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Global Energy Interconnection
Volume 8, Issue 2, Apr 2025, Pages 316-337
Efficient identification of photovoltaic cell parameters via Bayesian neural network-artificial ecosystem optimization algorithm☆
Abstract
Abstract Accurate identification of unknown internal parameters in photovoltaic (PV) cells is crucial and significantly affects the subsequent system-performance analysis and control.However, noise, insufficient data acquisition, and loss of recorded data can deteriorate the extraction accuracy of unknown parameters.Hence, this study proposes an intelligent parameter-identification strategy that integrates artificial ecosystem optimization(AEO)and a Bayesian neural network(BNN)for PV cell parameter extraction.A BNN is used for data preprocessing, including data denoising and prediction.Furthermore, the AEO algorithm is utilized to identify unknown parameters in the single-diode model (SDM), double-diode model (DDM), and three-diode model (TDM).Nine other metaheuristic algorithms(MhAs) are adopted for an unbiased and comprehensive validation.Simulation results show that BNN-based data preprocessing combined with effective MhAs significantly improve the parameter-extraction accuracy and stability compared with methods without data preprocessing.For instance, under denoised data, the accuracies of the SDM, DDM, and TDM increase by 99.69%, 99.70%, and 99.69%, respectively, whereas their accuracy improvements increase by 66.71%, 59.65%, and 70.36%, respectively.
0 Introduction
The widespread development and application of fossil fuels have become important factors in maintaining social operations over the recent centuries [1,2].Simultaneously,the energy-demand gap among countries is constantly expanding.Conventional power supplies rely on significant amounts of conventional fossil resources, which are both limited and nonrenewable [3,4].Additionally, excessive exploitation of fossil fuels has caused severe and even permanent damage to land,water sources,and ecosystems,thus resulting in numerous environmental issues with longterm and irreversible impacts [5,6].Hence, countries worldwide are actively promoting sustainable, diversified,and environmentally friendly renewable-energy development [7].
In this context, solar energy, which is one of the most representative clean and renewable energy sources, has been extensively utilized in various industries worldwide owing to its abundant reserve, low noise, and zero emission [8-10].During photovoltaic (PV) power generation,the output characteristics of PV models are related not only to the external temperature and solar irradiance but also to several critical operation parameters inside the PV cell [11-13].However, the internal operating parameters of PV models are not provided by the manufacturer and change over time.Therefore, these parameters inside a PV cell must be identified to achieve more accurate PV-cell modeling.Currently, three mathematical PV cell models are widely recognized and applied, namely, the single-diode model (SDM), double-diode model (DDM),and three-diode model (TDM).Because of their greater number of unknown parameters, the DDM and TDM are more complex and cost more for parameter identification as compared with the SDM; however, they provide more details regarding the operation characteristics of PV models [14,15].
As the parameter identification of PV cells is a nonlinear function-optimization problem with multiple variables and peaks,the optimal solutions cannot be easily obtained using analytical methods and conventional optimization algorithms.Nonetheless,metaheuristic algorithms(MhAs)can solve this problem owing to their advantages, such as high convergence speed and non-requirement for precise system models.As shown in [16-19], the parameters of various batteries are typically identified using MhAs, and some studies improved heuristic algorithms by adding certain mechanisms or strategies.In recent years, the parameter identification of PV models has been investigated extensively.For example, the authors of [20] proposed an iterative identification algorithm based on multiple innovation gradients that can identify unknown parameters easily and effectively.However, they did not fully investigate the effects of dataset noise and insufficient data on the accuracy of recognition results.The authors of[21]developed a PV model parameter-identification strategy based on a differential evolution algorithm that involves different evolutionary stages and exhibits favorable global optimizability and rapid convergence.However,this strategy requires optimization in terms of computational complexity and runtime.The authors of [22] designed an improved adaptive particle swarm optimization algorithm that balances the search relationship between global and local extrema by introducing asynchronous learning factors, thereby improving the accuracy of parameter identification.However, this algorithm may be limited when noisy data are involved.The authors of [23] proposed an MhA that combines data-prediction techniques and utilizes extreme-learning machines for data training, thus providing a more accurate fitness function.However,they did not consider the performance of the algorithm under noisy conditions.Meanwhile,the authors of[24]proposed a diversified Harris Hawk optimization strategy combined with a chaos drive that utilizes the unique drift of the chaotic local search mechanism to enhance the reinforcement ability of the original algorithm.However, the convergence speed of this strategy can be improved.The authors of [25] proposed an improved method based on a lion swarm optimization algorithm that combines tent chaotic mapping, adaptive parameters,and a chaotic search strategy.However, this method features high computational complexity and does not consider the effect of noise on the recognition results.The authors of [26] proposed an improved MhA based on sea spray swarm optimization for PV model-parameter identification.The entire search space was searched based on a randomly selected sea spray, and the search performance was improved using the optimal position of the sea spray.This improvement enhanced the exploratory ability of the algorithm while preventing premature convergence.Although this algorithm outperformed other algorithms in terms of convergence accuracy and stability, its performance in terms of computational complexity and runtime deteriorated.Meanwhile, the studies reported in [26-28] are similar in that they used mechanisms to improve the original algorithm, although at the expense of increased time cost.These algorithms outperform other algorithms in terms of convergence accuracy and stability while affording lower computational complexity and running times.
In summary, previous studies primarily used MhAs and their improved variants for PV cell parameter identification to improve the accuracy and speed of parameter identification.However, almost all previous studies have overlooked the important fact that not only is the external parameter-identification design important, but also the measurement dataset used directly affects the parameter-extraction performance.The dataset has some inherent defects, such as noisy data and insufficient data samples, which reduce the accuracy of the recognition results.Therefore, this study aims to address these issues and improve the accuracy and efficiency of parameter identification in PV models.Specifically, this study employs a Bayesian neural network (BNN) as a datapreprocessing tool and artificial ecosystem optimization(AEO) as an optimization algorithm for PV cell parameter extraction.The BNN can prevent overfitting, improve prediction flexibility, and provide uncertainty estimation,whereas the AEO offers high global-search ability, convergence speed, and adaptability.In detail, the BNN is applied in the first stage of data preprocessing to filter noisy data and enrich the dataset based on data prediction, which can compensate for the adverse effect on the identification results caused by insufficient datasets and noisy data.Subsequently, AEO is performed for unknown-parameter identification based on the processed datasets.The simulation results show that after data processing, both the identification accuracy and convergence improved.
The main contributions of this study are as follows:
1) Three commonly applied PV cell models and one PV module are established, and 10 different MhAs for parameter identification for all three models are discussed and compared.
2) The effects of insufficient data, data loss, and noisy data on the accuracy of PV cell parameter identification are investigated.Based on the data preprocessing of a BNN to reduce noise and predict datasets,more accurate output current-voltage (I-V) and power-voltage (P-V) characteristic curves are obtained, thereby improving the accuracy.
3) Based on a comparison of the simulation results obtained from 10 MhAs, the AEO outperformed the other MhAs under both denoised and predicted data.
The remainder of this paper is organized as follows:Section 1 introduces the PV modeling performed, which primarily includes four PV models and their objective functions; Section 2 introduces the principles of the BNN and AEO, as well as the parameter-identification framework using BNN-MhAs; Section 3 primarily analyzes the simulation results under the SDM, DDM, and TDM; Section 4 discusses the research presented in this article; and Section 5 concludes the paper.
1 PV cell modeling
1.1 Mathematical modeling
The most widely used PV cell models are the SDM,DDM, and TDM.In this study, a PV module model with 36 cells connected in series was constructed.Detailed descriptions of the abovementioned four models are presented in Table 1.
1.2 Objective function
To identify the parameters of the PV model, tone must determine an objective function to estimate the difference between the measured and calculated data.In this regard,the root mean square error(RMSE)is typically selected as the objective function primarily because of its significant advantages as an objective function in parameter identification, including its unit consistency, error sensitivity,and wide application.By contrast,the mean squared error(MSE) and mean absolute error (MAE) have certain limi-tations, such as the MSE being susceptible to outliers and the MAE being less sensitive to significant errors.The abovementioned indicators are expressed as follows [29]:
Table 1
Summary of SDM, DDM, TDM, and PV module model.
Model Model schematics Output I-V equation Extracted parameters SDM images/BZ_147_401_1668_994_1954.pngIL =Iph-Isd·[exp（q（VL+RSIL）α·k·T ）-1]-VL+RSILRsh Iph,Isd,RS,Rsh,α DDM images/BZ_147_401_1963_956_2309.pngIL =Iph-Isd1·[exp（q（V L+RSIL）α1·k·T ）-1]-Isd2·[exp（q（V L+RSIL）α2·k·T ）-1] -VL+RSIL Rsh Iph,Isd1,Isd2,RS,Rsh,α1,α2 TDM PV moduleimages/BZ_147_401_2391_1018_2673.pngimages/BZ_147_401_2796_1022_3059.pngIL =Iph-Isd1·[exp（q（V L+RSIL）α1·k·T ）-1]-Isd2·[exp（q（V L+RSIL）α2·k·T ）-1]-Isd3·[exp（q（V L+RSIL）α3·k·T ）-1]-V L+RSIL Rsh IL/Np=Iph-Isd·[exp（q（VL/NS+RSIL/Np）α·k·T ）-1]-VL/NS+RSIL/Np Rsh Iph,Isd1,Isd2,Isd3,RS,Rsh,α1,α2,α3 Iph,Isd,RS,Rsh,α
Table 2
Fitting results of various indicators.
Fitted value Metric Data N DN O P R2P RMSE 72.397383% 99.999837% 99.998944% 99.999835%MSE 72.397383% 99.999837% 99.998536% 99.999832%MAE 71.073809% 99.998551% 99.998518% 99.999758%R2I RMSE 71.524872% 99.999646% 99.998931% 99.998874%MSE 71.524872% 99.999646% 99.998476% 99.998872%MAE 69.865264% 99.997824% 99.998510% 99.998694%
where N is the number of experimental data points;x is the solution vector, and IL and V L represent the current and voltage measurement data of the PV model, respectively.
The overlap degrees of the I-V and P-V curves, i.e.,
and Rp2, respectively, are used as the evaluation criteria,which are calculated as follows:
where S is the number of input quantities;Iact and V act are the actual values of the output current and output voltage,respectively; ISC is the calculated value of the output current; and Iact is the actual average value of the output current.
Considering the SDM as an example, the MAE, MSE,and RMSE were used as indicators for parameter identification, and the obtained R2P and R2I are listed in Table 2.The symbols‘‘N”and‘‘DN”represent the result obtained using noised and denoised data, respectively.Meanwhile,the symbols ‘‘O” and ‘‘P” represent the original and predicted data,respectively.The results shows that the RMSE has the best-fitting effect for various data types.
The objective functions of the three PV models can be expressed as follows [30]:
1) SDM
2) DDM
Fig.1.BNN structure diagram.
3) TDM
4) PV module model
2 PV cell parameter-identification design
2.1 Principle of BNN
The BNN considers weights to follow the mean value μ and variance δ.The Gaussian distribution follows a different distribution for each weight, and BNNs optimize the mean and variance of the weights.A diagram of the BNN structure is shown in Fig.1.
In Bayesian methods,model parameters ω use the probability distribution description.First,based on prior experience, a prior distribution p（ω） is introduced to obtain possible ω values.Assuming that the experimental dataset D = {(x1, t1), (x2, t2),..., (xn, tn)} is known and applying the Bayesian theorem, the prior distribution p（ω） is updated based on [31]:
Fig.2.Result of BNN data prediction: (a) 33 
C; (b) 45 
C.
where xn and tn (n = 1,2,3,..., n)are the input and output data, respectively; n is the number of experimental data points; p（D|ω） is a likelihood function that contains parameters obtained from observations; p（ω|D） is the probability distribution of the posterior parameter ω of known data D, also known as the posterior distribution;p（D）is the distribution of experimental data,which serves as a normalized constant, thus ensuring that the posterior distribution is an effective probability density with a full space integral of 1.
The likelihood function p（D|ω） typically reflects the Gaussian distribution, i.e.,
Fig.3.Result of BNN data denoising: (a) 33 
C; (b) 45 
C.
where Δtn is the noise error related to the nth data; ω denotes all the parameters {α，bj，cj，dij}, which are the weights and deviations of the hidden layer,and the weights and deviations of the output layer, respectively; H is the number of hidden layer neurons; and I is the number of input quantities [32].
2.2 BNN for I-V data preprocessing
The BNN constructed in this study is based on[32],and the code required approximately 3 s to process.Owing to their complex operating environments, PV models are easily affected by noise.This type of noise can affect the parameter identification of PV models, thus resulting in inaccurate final identification results.This study adopts the BNN, which further associates and reduces noise in the data.The results of BNN denoising are shown in Fig.2.
In fact, the amount of data can affect the accuracy of parameter identification.However,the current and voltage data can be insufficient and difficult to obtain.Therefore,this study trains the existing I-V data using a BNN and then performs data prediction.The prediction results obtained using the BNN are shown in Fig.3.
2.3 Principle of AEO
AEO is an emerging metaheuristic optimization algorithm that replicates the energy flow in the Earth’s ecosystem.The algorithm simulates the production,consumption, and decomposition behaviors in an ecosystem using production, consumption, and decomposition operators to solve optimization problems [33].
Fig.4.Schematic diagram of PV parameter identification based on BNN-MhAs.
Fig.5.Intelligent parameter identification of PV cell based on BNN-AEO.
1) Production
In the AEO algorithm,the producer in the population is updated by searching for the upper and lower bounds of the space and the decomposer.The mathematical model for simulating producer behavior is established as follows:
where n is the number of individuals in a population; T is the maximum number of iterations;L and U are the lower and upper limits, respectively; r1 is a random number within the interval of [0,1]; r is a random vector within the interval of [0,1]; a is the linear weight coefficient; and xrand is the random position of an individual in the search space.
Table 3
Identification steps of BNN-AEO for PV parameter extraction.
1 Determine PV model;2 Collect measured I-V data of PV;3 Process measured I-V data by BNN;4 Initialize population and parameters of AEO;5 Set t = 0;6 WHILE t ≤t max 7 FOR1 i=1:n 8 Compute the fitness value of the ith individual using Eq.(1);9 END FOR1 10 Adjust the roles of all individuals based on their fitness values;11 FOR2 i=1:n 12 Update the solution of the ith individual based on its searching rule;13 END FOR2 14 Set t = t + 1;15 END WHILE 16 Output optimal parameters for PV.
2) Consumption
After producers provide food energy, they can randomly select consumers with lower energy levels, producers, or both to obtain food energy.To facilitate this, a simple,parameter-less random walk with Levy flight characteristics is adopted, which is directly adjusted via the consumption factor, which is expressed as
where N represents a normal distribution with a mean of 0 and a standard deviation of 1.
The consumption behavior of herbivores is expressed mathematically as follows:
Table 4
V-I data under variable temperature.
1000 W/m2, 33 images/BZ_152_407_1493_420_1527.pngC 1000 W/m2, 45 images/BZ_152_1025_1493_1038_1527.pngC V/V I/A V/V I/A-0.2057 0.764 0.1248 1.0315-0.1291 0.762 1.8093 1.03-0.0588 0.7605 3.3511 1.026 0.0057 0.7605 4.7622 1.022 0.0646 0.76 6.0538 1.018 0.1185 0.759 7.2364 1.0155 0.1678 0.757 8.3189 1.014 0.2132 0.757 9.3097 1.01 0.2545 0.7555 10.2163 1.0035 0.2924 0.754 11.0449 0.988 0.3269 0.7505 11.8018 0.963 0.3585 0.7465 12.4929 0.9255 0.3873 0.7385 13.1231 0.8725 0.4137 0.728 13.6983 0.8075 0.4373 0.7065 14.2221 0.7265 0.459 0.6755 14.6995 0.6345 0.4784 0.632 15.1346 0.5345 0.496 0.573 15.5311 0.4275 0.5119 0.499 15.8929 0.3185 0.5265 0.413 16.2229 0.2085 0.5398 0.3165 16.5241 0.101 0.5521 0.212 16.7987 -0.008 0.5633 0.1035 17.0499 -0.111 0.5736 -0.01 17.2793 -0.209 0.5833 -0.123 17.4885 -0.303 0.59 -0.21
Table 5
Specifications of RTC France and PWP-201.
PV module Datasheet values Pmax/W Vmpp/V Impp/A Voc/V Isc/A RTC France 0.7603 0.5728 0.6894 0.4507 0.3107 PWP-201 1.03163 16.7753 0.9162 12.6049 11.55
The consumption behavior of carnivores is expressed mathematically as follows:
The consumption behavior of omnivores is expressed mathematically as follows:
where r2 is a random number within the interval of [0,1].
3) Decomposition
Decomposers provide the necessary nutrients to the producers.The AEO algorithm allows the next position of each individual to propagate around the decomposer.The mathematical model for simulating the decomposition behavior is expressed as
Table 6
Main parameters of each MhA.
MhAs Parameter Value AEO C (Speed factor) 0.5 BOA P (Transition probability) 0.8 BWO P (Random dimension) 0.8 ILA n (Model quantity) 2 m1 (Percentage of Iterations in Phase 1) 0.33 m2 (Percentage of Iterations in Phase 1) 0.33 Bmin (Boundary minimum value) 0.4 Bmax (Boundary maximum value) 0.6 MVO Wmin (Maximum probability) 1 Wmax (Minimum probability) 2 PSO c1(Self-learning factor) 2 c2(Group learning factor) 2 w (Inertia weight) 0.6 SO c1(Constant) 0.5 c2(Constant) 0.5 c3(Constant) 2
Table 7
PV model operation and MhA parameter settings.
Types Parameter Value SDM, DDM, TDM Irradiance 1000 W/m2 Temperature 33 images/BZ_152_2206_2872_2219_2905.pngC PV module Irradiance 1000 W/m2 Temperature 45 images/BZ_152_2206_2955_2219_2988.pngC MhA Maximum iterations 500 Population size 30, 50, 70, 30 Run times 10
Table 8
Parameter-identification results of noisy and denoised data based on 10 MhAs for SDM.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A Rs/Ω Rsh/Ω α SDM AEO N 0.648 1.000E-06 0.000 100.000 1.615 1.858E-01 7.251 DN 0.760 3.204E-07 0.036 54.055 1.480 6.231E-04 7.169 BOA N 0.581 3.641E-07 0.021 67.234 1.494 2.184E-01 7.373 DN 0.818 1.655E-06 0.040 37.385 1.653 4.355E-02 7.297 BWO N 0.694 5.417E-07 0.000 29.869 1.539 1.918E-01 4.336 DN 0.784 1.000E-06 0.034 50.891 1.610 3.260E-02 4.323 ILA N 0.648 1.000E-06 0.000 100.000 1.614 1.858E-01 7.229 DN 0.762 4.489E-07 0.034 44.383 1.515 1.462E-03 7.120 MGO N 0.648 1.000E-06 0.000 100.000 1.615 1.858E-01 13.605 DN 0.760 8.155E-07 0.032 99.996 1.580 1.932E-03 13.679 PSO N 0.648 1.000E-06 0.000 100.000 1.614 1.858E-01 6.687 DN 0.760 8.175E-07 0.032 97.582 1.580 1.943E-03 6.746 SABO N 0.664 1.000E-06 0.000 40.628 1.611 1.865E-01 2.780 DN 0.777 1.000E-06 0.040 14.300 1.613 2.912E-02 2.778 SO N 0.648 1.000E-06 0.000 100.000 1.615 1.858E-01 3.477 DN 0.759 7.809E-07 0.032 100.000 1.577 1.985E-03 3.440 COA N 0.648 1.000E-06 0.000 100.000 1.615 1.858E-01 5.732 DN 0.760 9.720E-07 0.031 99.458 1.601 2.298E-03 5.878 ISSA N 0.643 2.329E-07 0.000 39.409 1.460 1.901E-01 3.838 DN 0.762 4.580E-07 0.034 41.487 1.517 1.648E-03 3.889
where r3 is a random number within the interval[0,1],De is the decomposition factor, and e and h are the weight coefficients.
Fig.6.Convergence curves of RMSEs obtained by 10 MhAs for SDM under noisy and denoised data: (a) noisy data; (b) denoised data.
2.4 Parameter-identification process
The parameter-identification process for PV models based on BNN-MhAs proposed in this study primarily includes three components: data acquisition, data preprocessing, and optimal parameter extraction, as shown in Fig.4.
Parameter identification is implemented as follows.First, the current and voltage data of the PV cell are obtained,based on which the BNN model is trained.Subsequently, the obtained data are denoised and predicted during data preprocessing to obtain more accurate and enriched datasets.Finally, MhAs are used to extract the optimal parameters based on the processed datasets.The specific steps are presented in Fig.5 and Table 3.
Fig.7.Boxplot of RMSEs obtained by 10 MhAs for SDM under noisy and denoised data.
Fig.8.AEO for fitting curves based on denoised data of SDM: (a) I-V curve; (b) P-V curve.
Fig.9.Convergence curves of RMSEs obtained by 10 MhAs for SDM under original and predicted data: (a) Predicted data; (b) original data.
Table 9
Parameter-identification results of original and predicted data based on 10 MhAs for SDM.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A Rs/Ω Rsh/Ω α SDM AEO O 0.760 3.221E-07 0.036 53.845 1.480 9.861E-04 7.252 P 0.760 3.344E-07 0.036 56.373 1.484 6.181E-04 16.757 BOA O 0.687 3.082E-08 0.025 71.264 1.290 5.928E-02 7.421 P 0.719 2.748E-07 0.022 65.614 1.469 3.696E-02 16.737 BWO O 0.748 7.044E-07 0.000 11.958 1.573 4.502E-02 4.346 P 0.760 6.308E-07 0.000 59.302 1.558 3.400E-02 9.852 ILA O 0.760 5.888E-07 0.033 56.364 1.544 1.817E-03 7.215 P 0.760 3.956E-07 0.035 62.937 1.501 7.034E-04 16.892 MGO O 0.760 8.390E-07 0.032 99.998 1.584 2.104E-03 14.208 P 0.760 8.315E-07 0.031 83.242 1.583 1.500E-03 33.946 PSO O 0.758 3.029E-07 0.037 100.000 1.474 1.735E-03 6.766 P 0.760 3.844E-07 0.035 56.864 1.498 6.806E-04 15.802 SABO O 0.720 9.774E-07 0.035 100.000 1.601 4.417E-02 2.709 P 0.777 7.180E-07 0.041 70.185 1.567 1.691E-02 6.476 SO O 0.761 1.642E-07 0.038 39.438 1.416 1.636E-03 3.447 P 0.761 7.183E-07 0.033 62.692 1.567 1.374E-03 7.924 COA O 0.760 1.447E-07 0.039 51.375 1.404 2.156E-03 5.922 P 0.760 9.068E-07 0.031 100.000 1.592 1.641E-03 13.546 ISSA O 0.766 6.836E-07 0.031 25.993 1.562 4.512E-03 3.878 P 0.760 2.504E-07 0.037 65.987 1.455 1.051E-03 8.523
Fig.10.Boxplot of RMSEs obtained by 10 MhAs for SDM under original and predicted data.
2.5 Comparative time-complexity analysis
When analyzing the time complexity of the BNN-AEO,tone must consider the costs of BNN training, algorithm initialization, and the main loop.The training cost of the BNN is high and is expressed as O(E*P*N^2), where E is the number of training epochs, P is the number of parameters, and N is the number of neurons per layer,which depends on the network structure and amount of training data.The initialization phase of the algorithm involves randomly selecting individuals and evaluating the objective function with time complexities of O(n*d)and O(n*C_obj), respectively.Here, n is the population size, d the dimension, and C_obj the cost of the objective function.The main loop is based on the maximum number of function evaluations.Therefore,the total time complexity of the BNN-AEO is approximately O(Max_Sun-Eval*(n*d + n*C_obj) + E*P*N^2).
3 Case studies
This section considers several conventional models such as the SDM, DDM, and TDM on commercial R.T.C.France silicon solar cells, as well as the PV module Photowatt-PWP201 with 36 polycrystalline silicon cells in a series system, were validated using two datasets.The specific data are listed in Table 4.Table 5 lists the prototype parameter values of the two PV models considered in this study.
Subsequently, the BNN and 10 MhAs including AEO,butterfly optimization algorithm(BOA)[34],beluga whale optimization (BWO) [35], intelligible-in-time logics algorithm (ILA) [36], mountain gazelle optimizer (MGO)[37], particle swarm optimization (PSO) [38], subtraction average based optimizer (SABO) [39], snake optimizer(SO) [40], crayfish optimization algorithm (COA) [41],improved salp swarm algorithm (ISSA) were used to extract the parameters of four PV cell models.The main parameters of each MhA are listed in Table 6.Before MhA-based parameter identification was performed, the trained BNN was used to predict, denoise, and add noise to the obtained data.Using the prediction, noisy,denoised, and original data, 10 MhAs were employed to identify the parameters of the four different PV models.Table 7 presents the operating conditions of the PV model.
3.1 SDM parameter extraction
3.1.1 Noise-reduction data
Table 8 summarizes the recognition results of the PV cell parameters obtained using 10 MhAs with different training data.The symbols ‘‘N” and ‘‘DN” represent theresult obtained using noised and denoised data, respectively.As shown in Table 8, after data denoising, the RMSE values of the MhAs were smaller than those under noisy data, among which AEO experienced the most significant decrease of 99.69%.By contrast, the decrease shown by the BOA was the least, (80.07%), followed by BWO (83.00%), the ILA (99.21%), the MGO (98.96%),PSO (98.95%), the SABO (84.39%), the SO (98.93%), the COA (98.76%), and the ISSA (99.13%).
Table 10
Parameter-identification results of noisy and denoised data based on 10 MhAs for DDM.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A ISD2/A Rs/Ω Rsh/Ω α1 α2 DDM AEO N 0.653 1.000E-06 1.000E-06 0.000 100.000 1.700 1.700 1.841E-01 12.406 DN 0.760 2.762E-07 2.726E-07 0.036 52.968 1.467 2.000 5.608E-04 12.531 BOA N 0.774 2.720E-07 8.895E-07 0.000 26.977 1.598 1.638 2.162E-01 11.751 DN 0.673 9.294E-08 5.972E-07 0.025 55.926 1.513 1.596 7.043E-02 11.549 BWO N 0.630 1.000E-06 1.000E-06 0.000 100.000 1.635 2.000 1.869E-01 6.598 DN 0.753 1.000E-06 1.000E-06 0.000 5.467 1.678 1.817 4.967E-02 6.535 ILA N 0.651 9.607E-07 3.755E-07 0.000 100.000 1.617 1.839 1.855E-01 12.699 DN 0.761 2.461E-07 2.315E-07 0.034 43.365 1.476 1.620 1.568E-03 12.535 MGO N 0.653 1.000E-06 1.000E-06 0.000 100.000 1.700 1.700 1.841E-01 24.897 DN 0.760 6.731E-07 4.187E-12 0.033 86.250 1.560 1.000 1.530E-03 23.342 PSO N 0.650 1.000E-06 1.000E-06 0.000 100.000 1.623 2.000 1.852E-01 11.202 DN 0.761 7.075E-07 2.819E-07 0.031 64.554 1.624 1.564 2.558E-03 11.180 SABO N 0.654 6.752E-07 1.000E-06 0.000 78.737 1.630 1.721 1.849E-01 4.736 DN 0.744 8.744E-07 3.488E-07 0.024 100.000 1.596 2.000 1.571E-02 4.882 SO N 0.645 9.378E-07 9.883E-07 0.000 100.000 1.711 1.672 1.842E-01 5.631 DN 0.760 7.509E-07 3.031E-07 0.033 82.345 1.600 1.542 2.570E-03 5.575 COA N 0.649 9.969E-07 1.573E-07 0.000 99.847 1.626 1.678 1.855E-01 9.295 DN 0.761 2.277E-07 5.984E-09 0.037 42.378 1.446 1.949 9.075E-04 9.279 ISSA N 0.651 1.000E-06 4.002E-08 0.000 45.955 1.616 1.996 1.860E-01 5.965 DN 0.762 2.448E-07 4.700E-07 0.034 39.869 2.000 1.522 2.052E-03 5.917
Fig.11.Convergence curves of RMSEs obtained by 10 MhAs for DDM under noisy and denoised data: (a) noisy data; (b) denoised data.
To comprehensively present the convergence performance throughout the iterations of all the algorithms,Fig.6 presents the RMSE convergence curves.The results obtained based on denoised data show higher convergence accuracy, stability, and speed than those obtained using noisy data.
To intuitively compare the optimization accuracy of all methods based on the result distributions between two different training datasets,the RMSE distribution map for 10 iterations was plotted in a boxplot graph, as shown in Fig.7.As shown, after data denoising, all the MhAs showed a smaller distribution range in terms of the lower and upper limits, and the outliers were either reduced or eliminated.This proves that all 10 MhAs exhibited higher optimization accuracy and stability in terms of parameter identification after data denoising.
Fig.8 shows the I-V and P-V characteristic curves obtained by fitting the MhA with the best iterative performance among the 10 MhAs under denoised data.The fitted data curve almost matched the actual data curve almost perfectly.The overlap degree of the I-V curve 
was 99.9996%, and that of the P-V curve R2P was 99.9998%, thus indicating that the parameteridentification results are consistent with expectations.
3.1.2 Insufficient data
Fig.12.Boxplot of RMSEs obtained by 10 MhAs for DDM under noisy and denoised data.
Fig.13.AEO for fitting curves based on denoised data of DDM:(a) I-V curve; (b) P-V curve.
Table 11
Parameter-identification results of original and predicted data based on 10 MhAs for DDM.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A ISD2/A Rs/Ω Rsh/Ω α1 α2 DDM AEO O 0.760 3.230E-07 0.000E+00 0.036 53.718 1.481 2.000 9.860E-04 12.342 P 0.760 3.344E-07 0.000E+00 0.036 56.373 1.484 1.878 6.181E-04 28.088 BOA O 0.729 8.867E-07 7.697E-07 0.049 86.230 1.627 1.737 5.882E-02 11.469 P 0.771 4.190E-08 8.662E-07 0.030 43.373 1.431 1.639 3.082E-02 26.692 BWO O 0.778 1.000E-06 1.000E-06 0.000 6.253 2.000 1.631 3.703E-02 6.633 P 0.749 1.000E-06 0.000E+00 0.018 34.817 1.604 2.000 1.494E-02 16.881 ILA O 0.762 0.000E+00 4.805E-07 0.034 64.455 1.291 1.522 1.699E-03 12.727 P 0.761 2.872E-07 3.567E-07 0.034 73.937 1.656 1.512 1.232E-03 28.496 MGO O 0.760 5.339E-10 8.079E-07 0.033 92.067 1.166 1.590 1.789E-03 23.397 P 0.760 8.508E-12 6.909E-07 0.033 74.340 1.000 1.565 1.189E-03 56.660 PSO O 0.761 7.449E-07 8.325E-07 0.029 99.721 1.610 1.740 3.261E-03 11.290 P 0.760 0.000E+00 1.000E-06 0.030 97.823 2.000 1.604 1.767E-03 26.157 SABO O 0.740 1.000E-06 9.479E-07 0.031 53.688 1.708 1.687 2.654E-02 4.786 P 0.771 5.948E-07 7.126E-07 0.047 79.957 1.585 1.689 2.575E-02 11.203 SO O 0.760 6.838E-07 0.000E+00 0.033 100.000 1.560 2.000 1.767E-03 5.616 P 0.760 1.000E-06 1.526E-07 0.035 70.660 1.775 1.438 1.121E-03 12.926 COA O 0.761 7.989E-07 3.863E-07 0.031 75.079 1.582 1.958 2.437E-03 9.374 P 0.760 1.790E-07 5.976E-07 0.032 68.930 1.846 1.549 1.237E-03 21.860 ISSA O 0.761 5.092E-07 2.768E-07 0.033 52.224 1.532 1.907 1.832E-03 5.942 P 0.760 7.468E-08 7.587E-07 0.032 90.627 1.724 1.575 1.479E-03 13.361
Table 9 summarizes the identification results of the PV cell parameters obtained using 10 MhAs with different training data.The symbols ‘‘O” and ‘‘P” represent the original and predicted data, respectively.Among them,the ISSA indicated the highest decrease of 76.71%,whereas the SO indicated the lowest decrease of 15.96%.Meanwhile, the decreases for the other eight MhAs were as follows: AEO, 37.31%; BOA, 37.65%; BWO, 24.48%;ILA, 61.30%; MGO, 28.71%; PSO, 60.79%; SABO,61.71%; and COA, 23.86%.
Fig.14.Convergence curves of RMSEs obtained by 10 MhAs for DDM under original and predicted data: (a) predicted data; (b) original data.
Based on Fig.9, the predicted data showed smaller errors than the original data.
As shown in Fig.10,except for the ILA,the upper and lower bounds as well as the median values of RMSE obtained based on the predicted data decreased compared with those under the original data.
3.2 DDM parameter extraction
3.2.1 Noise-reduction data
As shown in Table 10,the decrease shown by AEO was the highest, i.e., 99.70%, whereas that indicated by the BOA was the lowest, i.e., 67.43%.Meanwhile, the decreases indicated by the other eight MhAs were as follows: BWO, 73.43%; ILA, 99.15%; MGO, 99.17%; PSO,98.62%; SABO, 91.50%; SO, 98.60%; COA, 99.51%; and ISSA, y 98.90%.
Fig.15.Boxplot of RMSEs obtained by10n MhAs for DDM under original and predicted data.
Table 12
Parameter-identification results of noisy and denoised data based on 10 MhAs for TDM.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A ISD2/A ISD3/A Rs/Ω Rsh/Ω α1 α2 α3 TDM AEO N 0.653 0.000E+00 1.000E-06 1.000E-06 0.000 100.000 1.000 1.700 1.700 1.841E-01 16.134 DN 0.760 0.000E+00 1.390E-07 2.989E-07 0.036 54.407 2.000 2.000 1.474 5.646E-04 17.626 BOA N 0.632 6.916E-07 1.329E-06 9.0102-07 0.000 86.280 1.701 1.787 1.741 1.860E-01 17.2771 DN 0.778 4.117E-07 1.300E-06 8.575E-07 0.037 52.909 1.747 1.699 1.744 2.113E-02 17.1357 BWO N 0.653 1.000E-06 1.000E-06 9.986E-07 0.000 94.021 1.947 2.000 1.636 1.845E-01 9.1392 DN 0.744 1.000E-06 9.158E-07 1.000E-06 0.029 48.206 2.000 1.996 1.629 1.908E-02 9.222 ILA N 0.656 5.071E-07 9.807E-07 9.778E-07 0.000 91.643 1.697 1.756 1.719 1.837E-01 17.388 DN 0.762 3.405E-07 3.769E-07 2.618E-07 0.035 58.458 1.831 1.979 1.469 1.579E-03 17.782 MGO N 0.656 1.000E-06 1.000E-06 1.000E-06 0.000 100.000 1.755 1.755 1.755 1.833E-01 35.984 DN 0.760 3.015E-07 1.122E-08 9.999E-07 0.039 58.415 1.757 1.240 1.753 6.356E-04 34.640 PSO N 0.653 1.000E-06 0.000E+00 1.000E-06 0.000 100.000 1.699 2.000 1.702 1.841E-01 15.351 DN 0.761 6.894E-07 6.988E-07 1.156E-07 0.030 100.000 1.593 1.771 1.726 2.946E-03 15.4369 SABO N 0.653 9.889E-07 1.000E-06 1.000E-06 0.000 99.818 1.758 1.752 1.756 1.833E-01 6.723 DN 0.785 3.718E-07 5.895E-07 6.054E-07 0.038 38.229 1.678 1.666 1.642 2.089E-02 6.715 SO N 0.627 7.458E-07 1.000E-06 7.830E-07 0.000 100.000 1.704 1.803 1.723 1.814E-01 7.8044 DN 0.761 7.766E-07 0.000E+00 6.797E-07 0.029 100.000 1.698 1.670 1.613 3.161E-03 7.729 COA N 0.656 9.999E-07 9.999E-07 9.998E-07 0.000 99.9680 1.754 1.755 1.755 1.833E-01 12.871 DN 0.760 6.811E-07 1.210E-08 7.017E-07 0.029 92.876 1.595 1.801 1.729 3.049E-03 12.921 ISSA N 0.653 9.710E-07 7.971E-08 8.696E-07 0.000 90.3600 1.693 1.796 1.692 1.843E-01 8.116 DN 0.765 3.444E-07 4.923E-07 1.621E-07 0.035 25.913 1.990 1.825 1.427 3.291E-03 7.673
Fig.16.Convergence curves of RMSEs obtained by 10 MhAs for TDM under noisy and denoised data: (a) noisy data; (b) denoised data.
Fig.17.Boxplot of RMSEs obtained by 10 MhAs for TDM under noisy and denoised data.
Fig.11 shows that the RMSE based on the denoised data for 10 MhAs was smaller than that based on noisy data, although the curve convergence based on noisy data was faster.For example, the ILA converged in approximately 50 and 400 iterations under noisy and denoised data, respectively.
To obtain a clear comparison between the two different training datasets, the RMSE values for 10 iterations were plotted in a boxplot graph and the RMSE distribution map was obtained, as shown in Fig.12.The result shows that after data denoising, the upper and lower bounds as well as the median values of the RMSE for each MhA in the boxplot decreased significantly,thus proving that data denoising can improve the stability of MhAs in PV-cell DDM parameter recognition.
Based on Fig.13, the R2 I and R2P were 99.9997% and 99.9998%, respectively, thus indicating that the PV-cell DDM parameter-identification results satisfied the expectations.
Fig.18.AEO for fitting curves based on denoised data of TDM: (a) I-V curve (b) P-V curve.
3.2.2 Insufficient data
Fig.19.Convergence curves of RMSEs obtained by 10 MhAs for TDM under original data and predicted data: (a) predicted data; (b) original data.
Based on Table 11, BWO and BOA showed significant decreases; for example, BWO indicated the highest decrease of 59.65%, whereas SABO showed the lowest decrease of 2.96%.Meanwhile, the decreases shown by the other eight MhAs were as follows: AEO, 37.31%;BOA,47.60%;ILA,27.44%;MGO,33.52%;PSO,45.80%;SO, 36.57%; COA, 49.26%; and ISSA, 19.26%.
Table 13
Parameter-identification results of original and predicted data based on 10 MhAs for TDM.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A ISD2/A ISD3/A Rs/Ω Rsh/Ω α1 α2 α3 TDM AEO O 0.7599 8.4008E-11 0.0000E+00 1.0000E-06 0.0402 100.0000 1.0000 2.0000 1.6474 1.3757E-03 17.8341 P 0.7607 3.0790E-07 2.5168E-08 1.7873E-07 0.0363 56.5568 1.4778 1.9425 1.9955 6.0939E-04 39.1963 BOA O 0.7776 2.9042E-07 1.4524E-07 6.6232E-08 0.0451 14.7681 1.6443 1.5733 1.3916 2.8635E-02 18.1399 P 0.7652 4.8583E-07 2.8088E-08 3.2202E-07 0.0466 35.3181 1.5644 1.5886 1.6620 1.8721E-02 38.7092 BWO O 0.7225 0.0000E+00 9.8690E-07 0.0000E+00 0.0281 100.0000 2.0000 1.6092 2.0000 3.0360E-02 9.1846 P 0.7525 1.0000E-06 1.0000E-06 1.0000E-06 0.0215 100.0000 1.8713 1.9238 1.6392 8.9974E-03 21.0562 ILA O 0.7612 3.7901E-07 5.3462E-07 4.1114E-08 0.0356 68.1561 1.5288 1.9752 1.4290 1.9061E-03 18.3381 P 0.7625 2.9992E-07 6.9795E-07 5.5569E-07 0.0301 38.5255 1.5425 2.0000 1.6365 1.0790E-03 41.0415 MGO O 0.7606 4.6406E-07 2.9066E-09 1.1465E-11 0.0353 65.6720 1.5211 1.5780 1.0335 1.1757E-03 34.7492 P 0.7606 1.4029E-08 6.5618E-10 5.6943E-07 0.0348 65.2518 1.3526 1.2738 1.5650 8.4395E-04 77.7096 PSO O 0.7623 7.0856E-07 1.0257E-10 1.0257E-10 0.0325 46.9458 1.5654 1.9571 2.0000 2.4194E-03 15.5277 P 0.7606 0.0000E+00 3.2585E-07 9.7142E-07 0.0304 91.9058 2.0000 2.0000 1.6038 2.0232E-03 36.8249 SABO O 0.7832 9.9222E-07 1.0000E-06 7.3938E-07 0.0253 76.4142 1.6916 1.8599 1.7005 2.1420E-02 6.7815 P 0.7755 8.2211E-07 9.5508E-07 6.4100E-07 0.0335 36.1052 1.6465 1.7562 1.8627 3.0211E-02 16.0889 SO O 0.7603 0.0000E+00 2.9016E-07 2.9274E-07 0.0352 100.0000 1.8353 1.6970 1.4856 1.7645E-03 7.7318 P 0.7598 3.3174E-08 0.0000E+00 4.8143E-07 0.0349 100.0000 1.8174 1.7005 1.5229 1.2377E-03 18.2975 COA O 0.7624 7.6211E-07 1.3676E-07 4.5341E-08 0.0355 42.9115 1.8012 1.4594 1.4097 1.6660E-03 13.0061 P 0.7599 2.4555E-07 1.1510E-07 5.8732E-07 0.0324 82.6409 2.0000 1.7699 1.5495 1.4052E-03 29.2061 ISSA O 0.7636 7.4834E-07 6.5725E-07 5.6216E-07 0.0298 47.8516 1.5890 1.8675 1.9858 3.8309E-03 7.9199 P 0.7611 5.3803E-07 8.1735E-07 1.6849E-07 0.0297 69.4257 1.8522 1.5942 1.9339 2.1748E-03 17.5377
Fig.20.Boxplot of RMSEs obtained by 10 MhAs for TDM under original and predicted data.
Based on Fig.14,after data prediction,the RMSE of all the algorithms decreased.However,the convergence speed of the curve between the two scenarios differed minimally.Based on observation, the data processing improved only the accuracy.
Based on Fig.15, except for the abnormal upper and lower bounds of the RMSE in the BOA,the upper,lower,and median RMSE obtained from the other MhAs based on the predicted data decreased.
3.3 TDM parameter extraction
3.3.1 Noise-reduction data
Table 12 presents the parameter-recognition results of the TDM using 10 MhAs with different training data.Among them, AEO showed the highest decrease of 99.69%, whereas SABO showed the lowest decrease of 88.60%.Meanwhile, the decreases shown by the other eight MhAs were as follows: BOA, 88.64%; BWO,89.65%; ILA, 99.14%; MGO, 99.65%; PSO, 98.40%;SABO, 88.60%; SO, 98.26%; COA, 98.34%; and ISSA,98.21%.
As shown in Fig.16,the RMSE values of the 10 MhAs under denoised data were smaller and accurately reflected the performance differences among the MhAs.The curve values of AEO were the lowest,whereas the curves of each MhA did not differ significantly under noisy data.
The RMSE boxplots of the 10 MhAs shown in Fig.17 indicate that after data denoising, the upper and lower bounds as well as median values of the RMSE for each MhA decreased significantly.All 10 MhAs proved that data denoising can improve the accuracy of MhAs in PV-cell parameter recognition via the TDM.
As shown in Fig.18, the 
and 
were 99.9997% and 99.9998%, respectively, thus indicating that the identification results of the TDM parameters for PV models based on noise-reduction data are consistent with expectations without significant errors.
3.3.2 Insufficient data
Table 13 summarizes the recognition results of 10 MhAs for the TDM parameters of the PV models under different training data.Among them, BWO indicated the highest decrease of 70.36%, whereas PSO indicated the lowest decrease of 16.38%.The decreases indicated by the other eight MhAs were as follows: AEO, 55.70%;BOA, 34.62%; ILA, 43.39%; MGO, 28.22%; SABO,36.43%; SO, 29.86%; COA, 15.66%; and ISSA, 43.23%.
Table 14
Parameter-identification results of noisy and denoised data based on 10 MhAs for PV module.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1/A Rs/Ω Rsh/Ω α PV module AEO N 1.027 5.000E-05 0.000 2.910 1.738 1.685E-01 5.419 DN 1.028 4.266E-06 0.032 39.363 1.373 2.356E-03 5.460 BOA N 0.619 1.546E-06 0.052 596.092 1.593 4.447E-01 5.239 DN 0.868 2.827E-05 0.009 1337.431 1.785 1.942E-01 5.360 BWO N 0.939 5.000E-05 0.000 76.084 1.780 1.772E-01 2.998 DN 0.855 3.439E-05 0.000 1933.374 1.749 1.212E-01 2.870 ILA N 0.927 2.911E-05 0.000 690.665 1.703 1.764E-01 6.627 DN 1.037 2.838E-05 0.029 1837.995 1.683 1.532E-02 6.408 MGO N 1.027 5.000E-05 0.000 2.910 1.738 1.685E-01 14.021 DN 1.033 4.542E-05 0.022 1996.713 1.695 9.692E-03 11.677 PSO N 1.391 2.444E-06 0.000 0.405 50.000 3.257E-01 7.164 DN 1.034 5.000E-05 0.022 2000.000 1.779 1.012E-02 5.961 SABO N 1.353 1.179E-05 0.000 2000.000 1.565 4.650E-01 2.694 DN 0.885 4.579E-05 0.000 2000.000 2.109 3.933E-01 2.674 SO N 1.026 5.000E-05 0.000 2.912 1.804 1.685E-01 2.737 DN 0.759 7.809E-07 0.032 100.000 1.577 1.130E-01 2.480 COA N 1.391 4.458E-05 0.000 0.404 34.380 3.257E-01 5.381 DN 1.027 1.201E-05 0.028 1997.018 1.555 4.687E-03 5.145 ISSA N 0.902 4.066E-06 0.003 1817.042 1.436 1.932E-01 2.859 DN 1.016 4.081E-05 0.021 1990.204 1.747 1.442E-02 3.072
Fig.21.Convergence curves of RMSEs obtained by 10 MhAs for PV module under noisy and denoised data:(a)noisy data;(b)denoised data.
Fig.23.AEO for fitting curves based on denoised data of PV module:(a)I-V curve; (b) P-V curve.
Fig.22.Boxplot of RMSEs obtained by 10 MhAs for PV module under noisy and denoised data.
As shown in Fig.19, after data prediction, the RMSE achieved by all the algorithms decreased.Although the convergence of each MHA in either case differed minimally,the performance difference of each MHA was more pronounced in the predicted data, with no curve overlap.
As shown in Fig.20, except for the upper limit of the RMSE in BOA and the lower limit and median values of SABO, the upper limit, lower limit, and median values of the RMSE obtained from the other MhAs based on the predicted data decreased.
3.4 PV module under variable temperature
3.4.1 Noise-reduction data
Table 14 lists the parameter-recognition results of the PV module using 10 MhAs with different training data.COA indicated the highest decrease of 98.56%, whereas SABO indicated the lowest decrease of 15.43%.Meanwhile, the decreases indicated by the other eight MhAs were as follows: AEO, 98.19%; BOA, 56.33%; BWO,31.57%; ILA, 91.31%; MGO, 94.25%; PSO, 94.12%; SO,32.94%; and ISSA, 92.53%.
As shown in Fig.21,the RMSE values of the 10 MhAs under denoised data were smaller and reflected the performance differences among the MhAs more accurately.Under both noisy and denoised data, the curve values of AEO were significantly lower, and the convergence speed of AEO was higher under denoised data.
The RMSE boxplots of the 10 MhAs presented in Fig.22 show that after data denoising, the upper and lower bounds as well as the median values of the RMSE for each MhA decreased significantly.All 10 MhAs confirmed that data denoising can improve the accuracy of MhAs in PV module parameter recognition.
Table 15
Parameter-identification results of original and predicted data based on 10 MhAs for PV module.
Model MhAs Data Identified parameters RMSE Time/s Iph/A ISD1A Rs/Ω Rsh/Ω α PV module AEO O 1.027 4.625E-06 0.032 51.054 1.381 2.586E-03 5.418 P 1.032 2.302E-06 0.034 20.042 1.308 2.389E-03 13.738 BOA O 0.647 9.015E-06 0.082 328.618 1.862 4.058E-01 5.276 P 0.819 2.821E-05 0.044 1068.506 1.743 2.035E-01 8.818 BWO O 0.979 5.000E-05 0.000 27.371 1.797 6.087E-02 2.806 P 1.071 4.895E-05 0.000 267.506 1.730 9.982E-02 4.589 ILA O 1.028 1.649E-05 0.027 193.398 1.601 5.707E-03 5.650 P 1.028 1.811E-05 0.026 130.137 1.615 5.862E-03 9.045 MGO O 1.032 3.721E-05 0.023 1917.134 1.663 8.827E-03 11.385 P 1.027 1.908E-05 0.026 1968.716 1.561 5.814E-03 27.354 PSO O 1.434 0.000E+00 0.000 0.406 50.000 2.742E-01 5.834 P 1.030 5.000E-05 0.020 2000.000 1.779 8.873E-03 9.210 SABO O 0.642 1.636E-05 0.171 921.981 20.041 4.435E-01 2.654 P 1.134 5.000E-05 0.000 2000.000 1.805 1.523E-01 4.399 SO O 1.431 5.000E-05 0.000 0.405 25.572 2.742E-01 2.614 P 0.534 2.441E-15 0.000 1.093 7.785 2.606E-01 4.822 COA O 0.988 4.990E-05 0.000 27.601 1.795 5.994E-02 4.538 P 1.028 3.968E-05 0.021 1998.375 1.739 8.121E-03 7.491 ISSA O 1.034 3.997E-05 0.023 876.084 1.740 9.202E-03 3.874 P 1.025 7.894E-06 0.030 1679.504 1.500 3.886E-03 6.484
Fig.24.Convergence curves of RMSEs obtained by 10 MhAs for PV module under original and predicted data:(a)predicted data;(b)original data.
Based on Fig.23, the R2I and R2P were 99.9971% and 99.9982%, respectively, thus indicating that the identification results of the PV module parameters on the noisereduction data are consistent with expectations without significant errors.
3.4.2 Insufficient data
Table 15 summarizes the recognition results of 10 MhAs for the TDM parameters of the PV models under different training data.Among them, PSO indicated the highest decrease of 96.76%,whereas SO indicated the lowest decrease of 4.96%.Meanwhile,the decreases shown by the other eight MhAs were as follows: AEO, 20.91%;BOA, 49.84%; MGO, 34.13%; SABO, 65.65%; COA,86.45%; and ISSA, 57.76%.Among them, BWO and ILA showed significant instability, with the RMSE increasing by 63.98% and 2.72%, respectively.
As shown in Fig.24, after data prediction, the RMSE achieved by all the algorithms decreased.Under the two types of data, AEO offered the highest convergence speed and accuracy.
Fig.25 shows that the upper,lower,and median values of the RMSE obtained from the other MhAs based on the predicted data decreased.
3.5 Model sensitivity analysis
A robust metaheuristic algorithm model typically exhibits low sensitivity to parameters.Considering the SDM as an example, parameter sensitivity tests were conducted on the constructed AEO model using the control variable method while ensuring that the remaining parameters and training and testing data remained unchanged.The selected sensitivity testing parameters included C.The population size and maximum number of iterations were retrained and tested by floating them up and down by 20% to compare their changes in testing accuracy.The results are shown in Fig.26.Overall, the sensitivity of the AEO model to the aforementioned parameters was relatively weak, and even when the aforementioned parameters fluctuated by 20%, the RMSE of its testing fluctuated around 0.001.The RMSE of the first two parameters remained relatively constant.The larger the maximum iterations,the smaller was the RMSE,thus indicating that the number of iterations affects the AEO algorithm.
4 Discussion
Radar charts were used to rate each MhA to demonstrate its performance.The radar map was designed to display the five RMSE indicators obtained based on denoised data for 10 types of MhAs, namely, the minimum, maximum, standard deviation, median, and mean RMSEs.
Fig.25.Boxplot of RMSEs obtained by 10 MhAs for PV module under original and predicted data.
Fig.26.Comparison of testing accuracy of BNN-AEO key parameters with 20% fluctuations in SDM.
Fig.27.Radar maps of five RMSEs for 10 MhAs of four models under denoised data: (a) SDM; (b) DDM; (c) TDM; (d) PV module.
Fig.28.Radar maps of five RMSEs for 10 MhAs of four models under predicted data: (a) SDM; (b) DDM; (c) TDM; (d) PV module.
The higher the score, the lower was the RMSE value of the algorithm and the better was the parameter identification.Based on Fig.27(d), the maximum RMSE and standard deviation scores of AEO were not the highest.However,Fig.27(a),(b),and(c)show that the five RMSE indicators of AEO were lower than those of the other nine MhAs, thus indicating that AEO offers the highest accuracy in identifying the PV model parameters among the 10 MhAs.Similarly, based on the predicted data (see Fig.28), the AEO demonstrated absolute accuracy in PV parameter identification.In summary, after being processed by the BP neural network, the ability of AEO to identify the parameters of the PV models was much greater than that of the other MhAs.
5 Conclusion
A PV-system parameter-identification method for AEO based on a BNN was introduced herein.A BNN was used for data preprocessing to mitigate the adverse effects of data noise, insufficiency, and loss.Subsequently, AEO was performed to identify the parameters of the four PV models.The experimental results indicated that data denoising and prediction improved the accuracy of parameter recognition.Compared with the algorithms proposed in recent literature, AEO demonstrated the highest accuracy and reliability.Additionally, the purpose of extracting PV model parameters was to improve the optimization and control of actual solar energy systems.Under the predicted data, the RMSE values of the SDM, DDM, TDM, and PV modules generated by AEO were 6.1815E-04, 6.1815E-04, 6.0939E-04, and 2.3891E-03, respectively.
Therefore, in future studies, BNN-AEO can be used to process current and voltage data in PV cells, thereby improving the accuracy of parameter identification and establishing more stable PV models.
CRediT authorship contribution statement
Bo Yang:Writing -original draft. Ruyi Zheng: Writing- original draft. Yucun Qian: Conceptualization. Boxiao Liang: Data curation. Jingbo Wang: Writing - review &editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (62263014) and the Yunnan Provincial Basic Research Project (202301AT070443,202401AT070344).
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Fund Information
Author
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Bo Yang
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Ruyi Zheng
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Yucun Qian
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Boxiao Liang
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Jingbo Wang