## 2 Material and Methods
### Mathematical-Formulation Part
#### 2.2.3 Modeling of the Financial and Environmental Analysis  
The electricity production during the year (\(\text{E}_{\text{el}}\)) is calculated by integrating the power rate (\(\text{P}_{\text{el}}\)) for all the year:  
\[\text{E}_{\text{el}}=\int_{0}^{8760}\text{P}_{\text{el}}\cdot\text{dt} \tag{40}\]  
Similarly, the yearly solar energy (\(\text{E}_{\text{sol}}\)) is calculated as:  
\[\text{E}_{\text{sol}}=\int_{0}^{8760}\text{Q}_{\text{sol}}\cdot\text{dt} \tag{41}\]  
The system's capital cost (\(\text{C}_{0}\)) is estimated according to the next equation:  
\[\text{C}_{0}=\text{C}_{\text{cycle}}\cdot\text{P}_{\text{el,ref}}+\text{C}_{ \text{solar}}\cdot\text{A}_{\text{a}} \tag{42}\]  
The yearly cash flow (CF) is calculated as:  
\[\text{CF}=\text{K}_{\text{el}}\cdot\text{E}_{\text{el}}-(\text{O}\&\text{M}) \tag{43}\]  
The yearly operating and maintenance cost (O&M) is set to be 1% of the initial investment cost [39]:  
\[(\text{O}\&\text{M})=0.01\cdot\text{C}_{0} \tag{44}\]  
The simple payback period (SPP) of the examined configuration is defined as:  
\[\text{SPP}=\frac{\text{C}_{0}}{\text{CF}} \tag{45}\]  
The payback period (PP) of the examined configuration is calculated as:  
\[\text{PP}=\frac{\ln\left[\frac{\text{CF}}{\text{CF}-\text{C}_{0}\cdot\text{r} }\right]}{\ln(1+\text{r})} \tag{46}\]  
The internal rate of return (IRR) can be expressed as follows:  
\[\text{IRR}=\text{SPP}\cdot\left[1-\frac{1}{(1+\text{IRR})^{\text{N}}}\right] \tag{47}\]  
The calculation of the net present value (NPV) is done with the next equation:  
\[\text{NPV}=-\text{C}_{0}+\text{R}\cdot\text{CF} \tag{48}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Turbine isentropic efficiency & (\(\text{n}_{\text{ls,r}}\)) & 85\% \\ Pump isentropic efficiency & (\(\text{n}_{\text{ls,p}}\)) & 85\% \\ Electromechanical generator efficiency & (\(\text{n}_{\text{mg}}\)) & 97\% \\ Motor efficiency & (\(\text{n}_{\text{motor}}\)) & 80\% \\ Recuperator temperature difference & (\(\Delta\text{T}_{\text{rec}}\)) & 10 K \\ Temperature level in the condenser device & (\(\text{T}_{\text{con}}\)) & 298.15 K \\ \hline \end{tabular}
\end{table}
Table 2: Data of the thermodynamic cycle analysis [20,24].  
The equivalent project life (R) can be found as follows:  
\[\mathrm{R}=\frac{(1+\mathrm{r})^{\mathrm{N}}-1}{\mathrm{r}\cdot\left(1+\mathrm{r} \right)^{\mathrm{N}}} \tag{49}\]  
Table 3 gives the data of the economic analysis [40; 41; 42]. The nominal power of the system is 44.14 kW for this work, as it has been calculated by the results.  
Moreover, about the CO\({}_{2}\) emissions avoidance, the Greek energy mix for 2020 leads to approximately 600 kg CO\({}_{2}\) per MWh\({}_{\mathrm{el}}\) according to the DAPEEP [43]. Therefore, yearly CO\({}_{2}\) emissions avoidance is calculated as:  
\[\mathrm{M}_{\mathrm{CO2,y}}[\mathrm{kg}]=600\cdot\frac{\mathrm{E}_{\mathrm{ el}}[\mathrm{kWh}]}{1000} \tag{50}\]  
While the total CO\({}_{2}\) emissions avoidance for the life cycle of the project is calculated as:  
\[\mathrm{M}_{\mathrm{CO2,tot}}[\mathrm{kg}]=\mathrm{N}\cdot 600\cdot\frac{ \mathrm{E}_{\mathrm{el}}[\mathrm{kWh}]}{1000} \tag{51}\]