# Fault Detection and VSC-HVDC Network Dynamics Analysis for the Faults in Its Host AC Networks
## 2 Proposed Methodology
### Synchro-Squeezed Transform (SST)  
This is a mathematical tool used to analyze oscillatory signals. It is a compressed and squeezed sensing algorithm that works to obtain greater time-frequency plane information using the following features:  
* The dogma of signal processing techniques maintains that a signal must be sampled at a rate at least twice its highest frequency to represent and reconstruct the signal without any error.
* But in general practice, the data are soon compressed after sensing to reduce the computational burden and signal representation complexity (bits). This is a waste of valuable sensing resources.
* Over the past few years, a new theory of compressive sensing has begun to emerge in this field. In this theory, the signal is sampled and simultaneously compressed at a greatly reduced rate.
* The SST can accurately map time domain signals into their time-frequency representations.
* Compressed sensing.
* Compressive sampling.
* It has a well-grounded mathematical foundation that facilitates theoretical analysis.  
A clear time-frequency representation (TFR) makes signal characteristic extraction more accurate in comparison with the information provided by the algorithms that give time-scale plane representation [8; 9; 10]. By applying the SST on an oscillatory signal \(C_{i}\), the signal is decomposed as follows:  
\[C_{i}(t)=\sum_{n=1}^{k}A_{k}(t)cos(\theta_{k}(t))+\eta(t) \tag{2}\]where \(\eta\) (\(t\)) is the additive noise, The instantaneous amplitude and phase information of the \(k^{th}\) component are denoted as \(A_{k}\left(t\right)\) and \(\theta_{k}(t)\), respectively. The instantaneous frequency \(f_{k}(t)\) can be estimated from the instantaneous phase as follows:  
\[f_{k}(t)=\frac{1}{2\pi}\frac{d}{dt}\theta_{k}(t) \tag{3}\]  
The next step is to construct the time-scale map using a continuous wavelet (CWT) transform of the signal. Morlet was chosen as the mother wavelet:  
\[W_{s}(\alpha,\beta)=\frac{1}{\sqrt{p}}\int A_{k}(t)\xi^{*}\left(\frac{t-\beta }{\alpha}\right)dt \tag{4}\]  
where, \(\xi^{*}\) is the complex conjugate of the Morlet wavelet, \(\alpha\) is the scaling factor, \(\beta\) is the time shift, and \(W_{s}\) (\(\alpha\), \(\beta\)) is the time-frequency graph. The derivative of Equation (4) at any point (\(\alpha\), \(\beta\)) with respect to \(\beta\) for all \(w_{s}\) (\(\alpha\), \(\beta\)) \(\
eq\) 0 can be written as \(W_{s}\) (\(\alpha\), \(\beta\)):  
\[w_{s}(\alpha,\beta)=\frac{-j}{2\pi W_{s}(\alpha,\beta)}\frac{\partial W_{s}( \alpha,\beta)}{\partial\beta} \tag{5}\]  
The above obtained time-scale information is mapped into the time-frequency plane using the concept of synchro squeezing:  
\[S_{s}(\omega_{l},\beta)=\frac{1}{\Delta\omega}\sum_{\alpha_{k}:\omega(x_{k}, \beta)-\omega_{l}|\leq\Delta\omega/2}W_{s}(\alpha_{k},\beta)\alpha^{-3/2} \Delta\alpha_{k} \tag{6}\]  
For any value of \(\alpha_{k}\), \(\alpha\) and \(\beta\) are discrete in nature for the scaling step \(\Delta\alpha_{k}=\alpha_{k+1}-\alpha_{k}\). The transform is obtained at the center frequency \(\omega_{l}\) over the range of frequencies \([\omega_{l}-\Delta\omega/2,\omega_{l},\omega_{l}-\Delta\omega/2]\), with \(\omega_{l}\) as the center frequency.  
The mapping from the time-scale plane to the time-frequency plane \((\beta,\alpha)\rightarrow(\beta,\omega_{\text{ins}}(\beta,\alpha))\), mentioned in Equation (6), shows the synchro-squeezing process.