# Photovoltaic Cell and Module I-V Characteristic Approximation Using Bezier Curves
## 4 Results
### I-V Characteristic Approximation with Two Segments and a Quadratic Bezier Curve  
The first approximation implies five control points: \(P_{sc}(0,\,I_{sc})\), \(P_{a}(x_{a},y_{a})\), \(P_{b}(x_{b},y_{b})\), \(P_{c}(x_{c},y_{c})\), and \(P_{oc}(V_{oc},0)\) and is made of two segment lines \(\overline{I_{sc}P_{a}}\) and \(\overline{P_{b}V_{oc}}\) and one quadratic Bezier curve defined by the endpoints \(P_{a}\) and \(P_{b}\) and the control point \(P_{c}\) (Figure 4). It has already been proven [3] that the slopes of the lines can be written as (9) and (10):  
\[\frac{dI}{dV}\bigg{|}_{V=0}=-\frac{1}{R_{sld}} \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline
**Symbol** & **Description** & **Value** \\ \hline \(V_{oc,module,ref}\) & Cell open circuit voltage & 44.68 V \\ \(I_{sc,ref}\) & Short circuit current & 8.24 A \\ \(V_{mp}\) & Maximum power voltage & 37.66 V \\ \(I_{mp}\) & Maximum power current & 7.70 A \\ \(R_{sld}\) & Shunt resistance at \(I_{sc}\) & 316 \(\Omega\) \\ \(R_{s0}\) & Series resistance at \(V_{oc}\) & 130 m\(\Omega\) \\ \(k_{I}\) & Current temperature coefficient & 3.296 mA/K \\ \(k_{V}\) & Voltage temperature coefficient & \(-\)146.256 mV/K \\ \(n_{s}\) & Number of series cell & 72 \\ \hline \end{tabular}
\end{table} 
Table 2: MSMD290 AS-36_EU Module main specifications on STC (1000 W/m\({}^{2}\), AM 1.5, 25 \({}^{\circ}\)C). Figure 4: Projected PV cell I-V characteristic approximation with two straight line segments and one quadratic Bézier curve. \[\left.\frac{dI}{dV}\right|_{V=V_{ac}}=-\frac{1}{R_{s0}} \tag{10}\]  
Thus, the equation for first line is (11):  
\[I=I_{sc}-\frac{V}{R_{sh0}} \tag{11}\]  
By choosing \(x_{a}\) in the linear region (eg \(0.6V_{oc}\)), one can find \(y_{a}\) from the above equation, so \(P_{a}\) is completely defined. For the second line, the next equation is valid (12):  
\[I=\frac{V_{oc}-V}{R_{s0}} \tag{12}\]  
The \(P_{c}(V_{c},I_{c})\) control point has therefore the coordinates defined by (13):  
\[\left(x_{c}=V_{c}=\frac{V_{oc}R_{sh0}-I_{sc}R_{sh0}R_{s0}}{R_{sh0}-R_{s0}},y_{c }=I_{c}=\frac{I_{sc}R_{sh0}-V_{oc}}{R_{sh0}-R_{s0}}\right) \tag{13}\]  
For \(x_{b}\), it must be emphasized that its position is on the end of the curve, a realistic value being \(0.9V_{oc}\). The maximum power point is positioned on the second curve, so solving (14) gives \(t_{mp}\):  
\[t_{mp}^{2}(x_{a}+x_{b}-2x_{c})+2t_{mp}(x_{c}-x_{a})+x_{a}-V_{mp}=0 \tag{14}\]  
Replacing the positive solution for \(t_{mp}\) in (2) yields \(y_{b}\) as in (15):  
\[y_{b}=\frac{\left(2t_{mp}-t_{mp}^{2}-1\right)y_{a}+2t_{mp}\big{(}t_{mp}-1 \big{)}y_{c}+I_{mp}}{t_{mp}^{2}} \tag{15}\]  
Now all the control points of the plot are completely defined. The results are summarized in Table 3. The application code written for the coordinate finding can be retrieved from [30]. The final plot is represented in Figure 5, where one can observe an excellent correspondence between the actual PV cell I-V characteristic, represented with black dots and the \(\overline{P_{sc}P_{a}}\) segment (blue line), a fair correlation for the second range, approximated by the Bezier quadratic curve (red line) and some modest results in the third region (magenta line). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Point** & \(\mathbf{x}\) **coordinate (V)** & \(\mathbf{y}\) **coordinate (A)** \\ \hline \multicolumn{3}{c}{First line segment} \\ \(P_{sc}\) & 0 & 9.207 \\ \(P_{a}\) & 0.4893 & 9.2003 \\ \multicolumn{3}{c}{Quadratic Bézier Curve} \\ \(P_{a}\) & 0.4893 & 9.2003 \\ \(P_{c}\) & 0.6070 & 9.1987 \\ \(P_{b}\) & 0.6291 & 7.0181 \\ \multicolumn{3}{c}{Second line segment} \\ \(P_{b}\) & 0.6291 & 7.0181 \\ \(P_{oc}\) & 0.699 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The control point coordinates when using two lines and one quadratic Bézier curve. The same conclusion arises from Figure 6, where the relative error has been plotted.