1
1FUZZY
MODEL FOR REAL-TIME RESERVOIR OPERATION
Tanja Dubrovin,2 Ari Jolma,3 and Esko Turunen4
ABSTRACT
A fuzzy rule-based control model for multipurpose real-time reservoir operation is
constructed. A new, mathematically justified methodology for fuzzy inference, Total Fuzzy
Similarity is used and compared with the more traditional Sugeno-style method. Specifically
the seasonal variation in both hydrological variables and operational targets is examined. This
is done by considering the inputs as season-dependent relative values, instead of using
absolute values. The inference drawn in several stages allows a simple, accessible model
structure. The control model is illustrated using Lake Päijänne, a regulated lake in Finland.
The model is calibrated to simulate the actual operation, but also to better fulfill the new
multipurpose operational objectives determined by experts. Relatively similar results obtained
with the inference methods and the strong mathematical background of Total Fuzzy Similarity
put fuzzy reasoning on a solid foundation.
1
To be published in Journal of Water Resources Planning and Management
2
Laboratory of Water Resources Management, Helsinki University of Technology, Finland.
Current affiliation: Finnish Environment Institute, P.O. Box 140, 00251 Helsinki, Finland.
Email: tdubrovi@water.hut.fi
3
Laboratory of Water Resources Management, Helsinki University of Technology, P.O. Box
5200, 02015 HUT, Finland. Email:Ari.Jolma@hut.fi
4
Tampere University of Technology, P.O. Box 692, 33101 Tampere, Finland. Email:
Esko.Turunen@cc.tut.fi
2
INTRODUCTION
Real-time reservoir operation is a continuous decision-making process on the water level of a
reservoir and release from it. The operation is always based on operating policy and rules
defined and decided upon in strategic planning. The complexity of the real-time release
decision, considering all the time-dependent information, warrants the importance of real-time
operation. The operator's task in real-time reservoir operation is to fulfill the objectives as
well as possible while complying with legal and other constraints. Reservoir operation
involves uncertainty and inaccuracies. Uncertainty is involved in objectives in the sense that
the values and targets are usually subjective, and the relative emphases on different objectives
change with time. Evaluating all objectives in commensurate values is a complex and often
impossible task. Determination of the total net inflow into the reservoir and forecasting it is
both inaccurate and uncertain. The seasonal variation in both hydrological variables and
operational objectives brings uncertainty into the operation, since the seasons do not begin and
end on the same date every year. In many cases fuzzy logic may provide the most appropriate
methodological tool for modeling reservoir operation.
First introduced by Zadeh (1965), fuzzy logic and fuzzy set theory have been used e.g. in
modeling the ambiguity and uncertainty in decision-making. The basic idea in fuzzy logic is
simple: statements are not just ‘true’ or ‘false’, but partial truth is also accepted. Similarly, in
fuzzy set theory, partial belonging to a set, called a fuzzy set, is possible. Fuzzy sets are
characterized by membership functions. The demonstrated benefit of fuzzy logic in control
theory is in modeling human expert knowledge, rather than modeling the process itself.
Despite its indisputable successes, fuzzy logic suffers from a lack of solid mathematical
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foundations; e.g. in fuzzy IF-THEN inference systems there are a multitude of techniques in
the literature how to draw conclusions from partially true premises, although no logical
justification for such rules is given. Several approaches have been used to apply fuzzy set
theory to reservoir operation. These include fuzzy optimization techniques, fuzzy rule base
systems, and combinations of fuzzy approach with other techniques. Applications can be
found in the work of Fontane et al. (1997), Huang (1996), and Saad et al. (1996). Fuzzy rule
base control systems for reservoir operation are presented by Russell and Campbell (1996)
and Shrestra et al. (1996). The fuzzy rule base can be constructed on the basis of expert
knowledge or observed data. Methods for deriving a rule base from observations are presented
by Bardossy and Duckstein (1995) and Kosko (1992).
Russell and Campbell (1996) mentioned that as the number of inputs increases, a fuzzy rulebase system, specifically the number of rules, quickly becomes too large, unidentifiable, and
unmanageable. A similar problem, that of combining evidence, is solved in belief networks
using Bayesian updating. In Bayesian updating, evidence (premises) is incorporated one piece
at a time, assuming a conditional independence of different pieces of evidence (Russell and
Norvig, 1995).
The present paper describes a real-time fuzzy control model for multipurpose reservoir
operation. The position taken here is that first the number of rules does not become a problem
if expert knowledge is carefully studied and modeled; it is an advantage in all fuzzy inference
systems that the rule base may quite well be incomplete. Secondly, the modeling of expert
knowledge and the development of the fuzzy control model in general are facilitated using a
multistage model. Attention is specifically paid to circumstances in which the hydrological
conditions and water level targets change significantly within a year. In the calibration, the
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actual operation is used as a reference but it is also attempted to better meet the demands of
the re-evaluated objectives. Target levels for two interdependent variables, release and water
level, are considered. The effectiveness of a many-valued inference system based on the welldefined Lukasiewicz-Pavelka logic utilizing many-valued similarity and expert knowledge
(and only them!) is studied in the case of reservoir operation. The applied method, Total
Fuzzy Similarity, was introduced by Turunen (1999). Here the performance of Total Fuzzy
Similarity is compared with a more traditional fuzzy inference method known as Sugeno-style
fuzzy inference.
TOTAL FUZZY SIMILARITY
Recall that a fuzzy set X is an ordered couple (A,X), in which the reference set A is a nonvoid set and the membership function X: A [0,1] shows the degree to which an element a
A belongs to fuzzy set X.
The objective in what is called approximate reasoning in the fuzzy logic framework is to draw
conclusions from partially true premises. In a typical fuzzy inference machine, a control
situation comprehends a system S, an input universe of discourse IN (the IF-parts) and an
output universe of discourse OUT (the THEN-parts). We assume there are n input variables
and one output variable. The dynamics of S are characterized by a finite collection of IFTHEN rules; e.g.
Rule_1: IF x is A1 and y is B1 and z is C1
THEN w is D1
Rule_2: IF x is A2 and y is B2 and z is C2
THEN w is D2
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Rule_k: IF x is Ak and y is Bk and z is Ck
THEN w is Dk,
where A1,…,Dk are fuzzy sets. However, the outputs D1,…,Dk can also be crisp actions. All
these fuzzy sets are to be specified by the fuzzy control engineer. We avoid disjunction
between the rules by allowing some of the output fuzzy sets Di and Dk, i  j, to possibly be
equal. Thus, a fixed THEN part can follow various IF parts. Some of the input fuzzy sets may
also be equal (e.g. Bi = Bj for some values of i  j). However, the rule base should be
consistent; a fixed IF part precedes a unique THEN part. Moreover, the rule base can be
incomplete; if an expert is not able to define the THEN part of some combination of the form
'IF x is Ai and y is Bi and z is Ci' then this rule can simply be skipped.
Given an input (e.g. x = (x,y,z)), there is diversity in the literature about how to count the
corresponding output w. This procedure is called defuzzification. In Sugeno-style fuzzy
inference systems, for example, all the output fuzzy sets are fired partially, and weighted sums
or weighted average are calculated to calculate the output w.
Defuzzification, however, meets with resistance among mathematicians, since it does not
usually have any deeper mathematical justification. To establish fuzzy inference on solid
mathematical foundations, a method called Total Fuzzy Similarity was introduced by Turunen
(1999). The idea in the Total Fuzzy Similarity approach is to look for the most similar
premise, the IF part, and fire the corresponding conclusion, the THEN part. Moreover, the
degree of similarity may be composed of various partial similarities.
The following algorithm recounts how to construct a Total Fuzzy Similarity-based inference
system.
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Step 1. Create the dynamics of S, i.e. define the IF-THEN rules, give the shapes of the input
fuzzy sets (e.g. A1,…,Ck) and the shapes of the output fuzzy sets (e.g. D1,…,Dk).
Step 2. Give weights to various parts of the input fuzzy sets (e.g. m1, m2, m3 to Ai.s, Bi.s, and
Ci.s) to emphasize the mutual importance of the corresponding input variables.
Step 3. Put the IF-THEN rules in a linear order with respect to their mutual importance, or
give some criteria on how this can be done when necessary.
Step 4. For each THEN part i, give criteria for distinguishing outputs with equal degrees of
membership.
A general framework for the inference system is now ready. Assume then that we have input
value e.g. x = (x, y, z). The corresponding output value w is found in the following way.
Step 5. Compare the input value x separately with each IF part, in other words, count total
fuzzy similarities between the actual inputs and each IF part of the rule base; this simply
means counting the weighted means, e.g.
Similarity(x, Rule_1) = 1/M[m1A1(x)+m2B1(y)+m3C1(z)]
Similarity(x, Rule_2) = 1/M[m1A2(x)+m2B2(y)+m3C2(z)]
.
.
Similarity(x, Rule_k) = 1/M[m1Ak(x)+m2Bk(y)+m3Ck(z)]
where m1, m2, and m3 are the weights given in Step 2 and M = m1+m2+m3.
Step 6. Fire an output value w such that Di(w) = Similarity(x, Rule_i) corresponding to the
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maximal total fuzzy similarity; if such a Rule_i is not unique, use the mutual order given in
Step 3, and if there are several such output values w, use the criteria given in Step 4.
Of course, the algorithm can be specified by putting extra demands. e.g. in some cases the
degree of total fuzzy similarity of the best alternative should be greater than some fixed value
[0,1] before any action is taken, sometimes all the alternatives possessing the highest fuzzy
similarity should be indicated, or the difference between the best candidate and the second
best should be larger than a fixed value [0,1], etc. All this is dependent on expert choice. It
is worth noting that all the steps in our algorithm are based only on well-defined mathematical
concepts or on an expert's knowledge.
MODEL CONSTRUCTION
The model consists of two real-time submodels. The model structure is shown in Figure 1.
The first submodel sets up a reference water level (WREF) for each time step. Given this
reference level, the observed water level (W), and the observed inflow (I), the second
submodel makes the decision on how much should be released from the reservoir during the
next time step.
The seasonal variation is accounted for in the fuzzification phase. Instead of using absolute
observed values for inputs, season-dependent relative values are used. Hence, in fuzzification
the membership of the difference between the observed value and the season-dependent
reference value is determined, instead of the membership of the observed value alone. For
output, the same membership functions and absolute values are used throughout the year.
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Reference water level model
The output of the first submodel is the WREF value for each time step. The WREF value
relies on the water level targets, and it is an input into the second submodel, the release model.
The purpose of the reference water level model is to take snow depth observations and
seasonal variation of targets into consideration. Nonetheless, the aim is not to make the actual
water level exactly follow the WREF value every year. The actual water level can be above or
below the WREF value, depending on the hydrological conditions.
The year is divided into three seasons: the snow accumulation season, the snowmelt season,
and the rest of the year. In each season determination of the WREF value is performed
differently. For the ‘rest of the year’ season, the WREF values are individual for each time
step but do not change from year to year. For the snowmelt season, WREF value is dependent
on the snow water equivalent (SWE) and can be inferred for each time step with the fuzzy
rules:
IF SWE is smaller than average/average/larger than average/much larger than average
THEN WREF is high/middle/low/very low.
The premise in this rule is the difference between the observed SWE and the average SWE for
that time step. Average (or median) values can be calculated from historical data, or they can
be determined by an expert. During the snow accumulation season the WREF value is
reduced. The WREF value for the end of the season, i.e., the beginning of the next snowmelt
season, is inferred for each time step. Since the premise in the rules above is relative, the
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inference can be made again with the same rules and membership functions using the
observed and average SWE values for each time step. The submodel output is reduced linearly
toward the level inferred for each time step until the beginning of the next snowmelt period.
Release model
Premises in the inference system for the release are

Relative water level (Wrelative) at the beginning of that time step: The relative value is
determined by the difference between observed water level and WREF value, i.e. the
output of the first submodel for that time step.

Relative inflow (Irelative): The relative value is determined by the difference between
observed inflow and average inflow for a given month.
General rule formulation is as follows:
IF Wrelative is very low/low/objective/high/very high
AND Irelative is very small/small/large/very large
THEN
release
is
exceptionally
small/very
small/small/quite
small/quite
large/large/very large/exceptionally large.
During calibration it was found that in flood situations the rule base could not keep the water
level sufficiently below the critical flood level. Releases during floods could not be increased,
because it would have led to too large releases in less critical situations. Hence, an extra
release can be added to the system output when the water level is critical. Extra release is
determined again with a fuzzy rule:
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IF W is critical, THEN add extra release.
The membership of absolute water level belonging to a “critical” set is determined by a simple
membership function. Similarly, one membership function is used in defuzzification.
CASE STUDY: LAKE PÄIJÄNNE
Lake Päijänne is a 120 km long and 20 km wide regulated lake in central Finland (Figure 2).
The active storage volume is approximately 3000 Mm3 and annual inflow 7028 Mm3. The
release from Lake Päijänne runs through several smaller lakes and the River Kymijoki into the
Gulf of Finland. There are 12 hydropower plants along the River Kymijoki with total
hydropower generation potential of 200 MW. Other interests include agriculture, shoreline
real estate, recreation, forestry, fisheries, navigation, and ecology. The low shores of Lake
Päijänne, with their buildings and cultivated land, are exposed to flood damages. In addition
to conflicts between different interest groups there are conflicts between users of Lake
Päijänne and the River Kymijoki. If, for example, fluctuations in water level are controlled
only from the lake users’ point of view, flow in the River Kymijoki may be inadequate.
Changes in inflow into Lake Päijänne are fairly slow because its catchment area is rather large
and contains a large number of lakes.
There are legally binding constraints for the operation of Lake Päijänne. The regulation permit
for Lake Päijänne, which came into effect in 1954, defines flood protection as the key
objective of the regulation. The regulation permit defines sets of constraints for the operation
of Lake Päijänne. These include minimum and maximum water levels, maximum change in
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release, and target water levels. Target water levels are determined for the beginning of each
three-month period by inflow forecast and the target water level for the previous period. Thus
the target water level may change as the accuracy of the forecast increases and the final target
level is not known until the end of the three-month period.
New objectives for the regulation of Lake Päijänne have been developed in a recent study
(Marttunen and Järvinen, 1999). The report defines target water levels, or rather upper and
lower limits for target levels and objective releases for various seasons and hydrological
states. Some of the rules are already defined as IF-THEN rules. These guidelines were used as
expert knowledge in development of the fuzzy control model.
Policy regarding water level presented in the aforementioned study is in short:
January to April
The water level should be lowered by the beginning of the snowmelt season to avoid flooding.
The target water level is dependent on inflow predictions.
May
After the snowmelt season, the water level should be raised for natural production of pike. To
control overgrowth of reeds, the water level should be raised to 78.55 m.
June to August
An adequate summer water level for ecological and recreational objectives is about 78.30 –
78.60 m. The 78.55 m level should be reached by early July. A slight reduction after the peak
is advantageous for the ecology of the lake and its shores. If there is a risk of flooding or
drought, the reduction should be ignored.
September to December
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If there is no risk of flood, the water level should be raised to enable larger releases during the
coldest winter season in order to maximize hydropower benefit.
Water levels higher than 78.75 m in Lake Päijänne cause damage to buildings, agriculture,
industry, and forests. Correspondingly, the critical flow rate in the River Kymijoki is 480
m3/s. The optimum flow for hydropower generation in the River Kymijoki is 380 m3/s
(Marttunen and Järvinen, 1999).
MODEL APPLICATION
To apply the model to real-time operation of Lake Päijänne the release objectives were fitted
into the model structure described above. The time step used in the simulation was five days.
Daily water levels in Lake Päijänne, release and flow in the River Kymijoki, and the bimonthly SWE measurements in the catchment area of the lake during the snow accumulation
period were available. Net inflow time series were calculated using the water balance
equation. The SWE data were interpolated for each time step. Flows for each time step were
averaged from daily data.
Data from the years 1976-1985 were used for calibration and the years 1986-1995 for
validation. Reference values, i.e. monthly averages for inflow and SWE for each period were
calculated from the years 1970-1985.
For the reference water level model the snow accumulation season was assumed to last from
December to February. It was found necessary to set the snowmelt season to begin in the
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model somewhat earlier than it usually does in reality. The flood caused by snowmelt varies in
size and duration from year to year. To satisfy ecological and recreational objectives it is
desirable to raise the water level of the lake as early as possible if there is no risk of flooding.
March was always considered to be a part of the snowmelt season. If the SWE was larger than
average in April, the WREF value was kept low, otherwise its linear elevation toward higher
summer water levels was started. From early June onward, the same WREF curve was used
for each year, according to the water level targets in the operational guidelines.
The formulation of the rule base was made on the basis of both historical data and expert
knowledge. The method used in the calibration phase was Total Fuzzy Similarity. The
membership functions for premises and the rule base were set up first. Triangular membership
functions were used for all inputs and output. Relative snow depth in water equivalent
corresponded to the four fuzzy sets, relative water level to the five fuzzy sets, and relative
inflow to the four fuzzy sets. For each rule in the release model, the calibration data were
analyzed to find cases in which the premises were most similar to that rule. Corresponding
responses in observation data were used as a basis when forming membership functions for
output, which were then adjusted further by trial and error. Membership functions for relative
water level, relative inflow, and release are presented in Figure 3.
The model was calibrated using local and global optimization. In local optimization the
observed water level was used as input in each time step. Model response was compared with
observed release. In global optimization the model was allowed to operate ‘independently’,
i.e. the water level was simulated for each time step given the model output from the previous
time step. The release and simulated water level were compared with the observed values and,
in addition, with the new objectives. The objective of local optimization was to find a good
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match with the observed data and the model output, and the objective of global optimization
was to better meet the new demands set for the operation of Lake Päijänne.
Some further adjustments were made in the model. If the observed water level was less than
or equal to 78.60 m, the degree of membership in a fuzzy set ‘critical water level’ was 0, and
when the water level was greater than or equal to 78.95 m, the degree of membership was 1.
Membership values between these water levels were interpolated linearly. Extra release was
between 0 and 130 m3/s.
Calculated net inflow include much noise because a small error in water level reading may
cause a considerable error in storage volume value used in the water balance equation.
Calibration showed that a strong variation in the input led to undesirable variation in output in
the model. Therefore, relative inflow was filtered by averaging the values of the six previous
periods. The current release permit states that too rapid a change in release is not allowed. To
accommodate this, the total output was constrained so that the release was not allowed to
change more than 50 m3/s during two time steps (ten days).
Water level could be kept close to the targets if ‘narrow’ membership functions for relative
water level were used or if extreme releases were allowed. Yet the problem was too strong
fluctuations in the release. One challenge in developing an operational model for a
multipurpose reservoir is that both the water level and release are relevant, while there is
interdependence between them. Actual water level is dependent on previous decisions, i.e.,
releases, whereas the observed water level is the basis for the next decision. Since the aim of
reservoir operation is to fulfill the objectives of both release and water level, a compromise
between them was sought.
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The following is an example of the fuzzifying and defuzzifying process used in the release
model:
Assume the output from the reference water level model, WREF is 77.9 m and the water level
calculated with the result from previous time step is 78.09 m. Thus, the difference between
calculated water level and WREF is 0.19 m. From Figure 3 a) it can be seen that the water
level belongs to a fuzzy set ‘objective’ with a membership 0.367 and to the fuzzy set ‘high’
with a membership 0.633. Assume that inflow in previous time steps has been 10 m3/s larger
than average inflow in the current month. The Figure 3 b) shows that the membership in the
fuzzy set ‘small’ is 0.375 and the membership in the fuzzy set ‘large’ is 0.625. The
memberships in other sets are equal to zero.
The degrees of similarity for each rule in the rule base are calculated by counting an average
of the membership of water level in the rule-specific fuzzy set and the membership of inflow
in the rule-specific fuzzy set. Thus, the degrees of similarity for some of the rules are:
If Wrelative is objective and Irelative is very small then release is small.
0.18
If Wrelative is objective and Irelative is small then release is quite small.
0.37
If Wrelative is objective and Irelative is large then release is quite large.
0.50
If Wrelative is objective and Irelative is very large then release is large.
0.18
If Wrelative is high and Irelative is very small then release is quite small
0.32
If Wrelative is high and Irelative is small then release is quite large.
0.50
If Wrelative is high and Irelative is large then release is large.
0.63
If Wrelative is high and Irelative is very large then release is very large
0.32
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The rules not presented in this list have smaller similarities. The rule to be fired is the seventh
one, which has the largest degree of similarity. The defuzzified output will be release, which
belongs to a set ‘large’ at a membership 0.63. From Figure 3 c) we get two alternative values:
281 m3/s and 309 m3/s. Next, consider the rule that achieved the second largest similarity, the
sixth rule in the list. From two alternative values the one closer to the output of the sixth rule,
‘quite large’, will be chosen. Thus, the defuzzified output is 281 m3/s. If the output ‘very
large’ would also have had the second largest similarity, the value ‘closer to the middle’, 281
m3/s again, would have been chosen. This criteria for choosing one of the two alternative
values was found to be functional in this model, but other criterias can be devised and used,
depending on an expert’s choise.
RESULTS
The model was tested using data from the years 1985-1996. In the test phase global
optimization was used. Figure 4 shows the releases decided by the model when Total Fuzzy
Similarity was used. Correspondingly, the simulated water level is presented in Figure 5.
Figures also show the observed time series and WREF. The Sugeno method was chosen for
comparison against the Total Fuzzy Similarity, due to its simplicity and popularity. With both
methods the system was kept the same as much as possible. To apply the Sugeno method the
peak values of the triangular output membership functions were used as crisp values.
Combination of the premises was implemented by taking a minimum, and defuzzification was
performed using a weighted average.
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The performances of the two methods were almost indistinguishable. With the Total Fuzzy
Similarity the water level targets during the summer were sometimes better fulfilled, but the
release tended to fluctuate more, and the limitation on change in release was more relevant.
The comparison between releases with different methods in year 1990 is presented in Figure
6.
For the calculations it was assumed that no inflow forecast is available. In reality inflow
forecasts are available for the operations managers. To assess the effect of inflow forecasts,
simulations were run with perfect hindsight, i.e. average inflow values for the next six periods
were used as input instead of inflow values from previous periods. With perfect forecast the
release was more stable but no other major enhancement was seen in the model performance.
To estimate the model performance, the revenue from hydropower generation was calculated
for a test set. For this purpose the flow through the hydropower plants was calculated by
adding observed incremental flows to the model output. Energy prices cannot be exactly
determined in advance, but in general they are higher in winter than in summer. Prices of
38.18 euro/MWh for winter and 23.30 euro/MWh for summer were used. Monetary flood
damages on the shores of Lake Päijänne, other lakes, and the River Kymijoki were estimated.
For this purpose cost tables for flood damages for buildings, industry, and agriculture were
available. Revenues and losses are presented in Table 1.
DISCUSSION AND CONCLUSIONS
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A fuzzy control model for real-time reservoir operation was developed. A new method for
fuzzy inference, Total Fuzzy Similarity, was used and compared with a more traditional
method. The model performance was generally good, but the model did not capture expert
thinking in the most exceptional circumstances. This was evident in the calibration phase,
where the model could not manage the end of an unusually long flood as well as an expert.
However, this weakness could be solved by constructing a particular rule base which should
go off only under special circumstances. Another weakness of the model was its inability to
‘see’ forward and backward at several time steps. This inability led to excess variation in
release and flooding along the shores of the River Kymijoki.
Building up a fuzzy rule base for reservoir operation involves stressing the seasonal variation
inherent in both hydrological variables and operational targets. Under Finnish conditions, in
which snowmelt plays an important role in a hydrological year, good results in lake operation
cannot be reached with the same rules throughout the year. Due to differing predominant uses
of the lakes and reservoirs in different seasons, various kinds of rules must be effective in the
model for different seasons. Possible ways of handling seasonal variation are to use the time
of the year as model input or construct a separate rule base for each season. These require
calibration for each rule base or are subject to the ‘curse of dimensionality’.
In the present study, seasonal variation was considered in the fuzzification phase and in the
season-dependent WREF values. The seasonal change was smooth because there were no
rough changes from one rule base to another. With the help of a multistage model structure
the number of rules could be kept at a reasonable level and the functioning of the model easy
to comprehend. Moreover, this approach implements the release guidelines determined by
experts and the release permit. Target water levels play an important role in decision-making.
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The relatively similar results obtained with the Total Fuzzy Similarity method and the Sugeno
method indicate that the underlying concept in fuzzy control is many-valued similarity. The
strong mathematical background of the Total Fuzzy Similarity method puts fuzzy reasoning
on a more solid foundation.
The advantages of fuzzy logic are that calculation is straightforward and the model easy for
the operator to understand due to its structure which is based on human thinking. The system
can also be easily modified when necessary. In the fuzzy control model described, changes in
operation can be implemented by fine-tuning the reference water level submodel. Various
characteristics of different reservoirs can be encoded in rules, membership functions, and
WREF values. Application of the model requires expert knowledge and sufficient amounts of
historical data.
ACKNOWLEDGMENTS
Support for this work was partly provided by the Finnish Ministry of Agriculture and Forestry
(Grant No. 4855/421/98). The authors acknowledge Professor Pertti Vakkilainen for fruity
discussions and comments on the manuscript, Erkki A. Järvinen and Mika Marttunen for
generously sharing their expertise on the regulation of Lake Päijänne, and the two anonymous
reviewers for helpful suggestions.
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REFERENCES
Bárdossy, A., and Duckstein, L. (1995). Fuzzy rule-based modeling with applications to
geophysical, biological and engineering systems. CRC Press, Boca Raton.
Fontane, D. G., Gates, T. K., and Moncada, E. (1997). “Planning reservoir operations with
imprecise objectives.” Journal of Water Resources Planning and Management, ASCE,
123(3), 154-162.
Huang, W.-C. (1996). “Decision support system for reservoir operation.” Water Resources
Bulletin, 32(6), 1221-1232.
Kosko, B. (1992). Neural networks and fuzzy systems. Prentice Hall, Englewood cliffs, N.J.
Marttunen, M., and Järvinen, E. A. (1999). Development of regulation of Lake Päijänne Synthesis and recommendations. Finnish Environment Institute, Helsinki (in Finnish).
Russell, S. J., and Norvig, P. (1995). Artificial intelligence: a modern approach. Prentice
Hall, Englewood Cliffs (NJ).
Russell, S. O., and Campbell, P. F. (1996). “Reservoir operating rules with fuzzy
programming.” Journal of Water Resources Planning and Management, ASCE, 122(3), 165170.
21
Saad, M., Bigras, P., Turgeon, A., and Duquette, R. (1996). “Fuzzy learning decomposition
for the scheduling of hydroelectric power systems.” Water Resources Research, 32(1), 179186.
Shrestra, B. P., Duckstein, L., and Stakhiv, E. Z. (1996). “Fuzzy rule-based modeling of
reservoir operation.” Journal of Water Resources Planning and Management, ASCE, 122(4),
262-269.
Turunen, E. (1999). Mathematics behind fuzzy logic, Advances in soft computing. PhysicaVerlag, Heidelberg.
Zadeh, L. A., (1965). “Fuzzy Sets.” Information and Control, vol 8, 338-353.
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Figure captions:
Figure 1. Structure of fuzzy control model for reservoir operation.
Figure 2. Lake Päijänne
Figure 3. Membership functions for: (a) Water level; (b) Inflow; (c) Release (output).
Figure 4. Release with fuzzy logic and actual operation.
Figure 5. Water level with fuzzy logic and observed water level.
Figure 6. Release with Total Fuzzy Similarity and Sugeno in year 1990.
23
Table 1. Comparison of calculated monetary flood loss and hydropower revenue from actual
reservoir operation and fuzzy control model with alternative methods.
Years 1986-1995
Method
(1)
Monetary
Hydropower
flood loss,
revenue,
Million euros/year Million euros/year
(2)
(3)
Actual operation
0.19
41.65
Total Fuzzy Similarity,
0.22
41.72
0.18
41.93
0.22
41.73
no inflow forecast
Total Fuzzy Similarity,
perfect inflow forecast
Sugeno,
no inflow forecast
24
Average
SWE
Membership
functions
Observed SWE
Fuzzification
Fuzzy
inference
Defuzzification
Observed I
Observed W
Average I
Reference W
Membership
functions
Fuzzy
inference
Observed W
Fuzzification,
Fuzzy inference,
Defuzzification
Defuzzification
Extra release
Inferred release
Release
Figure 1.
Membership
functions
Fuzzification
Fuzzification
25
Figure 2.
26
Membership
1
very
low
objective
low
very
high
high
0
Wref-0.57m
Wref-0.3m
Wref+0.3m
Wref
Wref+0.57m
(a)
Membership
1
very
small
small
large
very
large
0
-160
-40
40
180
m3/s
(b)
Membership
1
exceptional very
small
small
small
quite
small
quite
large
large
very
large
exceptional
large
0
105
(c)
Figure 3.
130
160
190
265
290
340
380
3
m3/s
m
27
550
Observed
500
Fuzzy
450
3
Release [m /s]
400
350
300
250
200
150
100
86
87
88
95
94
93
92
91
90
89
Year
Figure 4.
79.3
WREF
Observed
Fuzzy
Water level [m]
78.8
78.3
77.8
77.3
86
87
88
89
90
91
Year
Figure 5.
92
93
94
95
28
400
Total Fuzzy Similarity
Sugeno
Release [m3/s]
350
300
250
200
150
100
01.01.90 01.03.90
Figure 6.
01.05.90
01.07.90
Date
01.09.90
01.11.90