# Impedance Modeling Based Method for Sub/Supsynchronous Oscillation Analysis of D-PMSG Wind Farm
## 2 Impedance Characteristics Analysis of D-PMSG
### Small-Signal Modeling of the PLL  
As described in Figure 4, phase voltages are transformed to \(dq\)-domain voltage \(v_{d}\) and \(v_{q}\) by Park's transformation. When there is a harmonic frequency \(f_{p}\) component in PCC voltage, the output angle \(\theta_{pll}\) will be \(\theta_{1}+\Delta\theta\). The Park's transformation could be broken into two parts as follows:  
\[\begin{array}{ll}\mathbf{T}(\theta_{pll})&=\frac{2}{3}\left[\begin{array}{cc} \cos\theta_{pll}&\cos(\theta_{pll}-2\pi/3)&\cos(\theta_{pll}+2\pi/3)\\ -\sin\theta_{pll}&-\sin(\theta_{pll}-2\pi/3)&-\sin(\theta_{pll}+2\pi/3)\\ \end{array}\right]\\ &=\left[\begin{array}{cc}\cos\Delta\theta&\sin\Delta\theta\\ -\sin\Delta\theta&\cos\Delta\theta\end{array}\right]\mathbf{T}(\theta_{1}(t))\end{array} \tag{1}\]  
where \(\mathbf{T}(\theta_{1})\) is the Park's transformation when \(\theta_{pll}=\theta_{1}=\omega_{1}t\).  
In the time domain, the phase voltage with a small-signal perturbation can be written as:  
\[v_{a}(t)=V_{1}\cos\omega_{1}t+V_{p}\cos(\omega_{p}t+\varphi_{up}) \tag{2}\]  
where \(V_{1}\) corresponds to the magnitude of the fundamental voltage at frequency \(f_{1}\), \(V_{p}\), with \(\phi_{\rm up}\) correspond to the magnitude and phase of the positive-sequence perturbation at frequency \(f_{p}\).  
Similar to phasor analysis, in the frequency domain, the voltage (1) could be written as:  
\[\mathbf{V}_{a}[f]=\left\{\begin{array}{c}\mathbf{v}_{1},f=f_{1}\\ \mathbf{V}_{p},f=f_{p}\end{array}\right. \tag{3}\]  
where \(\mathbf{V}_{1}=V_{1}\), \(\mathbf{V}_{\rm p}=V_{p^{*}}\)\(e^{i\phi_{up}}\) correspond to the fundamental and disturbance voltage in frequency domain.  
For linearization, apply \(\mathbf{T}(\theta_{1})\) to (3), and define \(\mathbf{V}_{dv}[f]\), \(\mathbf{V}_{qv}[f]\) as frequency domain d and q outputs:  
\[\mathbf{V}_{dv}[f]=\left\{\begin{array}{c}V_{1},\;\text{dc}\\ \mathbf{V}_{p},f=f_{p}-f_{1}\end{array}\right. \tag{4}\]  
\[\mathbf{V}_{qv}[f]=-j\mathbf{V}_{p},f=f_{p}-f_{1} \tag{5}\]  
According to (1), considering \(\Delta\theta\), \(v_{q}\) is easily found to be:  
\[v_{q}(t)\approx-\Delta\theta(t)v_{dq}(t)+v_{qv}(t) \tag{6}\]  
For voltage perturbation at \(f_{p}\), remove terms in \(\Delta\theta(t)\) proportional to second and higher orders by the harmonic linearization principle, then \(\Delta\theta(t)\) is given by  
\[\Delta\theta[f]=G_{p}(s)\mathbf{V}_{p},f=f_{p}-f_{1} \tag{7}\]  
Applying convolution to (4), (5), and (7), and neglecting terms in \(v_{q}\) proportional to the second order, the result of (6) in frequency domain is as follows:  
\[\mathbf{V}_{q}[f]=-G_{p}(s)V_{1}\mathbf{V}_{p}-j\mathbf{V}_{p},f=f_{p}-f_{1} \tag{8}\]  
Figure 4: Block diagram of PLL.  
Note that \(\Delta\Theta=H_{\text{pll}}(s)\)\(\mathbf{V_{q}}\), then \(G_{p}(s)\) can be solved from (8):  
\[G_{p}(s)=[-jH_{pll}(s)]/[1+V_{1}H_{pll}(s)],f=f_{p}-f_{1} \tag{9}\]  
To get the frequency errors of [PERSON]'s transformation, the final step is to obtain the response of \(\cos(\theta_{\text{pll}}(t))\). Since \(\cos(\theta_{\text{pll}}(t))=\cos(\theta_{1}(t)+\Delta\theta(t))=\cos\omega_{ 1}t-\sin(\omega_{1}t)\)\(\Delta\theta(t)\), apply convolution again:  
\[\begin{split}\cos\theta_{pll}[f_{p}]&=\frac{1}{2} \frac{H_{pll}(s-2\pi f_{1})}{1+V_{1}H_{pll}(s-2\pi f_{1})}\mathbf{V}_{p}=T_{p}( s)\mathbf{V}_{p}\\ \cos\theta_{pll}[2f_{1}-f_{p}]&=-T_{p}(s)\mathbf{ \widetilde{V}}_{p}\end{split} \tag{10}\]  
From above we can find that a voltage perturbation at \(f_{p}\) will result in both \(f_{p}\) and \(2f_{1}-f_{p}\) errors in Park's transformation, which gives a primary explanation of why sub/supsynchronous components usually exist together.  
A simulation experiment was conducted to verify the correctness of (10). Appendix A gives the setup parameters of the experiment. The voltages at PCC are \(V_{1}=566\) V, \(\mathbf{V}_{p}=V_{p}\)\(e^{i\phi}{}_{up}=20\cdot e^{j\pi/4}\). The response of \(\cos(\theta_{pll}(t))\) is depicted in Figure 5, compared with theoretical calculated results.