## 4 Results
### EMT Simulation Results  
The performance of the obtained controllers is validated in the described system in Section 2 using a detailed EMT simulation. Figure 11 shows the simulation results considering the \(H_{\infty}\)-tuned cascaded PR controller and the full \(\mathcal{H}_{\infty}\) controller. Note that the \(H_{\infty}\)-tuned PR is the only cascaded controller shown, as the results for the other two cascaded controllers (root locus PR and cascaded \(\mathcal{H}_{\infty}\)) are very similar to the results shown in Figure 11a). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**nfc Margin** & **PR** & **Cascaded \(\mathcal{H}_{\infty}\)** & **PR \(\mathcal{H}_{\infty}\)** & \(\mathcal{H}_{\infty}\) \\ \hline Islanded no-load & 0.06 & 0.14 & 0.12 & 0.32 \\ \hline Islanded rated-load & 0.05 & 0.14 & 0.10 & 0.12 \\ \hline Grid-connected (DR) & 0.01 & 0.06 & 0.02 & 0.07 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Normalised coprime factor uncertainty margin. Figure 10: Voltage and current result in the model used for design. The current \(I_{W}\) is scaled with a 0.3 factor in the case of DR-connected. (**a**) Cascaded \(\textit{PR}_{\mathcal{H}_{\infty}}\) controller. (**b**) \(\mathcal{H}_{\infty}\) voltage controller. To compare both current controller designs, changes in active power references are applied to each off-shore WPP. The changes in active power reference are ramped up, considering a ramp of 12.5 ms. Such a large slope is not realistic in practice, but it illustrates the behaviour of the different current controllers. From top to bottom, the first graph in Figure 11 shows the active power delivered by each WPP cluster, together with its reference. The second graph shows the reactive power being delivered by each WPP cluster. The third graph shows the WTG capacitor voltage (\(V_{c}\)) and its reference. The fourth graph shows the current magnitude of (\(I_{W}\)), and the frequency of each WPP cluster is shown in the fifth graph. Both controllers obtain a similar dynamic response because the outer droop control (see Figure 2) dominates the system dynamics. However, the cascaded voltage converter causes a higher overshoot in active power response. Moreover, the \(\mathcal{H}_{\infty}\) controller tracks the reference capacitor voltage \(V_{\text{C}}^{\ast}\) much better. Additionally, the active power response coupling between each WPP cluster is more negligible when using the full \(\mathcal{H}_{\infty}\) controller. It is worth pointing out that the current \(I_{W1}\) is higher than 1 pu for part of the transient. This is caused by the fact that \(WPP_{1}\) is required to supply 1 pu active power while contributing to the reactive power requirements of the overall off-shore grid. Therefore, the overall current magnitude is slightly higher than 1 pu. Voltage, current and reactive power ripple are the same for both cascaded controllers and full \(\mathcal{H}_{\infty}\) controllers. A comparative analysis shows the harmonic rejection capabilities  
Figure 11: Detailed EMT simulation of the DR-based off-shore WPP. (**a**) \(\mathcal{H}_{\infty}\) PR cascaded controller. (**b**) \(\mathcal{H}_{\infty}\) controller. of the designed controllers through a quantitative examination of voltage and current harmonic contents. Table 7 shows the 11 th and 13 th voltage and harmonic contents for each one of the considered controllers. The analysis has been carried out using Fourier analysis of the detailed EMT simulation results, considering the output power of each WPP cluster being \(P_{WT1}=1\) pu, \(P_{WT2}=0.85\) pu and \(P_{WT3}=0.75\) pu, as at the end of the first graph of Figure 2.  
The 11 th and 13 th harmonics have been selected, as they are the characteristic harmonics of the 12-pulse HVdc diode rectifier converter. The root locus-based PR controller offers the worst 11 th and 13 th voltage and current harmonic components of the four considered controllers.