## 3 Proposed Coordination Inertia Control Strategy
### Coordination Inertia Control Strategy of Wind Farms  
There are many control methods for direct-drive wind turbines to respond to grid frequency changes, and this paper focuses on inertial synchronous control. The mechanical equation of motion of the rotor of a permanent magnet synchronous motor is expressed as [22]:  
\[J\frac{d\overline{\omega_{r}}}{dt}=\overline{T}_{m}-\overline{T}_{e}-D(\overline {\omega}_{r}-\overline{\omega}_{0}) \tag{5}\]  
In Equation (5), \(J\) is the inertia of the PMSG. \(D\) is the damping coefficient. \(\overline{T_{m}}\) is the per unit mechanical torque, i.e., the input torque of the wind farms. \(\overline{T_{e}}\) is the per unit electromagnetic power. \(\overline{\omega_{r}}\) is the per unit rotor speed, and \(\overline{\omega_{0}}\) is the per unit rated speed. The DC bus voltage equation of the wind power converter is [23]:  
\[C_{w}\overline{\omega}_{d\omega 0}\frac{d\overline{\psi}_{d\omega}}{dt}= \overline{P}_{m}-\overline{P}_{WF} \tag{6}\]  
In Equation (6), \(\overline{P_{m}}\) is the per unit output power of the rectifier on the machine side; \(\overline{P_{WF}}\) is the per unit output power of the inverter on the grid side; \(\overline{\upsilon}_{d\omega 0}\) is the per unit rated DC voltage of the converter; \(\overline{\upsilon}_{d\omega}\) is the per unit DC voltage of the converter; and \(C_{w}\) is the capacitance value of the DC bus line. Ignoring the damping effect, the mathematical model of the SG rotor motion in Equation (5) and the DC-side capacitance in Equation (6) can be written as follows:  
\[2H_{J}\overline{\omega}_{0}\frac{d\overline{\omega}_{r}}{dt}=\overline{P}_{M} -\overline{P}_{E} \tag{7}\]  
\[2H_{C}\overline{\upsilon}_{d\omega 0}\frac{d\overline{\upsilon}_{d\omega}}{dt}= \overline{P}_{m}-\overline{P}_{WF} \tag{8}\]  
In the above, \(\overline{P_{M}}\) is the per unit input mechanical power of the prime mover of the permanent magnet synchronous motor, \(\overline{P_{E}}\) is the per unit output electromagnetic power of the permanent magnet synchronous motor; \(H_{J}\) is the inertia time constant of the permanent magnet synchronous motor, and \(H_{C}\) is the capacitance inertia time constant of the DC side. In the case of ignoring the loss and line resistance, can \(\overline{P_{WF}}\) be written as the following expressions [24]:  
\[\overline{P}_{E}=\frac{\overline{\omega}_{r}\overline{\psi}_{r}\overline{F}_{ g}}{\overline{X}_{G}}\sin\delta_{G} \tag{9}\]  
\[\overline{P}_{WF}=\frac{\overline{\upsilon}_{d}\overline{\upsilon}_{g} \overline{E}_{g}}{\overline{X}_{L}}\sin\delta_{WF} \tag{10}\]  
where \(\overline{\psi}_{t}\) is the per unit rotor flux of the permanent magnet synchronous motor; \(\overline{E_{g}}\) is the per unit amplitude of the grid voltage vector; \(\delta_{G}\) is the power angle of the synchronous motor; and \(\delta_{WF}\) is the angle between the voltage vector of the output of the grid-side converter and the voltage vector of the grid-side converter, which can be called the power angle of  
Figure 4: The overall control block diagram of the coordination inertia control strategy. the grid-side converter. \(\overline{v_{g}}\) is the per unit amplitude of the voltage vector at the output of the grid-side converter. \(\overline{X}_{G}\) is the per unit equivalent reactance of the permanent magnet synchronous motor. \(\overline{X}\) is the per unit sum of the line reactance and the filter reactance. From the similarity between Equations (7)-(10), it can be seen that the DC-side capacitor has similar dynamic characteristics with the speed of the synchronous motor. The increase in the power angle leads to an increase in the output power on the grid side, which, in turn, affects the DC voltage or speed. If the motor speed scale is equal to the DC-side capacitance unit value, the mapping relationship shown in Figure 5 can be formed. This mapping method makes the DC-side capacitor voltage and the frequency of the offshore MMC correlated, that is, the capacitor voltage can reflect the frequency change in the offshore MMC in real time.