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This paper presents Bilinear Time Frequency Distribution based on Wigner Ville Distribution for recognition and detection of power quality disturbances with certain related analysis. The proposed method would introduce the whole new power quality representation in term of windowed time versus frequency data. Since the occurrence of power quality disturbances is not at the same time for the same frequency there is a need to capture the identical pattern for certain power quality disturbances such as voltage sag, harmonics, and voltage swell in Time frequency representation. By obtaining the specific identical representation in windowed time versus frequency the accuracy and instantaneity of analysis on power quality disturbances would be rapidly increased and type of disturbances can be easily recalled.
Keywords-component; Power Quality (PQ) disturbances detection , Bilinear Time Frequency Distribution (BTFD) , Wigner Ville istribution (WVD),Total Harmonic Distortion (THD) .
Power quality is the most prolific and major concern in power industry especially for the power engineers, respectively. Maintaining the power quality at high level is always the aim for the power engineers so that the power delivered efficiently with lesser power loss. Voltage, current or frequency deviation manifesting in the power quality problem would result in failure or permanent disoperation of apparatus and load equipments. In addition to real power quality problems, there are also perceived power quality problems that may actually related to hardware, software, or control systems malfunctioning. For example, in power electronics, a capacitor is switching, which is quite common and normal on the utility system, but can cause transient over-voltages that disrupt manufacturing machinery . A repeated transient voltage may eventually fail due to low-magnitude events which would cause a severe degradation of the electronic components, over the time. There are four major reasons for power quality concerns ;
Sensitivity of the load equipments: Microprocessor based controls and power electronic devices in circuits tend to be more sensitive to disturbances since they are involved in low power limits.
Application of devices for power factor correction: The application of devices for power factor correction in order to reduce losses and increase power efficiency which consisted of inductive and capacitive properties has contributes to higher level of harmonics in the power system.
Awareness of power quality issues: The end users demanding for better power quality which provides ultrahigh availability of online power service and precision manufacturing systems which is efficient in terms of economics and energy consumption.
Interconnection in a network : Integrated processes of components in the interconnected network increase vulnerability to the failure that has much more impact consequences
Abjection of power quality in the power system usually caused by power-line and non linear load disturbances. Power quality disturbances such as voltage sag, swell, harmonics distortion, transients, spikes, flicker, notch and momentary disruption, tend to cause severe damage such as malfunction of the apparatus, overheating, power instabilities, shorter life span, failure of electrical equipments and much more. In an electrical distribution network system fault would cause voltage sag which is a reduction of AC Voltage at a given frequency for the duration of 0.5 cycles to 1 minute's time . Voltage swell which is an instantaneous voltage increase caused by single line ground failures (SLG), upstream failures, switching off a large load or capacitor ,for a brief of moment. While on the other hand, the usage of solid state switching devices and non linear load with power electronically switches such as rectifiers or inverter would contribute to harmonics distortions. Ferro resonances, transformer energization or capacitor switching would led to transients and lightning or surges would cause a severe spikes .
In a real time situation, in order to maintain power quality in the distribution system, these disturbance need to be identified as soon and accurate as possible before appropriate mitigation acts taken .Any delay of the right mitigation acts taken would cause the apparatus in power system to reach its lifespan, or disintegrate, which would lead to a disastrous losses, especially when involving multimillion dollar power line with millions of people depends on it. That's the solid reason why the detection should be in quick and precise manner.
A. TIME REPRESENTATION AND FREQUENCY DISTRIBUTION
In the most of basic engineering data representation, amplitude versus time domain is commonly used especially when involved continuous or discrete time as the x-axis, and usually generate graphical data such as sinusoidal, exponential, linear, hyperbolic data, et cetera. In statistics, a frequency distribution is an arrangement of the values that one or more variables take in a sample, which shows a summarize grouping of data based on frequency of the data. The problem in power quality analysis by using time domain is that we know what kind of disturbances signal happen but didn't precisely know in what time it's happen, by mean of frequency. Plus, when the distortion happens in a very long period of time, along with various of high frequency distortion involved, the analytic signal pattern ,in time domain varying signal, is usually hard to be recognized, whether what kind of distortion happened to the main signal. By doing a joint time-frequency distribution, the representation of the data would be no more only graphical, but a graphical statistical representation, rich with a lot of information, gathered in a more simplified representation. This could enhance the analysis of the PQ problem even further, more precise, and easier for detection.
B. TIME FREQUENCY REPRESENTATION (TFR)
In this paper these various types of disturbances would be analyzed and represented using bilinear time frequency distribution representation's spectrogram, which would able to generate accurate time frequency representation (TFR) pattern in instant manner. Time frequency representation, TFR would gather the frequencies in the time manner, so that the analytic signal would be represented in more compact and easier to analyze, with certain recognized recorded pattern. The TFRs can be sampled with respect to the type of disturbances and the identical TFR generated after disturbances can be compared to the most significant sampled TFR.
In order to identify the type of disturbance present in the power signal effectively, it is advisable to involve the usage of Digital Signal Processing (DSP) approaches. As an improvement of FFT techniques, Short Time Fourier Transform (STFT) was developed for power quality disturbances detection and characterization in time-frequency domain. However STFT has the limitation of fixed window width chosen appropriate and this would lead to limitations for low-frequency and high-frequency non-stationary signals analysis at the same time, so that the series may be lost or inaccurate.
The TFR approaches use the Fast Fourier Transform, FFT component, in its integrated formula. It can be designated in discrete form, DFT or continuous form. Thus it is possible to calculate the Total Harmonics Distortion through the TFR, via Power Spectral Density, PSD related formula.
The TFR itself can be categorized in Linear Time Frequency Distribution (LTFD), Bilinear Time Frequency Distribution (BTFD) and Multilinear Time Frequency Distribution (MTFD) . As the linearity of the time-frequency increase, the efficiency, accuracy and the cross term of the signal increase too. One of the examples for LTFD is power spectrum, commonly used in THD calculations. However, LTFD offers a limited frequency resolution compared to the BTFD.As for MTFD the cross term intensity is too high to analyze the simple analytic signal and need a lot of kernel parameters involved. Bilinear time-frequency distributions (BTFD) have been intensively used to characterize and analyze non stationary signals. The bilinear TFDs offer a good time and frequency resolution and are successfully applied to various real-life problems such as radar, sonar, seismic data analysis, biomedical engineering, and automatic emission. However, the TFDs suffer from the presence of cross terms interferences because of its bilinear structure. This inhibits interpretation of its TFR, especially when signal has multiple frequency components . In BTFD many types of representation introduced such as Wigner Ville, Choi Williams, Periodogram, Spectrogram and many more. In this paper Wigner Ville were chosen since the Wigner Ville Distribution given the best representation in term of power quality based on its characteristic kernel parameters, and the cross term generated is acceptable .
WIGNER VILLE DISTRIBUTION (WVD)
Wigner Ville Distribution is a Fourier transform acting on the delay variable of a properly symmetries (with respect to evaluation time, x(t) covariance function). In this distribution x(t) is the analytic signal or complex signal of s(t).The analytic signal, x(t) commonly represented in the power quality disturbance signal. It gives better temporal and frequency resolution, at the expense of many artifacts and the introduction of negative values than a previously known FFT TFR spectrogram. The analytic signal x(t) of the signal s(t) is defined as:
Where is the Hilbert transform of the signal s(t),which is for smoothing and imaging the pattern recognition.
Construction of a time-frequency distribution, Wigner Ville Distribution as is to represent precisely the energy, temporal and spectral characteristics of the signal.
Consider a monocomponent FM signal
The Wigner Ville Distribution representation is as below
Using central finite difference (CFD) approximation
The integral of the WVD w.r.t frequency is the instantaneous power.
The integral of the WVD w.r.t time is the energy spectrum. 
This can be used to generate Wigner Ville Spectrum similar to Power Spectral Density that useful to calculate total harmonic distortion, THD. Power spectral density, PSD can be related as:
B. POWER SPECTRAL DENSITY (PSD)
With applying limits, T, PSD can be calculated;
The Fourier transform normalized in a finite T
The PSD can be quoted as :
From the PSD calculated, the THD in harmonics disturbance case can be calculated based on fundamental frequency and power amplitude deviated, by percentage. Since power proportional to voltage, from PSD, power density,, which is also proportional to voltage, then related formula for THD to the n harmonics is as follow;
Where is the fundamental voltage, is the harmonic component voltage.
C. SIGNAL MODEL
The signal used for this paper is AC Voltage sinusoidal signal which consists in the power lines. The equation for the AC Voltage in time domain is noted as below:
Where is based on the sample number, n;
f=50Hz, Vpeak = 1p.u.
Power qualities (PQ) analysis comprises various kinds of electrical disturbances such as voltage sags, voltage swells, harmonic distortions and combination of either 2 or all of given disturbances. Using the time frequency localization property of the Wigner Ville distribution, those above power quality problems are analyzed and detected. The test of the PQ's done on the signals taken from the generated waveforms with certain characteristics using MATLAB. The chosen sampling rate is 50Hz, and the frequency were normalized equal to 0.1 at 50hz.The sample were 500 and the time is equal to 1 over sample, 1/n.
Pure sine wave:
The frequency used was 50Hz same as predominantly used frequency in electrical system. The signal were sampled at 500 samples
The pure sine wave function is defined by ;
With a=1 p.u.
The intensity of the line in Fig.2 were bold only at Frequency = 50Hz.During this period no disturbance was occurred.This WVD shows a clean sinusoidal signal of v(t) without any power quality disturbance.
Figure .1- Pure Sine wave
Figure - WVD for Pure sine wave
3RD harmonics in pure sine wave at Ts = [100,400]
3rd Harmonics Distortion: The problem occurs due to the non linear load in the households. The energization process of transformer also contributes to the triplen harmonics.
As in Fig.4 the intensity of frequency can be detected at 50Hz,100Hz and 150Hz.This shows that the signal were distorted with another frequency , it is,150Hz.The period of 3rd harmonics occurrence can be seen clearly in the interval of 100 to 400 s.
Figure - 3rd Harmonics
Figure - 3rd harmonics WVD
Voltage Sag in pure sine wave, v(t)
Voltage sag: This disturbance occurs due to a fault in line, starting of large motors or switching of heavy loads component which causing the amplitude of the voltage drops by 10 to 90 percent of the rated value .Fig. 9 below shows the 25% voltage sag in the voltage signal, v(t) .
The distortion of sag occur at sample time, Ts= [100,400] and the amplitude a=0.75
As in Fig.6 intensity of the line were high at t=100 but low at t=400, more over like a reversed intensity. This show that from high to low intensity during interval of t= [100,400] indicates voltage sag occurred, where amplitude of voltage dropped down. Note that the curved distorted at t=[100,150] and t=[350,400] shows that 100 out of 400 total frequency that equal to 25% of amplitude, indicates 25% sag of voltage amplitude.
Figure 5 - Voltage Sag at 25%
Figure 6- Voltage Sag 25% WVD
Sag with harmonics
Voltage sag with harmonics: Occurs when causes of the voltage sag involved non linear load, for example switching of heavy non linear loads. Figure 11 below shows the 45% voltage sag with 3rd harmonics component.
As in Fig.8 we can detect high intensity with curve at t=[100,150] and low intensity with curve at t=[350,400] indicates 25% voltage sag. The intensity also present at f=150Hz indicates that 3rd harmonics during t= [100,400]
Figure 7 -Voltage Sag with third harmonics
Figure 8-Third Harmonics Voltage sag WVD
Swell in pure sine wave
Voltage Swell: When a nominal voltage signal(t) increases to a range about 110 to 180 percent, according to IEEE 1159 standard ,the voltage swell was occurred. One of most causes for swell is Single line to ground fault, SLG fault. Fig. 14 shows the TFR of the WVD for voltage swell at sample time, Ts=[1380,3000] and the amplitude a=1.25
As in Fig 10 note that the low intensity with curve at t = [100,150] and high intensity at t= [350,400] indicates 25% voltage swell at the duration of t= [100,400].
Figure 9-Voltage Swell 25%
Figure 10-Voltage Swell 25% WVD
Voltage Swell with 3rd harmonics
Voltage swell with 3rd harmonics: Occurs when the causes for swell related with non linear loads.
As in Fig.12 high intensity with curve at t= [100,150] to low intensity with curve at t = [350,400] indicates voltage swell at 25%.The intensity of the line also can be seen during f=100Hz and f=150Hz shows that 3rd harmonics component is exist. The line for harmonics occur during t=[100,400].
Figure 11-Voltage Swell with 3rd harmonics
Figure 12-Voltage swell with 3rd harmonics WVD
Outage in pure sine wave
The distortion of swell occur at sample time, Ts=[1000,4000] and the amplitude a=0
As in Fig .12 we can see clearly absence of line intensity during t= [100,400].
Figure 13-Voltage Outage
Figure 14- Voltage Outage WVD
Table Type Styles
Analysis of signal disturbances
RMS Voltage, Vrms(V)
V=230, =1, Fl=50Hz, Ts=0.2ms
3rd Harmonics component
Fl=50Hz , F2=150Hz
Voltage sag with 3rd Harmonics component
Fl=50Hz , F2=150Hz
Voltage swell with 3rd Harmonics component
Fl=50Hz , F2=150Hz
In the conclusion, this paper introduces the use of Wigner Ville Distribution as a powerful and enhanced analysis tool. The property of multi resolution shows the ability of the technique to extract information from the analyzed distorted signal. The main advantage of the proposed WVD is that the ability to join the signal in both of frequency and time, so that the long and tedious signal can be compressed in certain pattern which identifies its type of disturbances for instantaneous detection. Using PSD the smoothened THD value can be calculated which represents less negative energy representation in spectrum.
I would like to thank my supervisor, Madam Zaiton Sharif and UiTM's Faculty of Electrical Engineering Staffs for the resources and guidance towards finishing this paper.