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## RMS and peak voltage techniques

Several classical techniques allow to determine the voltage amplitude. Herein, we interestby two techniques for voltage amplitude characterization, which are the root mean square (rms) and peak voltage techniques. The rms technique allows to obtain the amplitude by using the rms value of the voltage over a multiple of one half-cycle of the power system frequency. The rms voltage can be expressed in term of the instantaneous voltage, V (t), over a given period of time T.

### Fourier transform and its extensions

The Fourier transform (FT) is the most widely used computation algorithm for stationary signals by extracting spectrum at specific frequencies. The signal to be analysed is modeled by sum of sinusoids signals with different frequencies. The Fourier transform allows to estimate the frequency components without any information of the time at which the frequency components appear. Several papers and books are devoted to the Fourier transforms [57, 58].

In sampled data systems, the analysis of signal is performed by the discrete time Fourier transform (DTFT). The DTFT is usually implemented using the fast Fourier transform (FFT) over a short signal length. The FFT is computed at discrete steps in the frequency domain with an input signal that is sampled at discrete instants in the time domain. Let assume that the signal length contains the N measurements or samples from a sampled voltage or current signal in an equal interval tn.

#### Discussion: parametric methods versus non-parametric methods

The non-parametric methods are well used for the frequency and phasor estimation. These methods require a long data window for achieving a good resolution, however, this window leads to the leakage effect. In power quality applications, the recorded signal is finite that presents the major limitation of the non-parametric methods [13].

On the other hand, the parametric methods or based-model methods require a priori information about how the process is generated. In the cases where the used model is a close approximation of the real model, a highest estimation performances can be obtained over a significantly less data as compared to those in non-parametric methods. However, the limitation of these methods only depends on the matching between the true and theoretical signal model.

As highlighted previously, several power quality events are non-stationary signals. In such conditions, Fourier transform-based techniques are not suitable to analyse these non-stationary signals. Therefore, PQ estimation is usually performed by using nonstationary methods. These methods can be classified into two categories: non-parametric methods and parametric methods. Non-parametric methods include time-frequency or time-scale representation [61, 63–67, 70–84]. Parametric methods include non-stationary high-resolution techniques and non-stationary MLE techniques [2, 85, 86]. These techniques should use a sliding window in order to handle the non-stationarity of the signal. The signal is considered to be stationary over each window, which is a short duration of time. The estimation of the time-varying parameters can then be performed over each block by shifting the data window (block). Therefore, the estimated parameters are time-varying since they depend on the window number.

Several advanced techniques have been proposed to analyze the PQ disturbances under non-stationary conditions. Hilbert-Huang transform (HHT) is one of recent advances in signal processing techniques. HHT is based on Hilbert transform (HT) and Empirical Mode Decomposition (EMD) techniques [70]. Empirical Mode Decomposition is a time–frequency analysis method that decomposes the signal into number of intrinsic mode functions (IMFs) [71]. Then, the Hilbert transform analysis these IMFs to estimate the instantaneous frequency (IF), amplitude (IA), and phase (IP). HT main advantage is that it requires a single phase signal for extracting the instantaneous frequency and amplitude. The Hilbert transform can detect the presence of the fault, however it suffers from border effects that may lead to incorrect information of the event. Moreover, it can be difficult to interpret under fast signal modulation, i.e. when the Bedrosian condition is not satisfied. On the other hand, the EMD presents several drawbacks such as mode mixing that may lead to a wrong intrinsic mode functions (IMF) decomposition and border effect. A Hilbert-Huang transform based algorithm for PQ disturbance classification was proposed in [73]. It has a special focus on the the PQ events that are sag, swell, harmonics, spikes, notches, flicker, and transients. The probabilistic neural network (PNN) technique is used to classify the corresponding event. A PQ assessment approach expands a distorted signal into its intrinsic mode oscillations was proposed in [72]. A comparison is performed between the proposed technique and S-transform. The obtained results showed a good classification accuracy for detection of PQ disturbance, such as voltage spike and notch. Under non-stationary conditions, it is become imperative to separate between different components of frequency for identify the cause of PQ disturbances. For such purpose, a combination between EMD and Hilbert transform to detect the cause of voltage sag was proposed in [74]. Then, PNN classifier is used for identifying the voltage sag cause and type.

**Downhill Simplex Method**

For optimizing the one-dimensional function is to use the Downhill simplex optimization method. This method is due to Nelder and Mead [157] and it is commonly applied for finding the minimum or maximum of a one- or multi-dimensional cost-function. Downhill simplex is used to nonlinear optimization problems that do not require any derivative information. It is heuristic search method, which does not guaranteed to be optimal [158, 159]. This presented method could search a one-dimensional minimizer that finds a minimum for a problem specified by min w J (w) where, w1 < w < w2.

**Frequency error. Effect of the number of samples N**

Figure 2.5 presents the Average Frequency Error (FEmean) versus N. This figure clearly shows that the LS-NR estimator outperforms the other techniques. The FEmean obtained with the LS-fmin estimator is slightly above the FEmean obtained with the LS-NR method whatever the signal length is. For particular values of N, we observe that the LS-NR and DTFT-NR techniques have similar performance. Indeed, as demonstrated in Section II, the exact cost-function (2.16) is equal to the approximate cost-function (2.23) when N is equal to a multiple of the half-period or when it is large. This figure also shows that the accuracy requirement of the IEEE Standard [52, 56], which is set to 5 mHz, seems to be respected on average for the LS-NR and LS-fmin estimators whatever the signal length is.

**Frequency error. Effect of signal to noise ratio (SNR)**

Figure 2.6 illustrates the influence of the SNR on the estimator performances when N = 240 samples. We observe that the LS-NR estimator seems to outperform the other techniques whatever the SNR is. We also note that the FEmean of the proposed estimators decreases when the SNR increases. The LS-NR and LS-fmin estimators lead to a mean FE lower than 5 mHz when the SNR is greater than 70 dB. While, the DTFT estimator leads to a frequency error more than 5 mHz whatever the SNR is. It can be also noted that for a SNR lower than 70 dB, the mean FE is larger than the IEEE standard limit. For the low-SNR scenario, a larger number of samples N is required to meet the requirement of the IEEE standard.

**Table of contents :**

Abstract

Acknowledgements

List of Publications

List of figures

List of Tables

Glossary

General Introduction

**1 Power Quality Disturbances Characterization: State of the Art Review **

1.1 Introduction

1.2 Power Quality and Signal Processing

1.2.1 Definition

1.2.2 Voltage and Current Quality

1.2.3 Disturbances

1.2.3.1 Voltage variations

1.2.3.1.1 Frequency variations

1.2.3.1.2 Voltage amplitude variations

1.2.3.1.3 Waveform distortion: harmonic, inter-harmonic, and nonperiodic distortion

1.2.3.2 Events

1.2.3.2.1 Interruption

1.2.3.2.2 Voltage sag and swell

1.2.4 Monitoring

1.2.5 Standards

1.3 Phasor Measurement Units (PMUs)

1.3.1 Definition

1.3.2 Standards

1.3.3 Estimation Performances Evaluation

1.3.3.1 Frequency and ROCOF measurement evaluation

1.3.3.2 Total vector error (TVE) evaluation

1.4 Spectral Estimation Techniques for Power Quality Monitoring

1.4.1 Non-Parametric Methods

1.4.1.1 Zero-Crossing transform

1.4.1.2 RMS and peak voltage techniques

1.4.1.3 Fourier transform and its extensions

1.4.2 Parametric Methods

1.4.2.1 Discrete spectra

1.4.2.2 Continuous spectra

1.4.3 Discussion: parametric methods versus non-parametric methods

1.5 Power Quality Disturbances Classification Techniques

1.5.1 Classical Techniques

1.5.1.1 ABC classifier

1.5.1.2 Symmetrical component classifier

1.5.2 Pattern Recognition Techniques

1.5.2.1 Artificial neural network

1.5.2.2 Support vector machine

1.5.2.3 Fuzzy expert systems

1.5.2.4 Genetic algorithm

1.6 Conclusion

**2 Phasor and Frequency Estimations **

2.1 Introduction

2.2 Parameters Estimations

2.2.1 Study hypotheses

2.2.2 Signal model and phasor expression

2.2.3 Least squares estimations

2.2.3.1 Angular frequency estimation

2.2.3.2 Phasor estimation

2.2.4 Extension to non-stationary case

2.3 Cost-Function Optimization Method

2.3.1 Downhill Simplex Method

2.3.2 Newton-Raphson Method

2.3.2.1 Newton-Raphson iteration

2.3.2.2 Derivatives of the exact cost-function

2.3.2.3 Derivatives of the approximate cost-function

2.4 Simulation and Experimental Results

2.4.1 Monte Carlo Simulation Results

2.4.1.1 Frequency error. Effect of the number of samples N .

2.4.1.2 Frequency error. Effect of signal to noise ratio (SNR)

2.4.1.3 Total vector error. Effect of off-nominal frequency deviation

2.4.1.4 Total vector error with frequency estimation. Effect of off-nominal frequency deviation

2.4.1.5 Total vector error with frequency estimation. Effect of the number of samples N

2.4.1.6 Total vector error with frequency estimation. Effect of signal to noise ratio (SNR)

2.4.1.7 Total vector error with frequency estimation. Effect of harmonics and inter-harmonics

2.4.1.8 Computation complexity

2.4.1.9 Comparative study

2.4.2 Experimental Results

2.5 Conclusion

**3 Power Quality Disturbances Classification based on Model Order Selection **

3.1 Introduction

3.2 Signal and Phasor Models

3.2.1 Three-Phase Signals Model

3.2.2 Phasor Model and 4-Classes

3.2.3 General Vectorized Signal Model

3.3 Proposed Classifier based on Information Theoretic Criteria

3.3.1 Decision Tree

3.3.2 Proposed Classifier Expression

3.3.2.1 Expression of the log-likelihood function

3.3.2.2 Expression of γk

3.4 Simulation and Experimental Results

3.4.1 Monte Carlo Simulation Results

3.4.1.1 Probability of correct classification versus number of samples

3.4.1.2 Probability of correct classification versus noise variance .

3.4.1.3 Robustness of the proposed classifier

3.4.1.4 Computation complexity

3.4.1.5 Comparative study

3.4.1.5.1 Symmetrical component algorithm (SCA) .

3.4.1.5.2 Three-phases three-angles algorithm (TP-TAA) .

3.4.1.5.3 Numerical examples

3.4.2 Experimental Tests

3.4.3 Discussion

3.5 Conclusion

Conclusions and Perspectives

Appendices

Appendix A

Appendix B

Appendix C

**References**