Absfracl-This paper proposes the use of a complex Kalman filter for the estimation of positive and negative sequences from three phase voltages. A complex ...
Positive and Negative Sequence Estimation for Unbalanced Voltage Dips Rafael A. FloresS, Irene Y.H. Gus and Math H.J. Bollent Dept. of Electric Power Engineering §Dept. of Signals and Systems Chalmers University of Technology Gothenhurg, 41296, Sweden
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Absfracl-This paper proposes the use of a complex Kalman filter for the estimation of positive and negative sequences from three phase voltages. A complex voltage is obtained by applying the a~-transformfollowed by the dq-transform using a rotational operator. The algorithm for three phase voltages containing K harmonics is also given. In the conventional method, estimation of positive and negative sequences is performed through two steps: the magnitude and phase-angle in each individual phase of the voltages are first estimated and the symmetrical componenl transformation is then applied. The proposed method offers a direct estimation of the positive and negative sequences that may reduce the estimation errors. In addition, the proposed method has a reduced compntational cost since the number of state variables is reduced to 2/3 as compared to that in the conventional method. An experiment was performed on measured three phase voltage data. Results have shown that the proposed method offers a good estimation.
1. INTRODUCTION Positive and negative-sequence voltages were originally introduced to speed up calculations involving non-symmetrical faults, and as such are introduced in almost any text book on electric power systems. More recently their application lies in the diagnostics o i power systems during non-symmetrical operation including faults. A method for characterizing the unbalanced voltage dips from the positive and negative sequences has been proposed in [l] and [2]. The two-component method used in [2] is based on the idea that the positive and negativesequence source impedances are equal for static circuits. To achieved reliable and fast characterization and thereby a correct identification of voltage dips, a good estimation of the positive and negative sequence voltages is required. Several methods for estimating positive and negative sequences have been proposed: [3] has proposed the use of a weighted least squares, and [41 has proposed to use of Kalman filtcrs. In [4] the magnitudes and phase-angles were first estimated using three Kalman filters (one for each phase), and the positive and negative sequences were then obtained by using the symmetrical-component transformation, as shown in Fig.l(a). Since the negative sequence voltage and the phaseangle between the positive and negative sequence voltages are sensitive to estimation errors, these methods may suffer from inaccuracies due to the presence of noise and harmonic distortions. Note that the angle between positive and negativesequence voltage is an important parameter in determining the dip type in [Z]. Consequently, it may, in some cases, lead to wrong dip characterization. Therefore, more reliable
methods for estimating positive and negative sequences are desirable. Symmetrical component voltages and currents are also a suitable tool for describing load behavior during voltage dips and other disturbances [5]. In this paper we propose an improved approach for estimating the positive and negative sequences using a complex Kalman filter. Comparing with the conventional 2-step method used in [4], this method offers a direct estimation of the positive and negative sequences, and may reduce the estimation errors. As shown in Fig.l(b), the proposed method simplifies the structure used for the estimation: instead of three Kalman filters, only one Kalman filter is required. Further, the total number of variables to be estimated in Fig.l(b) is reduced to 2/3 of that required in Fig.l(a).
a)
e-jun
b)
Fig. I . (a) Conventional method (b) PN eslimator
In the proposed method, the sampled values of three phase voltages w,(n),ub(n) and uc(n) are &transformed to a complex voltage uafi(n). Then, this complex voltage vao(n) is dq-transformed, resulting in a complex voltage u d g ( n ) ,as it is shown in Fig.l(b). The result contains the embedded positive and negative sequences. The positive and negative sequences are modelled by the state-space equations, and the complex Kalman filter is used to estimate the state vector iteratively. In this paper the method will be used to estimate symmetrical-component voltages. The method can he applied in a similar way to estimate symmetrical-component currents.
11. ESTIMATION OF POSITIVE A N D NEGATIVE SEQUENCES
In this section. the cup-transform and the dq-tra~isfirni(or. the Pal-k's trai~rform)will he described. The complex Kalman filter for estimating the positive and negative scqucnces will also be described, where the three phase voltage is modelled as consisting of voltage of fundamental frequency (50Hz in Europe or 60Hz in USA) under harmonics and noise distortions. A. The cup-tranrforni and the dq-transform
of 2w. Any unbalance in the three phase system will appear as an non-zero rotating V, in the dq-space. The positive and negative sequcnces are estimated separately in the dq-space from (5) as detailed in next section. 6. Estimation of positive arid negative sequences Using state-space modelling, the positive and negative sequence voltages in ( 5 ) can he estimated hy a complex Kalman filter. The state equation and observation equation associated with a Kalman filter can be described as follows
Consider a three-phase system with the following voltages:
.,(n) = vb(n) = uc(lz)
=
X ( n ) = A(n - 1 ) X ( n- 1) + U ( n )
h v acos(wn + &) cos(wii v5K cos(wn
+4b)
Y(n) (1)
+ &)
Where w.(n), u b ( n ) and u J n ) are the sampled phase voltages. Va. Vh and V, are the rms or effective value. &, $6 and dC are the phase-angles, w is the discrete angular frequency, and n is the discrete time index. Define the cup-transform for the three-phase voltages as follows:
2 w ( n )= 3 [va(n)za
+ l J h ( n ) z h f Uc(n)'?c]
(2)
where Za = 1, z6 = eJ% = a and Fc = e - j % = uz a*, and a is a rotational operator representing a rotation over 120'. (2) can be interpreted as the projection of three voltages onto the cup-space. The dq-transform (or Park's transfirm) is then applied as follows vd9(n) = v,D(n)e-iWsn
=
H ( n ) X ( n )+ V ( n )
(6)
where X ( n ) is a complex state vector sized 2x1, A ( n ) is a transition matrix sized 2x2, U ( n ) is a vector containing model noise which is zero-mean with 0; variance, and V ( n ) is a vector containing Observation noise which is zero-mean with ut variance. I ) State-space modelling of the three-phase system: For the three phase system described in (I), define the following state vector
It follows that the state equation of the Kalman filter i n (6) associated with ( 5 ) is
and the observation equalion in (6) becomes
(3)
It should he noted that both v,o(n) and vdq(n)are complex voltages as a function of time. The transformation could be interpreted as a synchronization of the ol,O-space at angular frequency wo, set to be equal to the fundamental voltage frequency w in this paper. The symmetrical component voltages aTe defined from the complex phase voltages V, = Va&, V, = Vbe36. and V, = Kejm, as follows
The
negative
V, = Zje-jz*n, -
sequence
is
found
by
applying
2 ) Kalmari Filter Algorithm: The algorithm for iterativcly estimating the state vector is summarized as follows. a more detailed explanation can be found in [6]. The covariance matrix of model noise is 4 = E[lU(n)12]= u:I, where I , is the identity matrix of order n. The covariance matrix of observation noise is C = = u ~ I The . initial conditions are set as i(Ol0) = [0 01 and P(Ol0) = W I , where W is a relatively large number (chosen as 100 in this paper). VI = Vlej4l and VZ = Vzejm2 are the where V, = lroej@o,Prediction: symmetrical component voltages (zero, positive and negative b(n(n- 1) = A(n)f(n - l / n - 1) sequences). which are defined as complex phasers. Prediction error covafiance matrix: Re-writing (1) in the complex form and combining with (4) b(nln - 1 ) = A(n)P(n- lln - l)A(n)* 4 to calculate vd9(n).it follows, Kalman g$n matrix: K ( n ) = P ( n / n- l ) H T [ H P ( n I n 1)HT c1-1 udq(n)= d5 &*e-jZun) . (5) Filtering: i ( 7 1 ~ 1 1 )= b(n1n - 1) K ( n ) [ y ( n) Hf(nln - I)] For sinusoidal phase voltages as in (l),the first component V, Error covariance matrix a posteriori: in (5) is a complex constant number, while the 2nd component P ( l l l 1 1 ) = [I" - K(n)H]P(71111 1) V, in (5) is a complex number rotating with an angular speed
E[IV(n)J,,]
+
(5 +
~
+
~
2499
+
,
hi
SR I
os
56
...,
I
... ,,
.
~
Fig. 3.
C. Exierision to three-phase systems coniaining K harmonics The model used for the phase voltage in the preceding paragraphs is that of a non-distoncd sinusoid. In reality the power-system voltage is always distorted. This distortion is interpreted by the Kalman filter as a fast fluctuation in the amplitude and phase angle of the complex phase voltages. Mathematically-speaking this is not incorrect. However, the distortion of the voltages is generally described as the superposition of a fundamental and a number of harmonics at integer multiples of the fundamental frequency. This harmonic model can also be used as a basis for a Kalman filter estimating the symmetrical components. Such a model would not only estimate the fundamental complex voltages (either phase voltages or symmetrical components) but also the complex voltages for the harmonic components. Consider a three-phase system containing K harmonics, to be modelled through the following expression:
Angle phase estimated by Kdmm filter
I ) Model f o r the ihree-phase system containing K harmonics: Considering the system described in (IO). the state vector of the Kalman filter contains 2K elements and is defined as X ( n ) = [q
t ,
.2*K]
T
The state equation associated with the system in (10) becomes
X(n+l)=
[
::;
1 0 ._.
0 eJ(lC+l)WU
and the observation equation for the system in (IO) becomes
y(n)=&[l
... l]
[ I' ]
+U(.)
(15)
Z2h'
h'
Vb(71)
=
fix vt fix
cos(nWk
k= 1 K
Uc(n)
=
:1 '
+ 4;)
(10)
111. EXPERIMENTS A N D RESULTS
+ 6;)
cos(~Wk
k=l
Using the similar way as in the previous case, we may define the symmetrical components for each harmonic k as [7]
After some manipulations using ( 2 ) , ( 3 ) , (IO) and ( I I ) , and applying the dq-transform to the phase voltages modelled by (IO), it follows K
+xg*e-j(k+l)won k=
i=1
where the relation
Wk =
1
kwo is applied.
The negative sequence associated wilh the fundamental (k=l) and the harmonics ( k E [ 2 > K ]can ) he found by = z : * e - l z k n w u , k = ~ ... , ,I
