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

^{*}

## Abstract

**:**

## 1. Introduction

^{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].

## 2. Definition of the Bézier Curves

## 3. Materials and Methods

^{2}cell [25] and has the main characteristics summarized in Table 1.

## 4. Results

^{2}). 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

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

#### 4.3. Data Fitting Using the Least Squares Method

#### 4.4. Parameter Variation

#### 4.5. Final Validation

## 5. Discussion

## 6. Conclusions

## Supplementary Files

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Main Symbols | |

$a$ | Diode ideality factor |

$G\text{}$ | Actual irradiance |

${G}_{ref}$ | Reference irradiance, 1000 W/m^{2} |

$I$ | Output current |

${I}_{mp}$ | Output current at maximum power point |

${I}_{sc}$ | Short circuit current |

${I}_{sc,ref}$ | Short circuit current 25 °C |

$k$ | Boltzmann constant |

${k}_{I}$ | Current temperature coefficient, A/K |

${k}_{V}$ | Voltage temperature coefficient, V/K |

${k}_{{R}_{sh}}$ | ${R}_{sh}$ temperature exponent |

${n}_{s}$ | Number of series cells |

${P}_{mp}={V}_{mp}{I}_{mp}$ | Maximum output power |

$q$ | Electron charge |

${R}_{s}$ | Series resistance |

${R}_{s,ref}$ | Series resistance at 25 °C |

${R}_{s0}$ | Series resistance based on I-V characteristic slope close to ${V}_{oc}$ |

${R}_{sh}$ | Parallel (shunt) resistance |

${R}_{sh,ref}$ | Parallel (shunt) resistance, at 25 °C |

${R}_{sh0}$ | Parallel (shunt) resistance based on I-V characteristic slope close to ${I}_{sc}$ |

$T$ | Internal temperature, (K) |

${T}_{ref}$ | Reference temperature 298.15 K |

$\Delta T=T-{T}_{ref}$ | Temperature difference |

$V$ | Output voltage |

${V}_{oc}$ | Open circuit voltage |

${V}_{oc,ref}$ | Open circuit reference voltage at 25 °C |

${V}_{oc,cell}$ | Solar cell open circuit voltage |

${V}_{oc,cell,ref}$ | Solar cell open circuit reference voltage at 25 °C |

${V}_{mp}$ | Output voltage at maximum power point |

Abbreviations | |

AM | Air Mass |

MPPT | Maximum Power Point Tracking |

PV | Photovoltaic |

SAS | Solar Array Simulator |

STC | Standard Test Conditions (cell temp. 25 °C; irradiance 1000 W/m^{2}; air mass 1.5) |

Greek Symbols | |

${\alpha}_{Rs}$ | Series resistance temperature coefficient (linear law) |

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**Figure 1.**Typical photovoltaic (PV) module I-V characteristics, with 30 cells connected in series. (

**a**) at different temperatures (0–80 °C); (

**b**) at different irradiances (200–1000 W/m

^{2}).

**Figure 2.**A quadratic Bézier curve representation. ${P}_{0}$ and ${P}_{2}$ are the end points, the control point ${P}_{1}$ is approximated and the curve is tangent to $\overline{{P}_{0}{P}_{1}}$ and $\overline{{P}_{2}{P}_{1}}$ segments at ${P}_{0}$ and ${P}_{2}$ respectively.

**Figure 3.**A cubic Bézier curve representation. ${P}_{0}$ and ${P}_{3}$ are the end points, the control points ${P}_{1}$ and ${P}_{2}$ are approximated, and the curve is tangent to $\overline{{P}_{0}{P}_{1}}$ and $\overline{{P}_{2}{P}_{1}}$ segments at ${P}_{0}$ and ${P}_{3}$ respectively.

**Figure 4.**Projected PV cell I-V characteristic approximation with two straight line segments and one quadratic Bézier curve.

**Figure 5.**PV cell I-V characteristic approximation with two straight lines and one quadratic Bézier curve—results.

**Figure 8.**First Bézier curve with the associated control points. The slope is exaggerated for a better understanding.

**Figure 9.**PV cell I-V characteristic (black line, continuous) and the position of the 12 computed control points (red markers).

**Figure 10.**I-V characteristic of a PV cell modeled with three cubic Bézier curves, defined by 12 control points.

**Figure 11.**The relative error of our model compared with the actual data. Good performance can be observed in the 0–0.5 V interval and near ${V}_{mp}$. Higher errors occur near ${V}_{oc}$ for low output currents.

**Figure 13.**The relative error of the least squares method Bézier based approximation compared with the actual data. The absolute error $\Delta I={I}_{Bezier}-I$ is also indicated.

**Figure 14.**Bézier approximation of the I-V irradiance dependent characteristics for the MSMD290AS-36_EU monocrystalline PV module. The lines represent the computed curves, whereas the markers represent the actual data.

**Figure 15.**Bézier approximation of the I-V temperature dependent characteristics for the MSMD290AS-36_EU Monocrystalline PV module. The lines represent the computed curves, whereas the markers represent the actual data.

**Figure 16.**Bézier approximation of the I-V curves using the proposed method (left) and the least squares method (right). The lines represent the computed curves, whereas the markers represent the actual data. Control points are represented with black dots. (

**a**) Bosch M245 3BB Mono-Crystalline PV module; (

**b**) Kyocera KD135SX_UPU Poly-Crystalline PV module; (

**c**) Shell S36 Poly-Crystalline PV module; (

**d**) Solarex MSX-60 Poly-Crystalline PV Cell; (

**e**) Onyx Ref 10 amorphous Silicon PV Glass module; (

**f**) 6.5Wp L amorphous Silicon PV Cell.

**Figure 17.**Relative error (

**a**,

**b**) and curvature plots (

**c**,

**d**) for Shell ST40 Thin film PV module (a,c) and 6.5 Wp L amorphous Silicon PV Cell (b,d).

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 |

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 |

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 |

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 |

**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 |

No. | PV Type | Tech | ${\mathit{n}}_{\mathit{s}}$ | ${\mathit{V}}_{\mathit{o}\mathit{c}}$ $\left(\mathit{V}\right)$ | ${\mathit{V}}_{\mathit{m}\mathit{p}}$ $\left(\mathit{V}\right)$ | ${\mathit{I}}_{\mathit{m}\mathit{p}}$ $\left(\mathit{A}\right)$ | ${\mathit{I}}_{\mathit{s}\mathit{c}}$ $\left(\mathit{A}\right)$ | ${\mathit{k}}_{\mathit{V}}\text{}$ $\left(\mathit{V}/\mathit{K}\right)$ | ${\mathit{k}}_{\mathit{I}}\text{}$ $\left(\mathit{A}/\mathit{K}\right)$ |
---|---|---|---|---|---|---|---|---|---|

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 |

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 |

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 |

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 |

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## Share and Cite

**MDPI and ACS Style**

Szabo, R.; Gontean, A. Photovoltaic Cell and Module I-V Characteristic Approximation Using Bézier Curves. *Appl. Sci.* **2018**, *8*, 655.
https://doi.org/10.3390/app8050655

**AMA Style**

Szabo R, Gontean A. Photovoltaic Cell and Module I-V Characteristic Approximation Using Bézier Curves. *Applied Sciences*. 2018; 8(5):655.
https://doi.org/10.3390/app8050655

**Chicago/Turabian Style**

Szabo, Roland, and Aurel Gontean. 2018. "Photovoltaic Cell and Module I-V Characteristic Approximation Using Bézier Curves" *Applied Sciences* 8, no. 5: 655.
https://doi.org/10.3390/app8050655