We say that it is a possible description
because it satisfies all observations ex-
actly and it satisfies mass conservation.
However, if the flow descriptions repre-
sented by individual points on the hy-
perplane were examined you would
find that most of them are improbable
flows in the sense that they contain fea-
tures that are not normally observed in
the atmosphere. For example, it is pos-
sible for a flow field to contain an in-
tense vortex between the wind observ-
ing sites without violating either the
velocities measured at the wind stations
(Uj, Vj) or mass conservation.
Thus we need a means of using addi-
tional information, such as climatologi-
cal data or empirical knowledge of at-
mospheric motion, to diminish as much
as possible the highly underdetermined
nature of the flow specification. We pro-
pose to achieve this by assigning to
each of the possible flows a weight p,
which we call a probability, whose value
is a measure of how consistent that par-
ticular flow is with additional informa-
tion. The specific information that one
choses to use and the manner in which
the estimates of p are associated with it
are purely subjective decisions that be-
come an intrinsic part of the model in
which the wind fields are ultimately
used.
One method of estimating the flow
probabilities p, which is illustrated in the
report, makes use of observed kinetic
energy spectra. For example, aircraft
measurements made in the upper tro-
posphere indicate that both the u and v
components have energy spectral den-
sities that are proportional to the -5/3
power of the wave number for wave
numbers | Ik | greater than about 5.10~5
rad rrr1, i.e., wave lengths smaller than
about 500 km; and energy density pro-
portional to |k|~3 for longer wave
length perturbations. These observa-
tions lead us to the hypothesis:
The probability of occurrence of
atmospheric flow field described
by a given point on the hyper-
plane is directly proportional to
the degree to which the Fourier
transform associated with that
point satisfies the energy spectra
characteristics cited above.
Locating the points on the hyperplane
that satisfy the energy conditions best is
achieved by defining an optimization
problem whose solutions are the flow
fields we seek and applying methods of
operations research to solve the prob-
lems. The report describes this process
in some detail, showing how nonlinear
programming and related techniques
can be used to generate a collection of
points X,, i = 1,.. .I, from the flow field
hyperplane. To each point x,, which
represents a complete wind field de-
scription, a probability p; is assigned
whose value is determined by how well,
relative to the other (1-1) members of the
set, its energy spectrum conforms to the
given specifications.
The process just described yields an
ensemble of I wind descriptions appli-
cable to the single instant tv By repeat-
ing the process for each of the N hours,
say, that meteorological measurements
are available in a given period T, we ob-
tain N ensembles. By selecting one
member from each ensemble, we can
produce a sequence of functions to de-
scribe the temporal evolution of the
flow field. We assign probabilities to
each of the IN possible sequences under
the following hypothesis:
The probability of a given se-
quence of flow fields is directly
proportional to the degree to
which that sequence satisfies the
principle of momentum conserva-
tion.
The report describes in detail how dy-
namic programming and Bellman's
principle of optimality can be used to
implement this hypothesis. The final re-
sult is a set of function sequences and
their associated probability values that
provide descriptions of the winds (u, v)
over the given region during the period
T. An air pollution model driven by each
member of this set of flow descriptions
in turn will produce an ensemble of con-
centrations fields for each specified dis-
tribution of species sources. The differ-
ences that exist among the members of
the concentration ensemble reflect the
uncertainty created by our inability to
specify atmospheric motion exactly.
The underlying philosophy of the flow
field ensemble approach is that this in-
herent uncertainty should be dealt with
explicitly in air pollution models for
them to be used meaningfully in deci-
sion making studies.
Model Application and Game
Theory
Selecting sites for new industries that
produce air pollutants is a task that re-
quires crucial tradeoffs between capital
and operating costs on the one hand
and the requirement to satisfy air qual-
ity standards on the other hand. Ideally,
one would like to find plant locations
where air quality standards can be met
at minimal cost. In this study we focus
on the problem of siting power plants,
in which case the costs considered in-
clude fuel cost, the cost of transporting
fuel from given sources (such as coal
mines) to the given plant sites and the
cost of emissions controls (such as flue
gas desulfurization and fly ash precipi-
tation). Both primary and secondary air
quality standards on one or more pollu-
tants are considered. Optimal tradeoffs
among the air quality and cost objec-
tives are sought within the space of fea-
sible solutions using concepts of coop-
erative game theory.
Game theory has been developed for
both cooperative and noncooperative
games. In the latter, each player acts in-
dependently to maximize his own gain.
In cooperative games, players form
-------
coalitions under the expectation that
working together each can achieve an
outcome that is better than that achiev-
able in fully competitive play. The meas-
ure of success of cooperative play is
embodied in the concept of pareto-
optimality which involves a single ob-
jective function constructed from the
objectives of each player. This study
uses these concepts to develop a gen-
eral procedure for finding solutions to
air quality problems that involve multi-
ple objectives. The technique is demon-
strated using a simple, hypothetical
problem in which sites for two new
power plants are sought that jointly
minimize operating cost—fuel plus
emissions controls—and peak and
long-term averaged concentrations of
two pollutants, a total of five objectives
in all.
Conclusions
New methods have been developed
for use in two areas of air pollution
modeling: generation of wind fields for
regional scale models, and applications
of models in multi-criteria optimization
studies. The first technique differs fun-
damentally from existing methods in
that it produces ensembles of flows
rather than a single flow specification.
In this regard it is consistent with the
fact that physical laws and discrete me-
teorological data do not uniquely
specify the wind field. Moreover, it al-
lows total use of meteorological obser-
vations in contrast to the limited utiliza-
tion that is achieved in applications
where the bulk of the meteorological in-
puts are generated by a meterological
model.
The newly developed method of
treating multicriteria optimization prob-
lems should enhance the value of air
pollution models in cost/benefit, land
use and other studies where one is in-
terested in optimal tradeoffs among
policy options. Although the method is
quite general in its applicability, it can
be expensive to operate, computation-
ally, unless the air quality models are
expressible in closed forms.
-------
The EPA authors Robert G. Lamb and Saroj K. Hati are with the Atmospheric
Sciences Research Laboratory. Research Triangle Park. NC 27711.
The complete report, entitled "Applications of Decision Theory Techniques in Air
Pollution Modeling." (Order No. PB 86-216 793/AS; Cost: $ 11.95. subject to
change) will be available only from:
National Technical Information Service
5285 Port Royal Road
Springfield, VA 22161
Telephone: 703-487-4650
The EPA authors can be contacted at:
Atmospheric Sciences Research Laboratory
U.S. Environmental Protection Agency
Research Triangle Park, NC27711
United States
Environmental Protection
Agency
Center for Environmental Research
Information
Cincinnati OH 45268
Official Business
Penalty for Private Use $300
EPA/600/S3-86/036
0000329 PS
U S cNVIR PROTECTION AGENCY
REGION 5 LI8RARY
230 S DEARBORN STREFT
CHICAGO IL
------- |