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

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

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

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

Table 2.

The boundaries of extracted PV parameters for SDM and DDM

Parameter	Lower bound	Upper bound
I_ph (A)	0	1
I_sd, I_sd₁, I_sd₂, (μA)	0	1
R_s (Ω)	0	1
R_sh(Ω)	0	100
n, n₁, n₂	1	2

[i] DDM, double diode model; PV, photovoltaic; SDM, single diode model.

Table 3.

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

[i] AHO, artificial hummingbird optimization; DLMVO, Dynamic Levy Mutated Vortex Optimization; ILSA, Improved Learning Search Algorithm, GOA, Gazelle optimization algorithm; GOANM, Gazelle optimization-Nelder–Mead algorithms; IMFOL, improved multi-objective fitness optimised by Levy; RMSE, root mean square error; RTLBO, randomized teaching-learning-based optimization; SDM, single diode model; TSA, tunicate swarm algorithm; ELADE, Elite Learning Adaptive Differential Evolution, PSOCS, Particle Swarm Optimization and Cuckoo Search; BES, Bald Eagle Search algorithm; CGO-LS, Chaos Game Optimization-Least Squares algorithm; DE, Differential Evolution algorithm; ELADE, Elite Learning Adaptive Differential Evolution; IHGS, Improved Hunger Games Search; ISCA: Improved Sine Cosine Algorithm; ITSA, Improved Tunicate Swarm Algorithm; PSOCS, Particle Swarm Optimization with Cuckoo Search algorithm; RTC France solar cell, a standard silicon solar cell used for PV model validation an parameter extraction.

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

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

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

Table 4.

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

[i] WSO, HO, GOANM and GOA. AHO, artificial hummingbird optimization; DDM, double diode model; DLMVO, Dynamic Levy Mutated Vortex Optimization; GOA, Gazelle optimization algorithm; GOANM, Gazelle optimization-Nelder–Mead algorithms; IMFOL, improved multi-objective fitness optimized by Levy; RMSE, root mean square error; RTLBO, randomized teaching-learning-based optimization; TSA, tunicate swarm algorithm.

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

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

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

Table 5.

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

[i] DDM, double diode model; GOA, Gazelle optimization algorithm; GOANM, Gazelle optimization-Nelder–Mead algorithms; PV, photovoltaic; RMSE, root mean square error; SDM, single diode model; STD, standard deviation.

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DOI: https://doi.org/10.2478/pead-2025-0003 | Journal eISSN: 2543-4292 | Journal ISSN: 2451-0262

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Language: English

Page range: 41 - 59

Submitted on: Aug 30, 2024

Accepted on: Dec 29, 2024

Published on: Jan 27, 2025

Published by: Wroclaw University of Science and Technology

In partnership with: Paradigm Publishing Services

Keywords:

PV system,

Newton-Raphson method,

WSO-HO algorithms,

hybrid meta-heuristic algorithms

Related subjects:

Computer sciences,

Artificial intelligence,

Engineering,

Electrical engineering,

Electronics

© 2025 Ahmed Jeridi, Mohamed Hedi Moulahi, Hechmi Khaterchi, Abderrahmen Zaafouri, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 10 (2025): Issue 1 (January 2025)

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

Figures & Tables

Figure 1.

Table 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Table 2.

Table 3.

Figure 6.

Figure 7.

Figure 8.

Table 4.

Figure 9.

Figure 10.

Figure 11.

Table 5.

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