# Measurement and Modeling of 3D Solar Irradiance for Vehicle-Integrated Photovoltaic
## 2 Model
### 3D Irradiance Model  
For modeling the solar irradiance of the car roof, we assume a simple shading and scattering reflection model. The entire shading objects assume that the reflectance is 0.25, and the reflection is Lambertian. Both direct and scattered solar irradiance are assumed to be shaded by the shading objects as a function of the sun height and DNI (direct normal irradiance). The probability of the shading event calculates the shading probability and solid angle. Assuming that the diffused irradiance from the unshaded sky is uniform, the factor of the diffused sunlight relative to the unshaded sky is approximated by the Equation (1) using random numbers. Note that the shape of buildings and other shading objects are rectangular, they are not spherical triangles or spherical trapezoids. It is realistic to model it as a cylinder sky using Equation (1). \[\mathbf{\sigma_{\mathrm{s}}}(\theta_{h})=1-(1-R_{b})\Big{(}\frac{\theta_{h}}{90^{ \circ}}\Big{)} \tag{1}\]  
where \(\mathbf{\sigma_{\mathrm{s}}}(x)\) is a function of the ratio of the diffused sunlight from the sky excluding the reflection of the direct sunlight to the shading objects, \(R_{b}\) is the reflectance of the shading objects (fixed to 0.25 in this calculation), \(\theta_{h}\) is the average height of the shading objects scaled by the grazing angle (note that the maximum angle is 45\({}^{\circ}\) in this model). The direct sunlight was approximated by Equation (2). \[\mathbf{\sigma_{d}}(\eta,\theta_{h})=\left[ \begin{array}{cc}0&(\eta<0)\\ \frac{\eta}{2\theta_{h}}&(0\leq\eta\leq 2\theta_{h})\\ 1&(\eta>2\theta_{h})\end{array} \right. \tag{2}\]  
where \(\mathbf{\sigma_{d}}(x,y)\) is a function of the ratio of illumination of the direct sunlight with the sun height \(x\) and the average shading height \(y\). The scattered sunlight by the reflection of the building from the direct sunlight is approximated by Equations (3) and (4). Note that the scattered irradiation by the reflection of the direct sunlight is seen on one side of the car body (not both sides), and the impact is half of the reflection by scattered light. \[\mathbf{\sigma_{bsh}}(\eta,\theta_{h})=\frac{1}{2}R_{b}\Big{(}\frac{\theta_{h}}{90 ^{\circ}}\Big{)}\mathbf{\sigma_{d}}(\eta,\theta_{h})\cos\eta \tag{3}\]  
\[\mathbf{\sigma_{bsb}}(\eta,\theta_{h})=\frac{1}{4}R_{b}\Big{(}\frac{\theta_{h}}{90 ^{\circ}}\Big{)}\mathbf{\sigma_{d}}(\eta,\theta_{h})\cos\eta \tag{4}\]  
where \(\mathbf{\sigma_{bsh}}(x,y)\) and \(\mathbf{\sigma_{bsb}}(x,y)\) are functions of the ratio of illumination by the scattered reflection from the direct sunlight onto the horizontal and vertical planes with the sun height \(x\) and the average shading height \(y\). Note that the illumination by the wall of the building happens on a single side of the road. The directional cosine of the direct sunlight to the car side (vertical plane, random orientation) is approximated by Equation (5). \[\mathbf{\epsilon\theta_{\mathrm{side}}}(\omega)=\frac{\int_{0^{\circ}}^{360^{ \circ}}\cos(\gamma-\alpha(\omega))(\cos(\gamma-\alpha(\omega))>0)d\gamma}{360^ {\circ}}\cos(\eta(\omega)) \tag{5}\]where \(\mathbf{c}\mathbf{\Theta_{side}}(\omega)\) is a function of the directional cosine of the direct sunlight to the car body as a function of the hour angle \(\omega\). The sun height \(\eta\) and the azimuth angle \(\alpha\) are also the functions of \(\omega\). This equation also contains Boolean algebra in the parentheses that return 1 when the conditional equation is true and 0 when the conditional equation is false. Note that the azimuth angle is not needed to be considered in the calculation on the irradiance on the car roof (horizontal plane). The irradiance on the car roof is approximated using the direct normal irradiance _DNI_ and diffused irradiance from the sky _SI_ using Equation (6) with calculation results of Equations (1)-(3). \[CI_{roof}=\sigma_{s}SI+(\sigma_{d}\sin\eta+\sigma_{bsh})\cdot DNI \tag{6}\]  
where \(CI_{roof}\) is solar irradiance onto the car roof affected by surrounding shades.