## 5 Discussion and Conclusions  
This paper presents a methodology to design three different voltage controllers, using \(\mathcal{H}_{\infty}\) techniques with a resonant behaviour for grid-forming converters. The \(\mathcal{H}_{\infty}\) techniques use both cascaded controller design and a voltage controller without an inner current loop. Additionally, \(\mathcal{H}_{\infty}\) techniques are used to improve the performance and robustness of the cascaded control based on PR regulators via a structured PR \(\mathcal{H}_{\infty}\) design, which allows for a fair comparison between structured PR controllers, cascaded \(\mathcal{H}_{\infty}\) controllers and direct \(\mathcal{H}_{\infty}\) controllers. The controllers have been designed by considering the islanded no-load plant, which produces adequate results in islanded rated-load and DR-connected modes of operation. The paper provides directives for the input and output weights' selection to synthesise the controllers, considering the system's unstructured uncertainty and the controllers' desired performance. The obtained controllers using \(\mathcal{H}_{\infty}\) synthesis improve the performance, robustness and harmonic rejection in comparison with the controller achieved using classical control design techniques (root locus). The dynamic response of all controllers has been tested for the three considered operating modes (islanded no-load, islanded rated-load and DR-connected): first of all, considering a sinusoidal reference, and, secondly, as part of a detailed simulation of the complete WPP and the HVdc diode rectifier link. The dynamic response to a sinusoidal reference of all the controllers is reasonable for the islanded no-load condition (this is the plant used for all controller designs). However, the cascaded controllers' performance shows worse dynamics than the single \(\mathcal{H}_{\infty}\) controller when changing the mode of operation, particularly for the DR-connected case. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Harmonic** & \multicolumn{4}{c}{**Voltage (\%)**} & \multicolumn{4}{c}{**Current (\%)**} \\ \hline  & \(\mathbf{PR}\) & **Cascaded \(\mathcal{H}_{\infty}\)** & \(\mathbf{PR\mathcal{H}_{\infty}}\) & \(\mathbf{\mathcal{H}_{\infty}}\) & \(\mathbf{PR}\) & **Cascaded \(\mathcal{H}_{\infty}\)** & \(\mathbf{PR\mathcal{H}_{\infty}}\) & \(\mathbf{\mathcal{H}_{\infty}}\) \\ \hline
11 th & 1.02 & 0.86 & 0.88 & 0.60 & 0.59 & 0.32 & 0.32 & 0.25 \\
13 th & 0.91 & 0.80 & 0.85 & 0.56 & 0.66 & 0.37 & 0.37 & 0.21 \\ \hline \end{tabular}
\end{table} 
Table 7: Main voltage and current harmonic of the DR-based WPP. Concerning robustness, coprime factor uncertainty margins are better using \(\mathcal{H}_{\infty}\) controllers than using PR based on root locus design, as expected. Within the \(\mathcal{H}_{\infty}\) controllers, the best coprime factor uncertainty margin is shown by the single \(\mathcal{H}_{\infty}\) design. The single \(\mathcal{H}_{\infty}\) optimisation has no structural constraints in its design, at the expense of requiring more complicated current-limiting techniques than cascaded designs. Finally, the detailed EMT simulations of the complete system show that, even though all considered controllers' responses are reasonable, the \(\mathcal{H}_{\infty}\) controller shows the best dynamic performance of all controllers, with the lowest voltage and current harmonic distortion. It has been shown that all \(\mathcal{H}_{\infty}\)-based controllers offer better performance than the standard root locus-designed PR controller, with the best performance being obtained by the single \(\mathcal{H}_{\infty}\) controller. Therefore, even when considering a fixed PR structure, \(\mathcal{H}_{\infty}\) design offers clear advantages for the design of the voltage control of grid-forming converters. This work paves the way for further optimisation considering different grid configurations, which is particularly relevant for grid-forming converters used in DR-based off-shore WPPs, including the black-start operation of these WPPs.