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[1]FUZZY MODEL FOR REAL-TIME RESERVOIR OPERATION
Tanja Dubrovin,[2] Ari Jolma,[3] and Esko Turunen[4]
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.
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 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 rule-base 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 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 well-defined 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 setX is an ordered couple (A,X), in which the reference set A is a non-void set and the membership function X: A[0,1] shows the degree to which an element aA 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 IF-THEN 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
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.
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[m1A1(x)+m2B1(y)+m3C1(z)]
Similarity(x, Rule_2) = 1/M[m1A2(x)+m2B2(y)+m3C2(z)]
.
.
Similarity(x, Rule_k) = 1/M[m1Ak(x)+m2Bk(y)+m3Ck(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 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.
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 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:
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 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
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.