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Fuzzy Model for Real-Time Reservoir OperationTanja Dubrovin1; Ari Jolma2; and Esko Turunen3

Abstract: A fuzzy rule-based control model for multipurpose real-time reservoir operation is constructed. A new, mathemjustified methodology for fuzzy inference—total fuzzy similarity—is used and compared with the more traditional Sugeno-style mSpecifically, the seasonal variation in both hydrological variables and operational targets is examined by considering theseason-dependent relative values, instead of using absolute values. The inference drawn in several stages allows a simplemodel structure. The control model is illustrated using Lake Pa¨ijanne, a regulated lake in Finland. The model is calibrated to simulateactual operation, but also to better fulfill the new multipurpose operational objectives determined by experts. Relatively similaobtained with the inference methods and the strong mathematical background of total fuzzy similarity put fuzzy reasoning onfoundation.

DOI: 10.1061/~ASCE!0733-9496~2002!128:1~66!

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Introduction

Real-time reservoir operation is a continuous decision-makprocess of determining the water level of a reservoir and relefrom it. The operation is always based on operating policy arules defined and decided upon in strategic planning. The cplexity of the real-time release decision, considering all the timdependent information, warrants the importance of real-timeeration. The operator’s task in real-time reservoir operation isfulfill the objectives as well as possible while complying wilegal and other constraints. Reservoir operation involves untainty and inaccuracies. Uncertainty is involved in objectivesthe sense that the values and targets are usually subjectivethe relative emphases on different objectives change with tiEvaluating all objectives in commensurate values is a compand often impossible task. Determination of the total net inflinto the reservoir and forecasting it is both inaccurate and untain. The seasonal variation in both hydrological variables aoperational objectives brings uncertainty into the operation sithe seasons do not begin and end on the same date every yemany cases, fuzzy logic may provide the most appropriate modological tool for modeling reservoir operation.

First introduced by Zadeh~1965!, fuzzy logic and fuzzy settheory have been used, for example, in modeling the ambigand uncertainty in decision making. The basic idea in fuzzy lois simple: statements are not just ‘‘true’’ or ‘‘false,’’ but partia

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1Laboratory of Water Resources Management, Helsinki Univ. of Technology, Finland. Current affiliation: Finnish Environment Institute, P.O.Box 140, 00251 Helsinki, Finland. E-mail: [email protected]

2Laboratory of Water Resources Management, Helsinki Univ. of Technology, P.O. Box 5200, 02015 HUT, Finland. E-mail: [email protected]

3Tampere Univ. of Technology, P.O. Box 692, 33101 Tampere, Finland. E-mail: [email protected]

Note. Discussion open until June 1, 2002. Separate discussions mube submitted for individual papers. To extend the closing date by onmonth, a written request must be filed with the ASCE Managing EditorThe manuscript for this paper was submitted for review and possiblpublication on November 10, 2000; approved on March 26, 2001. Thipaper is part of theJournal of Water Resources Planning and Manage-ment, Vol. 128, No. 1, January 1, 2002. ©ASCE, ISSN 0733-9496/2002/1-66–73/$8.001$.50 per page.

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truth is also accepted. Similarly, in fuzzy set theory, partial blonging to a set, called a fuzzy set, is possible. Fuzzy setscharacterized by membership functions. The demonstrated beof fuzzy logic in control theory is in modeling human expeknowledge, rather than modeling the process itself. Despiteindisputable successes, fuzzy logic suffers from a lack of smathematical foundations. For example, in fuzzy IF-THEN infeence systems there are a multitude of techniques in the literafor how to draw conclusions from partially true premises,though no logical justification for such rules is given. Seveapproaches have been used to apply fuzzy set theory to reseoperation. These include fuzzy optimization techniques, furule base systems, and combinations of the fuzzy approachother techniques. Applications can be found in the work of Hua~1996!, Saad et al.~1996!, and Fontane et al.~1997!. Fuzzy rulebase control systems for reservoir operation are presented bysell and Campbell~1996! and Shrestha et al.~1996!. The fuzzyrule base can be constructed on the basis of expert knowledgobserved data. Methods for deriving a rule base from obsetions are presented by Ba´rdossy and Duckstein~1995! and Kosko~1992!.

Russell and Campbell~1996! mentioned that as the number oinputs increases, a fuzzy rule-based system, specifically the nber of rules, quickly becomes too large, unidentifiable, andmanageable. A similar problem—that of combining evidence—solved in belief networks using Bayesian updating, which edence~premises! is incorporated one piece at a time, assuminconditional independence of different pieces of evidence~Russelland Norvig 1995!.

The present paper describes a real-time fuzzy control mofor multipurpose reservoir operation. The position taken herethat, first, the number of rules does not become a problemexpert knowledge is carefully studied and modeled; it is anvantage in all fuzzy inference systems that the rule base may qwell be incomplete. Second, the modeling of expert knowledand the development of the fuzzy control model in generalfacilitated using a multistage model. Attention is specifically pato circumstances in which the hydrological conditions and walevel targets change significantly within a year. In the calibratithe actual operation is used as a reference, but it is also attemto better meet the demands of the reevaluated objectives. Ta

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levels for two interdependent variables—release and walevel—are considered. The effectiveness of a many-valued inence system based on the well-defined Lukasiewicz-Pavelka lusing many-valued similarity and expert knowledge~and onlythem!! is studied in the case of reservoir operation. The applmethod—total fuzzy similarity—was introduced by Turune~1999!. Here the performance of total fuzzy similarity is compared with a more traditional fuzzy inference method knownSugeno-style fuzzy inference.

Total Fuzzy Similarity

Recall that a fuzzy setX is an ordered couple (A,mx), in whichthe reference setA is a nonvoid set and the membership functiomx : A→@0,1# shows the degree to which an elementaPA be-longs to fuzzy setX.

The objective in what is called approximate reasoning in tfuzzy logic framework is to draw conclusions from partially trupremises. In a typical fuzzy inference machine, a control situatcomprehends a systemS, an input universe of discourse ‘‘IN’’~theIF-parts!, and an output universe of discourse ‘‘OUT’’~theTHEN-parts!. We assume there aren input variables and one out-put variable. The dynamics ofS are characterized by a finite collection of IF-THEN rules; for example:

Rule 1: IFx is A1 andy is B1 andz is C1 , THEN w is D1

Rule 2: IFx is A2 andy is B2 andz is C2 , THEN w is D2

Rule k: IF x is Ak andy is Bk andz is Ck , THEN w is Dk

whereA1 ,...,Dk are fuzzy sets. However, the outputsD1 ,...,Dk

can also be crisp actions. All these fuzzy sets are to be specby the fuzzy control engineer. We avoid disjunction betweenrules by allowing some of the output fuzzy setsDi and Dk , iÞ j , to possibly be equal. Thus, a fixed THEN part can follovarious IF parts. Some of the input fuzzy sets may also be eq~for example,Bi5Bj for some values ofiÞ j !. However, the rulebase should be consistent; a fixed IF part precedes a unTHEN part. Moreover, the rule base can be incomplete; ifexpert is not able to define the THEN part of some combinationthe form 8IF x is Ai andy is Bi andz8 is Ci8 , then this rule cansimply be skipped.

Given an input~for instance,x5(x,y,z)!, there is diversity inthe literature about how to count the corresponding outputw. Thisprocedure is called defuzzification. In Sugeno-style fuzzy infence systems, for example, all the output fuzzy sets are fipartially, and weighted sums or weighted average are calculato calculate the outputw.

Defuzzification, however, meets with resistance among maematicians since it does not usually have any deeper mathemajustification. To establish fuzzy inference on solid mathematifoundations, a method called total fuzzy similarity was introducby Turunen~1999!. The idea in the total fuzzy similarity approacis to look for the most similar premise, the IF part, and fire tcorresponding conclusion, the THEN part. Moreover, the degof similarity may be composed of various partial similarities.

The following algorithm recounts how to construct a totfuzzy similarity-based inference system.

Step 1: Create the dynamics ofS, that is, define the IF-THENrules, give the shapes of the input fuzzy sets~for example,A1 ,...,Ck! and the shapes of the output fuzzy sets~for example,D1 ,...,Dk!.

Step 2: Give weights to various parts of the input fuzzy s

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~for example,m1 , m2 , m3 to Ai•s, Bi•s, andCi•s! to emphasizethe mutual importance of the corresponding input variables.

Step 3: Put the IF-THEN rules in a linear order with respecttheir mutual importance, or give some criteria on how this candone when necessary.

Step 4: For each THEN parti, give criteria for distinguishingoutputs with equal degrees of membership.

A general framework for the inference system is now reaAssume then that we have input value, for example,x5(x,y,z). The corresponding output valuew is found in the fol-lowing way.

Step 5: Compare the input valuex separately with each IF partin other words, count total fuzzy similarities between the actinputs and each IF part of the rule base. This simply means coing the weighted means, for example

Similarity(x,Rule 1)51/M @m1•mA1(x)1m2•mB1(y)1m3•mC1(z)#Similarity(x,Rule 2)51/M @m1•mA2(x)1m2•mB2(y)1m3

•mC2(z)#Similarity(x,Rulek)51/M @m1•mAk(x)1m2•mBk(y)1m3

•mCk(z)#wherem1 , m2 , andm3 are the weights given in Step 2 andM5m11m21m3 .

Step 6: Fire an output valuew such that mDi(w)5Similarity(x,Rulei ) corresponding to the maximal total fuzzsimilarity; if such a Rulei is not unique, use the mutual ordegiven in Step 3, and if there are several such output valuesw, usethe criteria given in Step 4.

Of course, the algorithm can be specified by putting in exdemands. For example, in some cases the degree of total fsimilarity of the best alternative should be greater than some fivalue aP@0,1# before any action is taken. Sometimes all talternatives possessing the highest fuzzy similarity should bedicated, or the difference between the best candidate and theond best should be larger than a fixed valuebP@0,1#, and so on.All this is dependent on expert choice. It is worth noting thatthe steps in our algorithm are based only on well-defined maematical concepts or on an expert’s knowledge.

Model Construction

The model consists of two real-time submodels. The model stture is shown in Fig. 1. The first submodel sets up a referewater level ~WREF! for each time step. Given this referenclevel, the observed water level~W!, and the observed inflow~I!,the second submodel makes the decision on how much shoureleased from the reservoir during the next time step.

The seasonal variation is accounted for in the fuzzificatphase. Instead of using absolute observed values for inpseason-dependent relative values are used. Hence, in fuzzificthe membership of the difference between the observed valuethe season-dependent reference value is determined, insteadmembership of the observed value alone. For output, the smembership functions and absolute values are used througthe year.

Reference Water Level Model

The output of the first submodel is the WREF value for each tistep. The WREF value relies on the water level targets, and

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Fig. 1. Structure of fuzzy control model for reservoir operation

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an input into the second submodel, the release model. Thepose of the reference water level model is to take snow deobservations and seasonal variation of targets into consideraNonetheless, the aim is not to make the actual water level exafollow the WREF value every year. The actual water level canabove or below the WREF value, depending on the hydrologconditions.

The year is divided into three seasons: the snow accumulaseason, the snowmelt season, and the rest of the year. Inseason, determination of the WREF value is performed difently. For the ‘‘rest of the year’’ season, the WREF valuesindividual for each time step but do not change from year to yFor the snowmelt season, WREF value is dependent on the swater equivalent~SWE! and can be inferred for each time stewith the fuzzy rules:

IF SWE is smaller than average/average/larger than avermuch larger than average,

THEN WREF is high/middle/low/very low.

The premise in this rule is the difference between the obseSWE and the average SWE for that time step. Average~or me-dian! values can be calculated from historical data, or they candetermined by an expert. During the snow accumulation seathe WREF value is reduced. The WREF value for the end ofseason, that is, the beginning of the next snowmelt seasoinferred for each time step. Since the premise in the rules abo

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relative, the inference can be made again with the same rulesmembership functions using the observed and average SWE vues for each time step. The submodel output is reduced lineatoward the level inferred for each time step until the beginningthe next snowmelt period.

Release Model

Premises in the inference system for the release are

• Relative water level (Wrelative) at the beginning of that timestep: The relative value is determined by the difference btween observed water level and WREF value, that is, the oput of the first submodel for that time step.

• Relative inflow (I relative): The relative value is determined bythe difference between observed inflow and average inflow fa given month.General rule formulation is as follows:

IF Wrelative is very low/low/objective/high/very high,AND I relative is very small/small/large/very large,THEN release is exceptionally small/very small/small/quitesmall/quite large/large/very large/exceptionally large.During calibration it was found that in flood situations the rul

base could not keep the water level sufficiently below the criticflood level. Releases during floods could not be increased beca

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they would have led to too large releases in less critical situatioHence, an extra release can be added to the system output wthe water level is critical. Extra release is determined again withfuzzy rule:

IF W is critical, THEN add extra release.

The membership of absolute water level belonging to a ‘‘criticaset is determined by a simple membership function. Similarly, omembership function is used in defuzzification.

Case Study: Lake Pa¨ ijanne

Lake Paijanne is a 120 km long and 20 km wide regulated lakecentral Finland~Fig. 2!. The active storage volume is approximately 3000 Mm3 and annual inflow 7028 Mm3. The release fromLake Paijanne runs through several smaller lakes and the RivKymijoki into the Gulf of Finland. There are 12 hydropoweplants along the River Kymijoki with a total hydropower generation potential of 200 MW. Other interests include agricultureshoreline real estate, recreation, forestry, fisheries, navigation,ecology. The low shores of Lake Païjanne, with their buildingsand cultivated land, are exposed to flood damages. In additionconflicts between different interest groups there are conflicts b

Fig. 2. Paijanne-Kymijoki reservoir-river system. Small solid rect-angles denote dams, solid triangles denote water power plants,arrows denote tributaries.



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tween users of Lake Païjanne and the River Kymijoki. If, forexample, fluctuations in water level are controlled only from tlake users’ point of view, flow in the River Kymijoki may beinadequate. Changes in inflow into Lake Païjanne are fairly slowbecause its catchment area is rather large and contains anumber of lakes.

There are legally binding constraints for the operation of LaPaijanne. The regulation permit for Lake Païjanne, which cameinto effect in 1954, defines flood protection as the key objectivethe regulation. The regulation permit defines sets of constrafor the operation of Lake Païjanne; these include minimum anmaximum water levels, maximum change in release, and tawater levels. Target water levels are determined for the beginnof each three-month period by inflow forecast and the target wlevel for the previous period. Thus the target water level mchange as the accuracy of the forecast increases, and thetarget level is not known until the end of the three-month peri

New objectives for the regulation of Lake Païjanne have beendeveloped in a recent study~Marttunen and Ja¨rvinen 1999!. Thereport defines target water levels, or rather upper and lower limfor target levels and objective releases for various seasonshydrological states. Some of the rules are already defined asTHEN rules. These guidelines were used as expert knowledgdevelopment of the fuzzy control model.

In short, policy regarding water level presented in the afomentioned study is as follows:

January to April—The water level should be lowered by thbeginning of the snowmelt season to avoid flooding. The tarwater level is dependent on inflow predictions.

May—After the snowmelt season, the water level shouldraised for natural production of pike. To control overgrowthreeds, the water level should be raised to 78.55 m.

June to August—An adequate summer water level for ecologcal and recreational objectives is about 78.30–78.60 m. The 7m level should be reached by early July. A slight reduction athe peak is advantageous for the ecology of the lake andshores. If there is a risk of flooding or drought, the reductishould be ignored.

September to December—If there is no risk of flood, the waterlevel should be raised to enable larger releases during the cowinter season in order to maximize hydropower benefit.

Water levels higher than 78.75 m in Lake Païjanne cause dam-age to buildings, agriculture, industry, and forests. Correspoingly, the critical flow rate in the River Kymijoki is 480 m3/s. Theoptimum flow for hydropower generation in the River Kymijokis 380 m3/s ~Marttunen and Ja¨rvinen 1999!.

Model Application

To apply the model to real-time operation of Lake Païjanne, therelease objectives were fitted into the model structure descrabove. The time step used in the simulation was five days. Dwater levels in Lake Païjanne, release and flow in the RiveKymijoki, and the bimonthly SWE measurements in the catment area of the lake during the snow accumulation period wavailable. Net inflow time series were calculated using the wabalance equation, the SWE data were interpolated for eachstep, and flows for each time step were averaged from daily d

Data from the years 1976–1985 were used for calibrationthe years 1986–1995 for validation. Reference values, thamonthly averages for inflow and SWE for each period, were cculated from the years 1970–1985.

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For the reference water level model the snow accumulatseason was assumed to last from December to February. Itfound necessary to set the snowmelt season to begin in the msomewhat earlier than it usually does in reality. The flood causby snowmelt varies in size and duration from year to year.satisfy ecological and recreational objectives, it is desirableraise the water level of the lake as early as possible if there isrisk of flooding. March was always considered to be a part of tsnowmelt season. If the SWE was larger than average in Apthe WREF value was kept low, otherwise its linear elevation tward higher summer water levels was started. From early Jonward, the same WREF curve was used for each year, accorto the water level targets in the operational guidelines.

The formulation of the rule base was made on the basis of bhistorical data and expert knowledge. The method used incalibration phase was total fuzzy similarity. The membershfunctions for premises and the rule base were set up first. Trigular membership functions were used for all inputs and outpRelative snow depth in water equivalent corresponded to the ffuzzy sets, relative water level to the five fuzzy sets, and relatinflow to the four fuzzy sets. For each rule in the release modthe calibration data were analyzed to find cases in whichpremises were most similar to that rule. Corresponding responin observation data were used as a basis when forming memship functions for output, which were then adjusted furthertrial and error. Membership functions for relative water leverelative inflow, and release are presented in Fig. 3.

The model was calibrated using local and global optimizatioIn local optimization the observed water level was used as inin each time step. Model response was compared with obserrelease. In global optimization the model was allowed to oper‘‘independently,’’ that is, the water level was simulated for eactime step given the model output from the previous time step. Trelease and simulated water level were compared with theserved values and, in addition, with the new objectives. The ojective of local optimization was to find a good match with thobserved data and the model output, and the objective of glooptimization was to better meet the new demands set foroperation of Lake Pa¨ijanne.

Some further adjustments were made in the model. If the oserved water level was less than or equal to 78.60 m, the degof membership in a fuzzy set ‘‘critical water level’’ was 0, anwhen the water level was greater than or equal to 78.95 m,degree of membership was 1. Membership values between thwater levels were interpolated linearly. Extra release was betw0 and 130 m3/s.

Calculated net inflow include much noise because a smerror in water level reading may cause a considerable errorstorage volume value used in the water balance equation. Calition showed that a strong variation in the input led to undesiravariation in output in the model. Therefore, relative inflow wafiltered by averaging the values of the six previous periods. Tcurrent release permit states that too rapid a change in releasnot allowed. To accommodate this, the total output was costrained so that the release was not allowed to change more50 m3/s during two time steps~ten days!.

Water level could be kept close to the targets if ‘‘narrowmembership functions for relative water level were used orextreme releases were allowed. Yet the problem was too-strfluctuations in the release. One challenge in developing an optional model for a multipurpose reservoir is that both the watlevel and release are relevant, while there is interdependencetween them. Actual water level is dependent on previous de

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sions, that is, releases, whereas the observed water level isbasis for the next decision. Since the aim of reservoir operationto fulfill the objectives of both release and water level, a compromise between them was sought.

The following is an example of the fuzzifying and defuzzify-ing process used in the release model.

Assume the output from the reference water level modeWREF, is 77.9 m, and the water level calculated with the resufrom the previous time step is 78.09 m. Thus, the difference btween the calculated water level and WREF is 0.19 m. From Fig3~a! it can be seen that the water level belongs to a fuzzy s‘‘objective’’ with a membership 0.367 and to the fuzzy set ‘‘high’’with a membership 0.633. Assume that inflow in previous timsteps has been 10 m3/s larger than average inflow in the currentmonth. Fig. 3~b! shows that the membership in the fuzzy se‘‘small’’ is 0.375 and the membership in the fuzzy set ‘‘large’’ is0.625. The memberships in other sets are equal to zero.

The degrees of similarity for each rule in the rule base arcalculated by counting an average of the membership of the walevel ~W! in the rule-specific fuzzy set and the membership of thinflow ~I! in the rule-specific fuzzy set. Thus, the degrees of simlarity for some of the rules are as follows:

Fig. 3. Membership functions for~a! water level;~b! inflow; and~c!release~output!

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If Wrelative is objective andI relative is very small then release is small.0.18

If Wrelative is objective andI relative is small then release is quite small.0.37

If Wrelative is objective andI relative is large then release is quite large.0.50

If Wrelative is objective andI relative is very large then release is large. 0.18

If Wrelative is high andI relative is very small then release is quite small.0.32

If Wrelative is high andI relative is small then release is quite large. 0.50

If Wrelative is high andI relative is large then release is large. 0.63

If Wrelative is high andI relative is very large then release is very large. 0.32

Fig. 4. Release with fuzzy logic and actual operation

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The rules not presented in this list have smaller similaritieThe rule to be fired is the seventh one, which has the largdegree of similarity. The defuzzified output will be release, whicbelongs to a set ‘‘large’’ at a membership 0.63. From Fig. 3~c! weget two alternative values: 281 and 309 m3/s. Next, consider therule that achieved the second largest similarity, the sixth rulethe list. From two alternative values the one closer to the outof the sixth rule, ‘‘quite large,’’ will be chosen. Thus, the defuzzfied output is 281 m3/s. If the output ‘‘very large’’ would alsohave had the second largest similarity, the value ‘‘closer to tmiddle,’’ 281 m3/s again, would have been chosen. This criteriofor choosing one of the two alternative values was found tofunctional in this model, but other criteria can be devised aused, depending on an expert’s choice.

Results

The model was tested using data from the years 1985–1996the test phase global optimization was used. Fig. 4 shows



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releases decided by the model when Total Fuzzy Similarityused. Correspondingly, the simulated water level is presenteFig. 5. Figures also show the observed time series and WRThe Sugeno method was chosen for comparison againstfuzzy similarity, due to its simplicity and popularity. With botmethods the system was kept the same as much as possibapply the Sugeno method the peak values of the triangular oumembership functions were used as crisp values. Combinatiothe premises was implemented by taking a minimum, andfuzzification was performed using a weighted average.

The performances of the two methods were almost indisguishable. With total fuzzy similarity the water level targets ding the summer were sometimes better fulfilled, but the reletended to fluctuate more, and the limitation on change in relewas more relevant. The comparison between releases with dent methods in 1990 is presented in Fig. 6.

For the calculations it was assumed that no inflow forecasavailable. In reality inflow forecasts are available for the ope


anage. 2002.128:66-73.

72 / J

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tions managers. To assess the effect of inflow forecasts, simutions were run with perfect hindsight; that is, average inflow vaues for the next six periods were used as input instead of inflvalues from previous periods. With perfect forecast the releawas more stable, but no other major enhancement was seen inmodel performance.

To estimate the model performance, the revenue from hydpower generation was calculated for a test set. For this purpthe flow through the hydropower plants was calculated by addiobserved incremental flows to the model output. Energy priccannot be exactly determined in advance, but in general theyhigher in winter than in summer. Prices of 38.18 euro/MW/h fowinter and 23.30 euro/MW/h for summer were used. Monetaflood damages on the shores of Lake Pa¨ijanne, other lakes, andthe River Kymijoki were estimated. For this purpose cost tabl

Fig. 6. Release with total fuzzy similarity and Sugeno in year 199

OURNAL OF WATER RESOURCES PLANNING AND MANAGEMEN


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for flood damages for buildings, industry, and agriculture weavailable. Revenues and losses are presented in Table 1.

Discussion and Conclusions

A fuzzy control model for real-time reservoir operation was dveloped, and a new method for fuzzy inference, total fuzzy simlarity, was used and compared with a more traditional methoThe model performance was generally good, but the modelnot capture expert thinking in the most exceptional circumstancThis was evident in the calibration phase, where the model conot manage the end of an unusually long flood as well asexpert. However, this weakness could be solved by constructinparticular rule base that should go off only under special circustances. Another weakness of the model was its inability to ‘‘se

0

Table 1. Comparison of Calculated Monetary Flood Loss and Hdropower Revenue from Actual Reservoir Operation and Fuzzy Ctrol Model with Alternative Methods

Method

Years 1986–1995

Monetary flood loss~million euros/year!

Hydropower revenue~million euros/year!

Actual operation 0.19 41.65Total fuzzy similarity~no inflow forecast!

0.22 41.72

Total fuzzy similarity~perfect inflow forecast!

0.18 41.93

Sugeno~no inflow forecast!

0.22 41.73

T / JANUARY/FEBRUARY 2002

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forward and backward at several time steps. This inability ledexcess variation in release and flooding along the shores ofRiver Kymijoki.

Building up a fuzzy rule base for reservoir operation involvestressing the seasonal variation inherent in both hydrological vaables and operational targets. Under Finnish conditions, in whisnowmelt plays an important role in a hydrological year, gooresults in lake operation cannot be reached with the same ruthroughout the year. Due to differing predominant uses of thlakes and reservoirs in different seasons, various kinds of rumust be effective in the model for different seasons. Possibways of handling seasonal variation are to use the time of the yas model input or construct a separate rule base for each seaThese require calibration for each rule base or are subject to‘‘curse of dimensionality.’’

In the present study, seasonal variation was considered infuzzification phase and in the season-dependent WREF valuThe seasonal change was smooth because there were no rochanges from one rule base to another. With the help of a mustage model structure the number of rules could be kept areasonable level and the functioning of the model easy to coprehend. Moreover, this approach implements the release guilines determined by experts and the release permit. Target walevels play an important role in decision making.

The relatively similar results obtained with the total fuzzysimilarity and Sugeno methods indicate that the underlying cocept in fuzzy control is many-valued similarity. The strong mathematical background of the total fuzzy similarity method putfuzzy reasoning on a more solid foundation.

The advantages of fuzzy logic are that calculation is straighforward and the model easy for the operator to understand dueits structure, which is based on human thinking. The system calso be easily modified when necessary. In the fuzzy contrmodel described, changes in operation can be implementedfine-tuning the reference water level submodel. Various charactistics of different reservoirs can be encoded in rules, membersfunctions, and WREF values. Application of the model requireexpert knowledge and sufficient amounts of historical data.



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Acknowledgments

Support for this work was partly provided by the Finnish Ministof Agriculture and Forestry~Grant No. 4855/421/98!. The writersacknowledge Professor Pertti Vakkilainen for fruitful discussioand comments on the manuscript; Erkki A. Ja¨rvinen and MikaMarttunen for generously sharing their expertise on the regulaof Lake Paijanne; and the two anonymous reviewers for helpsuggestions.

References

Bardossy, A., and Duckstein, L.~1995!. Fuzzy rule-based modeling witapplications to geophysical, biological and engineering syste,CRC, Boca Raton, Fla.

Fontane, D. G., Gates, T. K., and Moncada, E.~1997!. ‘‘Planning reser-voir operations with imprecise objectives.’’J. Water Resour. PlanManage.,123~3!, 154–162.

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Kosko, B. ~1992!. Neural networks and fuzzy systems. Prentice-Hall,Englewood Cliffs, N.J.

Marttunen, M., and Ja¨rvinen, E. A.~1999!. Development of regulation oLake Paijanne—Synthesis and recommendations, Finnish Environ-ment Institute, Helsinki~in Finnish!.

Russel, S. J., and Norvig, P.~1995!. Artificial intelligence: A modernapproach, Prentice-Hall, Englewood Cliffs, N.J.

Russell, S. O., and Campbell, P. F.~1996!. ‘‘Reservoir operating ruleswith fuzzy programming.’’J. Water Resour. Plan. Manage.,122~3!,165–170.

Saad, M., Bigras, P., Turgeon, A., and Duquette, R.~1996!. ‘‘Fuzzy learn-ing decomposition for the scheduling of hydroelectric power stems.’’ Water Resour. Res.,32~1!, 179–186.

Shrestha, B. P., Duckstein, L., and Stakhiv, E. Z.~1996!. ‘‘Fuzzy rule-based modeling of reservoir operation.’’J. Water Resour. Plan. Manage.,122~4!, 262–269.

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