## 3 Design Concept and Operation of Enhanced-ADALINE Algorithm
### Working Principle of HSF Synchronizer Module  
Figure 3 provides a block diagram showing the control structure of an HSF synchronizer module. As the name indicates, the synchronizer module is developed by expanding the function of the existing  
Figure 2: Control structure of a generalized adaptive linear neuron (ADALINE) module [11; 19].  
HSF concept [18] so that it can track the operating phase of the source voltage \(v_{S}\) and then transform it into a synchronization signal \(sin(\omega t+\theta)\).  
The generation of synchronization signal is performed in two operating domains (three-phase \(abc\) and two-phase \(\alpha\beta\) domains) involving the use of Clarke's transformation. In two-phase \(\alpha\beta\) domain, fundamental component of source voltage \(v_{S}\) is extracted, and then in three-phase \(abc\) domain, the extracted fundamental component is transformed into synchronization signal. First, by using Clarke transform matrix, the measured source voltage \(v_{S}\) in three-phase domain is transformed into two-phase \(\alpha\beta\) domain according to the following expression.  
\[\left[\begin{array}{c}v_{\alpha}\\ v_{\beta}\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{cc}1&- \frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}[ ]{c}v_{S\alpha}\\ v_{Sb}\\ v_{Sc}\end{array}\right]. \tag{4}\]  
In two-phase \(\alpha\beta\) domain, due to harmonic distortion, the following relation holds for the transformed source voltage signal \(v_{\alpha\beta}\):  
\[\left[\begin{array}{c}v_{\alpha}\\ v_{\beta}\end{array}\right]=\left[\begin{array}{c}v_{\alpha,\;fund}+v_{ \alpha,\;har}\\ v_{\beta,\;fund}+v_{\beta,\;har}\end{array}\right], \tag{5}\]  
where \(v_{\alpha,\;fund}\) is the fundamental and \(v_{\alpha,\;har}\) is harmonic components in \(\alpha\) domain, and meanwhile \(v_{\beta,\;fund}\) and \(v_{\beta,\;har}\) represent the same relationship in \(\beta\) domain.  
At this stage, an HSF [18] is applied to remove the harmonic components (\(v_{\alpha,\;har}\) and \(v_{\beta,\;har}\)) so as to deliver the fundamental component (\(v_{\alpha,\;fund}\) and \(v_{\beta,\;fund}\)) to the subsequent processing stage. The overall operation of an HSF can be explained using the following expressions:  
\[v_{\alpha,\;fund}=\frac{K}{s}\big{(}v_{\alpha}-v_{\alpha,\;fund}\big{)}+\frac {2\pi f_{c}}{s}\big{(}-v_{\beta,\;fund}\big{)}, \tag{6}\]  
\[v_{\beta,\;fund}=\frac{K}{s}\big{(}v_{\beta}-v_{\beta,\;fund}\big{)}+\frac{2 \pi f_{c}}{s}\big{(}v_{\alpha,\;fund}\big{)}, \tag{7}\]  
where \(K\) is a constant gain parameter and \(f_{c}\) is the cutoff frequency. Analyses regarded to the selectivity of HSF have been reported in the literature [18; 21], where it is revealed that the selectivity of HSF improves with smaller value of \(K\). In addition, \(K=20\) and \(f_{c}=50\) Hz are also found to be the most suitable parameter settings for HSF [18; 22]. Hence, in this work, similar settings for HSF are applied.  
Subsequently, from the fundamental component obtained, inverse Clarke transform matrix as in Equation (8) is applied to revert the fundamental component in two-phase \(\alpha\beta\) domain (\(v_{\alpha,\;fund}\) and  
Figure 3: Control structure of a highly selective filter (HSF) synchronizer module.  
\(v_{\beta,\ fund}\)) back to its equivalent representation in three-phase \(abc\) domain \(v_{Sabc,\ fund}\). Finally, by using Equation (9), the resulting \(v_{Sabc,\ fund}\) is converted into its unity form which serves as the synchronization signal \(sin(\omega t\ +\ \theta)\).  
\[\left[\begin{array}{c}v_{Sa,fund}\\ v_{Sb,fund}\\ v_{Sc,fund}\end{array}\right]=\sqrt{\frac{2}{3}}\left[\begin{array}{cc}1&0 \\ -\frac{1}{2}&\frac{\sqrt{3}}{2}\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}v_{ \alpha,\ fund}\\ v_{\beta,\ fund}\end{array}\right], \tag{8}\]  
\[sin(\omega t\ +\ \theta)=\ \frac{v_{Sabc,\ fund}}{\sqrt{v_{\alpha,\ fund} ^{2}+v_{\beta,\ fund}}^{2}}. \tag{9}\]