Para metric harm o n ic a n a lysis

T.T. Nguyen

Indexing terms: Power systems, Harmonic levels, Harmonic analysis

Abstract: Harmonics in power systems is now a subject of wide ramifications. One particular aspect is that of capturing harmonic data at selected locations in a power network and processing it to identify harmonics and to quantify their magnitudes and arguments. Circumstances are encountered in practice for which the discrete Fourier transform (DFT) cannot be relied on to achieve valid harmonic component identification. These are where there are subharmonics, harmonics which are not integer multiples of the supply frequency, and where two or more harmonics have only small frequency separations between them. The paper reports a new procedure which fulfils the requirements of practical harmonic analysis. It avoids altogether the limitations of the DFT algorithm and is based on the nomination of a distorted waveform model expressed in terms of a sum of sinusoidal functions. Model parameters are the frequencies, magnitudes and arguments of the harmonics in the waveform it represents. The error between this model waveform and the actual one represented in captured form is minimised. At the minimum, the parameters of the model are those of the waveform for which harmonic analysis is required. A key advance in this parametric form of analysis is that of a partitioning of the data for the waveform to be analysed into a training set and a test set. This partitioned form of generalised parametric harmonic analysis is thus developed. Key concepts are clarified via a numerical example to illustrate how this approach can excel for the harmonic analysis in power systems.

List of principal symbols

m(t) = function representing the signal model t = independent continuous time variable s(t) = actual signal as a function of time N = number of frequency components in the signal

M = length of complete data file model

0 IEE, 1997 IEE Proceedings online no. 19970717 Paper received 29th January 1996 The author is with the Department of Electrical and Electronic Engineer- ing, Energy Systems Centre, The University of Western Australia, Ned- lands, Perth, Western Australia 6907, Australia

MI At = sampling time interval E, = training error E, = test error n = time step

1 Introduction

Direct measurements are now widely made of harmonic levels in power systems for the purposes of confirming the extent of waveform distortion. Often, these assist in locating harmonic sources which may be of particular concern. To meet all of the different circumstances that can be encountered in processing recorded harmonic data in practice, fully comprehensive and dependable generalised procedures are required to give reliable results in each processing task as it arises.

This paper reports a procedure that is based on a parametric form of harmonic analysis the elements of which have been given previously in the signal process- ing field [1-4]. A signal model is postulated in terms of a sum of sinusoidal functions. The parameters of these functions are their frequencies, magnitudes and argu- ments. An error is formed from the difference between this waveform representation and the actual waveform over a selected data record length for which harmonic analysis is to be carried out. This error is then mini- mised, whereupon, the signal model and the waveform to be analysed are the same within the tolerance allowed in minimisation. Therefore, the parameters of the model give the harmonic components of the wave- form for which harmonic analysis is required.

An acknowledged limitation of the method previ- ously put forward has been that of its sensitivity to the correct identification of the order of the signal model and the data record length in minimisation. As a result of this high sensitivity, there is always doubt as to the validity of the harmonic components that analysis iden- tifies. The key advance reported here is that of parti- tioning the total data set for analysis into a ‘training set’ and a ‘test set’ along lines analogous to those in neural networks. It is shown that the use of these parti- tions in sequence leads to a reliable harmonic analysis procedure which guarantees the validity of all of the harmonic components it identifies. The fully developed form of the procedure is therefore of general applica- tion in signal processing and advances previous work in that field. This is developed here specifically for practi- cal harmonic analysis in power systems in which it excels.

2

Today, facilities are available for online harmonic anal- ysis on site. These combine data acquisition and har-

= length of training data set

Harmonic data processing in power systems

21 IEE Proc.-Gene?. Trunsm. Distrib., Vol. 144, No. 1, January 1997

monic data processing in the one system. They can provide a continuous indication of individual harmonic levels at the point of data acquisition in a power net- work. Another arrangement, widely used in practice, is one in which data in recording systems in the field is saved and processed offline usually at a central loca- tion. In either arrangement the discrete Fourier trans- form (DFT) allows the discrete frequency spectrum of a periodic discrete-time waveform to be found when the time-window length in data processing is equal to the time-period of each waveform that is processed. When this is the case, no further questions of principle are likely to arise. Data processing based on the DFT algorithm should lead to an accurate identification of the harmonic content of waveforms defined in discrete- data form.

Not all harmonic analysis, however, is so straightfor- ward. Some nonlinear loads, including those which involve cycloconverters, can give rise to frequency com- ponents which are not integer multiples of the supply frequency. A way forward can sometimes be found when the harmonic frequencies are related to the sup- ply frequency by rational numbers. In this case, har- monic frequency, f,, is related to the supply frequency,

fa, by f , = fok,/m, where ki and mi are each integers. When this is fulfilled the time-window length to use in DFT analyses can be estimated from the time taken for trajectories in state-space axes to close on themselves. However, many cycles of data corresponding to data recordings of several seconds may be necessary for state-space trajectories to close. Sufficient data lengths for the procedure to be satisfactory are not always available. Moreover, the procedure breaks down when the ratio of to f a departs from that of a rational number.

In still other cases, subharmonic mode frequencies can be encountered. Subharmonic frequencies can be integer submultiples of supply frequency such as f 0 /2 or f J 3 , but the more general case is where subharmonic frequencies can take any value lower than that of the supply frequency. Most intractable of all are cases where waveforms, although bounded in value, do not repeat at all even for extended data recording periods. These are the chaos regimes of response in nonlinear circuits. Fortunately, such forms of response are rare in so far as can be discerned from published practical experience.

Expressed in its most general terms, the requirement in harmonic analysis is for data processing procedures that can be relied on to find all frequency components above and below supply frequency. The most general case is where there is no special relationship between the frequencies of individual components and the sup- ply frequency. They can take any value. The main idea of harmonic analysis in which harmonic frequencies are integer multiples of supply frequency then becomes a special case.

3 Parametric harmonic analysis

The procedure begins with a postulated waveform or signal model expressed in terms of a finite number of sinusoidal functions. In the continuous time domain the form of this representation is given by

N m(t) = mo + C [ A ~ coswkt + B~ sinwkt] (1)

k = l

In eqn. 1 t is the independent continuous time variable, and ma is the DC component. Ak and Bk are the magni- tudes of the cosine and sine functions of angular fre- quency, u k . N is the total number of harmonics.

When At is the sampling interval of the data acquisi- tion system that captures power system waveform data using either voltage or current quantities as may be available or convenient, the discrete form of the wave- form model is given as

m(n) = mo + C [ A k coswknAt + Bk sinwknAt] (2)

Here n identifies the time step in the discrete sample sequence and t = nht.

If the actual data recording of a system quantity for which harmonic analysis is to be carried out is denoted by s(n), the requirement is to find N, ma, o&, Ak and Bk such that the actual data sequence, s(n), matches the postulated one, m(n), as closely as possible. To achieve this matching between m(n) and s(n), the difference between them is formed and minimised.

Using a squared error form for the purpose, the error function to be minimised is expressed as

N

k=l

M _.

E = C [ m ( n ) - s(n) I2 ( 3 ) n=l

The upper limit of the summation in eqn. 3 is the time- window length of the method of harmonic analysis.

At the minimum point of E the values of the param- eters of the model (N, m,, colt, Ak and Bk for k = 1, 2, ..., i?) specify completely the individual frequency com- ponents of the harmonic data recording s(n). This is the principal basis of parametric harmonic analysis.

4 Data partitioning: training and test sets

If s(n) represents the whole of the waveform data sequence to be processed, there is the possibility of some or all the parameters found from minimising E not necessarily being valid even though a well-defined minimum for the error function E has been achieved. In particular, large values of N are conducive to achieving low error. There is, therefore, a need to develop a way of finding the value of N separately from the minimisation process being used to find the remaining parameter values. This is one of the princi- pal keys to the successful use of the harmonic analysis method proposed.

The starting point of the procedure developed in this paper for finding N is that of separating the data stream s(n) into two sets. Useful analogy can be drawn here with the training of neural networks. One of the sets is referred to as the ‘training set’ and is denoted by S(n). The remaining section of s(n) is referred to as the ‘test set’ and is denoted here by T(n). On that basis,

[s(n)lt = [S(7-)lt, [T(n)lt (4) The length of partition S(n) is denoted by M1. For compactness in the subsequent working, M2 is used for M1+ 1.

The first step of the overall procedure is to achieve error-function minimisation confined to the training set so that the function to be minimised is

(5) n=l

iL IEE Proc-Gener. Transm. Distrib., Vol. 144, No. 1, January 1997

Finding a minimum gives the parameter set ma, mk, Ak, Bk for k = 1, 2, ..., N for a nominated value of N . These parameters are then used to form m(n) over the range of samples M2 to M and a total test error is formed from

A4

>

n=M2 If ET 5 oT where oT is a nominated error-bound, the parameter set found from minimising E, is accepted as being valid. Otherwise, N is adjusted and a new set of parameters is found by minimising E,. With these new parameters, the test error ET is recalculated. The revised parameter set is accepted if this further test error is less than the upper bound nominated for it. Otherwise, the procedure is repeated successively until both E, and ET are simultaneously less than the toler- ances nominated for them.

The case for which ET > oT can be caused by either an incorrect value for N or for the choice of the win- dow length M1. There is also the possibility that both might be incorrect. A systematic procedure is required for adjusting M1 and N so as to arrive at correct values for them. A series of successive evaluations of E, and E, which achieves this is summarised in the flow dia- gram of Fig. l .

valid solution for N and

minimum requirement for M1 Wr @kr Ak, Bk k = l A J

initial values of parameters N. %, %. A,. E, k=1.2. ,N

p = p + I

find test error Er I

1 - increase MI

Fig. 'I p iteration counter pm maximum number of iterations MI data window length of training set M , maximum value of M1 N number of separate frequency components in the waveform for which har- monic analysis is carried out

Sequence for finding signal model parameters

For a given initial value of N, the minimisation starts from a low value of the training set window length, M1. If the test error is not less than the tolerance spec- ified for it, the window length A41 is successively

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 1. January 1997

increased in successive iterations until the test error sat- isfies the convergence criterion or until M1 reaches the maximum value. If the test error remains higher than its specified tolerance and M1 has reached its maxi- mum value, an adjustment for N is made in the subse- quent iteration.

Adjusting the value of N involves either increasing or decreasing it.

If the initial value for N used in the first iteration in the sequence of Fig. 1 is No, then initial adjustment of N involves decrementing it by 1. Decrementing by 1 is successively implemented in successive iterations until both E, and ET are less than a prespecified tolerance or until N = 1. The decrementing procedure ensures that there will be no spurious frequency components in the final frequency spectrum evaluated.

When N has reached the value of 1 and the conver- gence criteria for E, and ET are still not achieved, N is then incremented, starting from No + 1. The sequence in the minimisation and testing is then repeated until convergence is achieved.

Whenever N is adjusted, MI is reset to a low value.

5 Initial values

An initial frequency spectrum is found by DFT analy- sis. From this initial frequency spectrum, initial estima- tions for the number of frequency components, N , and parameters m,, Ak, Bk and for k = 1, 2, ..., N are available for starting the minimisation process.

6 Quasi-Newton method of minimisation

Of the different procedures that might be adopted for the purposes of minimising the total error function in eqn. 5 , a second-order method based on the quasi- Newton algorithm is used here. It requires the gradient of E, with respect to the unknown variables ma, Ak, Bk and mk. Differentiating eqn. 5 with respect to these gives the elements of the gradient:

M1

(7)

M1 d E S - =2 x[-nAtAk sinwknQt + nAtBk coswknAt]

n=l

7 Numerical example

The effectiveness of the harmonic analysis procedure developed is examined here for a distorted waveform that has a fundamental component s4(t) and harmonic components together with a DC component and is given by

The waveform of s(t) has a DC component, so, three subharmonic frequency components sl(t), sz(t) and s3(t), a base frequency component s4(t), and two com-

s ( t ) = so + Sl( t ) + sz ( t ) + ss( t ) + S4( t ) + s.5 ( t ) + S 6 ( t ) (11)

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ponents of frequency greater than the base frequency, ss(tj and s6(t).

For a base frequency of 50Hz, the subharmonic fre- quencies are 25Hz, (50/d2)Hz and {(50/d2) + 0.5)Hz. This gives a case for which the subharmonic frequency is an integer submultiple of the supply frequency (the 25 Hz component) together with subharmonic frequen- cies which have irrational number relationships to the supply frequency. Also, two of the subharmonic com- ponents are separated in frequency by only 0.5Hz. To separate these two components makes special demands on harmonic analysis. For the frequencies above the supply frequency, one is that of the third harmonic so that the harmonic frequency is an integer multiple of the supply frequency. The ratio of the frequency of the remaining component to the supply frequency is an irrational number.

The waveform proposed therefore has all of the com- ponents to test the harmonic analysis method devel- oped in each of the main aspects for which generalised harmonic analysis is required: subharmonics, non-inte- ger harmonics, and harmonics with closely similar fre- quencies. Expressions for the components of the test waveform are as follows: so = 0.1 DC component sl ( t ) = 0.3 cos(2x x (50/2)t + 70") subharmonic: 25Hz s2(t) = 0.2 cos(2x x (50/d2)t + 60") subharmonic:

s3(t) = 0.7 cos(2x x (50/d2 + 0.5)t + 80") subharmonic:

s4(t) = 1.0 cos(2x x 50t) fundamental: 50Hz ss(t) = 0.5 cos(2x x d3 x 50t + 90") harmonic: 86.6Hz s6(t) = 0.4 cos(2x x 3 x 50t + 40") third harmonic:

The first step of harmonic analysis is that of finding the initial values of N, m,, Ak, Bk and cuk for k = 1, 2, ..., N from which the optimisation method can start. It is taken that 600 equally-spaced data samples are avail- able corresponding to a time record length of 200ms for a sampling frequency of 3kHz. Initial values are found by DFT analysis over the complete record length of 600 samples. Given that two frequency components are separated by 0.5Hz, a frequency separation of 0.1 Hz was adopted leading to the frequency spectrum of Fig. 2.

35.35Hz

35.85 Hz

150Hz

n

0 25 50 75 100 125 150 175 200 frequency, Hz

Fig.2 Frequency spectrum of the test waveform as found from DFT analysis

On the basis of the peaks in the spectrum, DFT anal- ysis identifies the fundamental frequency component, and components at or close to 150Hz and 86Hz. It identifies the subharmonic component at 25Hz. It gives

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a further subharmonic component at 35.9Hz instead 2- identifying the two separate components at 35.35Hz and 35.85Hz. All of the numerous other frequency components of the spectrum of Fig. 2 are spurious and derive from spectral leakage. A complete summary of the results of initial DFT analysis is given in Table 1. As is generally the case, DFT analysis is least accurate in identifying phase angle values. However, the practi- cal significance of this usually reduces to those cases in which harmonic power levels are sought.

Table 1: Results of initial DFT analysis

Frequency Magnitude Phase

actual, Hz DFT, Hz actual DFT actual, deg DFT, deg

0.0 0.0 0.1 0.175 0.0 0.0

25.0 23.9 0.3 0.389 70.0 118.9 35.35 - - - - -

35.85 35.9 0.7 0.878 80.0 73.79

50.0 50.0 1.0 0.992 0.0 -2.44

86.6 86.5 0.5 0.467 90.0 91.61

150.0 150.2 0.4 0.396 40.0 30.46

Of the two subharmonic components that are sepa- rated in frequency by only 0.5Hz, the one that domi- nates is the component of frequency 35.85Hz and for which the magnitude is 0.7. From Table 1, DFT analy- sis gives a component of frequency 35.9Hz and magni- tude 0.878. The frequency error is small. The magnitude error is substantial (about 25%). There is no basis on which to decipher from the spectrum of Fig. 2 the subharmonic component of frequency 35.35Hz. This component is corrupted almost completely by spectral leakage in the DFT algorithm.

Harmonic analysis drawing on the error minimisa- tion procedure of Section 3 begins from the initial val- ues for m, and the A and B magnitude components for each harmonic frequency which DFT analysis discloses. In addition to the DC component, the spectrum of Fig. 2 gives five frequency components leading to the initial value N = 5 to begin the minimisation sequence.

The separation of the total data file into training and test partitions is at choice. As a rule, the preference is to choose the training set to be the larger of the two. On that basis, an initial choice is made here of 500 data samples in the training set leaving 100 samples for the test set. Therefore, the maximum window length for the training set is M1 = 500. That for the test set is M - M1 or 100.

Using the quasi-Newton procedure of Section 6, error minimisation for the training set led to conver- gence at low error. The window length for forming training error starts from a low value of 100. However, a high test error is encountered which indicates a need to revise the initial estimate for the number of fre- quency components, N. There is no indication from the test error magnitude as to whether N should be increased or decreased to lower the error.

Choosing first to decrement N to a value of 4 increases the error in both the training set minimisation and the test set minimisation. Successively reducing N to 3, then to 2, and then to I confirms a trend of increasing error in both the training set and test set minimisation.

Returning from this sequence to the initial choice of N = 5 and then incrementing to N = 6 leads to a low

IEE Proc -Gener Tvansm Distrib , Vol 144, No I , January 1997

error of in minimisation confined to the training set and for the test error. For checking purposes, increasing N to 7 and then to 8 confirms the level of test error similar to those for N = 4 or 3 . The proce- dure, therefore, correctly gives the value N = 6.

Turning to the choice of data window length, reduc- ing the window length for the training set from M1 = 500 confirms that, with the correct choice of N, analy- sis remains accurate for M1 reduced to as low as 300 samples. For window lengths shorter than 300, although the error from minimisation for the training set is low, the test error is high confirming an unsatis- factory choice of window length when N is correct. On the other hand, the errors are almost unchanged when M1 is increased from 300 up to 500. Therefore, any choice of M1 between 300 and 500 is satisfactory. Of course, the more data points there are in the test set, the more effective and reliable testing is likely to become. Overall, good choices in the present example might be M1 = 400, M - A41 = 200, or M1 = 300, M - M1 = 300.

The final results of harmonic analysis using the error minimisation method of the paper are summarised in Table 2.

Table 2: Final results of harmonic analysis

Frequency Magnitude Phase actual, minimisation actual minimisation actual, minimisation Hz analysis, Hz analysis deg analysis, deg

0.0 0.0 0.1 0.1 0.0 0.0 25.0 25.0 0.3 0.3 70.0 70.0 35.35 35.35 0.2 0.2 60.0 60.1 35.85 35.85 0.7 0.7 80.0 80.0 50.0 50.0 1.0 1.0 0.0 1.1~10-4

86.6 86.6 0.5 0.5 90.0 90.0 150.0 150.0 0.4 0.4 40.0 40.0

of the method to the lengths of the training set and test set provides an indication of the robustness of the method in practical use.

8 Conclusions

Although very widely used, the DFT algorithm has cer- tain limitations in its application to harmonic analysis in power systems. Spectral leakage can preclude some valid frequency components from being identified. There are circumstances in which it is not easy to dis- tinguish between valid and spurious components. Mag- nitude errors arise. Phase errors can be substantial.

By comparison, the parametric method developed in this paper overcomes fully these limitations. It consist- ently achieves high accuracy. It is free from spectral leakage errors. It achieves high resolution in identifying frequency components which are close together. The method is much less sensitive to the choice of data win- dow length than is the DFT. Monitoring test errors allows checks to be made of the minimum data window length required.

For the minimisation method, any second-order method should be satisfactory. The quasi-Newton method adopted here has been excellent for the pur- poses.

The proposal made in the paper to partition the data set into a training set and a test set is a key part of the new method and is emphasised. It has general applica- tion in harmonic analysis in signal processing and rep- resents an advance in parametric analysis in that field. It is reported here as an important development leading to a very sound method of generalised harmonic analy- sis that can be relied on in its application in power sys- tems.

9 Acknowledgments

The work of this paper has been carried out as part of a collaborative research program in which Western Power Corporation joins the University of Western Australia. The author expresses his appreciation to Western Power Corporation and to the University for permission to publish the paper. He wishes to thank Professor W. Derek Humpage and Western Power Transmission Branch personnel for discussions relating to the developments of the paper. He particularly expresses thanks to Professor Humpage for his many suggestions and contributions to the preparation of the paper.

Overall, the analysis method guarantees very low errors in finding harmonic frequencies and the magni- tude and phase values of all harmonic components. The accurate identification of the two subharmonic components that are separated by only 0.5 Hz indicates the high resolving power of the method. Spectral leak- age errors are avoided completely. The frequency spec- trum from analysis is that of Fig. 3. The low sensitivity

10 References

frequency, Hz Fig. 3 Frequency spectrum fYom parametric harmonic analysis with data partitioning

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2 CHAN, Y.T., and PLANT, J.B.: 'A parameter estimation approach to estimation of frequencies of sinusoids', IEEE Trans. Acoust. Speech & Signal Process., 1981, 29, (2), pp. 214219

3 CHICHARO, J.F , and NG, T.S.: 'Gradient-based adaptive IIR notch filtering for frequency estimation', IEEE Trans. Acoust. Speech & Signal Process., 1990, 38, (9, pp. 169-711

4 CHICHARO, J.F.: 'High resolution spectral estimation using a spe- cially constrained adaptive notch filter', Znt. J. Electron., 1992, 72, (I), pp. 57-66

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