## 2 Methodology Overview
### Cable System Structure Represented in the Tool  
Figure 1 shows a sketch of the structure of an AC coaxial cable composed of HTS tapes and considered by the authors in this work. The conductive phases are named 1, 2, and  
3 from the innermost to the outermost one. Each phase is composed of a number \(n_{l_{i}}\) of HTS tapes co-wound with one of top of the other. The number of tapes wound side by side on each layer is equal to the maximum number which can be accommodated around the circumference given by the inner radius of the HTS layer \(R_{i,j}\) and the winding pitch of the layer \(\alpha_{i,j}\). In the following, the subscript \(i\) will indicate the \(i\)th phase, and the subscript \(j\) will indicate the \(j\)th layer within that phase. The layers are numbered progressively from the innermost to the outermost one in a phase. Then, to describe the structure from the innermost to the outermost coaxial layer, the cable is composed of a central cylindrical and corrugated former, containing a flow of LN\({}_{2}\). The former has an inner radius \(R_{f}\) and a thickness \(t_{f}\). A leveling layer (not represented in Figure 1) of a thickness \(t_{lev}\) is inserted between the former and the first conductive layer to protect the latter against deformation. Subsequently, the tapes composing the first layer of phase 1 are arranged around the circumference \(R_{1,1}\) (in this example, phase \(1\to i=1\); layer \(1\to j=1\)). The thickness of each HTS layer corresponds to the thickness of the selected coated conductor \(t_{HTS}\). Eventual additional layers wound on top of the innermost one of the same phase are separated by a Kapton layer of a thickness \(t_{k}\). Then, between the outermost HTS layer of phase 1 and the innermost HTS layer of phase 2, there is a layer of insulating material of a thickness \(t_{ins_{1}}\), whose calculation is described in Section 3.5. The first insulating layer separates the conductive phases 1 and 2, the second insulating layer (with a thickness \(t_{ins_{2}}\)) is between conductive phases 2 and 3, and the third insulating layer (with a thickness \(t_{ins_{3}}\)) is between conductive phase 3 from the neutral copper layer. Wound around the third insulating layer is a neutral conductor layer with a thickness \(t_{n}\) which is made of copper. A core fixture layer (not represented in Figure 1) with a thickness \(t_{c}\) is then arranged around the neutral conductor to ensure mechanical stabilization during cable assembly and operations. Around the described combination of conductive and insulating layers, there is an annular duct with a thickness \(t_{an}\) acting as a return channel for the cryogenic fluid. Finally, all of the concentric layers described are contained withing a cryostat with a thickness \(t_{cryo}\). It is worth noting that the values of \(t_{f}\), \(t_{lev}\), \(t_{HTS}\), \(t_{k}\), \(t_{n}\), \(t_{c}\), and \(t_{cryo}\) are set by the user and are not subject to optimization. Figure 1: Illustration of the lateral view of a concentric HTS cable, showing some of the parameters selected as system variables (terms colored in bold). This figure is not to scale [38]. The conductor layout we just defined is integrated into the cable system, which additionally includes the cooling system. Figure 2 shows a sketch of the architecture of the cooling system adopted for the cable, called the _both-sided cooling option_[39]. The line length \(L_{line}\) is divided into a series of cable segments included between a couple of cooling stations. Each cable segment consists of two concentric tubes: the former and annular ducts. Both conduits carry the same mass flow of liquid nitrogen (\(m_{LN2}\)) in opposite directions so that it circulates within a closed loop through the cable. A termination cooling station is placed at both sides of the whole cable length; one manages the fluid coming from the former and transfers it into the annular gap, while the other handles the reverse flow.