## 4 Results and Discussion
### Air Expansion Stages and Superheating Stages  
Table 6 summarizes the effects of the air expansion stages and superheating stages on the overall thermodynamic and economic performance. Case A represents the four-stage air expansion and two-stage air superheating, as shown in Figure 1. Case B represents the three-stage air expansion and one-stage air superheating. This section carries out a comparative analysis of Case A and Case B to investigate the effect of air expansion stages on the coupled system performance. The RTEs of Case A and Case B decrease as the charging pressure varies from 12 MPa to 18 MPa. The maximum RTE of Case B is up to 47.58%, 0.14% lower than that of Case A. The minimum RTE of Case B is 44.10%, 1.66% higher than that of Case A. The ERTE of Case B decreases with an increase in the charging pressure. For the given charging pressure values, the highest ERTE in Case B is 69.91%, which is 0.18% higher than that of Case A. The SPC is similar in both cases, with 0.1855-0.2102 kWh of electricity consumed to generate 1 kg of LA. In Case B, the DPP and LCOE are 5.72-6.32 years and 0.0967-0.1032 USD-kWh\({}^{-1}\), respectively. When the charging pressure is 12 or 14 MPa, the difference in the LCOE between Case A and Case B is within 0.2. When the charging pressure is 16 or 18 MPa, the shorter DPP and lower LCOE are in Case A than in Case B. Figure 6 shows the equipment investment cost, materials investment cost, and LNG purchase cost for both Case A and Case B. At a charging pressure of 12, 14, 16, and 18 MPa. Case B offers a lower equipment investment cost, accounting for USD 28.23, 27.87, 27.63, and 27.34 million, respectively, which is less than that of Case A (USD 28.44, 28.07, 27.78, and 27.54 million, respectively). This can be attributed to the lower costs for the ATs and \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Charging**} & **RTE** & **ERTE** & **SPC** & **DPP** & **LCOE** \\
**Pressure (MPa)** & **(\%)** & **(\%)** & **(kWh\({}_{\text{e}}\cdot\)kg\({}_{\text{LA}}\)\({}^{-1}\))** & **(Year)** & **(USD-kWh\({}^{-1}\))** \\ \hline \multirow{2}{*}{12} & Case A & 47.72 & 69.74 & 0.1855 & 5.73 & 0.0965 \\  & Case B & 47.58 & 69.91 & 0.1855 & 5.72 & 0.0967 \\ \hline \multirow{2}{*}{14} & Case A & 46.40 & 69.51 & 0.1944 & 6.02 & 0.0989 \\  & Case B & 46.31 & 69.73 & 0.1944 & 6.00 & 0.0989 \\ \hline \multirow{2}{*}{16} & Case A & 45.24 & 69.33 & 0.2026 & 6.20 & 0.1011 \\  & Case B & 45.28 & 69.68 & 0.2026 & 6.12 & 0.1008 \\ \hline \multirow{2}{*}{18} & Case A & 42.73 & 69.17 & 0.2098 & 7.06 & 0.1067 \\  & Case B & 44.10 & 69.37 & 0.2102 & 6.32 & 0.1032 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Effect of the air expansion stages on the overall system performance. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Charging Pressure (MPa)** & **10** & **12** & **14** & **16** & **18** \\ \hline Liquefaction ratio (\%) & 83.44 & 86.49 & 85.94 & 85.38 & 84.97 \\ Total input power of the system (kW) & 30,546 & 32,085 & 33,412 & 34,588 & 35,649 \\ Total net output power of the system (kW) & 24,250 & 27,369 & 27,204 & 26,946 & 25,961 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Effect of the charging pressure on the liquefaction ratio, and the input and net output power. The annual total cost and revenue are shown in Figure 7. The power consumption in the ACs increases by over 1 MW for a 2 MPa increase in the charging pressure, and the annual electricity purchase cost increases correspondingly, resulting in an increase in the annual total cost. With constant on-peak and off-peak electricity prices, the power generation is reduced with an increase in the charging pressure, resulting in lower annual total revenue. Therefore, the NPV for Case B charging at 18 MPa is USD 47.43 million, which is less than that at 12 MPa of USD 55.60 million, as shown in Figure 8.  
By incorporating a configuration with the three-stage air expansion and one-stage air superheating, the economic advantages of the proposed system are enhanced.