method for extracting positive and negative sequence current

8/10/2019 Method for Extracting Positive and Negative Sequence current

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A Method for Extracting the Fundamental

Frequency Positive-Sequence Voltage Vector Basedon Simple Mathematical Transformations

F. Bradaschia, J. P. Arruda, H. E. P. Souza, G. M. S. Azevedo, F. A. S. Neves, M. C. Cavalcanti

Universidade Federal de Pernambuco/Departamento de Engenharia Eletrica e Sistemas de Potencia, Recife - PE, Brazil

AbstractIn this paper a novel scheme for obtaining thefundamental-frequency positive-sequence grid voltage is pro-posed. The method is based on four simple mathematicaltransformations, two of them in the stationary reference frame,which are able to eliminate odd harmonics from the originalsignals. The other two transformations are implemented in a

synchronously rotating reference frame in order to eliminate evenharmonics. The output of the last transformation block is input toa synchronous reference frame phase locked loop for detectingfrequency and position of the positive-sequence voltage vector.The proposed algorithm was verified through simulations andexperiments by applying distorted and unbalanced signals, con-taining positive and negative-sequence components. The resultsare in agreement with those theoretically predicted and indicatethat the proposed scheme has a great potential for using in gridconnected converters synchronization algorithms.

I. INTRODUCTION

Voltage oriented control (VOC) is one of the most used

techniques for the operation and control of any equipment

connected to the grid through a DC-AC controlled converter,such as a distributed power generation system, uninterrupt-

ible power supply or active filter. The correct value of

the fundamental-frequency positive-sequence voltage vector

of the electric grid is essential for achieving good control

performance when using VOC. This information is usually

obtained by using a phase locked loop (PLL). The PLL must

be capable of following the fundamental-frequency positive-

sequence voltage vector as fast as possible for eliminating the

impacts of the grid signal imperfections, even during voltage

sags or if the grid voltages are distorted and unbalanced [1][2].

The synchronous reference frame PLL (SRF-PLL) is fre-

quently used in three-phase systems and for balanced undis-

torted grid conditions, good results can be reached [3]. TheSRF-PLL can also operate correctly if only high order harmon-

ics are present in the grid voltages, by reducing its bandwidth

to attenuate these harmonics [4]. However, under unbalanced

conditions, the second harmonic content of the dq voltage

vector, caused by the negative-sequence components, makes

the reduction of the bandwidth an inefficient solution, since

the dynamics of the PLL would become very slow for such

a narrow bandwidth. A better performance under unbalanced

conditions can be achieved by separating the positive and

negative-sequence components of the voltage vector. This is

made by the double synchronous reference frame PLL (DSRF-

PLL) that uses a decoupling cell to isolate the positive and

negative-sequence components [5]. A cell for obtaining any

other harmonic component may be implemented and then

that component may be subtracted from the original voltages.

However, the computational cost greatly increases if many

harmonic components should be attenuated.

An alternative technique uses an enhanced PLL (EPLL)[6] for each phase. The phase voltages and their respective

values delayed by 900 detected by the EPLL are used forobtaining the positive-sequence voltages of the three-phase

system using the instantaneous symmetrical components (ISC)

method. Finally, a fourth EPLL is applied to the output of

the ISC method to estimate the positive-sequence voltage

phase angle. The fundamental-frequency negative-sequence

components of the grid voltages are eliminated by the EPLL

calculator. However, some harmonics will pass through the

EPLL and may be present in its output.

In the so called delayed signal cancellation method [7],

[8], the positive- and negative-sequence components of the

grid voltage are determined based on the voltage vector in astationary () reference frame and also on that voltage vectordelayed by a quarter of cycle. The method is suitable for un-

balanced undistorted grid voltages, but the calculated positive-

and negative-sequence voltages are sensitive to harmonics in

the grid voltages.

The PLL using a dual second order generalized integrator

(DSOGI) [9] is based on the ISC method in the stationary

() domain. The grid voltages are transformed to the reference frame and the and components delayed by900 are obtained by the DSOGI-QSG, where QSG meansquadrature signals generation. These signals are used as the

inputs of a positive-sequence calculator (PSC). After that, a

SRF-PLL is used to obtain the angle, frequency and position ofthe fundamental-frequency positive-sequence voltage vector.

This frequency is used to feedback the DSOGI-QSG.

A fast instantaneous method for sequence extraction from

a three-phase signal was proposed in [10]. It was shown that

the phase voltages can be manipulated through mathemati-

cal transformations to eliminate specific harmonics from the

original signals without using filters. The signal is passed

through some cascaded mathematical transformations in order

to eliminate odd harmonics. The even harmonic components

are not eliminated, although some of them may be attenuated.

This paper presents an original scheme that employs

only simple calculations to extract the fundamental-frequency

978-1-4244-1668-4/08/$25.00 2008 IEEE


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positive-sequence grid voltages, eliminating odd and even

harmonics. The method is based on different ways of obtaining

the positive- and negative-sequence components of the grid

voltage, by applying the symmetric components theory to

the dq voltage vector transformed into different referenceframes ( and synchronous with the fundamental positive-sequence voltage). It is shown how these transformations

affect each harmonic component of the grid voltage. The

extracted positive-sequence voltage is the input to a SRF-PLL

for detecting the frequency and the position of the desired

voltage vector.

I I . TRANSFORMATIONS TOE XTRACTO DD H ARMONICS

Using the method of symmetrical coordinates (1), it is

possible to obtain the symmetrical components (S+,S andS0) of a three-phase signal (current, voltage or flux:

Sa,

Sb

andSc) [11]. S+SS0= 1

3

1 1120 11201 1120 11201 1 1

[M]

SaSbSc (1)

The negative- and positive-sequence components can be

extracted in abc from: SaSbSc

=13

1 160 160160 1 160

160 160 1

[M] SaSbSc

(2)

and S+aS+bS+c=1

3

1 160 160160 1 160

160 160 1

[M+]

SaSbSc .(3)

As demonstrated in [10], operations (2) and (3) can be

written in different ways for obtaining the same responses

if the phase signals contain only fundamental- frequency

components. However, if the harmonics present in the orig-

inal signals are differently modified by each operation forobtaining the positive- or negative-sequence component. Four

of these transformations are used in the time domain (using

the instantaneous symmetrical components theory [12]), two

for obtaining the positive-sequence and two for the negative-

sequence components:

[s] =13{A1[s] + A2[s60] + A3[s60]} (4)

[s] = 1

3{B1[s] + B2[s90]} (5)

[s+] =13{C1[s] + C2[s60] + C3[s60]} (6)

[s+] = 1

3{D1[s] + D2[s90]} , (7)

where

A1 = C1 =

1 0 00 1 00 0 1

,A2 = C2 =

0 1 00 0 11 0 0

,

A3 = C3 =

0 0 11 0 00 1 0

, B1 = D1 =

1 12

12

12 1 1

212

12 1

,

B2 = D2 =

0 32 3232

032

32

32 0

, [s] = sasbsc

,

[s] = sa

sbsc

, [s+] = s+a

s+b

s+c , [s60] = sa60sb60

sc60 ,

[s60] =

sa60sb60sc60

, [s90] =

sa90sb90sc90

, [s90] =

sa90sb90sc90

.

The subscripts60 (60) and 90 (90) are used to indicatethe instantaneous signals delayed from the original ones by the

correspondent value in degrees. They can be implemented by

storing the original signals during a time interval correspond-

ing to the desired angles at the fundamental-frequency. For

discrete-time implementation, a determined number of samples

(N) of a fundamental period must be stored. For a delay of

90o the lastN/4samples are stored. To obtain a 90o advancedsignal, the 90o delayed signal with opposite signal is used.The delay of 60o can be generated by saving the last N/6samples. The advance of60o is produced by storing the lastN/3 samples, for obtaining a delay of120o, and multiplyingby -1.

The operations defined by (4)-(7) are called ofA, B, CandD, respectively. After applyingA,B ,CorD, the signalssa, sb and sc are modified. The operations A and B keepthe fundamental-frequency negative-sequence component, but

some harmonics are modified. On the other hand, the op-

erations C and D keep the fundamental-frequency positive-sequence component, changing the characteristics of other

components. Table I, an extended version of that presentedin [10], shows the gains (absolute values and phase shiftings)

of each operation A, B , C and D . The results of the cascadeapplication of operations AB and CD are also shown. Evenharmonics are not canceled, being only attenuated by these

operations.

III. PROPOSEDM ETHOD

Transforming the original signals to an arbitrary dq ref-

erence frame (stationary or rotating), operations capable of

producing the same results ofA, B, C and D are presentedand a way of using these transformations for eliminating odd

and even harmonics is proposed.


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Table IGAINS OF THE MATHEMATICAL OPERATIONS FOR THE ODD HARMONICS

Operation A B C D AB CD

1seq+ - - 10 10 - 10

1seq- 10 10 - - 10 -3seq+ - 10 - - - -3seq- - - - 10 - -5seq+ 10 - - 10 - -5seq- - 10 10 - - -7seq+ - 10 10 - - -7seq- 10 - - 10 - -9seq+ - - - 10 - -9seq- - 10 - - - -

11seq+ 10 10 - - 10 -11seq- - - 10 10 - 1013seq+ - - 10 10 - 1013seq- 10 10 - - 10 -15seq+ - 10 - - - -15seq- - - - 10 - -

A. Transformations in an arbitrary dq reference frame

From the Park transformation:

[sdq] = 2

3

cos cos(120) cos(+ 120) sin sin(120) sin(+ 120)

12

12

12

T

[s]

(8)

whereis thedaxis angular position and using (4) and (8), thenegative-sequence instantaneous components in an arbitrarydq

reference frame can be calculated as follows:

T[s] =13{TA1[s] + TA2[s60] + TA3[s60]} .

Thus,

[sdq] =1

3

TA1T

1[sdq] + TA2T1[sdq60] + TA3T

1[sdq60]

(9)

where

[sdq] =

sdsqs0

, [sdq] =

sdsqs0

,

[sdq60] = sd60sq60s060 , [sdq60] = sd60

sq60s060 .As T is not singular, the process of changing the referenceframe (An TAnT

1, n = 1, 2, 3) is a transformationof similarity. Some properties are common in the similar

matricesAn and TAnT1: the same characteristic polynomial

and eingevalues, algebraic and geometrical multiplicity [13].

Disregarding the zero sequence components, operation A inthe dq reference frame can be obtained from (9): sdsq

=

1

3A1dq

sdsq

+

1

6A2dq

sd60sq60

+

1

6A3dq

sd60sq60

.

(10)

Transformations of similarity analogous to (9) can be applied

to operations B , C and D to obtain them in any dq referenceframe, respectively:

sdsq = 12B1dq sdsq +12B2dq sd90sq90 , (11) s+ds+q

=

1

3C1dq

sdsq

+

1

6C2dq

sd60sq60

+

1

6C3dq

sd60sq60

,

(12) s+ds+q

=

1

2D1dq

sdsq

+

1

2D2dq

sd90sq90

. (13)

The transformed matrices in (10), (11), (12) and (13) are:

A1dq = B1dq =C1dq =D1dq =

1 00 1

,

A2dq = C2dq = 1 33 1

,A3dq = C3dq =

1

3

3 1

,B2dq =D2dq =

0 11 0

.

It is important to note that operations (10) - (13), named

Adq , Bdq , Cdq and Ddq , respectively, are identical to oper-ations A through D regarding to the effects over the har-monic components (Table I). This is proved now. Consider

a three-phase balanced positive-sequence signal of order h

(h= 1, 2, 3, . . .):

sah= Shcos(ht + h)sbh = Shcos(ht + h120o)sch= Shcos(ht + h+ 120

o). (14)

Transforming to the reference frame: sh = Shcos(ht + h)sh = Shsin(ht + h)

, (15)

and applying the formula of Euler in (15):

sh+ jsh = Shej(ht+h). (16)

Using the signal in (15) as the input to operationAdq in (10) and taking into account that an advance of 60

o

corresponds to a delay of120o with the opposite signal,

s =

Sh

3

cos(h) 12cos(h h120o) +

32 sin(h h120o)+

+12

cos(hh60o) +32

sin(hh60o)

s =

Sh

3

sin(h)

32

cos(h h120o) 12sin(hh120o)32 cos(hh60o) + 12sin(hh60o)

,

(17)

whereh = ht+h. Changing (17) for the complex domainand using the formula of Euler:

s +js

=She

j(ht+h) 1

3 1 + ej(1+h)120

o+ ej(1+h)60

o

.


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Then,

s +js

= (sh+ jsh)

1

3

1 + ej(1+h)120

o+ ej(1+h)60

o

GA+,

(18)where GA+ is the gain of the operation Adq for positive-sequence signals.

Considering a three-phase balanced negative-sequence sig-

nal of order h (h= 1,2,3, . . .), it can be shown that:

sjs = (shjsh)

1

3

1 + ej(1+h)120

o+ ej(1+h)60

o

GA

, (19)

where GA is the gain of the operation Adq for negative-sequence signals.

Using the signal in (15) as the input to operation Bdqin (11),

ss

=

Sh

2

cos(h) + sin(hh90o)sin(h)cos(hh90o)

, (20)

whereh= ht+h. Changing (20) for the complex domainand using the formula of Euler:

s +js

=She

j(ht+h) 1

2

1 + ej(1+h)90

o.

Then,

s +js

= (sh+ jsh)

1

2

1 + ej(1+h)90

o

GB+

, (21)

where GB+ is the gain of the operation Bdq for positive-sequence signals.

For negative-sequence signals (h= 1,2,3, . . .):

s js = (shjsh)

1

2

1 + ej(1+h)90

o

GB

, (22)

where GB is the gain of the operation Bdq .Results similar to (18) and (19) can be obtained for the

operation Cdq in (12):

s+js

= (sh+jsh)

1

3 1 + ej(1h)60

o

+ ej(1h)120o

GC+, (23)

s js = (shjsh)

1

3

1 + ej(1h)60

o+ ej(1h)120

o

GC,

(24)whereGC+ and GC are the gains of the operation Cdq forpositive- and negative-sequence signals, respectively.

Results similar to (21) and (22) can be obtained for the

operation Ddq in (13):

s +js

= (sh+ jsh)

1

2

1 + ej(1h)90

o

GD+

, (25)

s js = (shjsh)

1

2

1 + e

j(1h)90o

GD

, (26)

whereGD+ and GD are the gains of the operation Ddq for

positive and negative-sequence signals, respectively.

B. Implementation of the proposed method

After measuring the signals in abc and transforming them

into the reference frame, operations Cdq and Ddq areapplied in cascade. In this way, many odd harmonics are

canceled, but the even harmonics remain present in the signals.

The next step is to transform the output signals into asynchronous dq reference frame. Therefore, the fundamental

component becomes constant, the second harmonic becomes

of fundamental frequency and any harmonic of positive-

sequence with frequencyhin has its frequency decreasedby one in dq. The negative-sequence components have their

frequency increased by one, when they are transformed to thesynchronous dq reference frame. Then, the even harmonics

become odd and vice-versa. If the signals in the synchronous

dq reference frame go through the operations Adq and Bdq incascade, then odd harmonics of these signals (even harmonics

of the original signals) are eliminated and their constant

components contain the information about the fundamental-

frequency positive-sequence vector. The scheme is described

in the block diagram of Fig. 1.

Since the constant components do not pass integrally by the

operationsAdq andBdq, an adjustment must be made in orderto obtain the correct magnitudes of the fundamental frequency

positive-sequence signal insd and sq . Therefore, making a DC

signal go through Adq:

PI 1

s+-

v*q

= 0

vabc abc dq

vdDdq

vq

Cdq Adq Bdq MDCVv + vd q

2 2

1

Bdq++ff

LPF

LPF

dq

abc

vabc+

v +d

v +q

Figure 1. Block diagram of the proposed solution.


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sdsq

=

1

3A1dq

sDCdsDCq

1

6A2dq

sDCdsDCq

+

1

6A3dq

sDCdsDCq

.

Then, sdsq

=

1

3A3dq

sDCdsDCq

. (27)

The result goes through Bdq . Thus, sdsq

=

1

2B1dq

1

3A3dq

sDCdsDCq

+

1

2B2dq

1

3A3dq

sDCdsDCq

sdsq

=

1

6

(1 3) (1 + 3)(1 3) (1 3)

sDCdsDCq

. (28)

Then, the matrix [MDC]that must multiply the signal comingfrom the operation

Bdq is:

[MDC] =

1

6

(1 3) (1 + 3)(1 3) (1 3)

1[MDC] =

3

4

(1 3) (1 3)(1 +

3) (1 3)

. (29)

In the proposed solution, as it can be observed in Table I, the

11th negative-sequence and the 13th positive-sequence har-monic components and the fundamental-frequency positive-

sequence component are not eliminated by the cascaded Cdqand Ddq transformations. Since these components becomeeven after the to dq transformation, they cannot be

eliminated by the cascaded Adq and Bdq transformations andthey are present in the output signals. Analogously, the 12thpositive-sequence and the12thnegative-sequence componentsare not canceled by the cascaded Cdq andDdq transformationsand become the 11th positive-sequence and 13th negative-sequence components after the todq transformation. Thus,as it can be seen in Table I, these components are not

eliminated. All of the other harmonic components among the

above mentioned are canceled. In general, using the equations

shown in this paper, it is concluded that the positive-sequence

harmonics of order 12h and 12h + 1 as well as the negative-sequence harmonics of order 12h1and 12hare not canceledby the mathematical operations. However, the bandwidth of the

SRF-PLL may be reduced in such a way that the estimationof the frequency and the position are barely affected bythose harmonics.

To recover correctly the three-phase signal, it is advisable

to pass the estimated signalsv+d andv+q through filters that do

not affect the system dynamics, since they are not in the loop,

and have a high bandwidth. In this paper, filters with finite

impulse response (FIR) are used. The filters are based on the

windowHammingsmethod of order20 and cut-off frequencyof300 Hz.

To avoid the problems of offsets in the measurement and

acquisition signals, the following procedure is used: each

fundamental period, the last N samples of the input signal

are added and the result is divided by N. The result of thedivision is the offset. The offset is subtracted from the original

signal and this solution has a very small computational effort.

IV. SIMULATION ANDE XPERIMENTALR ESULTS

In order to verify the proposed algorithm effectiveness, sim-

ulations were carried out for obtaining the positive-sequence

fundamental-frequency voltages from unbalanced and dis-

torted input signals. The fundamental-frequency of the grid is

50 Hz. In all situations, v+1pf = 10p.u. is the pre-fault grid

voltage. In the first case, the unbalanced and distorted grid

voltage during the fault consisted ofv+1 = 0.747 14p.u.,v1 = 0.163 171.37p.u., v5 = 0.07 60p.u.,v+7 =0.0530p.u.. In the second case, v+1 = 2.5v1 = 10p.u.,v+h = vh = 0.6v

+1

h , h = 2, 3, . . . , 25. Figures 2 and 3

show the simulation results. In both figures, the first graphic

shows the input voltages, the second graphic shows the output

voltages, the third graphic shows the voltages (vd, vq) and thefourth graphic shows the error.

0 40 80 120 160 200 240-150

0

150

v a

bc

(%)

t(ms)

0 40 80 120 160 200 240

-100

0

100

v+ a

bc

(%)

t(ms)

0 40 80 120 160 200 240-50

0

50

100

150

v+ d

,v

+ q

(%)

t(ms)

0 40 80 120 160 200 240-1

-0.5

0

0.5

1

=+

-

+(ra

d)

t(ms)

Figure 2. Simulation results for unbalanced signals with low distortion.

Experiments are performed using a OMICRON-CMC256

programmable source to generate the same simulated input

voltages. The signals are acquired and processed using a

Texas Instruments TMS320F2812 DSP and measured after

passing through a digital-analog converter. Figures 4 and 5

show the experimental results for the same situations tested in

simulation. In both figures, the first graphic shows the input

voltages, the second graphic shows the output voltages and the

third graphic shows the voltages (vd, vq) and . It can be seen


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1120

0 40 80 120 160 200 240

-150

0

150

300

450

v a

bc

(%)

t(ms)

0 40 80 120 160 200 240

-100

0

100

v+ a

bc

(%)

t(ms)

0 40 80 120 160 200 240-50

0

50

100

150

v+ d

,v

+ q

(%)

t(ms)

0 40 80 120 160 200 240-1

-0.5

0

0.5

1

=+

-

+(r

ad)

t(ms)

Figure 3. Simulation results for unbalanced signals with high distortion.

a very good agreement between simulation and experiment,

showing that the theoretical assumptions used in simulationsare valid. The results illustrate the behavior of the proposed

method in two different conditions and verify its harmonic

rejection capability. In the first case, a realistic situation with

unbalanced signals and low distortion is considered. In the

second case, it is proved that the proposed method can handle

signals with high distortion.

Table II provides a quantitative comparison of the proposed

method with three existing synchronization methods, based on

the output total harmonic distortion (THD), response time and

offset. The methods used for comparison are the SRF, DSRF

and DSOGI. In Table II, (1) means the first case: unbalanced

signals with low distortion (Fig. 2), (2) means the second case:

unbalanced signals with high distortion (Fig. 3) and (3) meansthe third case: unbalanced signals without distortion (Fig. 2

without distortion). It is considered the highest THD among

the three phases. In the first and the second cases, the THD

of the input voltages is 14.34% and 66.71%, respectively. In

practice, the higher harmonic component of the THD that can

be calculated is equal to half of the sample frequency of the

input signals. In all simulations and experiments the sample

frequency is 18 kHz. Then, all THD results have harmoniccomponents up to 9 kHz.

The response time is the time necessary for that the absolute

error of stays between a tolerance range of1.5o (0.0262rad).In the cases in which the method does not guarantee the error

between the tolerance range, the response times in the table

are represented by - .Table II

COMPARISON OF THE SYNCHRONIZATION METHODS

SRF DSRF DSOGI Proposed

THD(1) 2.98% 1.16% 1.10% 0.01%

THD(2) 17.67% 23.04% 19.25% 0.24%

response time(1) - 49.44ms 42.78ms 32.06msresponse time(2) - - - 7.78msresponse time(3) - 36.67ms 42.72ms 31.89mscompense offset No No No Yes

V. CONCLUSION

In this paper simple mathematical transformations are

proposed for handling distorted and unbalanced three-phase

signals in order to eliminate part of the information (har-

monic and unbalanced content) and keep the positive-sequence

fundamental-frequency components. Using finite impulse re-

sponse filters, the components that are not eliminated by the

mathematical operations can be attenuated. These filters do

not affect the dynamic of the system since they are not in

the loop and they have a high bandwidth. Four situations

are tested to confirm the theoretical assumptions: unbalanced

signals with high distortion, unbalanced voltage sag with low

distortion, unbalanced voltage sag and balanced voltage with

offset. The results were obtained through simulations and also

using an experimental platform, showing that the proposed

scheme has a great potential for grid synchronization of power

converters, even during unbalanced faults and highly distorted

grid voltages.

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[2] V. Kaura and V. Blasko, Operation of Phase Locked Loop SystemUnder Distorted Utility Conditions, IEEE Transactions on IndustryApplications, vol. 33, no. 1, pp. 5863, January/February 1997.

[3] L. N. Arruda, S. M. Silva, and B. J. C. Filho, PLL Structuresfor Utility Connected Systems, Proceedings of Industry ApplicationsSociety Annual Meeting01, vol. 4, pp. 26552660, September/October2001.

[4] S.-K. Chung, A Phase Tracking System for Three Phase UtilityInterfece Inverters, IEEE Transactions on Power Electronics, vol. 15,

no. 3, pp. 431438, May 2000.[5] P. Rodrguez, J. Pou, J. Bergas, I. Candela, R. Burgos, and D. Boroye-

vich, Double Synchronous Reference Frame PLL for Power Convert-ers, Proceedings of IEEE Power Electronics Specialists Conference -PESC05, pp. 14151421, 2005.

[6] M. Karimi-Ghartemani and M. R. Iravani, A Method for Synchroniza-tion of Power Electronic Converters in Polluted and Variable-FrequencyEnvironment, IEEE Transactions Power Systems, vol. 19, no. 3, pp.12631270, August 2004.

[7] J. Svensson, M. Bongiorno, and A. Sannino, Practical Implementationof Delayed Signal Cancellation Method for Phase-Sequence Separation,IEEE Transactions on Power Delivery, vol. 22, no. 1, pp. 1826, January2007.

[8] G. Saccomando and J. Svensson, Transient Operation of Grid Con-nected Voltage Source Converter Under Unbalanced Voltage Condi-tions, Proceedings of IEEE Industry Applications Society - IAS01, pp.24192424, 2001.


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Figure 4. Experimental results for unbalanced signals with low distortion.

[9] P. Rodrguez, R. Teodorescu, I. Candela, A. Timbus, M. Liserre, andF. Blaabjerg, New Positive-Sequence Voltage Detector for Grid Syn-chronization of Power Converters Under Faulty Grid Conditions, Pro-ceedings of IEEE Power Electronics Specialists Conference - PESC06,pp. 17, June 2006.

[10] R. Cutri and L. M. Jr., A Fast Instantaneous Method for SequenceExtraction, Proceedings of Brazilian Power Electronics Conference -

Figure 5. Experimental results for unbalanced signals with high distortion.

COBEP07, pp. 16, October 2007.[11] C. L. Fortescue, Method of Symmetrical Co-ordinates Applied to

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[13] L. N. Trefethen and D. Bau, Numerical Linear Algebra. SIAM, 1997.

method for extracting positive and negative sequence current

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