Photovoltaic Cell and Module I-V Characteristic Approximation Using Bézier Curves

Abstract

The aim of this work was to introduce new ways to model the I-V characteristic of a photovoltaic (PV) cell or PV module using straight lines and Bézier curves. This is a complete novel approach, Bézier curves being previously used mainly for computer graphics. The I-V characteristic is divided into three sections, modeled with lines and a quadratic Bézier curve in the first case and with three cubic Bézier curves in the second case. The result proves to be accurate and relies on the fundamental points usually present in the PV cell datasheets: $V_{oc}$ (the open circuit voltage), $I_{sc}$ (the short circuit current), $V_{mp}$ (the maximum power corresponding voltage) and $I_{mp}$ (the maximum power corresponding current), and the parasitic resistances $R_{sh0}$ (shunt resistance at $I_{sc}$) and $R_{s0}$ (series resistance at $V_{oc}$). The proposed algorithm completely defines all the implied control points and the error is analyzed. The temperature and irradiance influence is also analyzed. The model is also compared using the least squares fitting method. The final validation shows how to use Bézier cubic curves to accurately represent the I-V curves of an extensive range of PV cells and arrays.

1. Introduction

The forecast of the total photovoltaic (PV) installations, offered by Bloomberg New Energy Finance (BNEF) predicts an optimistic growth at 111 GW in 2018, rising to 121 GW in 2019, along with a polysilicon factory growth boom and module prices drop to US$ 0.30/W [1,2]. This robust growth explains the high interest in PV research, modeling, and simulation–along with design and development of PV equipment.

The electrical characteristics of the PV cell and PV modules have been of interest for several decades, and different models have been proposed. Phang and Chan [3] were among the first to propose a solution for PV cell parameter extraction. Garrido-Alzar [4] uses a double exponential model to extract the PV cell parameters using the experimental I-V curve. Villalva et al. [5] developed an algorithm to find the parameters defining the I-V characteristic for the single diode model of a PV cell, using the Newton-Raphson method and imposing a minimum error threshold for the maximum power point. Babu and Gurjar [6] introduced a simplified two-diode model for the PV module, while Cubas et al. [7] used the Lambert W-Function for finding the solar panel equivalent circuit parameters and also proposed an LTSpice model according to their findings. Temperature influence has been studied by Chander et al. [8]. Ishaque and Salam [9] use differential evolution to find the PV modules parameters.

Franzitta et al. provided extensive studies of the most widely used models both in single diode case [10] and for the double-diode version [11], introducing a criterion for rating the accuracy and usability of the analyzed models. In our previous paper [12], we proposed a complete SPICE model including all the parameters variation and selfheating. In a recent work, Cuce et al. [13] claim a good accuracy for their electrical model for a PV module and they also discuss energy and exergy efficiency as a reliable substitute for the fill factor. All the aforementioned works use an electrical model to describe the behavior of the circuit and rely on a specific circuit to generate the I-V characteristic of the PV cell or module.

This paper introduces a new approach. This time, the cell or module are not involved at the electrical level, being defined by just the specific points $P_{sc},P_{mp},P_{oc}$ and by the parallel and series resistances $R_{sh0}$ and $R_{s0}$, specified at $I_{sc}$ and $V_{oc}$, respectively. Careful inspection of the typical I-V characteristic of the PV module or PV cell (Figure 1a,b) shows a similar pattern in all curves. Our aim was to find a way to model it using smooth curves and the datasheet information currently available.

Bernstein polynomials have been studied since the beginning of the 20^th century and they form the foundation for Bézier curves [14]. The core applications for graphics came first in 1959 when the French mathematician Paul de Casteljau developed an algorithm able to evaluate a family of specific curves at Citroën. In 1962 the French engineer Pierre Bézier also used them to design automobile bodies at Renault and afterwards they achieved wide acceptance [15].

Bézier curves are largely used in computer graphics [16,17] and also in the time domain for smoothing the trajectory of the robotic arms, for an accurate gluing or welding path or for trajectory generation [18]. Further development for shape representation was proposed by Jalba et al. [19].

The current proposal analyses the use of Bézier curves [20] in order to accurately represent the I-V characteristic of a PV cell or module. Although Bézier curves are commonly used as 2D curves (which are usually not functions) [21], our approach uses such curves to approximate an I-V characteristic (a function). A complete mathematical solution is provided, separately validated for a PV cell and a PV module and the error is analyzed. The results are also studied for different temperature and irradiances and finally compared with the ones offered by the least squares fitting method. In depth analysis of Bézier cubic curves fitting for 18 PV arrays and cells (from various manufacturers and different technologies) is also performed.

There are other possible approaches, for example spline fitting [22,23], well known Hermite interpolation, or spiral curve fitting [24]. Our aim was solely the Bézier curves.

The remainder of this paper is organized as follows: Section 2 briefly analyzes the definition of the quadratic and cubic Bézier curves and their equations, focusing on the basic knowledge needed in the subsequent paragraphs. Section 3 deals with the information usually provided by the PV cell or module manufacturers in their datasheets. The proposed models are covered in Section 4, with Section 4.1. introducing the approximation with two segments and one quadratic Bézier curve, while Section 4.2. deals with the better approximation based on three Bézier cubic curves. The models are rated at the reference temperature, 25 °C. The results are also compared with the least squares fitting method in Section 4.3. In Section 4.4. the influence of the external parameters is analyzed. The proposal is verified in Section 4.5. against a large range of PV cells and modules and the results show a good fit. Discussion and conclusions are provided in the next Sections.

2. Definition of the Bézier Curves

A quadratic Bézier curve (Figure 2) can be specified by three control points [15]: the curve goes through the ends $P_0$ and $P_2$ and approximates $P_1$.

The curve equation is as follows [20]:

$$B_2\left(t\right)={\left(1-t\right)}^2{P}_0+2t\left(1-t\right)P_1+t^2{P}_2$$

(1)

where $t$ varies between 0 and 1. Equation (1) can be expressed for the $x$ and $y$ coordinates:

$$\{\begin{array}{c}x\left(t\right)={\left(1-t\right)}^2{x}_0+2t\left(1-t\right)x_1+t^2{x}_2\\ y\left(t\right)={\left(1-t\right)}^2{y}_0+2t\left(1-t\right)y_1+t^2{y}_2\end{array}$$

(2)

The derivative of (1) is:

$$B_2^{\prime }\left(t\right)=2\left(1-t\right)\left(P_1-P_0\right)+2t\left(P_2-P_1\right)$$

(3)

At the end points, (3) becomes (4):

$$\{\begin{array}{c}{B_2^{\prime }\left(t\right)|}_{t=0}=2P_1-2P_0\\ {B_2^{\prime }\left(t\right)|}_{t=1}=2P_2-2P_1\end{array}$$

(4)

A cubic Bézier curve (Figure 3) can be specified by four control points [15]: the curve goes through the ends $P_0$ and $P_3$ and approximates $P_1$ and $P_2$. The analytical expression of the curve is a cubic polynomial. The curve is tangent at $P_0$ to $\overline{P_0{P}_1}$ and at $P_3$ to $\overline{P_3{P}_2}$.

The equation for the Bézier cubic curve is [20]:

$$B_3\left(t\right)={\left(1-t\right)}^3{P}_0+3t{\left(1-t\right)}^2{P}_1+3t^2\left(1-t\right)P_2+t^3{P}_3$$

(5)

The previous equation can be expressed for the $x$ and $y$ coordinates:

$$\{\begin{array}{c}x\left(t\right)={\left(1-t\right)}^3{x}_0+3t{\left(1-t\right)}^2{x}_1+3t^2\left(1-t\right)x_2+t^3{x}_3\\ y\left(t\right)={\left(1-t\right)}^3{y}_0+3t{\left(1-t\right)}^2{y}_1+3t^2\left(1-t\right)y_2+t^3{y}_3\end{array}$$

(6)

The derivative of (5) is:

$$B_3^{\prime }\left(t\right)=3{\left(1-t\right)}^2\left(P_1-P_0\right)+6t\left(1-t\right)\left(P_2-P_1\right)+3t^2\left(P_3-P_2\right)$$

(7)

At the end points, (7) becomes (8):

$$\{\begin{array}{c}{B_3^{\prime }\left(t\right)|}_{t=0}=3P_1-3P_0\\ {B_3^{\prime }\left(t\right)|}_{t=1}=3P_3-3P_2\end{array}$$

(8)

3. Materials and Methods

The first PV cell used in our work is a high efficiency, silicon monocrystalline 156 × 156 mm² cell [25] and has the main characteristics summarized in

Table 1. PV Cell main specifications on STC (1000 W/m², AM 1.5, 25 °C).

Symbol	Description	Value
$V_{oc,cell,ref}$	Cell open circuit voltage	0.699 V
$I_{sc,ref}$	Short circuit current	9.207 A
$V_{mp}$	Maximum power voltage	0.572 V
$I_{mp}$	Maximum power current	8.756 A
$R_{sh0}$	Shunt resistance at $I_{sc}$	73.19 Ω
$R_{s0}$	Series resistance at $V_{oc}$	3.8 mΩ

.

The MSMD290AS-36_EU Monocrystalline PV module formerly manufactured by MünchenSolar Germany [26] has been well documented and studied by Cubas et al. [6]. Its main electrical data is listed in

Table 2. MSMD290AS-36_EU Module main specifications on STC (1000 W/m², AM 1.5, 25 °C).

Symbol	Description	Value
$V_{oc,module,ref}$	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
$R_{sh0}$	Shunt resistance at $I_{sc}$	316 Ω
$R_{s0}$	Series resistance at $V_{oc}$	130 mΩ
$k_I$	Current temperature coefficient	3.296 mA/K
$k_V$	Voltage temperature coefficient	−146.256 mV/K
$n_s$	Number of series cell	72

and this information is used in Section 4.4. to evaluate the influence of the temperature and irradiance to our Bézier curves based model.

For studying and representing Bézier curves, an interesting application which allows draggable control points was developed by Mugnaini [27]. For computing the coordinates on the curves we used the Kronecker tensor product found as in [28]. An example for the Bézier least square fitting method is given in [29].

4. Results

It must be stressed that all the physical actual values involved in Section 4.1., Section 4.2., and Section 4.3. are specified at 25 °C, being reference values. The irradiance is also standard (1000 W/m²). As it is demonstrated in Section 4.4., the same method is suitable for different temperatures and irradiances.

4.1. I-V Characteristic Approximation with Two Segments and a Quadratic Bézier Curve

The first approximation implies five control points: $P_{sc}\left(0,I_{sc}\right),P_a\left(x_a,y_a\right),P_b\left(x_b,y_b\right),P_c\left(x_c,y_c\right),$ and $P_{oc}\left(V_{oc},0\right)$ and is made of two segment lines $\overline{I_{sc}{P}_a}$ and $\overline{P_b{V}_{oc}}$ and one quadratic Bézier 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):

$${\frac{dI}{dV}|}_{V=0}=-\frac{1}{R_{sh0}}$$

(9)

$${\frac{dI}{dV}|}_{V=V_{oc}}=-\frac{1}{R_{s0}}$$

(10)

Thus, the equation for first line is (11):

$$I=I_{sc}-\frac{V}{R_{sh0}}$$

(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):

$$I=\frac{V_{oc}-V}{R_{s0}}$$

(12)

The $P_c\left(V_c,I_c\right)$ control point has therefore the coordinates defined by (13):

$$\left(x_c=V_c=\frac{V_{oc}{R}_{sh0}-I_{sc}{R}_{sh0}{R}_{s0}}{R_{sh0}-R_{s0}},y_c=I_c=\frac{I_{sc}{R}_{sh0}-V_{oc}}{R_{sh0}-R_{s0}}\right)$$

(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 $t_{mp}$:

$$t_{mp}^2\left(x_a+x_b-2x_c\right)+2t_{mp}\left(x_c-x_a\right)+x_a-V_{mp}=0$$

(14)

Replacing the positive solution for $t_{mp}$ in (2) yields $y_b$ as in (15):

$$y_b=\frac{\left(2t_{mp}-t_{mp}^2-1\right)y_a+2t_{mp}\left(t_{mp}-1\right)y_c+I_{mp}}{t_{mp}^2}$$

(15)

Now all the control points of the plot are completely defined. The results are summarized in

Table 3. The control point coordinates when using two lines and one quadratic Bézier curve.

Point	$\mathit{x}$ coordinate (V)	$\mathit{y}$ coordinate (A)
First line segment
$P_{sc}$	0	9.207
$P_a$	0.4893	9.2003
Quadratic Bézier Curve
$P_a$	0.4893	9.2003
$P_c$	0.6070	9.1987
$P_b$	0.6291	7.0181
Second line segment
$P_b$	0.6291	7.0181
$P_{oc}$	0.699	0

. 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 Bézier quadratic curve (red line) and some modest results in the third region (magenta line).

The same conclusion arises from Figure 6, where the relative error has been plotted. It is worth mentioning that although the relative error is quite high above 0.64 V $\left(0.92V_{oc}\right)$, the absolute error is in fact less than 0.7 A in a region where the cell normally should not operate.

Looking for a more accurate model is the reason we came up with the second scenario, where the I-V characteristic is entirely modeled with cubic Bézier curves.

4.2. I-V Characteristic Approximation with Three Cubic Bézier Curves

In order to have a general solution, we analyzed the case where all three regions are covered with cubic Bézier 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 $P_{00}\left(0,I_{sc}\right),P_{01},P_{02,}$ 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 $P_{03x}=\raisebox{1ex}{$V_{oc}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. During various simulations we also discovered that all $P_{j,k}$ points can be evenly arranged, with $j=\left\{1,2\right\};k=\left\{1,2,3\right\}$. This leads to $P_{01x}=\raisebox{1ex}{$P_{03x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ and $P_{02x}=\raisebox{1ex}{$2P_{03x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$. For the y coordinates, $P_{0ky}=I_{sc}-\raisebox{1ex}{$P_{0kx}$}\!\left/ \!\raisebox{-1ex}{$R_{sh0}$}\right.$, with $k=\left\{1,2,3\right\}$. 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, $P_{1kx}=P_{10x}+\raisebox{1ex}{$k\left(P_{13x}-P_{10x}\right)$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$, with $k=\left\{1,2\right\}$. $P_{11}$ is also located in the linear region of the I-V curve, so $P_{11y}=P_{03y}-\raisebox{1ex}{$P_{11x}$}\!\left/ \!\raisebox{-1ex}{$R_{sh0}$}\right.$. 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 $P_{23}\left(V_{oc},0\right)$. Using the same assumptions as for the second curve, $P_{2kx}=P_{20x}+\raisebox{1ex}{$k\left(P_{23x}-P_{20x}\right)$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$, with $k=\left\{1,2\right\}$. 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 $P_{22y}=\raisebox{1ex}{$\left(V_{oc}-P_{22x}\right)$}\!\left/ \!\raisebox{-1ex}{$R_{s0}$}\right.$. This leaves $P_{21y}$ as an additional unknown at this step.

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):

$$3P_{13y}-3P_{12y}=3P_{21y}-3P_{20y}$$

(16)

Which means that:

$$P_{21y}=2P_{13y}-P_{12y}$$

(17)

The control point $P_{11}$ is placed on the second curve, so (18) can be written:

$$V_{11}=P_{11x}={\left(1-t_{11}\right)}^3{P}_{10x}+3t_{11}{\left(1-t_{11}\right)}^2{P}_{11x}+3t_{11}^2\left(1-t_{11}\right)P_{12x}+t_{11}^3{P}_{13x}$$

(18)

Solving the previous equation and keeping only the real solution for $t_{11}$, (19) is also valid:

$$I_{11}=P_{11y}={\left(1-t_{11}\right)}^3{P}_{10y}+3t_{11}{\left(1-t_{11}\right)}^2{P}_{11y}+3t_{11}^2\left(1-t_{11}\right)P_{12y}+t_{11}^3{P}_{13y}$$

(19)

Finally, the graph also goes through the maximum power point $P_{mp}\left(V_{mp},I_{mp}\right)$, yielding Equation (20):

$$V_{mp}={\left(1-t_{mp}\right)}^3{P}_{20x}+3t_{mp}{\left(1-t_{mp}\right)}^2{P}_{21x}+3t_{mp}^2\left(1-t_{mp}\right)P_{22x}+t_{mp}^3{P}_{23x}$$

(20)

Keeping only the real solution for $t_{mp}$, (21) is also valid:

$$I_{mp}={\left(1-t_{mp}\right)}^3{P}_{20y}+3t_{mp}{\left(1-t_{mp}\right)}^2{P}_{21y}+3t_{mp}^2\left(1-t_{mp}\right)P_{22y}$$

(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 control point coordinates when using three cubic Bézier curves.

Point	$\mathit{x}$ coordinate (V)	$\mathit{y}$ coordinate (A)
First Bézier cubic curve
$P_{00}$	0	9.207
$P_{01}$	0.1165	9.206
$P_{02}$	0.2330	9.204
$P_{03}$	0.3495	9.202
Second Bézier cubic curve
$P_{10}$	0.3495	9.202
$P_{11}$	0.4078	9.197
$P_{12}$	0.4660	9.210
$P_{13}$	0.5243	9.074
Third Bézier cubic curve
$P_{20}$	0.5243	9.074
$P_{21}$	0.5825	8.939
$P_{22}$	0.6408	8.616
$P_{23}$	0.6990	0

. 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.

Figure 10 shows the modeled characteristic (red, green, and blue lines) overlapping in most areas with the practical I-V characteristic (black markers). The application code can also be retrieved from [30].

The relative error of the Bézier modeled I-V characteristic against the actual data taken from [12] is shown in Figure 11. It must be emphasized that in the 0–0.94 $V_{oc}$ range, the relative error is below 1%. Above 0.94 $V_{oc}$ the absolute error is less than 72 mA, while the reference $I_{sc}=9.207\mathrm{A}$.

4.3. Data Fitting Using the Least Squares Method

Data fitting using the least squares method is a standard approach in data analysis [31,32]. A good overview of curve fitting using Bézier cubic curves in image processing is given by Shao et al. in [33], while Zhao et al. [34] extend this method using a genetic algorithm for parameter optimization for Bézier curve fitting. In Section 4.2. we have shown that for the studied PV cell, the best results arise when the $x$ coordinates of the middle end points are set at $0.5V_{oc}$ and $0.75V_{oc}$ respectively. A similar conclusion arises if the least squares method is used for the same cell modeling. Running the least squares method for the MSMD290AS-36_EU Monocrystalline PV module proved that the minimum error occurs when the control end points are set again at $0.5V_{oc}$ and $0.75V_{oc}$ respectively.

Table 5. Control point coordinates comparison. On the left, the least squares method is used for computation, on the right the same values as in Table 4 are listed.

	Least Squares Method		Proposed Method
Point	$x$ coordinate (V)	$y$ coordinate (A)	$x$ coordinate (V)	$y$ coordinate (A)
First Bézier cubic curve
$P_{00}$	0	9.207	0	9.207
$P_{01}$	0.1165	9.206	0.1165	9.206
$P_{02}$	0.2330	9.204	0.2330	9.204
$P_{03}$	0.3495	9.202	0.3495	9.202
Second Bézier cubic curve
$P_{10}$	0.3495	9.202	0.3495	9.202
$P_{11}$	0.4076	9.183	0.4078	9.197
$P_{12}$	0.4658	9.245	0.4660	9.210
$P_{13}$	0.5239	9.103	0.5243	9.074
Third Bézier cubic curve
$P_{20}$	0.5239	9.103	0.5243	9.074
$P_{21}$	0.5823	8.9646	0.5825	8.939
$P_{22}$	0.6406	8.6724	0.6408	8.616
$P_{23}$	0.6990	0.004	0.6990	0

summarizes the data fitting results for the same PV cell used in Section 4.1. and Section 4.2., where the results from the two approaches are very close. The graphical representation of the date fitting is given in Figure 12, where only the endpoints are represented.

Figure 13 depicts the relative error of the modeled I-V characteristic compared with the actual data taken from our previous work [12]. In the $0–0.96V_{oc}$ range, the relative error is below 2%. Furthermore, above $0.96V_{oc}$, the absolute error is less than 66 mA, while the reference short circuit current is 9.207 A.

4.4. Parameter Variation

In order to further validate the proposed method, in this section we analyze the temperature and irradiance influence for the MSMD290AS-36_EU Monocrystalline PV module. An extensive study of the parameter influence over the PV cell can be found in [12]. It is important to notice that the Bézier approximation is not related to any of the parameter variation, just to the specified points $P_{sc},P_{mp},P_{oc}$ and the parasitic resistances $R_{sh0}$ and $R_{s0}$ as already stated. The challenge becomes in this case the finding of the new position for the control points and the new values for the parasitic resistances.

Villalva et al. [5] accurately describe the short circuit current variation as in (22):

$$I_{sc}=\left(\frac{R_{sh}+R_s}{R_{sh}}{I}_{sc,ref}+k_I\Delta T\right)\frac{G}{G_{ref}}\approx \left(I_{sc,ref}+k_I\Delta T\right)\frac{G}{G_{ref}}$$

(22)

Ishaque and Salam [9] propose for the $V_{oc,cell}$ the following variation (23):

$$V_{oc,cell}=V_{0c,cell,ref}+a\frac{kT}{q}ln\frac{G}{G_{ref}}+k_v\Delta T$$

(23)

Equation (23) proved to be too conservative in this case, as larger $V_{oc,cell}$ variations were observed. A better approximation is the empirical law (24):

$$V_{oc}=29.579+2.1934\ln G$$

(24)

A possible way for defining $R_{sh}$ behavior is suggested in [12], as in (25) with $k_{Rsh}$ estimated as 8 for the best fit.

$$R_{sh}=R_{sh,ref}{\left(\frac{T_{ref}}{T}\right)}^{k_{Rsh}}$$

(25)

For $R_s$, a linear variation law (26) is given in [12] with ${\alpha }_{Rs}=-0.01{\mathrm{K}}^{-1}$, again for the best fit:

$$R_s=R_{s,ref}\left[1+{\alpha }_{Rs}\left(T-T_{ref}\right)\right]$$

(26)

Figure 14 shows the irradiance influence for the I-V module characteristic, where the approximated data using our proposed method is plotted with solid lines and the experimental data is represented with markers. $I_{sc},V_{oc},R_{sh},$ and $R_s$ were computed using (22), (24–26) respectively.

The temperature dependent Bézier curves resulted from our algorithm compared with the actual data are introduced in Figure 15. Once again, the results show a very good correlation between the modeled data and the actual data.

4.5. 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. 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].

No.	PV Type	Tech	${\mathit{n}}_{\mathit{s}}$	${\mathit{V}}_{\mathit{o}\mathit{c}}$ $\left(\mathit{V}\right)$	${\mathit{V}}_{\mathit{m}\mathit{p}}$ $\left(\mathit{V}\right)$	${\mathit{I}}_{\mathit{m}\mathit{p}}$ $\left(\mathit{A}\right)$	${\mathit{I}}_{\mathit{s}\mathit{c}}$ $\left(\mathit{A}\right)$	${\mathit{k}}_{\mathit{V}}$ $\left(\mathit{V}/\mathit{K}\right)$	${\mathit{k}}_{\mathit{I}}$ $\left(\mathit{A}/\mathit{K}\right)$
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 M245 3BB	Mono	60	37.8	30.11	8.14	8.72	−0.11718	0.002703
4	MSP300AS-36.EU	Poly	72	44.48	37.42	8.02	8.58	−0.14678	0.003432
5	Kyocera KG200GT	Poly	54	32.9	26.3	7.61	8.21	−0.123	0.00318
6	Kyocera KC85T	Poly	36	21.7	17.4	5.02	5.34	−0.0821	0.00212
7	Kyocera KD135SX_UPU	Poly	36	22.1	17.7	7.63	8.37	−0.08	0.00502
8	Kyocera KD245GH-4FB2	Poly	60	36.9	29.8	8.23	8.91	−0.133	0.00535
9	Sharp ND-224uC1	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	3.5	3.8	−0.08	0.003
12	Solarex MSX-60—cell	Poly	1	0.586	0.475			−0.00222
13	Amerisolar AS-6P 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 HDE4	HIT	60	43.6	35.5	6.77	7.37	−0.109	0.00221
16	Onyx 1200 × 600 Ref10	aSi glass	72	47	32	0.9	1.11	−0.0893	0.000999
17	Onyx 1200 × 600 Ref30					0.63	0.74		0.000666
18	6.5 Wp L Cel	aSi cell	1	2.2	1.6	4.09	5.1	−0.00836	0.00612

. 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. Computed values for main parameters of the analyzed PV cells and modules.

No.	${\mathit{R}}_{\mathit{s}}\left(\mathsf{\Omega }\right)$	${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathsf{\Omega }\right)$	${\mathit{I}}_0\left(\mathit{A}\right)$	${\mathit{I}}_{\mathit{p}\mathit{v}}\left(\mathit{A}\right)$	$\mathit{a}$	${\mathit{R}}_{\mathit{s}0}\left(\mathsf{\Omega }\right)$	${\mathit{R}}_{\mathit{s}\mathit{h}0}\left(\mathsf{\Omega }\right)$
1	0.506	74.30	6.57 × 10−10	4.732	1.022	0.691	95.27
2	0.109	284.83	2.17 × 10−8	8.703	1.234	0.233	304.09
3	0.378	220.45	2.55 × 10−10	8.735	1.012	0.535	266.54
4	0.142	192.59	5.23 × 10−10	8.586	1.023	0.372	202.92
5	0.308	193.05	2.15 × 10−9	8.223	1.076	0.463	225.66
6	0.277	439.46	1.63 × 10−9	5.343	1.071	0.437	502.34
7	0.19	51.83	1.51 × 10−9	8.401	1.067	0.3161	60.474
8	0.28	140.26	1.56 × 10−9	8.928	1.067	0.438	161.66
9	0.317	108.98	1.41 × 10−9	8.354	1.057	0.501	127.07
10	0.968	1.24E+06	3.41 × 10−10	2.3	1.022	1.332	151053
11	0.316	146.08	1.22 × 10−9	3.808	1.045	0.557	164.26
12	0.009	4.19	1.21 × 10−9	3.809	1.045	0.016	4.788
13	0.264	405.65	5.50 × 10−10	8.686	1.030	0.458	450.79
14	1.555	210.33	3.30 × 10−9	2.7	1.23	2.168	300.48
15	0.437	117.72	1.75 × 10−11	7.397	1.058	0.637	138.19
16	11.57	186.22	1.21 × 10−13	1.179	0.856	13.60	204.51
17	16.639	418.79	8.60 × 10−14	0.769	0.856	19.50	459.43
18	0.079	2.06	1.52 × 10−9	5.296	3.938	0.103	2.13

. For the last three cases, due to different technology, interesting values for the diode ideality factor $a$ occur.

Table 8. Control points coordinates of the analyzed PV cells and modules.

No.	x Coordinates (V)		y Coordinates (A)
	03	13	00	01	02	03	11	12	13	21	22
1	10.628	15.931	4.732	4.684	4.637	4.588	4.538	4.606	4.360	4.212	2.575
2	11.295	16.930	8.703	8.690	8.678	8.663	8.635	8.696	8.508	8.210	8.125
3	18.870	28.285	8.735	8.706	8.678	8.649	8.604	8.697	8.453	8.529	5.997
4	22.224	33.312	8.586	8.547	8.509	8.470	8.443	8.453	8.370	7.684	9.991
5	16.425	24.620	8.223	8.194	8.167	8.137	8.094	8.178	7.930	7.909	5.930
6	10.845	16.256	5.343	5.335	5.327	5.318	5.298	5.350	5.212	5.200	4.168
7	10.975	16.451	8.401	8.3301	8.260	8.189	8.128	8.178	7.946	7.918	6.273
8	18.415	27.603	8.928	8.883	8.841	8.796	8.749	8.812	8.595	8.581	7.029
9	18.228	27.362	8.354	8.298	8.243	8.187	8.133	8.191	7.967	7.954	6.288
10	10.940	16.399	2.300	2.300	2.300	2.299	2.287	2.331	2.226	2.180	1.367
11	10.527	15.779	3.808	3.784	3.760	3.736	3.715	3.733	3.649	3.599	3.241
12	0.293	0.438	3.809	3.786	3.763	3.739	3.718	3.738	3.652	3.608	3.204
13	22.336	33.479	8.686	8.667	8.649	8.630	8.606	8.650	8.525	8.411	8.209
14	11.565	17.335	2.700	2.680	2.666	2.642	2.606	2.669	2.275	1.868	0.969
15	21.720	32.557	7.397	7.336	7.274	7.213	7.168	7.185	7.063	7.200	5.950
16	23.500	35.225	1.110	1.0717	1.0343	0.995	0.971	0.978	0.775	0.578	0.289
17	23.500	35.225	0.740	0.723	0.707	0.689	0.677	0.686	0.542	0.403	0.201
18	1.100	1.649	5.100	4.928	4.760	4.584	4.456	4.506	3.949	3.372	1.786

lists the control points computed for Bézier 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. Computed errors for the analyzed PV cells and modules.

No.	Current ($\mathit{I}$) Error					Max. Power $\left({\mathit{P}}_{\mathit{m}\mathit{p}}\right)$ Error
	Avg.Rel. (%)	Coordinates		Abs. (mA)	Rel. (%)	Abs. (W)	Rel. (%)	Comp. (W)
		$\mathit{V}$ (V)	$\mathit{I}$ (A)
1	−0.11	18.423	3.247	16.63	0.52	−0.363	−0.52	70.32
2	−0.08	20.398	6.325	21.97	0.35	−0.246	−0.16	150.47
3	−0.21	32.952	6.615	59.85	0.90	−1.80	−0.74	246.90
4	0.10	34.638	8.319	92.81	1.12	−1.99	−0.66	302.10
5	−0.19	28.788	6.183	50.70	0.82	−1.223	−0.61	201.37
6	−0.20	19.077	4.105	33.65	0.82	−0.481	−0.55	87.83
7	−0.20	19.306	6.225	52.58	0.84	−0.227	−0.17	135.28
8	−0.20	32.432	6.801	57.57	0.85	−1.242	−0.51	246.50
9	−0.20	32.103	6.248	53.40	0.86	−1.308	−0.58	225.75
10	−0.13	18.548	1.675	9.200	0.55	−0.240	−0.67	36.21
11	−0.17	18.697	2.867	19.97	0.70	−0.215	−0.36	60.07
12	−0.17	0.519	2.872	20.60	0.72	−0.006	−0.38	1.669
13	−0.17	39.909	6.788	47.91	0.71	−0.914	−0.30	301.49
14	0.03	12.255	2.635	3.32	0.13	0.093	0.23	39.91
15	−0.27	38.253	5.730	67.47	1.18	−1.253	−0.52	241.59
16	0.04	30.966	0.925	0.94	0.10	0.015	0.05	28.79
17	0.04	30.816	0.650	0.79	0.12	0.010	0.05	20.15
18	0.014	1.578	4.142	7.09	0.17	0.01	0.16	6.53

Avg. Rel. = the average of the relative error, Abs. = absolute error, Rel. = relative error, Comp. = computed value

, 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%).

An additional plot of the relative error is introduced in Figure 17a,b for two of the analyzed PV devices. The same pattern arises as in Figure 11, with small errors up to $0.94V_{oc}$. The curvature plots from Figure 17c,d were computed using the Herman and Klette method [50], with 20 forward and backward points, using an adapted script from [51].

5. Discussion

In all studied cases, the $x$ coordinates can be evenly spaced. Both for the PV cells and for the PV modules, the first Bézier cubic curve was very close to a straight line and ended at $x_{03}=0.5V_{oc}$, while the middle curve ended at $x_{13}=0.75V_{oc}$. The relative error is less than or equal to 1.18% for all the studied PV devices, while neglecting the values in the $0.94V_{oc}-V_{oc}$ region, where the absolute error is tens of mA and the relative error is misleading, (Figure 11 and Figure 17).

The differences between the results obtained with the proposed method and the least squares method are negligible (less than 1% for all coordinates). $R_s$ and $R_{sh}$ can be easily derived from the manufacturer’s datasheet, using for example the method proposed by Vilallva et al. [5]. In most cases $R_{sh}$ is close to $R_{sho}$ (Table 7). Larger differences occur for the series parasitic resistance: for the same PV cell. For different irradiances and temperatures, Section 4.4 provides all the necessary formulae.

In most cases $V_{mp}>P_{13x}$. The method was also valid for Onyx Ref 10, Onyx Ref 30, and 6.5 Wp L Cell, where the previous relation was not satisfied.

In relation to the Maximum Power Point Tracking (MPPT), it must be emphasized that the Bézier curves are inherently smooth. This reduces the risk for the algorithm of becoming stuck in some false area/minimum of the curve. Furthermore, as the generation of the Bézier curves is so easy, the MPPT simulation can be further simplified.

6. Conclusions

A novel method for modeling a PV cell or a PV module I-V characteristic was introduced. To the best of our knowledge, Bézier curves have not been used to model the I-V characteristic of PV devices before. The method proved high accuracy and was validated both in the case of a single PV cell and in the case of a whole PV module, for different technologies and manufacturers. The method was also used in the case of varying irradiance and temperature. The proposed method can be used for implementing hardware solar array simulators, for teaching or remote study. It is far easier to use the proposed method to find the I-V characteristic of a PV cell or module when compared with solving the exponential equations associated with the single or double diode model largely used today. A common microcontroller can compute the points on the I-V curve with a minimum of resources, inherently increasing the computing speed and the response of the system.

The advantage of our method relies on the ease of I-V characteristic generation: if we exclude $V_{oc}$ and $I_{sc}$, only 10 different values ($P_{23x}=V_{oc}$, $P_{00y}=I_{sc}$, $P_{01y}$, $P_{02y}$, $P_{03y}$, $P_{11y}$ , $P_{12y}$, $P_{13y}$, $P_{21y}$, $P_{22y}$) have to be stored—SAS manufacturers usually use 1024 or more double points to accurately define the I-V characteristic. Alternate use is for any graphical plot of the I-V (and subsequently P-V) curves. Furthermore, little knowledge of the device itself is required, as only common data from the datasheet is needed.

As future work, fitting spiral segments will be investigated and compared with our current method.