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Energy Conversion and Management 48 (2007) 2129–2135

Frequency domain analysis of power system transientsusing Welch and Yule–Walker AR methods

Ahmet Alkan a,*, Ahmet S. Yilmaz b

a Department of Computer Engineering, Yasar University, 35500 Izmir, Turkeyb Department of Electrical and Electronics Engineering, Kahramanmaras Sutcu Imam University, 46050-9 Kahramanmaras, Turkey

Received 15 May 2006; received in revised form 11 December 2006; accepted 24 December 2006Available online 12 March 2007

Abstract

In this study, power quality (PQ) signals are analyzed by using Welch (non-parametric) and autoregressive (parametric) spectral esti-mation methods. The parameters of the autoregressive (AR) model were estimated by using the Yule–Walker method. PQ spectra werethen used to compare the applied spectral estimation methods in terms of their frequency resolution and the effects in determination ofspectral components. The variations in the shape of the obtained power spectra were examined in order to detect power system tran-sients. Performance of the proposed methods was evaluated by means of power spectral densities (PSDs). Graphical results comparingthe performance of the AR method with that of the Welch technique are given. The results demonstrate superior performance of the ARmethod over the Welch method.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Power quality; Power system transients; Spectral estimation; Welch; AR method

1. Introduction

Power quality (PQ) is defined as ‘‘the concept of power-ing and grounding sensitive equipment in a matter that issuitable to the operation of that equipment’’ in IEEEStd. 1100 [1]. Another definition is also given as the ‘‘setof parameters defining the properties of the power supplyas delivered to the user in normal operating conditions interms of continuity of supply and characteristics of voltage(symmetry, frequency, magnitude, and waveform) [2].However, PQ deals with not only voltage quality but alsocurrent quality. It includes the combination of voltageand current quality [3,4].

In practice, there are several types of PQ disturbances,such as voltage sag/swell/interruptions, switching tran-sients, flickers, harmonics, notches, etc., caused by faults,

0196-8904/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2006.12.017

* Corresponding author. Tel.: +90 232 461 41 11x316; fax: +90 232 46141 21.

E-mail addresses: ahmet.alkan@yasar.edu.tr (A. Alkan), asyilmaz@ksu.edu.tr (A.S. Yilmaz).

nonlinear loads and dynamic operating conditions [5].Especially, increasing the usage of nonlinear loads, suchas diode and thyristor rectifiers, lighting equipments, unin-terruptible power supplies (UPSs), arc furnaces and adjust-able speed motor drives, plays a major role in PQvalidations in industrial, commercial and residential powersystems. These loads disturb the current and voltage wave-form and change their magnitude and frequency. Also,they generate current harmonics and cause oscillatoryand impulsive transients when they are started andstopped. Mostly, pure sinusoidal currents and voltagescannot be provided to the customers.

The spectral estimation methods presented in this studyhave been mostly used for some power electronic applica-tions, such as flicker prediction, harmonic identification[6], switching converters [7] and resonant link inverters [8].

Spectrum estimation of discretely sampled processes isusually based on procedures employing the fast Fouriertransform (FFT). This approach is computationally effi-cient and produces reasonable results for a large class ofsignal processes. In spite of the advantages, there are

2130 A. Alkan, A.S. Yilmaz / Energy Conversion and Management 48 (2007) 2129–2135

several performance limitations of the FFT approach. Themost prominent limitation is that of frequency resolution,i.e. the ability to distinguish the spectral responses of twoor more signals. These procedures usually assume that onlysome harmonics are present and the periodicity intervalsare fixed, while the periodicity intervals in the presence ofinterharmonics are variable and very long, especially inthe case of power system transients. It is very importantto develop better tools of transient estimation to avoid pos-sible damages due to their influence. A second limitation isdue to windowing of the data. Windowing manifests itselfas ‘‘leakage’’ in the spectral domain. These two limitationsare particularly troublesome when analyzing short datarecords. Short data records occur frequently in practicebecause many measured processes are brief in duration orhave slowly time varying spectra that may be consideredconstant only for short record lengths. To alleviate the lim-itations of the FFT approach, many new spectral estima-tion methods, such as parametric methods, have beenproposed during the last decades [9–11].

Also, spectral estimation based methods such asESPRIT (estimation of signal parameters via rotationalinvariance techniques) [12] have been used for the analysisof PQ disturbances so that characterization and categoriza-tion of PQ signals were done in frequency domain analysis.

This paper proposes application of two methods to theanalysis of PQ signals. Impulsive and oscillatory transientdisturbances are analyzed by using Welch (non-parametric)and Yule–Walker AR (parametric) spectral estimationmethods. The main objective of this study is to illustratethe applicability of Welch and Yule–Walker-AR methodsto the analysis of PQ events. The Welch method, which isa classical method based on FFT, and the Yule–Walkermethod, which estimates the parameters of the ARmethod, are applied in the analysis of transient PQ distur-bances including different frequency components. Tran-sient disturbances are determined by evaluating the meanof the power spectral densities (PSDs).

Four cases including impulsive and oscillatory transientdisturbances are analyzed by the proposed methods. Gen-eration of the cases and all analyses are realized in Mat-labTM (The MathWorks Inc., Natick, MA) on an IBM PCwith 3 GHz Pentium IV processor, 1 GB of memory, andWindows-XPTM Professional operating system and SignalProcessing Toolbox [13,14].

2. Spectral estimation with Welch and Yule–Walker AR

methods

In this study, the PQ signals are analyzed by using theirPSDs. There are several methods to estimate the PSD.Non-parametric methods are those in which the estimateof the PSD is made directly from the signal itself. The sim-plest one is known as a ‘‘periodogram’’. The Welch methodis an improved version of the periodogram [14,15]. How-ever, parametric methods are those in which the signalwhose PSD we want to estimate is assumed to be the out-

put of a linear system driven by white noise. Estimation ofpower spectra is useful in a variety of applications, includ-ing the detection of signals buried in wide band noise [6,7].In order to model a signal, the properties of this signal areimportant and must be taken into account. For instance,an AR model is suitable for signals that have sudden peaksin their frequency spectrums [19]. Since power quality dis-turbance signals contain peaks at some frequencies, an ARmodel can be used by employing the Yule–Walkeralgorithm.

2.1. Welch method

The periodogram method used for determining thepower density of the frequency components in a signal isbased on the Fourier transform. To obtain the PSD of aPQ signal by the periodogram method, it was divided intoframes as 64, 128 and 256, which are powers of 2. To eval-uate the power spectrum, this method divides the data intoseveral overlapping segments, computes a power spectrumby using a FFT on each segment and then averages thesespectra [16].

The main disadvantage of nonparametric spectral esti-mation techniques, like the periodogram, is the impact ofoffside lobe leakage due to finite data sets. To overcomethis problem, Welch proposed an improved estimationmethod [15]. The method consists of four steps.

First, the time series were divided into segments (possi-ble overlapping). Then, the data is windowed in each seg-ment in order to smooth the edges of the signals (e.g.Hanning window). Next, a periodogram estimate is com-puted in each windowed segment. Finally, an average iscalculated over the estimates.

In addition to decrease the impact of side lobe leakage,the Welch method improves the statistical stability of thespectral estimate. The more segments are used, the morestable is the estimate. However, the signal length limitsthe number of segments used, but with overlapping seg-ments, the number of segments can be increased.

Averaging of the modified periodograms tends todecrease the variance of the estimate relative to a singleperiodogram estimate of the entire data record. Althoughoverlap between segments tends to introduce redundantinformation, this effect is diminished by use of a non-rect-angular window, which reduces the importance or weightgiven to the end samples of the segments (the samples thatoverlap).

The Welch method estimates the power density spec-trum by averaging modified periodograms. The ith modi-fied periodogram is:

bS ðiÞxx ðf Þ ¼T S

C:M

XM�1

n¼0

xiðnÞ � wðnÞ � e�j2pfn

����������2

ð1Þ

where f = fs is the normalized frequency variable havingunits of cycles per sample. The scaling factor Ts adjuststhe magnitude of the discrete time signal spectrum to be

A. Alkan, A.S. Yilmaz / Energy Conversion and Management 48 (2007) 2129–2135 2131

equal to the analog signal spectrum. M is the length of thesignal x(n). The windowing function is represented by thesamples w(n), and C is a normalizing constant defined as

C ¼ 1

M

XM�1

n¼0

w2ðnÞ ð2Þ

Finally, the estimate of the power density spectrum is:

bSwxxðf Þ ¼

1

L

XL�1

i¼0

bS ðiÞxx ðf Þ ð3Þ

However, as mentioned above, the combined use of shortdata records and non-rectangular windows results in re-duced resolution of the estimator. In summary, there isa tradeoff between variance reduction and resolution.One can manipulate the parameters in the Welch methodto obtain improved estimates relative to the periodo-gram, especially when the signal to noise ratio is low[17,18].

34.5 kV 10 MVA

Distribution

Network

CapacitorBank

DistributionLine

Load

450 V

Load

34.5/0.45 kV 2 MVA,Yg/Yg

MV PCC

Fig. 1. Single line diagram of sample distribution systems.

2.2. Yule–Walker AR method

The Yule–Walker AR method is commonly usedamong model based (parametric) methods that estimatethe PSD. This estimation method computes the ARparameters by forming a biased estimate of the signal’sautocorrelation function and solving the least square min-imization of the forward prediction error [14]. In the ARmodelling method, the amplitude of a signal at a givenperiod is obtained by summing the different amplitudesof the previous samples and adding the estimation error.The order of the model, namely, the filter, depends onthe number of AR coefficients. In the AR method, themodelling degree is identified according to different crite-ria. In this study, the modelling degree p = 10 was takenby using the Akaike Information Criteria (AIC) [19,20].The parametric methods are generally based on modelingthe data sequence x(n) as the output of a linear systemcharacterized by an all pole rational system. In thesemethods, estimation of the spectrum procedure has twosteps. In the first step, the parameters of the method areestimated from the given data sequence x(n),0 6 n 6 N � 1 [21,22]. It is assumed that the data {x(0),x(1), . . .,x(N � 1)} are observed. In the Yule–Walker ARmethod, or the autocorrelation method, as it is sometimescalled, the AR parameters are estimated by minimizing anestimate of the prediction error power,

variance ¼ q ¼ 1

N

X1n¼�1

xðnÞ þXp

k¼1

aðkÞ � xðn� kÞ�����

�����2

ð4Þ

The samples of the x(n) process that are not observed(i.e. those not in the range 0 6 n 6 N � 1) are set equalto zero in Eq. (4). The estimated prediction error poweris minimized by differentiating Eq. (4) with respect to thereal and imaginary parts of the a(k)’s. This may be doneby using the complex gradient to yield,

1

N

X1n¼�1

xðnÞ þXp

k¼1

aðkÞ � xðn� kÞ" #

� x�ðn� jÞ;

j ¼ 1; 2; . . . ; p ð5Þ

This set of equations in terms of autocorrelation functionestimates becomes

rp þ bRp � a ¼ 0 ð6Þwhere

rðkÞ ¼1N

PN�1�k

n¼0

x�ðnÞ � xðnþ kÞ k ¼ 0;1; . . . ;p

r�ð�kÞ k ¼ ð�pþ 1Þ; ð�pþ 2Þ; . . . ;�1

8<:ð7Þ

From Eq. (6), the AR parameter estimates are found as

a ¼ �bR�1p � rp ð8Þ

The estimate of the white noise variance r2 is calculated as

r2 ¼ rð0Þ þXp

k¼1

aðkÞ � rð�kÞ ð9Þ

From the estimates of the autoregressive parameters, thepower spectral density estimation is given as [23–25].

bP ðf Þ ¼ r2

1þPpk¼1

aðkÞ � e�j2pfk

���� ����2: ð10Þ

3. Spectral analysis of power system transients

In electric power systems, transients are known as tem-porary phenomena lasting less than one cycle. Mostly, theknown causes of power system transients are short circuitsand dynamic operation of power systems such as switchingof loads and capacitors. In this study, high frequency tran-sients caused by capacitor switching are considered.

Power systems transients are classified into impulsive andoscillatory transients. An impulsive transient is a sudden

2132 A. Alkan, A.S. Yilmaz / Energy Conversion and Management 48 (2007) 2129–2135

change in the steady state condition of voltage and/or cur-rent. It is quickly damped. Lightning is a most commoncause of impulsive transients. Oscillatory transients consistof damped oscillations that have large frequency intervalsfrom a few hundred Hz up to several MHz. Energizing acapacitor bank typically causes an oscillatory transient [12].

In the simulations, four sample waveforms for bothimpulsive and oscillatory transients due to switching ofcapacitors in medium voltage distribution networks areconsidered. The single line scheme of the considered distri-bution network is given in Fig. 1. This system is simulatedby using Matlab. Detailed parameters and values of theswitched capacitors are also given in Appendix A.

In all cases, the original voltage waveform at the MVbus is obtained during the capacitor switching. This signalbelongs to the phase to ground voltage. In the first case, asignal including impulsive transients is considered as seenin Fig. 2a. The spectral diagrams obtained by the Welchand Yule–Walker AR methods are given in the same fig-ure, the middle and bottom panels, respectively. The fun-damental frequency component at 50 Hz is obtained fromthe spectrums estimated by using both methods. Thiscomponent is clearer in the spectrum obtained by theYule–Walker AR method than in the spectrum obtainedby the Welch method. Also, another component isobserved with lower magnitude at about 1700 Hz. How-ever, the second one cannot be definite compared withthe fundamental frequency components in both methods.

0 0.005 0.01 0.015 0.02 0.0-4

-2

0

2

4x 10

4

Am

plitu

de (

V)

Time

0 200 400 600 800 10000

20

40

60

80

PS

D (

dB/H

z)

0 200 400 600 800 10000

20

40

60

80

PS

D (

dB/H

z)

Frequ

Fig. 2. (a) Considered original voltage signal waveform at PCC

This is caused by the disturbances in the original wave-form that have smaller magnitude than the fundamentalfrequency.

The original signal illustrated in Fig. 3a is obtained fromthe Matlab simulation during switching of the capacitorbank. The oscillations lasted one cycle. The power spectraldensities obtained by using the Welch and Yule–WalkerAR methods are given in Figs. 3b and c, respectively.The highest power density is at 50 Hz, observed clearlyby using both estimation methods. In addition to this,oscillations at 675 Hz can be seen in the spectrums. Whenthe methods were compared, it is clearly observed thatthe spectrum obtained by the Yule–Walker AR method issharper than the spectrum obtained by the Welch method.

In Fig. 4a, an oscillatory transient that lasts two cycles isconsidered during the capacitor switching in the test sys-tem. Power spectrums obtained by using the Welch andYule–Walker methods are given in the same figure. In thiscase, the disturbances are sharper and clearer in the Yule–Walker spectrum than those in the Welch spectrum. Thepeaks at 1050 Hz and 1770 Hz are clear in both spectra,but the Welch spectrum includes lower resolution.

In the final case, the considered current waveform of theswitched capacitor and PSD diagrams are shown in Fig. 5.As seen in this figure, the frequency of the observed oscil-lation is at 250 Hz. In the Welch spectrum, several spuriouspeaks can be observed at high frequencies. However, thereare no spurious peaks in the Yule–Walker AR spectrum.

25 0.03 0.035 0.04 0.045 0.05

(sec)

1200 1400 1600 1800 2000 2200

Welch

1200 1400 1600 1800 2000 2200

ency (Hz)

Yule-Walker AR

bus, (b) Welch spectrum, (c) Yule–Walker AR spectrum.

0 0.01 0.02 0.03 0.04 0.05 0.06-2

-1

0

1

2x 10

4

Am

plitu

de (

V)

Time (sec)

0 100 200 300 400 500 600 700 800 900 100020

40

60

80

PS

D (

dB/H

z)

Frequency (Hz)

Yule-Walker AR

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

PS

D (

dB/H

z)

Welch

Fig. 3. (a) Voltage waveform at PCC bus, (b) Welch spectrum, (c) Yule–Walker AR spectrum.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-4

-2

0

2

4x 10

4

Am

plitu

de (V

)

Time (sec)

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

20

40

60

80

PS

D (

dB/H

z)

Welch

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

20

40

60

80

PS

D (

dB/H

z)

Frequency (Hz)

Yule-Walker AR

Fig. 4. (a) Voltage waveform at PCC bus, (b) Welch spectrum, (c) Yule–Walker AR spectrum.

A. Alkan, A.S. Yilmaz / Energy Conversion and Management 48 (2007) 2129–2135 2133

Table A.1

MVArcapacitive

Switch on(ms)

Switch off(ms)

Bus barname

1.18 5.2 5.22 PCC0.36 10 – PCC0.18 5.2 – PCC

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-400

-200

0

200

400

Am

plitu

de (

A)

Time (sec)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

-40

-20

0

20

40

PS

D (

dB/H

z)

Frequency (Hz)

Yule-Walker AR

0 200 400 600 800 1000 1200 1400 1600 1800 2000-40

-20

0

20

40

PS

D (

dB/H

z)

Welch

Fig. 5. (a) Current waveform of capacitor, (b) Welch spectrum, (c) Yule–Walker AR spectrum.

2134 A. Alkan, A.S. Yilmaz / Energy Conversion and Management 48 (2007) 2129–2135

As a result of the simulations, the frequency componentsin the disturbances were obtained by using power spectraof the PQ signals, but certain and clear results can beobtained from the Yule–Walker spectrum. It can be con-cluded that the Yule–Walker method gives more definiteand true results than the Welch method to detect the powerquality disturbances.

4. Conclusions

For power quality assessment, it is important to deter-mine what the disturbances are in the current and voltagewaveforms. In this study, power system transients wereanalyzed by using the power spectral density that is oneof the signal processing methods aimed to determine thefrequency components of sample signals.

Welch and Yule–Walker AR methods were used todetermine the power spectrum. Also, these methods werecompared with each other according to frequency domainanalysis of the considered four sample cases.

As a result of the analyses, it appears that the Yule–Walker AR method gives more certain result (better fre-quency resolution) than the Welch method and can beeffectively used for analysis of power system transient sig-nals. The PSD has considerable effects to portray betterthe observing and assessment of power quality in powersystems. The proposed methods are investigated under dif-ferent conditions and found to be efficient tools for detec-tion of all power system transients in a signal.

Appendix A

A.1. Distribution network parameters

UN = 34.5 kV rated voltageFN = 50 Hz rated frequencySSC = 10 MVA short circuit power for utility networkRH = 1.1 ohm line resistance (single phase)LH = 27.2 ohm line reactance (single phase)STR = 2 MVA transformer rated power

(Yg/Yg connected)UTR = 34.5/0.45 kV transformer voltagesSLOAD1 = 2 MVA load 1SLOAD2 = 1030 kVA load 2

A.2. Switching conditions for all cases

Switching times and capacitor power are given in TableA.1.

A. Alkan, A.S. Yilmaz / Energy Conversion and Management 48 (2007) 2129–2135 2135

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