## 3 Impact of a Fault on the Grid-Connected SPV System  
This work considers a two-stage grid-connected SPV (Figure 1). The total power flow of the system can be defined as follows [29]:  
\[\mathrm{P_{spv}}=\mathrm{P_{DC~{}link}}+\mathrm{P_{g}} \tag{3}\]  
where \(\mathrm{P_{DC\_link}}\) is the DC link power, \(\mathrm{P_{g}}\) is the power inserted by the inverter to the grid, and \(\mathrm{P_{spv}}\) is the PV power. During normal operations:  
\[\mathrm{P_{spv}}=\mathrm{P_{g}}=3\mathrm{V_{gms}}\mathrm{I_{grms}} \tag{4}\]  
where \(\mathrm{V_{grms}}\) and \(\mathrm{I_{grms}}\) are the nominal RMS values of the phase voltage and phase current, respectively. Equation (4) shows that the DCL voltage can be maintained by keeping the PV array power (\(\mathrm{P_{spv}}\)) equal to the power inserted by the inverter to the grid (\(\mathrm{P_{g}}\)). We assume the power converter losses are neglected. However, under fault conditions, when voltage sag occurs on the PCC, it will not allow the power to be injected in the SPV system. The DC-DC boost converter using the MPPT control unit constantly obtains the maximum power from the SPV and injects it into the DCL. The inequality between the SPV-generated power and the power injected into the grid will sharply increase the DCL voltage. The fault condition is defined as follows:  
\[\mathrm{\left(P_{spv}-P_{gf}\right)}\Delta\mathrm{T}=\mathrm{P_{DCL}}\Delta \mathrm{T}= \frac{1}{2}\mathrm{C_{DC}}\mathrm{\left(V^{2}{}_{DCL\_f}-V^{2}{}_{DCL} \right)} \tag{5}\]  
where \(\mathrm{P_{gf}}\) is the grid power at the fault, \(\mathrm{V_{DCL}}\) is the DCL voltage before the fault, \(\mathrm{V_{DCL\_f}}\) is the DCL voltage at the fault period, \(\Delta\mathrm{T}\) is the fault period, and \(\mathrm{C_{DC}}\) is the DCL capacitance. \(\mathrm{V_{DCL\_f}}\) is given as:  
\[\mathrm{V_{DCL\_f}}=\sqrt{(2(\mathrm{P_{SPv}}-3\mathrm{V_{g}}\mathrm{I_{g}}) \Delta\mathrm{T}))/\mathrm{C_{DC}}+V_{DCL}^{2}} \tag{6}\]  
Equation (6) shows that the DCL voltage rise rate depends on the voltage sag level (\(\mathrm{V_{g}}\) goes low during a fault) and the fault period (\(\Delta\mathrm{T}\)). In the case of severe fault conditions, the voltage drops as the PCC decreases to a lower level, and longer fault duration causes a higher rise rate for the generated DCL voltage. Thus, without any protection scheme under any type of fault, it may violate the DCL over-voltage standard. Figure 2: Direct current–direct current (DC–DC) boost converter circuit. MOSFET: metal-oxide-semiconductor field-effect transistor. _Low Voltage Ride-Through (LVRT) Capability of the SPV System_  
According to German E.ON grid code shown in Figure 3a, the SPV system should not be disconnected from the grid when the PCC voltage falls down to zero percentage of the nominal voltage for a period less than or equal to 0.15 s, and should recover to 90% of the voltage from its pre-fault level within 1.5 s from the occurrence of the voltage sag [30]. In addition, SPV systems should inject reactive current under fault conditions, depending on the voltage sag range and the inverter current rating capacity, to provide dynamic grid-voltage support according to the profile given in Figure 3b. According to the grid code (GC), the amount of reactive current that should be injected into the grid during voltage sag can be calculated by the following:  
\[\mathrm{Iq}= \begin{cases}\mathrm{dead\ band},&0.9\ \mathrm{p.u}\ \leq\ \mathrm{V}\ \leq\ 1.1\ \mathrm{p.u}\\ \mathrm{k}\ \frac{\mathrm{V}-\mathrm{V}_{0}}{\mathrm{V}\mathrm{N}}\ \mathrm{I}_{\mathrm{N}}+\mathrm{I}_{\mathrm{q} \mathrm{o}},&0.5\ \mathrm{p.u}\ \leq\ \mathrm{V}<0.9\mathrm{p.u}\\ \mathrm{I}_{\mathrm{N}}+\mathrm{I}_{\mathrm{q}\mathrm{o}}&\mathrm{V}\ \leq\ 0.5\ \mathrm{p.u}\end{cases}  \tag{7}\]  
where \(\mathrm{V}\), \(\mathrm{V}_{\mathrm{o}}\) and \(\mathrm{V}_{\mathrm{N}}\) are the amplitude values of the grid instantaneous voltage, the initial voltage before the fault, and the nominal grid voltage, respectively.