Active Power Oscillation Property Classification of ... · tem and a practical simplified power...

Research ArticleActive Power Oscillation Property Classification ofElectric Power Systems Based on SVM

Ju Liu1 Wei Yao1 Jinyu Wen1 Haibo He2 and Xueyang Zheng1

1 The State Key Laboratory of Advanced Electromagnetic Engineering and TechnologyHuazhong University of Science and Technology Wuhan 430074 China

2Department of Electrical Computer and Biomedical Engineering University of Rhode Island Kingston RI 02881 USA

Correspondence should be addressed to Jinyu Wen jinyuwenhusteducn

Received 24 January 2014 Revised 21 April 2014 Accepted 23 April 2014 Published 6 May 2014

Academic Editor Hongjie Jia

Copyright copy 2014 Ju Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Nowadays low frequency oscillation has become a major problem threatening the security of large-scale interconnected powersystemsAccording to generationmechanism active power oscillation of electric power systems can be classified into two categoriesfree oscillation and forced oscillation The former results from poor or negative damping ratio of power system and externalperiodic disturbance may lead to the latterThus control strategies to suppress the oscillations are totally different Distinction fromeach other of those two different kinds of power oscillations becomes a precondition for suppressing the oscillations with propermeasures This paper proposes a practical approach for power oscillation classification by identifying real-time power oscillationcurves Hilbert transform is employed to obtain envelope curves of the power oscillation curves Twenty sampling points of theenvelope curve are selected as the feature matrices to train and test the supporting vector machine (SVM) The tests on the 16-machine 68-bus benchmark power system and a real power system in China indicate that the proposed oscillation classificationmethod is of high precision

1 Introduction

Damping ratio is a key factor in electric power oscillationsWith the interconnection of large-scale power grids via highvoltage long distance transmission lines coupling betweensynchronous generators becomes weaker which leads to pooror negative damping ratio of power systems [1 2] Poweroscillation has become a major problem threatening thesecurity of large-scale interconnected power systems [3 4] Inorder to safely operate the power system monitoring classi-fication and suppression of the electric power oscillations inpower grids has captured exponentially increasing attentionin power engineering community in the past decade

This paper focuses on the active power oscillation prop-erty classification problem According to generation mecha-nism active power oscillations of electric systems can be clas-sified into two categories [2] One is free oscillation resultedby negative damping ratio of power systems It could also becalled negative damping oscillation Worldwide acceptedexplanation mechanism of this kind of power oscillation is

based on the complex torque analysis method proposed byDeMello and Concordia in 1969 [5] When the reactance oftransmission system is large or power output of generators ishigh the negative damping torque produced by lagging phaseof quick excitation circuit counteracts the original positivedamping of generatorsrsquo damping windings This will lead tonegative damping ratio of power grids and cause a poweroscillation with increasing amplitude [6 7] Reducing trans-mission power of tie lines or installing PSS equipment whichenhances damping torque through phase compensation hasbeen proved to be important measure for suppressing nega-tive damping power oscillation [8 9]

The other kind of active power oscillation is forced poweroscillation which is explained by the resonance mechanism[10 11] Power resonance of generators will be provokedwhenthe frequency of a small periodic disturbance occurring at thepower system is equal or close to the systemrsquos natural fre-quency This kind of power oscillation is characterized byfast oscillation start and rapid decay after losing oscillationsource Some researchers have demonstrated that periodic

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 218647 9 pageshttpdxdoiorg1011552014218647

2 Journal of Applied Mathematics

disturbances of excitation circuit turbine speed governorsystem and active power load could stimulate forced poweroscillation [12ndash14] Separating the periodic disturbancesources from the system is the most effective countermeasureto eliminate its influence quickly [15]

Thus negative damping oscillation and forced oscillationhave different generation mechanisms with different cop-ing measures It is extremely important and necessary todistinguish them from each other However both of themhave similar oscillation forms with increasing amplitudes atinitial stage and probably develop into a constant amplitudeoscillation in the end Correct and rapid identification of theoscillation property becomes a difficult problem to be solvedNowadays researches on power oscillation classificationmostly concentrate on simulation after oscillation accidentsSystem response curves of simulation under negative damp-ing oscillation condition and forced oscillation conditionare compared with the actual oscilloscope records of powersystems to judge the oscillation property of power oscillationaccidents [16 17] An online oscillation property classificationmethod based on difference analysis of the oscillation curvesis proposed in [18] but there is a certain error of the dif-ferential calculation A fundamental theory of forced poweroscillation in a power system has been recommended in [19]which has proved that the forced power oscillation has dif-ferent envelop curve from the negative damping oscillationThus the active power oscillation property could be identifiedonline by distinguishing their envelop curves

Statistical learning theory and support vector machine(SVM) have given a systemic theoretical explanation aboutpattern recognition under circumstances of finite samplesMany problems like model-choosing overfitting nonlineardisaster of dimensionality and local minimum which havelong hindered the development of machine learning are nowsolved to a great extent [20 21] So far SVM has successfullybeen applied tomany fields like fault diagnosis speech recog-nition image recognition and text classification [22ndash24] In[25] two types of SVM were implemented to effectivelyclassify different kinds of power quality disturbances

This paper proposes a practical approach for power oscil-lation classification by recognition of real-time power oscil-lation curves utilizing SVM Hilbert transform is employedto obtain envelope curves of the power oscillation curvesTwenty sampling points on the envelope curve of power oscil-lation are selected for feature extraction Then forty poweroscillation curves are employed as samples to train the sup-porting vector machine At last three tests on the 16-machine68-bus benchmark system and a real power system in Chinaindicate that the proposed oscillation classification methodpossesses good precision

The rest of this paper is organized as follows In Section 2a Hilbert transform based oscillation feature extractionscheme is proposed Section 3 introduces a power oscillationproperty classification method utilizing SVM Case studyis undertaken on the 16-machine 68-bus benchmark sys-tem and a practical simplified power system of China toverify the effectiveness of the proposed oscillation propertyclassification method in Section 4 Conclusions are given inSection 5

2 Feature Extraction of Power Oscillations

21 Introduction of Power Oscillations When a free oscilla-tion happens in a power system the rotor angle rotationspeed of synchronous generators and relevant electric vari-ables (such as the power flow of transmission lines and busvoltages) will oscillate accordingly among which the powerflow of transmission line 119875

119894119895can be expressed as follows

119875119894119895=

119899

sum

119894=1

1198961198941199091198940119890120582119894119905 (1)

where 119896119894denotes the participation factor of oscillation pattern

119894 1199091198940is the initial value of state variable according to oscil-

lation pattern 119894 120582119894denotes the eigenvalue of the oscillation

pattern 119894 Generally 120582119894can be expressed as

12058212= 120572 plusmn 119895120596 (2)

In (2) 120572 indicates the damping performance of the oscilla-tion120596 denotes the frequency characteristic of the oscillationthe damping ratio of the oscillation is defined as

120585 =minus120572

radic1205722 + 1205962 (3)

The oscillation curve of any free oscillation mode can beexpressed as

119875119894119895= 1198611119890120572119905 sin (120596119905 + 120601

1) (4)

Therefore when any oscillation mode with 120585 lt 0 (ie120572 gt 0) exists in the power system any small disturbancewould stimulate thismode and result in a power oscillation ofthe tie line with increasing amplitude as shown in Figure 1(a)That is the so-called negative damping oscillation If all thedamping ratios of oscillation patterns are larger than zerothe small disturbance occurring at the power system wouldbe restrained by the positive damping

For any forced oscillation in the power system with anextreme disturbance like ℎ sin(120596

119889119905) it can be expressed as a

second-order system

+ 2120585120596 + 1205962119909 = ℎ sin (120596

119889119905) (5)

Solving (5) the oscillation curve of forced oscillation is

119875119894119895= 1198611119890120572119905 sin (120596119905 + 120601

1) + 1198612sin (120596

119889119905 + 1206012) (6)

Under the circumstance of 120585 gt 0 (ie 120572 lt 0) the freeoscillation part 119861

1119890120572119905 sin(120596119905+120601

1) in (6) will decay But power

resonance would be motivated and large oscillation wouldfollow as shown in Figure 1(b) if the oscillation frequency 120596

119889

of the extreme disturbance is equal or close to the systemrsquosnatural frequency 120596

From the analysis above it can be concluded that theenvelope curve of negative damping oscillation increases as1198611119890120572119905The envelope curve of forced oscillationwill increase at

the initial stage Then it transits to an oscillation as (ℎ2(120596 minus120596119889)120596)119890120572119905 sin((120596minus120596

119889)119905+1206012)+1198612at the transitory stage At last it

will die down to a constant as1198612at stable stage [19]Thus there

is obvious distinction between the envelope curves of the twokinds of power oscillations

Journal of Applied Mathematics 3

600

650

700

750

800Ac

tive p

ower

(MW

)

Time (s)0 5 10 15 20

Envelope curve

(a) Negative damping oscillation

Activ

e pow

er (M

W)

0 5200

300

400

500

Time (s)10 15 20

Envelope curve

(b) Forced oscillation

Figure 1 Power oscillation curve of tie line

22 Hilbert Transform Hilbert transform (HT) [26] offers aneffective approach to extract envelope curves as oscillationfeatures for different kinds of power oscillations For the fol-lowing power oscillation signal in a power grid

119906 (119905) = 119860 (119905) cos120593 (119905) (7)

HT could be applied to acquire its complex conjugate signal

V (119905) = 119867 [119906 (119905)] =1

120587int

+infin

minusinfin

119906 (120591)

119905 minus 120591119889120591 (8)

V(119905) is a sinusoidal signal similar to 119906(119905) It could beexpressed as V(119905) = 119860(119905) sin120593(119905) 119906(119905) and V(119905) constitute aHTpair which makes up the following HT analytic signal

119908 (119905) = 119860 (119905) cos120593 (119905) + 119895119860 (119905) sin120593 (119905) (9)

Namely

119908 (119905) = 119860 (119905) 119890119895120593(119905)

119860 (119905) = radic1199062(119905) + V2 (119905)

(10)

Therefore the amplitude 119860(119905) of HT analytic signalreflects global change trend of the signal 119860(119905) represents theenvelope of the oscillation signal and could be calculated byusing (8) when the analytic signal is a signal with intrinsicoscillation mode

23 Feature Extraction The envelope curve of power oscil-lation will be obtained by HT after the power oscillation inpower grid is detected by the wide area measurement system(WAMS) Then the envelope curve is normalized accordingto the steady value before oscillation happens Finally twentyevenly spaced points on the envelope curve are selected inevery 2119879 interval (119879 is oscillation period of the power oscil-lation) to constitute a group of feature matrices as shown inFigure 2

It is well known that the power oscillation in a power sys-tem generally combines various swing modes If there is onlyone dominant oscillation mode the oscillation amplitude ofother oscillationmodes is small compared with the dominantoscillationmodeThe influence of other oscillationmodes ondominant oscillation mode can be neglected If there are twoor more than two dominant oscillation modes the dominant

0 5 10 15 2009

095

1

105

11

Activ

e pow

er (p

u)

Time (s)

Figure 2 Extraction of the elements in the feature matrices

oscillation mode with positive damping ratio will decay afterseconds or minutes Then only the oscillation mode withnegative damping ratio retains Meanwhile the data used forpower oscillation feature extraction contains 40 cycles of theoscillation curve After several cycles only the oscillationmode with negative damping ratio will retain and be used forthe oscillation property classification

However the power oscillation with two negative damp-ing oscillation modes has similar expression with forcedpower oscillation The beat-frequency oscillation will also beperceived in the power oscillationwith two negative dampingoscillation modes It is hard to distinguish the forced poweroscillation and the power oscillationwith two negative damp-ing oscillation modes However the power oscillation withtwo negative damping oscillation modes is very unusual inpower system thus the negative damping power oscillation inthis paper is the power oscillation with only one negativedamping oscillation mode Under this premise the featurematrices can represent the power oscillation property of thesystem

Difference of adjacent points of the feature matricesdenotes the change direction of the envelope curve and thesecond-order difference denotes its change tendency As fornegative damping oscillation the variation rate of the enve-lope curves is generally positive with oscillation amplitudegrowing more and more rapidly This means that the first-order and second-order difference of adjacent points from thefeature matrices should also be positive However for forcedoscillations the envelope curve hasminimum andmaximumpoints This indicates that the first-order or second-orderdifference of adjacent points could be zero or negative


Optimalhyperplane

PositiveNegative

Optimalmargin

Figure 3 A classification example in a two-dimensional space basedon SVM [27 28]

Therefore the feature matrices obtained in this paper at leastcontain the oscillation characteristic of the power oscillationwhich could be utilized for power oscillation property clas-sification Besides the feature matrices extraction of poweroscillation curves will be the foundation of the followingwork

3 Oscillation Property ClassificationBased on SVM

31 Basic Theory of SVM SVM-based classification methodis established on the basis of structural risk minimizationprinciple and VC theory (Vapnik and Chervonenkis theory)[27] In order to obtain excellent capability of generalizationaccording to a certain amount of samples the SVM-basedclassificationmethod seeks the best compromise between thecomplication of themodel and its learning capacityThe prin-ciple idea of SVM is as follows it firstlymaps the input vectorsin sample space into some high or even infinite dimensionalfeature space through some nonlinear mapping function120601(119909) And then the nonlinear and nonclassifiable problemin the original sample space is transformed into a linear andclassifiable problem in the high dimensional feature space Atlast in this feature space a linear decision surface is con-structed with special properties that possess the largest sep-aration margin between the two classes just as shown inFigure 3

For the given linear and separable training patterns(119909119894 119889119894)119897

119894=1 119909119894isin 119877119899 119889119894isin 1 minus1 119909

119894denotes the 119899-

dimensional input vector and 119889119894denotes the class of the sam-

ple For further explanation 1 denotes the positive class andminus1 denotes the negative class These two sets are linearly sep-arable on the condition that there are a vector 120596 and scalar 119887which satisfy

120596 sdot 119909119894+ 119887 ge 1

if 119909119894isin positive sample set namely 119910

119894= 1

120596 sdot 119909119894+ 119887 le minus1

if 119909119894isin negative sample set namely 119910

119894= minus1

(11)

If the vector 120596 has the minimum norm then the hyper-plane 119910 = 120596 sdot 119909 + 119887 separates the training data with a maxi-mal margin It is the unique and optimal hyper plane forclassification of the train data

If the training data in the input space is nonlinear in orderto construct a hyper plane to classify the nonlinear samplesone first has tomap the input vector119909 into a higher dimensionfeature space by function 120601(119909) and then takes the sign of thefunction

119910 = 120596 sdot 120601 (119909) + 119887 (12)

Consider the casewhen the training data cannot be classi-fied without error Penalty constant 119862 and some nonnegativevariables 120585

119894are introduced to punish the incorrect classi-

fication Then this idea can be expressed formally as anoptimization problem as follows

min 1

21205962+ 119862

119897

sum

119894=1

120585119894

st 119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894ge 1

120585119894ge 0

(13)

The Lagrange function for this problem is

119871 (120596 119887 120585 ΛR)

=1

21205962

+ 119862

119897

sum

119894=1

120585119894minus

119897

sum

119894=1

120572119894(119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894minus 1)

minus

119897

sum

119894=1

119903119894120585119894

(14)

where the nonnegative multipliers Λ119879 = (1205721 1205722 120572

119897) and

119877119879= (1199031 1199032 119903

119897) arise from the constraint in (13)

Using the conditions for the minimum of this function atthe extreme point

120597119871

120597120596

10038161003816100381610038161003816100381610038161003816120596=1205960

= 1205960minus

119897

sum

119894=1

120572119894119889119894120601 (119909119894) = 0 (15)

120597119871

120597119887

10038161003816100381610038161003816100381610038161003816119887=1198870

=

119897

sum

119894=1

120572119894119889119894= 0

120597119871

120597120585119894

10038161003816100381610038161003816100381610038161003816120585119894=1205851198940

= 119862 minus 120572119894minus 119903119894= 0

(16)

From (15) 1205960can be obtained as

1205960=

119897

sum

119894=1

120572119894119889119894120601 (119909119894) (17)

Thus the decision function in (12) is transformed into thefollowing form

119910 =

119897

sum

119894=1

119889119894120572119894119870(119909 119909

119894) + 119887 (18)


Substituting the expressions for 1205960 1198870 and 120585

1198940to the

Lagrange function (13)

119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895) (19)

The original convex optimization problem is also trans-formed into a quadratic optimization problem

minΛ

119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895)

st119897

sum

119894=1

120572119894119889119894= 0

0 le 120572119894le 119862

(20)

Under this circumstance the optimal Lagrangersquosmultipli-cator 1205720

119894can be obtained by using the Kuhn-Tucker condition

in quadratic programming problem As for the optimal hyp-erplane algorithm the vector 120596 can be written as a combina-tion of the training data

1205960=

119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) (21)

According to (21) and (18) the classification function ofthe training data can be obtained as

119910 = sgn[119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) 120601 (119909) + 119887] (22)

From (12)ndash(22) 120585119894denotes the slack variable and

119870(119909119894 119909119895) is the kernel function 119870(119909

119894 119909119895) is the convolution

of dot product by 120601(119909119894) and 120601(119909

119895) in the feature space

119870(119909119894 119909119895) = 120601119879(119909119894) 120601 (119909

119895) (23)

The input parameters of the kernel function are thetraining data 119909

119894of power system and the kernel function for

the SVM is independent of the power system Any functionthat satisfies Mercerrsquos condition can be used as kernel func-tion [27] By employing different kinds of kernel functionsdifferent learning machines with arbitrary types of decisionsurface can be constructed The most commonly used threekernel function will be employed in this paper [28]

linear function

119870(119909119894 119909119895) = (119909

119894sdot 119909119895+ 1)119889

(24)

sigmoid function

119870(119909119894 119909119895) = tanh (119909

119894sdot 119909119895minus 120579) (25)

radial basis function

119870(119909119894 119909119895) = exp(minus

10038161003816100381610038161003816119909119894minus 119909119895

10038161003816100381610038161003816

2

21205902) (26)

Start

Input historical data andoscillation future extraction

Parameter initialization for theSVM model

Obtain the optimal parameter ofthe SVM model

End

Oscillation incidence detection

Classify the oscillation propertiesof the power oscillation curves

Property classification of theoscillation incident

No

Yes

Figure 4 Flow chart of the oscillation property classification

32 Procedures of the Oscillation Property ClassificationWhen power oscillation happens active power oscillations atsubstation tie line and generator terminal will be detected byWAMS The oscillation property can be well classified bytraining historical data with SVM algorithm Figure 4 givesthe flow chart of the oscillation property classification Thefollowing procedures will be employed

(1) Input historical data and oscillation future extractionCollect the power oscillation data of the power oscil-lation incidence that has happened And then extractthe oscillation feature of the power oscillation data bythe proposedmethod in Section 2 as the training datafor the SVM

(2) Parameter initialization for the SVMmodel Lagrangemultiplier 120572 and threshold 119887 are assigned with ran-dom number

(3) Obtain the optimal parameter of the SVM modelBased on training samples the objective function of(20) is established and the optimal problem is solvedusing Kuhn-Tucker condition to obtain the optimal 120572and 119887

(4) Power oscillation incidence detection judge whetherthe power oscillation amplitude of the fifth cycle islarger than 50MW or not for any 500 kV tie line andrecord the oscillation curve for the oscillation prop-erty classification

(5) Classify the oscillation properties of the power oscil-lation curves Classify the property of the power oscil-lation curves recorded by the WAMS from different


1

27

474640

2 25

53

2660 28 2961

24

21

16

19

20

57

56

58

22

23

59

15

14

13

1055

1211

3

4

56

547

8

9

18 17

30

363765

64

3233

34

35

45

4439

50

51

5268

49

48 38

31

42

41

67

66

6362

G1

G2G3

G4

G5

G6

G7

G8 G9

G10

G11

G12G13

G14

G15

G16

Figure 5 Simulation example of IEEE16-machine 68-bus benchmark system

buses at the power grid using the trained SVMmodelin procedure (3) As for the recognized result of theoscillation curves 1 represents the negative dampingoscillation and minus1 represents the forced oscillation

(6) Property classification of the oscillation incident ifthe quantity of 1 is muchmore than minus1 the oscillationincident is negative damping oscillation Else if thequantity of minus1 is much more than 1 the oscillationincident is forced oscillation

4 Simulation Verification

41 Tests on 16-Machine 68-Bus System The circuit diagramof IEEE16-machine 68-bus benchmark system is shown inFigure 5 Some forced oscillating sources with oscillationfrequency close to the natural oscillation frequency of thegenerators are applied at different nodes in the power gridand ninety-six samples of power oscillation curves areobtainedThen sixty-four power oscillation curves under thenegative damping condition are obtained by lowering the gainof power system stabilizer (PSS) Twenty forced poweroscillation curves (represented by C1) and negative dampingpower oscillation curves (represented byC2) respectively areselected for training and the remaining 120 oscillation curvesare used for testing Table 1 gives the simulation result forthe oscillation property classification based on the proposedmethod A 2 times 2 confusionmatrix is constructed to show theclassification performance for each case The diagonal ele-ments represent the correctly classified power oscillationtypes The off-diagonal elements represent the misclassifica-tions As we can see from Table 1 different kernel functionfor SVMmodel will bring different classification precision forpower oscillation curves When sigmoid function is selectedas kernel function themodel has highest accuracyThereforethis paper adopts this function as kernel function for SVMmodel

Since noise is omnipresent in electrical power systemGaussian white noise is considered in the classification ofpower oscillations Different levels of noises with the noise tosignal ratio values ranging from 5 to 10 were consideredAs the noise to signal ratio increases the adaptability of SVMmodel decreases and the precision of the power oscillationproperty classification degrades Table 1 shows that the pre-cision of the power oscillation property classifycation is stillabove 88 even with the noise to signal ratio rising up to10 Thus the SVM-based proposed method can effectivelyclassify different kinds of the power oscillation

42 Analysis of Forced Power Oscillation in a Real Incident In2008 a thermal power plant was asynchronously connectedto theCentral China PowerGrid through 110 kV transmissionlines Power oscillationswere observed inHenanHunan andJiangxi power gridsThewhole oscillation durationwas abouttwo minutes with 07Hz oscillation frequency at initial stageand then the oscillatory power of the tie line increasedgradually When the oscillation stabilized the oscillatoryfrequency of the power grid was 062Hz and the oscillatoryamplitude of the active power of tie line between Henan andHubei was about 250MW Subsequently the 110 kV transmis-sion line in the fault regionwas cut off and the oscillation dieddown quickly This power oscillation incident was classifiedas a typical forced power oscillation incident and the thermalpower plant was the oscillation source [29] The active poweroscilloscope record of Yao-Shao tie line is given in Figure 6Eleven additional active power oscilloscope records of differ-ent tie lines are selected for the oscillation property classifi-cation The support vector machine trained at Section 41 isemployed to identify the oscillation property of the elevenactive power oscilloscope records and the result is shown inTable 2 It can be seen that only one power oscillation curveis misidentified as negative damping oscillation and the otherten oscillation curves are all identified as forced oscillation


Table 1 Testing result of 16-machine 68-bus benchmark system

Kernel function Signal typeNo noise 5 noise 10 noise

Polynomial kernel function

C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40

Precision 9580 Precision 9250 Precision 8830

RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40


Sigmoid kernel function

C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41


Table 2 Identification result of oscilloscope records of forced oscillation incidence

Oscillation type C1(forced oscillation)

C2(negative damping oscillation) Identification result

Number 10 1 C1

Considering identification error forced oscillation is chosenas the final identification result which is consistent with theanalytical result of the incident [29]

43 Analysis of Negative Damping Oscillation in a RealIncident In 2005 a wide area power oscillation took place inCentral China PowerGrid and thewhole oscillation lasted forabout five minutes The power oscillation frequency wasabout 077Hz Large power fluctuations were detected at theThreeGorgesHydroelectric Power Plant and its nearby powerplants in Douli Jianglin Longquan and Wanxian Thenthe operator of the power grid increased the output reactivepower of the Three Gorges Hydroelectric Power Plant anddecreased the active power output of Huanglongtan PowerPlant in northwest Hubei Subsequently the oscillationdecayed gradually until it died outThe incident was a typicalnegative damping power oscillation [18] Figure 7 shows theactive power oscilloscope record of the second Long-Dou tieline during the power oscillation Twenty active poweroscilloscope records of different tie lines at the power gridare selected to identify the oscillation property by the SVMtrained in Section 41 The identification result is shown inTable 3 Taking identification error into consideration theresult in Table 3 indicates that the oscillation incident belongsto negative damping oscillation which is also consistent withthe analysis result [18]

5 Conclusions

Active power oscillation in power systems can be classifiedinto two types negative damping oscillation and forcedoscillation Although their oscillation curves are similar theadopted suppressing control strategies are totally different

Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)

Figure 6The active power oscilloscope record of Yao-Shao tie line

Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)

Figure 7The active power oscilloscope record of the second Long-Dou tie line

Thus the oscillation property classification is the premise forsuppressing the power oscillation To properly identify theoscillation property the paper proposed an oscillation prop-erty classification method utilizing the SVM Through thesimulation analysis it can be concluded that the envelopecurves of the negative damping oscillation and forced oscil-lation are different and can be used as the feature for classi-fication As noise is omnipresent in electrical power system


Table 3 Identification result of oscilloscope records of negative damping oscillation incidence



Number 2 18 C2

Gaussian white noise is considered in the research of poweroscillation classification With the increase of the noise tosignal ratio the adaptability of SVM model decreases andthe precision of the power oscillation property classificationdegrades However even with the noise to signal ratio reach-ing up 10 the precision of the power oscillation propertyclassification is still above 88 Moreover two real incidentsin the real power systems are used to verify the proposedclassification method Results show that the identified resultsfrom the proposed method is the same with the postincidentanalytical identification result while the postanalytical iden-tification is an offline time consuming method only appliedafter the oscillation happens The proposed method can beapplied online and provides guidance for the dispatcher toselect the correct suppressing control (the negative dampingone or the forced oscillating one) during the oscillationperiod to stop the propagation of oscillations

Conflict of Interests

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

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (nos 51177057 and 51228701)

References

[1] RThirumalaivasanM Janaki andN Prabhu ldquoInvestigation ofSSR characteristics of hybrid series compensated power systemwith SSSCrdquo Advances in Power Electronics vol 2011 Article ID621818 8 pages 2011

[2] L Chen Y Min and W Hu ldquoAn energy-based method forlocation of power system oscillation sourcerdquo IEEE Transactionson Power Systems vol 28 no 2 pp 828ndash836 2013

[3] A N Hussain F Malek M A Rashid L Mohamed and N AM Affendi ldquoOptimal coordinated design of multiple dampingcontrollers based on PSS and UPFC device to improve dynamicstability in the power systemrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 965282 15 pages 2013

[4] J H Chow J J Sanchez-Gasca H Ren and S Wang ldquoPowersystem damping controller designrdquo IEEE Control Systems Mag-azine vol 20 no 4 pp 82ndash90 2000

[5] F P DeMello and C Concordia ldquoConcepts of synchronousmachine stability as affected by excitation controlrdquo IEEE Trans-actions on Power Apparatus and Systems vol 88 no 4 pp 316ndash329 1969

[6] A A Shaltout and R T H Alden ldquoAnalysis of damping andsynchronizing torquesmdashIImdasheffect of operating conditions andmachine parametersrdquo IEEE Transactions on Power Apparatusand Systems vol 98 no 5 pp 1701ndash1708 1979

[7] A L S Pereira P B de Araujo and C T Miasaki ldquoAnalysis ofelectric torque in single machine-infinite bus system usingphase shiftersrdquo IEEE Latin America Transactions vol 6 no 5pp 389ndash394 2008

[8] Y Zhang D Yue and S Hu ldquoWide-area119867infincontrol for damp-

ing interarea oscillations with event-triggered schemerdquoMathe-matical Problems in Engineering vol 2013 Article ID 517608 11pages 2013

[9] T Hussein and A Shamekh ldquoPerformance assessment of fuzzylogic power system stabilizer on north benghazi power plantrdquoConference Papers in Engineering vol 2013 Article ID 635808 6pages 2013

[10] M A Magdy and F Coowar ldquoFrequency domain analysis ofpower system forced oscillationsrdquo IEE Proceedings C Genera-tion Transmission and Distribution vol 137 no 4 pp 261ndash2681990

[11] C D Vournas N Krassas and B C Papadias ldquoAnalysis offorced oscillations in amultimachine power systemrdquo inProceed-ings of International Conference on Control vol 1 pp 443ndash448Edinburgh UK March 1991

[12] Z-Y Han R-M He and Y-H Xu ldquoPower system low fre-quency oscillation of resonance mechanism induced by turbo-pressure pulsationrdquo Proceedings of the Chinese Society of Electri-cal Engineering vol 25 no 21 pp 14ndash18 2005 (Chinese)

[13] Y Yang G Li P Liu and H Liu ldquoMechanism analysis of forcedpower oscillation events based on WAMS measured datardquo inProceedings of the 2nd Annual Conference on Electrical andControl Engineering (ICECE rsquo11) pp 5328ndash5331 September 2011

[14] WZhu Y-Q ZhouX-Y Tan andY-J Tang ldquoMechanismanal-ysis of resonance-type low-frequency oscillation caused by net-works side disturbancerdquo Proceedings of the Chinese Society ofElectrical Engineering vol 29 no 25 pp 37ndash42 2009 (Chinese)

[15] W Hu T Lin Y Gao et al ldquoDisturbance source location offorced power oscillation in regional power gridrdquo in Proceedingsof the IEEE Power Engineering and Automation Conference(PEAM rsquo11) vol 2 pp 363ndash366 September 2011

[16] X Qiu Z Yu M Lei et al ldquoWAMS based statistics and assess-ment of low frequency oscillation in Shandong power gridrdquoAutomation of Electric Power Systems vol 32 no 6 pp 95ndash982008 (Chinese)

[17] Y Yu Y Min L Chen and P Ju ldquoThe disturbance sourceidentification of forced power oscillation caused by continuouscyclical loadrdquo in Proceedings of the 4th International Conferenceon Electric Utility Deregulation and Restructuring and PowerTechnologies (DRPT rsquo11) pp 308ndash313 July 2011

[18] Y LiW S Jia andW F Li ldquoOnline identification of power osci-llation properties based on the initial period of waverdquo Proceed-ings of the CSEE vol 33 no 25 pp 54ndash60 2013 (Chinese)

[19] Y Tang ldquoFundamental theory of forced power oscillation inpower systemrdquo Power System Technology vol 30 no 10 pp 29ndash33 2006

[20] S-F Jiang D-B Fu and S-Y Wu ldquoStructural reliability assess-ment by integrating sensitivity analysis and support vectormachinerdquo Mathematical Problems in Engineering vol 2014Article ID 586191 6 pages 2014


[21] B Yan Y Cui L Zhang et al ldquoBeam structure damage identi-fication based on BP neural network and support vector mach-inerdquoMathematical Problems in Engineering vol 2014 Article ID850141 8 pages 2014

[22] M Ferras C-C Leung C Barras and J-L Gauvain ldquoCom-parison of speaker adaptation methods as feature extraction forSVM-based speaker recognitionrdquo IEEE Transactions on AudioSpeech and Language Processing vol 18 no 6 pp 1366ndash13782010

[23] Y J Bao C J Song W S Wang T Yan L Wang and L YuldquoDamage detection of bridge structure based on SVMrdquoMathe-matical Problems in Engineering vol 2013 Article ID 490372 7pages 2013

[24] B P Li andM Q HMeng ldquoTumor recognition in wireless cap-sule endoscopy images using textural features and SVM-basedfeature selectionrdquo IEEE Transactions on Information Technologyin Biomedicine vol 16 no 3 pp 323ndash329 2012

[25] HHe and J A Starzyk ldquoA self-organizing learning array systemfor power quality classification based on wavelet transformrdquoIEEE Transactions on Power Delivery vol 21 no 1 pp 286ndash2952006

[26] N E Huang and S S Shen Hilbert-Huang Transform and ItsApplications World Scientific Press Singapore 1st edition2005

[27] V N Vapnik Statistical Learning Theory John Wiley amp SonsNew York NY USA 1998

[28] C Cortes and V Vapnik ldquoSupport-vector networksrdquo MachineLearning vol 20 no 3 pp 273ndash297 1995

[29] D J Yang J Y Ding H Q Shao H P Xu and J Q HuangldquoWAMS based characteristic discrimination of negative damp-ing low frequency oscillation and Forced oscillationrdquo Automa-tion of Electric Power Systems vol 37 no 13 pp 57ndash62 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


disturbances of excitation circuit turbine speed governorsystem and active power load could stimulate forced poweroscillation [12ndash14] Separating the periodic disturbancesources from the system is the most effective countermeasureto eliminate its influence quickly [15]

Thus negative damping oscillation and forced oscillationhave different generation mechanisms with different cop-ing measures It is extremely important and necessary todistinguish them from each other However both of themhave similar oscillation forms with increasing amplitudes atinitial stage and probably develop into a constant amplitudeoscillation in the end Correct and rapid identification of theoscillation property becomes a difficult problem to be solvedNowadays researches on power oscillation classificationmostly concentrate on simulation after oscillation accidentsSystem response curves of simulation under negative damp-ing oscillation condition and forced oscillation conditionare compared with the actual oscilloscope records of powersystems to judge the oscillation property of power oscillationaccidents [16 17] An online oscillation property classificationmethod based on difference analysis of the oscillation curvesis proposed in [18] but there is a certain error of the dif-ferential calculation A fundamental theory of forced poweroscillation in a power system has been recommended in [19]which has proved that the forced power oscillation has dif-ferent envelop curve from the negative damping oscillationThus the active power oscillation property could be identifiedonline by distinguishing their envelop curves

Statistical learning theory and support vector machine(SVM) have given a systemic theoretical explanation aboutpattern recognition under circumstances of finite samplesMany problems like model-choosing overfitting nonlineardisaster of dimensionality and local minimum which havelong hindered the development of machine learning are nowsolved to a great extent [20 21] So far SVM has successfullybeen applied tomany fields like fault diagnosis speech recog-nition image recognition and text classification [22ndash24] In[25] two types of SVM were implemented to effectivelyclassify different kinds of power quality disturbances

This paper proposes a practical approach for power oscil-lation classification by recognition of real-time power oscil-lation curves utilizing SVM Hilbert transform is employedto obtain envelope curves of the power oscillation curvesTwenty sampling points on the envelope curve of power oscil-lation are selected for feature extraction Then forty poweroscillation curves are employed as samples to train the sup-porting vector machine At last three tests on the 16-machine68-bus benchmark system and a real power system in Chinaindicate that the proposed oscillation classification methodpossesses good precision

The rest of this paper is organized as follows In Section 2a Hilbert transform based oscillation feature extractionscheme is proposed Section 3 introduces a power oscillationproperty classification method utilizing SVM Case studyis undertaken on the 16-machine 68-bus benchmark sys-tem and a practical simplified power system of China toverify the effectiveness of the proposed oscillation propertyclassification method in Section 4 Conclusions are given inSection 5

2 Feature Extraction of Power Oscillations

21 Introduction of Power Oscillations When a free oscilla-tion happens in a power system the rotor angle rotationspeed of synchronous generators and relevant electric vari-ables (such as the power flow of transmission lines and busvoltages) will oscillate accordingly among which the powerflow of transmission line 119875

119894119895can be expressed as follows

119875119894119895=

119899

sum

119894=1

1198961198941199091198940119890120582119894119905 (1)

where 119896119894denotes the participation factor of oscillation pattern

119894 1199091198940is the initial value of state variable according to oscil-

lation pattern 119894 120582119894denotes the eigenvalue of the oscillation

pattern 119894 Generally 120582119894can be expressed as

12058212= 120572 plusmn 119895120596 (2)

In (2) 120572 indicates the damping performance of the oscilla-tion120596 denotes the frequency characteristic of the oscillationthe damping ratio of the oscillation is defined as

120585 =minus120572

radic1205722 + 1205962 (3)

The oscillation curve of any free oscillation mode can beexpressed as

119875119894119895= 1198611119890120572119905 sin (120596119905 + 120601

1) (4)

Therefore when any oscillation mode with 120585 lt 0 (ie120572 gt 0) exists in the power system any small disturbancewould stimulate thismode and result in a power oscillation ofthe tie line with increasing amplitude as shown in Figure 1(a)That is the so-called negative damping oscillation If all thedamping ratios of oscillation patterns are larger than zerothe small disturbance occurring at the power system wouldbe restrained by the positive damping

For any forced oscillation in the power system with anextreme disturbance like ℎ sin(120596

119889119905) it can be expressed as a

second-order system

+ 2120585120596 + 1205962119909 = ℎ sin (120596

119889119905) (5)

Solving (5) the oscillation curve of forced oscillation is

119875119894119895= 1198611119890120572119905 sin (120596119905 + 120601

1) + 1198612sin (120596

119889119905 + 1206012) (6)

Under the circumstance of 120585 gt 0 (ie 120572 lt 0) the freeoscillation part 119861

1119890120572119905 sin(120596119905+120601

1) in (6) will decay But power

resonance would be motivated and large oscillation wouldfollow as shown in Figure 1(b) if the oscillation frequency 120596

119889

of the extreme disturbance is equal or close to the systemrsquosnatural frequency 120596

From the analysis above it can be concluded that theenvelope curve of negative damping oscillation increases as1198611119890120572119905The envelope curve of forced oscillationwill increase at

the initial stage Then it transits to an oscillation as (ℎ2(120596 minus120596119889)120596)119890120572119905 sin((120596minus120596

119889)119905+1206012)+1198612at the transitory stage At last it

will die down to a constant as1198612at stable stage [19]Thus there

is obvious distinction between the envelope curves of the twokinds of power oscillations


600

650

700

750

800Ac

tive p

ower

(MW

)

Time (s)0 5 10 15 20

Envelope curve


Activ

e pow

er (M

W)

0 5200

300

400

500

Time (s)10 15 20

Envelope curve




119906 (119905) = 119860 (119905) cos120593 (119905) (7)


V (119905) = 119867 [119906 (119905)] =1

120587int

+infin

minusinfin

119906 (120591)

119905 minus 120591119889120591 (8)


119908 (119905) = 119860 (119905) cos120593 (119905) + 119895119860 (119905) sin120593 (119905) (9)

Namely

119908 (119905) = 119860 (119905) 119890119895120593(119905)

119860 (119905) = radic1199062(119905) + V2 (119905)

(10)




0 5 10 15 2009

095

1

105

11

Activ

e pow

er (p

u)

Time (s)






Optimalhyperplane

PositiveNegative

Optimalmargin






119894=1 119909119894isin 119877119899 119889119894isin 1 minus1 119909




120596 sdot 119909119894+ 119887 ge 1


119894= 1

120596 sdot 119909119894+ 119887 le minus1


119894= minus1

(11)



119910 = 120596 sdot 120601 (119909) + 119887 (12)




min 1

21205962+ 119862

119897

sum

119894=1

120585119894

st 119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894ge 1

120585119894ge 0

(13)


119871 (120596 119887 120585 ΛR)

=1

21205962

+ 119862

119897

sum

119894=1

120585119894minus

119897

sum

119894=1

120572119894(119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894minus 1)

minus

119897

sum

119894=1

119903119894120585119894

(14)


119897) and

119877119879= (1199031 1199032 119903



120597119871

120597120596

10038161003816100381610038161003816100381610038161003816120596=1205960

= 1205960minus

119897

sum

119894=1

120572119894119889119894120601 (119909119894) = 0 (15)

120597119871

120597119887

10038161003816100381610038161003816100381610038161003816119887=1198870

=

119897

sum

119894=1

120572119894119889119894= 0

120597119871

120597120585119894

10038161003816100381610038161003816100381610038161003816120585119894=1205851198940

= 119862 minus 120572119894minus 119903119894= 0

(16)


1205960=

119897

sum

119894=1

120572119894119889119894120601 (119909119894) (17)


119910 =

119897

sum

119894=1

119889119894120572119894119870(119909 119909

119894) + 119887 (18)



1198940to the


119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895) (19)


minΛ

119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895)

st119897

sum

119894=1

120572119894119889119894= 0

0 le 120572119894le 119862

(20)




1205960=

119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) (21)


119910 = sgn[119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) 120601 (119909) + 119887] (22)






119870(119909119894 119909119895) = 120601119879(119909119894) 120601 (119909

119895) (23)




linear function

119870(119909119894 119909119895) = (119909

119894sdot 119909119895+ 1)119889

(24)

sigmoid function

119870(119909119894 119909119895) = tanh (119909

119894sdot 119909119895minus 120579) (25)


119870(119909119894 119909119895) = exp(minus

10038161003816100381610038161003816119909119894minus 119909119895

10038161003816100381610038161003816

2

21205902) (26)

Start




End




No

Yes









1

27

474640

2 25

53

2660 28 2961

24

21

16

19

20

57

56

58

22

23

59

15

14

13

1055

1211

3

4

56

547

8

9

18 17

30

363765

64

3233

34

35

45

4439

50

51

5268

49

48 38

31

42

41

67

66

6362

G1

G2G3

G4

G5

G6

G7

G8 G9

G10

G11

G12G13

G14

G15

G16












C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40


RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40



C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41





Number 10 1 C1



5 Conclusions


Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)


Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




600

650

700

750

800Ac

tive p

ower

(MW

)

Time (s)0 5 10 15 20

Envelope curve


Activ

e pow

er (M

W)

0 5200

300

400

500

Time (s)10 15 20

Envelope curve




119906 (119905) = 119860 (119905) cos120593 (119905) (7)


V (119905) = 119867 [119906 (119905)] =1

120587int

+infin

minusinfin

119906 (120591)

119905 minus 120591119889120591 (8)


119908 (119905) = 119860 (119905) cos120593 (119905) + 119895119860 (119905) sin120593 (119905) (9)

Namely

119908 (119905) = 119860 (119905) 119890119895120593(119905)

119860 (119905) = radic1199062(119905) + V2 (119905)

(10)




0 5 10 15 2009

095

1

105

11

Activ

e pow

er (p

u)

Time (s)






Optimalhyperplane

PositiveNegative

Optimalmargin






119894=1 119909119894isin 119877119899 119889119894isin 1 minus1 119909




120596 sdot 119909119894+ 119887 ge 1


119894= 1

120596 sdot 119909119894+ 119887 le minus1


119894= minus1

(11)



119910 = 120596 sdot 120601 (119909) + 119887 (12)




min 1

21205962+ 119862

119897

sum

119894=1

120585119894

st 119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894ge 1

120585119894ge 0

(13)


119871 (120596 119887 120585 ΛR)

=1

21205962

+ 119862

119897

sum

119894=1

120585119894minus

119897

sum

119894=1

120572119894(119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894minus 1)

minus

119897

sum

119894=1

119903119894120585119894

(14)


119897) and

119877119879= (1199031 1199032 119903



120597119871

120597120596

10038161003816100381610038161003816100381610038161003816120596=1205960

= 1205960minus

119897

sum

119894=1

120572119894119889119894120601 (119909119894) = 0 (15)

120597119871

120597119887

10038161003816100381610038161003816100381610038161003816119887=1198870

=

119897

sum

119894=1

120572119894119889119894= 0

120597119871

120597120585119894

10038161003816100381610038161003816100381610038161003816120585119894=1205851198940

= 119862 minus 120572119894minus 119903119894= 0

(16)


1205960=

119897

sum

119894=1

120572119894119889119894120601 (119909119894) (17)


119910 =

119897

sum

119894=1

119889119894120572119894119870(119909 119909

119894) + 119887 (18)



1198940to the


119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895) (19)


minΛ

119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895)

st119897

sum

119894=1

120572119894119889119894= 0

0 le 120572119894le 119862

(20)




1205960=

119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) (21)


119910 = sgn[119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) 120601 (119909) + 119887] (22)






119870(119909119894 119909119895) = 120601119879(119909119894) 120601 (119909

119895) (23)




linear function

119870(119909119894 119909119895) = (119909

119894sdot 119909119895+ 1)119889

(24)

sigmoid function

119870(119909119894 119909119895) = tanh (119909

119894sdot 119909119895minus 120579) (25)


119870(119909119894 119909119895) = exp(minus

10038161003816100381610038161003816119909119894minus 119909119895

10038161003816100381610038161003816

2

21205902) (26)

Start




End




No

Yes









1

27

474640

2 25

53

2660 28 2961

24

21

16

19

20

57

56

58

22

23

59

15

14

13

1055

1211

3

4

56

547

8

9

18 17

30

363765

64

3233

34

35

45

4439

50

51

5268

49

48 38

31

42

41

67

66

6362

G1

G2G3

G4

G5

G6

G7

G8 G9

G10

G11

G12G13

G14

G15

G16












C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40


RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40



C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41





Number 10 1 C1



5 Conclusions


Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)


Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




Optimalhyperplane

PositiveNegative

Optimalmargin






119894=1 119909119894isin 119877119899 119889119894isin 1 minus1 119909




120596 sdot 119909119894+ 119887 ge 1


119894= 1

120596 sdot 119909119894+ 119887 le minus1


119894= minus1

(11)



119910 = 120596 sdot 120601 (119909) + 119887 (12)




min 1

21205962+ 119862

119897

sum

119894=1

120585119894

st 119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894ge 1

120585119894ge 0

(13)


119871 (120596 119887 120585 ΛR)

=1

21205962

+ 119862

119897

sum

119894=1

120585119894minus

119897

sum

119894=1

120572119894(119889119894[120596 sdot 120601 (119909

119894) + 119887] + 120585

119894minus 1)

minus

119897

sum

119894=1

119903119894120585119894

(14)


119897) and

119877119879= (1199031 1199032 119903



120597119871

120597120596

10038161003816100381610038161003816100381610038161003816120596=1205960

= 1205960minus

119897

sum

119894=1

120572119894119889119894120601 (119909119894) = 0 (15)

120597119871

120597119887

10038161003816100381610038161003816100381610038161003816119887=1198870

=

119897

sum

119894=1

120572119894119889119894= 0

120597119871

120597120585119894

10038161003816100381610038161003816100381610038161003816120585119894=1205851198940

= 119862 minus 120572119894minus 119903119894= 0

(16)


1205960=

119897

sum

119894=1

120572119894119889119894120601 (119909119894) (17)


119910 =

119897

sum

119894=1

119889119894120572119894119870(119909 119909

119894) + 119887 (18)



1198940to the


119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895) (19)


minΛ

119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895)

st119897

sum

119894=1

120572119894119889119894= 0

0 le 120572119894le 119862

(20)




1205960=

119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) (21)


119910 = sgn[119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) 120601 (119909) + 119887] (22)






119870(119909119894 119909119895) = 120601119879(119909119894) 120601 (119909

119895) (23)




linear function

119870(119909119894 119909119895) = (119909

119894sdot 119909119895+ 1)119889

(24)

sigmoid function

119870(119909119894 119909119895) = tanh (119909

119894sdot 119909119895minus 120579) (25)


119870(119909119894 119909119895) = exp(minus

10038161003816100381610038161003816119909119894minus 119909119895

10038161003816100381610038161003816

2

21205902) (26)

Start




End




No

Yes









1

27

474640

2 25

53

2660 28 2961

24

21

16

19

20

57

56

58

22

23

59

15

14

13

1055

1211

3

4

56

547

8

9

18 17

30

363765

64

3233

34

35

45

4439

50

51

5268

49

48 38

31

42

41

67

66

6362

G1

G2G3

G4

G5

G6

G7

G8 G9

G10

G11

G12G13

G14

G15

G16












C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40


RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40



C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41





Number 10 1 C1



5 Conclusions


Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)


Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





1198940to the


119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895) (19)


minΛ

119865 (Λ) =

119897

sum

119894=1

120572119894minus1

2

119897

sum

119894=1

119897

sum

119895=1

120572119894120572119895119889119894119889119895119870(119909119894 119909119895)

st119897

sum

119894=1

120572119894119889119894= 0

0 le 120572119894le 119862

(20)




1205960=

119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) (21)


119910 = sgn[119897

sum

119894=1

1198891198941205720

119894120601119879(119909119894) 120601 (119909) + 119887] (22)






119870(119909119894 119909119895) = 120601119879(119909119894) 120601 (119909

119895) (23)




linear function

119870(119909119894 119909119895) = (119909

119894sdot 119909119895+ 1)119889

(24)

sigmoid function

119870(119909119894 119909119895) = tanh (119909

119894sdot 119909119895minus 120579) (25)


119870(119909119894 119909119895) = exp(minus

10038161003816100381610038161003816119909119894minus 119909119895

10038161003816100381610038161003816

2

21205902) (26)

Start




End




No

Yes









1

27

474640

2 25

53

2660 28 2961

24

21

16

19

20

57

56

58

22

23

59

15

14

13

1055

1211

3

4

56

547

8

9

18 17

30

363765

64

3233

34

35

45

4439

50

51

5268

49

48 38

31

42

41

67

66

6362

G1

G2G3

G4

G5

G6

G7

G8 G9

G10

G11

G12G13

G14

G15

G16












C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40


RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40



C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41





Number 10 1 C1



5 Conclusions


Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)


Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




1

27

474640

2 25

53

2660 28 2961

24

21

16

19

20

57

56

58

22

23

59

15

14

13

1055

1211

3

4

56

547

8

9

18 17

30

363765

64

3233

34

35

45

4439

50

51

5268

49

48 38

31

42

41

67

66

6362

G1

G2G3

G4

G5

G6

G7

G8 G9

G10

G11

G12G13

G14

G15

G16












C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40


RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40



C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41





Number 10 1 C1



5 Conclusions


Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)


Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014







C1 C2 C1 C2 C1 C2C1 72 1 C1 70 3 C1 66 4C2 4 43 C2 6 41 C2 10 40


RBF kernel function

C1 C2 C1 C2 C1 C2C1 74 2 C1 71 2 C1 66 4C2 2 42 C2 5 42 C2 10 40



C1 C2 C1 C2 C1 C2C1 75 1 C1 72 1 C1 68 3C2 1 43 C2 4 43 C2 8 41





Number 10 1 C1



5 Conclusions


Activ

e pow

er (M

W)

650

600

550

500175520 175540 175600 175620 175640 175700

Time (hhmmss)


Activ

e pow

er (M

W)

minus600

minus700

minus800

minus900

minus1000

21202210 2300 2350 2440 2530 2620Time (mmss)







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014







Number 2 18 C2




Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



Active Power Oscillation Property Classification of ... · tem and a practical simplified power...

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Transcript of Active Power Oscillation Property Classification of ... · tem and a practical simplified power...