## 3 Control Mechanism for LCC-HVDC with a Synchronous Condenser
### Application and Validation of the Area Ratio of Commutation Failure Probability
In order to indicate the severity of the three phases to ground fault, the minimum AC voltage RMS values at the LCC inverter side (without a SC) during the fault period, at different fault inductances, are summarized in Table 3.  
Table 4 summarizes the area of commutation failure probability and area ratio of commutation failure probability of the LCC-HVDC system with and without the SC under single-phase and three-phase inductive faults. Table 2 shows that when a single-phase fault was applied to the AC bus at the inverter side, the area of commutation failure probability and the area ratio of commutation failure probability both decreased continuously with the increasing number of SCs. The area ratio of commutation failure probability was 0.6939 and 0.5240 with one SC (Case 2) and two SCs (Case 3), respectively, which indicated an improvement in Case 3 as the area ratio of commutation failure probability was reduced by 0.1699 compared to Case 2. Considering the three-phase fault, the area ratio of commutation failure probability of the LCC-HVDC system with one SC at the inverter side was 0.4687. With two SCs linked to the LCC-HVDC system (Case 3), the area ratio of commutation failure probability was 0.1081, which by comparing with Case 2 revealed that an increasing number of SCs could effectively suppress the commutation failure under the three-phase fault. Thus it could be clearly stated that the proposed area ratio of commutation failure probability could comprehensively evaluate the effect of the SCs on the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Fault**Index**} & \multicolumn{2}{c}{**Area of Commutation Failure Probability**} & \multicolumn{2}{c}{**Area Ratio of Commutation Failure Probability**} \\ \cline{2-5}  & **Case 1 (no SC)** & **Case 2 (1 SC)** & **Case 3 (2 SCs)** & **Case 2 (1 SC)** & **Case 3 (2 SCs)** \\ \hline Single phase & 0.5949 & 0.4128 & 0.3117 & 0.6939 & 0.5240 \\ Three phase & 0.4348 & 0.2038 & 0.0470 & 0.4687 & 0.1081 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Values of area of commutation failure probability and area ratio of commutation failure probability under different types of faults. Figure 8: The probability of commutation failure (CF) in different cases under three phase fault. \begin{table}
\begin{tabular}{c c} \hline \hline
**Fault**Inductance/H** & **Minimum AC Voltage/p.u.** \\ \hline
0.7 & 0.863 \\
0.8 & 0.914 \\
0.9 & 0.925 \\
1.0 & 0.933 \\
1.1 & 0.940 \\
1.2 & 0.951 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The minimum AC voltage RMS values during the three phase to ground fault periods. commutation failure of the LCC-HVDC system under various fault types. In addition, the proposed index took into account various conditions of fault severities within a wider fault range, and was not limited to a specific fault. Comparing with existing indices such as the CFPI, the proposed index could quantitatively evaluate the impact of the SCs in suppressing the commutation failure of the LCC-HVDC system in a comprehensive and easy way. Under different system intensities indicated by the short circuit ratio (SCR), the effect of the SCs on the ability of the LCC-HVDC system to resist commutation failure in a certain fault range varied. Therefore, the single-phase to ground and the three-phase faults were applied respectively at the inverter bus, then the area ratio of commutation failure probability was used to measure the suppression effect of the SCs on commutation failure under different SCRs to obtain the results shown in Figures 9 and 10, respectively. It could be seen from Figure 9 that when a single-phase fault occurs at the AC bus of the inverter side, the area ratio of commutation failure probability in Case 2 and Case 3 were 0.4487 and 0.2893 respectively when SCR equalled 2; the area ratio of commutation failure probability in Case 2 and Case 3 were 0.8481 and 0.7246, respectively, when SCR equalled 6.