Study of Transients using Wavelet Transform

Abstract— This paper presents the study of power system transients using wavelet transform. Transients due to energization of capacitor banks and due to fault has been taken for consideration. The wavelet transform generates wavelet coefficients for the generated transients. Using Parsevals's theorem, energy and standard deviation are retrieved for transients. The wavelet transform is used to produce instantaneous frequency vectors of the signals, and then the energies of these vectors, obtained using Parseval's theorem, are utilized for the classification of different transients. The advantage of the proposed algorithm is its ability to distinguish different transients based on frequency change. The performance of this algorithm is bought bespeak by simulation of different events using MATLAB & SIMULINK software. The test results show that the new algorithm is very fast and accurate in identifying events. Keyword-: Transients due to fault, Oscillatory transients, Wavelet transform and Parseval's theorem.


II. INTRODUCTION TO WAVELET TRANSFORM
In both the STFT and the FT, the same window is used for the analysis of the entire signal. In the late 1970s J. Morlet came up with the wavelet in analyzing high frequency components with short time spans and low frequency components with long time spans, in different frequencies. The wavelet is a short duration waveform with a zero average value. Here the shortness refers to a finite length window function, i.e, compactly supported. Here the wave is meant for the oscillatory function only. The CWT is defined as given in equation 1 where τ is the translation parameter's and ψ(t) is the transforming function known as the mother wavelet.
The term mother implies that the function with different regions of support that are used in the transformation process, are derived from one main function, or the mother wavelet. The mother wavelet is a prototype for generating all the other window functions. In wavelet transform, the resemblance index i.e. the wavelet coefficient between the mother wavelet and the signal under consideration is calculated [7]. A multistage filter bank is used to decompose the signal in to various levels using the LP filter and the HP filter as shown in Fig 1. The LP filter will result in approximation coefficients of the original signal, and the HP filter in detailed coefficients of the signal.

A. Reasons for Using the Wavelet Transform
The FFT, which is a tool for high speed implementations of the FT uses sines and cosines as basic functions, which are non-local, and stretch out to infinity. They do a very poor job in detecting transients, and cannot provide the time localization of the frequency components in a signal. The wavelet transform has some advantages when compared to the FT. Wavelets have compact support, which means that wherever the function is not defined it will have a value of zero. It helps in speeding up the computations and also tells us about the locality of the wavelet in the time domain.

B. Selection of mother wavelets
The selection of the mother wavelet plays a significant role in the time frequency analysis. These have to be carefully selected to better approximate and capture the transient spikes of the faulty signal. The mother wavelet will not only determine how well we estimate the original signal, but also affect the frequency spectrum of the fault signal. The Haar wavelet is the simplest wavelet in the wavelet family. It was the first wavelet introduced, and is rarely used in practice, because it is discontinuous and lacks energy compaction. The Daubechies wavelet is one of the more advanced wavelets that was poved by several researchers. While the Haar does not allow for sharp transitions and fast attenuation, the Daubechies wavelets have continuous derivatives that respond well to discontinue. The Daubechies wavelets have a better frequency property than the Haar wavelets. When Haar is used to decompose the signal, the wavelets cannot efficiently separate the signal into low frequency and high frequency bands. The Daubechies wavelets allow the user to decide how fluctuation is acceptable in the high frequency bands.

III. METHODOLOGY OF THE PROPOSED WORK
 The methodology of the proposed work is shown in Fig. 2.  The transients due to energization of capacitor and due to fault are simulated using the MATLAB/SIMULINK software.  The wavelet transform is applied on the simulated transients.  The features extracted from WT are then processed to obtain the energy values using Parseval's theorem.  The transient events are then classified according to the energy value they contained.

IV. APPLICATION OF PARSEVAL'S THEOREM AND FEATURE EXTRACTION
The energy of signals that go through discrete wavelet transform decomposition can be described by (2) Where N is the sampling period. The first term on the right of above equation denotes the average power of the approximated version and the second term is for detailed version, whereas the left term represents the total energy. In the present study, the energy distributions for different transient events are divided in to one level for the approximate version and five for the detailed version. The coefficient cA of approximate version and coefficient cD of the detailed version are used to extract the features of signals [8].

V. INTRODUCTION TO TRANSIENTS
Transients are momentary changes in voltage or current that occur over a short period of time. Usually its duration is described as approximately 1/16 th of a voltage cycle or about 1ms. Transients are divided into three categories: impulsive transient, oscillatory transient and transient due to fault. When 77% of the peak-to-peak voltage of the pure component is of one polarity, the transient is then classified into the impulse category. A sudden, non-power frequency change in the steady state condition of voltage, current or both that includes positive and negative polarity values is an oscillatory transient. The transient due to fault is caused by fault conditions, the energization of large loads which require high starting currents or intermittent loose connections in power wiring. The transients are classified into any of these categories related to the frequencies they contained. Each type of transient can be associated with a group of phenomena occurring on the power system. The two types of transients are depicted in the following Fig. 3.a and 3.b. The capacitor switching oscillatory transients is considered here.

VI. SIMULATION RESULTS
The transients due to capacitor switching and due to fault on the system have been taken for consideration for the analysis.  Fig. 5a and Fig. 6a shows the generation of transients and Fig. 5b and Fig. 6b shows the amplitude of WT. Occurrence of transients due to energization of capacitance connected at the secondary of the transformer and due to fault are captured. Oscillatory phase voltages due to transients are obtained by voltage moment of the load side. The transient voltages shown in Fig. 5a and 6a processed to WT produce wavelet coefficients as shown in Fig. 5b and Fig. 6b respectively using Daubachies mother wavelet. Using Parseval's theorem, energy and STD are calculated for the duration of transients and they are tabulated in Table 1. These calculated energy of transients can be discriminated for different events of occurrence.

VI CONCLUSION
In this paper, two common types of transients such as transient due to fault and energization of capacitor are examined using wavelet transform. These transients that made distortion in alternating voltage waveform are analysed by using the energy values produced by Parseval's theorem. The obtained values of energy and standard deviation of transients are different for fault and switching conditions. The transients are classified from each other based on a threshold value set for each type of transient. Simulation has been performed on a sample power system model to demonstrate the effectiveness of the proposed approach for transient waveform classification. The proposed method is the reduction of data size as well indicating the main characteristics of signal without losing its distinguishing characteristics.