# Measurement and Modeling of 3D Solar Irradiance for Vehicle-Integrated Photovoltaic
## 2 Model
### Partial and Dynamic Shading Model  
The shading influence is complicated and varies by the position and the relative orientation (to the sun position) of the panel. In the case of the car roof PV, the position (orientation) of the panel cannot be predicted. One practical approach is to rely on the probability model, supposing that the distribution and height of the shading objects are randomly distributed by some statistical model and the orientation of the car is independent and random as well. Most of the PV on the car roof is curved, which induces the non-uniform illumination caused by areal variation of the local cosine loss and self-shading loss. Such non-uniform illumination caused by the curved surface is equivalent to the partial shading given by the non-uniform photo-current output from each solar cell. A ray-tracing simulation can estimate such non-uniformity (Figure 11). Figure 10: Correction factor vs. representative curve shape parameter (coving factor as \(f_{1}\)) in two extreme cases. One is the urban zone in Bangkok (N 13.7\({}^{\circ}\) and 4.87 kWh/m\({}^{2}\)/day global horizontal irradiance, GHI), drawn by a blue line, and the other is a rural zone of Omsk (N 54.9\({}^{\circ}\) and 3.34 kWh/m\({}^{2}\)/day GHI) drawn in an orange line: (**a**) curve-correction factor \(f\); (**b**) optical curve correction factor \(f_{2}\). The above approach simulates the distribution of the solar irradiance of the curved car roof PV panel. To do this, it is necessary to specify the orientation of the panel, sun height, and ratio of the direct and scattered components. It is not appropriate to use the statistical distribution of the incident angle data because the distribution depends on the direction of the direct sunlight. The distribution of the irradiance varies by both sun height and orientation angle. The orientation angle of the car roof PV varies by time. Such non-uniformity is observed by the measurement of the mini-module attached to various places on the car roof [23]. As was discussed in the previous sections, car roof PV is often partially shaded or receives equivalent non-uniform illumination caused by its basic curved shape. Both partial shading and non-uniform illumination can be simultaneously and quantitatively modeled, considering that both primarily effect variation of the short-circuit current of the cells. For simplicity, cells in a module are categorized into three types. 1. In the sun
2. Full shade
3. Partial shade  
Cells in the sun are supposed to be illuminated by all the sunlight, including direct sunlight; the shaded cells do not receive the direct component of the sunlight but instead receive diffused sunlight + reflected sunlight. Random numbers give the number of the shaded cells (both full shaded and partially shaded) with some constraints. Random numbers also give the ratio of the shaded area of each partially-shaded cell. Random numbers give the position of the sun that affects the angle of the sunbeam and the orientation of the car roof as well. Finally, the I-V curve of each cell is calculated by three randomly distributed parameters (normal distribution). The property of the probability variables used for the Monte Carlo simulation is summarized in Table 1. The irradiance and climate conditions corresponding to the above parameters are found in the METPV-11 [45] and METPV-ASIA [46] databases. Given these probability parameters, I-V curves of entire cells in the array affect different illumination conditions, and variations of characteristics are calculated, and the maximum power point is calculated for taking the simulated power output affected by the partial shading. Figure 11: Solar irradiance distribution on the curved surface of the PV panel. The light-green arrow lines correspond to the direct sunlight. The blue arrow lines correspond to the scattered sunlight. The color glaidiation on the curved surface indicates the non-uniformity of the irradiance on the curved surface. The darker color indicates the lower irradiance.