# Measurement and Modeling of 3D Solar Irradiance for Vehicle-Integrated Photovoltaic
## 2 Model
### Curve Correction Model
#### 2.5.1 Why the Curved PV Modules are Often Overestimated in Efficiency Measurements  
Before discussing the curve correction, let us clarify where the standard measurement method has a problem. The curved modules are often overestimated in efficiency measurements. It is because the input energy is often underestimated during the measurement by the indoor solar simulator.  
To explain this, let us return to the definition of efficiency measurement of the solar cells and modules (Equation (17)).  
\[\eta=\frac{P_{in}}{P_{out}} \tag{17}\]where \(\eta\) is the measured energy efficiency of the solar cell or module using the solar simulator. \(P_{in}\) is the input power to the solar cell or the module. \(P_{out}\) is the output power of the solar cell or the module. The measurement of \(P_{out}\) of the curved module is the same as the standard flat PV modules, and it is a simple electrical measurement. However, the trappy measurement is \(P_{in}\).  
In the standard measurement of the solar cell and module, \(P_{in}\) is calculated by Equation (18).  
\[P_{in}=A\cdot\mathit{Irr} \tag{18}\]  
where \(A\) is an aperture area. \(\mathit{Irr}\) is irradiance in the aperture window. In the solar simulator measurement, \(\mathit{Irr}\) is adjusted as 1 kW/m\({}^{2}\) and uniform in the entire area of the aperture area \(A\). For the standard flat PV module, the aperture area \(A\) is the same as the module active area. However, this definition is not applied to the curved PV module because the aperture area is defined as the window of the flat plane, and not the curved surface.  
The first cause of the overestimation is that the curved PV module often collects more light than is defined by the aperture window (Figure 4). Since the illumination area of the solar simulator is always more extensive than the PV module, the curved module receives more unexpected light coming from the outer region so that it generates more unexpected power. Alternatively, it is an excellent way to place an aperture mask on the curved PV module. However, it is often eliminated because the multiple reflections between the aperture mask and optics in the solar simulator disturbs the uniform illumination in the zone of the aperture window, and the solar simulator often and repeatedly requires time-confusion adjustment in the optics. In addition, there are no agreed standards in the position of the aperture mask for the curved PV module. Depending on the curved shape, the aperture mask interferes with the body of the curved module.  
The second cause is the aperture area; namely, the module area in the active region of the curved module is not defined clearly (Figure 5) and may have multiple definitions. It is a cause of fatal error in the estimation of the module efficiency in the dividing area of the module (Equations (17) and (18)).  
Figure 4: Illustration of the underestimation of the input energy in the measurement of the curved PV module.  
The third cause is the difference in the angular distribution in the indoor measurement (typical solar simulator) and outdoor operation. The curved surface generates more loss in the illumination by a higher incident angle (Figure 6). Unlike the case of the flat PV module, it cannot be applied to the indoor module efficiency to the outdoor operation. Correction by the curved shape is essential.