## 3 Advanced FRT Strategy
### Unbalanced Power Calculation for DC Voltage Control  
A DC capacitor can store surplus electrical power. MMC HVDC can relieve the burden of the DFIG turbine. When the fault occurs at an on-shore site, both MMC HVDC and the rotor will store surplus power in the DC capacitor and the rotor mass. Assuming the capacitance of the DC line is much smaller than that of the MMC converter, the initial energy stored in DC can be calculated as:  
\[W_{init}=\frac{1}{2}\Big{(}2\cdot C_{eq}\Big{)}\theta_{c\Sigma,m}^{2} \tag{6}\]  
If an on-shore fault occurs, the MMC on-shore side cannot handle the amount of output power because of low on-shore voltage, so the difference from the off-shore power transfer amount will cause DC voltage variation as:  
\[P_{in}-P_{out}=\frac{dW_{init}}{dt}=2C_{eq}v_{c\Sigma,m}\frac{v_{c\Sigma,m}}{dt} \tag{7}\]  
When the power difference is persistent, energy stored in the DC capacitor can be calculated as:  
\[\frac{1}{2}C_{eq}\Big{(}v_{dc,post}^{2}-v_{dc,pre}^{2}\Big{)}=\Delta W=\int_{0 }^{t}(P_{in}-P_{out})dt \tag{8}\]  
Absorbing the unbalanced power during the fault can be dangerous because it can easily induce DC overvoltage. To prevent such a disaster, calculating the unbalanced power and off-shore power reference is required. Typical MMC on-shore side power \(P_{on}\) is depicted in Figure 4. There is response time for power system dynamics and remaining power during the fault because of the remaining voltage. The main purpose of the proposed control strategy was absorbing as much power as possible during the fault to the DC link, but not reaching the overvoltage limit. To do this, maximum off-shore power \(P_{off}\) should be calculated first. However, there is communication latency between MMC converters and response time for deloading, so those time delays should be considered.  
To calculate off-shore power reference \(P_{off,ref}\), total energy during the fault should be calculated, and the energy can be calculated by summing time interval energies. The energies at each interval are expressed as:  
Figure 4: Studied system model of the off-shore wind farm.  
\[t_{1} \sim t_{2}:W_{\Delta 1}=\frac{1}{2}t_{\text{res}}(P_{wind}-P_{on})-\frac{1}{2 }\frac{(t_{\text{res}}-t_{com})^{2}}{t_{\text{res}}}\Big{(}P_{wind}-P_{off}\Big{)} \tag{9}\] \[t_{2} \sim t_{3}:W_{\Delta 2}=\frac{1}{2}\frac{t_{com}^{2}}{t_{\text{res}}} \Big{(}P_{wind}-P_{off}\Big{)}+\frac{1}{2}t_{com}\Big{(}P_{wind}-P_{off}\Big{)}\] (10) \[t_{3} \sim t_{4}:W_{\Delta 3}=\Big{(}t_{fault}-t_{com}\Big{)}\Big{(}P_{off}-P_{on }\Big{)}\] (11) \[t_{4} \sim t_{5}:W_{\Delta 4}=t_{com}\Big{(}P_{off}-P_{on}\Big{)}-\frac{1}{2 }(P_{wind}-P_{on})\frac{t_{com}^{2}}{t_{\text{res}}}\] (12) \[t_{5} \sim t_{6}:W_{\Delta 5}=\frac{1}{2}\Big{[}\Big{(}P_{off}-P_{on} \Big{)}-\frac{t_{com}}{t_{\text{res}}}(P_{wind}-P_{on})\Big{]}(t_{\text{res}}- t_{com}) \tag{13}\]  
The total energy during the fault can be calculated by summing (8)-(12):  
\[W_{\Delta}=\Big{(}t_{fault}+t_{\text{res}}\Big{)}\Big{(}P_{off}-P_{on}\Big{)}+ t_{com}\Big{(}P_{wind}-2P_{off}+P_{on}\Big{)} \tag{14}\]  
Finally, the off-shore power reference can be calculated through (13). The proposed control strategy cannot be performed when \(P_{off,ref}\) is negative, which means too large communication latency typically above 50 ms.  
\[P_{off,ref}=\frac{W_{\Delta max}+\Big{(}t_{fault}+t_{\text{res}}-t_{com} \Big{)}P_{on}-t_{com}P_{wind}}{\Big{(}t_{fault}+t_{\text{res}}-2t_{com}\Big{)}} \tag{15}\]  
If losses are considered, more power can be allocated to the HVDC. However, for safe operation, not including those losses can be a margin for not intruding into the DC overvoltage. Moreover, converter loss calculation is complicated research and outside the scope of this paper.