Positive and Negative Sequence Estimation forUnbalanced Voltage Dips

Rafael A. Flores§, Irene Y.H. Gu§ and Math H.J. Bollen‡§Dept. of Signals and Systems ‡ Dept. of Electric Power Engineering

Chalmers University of TechnologyGothenburg, 41296, Sweden

Abstract— This paper proposes the use of a complex Kalmanfilter for the estimation of positive and negative sequences fromthree phase voltages. A complex voltage is obtained by applyingthe αβ-transform followed by the dq-transform using a rotationaloperator. The algorithm for three phase voltages containing Kharmonics is also given.

In the conventional method, estimation of positive and negativesequences is performed through two steps: the magnitude andphase-angle in each individual phase of the voltages are firstestimated and the symmetrical component transformation is thenapplied. The proposed method offers a direct estimation of thepositive and negative sequences that may reduce the estimationerrors. In addition, the proposed method has a reduced compu-tational cost since the number of state variables is reduced to 2/3as compared to that in the conventional method. An experimentwas performed on measured three phase voltage data. Resultshave shown that the proposed method offers a good estimation.

I. INTRODUCTION

Positive and negative-sequence voltages were originallyintroduced to speed up calculations involving non-symmetricalfaults, and as such are introduced in almost any text book onelectric power systems. More recently their application liesin the diagnostics of power systems during non-symmetricaloperation including faults. A method for characterizing the un-balanced voltage dips from the positive and negative sequenceshas been proposed in [1] and [2]. The two-component methodused in [2] is based on the idea that the positive and negative-sequence source impedances are equal for static circuits.To achieved reliable and fast characterization and thereby acorrect identification of voltage dips, a good estimation of thepositive and negative sequence voltages is required.

Several methods for estimating positive and negative se-quences have been proposed: [3] has proposed the use ofa weighted least squares, and [4] has proposed to use ofKalman filters. In [4] the magnitudes and phase-angles werefirst estimated using three Kalman filters (one for each phase),and the positive and negative sequences were then obtained byusing the symmetrical-component transformation, as shown inFig.1(a). Since the negative sequence voltage and the phase-angle between the positive and negative sequence voltagesare sensitive to estimation errors, these methods may sufferfrom inaccuracies due to the presence of noise and harmonicdistortions. Note that the angle between positive and negative-sequence voltage is an important parameter in determiningthe dip type in [2]. Consequently, it may, in some cases,lead to wrong dip characterization. Therefore, more reliable

methods for estimating positive and negative sequences aredesirable. Symmetrical component voltages and currents arealso a suitable tool for describing load behavior during voltagedips and other disturbances [5].

In this paper we propose an improved approach for esti-mating the positive and negative sequences using a complexKalman filter. Comparing with the conventional 2-step methodused in [4], this method offers a direct estimation of the posi-tive and negative sequences, and may reduce the estimationerrors. As shown in Fig.1(b), the proposed method simplifiesthe structure used for the estimation: instead of three Kalmanfilters, only one Kalman filter is required. Further, the totalnumber of variables to be estimated in Fig.1(b) is reduced to2/3 of that required in Fig.1(a).

va(n)

vb(n)

vc(n)

a)

b)

Ac, φc

Aa, φaKalmanFilter

KalmanFilter

KalmanFilter

X V2

V1

V1

V0

V2

Ab, φb

KalmanFilter

T3

va(n)

vb(n)vc(n)

e−jωn

T1

vdq(n)vαβ(n)

Fig. 1. (a) Conventional method (b) PN estimator

In the proposed method, the sampled values of three phasevoltages va(n), vb(n) and vc(n) are αβ-transformed to acomplex voltage vαβ(n).

Then, this complex voltage vαβ(n) is dq-transformed, re-sulting in a complex voltage vdq(n), as it is shown in Fig.1(b).The result contains the embedded positive and negative se-quences. The positive and negative sequences are modelledby the state-space equations, and the complex Kalman filter isused to estimate the state vector iteratively.

In this paper the method will be used to estimatesymmetrical-component voltages. The method can be appliedin a similar way to estimate symmetrical-component currents.

II. ESTIMATION OF POSITIVE AND NEGATIVE SEQUENCES

In this section, the αβ-transform and the dq-transform (or,the Park’s transform) will be described. The complex Kalmanfilter for estimating the positive and negative sequences willalso be described, where the three phase voltage is modelledas consisting of voltage of fundamental frequency (50Hzin Europe or 60Hz in USA) under harmonics and noisedistortions.

A. The αβ-transform and the dq-transform

Consider a three-phase system with the following voltages:

va(n) =√

2Va cos(ωn + φa)vb(n) =

√2Vb cos(ωn + φb) (1)

vc(n) =√

2Vc cos(ωn + φc)

Where va(n), vb(n) and vc(n) are the sampled phase voltages,Va, Vb and Vc are the rms or effective value, φa, φb and φc

are the phase-angles, ω is the discrete angular frequency, andn is the discrete time index.

Define the αβ-transform for the three-phase voltages asfollows:

vαβ(n) =23

[va(n)�ea + vb(n)�eb + vc(n)�ec] (2)

where �ea = 1, �eb = ej 2π3 = a and �ec = e−j 2π

3 = a2 = a∗,and a is a rotational operator representing a rotation over 120◦.(2) can be interpreted as the projection of three voltages ontothe αβ-space.

The dq-transform (or Park’s transform) is then applied asfollows

vdq(n) = vαβ(n)e−jωon (3)

It should be noted that both vαβ(n) and vdq(n) are complexvoltages as a function of time. The transformation could beinterpreted as a synchronization of the αβ-space at angularfrequency ωo, set to be equal to the fundamental voltagefrequency ω in this paper.

The symmetrical component voltages are defined from thecomplex phase voltages Va = Vaejφa , Vb = Vbe

jφb and Vc =Vce

jφc as follows

V0

V1

V2

=

13

111

1aa2

1a2

a

Va

Vb

Vc

(4)

where V0 = V0ejφ0 , V1 = V1e

jφ1 and V2 = V2ejφ2 are the

symmetrical component voltages (zero, positive and negativesequences), which are defined as complex phasers.

Re-writing (1) in the complex form and combining with (4)to calculate vdq(n), it follows,

vdq(n) =√

2(V1 + V2

∗e−j2ωn). (5)

For sinusoidal phase voltages as in (1), the first component V1

in (5) is a complex constant number, while the 2nd componentV2 in (5) is a complex number rotating with an angular speed

of 2ω. Any unbalance in the three phase system will appearas an non-zero rotating V2 in the dq-space. The positive andnegative sequences are estimated separately in the dq-spacefrom (5) as detailed in next section.

B. Estimation of positive and negative sequences

Using state-space modelling, the positive and negative se-quence voltages in (5) can be estimated by a complex Kalmanfilter. The state equation and observation equation associatedwith a Kalman filter can be described as follows

X(n) = A(n − 1)X(n − 1) + U(n)Y (n) = H(n)X(n) + V (n) (6)

where X(n) is a complex state vector sized 2x1, A(n) is atransition matrix sized 2x2, U(n) is a vector containing modelnoise which is zero-mean with σ2

u variance, and V (n) is avector containing observation noise which is zero-mean withσ2

v variance.1) State-space modelling of the three-phase system: For the

three phase system described in (1), define the following statevector

X(n) = [x1 x2]Tn =

[V1 V2

∗e−j2ωn]T

(7)

It follows that the state equation of the Kalman filter in (6)associated with (5) is

[x1

x2

]

n

=[

10

0e−j2ω

] [x1

x2

]

n−1

+[

u1

u2

]

n

(8)

and the observation equation in (6) becomes

y(n) =√

2 [1 1][

x1

x2

]

n

+ v(n) (9)

The negative sequence is found by applyingV2 = x∗

2e−j2ωn.

2) Kalman Filter Algorithm: The algorithm for iterativelyestimating the state vector is summarized as follows, a moredetailed explanation can be found in [6]. The covariancematrix of model noise is Q = E[|U(n)|2] = σ2

uI , whereIn is the identity matrix of order n. The covariance matrixof observation noise is C = E[|V (n)|2] = σ2

vI . The initialconditions are set as x(0|0) = [0 0]T and P (0|0) = WI ,where W is a relatively large number (chosen as 100 in thispaper).

• Prediction:x(n|n − 1) = A(n)x(n − 1|n − 1)

• Prediction error covariance matrix:P (n|n − 1) = A(n)P (n − 1|n − 1)A(n)T + Q

• Kalman gain matrix:K(n) = P (n|n − 1)HT [HP (n|n − 1)HT + C]−1

• Filtering:x(n|n) = x(n|n − 1) + K(n)[y(n) − Hx(n|n − 1)]

• Error covariance matrix a posteriori:P (n|n) = [In − K(n)H]P (n|n − 1)

50 100 150 200 250 300 3500.8

0.85

0.9

0.95

1

1.05

1.1

msec

p.u.

va(n)

vb(n)

vc(n)

Fig. 2. Amplitude estimated by Kalman filter

C. Extension to three-phase systems containing K harmonics

The model used for the phase voltage in the precedingparagraphs is that of a non-distorted sinusoid. In reality thepower-system voltage is always distorted. This distortion isinterpreted by the Kalman filter as a fast fluctuation in theamplitude and phase angle of the complex phase voltages.Mathematically-speaking this is not incorrect. However, thedistortion of the voltages is generally described as the superpo-sition of a fundamental and a number of harmonics at integermultiples of the fundamental frequency. This harmonic modelcan also be used as a basis for a Kalman filter estimating thesymmetrical components. Such a model would not only esti-mate the fundamental complex voltages (either phase voltagesor symmetrical components) but also the complex voltages forthe harmonic components.

Consider a three-phase system containing K harmonics, tobe modelled through the following expression:

va(n) =√

2K∑

k=1

V ka cos(nωk + φk

a)

vb(n) =√

2K∑

k=1

V kb cos(nωk + φk

b ) (10)

vc(n) =√

2K∑

k=1

V kc cos(nωk + φk

c )

Using the similar way as in the previous case, we may definethe symmetrical components for each harmonic k as [7]

V k0

V k1

V k2

=

13

111

1aa2

1a2

a

V ka

V kb

V kc

(11)

After some manipulations using (2), (3), (10) and (11), andapplying the dq-transform to the phase voltages modelled by(10), it follows

vdq(n) =√

2

[K∑

k=1

V k1 ej(k−1)ω0n +

K∑k=1

V k2

∗e−j(k+1)ω0n

](12)

where the relation ωk = kω0 is applied.

50 100 150 200 250 300 35040

42

44

46

48

50

52

54

56

58

60

msec

o deg

rees

va(n)

vb(n)

vc(n)

Fig. 3. Angle phase estimated by Kalman filter

1) Model for the three-phase system containing K harmo-nics: Considering the system described in (10), the state vectorof the Kalman filter contains 2K elements and is defined as

X(n) = [x1 · · ·x2K ]T

=[V 1

1 (V21)

∗ejω0n · · ·V1

kej(K−1)ω0n (V2k)

∗ej(K+1)ω0n

]T

(13)The state equation associated with the system in (10)

becomes

X(n + 1) =

1...0

· · ·. . .· · ·

0...

ej(K+1)ω0

X(n) +

u1

...u2K

n

(14)

and the observation equation for the system in (10) becomes

y(n) =√

2 [1 . . . 1]

x1

...x2K

n

+ v(n) (15)

The negative sequence associated with the fundamental(k=1) and the harmonics (k ∈ [2,K]) can be found byapplying V2

k = xk∗2 e−j2knω0 , k=1, ... ,K.

III. EXPERIMENTS AND RESULTS

The proposed estimation method has been applied to thesampled voltages obtained from measurements in a publicmedium-voltage distribution network. The voltage dip record-ings were measured with a sample rate of fs=4800Hz in a10 kV network. The fundamental frequency during the dipwas slightly different from the ideal 50Hz, and was estimatedto be 49.93 Hz. The conventional estimator [4] was applied,amplitude estimation is shown in Fig.2 and phase angleestimation is in Fig.3 for three voltage phases. The appliedmethod models like (10) where the harmonic number was setto K=3.

Fig. 2 shows the standard dip in voltage amplitude dueto a non-symmetrical fault in the distribution network. Thecorresponding phase angles are shown in Fig. 3 and are inaccordance with theory [1]. The fast fluctuations in magnitudeand phase of the complex voltages are due to the distortion,

50 100 150 200 250 300 3500.64

0.66

0.68

0.7

0.72

0.74

msec

effe

ctiv

e va

lue

(p.u

.)

Fig. 4. Amplitude for the positive sequence |V1|

50 100 150 200 250 300 350−52

−50

−48

−46

msec

Phas

e an

gle(

degr

ees)

Fig. 5. Phase-angle for the positive sequence: V1

the distortion modelled is within the given model variance. Asmentioned before these can be eliminated by using a higherorder Kalman Filter. In this example only harmonics up toorder 3 were considered whereas a large amount of distortionis represented by higher harmonic orders. This can be doneby a straightforward extension of the method: increasing theorder of equations (13) through (15). This is however outsideof the scope of this paper.

The proposed estimation method was applied to the samesamples voltages to obtain the positive and negative sequencevoltages. The proposed estimation method does not estimatethe parameters (magnitudes and phases) of the complex phasevoltages, but directly estimates the complex positive andnegative sequence voltages. In Fig. 4 the amplitude of thepositive sequence voltage is plotted as obtained from theproposed estimation method. The proposed method also givesthe phase angle of the positive-sequence voltage as shown inFig. 5.

Fig. 4 and Fig. 5 show the dip in positive-sequence voltagedue to the fault. This dip is what affects, among others, theenergy transfer to induction motor load.

In Fig. 6 the amplitude of the negative sequence voltage isplotted and in Fig. 7 the negative sequence angle phase as re-sulting from the proposed estimation method. The appearanceof a negative-sequence voltage component during the dip isdue to the non-symmetrical nature of the fault.

50 100 150 200 250 300 3500

0.02

0.04

0.06

0.08

0.1

msec

effe

ctiv

e va

lue

(p.u

.)

Fig. 6. Amplitude for the negative sequence |V2|

50 100 150 200 250 300 350−200

−150

−100

−50

0

50

100

150

200

msec

Phas

e an

gle

(deg

rees

)

Fig. 7. Phase-angle for the negative sequence: V2

IV. CONCLUSIONS

A method is proposed for estimation of the positive andnegative-sequence voltages from the dq voltages. This methodmay give a smaller estimation error than the conventionalmethod using the complex phase voltages as an intermediatestep. The proposed method is computationally more efficientbecause the number of state variables is only 2/3 of that forthe conventional method.

The basic version of the proposed estimation method modelsthe phase voltages as a non-distorted sine wave. The harmonicdistortion of that is always present with measured voltagesleads to fast fluctuations in the estimated voltage. Althoughmathematically not incorrect, distortion is normally modelledthrough a sum of higher-frequency sinusoids superimposedon the fundamental frequency. The basic version has beenextended to include estimators for the harmonic components.

Examples are shown in which the method has been appliedto measured voltages obtained during a fault in a medium-voltage public distribution network.

ACKNOWLEDGEMENT

This project is sponsored by the Goteborg Energi researchfoundation.

REFERENCES

[1] Math H.J. Bollen, Understanding Power Quality Problems: Voltage sagsand interruptions, 1st ed. New York, USA: IEEE-Press, 2000.

[2] Lidong Zhang, Three phase unbalanced of Voltage dips, PhD the-sis. Gothenburg, Sweden: Department of Electric Power Engineering,Chalmers Technical University, 1999.

[3] H.S. Song and K. Nam, Instantaneous phase-angle estimation algorithmunder unbalanced voltage-sag conditions, IEE Proceedings on Genera-tion, Trasmission and Distribution, Vol 147, No 6, pp. 409-415, November2000.

[4] Emmanouil Styvaktakis, Irene Gu and Math H.J. Bollen, Voltage Dipdetection and Power System Transients, Power Engineering SocietySummer Meeting, 2001, Volume: 1 ,pp: 683 -688, 2001.

[5] G Yalcinkaya, M.H.J. Bollen and P.A. Crossley, Characterization ofvoltage sags in industrial distribution systems, IEEE Transactions onIndustry Applications, vol.34, no.4, July 1998, pp.682-688.

[6] Steven M. Kay, Fundamentals of Statistical Signal Processing: Estimationtheory, 1st ed. New Jersey, USA: Prentice Hall, 1993.

[7] Johan Lundquist, On Harmonic Distortion in Power Systems, PhD the-sis. Gothenburg, Sweden: Department of Electric Power Engineering,Chalmers Technical University, 2001.