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The Online Journal on Power and Energy Engineering (OJPEE) Vol. (1) – No. (2) Fault Diagnosis of Power Transformer Based on Fuzzy Logic, Rough Set theory and Inclusion Degree Theory Hossam A. Nabwey 3, E. A. Rady 1, A.M. Kozae 2, A. N. Ebady 3 (1) I.S.S.R, Cairo University, Cairo, Egypt. (2) Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt. (3) Department of Engineering Basic science, Faculty of Engineering, Menofia University, Egypt. Abstract- Power transformers are one of the most information, and these methods have different shortage. For expensive components of electrical power plants and the example, Petri network puts domain knowledge into a series failures of such transformers can result in serious power of producing rules; this can solve fault diagnosis problems. system issues, so fault diagnosis for power transformer is But when new fault or new information is coming, it will lead very important to insure the whole power system run to matching collision and combination blast because of the normally. Due to information transmission mistakes as slow speed of Petri network which resulted from vast rules. well as arisen errors while processing data in surveying and monitoring state information of transformer, In this paper, a new method aims to use fuzzy logic, rough uncertain and incomplete information may be produced. set theory and inclusion degree theory to infer useful rules for Moreover, real time is another important characteristic so power transformer fault diagnosis is presented. By using as to meet high speed diagnosis requirements. Based on fuzzy logic technique, the continuous attribute values are these points, this paper presents an intelligent fault transformed into the fuzzy values by automatically deriving diagnosis method of power transformer based on fuzzy membership functions from a set of data with similarity logic Rough set theory and inclusion degree theory. By clustering, then rough sets is applied to implement attributes using a fuzzy logic technique, the continuous attribute reduction and a simplified decision table is got, finally, values are transformed into the fuzzy values by inclusion degree theory is used for extracting rules. The automatically deriving membership functions from a set application to fault diagnosis of transformer shows the of data with similarity clustering, then rough sets is proposed algorithm can find more objective and effective applied to implement attributes reduction and a simplified diagnostic rules from the quantitative data and has yielded decision table is got, finally, inclusion degree theory is promising results. used for inducing logical rules from quantitative data. The practical results show that the method is an effective II. REVIEW OF ROUGH SET THEORY method for fault diagnosis of transformer and has yielded promising results. The Rough Set Theory is a new mathematical tool Keywords- Transformer, Fuzzy logic, inclusion degree presented to dispose incomplete and uncertainty problem by theory, rule induction, fault diagnosis, Rough Set theory Pawlak [3] in 1982. He defined the knowledge according to new point of view, and regarded it as partition of universe. I. INTRODUCTION The concept of a rough set can be defined quite generally by means of topological operation, interior and closure, called Power transformers are one of the most expensive approximations. Rough set theory can discover implicit components of electrical power plants and are vital to make knowledge and open out potentially useful rule by efficiently the whole power system run normally. The failures of such analyzing and dealing with all kinds of imprecise, incomplete transformers can result in serious power system issues, and disaccord information. particularly those come without warning, cause service disruptions and severe economic losses [4]. A diagnostic A. Decision system technique is needed to assess any degradation of the Definition 1): In rough set theory, an information system insulation materials on power transformer. Many artificial can be considered as system S =(U, A, V, f) , where U is the intelligence technologies such as neural network [11], universe; A C D is the sets of fault attribute, the subset Wavelet Analysis [2], gray clustering [7], decision tree [6], C and D are disjoint sets of fault symptoms attribute and fault Petri network [8], information fusion [5] have been applied to decision attributes respectively; V Va ,where Va is the transformer diagnosis and produced some results. But rR transformers are complex system with uncertainty factors and value set of fault symptoms attribute a ,is named the domain Reference Number: W09-0011 45 The Online Journal on Power and Energy Engineering (OJPEE) Vol. (1) – No. (2) of attribute a .Each attribute a A ; f is an information If D Bx /A x p 1 and i k the rule “if Ai then B j ” can f x,a Va function f : UxA V , and , in which x U . be obtained on condition that B ji can be found to meet the 1 B. Equivalent relations Definition 2): In decision system S = (U, A, V, f), every requirement of D B ji /A i D B j /Ai j 1 . attributes subset, an indiscernible relation (or equivalence relation) IND(B) defined in the following way: x,y Ux U a B, f x, a f y, a IV. FUZZIFICATION OF ATTRIBUTES IND(B) = (1) The family of all equivalence relation of IND(B), a partition The fuzzy logic theory was first proposed by Zade in determined by B, denoted by U/IND(B),[x]B can be 1965[12]. It is primarily concerned with quantifying and considered as equivalence classes, and defined as follows: reasoning using natural language in which words can have x B = y U a B, f x, a f y, a (2) ambiguous meanings. This can be thought of as an extension of traditional crisp sets in which each element must either be And in or not in a set. Fuzzy set concepts are often used to x IND ( B ) = x B (3) represent quantitative data expressed in linguistic terms and membership functions in intelligent systems because of its simplicity and similarity to human reasoning. They have been C. Reduction and Core applied to many fields such as manufacturing, engineering, Definition 3): In decision system S=(U, A, V ,f), Let diagnosis, and economics. In this paper, a fuzzy discretization b B and B A , if posB(D) = posB −{b}(D), attribute b is method is proposed. A learning method is given for redundant to B, which relatives to D, otherwise the attribute b automatically deriving membership functions from a set of is indispensable. data with similarity clustering [1]. If IND(B) = IND(A) and POSB (D) ≠ POSB −{b} (D) , then B For a m-dimensional attribute, the jth attribute value can be is called a reduction for information system S , are denoted as describes as (x1j, x2j…xnj), where n is the number of objects. RED( A) ; the intersection of these reduction sets is called The fuzzification method of continuous attributes proceeds as core, denoted as CORE = RED( A) . follows: Step1: Sort the attribute values in an ascending order. The modified order after sorting is then x1j, x2j…xnj. III. THE INCLUSION DEGREE THEORY Step2: Find the difference between adjacent data. The The inclusion degree theory was proposed by Zhang difference between adjacent data provides the information Wenxiu, a professor of Xi’an Jiaotong University, in 1995. It about the similarity between them. for each pair xij and x(i+1)j has been applied to several fields in recent years [9, 10]. ,(i = 1,2,…, n-1), the differences is diffij = (x(i+1)j- xij ). Suppose X is an object set, Ai X (i k ) is the partition Step3: Find the value of similarity between adjacent data. k In order to obtain the value of similarity between adjacent of X. That is, Ai I A j = (i j ) , and U Ai X . Ax is a i 1 data, we convert each distance diffij to a real number sij between 0 and 1 according to the following formula: partition of X, Ax Ai : i k and D is the total partition of diffij X. x and x are two partitions of X, Ax Ai : i k and A B 1 diffij C . ij sij C . ij (8) Bx Bi : j 1 . Bx depends on Ax Ax Bx if Ai B j . 0 diffij f C . ij D B j /Ai and D Bx /A x are the inclusion degree of X Where sij represents the similarity between xij and x(i+1)j, and D respectively. The definition of inclusion is: ij is the standard derivation of diff’s, and C is a control k 1 D Bx /A x D B j /Ai i 1 j 1 (4) parameter deciding the shape of the membership functions of similarity. A large C causes a grater similarity. A x Bx , D Bx /A x 1 (5) Step4: Cluster the data according to similarity. Here we use A x x Bx = Ai I B j : i k , j 1 (6) the cut of similarity to cluster the data. If sij then divide the two adjacent data into the same groups; else put them into Ax x Bx Ax , Ax x Bx B x (7) different groups. After the above operation, the data will be Reference Number: W09-0011 46 The Online Journal on Power and Energy Engineering (OJPEE) Vol. (1) – No. (2) clustered into the l j , where l j means the jth produced fuzzy from quantitative data for the power transformer. According to the historical fault data of the power transformer, the fault region. decision table is shown in Table1. Here, the condition attributes are concentrations (ppm by volume) of dissolved Step5: Determine membership functions. For simplicity, gases in the insulation oil, such as H2, CH4, C2H6, C2H4, and triangle membership functions are used here for each C2H2. The decision attribute (D) is the fault class of the linguistic variable. A triangle membership function can be transformer, where “0” represents the fault of local discharge, defined by a triad (b,a,c). For the hth fuzzy region, the “1” represents the fault of low-energy discharge, “2” parameters {bh, ah, ch} can be defined as: represents the fault of high-energy discharge, “3” represents the fault of low-temperature superheat, “4” represents the g 2 sij s(i 1) j fault of medium-temperature superheat and “5” represents the xij .sij x(i 1) j . x gj .s( g 1) j fault of high-temperature superheat. bh i 1 2 (9) g 2 sij s(i 1) j Table 1. A decision table for the transformer fault sij s( g 1) j i 1 2 U H2 CH4 C2H6 C2H4 C2H2 D bh xij X1 ah bh 68 8 3 8 11 2 (10) 1 h xij X2 49.7 18.5 4.4 60 12 1 X3 33.9 36.7 31.5 39.2 0 5 x gj bh X4 47 75.6 47.1 190.6 0 3 ch bh (11) X5 18.7 7.5 1.2 1.7 0 0 1 h x gj X6 24 27.9 24.3 30 0 5 Where X7 80 95.4 33.1 150 0 3 h xij h x gj min sij , s i 1 j , ...., s g -1 j , for 1th X8 X9 34 42 60.5 62 24.3 5 69.3 63 0 73 4 1 fuzzy region, h x b1 1 ; for the l j th fuzzy X10 55 57 37 90 0 3 region, h x bij 1 . The membership functions for each condition attribute are given according to previous algorithm. Take the attribute H 2 and C2H4 as an example, the membership functions are shown Step6: Find the membership value. The attribute value in Fig.1 and Fig.2. vij (i=1, 2, …., n; j=1, 2, …., m) can be described as : l 1 ij 2 ij ijj vij ....... (12) F1 j 2 Fj l Fj j h h Where F j is the hth fuzzy region of is c j , ij is the h F membership value of xi U in fuzzy region j . Fig.1 The membership functions of H2 V. POWER TRANSFORMER FAULT DIAGNOSIS In the normal operation of the transformer, the released gases are methane (CH4), ethane (C2H6), Hydrogen (H2), ethylene (C2H4), and acetylene (C2H2) and so on. When there is an abnormal situation such as occurring a fault, some specific gases are produced more than in the normal operation and the amount of them in the transformer oil increase. The increase in the amount of gases results in saturation of the transformer oil and no more gas can be dissolved in oil. Fig.2 The membership functions of C2H In this section, an example is given to show how the So the quantitative values of each object are transformed proposed method can be used to generate diagnostic rules into fuzzy sets. Take the attribute H2 in x3 as an example. The Reference Number: W09-0011 47 The Online Journal on Power and Energy Engineering (OJPEE) Vol. (1) – No. (2) value “33.9” is converted into a fuzzy set (0.28/L+0.42/M) Table 3. Simplified fuzzy decision table using the given membership functions. Results for all the objects are shown in Table2. U CH4 C2H6 D X1 1/L 0.99/L 2 Table 2. Fuzzy decision table X2 0.55/L 0.91/L 1 X3 0.55/L 0.07/M 1 U H2 CH4 C2H6 C2H4 C2H2 D X4 0.07/M 0.91/L 1 X1 0.81/H 1/L 0.99/L 0.99/L 1/M 2 X5 0.07/M 0.07/M 1 X2 0.71/M 0.55/L 0.91/L 0.98/M 1/M 1 X6 0.68/M 0.28/M 5 X3 0.71/M 0.55/L 0.07/M 0.98/M 1/M 1 X8 0.68/M 0.35/H 5 X4 0.71/M 0.07/M 0.91/L 0.98/M 1/M 1 X14 1/H 1/H 3 X5 0.71/M 0.07/M 0.07/M 0.98/M 1/M 1 X15 1/L 1/L 0 X6 0.28/L 0.68/M 0.28/M 0.09/L 1/L 5 X16 0.04/L 0.93/M 5 X7 0.28/L 0.68/M 0.28/M 0.57/M 1/L 5 X18 0.41/M 0.93/M 5 X8 0.28/L 0.68/M 0.35/H 0.09/L 1/L 5 X20 1/H 0.07/M 3 X9 0.28/L 0.68/M 0.35/H 0.57/M 1/L 5 X21 1/H 0.51/H 3 X10 0.42/M 0.68/M 0.28/M 0.09/L 1/L 5 X22 0.22/M 0.93/M 4 X11 0.42/M 0.68/M 0.28/M 0.57/M 1/L 5 X23 0.03/H 0.93/M 4 X12 0.42/M 0.68/M 0.35/H 0.09/L 1/L 5 X26 0.13/M 0.88/L 1 X13 0.42/M 0.68/M 0.35/H 0.57/M 1/L 5 X27 0.13/M 0.1/M 1 X14 0.87/M 1/H 1/H 1/H 1/L 3 X28 0.15/H 0.88/L 1 X15 0.87/L 1/L 1/L 1/L 1/L 0 X29 0.15/H 0.1/M 1 X16 0.66/L 0.04/L 0.93/M 0.36/L 1/L 5 X30 0.42/M 0.9/H 3 X17 0.66/L 0.04/L 0.93/M 0.48/M 1/L 5 X18 0.66/L 0.41/M 0.93/M 0.36/L 1/L 5 By application of the inclusion degree theory to the X19 0.66/L 0.41/M 0.93/M 0.48/M 1/L 5 Simplified fuzzy decision table shown in table 2, X20 1/H 1/H 0.07/M 1/H 1/L 3 The results of partitions of U divided by CH4 are as follows: X21 1/H 1/H 0.51/H 1/H 1/L 3 X22 0.27/L 0.22/M 0.93/M 0.82/M 1/L 4 X23 0.27/L 0.03/H 0.93/M 0.82/M 1/L 4 * {1, 9},{2, 3},{4, 5},{6, 7},{11},{14},{15} CH 4 , X24 0.42/M 0.22/M 0.93/M 0.82/M 1/L 4 {8, 12, 13},{10},{16, 17},{18, 19},{20} X25 0.42/M 0.03/H 0.93/M 0.82/M 1/L 4 X26 0.86/M 0.13/M 0.88/L 0.95/M 1/H 1 The results of partitions of U divided by C2H6 are as follows: X27 0.86/M 0.13/M 0.1/M 0.95/M 1/H 1 {1},{2, 4},{3, 5, 12},{6},{7},{8},{9},{13} X28 0.86/M 0.15/H 0.88/L 0.95/M 1/H 1 * X29 0.86/M 0.15/H 0.1/M 0.95/M 1/H 1 C2 H 6 X30 0.42/M 0.42/M 0.9/H 0.39/M 1/L 3 {10, 11, 14, 15},{17, 19},{16, 18},{20} X31 0.05/H 0.42/M 0.9/H 0.39/M 1/L 3 The results of partitions of U divided by D are as follows: In order to simplify the fuzzy decision table it is required to find the insignificant conditional attributes in the diagnoses, {1},{2, 3, 4, 5, 16, 17, 18, 19},{9}, i.e. it is required to find a minimum number of conditional D* attributes, but, are still able to diagnose the problem. This is {8, 12, 13, 20},{14, 15},{6, 7, 10, 11} done by number of conditional attributes should be removed each time and the decision table should be checked to make Then, all kinds of partitions of U divided by different sure any contradiction has not occurred. In this paper attributes are as follows: computing the reduct is done by using software called * * * * ROSETTA (a Rough Set tool kit for analysis of data). The U CH 4 , C2 H 6 , CH 4 xC2 H 6 GA is adopted in Reduction process. The simplified fuzzy * * * * decision table is shown in table 3. Since CH 4 f D , C2 H 6 f D then It is impossible to form decision rules only according to one of CH4 or C2H6. Reference Number: W09-0011 48 The Online Journal on Power and Energy Engineering (OJPEE) Vol. (1) – No. (2) But CH 4*xC2H6* D* shows that it is possible to REFERENCES divide D according to CH 4 and C2 H 6 . 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