## 3 Mathematical Modeling
### Multi-Objective Optimization
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Cost Balance Equation** & **Auxiliary Equation** \\ \hline Compressor & \(\dot{\text{C}}_{1}+\dot{\text{C}}_{\text{WC}}+\dot{\text{Z}}_{\text{C}}=\dot{ \text{C}}_{2}\) & \(c_{\text{wc}}=0.05\) \\ TES1 & \(\dot{\text{C}}_{2}+\dot{\text{C}}_{15}+\dot{\text{Z}}_{\text{TESI}}=\dot{ \text{C}}_{3}+\dot{\text{C}}_{16}\) & \(c_{2}=c_{3}\) \\ TES2 & \(\dot{\text{C}}_{3}+\dot{\text{C}}_{13}+\dot{\text{Z}}_{\text{TESI}}=\dot{ \text{C}}_{4}+\dot{\text{C}}_{14}\) & \(c_{3}=c_{4}\) \\ Condenser & \(\dot{\text{C}}_{4}+\dot{\text{Z}}_{\text{cond}}=\dot{\text{C}}_{5}\) & - \\ Valve & \(\dot{\text{C}}_{6}+\dot{\text{Z}}_{\text{valve}}=\dot{\text{C}}_{7}\) & - \\ Evaporator & \(\dot{\text{C}}_{7}+\dot{\text{Z}}_{\text{evap}}=\dot{\text{C}}_{8}\) & - \\ TES3 & \(\dot{\text{C}}_{8}+\dot{\text{C}}_{17}+\dot{\text{Z}}_{\text{TESI}}=\dot{ \text{C}}_{9}+\dot{\text{C}}_{18}\) & \(c_{18}=c_{17}\) \\ TES4 & \(\dot{\text{C}}_{9}+\dot{\text{C}}_{19}+\dot{\text{Z}}_{\text{TESI}}=\dot{ \text{C}}_{10}+\dot{\text{C}}_{20}\) & \(c_{20}=c_{19}\) \\ Turbine & \(\dot{\text{C}}_{10}+\dot{\text{Z}}_{\text{T}}=\dot{\text{C}}_{11}+\dot{\text{C}}_{ W_{t}}\) & \(c_{10}=c_{11}\) \\ Cooler & \(\dot{\text{C}}_{11}+\dot{\text{C}}_{21}+\dot{\text{Z}}_{\text{Cooler}}=\dot{ \text{C}}_{12}+\dot{\text{C}}_{22}\) & \(\dot{\text{C}}_{21}=0\), \(c_{12}=c_{11}\) \\ LST & \(\dot{\text{C}}_{5}+\dot{\text{Z}}_{\text{LST}}=\dot{\text{C}}_{6}\) & - \\ GST & \(\dot{\text{C}}_{12}+\dot{\text{Z}}_{\text{GST}}=\dot{\text{C}}_{1}\) & - \\ CWT & \(\dot{\text{C}}_{18}+\dot{\text{Z}}_{\text{CWT}}=\dot{\text{C}}_{13}\) & - \\ HWT & \(\dot{\text{C}}_{14}+\dot{\text{Z}}_{\text{HWT}}=\dot{\text{C}}_{17}\) & - \\ CST & \(\dot{\text{C}}_{20}+\dot{\text{Z}}_{\text{CST}}=\dot{\text{C}}_{15}\) & - \\ HST & \(\dot{\text{C}}_{16}+\dot{\text{Z}}_{\text{HST}}=\dot{\text{C}}_{19}\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cost balance modeling of each system component. reserving the parent elites, and proposing the crowded-comparison operator to ensure the diversity in the population. According to the reference, the corresponding optimization objectives in this paper are the system energy storage efficiency (\(\eta_{\text{Stor}}\)) and the unit power product cost (\(c_{\text{P,tot}}\)), and the optimization issues can be described as follows:  
\[ \begin{cases}Max(\eta_{\text{Stor}}(\overset{\rightarrow}{\mathbf{x}}))\\ Min\ (c_{\text{P,tot}}(\overset{\rightarrow}{\mathbf{x}}))\end{cases}  \tag{22}\]  
\[\overset{\rightarrow}{\mathbf{x}}=[P_{\text{liq}},P_{\text{evap}},\eta_{\text {C}},\eta_{\text{T}},\Delta\text{dT}_{1},\Delta\text{dT}_{2}]^{\text{T}} \tag{23}\]  
\[\overset{\rightarrow}{\mathbf{x}}\in[L,U] \tag{24}\]  
\(\overset{\rightarrow}{\mathbf{x}}\) is the variable vector for the optimization problem. \(P_{\text{liq}}\) refers to the liquid CO\({}_{2}\) pressure, \(\eta_{\text{C}}\) and \(\eta_{\text{T}}\) are the compressor isentropic efficiency and turbine isentropic efficiency, \(P_{\text{evap}}\) represents CO\({}_{2}\) evaporation pressure, \(\Delta\text{dT}_{1}\) and \(\Delta\text{dT}_{2}\) are the hot terminal temperature difference and cold terminal temperature difference in the TES2. The variation range of \(\overset{\rightarrow}{\mathbf{x}}\) is limited to the ceiling and floor array, while \(L\) means the minimum values and \(U\) means the maximum values.