## 3 Modeling of Hybrid MMC During DC Fault Conditions  
This section develops an equivalent representation for hybrid MMCs during DC fault scenarios. Considering that parameter design and protection coordination must account for worst-case current, this paper concentrate on the metallic short-circuit fault and propose a fault current analysis method. Diverging from conventional approaches, the proposed framework models the converter as an RL circuit coupled with a time-variant controlled voltage source. This formulation characterizes the influence of control parameters on DC fault current response.  
For each phase \(j\) (\(j=a,b,c\)), the governing equation depicted in Figure 1 is expressed as follows:  
\[L_{a}\frac{di_{pj}}{dt}+R_{a}\big{(}i_{pj}+i_{nj}\big{)}+L_{a}\frac{di_{nj}}{dt }=\big{(}u_{pj}+u_{nj}\big{)}-U_{dc} \tag{1}\]  
where \(L_{a}\) and \(R_{a}\) denote arm inductance and resistance, respectively, \(i_{pj}/i_{nj}\) indicate upper/lower arm currents, \(U_{dc}\) designates DC bus voltage, and \(u_{pj}/u_{nj}\) represent upper/lower arm voltages. Aggregating three phases, while incorporating the identity \(i_{pa}+i_{pb}+i_{pc}=i_{na}+i_{nb}+i_{nc}=i_{dc}\), (1) reduces to the following:  
\[\frac{2}{3}L_{a}\frac{di_{dc}}{dt}+\frac{2}{3}R_{a}i_{dc}=U_{e}-U_{dc} \tag{2}\]  
where \(i_{dc}\) is the current of DC side. The equivalent topology of the hybrid MMC is characterized by defining the composite voltage \(U_{e}=u_{pa}+u_{na}=u_{pb}+u_{nb}=u_{pc}+u_{nc}\), with the resultant configuration depicted in Figure 3. The equivalent parameters are expressed as follows:  
\[L_{e}=\frac{2L_{a}}{3},R_{e}=\frac{2R_{a}}{3} \tag{3}\]  
As depicted in Figure 3, \(U_{e}\) denotes the aggregate DC component of combined upper/lower arm output voltages, dynamically modulated by control systems. Given the decoupled regulation of AC and DC voltages in hybrid MMCs, DC current analysis requires exclusive consideration of the DC control loop response [30].  
Within the DC control infrastructure of hybrid MMCs, the outer loop establishes the DC current reference (\(I_{dc\prime}\)) through the following three operational modalities: DC voltage stabilization, active power regulation, or maintenance of mean SM capacitor voltage. Then, \(I_{dc\prime}\) undergoes comparative analysis with the monitored DC current (\(i_{dc\prime}\)), with the resultant error signal being processed by a PI controller to yield the DC modulation index  
Figure 3: Proposed DC-side equivalent model for hybrid MMCs under the DC short-circuit fault.  
(\(M_{dc}\))--a critical parameter for arm voltage reference computation. Governed by the hybrid MMC control strategy, \(M_{dc}\) is obtained as follows:  
\[M_{dc}=k_{pi}\Big{(}I_{dcref}-i_{dcpu}\Big{)}+k_{ii}\int\Big{(}I_{dcref}-i_{dcpu }\Big{)}dt \tag{4}\]  
As reported in [32], \(U_{e}\) is expressed as follows:  
\[U_{e}=U_{armdcn}M_{dc} \tag{5}\]  
where \(U_{armdcn}\) represents the DC voltage component of arm voltages, equivalent to the rated DC voltage level (\(U_{armdcn}=U_{dcn}\)). Subsequently, \(U_{e}\) can be further obtained as follows:  
\[U_{e}=U_{dcn}\Big{[}k_{pi}\Big{(}I_{dcref}-i_{dcpu}\Big{)}+k_{ii}\int\Big{(}I_ {dcref}-i_{dcpu}\Big{)}dt\Big{]} \tag{6}\]  
where \(k_{pi}\) and \(k_{ii}\) denote the proportional and integral coefficients of the DC current PI controller. Equation (6) reveals that \(U_{e}\) is influenced by the following two principal factors: the parametric configuration of the PI controller and outer-loop-regulated current references. Furthermore, fault current induces feedback modulation of \(U_{e}\). This bidirectional coupling establishes an analytical mapping between transient fault current characteristics and control system parameters during DC faults, thereby enabling the hybrid MMC equivalent model to quantify control-transient interdependencies.