## 7 Conclusions  
This paper presents a unified impedance-based modelling and analysis tool to predict unstable conditions resulting from the interactions between the wind power plant and the weak grid. The proposed approach allows to have advantages of both the dq domain modelling and sequence domain modelling. This enables transmission system operators (TSOs) to do stability analyses of large power systems more efficiently, and it enables wind turbine manufacturers to optimise the controller parameters to increase the stability margin under a weak grid connection. A 108 MW wind power plant has been studied and the short circuit ratio (SCR) of the grid is considered to be 1.25 and 1.5. The poles of the closed-loop system in the frequency-domain are calculated to find the frequency and damping of the oscillatory modes. Nyquist criteria of the open-loop system are plotted to predict easily the stability margin of the wind power plant connected to the weak grid. The results of a simulated wind power plant have shown that a wind power plant which is tuned for a strong grid can become unstable when connected to a weak grid. The proposed method is a valuable tool when adapting and retuning the control system to actual grid conditions. [PERSON] and [PERSON] proposed the initial methodology, [PERSON] proposed how to apply the method for weak grids, [PERSON] provided the data, and [PERSON] reviewed and edited the paper. This research was funded by \"Vestas Wind System A/S\" and \"Innovation Fund Denmark, grant number 8054-000239\". The authors declare no conflict of interest. Figure 12: Time-domain simulation for SCR = 1.25, \(X_{g}/R\) = 20 and \(X_{d}/X_{g}\) = 0 (an unstable case), (**a**) current, (**b**) fast Fourier transform (FFT) of the current before trip. Appendix A Relationship between dq-domain admittance/impedance and sequence-domain admittance/impedance. In this part, the relationship between dq-domain admittance/impedance and sequence-domain admittance/impedance is derived. Following Park's transformation is used to convert three-phase variables into a dq reference frame:  
\[\left[ \begin{array}{c}X_{d}\\ X_{q}\end{array} \right]=\frac{2}{3}\left[ \begin{array}{cc}\cos(\theta)&\cos( \theta-\frac{2\pi}{3})&\cos(\theta+\frac{2\pi}{3})\\ -\sin(\theta)&-\sin(\theta+\frac{2\pi}{3})&-\sin(\theta+\frac{2\pi}{3})\\ \end{array} \right]\left[ \begin{array}{c}X_{a}\\ X_{b}\\ X_{c}\end{array} \right] \tag{11}\]  
For \(\theta=\omega_{1}t\) in (11), three-phase perturbed voltages described in (29) can be transformed to the dq-domain as  
\[\left[ \begin{array}{c}\Delta v_{d}+\overline{V}_{d}\\ \Delta v_{q}+\overline{V}_{q}\end{array} \right]=\frac{2}{3}\left[ \begin{array} []{cc}\cos(2\pi f_{1}t)&\cos(2\pi f_{1}t-\frac{2\pi}{3})&\cos(2\pi f_{1}t+ \frac{2\pi}{3})\\ -\sin(2\pi f_{1}t)&-\sin(2\pi f_{1}t+\frac{2\pi}{3})&-\sin(2\pi f_{1}t+\frac{2 \pi}{3})\end{array} \right]\times \tag{12}\] \[\left[ \begin{array}{c}V_{1}\cos(2\pi f_{1}t+\phi_{v1})+\hat{ \sigma}_{p}\cos[2\pi(f_{p}+f_{1})t+\phi_{vp}]+\hat{\sigma}_{n}\cos[2\pi(f_{p}- f_{1})t+\phi_{vm}]\\ V_{1}\cos(2\pi f_{1}t+\phi_{v1}-\frac{2\pi}{3})+\hat{\sigma}_{p}\cos[2\pi(f_{p} +f_{1})t+\phi_{vp}-\frac{2\pi}{3}]+\hat{\sigma}_{n}\cos[2\pi(f_{p}-f_{1})t+ \phi_{vm}+\frac{2\pi}{3}]\\ V_{1}\cos(2\pi f_{1}t+\phi_{v1}+\frac{2\pi}{3})+\hat{\sigma}_{p}\cos[2\pi(f_{p} +f_{1})t+\phi_{vp}+\frac{2\pi}{3}]+\hat{\sigma}_{n}\cos[2\pi(f_{p}-f_{1})t+ \phi_{vm}-\frac{2\pi}{3}]\end{array} \right]\]  
Based on trigonometric formulas and some simplification, (12) can be simplified as  
\[\left[ \begin{array}{c}\Delta v_{d}+\overline{V}_{d}\\ \Delta v_{q}+\overline{V}_{q}\end{array} \right]=\left[ \begin{array}{c}V_{1} \cos(\phi_{v1})+\hat{\sigma}_{p}\cos(2\pi f_{p}t+\phi_{vp})+\hat{\sigma}_{n} \cos(2\pi f_{p}t+\phi_{vm})\\ V_{1}\sin(\phi_{v1})+\hat{\sigma}_{p}\sin(2\pi f_{p}t+\phi_{vp})-\hat{\sigma}_{n} \sin(2\pi f_{p}t+\phi_{vm})\\ \end{array} \right] \tag{13}\] \[=\left[