## 3 Proposed Coordination Inertia Control Strategy
### Coordination Inertia Control Strategy of MMC-HVDC  
The capacitance of the MMC-HVDC system is present in the individual sub-modules. Suppose the upper or lower arm of the MMC has \(N\) sub-modules for each phase, and the capacitance value of one sub-module is \(C_{sm}\). Therefore, the capacitance of the MMC submodule can be uniformly equivalent to the DC side, which can be regarded as a capacitor with a voltage of \(U_{dcm}\) and a size of \(12C_{sm}/N\). This capacitance \(C_{eq}\) is the equivalent  
Figure 5: Mapping in inertial synchronization control.  
capacitance on the DC side of the converter station at the receiving end of the MMC structure. The dynamic characteristics of DC capacitors can be expressed as:  
\[C_{eq}\overline{U}_{dcm0}\frac{d\overline{U}_{dcm}}{dt}=\overline{P}_{WF}- \overline{P}_{gm} \tag{13}\]  
\(\overline{U}_{dcm0}\) is the per unit rated DC voltage of the MMC-HVDC system; \(\overline{U}_{dcm}\) is the per unit DC voltage of the MMC-HVDC system; \(\overline{P}_{WF}\) is the per unit active power emitted by the sender; and \(\overline{P_{sd}}\) is the per unit active power transmitted to the grid side. To simulate the inertial time constant of a synchronous generator, as shown in Equation (7), \(\overline{P}_{WF}\) is similar to the mechanical power of the generator. \(\overline{P_{sd}}\) is similar to the electromagnetic power of a generator. So, the varying DC voltage is similar to the varying speed of a synchronous machine. Equation (14) can be obtained by combining Equations (7) and (13).  
\[C_{eq}\overline{U}_{dcm0}\frac{d\overline{U}_{dcm}}{dt}=2H_{dc}\frac{df}{dt} \tag{14}\]  
\(H_{dc}\) is the equivalent time constant of the inertia provided by the MMC-HVDC; and \(\overline{f}\) is the frequency of the system. Both ends of Equation (14) can be integrated to obtain Equation (15).  
\[\frac{C_{eq}\Big{(}\overline{U}_{dcm}^{2}-\overline{U}_{dcms}^{2}\Big{)}}{2}= 2H_{dc}\Big{(}\overline{f}-\overline{f_{s}}\Big{)} \tag{15}\]  
\(\overline{U}_{dcms}\) and are the equilibrium points for the DC voltage and grid frequency, respectively. Since the DC voltage fluctuates in a small range, the first-order Taylor deployment is used at \(\overline{U}_{dcms}\), which has the following:  
\[C_{eq}\overline{U}_{dcms}\Delta\overline{U}_{dcm}=2H_{dc}\Delta\overline{f} \tag{16}\]  
Therefore, the GSMMC can use DC voltage slip control:  
\[\overline{U}_{dcm}^{*}=K_{1}\Delta\overline{f}+\overline{U}_{dcm0} \tag{17}\]  
where \(K_{1}\) is the control parameter, which can be artificially selected based on the inertia time constant provided, \(\overline{U}_{dcm}^{*}\) is the reference value of the DC voltage in the GSMMC control link.  
The proposed coordinated control strategy necessitates the wind farm's ability to perceive changes in the grid-side frequency. However, direct communication of frequency signals is unreliable for long-distance transmission in offshore wind farms. The WFMMC can work under frequency variation control, so through the DC voltage of the WFMMC, the frequency of the WFMMC and the frequency of the grid can establish a certain connection and realize the artificial coupling relationship. Therefore, the frequency variation control of the WFMMC can be written as:  
\[\overline{f}_{WF}^{*}=K_{2}\Delta\overline{U}_{dcm}+\overline{f}_{WF0} \tag{18}\]  
where \(K_{2}\) is the control parameter; and \(\overline{f}_{WF}^{*}\) and \(\overline{f}_{WF0}\) are the per unit frequency reference and initial frequency value of the WFMMC.