Parameters Identification of PV TripleDiode Model Using Improved Generalized Normal Distribution Algorithm
Abstract
:1. Introduction
 ❖
 Relating its exploitation capability with the average of the current mean position of the population, the bestsofar solution, and the position of the current individual, and that may cause low convergence toward the bestsofar solution for reaching better solutions quickly whether the bestsofar solution is not a local minimum one.
 ❖
 Relating its exploration capability with three solutions selected randomly from the population and may make the algorithm explore regions that may already have been explored.
 ❖
 A novel rankingbased position updating method (RUM) to help the algorithm in exploring as many regions as possible; and
 ❖
 A premature convergence method (PCM) to help accelerate its convergence speed toward the nearoptimal solution.
 Improving the GNDO by the novel RUM and the premature convergence method (PCM) to produce a new variant called RGNDO for tackling the parameter estimation of the TDM.
 Comparing the performance of RGNDO with some wellestablished parameter estimation techniques, in addition to the standard GNDO, on five wellknown commercial PV modules confirms the superiority of RGNDO over these compared algorithms in terms of convergence speed and final accuracy, in addition to its competitivity for the computational cost.
2. Mathematical Descriptions of the TripleDiode Model
3. The Standard Algorithm: Generalized Normal Distribution Optimization
3.1. Local Exploitation
3.2. Global Exploration
4. The Proposed Algorithm: RGDNO
4.1. Initialization
4.2. The Objective Function
4.3. RankingBased Novel Updating Method (RUM)
4.4. Premature Convergence Method (PCM)
 Utilizing each individual in the population through the optimization process by the RUM to help in exploring more regions within the search space as possible. The RUM here aids the standard GNDO to improve the exploration operator at the start of the optimization process as an attempt to prevent stuck into local minima, while, with increasing the current function evaluation, the exploration operator is gradually converted into exploitation to search around the bestsofar solution to promote the convergence speed.
 Highly stable due to using the PCM that helps in steering the convergence speed in the right direction of the bestsofar solution to explore the promising regions that appear within the optimization process.
5. Experimental Results
5.1. Parameter Settings
5.2. Dataset Descriptions
6. Results and Discussion
6.1. Test Case 1: RTC France Cell
6.2. Test Case 2: Kyocera KC200GT—204.6 W Module
6.3. Test Case 3: Ultra 85P
6.4. Test Case 4: STP6120/36 Module
6.5. Comparison between GNDO and RGNDO
6.6. CPU Time
6.7. Wilcoxon Rank Sum Test
6.8. Various SteadyState Characteristics under Varied Operating Conditions
7. Conclusions and Future Work
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
 Alam, D.; Yousri, D.; Eteiba, M. Flower pollination algorithm based solar PV parameter estimation. Energy Convers. Manag. 2015, 101, 410–422. [Google Scholar] [CrossRef]
 AbdelBasset, M.; Mohamed, R.; Mirjalili, S.; Chakrabortty, R.K.; Ryan, M.J. Solar photovoltaic parameter estimation using an improved equilibrium optimizer. Sol. Energy 2020, 209, 694–708. [Google Scholar] [CrossRef]
 Selem, S.I.; ElFergany, A.A.; Hasanien, H.M. Artificial electric field algorithm to extract nine parameters of triplediode photovoltaic model. Int. J. Energy Res. 2020, 45, 590–604. [Google Scholar] [CrossRef]
 Abbassi, R.; Abbassi, A.; Heidari, A.A.; Mirjalili, S. An efficient salp swarminspired algorithm for parameters identification of photovoltaic cell models. Energy Convers. Manag. 2019, 179, 362–372. [Google Scholar] [CrossRef]
 Ishaque, K.; Salam, Z.; Mekhilef, S.; Shamsudin, A. Parameter extraction of solar photovoltaic modules using penaltybased differential evolution. Appl. Energy 2012, 99, 297–308. [Google Scholar] [CrossRef]
 ElHameed, M.A.; Elkholy, M.M.; ElFergany, A.A. Threediode model for characterization of industrial solar generating units using Mantarays foraging optimizer: Analysis and validations. Energy Convers. Manag. 2020, 219, 113048. [Google Scholar] [CrossRef]
 Long, W.; Cai, S.; Jiao, J.; Xu, M.; Wu, T. A new hybrid algorithm based on grey wolf optimizer and cuckoo search for parameter extraction of solar photovoltaic models. Energy Convers. Manag. 2020, 203, 112243. [Google Scholar] [CrossRef]
 Li, S.; Gu, Q.; Gong, W.; Ning, B. An enhanced adaptive differential evolution algorithm for parameter extraction of photovoltaic models. Energy Convers. Manag. 2020, 205, 112443. [Google Scholar] [CrossRef]
 Qais, M.H.; Hasanien, H.M.; Alghuwainem, S.; Nouh, A.S. Coyote optimization algorithm for parameters extraction of threediode photovoltaic models of photovoltaic modules. Energy 2019, 187, 116001. [Google Scholar] [CrossRef]
 Allam, D.; Yousri, D.; Eteiba, M. Parameters extraction of the three diode model for the multicrystalline solar cell/module using MothFlame Optimization Algorithm. Energy Convers. Manag. 2016, 123, 535–548. [Google Scholar] [CrossRef]
 Ahmad, T.; Sobhan, S.; Nayan, M.F. Comparative analysis between single diode and double diode model of PV cell: Concentrate different parameters effect on its efficiency. J. Power Energy Eng. 2016, 4, 31–46. [Google Scholar] [CrossRef] [Green Version]
 Khanna, V.; Das, B.; Bisht, D.; Singh, P. A three diode model for industrial solar cells and estimation of solar cell parameters using PSO algorithm. Renew. Energy 2015, 78, 105–113. [Google Scholar] [CrossRef]
 AbdelBasset, M.; Chang, V.; Mohamed, R. HSMA_WOA: A hybrid novel Slime mould algorithm with whale optimization algorithm for tackling the image segmentation problem of chest Xray images. Appl. Soft Comput. 2020, 106642. [Google Scholar] [CrossRef] [PubMed]
 AbdelBasset, M.; Elshahat, D.; Elhoseny, M.; Song, H. EnergyAware Metaheuristic algorithm for Industrial Internet of Things task scheduling problems in fog computing applications. IEEE Internet Things J. 2020, 1. [Google Scholar] [CrossRef]
 AbdelBasset, M.; Mohamed, R.; Elhoseny, M.; Bashir, A.K.; Jolfaei, A.; Kumar, N. EnergyAware Marine Predators Algorithm for Task Scheduling in IoTbased Fog Computing Applications. IEEE Trans. Ind. Inform. 2020, 17, 5068–5076. [Google Scholar] [CrossRef]
 Ezugwu, A.E.; Pillay, V.; Hirasen, D.; Sivanarain, K.; Govender, M. A Comparative study of metaheuristic optimization algorithms for 0–1 knapsack problem: Some initial results. IEEE Access 2019, 7, 43979–44001. [Google Scholar] [CrossRef]
 Fathy, A.; Rezk, H. Robust electrical parameter extraction methodology based on Interior Search Optimization Algorithm applied to supercapacitor. ISA Trans. 2020, 105, 86–97. [Google Scholar] [CrossRef]
 Elazab, O.S.; Hasanien, H.M.; Alsaidan, I.; Abdelaziz, A.Y.; Muyeen, S. Parameter estimation of three diode photovoltaic model using grasshopper optimization algorithm. Energies 2020, 13, 497. [Google Scholar] [CrossRef] [Green Version]
 Yousri, D.; Thanikanti, S.B.; Allam, D.; Ramachandaramurthy, V.K.; Eteiba, M. Fractional chaotic ensemble particle swarm optimizer for identifying the single, double, and three diode photovoltaic models’ parameters. Energy 2020, 195, 116979. [Google Scholar] [CrossRef]
 Diab, A.A.Z.; Sultan, H.M.; Do, T.D.; Kamel, O.M.; Mossa, M.A. Coyote optimization algorithm for parameters estimation of various models of solar cells and PV modules. IEEE Access 2020, 8, 111102–111140. [Google Scholar] [CrossRef]
 Ibrahim, I.A.; Hossain, M.; Duck, B.C.; Nadarajah, M. An improved wind driven optimization algorithm for parameters identification of a triplediode photovoltaic cell model. Energy Convers. Manag. 2020, 213, 112872. [Google Scholar] [CrossRef]
 Qais, M.H.; Hasanien, H.M.; Alghuwainem, S. Parameters extraction of threediode photovoltaic model using computation and Harris Hawks optimization. Energy 2020, 195, 117040. [Google Scholar] [CrossRef]
 Chenouard, R.; ElSehiemy, R.A. An interval branch and bound global optimization algorithm for parameter estimation of three photovoltaic models. Energy Convers. Manag. 2020, 205, 112400. [Google Scholar] [CrossRef]
 Liang, J.; Ge, S.; Qu, B.; Yu, K.; Liu, F.; Yang, H.; Wei, P.; Li, Z. Classified perturbation mutation based particle swarm optimization algorithm for parameters extraction of photovoltaic models. Energy Convers. Manag. 2020, 203, 112138. [Google Scholar] [CrossRef]
 Long, W.; Wu, T.; Jiao, J.; Tang, M.; Xu, M. Refractionlearningbased whale optimization algorithm for highdimensional problems and parameter estimation of PV model. Eng. Appl. Artif. Intell. 2020, 89, 103457. [Google Scholar] [CrossRef]
 Ridha, H.M.; Gomes, C.; Hizam, H. Estimation of photovoltaic module model’s parameters using an improved electromagneticlike algorithm. Neural Comput. Appl. 2020, 32, 12627–12642. [Google Scholar] [CrossRef]
 Ridha, H.M.; Heidari, A.A.; Wang, M.; Chen, H. Boosted mutationbased Harris hawks optimizer for parameters identification of singlediode solar cell models. Energy Convers. Manag. 2020, 209, 112660. [Google Scholar] [CrossRef]
 Ram, J.P.; Pillai, D.S.; Rajasekar, N.; Chinnaiyan, V.K. Flower Pollination Based Solar PV Parameter Extraction for Double Diode Model. In Intelligent Computing Techniques for Smart Energy Systems; Springer: Berlin/Heidelberg, Germany, 2020; pp. 303–312. [Google Scholar]
 Hassan, K.H.; Rashid, A.T.; Jasim, B.H. Parameters estimation of solar photovoltaic module using camel behavior search algorithm. Int. J. Electr. Comp. Eng. 2021, 11, 788–793. [Google Scholar] [CrossRef]
 Kashefi, H.; Sadegheih, A.; Mostafaeipour, A.; Omran, M.M. Parameter identification of solar cells and fuel cell using improved social spider algorithm. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 2020. [Google Scholar] [CrossRef]
 Li, S.; Gong, W.; Yan, X.; Hu, C.; Bai, D.; Wang, L.; Gao, L. Parameter extraction of photovoltaic models using an improved teachinglearningbased optimization. Energy Convers. Manag. 2019, 186, 293–305. [Google Scholar] [CrossRef]
 Premkumar, M.; Jangir, P.; Sowmya, R.; Elavarasan, R.M.; Kumar, B.S. Enhanced chaotic JAYA algorithm for parameter estimation of photovoltaic cell/modules. ISA Trans. 2021. [Google Scholar] [CrossRef] [PubMed]
 Ismaeel, A.A.; Houssein, E.H.; Oliva, D.; Said, M. Gradientbased optimizer for parameter extraction in photovoltaic models. IEEE Access 2021, 9, 13403–13416. [Google Scholar] [CrossRef]
 Mokeddem, D. Parameter Extraction of Solar Photovoltaic Models Using Enhanced Levy Flight Based Grasshopper Optimization Algorithm. J. Electr. Eng. Technol. 2021, 16, 171–179. [Google Scholar] [CrossRef]
 Ramadan, A.; Kamel, S.; Korashy, A.; Yu, J. Photovoltaic cells parameter estimation using an enhanced teaching–learningbased optimization algorithm. Iran. J. Sci. Technol. Trans. Electr. Eng. 2020, 44, 767–779. [Google Scholar] [CrossRef]
 Kumar, C.; Raj, T.D.; Premkumar, M.; Raj, T.D. A new stochastic slime mould optimization algorithm for the estimation of solar photovoltaic cell parameters. Optik 2020, 223, 165277. [Google Scholar] [CrossRef]
 Huynh, D.C.; Ho, L.D.; Dunnigan, M.W. Parameter Estimation of Solar Photovoltaic Cells Using an Improved Artificial Bee Colony Algorithm. In Proceedings of the International Conference on Green Technology and Sustainable Development, Ho Chi Minh City, Vietnam, 27–28 November 2020; pp. 281–292. [Google Scholar]
 Ćalasan, M.; Jovanović, D.; Rubežić, V.; Mujović, S.; Đukanović, S. Estimation of singlediode and twodiode solar cell parameters by using a chaotic optimization approach. Energies 2019, 12, 4209. [Google Scholar] [CrossRef] [Green Version]
 Nayak, B.; Mohapatra, A.; Mohanty, K.B. Parameter estimation of single diode PV module based on GWO algorithm. Renew. Energy Focus 2019, 30, 1–12. [Google Scholar] [CrossRef]
 Zhang, Y.; Jin, Z.; Mirjalili, S. Generalized normal distribution optimization and its applications in parameter extraction of photovoltaic models. Energy Convers. Manag. 2020, 224, 113301. [Google Scholar] [CrossRef]
 Askarzadeh, A.; Rezazadeh, A. Parameter identification for solar cell models using harmony searchbased algorithms. Sol. Energy 2012, 86, 3241–3249. [Google Scholar] [CrossRef]
 Fossum, J.G.; Lindholm, F.A. Theory of grainboundary and intragrain recombination currents in polysilicon pnjunction solar cells. IEEE Trans. Electron Devices 1980, 27, 692–700. [Google Scholar] [CrossRef]
 KoohiKamali, S.; Rahim, N.; Mokhlis, H.; Tyagi, V. Photovoltaic electricity generator dynamic modeling methods for smart grid applications: A review. Renew. Sustain. Energy Rev. 2016, 57, 131–172. [Google Scholar] [CrossRef]
 Nunes, H.; Pombo, J.; Mariano, S.; Calado, M.; De Souza, J.F. A new high performance method for determining the parameters of PV cells and modules based on guaranteed convergence particle swarm optimization. Appl. Energy 2018, 211, 774–791. [Google Scholar] [CrossRef]
 AbdelBasset, M.; Mohamed, R.; Elhoseny, M.; Chakrabortty, R.K.; Ryan, M. A Hybrid COVID19 Detection Model Using an Improved Marine Predators Algorithm and a RankingBased Diversity Reduction Strategy. IEEE Access 2020, 8, 79521–79540. [Google Scholar] [CrossRef]
 Yousri, D.; Rezk, H.; Fathy, A. Identifying the parameters of different configurations of photovoltaic models based on recent artificial ecosystembased optimization approach. Int. J. Energy Res. 2020, 44, 11302–11322. [Google Scholar] [CrossRef]
 Elazab, O.S.; Hasanien, H.M.; Elgendy, M.A.; Abdeen, A.M. Parameters estimation of singleand multiplediode photovoltaic model using whale optimisation algorithm. IET Renew. Power Gener. 2018, 12, 1755–1761. [Google Scholar] [CrossRef]
 Shell PowerMax Solar Modules for OffGrids Markets. Available online: http://www.effectivesolar.com/PDF/shell/SQ8085P.pdf (accessed on 5 March 2021).
 Gao, X.; Cui, Y.; Hu, J.; Xu, G.; Wang, Z.; Qu, J.; Wang, H. Parameter extraction of solar cell models using improved shuffled complex evolution algorithm. Energy Convers. Manag. 2018, 157, 460–479. [Google Scholar] [CrossRef]
 Haynes, W. Wilcoxon rank sum test. In Encyclopedia of Systems Biology; Springer: New York, NY, USA, 2013; pp. 2354–2355. [Google Scholar]
Algorithm  Year  PV Model  Contributions and Limitations. 

Classified Perturbation Mutation Based PSO Algorithm (CPMPSO) [24]  2020  SDM, and DDM 

Enhanced Adaptive Differential Evolution [8]  2020  SDM, and DDM 

GOA [18]  2020  TDM 

Whale Optimization Algorithm (WOA) based Reflecting Learning (RLWOA) [25]  2020  SDM 

Improved equilibrium optimizer (IEO) [2].  2020  SDM, and DDM 

Improved Electromagnetismlike algorithm [26]  2020  SDM 

Grey Wolf Optimizer (GWO) And Cuckoo Search (CS): GWOCS [7]  2020  SDM, and DDM 

Boosted Harris Hawk’s Optimization (BHHO) [27]  2020  SDM 

FPA [28].  2020  DDM 

Camel behavior search algorithm (CBSA) [29].  2020  SDM 

Improved social spider algorithm [30]  2020  SDM, and DDM 

Improved TeachingLearningBased Optimization (ITLBO) [31]  2019  SDM, and DDM 

Chaotic JAYA (CJAYA) [32]  2021  SDM, and DDM 

Gradientbaed optimizer (GBO) [33].  2021  SDM, DDM, and TDM 

Improved levy flightbased grasshopper optimization algorithm [34]  2020  SDM, and DDM 

Enhanced teaching–learningbased optimization (ETLBO) [35].  2020  SDM, and DDM 

Slime mould algorithm (SMA) [36]  2020  SDM, and DDM 

Improved Artificial Bee Colony Algorithm (IABC) [37]  2020  SDM 

Chaotic optimization approach [38]  2019  SDM, and DDM 

GWO [39]  2019  SDM 

Parameter  L  U 

${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$  $0.9{I}_{SC}$  $1.1{I}_{SC}$ 
${\mathit{I}}_{\mathit{s}\mathit{d}\mathit{i}}\left(\mathit{A}\right),\mathit{i}\mathsf{\in}\mathbf{1}:\mathbf{3}$  $1\mathrm{n}A$  $10\mathsf{\mu}A$ 
${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$  $0$  $0.5$ 
${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$  $0$  $500$ 
$\mathit{a}\mathbf{1}$  $1$  $2$ 
$\mathit{a}\mathbf{2}$  $1.2$  $2$ 
$\mathit{a}\mathbf{3}$  $1.4$  $2$ 
${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$  ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$  ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$  $\mathit{a}1$  $\mathit{a}2$  $\mathit{a}3$ 

0.720205  3.87 × 10^{−7}  9.43 × 10^{−9}  1.49 × 10^{−8}  0.03571  69.93044  1.90020  1.29812  1.68252 
Output: return${X}^{*}$ 1. Input: N, ${t}_{max}$, and NCG 2. $t=0$ 3. RK: a vector of size N and initialized with 0’s value. 4. Initialize a population of N individuals using Equation (12) 5. While $t<{t}_{max}$ 6. For $i=1:N$ 7. Create two random numbers $\alpha $, ${\alpha}_{1}$ within [0, 1] 8. If $\alpha >{\alpha}_{1}$ 9. Calculate the mean of the population M using Equation (6) 10. Compute ${\mu}_{i},{\delta}_{i},and\eta $ 11. Compute ${T}_{i}{}^{t}$ using Equation (4). 12. If $f({T}_{i}{}^{t})<f\left({X}_{i}{}^{t}\right)$ 13. ${X}_{i}{}^{t}={T}_{i}{}^{t}$ 14. $R{K}_{i}=0;$ 15. Else 16. $R{K}_{i}++$ 17. End 18. Else 19. // global exploration 20. Compute ${T}_{i}{}^{t}$ according to Equation (9). 21. If $f({T}_{i}{}^{t})<f\left({X}_{i}{}^{t}\right)$ 22. ${X}_{i}{}^{t}={T}_{i}{}^{t}$ 23. $R{K}_{i}=0;$ 24. Else 25. $R{K}_{i}++$ 26. End 27. Applying the ranking method depicted in Figure 2 28. End 29. $t++$; 30. End 31. /// applying the premature convergence method. 32. Generate two random numbers ${\alpha}_{1}$ and ${\alpha}_{2}$ within [0, 1]. 33. If ${\alpha}_{1}<{\alpha}_{2}$ 34. For $i=1:N$ 35. Compute ${T}_{i}{}^{t}$ using Equation (16). 36. If $f({T}_{i}{}^{t})<f\left({X}_{i}{}^{t}\right)$ 37. ${X}_{i}{}^{t}={T}_{i}{}^{t}$ 38. $R{K}_{i}=0;$ 39. Else 40. $R{K}_{i}++$ 41. End 42. $t++$; 43. End 44. End 45. End 
Algorithms  ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$  ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$  ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$  $\mathit{a}1$  $\mathit{a}2$  $\mathit{a}3$  RMSE 

AEO [46]  0.760205  3.87 × 10^{−7}  9.43 × 10^{−9}  4.49 × 10^{−8}  0.0357  69.9304  1.5002  1.9981  1.8825  9.899220431 × 10^{−4} 
ITLBO [31]  0.760500  2.98 × 10^{−8}  9.17 × 10^{−7}  1.86 × 10^{−9}  0.0381  59.7254  1.3101  1.7186  1.6611  7.618033553 × 10^{−4} 
ISA [17]  0.760500  1.21 × 10^{−7}  1.00 × 10^{−9}  1.68 × 10^{−6}  0.0377  59.5672  1.3995  1.9936  2.0000  7.534445387 × 10^{−4} 
HHO [22]  0.759740  1.75 × 10^{−7}  2.77 × 10^{−7}  9.10 × 10^{−7}  0.0342  127.1454  1.4533  1.7284  1.8222  1.546454764 × 10^{−3} 
WOA [47]  0.760010  2.86 × 10^{−9}  6.62 × 10^{−7}  6.64 × 10^{−7}  0.0303  353.9084  1.5664  1.6037  1.6904  2.556963482 × 10^{−3} 
CPMPSO [24]  0.760500  9.62 × 10^{−8}  3.73 × 10^{−7}  1.67 × 10^{−6}  0.0379  61.1542  1.3812  1.9995  1.9993  7.508298630 × 10^{−4} 
GNDO [40]  0.760499  1.02 × 10^{−6}  4.43 × 10^{−7}  1.40 × 10^{−7}  0.0374  59.0192  1.9912  2.0000  1.4112  7.557191951 × 10^{−4} 
RGNDO  0.760500  9.08 × 10^{−8}  1.96 × 10^{−6}  1.58 × 10^{−7}  0.0380  61.3221  1.3766  2.0000  2.0000  7.506838880 × 10^{−4} 
Method  AEO [46]  ITLBO [31]  ISA [17]  HHO [22]  WOA [47]  CPMPSO [24]  GNDO [40]  RGNDO 

Best  9.899220 × 10^{−4}  7.618033 × 10^{−4}  7.534445 × 10^{−4}  1.546454 × 10^{−3}  2.556963 × 10^{−3}  7.508298 × 10^{−4}  7.557192 × 10^{−4}  7.506838 × 10^{−4} 
Worst  4.845654 × 10^{−3}  2.006802 × 10^{−3}  3.193321 × 10^{−3}  9.090638 × 10^{−3}  1.140435 × 10^{−2}  7.797626 × 10^{−4}  1.457815 × 10^{−3}  7.663392 × 10^{−4} 
Avg  2.480973 × 10^{−3}  1.001097 × 10^{−3}  1.568473 × 10^{−3}  6.079471 × 10^{−3}  8.282383 × 10^{−3}  7.622312 × 10^{−4}  8.259549 × 10^{−4}  7.529015 × 10^{−4} 
SD  9.316490 × 10^{−4}  3.767089 × 10^{−4}  6.760342 × 10^{−4}  2.146342 × 10^{−3}  2.002442 × 10^{−3}  8.744482 × 10^{−6}  1.434043 × 10^{−4}  3.933168 × 10^{−6} 
Rank  6  4  5  7  8  2  3  1 
Algorithms  ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$  ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$  ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$  $\mathit{a}1$  $\mathit{a}2$  $\mathit{a}3$  RMSE 

AEO [46]  8.1614  1.13 × 10^{−9}  2.42 × 10^{−8}  2.67 × 10^{−9}  0.0038  5.9997  1.7205  1.2159  1.7762  0.04384316 
ITLBO [31]  8.1037  9.29 × 10^{−9}  5.97 × 10^{−7}  6.13 × 10^{−7}  0.0040  352.8323  1.1612  1.9926  1.8737  0.04596226 
ISA [17]  8.1797  1.00 × 10^{−9}  1.19 × 10^{−9}  2.50 × 10^{−9}  0.0046  3.1251  1.0468  2.0000  1.6340  0.02897981 
HHO [22]  8.1384  9.00 × 10^{−8}  4.29 × 10^{−8}  1.00 × 10^{−9}  0.0033  23.2043  1.3046  1.5244  1.5825  0.05640261 
WOA [47]  8.1265  1.02 × 10^{−9}  3.47 × 10^{−6}  1.02 × 10^{−9}  0.0041  152.0232  1.0546  1.8552  1.4212  0.04680127 
CPMPSO [24]  8.1888  1.65 × 10^{−9}  1.49 × 10^{−9}  9.70 × 10^{−9}  0.0044  3.1390  1.0742  1.2009  1.9451  0.03042386 
GNDO [40]  8.2002  1.00 × 10^{−9}  1.00 × 10^{−9}  1.04 × 10^{−9}  0.0046  2.6505  1.0469  1.8270  1.6336  0.02822634 
RGNDO  8.2011  1.00 × 10^{−9}  1.00 × 10^{−9}  1.00 × 10^{−9}  0.0046  2.6410  1.0469  2.0000  2.0000  0.02821281 
Algorithms  AEO [46]  ITLBO [31]  ISA [17]  HHO [22]  WOA [47]  CPMPSO [24]  GNDO [40]  RGNDO 

Best  0.0438431608  0.0459622563  0.0289798147  0.0564026070  0.0468012666  0.0304238578  0.0282263443  0.0282128080 
Worst  0.0934460402  0.1163438794  0.0867319890  0.1359284618  0.2418484379  0.0683562899  0.0683562899  0.0683562899 
Avg  0.0654501748  0.0719622807  0.0581298639  0.1028267389  0.1369643198  0.0434628437  0.0422750429  0.0406449525 
SD  0.0100639011  0.0163810072  0.0113337942  0.0229987194  0.0421336396  0.0100268403  0.0119265384  0.0145287520 
Rank  5  6  4  7  8  3  2  1 
Algorithms  ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$  ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$  ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$  $\mathit{a}1$  $\mathit{a}2$  $\mathit{a}3$  RMSE 

AEO [46]  5.226139  2.95 × 10^{−6}  8.21 × 10^{−6}  6.50 × 10^{−6}  0.0112  3.9298  1.4669  1.7847  1.7682  2.455842651 × 10^{−3} 
ITLBO [31]  5.226022  8.88 × 10^{−6}  2.50 × 10^{−6}  1.00 × 10^{−5}  0.0112  3.9630  1.9046  1.4497  1.7691  2.431633915 × 10^{−3} 
ISA [17]  5.226719  3.45 × 10^{−6}  9.23 × 10^{−7}  9.28 × 10^{−6}  0.0111  3.8525  1.4903  1.6419  1.7129  2.497373210 × 10^{−3} 
HHO [22]  5.190855  5.21 × 10^{−6}  4.44 × 10^{−6}  3.45 × 10^{−6}  0.0113  7.5484  1.5177  1.7167  1.7359  1.076041865 × 10^{−2} 
WOA [47]  5.198240  4.20 × 10^{−6}  5.01 × 10^{−6}  3.79 × 10^{−7}  0.0116  5.9580  1.4955  1.6688  1.6537  1.032542474 × 10^{−2} 
CPMPSO [24]  5.225747  1.90 × 10^{−6}  9.98 × 10^{−6}  9.78 × 10^{−6}  0.0113  3.9926  1.4273  1.7946  1.8201  2.423466909 × 10^{−3} 
GNDO [40]  5.226051  1.00 × 10^{−5}  2.76 × 10^{−6}  9.91 × 10^{−6}  0.0112  3.9679  1.7967  1.4552  1.9194  2.428164856 × 10^{−3} 
RGNDO  5.225629  6.45 × 10^{−7}  1.00 × 10^{−5}  1.00 × 10^{−5}  0.0113  4.0252  1.3519  1.7529  1.7439  2.417084253 × 10^{−3} 
Algorithms  AEO [46]  ITLBO [31]  ISA [17]  HHO [22]  WOA [47]  CPMPSO [24]  GNDO [40]  RGNDO 

Best  0.002470471  0.002443520  0.002679316  0.019364346  0.010087377  0.002417985  0.002426150  0.002417084 
Worst  0.018785517  0.017193050  0.017503896  0.039575676  0.049733913  0.005058644  0.011573784  0.002492268 
Avg  0.004108074  0.003789465  0.007121044  0.027427524  0.027609976  0.002573152  0.002819667  0.002446177 
SD  0.003918587  0.003717768  0.004665595  0.005092069  0.008970604  0.000482496  0.001656137  0.000025994 
Rank  6  5  7  8  9  2  3  1 
Algorithms  ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}1}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}2}\left(\mathit{A}\right)$  ${\mathit{I}}_{\mathit{s}\mathit{d}3}\left(\mathit{A}\right)$  ${\mathit{R}}_{\mathit{s}}\left(\mathit{\Omega}\right)$  ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathit{\Omega}\right)$  $\mathit{a}1$  $\mathit{a}2$  $\mathit{a}3$  RMSE 

AEO [46]  7.475257  6.02 × 10^{−9}  1.85 × 10^{−6}  2.26 × 10^{−6}  0.004677  17.4376  1.9961  1.2418  1.7719  1.389396490646 × 10^{−2} 
ITLBO [31]  7.476115  1.90 × 10^{−6}  1.77 × 10^{−8}  1.00 × 10^{−9}  0.004694  15.1633  1.2437  1.3065  1.4249  1.379885388914 × 10^{−2} 
ISA [17]  7.476936  1.00 × 10^{−9}  1.88 × 10^{−6}  1.00 × 10^{−9}  0.004703  14.3643  1.9907  1.2424  1.5690  1.380086028210 × 10^{−2} 
HHO [22]  7.458183  2.18 × 10^{−6}  3.64 × 10^{−9}  2.56 × 10^{−9}  0.004653  248.4131  1.2545  1.2125  1.4412  1.424187705506 × 10^{−2} 
WOA [47]  7.464125  1.82 × 10^{−6}  1.62 × 10^{−6}  9.32 × 10^{−6}  0.004575  337.8192  1.9703  1.2357  1.7318  1.493293998738 × 10^{−2} 
CPMPSO [24]  7.476213  5.09 × 10^{−8}  1.88 × 10^{−6}  1.00 × 10^{−9}  0.004692  15.1426  1.2443  1.2443  2.0000  1.379827332710 × 10^{−2} 
GNDO [40]  7.476214  1.93 × 10^{−6}  1.01 × 10^{−9}  1.00 × 10^{−9}  0.004692  15.1424  1.2443  1.2442  2.0000  1.379827333205 × 10^{−2} 
RGNDO  7.476213  1.93 × 10^{−6}  1.02 × 10^{−9}  1.00 × 10^{−9}  0.004692  15.1427  1.2443  1.2443  2.0000  1.379827332701 × 10^{−2} 
Algorithms  AEO [46]  ITLBO [31]  ISA [17]  HHO [22]  WOA [47]  CPMPSO [24]  GNDO [40]  RGNDO 

Best  0.013893964  0.013798853  0.013800860  0.014241877  0.014932940  0.013798273  0.013798273  0.013798273 
Worst  0.028970100  0.014295495  0.023508622  0.049436644  0.141388822  0.014659372  0.014863306  0.013799111 
Avg  0.016038025  0.013925848  0.014629957  0.025279469  0.041237117  0.013899188  0.013882239  0.013798325 
SD  0.003578909  0.000126236  0.001770716  0.009375668  0.026679451  0.000211721  0.000224301  0.000000149 
Rank  6  4  5  7  8  3  2  1 
Algorithms  RTC France  KC200GT  Ultra 85P  STP6120/36  

h  pValue  h  pValue  h  pValue  h  pValue  
RGNDO vs. AEO  1  3.0199 × 10^{−11}  1  2.5473 × 10^{−12}  1  1.2057 × 10^{−10}  1  3.0199 × 10^{−11} 
RGNDO vs. ITLBO  1  4.5043 × 10^{−11}  1  2.6537 × 10^{−13}  1  5.0922 × 10^{−8}  1  3.3384 × 10^{−11} 
RGNDO vs. ISA  1  8.1527 × 10^{−11}  1  1.1737 × 10^{−9}  1  3.0199 × 10^{−11}  1  3.0199 × 10^{−11} 
RGNDO vs. HHO  1  3.0199 × 10^{−11}  1  1.6998 × 10^{−16}  1  3.0199 × 10^{−11}  1  3.0199 × 10^{−11} 
RGNDO vs. WOA  1  3.0199 × 10^{−11}  1  3.5254 × 10^{−17}  1  3.0199 × 10^{−11}  1  3.0199 × 10^{−11} 
RGNDO vs. CPMPSO  1  4.1178 × 10^{−6}  1  2.2893 × 10^{−4}  1  5.5611 × 10^{−4}  1  8.8411 × 10^{−7} 
RGNDO vs. GNDO  1  4.0772 × 10^{−11}  1  4.1782 × 10^{−3}  1  1.0907 × 10^{−5}  1  1.8916 × 10^{−4} 
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. 
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
AbdelBasset, M.; Mohamed, R.; ElFergany, A.; Abouhawwash, M.; Askar, S.S. Parameters Identification of PV TripleDiode Model Using Improved Generalized Normal Distribution Algorithm. Mathematics 2021, 9, 995. https://doi.org/10.3390/math9090995
AbdelBasset M, Mohamed R, ElFergany A, Abouhawwash M, Askar SS. Parameters Identification of PV TripleDiode Model Using Improved Generalized Normal Distribution Algorithm. Mathematics. 2021; 9(9):995. https://doi.org/10.3390/math9090995
Chicago/Turabian StyleAbdelBasset, Mohamed, Reda Mohamed, Attia ElFergany, Mohamed Abouhawwash, and S. S. Askar. 2021. "Parameters Identification of PV TripleDiode Model Using Improved Generalized Normal Distribution Algorithm" Mathematics 9, no. 9: 995. https://doi.org/10.3390/math9090995