# Estimation of Single-Diode and Two-Diode Solar Cell Parameters by Using a Chaotic Optimization Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (a)
- (b)
- estimation based on experimental data [8].

- (a)
- (b)
- numerical extraction [13] and
- (c)

## 2. Mathematical Modeling of Single and Double Solar Cells

_{pv}is the photo-generated current, R

_{s}the series parasitic resistance, R

_{p}the parallel parasitic resistance, I

_{0}the saturation current, n is ideality factor and V

_{th}= k

_{B}T/q is the thermal voltage (k

_{B}is Boltzmann constant equal to 1.38 × 10

^{−23}J/K, T the temperature and q the electron charge equal to 1.602 × 10

^{−19}C).

_{o1}and I

_{o2}are the diffusion and saturation currents, whereas n

_{1}and n

_{2}are the diffusion and recombination diode ideality factors [48]. The ideality factor is discussed in [69,70], whereas [71] presents a method for ideality factor calculation.

## 3. COA and Objective Function

_{s}, R

_{p}, I

_{pv}, I

_{o}, n], and for DDM, X = [R

_{s}, R

_{p}, I

_{pv}, I

_{o1}, I

_{o2}, n

_{1}, n

_{2}]. Therefore, in general, vector X = [x

_{1}, x

_{2}, ... x

_{n}] contains variables limited to the lower (LV) and upper (UV) permitted value, i.e, ${x}_{i}\in \left[{L}_{i},{U}_{i}\right]$. On the other side, the OF for SDM is

_{t}and I

_{t}represent the voltage and current value of pair t.

_{est,k}and I

_{meas,k}represent the estimated and the measured values of solar output current in point k, respectively.

## 4. Simulation Results

_{of}p)N

_{i}), where O is the big O notation, n is the dimension of the parameter space, p is the population size, N

_{i}is the number of iterations and C

_{of}is the complexity of the OF. The complexity of COA is O(QC

_{of}), where Q is the number of points in the parameter space in which the OF is calculated. Therefore, the proposed COA-based estimation has significantly lower computational complexity than evolutionary algorithms.

^{−4}). All computer simulations were carried out on a PC with Intel(R) Core (TM) i3-7020U CPU @ 2.30 GHz and 4 GB RAM. The obtained results, i.e., the mean, maximal and minimal required time per one iteration, obtained over 20 runs, are presented in Table 3. Clearly, the COA-based algorithm is the most efficient method, as it is characterized by the lowest value of required time per iteration. Note, in order to draw a fair comparison between the considered algorithms, MATLAB implementation follows the same rules for each algorithm (e.g., avoiding loops and using array operations such as dot product and matrix product whenever possible).

## 5. Experimental Results and Analysis

- two solar modules and one module: 4 solar cells, 400 mW, 2 V, 0.5 A,
- TES 1333R data logging Solar power meter—instrument with range of 2000 W/m
^{2}, high resolution (0.1 W/m^{2}), and wide spectral resolution (400–1100 nm), etc. - lamp—special double spotlight lamp that simulates sunlight. It provides the optimal light spectrum for the solar module.
- USB Data Monitor—used for data acquisition. Also, it is connected to the computer and software through the USB port.
- load—simulates electric consumer load.
- software—designed to facilitate system control, parameter monitoring, data acquisition and graphical representation of the collected data.

^{2}and temperature of 42 °C. For the measured I-V pairs, we determined single and double diode solar cell parameters (see Table 5). The parameter ranges for SDM estimation were ${R}_{s}(\Omega )\in \left[0.1,\text{}0.4\right]$, ${I}_{pv}\left(A\right)\in \left[0.2,\text{}0.4\right]$, ${I}_{o}\left(A\right)\in \left[5\times {10}^{-8},\text{}15\times {10}^{-8}\right]$, ${R}_{p}(\Omega )\in \left[200,\text{}600\right]$ and $n\in \left[0.2,\text{}1\right]$, whereas for DDM were ${R}_{s}(\Omega )\in \left[0.1,\text{}0.4\right]$, ${R}_{p}(\Omega )\in \left[600,\text{}900\right]$, ${n}_{1}\in \left[0.2,\text{}1\right]$, ${n}_{2}\in \left[1.95,\text{}2\right]$, ${I}_{pv}\left(A\right)\in \left[0.2,\text{}0.4\right]$, ${I}_{o1}\left(A\right)\in \left[5\times {10}^{-8},\text{}15\times {10}^{-8}\right]$, and ${I}_{o2}\left(A\right)\in \left[5\times {10}^{-8},\text{}15\times {10}^{-8}\right]$. Then we measured the I-V and P-V characteristics for different values of insolation and temperature. The corresponding simulated characteristics were determined by taking into account the change of parameters with insolation and temperature (see [13]). The measured and estimated I-V and P-V characteristics for different values of insolation and temperature are presented in Figure 7, Figure 8, Figure 9 and Figure 10. The agreement between the measured and estimated characteristics is evident (see zoomed parts in these figures). Finally, we repeated the estimation procedure on all measured I-V characteristics. The estimated values of parameters were in range of ±4% of the initially observed, which confirms that we can use any of the measured characteristics for parameter estimation. On the other hand, by observing the data provided in Table 5, even for this module, it is evident that DDM is more accurate than SDM.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 7.**Current-voltage characteristics for two different insolation values and for temperature T = 42 °C.

**Figure 8.**Power-voltage characteristics for two different insolation values and for temperature T = 42 °C.

No. | Algorithm | Reference | First Author, Year | I_{pv} (A) | I_{0} (μA) | n | R_{s} (Ω) | R_{p} (Ω) | RMSE |
---|---|---|---|---|---|---|---|---|---|

Proposed Method—COA | 0.7607745 | 0.3230018 | 1.4811774 | 0.0363775 | 53.73 | 9.860221 × 10^{−4} | |||

1. | HISA * | [15] | Dhruv, 2019 | 0.7607078 | 0.3106845918 | 1.47726778 | 0.03654694 | 52.88979426 | 9.8911 × 10^{−4} |

2. | HCLPSO * | [16] | Dalia, 2019 | 0.76079 | 0.31062 | 1.4771 | 0.036548 | 52.885 | 1.12009 × 10^{−3} |

3. | OBWOA * | [17] | Abd, 2018 | 0.76077 | 0.3232 | 1.5208 | 0.0363 | 53.6836 | 1.1417 × 10^{−3} |

4. | MPSO * | [18] | Manel, 2018 | 0.760787 | 0.310683 | 1.475262 | 0.036546 | 52.88971 | 7.33007 × 10^{−3} |

5. | ER-WCA | [19] | Kler D, 2017 | 0.760776 | 0.322699 | 1.481080 | 0.036381 | 53.69100 | 9.8609 × 10^{−4} |

6 | MSSO | [20] | Lin P, 2017 | 0.760777 | 0.323564 | 1.481244 | 0.036370 | 53.742465 | 1.0599 × 10^{−3} |

7 | BPFPA * | [21] | Ram JP, 2017 | 0.7600 | 0.3106 | 1.4774 | 0.0366 | 57.7151 | 1.2536 × 10^{−3} |

8 | ICA | [22] | Fathy A, 2017 | 0.7603 | 0.14650 | 1.4421 | 0.0389 | 41.1577 | 1.1582 × 10^{−1} |

9 | GOTLBO | [23] | Chen X, 2016 | 0.760780 | 0.331552 | 1.483820 | 0.036265 | 54.115426 | 9.8744 × 10^{−4} |

10 | CSO | [24] | Guo L, 2016 | 0.76078 | 0.3230 | 1.48118 | 0.03638 | 53.7185 | 9.8612 × 10^{−4} |

11. | NM-MPSO | [25] | Hamid N, 2016 | 0.76078 | 0.32306 | 1.48120 | 0.03638 | 53.7222 | 9.8620 × 10^{−4} |

12. | PCE | [26] | Zhang Y, 2016 | 0.760776 | 0.323021 | 1.481074 | 0.036377 | 53.718525 | 1.0606 × 10^{−3} |

13. | TONG | [27] | Tong NT, 2016 | 0.7610 | 0.3635 | 1.4935 | 0.03660 | 62.574 | 2.3859 × 10^{−3} |

14. | MABC | [28] | Jamadi M, 2016 | 0.760779 | 0.321323 | 1.481385 | 0.036389 | 53.39999 | 2.7610 × 10^{−3} |

15. | MVO | [29] | Ali EE, 2016 | 0.7616 | 0.32094 | 1.5252 | 0.0365 | 59.5884 | 1.2680 × 10^{−1} |

16. | DET | [30] | Chellaswamy C, 2016 | 0.751 | 0.315 | 1.487 | 0.036 | 54.532 | 2.4481 × 10^{−2} |

17. | WCA | [31] | Jordehi AR, 2016 | 0.760908 | 0.4135540 | 1.504381 | 0.035363 | 57.669488 | 7.6069 × 10^{−3} |

18. | TLBO | 0.760809 | 0.312244 | 1.47578 | 0.036551 | 52.8405 | 7.2723 × 10^{−3} | ||

19. | GWO | 0.760996 | 0.2430388 | 1.451219 | 0.037732 | 45.116309 | 7.2845 × 10^{−3} | ||

20. | TVACPSO | 0.760788 | 0.3106827 | 1.475258 | 0.036547 | 52.889644 | 7.3438 × 10^{−3} | ||

21. | PPSO | [32] | Ma J, 2016 | 0.7608 | 0.3230 | 1.4812 | 0.0364 | 53.7185 | 9.9161 × 10^{−4} |

22. | CARO | [33] | Yuan X, 2015 | 0.76079 | 0.31724 | 1.48168 | 0.03644 | 53.0893 | 8.1969 × 10^{−3} |

23. | LI | [34] | Lim LHI, 2015 | 0.7609438 | 0.3456572 | 1.48799169 | 0.03614233 | 49.482205 | 1.3462 × 10^{−3} |

24. | MBA | [35] | El-Fergany A. 2015 | 0.7604 | 0.2348 | 1.4890 | 0.0388 | 44.61 | 1.1672 × 10^{−1} |

25. | FPA * | [36] | Alam DF, 2015 | 0.76079 | 0.310677 | 1.47707 | 0.0365466 | 52.8771 | 1.2121 × 10^{−3} |

26. | LMSA | [37] | Dkhichi F, 2014 | 0.76078 | 0.31849 | 1.47976 | 0.03643 | 53.32644 | 9.8649 × 10^{−4} |

27. | DE | [38] | Niu Q, 2014 | 0.76068 | 0.35515 | 1.49080 | 0.03598 | 56.5533 | 1.0035 × 10^{−3} |

28. | BBO | 0.76098 | 0.86100 | 1.58742 | 0.03214 | 78.8555 | 2.3929 × 10^{−3} | ||

29. | BBO-M | 0.76078 | 0.31874 | 1.47984 | 0.03642 | 53.36227 | 9.8656 × 10^{−4} | ||

30. | STLBO | [39] | Niu Q, 2014 | 0.76078 | 0.32302 | 1.48114 | 0.03638 | 53.7187 | 9.9763 × 10^{−4} |

31. | TLBO | 0.76074 | 0.32378 | 1.48136 | 0.03641 | 54.4029 | 1.0016 × 10^{−3} | ||

32. | ABC | [40] | Oliva D, 2014 | 0.7608 | 0.3251 | 1.4817 | 0.0364 | 53.6433 | 1.0967 × 10^{−3} |

33. | HPEPD | [8] | Laudani A, 2014 | 0.7607884 | 0.3102482 | 1.4769641 | 0.03655304 | 52.859056 | 1.1487 × 10^{−3} |

34. | MPCOA | [41] | Yuan X, 2014 | 0.76073 | 0.32655 | 1.48168 | 0.03635 | 54.6328 | 2.3131 × 10^{−3} |

35. | TLBO | [42] | Patel SJ, 2014 | 0.7608 | 0.3223 | 1.4837 | 0.0364 | 53.76027 | 9.6960 × 10^{−3} |

36. | BMO | [43] | Askarzadeh A, 2013 | 0.76077 | 0.32479 | 1.48173 | 0.03636 | 53.8716 | 9.8622 × 10^{−4} |

37. | ABSO | [44] | 0.76080 | 0.30623 | 1.47583 | 0.03659 | 52.2903 | 9.9125 × 10^{−4} | |

38. | IADE | [45] | Jiang LL, 2013 | 0.7607 | 0.33613 | 1.4852 | 0.03621 | 54.7643 | 9.9076 × 10^{−4} |

39. | CS | [46] | Ma J, 2013 | 0.7608 | 0.323 | 1.4812 | 0.0364 | 53.7185 | 9.9161 × 10^{−4} |

40. | ABSO | [47] | Hachana O, 2013 | 0.76080 | 0.30623 | 1.47986 | 0.03659 | 52.2903 | 1.4169 × 10^{−2} |

41. | ABCDE | 0.76077 | 0.32302 | 1.47986 | 0.03637 | 53.7185 | 4.8548 × 10^{−3} | ||

42. | DE | 0.76077 | 0.32302 | 1.48059 | 0.03637 | 53.7185 | 2.3423 × 10^{−3} | ||

43. | MPSO | 0.76077 | 0.32302 | 1.47086 | 0.03637 | 53.7185 | 3.9022 × 10^{−2} | ||

44. | GGHS | [48] | Askarzadeh A, 2012 | 0.76092 | 0.32620 | 1.48217 | 0.03631 | 53.0647 | 9.9089 × 10^{−4} |

45. | HS | 0.76070 | 0.30495 | 1.47538 | 0.03663 | 53.5946 | 9.9515 × 10^{−4} | ||

46. | IGHS | 0.76077 | 0.34351 | 1.48740 | 0.03613 | 53.2845 | 1.0335 × 10^{−3} | ||

47. | PS | [49] | AlHajri MF, 2012 | 0.7617 | 0.9980 | 1.6000 | 0.0313 | 64.10256 | 1.4936 × 10^{−2} |

48. | SA | [50] | El-Naggar KM, 2012 | 0.7620 | 0.4798 | 1.5172 | 0.0345 | 43.10345 | 1.8998 × 10^{−2} |

49. | GA | [51] | AlRashidi MR, 2011 | 0.7619 | 0.8087 | 1.5751 | 0.0299 | 42.37288 | 1.9078 × 10^{−2} |

50. | PSO | [52] | Ye M, 2009 | 0.760798 | 0.322721 | 1.48382 | 0.0363940 | 53.7965 | 9.6545 × 10^{−3} |

No. | Algorithm | Ref. | First Author, Year | I_{pv}(A) | I_{o1}(μA) | I_{o2}(μA) | R_{s}(Ω) | R_{p}(Ω) | n_{1} | n_{2} | RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|

Proposed Method—COA | 0.76078105 | 0.2259742 | 0.749346 | 0.03674043 | 55.4854236 | 1.45101673 | 2 | 9.82484852 × 10^{−}^{4} | |||

1. | GOFPANM | [53] | X Shuhui, 2017 | 0.7607811 | 0.7493476 | 0.2259743 | 0.0367404 | 55.485449 | 2 | 1.4510168 | 9.82485 × 10^{−4} |

2. | SATLBO | [54] | Y Kunjie, 2017 | 0.76078 | 0.25093 | 0.545418 | 0.03663 | 55.117 | 1.45982 | 1.99941 | 9.82941 × 10^{−4} |

3. | MSSO | [20] | P Lin, 2017 | 0.760748 | 0.234925 | 0.671593 | 0.036688 | 55.714662 | 1.454255 | 1.995305 | 1.059101 × 10^{−3} |

4. | WDO | [55] | M Derick, 2017 | 0.7606 | 0.2531 | 0.0482 | 0.037433 | 52.6608 | 151.162 | 1.38434 | 1.095213 × 10^{−3} |

5. | CSO | [24] | L Guo, 2016 | 0.76078 | 0.22732 | 0.72785 | 0.036737 | 55.3813 | 1.45151 | 1.99769 | 9.82532 × 10^{−4} |

6. | GOTLBO | [23] | X Chen, 2016 | 0.760752 | 0.800195 | 0.220462 | 0.036783 | 56.0753 | 1.999973 | 1.448974 | 9.83152 × 10^{−4} |

7. | PCE | [26] | Y Zhang, 2016 | 0.760781 | 0.226015 | 0.749340 | 0.03674 | 55.483160 | 1.450.923 | 2 | 9.8248 × 10^{−4} |

8. | MABC | [28] | M Jamadi, 2016 | 0.7607821 | 0.24102992 | 0.6306922 | 0.03671215 | 54.7550094 | 1.4568573 | 2.0000.538 | 9.8276 × 10^{−4} |

9. | FPA | [36] | DF Alam, 2015 | 0.760795 | 0.300088 | 0.166159 | 0.0363342 | 52.3475 | 1.47477 | 2 | 1.24239 × 10^{−3} |

10. | BMO | [43] | A. Askarzadeh, 2013 | 0.76078 | 0.2111 | 0.87688 | 0.03682 | 558.081 | 1.44533 | 1.99.997 | 9.82661 × 10^{−4} |

11. | ABSO | [44] | A. Askarzadeh, 2013 | 0.73078 | 0.26713 | 0.38191 | 0.03657 | 54.6219 | 1.46512 | 1.98152 | 9.8359 × 10^{−}^{04} |

12. | IGHS | [48] | A. Askarzadeh, 2012 | 0.76079 | 0.97310 | 0.16791 | 0.03690 | 56.8368 | 1.92126 | 1.42814 | 9.86572 × 10^{−4} |

Algorithm | Mean Value of Requested Time (s) | Maximal Value of Requested Time (s) | Minimal Value of Requested Time (s) |
---|---|---|---|

COA | 0.016416 | 0.017023 | 0.015871 |

ER-WCA [19] | 0.021063 | 0.024145 | 0.019492 |

CS [46] | 0.029179 | 0.037177 | 0.027130 |

HS [48] | 0.021103 | 0.023264 | 0.020393 |

Parameter | Analytical Method [13] | Numerical Method [13] | Iteration Method [57] | Newton Method [58] | COA |
---|---|---|---|---|---|

I_{pv} (A) | 3.8752 | 3.8046 | 3.8 | 3.8084 | 3.8418 |

I_{o1} (A) | 3.6129 × 10^{−10} | 3.9901 × 10^{−10} | 4.704 × 10^{−10} | 4.8723 × 10^{−10} | 4.95821 × 10^{−8} |

I_{o2} (A) | 9.3773 × 10^{−6} | 4.033 × 10^{−6} | 4.704 × 10^{−10} | 6.1528 × 10^{−10} | 9.54961 × 10^{−9} |

R_{s} (Ω) | 0.3084 | 0.3397 | 0.35 | 0.3692 | 0.2495 |

R_{p} (Ω) | 280.6449 | 280.2171 | 176.4 | 169.0471 | 267.57 |

n_{1} | 1 | 0.99859 | 1 | 1.0003 | 1.2569 |

n_{2} | 2 | 2.0014 | 1.2 | 1.9997 | 1.9345 |

RMSE | 0.0358 | 0.0517 | 0.1211 | 0.1636 | 0.0194 |

SDM | DDM | ||
---|---|---|---|

R_{s} (Ω) | 0.2283 | R_{s} (Ω) | 0.2513 |

R_{sh} (Ω) | 439.55 | R_{sh} (Ω) | 782.9911 |

I_{o} (A) | 10.56 × 10^{−8} | I_{o1} (A) | 6.8452 × 10^{−8} |

I_{pv} (A) | 0.2987 | n_{1} | 0.3342 |

n | 0.3441 | I_{pv} (A) | 0.2972 |

RMSE | 4.3418 × 10^{−4} | I_{o2} (A) | 6.0643 × 10^{−8} |

n_{2} | 1.9906 | ||

RMSE | 4.146 × 10^{−4} |

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**MDPI and ACS Style**

Ćalasan, M.; Jovanović, D.; Rubežić, V.; Mujović, S.; Đukanović, S.
Estimation of Single-Diode and Two-Diode Solar Cell Parameters by Using a Chaotic Optimization Approach. *Energies* **2019**, *12*, 4209.
https://doi.org/10.3390/en12214209

**AMA Style**

Ćalasan M, Jovanović D, Rubežić V, Mujović S, Đukanović S.
Estimation of Single-Diode and Two-Diode Solar Cell Parameters by Using a Chaotic Optimization Approach. *Energies*. 2019; 12(21):4209.
https://doi.org/10.3390/en12214209

**Chicago/Turabian Style**

Ćalasan, Martin, Dražen Jovanović, Vesna Rubežić, Saša Mujović, and Slobodan Đukanović.
2019. "Estimation of Single-Diode and Two-Diode Solar Cell Parameters by Using a Chaotic Optimization Approach" *Energies* 12, no. 21: 4209.
https://doi.org/10.3390/en12214209