Research Article A Novel Power System Reliability Predicting Model Based …downloads.hindawi.com/journals/mpe/2013/648250.pdf · 2019-07-31 · A Novel Power System Reliability Predicting

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 648250 6 pageshttpdxdoiorg1011552013648250

Research ArticleA Novel Power System Reliability Predicting ModelBased on PCA and RVM

Yuping Zheng1 Guoqiang Sun12 Zhinong Wei2 Feifei Zhao2 and Yonghui Sun2

1 NARI Technology Development Co Ltd Nanjing 210061 China2 College of Energy and Electrical Engineering Hohai University Nanjing 210098 China

Correspondence should be addressed to Yonghui Sun sunyonghui168gmailcom

Received 9 January 2013 Accepted 4 February 2013

Academic Editor Yang Tang

Copyright copy 2013 Yuping Zheng et al This 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

The power system reliability is an important index to evaluate the ability of power supply According to the characteristics of thepractical grid operation this paper trains and sets up power grid reliability predicting model based on relevance vector machinetaking the load supplying capacity of power grid and natural calamities as input variables and the outage time of power grid failureaffecting the reliability of the power supply as output variables In the modeling process through principal component analysis ofthe training sample set of relevance vector machine the input factor number of sample is improved the input number of networkis reduced the network structure is simplified and the predicting accuracy is increased Simulation results are provided to verifythe effectiveness of the proposed algorithm which show that it provides a new way for power system reliability predicting

1 Introduction

The power system reliability is the ability of power system toprovide continuous power supply for all the consumersWiththe development of society and the increasing of peoplersquosliving level the requirement for power system reliability isbecoming more and more important in recent years Thehigher reliability of power system is not only the need ofthe consumers but also of importance for the developmentof power companies Therefore in the past few decades thepower system reliability has attracted an increasing attentionby lots of researchers inland or abroad several different wayshave been proposed to improve the power system reliabilityby analyzing affecting factors [1]

The traditional predicting assessment methods for powersystem reliability require the historical data with accuratenetwork structure and the reliable elements [2] In [3] theauthors applied the Monte Carlo method in the assessmentof the power system reliability where the calculation struc-ture was simple but the calculation error was inverselyproportional to the square root of experiment frequencywhich reduced errors at the expense of computing timeIn [4] a grid reliability algorithm based on the sensitivity

analysis was proposed in which a more important keyelement information was obtained through the sensitivityanalysis of element reliability parameters In [5] a power gridreliability evaluation model based on radical basis function(RBF) neural network has been proposed where the statesof power system could be firstly classified through theadaptive algorithm of RBF neural network then the speed ofidentifying grid failure state could be improved by computingthe power grid reliability In [6] the BP neural network wasused to predict the power supply reliability of a city gridhowever due to the less influential factors the predictingperformance was not good and the speed was slow andsometimes it even fall into local extremism The authors in[7] proposed a power system reliability predicting assessmentalgorithm based on the actual operation constraint whichused the actual power grid operation parameters to predictthe reliability index of planned distribution network so thatthe reliability index could be decomposed and the reliabilitymanagement could be arranged In [8] the Markov cut-setalgorithm was employed to evaluate power system reliability

However the grid structure is too complex and theamount of data is too huge and it is often difficult to deter-mine the structure so the traditional predicting methods

2 Mathematical Problems in Engineering

cannot be used to predict the power system reliability Onthe other hand in practical applications if all the influencingfactors are taken as the inputs into the sample set it willinevitably lead to dimension expanding of the sample set andthen affecting the predicting accuracy and the generalizationcapability

Based on the above discussion this paper presents powersystem reliability predicting model based on relevance vectormachine (RVM) the principle component analysis (PCA)method is used to extract sample sets and then it is taken intothe RVM model after eliminating the correlation betweenvariables The proposed approach not only considers thefeature extracting ability of PCA but also takes advantageof the nonlinear approximating ability of RVM and thusthe predicting accuracy of the predicting model and thegeneralization ability can be improved greatly

The rest of this paper is organized as follows In Sections2 and 3 the theory of PCA and the principle of RVM areintroduced respectively In Section 4 the reliability pre-dicting model is proposed and the corresponding algorithmis presented In Section 4 an illustrative example about aregional grid is provided to illustrate the effectiveness ofthe developed results At last this paper completes with aconclusion

2 The Theory of PCA

In this section the theory of PCA is briefly introduced Itfirstly calculates the correlation matrix of the data matrixderived from many sample data then gets the accumulatedvariance contribution according to the eigenvalues of thecorrelation matrix and finally determines the principal com-ponents from the eigenvectors of the correlation matrix Themain results can be divided into the following five steps [9ndash11]

Step 1 (standardization of the original data) The main pur-pose is to eliminate the effects of different dimensions of theoriginal variables and large numerical difference that is

119909lowast

119895=

119909119895minus 119864 (119909

119895)

radicVar (119909119895)

(1)

where 119909119895is the 119895th column of the original data X

119899times119901 119899 is

the class number of the sample data 119901 is the number ofevaluation indexes and119864(119909

119895) and Var(119909

119895) stand for themean

and covariance of 119909119895 respectively

Step 2 Calculation of the correlation coefficient matrix R =(119903119894119895)119901times119901

where 119903119894119895(119894 119895 = 1 2 119901) is the correlation

coefficient of variables 119909119894and 119909

119895 119903119894119895= 119903119895119894 and the formula

is given by

119903119894119895=

sum

119899

119896=1

(119909119896119894minus 119909119894) (119909119896119895minus 119909119895)

radicsum

119899

119896=1

(119909119896119894minus 119909119894)2

sum

119899

119896=1

(119909119896119895minus 119909119895)

2

(2)

Step 3 (calculation of the eigenvalues and eigenvectors)Firstly deriving the characteristic values 120582

119894(119894 = 1 2 119901)

by solving the characteristic equation |120582119868 minus 119877| = 0 thenarranging them in order of size 120582

1ge 1205822ge sdot sdot sdot ge

119901ge 0

and finally calculating the eigenvectors 119890119894(119894 = 1 2 119901)

corresponding to the eigenvalues 120582119894

Step 4 Calculation of the principal component contributionrate and cumulative contribution rate The principal compo-nent contribution rate () is derived by

120578119894=

120582119894

sum

119901

119894=1

120582119894

times 100 (3)

and the cumulative contribution rate () is derived by

120578Σ(119896) =

119896

sum

119894=1

120578119894 (4)

The number of principle component is selected depend-ing on the cumulative contribution the cumulative contri-bution rate is usually set more than 85ndash90 and then thecorresponding first 119896main ingredients comprise most of theinformation provided by the119901 original variablesThenumberof principle component is 119896

Step 5 Calculation of the principle component load by

119897119894119895= radic120582

119895119890119894119895(119894 = 1 2 119901 119895 = 1 2 119896) (5)

Thus the principle component score is further obtainedby

119885 = (119911119894119895)119899times119896

(6)

3 The Principle of RVM

RVM is a learning algorithm derived from the theory ofBayesian learning algorithm which is based on the supportvector machine [12ndash14] This algorithm combines severaltheories such as the Markov chain the Bayesian theoremautomatic related decision prior and the maximum likeli-hood and thus it has the following advantages (1) the highsparsity (2) the shorter training timedue to only the setting ofthe kernel function (3) the flexible choosing of the the kernelfunction for it does not need to satisfy the Mercer condition

For a given training sample set 119909119899119873

119899=1 the output set

is defined as 119905119899119873

119899=1 and the RVM regression model can be

obtained by

119905119894=

119873

sum

119894=1

119908119894119870(119909 119909

119894) + 1199080+ 120598 (7)

where 120598 sim 119873(0 1205902

) is the independent sample error119908119894(119894 = 0 1 119873) is the weight coefficient119870(119909 119909

119894) is kernel

function and119873 is the sample quantity

Mathematical Problems in Engineering 3

For the independent output set the likelihood function ofthe whole sample is

119901 (119905 | 119908 1205902

) =

119873

prod

119894=1

(119905119894| 119910 (119909119894 119908) 120590

2

)

= (21205871205902

)

minus1198732

exp(minus119905 minus Φ (119909)1199082

21205902

)

(8)

where 119905 = (1199051 1199052 119905

119873) 119908 = [119908

0 1199081 119908

119873]T

Φ = [120601(1199091) 120601(1199092) 120601(119909

119873)]T 120601(119909

119873) =

[1 119870(1199091 119909119873) 119896(1199092 119909119873) 119870(119909

119899 119909119873)]T and (119905

119894|

119910(119909119894 119908) 120590

2

) is the Gaussian distribution functionWhen employing the maximum likelihood method to

solve119908 and1205902 directly it usually results in serious over-fittingproblem and in order to avoid this phenomenon 119908 is set asthe zero-meanGaussian prior distribution by using the sparseBayesian principle

119901 (119908 | 120572) =

119873

prod

119894=0

(119908119894| 0 120572minus1

119894) (9)

where 120572 is the corresponding hyper parameter of weight119908 ifevery weight is corresponding to one hyper parameter thenit can control the influences of the prior distribution on eachparameter and thus the sparse characteristics of the RVMcanbe realized

After defining the prior probability distribution and thelikelihood distribution according to the Bayesian theoremthe posterior probability distribution of all the unknownparameters can be derived by

119901 (119908 | 119905 120572 1205902

)

= (2120587)minus(119873+1)2

|Ψ|minus12 exp minus1

2

(119908 minus 120583)TΨ

T(119908 minus 120583)

(10)

where the posterior covariance matrix is Ψ =

(120590minus2

ΦTΦ + 119860)

minus1 120583 = 120590minus2

ΨΦT119905 and 119860 =

diag(1205720 1205721 120572

119873)

Finally the maximum likelihood method is used toestimate the hyper parameter 120572 and variance 1205902 Given thenew input sample 119909 the corresponding output probabilitydistribution will follow Gaussian distribution and the corre-sponding predicting value is derived by 119910 = 120583T120601(119909)

4 The Reliability Predicting Assessment

41 Selection of the Input Variables It is well known thatpower system reliability highly depend on the grid supplycapacity and the natural environment of the grid thereforein this paper some main factors reflecting supply capabilityof the grid itself such as the available efficient of the gridequipment the ratio of the power system reliability powersupply radius capacity-load ratio the loss of electricity as forpower rationing the unit of the new substation capacity andhigh temperature lightning strikes strong winds and heavy

rain which has a larger impact on the safe operation of powersystem are all taken as the input of the model The availableefficient of the grid equipment expresses the ratio of theavailable hours of major equipment to hours during surveyperiodThe ratio of the power system reliability stands for theratio of the power supply total hours to hours during surveyperiod Power supply radius means the physical distances ofpower supply circuit between power point and the farthestpoint of the power supply Capacity-load ratio expresses theratio of the city substation capacity to the correspondingload on the basis of which meet with the power supplyreliability The loss of electricity as for power rationing showsthe power supply loss value caused by power rationing duringsurvey period The unit new substation capacity illustratesthe corresponding values to the new substation capacity ofdifferent voltage levels when increasing the unit load duringsurvey period

42 Selection of theOutput Variables In the daily productionthere are two main aspects affecting the power systemreliability one is the fault power outages and the otheris the preliminary arrangement power It is known thatgrid system is made up of a large number of transmissionlines transformers switch equipment sorts of reactive powercompensation equipment and power points The final directrepresentation ismainly the grid blackout due to these factorson the grid Therefore in this paper the grid average faultpower failure time is taken as the output of the powerreliability predicting model

43 RVM Modeling and Structure Model The basic idea ofthe hybrid kernel function is briefly introduced in advancefirstly the principal component analysis is performed onthe original input variables then adopting RVM to modelthe new matrix and finally we get the result of predictingThe selection of kernel function of RVM does not needto satisfy Mercer condition therefore there have a certaindegree of freedom in the selection of the kernel function andintegrating the different characteristics of the kernel functionwe will get the nuclear function which has better properties

In this paper the linear combination of the RBF kernelfunction and polynomial function is chosen as the kernelfunction of RVM which can be derived by

119870(119909119894 119909119895) = 120582119866 (119909

119894 119909119895) + (1 minus 120582) 119875 (119909

119894 119909119895)

119866 (119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

1205902

)

119875 (119909119894 119909119895) = [(119909

119894sdot 119909119895) + 1]

2

(11)

where 119866(119909119894 119909119895) is the radial basis function 119875(119909

119894 119909119895) is the

binomial kernel function and 120582 is the weight of kernelfunction It is a single kernel function if 120582 = 0 or 120582 = 1 120590is the width of kernel function 120582 and 120590 are the parameters tobe optimized where the grid search algorithm is employed

Figure 1 is the structure diagram of the power supplyreliability predicting model The unit load new substation


Table 1 The Eigenvalue and principal component contribution

Principal component Eigenvalue Contribution () Cumulative contribution ()1 9458 8 4090 40902 7216 5 2625 67153 5795 5 1437 81524 3551 6 966 91185 1254 6 706 98246 0753 0 176 100007 0 0 10000

Table 2 Principal component

Number Category Component1198851

1198852

1198853

1198854

1198855

1 Overhead available coefficient of 220 kV 0100 7 minus0606 6 minus0074 7 minus0774 2 minus0037 02 Overhead available coefficient of 110 kV 0166 4 0709 5 0571 8 minus0068 7 0326 83 Transformer available coefficient of 220 kV minus0573 5 0494 5 0602 5 0150 2 0180 84 Transformer available coefficient of 110 kV minus0886 7 minus0011 0 0444 6 minus0088 3 minus0089 25 Circuit breaker available coefficient of 220 kV minus0876 3 0137 1 minus0437 6 0139 8 minus0020 66 Circuit breaker available coefficient of 110 kV 0125 5 0760 8 minus0606 0 minus0080 0 minus0160 57 Capacity-load ratio of 220 kV 0807 5 0017 5 minus0332 3 0331 2 0303 78 Capacity-load ratio of 110 kV 0871 4 0370 4 minus0050 7 0271 6 0158 59 Supply radius of 220 kV 0980 8 0026 6 0165 9 0084 0 minus0040 210 Supply radius of 110 kV minus0341 3 0508 3 minus0554 2 0200 5 0504 211 Substation capacity of 220 kV supply load per unit minus0612 8 0521 2 minus0532 1 0104 8 0173 512 Substation capacity of 110 kV supply load per unit 0323 5 0763 6 0149 8 0374 7 minus0369 513 Rural power supply stability minus0966 7 minus0170 4 0016 7 minus0142 2 minus0007 614 City power supply stability minus0978 6 minus0151 2 minus0019 2 minus0120 9 0012 715 The loss of switch out limit electric power 0024 4 0620 0 minus0027 2 0237 3 minus0746 116 Lightning days minus0324 9 minus0 426 4 0442 9 0697 4 0088 417 Rainstorm days 0284 7 minus0839 5 minus0445 8 0095 9 0078 118 Storm days 0340 6 minus0826 3 0174 2 minus0190 1 minus0146 419 High temperature days 0828 9 0066 5 0312 5 minus0413 3 0186 0

capacity power supply radius and capacity-load ratio aredivided into two levels of 220 kV and 110 kV The rate ofpower supply reliability is divided into rural and urbanThe available coefficients of the equipment are divided into220 kV and 110 kV overhead line available coefficient 220 kVand 110 kV transformer available coefficient and 220 kV and110 kV breaker available coefficient

5 Numerical Analysis

In this section a regional grid is taken as an example toillustrate the effectiveness of the proposed algorithm Thisregion mainly relies on the 500 kV and 220 kV grid thetransformer substations are mainly 220 kV and 110 kV andthere are more 220 kV and 110 kV lines The average faultpower failure time from year 1998 to year 2008 and 19influence factors of historical data are taken as the trainingsample and the actual data of year 2009 (true value is 1213465per house) is taken as the testing sample The power supply

reliability predicting model of this region can be establishedin the MATLAB environment

By using the PCA discussed in Section 3 for the originalinput sample X

12 times 19 the eigenvalues and principal compo-

nent contribution of principal component can be derived andthe results are provided in Table 1 It follows from Table 1that the cumulative contribution rate of the first five principalcomponents already achieves 9824 which is larger than95 that is the first five principal components can providesufficient information so these five irrelevant new variablescan replace the original nineteen variables Firstly we get theeigenvectors of the five new principal components and thencalculate the load of variables in the principal componentand the calculation results are given in Table 2

Finally based on the five principle components the RVMapproach is used to predict the new sample space and somepredicting results are shown in Table 3 It can be seen fromTable 3 that the predicting accuracy of RVMmethod is betterthan that used by the BP neural network and has a shortertraining time The new approach proposed combines the


Table 3 Prediction result analysis

Predicting method Predict valueshhousehold Relative error Variable dimension Training time (s)BP 12983 16 0069 9 19 412RVM 12742 68 0050 1 19 243PCA + RVM 12413 04 0022 9 5 124

Hig

h te

mpe

ratu

re d

ays

Stor

m d

ays

Rain

storm

day

s

Ligh

tnin

g da

ys

The l

oss o

f sw

itch

out l

imit

elec

tric

pow

er

Capa

city

-load

ratio

Supp

ly ra

dius

Pow

er su

pply

stab

ility

Equi

pmen

t ava

ilabi

lity

fact

or

Subs

tatio

n ca

paci

ty su

pply

load

per

unit

PCA

RVM model

Average annual outage time

Figure 1 Schematic diagram of power supply reliability structurebased on PCA and RVM

RVMmethod and the PCAmethod together which does notchange the structure of the sample data but can reduce thenumber of input variables and simplify the network structurethus the learning rate and performance of the network canbe improved due to the elimination of the correlation of thenetwork input factors therefore the predicting performancecan be improved greatly and can get a shorter training time

6 Conclusions

In this paper a novel power system reliability predictingmodel based on PCA and RVM has been proposed By usingthe PCAmethod a variety of factors affecting the reliability ofpower supply were analyzed and the new derived matrix hasbeen taken as the input of RVM for training and predictingThis lower-dimensional matrix could eliminate the redun-dant information and reduce the number of the dimensionsof the sample space thus the predicting accuracy of thenetwork could be greatly improved At last simulation resultson a regional grid were provided to show the usefulnessand effectiveness of the proposed predicting approach Theproposed approach not only provides a scientific referencefor improving power system reliability but also provides aneffective method for the grid investment modeling

Furthermore for testing the robustness of the proposedmethod it would be better to consider the impact of noise onprediction for future research [15ndash18]

Acknowledgments

This work was supported in part by the National Natural Sci-ence Foundation of China under Grants 51277052 51107032and 61104045 and in part by the Fundamental ResearchFunds for the Central Universities of China under Grant2012B03514

References

[1] Y M Liu and J Liang ldquoInfluence of planned interruption onpower supply reliability of distribution systemrdquo North ChinaElectric Power vol 2 pp 52ndash542 2011

[2] Y T Song Y J Guo and L Cheng ldquoStatistical anlysis ofreliability data for power system unitsrdquo Relay vol 30 no 7 pp14ndash16 2002

[3] Z Bie and XWang ldquoThe application ofMonte Carlomethod toreliability evaluation of power systemsrdquo Automation of ElectricPower Systems vol 21 no 6 pp 68ndash73 1997

[4] K G Wu and Z F Wu ldquoReliability evaluation algorithm ofelectrical power systems using sensitivity analysisrdquo Proceedingsof the CSEE vol 23 no 4 pp 53ndash56 2003

[5] K G Wu S Wang A B Zhang and J Q Zhou ldquoThe studyon reliability assessment of electrical power systems using RBFneural networkrdquoProceedings of the CSEE vol 20 no 6 pp 9ndash122000

[6] Y T Song J-L Wu D Peng et al ldquoA BP neural network basedmethod to predict supply reliability of urban power networkrdquoPower System Technology vol 10 no 32 pp 56ndash59 2008

[7] Q Gu A Luo JWang andM-J Qi ldquoA practicable algorithm toforecast and evaluate reliability of power supplyrdquo Power SystemTechnology vol 12 no 27 pp 76ndash79 2003

[8] Y Liu and C Singh ldquoReliability evaluation of composite powersystems using markov cut-set methodrdquo IEEE Transactions onPower Systems vol 25 no 2 pp 777ndash785 2010

[9] Z P Wang and X Wang ldquoInrush current recognition in powertransformer based on modified principal component analysisrdquoPower System Protection and Control vol 39 no 22 pp 1ndash62011

[10] B Liu and R Yang ldquoShort-term load forecasting model basedon LS-SVM with PCArdquo Electric Power Automation Equipmentvol 28 no 11 pp 13ndash17 2008

[11] J C Li D X Niu J Y Li and Z D Wang ldquoEvaluation of thepower grid development coordination based on the GRA andPCAmethodrdquo Power System Protection and Control vol 38 no18 pp 49ndash53 2010

[12] M E Tipping ldquoSparse Bayesian learning and the relevancevector machinerdquo Journal of Machine Learning Research vol 1no 3 pp 211ndash244 2001


[13] Z G SunW-X ZhaiW-L Li and Z-NWei ldquoShort-term loadforecasting based on EMD and RVMrdquo Proceedings of the CSU-EPSA vol 23 no 1 pp 92ndash97 2011

[14] G Y Lu C M Wu W M Sun X D Wang and X S CaildquoDetection and identification of voltage sags based on RVMandANFrdquo Power System Protection and Control vol 39 no 12 pp65ndash68 2011

[15] E Tian D Yue and C Peng ldquoQuantized output feedbackcontrol for networked control systemsrdquo Information Sciencesvol 178 no 12 pp 2734ndash2749 2008

[16] E Tian D Yue and C Peng ldquoReliable control for networkedcontrol systems with probabilistic actuator fault and randomdelaysrdquo Journal of the Franklin Institute vol 347 no 10 pp 1907ndash1926 2010

[17] Y Tang W Zou J Q Lu and J Kurth ldquoStochastic resonance inan ensemble of bistable systems under stable distribution noisesand heterogeneous couplingrdquo Physical Review E vol 85 ArticleID 046207 2012

[18] Y Tang H J Gao W Zou and J Kurths ldquoDistributed synchro-nization in networks of agent systems with nonlinearities andrandom switchingsrdquo IEEE Transactions on Systems Man andCybernetics vol 13 no 1 pp 358ndash370 2013

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cannot be used to predict the power system reliability Onthe other hand in practical applications if all the influencingfactors are taken as the inputs into the sample set it willinevitably lead to dimension expanding of the sample set andthen affecting the predicting accuracy and the generalizationcapability

Based on the above discussion this paper presents powersystem reliability predicting model based on relevance vectormachine (RVM) the principle component analysis (PCA)method is used to extract sample sets and then it is taken intothe RVM model after eliminating the correlation betweenvariables The proposed approach not only considers thefeature extracting ability of PCA but also takes advantageof the nonlinear approximating ability of RVM and thusthe predicting accuracy of the predicting model and thegeneralization ability can be improved greatly

The rest of this paper is organized as follows In Sections2 and 3 the theory of PCA and the principle of RVM areintroduced respectively In Section 4 the reliability pre-dicting model is proposed and the corresponding algorithmis presented In Section 4 an illustrative example about aregional grid is provided to illustrate the effectiveness ofthe developed results At last this paper completes with aconclusion

2 The Theory of PCA

In this section the theory of PCA is briefly introduced Itfirstly calculates the correlation matrix of the data matrixderived from many sample data then gets the accumulatedvariance contribution according to the eigenvalues of thecorrelation matrix and finally determines the principal com-ponents from the eigenvectors of the correlation matrix Themain results can be divided into the following five steps [9ndash11]

Step 1 (standardization of the original data) The main pur-pose is to eliminate the effects of different dimensions of theoriginal variables and large numerical difference that is

119909lowast

119895=

119909119895minus 119864 (119909

119895)

radicVar (119909119895)

(1)

where 119909119895is the 119895th column of the original data X

119899times119901 119899 is

the class number of the sample data 119901 is the number ofevaluation indexes and119864(119909

119895) and Var(119909

119895) stand for themean

and covariance of 119909119895 respectively

Step 2 Calculation of the correlation coefficient matrix R =(119903119894119895)119901times119901

where 119903119894119895(119894 119895 = 1 2 119901) is the correlation

coefficient of variables 119909119894and 119909

119895 119903119894119895= 119903119895119894 and the formula

is given by

119903119894119895=

sum

119899

119896=1

(119909119896119894minus 119909119894) (119909119896119895minus 119909119895)

radicsum

119899

119896=1

(119909119896119894minus 119909119894)2

sum

119899

119896=1

(119909119896119895minus 119909119895)

2

(2)

Step 3 (calculation of the eigenvalues and eigenvectors)Firstly deriving the characteristic values 120582

119894(119894 = 1 2 119901)

by solving the characteristic equation |120582119868 minus 119877| = 0 thenarranging them in order of size 120582

1ge 1205822ge sdot sdot sdot ge

119901ge 0

and finally calculating the eigenvectors 119890119894(119894 = 1 2 119901)

corresponding to the eigenvalues 120582119894

Step 4 Calculation of the principal component contributionrate and cumulative contribution rate The principal compo-nent contribution rate () is derived by

120578119894=

120582119894

sum

119901

119894=1

120582119894

times 100 (3)

and the cumulative contribution rate () is derived by

120578Σ(119896) =

119896

sum

119894=1

120578119894 (4)

The number of principle component is selected depend-ing on the cumulative contribution the cumulative contri-bution rate is usually set more than 85ndash90 and then thecorresponding first 119896main ingredients comprise most of theinformation provided by the119901 original variablesThenumberof principle component is 119896

Step 5 Calculation of the principle component load by

119897119894119895= radic120582

119895119890119894119895(119894 = 1 2 119901 119895 = 1 2 119896) (5)

Thus the principle component score is further obtainedby

119885 = (119911119894119895)119899times119896

(6)

3 The Principle of RVM

RVM is a learning algorithm derived from the theory ofBayesian learning algorithm which is based on the supportvector machine [12ndash14] This algorithm combines severaltheories such as the Markov chain the Bayesian theoremautomatic related decision prior and the maximum likeli-hood and thus it has the following advantages (1) the highsparsity (2) the shorter training timedue to only the setting ofthe kernel function (3) the flexible choosing of the the kernelfunction for it does not need to satisfy the Mercer condition

For a given training sample set 119909119899119873

119899=1 the output set

is defined as 119905119899119873

119899=1 and the RVM regression model can be

obtained by

119905119894=

119873

sum

119894=1

119908119894119870(119909 119909

119894) + 1199080+ 120598 (7)

where 120598 sim 119873(0 1205902

) is the independent sample error119908119894(119894 = 0 1 119873) is the weight coefficient119870(119909 119909

119894) is kernel

function and119873 is the sample quantity



119901 (119905 | 119908 1205902

) =

119873

prod

119894=1

(119905119894| 119910 (119909119894 119908) 120590

2

)

= (21205871205902

)

minus1198732

exp(minus119905 minus Φ (119909)1199082

21205902

)

(8)

where 119905 = (1199051 1199052 119905

119873) 119908 = [119908

0 1199081 119908

119873]T

Φ = [120601(1199091) 120601(1199092) 120601(119909

119873)]T 120601(119909

119873) =

[1 119870(1199091 119909119873) 119896(1199092 119909119873) 119870(119909

119899 119909119873)]T and (119905

119894|

119910(119909119894 119908) 120590

2



119901 (119908 | 120572) =

119873

prod

119894=0

(119908119894| 0 120572minus1

119894) (9)



119901 (119908 | 119905 120572 1205902

)

= (2120587)minus(119873+1)2


2

(119908 minus 120583)TΨ

T(119908 minus 120583)

(10)


(120590minus2

ΦTΦ + 119860)

minus1 120583 = 120590minus2

ΨΦT119905 and 119860 =

diag(1205720 1205721 120572

119873)








119870(119909119894 119909119895) = 120582119866 (119909

119894 119909119895) + (1 minus 120582) 119875 (119909

119894 119909119895)

119866 (119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

1205902

)

119875 (119909119894 119909119895) = [(119909

119894sdot 119909119895) + 1]

2

(11)


119894 119909119895) is the








1198852

1198853

1198854

1198855













Hig

h te

mpe

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Stor

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ays

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Ligh

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pow

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-load

ratio

Supp

ly ra

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Pow

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pply

stab

ility

Equi

pmen

t ava

ilabi

lity

fact

or

Subs

tatio

n ca

paci

ty su

pply

load

per

unit

PCA

RVM model




6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901 (119905 | 119908 1205902

) =

119873

prod

119894=1

(119905119894| 119910 (119909119894 119908) 120590

2

)

= (21205871205902

)

minus1198732

exp(minus119905 minus Φ (119909)1199082

21205902

)

(8)

where 119905 = (1199051 1199052 119905

119873) 119908 = [119908

0 1199081 119908

119873]T

Φ = [120601(1199091) 120601(1199092) 120601(119909

119873)]T 120601(119909

119873) =

[1 119870(1199091 119909119873) 119896(1199092 119909119873) 119870(119909

119899 119909119873)]T and (119905

119894|

119910(119909119894 119908) 120590

2



119901 (119908 | 120572) =

119873

prod

119894=0

(119908119894| 0 120572minus1

119894) (9)



119901 (119908 | 119905 120572 1205902

)

= (2120587)minus(119873+1)2


2

(119908 minus 120583)TΨ

T(119908 minus 120583)

(10)


(120590minus2

ΦTΦ + 119860)

minus1 120583 = 120590minus2

ΨΦT119905 and 119860 =

diag(1205720 1205721 120572

119873)








119870(119909119894 119909119895) = 120582119866 (119909

119894 119909119895) + (1 minus 120582) 119875 (119909

119894 119909119895)

119866 (119909119894 119909119895) = exp(minus

10038171003817100381710038171003817119909119894minus 119909119895

10038171003817100381710038171003817

2

1205902

)

119875 (119909119894 119909119895) = [(119909

119894sdot 119909119895) + 1]

2

(11)


119894 119909119895) is the








1198852

1198853

1198854

1198855













Hig

h te

mpe

ratu

re d

ays

Stor

m d

ays

Rain

storm

day

s

Ligh

tnin

g da

ys

The l

oss o

f sw

itch

out l

imit

elec

tric

pow

er

Capa

city

-load

ratio

Supp

ly ra

dius

Pow

er su

pply

stab

ility

Equi

pmen

t ava

ilabi

lity

fact

or

Subs

tatio

n ca

paci

ty su

pply

load

per

unit

PCA

RVM model




6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1198852

1198853

1198854

1198855













Hig

h te

mpe

ratu

re d

ays

Stor

m d

ays

Rain

storm

day

s

Ligh

tnin

g da

ys

The l

oss o

f sw

itch

out l

imit

elec

tric

pow

er

Capa

city

-load

ratio

Supp

ly ra

dius

Pow

er su

pply

stab

ility

Equi

pmen

t ava

ilabi

lity

fact

or

Subs

tatio

n ca

paci

ty su

pply

load

per

unit

PCA

RVM model




6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Hig

h te

mpe

ratu

re d

ays

Stor

m d

ays

Rain

storm

day

s

Ligh

tnin

g da

ys

The l

oss o

f sw

itch

out l

imit

elec

tric

pow

er

Capa

city

-load

ratio

Supp

ly ra

dius

Pow

er su

pply

stab

ility

Equi

pmen

t ava

ilabi

lity

fact

or

Subs

tatio

n ca

paci

ty su

pply

load

per

unit

PCA

RVM model




6 Conclusions



Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Novel Power System Reliability Predicting Model Based …downloads.hindawi.com/journals/mpe/2013/648250.pdf · 2019-07-31 · A Novel Power System Reliability Predicting