## 3 Control Mechanism for LCC-HVDC with a Synchronous Condenser
### Application and Validation of the Area Ratio of Commutation Failure Probability  
In order to evaluate the suppression effect of the SC on the commutation failure of the LCC-HVDC system within a fault range, and to verify the proposed area ratio of commutation failure probability index under single-phase and three-phase inductive faults, the simulations were conducted when considering different numbers of SCs and different short circuit ratios (SCRs) at the inverter side, based on the electromagnetic transient model of the LCC-HVDC system with SCs. The area of commutation failure probability in the three cases mentioned above were calculated using complex trapezoidal formula as in (2) and (3):  
\[S=\frac{h}{2}[f(L_{\text{no\_sc}})+f(L_{\text{sc}})]+h\sum_{k=1}^{n-1}f(L_{k}) \tag{2}\]  
Figure 6: Schematic presentation of area of commutation failure probability with and without a SC. \[L_{k}=L_{\text{sc}}+k*h_{i}\left(k=1,2.\ldots\ldots n-1\right) \tag{3}\]  
where,  
_h_: step change of fault inductance in the simulation;  
\(L_{k}\): fault inductance in [\(l_{\text{sc}}\), \(L_{\text{no\_sc}}\)] interval;  
\(f(L_{k})\): the commutation failure probability corresponding to the fault inductance \(L_{\text{k}}\). \(S\): the area of commutation failure probability;  
\(n\): the total number of fault inductance level in [\(L_{\text{sc}}\), \(L_{\text{no\_sc}}\)];  
The LCC-HVDC system was assumed to be operating at a rated nominal value (DC power = 1 p.u., DC voltage =1 p.u.). The short circuit ratios (SCRs) of the AC system at the rectifier and the inverter side of the LCC-HVDC system were both set at 2.5. The curve for the commutation failure probability of the LCC-HVDC system considering the three cases mentioned above during the single-phase inductive fault with changing inductance level is depicted in Figure 7. It is obvious from Figure 7 that in Case 3 with two SCs at the inverter side, the commutation failure probability was 100% for the single-phase fault with an inductance level of 0.25 H. Whereas in Case 2, the commutation failure probability was 100% for the fault inductance level of 0.35 H. Considering Case 1 without the SC, the commutation failure probability was 0% for the single-phase inductive fault with a 0.97 H inductance value. Using these values, the area ratio of commutation failure probability for the three cases under single-phase inductive fault was calculated using Equations (1)-(3). In order to indicate the severity of a single phase to ground fault, the minimum AC voltage RMS values at LCC inverter side (without a SC) during the fault period, at different fault inductances, are summarized in Table 2.  
Figure 8 shows the curve built for the commutation failure probability of the LCC-HVDC system under the three phase-ground inductive faults with an inductance range between 0.7 H and 1.4 H while assuming the three cases. It could be depicted that with two SCs, the commutation failure probability \begin{table}
\begin{tabular}{c c} \hline \hline
**Fault Inductance/H** & **Minimum AC Voltage/p.u.** \\ \hline
0.2 & 0.868 \\
0.3 & 0.904 \\
0.4 & 0.920 \\
0.5 & 0.934 \\
0.6 & 0.943 \\
0.7 & 0.949 \\
0.8 & 0.953 \\
0.9 & 0.956 \\
1.0 & 0.960 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The minimum AC voltage RMS values during the single phase to ground fault period. Figure 7: The probability of commutation failure (CF) in different cases under single phase to ground fault. was 100 % under a fault inductance level of 0.78 H. The commutation failure probability with one SC (Case 2) was 100% for a fault inductance value of 0.91 H, whereas the commutation failure probability without the SC was 0% for a fault inductance level of 1.28 H. Considering these values in the three different cases, the area ratio of the commutation failure probability of the LCC-HVDC system under three-phase fault was calculated by applying Equations (1)-(3).