## 2 Theoretical Model
### Heat Transfer Model  
Non-uniform illumination not only affects photoelectric performance but also affects the temperature distribution of the module receiver. ANSYS is used to simulate finite element coupled multiple physics fields and calculate the temperature distribution of the receiver at different positions along the optical axis. The receiver consists of a multijunction solar cell, DBC (Direct Bonded Copper) and heat sink. DBC is used to maintain the electrical insulation between the cell and the base plate has good performance on thermal conductivity. Figure 3 shows the heat dissipation diagram of the receiver.  
The receiver is divided into five layers according to the types of materials. The top layer is the triple-junction solar cell. As the main component of the cell is Ge, the cell material can be set as Ge in the calculation of heat dissipation model. The DBC layer is divided into three parts: the upper layer and lower layer are copper and the middle layer is ceramic. The upper copper layer is also used as an electrode of the cell, and the ceramic layer is used for electrical insulation. The bottom layer is aluminum for heat dissipation [36]. Table 1 lists the material and thickness of each layer of the receiver.  
In the photovoltaic module, the sunlight converges to the cell surface through the lens to join in photoelectric conversion. Part of the remaining light energy is reflected by the cell surface and part is converted into internal energy of the cell [37]. The illumination energy converted into the internal energy is used as the boundary heat source [38] flowing into the cell. Because the cell efficiency is related to temperature, the boundary heat source can be considered as a function of temperature. In passive cooling environment, for the cell layers, the exposed surface of the DBC layer and heat dissipation aluminum layer, the boundary equations can be expressed as follows:  
\[-\mathrm{k_{i}}\bigg{(}\frac{\partial\mathrm{T_{i}}}{\partial y}\bigg{)}= \mathrm{Q_{li}}+\mathrm{Q_{2i}}, \tag{12}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category** & **Thickness (mm)** & **Length (mm)** & **Width (mm)** \\ \hline Cell & 0.19 & 5.5 & 5.5 \\ DBC upper layer & 0.25 & 23 & 25 \\ DBC ceramic layer & 0.32 & 24.5 & 26.5 \\ DBC lower layer & 0.25 & 24 & 26 \\ Aluminum heat spreader & 0.6 & 90 & 90 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of each layer material of the receiver.  
Figure 3: Heat dissipation diagram of the receiver.  
where \(\rm k_{i}\) and \(\rm T_{i}\) are thermal conductivity and temperature of each layer, \(\rm Q_{li}\) refers to the radiation heat transfer with surrounding environment and \(\rm Q_{2i}\) represents the convection heat transfer with air [39]. They can be expressed as the following equations:  
\[\rm Q_{li}=\varepsilon_{i}\cdot A_{i}\cdot\left(T_{i}^{4}-T_{s}^{4}\right), \tag{13}\]  
\[\rm Q_{2i}=h_{conv(i)}(T_{i}-T_{amb}), \tag{14}\]  
where \(\varepsilon_{i}\) and \(A_{i}\) are the emissivity and heat exchange area of each layer; \(\rm h_{conv(i)}\) is the convective heat transfer coefficient of each layer; and \(T_{s}\) is the sky temperature. For two adjacent materials (i, j), the heat is transferred in the way of heat conduction. The boundary condition of them can be expressed as follows:  
\[\rm k_{i}\frac{\partial T_{i}}{\partial y}=\rm k_{j}\frac{\partial T_{j}}{ \partial y}, \tag{15}\]