## 2 Short-Circuit Calculations
### Grid-Following Inverter Constraints  
The following Subsections describe the additional constraints that inverters impose on the short-circuit flow constraints introduced in Section 2.1. For the purposes of short-circuit calculations, grid-following and grid-forming inverters are distinguished. Typically, grid-following inverters (Section 2.3) operate as distributed energy resources (e.g., residential or community-scale solar installations). Grid-forming inverters (Section 2.4), on the other hand, are intended for standalone operation (e.g., used in off-grid solar systems) or are designed specifically for microgrids that are capable of islanded operation. Grid-following inverters behave as a constant-power source under normal operation, while grid-forming inverters behave as a constant-voltage source [67; 68; 69]. Grid-following inverters can have a non-negligible contribution to fault currents in networked microgrids connected to the bulk power system through a high impedance or in islanded microgrids [70]. Design and operating principles of a grid-following inverter are described in detail in [71]. While they nominally operate as a constant-power source, during fault conditions the output terminal voltage may drop to sufficiently low limits so that the inverter is unable to meet its power setpoint without exceeding its current limits. These current limits are determined by the thermal rating of the power semiconductor switches in the inverter. Due to the low thermal time-constant of power semiconductors compared to rotating generation, the inverter will switch to current-limiting mode within cycles of a fault appearing. The introduced formulation assumes that the inverter supplies only real power and positive-sequence current. It is also assumed that the inverter will switch between normal and current limiting operation depending on grid conditions. First the complex term \(\alpha\) is defined, which is needed to calculate the sequence components:  
\[\alpha =e^{\mathsf{j}2\pi/3} \alpha^{2} =e^{-\mathsf{j}2\pi/3}\] \[\alpha_{r} =\Re(\alpha)=-\frac{1}{2} \alpha_{i} =\Im(\alpha)=\frac{\sqrt{3}}{2}\] \[\alpha_{r}^{2} =\Re(\alpha)^{2}=-\frac{1}{2} \alpha_{i}^{2} =\Im(\alpha)=-\frac{\sqrt{3}}{2},\]  
Figure 5: The IEEE 123-Node System [66]. where the notation \(\Re\) and \(\Im\) is used to denote the real and imaginary parts of a complex number. The term \(\alpha\) is commonly used within the transformation matrices \(A\) and \(A_{i}\) but for this work the scalar representation is used instead. Once the transients of the fault subdue, the inverter injects positive-sequence current resulting in three sets of constraints applied to the sequence components of grid-following inverters:  
\[\forall f\in\mathcal{F},\;\forall g\in\mathcal{G}_{pqi}:\]  
\[\sum_{\phi\in\Phi}I_{gf}^{\phi}=0\]  
\[0=I_{gf}^{a}+\alpha^{2}I_{gf}^{b}+\alpha I_{gf}^{c}\]  
\[3I_{gf}^{+}=I_{gf}^{a}+\alpha I_{gf}^{b}+\alpha^{2}I_{gf}^{c},\]  
where (26) is the zero sequence constraint, (27) is the negative sequence constraint, and (28) is the positive sequence constraint. In constraints (27) and (28), \(\alpha\) is used to transform from the phase reference frame to the sequence component frame. During a fault, the only injected current from the grid-following inverter is the positive sequence component, so the constraints are based on the positive sequence voltage and current.