# Accelerating Convergence for the Parameters of PV Cell Models

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- $c\leftarrow $ optimum values from minimizing the sums of the residuals with classical vertical offsets (${\sum}_{i=1}^{m}{({y}_{i}-f({x}_{i};c))}^{2}$);
- Minimize($g,c$).

## 3. Case Studies

- Pink (n = 17): (1132, 1.147), (1110, 1.187), (1080, 1.257), (1038, 1.312), (1010, 1.362), (973, 1.406), (930, 1.480), (900, 1.493), (845, 1.556), (772, 1.609), (703, 1.672), (593, 1.742), (493, 1.776), (405, 1.785), (332, 1.812), (254, 1.821), and (163, 1.834).

- Blue (n = 8): (0.1019, 0.1011), (0.1997, 0.1015), (0.3006, 0.1007), (0.3994, 0.0974), (0.4680, 0.0830), (0.4992, 0.0597), (0.5169, 0.0395), and (0.5294, 0.0189);
- Gray (n = 13): (0.1007, 0.5565), (0.1503, 0.5546), (0.2008, 0.5498), (0.2503, 0.5413), (0.3009, 0.5318), (0.3266, 0.5242), (0.3504, 0.5137), (0.3742, 0.4985), (0.3990, 0.4671), (0.4219, 0.4253), (0.4505, 0.3482), (0.4743, 0.2579), (0.4981, and 0.1504).

## 4. Results and Discussion

#### 4.1. Pink Dataset

#### 4.2. Blue and Gray Datasets

## 5. Conclusions and Forthcoming Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PV | Photovoltaic |

RSS | Residual sum of squares |

F | Fisher’s F value |

r${}_{\mathrm{adj}}^{2}$ | Ajusted determination coefficient |

MSE | Mean square error |

## References

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**Figure 4.**The evolution of coefficient ${c}_{1}$ of Equation (8) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 5.**The evolution of coefficient ${c}_{2}$ of Equation (8) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 6.**The evolution of coefficient ${c}_{3}$ of Equation (8) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 7.**The evolution of coefficient ${c}_{4}$ of Equation (8) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 8.**The evolution of coefficient ${c}_{1}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 9.**The evolution of coefficient ${c}_{2}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 10.**The evolution of coefficient ${c}_{3}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{17}{h}_{i}^{2}$ to minimum) for the Pink dataset.

**Figure 11.**The evolution of coefficient ${c}_{1}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{8}{h}_{i}^{2}$ to minimum) for the Blue dataset.

**Figure 12.**The evolution of coefficient ${c}_{2}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{8}{h}_{i}^{2}$ to minimum) for the Blue dataset.

**Figure 13.**The evolution of coefficient ${c}_{3}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{8}{h}_{i}^{2}$ to minimum) for the Blue dataset.

**Figure 14.**The evolution of coefficient ${c}_{1}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{13}{h}_{i}^{2}$ to minimum) for the Gray dataset.

**Figure 15.**The evolution of coefficient ${c}_{2}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{13}{h}_{i}^{2}$ to minimum) for the Gray dataset.

**Figure 16.**The evolution of coefficient ${c}_{3}$ of Equation (9) from vertical to perpendicular offset optimum points (${\Sigma}_{i=1}^{13}{h}_{i}^{2}$ to minimum) for the Gray dataset.

Function | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ |
---|---|---|

Parameters | ${c}_{1}=3346.61$, | ${c}_{1}=1.82577$ |

${c}_{2}=2688.48$, | ${c}_{2}=22.3764$, | |

${c}_{3}=475.487$, | ${c}_{3}=3.12554$, | |

${c}_{4}=1.92715$ | ||

Statistics | $m=4$ | $m=3$ |

${r}_{\mathrm{adj}}^{2}=0.9977$, | ${r}_{\mathrm{adj}}^{2}=0.9987$, | |

$F=2530$, | $F=9543$, | |

$RS{S}_{\perp}=0.0013137$ | $RS{S}_{\perp}=0.0011110$ |

Function | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ |
---|---|---|

Parameters | ${c}_{1}=3346.60$, | ${c}_{1}=1.82568$ |

${c}_{2}=2688.50$, | ${c}_{2}=22.3764$, | |

${c}_{3}=475.444$, | ${c}_{3}=3.12553$, | |

${c}_{4}=1.92679$ | ||

Statistics | $m=4$ | $m=3$ |

${r}_{\mathrm{adj}}^{2}=0.9977$, | ${r}_{\mathrm{adj}}^{2}=0.9987$, | |

$F=2529$, | $F=9543$, | |

$RS{S}_{\perp}=0.0013087$ | $RS{S}_{\perp}=0.0011110$ |

n | y | ${\mathit{f}}_{1}\left(\mathit{x}\right)$ | x | ${\mathit{f}}_{1}^{-1}\left(\mathit{y}\right)$ |
---|---|---|---|---|

1 | 1.147 | 1.159 | 1132 | 1139 |

2 | 1.187 | 1.196 | 1110 | 1115 |

3 | 1.257 | 1.245 | 1080 | 1073 |

4 | 1.312 | 1.311 | 1038 | 1037 |

5 | 1.362 | 1.353 | 1010 | 1004 |

6 | 1.406 | 1.405 | 973 | 972 |

7 | 1.48 | 1.462 | 930 | 915 |

8 | 1.493 | 1.499 | 900 | 905 |

9 | 1.556 | 1.559 | 845 | 848 |

10 | 1.609 | 1.626 | 772 | 792 |

11 | 1.672 | 1.676 | 703 | 709 |

12 | 1.742 | 1.734 | 593 | 574 |

13 | 1.776 | 1.77 | 493 | 474 |

14 | 1.785 | 1.794 | 405 | 441 |

15 | 1.812 | 1.809 | 332 | 314 |

16 | 1.821 | 1.822 | 254 | 259 |

17 | 1.834 | 1.834 | 163 | 162 |

n | x | ${\mathit{f}}_{2}\left(\mathit{y}\right)$ | y | ${\mathit{f}}_{2}^{-1}\left(\mathit{x}\right)$ |
---|---|---|---|---|

1 | 1132 | 1136 | 1.147 | 1.154 |

2 | 1110 | 1114 | 1.187 | 1.194 |

3 | 1080 | 1073 | 1.257 | 1.246 |

4 | 1038 | 1039 | 1.312 | 1.314 |

5 | 1010 | 1006 | 1.362 | 1.356 |

6 | 973 | 974 | 1.406 | 1.407 |

7 | 930 | 915 | 1.48 | 1.462 |

8 | 900 | 904 | 1.493 | 1.498 |

9 | 845 | 846 | 1.556 | 1.557 |

10 | 772 | 788 | 1.609 | 1.623 |

11 | 703 | 706 | 1.672 | 1.674 |

12 | 593 | 581 | 1.742 | 1.737 |

13 | 493 | 492 | 1.776 | 1.776 |

14 | 405 | 462 | 1.785 | 1.799 |

15 | 332 | 326 | 1.812 | 1.811 |

16 | 254 | 231 | 1.821 | 1.819 |

17 | 163 | 278 | 1.834 | 1.824 |

Function | ${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ |
---|---|---|

Initial parameters | ${I}_{\mathrm{sc}}=1.85030$ mA, | ${I}_{\mathrm{sc}}=1.82568$ mA |

${V}_{\mathrm{oc}}=1736.56$ mV, | ${V}_{\mathrm{oc}}=1559.00$ mV, | |

${I}_{\mathrm{mpp}}=1.38706$ mA, | ${I}_{\mathrm{mpp}}=1.38315$ mA, | |

${V}_{\mathrm{mpp}}=985.638$ mV, | ${V}_{\mathrm{mpp}}=990.666$ mV, | |

${P}_{\mathrm{mpp}}=1.36714$ mW | ${P}_{\mathrm{mpp}}=1.37024$ mW | |

Final parameters | ${I}_{\mathrm{sc}}=1.85009$ mA, | ${I}_{\mathrm{sc}}=1.82568$ mA |

${V}_{\mathrm{oc}}=1736.88$ mV, | ${V}_{\mathrm{oc}}=1559.04$ mV, | |

${I}_{\mathrm{mpp}}=1.38726$ mA, | ${I}_{\mathrm{mpp}}=1.38315$ mA, | |

${V}_{\mathrm{mpp}}=985.930$ mV, | ${V}_{\mathrm{mpp}}=990.688$ mV, | |

${P}_{\mathrm{mpp}}=1.36774$ mW | ${P}_{\mathrm{mpp}}=1.37027$ mW |

Parameter | | Offsets (Initial) | ⊥ Offsets (Final) | ||
---|---|---|---|---|

Blue PV | Gray PV | Blue PV | Gray PV | |

${c}_{1}$ | 0.10100 | 0.55048 | 0.10101 | 0.55076 |

${c}_{2}$ | −5.11120 | −3.89420 | −5.11113 | −3.79040 |

${c}_{3}$ | 11.9658 | 6.89320 | 11.9659 | 6.76415 |

${I}_{\mathrm{sc}}$ [A] | 0.10100 | 0.55048 | 0.10101 | 0.55076 |

${V}_{\mathrm{oc}}$ [V] | 0.53862 | 0.52124 | 0.53863 | 0.52281 |

${I}_{\mathrm{xp}}$ [A] | 0.09321 | 0.48074 | 0.09322 | 0.47982 |

${V}_{\mathrm{xp}}$ [V] | 0.43479 | 0.38626 | 0.43480 | 0.38614 |

${P}_{\mathrm{xp}}$ [W] | 0.04053 | 0.18569 | 0.04053 | 0.18528 |

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**MDPI and ACS Style**

Jäntschi, L.; Louzazni, M.
Accelerating Convergence for the Parameters of PV Cell Models. *Math. Comput. Appl.* **2024**, *29*, 4.
https://doi.org/10.3390/mca29010004

**AMA Style**

Jäntschi L, Louzazni M.
Accelerating Convergence for the Parameters of PV Cell Models. *Mathematical and Computational Applications*. 2024; 29(1):4.
https://doi.org/10.3390/mca29010004

**Chicago/Turabian Style**

Jäntschi, Lorentz, and Mohamed Louzazni.
2024. "Accelerating Convergence for the Parameters of PV Cell Models" *Mathematical and Computational Applications* 29, no. 1: 4.
https://doi.org/10.3390/mca29010004