## 7 Results and Discussion
### Daily Heliostat Efficiency (DHE)  
To start integrating all these data, an hourly heliostat efficiency (HHE) can be defined:  
\[\mathrm{HHE}=1-S-B \tag{11}\]  
where \(S\) and \(B\) represent the fraction of shading and blocking occurring at that hour. When the altitude is negative (night-time), \(\mathrm{HHE}=0\). This quantity in turn can be used to define a daily heliostat efficiency (DHE):  
\[\mathrm{DHE}=\frac{0.5\mathrm{HHE_{8}}+\sum_{i=9}^{15}\mathrm{HHE_{i}}+0.5 \mathrm{HHE_{16}}}{8} \tag{12}\]  
DHE can be plotted over the year, as shown in Figures 10-13, for Julich and Protaras and for rectangular and square heliostat configurations. Each figure shows independent curves for the six representative heliostats (colored green in Figure 9) simulated in our work.  
In all four of these configurations, efficiency is close to 1.0 for the front-row heliostats (SW, S, and SE) but drops significantly for the other heliostats, even in summer, due to shading/blocking events. This was expected because front heliostats cannot be blocked and can only suffer some east-west shading, mainly in spring and autumn, when the sun's azimuth is closer to the east (90\({}^{\circ}\)) or west (270\({}^{\circ}\)), because in winter, the sun comes more from south (the azimuth is closer to 180\({}^{\circ}\)).  
Clearly, the three back heliostats (NW, N, and NE) seem to be much more representative of the average heliostat functionality for the whole field than the front heliostats. Comparing only the three back heliostats, for all four configurations shown here, the NE heliostat has the lowest efficiency, and the N heliostat has the highest. This happens because the sun altitude in Julich and Protaras is always higher at 08:00 than at 16:00 LCT, as these locations are in the easterly side of the time zone. The NE heliostat cannot be shaded at 8 h (clear view to the sun) but is significantly shaded at 16 h. The NW heliostat is shaded at 8 h but not at 16 h (clear view to the sun), so the NW efficiency is better than for NE. The central N heliostat is geometrically closer to the target than the corner NW and NE heliostats, so the ray angles from the N heliostat to the target are substantially higher, thus reducing blocking events.  
As expected, efficiency losses are much higher in Julich than in Protaras, particularly in winter, as the altitude of sun trajectories decreases rapidly with latitude, resulting in more shading events.  
In Julich and Protaras, efficiency is much higher for rectangular than for square heliostats. This happens because our square heliostats are taller (2.0 m) than the rectangular heliostats (1.6 m) with the same area (4 m\({}^{2}\)), making the occurrence of blocking events in the upper part of front heliostats more probable. Moreover, to guarantee that heliostat mirrors are 1.0 m apart, even when rotated 90\({}^{\circ}\), the north-south distance between the rectangular heliostats is larger (1.90 m) than in square heliostats (1.0 m), thus increasing the space around each heliostat and decreasing the probability of shading/blocking events.  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **JSF\({}_{1}\)** & **JSF\({}_{2}\)** & **JSF\({}_{3}\)** & **PRS\({}_{123}\)** \\ \hline AHE & 0.8052 & 0.7822 & 0.7985 & 0.9989 \\ ATE & 0.3600 & 0.3625 & 0.3501 & 0.5498 \\ \hline \end{tabular}
\end{table}
Table 4: AHE and ATE for improved configurations: JSF\({}_{1}\) (JSF with heliostat distance of 2 m), JSF\({}_{2}\) (JSF with tower height of 20 m), JSF\({}_{3}\) (JSF with slope of 20\({}^{\circ}\)), and PRS\({}_{123}\) (PRS with the three previous improvements combined). Results for the original configurations (JSF and PRS) can be seen in Table 2 (AHE) and Table 3 (ATE).