## 4 DC Fault Current Analysis of Hybrid MMC Considering FRT Control
### Current Control Mode  
Combined with the hybrid MMC equivalent model, the fault current satisfies, as follows:  
\[L\frac{di_{dc}}{dt}+Ri_{dc}=U_{e}-U_{f} \tag{7}\]  
\[L=L_{e}+L_{dc},R=R_{e}+R_{dc} \tag{8}\]  
Figure 4: Hybrid MMC FRT control strategies: (**a**) current control; (**b**) voltage control. where \(U_{f}\) denotes the fault point DC voltage. Subject to bolted pole-to-pole DC line faults, \(U_{f}\) is immediately reduced, resulting in a step disturbance to the DC system. Thus, \(U_{f}\) is given as:  
\[U_{f}(s)=-\frac{U_{dcn}}{s} \tag{9}\]  
Under current-controlled operation, the outer-loop response becomes negligible. The current reference \(I_{dcref}\) is explicitly prescribed as depicted in Figure 4a. Through the complex frequency domain synthesis of (6)-(8), the representation of the hybrid MMC control loop is derived (Figure 5). Defining \(I_{dcn}\) as the rated DC current, \(U_{e}\) is characterized by:  
\[U_{e}(s)=\left[\frac{I_{dcref}}{s}-\frac{I_{dc}(s)}{I_{dcn}}\right]\left(k_{ pi}+\frac{k_{ii}}{s}\right)U_{dcn} \tag{10}\]  
According to Figure 5 and considering the initial DC current \(I_{0}\), the complex frequency domain representation of DC fault current is derived (see Appendix A.1). Subsequent inverse Laplace transformation yields the time domain expression for \(i_{dc}\). This study prioritizes low-impedance fault scenarios due to their paramount significance in control parameter design, where such faults generate the most severe current transients. Therefore, \(R\) is ignored in this analysis. Considering the transient characteristic of the fault circuit, the analytical expression of \(i_{dc}\) is discussed depending on different conditions. 1. When \(k_{ii}/k_{pi}^{2}>U_{dcn}/(4 LI_{dcn})\), the fault circuit shows underdamped characteristic, and \(i_{dc}\) is obtained as follows: \[i_{dc}(t)=I_{dcref}I_{dcn}+A_{1}e^{-\delta_{c}t}\sin(\omega_{c}t+\theta_{c})\] (11) \[\left\{ \begin{array}{l}A_{1}=2\sqrt{\frac{I_{dcn}(k_{ii}LX^{2}+U_{dcn}k_{pi}X +P_{dcn})}{4|k_{ii}L_{dcn}-k_{pi}^{2}U_{dcn}|}}\\ X=I_{dcn}I_{dcref}-I_{0},P_{dcn}=U_{dcn}I_{dcn}\\ \delta_{c}=\frac{k_{pi}U_{dcn}}{2 LI_{dcn}},\omega_{c}=\frac{1}{2 LI_{dcn}} \sqrt{U_{dcn}\Big{|}4k_{ii}LI_{dcn}-k_{pi}^{2}U_{dcn}\Big{|}}\\ \theta_{c}=\arctan\frac{-\omega_{c}k_{pi}}{\delta_{c}(k_{pi}X+2I_{dcn})}\end{array} \right.\] (12)
2. When \(k_{ii}/k_{pi}^{2}<U_{dcn}/(4 LI_{dcn})\), the fault circuit shows overdamped characteristic, and \(i_{dc}\) is obtained as follows: \[i_{dc}(t)=I_{dcref}I_{dcn}+A_{2}e^{(-\delta_{c}+\omega_{c})t}+A_{3}e^{(-\delta _{c}-\omega_{c})t}\] (13) \[\left\{ \begin{array}{l}A_{2}=\frac{1}{2}[\frac{\delta_{c}}{\omega_{c}}(X+ \frac{2I_{dcn}}{k_{pi}})-X]\\ A_{3}=-\frac{1}{2}[\frac{\delta_{c}}{\omega_{c}}(X+\frac{2I_{dcn}}{k_{pi}})+X] \end{array} \right.\] (14)  
Figure 5: Hybrid MMC control representation in current-regulated operation during DC faults. When \(k_{ii}/k_{pi}^{2}=U_{dcn}/(4 LI_{dcn})\), the fault circuit shows critical damped characteristic, and \(i_{dc}\) is obtained as follows:  
\[i_{dc}(t)=I_{dcref}I_{dcn}-Xe^{-\delta_{c}t} \tag{15}\]  
From (11)-(15), the DC fault current is composed of steady component \(I_{dcref}I_{dcn}\) and transient component. Taking the underdamped case as an example, Figure 6 illustrates the fault currents under different \(I_{dcref}\). The fault current rise rate and peak magnitude is proportional to \(I_{dcref}\). Post-fault, \(M_{dc}\) undergoes nonlinear attenuation. As the hybrid MMC continuously injects fault current until \(M_{dc}\) decays to zero, current peaking coincides with \(M_{dc}\) nullification. Reduced \(I_{dcref}\) accelerates \(M_{dc}\) attenuation, consequently suppressing fault current transients. Furthermore, fault clearance is achieved by setting \(I_{dcref}=0\). Under the \(I_{dcref}=0\) condition, fault current responses are modulated by the proportional and integral gains of the DC current PI controller.