# Measurement and Modeling of 3D Solar Irradiance for Vehicle-Integrated Photovoltaic
## 2 Model
### Curve Correction Model
#### 2.5.2 Examples of the Curve-Correction Calculations  
A PV module with a curved surface is different from a flat plate module on power output. Specifically, the self-shielding effect of shielding a part of the incident from a shallow angle on its convex surface, the local cosine loss at each point of the solar cell is not constant, and due to the above factors, a mismatching loss further reduces the power output. That is, the output of the curved module has a different output value from the performance test with a conventional solar simulator designed for a flat plate, as we discussed in the previous section. Nonetheless, the exact test method of synthesizing the separately measured outputs by decomposing the curved surface into little surface elements (approximating each surface element to a very flat plate) is scarce in reality [25]. Therefore, a method of considering the power generation output on the 3D curved surface by multiplying the value measured by assuming the curved module as a 2D flat plate by the curve correction factor was studied. This curve correction factor is convenient because it can be uniquely determined for each curved surface given the incidence angle characteristics of the module and the incidence angle distribution of the solar cell if the mismatching loss is neglected. The curve correction factor can be calculated by numerical, geometric calculation [35] or ray-tracing simulation [1]. For the geometrical calculation, it is important to add two conditions for avoiding complexity. First, the module does not absorb the light in the backside. Second, the curved surface is  
Figure 5: Illustration of the multiple definitions of the module area. Figure 6: Illustration of the multiple definitions of the module area: (**a**) principle ray direction in testing, although the illumination by the typical solar simulators is not collimated, and the field of view is typically \(\pm 10^{\circ}\) to \(\pm 45^{\circ}\); (**b**) principle ray direction in the outdoor operation. The outdoor illumination is the mixture of the collimated light (direct sunlight) and diffused sunlight (illumination from the sky and reflection by surroundings). The ratio of the collimated and diffused light varies by climate; (**c**) example of the distribution level in the outdoor operation calculated by the ray-tracing simulation. The light-green arrow lines correspond to the direct sunlight. The blue arrow lines correspond to the scattered sunlight. The color gladiation on the curved surface indicates the non-uniformity of the irradiance on the curved surface. The darker color indicates lower irradiance. simple convex (no two or more peaks), namely the partial derivative functions of the profile function of the curved surface concerning \(x\) and \(y\) have no more than two points of the zero-crossing. The curve correction factor \(f\) is defined as Equation (19). \[P=AIf\eta \tag{19}\]  
where \(P\) is the output of the module, \(A\) is a flattened area of the panel (not projected area), \(I\) is irradiance of the curved roof, \(f\) is the curve-correction factor, and \(\eta\) is the module efficiency measured by the flat condition by the conventional solar simulator. The curve correction factor \(f\) is a unique value depending on the 3D curve shape of the panel if the angular distribution of the solar irradiance is given. It is typically calculated by a numerical calculation (calculation algorithm should be open, transparent, and repeatable within some acceptable numerical errors, i.e., use of the Monte Carlo method). It should apply to the 3D CAD (computer-aided design) interface (most of the car roof 3D shapes were not simple polynomials but segmented smooth functions). The curve correction factor \(f\) can be expressed as the product of two parameters (Equation (20)).