# Photovoltaic Cell and Module I-V Characteristic Approximation Using Bezier Curves
## 4 Results
### Final Validation  
In order to definitely test whether the proposed method is applicable to common PV cells and modules, a selection of 18 cases were analyzed at reference conditions, based on information found in their technical data [35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49], synthesized in Table 6. The selection includes three mono-crystalline PV arrays (1-3), nine poly-crystalline arrays (4-11 and 13) and one poly-crystalline PV cell (12), a thin-film array (14), one Heterojunction with Intrinsic Thin layer (HIT, 15) two amorphous silicon glass arrays (with different see-through degrees, respectively 10% and 30%, 16 and 17) and eventually an amorphous silicon cell (18). Using the Villalva algorithm [5], the main parameters were computed and are listed in Table 7. For the last three cases, due to different technology, interesting values for the diode ide \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**No.** & **PV Type** & **Tech** & \(n_{s}\) & \(V_{\infty}\) & \(V_{mp}\) & \(I_{mp}\) & \(I_{sc}\) & \(k_{V}\) & \(k_{I}\) \\  & & & & (\(V\)) & (\(V\)) & (\(A\)) & (\(A\)) & (\(V/K\)) & (\(A/K\)) \\ \hline
1 & Shell SP-70 & Mono & 36 & 21.4 & 16.5 & 4.24 & 4.7 & \(-\)0.076 & 0.002 \\
2 & Isofoton I150 InDach & Mono & 36 & 22.6 & 18.5 & 8.12 & 8.7 & \(-\)0.1026 & 0.00365 \\
3 & Bosch M24S 3 BB & Mono & 60 & 37.8 & 30.11 & 8.14 & 8.72 & \(-\)0.11718 & 0.002703 \\
4 & MSF300A5-36.EU & Poly & 72 & 44.48 & 37.42 & 8.02 & 8.58 & \(-\)0.14678 & 0.003432 \\
5 & Kycorea K5200 GT & Poly & 54 & 32.9 & 26.3 & 7.61 & 8.21 & \(-\)0.123 & 0.00318 \\
6 & Kycorea K485T & Poly & 36 & 21.7 & 17.4 & 5.02 & 5.34 & \(-\)0.0821 & 0.00212 \\
7 & Kycorea KCD135 SK\_UPU & Poly & 36 & 22.1 & 17.7 & 7.63 & 8.37 & \(-\)0.08 & 0.00502 \\
8 & Kycorea KCD245 GH-4 FB2 & Poly & 60 & 36.9 & 29.8 & 8.23 & 8.91 & \(-\)0.133 & 0.00535 \\
9 & Sharp ND2-24 aC1 & Poly & 60 & 36.6 & 29.3 & 7.66 & 8.33 & \(-\)0.13176 & 0.004415 \\
10 & Shell S36 & Poly & 36 & 21.4 & 16.5 & 2.18 & 2.3 & \(-\)0.076 & 0.001 \\
11 & Solarex MSX-60 & Poly & 36 & 21.1 & 17.1 & 1.3 & & \(-\)0.08 & \\
12 & Solarex MSX-60—cell & Poly & 1 & 0.586 & 0.475 & 3.5 & 3.8 & \(-\)0.00222 & 0.003 \\
13 & AmeriosiSA-6F 300W & Poly & 72 & 44.7 & 36.7 & 8.19 & 8.68 & \(-\)0.14751 & 4.86E-03 \\
14 & Shell ST40 & Thin-Film & 36 & 23.3 & 16.6 & 2.41 & 2.68 & \(-\)0.1 & 0.00035 \\
15 & Sanyo HIT-240 HDR4 & HIT & 60 & 43.6 & 35.5 & 6.77 & 7.37 & \(-\)0.109 & 0.00221 \\
16 & Onyx 1200 \(\times\) 600 Ref10 & aSI & 72 & 47 & 32 & 0.9 & 1.11 & \(-\)0.0893 & 0.000999 \\
17 & Onyx 1200 \(\times\) 600 Ref30 & glass & 72 & 47 & 32 & 0.63 & 0.74 & \(-\)0.0893 & 0.000666 \\
18 & 6.5 WP L Cel & aSI cell & 1 & 2.2 & 1.6 & 4.09 & 5.1 & \(-\)0.00836 & 0.00612 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Datasheet technical data for the analyzed 18 PV cells and modules [35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49]. Figure 15: Bézier approximation of the I-V temperature dependent characteristics for the MSMD290 AS-36_EU Monocrystalline PV module. The lines represent the computed curves, whereas the markers represent the actual data. Table 8 lists the control points computed for Bezier curve fitting, where in all cases the control end points are set at \(0.5V_{oc}\) and \(0.75V_{oc}\) respectively. Only two \(x\) coordinates are presented, as the others are evenly spaced and can be easily computed. Selected plots of the PV devices are presented in Figure 16.  
In the second column of Table 9, the average of the current (\(I\)) relative error is displayed in order to evaluate the fit quality. Maximum current error (absolute and relative, listed as positive values) is associated in each case with the coordinates where it appears. The maximum power point is also investigated as an absolute and relative error and finally the computed value is listed. The current (\(I\)) relative error is below 1.18% in all cases, and the \(P_{mp}\) relative error is even better (lower than 1%).