## 2 Fault Analysis of Grid-Forming PV Systems  
The topology of a grid-forming PV inverter with its control structure is illustrated in Figure 1[22; 23]. Within inner-loop control, a selector enables the system to operate under either droop control (Dcontrol) or VSG control (Vcontrol). The key parameters in Figure 1 are defined as follows:  
* \(u_{\text{dc}}\): DC-link voltage of the PV inverter. * \(\text{C}_{\text{d}}\): DC-link capacitor of the PV inverter. * \(i_{\text{abc}}\): Output current (three-phase) of the PV inverter. * \(u_{\text{gabc}}\): Output voltage (three-phase) of the PV inverter. * \(u_{\text{abc}}\): Voltage (three-phase) at the PCC. * \(R_{\text{c}}\): Equivalent resistance between the inverter terminal and the PCC. * \(L_{\text{c}}\): Equivalent inductance between the inverter terminal and the PCC. * \(R_{\text{g}}\): Equivalent resistance of the grid side. * \(L_{\text{g}}\): Equivalent inductance of the grid side. Within the VSG control, the active power (\(P\))-frequency control loop emulates the rotor swing equation of synchronous generators to provide damping and inertia, which then delivers the phase angle (\(\theta_{\text{ref}}\)) of the virtual internal EMF. Correspondingly, the reactive power (\(Q\))-voltage control loop regulates reactive power and voltage characteristics by mimicking the excitation system of synchronous generators, outputting the magnitude (\(E_{\text{ref}}\)) of the virtual internal EMF. The control equations are illustrated below:  
\[\left\{ \begin{array}{l}P_{ref}-P_{e}-D\cdot\Delta\omega\ =\ J\frac{d^{2}\theta_{ref}}{ dt^{2}}\\ k_{q}(Q_{ref}-Q_{e})\ =\frac{d(E_{ref}-L_{\text{g}})}{dt}\end{array} \right. \tag{1}\]  
where \(k_{q}\) denotes the droop coefficient. The topology schematic of a grid-connected PV system is shown in Figure 2. The DC power from the PV source is inverted to AC by a grid-forming inverter, stepped up to 220 kV voltage level via a transformer, and finally delivered to the main grid through the transmission line. Figure 1: The topology and control schematics of a grid-forming PV inverter. Currently, various studies have analyzed the fault response of grid-forming PV inverters and derived analytical expressions for fault currents, particularly transient components [24; 25], which benefits the optimization of inverter controls and the protection adaptability analysis for transmission lines. By formulating the circuit equations alongside the equations derived from the control loops, the analytical expressions for short-circuit currents under both symmetrical and unsymmetrical faults are respectively shown in Equations (2) and (3) in the dq reference frame:  
\[ \begin{cases}i_{d}\ =\ k_{d0}\ +\ k_{d1}\ \times\ cos(-\omega t\ +\ \theta_{d1})e^{-t/ \tau_{\rm{v}}}\ +\ k_{d2}e^{-t/\tau_{\rm{1}}}\ +\ k_{d3}\ \times\ cos(-\omega t\ +\ \theta_{d2})e^{-t/\tau_{\rm{v}}}\\ i_{q}\ =\ k_{q0}\ +\ k_{q1}\ \times\ sin\bigl{(}-\omega t\ +\ \theta_{q1})e^{-t/ \tau_{\rm{v}}}\ +\ k_{q2}e^{-t/\tau_{\rm{1}}}\ +\ k_{q3}\ \times\ sin\bigl{(}-\omega t\ +\ \theta_{q2})e^{-t/ \tau_{\rm{v}}}\\ \end{cases}  \tag{2}\]  
\[ \begin{cases}i_{d}\ =\ N_{d1}e^{-m_{1}t}\ +\ N_{d2}+N_{d3}e^{-\tau_{\rm{v}}t} cos(\omega t+\theta_{1})\\ i_{q}\ =\ N_{q1}e^{-m_{1}t}\ +\ N_{q2}+N_{q3}e^{-\tau_{\rm{v}}t}sin(\omega t+ \theta_{2})\\ \end{cases}  \tag{3}\]  
where \(k_{d0}\sim k_{d3}\), \(k_{q0}-k_{q3}\) denote the current magnitude in the dq reference frame; \(\theta_{d1,2}\), \(\theta_{d1,2}\) denote the initial phase angle in the dq reference frame; and \(\tau_{\rm{v}}\), \(\tau_{1}\) denote the decay time constants related to the virtual impedance (\(Z_{\rm{v}}\)). \(m_{1}\), \(N_{d1}\sim N_{d3}\), \(N_{d1}\sim N_{d3}\), \(\theta_{1}\), \(\theta_{2}\) are expressions composed of factors such as virtual impedance inductance-to-resistance ratio, PCC voltage phase angle, etc., and will not be shown here due to their complexity. By performing the dq/abc coordinate transformation on Equations (2) and (3), the three-phase short-circuit currents are obtained.