Time-Frequency Analysis of Non-stationary Phenomena in Electrical Engineering Antonio Bracale, Guido Carpinelli Universita degli Studi di Napoli “Federico

Time-Frequency Analysis of Non-stationary

Phenomenain Electrical Engineering Antonio Bracale, Guido Carpinelli

Universita degli Studi di Napoli “Federico II”

Zbigniew Leonowicz, Tomasz Sikorski, Krzysztof Wozniak

Wroclaw University of Technology, Poland

Modern Electrical Power Systems - MEPS 06 September 06-08, 2006, Wroclaw, Poland

Contents of presentation

• Motivations for applying time-frequency analysis in electrical engineering

• How to obtain time-frequency representations?

• Mathematical backgrounds of applied tools• Investigated non-stationary phenomena• Results of investigations• Conclusions

Motivations

• Increasing level of non-stationary phenomena in contemporary power systems and its influence on power quality

Converter systems generate a wide range of characteristic harmonics typical for the ideal converter operations, but also in some cases they become a source of non-characteristic harmonics.

The duration time of some transient states can reach values up to 5-10 periods of basic component.

• Limitation of one-dimensional Fourier spectrum and new trends in signal processing for designing comprehensive and adaptive algorithms

Classical Fourier spectrum loses the information about transient character of investigated phenomena. Non-stationarity is spread out over the whole frequency domain.

Fourier algorithm with sliding window has a inseparable tradeoff between window width and time-frequency resolution.

To know how to merge two dimensions in one

Spectrograms:- Gabor- STFT- S-transform

Scalograms:- Wavelet transform

Spectrograms:Bilinear:Cohen's class (TFC):- Wigner- Wigner-Ville- Page- Levin- Margenau-Hill- Rihaczek- Born-Jordan- Choi-Williams- Zhao-Atlas-Marks- Butterworth

Scalograms:Affine Wigner:- Bertrand-Bertrand- Rioul-FlandrinQ distribution:- Eichman-Marinovic- Altes

How to obtain joint time-frequencyrepresentations?

Sliding window and differentspectrum estimation methods

Two-dimensional non-parametric equations

Linear Non-Linear

Mathematical backgrounds – STFT

• STFT is classical method of time frequency analysis

• Involves both time and frequency and allows a time-frequency analysis or in other words a signal representation in the time-frequency plane

• The width of analysis window is fixed = constant time-frequency resolution for all frequency components

• Time-frequency resolution is dependent of analysis window width

• Wide window good frequency resolution, poor time resolution

• Narrow window good time resolution, poor frequency resolution

* 2STFT( , ) x( ) e di ftf t t t

Short-Time Fourier Transform-Spectrogram Hamming, width 0.25s

Time [s]

Fre

qu

en

cy

[H

z]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

Short-Time Fourier Transform- SpectrogramHamming, width 1s

Time [s]

Fre

qu

en

cy

[H

z]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

Short-Time Fourier Transform - SpectrogramHamming, width 0.02s

Time [s]

Fre

qu

en

cy

[H

z]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

STFT cannot be used successfully to analyze transient signals which contain high and low frequency components simulatneously

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time

am

plitu

de

Non-stationary signal with freguency components occure in different intervals time100Hz 0-270ms ; 10Hz 270-500ms; 25Hz 500-760ms; 50Hz 760-100ms

Mathematical backgrounds – S-Transform

• S transform is conceptually a hybrid of STFT and wavelet analysis, containing elements of both but falling entirely into neither category

• S transform uses a moving analyzing window but unlike STFT the width of the window is scaled with frequency as in wavelets

• The width of analysis window is the inverse of the frequency = frequency-dependent resolution

• S transform performs multi-resolution analysis on the signal, gives high time resolution at high frequencies and high frequency resolution at low frequencies

2 2( / 2 )1( ) ,

2t k

t ef

2 2 2(( ) / 2 ) 2( , )2

t f k i ftfS f x t e e dt

k

2S( , , ) ( , ) e di ftf x t t t

tim e

frequency

Resolution of time-frequency plane w hen the signalis view ed in the S-Transform representation,

having frequency dependent resolution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time

am

plitu

de

Non-stationary signal with freguency components occure in different intervals time100Hz 0-270ms ; 10Hz 270-500ms; 25Hz 500-760ms; 50Hz 760-100ms

S Transform

Time [s]

Fre

qu

en

cy

[H

z]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

Short-Time Fourier Transform-Spectrogram Hamming, width 0.25s

Time [s]

Fre

qu

en

cy

[H

z]

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

Mathematical backgrounds – Cohen’s Class

General comments:The equation leads to two-dimensional time-varying spectrum which represents

the energy changes of frequency components, here called auto-terms (a-t). Unfortunately, bilinear nature of discussed transformations manifests itself in existing of undesirable oscilating components, called cross-terms (c-t).

TFC , x x , e e e d d d2 2

j t j j ux tt u u u

- time-frequency representation belonging to Cohen’s class

,t - kernel function of chosen time-frequency representation

x t - investigated signal- time variable- time shift- additional integral time variable

- angular frequency- angular frequency shift

Legend:

tu

TFC ,x t

instantaneousautocorrelation function

kernelfunction

Mathematical backgrounds – Cohen’s ClassBasic level of adaptation for signal analysis –selection of the kernel

function:WIGNER-VILLE(constant kernel)

CHOI-WILLIAMS(Gaussian kernel)

ZHAO-ATLAS-MARKS(cone-shaped kernel)

, 1t

2

,t e

sin

2,

2

t h

CWD ,x t

ZAMD ,x t

WVD ,x t

Additional level of adaptation for signal analysis – applying smoothing function:

x tSignal

Pseudo-time-frequencyrepresentation

h

Smoothed pseudo-time-frequencyrepresentation

g t

smoothing effect along frequency axis smoothing effect along time axis

PTFC ,x t SPTFC ,x t

Investigated phenomena – switching of capacitor banks

T – transformer HV/MV, Δ-Y connected,25 MVA, 110kV/15kV

First capacitor: 900kVar, 0.2km from the station, switching on at 0.03sSecond capacitor: 1200kVar, 1.2km from the station, switching on at 0.09s

• One-phase diagram of simulated distribution system • Fragment of current waveform at

MV busbar (a) and its spectrum (b)

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3x 10

5

frequency (Hz)

ener

gy d

ensi

ty s

pect

rum

(J)

200 300 400 500 6000

0.5

1

1.5

2

2.5

3

3.5

4x 104

frequency (Hz)

ener

gy d

ensi

ty s

pect

rum

(J)

Zoom

b)

0 0.04 0.08 0.12 0.16 0.2-8

-6

-4

-2

0

2

4

6

8

10

time (s)

curr

ent (

A)

x10 3

switching on1200kVar

switching on900kVar

a)

HV MV

0.2kmLine

TransformerDY 110/15kV

25MVA

measurement

1.0kmLine

900kVar 1200kVar

0.03s 0.09s Load

a) |S-transform|

600

500

400

200

100

00 0.05 0.10 0.15

time (s)

300

700

freq

uenc

y (H

z)

0.20

270Hz

475Hz

50Hz

Fig. 2a. Time-varying spectrum of switching on the capacitor banks phenomena obtained using S-Transform.

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

5

time (s)

|ene

rgy|

(J)

Dynamism of tracking the energy changes -475Hz component

Short-Time FourierTransform with Hammingwindow, width 0.04s

S-Transformt=0.03s

switching on900kVar

b)

Fig. 2b. Comparison of STFT with S-Transform for tracking capability.

Time-frequency analysis of investigated phenomena – switching on the capacitor banks

Fig. 1. Time-varying spectrum of switching on the capacitor banks phenomena obtained using STFT.

• Short-Time Fourier Transform vs S-Transform

0.20

600

500

400

freq

uenc

y (H

z)

200

100

00 0.05 0.10 0.15

time (s)

300

700

270Hz

475Hz

50Hz

h |Short-Time Fourier Transform|- Spectrogram - Hamming, width - 0.04s

Fig. 4a. Time-varying spectrum of switching on the capacitor banks phenomena obtained using Wigner-Ville Distribution.

a)

600

500

400

freq

uenc

y (H

z)200

100

00 0.05 0.10 0.15

time (s)

300

700

270Hz(a-t)

475Hz(a-t)

50Hz(a-t)

372.5Hz(c-t)

262.5Hz(c-t)

|Wigner-Ville Distribution|

160Hz(c-t)

0.20

(a-t): auto-terms(c-t): cross-terms

600

500

400

freq

uenc

y (H

z)200

100

00 0.05 0.10 0.15

time (s)

300

700

sup

pre

ssio

n e

ffec

to

f t

he

(c-t

)re

gar

din

g t

o k

ern

elfu

nct

ion

270Hz(a-t)

475Hz(a-t)

50Hz(a-t)

|Choi-Williams Distribution| - Gaussian kernel function, 0.05 b)

0.20


Fig. 4b. Time-varying spectrum of switching on the capacitor banks phenomena obtained using transformation with Gaussian kernel.

600

500

400

freq

uenc

y (H

z)200

100

00 0.05 0.10 0.15

time (s)

300

700

270Hz(a-t)

475Hz(a-t)

50Hz(a-t)

c)

0.20

|Zhao-Atlas-Marks Distribution| - Cone-Shaped kernelfunction based on Hamming, width - 0.04s


sup

pre

ssio

n e

ffec

to

f t

he

(c-t

)re

gar

din

g t

o k

ern

elfu

nct

ion

Fig. 4c. Time-varying spectrum of switching of the capacitor banks phenomena obtained using transformation with cone-shaped kernel.


Fig. 3. Time-varying spectrum of switching on the capacitor banks phenomena obtained using STFT.

• Short-Time Fourier Transform vs Cohen’s class of transformation

0.20

600

500

400

freq

uenc

y (H

z)

200

100

00 0.05 0.10 0.15

time (s)

300

700

270Hz

475Hz

50Hz

h |Short-Time Fourier Transform|- Spectrogram - Hamming, width - 0.04s


Fig. 5. Comparison of STFT with Cohen’s class on account of tracking ability, when transient 475Hz component (a) and basic 50Hz component (b) are detected.

• Short-Time Fourier Transform vs Cohen’s class of transformation

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

time (s)

|ene

rgy|

(J)

Short-Time Fourier Transform withHamming window, width 0.04s

Choi-Williams Distribution withGaussian kernel function,

Zhao-Atlas-Marks Distribution withcone-shaped kernel function basedon Hamming window, width 0.04s

0.05


t=0.03sswitching on

900kVar

a)

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3

3.5x 10

5

time (s)

|ene

rgy|

(J)

Zhao-Atlas-Marks Distribution withcone-shaped kernel function basedon Hamming window, width 0.04s

Short-Time Fourier Transform withHamming window, width 0.04s

Choi-Williams Distribution withGaussian kernel function, 0.05


t=0.03sswitching on

900kVar

t=0.09sswitching on

1200kVar

b)

General comment: Selection of appropriate kernel function allows to achieve sharp detection of the beginning of transient states.


Fig. 6. Comparison of STFT (a) with smoothed version of Cohen’s class of transformation (b), on account of tracking tracking ability, when different width of smoothing window are applied.

• Short-Time Fourier Transform vs Smoothed version of Cohen’s class of transformation

0 0.05 0.1 0.15 0.20

1

2

3

4

5

6

7x 10

5

time (s)

|ene

rgy|

(J)

a) Influence of the window width on dynamism of Short-Time Fourier Transform



t=0.09sswitching on

1200kVar

270Hz

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

5

time (s)

|ene

rgy|

(J)

b)Smoothed Pseudo-Choi-WilliamsDistribution with Gaussian kernelfunction, - Hamming, width 0.04s - Hamming, width 0.04s

h g t

0.05

Smoothed Pseudo-Choi-WilliamsDistribution with Gaussian kernelfunction, - Hamming, width 0.1s - Hamming, width 0.04s

h g t

0.05

t=0.09sswitching on

1200kVar

Influence of the window width on dynamism ofSmoothed Choi-Williams Distribution

270Hz

General comment: Applying additional smoothing windows in Cohen’s equation allows to control time and frequency resolution separately. In STFT the relationship between window width and time-frequency resolution is inseparable

Conclusions

• Time-frequency representations can be treated as a comprehensive tool for analysis of transient states. Simultaneous representation of two dimensions delivers much more information about character of investigated phenomena than known tools.

• Proposed S-Transform and transformation from Cohen’s family have adaptability to investigated phenomena:

in case of S-Transform, moving and scalable Gaussian window uniquely combines frequency dependent resolution with desirable information about time-varying spectrum,

referring to Cohen’s generalization, selection of appropriate kernel function enables adaptation of the representation method to investigated non-stationarity.

Undesirable cross-term components, characteristic for bilinear nature of Cohen’s family, can be successfully reduced through the selection of appropriate kernel function as well as application of additional smoothing functions.

Conclusions

• In comparison with classical spectrogram, proposed methods break of the inherent relation between time-frequency resolution and width of the window.

Smoothed version of Cohen’s transformation allows to control the time and frequency resolution independently.

In case of S-Transform, achieved time-frequency resolution comes from combination of scalable localizing Gaussian function and width of spectrum.

• Allows detection of the begging of distortion with better precision than classical spectrogram.

Documents

Time-Frequency Analysis of Non-stationary Phenomena in Electrical Engineering Antonio Bracale, Guido Carpinelli Universita degli Studi di Napoli “Federico