## 2 System Description and Problem Statement
### Voltage Control \(K(z)\) Tuning  
The grid-side converter control \(\mathbf{K}(z)\) shown in Figure 2 is  
\[V_{W\alpha\beta}^{*}(z)=\mathbf{K}(z)\left(\begin{array}{c}V_{C\alpha\beta}^ {*}(z)-V_{C\alpha\beta}(z)\\ I_{W\alpha\beta}(z)\end{array}\right) \tag{6}\]  
where the \((z)\) notation indicates that it will be implemented in discrete time, with a sampling period of \(T_{s}=2.5\times 10^{-4}\) s.  
This paper has included a standard control procedure as a benchmark for the proposed controllers. Figure 4 shows the considered control scheme. Incorporating a cascaded control structure with voltage and current control in the outer and inner loops; the design of both controllers aligns with the principles outlined in [22]. The voltage and current controllers are designed as PI controllers within a rotating frame of reference,  
Figure 3: Frequency response of each operating mode seen from a WTG.  
settling time and overshoot criteria. Subsequently, a frequency shift to a PR structure is applied using (7), as detailed in [23; 24]:  
\[G_{AC}(s)=G_{DC}(s-j\omega)+G_{DC}(s+j\omega) \tag{7}\]  
Figure 4 shows the PR control structure with the considered plant. The effects of the PWM and the measurement delay are considered in \(G_{PWM}\). \(G_{I_{W}}\) and \(G_{V_{C}}\) models the inductor and capacitor filter dynamics. \(G_{PWM}\), \(G_{I_{W}}\) and \(G_{V_{C}}\) are  
\[G_{PWM}(s)=\frac{1}{1+1.5T_{s}s} \tag{8}\]  
\[G_{I_{W}}(s)=\frac{1}{L_{\text{WS}}+R_{W}} \tag{9}\]  
\[G_{V_{C}}(s)=\frac{1}{C_{s}s} \tag{10}\]  
The tuning of the inner current controller is performed to attain a settling time of \(7.5\,\text{ms}\) and ensure an overshoot below 10%, considering PWM converter delays (\(G_{PWM}\) used as discretisation artifact [22]).  
The outer voltage controller's design considers the current loop's closed-loop transfer function, targeting a settling time lower than 25 ms and an overshoot lower than 10%.  
The equivalent PR controller is obtained by applying a frequency shift to the voltage and current controllers (7). The current and voltage PR controller parameters are listed below.  
Therefore, the PR based voltage controller \(\textbf{K}(s)\) is, in transfer matrix form,  
\[K_{PR}=\left[\begin{array}{cc}1-G_{PR,V}(s)*G_{PR,I}(s)&-G_{PR,I}(s)\end{array}\right] \tag{11}\]  
where \(G_{PR,V}(s)\) and \(G_{PR,V}(s)\) are  
\[G_{PR,V}(s)=K_{P,V}+\frac{K_{R,V}\cdot s}{s^{2}+\omega_{0}^{2}} \tag{12}\]  
\[G_{PR,I}(s)=K_{P,I}+\frac{K_{R,I}\cdot s}{s^{2}+\omega_{0}^{2}} \tag{13}\]  
A discretisation of the obtained controller \(K_{PR}(s)\) is carried out using the bilinear approximation with a sample time of 0.25 ms and prewarping at 50 Hz.  
The controller parameters are shown in Table 1.  
\begin{table}
\begin{tabular}{c} \hline \hline \multicolumn{1}{c}{**PR Tuned Using Root Locus**} \\ \hline Voltage control: \(G_{PR,V}\): \(K_{P,V}=0.08\), \(K_{R,V}=150.0\) \\ \hline Current control: \(G_{PR,I}\): \(K_{P,I}=0.92\), \(K_{R,I}=750.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Proportional–resonant control parameters of the PR benchmark controller.  
Figure 4: Benchmark PR-based voltage control of the grid-side converter.