1.1. Background

For many years, the global energy structure has predominantly relied on non-renewable fossil fuels. However, with the increasing severity of resource depletion and environmental pollution, countries worldwide have progressively shifted their focus towards developing low-carbon and even zero-carbon renewable energy sources ^1,2. Currently, new energy generation has secured a significant share within global power systems ³. Among these, photovoltaic (PV) power generation stands out for its cost-effectiveness, high efficiency, and abundant resource base, establishing itself as a prominent form of renewable distributed energy worldwide. Furthermore, the rise of the “net zero carbon” and “carbon neutral” concepts has accelerated the rapid advancement of distributed PV technologies ^4,5. Distributed PV systems involve the installation of relatively small-scale PV power generation units near end users, allowing for local electricity consumption and integration of surplus electricity into the grid, thereby alleviating the supply pressure on the grid ⁶. However, PV power generation is highly susceptible to fluctuations due to external meteorological changes, which can lead to voltage variations and localized power quality issues during grid integration, affecting the stability of the power system ⁷. Achieving accurate forecasting of PV power generation would enable better utilization of solar energy and more efficient scheduling and allocation of electrical resources ⁸.

1.2. Related Works

Common methods for PV power forecasting can be broadly categorized into two types: physical modeling methods and data-driven methods. Physical modeling methods predict the output power of PV systems based on fundamental physical principles, such as the physical characteristics of PV cells and circuit theory. For instance, in ⁹, Malik and Chandel developed a novel integrated single-diode solar cell model based on various solar cell parameters for PV power prediction. In ¹⁰, Kumar et al. introduced a new analytical technique to refine solar cell models with different parameters, thus enhancing the accuracy of PV system power forecasting. In ¹¹, Chaibi et al. analyzed equivalent circuit models of PV cells under varying irradiance and temperature conditions, and fusion between two equivalent circuit configurations (single-diode and double-diode models) based on actual meteorological inputs. In ¹², Villemin et al. modeled the thermal behavior of PV systems, estimating the temperature of PV panels using meteorological parameters and employing Monte Carlo methods to probabilistically interpret energy balance, thereby predicting PV power output based on the relationship between electrical efficiency and temperature.

Due to the extensive variety of feature parameters that physical models need to identify, their application in PV power forecasting is relatively limited. In contrast, data-driven methods offer greater flexibility and can automatically learn the intricate relationships within the data, making them more widely used in the field of PV power forecasting ¹³. Data-driven methods can be broadly classified into two categories: traditional statistical methods and modern machine learning techniques. Traditional statistical models are structurally simple and have relatively low computational complexity. For example, in ¹⁴, Chu et al. developed an auto-regressive moving average (ARMA) model to perform hourly power forecasting for a 48-megawatts PV power plant. In ¹⁵, Zhang et al. proposed an adaptive hybrid model by integrating auto-regressive integrated moving average (ARIMA) to forecast the PV power output for the following day. In ¹⁶, a multiple linear regression model incorporating performance ratio factors was introduced for forecasting hourly PV power output. In ¹⁷, Li et al. employed a generalized ARIMA model, incorporating meteorological data such as temperature and humidity as exogenous inputs to forecast PV power output. The proposed model is more versatile and flexible compared to traditional ARIMA models, making it suitable for practical applications. In ¹⁸, Kim et al. segmented the input data by season and employed seasonal ARIMA (SARIMA) to forecast PV output. In ¹⁹, Bouzerdoum and Mellit combined SARIMA with support vector machines (SVM) to forecast the power output of PV plants integrated into the grid. In ²⁰, Lotfi et al. proposed a kernel density estimation method that relies solely on weather forecasts and historical power generation data to predict distributed generation from renewable energy sources.

Compared to traditional statistical models, modern machine learning methods offer superior prediction accuracy and adaptability. They are capable of constructing more complex mathematical models to capture intricate patterns within the data, making them particularly well-suited for handling large-scale and high-dimensional datasets ²¹. In ²², a PV power forecasting learner based on decision trees was proposed, utilizing an enhanced attribute selection algorithm to effectively mitigate the impact of noisy data. In ²³, Brester et al. suggested that historical weather observation data can be employed to train an artificial neural network model as an alternative to physical models for daily PV power forecasting. In ²⁴, Wang et al. employed an enhanced empirical mode decomposition (EMD) method and particle swarm optimization (PSO) method. They transformed the PV power output into intrinsic mode functions (IMFs), which were then predicted individually. In ²⁵, Hossain and Mahmood applied the \(K\)-means algorithm to classify historical irradiance data, subsequently using the data from each category to train a long short-term memory (LSTM) neural network. In ²⁶, Li et al. utilized wavelet packet decomposition (WPD) to decompose the original PV power time series data into multiple subsequences. Each subsequence was independently predicted using an LSTM network, and the results were then reconstructed to achieve a 5-minutes interval forecast of PV power. In ²⁷, Lee and Kim attempted to predict the peak PV power output by using a gated recurrent unit (GRU) model, taking only past meteorological information and the PV power output from the previous period as inputs.

In comparison with individual prediction models, ensemble models have been more widely applied in diverse PV power generation scenarios due to their enhanced generalization capabilities. In ^28,29, a combination of LSTM and recurrent neural networks (RNN) was applied to forecast PV power generation under various time intervals and weather conditions. In ³⁰, a novel LSTM-ARMA hybrid model was proposed, where the LSTM component captures long-term dependencies, and the ARMA model addresses short-range dependencies. In ³¹, Zhang et al. developed a hybrid interval forecasting model by combining fuzzy information granulation, an improved long and short-term memory network (ILSTM), and a differential ARIMA model, aimed at predicting the range of PV power output. In ^32,33, convolutional neural networks and LSTM models were combined to extract both temporal and spatial features affecting PV power generation, thereby achieving precise forecasting of PV output. Although the aforementioned models achieve relatively high prediction accuracy, their interpretability tends to decrease correspondingly with the increasing complexity of such models.

The fuzzy inference method proposed by Zadeh in 1965 employs fuzzy rules and membership functions to articulate the behavior of a system, offering greater interpretability compared to machine learning methods often referred to as “black-box models” ³⁴. In recent years, the fuzzy method has garnered increasing attention in various research directions related to modeling and control. Some researchers have progressively applied fuzzy methodologies to the domain of PV power forecasting ³⁵. In ³⁶, Fujita and Kanzawa proposed three fuzzy clustering algorithms for sequential data based on shape distance, which demonstrated high clustering accuracy across multiple benchmark datasets. In ³⁷, Calinao Jr. et al. proposed a fault detection method for PV systems that integrates fuzzy logic with a radial basis function neural network. This approach enables high-precision fault classification in grid-connected PV systems and facilitates the effective identification of anomalous data in PV power generation forecasting. In ³⁸, Liao et al. proposed a PV power forecasting method based on a T–S fuzzy neural network, approaching the problem from a nonlinear perspective. In ³⁹, a combination of wavelet decomposition and adaptive neuro-fuzzy inference system (ANFIS) was utilized for short-term PV output forecasting. In ⁴⁰, Monfared et al. developed a novel universal technique that inputs the relevant statistical features from PV data into a fuzzy predictor, achieving accurate ultra-short-term forecasting with limited training data. In ⁴¹, Li et al. proposed an enhanced fuzzy \(C\)-means (FCM) clustering method based on multiple meteorological features, reducing the impact of clustering results influenced by a single variable. In ⁴², a method utilizing monotonic fuzzy systems for forecasting PV output was proposed. In ⁴³, a hybrid fuzzy-PSO forecasting model incorporating novel fuzzy input variables was proposed.

1.3. Motivations and Contributions

Improving prediction accuracy is often achieved by either increasing the number of input variables or using hybrid models that combine various types of models. However, in traditional fuzzy methods, an excessive number of input variables can result in an exceedingly complex model structure, leading to issues such as an overabundance of training parameters and difficulties in training, and it may even cause rule explosion. This elucidates the reason for the limited application of fuzzy methods in multi-variable PV power forecasting tasks, despite their considerable interpretability.

In addition, all of the aforementioned studies focused primarily on the global prediction performance of the models. Although some methods were proposed to enhance prediction accuracy, their improvements are typically assessed from a global perspective, often overlooking the potentially significant local prediction errors, especially at some peak or valley points.

To simplify the fuzzy model structure and decrease large local prediction errors commonly faced in PV power forecasting, this paper first proposes a distributed function-weighted fuzzy method (DFWFM), and then develops a data-driven multi-scale fusion model (DMFM). The primary innovations and contributions of this study are as follows:

• To address the rule explosion problem commonly encountered in traditional fuzzy methods, this study proposes the DFWFM. For each input variable, a single-input fuzzy module (SIFM) is constructed. The output of the DFWFM is determined based on the outputs of these SIFMs and the corresponding multivariate functional weights. This approach can significantly reduce the number of fuzzy rules, and can simplify the model structure.

• To assure the forecasting performance, one iterative learning method is presented for the DFWFM. In this method, the regularized least squares method is employed to train the functional weights and consequent parameters in the fuzzy rules iteratively. Furthermore, the regularization mechanism in this learning method can effectively help to reduce the risk of overfitting.

• To address the issue of substantial local prediction errors, a DMFM is developed. After training a global DFWFM, a distinctive encoding mechanism is established to identify and mark regions with significant errors. Data from these regions are then used to train local DFWFMs. The DMFM is constructed by combining the global model with these local models. The DMFM improves the global model performance by enhancing local prediction accuracy. To the best of the authors’ knowledge, this is the first time to focus on the local prediction errors in PV forecasting.

• The proposed model was applied to two real-world PV power prediction scenarios, with detailed comparisons provided. Experimental results demonstrate that the model offers the best prediction performance when considering both training speed and accuracy, thereby validating the effectiveness of this model in PV power forecasting.

The remainder of the paper is organized as follows. In Section 2, the proposed DFWFM is introduced. In Section 3, the proposed DMFM is presented. In Section 4, two experiments are conducted and compared. Finally, the conclusions are presented in Section 5.

2.1. Structure of the Proposed DFWFM

In this DFWFM, a SIFM is independently constructed for each input variable. The output of each SIFM is then weighted and integrated to obtain the total output of the DFWFM.

For each input variable \(x_i\), the fuzzy rules of the corresponding \(\mathit{SIFM}\)-\(i\) can be represented as

where \(m\) represents the number of fuzzy rules in the \(\mathit{SIFM}\)-\(i\); \(A_i^1, A_i^2, \dots, A_i^m\) denote the fuzzy sets for the variable \(x_i\); \(c_i^1, \dots, c_i^m\) and \(d_i^1, \dots, d_i^m\) are the consequent parameters in the consequent parts of the fuzzy rule \(R_i^1, R_i^2, \dots, R_i^m\).

Utilizing the singleton fuzzification and weighted defuzzification methods, the fuzzy inference output of \(\mathit{SIFM}\)-\(i\) can be expressed as

where \(\mu_{A_i^j}\) is the membership degree reflecting the extent to which the element \(x_i\) belongs to the fuzzy set \(A_i^j\).

The final output \(\hat y\left( {\mathbf{x}} \right)\) of the DFWFM is obtained by weighting and aggregating the outputs of the \(n\) SIFMs, and can be expressed as

where \({\omega _i}\left( {\mathbf{x}} \right) = \omega _i^0 + \omega _i^1{x_1} + \dots + \omega _i^n{x_n}\) is a multivariable function of \(\mathbf{x} = (x_1, x_2, \dots, x_n)\), which represents the dynamic weight or dynamic significance of the \(\mathit{SIFM}\)-\(i\).

Since each SIFM is a standalone single-input single-output fuzzy model and each input variable has \(m\) fuzzy sets, the number of fuzzy rules of DFWFM is \(m \times n\). Consequently, DFWFM can effectively reduce the number of rules in multi-variable fuzzy time series prediction tasks.

2.2. Data-Driven Design of the Proposed DFWFM

In this subsection, a detailed parameter learning method based on the regularized least square is provided.

2.2.2. Parameter Learning Based on the Regularized Least Squares Method

Assuming there are \(L\) input–output data pairs \(({\mathbf{x}}^1, {y^1}), ({\mathbf{x}}^2, {y^2}), \dots, ({\mathbf{x}}^L, {y^L})\) in the training data set, where \({{\mathbf{x}}^l} = ( {x_1^l,x_2^l, \dots ,x_n^l} )\), theoretically, the optimal solutions for the functional weights and the consequent parameters in the fuzzy rules of the DFWFM can be determined by the least squares method. The loss function for the least squares method can be expressed as

\begin{equation} \label{eq:8} E\left( {{\mathbf{w}},{\mathbf{c}}} \right) = \displaystyle\sum\limits_{l = 1}^L {{{\left( {\hat y\left( {{{\mathbf{x}}^l},{\mathbf{w}},{\mathbf{c}}} \right) - y^l} \right)}^2}}, \end{equation}

where \(\hat y( {{{\mathbf{x}}^l},{\mathbf{w}},{\mathbf{c}}} )\) represents the inference output of the DFWFM, and \(y^l\) is the actual output.

However, in practical applications, it becomes evident that using only the least squares method to determine the optimal solution of the model can make it susceptible to multicollinearity and overfitting. Multicollinearity can cause instability in the regression coefficients, resulting in poor prediction performance on new data. Similarly, overfitting can diminish the model’s generalization capability. To address these issues, the L2 regularization constraint is always adopted in the parameter learning of the DFWFM.

The L2 constraint limits the magnitude of the regression coefficients by adding a regularization term to the loss function, which smooths the coefficients and thereby can reduce overfitting to the training data. This enhances the model’s prediction performance on new data. Incorporating the L2 regularization constraint, the loss function expression according to Eq. (6) can be rewritten as

\begin{align} \label{eq:9} E\left( {{\bf{w}},{\bf{c}}} \right) &= \sum\limits_{l = 1}^L {{{\left( {g{{\left( {{{\bf{x}}^l},{\bf{c}}} \right)}^{\rm{T}}}{\bf{w}} - {y^l}} \right)}^2}} \nonumber\\ &\phantom{=~}+ {\lambda _{\bf{w}}}\left( {{{\left( {\omega _1^0} \right)}^2} + \dots + {{\left( {\omega _n^n} \right)}^2}} \right)\nonumber\\ &= \left\| {\Phi \left( {\bf{c}} \right){\bf{w}} - {\bf{y}}} \right\|_2^2 + {\lambda _{\bf{w}}}\left\| {\bf{w}} \right\|_2^2, \end{align}

where

\begin{align} \Phi \left( {\mathbf{c}} \right) &= {\left[ {g\left( {{{\mathbf{x}}^1},{\mathbf{c}}} \right),g\left( {{{\mathbf{x}}^2},{\mathbf{c}}} \right), \dots ,g\left( {{{\mathbf{x}}^L},{\mathbf{c}}} \right)} \right]^{\rm{T}}}, \label{eq:10} \\ \end{align}

\begin{align} {{\mathbf{y}}} &= {\left[ {y^1,y^2, \dots ,y^L} \right]^{\rm{T}}}. \label{eq:11} \end{align}

Alternatively, according to Eq. (7), incorporating the L2 regularization constraint, the loss function expression can also be rewritten as

\begin{align} \label{eq:12} E\left( {{\bf{w}},{\bf{c}}} \right) &= \sum\limits_{l = 1}^L {{{\left( {f{{\left( {{{\bf{x}}^l},{\bf{w}}} \right)}^{\rm{T}}}{\bf{c}} - {y^l}} \right)}^2}} \nonumber\\ &\phantom{=~}+ {\lambda _{\bf{c}}}\left( {{{\left( {c_1^1} \right)}^2} + {{\left( {d_1^1} \right)}^2} + \dots + {{\left( {c_n^m} \right)}^2} + {{\left( {d_n^m} \right)}^2}} \right)\nonumber\\ &= \left\| {\Psi \left( {\bf{w}} \right){\bf{c}} - {\bf{y}}} \right\|_2^2 + {\lambda _{\bf{c}}}\left\| {\bf{c}} \right\|_2^2, \end{align}

where

\begin{equation} \label{eq:13} \Psi \left( {\mathbf{w}} \right) = {\left[ {f\left( {{{\mathbf{x}}^1},{\mathbf{w}}} \right),f\left( {{{\mathbf{x}}^2},{\mathbf{w}}} \right), \dots ,f\left( {{{\mathbf{x}}^L},{\mathbf{w}}} \right)} \right]^{\rm{T}}}. \end{equation}

It is challenging to simultaneously determine the optimal solutions for \(\mathbf{w}\) and \(\mathbf{c}\) that minimize the loss function. However, as shown in Eq. (9), if the parameter vector \(\mathbf{c}\) is known, the optimal solution for the parameter vector \(\mathbf{w}\) can be determined. Similarly, as indicated in Eq. (12), if the parameter vector \(\mathbf{w}\) is known, the optimal solution for the parameter vector \(\mathbf{c}\) can be found. Therefore, the optimal solutions for \(\mathbf{w}\) and \(\mathbf{c}\) can be sequentially determined by alternately solving the following equations:

\begin{equation} \label{eq:14} \left\{ \begin{array}{l} \mathop {\min }\limits_{\mathbf{w}} \left\| {\Phi \left( {\mathbf{c}} \right){\mathbf{w - }}{{\mathbf{y}}}} \right\|_2^2 + {\lambda _{\mathbf{w}}}\left\| {\mathbf{w}} \right\|_2^2,\\ \mathop {\min }\limits_{\mathbf{w}} {\left\| {\mathbf{w}} \right\|_2}, \end{array} \right. \end{equation}

and

\begin{equation} \label{eq:15} \left\{ \begin{array}{l} \mathop {\min }\limits_{\mathbf{c}} \left\| {\Psi \left( {\mathbf{w}} \right){\mathbf{c - }}{{\mathbf{y}}}} \right\|_2^2 + {\lambda _{\mathbf{c}}}\left\| {\mathbf{c}} \right\|_2^2,\\ \mathop {\min }\limits_{\mathbf{c}} {\left\| {\mathbf{c}} \right\|_2}. \end{array} \right. \end{equation}

Based on the derived formulas, we propose a data-driven iterative learning algorithm to learn the parameters of DFWFM. The specific iterative learning process is outlined in Algorithm 1 in pseudocode format.

3.1. Overall Process of the Proposed DMFM

As previously mentioned, the global prediction performance improvements are often achieved by increasing the number of input variables or by combining different models. However, these methods are not entirely suitable for enhancing PV power prediction. This is because, during prediction, significant deviations often occur during peak periods of concentrated PV power generation. Ignoring these critical periods of prediction error and focusing solely on global model improvements make it difficult to achieve precise PV power forecasts. Therefore, to address the issue of substantial prediction errors in peak regions and other local areas during PV power forecasting, this section proposes the DMFM, which focuses on enhancing both the global and local prediction performance. The overall process of the proposed DMFM, based on the DFWFM, is illustrated in Fig. 1 and involves the following main steps:

• Step 1: Train the DFWFM using the PV power dataset to establish a global model.

• Step 2: Develop a partitioning mechanism for the input space, dividing the complete input space into a series of local input regions.

• Step 3: Compare the global model’s predictions with actual values to identify and label local regions with poor prediction performance.

• Step 4: Use the training data from the labeled local regions to train local DFWFMs, while updating the global DFWFM with the training data that were not labeled.

• Step 5: Combine the local DFWFMs with the updated global DFWFM to form the DMFM.

The following subsections provide a detailed implementation methodology for these steps.

3.2. Input Space Partitioning

The input space partitioning of the global DFWFM consists of a set of non-overlapping intervals ⁴⁴. The input domain of variable \(x_i\) can be divided into \(m-1\) regions by the \(m\) fuzzy sets, as illustrated in Fig. 2, where \({\mathbf{P}}_i^{k_i}\) (\(i = 1,2, \dots, n; k_i = 1, 2, \dots, m-1\)) represents the input domain corresponding to the \(k\)-th partition of the input variable \(x_i\).

Let \({{\mathbf{P}}^{{k_1},{k_2}, \dots ,{k_n}}}\) denote the set of the local input domains \({\mathbf{P}}_i^{{k_i}}\left( {i = 1,2, \dots ,n} \right)\). In other words, \({{\mathbf{P}}^{{k_1},{k_2}, \dots ,{k_n}}}\) can be represented as:

where \({k_i} = 1,2, \dots ,m - 1\), \(i = 1,2, \dots ,n\).

It is worth noting that each input \({\mathbf{x}} = ( {{x_1},{x_2}, \dots ,{x_n}} )\) should belong to only one local input domain \({{\mathbf{P}}^{{k_1},{k_2}, \dots ,{k_n}}}\).

3.3. Fusion of the Global and Local DFWFMs

In the global DFWFM, some regions often exhibit substantial errors, indicating that the existing fuzzy rules may not be suitable for predicting the PV power in these local regions. The proposed DMFM identifies the training data within these high-error regions and retrains new fuzzy rules based on these data patterns to construct corresponding local DFWFMs. Subsequently, the remaining training data are used to update the global DFWFM. In essence, the proposed DMFM is a combination of local models and the updated global model. The flowchart for constructing the DMFM based on the DFWFM is illustrated in Fig. 3. Detailed steps are as follows:

• Step 1: Utilize the global DFWFM to predict the PV power.

• Step 2: Calculate the sum of squared errors (SSE) for the training data in all \({\rm{P}}^{k_1,k_2,\dots,k_n}\) (\(k_i = 1, 2, \dots, m-1; i = 1,2,\dots,n\)). The SSE is employed to quantify the discrepancy between the predicted values and the actual values of the DFWFM, which is given by \(\mathit{SSE} = \sum{l = 1}^L {{{ ( {\hat y^l - y^l} )}^2}}\), where \(L\) denotes the total number of input–output data pairs, \({\hat y^l}\) represents the predicted value, and \(y^l\) denotes the true value.

• Step 3: Sort the SSE results in the descending order, and the top \(N\) values of SSEs correspond to the local input domains that need to be marked.

• Step 4: Label the training data that fall into the \(N\) local input domain ranges.

• Step 5: Train the \(N\) local models using the labeled training data, and utilize the remaining training data, which do not fall into any local model and have not been labeled, to update the global model.

• Step 6: If \(\mathbf{x}\) is within the local input domain of the \(I\)-th local model, the corresponding local model is employed to make the prediction, yielding the predicted value \({\hat y^{\left( I \right)}}\). Conversely, if \(\mathbf{x}\) does not fall within the input domain range of any local model, the updated global model is used for prediction, resulting in the predicted value \({\hat y^{\left( 0 \right)}}\). In other words, the final predicted output of DMFM can be computed as
  \begin{equation} \label{eq:17} \hat y\left( {\bf{x}} \right) = \left\{ {\begin{array}{*{20}{l}} {{{\hat y}^{\left( I \right)}}},\\ {{{\hat y}^{\left( 0 \right)}}}, \end{array}} \right. \end{equation}
  where \({{\hat y}^{\left( I \right)}}\) represents \({\bf{x}}\) that falls into the \(I\)-th local input domain \({\left( {I = 1,2, \dots ,N} \right)}\), \({{{\hat y}^{\left( 0 \right)}}}\) represents \({\bf{x}}\) that does not fall into the \(N\) local input domains.

4.1. Comparative Methods

To demonstrate the superiority of the proposed model, we select ANFIS, fuzzy neural network (FNN), support vector regression (SVR), back propagation neural network (BPNN) , and GRU as the comparative models. We now provide a brief overview of such comparative methods.

ANFIS is a hybrid model that integrates fuzzy inference systems with neural networks ⁴⁵. It utilizes the neural network framework to extract fuzzy rules from data through an adaptive hybrid algorithm, continuously adjusting the antecedent and consequent parameters of the rules to obtain the optimal solution. ANFIS combines the advantages of fuzzy inference systems and neural networks while addressing their individual limitations, making it one of the most commonly used fuzzy time series prediction models today.

FNN is a hybrid model that integrates fuzzy logic with neural networks ⁴⁶. It first fuzzifies the input data and then applies fuzzy rules for inference, making it well-suited for handling complex tasks characterized by uncertainty and vagueness. The FNN not only possesses the learning capabilities of neural networks but also benefits from the strong interpretability of fuzzy logic, which makes it widely used in fields such as control and prediction.

SVR is a variant of SVM, specifically designed for regression problems ⁴⁷. SVR excels in handling high-dimensional feature spaces and nonlinear regression, demonstrating significant advantages in these areas. Additionally, it exhibits strong robustness in the presence of noise and outliers, making it widely applicable in fields such as time series forecasting.

BPNN is one of the most fundamental neural network structures ⁴⁸, capable of approximating any continuous nonlinear function. As the complexity of the network architecture increases, BPNN can learn more intricate features, thereby addressing more complex problems. However, it is important to be cautious of overfitting due to excessive parameters. BPNN is among the most commonly used neural networks and finds extensive application in areas such as speech processing and time series forecasting.

GRU is an advanced variant of RNN ⁴⁹, specifically designed for handling sequential data. By incorporating update and reset gates, GRU effectively retains crucial information over extended sequences while reducing the number of parameters. As a significant variation of RNN, GRU excels in applications such as natural language processing and time series forecasting.

4.2. Evaluation Indices

The forecasting performance of all models is evaluated using three metrics: the mean absolute error (MAE), the root mean squared error (RMSE), and the coefficient of determination (\({R^2}\)). The formulas for calculating these metrics are as follows:

Lower values of MAE and RMSE suggest that the predicted values are closer to the actual values, thereby indicating superior prediction performance of the model. The \({R^2}\) value ranges from 0 to 1, with values closer to 1 reflecting greater prediction accuracy.

4.3. PV Generation Prediction Without Meteorological Factors

4.3.3. Experimental Results and Comparative Analysis

In this experiment, the number of local models in the DMFM is set to be 0, 1, 2, or 3, respectively. When the number of local models is set to be 0, it indicates that there is only one global model with no local models, meaning all input variables fall within the domain of the global model. In this case, the prediction models are the traditional ones. Additionally, LM1, LM2, and LM3 are used to represent the scenarios with 1, 2, and 3 local models, respectively.

To demonstrate the improvement performance of the proposed model, Table 1 lists the MAE, RMSE, \({R^2}\) for each model during both training and testing phases, along with the training time and the number of fuzzy rules (NoFR) or neuro nodes (NoNN).

From Table 1, it can be evident that compared to other fuzzy models, DFWFM consistently exhibits the fastest training speed and the fewest fuzzy rules. Moreover, this model maintains relatively optimal prediction performance across various numbers of local models. This indicates that the proposed model structure is relatively simple, allowing it to reduce model complexity while preserving prediction accuracy.

Additionally, Table 1 shows that as the number of local models in the DMFM increases, the training performance of the six prediction models improves. At the same time, since each additional local model requires the fuzzy rules or neuro nodes of the original base model to be retrained, the computational resource consumption also increases. From a theoretical perspective, in the absence of computational resource constraints, increasing the number of local models should further enhance the prediction performance of the DMFM. However, as shown in Table 1, nearly every model exhibits optimal testing or generalization performance when the number of local models in the DMFM is set to be 2, and further increasing the number of local models may actually degrade the model’s generalization performance or testing accuracy. This may be due to the atypical nature of PV power generation data or insufficient data for training local models, leading to overfitting.

Figure 4 illustrates the evaluation metrics of the six prediction models when two local models are employed. It can be observed from this figure that the DFWFM has nearly the smallest MAE and RMSE values, as well as an \({R^2}\) value that is closest to 1. This indicates that the DFWFM demonstrates superior prediction performance among the assessed models.

Figure 5 presents part of the prediction errors of the six prediction models using two local models, displayed in both two-dimensional and three-dimensional forms. The prediction errors in this figure represent the absolute differences between the predicted data and the real data at each sample point. From this, we can observe that after considering local models, all prediction models exhibit relatively small prediction errors. Furthermore, it is clear from this figure that the maximum prediction error of DFWFM_LM2 is significantly lower than that of the other models, indicating that the proposed model has better prediction performance.

To intuitively demonstrate the superiority of the proposed method in reducing local prediction errors, Fig. 6 presents part of the prediction results of the DFWFM-DMFM. As can be observed, compared to using only the global model, adding extra local models can significantly reduce local prediction errors, particularly around peak points where the predicted values are closer to the actual values. This indicates that the proposed model can capture the dynamics of PV power generation more accurately.

4.4. PV Power Generation Prediction Considering Meteorological Factors

4.4.2. Configuration of the Prediction Models

In this experiment, 10 input variables-comprising the current meteorological data (ambient temperature, surface solar radiation, direct normal solar radiation, and diffuse solar radiation) and the historical PV power generation data from the previous six time steps-are used to predict future PV power generation. The parameter settings for the DMFM based on DFWFM, ANFIS, FNN, SVR, BPNN, and GRU are identical to those used in the previous experiment.

4.4.3. Experimental Results and Comparative Analysis

To illustrate the enhanced performance of the proposed models, Table 2 presents the MAE, RMSE, \({R^2}\) for each model during both training and testing phases, along with the training time and the NoFR or NoNN.

It is noteworthy that the number of input variables involved in this experiment has significantly increased, with all prediction models having 10 input variables and one output variable. ANFIS and FNN encountered a “rule explosion” problem due to the large number of fuzzy rules generated by the 10 input variables, which led to training failures for these models.

Table 2 shows that DFWFM consistently exhibits relatively fast training speed and optimal prediction performance in every case. Notably, its number of fuzzy rules is significantly lower than that of ANFIS and FNN. This indicates that increasing the number of input variables enhances the prediction performance of DFWFM without compromising the model’s structure. Therefore, for high-dimensional input prediction tasks, DFWFM is more suitable than traditional fuzzy inference methods.

The output power of PV systems is influenced by various meteorological factors. Accurately integrating these meteorological data allows for a more precise simulation of the actual generation environment, thereby significantly enhancing the accuracy of the prediction models. Neglecting these meteorological factors could result in predictions that deviate more significantly from reality. Thus, the inclusion of meteorological factors is one of the key reasons for the absence of overfitting in this case.

Compared to the first experiment, this one additionally incorporates meteorological data. From Table 2, it is evident that the performance of all models in both training and testing significantly improves with the increasing number of local models in DMFM. Specifically, the MAE and RMSE values for each model decrease progressively on both the training and test sets, while the \({R^2}\) values increase correspondingly. When the number of local models in the DMFM is set to be 3, nearly every model demonstrates optimal prediction performance.

Figure 7 presents the evaluation metrics for the four prediction models utilizing three local models. It is apparent that the DFWFM exhibits the best prediction performance.

Figure 8 presents part of the prediction errors of the four prediction models using three local models, displayed in both two-dimensional and three-dimensional forms. It can be observed that all prediction models considering local models exhibit relatively small prediction errors. Furthermore, the maximum prediction error of DFWFM is significantly lower than that of the other models.

To intuitively demonstrate the superiority of the proposed method, Fig. 9 presents part of the prediction results of the DFWFM-DMFM with three additional local models. As can be observed, the DMFM with three local models aligns more closely with the actual values, particularly showing a significant reduction in prediction errors in the peak regions. This further validates the effectiveness of additional local models in enhancing local prediction accuracy.

4.5. In-Depth Analysis of the DMFM

4.5.1. Interpretability of the DFWFM

A model or system can be defined as interpretable if it presents its decision-making process and outcomes in a manner comprehensible to humans. For instance, linear models exhibit interpretability because their coefficients clearly characterize the linear influence of input variables on the output results. Moreover, fuzzy models provide a transparent reasoning process by utilizing fuzzy rules to articulate the relationships between inputs and outputs, thereby enhancing decision interpretability.

After the learning of the DFWFM-DMFM, all fuzzy rules constructed in the global and local DFWFMs can be obtained. The fuzzy models in the first and the second experiments are listed in Tables 3 and 4, respectively.

By examining the activation strengths of these fuzzy rules presented in these two tables, we can accurately comprehend the activation effects of each fuzzy rule. The triggered rules and the data that activated the corresponding rules can be clearly displayed, providing valuable insights. Furthermore, the parameters within the fuzzy rules can also be interpreted through fuzzy or qualitative knowledge. This further substantiates the strong interpretability of the proposed DFWFM-DMFM.