EEE508 GÜÇ SİSTEMLERİNDE SİNYAL İŞLEME

Transcription

1 EEE508 GÜÇ SİSTEMLERİNDE SİNYAL İŞLEME Signal Processing for Power System Applications Frequency Domain Analysis Techniques Parametric Methods for Line Spectra (Week-5-6) Gazi Üniversitesi, Elektrik ve Elektronik Müh. Böl Güz Dönemi Prof. Dr. Özgül SALOR DURNA 1

2 SIGNAL MODEL General sinusoidal model for many signal applications: The sequence {e(t)} is complex (or circular) white noise and hence it satisfies: Note that σ 2 = E { e(t) 2 } is the variance (or power) of e(t). 2

3 Initial phases (φ k ) are independent random variables uniformly distributed on [π, - π] 3

4 SIGNAL MODEL Noise signal: If the correlation condition above is not satisfied, but we know the shape of the noise spectrum, we can filter y(t) by a linear whitening filter which makes the noise component at the filter output white; the sinusoidal components remain sinusoidal with the same frequencies, and with amplitudes and phases altered in a known way. We need to specify the sign of α k otherwise we are left with a phase ambiguity (α k, ω k, φ k ) and (-α k, ω k, φ k +π) are the same signals. Therefore we assume that α k > 0. 4

5 LINE SPECTRUM r(k) is the covariance function of y(t) HW 5: Using the signal definition on Slide 3, show that r(k) is as given on the left. FT of covariance function gives the power spectrum density: 5

6 LINE SPECTRUM Many applications have signals with (near) sinusoidal components. Examples: communications radar, sonar geophysical seismology power analysis Power Spectral Density OR 6

7 PARAMETER ESTIMATION 7

8 COVARIANCE MATRIX MODEL Define: n, # of sines m, # of data points A A is a Vandermonde matrix and hence: 8

9 COVARIANCE MATRIX MODEL Arrange data as: 9

10 COVARIANCE MATRIX MODEL Covariance matrix of y-(t) (HW 6: Show that R is as given below) 10

11 NONLINEAR LEAST SQUARES ESTIMATION OF LINE SPECTRUM Minimize the square of the error function for a data of length N: 11

12 NONLINEAR LEAST SQUARES ESTIMATION OF LINE SPECTRUM Error function can be rewritten as: Make the first term zero for β and minimize the last term: 12

13 NONLINEAR LEAST SQUARES ESTIMATION OF LINE SPECTRUM It can be shown that, as N tends to infinity, ω- obtained as above converges to ω (i.e., ω- is a consistent estimate). Unfortunately, the good statistical performance associated with the NLS method of frequency estimation is difficult to achieve. The function F has a complicated multimodal shape with a very sharp global maximum corresponding to ω-. Hence, finding ω- by a search algorithm requires very accurate initialization. 13

14 MUSIC METHOD Multiple Signal Classification (MUSIC) Method Let {λ 1 λ 2 λ m } denote the eigenvalues of R arranged in nonincreasing order, and let {s 1, s 2, s n } be the orthonormal eigenvectors associated with {λ 1 λ 2 λ n }, and {g 1, g n-m } set of orthonormal eigenvectors corresponding to {λ n+1 λ n+2 λ m }. n, # of sines m, # of data points A 14

15 MUSIC METHOD Hence: 15

16 MUSIC ALGORITHM Compute the covariance matrix: where Obtain the eigendecomposition (orthonormal eigenvectors) Sort eigenvalues and corresponding eigenvectors in decending order Take the smallest m-n eigenvectors to form mx(m-n)g matrix Obtain n highest peaks of the function: 16

17 PISARENKO METHOD Estimation accuracy of MUSIC increases with increasing m Pisarenko method decreases the computational complexity while sacrificing estimation accuracy. 17

18 OTHER PARAMETRIC METHODS ESPRIT (Estimation of Signal Parameters via Rotational Invariance Techniques) Min-Norm method High-Order Yule-Walker (HOYW) Method You can go to and download MATLAB functions for MUSIC, ESPRIT, Min-Norm and HOYW methods Reference for slides 2-17: Introduction to Spectral Analysis, Stoica and Moses, Prentice Hall. 18

19 EXAMPLES Synthetic signals with typical even and odd harmonic magnitudes are used 19

20 EXAMPLE 1: MUSIC METHOD 20

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22 EXAMPLE 2: MUSIC METHOD 22

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24 EXAMPLE 3: MUSIC METHOD 24

25 EXAMPLE 3: MUSIC METHOD 25

26 EXAMPLE 4: MUSIC METHOD 26

27 EXAMPLE 4: MUSIC METHOD 27

28 EXAMPLE 5: MUSIC METHOD 28

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30 PREPROCESSING BEFORE MUSIC Practically, the sharp contrast between the strong power at the power system fundamental frequency and the relative weak power at the harmonic or interharmonic frequencies will make the MUSIC algorithm less accurate in estimating the harmonics. Preprocessing is needed before applying the MUSIC algorithm. A simple preprocessing used in the above examples (where the data were obtained from the measurements) consists of applying the DFT, removing several spectral lines around the 50-Hz component, followed by an inverse DFT. The main disadvantage is that this may result in a pseudoharmonic at the cutoff frequency (in the above examples, at around 90 Hz) in the MUSIC estimation. The reason is that such a simple preprocessing is equivalent to applying an ideal filter with a rectangular shaped transfer function. Alternatively, a notch filter centered around the power system frequency or a high-pass filter with a stop band including the power system frequency can be applied. 30

31 ESPRIT METHOD RESULTS 31

32 COMPARISON OF RESULTS FROM ESPRIT AND MUSIC 32

33 DISCUSSION: MUSIC AND ESPRIT 33

34 HOMEWORK Covariance matrix of y-(t) on Slide 9 (HW 6: Show that R is as given below) r(k) is the covariance function of y(t) HW 5: Using the signal definition on Slide 3, show that r(k) is as given on the left. Due on May 1 st in class at