Chapter 3 Spectral Analysis using Pattern Classification

Transcription

1 36 Chapter 3 Spectral Analysis using Pattern Classification 3.. Introduction An important application of Artificial Intelligence (AI) is the diagnosis of fault mechanisms. The traditional approaches to the problem of fault diagnosis require the construction of a heuristic, rulebased system, embodying a portion of the compiled experience of a human expert. These systems perform diagnosis by mapping fault symptoms to generated hypothesis and arrive at diagnostic conclusions. The knowledge acquisition and search process involved in expert systems is exhaustive and time consuming. In addition, the simulation of models is usually too slow for applying in a real time environment. ANNs are found to be suitable for the above requirements. They are massively parallel interconnected networks of simple adaptive elements and their hierarchical organizations intend to interact with the objects of the real world in the same way as the biological counterparts. Neural networks have wide applications in parallel distributed processing and real time environments [9, 4 and 5]. Neural networks have considerable advantages over expert systems in terms of knowledge acquisition, addition of new knowledge, performance and speed [8]. Recently, interest in the application of associative memories and neural networks to the problems encountered in the development of diagnostic expert systems has

2 37 increased. The features of neural networks coincide with the requirements of pattern based diagnosis [, and 9]. An important feature of fault diagnosis using neural networks is the interpolation of the training data to give an appropriate response for the cases described by neighbouring or noisy input data. This work describes the design and simulation of a neural network for fault detection and diagnosis of power systems. In this work, fault diagnosis is conceptualized as a pattern classification problem, which involves the association of patterns of input data representing the behaviour of the power system to one or more fault conditions. The neural network is trained off-line with different fault situations and used on-line. The diagnostic system was able to detect and diagnose the fault component corresponding to the input pattern consisting of switching status of relays and circuit breakers Neuro Computing: A Pattern Recognition Perspective Neuro computing is one of the rapidly expanding areas of the present research, attracting research scholars from a wide variety of disciplines. Pattern recognition in one form or the other dominates the application of neural networks. Pattern recognition is generally defined as an abstract formulation of the categorization tasks in pattern classification. For a given input, it is possible to analyze the input to provide a meaningful categorization of it s data content. The input to the pattern recognition system is a pattern vector and the output is the decision related to the category or the class to which the

3 38 input pattern belongs. A pattern recognition system is generally considered as a two stage device. The first stage is feature extraction and the second stage is classification. Feature extraction corresponds to selection of a definite characteristic of the input pattern. The classifier is supplied with a list of measured features. The task is to map these input patterns to a classification state, that is, given the input features, the classifier decides the type of the class category it must match closely. The classifiers typically rely on distance metrics and probability theories to perform the above task. Once the feature extractor is fixed, the feature vector X is obtained. The next stage is to design an optimal decision rule so that by observing X, the classifier can make a correct decision related to it s class membership. Often it is not possible to derive such rules analytically. The classifier is then designed to learn the proper decision rule using a training set. The training set consists of feature vectors of known classification. During the training phase, the system is fed with the feature vectors one after the other and informed regarding the type of the classification. The system uses this information in a learning algorithm to learn the required decision rules [, 2, and 6]. Pattern classification serves as a general framework in the learning phenomena of neural networks. In general terms, template matching may be interpreted as a special case of the feature extraction approach, where the templates are stored in terms of the measurements, and a special classification criterion (matching) is used for the classifier.

4 Pattern Classification Pattern classification techniques are categorized into numeric and non-numeric methods [22]. Numeric techniques include the deterministic and statistical measures. These measures are made on the geometric space. Non-numeric techniques deal with the domain of symbolic processing. Numeric techniques have large influence on the concept of pattern recognition approach to neural networks. The concept of pattern classification may be expressed in terms of feature space or mapping from feature space ( ) to decision space ( ). The input pattern consists of N measured patterns, where each set of N features can be considered as a vector X, known as the feature or measurement vector, or a point in the N-dimensional feature space. The problem of classification is to assign each possible vector or point in the feature space to a proper pattern class. This corresponds to the partitioning of the feature space into mutually exclusive regions and each region corresponds to a particular class Behavioural Representation Behavioural data is applied to the inputs of the networks to train and use neural networks. Behavioural data is represented by two methods. In the first method, simulation data is applied at the input layer of the neural network as a vector of continuous real variables. In the second method, parameter behaviour is represented in a binary form. The performance of the power system under fault conditions is represented by a field of s and s corresponding to the off and on

5 4 states of the circuit breakers and relays of the power system. The field is represented by a single binary vector. Input Pattern Information from relays of power system components Input Layer I Hidden Layer J Wij Output Layer K Wjk Fault components Figure 3.: Neural network topology. The single binary vector is applied as input to the neural network. The performance of the neural network in terms of discriminating power is found to be more suitable in the second method as the behaviour of the power system is exploited completely in training the network Neural Network Performance ANNs are associated with the inherent properties of learning, generalization, abstraction and applicability and hence it s performance is measured in terms of the above features. ANNs modify their response to the environment. ANNs self organize to produce

6 4 consistent properties for a given set of inputs along with the desired outputs. Once trained, the networks response can be insensitive to minor variations in it s input. ANN generalizes automatically due to it s structure. ANNs abstracts the essence of a set of inputs. They can be trained to produce new outputs. They are best suited for large class of pattern recognition tasks through the connectivity property Spectral Fault Analysis Introduction Fault location estimation is a very important issue in power system engineering in order to clear faults quickly and restore power supply as soon as possible with minimum interruption. This is necessary for health of power equipment and satisfaction of customer. In the past, several methods have been used for estimating fault location with different techniques such as line impedance based numerical method, traveling wave methods and Fourier analysis [26].Now-a-days, high frequency methods instead of traditional methods have been used[27]. Fourier transform is used to abstract fundamental frequency component but it has been shown that Fourier transform based analysis sometimes are not exactly enough. Recently wavelet transform has been used extensively for estimating fault location accurately. The most important characteristic of wavelet transform is to analyze the waveform on time scale rather than frequency. The wavelet transform is a mathematical tool like Fourier transform in analyzing a signal that decomposes a signal into different scales with different levels of resolution. Santoso etal [28] proposed

7 42 wavelet transform technique for the detection and localization of the actual power quality disturbances. They explored the potential of wavelet transform as a new tool for automatically classifying power quality disturbances. The power system [] demand for quality power has been increasing in the past several years. The main reason is the increased use of microelectronic processors in various types of equipment such as computer terminals, programmable logic controllers and diagnostic systems. Most of these systems are quite susceptible to disturbances in the supply voltage. Now-a-days the amount of waveform distortion is more significant due to the wide applications of non-linear electronic devices in power apparatus and systems [2]. Electric utilities cannot adopt suitable strategies to provide a better service without determining the existing levels of power quality. Therefore an efficient approach to justify the electric power quality disturbances [3] is motivated. Several research studies regarding the power quality have been conducted. Their aims are often concentrated on the collection of raw data for further analysis, so that the impacts of various disturbances can be investigated [3]. Sources of such disturbances can be located or further mitigated. However, the amount of acquisition data is massive in their test cases. Such an abundance of data is time consuming for the inspection of possible culprits. Thus a more efficient approach is required in the power quality assessment. The implementation of the DFT by various algorithms is the basis for modern spectral analysis. Such transforms

8 43 are successfully applied to stationary signals where the properties of signals do not evolve in time. However, for non-stationary signals an abrupt change spreads over the whole frequency axis. In this situation, the Fourier transform is less efficient in tracking the signal dynamics. A point to point comparison scheme has been proposed to identify the dissimilarities between consecutive cycles. This approach is feasible in detecting certain types of disturbance but fails to detect the disturbance occurring periodically. With the introduction of new network topologies and improved training algorithms, neural network technologies have demonstrated their effectiveness in several power system applications [4, 5, and 6]. Once the networks have been trained well, the disturbances that correspond to the new scenario can be identified in a very short time. This technique has been applied in the power system applications. However, it can only be applied to detect a particular type of disturbance. With different disturbances, the power system network structure has to be reorganized and the training process must be restarted. A method of detecting power quality disturbances based on neural networks and wavelets has been proposed. In this method, the fundamental component is removed using wavelets and the remaining signal corresponding to disturbances is processed and given as input to ANN. However, this method fails to detect voltage sag/swell and also new ANNs have to be developed for different rated load voltages and sampling frequencies. Recently the emergence of wavelets has paved a unified framework for signal processing and it s applications. Fourier Transforms are based

9 44 on a single window for the spread of frequencies. Wavelet Transforms apply different lengths of windows depending on the signal frequencies. Features of non-stationary disturbances are closely monitored by the wavelets. The transient behaviour of cavities and disruption of the signals can be investigated by wavelet transformations. For example, if there is an immediate impulse disorder occurring at a certain time interval, it contributes to Fourier Transform, but it s position on the time axis is lost. Information is obtained by wavelets in time domain and frequency domain. In other words, Wavelet Transform is more local. Instead of transforming a pure time domain into a pure frequency domain processing, wavelet transforms find a good compromise in time - frequency domain. In this work an algorithm which overcomes all these difficulties and detects and classifies the disturbance in the signal accurately is developed. This method is independent of the load voltage and can be easily customized for different sampling frequencies. In this approach, a particular wavelet is used to detect each disturbance. The method uses wavelet filter banks in an effective way and performs multiple filtering to detect the disturbances. The fundamental idea behind spectral analysis is to analyze the recorded current signal at different scales or resolutions, which is called multi resolution analysis. For the analysis of current signal in multi spectral domain frequency domain transformation and analysis is carried out. For the decomposition of current signal pulse into individual resolution for analysis is carried out using advanced signal

10 45 transformation technique called Wavelet transformation. Wavelets are a class of functions used to localize a given signal in both space and scaling domains. Compared to Windowed Fourier analysis, a wavelet is stretched or compressed to change the size of the diagnosis window. In this way, wavelets give an approximate better analysis of the signal, while smaller and smaller wavelets explore the details of the signal. Wavelets automatically adapt to both the high frequency and the lowfrequency components of a signal by different sizes of windows. Any small change in the wavelet representation produces a correspondingly small change in the measured measured signal, which means a local mistake does not influence the entire transform. With these property wavelet transform is best suited for the analysis of non-stationary current signals, which are very brief signals and with interesting components at different scales. To diagnose the measured online current pulse in this work, advanced wavelet transformation using spline representation is used for spectral signal analysis and synthetic representation Signal Representation The fundamental idea behind wavelets is to analyze the signal at different scales or resolutions, known as multi resolution. Wavelets are a class of functions localizing a given signal in both space and scaling domains. A family of wavelets can be constructed from a mother wavelet. When compared with windowed Fourier analysis, a mother wavelet is stretched or compressed to change the size of the window. In this way, big wavelets give an approximate image of the

11 46 signal, and small wavelets zoom on details. Therefore, wavelets automatically adapt to both the high frequency and the low frequency components of a signal by different sizes of windows. A small change in the wavelet representation produces a correspondingly small change in the original signal, which means local mistakes do not influence the entire transform. The Wavelet Transforms suits for nonstationary signals, such as very brief signals and signals with interesting components at different scales. Wavelets are functions generated from a single function ψ known as mother wavelet, by dilations and translations. xb )x( a 2 b,a a where ψ must satisfy the equation ψ (x) dx =. A wavelet transform represents an arbitrary function f as a decomposition of the wavelet basis or f is represented by an integral over a and b of ψa,b. Let a m m a,bnb o a with m, n integers, and a>,b> fixed. Then the wavelet decomposition is f c n,m )f( n,m In power analysis, the sampled data is discrete in time. It is required to have discrete representation of time and frequency known as Discrete Wavelet Transform (DWT). Wavelet Transform (WT) analyzes non-stationary signals, i.e., frequency response vary with time. Although the time and frequency

12 47 resolution problems are a result of a physical phenomenon and exist irrespective of the transform used, it is possible to analyze any signal by using an alternative approach, Multi Resolution Analysis (MRA). MRA analyzes the signal at different frequencies with different resolutions. MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. This approach is useful when the signal considered has high frequency components for short durations and low frequency components for long durations which are basically used in practical applications Spectral Analysis For the representation of a given current pulse in its spectral domain a wavelet transform is applied to the current pulse defined by: *, s f t t dt st where * denotes a complex conjugation. A function f(t) is decomposed into a set of basis functions, Ψs,τ(t) called the wavelet function. The variables s and t, are the scale and translation, with new dimensions after the transformation. The coefficients are generated from a single basic wavelet (t), the socalled prime wavelet, by repetitive scaling and translation: t t s s st where s is the scale factor which is the translation factor and the factor s-/2 is used for energy normalization across different

13 48 frequency scales. It is important to note that in equations 3.3 and 3.4 the wavelet basis functions are not specified. This is a difference between the wavelet transform and the Fourier transform, or other transforms. The wavelet transforms deals with the general properties of the wavelets and wavelet transforms for the observatory signal only. The wavelet transform has to satisfy three properties that make it difficult to use directly on measured signal. The first is the redundancy of the CWT.The wavelet transform is calculated by continuously shifting a continuously scalable function over a signal and calculating the correlation between the two. It is observed that these scaled functions are nowhere near an orthogonal basis and the obtained wavelet coefficients is therefore be highly redundant. For most practical applications this redundancy is removed. Even without the redundancy of the CWT one still have an infinite number of wavelets in the wavelet transform and would like to see this number reduced to a more manageable count. This is the second problem. The third problem is that for most functions the wavelet transforms have no analytical solutions and they can be calculated only numerically increasing the complexity of implementation. Fast algorithms are hence needed to be able to exploit the power of the wavelet transform and it is in fact the requirement of these fast algorithms that have put wavelet transforms for the improvement and made the requirement for real time applications.

14 49 As mentioned before the transformation maps a one-dimensional signal to a two-dimensional time-scale joint representation that is highly redundant. The time-bandwidth product of the transformation is the square of that of the signal and for most applications, which seek a signal description with as few components as possible, which is not efficient. To overcome this problem discrete transformations have been introduced. Discrete transformations are not continuously scalable and translatable but can only be scaled and translated in discrete steps. This is achieved by modifying the equation 3.4 to obtain, t k s j o jk, t j j s s Although it is called a discrete wavelet, it normally is a (piecewise) continuous function. In equation 3.5, j and k are integers and s > is a fixed dilation step. The translation factor τ depends on the dilation step. The effect of discretizing the wavelet is that the timescale space is now sampled at discrete intervals. It is usually chosen s = 2 so that the sampling of the frequency axis corresponds to dyadic sampling. This is a very natural choice for computers, the human ear and music for instance. For the translation factor the value is usually chosen τ = so that a dyadic sampling of the time axis is obtained. When discrete transformation are used to transform a continuous signal the result is be a series of wavelet coefficients, and it is referred to as the wavelet series decomposition. An important issue in such a

15 5 decomposition scheme is of course the question of reconstruction. It is all very well to sample the timescale joint representation on a dyadic grid, but if it is not be possible to reconstruct the signal it is not be of great use. As it turns out, it is indeed possible to reconstruct a signal from its wavelet series decomposition. A generic representation of a current pulse and its spectral decomposition is as illustrated in figure 3.2, Figure 3.2: Original phase current pulse without disturbances and the Spectral coefficient representation of each phase signal in frequency domain. The spectral decomposition illustrates the coefficient domination and its coefficient density per band. It could be observed that with the increase in the band level the coefficient density of the fundamental frequency content get reduced and it is more dominantly been observed in detail band (d) as compared to d3. This feature very effectively gives the option of selecting required coefficients for further

16 5 processing which effectively improves the response time of a controlling system Spectral Filter Design With the redundancy removed, there are still two hurdles to take before, the transformation in a practical form. Continuing by trying to reduce the number of transformation needed in the wavelet transform and save the problem of the difficult analytical solutions for the end. Even with discrete transformation it still needs an infinite number of scalings and translations to calculate the wavelet transform. The easiest way to tackle this problem is simply not to use an infinite number of discrete transformations. Of course this poses the question of the quality of the transform. Is it possible to reduce the number of transformation to analyze a signal and still have a useful result? The translations of the transformation are of course limited by the duration of the signal under investigation so that have an upper boundary for the transformation. It is observed that wavelet has a band-pass like spectrum. From Fourier theory the compression in time is equivalent to stretching the spectrum and shifting it upwards: F f at F a a This means that a time compression of the wavelet by a factor of 2 is stretch the frequency spectrum of the wavelet by a factor of 2 and also shifts all frequency components up by a factor of 2. Using this insight one can cover the finite spectrum of our signal with the spectra of dilated transformation in the same way as that one covered

17 52 our signal in the time domain with translated transformation. To get a good coverage of the signal spectrum the stretched wavelet spectra should touch each other, as if they were standing hand in hand. This can be arranged by correctly designing the transformation. Figure 3.3: Multi Band wavelet Spectra resulting from scaling of prime wavelet in the time domain. The frequency response for such filter coefficient is as illustrated in figure 3.4, Original scaling filter Decomposition low-pass filter Lo_D Decomposition high-pass filter Hi_D Reconstruction low-pass filter Lo_R Transfer modulus: lowpass (Lo_D or Lo_R Reconstruction high-pass filter Hi_R Transfer modulus: highpass (Hi_D or Hi_R Figure 3.4: Frequency response plot for filter coefficients used for spectral decomposition. A time-scale representation of a digital signal is obtained using digital filtering Techniques. The transformation is a correlation between a wavelet at different scales and the signal with the scale (or

18 53 the frequency) being used as a measure of similarity. The continuous wavelet transform was computed by changing the scale of the analysis window, shifting the window in time, multiplying by the signal, and integrating over all times. In the discrete case, filters of different cutoff frequencies are used to analyze the signal at different scales. The signal is passed through a series of high pass filters to analyze the high frequencies, and it is passed through a series of low pass filters to analyze the low frequencies. The resolution of the signal, which is a measure of the amount of detail information in the signal, is changed by the filtering operations, and the scale is changed by up sampling and down sampling (sub sampling) operations. Sub sampling a signal corresponds to reducing the sampling rate, or removing some of the samples of the signal. For example, sub sampling by two refers to dropping every other sample of the signal. Sub sampling by a factor n reduces the number of samples in the signal n times. Up sampling a signal corresponds to increasing the sampling rate of a signal by adding new samples to the signal. For example, up sampling by two refers to adding a new sample, usually a zero or an interpolated value, between every two samples of the signal. Up sampling a signal by a factor of n increases the number of samples in the signal by a factor of n. Although it is not the only possible choice, DWT coefficients are usually sampled from the transformation on a dyadic grid, i.e., s = 2 and =, yielding s=2 j and =k*2 j. Since the signal is a discrete

19 54 time function, the terms function and sequence is be used interchangeably in the following discussion. This sequence is be denoted by x[n], where n is an integer. The procedure starts with passing this signal (sequence) through a half band digital low pass filter with impulse response h [n]. Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter. The convolution operation in discrete time is defined by: * x n h n x k h n k k A half band low pass filter removes all frequencies that are above half of the highest frequency in the signal. For example, for a current signal having a maximum of 5 Hz component, then half band low pass filtering removes all the frequencies above 25 Hz. The spectral representation of such a processing on a measured current signal is as observed below, Figure 3.5: The frequency and energy spectral plot for the detail coefficients (d) for the phase current. The unit of frequency is of particular importance at this time. In discrete signals, frequency is expressed in terms of radians.

20 55 Accordingly, the sampling frequency of the signal is equal to 2 radians in terms of radial frequency. Therefore, the highest frequency component that exists in a signal is be radians, if the signal is sampled at Nyquist s rate (which is twice the maximum frequency that exists in the signal); that is, the Nyquist s rate corresponds to rad/s in the discrete frequency domain. Therefore using Hz is not appropriate for discrete signals. However, Hz is used whenever it is needed to clarify a discussion, since it is very common to think of frequency in terms of Hz. It should always be remembered that the unit of frequency for discrete time signals is radians. After passing the signal through a half band low pass filter, half of the samples can be eliminated according to the Nyquist s rule, since the signal now has a highest frequency of /2 radians instead of radians. Simply discarding every other sample is sub sample the signal by two, and the signal is then have half the number of points. The scale of the signal is now doubled. Note that the low pass filtering removes the high frequency information, but leaves the scale unchanged. Only the sub sampling process changes the scale. Resolution, on the other hand, is related to the amount of information in the signal, and therefore, it is affected by the filtering operations. Half band low pass filtering removes half of the frequencies, which can be interpreted as losing half of the information. Therefore, the resolution is halved after the filtering operation. Note, however, the sub sampling operation after filtering does not affect the resolution, since removing half of the spectral components from the signal makes

21 56 half the number of samples redundant anyway. Half the samples can be discarded without any loss of information. In summary, the low pass filtering halves the resolution, but leaves the scale unchanged. The signal is then sub sampled by 2 since half of the number of samples is redundant. This doubles the scale. This procedure can mathematically be expressed as y n h k. x 2n k k Having said that, now looking at how the DWT is actually computed: The DWT analyzes the signal at different frequency bands with different resolutions by decomposing the signal into coarse approximation and detail information. DWT employs two sets of functions, called scaling functions and wavelet functions, which are associated with low pass and high pass filters, respectively. The decomposition of the signal into different frequency bands is simply obtained by successive high pass and low pass filtering of the time domain signal. The measured signal x[n] is first passed through a half band high pass filter g[n] and a low pass filter h[n]. After the filtering, half of the samples can be eliminated according to the Nyquist s rule, since the signal now has a highest frequency of /2 radians instead of. The signal can therefore be sub sampled by 2, simply by discarding every other sample. This constitutes one level of decomposition and can mathematically be expressed as follows:

22 where yhigh [k] and ylow [k] are the outputs of the high pass and low pass filters, respectively, after sub sampling by 2. This decomposition halves the time resolution since only half the number of samples now characterizes the entire signal. However, this operation doubles the frequency resolution, since the frequency band of the signal now spans only half the previous frequency band, effectively reducing the uncertainty in the frequency by half. The above procedure, which is also known as the decomposition coding, can be repeated for further decomposition. At every level, the filtering and sub sampling is result in half the number of samples and hence half the time resolution and half the frequency band spanned and hence double the frequency resolution. For the spectral decomposition Coding Algorithm as an example, suppose that the measured signal x[n] has 52 sample points, spanning a frequency band of zero to rad/s. At the first decomposition level, the signal is passed through the high pass and low pass filters, followed by sub sampling by 2. The output of the high pass filter has 256 points (hence half the time resolution), but it only spans the frequencies /2 to rad/s (hence double the frequency resolution). These 256 samples constitute the first level of DWT coefficients. The output of the low pass filter also has 256 samples, but it spans the other half of the frequency band, frequencies from

23 58 to /2 rad/s. This signal is then passed through the same low pass and high pass filters for further decomposition. The output of the second low pass filter followed by sub sampling has 28 samples spanning a frequency band of to /4 rad/s, and the output of the second high pass filter followed by sub sampling has 28 samples spanning a frequency band of /4 to /2 rad/s. The second high pass filtered signal constitutes the second level of DWT coefficients. This signal has half the time resolution, but twice the frequency resolution of the first level signal. In other words, time resolution has decreased by a factor of 4, and frequency resolution has increased by a factor of 4 compared to the measured signal. The low pass filter output is then filtered once again for further decomposition. This process continues until two samples are left. For this specific example there would be 8 levels of decomposition, each having half the number of samples of the previous level. The DWT of the measured signal is then obtained by concatenating all coefficients starting from the last level of decomposition (remaining two samples, in this case). The DWT is then having the same number of coefficients as the measured signal. The frequencies that are most prominent in the measured signal is appear as high amplitudes in that region of the DWT signal that includes those particular frequencies. The difference of this transform from the Fourier transform is that the time localization of these frequencies is not be lost. However, the time localization is have a resolution that depends on which level they appear. If the main

24 59 information of the signal lies in the high frequencies, as happens most often, the time localization of these frequencies is be more precise, since they are characterized by more number of samples. If the main information lies only at very low frequencies, the time localization is not be very precise, since few samples are used to express signal at these frequencies. This procedure in effect offers a good time resolution at high frequencies, and good frequency resolution at low frequencies. Most practical signals encountered are of this type. Two of the three problems mentioned in above section have now been resolved, but one still does not know how to calculate the wavelet transform. If regarded the wavelet transform as a filter bank, then considering the wavelet transforming a signal as passing the signal through this filter bank. The outputs of the different filter stages are the wavelet and scaling function transform coefficients. Analyzing a signal by passing it through a filter bank is not a new idea and has been around for many years under the name band coding. It is used for instance in computer vision applications. Figure 3.6: Splitting the signal spectrum with an iterated filter bank.

25 6 The filter bank needed in coding can be built in several ways. One way is to build many band pass filters to split the spectrum into frequency bands. The advantage is that the width of every band can be chosen freely, in such a way that the spectrum of the signal to analyze is covered in the places where it might be interesting. The disadvantage is that to design every filter separately and this can be a time consuming process. Another way is to split the signal spectrum in two (equal) parts, a low pass and a high-pass part. The high-pass part contains the smallest details that are interested in and could stop here. However, the low-pass part still contains some details and therefore it can be split again. And again, until a satisfactory number of bands are have created. In this way an iterated filter bank can be created. Figure 3.7: Implementation of one stage iterated filter banks. Usually the number of bands is limited by for instance the amount of data or computation power available. The process of splitting the spectrum is shown in figure 3.6. The advantage of this scheme is to design only two filters for complete spectrum coverage. A analysis on the measured current phase current using such filter banks and its energy spectrum domination could be observed in following illustration,

26 6 Figure 3.8: Spectral plot for the original signal. Figure 3.9: Spectral plot of approximate (a) coefficient with lowest frequency band.

27 62 Figure 3.: Spectral plot for the detail coefficient band-2 (d2). Figure 3.: Spectral plot of detail coefficient of band-3 (d3).

28 63 Figure 3.2: Energy spectrum representation of the detail band- (d) with maximum fundamental frequency content. Figure 3.3: Energy spectrum representation of the approximate and (a) with minimum fundamental frequency content. These properties are very useful towards the utilization of coefficient reduction in signal representation; the coefficients with

29 64 dominant representations per band are extracted and passed to the learning system for developing the decision faster than the conventional system. The processing system for decision making is based on neural network architecture, as neural network provides high accuracy in decision making than the conventional system System Design The first step of the detection module obtains the transmitting voltage and current samples. The current samples are normalized and sent to DWT to obtain frequency resolved coefficients. The fault detection is performed by analyzing the current wavelet coefficient energy. When the faults do not occur data is not transferred. The power quality deviations are caused by short circuits, harmonic distortions, voltage sags and voltage swells etc. In order to correct such problems the disturbances must be detected and identified. Whenever the disturbance lasts for few cycles, a simple observation of the waveform in a bus bar is not sufficient to identify whether the problem exists or not and if the problem exists it is difficult to identify the nature of the problem. DWT has been applied to analyze the currents during short duration disturbances in the transmission line. DWT is one of the three forms of wavelet transform. It moves a time domain discretized signal into it s corresponding wavelet domain by a process known as sub-band coding, which uses digital filter techniques. The line current signals obtained from the bus bar are

30 65 applied to the wavelet filters to evaluate the frequency resolution coefficients by sending through high pass filter and low pass filter. Figure 3.4: Wavelet decomposition of electrical current pulse. To filter the given current signal f(n), the signal is convoluted at each filtration using a defined filter coefficient hd and gd. A recursive multiplication and accumulation operation is performed on the given current signal to obtain the detail and approximate coefficient as given in equations 3.3, 3.4. )n(ac d(h)n(f k )n2 k )n(dc d(g)n(f k )n2 k The f(n) signal is passed through a low pass digital filter (hd(n)) and a high pass digital filter (gd(n)). The obtained coefficients are decimated by 2 i.e., half of the signal samples are eliminated. The DWT operation is performed in two stages. The first stage consists of determination of the wavelet coefficients. These coefficients represent the given signal in the wavelet domain. From these coefficients, the second stage is achieved by calculating the approximated and the detailed version of the original signal in different levels of resolutions in the time domain. At the end of the

31 66 first level of signal decomposition, the resulting vectors yh(k) and yg(k) are the level- approximation wavelet coefficients and detail wavelet coefficients. The fault detection rules are formulated by analyzing the current waveforms in the time domain and in the first decomposition level of the DWT. This level contains the highest frequency components. To calculate the wavelet coefficient, DWT based decomposition is implemented based on the following steps; Step: Evaluation of the wavelet coefficients of the signal. Step2: Evaluation of the square of the wavelet coefficients. Step3: Calculation of the distorted signal energy in each wavelet coefficient level. The energy mentioned in step3 is based on the Parse Val s theorem which states that: the energy contained by a time domain function equals the sum of the energies concentrated in the different resolution levels of the corresponding wavelet transformed signal. This is expressed mathematically as N N JN 2 2 )n(f j)n(a j)n(d n n j n f(n): Time domain signal under consideration. N: Total number of samples of the signal. N 2 )n(f : n Total energy of the signal )n(f

32 67 N 2 j n jn :)n(atotal energ conc ed in the lev 'j' of ap e version of the signal 2 j j n :)n(dtotal energ conc ed in the det aile ve of si from levels ''to''j The energy obtained for the given current signal is considered as the feature for training the neural network. In order to classify faults, a feed forward back propagated neural network architecture is trained before testing the proposed method. The learning database contains a variety of fault scenarios to improve the ANN s generalization capability. By using this strategy, ANN classifies simulated and real faults in transmission line. The output of the ANN indicates type of the fault related to the actual input pattern. Hence, binary coding is used for the ANN s outputs such that a fault is characterized by presence () or absence () as shown in Table 3.., where the term no fault indicates that the input pattern is not related to a fault. After the ANN learns, the fault classification [4] is carried out by analyzing each window obtained from windowing process. This means that the most identified fault type prevails. By using this strategy, even if the ANN makes a mistake for some windows, the fault classification will be correct.

33 68 Fault type Phase A Output Phase B Output 2 Phase C Output 3 Ground Output 4 AG BG CG AB AC BC BG ACG BCG ABC No fault Table 3.: Binary coding of ANN output The neural network is passed with the above stated fault outputs with their trained fault current wavelet features. On testing, these features are used as knowledge by the neural network for the accurate classification of fault type and their occurrences. For the implementation of the neural network, feed forward back propagation architecture is developed with 3 hidden nodes and tangential sigmoid function for each hidden node. The training of this network is carried out by using Least Mean Square (LMS) algorithm. The network is trained with the possible input patterns obtained for each fault case, with the minimum and maximum range of inputs and outputs with an epoch limit of 8 iterations, probability of error as. with a learning rate of.. The operational model of the suggested approach is as defined below.

34 69 Get the voltage & current samples from the record Normalize the current samples Detection Module Compute the DWT for the current samples Compute the energy of the current wavelet coefficients Is it a Fault? No Avoid the record transfer Yes Identify the voltage and current samples related to the fault clearing time Normalize the voltage and current samples Classification Module Resample the wave forms Perform the windowing process Analyze each window using ANN Report the diagnosis & allow record transfer Figure 3.5: Operational flow chart for detection and classification method Power Quality Problem Classification There are various types of events that degrades power quality and results in identification problems being often elusive and difficult. A classification algorithm has been developed based on disturbance events from six major categories, namely harmonics, capacitor high frequency switching, capacitor low frequency switching, sudden voltage sag, gradual voltage sag and voltage swell. Figure 3.6 shows the sample disturbance signals. Figure 3.6(a) represents the waveform affected by harmonics. Harmonic distortion is a significant

35 7 power quality problem [5]. Due to the increase in the popularity of electronic and other non-linear loads, such as adjustable speed drives, arc furnaces and induction furnaces, perfect sinusoid waveforms are frequently distorted. Current harmonics causes increased losses in the customer s and utility s power system components and voltage harmonics affects not only sensitive electronic loads but also electric motors and capacitor banks. Transients caused by capacitor switching are one of the most common sources of degradation in utility systems [5]. Two sample signals that correspond to capacitor high frequency switching and capacitor low frequency switching are shown in the figures 3.6(b) and 3.6(c). The frequency of a transient is determined by capacitance and inductance of the system. Although capacitors have the drawback of producing the oscillatory transients while switching, they are still used in power systems to correct the power factor. A momentary voltage dip that lasts for a few seconds or less is classified as voltage sag.

36 Power Quality Disturbance Classes Figure 3.6: Sample disturbance signals: a) harmonics b) fast switching transient c) slow switching transient d) sudden voltage sag e) gradual voltage sag f) voltage swell. Voltage sag is caused by faults on the power system during starting of large loads, such as motors. Figure 3.6(d) shows an example of sudden voltage sag. A gradual voltage decay sag signal is shown in the figure 3.6(e). Process industry equipments are particularly susceptible to problems with voltage sags Multilayer Neural Network Artificial neural networks (ANN), which are parallel distributed information processing units with different connection structures and processing mechanism, are particularly suitable to link the different variables of a physical system where the relationship between the independent and the dependent variables cannot be obtained easily.

37 72 The structure of Multi Layer Neural Networks (MLNN) is shown in the figure 3.7. It has 3 layers input, hidden and output layers. The input layer has 2 nodes represented by the levels of ΔStd_MRA. The hidden layer has 2 nodes. The output layer has 5 nodes representing normal sine wave and 4 types of disturbances: voltage sag, voltage swell, harmonic distortion and interruption. The combined wavelet transformation with ANN is run by using MATLAB program with following data: No. of input nodes = 2 No. of output nodes = 5 No. of hidden nodes = 2 No. of iterations = 3 Alpha =.8 Eta =.2 Tolerance =. No. of training patterns = Disturbances such as sag, swell and interruption are generated by MATLAB program with different percentages, durations and instants of disturbances. Patterns of harmonics are generated by randomly selecting harmonics of various percentages as compared to fundamental component and for different duration. In total, a set 4 patterns are generated. Training of the MLNN is performed with a data set of patterns with 25 patterns in each type of disturbance and pattern of pure sine wave signal. Testing is performed with the remaining 3 patterns of the disturbances and pattern of the normal sine wave. Simulation results of the combined wavelet transformation and MLNN, to detect and classify the data into normal sine wave and other 4 types of power quality problems has been presented.

38 73 Figure 3.7: Multilayer neural network structure used for power quality classification. Table 3.2: Percentage classification accuracy of single multilayer neural network for PQ problem. All the values are indicated in percentages. The model presented is capable of classifying the dataset but the classification accuracy is less. The overall accuracy of the combined WT and MLNN is the average of diagonal elements as shown in the table 3.2 and it is 9.8%.

39 Modular Neural Network Inorder to improve the accuracy of the neural network in classifying PQ disturbances, the complex task is divided into subtasks resulting in Modular Neural Network (MNN). MNN consists of more than one neural network known as modules, to handle each subtask. Each module is independent and domain specific. Hence modules respond to a particular set of data input for which it is intended. The solution of the overall task is achieved by combining the result of each module. The structure of the MNN is shown in the figure 3.8. The advantage of modular structure is the individual model responds to a given input faster than a complex monolithic system. Such a modular structure can be imbibed in different types of neural networks, including MLNN. An approach of MNN is exploited to classify PQ disturbance signals. Simulation results are presented and the performances of the MNN and MLNN are compared. Figure 3.8: Model of modular neural network technique. The structure of MNN for PQ classification is designed with 5 modules for identification of pure sine wave and 4 disturbances as

40 75 shown in the figure 3.9. Each module has 2 input nodes, 2 hidden nodes and output node. Hence each module is a multi layer network. Training is performed with delta-std indices, obtained from 2 levels of WT of the voltage signals. These inputs are applied simultaneously to all modules. The output of the module corresponding to the disturbance under consideration is set to and output of remaining modules is set to during training. Training of the MNN is performed with patterns (25 patterns for each disturbance and pattern of pure sine wave). The 2 indices of a particular disturbance are applied to all modules (pure sine wave, sag, swell, harmonics and interruption modules) simultaneously. Then the output of all modules is combined by maximum operation. The largest of the output nodes is considered as the output of MNN and determines the disturbance.

41 76 Figure 3.9: Implementation of modular neural network for PQ classification Classification Algorithm Figure 3.2 shows the proposed classification algorithm. Each voltage signal to be identified consists of five cycles of voltage waveform sampled 256 times per cycle. The target of the feature extraction is to generate a N-point feature vector from the original 28-point voltage signal. The feature vectors are the inputs of the neural network for classification.

42 77 Figure 3.2: Classification algorithm Fuzzy Classification Approach A classification algorithm for power quality disturbance signals is proposed. However, the idea behind the proposed algorithm is to classify the disturbance signal into one of the disturbance classes, named as Crisp Classification Approach. It is assumed that there is only one type of disturbance in a 5 cycle waveform. In some cases, however, there are multiple types of disturbances at the same time. The crisp classification strategy does not work very well for these combined events. Based on the same feature extraction scheme, a fuzzy classification approach that provides more accurate, more comprehensive and more useful information for the power system under consideration is developed. The basic idea is to provide a soft evaluation of the disturbance component in a given part of 5-cycle disturbance waveform. Specifically, the grade (i.e. membership

43 78 function) corresponding to each disturbance class is determined. The grade of an arbitrary class A shows the extent to which the waveform includes the disturbance of class A. For example, a 5-cycle disturbance waveform input provides output as harmonics 2.5, capacitor fast switching 7.5, capacitor slow switching 5.5, sudden sag.5, sag gradual decay 3, swell.5. Thus the short duration disturbance is mostly a capacitor switching event. But it also has slight harmonics and gradual sag decay components. This strategy serves better the goal of monitoring, analyzing and evaluating the power quality of a given power system Results The proposed wavelet and ANN based fault detection and classification architecture is tested on a randomly distributed network as shown below with following specifications.

44 79 Figure 3.2: Implemented distributed power system architecture. The faults are simulated on a 25 MV, 3.2 kv, and km transmission line system. The feature table for training the neural network is in appendix-a. The simulation results are as illustrated below; Figure 3.22: Menu generated for the selection of test conditions, neural network operation and fault testing conditions.

45 8 Performance is , Goal is Epochs 4 Figure 3.23: Learning plot for the validation, testing, and training of the generated neural network. Figure 3.24: Simulation result of the training process of the neural network.

46 8 Figure 3.25: The input test values entered for the simulation of the electrical system. 2 Current (Amp) Current (Amp) Current (Amp) Phase A Line Current (Amp) Time (sec) Phase B Line Current (Amp) Time (sec) Phase C Line Current (Amp) Time (sec) Figure 3.26: Three line currents generated for transmission.

47 b A Wavelet coefficient for Phase A b B Time (sec) Wavelet coefficient for Phase B Time(sec) b C Wavelet coefficient for Phase C Time(sec) Figure 3.27(a),(b),(c) :Wavelet coefficients of the three line currents. Figure 3.28: ANN output for the normal testing.

48 83 Phase A Line Current due to L-G fault in Phase A 5 Current (Amperes) Time (sec) Figure 3.29: Fault current at phase-a for L-G fault. 8 Wavelet coefficient for Phase A b A Time (sec) Figure 3.3: Wavelet coefficient generated for the fault current in phase A.

49 84 Figure 3.3: ANN test output for L-G fault in phase A. 4 2 Current (Amperes) -2-4 s 4 2 Current (Amperes) -2-4 Phase A Line Current due to L-L fault between phases A, B Time (sec) Phase B Line Current due to L-L fault between phases A, B Time (sec) Figure 3.32: Fault currents generated for L-L fault between phases A-B.

50 85 Figure 3.33: ANN output for L-L-L fault.

51 86 Phase A Current (Amperes) Time (sec) Phase B Current (Amperes) Time (sec) Current (Amperes) Phase C Time (sec) Figure 3.34: Line current for three phases under sag testing.

52 87 Phase A Current (Amperes) Time (sec) Phase B Current (Amperes) Time (sec) Phase C Current (Amperes) Time (sec) Figure 3.35: Sag current generated in phase B.

53 88 6 Compensation Current 4 2 Current (Amperes) Time (sec) Figure 3.36: Compensation current generated to compensate sag. Phase A Current (Amperes) Time (sec) Phase B Current (Amperes) Current (Amperes) Time (sec) Phase C Time (sec) Figure 3.37: Current after sag compensation.