###### Abstract  
Most equations and models for photovoltaics are based on the assumption that photovoltaic (PV) devices are flat. Therefore, the actual performance of nonplanar PV devices should be investigated and developed. In this study, two algorithms were developed and defined using vector computations to describe a curved surface based on differential geometry and the interaction with non-uniform solar irradiance (i.e., non-uniform shading distribution in the sky). To validate the computational model, the power output from a commercial curved solar panel for the Toyota Prius 40 series was monitored at four orientation angles and in various climates. Then, these were compared with the calculation results obtained using the developed algorithm. The conventional calculation used for flat PV devices showed an overestimated performance due to ignorance of inherent errors due to curved surfaces. However, the new algorithms matched the measured trends, particularly on clear-sky days. The validated computation method for curved PV devices is advantageous for vehicle-integrated photovoltaic devices and PVs including building-integrated photovoltaics (BIPVs), drones, and agribotovoltaics. V 2024  
Due to the amount of energy yield of vehicles with a limited installation area, the power conversion efficiency of PV devices is crucial, and [PERSON] discussed the possibilities and future development [31]. A VIPV is a rather sophisticated PV panel installed on the body or roof of a car. Several technological challenges should be addressed [14]. A few of these issues are related to the structural constraints of automobiles and require new design considerations. Many new developments have been made, such as lightweight, robust, flexible, and curved surfaces for both car bodies and VIPVs [32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. Recently, Toyota Motors developed a lightweight plastic-covered VIPV with sufficient strength as the exterior of a car [57]. In addition to these mechanical issues, the VIPV should enclose the curved surface of the car body. In general, PV panels are flat and not designed to enclose curved surfaces. Therefore, covering curved surfaces causes problems such as mechanical stress and mismatching losses [58; 59; 60]. Several innovative PVs suitable for enclosing curved surfaces have been developed recently [61; 62]. VIPVs were evaluated recently, including the curved surface and impact from temperature [63; 64]. In addition, our group proposed correcting the curved surface effect using a unique curve correction factor [65; 66; 67]. The shape of the curved surface varies significantly according to the automobile product line. This impacts the power generation performance and durability [68]. For a general discussion on VIPVs, it is necessary to understand the various curve shapes and general rules that affect the optical, thermal, and mechanical properties of PV panels based on global and local differences. Quantitatively comprehending the quantity and type of curved surface an actual automobile body has would help in developing versatile VIPV technology and measurement protocols [69]. It would also be effective for quantifying the advantages of VIPVs, such as the stability of the energy source for various VIPVs with various curved surface shapes during actual driving and the decrease in the frequency of charge replenishment [70; 71]. PV technologies rely on the assumption that all solar panels have flat surfaces. Many PV-related equations are implicitly based on this assumption. The curved PVs necessary for VIPVs override this assumption. We need to identify a new technology. Our approach uses a vector-based computation of curved PV modules from a flat PV framework. Furthermore, it is effective in helping us to understand the general behavior of PV devices resulting from the curved surface of vehicle bodies by evaluating modeling algorithms.