## 7 Results and Discussion
### Mechanical Rotation Issues
At Protaras, the \(\alpha\) angle in the back row heliostats ranges only from \(-\)93.02\({}^{\circ}\) to \(+\)100.81\({}^{\circ}\) (still a large range) throughout the whole year. The land slope of 10\({}^{\circ}\) used throughout this work (see Section 5, Computational Details) also increases the height of the back row heliostats, emphasizing these differences in the \(\alpha\) angles. As expected, these issues work differently in the TA/TR model. At Protaras on 28 February 2024, both \(\alpha\) and \(\beta\) angles change from negative to positive values around 09:27 LCT (for heliostat S, see Figure 8), where the two angles reach a combined minimum, with \(\alpha=0.006^{\circ}\) and \(\beta=-0.038^{\circ}\). At Julich on 31 March 2024, both \(\alpha\) and \(\beta\) angles again change from negative to positive values around 09:12 LCT (also for heliostat S), reaching a combined minimum with \(\alpha=0.004^{\circ}\) and \(\beta=0.135^{\circ}\). At both instances, the Sun is exactly aligned with the target (as seen from heliostat S), the model performs well, and the flux radiation image collected at the target is accurate. The angular range needed to align the six heliostats (see Table 3) throughout the whole year is surprisingly small for both \(\alpha\) and \(\beta\) angles: \([-16.89^{\circ},46.79^{\circ}]\) and \([-46.62^{\circ},51.87^{\circ}]\), respectively (see Table 1). Again, the front row heliostats require more negative \(\alpha\) angles than the back row heliostats (see the working range required for the \(\alpha\) angle, with the TA/TR model, in Table 3). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Angle**} & \multicolumn{6}{c}{**Heliostat**} \\ \cline{3-8}  & & **SW** & **S** & **SE** & **NW** & **N** & **NE** \\ \hline \multirow{2}{*}{Jülich} & \(\alpha\) Min (\({}^{\circ}\)) & \(-180.00\) & \(-180.00\) & \(-180.00\) & \(-180.00\) & \(-101.12\) & \(-179.60\) \\  & \(\alpha\) Max (\({}^{\circ}\)) & 180.00 & 180.00 & 180.00 & 179.70 & 102.19 & 179.90 \\ \hline \multirow{2}{*}{Protaras} & \(\alpha\) Min (\({}^{\circ}\)) & \(-180.00\) & \(-180.00\) & \(-180.00\) & \(-93.02\) & \(-88.56\) & \(-83.65\) \\  & \(\alpha\) Max (\({}^{\circ}\)) & 180.00 & 180.00 & 180.00 & 88.41 & 94.14 & 100.81 \\ \hline \end{tabular}
\end{table} 
Table 2: Minimum and maximum \(\alpha\) angles for the TA/AE model, for Julich and Protaras, and for the six representative heliostats: SW, S, SE, NW, N, and NE (see Figure 8). The angular range is larger for the front row and for Julich. Figure 18: For simplicity, it is assumed that this image represents the heliostat field for TA/AE model when daily working period starts (at 08:00 LCT in our simulations). Initially, all heliostats are at parking orientation (\(\alpha=\beta=0^{\circ}\)), with the mirrors pointing to the target (defined by vector \(\overset{\rightarrow}{t}\)). The front row heliostats see the Sun (defined by vector \(\overset{\rightarrow}{s}\)) below the target, so they must rotate down (\(\beta<0^{\circ}\)) to align. The back row heliostats see the Sun above the target, so they must rotate up (\(\beta>0^{\circ}\)) to align. The land slope is set to \(0^{\circ}\) for simplicity. The issues described above for the TA/AE model could be solved, for example, using the following two approaches: (1) every time the \(\alpha\) angle is outside the \([-90^{\circ},90^{\circ}]\) range (likely because the \(\beta\) angle is negative, necessitating a large \(\alpha\) rotation), the direction of the \(\beta\) rotation could be inverted (inverting the axis b); (2) the initial (parking) angle \(\beta=0^{\circ}\) could be redefined to point to the sky instead of pointing to the target, ensuring that the \(\beta\) angle is always positive (with the x-axis pointing easterly, as in the other azimuth-elevation models).