data-driven fault diagnosis method for power transformers … · 2018. 12. 12. · researcharticle...

Research ArticleData-Driven Fault Diagnosis Method for Power TransformersUsing Modified Kriging Model

Yu Ding and Qiang Liu

School of Information and Control Engineering Liaoning Shihua University Fushun Liaoning 113001 China

Correspondence should be addressed to Qiang Liu neuliuqiang163com

Received 25 May 2017 Accepted 20 September 2017 Published 22 October 2017

Academic Editor Wanquan Liu

Copyright copy 2017 Yu Ding and Qiang Liu This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A data-driven fault diagnosis method that combines Kriging model and neural network is presented and is further used for powertransformers based on analysis of dissolved gases in oil In order to improve modeling accuracy of Kriging model a modifiedmodel that replaces the global model of Kriging model with BP neural network is presented and is further extended using linearityweighted aggregation method The presented method integrates characteristics of the global approximation of the neural networktechnology and the localized departure of theKrigingmodel which improvesmodeling accuracy Finally the validity of thismethodis demonstrated by several numerical computations of transformer fault diagnosis problems

1 Introduction

Transformer is one of the most important equipment inpower system [1] which is mainly used to transfer electricalenergy between two or more circuits through electromag-netic induction In the course of using this equipment somefactors such as electrical thermal and mechanical stressesmay lead to irreversible damage to the insulating material[2] In order to improve the reliability of power supply faultdiagnosis for transformer has drawn much attention fromresearchers and many fault diagnosis methods have beenwidely proposed during the past decades

At present Dissolved Gas-in-oil Analysis (DGA) is acommonly used method to identify incipient failures oftransformer fault [3] With the development of artificialintelligence and computer technology many fault diagnosisalgorithmshave beenproposed based onDGA such as neuralnetwork [4 5] fuzzy logic [6 7] expert system [8] supportvector machine [2 9] and rough set theory [10] Given thatexisting methods have their own characteristics and somelimitations effective fault diagnosis methods that integrateadvantages of existing technologies to improve the modelingaccuracy still remain an open area of research

As a classic modeling technology Kriging model com-bines a global model plus localized departures to construct

approximation from sample data It has been widely usedin the field of Computer-Aided Engineering (CAE) [1112] On the other hand neural network technology is awell-known information processing paradigm and has beenwidely applied in various areas due to its advantages such asadaptive learning In this paper a data-driven fault diagnosismodel based on Kriging model and BP neural network isconstructed and then is used for transformer fault diagnosisproblems based on DGA In order to improve the modelingaccuracy of Kriging model BP neural network is used tomodify the global model of Kriging model where the modi-fied formula is given in detail The modified Kriging modelcombines global and adaptive learning abilities of neuralnetwork technology and retains the localized departures ofthe Kriging model in the meantime It integrates the advan-tages of both methods which thus effectively improve themodeling accuracy Finally some examples of transformerfault diagnosis show that the proposed method is effectiveand feasible

2 Overall Design of Modified Kriging ModelA Hybrid Model

TheKrigingmodel is an unbiased estimationmodel based onthe minimum variance estimation of sample points and their

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 3068548 5 pageshttpsdoiorg10115520173068548

2 Mathematical Problems in Engineering

Begin

Output end

Sample and testdata

Globalmodel

Localizeddepartures

BP neuralnetwork

MPSO

Modifiedglobal model

Modified Kriging model(hybrid model)

Kriging model

Figure 1 Overall design of hybrid model

response values The output can be viewed as a combinationof a regression model and a stochastic process [13 14] Theregression model is equivalent to the global simulation of thesample space and the stochastic process is equivalent to localdeviation

In order to further improve the accuracy of the Krigingmodel this paper proposes a transformer fault diagnosismethod that combines Kriging model and neural networktechnology The structure of this hybrid model is shown inFigure 1 The main steps for constructing the hybrid modelare as follows

(1) Determine characteristic variables and fault typesbased onDGAmethod and collect sample data and test data

(2) According to sample data construct BP networkmodel

(3) Construct the Kriging model based on sample data(4) The global simulation of Kriging model is modified

and updated by neural network and then the hybrid modelis constructed

3 Modeling Method of Hybrid Model

31 Kriging Model and Parameter Optimization The Krigingmodel contains global simulation plus localized departuresand the basic principles of which can be briefly given asfollows [13 14] set approximate function as 119910(119909) and thefunction between the response value and the independentvariable of the system can be formulated as follows

119910 (119909) = 119865 (120573 119909) + 119885 (119909) (1)

where 119910(119909) is the unknown function of interest 119865(120573 119909) is theregression model that is equivalent to the global simulationand 120573 is regression parameter 119885(119909) is a normal stochastic

process in which the mean value is 0 and the variance isdenoted as 1205902 It reflects the randomness of the response andis equivalent to partial divergence

The covariance matrix of 119885(119909) is formulated as follows

cov [119885 (119909119894) 119885 (119909119895)] = 1205752R [119877 (120579 119909119894 119909119895)] (2)

where R is correlation matrix the order of matrix is 119872 119894 119895 =1 2 119872 119872 is the number of sample points 119909119894 and 119909119895 arethe 119894th and the 119895th sample points119877(120579 119909119894 119909119895) is the correlationfunction between 119909119894 and 119909119895 In this paper we utilize theGaussian correlation function

119877 (120579 119909119894 119909119895) = exp(minus 119873sum119896=1

12057911989610038161003816100381610038161003816119909(119896)119894 minus 119909(119896)119895 100381610038161003816100381610038162) (3)

where119873 is the dimension of the problem 119909(119896)119894 and 119909(119896)119895 are the119896th dimensional components of the 119894th and the 119895th samplepoints respectively 120579119896 is the unknown related parameters ofthe interpolation model

In general 120579119896 can be replaced with a scalar 120579 Thusformula (3) can be formulated as

119877 (120579 119909119894 119909119895) = exp(minus120579 119873sum119896=1

10038161003816100381610038161003816119909(119896)119894 minus 119909(119896)119895 100381610038161003816100381610038162) (4)

Therefore the estimated value of test point 119909 can be givenby the following equation

119910 (119909) = 120573 + r119879 (119909)Rminus1 (y minus f120573) (5)

where 120573 is the estimate of the global simulation y is sampledata response f is119872 column vectors r119879(119909) is the correlationvector between observation point 119909 and sample data whichcan be formulated as follows

r119879 (119909) = (119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119872))119879 (6)

When 119891(119909) is a constant 120573 can be simplified andestimated by the following equation

120573 = (f119879Rminus1f)minus1 (f119879Rminus1y) (7)

The parameter 120579 determines the accuracy of the Krigingmodel which can be solved by the following optimizationproblem

min 119871 (120579) = |R|1119872 1205752 (8)

where 1205752 is variance estimation which can be determined bythe following equation

1205752 = 1119872 (y minus f120573)119879Rminus1 (y minus f120573) (9)

To optimize parameter 120579 intelligent optimization algo-rithms are commonly used In this paper a modified particleswarm optimization (MPSO) [15] is used to optimize theparameters of Kriging model The key points of applying

Mathematical Problems in Engineering 3

MSPO to perform optimization are threefold (1) make 120579as encoding in real numbers (2) take (8) as the objectivefunction and (3) with respect to the constraint conditionof 120579 gt 0 the commonly used penalty function method isapplied

The inertia weight and learning factors in MPSO [15] areupdated as follows

120596 = (120596119904 minus 120596119890) ( 119905119879) + (120596119890 minus 120596119904) (2119905119879 ) + 120596119904 (10)

1198881 = (1198881119904 minus 1198881119890) ( 119905119879) + (1198881119890 minus 1198881119904) (2119905119879 ) + 1198881119904 (11)

1198882 = (1198882119904 minus 1198882119890) ( 119905119879) + (1198882119890 minus 1198882119904) (2119905119879 ) + 1198882119904 (12)

where 120596119904 and 120596119890 are the initial value and the final value ofinertia weight 120596 respectively 1198881119904 and 1198881119890 and 1198882119904 and 1198882119890 arethe initial value and the final value of learning factors 1198881 and1198882 respectively 119879 is the maximum number of iterations 119905 isthe current iteration number

32 Combinations of Neural Network and Kriging ModelThe mapping relationship between the characteristic vari-ables and fault types of transformer is very complex whichincreases difficulty in improving high accuracy of Krigingmodel On the other hand neural network technology forexample BP network is a well-known information process-ing paradigmwith some advantages such as adaptive learningand strong adaptabilityThe basic principle of BP network canbe found inmany references such as [4 5 16] details of whichare not introduced here In general the output of BP networkcan be formulated as follows

119900119894 = 119891( 119899sum119894=0

120596119894119895119909119895 + 119902119894) (13)

where 119900119894 are the outputs of BP network119891 is transfer function120596119894119895 are network weights 119902119894 are network thresholds and 119909119895 arethe outputs of the upper layer node

To improvemodeling accuracy amodifiedKrigingmodel(hybrid model) is constructed by combining Kriging modeland BP neural network technology and the overall designof which has been shown in Figure 1 More specifically theglobal model of Kriging model is replaced with BP neuralnetwork which is given by (14)

119910 (119909) = 119900119894 + r119879 (119909)Rminus1 (y minus f120573) (14)

Further this modified method can be extended usinglinearity weighted aggregation method which is formulatedby (15)

120582 = 1205741119900119894 + 1205742120573 (15)

where 120582 is the modified global model and 1205741 and 1205742 areweighting coefficients

Thus the final output of the hybrid model can be given asfollows

119910 (119909) = 120582 + r119879 (119909)Rminus1 (y minus f120573) (16)

Table 1 Data distribution of each fault type

Fault type Coding mode Sample data Test dataNormal 1198841 15 15High temperatureoverheating 1198842 25 25

Medium temperatureoverheating 1198843 15 15

Low temperatureoverheating 1198844 9 9

Partial discharge 1198845 9 8Low energy discharge 1198846 20 16High energy discharge 1198847 20 23

Table 2 Comparisons of diagnostic results

Test method Average calculation time Accuracy rateBP network 506 s 7207SVM 621 s 7567Kriging model 249 s 8108Hybrid method 784 s 9189

Obviously the hybridmodel is of generalityWhen 1205741 = 0and 1205742 = 1 the hybridmodel becomes original Krigingmodeland (16) can be rewritten by (5) When 1205741 = 1 and 1205742 = 0the hybrid model can be formulated by (14) where the globalmodel of Kriging model is replaced with BP neural network

4 Application of Transformer Fault Diagnosis

41 Feature Variable and Fault Type In general the con-centrations of five gases (H2 CH4 C2H6 C2H4 and C2H2)dissolved in transformer oil can be selected as characteristicvariables based on DGA data samples The correspondingfault types of the characteristic variables contain normal hightemperature overheating medium temperature overheatinglow temperature overheating partial discharge low energydischarge and high energy discharge In this paper we selectsome DGA data published in [17 18] The distributions andcoding of these fault data are shown in Table 1

42 Parameter Setting In this paper BP network structureis set as three-layer network The selected sample data have 5characteristic variables thus the number of nodes in the inputlayer of the network is set as 5 the number of nodes in theoutput layer is set as 1 the number of hidden layer nodes isset as 8 by trial and error The initial weights and thresholdsof the network are randomly initialized and Log-sigmoid isselected as function transfer function

Parameters of MPSO are set as follows 120596119904 = 095 120596119890 =055 1198881119904 = 1198882119890 = 225 1198881119890 = 1198882119904 = 075 119879 = 40 andpopulation size 119875 = 20 Figure 2 shows the MPSO conver-gence curve of objective function

43 Analysis of Examples (1) Table 2 shows the comparisonsbetween the proposed method and other methods As faras these test examples are concerned the proposed method


Table 3 Some test results (120583LL)Serial number H2 CH4 C2H6 C2H4 C2H2

Fault typeActual fault Fault analysis

(1) 93 58 43 37 0 Middle temperature overheating Middle temperature overheating(2) 139 52 68 63 96 Middle and low temperature overheating Middle temperature overheating(3) 196 3207 2792 5747 0 High temperature overheating High temperature overheating(4) 457 79 54 190 24 High temperature overheating High temperature overheating(5) 279 41 181 42 318 High energy discharge High energy discharge(6) 1467 368 1054 271 02 Normal Normal(7) 345 11225 275 515 5875 Low energy discharge Low energy discharge(8) 2175 40 49 518 675 High energy discharge High energy discharge(9) 181 262 210 528 0 Middle and low temperature overheating Low temperature overheating(10) 443 173 36 233 104 High energy discharge High energy discharge(11) 1729 3341 1729 8125 377 Partial discharge Partial discharge(12) 1226 886 1395 1821 0 Normal Low energy discharge(13) 56 78 18 173 0 Low temperature overheating Low temperature overheating

Convergence curve of objective function

00258

00258

00258

00258

00258

00258

00258

00258

Solu

tion

of o

bjec

tive f

unct

ion

5 10 15 20 25 30 35 400Iteration number

Figure 2 Convergence curve of MPSO

can effectively improve the accuracy of transformer faultdiagnosis

(2) Tables 2 and 1 also show that the calculation timeof presented method is averagely about 784 s for test dataincluding 111 sample points (hardware configuration CPU i5RAM 4G programming software Matlab) which demon-strates the efficiency of the presented method

(3) Table 3 lists some test results using the presentedmethod (due to limitation of paper length the completeresults are not listed here) The results show that diagnosisaccuracy is basically satisfactory

5 Conclusions

In this paper a data-driven fault diagnosis model based onKrigingmodel and neural network is proposedTheproposedmodel is based on the Kriging model and integrates neuralnetwork technology Meanwhile the localized departures ofKriging model are retained

The presented hybrid model is further used for powertransformer fault diagnosis problems based onDGAmethodSome numerical computations of transformer fault diagnosisproblems are conducted and the results show the feasibilityand efficiency of the proposed method In addition thepresented modified Kriging model is of some potentialapplication value in other areas such as power system andengineering machinery

Conflicts of Interest

The authors declare that the mentioned received funding didnot lead to any conflicts of interest regarding the publicationof this manuscript and there are not any possible conflicts ofinterest in the manuscript

Acknowledgments

This study is supported by Program for Liaoning ExcellentTalents in University (Grant no LJQ2014037) and NaturalScience Foundation of Liaoning Province of China (Grant no20170540589)

References

[1] J Pleite C Gonzalez J Vazquez and A Lazaro ldquoPowerTransfomer Core Fault Diagnosis Using Frequency ResponseAnalysisrdquo in Proceedings of the MELECON 2006 - 2006IEEE Mediterranean Electrotechnical Conference pp 1126ndash1129Benalmadena Spain

[2] S-W Fei and X-B Zhang ldquoFault diagnosis of power trans-former based on support vector machine with genetic algo-rithmrdquo Expert Systems with Applications vol 36 no 8 pp11352ndash11357 2009

[3] S S M Ghoneim and I B M Taha ldquoA new approach ofDGA interpretation technique for transformer fault diagnosisrdquoInternational Journal of Electrical Power amp Energy Systems vol81 pp 265ndash274 2016

[4] Y J Sun S Zhang C X Miao et al ldquoImproved BP neuralnetwork for transformer fault diagnosisrdquo Journal of China


University of Mining amp Technology vol 17 no 01 pp 138ndash1422007

[5] S Seifeddine B Khmais and C Abdelkader ldquoPower trans-former fault diagnosis based on dissolved gas analysis by artifi-cial neural networkrdquo in Proceedings of the 2012 1st InternationalConference on Renewable Energies and Vehicular TechnologyREVET 2012 pp 230ndash236 March 2012

[6] S M Islam T Wu and G Ledwich ldquoA novel fuzzy logicapproach to transformer fault diagnosisrdquo IEEE Transactions onDielectrics and Electrical Insulation vol 7 no 2 pp 177ndash1862000

[7] Y-C Huang and H-C Sun ldquoDissolved gas analysis of mineraloil for power transformer fault diagnosis using fuzzy logicrdquoIEEE Transactions on Dielectrics and Electrical Insulation vol20 no 3 pp 974ndash981 2013

[8] C E Lin J M Ling and C L Huang ldquoAn expert system fortransformer fault diagnosis using dissolved gas analysisrdquo IEEETransactions on Power Delivery vol 8 no 1 pp 231ndash238 1993

[9] Y-C Xiao X-H Chen and H-J Zhu ldquoApplication of geneticsupport vector machine in power transformer fault diagnosisrdquoShanghai Jiaotong Daxue XuebaoJournal of Shanghai JiaotongUniversity vol 41 no 11 pp 1878ndash1886 2007

[10] A-H Zhou H Song H Xiao and X-H Zeng ldquoPowertransformer fault diagnosis based on integrated of rough settheory and neural networkrdquo in Proceedings of the 2nd Interna-tional Conference on Intelligent Systems Design and EngineeringApplications ISDEA 2012 pp 1463ndash1465 January 2012

[11] S Jeong M Murayama and K Yamamoto ldquoEfficient optimiza-tion design method using kriging modelrdquo Journal of Aircraftvol 42 no 2 pp 413ndash420 2005

[12] T W Simpson T M Mauery J J Korte and F MistreeldquoKriging models for global approximation in simulation-basedmultidisciplinary design optimizationrdquo AIAA Journal vol 39no 12 pp 2233ndash2241 2001

[13] S Dey T Mukhopadhyay and S Adhikari ldquoStochastic freevibration analyses of composite shallow doubly curved shells- A Kriging model approachrdquo Composites Part B Engineeringvol 70 no 5 pp 99ndash112 2015

[14] Z Chen S Peng X Li et al ldquoAn important boundary samplingmethod for reliability-based design optimization using krigingmodelrdquo Structural and Multidisciplinary Optimization vol 52no 1 pp 55ndash70 2015

[15] Q Liu and CWang ldquoPipe-assembly approach for aero-enginesby modified particle swarm optimizationrdquo Assembly Automa-tion vol 30 no 4 pp 365ndash377 2010

[16] H M Guo and A Yang ldquoDiagnosis of transformer fault basedon GA-BP neural networkrdquo Coal MineMachinery vol 36 no 7pp 318ndash320 2015

[17] Y B Tang Data-Driven Fault Diagnosis and Prediction forLarge-Scale Power Transformer Central South UniversityChangsha China 2013

[18] W Gao Study on Fault Diagnosis for Power Transformer Basedon Support Vector Machine of Artificial Immune AlgorithmTaiyuan University of Technology Taiyuan China 2012

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Begin

Output end

Sample and testdata

Globalmodel

Localizeddepartures

BP neuralnetwork

MPSO

Modifiedglobal model

Modified Kriging model(hybrid model)

Kriging model

Figure 1 Overall design of hybrid model

response values The output can be viewed as a combinationof a regression model and a stochastic process [13 14] Theregression model is equivalent to the global simulation of thesample space and the stochastic process is equivalent to localdeviation

In order to further improve the accuracy of the Krigingmodel this paper proposes a transformer fault diagnosismethod that combines Kriging model and neural networktechnology The structure of this hybrid model is shown inFigure 1 The main steps for constructing the hybrid modelare as follows

(1) Determine characteristic variables and fault typesbased onDGAmethod and collect sample data and test data

(2) According to sample data construct BP networkmodel

(3) Construct the Kriging model based on sample data(4) The global simulation of Kriging model is modified

and updated by neural network and then the hybrid modelis constructed

3 Modeling Method of Hybrid Model

31 Kriging Model and Parameter Optimization The Krigingmodel contains global simulation plus localized departuresand the basic principles of which can be briefly given asfollows [13 14] set approximate function as 119910(119909) and thefunction between the response value and the independentvariable of the system can be formulated as follows

119910 (119909) = 119865 (120573 119909) + 119885 (119909) (1)

where 119910(119909) is the unknown function of interest 119865(120573 119909) is theregression model that is equivalent to the global simulationand 120573 is regression parameter 119885(119909) is a normal stochastic

process in which the mean value is 0 and the variance isdenoted as 1205902 It reflects the randomness of the response andis equivalent to partial divergence

The covariance matrix of 119885(119909) is formulated as follows

cov [119885 (119909119894) 119885 (119909119895)] = 1205752R [119877 (120579 119909119894 119909119895)] (2)

where R is correlation matrix the order of matrix is 119872 119894 119895 =1 2 119872 119872 is the number of sample points 119909119894 and 119909119895 arethe 119894th and the 119895th sample points119877(120579 119909119894 119909119895) is the correlationfunction between 119909119894 and 119909119895 In this paper we utilize theGaussian correlation function

119877 (120579 119909119894 119909119895) = exp(minus 119873sum119896=1

12057911989610038161003816100381610038161003816119909(119896)119894 minus 119909(119896)119895 100381610038161003816100381610038162) (3)

where119873 is the dimension of the problem 119909(119896)119894 and 119909(119896)119895 are the119896th dimensional components of the 119894th and the 119895th samplepoints respectively 120579119896 is the unknown related parameters ofthe interpolation model

In general 120579119896 can be replaced with a scalar 120579 Thusformula (3) can be formulated as

119877 (120579 119909119894 119909119895) = exp(minus120579 119873sum119896=1

10038161003816100381610038161003816119909(119896)119894 minus 119909(119896)119895 100381610038161003816100381610038162) (4)

Therefore the estimated value of test point 119909 can be givenby the following equation

119910 (119909) = 120573 + r119879 (119909)Rminus1 (y minus f120573) (5)

where 120573 is the estimate of the global simulation y is sampledata response f is119872 column vectors r119879(119909) is the correlationvector between observation point 119909 and sample data whichcan be formulated as follows

r119879 (119909) = (119877 (119909 1199091) 119877 (119909 1199092) 119877 (119909 119909119872))119879 (6)

When 119891(119909) is a constant 120573 can be simplified andestimated by the following equation

120573 = (f119879Rminus1f)minus1 (f119879Rminus1y) (7)

The parameter 120579 determines the accuracy of the Krigingmodel which can be solved by the following optimizationproblem

min 119871 (120579) = |R|1119872 1205752 (8)

where 1205752 is variance estimation which can be determined bythe following equation

1205752 = 1119872 (y minus f120573)119879Rminus1 (y minus f120573) (9)

To optimize parameter 120579 intelligent optimization algo-rithms are commonly used In this paper a modified particleswarm optimization (MPSO) [15] is used to optimize theparameters of Kriging model The key points of applying




120596 = (120596119904 minus 120596119890) ( 119905119879) + (120596119890 minus 120596119904) (2119905119879 ) + 120596119904 (10)

1198881 = (1198881119904 minus 1198881119890) ( 119905119879) + (1198881119890 minus 1198881119904) (2119905119879 ) + 1198881119904 (11)

1198882 = (1198882119904 minus 1198882119890) ( 119905119879) + (1198882119890 minus 1198882119904) (2119905119879 ) + 1198882119904 (12)



119900119894 = 119891( 119899sum119894=0

120596119894119895119909119895 + 119902119894) (13)



119910 (119909) = 119900119894 + r119879 (119909)Rminus1 (y minus f120573) (14)


120582 = 1205741119900119894 + 1205742120573 (15)



119910 (119909) = 120582 + r119879 (119909)Rminus1 (y minus f120573) (16)



















00258

00258

00258

00258

00258

00258

00258

00258

Solu

tion

of o

bjec

tive f

unct

ion






5 Conclusions





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







120596 = (120596119904 minus 120596119890) ( 119905119879) + (120596119890 minus 120596119904) (2119905119879 ) + 120596119904 (10)

1198881 = (1198881119904 minus 1198881119890) ( 119905119879) + (1198881119890 minus 1198881119904) (2119905119879 ) + 1198881119904 (11)

1198882 = (1198882119904 minus 1198882119890) ( 119905119879) + (1198882119890 minus 1198882119904) (2119905119879 ) + 1198882119904 (12)



119900119894 = 119891( 119899sum119894=0

120596119894119895119909119895 + 119902119894) (13)



119910 (119909) = 119900119894 + r119879 (119909)Rminus1 (y minus f120573) (14)


120582 = 1205741119900119894 + 1205742120573 (15)



119910 (119909) = 120582 + r119879 (119909)Rminus1 (y minus f120573) (16)



















00258

00258

00258

00258

00258

00258

00258

00258

Solu

tion

of o

bjec

tive f

unct

ion






5 Conclusions





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









00258

00258

00258

00258

00258

00258

00258

00258

Solu

tion

of o

bjec

tive f

unct

ion






5 Conclusions





Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




data-driven fault diagnosis method for power transformers … · 2018. 12. 12. · researcharticle...

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