## 2 Mathematical Modeling
### Economic Analysis  
An economic analysis can be employed to provide the economic performance of a system. To evaluate the economic profitability of two systems, capital investment costs and installation costs were obtained. The total cost of the one system was calculated by the following equation [53]:  
\[Z_{total}=Z_{ins}+Z_{ini}+Z_{OM}+Z_{if}+Z_{cont}+Z_{dec}+Z_{lab} \tag{32}\]  
In which, the subscripts installation, initial, OM, if, cont, dec, and lab denote installation, initial, operation and maintenance, indirect factor, unexpected technological and regulatory issues, decommissioning at the end of the project lifetime, and labor cost. The cost associated with installation cost was about 20% of the initial cost. Operation and maintenance costs were about 4% of the initial cost. The indirect factor, unexpected technological and regulatory issues, and decommissioning at the end of project lifetime were assumed to be 5%, 10%, and 5% of the initial cost of the system [49; 54]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Component** & **Exergy Efficiency** & **Exergy Destruction Rate** \\ \hline \hline
1 & Receiver & \(\frac{\dot{Q}_{av}\big{(}1-\frac{T_{0}}{T_{nr}}\big{)}}{\dot{\text{m}}(ex_{ 2}-ex_{3})+\dot{E}x_{Q}}\) & \(\dot{\text{m}}(ex_{2}-ex_{3})+\dot{Q}_{\text{loss}}\big{(}1-\frac{T_{0}}{T_{nr} }\big{)}\) \\ \hline
2 & Pump & \(\frac{\dot{\text{m}}_{1}(ex_{2}-ex_{1})}{\dot{\text{W}}_{p}}\) & \(\dot{\text{m}}(ex_{1}-ex_{2})+\dot{\text{W}}_{p}\) \\ \hline
3 & Storage Tank & \(\frac{\dot{\text{m}}(ex_{3}-ex_{1})}{\dot{\text{m}}(ex_{3}-ex_{3})}\) & \(\dot{\text{m}}(ex_{3}-ex_{1})-\dot{\text{m}}(ex_{6}-ex_{5})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Equations of exergy balance and exergy destruction rate for the HSR cycle [33; 34; 35]. The labor costs for the system is given as follows [48]:  
\[Z_{lab}=1.5\sum\limits_{staff}sal_{stf}N_{stf}N_{years} \tag{33}\]  
In which, \(sal_{stf}\), \(N_{stf}\), and \(N_{years}\) respectively represent the salary of staff, numbers of staff, and project lifetime that is considered in 20 years. The number 1.5 was considered for extra unpredictable costs. For an integrated system (a), GT-HRSG, following Table 7 is presented for cost functions of installation and capital investment. In Table 7, P, W, and in denote pressure, power rate, and mass flow rate. Subscripts GT, comp, BC, T, and P respectively define the gas turbine, the compressor, the booster compressor, the turbine, and the pump. Moreover, \(A_{land}\), \(A_{hel}\), and \(A_{re}\) are respectively the area of the land, heliostat, and receiver. \(N_{hel}\) defines the number of heliostats. \(N_{hel,cell,i}\) defines the number of the heliostat in cell \(i\), as the field is divided into 100 cells [62]. \(\rho_{cell,i}\) denotes the heliostat density in \(A_{cell,i}\), which is the area of cell \(i\). \(r_{cell,i}\) represents the distance of the cell's center and the receiver. Further information can be obtained from Reference [47; 57; 62; 63; 64]. ETES is the amount of energy stored in the storage tank [60]. Other subscripts int and outer define the interior and outer to express the diameter (D) of concentric piping. For more information, Reference [23; 48; 59] can be used.