## 2 Short-Circuit Calculations
### Grid-Forming Inverter Constraints  
As mentioned before, grid-forming inverters nominally operate as a constant-voltage source [70]. The implication here is that while grid-following inverters are constrained to inject real power and positive-sequence current (see Section 2.3), grid-forming inverters do not have these constraints. Additionally, these inverters toggle between normal and current-limiting operation on a per-phase basis. The grid-forming inverter constraints are the following:  
Subject to:  
\[\hat{\varPsi}\in\mathcal{F},\ \forall g\in\mathcal{G}_{\varPi},\ \forall\phi\in\Phi: \tag{37}\]  
\[\hat{\varPhi}_{\mathit{gf}}\leq\overline{\hat{\varUpsilon}}_{\mathit{gf}} \tag{38}\]  
\[(\hat{\varPhi}_{\mathit{gf}}-\overline{\hat{\varUpsilon}}_{\mathit{gf}})z_{ \mathit{gf}}^{\phi}\geq 0 \tag{39}\]  
\[\left|\sum_{\phi\in\Phi}p_{\mathit{gf}}^{\phi}\right|\leq\left|\Re(V_{\mathit{gf }}(I_{\mathit{gf}})^{*})\right| \tag{40}\]  
\[|\hat{\varPhi}_{\mathit{gf}}|\leq\overline{\hat{\varUpsilon}}_{\mathit{gf}} \tag{41}\]  
\[V_{\mathit{gf}}^{\Phi}=V_{\mathit{gf}}^{\Phi\mathit{set}}-Z_{\mathit{gf}}^{ \Phi\mathit{virtual}}I_{\mathit{gf}}^{\Phi}, \tag{42}\]  
where \(Z_{\mathit{gf}}^{\Phi}\) is a set of decision variables that are based on the current limiting indicator variable \(z_{\mathit{gf}}^{\Phi}\) and allow the inverter's terminal voltage to adjust to match the system. \(Z_{\mathit{gf}}^{\Phi}\) is defined as:  
\[Z_{\mathit{gf}}^{\Phi\mathit{virtual}}=r_{\mathit{gf}}^{\Phi}+jx_{\mathit{gf}}^ {\Phi} \tag{43}\]  
\[0\leq r_{\mathit{gf}}^{\Phi}\leq\hat{\varPhi}_{\mathit{g}}\qquad\qquad 0\leq x_{ \mathit{gf}}^{\Phi}\leq\hat{\varPhi}_{\mathit{g}},\]  
where the parameters \(\hat{\varPhi}_{\mathit{g}}\) and \(\hat{\varPhi}_{\mathit{g}}\) set the upper limit of the virtual resistance and impedance.  
\[V_{\mathit{gf}}^{\Phi\mathit{set}}\leq V_{\mathit{gf}}^{\Phi\mathit{ref}}+ \boldsymbol{M}_{\mathit{g}}z_{\mathit{gf}}^{\phi} \tag{44}\]  
\[V_{\mathit{gf}}^{\Phi\mathit{set}}\geq V_{\mathit{gf}}^{\Phi\mathit{ref}}- \boldsymbol{M}_{\mathit{g}}z_{\mathit{gf}}^{\phi}. \tag{45}\]  
Constraint (37) is the per-phase current limiting constraint, and constraint (38) is a similar per-phase lower limit constraint for current-limiting mode of grid-forming inverters. Constraint (39) defines that the total power across all phases must not exceed the power  
Figure 6: Graphical illustration of constraints for a grid-following inverter.  
supplied by the source at the dc-bus (regardless if it is PV solar, battery, diesel generator, etc.), while constraint (40) restricts the magnitude of the actual current to be less than the rated current. Constraint (41) is a voltage drop constraint on the voltage setpoints that account for virtual impedances used for power sharing. Constraints (43) and (44) allow for the reference voltage \(V_{gf}^{qref}\) to adjust when the inverter reaches the current limits, while also ensuring that the set point of the inverter is equal to the reference voltage when not in current limiting mode.  
Constraints (37)-(44) are added to the formulation to aid in calculating fault currents. The relationships between the inverter terminal voltage, current and the Thevenin admittance of network resulting from these constraints are illustrated graphically in Figure 7.