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Application of Parameter Estimation Theory to Power Systems Analysis
Cop yrig ht © I L\ C Id e ntifica tion a nd S'·ste m Para mete r Est im atio n . Beij ing. PRC f
APPLICATION OF PARAMETER ESTIMATION THEORY TO POWER SYSTEMS ANALYSIS S. Bergman* and S. Ljung** *ASIC Design Cnl ter, Asea B rown B m'eri Corporate R esm rrh, \' iistenls. Sll'erlm ** AsNI B rowlI Bm'el1 R elays AB. Viistnas, Sll'edell
ABSTRACT. Models for protection of electric transmission lines are studied. When a fault occurs in a steady-state system, the signals are influenced more or less seriously, depending on the properties of the fault . The crucial problem should be to analyse the individual signals in order to determine when the event has occurred, to classify it, to perform phase selection and in some cases, to determine the location of it. The systems are inherently non-linear, but they may be approximated with sufficient accuracy by time-series or Fourier decomposition. This implies that the system is modelled as a black-box, which has a unique structure, but where the parameters may be time dependent. The parameters are determined by applying a recursive least-squares estimation algori thms. In conjunction to this algorithm an accurate frequen c y estimation procedure is attached. Detection of abrupt changes in the system are used t o establish logic for protection schemes and for reinitializing the parameter estimation routine . The algorithms are tested by simulations.
KEYWORDS . Adaptive systems; Fourier analysis; frequency control; harmonic identification; parameter estimation; power transmission; signal processing.
INTRODUCTION
analysis;
crucial problem should be to analyse the individual s i gnals in order to determine when the abrupt event has occurred, to classify it, to perform phase selection and in some cases, to determine the location of the fault. A general model structure for a protection scheme should thus be built up around an initial signal extraction, followed by a logic part, where discrimination of the signals after the event is performed.
At the present time, a majority of new approaches for the protection of electrical power transmission lines are based on digital techniques. Due to this fact, more &ophisticated mathematical tools can be utilized when implementing protective schemes and hence a more refined treatment of the measured signals compared to entirely analogue implementations is achievable. Mostly well-known models of the system to be studied are applied. Usually the parameters of the model structure, such as the line impedance per unit length, are determined In advance from empirical considerations or theoretical reasoning based on conventional relations from circuit theory. Because of this reason, the models are kept constant once the relay has been Initialized. However, In reality the system dynamics may exhibit a slowly time Invariant behaviour in such a way that the parameters of a given representation may vary with the properties of the surrounding environment, e.g. the prevailing wheather conditions or structural changes in the network . There are thus advantages in utilizing techniques, which take these phenomena into consideration by consecutively adjusting the parameters of the model In relation to the changing conditions. This can be accomplished In many ways. Within the signal processing literature the problem is called identification. An adaptive regulator, for example, necessarily contains such a feature for tracking variations in the system to be controlled. The main idea should here be to decide upon a model structure, which uniquely define& the system except for a fini te set of parameters. Small time variations of the parameters are modelled as drifts of the elements of that set, and the values are thus allowed to continuously belonging to a severed part of the parameter space rather than prescribing a fixed number of discrete values.
:K YI/1l!fJ!J!JJJJ ;)
1 0',~
2000
3000
4000
::~ o
1000
2000 .
3000
Fig. 1 Voltages and currents for an RN-fault The fundamental idea of a microprocessor protection relay can be based on applying modern concepts from signal theory, modelling of dynamical systems and numerical analysis. Studies along these lines have been performed e.g. by Girgis et al (1985), who apply recursive Kalman filteri ng techniques for estimating the unknown parameters. Related approaches have been investigated by Girgis et al (1985) and by Sachdev et al (1979), where adaptive parameter estimation techniques are considered, which are based on a fixed size moving time window and where the pseudo-inverse is applied to the solution of the subsequent system of equations, defining the Fourier coefficients. On the other hand, adaptation in each sample can be accomplished by the celebrated technique of exponential forgetting. The main feature of the method amounts to discard such information that lies sufficiently far apart from the current sample instant. In this way, slow variations of the parameters are treated. An abrupt change of the system dynamics
When a fault, e.g. a shortcircuit, occurs in a steady state electrical system, the signals are influenced more or less seriously, depending on the properties of the fault. Figure 1 depicts a typical behaviour of the voltages and currents around the time of a fault. The
1209
S. lkrgman and S. Lillng
1210
inevitably leads to greater changes of the correct parameters. This implies that a method for adjusting the estimator to fast changes must be considered as well. Many algorithms have been investigated in this respect, e.g. by Bassevllle (1981>, Phadke et al (1985) and Isaksson (1988). Algorithms for adaptive systems have been extensively described, e.g. by Ljung et al (1983), wi th respect to speed, convergence and numerical behaviour. Robust and reliable on-line algori thms have been developed through the years. Recursive least squares algorithms have proven to be versatile tools for adaptive applications of changing industrial processes (cf. Astr~m (1987». Due to this fact, an appropriate approach would be to study the applicability of recursive techniques for different protective relay schemes. The main issue of the present paper is devoted to that task. It will especiall y be accomplished by studying some models for a power line system and to show how parameter estimation algorithms can be utilized to extract optimal adjustments of these models according a given loss function.
MODEL OF THE SYSTEM Parameter estimation methods themselves do not prescribe a structure of the system to be analyzed. They are rather tools for performing a best fit of data to a member of a given model set. The set is defined in terms of some ini tiall y unknown parameters. Usually physical or empirical observations are used to model the power line. In the literature many models are fundamentally based on a description of the system according to a set of differential equations: (1)
y(t) = g(x,u,t)
(2 )
Sufficiently close to the individual singular pOints of the function f, linear approximations of (1) and (2) apply. Different approaches may be used to describe the linearized equations. The complete system according to (1) can eventually be treated by combining the linearized models around the singular points. The following set of difference equations describes the corresponding discrete time representation:
x(t+ll
Cx(t) + Du(t)
(3)
y(t)
Ex(t) + Fu(t)
(4)
The transfer function of the syste_"t is given in terms of the matrices in (3) and (4) by q y(t)=y(t-T), where T is the sampling time, defining the shift operator:
(8)
Another approach would be to express the solution of (ll as a linear combination of some functions, e.g. trigonometric. In the sequel of the section, some approaches are treated in the scope of the described techniques . In order to show the applicability of parameter estimation approaches, we here study three frequentl y occurring models for power line applications.
PI Equivalent Circuit The approach described here implies that a short-circuited transmission line can be represented by a lumped impedance model with a resistance and an inductance in parallel with a capacitance. The resistance of the capacitive sub-line is neglected. The relation between the current and the voltage at the end of the line is described by a second order differential equation as: u = Ri + Li' - C(Ru' + Lu")
(9)
where Rand L are the series resistance and inductance, respectively, whereas C denotes the shunt capacitance. With this technique, series capacitances may also be considered by extending the model (9). However, then the order of the system increases with one unit. Equation (9) should be regarded as a combination of a set of differential equations. An assumption of a uniform distribution of the impedance makes it possible to arrive at one single scalar equation expressed in lumped system parameters. The model has been extensively applied in the case of known values of the impedance quantities. As initial values such a 'guess' may do. On the other hand, for long-run estimation experiments the dynamics of the system can vary to such an extent that the model parameters should be be successi vel y adjusted. Equation (9) may be discretized by applying a numerical scheme directly on the deri vati ves. However, due to more refined numerical behaviour, it is advantageous to firstly integrate the equation analytically and then using some quadrature formula . Here two cases should be separated. The first allows the presence of a shunt capacitance. Integrating equation (9) twice yields the integral equation:
(5)
t
t
This implies that the output signal can be equivalently expressed with the difference equation:
J(t-o)u(O)dO
A(q -1)y(t) = B(q -1)u(t) + C(q -1)e(t)
to
(6)
or in a more compact form: y(t) = 9 T
t
RJ(t-O)i(O)dO + L[Ji(O)do-
to
to
-(t-tO)i(t )] - C{L(U(t)-U(t » O o (7)
t
where E(t) is the noise function with a given statistical distribution. The parameters of the model are collected in 9 and the regression vector, q>( t), contains the signal values:
eT
=[a
q> T(t)
and
a ···a b b .. ·b ] n 1 2 n 1 2
= [y(t) y(t-ll. .. y(t-n ) u(t) u(t-1> ... u(t-n )] a b
( 10)
-(t-tO)[i(tOl+Lu'(t )] + Ji(O)dO} O
to In the second case C equals zero. Hence, equation (9) should be integrated only once. This results in the following expression:
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Parameter Estimation TheorY. and PO\\T r SYstems .-\nah·sis . . A straightforward elaboration of (14) gives:
t
+ L- 1 Sexp(-R(t-O)/L)u(O)dO
(11 ) ( 16)
to where After discretization of (10) difference equations arise:
and
(Il>,
the
following
u(t) + alu(t-l> + a u(t-2) = bOHll + bIHt-l> 2
(12)
Hll + Cl Ht-l> = dOu(ll + d u(t-l> l
( 13)
The parameters (al , a2,bO,bl,cI,dO) are usually unknown In advance. Given sampled values of the current and voltage, the parameters should thus be determined as best as possible according to the given models (12) and (13)' Apparently, the models have the same structure as described by (7)-(8) . The properties of the signals in (12)-(13) corresponding to the B-polynomial are very important whell applying an ARMA-model (ct. Astrom (1987». The signal should with necessity be persistently exciting of order deg(A) + deg
( 17)
"Mcos~M,aMsin~M)
( 18)
The vector 8 thus contains information about the system, whereas
= arg (8
12
(19)
,
(20)
_ , 8 / 2 2j l
A scheme for decision logic may be partly based on these quantities as will be explained in a later section.
ESTIMATION ALGORITHMS General Fourier Based Model sufficient information about the system to be protected is available, the crucial relay problem can be transferred to applying suitable methods from the signal processing theory. For this purpose, a black-box model describes the system according to the prev ious section. The model is defined by analytical expressions, which are given as function of some parameters to be determined. Numerical values of the parameters can be extracted from measured data. Since the system in steady-state conditions may be subjected to different types of variations, adaptive estimation of the unknown parameters is a necessity. Also methods for detecting abrupt changes in connection with events must be considered as well as procedures for frequency estimation.
If
For a harmonic signal a more general approach would be to base the signal structure on Fourier decomposition. This is convenient since the voltages and currents of the system are usually periodic in stationari ty. Also the transient behaviour can then be modelled in the same manner with sufficient accuracy by assuming the envelope of the signal to be exponentially decaying immediately after an abrupt event in the system. The following relation thus applies: 11
y(t)
=
aOexp(-~ot)
+
~a.sin(w . t+~ . )
1 J
J
( 14 )
J
However, the exponential term usually peters out after some cycles, which implies that the stationary behaviour should be described by the simplified model: N
y(t)
= a
O
+
~a . sin(w , t+~ . )
1 J
J
( 15)
J
The model_fhould be changed when the current ~~e ex~eds ~ 1nl'), for some small value of I') (0: 10 10 ). Relation (IS) is achieved by setting ~O equal to zero in (14)' The coefficients of the expressions are identical except for the mean values. In model (14) M should be chosen greater than N in model (IS). Theoretically M may take infinitely large values, but due to the necessity to keep the time complexity below a moderate level in real time applications, M should be restricted to the first superharmonic components. Of course, the accuracy of the estimated parameters depends on the truncation of the Fourier series. This is not a serious restriction since the power line signals have a limited amount of harmonic components also after a major change of the dynamics. Subharmonic components can be incorporated in the models (14) and (15) analogously. Furthermore, band-pass prefiltering may be utilized to extract suitable information from the real-time signal.
Parameter Estimation The problem of determining the system parameters from successively measured values of y(ll and u(t> in (7) is usually performed according to the least squares method, (Ljung et ai, (1983», i.e. the parameters are c hosen to minimize the criterion: (2l>
" Elll=y(ll-y(t).
where minimization is:
" = arg min 8(1l
(V
The estimate according to:
The
general
solution
of
(22)
(8,t» N
of
8(t)
the
can
be
recursively
updated
9(t) = S(t- l> T
(23)
" = 8(t-l> " 8(t> + L(t)e:(t)
(24)
where the gain vector, following algorithm: r(t) = P(t-l>
L(t>,
is computed from
the
(25)
S. Bergman and S. Ljung
1212
4Y2 j !Y 1Y3 - Y;j 2 y +y +2Y * Y+Y 3-2Y *Y 3 1 1 2 2 !
(26) L = P(t-1>cp(t)/d(t)
(27)
PIt) = (P{t-1>-r(t)L T) lA
(28)
P
" =9 = <1/6>1, 9(Q) 0
(29)
The implementation of the algorithm can easily be performed since it bases entirely on matrix and vector computations (cf. Bergman et al (1988». The quantity 6 is chosen sufficiently small in order to avoid problems due to singular matrices and to allow a fast adaptation immediately after an initialization. Usually the restart of the parameter estimation procedure should be performed in such a way that the updating of the P(t)-matrix takes its initial value from (29). This recursive algorithm applies when estimating the coefficients of (14)-( 15) since the magnitude of the elements of the regression vector are then limited to 1However, for lhe ARMA-models (12) and (13), the regressors may lake values of arbitrary magnilude. Due to this fact, the covariance matrix should ~ decomposed inlo its squ~re root components, P=UDU • The initial setling of 9, 9 can either be chosen 0 arbitrarily (e.g. all components are equal to zero) or empirical model values may be initially imposed unlil the updaling of lhe covariance matrix has slabilized. After a restart, the eslimaled parameter vector prior to lhe evenl is a better ini tializalion strategy lhan equalizing the entries to zero. The factor of exponential forgetting causes an adjustment of the algorithm to slow variations of the dynamics. A smaller value of A will give a more alert algorithm in such a way that faster tracking is made possible. However, the factor may not be chosen too small since that would cause the quanti ties of the algori thm to take too large numerical values, a fact that necessitates the incorporation of a method, which can manipulate abrupt changes in the system. Once a major change has been detected, a reintialization of the updating of the covariance matrix have to be performed. The same event should also be utilized as indication of an abnormal situation in the identified system and classification can be accomplished. Methods for detecting abrupt changes are described by Basseville (1981>. In addition, the loss function VN(8, t) can be used wi th satisfactory results in most situations. A discussion of this possibility is performed in the next section.
Estimation of Fundamental Frequency From (18) it comes out that an accurate estimate of the angular frequency of the studied signal is needed. Because of this reason, a frequency estimation algorithm is appended to the previous scheme. As an alternative to the approach in the previous section, we here describe a procedure for estimating the fundamental frequency of a given harmonic signal. The fundamental component of the signal models of (14)-<15) is in stationarity y (t)=a' sin(w' t). This expression contains the unknown ~nti ties a, wand t. By sampling three consecuti ve val ues y l' Y2 and y 3' wi th time separation h, i.e. : Yl = Ye(t-h),
Y2 = Ye(t),
Y3 = Ye(t+h)
(30)
makes it possible to estimate a and w a'70rding to the following technique. By introducing b=a and applying straightforward trigonometrical relations it turns out that:
b
(31)
The last equality in (31) may be favourable if the signal can take values with large magnitudes. Equation (31) is taken as the estimate of b. Gi ven the sampled signal values and the estimate of b, the estimate of the angular frequency, x(t), now turns out to be: z = b-1(Y3(b_y~)1I2_y~(b_y;)II2) = b-l{Y2(b-y~)1I2_Yl(b-y~)112) x(t) = h -larcsin(z)
(32) (33)
The input parameters for computing the angular frequency are y , y , YJ and b. Since the amplitude of the signal cad b~ obtained from the parameter estimation procedure, the computation of b in <31> may be eliminated, which implies that only equations (32)-(33) for computing the frequency should be added to the parameter estimation scheme. Equivalently, by elaborating the definitions of Yl' Y2 and Y3 in (30) it comes out that: 2112 . y 1 = Y2coswh + (b-y 2) slOwh
(34)
2 1/2 . Y3 = Y2coswh - (b- Y2) slOwh
(35)
Solving for w yields the following expression as the instantaneous estimate of the angular frequency: x(t) = arccos«Yl+Y3)/(2Y2»
(36)
If the frequency estimator is coupled to the estimation
of the Fourier coefficients, the filtered signal can be used in (36) . In order to suppress too large deviations w-om the actual frequency, the moving average value, wIt), of the instantaneously computed estimate of the frequency should be considered as: (37)
"
Wo
(38)
where w
is an initial guess of the angular frequency.
w
Normall~ it is chosen to correspond to the nominal frequency of the system. Additionally, a further precaution should be considered. If the instantaneously computed estimate of the frequency differs too much from the moving average value, it should be ignored. There are also advantages in ignoring such signal values, that differ sufficiently much from the previous one. In order to decrease the number of false estimates, the input to the frequency estim~tion algorithm should thus be the filtered values y(t), defined in (23 ).
DECISION LOGIC The decision logic can be separated into two different parts, a transient one and a stationary. Immediately after a fault, there should be some quantity which rapidly detects the event. The quantity can be based on the fact that the model parameters prior to an abrupt change of the power system dynamics are not valid after the inception, which implies the residual functions E(t) of the currents and voltages to have large magni tudes. Since the algori thm tracks changes coming from the new situation, it will adapt to the best fit for new system dynamics. Meanwhile the residuals thus tend to decay to a level they had before the event. In most cases the sign of the residuals also contains information about the direction to the fault from the measuring unit. Because of these facts the deci sion logic is based on a combination of the magnitude of E(t), the cost function V (9,t) and the N current estimated signal values . In this respect, the
Parameter Estimation Theory and Power Systems Anal ysis currents are most appropriate to be studied since variations of the current signal in the faulty phase are more pronounced than those of the corresponding voltage signal. As a suggestion of an appropriate cost function we can use (21) with VN(S,O)=O. The updating of the function is performed for each signal (totally 6 for a three phase system). How fast V (S,t) r~nds to changes of £(t) depends on the magnit'ude of £ (t) as well as A. In order to detect high impedance faults, an additional checking of the level of I Elt) I is necessary. This should be performed in combination with a study of the loss function, which may be performed In many different ways. However, generally It is suitable to apply statistical analysis to achieve a more reliable extraction of information about trends that occur. There are numerous facilities for this purpose. Just to mention some of them: Hinkley detector, hypotheses testing and Magill predictor. Further information on this subject is given by Basseville (1981). A further measure to increase the sensitivity for high impedance fault detection is to estimate the Fourier coefficients of the symmetrical components rather than of the phase quantities. The approach described above can be used for investigating transient behaviour. For slowly occurring faults as well as for cases when the algorithm is initially applied to estimating the dynamics of a faulty system, the transients may not be used any longer for the analysis. In addition to the transient treatment, a stationary analysis should thus be applied in order to detect slow variations, which are not considered by studying the transients. At this, impedance anai ysis or treatment of symmetrical components of the individual voltages and currents can serve as discriminating quantities. As a further feature of the described considerations serves the possibility to separate conventional faults from special situations, e.g. (dis)connections of lines. This separation can be accomplished by Investigating the amplitudes and phases of the symmetrical currents. The analysis of the impedance plane can be based on an appropriate demarcation of the plane into a domain, where the Impedance vector lies in case of a fault. Figure 2 elucidates the signai flow between the parameter estimation routine and the different blocks of the logic procedure. The output from the parameter estimation routine are the instantaneous residuals, £(t), define4. in (7) and the estimated Fourier coefficients, S(t). The loglcai scheme can be separated In three different states as depicted in figure 3. State 1 represents normal situations in steady state when no fauits are present in the system. State 2 corresponds to situations when speciai conditions has caused the abrupt change, e.g. a connection or disconnection of a line, whereas the algorithm is in state 3 when a conventional fault has been detected. Transitions between the different states are determined in the transient and stationary analysis with the aid of the parameter vector, the residuals of the individual signals and the loss function.
Fig. 3 The states of the logical scheme
RESULTS The described estimation algorithms have been verified by simulation studies partly in situations where the signals are directly generated In the computer programs and partly to analyse signals coming from systems as defined by the ElectroMagnetic Transients Program (EMTP). This program is utilized for the simulation of electromagnetic transients and related phenomena in power networks. Time Series Analysis In order to investigate the model (11) for a transmission line, the current is supposed to have the structure:
PHASE SEl. ••
. . .LYSIS
DETERS.
aESlDUALS
OF"
DIaECTIOIf
Fig. 2 The complete protective scheme
(39)
I
where w=2nf=100n and the distribution of the noise was ReW,O.I). The step size was chosen to be 0.1 millisecond and the estimates of the R- and L-quantities are monitored for each 10 sample. The ini tial parameters of A-he estimation algori thm are: P(0l=1000' I, ,,=0.98 and 8W)=0. In figure 4 two examples are illustrated, where the correct parameters are (L=O.OI, R=100l and (L=O.l, R=10), respectively. As comes out from the figure the estimated quantities converge rapidly to the exact values.
·. DO..[d 0 . 101
' .114
....
t.
0.'
10 .
O.
le.
0
fiO
"·.D '°. [J •• .
...
•
t
IWI
Fig. 4 Estimated R- and L-values Harmonic Signal Analysis The Fourier-series model (14) has been used to estimate properties of signals as directly generated in the simulation program on one hand and as generated by the EMTP program on the other. In the first case the signal was given by:
0.5 + 1 . 5coswt + 2.5sinwt +
yet> nAIfSIDfT
12 13
+
1.75cos2wt
+
0.75 sin2wt
( 40)
where the noise was NW,0.25) and w=100n. The UD-algorithm was applied with 0(0)=1000 ' I and ,,=0.98. The sampling time was 1 mill i second. According to figure 5, the estimates of the parameters are accurate after 15 samples. The rate of convergence depends among other things on the noise function and the forgetting factor . Figure 1 depicts the currents and voltages of a three-phase transmission line including an RN-fault as simulated with the EMTP program. Figure 6 represents the same signals after first having passed through a low-pass analogue fll ter wi th a
S. Bergman and S. Ljung
1214
cut-off frequency of 1000 radians/second in order to eliminate the transient effects. Noise with distribution N(O,O.OO6) was then imposed on the current signals and N(0,12.5) on the voltage signals. The fundamental frequency in the system was SO Hz and the sampling frequency 4 kHz. Figure 7 shows the estimated parameters when applying the UP-balled estimation algorithm with D(o)=l00' I, ~=O.98 and 8(0)=0. Also in this case the estimated coefficients converge rapidly to their correct values. By comparing the curves in figure 7, the necessity to reinitialize the estimation algorithm when a fault has occurred is immediate. However, also when reinitializing it turns out that the estimates are oscillating after the fault . This depends on the fact that the signal contains harmonic components, which are not modelled.
Fig. 5 Estimated Fourier coefficients for (40)
Fig. 6 Filtered currents and voltages
f\...
CONCLUSIONS It has been descri bed how an arbitrary dynamical system may be modelled as a non-linear differential equation, which can be discretized to yield a difference equation. When the signals are known to be harmonic, Fourier series analysis can be applied to model the system. In common for tbe models is the possibility to use them as black-box representations of the treated system. The main problem tbus consists of determining a number of quantities connected to tbe chosen structures. Recursive parameter estimation algorithms have been utilized to extract information from measured values of important signals. A new algorithm for determining tbe fundamental angular frequency of a harmonic signal has been described in combination to the estimation scheme. The algorithm utilizes the filtered signals coming from the Fourier filter to perform an exact computation of the frequency. Requirements for an additional digital filter are tben removed. Logic scbemes for protective relaying can also be connected to the parameter estimation routine. The procedure is fed with the instantaneous residuals, tbe loss function and the estimated Fourier coefficients from the parameter estimation algorithm. Based on this information, tbe logic algorithm performs transient as well as stationary analysis. The model. are verified by simulation results. A complete relay protection scheme can thus be be tailored by utilizing quantities from the complete procedure. The transient analysis also applies to determining when to reinitialize the parameter estimator itself. REFERENCES
f{
1\
Fig. 8 Estimated fundamental frequency
Astr5m, K.J., Adaptive F_dback Control, Proceedings of the IEEE, Vol. 75, No. 2 (1987)
.
.M
\-.
-. -
Fig. 7 Estimated Fourier coefficients (R-current> Frequency Estimation By applying the frequency algorithm together with the estimation algorithm (Initialization as in previous section), the fundamental frequency was estimated for the signal (R-phase current> in figure 6. The estimated frequency has been plotted in figure 8 for some different initial values of the frequency and for two different values of ~, 0.80 and 0.95, respectively. The convergence is evidently faster for a small value of ~ whereas the algorithm becomes more sensitive to fluctuations of the filtered signal. By studying the different currents and voltages in figure 6 It must be concluded that the frequency should be estimated by applying the algorithm to the voltages rather than to the currents since they are less influenced by a change of the system. Sometimes the change depends on that a line is disconnected, which results in a signal with zero amplitude. This implies that a logic scheme should be implemented for deciding whicb voltage can be used for estimating the frequency. The signal is naturally chosen as the first in the sequence (A,B,C) that has a non-vanisbing amplitude.
Basseville, M., Edge Detection Using Sequential Methods for Change in Level --- Part II: Sequential Detection of Change in Mean, IEEE Trans. on ASSP. Vol. ASSP-29 No. 1 (1981) Bergman, S. and Ljung, S., Implementation of Parameter Estimation Algorithms, IFAC WorkshoD on Adapti ve Control. Newcastle, Australia (1988) Girgis, A.A. and Brown, R.G., Adaptive Kalman Filtering in Computer Relaying: Fault Classification Using Voltage Models, IEEE Trans. on PAS. Vol. PAS-I04, No. 5 (1985) Isaksson, A., Di gital Protective Relaying Througb Recursive Least Squares Identification; Submitted for publication to lEE Proceedings. Part C (1988) Ljung, L. and SMerstr5m, T., Theory and Practice of Recursive Identification, Cambridge, Mass.: MIT Pre.. (1983) Phadke, A. G. and Lu Jihuang, A New Computer Based Integrated Distance Relay for Parallel Transmission Lines, IEEE Trans. on PAS. Vol. PAS-I04, No. 2 (1985) Sachdev, M.S. and Baribeau, M.A., A New Algorithm for Digital Impedance Relays, IEEE Tuns. on PAS, Vol. P AS-98, No. 6 (1979)
APPLICATION OF PARAMETER ESTIMATION THEORY TO POWER SYSTEMS ANALYSIS S. Bergman* and S. Ljung** *ASIC Design Cnl ter, Asea B rown B m'eri Corporate R esm rrh, \' iistenls. Sll'erlm ** AsNI B rowlI Bm'el1 R elays AB. Viistnas, Sll'edell
ABSTRACT. Models for protection of electric transmission lines are studied. When a fault occurs in a steady-state system, the signals are influenced more or less seriously, depending on the properties of the fault . The crucial problem should be to analyse the individual signals in order to determine when the event has occurred, to classify it, to perform phase selection and in some cases, to determine the location of it. The systems are inherently non-linear, but they may be approximated with sufficient accuracy by time-series or Fourier decomposition. This implies that the system is modelled as a black-box, which has a unique structure, but where the parameters may be time dependent. The parameters are determined by applying a recursive least-squares estimation algori thms. In conjunction to this algorithm an accurate frequen c y estimation procedure is attached. Detection of abrupt changes in the system are used t o establish logic for protection schemes and for reinitializing the parameter estimation routine . The algorithms are tested by simulations.
KEYWORDS . Adaptive systems; Fourier analysis; frequency control; harmonic identification; parameter estimation; power transmission; signal processing.
INTRODUCTION
analysis;
crucial problem should be to analyse the individual s i gnals in order to determine when the abrupt event has occurred, to classify it, to perform phase selection and in some cases, to determine the location of the fault. A general model structure for a protection scheme should thus be built up around an initial signal extraction, followed by a logic part, where discrimination of the signals after the event is performed.
At the present time, a majority of new approaches for the protection of electrical power transmission lines are based on digital techniques. Due to this fact, more &ophisticated mathematical tools can be utilized when implementing protective schemes and hence a more refined treatment of the measured signals compared to entirely analogue implementations is achievable. Mostly well-known models of the system to be studied are applied. Usually the parameters of the model structure, such as the line impedance per unit length, are determined In advance from empirical considerations or theoretical reasoning based on conventional relations from circuit theory. Because of this reason, the models are kept constant once the relay has been Initialized. However, In reality the system dynamics may exhibit a slowly time Invariant behaviour in such a way that the parameters of a given representation may vary with the properties of the surrounding environment, e.g. the prevailing wheather conditions or structural changes in the network . There are thus advantages in utilizing techniques, which take these phenomena into consideration by consecutively adjusting the parameters of the model In relation to the changing conditions. This can be accomplished In many ways. Within the signal processing literature the problem is called identification. An adaptive regulator, for example, necessarily contains such a feature for tracking variations in the system to be controlled. The main idea should here be to decide upon a model structure, which uniquely define& the system except for a fini te set of parameters. Small time variations of the parameters are modelled as drifts of the elements of that set, and the values are thus allowed to continuously belonging to a severed part of the parameter space rather than prescribing a fixed number of discrete values.
:K YI/1l!fJ!J!JJJJ ;)
1 0',~
2000
3000
4000
::~ o
1000
2000 .
3000
Fig. 1 Voltages and currents for an RN-fault The fundamental idea of a microprocessor protection relay can be based on applying modern concepts from signal theory, modelling of dynamical systems and numerical analysis. Studies along these lines have been performed e.g. by Girgis et al (1985), who apply recursive Kalman filteri ng techniques for estimating the unknown parameters. Related approaches have been investigated by Girgis et al (1985) and by Sachdev et al (1979), where adaptive parameter estimation techniques are considered, which are based on a fixed size moving time window and where the pseudo-inverse is applied to the solution of the subsequent system of equations, defining the Fourier coefficients. On the other hand, adaptation in each sample can be accomplished by the celebrated technique of exponential forgetting. The main feature of the method amounts to discard such information that lies sufficiently far apart from the current sample instant. In this way, slow variations of the parameters are treated. An abrupt change of the system dynamics
When a fault, e.g. a shortcircuit, occurs in a steady state electrical system, the signals are influenced more or less seriously, depending on the properties of the fault. Figure 1 depicts a typical behaviour of the voltages and currents around the time of a fault. The
1209
S. lkrgman and S. Lillng
1210
inevitably leads to greater changes of the correct parameters. This implies that a method for adjusting the estimator to fast changes must be considered as well. Many algorithms have been investigated in this respect, e.g. by Bassevllle (1981>, Phadke et al (1985) and Isaksson (1988). Algorithms for adaptive systems have been extensively described, e.g. by Ljung et al (1983), wi th respect to speed, convergence and numerical behaviour. Robust and reliable on-line algori thms have been developed through the years. Recursive least squares algorithms have proven to be versatile tools for adaptive applications of changing industrial processes (cf. Astr~m (1987». Due to this fact, an appropriate approach would be to study the applicability of recursive techniques for different protective relay schemes. The main issue of the present paper is devoted to that task. It will especiall y be accomplished by studying some models for a power line system and to show how parameter estimation algorithms can be utilized to extract optimal adjustments of these models according a given loss function.
MODEL OF THE SYSTEM Parameter estimation methods themselves do not prescribe a structure of the system to be analyzed. They are rather tools for performing a best fit of data to a member of a given model set. The set is defined in terms of some ini tiall y unknown parameters. Usually physical or empirical observations are used to model the power line. In the literature many models are fundamentally based on a description of the system according to a set of differential equations: (1)
y(t) = g(x,u,t)
(2 )
Sufficiently close to the individual singular pOints of the function f, linear approximations of (1) and (2) apply. Different approaches may be used to describe the linearized equations. The complete system according to (1) can eventually be treated by combining the linearized models around the singular points. The following set of difference equations describes the corresponding discrete time representation:
x(t+ll
Cx(t) + Du(t)
(3)
y(t)
Ex(t) + Fu(t)
(4)
The transfer function of the syste_"t is given in terms of the matrices in (3) and (4) by q y(t)=y(t-T), where T is the sampling time, defining the shift operator:
(8)
Another approach would be to express the solution of (ll as a linear combination of some functions, e.g. trigonometric. In the sequel of the section, some approaches are treated in the scope of the described techniques . In order to show the applicability of parameter estimation approaches, we here study three frequentl y occurring models for power line applications.
PI Equivalent Circuit The approach described here implies that a short-circuited transmission line can be represented by a lumped impedance model with a resistance and an inductance in parallel with a capacitance. The resistance of the capacitive sub-line is neglected. The relation between the current and the voltage at the end of the line is described by a second order differential equation as: u = Ri + Li' - C(Ru' + Lu")
(9)
where Rand L are the series resistance and inductance, respectively, whereas C denotes the shunt capacitance. With this technique, series capacitances may also be considered by extending the model (9). However, then the order of the system increases with one unit. Equation (9) should be regarded as a combination of a set of differential equations. An assumption of a uniform distribution of the impedance makes it possible to arrive at one single scalar equation expressed in lumped system parameters. The model has been extensively applied in the case of known values of the impedance quantities. As initial values such a 'guess' may do. On the other hand, for long-run estimation experiments the dynamics of the system can vary to such an extent that the model parameters should be be successi vel y adjusted. Equation (9) may be discretized by applying a numerical scheme directly on the deri vati ves. However, due to more refined numerical behaviour, it is advantageous to firstly integrate the equation analytically and then using some quadrature formula . Here two cases should be separated. The first allows the presence of a shunt capacitance. Integrating equation (9) twice yields the integral equation:
(5)
t
t
This implies that the output signal can be equivalently expressed with the difference equation:
J(t-o)u(O)dO
A(q -1)y(t) = B(q -1)u(t) + C(q -1)e(t)
to
(6)
or in a more compact form: y(t) = 9 T
t
RJ(t-O)i(O)dO + L[Ji(O)do-
to
to
-(t-tO)i(t )] - C{L(U(t)-U(t » O o (7)
t
where E(t) is the noise function with a given statistical distribution. The parameters of the model are collected in 9 and the regression vector, q>( t), contains the signal values:
eT
=[a
q> T(t)
and
a ···a b b .. ·b ] n 1 2 n 1 2
= [y(t) y(t-ll. .. y(t-n ) u(t) u(t-1> ... u(t-n )] a b
( 10)
-(t-tO)[i(tOl+Lu'(t )] + Ji(O)dO} O
to In the second case C equals zero. Hence, equation (9) should be integrated only once. This results in the following expression:
1211
Parameter Estimation TheorY. and PO\\T r SYstems .-\nah·sis . . A straightforward elaboration of (14) gives:
t
+ L- 1 Sexp(-R(t-O)/L)u(O)dO
(11 ) ( 16)
to where After discretization of (10) difference equations arise:
and
(Il>,
the
following
u(t) + alu(t-l> + a u(t-2) = bOHll + bIHt-l> 2
(12)
Hll + Cl Ht-l> = dOu(ll + d u(t-l> l
( 13)
The parameters (al , a2,bO,bl,cI,dO) are usually unknown In advance. Given sampled values of the current and voltage, the parameters should thus be determined as best as possible according to the given models (12) and (13)' Apparently, the models have the same structure as described by (7)-(8) . The properties of the signals in (12)-(13) corresponding to the B-polynomial are very important whell applying an ARMA-model (ct. Astrom (1987». The signal should with necessity be persistently exciting of order deg(A) + deg
( 17)
"Mcos~M,aMsin~M)
( 18)
The vector 8 thus contains information about the system, whereas
= arg (8
12
(19)
,
(20)
_ , 8 / 2 2j l
A scheme for decision logic may be partly based on these quantities as will be explained in a later section.
ESTIMATION ALGORITHMS General Fourier Based Model sufficient information about the system to be protected is available, the crucial relay problem can be transferred to applying suitable methods from the signal processing theory. For this purpose, a black-box model describes the system according to the prev ious section. The model is defined by analytical expressions, which are given as function of some parameters to be determined. Numerical values of the parameters can be extracted from measured data. Since the system in steady-state conditions may be subjected to different types of variations, adaptive estimation of the unknown parameters is a necessity. Also methods for detecting abrupt changes in connection with events must be considered as well as procedures for frequency estimation.
If
For a harmonic signal a more general approach would be to base the signal structure on Fourier decomposition. This is convenient since the voltages and currents of the system are usually periodic in stationari ty. Also the transient behaviour can then be modelled in the same manner with sufficient accuracy by assuming the envelope of the signal to be exponentially decaying immediately after an abrupt event in the system. The following relation thus applies: 11
y(t)
=
aOexp(-~ot)
+
~a.sin(w . t+~ . )
1 J
J
( 14 )
J
However, the exponential term usually peters out after some cycles, which implies that the stationary behaviour should be described by the simplified model: N
y(t)
= a
O
+
~a . sin(w , t+~ . )
1 J
J
( 15)
J
The model_fhould be changed when the current ~~e ex~eds ~ 1nl'), for some small value of I') (0: 10 10 ). Relation (IS) is achieved by setting ~O equal to zero in (14)' The coefficients of the expressions are identical except for the mean values. In model (14) M should be chosen greater than N in model (IS). Theoretically M may take infinitely large values, but due to the necessity to keep the time complexity below a moderate level in real time applications, M should be restricted to the first superharmonic components. Of course, the accuracy of the estimated parameters depends on the truncation of the Fourier series. This is not a serious restriction since the power line signals have a limited amount of harmonic components also after a major change of the dynamics. Subharmonic components can be incorporated in the models (14) and (15) analogously. Furthermore, band-pass prefiltering may be utilized to extract suitable information from the real-time signal.
Parameter Estimation The problem of determining the system parameters from successively measured values of y(ll and u(t> in (7) is usually performed according to the least squares method, (Ljung et ai, (1983», i.e. the parameters are c hosen to minimize the criterion: (2l>
" Elll=y(ll-y(t).
where minimization is:
" = arg min 8(1l
(V
The estimate according to:
The
general
solution
of
(22)
(8,t» N
of
8(t)
the
can
be
recursively
updated
9(t) = S(t- l> T
(23)
" = 8(t-l> " 8(t> + L(t)e:(t)
(24)
where the gain vector, following algorithm: r(t) = P(t-l>
L(t>,
is computed from
the
(25)
S. Bergman and S. Ljung
1212
4Y2 j !Y 1Y3 - Y;j 2 y +y +2Y * Y+Y 3-2Y *Y 3 1 1 2 2 !
(26) L
(27)
PIt) = (P{t-1>-r(t)L
(28)
P
" =9 = <1/6>1, 9(Q) 0
(29)
The implementation of the algorithm can easily be performed since it bases entirely on matrix and vector computations (cf. Bergman et al (1988». The quantity 6 is chosen sufficiently small in order to avoid problems due to singular matrices and to allow a fast adaptation immediately after an initialization. Usually the restart of the parameter estimation procedure should be performed in such a way that the updating of the P(t)-matrix takes its initial value from (29). This recursive algorithm applies when estimating the coefficients of (14)-( 15) since the magnitude of the elements of the regression vector are then limited to 1However, for lhe ARMA-models (12) and (13), the regressors may lake values of arbitrary magnilude. Due to this fact, the covariance matrix should ~ decomposed inlo its squ~re root components, P=UDU • The initial setling of 9, 9 can either be chosen 0 arbitrarily (e.g. all components are equal to zero) or empirical model values may be initially imposed unlil the updaling of lhe covariance matrix has slabilized. After a restart, the eslimaled parameter vector prior to lhe evenl is a better ini tializalion strategy lhan equalizing the entries to zero. The factor of exponential forgetting causes an adjustment of the algorithm to slow variations of the dynamics. A smaller value of A will give a more alert algorithm in such a way that faster tracking is made possible. However, the factor may not be chosen too small since that would cause the quanti ties of the algori thm to take too large numerical values, a fact that necessitates the incorporation of a method, which can manipulate abrupt changes in the system. Once a major change has been detected, a reintialization of the updating of the covariance matrix have to be performed. The same event should also be utilized as indication of an abnormal situation in the identified system and classification can be accomplished. Methods for detecting abrupt changes are described by Basseville (1981>. In addition, the loss function VN(8, t) can be used wi th satisfactory results in most situations. A discussion of this possibility is performed in the next section.
Estimation of Fundamental Frequency From (18) it comes out that an accurate estimate of the angular frequency of the studied signal is needed. Because of this reason, a frequency estimation algorithm is appended to the previous scheme. As an alternative to the approach in the previous section, we here describe a procedure for estimating the fundamental frequency of a given harmonic signal. The fundamental component of the signal models of (14)-<15) is in stationarity y (t)=a' sin(w' t). This expression contains the unknown ~nti ties a, wand t. By sampling three consecuti ve val ues y l' Y2 and y 3' wi th time separation h, i.e. : Yl = Ye(t-h),
Y2 = Ye(t),
Y3 = Ye(t+h)
(30)
makes it possible to estimate a and w a'70rding to the following technique. By introducing b=a and applying straightforward trigonometrical relations it turns out that:
b
(31)
The last equality in (31) may be favourable if the signal can take values with large magnitudes. Equation (31) is taken as the estimate of b. Gi ven the sampled signal values and the estimate of b, the estimate of the angular frequency, x(t), now turns out to be: z = b-1(Y3(b_y~)1I2_y~(b_y;)II2) = b-l{Y2(b-y~)1I2_Yl(b-y~)112) x(t) = h -larcsin(z)
(32) (33)
The input parameters for computing the angular frequency are y , y , YJ and b. Since the amplitude of the signal cad b~ obtained from the parameter estimation procedure, the computation of b in <31> may be eliminated, which implies that only equations (32)-(33) for computing the frequency should be added to the parameter estimation scheme. Equivalently, by elaborating the definitions of Yl' Y2 and Y3 in (30) it comes out that: 2112 . y 1 = Y2coswh + (b-y 2) slOwh
(34)
2 1/2 . Y3 = Y2coswh - (b- Y2) slOwh
(35)
Solving for w yields the following expression as the instantaneous estimate of the angular frequency: x(t) = arccos«Yl+Y3)/(2Y2»
(36)
If the frequency estimator is coupled to the estimation
of the Fourier coefficients, the filtered signal can be used in (36) . In order to suppress too large deviations w-om the actual frequency, the moving average value, wIt), of the instantaneously computed estimate of the frequency should be considered as: (37)
"
Wo
(38)
where w
is an initial guess of the angular frequency.
w
Normall~ it is chosen to correspond to the nominal frequency of the system. Additionally, a further precaution should be considered. If the instantaneously computed estimate of the frequency differs too much from the moving average value, it should be ignored. There are also advantages in ignoring such signal values, that differ sufficiently much from the previous one. In order to decrease the number of false estimates, the input to the frequency estim~tion algorithm should thus be the filtered values y(t), defined in (23 ).
DECISION LOGIC The decision logic can be separated into two different parts, a transient one and a stationary. Immediately after a fault, there should be some quantity which rapidly detects the event. The quantity can be based on the fact that the model parameters prior to an abrupt change of the power system dynamics are not valid after the inception, which implies the residual functions E(t) of the currents and voltages to have large magni tudes. Since the algori thm tracks changes coming from the new situation, it will adapt to the best fit for new system dynamics. Meanwhile the residuals thus tend to decay to a level they had before the event. In most cases the sign of the residuals also contains information about the direction to the fault from the measuring unit. Because of these facts the deci sion logic is based on a combination of the magnitude of E(t), the cost function V (9,t) and the N current estimated signal values . In this respect, the
Parameter Estimation Theory and Power Systems Anal ysis currents are most appropriate to be studied since variations of the current signal in the faulty phase are more pronounced than those of the corresponding voltage signal. As a suggestion of an appropriate cost function we can use (21) with VN(S,O)=O. The updating of the function is performed for each signal (totally 6 for a three phase system). How fast V (S,t) r~nds to changes of £(t) depends on the magnit'ude of £ (t) as well as A. In order to detect high impedance faults, an additional checking of the level of I Elt) I is necessary. This should be performed in combination with a study of the loss function, which may be performed In many different ways. However, generally It is suitable to apply statistical analysis to achieve a more reliable extraction of information about trends that occur. There are numerous facilities for this purpose. Just to mention some of them: Hinkley detector, hypotheses testing and Magill predictor. Further information on this subject is given by Basseville (1981). A further measure to increase the sensitivity for high impedance fault detection is to estimate the Fourier coefficients of the symmetrical components rather than of the phase quantities. The approach described above can be used for investigating transient behaviour. For slowly occurring faults as well as for cases when the algorithm is initially applied to estimating the dynamics of a faulty system, the transients may not be used any longer for the analysis. In addition to the transient treatment, a stationary analysis should thus be applied in order to detect slow variations, which are not considered by studying the transients. At this, impedance anai ysis or treatment of symmetrical components of the individual voltages and currents can serve as discriminating quantities. As a further feature of the described considerations serves the possibility to separate conventional faults from special situations, e.g. (dis)connections of lines. This separation can be accomplished by Investigating the amplitudes and phases of the symmetrical currents. The analysis of the impedance plane can be based on an appropriate demarcation of the plane into a domain, where the Impedance vector lies in case of a fault. Figure 2 elucidates the signai flow between the parameter estimation routine and the different blocks of the logic procedure. The output from the parameter estimation routine are the instantaneous residuals, £(t), define4. in (7) and the estimated Fourier coefficients, S(t). The loglcai scheme can be separated In three different states as depicted in figure 3. State 1 represents normal situations in steady state when no fauits are present in the system. State 2 corresponds to situations when speciai conditions has caused the abrupt change, e.g. a connection or disconnection of a line, whereas the algorithm is in state 3 when a conventional fault has been detected. Transitions between the different states are determined in the transient and stationary analysis with the aid of the parameter vector, the residuals of the individual signals and the loss function.
Fig. 3 The states of the logical scheme
RESULTS The described estimation algorithms have been verified by simulation studies partly in situations where the signals are directly generated In the computer programs and partly to analyse signals coming from systems as defined by the ElectroMagnetic Transients Program (EMTP). This program is utilized for the simulation of electromagnetic transients and related phenomena in power networks. Time Series Analysis In order to investigate the model (11) for a transmission line, the current is supposed to have the structure:
PHASE SEl. ••
. . .LYSIS
DETERS.
aESlDUALS
OF"
DIaECTIOIf
Fig. 2 The complete protective scheme
(39)
I
where w=2nf=100n and the distribution of the noise was ReW,O.I). The step size was chosen to be 0.1 millisecond and the estimates of the R- and L-quantities are monitored for each 10 sample. The ini tial parameters of A-he estimation algori thm are: P(0l=1000' I, ,,=0.98 and 8W)=0. In figure 4 two examples are illustrated, where the correct parameters are (L=O.OI, R=100l and (L=O.l, R=10), respectively. As comes out from the figure the estimated quantities converge rapidly to the exact values.
·. DO..[d 0 . 101
' .114
....
t.
0.'
10 .
O.
le.
0
fiO
"·.D '°. [J •• .
...
•
t
IWI
Fig. 4 Estimated R- and L-values Harmonic Signal Analysis The Fourier-series model (14) has been used to estimate properties of signals as directly generated in the simulation program on one hand and as generated by the EMTP program on the other. In the first case the signal was given by:
0.5 + 1 . 5coswt + 2.5sinwt +
yet> nAIfSIDfT
12 13
+
1.75cos2wt
+
0.75 sin2wt
( 40)
where the noise was NW,0.25) and w=100n. The UD-algorithm was applied with 0(0)=1000 ' I and ,,=0.98. The sampling time was 1 mill i second. According to figure 5, the estimates of the parameters are accurate after 15 samples. The rate of convergence depends among other things on the noise function and the forgetting factor . Figure 1 depicts the currents and voltages of a three-phase transmission line including an RN-fault as simulated with the EMTP program. Figure 6 represents the same signals after first having passed through a low-pass analogue fll ter wi th a
S. Bergman and S. Ljung
1214
cut-off frequency of 1000 radians/second in order to eliminate the transient effects. Noise with distribution N(O,O.OO6) was then imposed on the current signals and N(0,12.5) on the voltage signals. The fundamental frequency in the system was SO Hz and the sampling frequency 4 kHz. Figure 7 shows the estimated parameters when applying the UP-balled estimation algorithm with D(o)=l00' I, ~=O.98 and 8(0)=0. Also in this case the estimated coefficients converge rapidly to their correct values. By comparing the curves in figure 7, the necessity to reinitialize the estimation algorithm when a fault has occurred is immediate. However, also when reinitializing it turns out that the estimates are oscillating after the fault . This depends on the fact that the signal contains harmonic components, which are not modelled.
Fig. 5 Estimated Fourier coefficients for (40)
Fig. 6 Filtered currents and voltages
f\...
CONCLUSIONS It has been descri bed how an arbitrary dynamical system may be modelled as a non-linear differential equation, which can be discretized to yield a difference equation. When the signals are known to be harmonic, Fourier series analysis can be applied to model the system. In common for tbe models is the possibility to use them as black-box representations of the treated system. The main problem tbus consists of determining a number of quantities connected to tbe chosen structures. Recursive parameter estimation algorithms have been utilized to extract information from measured values of important signals. A new algorithm for determining tbe fundamental angular frequency of a harmonic signal has been described in combination to the estimation scheme. The algorithm utilizes the filtered signals coming from the Fourier filter to perform an exact computation of the frequency. Requirements for an additional digital filter are tben removed. Logic scbemes for protective relaying can also be connected to the parameter estimation routine. The procedure is fed with the instantaneous residuals, tbe loss function and the estimated Fourier coefficients from the parameter estimation algorithm. Based on this information, tbe logic algorithm performs transient as well as stationary analysis. The model. are verified by simulation results. A complete relay protection scheme can thus be be tailored by utilizing quantities from the complete procedure. The transient analysis also applies to determining when to reinitialize the parameter estimator itself. REFERENCES
f{
1\
Fig. 8 Estimated fundamental frequency
Astr5m, K.J., Adaptive F_dback Control, Proceedings of the IEEE, Vol. 75, No. 2 (1987)
.
.M
\-.
-. -
Fig. 7 Estimated Fourier coefficients (R-current> Frequency Estimation By applying the frequency algorithm together with the estimation algorithm (Initialization as in previous section), the fundamental frequency was estimated for the signal (R-phase current> in figure 6. The estimated frequency has been plotted in figure 8 for some different initial values of the frequency and for two different values of ~, 0.80 and 0.95, respectively. The convergence is evidently faster for a small value of ~ whereas the algorithm becomes more sensitive to fluctuations of the filtered signal. By studying the different currents and voltages in figure 6 It must be concluded that the frequency should be estimated by applying the algorithm to the voltages rather than to the currents since they are less influenced by a change of the system. Sometimes the change depends on that a line is disconnected, which results in a signal with zero amplitude. This implies that a logic scheme should be implemented for deciding whicb voltage can be used for estimating the frequency. The signal is naturally chosen as the first in the sequence (A,B,C) that has a non-vanisbing amplitude.
Basseville, M., Edge Detection Using Sequential Methods for Change in Level --- Part II: Sequential Detection of Change in Mean, IEEE Trans. on ASSP. Vol. ASSP-29 No. 1 (1981) Bergman, S. and Ljung, S., Implementation of Parameter Estimation Algorithms, IFAC WorkshoD on Adapti ve Control. Newcastle, Australia (1988) Girgis, A.A. and Brown, R.G., Adaptive Kalman Filtering in Computer Relaying: Fault Classification Using Voltage Models, IEEE Trans. on PAS. Vol. PAS-I04, No. 5 (1985) Isaksson, A., Di gital Protective Relaying Througb Recursive Least Squares Identification; Submitted for publication to lEE Proceedings. Part C (1988) Ljung, L. and SMerstr5m, T., Theory and Practice of Recursive Identification, Cambridge, Mass.: MIT Pre.. (1983) Phadke, A. G. and Lu Jihuang, A New Computer Based Integrated Distance Relay for Parallel Transmission Lines, IEEE Trans. on PAS. Vol. PAS-I04, No. 2 (1985) Sachdev, M.S. and Baribeau, M.A., A New Algorithm for Digital Impedance Relays, IEEE Tuns. on PAS, Vol. P AS-98, No. 6 (1979)