## 2 Mathematical Modeling
### Economic Analysis
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Component** & **Cost Function (\$)** & **Reference** \\ \hline \multicolumn{4}{c}{**Gas Cycle**} \\ \hline
1 & Combustion Chamber & \(\frac{28.98\text{m}_{2}}{0.995-\frac{P_{1}}{P_{2}}}(1+exp(0.015T_{3}-1540)\) & [55] \\ \hline
2 & Gas Turbine & \(\frac{301.45\text{m}_{2}}{(0.944-10\text{cm}_{3})}ln\big{(}\frac{P_{1}}{P_{2}} \big{)}(1+\) & [55] \\  & \(exp(0.025T_{3}-1570)\) & \\ \hline
3 & Compressor & \(\frac{44.71\text{m}_{2}}{0.95-\frac{P_{1}}{P_{2}}}\big{(}\frac{P_{1}}{P_{1}} \big{)}ln\big{(}\frac{P_{1}}{P_{1}}\big{)}\) & [55] \\ \hline
4 & Booster Compressor & \(\frac{44.71\text{m}_{2}}{0.95-\frac{P_{1}}{P_{2}}}\big{(}\frac{P_{1}}{P_{1}} \big{)}ln\big{(}\frac{P_{1}}{P_{1}}\big{)}\) & [55] \\ \hline \multicolumn{4}{c}{**Rankine cycle**} \\ \hline
5 & HRSG & \(4745\Big{(}\frac{\text{m}_{6}(\text{m}_{7}-1\text{m}_{5})}{\text{m}_{6}(\text{ m}_{7}-1\text{m}_{5})}\Big{)}^{0.8}+11820\text{m}_{6}+685\text{m}_{4}\) & [55] \\ \hline
6 & Turbine & \(\Big{(}\frac{301.45\text{m}_{2}}{0.94-\text{m}_{7}}\Big{)}ln\big{(}\frac{P_{2} }{P_{2}}\big{)}(1+\exp(0.025T_{7}-1570))\) & [55] \\ \hline
7 & Condenser & \(1773\text{m}_{8}\) & [55] \\ \hline
8 & Pump & \(3540\Big{(}\dot{\text{W}}_{P}\Big{)}^{0.71}\) & [55] \\ \hline \multicolumn{4}{c}{HSR cycle} \\ \hline
9 & Land & \(0.62\Big{(}1.5A_{land}+1.8\times 10^{5}\Big{)}\) & [47; 48; 56; 57] \\ \hline
10 & Mirror & \(126N_{hel}A_{hel}\) & [47; 48; 56; 57] \\ \hline
11 & Wire & \(\sum\limits_{i=1}^{100}N_{hel,cell,i}\Big{[}0.031r_{cell,i}+24\sqrt{\frac{A_{hel} }{\rho_{cell,i}}}\Big{]}\) & [48; 56] \\ \hline
12 & Tower & \(0.78232\times 10^{6}exp(0.01130H_{tow})\) & [56; 58] \\ \hline
13 & Piping & \(\Big{[}3600\frac{D_{tow}}{1.31}+420\frac{D_{tow}}{0.87}\Big{]}H_{tow}+90000 \frac{D_{tow}}{0.87}\) & [23; 48; 59] \\ \hline
14 & Receiver & \(23500A_{re}\) & [23; 48; 59] \\ \hline
15 & Storage tank & \(1.13\Big{(}155.6E_{TES}^{-0.3540}+10.55\Big{)}\) & [60; 61] \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Cost functions of installation and capital investment. \(\rho_{cell,i}\) is calculated as follows [48; 56]:  
\[\rho_{cell,i}=\frac{N_{hd,cell,i}A_{hel}}{A_{cell,i}} \tag{34}\]  
The simple payback period of a system (SPP) is given as [40; 65]:  
\[SPP=\frac{Z_{total}}{ZF} \tag{35}\]  
In which, ZF defines the cash flow of a system per year. For the cash flow of both systems per year following equation can be used [40; 65]:  
\[ZF=Y_{electrical}k_{electrical} \tag{36}\]  
Here, the \(k_{electrical}\) denotes the specific cost of electricity produced by the systems that is considered equal to 0.22 US/kWh [65; 66]. \(Y_{electrical}\) represents the annual electricity production capacity. It should be mentioned that the price of natural gas in this study was 0.07 US/kWh [67]. For both systems, the payback period (PP) equation can be written as [40; 65]:  
\[PP=\frac{ln\Big{(}\frac{ZF}{ZF-rZ_{total}}\Big{)}}{ln(1+r)} \tag{37}\]  
In which, the r is the factor of discount considered as 3% in this study [40; 65]. The internal rate of return (IRR) for a system is given as [40; 65]:  
\[IRR=\frac{ZF}{Z_{total}}\Bigg{[}1-\frac{1}{\left(1+IRR\right)^{N}}\Bigg{]} \tag{38}\]  
Here, N defines the lifetime of a system that is 20 years in this study. Additionally, the net present value (NPV) of a system can be formulated as follows [40; 65]:  
\[NPV=ZF\frac{\left(1+r\right)^{N}-1}{r\left(1+r\right)^{N}}-Z_{total} \tag{39}\]