## 1 Introduction
The outdoor performance of multi-junction solar cells for CPV application has been intensively analyzed, and the most significant loss is known as the spectrum-mismatching loss [29; 30; 31; 32; 33; 34; 35; 36; 37]. This is caused by the fact that the solar spectrum is not always the same as the designed one (typically, ASTM G173 AM1.5D spectrum for CPV application). The sub-cells in the multi-junction cells are electrically connected in series. The spectrum shift hampers the balance of the output current from the sub-cells, and the sub-cell with the smallest output current constrains the total output current according to Kirchhoff's law. This type of loss is called \"spectrum-mismatching loss.\" Spectrum-mismatching loss is an inherent loss for all types of multi-junction or multi-junction solar cell, regardless of CPV or normal flat-plate application, except for more than 3 terminal configurations where the output of the sub-cells is individually connected to the load. Please note that in every type of installation there is a variation of the solar spectrum by sun height and fluctuation of scattering and absorption of the air by seasonal effect, but its influence can be minimized by the improvement of the solar cell design [38; 39; 40; 41; 42; 43]. Research into the robustness to spectrum change has been made in the past 20 years, including a computer model named Syracuse by Imperial College London [44; 45; 46]. For CPV applications, it was understood that the chromatic aberration of the concentrator optics enhanced the spectrum-mismatching loss [44; 45; 46; 47; 48; 49; 50; 51; 52; 53]. However, such loss, coupled with the concentrator optics, could be solved by the innovation of optics, including homogenizers and the secondary optical element (SOE) [54; 55]. The remaining problems of the spectrum-mismatching loss have been overcome by the adjustment of the absorption spectrum of each sub-cell, including overlapping the absorption spectrum and broadening the absorption band to the zone of massive fluctuation. Recently, a new configuration enhancing the radiative coupling among the sub-cells has been found useful for solving this inherent loss of the multi-junction cells. The first study was presented by [PERSON] in 2002 [56]. However, his model was too simplified and overlooked the most important factor, namely a variation of atmospheric parameters. Later, [PERSON] developed a power-generation model considering the variation of atmospheric parameters and quantitatively anticipated that the radiation coupling would be adequate for suppressing the spectrum-mismatching loss [57; 58; 59; 60]. This idea was further developed by a group of authors [61; 62; 63; 64]. However, the work of these authors was limited to the application of CPV because of the simplicity of spectrum and performance modeling. Radiative recombination was also identified to impact the performance of the multi-junction cell, even in operation under standard testing conditions (not a dynamically changing spectrum like the outdoor spectrum). Taking an example of the research on Fraunhofer ISE [65], and later, by use of a rear-side mirror for the use of the recycled photon by radiative recombination, high open-circuit voltage, and 28.8% efficiency under 18.2 W/cm\({}^{2}\) concentrated irradiance was obtained [66]. The measurement and identification of radiative coupling and photon recycling was done in several types of solar cells, including GaAs cells [67], strain-balanced quantum well cells [68], and even emerging solar cells such as Perovskite solar cells [69]. Radiative coupling also affects the measurement of multi-junction solar cells, and it is often called luminescence coupling [70; 71; 72]. Recently, multi-junction solar cells have been considered for use in non-concentrating applications, including car-roof photovoltaics (PV) [1; 9; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85]. It was considered that most electric vehicles might be able to run by solar energy using a solar cell mounted on the car roof [1].