# Measurement and Modeling of 3D Solar Irradiance for Vehicle-Integrated Photovoltaic
## 2 Model
### Curve Correction Model
#### 2.5.3 Curve-Correction Calculation Based on Ray-Tracing Simulation  
The basic idea of the curve correction factor is the ratio of the absorbed flux and that of the light source with the area of the projected area of the curved PV panel that is placed just on the curved PV panel (Figure 7). The line segment around the light source (semi-transparent light-blue area) corresponds to the rays emitted from the light source. The short line segments correspond to the ray that does not reach to the curved surface. The longer line segments correspond to the one that hits the curved PV panel (curved opaque gray surface), namely the ray that is absorbed by the PV. Note that some rays just above the curved PV panel do not hit the curved module due to its curvature. To avoid the situation, the real light-source was placed on the surface of the virtual source plane (semi-transparent light-blue area) with a ten times larger area than the virtual light source. A detector is placed on the virtual light source and measures the total flux that passes the aperture zone (corresponding to the projected area) and compares the total flux absorbed by the curved PV surface (opaque gray surface). Note that the absorption at the backside surface is disabled in this calculation. The angular characteristics of the absorption are assumed as a Lambertian surface. The curve correction factors of the various shape of the car roof are calculated and plotted in Figure 8; each corresponding marker in the plot was given by the random numbers. The curve shape of the car roof widely varies. Random numbers model each surface with an approximation of the bi-cubic spline function structured by the power function of the tangent lines (Figure 1). The curve shape profile is approximated as the segmented (symmetrical to left-right direction, but asymmetrical to front-back direction) power function constrained by the maximum slope at the edges. Specifically, it keeps symmetry in the X direction, but allows asymmetry in the Y direction, considering that the curve shape of the car roof may vary in the front and tail of the car body. The synthesis calculation is illustrated in Figure 9.  
The curve correction factor decreases by the increase of curvature of the surface [20], corresponding to the ratio of the projected area by surface area (Figure 8). The geometrical factor \(f_{1}\) dominates it, but the optical curve correction also decreases by the increase of the curvature as well. However, the optical influence depends on the surface shape, and it is essential to calculate its factor to every curve surface. The above approach can simulate 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  
Figure 8: Curve correction factor vs. representative curve shape parameter (coving factor as \(f_{1}\)). Figure 7: Geometrical relation between the curved PV panel (curved detector, opaque gray surface) and light source (corresponding to the projected area, semi-transparent light-blue area). The line segment around the light source (semi-transparent light-blue area) corresponds to the rays emitted from the light source. The short line segments correspond to the ray that does not reach the curved surface. The longer line segments correspond to the one that hits the curved PV panel (curved opaque gray surface), namely the ray that is absorbed by the PV. Note that rays just above the module do not always hit the module due to its curvature [1]. Figure 9: Modeling the curved surface of the car roof PV using random numbers. 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. Unlike the BIPV, the orientation angle of the car roof PV varies by time.