Abstract
Integration renewable energy sources into current power generation systems necessitates accurate forecasting to optimize and preserve supply–demand restrictions in the electrical grids. Due to the highly random nature of environmental conditions, accurate prediction of PV power has limitations, particularly on long and short periods. Thus, this research provides a new hybrid model for forecasting short PV power based on the fusing of multifrequency information of different decomposition techniques that will allow a forecaster to provide reliable forecasts. We evaluate and provide insights into the performance of five multiscale decomposition algorithms combined with a deep convolution neural network (CNN). Additionally, we compare the suggested combination approach's performance to that of existing forecast models. An exhaustive assessment is carried out using three gridconnected PV power plants in Algeria with a total installed capacity of 73.1 MW. The developed fusing strategy displayed an outstanding forecasting performance. The comparative analysis of the proposed combination method with the standalone forecast model and other hybridization techniques proves its superiority in terms of forecasting precision, with an RMSE varying in the range of [0.454–1.54] for the three studied PV stations.
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Introduction
Solar photovoltaic (PV) energy has recently emerged as a viable alternative to conventional forms of electrical power production, such as fossil fuels^{1} power production. There has been a significant increase in the amount of PV capacity that has been installed throughout the world because solar energy is clean, economical, renewable, and beneficial to the environment. Therefore, photovoltaic (PV) systems provide an efficient alternative to supply distant locations by power, pumping water, and according to gridconnected PV plants, reducing electricity expenses.
PV power production is very sensitive to variations in solar irradiation as well as other factors of the local environment, in contrast to the traditional sources, where electrical power production can be simply managed. As a result, the integration of PV electricity into the electrical grid is a major challenge. Renewable power producers, like other energy producers, should respect the expected electrical power for the market session^{2}. Precise PV power assessment is vital for (1) integrating PV power into the grid, (2) selling PV power in the electricity markets, and (3) planning PV plant maintenance.
The most current research on PV power forecasting includes a large number of different forecasting approaches, the choice of which is dependent on the adopted time horizon of the forecast, the utilized inputs, and the forecasting strategy that is chosen. The PV power forecasting techniques may be classified into^{3}: (1) physical methods, relying on satellite/sky imagery or numerical weather prediction (NWP) that need postprocessing to transform their output to PV power, and (2) datadriven approaches that relate the output of the PV plant to external factors^{4}. In the first category, the modeling process necessitates the collection of a substantial amount of data and the execution of several complicated analyses to estimate the parameters of the physical model, which restricts its applicability to the real world. On the other hand, datadriven approaches are extensively utilized in PV generation forecasts due to the fast growth of data mining, machine learning, and deep learning methods in recent years.
In a mathematical sense, the datadriven forecasting category may be classified into three subclass: linear, nonlinear, and hybrid models. The linear models are the simplest kind of these classes, and the most wellknown linear forecasting models include the autoregression (AR), autoregressive integrated moving average (ARIMA)^{5}, and autoregressive moving average extrapolation (ARMAX)^{6}. For onestep forecasting tasks, previous research has revealed that linear forecasting models may usually provide reliable outcomes. Despite this, it is apparent that the PV generation series is not linear. Thus, the majority of academics are concentrating their efforts on the creation of nonlinear forecasting models such as artificial neural networks (ANN). Artificial Neural Networks (ANNs) have been shown in prior studies to be an effective tool for accurate predicting. It has been shown that employing ANNs to simulate linear trends may provide mixed results and it is not advised to use ANNs naively in any form of data^{7}. Extreme learning machines, fuzzy systems, knearest neighbor, support vector machines, and their hybridization have dominated current PV forecasting literature. The behavior of a single model does not result in a more precise forecast of the amount of power produced by PV in a variety of scenarios. One possible explanation for this issue is that the standalone approach has certain shortcomings. In such situations, the ideal solution is a hybrid model, which combines two or more methodologies. These sorts of the model have been utilized for numerous forecasting applications to reach improved accuracy. One of the key goals of these models is to study the various combinations of different topologies in enhancing prediction accuracy. By using the advantages of each topology separately, hybrid models may boost forecasting performance. Solving the PV power forecasting issue using hybrid models has shown excellent performance compared with standalone models. Hybrid methods for time series forecasting have recently developed as a dynamic and active research area^{8} using different techniques such as metaheuristic algorithms^{9}, sparse representation theory^{10,11,12} and decomposition ensemble learning approaches^{13}. A review of various forecasting techniques for solar irradiation and PV power is provided in^{13,14}. A trending technique in the field of PV and other shortterm forecasting domains is the use of deep learning techniques. Deep learning techniques are a complex and advanced type of machine learning approach. It can derive deep information from the PV power time series and produce high predictive results than other conventional models^{15}.
Lulu Wen et al.^{16} developed a DL technique for solar power forecasting on an hourly scale. The forecasting outcomes prove that the DL model provides high precision than MLP and SVM models. A combined variational autoencoder VAE with a deep LSTM model provides low forecasting error (RMSE = 5.471) for shortterm forecasting of PV power on different PV systems compared with different machine learning techniques^{17}. The LSTM neural network model was proposed in^{18}, for daily forecasting PV power. The outcomes of the developed model exhibit high forecasting performance compared to MLP, LR, and persistence methods. The proposed model present also 42.9% RMSE skill compared to the benchmarking models for 1year testing data. Narvaez et al.^{19} used deep learning techniques for daily and weekly scale forecasting of PV power. The obtained results showed that the DL model performed 38% better than the traditional forecasting method. The integration of the CNN model with LSTM^{20} shows excellent forecasting accuracy compared to other methods for the hourly scale of PV power forecasting. The developed mechanism has yielded the lowest forecasting error in terms of MAE, MAPE, RMSE, 0.0506,13.42, and, and 0.0987, respectively.
Zhen et al. proposed a new combination methodology based on the BILSTM model with a genetic algorithm (GA) to improve PV power forecasting^{21}. AbdelBasset et al.^{22}, introduce a new learning model PVNet for shortterm forecasting of PV power by reconfiguring the gates of the GRU model utilizing convolution layers. The achieved results show that the proposed PVNet can extract hidden features from historical PV data and provide high forecasting accuracy. Wang et al.^{23} propose the use of the new deep learning model generative adversarial model networks (GANs) for weather classification employing Wasserstein GAN with gradient penalty and deep CNNmodel. Research findings demonstrate the proposed WGANGP offers enhanced precision compared to reference models. Bendaoud et al.^{24} developed a conditional generative adversarial model network (CGAN) for shortterm PV power forecasting using exogenous data. Huang et al.^{25}, proposed a new combined model for hourly PV power forecasting. In their work, they developed a new time series conditional generative adversarial model TSFCGAN, the proposed model is built by CNN and the BiLSTM model. The proposed combination methodology proves its performance against standalone models such as SVM, LSTM, RNN, and MLP models. Recent research has shown that the decompositionlearning strategy enhances the forecasting performance of standalone models significantly, and several time–frequency methodologies have been introduced for nonstationary signals analysis.
Wavelet Transform (WT)^{26}, Empirical Mode Decomposition (EMD)^{27}, Ensemble Empirical Mode Decomposition (EEMD)^{28}, Multivariate Empirical Mode Decomposition (MEMD)^{29}, Ensemble Empirical Mode Decomposition (EEMD)^{30}, a modified variant of the conventional EMD (CEEMDAN)^{31,32} and Iterative Filtering decomposition method (IF)^{2}. In this paper, five different decomposition learning approaches were independently investigated before being combined for shortterm PV power forecasting, using one dimensional CNN model as an essence regressor owing to its capacity to extract more relevant features from the supplied input data.
The remaining parts of the work are structured as described below. In the second section, a detailed literary analysis of the different decomposition ensemble learning algorithms for PV power forecasting is presented. "Contributions of the paper" section presents the paper's contribution. Case studies, data collection, are presented in "Material and monitoring" section. The fundamental structure of the proposed model is presented in "Methodology" section. "Performance metrics" and "Results and discussion" sections discuss the model assessment and the key findings of this study, respectively.
Literature review
Scientists have conducted various research on sun radiation evaluation so far, with interesting results. In this section, hybrid decomposition approaches are classified according to the used decomposition method:
Empirical mode decomposition
Shang and Wei^{33} proposed a multistage PV power forecasting model in four regions in USE. They proposed an enhanced version of empirical model decomposition EEMD combined with the SVM regression model for 15min and daily solar radiation forecasting. Maximize Relevancy Minimize Global Redundancy MRMGR feature selection technique is employed to reduce the huge numbers of decomposed components of endogenous and exogenous in different sites (400–600 feature vectors), after selecting the best candidate feature sets the improved SVM model is coupled with PSO technique for hyperparameters tuning. The provided results show high forecasting accuracy compared with benchmarking models (MLP, RBF, ANFIS, NNPSO, MLP, WTRBF, WTANFIS, and WTPSO).
Eseye et al.,^{34} suggested the use of wavelet transform coupled with POS and SVM models for multihour PV power forecasting using previous PV power with SCADA records and physical data. Behera et al.^{35}, suggested a new hybrid model for shortterm PV power forecasting. The developed model consists of three main steps: firstly, the historical PV power is split into a multifrequency band using the EMD algorithm. The decomposed component derived from EMD is then fed into the ELM model to generate the specified PV power. The sine cosine algorithm (SCA) is utilized for ELM hyperparameter tuning. The proposed EMDSCAELM method is compared to its counterparts models, SCAELM, EMDELM, and standalone ELM model. The comparison in term of statistical metrics shows that the suggested appraoch provides better performance for multistep forecasting.
Wang et al.^{36}, proposed the use of ICEEMDAN decomposition techniques with a modified version of particular swarm optimization (MPSO) for hyperparameter selection of the SVM model. The suggested strategy is more suitable for other common approaches for PV power forecasting.
Wang et al.^{37}, proposed the combination of the variableweight combination model with ensemble empirical mode decomposition (EEMD) for PV power forecasting. Firstly, EEMD is employed for decomposing PV power data into different sets of frequency ranges. The decomposed components are classified into three main categories, highfrequency, and intermediatefrequency. These three categories are separately forecasted using a parametersweight integration model and summed to get the actual PV signal. Experimental results show that an individual forecast strategy provides better precision than a direct forecasting method.
Zhou et al.^{38}, proposed a new integrated model for PV power forecasting, the developed model was validated on three different databases in SafiMorocco. The combined model consists of using the CEEMDAN algorithm, multiobjective chameleon swarm algorithm (MOcsa), and four ML and DL models. The CEEMDAN is utilized for PV power decomposition, and the MOcsa strategy is employed for determining the weight coefficient of the used subsystems (Enn, LStm, Cnn, and BiLStm). The proposed forecasting strategy is validated on three data sets with different panel technology (May 25 to July 15, 2018, 5 min time scale) including one polycrystalline (pSi), one amorphous (aSi) technique, and eight monocrystalline (mSi), and proves its prediction performance against different deep learning and machine learning technique in terms of mean relative error (MRE), mean absolute error (MAE), and Symmetric mean absolute percentage error (SMAPE).
Lin et al.^{39} developed a new cascade decomposition using EEMD and VMD algorithms, in which EEMD is used firstly to decompose the given time series into a set of intrinsic mode functions, then VMD is employed to split the first component IMF1 from the EEMD method to obtain more stable components. All the obtained IMFs components are then fed into the BILSTM model for PV power forecasting. The developed approach proves its performance compared to other forecasting models (EEMDBILSTM, EEMDLSTM, ANN, LSTM, VMDGRU, EEMDANN, and BILSTM) models.
Niu et al.^{40}, proposed multistages for short PV power forecasting. In the first stage, the random forest algorithm RF is used to rank the most important exogenous input and then remove the irrelevant data, then the selected factors are moved as weighted values to the improved grey (IGIVA) to select days with identical weather patterns. Time series PV power is decomposed using complementary ensemble empirical mode decomposition (CEEMD). In the modeling stage, the backpropagation neural network (BPNN) is used to forecast the desired output. The proposed RFCEEMDDIFPSOBPNN model shows its forecasting performance and its stability under different weather conditions compared to other studied models.
Zhang et al.^{41}, proposed a new integrated model for PV power forecasting which include time series decomposition using improved empirical mode decomposition (IEMD), feature selection strategy using Maximize RelevancyMinimize Global Redundancy, then PSO–SVM as a forecasting model in which PSO algorithm serves for hyperparameters selection.
Wavelet decomposition
Random Vector Functional Link (RVFL)—Seasonal Autoregressive Integrated Moving Average (SARIMA)—model combined with Maximum Overlap DWT was proposed in^{42}, for three stepsahead PV power forecasting in IIT Gandhinagar, India. In their work, the DWT was used to split the PV data into 5 subseries for each decomposed component both forecasting models were applied, then a convex combination was utilized for the final forecast. The combination of these two models with DWT delivers excellent precision compared to the single models, and the combination of DWTREVL, DWTSARIMA.
De Giorgi et al.^{43} proposed the use of wavelet transform WT with least square support vector machine LSSVM for hourly PV power forecasting 24 h ahead in Italy. In their work, they conducted many experiments and observed that the pairing of LSSVM and ANN with WT delivers high performance, and for long time horizons WTLSSVM reaches the highest performance.
Wan et al.^{44} proposed the use of wavelet transform WT coupled with a deep CNN model to build a WTDCNN forecasting model for hourly PV power in Belgium. In their work, WT is used to split the PV data into a set of different frequency ranges. The framework consists of a 1Dto ato2DImage layer, convolution layer, pooling layer, 2Dto a 1Dlayer, and finally logistic regression layer for each decomposition. Then a wavelet reconstruction phase is employed for the final forecast of the PV power signal. The achieved results show that the hybridization strategy performs better than the standalone (SVM and MLP) model and the hybrid WTSVM in terms of forecasting criteria for all forecasting horizons in the two studied regions.
Chen et al.^{45}, proposed a new fusing strategy for PV power forecasting using endogenous data on an hourly scale. In their work, the DWT algorithm was employed with an adaptive neurofuzzy inference system (ANFIS). The decomposed PV power signal is fed into the ANFIS model for outputting shortterm PV power; different functions of wavelet mother were used (Haar, Daubechies, Coiflets, and Symlets). The achieved results demonstrate that the developed combination delivers high performance. Furthermore, they found that coif2ANFIS and sym4ANFIS offer low forecasting errors compared to all studied models.
Zhang et al.^{46} utilized a hybrid deep learning model coupled with WD and Artificial neural networks (ANN) to improve solar power plant output forecasts. They used WD as the network's transfer function and treats the modelweight, scalingfactor, and translationfactor as genetic individuals. The network's parameters are then obtained through independent optimization using a genetic algorithm and imported into the network. The authors found that the GAWNN network performance outperforms conventional ANN models.
A hybrid deep LSTM network integrating wavelet packet decomposition (WPD) is proposed by Pengtao et al.^{47}. Authors have utilized hybrid deep learning for 1hahead PV power forecasting. The WPD algorithm is used to subdivide the initial PV power series. Four LSTM networks model are created for these subseries. The performance of the WPD–LSTM method was demonstrated with a case study using data collected from an actual PV system in DKASC, Alice Springs, Australia. compared to individual LSTM, RNN, GRU, and MLP models, the WPD–LSTM model performs better in terms of predicting stability and accuracy.
Variational mode decomposition
Zang et al.^{48}, developed a new hybridization method based on variational mode decomposition (VMD) integrated with the DCNN model for PV power forecasting on a daily and hourly scale. Each component of the VMD technique, which originated from PV power, is transformed into a 2D image for further training in parallel channels of the DCNN model. The proposed forecasting methodology 2DVMDDCC proves its ability in forecasting PV power with high precision compared to SVM, GPR, MLP, VMDSVM, SMDGPR, and 1DVMDCNN. Netsanet et al.^{49} proposed a multistage forecasting model for daily PV power forecasting in China. The proposed methodology involves five main stages, firstly the time series of PV power is decomposed through the VMD algorithm into different components, then mutual information MI is employed to select the most pertinent input data. In the forecasting steps, the ANN model is employed with the ant colony optimization (ACO) technique for hyperparameters tuning to output the desired PV power, another ANN model is utilized as a cascade forecasting using the previously estimated PV power as input for the final forecast. The forecasting ability of the proposed VMDACOANN was compared to the single ANN model and the combined ANN with the genetic algorithm GAANN model. The forecasting performance of the proposed technique in terms of statistical metrics outperforms the benchmarking models.
Wu et al.^{50}, proposed a new hybridization methodology for very shortterm PV power forecasting in Australia. In the first stage, the VMD technique is used to decompose the PV signal into different frequency bands, then the fast correlationbased filter (FCBF) algorithm is employed to select relevant input features, then a BILSTM model is used as the essence regressor. The proposed VMDFCBFBILSTM model proves its forecasting ability against conventional models.
A new decompositioncombined model is proposed by Xie et al.^{51}, the proposed combination method is based on the use of the VMD algorithm to decompose the PV power data into different frequency components, then Deep Belief Network DBN model is used to forecast the highfrequency components and Autoregressive model is used to forecast lowfrequency components. The achieved results prove that the developed technique gives excellent performance compared to the basic models.
Korkmaz et al.^{52}, proposed a new forecasting model SolarNet for short PV power forecasting in Australia Solar Centre. The proposed SolarNet is dependent on the use of the VMD decomposition algorithm and CNN model. The input data are; solar radiation, active power data, temperature, and humidity. The proposed CNN model consists of two structures, maxpooling, and pooling blocks to boost the forecasting effectiveness. The achieved results show that the proposed model outperforms other deep learning techniques for 1h ahead forecasting.
Zhang et al.^{53} developed an integrated model for PV power assessment through the hybridization of the VMD algorithm, CNN, and BiGRU. The VMD algorithm decomposed the PV power series into several submodes. An input feature matrix is built using the antecedent values and various environmental parameters. The CNN technique is then used to model the fundamental input–output relation between the feature matrix and the target value. Next, the BiGRU network is used to forecast the next PV power. Combining the forecasting values of the submodes yields the exact forecasted PV power result. The results showed that the VMDCNNBiGRU model outperformed the EEMDCNNBiGRU model.
Korkmaz et al.^{54} proposed the use of SolarNet which is a onedimension convolutional neural network model for shortterm PV output power forecasting over various weather conditions. The experiments are carried out as a case study utilizing a 23.40 kW PV dataset from the Desert of Australia. The measured meteorological data are used as input parameters to the SolarNet model. To create deep input feature maps, the power data is split into subcomponents using the VMD approach, followed by a data preparation and reconstruction procedure. The achieved results show that the VMDCNN forecasting technique delivers excellent performance.
Contributions of the paper
Based on the analysis of relevant literature, we may infer that decomposition ensemblelearning techniques for PV power forecasting provide promising outcomes for further research. Firstly, the majority of developed models employ only one decomposition technique as a preprocessing step in the forecasting process of PV power. Secondly, instead of utilizing the decomposed PV power time series directly, the use of feature selection is a necessary step in improving prediction performance. Decomposition models are a feasible way of enhancing forecasting precision, based on the advantages and limits indicated in the current research. This research makes use of the concept of hybrid forecasting and introduces a new way of predicting PV power to fill the gap in the existing literature on this problem. The proposed hybridization mechanism is based mainly on the following combination: ensemble decomposition methods, multiscale feature fusing, feature selection approach, and deep learning model.
The following section outlines the work's main novelty:

A new fusion strategy IFVMDFSRACNN for short PV power assessment is optimized to generalize and improve predicting efficiency in a variety of meteorological conditions.

The assessment and interpretation of several decomposition methods for shortterm PVplants power forecasting.

The fusing of multiscale features from the employed decomposition techniques for boosting the forecasting precision of PV power output.

Using a feature selection technique allows users to choose the most meaningful intrinsic function IMFs from various multiscale methodologies, which improves forecasting ability and minimizes the dimensionality of the data.

The deep CNN model is capable of deriving deeper characteristics from the fused multiscale information to provide the appropriate PV output power.

Several tests were conducted, and the findings show that the suggested fusion mechanism produces more precise forecast outcomes than its counterparts models.
In comparison to previous research, the suggested forecasting methodology's efficiency is confirmed and validated on three gridconnected PV power systems in Algeria with various climate conditions. The suggested solution that utilizes hybrid approaches has yielded the lowest forecasting error.
Material and monitoring
Summary of data
Data from three photovoltaic power installed in three distinct locations covering different climatic conditions in Algeria were used in the study. The map in Fig. 1 depicts the geographic coverage of solar PV power plants employed in this research. The study area covers the three selected sites. The first PV plant station is Lekhneg, in the Laghouat region (33° 43′ 26.74″ N, 2° 48′ 45.27″ E)with a capacity of 60 MWp located in the south of Algeria^{55}. The second gridconnected PV power is Oud Nechou in the Ghardaia region^{56,57}, (32° 36′ 1.46″ N, 3° 42′ 3.42″ E) located also in the south of Algeria with a capacity of 1.1MWp. The third PV power station is Dhaya of Sidi BelAbbes (34° 41′ 32.23″ N, 0° 36′ 2.89″ O) located in the west of Algeria with a capacity of 12 MWp^{58}.
The photovoltaic solar power plants of Lahgouat, Ghardaa, and Sidi BelAbbes were commissioned in (April 2015/2016), as part of the national renewable energy program, and are one of 21 identical stations installed across the country to produce 400 MW. The information regarding the type of photovoltaic modules, the area of photovoltaic plants, and the data period of measured PV power in each PV station are given in Table 1. The photographs of the studied PV plants can be viewed in Fig. 2.
Methodology
Complete ensemble empirical mode decomposition with adaptive noise (CEEMD)
CEEMDAN is an extension of EMD. Modal aliasing could be further reduced by using adaptive white noise to increase decomposition effectiveness. It is possible to utilize CEEMDAN to analyze and handle nonstationary signals in time and frequency^{59}. It can generate signal wave patterns of various sizes and create a series of data sequences with local features at various periods, where each component is a stationary IMF. Two criteria must be satisfied by each IMFs component: (1) the mean value of the envelope formed by its local maximum and minimum must be zero at any moment, and (2) the number of extreme points in the entire dataset must be the same as zerocrossing points or vary by at most one.
For a timeseries sequence \(\left(t\right)\) , the key steps of CEEMDAN decomposition are as follows:

At time m, we introduce white noise \({W}^{m}\left(t\right)\) to the original signal \({X}^{m}\left(t\right)\).
$${X}^{m}\left(t\right)=X\left(t\right)+{\varepsilon }_{0}{W}^{m}\left(t\right),m=\mathrm{1,2},\dots ,M$$(1)where M represents the number of realizations.

The first EMF \(\overline{{IMF }_{1}} \left(t\right)\) is calculated by taking the average of the EMD:
$$\overline{{IMF }_{1}} \left(t\right)=\frac{1}{M}\sum_{i=1}^{M}{IMF}^{1}\left(t\right)$$(2)Then, the residual components are computed as:
$${{\text{r}}}_{1}\left(t\right)=X\left(t\right)\overline{{IMF }_{1}} \left(t\right)$$(3) 
The signal \({r}_{1}\left(t\right)+{\varepsilon }_{1}{EMD}_{1}\left({W}^{m}\left(m\right)\right)\) is decomposed in its turn by the EMD algorithm to produce the second IMF and the corresponding residual as below:
$$\overline{{IMF }_{2}}=\frac{1}{M}\sum_{i=1}^{M}{EMD}_{1}\left({r}_{1}\left(t\right)+{\varepsilon }_{1}{EMD}_{1}\left({W}^{i}\left(t\right)\right)\right)$$(4)$${r}_{2}\left(t\right)={r}_{1}\left(t\right)\overline{{IMF }_{2}}$$(5) 
Repeat these steps to obtain the kth residual and k + 1th IMF component:
$${r}_{k}\left(t\right)={r}_{k1}\left(t\right)\overline{{IMF }_{2}}\left(t\right)$$(6)$$\overline{{IMF }_{k+1}}\left(t\right)=\frac{1}{M}\sum_{i=1}^{M}{EMD}_{1}\left({r}_{k}\left(t\right)+{\varepsilon }_{k}{EMD}_{k}\left({W}^{i}\right)\right)$$(7)where \({EMD}_{k}\) (■) indicates the kth IMF mode.

Repeat Eqs. (6) and (7) until the remaining signal is unsuitable for being further decomposed through the EMD algorithm and then terminate the algorithm.
Finally, the time series X(t) decomposition result may be written as follows:
Variation mode decomposition (VMD)
The VMD technique is one of the most recent signal processing methods.. The VMD method is a nonrecursive and adaptive decomposition process proposed for nonstationary signals. The VMD algorithm is similar to an adaptive Winer filter bank, which can split a given signal into a set of center frequencies with a restricted frequency range different from the EMD algorithm. The essence of the VMD technique is to decompose a given input signal into several submodels that have a restricted bandwidth and a certain level of sparseness in bandwidth^{60}.
Constructing and resolving variation issues is the primary focus of this approach. Solving constrained variation optimization issues is stated as:
Wher \({u}_{k}\left(t\right)\) represent the model function of the signal,\(\left\{uk\right\}\) is the model group {\(u1\) ,\(u2\) ,…,\(u3\) },\(wk\) is the central frequency of the kth mode of the signal,\(\left\{uk\right\}\) indicate the center frequency of these decomposed modes,f(t) and \(\delta \left(t\right)\) represent the input signal and unit pulse function respectively.
Utilizing quadratic penalty factor and Lagrangian multiplier Eq. (1) can be developed as:
The changing orientation multiplication technique (ADMM) is employed to solve Eq. (10). The \({u}_{k}\) can be described as:
where n represents the iteration numbers, \(\widehat{f}\left(w\right)\), \({\widehat{u}}_{k}^{n+1}\left(w\right)\), \({\widehat{u}}_{i}\left(w\right)\) and \({\stackrel{`}{\lambda }}\left(w\right)\) indicate the form after \(f\left(wt\right)\), \({u}_{k}^{n+1}\left(t\right)\), \({u}_{k}\left(t\right)\), and \(\lambda \left(t\right)\) Fourier transform^{61}.
Wavelet packet decomposition (WPD)
The WPD technique is traditional signal processing, which split a given signal into its adequate and detailed elements. Wavelet basis and decomposition levels have a significant impact on the WPD's performance. There are two kinds of WPD: the wavelet transform in its continuous and discrete forms. The following is a description of the continuous wavelet transform^{62}:
where f(t) and \({\psi }\left(t\right)\) represent the input signal and the mother wavelet, respectively. The term b indicates the translation factor and a represents the scale factor and * denotes the conjugate complex factor. The discrete form of a and b in DWT can be summarized as follows:
Iterative filtering decomposition method (IF)
Recently, the iterative filtering technique is proposed to be an alternative to the EMD method and its variants^{2} for signal processing with adaptive filtering to enforce the convergence of IF. The primary variance between the IF approach and EMD and its alternative is that the moving average of a specific signal \(f\left(x\right),x\in {\mathbb{R}}\) is accomplished by employing the convolution of \(f\left(x\right)\) with a specific law pass filter^{63}. Let's consider a signal \(\left(x\right),x\in {\mathbb{R}}\) and \(\mathcal{L}\left(f\right)\) represent the moving average of the input signal which can be estimated by^{64}:
The term \(a\left(t\right)\) indicate the double average filter given by:
In IF methodology the operator \(\mathcal{L}\left(f\right)\) is obtained by the convolution of the signal \(f\left(x\right)\) by some specific filter \(W\).
Let’s define our operator as: \({S}_{1,n}\left({f}_{n}\right)\)=\({f}_{n}{\mathcal{L}}_{1,n}\left({f}_{n}\right)={f}_{n+1}\) which shows the alteration part of \({f}_{n}\), the first IMF_{1} can be obtained by Eq. (17):
Apply the same method to the rest \(f{IMF}_{1}\) until it becomes a trend signal, which implies it has just one local maximum or minimum. All IMF components are driven by two loops in the proposed IF methodology: an inner and an outer loop.
Timevarying filterbased empirical mode decomposition algorithm (TVFEMD)
Timevarying filterbased empirical mode decomposition (TVFEMD) is a newly established variant of the classic (EMD) suggested by Li et al.^{65}. By adopting a Bspline approach filter to cope with shifting processes, the suggested TVFEMD demonstrates its efficacy in resolving the shortcomings of the standard EMD algorithm. The fundamental concept of the suggested TVFEMD relies on the fixed cutoff frequency and then follows with a timevarying filtering technique. The following are the key phases of the proposed TVFEMD^{65,66}:

1: Estimate the maximum point and identify it for a given signal \(x\left(t\right)\) as :

2: Determine all instances of intermittency that meet the following criteria:
\(\frac{max\left({\varphi }_{bis}{\prime}\left({u}_{i}:{u}_{i+1}\right)\right)min({\varphi }_{bis}{\prime} ({u}_{i}:{u}_{i+1} ))}{min\left({\varphi }_{bis}{\prime}({u}_{i}:{u}_{i+1})\right)}>\rho \), where the position of \({u}_{i}\) can be considered as an intermittence :\({e}_{j}\left(j=\mathrm{1,2},3,\dots \right),{e}_{j}={u}_{i}\).

3: For each state of \({e}_{j}\) , there are two alternative positions placed on the falling edge or rising of \({\varphi }_{bis}{\prime}\left(t\right)\), if \({\varphi }_{bis}{\prime}\left({u}_{i+1}\right)>{\varphi }_{bis}{\prime}\left({u}_{i}\right)\),\({\varphi }_{bis}{\prime}\left({u}_{i}\right)\) can be viewed as a floor. Although, if \({\varphi }_{bis}{\prime}\left({u}_{i+1}\right)<{\varphi }_{bis}{\prime}\left({u}_{i}\right)\), then \({\varphi }_{bis}{\prime}\left({u}_{i}\right)\) can be found on its falling edge.

4: Interpolatebetween the maxima to perform the adaptation of the local cutoff frequency.
where \(\left({\varphi }_{bis}{\prime}\left(t\right)\right)\) specifies the cutoff frequency; a more detailed explanation may be found in^{65}. Furthermore, the ending condition mentioned in the sifting step is as follows:
where \({B}_{Loughlin}\left(t\right)\) represent Loughlin instantaneous bandwidth and \({\varphi }_{avg}\left(t\right)\) displays the weighted average of the different components' instantaneous frequencies^{65,66}. Actually, with the stated bandwidth cutoff \(\varepsilon \), the term "signal local narrowband" is described as:\(\theta \left(t\right)\le \varepsilon \) , in which TVFEMD spliting performance will be adequately adjusted.
Convolution neural network (CNN)
Deep learning techniques as a subfield of machine learning (ML) have received considerable attention from the scientific community in recent years. The CNN model, which is a significant deeplearning architecture inspired by the natural visual system of mammals is developed by LeCun^{67,68,69}proposed. Deep CNNs can be built using the backpropagation technique to perform tasks such as classifying handwritten digits. Deep CNNs have dominated various computer vision and image processing applications, and recent applications of CNNs for time series prediction have generated highly promising results^{70,71}. Convolution, pooling layer, and fully connected layers are the main building blocks that CNN's use to perform spatial hierarchies of features. The deep CNN model is mostly well adopted with time series signals for various regression and classification problems as introduced in^{72}. The CNN models perform the optimization performance with less memory consumption, there is already a connection between the neurons in the adjacent layer of the fully connected networks. The hyperparameters of Deep CNN model can be reduced efficiently; also, CNN models are able to extract special features with different lengths. Figure 5 displays an illustration of a CNN model used for time series forecasting employing univariate timeseries data as input, we note that CNN's are more suitable suited for multivariate time series with features extracted through convolution and pooling layers.
Forward stepwise regression algorithm (FSRA)
Forward Stepwise Regression Algorithm (FSRA) is a systematic method used in statistical modeling to select the most relevant variables for a predictive model. This algorithm falls under the category of stepwise regression, which includes both forward selection and backward elimination methods. FSRA specifically uses the forward selection approach, where the model building process starts with no variables and then adds them one at a time based on specific criteria. The main steps of FSRA are as follow:

1.
Initialization Begin with a model containing no predictors. This means starting with a simple model that only includes the intercept .

2.
Variable Selection At each step, consider adding each variable that is not already in the model. The choice of which variable to add is based on a predetermined criterion, usually the one that results in the most significant improvement to the model. Common criteria include the lowest pvalue in a ttest for the coefficient, the largest increase in R^{2}, or a significant decrease in the Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC).

3.
Model Evaluation After adding the new variable, evaluate the model to ensure it significantly improves the model's performance. This can involve checking for statistical significance, improvements in goodnessoffit measures, or crossvalidation performance.

4.
Iteration Repeat the variable selection and model evaluation steps, adding one variable at a time, until no further significant improvements can be made to the model.

5.
Final Model The result is a model that includes a subset of the available variables that best explain the variation in the response variable, based on the criteria used for variable selection.
The hybridforecasting model
The main configuration of the suggested fusion strategy is presented in Fig. 3. Furthermore, the following are the major phases involved in the construction of the combined IFVMDFSRACNN forecasting model:

Step 1 Collect and analyze PV power data to generate training and testing sets. The training sets are utilized for model tuning (Hyperparameter setting), while the testing sets are used for model assessment.

Step 2 Using the decomposition techniques, divide the training and testing sets into a varied number of IMFs elements. The nonlinearity and nonstationarity characteristic of the actual data might be efficiently used addressed using these decomposition methodologies.

Step 3 Individual use of each decomposition method for PV power forecasting and then fusing them to increase the forecasting accuracy.

Step 4 A feature selection technique is used to identify and rank the most important IMFs components for PV power forecasting.

Step 5 The identified IMFs sequence is deployed to train the deep CNN forecasting model as input variables.

Step 6 After that, the fully trained CNN model is utilized to evaluate forecasting efficiency on the test dataset.

Step 7 Examine the findings and assess the effectiveness of the three gridconnected under consideration.
It is feasible to demonstrate that the suggested integrated approach is intended to deal with a variety of qualities found that occur in the real world. Particularly, decomposition strategies were used to cope with the data's nonstationarity. The noisy and unnecessary IMFs components are then removed using a feature selection approach. The nonlinear CNN model is then used to model the PV power output.
Performance metrics
Five quality evaluations were chosen to quantify the impact of the suggested forecasting approach^{8,69,73,74,75} expressed in Table 2 as:
Results and discussion
Precise shortterm assessments of PV power are crucial for ensuring the production and stability of necessary power system capacity. This section intends to present the simulation results for the suggested technique using multiple techniques. These strategies are further divided into three phases. The primary concept is to create a combination of different decomposition strategies with a deep learning approach for PV power forecasting, and then apply the suggested strategy to diverse time series forecasting. We start by assessing the potential of extracting multifrequency features with the suggested decomposition techniques and evaluate each decomposition method separately on three gridconnected photovoltaic stations. The suggested combination methodologies have been adjusted for a horizon of 15 min, and a half hours ahead of PV power output. The suggested combination technique is assessed and verified on three separate PV power data sets, with 50% of each analyzed data set employed for training and the remainder for assessing forecasting ability. Solar irradiation, temperature, and the angle at which PV arrays are installed all influence PV power production. The association between previous PV power and future output PV power was the only focus of the present research. The concept of trial and error was used to evaluate the effect of different delays and determine the acceptable number of delays.
The PV power and its previously decomposed elements with optimum lag identification are the target outputs for training the supervised decompositiondeep learning method in this work. For each analyzed region, data preprocessing is the same. To validate the integrated model's forecasting ability, the hybridization approaches were compared to standalone models (GPR, LSSVM, ELM). All models were trained and tested using the same methodology. The quantitative findings of the assessment criteria for each decomposition strategy and their counterpart models are depicted in Tables 3, 4, and 5 for Laghouat, Sidi Bel Abbés, and Ghardaia stations, respectively. Bold numbers reflect the most accurate estimation. during the forecasting process. According to the findings reported, in Tables 3, 4, and 5, various observations may be inferred. Starting with the results delivered by hybrid approaches, where we can see that these latter significantly exceed the basic models in terms of statistical metrics^{75}. Moreover, the hybrid approaches generate PV power output with high precision. It observed that the nRMSE error generated by all decomposition techniques is less than 10% in all analyzed areas whereas the nRMSE of individual models ranged from 22 to 39%. It can be also observed that all decomposition algorithms provide different forecasting errors which vary within the range of nRMSE = [0.61–7.44] in all studied stations. That is, the proposed combination can generate a 30min and 15min ahead PV power output with acceptable accuracy better than standalone models. The proposed VMDCNN and IFCNN models, followed by TVFCNN, WPDCNN, and CEEMDANCNN models achieve the best forecasting precision in all studied regions. The VMDCNN and IFCNN models provide the highest average PV power production, which will be evaluated as the best candidate algorithm for multiscale fusing. For a more comprehensive view of the model's precision, a statistical representation is also required in order to have a better comprehension of model precision.
As shown in Fig. 4, the VMDCNN model ranked as the best forecasting model for Laghouat and Sidi bel abbes PV station followed by the IFCNN model. In the case of Ghardaia PV station, we observe that the IFCNN model provides the best forecasting performance followed by the VMDCNN model. All hybrid models, which have satisfactory convergence, perform better than individual models. On the contrary, the results achieved by single models (LSSVM, GPR, and ELMmodel) illustrate that these models are ineffective as compared to hybrid models.
Figures 5, 6, and 7 show the actual and estimated PV power output on sunny and cloudy days patterns for the three studied regions. One can observe from the comparison of PV power forecasting that the forecasted PV power curve of standalone models (GPR, ELM, and LSSVM model) show a remarkable gap between true and modeled values of PV power under different weather conditions. In both specific cases (sunny, and cloudy days), the results produced using hybrid decomposition ensemble learning techniques outperform those obtained with standalone models. Another relevant point is that the forecasting trend of the five developed hybrid models is identical to the actual data. For the threestudied PV station, the IFCNN model and VMDCNN model generate the closest PV power estimate to the real PV power data. In addition, the subfigures show that the proposed VMDCNN model and IFCNN model can accurately match the trend and features of real PV power than the benchmarking models. The same figures may be used to infer additional information.
The outcomes are analyzed by considering all information investigated with all forecasting ranges. Because of the large number of numerical results in Tables 3, 4 and 5, it is necessary to investigate several elements of the forecasting model efficiency. On the resulting nRMSE metric, the Friedman and Nemenyi hypothesis testing^{76} was employed. Friedman test is a statistical method used to rank various models in classification or regression tasks. This test is particularly useful when these models exhibit inconsistent performance, i.e., their effectiveness varies from one dataset to another, making it challenging to determine the best model for a specific application. The FriedmanNemenyi test is an effective statistical approach that aids in the ranking and comparison of the models used.
Friedman's test has been employed to score the proposed models throughout all forecasting horizons and analyzed PV plants. If the order difference is statically important, then a Nemenyi hypothesis test is conducted by comparing all forecasting models pairbypair. The RMSE, as a dependent scale statistic, cannot be used for model comparison across several datasets^{77}. The previous study on time series forecasting has reported a similar concept, which can arise when alternative error metrics are utilized^{78}. As a result, the test is conducted totally with the nRMSE metric, and the results are shown in Fig. 8.
As can be shown, after employing the FriedmanNemenyi test statistic with 95% certainty, the suggested hybridization approach VMDCNNmodel delivers the best overall forecasting performance, outperforming hybrid and classic models when alltime series data is used.
The second part of our experiment investigates the fusing of multiscale features two by two to improve the effectiveness of the forecasting performance of PV power output. Tables 7, 8, and 9, show the achieved results of the suggested strategy. The forecasting combination’s performance was examined using distinct day types. The results of different fusing combinations are shown in Tables 6, 7, and 8, the best outcomes are shown in bold. According to the numerical indicators, all proposed combination methods succeed in increasing the forecasting accuracy compared to the conventional forecasting model and individual decomposition techniques combined with the deepCNN model, resulting in a reduced nRMSE = [0.44%4] for all studied regions.
Another important remark is that combining decomposition approaches results in a significant improvement compared with the case of using each decomposition technique individually. As expected the integration of IF and VMD algorithm with deepCNNmodel provides high forecasting precision for all gridconnected PV stations. These outcomes show clearly that combining multiscale features is highly beneficial for improving the forecasting precision of PV power^{87,88}.
Another important observation is that VMDCEEMDANCNN, VMDWPDCNN, and VMDIFCNN models are ranked as the best forecasting models for PV power output production, with a significant improvement achieved by the VMDIFCNN model for all studied PV power stations.
Figures 9, 10 and 11 exhibits the forecasting outcomes of all suggested multiscale approach combinations in terms of the nRMSE indicator. According to the findings shown in Fig. 9, all suggested combinations produce strong forecasting results and increase the forecasting precision of individual decomposition approaches; in which the nRMSE error of the proposed hybridization technique varies within the range of [0.44–4].
As can be observed, after the employment of the Friedman–Nemenyi statistical test with 95% confidence, it was discovered that the suggested VMDIFCNNmodel combination approach delivers the best forecasting performance, and outperforms its counterpart hybrid models for used time series data (See Fig. 12).
Furthermore, an effective PV powerforecasting algorithm should not be timeconsuming; however, the suggested forecasting strategy is based on a deep learning technique, which requires time during the training phase. This indicates that training the deep CNN model may take a long time if a large number of similar multiscale samples are employed. In this respect, Forward Stepwise Regression Algorithm (FSRA) is utilized for selecting the most appropriate IMFs input for PV power forecasting. The primary purpose of identifying the most significant IMFs inputs is to eliminate redundant features, which reduces the computation costs and model complexity^{89}. The threshold between approved and rejected features was empirically established during the feature selection process. The developed models are assessed and trained by sequential feature sets obtained by including the next essential IMF elements continuously. The statistical forecasting metrics of the three analyzed PV stations are obtained by two scenarios, the suggested IFVMDCNN model with all IMFs elements, and the proposed IFVMDFSRACNN model combined with a feature selection approach (FSRA), which are illustrated in Table 9 for the three gridconnected photovoltaic plants. The benefit of combining our forecasting technique with a feature selection approach on forecasting accuracy and information space saving is observed. For all analyzed PV power stations, the FSRA algorithm was very useful in minimizing the amount of IMFs inputs necessary for training our deep neural network model while maintaining the forecasting quality of PV power output.
The undesired IMF features differ slightly by regional location, and forecasting time step, taking the case of the Laghouat PV station 30 min ahead forecasting, the selected IMFs components are 120 IMFs, and the unwanted IMFs numbers are 42. For Sidi Bel Abbés station, the FSRA algorithm removes 46 unneeded IMFs elements from the entire dataset while maintaining the forecasting effectiveness of the IFVMDCNN model. The FSRA technique at Ghardaia station removes 84 redundant IMFs elements from the entire dataset while retaining the forecasting ability of the proposed IFVMDCNN model. The FSRA algorithm was useful in reducing the amount of IMFs elements for training our model while preserving forecasting effectiveness. This reveals the findings that FS techniques can improve the accuracy of PV power forecasting methods. It is concluded that using a feature selection technique in training the suggested methodology improves forecasting accuracy and computation efficiency. The efficacy of FS methods is highly dependent on the dataset employed since the appropriate quantity of input varies from one PV plant to another. In addition, for each forecasting time step in each analyzed location, a distinct quantity of IMFs was employed as inputs. The findings clearly show that the proposed technique not only provided a remarkable precision rate but also ensured precise forecasting results throughout all periods and regions examined.
An additional evaluation was carried out to compare our proposed model with existing stateoftheart models in PV power forecasting. The complexity of directly comparing our model to those from previous studies arises from variations in data duration, input variables, climate conditions, and the fact that most studies focus solely on onestepahead forecasting at a specific time scale. According to Table 10, our hybridization approach achieves superior forecasting performance relative to numerous preceding efforts. The IFVMDFSRACNN method we propose demonstrates enhanced suitability for forecasting PV power output, highlighting its efficacy and potential advancements in the field.
Limitations of the study
Our model demonstrates considerable potential in forecasting photovoltaic power yet encounters several challenges. These challenges encompass reliance on the quality of data, a propensity for overfitting when using small datasets, questions about its adaptability to various geographical locations and types of PV systems, and the need for significant computational resources due to its complexity.
In this research, our focus was primarily on the three region, constrained by the availability of data. It is imperative for future studies to broaden their scope to encompass additional regions characterized by diverse weather conditions, especially areas prone to cloudy skies. Examining the efficacy of different PV panel technologies across these varied climates could shed light on the versatility and effectiveness of forecasting models in differing environmental settings.
Moreover, although we employed deep learning techniques such as CNNs, LSTMs, which have shown encouraging outcomes, there remains room for exploration of other sophisticated deep learning architectures. Techniques like the Transformer and Informer, known for their prowess in time series forecasting, present promising avenues for future work. Investigating these models could uncover new strategies to enhance the accuracy and efficiency of PV power forecasting models.
Conclusion
The present study investigates the performance assessment and shortterm forecasting of three PV systems of a 73.1 MW located in Algeria. The proposed forecasting method, VMDIFFSRACNN is presented to deal with dynamic changes in PV power production 15–30 min ahead. It was tested on three gridconnected PV power plants, and the results were compared to hybrid and standalone models. The findings show that the VMDIFFSRACNN model outperforms the stateoftheart techniques and exhibits high forecasting accuracy for shortterm prediction of PV power. Therefore, this work concentrates on the issue of forecasting PV power in the absence of climatic data, using few historical PV plant data as inputs to the suggested model. to estimate the future PV power output. An indepth examination of several decomposition approaches coupled with a deep learning algorithm was carried out to get a better insight into what drives the performance of the PV dataset. This study indicates that the proposed combination mechanism is more suitable for multisite time series forecasting, and could be used for other gridconnected PV power plants.
For future work, we aim to enhance our forecasting models by exploring more sophisticated deep learning methods in conjunction with sky imager data for PV power forecasting. This direction is intended to improve the accuracy and adaptability of our forecasting models by incorporating detailed visual observations of the sky. Such advancements will not only refine solar irradiance predictions but also broaden the models' applicability across diverse geographical locations and environmental conditions. Our ongoing research efforts will focus on developing these integrated models to further optimize the utilization of renewable energy resources, contributing to significant progress in the field of PV power forecasting.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Abbreviations
 ACO:

Ant colony optimization
 ANFIS:

Adaptive neurofuzzy inference system
 ANN:

Artificial neural networks
 AR:

Autoregression
 ARIMA:

Autoregressive integrated moving average
 ARMAX:

Autoregressive moving average extrapolation
 aSi:

One amorphous
 BiGRU:

Bidirectional gated recurrent unit
 BILSTM:

Bidirectional long shortterm memory
 BPNN:

Back propagation neural network
 CEEMD:

Complementary ensemble empirical mode decomposition
 CGAN:

Conditional generative adversarial model network
 CNN:

Convolutional neural network
 DBN:

Deep belief network
 DCNN:

Deep convolutional neural network
 DL:

Deep learning
 EEMD:

Empirical mode decomposition
 ELM:

Extreme learning machine
 EMD:

Empirical mode decomposition
 FCBF:

Fast correlationbased filter
 GA:

Genetic algorithm
 GANs:

Generative adversarial model networks
 GPR:

Gaussian process regression
 GRU:

Gated recurrent unit
 HSV:

Hue saturation value
 ICEEMDAN:

Improved complete ensemble empirical mode decomposition with adaptive
 IEA:

International energy agency
 IEMD:

Improved empirical mode decomposition
 IF:

Iterative filtering
 IGIVA:

Improved grey ideal value approximation
 LR:

Linear regression
 LSSVM:

Least squares support vector machines
 LSTM:

Long shortterm memory
 MAE:

Mean absolute error
 MAPE:

Mean absolute percentage error
 MEMD:

Multivariate empirical mode decomposition
 MLP:

Multilayer perceptron
 MOcsa:

Multiobjective chameleon swarm algorithm
 MODWT:

Maximum overlap discrete wavelet transform
 MPSO:

Modified particle swarm optimization
 MRE:

Mean relative error
 MRMGR:

Maximize relevancy minimize global redundancy
 mSi:

Eight monocrystalline noise
 NWP:

Numerical weather prediction
 PR:

Performance ratio
 pSi:

One polycrystalline
 PSO:

Particle swarm optimization
 PV:

Photovoltaic
 RBF:

Radial basis function
 RMSE:

Root mean square error
 RNN:

Recurrent neural network
 RVFL:

Random vector functional link
 SARIMA:

Seasonal autoregressive integrated moving average
 SCA:

Sine cosine algorithm
 SMAPE:

Symmetric mean absolute percentage error
 SPVP:

Solar photovoltaic plant
 SVM:

Support vector machine
 TSFCGAN:

Time series conditional generative adversarial model network
 VAE:

Variational autoencoder
 VMD:

Variational mode decomposition
 WGANGP:

Wasserstein GAN with gradient penalty
 WPD:

Wavelet packet decomposition
 WT:

Wavelet transform
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The researchers would like to acknowledge the Deanship of Scientific Research, Taif University for funding this work.
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This work is funded and supported by the Deanship of Scientific Research, Taif University.
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M.G., A.F., Z.M., A.R.: Conceptualization, Methodology, Software, Visualization, Investigation, Writing—Original draft preparation. K.F., N.B., S.B., A.R.: Data curation, Validation, Supervision, Resources, Writing—Review & Editing. E.A., M.B., S.A.D.M. and S.S.M.G.: Project administration, Supervision, Resources, Writing—Review & Editing.
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Guermoui, M., Fezzani, A., Mohamed, Z. et al. An analysis of case studies for advancing photovoltaic power forecasting through multiscale fusion techniques. Sci Rep 14, 6653 (2024). https://doi.org/10.1038/s4159802457398z
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DOI: https://doi.org/10.1038/s4159802457398z
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