# Photovoltaic Cell and Module I-V Characteristic Approximation Using Bezier Curves
## 4 Results
### I-V Characteristic Approximation with Three Cubic Bezier Curves  
In order to have a general solution, we analyzed the case where all three regions are covered with cubic Bezier curves. This implies 12 control points (Figure 7), i.e. 24 coordinates to be found. The first curve, represented in Figure 8, is described by the control points \(P_{00}(0,I_{sc}),\ P_{01},\ P_{02},\) and \(P_{03}\). It turns out that the linear approximation of the first region of the I-V curve has an error below 0.5% if \(P_{03x}=\frac{V_{0x}}{2}\). During various simulations we also discovered that all \(P_{j,k}\) points can be evenly arranged, with \(j=\{1,2\}\); \(k=\{1,2,3\}\). This leads to \(P_{01x}=\frac{P_{03x}}{3}\) and \(P_{02x}=\frac{2P_{03x}}{3}\). For the \(y\) coordinates, \(P_{0\text{t}y}=I_{sc}-\frac{P_{03x}}{R_{040}}\), with \(k=\{1,2,3\}\). Now the first curve is completely defined. The second curve (Figure 7) is described by the control points \(P_{10}=P_{03},\ P_{11},\ P_{12},\) and \(P_{13}\). We observed that \(P_{13x}=0.75V_{oc}\) offers a very good fit of the curve for this type of PV cell. With the same evenly arrangement for the \(x\) coordinates, \(P_{1\text{k}x}=P_{10x}+\frac{k(P_{13x}-P_{10x})}{3}\), with \(k=\{1,2\}\). \(P_{11}\) is also located in the linear region of the I-V curve, so \(P_{11y}=P_{03y}-\frac{P_{11y}}{R_{040}}\). This leaves \(P_{12y}\) and \(P_{13y}\) as unknowns at this stage. The third curve (Figure 7) is described by the control points \(P_{20}=P_{13},\ P_{21},\ P_{22},\) and \(P_{23}(V_{oc},0)\). Using the same assumptions as for the second curve, \(P_{2 kx}=P_{20x}+\frac{k(P_{20x}-P_{03y})}{3}\), with \(k=\{1,2\}\). It is obvious that \(P_{20x}=P_{13x}\) and \(P_{23x}=V_{oc}\). The segment \(\overline{P_{22}P_{23}}\) is tangent to the curve at the point \(P_{23}\), so \(P_{22y}=\frac{(V_{oc}-P_{23x})}{R_{040}}\). This leaves \(P_{21y}\) as an additional unknown at this step. Figure 8: First Bézier curve with the associated control points. The slope is exaggerated for a better understanding. Figure 7: Projected PV cell I-V characteristic approximation with three cubic Bézier curves. For continuity reasons, \(\overline{P_{12}P_{13}}\) and \(\overline{P_{20}P_{21}}\) segments belong to the same line. This implies that the derivatives of the second curve at \(P_{13}\) and of the third curve at \(P_{20}\) are equal (16):  
\[3P_{13y}-3P_{12y}=3P_{21y}-3P_{20y} \tag{16}\]  
Which means that:  
\[P_{21y}=2P_{13y}-P_{12y} \tag{17}\]  
The control point \(P_{11}\) is placed on the second curve, so (18) can be written:  
\[V_{11}=P_{11x}={(1-t_{11})}^{3}P_{10x}+3t_{11}{(1-t_{11})}^{2}P_{11x}+3t_{11}^{ 2}{(1-t_{11})}P_{12x}+t_{11}^{3}P_{13x} \tag{18}\]  
Solving the previous equation and keeping only the real solution for \(t_{11}\), (19) is also valid:  
\[I_{11}=P_{11y}={(1-t_{11})}^{3}P_{10y}+3t_{11}{(1-t_{11})}^{2}P_{11y}+3t_{11}^{ 2}{(1-t_{11})}P_{12y}+t_{11}^{3}P_{13y} \tag{19}\]  
Finally, the graph also goes through the maximum power point \(P_{mp}\big{(}V_{mp},I_{mp}\big{)}\), yielding Equation (20):  
\[V_{mp}={\big{(}1-t_{mp}\big{)}}^{3}P_{20x}+3t_{mp}{\big{(}1-t_{mp}\big{)}}^{2} P_{21x}+3t_{mp}^{2}{\big{(}1-t_{mp}\big{)}}P_{22x}+t_{mp}^{3}P_{23x} \tag{20}\]  
Keeping only the real solution for \(t_{mp}\), (21) is also valid:  
\[I_{mp}={\big{(}1-t_{mp}\big{)}}^{3}P_{20y}+3t_{mp}{\big{(}1-t_{mp}\big{)}}^{2} P_{21y}+3t_{mp}^{2}{\big{(}1-t_{mp}\big{)}}P_{22y} \tag{21}\]  
The linear system made of equations (17), (19), and (21) give the last three unknown coordinates \(P_{12y}\), \(P_{13y}\), and \(P_{21y}\). The results are summarized in Table 4. The application code written for coordinate finding can be retrieved from [30]. Figure 9 shows the location of the control points with respect to the I-V characteristic of the PV cell. The control points \(P_{00},\ P_{01},\ P_{02},\ P_{03}=P_{10}\) and \(P_{11}\) are collinear and with \(P_{mp}\), are all placed on the I-V characteristic.