###### Abstract
This approach enables the rapid transmission of the grid frequency to the wind farm, preventing the issue where MMC-HVDC decoupling isolates the wind farm from providing inertia support to the grid. Additionally, due to capacitor voltage variations, this method allows MMC-HVDC and wind farm capacitors to contribute frequency support to the grid while transmitting grid frequency information. Furthermore, an inertia transfer mechanism is introduced in the wind turbine's machine-side converter to enhance its frequency support capability. A PSCAD/EMTDC-based simulation model was used to compare the proposed coordinated inertia control strategy and existing uncoordinated inertia support methods. It was verified that the proposed coordinated inertia support strategy has stronger frequency support capacity, a faster response time, and the ability to suppress oscillations compared to alternative methods. Although the authors conducted a preliminary study on this method in the original conference paper [20], this paper delves into the transient characteristics and protection strategy under grid-side faults. Compared to [20], this study provides a comprehensive and detailed analysis of offshore wind farms connected to MMC-HVDC systems, examining them from multiple dimensions, including steady-state operation, fault transient characteristics, and inertia response control. The Basic Structure and Traditional Control Strategy of Offshore Wind Farms Connected with MMC-HVDC Systems  
Compared with a doubly fed induction generator (DFIG), a permanent magnetic synchronous generator (PMSG) does not need a gearbox and has better low-voltage ride-through performance. Therefore, they are widely used in the field of offshore wind power generation. An MMC-HVDC system consists of a DC cable, a wind-farm-side MMC (WFMMC), and a grid-side MMC (GSMMC), each of which contains an MMC and a connecting transformer. Figure 1 shows the structure of a PMSG-based wind farm connected to the grid through an MMC-HVDC system. The traditional grid-following control strategy of PMSG is described below [21]. The control goal of the grid-side converter are to maintain a constant DC-side capacitor voltage and ensure controllable reactive power of PMSG. The vector control method of grid voltage orientation is used to decouple the active current from the reactive current. The grid voltage vector, \(e_{g}\), is set on the \(d\)-axis of the two-phase synchronous rotation \(d/q\) coordinate system. The \(q\)-axis is obtained by rotating the \(d\)-axis 90 degrees counterclockwise. The following relationships exist:  
\[\left\{ \begin{array}{l}e_{gd}=E_{g}\\ e_{gq}=0\end{array} \right. \tag{1}\]  
where \(E_{g}\) is the amplitude of the grid voltage vector, and \(e_{g}\), \(e_{gd}\), and \(e_{gq}\) are the \(d\)-axis and \(q\)-axis components of the power grid voltage, \(e_{g}\), respectively. Figure 1: Structure of the PMSG-based wind farm connected to the grid through the MMC-HVDC system. The voltage equation of the grid-side converter based on voltage orientation in a synchronous rotating coordinate system can be obtained. \[\left\{ \begin{array}{l}v_{gd}=-i_{gd}R_{f}-L_{f}\frac{\mathrm{d}i_{gd}}{\mathrm{ d}t}+\omega_{ac}L_{f}i_{gq}+E_{g}\\ \\ v_{gq}=-i_{gq}R_{f}-L_{f}\frac{\mathrm{d}i_{gq}}{\mathrm{d}t}-\omega_{ac}L_{f} i_{gd}\end{array} \right. \tag{2}\]  
where \(v_{gd}\) and \(v_{gq}\) are the \(d\)-axis and the \(q\)-axis components of the output voltage of the grid-side converter, \(v_{g}\), respectively, \(i_{gd}\) and \(i_{gq}\) are the \(d\)-axis and \(q\)-axis components of the power grid current, \(i_{g}\), respectively, \(R_{f}\) and \(L_{f}\) are the equivalent resistance and equivalent inductance between the grid-side converter and the grid, respectively, and \(\omega_{ac}\) is the angular frequency of the AC collection line. At this time, the active power and reactive power output to the grid by the grid-side converter can be written as follows:  
\[\left\{