A New Analysis Method of Inter-harmonic in

Power System Using Orthogonal Basis Neural

Network AlgorithmYuan PEI, Zhe-Zhao ZENG

College of Electrical & Information Engineering, Changsha University of Science & TechnologyChangsha 410076,Hunan, ChinaE-mail:hncs6699gyahoo.com.cn

Abstract- It is well known that the inter-harmonic is theharmonic which its frequency is not an integer of thefundamental frequency. It has serious impacts on the design ofharmonic compensation equipments. Therefore it is importantto measure accurately inter-harmonic. The neural networkalgorithm with the orthogonal basis functions was presented bywhich the amplitudes and phase of harmonic and inter-harmonic could be obtained very fast. To validate thealgorithm's validity, the simulation example of inter-harmonicanalysis was given. The results show that the proposed inter-harmonic analysis approach has high precision and fasttraining speed, so it could be applied to power system inter-harmonic analysis.

I. INTRODUCTION

In recent years, the pollution problem about inter-harmonic is getting more and more serious with the broaduse of non-linear components in electric and electronicdevices. The inter-harmonic that the frequency is lower thanfundamental frequency (namely sub-harmonic) can causethe photo-electricity flickering, exceptional run of the lowfrequency relay and the over current turn off of the passivepower filter and so on. The inter-harmonic that thefrequency is higher than fundamental frequency can disturbthe audio equipment, may cause the noise and vibration ofthe induction motor. So it is very necessary to pay attentionto the inter-harmonic phenomenon.

The inter-harmonic analysis of power system isachieved usually using the fast Fourier transform (FFT) [1However, because of the grid effect and energy leakage ofFFT, the calculated signal parameters, including frequencies,amplitudes and especially the phases, are not precise andcannot meet the need of inter-harmonic measurement.References [6-9] proposed an interpolation algorithm thatcould rectify the calculated result of FFT and effectivelyimprove the computing precision, but the computationquantity is too big and it is unable to satisfy the requestwhich real-time monitor the inter-harmonic in power system.Furthermore, there are many inter-harmonic analysismethods such as Prony algorithm [10-11]ISVD [12] WTE13-14]whose results have high precision, but the hardware designis quite difficult.The paper proposed an algorithm of neural network based

on orthogonal basis functions and researched theconvergence property of the algorithm. The amplitudes andphases of harmonic and inter-harmonic could be obtained bythe approach. The simulation results show that the proposedinter-harmonic analysis method was very accurate and the

training speed was very fast, so it could be applied to inter-harmonic analysis ofpower system

II. INTER-HARMONIC ANALYSIS USING NEURAL NETWORKALGORITHM

A. Neural networkmodel with Fourier basisfunctionThe signal with harmonic and inter-harmonic is expressed

as followsN

y(t) = E An sin(2Tnt + pnp)n=O (1)

+ZEBm sin(24mt+Qm)ml

Where the fn , An and qon is respectively the frequency,amplitude and phase of the nth harmonic. N is the highestdegree of the harmonic. The fm , Bm and POm isrespectively the frequency, amplitude and phase of the mthinter-harmonic. M is the highest order of the inter-harmonic. Let t = kT , then we can obtain from theformula (1):y(k) Ao + E [A sin qj cos(jc9okT) + Aj cos q j sin(jc9okT)] (2)

M+ [Bi sin qi cos(6bjkT) + Bi cos oi sin(dbikT)]

where C)0 is the fundamental wave's angle frequency,

Coo = 2TfO , fo is the fundamental wave's frequency, j is

the order of harmonic, AO is direct current component, (01

is the angle frequency of the jth harmonic; C)i is the anglefrequency of the ith inter-harmonic, k is the number of thesample data, T is the sampling interval.Let

wo = AO,w1 = Ai sin oj,W1jN = Ai cosqj (j =

{ 1i = Bi sin (p,W,+, = B, cosq' (i = 1,2,. ,M)UW = [w0,wI, , W2N 1,W . W2M]

1,2,, N)

(3)

1-4244-0549-1/06/$20.00 (2006 IEEE.

j= 0,1,..., N)(j = N+1,N+2,...,2N)

= 1,2,...,M)

(i=M+1,M+2,...,2M)(4)

AndCk = C

TCk= C (k), c, (k), c,C2 (k), c^ (k), c^ (k), .. C2M (k)]then formula (2) is expressed as follows

2N 2M

y(k) = wWcT (k) + E wicv (k) W Ck (5)

YO iktFigurel indicates the model of neural network based on

orthogonal basis functions. The output of neural network isexpressed by

2N 2M

y(k) = wWcT (k) + E wicv (k) W Ck (6)

YO i k6where y(k) is the output of the neural network, Yd iS the

desired output of the neural network , WT is the weightvector of the neural network, Ck is the activation vector ofthe hidden units of neural network, and

{k, Yd (k) = 0,1,J , p} is the sample set of the neural

network..We define an objective function J as follows:

P

J = - Ee (k)' k=O

(7)

wheree(k) = Yd (k) - y(k) (k =O,1,...,p) (8)

To minimize J, the Wk is recursively calculated via usinga simple gradient descent rule as

Wk+l =wk 7a

k (9)

where 0 < 7 < 1 is a learning rate. After differentiating Eq.(7), we have

AW=k=7a

= 7%Wk ae(k)

ae(k) ay(k)ay(k) aWk

= 7e(k)CkSubstituting Eq. (10) into Eq.(9), we have

Wk+l = Wk + 7e(k)Ck (11)

If the run work frequency of the power system was known,the weights of the neural network can be obtained throughabove algorithm. Therefore the amplitudes and phases ofharmonic and inter-harmonic in power system can be alsoobtained.

Figure 1 The model of neural network

B. Convergence property ofNeural Network with FourierBasis Function

In order to ensure the convergence of neural network, it isimportant to select a proper learning rate 77. In the section,we'll present and prove the convergence theorem of neuralnetwork algorithm as follows:Theorem 1 Only when learning rate 77 satisfies

20 < < ,2M+1 the algorithm of the neural

network is convergent, where 77 is learning rate,2N + 2M + 1 is the number ofhidden nodes.Proof Define a Lyapunov function:

V(k) 2 (k)2

Therefore, we have1 2 12

AV(Ak)= e (k+l')- e(k

2 2

e(k +1) = e(k) + Ae(k) = e(k) + ( Tk) I

Consider

AWk = -Re(k) ae(k) ae(k)aWk Wk

(10) Ae(k) = -7e(k)(aW(k ae

(12)

(13)

AWk

(14)

= -Ck, then we have

(k) a- e(k) 2

yk -77e(k) aW k

(15)n

where ||X|| = Xi 2, which is the square of Euclideani=l

norm. X = [x1, xX2, ... , xn]T . According to Eq.(14) and

Eq.(15), we have

AV(k) = [e(k) + Ae(k)]2 - 1 e2(k)2 2

= Ae(k)Le(k) + IAe(k)= )2ae(k) 1 ae(k)

= -7e(e(k)(ak) e(k) 177e(k) ek)

2 aW

tcj (k) = cos(jwokT) (

cj (k) = sin[(j - N)coo kT, ]ci (k) = cos(1kTs ) (i =

p1(k) = sin(d)1 MkT) ( ,

ae(k) 2(k+_ + 1 2 ae(k) 2>)aWk 2' aWk

Because ae(k) >0, e2 (k) > 0, if the algorithm is

convergent, i.e. AV(k) < 0, then it is easy to see fromEqu.(16) that

Because 7 > 0, we ha

We have from Equ.(5)

12 ae(k) 2

2

[ve

ae(k) 2

aWk

ae(k) ck

(17)

(18)

aWk

ae(k) 2 2C 2Wk -ck ck

2N 2 2MZ cj(k) +Zacj(k)2i=° i=l

< 2N + 2M +1Apparently, if learning rate 77 satisfies

0 < 7 < we have AV(k) < 0, therefore,2N+2M+I

the algorithm is convergent. The theorem is provedcompletely.

According to the above neural network algorithm, we canobtain the weight vectorW of the neural network throughthe neural network training, then the amplitudes and phasesof harmonic and inter-harmonic can be obtained throughtheW. The parameters of harmonic and inter-harmonic areas follows:Harmonic amplitudes:

A = W + W2j N+j

Harmonic phases:-j= arctan(wj / WN+j )

Inter-harmonic amplitudes:Bj 2+=

Inter-harmonic phases:qp = arctan(wi^ / WM+j )

III. RESULT OF SIMULATION ANALYSIS

In order to verify the validity of the neural networkalgorithm, the simulation analysis was given with the signalfrom reference [7].Simulation 1 The expression of the signal in reference [7] is:

(19)

9

y(k) = E Am CoS(24mkT, + 9m )m=l

(23)

Let the sampling frequency be 1900 Hz and the length ofdata be 1024. All the parameters are shown in TABLE I.

TABLE IPARAMETERS OF SIGNAL

Parameters Frequencies Amplitudes phases/( )

25 2.28 2050 380 10

Harmonic 150 19 25Harmonic 175 1.9 30

inter- 250 15.2 100

harmonic 330 1.52 120350 11.4 150380 1.14 180450 7.6 210

First calculate the frequencies according to reference [7].Because of the highest time of harmonic is nine, topology ofneural network is Ix 19 x 1 . Suppose that the samplingfrequency is 1900 Hz and the length of data is 1024,

Tol =10-5 77 1.2 1- 2= 0.068

2N+2M+1 19Produce the random weights, then after network training fortwo times, we can obtain the accumulative square error, i.e.

J=2.6182X10-, the amplitudes and phases of theharmonic and inter-harmonic are shown in TABLE II.

TABLE IIANALYSIS RESULTS WITH THE PAPER'S ALGORITHM

The results of the inter-harmonicRelative Relative

Amplitudes error(%) Phases/( error(%)x10'3 X1012

2.2800 280.09 20.0000 465.94

1.9000 -1726.2 30.0000 393.02

1.5200 -38887 120.0000 -397.68

1.1400 13200 180.0000 -22.832

The results of the harmonic380.00 1.6455 10.0000 -7.7272

19.000 138.56 25.0000 -51.088

15.200 -5.8433 100.0000 -29.800

11.400 -1103.1 150.0000 207.78

7.6000 319.04 210.0000 16.972

(20)The simulation results show that the proposed analysis

method of inter-harmonics was very accurate. Compared to(21) results of the Rife-Vincent window in reference [7], the

precision of the amplitudes improved seven magnitudes, and(22) the phases improved ten magnitudes.

IV. CONCLUSION

The simulation results show that the proposed inter-harmonics analysis method was very accurate and thetraining speed was very fast. Compared to reference [7], theprecision of the amplitudes improved seven magnitudes, andthe phases improved ten magnitudes. Therefore it could beapplied to inter-harmonic analysis of power system.

Certainly, the method has also the certain limitation. Whenthe fundamental and harmnonic frequencies were unknown,the method is invalid.

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