## 2 Materials and Methods
### Theoretical Analysis of the Short-Circuit Current  
For verifying the dynamic and thermal effects of short-circuit current and identifying whether it will damage the equipment and destroy the system safety, it is only necessary to verify the situation that the short-circuit current is the maximum, i.e., the line suddenly experiences a short-circuit fault when there is no load. Figure 1: Diagram of a DC power distribution system in a building. Figure 2: Equivalent model with lumped parameters. In this subsection, a prospective short-circuit current is reviewed while ignoring the influence of protective devices. Protective devices reduce the amplitude of the fault current. As a result, the prospective short-circuit current is a necessary basis for Section 3. Two-level voltage-source AC-DC converters (VSC) are commonly used in the DC distribution in buildings, and a protective circuit blocks the insulated gate bipolar transistor (IGBT) within 1 ms when a short circuit occurs. For a DC system, the short-circuit current is the highest and the situation is the most serious when a pole-pole cable short circuit occurs, and the equivalent model of a VSC and fault circuit is shown in Figure 3.  
R\({}_{g}\) and L\({}_{g}\) are the total resistance and inductance of the AC-link, including the resistance and inductance of the grid and transformer, as well as the filter inductance of the VSC. C is the capacitor on the DC-link of the VSC. R and L are the total resistance and inductance of the DC-link conductors within the fault circuit, including the bus-bars and cables. The variation process of the short-circuit current i\({}_{s}\) goes through 2 or 3 stages depending on the resistance, capacitance, and inductance of the fault circuit [21]. 3.1 Under the Condition R \(<\) 2\(\sqrt{\frac{\text{L}}{\text{C}}}\), When It Is Under-Damped, the Short Circuit Current Goes through Three Stages [21]  
Stage 1: At the moment in which a short circuit occurs, as the impedance of the DC-link plummets, the AC-link voltage is lower than the DC-link voltage, and the diodes are cut off. The capacitor is discharged in an oscillation. The equivalent circuit is shown in Figure 4a. It can be seen that the capacitor discharging current i\({}_{c}\) (i.e., i\({}_{s}\)) rapidly increases, and the capacitor voltage decreases. The state equation and its solution are shown in the following equations:  
\[\text{LC}\frac{\text{d}^{2}\text{V}\text{c}}{\text{dt}^{2}}+\text{RC}\frac{ \text{d}\text{V}\text{c}}{\text{dt}}+\text{V}\text{c}=0 \tag{1}\]  
\[\text{i}_{\text{C}}=\text{C}\frac{\text{d}\text{V}\text{c}}{\text{dt}}= \frac{\text{V}_{0}}{\omega\text{L}}\text{e}^{-\delta\text{t}}\text{sin}( \omega\text{t}) \tag{2}\]  
\[\delta=\frac{\text{R}}{2\text{L}},\ \omega^{2}=\frac{1}{\text{LC}}-\left( \frac{\text{R}}{2\text{L}}\right)^{2}\]  
where V\({}_{0}\) is the initial voltage of the capacitor. The time for the discharge current to reach its peak is:  
\[\text{t}_{\text{P}}=\beta/\omega,\ \beta=\arctan\!\left(\frac{\omega}{\xi} \right) \tag{3}\]  
Figure 3: Equivalent model of a VSC and short-circuit fault circuit. The peak of the discharge current \(\rm i_{p}\), i.e., the peak of the short-circuit current, can be obtained by bringing Equation (3) into Equation (2). Stage 2: When the capacitor voltage drops to zero, the diodes are turned on under the action of inductor freewheeling. The AC-link is equivalent to a three-phase short circuit, and the DC-link is an R-L first-order circuit; it is noted that the capacitor voltage at the moment is zero. The equivalent circuit in this stage is shown in Figure 4b. How long this stage lasts depends on the attenuation of the inductor freewheeling; this is difficult to calculate with a formula and can only be roughly determined through simulation. Stage 3: The equivalent circuit of the entry into a state of uncontrolled rectification and a tendency toward the steady state is shown in Figure 4c; this is a third-order circuit.