**2.2 Wavelet Transform**

158 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

high accuracy in assessing of active power unbalance of system and minimal underfrequency shedding, that is, operating of under-frequency protective devices. Furthermore, if a system is compact and we know the total system inertia, it becomes possible to estimate total unbalance of active power in the system using angle or frequency measuring in any system's part by directly assessing of rate of change of a weighted average frequency (frequency of the centre of inertia) using WT. In order to avoid bigger frequency drop and eventual frequency instability, identification of the frequency of the centre of inertia rate of change should be as quick and unbalance estimate as accurate as possible. Given the oscillatory nature of the frequency change following the disturbance, a quick and accurate estimate of medium value is not simple and depends on the system's characteristics, that is,

This chapter presents possibilities for application of Discrete Wavelet Transformation (DWT) in estimating of the frequency of the centre of inertia rate of change (*df/dt*). In physics terms, low frequency component of signal voltage angle or frequency is very close to the frequency of the centre of inertia rate of change and can be used in estimating *df/dt*, and therefore, can also be used to estimate total unbalance of active power in the system. DWT was used for signal frequency analysis and estimating *df/dt* value, and the results were

Wavelet theory is a natural continuation of Fourier transformation and its modified shortterm Fourier transformation. Over the years, wavelets have been being developed independently in different areas, for example, mathematics, quantum physics, electrical engineering and many other areas and the results can be seen in the increasing application in signal and image processing, turbulence modelling, fluid dynamics, earthquake predictions, etc. Over the last few years, WT has received significant attention in electric power sector since it is more suitable for analysis of different types of transient wavelets

From a historical point of view, wavelet theory development has many origins. In 1822, Fourier (Jean-Baptiste Joseph) developed a theory known as Fourier analysis. The essence of this theory is that a complicated event can be comprehended through its simple constituents. More precisely, the idea is that a certain function can be represented as a sum of sine and cosine waves of different frequencies and amplitudes. It has been proved that every 2π periodic integrible function is a sum of Fourier series 0 *k k* cos sin *<sup>k</sup> a a kx b kx* , for corresponding coefficients *<sup>k</sup> a* i *<sup>k</sup> b* . Today, Fourier analysis is a compulsory course at every technical faculty. Although the contemporary meaning of the term 'wavelet' has been in use only since the 80s', the beginnings of the wavelet theory development go back to the year 1909 and Alfred Haar's dissertation in which he analysed the development of integrable functions in another orthonormal function system. Many papers were published during the 30s'; however, none provided a clear and coherent theory (Daubechies, 1996;

compared with a common *df/dt* estimate technique, the Method of Least Squares.

total inertia of the system (Madani et al., 2004, 2008).

**2. Basic wavelet theory** 

when compared to other transformations.

**2.1 Development of wavelet theory** 

Polikar, 1999).

Development of WT overcame one of the major disadvantages of Fourier transformation. Fourier series shows a signal through the sum of sines of different frequencies. Fourier transformation transfers the signal from time into frequency domain and it tells of which frequency components the signal is composed, that is, how frequency resolution is made. Unfortunately, it does not tell in what time period certain frequency component appears in the signal, that is, time resolution is lost. In short, Fourier transformation provides frequency but totally loses time resolution. This disadvantage does not affect stationary signals whose frequency characteristics do not change with time. However, the world around us mainly contains non-stationary signals, for whose analysis Fourier transformation is inapplicable. Attempts have been made to overcome this in that the signal was observed in segments, that is, time intervals short enough to observe non-stationary signal as being stationary. This idea led to the development of short-time Fourier transformation (STFT) in which the signal, prior to transformation, is limited to a time interval and multiplied with window function of limited duration. This limited signal is then transformed into frequency area. Then, the window function is translated on time axis for a certain amount (in the case of continued STFT, infinitesimal amount) and then Fourier transformation is applied (Daubechies, 1992; Vetterli & Kovacevic, 1995; Mallat, 1998; Mertins, 1999).

The process is repeated until the window function goes down the whole signal. It will result in illustration of signals in a time-frequency plane. It provides information about frequency

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 161

WT is based on a rather complex mathematical foundations and it is impossible to describe all details in this chapter of the book. The following chapters will provide basic illustration of Continuous WT (CWT) and Discrete WT (DWT), which have become a standard research

In 1946, D. Gabor was the first to define time-frequency functions, the so-called Gabor wavelets (2005/second reference should be Radunovic, 2005). His idea was that a wave,

keep just one of them. This wavelet contains three information: start, end and frequency content. Wavelet is a function of wave nature with a compact support. It is called a wave because of its oscillatory nature, and it is small because of the final domain in which it is different from zero (compact support). Scaling and translations of the *mother wavelet*

> , <sup>1</sup> , 0. *a b x b x a a a*

The choice of scaling parameter *a* and translation *b* makes it possible to represent smaller fragments of complicated form with a higher time resolution (zooming sharp and short-term peaks), while smooth segments can be represented in a smaller resolution, which is

CWT is a tool to break down for mining of data, functions or operators into different components and then each component is analysed with a resolution which fits its scale. It is

<sup>1</sup> \* (,) *x b CWT <sup>f</sup> a b <sup>f</sup> x dx*

where asterix stands for conjugate complex value, *a* and *b ab R* , are scaling parameters

CWT is function of scale *a* and position *b* and it shows how closely correlated are the wavelet and function in time interval which is defined by wavelet's support. WT measures

factor. By choosing values *a bR* 0, , mother wavelet provides other wavelets which, when compared to the mother wavelet, are moved on time axis for value *b* and 'stretched' for scaling factor *a* (when *a*>1). Therefore, continued wavelet transformation of signal *f(x)* is calculated so that the signal is multiplied with wavelet function for certain *a* and *b*, followed by integration. Then parameters *a* and *b* are infinitesimally increased and the process is repeated. As a result we get wavelet coefficients *CWT (a,b)* which represent the signal in

*a a*

 

 

*x dx* <sup>0</sup>

and it represents wave function of limited duration for which the following is applicable:

should be divided into segments and should

, (1)

. (2)

(3)

*x* is called mother wavelet, *a* – scaling factor, *b* – translation

*a b*, *x* in time-frequency

*x*

*x* 

wavelet's good trait (basis functions are time limited).

defined by a scale multiplication of function and wavelet basis:

(He & Starzyk, 2006; Avdakovic et al. 2010, Omerhodzic et al. 2010).

the similarity of frequency content of function and wavelet basis

tool for engineers processing signals.

whose mathematical transcript is cos

(mother) define wavelet basis,

domain. In *a*=1 and *b*=0,

components of which the signal is composed and time intervals in which these components appear. However, this illustration has a certain disadvantage whose cause is in Heisenberg's uncertainty principle which in this case can be stated as: *'We cannot know exactly which frequency component exists at any given time instant. The most we can know is the range of the frequency represented in a certain time interval, which is known as problem of resolution.'* 

Generally speaking, resolution is related to the width of window function. The window does not localize the signal in time, so there is no information about the time in frequency area, that is, there is no time resolution. With STFT, the window is of definite duration, which localizes the signal in time, so it is possible to know which frequency components exist in which time interval in a time-frequency plane, that is, we get a certain time resolution. If the window is narrowed, we get even better time localisation of the signal, which improves time resolution; however, this makes frequency resolution worse, because of Heisenberg's principle.

Fig. 2. Relation between time and frequency resolution with multiresolution analysis

*Δt* i *Δf* represents time and frequency range. These intervals are resolution: the shorter the intervals, the better the resolution. It should be pointed out that multiplication *Δt\*Δf* is always constant for a certain window function. The disadvantage of time-limited Fourier transformation is that by choosing the window width, it defines the resolution as well, which is unchangeable, regardless of whether we observe the signal on low or high frequencies. However, many true signals contain lower frequency components during longer time period, which represent the signal's trend and higher frequency components which appear in short time intervals.

When analysing these signals, it would be beneficial to have a good frequency resolution in low frequencies, and good time resolution in high frequencies (for example, to localise highfrequency noise in the signal). The analysis which meets these requirements is called multiresolution analysis **(**MRA) and leads directly to WT. Figure 2 illustrates the idea of multiresolution analysis: with the increase of frequency *Δt* decreases, which improves time resolution, and *Δf* increases, that is, frequency resolution becomes worse. Heisenberg's principle can also be applied here: surfaces *Δt\*Δf* are constant everywhere, only *Δt* and *Δf* values change.

components of which the signal is composed and time intervals in which these components appear. However, this illustration has a certain disadvantage whose cause is in Heisenberg's uncertainty principle which in this case can be stated as: *'We cannot know exactly which frequency component exists at any given time instant. The most we can know is the range of the* 

Generally speaking, resolution is related to the width of window function. The window does not localize the signal in time, so there is no information about the time in frequency area, that is, there is no time resolution. With STFT, the window is of definite duration, which localizes the signal in time, so it is possible to know which frequency components exist in which time interval in a time-frequency plane, that is, we get a certain time resolution. If the window is narrowed, we get even better time localisation of the signal, which improves time resolution; however, this makes frequency resolution worse, because

*frequency represented in a certain time interval, which is known as problem of resolution.'* 

Fig. 2. Relation between time and frequency resolution with multiresolution analysis

*Δt* i *Δf* represents time and frequency range. These intervals are resolution: the shorter the intervals, the better the resolution. It should be pointed out that multiplication *Δt\*Δf* is always constant for a certain window function. The disadvantage of time-limited Fourier transformation is that by choosing the window width, it defines the resolution as well, which is unchangeable, regardless of whether we observe the signal on low or high frequencies. However, many true signals contain lower frequency components during longer time period, which represent the signal's trend and higher frequency components

*Δf*

*Δt*

When analysing these signals, it would be beneficial to have a good frequency resolution in low frequencies, and good time resolution in high frequencies (for example, to localise highfrequency noise in the signal). The analysis which meets these requirements is called multiresolution analysis **(**MRA) and leads directly to WT. Figure 2 illustrates the idea of multiresolution analysis: with the increase of frequency *Δt* decreases, which improves time resolution, and *Δf* increases, that is, frequency resolution becomes worse. Heisenberg's principle can also be applied here: surfaces *Δt\*Δf* are constant everywhere, only *Δt* and *Δf*

of Heisenberg's principle.

which appear in short time intervals.

values change.

WT is based on a rather complex mathematical foundations and it is impossible to describe all details in this chapter of the book. The following chapters will provide basic illustration of Continuous WT (CWT) and Discrete WT (DWT), which have become a standard research tool for engineers processing signals.

In 1946, D. Gabor was the first to define time-frequency functions, the so-called Gabor wavelets (2005/second reference should be Radunovic, 2005). His idea was that a wave, whose mathematical transcript is cos*x* should be divided into segments and should keep just one of them. This wavelet contains three information: start, end and frequency content. Wavelet is a function of wave nature with a compact support. It is called a wave because of its oscillatory nature, and it is small because of the final domain in which it is different from zero (compact support). Scaling and translations of the *mother wavelet x* (mother) define wavelet basis,

$$
\psi\_{a,b}(\mathbf{x}) = \frac{1}{\sqrt{a}} \psi\left(\frac{\mathbf{x} - b}{a}\right), \qquad a > 0. \tag{1}
$$

and it represents wave function of limited duration for which the following is applicable:

$$\int\_{-\infty}^{\infty} \boldsymbol{\nu} \left( \mathbf{x} \right) d\mathbf{x} = \mathbf{0} \,. \tag{2}$$

The choice of scaling parameter *a* and translation *b* makes it possible to represent smaller fragments of complicated form with a higher time resolution (zooming sharp and short-term peaks), while smooth segments can be represented in a smaller resolution, which is wavelet's good trait (basis functions are time limited).

CWT is a tool to break down for mining of data, functions or operators into different components and then each component is analysed with a resolution which fits its scale. It is defined by a scale multiplication of function and wavelet basis:

$$\text{CWT}\_{\text{y}}f(a,b) = \frac{1}{\sqrt{|a|}} \int\_{-\infty}^{+\infty} f(\mathbf{x}) \boldsymbol{\wp}^\* \left(\frac{\mathbf{x}-b}{a}\right) d\mathbf{x} \tag{3}$$

where asterix stands for conjugate complex value, *a* and *b ab R* , are scaling parameters (He & Starzyk, 2006; Avdakovic et al. 2010, Omerhodzic et al. 2010).

CWT is function of scale *a* and position *b* and it shows how closely correlated are the wavelet and function in time interval which is defined by wavelet's support. WT measures the similarity of frequency content of function and wavelet basis *a b*, *x* in time-frequency domain. In *a*=1 and *b*=0, *x* is called mother wavelet, *a* – scaling factor, *b* – translation factor. By choosing values *a bR* 0, , mother wavelet provides other wavelets which, when compared to the mother wavelet, are moved on time axis for value *b* and 'stretched' for scaling factor *a* (when *a*>1). Therefore, continued wavelet transformation of signal *f(x)* is calculated so that the signal is multiplied with wavelet function for certain *a* and *b*, followed by integration. Then parameters *a* and *b* are infinitesimally increased and the process is repeated. As a result we get wavelet coefficients *CWT (a,b)* which represent the signal in time-scale plane. The value of certain wavelet coefficient *CWT (a,b)* points to the similarity between the observed signal and wavelet generated by shifting on time axis and scaling for values *b* and *a*. It can be said that wavelet transformation shows signal as infinite sum of scaled and shifted wavelets, in which wavelet coefficients are weight factors. Using wavelets, time analysis is done by compressed, high-frequency versions of mother wavelet, since it is possible to notice fast changing details on a small scale.

Frequency analysis is done by stretched high-frequency versions of the same wavelet, because a large scale is sufficient for monitoring slower changes. These traits make wavelets an ideal tool for analysis of non-stationary functions. WT provides excellent time resolution of high-frequency components and frequency (scale) resolution of low-frequency components.

CWT is a reversible process when the following condition (admissibility) is met:

$$C\_{\nu} = \int\_{-\infty}^{\infty} \frac{\left| \Psi \left( \rho \right) \right|^{2}}{\alpha} d\rho < \infty \tag{4}$$

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 163

are used to describe rapid changing segments of signal, while stretched, sparsely distributed

DWT is the most widely used wavelet transformation. It is a recursive filtrating process of input data set with lowpass and highpass filters. Approximations are low-frequency components in large scales, and details are high-frequency function components in small scales. Wavelet function transformation can be interpreted as function passing through the filters bank. Outputs are scaling coefficients *<sup>j</sup>*,*<sup>k</sup> a* (approximation) and wavelet coefficients *<sup>j</sup>*,*<sup>k</sup> b* (details). Signal analysis which is done by signal passing through the filters bank is an old idea known as *subband coding*. DTW uses two digital filters: lowpass filter *hn n Z* , *,*

*x* . Filters *h(n) and g(n)* are associated with the scaling function and wavelet

 

 

*n*

2 2

22 ,

*g n* , and <sup>2</sup>

It is possible to reconstruct any input signal on the basis of output signals if filters are observed in pairs. High frequency filter is associated to low frequency filter and they become Quadrature Mirror Filters (QMF). They serve as a mirror reflection to each other.

DWT is an algorithm used to define wavelet coefficients and scale functions in dyadic scales and dyadic points. The first step in filtering process is splitting approximation and discrete signal details so to get two signals. Both signals have the length of an original signal, so we get double amount of data. The length of output signals is split in half using compression, that is, discarding all other data. The approximation received serves as input signal in the following step. Digital signal *f(n)*, of frequency range 0-*Fs*/2, (*Fs* – sampling frequency), passes through lowpass *h(n)* and highpass *g(n)* filter. Each filter lets by just one half of the frequency range of the original signal. Filtrated signals are then subsampled so to remove any other sample. We mark *cA1(k)* and *cD1(k)* as outputs from *h(n)* and *g(n)* filter, respectively. Filtrating process and subsampling of input signal can be represented as:

> <sup>1</sup> 2 *n*

> <sup>1</sup> 2 *n*

where coefficients *cA1(k)* are called approximation of the first level of decomposition and represent input signal in frequency range 0-*Fs*/4 Hz. By analogy, *cD1(k)* are coefficients of details and represents signal in range *Fs*/4 - *Fs*/2 Hz. Decomposition continues so that approximation coefficients *cA1(k)* are passed through filters *g(n)* and *h(n)* that is, they are split to coefficients *cA2(k)* which represent signal in range 0- *Fs*/8 Hz and *cD2(k),* range

*n*

*n*

*x* and highpass filter *gn n Z* , *,* defined by wavelet

*x hn x n* (7)

*x gn x n* (8)

*h n* and 0 .

*cA k f n h k n* (8)

*cD k f n g k n* (9)

*n g n*

wavelets are used to describe slow changing segments of signal (Mei et al., 2006).

*n*

*h n* and <sup>2</sup> <sup>1</sup>

defined by scaling function

and equals to: <sup>2</sup> 1 *n*

function, respectively (He & Starzyk, 2006):

function

where is Fourier transformation of basis function *x* . Inverse wavelet transformation is defined by:

$$f\left(\mathbf{x}\right) = \frac{1}{C\_{\varphi}} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \text{CV}\mathcal{T}\_{f}\left(a, b\right) \wp\_{a, b}\left(\mathbf{x}\right) \frac{da \, db}{a^{2}} \tag{5}$$

where it is possible to reconstruct the observed signal through CWT coefficient.

CWT is of no major practical use, because correlation of function and continually scaling wavelet is calculated (*a* and *b* are continued values). Many of the calculated coefficients are redundant and their number is infinite. This is why there is discretization – time-scale plane is covered by grid and CWT is calculated in nodes of grid. Fast algorithms are construed using discrete wavelets. Discrete wavelets are usually a segment by segment of uninterrupted function which cannot be continually scaled and translated, but merely in discrete steps,

$$\nu\_{j,k}\left(\mathbf{x}\right) = \frac{1}{\sqrt{a\_0^j}} \nu\left(\frac{\mathbf{x} - kb\_0 a\_0^j}{a\_0^j}\right) \tag{6}$$

where *j*, *k* are whole numbers, and 0 *a* 1 is fixed scaling step. It is usual that 0*a* 2 , so that the division on frequency axis is dyadic scale. 0 *b* 1 is usually translation factor, so the division on time axis on a chosen scale is equal,

$$\Psi\_{j,k}(\mathbf{x}) = 2^{-j/2}\nu\left(2^{-j}\mathbf{x} - k\right), \text{ i } \Psi\_{j,k}(\mathbf{x}) \neq 0 \text{ za } \mathbf{x} \in \left[2^j k, 2^j \left(k+1\right)\right].$$

Parameter *a* is duplicated in every level compared to its value at the previous level, which means that wavelet doubles in its width. The number of points in which wavelets are defined are half the size compared to the previous level, that is, resolution becomes smaller. This is how the concept of *multiresolution* is realised. Narrow, densely distributed wavelets

time-scale plane. The value of certain wavelet coefficient *CWT (a,b)* points to the similarity between the observed signal and wavelet generated by shifting on time axis and scaling for values *b* and *a*. It can be said that wavelet transformation shows signal as infinite sum of scaled and shifted wavelets, in which wavelet coefficients are weight factors. Using wavelets, time analysis is done by compressed, high-frequency versions of mother wavelet,

Frequency analysis is done by stretched high-frequency versions of the same wavelet, because a large scale is sufficient for monitoring slower changes. These traits make wavelets an ideal tool for analysis of non-stationary functions. WT provides excellent time resolution of high-frequency components and frequency (scale) resolution of low-frequency

<sup>2</sup>

*j*

*x za x k k* .

*x kb a*

(5)

(4)

*x* . Inverse wavelet

(6)

 , <sup>2</sup> <sup>1</sup> , *f ab da db f x CWT a b x*

CWT is of no major practical use, because correlation of function and continually scaling wavelet is calculated (*a* and *b* are continued values). Many of the calculated coefficients are redundant and their number is infinite. This is why there is discretization – time-scale plane is covered by grid and CWT is calculated in nodes of grid. Fast algorithms are construed using discrete wavelets. Discrete wavelets are usually a segment by segment of uninterrupted function which cannot be continually scaled and translated, but merely in

> <sup>0</sup> <sup>0</sup> , 0 0 <sup>1</sup> ,

where *j*, *k* are whole numbers, and 0 *a* 1 is fixed scaling step. It is usual that 0*a* 2 , so that the division on frequency axis is dyadic scale. 0 *b* 1 is usually translation factor, so the

 

*a a*

 *x x <sup>k</sup>* i , 0 2 ,2 1 *j j j k* 

Parameter *a* is duplicated in every level compared to its value at the previous level, which means that wavelet doubles in its width. The number of points in which wavelets are defined are half the size compared to the previous level, that is, resolution becomes smaller. This is how the concept of *multiresolution* is realised. Narrow, densely distributed wavelets

*j k <sup>j</sup> <sup>j</sup>*

*x*

 <sup>2</sup> , 22 , *j j*

 

division on time axis on a chosen scale is equal,

*j k* 

*a*

CWT is a reversible process when the following condition (admissibility) is met:

is Fourier transformation of basis function

where it is possible to reconstruct the observed signal through CWT coefficient.

*C*

*C d*

 

since it is possible to notice fast changing details on a small scale.

components.

where

discrete steps,

transformation is defined by:

are used to describe rapid changing segments of signal, while stretched, sparsely distributed wavelets are used to describe slow changing segments of signal (Mei et al., 2006).

DWT is the most widely used wavelet transformation. It is a recursive filtrating process of input data set with lowpass and highpass filters. Approximations are low-frequency components in large scales, and details are high-frequency function components in small scales. Wavelet function transformation can be interpreted as function passing through the filters bank. Outputs are scaling coefficients *<sup>j</sup>*,*<sup>k</sup> a* (approximation) and wavelet coefficients *<sup>j</sup>*,*<sup>k</sup> b* (details). Signal analysis which is done by signal passing through the filters bank is an old idea known as *subband coding*. DTW uses two digital filters: lowpass filter *hn n Z* , *,* defined by scaling function *x* and highpass filter *gn n Z* , *,* defined by wavelet function *x* . Filters *h(n) and g(n)* are associated with the scaling function and wavelet function, respectively (He & Starzyk, 2006):

$$\varphi(\mathbf{x}) = \sum\_{n} h(n) \sqrt{2} \varphi(2\mathbf{x} - n) \tag{7}$$

$$\Psi'(\mathbf{x}) = \sum\_{n} \mathbf{g}(n) \sqrt{2} \phi(2\mathbf{x} - n) \, \tag{8}$$

and equals to: <sup>2</sup> 1 *n h n* and <sup>2</sup> <sup>1</sup> *n g n* , and <sup>2</sup> *n h n* and 0 . *n g n*

It is possible to reconstruct any input signal on the basis of output signals if filters are observed in pairs. High frequency filter is associated to low frequency filter and they become Quadrature Mirror Filters (QMF). They serve as a mirror reflection to each other.

DWT is an algorithm used to define wavelet coefficients and scale functions in dyadic scales and dyadic points. The first step in filtering process is splitting approximation and discrete signal details so to get two signals. Both signals have the length of an original signal, so we get double amount of data. The length of output signals is split in half using compression, that is, discarding all other data. The approximation received serves as input signal in the following step. Digital signal *f(n)*, of frequency range 0-*Fs*/2, (*Fs* – sampling frequency), passes through lowpass *h(n)* and highpass *g(n)* filter. Each filter lets by just one half of the frequency range of the original signal. Filtrated signals are then subsampled so to remove any other sample. We mark *cA1(k)* and *cD1(k)* as outputs from *h(n)* and *g(n)* filter, respectively. Filtrating process and subsampling of input signal can be represented as:

$$cA\_1(k) = \sum\_{n} f\left(n\right) h\left(2k - n\right) \tag{8}$$

$$cD\_1(k) = \sum\_{n} f(n)g(2k - n) \tag{9}$$

where coefficients *cA1(k)* are called approximation of the first level of decomposition and represent input signal in frequency range 0-*Fs*/4 Hz. By analogy, *cD1(k)* are coefficients of details and represents signal in range *Fs*/4 - *Fs*/2 Hz. Decomposition continues so that approximation coefficients *cA1(k)* are passed through filters *g(n)* and *h(n)* that is, they are split to coefficients *cA2(k)* which represent signal in range 0- *Fs*/8 Hz and *cD2(k),* range *Fs*/8 - *Fs*/4 Hz. Since the algorithm is continued, that is, since it goes towards lower frequencies, the number of samples decreases which worsens time resolution, because fewer number of samples stand for the whole signal for a certain frequency range. However, frequency resolution improves, because frequency ranges for which the signal is observed are getting narrower.

Therefore, multiresolution principle is applicable here. Generally speaking, wavelet coefficients of *j* level can be represented through approximation coefficients of *j*-1 level as follows:

$$cA\_j(k) = \sum\_{n} h(2k - n)cA\_{j-1}(n) \tag{10}$$

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 165

Stability of power system refers to its ability to maintain synchronous operation of all connected synchronous generators in stationary state and for the defined initial state after disturbances occur, so that the change of the variables of state in transitional process is limited, and system structure preserved. The system should be restored to initial stationary state unless topology changes take place, that is, if there are topological changes to the system, a new stationary state should be invoked. Although the stability of power system is its unique trait, different forms of instability are easier to comprehend and analysed if stability problems are classified, that is, if "partial" stability classes are defined. Partial stability classes are usually defined for fundamental state parameters: transmission angle, voltage and frequency. Figure 4. shows classification of stability according to (IEEE/CIGRE, 2004). Detailed description of physicality of dynamics and system stability, mathematical models and techniques to resolve equations of state and stability aspect analysis can be

> Power System Stability

Frequency Stability

Voltage Stability

> Small Disturbance Voltage Stability

Long Term

Large Disturbance Voltage Stability

> Short Term

**3. Frequency stability of power system – An estimation of active power** 

**unbalance** 

found in many books and papers.

Short Term

Angle Stability

formed in this way.

Small Signal Stability

Fig. 4. Classification of "partial" stability of electric power system

Short Term

Transient Stability

Frequency stability is defined as the ability of power system to maintain frequency within standardized limits. Frequency instability occurs in cases when electric power system cannot permanently maintain the balance of active powers in the system, which leads to frequency collapse. In cases of high intensity disturbances or successive interrelated and mutually caused (connected) disturbances, there can be cascading deterioration of frequency stability, which, in the worst case scenario, leads to disjunction of power system to subsystems and eventual total collapse of function of isolated parts of electric power system

Long Term

$$cD\_j(k) = \sum\_n g\left(2k - n\right) cA\_{j-1}(n) \tag{11}$$

The result of the algorithm on signals sampled by frequency *Fs* will be the matrix of wavelet coefficients. At every level, filtrating and compression will lead to frequency layer being cut in half (subsequently, frequency resolution doubles) and reducing the number of sampling in half.

Eventually, if the original signal has the length 2*<sup>m</sup>* , DWT mostly has *m* steps, so at the end we get approximation as the signal with length one. Figure 3 illustrates three levels of decomposition.

Fig. 3. Wavelet MRA (Avdakovic et al., 2010)

We get DWT of original signal by connecting all coefficients starting from the last level of decomposition, and it represents the vector made of output signals 1 , ,...., *Aj j D D* . Assembling components, in order to get the original signal without losing information, is known as reconstruction or synthesis. Mathematical operations for synthesis are called *inverse discrete wavelet transformation* (IDTW). Wavelet analysis includes filtering and compression, and reconstruction process includes decompression and filtering.

*Fs*/8 - *Fs*/4 Hz. Since the algorithm is continued, that is, since it goes towards lower frequencies, the number of samples decreases which worsens time resolution, because fewer number of samples stand for the whole signal for a certain frequency range. However, frequency resolution improves, because frequency ranges for which the signal is observed

Therefore, multiresolution principle is applicable here. Generally speaking, wavelet coefficients of *j* level can be represented through approximation coefficients of *j*-1 level as

<sup>1</sup> 2 *j j*

<sup>1</sup> 2 *j j*

The result of the algorithm on signals sampled by frequency *Fs* will be the matrix of wavelet coefficients. At every level, filtrating and compression will lead to frequency layer being cut in half (subsequently, frequency resolution doubles) and reducing the number of sampling

Eventually, if the original signal has the length 2*<sup>m</sup>* , DWT mostly has *m* steps, so at the end we get approximation as the signal with length one. Figure 3 illustrates three levels of

down sampling by 2

N/2 samples

N/4 samples

HPF

LPF

D1

D2

D3

A3

HPF

LPF

We get DWT of original signal by connecting all coefficients starting from the last level of decomposition, and it represents the vector made of output signals 1 , ,...., *Aj j D D* . Assembling components, in order to get the original signal without losing information, is known as reconstruction or synthesis. Mathematical operations for synthesis are called *inverse discrete wavelet transformation* (IDTW). Wavelet analysis includes filtering and

compression, and reconstruction process includes decompression and filtering.

*cA k h k n cA n* (10)

*cD k <sup>g</sup> k n cA n* (11)

*n*

*n*

Wavelet function

(HPF)

(LPF)

Scaling function

Signal

Fig. 3. Wavelet MRA (Avdakovic et al., 2010)

are getting narrower.

follows:

in half.

decomposition.
