## 4 Cost Function to Be Minimized
### Cost of the Superconducting Material  
The capital cost for purchasing the HTS tapes needed for the cable is calculated as follows:  
\[C_{HTS}{}^{\%}=\sum_{i}^{3}\left\{\left[c_{HTS}{}_{i}{}^{\%}\big{(}I_{c\,77\, \,\kappa_{i}}\big{)}\right]\sum_{j}^{nl_{i}}\,\,Tapes_{i,j}\frac{L_{line}}{ \left|cos\big{(}\alpha_{i,j}\big{)}\right|}\right\} \tag{16}\]  
\[Tapes_{i,j}=round\,\left(\frac{2\pi\,\,R_{i,j}\,\left|cos\big{(}\alpha_{i,j} \big{)}\right|}{w_{HTS}}\right) \tag{17}\]  
where \(c_{HTS}{}_{i}{}^{\%}\) is the cost per unit length of the HTS tapes of the \(i\)th phase, discounted with respect to the total length of tape acquired for that phase, and \(Tapes_{i,j}\) is the number of HTS tapes constituting the \(j\)th HTS layer of the \(i\)th phase, having a width \(w_{HTS}\). In Equation (15), the dependency of the conductor cost per unit length on the value of \(I_{c\,77\,\,\kappa}\) is made explicit. This follows the approach outlined in Section 2.2. For clarity, the mathematical steps leading to the determination of \(I_{c\,77\,\,\kappa_{i}}\) and consequently to \(c_{HTS}{}_{i}{}^{\%}\) are reported here as implemented within the optimization algorithm:  
1. At each algorithm step, a specific set of system variables is examined, including the \(\sqrt{2}\left|\frac{I_{tape_{i,1}}}{\left|I_{c}\right|}\right/I_{c}\left(T_{ HTS_{max}},\,\left|B\right|_{i,1},\,\theta_{i,1}\right)\) ratios. 2. Based on each \(\sqrt{2}\big{|}\frac{I_{tape_{i,1}}}{\left|I_{c}\right|}\big{/}I_{c}\left(T_{ HTS_{max}},\left|B\right|_{i,1},\theta_{i,1}\right)\) ratio and the calculation of \(\big{|}\big{|}\frac{I_{tape_{i,1}}}{\left|I_{c}\right|}\) for the innermost layer of each phase (as detailed in Section 3.2), the values of \(I_{c}\Big{(}T_{HTS_{max}},\left|B\right|_{i,1},\theta_{i,1}\Big{)}\) are derived. These correspond to the lowest critical current values for the tapes belonging to the innermost layer of each phase occurring in a cable segment. 3. From the results of the thermal model (Section 3.1) and the magnetic model (Section 3.3), the higher temperature and magnetic field values within a cable segment are determined for the innermost layer of each phase. These represent the most stressful scenarios for the coated conductor in each phase, where the upper limit of the ratios between the transport currents and the critical currents are conservatively determined. 4. Then, the characteristic curves of the chosen coated conductor \(I_{c}\big{(}T_{HTS},\,\left|B\right|,\,\theta\big{)}\) are considered. These curves are generally provided by the manufacturer for a reference tape and are not always reported in dimensionless terms. These curves enable calculation of the critical current values of the reference tape under the same temperature and magnetic field conditions identified in the previous step, with one for each phase, called \(I_{c_{ref}T_{HTS_{max}},\,\left|B_{i,1},\,\theta_{i,1}\right|}\). 5. Similarly, using the manufacturer's characteristic curves, the critical current value of the reference tape at 77 K and self-field is known, called \(I_{c_{ref}/77\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\  
It is assumed that the same characteristic curves valid to the reference tape can be scaled for the actual tape adopted in each phase of the optimized structure. By solving a simple proportion for each phase, the corresponding values of \(I_{c\,77\,\,K_{i}}\) can be obtained: \[I_{c\,77\,\,K_{i}}=I_{c}\Big{(}T_{HTS_{max}},\,\,|B|_{i,1},\,\theta_{i,1}\Big{)} \,\frac{I_{c_{effT\,\,K}}}{I_{c_{effT_{HTS_{max}},\,\,|B|_{i,1},\,\theta_{i,1}}}}\] (18)
7. Then, the \(c_{HTS_{i}}\) values (before quantitative discounting) are computed by multiplying the \(I_{c\,77\,\,K_{i}}\) values for the cost per unit length and per kiloampere of current carried by the tape at 77 K and the self-field tape (\(c_{HTS_{i}}\)), for which statistical data can be gathered from the literature: \[c_{HTS_{i}}\Big{[}\frac{\mathbf{\xi}}{m}\Big{]}=c_{HTS}{}^{kA}\Big{[}\frac{\mathbf{ \xi}}{kA\cdot m}\Big{]}\cdot I_{c\,77\,\,K_{i}}[kA]\] (19)
8.