Stochastic Optimization Models
for Lake Eutrophication
Management
Somlyody, L. and Wets, R.J.-B.
IIASA Collaborative Paper
April 1985
Somlyody, L. and Wets, R.J.-B. (1985) Stochastic Optimization Models for Lake Eutrophication Management.
IIASA Collaborative Paper. IIASA, Laxenburg, Austria, CP-85-016 Copyright © April 1985 by the author(s).
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STOCHASTIC OPTIMIZATION MODELS
FOR LAKE EUTROPHICATION MANAGEXENT
Ldszld Somly6dy*
Roger J-B Wets**
April 1985
CP-85-16
*
Research Center f o r Water Resources Development, VITUKI,
Budapest, Hungary
** IIASA and University of California
(Supported in p a r t by a g r a n t of t h e National Science Foundation)
Cottaborative Pepers r e p o r t work which h a s not been performed
solely at t h e International Institute f o r Applied Systems Analysis
and which h a s received only limited review. Views or opinions
e x p r e s s e d herein do not necessarily r e p r e s e n t those of t h e Institute, i t s National Member Organizations, or o t h e r organizations
supporting t h e work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS
2361 Laxenburg, Austria
The development of stochastic optimization techniques, r e l a t e d
software, and i t s application, w a s a major p a r t among t h e activities of t h e
System and Decision Sciences P r o g r a m during t h e p a s t f e w y e a r s .
This p a p e r d e s c r i b e s r e s u l t s of t h e application of stochastic programming to water quality management. I t provides a n example of both important
issues f o r investigation: a r e a l i s t i c problem with inherent stochasticity,
and a valuable test problem f o r t h e algorithms under development. I t a l s o
gives some insights into t h e n a t u r e of solutions of c e r t a i n classes of stochastic programming problems, and t h e justification f o r t h e consideration
of randomness in decision models.
Alexander B. Kurzhanski
Chairman
System and Decision Sciences
Program
ABSTRACT
W e develop a general framework f o r t h e study and t h e control of t h e
eutrophication p r o c e s s of (shallow) lakes. The randomness of t h e environment (variability in hydrological and meteorological conditions) i s a n
intrisic c h a r a c t e r i s t i c of such systems t h a t cannot b e ignored in t h e
analysis of t h e p r o c e s s or by management in t h e design of control measures.
The models t h a t w e suggest t a k e into account t h e stochastic a s p e c t s of t h e
eutrophication process. An algorithm, designed to handle t h e resulting stochastic optimization problem, is described and i t s implementation i s outlined. A second model, based on expectation-variance considerations, t h a t
approximates t h e "full" stochastic model can b e handled by s t a n d a r d linear
o r nonlinear programming packages. A case study illustrates t h e approach;
w e compare t h e solutions of t h e stochastic models, and examine t h e effect
t h a t randomness h a s on t h e design of good management programs.
Key Words: Lake eutrophication, eutrophication management, stochastic
programming, r e c o u r s e model, water quality management, probabilistic constraints.
CONTENTS
INTRODUCTION
1. MANAGEMENT GOALS AND OBJECTIVES
2.
T H E APPROACH
3.
FORMULATION O F T H E STOCHASTIC MODELS LEMP AND NLMPZ
3.1. A g g r e g a t e d L a k e E u t r o p h i c a t i o n m o d e l (LEMP)
3.2 N u t r i e n t L o a d M o d e l (NLMPZ)
4.
CONTROL VARIABLES, COST FUNCTIONS AND CONSTRAINTS
4 .I.C o n t r o l V a r i a b l e s
4.2 C o s t F u n c t i o n s
4.3 C o n s t r a i n t s
5.
FORMULATION OF T H E EUTROPHICATION MANGEMENT
OPTIMIZATION MODEL: EMOM
6.
EMOM: AN EXPECTATION-VARIANCE MODEL
7.
SOLVING EMOM: STOCHASTIC VERSION
8.
APPLICATION TO LAKE BALATON
8.1. B a c k g r o u n d f o r L a k e B a l a t o n
8.1.1. D e s c r i p t i o n o f the l a k e , its w a t e r s h e d
a n d possible c o n t r o l m e a s u r e s
8.1.2. S p e c i f i c a t i o n o f e l e m e n t s o f EMOM f o r L a k e B a l a t o n
8.2. R e s u l t s o f t h e e x p e c t a t i o n - v a r i a n c e m o d e l
8.3. R e s u l t s o f t h e Stochastic R e c o u r s e M o d e l
9.
8.4 Comparison of the Deterministic and Stochastic Solutions
Sensitivity Analysis
-
77
SUMMARY
83
STOCHASTIC OPTIMIZATION MODELS FOR LAKE EUTROPHICATION
MANAGEIIIENT
b?'
Ldszld Somly6dy and Roger J-B Wets
INTRODUCTION
Yan-made ( o r artificial) eutrophication h a s been considered as one of
t h e most serious water quality problems of lakes during t h e last 10-20
years. Increasing discharges of domestic and industrial waste water and
t h e intensive use of c r o p fertilizers
of t h e recipients
--
- all leading t o growing nutrient loads
can b e mentioned among t h e major causes of this
undesirable phenomenon.
The typical symptoms of eutrophication a r e
among o t h e r s sudden algal blooms, water coloration, floating water plants
and debris, excreation of toxic substances causing taste and odor problems
of drinking water and fish kills. These symptoms can easily result in limitations of water use f o r domestic, agricultural, industrial o r recreational
purposes.
One of t h e major f e a t u r e s of artificial eutrophication is t h a t although
t h e consequences a p p e a r within t h e lake, t h e cause
- t h e gradual increase
of nutrients (various phosphorous and nitrogen compounds) reaching t h e
lake
- and most
of t h e possible control measures lie in t h e region. Conse-
quently, eutrophication management r e q u i r e s analysis of complex interactions between t h e water body and i t s surrounding region. In t h e lake, diff e r e n t biological, chemical and hydrophysical processes
and space dependent, furthermore non-linear
-- all
being time
- are important, while in t h e
region one must t a k e into account human activities generating nutrient residuals and control measures determining t h a t portion of t h e emission which
r e a c h e s t h e water body.
Eutrophication management r e q u i r e s a sound understanding of all t h e s e
processes and activities which, in fact, belong t o quite diverse disciplines.
Additionaily, various uncertainties and stochastic f e a t u r e s of t h e problem
have t o be also taken into account, f o r example, t h e estimation of loads
from infrequent observations and t h e dependence of water quality on hydrologic and meteorologic f a c t o r s , respectively. The f a c t t h a t we a r e dealing
with a stochastic environment, i s especially important f o r shallow lakes, due
primarily t o t h e absence of thermal stratification which predicates a much
more definite response t o randomness as would b e t h e c a s e f o r deep lakes.
Models developed with t h e aim of analyzing eutrophication can b e classified into two broad groups:
(a) Dynamic simulation models (ecological models) which intend t o
describe t h e temporal and spatial changes of various groups of
species and nutrient fractions (algae, zooplankton, phosphorus,
and nitrogen components, etc.). These models a r e being designed
primarily f o r r e s e a r c h purposes.
(b) Management optimization models being more applications oriented
which have as objective of t h e determination of t h e best o r
"optimal" combination of alternative control measures.
There is a l a r g e l i t e r a t u r e devoted t o descriptive models (see f o r
example, Canale, 1976; Scavia and Robertson, 1979; Dubois, 1981; Orlob,
1983), but only relatively few management models have been proposed
(Thomann, 1972; Biswas, 1981; Loucks et al. 1981; Bogdrdi et al., 1983).
Although changes in t h e ambient lake water quality should b e an important
element in decision making, t o o u r knowledge, no lake is mentioned in t h e
l i t e r a t u r e f o r which a p r o p e r description of t h e in-lake processes has been
taken into account in t h e management model; i.e., t h e interaction between
t h e models of type (a) and (b) is missing. The reason f o r this is at least
threefold:
(i)
Models (a) are too complex t o be involved directly in optimization
and no well-based methodology exists in this r e s p e c t .
(ii) In general, f o r management purposes, i t suffices t o use some
aggregated f e a t u r e s of water quality. A s will b e seen, microscopic
details offered by models (a) can b e often ruled out, but i t is not
easy t o determine t h a t portion of information offered by t h e
eutrophication model (a) which should b e maintained in o r d e r t o
a r r i v e at a scientifically well-based management model.
(iii) In most of t h e cases, not enough time and money are available t o
perform a systematic analysis including t h e joint development of
both models.
Consequently, t h e r e is a gap between the application of descriptive and
management models; the objectives of decision models a r e formulated in
t e r m s of nutrient loads r a t h e r than of lake water quality, which is only
acceptable f o r relatively simple situations. The present paper
- after con-
sidering management goals in Section 1 - offers an approach which allows
the combined use of descriptive, simulation and management optimization
models. This is discussed in Section 2. The derivation of the aggregated
lake and planning type nutrient load models t o be used in t h e management
model is the subject of Section 3. Both models a r e stochastic, special
emphasis is given t o shallow lakes. Section 4 discusses f u r t h e r elements of
the management model (cost functions, constraints, etc.).
Alternative
management models a r e formulated in Sections 5 and 6. Two of them were
implemented: a "true" stochastic model (which uses a s t h e starting point of
the iterative solution procedure the corresponding deterministic model)
and a linear programming approach capturing stochastic features of the
problem through expectation and variance. Sections 6 and 7 give details of
these models. Finally, in Section 8 t h e methods developed a r e used primarily f o r Lake Balaton that plays the role of a major case study.
1. MANAGEMENT GOALS AND OBJECTIVES
A s mentioned in t h e Introduction, artificial eutrophication leads t o
water quality changes which then restricts the use of water. The objective
of a manager when considering an eutrophication problem very much
depends on the features of t h e particular system. In most of the cases, however, the wish of managers can be formulated in quite general t e r m s . The
basis f o r this is the definition of trophic classes.
Trophic state classification i s t h e subject of limnology. In p r a c t i c e
f o u r major classes: oligotrophic, mesotrophic, eutrophic and hypertrophic
categories are used (OECD, 1982), where oligotrophic indicates relatively
"clean" water, while hypertrophic r e f e r s to a l a k e r i c h in nutrients and a n
advanced s t a g e of eutrophication. To specify t r o p h i c classes, water quality
components (concentrations of phosphorus and nitrogen fractions, oxygen
content, algal biomass, etc., see f o r example, OECD, 1982) are applied as
indicators. Based on observations and studies performed on many shallow
and deep lakes t r o p h i c classes are specified quantitatively by c e r t a i n
r a n g e s of t h e indicators. One of t h e m o s t widely used indicator i s t h e annual
mean or annual peak chlorophyll-a concentration, (Chl-a), a measure of
algal biomass. The chlorophyll content affects t h e c o l o r of water and thus
i t c h a r a c t e r i z e s f o r example t h e r e c r e a t i o n a l value of t h e lake. In t e r m s of
concentration (Chl -a),
t h e r a n g e s of classes oligotrophic ...hypertrophic
are as follows: 0-15, 10-25, 25-75 and 75- (see OECD, 1982).
A s t h e s e classes are closely r e l a t e d to t h e use of water, decision mak-
ers' objective i s often to shift a lake, say, f r o m hypertrophic t o a n oligot r o p h i c state and water quality goals are specified accordingly. Note t h a t
t h e definition of t r o p h i c classes (with fixed boundaries) i s not unambiguous,
which i s certainly not a s u r p r i s e as t h e trophic changes are caused by complex ecological processes. Still, however, decision makers r e q u i r e guidelines easy t o understand and apply, and in t h i s s e n s e t h e use of t r o p h i c
state classification i s inevitable in eutrophication management.
F o r many lakes t h e r e i s a spatial variation in t h e water quality. The
t y p e of water usage c a n also b e different in one area of t h e lake than in
o t h e r s , e.g., agricultural, recreational o r industrial. For this r e a s o n s t h e
objective of management can b e different f o r different segments o r basins
of t h e water body. Thus spatial segmentation is a major component of
management. Similarly, t h e decision makers should decide how important
t h e random fluctuations in water quality (from y e a r t o year) are as compared t o expectations. Would h e f o r instance b e satisfied with drastic
reduction in t h e "average" water quality without excluding t h e occurence of
extreme situations o r r a t h e r would he p r e f e r t o achieve a modest improvement in t h e mean provided t h a t he is now able t o limit t h e range of possible
fluctuations? Moreover, how would h e judge t h e situation if fluctuations
also strongly vary in space?
Simultaneously with developing his judgement on t h e lake basins and
management goals, t h e decision maker has t o perform a careful analysis as
t o where (and when) h e can control nutrient loads in t h e watershed t h a t
determine t h e trophic state of t h e lake? How effective are t h e control
measures, what are t h e costs, benefits and associated constraints? How
would a control measure taken in a subwatershed of a basin influence t h e
water quality of all t h e basins (including both expectations and variances)?
Finally, how would h e select among alternative combinations of projects?
The methodology developed in t h e next sections is aimed at answering such
questions, and thus provide technical support t o t h e decision process.
2.
THE APPROACH
The approach t o eutrophication and eutrophication management is
based on t h e idea of decomposition and aggregation (Somly6dy, 1982 and
1983a). The f i r s t s t e p is t o decompose t h e complex system into smaller,
tractable units forming a hierarchy of issues (and models), such as
biological and chemical processes in the lake, sediment-water interaction,
water circulation and m a s s exchange, nutrient loads, watershed processes
and possible control measures; influence of natural, noncontrollable
meteorological factors, etc. One can make detailed investigations of each
of these issues. This step is followed by aggregation, t h e aim of which is t o
preserve and integrate only the issues that a r e essential f o r t h e higher
level of t h e analysis and hierarchy, ruling out t h e unnecessary details. In
this way a sequences of corresponding detailed and aggregated (mainly
descriptive type) models are developed. Only aggregated models a r e coupled in an on-line fashion (the approach is off-line f o r t h e detailed models)
thus allowing their incorporation in a management optimization model a t the
highest stratum of the hierarchy. The procedure f o r deriving t h e eutrophication management optimization model (EMOM) is illustrated in Figure 1 (for
f u r t h e r details, s e e Somlybdy, 1983b; and Somlyddy and van Straten, 1985)
and consists of four stages:
Phase 1
This i s t h e development phase of t h e dynamic, descriptive lake eutrophication model (LEM) which has two sets of inputs: controllable inputs
(mainly artificial nutrient loads) and noncontrollable inputs (meteorological
factors, such as temperature, solar radiation, wind, precipitation). The
output of t h e model i s t h e concentrations vector y of a number of water
quality components as a function of time (on a daily basis) t , and space r :
y (t , r ) . LEM is calibrated and validated by relying on historical data; t h e
inputs of LEM are the recorded observations. Because of methodological
and computational difficulties on one hand, and uncertainties in knowledge
Loads
Met
LEM
Development phase
~ ( trl,
-NLMP 1
k-
LEM
-4
METG
Planning mode usage
-
Yi(t)
'i
I
I
Water quality indicator(s)
for management.
I
I
I
I
I
r----- I-+
I
I
I
I
I
I
I
I
Y+
-I
-.
---1..
I
I
I
I
Management
(optimization)
phase
Checking phase
FIGURE 1.S t r u c t u r e of t h e Analysis.
and observational d a t a on t h e o t h e r hand, most of t h e models are not "continuous" in space, but r a t h e r they describe t h e a v e r a g e water quality of
lake segments ( b a s i n s ) having differing characteristics.
For details,
r e a d e r s are r e f e r r e d t o t h e l i t e r a t u r e on ecological modeling (e.g,
Somlyddy and van S t r a t e n , 1985).
Phase 2
In o r d e r t o apply LEM at a l a t e r s t a g e f o r solving t h e management
problem, two important s t e p s must b e taken:
(i)
A decision has t o b e made as t o t h e kinds of loads and meteorologic a l inputs t o b e used in planning scenarios. For r i v e r water quality planning problems deterministic "critical" scenarios can b e
generally used, but f o r lake eutrophication problems (especially
f o r shallow water bodies), t h e inputs should b e considered stochastic functions. In Figure 1 NLMPl and METG indicate models
which g e n e r a t e t h e loads and meteorological f a c t o r s in a random
fashion based on t h e analysis of historical data. The p r o p e r time
scale of NLMPl and METG should b e established in Phase 1 ; i t generally suffices t o use a month as a basis (Somlybdy, 1983b).
(ii) The planning type nutrient load model (NLMPl) incorporates
aggregated control variables, f o r deriving t h e ' loads of individual
lake basins ( 1 S i
S
N). They are used t o determine classes of
aggregate load scenarios with different expectations and fluctuations f o r each individual basin, but they do not e x p r e s s t h e way in
which a scenario i s actually realized in t h e watershed,
The LEM can then be run systematically under stochastic inputs with different control variable vectors zi yielding t h e stochastic vectors y i ( t )
Pw
that describe t h e temporal changes in various water quality components f o r
t h e basins (1 S i
S
N) (in contrast to t h e y i ( t ) of Phase 1which a r e deter-
ministic and noncontrolled). These changes are of interest f o r understanding t h e eutrophication process, but f o r decision making purposes, various
indicators that express t h e global behavior of t h e system can be used (see
Section 1).
Experience has shown (OECD, 1982; Somlyl&y, 1983b) t h a t because of
t h e cumulative nature of eutrophication, t h e indicators primarily depend on
t h e annual average nutrient load with t h e (annual) dynamics of secondary
importance, at least f o r lakes whose retention time is not too small. This
means that time can be disregarded and w e can work with (vector-) indicators,
s*,
(3, (t), 1S i
N
that
S N).
are
derived
from
the
concentration
vectors
This also implies t h a t t h e LEM can be used in an off-line
fashion.
There are various ways in which t h e LEM can be involved in t h e optimization model, and implications and conclusions requires careful analysis.
For example, t h e
xi indicators obtained from systematic computer experi-
ments can be stored as a function of t h e corresponding aggregated control
vectors. These form a "surface" on which t h e "optimal" solution is built
later. The o t h e r possibility, t h a t w e follow here, i s to parametrize t h e
*
Note that the number of indicators i s generally less than that of the water quality components, and often a single (scalar) indicator i s employed (see later).
outputs of LEM (obtained under various control vectors
& ) and t o a r r i v e at
an analytical expression which then can be included in t h e optimization
model. This i s the LEMP model, cf. Figure 1.
As indicated in Figure 1,t h e success of this procedure i s not a p r i o r i
obvious and depends mainly on t h e complexity and major f e a t u r e s of t h e sys-
tem. Still, observations and model results have shown f o r several lakes that
a linear relationship holds between t h e indicator (e.g., ((Chl-a),,)
and t h e
annual average phosphorus load (OECD, 1982; Somlybdy, 1983b; Lam and
Somlybdy, 1983). Van S t r a t e n (1983) has proven analytically by using a simplified LEM, t h a t f o r nutrient limited, turbid shallow lakes the maximum
algal level is practically proportional t o t h e phosphorus load. Based on
these findings, w e assume t h a t a n analytically expression, moreover a linear
model, can be used as LEMP f o r a l a r g e variety of water bodies.
LEMP is an aggregated version of the LEM t h a t can b e employed
directly f o r planning purposes. I t describes approximately t h e indicators
5 as a function of t h e nutrient load, including stochastic variability.
LEMP
is connected t o a nutrient load model, NLMP2, which includes the control
variables, zil ( 1 5 I
5
NLf ; 15 i
5
N). Compared with NLMP1, the NLMPZ
model exhibits t h r e e significant differences:
(i)
i t covers only t h a t p a r t of t h e load t h a t i s thought t o be controllable on t h e basis of available, realistic measures;
(ii) i t contains more details about t h e subwatersheds of each basin
(location of pollution discharges and control measures, etc.); and
(iii) i t is aggregated with r e s p e c t t o time; as f o r LEMP t h e annual
average load can generally b e used.
The coupled N L M P 2 - L E M P m o d e l s a r e then put through an optimization
procedure, which generates (depending on the formulation of EMOM, the
Eutrophication Management- Optimization Model) f o r example, "optimal"
values f o r the control variables, z i t , the associated indicators, ; '?
and-
say-the total annual costs, TAC, needed f o r carrying out t h e project o r
fixed by budgetary considerations.
Phase 4
In the course of this procedure various simplifications and aggregations a r e made without a quantitative knowledge of t h e associated e r r o r s .
Accordingly, the last step in the analysis is validation. That is, the L E M can
be run with the "optimal" load scenario as indicated in Figure 1 by the
dashed line, and the
5
"accurate" and
"approximate" solutions can be
compared.
3.
FORMULATION OF THE STOCHASTIC MODELS LEKP AND NUiP2
The modeling procedure w a s outlined in Section 2 in general terms.
Here we continue the discussion detailing assumptions and limitations. The
reason f o r these is threefold:
-
All the models used in the field of eutrophication a r e system
specific;
-
Consequently o u r objective cannot be more than to capture some
of t h e major features of eutrophication management problems;
and
-
W e wish t o avoid generalization beyond o u r experiences.
The major assumptions we make a r e as follows:
(i)
t h e lake is shallow, t h e water quality of which is vertically uniform:
(ii) t h e lake can be subdivided into basins which are sequentially connected, see Figure 2;
(iii) t h e lake is phosphorus limited (like most water bodies) and thus
nutrients o t h e r than P (phosphorous) are not involved in the
analysis;
(iv) a single water quality indicator, t h e (Chla),,,
concentration is
used f o r defining trophic s t a t e and t h e goal of management (see
Section 1);
(v) as discussed before linear relationship holds f o r individual basins
between (Chl-),,,
and P load;
(vi) s h o r t t e r m (a f e w years) management is considered, t h a t i s t h e
renewal processes in t h e lake and its sediment layer following
external load reduction, and t h e scheduling of t h e investments are
out of t h e scope of t h e present effort;
(vii) only certain types of P sources and associated control alternatives are taken into account.
A t t h e end of this section, w e w i l l show t h a t some of these conditions can be
relaxed if needed, and in fact the models t o be introduced have a broader
range of applicability than suggested by t h e above assumptions.
3.1. Aggre~atedLake Eutrophication model (LEW)
Based on Section 2, t h e assumptions made and t h e insight gained from
t h e study on Lake Balaton (Section 8), the s h o r t term response of water
quality t o load reduction can be written as follows (Somlybdy, 1983b):
where t h e elements of the N-vector
water quality in t h e N basins
- in
represent the (to be controlled)
t e r m s of (Chl-a),,,
[mg/m 3I
- and Y,
refers t o t h e (noncontrolled) nominal state; E i s expectation. In the equation, t h e N-vector
4 expresses t h e change in load due t o control
where t h e elements of
L are the annual mean volumetric
able P load, BAP [mg/m3d] in each basins
biologically avail-
(LJ = LJa/h.h e r e
L_P
i s the
"absolute" load [mg/d] and Vi is the volume [m33). The BAP load covers t h e
P fractions that can be taken up directly by algae and thus determine t h e
short term response of t h e water body. Stochastic variables and stochastic
parameters are bold faced. The random N-vector lt;l represents t h e random
changes of water quality caused by noncontrollable meteorological factors.
Finally t h e elements of t h e square NxN-matrix D and the vector d are
derived from t h e analysis and simulation in Phase 2 (see Figure 1).
The elements of matrix D are the reciprocals of lumped reaction rates.
The main diagonal comprises primarily t h e effect of biological and biochemical processes, while t h e o t h e r elements refer t o those of interbasin
exchange due t o hydrological throughflow and mixing.
These elements
express that due t o water motion a control measure taken on subwatershed
of basin i will affect t h e water quality of o t h e r basins. A s will b e seen in
Section 8, the diagonal elements dii of the matrix D (the slopes of linear
load response relationships) are such that (Chl-a),,,
approach z e r o ( o r a relatively small value) if
t h a t (Chl-a),,,
Li
does not necessarily
goes to 0. The reason is
i s linearly related t o t h e sum of external and internal loads.
The internal load is t h e P release of t h e sediment (a consequence of P accumulation during preceding y e a r s and decades) which practically cannot b e
controlled. Since external and internal loads are coupled (a reduction in
external load generates a time-lagged reduction in t h e internal load) t h e
long-term improvement of water quality is generally l a r g e r than given by
Equation (3.1). The memory effect and renewal of sediment are however
poorly understood (see f o r example, Lijklema et al, 1983), this is one of t h e
reasons why w e concentrate on short-term control. Note t h a t due to t h e
definition of ,,LA
(3.1) also yields t h e random variations of water quality in
t h e (noncontrolled) nominal state. The c h a r a c t e r of slopes di is similar t o
t h a t of t h e diagonal elements d i i , t h e t e r m di zui hLI
expresses a change,
linear in hLNf, in t h e random component of the water quality indicator in
basin i. Of course, t h e effect of t h e random fluctuations E~ caused by
meteorology decreases if t h e sum of external and internal loads diminishes.
This also means t h a t with new control measures t h e water quality of a lake
may approach a "new equilibrium" via m a j o r fluctuations, as observed in
nature.
In view of (3.1) water quality varies randomly f o r t h r e e reasons:
(i)
random changes in meteorological f a c t o r s (primarily s o l a r radiation and temperature)
(the distributions of t h e UJ~, typically
skewed, are obtained in Phase 2 of t h e procedure (Figure 1 ) and
can generally b e approached by three-parameter gamma distributions);
(ii) stochastic changes and uncertainties in t h e loads (see below); and
(iii) t h e combined effect of climatic and load factors.
Relation (3.1) gives t h e aggregated lake eutrophication model. The
model takes into account on a macroscopic level t h e effect of biological and
biochemical processes, interbasin mass exchanges, t h e sediment furthermore t h e influence of stochastic factors and uncertainties.
3.2 N u t r i e n t Load Model (NTXP2)
W e consider t h r e e P sources as indicated in Figure 2:
(i)
d i r e c t sewage P load, LS;
(ii) indirect sewage P load when t h e recipient i s a tributary of the
lake, LSN (both LS and LSN can be considered biologically available and deterministic); and
(iii) tributary load t o which contribute various point sources (sewage
discharges) and non-point sources of t h e watershed.
The biologically available portion of t h e tributary load is
L
a
=L& + b(LT -&J,),
where L
& i s t h e dissolved reactive P load,
LTtotal P load and b availability
r a t i o of t h e particulate (not dissolved) P load (the difference of
L&), b i s a%out 0.2.
(3.3)
kT and
Direct sewage
1
l ndirect
sewage
FIGURE 2. Development of models LEMP and NLMPZ.
Now, l e t us consider t h e basic control options from t h e above P loads:
(i) and (ii)P precipitation by sewage treatment plants, and
(iii) pre-rersevoir systems established on r i v e r s before they e n t e r t h e
lake. These consist of two p a r t s (Figure 2): t h e removal of particulate P through sedimentation in t h e f i r s t segment, and t h e removal of dissolved P (benthic eutrophication in reed-lakes, sorption,
etc.) in t h e second p a r t . The corresponding control variables are
z s , x ~ , z p and
,
zg as indicated in Figure 2. All of them can be
thought as removal coefficients, with
where r
- and
r + a r e lower and upper limits respectively, and if
z = 0,no action i s taken.
Now consider t h e simple situation given in Figure 2 f o r t h e i t h basin of
t h e lake. The original, uncontrolled load,
where L
a
Lican be expressed as follows:
is t h e portion of t h e load t h a t is beyond t h e controls considered
(e.g., atmospheric pollution); f o r the s a k e of simplicity, w e d r o p t h e index
i from t h e r i g h t hand side of Equation (3.5). The controlled load of t h e 2-th
basin is
where r t is t h e retention coefficient. The expression ( 1
- r t ) replaces a
r i v e r P t m n s p o r t model and i t defines t h a t portion of P t h a t r e a c h e s t h e
lake from an indirect sewage discharge at a given point on t h e tributary
(rt
= 0 means no retention).
I t is apparent from Equation (3.6) t h a t the tri-
butary load can b e controlled by P precipitation and/or pre-reservoirs.
The latter influence linearly both expectation and variance of
ki,while P
precipitation only influences the expectation, thus t h e r e is an obvious
trade-off between these alternatives. With equations (3.5) and (3.6) w e can
now obtain
hNi
(recall t h a t
bL1
= ALL/Q):
Terms
@
and
@
express reduction in t h e expectation of t h e r i v e r s dis-
solved P load; term
@ represents t h e effects of
t h e fluctuations of the L_D
@ gives t h e modification in the particulate P load of t h e river,
while term @ exhibits influence of direct sewage control. If we set all t h e
load: term
control variables x t o zero in Equation (3.7), w e obtain t h e fluctuations in
t h e original, noncontrolled load, t h e expectation of which i s zero.
Equation (3.7) is nonlinear in t h e control variables because of t h e product term XD
m
xm.which may
cause difficulties when this relation is used in
an optimization scheme. Several possibilities a r e available t o overcome
this nonlinearity. For example, a new variable
can be introduced, a linear function of XD and XSN, which i s then included in
t h e constraint equation, see e.g.,Loucks et al., 1981. In such a case t h e
optimization requires a parametric analysis involving this variable.
Another possibility i s offered by t h e surface dependent c h a r a c t e r of
t h e P removal in t h e pre-reservoir (second element). Generally, f o r a r e e d
lake one cannot estimate more than t h e P removal p e r unit of surface-area,
independent of t h e inflow concentration. Under this approximation, howe v e r , XD can be defined in terms of t h e original uncontrolled load of t h e
inflow (in c o n t r a s t t o t h e t h e conventional definition: actual inflow minus
outflow divided by t h e actual inflow) as a variable which i s not influenced by
indirect sewage control. The p r i c e f o r such a n elimination of nonlinearity
is twofold:
(i)
An u p p e r limit should b e specified f o r zg which states t h a t no
more nutrients can b e removed than those t h a t r e a c h t h e lake via
t h e p a r t i c u l a r t r i b u t a r y . In terms of expectations t h e constraint
equation can b e written as
(we note t h a t in a more precise sense t h e above condition should
actually b e fulfilled f o r all t h e realizations of Ld).
(ii) A new variable z$ should b e introduced with zg 5 z$ t o t a k e into
account t h e f a c t t h a t t h e impact of t h e r e s e r v o i r on t h e fluctuation (Equation (3.7): term
@
is not r e s t r i c t e d by t h e (physical)
constraints introduced by (3.9).
For a more general situation than illustrated in Figure 2, when t h e i t h
lake basin is fed by N i d i r e c t sewage discharges ( 1
t a r i e s (1 s m
S
N2), each with
Equation (3.7) becomes:
4 indirect sewage
<n
5
N1) and N2 tribu-
discharges (1 9 1 S M,),
Equation (3.10) is actually t h e final NLMP2 model e x c e p t t h a t w e have not
yet discussed t h e derivation of t h e stochastic load components
kT and La,
which is in itself a difficult problem because insufficient (infrequent) observations, s h o r t historical data, and o u r lack of understanding (see e.g.
Haith, 1982; Beck, 1982). Since t h e annual means are used f o r I,T and L
a
and t h e dynamics are less important, t h e recommendation is t o derive t h e
loads from a regression type analysis as functions of t h e major hydrologic
and watershed parameters ( e r r o r terms should also b e included).
Observations and careful analysis of t h e composition of t h e load (point
vs non-point s o u r c e contributions) and watershed are required f o r such a
procedure. In general
LTand La
have (positive) lower bounds and can b e
characterized by strongly skewed distributions. V e r y often they can b e
expressed as simple functions of t h e annual mean streamflow r a t e s ,
&,
the
statistics of which are generally known from much longer r e c o r d s than
those available f o r loads. This way t h e basic stochastic influence of hydrologic regime can b e involved. In many cases, annual means of
and Q are
estimated from scarce observations. The uncertainty associated can b e
investigated from a Monte Carlo type analysis o r basic statistical considerations (Cochran, 1962: SomlyWy and van S t r a t e n , 1985). Taking into account
all these f a c t o r s , t h e loads can have r a t h e r complex distributions composed
of various normal, log-normal, gamma,. .. distributions.
The model LEMP will b e used together with NLMPZ given by Equation
(3.10) as major components of t h e eutrophication management-optimization
model, EMOM. Let us note t h a t after introducing (3.8) o r (3.9) into NLMPZ,
t h e water quality indicator will b e expressed by t h e coupled model NLMP2LEMP as a l i n e a r function of t h e decision variables, z t l .
To conclude this section, we r e t u r n t o assumptions (i)-(vii) and check
t h e i r "rigour" in t h e light of t h e knowledge gained in t h i s section:
(i)
shallowness i s not r e s t r i c t i v e from t h e point of view of using
NLMPZ;
(ii) non-sequential connection of t h e basins would simply involve a
change in t h e s t r u c t u r e of D;
(iii) P is practically t h e only element t o b e controlled even in c a s e s
when e.g., nitrogen i s limiting algal growth (OECD, 1982; Herodek,
1983); and
(vi) short-term
management
basically
determines
the
long-term
behavior of t h e lake water quality, thus t h e r e s u l t s h e r e have
strong implications f o r long-term management;
(vii) controls disregarded h e r e (e.g., sewage diversion) have similar
f e a t u r e s from a methodological viewpoint than t h o s e handled in
t h e present paper.
- 23 4.
CONTROL VARIABLES, COST FUNCTONS AND CONSTRAINTS
We now consider t h e management options t h a t a r e available t o control
t h e eutrophication p r o c e s s and t h e restrictions t h a t limit t h e i r choice.
4.1. C o n t r o l V a r i a b l e s
We already touched upon some of t h e decision variables as removal
rates of sewage treatment plants and r e s e r v o i r s . Here we give a more
detailed discussion of possible control variables, including t h e i r c h a r a c t e r
and associated bounds.
Sewage
Treatment
(i) If we consider artificial eutrophication as a problem of t h e lakewatershed system and w e are not interested in t h e details of t h e engineering
design of each treatment plant, zs and z~/vshould b e handled as real-valued
variables; otherwise all t h e elements of t h e technological process of all t h e
plants should have been taken into account with t h e aid of f0,11 variables.
Lower bounds f o r zs and XSN often exist, since in many countries t h e
effluent P concentration is fixed by standards (between 0.5-2 g/m3) specified by environmental agencies.
(ii) Because of historical reasons
- at least in developed countries -
P precipitation (chemical treatment) is going t o b e realized in existing
plants designed originally f o r biological treatment. If t h e efficiency of h e
biological treatment is unsatisfactory, which is often t h e case, i t should b e
upgraded p r i o r to introducing chemical treatment.
If decision variable
associated t o upgrading i s z
,, this type of treatment plant management can
- 24 b e taken into account by t h e condition
sgn zu 2 sgn zs,
if z
, is a real-valued variable.
(iii) If t h e problem incorporates not only decision about P precipitation and t h e associated upgrading, but also t h e design of t h e sewer network,
this would again lead t o a n engineering type planning task (e.g., Kovdcs et
a l , 1983) t h a t differs from o u r present considerations. If, however, w e
wanted t o involve sewer network design in t h e eutrophication management in
a simple way, t h e network of t h e subregions could b e o r d e r e d s o as t o
correspond t o treatment plants and handled similarly as explained in item
(ii), after introducing decision variables zm.
(iv) Finally, a rough p i c t u r e on t h e spatial distribution of treatment
plants can b e obtained defining corresponding f 0 , l j variables (0 means
again no action).
(i) A s indicated in Section 3, basic P removal processes are s u r f a c e
dependent, thus control variables zg and zp specify t h e size of r e s e r v o i r s .
Lower bounds are generally given by t h e smallest reasonable retention time
t h a t corresponds t o a given size.
(ii) Variables ZD and zp must often b e handled as 10,1{variables.
4.2. Cost Functions
Costs and cost functions a r e discussed in t h e same o r d e r as decision
variables above.
Sewage Weatment
(i) The costs of P precipitation (including investment and operation
costs being site specific) grow exponentially with increasing removal rates
and decreasing effluent concentrations (see e.g., OECD, 1982; Monteith et
al., 1980; Schiissler, 1981). The most straightforward method t o approach
this type of cost functions is piecewise linearization (see e.g., Loucks et al.,
1981) which requires t h e introduction of dummy variables.
(ii)-(iv) They are fixed costs.
Pre-reservoirs
Cost functions of r e s e r v o i r systems are strongly depending on which
process determines primary P removal, construction and operation conditions. Very often not enough knowledge i s available t o define a cpst function o t h e r than linear and usually running costs can be neglected o r can be
assumed t o be compensated by the benefits of t h e r e s e r v o i r s (e.g., utilization of harvested reeds). A s a summary, w e can conclude t h a t costs can be
expressed as piecewise linear functions of t h e decision variables, an important feature from t h e viewpoint of building up t h e management model EMOM.
4.3. Constraints
A s we already discussed most of the physical, technological and logical
constraints
- cf. (3.4), (3.8), (3.9),
(4.1) and (4.2) and related explanations
- we consider h e r e only t h e budgetary constraint which is perhaps t h e most
important one.
In o r d e r t o select among management alternatives of different investment costs (IC), and operational, maintenance and r e p a i r costs (OC), the
total annual cost (TAC) t e r m is used
TAC =
C OC1 + C al ICl
where al is t h e capital recovery factor that depends on t h e discount rate
and t h e lifetime of t h e project (see e.g.. Loucks et al., 1981). This factor
can be different f o r "small" and "large" projects (e.g., introduction of P
precipitation in treatment plants and creation of reservoirs of considerable
size, respectively) as f o r '7arrge " investments governments of ten guarantee
finance a t low ("pure") interest rates. For this reason, as pointed out by
Thomann (1972), al should be considered as a model parameter of a certain
range and its influence on model performance should be tested.
In most cases t h e TAC i s limited by budgetary considerations
TAC
S
fl
(4.4)
o r reexpressing this in t e r m s of the control variables
This constraint, which involves t h e piecewise linear functions cil, can be
replaced by a linear constraint by substituting x $ , x i ... f o r t h e variable
- 27 ztl, each z&corresponding t o a piece of linearity of cil .
5.
FORMULATION OF TEIE EUTROPEFTCATION HANGEBIENT OPTIMIZATION
YODEL: EMOH
A s already indicated in Section 1 , t h e r e are a number of variants avail-
able in t h e building of t h e management optimization model t h a t allow us t o
c a p t u r e t h e stochastic f e a t u r e s of t h e water quality management problem.
The NLMP2-LEMP model yields t h e following description of t h e eutrophication process:
z = E
&=E
fzo!
+%
-((D
+ d ~ UN.
)
IkoI - k ,
(3.1)
(3.2)
where f o r t h e s a k e of convenience, t h e water quality indicator is denoted by
y , and f o r i
= 1,...,N ,
-
La,i = bpi
+ a ( L , ~ , i L4,t )
+
Ls,~ + LsN,~ + L x , i
(3.5)
t h e last relation being valid under some additional constraints, see (3.9) and
t h e related comments.
Substituting, regrouping terms and renumbering
(reindexing) t h e control variables, we obtain a n affine relation f o r t h e
water quality indicators y i (1 S i
S
N) of the type
N
y
N
where
kt
quality
=
cz - h
(5.1)
N
incorporates all t h e noncontrollable factors that affect t h e water
zi
in basin i , and t h e random coefficients associated t o t h e z-
variables in (3.10) determine t h e entries of t h e random matrix
t h e transformation: ( D
+ dzu)ALN.
through
To simplify t h e presentation of what fol-
lows, w e string t h e decision variables in an n-vector (xi,. . . , z,),
each zj
corresponding t o a specific control measure affecting t h e load in some
basin i .
is thus a N
X
n-matrix and
is a n N-vector. We also write
f o r t h e preceding equation, t h e notation y(z ,ul)is used t o stress t h e depenN
dence of t h e water quality indicators y i (1 S i
S
N) on t h e decision vari-
N
ables z and on t h e existing environmental conditions, denoted by w (controllable and noncontrollable) determining the entries of T and h .
The distribution function Gy ( z , -) of the random vector y ( z , *) depends
on t h e choice of t h e control measures x i , ...,z,.
W e could view o u r objec-
tive as finding z * t h a t satisfies t h e constraints and such t h a t f o r every
o t h e r feasible z
i.e., such that f o r all z
E
R~
prob. [ y ( z S , -1 < z ] r p r o b . [ y ( z ,
m)
<z] .
If such a n z * existed it would, of course, be t h e "absolute" optimal solution,
since it guarantees t h e best water quality whatever be t h e actual realization of t h e random environment. There always exists such a solution if
t h e r e are no budgetary limitations: simply build all possible projects t o
t h e i r physical upper bounds ! If this were t h e case, t h e r e would be no need
f o r this analysis and i t is precisely because t h e r e are budgetary limitations
t h a t w e are led t o choose a restricted number of treatment plants and/or
pre-reservoirs, and unless t h e problem is very unusual, t h e r e will be no
choice of investment program t h a t will dominate all o t h e r feasible programs
in terms of t h e preference ordering suggested by (5.3).
W e are thus forced t o examine somewhat more carefully t h e objectives
w e want t o achieve. We could, somewhat unreasonably, s e e t h e goal as
bringing t h e water quality indicator t o a n e a r zero level (depending on t h e
internal load) in all basins. This would ignore t h e individual characteristics
of each basin, as w e l l a s t h e user-oriented c r i t e r i a , a s f o r example r e c r e a tional versus agricultural. A much more sensible approach, as discussed in
Section 1, is t o choose t h e control measures s o as t o achieve certain
desired trophic states. Let
be water quality goals expressed in terms of t h e selected indicator,
(Chla),,,,
each yi corresponding t o t h e particular use of basin i . The
sensitivity of t h e solution t o these fixed levels yi would have t o be a p a r t of
t h e overall analysis of t h e system. W e a r e thus interested in t h e quantities:
C Y ~ ( ~ ,w ) - 7 i I +
for i
= 1,..., N ,
t h a t measure t h e deviations between t h e realized water quality and the
fixed goals y i , where [@I+ denotes t h e nonnegative p a r t of @:
The vector
is random with distribution function G ( z ,
0)
defined on R ~and
, the problem
is again t o choose among all feasible control measures z l,
. . . ,z,
, i.e., that
satisfy all technological and budgetary constraints, a program z s that gen-
erates the "best" distribution G ( z s , e) by which once could again mean
f o r all
2 E
A s already mentioned earlier, such an z s exists only in
very unusual circumstances, and thus we must find a way t o compare the distribution functions that takes into account their particular characteristics
but leads t o a measure that can be expressed in t e r m s of a scalar functional.
5.1. The first possibility would be t o introduce a pure r e l i a b i l i t y criterion, i.e., t o fix, in consultation with t h e decision maker, certain reliabil-
ity coefficients t o guide in the choice of an investment program. More
specifically we would fix 0
< a S 1, so that among
all feasible z w e should
r e s t r i c t ourselves t o those satisfying
where 7 = ( 7 1 , . . . , y , ~ . O r preferably, if w e take into account t h e fact that
each basin should be dealt with separably, w e would fix the reliability coeff icients a t , i
= 1,...,m Z, and impose t h e constraints
prob. [ y i ( z ,
0)
< y i ] r a t , i = 1,...,mo.
(5.5)
The scalars a o r ( a i , i = l , , . . , m z ) being chosen sufficiently large s o that
we would observe the unacceptable concentration level only on r a r e
In terms of t h e distribution function G , these constraints
occasions.
become
f o r (5.4), and
Gi(z,O) 2 at
for i
= 1,...,N ,
(5.7)
f o r (5.5) where t h e Gi ( z , *) are t h e marginal distributions of t h e random
variables [ y i ( z , 0)
- y i ] +. I n t h e parlance of stochastic optimization models
these are p r o b a b i l i s t i c (or chance) c o n s t r a i n t s ; one refers t o (5.4) as a
j o i n t probabilistic constraint. To find a measure f o r comparing t h e distri-
butions f G ! z , 0). z feasible j w e have a simple accept/reject criterion,
namely if a t 0 G ( z , 0) is e i t h e r l a r g e r than o r equal t o a , o r f o r each
i
= 1 ,...,N, Gi ( z , 0) i s l a r g e r than o r equal t o a t , t h e investment program z
is acceptable and otherwise i t is rejected. This means that w e "compare"
t h e possible distributions f G ( z , a), z feasible j at 1point, but w e do not consider any systematic ranking. Assuming w e opt f o r t h e more natural separable version of t h e probabilistic constraints (5.5), w e would r e l y on t h e following model f o r t h e policy analysis:
E Rn
find z
-
rj
such t h a t
j =I,... , n ,
s z j s ff+,
CT=la i j z j
prob
and z
bi *
[zj'=l
= z.j?=l
I
t i j (W )zj
- hi (W ) < 7 i 1I 2 a t ,
...,N,
1 51,
cj ( z j ) is minimized
where as before t h e vector r
inequalities
i =I, ...,m ,
EGl aij z j
S
bi
- and
r
+
are upper and lower bounds on z , t h e
describes t h e technological constraints,
including (3.9), and f o r every j
is t h e cost function associated t o project j , see (4.5). The overall objective would thus be t o find t h e smallest possible budget that would guarantee
meeting t h e present goals yi at least a portion at of t h e time.
W e did not pursue this approach because i t did not a l l o w u s t o distinguish between situations where w e almost m e t t h e p r e s e t goals yi and those
t h a t generate "catastrophic" situations, i.e., when some of t h e values of t h e
(pi ( x ,w ), i
= 1,...,N) would exceed by f a r (yi, i = 1 , . . . ,N) and f o r pur-
poses of analysis of this eutrophication model this i s a serious shortcoming.
Let us also point out that t h e r e are also major technical difficulties t h a t
would have t o be overcome. Probabilistic constraints involving &fine functions with random coefficients are difficult t o manage. W e have only very
limited knowledge about such constraints, and then only if t h e random coefficients ((ti,
hi (a)) are jointly normally distributed, cf. Section 1of
Wets, 1983a f o r a survey of t h e available results and t h e relevant references. Since in environmental problems the coefficients are generally not
normally distributed random variables w e could not even use t h e f e w results
that are available, except possibly by replacing t h e probabilistic constraints by approximates ones using Chebyshev's inequality, as suggested by
Sinha, cf. Proposition 1.26 in Wets, 1983a.
5.2. A second possibility i s t o recognize t h e f a c t that one should distinguish between situations that barely violate t h e desired water quality o r
levels (yi, i
norms.
= 1 ,...,N) and those that deviate substa.ntially from these
This suggests a formulation of o u r objective in terms of a
penalization
[vi (z ,w )
that
- yi ]
+
would
for i
take
into account t h e observed values of
= , . - . ,N. We expect such a function
q : R~ 4 R
t o have t h e following properties:
(i)
4' is nonnegative,
(ii) q ( z )
= 0 if zi
5
0
i = l?....N,
(iii) 4' is separable, i.e., q ( z )
=
zrZlqi
(zi).
This last property comes from t h e fact t h a t t h e objectives f o r each basin
are o r may be different and t h e r e are essentially no "joint rewards" t o be
accrued from having given concentration levels in neighboring basins, t h e
interconnections between t h e basins being already modeled through t h e
Equation (3.1). A more sophisticated model, would still work with s e p a r a t e
I
penalty functions ql(z
..-
. 'PN(zN)]but instead of simply summing these
penalties, would treat them as multiple objectives. A solution t o such a
problem would eventually assign specific weights t o each basin, making i t
equivalent t o a n optimization problem with single objective function. W e
shall assume t h a t these weighting f a c t o r s have been made available t o o r
have been discovered by t h e model builder, and have been incorporated in
t h e functions qi themselves; note however that t h e methodology developed
h e r e would apply equally w e l l t o a multiple objective version of t h e model.
In addition, t o (i)-(iii) w e would expect t h e following properties:
i = l , - - . ,N,
(iv) qi is differentiable, with derivative
qi.
for
(v)
(vi)
+; is monotone increasing, i.e., +* is convex,
+; ( z t ) > 0 whenever zt > 0,
-
relatively small if zt i s "close" t o 0,
-
leveling off when zt is much "larger" than 0.
A couple of possibilities, both with Ot ( z * )= 0 if z f 5 0, are
+ * ( z * )=
with
p*
2:
If
Z*
P O ,
pi > 0,
+ * ( z t ) = pt(ez' - z t
also with pi
-1)
if zi 2 0 ,
> 0.
A s w e see, t h e r e is a wide variety of functions t h a t have t h e desired
p r o p e r t i e s , what i s at s t a k e h e r e is t h e creation of a (negative) utility function t h a t measures t h e socio-economic consequences of t h e deterioration of
t h e environment. W e found t h a t t h e following class of functions provided a
flexible tool f o r t h e analysis of t h e s e factors. Let 8 : R -, R + b e defined by
This
(+*, i
is a
piecewise
linear-quadratic-linear
function.
The functions
= 1,...,N ) are defined through:
Or(Zr)
= Q* ei 8 ( e c i z t ) f o r = 1,...,N ,
(5.10)
where qt and et are positive quantities t h a t allow us t o scale each function
+* in t e r m s of slopes and t h e r a n g e of i t s quadratic component.
By varying
t h e parameters et and qt w e are a b l e t o model a wide r a n g e of p r e f e r e n c e
relationships and study t h e stability of t h e solution under perturbation of
these scaling parameters.
FIGURE 3. Criteria functions.
The objective i s thus to find a program that in the average minimizes
the penalties associated with exceeding the desired concentration levels.
This leads us to the following formulation of the water quality management
problem:
find z
E Rn
such t h a t
rj- s zj s rj+ ,
and
2
=E
[z!=l qi ei
6 ( e P [ y i (w )
- 7i 1)
1
is minimized
where B is t h e available budget. This type of stochastic optimization problem goes under t h e name of stochastic program w i t h recourse: a decision
z (the investment program) must b e chosen before w e can observe t h e outcome of t h e random events (the environment modeled h e r e by t h e random
quantities tij(w),hi (w)) at which t i m e a r e c o u r s e decision is selected s o as
t o make up whatever discrepancies t h e r e may be; t h e variables yi are just
measuring t h e difference between
zj tijxj and
hl . One refers t o (5.11) as
a program with simple recourse in that t h e r e c o u r s e decision is uniquely
determined by t h e first-stage decision z and t h e values taken on by t h e random variables.
It is very important t o note that no attempt has been made at combining
budgetary considerations and t h e penalty functions t h a t measure t h e deviations from t h e desired concentration levels in a single objective function,
although t h e r e are financial considerations t h a t may affect t h e choice of
t h e coefficients qi and ei of t h e penalty terms. In o u r approach w e handle
these two criteria separately, w e rely on a (discrete) parametric analysis
of the solution of (5.11) as a function of
B, the available budget. An essen-
tially equivalent approach would have been t o formulate (5.11) as a multiobjective program, one objective corresponding t o the penalizations terms.
t h e o t h e r t o t h e cost function.
In terms of t h e distribution functions f G ( z , a), z feasible1 t h e entire
'tail" of t h e distributions e n t e r s into t h e comparison not just t h e value of
G ( z , *) at 0, as w a s t h e c a s e in model ( 5 . 8 ) with probabilistic constraints.
Indeed, the objective function can now b e expressed as:
5.3. A third possibility is t o essentially ignore t h e stochastic aspects
of t h e eutrophication model and replace t h e random variables t h a t a p p e a r
in t h e formulation of t h e water quality management problem by fixed quantities. This would lead us t o t h e following d e t e r m i n i s t i c o p t i m i z a t i o n prob-
lem:
find z
E
Rn such t h a t
rj- s z j s r;
C?=I % j z j
bi
C&1 c j ( z j )
S
Cj%1
tij z j -vi
and r
=
x:=,
j=1,
. . . ,n ,
i=1,
-
,m ,
B
pi
i = l , . . -N ,
=&
ei B [ei-l[yi - T ~ ]is] minimized.
The choice of t h e paramaters
fij
and
&
is left t o t h e model builder. One
possibility is t o choose
i.e., replace t h e random quantities by t h e i r expectations. Without accepting t h e solution of ( 5 . 1 2 ) , w e could always use i t as p a r t of an initialization
scheme f o r solving t h e stochastic optimization problem (5.11), and this i s
actually how t h e algorithm proceeds, see Section 7.
5.4. A fourth model
- in which reliability considerations again occupy
a central r o l e but in which t h e shapes of t h e distribution functions
)Gi ( x , a ) , i
= 1,..., m2j
values at one point
( a i , i =1,
play a much more important r o l e than just their
- allows f o r variables concentration levels.
Again let
, N ) be scalars that correspond t o desired reliability level.
The objective is t o find an investment program x such t h a t
but now t h e ( v i , i
= 1,...,N ) are also decision variables that w e like t o
choose a s low a s possible. There is a variety of ways measuring "as low a s
possible", f o r example by minimizing
where the qi are nonnegative scalars that assign different importance t o
meeting t h e desired water quality goals ( y i , i
= 1,...,N) in t h e various
basins, o r by minimizing
max v i
(5.15)
i
i.e., by bringing t h e overall concentration level as f a r down as possible (at .
least a certain portion of t h e time determined by t h e a; s), o r by minimizing
as in model (5.11) t h e function
which penalizes t h e deviations from yi in a nonlinear manner, cf. Figure 3,
o r still t o handle t h e minimization of t h e ( v i , i
= 1,...,N ) a s a multiple objec-
tive optimization problem, each coordinate of v corresponding t o a n objective that w e seek to minimize.
We shall h e r e formulate o u r optimization problem in terms of t h e objective (5.14) but of c o u r s e any of t h e o t h e r variants could o r should also b e
considered.
The optimization problem again involves probabilistic con-
s t r a i n t s but i t s s t r u c t u r e now resembles much more t h e stochastic program
with r e c o u r s e (5.11) than t h e f i r s t model (5.0) involving probabilistic constraints. We obtain
find z
-
rj
E Rn
4 zj 4
(5.17)
such t h a t
j =I ,... ,n
rt,
i = I , ...,m
C?=l aij zj bi
C?=l c j ( 4 )4 B
1
prob. ~ ~ & l t i j ( w ) z j - h i ( w ) - v i <J +
Oai,
and z =
EL1qi [ v i - yi ]
+
i = l ,....N
is minimized .
A t this point i t may b e worthwhile t o observe t h a t (5.14) i s just a limit c a s e
of (5.16). Recall t h a t t h e r a n g e o v e r which qi ei 8 (ei-'(*
-yi))
i s quadratic
i s [O,ei], cf. Figure 3 . If we shrink this interval t o 1 point, we are l e t with
t h e piecewise linear function gi
[a
-yi
I+.
A s f o r o u r e a r l i e r models, w e should study t h e solution as a function of
8, t h e available budget. However, solving (5.17) p r e s e n t s all t h e technical
challenges mentioned in connection with t h e f i r s t model ( 5 . 8 ) involving probabilistic constraints, t h e presence of t h e ( v i , i
= 1. ....m2)h a s in no way
simplified t h e problem. W e d o not know of any d i r e c t method f o r solving
(5.17). One possibility is t o find a n approximation of (5.17) t h a t could b e
handled by available linear o r nonlinear programming techniques.
r e t u r n t o this in t h e next section.
We
6.
EXOY: AN EXPECTATION-VARIANCEMODEL
In the formulation of t h e eutrophication management optimization model
(EMOM) t h e objective h a s always been chosen s o as t o measure in t h e most
realistic fashion possible t h e deviations of t h e observed concentrations
indicators from t h e water quality goals. This led us t o t h e stochastic program with recourse model (5.11) with t h e associated solution procedure t o
be discussed in Section 7 . Here w e introduce a line*
programming model
(Somly6dy, 1983b) t h a t is based on expectation-variance considerations
(for t h e water quality indicators). The justification of t h e model relies on
t h e validity of certain approximations and thus in some situations one
should accept t h e so-generated optimal solution with some circumspection.
However, as shall b e argued, its solution always points in t h e right direction
and is usually far superior t o t h a t obtained by solving a "deterministic"
problem such as (5.12). In t h e Lake Balaton case study t h e results f o r both
this expectation-variance model and t h e stochastic programming model
(5.11) lead t o remarkably similar investment decisions as shown by t h e
analysis of t h e results in Section 8.
A s a starting point f o r the construction of this model, let us consider
t h e following objective function:
where, as in Section 5 yi (z,w ) is t h e water quality level characterized by
t h e selected indicator in basin i given t h e investment program z and t h e
environmental conditions w , 7( t h e goal set f o r basin i and qi a weighting
f a c t o r associated t o basin i . The objective being quadratic in t h e area of
i n t e r e s t , and t h e distribution functions Gi ( z ,*) of t h e y i ( z ,*) not being too
f a r from normal, one should b e able t o r e c a p t u r e t h e essence of i t s effect
on t h e decision p r o c e s s by considering just expectations and variances.
This observation, and t h e "soft" c h a r a c t e r of t h e management problem
which in any case means t h a t t h e r e is a l a r g e d e g r e e of flexibility in t h e
choice of t h e objective, suggest t h a t w e could substitute
f o r (6.1): where 8 i s a positive scalar (usually between 1 and 2.5),
- = E{yoi j
yo,
i s t h e expected nominal state of basin i , and u denotes stan-
d a r d deviation,
Since f o r each i =I,
. . ,N,t h e
y i are affine (linear plus a constant term)
with r e s p e c t t o z , t h e expression f o r
as a function of z i s easy t o obtain from equations (3.1) and (3.10). The k j
a r e t h e expectations of t h e coefficients of t h e z j and t h e 4, t h e expectation of t h e constant term. Unfortunately t h e same does not hold f o r t h e
standard deviation a ( y i ( z ,*)
- c,).
Equations (3.1) and (3.10) suggest t h a t
where ail i s t h e p a r t of t h e standard deviation t h a t c a n b e influenced by t h e
decision variable z l ; f o r example, t h e standard deviation of t h e t r i b u t a r y
load LND. Cross terms are f o r all practical purposes i r r e l e v a n t in this situa-
tions since t h e total load in basin i is essentially t h e r e s u l t of a sum of t h e
loads generated by various sources t h a t a r e independently controlled. This
justifies using
instead of (6.2) as a n objective f o r t h e optimization problem. This function
i s convex and differentiable on R? except at z
= 0,
and conceivably one
could use a nonlinear programming package t o solve t h e optimization problem :
find z
E Rn
such t h a t
r j s z j s r,+
C?=lW Z j bi
CYZlcj ( 4 )S 8
I
and z = CrZ1
qi
...
j=1, ,n
i=1,
,m
hj z j + Q [~j'=~ ofj
Assuming t h a t t h e c o s t functions have been linearized, i.e. with each c j
piecewise linear, t h e MINOS package would b e a n excellent choice since t h e
solution is clearly bounded away from 0.
We c a n go one s t e p f u r t h e r in simplifying t h e problem t o b e solved,
namely by replacing t h e term.
in t h e objective, by t h e linear (inner) approximation
On each axis of R y , no e r r o r is introduced by relying on t h i s l i n e a r approximation, otherwise w e are over-estimating t h e effect a c e r t a i n combination
of t h e z j ' s will have on t h e variance of t h e concentration levels. Thus, at a
given budget level w e shall have a tendency to start p r o j e c t s t h a t affect
more strongly t h e variance if w e use t h e linear approximation, and this is
actually what w e observed in p r a c t i c e (see Section 8). Assuming t h e cost
functions c j are piecewise linear, w e have t o solve t h e l i n e a r program:
find z
-
E Rn
such t h a t
r j s zj s r;,
(6.7)
j = l , - a . ,n
i=
,I
- . ,m
C?=l q j z j S bi
CT=1c j ( z j ) S B
and t = CiN qi CT=l( h j + 8 q j ) 4
1
is minimized.
W e r e f e r t o t h i s problem as t h e (linearized) expectation-variance model
(see also Somlybdy, 1983b; and Somlyddy and van S t r a t e n , 1985).
For t h e s a k e of illustration, let us consider t h e i - t h basin of Figure 2
and suppose t h a t t h e r e i s no m a s s exchange with neighboring basins. Then
from equations (3.1) and (3.7), recalling t h a t &
= ha/h, w e obtain
To obtain a linear form in t h e z j , w e proceed as indicated in Section 3, see
equation (3.9). To d e r i v e t h e remaining term in t h e objective of (6.7) we
only need t o consider t h e controllable portion of
gt ( z ,=)
- w e r e w r i t e (3.1) as follows
- yo(,
VJz,=)
Let us write
from which w e obtain
- = z f-(dii
-Yof
+ d f ~ f )
&
L
t h e variance of
$ ( AN
w ~ )= d:(
I
1 + d: uyt [ ~ ( f &+~E) ' I ~ &1, ~
(6.8)
Now from equations ( 3 . 5 ) , ( 3 . 6 ) and ( 3 . 7 ) w e have t h a t
This would lead t o a n expression f o r u ( A y i ) t h a t would b e nonlinear in t h e x
N
variables. To avoid t h e nonlinearities w e specify ua ( b y i ) and ua (A&,)
as
N
t h e linear combination of t h e additive terms in ( 6 . 8 ) and ( 6 . 9 )
ua(AWi):
N
'
= d i C u a ( & & ) + d i u E t 1(ua(A&Ni) +EfUNiLilll
(6.10)
and
Note t h a t in equations ( 6 . 8 ) and ( 6 . 9 ) all t h e coefficients are positive and
t h e behavior of t h e "new" ua is similar to t h e standard deviations as defined
through ( 6 . 8 ) and ( 6 . 9 ) . Substituting (6.11) in (6.10) yields.
Collecting terms w e obtain t h e coefficients u i j t h a t a p p e a r in t h e objective
of t h e linear program ( 6 . 7 ) . (A more detailed, but similar, derivation also
yields t h e expression f o r t h e standard derivation when t h e r e is mass
exchange between neighboring basins).
The arguments t h a t we have used t o justify t h e expectation-variance
model are mostly of a heuristic nature, in t h a t they r e l y on a good understanding of t h e problem at hand and "engineering" intuition. In t h e formulation of t h e EMOM m o d e l s , in Section 5, t h e objective h a s usually been formulated in terms of finding control measures such t h a t t h e observed concentration levels (water quality indicators) are not too far from pre-set goals
(given trophic states). If by 'hot too far" w e mean t h a t
should b e as small as possible, w e could also reformulate t h e problem, in
terms of t h e nominal concentration levels. Indeed, with
*A:
where as in Section 3:
--
-3
f
zotdenotes t h e nominal
state in basin i. Then
instead of minimizing (6.12), w e could maximize
and this should give about t h e same results. This is actually t h e motivation
behind t h e formulation of (6.7), see Figure 4.
There is however a n o t h e r approach t h a t does not r e l y s o extensively
on heuristic considerations, which leads us t o t h e model (6.6), i.e., t h e nonlinear version of t h e expectation-variance model.
The fourth model,
described in Section 5, which integrates both reliability considerations and
penalties f o r fixing t h e reliability levels, led us t o t h e nonlinear program
find z
E Rn
such t h a t
(5.17)
.-
rj-szj s r f ,
j=1,.
CT=l~ fzjj bi
CT=lc j (xj ) s 8
i = 1 , . . . ,m
prob. I[ C T = l t i j ( ~ ) z j- h i ( w ) - y i
and z
= ELl qi [wi
- yi ]+
< O Il + a i ,
is minimized
,n,
i=l:.-.N
.
Because t h e s e probabilistic constraints are v e r y difficult t o handle, w e may
consider finding a n approximate solution by replacing t h e probabilistic
constraints by
where
and f o r j =O,.. ,n and k =O
I
,...,n
I
If t h e random variables t y (a), j=O, ...,n are jointly normal, then t h e restrictions generated by t h e deterministic constraints (6.14) are exactly t h e
same as those imposed by t h e probabilistic constraints, but in general they
are more r e s t r i c t i v e , cf. Propositions 1.25 and 1.26 in Wets, 1983a. Without
going into t h e details, w e can see t h a t (6.14) is obtained by applying
Chebyshev's inequality and this, in general, determines a n u p p e r bound f o r
t h e probabilistic event
FIGUP& 4. Objective of expectation-variance model.
S o if we can justify a n e a r normal behavior f o r t h e random variable (for
fixed x )
we can use the constraints (6.14) instead of the probabilistic constraints t o
obtain an approximate solution of (5.17). Note t h a t in this setting, 'hear
normalitp" of t h e y i (x ,*) is a much more natural, and weaker, assumption
N
than normality of t h e
Lij. Assuming
obtain the nonlinear program:
t h a t we proceed in this fashion, w e
find z E R n such t h a t
rj-
.
j=o, . . - n
s zj s rj+,
C=
;1 %j zj bi *
C&1 cj (4) 5 B
i=1,
-
-
,m
and
I
= CIN.1q i [Cf=o & j X j
2
+ ( 1 -ai
where r{
) -2
C h o peO uijk zjz k )% - yi jI
+
i s minimized
= ro+= 1. W e have eliminated t h e variables ( p i , i =1, - . . ,N)
from t h e formulation of t h e problem by using t h e f a c t t h a t t h e optimal y;
can always b e chosen s o t h a t (6.14) is satisfied with equality. Moreover, if
t h e desired concentration levels yi are low enough, then w e know t h a t t h e
optimal solution will always have y;
> yi
and thus w e c a n r e w r i t e (6.15) as
follows:
find z
-
rj
E
Rn such t h a t
j=O,...,n
s zj 5 7:;
cfnLl aij zj
5
bi ,
i = I , ...,m
Cr"=lcj ( z j ) s B
and
2
= C&l
+ ( 1-ai
i
[C?=O
& j zj
) -2 (C?=, Cr=,uijk
p.,
zj zk)%-yi
1
is minimized.
The objective of t h i s optimization problem is sublinear, i.e., convex and
positively homogeneous. Assuming t h a t t h e cost functions c, are linear, o r
more realistically have been linearized, see Section 4.3, w e are thus confronted with a nearly l i n e a r program t h a t w e could solve by specially
designed subroutines (nondifferentiability at 0), o r by a linearization
scheme t h a t would allow us t o use linear programming packages.
Now
observe that t h e nonlinear program (6.16) is exactly of t h e same type as
(6.6) if w e make t h e following adjustments:
(i)
in
the
objective
of
(6.16)
replace
the
z&oxt=o
covariance
uijk xk xj by the sum of t h e variances
xFZiuij,
term
zp;
and
(ii) if f o r all i =1, ...,N , the ai a r e t h e same set 8
wise we replace Q by 8
= (l-ai
= (l-ai
)-I,
other-
) *in (6.6).
To justify (i), w e appeal t o (6.4).
In t h e derivation t h a t led us from (5.17) t o (6.16), w e stressed t h e fact
that t h e solution of (6.16) and thus equivalently of (6.6), would be feasible
f o r t h e original program (5.17), and that in fact it would more than meet t h e
probabilistic constraints specified in (5.17). The f u r t h e r linearization of
t h e objective bringing us from (6.6) t o (6.7) overstresses (possibly only
slightly) the r o l e that t h e variance will play in meeting t h e prescribed reliability levels. In terms of model (5.17), w e can thus view t h e solution of
(6.7) as a "conservative" solution that overestimates t h e importance t o be
given to t h e stochastic aspects of t h e problem. In t h a t sense, t h e solution
of (6.7), especially in comparison to that of t h e deterministic problem
(5.12). always indicates how w e should adjust t h e decisions s o as t o take into
account the stochastic features of t h e problem.
In our analysis (see Section 8), w e have used t h e linear programming
version (6.7) of this expectation-variance model; t h e wide availability of
reliable linear programming packages makes it easy t o implement, and thus
a n attractive approach, provided one keeps in mind t h e reservations
expressed e a r l i e r .
7.
S O L W G EMOM: STOCHASTIC YERSION
W e outline h e r e a method f o r solving t h e "full" stochastic version of
t h e eutrophication management optimization model (5.11).
W e recognize
(5.11) as a stochastic program with simple r e c o u r s e , with technology matrix
T and r i g h t hand sides h stochastic. When only h is stochastic and t h e
objective function ( r e c o u r s e cost function) is piecewise linear, efficient
procedures are available f o r stochastic programs with simple r e c o u r s e , cf.
Wets, 1983b (implemented by Kahlberg and Kusy, 1976); f o r a r e p o r t on
numerical experimentation with this code, see Kusy and Ziemba, 1983; and
Nazareth, 1984; f o r a different algorithmic approach and a p r o g r e s s r e p o r t
on i t s implementation.
But h e r e w e have t o deal as with nonlinear (convex)
r e c o u r s e costs. In o r d e r t o deal with this class of problems, a new proc e d u r e was developed which exploits t h e p r o p e r t i e s of a dual associated t o
problem (5.11).
Replacing t h e budget constraint by t h e equivalent linear system, cf
(4.5) and t h e r e l a t e d discussion, substituting cj
+ rj-
f o r z j , defining
and dropping t h e -, w e obtain a version of (5.11) t h a t f i t s into t h e following
class of problems:
find z
and z
E Rn
such t h a t
= ZG1[ c j x j +
]+E
j
{
qf e 0 [ef'vf
(w )
t o which one r e f e r s as q u a d r a t i c stochastic programs w i t h simple
recourse.
m , = N , cj = d j = 0 ,
Here
(CTzl a f j S bf , i =l,..., m l )
and
has been expanded ( m l
>m )
system
the
t o include t h e
budgetary constraints. The 8 function i s defined by (5.9). For problems of
this type, in fact with this application in mind, an algorithm i s developed in
Rockafellar and Wets, 1983, which relies on t h e properties of a dual prob-
l e m which can be associated t o (7.1). In particular it i s shown that t h e following problem:
find y
E RT
and z
(e):
Q -.
i =I, -
OSzf(w)sqi,
and
CFA~y i
-xI;L=lr j d j
bi
(7.2)
measurable such that
. ,m2
ei
+-
- CT2: E
2%
8 (dj-l u j) is maximized ,
is dual to t h e original problem, provided that f o r i =l ,...,m 2 , t h e ei and qi
are positive (and that i s t h e case h e r e ) and f o r j = 1,...,n , t h e d j
> 0 which
i s not the case. This can be taken c a r e of in two different manners.
First, one should observe that t h e Duality Theorem proved in Rockafel-
lar and Wets, 1983, remains valid f o r d j = 0 provided that t h e function
uj
+ rjdje(dj-luj)
be replaced by
uj
when d j
= 0 , where
-. rj O - ( u j )
(the function r j Q '(4
i s t h e limit of t h e collection rj d j 8 (dj-I m
) as d j tends
t o zero). The same proof applies. An algorithm f o r solving this problem
i.e.. (6.1) but with r j d, Q (d;'
- could
-
wj), replaced by r j Bm(wj) in t h e objective
be constructed along t h e same lines as those of t h e method t o b e
described h e r e below. The basic difference being t h a t Q " i s not differentia b l e and t h a t needs t o b e handled appropriately.
Second, to introduce in t h e objective of (5.11) a n artificial perturbation of t h e objective t h a t would result in a problem of t y p e (7.1) with the
prescribed properties. In t h e algorithm to be described such quadratic
terms are added in a most natural fashion. The overall s t r a t e g y i s t h a t of
t h e proximal point algorithm, studied by Rockafellar, 1976.
sequence of feasible solutions f x', v
= 1, -
-
W e generate a
of (6.1) with each x v t h e
(unique) optimal solution of (7.1) where c j and d j a r e replaced by
and
d,"
t h e scalars p, r 0 with p,
p,
> 0.
= d j + p,
rj ;
Since f o r each v, d r
> 0), w e can apply t h e Duality Theorem of
>0
(the rj > O and
Rockafellar and Wets, 1983 (as
stated), t h e dual being exactly (7.2) with c j and d j replaced by c y and djv
for j = 1,...,n . This dual problem is then solved and t h e (optimal) multipliers associated t o t h e equations
give us x Vt h e optimal solution of problem (7.1) with t h e adjusted coefficients c," and d,".
W e then generate c,"+I and d,"+l and again t h e resulting
dual is solved. W e r e p e a t this until a n e r r o r bound (involving t h e evaluation
of the objective function at zV)is below a preset level. A detailed description of the method (in a somewhat more general setting), its convergence
and various e r r o r bounds can b e found in Rockafellar and Wets, 1985.
Every iteration of t h e algorithm, from z t o z
+
l, involves t h e solu-
tion of a problem of type (7.2). which can almost be viewed as a nonstochast i c problem except f o r t h e simple constraints
The method that w e use relies on a finite element representation of t h e zi (a)
functions, that is chosen s o that t h e above constraints a r e automatically
satisfied. Let
til(a), I = 1,...,L
be a finite collection of functions such t h a t
f o r all 2 ,
If this collection of functions is rich enough, o r if chosen s o that t h e
optimal z;(*) t h a t i s p a r t of an optimal solution of (7.2), lies close t o t h e i r
linear span, w e write
where t h e Ail are weights associated t o t h e functions
Cf=l Ail
= 1,
2
f o r all I
til(a) such t h a t
.
The generated functions zi(*) a r e always between 0 and qi and t h e constraints (7.4) can be ignored. With t h e substitution (7.5), t h e dual problem
(7.2) becomes a f i n i t e dimensional quadratic program
and f o r i = l .....m 2 . I=1. ...L , A ,
find y E
l
?
:
'
= cj
- zF=la i j yi - xf=lAil Gl ,
a such t h a t
(7.6)
j =1,....n and
and
+
zT=l
r j d j f3(dl-'uj ) is minimized
where
The only question is t o know if t h e collection ( ( i l ( * ) ,I = I , ...,L j is rich
enough t o yield a solution of (7.1) by solving (7.6). This is resolved in t h e
following fashion. Let (y^, z^(*))
with 2(*)=
EL1 xil til(*)
b e t h e solution
generated by solving (7.6). This is a feasible solution of (7.2), but not
necessarily a n optimal solution. W e then solve t h e following problems: f o r
i = 1,...,m 2
find
E argmin
I C(y^,O10 r < r pi 1I
I
where 9(y^,*)is t h e objective function of (7.2) obtained by setting y
(7.7)
c.
= y.
This is a v e r y simple optimization problem, in f a c t i t s solution is given by
t h e formula: f o r i
= 1,...,m 2
\
where
f are t h e multipliers associated t o t h e equation (7.3),f o r t h e optimal
solution (y^
,XI)
of (7.6) and O ' is t h e (particularly simple) derivative of 8 . If
w e add t h e s e functions t o those used t o r e p r e s e n t z (a), by solving t h e aug-
+ 1 , instead
mented problem (7.6) with L
of L , w e are guaranteed t o obtain
a n improved solution of (7.2) unless (y^ ,z^(a)) i s already optimal f o r (7.2).
For f u r t h e r details, including a n analysis of t h e convergence r a t e , consult
Rockafellar and Wets, 1985.
An experimental version of this algorithm w a s implemented at IIASA by
A. King (and is available through IIASA as p a r t of a collection of codes f o r
solving stochastic programs), see King, 1985. W e start t h e procedure by
solving t h e deterministic problem (5.12) with
gives us a v e c t o r x with v
Li
-
= hi and
A
-
t f j = t f j . This
= 1t h a t is used t o define t h e v e c t o r s c Vand
d V,
and f o r i=1,. . . .m2:
In addition t o this function
finite elements
t"
=
tl, w e
0 and
also include in o u r s t a r t i n g collection of
to= q . The optimal solution of t h e quadratic
program (7.6) and t h e associated multipliers are then used t o g e n e r a t e
through (7.8) a new function
(leading up t o a l a r g e r version of (7.6)) o r
t h e solution is recognized as optimal ( o r more exactly &-optimal). In this
latter case, w e use t h e (optimal) value of t h e multipliers zV+l
associated t o
t h e equations t h a t define u,,j
= 1, . . ,n as t h e s t a r t i n g point of a new
major cycle, redefining cVfl and dV+' and proceeding from t h e r e as indicated above. The algorithm is stopped when t h e gap, between t h e objective
of t h e original problem (7.1) evalued at z
dual evaluated at y
is sufficiently small.
= xV+' and t h e objective of i t s
and zV+'(a) obtained from t h e (last) solution of (7.6),
The distribution of t h e random elements ti, (-1 and hi (*I i s replaced by a
discrete distribution obtained from t h e output of NLMP2 and LEMP by s a m pling. For a number of reasons including numerical stability considerations
and t h e speed-up of t h e computation, i t i s recommended t o start with a relatively small sample increasing its size only f o r verification purposes. W e
have observed t h a t a relatively s m a l l sample, about 50, will give surprisingly good results.
To solve t h e quadratic program (6.6) w e used MINOS (Murtagh and
Saunders, 1983). Much could b e done t o improve t h e performance of that
p a r t of t h e algorithm, a specially designed quadratic programming subroutine should be able t o exploit not only t h e s t r u c t u r e of this quadratic program but also t h e fact t h a t when introducing an additional finite element w e
can use t h e previous solution t o simplify t h e s e a r c h f o r a new basis, etc.
8.
APPLICATION TO LAKE BALATON
8.1. Background for Lake Balaton
8-1.1. Description of the lake. its watershed and possible control
measures
Lake Balaton (Figure 5), one of t h e largest shallow lakes of t h e world,
which i s also t h e c e n t e r of the most important recreational a r e a in Hungary, has recently exhibited the unfavorable signs of artificial eutrophication. An impression of t h e major features of the lake-region system, of the
main processes and activities, about t h e underlying research, data availability and control alternatives can be gained from Figure 5 and Table 1 (for
details, s e e Somlyddy e t al, 1983; and Somlyddy and van Straten, 1985).
Four basins of different water quality can be distinguished in the lake (Figu r e 5) determined by t h e increasing volumetric nutrient load from east t o
west* (the biologically available load, BAP, is about ten times higher in
Basin I than in Basin IV, Table 1, line 7). The latter is associated t o the
asymetric geometry of t h e system, namely t h e smallest western basin drains
half of the total watershed, while only 5% of t h e catchment a r e a belongs t o
t h e l a r g e r basin (Table 1,lines 2 and 4).
Based on observations f o r t h e period 1971-1982 the average deterioration of water quality of t h e entire lake is about 20% (in terms of Chl-a).
According t o t h e OECD classification, t h e western p a r t of t h e lake is in a
hypertrophic, while t h e eastern portion of i t is in an eutrophic stage (Table
1,line 9).
The absolute loads are roughly equal f o r t h e four basins
TABLE 1. Major features of t h e lake and its watershed.
Basin
I
TI
111
IV
Whole lake
Watershed a r e a [km 1
2750
1647
534
249
5180
U s e of t h e watershed
agriculture and intensive tourism (main season:
July and August)
2
Lake surface a r e a [km
2
Volume [106m31
Depth [m]
BAP load [mg/m3dl
Climatic influences
Water quality
3
(Chi a
)
,
,
,[mg/m 1
no stratification; large fluctuation in temperat u r e (up t o 25-28%);
2-month ice cover; strong wind action
(a) 75
38
28
20
late seventies
(b) 150 90
60
35
1982
hyper-eutrophic state; P limitation till t h e end
of the seventies; large year-to-year fluctuation
in Chl-a depending on meteorology and hydrology; 20% p e r y e a r increase in Chl-a during
1971-1982; marked longitudinal gradient
10.
Sediment
internal load nowadays is roughly equal t o t h e
external BAP load
11.
Data
long hydrological and wheater records; regular water quality and load survey since 1971
and 1975, respectively
12.
Research
increasing activity in Hungary in various institutes during t h e past 30 years; joint study of
TIASA, t h e Hungarian Academy of Sciences and
the Hungarian National Water Authority, 19781982, see Somlyddy et al. (1983).
13.
Models developed
various alternative models f o r t h e components
indicated in Figure 1, see Somly6dy e t al.
(1983)
14.
Methodologies
OD and PDE models, regression analysis, Kalman filtering, t i m e s e r i e s analysis, Monte Carlo
simulations, uncertainty analyses, optimization
techniques
15.
Measures of short-term control
P precipitation on existing treatment plants;
pre-reservoirs
16.
Policy making
Government decision in 1983: P control is being under realization
The lakes' t o t a l P, LT is in a n a v e r a g e 315t / yr (the BAP load is
170t / y r ) , but depending on t h e hydrologic regime i t c a n r e a c h 550t / y r .
53% of
LT i s c a r r i e d by t r i b u t a r i e s
(30% of which is of sewage origin
-
indirect load, see e.g., t h e l a r g e s t city of t h e region, Zalaegerszeg in Figu r e 5 ) , 17% is associated t o d i r e c t sewage discharges ( t h e recipient i s t h e
lake). Atmospheric pollution if responsible f o r 8%of t h e l a k e ' s TP load and
t h e rest is formed by d i r e c t runoff (urban and agricultural). Tributary load
increases from east t o west, while t h e change in t h e d i r e c t sewage load goes
in t h e opposite direction. The sewage contribution (direct and indirect
loads) t o LT i s 30%, while about 52% t o LBRP (the load of a g r i c u l t u r a l origin
c a n b e estimated as 47 and 33%,respectively) suggesting t h e importance of
sewage load from t h e viewpoint of t h e s h o r t t e r m eutrophication control.
Figure 5 indicates a l s o t h e loads of sewage discharges and t r i b u t a r i e s which
were involved in t h e management optimization model. These c o v e r about 85%
of t h e nutrient load* which w e consider controllable on t h e s h o r t t e r m (e.g.
atmospheric pollution and d i r e c t runoff are excluded).
Control a l t e r n a t i v e s are sewage treatment (upgrading and P precipitation) and t h e establishment of p r e - r e s e r v o i r s as indicated in Figure 5 (see
e.g. t h e Kis-Balaton r e s e r v o i r system planned f o r a s u r f a c e a r e a of about
75 km2). Besides Hungarian r e s e a r c h activities, t h e problem of Lake Bala-
ton was studied in t h e framework of a four-year cooperative r e s e a r c h project on Lake Balaton involving t h e International Institute f o r Applied Sys-
t e m s Analysis, IIASA (Laxenburg, Austria), t h e Hungarian Academy of Sciences, and t h e Hungarian National Water Authority (Somlybdy, 1982 and
* The rest represented by
sake of simplicity.
several small creeks and sewage outlets were neglected for the
1983a; Somly6dy et al, 1983). The development of t h e management model t o
be discussed h e r e formed a p a r t of t h e Case Study. The results achieved
were then utilized in 1982* in t h e policy making procedure associated with
t h e Lake Balaton water quality problem which was completed by a governmental decision in 1983 (Ldng, 1985).
8.1.2.
Specification of elements of EBIOBI for Lake Balaton
(a) Lake e u t r o p h i c a t i o n model, L E W
Starting from a four-compartment, four-box dynamic phosphorus cycle
model (van Straten, 1981), LEM, t h e aggregated model LEMP of a s t r u c t u r e
defined by Equation (3.1) was derived by a systematic analysis a s described
in Section 2. A s seen from Figure 6, illustrating t h e deterministic version
of LEMP, t h e
= (Chl-a)ma,~i indicators (1 S i
S 4)
are really linearly
dependent on t h e sum of t h e external and internal loads (see also line 1 0 in
Table 1). The matrix D takes a specific bloc-diagonal s t r u c t u r e in this case,
since neighboring basins a r e only related in a unidirectional way from west
t o east, which indicates t h a t any management actions performed on t h e
eastern subwatersheds have no effect on water quality of t h e western basins
(dii
range between 20 and 35, while elements related t o interbasin
exchange a r e smaller by an o r d e r of magnitude).
A t t h a t time only t h e results of t h e expectation-variance model were available, the
development of which was the fastest.
LYp (mg/m3d)
L;,,
(mg/m3d)
External load
l nternal load
FIGURE 6. Aggregated lake eutrophication model: deterministic version of
LEM?.
The stochastic version of LEMP w a s derived from a Monte-Carlo type
usage of LEM under synthetic time s e r i e s generated by models METG and
NLMPl (Figure 1). The analysis h a s shown t h a t in t h e non-controlled c a s e
(Chl-a),,
f o r Basin I can r a n g e between 30 and 90 mg/m3 (i40X around t h e
mean) depending solely on meteorological factors; a strikingly wide domain.
Such fluctuations can mask t h e effect of considerable load reductions. The
sensitivity of t h e o t h e r basins in t h e lake is smaller: t h e coefficients of
variation of (Chl-a),,,
f o r Basin I1
... IV
are 10, 6 and 5%(it i s 13%f o r
Basin I).
If stochastic variations in loads are also taken into account, t h e coefficient of variation f o r Basin I goes t o 20% and t h e extreme value of
(Chl-a),,
can r e a c h 150 mg/m3*.
Upper extremes f o r subsequent basins
are 60, 35 and 25 mg/m3, respectively.
*
A s a p p a r e n t from Figure 7,
A value observed in 1982 (Table 1, line 9) when the prognosis was already available.
referring t o Basin I under pre-reservoir control, linearity is preserved a s
before, and not only f o r t h e mean, but also f o r statistical properties of the
typically skewed distributions (standard deviations and extreme values).
Basin I
l
6
1 '
/
O
1
*
3: mean; 2 and 4:
standard deviation; 1and 5: extremes; E
is t h e o p e r a t o r of expectation.
FIGURE 7 . Aggregated lake eutrophication model: stochastic version.
The analysis outlined h e r e led t o t h e specification of Equation (3.1) f o r
Lake Balaton; f o r details t h e r e a d e r is r e f e r r e d t o Somlyddy (1983b) and
SomlyCldy and van S t r a t e n (1985).
(?I)
Nutrient Load model, N L M R
The nutrient load model f o r Lake Balaton can b e derived on t h e basis of
Figure 5 from relation (3.10). The t r i b u t a r y loads,
LT and La
are computed
from regression models (Somly6dy and van S t r a t e n , 1985)
where Q is t h e stream flow r a t e ,
& is
t h e residual, and t h e variable #
N
accounts f o r t h e influence of infrequent sampling ( # - is t h e lower bound).
The most detailed d a t a set including 25 y e a r s long continuous r e c o r d s f o r Q
and 5 y e a r s long daily observations f o r t h e loads w a s available f o r t h e Zala
River* (Figure 5) draining half of t h e watershed and representing practically t h e total load of Basin I. For t h e Zala River
normal distribution, while
QN
Lp was found t o have a
was approached by a lognormal distribution.
The time s t e p of t h e original model employed in NLMP 1 (Figure 1 ) w a s a
month and t h a t of the aggregated version used in NLMPZ a y e a r . The loads
of o t h e r t r i b u t a r i e s w e r e established on t h e basis of much more scarce
observations. F o r modeling t h e uncertainty component of #, f i r s t a MonteN
Carlo analysis w a s performed on t h e Zala River d a t a by assuming various
sampling strategies. Subsequently, t h e conclusions were extended t o t h e
o t h e r r i v e r s and t h e p a r a m e t e r s of t h e (assumed) gamma distributions of #
N
were estimated.
I t should b e noted t h a t t h e nonlinearity r e l a t e d to t h e product terms
(zD axrn) in NLMPZ w a s handled in both ways, see (3.8) and (3.9), indicated
in Section 3 in t h e expectation-variance model, while through t h e constraint
equation (3.9) in o t h e r management models discussed in Section 5.
* I t s annual load estimated from daily data can be considered accurate.
(c)
Cbntrol v a r i a b l e s a n d cost f u n c t i o n s
A l l t h e optimization models implemented use real-valued control vari-
ables. Integer { O , l j variables f o r t h e two reservoir systems (see Figure 5)
were also used by simply fixing t h e variable values a t 0 o r 1 a s p a r t of the
input. The elaboration of cost functions was based on analyzing a variety of
technological process combinations (leading t o different removal efficiencies) f o r treatment plants included in t h e analysis (Figure 5). A s an example, t h e cost function f o r t h e largest treatment plant, Zalaegerszeg (see
Figure 5) t h e capacity of which i s Q,
= 1 5 0 0 0 m 3 / d , i s given in Figure
8*.
Three groups of expenses are illustrated in t h e figure:
( i
Investment cost required f o r upgrading biological treatment;
(ii) Investment cost of P precipitation which increases rapidly with
increasing requirements. The use of piecewise linear cost functions required t h e introduction of t h r e e dummy variables f o r each
treatment plants.
(iii) Running cost
Roughly U S 81 is equivalent t o 50 Forints (Pt).
-
50
.
40
(ii)
30 -
Y
LL
w
-0
C
i;
20
-
0
10 I - - - - - - - - - - - - -
-
I
0
(iii)
I
I
I
0.5
I
I
I
'SNI
I
I
'SN~
FIGURE 8. Costs of sewage treatment (Zalaegerszeg).
For pre-reservoirs linear cost functions of the surface area (and control variable) were used, see Section 4.2.
8.2. Results of the expectation-variance model
In o r d e r t o gain a n impression of the c h a r a c t e r of t h e problem and the
behavior of t h e solution, f i r s t w e specify a "basic situation" (which is close
t o t h e r e a l case) having t h e following features and with t h e following
assumptions (Somlyddy , 1983b):
(i)
control variables a r e continuous;
(ii) no effluent standard prescription is given;
(iii) no P retention t a k e s place in r i v e r s ( r p = 0, s e e Equation
(3.10));
(iv) t h e capital recovery f a c t o r is equal f o r all t h e projects,
a~ = a = 0.1 and
(v)
equal weighting is adopted (see qi and 6 in Section 6).
With these assumptions optimization w a s performed under different
budgetary conditions (TAC
S
@
= 0.5-25.10'~t / y r ) . Statistical parame-
t e r s (expectation, standard deviation and extremes) of the water quality
indicators gained from Monte Carlo procedure* a r e illustrated in Figure 9
f o r t h e Keszthely basin as a function of t h e total annual cost, TAC
**.
In Figure 10, w e r e c o r d t h e changes in t h e two major control variables
(zmland zDi)associated t o t h e treatment plant of Zalaegerszeg and t h e
r e e d lake segment of t h e Kis-Balaton system (see Figure 5). There is a significant trade-off between t h e s e two variables. For decision making purposes, i t i s important t o observe t h a t t h e r e are four ranges of possible
values of @ (the budget), in which t h e solution h a s different characteristics.
* 1000 simulations were performed in each case.
** Running cost i s about ten times larger than TAC.
Expectation varianat model, Eq. (6.7)
X Stochastic model with recourse, Eq.
(5.11)
y, = (48,28,24, 181, i = ( 1 ,..., 4)
e . = 5 . 0 , qiSlO.O, i = ( 1 ,
4)
...,
Basin I
95% confidence level
. . -.
%x
min {Y
A
0
.a
c.
10
v
w
dylT
FIGURE 9. Water quality indicator (Chl -a),
nual cost.
b
0 > TAC
20
[lo7 Ftlyr]
as a function of the total an-
-
Expectation variance model
---
ei = 5.0, qi = 1.0, i = (1 ,..., 4)
- Stochastic model with recourse
Basin I
--
Typical
domains
0 0
0
-
FIGURE 10. Change of major decision variables.
4 > TAC
(i)
In t h e range of
6 = 0.5-5.10~-
/ y r , i t appears that sewage
treatment can be intensified and tertiary treatment introduced.
Expectation of t h e concentration levels will decrease considerably, but not t h e fluctuations. Under very s m a l l costs (4.3.107rt
investment costs, IC) i t turns out that only the sewage of Zalaegerszeg (Figure 5) should be treated. Under increasing budget,
potential treatment plants are built, going from west t o east.
(ii) If /3 is between 5.10~and 10.10~FY/ y r , the effectiveness of
sewage treatment cannot be increased f u r t h e r but reservoir syst e m are still too expensive.
(iii) A t about /3 = 15.1o7FY / yr the solution is a combination of tertia r y treatment and reservoirs. Fluctuations in water quality are
reduced by the latter control alternatives.
(iv) Finally around /3 = 20.10~- / y r , tertiary treatment is dropped in
regions where reservoirs can be built. A f t e r constructing all the
reservoirs no f u r t h e r water quality improvement can be achieved.
Concerning the model sensitivity on major parameters, the following conclusions can be drawn (for details see Somlyddy, 1983b; and Somlyddy and van
Straten ,1985):
(i)
Fixed water quality standard not reflecting t h e properties of the
system (spatial non-uniformities) can result in a strategy far from
the optimal one, since the distribution of a portion of the budget is
a p r i o r i determined by the pre-set standard.
(ii) Under increasing P retention in rivers the improvement in water
quality is less remarkable in t h e budget range 0-10.10'rt
/ yr
than in t h e basic case. The worst
tic
- situation
- nevertheless nearly unrealis-
i s if all t h e phosphorus was removed along t h e
r i v e r and still treatment has t o be performed: t h e budget should
be partially allotted f o r investments having no influence on t h e
lake's load.
(iii) If only deterministic effects are considered (8
= O), r e s e r v o i r
projects e n t e r t h e solution under much l a r g e r budget values, see
Section 8.4.
(iv) If t h e capital recovery factor i s s m a l l e r f o r r e s e r v o i r projects
than f o r sewage treatment plants, s e e Section 4.3, r e s e r v o i r projects start t o be feasible at smaller budgets. E r r o r s in t h e efficiency o r in costs of reservoirs cause similar shifts in t h e solution.
(v) When selecting properly t h e model parameters, t h e combination of
t h e absolute load reductions f o r t h e four basins is maximized by
t h e model (as i t is suggested most frequently in t h e literature, see
t h e Introduction). Since, however, t h e absolute loads alone do not
reflect t h e spatial changes in water quality, t h e policy drastically
differs from t h e optimal one.
Subsequently w e give t h e "realistic" solution f o r t h e Lake Balaton
management problem by using
-
actual retention coefficients (ranging between 0.3
-
upper limits 0.9 f o r t h e P removal rate of reservoirs; and
-
fixed variables f0,0.9] f o r t h e Kis-Balaton r e s e r v o i r system.
- 0.5)
Figure l l * , which r e f e r s again t o t h e Keszthely bay, shows remarkable
differences as compared t o Figure 9. First of all t h e d r a s t i c effect of
r e s e r v o i r s upon expectation but even s t r o n g e r upon fluctuation of water
quality i s stressed.
Reservoirs e n t e r t h e solution between 15.10~and
1 7 . 5 . 1 0 ~ m/ yr total annual cost resulting in a reduction in t h e mean
(Chla),,,
concentration from about 55 t o 35 mg/m
and in t h e extreme
values from more than 100 to about 60 mg/m 3.
ExpecUtiomnunw tmcbl
X Stochastic model wlth ncwm
7,= (48.28.24.18). i = (1,..., 4)
ei = 5.0, qi = 10.0, i = (1,...,4)
0
-
1
From dytumic lake eutrophiution model
i
Hypertrophic
Eutrophic
1 I Mezotrophic
I
0
I
I
10
I
20
Oligotrophic
3,
> TAC 110'
Ft/yrl
2: 95%confidence level; 1and 6:
4: expectation; 3 and 5:
extremes.
FIGURE 11. Solutions of EMOM f o r lake Balaton, Basin I.
*
In t h e Figure f standard d e v i a t i o n and t h e upper 95% confidence level a r e a l s o i l l u s t r a t c o n c e n t r a t i o n s and t h e lower 95% confie d ( t h e d i s t r i b u t i o n s a r e bound t o w a r d s m a l l
dence l e v e l v a l u e s a r e c l o s e t o t h e minimum).
5
While Figure 9 o f f e r s several solutions f o r a decision maker depending
on t h e budget available, on t h e basis of Figure 11, only two feasible alternatives come t o mind:
(i)
If total annual cost of about 2.5.107m / yr is available, all t h e
sewage projects can and should b e realized (going from w e s t t o
east).
Through
XI = (Chla),,
this
alternative
the
is reduced to about 55 mg/m
expectation
of
( t e r t i a r y treatment
affects t h e water quality at a slightly smaller extent than in t h e
basic case due t o P retention of tributaries) but still extremes
l a r g e r than 110 mg/m
can occur (hypertrophic domain according
t o t h e classification of OECD, 1982). F u r t h e r increase in t h e
budget (up to 1 0 . 1 0 7 /~y r ) has no impact on water quality (under
t h e alternatives included in t h e analysis).
(ii) If budget around 20.107Ff / yr is given (not only t h e Kis-Balaton,
but) all t h e r e s e r v o i r s can b e established and t e r t i a r y treatment
can be realized f o r d i r e c t sewage sources. The m e a n (Chla),,,
concentration i s about 35 mg/m
while t h e maximum about 60 mg/m
(eutrophic stage).
In Figure 11 t h e r e s u l t s of a detailed simulation model f o r t w o optimal
solutions (TAC
= 2.5.107R and 2 0 . 1 0 ~ ~a r) e also given. The agreement
between t h e calculated concentration indicators suggests t h a t t h e aggregated lake eutrophication model i s quite appropriate f o r our present purpose (Figure l , Phase 4).
.
Figure 12 compares t h e typically skewed probability density functions
of
t w o considerably different solutions ( 8 = 2 . 5 . 1 0 ~ ~and 2 0 . 1 0 ~ ~
respectively) f o r f o u r basins: derived from Monte Carlo simulations (the
non-controlled state i s also given in this figure). A l s o from t h i s figure w e
can conclude t h a t t e r t i a r y treatment i s more effective than r e s e r v o i r s
(when both alternatives are available) f o r controlling t h e m e a n concentration, but fluctuation c a n b e controlled by r e s e r v o i r s only. In t h e case
(6
= 2.5.107rt/yr)
eutrophic,
(6
whilst
@
Basin I remains hypertrophic, Basins I1 and I11
Basin
IV
mesotrophic.
In
the
second
situation
= 20.107rt / y r ) t h e spatial differences and stochastic changes are much
smaller: Basin I...III are eutrophic and Basin IV mesotrophic (the long-term
improvement of water quality i s certainly l a r g e r than t h e s h o r t term one
discussed h e r e ) .
From all what w e learned through t h e management model, it follows t h a t
in o r d e r t o realize t h e optimal s h o r t term s t r a t e g y of eutrophication
management
-
t e r t i a r y treatment of d i r e c t sewage discharges should b e introduced (from west to east);
-
depending on t h e budget available t e r t i a r y treatment of indirect
sewage discharges of pre-reservoirs (again from west to e a s t )
should b e realized.
For f u r t h e r details of t h e management s t r a t e g y worked out f o r Lake Balaton
and o t h e r management models not discussed in t h i s p a p e r , t h e r e a d e r i s
r e f e r r e d to Somlyddy (1983b) and Somlyddy and van S t r a t e n (1985).
I
( 1 ) 0 = 2.5. lo7 Ftlyr
Basin IV
f
Basin I I I
Basin I I
hEWi) 2 o(yi);
noncontrolled case
Basin I V I l l I I I
Basin I V
Basin I I I
Basin I I
I
0
50
100
_Yi = (Chl-a),,
[rng/rn31
FIGURE 12. Probability density functions for two different situations
(from 1000 Monte Carlo simulations).
8.3 Results of the Stochastic Recourse Model
A s seen from Table 1(line 9). t h e nominal s t a t e of water quality i s given
by t h e indicator vector YOi
fied by y i
= (75,38,28,20), (i =(I,. ..,4)). G o a l s were speci-
= (48,28,24,18) expressing t h e desire that Basin I should be
shifted t o t h e eutrophic, and o t h e r segments t o t h e mesotrophic state (see
Figure l l ) , but without forcing a completely homogeneous water quality in
t h e entire lake on t h e s h o r t term, which would be unrealistic.
The definition of these goals, however, means t h a t t h e improvement
intended t o be achieved i s quite uniform f o r t h e four basins in a relative
sense: as compared to t h e maximal possible reduction in t h e water quality
indicator on the s h o r t t e r m (see Figure 6). w e plan 50-60% improvement f o r
Basins I, 11, and 111. Basin IV (with 20%) is t h e only exception as its water
quality is presently quite good, but this segment plays a secondary r o l e
from t h e viewpoint of t h e management problem.
Other parameters of t h e objective function (see (5.11) and Figure 3).
ei
and q i , w e r e selected uniformly f o r t h e four basins: ei
qi
= 10, i =1, . - ,4. This corresponds, in the region
=5
and
zi S 5, t o a "variance
formulation" of the objective function (being similar t o (6.1)) as q / 2e
= 1.
With these parameter values, t h e quadratic portion of t h e utility function i s
predominant in Basins 11, 111, and IV, while f o r Basin I t h e upper linear portion of t h e utility functional i s also of importance.
Results of t h e stochastic optimization model with recourse are also
illustrated in Figures 9-11, in comparison with that of t h e expectationvariance model. A s seen from Figures 9-11, t h e two models produce practically t h e same results in terms of t h e water quality indicator (including also
i t s distribution).
With r e s p e c t t o details t h e r e a r e minor deviations.
According t o Figure 1 0 , t h e expectation-variance
model gives more
emphasis t o fluctuations in water quality (see also f u r t h e r on) and consequently t o r e s e r v o i r projects, than t h e stochastic r e c o u r s e model (with t h e
parameters specified above), and this is in accordance with t h e remarks
made in Section 6, i.e., t h e r o l e of t h e variance is o v e r s t r e s s e d in t h e
expectation-variance model.
From t h i s quick comparison of t h e performances of t h e two models, we
may conclude t h a t t h e more precise stochastic model validates t h e use of
t h e expectation-variance model in t h e case of Lake Balaton.
For a more systematic comparison of t h e two models, t h e difference in
t h e objective functions should be k e p t in mind: t h e stochastic model has
more parameters than t h e expectation-variance model. In p a r t i c u l a r , t h e
exclusion of t h e water quality goal from t h e expectation-variance model
version plays an important role; t h e goal can b e included only indirectly by
modifying t h e weighting f a c t o r s in (6.7). Figure 1 0 illustrates clearly t h a t
t h e prescription of t h e goal close t o t h e lowest realizable value f o r Basin I
(see Figure 6, 90%of t h e possible improvement in t h e s h o r t term) leads t o a
s t r o n g e r emphasis on r e s e r v o i r s as compared, not only t o t h e original c a s e ,
but also t o t h e results of t h e expectation-variance model.
increase in z~~(as a function of
d e c r e a s e in zml
The f a s t e r
8, s e e Figure 1 0 ) is associated with a
- as expected - in addition t o smaller budget allocations
f o r t h e o t h e r basins. Depending on t h e value of yl, in t h e r a n g e (35, 48),
t h e solutions lay in t h e shadowed areas indicated in Figure 1 0 and t h e solution of t h e expectation-variance model is located in t h e "center" of these
domains.
A s mentioned before, t h e expectation-variance model gives more
weight t o variance than t h e stochastic r e c o u r s e model and f o r computational justification w e compared c u r v e s (A) and (B) in Figure 10. The
rationality f o r t h i s comparison is t h a t in lack of water quality goals and
with equal weighting, t h e expectation-variance model follows t o some extent
t h e principle of "equal relative" water quality improvement in all basins
(other f a c t o r s
- e.g. t h e distribution of costs f o r basins - play also impor-
t a n t roles), and in t h i s sense i t s solution c a n b e s t b e compared t o solution
(B) of t h e stochastic method.
Further discussion on t h e r o l e of parameters y f ,ef and qf , and on t h e
comparison of t h e deterministic model (5.12) and t h e stochastic model is
given in t h e subsequent section.
8.4 Comparison of the Deterministic and Stochastic Solutions
-- Sensitivity Analysis
In o r d e r t o gain experience with systems different from Lake Balaton,
w e changed some of t h e parameters of t h e Balaton problem and continue o u r
presentation on t h e application of EMOM with t h i s modified example. W e call
t h e hypothetical water body: Lake Alanton.
Lake Alanton differs from Balaton in t h e following respects:
(i)
The volume of Basin IV i s only 6 0 - 1 0 ~ m(see
~ Table 1 , line 5);
(ii) Two d i r e c t sewage loads are increased in t h e region of Basin IV
(see Figure 5 ) t o 50 and 70 kg/d, resulting in t h e same absolute
BAP load than t h a t of Basin I (the volumetric load is however
l a r g e r due to (i); 2.3 mg/m3/d, see Table 1 , line 7);
(iii) The nominal water quality indicator and t h e slope of t h e response
line was modified f o r Basin IV in such a manner, that response
lines of Basins I and IV coincide (see Figure 6);
(iv) Cost functions of the two treatment plants were modified (they
became similar t o t h e one illustrated in Figure 9).
A s a consequence of these changes, t h e water quality of Lake Alanton is
approximately equally "bad" a t i t s two ends. Still, however, a n important
difference exists between Basins I and IV: t h e load of Basin I is governed by
"stochastic" tributary load, while t h a t of Basin IV, is given by "deterministic" sewage discharges. This way t h e low water quality of Basin I is associated with large fluctuations (as seen in previous Sections), but randomness
is of secondary importance for Basin IV.
The longitudinal distribution of t h e water quality is now quite different
from t h a t of Lake Balaton, and a manager may have t h e intention t o establish a uniform quality by control decisions. Accordingly we fixed the goals
to 71 = 30 mg/m3 i =I.
. . ,4,and maintained
t h e same parameter values as
used in Section 8.3. Results f o r t h e "basic situation" (Section 8.2) a r e given
in Figures 1 3 and 14.
From Figure 1 3 t h e same conclusions can be drawn f o r Basin I than
from Figure 9. The only difference i s t h a t at a fixed budget t h e water quality improvement i s slightly smaller than for Lake Balaton as a p a r t of t h e
budget is utilized for Basin IV. A s seen f r o m t h e figure, P precipitation i s an
effective tool for improving t h e water quality of Basin IV: t h e concentration (Chla),,
i s reduced from 80 to about 40 mg/m3 already a t a budget of
2 . 5 . 1 0 ' ~/ yr. The increase of @ results in nearly no f u r t h e r change and
,
Basin I
-
min { Y 1 )
Basin IV
m x
-
-
-
e
EW4I
min
I
I
I
10
20
30
Iy41
{y4}
b
fl > TAC [ 1o7 F t l ~ r I
FIGURE 13. Lake Alanton: water quality indicator as a function of t h e total annual cost.
Basins I and I V
-.-.-
Stochastic model with recourse, Eq. (5.1 1)
Deterministic model, Eq. (5.12)
XS'l
FIGURE 14. Lake Alanton: major decision variables.
Basins I and IV
FIGURE 15. Sensitivity with r e s p e c t to water quality goals.
t h e fluctuations also remain approximately constant, nevertheless s m a l l in
comparison t o Basin I.
In Figure 14, xss belongs t o t h e largest treatment plant in t h e region of
Basin IV. The c h a r a c t e r of xD1(@)and xsN1(@) gained from t h e stochastic
m o d e l i s t h e same as f o r Lake Balaton (Figure l o ) , while xsq is practically
constant above @
= 2.5.107R / yr . The most important conclusion of this
figure can b e drawn from t h e comparison of t h e stochastic and deterministic
solutions: t h e deterministic quadratic model (5.12) excludes (incorrectly)
t h e reed-lake p a r t of t h e l a r g e s t r e s e r v o i r p r o j e c t even under t h e l a r g e r
budget values, which is a consequence of neglecting t h e random elements of
t h e problem. The (Chl-a),,,
concentration is reduced on a n a v e r a g e t o
about 40 mg/m3; however, extreme values still can exceed 1 0 0 mg/m3; as
contrasted t o t h e solution of t h e stochastic m o d e l when t h e maximum i s
about 60 mg/m3 This example shows clearly if t h e analyst is not able t o
recognize t h e stochastic f e a t u r e s of t h e problem, t h e control s t r a t e g y
worked out may lead t o serious failures.
The sensitivity of t h e solution with r e s p e c t t o t h e water quality goals i s
illustrated in Figure 15 r e f e r r i n g t o a typical budget 1 5 . 1 0 /~yr.
~ From
t h e analysis performed and t h e figure, t h e following conclusions can b e
drawn:
(i)
If t h e goal is set unrealistically f a r from t h e nominal value and
from t h e values which can b e realized by t h e available control
measures (e.g. 0 o r 200 mg/m3 in t h i s case), t h e penalty function
h a s nearly no influence and thus t h e solution i s equivalent t o t h e
deterministic one (xD1
= 0);
(ii) The solution is quite sensitive on t h e choice of t h e y i . The variable z~~ has a maximum at about 40 mg/m3. If t h e goals uniformly
o r individually f o r Basins I and IV are close t o 75-80 mg/m 3 t h e
corresponding major decision variables. are close to z e r o (no
action is taken).
In summary, w e can state t h a t t h e water quality goal h a s a major influence on t h e solution, i t f o r c e s indeed t h e solution towards t h e desired levels of water quality in t h e different lake basins. This f e a t u r e is t h e primary
advantage of t h e stochastic model with r e c o u r s e as contrasted to t h e
expectation-variance model f o r t h e Lake Balaton c a s e and i t s variant Lake
Alanton.
A s t o t h e r o l e of t h e o t h e r parameters of t h e penalty function of t h e
stochastic model is concerned, t h e systematic analysis performed did not
lead t o unambiguous conclusions. For both examples, Lake Balaton and Lake
Alanton, t h e influence of ei and qi is of secondary importance as compared
t o t h e effect of yi . In general, i t c a n b e said t h a t t h e influence is minor f o r
small and l a r g e budgets.
In t h e middle budget r a n g e i t is difficult t o
s e p a r a t e t h e impact of ei and q i , especially because it strongly depends
also on t h e p r e s e t goals y i , and t h e stochastic f e a t u r e s of the water quality
f o r t h e different basins.
The experience gained in t h e frame of t h e p r e s e n t study suggests t h e
choice of a piecewise linear-quadratic utility function with q / 2 e = 1 as a
f i r s t step, see (5.9) and Figure 3, and t o perform a thorough sensitivity
analysis on t h e p a r a m e t e r s y i , ei and qi in t h e subsequent steps.
W e complete this section with t h e following conclusions:
(i)
The stochastic optimization model with r e c o u r s e justified t h e
applicability of t h e much simpler expectation-variance model f o r
Lake Balaton;
(ii) Deterministic version of t h e stochastic objective function and t h e
solution of t h e corresponding deterministic quadratic optimization
problem leads t o strikingly different and i n c o r r e c t management
s t r a t e g y as compared t o t h e stochastic model;
(iii) The major p a r a m e t e r of t h e stochastic optimization model with
r e c o u r s e i s t h e water quality goal p r e s c r i b e d f o r different basins.
The inclusion of t h e goal in t h e objective function i s t h e primary
advantage in comparison with t h e expectation-variance model.
The advantage of t h e latter model is, of course, simplicity and f a s t
implementation;
(iv) F u r t h e r experimentation i s needed in t h e selection of p a r a m e t e r s
ei and qi (in t h e objective function of t h e stochastic model).
In this p a p e r w e d e a l t with t h e development and application of stochast i c optimization models f o r Lake eutrophication management. W e considered
primarily shallow l a k e s which are strongly influenced by hydrologic and
meteorologic f a c t o r s and thus stochasticity should b e a key component of
water quality control.
Major elements of t h e study performed are as follows:
(i)
Identification of important s t e p s of eutrophication management in
practice,
(ii) Based on t h e principle of decomposition and aggregation, an
approach is presented how t o develop a eutrophication management optimization model, EMOM, which p r e s e r v e s t h e scientific
details of diverse in-lake and watershed processes needed at t h e
decision-making level.
The procedure combines simulation and
optimization in t h e framework of EMOM.
(iii) W e describe t h e proposed stochastic, planning type lake eutrophication and nutrient load models, LEMP and NLMP2, respectively,
which are major components of EMOM.
(iv) W e discuss control variables, c o s t functions and various cons t r a i n t s t o b e used in EMOM.
(v) Alternative management optimization models are formulated which
use t h e same LEMP, NLMP2, control variables etc., and differ primarily in t h e objective function and solution technique t o b e
adopted. Three of them were selected f o r implementation: a full
stochastic model, a n expectation-variance model, and t h e deterministic (quadratic) version of t h e full stochastic method. For t h e
f i r s t one a new stochastic programming procedure had t o b e
developed, while f o r t h e o t h e r two standard packages could be
employed. The objective function of t h e full stochastic model h a s
one more parameter as compared t o t h e expectation-variance
model: including t h e possibility of selecting t h e water quality goal
t o be achieved by t h e management, which makes this model espe-
cially attractive.
(vi) Application t o Lake Balaton. This p a r t of the study had a d i r e c t
impact on t h e policy making procedure performed in Hungary in
1982 which ended up with a government decision in 1983.
(vii) Comparison of t h e two stochastic models, furthermore deterministic and stochastic approaches on t h e example of Lake Balaton and
on a modified, hypothetical system (the comparison w a s associated
with a detailed sensitivity analysis).
ACKNOWLEDGEMENT
In addition to t h e development of t h e computer code f o r t h e method
described in Section 7, Alan King (University of Washington, Seattle) h a s
contributed much to this p r o j e c t by his pertinent comments concerning
algorithmic implementation and d a t a analysis. W e hope h e finds a small sign
of o u r appreciation in seeing t h e variant of Lake Balaton named in his
honor.
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