Research Article Application of Extension Theory with ...

Research ArticleApplication of Extension Theory with Chaotic SignalSynchronization on Detecting Islanding Effect of PhotovoltaicPower System

Meng-Hui Wang1 Mei-Ling Huang2 and Kang-Jian Liou1

1Department of Electrical Engineering National Chin-Yi University of Technology Sec 2 57 Chung Shan RoadTaiping Taichung Taiwan2Department of Industrial Engineering amp Management National Chin-Yi University of Technology Sec 257 Chung Shan Road Taiping Taichung Taiwan

Correspondence should be addressed to Mei-Ling Huang huangmlncutedutw

Received 15 December 2014 Revised 5 March 2015 Accepted 5 March 2015

Academic Editor Leonardo Palmisano

Copyright copy 2015 Meng-Hui Wang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The detection of islanding effect is a highly important topic for photovoltaic (PV) power system The islanding effect occurs whenthe distributed power source is disconnected from the main supply while the power is still supplied in partial load area which mayinjure the set maintenance personnel or damage the equipment Combining chaotic synchronization and extension theory thisresearch is to propose a novel detection method to distinguish the occurrence of islanding effect based on nonautonomous Chuarsquoscircuit To demonstrate the effectiveness of the proposed method this paper applies PSIM to simulate the PV power system Theexperimental results show that the accuracy of the proposed method achieves 98 on islanding effect

1 Introduction

In recent years Taiwanrsquos rapid industrial development has ledto increasing demand on fossil energy Topics of the envi-ronmental protection sustainable resources application andgreen energy industry have drawn more attention [1] Interms of green energy the photovoltaic (PV) and wind powerdevelopment is relatively mature Taiwan as a small islandhas limited natural resources to develop wind power First itis more difficult to install wind turbines on the sea as com-pared to install PV power system on the land [2] Secondthe peak period of energy consumption is in the summerwhile it is the lowest power generation season for wind powerThe location of Taiwan is on latitude of 235 degrees northwith abundant sunshine in the summer and is suitable fordeveloping PV power systemThe annual power generation isabout 48917MWh in Taiwan [3] Recently Taiwan has been

promoting PV power system and the total installed capacityreached 296MW in 2014

The output of PV system is DC which must be convertedinto AC by inverter before it is connected parallel to the mainsupply The phenomenon is called islanding effect when boththe load and PV power system are disconnected from theload simultaneously and the PV power system still suppliespower to the loadOnce the islanding effect occurs and the PVpower system is not disconnected from the load instantly itwill result in the casualties ofmaintenance personnel or causethe heavy damage on power users power supply system orelectrical equipment With the construction of different dis-tributed power source and power network parallel systemsonce the mains supply is disconnected by fault the hazardprobability of islanding effect increases relatively Thereforethe prevention and detection of islanding effect are crucial onthe application of renewable energy

Hindawi Publishing CorporationInternational Journal of PhotoenergyVolume 2015 Article ID 756313 10 pageshttpdxdoiorg1011552015756313

2 International Journal of Photoenergy

There are two main categories for islanding detectionactive and passive methods The active detection method isdesigned to directly and actively detect the part of the net-work and then distinguish the signal changes while the pas-sive detection method monitors system parameters to dis-connect from the load when nonnormal condition occurswithout directly interactingwith the systemCompared to thepassive method active method is more effective Howeverthe effectiveness of active method may become the adversityto the system and besides it requires higher implementationcost thanpassivemethod [4 5]Many recent studies have pro-posed islanding operation preventionmethods such as phasejump detection method [6] voltage harmonic detectionmethod [7] and power variation rate detection method [8]Those methods are relatively easy to be implemented How-ever when the output power of electric governor and theload power of PV systems approach balance the voltage andfrequency changes of PV power system are insignificantand the passive detection method often fails All the abovemethods face the problem of nondetection zone and cannotindependently detect the islanding effect

Combining chaotic synchronization and the extensiontheory this research is to propose a novel detectionmethod todistinguish the occurrence of islanding effect based on nona-utonomous Chuarsquos circuit [9] With simple structure thenonautonomousChuarsquos circuit is easy to be implementedThechaotic synchronization detection method and the extensiontheory have high computing speed and both are likely to beimplemented by chips not only to enhance the performanceof nonautonomous Chuarsquos circuit detection but also to inc-rease the accuracy of islanding detection

2 Summary of Islanding Effect

The islanding effect is a phenomenon when the PV powersystem and the power network are connected in operation[10] The connection between PV power system and powernetwork is shown in Figure 1 The islanding effect refersto independent power supply of partial network when theelectricity is interrupted and the power network has faultsor powers off but the PV power system fails to detect itimmediately and the protection relay has not been cut offthe system [11] IEEE 1547 IEC 61727 and NEC 690 are theregulations for the parallel connection between distributedpower source of power system and main supply As moreand more distributed power sources are connected parallelto the main supply network the potential islanding problembecomes severe increasingly Due to disconnection frommain supply network the distributed power source loses thereference of power supply from power company Under inde-pendent operation of distributed power source the powercompany cannot monitor the power system and the damageof household appliances or sensitive machines occurs

When the islanding effect occurs the voltage and fre-quency of distributed power source may be abnormal andcannot be monitored A reliable PV power system shoulddetect the occurrence of islanding effect and function intime to avoid damaging the load or injuring the maintenance

DCACinverter

Grid system

Photovoltaicpower system

AC load

DCDCconverter

Transformer

Figure 1 Photovoltaic power system links with grid system

workers The traditional islanding operation preventionmethods are described below

21 Phase Jump Detection Method When the distributedpower source is disconnected from main supply the outputrequirement of distributed power source for load is unbal-anced so the inverter output current and load end voltagehave phase difference with respect to different forms of loadThe system determines whether the islanding effect is gener-ated or not according to the phase difference However if theload is resistive thismethod cannot recognize islanding effectsuccessfully [6] In addition some loads such as motorscause temporary phase deviation at the moment of startupand will reduce the accuracy rate

22 Voltage Harmonic Detection Method The voltage har-monic detection method checks whether the voltage thirdharmonic distortion of load exceeds the threshold If it doesthe protection relay is cut off for islanding detection Thelimitation of the method is due to nonlinear load and powerquality in practice and the threshold range of harmonicdetection method is difficult to be determined thus theaccuracy detection rate decreases [7]

23 Power Changing Detection Method Power changingdetectionmethod identifies islanding effect based on the out-put power changing of inverter When the main supply is dis-connected from the distributed power source the outputpower of inverter is significantly different from normal par-allel connection and the power variation rate is easily mea-sured by power detection instrument When the outputpower of PV power system approaches the load consumedpower the power of load is still in equilibrium state Whenthe PV power system cuts off the main supply the outputvoltage still keeps the magnitude and frequency variation ofnormal main supply and it will result in misoperation of thisdetection method [8]

3 Acquisition Method of Characteristic Signal

NonautonomousChuarsquos circuit (Figure 2)whichwe proposedin this study is a characteristic signal acquisition platformwhich connects to the utility power with PV power systemto obtain the voltage waveforms 119881

119860and 119881

119861 The waveforms

are used as the inputs to chaotic signal synchronization andextension detectionmethod to determinewhether the island-ing effect occurs to instantly initiate the protection relays

International Journal of Photoenergy 3

Photovoltaic power system

NonautonomousChuarsquos circuit

Chaotic signalsynchronization

Protection relays

Extensiondetectedmethod

AC load

Grid system

Transfomer

Transfomer

DCDCconverter

DCACinverter

VA

VBVin

Figure 2 The proposed method for detecting islanding effect

Given the abovedescribed background this study utiliz-ing the dynamic trajectories of a chaotic system to convertthe disturbance waveforms of power systems extracts fewercharacteristics and increases the detection accuracy based onthe sensitive characteristics of chaos Specifically this studydesigns a chaotic synchronization detector to convert theinput signal waveform and extracts prominent characteristicsfrom the waveformThe extension theory in pattern recogni-tion will be used to identify the type of the power disturbancesignals The overall scheme is shown in Figure 3

31 Basic Introduction of Chuarsquos Circuit In this paper Chuarsquoscircuit is a simple nonautonomous circuit designed and deve-loped by Professor Chua in 1983 [12 13] (Figure 4) It consistsof three active components capacitors inductors resistorsand a nonlinear resistor composition 119877119871 to Chua diodeAccording toKirchhoff rsquos circuit laws Chuarsquos circuit equationsare as

1198621

1198891198811198881

119889119905= minus119892119881

1198881+ 1198941198711minus 1198941198712

1198622

1198891198811198882

119889119905= 1198941198711minus 119894119877119871

1198711

1198891198941198711

119889119905= minus1198811198881minus 11989411987111198771+ 119881119894119899sin (2120587119896119891119905)

1198712

1198891198941198712

119889119905= 1198811198881minus 1198811198882minus 11989411987121198775

(1)

where 119896 is the number of higher harmonics 119894119877119871

is defined as

119894119877119871= 1198661198861198811198882+1

2(119866119887minus 119866119886) [10038161003816100381610038161198811198882 + 119864119886

1003816100381610038161003816 minus10038161003816100381610038161198811198882 minus 119864119886

1003816100381610038161003816] (2)

where 119866119886and 119866

119887are the slopes and 119864

119886is breakpoint

Normal conditions voltage waveforms of nonautonomousChuarsquos circuit 119881

119860and 119881

119861are as shown in Figure 5

32 Chaotic Synchronization Pecora andCarroll showed thatwhen the signs of the Lyapunov exponents for the subsystems

are all negative the chaotic system will synchronize andconstructed a real set of chaotic synchronizing circuits [14]In chaotic systems the initial state has subtle change whichis gradually enlarged and leads to significant difference even-tually Generally the two synchronous chaotic systems arecalledmaster systemand slave systemWhen the initial valuesofmaster and slave systems are different the operation trajec-tories of the two chaotic systems are different The basicformula for the chaotic synchronization is shown as in

lim119905rarrinfin

1003817100381710038171003817119884slave119894(119905) minus 119883master119894(119905)1003817100381710038171003817 119894 = 1 2 119899 (3)

where 119884slave is slave system and119883master is main systemThis paper uses chaotic synchronization to analyze char-

acteristics signal of system [15] Master and slave chaoticsystem are as in the following equations

Master

1= 1198911(1199091 1199092 119909

119899)

2= 1198912(1199091 1199092 119909

119899)

119899= 119891119899(1199091 1199092 119909

119899)

(4)

Slave

1199101= 1198911(1199101 1199102 119910

119899)

1199102= 1198912(1199101 1199102 119910

119899)

119910119899= 119891119899(1199101 1199102 119910

119899)

(5)

where 119891119894(119894 = 1 2 119899) are non-linear functions Equation

(6) dynamic error equation are formed from (4) and (5) bysubtracting the error status

1198901= 1199101minus 1199091

1198902= 1199102minus 1199092

119890119899= 119910119899minus 119909119899

1198901= 1198911(1199091 1199092 119909

119899) minus 1198911(1199101 1199102 119910

119899) = 119867

1

1198902= 1198912(1199091 1199092 119909

119899) minus 1198912(1199101 1199102 119910

119899) = 119867

2

119890119899= 119891119899(1199091 1199092 119909

119899) minus 119891119899(1199101 1199102 119910

119899) = 119867

119899

(6)

where 119867119894(119894 = 1 2 119899) are nonlinear equations and the

dynamic error equation is a chaotic system This paper usesthe chaotic dynamic trajectory tomimic various system oper-ating states including periodic nonperiodic and random


Nonautonomous Chaotic signalsynchronization


x10 c a b d

Normal

Dips

Swells

Harmonic

Islanding

Chuarsquos circuit

VA

VB K(x)

minus1

Figure 3 The proposed islanding detection system

Vin

VA VB

R1

R2

R3

R5

R6C2C1 RL

R4

L1

L2

minus15V

+15V

minus

+

Figure 4 Chuarsquos circuit diagram

20

15

10

5

0

10 2 3 4 5

VA

VB

minus15

minus20

minus10

minus5

V(V

)

t (s)times10

minus2

Figure 5 Output voltage waveforms in Chuarsquos circuit

states in the time domain so as to identify the disturbancestate of PV power systems As a test framework this studyuses two Lorenz chaotic systems one is themaster system andthe other is the slave system expressed as in (7) and (8) Thedynamic error state equation is worked out and expressed inmatrix form as in (9)

Master

1= 120572 (119909

2minus 1199091)

2= 1205731199091minus 11990911199093minus 1199092

3= 11990911199092minus 1205741199093

(7)

Slave

1199101= 120572 (119910

2minus 1199101)

1199102= 1205731199101minus 11991011199103minus 1199102

1199103= 11991011199102minus 1205741199103

(8)

[[

[

1198901

1198902

1198903

]]

]

=[[

[

minus120572 120572 0

120573 minus1 0

0 0 minus120574

]]

]

[[

[

1198901

1198902

1198903

]]

]

+[[

[

11991021199103minus 11990921199093

minus11991011199103+ 11990911199093

11991011199102minus 11990911199092

]]

]

(9)

where 1198901 1198902 and 119890

3are the chaotic dynamic error values of

the main system and slave system 120572 120573 and 120574 are adjustmentparameters This research simulates five waveforms of thePV power systems including normal islanding harmonicsvoltage swells and voltage dips This paper use two errorvalues 119890

1and 1198902to generate chaos error scatter patterns for

islanding analysis

4 The Proposed Extension Detection Method

PSIM and Matlab simulate nonautonomous Chuarsquos circuitand power grid in this study In order to detect slight changesin the PV power system this study imported the originaldata into the chaotic synchronization-based detector moduleto form the chaos error scatter pattern However there weremany error distribution points in the diagram the centroidpattern was used as the characteristic of islanding detectionto set the feature range and effectively reduce the quantity ofthe extracted features Figure 6 shows the typical chaos errorscatter patterns under normal condition where the triangleis the center of chaos scatter patterns and the four centercharacteristics (eight values from 119881

1to 1198818) are used as the

input patterns of the proposed extension detection methodIn order to precisely distinguish genuine islanding effect

from the power system quality issues five PV power systemsincluding normal islanding harmonics voltage swells andvoltage dips are simulated in this study The specific charac-teristics are (1) normal (2) islanding is when the grid systemand the photovoltaic system cut off (3) harmonics is thepower system have contain high frequency (4) voltage swellsoccur between reference voltage 11 pu and 18 pu (5) voltagedips occur between reference voltage 01 pu and 09 pu

The proposed extension detection method is based onthe extension theory The extension set extends the fuzzy setfrom [0 1] to (minusinfininfin) So it allows the researcher to define


4

2

0

0 2 4 6

0

5000

0 5000 10000

times105

times104

e 2e 2

minus2

minus4

minus5000

minus5000

minus10000

V2

V4

V8

V6

V5 V7

V3

V1 e1

e1

Figure 6 Typical chaos error scatter patterns

a set that includes any data in the domain According tothe extension theory 119877 = (119873 119862 119881) is a multidimensionalmatter-element 119862 = [119888

1 1198882 119888

119899] is a characteristic vector

and 119881 = [V1 V2 V

119899] is a value vector of 119862 then a multidi-

mensional matter-element is defined as

119877 =

[[[[[[

[

119873 1198881 V1

1198882 V2

119888119899 V119899

]]]]]]

]

=

[[[[[[

[

1198771

1198772

119877119899

]]]]]]

]

(10)

where 119877119894= (119873 119888

1 V1) (119894 = 1 2 119899) is defined as the sub-

matter-element of 119877 and can be simplified as follows

119877 = (119873 119862 119881) (11)

According to extension theory [16 17] and the testingresults of this study the upper and lower limits for eachclassical domain are set by the center of the chaos error scatterpatterns and the maximum and minimum values of theclassical domain are shown in Table 1 The various numericvalues about 119888

119894change at the condition of different power

Table 1 The matter element models of different categories

Category Matter element

Normal 1198771=

11987311198881 ⟨minus13 minus11⟩

1198882 ⟨43 49⟩

1198883 ⟨27 30⟩

1198884 ⟨minus131 minus75⟩

1198885 ⟨minus16 minus15⟩

1198886 ⟨34 36⟩

1198887 ⟨6 63⟩


Islanding 1198772=

11987321198881 ⟨minus12 minus11⟩

1198882 ⟨57 6⟩

1198883 ⟨74 84⟩



1198886 ⟨07 08⟩

1198887 ⟨27 31⟩


Harmonics 1198773=

11987331198881 ⟨minus15 minus1⟩

1198882 ⟨57 4⟩

1198883 ⟨28 296⟩

1198884 ⟨minus15 minus101⟩


1198886 ⟨33 36⟩

1198887 ⟨6 63⟩


Voltage swells 1198774=

11987341198881 ⟨minus12 minus13⟩

1198882 ⟨41 5⟩

1198883 ⟨32 786⟩

1198884 ⟨minus435 minus91⟩


1198886 ⟨4 83⟩

1198887 ⟨65 88⟩



Table 1 Continued


Voltage dips 1198775=

11987351198881 ⟨minus12 minus09⟩

1198882 ⟨01 48⟩

1198883 ⟨13 236⟩

1198884 ⟨minus116 minus03⟩

1198885⟨minus15 minus04⟩

1198886 ⟨004 39⟩

1198887 ⟨05 65⟩

1198888 ⟨minus58 minus007⟩

systemThemethodology of deciding the values of 119888119894is based

on the simulation resultsThe best values of 119888119894will be selected

and therefore implementing those values to chipsThe neigh-borhood domain is set by all classical domain maximum andminimum values the values of neighborhood domain are setas

119877119901=

1198731199011198881 ⟨minus16 minus08⟩

1198882 ⟨01 8⟩

1198883 ⟨12 80⟩



1198886 ⟨0 85⟩

1198887 ⟨04 9⟩

1198888 ⟨minus98 0⟩

(12)

where 1198881to 1198888are eight input characteristics After the ele-

ment-matter model of islanding detection is formulated thesystem detection of PV power systems can be initiated Theproposed extension detection algorithm is as follows [18]

Step 1 Establish the matter-element of each category such as

119877119894=

[[[[[[

[

11987711989411988811198811198941

11988821198811198942

11988881198811198948

]]]]]]

]

119894 = 1 2 5 (13)

where 119881119894119895= ⟨119886119894119895 119887119894119895⟩ 119895 = 1 2 8 are the upper and lower

characteristic values of classical domain in the 119894th categoryand the detail setting is shown in Table 1

Step 2 Input a tested matter-element

119877119905= (119877119905 119862 119881119905) =

[[[[[[

[

1198771199051198881V1199051

1198882V1199052

1198888V1199058

]]]]]]

]

(14)

where V1199051to V1199058are characterized values of 119888

1to 1198888

Step 3 Calculate the relation degree of the tested elementwith the category characteristic using

119870119894119895(V119905119895) =

minus05120588 (V119905119895 119881119894119895)

10038161003816100381610038161003816119881119894119895

10038161003816100381610038161003816

V119905119895isin 119881119894119895

120588 (V119905119895 119881119894119895)

120588 (V119905119895 119881119901119895) minus 120588 (V

119905119895 119881119894119895)

V119905119895notin 119881119894119895

119894 = 1 2 5 119895 = 1 2 8

(15)

where

120588 (V119905119895 119881119894119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119894119895+ 119887119894119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119894119895minus 119886119894119895

2

120588 (V119905119895 119881119901119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119901119895+ 119887119901119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119901119895minus 119886119901119895

2

(16)

119886119894119895 119895th lower characteristic values of classical domain in the

119894th category 119887119894119895 119895th upper characteristic values of classical

domain in the 119894th category 119881119901119895 119895th characteristic values of

neighborhood domain or 119881119901119895

= ⟨119886119901119895 119887119901119895⟩ 119886119901119895 119895th lower

characteristic values of neighborhood domain 119887119901119895 119895th upper

characteristic values of neighborhood domain

Step 4 Set the weights of the characteristics11988211198822 119882

8

according to the importance of every characteristic in thedetection process In order to reach high detection accuracythe weights of design order from 119888

1to 1198888are 01 02 02 01

01 01 01 and 01 in this study

Step 5 Calculate the relation coefficients for each category 120582119894

as

120582119894=

8

sum

119895=1

119882119895119870119894119895 119894 = 1 2 5 (17)

Step 6 Normalize the values of the relation coefficients intoan interval between 1 and minus1 as

1205821015840

119894=2120582119894minus 120582min minus 120582max120582max minus 120582min

119894 = 1 2 5 (18)

Step 7 Ranking the normalized relation coefficients to detectthe category of test element the detection rule is as

IF (1205821015840

119896= 1) than the type of 119877

119905is 119873119896 (19)


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000

Figure 7 Chaos error scatter patterns under normal condition

Themajor category is119873119896as shown inTable 1 when1205821015840

119896= 1

and is impossible to classify to 119873119896when 1205821015840

119896= minus1 Other

categories can be justified depending on the values of therelation coefficients Larger relation coefficients demonstratehigher possibility to this category otherwise lower

Step 8 Go back to Step 2 for the next test data until all thewhole sets have been done

5 Simulation Results

To demonstrate the effectiveness of the proposed method500 sets of tested data are simulated using PSIM softwareThis paper uses a 66-PV (w) panel to simulate the distributedpower source and parallel connection to 110V 60Hz mainsupply The specifications of the PV panels for open-circuitvoltage and short-circuit current are 217 V and 345ArespectivelyThe ideal electric power supply only contains thefundamental frequency component But when it is deliveredto the client-side the voltage waveform possesses harmoniccomponents due to long-distance delivery nonlinear loads

0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000

Figure 8 Chaos error scatter patterns under islanding condition

nonlinear electricity or electronic equipmentWe set 1sim10harmonic components in the study Voltage swells are theroot-mean-square value of the voltage and are within therange of 11ndash18 pu and continuing 05sim30 cycles voltagedips are the root-mean-square value of the fundamental fre-quency voltage and are within the range of 01ndash09 pu andcontinuing 05sim30 cycles When the voltage is lower than01 it is referred to power interruption and the PV systemwill produce the islanding condition Figures 7 to 11 showthe chaotic scatter patterns under different test conditionsThe conditions include normal islanding effect harmonicvoltage swells and voltage dips

According to the centroid of chaos scatter diagram inFigure 7 the islanding effect can be easily detected which issignificantly different from other conditions So the protec-tion relay can function in time to cut off the main supply toeliminate the casualties of maintenance personnel or causethe heavy damage on power users power supply systemor electrical equipment Moreover the harmonic conditionis close to normal condition in a few cases so the systemmay misrecognize its state However when the islandingeffect occurs the waveform and centroid are apparently


0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1

Figure 9 Chaos error scatter patterns under harmonics condition

different from normal and other power quality issues There-fore the islanding operation can be identified accurately andthe system can be immediately shut off

Table 2 shows the typical patterns of the proposed meth-od 10 sample data are selected from 500 test data randomlyin which sample data number 8 is harmonic state and sampledata number 9 is normal state The values are very closeto each other according to the centroid characteristics butboth are significantly different from islanding state Thus thepower quality problem can be eliminated and the protectionrelay correctly functions

Using the proposed method the partial detection resultsare shown in Table 3 and the classification results for all 500data sets are shown in Table 4 While the relation degreewith the islanding condition equals 1 (the maximum value)sample number 1 is classified as in islanding state Moreoversamples number 9 is classified as in normal state as the rela-tion degree with the normal condition equaling 1 Althoughthe detection result is normal it can also provide useful

0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1

Figure 10 Chaos error scatter plot under voltage swells

information for future analysis when the harmonic correla-tion grade ranks the second On the contrary the sampleisland correlation grade is minus1 meaning that it is unlikely tobe islanding effectTherefore the proposed detectionmethodcan efficiently and effectively distinguish islanding effect fromthe power quality problem

Various detection methods are also used to test the samePV power system The testing time and accuracy are shownin Table 5 The detection times of phase jump and voltageharmonic methods are relatively proposed method long Thephase jumpdetectionmethod has high accuracy for nonresis-tive loads but it fails when the load is resistive [19] Thevoltage harmonic detection method is unable to resist theinterference of power quality andmisrecognition occursThepower changing method is faster and more accurate thanvoltage harmonic method However when the output powerof PV power system approaches the load consumed powerthis method cannot detect islanding effect Among them allour proposedmethod has the shortest detection time and canprecisely identify the islanding effect


Table 2 Typical patterns of proposed method (partial results)

Sample V1

V2

V3

V4

V5

V6

V7

V8

Actual category1 minus119 578 783 minus478 minus138 079 285 minus129 Islanding2 minus118 585 823 minus582 minus138 079 298 minus138 Islanding3 minus119 578 8 minus515 minus140 079 291 minus132 Islanding4 minus095 016 139 minus043 minus043 004 049 minus008 Dips5 minus12 435 4826 minus1881 minus116 737 898 minus1042 Swells6 minus143 508 2859 minus1159 minus154 345 627 minus531 Harmonic7 minus136 581 2851 minus1379 minus155 346 612 minus514 Harmonic8 minus143 508 2864 minus1161 minus154 345 629 minus532 Harmonic9 minus121 448 2780 minus11 minus151 348 622 minus527 Normal10 minus121 449 2783 minus1135 minus150 352 613 minus518 Normal

Table 3 Testing results of the proposed extension detection method (partial results)

Samples Normalcorrelation

Islandingcorrelation

Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory

1 minus1 1 minus045 minus090 049 Islanding Islanding2 minus1 1 minus039 minus090 054 Islanding Islanding3 minus1 1 minus040 minus092 058 Islanding Islanding4 minus1 minus098 minus097 minus1 1 Dips Dips5 minus092 minus083 minus097 1 minus1 Swells Swells6 016 minus1 1 minus041 minus012 Harmonic Harmonic7 033 minus1 1 minus012 020 Harmonic Harmonic8 031 minus1 1 minus007 017 Harmonic Harmonic9 1 minus1 057 031 035 Normal Normal10 1 minus1 065 036 038 Normal Normal

Table 4 Classification table

Actual group Predicted groupNormal Islanding Harmonics Voltage swells Voltage dips Total

Normal 98 0 2 0 0 100Islanding 0 100 0 0 0 100Harmonics 8 0 92 0 0 100Voltage swells 0 0 0 100 0 100Voltage dips 0 0 0 0 100 100Total 106 100 94 100 100 500

Table 5 Detection performances of different methods

Detection methods Detection timesms Accuracy

Phase jump method 17 95Voltage harmonic method 17 85Power changing method 14 90Proposed method 12 98


The islanding effect is an inevitable problem in distributedpower source and it causes the damage on maintainers or

electrical equipment Therefore this paper proposes a noveldetection method using the chaotic signal synchronizationwith extension theory to analyze the islanding effect of thePV power systemThe simulation conditions include normalislanding harmonic voltage swell and voltage dip Testresults show that the proposedmethod cannot only detect themain states of PVpower systems but also provide useful info-rmation for future analysis by the relative relation degreesBased on the results the following conclusions are proposed

(1) Proposed method combining Chuarsquos circuit andchaotic signal synchronization is effective on island-ing detection and the architecture is easy to be imple-mented in hardware circuit


0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4

Figure 11 Chaos error scatter patterns under voltage dips

(2) The algorithm combining chaotic signal synchroniza-tion with extension detection is easy to be imple-mented by chip for DCAC inverter and it effectivelyincreases the accuracy of detecting islanding effect

(3) Experimental results show that our proposedmethodsignificantly achieves high degree of detection accu-racy

Conflict of Interests

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

References

[1] Y KWu andH J Lin ldquoA research review of small wind turbinesin urban areasrdquoMonthly Journal of Taipowers Engineering vol773 pp 59ndash73 2014

[2] C-D Yue C-M Liu and E M L Liou ldquoA transition towarda sustainable energy future feasibility assessment and develop-ment strategies of wind power in Taiwanrdquo Energy Policy vol 29no 12 pp 951ndash963 2001

[3] J S You Y T Cheng andM T Tseng ldquoDesign and installationof a thin-film 10 kWp demonstration PV system at Taipowerheadquartersrdquo Monthly Journal of Taipowerrsquos Engineering vol775 pp 69ndash84 2013

[4] L A C Lopes and H Sun ldquoPerformance assessment of activefrequency drifting islanding detection methodsrdquo IEEE Transac-tions on Energy Conversion vol 21 no 1 pp 171ndash180 2006

[5] X Ding P A Crossley and D J Morrow ldquoIslanding detectionfor distributed generationrdquo Journal of Electrical Engineering ampTechnology vol 2 no 1 pp 19ndash28 2007

[6] G-K Hung C-C Chang and C-L Chen ldquoAutomatic phase-shift method for islanding detection of grid-connected photo-voltaic invertersrdquo IEEE Transactions on Energy Conversion vol18 no 1 pp 169ndash173 2003

[7] F de Mango M Liserre A DellrsquoAquila and A Pigazo ldquoOver-view of anti-islanding algorithms for PV systems Part Ipassive methodsrdquo in Proceedings of the 12th International PowerElectronics and Motion Control Conference pp 1878ndash1883September 2006

[8] M A Redfern O Usta and G Fielding ldquoProtection againstloss of utility grid supply for a dispersed storage and generationunitrdquo IEEETransactions onPowerDelivery vol 8 no 3 pp 948ndash954 1993

[9] A G Jha A P Das and A Kumar ldquoEffects of electromagneticinterference on non-autonomous chaotic circuitsrdquo in Proceed-ings of the 4th International Conference on Computers andDevices for Communication pp 1ndash4 December 2009

[10] Y K Wu Y Q Huang and W G Chang ldquoEstablishment ofcontrol technologies and simulation platform under variousoperation modes of micro gridsrdquoMonthly Journal of TaipowerrsquosEngineering vol 776 pp 54ndash69 2013

[11] M Liserre A Pigazo A DellrsquoAquila and V M Moreno ldquoAnanti-islanding method for single-phase inverters based on agrid voltage sensorless controlrdquo IEEE Transactions on IndustrialElectronics vol 53 no 5 pp 1418ndash1426 2006

[12] S R Huang Y H Ma J H Chou et al ldquoApplication Wignerville distribution (WVD) algorithm analysis and identify thechaotic signals of non-autonomous Chuarsquos circuit for islandingdetectionrdquo in Proceedings of the 34th Symposium on ElectricalPower Engineering pp 1691ndash1696 Taipei Taiwan December2012

[13] L O Chua and G N Lin ldquoCanonical realization of Chuarsquoscircuit familyrdquo IEEE Transactions on Circuits and Systems vol37 no 7 pp 885ndash902 1990

[14] L M Pecora and T L Carroll ldquoSynchronization in chaotic sys-temsrdquo Physical Review Letters vol 64 no 8 pp 821ndash824 1990

[15] H Huijberts H Nijmeijer and R Willems ldquoSystem identifi-cation in communication with chaotic systemsrdquo IEEE Trans-actions on Circuits and Systems I Fundamental Theory andApplications vol 47 no 6 pp 800ndash808 2000

[16] M-H Wang K-H Chao G J Huang and H-H Tsai ldquoAppli-cation of extension theory to fault diagnosis of automotiveenginerdquo ICIC Express Letters vol 5 pp 1293ndash1299 2011

[17] M H Wang ldquoApplication of extension theory to vibrationfault diagnosis of generator setsrdquo IEE Proceedings-GenerationTransmission andDistribution vol 151 no 4 pp 503ndash508 2004

[18] M H Wang and H H Tsai ldquoFuel cell fault forecasting systemusing grey and extension theoriesrdquo IET Renewable Power Gen-eration vol 6 no 6 pp 373ndash380 2012

[19] C-C Hou and Y-C Chen ldquoActive anti-islanding detectionbased on pulse current injection for distributed generationsystemsrdquo IETPower Electronics vol 6 no 8 pp 1658ndash1667 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Carbohydrate Chemistry

International Journal of


Journal of

Chemistry


Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of


Chromatography Research International


Applied ChemistryJournal of



Theoretical ChemistryJournal of


Journal of

Spectroscopy

Analytical ChemistryInternational Journal of


Journal of


Quantum Chemistry


Organic Chemistry International

ElectrochemistryInternational Journal of



CatalystsJournal of


There are two main categories for islanding detectionactive and passive methods The active detection method isdesigned to directly and actively detect the part of the net-work and then distinguish the signal changes while the pas-sive detection method monitors system parameters to dis-connect from the load when nonnormal condition occurswithout directly interactingwith the systemCompared to thepassive method active method is more effective Howeverthe effectiveness of active method may become the adversityto the system and besides it requires higher implementationcost thanpassivemethod [4 5]Many recent studies have pro-posed islanding operation preventionmethods such as phasejump detection method [6] voltage harmonic detectionmethod [7] and power variation rate detection method [8]Those methods are relatively easy to be implemented How-ever when the output power of electric governor and theload power of PV systems approach balance the voltage andfrequency changes of PV power system are insignificantand the passive detection method often fails All the abovemethods face the problem of nondetection zone and cannotindependently detect the islanding effect

Combining chaotic synchronization and the extensiontheory this research is to propose a novel detectionmethod todistinguish the occurrence of islanding effect based on nona-utonomous Chuarsquos circuit [9] With simple structure thenonautonomousChuarsquos circuit is easy to be implementedThechaotic synchronization detection method and the extensiontheory have high computing speed and both are likely to beimplemented by chips not only to enhance the performanceof nonautonomous Chuarsquos circuit detection but also to inc-rease the accuracy of islanding detection

2 Summary of Islanding Effect

The islanding effect is a phenomenon when the PV powersystem and the power network are connected in operation[10] The connection between PV power system and powernetwork is shown in Figure 1 The islanding effect refersto independent power supply of partial network when theelectricity is interrupted and the power network has faultsor powers off but the PV power system fails to detect itimmediately and the protection relay has not been cut offthe system [11] IEEE 1547 IEC 61727 and NEC 690 are theregulations for the parallel connection between distributedpower source of power system and main supply As moreand more distributed power sources are connected parallelto the main supply network the potential islanding problembecomes severe increasingly Due to disconnection frommain supply network the distributed power source loses thereference of power supply from power company Under inde-pendent operation of distributed power source the powercompany cannot monitor the power system and the damageof household appliances or sensitive machines occurs

When the islanding effect occurs the voltage and fre-quency of distributed power source may be abnormal andcannot be monitored A reliable PV power system shoulddetect the occurrence of islanding effect and function intime to avoid damaging the load or injuring the maintenance

DCACinverter

Grid system

Photovoltaicpower system

AC load

DCDCconverter

Transformer

Figure 1 Photovoltaic power system links with grid system

workers The traditional islanding operation preventionmethods are described below

21 Phase Jump Detection Method When the distributedpower source is disconnected from main supply the outputrequirement of distributed power source for load is unbal-anced so the inverter output current and load end voltagehave phase difference with respect to different forms of loadThe system determines whether the islanding effect is gener-ated or not according to the phase difference However if theload is resistive thismethod cannot recognize islanding effectsuccessfully [6] In addition some loads such as motorscause temporary phase deviation at the moment of startupand will reduce the accuracy rate

22 Voltage Harmonic Detection Method The voltage har-monic detection method checks whether the voltage thirdharmonic distortion of load exceeds the threshold If it doesthe protection relay is cut off for islanding detection Thelimitation of the method is due to nonlinear load and powerquality in practice and the threshold range of harmonicdetection method is difficult to be determined thus theaccuracy detection rate decreases [7]

23 Power Changing Detection Method Power changingdetectionmethod identifies islanding effect based on the out-put power changing of inverter When the main supply is dis-connected from the distributed power source the outputpower of inverter is significantly different from normal par-allel connection and the power variation rate is easily mea-sured by power detection instrument When the outputpower of PV power system approaches the load consumedpower the power of load is still in equilibrium state Whenthe PV power system cuts off the main supply the outputvoltage still keeps the magnitude and frequency variation ofnormal main supply and it will result in misoperation of thisdetection method [8]

3 Acquisition Method of Characteristic Signal

NonautonomousChuarsquos circuit (Figure 2)whichwe proposedin this study is a characteristic signal acquisition platformwhich connects to the utility power with PV power systemto obtain the voltage waveforms 119881

119860and 119881

119861 The waveforms

are used as the inputs to chaotic signal synchronization andextension detectionmethod to determinewhether the island-ing effect occurs to instantly initiate the protection relays





Protection relays


AC load

Grid system

Transfomer

Transfomer

DCDCconverter

DCACinverter

VA

VBVin




1198621

1198891198811198881

119889119905= minus119892119881

1198881+ 1198941198711minus 1198941198712

1198622

1198891198811198882

119889119905= 1198941198711minus 119894119877119871

1198711

1198891198941198711

119889119905= minus1198811198881minus 11989411987111198771+ 119881119894119899sin (2120587119896119891119905)

1198712

1198891198941198712

119889119905= 1198811198881minus 1198811198882minus 11989411987121198775

(1)


is defined as

119894119877119871= 1198661198861198811198882+1

2(119866119887minus 119866119886) [10038161003816100381610038161198811198882 + 119864119886

1003816100381610038161003816 minus10038161003816100381610038161198811198882 minus 119864119886

1003816100381610038161003816] (2)

where 119866119886and 119866


119886is breakpoint


119860and 119881




lim119905rarrinfin




Master

1= 1198911(1199091 1199092 119909

119899)

2= 1198912(1199091 1199092 119909

119899)

119899= 119891119899(1199091 1199092 119909

119899)

(4)

Slave

1199101= 1198911(1199101 1199102 119910

119899)

1199102= 1198912(1199101 1199102 119910

119899)

119910119899= 119891119899(1199101 1199102 119910

119899)

(5)



1198901= 1199101minus 1199091

1198902= 1199102minus 1199092

119890119899= 119910119899minus 119909119899

1198901= 1198911(1199091 1199092 119909

119899) minus 1198911(1199101 1199102 119910

119899) = 119867

1

1198902= 1198912(1199091 1199092 119909

119899) minus 1198912(1199101 1199102 119910

119899) = 119867

2

119890119899= 119891119899(1199091 1199092 119909

119899) minus 119891119899(1199101 1199102 119910

119899) = 119867

119899

(6)






x10 c a b d

Normal

Dips

Swells

Harmonic

Islanding

Chuarsquos circuit

VA

VB K(x)

minus1


Vin

VA VB

R1

R2

R3

R5

R6C2C1 RL

R4

L1

L2

minus15V

+15V

minus

+


20

15

10

5

0

10 2 3 4 5

VA

VB

minus15

minus20

minus10

minus5

V(V

)

t (s)times10

minus2



Master

1= 120572 (119909

2minus 1199091)

2= 1205731199091minus 11990911199093minus 1199092

3= 11990911199092minus 1205741199093

(7)

Slave

1199101= 120572 (119910

2minus 1199101)

1199102= 1205731199101minus 11991011199103minus 1199102

1199103= 11991011199102minus 1205741199103

(8)

[[

[

1198901

1198902

1198903

]]

]

=[[

[

minus120572 120572 0

120573 minus1 0

0 0 minus120574

]]

]

[[

[

1198901

1198902

1198903

]]

]

+[[

[

11991021199103minus 11990921199093

minus11991011199103+ 11990911199093

11991011199102minus 11990911199092

]]

]

(9)

where 1198901 1198902 and 119890




islanding analysis








4

2

0

0 2 4 6

0

5000

0 5000 10000

times105

times104

e 2e 2

minus2

minus4

minus5000

minus5000

minus10000

V2

V4

V8

V6

V5 V7

V3

V1 e1

e1



1 1198882 119888


and 119881 = [V1 V2 V



119877 =

[[[[[[

[

119873 1198881 V1

1198882 V2

119888119899 V119899

]]]]]]

]

=

[[[[[[

[

1198771

1198772

119877119899

]]]]]]

]

(10)

where 119877119894= (119873 119888



119877 = (119873 119862 119881) (11)





Normal 1198771=

11987311198881 ⟨minus13 minus11⟩

1198882 ⟨43 49⟩

1198883 ⟨27 30⟩

1198884 ⟨minus131 minus75⟩


1198886 ⟨34 36⟩

1198887 ⟨6 63⟩


Islanding 1198772=

11987321198881 ⟨minus12 minus11⟩

1198882 ⟨57 6⟩

1198883 ⟨74 84⟩



1198886 ⟨07 08⟩

1198887 ⟨27 31⟩


Harmonics 1198773=

11987331198881 ⟨minus15 minus1⟩

1198882 ⟨57 4⟩

1198883 ⟨28 296⟩

1198884 ⟨minus15 minus101⟩


1198886 ⟨33 36⟩

1198887 ⟨6 63⟩



11987341198881 ⟨minus12 minus13⟩

1198882 ⟨41 5⟩

1198883 ⟨32 786⟩

1198884 ⟨minus435 minus91⟩


1198886 ⟨4 83⟩

1198887 ⟨65 88⟩



Table 1 Continued



11987351198881 ⟨minus12 minus09⟩

1198882 ⟨01 48⟩

1198883 ⟨13 236⟩

1198884 ⟨minus116 minus03⟩


1198886 ⟨004 39⟩

1198887 ⟨05 65⟩

1198888 ⟨minus58 minus007⟩




119877119901=

1198731199011198881 ⟨minus16 minus08⟩

1198882 ⟨01 8⟩

1198883 ⟨12 80⟩



1198886 ⟨0 85⟩

1198887 ⟨04 9⟩

1198888 ⟨minus98 0⟩

(12)




119877119894=

[[[[[[

[

11987711989411988811198811198941

11988821198811198942

11988881198811198948

]]]]]]

]

119894 = 1 2 5 (13)




119877119905= (119877119905 119862 119881119905) =

[[[[[[

[

1198771199051198881V1199051

1198882V1199052

1198888V1199058

]]]]]]

]

(14)


1to 1198888


119870119894119895(V119905119895) =

minus05120588 (V119905119895 119881119894119895)

10038161003816100381610038161003816119881119894119895

10038161003816100381610038161003816

V119905119895isin 119881119894119895

120588 (V119905119895 119881119894119895)

120588 (V119905119895 119881119901119895) minus 120588 (V

119905119895 119881119894119895)

V119905119895notin 119881119894119895

119894 = 1 2 5 119895 = 1 2 8

(15)

where

120588 (V119905119895 119881119894119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119894119895+ 119887119894119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119894119895minus 119886119894119895

2

120588 (V119905119895 119881119901119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119901119895+ 119887119901119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119901119895minus 119886119901119895

2

(16)





= ⟨119886119901119895 119887119901119895⟩ 119886119901119895 119895th lower




8


1to 1198888are 01 02 02 01



as

120582119894=

8

sum

119895=1

119882119895119870119894119895 119894 = 1 2 5 (17)


1205821015840


119894 = 1 2 5 (18)


IF (1205821015840

119896= 1) than the type of 119877

119905is 119873119896 (19)


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000



119896= 1







0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000





0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of





Protection relays


AC load

Grid system

Transfomer

Transfomer

DCDCconverter

DCACinverter

VA

VBVin




1198621

1198891198811198881

119889119905= minus119892119881

1198881+ 1198941198711minus 1198941198712

1198622

1198891198811198882

119889119905= 1198941198711minus 119894119877119871

1198711

1198891198941198711

119889119905= minus1198811198881minus 11989411987111198771+ 119881119894119899sin (2120587119896119891119905)

1198712

1198891198941198712

119889119905= 1198811198881minus 1198811198882minus 11989411987121198775

(1)


is defined as

119894119877119871= 1198661198861198811198882+1

2(119866119887minus 119866119886) [10038161003816100381610038161198811198882 + 119864119886

1003816100381610038161003816 minus10038161003816100381610038161198811198882 minus 119864119886

1003816100381610038161003816] (2)

where 119866119886and 119866


119886is breakpoint


119860and 119881




lim119905rarrinfin




Master

1= 1198911(1199091 1199092 119909

119899)

2= 1198912(1199091 1199092 119909

119899)

119899= 119891119899(1199091 1199092 119909

119899)

(4)

Slave

1199101= 1198911(1199101 1199102 119910

119899)

1199102= 1198912(1199101 1199102 119910

119899)

119910119899= 119891119899(1199101 1199102 119910

119899)

(5)



1198901= 1199101minus 1199091

1198902= 1199102minus 1199092

119890119899= 119910119899minus 119909119899

1198901= 1198911(1199091 1199092 119909

119899) minus 1198911(1199101 1199102 119910

119899) = 119867

1

1198902= 1198912(1199091 1199092 119909

119899) minus 1198912(1199101 1199102 119910

119899) = 119867

2

119890119899= 119891119899(1199091 1199092 119909

119899) minus 119891119899(1199101 1199102 119910

119899) = 119867

119899

(6)






x10 c a b d

Normal

Dips

Swells

Harmonic

Islanding

Chuarsquos circuit

VA

VB K(x)

minus1


Vin

VA VB

R1

R2

R3

R5

R6C2C1 RL

R4

L1

L2

minus15V

+15V

minus

+


20

15

10

5

0

10 2 3 4 5

VA

VB

minus15

minus20

minus10

minus5

V(V

)

t (s)times10

minus2



Master

1= 120572 (119909

2minus 1199091)

2= 1205731199091minus 11990911199093minus 1199092

3= 11990911199092minus 1205741199093

(7)

Slave

1199101= 120572 (119910

2minus 1199101)

1199102= 1205731199101minus 11991011199103minus 1199102

1199103= 11991011199102minus 1205741199103

(8)

[[

[

1198901

1198902

1198903

]]

]

=[[

[

minus120572 120572 0

120573 minus1 0

0 0 minus120574

]]

]

[[

[

1198901

1198902

1198903

]]

]

+[[

[

11991021199103minus 11990921199093

minus11991011199103+ 11990911199093

11991011199102minus 11990911199092

]]

]

(9)

where 1198901 1198902 and 119890




islanding analysis








4

2

0

0 2 4 6

0

5000

0 5000 10000

times105

times104

e 2e 2

minus2

minus4

minus5000

minus5000

minus10000

V2

V4

V8

V6

V5 V7

V3

V1 e1

e1



1 1198882 119888


and 119881 = [V1 V2 V



119877 =

[[[[[[

[

119873 1198881 V1

1198882 V2

119888119899 V119899

]]]]]]

]

=

[[[[[[

[

1198771

1198772

119877119899

]]]]]]

]

(10)

where 119877119894= (119873 119888



119877 = (119873 119862 119881) (11)





Normal 1198771=

11987311198881 ⟨minus13 minus11⟩

1198882 ⟨43 49⟩

1198883 ⟨27 30⟩

1198884 ⟨minus131 minus75⟩


1198886 ⟨34 36⟩

1198887 ⟨6 63⟩


Islanding 1198772=

11987321198881 ⟨minus12 minus11⟩

1198882 ⟨57 6⟩

1198883 ⟨74 84⟩



1198886 ⟨07 08⟩

1198887 ⟨27 31⟩


Harmonics 1198773=

11987331198881 ⟨minus15 minus1⟩

1198882 ⟨57 4⟩

1198883 ⟨28 296⟩

1198884 ⟨minus15 minus101⟩


1198886 ⟨33 36⟩

1198887 ⟨6 63⟩



11987341198881 ⟨minus12 minus13⟩

1198882 ⟨41 5⟩

1198883 ⟨32 786⟩

1198884 ⟨minus435 minus91⟩


1198886 ⟨4 83⟩

1198887 ⟨65 88⟩



Table 1 Continued



11987351198881 ⟨minus12 minus09⟩

1198882 ⟨01 48⟩

1198883 ⟨13 236⟩

1198884 ⟨minus116 minus03⟩


1198886 ⟨004 39⟩

1198887 ⟨05 65⟩

1198888 ⟨minus58 minus007⟩




119877119901=

1198731199011198881 ⟨minus16 minus08⟩

1198882 ⟨01 8⟩

1198883 ⟨12 80⟩



1198886 ⟨0 85⟩

1198887 ⟨04 9⟩

1198888 ⟨minus98 0⟩

(12)




119877119894=

[[[[[[

[

11987711989411988811198811198941

11988821198811198942

11988881198811198948

]]]]]]

]

119894 = 1 2 5 (13)




119877119905= (119877119905 119862 119881119905) =

[[[[[[

[

1198771199051198881V1199051

1198882V1199052

1198888V1199058

]]]]]]

]

(14)


1to 1198888


119870119894119895(V119905119895) =

minus05120588 (V119905119895 119881119894119895)

10038161003816100381610038161003816119881119894119895

10038161003816100381610038161003816

V119905119895isin 119881119894119895

120588 (V119905119895 119881119894119895)

120588 (V119905119895 119881119901119895) minus 120588 (V

119905119895 119881119894119895)

V119905119895notin 119881119894119895

119894 = 1 2 5 119895 = 1 2 8

(15)

where

120588 (V119905119895 119881119894119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119894119895+ 119887119894119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119894119895minus 119886119894119895

2

120588 (V119905119895 119881119901119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119901119895+ 119887119901119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119901119895minus 119886119901119895

2

(16)





= ⟨119886119901119895 119887119901119895⟩ 119886119901119895 119895th lower




8


1to 1198888are 01 02 02 01



as

120582119894=

8

sum

119895=1

119882119895119870119894119895 119894 = 1 2 5 (17)


1205821015840


119894 = 1 2 5 (18)


IF (1205821015840

119896= 1) than the type of 119877

119905is 119873119896 (19)


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000



119896= 1







0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000





0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of




x10 c a b d

Normal

Dips

Swells

Harmonic

Islanding

Chuarsquos circuit

VA

VB K(x)

minus1


Vin

VA VB

R1

R2

R3

R5

R6C2C1 RL

R4

L1

L2

minus15V

+15V

minus

+


20

15

10

5

0

10 2 3 4 5

VA

VB

minus15

minus20

minus10

minus5

V(V

)

t (s)times10

minus2



Master

1= 120572 (119909

2minus 1199091)

2= 1205731199091minus 11990911199093minus 1199092

3= 11990911199092minus 1205741199093

(7)

Slave

1199101= 120572 (119910

2minus 1199101)

1199102= 1205731199101minus 11991011199103minus 1199102

1199103= 11991011199102minus 1205741199103

(8)

[[

[

1198901

1198902

1198903

]]

]

=[[

[

minus120572 120572 0

120573 minus1 0

0 0 minus120574

]]

]

[[

[

1198901

1198902

1198903

]]

]

+[[

[

11991021199103minus 11990921199093

minus11991011199103+ 11990911199093

11991011199102minus 11990911199092

]]

]

(9)

where 1198901 1198902 and 119890




islanding analysis








4

2

0

0 2 4 6

0

5000

0 5000 10000

times105

times104

e 2e 2

minus2

minus4

minus5000

minus5000

minus10000

V2

V4

V8

V6

V5 V7

V3

V1 e1

e1



1 1198882 119888


and 119881 = [V1 V2 V



119877 =

[[[[[[

[

119873 1198881 V1

1198882 V2

119888119899 V119899

]]]]]]

]

=

[[[[[[

[

1198771

1198772

119877119899

]]]]]]

]

(10)

where 119877119894= (119873 119888



119877 = (119873 119862 119881) (11)





Normal 1198771=

11987311198881 ⟨minus13 minus11⟩

1198882 ⟨43 49⟩

1198883 ⟨27 30⟩

1198884 ⟨minus131 minus75⟩


1198886 ⟨34 36⟩

1198887 ⟨6 63⟩


Islanding 1198772=

11987321198881 ⟨minus12 minus11⟩

1198882 ⟨57 6⟩

1198883 ⟨74 84⟩



1198886 ⟨07 08⟩

1198887 ⟨27 31⟩


Harmonics 1198773=

11987331198881 ⟨minus15 minus1⟩

1198882 ⟨57 4⟩

1198883 ⟨28 296⟩

1198884 ⟨minus15 minus101⟩


1198886 ⟨33 36⟩

1198887 ⟨6 63⟩



11987341198881 ⟨minus12 minus13⟩

1198882 ⟨41 5⟩

1198883 ⟨32 786⟩

1198884 ⟨minus435 minus91⟩


1198886 ⟨4 83⟩

1198887 ⟨65 88⟩



Table 1 Continued



11987351198881 ⟨minus12 minus09⟩

1198882 ⟨01 48⟩

1198883 ⟨13 236⟩

1198884 ⟨minus116 minus03⟩


1198886 ⟨004 39⟩

1198887 ⟨05 65⟩

1198888 ⟨minus58 minus007⟩




119877119901=

1198731199011198881 ⟨minus16 minus08⟩

1198882 ⟨01 8⟩

1198883 ⟨12 80⟩



1198886 ⟨0 85⟩

1198887 ⟨04 9⟩

1198888 ⟨minus98 0⟩

(12)




119877119894=

[[[[[[

[

11987711989411988811198811198941

11988821198811198942

11988881198811198948

]]]]]]

]

119894 = 1 2 5 (13)




119877119905= (119877119905 119862 119881119905) =

[[[[[[

[

1198771199051198881V1199051

1198882V1199052

1198888V1199058

]]]]]]

]

(14)


1to 1198888


119870119894119895(V119905119895) =

minus05120588 (V119905119895 119881119894119895)

10038161003816100381610038161003816119881119894119895

10038161003816100381610038161003816

V119905119895isin 119881119894119895

120588 (V119905119895 119881119894119895)

120588 (V119905119895 119881119901119895) minus 120588 (V

119905119895 119881119894119895)

V119905119895notin 119881119894119895

119894 = 1 2 5 119895 = 1 2 8

(15)

where

120588 (V119905119895 119881119894119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119894119895+ 119887119894119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119894119895minus 119886119894119895

2

120588 (V119905119895 119881119901119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119901119895+ 119887119901119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119901119895minus 119886119901119895

2

(16)





= ⟨119886119901119895 119887119901119895⟩ 119886119901119895 119895th lower




8


1to 1198888are 01 02 02 01



as

120582119894=

8

sum

119895=1

119882119895119870119894119895 119894 = 1 2 5 (17)


1205821015840


119894 = 1 2 5 (18)


IF (1205821015840

119896= 1) than the type of 119877

119905is 119873119896 (19)


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000



119896= 1







0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000





0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


4

2

0

0 2 4 6

0

5000

0 5000 10000

times105

times104

e 2e 2

minus2

minus4

minus5000

minus5000

minus10000

V2

V4

V8

V6

V5 V7

V3

V1 e1

e1



1 1198882 119888


and 119881 = [V1 V2 V



119877 =

[[[[[[

[

119873 1198881 V1

1198882 V2

119888119899 V119899

]]]]]]

]

=

[[[[[[

[

1198771

1198772

119877119899

]]]]]]

]

(10)

where 119877119894= (119873 119888



119877 = (119873 119862 119881) (11)





Normal 1198771=

11987311198881 ⟨minus13 minus11⟩

1198882 ⟨43 49⟩

1198883 ⟨27 30⟩

1198884 ⟨minus131 minus75⟩


1198886 ⟨34 36⟩

1198887 ⟨6 63⟩


Islanding 1198772=

11987321198881 ⟨minus12 minus11⟩

1198882 ⟨57 6⟩

1198883 ⟨74 84⟩



1198886 ⟨07 08⟩

1198887 ⟨27 31⟩


Harmonics 1198773=

11987331198881 ⟨minus15 minus1⟩

1198882 ⟨57 4⟩

1198883 ⟨28 296⟩

1198884 ⟨minus15 minus101⟩


1198886 ⟨33 36⟩

1198887 ⟨6 63⟩



11987341198881 ⟨minus12 minus13⟩

1198882 ⟨41 5⟩

1198883 ⟨32 786⟩

1198884 ⟨minus435 minus91⟩


1198886 ⟨4 83⟩

1198887 ⟨65 88⟩



Table 1 Continued



11987351198881 ⟨minus12 minus09⟩

1198882 ⟨01 48⟩

1198883 ⟨13 236⟩

1198884 ⟨minus116 minus03⟩


1198886 ⟨004 39⟩

1198887 ⟨05 65⟩

1198888 ⟨minus58 minus007⟩




119877119901=

1198731199011198881 ⟨minus16 minus08⟩

1198882 ⟨01 8⟩

1198883 ⟨12 80⟩



1198886 ⟨0 85⟩

1198887 ⟨04 9⟩

1198888 ⟨minus98 0⟩

(12)




119877119894=

[[[[[[

[

11987711989411988811198811198941

11988821198811198942

11988881198811198948

]]]]]]

]

119894 = 1 2 5 (13)




119877119905= (119877119905 119862 119881119905) =

[[[[[[

[

1198771199051198881V1199051

1198882V1199052

1198888V1199058

]]]]]]

]

(14)


1to 1198888


119870119894119895(V119905119895) =

minus05120588 (V119905119895 119881119894119895)

10038161003816100381610038161003816119881119894119895

10038161003816100381610038161003816

V119905119895isin 119881119894119895

120588 (V119905119895 119881119894119895)

120588 (V119905119895 119881119901119895) minus 120588 (V

119905119895 119881119894119895)

V119905119895notin 119881119894119895

119894 = 1 2 5 119895 = 1 2 8

(15)

where

120588 (V119905119895 119881119894119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119894119895+ 119887119894119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119894119895minus 119886119894119895

2

120588 (V119905119895 119881119901119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119901119895+ 119887119901119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119901119895minus 119886119901119895

2

(16)





= ⟨119886119901119895 119887119901119895⟩ 119886119901119895 119895th lower




8


1to 1198888are 01 02 02 01



as

120582119894=

8

sum

119895=1

119882119895119870119894119895 119894 = 1 2 5 (17)


1205821015840


119894 = 1 2 5 (18)


IF (1205821015840

119896= 1) than the type of 119877

119905is 119873119896 (19)


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000



119896= 1







0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000





0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


Table 1 Continued



11987351198881 ⟨minus12 minus09⟩

1198882 ⟨01 48⟩

1198883 ⟨13 236⟩

1198884 ⟨minus116 minus03⟩


1198886 ⟨004 39⟩

1198887 ⟨05 65⟩

1198888 ⟨minus58 minus007⟩




119877119901=

1198731199011198881 ⟨minus16 minus08⟩

1198882 ⟨01 8⟩

1198883 ⟨12 80⟩



1198886 ⟨0 85⟩

1198887 ⟨04 9⟩

1198888 ⟨minus98 0⟩

(12)




119877119894=

[[[[[[

[

11987711989411988811198811198941

11988821198811198942

11988881198811198948

]]]]]]

]

119894 = 1 2 5 (13)




119877119905= (119877119905 119862 119881119905) =

[[[[[[

[

1198771199051198881V1199051

1198882V1199052

1198888V1199058

]]]]]]

]

(14)


1to 1198888


119870119894119895(V119905119895) =

minus05120588 (V119905119895 119881119894119895)

10038161003816100381610038161003816119881119894119895

10038161003816100381610038161003816

V119905119895isin 119881119894119895

120588 (V119905119895 119881119894119895)

120588 (V119905119895 119881119901119895) minus 120588 (V

119905119895 119881119894119895)

V119905119895notin 119881119894119895

119894 = 1 2 5 119895 = 1 2 8

(15)

where

120588 (V119905119895 119881119894119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119894119895+ 119887119894119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119894119895minus 119886119894119895

2

120588 (V119905119895 119881119901119895) =

100381610038161003816100381610038161003816100381610038161003816

V119905119895minus

119886119901119895+ 119887119901119895

2

100381610038161003816100381610038161003816100381610038161003816

minus

119887119901119895minus 119886119901119895

2

(16)





= ⟨119886119901119895 119887119901119895⟩ 119886119901119895 119895th lower




8


1to 1198888are 01 02 02 01



as

120582119894=

8

sum

119895=1

119882119895119870119894119895 119894 = 1 2 5 (17)


1205821015840


119894 = 1 2 5 (18)


IF (1205821015840

119896= 1) than the type of 119877

119905is 119873119896 (19)


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000



119896= 1







0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000





0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


4

4 6

2

2

0

0

times105

times104

e 2

minus2

minus4

e1

e1

0

5000

0 5000 10000

e 2

minus5000

minus5000

minus10000



119896= 1







0 05 1 15 2 25

4

6

2

0

times104

times104

e 2

minus2

minus4

minus6

e1

e1

0

2000

0 2000 4000 6000

e 2

minus2000

minus2000

minus4000





0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 2 4 6 8

5

0

0

1

0 5000 10000

05

minus5

minus15

minus1

minus05

times104

times104

times105

e 2e 2

e1

e1





0 5000 10000 15000

4

6

2

1

0

0

times104

times104

e 2e 2

minus2

minus4

minus60 05 1 15 2 25

times104e1

minus2

minus1

e1






Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of



Sample V1

V2

V3

V4

V5

V6

V7

V8





Harmoniccorrelation

Swellscorrelation

Dipscorrelation

Actualcategory

Detectioncategory













0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of


0 05 1 15 2 25

4

6

2

0

0

0

1

2

2000 4000 6000

3

times104

times103

times104

e 2

minus2

minus4

minus6

e1

e1

e 2

minus1

minus2

minus2000

minus3

minus4






References





























Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of










Journal of

Chemistry


Advances in

Physical Chemistry



Journal of

Volume 2014














Journal of

Spectroscopy



Journal of


Quantum Chemistry






CatalystsJournal of

Date post:	04-Oct-2021
Category:	Documents
Upload:	others
View:	3 times
Download:	0 times

Research Article Application of Extension Theory with ...

Documents