## 4 Discussion
### Mathematical Tools--A Further Research Avenue  
In the field of PQ, the application of many AI algorithms (e.g., Fourier transform, Kalman filter, wavelet transform, S-transform, Hilbert-Huang transform, Gabor transform, etc.) were reviewed by [72], and this still remains an open field of research. However, since [72] focuses on PQ disturbance classification and feature extraction, it does not cover two important aspects of analysis of voltage sag behavior: (1) strong interconnection between PQ events (the idea has been raised for the first time in [5]); (2) the method of the voltage sag matrix. Some cases of voltage unbalance, caused by a voltage sag, were investigated in Section 3; thus, let us now have a closer look into transients caused by voltage sags, through the prism of wavelet transform. It is a recent but promising tool in many fields, such as audio or video signal processing for noise suppression, also in biomedical signal processing (e.g., electrocardiogram analysis). According to the general  
Figure 77: Voltage sag mitigation efficiency comparison of capacitor bank and flywheel systems on the 10 kV side of a 110/10 kV substation, when the PQ mitigation device is installed on the same 10 kV line at the 10/0.4 kV substation. The voltage sag is caused by a three-phase short-circuit which occurs at the selected farther node of the 110 kV grid. and well-known definition, the relationship of the mother wavelet \(\psi\) to the daughter wavelet \(\psi_{a,\tau}\) is the following:  
\[\psi_{a,\tau}(t)=\frac{1}{\sqrt{a}}\psi\bigg{(}\frac{t-\tau}{a}\bigg{)}, \tag{36}\]  
where: \(a\)--scaling factor (positive real); \(\tau\)--shift factor. Wavelet transform is based on wavelet function convolution with input signal \(x(t)\) as follows:  
\[X(a,b)=\frac{1}{\sqrt{a}}\int\limits_{-\infty}^{\infty}x(t)\psi^{*}\bigg{(} \frac{t-\tau}{a}\bigg{)}dt, \tag{37}\]  
where: \(\psi^{*}\)--the complex conjugate of the mother wavelet. Let us introduce an investigation example with one of the most famous Haar wavelet family. Haar wavelet's mother wavelet function is as follows:  
\[\psi(t)=\left\{ \begin{aligned} & 1,& 0\leq t<\frac{1}{2},\\ &-1,&\frac{1}{2}\leq t<1.\end{aligned} \right. \tag{38}\]  
The transient of phase C (cyan curve) of Figure 7b (Scenario No. 2, three-phase fault, phase-to-ground voltage of M1) is selected for the investigation. The region of direct interest is 0.04-0.10 s because the transient begins at 0.06 s: the array consists of 60,000 samples (sampling time is 10\({}^{-6}\) s), and, respectively, the element number, corresponding to the moment the event begins, is 20,000. At the first level of discrete wavelet transform, the transient signal is simultaneously passed through low-pass and high-pass filters, and both arrays are decimated by 2. Output from the high-pass filter gives the detail coefficients, while output from the low-pass filter--approximation coefficients. At the next level, the initial signal is replaced by the approximation signal, and the procedure is repeated. The process is usually represented with a wavelet decomposition tree. The details coefficients by decomposition level are given in Figure 78. Since the 50 Hz component has not been removed, levels 8-12 should be analyzed critically, because the involvement of a 50 Hz signal is visible even before the transient, i.e., during 0.04-0.06 s. At the initial period of the transient, some high coefficients can be observed at the lower levels. The differences between levels (determined by the scaling factors) are clearly seen in Figure 79, where the signal details are given in superimposed mode. The next important avenue for further research is the development (extension) of the voltage sag matrix method. In this paper, the analysis of voltage sag matrixes is not very complex due to the following reasons: (1) all test schemes are incomparably smaller than real systems; (2) mainly radial topology has been investigated with the specially adapted numeration for the simplicity of analysis.