## 3 Methods
### Thermodynamic Analysis  
Energy- and exergy-based analyses are conducted to investigate the thermodynamic performance of the proposed system. Specific energy and exergy equations for key components are included in Appendix A, Table A1.  
The round-trip efficiency (RTE) reflects the proportion of a given electricity input that can be recovered, and is expressed as the ratio of the electricity net output to the sum of the electricity consumption and low heating value (LHV) of the LNG fuel [32].  
\[\text{RTE}=\frac{(P_{\text{AT}}+P_{\text{GT}}+P_{\text{CT}}-P_{\text{LNGP}}-P_ {\text{LAP}})\cdot t_{\text{dis}}}{(P_{\text{AC}}+\text{LHV}_{27}\cdot m_{27}) \cdot t_{\text{ch}}} \tag{1}\]  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Values** \\ \hline Air mass flow rate \(m_{1}/\text{kg}\cdot\text{h}^{-1}\) & 200,000 \\ Air inlet temperature \(T_{1}/\text{K}\) & 298.1 \\ Air inlet pressure \(p_{1}/\text{MPa}\) & 0.1013 \\ Air outlet pressure \(p_{26}/\text{MPa}\) & 0.1 \\ Liquid air storage pressure \(p_{14}/\text{MPa}\) & 0.1 \\ Liquid air regasification pressure \(p_{16}/\text{MPa}\) & 8 \\ LNG feed temperature \(T_{27}/\text{K}\) & 111.4 \\ LNG feed pressure \(p_{27}/\text{MPa}\) & 0.1 \\ LNG regasification pressure \(p_{28}/\text{MPa}\) & 7 \\ Charging duration \(t_{\text{ch}}/\text{h}\) & 8 \\ Discharging duration \(t_{\text{dis}}/\text{h}\) & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Process design basis [27].  
where \(P_{\rm{AT}}\) refers to the power produced by air turbines (kW), \(P_{\rm{GT}}\) refers to the power produced by the flue gas turbine (kW), \(P_{\rm{CT}}\) refers to the power produced by the cryoturbine (kW), \(P_{\rm{LNCP}}\) refers to the power consumed by the LNG pump (kW), \(P_{\rm{LAP}}\) refers to the power consumed by the liquid air pump (kW), \(t_{\rm{dis}}\) is the discharging period (h), \(P_{\rm{AC}}\) refers to the power consumed by the air compressors (kW), \(L{\rm{HV}}_{\rm{27}}\) refers to the low heating value of the LNG fuel (kJ/kg), \(m_{\rm{27}}\) refers to the mass flow rate of the LNG fuel (kg\(\cdot\)h\({}^{-1}\)), and \(t_{\rm{ch}}\) is the charging period (h).  
Exergy analysis is used to evaluate the energy conversion with different types of energy inputs and outputs, considering the quality of various energy streams [32]. According to the second law of thermodynamics, the overall exergy round-trip efficiency (ERTE) is calculated as Equation (2),  
\[{\rm{ERTE}}=\frac{(P_{\rm{AT}}+P_{\rm{GT}}+P_{\rm{CT}}+Q_{\rm{AC}}+Ex_{\rm{26} }+Ex_{\rm{42}}+Ex_{\rm{43}}+Ex_{\rm{44}}+Ex_{\rm{45}}+Ex_{\rm{46}})\cdot t_{ \rm{dis}}}{P_{\rm{AC}}\cdot t_{\rm{ch}}+(P_{\rm{LNGP}}+P_{\rm{LAP}}+Ex_{\rm{27 }}+{\rm{LHV}}_{\rm{27}}\cdot m_{\rm{27}})\cdot t_{\rm{dis}}} \tag{2}\]  
where \(Q_{\rm{AC}}\) is the stored compression heat (kW), and \(Ex_{\rm{26}}\), \(Ex_{\rm{27}}\), \(Ex_{\rm{42}}\), \(Ex_{\rm{43}}\), \(Ex_{\rm{44}}\), \(Ex_{\rm{45}}\), and \(Ex_{\rm{46}}\) are the exergy of stream 26, 27, 42, 43, 44, 45, and 46 (kW), respectively.  
Specific power consumption (SPC, kWh\({}_{\rm{e}}\)-kg\({}_{\rm{LA}}\)\({}^{-1}\)), namely the electricity required to produce a specific mass of liquid air, is a valid tool for comparing the energy storage efficiency of various LAES systems [3],  
\[{\rm{SPC}}=P_{\rm{AC}}/m_{\rm{LA}} \tag{3}\]  
where \(m_{\rm{LA}}\) is the mass flow rate of the LA (kg\(\cdot\)h\({}^{-1}\)).  
The liquefaction ratio (%), Y, is defined as the ratio of the mass flow of LA to the mass flow of compressed air, \(m_{1}\) (kg\(\cdot\)h\({}^{-1}\)).  
\[{\rm{Y}}=m_{\rm{LA}}/m_{1} \tag{4}\]