Research Article Multivariate Statistics and Supervised Learning …downloads.hindawi.com/journals/acisc/2016/3684238.pdf · 2019. 7. 30. · Prevalent anti-islanding methods include

Research ArticleMultivariate Statistics and Supervised Learning forPredictive Detection of Unintentional Islanding in Grid-TiedSolar PV Systems

Shashank Vyas1 Rajesh Kumar1 and Rajesh Kavasseri2

1Centre for Energy and Environment Malaviya National Institute of Technology Jawaharlal Nehru Marg Jaipur 302017 India2Department of Electrical and Computer Engineering North Dakota State University 1340 Administration Avenue FargoND 58102 USA

Correspondence should be addressed to Rajesh Kumar rkumareegmailcom

Received 28 October 2015 Accepted 6 March 2016

Academic Editor Francesco Carlo Morabito

Copyright copy 2016 Shashank Vyas 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

Integration of solar photovoltaic (PV) generation with power distribution networks leads to many operational challenges andcomplexities Unintentional islanding is one of them which is of rising concern given the steady increase in grid-connected PVpower This paper builds up on an exploratory study of unintentional islanding on a modeled radial feeder having large PVpenetration Dynamic simulations also run in real time resulted in exploration of unique potential causes of creation of accidentalislands The resulting voltage and current data underwent dimensionality reduction using principal component analysis (PCA)which formed the basis for the application of 119876 statistic control charts for detecting the anomalous currents that could island thesystem For reducing the false alarm rate of anomaly detection Kullback-Leibler (K-L) divergence was applied on the principalcomponent projections which concluded that 119876 statistic based approach alone is not reliable for detection of the symptoms liableto cause unintentional islanding The obtained data was labeled and a 119870-nearest neighbor (119870-NN) binomial classifier was thentrained for identification and classification of potential islanding precursors from other power system transients The three-phaseshort-circuit fault case was successfully identified as statistically different from islanding symptoms

1 Introduction

The distribution segment of the electricity supply network isalways under stress given the variable consumption patternsin complex geographical spread Since the power grid wasnot designed for the inclusion of sources in its distributionpathway the integration of distributed energy resources is atechnically complex issue When the sources are renewable-energy based their stochastic nature and the variability inavailability of generation induce extra randomness in thesystemoperationThedistributed generators (DGs) challengethe conventional functioning of the distribution feeders andlead to operational issues affecting the power quality stabilityand protection aspects [1]

Since most of the dispersed solar PV integration is takingplace on the distribution side [2] especially in the form ofrooftop systems the vulnerability of such a system to supply

interruptions and shutdowns becomes an important concernInterconnecting a solar PV system thus becomes challenginghowever the continued feeding of loads in the vicinity of thePV when the mains suddenly go off is a situation that mustbe avoided

Most of the DGs including PV inverters operate inconstant PQ control mode or constant power (active andreactive) control mode This means they are commanded togive output power in synchronism with the grid based onthe provided power set points PV is generally operated atunity power factor [3] because this is a strategy ofmaximizingthe energy yield from the array through maximum powerpoint tracking (MPPT) The inverter thus cannot adjust itsactive or reactive power output accordingly to regulate thegrid frequency and voltage

Islanding is said to occur when the DG continues pro-viding power (with utility level voltage and frequency) to

Hindawi Publishing CorporationApplied Computational Intelligence and So ComputingVolume 2016 Article ID 3684238 10 pageshttpdxdoiorg10115520163684238

2 Applied Computational Intelligence and Soft Computing

DG

Island

PCC

Load

Figure 1 Creation of a power island

a segment containing certain loads even after that portion ofthe network including the point of common coupling (PCC)gets disconnected from the main power system This is clearfrom a schematic diagram given in Figure 1 Based on thereasons for disconnection islanding can be categorized intointentional or unintentional When the distribution systemoperator knowingly separates certain sections from the maingrid with an intention to secure critical loads this practiceis called intentional islanding The move is preplanned andis generally required in situations of network congestion or alarge power system blackout On the other hand accidentalor unplanned disconnection of a loads-DG portion from themains and continuation of the DG in grid-connected modeof operation maintaining grid level voltage and frequencyare known as unintentional islanding The creation andmaintenance of an unplanned island are harmful to systemhealth Basically such an islanded network operates as anindependent autonomous entity without the regulation fromthe grid Since the DG continues to operate in constant PQcontrol mode it cannot adjust its supply to maintain voltageand frequency according to the loads hence leading to poorpower quality Sudden resumption of grid due to action ofautomatic reclosers can cause circulating currents to flow ifthe utility and the island are out of phase Loss of effectivegrounding in the islanded portion and transient overvoltagesare other issuesThere is also a constant safety threat to utilityrepair personnel due to a live portion existing on a deadpower network

Although internationally documented experiences ofunintentional islanding events are limited some real eventsnoted in [5 6] indicate the potential threat expected tointensify in the scenario of rising DG penetration A recentlypublished survey in [7] reaffirms the same concern amongdistribution utilities that feel that unintentional islanding willimpact their network the most after DG interconnection

Prevalent anti-islanding methods include classical meth-ods involving passive active and hybrid techniques Theselocal techniques either monitor the parameters around theDG and sense any threshold-exceeding changes in them todetect islanding (passive) or force the parameters out of thesafe range (active) or combine both strategies (hybrid) Theproblems of threshold selection causing large nondetectionzones (NDZs) in passive and power quality disturbance fromactive techniques impact their use in high DG penetrationMany computational intelligence (CI) based techniques havealso been reported in the literature [8] but they are funda-mentally based on classical techniques and seem to reinforcethe reactive strategy of detecting the island formation and

then disconnecting the DG This practice is not expected toremain in the future smart grid that will accommodate a largeshare of renewable-energy based DG power which cannot bewasted even for a few cycles Hence a predictive approach toislanding detection can prove to be a robust solution

Early works like [9] contemplated predicting PV inverterunintentional islanding in distribution grids considering thepredictable utility supply and the load and PV generationprofiles and utilizing analytical modeling of the early self-commutated inverters [10 11] More recently data miningtechniques were used on real [12] and simulated [13] phasormeasurement unit (PMU) data to predict islanding in bulktransmission networks Other works predicted the param-eters at which a possible island could form Traditionalconcepts of limit cycle behavior and small signal stabilityand describing function methods [14] and analysis of variousinverter control functions [15] were used

This paper describes the application of anomaly detectiontechniques multivariate statistical and supervised learningbased for predictive detection of an unintentional islandingevent The discovered anomalous currents occur before theislanding event on a modeled distribution feeder with a solarPV interconnection This is a highlight of the paper whichbegins with a description of the system model in Section 2Section 3 describes the anomalous precursors obtained fromthe dynamic simulations as part of the exploratory studySection 4 describes data reduction using PCA and applicationof 119876 statistics for detecting the islanding precursors fromother signals An improved detection accuracy obtainedusing K-L divergence is detailed in Section 5 Section 6describes labeling of the data points for training a 119870-NNclassifier and reports its performance for the same test datasets as used in the previous two sections Section 7 concludesthe paper

2 Power System Model

The distribution network modeled in this study is based onthe benchmark IEEE 13 node test radial feeder [16] and wasmodeled in MATLAB-Simulink Some modifications weremade to the original feeder model in order to carry outthe islanding studies as required First of all the substationautomatic voltage regulator (AVR) at node 650 was notincluded in the model This was done so as to avoid anypossible interaction of PV with the tap-changing controlsof the AVR [17] that could mask any signature related toislanding on the system A 1007 kWp solar PV array wasintegrated at node 692 through a three-phase inverter thusmaking this node the PCC These changes are visible in themodified feeder shown in Figure 2

Apart from the changes mentioned previously the con-stant current load at node 675was removed and the active andreactive power demands119875 and119876 respectively of the constantcurrent load at 692 were modified so as to make section 671-692 the islandable feeder sectionThe PV inverter operates onunity power factor and thus the loads on lateral 671-675 hadto be scaled according to the fixed PV penetration and feedercapacitor bank size for attaining 119875-119876 balance required for

Applied Computational Intelligence and Soft Computing 3

650

ACDC

PV inverter

v

646

611

645 632 633

684 671 692

680

634

675

652

Figure 2 The modified IEEE 13 node feeder

the exploratory islanding study described in the next sectionThe details of the system component modeling are given inour earlier work [18] from which the same model is takenhere This section only produces the results that describe thesystem model functioning and verification

To verify that the system performs according to theory itwas required to test the voltage and frequency at any point onsection 671-692 when it is islanded with the PV inverter Thevoltage and frequency in an unintentional island depend onthe mismatch of 119875 and119876 between the loads and the source(s)in that network Accordingly an island was forced to formby opening the islanding switch (circuit breaker in Figure 2)from 119905 = 045 s to 119905 = 048 s The solar irradiance in thiscase and for all cases discussed in this paper was kept fixedat 1000Wm2 in order to operate the PV array at standardtest conditions (STC) for a fixed rated output The MPPT isswitched on at 119905 = 040 s from the start of the simulation andreaches the MPP at 042 s after a few transients

The situation of 119875-119876 mismatch between the island loadsand sources was created for which the values were set asfollows 119875load = 90 kW 119876load = 151 kVAr 119875PV = 100 kW(effective three-phase AC inverter output) and 119876PV = 0 119876was supplied by the feeder capacitor bank and the inverterrsquosfilter circuit coming in the islanded network thus 119876supply =

119876capbank + 119876invfilterckt = 600 kVAr + 10 kVAr = 610 kVArThe island load values are for a single-phase load while thesupply values of 119875 and 119876 come from three phase sourcesThe voltage magnitude and frequency in the resulting islandmeasured at the PCC are plotted along with many otherquantities in Figure 3 The PV array continues operating inthe preprogrammed constant 119875-119876 control mode and hencethe undervoltage and underfrequency are evident in thefigure The low value of voltage magnitude and frequency isconsistent with the given amount of 119875-119876mismatch inside theisland [19]

The harmonics observable in the displayed voltage andcurrent are obvious when a PV system is integrated Howeverthe current harmonics shown are those in the grid-sidecurrent and not the inverter output The simulation runs themodel using a discrete solver which samples the voltagesand currents at the rate of 1MHz Similar kinds of waveshapeswere obtained fromfield tests carried out on amediumvoltage distribution feeder section in Spain [4] The resulting

PCC region voltage (V)

PCC region current (A)

PV array DC power output (kW)

Grid-side current in phase C of node 692 (A)

Solar irradiance fixed at 1000Wm2

minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501Time offset 0266 s (graph origin at 0266 s)





0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

Figure 3 Island parameters in 119875-119876 mismatch case

Three-phase voltage (V)

Three-phase current (A)

10msdivision

10msdivision

CH1ndashCH3 12500Adiv CH4 12500Adiv

Figure 4 Field-test results as in [4]

island three-phase voltages and currents are reproduced withpermission in Figure 4 Here overvoltages are observed inthe island The time scale of observations in the field-testresults is 10msdivision Some kinds of similarities in thesetwo figures provide some confidence about the modelingapproach although comparing simulation results with field-test results might not be completely appropriate

3 Exploration of Anomalous Precursors toUnintentional Islanding

The level of 119875-119876 mismatch between the loads and the PVinverter has a close associationwith island formation and sus-tenance as was seen in the previous section Many practicalstudies documented in [20 21] have characterized invertersrsquo


islanding behavior for different levels of 119875-119876 mismatchHowever in the expected scenario of rising PV penetrationlevels on distribution feeders complete 119875-119876 match is not aremote possibility Internationally accepted reports like [22]have acknowledged that 119875-119876 balance between PV inverterand loads is a quantifiable possibility Field tests in [5] andlaboratory tests in [23] have studied the invertersrsquo anti-islanding capabilities for complete match Also a case studydone for India [24] has estimated the risk of unintentionalislanding in a spot distribution network based on the numberof hours for which such a condition occurs These docu-mented practices have highlighted the significance of powermismatch to islanding but after the occurrence of the eventThis study explores the impact of complete 119875-119876 match caseon the possibility of building up of an imminent islandingcondition

Such dynamic load-PV interactions on a high PV pene-tration radial feeder for different grid conditions can throwup interesting results related to system islanding In anattempt to explore such patterns two types of disturbanceswere programmed to occur from the substation undervolt-age and overvoltage in concurrence with exact 119875-119876 balanceThese two types of disturbances are commonly used inpractice for islanding related studies [25] In either case thefollowing values were set to implement 119875-119876 match betweenthe 1-phase load and the PV and 119876 sources on the islandablefeeder section 119875load = 2333 kW and 119876load = 20333 kVAr A10 kW resistor in parallel with this loadmodel forces the exact119875 match as 119875 and 119876 demands of the inverter filter circuit arenegligible The values of 119875PV and 119876supply remain the same asbefore

An undervoltage disturbance of 07 per unit (pu) of thenominal voltage amplitude was forced from the grid side fora period of 30ms from 119905 = 045 s to 119905 = 048 s The grid-side current flowing in phase C of section 671-692 has manyanomalous peaks during the voltage disturbance and after thedisturbance ends as shown in Figure 5 The 30ms windowduring and after the disturbance is important for this studyfrom the perspective of data extraction The circled peak isan anomaly whose severity to cause islanding is explainedand verified in [26] which also verifies the occurrence of asimilar peak for the same set of conditions on a single-phasesingle bus system implemented in emulators and relatedhardware An overvoltage disturbance symmetrically 13 puof the nominal voltage amplitude was programmed to occurfrom the grid side for 30ms from 119905 = 045 s to 048 s Theresulting grid-side anomalous current shown in Figure 6 wasnot proved to be severe enough to lead to section islandingThe complete system was remodeled and simulated for 05 sin real time on a real-time digital simulator (RTDS) for bothdisturbance cases This was to validate the model robustnessand the real-time results similar to Simulink results are shownin Figures 7 and 8 for both disturbance cases respectivelyThelarge peaks are not noteworthy since they are due to initial PVintegration transientsThe 30mswindow of each disturbancecase contains data points sampled at 1MHz The voltage andcurrent samples of phase C taken around the PCC collectedin this window are used as one of the data sets in each of






minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501


Figure 5 Anomalous peak as islanding precursor

the different statistical and CI techniques discussed in thenext three sections

A 30ms window after each disturbance ends is alsocaptured during the simulation run For the case of Figure 7previously this window contains data points correspondingto the anomalous current liable to cause islanding and henceis used as a part of the composite training data set used forthe119870-NN classifier training More details about data sets andapplications follow in the next section

4 Data Handling and AnomalyDetection Using PCA

The simulation for each of the two grid disturbances wasrun for 05 seconds at a sampling rate of 1MHz as discussedaboveThe data of the voltage and current samplesmentionedabove was collected and bifurcated into different data setsaccording to three conditions found in each of the two casesnamely normal during disturbance and after disturbanceThe normal condition was common for both the disturbancecase simulations and contained 30484 samples The normalcondition corresponds to that where no extra disturbancesapart from the PV induced harmonics are present in thesystem

PCAwas used to reduce the dimensionality of the data setcorresponding to the normal system operation case as voltageand current are correlated quantities The standard singularvalue decomposition (SVD) approach was used in MATLABto find the principal componentmatrixThe two-dimensional







minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

Figure 6 Overvoltage with 119875-119876 match

times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)

Figure 7 RTDS result for undervoltage with 119875-119876 match

data resulted in two principal components (PCs) Based onthe variance of the projections onto the two PCs the 1st PCwas retained for all analysesThe latent matrix containing thevalues of variances onto the 2 PCs called the scores is shownin Table 1 and makes the choice of PC selection clear

The PCA model created was used for detecting anyabnormal occurrence using statistical process control strategyfor anomaly detection The data belonging to different casessimulated was preprocessed and projected onto the 1st PCof the reference PCA model The aim of this strategy is

times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)

Figure 8 RTDS result for overvoltage with 119875-119876 match

Table 1 Latent values

On PC 1 On PC 2537 times 10

6 4866

to differentiate a condition that can cause unintentionalislanding on the modeled feeder from conditions like faultsand other transients which appear close to islanding andthus are tricky to detect and identify correctly The existingliterature describes techniques that detect an islanding con-dition among other transients like surges load and capacitorswitching and faults after the island has been formed Thisstudy initiates efforts towards exploring possible practicalcauses of the event and detecting such conditions from theones that appear close enough to fool the inverterHence onlythe four cases resulting from the two grid-side disturbancesas described previously and a 3-phase short-circuit fault casehave been simulated

Case 1 is the normal system operation case which hasbeen described previously Each of the grid-side undervoltageand overvoltage disturbance conditions gives rise to twocases A three-phase line-to-line-to-line-to-ground (L-L-L-G) fault at the PCC is designated as case 4 All the event basedcases simulated are summarized in Table 2

For 119899 number of data sample vectors 119909 isin 119877119898 stacked

above one another to form a data matrix 119883119899times119898 application

of PCA on 119883 leads to a 119901 times 119901 coefficient matrix 119875 If an119903 le 119898 number of PCs are retained based on latent valuesthen119883 can be resolved as a PCAmodel and a residual modelas 119883 = 119883pca + 119883res The projection onto the PC or loadingmatrix leads to formation of a scorematrix119879

119899times119898The originaldata matrix119883 can be reconstructed using score matrices 119879pcaand 119879res and loadings 119875pca and 119875res as 119883 = 119879pca119875

119879

pca + 119879res119875119879

reswhere 119875res and 119879res are of 119899 times 119898 minus 119903 dimension [27]

The statistical process control method is widely usedin industrial engineering for quality control purposes Ithas found other applications in many domains for outlierdetection by checking whether the process variables are in


Table 2 Cases simulated

Case Number of samples EventCase 1 30484 Normal

Case 2 30584 UV + 119875-119876 match(during disturbance)

Case 3 30622 UV + 119875-119876 match(after disturbance)

Case 4 30585 L-L-L-G fault at PCC

Case 5 30656 OV + 119875-119876 match(during disturbance)

Case 6 30644 OV + 119875-119876 match(after disturbance)

control or not Any process variable is indicated as out ofcontrol when a certain statistic associated with it crosses itsupper limit PCA has two multivariate statistics associatedwith it Hotellingrsquos 119879

2 statistic and 119876 statistic Both have anupper control limit (UCL) defined and when both of themare crossed by the corresponding statistics of a data point ordata set this indicates an anomalous and abnormal behavior

Hotellingrsquos 1198792 statistic is a multivariate distance for a set

of data points from a target value indicating variance insidethe PCA model If 119891 is a mean-centered (scaled) sample datavector then 119905pca = 119891119875pca is a score vector The 119879

2 statisticfor 119891 is defined as 119879

2= 1199051015840and 119905 where and is a diagonal matrix

having 119903 eigenvalues of datamatrix119865119899times119898 for 119903 le 119898 number of

retained PCs The UCL for the statistic is defined as 1198792120572 If all

data points are linear and normally distributed 1198792120572follows an

119865 distribution and is given as 1198792

120572= 119903((119899

2minus 1)119899(119899 minus 119903))119865

119903119899minus119903

at a given level of confidence 120572The 119876 statistic is a measure of deviation of the original

data points from the projection onto the PC axes Henceit measures variance among data points inside the residualsubspaceThe119876 statistic is calculated using residuals and fora residual vector 119890 of a scaled sample vector 119891 119876 statistic isgiven as 119876 = 119890

119879119890 = 119891

119879(119868 minus 119875pca119875

119879

pca) where 119868 is an identitymatrix For normally distributed linear data points the 119876

statistic follows a central 1205942 distribution and its UCL is givenas119876120572= (12059022120583)times120594

2(212058321205902) at a given level of confidence 120572

Here 120583 and 1205902 are the mean and variance of the 119876 statistic

Recently PCA based process control strategy has beenapplied for detecting the occurrence of islanding and distin-guishing it from several nonislanding events PMU record-ings of frequency measurements on 6 different sites in theUK power grid were used as reference data for implementing1198792 and 119876 statistic based islanding detection in [28] The

occurrence of an islanding situation was evident only when119876120572was crossed in addition to the crossing of 119879

2

120572by the

corresponding multivariate statistics for a test event data setSince the power system is a dynamically changing system thesystem variables used for creating the reference PCA modelchange dynamically causing it to change with time also Totackle this issue a recursive PCA algorithm was developedin [29] for the same UK power system case The referencePCA model was updated in every iteration and the detection

Table 3 Event detection results using 119876 statistic

Case 119876 119876 gt 119876120572

Islanding precursor2 157 times 10

7 No No3 405 times 10

7 Yes Yes4 123 times 10

9 Yes No5 857 times 10


7 Yes No

results for abnormal transients verified its effectiveness overthe simple PCA approach This study has made use of theusual SVD for creating the reference PCA model since thereference data does not change from one event to anotheras the simulation has been performed for fixed settings toobserve some unique changes that occur in fixed windows asdescribed previously

Each new test data set 119883samplestimes2 underwent scaling to

make the mean along the columns zero The mean-centereddata set 119883mc was projected onto the 1st PC of the referencePCA model by 119883mc times 119875pca Correspondingly the 119879

2 and 119876

statistics were calculated Each of the remaining 5 cases wasused as the test case Since crossing of the 119879

2

120572limit for the

reference case by the1198792 statistic of any test case indicates onlya faulty or out-of-control event the119876 statistic was used as theonly parameter for detection 119876 statistic measures deviationinside the residual subspace and hence is a strong indicatorof any abnormal or anomalous condition

Following the same the 5 test cases were subjected tomean-centering as before and were projected onto the 1st PCof the reference PCA model The 119876 statistics for each caseprojected data matrix were found and compared with theUCL 119876

120572of the reference case score 119876

120572at 98 confidence

level was calculated to be = 3846 times 107 The results of this

multivariate statistics based detection are given in Table 3As seen in Table 3 this approach identifies the anomalous

case correctly It also identifies the disturbance event in case2 correctly as not an anomaly that can island the systemHowever cases 4 5 and 6 are incorrectly identified Thisshows that the 119876 statistic based statistical process controlapproach is not completely reliable for detecting the anoma-lous currents that can lead to islanding on the system Toimprove upon the false detection rate the Kullback-Leibler(K-L) divergence based approach using the PCA model ispresented in the next section

5 K-L Divergence Based Detection

K-L divergence also known as relative entropy is an impor-tant statistical measure coming from information theory Ithas shown a great potential for application in fault detectionand diagnosis (FDD) It has been aptly used for incipientfault detection in mechanical and electrical systems in [30]and has also been widely used in multimedia security andneuroscience However the application of K-L divergence inislanding detection related studies could not be confirmed inthe literature This section details the use of K-L divergence


involving the PCA model for improved accuracy of anomalydetection

K-L divergence is basically a measure of dissimilaritybetween two probability distributions If two data samples aredrawn from two populations having the same distributiontheir K-L divergence will be zero For two continuous prob-ability density functions (PDFs) 119891(119909) and 119892(119909) of a randomvariable 119909 the K-L Information (KLI) is defined as 119868(119891 119892) =

int119891(119909) log(119891(119909)119892(119909))119889119909TheK-L divergence is then given asK-LD(119891 119892) = 119868(119891 119892) + 119868(119892 119891) a symmetric operation ofKLI For discrete distributions K-LD is defined as the meanvalue of the log-likelihood ratio of the two distributions

For an anomalous behavior or a sudden change in aprocess the PDF of the corresponding data set changes fromthe reference case and if it goes beyond the safe threshold120598 it can be statistically detected For two normal (Gaussian)probability densities 119891 and 119892 having means and variances as1205831 1205832and 120590

2

1 12059022 respectively the K-L divergence between

them can be given by a simple expression

K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)

In our case we find the divergence between two distributionsprojection of different event (test) cases onto the 1st PCand the reference PCA score or projection of case 1 onthe 1st PC Nonparametric kernel-density estimation hasbeen used to approximate each of these two distributionsas normal distributions graphically Since mean-centering ofdata samples is a part of PCA the means of the projectionsfor both test cases and reference case are zero Since the PCscores are linear combinations of the original data samplesthey are assumed to be fairly normally distributed [31]Taking this assumption into consideration we have used thefollowing formula to calculate K-LD between a test case andthe reference case

K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)

Here (120583ref minus 120583test)2= 0 and 120590

2

ref is nothing but the varianceof projection on the 1st PC which is in the first column ofTable 1The other variances are those of the projections of thedifferent test cases onto the 1st PC

Using (2) test cases 2 to 6 were used as the seconddistribution and case 1 was taken as the reference distributionThe values of K-L divergence calculated for different casesare given in Table 4 The results from Table 4 throw animportant picture All those cases which had 119876 gt 119876

120572and

were wrongly detected seem to have been differentiated bytheir K-L divergence values It can be seen clearly that cases4 5 and 6 do not fall in the same category as had beenpreviously clubbed by the119876 statistic approachThe extremelylarge and small values of cases 4 and 6 respectively segregatethem into different category of events however the similarorders of values for case 3 and case 6 do not give a clearboundary

The kernel-density estimated normal PDFs for cases 34 and 6 and their divergence from that of case 1 are shown

Table 4 Event detection results using K-LD

Case 1205902 K-LD Islanding precursor

2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4

Score case 1Score case 3

0 20001000 3000minus2000minus3000 minus1000

Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity

Figure 9 Divergence case 3 versus case 1

in Figures 9 10 and 11 respectively The more the K-LDthe more the gap between the densities The L-L-L-G faultcase has the least variance and hence it has the largest K-L divergence among all cases Physically this event createssuch low voltages for a given short-circuit capacity of thefeeder that the PV inverter itself trips thus avoiding islandingand this fact is brought out by its large divergence from thereference case PDF However after looking at the divergencevalues of case 3 and case 6 setting the correct 120598 for an event tobe identified as the anomalous islanding precursor seems tobe the problem with this approach although the false alarmdetection rate has reduced to 15 from 35 in the previoussection To tackle this issue of threshold selection amachine-learning based approach to detect anomalous events correctlyhas been presented in the next section

6 119870-NN Classifier Based Approach

Moving from the statistical techniques presented in theprevious sections this section describes application of asupervised learning technique called 119870-nearest neighbors(119870-NN) classification 119870-NN methods are instance basedlearningmethods that classify a new test instance based on itssimilarity to the training data points stored Such techniquesare also called lazy-learning techniques because they do notcreate a model for classification of test data but rather they



0

002

004

006

008

01

012

014

Den

sity

0 1000 2000 3000minus1000minus2000minus3000

Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity

0 20001000 3000 4000minus3000 minus1000minus2000

Data


carry out classification only when a new instance comes upAs a result they look into the database of training data pointssimilar to the test point based on some distance measureand assign a class label to the new point usually based on amajority vote among the training points in its neighborhood[32] The algorithm used here is described as follows

Training Algorithm (i) Add each training example (119909 119891(119909))

to the list of training examples Here119891(119909) is a binary encodedindicator response variable storing the class label for 119909

Classification Algorithm Given a query point 119909119902 to be classi-fied

(i) let 1199091 1199092 119909

119896be the 119896 training instances nearest to

119909119902based on a distance measure

(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)

For this study this algorithm was selected because theanomalous instances liable to cause islanding in case 3 wereall consecutively located in the data set Two class labels wereused for binary classification of the test data points Class label0 = all data points isin set of cases that cannot cause islandingwhile class label 1 = all data points isin set of cases that cancause islanding Physically this corresponds to all currents 119868 gt

01 kA as decided from the static fault study explained in [26]Three training data sets were used as a composite trainingdata set

First 20000 Points from Case 1 All Data Points from Case 2and 555 Anomalous Data Points from Case 3 The number ofneighbors 119896 was set = 5 and Euclidean distance measure wasusedThe classifier was trained and tested with three test datasets

Test set I last 10484 points of case 1Test set II data points isin case 4Test set III data points isin case 6

The average cross-validated classification error or 10-foldloss on the training data was 00070 indicating high trainingaccuracyThe classifier performancewas tested for each of thetest cases using the majority vote with nearest point tie-breakrule For test set II the classifier identified all data points toisin class 0 This pertains to a 100 accuracy in this case Thevariance of data points in case of the 3-120601 fault is the leastamong all cases Fault also causes very low voltagewhich itselfcan trip a PV inverter and thus it is detected naturally andcannot be labeled as an anomalous precursor This confirmsthe correctness of the classifier in assigning label 0 to thiscase The classifier accuracy for test sets I and III was foundto be 9742 and 9012 respectively aftermultiple runsTheconfusion matrices for test sets I and II are shown in Tables5 and 6 respectively Case 6 comes very close to the case ofactual islanding precursors discovered in case 3 and hence alarge number of data points were assigned label 1The averageclassifier accuracy can be reported as 9575 The classifiertakes an average time of 294ms in classifying a new test datapoint As 119896 was reduced till 1 the time taken remained thesame but the accuracy improved even for testing on the thirdtraining data set 3 Clearly in this approach also the three-phase short-circuit fault case is identified to be different fromall other cases with 100 accuracy

A comparison of the performances of the three methodsdiscussed in this paper is presented in Table 7

7 Conclusions

This paper has contributed an exploratory study towardsunderstanding discovering and analyzing the possible rea-sons that can unintentionally island a modified IEEE 13 bus


Table 5 Confusion Matrix I

Class 0 Class 1Class 0 10214 270Class 1 0 0

Table 6 Confusion Matrix III


Table 7 Performance comparison of the three methods

Method Accuracy Issues119876 statistic based 40 High false alarm rateK-L divergence 80 Setting correct 120598119870-NN classifier 9575 119896 and time trade-off

system with large PV penetration on a segment The appli-cation of multivariate statistical techniques and a supervisedlearning technique to identify anomalous signatures liable tocause islanding from other transients has been detailed Thethree-phase short-circuit fault is clearly identified as not anislanding precursor case by both the K-L divergence and 119870-NN classification methods The paper also shows that usingPCA based process control strategy alone is not sufficientfor predictive islanding detection The classifier gives thebest accuracy among all and it can be concluded that itsimplementation should detect the precursor and trip the PVinverter before the utility PCC relay The classification timecan be equal to the relay delay if not less than it

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors acknowledge the Ministry of New and Renew-able Energy of the Government of India for supportingthis work under the National Renewable Energy FellowshipScheme The support provided by Dr Deepak Fulwani andMr Vinod Kumar of the Indian Institute of TechnologyJodhpur for helping with the RTDS study is equally acknowl-edged

References

[1] R C Dugan and T E McDermott ldquoDistributed generationrdquoIEEE Industry Applications Magazine vol 8 no 2 pp 19ndash252002

[2] L Mihalache S Suresh Y Xue and M Manjrekar ldquoModelingof a small distribution grid with intermittent energy resourcesusing MATLABSIMULINKrdquo in Proceedings of the IEEE Powerand Energy Society General Meeting pp 1ndash8 San Diego CalifUSA July 2011

[3] C Greacen R Engel and T Quetchenbach ldquoA guidebook oninterconnection and islanded operation of mini grid powersystems up to 200 kWrdquo Tech Rep Lawrence Berkeley NationalLaboratory Berkeley Calif USA 2013

[4] F J Pazos ldquoPower frequency overvoltages generated by solarplantsrdquo in Proceedings of the 20th International Conference onElectricity Distribution (CIRED rsquo09) pp 159ndash159 Prague CzechRepublic June 2009

[5] F J Pazos ldquoOperational experience and field tests on islandingevents caused by large photovolatic plantsrdquo in Proceedings of the21st International Conference on Electricity Distribution (CIREDrsquo11) Frankfurt Germany June 2011

[6] C Li J Savulak and R Reinmuller ldquoUnintentional islandingof distributed generation-Operating experiences from naturallyoccurred eventsrdquo IEEE Transactions on Power Delivery vol 29no 1 pp 269ndash274 2014

[7] R F Arritt and R C Dugan ldquoReview of the impacts of dis-tributed generation on distribution protectionrdquo in Proceedingsof the 59th IEEE Annual Conference on Rural Electric PowerConference (REPC rsquo15) pp 69ndash74 Asheville NC USA April2015

[8] J A Laghari H Mokhlis M Karimi A H A Bakar and HMohamad ldquoComputational intelligence based techniques forislanding detection of distributed generation in distributionnetwork a reviewrdquo Energy Conversion andManagement vol 88pp 139ndash152 2014

[9] G J Vachtsevanos andH Kang ldquoSimulation studies of islandedbehavior of grid-connected photovoltaic systemsrdquo IEEE Trans-actions on Energy Conversion vol 4 no 2 pp 177ndash183 1989

[10] J Stevens R Bonn J Ginn and S Gonzalez ldquoDevelop-ment and testing of an approach to anti-islanding in utility-interconnected photovoltaic systemsrdquo Tech Rep SAND 2000-1939 Sandia National Laboratories 2000

[11] M A Eltawil and Z Zhao ldquoGrid-connected photovoltaicpower systems technical and potential problemsmdasha reviewrdquoRenewable and Sustainable Energy Reviews vol 14 no 1 pp 112ndash129 2010

[12] V Vittal T Merho M Kezunovic et al ldquoData mining to char-acterize signatures of impending system events or performancefromPMUmeasurementsrdquo Tech Rep Arizona StateUniversityand Texas AampM University 2013

[13] Z Pakdel Intelligent instability detection for islanding prediction[PhD dissertation] Virginia Polytechnic Institute and StateUniversity 2011

[14] S J Ranade N R Prasad S Omick and L F Kazda ldquoA study ofislanding in utilityndashconnected residential photovoltaic systemspart I-models and analytical methodsrdquo IEEE Transactions onEnergy Conversion vol 4 no 3 pp 436ndash445 1989

[15] G A Smith P A Onions and D G Infield ldquoPredicting island-ing operation of grid connected PV invertersrdquo IEE ProceedingsElectric Power Applications vol 147 no 1 pp 1ndash6 2000

[16] W Kersting ldquoRadial distribution test feedersrdquo in Proceedings ofthe IEEE Power Engineering Society Winter Meeting vol 2 pp908ndash912 February 2001

[17] M S Elnozahy and M M A Salama ldquoTechnical impacts ofgrid-connected photovoltaic systems on electrical networksmdasha reviewrdquo Journal of Renewable and Sustainable Energy vol 5no 3 Article ID 032702 2013

[18] S Vyas R Kumar and R Kavasseri ldquoObservations from studyof pre-islanding behaviour in a solar PV system connectedto a distribution networkrdquo in Proceedings of 4th International


Conference on Advances in Computing Communications andInformatics (ICACCI rsquo15) pp 645ndash650 Kochi India August2015

[19] T Tran-Quoc TM C Le C Kieny and S Bacha ldquoBehaviour ofgrid-connected photovoltaic inverters in islanding operationrdquoin Proceedings of the IEEE PES TrondheimPowerTechThe Powerof Technology for a Sustainable Society (POWERTECH rsquo11) pp1ndash8 Trondheim Norway June 2011

[20] M Dymond ldquoPractical results from islanding tests on the20 kW Kalbarri PV systemrdquo in Proceedings of the Solar rsquo97ldquoSustainable Energyrdquo 35th Annual Conference of the Australianand New Zealand Solar Energy Society December 1997

[21] M Ross C Abbey Y Brissette and G Joos ldquoPhotovoltaicinverter characterization testing on a physical distributionsystemrdquo in Proceedings of the IEEE Power and Energy SocietyGeneral Meeting (PES rsquo12) pp 1ndash7 San Diego Calif USA July2012

[22] B Verhoeven ldquoProbability of islanding in utility neyworks dueto grid connected photovoltaic power systemsrdquo IEA Tech RepIEA-PVPS T5-072002 2002

[23] R J Bravo S A Robles and E Muljadi ldquoAssessing solar PVinvertersrsquo anti-islanding protectionrdquo in Proceedings of the 40thIEEE Photovoltaic Specialist Conference (PVSC rsquo14) pp 2668ndash2671 IEEE Denver Colo USA June 2014

[24] K A Joshi and N M Pindoriya ldquoRisk assessment of uninten-tional islanding in a spot network with roof-top photovoltaicsystemmdasha case study in Indiardquo in Proceedings of the IEEEInnovative Smart Grid TechnologiesmdashAsia (ISGT Asia rsquo13) pp1ndash6 Bangalore India November 2013

[25] S Srivastava K S Meera R Aradhya and S Verma ldquoPerfor-mance evaluation of composite islanding and loadmanagementsystem using real time digital simulatorrdquo in Proceedings of15th National Power Systems Conference pp 242ndash247 MumbaiIndia 2008

[26] S Vyas R Kumar and R Kavasseri ldquoExploration and inves-tigation of potential precursors to unintentional islanding ingrid-interfaced solar photo voltaic systemsrdquo in Proceedingsof the IEEE International Conference on Energy Systems andApplications (ICESA rsquo15) pp 579ndash584 Pune India October2015

[27] J Tang W Yu T Chai and L Zhao ldquoOn-line principalcomponent analysis with application to process modelingrdquoNeurocomputing vol 82 no 2012 pp 167ndash178 2012

[28] X Liu D M Laverty R J Best K Li D J Morrow and SMcLoone ldquoPrincipal component analysis of wide-area phasormeasurements for islanding detectionmdasha geometric viewrdquo IEEETransactions on Power Delivery vol 30 no 2 pp 976ndash985 2015

[29] Y Guo K Li D M Laverty and Y Xue ldquoSynchrophasor-based islanding detection for distributed generation systemsusing systematic principal component analysis approachesrdquoIEEE Transactions on Power Delivery vol 30 no 6 pp 2544ndash2552 2015

[30] J Harmouche Statistical incipient fault detection and diagnosiswith kullback-leibler divergence from theory to applications[PhD thesis] Supelec 2014

[31] A Youssef C Delpha and D Diallo ldquoPerformances theoreticalmodel-based optimization for incipient fault detection withKL Divergencerdquo in Proceedings of the 22nd European SignalProcessing Conference (EUSIPCO rsquo14) pp 466ndash470 September2014

[32] T M MitchellMachine Learning McGraw Hill New York NYUSA 1997

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



DG

Island

PCC

Load

Figure 1 Creation of a power island

a segment containing certain loads even after that portion ofthe network including the point of common coupling (PCC)gets disconnected from the main power system This is clearfrom a schematic diagram given in Figure 1 Based on thereasons for disconnection islanding can be categorized intointentional or unintentional When the distribution systemoperator knowingly separates certain sections from the maingrid with an intention to secure critical loads this practiceis called intentional islanding The move is preplanned andis generally required in situations of network congestion or alarge power system blackout On the other hand accidentalor unplanned disconnection of a loads-DG portion from themains and continuation of the DG in grid-connected modeof operation maintaining grid level voltage and frequencyare known as unintentional islanding The creation andmaintenance of an unplanned island are harmful to systemhealth Basically such an islanded network operates as anindependent autonomous entity without the regulation fromthe grid Since the DG continues to operate in constant PQcontrol mode it cannot adjust its supply to maintain voltageand frequency according to the loads hence leading to poorpower quality Sudden resumption of grid due to action ofautomatic reclosers can cause circulating currents to flow ifthe utility and the island are out of phase Loss of effectivegrounding in the islanded portion and transient overvoltagesare other issuesThere is also a constant safety threat to utilityrepair personnel due to a live portion existing on a deadpower network

Although internationally documented experiences ofunintentional islanding events are limited some real eventsnoted in [5 6] indicate the potential threat expected tointensify in the scenario of rising DG penetration A recentlypublished survey in [7] reaffirms the same concern amongdistribution utilities that feel that unintentional islanding willimpact their network the most after DG interconnection

Prevalent anti-islanding methods include classical meth-ods involving passive active and hybrid techniques Theselocal techniques either monitor the parameters around theDG and sense any threshold-exceeding changes in them todetect islanding (passive) or force the parameters out of thesafe range (active) or combine both strategies (hybrid) Theproblems of threshold selection causing large nondetectionzones (NDZs) in passive and power quality disturbance fromactive techniques impact their use in high DG penetrationMany computational intelligence (CI) based techniques havealso been reported in the literature [8] but they are funda-mentally based on classical techniques and seem to reinforcethe reactive strategy of detecting the island formation and

then disconnecting the DG This practice is not expected toremain in the future smart grid that will accommodate a largeshare of renewable-energy based DG power which cannot bewasted even for a few cycles Hence a predictive approach toislanding detection can prove to be a robust solution

Early works like [9] contemplated predicting PV inverterunintentional islanding in distribution grids considering thepredictable utility supply and the load and PV generationprofiles and utilizing analytical modeling of the early self-commutated inverters [10 11] More recently data miningtechniques were used on real [12] and simulated [13] phasormeasurement unit (PMU) data to predict islanding in bulktransmission networks Other works predicted the param-eters at which a possible island could form Traditionalconcepts of limit cycle behavior and small signal stabilityand describing function methods [14] and analysis of variousinverter control functions [15] were used

This paper describes the application of anomaly detectiontechniques multivariate statistical and supervised learningbased for predictive detection of an unintentional islandingevent The discovered anomalous currents occur before theislanding event on a modeled distribution feeder with a solarPV interconnection This is a highlight of the paper whichbegins with a description of the system model in Section 2Section 3 describes the anomalous precursors obtained fromthe dynamic simulations as part of the exploratory studySection 4 describes data reduction using PCA and applicationof 119876 statistics for detecting the islanding precursors fromother signals An improved detection accuracy obtainedusing K-L divergence is detailed in Section 5 Section 6describes labeling of the data points for training a 119870-NNclassifier and reports its performance for the same test datasets as used in the previous two sections Section 7 concludesthe paper

2 Power System Model

The distribution network modeled in this study is based onthe benchmark IEEE 13 node test radial feeder [16] and wasmodeled in MATLAB-Simulink Some modifications weremade to the original feeder model in order to carry outthe islanding studies as required First of all the substationautomatic voltage regulator (AVR) at node 650 was notincluded in the model This was done so as to avoid anypossible interaction of PV with the tap-changing controlsof the AVR [17] that could mask any signature related toislanding on the system A 1007 kWp solar PV array wasintegrated at node 692 through a three-phase inverter thusmaking this node the PCC These changes are visible in themodified feeder shown in Figure 2

Apart from the changes mentioned previously the con-stant current load at node 675was removed and the active andreactive power demands119875 and119876 respectively of the constantcurrent load at 692 were modified so as to make section 671-692 the islandable feeder sectionThe PV inverter operates onunity power factor and thus the loads on lateral 671-675 hadto be scaled according to the fixed PV penetration and feedercapacitor bank size for attaining 119875-119876 balance required for


650

ACDC

PV inverter

v

646

611

645 632 633

684 671 692

680

634

675

652












minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501




10msdivision

10msdivision















minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501














minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501


times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




6 4866







119879

pca + 119879res119875119879





















120572= 119903((119899

2minus 1)119899(119899 minus 119903))119865

119903119899minus119903



119879119890 = 119891

119879(119868 minus 119875pca119875

119879







2

120572by the



Case 119876 119876 gt 119876120572






7 Yes No




2 and 119876


2

120572limit for the















2



K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)


K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)


2



120572and





2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4



Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity







0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




650

ACDC

PV inverter

v

646

611

645 632 633

684 671 692

680

634

675

652












minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501




10msdivision

10msdivision















minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501














minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501


times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




6 4866







119879

pca + 119879res119875119879





















120572= 119903((119899

2minus 1)119899(119899 minus 119903))119865

119903119899minus119903



119879119890 = 119891

119879(119868 minus 119875pca119875

119879







2

120572by the



Case 119876 119876 gt 119876120572






7 Yes No




2 and 119876


2

120572limit for the















2



K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)


K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)


2



120572and





2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4



Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity







0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in












minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501














minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501


times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




6 4866







119879

pca + 119879res119875119879





















120572= 119903((119899

2minus 1)119899(119899 minus 119903))119865

119903119899minus119903



119879119890 = 119891

119879(119868 minus 119875pca119875

119879







2

120572by the



Case 119876 119876 gt 119876120572






7 Yes No




2 and 119876


2

120572limit for the















2



K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)


K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)


2



120572and





2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4



Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity







0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









minus5000

05000

200100

0minus100minus200

10050

0

1000500

0

10050

0minus50minus100

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501






0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501

0005 02 0250 01501


times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




times105

41 32 50 1505 25 35 45

Sample number

minus200

minus150

minus100

minus50

0

50

100

150

Curr

ent (

A)




6 4866







119879

pca + 119879res119875119879





















120572= 119903((119899

2minus 1)119899(119899 minus 119903))119865

119903119899minus119903



119879119890 = 119891

119879(119868 minus 119875pca119875

119879







2

120572by the



Case 119876 119876 gt 119876120572






7 Yes No




2 and 119876


2

120572limit for the















2



K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)


K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)


2



120572and





2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4



Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity







0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





















120572= 119903((119899

2minus 1)119899(119899 minus 119903))119865

119903119899minus119903



119879119890 = 119891

119879(119868 minus 119875pca119875

119879







2

120572by the



Case 119876 119876 gt 119876120572






7 Yes No




2 and 119876


2

120572limit for the















2



K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)


K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)


2



120572and





2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4



Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity







0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in








2



K-LD =1

2[1205902

2

12059021

+1205902

1

12059022

+ (1205831minus 1205832)2

(1

12059021

+1

12059022

) minus 2] (1)


K-LD =1

2[1205902

test case1205902ref

+1205902

ref1205902test case

minus 2] (2)


2



120572and





2 195 times 106 0554 No

3 492 times 106 0004 Yes

4 553 times 103 48555 No

5 842 times 106 01023 No

6 512 times 106 00012 No

times10minus4



Data

04

06

08

1

12

14

16

18

2

22

24

Den

sity







0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





0

002

004

006

008

01

012

014

Den

sity


Data


times10minus4


02

04

06

08

1

12

14

16

18

2

22

Den

sity


Data






(i) let 1199091 1199092 119909



(ii) return

119891(119909119902) larr997888

sum119896

119894=1119891 (119909119894)

119896 (3)







7 Conclusions










Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in











Competing Interests


Acknowledgments


References










































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Research Article Multivariate Statistics and Supervised Learning …downloads.hindawi.com/journals/acisc/2016/3684238.pdf · 2019. 7. 30. · Prevalent anti-islanding methods include