## 2 Review of Synchrosqueezing Wavelet Transform  
As with the demodulation of vibration signals resulted from rolling element bearing defects in mechanical engineering [29; 30], TF analysis also can be applied to detect the frequency and magnitude of non-stationary signals in power systems [31]. Regarding applications in voltage or current waveform measurement, TF analysis (TFA) has been exploited for PQ assessment in recent years. In [23], the authors proposed the synchrosqueezing wavelet transform method by resetting the time-frequency spectra of SSWT. The time-frequency ridges are substantially improved. SSWT is an empirical mode decomposition (EMD)-like method [32]. It is the TF signal analysis algorithm designed to decompose the signal \(s(t)\) of Equation (1) into several IMF. \[s(t)=\sum_{k=1}^{K}s_{k}(t)+e(t) \tag{1}\]where the k-\(th\) component \(s_{k}(t)=A_{k}(t)\cos\theta_{k}(t)\) is a Fourier-like non-stationary mode with time-varying magnitude, \(A_{k}(t)\), and phase, \(\theta_{k}(t)\). \(e(t)\) represents the noise or measurement error; the instantaneous frequency is \(f_{k}(t)=\theta_{k}^{\prime}(t)/2\pi\). The goal is to recover the magnitude \(A_{k}(t)\) at the instantaneous frequency \(f_{k}(t)\) for each component \(k\). The SSWT starts from the complex CWT of the signal \(s(t)\) and is defined by Equation (2). \[W_{s}(a,b)=\left\langle s(t),\psi_{a,b}(t)\right\rangle=\frac{1}{\sqrt{a}}\int s (t)\psi^{*}(\frac{t-b}{a})dt \tag{2}\]  
where \(a\) and \(b\) are scaling and localization parameters of the mother wavelet, and \(\psi\) is a properly selected complex mother wavelet in which the lognormal wavelet is adopted. Take a wavelet \(\psi\) that is concentrated on the positive frequency axis; we have the Fourier transform of the wavelet at \(\varsigma\), \(\hat{\psi}(\varsigma)\approx 0\) for \(\varsigma<0\). By [PERSON]'s theorem, we can rewrite \(W_{s}(a,b)\), the CWT of \(s(t)\) with respect to \(\psi\), as  
\[ \begin{array}{ll}W_{s}(a,b)&=\ \int\hat{s}(\varsigma)\frac{\sqrt{a}}{2\pi} \hat{\psi}^{*}(a\varsigma)e^{ib\varsigma}d\varsigma\\ &=\ \frac{A}{4\pi}\int[\delta(\varsigma-\omega)+\delta(\varsigma+\omega)]\sqrt{a }\,\hat{\psi}^{*}e^{ib\varsigma}d\varsigma\\ &=\ \frac{A}{4\pi}\sqrt{a}\hat{\psi}^{*}(a\varsigma)e^{ib\omega}\end{array}  \tag{3}\]  
\(W_{s}(a,b)\) will be concentrated around \(a=\omega_{0}/\omega\). However, \(W_{s}(a,b)\) will be spread out over a region around the horizontal line \(a=\omega_{0}/\omega\) on the time-scale plane. Assume that \(W_{a}(a,b)\
eq 0\) for any \((a,b)\), the instantaneous frequency \(\omega_{s}(a,b)\) can be obtained from Equation (4). \[\omega_{s}(a,b)=-i(2\pi W_{s}(a,b))^{-1}\frac{\partial}{\partial b}W_{s}(a,b) \tag{4}\]  
Then, the information is transferred from the time-scale plane to the time-frequency plane by relocating points at \((b,a)\) to \((b,\omega_{s}(a,b))\). The resulting SSWT representation \(T_{s}(\omega_{l},b)\) is given by  
\[T_{s}(\omega_{l},b)=(\Delta\omega)^{-1}\sum_{a_{k}:|\omega(a_{k},b)-\omega_{l }|\leq\Delta\omega/2}W_{s}(a_{k},b)a_{k}{}^{-3/2}\Delta a_{k} \tag{5}\]  
where (5) is calculated only at discrete central frequencies, \(\omega_{l}\), separated by a frequency step, \(\Delta\omega\), i.e., (5) is performed at the center \(\omega_{l}\) of the interval \([(\omega_{l}-\Delta\omega/2),(\omega_{l}+\Delta\omega/2)]\). In a similar manner, since the scale \(a\) and time \(b\) are discrete values, a scaling step \(\Delta a_{k}=a_{k}-a_{k-1}\), is presented, separating discrete scales \(a_{k}\) for which the wavelet decomposition of (3) is calculated [23]. \(T_{s}(\omega_{l},b)\) presents the time-frequency rigid (TFR) of the signal \(s(t)\) and is synchrosqueezed along the frequency axis. As with CWT, the SSWT is reversible. It can be shown that the signal can be reconstructed after the synchrosqueezing and is approximated by Equation (6) in the piecewise constant approximation corresponding to the binning in \(a\)[23].