TEHNI�KA DIJAGNOSTIKA (BROJ 4 • 2007) 3

UDC: 621.311.1.018/.015:621.317.001.573 ORIGINALNI NAU�NI RAD

AN ADAPTIVE RESONATOR BASED METHOD FOR REAL TIME POWER SYSTEM HARMONIC ANALYSIS

ADAPTIVNI ALGORITAM ZA HARMONIJSKU ANALIZU U REALNOM VREMENU BAZIRAN NA REZONATORIMA

Dr Miodrag Kušljevi�

TERMOELEKTRO ENEL AD, Uralska 9, Beograd

ABSTRACT A new technique for harmonic estimation from either a voltage or current signal is presented. In this pa-per, an implementation of the finite-impulse-response (FIR) and infinite-impulse-response (IIR) filter trans-fer functions is used. This implementation is based on a recently introduced common structure for recursive discrete transforms. This structure consists of digital resonators in a common negative feedback loop. The structure of the estimation algorithm consists of two decoupled modules: the first for adaptive filtering of input signals with the harmonic amplitude and phase calculation, the second for the external frequency estimation. A very suitable algorithm for frequency and harmonic phasors estimation in a wide range of the frequency variations is obtained. To demonstrate the performance of the developed algorithm, computer-simulated data records are processed. Simulation results show that this algorithm is applicable to detect the amplitude of steady-state, varying and decaying sinusoidal signals. It has been found that the proposed method really meets the needs of online applications. This technique provides accurate amplitude estimates in about one period. Key words: Harmonics, power quality, harmonic analysis, phasor estimation, frequency estimation, recursive algorithm, adaptive filtering, finite-impul-se-response (FIR) filter, infinite-impulse-response (IIR) filter, digital resonator.

REZIME U radu je prikazana jedna rekurzivna metoda za merenje harmonika napona i/ili struje elektro-energetske mreže. Koriš�ena je efikasna metoda za implementaciju funkcija prenosa digitalnih filtera sa kona�nim impulsnim odzivom (FIR) i beskona�nim impulsnim odzivom (IIR). Implementacija se bazira na nedavno uvedenoj paralelnoj strukturi rezonatora sa zajedni�kom povratnom vezom. Struktura algo-ritma za estimaciju se sastoji od dva dekuplovana modula: prvog, koji služi za adaptivno filtriranje ulaznog signala sa ra�unanjem amplituda i faza harmonika i drugog, koji služi za eksternu estimaciju frekvencije. Dobijen je veoma pogodan algoritam za estimaciju frekvencije signala i fazora harmonika u širokom opsegu promene frekvencija. U cilju pro-cene performansi algoritma izvršene su ra�unarske simulacije i dati njihovi rezultati. Zahvaljuju�i sma-njenim zahtevima predloženog algoritma za ra�u-narskim resursima, omogu�eno je procesiranje po-dataka u realnom vremenu. Predloženi algoritam omogu�ava da se spektar meri trenutno, tako da se harmonijski sadržaj može detektovati sa promenom signala. Klju�ne re�i: Harmonici, kvalitet elektri�ne ener-gije, harmonijska analiza, ra�unanje fazora, mere-nje frekvencije, filter sa kona�nim impulsnim odzi-vom (FIR), filter sa beskona�nim impulsnim odzivom (IIR), digitalni rezonator.

1. INTRODUCTION It goes without saying that harmonic analysis is a

very important subject in power systems. With the increasing use of nonlinear loads in power systems, harmonic distortion becomes more and more serious. The detailed frequency analysis method has been required to overcome these problems.

The harmonics measurement is important for deri-ving power quantities of power quality indices. Ho-wever, despite the fact that harmonics are steady-

state phenomena, in practice the measurements have to be performed in dynamic conditions as well. A good parameters estimation method should yield accurate estimates in the presence of distortion and noise as well as track relatively fast frequency varia-tions. In some cases, in the presence of harmonic fluctuation, the measurement process should be fast enough to measure the spectrum instantaneously, so that the frequency content at a point in time can be detected when the signal is changing, [1, 2]. An

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example includes driving active filters for harmonic suppression or monitoring the current harmonic fluctuation on speed variable electrical drives, [3]. Fast tracking is also critical for applications such as under or over frequency protection relays, which require accurate estimates in a period of a few cycles or even a fraction of a cycle.

Electrical parameters measurement of a fixed-fre-quency signal is a straightforward task. However, if the frequency is not known a priori, it becomes a very difficult task to measure the amplitude accura-tely. Various numerical algorithms for power measu-rements are sensitive to frequency variations. Typi-cal examples are algorithms based on the fast Fouri-er transform (FFT), or algorithms based on the least mean squares technique (LMS) and on the assum-ption that system frequency is known in advance and constant (50 or 60Hz), [4,5]. Generators and their step-up transformers may operate in comparatively wide range of frequencies. Most of the digital measuring algorithms as used directly in numeri-cal relays are sensitive to frequency swings what affects the accuracy of the measurement, and consequently, the overall performance of protec-tive relays. The problem touches both the gene-rator protection due to large frequency deviations and the transformer protection due to the multi-plying effect of the frequency variation on the fre-quency shift of the 2nd and 5th harmonics, [6].

The discrete Fourier transform (DFT) is a digital filtering algorithm that computes the amplitude and phase at discrete frequencies of a discrete time sequence. The DFT as phasor-based algorithm includes two linear filters for eliminating undesi-rable harmonics and providing necessary phase shift at the enhanced frequency. The outputs of these two filters are further processed to extract the amplitude and phase angle of the fundamental frequency and harmonic signals. The FFT is computationally efficient algorithm for computing DFT.

The FFT is the most widely used computation algorithm for harmonic analysis. However, leakage effect, picket-fence effect, and aliasing effect make FFT suffer from specific restrictions. Some methods [4, 7] have also been provided to improve these drawbacks. Besides disadvantages related to the syn-chronization of the sampling frequency with freque-ncy of the signal, the FFT has disadvantages caused by frame implementation. Thus, the FFT processes entire frames of data and cannot provide in-between data. If the calculation is done in sliding mode, i.e. the FFT is repeatedly applied to a frame, much more calculations are needed. The computational effort to compute the N/2 Fourier coefficients from N sam-ples is proportional to Nlog2(N). If the calculation is

done in sliding mode, i.e. the FFT is repeatedly ap-plied to a frame of N elements consisting of the last N–1 shifted elements of the previous frame and a single new element, N2log2(N) calculations are nee-ded, [8].

An estimation of power system parameters in the wide range of frequency deviations requires the design of new filters (or an adaptation of their parameters) during estimation. Orthogonal FIR di-gital filters used in papers [9, 10] are used to minimize the noise effect and are not affected by the presence of harmonics. But, the design algorithm involves multiplication and inversion of matrices and requires considerable amount of computations that may not be completed within available time, i.e. one sampling interval.

An efficient method for on-line designing of digital filters of sinusoidal signals is used in [11, 12]. This method describes closed forms for calculating filter coefficients. In general, idea is to search the second-order subsections that eliminate dc compo-nent and all harmonic frequencies except measured one for which has to have unity gain. The complete filter can be realized as cascade of all these sub-sections. The harmonic analysis method proposed in these papers is composed of three different parts: an accurate frequency estimation algorithm, an adaptive cascade FIR filter for filtering out specific signal components and an accurate amplitude estimation algorithm. The chosen model is linear and a linear algorithm for parameter estimation is used. It has been found that the proposed method really meets the need of off-line applications.

In this paper, a simple algorithm for harmonic estimation with benefits in reduced complexity and computational efforts estimation is prescribed. An implementation based on a recently introduced common structure for recursive discrete transforms which contemplated as an implementation of FIR and IIR filter transfer functions with the aim to avoid above mentioned disadvantages. This structure consists of digital resonators in a feedback loop. Computer simulation results tests are quoted to confirm the validity and the performance of the proposed algorithm. Results obtained by algorithm proposed in this paper confirm high accuracy with fast convergence rate.

The organization of this paper is as follows. The description of the existing algorithms for an ortho-gonal filtering and the proposed new implementation based on digital resonators in a feedback loop are given in the part two. The simulation results that confirm validity and performance of the proposed algorithm are presented in the section three. Finally, the conclusion is given in the section four.

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2. MATHEMATICAL MODELS AND PROPOSED ALGORITHM

2.1 FIR filtering of harmonics with on-line adaptation

A number of measurement algorithms apply orthogonal signal components obtained by two ort-hogonal FIR filters. The frequency response of the filters must have nulls at the higher order harmonic frequencies that are expected to be present in the signal and must have a unity gain at the main harmonic frequency. In a case of time-varying frequency, the filter parameters have to be adapted during frequency estimation.

The cascade structure is a particularly useful one for implementing FIR filters. In general, the idea is to search the second-order subsections that eliminate the dc component and all harmonic frequencies except the measured one for which has to have unity gain [11, 12]. The complete filter can be realized as cascade of all these subsections. The block diagram of the algorithm is shown in Fig. 1. Four parts could be noticed. The first part is prefiltering using in order of denoising and antialiasing. The other three parts are the adaptive filtering, the frequency estimation and the amplitude estimation of the basic and higher harmonics.

Fig. 1. Block diagram of the FIR filter based estimation

algorithm The second-order subsection that eliminates the dc

component and frequency 2S� and has a unity gain at the frequency of the mth harmonic 1�� mm � is given by the following transfer function:

� �2

2

011

�

�

�

��

mm

zzzH (1)

where � �tmzm ��� �1

2 sin21 � , Sffj

ez�21 �� � ,

f�� 2� , fs is the sampling frequency, 1 St f� � ,

and Sfmfj

m ez121 ��� � , 11 2 f�� � , f1 is the fundamen-

tal frequency (the frequency of the first harmonic). The subsection that eliminates the harmonic i�

and has a unity gain at the frequency 1�� mm � is shown as follows:

� � � �� � 21

21

cos21cos21

��

��

��

���

mmi

iim

zztzztzH

��

(2)

where the gain � � � � � �1 2

1 11 2cos 2 cos cosi m mt z z m t i t� � �� �� � � � � �

i=1,2,3,…,M, is used to adjust the gain for the mth harmonic.

12SfM f

�� � � �

denotes the maximum integer

part of 12

Sff

. It is equal to the number of subsections

in the cascade. In general, the transfer function of the filter for the

mth harmonic is given as follows:

� � � � � �01

M

m m imii m

H z H z H z��

� � (3)

It can be seen that calculation of the filter coefficients can be easily performed if the funda-mental frequency 1� is known. Therefore, this de-sign algorithm is convenient for the usage in the algorithms where the frequency estimation is in-volved. Algorithms for the frequency measurement where � �1tan t� � or � �1cos t� � are estimated directly, like in [9, 10, 22, 23], are especially convenient. In this case, the cos( )m t�� can be easily calculated using trigonometric formulas.

Due to its cascade structure, the filter (3) is convenient for designing different variations by adding or omitting certain subsections.

The frequency responses of the filters for the first harmonic for different fundamental frequencies and the sampling frequency fs=600 Hz (12 samples per period T0=1/50=0.02s) are shown in the Fig. 2.

Fig. 2. Frequency responses of the first harmonic filter for

different fundamental frequencies and sampling frequency fs=600Hz

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It has been found that the proposed method really meets the need of off-line applications. By using parallel computation algorithms this method should meet the need of on-line applications and should be more practical.

2.2 Frequency sampling based filters A computationally efficient structure for imple-

menting the transfer function (3) may be derived thr-ough use of the Lagrange interpolation formula, [13]:

� � � � 11m

m Wm

AH z H zz z��

� (4)

where

� � � �1

1

0

1N

W nn

H z z z�

�

�

� �� (5)

is a so-called whitening filter (it cancels all harmonics), [14], and

� ���

��

��� 1

0

11

)(N

mnn

mn

mm

zz

zHA . (6)

The realization (4) is convenient since the whitening filter � �zHW is common for all harmonic frequencies and it should be calculated only once at each iteration.

The {zn}, M n M� � � are 2M+1 arbitrary points in the z-plane at which the z-transform is evaluated to give the values H(zn). The resulting structure is seen from (4) to consist of a cascade of 2M+1 first-order sections (containing zeros at z=zn,

= - , ... , 0, ... , n M M ) in cascade with first-order section containing a pole at z=zm.

This structure has been called the frequency sampling structure since its basic coefficients

� �� �nH z are the values of the filter’s frequency

response � �� ��jeH sampled at points equally spaced around the unit cycle.

This frequency sampling structure has several interesting properties. For example, using finite precision arithmetic to represent the structure (4) the pole of the filter will not exactly cancel the corresponding zero of the structure. Thus, resulting structure will have both poles and zeros, and its impulse response will not be of finite duration.

2.3 Resonator based filters Recently, a common structure for recursive

discrete transforms has been suggested, [15], which seems to be suitable to form a common base for every linear filtering-like signal processing opera-

tion. The derivation of this structure and its parame-ters is based on the state-variable formulation, and the results of the observer theory, [16], while its applicability to FIR and IIR filtering operations co-mes from the generalization of the frequency sampling method. A model-based structure was proposed for recursive discrete transforms, which was proven especially suitable for calculating the discrete Fourier transformation. The structure is nothing but an observer containing an embedded model of the signal to be measured, [17].

The block diagram of the structure is given in Fig.3. This structure is based on harmonically related parallel resonators embedded into a common negative feedback. The number of resonator pairs (the number of formed harmonics M) depends on the fundamental frequency �1. Due to this feedback, the properties of this structure substantially differ from that of the well-known above described Lagrange or frequency sampling structures.

Fig. 3. Block diagram of the resonators based

estimation algorithm

In this paper we will concentrate on the time-invariant version of the common transformer structure (see Fig.3), and for simplicity we do not consider the case of multiple resonator poles, thus we have in every channel of the structure as an internal transfer function

� �1

11m

mm

r zH zz z

�

���

, (7)

1 = - , ... , 0, ... , , m M M M� �� ,

which is usually a function of complex coefficients having a pole on the unit circle. The global transfer function for every channel has the form of

TEHNI�KA DIJAGNOSTIKA (BROJ 4 • 2007) 7

� � � �

� �1

01

mm N

nn

H zT z

H z�

�

��

, (8)

= - , ... , 0, ... , m M M . In this paper we will suppose that the poles and

zeros of Tm(z) ( = - , ... , 0, ... , m M M ) are real numbers or occur in complex conjugate pairs, i.e.

m mz z��� . It is a very interesting property of this structure

that at the resonator pole positions

� � 10m n

za n mT z

za n m��

� � �� (9)

= - , ... , 0, ... , m M M , n = - , ... , 0, ... , M M from which we can deduce that the sensitivity of the transfer function at z=zm = - , ... , 0, ... , m M M is zero with respect to any Hn(z), and zn ( = - , ... , 0, ... , n M M ) except the n=m case, [17]. This property is due to the infinite loop gain at these frequencies providing complete independency of the coefficients within the feedback loop.

It is very important to note that this common feedback implements a perfect pole-zero cancella-tion mechanism. The poles in (7), which are not necessarily located on the unit circle in a general case, will be transformed by the feedback loop into zeros, which automatically, and perfectly, cancel their "generators" in expression (8). This is the reason why the application of ideal resonators does not cause implementational problems since every resonator pole is cancelled by a zero generated by the overall common feedback from the very same resonator pole, and this is true even if implemen-tational errors occur, [17].

The design of the filter parameters for this structure is rather straightforward. If the resonator positions are known, the weighting coefficients are given by (4), while if {pn}, n=0,l, ... , N-1, are the poles of the filter, [15]:

� �

� �

11

01

1

0

1

1

N

n mn

m m N

n mnn m

p zr z

z z

��

��

�

��

��

�

�

� (10)

The resonator poles, at the price of some redundancy, can be located arbitrarily. Excellent pass-band behavior can be achieved if the resonator poles are distributed in the pass-band. For elliptic filters, to provide a good stop-band behavior, and spare weighting coefficients, the resonator poles can be located to the transmission zero positions of the

filter transfer function. For a dead-beat observer, the coefficients are calculated as follows:

� �1

1

0

1

mm N

n mnn m

zrz z

��

��

���

(11)

In this case the system has poles only at the origin and operates as a dead-beat observer with transient length equal to the total number of resonators 2M+1.

The amplitude and phase of the mth harmonic are calculated from the harmonic component as the following

� � � � � �2 2Re ( ) Im ( )m m mV k v k v k� (12)

Im ( )( ) arctan Re ( )m

mm

v kk v k� � (13)

2.4 Frequency estimation

The coefficients (11) of the resonator based filters are given by explicit formulas and thus, can be cal-culated on-line or can be tabulated for the frequency range. The frequency of the basic harmonic compo-nent is either calculated by recursive adaptation or can be tuned by an external signal. Due to conver-gence problems that can be present in recursive adaptation, [18, 19], a decoupled module for the frequency estimation has been used in this paper.

Conventional methods assume that the power system voltage waveform is purely sinusoidal, and therefore the time between two zero crossings is an indication of system frequency. Use of zero crossing detection and calculation of the number of cycles that occur in a predetermined time interval, [21], is a simple and well-known methodology. Discrete Fourier Transformation, Least Error Squares and Kalman Filter are known signal processing techniques, used for the frequency measurement of power system signals, [22-24]. A nonlinear filter based on the Extended Kalman Filter principle to estimate the frequency of the signal corrupted with white noise has been proposed in [24]. In this paper, the proposed method deals with the measurement of the parameters of power system signal which is usually perturbed with noise and high disturbances.

Novel recursive methods for the measurement of the local system frequency, where three consecutive samples are used to update the associated weights recursively, have been described in [24, 25]. These methods are convenient since their linear models and direct estimation of cos( )t�� can be used for the calculation of digital filter parameters. Even more, the proposed techniques are suitable for the

TEHNI�KA DIJAGNOSTIKA (BROJ 4 • 2007)8

frequency estimation in a wide range of frequency variations. The algorithm for the frequency estimation from [25] has been used and will be described shortly.

Let us assume the following observation model of the measured signal (arbitrary voltage or current), digitized at the measurement location and uniformly sampled at the frequency 2s t� �� � and filtered by the filter (3),

)sin()( �� �� tkVkv (14) where

v(k) instantaneous signal value; V amplitude; � radian frequency; �t sempling time; Three consecutive samples are connected with the

following equation, [14,16]:

)cos()1(2

)2()( tkvkvkv���

� � (15)

or )()()( kxkzky � . (16)

where

2)2()()( �

�kvkvky , )cos()( tkx �� � ,

)1()( �� kvkz . (17) The x(k) can be calculated using the following recursive formula:

�

�

�

�

�

�

� k

i

ik

k

i

ik

iz

iyizkx

0

2)(2

0

)(2

)(

)()()(

�

� (18)

i.e.

22

2

)()1()()()()1()(

kzkSumZZkSumZZkykzkSumZYkSumZY

����

�� (19)

( ) arccos( ( ))( ) ; ( )( ) 2

SumZY k x kx k f kSumZZ k t�

� ��

(20)

where 0<�2<1 is the forgetting factor. In the iterative procedure (18), the factor �2 is

assumed to be a constant. As shown in [25], an adaptation of the forgetting factor �2 can be a convenient way to improve performance of the estimating procedure. We can define a covariance of the estimation error as follows:

)()( ZxyZxy ��� TR . (21) Using the form of matrices y, Z and x given in (16) we obtain

� �22 )()()()1()1()( kxkzkykRkkR ����� (22) The forgetting factor �2 can be calculated as follows:

� � p

RkR

k

0

minmaxmin2

)(1

1)(

�� ���� (23)

or � �p

RkR

ek 0

)(

minmaxmin2)(

��� ���� (24)

where min� , max� , R0, and p are the chosen values.

3. PERFORMANCE EVALUATION THROUGH SIMULATION

The performance of the proposed technique has been evaluated using simulated waveforms.

The time responses of the algorithm with the algorithm proposed in this paper for the fundamental harmonic amplitude step change from 1 to 0,9 p.u. and the frequency step change from 50Hz to the 49,8Hz are shown in Fig. 4. and Fig. 5., respectively. The additive white zero-mean Gaussian noise with SNR=60dB is present. Fluctuations around the true value are due to noise in the input data. If better accuracy is desirable, this effect of noise could be further reduced by the pole placement of the structure by the coefficients. However, it must be noted that these actions will increase the measurement time.

Fig. 4. Estimation error for f=50Hz and V1=1 p.u. for

t<0s and V1=0.9 p.u. for t>0s, with SNR=60 dB and with harmonics presence.

Fig. 5. Estimation error for f=50Hz for t<0s and f=49,8

Hz for t>0s with SNR=60 dB and with harmonics presence.

The ability of the frequency estimation over a

wide range of frequency changes is investigated using sinusoidal test signal with the following time dependence: f(t)=50+0,5sin(10�t). The estimation is computed for the proposed algorithm, as shown in Fig.6. Good dynamic responses of the proposed algorithm can be noticed.

TEHNI�KA DIJAGNOSTIKA (BROJ 4 • 2007) 9

Fig. 6. Amplitude estimation error for f(t) =

50+0.5sin(10�t), SNR=60 dB and harmonics presence

The effect of presence of noise in the signals was

studied by estimating the frequency and harmonic amplitudes of signals that contain noise. Sinusoidal 50-70Hz input test signal with the superimposed additive white zero-mean Gaussian noise was used as input for the test. The random noise was selected to obtain a prescribed value of the Signal-to-Noise-Ratio (SNR), which is defined as 20log

2ASNR�

�

where A is the amplitude of the signal fundamental harmonics and � is the noise standard deviation. Fig. 7. shows the maximum errors observed in frequency and harmonic amplitude estimates when input signals of 30, 50 and 70 Hz having SNRs of 40, 50, 60, 70 and 80 dB were used. It should be noted that, in practice, the SNR of voltage signal obtained from a power system ranges between 50 dB and 70 dB. At this level of noise, very little errors are expected with the proposed technique, as shown in Fig. 7.

Fig. 7. Maximum estimation errors for noisy input signals

4. CONCLUSION The proposed algorithm for harmonic phasors

estimation uses recursive model-based computation techniques, which enable the reduction of both the computational cost and the memory requirements of the algorithm. This method is easy to implement and very flexible. Due to its recursive form, easy cal-culation, and high accuracy, the proposed algorithm can be very useful for real-time digital systems. The proposed technique is suitable for the harmonic analyzing in a wide range of frequency variations. The simulations and experimental results have shown that proposed technique provides accurate estimates. This algorithm really meets the need of online applications. This technique provides accura-te amplitudes estimates in about one period.

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