COMPUTATION OF THE NETWORK HARMONIC IMPEDANCE WITH CHIRP-Z TRANSFORM

Transcription

1 METROLOGY AND MEASUREMENT SYSTEMS Index , ISSN COMPUTATION OF THE NETWORK HARMONIC IMPEDANCE WITH CHIRP-Z TRANSFORM Krzysztof Duda, Dariusz Borkowski, Andrzej Bień AGH University of Science and Tecnology, Department of Measurement and Instrumentation, Al. Mickiewicza 30, Kraków, Poland ( kduda@ag.edu.pl, , borkows@ag.edu.pl, abien@ag.edu.pl). Abstract An innovative tecnique for power system armonic impedance computation is presented in te paper. We present a brief introduction to armonic impedance estimation problems as well as a detailed description of te proposed metod. Frequency analysis wit DFT is strongly affected by frequency leakage and picket fence errors. A number of metods was developed to reduce te impact of tese penomena including ardware solutions like PLL-based sampling syncronization, or software solutions like time- or frequency interpolation. Our work presented in te paper concerns new software-based metod for reducing frequency leakage and picket fence errors and an application of tis metod of power system armonic impedance estimation. Te proposed metod of armonic impedance computation is based on Cirp-Z transform (CZT) instead of discrete Fourier transform (DFT). CZT enables arbitrary sampling of te frequency axis wen computing spectra of discrete signals. We apply CZT to fundamental frequency estimation and for evaluating voltage and current frequency bins for iger armonics. Te spectral leakage is reduced by a standard Hanning window. Te accuracy of te proposed metod was verified by laboratory experiments and compared wit te results of DFTbased computation. Te proposed metod is robust against non-syncronous sampling, tus it is dedicated for widely used data acquisition systems wit a fixed sampling frequency. Te proposed metod is more accurate ten DFT-based computations in case of voltage and current signals sampled at a fixed frequency. Te advantages of te proposed metod are evident for signals wit off-nominal frequencies especially for iger armonics. Keywords: Cirp-Z transform, Fourier transform, impedance, armonic analysis, power quality Polis Academy of Sciences. All rigts reserved 1. Introduction Te armonic impedance of te power system is an important quantity describing te state of te power system. Te term armonic impedance means te equivalent spectral impedance of te power system, as seen from te measurement point towards te supply source of te system [1]. Te knowledge of te armonic impedance value allows one to evaluate te quality of te supply line i.e. its ability for armonics transfer [2], te risk of failure in case of overload and te risk of resonance between te system and te loads [3]. It is also elpful in making te decision weter it is safe or not to connect a new load to te existing power line or weter to redesign and rebuild te line. Moreover, te armonic impedance value is used for designing passive armonic filters and for controlling te active ones [4]. Some metods of armonic source location also utilize te armonic impedance value [5]. 2. Network Harmonic Impedance Bot measurements and computations of armonic impedance are difficult tasks witout general solutions. Recommendations for measurements and computations of armonic impedance are given in [1]. It is important to take notice tat armonic impedance is a time-

2 varying quantity. It varies due to reconfiguration of te power system and due to variation of loads connected on te measured side of te system. Neverteless te assumption of time invariance of te system parameters during te measurement is typically made. Measurement and computation of armonic impedance is a part of power system identification. Numerous metods of power system identification were developed over about 20 years of interest in te field. Most of tem utilize te Tevenin model of te power system (e.g. [4, 6-12]). Te Tevenin model includes an equivalent voltage source in series wit an equivalent impedance. It can also be generalized to include 3-pase systems. Te equivalent Tevenin circuit sown in Fig. 1 corresponds to a single frequency of interest. Te superposition (under te assumption of system linearity) of many equivalent circuits, identified for te range of armonic frequencies, gives a complete description of te power system dynamics. Tus te equivalent impedance Z sown in Fig. 1 is a function of frequency and is called network armonic impedance ( is te armonic s order). As te equivalent voltage source value is unknown, te armonic impedance as to be calculated from voltage and current armonics canges U and I respectively [1] Z = U / I, (1) were U and I stand for te differences of armonic currents and armonic voltages respectively, caused by te cange of te power system state. Tey are complex quantities, obtained using DFT (discrete Fourier transform) [1]. Harmonics canges U and I are te results of power system state cange, wic is caused by a load cange, at te assumption of constant source voltage and armonic impedance values during te measurement. Power system identification experiments can be divided into tree groups depending on te excitation source: 1) Harmonic current injection (invasive experiment). Te state cange is intentionally caused by so-called armonic generation devices [10], wic inject current armonics into te system. Te frequencies of te injected current sould be close to but different from te armonics existing in te system [7, 13], e.g. tose caused by nonlinear loads. Tis kind of experiment is te most common one, because it offers te igest accuracy. On te oter and it is difficult to conduct it in MV and HV networks, because it is necessary to consume/dissipate very ig energy in a very sort time in order to significantly disturb a MV or HV system. For example te disturbing load designed for identification of a 132 kv network, built by te Electricity Council Centre in England, occupies a specialized truck and its rated power equals 180 kw [13]. 2) Planned switcing of network equipment. Tis kind of experiment utilizes planned switcing or system reconfiguration as an excitation source. It includes capacitor bank switcing [8] and transformer energization, power line disconnection [14] or starting a big load [15]. Suc a switcing operation causes transients and delivers a sufficient amount of information about te system, tus it offers ig accuracy. Te disadvantage is tat suc an experiment cannot be performed anytime and usually cannot be repeated. It sould be well planned and agreed wit te utilities. 3) Observation of voltage and current canges caused by natural load variations (noninvasive experiment). Suc an experiment teoretically may be conducted at any point of te system and at any time, because it does not disturb te system. However significant load variations at te point of measurement are required to acieve satisfactory accuracy. In most cases suc an experiment requires long observation and offline data processing (load cange detection, averaging, etc.) [12, 16]. A relatively small number of publications concerns non-invasive experiments [11, 24].

3 Fig. 1. Measurement of armonic impedance. Tere are some issues connected wit a non-invasive experiment wic complicate tis apparently simple measurement problem. Tey are: 1) Insufficient excitation. Tis problem is common in non-invasive experiments, because current and voltage canges are often small (especially at iger frequencies) in comparison to noise (low SNR problem) [12]. 2) Syncronization problems. Te lack of sampling syncronization wit varying fundamental frequency of te signals plays an important role in DFT computations. It results in spectral leakage and calculation of armonic impedance for nonexistent components of te signals [17]. Te lack of syncronization between two states of te power system leads to wrong pase angles of te voltage and current increments and increases errors of armonic impedance estimation. 3) Time variability of te identified power system (i.e. variability of equivalent voltage source value or equivalent impedance). Time variability leads to erroneous identification results. For example, if te variation of te identified side of te system (from te measurement location) is greater tan te load variation, ten te measurement results will be closer to te equivalent load impedance [2]. Time variability enforces proper selection of te measurement location (e.g. at te secondary winding of a substation transformer [12]) as well as use of relatively sort measurement periods. Te researcers took into account various aspects of armonic impedance measurement. General recommendations for measurements and computations of armonic impedance in various cases are given in [1]. Time domain ARMA modeling and frequency domain modeling were analyzed in [6, 18]. Continuous, real-time LV power line impedance evaluation is proposed in [10, 19]. Spectral leakage and syncronization problems are solved in many ways, because tese problems are important also in oter kinds of measurements. Most popular are ardware PLL- (Pase-Locked Loop) [20] and software PLL- [21] based metods wic control te sampling rate of te signals. Oter autors use frequency domain [22] and time domain [17] interpolation metods or bot [23]. Te proposed cirp-z transform- (CZT) based metod for armonic impedance computation belongs to te group of software-based metods. In practice, evaluation of (1) is based on digital signals processed wit DFT and tus te results are strongly influenced by sampling conditions. Te standard [25] recommends syncronous sampling of network voltages and currents tat may be realized by a DAQ (Data Acquisition) system wit PLL as a syncronization circuit [20]. Te lack of sampling syncronization wit varying fundamental frequency of te power system is te reason of spectral leakage in spectrum computations and causes significant errors of armonic impedance estimation using (1). Te lack of sampling syncronization may also be te reason of misinterpretation of armonic frequencies. Te frequency step in DFT computations equals 5 Hz (for 50 Hz systems) [25] tus nominal (i.e. integer multiplies of 50 Hz) armonics are

4 evaluated. In case wen te fundamental frequency canges by a small amount of df Hertz, eac armonic canges times more i.e. for df Hertz, were is te number of te armonic. For example, a 0.25 Hz cange in te fundamental frequency canges te 20-t armonic frequency by 5 Hz, but DFT still considers 1000 Hz as te 20-t armonic, altoug it actually equals 995 Hz (for Hz fundamental frequency) or 1005 Hz (for Hz fundamental frequency) and 1000 Hz is not even present in te signal. Spectral leakage and insufficient frequency resolution are important sources of errors of armonic impedance computed by DFT. Tose errors are typically reduced by ardware (sampling wit PLL) or software (signal resampling [12]) solutions. In te following sections, we propose a new metod for network armonic impedance computation tat is robust against non-syncronous sampling. Te metod is based on application of CZT instead of DFT for armonic impedance computations. In case of syncronous sampling results of armonic impedance computation based on CZT and DFT are te same, but for non-syncronous sampling te accuracy of te CZT-based algoritm is significantly better. Te algoritm consists of two stages: 1) fundamental frequency estimation, and 2) spectrum evaluation for actual armonic frequencies tat may differ from nominal values. Bot stages are based on CZT and exploit its property of arbitrary sampling of te frequency axis. Te spectral leakage is reduced by a standard Hanning window [25]. 3. Cirp-Z Transform Te CZT algoritm was introduced in [26] wit te early applications in ig- resolution, narrowband frequency analysis and time interpolation of data from one sampling rate to anoter sampling rate. Since ten, only few papers appeared concerning te application of CZT to power system signals. In [27], [28] CZT is used for te accurate estimation of te power system fundamental frequency wic is next used for syncronization in PLL. In [29] CZT is used for te armonic analysis of te supply current of an induction motor. Te results reported in [27-29] confirm te superior accuracy of CZT performance over te classical DFT approac for analysis of power system signals. In te proposed metod we use CZT for fundamental frequency estimation and next for armonic frequency computation. Te continuous spectrum of te discrete time signal x[n], n = 0, 1,...N - 1 is defined by Fourier Transform (FT) as X ( e N 1 j ω jω n ) = x[ n] e. (2) n= 0 For te special case wen te frequency axis is sampled wit te step ω = 2π/N te DFT obtained from (2) is X[ k] = N 1 n= 0 x[ n] e j(2π / N ) kn, 0 N 1, 0 k N 1 n. (3) Te useful property of te CZT is tat te frequency axis in (2) may be arbitrarily sampled. It means tat one can coose any starting frequency ω 0, frequency step ω and te number of computed frequency bins M. For te special case wen ω 0 = 0, ω = 2π/N and M = N, te CZT equals DFT. Te algoritm for computing CZT on te unit circle in te Z transform plane is presented in Fig. 2. Te CZT parameters are defined as follows

5 ω = ω + k ω, k = 0,1,... M 1, k 0 j ω W = e, 2 n / 2 W, [ n] = 0, ( N 1) n M 1. (4) oterwise In practice, convolution from Fig. 2 is computed wit tree FFTs (Fast Fourier Transforms). If CZT parameters are fixed during filtering te spectrum of [n] is computed only once and te filtration is realized by two FFTs. Fig. 2. Computation of CZT by convolution in te time domain. Fig. 4a-b sows te spectra of te voltage signal acquired in laboratory experiment computed by FT, DFT and CZT. Te fundamental frequency equals 49.6 Hz. Te frequency step in DFT equals 5 Hz [6] and for CZT te frequency step is set to Hz. Te presented example illustrates ow te lack of sampling syncronization influences te amplitude spectrum of te DFT. On te oter and, computation of te spectrum from te same samples wit CZT is robust against non-syncronous sampling, in te sense tat one can always compute amplitude and pase values for te frequencies of interest (49.6 Hz in te example). Te spectra presented in Fig. 4c, d sow tat frequency bins computed by CZT are very accurate, wereas DFT bins computed wit a fixed 5 Hz step, may contain significant errors, as in te case depicted in Fig. 4d for te 7-t armonic. 4. Computation of armonic impedance Te proposed metod for armonic impedance computation is based on load variability. Te algoritm detects significant voltage and current canges in order to select fragments of signals comprising a sufficient level of information about te estimated impedance. Te algoritm simultaneously estimates te fundamental frequency, in order to establis CZT parameters for iger armonics analysis as well as to ensure a constant fundamental frequency during load canges. A block diagram of te proposed computation metod is presented in Fig. 3. Digital voltage and current signals u[n] and i[n] are inputs to te CZT 1. Parameters of CZT 1 are fixed, i.e. starting frequency f 1s = 47.5 Hz, frequency step f = Hz and number of bins M = Te outputs of CZT 1 are estimated frequencies and amplitudes of fundamental components of voltage and current signals computed as mean values for applied time window lengt, wic as te duration of 0.2 s. In eac iteration, te time window is sifted by alf of its lengt along te signal. We used te Hanning time window [25]. Voltage and current amplitude estimates A 1 are used for automatic detection of power system state canges. Frequencies and amplitudes of voltage and current signals for iger armonics are estimated by CZT 2. Frequency step in CZT 2 equals f 1 as estimated by CZT 1. CZT 1 and CZT 2 are computed sequentially; first CZT1 wit te objective of estimation of te actual fundamental frequency and ten CZT 2 for computing frequency bins for iger armonics. CZT2 - was set to compute M = 25 frequency bins (only for armonic frequencies) wit te step f equal to te actual value of te fundamental frequency and starting frequency f 2s also

6 equal to te actual value of te fundamental frequency, tus f and f 2s depend on te value of te fundamental frequency estimated by CZT1. Spectra computed by CZT 1 and CZT 2 for signals recorded in a laboratory experiment are presented in Fig. 4. Te fundamental frequency is estimated by CZT 1 as 49.6 Hz, next frequency bins for armonics are computed by CZT 2 for frequencies 49.6, = 2, 3, 4,...25, (e.g., te frequency of 7-t armonic equals = Hz, and not 7 50 = 350 Hz as it would be assumed in DFT computations). For comparison, Fig. 4 depicts spectra computed by FT and DFT. Te frequency step in DFT is fixed and equals 5 Hz [25], tus te value of te 7-t armonic is not even cecked. Fig. 3. Block diagram for computation of armonic impedance. a) b) c) d) Fig. 4. Exemplary results: a) estimation of fundamental frequency by CZT 1, b) zoomed view on fundamental frequency, c) frequency bins for iger armonics from 2 to 25 (only odd armonics are present), d) zoomed view on 7-t armonic.

7 A sliding time window may select fragments of signal wit different pase before and after power system state cange. Proper alignment of tose pases is obtained in te frequency domain by using te time sifting property of te Fourier Transform and te pase difference φ between U 1 and U 2 is taken into account in computation of armonic impedance as follows Z U = I U = I 1 ( e ( e jω jω ) e ) I jϕ 2 U ( e 1 jω ( e jω ) e jϕ ) 2. (5) 5. Laboratory experiment A laboratory model of te power system was built in order to verify te correctness of te proposed metod for armonic impedance computation in case of non-syncronous sampling. Te model, sown in Fig. 5, consisted of a programmable voltage source, te measurement system and a variable load. Te programmable Agilent 6812B voltage source allowed us to adjust its internal resistance, inductance, fundamental frequency, RMS voltage and armonics. Te measurement system consisted of a PC equipped wit a 16-bit DAQ card PCI-6036E from National Instruments working wit a sampling rate of Hz connected to a first-order antialiasing RC filter. A laboratory current transformer of class 0.5 wit a class 0.1 sunt resistor and voltage transformer of class 0.5 were used. Te load part consisted of a bulb, a current-limiting reactor (coke), a fluorescent lamp and an autotransformer loaded wit a resistance. Te internal impedance Z = R + jω L (R = 1 Ω and L = 1 mh) of te voltage source was estimated. Load variation was controlled manually, i.e. te autotransformer ratio was adjusted and te fluorescent lamp was turned on and off at random moments. Te fundamental frequency of te source was also manually adjusted during te experiments. Fig. 5. Laboratory model of te power system. 6. Results Te known armonic impedance of te laboratory power system model (Fig. 5) was estimated using voltage and current signals acquired during a 1-minute-long experiment. Te measurement point was placed between te identified part of te model and te load. Te analysis was performed off-line in an Matlab environment. Fig. 6 sows te results computed by CZT 1 from te block diagram in Fig. 3. Fig. 6a depicts canges of fundamental

8 component amplitudes of voltage and current signals acquired during te experiment. Fig. 6b sows fundamental frequency values of voltage and current. Power system state canges tat are necessary for armonic impedance estimation result from load canges. Terefore signal fragments, used in (5), describing two different power system states, sould be located before and after significant load cange. For detection of power system state canges, a simple algoritm based on te difference of two successive values of te voltage amplitude computed by CZT 1 was used. For clear presentation of te proposed metod te fragment of te signal presented in Fig. 7 was selected. Fig. 7 sows state canges, marked by circles and described by numbers 1 to 6, used for impedance computation according to (5). Te same plot sows te value of te fundamental frequency as (5) is only valid for constant fundamental frequency. For te analyzed load canges, te fundamental frequency was equal to 50 Hz (cange 1-2), 50.2 Hz (cange 3-4) and 50.4 (cange 5-6). Harmonic impedance estimates for CZT- and DFT- based computations as well as actual impedance values are presented in Fig. 8. Reference impedance values denoted in plots as Z ref were computed from known parameters of te load. Fig. 8a, c sow results obtained for CZT and DFT respectively for odd armonics from 1 to 25, as only odd armonics ad significant presence in te signal. Fig. 8b, d and Tables 1-2 present estimation errors computed as err abs( Z ) abs( Z ) 100% ref =. (6) abs( Zref ) Numbers in te legends in Fig. 8 denote power system state canges used for armonic impedance computation depicted in Fig. 7. Te constant fundamental frequency during te state cange was an additional constraint for impedance estimation. Te fundamental frequency during state cange denoted in Fig. 7 as 1-2 equals 50 Hz, wic means tat signals are syncronously sampled. In tis case te fixed 5 Hz frequency step in DFT ensures correct results of impedance computation. Te impedance estimation errors for CZT and DFT metods are practically te same. For 3-4 load cange te fundamental frequency equals 50.2 Hz. Tis is te case of nonsyncronous sampling. CZT results are still correct. As seen from Tab.1 te impedance is estimated for multiplies of te actual (measured) fundamental frequency and not for multiplies of te nominal 50 Hz frequency like in te DFT case. Results obtained from DFT computations are difficult for pysical interpretation because te impedance is estimated for frequencies at wic te excitation (current armonics) does not exist. For example te impedance is calculated for 150 Hz instead of te actual armonic frequency equal to Hz. Analog voltage and current signals do not contain components of suc frequency, so te impedance for tis frequency sould not be calculated. Te nonzero numerical results obtained wit te DFT for suc frequency are te effect of te convolution of te signal spectrum wit te spectrum of te time window. Tis is also te reason wy frequency lobes ave a widt of a few Hz, (as seen in Fig. 4) and tis ensures reasonably-looking results (small impedance errors). For 5-6 load cange te fundamental frequency equals 50.4 Hz. CZT results are still correct and for DFT significant errors appear for iger armonics. DFT errors are caused by misinterpretation of armonic frequencies e.g., DFT incorrectly interprets te 19-t armonic as 950 Hz wereas te actual value equals Hz. Tis case illustrates te advantage of using CZT instead of DFT for impedance computation. In real-world power systems te fundamental frequency deviates sligtly around a nominal value, tus ardware or software syncronization is a must. Te proposed software solution is an alternative for syncronous sampling; it can be as well used for off-line analysis.

9 a) b) Fig. 6. Computation results for signals acquired in laboratory experiment, obtained from CZT 1 (see Fig. 3): a) voltage and current amplitudes of fundamental frequency components, b) voltage and current fundamental frequency. Te fundamental frequency was manually adjusted in te power source during te experiment, tus its values sow rapid canges wic are not te result of variable power demand. Fig. 7. Power system state canges; numbered circles stand for measurements taken for impedance computation.

10 a) b) c) d) Fig. 8. Harmonic impedance estimates: a), b) proposed metod and estimation errors, c), d) DFT metod and estimation errors. Denotations: Z ref - reference impedance values computed from parameters of te load, numbers stand for state canges depicted in Fig Conclusions Te paper presents a new metod for armonic spectral impedance computation dedicated to digital signals sampled at a fixed frequency. Te novelty of our approac relies on using CZT instead of DFT for computing voltage and current spectra. CZT is used for fundamental frequency estimation and ten iger armonic estimation. Higer armonic spectra are computed at exact armonic frequencies even in te case wen te fundamental frequency sligtly differs from its nominal value. Tis feature is not available in te standard DFT metod because of its fixed frequency step. Te proposed metod was tested in laboratory experiments wit satisfactory results. It can be noticed from te results tat te metod may be especially useful for armonic impedance measurements in case of lack of ardware sampling syncronization or for off-line analysis of signals already acquired wit a fixed sampling frequency. Te obtained results sow tat te proposed metod offers better accuracy of armonic impedance estimation tan DFT-based computations especially for iger armonics (an example of a well-suited application may be te estimation of power network armonic impedance for power line communications) in te case wen te fundamental frequency differs from its nominal value. An additional advantage of te proposed CZT-based metod is monitoring of power system fundamental frequency.

11 Te computational cost of te proposed metod can be rougly evaluated by te number of FFTs executed during computations. Te standard metod based on syncronous sampling needs one FFT for computing te wole spectrum of te signal, wile te proposed metod requires two FFTs for te first CZT wit fixed parameters and tree FFTs for te second CZT wit te frequency step dependent on fundamental frequency tat varies in te power system. Bot CZT are computed for te same discrete signal, tus overall computations require 4 FFTs. Table 1. Harmonic impedance estimation errors for CZT f [Hz] err [%] f [Hz] err [%] f [Hz] err [%] References Table 2. Harmonic impedance estimation errors for DFT f [Hz] err [%] f [Hz] err [%] f [Hz] err [%] [1] A. Robert, T. Deflandre, Joint CIGRE/CIRED Working Group CC02: A guide for assessing te network armonic impedance. Int. Conf. Electricity Distribution (CIRED 97), June 2 5, 1997, Paper 2.3. [2] O. Gonbeau, L. Bertet, J.L. Javerzac, D. Boudou: Metod to determine contribution of te customer and te power system to te armonic disturbance. Proc. of XVII International Conference on Electricity Distribution CIRED 2003, Barcelona, 2003.

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