## 4 Discussion  
We presented two types of vector computation algorithms for curved PV devices affected by non-uniform solar irradiance (Figures 1 and 2). Figure 1 shows a rigorous model, and Figure 2 shows a simplified model that was applied to the following cases:  
* The attitude and position are unaltered during the timeframe of the power measurement. * The solar irradiance angular distribution is unaltered during the power measurement timeframe. * The surface is a simple concave shape (not saddle-shaped or with multiple peaks, such as a twin bubble roof) [101; 102]. Figure 11: Comparison of the outdoor power output using the indoor test result on cloudy days in four orientations of the module with (1) conventional power calculation assuming the PV device is flat (purple line), (2) vector computation using the algorithm in Figure 2 considering the curved surface (green line), and (3) measured outdoor power (blue region). * All the solar cells are connected in series. * The nonlinear effects in equivalent circuits in the solar cells are negligible [103; 104]. * The spectrum mismatching loss is negligible [105; 106; 107]. * The dynamic and partial shadings are negligible [108; 109; 110]. With respect to its application to VIPVs, it can be used for solar panels on a car roof (but not for advanced aero-designed roofs such as double bubbles). For solar panels on the engine or side of the car, the rigorous algorithm shown in Figure 1 should be used. In this study, the computational model was validated using the algorithm shown in Figure 2. Therefore, the indoor measurements of curved VIPV products for car roofs can be used to predict their outdoor performance. Although this method was developed for the power estimation of curved PV devices, it can also be used to estimate the energy yield because the annual energy yield correlates with the energy yield's actual value. To calculate the energy yield, the power of the photovoltaic device is typically calculated every hour based on a solar database such as the METPV series [111; 112] considering the position of the sun relative to the device. Its power value is then integrated, multiplied, and weighted by the timeframe (typically 1 h). The models shown in Figures 1 and 2 can be used for other advanced integrations of photovoltaic devices:  
* Wavy PV arrays that are installed in hilly areas where the performance ratio decreases in winter owing to self-shading by the PV arrays and irregular skylines can use the algorithm shown in Figure 1;
* Aircraft including drones and high-altitude platform systems (HAPS) can use the algorithm shown in Figure 1;
* BIPVs with multiplexed wall shading and self-shading by irregular or nonplanar PV arrays can use the algorithm shown in Figure 1;
* In flexible and film-type PV devices enclosing non-planar structures (Figures 1 and 2);
* In photosynthetic photon flux density (PPFD) distribution calculations for crops grown below agrib photovoltaic systems. This study, in particular, considered how PV devices behave and are affected by the surface profile (inherent properties). However, the performance of PV devices is affected by external factors like shading, including partial and dynamic shading. These factors are not typically used to describe product performance but are referenced in the energy yield discussion. Such evaluations were discussed in international standardization activities [113; 114]. These algorithms and the testing and validation procedures, including safety issues with installing PV devices on the car body, are also discussed within the framework of international standardization [113; 114]. The story of international standardization activity started in July 2017 with a web meeting of a group of researchers and testing engineers in PV industries to seek a solution for the measurement errors and different behaviors of curved PV devices from flat ones. The group gradually expanded, inviting experts from testing laboratories and engineers from the automotive industry seeking applications for automobile PV devices.