## 3 Main Simplifying Hypotheses Assumed in the Optimization Tool  
### Thermo-Hydraulic Model  
To effectively model a complex system like a superconducting cable within a versatile tool, several simplifying assumptions must be imposed. Particularly relevant are those related to the thermal model of the coolant flows and the cable layers. As anticipated in the previous paragraph, and as will be more evident later in this document when the dependencies of some techno-economic parameters on the cable temperatures will be explained, the fluid dynamic conditions of the fluids, along with the temperature of the HTS tapes, significantly influence the cable system's design. Assuming modularity in the cable structure implies that the coolants absorb the same amount of heat within their respective ducts across each cable segment, named \(Q_{f}\) and \(Q_{an}\) for the LN\({}_{2}\) flows in the former and annular gaps, respectively, as depicted in Figure 2. Additionally, these thermal inputs are approximated to be uniformly absorbed along the length of a segment. This assumption, combined with the effect on the temperature and pressure due to friction with the ducts' walls (also assumed to be uniform along the cable length), allows considering the linear temperature increases and linear pressure drops of the fluids from their initial fluid dynamic conditions. Figure 3 illustrates the temperature trends of the two counterflows of LN\({}_{2}\) within a generic cable segment. As assumed in this study, both flows exit each intermediate or termination cooling station under identical initial temperature and pressure conditions, named \(T_{0}\) and \(P_{0}\), respectively. The directions of the flows and the proportion between their maximum temperatures depicted in the figure are arbitrary. It is worth noting that the heat source \(Q_{cl}\) is not present in Figure 3, instead appearing in Figure 2. Indeed, this corresponds to the power dissipated by the current leads. It is assumed that this heat is individually managed by the termination cooling stations, where the current leads are located, using a cooling flow distinct from the one circulating through the rest of the cable. In equation terms, the linear variations in temperature and pressure of the fluid in one of the two pipes along a cable segment of a length \(d_{stat}\)\(\Delta T\) and \(\Delta P\), respectively, are calculated using the simple set of analytic equations in Equations (1)-(5), which were adapted for this model from [40]. To avoid the redundancy of writing the same equations for both fluids, the formulae for a single generic pipe having the subscript \(h\) are reported. For each fluid flow, readers can replace these subscripts with the one referring to the former or the annular gap:  
\[\Delta P_{h}=d_{stat}\left(\frac{\rho_{LN2}\;{\
u_{LN2_{h}}}^{2}\;f_{LN2_{h}}} {4\;R_{h}}\right)+\left(g\rho_{LN2}\Delta height\right) \tag{1}\]  
\[\Delta T_{h}=d_{stat}\left(\frac{{\
u_{LN2_{h}}}^{2}\;f_{LN2_{h}}}{  
\[f_{LN2_{k}}=\left\{-1.8\,\ln\left[\left(\frac{\varepsilon_{h}}{7.4\ R_{h}}\right)^{1. 11}+\left(\frac{6.9}{Re_{h}}\right)\right]\right\}^{-2} \tag{4}\]  
\[Re_{h}=\frac{2\,R_{h}\ \
u_{LN2_{k}}\ \rho_{LN2_{h}}}{\mu_{LN2}} \tag{5}\]  
where \(\rho_{LN2}\), \(\
u_{LN2_{k}}\), \(cp_{LN2}\), and \(\mu_{LN2}\) are the density, velocity, heat capacity, and dynamic viscosity of LN\({}_{2}\), respectively. These values would depend on the fluid temperature and pressure as well but are approximated here as constants by using their average values within the chosen operating temperature and pressure ranges (i.e., 67-77 K). Here, \(\varepsilon_{h}\) is the roughness of the pipe \(h\), \(g\) is gravity's acceleration, and \(\Delta\)_height_ is the height difference between the two ends of a cable segment (assumed equal for all cable segments) [41], while \(f_{LN2_{k}}\) and \(Re_{h}\) are the friction factor and the Reynolds number of the nitrogen flowing in the pipe \(h\).