## 7 Results and Discussion
### Mechanical Rotation Angles
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Location** & **Model** & \(\alpha\) **Min** & \(\alpha\) **Max** & \(\beta\) **Min** & \(\beta\) **Max** & \(\alpha\) **Total** & \(\beta\) **Total** \\  & & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) \\ \hline \multirow{7}{*}{Jülich} & AE & \(-67.19\) & \(61.40\) & \(0.00\) & \(82.25\) & \(49\) \(674.96\) & \(50\) \(661.70\) \\  & AE/TA & \(-69.38\) & \(59.35\) & \(0.00\) & \(82.25\) & \(49\) \(675.73\) & \(50\) \(661.70\) \\  & TR & \(0.00\) & \(80.16\) & \(-53.32\) & \(46.83\) & \(43\) \(093.39\) & \(41\) \(960.26\) \\  & TR/TA & \(0.00\) & \(81.65\) & \(-51.62\) & \(46.52\) & \(46\) \(789.12\) & \(39\) \(575.73\) \\  & TA/AE & \(-180.00\) & \(180.0\) & \(0.00\) & \(53.77\) & \(115\) \(182.43\) & \(38\) \(581.16\) \\  & TA/TR & \(-16.89\) & \(42.77\) & \(-46.62\) & \(51.87\) & \(15\) \(496.01\) & \(39\) \(663.53\) \\ \hline \multirow{7}{*}{Protaras} & AE & \(-71.50\) & \(68.65\) & \(0.00\) & \(79.56\) & \(53\) \(776.67\) & \(50\) \(328.07\) \\  & AE/TA & \(-72.50\) & \(64.99\) & \(0.00\) & \(79.56\) & \(53\) \(776.67\) & \(50\) \(328.07\) \\ \cline{1-1}  & TR & \(0.00\) & \(76.33\) & \(-51.56\) & \(47.49\) & \(40\) \(061.38\) & \(43\) \(107.48\) \\ \cline{1-1}  & TR/TA & \(0.00\) & \(78.66\) & \(-47.91\) & \(43.72\) & \(44\) \(306.41\) & \(40\) \(191.39\) \\ \cline{1-1}  & TA/AE & \(-180.00\) & \(180.00\) & \(0.00\) & \(51.87\) & \(107\) \(184.19\) & \(37\) \(739.69\) \\ \cline{1-1}  & TA/TR & \(-11.51\) & \(46.79\) & \(-43.90\) & \(48.12\) & \(18\) \(529.54\) & \(40\) \(318.35\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Minimum and maximum \(\alpha\), \(\beta\) rotation angles, plus total accumulated \(\alpha\), \(\beta\) angular rotations, averaged for the 6 representative heliostats (SW, S, SE, NW, N, NE), from 1 January to 31 December 2024, with rectangular heliostats and 10\({}^{\circ}\) slope, for Julich and Protaras, for the six mechanical models: AE and AE/TA, TR and TR/TA, TA/AE and TA/TR (see Section 2. Heliostat Orientation). The results reported in Table 1 for the AE/TA and TR/TA radial-aligned models are very similar (AE) or even slightly worse (TR) than those reported for the matrix-aligned AE and TR models. Given the increased complexity of these aligned models, where the y-axis is aligned with the target direction instead of simply pointing south-north, as in the matrix-aligned AE and TR models, it seems safe to conclude that they do not represent an improvement over the commonly used AE and TR models. It is important to mention that every morning, the heliostats start from the rest orientation (\(\alpha=0\), \(\beta=0\)) at 08:00 LCT and return to this orientation at the end of the day, at 16:00 LCT. These parking rotations are a significant part of the total rotations performed by each heliostat during the day and therefore contribute to the total rotations reported in Table 1. In Protaras, when parking is not applied, the total annual \(\alpha\)-\(\beta\) rotations decrease by 0.1-77.4% (AE), 0.1-77.4% (AE/TA), 1.4-45.4% (TA/AE), 90.3-0.1% (TR), 70.2-0.1% (TR/TA), and 29.3-0.1% (TA/TR). Clearly, parking primarily affects the axis controlling the altitude: the secondary axis, b, in azimuth-elevation models (AE, AE/TA, and TA/AE) and the primary axis, a, in tilt-roll models (TR, TR/TA, and TA/TR). To show how the various heliostats behave throughout the year, Figures 14-17 represent the total (\(\alpha+\beta\)) daily rotation for the six representative heliostats (SW, S, SE, NW, N, and NE; see Figure 8) during the year 2024, for the azimuth-elevation (AE) and tilt-roll (TR) models (arguably the heliostat models most commonly used in research and production) in Julich and Protaras. Although the general trends are similar for Julich and Protaras, rotations are larger in Protaras for the AE model (Figures 14 and 15).