# Estimation of Parameters of Triple Diode Photovoltaic Models Using Hybrid Particle Swarm and Grey Wolf Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}Benban Solar Park in Egypt, which reaches around 1.8 GW [6]. To study these systems thoroughly, an accurate model for the PV cell model is needed under different environmental conditions to obtain data on several aspects, such as Maximum Power Point Tracking (MPPT) and various grid operations [7]. All losses inside the cell and I–V characteristics should be considered in order to accurately model PV cells during the study. The ideal PV cell is represented by a photo-generated current supply (Iph) affected by the solar irradiance (G) landing on the cell [8,9]. Furthermore, different losses such as optical losses in the PN (positive–negative) junction of the cell (represented in the single diode model) [10], recombination losses due to the space charge region (SCR), which is defined in the double-diode (DD) model [11,12], and losses in the defect region and grain boundaries (described in the triple-diode (TD) model) [13,14] must be taken into consideration.

_{ph}), ideality factor (a), the diode cut-off region current (I

_{o}), which represents the diode parameters, along with series resistance (R

_{s}) which indicates the summation of the resistance of the terminals on the external surfaces, and the resistances of the bulk and diffuse layers for the PN junction on the outer sides. Additionally, parallel resistance (R

_{p}) arises from the PV surface and bulk irregularities, along with current losses across the edge of the cell [10]. Another couple of variables representing the second diode are added for the DD model. In comparison, another two variables for the third diode (TD) are added to the model, making it nine parameters in total for this model.

_{sc}, V

_{oc}, I

_{m}, V

_{m}) are used to extract the parameters at standard conditions (G = 1000 W/m

^{2}and T = 25 °C) using analytical, iterative, and meta-heuristic optimization strategies.

## 2. Modeling of PV Module

#### 2.1. PV Models

#### 2.1.1. Single Diode Model

_{s}, while the shunt resistance, R

_{p}, represents the leakage current in the P–N junction. The output current of this model is calculated as follows [39]:

_{pv}represents the cell photocurrent, I

_{o}represents the reverse saturation current, a is the ideality factor and V

_{th}= N

_{s}kT/q represents the thermal voltage. N

_{s}is the number of the series-connected cells in the module, k is the Boltzmann constant, and q is the electron charge. The five needed parameters for this model are I

_{pv}, I

_{o}, R

_{p}, R

_{s}, and a.

#### 2.1.2. Double Diode Model

_{o}

_{1}and I

_{o}

_{2}. I

_{o}

_{2}stands for the extra recombination losses in this model. The ideality factors for the two diodes are a

_{1}and a

_{2}, respectively. Despite its higher accuracy, this model needs the extraction of the SD parameters and two additional parameters, which are I

_{o}

_{2}, and a

_{2}.

#### 2.1.3. Triple Diode Model

_{pv}, I

_{o}

_{1}, I

_{o}

_{2}, I

_{o3}, R

_{p}, R

_{s}, a

_{1}, a

_{2}, a

_{3}.

#### 2.2. Parameters Variation

_{g}= E

_{gn}(1 − 0.0002677 ΔT)

_{pvn}, E

_{gn}, G

_{n}, R

_{shn}, and T

_{n}stand for the photo-generated current, energy gap, irradiance, parallel resistance, and cell temperature at standard conditions. K

_{i}is the short-circuit current coefficient, E

_{gn}equals 1.21 eV [44] for silicon, and ΔT denotes the deviation between T and T

_{n}. These equations are implemented to interpret the I–V characteristics of the SD, DD, and TD PV module models at various temperatures and solar irradiation levels. Calculating the parameters of the models above from the I–V curves, an interpretation for the fitness function is necessary to utilize the optimization techniques.

## 3. Hybrid Particle Swarm–Grey Wolf Optimization

#### 3.1. Grey Wolf Optimization Algorithm

_{1}, r

_{2}are generated at random in the range [0, 1]. Over a number of iterations, the value of $\overrightarrow{u}$ declines uniformly between 2 and 0. The best candidates for the solution in the grey wolf hunting process are alpha, beta, and delta, who are depicted to be aware of the prey’s likely location. As a result, the three best solutions found for a given iteration are preserved, forcing other wolves to adjust their locations in the hunting space to match the optimal spot. The following is the approach for updating positions [46]:

#### 3.2. Particle Swarm Optimization

#### 3.3. The Applied Hybrid PSOGWO Algorithm

## 4. Results

^{2}and T = 25 °C) are presented in Table 1.

_{sc}, V

_{oc}, I

_{m}, V

_{m}, P

_{m}) are acquired at standard conditions, as clarified in Table 1.

## 5. Discussion

#### 5.1. Kyocera KC200GT

^{−10}. The absolute error for the current is compared to the WOA [18] and the SFO [49] for the KC200GT module. Furthermore, the suggested approach’s I–V and P–V curves are compared with experimental curves under varied T and G to prove its effectiveness [50,51]. The I–V and P–V curves are produced and verified using actual data at a fixed temperature of 25 °C and variable irradiances ranging from 200 to 1000 W/m

^{2}in Figure 3. The given approach is evaluated in the KC200GT module at fixed irradiance of 1000 W/m

^{2}and various temperatures of (25, 50, and 75 degrees Celsius), after which the illustrated I–V and P–V curves are verified with observed outcomes, as depicted in Figure 4. We may conclude that the results for the proposed TD model’s optimized parameters showed better outcomes, more realistic curves regarding the experimental readings, and good accuracy when compared to those obtained using existing meta-heuristic optimization procedures.

#### 5.2. Canadian Solar Cell CS6K-280M

^{2}) are illustrated in Figure 5 and then justified using observed data. Also presented are the I–V and P–V curves with a fixed irradiance of 1000 W/m

^{2}and varied temperatures (25, 45, and 65 degrees Celsius), which are then validated with observed data in Figure 6. The offered method is verified by comparing its absolute current error to the WOA and SFO [49] methods in Figure 7. This comparison confirms the effectiveness of the results for the proposed approach, with the error within the results being less than 0.4%, reaching 0.07% with a fitness value of 1.59 × 10

^{−10}. Results were of excellent accuracy compared to the other mentioned techniques and the experimental results.

## 6. Conclusions

_{sc}, V

_{OC}, etc.). The objective of using the optimization technique is to minimize the current deviation. Aiming for a realistic study, the proposed technique is applied to two well-known PV modules, Kyocera and Canadian. The obtained parameters are validated and compared with those obtained from other recent algorithms such as SFO, GA, and WOA. The simulation outcomes ensure the competitiveness and robustness of the proposed model over the other techniques under comparison.

^{−10}in the case of Kyocera and 1.59 × 10

^{−10}in case of the Canadian solar cell. Accordingly, with the help of the hybrid PSOGWO algorithm, an accurate PV model was obtained. This model may also be useful for power electronics studies that need an accurate, efficient, and dependable PV model, and in studies for grid-connected PV systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Ideality factor of diode |

E_{g} | Band gap energy (eV) |

G | Solar Irradiance(W/m2) |

I_{d} | Diode Current (A) |

I_{m} | Maximum output current of PV Array (A) |

I_{o} | Reverse saturation current of diode (A) |

I_{ph} | Photo-generated current (A) |

I_{sc} | Short circuit current of PV module (A) |

k | Boltzmann constant (1.38065 × 10^{−23} J/K) |

K_{i} | Short-circuit current coefficient |

N_{s} | Number of the series-connected cells in the module |

P | Output power of PV module (W) |

P_{m} | Maximum Output power of PV Module (W) |

q | Electron charge (1.6022 × 10^{−19} C) |

R_{s} | Series resistance (Ω) |

R_{p} | Shunt resistance (Ω) |

T | Cell Temperature (K) |

V_{m} | Maximum output voltage of PV Array (V) |

V_{oc} | Open circuit voltage of PV module (V) |

V_{th} | Thermal voltage (V) |

Abbreviations | |

PV | Photovoltaic |

CSP | Concentrated Solar Power |

SPS | Solar Power Satellite |

NREA | New and Renewable Energy Agency |

MPPT | Maximum Power Point Tracking |

SCR | Space charge region |

WOA | Whale optimization algorithm |

SFO | Sunflower optimization algorithm |

SD | Single Diode |

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**Figure 3.**Simulation and practical outcomes of the Kyocera KCG200T module under various irradiances at 25 °C. (

**a**) I–V Curves. (

**b**) P–V Curves.

**Figure 4.**Simulation and practical outcomes of the Kyocera KCG200T under various temperatures, at 1000 W/m

^{2}. (

**a**) I–V Curves. (

**b**) P–V Curves.

**Figure 5.**Simulation and practical outcomes of the Canadian CS6K-280M cell under various irradiances at 25 °C. (

**a**) I–V Curves. (

**b**) P–V Curves.

**Figure 6.**Simulation and practical outcomes of the Canadian CS6K-280M cell under various temperatures, at 1000 W/m

^{2}. (

**a**) I–V Curves. (

**b**) P–V Curves.

Manufacturer | Kyocera | Canadian Solar |
---|---|---|

Model | KC200GT | CS6K-280M |

Cell Type | Multicrystal | Monocrystal |

P_{m} (W) | 200 | 280 |

V_{m} (V) | 26.3 | 31.5 |

I_{m} (A) | 7.61 | 8.89 |

V_{oc} (V) | 32.9 | 38.5 |

I_{sc} (A) | 8.21 | 9.43 |

Number of series cells | 54 | 60 |

K_{i} | 0.00318A/°C | 0.053%/°C |

K_{v} | −0.123 V/°C | −0.31%/°C |

KC200GT | CS6K-280M | |||
---|---|---|---|---|

Minimum | Maximum | Minimum | Maximum | |

I_{Ph} (A) | 8.1 | 8.3 | 9.4 | 9.7 |

R_{p} (Ω) | 200 | 500 | 1000 | 50,000 |

R_{s} (Ω) | 0.2 | 0.4 | 0 | 0.25 |

n_{1} | 1.1 | 1.5 | 1 | 1.9 |

n_{2} | 1.2 | 1.5 | 1.5 | 1.9 |

n_{3} | 1 | 1.5 | 1.5 | 1.9 |

I_{o}_{1} (A) | 1 × 10^{−12} | 1 × 10^{−6} | 1 × 10^{−12} | 1 × 10^{−5} |

I_{o}_{2} (A) | 1 × 10^{−12} | 1 × 10^{−6} | 1 × 10^{−12} | 1 × 10^{−5} |

I_{o}_{3} (A) | 1 × 10^{−12} | 1 × 10^{−6} | 1 × 10^{−12} | 1 × 10^{−5} |

KC200GT | CS6K-280M | |
---|---|---|

I_{Ph} (A) | 8.1705 | 9.628 |

R_{p} (Ω) | 495.55 | 30.202 × 10^{3} |

R_{s} (Ω) | 0.2375 | 0.071834 |

n_{1} | 1.2764 | 1.83024 |

n_{2} | 1.496 | 1.8212 |

n_{3} | 1.2159 | 1.16302 |

I_{o}_{1} (A) | 1.54 × 10^{−8} | 6.00 × 10^{−6} |

I_{o}_{2} (A) | 3.85 × 10^{−10} | 6.00 × 10^{−6} |

I_{o}_{3} (A) | 3.67 × 10^{−10} | 3.03 × 10^{−10} |

GA | SA | WOA | SFO | Hybrid PSOGWO | |
---|---|---|---|---|---|

I_{Ph} (A) | 8.143 | 8.25 | 8.231 | 8.212 | 8.21 |

R_{p} (Ω) | 311.8 | 327,597 | 341.387 | 606.12 | 495.55 |

R_{s} (Ω) | 0.3614 | 0.378 | 0.3421 | 0.23796 | 0.2375 |

n_{1} | 1.189 | 1.199 | 1.32 | 1.2481 | 1.2764 |

n_{2} | 1.495 | 1.2 | 1.236 | 1.991 | 1.4957 |

n_{3} | 1.38 | 1.48 | 1.0216 | 1.8421 | 1.2159 |

I_{o}_{1} (A) | 1.52 × 10^{−8} | 1.78 × 10^{−8} | 2.692 × 10^{−}^{8} | 4.3 × 10^{−8} | 1.54 × 10^{−8} |

I_{o}_{2} (A) | 4.58 × 10^{−10} | 3.76 × 10^{−10} | 4.678 × 10^{−10} | 2.22 × 10^{−10} | 3.85 × 10^{−10} |

I_{o}_{3} (A) | 1.019 × 10^{−10} | 4.62 × 10^{−10} | 4.927 × 10^{−10} | 1.35 × 10^{−6} | 3.67 × 10^{−10} |

MLE | WOA | SFO | Hybrid PSOGWO | |
---|---|---|---|---|

I_{Ph} (A) | 9.46 | 9.516574724 | 9.440369 | 9.628 |

R_{p} (Ω) | 599.99 | 1.50 × 10^{3} | 2.16× 10^{4} | 30.202 × 10^{3} |

R_{s} (Ω) | 0.168 | 0.01351602 | 0.2 | 0.0718 |

n1 | 1.108 | 1.841521006 | 2 | 1.830 |

n2 | - | 1.750541567 | 2 | 1.821 |

n3 | - | 1.623788571 | 1.1913 | 1.163 |

I_{o}_{1} (A) | 10 × 10^{−10} | 6.03528 × 10^{−6} | 1.00 × 10^{−12} | 6.00 × 10^{−6} |

I_{o}_{2} (A) | - | 3.21299 × 10^{−6} | 1.00 × 10^{−12} | 6.00 × 10^{−6} |

I_{o}_{3} (A) | - | 1 × 10^{−12} | 7.46 × 10^{−9} | 3.03 × 10^{−10} |

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**MDPI and ACS Style**

Ellithy, H.H.; Taha, A.M.; Hasanien, H.M.; Attia, M.A.; El-Shahat, A.; Aleem, S.H.E.A.
Estimation of Parameters of Triple Diode Photovoltaic Models Using Hybrid Particle Swarm and Grey Wolf Optimization. *Sustainability* **2022**, *14*, 9046.
https://doi.org/10.3390/su14159046

**AMA Style**

Ellithy HH, Taha AM, Hasanien HM, Attia MA, El-Shahat A, Aleem SHEA.
Estimation of Parameters of Triple Diode Photovoltaic Models Using Hybrid Particle Swarm and Grey Wolf Optimization. *Sustainability*. 2022; 14(15):9046.
https://doi.org/10.3390/su14159046

**Chicago/Turabian Style**

Ellithy, Hazem Hassan, Adel M. Taha, Hany M. Hasanien, Mahmoud A. Attia, Adel El-Shahat, and Shady H. E. Abdel Aleem.
2022. "Estimation of Parameters of Triple Diode Photovoltaic Models Using Hybrid Particle Swarm and Grey Wolf Optimization" *Sustainability* 14, no. 15: 9046.
https://doi.org/10.3390/su14159046