## 3 Received Power Model of a PV Array with Experimental Validation  
### Modeling of Received Power at PV Array  
[PERSON] and [PERSON] proposed a PV array model under the Gaussian laser beam condition [31]. In their propose model, the PV array is a network of dimensions \(m\times n\), where \(m\) and \(n\) represents the number of PV cells in the column and row, respectively. Figure 1a illustrates the arrangement of individual PV cells, and Figure 2a shows the calculated normalized irradiance profile model on each numbered PV cell in a \(5\times 5\) PV array. Based on [PERSON]'s PV array model and the properties of 5S1010.4-A555555 of MHGP's standard VMJ PV array product [32], we modified and added some definitions to cover the off-centered position and the electrical output power of the PV array, as illustrated in Figures 1b-d [33; 34; 35]. Figures 2a-d also show the distribution of normalized irradiance with an off-centered position and different laser beam spot radii in a \(5\times 5\) PV array according to Table 1.  
\(PV(0,0)\) is the position of the cell with the highest irradiance in the PV array, and each \(PV(i,j)\) represents an individual PV cell, where index \(i\) and \(j\) represent the distances from \(PV(0,0)\) in the PV array. The distance \(D_{i,j}\) between \(PV(i,j)\) and \(PV(0,0)\) can be defined by Equation (1).  
\[D_{i,j}=\sqrt{i^{2}+j^{2}}. \tag{1}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Setup** \#1 & **Setup** \#2 & **Setup** \#3 & **Setup** \#4 \\ \hline Distance between & \multirow{2}{*}{7 m} & \multirow{2}{*}{7 m} & \multirow{2}{*}{7 m} & \multirow{2}{*}{7 m} \\ laser and PV array & & & & \\ \hline Spot radius on & 1.45 cell & 1.45 cell & 1.45 cell & 1.45 cell \\ PV array surface & (29 mm) & (29 mm) & (41 mm) & (62 mm) \\ \hline Laser power & 6 W & 6 W & 6 W & 6 W \\ \hline \multirow{3}{*}{Location of highest irradiation cell} & \multirow{3}{*}{Center (Figure 1a)} & Upper & Bottom & Bottom \\  & & left side & center side & right side \\ \cline{1-1}  & & & (Figure 1b) & (Figure 1c) & (Figure 1d) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Four experimental setups to validate a mathematical model.  
Figure 2: Distribution of normalized irradiance profile model for a \(5\times 5\) PV array: highest irradiance cell is located at (**a**) center, (**b**) upper left side, (**c**) bottom center side, and (**d**) bottom right side.  
In each PV cell, the normalized irradiance \(\big{(}\frac{G_{ij}}{G_{0,0}}\big{)}\) ranges from 0 to 1 and can be specified using Equation (2).  
\[\frac{G_{ij}}{G_{0,0}}=e^{-\frac{2D_{ij}^{2}}{\
u_{0}^{2}}}. \tag{2}\]  
\(\omega_{0}\) is the spot radius of the laser beam from PTX on the target surface, as defined by Equation (3).  
\[\omega_{0}=\frac{D_{L}+\alpha_{T}R_{T}}{\cos\theta_{T}}, \tag{3}\]  
where the entities in Equation (3) are as follows:  
* \(D_{L}\): diameter of the transmitted laser beam (m);
* \(\alpha_{T}\): divergence angle of the transmitted laser beam (rad);
* \(R_{T}\): distance between PTX and PRX (m);
* \(\theta_{T}\): incident angle of the laser beam on the PV array (\({}^{\circ}\)).  
\(G\) is the sum of the irradiance of each cell in the PV array with transmitted power PTX, as defined by Equation (4).  
\[G=\sum_{i=n_{0}}^{i=m}\sum_{j=n_{0}}^{j=n}G_{ij}=G_{0,0}\sum_{i=m_{0}}^{i=m} \sum_{j=n_{0}}^{j=n}\frac{G_{ij}}{G_{0,0}}=\frac{2P_{TX}}{\pi\omega_{0}^{2}} \sum_{i=m_{0}}^{i=m}\sum_{j=n_{0}}^{j=n}e^{-\frac{2D_{ij}^{2}}{\
u_{0}^{2}}}. \tag{4}\]  
The received power will be converted from the irradiated energy (\(G\)) on a given target surface (\(A\)) of the PV array with a photovoltaic power conversion (PPC) rate (\(\tau_{PPC}\)) and can be obtained from Equation (5).  
\[P_{RX}=G\times A\times\tau_{PPC}=A\tau_{PPC}\frac{2P_{TX}}{\pi\omega_{0}^{2}} \sum_{i=m_{0}}^{i=m}\sum_{j=n_{0}}^{j=n}e^{-\frac{2D_{ij}^{2}}{\omega_{0}^{2}}}. \tag{5}\]