## 5 Simulation Verification
### Verification with the Current FRT Control  
During electromagnetic transient simulations, a predefined DC bus fault (\(F_{1}\)) is imposed at MMC\({}_{1}\)'s output port. The hybrid MMC\({}_{1}\) adopts a current regulation strategy with reference parameter \(I_{\text{dcrf}}\) = 1.0. Figure 13 shows the calculated results of \(I_{1}\) with constant \(I_{\text{dcrf}}\) and different \(k_{pi}\) and \(k_{ii}\). They are compared with the simulation results. The values of \(k_{pi}\) and \(k_{ii}\) are set to 1 and 40 (overdamped circuit), 0.5 and 30 (underdamped circuit), and 0.7 and 44 (critical damped circuit), respectively.  
Figure 12: Diagram of two-terminal monopolar hybrid MMC-HVDC system.  
Figure 13: Comparative profiles of calculated versus simulated \(I_{1}\) under constant \(I_{\text{dcrf}}\) with varying PI gains (\(k_{pi}\), \(k_{ii}\)) during DC fault \(F_{1}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Converter** & **Hybrid MMC\({}_{1}\)** & **Hybrid MMC\({}_{2}\)** \\ \hline Rated DC voltage (kV) & 500 & 500 \\ Rated DC current (kA) & 3 & 3 \\ Arm inductance (mH) & 100 & 50 \\ Current limiting reactor (mH) & 200 & 200 \\ SM capacitor (mF) & 15 & 15 \\ Number of SMs per arm & 244 & 244 \\ PI coefficients of DC current controller & 1(\(k_{pi}\)), 40(\(k_{ii}\)) & 0.2(\(k_{pi}\)), 20(\(k_{ii}\)) \\ PI coefficients of DC voltage controller & - & 2(\(k_{pi}\)), 100(\(k_{ii}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Hybrid MMC parameters.  
As demonstrated in Figure 13, the developed method achieves high-fidelity representation of DC fault transient characteristics. This verification substantiates the theoretical foundations of the quantitative fault current calculation methodology.  
The validity of the proposed framework under FRT control strategy is further substantiated through systematic analysis. Under steady-state operating conditions, MMC\({}_{1}\) maintains active power regulation functionality. Following a 2 ms delay in fault detection, the current FRT control mechanism is initiated, prompting the converter to transition into closed-loop fault current regulation mode with its current reference value nullified. Figure 14 presents a comparative analysis between the simulated and estimated \(I_{1}\) within the initial 50 ms timeframe. The proposed model is also compared with the model without considering control saturation and the conventional models discussed in Section 2 to verify its performance and improvement over other methods. Figure 15 illustrates the corresponding responses of both the DC current PI controller and modulation index \(M_{dc}\).  
The results demonstrate superior alignment between calculation values obtained by proposed method and detailed simulations, as evidenced in Figure 14. Conversely, conventional modeling approaches yield substantial errors in fault current estimation owing to their inability to represent the FRT control response of hybrid MMC. The activation of FRT control induces a rapid saturation of \(M_{dc}\) in the hybrid MMC, as depicted in Figure 15. This transient phase coincides with peak fault current magnitude, followed by progressive current attenuation attributable to diminished MMC output voltage. Within 36 ms after the FRT activation, the fault current exhibit controller-independent characteristics until \(M_{dc}\) re-enters its operational domain. Subsequent to this recovery phase, the current controller effectively drives the residual fault current to complete extinction. A comparative analysis reveals that neglecting control saturation constraints in the calculation process leads to  
Figure 14: Calculation and simulation values of \(I_{1}\) considering current FRT control. \(I_{dcrf}\) is set to 0 after the current FRT control is activated.  
Figure 15: Outputs of the DC current PI controller corresponding to Figure 14.  
significant discrepancies in fault current calculations. This verification underscores the importance of incorporating control saturation in accurate fault current prediction.