## 2 Methodology Overview
### System Variables Selected for the Optimization and Grading Approach  
To achieve high computational efficiency, it is advisable to focus on a key subset of variables considered most significant to the cable system's cost (i.e., the objective function for the solver). This approach facilitates the identification of correlations between these variables and the overall system cost while minimizing the solver's computational burden. The remaining parameters are either fixed or derived from the system variables, although users can adapt the proposed approach to include them in the optimization for dedicated parametric analyses. The parameters selected as system variables are highlighted in Figures 1 and 2 as bold terms. These are the inner radius of the former duct \(R_{f}\), the mass flow rate of nitrogen flowing in both the former and the annular duct \(m_{LN2}\), the number of HTS layers of each phase \(m_{i}\), the winding angle of each HTS layer of each phase \(\alpha_{i,j}\), the thickness of the annular duct \(t_{an}\), the number of intermediate cooling stations \(n_{stat}\), and the ratios  
Figure 2: Illustration of a both-sided cooling option with a single intermediate cooling station. The red colored arrows represent the heat sources entering the two cryogenic fluid flows which circulate through the cable. between the maximum currents flowing in the tapes of the innermost HTS layers of each phase (\(I_{tape_{i,1}}\), the sinusoidal current of the tapes in the \(i\)th layer of the \(j\)th phase) and their corresponding critical current under the most stressful conditions in terms of temperature and magnetic field along a cable segment (\(\sqrt{2}\left|I_{tape_{i,1}}\right|/L_{\xi}\left(T_{HTS_{max}},\;|B|_{i,1},\; \theta_{i,1}\right)\)). The introduction of the last variable changes the approach compared with earlier versions of OSCaR, in which an HTS tape with uniform characteristics was set across all phases. Indeed, it was already observed for medium-voltage coaxial cables [36] that this strategy can result in certain layers exhibiting a rather low ratio between \(I_{tape}\) and \(I_{c}\). While not inherently posing operational issues, this scenario implies an \"inefficient\" expenditure in tape acquisition, wherein tapes with excellent electrical properties might be extremely underloaded. This is particularly true for the outermost phases of the coaxial structure, which require more tapes wound side-by-side around corresponding larger circumferences. The use of high voltages exacerbates this aspect, as thicker layers of insulation are interposed between the phases, leading to much larger circumferences for the outer HTS layers across the conductor cross-section and requiring a corresponding greater number of tapes, while the total phase current remains the same. To overcome this problem, the solver can control the values of \(\sqrt{2}\left|I_{tape_{i,1}}\right|/I_{c}\left(T_{HTS_{max}},\;|B|_{i,1},\; \theta_{i,1}\right)\) for the three phases, optimizing them within the maximum and minimum limits. While the numerator of the ratio is derived from the operating current and the geometry of the HTS layer using the electrical model described in Section 3.2, the denominator relies on the coated conductor's characteristic curves, concerning the local temperature and magnetic field conditions. Thus, a dimensionless characteristic function \(I_{c}\left(T_{HTS},\;|B|,\;\theta\right)\) is implemented in the tool, referring to a selected tape, where the user can freely modify it before running the code. Then, albeit not explicitly, the solver operates on the multiplication factor to apply to this function equivalent to the critical current value at 77 K and the self-field (hereafter called \(I_{c_{77\,K}}\)). Typically, manufacturing companies provide customers with this parameter to characterize their product, often corresponding to the minimum value across the batch length required by the customer.