## 3 Results and Discussion
### Simple Parametric Investigation
Figure 8 includes the results of the influence of solar beam irradiation on collector efficiency. Firstly, it has to be said that the cycle efficiency is not affected by the solar irradiation level. On the other hand, the collector efficiency enhances with the solar potential increase and consequently the system efficiency and the electricity improves. It is remarkable to state that the impact of the solar irradiation on the collector efficiency makes it increase from 63.90% to 75.62%, but the electricity production presents a high increase from 19.16 to 56.66 kW which is  
Figure 10: Cycle efficiency for different combinations of temperature and pressure levels in the turbine inlet. Figure 9: Collector thermal efficiency for different combinations of temperature and pressure levels in the turbine inlet. Figure 11: System efficiency for different combinations of temperature and pressure levels in the turbine inlet. Figure 3:0 Electricity production for different combinations of temperature and pressure levels in the turbine inlet. increase in the pressure level. Figure 12 shows the electricity production results which are in total accordance with the shape of the curves in Figure 11 about the system efficiency. This is a reasonable result because electricity production is the product of the system efficiency and solar energy input. The solar energy input is the same for all the cases in this parametric investigation, and therefore, the electricity production and the system efficiency are proportional parameters. Figure 13 illustrates the collector exergy efficiency and Figure 14 illustrates the system exergy efficiency. The collector exergy efficiency is maximized for TIT in the range of 800 to 850 K, as well as the system exergy efficiency which has similar optimum TIT values. Therefore, it is very interesting to state that the collector exergy map is the critical parameter for the determination of the optimal operating range of the system. In other words, the optimum value of the TIT which maximizes the collector exergy efficiency makes the total system to be optimized in this temperature range. The exergetic performance of the system is lower than the collector exergy efficiency which is a reasonable result, and it is explained by the irreversibilities of the examined real thermodynamic cycle. The collector exergy efficiency assumes that the PTC feeds an ideal Carnot cycle and thus the collector exergy efficiency is an upper limit for the system exergy efficiency if the operation of the PTC is taken into account. In any case, the most important conclusion by the analysis of Figures 13 and 14 together is that designing the system with the optimum TIT according to the collector exergy efficiency is able to lead to the overall optimum designs for all the systems. Moreover, these results indicate that the selection of PTC in the present cycle is a proper choice due to the exergetic compatibility of the PTC and transcritical CO\({}_{2}\) Rankine cycle. In the last part of Section 3.3, two approximating formulas about the system efficiency are given. Equation (53) indicates the optimum values of the TIT which maximizes the system efficiency as a function of the pressure level and the solar beam irradiation. The suggested approximation has a mean absolute percentage error (MAPE) of 0.40% and an R\({}^{2}\) of 99%. These are acceptable and proper values that show a suitable approximation. Figure 15 shows the results about the validity of Equation (53) by comparing the real values calculated by EES and the respected value calculated by Equation (53). Figure 3:1The optimum results of the TIT lead to an optimum system efficiency which is presented in Equation (54). Equation (54) has the high pressure and the solar beam irradiation levels as inputs, while the TIT has the respective optimum value according to Equation (53). This formula has \(\rm R^{2}\) equal to 99.60% and MAPE equal to 0.87% which are acceptable values.