**1. Introduction**

Power system is a complex, dynamic system, composed of a large number of interrelated elements. Its primary mission is to provide a safe and reliable production, transmission and distribution of electrical energy to final consumers, extending over a large geographic area. It comprises of a large number of individual elements which jointly constitute a unique and highly complex dynamic system. Some elements are merely the system's components while others affect the whole system (Machowski, 1997). Securing necessary level of safety is of great importance for economic and reliable operation of modern electric power systems.

Power system is subject to different disturbances which vary in their extent, and it must be capable to maintain stability. Various devices for monitoring, protection and control help ensure reliable, safe and stable operation. The stability of the power system is its unique feature and represents its ability to restore the initial state following a disturbance or move to a new steady state. During transient process, the change of the parameters should remain within the predefined limits. In the case of stability loss, parameters either increase progressively (power angles during angle instability) or decrease (voltage and frequency during voltage and frequency instability) (Kundur, 1994; Pal & Chaudhuri, 2005). Accurate and fast identification of disturbances allows alerting the operator in a proper manner about breakdowns and corrective measures to reduce the disturbance effects.

Several large blackouts occurred worldwide over the past decade. The blackout in Italy (28th Sept. 2003) which left 57 million people in dark is one f the major blackouts in Europe's history ever. The analyses show that the most common causes are cascading propagation of initial disturbance and failures in the power system's design and operation, for example, lack of equipment maintenance, transmission congestion, an inadequate support by reactive power, system operating at the margin of stability, operators' poor reactions, and low or no coordination by control centres (Madani et al., 2004). It would, therefore, be beneficial to have automatic systems in electric power systems which would prevent propagation of effects of initial disturbance through the system and system's cascade breakdown. In order to prevent the already seen major breakdowns, the focus has been placed on developing algorithms for monitoring, protection and control of power system in real time. Traditionally, power system monitoring and control was based on local measurements of

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 157

Electromagnetic transients are usually a consequence of the change in network configuration due to switching or electronic equipment, transient fault, etc. Electromechanical transients are slower (systematic) occurrences due to unbalance of active power (unbalance in production and consumption of active power) and are a consequence of mechanical nature of synchronous machines connected to the network. Such systems have more energy storages, for example, rotational masses of machines which respond with

Table 1. Typical Frequency Ranges for Transients Phenomena in Power System (Henschel,

If electric power system has an initial disturbance of 'higher intensity', it can lead to a successive action of system elements and cascade propagation of disturbance throughout the system. Usually the tripping of major generators or load busses results in under-voltage or under-frequency protective devices operation. This disturbance scenario usually results in additional unbalance of system power. Moreover, power flow in transmission lines is being re-distributed which can lead to their tripping, further affecting the transmission

Frequency instability occurs when the system is unable to balance active power which results in frequency collapse. Monitoring *df/dt* (the rate-of-change of frequency) is an immediate indicator of unbalance of active power; however, the oscillatory nature of *df/dt*

Given its advantages over other techniques for signal processing, WT enables direct assessment of rate of change of a weighted average frequency (frequency of the centre of inertia), which represents a true indicator of active power unbalance of power system (Avdakovic et al. 2009, 2010, 2011). This approach is an excellent foundation for improving existing systems of under-frequency protection. Namely, synchronised phasor measurements technique provides real time information on conditions and values of key variables in the entire power system. Using synchronised measurements and WT enables

can lead to unreliable measuring (Madani et al., 2004, 2008).

oscillations to a slightest unbalance. (Henschel, 1999).

1999)

network structure.

process parameters (voltage, power, frequency). Following major breakdowns from 2003., extensive efforts were made to develop and apply monitoring, protection and control systems based on parameters, the so-called Wide Area Monitoring Protection and Control systems (WAMPC). These systems are based on systems for measuring voltage phasors and currents in those points which are of special importance for power system (PMU devices - Phasor Measurement Unit). This platform enables more real and dynamic view of the power system, more accurate measurement swift data exchange and alert in case of need. Traditional "local"devices cannot achieve optimal control since they lack information about events outside their location (Novosel et al., 2007; Phadke & Thorp, 2008).

On the other hand, wavelet transformation (WT) represents a relatively new mathematical area and efficient tool for signal analysis and signal representation in time-frequency domain. It is a very popular area of mathematics applicable in different areas of science, primarily signal processing. Since the world around us, both nature and society, is constantly subjected to faster or slower, long or short-term changes, wavelets are suitable for mathematical tools to describe and analyse complex process in nature and society. A special problem in studying and analysing these processes are 'non-linear effects' characterised by quick and short changes, thus wavelets are an ideal tool for their analysis.

Historically, the WT development can be tracked to 1980s' and J.B.J. Fourier (Fig. 1a). Namely, in 1988, Belgian mathematician Ingrid Daubechies (Fig. 1b) presented her work to the scientific community, in which she created orthonormal wavelet bases of the space of square integrable functions which consists of compactly supported functions with prescribed degree of smoothness.

Fig. 1. a) Jean B. J. Fourier (1768 –1830) (http://en.wikipedia.org) and b) Ingrid Daubechies (August 17, 1954 in Houthalen, Belgium) (http://www.pacm.princeton.edu)

Today, this is considered to be the end of the first phase of WT development. Since it has many advantages, when compared to other signal processing techniques, it is receiving huge attention in the field of electrical engineering. Over the past twenty years, many valuable papers have been published with focus on WT application in analysis of electromagnetic transients, electric power quality, protection, etc., as well as a fewer number of papers focusing on the analysis of electromechanic oscillations/transients in power system. In terms of time and frequency, transients can be divided into electromagnetic and electromechanic. Frequency range for transients phenomena is provided in Table 1.

process parameters (voltage, power, frequency). Following major breakdowns from 2003., extensive efforts were made to develop and apply monitoring, protection and control systems based on parameters, the so-called Wide Area Monitoring Protection and Control systems (WAMPC). These systems are based on systems for measuring voltage phasors and currents in those points which are of special importance for power system (PMU devices - Phasor Measurement Unit). This platform enables more real and dynamic view of the power system, more accurate measurement swift data exchange and alert in case of need. Traditional "local"devices cannot achieve optimal control since they lack information about

On the other hand, wavelet transformation (WT) represents a relatively new mathematical area and efficient tool for signal analysis and signal representation in time-frequency domain. It is a very popular area of mathematics applicable in different areas of science, primarily signal processing. Since the world around us, both nature and society, is constantly subjected to faster or slower, long or short-term changes, wavelets are suitable for mathematical tools to describe and analyse complex process in nature and society. A special problem in studying and analysing these processes are 'non-linear effects' characterised by

Historically, the WT development can be tracked to 1980s' and J.B.J. Fourier (Fig. 1a). Namely, in 1988, Belgian mathematician Ingrid Daubechies (Fig. 1b) presented her work to the scientific community, in which she created orthonormal wavelet bases of the space of square integrable functions which consists of compactly supported functions with

a) b) Fig. 1. a) Jean B. J. Fourier (1768 –1830) (http://en.wikipedia.org) and b) Ingrid Daubechies

Today, this is considered to be the end of the first phase of WT development. Since it has many advantages, when compared to other signal processing techniques, it is receiving huge attention in the field of electrical engineering. Over the past twenty years, many valuable papers have been published with focus on WT application in analysis of electromagnetic transients, electric power quality, protection, etc., as well as a fewer number of papers focusing on the analysis of electromechanic oscillations/transients in power system. In terms of time and frequency, transients can be divided into electromagnetic and electromechanic. Frequency range for transients phenomena is provided in Table 1.

(August 17, 1954 in Houthalen, Belgium) (http://www.pacm.princeton.edu)

events outside their location (Novosel et al., 2007; Phadke & Thorp, 2008).

quick and short changes, thus wavelets are an ideal tool for their analysis.

prescribed degree of smoothness.

Electromagnetic transients are usually a consequence of the change in network configuration due to switching or electronic equipment, transient fault, etc. Electromechanical transients are slower (systematic) occurrences due to unbalance of active power (unbalance in production and consumption of active power) and are a consequence of mechanical nature of synchronous machines connected to the network. Such systems have more energy storages, for example, rotational masses of machines which respond with oscillations to a slightest unbalance. (Henschel, 1999).


Table 1. Typical Frequency Ranges for Transients Phenomena in Power System (Henschel, 1999)

If electric power system has an initial disturbance of 'higher intensity', it can lead to a successive action of system elements and cascade propagation of disturbance throughout the system. Usually the tripping of major generators or load busses results in under-voltage or under-frequency protective devices operation. This disturbance scenario usually results in additional unbalance of system power. Moreover, power flow in transmission lines is being re-distributed which can lead to their tripping, further affecting the transmission network structure.

Frequency instability occurs when the system is unable to balance active power which results in frequency collapse. Monitoring *df/dt* (the rate-of-change of frequency) is an immediate indicator of unbalance of active power; however, the oscillatory nature of *df/dt* can lead to unreliable measuring (Madani et al., 2004, 2008).

Given its advantages over other techniques for signal processing, WT enables direct assessment of rate of change of a weighted average frequency (frequency of the centre of inertia), which represents a true indicator of active power unbalance of power system (Avdakovic et al. 2009, 2010, 2011). This approach is an excellent foundation for improving existing systems of under-frequency protection. Namely, synchronised phasor measurements technique provides real time information on conditions and values of key variables in the entire power system. Using synchronised measurements and WT enables

Wavelet Theory and Applications for Estimation of Active Power Unbalance in Power System 159

First papers on wavelet theory are the result of research by French geophysicist and engineer, Jean Morlet, whose research focused on different layers of earth, and reflection of acoustic waves from the surface. Without much success, Morlet attempted to resolve the problem using localization technique put forward by Gabor in 1946. This forced him to 'make up' a wavelet. In 1984, Morlet and physicist Alex Grossmann proved stable decomposition and function reconstruction using wavelets coefficients. This is considered to

Grosman made a hypothesis that Morlet's wavelets form a frame for Hilbert's space, and in 1986 this hypothesis was proved accurate by Belgian mathematician Ingrid Daubechies. In 1986, mathematician Ives Meyer construed continuously differentiable wavelet whose only disadvantage was that it did not have a compact support. At the same time, Stephane Mallat, who was dealing with signal processing and who introduced auxiliary function which in a certain way generates wavelet function system, defined the term 'multiresolution analysis' (MRA). Finally, the first stage in the wavelet theory development was concluded

She created orthonormal wavelet bases of the space of square integrable functions which consists of compactly supported functions with prescribed degree of smoothness. Compact support means that the function is identically equal to zero outside a limited interval, and therefore, for example, corresponding inappropriate integrals come down to certain integrals. Daubechies wavelets reserved their place in special functions family. The most important consequence of wavelet theory development until 1990 was the establishment of a common mathematical language between different disciplines of applied and theoretical

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;

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

be the first paper in wavelet theory (Teofanov, 2001; Jaffard, 2001).

with Ingrid Daubechies' spectacular results in 1988 (Graps, 1995).

Vetterli & Kovacevic, 1995; Mallat, 1998; Mertins, 1999).

mathematics.

**2.2 Wavelet Transform** 

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

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