## 3 Methods
### Economic Analysis
#### 3.3.2 Economic Performance Indicators
\begin{table}
\begin{tabular}{l l} \hline
**Equipment** & **Model (USD)** \\ \hline Air compressor (AC) [26; 33] & \(Z_{\text{AC}}=7900\,(P_{\text{AC}})_{0.81}^{0.62}\) \\ Air turbine (AT) [26; 33] & \(Z_{\text{AT}}=1100\,(P_{\text{AT}})_{0.81}^{0.81}\) \\ Cryo-turbine (CT) [26; 33] & \(Z_{\text{CT}}=1100\,(P_{\text{CT}})_{0.81}^{0.6}\) \\ Inter-cooler (IC) [6] & \(Z_{\text{IC}}=12,000(A_{\text{IC}}/100)^{0.6}\) \\ Condenser (CON) [6] & \(Z_{\text{CON}}=32,800(A_{\text{CON}}/80)^{0.68}\) \\ Evaporator (EVA) [6] & \(Z_{\text{FA}}=32,800(A_{\text{FAW}}/80)^{0.68}\) \\ Re-heater (RH) [6] & \(Z_{\text{RH}}=12,000(A_{\text{RH}}/100)^{0.6}\) \\ Superheater (SH) [6] & \(Z_{\text{SH}}=12,000(A_{\text{SH}}/100)^{0.6}\) \\ Liquid air tank (LAT) [6] & \(Z_{\text{LAT}}=320\times V_{\text{LAT}}\) \\ Liquid methanol tank (LMT) [33] & \(Z_{\text{LMT}}=572\times V_{\text{LMT}}\) \\ Liquid propane tank (LPT) [33] & \(Z_{\text{LPT}}=1326\times V_{\text{LPT}}\) \\ Liquid thermal oil tank (LOT) [33] & \(Z_{\text{LOT}}=423\times V_{\text{LOT}}\) \\ Liquid air pump (LAP) [33] & \(Z_{\text{LAP}}=483\,\text{LP}_{\text{LAP}}\) \\ LNG pump (LNGP) [28] & \(Z_{\text{LNGP}}=1120\,(P_{\text{LNGP}})^{0.8}\) \\ Combustion chamber (CRV) [34] & \(Z_{\text{CRV}}=\frac{25.65\,\text{erg}_{\text{CRV}}}{0.995^{-1}-\text{CRV}_{ \text{LNF},\text{in}}}\left(1+e^{(0.018\cdot T_{\text{CW},\text{at}}-26.4)}\right)\) \\ Flue gas turbine (GT) [34] & \(Z_{\text{GT}}=1100\,(P_{\text{GT}})^{0.81}\) \\ Air throttle valve (ATV) [6] & \(Z_{\text{ATV}}=114.5\,(m_{\text{ATV}})^{0.67}\) \\ Separator (SEP) [6] & \(Z_{\text{SEP}}=114.5\,(m_{\text{SEP}})^{0.67}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Investment cost models for the equipment. \begin{table}
\begin{tabular}{l l} \hline
**Materials** & **Mathematical Model (USD)** \\ \hline Thermal oil [35] & \(Z_{\text{oil}}=326.3(t_{\text{ch}}\times m_{\text{oil}}/1000)\) \\ Methanol [35] & \(Z_{\text{met}}=352(t_{\text{ch}}\times m_{\text{met}}/1000)\) \\ Propane [35] & \(Z_{\text{pro}}=649.6(t_{\text{ch}}\times m_{\text{pro}}/1000)\) \\ \hline \end{tabular}
\end{table} 
Table 3: Investment cost models for the materials. where \(n\) refers to the considered year, SL refers to the service life of the proposed system, and \(i\) refers to the discount rate. The dynamic payback period (DPP) represents the payback years that the discounted NPV recovers the initial investment cost. The DPP method takes into account the influence of the time factor on the monetary value, which is defined as [26]:  
\[\text{DPP}=(j-1)+\frac{\left|\text{NPV}_{j-1}\right|}{AP_{\text{t}}} \tag{12}\]  
where \(j\) represents the year of the first positive NPV harvest, and \(\left|\text{NPV}_{j-1}\right|\) is the absolute value of the NPV at the \((j-1)\) th year (USD). The levelized cost of electricity (USD-kWh\({}^{-1}\)), LCOE, allows for comparing the electricity generation costs based on the same criteria. It is as described by the following formula [14; 36; 37]:  
\[\text{LCOE}=\frac{R\times Z_{\text{IIC}}+\sum_{n=1}^{\text{SL}}\frac{AC_{\text {t}}}{(1+f)^{n}}}{\sum_{n=1}^{\text{SL}}\frac{E_{\text{out}}}{(1+f)^{n}}}= \frac{R\times Z_{\text{IIC}}+\sum_{n=1}^{\text{SL}}\frac{AC_{\text{t}}}{(1+f)^{ n}}}{\sum_{n=1}^{\text{SL}}\frac{P_{\text{out}}\cdot t_{\text{alg}}\cdot 365}{(1+f)^{ n}}} \tag{13}\]  
\[R=\frac{R_{\text{ai}}\times\left(1+R_{\text{ai}}\right)^{\text{SL}}}{(1+R_{ \text{ai}})^{\text{SL}}-1} \tag{14}\]  
where \(R_{\text{ai}}\) is the annual interest rate, \(R\) is the capital recovery factor, which is determined by \(R_{\text{ai}}\), \(f\) is the inflation rate, and \(E_{\text{out}}\) (kWh) represents the electricity generation in year \(n\), which equals the output power multiplied by the discharge periods in year \(n\)th.