## 2 Theoretical Model
### Optical-Electrical-Thermal Model of Non-Uniform Concentration  
Under high concentration, the model is equipped with the triple-junction solar cells, type 3C44C produced by AZUR SPACE Solar Power GmbH [30]. The solar cell consists of GaInP/GaInAs/Ge. All sub cells are connected by tunnel diodes. The sub cells are simulated with the two-diode model which uses two diodes connected in parallel with a current source. According to the equivalent circuit, the current-voltage equation of three junctions is:  
\[\mathrm{I}=\mathrm{I}_{\mathrm{Li}}-\mathrm{I}_{\mathrm{oi1}}\bigg{(}\exp \frac{\mathrm{qV_{i}}}{\mathrm{kT}}-1\bigg{)}-\mathrm{I}_{\mathrm{oi2}}\bigg{(} \exp\frac{\mathrm{qV_{i}}}{\mathrm{2 kT}}-1\bigg{)}, \tag{5}\]  
where \(\mathrm{I}_{\mathrm{Li}}\) (i = 1, 2, 3) represents the photogenerated current of the upper, middle and lower diodes. \(\mathrm{I}_{\mathrm{oi1}}\), \(\mathrm{I}_{\mathrm{oi2}}\) (i = 1, 2, 3) are the reverse saturation currents of the upper, middle and lower layer diodes, respectively. The expression of the variation with temperature is as follows [31]:  
\[\mathrm{I}_{\mathrm{oi1}}(\mathrm{T})=\mathrm{A}\cdot\mathrm{k}_{\mathrm{oi1}} \cdot\mathrm{T}^{3}\bigg{[}\exp\bigg{(}-\frac{\mathrm{E}_{\mathrm{gi}}( \mathrm{T})}{\mathrm{k}_{\mathrm{B}}\mathrm{T}}\bigg{)}\bigg{]}, \tag{6}\]  
\[\mathrm{I}_{\mathrm{oi2}}(\mathrm{T})=\mathrm{A}\cdot\mathrm{k}_{\mathrm{oi2}} \cdot\mathrm{T}^{\frac{3}{2}}\bigg{[}\exp\bigg{(}-\frac{\mathrm{E}_{\mathrm{gi }}(\mathrm{T})}{\mathrm{2k}_{\mathrm{B}}\mathrm{T}}\bigg{)}\bigg{]}, \tag{7}\]  
\[\mathrm{E}_{\mathrm{gi}}(\mathrm{T})=\mathrm{E}_{\mathrm{gi}}(0)-\alpha\mathrm{ T}^{2}/(\mathrm{T}+\beta), \tag{8}\]  
where A is the area of the triple-junction cell; \(\mathrm{k}_{\mathrm{oi1}}\) and \(\mathrm{k}_{\mathrm{oi2}}\) are constants related to the cell material and can be calculated by fitting Equation (5), where the data are from irradiance-V\({}_{\mathrm{OC}}\) measurements [32]; \(\mathrm{E}_{\mathrm{gi}}(\mathrm{T})\) indicates the band gap width at temperature T; \(\alpha\) and \(\beta\) are material-dependent constants; and \(\mathrm{k}_{\mathrm{B}}\) is the Boltzmann constant. Considering that the illumination from the Fresnel lens is non-uniform, the photoelectric conversion of solar cells is calculated by circuit network model [30; 33; 34; 35]. As shown in Figure 2a,b, the multijunction solar cell is divided into several triple junction cells. Each sub cell is simulated by an electrical circuit based on the two-diode model. These two diodes represent the losses by recombination in the neutral and in the depletion region of the p-n junction. Because it is a 3D model, all of them have the lateral series resistance including both transverse and longitudinal resistance (\(\mathrm{R}_{\mathrm{1x}}\)-\(\mathrm{R}_{\mathrm{6x}}\), \(\mathrm{R}_{\mathrm{1y}}\)-\(\mathrm{R}_{\mathrm{6y}}\)). Taking the GaInP sub cell as an example, the emission layer and the window layer are regarded as a whole and their lateral resistance in x, y direction is \(\mathrm{R}_{\mathrm{1x}}\), \(\mathrm{R}_{\mathrm{1y}}\). Similarly, the base layer and the back surface field are also regarded as a whole and their resistance is \(\mathrm{R}_{\mathrm{2x}}\), \(\mathrm{R}_{\mathrm{2y}}\), respectively. The sub cells are connected in tunnel diodes which in this model are fixed value resistors(R\({}_{\rm sh1}\)-R\({}_{\rm sh3}\)). \(R_{c}\) represents metal-semiconductor contact resistor and R\({}_{\rm f}\) is the resistance of the grid finger. Since it is difficult to directly measure various resistance values of the triple-junction solar cells and the basic production process is almost similar, the simulation used the resistance data values in the literature [32]. It is worth noting that when the grid is divided according to the grid lines of the cell, there are two situations: One is the area that is not blocked by the metal grid line where the illumination can reach the cell surface to produce photoelectric effect. This is simulated by one current source and two diodes.