## 6 Conclusions  
This paper investigated the concept of performing short-circuit analysis within an optimization problem formulation. An optimization formulation for short-circuit currents on unbalanced distribution networks was derived and implemented in the PowerModels.jl [58] ecosystem in the Julia programming language. This formulation was expressed as a feasibility problem that explicitly represents voltages and currents during short-circuit conditions. It is based on the rectangular coordinates of the voltages and currents of the model, with a current balancing constraint based on Kirchoff's Current Law coupling the model together and providing an expression for determining the fault current at a particular bus. Results from the IEEE 123-Node System showed that the optimization-based approach was able to produce fault currents within 5% of the results produced by OpenDSS and the published results of the IEEE Power and Energy Society. Constraints were developed for the short-circuit contributions of both grid-following and grid-forming inverters. These constraints not only represent the current and power limitations of the inverters, but also the operational constraints based on the inverter type for both normal and faulted conditions. The developed formulation as a whole and the individual inverter short-circuit models were validated against existing models: Experimental validation of the grid-following inverter constraints in PMsp was performed against OpenDSS on the IEEE 123-Node System with added inverter-interfaced generation; reported fault current magnitudes were within 5% of those calculated by OpenDSS. Experimental validation of the grid-forming inverter constraints in PMsp was performed against a built dynamic simulation model implemented in MATLAB(r)/Simulink; fault currents calculated by PMsp and Simulink agreed closely on high-impedance faults, however, for the case of low impedance faults, results showed noticeable differences that can be attributed to the present level of detail in transformer modeling under PowerModelsDistribution.jl [59]. In addition, experiments were performed on the grid-forming inverter in both grid-connected and islanded modes in order to demonstrate the ability of the constraints to conform to the inverters supporting the system voltage or being the main source of voltage in an island. The use of the constraints presented in this paper in optimization problems has the potential to simplify short-circuit analysis involving inverter-interfaced networked microgrids and to assist optimal operation formulations to select solutions that provide protection coordination. Modeling inverter-interfaced generation involves constraints that are not straightforward to put into a traditional, Thevenin equivalent based short-circuit solver, and furthermore requires nested iterative solution methods. This concept was implemented and demonstrated in PMsp, and tested on the IEEE 123-Node System for both synchronous and inverter-interfaced generations. Optimization-based short-circuit analysis offers the potential to integrate operational switching problems within the process of verifying protection coordination constraints by introducing a coupled multi-network formulation, similar to that employed by PowerModelsGMD.jl [61]. Such an approach allows one to perform optimal operational switching problems with switching constraints on larger microgrids, where the number of switch states precludes enumerating the protection-feasible switch-states a priori with a traditional fault study. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fault Resistance (\(\Omega\))** & **Fault Phasing** & **Simulink Fault Current (A)** & **PMsp Fault Current (A)** & **Difference (\%)** \\ \hline
20.0 & LG & 13.79 & 13.73 & \(-\)0.44 \\
20.0 & LL & 11.94 & 11.89 & \(-\)0.42 \\
20.0 & 3P & 13.79 & 13.73 & \(-\)0.44 \\
5.0 m & LG & 103.4 & 114.6 & \(-\)10.8 \\
5.0 m & LL & 60.55 & 57.29 & \(-\)5.38 \\
5.0 m & 3P & 69.93 & 66.15 & \(-\)5.41 \\ \hline \hline \end{tabular}
\end{table}