## 2 Methods
### Simplified Calculation for VIPVs or Other Loosely Curved PV Devices  
A simplified flowchart of the computation for the PV module on a car roof is shown in Figure 2. It can be used under the following conditions:  
1. The attitude and position remain constant during the timeframe of the power measurement.
2. The solar irradiance angular distribution remains constant during the power measurement timeframe.
3. The surface is a simple concave shape (not saddle-shaped or with multiple peaks, such as a twin bubble roof) [101; 102].
4. All the solar cells are connected in series.
5. The nonlinear effects in the equivalent circuits in solar cells are negligible [103; 104].
6. The spectrum mismatching loss is negligible [105; 106; 107].
7. The dynamic and partial shadings are negligible [108; 109; 110].  
The stress calculation substantially impacts the mechanical robustness (cell crack) and reliability (impact strength, vibration, and fatigue degradation) but does not significantly impact the initial performance unless it is heavily deflected, leading to the cell cracking [14].  
The impact of dynamic and partial shading varies dramatically depending on the relative position of shading objects and the sun, as well as the vehicle's orientation. Several publications have quantitatively analyzed these impacts [108; 109; 110].  
The percentile function (_v,p_) in Figure 2 returns a value below which p percent of the data are recorded in \(v\) (vector). _Iel_ in Figure 2 represents the vector data of the _Isc_ of each cell; _Ield_ in Figure 2 corresponds to the _Isc_ of each cell generated by direct solar irradiance; and _Iels_ represents that of diffused sunlight. The vector or matrix \(c\) is the inner product of the normal vector of the cell and the ray vector (the directional vector of direct sunlight). This is calculated using the areal integration of Equations (9) and (10) for the entire area of each solar cell divided by the total area of the cell calculated by the sum or integration in Equation (6).  
In this situation, the time-saving advantage of computing using a four-tensor, similar to the one in Figure 1, is not apparent. The detection of the hidden surface generally observed in saddle-shaped aircraft wings and double-bubble roofs in superior aerodynamic vehicles (which requires a considerable amount of computation) is not required. Equations (9) and (10) are used to calculate the entire angular response without hidden surfaces in any surface element. These are simple inner products of vectors, and their nonzero discrimination and computation time are moderate. Therefore, we can directly apply Equations (9) and (10) without interpolating the four-tensor described in Figure 1.  
Computing the power output from each cell by considering the angular response is not necessary. Rather, a single solar cell with a specific percentile value of the order of the cosine value to the ray vector of direct sunlight is used to determine the total power output from the module. Unless the mismatch loss is large (which causes the maximum power point peak to shift to another value and occurs frequently in the case of partial or dynamic shading), the overall shape of the \(I\)-\(V\) curve does not vary dramatically. Moreover, the solar cell that controls the position of the maximum power point remains unaltered. This study did not solve differential or partial differential equations, numerically or analytically, but relied on the calculation of geometries (differential geometry). Although we frequently used tensor, matrix, and vector computations, we were not required to define boundary conditions.  
In certain cases, the indoor test is performed on the test bed in testing laboratories. It has a different angle of placement relative to the ground on the actual car roof. In such cases, the group of normal vectors of the surface element is transformed by coordinate rotation by applying rotation matrices.