## 2 Methods
### Impact of the Non-Uniform Solar Irradiance  
Building-integrated photovoltaics (BIPVs) and vehicle-integrated photovoltaics (VIPVs) receive solar irradiance through non-uniform shading objects. Standard scalar calculations cannot accurately determine the solar irradiance of BIPV and VIPV systems. This study proposes a matrix model using an aperture matrix to accurately calculate the horizontal and vertical planes affected by non-uniform shading objects. Applying a local coordinate system can extend this to the solar irradiance on a VIPV. Fish-eye videos were used to measure the non-uniform solar irradiance of the vehicle's surroundings. With the help of image processing calculations, the shaded objects and sky were discriminated using the binarizing algorithm [71]. The 3D model was validated by simultaneously measuring five orientations (roof and four sides: front, left, tail, and right) of solar irradiance on a car body [71]. The shaded objects distributed around the VIPV are essential for performance. The aperture matrix \(\mathbf{E}\) was defined using a 2D histogram of the angular distribution of the shaded objects. The matrix elements are real numbers ranging from zero to one (zero: shaded; one: unshaded). Shading refers to the shading of a 2D bin in a hemispherical sky. The \((i,j)\) elements, \(\mathbf{E}_{ij}\), are the probabilities of the unshaded 2D bin of the elevation angles [\(i^{\circ}\), \((i+1)^{\circ}\)] (\(i\) = 0, 1,.... 89) and orientation angles [\(4j^{\circ}\), \(4(j+1  
Many vehicles have curved roofs, and typical VIPVs have curved surfaces. However, curved PVs typically perform less efficiently than flat PVs because of two factors. First, the aperture area of a curved surface is smaller than that of a flat surface. The input power to the PV devices is the product of the aperture area and irradiance. Therefore, the smaller the aperture of the curved PV devices, the less the input power that these devices receive. Another factor is that the cosine loss of a curved surface varies at different points, thereby generating a mismatch loss. There are two degrees of freedom in the 3D space direction, and two indices can describe the surface. The 2 \(\times\) 2 combination generates a nested matrix or four-tensor, as described by Equations (1) and (2) [71]:  
\[\mathbf{M}\mathbf{M}= \begin{bmatrix}M_{0,0}&M_{0,1}&M_{0,2}&&M_{0,l}&&M_{0,NN-1}\\ M_{1,0}&M_{1,1}&M_{1,2}&\cdots&M_{1,l}&\cdots&M_{1,NN-1}\\ M_{2,0}&M_{2,1}&M_{2,2}&&M_{2,l}&&M_{2,NN-1}\\ \vdots&\vdots&\vdots&\ddots&&&&&\\ M_{k,0}&M_{k,1}&M_{k,2}&&M_{k,l}&\cdots&M_{k,NN-1}\\ \vdots&\vdots&\vdots&\cdots&\vdots&\ddots&\vdots\\ M_{NN-1,0}&M_{NN-1,1}&M_{NN-1,2}&&M_{NN-1,l}&\cdots&M_{NN-1,NN-1}\\ \end{bmatrix}  \tag{1}\]  
\[M_{k,l}= \begin{bmatrix}m_{0,0}&m_{0,1}&m_{0,2}&&m_{0,j}&&m_{0,N-1}\\ m_{1,0}&m_{1,1}&m_{1,2}&\cdots&m_{1,j}&\cdots&m_{1,N-1}\\ m_{2,0}&m_{2,1}&m_{2,2}&&m_{2,j}&&m_{2,N-1}\\ \vdots&\vdots&\vdots&\ddots&&&&&\\ m_{i,0}&m_{i,1}&m_{i,2}&&m_{i,j}&\cdots&m_{i,N-1}\\ \vdots&\vdots&\vdots&\cdots&\vdots&\ddots&\vdots\\ m_{N-1,0}&m_{N-1,1}&m_{N-1,2}&&m_{N-1,j}&\cdots&m_{N-1,N-1}\\ \end{bmatrix}  \tag{2}\]  
The four-tensor \(\mathbf{M}\mathbf{M}\) has a nested matrix element \(\mathbf{M}\). \(M_{k,l}\) are matrices with \(\mathbf{m}_{ij}\) scalar elements. The matrix \(\mathbf{m}\) is the 2D angular response of the surface elements positioned at (\(k\),\(l\)). In general, the four-tensor has a substantial number of elements. If both matrix \(\mathbf{M}\mathbf{M}\) in Equation (1) and \(\mathbf{M}\) in Equation (2) are 100 \(\times\) 100, the total number of elements would be 1004 = 100,000,000. Therefore, it is ineffective to recalculate or regenerate the four-tensor