# An Integration Model for Flux Density Distribution Formed by a Heliostat
## Appendix A Location Determination
### Calculation of Geometrical Optics  
Then, the reflected ray can be calculated according to the incident vector \((u_{sun},v_{sun},w_{sun})\) and the normal vector of the heliostat grid point \((u_{ht},v_{hm},w_{ht})\):  
\[u_{ray}=2(u_{ht}u_{sun}+v_{hn}v_{sun}+w_{hn}w_{sun})u_{hm}-u_{sun} \tag{17}\]  
\[v_{ray}=2(u_{ht}u_{sun}+v_{hn}v_{sun}+w_{hn}w_{sun})v_{hn}-v_{sun} \tag{18}\]  
\[w_{ray}=2(u_{ht}u_{sun}+v_{hn}v_{sun}+w_{hn}w_{sun})w_{hm}-w_{sun} \tag{19}\]  
The receiver point, which coordinates in the global coordinate system, is formed when a reflected ray hits the receiver plane:  
\[t=\frac{z_{receiver}w_{rn}+y_{receiver}v_{rn}+x_{receiver}u_{rn}-u_{rn}x_{h}-v_{ rn}y_{h}-w_{rn}z_{h}}{u_{rn}u_{ray}+v_{rn}v_{ray}+w_{rn}w_{ray}}, \tag{20}\]  
\[x_{tf}=x_{h}+u_{ray}\times t \tag{21}\]  
\[y_{tf}=y_{h}+v_{ray}\times t \tag{22}\]  
\[z_{tf}=z_{h}+w_{ray}\times t \tag{23}\]  
The receiver point \(O_{3}\) on the receiver is converted from the global coordinate system to the receiver plane coordinate system:  
\[(x_{rf},y_{rf},z_{rf})_{receiver}=(x_{rf}-x_{receiver},y_{rf}-y_{receiver},z_{rf}-z_{receiver})_{global}\times AR. \tag{24}\]  
Heliostat grid point coordinates are converted from the global coordinate system to the receiver plane coordinates:  
\[(x_{h},y_{h},z_{h})_{receiver}=(x_{h}-x_{receiver},y_{h}-y_{receiver},z_{h}-z_{receiver})_{global}\times AR \tag{25}\]  
The \(O_{1}O_{2}\) vector is:  
\[u_{O_{1}O_{2}}=-(x_{r}-x_{h}) \tag{26}\]  
\[v_{O_{1}O_{2}}=-(y_{r}-y_{h}) \tag{27}\]  
\[w_{O_{1}O_{2}}=-(z_{r}-z_{h}) \tag{28}\]  
The angle between \(O_{1}O_{2}\) and the heliostat normal vector is:  
\[\beta_{1}=acos(\frac{u_{O_{1}O_{2}}u_{hn}+v_{O_{1}O_{2}}v_{hn}+w_{O_{1}O_{2}} w_{hm}}{\sqrt{u_{O_{1}O_{2}}^{2}+v_{O_{1}O_{2}}^{2}+w_{O_{1}O_{2}}^{2}}}) \tag{29}\]  
The angle between \(O_{1}O_{2}\) and the receiver normal vector is:  
\[\beta_{2}=acos(\frac{u_{O_{1}O_{2}}u_{m}+v_{O_{1}O_{2}}v_{m}+w_{O_{1}O_{2}} w_{rn}}{\sqrt{u_{O_{1}O_{2}}^{2}+v_{O_{1}O_{2}}^{2}+w_{O_{1}O_{2}}^{2}}}) \tag{30}\]Firstly, we calculated the distances between points in the coordinate system of the receiver plane:  
(1) The distance between receiver point \(O_{3}\) and heliostat grid point \(O_{1}\) is:  
\[l_{hrf}=\sqrt{(x_{h}-x_{rf})^{2}+(y_{h}-y_{rf})^{2}+(z_{h}-z_{rf})^{2}}\] (A31)  
and \(l_{hrf}\) in the X direction and the Y direction are, respectively:  
\[l_{hrfx}=\sqrt{(x_{h}-x_{rf})^{2}+(z_{h}-z_{rf})^{2}}\] (A32)  
\[l_{hrfy}=\sqrt{(y_{h}-y_{rf})^{2}+(z_{h}-z_{rf})^{2}}\] (A33)  
(2) The distance between receiver point \(O_{3}\) and receiver plane grid point \(O_{2}\) is:  
\[l_{rfr}=\sqrt{(x_{r}-x_{rf})^{2}+(y_{r}-y_{rf})^{2}+(z_{r}-z_{rf})^{2}}\] (A34)  
and \(l_{rfr}\) in the X direction and the Y direction are, respectively:  
\[l_{rfrx}=\sqrt{(x_{r}-x_{rf})^{2}+(z_{r}-z_{rf})^{2}}\] (A35)  
\[l_{rfry}=\sqrt{(y_{r}-y_{rf})^{2}+(z_{r}-z_{rf})^{2}}\] (A36)  
(3) The distance between heliostat grid point \(O_{1}\) and receiver plane grid point \(O_{2}\) is:  
\[l_{rh}=\sqrt{(x_{r}-x_{h})^{2}+(y_{r}-y_{h})^{2}+(z_{r}-z_{h})^{2}}\] (A37)  
and \(l_{rh}\) in the X direction and the Y direction are, respectively:  
\[l_{rhx}=\sqrt{(x_{r}-x_{h})^{2}+(z_{r}-z_{h})^{2}}\] (A38)  
\[l_{rhy}=\sqrt{(y_{r}-y_{h})^{2}+(z_{r}-z_{h})^{2}}\] (A39)  
Then, the included angle between the reflected ray and \(O_{1}O_{2}\) of each grid can be calculated, as shown in Figures A1-A3:  
\[\theta_{x}=acos(\frac{l_{rhy}^{2}+l_{hrfx}^{2}-l_{rfx}^{2}}{2l_{hrx}l_{hrfx}})\] (A40)  
\[\theta_{y}=acos(\frac{l_{rhy}^{2}+l_{hrfy}^{2}-l_{rfy}^{2}}{2l_{rhy}l_{hrfy}})\] (A41)  
\[\theta=acos(\frac{l_{rh}^{2}+l_{hrf}^{2}-l_{rfr}^{2}}{2l_{rh}l_{hrf}})\] (A42)Figure 17: The included angle between the reflected ray and \(O_{1}O_{2}\) of each grid under the circular Gaussian model.