# Zero-Voltage Ride-Through Capability of Single-Phase Grid-Connected Photovoltaic Systems
## 3 Overview of PLL Synchronization Techniques
### Second-Order Generalized Integrator-Based PLL (SOGI-PLL)  
The second-order generalized integrator-based PLL (SOGI-PLL) is an effective approach to implement the OSG-based PD to generate the orthogonal signals, and then the conventional synchronization can be adopted [29]. A general structure of a SOGI-PLL is shown in Figure 5. The quadrature signal generation (QSG) contains a pair of 90\({}^{\circ}\)-shifted output signals, \(v_{\alpha}\) and \(v_{\beta}\), which are generated by a second-order adaptive filter, also known as a sinusoidal integrator (more specific, a second-order generalized integrator (SOGI)). The OSG-based PD unit is followed by Park transformation that gives the \(dq\)-voltages, \(v_{d}\) and \(v_{q}\). The SOGI-based PD has the advantages of frequency adaptability and harmonic rejection [26; 29]. The characteristic transfer function of the adaptive filtering based on the SOGI is given as  
\[G_{\text{SOGI}-\text{AF}}(\text{s})=\frac{\omega^{\prime}s}{s^{2}+\omega^{ \prime 2}} \tag{7}\]  
in which \(\omega^{\prime}\) is the estimated grid angular frequency. Further analyzing the SOGI-OSG-based PD block in Figure 5, the closed-loop transfer functions can be derived as  
\[\left\{ \begin{array}{l}G_{\alpha}=\frac{v_{\alpha}(s)}{v_{\text{m}}(s)}=\frac {k\omega^{\prime}s}{s^{2}+k\omega^{\prime}s+\omega^{\prime 2}}\\ G_{\beta}=\frac{v_{\beta}(s)}{v_{\text{m}}(s)}=\frac{k\omega^{\prime 2}}{s^{2}+k \omega^{\prime}s+\omega^{\prime 2}}\end{array} \right. \tag{8}\]  
where \(k\) is the control parameter that can regulate the bandwidth and dynamic response of the estimated output signals. Considering the grid voltage input signal as \(v_{\text{in}}=V_{\text{gm}}\cos(\omega t+\phi_{i})\), the time domain outputs can be derived as  
\[\left\{ \begin{array}{l}v_{\alpha}(t)=V_{\text{gm}}\cos(\omega t+\phi_{i})+A_ {\alpha}\cos(\omega\sqrt{1-{(k/2)}^{2}}t+\phi_{\alpha})e^{-\frac{k\omega}{2} t}\\ v_{\beta}(t)=V_{\text{gm}}\sin(\omega t+\phi_{i})+A_{\beta}\sin(\omega\sqrt{1-{(k/2)}^{2}}t+ \phi_{\beta})e^{-\frac{k\omega}{2}t}\end{array} \right. \tag{9}\]  
in which \(A_{\alpha}\), \(A_{\beta}\), \(\phi_{\alpha}\), and \(\phi_{\beta}\) are functions of \(V_{\text{gm}}\), \(\phi_{i}\), and \(k\). From (9), assuming that the instantaneous phase angle is defined as \(\omega t+\phi_{i}=\phi_{i}\), and the output estimated signal phase angle is defined as \(\theta_{o}\), applying the Park transformation yields the signals in the \(dq\)-reference frame as [26]  
\[\left\{ \begin{array}{l}v_{d}(t)=V_{\text{gm}}\cos(\theta_{i}-\theta_{o}{}^{ \prime})\\ +[A_{\alpha}\cos(\omega\sqrt{1-{(k/2)}^{2}}t+\phi_{\alpha})e^{-\frac{k\omega} {2}t}\cos{\theta_{o}}^{\prime}+A_{\beta}\sin(\omega\sqrt{1-{(k/2)}^{2}}t+\phi_ {\beta})\sin{\theta_{o}}^{\prime})]e^{-\frac{k\omega}{2}t}\\ v_{q}(t)=V_{\text{gm}}\sin(\theta_{i}-\theta_{o}{}^{\prime})\\ -[A_{\alpha}\cos(\omega\sqrt{1-{(k/2)}^{2}}t+\phi_{\alpha})e^{-\frac{k\omega} {2}t}\cos{\theta_{o}}^{\prime}-A_{\beta}\sin(\omega\sqrt{1-{(k/2)}^{2}}t+ \phi_{\beta})\sin{\theta_{o}}^{\prime})]e^{-\frac{k\omega}{2}t}\end{array} \right. \tag{10}\]  
At the steady state, the high-frequency fluctuating terms in (9) and (10) with the time constant of \(2/k\omega\) will be zero. And thus,  
Figure 5: General structure of the Second-Order Generalized Integrator-based PLL (SOGI-PLL): (**a**) entire PLL structure; (**b**) the orthogonal signal generator (OSG)-based Phase Detector (PD). \[\left\{ \begin{array}{l}v_{d}(t)=V_{\rm gm}\cos\Delta\theta\approx V_{\rm gm}\\ v_{q}(t)=V_{\rm gm}\sin\Delta\theta\approx V_{\rm gm}\Delta\theta\end{array} \right. \tag{11}\]  
which shows that \(v_{d}\) estimates the amplitude of the input grid voltage, and the phase angle detected by the PLL is provided by \(v_{q}\).