Adaptive control of power factor correctionconverter using nonlinear system identification

K.M. Tsang and W.L. Chan

Abstract: PID controllers have been very successful in industrial applications. However, fornonlinear processes, their performances are not very satisfactory. Conventional active power factorcorrection (PFC) converters use classical PI controllers to carry out output voltage regulation andpower factor correction. Since the process is highly nonlinear, an adaptive controller has beendeveloped to improve the performance of PFC converters. An orthogonal least squares estimationalgorithm was incorporated for the identification of boost converters. A parsimonious nonlinearmodel for power converters was obtained based on the output prediction and the error reductionratio test. The least mean squares algorithm is then implemented for the online tracking of thesystem parameters and the adaptive controller was derived based on the fitted model. Experimentalstudies were carried out to demonstrate the effectiveness of the proposed identification and controlalgorithm for power factor correction and harmonic elimination.

1 Introduction

Electric power usage today is moving from simple linearloads to electronic ones, such as solid-state motor drives,personal computers and energy-efficient ballasts. Theelectric current drawn by these new devices is nonsinusoidaland causes problems in the power system. Typical switched-mode power supplies employ diode rectifiers for the AC toDC conversion. They draw input current in short pulsesrather than in smooth sine waves [1]. This puts stress on thewiring, circuit breakers and even the distribution equipmentprovided by utilities. This type of utility interface generatesharmonics and both of the input power factor (PF) andtotal harmonic distortion (THD) are poor [2]. To minimisethe stresses and maximise the power handling capabilities,power factor correction (PFC) circuitry [3, 4] can be addedto improve the shape of the input current. Moreover, thereare international regulations IEC 61000-3-2 [5] that limit theinput harmonic content and products in the EuropeanUnion. It establishes limits on harmonics of the inputcurrent. To comply with today’s standards, such as IEC61000-3-2 and IEEE 519 [6], the design of the switchingpower supply requires features such as lower input currentharmonics to meet the harmonic limits and a high inputpower factor to minimise reactive requirements. Conven-tional PFC converters use classical PI controllers to carryout the output voltage regulation and power factorcorrection. As the voltage input to the converter isfluctuating, linear PI controllers [7] may not provide theoptimal solution for the output voltage regulation andpower factor correction. Moreover, a conventional con-troller will produce rather poor performance if the system is

operated away from the linearised operating point. Hence, anonlinear model which can capture the nonlinear phenom-ena and the system characteristics over a wide range ofoperating points is more desirable. In the present approach,a multiple input single output (MISO) polynomial type ofnonlinear difference equation is fitted to a collection ofinput and output records using the orthogonal least squaresestimation algorithm [8–10]. To be able to achieve onlineadaptive control, the number of system parameters has tobe minimised such that the computing resource requirementwill be less and the sampling and switching frequencies canbe maintained as high as possible. Therefore, the errorreduction test [8–10] has been incorporated to theorthogonal least squares estimation algorithm to detectthe structure of the final fitted model, such that aparsimonious model results. The goodness of fit is furtherverified by the output prediction of the fitted model. Anonlinear predictive controller is then derived based on thefitted model and a least mean squares algorithm isimplemented for the tracking of the model parameters.Experimental studies are included to demonstrate theperformance of the proposed adaptive controller.

2 Power factor correction (PFC) converters

A typical implementation of a PFC-boosted converter [11]is shown in Fig. 1. The output voltage is regulated viafeedback to an operational amplifier. The sensed inputvoltage Vi will be in the form of a rectified sine wave, whichaccurately reflects the instantaneous value of the input AC

RL

L

C

Rs

AC

VoILVi

gatedrive

Fig. 1 PFC-boosted converter

The authors are with the Department of Electrical Engineering, The HongKong Polytechnic University, Hung Hom, Kowloon, Hong Kong

E-mail: steve.tsang@polyu.edu.hk

r IEE, 2005

IEE Proceedings online no. 20045058

doi:10.1049/ip-epa:20045058

Paper first received 16th June and in revised form 19th October 2004. Originallypublished online: 8th April 2005

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 627

voltage. This signal is used as an input to a multiplier, alongwith the output error voltage, to formulate a voltage that isproportional to the desired current. This signal is thencompared with the sensed actual converter current to formthe error signal that drives the converter switch. The result isthe input current waveform which tracks the AC inputvoltage waveform. The active boost circuit will correct fordeficiencies in both the power factor and harmonicdistortion. The converter is controlled by two feedbackloops based on linear PI controllers as shown in Fig. 2. Theoutput DC voltage is regulated by an ‘outer loop’, whereasthe ‘inner loop’ shapes the inductor current. When theswitching frequency is very high, it has been assumed thatthe input voltage will have very little effect on the transferratio between the inductor current and the switching dutyratio, and linear PI controllers are normally implementedfor the control of the current loop. In actual fact, thefluctuated rectified input voltage does have an effect on thetransfer ratio between the inductor current and the switch-ing duty ratio, especially when the switching frequency isnot sufficiently high. The relationship between the inputvoltage Vi and the output DC voltage Vo is given by

Vo

Vi¼ 1

1� dð1Þ

where d is the duty ratio. Hence, for a fixed output voltageVo, the input voltage will actually affect the duty ratiorequired. If the controller is tuned at a low input voltagelevel, it may produce an oscillatory response when the inputvoltage rises to a higher value. From (1), the duty ratiorequired at low input voltage level will be much higher thanthe duty ratio required at high input voltage level for thesame output voltage. Hence, the controller gain required atlow input voltage level will be larger than the controller gainrequired at high input voltage level. However, if thecontroller is tuned at a high input voltage level, it mayproduce a sluggish response when the input voltage drops toa lower value. In order to achieve a better performance, anonlinear controller is required. The availability of anonlinear model for the current loop will help the designof a better controller for the current control loop.

3 Nonlinear modelling of current loop

Recent results in approximation and realisation theoryproduce a nonlinear difference equation model that issuitable as a basis for the modelling of power converters,and Leontaritis and Billings [12] shows that any discrete

time nonlinear system can be represented by the NARMAX(nonlinear ARMAX) model

yðkÞ ¼F l½yðk � 1Þ; . . . ; yðk � nyÞ; u1ðk � 1Þ; . . . ;

u1ðk � nu1Þ; . . . ; umðk � 1Þ; . . . ; umðk � numÞ�in a region around the equilibrium point where Fl[.] is apolynomial type nonlinear function, y(k) is the output of themodel, u1(k),y, um(k) are the m inputs, ny, nu1 ,y, num arethe orders of the output and m inputs, respectively. For thePFC boost converter, the inductor current IL can becharacterised by the rectified supplied voltage Vi, the dutyratio d and the output DC voltage Vo as

ILðkÞ ¼F l½ILðk � 1Þ; . . . ; Viðk � 1Þ; . . . ;

dðk � 1Þ; . . . ; Voðk � 1Þ; . . .�ð2Þ

For example, a first-order model with a second-ordernonlinearity will give

ILðkÞ ¼y1ILðk � 1Þ þ y2Viðk � 1Þ þ y3dðk � 1Þþ y4Voðk � 1Þ þ y5I2Lðk � 1Þþ y6ILðk � 1ÞViðk � 1Þ þ y7ILðk � 1Þdðk � 1Þþ y8ILðk � 1ÞVoðk � 1Þþ y9V 2

i ðk � 1Þ þ y10Viðk � 1Þdðk � 1Þþ y11Viðk � 1ÞVoðk � 1Þþ y12d2ðk � 1Þ þ y13dðk � 1ÞVoðk � 1Þþ y14V 2

o ðk � 1Þwhere yi is the model parameter.

4 Nonlinear system identification

A normal expansion of (2) may involve hundreds orthousands of linear and nonlinear terms if the order ofdynamics and order of nonlinearity are high and a lot ofthem may be redundant. To be able to utilise theNARMAX model for online adaptive control, thesize has to be reduced. Hence, an estimation algorithmwhich involves procedures for the selection of significantcandidate terms and the estimation of the correspondingcoefficients is therefore required. The orthogonal leastsquares estimation algorithm [8–10] has been found to be anefficient tools for the estimation of nonlinear systems. Theerror reduction ratio [8–10], which is a byproduct of theestimation algorithm, provides information regarding thesignificance of individual linear and nonlinear terms.

supply voltage

referencecurrent

inductorcurrent

lowpass filter

duty ratio

boostconverter

absolute value

multiplier

voltage-loopcontroller

current-loopcontroller

reference DCvoltage

output DCvoltage

Vo

Vr

Vi

d

IL

Ir

Go(z)

Gi (z)Gv (z) 1.1+−

+−

Fig. 2 Block diagram of controlled PFC-boosted converter

628 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005

Representing the NARMAX model equation (2) with theregression equation

ILðkÞ ¼XMi¼1

piðkÞyi þ eðkÞ ð3Þ

where M is the number of the unknown parameters, e(k) isthe estimation error, pi(k) represents a term in theNARMAX model and no two pi(k)s are identical, and yi

is the unknown parameter. The objective of the orthogonalleast squares estimation algorithm is to minimise the costfunction

J ¼ 1

N

XN

k¼1e2ðkÞ ð4Þ

where e(k) is the estimation error of the fitted model and Nis the number of data records, by transforming (3) into anequivalent orthogonal equation

ILðkÞ ¼XMi¼1

giwiðkÞ þ eðkÞ ð5Þ

using the procedures

w1ðkÞ ¼p1ðkÞ

wiðkÞ ¼piðkÞ �Xi�1j¼1

ajiwkðkÞ; i ¼ 1; . . . ; M

aji ¼1N

PNk¼1

wjðkÞpiðkÞ

1N

PNk¼1

w2j ðkÞ

;j ¼ 1; . . . ; i� 1

i ¼ 1; . . . ; M

�

gi ¼1N

PNk¼1

ILðkÞwiðkÞ

1N

PNk¼1

w2i ðkÞ

; i ¼ 1; . . . ; M

ð6Þ

and N is the number of data records for the estimation. Theoriginal system parameters yi can then be recovered as

yM ¼gM

yj ¼gj �XM

i¼jþ1ajiyi; j ¼ M � 1; . . . ; 1

ð7Þ

The relative contribution of each individual orthogonal dataset wi(k) can be determined using the error reduction ratiodefined as

eRRi ¼1N

PNk¼1

g2i w2

i ðkÞ

1N

PNk¼1

I2LðkÞ; i ¼ 1; . . . ; M ð8Þ

A large value of eRRi indicates the significance orimportance of a particular term to the output, whereas asmall value indicates the term is insignificant. Hence,incorporating the error reduction ratio test with the ortho-gonal estimation algorithm to sort through all the linear andnonlinear candidate terms will result in a parsimoniousmodel where all redundant terms are eliminated.

4.1 Experimental resultsIn order to demonstrate the effectiveness of the orthogonalleast squares algorithm and the error reduction test in theidentification of the boost converter, a simple experimentalconverter was implemented with a VIA Eden 400 32-bitmicrocontroller. A boost converter with L¼ 1.85mH,

C¼ 300mF and RL¼ 165O was built. The PWM carrieror switching frequency was 20kHz, which was derived froma crystal oscillator of frequency 5.0688MHz [13]. The rangeof available duty ratio command was 1–253 (or 0.00395–1.0). It is possible to increase the resolution of the duty ratioby simply increasing the oscillator frequency or reducing theswitching frequency. However, experimental results demon-strated that the existing eight-bit resolution was sufficient.The data acquisition subsystem of the controller was ratherstandard [14]. Three channels of a 12-bit, 10-V bipolarinput, analogue-to-digital converter were used. The threeantialiasing filters were second-order Butterworth lowpassfilters using Sallen and Key implementation. The -3 dB cut-off frequencies were all at 3410Hz. The required samplingfrequency was 10kHz per channel and total throughput forthe analogue-to-digital converter should be over 30kHz.The throughput of the analogue-to digital converter usedwas 100kHz. The controller was implemented in Clanguage and the most important routine was the periodictime interrupt routine, which activated every 0.00001 s. Themicrocontroller sampled the three channels in sequence, thefirst one was the AC supply voltage Vi, the second was theinductor current IL and the last was the output voltage Vo.Initially, PI controllers were implemented for the control ofthe boost converter. The voltage loop PI controller wasgiven by

GvðzÞ ¼0:00036ðz� 0:9938Þ

z� 1ð9Þ

and the smoothing output filter was given by

GoðzÞ ¼0:005

z� 0:995ð10Þ

The current loop PI controller was given by

GIðzÞ ¼0:075ðz� 0:2667Þ

ðz� 1Þ ð11Þ

The supply voltage was 114.6Vrms and the referenceoutput DC voltage was set to 200V. 400 sets of inductorcurrent IL, duty ratio d, supply voltage Vi and outputvoltage Vo were collected for the identification of the currentdynamics, and randomness was introduced into the

curr

ent,

A

time,sa

time,sc

0 0.02 0.04

0 0.02 0.04

0

2

4

0

100

−100

200

−200

time,sb

duty

rat

io

0 0.02 0.040

0.5

1.0

volta

ge,v

time,sd

0 0.02 0.04150

200

250

volta

ge,v

Fig. 3 Collected experimental data setsa Inductor currentb Duty ratioc Supply voltaged Output voltage

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 629

closed-loop control system by inserting a random duty ratioto the output of the current loop PI controller to make surethat the collected data were information rich. Figure 3shows the 400 collected data set. Initially a second-orderlinear model was fitted to the data set and the fitted modelwas given by

ILðkÞ ¼1:2340ILðk � 1Þ � 0:3116ILðk � 2Þþ 0:8336dðk � 1Þ þ 0:4922dðk � 2Þþ 0:0049Viðk � 1Þ � 0:0015Viðk � 2Þ� 0:0039Voðk � 1Þ � 0:0016Voðk � 2Þ þ 0:3818

ð12Þand Fig. 4 shows the model predicted output currentsuperimposed on the converter inductor current. Clearly,the linear model failed to capture the nonlinear character-istics of the system because the predicted output current fellbelow zero by a large margin in some regions.

To capture the nonlinear characteristics, a nonlinearmodel with second-order dynamics with third-order non-linearity was specified for the identification of the converter.The initial specification contained 165 candidate terms.When the orthogonal least squares estimation algorithmcoupled with the error reduction ratio test sorted throughall possible candidate terms, 160 terms were eliminated after

five stages of orthogonalisation and the resultant model hadonly five terms given by

ILðkÞ ¼0:1870ILðk � 1Þ þ 0:0245d2ðk � 2ÞViðk � 1Þ½0:9925� ½0:0021�þ 0:0041ILðk � 1Þdðk � 1ÞViðk � 2Þ

½0:0022��0:0155I3Lðk � 1Þ þ 0:0113ILðk � 1Þdðk � 2ÞViðk � 1Þ½0:0005� ½0:0005�

ð13ÞThe square-bracketed value beneath the individual para-meters indicates the error reduction ratio of each corre-sponding candidate term. The sum of the error reductionratio captured by the five terms was 0.9977 and Fig. 5shows the model-predicted output current superimposed onthe converter inductor current. Even with fewer terms thanthe linear model, the fitted nonlinear model captured thedynamics of the converter very well. This result clearlydemonstrated the effectiveness of the orthogonal leastsquares algorithm and the error reduction ratio test in theidentification of the nonlinear system.

5 Adaptive control of PFC boost converter

Since there will still be deficiency in the fitted model,adaptive control law is implemented for the control of theboost converter. If online identification of (2) is given by

IILðkÞ ¼FF l½ILðk � 1Þ; . . . ; Viðk � 1Þ; . . . ;

dðk � 1Þ; . . . ; Voðk � 1Þ; . . .�ð14Þ

where FF l½�� is the estimated system model, the one-stepahead prediction is given by

IILðk þ 1Þ ¼FF l½ILðkÞ; . . . ; ViðkÞ;. . . ; dðkÞ; . . . ; VoðkÞ; . . .�

ð15Þ

Setting the one-step ahead prediction IILðk þ 1Þ ¼ Irðk þ 1Þto follow the reference current, the control law becomes

dðkÞ ¼G½Irðk þ 1Þ; ILðkÞ; :::;ViðkÞ; :::; dðk � 1Þ; :::; VoðkÞ�

ð16Þ

where G [ � ] is a nonlinear function derived from (15).In order to apply adaptive control of boosted converters,

there are a number of things that have to be considered suchthat the converters can be operated at high switchingfrequency. First, the fitted model has to be as compact aspossible such that less time will be spent on the identifica-tion of the model parameters. This can be solved by theorthogonal least squares estimation algorithm coupled withthe error reduction ratio test. Secondly, the onlineadaptation algorithm has to be simple and efficient. Theleast mean squares (LMS) algorithm given by

Yðk þ 1Þ ¼ YðkÞ þ 2meðkÞP ðkÞ ð17Þ

where YðkÞ ¼ ½ y1ðkÞ y2ðkÞ � � � �T , eðkÞ ¼ ILðkÞ � IILðkÞ,P ðkÞ ¼ ½p1ðkÞ p2ðkÞ � � � �T and m is the step size, isselected for the online tracking of the model parameters.

5.1 Experimental resultsFrom (13), the update of model parameters used was givenby

Yðk þ 1Þ ¼ YðkÞ þ 0:00001eðkÞP ðkÞ

where YðkÞ ¼ ½ y1ðkÞ y2ðkÞ y3ðkÞ y4ðkÞ y5ðkÞ �T ,Yð0Þ ¼ ½ 0:1870 0:0245 0:0041 �0:0155 0:0113 �

curr

ent,

A

converter outputmodel output

0

0

2

4

−20.01 0.02 0.03 0.04

time, s

Fig. 4 Linear model predicted output superimposed on true systemoutput

curr

ent,

A

00

1

2

3

4

0.01 0.02 0.03 0.04

converter output model output

time, s

Fig. 5 Nonlinear model predicted output superimposed on truesystem output

630 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005

and

PðkÞ ¼

p1ðkÞp2ðkÞp3ðkÞp4ðkÞp5ðkÞ

266664

377775 ¼

ILðk � 1Þd2ðk � 2ÞViðk � 1Þ

ILðk � 1Þdðk � 1ÞViðk � 2ÞI3Lðk � 1Þ

ILðk � 1Þdðk � 2ÞViðk � 1Þ

266664

377775

and the control law was given by

dðkÞ ¼Irðk þ 1Þ � y1ðkÞp1ðkÞ � y2ðkÞp2ðkÞ � y4ðkÞp4ðkÞ � y5ðkÞp5ðkÞ

y3ðkÞILðkÞViðk � 1Þð18Þ

where Ir(k) is the reference current. To avoid division byzero, a safety precaution had been added to the denomi-nator such that

y3ðkÞILðkÞViðk � 1Þ ¼ x if y3ðkÞILðkÞViðk � 1Þox

and x¼ 10�8. The same system as in Section 4.1 was used todemonstrate the effectiveness of the proposed controlscheme. The supply voltage was a 114.6Vrms and THDof the supply voltage was 2.1% obtained from a Fluke 41Bpower harmonics analyser. The switching frequency was setto 20kHz and the control loops were implemented digitallywith a sampling frequency of 10kHz. When the converterwas switched off with no control, the power factor and thecurrent THD obtained from the Fluke 41B powerharmonics analyser was 0.68 and 99.6%, respectively.Figure 6 shows the performance of the PFC converter

+5

−5

0

360o180o0o31 5 7 9 11 13 15

a b

c d

100

50

0

%R

49.9 HZ 8o

Fig. 6 Performance of PI controller obtained from Fluke 41Bpower harmonics analysera Current waveformb Current spectrumc THD of supply currentd Power factor

+5

−5

0

360o180o0o31 5 7 9 11 13 15

a b

c d

100

50

0

%R

49.9 HZ 2o

Fig. 7 Performance of nonlinear adaptive controller obtained fromFluke 41B power harmonics analysera Current waveformb Current spectrumc THD of supply currentd Power factor

+5

−5

0

360o180o0o31 5 7 9 11 13 15

a b

c d

49.9 HZ 2o100

50

0

%R

Fig. 8 Performance of nonlinear adaptive controller with added25% load obtained from Fluke 41B power harmonics analysera Current waveformb Current spectrumc THD of supply currentd Power factor

100

50

0

+5

−5

0

360o180o0o31 5 7 9 11 13 15

50.0 HZ 7o

a b

c d

%R

Fig. 9 Performance of PI controller with added 25% load obtainedfrom Fluke 41B power harmonics analysera Current waveformb Current spectrumc THD of supply currentd Power factor

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 631

when conventional PI controllers of (9) and (11) wereapplied. The current reference was derived from the rectifiedsupply voltage. The power factor and the current THDobtained from the Fluke 41B power harmonics analyserwere 0.99 and 8.6%, respectively. Figure 7 shows theperformance of the PFC converter when the current loop PIcontroller was replace by the nonlinear control law of (18).The power factor and the current THD obtained from theFluke 41B power harmonics analyser were 1.0 and 4.7%,respectively. Clearly a far superior result had been achieved.

To illustrate the robustness of the nonlinear adaptivecontroller against system changes, the load of the systemwas increased by 25% by changing RL to 132O. Figure 8shows the performance of the nonlinear adaptive controller.The power factor and the current THD obtained from theFluke 41B power harmonics analyser were 0.99 and 5.4%,

respectively. Figure 9 shows the performance of the PFCconverter when the PI controller was used. The powerfactor and the current THD became 0.99 and 8.3%,respectively. These clearly demonstrated the effectivenessand robustness of the proposed nonlinear adaptivecontroller for both power factor correction and harmonicelimination.

To further demonstrate the robustness of the nonlinearadaptive controller, the rms value and THD of the supplyvoltage were varied. Figures 10 and 11 show theperformances of the nonlinear adaptive controller whenthe supply voltage was 100.4V rms and 120V rms,respectively. The power factor obtained from the Fluke41B power harmonics analyser stayed at 0.99–1 and thecurrent THD stayed around 4.1–4.2%. Figure 12 shows the

+5

−5

0

360o180o0o 31 5 7 9 11 13 15

50.1 HZ 2o

a b

c d

100

50

0

%R

Fig. 10 Performance of nonlinear adaptive controller with 100.4 Vrms supply obtained from Fluke 41B power harmonics analysera Current waveformb Current spectrumc THD of supply currentd Power factor

+5

0

−5

c d

31 5 7 9 11 13 15

%R

100

50

0360o180o0o

a b

Fig. 11 Performance of nonlinear adaptive controller with 120 Vrms supply obtained from Fluke 41B power harmonics analysera Current waveformb Current spectrumc THD of supply currentd Power factor

+200

−200

0

+5

−5

0

180o 360o0o

a b c

d e f

180o 360o0o31 5 7 9 11 13 15

50.1 HZ 4o100

50

0

%R

Fig. 12 Performance of nonlinear adaptive controller with 3.7% supply THD obtained from Fluke 41B power harmonics analysera Supply voltage and THDb Voltage waveformc THD of supply currentd Current waveforme Current spectrumf Power factor

632 IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005

performance of the nonlinear adaptive controller when thesupply THD was increased from 2% to 3.7%. The powerfactor remained at 1 and there was a slight increase in thecurrent THD to 5.9%. Since the current reference wasderived from the supply voltage, a poor supply THD wouldbring forward the THD to the generation of the supplycurrent. Figure 13 shows the performance of the PIcontroller when the supply THD increased to 3.7%. Thepower factor dropped to 0.98 and the current THDincreased to 9.7%. These clearly demonstrated the effec-tiveness and robustness of the proposed nonlinear adaptivecontroller for both power factor correction and harmonicelimination under changing environments.

6 Conclusions

An adaptive controller based on nonlinear system identifi-cation has been successfully implemented for controllingPFC-boosted converters. The responses obtained werebetter than the linear PI controller. The orthogonal leastsquares estimation algorithm coupled with the errorreduction ratio test has been successfully applied for thefitting of the NARMAX model for PFC-boost converters.The fitted nonlinear model is compact and captures thecharacteristics of the converter in a far superior mannerthan the linear model. The one-step ahead predictivecontrol derived from the compact fitted model outperformsclassical PI controllers with a better power factor and lowertotal harmonic distortion.

7 Acknowledgments

The authors gratefully acknowledges the support of theHong Kong Polytechnic University.

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+200

−200

0

+5

−5

0

180o 360o0o

a b c

d e f

180o 360o0o31 5 7 9 11 13 15

50.1 HZ 11o100

50

0

%R

Fig. 13 Performance of PI controller with 3.7% supply THD obtained from Fluke 41B power harmonics analysera Supply voltage and THDb Voltage waveformc THD of supply currentd Current waveforme Current spectrumf Power factor

IEE Proc.-Electr. Power Appl., Vol. 152, No. 3, May 2005 633