## 2 Short-Circuit Calculations
### Short-Circuit Flow Constraints  
This subsection describes the fault-induced current flows in the protection scheme formulation. Given a set of faults \(\mathcal{F}\), the formulation calculates a fault current and voltage for each \(f\in\mathcal{F}\) fault. The fault current is used to determine whether the scheme responds correctly and restricts the networked microgrid to operating points that are protected during each fault. During a fault, synchronous generators are one of the largest sources of injected currents. The current that a certain generator supplies is a function of the pre-fault voltage and the unit's transient reactance. The following constraint defines this:  
\[\forall f\in\mathcal{F},\;\forall g\in\mathcal{G}_{i}: \tag{1}\]  
where \(V_{g}^{\Phi}\) is the internal (pre-fault) voltage and \(x_{g}\) is the transient reactance of \(g\) generator. This constraint defines the terminal voltage of \(g\) as equal to the internal voltage minus the voltage drop created by the injection of the unit and its transient reactance. In the following, this \(I_{gf}^{\Phi}\) current is constrained by grid following and grid-forming inverter limits. Each \(f\in\mathcal{F}\) fault is defined by the bus where it occurs, \(i_{f}\), and by its admittance matrix, \(\mathcal{G}_{f}\). The elements of \(\mathcal{G}_{f}\) depend on the fault type: phase-to-ground or phase-to-phase. The general constraint for calculating fault current at bus \(i_{f}\):  
\[I_{is_{f}f}^{\Phi}=\mathcal{G}_{f}V_{i_{f}}^{\psi}. \tag{2}\]Next, the specifics of \(\mathbf{G}_{f}\) for each fault type are discussed. For a single line-to-ground fault, \(\mathbf{G}_{f}\) is defined as:  
\[\mathbf{G}_{f}= \begin{bmatrix}\mathbf{\mathrm{g}}_{f}&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix} . \tag{3}\]  
where \(\mathbf{\mathrm{g}}_{f}\) is the admittance of the fault between the phase and ground. For a line-to-line fault, \(\mathbf{G}_{f}\) is defined as:  
\[\mathbf{G}_{f}= \begin{bmatrix}\mathbf{\mathrm{g}}_{f}&-\mathbf{\mathrm{g}}_{f}&0\\ -\mathbf{\mathrm{g}}_{f}&\mathbf{\mathrm{g}}_{f}&0\\ 0&0&0\end{bmatrix} , \tag{4}\]  
where \(\mathbf{\mathrm{g}}_{f}\) is the admittance of the fault between the phases. For a three-phase ungrounded fault, illustrated in Figure 1, \(\mathbf{G}_{f}\) is based on the delta connection, that is, defined as:  
\[\mathbf{G}_{f}=3 \begin{bmatrix}2\mathbf{\mathrm{g}}_{f}&-\mathbf{\mathrm{g}}_{f}&-\mathbf{ \mathrm{g}}_{f}\\ -\mathbf{\mathrm{g}}_{f}&2\mathbf{\mathrm{g}}_{f}&-\mathbf{\mathrm{g}}_{f}\\ -\mathbf{\mathrm{g}}_{f}&-\mathbf{\mathrm{g}}_{f}&2\mathbf{\mathrm{g}}_{f}\end{bmatrix} , \tag{5}\]  
where \(\mathbf{\mathrm{g}}_{f}\) is the admittance of the fault between the phases. For a line-line-ground fault, illustrated in Figure 2, a star-mesh transformation is performed in order to calculate the values of elements of \(\mathbf{G}_{f}\), based on the resistances between the phases (\(\mathbf{\mathrm{g}}_{p}\)) and the ground (\(\mathbf{\mathrm{g}}_{g}\)), where  
\[\mathbf{\mathrm{g}}_{total}=2\mathbf{\mathrm{g}}_{p}+\mathbf{\mathrm{g}}_{g} \tag{6}\]  
\[\mathbf{\mathrm{g}}_{pp}=\frac{\mathbf{\mathrm{g}}_{p}^{2}}{\mathbf{\mathrm{g}}_{total}} \tag{7}\]  
\[\mathbf{\mathrm{g}}_{pg}=\frac{\mathbf{\mathrm{g}}_{p}\mathbf{\mathrm{g}}_{g}}{\mathbf{\mathrm{ g}}_{total}}. \tag{8}\]  
The \(\mathbf{G}_{f}\) matrix for a line-line-ground fault is then defined as:  
\[\mathbf{G}_{f}= \begin{bmatrix}\mathbf{\mathrm{g}}_{pp}\mathbf{\mathrm{g}}_{pg}&-\mathbf{ \mathrm{g}}_{pp}&0\\ -\mathbf{\mathrm{g}}_{pp}&\mathbf{\mathrm{g}}_{pp}\mathbf{\mathrm{g}}_{pg}&0\\ 0&0&0\end{bmatrix} . \tag{9}\]  
Figure 1: Star-mesh transformation for a three-phase ungrounded fault.