New dynamic filter based on least absolute value algorithm for on-line tracking of power system harmonics

S.A. Soliman A.M. AI-Kandari K. El-Nagar ME.-El-Hawary

Indexing terns: Power system harmonics, Time-varying models

Abstmct: The authors present the application of the recently developed technique of least absolute value (LAV) dynamic filtering to optimal tracking of power system harmonics. The proposed tech- nique uses digitised samples of the voltage and current waveform at a power system bus, where a harmonics standard is contemplated. The pro- posed technique can easily handle time-varying harmonic parameters. Two models are developed and tested. In the first, the measurements matrix varies with time, with an identity transition matrix. In the second model, the state transition matrix is a function of the sampling rate and the number of harmonics chosen. The algorithm is tested using simulated and actual recorded data set. A sample set of results is reported.

1 Introduction

Power system harmonics frequently occur owing to the widespread use of electronic equipment, arcing devices and equipment with saturable ferromagnetic cores. The identification of harmonics is important where harmonic standards are to be adopted. It may also be used to alloc- ate loads that exceed specified harmonic current limits. Furthermore, it is needed for designing harmonic filters. Clearly an accurate method is required to carry out the identification process. A bibliography of power system harmonics is available in Reference 5, and Reference 8 offers an overview of relevant issues.

Most frequency-domain harmonic analysis algorithms are based on either the discrete Fourier transform (DFT) or the fast Fourier transform (FFT) to obtain the voltage and frequency spectra of discrete time samples [3, 91. However, misapplication of the FFT algorithm can lead to inaccurate results [14]. In applying FFT the phenom- ena of aliasing, leakage and picketfence effects may lead to inaccurate estimates of harmonics magnitude [14].

0 IEE, 1995 Paper 1587C (P7, Pll), fyst received 2nd February 1993 and in h a 1 revised form 2nd September 1994 S A Soliman and A.M. AI-Kandari are with the Faculty of Technical Studies, Electrical Engineering Department, PO Box 42325, Shuwaikh, 70654, Kuwait K. El-Nagar is with the Electrical Power and Machines Department, Ain Shams University, Abbassia, Cairo, Egypt M E El-Hawary is with the Ekctrical Engineering Department, Techni- cal University of Nova Scotia, Halifax, Nova Scotia, Canada B3J 2x4

IEE Rcc-Gem. T r a m . Disrrib, Vol. 14Z9 No. I, Janumy I995

Damped high-frequency transients are generated when power semiconductors are switched on. These transients are nonstationary and their frequencies are not multiples of 60 Hz [14]. The voltage and current waveforms in a power system may consist of a combination of the funda- mental frequency, harmonics and high-frequency tran- sients. This produces a time-varying amplitude for the voltage-current waveforms; an optimal estimation tech- nique is needed to track harmonics with time-varying amplitudes.

The application of Kalman filtering for online measur- ing of nonstationary transients has been given earlier by the authors. Girgis et al. [9] and George and Bones [13] applied the same technique to power system harmonics analysis, offering a comparison of Kalman filtering and well-known DFT and FFT procedures. The implementa- tion of linear Kalman filter models is relatively simple. However, state equations, measurement equations and covariance matrices need to be defined correctly [14].

This paper presents the application of a newly devel- oped algorithm for the optimal tracking of power system harmonics. The technique is based on the weighted least absolute value (WLAV) dynamic filtering; two models are developed and tested. In the first model a time- varying measurements matrix and a unity state transition matrix are employed. In the second model the state trans- ition matrix is a function of the sampling rate and the number of harmonics chosen.

2

A complete derivation of the proposed filter equations is beyond the scope of this paper. The interested reader can consult Reference 15 for details. Only the filter recursive equations are presented, with a brief description of their

Optimal filtering based on IAV

use.

2.1 The system model This is described in state form by

X ( k + 1) = #k)X(k) + r(k)u(k) (1) where X(k) is the n x 1 state vector of estimates at step k, &k) is the n x n state transition matrix, r ( k ) is the n x s control matrix, and u(k) is the s x 1 control vector at step k. The initial condition X(0) is a Gaussian random vector with the following statistics:

E[X(O)] = B(0) E[{X(O) - R(O)}{X(O) - B(O)}T-J = M(0) (2)

37

For m observations, eqn 20 can be written in ~ ~ c t o r

(21)

h a s

Z(k At) = H(k AtMk) + 4 k ) W k

Z ( k A t ) = m x 1 measurement vector taken ~ v e r the window size

H(k At) = m x (2N + 1) matrix, giving the ideal con- nection between 4kAtbt) and 8# in the b n c e of noise 4 k )

@ ( k ) = ( 2 N + l ) x l s ta tevac tor tobeesha td w(k)= m x 1 mise vcctor associated with the

measuremeDts to beminimkd.

Eqn. 21 describes the measurement system equation at time k A t . The state-qace variable equation for this modd may be expressed as:

Eqn. 22 can be written for N harmonies in vector forsn

(223)

&k) = (2N + 1) x (2N + 1) identity state transition

E(k) = 2N x 1 error vector associated with the states

For model 1 the matrix H(k) is varies with time, whereas the state transition matrix is a constant identity matrix

as:

@(k + 1) = &kMk) + 4k) Where

matrix

4 Variable state transition matrix harmonice model 2

In this model the state transition matrix &k) depends on the sampling rate and the number of samples chosen. Define thc fdlowing sets of states:

dt) = Pdt) cos (#i + iot)

yir) = Pdt) sin + iot) At t = k At we write:

Note that the values of x and y in fhe above quation ace expressed as maximum &S. A $atmonic source is said to exist when the bus voltage .($I a d the bus current 48) are such that:

CPAm" > 0 t323 In other words, the average injected power ;associated with the hfrmonic fraq3lency jq j = Q, 2, . ..# N at any time step k other than the powex frssuency, is positive. Also, the distortion @A] is the m m of the prodwcts of Ule em3Eeieimts of the voltage and eurreat terms of dissimilar frequencies [q.

6 Softwars implementation

A software package has been developed to analyse digit- ised current and voltage waveforms. This package has been tested on simulated data sets, as well as m an actual recorded date set. The p r o g r a m s of this paekage haw been written in Fortran 77, and they compute the voltage and cumnt harmonics magnitude, the voltage and mrreut harmonics phase angles, an$ the fundamen- tal power and hannanics pow+r.

6.1 InitiaJisation of proposed filter To initialise the reclusive process of the proposed filter, with an initial process rector and covariance matrix P, a simple deterministic procedure uses the static least squares error estimate of previous measurements. Thus, the initial process vector may be computed as:

and the corresponding covariance error matrix is:

Po = rHTH-J-1 (34) when? H is an m x rn matrix of measurements, and z is an m x 1 vector of previous measurements, However, for online application with no prior measurements, the initial procsss veetor may be selected to be zero, and the. 6cst

39

few milliseconds are considered to be the initialisation period [14].

62 Testing proposed algorithm using simulated data The proposed algorithm and the two models were tested using a voltage signal waveform of known harmonic con- tents described as [!SI:

v(t) = 1 cos (wt + lo") + 0.1 cos (3wt + 20")

+ 0.08 cos (5wt + 30") + 0.08 cos (9wt + 40") + 0.06 cos (l lwt + SO0) + 0.05 cos (13wt + 60") + 0.03 cos (19wt + 70")

The data window size is two cycles, with sampling fre- quency of 64 samples/cycle. That is, the total number of samples used is 128 samples, and the sampling frequency is 3840 Hz

For this simulated example we have the following results.

Using the two models, the proposed filtering algorithm estimates exactly the harmonic content of the voltage waveform both magnitudes and phase. angles and the two proposed models produce the same results.

The steady-state gain of the proposed filter is periodic with a period of 1/60 s. This time variation is due to the timevarying nature of the vector states in the measure- ment equation. Fig. 1 gives the proposed filter gain for X , and Y,.

1 0 25 50 75 I00 125 150

k time step Fig. 1

~ x , y,

Gain of the propasedfilter for X , and Y, us~ng models I and 2 OoeUW step = 0261 ms

_ _ _ _

The gain of the proposed filter reaches the steady-state value in a very short time, since the initialisation of the recursive prosess, as explained in the p r d n g section, was sficiently accurate.

The effects of frequency drift on the etimate are also considered. We assume small and large values for the fre- quency drift: Af= -0.10 Hz and Af= -1.0 Hz, respect- ively. In this study the elements of the matrix H(k) are calculated at 60 Hz, and the voltage signal is sampled at w (0 = 2 d f= 60 + 4). Figs. 2 and 3 give the results obtained for these two frequency deviations for the fun- damental and the third harmonic. Fig. 2 gives the estim- ated magnitude, and Fig. 3 gives the estimated phase angles.

Examination of these two curves reveals the following. For a small frequency drift, Af = - 0.10 Hz, the funda-

mental magnitude and the third harmonic magnitude do not change appreciably; whereas for a large frequency drift, 4= - 1.0 Hz, they exhibit large relative errors, ranging from 7% for the fundamental to 25% for the third harmonics.

On the other hand, for the small frequency drift the fundamental phase angle and the third harmonic phase 40

angle do not change appreciably, whereas for the large frequency drift both phase angles have large changes and the estimates produced are of bad quality.

exact value

0.8

0 50 100 150 k time step

Fig. 2 drifts using models I and 2

~ iundamentalq= - 1

Estimated magnitude of 60 Hz Md third harmonic fmfrequency

One time step = 0.261 ms

fundamenlal, 4 = -0.1 . . . . . . . third harmonic, 4 f = -0.1 _ _ _ _ third harmonic, 4 f = - 1

30r exact value

exact value L

In . : -.___ a

' ........... .. ._._.............

-101 0 50 100 150

k time step

Fig. 3 One time step = 0261 ms, data Window sip - 1 cycle, sampling rate = 3840 Hz ~ rundamental,q- -0.1

fundamental, 4 = - 1 _ _ _ _ third harnumk 4 = -0.1 . . . . . . . third harmonic, 4 = - 1

Estimated phase angles forfrequency drifts using models I and 2

To overcome this drawback, it has been found through extensive runs that if the elements of the matrix H(k) are calculated at the same frequency of the voltage signal waveform, good estimates are produced and the frequency drift has in this case no effect. Indeed, to perform this modification the proposed algorithm needs a frequency-measurement algorithm before the estima- tion process is begun [19].

It has been found, through extensive runs, that the filter gains for the fundamental voltage components, as a case study, do not change with the frequency drifts. Indeed, that is true since the filter gain K(k) does not depend on the measurements (eqn. 8). As the state transition matrix for model 2 is a full

matrix, it requires more computation than model 1 to update the state vector. Therefore in the rest of this study only model 1 is used.

6.3 Testing on actual recorded data The proposed algorithm is implemented to identify and measure the harmonics content for a practical system in operation. The data for this system are from Alberta Power Limited (APL). The system under study consists of a variable-frequency drive that controls a 3000 HP, 25 kV induction motor connected to an oil pipeline com- pressor. The waveforms of the three phase. currents are

IEE Proc-Gener. Transm Disnib., Vol. 14.2, No. I , January 1995

given in Fig. 4. It has been found for this system thqt the waveforms of the phase voltages are nearly pure sinus- oidal wavefom A careful examination of the current

-2.0 0 5 - 1 . 6 ~ IO 15 20 25 30 35 CO

time.ms Fig. 4

~ 1, ', IC

Actual recorded ament waw$onn of p k s A, Band C

_ _ _ _ .......

waveforms revealed that the waveforms consist of har- monics of 60Hz, decaying period high-frequency tran- sients, and harmonics of less than 60 Hz (subharmonics). The waveform was originally sampled at a 118 ms time interval and a sampling frequency of 8.5 kHz. A com- puter program was written to change this sampling rate in the analysis.

Figs. 5 and 6 show the recursive estimation of the magnitude of the fundamental, second, third and fourth

1.01 0 50 100 150 200 250 300

k lime step Fig. 5 Estimatedjhfmmtal voltage Onc time stcp=O.l88ms. sampiii rate=8474.57Hz, number of har- monics = 15, window sip = 2 cydes

-0.01

k time step Fig. 6

m o k = IS. window airc = 2 cydcs

Estimated d t a g e krrmonics for V, Onc time s C C p = O . l 8 8 m ~ . sampling rate= 8414.57Hz, number of har-

~ savndharmooic _ _ _ _ thkdbumoo* ....... fourth harmonic

IEE Roc.-Gem. Ti- Distrib- Vol. 142. No. I , January 1995

harmonics for the voltage of phase A. Examination of these curves reveals that the highestenergy harmonic is the fundamental, 60 Hz, and the magnitude of the second, third and fourth harmonics are very small. However, Fig. 7 shows the recursive estimation of the fundamental, and

- n g U E, 14 c 0.2 I

I I I I r

1.01 1 , I , 1 0 50 100 150 200 250 300

k time step Fig. 7 Estimatedjimdamntal current I , One time stcp=0.188ms, sampling ratc=8474.57Hz, numbcr of har- monics = 15

Fig. 8 shows the recursive estimation of the second, fourth and sixth harmonics for the current of phase A at different data window sizes. Indeed, we can note that the

s 0.2 ..........

E

C

I b -0.2 -0.3r , I , I , , I I 1

0 50 100 150 200 250 300 k time step

Fig. 8 Harmonics magnitude of I , against time steps ai wrim window sizes One time step-0.188ms. sampling rate=8474.57Hz, number of har- monics = 15 - skondharmonic _ _ _ _ fourth hannonic ....... sixth harmonic

magnitudes of the harmonics are time-varying since their magnitudes change from one data window to another, and the highest energy harmonics are the fourth and sixth. On the other hand, Fig. 9 shows the estimate of the phase angles of the second, fourth and sixth harmonics, at different data window sizes. It can be noted from this Figure that the phase angles are also time-varying because their magnitudes vary from one data window to another.

Furthermore, Figs. 10-12 show the recursive estima- tion of the fundamental, fourth and the sixth harmonics power, respectively, for the system under study (the factor 2 in these figures is due to the fact that the maximum values for the voltage and current are used to calculate this power). Examination of these curves reveals the fol- lowing results.

The fundamental power and the fourth and sixth har- monics are time-varying.

41

For this system the highestaergy harmonic com- ponent is the fundamental power, the power due to the fondamental voltage and current.

480r .... ............... ..................

Izo ws.= I. cycle I

0.0 J I J I w.s.=z.b cycles I

-120 I , I , , I I

0 50 100 150 200 250 300 k time step

Fig. S Hmkronics phase angles of I , against time steps at various wi&wsizes one lime stcp-O.188ms sam* rate- 8474.57&, n u m k of har- mwia=15 - aeeond b.rmonie _ _ _ - fouah harmonic ....... rhth l l umdc

- . I , i , I I

0 50 100 150 200 250 k time step

Fig. 10 One time step = 0 . 1 8 8 ~ samm rate= 8474.57m number mmig= IS - p.4

p . pc

F ~ u u b m t d powers agm time steps

_ _ _ - .......

--I

300

of har-

............. ...

s - 0 50 100 150 200 250 300

k time step Fig. 11 F w h hamonk power in the three phases against time steps U t t o o r i o u r W s i u s ooc tims 9cp -0.188mq sam* rate-8474.57&, number of har- monics = 15

P A

p . pc

~

....... ___ -

The fundamental power, in the three phases, are unequal; i.e. the system is unbalanced.

The fourth harmonic of phase C, and later after 1.5 cycles of phase A, are absorbing power from the supply, whereas those for phase B and the earlier phase A are supplying power to the network.

42

The sixth harmonic of phase B is absorbing power from the network, whereas the six harmonics of phases A and C are supplying power to the network; but the total power is still the sum of the three-phase power.

I , I , , I , I , 0 U) 100 150 200 250 300

k time step

Fig. 12 Sixth harmonic power in the three phases agana time steps clf various window sizes Onc time stcp=O.I88ms, sampling rate = 8474.57&, number of har- monics = 15

P A p . PC

- _ _ _ _ .......

The fundamental power and the fourth and sixth har- monics power are changing from one data window to another.

6.4 Comparison with the Kalman filter (KF) algorithm The proposed algorithm is compared with KF algorithm. Fig. 13 gives the results obtained when both filters are

d 0.1 o'2[

v) -0.1 W.S..I.O cycles U I E W.S = 1.5 cycle+ ._

-0.2 I W.5.I 2-0 cycles I - 2 -0.31 , I , I , , I , I ,

0 U) 100 150 200 250 300 k time step

Fig. 13 One tiw stcp-O.l88ms, apmplias rate = 8474.57Hz number of har- monica = 15 - KF _--- w u v

Estimated second harmonic magnhde using KF cad WLAV

implemented to estimate the second harmonic com- ponents of the current in phase A, at different data window sizes and when the considered number of harmonics is 15. Examination of this Figure reveals the following: both filters produce almost the same estimate for the second harmonic magnitude; and the magnitude of the estimated harmonic varies from one data window to another.

7 Effects of outliers

In this Section the effects of outliers (unusual events on the system waveforms) are studied, and we compare the new proposed filter and the well-known Kalman filtering algorithm. In the h t Subsection we compare the results obtained using the simulated data set of Section 6.2, and in the second Subsection the actual recorded data set is Used.

IEE Proc.-Gener. Ti- Disnib., Yd. 142, No. I , January 1995

7.1 Simulated data The simulated data set of Section 6.2 has been used in this Section, where we assume (randomly) that the data set is contaminated with gross error, we change the sign for some measurements or we put these measurements equal to zero. Fig. 13 shows the recursive estimate of the fundamental voltas magnitude using the proposed filter and the well-known Kalman filtering algorithm. Careful examination of this curve reveals the following results.

The proposed dynamic filter and the Kalman filter produce an optimal estimate to the fundamental voltage magnitude, depending on the data considered. In other words, the voltage waveform magnitude in the presence of outliers is considered as a time-varying magnitude instead of a constant magnitude.

The proposed filter and the Kalman filter take approx- imately two cycles to reach the exact value of the funda- mental voltage magnitude. When the technique of Reference 17 is used to correct

these outliers, Fig. 14 is obtained. The new proposed

exact

1

0 30 60 90 120 150 k time step

Eflects of bad data on the estimated fundamental voltage Fig. 14 Onetime stcp = 0.188 rn ~ KF . . . . . . . WLAVF

filter almost produces the exact value of the fundamental voltage during the recursive process, and the effects of the outliers are greatly reduced.

7 2 Actual recorded data In this Section the actual recorded data set that is avail-

is tested for outliers contamination. Fig. 15 shows

r exact

0 30 60 90 120 150 k time step

Fig. 15 cowectwn for outliers One tune step = 0.188 ms __ beforecorrection

Estimated fundamental voltage magnitude before and after

after a m t i o n . .....

the recursive estimate of the fundamental current of phase A using the proposed filter, as well as Kalman filter algorithms. Indeed, both filters produce an optimal IEE h . - G e n e r . T r a m Distrib., Vol. 142, No. I , January 1995

estimate according to the data available. However, if we compare this figure with Fig. 7, we can nob that both filters produce an estimate different from what it should be.

Fig. 16 shows the recursive estimates using both algo- rithms when the outliers are corrected, as explained in

0 30 60 90 120 150 k time step

Fig. 16 Estimated f i a l current when the data set is contmn- inated with outliers One time step = 0.188 ms, sampling rate = 8474.57 Hz number of harmonica = 7, window size = I cycle - KFbefore

WLAV after _ _ _ _

References 17 and 18. Indeed, the proposed filter pro- duces an optimal estimate similar to what it should be, which is given in Fig. 17.

0.61 I 0 30 60 90 120 150

k tlmQ step

Fig. 17 Estimatedfundamental currenf w o r e and clfter correction for outliers One time step = 0.188 ms, sampling rate = 8414.57 Ha, numbcr of harmonica = 7, window size = I cyck ~ WLAV after

KF before _ _ _ _

8 Conclusion

The contributions of this paper can be summarised as follows.

The proposed filter algorithm can easily handle the parameters of harmonics with time-varying magnitudes.

The proposed filter and KF produce the same estim- ates if the measurement set is not contaminated with bad data.

The proposed filter is able to identify and correct bad data, whereas the KF algorithm needs prefiltering to identify and eliminate this bad data.

The proposed filter produces the optimal recursive estimate according to the data available, and when gross errors (bad data) contaminate a wave with a constant

43

7- 1

harmonic magnitude, this wave is transformed to have a time-varying harmonics magnitude.

If the proposed algorithm is used to estimate har- monics of constant magnitude, which are contaminated with outliers, the estimates produced will be of poor quality. However, if these outliers are corrected using the technique explained in References 17 and 18, the pro- posed filter algorithm produces good estimates.

It has been shown that if the waveform is nonsta- tionary, the estimated parameters are affected by the size of the data window.

It has been pointed out in the simulated results that the harmonic filter is sensitive to the deviations of fre- quency of the fundamental component. An algorithm to measure the power system frequency should precede the harmonics filter [19].

9 References

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2 BREUER, GD., st al.: ‘HVDGAC harmonic interaction: Part I, Development of a harmonic measurement system hardware and software’, IEEE Trans. P o w A p p . Syst, 1982, 101, (3), pp. 701-7023

3 KRAFT, L.A., and HYDT, G.T.: ‘A method to analyze voltage res- onancc in power system’, IEEE Trans. Power A p p . Syst., 1984, 103, (5), pp. 1033-1037

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6 TURANLI, H.M., MAHMOUD, AA., TANG, Y., and TRAHAN, RE.: ‘Analysk of harmonics in an EHV transmission system with three-phase representation’, Int. J . Energy Syst, 1989, 9, (3), pp. 157-161

44

7 HEYDT, G.T.: ‘Identihtion of harmotic sou~ce8 by a state eatimation technique’, IEEE Trans. Power Deliu., 1988, 4 pp. 569- 576

MONICS: ‘Power system harmonics: an ove+View’, IEEE Trmu.

9 GIRGIS, A.A., et al.: ‘Measuremart and characterktion of har- monic and high frequency distortion for a luge industrial load‘, IEEE Trans. Power Delia., 1990,S, (1). pp. 427-434

10 WRTY, Y.V.V.S., and SMOLINSKI, WJ.: ‘A Kalman filter bascd digital percentage differential and ground fault rday for a >phase power transformer: IEEE Trans. Power Deliu., 1999 S, (3), pp. 1299-1307

11 CHOWDHURY, F.N., et al.: ‘Power system fault detection and state estimation using Kalman filter with hypothesis testing’, IEEE Trans. Power Deb . , 1991,6, (3), pp. 1025-1030

12 HEYDT, G.T.: The identification and analysis of harmonic signal in electric power systems’, h t . J . Energy Syst., 1991, 11, (l), pp. 20-24

13 GEORGE, T.A., and BONES, D.: ‘Harmonic power determination using the fast Fourier transform’, IEEE T i m . Power Deliu., 1991,6, (2), pp. 530-535

14 GIRGIS, A.A., CHANG, W.B., and MAKRAM, E.B.: ‘A digital recursive measurement scheme for on-line tracking of powex system harmonics’, IEEE Trans. Power Deliu., 1991,6, (3), pp. 1153-1160

15 CHRISTENSEN, G.S., and SOLIMAN, S.A.: ‘Optimal filtering of linear discrete dynamic systems b a d on least absolute approx- imation’, Automnfica, 1990,26, (Z), pp. 389-395

16 BROWN, R.G.: ‘Introduction to random signal analysis and Kalman filtering’ (John Wdey & Sons, 1983)

17 CHRISTENSEN, G.S., SOLIMAN, SA., and ROUHI, A.: ‘Dis- cussion of “An example showing that a new technique for LAV esti- mation breaks down in artain cases”’, Cmput. Stat. Data Anal., 1991,19, pp. 203-313

18 SOLIMAN, S.A., CHRISTENSEN, G.S., and ROUHI, A.: ‘A mw technique for unconstrained and constrained linear LAV parame& estimation’, Can. Electr. Comput. Eng. J., 1989,l4 (I), pp. 24-30

19 SOLIMAN, S.A., CHRISTENSEN, G.S., KELLY, D.H., and EL- NAGGAR, K.M.: ‘Least absolute value based on linear programing algorithm for measuring of power system frequency from a distorted bus voltage signals: Electr. Mach. Power Syst., 1992, 20, pp. 549- 568

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