## 4 Cost Function to Be Minimized
### Capital and Operating Costs for the Cooling Stations  
The operating and capital costs for cooling stations are conveniently distinguished in Equation (14) to simplify their exposition in this section. Regarding the capital costs for purchasing all the cooling stations required for the whole cable length (whose total number corresponds to \(n_{stat}+2\)), these are equal to  
\[C_{cool_{cap}}{}^{\%_{c}}=\sum_{k}^{n_{tgt}+2}C_{cool_{find}}+\left[\left(c _{cyvocooler}{}^{\%_{c}}Q_{c_{k}}\ \right)\beta_{Q}\right]\ \beta_{r} \tag{23}\]  
\[Q_{c_{k}}= \begin{cases}Q_{f}+Q_{an}&\text{if $k=intermediate$ {station}}\\ Q_{an}+Q_{cl}&\text{if $k=left$ termination station}\\ Q_{f}+Q_{cl}&\text{if $k=right$ termination station}\end{cases}  \tag{24}\]  
where \(c_{cyvocooler}{}^{\%_{c}}\) is the cost of each cryocooler system per unit of cooling power to extract (\(Q_{k}\)), which is discounted following the approach described in Section 4.1 according to the total number of cooling stations, \(\beta_{Q}\) serves as a markup factor to accommodate potential heat peaks and address the practical difficulty of finding cooling devices precisely matching the required sizes in the market (in these cases, the superior size is selected), \(\beta_{r}\) is a markup factor applied to consider the cost of those cooling station components for which no parametric function has been identified (circulation pumps, LN\({}_{2}\) tank, and accessory components), assumed to correspond to a certain percentage of the cryocooler cost, \(C_{cool_{find}}\) accounts for the fixed costs required for each cooling station aside from the device cost, such as installation, maintenance, and disposal, and \(Q_{c_{k}}\) is the sum of the heat contributions which must be removed by the \(k\)th cooling station from the LN\({}_{2}\) inlet flows it has to handle. A detailed explanation of the terms \(Q_{f}\), \(Q_{an}\), and \(Q_{cl}\) will be provided in Section 5. From Equation (24), it can be understood that the only differentiation between the capital costs of the various cooling stations concerns the intermediate and termination ones, with the latter being distinguished further between the two cable sides (left and right), as these are designed to manage different nitrogen flows or additional heat inputs, such as \(Q_{cl}\). Moreover, it is important to highlight that the cooling system considered in this study is not designed with redundancy. Nonetheless, an additional markup factor could be included in Equation (23) to accommodate this potential design choice. Concerning the costs to operate all the cooling stations, these correspond to  
\[C_{cool_{pp}}=\sum_{k}^{n_{stat}+2}\left\{\sum_{z}^{yr}c_{kWh}h_{y}\left[\left( \frac{Q_{c_{k}}}{\eta_{cry}}\frac{(T_{a}-T_{0})}{T_{0}}\right)+\frac{Q_{p_{k}} }{\eta_{p}}\right]\frac{1}{(1+\psi)^{2}}\right\} \tag{25}\]  
\[Q_{p_{k}}= \begin{cases}\begin{array}{ll}m\dot{m_{LN2}}\ \frac{\Delta p_{f}+ \Delta p_{m_{k}}}{\rho_{LN2}}&\text{if $k=intermediate$ {station}}\\ \frac{m\dot{m_{LN2}}\ \Delta p_{m_{k}}}{\rho_{LN2}}&\text{if $k=left$ termination station}\\ \frac{m\dot{m_{LN2}}\ \Delta p_{f}}{\rho_{LN2}}&\text{if $k=right$ termination station}\end{array}\end{cases}  \tag{26}\]  
where \(yr\) represents the expected years of service for the cable system, \(h_{y}\) is the annual operational hours, \(c_{kWh}\) is the actual energy cost for non-residential consumers in the region where the cable is planned to be installed, \(Q_{c_{k}}\) is computed from Equation (24), (In Equation (25), the temperature ratio involving the LN\({}_{2}\) inlet temperature \(T_{0}\) and the average ambient temperature during the year at the cable installation site \(T_{a}\) corresponds to the inverse Carnot efficiency.) and \(Q_{p_{k}}\) stands for the hydraulic power supplied by the circulation pumps to the liquid nitrogen flows to restore their initial pressure \(P_{0}\) after their corresponding drops, computed with Equation (1).