# Zero-Voltage Ride-Through Capability of Single-Phase Grid-Connected Photovoltaic Systems
## 3 Overview of PLL Synchronization Techniques
### Enhanced PLL (EPLL)  
The enhanced PLL (EPLL) improves the performance of the basic PLL structure by employing an ANF-based PD system [3]. A general structure of an EPLL is shown in Figure 4. The control loop achieves phase-angle tracking and amplitude estimating, and the aforementioned major drawbacks of the standard PLL are overcome. Assuming that the input grid voltage \(v_{\text{in}}\) is a purely sinusoidal waveform at the rated grid frequency, which can be defined as \(v_{\text{in}}=V_{\text{gm}}\) sin \(\theta_{i}\), and the output  
Figure 3: Basic structure of a phase-locked loop (PLL) system. Figure 2: Current control block diagram of the LCL-filtered grid-connected inverter. estimated signal can be defined as \(v_{\text{out}}=V_{\text{gm}}{}^{\prime}\sin\theta_{o}{}^{\prime}\). Therefore, the phase error output of the PD can be expressed as  
\[ \begin{array}{l}\varepsilon=e\cos\theta_{o}{}^{\prime}=(V_{\text{gm}}\sin \theta_{i}-V_{\text{gm}}{}^{\prime}\sin\theta_{o}{}^{\prime})\cos\theta_{o}{}^{ \prime}\\ =\frac{V_{\text{gm}}}{2}\sin(\theta_{i}-\theta_{o}{}^{\prime})+\frac{V_{\text {gm}}{}^{\prime}}{2}\sin(\theta_{i}+\theta_{o}{}^{\prime})-\frac{V_{\text{gm} }}{2}\sin(2\theta_{o}{}^{\prime})\end{array}  \tag{2}\]  
Meanwhile, in order to estimate the amplitude of the grid voltage, the output signal \(x\) can be expressed as  
\[ \begin{array}{l}x=e\sin\theta_{o}{}^{\prime}=(V_{\text{gm}}\sin\theta_{i}-V_ {\text{gm}}{}^{\prime}\sin\theta_{o}{}^{\prime})\sin\theta_{o}{}^{\prime}\\ =\frac{V_{\text{gm}}}{2}\cos(\theta_{i}-\theta_{o}{}^{\prime})-\frac{V_{\text {gm}}{}^{\prime}}{2}+\frac{V_{\text{gm}}{}^{\prime}}{2}\cos(2\theta_{o}{}^{ \prime})-\frac{V_{\text{gm}}}{2}\cos(\theta_{i}+\theta_{o}{}^{\prime})\end{array}  \tag{3}\]  
Considering the steady state, the estimated outputs \(V_{\text{gm}}{}^{\prime}\), \(\theta_{o}{}^{\prime}\) can then be approximated as \(V_{\text{gm}}=V_{\text{gm}}{}^{\prime}\), \(\theta_{i}=\theta_{o}{}^{\prime}\). Consequently, the double-frequency term approaches zero. The phase angle, frequency, and amplitude of the estimated signals will contain no double-frequency ripples [28]. As aforementioned, compared with the conventional PLL structure, the EPLL can track the major information both in phase and in amplitude with no double-frequency. In order to evaluate the performance of the accuracy and stability of the EPLL, from Figure 4, the transfer functions of the EPLL for amplitude and phase-angle estimations (denoted as \(G_{A}(s)\) and \(G_{\Theta}(s)\), respectively) can be derived as  
\[G_{A(\text{EPLL})}(s)=\frac{\frac{k_{p}V_{\text{gm}}{}^{\prime}}{2}}{s+\frac{ k_{p}V_{\text{gm}}{}^{\prime}}{2}}=\frac{1}{s+\mu} \tag{4}\]  
\[G_{\Theta(\text{EPLL})}=\frac{V_{\text{gm}}{}^{\prime}}{2}\frac{k_{p}s+k_{i}} {s^{2}+\frac{k_{p}V_{\text{gm}}{}^{\prime}}{2}s+\frac{k_{i}V_{\text{gm}}{}^{ \prime}}{2}}=\frac{V_{\text{gm}}{}^{\prime}}{2}\frac{2\sigma\omega_{r}s+ \omega_{r}^{2}}{s^{2}+2\sigma\omega_{r}s+\omega_{r}^{2}} \tag{5}\]  
where \(k_{p}\), \(k_{p}\), and \(k_{i}\) are the parameters for adjusting the accuracy and the stability of the EPLL, and \(2\sigma\omega_{r}=k_{p}V_{\text{gm}}{}^{\prime}/2\), \(\omega_{r}^{2}=k_{i}V_{\text{gm}}{}^{\prime}/2\), \(\mu=k_{v}V_{\text{gm}}{}^{\prime}/2\). The characteristic transfer function shows that the amplitude and phase-angle estimations are a type-II control system. Therefore, the settling time is measured from the start time to the time in which the system stays within 5% of the steady-state errors. The settling time \(t_{\text{s}}\) can approximately be calculated as  
\[t_{\text{s}}\approx\frac{3.5}{\sigma\omega_{r}}=\frac{14}{k_{p}V_{\text{gm}}{ }^{\prime}} \tag{6}\]  
From (4) to (6), it is clear that the control parameters \(k_{p}\) and \(k_{i}\) mainly affect the convergence rate of the frequency and phase angle. The control parameter \(k_{v}\) determines the performance of the amplitude estimation process in the PD.