WO1997008562A1

WO1997008562A1 - Method of locating a single-phase ground fault in a power distribution network

Info

Publication number: WO1997008562A1
Application number: PCT/FI1996/000457
Authority: WO
Inventors: Reijo Rantanen; Janne Suontausta
Original assignee: ABB Research Ltd Switzerland
Current assignee: ABB Research Ltd Switzerland
Priority date: 1995-08-23
Filing date: 1996-08-23
Publication date: 1997-03-06
Anticipated expiration: 1998-02-23
Also published as: PL324885A1; FI953970A7; PL181121B1; RU2159445C2; NO980724L; FI102700B; FI953970A0; CN1200177A; CN1070612C; FI102700B1; AU6702996A; EP0846271A1; NO980724D0

Abstract

The present invention concerns a method of locating a ground fault in a power distribution network, in which method the start instant of the fault transient is determined from the change of the neutral point voltage, the current and voltage transient signals of the faulty phase are filtered, the duration of the transient is determined, the frequency of the fault transient waveform is estimated, the measured voltage and current transient signals are low-pass filtered, the spectra U(φ) and I(φ), of the voltage and current transients are computed, the impedance spectrum (1) is computed, and the estimate of the fault distance is computed for a discrete angular frequency φk from the equation (2). According to the invention, the voltage and current signals of the faulty phase are filtered using a comb filter, the measured voltage and current signals are low-pass filtered in two directions, the frequency of the charge/discharge transient is estimated from the autocorrelation function of the transient, and the complex value spectra U(φ) and I(φ) of the voltage and current transients are computed using parametric spectral estimation methods.

Description

Method of locating a single-phase ground fault in a power distribution network

The present method concerns a method of locating a single-phase ground fault in a power distribution network.

The invention is particularly related to a computational method of locating a single-phase ground fault in medium-voltage overhead and underground-cable networks.

In Finland, for instance, electric power transmission and distribution is implemented using a three-phase AC system with 50 Hz nominal frequency. The nominal, or principal voltage of an AC transmission system indicates the phase-to-phase voltage. The power transmission and distribution system may be hierarchically classified into the interconnecting transmission network, the medium-voltage or intermediate-voltage distribution network and the low-voltage secondary-network system. The interconnecting network comprises transformers and lines operated at 123 kV, 245 kV and 420 kV nominal voltages serving to deliver bulk electric energy from power plants to large load centers. In Finland, medium-voltage distribution networks are typically operated at nominal voltages of 10 and 20 kV. Distribution networks in rural regions are chiefly 20 kV overhead line networks, while 10 or 20 kV underground cable networks are used in urban areas. The medium-voltage network serves to transmit electricity to both large commercial loads at the secondary distribution level and to 20/0,4 kV distribution transformer substations, from which power is delivered to residential consumers in a 0.4 kV low- voltage secondary-network system.

Due to the disturbances and hazards caused by a ground fault resulting from an insulation failure between a phase and ground in the network, rapid indication of such a failure and quick tripping ofthe faulty line section is crucial. This requires a ground-fault protection with quick and reliable operation. To minimize service interruptions in power distribution, also immediate action must be undertaken to locate and repair the fault. Previously, the location of the fault has been first coarsely estimated by means of switching operations carried out about the faulty section of the network and then the actual site has been identified by patrolling in the terrain, which may become a time- consuming task. In order to improve the quality and availability of electricity supply, a need has arisen for computational fault location methods that could determine the distance from the substation to the location of the fault. Then, the fault could be located and repaired faster than by prior-art methods This topic has been investigated to some extent during the last decade. However, no really reliable method has so far been devised.

A single-phase ground fault causes in the distribution network a transient disturbance in which the voltage of the faulty phase falls and its phase-to-ground capacitances are discharged resulting in a discharge current transient. Simultaneously, the voltages of the intact phases rise and their phase-to-ground capacitances are charged resulting in a charge current transient.

A single-phase ground fault can be located by computational means from the charge/discharge current transients. The required measurement signals comprise the waveforms of the current and voltage of the faulty phase and the neutral point potential ofthe line as measured at the substation. The phase voltage is measured from the voltage measurement cell and the phase current from current measurement cell of the faulty feeder line, or alternatively, from primary side cell of the substation. In the method developed for the estimation of distance to the fault location, the faulty phase is modelled using a first-order differential equation.

In the prior art, ground faults have been located using, among other methods, numerical integration, computing the energy spectral density function of the measurement signals by a Fourier transform and using a damped-signal model.

A drawback of the numerical integration method is that at near-zero values of the denominator, noise present even at the smallest levels causes a large error in the computed end result. Furthermore, the method is unreliable at high values of ground fault resistance after the transient has died away. Methods based on a Fourier transform are hampered by the large number of samples required for a reliable estimate of the signal spectrum. Moreover, the signal must be assumed to be stationary when using the periodogram spectrum. Due to the instantaneous and nonstationary nature ofthe transient (allowing only a small number of samples), the computed spectrum may not necessarily give reliable results. In fact, low reliability of a single spectral value is the weakness of the periodogram analysis. Additionally, the weighting of samples used in the computation of the FFT, that is, multiplication by a window function, distorts the computed spectrum, because the shape ofthe window function can be seen in the final estimate ofthe spectrum.

In the damped sinusoidal signal model, noise corrupting the measurement signal causes a major problem in the use of the Prony method. No separate noise model is included in the Prony method. The noise may have a very wide spectrum, whereby the high- frequency components of the noise are folded over on the lower frequencies. By using a model of higher order, the reliability of the parameter estimation process may be improved also when applied to a noisy signal. However, reliable separation of the actual signal from the noise requires that an order estimate of the background process is available. Computation using such a higher order can cause a singularity or near- singularity of the data matrix being processed, whereby correctness ofthe results may be lost. Also bandwidth-limiting filtration can be applied to reduce the noise power of the signal. It must be noted herein, however, that noise power over the filter passband will not be reduced.

In the described method, the instantaneous values representing the line inductance between the distribution station and the fault location are computed only from a few waveform samples of the phase voltage transient. Therefore, this method is extremely sensitive to noise and modelling errors, and particularly to fault resistances larger than 50 ohms, the estimates of distance to fault location are highly erroneous or no estimate for the distance can be computed.

In the method disclosed in publication Igel, M. 1990. Neuartige Verfahren fur den

Erdschluβdistanzschutz in isoliert und kompensiert betriebenen Netzen. Signale und Algorithmen in Frequenzbereich. Dissertation. Saarbriicken, Universitάt des Saarlandes. 181 pp., an FIR high-pass filter of 127th order was used to suppress the fundamental frequency from the measurement signals and Butterworth-type IIR low- and high-pass filters of 10th order were used to filter away spectral components below and above the frequency of the estimated line charge/discharge transient. In spite of the high order of the FIR high-pass filter, the flanks of the filter passband curve cannot be made sufficiently steep to attain complete suppression of the fundamental frequency component. Moreover, the use of IIR filters causes a nonlinear phase delay on the filtered signals unless the signals are not filtered bidirectionally.

In the method described in publication Lehtonen, M. 1992. Transient analysis for ground fault distance estimation in electrical distribution networks. Doctoral thesis. Espoo, Technical Research Centre of Finland. 182 pp, an FIR low-pass filter of 20th order based on the Kaiser window was used to filter away all spectral components above the frequency of the charge/discharge transient from the measurement signals. A disadvantage of such a filter is its relatively broad passband giving a nonoptimal filtration result. Moreover, the passband ofthe filter is not flat, which may distort the shape of the filtered transient signal.

It is an object of the present invention to overcome the drawbacks of the above- described techniques and to achieve an entirely novel type of method for locating a ground fault in a power distribution network.

The goal of the invention is achieved by determining the inductance of the line section between the distribution station and the fault location through a signal processing procedure comprising first filtering the measured charge/discharge transient by a comb filter and then processing the filtered signal by the least squares Prony method and examining Prony spectrum computed using the singular value decomposition theorem.

The invention offers significant benefits.

The fault distance estimation process uses only such data that have been gathered during the charge/discharge transient. Using the autocorrelation function, the frequency estimate of the transient is obtained with a higher accuracy than that achievable by means of a Fourier transform. Furthermore, the effect of noise on the results can be reduced with the help of singular value decomposition of the data matrix. In fact, the Prony method is particularly suited for modelling a damped sinusoidal signal. With the help of the parametric model, the energy spectral density estimate can be computed as densely as desired with an arbitrary choice ofthe frequency points.

In the following, the invention is examined with the help of exemplifying embodiments illustrated in the annexed drawing in which

Figure 1 is the amplitude characteristic curve of a comb filter suited for use according to the invention, plotted for 50 Hz fundamental frequency;

Figure 2 is a plot illustrating the principle according to the invention of determining the end instant ofthe transient;

Figure 3 is a plot of an autocorrelation function of a transient signal processed according to the invention;

Figure 4 is a plot ofthe current and voltage waveform of a faulty phase during a ground fault;

Figure 5 is a plot ofthe signals of Fig. 4 after filtration according to the invention;

Figure 6 is a plot of a portion of the autocorrelation function of the current signal of Fig. 5;

Figure 7 is an autoregressive (AR)-type plot of fault current and voltage signal spectra computed at 20 Hz frequency increments;

Figure 8 is a plot of the Prony spectra of fault current and voltage signals computed according to the invention at 20 Hz frequency increments; and

Figure 9 is another plot of the Prony spectra of fault current and voltage signals computed according to the invention. The method according to the invention for estimation of fault distance based on the Prony spectra ofthe fault transient signals is outlined as follows:

1. The start instant of the transient is determined from the change of the neutral point voltage. 2. The current and voltage waveforms of the faulty phase are filtered using a comb filter.

3. The duration ofthe transient is determined.

4. The frequency of the charge/discharge transient is estimated from the autocorrelation function ofthe transient. 5. The measured current and voltage signals are low-pass filtered in both directions using an IIR filter of at least fourth order. To obtain a reliable filtration result, the train of transient waveform samples must contain measurement values not belonging to the actual transient, whereby the response of the filter itself to a transient signal input cannot corrupt the filtered transient signal.

6. The complex-value spectra U{ω) and l ω) of the current and voltage transients are obtained by means of the least squares Prony method and Prony spectrum computed using the singular value decomposition theorem.

7. The impedance spectrum is computed from the equation

Z(ω) = ^ = Re(Z(ω)) ₊ jIm(Z{ω)) . (1)

8. An estimate of the fault distance is computed for a discrete angular frequency ω_k from the equation

9. The final value of fault distance is computed as follows: Ifthe maxima ofthe current and voltage spectra

and

occur at the same frequency, the fault distance is computed from the impedance spectrum at said frequency. Otherwise, the final value of fault distance is computed as a weighted average of the values obtained at the frequency corresponding to the maximum ofthe spectrum ofthe current transient and two frequency points (that is, totally at max. 3 frequency points) about said frequency using the equation given below: n

∑^w*^α(^ω _*)

« = ^Jah; , (3)

where the weighting coefficients w_k are

|/(oj_d)| is the value of the spectrum at a frequency ω_d , giving the global maximum amplitude of the current transient spectrum. A precondition for the use of the frequency points adjacent to this frequency of current spectrum maximum is that the spectral amplitude at the adjacent frequency points is at least 80 % of the maximum amplitude of the current signal spectrum.

In principle, the spectrum can be computed at a desired number of frequency points.

As both the fault distance estimation methods disclosed in the literature of the art and those developed in conjunction with the present invention are based on computing the inductance of the line section between the distribution station and the location of the fault from the charge/discharge transient waveform, filtration must be applied to extract the transient waveforms from the measured current and voltage signals. The fundamental frequency and its harmonics are filtered by means of a comb filter, while the frequency components higher than that of the charge transient are eliminated by means of a low- pass filter. In principle, the signals should after the filtering steps contain the charge/discharge transient alone. In practice, complete elimination of noise from the signals is not possible due to the nonideal behaviour ofthe filters used.

For the separation ofthe charge/discharge transient from other components of the meas- ured signal, the start instant of the fault, that is, the first measurement point representing the transient process must be known. The identification of the start instant of the transient can be performed by monitoring a change in the neutral point potential U₀ measured at the substation. The only method today applicable for this purpose is to set a suitable limit for the neutral point potential whose violation is considered indicative of a ground fault.

The measured phase current waveform comprises a stationary fundamental frequency component and a nonstationary transient, while the measured phase voltage waveform comprises a nonstationary fundamental frequency component and a nonstationary transient. As soon as the start instant of the fault is identified, the current and voltage waveforms can be filtered.

The fundamental frequency component and its harmonics are subjected to filtration:

y[n] = x[n] - x[n + f I f] , (5)

where n is a discrete time index, yfnj is the filter output signal and x[n] is the filter input signal, f_s is the sampling rate and / is the fundamental frequency of the signal. The

full cycle ofthe fundamental frequency component in the signal, and it must be an integral number. As the filter output state at instant n is dependent on the filter input state at instant n+f/f, this type of filtration cannot be carried out in real time, but instead, only after the measurement ofthe transient waveform is completed.

In Figure 1 is shown a portion of the amplitude characteristic of the comb filter defined by Eq. (5) when the fundamental frequency is 50 Hz. As can be seen from the graph, the zero points of the passband coincide exactly with the fundamental frequency and its harmonics. Because of the nonstationary nature of a transient, the filter amplitude characteristic shown in Fig. 1 does not tell the effect ofthe filtration process on the transient waveform that is dependent on the damping coefficient. With a large value of the damping coefficient, the transient will die out in a shorter time than the cycle time of the fundamental frequency component, whereby the filter will not distort the transient waveform. In real cases of ground faults described in the literature ofthe art, the smallest values ofthe transient signal damping coefficient are quoted to be in the order of approx. 300 1/s. Then, the amplitude error caused by the filter is 0.0025A, where A is the amplitude of the transient. Because the amplitude characteristic shown in Fig. 1 is valid for stationary components ofthe input signal only, the filter will not distort such damped transients whose frequency is a harmonic of the fundamental frequency or which decay to zero in a shorter time than the cycle time of the fundamental frequency. In practice, the transient can be assumed to decay away during one cycle of the fundamental frequency. When desired, the delay of the filter may be designed longer, e.g., doubled, whereby the distortion caused by the filter will be further reduced.

Use of a filter characterized by Eq. (5) presumes a priori knowledge of the network fundamental frequency. If the fundamental frequency is deviated from its nominal value, the comb filter will lose its efficiency in the filtration of possible harmonic components occurring in the measured current and voltage waveforms. Table 1 below illustrates the effect of changes in the fundamental frequency on the filtration result when the comb filter is designed for a 50 Hz signal. The filter gain is listed up to the 8th harmonic.

Table 1. Change of comb filter gain for different harmonics with a change of input signal fundamental frequency.

Fundamental Gain Gain Gain Gain Gain Gain Gain Gain frequency l f 2 f 3 f 4 f 5 f 6 f 7 f 8 f f [Hz]

49.5 0.1 0.18 0.22 0.3 0.34 0.38 0.42 0.55

49 0.18 0.3 0.42 0.57 0.65 0.8 0.95 1.18 It can be seen from Table 1 above that the attenuation of the filter for higher harmonics becomes rather inefficient when the fundamental frequency is offset. In fact, the filter begins to amplify harmonic components higher the 8th harmonic when the fundamental frequency is deviated to 49 Hz. Although the values given in the table are approximate, they give a clear picture of the effect of changes in the fundamental frequency on the filtered signal.

If deviations in the fundamental frequency from its nominal value are considered to cause significant difficulties in the filtration of harmonic components, the fundamental frequency of the network can be estimated from the steady-state measurement of the phase current performed after the decay of the fault transient. Frequency estimation of a single sinusoidal signal can be made using the maximum likelihood estimation (MLE) method. In this method, the frequency is sought at which the periodogram of the signal reaches its maximum value, that is, the value ofthe equation below is maximized:

In practice the fundamental frequency can be estimated by computing the MLE values for frequencies about 50 Hz and then selecting the frequency corresponding to the largest value.

Determination of the transient end instant is necessary, because the spectral estimate computed from the signal samples of the transient will be significantly corrupted, unless specifically computed over the duration ofthe charge/discharge transient.

The end instant of the transient may be determined after the low-pass filtration using a relatively simple procedure. From the data gathered after the start ofthe transient for one full cycle ofthe network fundamental frequency, the signal value of maximum magnitude is selected. This value is then inteφreted as the maximum value of noise associated with the measurement. Next, the values of this signal sample sequence are compared proceeding from the end toward the start of the sequence until a limit value is reached that is, e.g., 10 % larger than the maximum value of the noise component in the measurement. Finally, the first signal sample exceeding the limit value thus defined is chosen as the end instant of the transient. The method is illustrated in Fig. 2 showing a simulated transient with superimposed noise. In the diagram is drawn by horizontal dashed lines a corridor within which the instantaneous values of the measurement signal fall after the decay ofthe transient.

In the fault distance estimation method developed in this invention, the measurement signals are assumed to contain only the charge/discharge transient occurring at the onset of a ground fault. Since the Prony method is sensitive to noise, the transient signal must be filtered free from other frequency components. The filtration method chosen in this invention is low-pass filtration. The design of a low-pass filter comprises the selection of the order and cutoff frequency ofthe filter. The filter cut-off frequency is chosen on the basis ofthe estimated frequency ofthe transient waveform.

Estimation of the transient frequency can be performed using the autocorrelation function which is used for examination of the dynamic changes in the measurement signals. The autocorrelation function of a discrete data signal is computed using the equation below:

where N is the number of samples and the delay is k = 0,1, ...,m, where m is the largest value ofthe delay. The autocorrelation function is computed for the data extending from the start instant to the end instant ofthe transient. The time value of the autocorrelation function at which the function reaches a minimum corresponds to the duration of half- cycle in the decaying sinusoidal input signal. The frequency estimate of the transient is obtained by dividing half the sampling rate with the delay time corresponding to minimum value of the autocorrelation function. In Fig. 3 is shown a portion of the autocorrelation function ofthe simulated transient.

Accordingly, the cutoff frequency of the low-pass filter is automatically determined by the frequency of the charge/discharge transient. A practical realization of a low-pass filter cannot have a zero-width passband. Hence, a safety margin of a few hundred hertz must be added to the estimated cutoff frequency to avoid distortion to the waveform of the transient by the filter.

The low-pass filter can be selected to be, e.g., a Butterworth-type IIR (Infinite Impulse Response) filter of at least fourth order. A benefit of IIR filters is that a relatively steep flank ofthe passband can be obtained already with a filter of a very low order. A further beneficial characteristic of Butterworth low-pass filter is its flat-topped passband.

However, IIR filters are hampered by an infinite impulse response, which means that their phase delay is nonlinear. Such a nonlinear phase delay can be eliminated by filtering the measurement signal bidirectionally (two times), first forward from the beginning of the data sequence to its end and then backward from the end to the beginning. The amplitude distortion caused by the filtration is very small in practice. The initial transient response ofthe filter can be minimized by selecting proper initial values for the filter and then adding to the filter input a short inverted portion of the original input signal sequence. The best possible filtration result is obtained when the length of the signal sequence to be filtered is at least three-fold relative to the order ofthe filter and when the signal values at the start and end ofthe data sequence approach zero.

Spectral estimation methods based on a parametric signal model are called parametric methods. The goal of parametric methods is to find a linear difference equation model capable of describing the signal. The use of parametric models in describing a signal presumes that the model of the signal is known, or alternatively, the signal is known to obey a model having a finite number of parameters which are independent from the number of samples in the data sequence. In the adaptation of the model, the coefficients of the difference equation and the order of the model are optimized. The use of parametric models in estimation of a spectrum comprises three steps:

1. Selecting a suitable model for the measurement signal.

2. Estimating the parameters for the selected model..

3. Inserting the estimated parameters in the energy spectral density equation corresponding to the selected model. Parametric methods achieve a better estimate of energy spectral density than that available by conventional means such as those based on either the autocorrelation function or directly on the Fourier transform of the measurement data. One of such benefits is the higher accuracy of the computed spectrum. When estimating the energy spectral density function as a periodogram, that is, computing a Fourier transform directly for the data, an assumption is made that the data are zero outside the range of computation. Normally, this is an unrealistic assumption and causes distortion to the computed spectral estimate, because also the frequency response ofthe window function itself used for limiting the range of computation will be reflected in the spectral estimate. When estimating the energy spectral density function on the basis of the parametric model, the signal is assumed to be compliant with the model also outside the data range used in the computation. This method dispenses with the weighting of the signal data with a window function, whereby also the distortion caused by the window function on the spectrum is avoided. Additionally, a certain reliability level ofthe computed spectrum is attained with an appreciably shorter data length than is required in Fourier transform methods. However, a disadvantage of this method is that the reliability assessment ofthe computed spectrum is not as easy as in the Fourier transform methods. The degree of improvement in the resolution and reliability of the computed spectrum is dependent on the suitability ofthe model for use in conjunction with the signal being processed and its ability to fit the coefficients ofthe model to the measurement data or its autocorrelation function. As normally some background information is available on the process producing the signal to be examined, this information can be used in the selection of a suitable type of model. A further benefit of parametric estimation methods of energy spectral density functions is that the signal spectrum may be computed at desired discrete points of frequency, thus giving an arbitrarily high plotting resolution ofthe spectrum.

Next, the estimation ofthe fault distance is examined by way of modelling the waveforms of the current and voltage transients using both the autoregressive, or AR model, and a model based on the least squares Prony method. The AR-model is selected as the basis of the examination, because it is more suitable for particularly short data sequencies than the Fourier transfer methods and it is capable of extracting frequency components from which a signal sample sequence of less than a full cycle can be retrieved. Due to these reasons, the AR model may also possibly be used for modelling nonstationary signals such as charge/discharge transients. The model based on the Prony method is preferentially investigated herein, because it is particularly developed for modellling a damped sinusoidal signal.

The autoregressive (AR) model ofthe signal computes the expected present value of the signal as a weighted sum of its preceding values. An autoregressive model AR(p) of order p is described by the equation:

where x[n] is the output of the autoregressive filter at instant n , e[n] is a process generating at the filter input a white noise signal with zero average value and variance p² and a_k ,k - \,...,p are the coefficients ofthe model.

The energy spectral density function ofthe signal as given by the AR model is:

Since the estimation ofthe fault distance is made in the complex plane, it is sufficient that the spectra of current and voltage transients required in the computation of the impedance spectrum are processed in the form:

Use of the AR model in a fault distance estimation method based on the spectra of the current and voltage transients of the faulty phase is favoured by its simple form and the easy interpretation of the spectrum produced by the method. The spectral peaks corre¬ sponding to the poles of the AR-spectral estimate are narrow, and owing to the zero order ofthe denominator, the zeros ofthe AR estimate are seen as smooth valleys. To adapt parametric methods to the modelling of a signal, the model must be compatible with the process being modelled. In fact, this is problem in the use of the AR model for fault distance estimation, because in the AR model the input signal is assumed to be white noise, while the charge/discharge transient caused by a ground fault is closer to an impulse function. Moreover, the input signal to the AR model is assumed to be stationary or slowly changing, which is not true in the present case.

The Prony method is sensitive to noise. Hence, ifthe input signal contains additive noise by a significant amount, the parameters of the original transient waveform cannot be determined correctly any more. In particular, the values of the damping coefficients become extremely unreliable and typically greater than the actual values. In the present invention, the effect of additive noise on the signal is reduced by using a higher order of the exponential model and low-pass filtration of the transient signal. Significant additional reduction of the noise effect can be attained by using a further developed version of the least squares Prony method in which the data matrix formed from the signal sample sequence is divided on the basis of the singular values of the matrix into two parts so that the signal components of appreciably larger singular values are considered to represent the original transient, while the rest ofthe signal components are taken as noise. Ifthe transient is strongly damped or the signal-to-noise ratio ofthe data is poor, the separation of the actual signal from noise may sometimes be difficult even when using the singular value decomposition theorem.

Prony's methods are described in greater detail in the publication Marple, S.L r. 1987. Digital Spectral Analysis with Applications. New Jersey, USA, Prentice-Hall Inc. 492 pp.

The definition of the Prony spectrum makes an initial assumption that the measured transient is symmetrical about the origin. Then, the transient can be modelled using a two-sided function:

where z_k = exp(a _kT + J2πf_kT) and z^^* = exp(-a_kT + j2qf_kT). Modelling of the transient by means of Eq. (5) gives a more accurate spectral estimate than is achievable by assuming the existence of the transient for positive values of time only. The Z- transform ofthe paired Eq. (11) is:

By assuming all damping coefficients a_k ,k = \,...,p to be negative, that is,

< and substituting ∑ = exp{j2πfT) , a discrete Fourier transform of the two-sided exponential function is obtained:

(13)

The final form ofthe estimate ofthe Prony energy spectral density is:

P{f) = P_β(f) (14)

The width of the Prony spectrum peak is determined by the magnitude of the damping coefficient. The height of the peak in the Prony spectrum computed according to

Eq. (14) is (2A_k I a_k)^~ and its -6 dB bandwidth is al π. Hence, the resolution of the spectrum varies as a function of the damping coefficient. For a large value of the damping coefficient, wide spectral peaks are obtained, while a small value ofthe damping coefficient results in narrow spectral peaks.

Since the Prony method was particularly developed for modelling a damped sinusoidal signal, this type of model is extremely well suited for modelling a charge/discharge transient occurring in the beginning of a ground fault. Hence, the estimate of the energy spectral density function obtained by virtue of this method is more reliable than that computed by other Fourier transform methods. Analogously to other parametric spectral estimation methods, the Prony spectrum of a signal can be computed for as many frequency points as is desired independently from the length of the data sequence used for the determination ofthe parameters.

A disadvantage of the method is that the computation of the model parameters is rather laborious due to large size of the data matrices processed. In turn a reduction of the computation task can be accomplished by focusing the computation of the spectrum to the frequency range of interest, that is, about the estimated frequency of the charge/discharge transient.

The total estimation process according to the invention for locating ground faults on the basis of transient spectra is outlined as follows:

1. The start instant of the transient is determined from the change of the neutral point ofthe network. 2. The measurement signals ofthe faulty phase current and voltage are filtered using a comb filter defined by Eq. (5).

3. The duration ofthe transient is determined.

4. The frequency of the charge/discharge transient is estimated from the autocorrelation function (7) ofthe transient. 5. The current and voltage measurement signals are low-pass filtered using, e.g., an ILR filter of 4th order. 6. Depending on the selected method, the complex- value spectra U(ω) and l(ω) of the current and voltage transients are computed using either Eq. (10) or Eq. (13).

7. The impedance spectrum defined by Eq. (1) is computed. 8. An estimate of the fault distance is computed according to Eq. (2) for a discrete angular frequency ω_k.

9. The final value ofthe fault distance is computed followingly:

Ifthe maxima ofthe voltage and current spectra

and |/(fi>)| occur at the same frequency, the fault distance is computed from the impedance spectrum at said frequency. Otherwise, the final value of fault distance is computed as a weighted average of the values obtained at the frequency corresponding to the maximum ofthe spectrum ofthe current transient and two frequency points (that is, totally at max. 3 frequency points) about said frequency using Eq. (3). A precondition for the use ofthe frequency points adjacent to this frequency of current spectrum maximum is that the spectral amplitude at the adjacent frequency points is at least 80 % of the maximum amplitude ofthe signal spectrum.

In the following, the principles of the computational procedures are elucidates with the help of an example. The signal recordings used in the example were measured at a substation having an artificial ground fault of zero ohms made to a 20 kV medium- voltage network operated with its neutral point isolated from ground. The actual distance to the fault was 14.2 km from the substation.

Referring to Fig. 4, the current and voltage transient waveforms of the faulty phase are shown at the occurrence ofthe ground fault. The fault has occurred just prior to the time instant 0.04 s ofthe plot.

In the first step of the procedures, the start instant of the transient is determined on the basis of the time instant corresponding to the change of the neutral point voltage. The ground fault can be assumed to have occurred when the neutral point voltage exceeds a preset limit. In practice, the tripping instant ofthe neutral point voltage relay is taken the start instant ofthe ground fault.

In the second step, the measurement signals are filtered by means of a comb filter defined by Eq. (5) to eliminate the fundamental frequency and harmonic components. The signals of Fig. 4 are shown in Fig. 5 after filtration by the comb filter. As can be seen from the graph, the fundamental frequency component of the signal existing prior to the occurrence of the fault cannot be removed entirely from the current signal due to the nonideal performance ofthe filter. However, this has no meaning to the estimation ofthe fault distance, because the start instant of the transient is known from the change of the neutral point voltage and the estimation method of fault distance uses only signal sample sequences following the start instant ofthe ground fault.

After the filtration step with the comb filter, the duration of the transient is determined. Using the procedure described above, the duration of the transient is determined as approximately 150 data samples. In real time this corresponds to 7.5 ms when the sampling rate has been 20 kHz.

The frequency of the charge/discharge transient is estimated from the current transient, because typically the fault transient is most evident in the current measurement signal. Fig. 6 shows a portion ofthe autocorrelation function computed according to Eq. (7) for the current transient waveform of Fig. 5. With the 20 kHz sampling rate used and the first minimum ofthe autocorrelation function coinciding with a delay of 27 samples, the estimate ofthe transient frequency is 10000 Hz / 27 « 370 Hz.

Prior to the final step of fault distance estimation, the measurement signals are low-pass filtered. According to the invention, a Butterworth-type IIR filter of 4th order is used herein. The filter cutoff frequency is selected on the basis of the frequency estimate obtained for the charge/discharge current transient as described above. To prevent the low-pass filtration from distorting the waveform of the transient, the filter cutoff frequency is set a few hundred hertz above the estimated frequency fo the charge/discharge transient. In the method according to the invention, the fault distance is estimated based on the spectra ofthe current and voltage transients. Two modelling alternatives were described above: the AR model and the damped sinusoid model defined by the Prony method using a parameter solution utilizing the singular value decomposition theorem for processing the data matrix.

Referring to Fig. 7 the spectra of the current and voltage transients computed according to Eq. (10) are shown for 20 Hz spacing ofthe frequency points. When the final value of the fault distance is computed according to steps 8-9 of the invention, the result is 15.3 km.

The order of the AR model for computation of the results given below was set as 20. This order ofthe model was arrived at experimentally by comparing the results computed using different data and different order ofthe model.

Referring to Fig. 7, the maxima of the current and voltage transient spectra are seen to occur at different frequencies. As the transients are measured at different points (voltage from the voltage measurement cell and current from the out cell ofthe faulty feeder), the transient waveforms need not necessarily have exactly the same frequency and damping coefficient. An additional error is caused therefrom that the AR model is not initially intended for the modelling of fast-changing signals ofthe impulse type.

Referring to Fig. 8, the spectra ofthe current and voltage transients computed according to Eq. (13) are shown for 20 Hz spacing ofthe frequency points. When the final value of the fault distance is computed according to steps 8-9 of the invention, the result is 13.7 km.

The order of the exponential function model used for modelling the transients was set a 6, whereby two of the exponential functions were used to represent the actual signal Then, the model contains only one frequency intended to correspond to the charge/discharge transient. Four of the exponential functions were set to represent the additive noise corrupting the actual signal. The choise of the model order was based on tests in which noise of normal distribution was added on a simulated transient, after which the parameters ofthe transient were estimated.

Table 2 below gives the singularity values of the data matrices formed from the sampled current and voltage measurements illustrated in Fig. 8. The table shows that the two largest singularities of the data matrices corresponding to the current and voltage tran¬ sients are clearly larger than the other singularities. Hence, the values of the singularities prove a successful selection of the model and the measured transients contain only one damped sinusoid corresponding to the actual transient waveform. Also when tested with other measured transients, the selection of the model was successful as evaluated on the basis ofthe relative magnitudes ofthe singularity values in the data matrices representing the transients.

Table 2 Singularity values of data matrices formed from current and voltage transient measurement. Measurement site Tuovila, fault distance 14.2 km and fault resistance 0 ohms.

Current Voltage singularity singularity

452.0 40666

87.3 7924

2.0 121

0.1 11

0 1

0 0

As is evident from Fig. 8, the spectra of current and voltage transients are seen to occur at the same frequency. The frequency ofthe current transient was computed as 362.4 Hz and the damping coefficient as 211.4 1/s. The corresponding values for the voltage transient were 363.5 Hz and 428.3 1/s. In this case, the frequencies were almost equal, but the voltage transient was much more effectively damped than the current transient. Because the Prony method was particularly developed for modelling damped sinusoids, the spectral estimates given in Fig 8 can be considered reliable. Comparison of Fig. 7 with Fig. 8 shows that in both graphs the current maximum occurs at the same frequency. In the modelling of the voltage transient, the AR model is sligthly unsuccessful, probably due to the stronger damping ofthe transient.

According to the invention, the current transient may also be measured from the primary side cell cell instead of the feeder side cell. The end instant of the transient can also be determined with the help of the autocorrelation function of the transient when the function is computed over a longer time (e.g., comprisng a full cycle of the fundamental frequency after the occurrence ofthe fault). In this manner, the duration of the transient and the frequency estimate of the could be determined in a single step. In the computation of the Prony spectrum, the transient may alternatively be defined for positive instants of time only in the following manner:

x[n] = ∑h_kz_k",n ≥ 0 k=\

0, n < 0.

Then, the Z-transform ofthe signal is:

and its spectrum is:

Also other methods besides the Prony method can be contemplated for the modelling of a damped sinusoid and spectral computation.

Claims

Claims:

1. A method of locating a ground fault in a power distribution network, in which method

- the start instant of the fault transient is determined from the change of the neutral point voltage,

- the current and voltage transient signals ofthe faulty phase are filtered,

- the duration ofthe transient is determined,

- the frequency ofthe fault transient waveform is estimated,

- the measured voltage and current transient signals are low-pass filtered, - the spectra U{ω) and l( ) , of the voltage and current transients are computed,

- the impedance spectrum Z(ω) = - = Re(Z(<s>)) + j Im(Z(ω)) (1)

is computed, and

- the estimate of the fault distance is computed for a discrete angular frequency ω_k from the equation

c h a r a t e r i z e d in that

the voltage and current signals of the faulty phase are filtered using a comb filter, - the measured voltage and current signals are low-pass filtered in two directions,

- the frequency of the charge/discharge transient is estimated from the autocorrelation function ofthe transient, and - the complex value spectra U(ω) and /(try) of the voltage and current transients are computed using parametric spectral estimation methods.

2. A method as defined in claim 1, c h a r a c t e r i z e d in that the final value of the fault distance is computed in the following manner:

- if the maxima of the current and voltage spectra

and

occur at the same frequency, the fault distance is computed from the impedance spectrum at said frequency. Otherwise, the final value of fault distance is computed as a weighted average of the values obtained at the frequency corresponding to the maximum of the spectrum of the current transient and two frequency points (e.g., totally at max. 3 frequency points) about said frequency using the equation given below:

where the weighting coefficients w_k are

in which | (ω_d)| is the value ofthe spectrum at a frequency ω_d , giving the global maximum amplitude of the current transient spectrum, whereby a precondition for the use of said frequency points adjacent to said maximum of the of current spectrum is that the spectral amplitude at the adjacent frequency points is at least 80 % of the maximum amplitude of the current signal spectrum.

3. A method as defined in claim 1, characterized in that the complex value spectra U(ω) and l(ω) ofthe current and voltage transients are obtained by means of the least squares Prony method and Prony spectrum computed using the singular value decomposition theorem.