# A Novel High Accuracy PV Cell Model Including Self Heating and Parameter Variation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Classical PV Cell Model

## 3. Materials, Methods and Equipment

- The claimed maximum power (5.21 W) differs from ${V}_{mp}{I}_{mp}=5.01\text{}\mathrm{W}$. This latter value will be considered subsequently.
- The claimed fill factor (81.90%) differs from the standard definition $FF=\frac{{V}_{mp}{I}_{mp}}{{V}_{oc,ref}{I}_{oc,ref}}=77.82\%$.

^{2}irradiance. The accuracy of the temperature sensor for the ambient temperature and PV cell backside coating is 0.6 °C in the measurement range of 0–60 °C. The FLIR E8 infrared camera has an accuracy of 2 °C with thermal sensitivity below 0.06 °C. The accuracy of the FLIR infrared camera is lower than the measurements taken with the temperature sensors but offers the advantage of contactless measurement which is important for PV cell front side measurements. However, the existence of hot spots can be easily observed with this method. The electrical noise was minimized using the following digital busses for communication: Modbus for irradiance and PV cell back side temperature measurement, USB for ambient and PV cell front side temperature measurements.

## 4. Classical Model Solving

#### 4.1. Solving the Equations for the Classical PV Model

- Compute ${a}_{1}$
- For validation purposes determine the limits ${R}_{s,max}$ and ${R}_{sh,min}$
- For all values between $\left[0,{R}_{s,max}\right]$ with ${R}_{s,inc}$ as increment, numerically solve (1) for the MPP.
- When the maximum power error is below the imposed threshold error, ${R}_{s}$ is established and ${R}_{sh}$ can be computed.

#### 4.2. Parameters Variation for Different Conditions

#### 4.2.1. Diode Saturation Current—${I}_{o1}$

#### 4.2.2. Band Gap Energy and Bandgap Voltage—${E}_{g},{V}_{g}$

#### 4.2.3. Series Resistance—${R}_{s}$

- An 8-parameter model, where Equation (1) describes the output current
- A 5-parameter model that neglects ${D}_{2}$ in Figure 1 and the value of the shunt resistor is infinite.

#### 4.2.4. Parallel Resistance—${R}_{sh}$

#### 4.2.5. Ideality Diode Factor—${a}_{1}$

#### 4.2.6. Self Heating Phenomenon

#### 4.2.7. Open Circuit Voltage—${V}_{oc}$

#### 4.3. The New Proposed PV Cell Model

## 5. Experimental Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Main Symbols | |

${a}_{1}$ | Diode ideality factor |

${a}_{1,ref}$ | Diode ideality factor at 25 °C |

${R}_{th}$ | Thermal capacitance of the cell, a lumped parameter |

${E}_{g}$ | Bandgap energy |

$FF$ | Fill factor |

$G$ | Actual irradiance on cell surface |

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

$I$ | Solar cell current |

${I}_{o1}$ | Saturation current of the modeled diode, due to diffusion |

${I}_{o1,ref}$ | Saturation current of the modeled diode, due to diffusion, at 25 °C |

${I}_{mp}$ | Current at maximum power point |

${I}_{ph}$ | Photo generated current |

${I}_{ph,ref}$ | Photo generated reference current at 25 °C |

${I}_{sc}$ | Short circuit current of the solar cell |

${I}_{sc,ref}$ | Short circuit current of the solar cell at 25 °C |

$k$ | Boltzmann constant |

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

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

${k}_{P}$ | Power temperature coefficient, W/K |

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

${k}_{Rs}^{\prime}$, ${k}_{Rsh}^{\prime}$ | ${R}_{s},{R}_{sh}$ temperature exponent in Matlab |

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

${n}_{p}$ | Number of parallel cells |

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

$q$ | Electron charge |

${R}_{s}$ | Cell series resistance |

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

${R}_{so}$ | Cell series resistance based on slope close to ${V}_{oc}$ |

${R}_{sh}$ | Cell parallel (shunt) resistance |

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

${R}_{sho}$ | Cell parallel (shunt) resistance based on slope close to ${I}_{sc}$ |

${R}_{th}$ | Thermal resistance of the cell, a lumped parameter |

$T$ | Solar cell temperature, K |

${T}_{ref}={T}_{25}$ | Reference temperature 298 K |

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

${T}_{amb}$ | Ambient/air temperature, °C |

${T}_{cell}$ | Internal PV cell temperature, °C |

$V$ | Solar cell voltage |

${V}_{oc}$ | Solar array open circuit voltage |

${V}_{oc,ref}$ | Solar array 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 |

${V}_{mp}$ | Voltage at maximum power point |

${V}_{g}$ | Bandgap voltage |

${V}_{T}=kT/q$ | Diode thermal voltage |

Abbreviations | |

AM | Air Mass |

KCL | Kirchhoff’s current law |

MPP | Maximum power point |

MPPT | Maximum Power Point Tracking |

NOCT | Normal Operating Cell Temperature |

PV | Photovoltaic |

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 7.**${V}_{oc,cell}$ vs. Irradiance for different temperatures. Solid and dashed lines are given by (33), while symbols correspond to experimental data.

**Figure 11.**I-V and P-V curves—simulation and experiments. (

**a**) I-V curves at 25 °C (upper lines) and with all parameters variation included (lower lines)—internal temperature is 54 °C. (

**b**) P-V curves at 25 °C (upper lines) and with all parameters variation included (lower lines)—internal temperature is 54 °C.

**Figure 12.**${R}_{s}$ and ${R}_{sh}$ influence and the performance. (

**a**) ${R}_{s}$ increase with temperature (25 °C to 54 °C) determines an increase in the output current and power (

**b**) ${R}_{sh}$ has no significant influence on the performance.

**Figure 13.**Final model validation by comparison for the MSMD290AS-36.EU monocrystalline PV cells. (

**a**) I-V curves at 25 °C; (

**b**) P-V curves at 25 °C (

**c**) I-V curves at 54 °C (

**d**) P-V curves at 54 °C.

Symbol | Description | Value |
---|---|---|

${V}_{oc,cell,ref}$ | Cell open circuit voltage | 0.699 V |

${I}_{sc,ref}$ | Short circuit current | 9.206 A |

${V}_{mp}$ | Maximum power voltage | 0.572 V |

${I}_{mp}$ | Maximum power current | 8.756 A |

$\left({P}_{mp}\right)$ | Maximum power ${P}_{mp}={V}_{mp}{I}_{mp}$ | (5.21 W) |

$FF$ | Fill factor | (81.90%) |

${k}_{I}$ | Short circuit temperature coefficient | 0.035 %/K |

${k}_{V}$ | Open circuit voltage temperature coefficient | −0.25 %/K |

${k}_{P}$ | Maximum power temperature coefficient | −0.41 %/K |

Symbol | Description | Results |
---|---|---|

${I}_{ph,ref}$ | Photo generated current | 9.207 A |

${I}_{o,ref}$ | Reverse diode current | 1.39427 nA |

${R}_{s,max}$ | Maximum ${R}_{s}$ value (initial guess) | 14 mΩ |

${R}_{s,ref}$ | Series resistance | 3.8 mΩ |

${R}_{sh,min}$ | Minimum ${R}_{sh}$ value (initial guess) | 1.25 Ω |

${R}_{sh,ref}$ | Parallel resistance | 73.19 Ω |

${a}_{1,ref}$ | Ideality factor | 1.2034 |

${E}_{g,ref}$ | Bandgap energy | 1.795 × 10^{−9} V |

${V}_{g,ref}$ | Bandgap voltage | 1.121 V |

${\mathit{a}}_{\mathbf{1}}$ Accepted Range | Phang, Equation (25) | De Blas, Equation (26) | Saloux, Equation (27) | Villalva, Equation (28) |
---|---|---|---|---|

1–1.5 | 1.1952 | 1.2016 | 1.6377 | 1.2034 |

Symbol | Description | Datasheet Value | Proposed Model | Model Error vs. Datasheet (%) | Experimental Values | |
---|---|---|---|---|---|---|

Results | Error vs. Datasheet (%) | |||||

${V}_{oc,cell,ref}$ | Cell open circuit voltage | 0.699 V | 0.6985 V | −0.07% | 0.693 V | −0.86 |

${I}_{sc,ref}$ | Short circuit current | 9.206 A | 9.206 A | 0% | 9.221 A | 0.16 |

${V}_{mp}$ | Maximum power voltage | 0.572 V | 0.575 V | 0.52% | 0.569 V | −0.52 |

${I}_{mp}$ | Maximum power current | 8.756 A | 8.705 A | −0.58% | 8.731 A | −0.29 |

$\left({P}_{mp}\right)$ | Maximum power ${P}_{mp}={V}_{mp}{I}_{mp}$ | 5.01 W | 5.005 W | −0.06% | 4.968 W | −0.81 |

$FF$ | Fill factor | 77.83% | 77.84% | 0.01% | 77.52% | −0.40 |

PV Type | ${\mathit{n}}_{\mathit{s}}$ | ${\mathit{V}}_{\mathit{o}\mathit{c}}$ (V) | ${\mathit{V}}_{\mathit{m}\mathit{p}}$ (V) | ${\mathit{I}}_{\mathit{m}\mathit{p}}$ (A) | ${\mathit{I}}_{\mathit{s}\mathit{c}}$ (A) | ${\mathit{k}}_{\mathit{V}}$ (mV/K) | ${\mathit{k}}_{\mathit{I}}$ (mA/K) |
---|---|---|---|---|---|---|---|

Shell SP-70 | 36 | 21.4 | 16.5 | 4.24 | 4.7 | −76 | 2 |

MSMD290AS-36.EU | 72 | 44.68 | 37.66 | 7.7 | 8.24 | −138.508 | 3.296 |

MSP290AS-36.EU | 72 | 44.32 | 37.08 | 7.82 | 8.37 | −146.256 | 3.348 |

KG200GT | 54 | 32.9 | 26.3 | 7.61 | 8.21 | −123 | 3.18 |

Sharp ND-224uC1 | 60 | 36.6 | 29.3 | 7.66 | 8.33 | −131.76 | 4.4149 |

PV Type | Solution | ${\mathit{a}}_{\mathbf{1}}$ | ${\mathit{R}}_{\mathit{s}}$ (mΩ) | ${\mathit{R}}_{\mathit{s}\mathit{h}}$ (Ω) | ${\mathit{I}}_{\mathit{p}\mathit{v}}$ (A) | ${\mathit{I}}_{\mathit{o}}$ (nA) |
---|---|---|---|---|---|---|

Shell SP-70 | Ishaque * [35] | 1 & 2.2 | 510 | 91 | 4.7 | 0.421; 0.421 |

Proposed | 1.022 | 505 | 73.85 | 4.732 | 0.657 | |

MSMD290AS-36.EU | Cubas [31] | 1.1 | 130 | 316 | 8.24 | 2.36 |

Proposed | 1.0 | 159 | 194 | 8.247 | 0.243 | |

MSP290AS-36.EU | Cubas [31] | 1.1 | 162 | 331 | 8.37 | 2.86 |

Proposed | 1.02 | 191 | 230 | 8.377 | 0.513 | |

KG200GT | Ishaque * [35] | 1 & 2.2 | 320 | 160.5 | 8.21 | 0.422; 0.422 |

Sumathi et al. [5] | 1.3 | 221 | 415.4 | 8.214 | 98.25 | |

Proposed | 1.08 | 305 | 186 | 8.223 | 2.15 | |

Sharp ND-224uC1 | Proposed | 1.06 | 316 | 108 | 8.354 | 1.41 |

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**MDPI and ACS Style**

Gontean, A.; Lica, S.; Bularka, S.; Szabo, R.; Lascu, D.
A Novel High Accuracy PV Cell Model Including Self Heating and Parameter Variation. *Energies* **2018**, *11*, 36.
https://doi.org/10.3390/en11010036

**AMA Style**

Gontean A, Lica S, Bularka S, Szabo R, Lascu D.
A Novel High Accuracy PV Cell Model Including Self Heating and Parameter Variation. *Energies*. 2018; 11(1):36.
https://doi.org/10.3390/en11010036

**Chicago/Turabian Style**

Gontean, Aurel, Septimiu Lica, Szilard Bularka, Roland Szabo, and Dan Lascu.
2018. "A Novel High Accuracy PV Cell Model Including Self Heating and Parameter Variation" *Energies* 11, no. 1: 36.
https://doi.org/10.3390/en11010036