# Impedance Modeling Based Method for Sub/Supsynchronous Oscillation Analysis of D-PMSG Wind Farm
## 3 Impedance-Based Stability Analysis of SSO
### SSO Simulation and Theoretical Explanation  
The structure diagram of the studied system is shown in Figure 14. The converter side parameters are set as Appendix A, the grid side parameters set in simulation are as Table 2. The SCR decreases from 5.37 to 2.13 at 3 s, and decreases to 1.73 at 4 s.  
Furthermore, the power waveform and current spectrum analysis are shown in Figure 15. From (a) we can find that the output power of D-PMSG oscillates twice at the SCR changing moments. The first oscillation quickly stabilized, but the second oscillation at 24 Hz gradually diverges until the system crashes. The magnitude of harmonic components at 5.5 s in output current is shown in Figure 15b, the 26/74 Hz sub/supsynchronous currents exist simultaneously.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Item** & **Value** \\ \hline \(V_{s}\) & System Voltage & 0.82 kV (L-L, RMS) \\  & \(L_{g1}\): Before 3 s & 0.2 mH \\ \(L_{g}\) & \(L_{g2}\): 3–4 s & 0.495 mH \\  & \(L_{g3}\): After 4 s & 0.62 mH \\ \(R_{f}\) & Filter resistance & 0.02 \(\Omega\) \\ \(C_{f}\) & Filter capacitance & 500 μF \\ \hline \hline \end{tabular}
\end{table}
Table 2: Grid side impedance parameters.  
Figure 14: Diagram of subsynchronous oscillation (SSO) studied simulation system.  
Figure 13: Impedance sensitivity of current controller parameters (Increase). (**a**) Sensitivity of proportional constant \(K_{pi}\) in \(H_{i}\)(s); (**b**) Sensitivity of integral constant \(K_{ii}\) in \(H_{i}\)(s).  
Theoretical analysis is conducted to explain this simulation result. Since the current reference does not change, from Figure 6 we can know that the voltage at the PCC will drop along while the grid inductance \(L_{g}\) is increasing. So, the output power is also decreasing. Quantitative stability analysis should be done to figure out why the second oscillation is unstable. In these two SCR condition, calculate the power flow, then the total impedance of D-PMSG and grid \(Z_{PMSG}+Z_{GRID}\) could be obtained as follows:  
\[\begin{array}{ll}Z_{total}(s)&=Z_{PMSG}+Z_{GRID}=R_{PMSG}+R_{GRID}+j(X_{PMSG} +X_{GRID})\\ &=\frac{j\omega_{1}L-sL-H_{i}(s-j\omega_{1})}{H_{i}(s-j\omega_{1})_{1}I_{1}T_{g }(s)e^{i\pi_{1}}+V_{1}T_{1}(s)-1}+(R_{f}+\frac{1}{sC_{f}})//(sL_{g})\end{array} \tag{25}\]  
Then, according to the developed RLC circuit-based stability criterion mentioned in the beginning of this section, the system will be unstable if:  
\[R_{total}(f_{r})<0,\text{ {when }}X_{total}(f_{r})=0 \tag{26}\]  
This criterion is easier to implement by plotting them in the same figure and the direction of \(R_{total}\) at the resonant frequency is obviously to see. The analysis shown in Figure 16 gives the impedance comparison of D-PMSG and grid when SCR = 2.13 and 1.73.  
Figure 16: Impedance comparison of \(Z_{PMSG}\) and \(Z_{Grid}\).  
Figure 15: Power waveform and current spectrum analysis of SSO. (**a**) Output power oscillation frequency; (**b**) Harmonic current components at 5.5 s.  
The green vertical line indicates that the total resistance at the resonant frequency is positive when SCR = 2.13, which means this system is stable. This is consistent with the power waveform after 3 s in Figure 15a.  
When SCR decreases to 1.73, the total resistance at the resonant frequency 72.77 Hz is \(-\)0.04, and this system is judged unstable under this operating condition. The \(f_{p}\) supsynchronous oscillation current will be captured by the GSC controller and introduces a (\(2f_{1}-f_{p}\)) subsynchronous current in return, as discussed before, about 27 Hz and 73 Hz, respectively. Moreover, the power will oscillate at a frequency of 23 Hz (\(|f_{p}-f_{1}|\)), which is very close to the simulation results after 4 s, considering the calculating and modeling errors. It can be concluded that the mechanism of SSO when SCR decreases is theoretically revealed.