# Enhanced Quasi Type-1 PLL-Based Multi-Functional Control of Single-Phase Dynamic Voltage Restorer
## 3 Reference Signal Generator
### Tuning-Free Fixed-Frequency Orthogonal Signal Generator  
Once the DC offset is eliminated, the signal \(v_{g}^{\varnothing}\) can be used to generate an orthogonal signal component. In this study, we are considering an all-pass filter (APF) [26; 36]. APF is a first-order filter, and it can be used to generate orthogonal signals without any tuning gain. This filter can be used either as frequency-adaptive or non-adaptive configurations. In this study, a frequency-fixed operation is considered same as the DC offset rejection method, as highlighted in Section 3.1. The frequency-fixed APF transfer function is given by the following:  
\[\frac{v_{g}^{\varnothing\bot}}{v_{g}^{\varnothing}}(s)=\text{APF}(s)=\frac{ \omega_{n}-s}{\omega_{n}+s}, \tag{14}\]  
where superscript \(\bot\) indicates orthogonal signals. A time-domain block diagram of APF is provided in Figure 2. Similarly to Section 3.1, the frequency-fixed operation of the APF will introduce amplitude and phase attenuation in the estimated orthogonal signal. As such, characterization of the APF is needed to determine the necessary compensation mechanism. Phase-angle of the transfer function (14) is given by the following.  
\[\angle\text{APF}(s)=\tan^{-1}\Biggl{(}\frac{2\omega_{g}\omega_{n}}{\bigl{(} \omega_{g}-\omega_{n}\bigr{)}\bigl{(}\omega_{g}+\omega_{n}\bigr{)}}\Biggr{)}. \tag{15}\]  
For small frequency drift, i.e., \(\delta_{\omega}\approx 0\), it can be assumed that \(\omega_{g}+\omega_{n}\approx 2\omega_{g}\). Then, the phase angle (15) can be approximated as follows.  
\[\angle\text{APF}(s) \approx\tan^{-1}\Biggl{(}\frac{1}{\frac{\omega_{g}}{\omega_{n}}- 1}\Biggr{)}.\] \[\approx-\frac{\pi}{2}-\tan^{-1}\biggl{(}\frac{\omega_{g}}{\omega_ {n}}-1\biggr{)},\] \[\approx-\frac{\pi}{2}-\Biggl{(}-\frac{\pi}{4}+\frac{\frac{\omega_ {g}}{\omega_{n}}}{2}+\frac{\Bigl{(}\frac{\omega_{g}}{\omega_{n}}\Bigr{)}^{2}} {2!}+\cdots\Biggr{)}. \tag{16}\]  
Figure 2: Time-domain implementation of frequency-fixed APF.  
By ignoring the high-order (2 nd and above) terms from the Taylor series expansion in (16), this equation can be simplified as follows.  
\[\angle\text{APF}(s) \approx-\frac{\pi}{2}-\left(-\frac{\pi}{4}+\frac{\omega_{g}}{2 \omega_{n}}\right),\] \[\approx-\frac{\pi}{4}-\frac{\omega_{g}}{2\omega_{n}},\] \[\approx-\frac{\pi}{4}-\frac{(\omega_{n}+\delta\omega)}{2\omega_{ n}},\] \[\approx-\frac{\pi}{4}-\frac{1}{2}-\frac{\delta_{\omega}}{2\omega _{n}}. \tag{17}\]  
As shown in (17), the first two terms are frequency independent while the third term depends on frequency variation. This term, i.e., \(k^{\prime}_{\phi}=1/2\omega_{n}\), needs to compensated.  
It is well known that single-phase grid voltage can be represented by an unbalanced two phase-system. It was shown in [36] that single-stage frequency-fixed APF cannot effectively remove the unbalanced component as the bandwidth of the unbalanced component rejection part is very narrow. As such, the unbalanced component will appear as double the fundamental frequency component after Park transformation. Thus, in order to eliminate this error, the double frequency component needs to be rejected. This issue can be solved by increasing the bandwidth of the notch component. In the literature, a two-stage APF has been suggested for this purpose. The two-stage APF effectively increases notch bandwidth and enables frequency-fixed APFs to reject off-nominal frequency unbalanced components. However, the two-stage APF will double the phase delay in the off-nominal frequency condition. As such, the required phase compensation value is computed by \(k_{\phi}=2k^{\prime}_{\phi}=1/\omega_{n}\).