## 3 Improvement of DSOGI-PLL
### Drawbacks of DSOGI-PLL Under Complex Faults  
The asymmetrical grid voltage vabc can be decomposed into its positive-sequence, negative-sequence, and zero-sequence components. The formulae for the positive-sequence and negative-sequence components are given below [21]. \[ \begin{split} v^{+}_{abc}&=\left[\begin{array}{cc}v^{+} _{a}&v^{+}_{b}&v^{+}_{c}\end{array}\right]^{T}=C^{+}v_{abc}\\ v^{-}_{abc}&=\left[\begin{array}{cc}v^{-}_{a}&v^{-}_{b}&v^{-}_{c}\end{array} \right]^{T}=C^{-}v_{abc}\end{split}  \tag{8}\]  
where:  
\[C^{+}=\frac{1}{3} \begin{bmatrix}1&\alpha&\alpha^{2}\\ \alpha^{2}&1&\alpha\\ \alpha&\alpha^{2}&1\end{bmatrix} ;\;C^{-}=\frac{1}{3} \begin{bmatrix}1&\alpha^{2 }&\alpha\\ \alpha&1&\alpha^{2}\\ \alpha^{2}&\alpha&1\end{bmatrix} ;\;\alpha=e^{2\pi/3}\]  
After the grid voltage undergoes the Clark transformation, the positive and negative sequence components are expressed as follows after passing through the DSOGI-Filter:  
\[ \begin{split} v^{+}_{a\beta}&=\left[\begin{array}{cc}v^{+}_{ \alpha}&v^{+}_{\beta}\end{array}\right]^{T}=\frac{1}{2}\begin{bmatrix}1&-q\\ q&1\end{bmatrix}v^{\beta}_{\alpha}\\ v^{-}_{a\beta}&=\left[\begin{array}{cc}v^{-}_{\alpha}&v^{-}_{\beta}\end{array} \right]^{T}=\frac{1}{2}\begin{bmatrix}1&q\\ -q&1\end{bmatrix}v_{\alpha}\end{split}  \tag{9}\]  
Based on the previous analysis, it is clear that to separate the positive and negative sequence components of the grid voltage, one must initially obtain the grid voltage components in the two-phase stationary coordinate system. However, when the grid voltage contains significant harmonics, the filtering capability of the SOGI alone is insufficient. A comprehensive analysis is presented below. The three-phase voltage during complex asymmetric faults with harmonics can be expressed as:  
\[ \begin{split} v_{i}&=V^{+}\cos(\omega_{0}t+\phi^{+}+k_{i} \frac{2\pi}{3})+V^{-}cos(-\omega_{0}t+\phi^{-}+k_{i}\frac{2\pi}{3})+\\ &\sum_{m=1}^{\infty}V^{n}cos(n\omega_{0}t+\phi^{n}+k_{i}\frac{2\pi}{3}) \end{split}  \tag{10}\]  
In the equation: \(k_{i}=0\), \(+1\), \(-1\); \(V^{+}\), \(V^{-}\), and \(V^{n}\) represent the amplitudes of the positive-sequence, negative-sequence, and nth harmonic components of the grid voltage, respectively; \(\phi^{+}\), \(\phi^{-}\), and \(\phi_{n}\) represent the initial phase angles of the positive-sequence, negative-sequence, and nth harmonic components of the grid voltage, respectively; and \(\omega_{0}\) is the fundamental frequency of the grid voltage. Equation (10), transformed into the two-phase stationary coordinate system, is expressed as:  
\[v_{a\beta}= \begin{bmatrix}v_{\alpha}\\ v_{\beta}\end{bmatrix} =v^{+}_{a\beta}+v^{-}_{a\beta}+\sum_{m=1}^{\infty}v^{n} _{a\beta} \tag{11}\]  
where \(v^{+}_{a\beta}= \begin{bmatrix}v^{+}_{\alpha}&v^{+}_{\alpha}\end{bmatrix} ^{T}=V ^{+} \begin{bmatrix}\cos(\omega_{0}t+\phi^{+})\\ \sin(\omega_{0}t+\phi^{+})\end{bmatrix} v^{-}_{a\beta}= \begin{bmatrix}v^{-}_{ \alpha}&v^{-}_{\alpha}\end{bmatrix} ^{T}=V^{-} \begin{bmatrix}\cos(-\omega_{0}t+ \phi^{-})\\ \sin(-\omega_{0}t+\phi^{-})\end{bmatrix} \); \(v^{n}_{a\beta}= \begin{bmatrix}v^{n}_{\alpha}&v^{n}_{\alpha}\end{bmatrix} ^{T}=V ^{n} \begin{bmatrix}\cos(n\omega_{0}t+\phi^{n})\\ \sin(n\omega_{0}t+\phi^{n})\end{bmatrix} \). Taking this as the input signal of the SOCI, and combining Equations (9) and (11), the output signal of the SOCI can be obtained as:  
\[\left\{ \begin{array}{l}v_{\alpha\alpha}=v_{\alpha}^{+}+v_{\alpha}^{-}+v_{ \alpha}^{\prime*}\\ qv_{\alpha\alpha}=v_{\beta}^{+}-v_{\beta}^{-}+\frac{1}{n}v_{\beta}^{\prime*}\\ v_{\beta\alpha}=v_{\beta}^{+}+v_{\beta}^{-}-v_{\beta}^{\prime*}\\ qv_{\beta\alpha}=-v_{\alpha}^{+}+v_{\alpha}^{-}-\frac{1}{n}v_{\alpha}^{\prime*} \end{array} \right. \tag{12}\]  
In the equation: \(v_{\alpha\alpha}{}^{n^{*}}\) and \(v_{\beta\alpha}{}^{n^{*}}\) represent the output values of the nth harmonic after passing through the SOCI.