## 2 System Description and Problem Statement
### Wind Turbine Model  
Figure 2 shows the WTG grid-side converter model and its control strategy. WTG's grid-side converter consists of a three-phase h-bridge converter, an LC filter and a transformer.  
The LC filter and the WTG transformer dynamics of the WTG in Figure 2 are described in Equations (1) and (2):  
\[\begin{pmatrix}I_{W}^{i}\\ V_{\text{C}}\\ I_{T}\end{pmatrix}=\begin{pmatrix}-\frac{R_{W}}{L_{W}}&0&-\frac{1}{L_{W}}\\ \frac{1}{C_{W}}&0&-\frac{1}{C_{W}}\\ 0&\frac{1}{L_{T}}&-\frac{R_{T}}{L_{T}}\end{pmatrix}\begin{pmatrix}I_{W}\\ V_{\text{C}}\\ I_{T}\end{pmatrix}+\begin{pmatrix}\frac{1}{L_{W}}&0\\ 0&0\\ 0&-\frac{1}{L_{T}}&\frac{0.69}{66}\end{pmatrix}\begin{pmatrix}V_{W}\\ V_{\text{F}}\end{pmatrix} \tag{1}\]  
\[\begin{pmatrix}I_{W}\\ V_{\text{C}}\\ I_{T}\end{pmatrix}=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}I_{W}\\ V_{\text{C}}\\ I_{T}\end{pmatrix}+\begin{pmatrix}0&0\\ 0&0\\ 0&0\end{pmatrix}\begin{pmatrix}V_{W}\\ V_{\text{F}}\end{pmatrix} \tag{2}\]  
Figure 1: DR-based off-shore WPP [13].  
where \(R_{W}\) and \(L_{W}\) are the resistance and the inductance values of the inductor LC filter, \(C_{W}\) is the Capacitor LC filter and \(R_{T}\) and \(L_{T}\) are the resistance and the inductance value of the WTG transformer. \(V_{W}\) and \(I_{W}\) are the h-bridge converter output voltage and current, \(V_{C}\) is the capacitor voltage and \(V_{F}\) and \(I_{T}\) are the high-side transformer voltage and the low-side transformer current.  
Note that the current output \(I_{T}\) should be multiplied by \(0.69/66\) to obtain the high-voltage-side transformer current. The machine-side converter of each WTG is modelled as a DC voltage source for simplification.  
The grid-forming converter control loop should perform adequately in islanded operation (no-load and rated-load) and deliver full power to the DR-HVdc station. These three operational modes can be considered by changing \(R_{T}\) and \(L_{T}\) values:  
1. Islanded no-load: This mode considers a WTG before connecting to the off-shore AC grid. The WTG operates in an open circuit, and the resistance transformer value can be considered \(R_{T}=\infty\). The inductance transformer value \(L_{T}\) remains the same;
2. Islanded rated-load: This mode considers a WTG before connecting to the off-shore AC grid with an equivalent resistive load equal to the rated capacity. In this mode, the equivalent value of the resistance transformer in pu is \(R_{T}=1\). The inductance transformer value \(L_{T}\) remains the same;
3. DR-connected: This mode considers the WTG, while the power transmission is via the DR-HVdc link. In this mode, the equivalent values of the resistance and inductance transformer in pu are \(R_{T}=1\), \(L_{T}=L_{T}+L_{DR}\).  
Figure 3 shows the frequency response of each considered operating mode. As Figure 3 shows, both the resonant peak amplitude and frequency depend on the case being considered. Moreover, an anti-resonant peak (at 640 Hz) in the current's amplitude diagram appears when the system is DR-connected. On the other hand, both resonant and anti-resonant peaks are much smaller when considering islanded operation at rated-load, as the considered resistive load would introduce additional damping.  
Figure 2: Control strategies for the grid-side converter of each grid-forming WTG [13].