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CHAPTER 7

PATTERN RECOGNITION OF POWER SIGNAL

DISTURBANCES USING S-TRANSFORM AND

TT-TRANSFORM

7.1 INTRODUCTION

This chapter deals with the identification of power signal

disturbances using the S-Transform and TT-Transform. The various power

signal disturbances are simulated using MATLAB. These power signal

disturbances are subjected to S-Transform and TT-Transform. The results of

the transformation are generated as a pattern. It was found that the patterns

obtained for each of the power signal disturbance is unique in nature. Because

of this uniqueness of the pattern generated for each disturbances the

identification of the disturbance is done accurately.

The S-transform of various power signal disturbances are obtained.

The power signal disturbances in time-time transformation are derived from

the S-transform of power signals. TT-Transform is the two dimensional time-

time representation of a one dimensional time series based upon the

S-transform. One of the major utilities of the TT-Transform is the time-local

view, through the scaled windows of the primary time series. This helps in the

interpretation of the S-transform. The S-transform and TT-transform be

always looked at as a side by side pair, thus avoiding the interpreting artifacts

as real information and vice versa. In this paper S-Transform and

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TT-transform are applied to power signal disturbances and the pattern

obtained is shown in various figures. The chapter is organized as follows.

Section 7.2 gives an overall view of S-transform. In Section 7.3 TT-transform

is presented. In Section 7.4 the various power signal disturbances are

subjected to S-transform and TT-transform. Section 7.5 concludes the work.

7.2 THE S-TRANSFORM

Short Time Fourier Transform has the disadvantage of the fixed

width and height. This results in misinterpretation of signal components with

periods longer than the window width. The finite width limits the time

resolution of high frequency signal components. Unlike STFT the S-transform

has a window whose height and width vary with frequency. The expression

for the S-transform is

diftffhftS )2exp(

2)(exp

2)(),(

22

(7.1)

Equation (7.1) is similar to the STFT. The window of STFT is

a scaled Gaussia -

transform may be considered as a set of localized Fourier coefficients,

obtained by considering only the portion of the primary function lying within

-t. The scaled contraction of w causes the

S may be alternately expressed as

.)2exp()2exp()(),( 2

22

dtif

fHftS (7.2)

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S is fully invertible, since integration of S over all t yields H from

which h is obtained,

).(),( fHdtftS (7.3)

If h and w are both real-valued functions replacing f with –f in (1)

leads to

),(),( ftSftS . (7.4)

Hence the amplitude spectrum is displayed for non-negative f only.

7.3 THE TT-TRANSFORM

The general expression for STFT is given by

diftwhftSTFT )2exp()()(),( . (7.5)

By applying the inverse Fourier transform of (7.1) results in

.)2exp(),()()( dfifftSTFTtwh (7.6)

When all values of t are considered, the windowed function

)()( twh becomes a two-dimensional function, denoted by STFTTT as in

equation (7.6) Can be rewritten as

.)2exp(),(),( dfifftSTFTtSTFTTT (7.7)

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STFTTT is a time-time distribution. TT-Transform is obtained from

the S-Transform as given by equation below

.)2exp(),(),( dfifftStTT (7.8)

The time-local function is different from the windowed functions of

the STFT. The scaling properties of S lead to higher amplitudes of high

From (7.3) and (7.8)

)(),( hdttTT , (7.9)

Hence like the S-transform, the TT-transform is invertible.

7.4 APPLICATION OF S-TRANSFORM AND TT-TRANSFORM

IN POWER SIGNAL DISTURBANCES

The power signal disturbances used in this analysis are i) a signal

with sag, ii) a signal with swell, iii) a signal with momentary interruption iv) a

signal with transient v) a signal with notches. These power signal disturbances

are subjected to S-Transform and TT-Transform. The S-Transform and TT-

Transform are transformed into wireframe parametric surface. The vectors

X(i,j) and Y(I,j) of S-Transform and TT-Transform are converted to

wireframe mesh such that the three coordinattes are the intersections of the

wireframe grid lines. The transformations are converted into 3D figure by

using the linear transformation on the data.

The transformations are applied to a pure signal. Then a signal with

various power quality disturbances are subjected to the above transformations

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and the 3D figure is generated for comparison. Each of the disturbances after

transformation is unique. ThiS-Transformation of a pure signal is shown in

Figure 7.1. A signal with sag is as in Figure 7.2. The time at which the

disturbance has occurred and the magnitude of the sag signal could be viewed

from the Figure 7.2.The next PQ disturbance used is a signal with swell. This

is given in Figure 7.3. From the figure it could be viewed that the instant at

which the disturbance has occurred and the magnitude of the disturbance

could be visualized. The power signal with notches is shown in Figure 7.4.

The pattern generated for the disturbance region is different from the rest of

the other PQ disturbances. The power signal disturbance with transients

subjected to this 3D transformation resulted in Figure 7.5. Again the patterns

generated for the disturbance is different from the other patterns. Along X,

Y,Z axis the magnitude of voltage, time of occurrence of disturbance and the

frequency are obtained by S-Transform.

Figure 7.1 3D Transformation of a Pure Signal Subjected to

S-Transform

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Figure 7.2 3D Transformation of a Signal with Sag Subjected to

S-Transform

Figure 7.3 3D Transformation of a Signal with Swell Subjected to

S-Transform

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Figure 7.4 3D Transformation of a Signal with Notches Subjected to

S-Transform

Figure 7.5 3D Transformation of a Signal with Transients Subjected to

S-Transform

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Both the S-Transform and TT-Transform resulted in X and Y

coordinates. This is given a transformation as XYZ coordinates along with the

color representation. From the Figures 7.1- 7.5 it could be seen that the each of the

signal has its own pattern. Hence identification of the disturbances is easy. The X

coordinates from the S-Transform, Y coordinates from the S-Transform and XYZ

coordinates of S-Transform for a pure signal is plotted in Figure 7.6.The

X coordinates from the TT-Transform, Y coordinates from the TT-Transform and

XYZ coordinates of TT-Transform for a pure signal is plotted in Figure 7.7. The

transformation of X, Y and XYZ coordinates of S-Transform and TT-Transform

of a signal with sag are given in Figures 7.8 and 7.9.The transformations obtained

for a signal with swell is shown in Figures 7.10 and 7.11. The transformations

obtained for a signal with transients is shown in Figures 7.12 and 7.13. The

transformations obtained for a signal with momentary interruption is shown in

Figures 7.14 and 7.15.The transformations obtained for a signal with notch is

shown in Figures 7.16 and 7.17.

Figure 7.6 Transformation of the S-Transform Coordinates of a Pure

Sine Signal

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Figure 7.7 Transformation of the TT-Transform Coordinates of a Pure

Sine Signal

Figure 7.8 Transformation of the S-Transform Coordinates of a Signal

with Sag

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Figure 7.9 Transformation of the TT-Transform Coordinates of a

Signal with Sag

Figure 7.10 Transformation of the S-Transform Coordinates of a Signal

with Swell

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Figure 7.11 Transformation of the TT-Transform Coordinates of a

Signal with Swell

Figure 7.12 Transformation of the S-Transform Coordinates of a Signal

with Transients

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Figure 7.13 Transformation of the TT-Transform Coordinates of a

Signal with Transients

Figure 7.14 Transformation of the S-Transform Coordinates of a Signal

with Momentary Interruption

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Figure 7.15 Transformation of the TT-Transform Coordinates of a

Signal with Momentary Interruption

Figure 7.16 Transformation of the S-Transform Coordinates of a Signal

with Notch

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Figure 7.17 Transformation of the TT-Transform Coordinates of a

Signal with Notch

From the above Figures 7.6-7.17 it could be seen that each of the

disturbance is characterized by its unique pattern. Hence identification of

disturbance without any misinterpretation is possible.

7.5 CONCLUSION

The power signal disturbances are subjected to S-Transform and

TT-Transform. The patterns obtained for a pure sine signal and the signal

with disturbances are compared for identification of the signal with

disturbance. The pattern obtained for a signal with sag is different from the

patterns obtained for the signal with the other disturbances. This uniqueness

of the patterns obtained for each disturbance helps in accurate identification of

the power signal disturbances.