WO2010037413A1

WO2010037413A1 - A method and a device for constructing new sample values used in a fir algorithm

Info

Publication number: WO2010037413A1
Application number: PCT/EP2008/063063
Authority: WO
Inventors: Magnus Akke; Björn WESTMAN; Henrik ÅSHUVUD
Original assignee: Abb Technology Ag
Priority date: 2008-09-30
Filing date: 2008-09-30
Publication date: 2010-04-08

Abstract

The present invention relates to the method and a device for constructing new sample values to be used in a Finite Impulse Response, denoted FIR, algorithm, wherein a sampled analog signal contains at least a periodic fundamental component that has an actual period length that may deviate from a nominal period length, the device (3) comprises: a measuring unit (7) configured to continuously measure an actual fundamental frequency of the analog signal, a sampling unit (1 ) for sampling the analog signal at controlled time steps, a first storage (2) for storing the sample values as original sampling values, a system period length determination module (8) configured to continuously determine the actual fundamental period length based on the measured the actual fundamental frequency, a calculating unit (6) configured, to determine a new sampling time step based on the actual fundamental period length such that the number of new sampling time steps in the actual fundamental period is the same as the number of original sampling time steps in the nominal fundamental period, to calculate, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step, wherein each of new sample values is calculated based on at least two original sample values, and a second storage unit (4) for storing the new calculated sample values.

Description

A METHOD AND A DEVICE FOR CONSTRUCTING NEW SAMPLE VALUES USED IN A FIR ALGORITHM

FIELD OF TH E INVENTION

The present invention relates to a method and a device for constructing new sample values to be used in a FI R algorithm. Such an algorithm can be implemented in a FI R filter. The invention is suitable for a digital signal processing application in which a FI R filter is used. For example, various devices for metering, control and protection of an electrical power system use FI R filters to estimate operating quantities, such as fundamental frequency voltage and current magnitudes. Another application could be to determine the magnitude of a specific vibration in a mechanical system.

PRIOR ART

A finite impulse response, denoted FIR, algorithm can be implemented as a digital filter used in digital signal processing for many industrial applications due to its many advantages over an infinite impulse response, de- noted H R, filter.

The inputs for a FI R are digital samples of an analog signal. The analog signal can be, for example, in form of alternating voltage or alternating current coming from an electrical power system. Such an analog signal is a periodic signal including a fundamental component and has an actual fundamental frequency that may deviate from a nominal fundamental frequency.

A nominal fundamental frequency is the fundamental tone of the analog signal at a nominal frequency and the actual fundamental frequency means the fundamental tone of the analog signal at an actual frequency. Correspondingly, a nominal fundamental period length is the pitch length of the fundamental tone of the nominal frequency and an actual fundamental period length is the pitch length of the fundamental tone of the actual frequency when a time domain is used. Commonly, a FI R filter is designed to rejects harmonics, typically a third harmonics. It is often optimally designed for an input signal with a nominal fundamental frequency, for example, 50 Hz in Europe or 60 Hz in USA, which means that the FI R filter works perfectly when the input signal is sampled at the nominal fundamental frequency. However, when an actual fundamental frequency deviates from the nominal fundamental frequency, such an optimally designed property is changed, which means at the off- nominal fundamental frequency, the total rejection of selected harmonics, such as a third harmonics, is lost.

Consequently, the applications using such a FI R filter may not able to sustain the frequency response at off-nominal fundamental frequency, which results in erroneous and poor performance of the applications. For example, in an electrical power system, a digital protective relay uses two or- thogonal FI R filters to calculate phasors in order to provide protection for power system equipments against abnormal conditions. However, such two filters are designed as a pair with prearranged characteristics. The filter pair should be orthogonal in the sense that the frequency response has unit magnitude and 90 degrees phase shift at a nominal fundamental fre- quency. These properties are used to calculate phasors. One of the filters gives the real part and the other gives the imaginary part. However, at the off-nominal fundamental frequency, the orthogonal property is not ideal and results in distortion of the estimated phasor because the filters are only orthogonal at exactly the nominal fundamental frequency. In other words, relay performance will improve with better phasor estimation at off nominal fundamental frequency. At dynamic conditions, such a deviation can be in a range of ±10% off-nominal fundamental frequency.

To enable a FIR filter preserve its optimal design properties, one possible solution is to resample samples at a rate which is a multiple of the actual fundamental frequency. For example, an analog-digital convertor operates at a higher fixed sampling frequency, for example, 8 kHz, and then samples are re-sampled at a lower sampling frequency, typically in the range from 1440 to 1760 Hz, which is a factor of the estimated frequency. The drawback with over-sampling and re-sampling is the additional burden to hardware and processor calculation. A method for improving the accuracy of digital sampling is presented in a paper, entitled "A new algorithm for improving the accuracy of periodic signal analysis", I EEE Transactions on Instrumentation and Measurement, Vol. 45, No 4, August. 1996, pp. 827-830. The basic idea of the proposed method is to modify the actual sampled sequence such that it becomes an ideal sample sequence which is synchronized with the signal subjected to sampling. The algorithm for modifying the sampled sequence is derived based on interpolation. However, the method in the paper uses a first or- der approximation, which may not be accurate enough for some applications, in comparison with a higher order approximation. Secondly, the approximation is based on the periodicity of sampled values. To determine whether a current sample is needed to be corrected or not, the current sample is compared with the corresponding sample from the nominal fun- damental period. If there is a deviation between the current sample and the corresponding previous sample, which means there is a deviation between the actual fundamental frequency and the nominal fundamental frequency, then the current sample will be corrected by the following equation, x_comcted (*) - x(k) + A [x(k + N) - x(k)] for k = 0, 1, 2, ... N- 1

where N is the number of samples corresponding to one nominal fundamental period. Because the presented algorithm is based on steady state samples of an entire period , a delay in fault detection may occur when such a sample sequence is used in a FI R filter which in turn is used in a signal processing application . For an application which is highly expected to have a quick response time to a detected fault, such a delay is unde- sired . For example, a digital protective relay is required to make a trip as soon as possible when a fault is detected .

Therefore it is desired to provide a FI R filter sample values which are able to keep the optimal design properties of a FI R filter at the off-nominal fundamental frequency, and at the same time, to ensure the FI R filter to response a detected fault quickly. OBJECTS AND SUMMARY OF THE INVENTION

The object of the present invention is to provide an improved method for constructing new sample values to be used a FI R algorithm.

This object is achieved by a method as defined in claim 1 .

Such a method is characterized in that the method comprises continuously measuring an actual fundamental frequency of the analog signal, sampling the analog signal at controlled time steps, storing the sample values as original sampling values, continuously determining the actual fundamental period length based on the measured actual fundamental frequency, determining a new sampling time step based on the actual fundamental period length such that the number of new sampling time steps in the actual fundamental period is the same as the number of original sampling time steps in the nominal fundamental period, calculating, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step, each of new sample values being calculated based on at least two original sample values, storing the new calculated sample values.

To achieve the object of the invention, First of all, a new sampling time step is determined based on the actual fundamental period length such that the number of new sampling time steps in the actual fundamental pe- riod is the same as the number of original sampling time steps in the nominal fundamental period. The determination of the new sampling time step is based on a measured actual fundamental frequency in proportion to the nominal fundamental frequency and the nominal fundamental period length. Then new sample values are constructed each by each at each new sampling time step. The construction of a new sample value is based on at least two original sample values. As a result, a finite number of new sample values in the actual fundamental period are constructed.

According to the invention, the method gives a highly expected accuracy when the maximum frequency deviation is 10%. In this case the finite number of the original samples is N_ = N_-1 xI≡≡±fL . ^_{-1 is} the

J actual number of original samples sampled during the nominal fundamental period, f_noπanal and f_actual are the nominal fundamental frequency and the actual fundamental frequency respectively.

Due to the fact that the method is able to provide a FIR filter with new calculated sample values so it looks like that the signal has been re-sampled at a multiple of the nominal fundamental frequency, it ensures that the FI R filter can preserve its optimally designed properties at an off-nominal fun- damental frequency and maintain the frequency response. This is a highly desired feature by many industrial applications, for example, when a FI R filter is used to suppress harmonics and when used in conjunction with a digital protective relay, the performance of the digital protective relay is improved by better phasor estimation at the off-nominal fundamental fre- quency.

According to the invention, yet another advantage, compared to the above- mentioned prior art, is that the present method is not dependent on a period of time during which the input signal is in steady-state. Therefore, no extra delay in the final FI R filter response is introduced by the method, which means that a quick response time for a detected fault when the FI R filter is used in a digital protective relay.

There is another advantage of the method. The method adjusts a new sampling time step within an algorithm in case that an actual fundamental frequency deviates from a nominal frequency. Therefore the method is not restricted, for the analog signal, by the sampling rate of hardware, which means the signal can be sampled in a fixed rate. With a fixed sampling rate, computation is efficient, and hardware design of a sampling device is simple comparing with an over-sampling or a down-sampling method.

According the invention, a further advantage is that the method can be implemented as a common algorithm to be used by all FI R filters within the same digital protective relay device. For example, a relay may include three FI R filters, a pair of FI R filters used to calculate the phasors, a third FI R filter may be used to detect second harmonics for stabilization of transformer differential protection. In this case, the new calculated sample values may be stored in memory storage reachable by all three FI R filters.

Yet another advantage is the method is very suitable for FI R filters with variable window lengths. This is because that the construction of a number of new samples is made before they are used by FI R filters. Because the construction is made only once and then the new calculated samples are ready for use by any FIR filter, thus the construction of new samples is not depending on the length of a filter, or filter coefficients.

According to an embodiment of the invention, each of the new sample values is calculated by a curve fitting method. With new sample values curve- fitted to a sinusoidal waveform, a high accuracy of the new calculated samples is achieved.

According to a specific embodiment of the invention, the curve fitting method is spline-curve fitting method. Spline interpolation means that each new sample is calculated separately based on its own dedicated curve, therefore, the spline interpolation combined with curve-fitting means that the new constructed samples may have high accuracy.

In an embodiment, each new sample is calculated based on three consecutive original sample values. This improves the accuracy of the new samples significantly comparing with the one where only two original samples are used to calculate a new sample.

In an embodiment of the invention, the method is implemented in FIR filters of a digital protective relay device used in an electrical power system. The analog signal is a voltage signal or a current signal receiving from an electrical power system. Such a protective relay uses a pair of FIR filters to calculate phasors in order to protect the power system against abnormal conditions. Furthermore the relay may use other FI R filters for other protection purposes. According to an embodiment of the invention, the present method is implemented by a computer program product defined in claim 7.

Such a computer program product comprises receiving a measured actual fundamental frequency of the analog signal, receiving sampled values of the analog signal, storing the sample values as original sampling values, continuously determining the actual fundamental period length based on the measured the actual fundamental frequency, determining a new sampling time step based on the actual fundamental period length such that the number of new sampling time steps in the actual fundamental period is the same as the number of original sampling time steps in the nominal fundamental period, calculating, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step, each of new sample values being calculated based on at least two original sample values, and storing the new calculated sample values.

Such a computer program can be loaded from a readable medium into the memory of a computing processor, for example, perform the above- mentioned steps to provide new calculated sample values for FIR filters.

The present method is carried out by a device defined in claim 9.

Such a device comprises a measuring unit configured to continuously measure an actual fundamental frequency of the analog signal, a sampling unit for sampling the analog signal at a controlled time step, a first storage for storing the sample values as original sampling values, a system period length determination module configured to continuously determine the actual fundamental period length based on the measured the actual fundamental frequency, a calculating unit configured to determine a new sam- pling time step based on the actual fundamental period length, and to calculate, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step such that the number of new sample values in the actual fundamental period is the same as the number of original sample values in the nominal fundamental pe- riod, and a second storage unit for storing the calculated sample values. Such a device can be integrated into a digital protective relay used in an electrical power system without changing any hardware design of the relay, wherein the digital protective relay may comprise a plurality of FI R filters for different protection purposes.

BRI EF DESCRI PTION OF TH E DRAW I NGS

The invention will now be explained more closely by the description of different embodiments of the invention and with reference to the appended figures.

Figure 1 shows a block diagram of a device for constructing new sample values, according to an embodiment of the invention .

Figure 2 is a flow chart illustration of the method and the computer program product according to an embodiment of the invention .

Figure 3a shows a setup for a simulation of two instances of a full cycle

DFT algorithm.

Figure 3b shows the results of the simulation shown in Figure 3a.

Figure 3c shows the plot view of the maximum relative magnitude error versus frequency for the simulation shown in Figure 3a.

Figure 4a shows simulation results of two instances of a half cycle DFT algorithm.

Figure 4b shows the plot view of the maximum relative magnitude error versus frequency for the simulation shown in Figure 4a.

Figure 5a shows an input signal represents a current during a fault sequence.

Figure 5b shows simulation results of the variable window FI R algorithm , the algorithm is fed with two different inputs, one is the original signal shown in Figure 5a and the other is the new calculated signal.

Figure 6 shows a block diagram, in which a device of the invention is in- tegrated in a protective relay device, according to an embodiment of the invention.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS OF THE INVENTION

In the following the method of the invention will be explained in conjunction with a nominal fundamental frequency 50 Hz, however the invention is also applicable to another nominal fundamental frequency, for example, 60 Hz.

Figure 1 shows a block diagram of a device 3 for constructing new sample values, according to an embodiment of the invention. The device comprises a sampling unit 1 , a first storage 2 and a second storage 4, a calculating unit 6, a sampling clock 5, a measuring unit 7 and a system period length determination module 8. The function of the sampling unit 1 is to sample a sinusoidal analog signal which has a nominal fundamental frequency, for example, 50 Hz, but the actual fundamental frequency of the signal may deviate from the nominal fundamental frequency. The input analog signal may be in form of a voltage V_1n or a current I_1n. A sampling unit can be, for example an A/D converter. The function of the sampling clock 5 is to control the sampling frequency of the sampling process. The analog signal can be sampled at either varied or fixed rate as long as it satisfies the Nyquist-Shannon sampling theorem which states the sampling frequency must be greater than twice the highest frequency to be sampled. When the signal is sampled, the original sample values X_n, X_n-i, ■■■ Xi are sent to the first storage either through program control or through a direct access channel. The first storage can be, for example, in form of RAM. The function of measuring unit 7 is to continuously measure the actual fundamental frequency factual of the analog signal V_1n or I_1n. Such a measur- ing unit is, in most cases, a micro-computing processor with a numerical algorithm that is based on correlation between sampled analog signal V_1n or /,„ and constant reference signals. The system period length determination module 8 is configured to determine the actual fundamental period length L_actuai- The calculating unit 6 is configured to calculate a new sampling time step and then calculate new sample values Y_n, Y_n-i, ... Yi at each new sampling time step. The inputs to the calculating unit 6 are the actual fundamental period length L_actuai and the original sample values X_n, X_n.i, ... Xi. The calculating unit 6 can be for example a micro-computing processor, a digital signal processor, a field-programmable gate array, or a standard computer. The output of the calculating unit 6 are the new cal- culated samples Y_n, Y_n^, ... Y₁ which are stored in the second storage 4. The second storage can also be, for example, a form of RAM.

Figure 2 is a flow chart illustration of the method and the computer program product according to an embodiment of the invention. It will be under- stood that each block of the flow chart can be implemented by computer program instructions.

Starting with receiving an actual fundamental frequency factual of the analog signal, block 10, which is measured by the measuring unit 7, the sampled values of the analog signal are received and stored as original sample values X_n, X_n. i, ... Xi in the first buffer, block 1 1 and 12. The analog signal is sampled at controlled time steps depending on the sampling rate of the sampling unit 1 .

The length of the first buffer is based on the nominal fundamental length of a filter window for the nominal fundamental tone and the deviation between the actual fundamental frequency and the nominal fundamental fre- quency, that is N_ = N_-1 x -^≡≡^ , N_-1 is the number of samples

J actual during the nominal fundamental frequency, For example, ^_MX ⁼ 2Ox — = 25 is needed if the actual fundamental frequency is 80% of 0.8 the nominal fundamental frequency.

Upon receiving the measured actual fundamental frequency factual of the analog signal, the actual fundamental period length L_actuai can be deter- mined in a known manner, block 13. Then a new sampling time step Δs is determined based on a measured actual fundamental frequency factual in proportion to the nominal fundamental frequency and the nominal fundamental period length such that the number of new sampling time steps in the actual fundamental period is the same as the number of original sampling time steps in the nominal fundamental period, block 14. For example, when the actual fundamental frequency is higher than a nominal fundamental frequency, the new sampling time step is shorter than the original sampling time step and each new sample value is approximated at each new sampling time step.

When an actual fundamental frequency is the same as a nominal fundamental frequency, then the number of the original samples for the actual fundamental period is the same as the signal is sampled during the nomi- nal fundamental period. For example, with a sampling rate as 1 kHz, 20 samples are expected for a nominal fundamental period, or a full-cycle of the nominal fundamental tone. However, when the actual fundamental frequency deviates from the nominal fundamental frequency, with the same sampling rate, although the same number of the original samples is still 20, they either cannot cover for a whole period of the actual fundamental tone or they cover more than one period of the actual fundamental tone.

With the new determined sampling time step Δs, new sample values Y_n, Y_n. 1, ... Y-I are calculated, block 15. To calculating a new sample value, a few number of consecutive original sample values are used to approximate the new value, for example three consecutive original sample values can be used to calculate a new sample value. The approximate is a curve-fitting based interpolation, which provides FI R filters with higher accurate new sample values and quick frequency response time. At the last step, new calculated sample values Y_n, Y_n.i, ... Y₁ are stored in the second buffer, block 16, to be ready used by the FI R filters.

The calculation of new sample values is a spline approximation with a second order polynomial in order to curve-fit the signal. At an instant, an analog signal is not a pure sinusoidal waveform, using only one function is impossible to approximate original samples for an entire nominal/actual fundamental period length. Therefore for one new sample value to be calculated, one curve is used, which means a plurality of curves are used for calculating the new samples for whole actual fundamental period length. Depending on the needs of FI R filters, it may be a case that only the samples of a half-cycle are needed.

Each part of a curve is a second order polynomial as follows, y(t) = a₀ + a_γt + a₂t² (1 ) where y presents a new sample value to be calculated at time point t. This polynomial approximates the signal x(t) representing by the original samples.

The notation x is used for the original sampled signal and y is used for the adjusted signal. Let the most recent sample value be x(t₀) , sampled at present time t₀ . The most recent value in the buffer goes through unchanged, that is y(t_o) = ^χ(t_o) (₂)

Because the polynomial curve is expected to have the same derivatives as the signal, so the first and second derivatives are calculated as,

dy(t, ) „ {-i \

- U₁ + z.a₂τ_k at d^2y(t^ = 2a₂ W dt

The first and second derivatives can further be calculated as follows, a_ϋ = y(t₀ ) = x(t₀) (5)

O₁ = ^- = \.5x(t_k) + 2x(t_k__l) - 0.5(t_k_₂) tø at a₂ = /2 = > = 0.5x(_(l )-x(v, ) ₊ 0.5xft_₂) ∞ at I at

For all frequencies, Ah is the magnitude of the relative slip. Ah = L no min al f

- 1 J nominal - J f a actual m f. actual /aacctual

where /jj_5ram≤

are the nominal fundamental frequency and actual fundamental frequency respectively, and the unit for Ah is per unit.

The calculation is slightly different dependent on if the system frequency is above or below the nominal fundamental frequency.

For an actual fundamental frequency below a nominal fundamental fre- quency, the adjusted values y(t₀ - k - h) , for£ = l, 2,...N , are calculated as

, _{I N} , , _{I N} dx(t_n - k - h) ₄ d²x(t_n - k - h) _{4 2} rv, j(t₀ - k - h) = x(t₀ - k - h) + — ^ -Δt + ^₁ W

Jt Jt where Δt is the accumulated time skew, calculated as At = Jt - Ah , meaning that a new sampling time step Δs has a factor of 1 + ΔA in relation with the original sampling time step.

For an actual fundamental frequency above a nominal fundamental frequency, the adjusted values y(t₀ - k - h) , for ^ = I, 2,...N , are calculated as do)

where Δt is the accumulated time skew, calculated as At = \ -k - Ah , meaning that a new sampling time step Δs has a factor of 1 -Δ/z in relation with the original sampling time step.

This is now demonstrated with a detailed calculation example where the actual fundamental frequency is below the nominal fundamental frequency that is 50 Hz. Let say the latest sample has index 10, that is t_o=1O and the oldest sample has index 1 and note y_k and x_k are the new sample to be calculated and the original sample with index k for simplicity. According to the formula 9, the first sample in the second buffer is

For the second sample in the second storage, the delay is k=1 , thus

dx

— - = -1.5x₉ + 2x₈ - 0.5x₇ dt

^-^ = 0.5x₉-x₈ + 0.5x₇

Substituting the derivatives gives y_g = x_g+At(-l.5x_g + 2x_ii-O.5x₇) + At²(θ.5x_g -x₈+0.5x₇)

The above formula can be reformulated as a FIR filter y_g = (1 - 1.5At + 0.5Δ^²)x₉ + (2At - Δ^²)x₈ + (0.5At - 0.5Δ^²)x₇

The accumulated time step for k=1 \sAt = Ah, thus y_g = (1 - 1.5ΔA + 0.5Ah²)x_g + (2Ah - Δ/*²)x₈ + (0.5Ah - 0.5Ah²)x₇

The accumulated time step for k=2 is At = 2Ah, thus y, = (1 - 3Ah + 2Δ/*²)x₈ + (4Ah - 4Ah²)x₇ + (Ah - 2Ah²)x₆

The general equation to calculate y_g,y_%,...y_l is y_w__k=b_o(k,Ah)x_W-_k+b_λ(k,Ah)x₉__k+b₂(k,Ah)x_%-_k &τk = 1,2,...9 where O₀₅O₁ and b₂ are calculated as follows, b₀ (k, Ah) = 1 - 1 MAh + OM²Ah² b_l(k,Ah) = 2kAh-k²Ah² b₂(k,Ah) = 0MAh-0M²Ah²

Let assume 45.45 Hz is an actual fundamental frequency, which gives the slip Ah ~ 0.10, and it results in

J₁₀ = X₁₀ y_g = 0.855x₉ + 0.19x₈ + 0.045x₇ y_& = 0.72Ox₈ + 0.36Ox₇ + 0.08Ox₆ y_η = 0.595x₇ + 0.51Ox₆ + 0.105x₅ y₆ = 0.48Ox₆ + 0.64Ox₅ + 0.12Ox₄ y₅ =0.375x₅+0.750x₄ + 0.125x₃ y_A = 0.28Ox₄ + 0.84Ox₃ +0.12Ox₂ j₃ =0.195x₃ + 0.910x₂ + 0.105x₁ y₂ = 0.12Ox₂ + 0.96Ox₁ + 0.08Ox₀ y_x = 0.055X₁ + 0.99Ox₀ + 0.045X_-1 Similar formulas for the case where the actual fundamental frequency is above the nominal fundamental frequency can be derived from the formula 10.

Figure 3a shows a setup for a simulation of two instances of a full cycle Discrete Fourier Transformation, denoted DFT, algorithm, in which the first DFT is fed by the original samples and the second DFT is fed by the new calculated samples and the test signal is an ideal sinusoidal with fixed amplitude and frequency is at 45Hz.

In this example the test signal is an ideal sinusoidal with fixed amplitude and constant frequency at 45 Hz. The behavior of the well-known full cycle DFT algorithm is simulated. A first instance of the algorithm is fed by the original signal x_vec20 and the second instance is fed by the recalculate signal y_vec20.

Figure 3b shows the results of the simulation shown in Figure 3a. The figure shows that the recalculated input gives a DFT, with a fix magnitude that is very close to 1 , which is the correct value. Moreover a drift in phase angle is observed, which is a natural and predictable phenomenon, in addition, there is no oscillation in phase angle. This becomes obvious when differentiating the phase angle, the derivative is constant and equals to the frequency deviation.

Figure 3c shows the plot view of the maximum relative magnitude error versus frequency for the simulation shown in Figure 3a. The plot shows that for the frequency range from 45 to 54 Hz the maximum relative magnitude error is less than 0.1 %. For frequencies between 54 to 55 Hz the er- ror is bounded to be less than 0.35%. This means that for almost all practical conditions the magnitude errors cause by the DFT is less than the tolerances of a measurement device, for example a measurement transformers.

Figure 4a shows simulation results of two instances of a half cycle DFT algorithm, in which the first DFT is fed by the original samples and the second DFT is fed by the new calculated samples and the test signal comprises a third harmonic with frequency 135 Hz corresponding to 45 Hz fundamental. The simulations are repeated for frequencies from 135 Hz to 165 Hz respectively, and the plot view of the maximum relative magnitude error versus frequency for the simulations is shown in Figure 4b. The plot shows that the rejection of the third harmonics is significantly improved with the new calculated values.

Figure 5a shows an input signal that represents a current during a fault sequence. The fundamental frequency of the input signal is 45 Hz and it contains a third harmonics with amplitude 5%, white noise with standard deviation 1 % and some exponentially decaying DC component. At the fault point the fundamental signal amplitude and phase change instantaneously.

Figure 5b shows simulation results of the variable window FI R algorithm, the algorithm is fed with two different inputs, one is the original signal shown in Figure 5a, and the other is the new calculated signal. It shows that the new calculated input gives better output, in the sense that the magnitude shows much less oscillation comparing with the one when the algorithm is fed by the original input signal.

Figure 6 shows a block diagram, in which a device 3 for constructing new sample values, according to an embodiment of the invention, is integrated in a protective relay device 40. In this arrangement, the digital protective relay 40 comprises an analog filter unit 22, a device 3 for constructing new sample values, three FI R filters 30, 30' and 30", a digital input unit 24, a relay setting unit 36, a relay logic unit 34 and a digital output unit 38. The components of the device 3 correspond to those in Figure 1 and have been given the same reference numerals, and will not be described in more detail here.

The input signal to the analog filter unit 22 can be either a voltage or a current analog signal received from an electrical power system. The analog filter unit 22 is configured to remove undesired frequency components and potentially damaging surges. The filtered signal is further sampled by the sampling unit 1 and sample values are stored in the first buffer, which can be in form of RAM. The new sample values are calculated by the computing unit 6 as described with reference to Figure 1 . The new calculated samples are further on provided to the FIR filters 30, 30' and 30". The functions of the FI R filters are to estimate the appropriate relaying quanti- ties, for example, a root mean square, or RMS value of a current, voltage or current phasor, or apparent impedance. For example, in this arrangement, the FI R filters 30 and 30' may be configured as a pair to estimate phasors of the voltage or current signal, while the third FI R filter 30" may be configured to detect a second harmonics for stabilization of a trans- former differential protection. The output from the FI R filter is sent to the relay logic unit 10. The digital input unit 24 is configured to provide the relay with contact position or voltage-sensing information which the relay need, the number of inputs may be in a range of 5 to 10. The digital input signals may also include transient voltages that must be buffered to pro- tect the digital device. The relay setting unit 36 is configured to store prearranged relay setting parameters. The function of the relay logic unit 34 is to compare estimated relaying quantities with pre- arranged relay parameters, perform a protective function and initialize the appropriate control action depending on the result of executing protective function. Such a control decision, for example may be a trip command. The digital output unit 38 is configured to send a control decision, in form of a contact digital signal to the substation equipment, for example, a trip signal is sent to a circuit breaker which executes the command. The control decision is, in many cases, communicated via a communication line 38 to another device within a substation.

This example shows one possible arrangement of the device according to invention; however other arrangements are also possible.

Claims

1 . A method for constructing new sample values to be used in a Finite Impulse Response, denoted FI R, algorithm, wherein a sampled analog signal contains at least a periodic fundamental component that has an actual period length that may deviate from a nominal period length, the method comprises:

- continuously measuring an actual fundamental frequency of the analog signal, - sampling the analog signal at controlled time steps,

- storing the sample values as original sampling values,

- continuously determining the actual fundamental period length based on the measured the actual fundamental frequency,

- determining a new sampling time step based on the actual fundamen- tal period length such that the number of new sampling time steps in the actual fundamental period is the same as the number of original sampling time steps in the nominal fundamental period,

- calculating, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step, each of new sample values being calculated based on at least two original sample values,

- storing the new calculated sample values.

2. The method according to claim 1 , wherein said new sample values are calculated by a curve fitting method.

3. The method according to claim 2, wherein said curve fitting method is spline-curve fitting method.

4. The method according to claim 1 , wherein each of new sample values is calculated based on three consecutive original sample values.

5. The method according to claim 1 , wherein said analog signal is a voltage signal or a current signal receiving from an electrical power system.

6. The method according to any of previous claims, wherein said FI R algorithm is implemented in a FI R filter in a digital protective relay device used in said electrical power system.

7. A computer program product for constructing new sample values to be used in a Finite Impulse Response, denoted FI R, algorithm, wherein a sampled analog signal contains at least a periodic fundamental component that has an actual period length that may deviate from a nominal period length and an actual fundamental frequency of the analog signal is con- tinuously measured, wherein the computer program product is directly loadable into the internal memory of a computer, comprising software to perform the steps,

- receiving a measured actual fundamental frequency of the analog signal, - receiving sampled values of the analog signal ,

- storing the sample values as original sampling values,

- calculating, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step, each of new sample values being calculated based on at least two original sample values, and

- storing the new calculated sample values.

8. A computer readable medium, having a program recorded thereon, where the program is to make a computer perform the steps of the claim 7, when said program is run on the computer.

9. A device for constructing new sample values to be used in a Finite Impulse Response, denoted FI R, algorithm, wherein a sampled analog signal contains at least a periodic fundamental component that has an actual pe- riod length that may deviate from a nominal period length, characterized in that the device (3) comprises:

- a measuring unit (7) configured to continuously measure an actual fundamental frequency of the analog signal, - a sampling unit (1 ) for sampling the analog signal at controlled time steps,

- a first storage (2) for storing the sample values as original sampling values,

- a system period length determination module (8) configured to con- tinuously determine the actual fundamental period length based on the measured the actual fundamental frequency,

- a calculating unit (6) configured,

• to determine a new sampling time step based on the actual fundamental period length such that the number of new sampling time steps in the actual fundamental period is the same as the number of original sampling time steps in the nominal fundamental period,

• to calculate, for a finite number of stored original sample values, a finite number of new sample values, using said new sampling time step, wherein each of new sample values is calculated based on at least two original sample values, and

- a second storage unit (4) for storing the new calculated sample values.

10. The device according to claim 9, wherein said calculating unit is con- figured to calculate new sample values using a curve fitting method.

1 1 . The device according to claim 10, wherein said curve fitting method is spline-curve fitting method.

12. Use of the device according to any of the claims 9-1 1 in a digital protective relay device for detecting faults in an electrical power system, wherein the relay device comprises a FI R filter.

13. A digital protective relay device for detecting faults in an electrical power system, wherein the relay device includes a FIR filter, comprises a device for constructing new sample values for a FI R filter according to any of the claims 9-1 1 .