## 7 Results and Discussion
### Mechanical Rotation Issues  
Close inspection of Table 1 shows some interesting results. Models AE and AE/TA show exactly the same yearly total rotations. These models differ only in the azimuth (\(\alpha\)) angle, so this equality was expected for the altitude (\(\beta\)) angle. However, detailed software testing shows that the total \(\alpha\) rotations (updating every 30 s) for a workday (from 08:00 to 16:00 LCT) across all 366 days of 2024 are exactly equal for the AE and AE/TA models (within six decimal places), as differences in the morning are compensated for in the afternoon. The small difference observed for Julich in Table 1 is due to the fact that it is night at 16:00 LCT in winter (from 3 November to 21 January), so no rotations occur at that time, resulting in a break in the daily symmetry. For 30 s at Julich (31 March 2024 at 09:11 LCT) and 60 s at Protaras (28 February 2024 at 09:26 and 09:27 LCT), the model TA/AE is unable to align the SW heliostat (see Figure 8). No such misfit was registered for any other heliostat or mechanical model during the whole year. This occurs because at these precise moments, the \(\beta\) angle tends to 0\({}^{\circ}\). At Protaras, as time progresses, \(\beta\) keeps decreasing until it reaches 0.152\({}^{\circ}\), then the misfit occurs, and then \(\beta\) starts increasing again, starting from 0.044\({}^{\circ}\). In Julich, \(\beta\) keeps decreasing until it reaches 0.084\({}^{\circ}\) and stops aligning, and then \(\beta\) starts increasing again, starting from 0.193\({}^{\circ}\). In the AE and AE/TA models, this problem cannot happen because an aligned heliostat cannot point to the sky (\(\beta\) = 0\({}^{\circ}\)). However, in the TA/AE model, this is possible when the sun is exactly at the target position as seen from the heliostat (when the vector sun, \(\overrightarrow{s}\), and the vector target, \(\overrightarrow{t}\), are aligned; see Figure 4a). Admittedly, this misfit is partially due to the very strict convergence criterion used throughout this work: both the \(\alpha\) and \(\beta\) angles must converge to less than 10\({}^{-10}\) rad in fewer than 1000 iterations for the mirror orientation task to be successfully accepted. Easing the convergence criterion to 10\({}^{-9}\) rad resolves the problem (for the time instances mentioned above, the \(\beta\) angle becomes 0.135\({}^{\circ}\) at Julich and 0.093\({}^{\circ}\) and 0.038\({}^{\circ}\) at Protaras). Increasing the number of iterations, even to 100 000, does not resolve the problem. In all azimuth-elevation models (AE, AE/TA, and TA/AE), it is assumed that the \(\beta\) angle ranges from 0\({}^{\circ}\) to some positive value, usually not exceeding 90\({}^{\circ}\) (corresponding to the polar angle in spherical coordinates). In AE and AE/TA models, this angle cannot be negative; otherwise, the heliostat would be pointing away from the target. In the TA/AE model, when the sun is above the target (in summer, around 12:00 LCT), the heliostat needs to rotate upward, making the \(\beta\) angle positive (the x-axis points westerly; see Equation (6) and Figure 4a), and relatively small \(\alpha\) rotation angles are needed to achieve the required orientation. However, when the Sun is below the target center (in winter, around 08:00 and 16:00 LCT), the heliostat needs to rotate downwards, resulting in a negative \(\beta\) angle. As the \(\beta\) angle is always assumed to be positive (as if the mechanical axis only rotates to one side), the \(\alpha\) angle needs to rotate close to 180\({}^{\circ}\) to meet the required orientation. This is why the \(\alpha\) angle needs to rotate so widely in this TA/AE model, from \(-\)180\({}^{\circ}\) to \(+\)180\({}^{\circ}\). This problem affects the front row heliostats more (see the working range required for the \(\alpha\) angle with the TA/AE model in Table 2), as they are more prone to see the sun below the target (see Figure 18).