research article a fault diagnosis method of power systems...

Research ArticleA Fault Diagnosis Method of Power Systems Based onGray System Theory

Huang Darong Tang Jianping and Zhao Ling

College of Information Science and Engineering Chongqing Jiaotong University Chongqing 400074 China

Correspondence should be addressed to Huang Darong hcx1978163com

Received 22 August 2014 Revised 9 November 2014 Accepted 13 November 2014

Academic Editor XiaoSheng Si

Copyright copy 2015 Huang Darong et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

To provide some decision-making suggestions for fault diagnosis in power systems a new model for identifying fault componentis constructed by using Gray theory Firstly the basic concepts of Gray theory are introduced and explained in detail And thenthe recognition algorithm of the power supply interrupted districts and the assignment principle of fault state vectors are depictedaccording to the working principle of protective relays (PRs) and circuit breakers (CBs) Secondly based on the concept of the Graycorrelation degree the fault information explanation degree model is constructed and the judging method of malfunction andrejection for PRs and CBs is established Meanwhile to achieve the goal of the fault diagnosis the fault diagnosis procedure thatdetermined which components malfunction is designed for power systems Finally some simple experiments have already verifiedthat the proposed method and model are effective and reasonable and the trend of further research is analyzed and summarized

1 Introduction

Fault diagnosis of power systems is a method which usesthe information collected from protective relays and circuitbreaker to recognize fault component and malfunctioned ortripped PRs and CBs It is generally known that the faultcomponent recognition is the crucial problem in engineeringapplication [1] In recent year many fault diagnosis methodsof power systems are proposed including expert system [2]artificial neural network [3] optimization technology [4]rough set theory [5 6] Petri net [7] and Bayesian network[6] In order to meet certain preconditions in the existingdiagnosismethods some assumptions aremade for them Butin some cases these assumptions may be contrary to realityand may even cause error diagnosis results For examplewhen the information of protective device is incompletethe fault diagnosisrsquo conclusion may be incorrect So thecrucial difficulty of fault diagnosis is how to guarantee thecorrective and effective diagnosis results for power systemswith incomplete information [8] In other words if we wantto get the corrective and effective diagnosis results we have

to know how to obtain the unknown information by utilizingthe known information under the condition of incompleteinformation [9ndash11]

Fortunately the Gray system theory can simulate anddetermine the unknown information according to the knowninformation of systems In addition using the Gray systemstheory to construct the fault diagnosis model involves asmaller sample and little informationThe process is easier tobe operated However the anomalies contrary to qualitativeanalysis will not be produced in the process of Gray corre-lation analysis Hence under the condition of informationincomplete and not explicit the Gray system theory has itsunique superiority that is it is an effective method in case ofincomplete and uncertain information

Based on the above application of the Gray theory to thefault diagnosis of power systems is proposed in this paperThe layout of the rest of the paper is arranged as followsin Section 2 the Gray system theory and Gray correlationanalysis were introduced briefly Section 3 described the faultdiagnosismethod andmodel of power systems based onGraysystem theory and designed the diagnostic procedure for fault

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 971257 11 pageshttpdxdoiorg1011552015971257

2 Mathematical Problems in Engineering

components In Section 4 we have discussed the simulativeresults by experiments Finally Section 5 concluded thispaper with inferences and directions for future work

2 Basis Introduction of Gray System Theoryand Gray Correlation Analysis

21 Basal Principle of Gray System Theory Gray systemtheory which is established by Chinese scholar ProfessorDeng Ju-Long in 1982 has been used to research uncertainproblem with lack of data and information In generalspeaking some information is known and other pieces ofinformation are unknown in uncertain data systems Themain idea of Gray system theory is to describe correctlythe systemsrsquo evolution law and to monitor effectively theirrunning behavior by extracting valuable information fromthe known information Over forty years later the structuresystem of Gray system theory has basically taken shapeThe theoretical basis includes Gray Matrix Gray Algebraand Gray Equation And the Gray model (GM) may alsoimplement the analysis evaluation prediction and controldecision of uncertain data systems by using spatial asso-ciation rule and sequence generation method For futureunderstanding of the advantages of Gray system theory sixbasic principles are introduced as follows

211 Differential Information Principle There is the differ-ence among differential information In other words thedifference is the information For example if there are twodifferential objects or systems everyone has unique informa-tion that is not similar to another piece of informationOne ofthemost basic pieces of informationwhich the human societyperceives is that the world comes from the difference betweenmatter and matter

212 Nonuniqueness of Solutions The solution is nonuniqueunder the condition of information being incomplete andvague The uncertainty of systems leads to the existence ofuncertain information and then causes nonuniqueness ofsolutions

213 Smallest Information Principle The basic idea of Graysystem theory is to utilize the smallest information achievingfrom known knowledge data to accomplish a given task Thesmallest information getting by researching the uncertainproblem with the small samples and poor information is thefundamental basis of making a distinction between Gray areaand no-Gray

214 Cognitive Foundation Principle

The Foundation of Cognition Is Information The accurate andcomplete cognition is determined according to definite andprecise knowledge the uncertain and incomplete knowledgemay also lead to vague cognition Correspondingly if thereis no information of systems the cognition of systems is notalso completed Thus the cognition should be studied basedon information

215 Innovation Priority Principle The function of newinformation is more important than old information forcognitive behaviors because the new information directlyreflects the current states of the system andmainly influencesthe future trend of the development

216 Gray Indestructibility Principle Notably the incom-plete and uncertain information is very much stronglyentrenched in real systems As new information is continu-ously generated in real engineering the cognition conclusionis improved gradually and the level of cognition will tend torationality and correctness Accordingly the Gray systems donot disappear

Based on the above principles the theoretical modelimplementing a given task may be constructed according tosmall sample and poor information However the runningstates and trend of systems are determined by many factorsin real application So we need to discern the primary factorsand lesser factors And fortunately the Gray correlation anal-ysis which is the important part of Gray system theory canjudge the connection in accordance with the approximationdegree of the two-dimensional curve in time domain andfrequencies domain The higher the similarity is the greaterthe correlation is For further analysis the Gray relationalaxioms are introduced firstly

22 Gray Correlation Axiom Suppose that the behaviorsequence of the system is119883

0= (1199090(1) 1199090(2) 119909

0(119899)) thus

the corresponding factor series is as follows

1198831= (1199091 (1) 1199091 (2) 1199091 (119899))

119883119894= (119909119894(1) 119909

119894(2) 119909

119894(119899))

119883119898

= (119909119898(1) 119909

119898(2) 119909

119898(119899))

(1)

For a given real 119903(1199090(119896) 119909119894(119896)) if 119903(119883

0 119883119894) =

(1119899)sum119899

119896=1119903(1199090(119896) 119909119894(119896)) satisfies

(1) normalization 0 lt 119903(1198830 119883119894) le 1 and 119883

0= 119883119894rArr

119903(1198830 119883119894) = 1

(2) integrity let 119883119894 119883119895isin 119883 = 119883

119904| 119904 = 0 1 119898 119898 ge

2 if 119894 = 119895 then 119903(119883119894 119883119895) = 119903(119883

119895 119883119894)

(3) even symmetry if 119883 = 119883119894 119883119895 119883 = 119883

119894 119883119895 hArr

119903(119883119894 119883119895) = 119903(119883

119895 119883119894)

(4) accessibility the value of the Euclid distance|1199090(119896) minus 119909

119894(119896)| is inversely proportional to the value

of the given real 119903(1199090(119896) 119909119894(119896))

Thus 119903(1198830 119883119894) = (1119899)sum

119899

119896=1119903(1199090(119896) 119909119894(119896)) is called as Gray

relational degree where 119903(1198830 119883119894) represents the correlation

coefficient of each pair of variables 119883119894and 119883

119895 Four condi-

tions (1ndash4) are also regarded as Gray relational four axioms

Mathematical Problems in Engineering 3

Notice that four conditions just do positively meanthese things Normalization illustrates that there is a corre-lation between two arbitrary behavior sequences Integritydescribes that the Gray relational degree is influenced by theexternal environment If the outside environment is changedthe relational degree is also varied So the symmetry principleis not necessarily trueMeanwhile even symmetry representsthat the symmetry principle is true while the set of factorscontains just two factors Accessibility may constrain therelational quantization

23 Gray Correlation Analysis To get the computing formulaof Gray related degree the distance measure between vectors1199090(119896) and 119909

119894(119896) is defined as follows

Δ0119894119896

=10038171003817100381710038171199090 (119896) minus 119909

119894(119896)

1003817100381710038171003817 (2)

And suppose that

Δmax = max119894

max119896

Δ0119894119896

Δmin = min119894

min119896

Δ0119894119896

(3)

According to the formulas (2)-(3) the correlation coef-ficient between vectors 119909

0(119896) and vector 119909

119894(119896) is defined as

follows

119903 (1199090 (119896) 119909119894 (119896)) =

Δmin + 120585ΔmaxΔmin + 120585Δmax

(4)

that is

119903 (1199090 (119896) 119909119894 (119896))

=min119894min119896

10038161003816100381610038161199090 (119896) minus 119909119894(119896)

1003816100381610038161003816 + 120585max119894max119896

10038161003816100381610038161199090 (119896) minus 119909119894(119896)

100381610038161003816100381610038161003816100381610038161199090 (119896) minus 119909

119894 (119896)1003816100381610038161003816 + 120585max

119894max119896

10038161003816100381610038161199090 (119896) minus 119909119894 (119896)

1003816100381610038161003816

(5)

where 120585 is called the resolution coefficients and the values of120585 are usually restricted to a certain range (0 1)

Notice that the discriminatory power varies dependingon the different correlation coefficients the smaller the120585 is the higher the differences between two correlationcoefficients are and the stronger the discriminatory power is

Let 119903(1199090(119896) 119909119894(119896)) = 119903

0119894(119896) and then we define

119903 (1198830 119883119894) =

1

119899

119899

sum

119896=1

119903 (1199090 (119896) 119909119894 (119896)) =

1

119899

119899

sum

119896=1

1199030119894 (119896) (6)

and 119903(1198830 119883119894) is considered as the Gray correlation degree

between reference sequence 1198830and compare sequence 119883

119894

Obviously 119903(1198830 119883119894) satisfy Gray relational four axioms (1ndash4)

24 Computing Algorithm of Gray Correlation Degree By thedefinition ofGray correlation degree the computational stepsof Gray correlation are made as follows

(1) Collect the evaluation data on the evaluation indexsystem then the sequences of data may be stated in matrixform as follows

(1198830 1198831 119883

119898) = (

1199090(1) 119909

1(1) sdot sdot sdot 119909

119898(1)

1199090(2) 119909


119898(2)

sdot sdot sdot

1199090(119899) 119909

1(119899) sdot sdot sdot 119909

119898(119899)

) (7)

where 119899 represents the number of indexes And 119883119894

=

(119909119894(1) 119909119894(2) 119909

119894(119899)) 119894 = 1 2 119898

(2) Apply dimensionless method to the original datasequences let the dimensionless model be

1199091015840

119894(119896) =

119909119894 (119896)

(1119899)sum119899

119896=1119909119894(119896)

119894 = 0 1 119898 119896 = 1 2 119899

(8)

Thus the new data sequences processed by dimensionlessmodel may be rewritten as

(1198830 1198831 119883

119898) = (

(

1199091015840

0(1) 119909

1015840


1015840

119898(1)

1199091015840

0(2) 119909

1015840


1015840

119898(2)

sdot sdot sdot

1199091015840

0(119899) 119909

1015840

1(119899) sdot sdot sdot 119909

1015840

119898(119899)

)

)

(9)

(3) Define the reference sequence 1198831015840

0 The reference

sequence consists of the most optimal value or the worstvalue of every index That is 1198831015840

0= (1199091015840

0(1) 1199091015840

0(2) 119909

1015840

0(119899))

Accordingly the rest of the data is as compare sequence(4) Compute the distance measure between the cor-

responding elements of the reference sequence 1198831015840

0(119896) and

compare sequence1198831015840

119894(119896) that is Δ

0119894119896= 1199091015840

0(119896) minus 119909

1015840

119894(119896)

(5) Calculate Δmax Δmin using the formula (3) that is

Δmax = max119894

max119896

100381710038171003817100381710038171199091015840

0(119896) minus 119909

1015840

119894(119896)

10038171003817100381710038171003817

Δmin = min119894

min119896

100381710038171003817100381710038171199091015840

0(119896) minus 119909

1015840

119894(119896)

10038171003817100381710038171003817

(10)

(6) Compute the correlation coefficient between vector1199091015840

0(119896) and vector 1199091015840

119894(119896) by formulas (4) and (5) that is

119903 (1199091015840

0(119896) 119909

1015840

119894(119896)) =

Δmin + 120585ΔmaxΔ0119894119896

+ 120585Δmax (11)

Notice that 120585 is the resolution coefficients and the values of 120585are usually restricted to a certain range (0 1) To keep thingssimple put 120585 = 05 in this paper

(7) Compute the correlation degree between referencesequence 119883

0and compare sequence 119883

119894by formula (6) and

then obtain the evaluation conclusion by comparing the sizeof correlation degree


3 Model and Algorithm of FaultDiagnosis of Power Systems Basedon Gray System Theory

31 Overall Description of Fault Diagnosis of Power SystemsFirstly when some relevant components of power systemsmalfunction the corresponding protective relays and circuitbreakers will work actively and the fault components will bedisconnected with the power supplier So the power supplyinterrupted districts will be formed And then analyzing andeliminating breakdownwill be alsomade in the power supplyinterrupted districts Secondly in the blackout area theworking principle of protective relays and circuit breakerswillbe analyzed in detail and the information of the protectiverelays and circuit breakers which are in fault state may bededuced And based on what is mentioned above we mayestablish the state vector of the fault modes and quarantinedmodes of the components and give the assignment principleof the value for each member of the vector Finally thecorrelation degree between the reference component andcompare component will be calculated and sorted Thediscriminant criterion is that the component with maximumcorrelation value is as the fault component Moreover thenumbers of fault components increase constantly Whenthe information explanation degree of fault componentsreaches a specified threshold and the numbers of protectiverelays and circuit breakers which fail to operate or refusedoperation do not increase the number of fault componentsis determined According to the order of correlation degreethose components which have same account of maloperationand refused operation are regarded as the fault component Sothe ultimate goal of the fault diagnosis is also accomplished

Obviously some basic concepts should be explained suchas the power supply interrupted districts fault informationexplanation degree and judging rule of maloperation andrefused operation Next these related concepts and modelwill be explained step by step

32 Quick Recognition for the Power Supply InterruptedDistricts The formation mechanism of the power supplyinterrupted districts is established on the difference of thetopological structures of power systems before and after faultoccurrence When the fault happens the protection actionwill be implemented to trip the relevant circuit breakersand the fault components can be isolated from the systemsfor avoiding the expansion of accident On the basis of thereal time information of the circuit breaker the topologiesof systems before and after fault occurrence are recognizedby the real time topology analysis on power systems Andthe fault components can automatically form some passivenetworks according to the difference of the information ofthe topological structures So these passive networks areregarded as the power supply interrupted districts Thusthe recognition of fault components may be limited to theblackout area

The specific steps [12] are shown as follows

(1) Based on topology analysis for the normal powersystems the corresponding equivalent power

or the generator are remarked as the activenodes

(2) The topology analysis is made for the faulty systemagain and several subsystems are achieved at the sametime according to the different information of thetopological structures

(3) The nodes of each subsystem are searched for one byone and every component connected with each nodeis active or is not judged If the component is activethe subsystem is in the normal state and the searchends otherwise if all components in a subsystem aresearched for and no active components are found thesubsystem is regarded as the power supply interrupteddistrict

33 Construction for Fault State Vectors and Assignment Prin-ciple for Vector Elements The state vectors are established bymeans of the protective relays and circuit breakers If there are119899 circuit breakers and 119898 protective relays the form of statevectors can be expressed as

119865 (119896) = (119877 119862) = (1199031 1199032 119903

119898 1198881 1198882 119888

119898)

119896 = 1 2 119899

(12)

where119877 is the state value of protective relays and119862 is the statevalue of circuit breakers The assignment principle of vectorelements is as follows

(1) If the protective relay run in active mode then 119903119894= 1

119894 = 1 2 119898 otherwise 119903119894= 0

(2) Analogously if the circuit breaker is kept in off statethen 119888

119894= 1 119894 = 1 2 119899 otherwise 119888

119894= 0

(3) When the state of the element possesses two modessimultaneously the information of the protectiverelays conflicts with the information of the circuitbreakers That is to say the value of state is both 1 and0 In that case the value of119877 and119862 is given as 05 thatis 119903119894= 05 119894 = 1 2 119898 119888

119894= 05 119894 = 1 2 119899

34 Modeling the Fault Information Explanation DegreeFault information explanation degree (FIED) is the matchingdegree of quarantined state vector and fault state vectorof multicomponent system In the other words it is usedfor explaining the behavior of protective relays and circuitbreakers when the fault of multicomponent systems happens

Let 119865119860 119865119861 119865

119883 whose sizes of dimension are all 1 times

(119898 + 119899) be the fault state vector respectively to single-faultcomponents 119860 119861 119883 119865

0is quarantined state vector with

1 times (119898 + 119899) dimensionTherefore the fault information explanation degree

(FIED) of single component can be expressed by the follow-ing formula

FIED = ((the number of elements whose values are all

greater than 1 among (119865119860+ 1198650))


times (the number of elements

whose values are all equal to 1)minus1) times 100

(13)

If there aremore than one faulty component in power systemFIED can be presented as follows

FIED = ((the number of elements whose values are

all greater than 1

in [(119865119860+ 1198650) or (119865

119861+ 1198650) or (119865

119862+ 1198650) or sdot sdot sdot ])



(14)

where the symbol ldquoorrdquo denotes that the greater value isselected for the values of corresponding elements

To locate out the fault components in systems we need tocompare the size of FIED Once FIED satisfies FIED = 100the fault components may be determined by formulas (13)and (14)

35 Judging Method of Malfunction and Rejection for Protec-tive Relays andCircuit Breakers According to the assumptionand analysis above the judging rule is designed as follows

(1) If the value of element of the vector (119865119860

minus 1198650) is

negative then the corresponding protection of protectiverelays or action of circuit breakers is regarded as malfunctionfor single-component fault Otherwise the protection ofprotective relays or trip of circuit breakers is regarded asrejecting action

(2) If there is more than one fault component in thesystem the judging rule is as follows The protection ofprotective relays or action of circuit breakers is regarded asmalfunctionwhile the value of the vector ((119865

119860or119865119861or119865119862orsdot sdot sdot )minus

1198650) is negative otherwise the action is regarded as rejecting

actionNotice that when failures happen the main protection

of PRs is first implemented to trip the circuit breakers Ifthe main protective relays do not work then the backupprotective relays begin to work to trip the circuit breakersTherefore the situation of the backup PRs deduced by (1) and(2) was the rejecting action it may be that the action of PRshas already tripped the action of CBs So we concern onlythe main protective relays and the circuit breakers and thengive priority tomain PRs when themalfunction and rejectingaction are considered in this paper

36 Fault DiagnosisModel of Single Component Theproblemof recognition for single-fault component can be solved byseeking for the fault assumptions which can best explain the

alarming information and can be expressed by the followingmaximization problem

119877119894=

1

119899

119899

sum

119896=1

((min119894

min119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816)

times (10038161003816100381610038161198650 (119896) minus 119865

119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894 (119896)

1003816100381610038161003816)

minus1

)

(15)

where 119894 stands for the total number of the single-fault andits value is determined by adding the number of componentsand the number of lines 119899 denotes the number of vectorelements and its value is the number of protective relays andcircuit breakers 120585 is on behalf of the resolution coefficient1198650(119896) indicates the quarantined faulty state vector 119865

119894(119896)

means the fault state vector of single-line or single componentand also is called the single-fault state vector119877

119894represents the

similarity degree between the quarantined fault state vectorand single-fault state vector and also is called correlationdegree

Notice that if the state vector of single-fault component119860 119861 119883 is119865

119860 119865119861 119865

119883 respectively then the dimension

of these state vectors is 1 lowast (119898 + 119899) and the dimensionof the quarantined fault state vector 119865

0is also 1 lowast (119898 +

119899) Meanwhile we may calculate the correlation degree andsort the calculated results Based on above discussion therecognition rule of fault component is given as follows

Rule 1 If the correlation degree between the reference statevector 119865

0and the compare state vector 119865

119894(119894 = 119860 119861 119883) is

the largest the corresponding part is fault component or faultline

Rule 2 Let the threshold of fault diagnosis be the meanof correlation degree If the correlation degree between thereference state vector 119865

0and new compare state vector 119865

119873

is more than the threshold the part is regarded as the faultcomponent or fault line

Therefore the possibility that fault happens may bedetermined for every single component according to Rules1 and 2

37 Diagnostic Procedure for Fault Components In the actualproject there is more than one fault component when faultsoccur But the diagnostic procedure may be designed on thebasis of the fault diagnosis model of single component So thespecific steps of fault diagnosis are shown as follows

371 Data Preprocessing The data which contains the infor-mation of protective relays and circuit breakers and thetopology of the power system is written in the system


Start

Recognition for power failure area

Analyzing of the workingprinciple of protective

relays and circuit breakers

Construction of fault state vector

Calculating and sorting of the correlation degree

Initializing the number of fault components

Calculating of the fault information explanation

degree (FIED)

The number of faultcomponents is FEN

End

FEN = FEN + 1

Number of primary protections of malfunction

and tripping increase

The number of faultcomponents is FEN-1

Number of primary


No

No

No

NoYes

Yes

Yes Yes

protections of malfunction

FEN = 1

FEN = 1FIED = 100

Figure 1 The flowchart for fault diagnosis

372 Recognition for the Power Supply Interrupted DistrictsThe blackout area of power systems is recognized and thetripping principle of protective relays and circuit breaker inblackout area is listed

373 Recognition of State Information Under single-faultstate the state information of protective relays and circuitbreaker is obtained in accordance with the working principleof protective relays and circuit breakers

374 Construction for Fault State Vectors The state vector offault mode for each component is given by the assignmentprinciple of vector elements and then the quarantined faultstate vector is loaded

375 Calculating and Sorting the Correlation Degree Thecorrelation coefficients are calculated respectively and thenthese results are sorted If the correlation degree betweenquarantined state vector and reference state vector is thehighest the probability that component malfunctions is alsothe largest

376 Computing Fault Information Explanation DegreeFIED is calculated by formulas (13) and (14) and is sorted IfFIED of some fault components satisfies FIED = 100 go tonext step

377 Fault Diagnosis Analysis According toRules 1 and 2 thefault component can be recognized and diagnosed And thenjudge the malfunction and rejection for protective relays andcircuit breakers

The specific flowchart for fault diagnosis is shown as inFigure 1

CB1 CB2 CB3 CB4 CB5

A BL1 L2

Figure 2 Framework of simple circuit structure

4 Experiment and Its Results Analysis

Firstly regarding the simple circuit structure shown inFigure 2 as the power supply interrupted districts the follow-ing sections will explain and demonstrate in detail how to usetheworking principle of protection action to identify the statevectors of fault components

Obviously the simple system shown in Figure 2 consistsof 4 elements (ie A B L1 and L2) and 5 circuit breakers (ieCB1 CB2 CB3 CB4 and CB5) So there are 14 protectionactions respectively that is Am Bm L1Am L1Bm L2BmL2Cm L1Ap L1Bp L2Bp L2Cp L1As L1Bs L2Bs and L2Cs Tomake things simple these protection actions are representedsymbolically by 119903

1 1199032 119903

14 Note that in this example A

and B are Bus L represents Line the subscript119898 denotes theprimary protection 119904 stands for the first backup protectionand 119901 is on behalf of the second backup protection Throughanalysis of the circuit structure the working principle of theprimary protection and the backup protection is listed inTable 1

According to the working principle of primary protectionand backup protection shown inTable 1 the state informationof protective relays and circuit breakers is deduced as shownin Table 2

Let 119888119894(119894 = 1 2 5) denote the action of five circuit

breakers Thus by combining with the state informationshown in Table 2 the fault state vector of every componentmay be determined using the assignment principle intro-duced in Section 33 Therefore the fault state vector of everyelement is shown in Table 3


Table 1 Working principle of protective relays and circuit breakers

Number Name Corresponding action of protectiverelays and circuit breakers

1199031

A119898

Fault A action A119898 tripping CB1 and

CB2

1199032

B119898

Fault B action B119898 tripping CB3 and

CB41199033

L1A119898

Fault L1 action L1A119898 tripping CB2

1199034

L1B119898

Fault L1 action L1B119898 tripping CB3

1199035

L2B119898


1199036

L2C119898

Fault L2 action L2C119898 tripping CB5

1199037

L1A119901

Fault L1 unaction L1A119898 action

L1A119901 tripping CB2

1199038

L1B119901

Fault L1 unaction L1B119898 action L1B

119901

tripping CB3

1199039

L2B119901

Fault L2 unaction L2B119898 action

L2B119901 tripping CB4

11990310

L2C119901

Fault L2 unaction L2C119898 action

L2C119901 tripping CB5

11990311

L1A119904

Fault B untripping CB3 action L1A119904

tripping CB2or fault L2 untripping CB3 and CB4action L1A

119904 tripping CB2

11990312

L1B119904

Fault A untripping CB2 action L1B119904

tripping CB3

11990313

L2B119904

Fault C untripping CB5 action L2B119904

tripping CB4

11990314

L2C119904

Fault B untripping CB4 actionL2C119904 tripping CB5

or fault L1 untripping CB3 and CB4action L2C

119904 tripping CB5

Table 2 State information of protective relays and circuit breakers

Fault nodes The state information for protective relays andcircuit break

A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1

If a quarantined fault state vector obtained by alarminginformation is (119903

1 1199035 1199036 1198881 1198882 1198884 1198885) and it needs to determine

which node is in fault we need to compare the size ofcorrelation coefficient between quarantined fault state andreference state vector According the data in Table 3 thecorrelation degree may be computed by formula (15) that is

119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)

The sorting result is

119903L2 gt 119903A gt 119903B gt 119903L1 (17)

So we can infer that the probability that the Line L2malfunctions is maximum and then the number of faultelements (FEN) is 1 that is there is only one fault componentL2 in systems In this case we may get

FIED = (5 divide 7) times 100 = 714 (18)

Obviously the fault component cannot be completelydetermined Through analysis of the information data whenthere are two fault components L2 and A FIED is


Therefore the quarantined state vector indicates that twocomponents L2 and A are failure

Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)

By the judging rule of malfunction and rejection the pri-mary protection which corresponds to 119903

5and 119903

6and the

action 1198884which corresponds to the circuit breaker CB4 is

regarded as malfunction Note that the alarming informationis (1199031 1199035 1199036 1198881 1198882 1198884 1198885) and the action 119888

3shown in formula

(21) is a rejection In other words the information whichcorresponds to the action of the circuit breaker CB3 has lostNamely the more accurate diagnosis results can be obtainedby the algorithm even if some information has lost in systems

The above computing process has primarily illustrated thecalculated steps of fault diagnosis algorithm based on Graysystems theory And the results indicate that the algorithmis effective and reasonable To further explain and verify therationality and effectiveness of the method mentioned in thispaper we take the classic system structure of local power relayprotection shown by [9] which contains 28 components84 protective relays and 40 circuit breakers to analyze thediagnosis procedureThe structure of the local power systemsis shown in Figure 3

Where A and B denote Bus L is Line T representsTransformer and CB is circuit breaker

As can be seen in Figure 3 28 components are A1 A2A3 A4 T1 T2 T3 T4 T5 T6 T7 T8 B1 B2 B3 B4 B5B6 B7 B8 L1 L2 L3 L4 L5 L6 L7 and L8 84 protectionsare consisting of 36 primary protections and 48 backupprotections where 36 primary protections are respectivelyA1m B1m B2m B3m B4m B5m B6m B7m B8m L1Sm L1RmL2Sm L2Rm L3Sm L3Rm L4Sm L4Rm L5Sm L5Rm L6SmL6Rm L7Sm L7Rm L8Sm and L8Rm 48 backup protectionsare as follows T1p T2p T3p T4p T5p T6p T7p T8p T1s T2sT3s T4s T5s T6s T7s T8s L1Sp L1Rp L2Sp L2Rp L3Sp L3RpL4Sp L4Rp L5Sp L5Rp L6Sp L6Rp L7Sp L7Rp L8Sp L8RpL1Ss L1Rs L2Ss L2Rs L3Ss L3Rs L4Ss L4Rs L5Ss L5Rs L6SsL6Rs L7Ss L7Rs L8Ss and L8Rs


Table 3 State vectors of fault component

Fault node The state vectors (1199031 1199032 119903

141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5

Figure 3 Classic system structure of the local power relay protection

And the meaning of these symbols is respectively asfollows S is Sending End of Line and R is Receiving End119898 isprimary protection 119901 represents first backup protection and119904 is second backup protectionThe working principle of theseprotections may be seen in the Resources [11 13] section

By comparing themethod proposed in this paperwith themethods in [13 14] we are testing out 4 most complicatedcircumstances in case of incomplete information and com-plete information to verify the effectiveness of the methodThe diagnosis results are listed as in Table 4

As is shown in Table 4 in case of complete informationthe diagnosis result for test sequence 1 and sequence 2 isidentical to the results in [11 12] in case of incompleteinformation the diagnosis result for test sequence 3 and testsequence 4 is identical to the results in [11] but it is notidentical to the result in [12] This is because there are somelost information in test sequence 3 and test sequence 4

Therefore to further test this algorithm proposed inthis paper we apply the presented algorithm in this paperand the Bayesian algorithm in [12] to diagnose the powersupply in intelligent traffic systems (ITS) The detectinginformation of the power supply in ITS contains the PTbreak-phase CT signal volts dc and current synchronoussignal temperature of sensors and the Airborne capacityIn real application the fitting curve between using the twomethods is as in Figure 4

Figure 4(a) showed the whole effect of diagnosis proce-dure Correspondingly Figures 4(b) and 4(c) described thepart effect of diagnosis algorithmThis indicates that the errorbetween using the two methods is very small This indicatesthat the algorithm presented in this paper is reasonable

However the running stability of real system needs to beensured for any algorithm whether or not the algorithm iseffective and reasonable That is to say the stability of thediagnosis procedure should to be analyzed For verifying therationality of the designed algorithm the anticipant precisionthreshold is set as 001 and every running test time is set as1000 epochs The total running performance is shown as inFigure 5

The running effect of diagnosis system indicated that theabnormal phenomenon of the security running took placeat 20th second and then the parameter of systems needs tobe adjusted to ensure the systems normal running Figure 5displays that the abnormal phenomena of the designeddiagnosis systems seem to change small in warning and yetthe running performance is very stable during the wholerunning process To find out why the danger happens wetake the running performance chart from 0th second to 40thsecond as in Figure 6

Figure 6 displays that the interior change of diagnosissystem was quite rapid Simultaneously there exists a glacisfrom the initial warning to normal running As a result the


Table 4 Part of the test results

Testsequences Relay action information Missing

informationDiagnosisresult

Judgment formalfunction and

rejection

Diagnosis resultin [11]


1Behavior T5s T6s

tripping C21 C22 C23 C24and C25

unexisting A3Malfunction A3

119898

rejection (tripping)C21 C24 and C25

A3 A3

2Behavior B1m L2Rs

L4Rs C4 tripping C5 C7C9 C12 and C27

unexisting B1 Rejection (tripping) C6 B1 B1

3Behavior B1m L4Rs


L2Rs C5 B1 Rejection (tripping) C5and C6

B1

4

Behavior B1119898 T1119898 T2119898

L1s119898 L2RP tripping C2

C3 C4 C5 C7 C9 C11 andC28

T7s C6 L1 B1 T1 T2Rejection (tripping)

L1R119898 C6 malfunction

C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)

Figure 4 (a) The fitting curve of test sequence (0ndash780) using the two methods (b) The fitting curve of test sequence (0ndash301) using the twomethods (c)The fitting curve of test sequence (301ndash500) using the twomethods (Note + indicates the test sequence using Bayesian algorithmin [12] I indicates the test sequence using Gray theory algorithm)


0 100 200 300 400 500 600 700 800 900 10001000 epochs

Performance is 00510264 goal is 001

Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training

Figure 5The total running performance chart of fault diagnosis forthe power supply in ITS

Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training

Figure 6The part running performance chart of fault diagnosis forthe power supply in ITS

running effect is in accord with the practical situationThis isbecause certain period of time is used to adjust parameter ofthe power supply for ensuring the final stable running of thealgorithm

5 Conclusion

Based on the theory of Gray system a fault diagnosismethod in power systems is proposed with respect to thediagnosis problem of incomplete information for protectiverelays devices in power system and the analysis of overallarchitecture and the diagnosis process are conducted Somesimple experiments show that the method presented in thispaper is effective and can also carry out the same faultdiagnosis task that another traditional diagnosis methodcompleted Through comparing these diagnosis results weknow that the method may diagnose the fault in case ofincomplete and complete information In addition throughthe practical engineering application in ITS the stabilityof diagnosis procedure was also analyzed The simulationresult demonstrated that the running stability of the diagnosis

procedure is practical and consistent with the operationrequirements

However owing to the fact that information data comesfrom different parts of power supply systems in ITS the run-ning situation of diagnosis systems was influenced Therebythe investigation about how to logically distribute the infor-mation of power supply device in ITS remains an interestingarea for further research Meanwhile the method and algo-rithm presented in this paper can only be used to diagnoseand determine which component malfunctions after theabnormal phenomenon of power systems has occurred Butin practical application of power supply devices in ITS theusers always hope that the diagnosis systems can forecastthe faults before the fault states of components will happenSo how to fuse the incomplete and complete information ofprotective relays and circuit breakers to forecast the faultsby GM model which provides the maintaining decision forthe components and lines is also very important problem inpower systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National Nature Science Foun-dation (nos 61004118 61304104) and Program for ExcellentTalents of Chongqing Higher School (no 2014-18)

References

[1] W Fu-Shuan H Zhen-Xiang T Lei et al ldquoAn analytic modeland genetic algorithm based methods for fault diagnosis inpower systems part 1 the model and methodrdquo Proceedings ofthe CSU-EPSA vol 10 no 3 pp 1ndash7 1998

[2] G Cardoso Jr J G Rolim and H H Zurn ldquoIdentifyingthe primary fault section after contingencies in bulk powersystemsrdquo IEEE Transactions on Power Delivery vol 23 no 3pp 1335ndash1342 2008

[3] MWang and F Long ldquoFault diagnosis of rectifying circuit usingANNrdquo Journal of Wuhan University of Technology (Transporta-tion Science and Engineering) vol 37 no 3 pp 578ndash580 2013

[4] W Guo F Wen G Ledwich Z Liao X He and J Liang ldquoAnanalytic model for fault diagnosis in power systems consideringmalfunctions of protective relays and circuit breakersrdquo IEEETransactions on Power Delivery vol 25 no 3 pp 1393ndash14012010

[5] Q Li Z-B Li and Q Zhang ldquoResearch of power transformerfault diagnosis system based on rough sets and BayesianNetworksrdquo Advanced Materials Research vol 320 pp 524ndash5292011

[6] Z Xin Power System Fault Diagnosis Based on Rough SetTheoryand Bayesian Network Shandong University Shandong China2010

[7] S-H Liu and X Li ldquoPower system fault diagnosis by use ofprediction petri net modelsrdquo Proceedings of the CSU-EPSA vol25 no 4 pp 162ndash166 2013


[8] Z J Zhou C H Hu J B Yang D L Xu and D H ZhouldquoOnline updating belief-rule-base using the RIMER approachrdquoIEEE Transactions on Systems Man and Cybernetics Part ASystems and Humans vol 41 no 6 pp 1225ndash1243 2011

[9] B-C Zhang X-X Han Z-J Zhou L Zhang X-J Yin and Y-W Chen ldquoConstruction of a new BRB based model for timeseries forecastingrdquo Applied Soft Computing Journal vol 13 no12 pp 4548ndash4556 2013

[10] H Dong Z Wang and H Gao ldquoDistributed 119867infin

filteringfor a class of markovian jump nonlinear time-delay systemsover lossy sensor networksrdquo IEEE Transactions on IndustrialElectronics vol 60 no 10 pp 4665ndash4672 2013

[11] Z Wang H Dong B Shen and H Gao ldquoFinite-horizon 119867infin

filtering with missing measurements and quantization effectsrdquoIEEE Transactions on Automatic Control vol 58 no 7 pp 1707ndash1718 2013

[12] FWen Y Qian and ZHan ldquoATabu search b Based approach tofault section estimation and state identification of unobservedprotective relays in power systems using information fromprotective relays and circuit breakersrdquo Transactions of ChinaElectrotechnical Society vol 13 no 5 pp 1ndash8 1998

[13] W Xin G Chuang-xin and C Yi-jia ldquoA new fault diagnosisapproach of power system based on Bayesian network andtemporal order informationrdquo Proceedings of the CSEE vol 25no 13 pp 14ndash18 2005

[14] L Tang H-B Sun and B-M Zhang ldquoOnline fault diagnosisfor power system based on information theoryrdquo Proceedings ofthe CSEE vol 23 no 7 pp 5ndash11 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


components In Section 4 we have discussed the simulativeresults by experiments Finally Section 5 concluded thispaper with inferences and directions for future work

2 Basis Introduction of Gray System Theoryand Gray Correlation Analysis

21 Basal Principle of Gray System Theory Gray systemtheory which is established by Chinese scholar ProfessorDeng Ju-Long in 1982 has been used to research uncertainproblem with lack of data and information In generalspeaking some information is known and other pieces ofinformation are unknown in uncertain data systems Themain idea of Gray system theory is to describe correctlythe systemsrsquo evolution law and to monitor effectively theirrunning behavior by extracting valuable information fromthe known information Over forty years later the structuresystem of Gray system theory has basically taken shapeThe theoretical basis includes Gray Matrix Gray Algebraand Gray Equation And the Gray model (GM) may alsoimplement the analysis evaluation prediction and controldecision of uncertain data systems by using spatial asso-ciation rule and sequence generation method For futureunderstanding of the advantages of Gray system theory sixbasic principles are introduced as follows

211 Differential Information Principle There is the differ-ence among differential information In other words thedifference is the information For example if there are twodifferential objects or systems everyone has unique informa-tion that is not similar to another piece of informationOne ofthemost basic pieces of informationwhich the human societyperceives is that the world comes from the difference betweenmatter and matter

212 Nonuniqueness of Solutions The solution is nonuniqueunder the condition of information being incomplete andvague The uncertainty of systems leads to the existence ofuncertain information and then causes nonuniqueness ofsolutions

213 Smallest Information Principle The basic idea of Graysystem theory is to utilize the smallest information achievingfrom known knowledge data to accomplish a given task Thesmallest information getting by researching the uncertainproblem with the small samples and poor information is thefundamental basis of making a distinction between Gray areaand no-Gray

214 Cognitive Foundation Principle

The Foundation of Cognition Is Information The accurate andcomplete cognition is determined according to definite andprecise knowledge the uncertain and incomplete knowledgemay also lead to vague cognition Correspondingly if thereis no information of systems the cognition of systems is notalso completed Thus the cognition should be studied basedon information

215 Innovation Priority Principle The function of newinformation is more important than old information forcognitive behaviors because the new information directlyreflects the current states of the system andmainly influencesthe future trend of the development

216 Gray Indestructibility Principle Notably the incom-plete and uncertain information is very much stronglyentrenched in real systems As new information is continu-ously generated in real engineering the cognition conclusionis improved gradually and the level of cognition will tend torationality and correctness Accordingly the Gray systems donot disappear

Based on the above principles the theoretical modelimplementing a given task may be constructed according tosmall sample and poor information However the runningstates and trend of systems are determined by many factorsin real application So we need to discern the primary factorsand lesser factors And fortunately the Gray correlation anal-ysis which is the important part of Gray system theory canjudge the connection in accordance with the approximationdegree of the two-dimensional curve in time domain andfrequencies domain The higher the similarity is the greaterthe correlation is For further analysis the Gray relationalaxioms are introduced firstly

22 Gray Correlation Axiom Suppose that the behaviorsequence of the system is119883

0= (1199090(1) 1199090(2) 119909

0(119899)) thus

the corresponding factor series is as follows

1198831= (1199091 (1) 1199091 (2) 1199091 (119899))

119883119894= (119909119894(1) 119909

119894(2) 119909

119894(119899))

119883119898

= (119909119898(1) 119909

119898(2) 119909

119898(119899))

(1)

For a given real 119903(1199090(119896) 119909119894(119896)) if 119903(119883

0 119883119894) =

(1119899)sum119899

119896=1119903(1199090(119896) 119909119894(119896)) satisfies

(1) normalization 0 lt 119903(1198830 119883119894) le 1 and 119883

0= 119883119894rArr

119903(1198830 119883119894) = 1

(2) integrity let 119883119894 119883119895isin 119883 = 119883

119904| 119904 = 0 1 119898 119898 ge

2 if 119894 = 119895 then 119903(119883119894 119883119895) = 119903(119883

119895 119883119894)

(3) even symmetry if 119883 = 119883119894 119883119895 119883 = 119883

119894 119883119895 hArr

119903(119883119894 119883119895) = 119903(119883

119895 119883119894)

(4) accessibility the value of the Euclid distance|1199090(119896) minus 119909

119894(119896)| is inversely proportional to the value

of the given real 119903(1199090(119896) 119909119894(119896))

Thus 119903(1198830 119883119894) = (1119899)sum

119899

119896=1119903(1199090(119896) 119909119894(119896)) is called as Gray

relational degree where 119903(1198830 119883119894) represents the correlation

coefficient of each pair of variables 119883119894and 119883

119895 Four condi-

tions (1ndash4) are also regarded as Gray relational four axioms





Δ0119894119896

=10038171003817100381710038171199090 (119896) minus 119909

119894(119896)

1003817100381710038171003817 (2)

And suppose that

Δmax = max119894

max119896

Δ0119894119896

Δmin = min119894

min119896

Δ0119894119896

(3)


0(119896) and vector 119909

119894(119896) is defined as

follows

119903 (1199090 (119896) 119909119894 (119896)) =


(4)

that is

119903 (1199090 (119896) 119909119894 (119896))

=min119894min119896

10038161003816100381610038161199090 (119896) minus 119909119894(119896)

1003816100381610038161003816 + 120585max119894max119896

10038161003816100381610038161199090 (119896) minus 119909119894(119896)

100381610038161003816100381610038161003816100381610038161199090 (119896) minus 119909

119894 (119896)1003816100381610038161003816 + 120585max

119894max119896

10038161003816100381610038161199090 (119896) minus 119909119894 (119896)

1003816100381610038161003816

(5)



Let 119903(1199090(119896) 119909119894(119896)) = 119903


119903 (1198830 119883119894) =

1

119899

119899

sum

119896=1

119903 (1199090 (119896) 119909119894 (119896)) =

1

119899

119899

sum

119896=1

1199030119894 (119896) (6)



119894




(1198830 1198831 119883

119898) = (

1199090(1) 119909


119898(1)

1199090(2) 119909


119898(2)

sdot sdot sdot

1199090(119899) 119909

1(119899) sdot sdot sdot 119909

119898(119899)

) (7)


=

(119909119894(1) 119909119894(2) 119909

119894(119899)) 119894 = 1 2 119898


1199091015840

119894(119896) =

119909119894 (119896)

(1119899)sum119899

119896=1119909119894(119896)

119894 = 0 1 119898 119896 = 1 2 119899

(8)


(1198830 1198831 119883

119898) = (

(

1199091015840

0(1) 119909

1015840


1015840

119898(1)

1199091015840

0(2) 119909

1015840


1015840

119898(2)

sdot sdot sdot

1199091015840

0(119899) 119909

1015840

1(119899) sdot sdot sdot 119909

1015840

119898(119899)

)

)

(9)


0 The reference


0= (1199091015840

0(1) 1199091015840

0(2) 119909

1015840

0(119899))



0(119896) and


119894(119896) that is Δ

0119894119896= 1199091015840

0(119896) minus 119909

1015840

119894(119896)


Δmax = max119894

max119896

100381710038171003817100381710038171199091015840

0(119896) minus 119909

1015840

119894(119896)

10038171003817100381710038171003817

Δmin = min119894

min119896

100381710038171003817100381710038171199091015840

0(119896) minus 119909

1015840

119894(119896)

10038171003817100381710038171003817

(10)


0(119896) and vector 1199091015840


119903 (1199091015840

0(119896) 119909

1015840

119894(119896)) =

Δmin + 120585ΔmaxΔ0119894119896

+ 120585Δmax (11)

















119865 (119896) = (119877 119862) = (1199031 1199032 119903

119898 1198881 1198882 119888

119898)

119896 = 1 2 119899

(12)



119894 = 1 2 119898 otherwise 119903119894= 0


119894= 1 119894 = 1 2 119899 otherwise 119888

119894= 0


119894= 05 119894 = 1 2 119899


Let 119865119860 119865119861 119865











(13)



all greater than 1

in [(119865119860+ 1198650) or (119865

119861+ 1198650) or (119865

119862+ 1198650) or sdot sdot sdot ])



(14)





minus 1198650) is









119877119894=

1

119899

119899

sum

119896=1

((min119894

min119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816)

times (10038161003816100381610038161198650 (119896) minus 119865

119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894 (119896)

1003816100381610038161003816)

minus1

)

(15)


119894(119896)





119860 119865119861 119865







119894(119894 = 119860 119861 119883) is




119873






Start








degree (FIED)


End

FEN = FEN + 1




Number of primary


No

No

No

NoYes

Yes

Yes Yes


FEN = 1

FEN = 1FIED = 100









CB1 CB2 CB3 CB4 CB5

A BL1 L2





1 1199032 119903









1199031

A119898


CB2

1199032

B119898


CB41199033

L1A119898


1199034

L1B119898


1199035

L2B119898


1199036

L2C119898


1199037

L1A119901



1199038

L1B119901


119901

tripping CB3

1199039

L2B119901



11990310

L2C119901



11990311

L1A119904



119904 tripping CB2

11990312

L1B119904


tripping CB3

11990313

L2B119904


tripping CB4

11990314

L2C119904



119904 tripping CB5



A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1




119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)


119903L2 gt 119903A gt 119903B gt 119903L1 (17)






Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)


5and 119903

6and the



3shown in formula








141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Δ0119894119896

=10038171003817100381710038171199090 (119896) minus 119909

119894(119896)

1003817100381710038171003817 (2)

And suppose that

Δmax = max119894

max119896

Δ0119894119896

Δmin = min119894

min119896

Δ0119894119896

(3)


0(119896) and vector 119909

119894(119896) is defined as

follows

119903 (1199090 (119896) 119909119894 (119896)) =


(4)

that is

119903 (1199090 (119896) 119909119894 (119896))

=min119894min119896

10038161003816100381610038161199090 (119896) minus 119909119894(119896)

1003816100381610038161003816 + 120585max119894max119896

10038161003816100381610038161199090 (119896) minus 119909119894(119896)

100381610038161003816100381610038161003816100381610038161199090 (119896) minus 119909

119894 (119896)1003816100381610038161003816 + 120585max

119894max119896

10038161003816100381610038161199090 (119896) minus 119909119894 (119896)

1003816100381610038161003816

(5)



Let 119903(1199090(119896) 119909119894(119896)) = 119903


119903 (1198830 119883119894) =

1

119899

119899

sum

119896=1

119903 (1199090 (119896) 119909119894 (119896)) =

1

119899

119899

sum

119896=1

1199030119894 (119896) (6)



119894




(1198830 1198831 119883

119898) = (

1199090(1) 119909


119898(1)

1199090(2) 119909


119898(2)

sdot sdot sdot

1199090(119899) 119909

1(119899) sdot sdot sdot 119909

119898(119899)

) (7)


=

(119909119894(1) 119909119894(2) 119909

119894(119899)) 119894 = 1 2 119898


1199091015840

119894(119896) =

119909119894 (119896)

(1119899)sum119899

119896=1119909119894(119896)

119894 = 0 1 119898 119896 = 1 2 119899

(8)


(1198830 1198831 119883

119898) = (

(

1199091015840

0(1) 119909

1015840


1015840

119898(1)

1199091015840

0(2) 119909

1015840


1015840

119898(2)

sdot sdot sdot

1199091015840

0(119899) 119909

1015840

1(119899) sdot sdot sdot 119909

1015840

119898(119899)

)

)

(9)


0 The reference


0= (1199091015840

0(1) 1199091015840

0(2) 119909

1015840

0(119899))



0(119896) and


119894(119896) that is Δ

0119894119896= 1199091015840

0(119896) minus 119909

1015840

119894(119896)


Δmax = max119894

max119896

100381710038171003817100381710038171199091015840

0(119896) minus 119909

1015840

119894(119896)

10038171003817100381710038171003817

Δmin = min119894

min119896

100381710038171003817100381710038171199091015840

0(119896) minus 119909

1015840

119894(119896)

10038171003817100381710038171003817

(10)


0(119896) and vector 1199091015840


119903 (1199091015840

0(119896) 119909

1015840

119894(119896)) =

Δmin + 120585ΔmaxΔ0119894119896

+ 120585Δmax (11)

















119865 (119896) = (119877 119862) = (1199031 1199032 119903

119898 1198881 1198882 119888

119898)

119896 = 1 2 119899

(12)



119894 = 1 2 119898 otherwise 119903119894= 0


119894= 1 119894 = 1 2 119899 otherwise 119888

119894= 0


119894= 05 119894 = 1 2 119899


Let 119865119860 119865119861 119865











(13)



all greater than 1

in [(119865119860+ 1198650) or (119865

119861+ 1198650) or (119865

119862+ 1198650) or sdot sdot sdot ])



(14)





minus 1198650) is









119877119894=

1

119899

119899

sum

119896=1

((min119894

min119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816)

times (10038161003816100381610038161198650 (119896) minus 119865

119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894 (119896)

1003816100381610038161003816)

minus1

)

(15)


119894(119896)





119860 119865119861 119865







119894(119894 = 119860 119861 119883) is




119873






Start








degree (FIED)


End

FEN = FEN + 1




Number of primary


No

No

No

NoYes

Yes

Yes Yes


FEN = 1

FEN = 1FIED = 100









CB1 CB2 CB3 CB4 CB5

A BL1 L2





1 1199032 119903









1199031

A119898


CB2

1199032

B119898


CB41199033

L1A119898


1199034

L1B119898


1199035

L2B119898


1199036

L2C119898


1199037

L1A119901



1199038

L1B119901


119901

tripping CB3

1199039

L2B119901



11990310

L2C119901



11990311

L1A119904



119904 tripping CB2

11990312

L1B119904


tripping CB3

11990313

L2B119904


tripping CB4

11990314

L2C119904



119904 tripping CB5



A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1




119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)


119903L2 gt 119903A gt 119903B gt 119903L1 (17)






Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)


5and 119903

6and the



3shown in formula








141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119865 (119896) = (119877 119862) = (1199031 1199032 119903

119898 1198881 1198882 119888

119898)

119896 = 1 2 119899

(12)



119894 = 1 2 119898 otherwise 119903119894= 0


119894= 1 119894 = 1 2 119899 otherwise 119888

119894= 0


119894= 05 119894 = 1 2 119899


Let 119865119860 119865119861 119865











(13)



all greater than 1

in [(119865119860+ 1198650) or (119865

119861+ 1198650) or (119865

119862+ 1198650) or sdot sdot sdot ])



(14)





minus 1198650) is









119877119894=

1

119899

119899

sum

119896=1

((min119894

min119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816)

times (10038161003816100381610038161198650 (119896) minus 119865

119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894 (119896)

1003816100381610038161003816)

minus1

)

(15)


119894(119896)





119860 119865119861 119865







119894(119894 = 119860 119861 119883) is




119873






Start








degree (FIED)


End

FEN = FEN + 1




Number of primary


No

No

No

NoYes

Yes

Yes Yes


FEN = 1

FEN = 1FIED = 100









CB1 CB2 CB3 CB4 CB5

A BL1 L2





1 1199032 119903









1199031

A119898


CB2

1199032

B119898


CB41199033

L1A119898


1199034

L1B119898


1199035

L2B119898


1199036

L2C119898


1199037

L1A119901



1199038

L1B119901


119901

tripping CB3

1199039

L2B119901



11990310

L2C119901



11990311

L1A119904



119904 tripping CB2

11990312

L1B119904


tripping CB3

11990313

L2B119904


tripping CB4

11990314

L2C119904



119904 tripping CB5



A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1




119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)


119903L2 gt 119903A gt 119903B gt 119903L1 (17)






Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)


5and 119903

6and the



3shown in formula








141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(13)



all greater than 1

in [(119865119860+ 1198650) or (119865

119861+ 1198650) or (119865

119862+ 1198650) or sdot sdot sdot ])



(14)





minus 1198650) is









119877119894=

1

119899

119899

sum

119896=1

((min119894

min119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894(119896)

1003816100381610038161003816)

times (10038161003816100381610038161198650 (119896) minus 119865

119894(119896)

1003816100381610038161003816

+120585max119894

max119896

10038161003816100381610038161198650 (119896) minus 119865119894 (119896)

1003816100381610038161003816)

minus1

)

(15)


119894(119896)





119860 119865119861 119865







119894(119894 = 119860 119861 119883) is




119873






Start








degree (FIED)


End

FEN = FEN + 1




Number of primary


No

No

No

NoYes

Yes

Yes Yes


FEN = 1

FEN = 1FIED = 100









CB1 CB2 CB3 CB4 CB5

A BL1 L2





1 1199032 119903









1199031

A119898


CB2

1199032

B119898


CB41199033

L1A119898


1199034

L1B119898


1199035

L2B119898


1199036

L2C119898


1199037

L1A119901



1199038

L1B119901


119901

tripping CB3

1199039

L2B119901



11990310

L2C119901



11990311

L1A119904



119904 tripping CB2

11990312

L1B119904


tripping CB3

11990313

L2B119904


tripping CB4

11990314

L2C119904



119904 tripping CB5



A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1




119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)


119903L2 gt 119903A gt 119903B gt 119903L1 (17)






Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)


5and 119903

6and the



3shown in formula








141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start








degree (FIED)


End

FEN = FEN + 1




Number of primary


No

No

No

NoYes

Yes

Yes Yes


FEN = 1

FEN = 1FIED = 100









CB1 CB2 CB3 CB4 CB5

A BL1 L2





1 1199032 119903









1199031

A119898


CB2

1199032

B119898


CB41199033

L1A119898


1199034

L1B119898


1199035

L2B119898


1199036

L2C119898


1199037

L1A119901



1199038

L1B119901


119901

tripping CB3

1199039

L2B119901



11990310

L2C119901



11990311

L1A119904



119904 tripping CB2

11990312

L1B119904


tripping CB3

11990313

L2B119904


tripping CB4

11990314

L2C119904



119904 tripping CB5



A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1




119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)


119903L2 gt 119903A gt 119903B gt 119903L1 (17)






Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)


5and 119903

6and the



3shown in formula








141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199031

A119898


CB2

1199032

B119898


CB41199033

L1A119898


1199034

L1B119898


1199035

L2B119898


1199036

L2C119898


1199037

L1A119901



1199038

L1B119901


119901

tripping CB3

1199039

L2B119901



11990310

L2C119901



11990311

L1A119904



119904 tripping CB2

11990312

L1B119904


tripping CB3

11990313

L2B119904


tripping CB4

11990314

L2C119904



119904 tripping CB5



A 1199031= 1 CB1 = 1 and CB2 = 1 or 119903

12= 1 CB2 = 0

and CB3 = 1

B 1199032= 1 CB3 = 1 and CB4 = 1 or 119903

11= 1 CB3 = 0

and CB2 = 1 or 11990314= 1 CB4 = 0 and CB5 = 1

L11199033= 1 and CB2 = 1 or 119903

4= 1 and CB3 = 1 or 119903

7=

1 1199033= 0 and CB2 = 1 or 119903

8= 1 1199034= 0 and CB3

= 1 or 11990314= 1 CB3 = 0 CB4 = 0 and CB5 = 1

L21199035= 1 and CB4 = 1 or 119903

6= 1 and CB5 = 1 or 119903

9=

1 1199035= 0 and CB4 = 1 or 119903

10= 1 1199036= 0 and CB5

= 1 or 11990311= 1 CB3 = 0 CB4 = 0 and CB2 = 1




119903A = 08333 119903B = 08247

119903L1 = 07807 119903L2 = 08509

(16)


119903L2 gt 119903A gt 119903B gt 119903L1 (17)






Meanwhile

119865L2 or 119865A = (11990311199035

21199036

2 1199039 11990310 11990311 11990312 1198881 1198882 11988831198884

2 1198885) (20)

119865L2 or 119865A minus 1198650= (minus

1199035

2 minus

1199036

2 1199039 11990310 11990311 11990312 1198883 minus1198884) (21)


5and 119903

6and the



3shown in formula








141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












141198881 1198882 119888

5) Simple forms

A (1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 05 1 0 0) (1199031 11990312 1198881 11988822 1198883)

B (0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 05 05 1) (1199032 11990311 11990314 1198882 11988832 11988842 1198885)

L1 (0 0 05 05 0 0 1 1 0 0 0 0 0 1 0 1 05 0 1) (11990332 11990342 1199037 1199038 11990314 1198882 11988832 1198885)

L2 (0 0 0 0 05 05 0 0 1 1 1 0 0 0 0 1 0 05 1) (11990352 11990362 1199039 11990310 11990311 1198882 11988842 1198885)

A4A3

CB2

CB1

CB3

A1

T1

T2T4

T3

T8

T7T5

CB6

CB19

CB4

CB5

B1

CB7

B2

L1

CB9

CB8

CB11

B4

CB10

L2CB12

CB35

CB20

B3

CB13

CB14 CB16

CB15 CB17

A2CB18

T6

L5L6

CB33

CB34 CB36CB31

CB32

CB37

CB38

CB39

CB40

CB21

CB22

CB23

CB24

CB25

CB26

CB27

CB28

CB29

CB30

B6 B7B6

L8

L4 L3

L7

B5















rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














rejection



1Behavior T5s T6s



119898


A3 A3

2Behavior B1m L2Rs



3Behavior B1m L4Rs



B1

4

Behavior B1119898 T1119898 T2119898





C28

L1 B1 T1 T2

0 100 200 300 400 500 600 700 800minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)

(a)

0 50 100 150 200 250 300minus5

0

5

10

15

20

25

30

Time (epoch)

Dia

gnos

is an

alys

is

(b)

minus5

0

5

10

15

20

25

30

Dia

gnos

is an

alys

is

Time (epoch)300 350 400 450 500

(c)



0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900 10001000 epochs


Trai

ning

blu

e goa

l bl

ack

10minus3

10minus2

10minus1

100

Stop training


Trai

ning

blu

e goa

l bl

ack

50 10 15 20 25 30 35 4040 Epochs


10minus3

10minus2

10minus1

100

Stop training



5 Conclusion






Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a fault diagnosis method of power systems...

Documents