## 3 Main Simplifying Hypotheses Assumed in the Optimization Tool
### Time-Varying Power Transferred by the Line  
The choice of a cable's rated power is crucial in defining its resulting design. This work focuses on HVAC cables capable of transferring extremely high-power loads. Some of the values proposed in this study may be considered speculative, given the current lack of network infrastructure to accommodate lines with power values exceeding 1-2 GW. Nonetheless, these high-power lines could potentially be required in the future, especially if their techno-economic viability is demonstrated, considering the growing scale of single-site power plants and the rising demand for localized energy-intensive loads [48; 49; 50].  
Furthermore, it is important to note that a cable may not operate at its full capacity throughout its whole expected lifespan. Power curves for the following hours, months, and years can be hypothesized to depict fluctuating power demands over time and the specifics of a network. Figure 4 shows the time-varying power trend adopted in this study for any optimized superconducting line (in blue) over the course of one year. The power values are normalized, allowing for the curve to be scaled by the selected peak power. Notably, the curve does not reflect the actual progression of power from the beginning to the end of the year, as in reality, power fluctuates on a daily basis. In Figure 4, the time durations throughout the year during which a specific power level is applied are aggregated, and these power levels are then arranged conveniently in descending order. It is considered that the same power curve can be applied identically for all years of the lifespan expected for the designed line.  
In the optimization tool, the maximum power during the year is taken as a benchmark. This is used, for example, to calculate the amount of superconducting tape needed, the maximum instantaneous losses, and the corresponding temperature increase in the coolant flows. However, for an accurate computation of the energy dissipation over the line's lifespan--upon which the operating costs of the cooling system depend (as will be detailed in Section 4.6)--the actual power curve should be used. Nonetheless, this approach would substantially increase the computation time within the optimization algorithm proposed. Indeed, this would require repeating the numerical calculations to determine the thermal inputs, temperatures, and operating costs for every individual of a single generation of the genetic algorithm and for all the time intervals discretized from the curve in Figure 4. Moreover, this time-consuming numerical resolution must then be iterated at each step from one generation of the genetic algorithm to the next one until the convergence criteria are met.  
Hence, it is deemed practical to optimize a cable by calculating the cost function as if the cable were transferring its maximum power throughout the year. This scenario is represented by the dashed black curve in Figure 4. This enables performing the necessary numerical computation only once for each individual in a generation of the genetic algorithm, thereby significantly reducing the calculation times. However, this approach tends to overestimate the total losses and the corresponding operating costs over the life of the cable. The area highlighted in yellow in Figure 4 represents the overestimation of the cable's energy consumption assumed during the optimization process, following the use of the dotted horizontal line compared with the more realistic power curve. To mitigate this approximation, once the optimal system parameters and the corresponding cable design are determined while assuming a constant peak power, only the operating costs are re-calculated, considering the real power curve. This recalibration process is executed just once when the optimization procedure is concluded, thereby maintaining the computational efficiency of the algorithm.