## 4 Results and Discussion
### Multi-Objective Optimization  
In the previous section, we analyzed three performance parameters as specific work, overall efficiency, and cost of core equipment. For the S-CO\({}_{2}\) power cycle using solar energy, we were concerned about the net power--that the system could provide in the operation condition and the purchase cost of the system during the construction period. Thus, we chose the specific work and cost of core equipment during the optimization. The objective of this study is to maximize the system net power and, in the meantime, minimize the investment cost. Table 5 gives the selected decision variables and its corresponding range. The multi-objective optimization process is conducted with GA and the solver is MATLAB R2019b. The basic steps of genetic algorithm include coding, population initialization, fitness calculation, evolution (composed of selection, crossover, mutation), and iteration. In this work, the optimal setting of the GA algorithm is set to 50 for each generation of population at population initialization. Moreover, when the genetic algebra reaches 400, the calculation terminates, which can limit the number of iterations. The reason for this set is--we find that if the total number of calculations is less than this set, the results have much more randomness. When we chose these two parameters, the results of the multiple calculations had little change. Based on the three research cycles, the optimal Pareto fronts are presented in Figures 10-12. The optimal Pareto fronts show the trade-off between the cost of core equipment and the specific work under the optimum conditions. In addition, the scatter points in the figures are for each working condition and the optimization results of the calculation process. It is worth noting that there is a significant discontinuity in the Pareto optimization results in Figures 10-12, especially when the specific work is low. The results from the algorithm and its capability of evolution can first analyze the first-generation cluster. Then, the algorithm keeps the representative individuals and proceeds. These representative individuals are selected from the clusters of best performance. When a series of results barely improve through optimization, a few elite individuals are further selected. Hence, this results in a smaller portion of the global Pareto front with fewer solutions condensing. However, this discontinuity has no effect on the analysis of the results and the comparison of the systems. From Figure 10, the values of the optimization objectives for a simple recuperated cycle increased from 30 to 70 kW/kg for specific work, and from $155,000 to $210,000 USD for the cost of core equipment. Outside these ranges, we can conclude that--for the poorer performance solutions--the cost cannot reduce, or, due to thermodynamic limits, the system net power cannot increase. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Decision Variables** & **Range** & **Unit** \\ \hline Turbine inlet temperature & 300–400 & \({}^{\circ}\)C \\ Turbine inlet pressure & 14–20 & MPa \\ Compressor inlet pressure & 7.5–9.5 & MPa \\ Compressor inlet temperature & 32–45 & \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Set of decision variables and their range of variation. Figure 11 presents the result of intercooling configuration. The values of the optimization objectives extend between 27 and 65 kW/kg for specific work and from $255,000 to $365,000 USD for the cost of core equipment. Compared with a simple recuperated cycle, the economic cost of the intercooling cycle is too expensive and the specific work has not been significantly improved. Moreover, it can be concluded that the optimization result of the range of cost of the core equipment in the intercooling cycle is larger than the optimization result range of the simple recuperated cycle. Hence, for this cycle, reasonable selection of system operation parameters is more important for the cost reason. Figure 12 shows the optimization result of the recompression cycle.