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Electrical Power and Energy Systems 28 (2006) 723–728

Power quality analysis based on fuzzy estimation algorithm:Voltage flicker measurements

A.M. Al-Kandari a, S.A. Soliman b,*, R.A. Alammari c

a College of Technological Studies, Electrical Engineering Technology Department, Shewiekh, Kuwaitb Ain Shams University, Electric power and Machines Department, Abbassia, Cairo, Egypt

c University of Qatar, Electrical Engineering Department, P.O. Box 2713, Doha, Qatar

Received 16 April 2004; received in revised form 16 December 2005; accepted 23 March 2006

Abstract

This paper presents a method based on fuzzy linear estimation for voltage flicker measurements. The proposed algorithm uses thedigitized samples of the voltage signal at the location where the power quality standards are implemented. The voltage signal is modeledas a fuzzy linear parameter estimation problem, where the coefficients are assumed to be fuzzy having certain middle and spread. A tri-angular membership is assumed. The linear programming based simplex method is used to solve the resulting linear optimization prob-lem. Results for simulated examples are given in the text.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Voltage flicker; Fuzzy linear regression; Power quality analysis

1. Introduction

Voltage flicker is a common term describing voltagefluctuations. The most affected element by flicker is thelight of a lamp. Voltage flicker can be divided into twotypes: cyclic and noncyclic. Cyclic voltage flicker resultsfrom periodic voltage fluctuations, such as that caused bythe operation of a reciprocating compressor or an electricfurnace. Noncyclic flicker corresponds to occasional volt-age fluctuations, such as starting of large motors or theoperation of a welder.

Electric utility companies may have limits for an individ-ual customer, such as a large fluctuating industrial load,due to the impact on the quality of power in the serviceto other customers. Thus, an accurate method is requiredto measure the level of the flicker in the signal.

Ref. [1] implements a digital processing technique calledthe ‘‘continuo Wavelet transform’’ for power quality anal-

0142-0615/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2006.03.010

* Corresponding author.E-mail address: solimans@qu.edu.qa (S.A. Soliman).

ysis. The proposed technique appears to be reliable fordetecting and measuring voltage sags, flicker and transientsin power quality analysis. However, a good estimate for themother wavelet is required to obtain accurate results.

Ref. [2] installs an adaptive VAR compensator (AVC) toreduce the flicker, and a detailed computer model of thegiven system has also been developed, including the AVCas well as the UIE/IEC flicker meter to determine the effec-tiveness of the AVC in an objective manner. The simulatedsystem response is compared with the actual system mea-surements, revealing good agreement. The modeling meth-odology provides an efficient and convenient means ofevaluating complex flicker resulting in a simple measureof human irritability.

Ref. [3] presents a virtual instrument to measure theflicker based on the implementation in the frequencydomain of the weighting flicker block of the eye-brain sim-ulation chain. The proposed instrument uses the voltagesamples and furnishes the instantaneous flicker level andother severity coefficients. The virtual flicker meter canimplement different weighting curves in the case of varia-tions in the filter transfer function definition.

724 A.M. Al-Kandari et al. / Electrical Power and Energy Systems 28 (2006) 723–728

Ref. [4] presents a digital algorithm to analyze voltageflicker. Special attention is given to the effect of the systemfrequency deviation on the voltage flicker measurement bydirect FFT spectrum analysis of the voltage waveform, thedc component leakage effect on the rms voltage and thewindowing effect on the data acquisition of the voltage sig-nal. It has been found that the aliasing, picket fence andleakage have a great effect on the measurement of voltageflicker.

Ref. [5] presents an EMTP-based arc furnace model forthe evaluation of flicker concerns associated with supplyingpower to a large integrated steel-mill. The model includes adynamic arc representation. The model developed includesthe nonlinear relationship between voltage and current forharmonic investigation, the control of arc voltage regula-tor, the electrode control, and the random variations inthe arc characteristics. It has also been found that theactual flicker levels are less than the values predicted formost conditions.

Linear and nonlinear Kalman filtering algorithms arepresented in Refs. [6,7] for voltage flicker measurements.The voltage flicker amplitude and frequency are estimatedin two steps. In the first step, a linear Kalman filter isimplemented to estimate the modulated amplitude andphase angle of the 60 Hz voltage signal, a two state linearKalman filter is used in this step. In the second step, anextended nonlinear Kalman filtering algorithm is used toextract the voltage flicker amplitude and frequency fromthe estimated 60 Hz modulated amplitude. The techniquedoes not show the effects of the nominal frequency drifton the estimated parameters.

A digital multi-rate-processing algorithm for activepower, voltage, current and flicker measurements is pre-sented in Ref. [8]. The multi-rate flicker meter algorithmis completed with a method that allows the evaluation ofindividual flicker sources of the power system, while othersources remain active, but they stay inactive in terms offlicker generation.

Ref. [9] presents a digital algorithm for the measurementof the voltage flicker magnitude and frequency. The algo-rithm proposed is based on least error squares (LES) esti-mate. The main advantage in this algorithm is that it isnot an iterative algorithm and can easily estimate the volt-age flicker magnitude as well as the flicker frequency atwhich it occurs.

An application of continuous wavelet transform (CWT)for the analysis of voltage flicker-generated signals is pro-posed in Ref. [10]. With the time-frequency localizationcharacteristics embedded in the wavelets, the time and fre-quency information of a waveform can be integrally pre-sented. Voltage flicker and harmonics are introduced tothe power system as a result of the arc furnace operation,and power utilities are concerned about their effects. Assuch an accurate model for the voltage flicker is needed.An arc furnace model that implemented in the Simulinkenvironment by using chaotic and deterministic elementsis presented in Ref. [11]. This model is obtained by solving

the corresponding differential equation, which yieldsdynamic and multi-valued v-i characteristics of the arc fur-nace load. In order to evaluate the flicker in the simulatedarc furnace voltage, the IEC flicker meter is implementedbased on the IEC 1000-4-15 standard in Matlab environ-ment. Ref. [12] presents an approach to estimate the volt-age flicker components magnitudes and frequencies,based on Lp norms (p = 1, 2 and 1) and Taylor series lin-earization. It has been found that it is possible to design anLp estimator to identify flicker frequency and amplitudefrom time series measurements. The Teager energy opera-tor (TEO) and the Hilbert transform (HT) are introducedin Ref. [13] as effective approaches for tracking the voltageflicker levels. It has been found that TEO and HT are capa-ble of tracking the amplitude variations of the voltageflicker and supply frequency in industrial systems with anaverage error of 3% but the technique cannot estimatethe voltage flicker amplitude and frequency for each com-ponent. Ref. [14] presents a control technique for flickermitigation. This technique is based on the instantaneoustracking of the measured voltage envelope. The ADALINE(ADAptive LINear) neuron algorithm and the recursiveleast square (RLS) algorithm are introduced for the flickerenvelope tracking. Presented in Ref. [15] is an algorithm fortracking the voltage envelope based on calculating theenergy operator of a sinusoidal waveform. It is assumedthat the frequency of the sinusoidal waveform is knownand a lead-lag network with unity gain is used.

Ref. [16] develops an enhanced method for the electricarc DV10 estimating the voltage fluctuation furnace(EAF). The method proposed considers the reactive powervariation and also the active power values of ac and dcEAFs. It DV10 variation in calculating is well known thatthe voltage flicker is a random slow variation in the powersystem voltage signal. Hence a method is needed to takethis randomness into account. An algorithm based onfuzzy linear parameter estimation is presented in this paperfor voltage flicker. The proposed algorithm uses the digi-tized samples of the voltage signal at the location wherethe power quality standards are required to be met. Theproblem of evaluation of the voltage signal is modeled asa linear fuzzy parameter estimation problem, where thecoefficients of the model are assumed to be fuzzy havingcertain middle and certain spread, where a triangular fuzzymembership function is assumed. The linear programmingbased on simplex method is used to solve the resulting lin-ear optimization problem. Results of the synthetic signalsare given in the text.

2. Fuzzy linear regression [17–20]

The fuzzy parameters linear estimation model, or fuzzyregression, can be described by the following relationbetween the output dependant Y, the independent variablexi and the fuzzy parameter Ai

Y ¼ f ðx;AÞ ¼ A1x1 þ A2x2 þ � � � þ Anxn ð1Þ

A.M. Al-Kandari et al. / Electrical Power and Energy Systems 28 (2006) 723–728 725

For an observation j, j = 1,2, . . .,m, Eq. (1) can be writtenas

Y j ¼ f ðx;AÞ ¼ A1x1j þ A2x2j þ � � �Anxnj; j ¼ 1; 2; . . . ;m

ð2ÞIn fuzzy regression, the difference between the observed

and the estimated values is assumed to be due to the ambi-guity inherently present in the system. Therefore, the abovefuzzy regression model is built in terms of the possibility,and evaluates all observed values as possibilities whichthe system should contain. The model in Eq. (1) is referredto as a possibilistic regression model. In this model Yj is theobservation measurement j. This output observation maybe a non-fuzzy or a fuzzy observation, Ai, wherei = 1,2, . . .,n are the fuzzy coefficients of the model expressin the form of (pi,ci), where pi is the middle and ci is thespread, or it may take the form of an LR-type aspi; c

Li ; c

Ri

� �and xij is the input to the model i = 1, . . .,n

and j = 1,2, . . .,m. In this section, three cases for the outputYj are studied.

2.1. Non-fuzzy output Yj = (yj, 0)

In this model, the output Yj is assumed to be a non-fuzzy observation, but the model coefficients Ai, i = 1,. . .,n are fuzzy either in the form Ai = (pi,ci), or Ai ¼ðpi; c

Li ; c

Ri Þ for the LR-type and the input xij is a non-fuzzy

input. For a triangular fuzzy number, the membershipfunction is expressed as

lA ¼1�jpj � aij

cipj � cj 6 ai 6 pj þ cj

0 otherwise

8<: ð3Þ

While the membership function of Ai of an LR-type fuzzynumber is assumed to be a trapezoidal described as

lAi¼

L pj �xcL

j

!for x 6 pj

R x�pj

cRj

!for x P pj

8>>>>><>>>>>:

ð4Þ

Here, pj is called the middle of Aj or the mean, cLj is the left

spread and cRj is the right spread. For the first type of fuzzy

coefficients, Eq. (1) can be written as

Y j ¼ p1; c1ð Þx1j þ p2; c2ð Þx2j þ pn; cnð Þxnj; j ¼ 1; 2; . . . ;m

ð5ÞThe second type of fuzzy coefficients is given as

Y j ¼ p1; cL1 ; c

R1

� �x1j þ p2; c

L2 ; c

R2

� �x2j þ pn; c

Ln ; c

Rn

� �xnj;

j ¼ 1; 2; . . . ;m ð6Þ

For the non-fuzzy output data regression problemdescribed by Eqs. (5) and (6) we seek to find the coefficientsAi = (pi,ci) or Ai ¼ pi; c

Li ; c

Ri

� �to minimize the spread of the

fuzzy output for all data sets. In mathematical form, thiscan be described as

Minimize J ¼Xm

j¼1

Xn

i¼1

cixij ð7Þ

Such that the fuzzy regression model could contain all theobserved data in the estimated fuzzy numbers which re-sulted from the model. This can be expressed as

yj PXn

i¼1

pixij � ð1� kÞXn

i¼1

cixij; j ¼ 1; 2; . . . ;m ð8Þ

yj 6

Xn

i¼1

pixij þ ð1� kÞXn

i¼1

cixij; j ¼ 1; 2; . . . ;m ð9Þ

Note that the first term in the right hand side of Eqs. (8)and (9) represents the estimated middle of the fuzzy coeffi-cients, while the second term represents the estimatedspread of these coefficients and k is the level of fuzzinessspecified by the user.

For the fuzzy coefficients of the LR-type, the cost func-tion to be minimized is

Minimize J ¼Xm

j¼1

Xn

i¼1

2mj � 2pixij þ cLi xij � cR

i xij

� �ð10Þ

Subject to satisfying the following two constraints on eachdata point

yj PXn

i¼1

pixij � ð1� kÞXn

i¼1

cLi xij; j ¼ 1; 2; . . . ;m ð11Þ

yj 6

Xn

i¼1

pixij þ ð1� kÞXn

i¼1

cRi xij; j ¼ 1; 2; . . . ;m ð12Þ

The problem formulated in Eqs. (7)–(9) and that formu-lated in Eqs. (10)–(12) are linear optimization problems,which can be solved by linear programming.

3. Flicker voltage simulation

The power system voltage during flicker can be consid-ered as an amplitude modulated waveform signal. Thiscan be expressed mathematically as [1]

vðtÞ ¼ V 0 þXM

i¼1

V fi cos xf it þ uf ið Þ" #

cosðx0t þ u0Þð13Þ

where V0 is the amplitude of the power voltage, x0 thepower frequency, and /0 its phase angle. Furthermore,Vfi is the amplitude of the flicker voltage, xfi its frequencyand /fi its phase angle. The term between the square brack-ets is the modulated amplitude of the power system volt-age. Eq. (13) can be expanded as

vðtÞ ¼ V 0 cos x0t þ u0ð Þ

þXM

i¼1

V f i cos xfit þ uf ið Þ cos x0t þ u0ð Þ ð14Þ

726 A.M. Al-Kandari et al. / Electrical Power and Energy Systems 28 (2006) 723–728

Without loss of generality, we assume that i = 1, and thefollowing steps are generally enough to apply for any num-ber of waveforms. Eq. (14) in this case becomes

vðtÞ ¼ V 0 cosðx0t þ u0Þ þ V f1 cosðxf1t þ uf1Þ cosðx0t þ u0Þð15Þ

Using the trigonometric identity, as well as the equivalenceof the product of two cosine terms, Eq. (15) becomes

vðtÞ ¼ V 0 cos u0 cos x0t � V 0 sin u0 sin x0t

þ 0:5V f1 cos xf1 þ x0ð Þt þ u0 þ uf1ð Þ½ �þ 0:5V f1 cos xf1 � x0ð Þt þ u0 � uf1ð Þ½ � ð16Þ

Assuming that the system nominal frequency x0 is known,50/60 Hz, and using the first order Taylor series expansionfor cosxfit and sinxfit around x0

f i, Eq. (16) can be writtenas

vðtÞ ¼ A1 cos x0t � A2 sin x0t þ A3 cos x0t cos x0f1t

� A4 cos x0t sin x0f1t � A5t cos x0t sin x0

f1t

� A6t cos x0t cos x0f1t � A7 sin x0t cos x0

f1t

þ A8 sin x0t sin x0f1t þ A9t sin x0t sin x0

f1t

þ A10t sin x0t cos x0f1t ð17Þ

where we define the model parameters as the fuzzy coeffi-cients A1 to A10 defined by

A1 ¼ V 0 cos /0 ð18ÞA2 ¼ V 0 sin /0 ð19ÞA3 ¼ V f1 cos /0 cos /f1 ð20ÞA4 ¼ V f1 cos /0 sin /f1 ð21ÞA5 ¼ V f1Dxf1 cos /0 cos /f1 ð22ÞA6 ¼ V f1Dxf1 cos /0 sin /f1 ð23ÞA7 ¼ V f1 sin /0 cos /f1 ð24ÞA8 ¼ V f1 sin /0 sin /f1 ð25ÞA9 ¼ V f1Dxf1 sin /0 cos /f1 ð26ÞA1 ¼ V f1Dxf1 sin /0 sin /f1 ð27Þ

the above model is used for one voltage flicker term and thefirst order Taylor’s series expansion is referred to as the 10state model, but for two term voltage flicker, the modelproduced will be 18 state model, and so on. Hence, Eq.(17) can be written as

vðtÞ ¼ A1X 1 þ A2X 2 þ A3X 3 þ A4X 4 þ A5X 5 þ A6X 6

þ A7X 7 þ A8X 8 þ A9X 9 þ A10X 10 ð28Þ

where the non-fuzzy time dependents X1 to X10 are definedas

X 1 ¼ cos x0t

X 2 ¼ � sin x0t

X 3 ¼ cos x0t cos x0f1t

X 4 ¼ � cos x0t sin x0f1t

X 5 ¼ �t cos x0t sin x0f1t

X 6 ¼ �t cos x0t cos x0f1t

X 7 ¼ � sin x0t cos x0f1t

X 8 ¼ sin sx0t sin x0f1t

X 9 ¼ t sin x0t sin x0f1t

X 10 ¼ t sin x0t cos x0f1t ð29Þ

If the fuzzy parameters are assumed to have a middle pi anda spread ci, each i = 1,2,10, then Eq. (28) becomes

vðtÞ ¼ ðp1; c1ÞX 1 þ ðp2; c2ÞX 2 þ ðp3; c3ÞX 3

þ ðp4; c4ÞX 4 þ ðp5; c5ÞX 5 þ ðp6; c6ÞX 6

þ ðp7; c7ÞX 7 þ ðp8; c8ÞX 8 þ ðp9; c9ÞX 9

þ ðp10; c10ÞX 10 ð30Þ

Now, given the m samples of v(t) which may be non-fuzzy,v(t)j = (yj, 0), or fuzzy samples having certain spread,v(t)j = (yj,ej). It is required to find the fuzzy parametersA1 to A10. This is a fuzzy linear regression problem similarto the type explained in the previous section. It is now re-quired to estimate the parameters Ai = (pi,ci), i = 1,2,. . ., 10 that minimize the spread of each coefficient all overthe voltage samples. This can be expressed mathematicallyas

J ¼Xm

j¼1

X10

i¼1

cjX ij

�� �� ð31Þ

Subject to satisfying the following two constraints for eachmeasurement sample to insure that the estimated middleand spread are included in the membership functions. Inthis analysis we assume a triangular membership function.Thus, for non-fuzzy samples, these constraints are

yj PX10

i¼1

piX ij � ð1� kÞX10

i¼1

ciX ij; j ¼ 1; 2; . . . ;m ð32Þ

yj 6

X10

i¼1

piX ij þ ð1� kÞX10

i¼1

ciX ij; j ¼ 1; 2; . . . ;m ð33Þ

While for the fuzzy samples vj = (yj,ej), these constraintsare

yj � ð1� kÞej PX10

i¼1

piX ij � ð1� kÞX10

i¼1

ciX ij;

j ¼ 1; 2; . . . ;m ð34Þ

yj þ ð1� kÞej 6

X10

i¼1

piX ij þ ð1� kÞX10

i¼1

ciX ij;

j ¼ 1; 2; . . . ;m ð35Þ

The problem formulated in Eqs. (31)–(35) is a linearoptimization problem that can be solved using the linearprogramming simplex technique available in the IMSL/STAT library. Having obtained (pi,ci), i = 1, . . ., 10, theparameters of the power system voltage as well as the volt-age flicker can be obtained as

V 20 ¼ A1 � A1 þ A2 � A2 ð36Þ

A.M. Al-Kandari et al. / Electrical Power and Energy Systems 28 (2006) 723–728 727

and the phase angle is

tan u0 ¼ A2 � A1 ð37ÞThe voltage flicker signal parameters are as follows:

• The frequency deviation of the signal can be calculatedfrom one of the following equations:

Dxf1 ¼ A5 � A3 or

Dxf1 ¼ A6 � A4 or

Dxf1 ¼ A9 � A7 or

Dxf1 ¼ A10 � A8 or

ð38Þ

• The voltage flicker amplitude can be calculated from thefollowing fuzzy equation:

V 2f1 ¼ A3 � A3 þ A4 � A4 þ A7 � A7 þ A8 � A8 ð39Þ

• The voltage flicker phase angle can be calculated usingone of the following equations:

tan uf1 ¼ A4 � A3 or

¼ A6 � A5 or

¼ A8 � A7 or

¼ A10 � A9

ð40Þ

The arithmetic operations in the above equations are fuzzyoperations.

4. Testing the algorithm on simulated data

The proposed algorithm and the voltage flicker modelexplained in the previous section are tested using a simu-lated example, where the voltage signal that contains aflicker is assumed to be

vðtÞ ¼ ½1þ 0:1 cosð2p2tÞ� cosð2p50tÞThis voltage signal is sampled at 1000 Hz and 20 samples,one cycle, are used to estimate the signal parameters, theparameters of the power system voltage and the parametersof the voltage flicker.

4.1. Non-fuzzy voltage signal (mj,0)

In this mode, we assume that the model parameters arefuzzy, while the samples are non-fuzzy. The linear pro-gramming based simplex method available in the IMSL/STAT library is used to solve the resulting fuzzy optimiza-tion problem. The following parameters are obtained

A1 ¼ ð1:0; 0:0Þ; A2 ¼ 4:5� 10�10; 0� �

;

A3 ¼ ð0:099973; 0Þ; A4 ¼ 6:096� 10�6; 0� �

and the rest of the parameters are zeros. It can be noted,from these results, that the estimated coefficients are crisp,since the spread of the parameters is zero. Using the equa-tions explained above to calculate the signal parameters,the following results are obtained:

• The power system voltage

V 0 ¼ ð1:0; 0:0Þ and /0 ¼ ð0:0; 0:0Þ:• The flicker voltage signal

V f1 ¼ ð0:099973; 0:0Þ; Dxf1 ¼ ð0:0; 0:0Þ;uf1 ¼ ð0:0; 0:0Þ:

These results show that the proposed fuzzy model algo-rithm successfully estimated the parameters of the voltagesignal.

4.2. Fuzzy voltage signal (mj, ej)

In this test, the voltage signal samples are contaminatedwith 10% noise on each sample, (ej = 0.1 mj). The linearproblem for fuzzy input data formulated in the previoussection is solved. The following fuzzy coefficients areobtained:

A1 ¼ ð0:99931; 0:11Þ; A3 ¼ ð0:09642; 0:0Þ;A4 ¼ ð2:304� 10�4; 1:02� 10�3ÞA8 ¼ ð�1:292� 10�3; 0:0Þ; A10 ¼ ð0:014; 0:0Þ

It can be noted from these results that the parameters A1

and A4 are the only fuzzy parameters among the modelparameters, while the rest of the parameters are zeros.Using these values the signal parameters can be computedas follows:

• The supply voltage signal V0 = (0.99931, 0.11) which isfuzzy amplitude. The phase angle is found as/0 = (0.0,0.0), which is not fuzzy.

• The flicker parameters

V f1 ¼ ð0:09642; 0:00102Þ; Dxf1 ¼ ð0:0; 0:0Þ;uf1 ¼ ð0:140; 0:60Þ

It can be noted that the proposed fuzzy algorithm success-fully estimated the voltage signal parameters. The esti-mated parameters are within the assumed triangularmembership function.

5. Conclusions

This paper presented an application of fuzzy regressionalgorithm to estimate the parameters of the voltage signalcontaminated with voltage flicker, for power quality analy-sis. These parameters were successfully estimated by theproposed algorithm when the input samples are non-fuzzysamples and when they are fuzzy samples. In this analysis,we assumed that the voltage flicker contains only one volt-age signal with a low frequency and a low amplitude com-pared to the main power system voltage. For more thanone flicker signal, the method is also applicable for the esti-mation of the voltage flicker amplitude and frequency, butwith more terms for each component.

728 A.M. Al-Kandari et al. / Electrical Power and Energy Systems 28 (2006) 723–728

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