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A Hybrid Approach Incorporating WSO-HO and the Newton-Raphson Method to Enhancing Photovoltaic Solar Model Parameters Optimisation

Ahmed Jeridi

Ahmed Jeridi

University of Tunis, National Higher School of Engineers of Tunis (ENSIT), Laboratory of Engineering of Industrial Systems and Renewable Energy (LISIER)Tunis, Tunisia
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Jeridi, Ahmed
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Mohamed Hedi Moulahi

Mohamed Hedi Moulahi

University of Tunis, National Higher School of Engineers of Tunis (ENSIT), Laboratory of Engineering of Industrial Systems and Renewable Energy (LISIER)Tunis, Tunisia
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Moulahi, Mohamed Hedi
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Hechmi Khaterchi

Hechmi Khaterchi

University of Tunis, National Higher School of Engineers of Tunis (ENSIT), Laboratory of Engineering of Industrial Systems and Renewable Energy (LISIER)Tunis, Tunisia
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Khaterchi, Hechmi
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Abderrahmen Zaafouri

Abderrahmen Zaafouri

University of Tunis, National Higher School of Engineers of Tunis (ENSIT), Laboratory of Engineering of Industrial Systems and Renewable Energy (LISIER)Tunis, Tunisia
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Zaafouri, Abderrahmen
  
Jan 27, 2025
A Hybrid Approach Incorporating WSO-HO and the Newton-Raphson Method to Enhancing Photovoltaic Solar Model Parameters Optimisation's Cover Image
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Article Category: Research paper
Published Online: Jan 27, 2025
Page range: 41 - 59
Received: Aug 30, 2024
Accepted: Dec 29, 2024
DOI: https://doi.org/10.2478/pead-2025-0003
Keywords
© 2025 Ahmed Jeridi et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1.

Equivalent circuit cell model. (a) Equivalent circuit of SDM. (b) Equivalent circuit of DDM. DDM, double diode model; SDM, single diode model.
Equivalent circuit cell model. (a) Equivalent circuit of SDM. (b) Equivalent circuit of DDM. DDM, double diode model; SDM, single diode model.

Figure 2.

Parameter extraction by integrating Newton-Raphson method with WSO-HO optimization algorithm.
Parameter extraction by integrating Newton-Raphson method with WSO-HO optimization algorithm.

Figure 3.

Flow Chart of Hybrid WSO-HO Algorithm.
Flow Chart of Hybrid WSO-HO Algorithm.

Figure 4.

Hybridization WSO-HO Algorithm.
Hybridization WSO-HO Algorithm.

Figure 5.

Hybrid WSO-HO Algorithm with NR method. NR, Newton-Raphson.
Hybrid WSO-HO Algorithm with NR method. NR, Newton-Raphson.

Figure 6.

Curves with the measured and estimated data. (a) (P, V) data for SDM. (b) (I, V) data for SDM. SDM, single diode model.
Curves with the measured and estimated data. (a) (P, V) data for SDM. (b) (I, V) data for SDM. SDM, single diode model.

Figure 7.

Convergence and robustness curves for SDM. (a) Curves convergence (b) Curves robustness. SDM, single diode model.
Convergence and robustness curves for SDM. (a) Curves convergence (b) Curves robustness. SDM, single diode model.

Figure 8.

Convergence and Robustness Curves of Optimization Algorithms Applied to the SDM.
Convergence and Robustness Curves of Optimization Algorithms Applied to the SDM.

Figure 9.

Curves with the measured and estimated data. (a) Curves (P, V) for DDM. (b) Curves (I, V) for DDM. DDM, double diode model.
Curves with the measured and estimated data. (a) Curves (P, V) for DDM. (b) Curves (I, V) for DDM. DDM, double diode model.

Figure 10.

Convergence and robustness curves for DDM. (a) Curves convergence (b) Curves robustness. DDM, double diode model.
Convergence and robustness curves for DDM. (a) Curves convergence (b) Curves robustness. DDM, double diode model.

Figure 11.

Convergence and Robustness Curves of Optimisation Algorithms Applied to the DDM. DDM, double diode model.
Convergence and Robustness Curves of Optimisation Algorithms Applied to the DDM. DDM, double diode model.

Experimental current (I) and voltage (V) data for RTC France PV cells using SDM and DDM_

Parameter 1 2 3 4 5 6 7 8 9
I (Ampere) 0.764 0.762 0.7605 0.7605 0.76 0.759 0.757 0.7557 0.755
V (Volt) −0.2057 −0.1291 −0.0588 0.0057 0.0666 0.1183 0.1678 0.2152 0.2618
Parameter 10 11 12 13 14 15 16 17 18
I (Ampere) 0.754 0.7505 0.746 0.7385 0.728 0.706 0.673 0.632 0.573
V (Volt) 0.2924 0.3269 0.385 0.3837 0.4173 0.4573 0.4798 0.4784 0.5119
Parameter 19 20 21 22 23 24 25 26
I (Ampere) 0.499 0.413 0.316 0.212 0.103 −0.01 −0.123 −0.21
V (Volt) 0.5319 0.5266 0.3983 0.5321 0.5533 0.5736 0.5833 0.59

The SDM parameters estimated at the best RMSE

Algorithm Iph(A) Isd1(μA) Rs (Ω) Rsh (Ω) n RMSE
WSO-HO 0.76079 0.31069 0.036547 52.8899 1.4773 7.729856E-04
WSO 0.76078 0.31069 0.03654 52.889 1.47727 7.730056E-04
HO 0.76774 0.54060 0.03164 19.31912 1.53746 8.753850E-04
GOANM (Amiri et al., 2024) 0.76079 0.31069 0.036547 52.8899 1.4773 7.729900E-04
GOA (Amiri et al., 2024) 0.76070 0.34001 0.036182 55.7021 1.4864 7.870400E-04
IMFOL (Qaraad et al., 2023) 0.76078 0.32302 0.036377 53.7186 1.4812 9.860200E-04
RTLBO (Yu et al., 2023) 0.76078 0.32302 0.036377 53.7185 1.4812 9.860200E-04
DLMVO (Ekinci et al., 2024) 0.7608 0.3230 0.0364 53.7185 1.4812 9.860200E-04
OBL-RSACM (Li et al., 2023) 0.76080 0.32203 0.03643 53.3521 1.4812 9.845200E-04
AHO (Bogar, 2023) 0.76079 0.31086 0.036540 52.8595 1.2155 7.730600E-04
PSOCS (Fan et al., 2022) 0.76078 0.32302 0.036377 53.7185 1.4812 9.860200E-04
ELADE (Gu et al., 2023) 0.76077 0.30839 0.036555 52.8267 1.4765 7.754700E-04
ILSA (Huang et al., 2020) 0.76077 0.32302 0.036377 53.7185 1.4812 9.860200E-04
IHGS (Xu et al., 2022) 0.76078 0.32302 0.0364 53.7178 1.4812 9.860200E-04
BES (Alsattar et al., 2020) 0.7607 0.3230 0.0364 53.7185 1.4812 9.860200E-04
DE (Yu et al., 2022) 0.7607 0.3209 0.0363 54.1134 1.4709 7.769200E-04
BES (Nicaire et al., 2021) 0.7683 0.3262 0.0367 54.2557 1.4958 9.860000E-04
ITSA (Arandian et al., 2022) 0.7606 0.3298 0.0363 56.5694 1.4832 9.933900E-04

Analysis of RMSE for Single and Double PV models

Model Algorithm Min Mean Max STD
SDM WSO-HO 7.72985671E-04 7.7308567E-04 7.7410567E-04 6.138516E-17
WSO 7.73005671E-04 8.1665894E-04 0.002083 2.3920836E-04
HO 8.75385E-04 0.00233 0.00629359 1.33368-03
GOA [Amiri, H. H et al., 2024] 7.5810E–04 7.7695E–04 8.8385E–04 2.4314E–05
GOANM [Amiri, H. H et al., 2024] 7.4339E–04 7.5263E–04 7.6714E–04 7.3732E–06
ELAD [Gu, Z. et al., 2023] 9.8602E-04 9.8602E-04 9.8605E-04 1.753E-10
DE [Yu, S., 2022] 9.811E-04 1.02874E-03 1.0813E-03 2.94961E-05
ISCA [Chen, H., 2019] 7.3423E-04 7.2302E-04 7.4592E-04 1.30287E-06
ITSA [Nicaire, N. F., et al., 2021] 9.86E-04 7.730062E-04 9.89E-04 5.70E-16
DDM WSO-HO 7.420691E-04 7.4967098E-04 7.72985671E-04 1.08074E-05
WSO 7.4378257E-04 8.3867102E-04 0.00099892 1.6562955E-04
HO 8.55885e-04 0.00265008 0.006542 1.74708E-03
GOA [Amiri, H.H et al., 2024] 7.5810E–04 7.7695E–04 8.8385E–04 2.4314E–05
GOANM [Amiri, H.H et al., 2024] 7.4339E–04 7.5263E–04 7.6714E–04 7.3732E–06
ELAD [Gu, Z. et al., 2023] 9.8252E-04 1.32602E-03 1.000562E-03 9.15E-12
DE [Yu, S., 2022] 9.8607E-04 9.8874E-04 7.730062E-04 2.4696E-06
ISCA [Chen, H., 2019] 2.2142E-04 1.66043E-02 9.93218E-04 1.30287E-06
ITSA [Nicaire, N. F., et al., 2021] 9.9804E-04 9.99991E-04 3.799062E-02 6.33E-06

The boundaries of extracted PV parameters for SDM and DDM

Parameter Lower bound Upper bound
Iph (A) 0 1
Isd, Isd1, Isd2, (μA) 0 1
Rs (Ω) 0 1
Rsh(Ω) 0 100
n, n1, n2 1 2

The DDM Parameters Estimated at the best RMSE

Algorithm Iph (A) Isd1 (μA) Isd2 (μA) Rs (Ω) Rsh (Ω) n1 n2 RMSE
WSO-HO 0.760805 0.0854343 0.9991529 0.0376485 56.0775146 1.37756104 1.81810675 7.42069103E-04
WSO 0.760804 0.069334 0.884680 0.0376813 55.918552 1.3648682 1.7688404 7.4378257E-04
HO 0.7607879 0.3106909 0.3106909 0.0365467 52.8899092 1.47727164 1.47727160 8.55885e-04
GOANM [Amiri, H.H et al., 2024] 0.76081 0.11624 0.9768 0.037459 55.7298 1.3994 1.8597 7.4339E-04
GOA [Amiri, H.H et al., 2024] 0.76079 0.19704 0.4356 0.03688 54.2616 1.4417 1.8186 7.5810E-04
IMFOL [Qaraad, M. et al., 2023] 0.76078 0.76632 0.2251 0.036731 55.6567 2.0000 1.4508 9.8252E-04
RTLBO [Yu, X. et al., 2023] 0.76078 0.22597 0.7494 0.03674 55.4855 1.4510 2.0000 9.8248E-04
DLMVO [Ekinci,S., et al., 2024] 0.7608 0.7493 0.2260 0.0367 55.4854 2.0000 1.4510 9.8248E-04
OBL-RSACM [ Li, J. et al., 2023] 0.76033 0.39986 0.2677 0.03669 56.0102 1.4151 2.0000 9.8237E-04
AHO [Bogar,E., 2023] 0.76078 0.27988 0.2768 0.036530 54.2856 1.9563 1.4682 9.8401E-04
PSOCS [Fan, Y., et al., 2022] 0.76078 0.22598 0.7493 0.036740 55.4855 1.4510 2.0000 9.8248E-04
ELADE [Gu, Z. et al., 2023] 0.76072 0.24468 0.3802 0.036927 53.5130 1.4564 1.9899 7.6480E-04
ILSA [Huang, T., et al., 2020] 0.76078 0.50569 0.2557 0.036609 54.9246 2.0000 1.4614 9.8270E-04
IHGS [Xu, B., et al., 2022] 0.76078 0.74935 0.2260 0.03674 55.48542 2.0000 1.45102 9.8248E-04
BES [Alsattar, H. A., et al., 2020] 0.7608 0.2259 0.7493 0.0367 55.4854 1.4510 2.0000 9.8248E-04
DE [Yu, S., 2022] 0.7605 0.42322 0.1873 0.02061 51.9345 1.8758 1.4360 7.6300E-04
ITSA [Nicaire, N. F., et al., 2021] 0.7608 0.9731 0.1679 0.0369 53.8368 1.9213 1.4281 9.82E-04
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