# HCL Control Strategy for an Adaptive Roadway Lighting Distribution
## 2 Materials and Methods
### Zonal Flux  
An imaginary spherical surface is used for calculating luminous flux, as shown in Figure 2, with point O as the center of the sphere and radius \(R\). If luminous flux in a certain direction is required, it can be calculated by multiplying the luminous intensity with the solid angle from that particular direction. Therefore, when a certain direction is \((\theta,\varphi)\), light intensity is \(I(\theta,\varphi)\), spherical projection area is \(ds\), solid angle is \(d\Omega\), luminous flux is \(d\varphi\) and the total zonal flux of the sphere is \(\phi\), the equations are shown in Formula (1)-(4). If the beam pattern is display in a symmetrical light distribution condition, that is, \(I(\theta,\varphi)=I(\theta)\), then the total zonal flux can be calculated by Formula (5) [36]:  
\[dS=Rd\theta\times R\sin\theta d\varphi=R^{2}\sin\theta d\theta d\varphi \tag{1}\]  
\[d\Omega=ds/R^{2}=\sin\theta d\theta d\varphi \tag{2}\]  
\[d\varphi=I(\theta,\varphi)d\Omega=I(\theta,\varphi)\sin\theta d\theta d\varphi \tag{3}\]  
\[\phi=\int_{\varphi=0}^{2\pi}\int_{\varphi=0}^{\pi}\int_{\theta(\varphi,\varphi )\sin\theta d\theta d\varphi}^{\pi} \tag{4}\]  
\[\phi^{\prime}=2\pi\int_{0}^{\pi}I(\theta)\sin\theta d\theta \tag{5}\]  
If the light source \(I(\theta)\) is distributed irregularly and cannot be expressed by a mathematical function, then, when calculating the total zonal flux, Formula (6) should be applied instead, where \(C(\theta)\) is used as the solid angle of \(\Delta\theta\), which is also regarded as the ring coefficient. If the required solid angle \(\Delta\theta\) is between the vertical angle \(\theta_{a}\) to \(\theta_{b}\) of the light distribution, then the ring coefficient \(C(\theta_{ab})\) and the zonal flux \(\phi_{ab}\) can be obtained using Formula (7) and (8) as shown in Figure 3. Table 1 is a list of ring coefficients that correspond to angles less than \(90^{\circ}\) and are calculated at \(10^{\circ}\) intervals.  
Figure 4 is a standard Lambertian LED light distribution and its total luminous flux is 100 lm. At intervals of \(10^{\circ}\), we use the above zonal flux formula to obtain the luminous flux of each zone, as shown in Table 2. Moreover, the luminous flux for the whole circumference is confirmed with a total of 100 lm. As is evident, while the LED light source might have the highest central luminous intensity, the luminous flux is only about 3% within the vertical angle of \(0^{\circ}\)-\(10^{\circ}\). This means that high luminous intensity is not equivalent to high luminous flux. Since the secondary optical design uses luminous flux distribution to adjust the output beam angle, zonal flux analysis is of considerable importance for the light source and targeted beam pattern.  
Figure 2: Zonal flux calculation of a 3D sphere.  
Figure 4: LED Lambertian light distribution (100 lm).  
Figure 3: Schematic diagram of zonal flux calculation.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(\Delta\theta_{ab}\) & \(C(\theta)\) \\
**Zonal Zone** & **Zonal Coefficient** \\ \hline
0–10 & 0.095 \\ \hline
10–20 & 0.283 \\ \hline
20–30 & 0.463 \\ \hline
30–40 & 0.628 \\ \hline
40–50 & 0.774 \\ \hline
50–60 & 0.897 \\ \hline
60–70 & 0.993 \\ \hline
70–80 & 1.058 \\ \hline
80–90 & 1.091 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Zonal coefficients (vertical angle).