# Impedance Modeling Based Method for Sub/Supsynchronous Oscillation Analysis of D-PMSG Wind Farm
## 2 Impedance Characteristics Analysis of D-PMSG
### Impedance Modeling of D-PMSG  
#### 2.3.1 Harmonic Components in Control Loop  
It is easily to get from Figure 3 that the modulating signals, current, and PCC voltage satisfy:  
\[L\frac{d}{dt}\left[\begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\end{array}\right]=\left[\begin{array}{c}v_{ga}\\ v_{gb}\\ v_{gc}\end{array}\right]-\left[\begin{array}{c}v_{a}\\ v_{b}\\ v_{c}\end{array}\right] \tag{11}\]  
In order to solve (11) for impedance in the frequency domain, the sequence components in the modulating signals should be found as functions of the voltage and current perturbations. In the previous section, we have initially reached a conclusion that sub/supsynchronous always appear together. Therefore, for the voltage of (2), the current can be assumed as:  
\[i_{a}(t)=I_{1}\cos(\omega_{1}t+\varphi_{11})+I_{p}\cos(\omega_{p}t+\varphi_{ ip})+I_{p2}\cos((2\omega_{1}-\omega_{p})t+\varphi_{ip2}) \tag{12}\]  
where the annotations similar to (2). Other phase currents can be inferred.  
Recall that currents _id_ and _iq_ are outputs of a _dq_-domain transformation, which in the frequency domain involves a convolution of the frequency components in the phase currents, with the frequency components in Park's transformation. Currents in frequency domain can be defined as (3), and frequency  
Figure 5: Frequency response of cosine of PLL angle to perturbation in phase a voltage. (**a**) Magnitude comparison; (**b**) Phase comparison. Solid lines: model prediction; Marking points: simulation results.  
components of Park's transformation are described in (10). Table 1 shows the possible combinations to consider in the convolution, it is worth noting that second and higher orders small signals are neglected. The _dc_ components are not listed here, because _id_ and _iq_ would be exactly equal to _I\({}_{dref}\)_ and _I\({}_{dref}\)_ at the stable operating point.  
Combine the frequency components in Table 1 noting that \(\mathbf{I_{d}}\) [\(f_{p}-f_{1}\)] is the conjugate of \(\mathbf{I_{d}}\) [\(f_{1}-f_{p}\)]:  
\[\begin{array}{l}\mathbf{I_{d}}[f_{p}-f_{1}]=-2 jT_{p}(j2\pi f_{p})\mathbf{V}_{ p}I_{1}\sin\varphi_{i1}+\mathbf{I_{p}}+\overline{I}_{p2}\\ \mathbf{I_{q}}[f_{p}-f_{1}]=+2 jT_{p}(j2\pi f_{p})\mathbf{V}_{p}I_{1}\cos\varphi _{i1}-j\mathbf{I_{p}}+\overline{I}_{p2}\end{array} \tag{13}\]  
Figure 3 depicts a _dq_-domain current controller, according to its diagram, the _dq_ reference voltages are given by:  
\[\left\{\begin{array}{l}\mathbf{V}_{dref}=H_{i}(s)(I_{dref}-\mathbf{I_{d}})- \omega_{1}L\mathbf{I_{q}}\\ \mathbf{V}_{qref}=H_{i}(s)(I_{qref}-\mathbf{I_{q}})+\omega_{1}L\mathbf{I_{d}} \end{array}\right.,H_{i}(s)=K_{pi}+K_{ii}/s \tag{14}\]  
It is easy to find that the _dq_ reference voltages have _dc_ and (_fp_ -_f_1) components. Moreover, (14) can be further described as:  
\[V_{dref}[dc]=V_{1}-\omega_{1}LI_{1}\sin\varphi_{i1} \tag{15}\]  
\[\begin{array}{l}\mathbf{V}_{dref}[f_{p}-f_{1}]=-H_{i}(s)\mathbf{I_{d}}[f_{p }-f_{1}]-\omega_{1}LI_{q}[f_{p}-f_{1}]\\ \mathbf{V}_{qref}[f_{p}-f_{1}]=-H_{i}(s)\mathbf{I_{q}}[f_{p}-f_{1}]+\omega_{1} L\mathbf{I_{d}}[f_{p}-f_{1}]\end{array} \tag{16}\]  
These signals are convoluted with inverse Park's transformation to generate their phase-domain counterparts. Counting equations (10) and (13-16) together, the harmonics of phase-domain voltage references are as followings:  
\[\begin{array}{l}\mathbf{V}_{aref}[f_{p}]=H_{i}(s-j\omega_{1})I_{1}T_{p}(s) e^{j\varphi_{i2}}\mathbf{V}_{p}+V_{1}T_{p}(s)\mathbf{V}_{p}\\ \hskip 14.226378 pt+[j\omega_{1}L-H_{i}(s-j\omega_{1})]\mathbf{I_{p}}\\ \mathbf{V}_{aref}[2f_{1}-f_{p}]=-H_{i}(s-j\omega_{1})I_{1}T_{p}(s)e^{j\varphi_ {i1}}\overline{\mathbf{V}}_{p}-V_{1}T_{p}(s)\overline{\mathbf{V}}_{p}\\ \hskip 14.226378 pt+[j\omega_{1}L-H_{i}(s-j\omega_{1})]\mathbf{I_{p2}}\end{array} \tag{17}\]