Electrical Power System Harmonic Analysis Using Adaptive BSS Algorithm

Transcription

1 Sensors & ransducers 2013 by IFSA Electrical Power System Harmonic Analysis Usin Adaptive BSS Alorithm 1,* Chen Yu, 2 Liu Yuelian 1 Zhenzhou Institute of Aeronautical Industry Manaement, Henan, Zhenzhou, , China 2 Department of Electrical Enineerin, Beijin University of Posts and elecommunications, Beijin, , China * chenyu@zzia.edu.cn Received: 9 ovember 2013 /Accepted: 25 ovember 2013 /Published: 30 ovember 2013 Abstract: his paper used the radient blind sinal separation alorithm to carry out the electrical power system harmonic sinal analysis throuh the simulation experiment. he experiment result proved that the electrical power system harmonic sinal separation based on the radient alorithm is accurate. hrouh comparin the separation performance by usin the different step size, the adaptive chanin step size natural radient blind separation alorithm for the electrical power system harmonic sinal blind source separation can effectively separate each frequency harmonic component of the observation blind mixed sinal. his paper mainly analyzed the alorithm performance of converence speed and steady state error under the selection of different step size. Copyriht 2013 IFSA. Keywords: Blind sinal separation, atural radient alorithm, Electrical power system, Harmonic analysis, Step size selection. 1. Introduction Blind Source Processin (BSP) is a quickly developed new method of many research fields in the late 1980s. It is a new subject combinin the statistical sinal processin, information theory and artificial neural network. [1, 2] Blind source separation technoloy [3] is refers to the source sinal is unknown and in blind mixed process under the condition is unknown, and separates out the source sinal from the mixin blind sinal. he core issue of blind source separation is the research to the separation matrix learnin alorithm. Blind sinal separation alorithm mainly includes: the information maximization (Informax) alorithm [4], the natural radient alorithm, the Equivariant Adaptive Blind Separation (EASI) alorithm [5], fixed point Fast trainin (Fast ICA) alorithm [6], and so on. Blind source separation is an optimization process based on information theory, hiher order statistics theory, to establish a cost function, and a kind of optimization alorithm is applied to the objective function. [7] Amon them, the radient alorithm is a kind of classic unconstrained optimization alorithm, which has an advancement of simple principle, easy to implement, therefore, can realize the online calculation sinal, especially the classical natural radient alorithm has been widely used in many blind source separation field. Especially in the biomedical sinal processin, speech sinal, imae processin, communications, radar, and underwater acoustic sinal processin, and other fields, which has obtained the ood application effect. In this paper, we study a kind of variable step size adaptive natural radient blind source separation alorithm [8] applyin into the power system harmonic analysis, and study the role of step in the alorithm performance. 364 Article number P_RP_0016

2 2. Blind Separation Model As there is no prior knowlede of source sinals and mixin matrix can be used, we must make some additional assumptions of source sinals and mixed matrix, the assumption of different practical problems and alorithms to the source sinals and mixin matrix are not the same, but the basic premises are the same. amely the each component si () t of source sinals s() t is statistically independent each other, and most can only have one component obey the Gaussian distribution, after multiple mixed Gaussian process is still a Gaussian process, so it can't be separated. [9] When carryin the blind sinal separation, first of all, modelin the nonlinear system based on the study of the problem, then, accordin to the information theory and statistical theory, establishin an objective function JW accordin to the chanin factor W. Finally, findin an effective alorithm to calculate W. If a W can be make to achieve maximum (or minimum), namely the W as requested. Buildin the objective function has variety ways, the most basic way respectively is: neative entropy, KL diverence and maximum likelihood taret function [10]. Blind sinal separation model is shown as Fi. 1. s(n) Mix matrix A x(n) Albino sinal u(n) Blind separation alorithm Fi. 1. Blind separation model. Solute mix system y(n) he optimization methods are often used in the blind source separation. In terms of optimization way, there are Infomax alorithm, natural radient alorithm and EASI alorithm. he converence speed is linear, and speed is slihtly slower, but they belon to the adaptive method, which has a real-time online processin ability. But Fast ICA alorithm is a method in hih speed and numerical stability, which has super linear converence speed than natural radient alorithm. Its converence speed is usually much faster. Comprehensive considerin the radient alorithm is a method of online processin, which has a suitable for on-line blind source separation and predictin dianosis. herefore, we carry out the analysis to the power system harmonic voltae based on the variable step size adaptive natural radient alorithm. 3. BSS Alorithm Analysis and Desin Informax radient alorithm is a usual way to calculate the extremum value of the objective function L(W). Its main principle is: firstly, selectin a initial solution mixed matrix, brinin it to the objective function and ettin the radient of L(W) at position W(0), and then addin a suitable step size to calculate new separation matrix W(1) in the neative radient direction (if solvin the maximum, then takin positive radient direction), and repeatin the process. So the calculation formula to solvin the random radient alorithm for solvin W is: LW W( k + 1) = W( k) + u( k) W = W k W (1) Amon the formula (1), k is the number of iterations, u(k) is learnin step size, and makin W( k + 1) W( k) = W.Without considerin k, the formula (1) can be expressed as LW W = u( k). W When selectin different objective function L(W), we can et different iteration formula. - W( t+ 1) = W() t + u()[ t W - ( y()) t x() t ] (2) Amon the formula (2), µ (t) is step size, () i is a nonlinear function, which is a function closely related to the source sinal probability density 3 function. Usually, to sub-gaussian sinal, () i = y. And to super Gaussian sinal, ( i ) = tanh( y). Infomax alorithm can effectively separate the multiple super Gaussian distribution of the source sinal. he main disadvantaes of this alorithm are: slow converence speed, and at the same time, because of the calculatin the separation matrix inversion W, once the condition of W ets worse durin the update process, the alorithm could have diverence. We can et the natural radient alorithm by riht multiplication W W, which is shown as followin formula (3): Wt ( + 1) = Wt + ut [ I- f( yt ) y( twt )] (3) In the radient alorithm, the best step size selection has been one of the problems of blind sinal separation, its choice of alorithm converence plays a key role. he simplest approach is to use a fixed step size, the same as the eneral radient alorithm. And its disadvantaes respectively are that: if the step size is bi, the alorithm s converes is fast, but the sinal s steady state performance is poor; on the other hand, the steady state performance is ood, but the alorithm converes becomes slowly. From the formula (3) we can see, the function of step size u is controllin amplitude of separation matrix iteration update, so, properly selectin step 365

3 size is very important for the performance of the blind sinal separation alorithm. Any time-varyin step size process is the purpose of increasin step size into a bi stable value and achieves the fastest converence, when achievin the best converence point's neihborhood, the correspondin step size should be reduced, and the steady-state error becomes little. If the step size is a fixed value, the converence speed is bound to limit, and appear steady-state miss-adjustment problems, which produces the contradiction between the converence speed and steady state performance. When the step size is small, alorithm of steady state performance is ood, but the alorithm converence is slow. On the other hand, the converence is fast, but the steady state performance is poor. So that the alorithm in trackin performance in non-stationary environments. his paper used a kind of Variable step adaptive blind source separation alorithm, ive consideration to the converence speed and steady state performance. When selectin the step size, we use the followin formula (4) to renew the step size. [11] Jt ( 1)) ut ( + 1) = ut ρ + ut () Amon the formula (4), ρ >0, and it is a small factor. hrouh the induction of ρ, we can et the step size alorithm formula (5). (4) simulation way to obtain electrical power system voltae sinal includin the fundamental wave of 50 Hz, 3 times harmonic, 5 times harmonic and noise waveform sinal. And samplin 4000 points to observation and analysis, the sinal is shown as Fi. 2. Fi. 2. Initial electrical power system voltae waveform sinal. hrouh random matrix blind mixin, the observation sinal waveform is shown as Fi. 3. We can find the mixed sinal is hard to identification. ut ( + 1) = ut ρtrace{ f( yt ( + 1)) x ( t+ 1) W ()[ t I y() t f ( y())]} t (5) In addition, the performance of blind source separation alorithm usually has the two ways, one is the similarity coefficient expressed as similarity deree between the separated sinals and source sinal. Another way is usin crosstalk error as the independence of elements based on the lobal transfer matrix elements. he crosstalk error expressed the deviation deree between the inverse matrix of the separation matrix W and the mixed matrix A. he crosstalk error is defined as bellow formula (6). E = {( max ik 1) + ( i= 1 i= 1 j ij k = 1 max ki j ji 1)} (6) As formula (6), max j ij is the absolute maximum value of the i th row element. When the separated sinal y with the source sinal s waveform is completely same, then E=0. 4. Simulation and Experiment he electrical power system voltae sinal is composed of fundamental wave, harmonic and noise, the samplin frequency is 1 MHz. We carry out Fi. 3. Blind mixin waveform sinal. When carryin out the blind separation to the observation sinal, first of all, we use meanin and bleachin process, etc to the power system voltae waveform sinal. hen, use the improved variable step size adaptive natural radient alorithm with the 7 initial step size of and ρ = In the alorithm, we select the nonlinear function of y 3. he sinal waveform of separation result is shown as Fi. 4. From the Fi. 4, we can see, this alorithm can accurately separate each harmonic component. We can carry out the crosstalk error to the separation waveform, and the result of crosstalk error is shown as Fi

4 Fi. 4. Separation waveform. (a) Before the sinal separation. (b) After the sinal blind separation. Fi. 6. Spectrum diaram. Fi. 5. he crosstalk error. he crosstalk error is shown as Fi. 5. Alorithm can realize converence within the number of 500. If we carry out the spectrum analysis to the separation sinal. he result of the spectrum before the sinal separation and after the sinal blind separation respectively is shown as Fi. 6. From the spectrum diaram we can see, source sinal can be better separated form the blind mixed observation sinal, and includes 50 Hz fundamental wave, 150 Hz 3 times harmonic, 250 Hz 5 times harmonic and noise sinal, etc. However, when not to blind source separation, blind mixed sinal spectrum is shown in Fi. 6, which can not see the detailed information of each component from it. herefore, the adaptive natural radient blind source separation alorithm this paper presented can separate the sinal well. In order to study the effect of step size on the separation performance, we take the initial step size of and he crosstalk error respectively is shown as Fi. 5. From the Fi. 7 of crosstalk error, the step size is 0.001, the iterative around 2500 times alorithm realizes converence, which showin choosin a small step size, the converence is slow, but steady state performance is better; on the contrary, when choosin a lare step size of 0.01, the iterative 250 times alorithm can realize converence, but the steady state performance is poorer. (a) Initial step size µ = (b) Initial step size µ =0.01 Fi. 7. he crosstalk error. 367

5 hus, the natural radient alorithm is sensitive to step size selection, and the tiny difference can brin different separation result, throuh simulation we can see, step size selection is a key problem. A fixed step size may not apply to all of the blind source separation problem. herefore, we must adopt the method of multiple trials, in order to look for a suitable step size to achieve the best separation effect. 5. Conclusions In view of the radient blind sinal separation alorithm, it is an online alorithm, and can realize online blind sinal separation, and, throuh the experimental analysis, the natural radient alorithm of blind source separation can well separate the blind mixed sinals, and can accordin to the properties of model system to adjust the value of separation matrix, so as to et better separation result. Because the natural radient alorithm has the equivalent chane. he amount of calculation, converence speed and steady state performance are mutually contradictory, so, we need to carry on the comprehensive consideration on the selection of step size. Acknowledments his paper is supported by the Science and echnoloy Development Plan Projects of Henan Province in (o ) References [1]. Zhan Faqi, Zhan Bin, Zhan Xibin, Blind Sinal Process and Application, Xian University of Electronic Science and echnoloy Press, Xi an, [2]. Shi Xizhi, Blind Sinal Process-heory and Practice, Shanhai Jiaoton University Press, Shanhai, [3]. Ma Jiancan, iu Yilon, Chen Haiyan, Blind Sinal Process, ational Defence Industry Press, Beijin, [4]. Lee. W., Girolami M., Sejnowski. J., Independent Component Analysis Usin Extended Infomax Alorithm for Mixed Sub-Gaussian and Super-Gaussian Source, eural Computation, Vol. 14, o. 9, 1998, pp [5]. Cardoso J. F., Laheld B. H., Equivariant Adaptive Source Separation, IEEE rans, Sinal Processin, Vol. 44, o. 10, pp [6]. Hyvärinen A., Oja E., A Fast Fixed-Point Alorithm for Independent Component Analysis, eural Computation, Vol. 9, o. 7, 1997, pp [7]. Yan Fushen, Hon Bo, Principle and Application of ICA, sinhua University Press, Beijin, [8]. Gao Yin, Li Yue, Yan Baojun, Overview on Variable Step Size echniques for On-line Blind Source Separation, Computer Enineerin and Applications, Vol. 43, o. 19, 2007, pp [9]. Cardoso J F., Blind Beamformin for on-gaussian Sinals, IEE Proceedins F, Vol. 140, o. 6, 1993, pp [10]. M. Babaie-Zadeh, C. Jutten, A General Approach for Mutual Information Minimization and Its Application to Blind Source Separation, Sinal Processin, Vol. 85, o. 5, 2005, pp [11]. Li Zhuchen, Zhan Liyi, A ew Adaptive Step Size Alorithm for Blind Source Separation, Modern Electronics echnique, o. 24, 2005, pp Copyriht, International Frequency Sensor Association (IFSA). All rihts reserved. ( 368