## 2 Tracking Error Model of Azimuth Axis Tilt Influence
### Tracking Error Model of the SDC System  
An SDC system controls the output angle of a DATD by a solar position algorithm (SPA) and tracks the sun in real time from the initial tracking position. However, after the azimuth axis inclines, the SDC system will produce a tracking error. In this paper, an SPA [28] is used to derive the information of the sun's position (the azimuth angle and elevation angle) at any time and any place. At any time, the sun's ray vector is expressed by \(\text{N}_{\text{s}}\), which can be calculated by Equation (5), where \(\alpha\) and \(\beta\) are the sun's azimuth and elevation angles, respectively. In the process where an SDC system tracks the sun, the focal axis needs to rotate at a certain angle around the azimuth axis, and then rotate at a certain angle around the elevation axis so that the focal axis vector of the concentrator aiming at the sun's center can be obtained. The main steps are as follows:  
\[\text{N}_{\text{s}}=(-\text{cos}\alpha\text{sin}\beta,\text{ sin}\alpha\text{cos}\beta,\text{ sin}\beta) \tag{5}\]  
(1) The elevation axis vector \(\text{N}_{\text{e1}}\) rotates the azimuth axis vector \(\text{N}_{\text{a1}}\) around \(\alpha\) angle and the elevation axis vector \(\text{N}_{\text{e2}}\) after tracking the sun is obtained, which can be calculated by Equation (6). In this study, it is assumed that the elevation axis and the concentrator focal axis have no initial axis system error, i.e., \(\text{N}_{\text{e1}}=\text{N}_{\text{e}}\) and \(\text{N}_{\text{f1}}=\text{N}_{\text{f}}\). \[\text{N}_{\text{e2}}=\text{N}_{\text{e1}}\cdot\text{Rot}(\text{N}_{\text{a1} },\alpha) \tag{6}\]  
\[\text{Rot}(\text{n},\theta)=\left[ \begin{array}{cc}\text{cos}\theta+\text{ n}_{\text{e}}^{2}(1-\text{cos}\theta)&\text{n}_{\text{e}}\text{n}_{\text{e}} \text{n}_{\text{f}}(1-\text{cos}\theta)+\text{n}_{\text{e}}\cdot\text{sin} \theta&\text{n}_{\text{e}}\text{n}_{\text{e}}(1-\text{cos}\theta)-\text{n}_{ \text{e}}\cdot\text{sin}\theta\\ \text{n}_{\text{e}}\text{n}_{\text{e}}(1-\text{cos}\theta)-\text{n}_{\text{e}} \cdot\text{sin}\theta&\text{cos}\theta+\text{n}_{\text{e}}^{2}(1-\text{cos} \theta)&\text{n}_{\text{e}}\text{n}_{\text{e}}(1-\text{cos}\theta)+\text{n}_{ \text{e}}\cdot\text{sin}\theta\\ \text{n}_{\text{e}}\text{n}_{\text{e}}(1-\text{cos}\theta)+\text{n}_{\text{e}} \cdot\text{sin}\theta&\text{n}_{\text{e}}\text{n}_{\text{e}}(1-\text{cos} \theta)-\text{n}_{\text{e}}\cdot\text{sin}\theta&\text{cos}\theta+\text{n}_{ \text{e}}^{2}(1-\text{cos}\theta)\end{array} \right] \tag{7}\]  
where \(\text{Rot}(\text{n},\theta)\)[16] is the function to ensure that the vector \(\text{P}\in\mathbb{R}^{1\times 3}\) is rotated by angle \(\theta\) around the unit vector \(\text{n}=[\text{n}_{\text{x}},\text{n}_{\text{y}},\text{n}_{\text{z}}]\). Simplifying the substitution of Equation (4) into Equation (6), we get Equation (8):  
\[\text{N}_{\text{e2}}=\left[ \begin{array}{c}\theta_{1}^{2}(1-\text{cos} \alpha)\cdot\text{sin}\phi_{1}\text{cos}\phi_{1}+\text{sin}\alpha\\ \theta_{1}^{2}(1-\text{cos}\alpha)\cdot\text{cos}^{2}\phi_{1}+\text{cos}\alpha \\ \theta_{1}\cdot[\text{sin}(\phi_{1}+\alpha)-\text{sin}\phi_{1}]\end{array} \right]^ {\text{T}} \tag{8}\]  
(2) The concentrator focal axis vector \(\text{N}_{\text{f1}}\) rotates \(\alpha\) angle around the azimuth axis vector \(\text{N}_{\text{a1}}\), which can be calculated by Equation (9). It then rotates \(\beta\) angle around the elevation axis to obtain the focal axis vector \(\text{N}_{\text{f2}}\) after tracking the sun, and can be calculated by Equation (11). \[\text{N}^{\prime}_{\text{f2}}=\text{N}_{\text{f1}}\cdot\text{Rot}(\text{N}_{ \text{a1}},\alpha) \tag{9}\]Simplifying the substitution of Equation (4) into Equation (9), we get Equation (10):  
\[\mathrm{N^{\prime}}_{\mathrm{f2}}=\left[ \begin{array}{c}\theta_{1}\cdot[\mathrm{ sin}\phi_{1}-\mathrm{sin}(\phi_{1}+\alpha)]\\ \theta_{1}\cdot[\mathrm{cos}\phi_{1}-\mathrm{cos}(\phi_{1}+\alpha)]\\ -1\end{array}