## 3 Analysis of High Resistance Fault  
The voltage source converter is an important component of a DC microgrid. If a DC line fault occurs, it is propagated through the capacitor discharge, diode free conduction and power grid side-feed flow stages. Most HRFs are single-pole grounding faults; thus, one should focus on these faults. A schematic of the capacitor discharge stage is shown in Figure 1.  
As in the capacitor discharge phase, capacitor \(C\) discharges after a fault occurs; the initial voltage \(V_{p}\) is equal to the normal line-end voltage; \(I_{c}\) is the capacitor discharge current; \(R\) and \(L\) are the equivalent resistance and line equivalent inductance from the capacitor to the fault point, respectively; \(R_{\text{f}}\) is the fault resistance; and \(I_{\text{f}}\) is the current flow. The fault resistance current at this stage is the capacitive discharge current. The time-domain expression of the discharge current is given by:  
\[i(t)=\frac{V_{c}(0)}{L(s_{2}-s_{1})}[e^{-s_{1}t}-e^{-s_{2}t}]+\frac{i_{L}(0)}{ s_{2}-s_{1}}[-s_{1}e^{-s_{1}t}+s_{2}e^{-s_{2}t}] \tag{8}\]  
\(s_{1},s_{2}\) is given by:  
\[s_{1,2}=-\alpha\pm\sqrt{\alpha^{2}-\omega_{0}^{2}} \tag{9}\]  
The damping coefficient \(\alpha\) and resonance frequency \(\omega_{0}\) are given by Equations (10) and (11), respectively:  
\[\alpha=\frac{R+R_{\text{f}}}{2L} \tag{10}\]  
\[\omega_{0}=\frac{1}{\sqrt{LC}} \tag{11}\]  
Substituting arc resistance into the Equation (10), yields Equation (12):  
\[\alpha=\frac{R+\text{e}^{\frac{1}{\alpha}\int{(1-\frac{\pi\epsilon^{2}}{E^{2 }})\text{d}t}}}{2L} \tag{12}\]  
After HRFs, the fault response curves are overdamped, the transient response times are short, and the peak responses are small, unlike after metallic and low-resistance faults. During the grid-side current-feeding stage, the AC side transmits current via the voltage source converter bridge arm to the DC fault; the equivalent circuit is shown in Figure 2.  
Figure 1: Schematic diagram of the capacitor discharge stage during high resistance fault. (**a**) capacitor discharge stage circuit; (**b**) equivalent diagram.  
Figure 2: Schematic diagram of the power grid side feeding stage.  
If an HRF persists, the converter will lock up, then becoming equivalent to an uncontrolled rectifier bridge. The current at the fault point is the sum of the diode bridge arm currents, which can be written as:  
\[i_{VSC}=i_{D1}+i_{D2}+i_{D3}=i_{\text{ga}}+i_{\text{gb}}+i_{\text{gc}} \tag{13}\]  
where \(i_{\text{ga}}\), \(i_{\text{gb}}\), and \(i_{\text{gc}}\) are the three-phase currents of AC, respectively, i.e., the current passing through the diode bridge arm when a positive value is selected. Taking phase A as an example, Equations (14)-(17) are applied:  
\[i_{\text{ga}}=I_{\text{g}}\sin(\omega_{\text{s}}t+\alpha-\varphi)+I_{\text{gn} }e^{\frac{-t}{\tau}} \tag{14}\]  
\[\varphi=\arctan\biggl{[}\frac{\omega_{\text{s}}(L_{\text{ac}}+L)}{R+R_{\text{ f}}}\biggr{]} \tag{15}\]  
\[\tau=\frac{L_{\text{ac}}+L}{R+R_{\text{f}}} \tag{16}\]  
\[I_{\text{gn}}=\Bigl{[}I_{\text{g}|0}\sin(\alpha-\varphi_{0})-I_{\text{g}}\sin (\alpha-\varphi)\Bigr{]} \tag{17}\]  
where \(I_{\text{g}|0}\) is the amplitude of the initial grid current; \(\varphi_{0}\) is the impedance angle; \(L_{\text{ac}}\) is the AC-side inductance; \(\alpha\) is the AC phase angle; and \(I_{\text{g}}\) is the amplitude of the periodic component of the short-circuit current. After an HRF, the zero-mode current is extracted, and the transient zero mode current is calculated as follows:  
\[i_{0}=\frac{\sqrt{2}}{2}(i_{\text{1p}}+i_{\text{1n}}) \tag{18}\]  
The frequency domain component of the current in the Cassie DC arc model decreases with increasing frequency. After the arc current is superimposed, the amplitude of the corresponding frequency band changes. Thus, phase decomposition of the low frequency band reveals the amplitude change, allowing arc resistance fault detection with a low arc current.