This thesis focuses on measurement and non-parametric estimation of periodic signals
in low voltage distribution systems. In the introduction, detailed overview of estimation
methods for frequency, amplitude, and phase of periodic signal are presented.
Signal analysis can be performed either in time domain or in frequency domain. Parameters
estimation in time domain is based on three- or four-parametric sine fitting
algorithm. Parametric approach is computational extensive, but it has a very good statistical
efficiency. Non-parametric spectral analysis uses Fourier transform. Fast Fourier
transform is probably one of the most useful tool in the signal processing. In frequency
domain analysis there are two main methods: energy based and interpolated. Energy
based method computes energy within selected frequency band. Usually DFT coefficients
from windows main lobe are taken under investigation. Non-parametric approach
requires less computational complexity than parametric estimation algorithms.
Sampling of signals in time domain is a finite process. When the signal is sampled in
the non-coherent condition, e.g. the sampling system is not fully synchronized with
the signal, spectral leakage occurs in the frequency domain. To reduce spectral leakage
along complete frequency axis, selection of the proper window function is needed.
In this thesis, families of cosine sum windows are presented. Two main classes
are introduced: maximum side lobe decay windows (known also as Rife-Vincent class
1 windows) and minimum side lobe level windows (known also as Rife-Vincent class
2 windows). Non-parametric signal analysis can only be performed analytically when
dealing with maximum side lobe decay windows. In case of other windows, parameter
estimation can only be performed by means of numerical approach.
Classical interpolation algorithms uses only two the highest spectral lines from frequency
domain. Using more than two amplitude spectrum coefficients from DFT, spectral
leakage on parameter estimation can be additionally reduced. When working with
multi-point interpolated DFT algorithms there is always present weighted summation
of DFT coefficients. Multi-point interpolation in this thesis uses odd numbers of locally
highest spectral DFT coefficients. Systematic errors of algorithms are given for
different orders of cosine sum windows and different numbers of DFT coefficients in
interpolation algorithms. Complete noise analysis is also performed.
In many test and measurement applications processing of two signals with the same
frequency is required. In multi-signal analysis, simultaneously sampling is required.
Parameter estimation of parametric approach is based on extension of classical threeor
four-parametric sine-fitting algorithms to six- or seven-parameter matrix form. Nonparametric estimation methods can estimate all parameter for both signals, but usually
only a few parameters are needed to estimate desired electrical quantity. Nonparametric
estimation of amplitude quotient and phase difference of two signals is presented.
Results are given for selected test frequency. Complete systematic errors and
noise analysis are performed using two main cosine sum windows and multi-point
DFT interpolated algorithms.Measurements on low voltage electrical installations can be performed in conditions of low signal-to-noise ratio or in presence of higher harmonic distortion. Measurements of earthing resistances and earth fault loop impedances are usually performed in such noisy circumstances. Case study shows the measurement of earth fault loop impedance. Amplitude ratio and phase difference is estimated at the very low signal-to-noise ratio of the voltage signal. For selected test frequency amplitude quotient and phase difference between voltage and current is calculated.
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