## 3 Superconducting Fault Current Limiter and Shunt Resistor Numerical Model  
The r-SFCL devices considered in this study were used with AMSC-8602 superconducting tape, as shown in Figure 1. The tape was composed of several layers, each modeled by a current-dependent resistor. The operation of the r-SFCLs in the superconducting state was guaranteed if their current and temperature were less than the critical values for the tape. In this case, the entirety of the current flowed in the superconducting layer, which had zero resistivity. The geometry of the tape and the materials used in its construction governed its critical current and temperature. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Quantity** & **Value** \\ \hline AC source & Three-phase & 60 Hz, 25 kV, 0.892 \(\Omega\) source resistance \\ Step-up transformer & Three-phase & 25/33 kV, 10 MVA \\ Transmission lines & Three-phase & 33 kV, 0.06 \(\Omega\)/km \\ r-SFCL devices & Three units & AMSC-8602, 300 m \\ Load & Three-phase & 60 Hz, 5.7 MW \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of AC system. Figure 2: Configuration of considered power system with integration of r-SFCL units. Once a fault occurs, the current increases beyond the critical value, resulting in a sudden increase in the resistivity of the superconducting layer. As a result, the current is redistributed among the various layers of the superconducting tape and the shunt resistor. The superconducting layer resistivity (\(\rho_{SC}\)) is obtained in terms of the current density (\(J\)) as follows:  
\[\rho_{SC}=\frac{E_{c}}{J_{c}(T)}(\frac{J}{J_{c}(T)})^{N-1}\hskip 56.905512 ptJ>J_{C}, \ T<T_{c} \tag{2}\]  
where \(E_{c}\) is the critical electrical field, \(J_{C}\) is the critical current density, \(T_{C}\) is the critical temperature, and \(N\) is a constant ranging between 21 and 30 [30]. The critical current density is a function of the critical temperature as follows:  
\[J_{c}(T)=J_{co}\left(\frac{\left(T_{c}-\ T(t)\right)^{a}}{\left(T_{c}-\ T_{o} \right)^{a}}\right)\hskip 56.905512 ptT_{o}<T<T_{c} \tag{3}\]  
where \(J_{co}\) is the critical current density at the operating temperature (\(T_{o}\)), which equals 77 K in this case, and the exponent \(\alpha\) is the density exponent, equal to 1.5 [31]. The resistivity (\(\rho_{x}\)) of any other layer (\(x\)) is temperature-dependent and can be obtained from [32; 33; 34]. Then, the resistance of the layer itself can be obtained using (4):  
\[R_{x}=\rho_{x}\ \frac{l}{A_{x}} \tag{4}\]  
where \(l\) is the tape length and \(A_{x}\) is the layer's cross-sectional area. The total resistance of the r-SFCL and the shunt resistor can now be determined using (5):  
\[\frac{1}{R_{SFCL}}=\frac{1}{R_{YBCO}}+\frac{1}{R_{SS}}+\frac{1}{R_{Ag}}+\frac{ 1}{R_{NiW}}+\frac{1}{R_{Shunt\ Resistor}} \tag{5}\]  
The electrical representation of the r-SFCL, which contains YBCO, stainless steel, silver, and NiW layers, in addition to the shunt resistor, is given in (5). The dependence of the electrical resistivity of each layer on temperature differs based on the temperature coefficient of the layer's material. Consequently, it is important to precisely calculate the temperature of the r-SFCL. The temperature of the superconductor tape can be obtained as follows:  
\[T(t)=T_{o}+\frac{1}{C_{p}}\int_{0}^{t}Q_{sc}(t)dt \tag{6}\]  
where \(C_{p}\) is the heat capacity of the superconducting material and \(Q_{sc}\) is the net power of the tape, which is given by the difference between the power dissipated in the tape (\(P_{diss}\)) and the cooling power (\(P_{cooling}\)) calculated as follows:  
\[Q_{sc}(t)=P_{diss}(t)-P_{cooling}(t) \tag{7}\]  
\(P_{diss}\) (\(t\)) and \(P_{cooling}\) (\(t\)) are calculated by (8) and (9), respectively:  
\[P_{diss}(t)=i(t)^{2}R_{sc}(t) \tag{8}\]  
\[P_{cooling}(t)=hA(T(t)-T_{o}) \tag{9}\]  
where \(A\) is the area exposed to the LN\({}_{2}\) coolant and \(h\) is the heat transfer coefficient.