## 3 Control Mechanism for LCC-HVDC with a Synchronous Condenser
### Proposed Area Ratio of Commutation Failure Probability  
Previous studies have shown that different numbers of SCs can improve the ability of the LCC-HVDC system to resist commutation failure, but the method for quantitatively evaluating the suppression effect of SCs needs further study. A single- or three-phase fault occurring at the AC bus of the inverter side may cause a commutation failure to occur in the LCC-HVDC system. The occurrence of a commutation failure is not only sensitive to the severity of the AC fault, but also to the time-instant of when the fault occurs during one AC cycle. Thus, the commutation failure probability index (CFPI) in [11] is used to assess the chances of commutation failure occurrence in the LCC-HVDC system during one cyclic frequency. To obtain the commutation failure probability curve, different fault levels under specific fault times were considered. A single- or three-phase fault with inductance was applied at the AC busbar and the fault level was varied by changing the inductance value. A simultaneous fault level in the range of 100 fault points was applied while changing the fault time within one AC cycle duration (0.02 s). The multiple run mechanism in the PSCAD/EMTDC examined the occurrence of commutation failure and enlisted all related data. The probability of commutation failure for a specific fault was then evaluated by taking the ratio of faults that could cause commutation failure to the total equivalent fault points in one AC cycle. Multiple simulations were carried out to evaluate the probability for each fault level and to obtain the commutation failure probability curve. However, when the CFPI was used to quantitatively evaluate the effect of the SC on the commutation failure of the LCC-HVDC system, the evaluation results were only based on a specific fault. It could not fully and intuitively reflect the effect of the SC on the commutation failure of the LCC-HVDC system in a wider fault range and thus could not provide a more comprehensive evaluation for the commutation failure mitigation effect. By comparing the curves of the commutation failure probability before and after the implementation of the SC at the inverter side, it was found that the area surrounded by the curve of the commutation failure probability with the SC was smaller than that without the SC within the same fault range. Considering this, an area ratio of commutation failure probability that could cope with the shortcomings of CFPI was proposed. The flow-chart for calculating the area ratio of commutation failure probability is shown in Figure 5.  
Figure 5: Flow-chart of the area ratio of the commutation failure probability computing method. The detailed steps for calculating the area ratio of commutation failure probability are as follows:  
(1) At the inverter end, the fault with a setting inductance value was applied to the AC busbar and the initial fault time was set with the fault clearance duration. (2) When no SC was considered, the commutation failure probability of the LCC-HVDC system under a different fault inductance \(L\) was obtained. The fault inductance value was defined as \(L_{\text{no\_sc}}\), under which the commutation failure probability of the LCC-HVDC system was exactly 0 %. (3) Considering that SC was linked at the inverter end, the commutation failure probability of the LCC-HVDC system under a different fault inductance \(L\) was obtained. The fault inductance value was obtained as \(L_{\text{sc}}\), under which the commutation failure probability of the LCC-HVDC reached 100%. (When there were multiple SCs, the \(L_{\text{sc}}\) that could take in the maximum number of SCs was chosen). (4) With the fault inductance \(L\) as abscissa and the commutation failure probability as ordinate, the curve of commutation failure probability with and without SCs was formed as graphically presented in Figure 6. The area between the commutation failure probability curve and horizontal axis [\(L_{\text{sc}}\), \(L_{\text{no\_sc}}\)] was calculated.