## 2 Methods
### Expression of Curved Surfaces of PVs  
In a curved PV, the shade owing to self-shadowing and cosine loss differs depending on the panel's PV location. Therefore, each cell is not irradiated with uniform sunlight. That is, the total amount of solar radiation and the amount of solar radiation absorbed by the PV system do not match. Moreover, the degree of solar radiation varies in a complex manner depending on the curved surface shape, orientation of the automobile, sun, and distribution of the surrounding structures. Through this research, we formulated a method for determining the distribution of cosine loss, self-shadow, direction vector of face elements, and area of surface elements at each curved surface. We also developed a method for determining the distribution of cosine loss, self-shadow, direction vector of surface elements, and surface elements' area at each curved surface. For example, if a 3D measurement result exists for the shape of the car body roof, it is also feasible to calculate the solar radiation distribution in each part. The non-uniform distribution of solar radiation intensity on the curved surface induces a mismatch loss in the module. This mismatch loss is a significant loss factor in VIPVs. Symptomatic treatments such as mounting multiple MPPTs and developing string configurations are known to cause mismatch losses. For a quantitative analysis of the mismatching phenomena and design optimization, each solar cell that comprises the module should determine the amount of incident light. Therefore, it is essential to calculate the short-circuit current of each cell by dividing the area of sunlight irradiating the surface integral in each cell region and calculating various quantities based on the curved surface geometry. Each face surface element, normal vector, and first derivative in each direction are required. The parametric format shown in Equation (3) is convenient:  
\[S(u,v)= \begin{bmatrix}x(u,v)\\ y(u,v)\\ z(u,v)\end{bmatrix} , \tag{3}\]  
The \(uv\) plane is a matrix in which grid-point position information is arranged. The actual curved surface exists in the \(XYZ\) space, and Equation (3) represents the mapping  
Figure 1: The algorithm flowchart for computing the power output of a curved PV module affected by non-uniform solar irradiance and subjected to frequent movements in the local coordinate system [68]. from the \(uv\) plane to the \(XYZ\) space. It is an image of a square rubber film on a flat surface that is stretched or twisted and placed in a 3D space. The grid points do not need to be squares. It is necessary to make these non-uniform and rectangular because the approximation would be higher if the grid points are set finer in the region where the curve is steep. Therefore, as shown in Equation (3), it is necessary to expand the mapping target in the \(z\)-direction profile and the \(x\) and \(y\) directions. The derivative was calculated using numerical differentiation based on the central difference, as shown in Equations (4) and (5):  
\[\mathbf{S_{u}}_{i,j}= \begin{bmatrix}\mathbf{S_{0}}_{i,j}-\mathbf{S_{0}}_{i+2,j}\\ \mathbf{S_{1}}_{i,j}-\mathbf{S_{1}}_{i+2,j}\\ \mathbf{S_{2}}_{i,j}-\mathbf{S_{2}}_{i+2,j}\end{bmatrix}  \tag{4}\]  
\[\mathbf{S_{v}}_{i,j}= \begin{bmatrix}\mathbf{S_{0}}_{i,j}-\mathbf{S_{0}}_{i,j+2}\\ \mathbf{S_{1}}_{i,j}-\mathbf{S_{1}}_{i,j+2}\\ \mathbf{S_{2}}_{i,j}-\mathbf{S_{2}}_{i,j+2}\end{bmatrix}  \tag{5}\]  
The \(\mathbf{S_{u}}\) and \(\mathbf{S_{v}}\) 3 \(\times\) 1 vectors nest a matrix with rows and columns corresponding to the number of lattice points as elements. These display the partial differentials in the \(u\) and \(v\) directions in the parameter space (\(uv\) plane). In addition, \(\mathbf{S}\) represents the curved surface coordinates in the \(XYZ\) space. The values of \(x\), \(y\), and \(z\) in the subscripts (row number, column number) \(i\) and \(j\) corresponding to the lattice points in the \(uv\) plane are stored as a matrix. It is a 3 \(\times\) 1 vector element.