## 2 Methodology
### Cost Analysis  
As is well-known, working with configurations that offer very low power consumption does not imply that they are the most economically feasible. For that reason, this subsection covers the analysis of the costs associated with the process of compression and storage of the two working gases. The aim of this analysis is to obtain mathematical expressions for the cost of the equipment involved in the process as well as for the operational costs. It must be mentioned that the storage pressure again keeps a constant value of 20 bar to evaluate the different cost expressions in the first instance. There are several methodologies to estimate the investment needed for the whole system. However, in this research the methodology used is based on a percentage of the procurement costs of the equipment needed. This investment method was selected due to it being used in preliminary cost estimations where little cost-related data are available. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Carbon Dioxide**} & \multicolumn{2}{c}{**Hydrogen**} \\  & **Value** & **Unit System** & **Value** & **Unit System** \\ \hline Methane Power & 1000 & kW & - & - \\ LHV Methane & 50,030 & kJ/kg & - & - \\ Methane mass & 0.02 & kg/s & - & - \\ Carbon dioxide mass & 0.055 & kg/s & - & - \\ Days per week & 2 & day & 2 & day \\ Compression ratio & 1-4 & - & 1-4 & - \\ Hydrogen power & - & - & 1000 & kW \\ Electrolyser efficiency & - & - & 70 & \% \\ LHV Hydrogen & - & - & 120,000 & kJ/kg \\ Hydrogen mass & - & - & 0.0058 & kg/s \\ Inlet gas pressure & & 1 bar & & \\ Storage pressure & & 20 bar & & \\ Temperature between serial & & & 60 \({}^{\circ}\)C & & \\ heat exchangers & & & 30 \({}^{\circ}\)C & & \\ Inlet gas temperature & & & 85\% & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation hypothesis. Figure 1: General scheme of the simulation. For that reason, it should be mentioned that the uncertainty associated with this method is approximately 20% to 30% [25], which is usual for economic analyses in the literature. The additional elements involved in the initial investment are estimated on a percentage average of the equipment costs, as shown in Table 2 and Equation (3) [25], where f\({}_{i}\) are factors that represent piping, electric costs, control equipment, etc. The average percentages were obtained from [PERSON] et al. [25], for the calculation of industrial plant costs where the whole process involves fluid-type components. Provided the industrial equipment cost varies in time and considering that this study has been developed in October 2021, an update rate, based on the variation in the Chemical Engineering Plant Cost Index [26], was considered to actualize the overall costs. \[\text{CAPEX}=\sum(E+f_{1}E+f_{2}E+\ldots+f_{n}E)=E\ \sum(1+f_{1}+f_{2}+\ldots+f_{n}) \tag{3}\]  
For the compressors, heat exchangers, pressure storage vessels, and equipment procurement costs the following equipment cost expressions were used [25]:  
\[Cost_{C}(W)=-0.1288*W^{2}+500.04*W+43.997 \tag{4}\]  
\[Cost_{HE}(H)=-0.038*H^{2}+149.18*H+12.849 \tag{5}\]  
\[ \begin{array}{c}Cost_{T}(V,\ M,\ P)=VF*MF*PF\\ =\left(0.0811*\ V^{2}+167.42*V+13529\right)*\ (0.0365*P+1.227)\\ E_{i}=(n_{C}*Cost_{C}+n_{HE}*\ Cost_{HE}+n_{T}*Cost_{T})\end{array}  \tag{6}\]  
For the operational costs, a fixed energy cost price was assumed (Iberian Electricity Market) [27]. For electricity, an average cost of 0.106 \(\upepsilon\)/kWh was assumed as representative and 0.0351 \(\upepsilon\)/kWh for natural gas. To calculate the value of the operational cost and the savings for the utilization of the waste energy in the heat exchangers the following expressions were used:  
\[\text{OPEX}=\left(W_{n}\ [kW]*COE\ \left[\frac{\upepsilon}{kWh}\right]*T_{o} \ \left[\frac{h}{year}\right]\right)-U_{e}\ \left[\frac{\upepsilon}{year}\right] \tag{7}\]  
\[U_{e}=\frac{H_{r}}{\eta_{HE}}[kW]*COG\left[\frac{\upepsilon}{kWh}\right]*\ T_{o} \left[\frac{h}{year}\right] \tag{8}\]