## 1 Introduction  
Advanced control design and accurate modelling and simulation of wind power plants (WPP) are key to increasing the efficiency and robustness of this class of systems. Controllers based on \(\mathcal{H}_{\infty}\) synthesis have been used in Voltage Source Converter (VSC) control [1, 2]. Nevertheless, earlier studies overlooked the use of \(\mathcal{H}_{\infty}\) controllers for grid-forming converters. Therefore, they did not consider the voltage control design nor the cascaded voltage and current control design. The use of HVdc diode rectifier (DR) stations reduces the investment cost of the off-shore WPP and increases the systems' resilience and efficiency [3]. Siemens asserts that using HVdc DR stations can potentially decrease the connection cost of WPP by as much as 30% [4, 5]. Grid-forming capability is a requirement for DR-based HVdc links. Several control alternatives have been proposed for this purpose [6]. The authors in [7] proposed a cascade control loop using \(\mathcal{H}_{\infty}\) synthesis for a micro-source inverter, but just considered uncertainty in the PWM LC filter parameters. In [8], the authors considered a sliding-mode control for the current loop, while the voltage outer loop controller was based on mixed \(\mathcal{H}_{2}/\mathcal{H}_{\infty}\). Still, the controller was not designed considering the operating modes with which the WTG was going to operate. In [9], the authors proposed a grid-forming controller that can synchronise with the ac grid for DR-based off-shore WPPs, but the operating modes of the WTGs were not considered during the controller design procedure. In [10], an \(\mathcal{H}_{\infty}\)-based grid-forming controller was proposed but for grid-following converters. In [11], the authors used passivity theory to design a grid-forming controller, but it was not considered for DR-based WPPs nor did they consider the operating modes of the converter. The authors in [12]proposed validating grid-forming converters using a co-simulation approach between the mosaic framework and a power system. In [13], the authors used \(\mu\)-synthesis with the purpose of optimising a grid-forming controller by considering the operating modes of the WPP, but just a single-loop voltage control was considered. In [14], the influence in the stability of the parameters of the voltage control for a grid-forming WTG was analysed, but just for one of the operating modes of the grid-forming converter. The authors in [15] implemented the typical P/\(\omega\) Q/V droops in the WTGs, proposing a transmission active power control for the off-shore WPP, where the on-shore MMC modifies the HVdc voltage to maintain the off-shore ac grid frequency. The problem is that the power transmission requires fast communication for its operation. In [16], the authors proposed a grid-following control for the WTGs of a DR-based WPP. The main problems were the islanded operation and the black-start operation of the WPP. Such grid-forming wind turbine converters need to show adequate performance during islanding operation at no-load (ISL no-load), during power transmission through the DR-HVdc link (DR-connected) and also during islanding operation at rated-load if the WPP is used for black-start or to supply local loads (ISL rated-load). Any control strategy for such grid-forming converters should consider these three different modes of operation. This paper evaluates current control alternatives that use \(\mathcal{H}_{\infty}\)-based design [17; 18] to improve the grid-forming WTG dynamic performance. Three designs based on \(\mathcal{H}_{\infty}\) synthesis are calculated. The first consists of a cascaded control based on \(\mathcal{H}_{\infty}\). Therefore, two controllers are calculated (voltage controller as outer loop and current controller as inner loop). The second design uses the cascaded control based on proportional-resonant regulators, with the controller parameters tuned using \(\mathcal{H}_{\infty}\) synthesis for fixed-structure controllers.