ABSTRACT - This paper applies to a practical test case two schemes for determining harmonics in power systems that are not limited to stationary waveforms, ...
IEEE Transactions on Power Delivery, Vol. 11, No. 4, October 1996
2020
n-Line Tracking of Chan Svstems: Application to
onics in Stress ro-Qu6bec Networ
W
I. KAMWA, MEMBE.R
D. MCNABB, NON MEMBER
R. GRONDIN, MEMBER
Etudes de Re'seaux, Hydro-Que'hec
Imtitut de recherche d'Hydro-Que'hec IREQ, 1800 m o d e Ste-Julie, Varennes, Canada J I X IS1
ABSTRACT- This paper applies to a practical test case two schemes for determining harmonics in power systems that are not limited to stationary waveforms, but can equally estimate harmonic phasors in waveforms with time-varying amplitudes and changing fundamental frequency. The first scheme relies on the short-time Fourier analysis performed using the Fast Fourier 'Ikansform (FET) with time-domain windowing and frequencydomain interpolation. Another scheme combines fundamental-frequency tracking with a Kalman filter based harmonic analyzer, yielding a uniformly sampled, self-synchronizing harmonics tracker. The performance of these advanced measurement schemes for changing harmonics is illustrated convincingly on nonstationary waveforms generated in the EMTP using a realistic series-compensated transmission network. Keywords: series-compensated network, short-time Fourier transform, Kalman jiltel; spectral interpolation
I. INTRODUCTION Awareness of nonstationary harmonics in power systems has been on the rise recently [ I ,2]. In our case, three facts attracted our attention to this problem: 0
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Most electronically switched industrial loads found in mining, refining and melting processes, paper-mills, etc., are dynamic in nature. In normal operation, repeated stopktart and braking/acceleration cycles tend to generate significant speed variations resulting in time-varying current amplitudes [ 1-41, An increasing number of high-voltage transmission systems now comprise static var compensators, which can experience h m o n i c resonance [22], during geomagnetic storms for example. In fact, constant advances in FACTS suggest that future power systems will depend more and more on thyristor-switched controllers such as the GTO-based STATCON and lJPF devices [5]. Moreover, despite the amount of harmonic filtering they usually have, high-voltage directcurrent systems can still generate current and voltage harmonics at meaningful levels, under constraining operating conditions [6]. 96 WM 126-3 PWRD A paper recommended and approved by the IEEE Transmission and Distribution Committee of the IEEE Power Engineering Society for presentation at the 1996 IEEEPES Winter Meeting, January 2125, 1996, Baltimore, MD. Manuscript submitted July 14, 1995; made available for printing December 8, 1995.
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855 Ste-Cathe'rine E., Place Dupuis Montre'al, Canada H2L 4P5
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Finally, series compensation yields a sub-synchronous resonance with very low frequency that interacts with static var compensators to produce amplitude-modulated harmonics and interharmonics during transformer saturation [7]. Severe disturbances further complicate this phenomenon by modulating the fundamental frequency, which ultimately yields harmonics with changing amplitudes and frequencies.
Accurate measurement of harmonic levels is essential for several reasons. 0
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Improving the design and fine-tuning the harmonic filters necessary near power converter stations [6]. Monitoring, in real-time, the stress to which these devices are subjected under severe disturbances [8]. Specifying digital and adaptative filtering requirements for advanced phasor measurement units, with a particular interest for series-compensated networks [7].
The paper is organized as follows. In Section I1 and 111, we take another look at harmonic estimation by the short-time Fonrier transform (STFT) and Kalman filtering [16], with a view to improving their performance for nonstationary waveforms. This is achieved by first estimating the fundamental frequency, then using this to correct raw harmonic estimates by spectral interpolation [9- 111 or centre-frequency adaptation. The next section compares these two approaches to nonstationary harmonics, based on waveforms arising from EMTP simulations of Hydro-QuCbes's series-compensated network. Section V discusses the results of this study in light of implementation.
11. IMPROVED FFI'-BAsEDHARMONICANALYSIS
2.1 Spectral Interpolation: Why and How ? It has been demonstrated extensively that the use of spectral windows improves harmonic estimation by mitigating leakage and picket-fence effects. With no windowing, a fundamental frequency offset of 0.4 Hz produces an error of 10% on the amplitude of the fifth harmonic, while a Hanning window reduces this error to 2% [IO]. Under stationary conditions, optimized flat-top windows yield an even more spectacular performance, reducing spectral leakage altogether and amplitude losses to insignificant levels [3]. However, this performance has a cost: long measurement times (generally more than 3 cycles of the fundamental) are necessary for acceptable spectral separation. By contrast, nonstationary conditions require short averaging windows to keep track of fast changes. This is achieved 00 0 1996 IEEE
2021 using high-resolution spectral windows [ 121. But as expected, these highly selective windows are subject to strong amplitude loss, because their magnitude response drops sharply for frequencies just slightly away from the peak (Figql).A classical remedy to this problem consists first in accurately determining the dominant frequency in the signal, and then correcting the phasor estimate according to the frequency response of the weighting window. The whole procedure is known as spectral interpolation [ 11,121.
N=M=96 and K=1[23]. Selected bells of the corresponding filter bank are shown in Fig. l. Thrw-cycle Taylor window
Frequency estimation is usually performed in two steps: Coarse search: the fundamental frequency is found at the bin with the maximum spectral amplitude. For instance, if Npl = argmax(/Xi) ( Z E {O ,..., ~ - 1 } ) , t h e n a c o a r s e e s t i mate of the fundamental will be: fp I = (FsNp ) / N in Hertz. Fine search: the coarse estimate is accurate only if the l?W was performed on an integer number of cycles of the fundamental, which is hardly the case, especially under non stationarity. A correction factor based on parabolic fitting through three adjacent spectral samples has been proposed for improving this estimate [ 141:
0 Hz
FIGURE 1. A Taylor-window basedfilter bank f o r STFX Following the above interpolation procedure, analysis of the benchmark signal defined in appendix yields the remarkable performance depicted in Fig. 2. The frequency value obtained by interpolation is quite accurate (Fig. 2a), although slightly lagging the true value. Sirnilar comments apply to the amplitude deviation in Fig. 2c. la) Fwuencv deviation
where lii2is the output of the magnitude squared phasor at the I-th bin. The correction factor applied to the coarse bin number gives an improved estimate of the fundamental frequency: "0
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1
15
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25
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I
I
I
5%
(b) Correction factor
When the line frequency is well-known, the initial amplitude of the phasor can be corrected by taking due account of the frequency response of the spectral analysis window, since it constitutes the main source of amplitude distortion [ll]. To outline how this can be achieved in practice, let us assume that the spectral window belongs to the multi-cos family, i.e. its impulse response can be computed according to the following law:
w ( n ) = 1t x d , c o s ( 2 l i i & ) , n
= O,1,
...M - 1
(EQ3)
1 0 4 / - - - 1 7 -
1
I
SeC
(c) Amplitude deviation I
-2 0-
0
05
1
I
'
15
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25
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SeC
i= 1
According to Rife and Vincent [13], the correction factor based on (3) is given by:
I~N~,I
, where the coarse estimate of the fundamental amplitude, should be multiplied by this correction factor in order to obtain a more accurate approximation of the amplitude:
A
=
K~I-%, I
(EQ 5 )
To illustrate this procedure, we have chosen a three-cycle Taylor window [13] with the following settings of STET:
FIGURE 2. Assessing the interpolation procedure at fundamental frequency. A c t u a l ; -----Estimated.
In fact, the correction factor (Fig. 2b) is an image of the amount of error incurred by a standard uncorrected Fourier transform. At peak frequency deviation, this error is nearly 3% of the 1 p.u. full scale which, when reported on a 1% peak deviation, means a blaring 300'% error!
2.2 Generalizing interlpolation to harmonics Since interpolation works so well at fundamental frequency, it may be useful to extend the same procedure to harmonic frequencies. Each time the fractional bin correction is known at fundamental frequency, the implied correction at harmonic frequencies is given by 6, =- h6 , with h the harmonic index. It follows. then, that the raw harmonic amplitudes yielded by the
2022 STIT can be corrected according to (4) simply by replacing 6 with 6 , . We applied this scheme to the benchmark signal and obtained the harmonic-correction factors portrayed in Fig. 3 . (a) 2nd, 3rd and 4th Harmonics 1.21
With the help of these factors it was possible to improve the STFT-based harmonics (Fig. 4) to such an extent that the resulting amplitude estimates are very close to the exact ones (cf. Appendix, Fig. 18).
I
111. IMPROVED KALMAN FILTER-BASED HARMONIC ANALYSIS
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sec (a) 5tk 6th and 7th Harmonics
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It is well known that a Kalman filter can be used for harmonic estimation when the fundamental frequency is fixed [4,16]. However, although the analyzer can track time variations, its capability in this respect is restricted to amplitude variations and phase jumps. During frequency changes, it is bound to fail (cf. Appendix, Fig. 18), because it cannot automatically retune itself to the new incoming frequency. The only way to solve this problem is to allow for centre-frequency trackmg, using for instance, the easy-to-implement computational structure illustrated in Fig.5.
3.5
SBC
3. STFTcorrection factors: the benchmark signal. (a)-2nd; ----3rd;.....4th; ( b ) -5th; ----6Eh;.....7th.
I
i Frequency Estimation
Kalman-Filte Based Harmonic
Analyzer
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