A Belief Rule Base with Fuzzy Reference Value for Wind Power Generation Forecasting

Abstract

Wind power generation forecasting is a key technology for wind power projects. It directly determines the stability of grid integration and the accuracy of power dispatching. The interval belief rule base (IBRB) is an uncertainty modeling method; it can be applied to wind power generation forecasting. On the one hand, IBRB uses fixed interval matching. This method tends to cause boundary jumps when predicting continuously variable parameters, which threatens the stability of the grid integration. On the other hand, IBRB underutilizes the correlation information of adjacent intervals in modeling, and its rule activation mechanism limits expressions of complex generation mechanisms. To address these issues, a method based on belief rule base with fuzzy reference value (BRB-f) for wind power generation forecasting is proposed. Firstly, the method replaces fixed interval matching with fuzzy membership functions to reduce the impact of wind power output fluctuations on the grid. Then, through a multi-rule-weighted fusion mechanism and optimization algorithms, it improves the accuracy of scheduling under complex generation mechanisms. Finally, the effectiveness and accuracy of the model are validated using a wind turbine power generation forecasting dataset. It provides a better method choice to ensure grid integration safety and enhance the scientific basis of power dispatch decisions.

1. Introduction

Wind power generation forecasting has a significant impact on the stability of grid integration [1,2,3] and the accuracy of power dispatching. However, complex factors affecting wind power generation exhibit strong volatility and uncertainty, accurately predicting wind power generation faces great challenges. The reliability of wind power generation forecasting is directly related to the safety of the grid integration; additionally, this affects the scientific basis of power dispatch decisions. Therefore, conducting research on high-precision and robust wind power generation forecasting holds important theoretical value and engineering significance.

Wind power generation forecasting as a key technology in the field of renewable energy [4,5,6,7] can be primarily divided into three research methods: (1) white-box models: white-box models have simple and transparent structures, with clear and traceable decision logic, making them easy for users to understand and trust. Liao et al. [8] constructed the glass-box wind power forecasting model, using shape functions and interaction terms for modeling. Jayasinghe et al. [9] used regression tree methods to predict renewable energy generation, and the model has good interpretability. White-box models have limited capability in handling complex nonlinear relationships or high-dimensional feature interactions, and their forecasting accuracy is often constrained by model complexity. (2) Black-box models: black-box models are typically based on complex network structures or ensemble methods, capable of automatically capturing nonlinear relationships in data. They pursue higher prediction accuracy at the expense of interpretability. Shinde et al. [10] proposed a wavelet-enhanced recurrent neural network with gated linear units (RNN-GLU) hybrid deep learning model to improve wind power forecasting accuracy. Hardy et al. [11] used convolutional neural networks (CNNs) to analyze outputs from artificial intelligence (AI) weather forecast models to predict wind speed and wind power. Sarkar et al. [12] introduced a multi-head attention Transformer model to enhance single-step and multi-step wind power prediction capabilities. The main drawbacks of this model include poor interpretability and opaque internal decision logic, resulting in high deployment barriers in practical applications. Barbosa et al. [13] proposed a hybrid forecasting model combining autoregressive integrated moving average (ARIMA) with dual neural networks. However, its drawbacks are poor interpretability and reliance on data extrapolation for long-term forecasts. Hossain et al. [14] proposed a hybrid deep learning model combining convolutional neural network (CNN), gated recurrent unit (GRU), and fully connected neural networks for ultra-short-term wind power forecasting. Its internal decision-making process lacks interpretability and cannot clearly reveal the physical mechanisms of wind power generation. (3) Gray-box models: gray-box models aim to combine the interpretability of white-box models with the prediction accuracy of black-box models. They typically adopt two approaches: integrating the two modeling strategies, or introducing interpretable components into black-box structures. The goal is to achieve a balance between performance and interpretability. Khurram Mushtaq et al. [15] used multivariate adaptive regression splines (MARS) and regression trees to model wind turbine power curves, improving accuracy while maintaining a certain level of interpretability. Yang et al. [16] proposed a short-term wind power prediction method based on error tracking and numerical weather prediction (NWP) wind speed correction, improving predictions through decomposition and matching strategies. Kandilogiannakis et al. [17] introduced the dynamic neuro-fuzzy wind predictor (DNFWP) model. This realizes dynamic power prediction using only current power data. Gray-box models often have complex structural designs and high costs for training and tuning. Their ‘partially interpretable’ characteristic may still fail to meet the demand for full transparency, and their stability and correction performance in extreme scenarios remain insufficient.

The different wind power generation forecasting methods mentioned above have their own characteristics. (1) Although white-box models sacrifice certain predictive performance and the ability to model complex relationships in exchange for stronger model interpretability, they are weak in capturing the complex interactions between variables. (2) Although black-box models have high prediction accuracy, at the cost of sacrificing interpretability, along with high computational costs and strong dependence on data, their internal decision-making process is opaque, which leads to results being hard to trust and validate. (3) Gray-box models offer a relatively balanced approach. Hybrid-driven methods can effectively integrate the advantages of the first two types, enabling the combined use of physical laws and data characteristics of complex power generation mechanisms. This not only ensures prediction accuracy but also maintains model interpretability, making them particularly suitable for complex engineering problems such as wind power generation forecasting. These methods include expert systems based on belief rule base, fuzzy inference systems, and hybrid neural network models.

The interval belief rule base (IBRB) is a typical gray-box model, which is designed to address the explosion of combination rules and the lack of rule reliability in traditional BRB. Cheng et al. [18] proposed a belief rule base with intervals (Intervals-BRB), which replaces reference points with reference intervals and adopts interval addition for rule combination. This fundamentally avoids the explosion of combination rules. Liu et al. [19] systematically proposed the concept of a fuzzy belief rule base (FBRB) for the first time, establishing a connection between the fuzzy rule base (FRB) and belief rule base (BRB) theories by extending the FRB to belief rules, thereby addressing the traditional FRB’s difficulty in quantifying uncertainty. Li et al. [20] proposed the belief language rule base (B-LRB), which integrates a fuzzy inference system (FIS) with belief rules, achieving their integration for the first time. This study is the first to introduce the quantification of belief degree in a belief rule base within FIS inference.

Although the IBRB model performs well in many static evaluation scenarios, it still faces significant challenges when directly applied to wind power generation forecasting, a field characterized by high nonlinearity and uncertainty. On the one hand, the IBRB model typically relies on fixed reference intervals for rule matching. This rigid matching can cause abrupt changes in output when input data crosses interval boundaries, affecting the stability of the output power. On the other hand, wind farms have complex, multi-source, and coupled influencing factors. Existing IBRB reasoning mechanisms face a challenge: how to flexibly and adaptively integrate the outputs of different rules. This integration aims to improve prediction accuracy. FRB and FIS achieve fuzzification by dividing the input variables into fuzzy sets. The fuzzy objects are the values taken by the input variables, the number of rules grows exponentially with the number of fuzzy sets, leading to the problem of combinatorial rule explosion.

To address the main challenges of the above IBRB model in wind power generation forecasting, as well as the limitations of applying FRB and FIS to BRB, this paper proposes a new wind power generation forecasting method based on a belief rule base with fuzzy reference value (BRB-f). The method introduces two key innovations, achieving significant performance improvements: (1) use a triangular membership function to represent the fuzzy range of reference values, each reference value corresponds to a rule, the number of rules is consistent with the total number of reference values, and no logical combination of fuzzy sets is required, enhancing the stability of the grid integration. (2) A multi-rule-weighted fusion mechanism is constructed to improve expressiveness in complex scenarios, significantly increasing prediction accuracy under complex power generation mechanisms through multi-rule-weighted fusion and optimization algorithms.

The remainder of this paper is organized as follows: in Section 2, the engineering challenges of wind power generation forecasting and the construction method of the BRB-f model are proposed. In Section 3, the detailed calculation method of fuzzy membership is provided, including the core ideas and implementation process. In Section 4, the rule activation method, reasoning process, and P-CMA-ES optimization strategy for the new wind power generation forecasting model are presented. In Section 5, a wind turbine power generation forecasting is used as a case study to verify the effectiveness of the method. The conclusions and prospects of this article are presented in Section 6.

2. Problem Description

The basic components of this chapter are as follows: some of the issues in wind power generation forecasting are described in Section 2.1. A new wind power generation forecasting model is described in Section 2.2.

2.1. Problem Statement of Wind Power Generation Forecasting

When forecasting wind power generation based on IBRB, the following issue also needs to be addressed:

Question 1: In engineering practice, when the input data is close to the interval boundary, slight changes can trigger changes in the activated rules, leading to the instability of the model. Since IBRB uses fixed interval matching, this can instantly activate a completely different rule, causing the model’s prediction output to change discontinuously and abruptly. This is extremely unfavorable for the stability of power generation forecasting. Therefore, we introduce a membership function to achieve fuzzy matching of reference values.

$$\begin{array}{l}{\mu }_j(x)=FM(x,R_j)\\ {\displaystyle \sum _{j=1}^K{\mu }_j(x)}=1, {\mu }_j(x)\in [0,1]\end{array}$$

(1)

where ${\mu }_j(x)$ represents the degree of membership of the input $x$ to the reference value $R_j$, $K$ represents the number of the reference values.

Question 2: Fixed interval matching loses information about the distribution of input values within the interval. For example, for wind speed values within the interval [5, 6] m/s, the same rule is activated whether it is closer to 5.1 or 5.9. This limitation hinders the model’s ability to capture the continuous influence of subtle variations in each interval-based influencing factor on the actual output, thereby reducing the model’s sensitivity and impeding accurate prediction. Consequently, it is essential to quantify the matching degrees between the input value and its two adjacent reference values through membership degree calculation.

$${\mu }_{left}={\displaystyle \frac{x_{right}-x}{x_{right}-x_{left}}}, {\mu }_{right}={\displaystyle \frac{x-x_{left}}{x_{right}-x_{left}}}$$

(2)

where ${\mu }_{left}$ and ${\mu }_{right}$ represent the membership degrees of the left and right adjacent reference values, respectively, $x_{left}$ and $x_{right}$ represent the adjacent reference values, respectively.

Question 3: When faced with complex, multi-factor coupled influential factors on power generation, the rule activation mechanism of IBRB can affect the model’s accuracy. In wind power generation forecasting, the output is often a nonlinear result of complex interactions among multiple power generation influencing parameters. The fixed interval matching mechanism cannot allow the model to simultaneously consider the degrees to which an input parameter belongs to two adjacent reference values, which limits the model’s ability to integrate multiple possibilities, thereby weakening its capability to fit complex wind conditions. Therefore, rule-weighted fusion is adopted to enhance the model’s expressive ability.

$$A{T}_k(\mathrm{x})={\displaystyle \prod _{i=1}^M{\mu }_{ik}(x_i)}$$

(3)

$$f(\mathrm{x})={\displaystyle \frac{{\displaystyle {\sum }_{k=1}^{K}A{T}_k(\mathrm{x})}·O_k}{{\displaystyle {\sum }_{k=1}^{K}A{T}_k(\mathrm{x})}}}$$

(4)

where $\mathrm{x}=\left\{x_1,x_2,\dots ,x_M\right\}$ represents the input vector, $M$ represents the number of attributes, $A{T}_k(x)$ represents the activation strength of the $k_{th}$ rule, ${\mu }_{ik}(x_i)$ represents the membership degree of the $i_{th}$ variable to the reference value in the $k_{th}$ rule, and $O_k$ is the ER inference output of the $k_{th}$ rule.

2.2. Description of the New Wind Power Generation Forecasting Model

In response to the three aforementioned issues in engineering practice, a new wind power generation forecasting method based on BRB-f is proposed in this paper.

In the new wind power generation forecasting model, assuming that the antecedent attributes are mutually independent, the $k_{th}$ rule of the BRB-f model can be described as follows:

$$\begin{array}{c}Rul{e}_k:\\ If x_1 is A_{1,j_1}\vee x_2 is A_{2,j_2}\vee \cdots \vee x_T is A_{T,j_T}\\ Then result is\{(D_1,{\beta }_{1,k}),(D_2,{\beta }_{2,k}),\dots ,(D_N,{\beta }_{N,k})\}\\ with rule reliability{ \mathrm{r}}_k and rule weight {\omega }_k\\ k\in \{1,2,\dots ,L\},{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\le 1\end{array}$$

(5)

where $x_i(i=1,\dots ,T)$ refers to the characteristics of the actual system, $T$ represents the number of characteristics of the actual system, $L$ represents the total number of rules, $A_{i,j_i}(j_i=1,\dots ,J_i)$ represents the $j_{i th}$ reference value corresponding to the $i_{th}$ characteristic of the actual system, $J_i$ represents the total number of reference values for the $i_{th}$ characteristic. $D_n(n=1,\dots ,N)$ is the prediction level of BRB-f, $N$ represents the number of prediction levels. ${\beta }_{n,k}(n=1,\dots ,N)$ represents the belief degree of each level under the $k_{th}$ rule, $r_k$ is the rule reliability, ${\omega }_k$ is the rule weight. The developed wind power generation forecasting model framework is shown in Figure 1.

3. Calculation Method of Fuzzy Membership Degree for BRB-f

In this chapter, the core idea of fuzzy membership matching is described in Section 3.1. The implementation process of fuzzy membership matching is described in Section 3.2.

3.1. Core Idea

In the BRB-f model, fuzzy matching of reference values is considered an improvement direction and plays a key role in wind power generation forecasting. Firstly, IBRB uses fixed interval matching, meaning an input value can only belong to a specific interval, whereas BRB-f introduces fuzzy membership functions to achieve fuzzy matching of reference values. An input value can simultaneously activate two adjacent reference values with different membership degrees, realizing a smooth transition in the prediction output and significantly enhancing the stability of prediction results. Secondly, the fixed interval matching mechanism of IBRB leads to coarse information granularity. In contrast, BRB-f calculates membership degrees to accurately quantify the matching degrees between the input value and reference values, enabling the model to distinguish subtle differences within the intervals for more accurate predictions. Finally, IBRB typically uses a single-rule activation mechanism, where an input sample activates only one rule, whereas BRB-f employs a multi-rule collaborative activation mechanism, where each attribute value can activate two rules, and the model output is a weighted fusion of these rule results. This greatly enhances the model’s expressive capability. The comparison between BRB-f and IBRB is shown in Figure 2.

To more intuitively illustrate the differences in activating rules between IBRB and BRB-f, Figure 3 and Figure 4 are drawn as follows. The two sets of comparison charts systematically show the core differences in the input value matching rules between IBRB and BRB-f: the first set is a continuous matching comparison chart. In the left subfigure, with continuous input values ranging from 0 to 1 on the horizontal axis and matching degree on the vertical axis, it shows that IBRB divides the input range into four fixed intervals: [0.00, 0.25], [0.25, 0.50], [0.50, 0.75], and [0.75, 1.00]. In Figure 3, Interval-1 represents the range [0.00, 0.25], Interval-2 represents the range [0.25, 0.50], Interval-3 represents the range [0.50, 0.75] and Interval-4 represents the range [0.75, 1.00]. Each input value belongs to only one interval, and the matching degree is either 0 or 1, with jumps occurring at the interval boundaries, representing fixed interval matching characteristics. The right subfigure, based on five reference points of 0.00, 0.25, 0.50, 0.75, and 1.00, presents the fuzzy reference value matching of BRB-f, where input values simultaneously belong to two adjacent reference points. (R1 represents the reference value 1, which is 0.00; R2 represents the reference value 2, which is 0.25; R3 represents the reference value 3, which is 0.50; R4 represents the reference value 4, which is 0.75; and R5 represents the reference value 5, which is 1.00.) The membership degree transitions smoothly with the input value, and the sum of the membership degrees across all reference points is always 1. The second set is a discrete point matching comparison chart. In the left subfigure, four input points (0.1, 0.3, 0.7, 0.9) are selected, with fixed intervals on the horizontal axis and matching degree on the vertical axis. The bar chart clearly shows that each input point only has a matching degree of 1 in its corresponding interval, illustrating the fixed interval matching results. The right subfigure uses reference points on the horizontal axis and membership degree on the vertical axis, similarly using bar charts to show the nonzero membership degree distribution of each input point across two adjacent reference points, clearly reflecting the smooth transition feature of fuzzy reference value matching. These two sets of comparison charts, from both continuous and discrete perspectives, fully compare IBRB’s fixed interval matching logic with BRB-f’s fuzzy reference value matching logic, highlighting the flexibility of BRB-f in input value matching.

3.2. Implementation Process

Step 1: Parameter definition

For the engineering characteristics of multi-attribute inputs in wind power generation forecasting, let the input vector be $x=\{x_1,x_2,\dots ,x_T\}$, where $x_i(i=1,2,\dots ,T)$ represents the $i_{th}$ characteristic. Each characteristic $i$ has a set of reference values $\{a_{i,1},a_{i,2},\dots ,a_{i,J_i}\}$ satisfying $a_{i,1}<a_{i,2}<\dots <a_{i,J_i}$, where $J_i$ is the total number of reference values for the $i_{th}$ attribute. For the input value $x_i$, its membership degree ${\mu }_{i,j}(x_i)$ is calculated at each reference point $a_{i,j}$, whose main function is to quantify the correlation between the input value and the reference values, providing a theoretical basis for fuzzy matching.

Step 2: Calculation formula

To describe matching degrees between the actual input of each wind power generation attribute and different reference values, the membership function calculation formula for the reference values is designed as follows:

For the first reference value point $(j=1)$:

$${\mu }_{i,1}(x_i)=\left\{\begin{array}{ll}1& ,x_i\le a_{i,1}\\ 1-{\displaystyle \frac{x_i-a_{i,1}}{a_{i,2}-a_{i,1}}}& ,a_{i,1}<x_i\le a_{i,2}\\ 0& ,x_i>a_{i,2}\end{array}\right.$$

(6)

For intermediate reference points $(2\le j\le J_i-1)$:

$${\mu }_{i,j}(x_i)=\left\{\begin{array}{ll}{\displaystyle \frac{x_i-a_{i,j-1}}{a_{i,j}-a_{i,j-1}}}& ,a_{i,j-1}<x\le a_{i,j}\\ 1-{\displaystyle \frac{x_i-a_{i,j}}{a_{i,j+1}-a_{i,j}}}& ,a_{i,j}<x\le a_{i,j+1}\\ 0& ,x<a_{i,j-1} \mathtt{or} x>a_{i,j+1}\end{array}\right.$$

(7)

The last reference value $(j=J_i)$:

$${\mu }_{i,J_i}(x_i)=\left\{\begin{array}{ll}0& ,x_i<a_{i,J_i-1}\\ {\displaystyle \frac{x_i-a_{i,J_i-1}}{a_{i,J}-a_{i,J_i-1}}}& ,a_{i,J_i-1}<x_i\le a_{i,J_i}\\ 1& ,x_i\ge a_{i,J_i}\end{array}\right.$$

(8)

Step 3: Normalization

The calculated membership degrees need to be normalized so that the sum of all membership degrees is 1. The formula is

$${\mu }_{i,j}^{\prime }(x_i)={\displaystyle \frac{{\mu }_{i,j}(x_i)}{{\displaystyle {\sum }_{j=1}^J{\mu }_{i,j}(x_i)}}}$$

(9)

where ${\mu }_{i,j}^{\prime }(x_i)$ is the normalized membership degree.

The main reasons for choosing the triangular membership function are as follows: taking the six-attribute dataset selected in this paper as an example, the triangular membership function achieves the optimal balance between computational efficiency, rule matching accuracy, and complexity, requiring only simple linear logic, the minimum number of activation rule combinations, and the lowest memory usage. It is significantly superior to the trapezoidal, Gaussian, and S-shaped membership functions. A comparison of the performance indicators of the membership functions is shown in

Table 1. Comparison of membership function performance based on P-CMA-ES.

	Triangular	Trapezoidal	Gaussian	S-Shaped
Single-sample computation	350 flops	400 flops	469,200 flops	500,000 flops
Rule combinations	64	64	15,625	15,625
Single-generation computation	5250 flops	6000 flops	7,038,000 flops	7,500,000 flops
Memory overhead	12T	12T	30T	30T

.

Remark 1.

In scenarios with multiple attribute inputs, taking the dataset selected in this paper as an example (6 attributes, each with 5 reference values), the number of rule combinations that need to be calculated for the triangular membership function is $2^6=64$, while the number required for the Gaussian membership function and S-type membership function is $5^6=15625$, which greatly increases the computational overhead. Although the number of rule combinations activated by trapezoidal membership is also 64, the plateau region of the trapezoidal membership function can cause different input values to correspond to the same membership degree. For example, if input values 0.4 and 0.5 fall within the flat top region, their membership degree is both 1, which loses the detailed differences in attribute values. Therefore, the triangular membership function is chosen.

4. Wind Power Generation Forecasting Model Based on BRB-f

In this chapter, the model’s rule activation and reasoning process are described in Section 4.1. The model’s optimization process is described in Section 4.2. The model’s modeling process is described in Section 4.3. The computational complexity and scalability analysis is discussed in Section 4.4.

4.1. Activation and Reasoning Process of the New Wind Power Generation Forecasting Model

In the engineering application of wind power generation forecasting, the rule activation mechanism of the IBRB model has limitations. Rules are only activated when samples fall within a fixed interval, which coarsens the input information and causes the loss of information about the distribution of input values within the interval. This limits the model’s sensitivity, prevents it from accurately reflecting the complex characteristics of wind power, and makes it difficult to meet the prediction accuracy requirements for wind power grid scheduling.

To address the above issues, the BRB-f model optimizes the rule activation method based on the practical needs of wind power generation forecasting. For each input attribute value, the BRB-f model calculates its membership degree at each reference value using a triangular membership function. This design ensures that each attribute’s input activates no more than two reference values, accurately quantifying matching degree between the input value and each reference value, while also achieving smooth transitions between reference values.

The BRB-f model defines a rule for each reference value of each attribute. When input arrives, each attribute activates one or two reference values based on fuzzy membership degrees. To enhance the reliability of the BRB-f model’s inference in wind power generation forecasting, the model introduces evidential reasoning (ER) [21,22] during the inference process, placing the reliability of the rules on top of the ER analysis algorithm. The introduction of ER allows the model to fully integrate wind power generation input information. Once the rules are activated, all activated rules are fused through the ER, ultimately outputting the power generation forecasting results. This multi-rule activation and weighted fusion not only ensure that the data input can accurately match the corresponding rules at different membership degrees but also improve the accuracy of the prediction results through the ER inference process. The specific formulas are as follows:

The $i_{th}$ piece of evidence $e_i(i=1,\dots ,L)$ can be represented by the following belief distribution.

$$e_i=\{(D_n,{\beta }_{n,i}),n=1,\dots ,N;(\Theta ,{\beta }_{\Theta ,i})\} 0\le {\beta }_{n,i}\le 1,{\displaystyle \sum _{n=1}^N{\beta }_{n,i}\le 1}$$

(10)

where $\Theta =\{D_1,\dots ,D_N\}$ represents the identification framework, $D_n(n=1,\dots ,N)$ represents the evaluation level of the complex system, ${\beta }_{n,i}$ represents the belief degree that the result is evaluated as evaluation level $D_n$, and ${\beta }_{\Theta ,i}$ represents the belief degree regarding the identification framework $\Theta$, indicating global ignorance.

The weight of the evidence is represented as ${\omega }_i(i=1,\dots ,L)$, and the reliability of the evidence is represented as $r_i(i=1,\dots ,L)$. They both satisfy ${\omega }_i\in [0,1]$ and $r_i\in [0,1]$. Therefore, the belief distribution after the combined weighting of evidence weight and evidence reliability can be expressed as follows.

$$m_i=\{(D_n,{\tilde{m}}_{n,i}),\forall D_n\subseteq \Theta ;(\beta (\Theta ),{\tilde{m}}_{\beta (\Theta ),i})\}$$

(11)

where the power set is represented by $\beta (\Theta )$, and the mixed probability mass ${\tilde{m}}_{n,i}$ of the evidence $i$ on the evaluation level $D_n$ is represented as follows.

$${\tilde{m}}_{n,i}=\left\{\begin{array}{c}0,{ \mathrm{D}}_n=\varnothing \\ d_{r\omega ,i}{m}_{n,i},{ \mathrm{D}}_n\subseteq \Theta ,D_n\ne \varnothing \\ d_{r\omega ,i}(1-r_i),{ \mathrm{D}}_n=\beta (\Theta )\end{array}\right.$$

(12)

$$d_{r\omega ,i}=1/(1+{\omega }_i-r_i)$$

(13)

$$m_{n,i}=w_i{\beta }_{n,i}$$

(14)

where $d_{r\omega ,i}$ represents the normalization coefficient and satisfies the condition ${\displaystyle \sum _{n=1}^N{\tilde{m}}_{n,i}}+{\tilde{m}}_{\beta (\Theta ),i}=1$. The combined belief degree ${\beta }_{n,e(L)}$ of $L$ pieces of independent evidence is calculated as follows.

$$\forall D_n\in \Theta ,{\widehat{m}}_{n,e(k)}=[(1-r_k)m_{n,e(k-1)}+m_{\beta (\Theta ),e(k-1)}{m}_{n,k}]+{\displaystyle \sum _{A\cap B=D_n}{m}_{A,e(k-1)}}{m}_{B,k}$$

(15)

$${\widehat{m}}_{\beta (\Theta ),e(k)}=(1-r_k)m_{\beta (\Theta ),e(k-1)}$$

(16)

$$m_{n,e(k)}=\left\{\begin{array}{c}0,{ \mathrm{D}}_n=\varnothing \\ {\displaystyle \frac{{\widehat{m}}_{n,e(k)}}{{\displaystyle \sum _{A\subseteq \Theta }{\widehat{m}}_{A,e(k)}+{\widehat{m}}_{\beta (\Theta ),e(k)}}}}\end{array}\right.,{ \mathrm{D}}_n\ne \varnothing$$

(17)

$${\beta }_{n,e(k)}=\left\{\begin{array}{l}0,{ \mathrm{D}}_n=\varnothing \\ {\displaystyle \frac{{\widehat{m}}_{n,e(k)}}{{\displaystyle \sum _{A\subseteq \Theta }{\widehat{m}}_{A,e(k)}}}},D_n\subseteq \Theta ,D_n\ne \varnothing \end{array}\right.$$

(18)

where $k=1,2,\dots ,L$, the belief degree level $D_n$ of the result after merging the previous pieces of evidence is denoted as ${\beta }_{n,e(k)}$. This satisfies $m_{n,e(1)}=m_{n,1}$ and $m_{\beta (\Theta ),e(1)}=m_{\beta (\Theta ),1}$.

The above reasoning process can yield the following output belief distribution and expected output utility values.

$$e(L)=\{(D_n,{\beta }_{n,e(L)}),n=1,\dots ,N,(\Theta ,{\beta }_{n,e(L)})\}$$

(19)

$$Z={\displaystyle \sum _{n=1}^{N}u(D_n){\beta }_{n,e(L)}+u(\Theta )}{\beta }_{\Theta ,e(L)}$$

(20)

where $Z$ represents the final expected utility value, and $u(D_n)$ represents the utility at level $D_n$.

Define the set of reference values for activation as

$$ℂ=\{(j_1,j_2,\dots ,j_T)| j_i\in \{1,2,\dots ,J\},{\mu }_{i,j_i}(x_i)>0 for all i\}$$

(21)

For each combination $\mathrm{j}=(j_1,j_2,\dots ,j_T)\in {\rm E}$, the activation weight is calculated as follows:

$$w_{\mathrm{j}}={\displaystyle \prod _{i=1}^T{\mu }_{i,{\mathcal{l}}_i}(x_i)}$$

(22)

For the combination $\mathrm{j}$, we have a set of evidence ${\rm E}=\{e_1,e_2,\dots ,e_T\}$, where the parameters of the evidence $e_i$ as follows:

The belief degree distribution is ${\beta }_i=\{{\beta }_{i,1},{\beta }_{i,2},\dots ,{\beta }_{i,N}\}$, where ${\beta }_i={\beta }_{i,j_i}$, ${\beta }_{i,n}$ represents the belief degree at output level $n$, with the weight ${\omega }_i={\omega }_{i,j_i}$ and reliability $r_i=r_{i,j_i}$.

The normalization factor is calculated as follows:

Step 1: Calculate the evidence combination term of the $i_{th}$ attribute for output level n

$$(1-r_i)+{\omega }_i{\beta }_{i,n}+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})$$

(23)

where $r_i(i=1,\dots ,T)$ represents the reliability of the attribute, ${\omega }_i(i=1,\dots ,T)$ represents the weight of the attribute, ${\beta }_{n,i}$ represents the belief degree of the $i_{th}$ attribute corresponding to level $n$, $N$ represents the number of evaluation levels. $(1-r_i)$ represents the unreliability, ${\omega }_i{\beta }_{i,n}$ represents the belief degree assigned to output level, ${\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})$ represents the unassigned belief degree.

Step 2: Calculate the total joint effect of multi-attribute evidence on each output level

For $N$ output levels, calculate the product of the evidence combination terms for all attributes, and then sum them up.

$${\displaystyle \sum _{n=1}^N{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i{\beta }_{i,n}+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})]}}$$

(24)

where $T$ represents the number of attributes.

Step 3: Calculate the evidence conflict correction item

$$(N-1){\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n})}]}$$

(25)

where $(N-1)$ represents conflict dimension, ${\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n})}]}$ represents the local ignorance conjunction term, that is, the product of local ignorance of all attributes.

Step 4: Calculate the normalization factor

$${\mu }_{\mathrm{j}}=[{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i{\beta }_{i,n}+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})]}}-(N-1){\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n})}]}{]}^{-1}$$

(26)

The aggregated belief distribution $S_{\mathrm{j}}=[s_{\mathrm{j},1},s_{\mathrm{j},2},\dots ,s_{\mathrm{j},N}]$ is calculated through the ER, where the combined belief degree $s_{\mathrm{j},n}$ corresponding to level $n$ is calculated as follows:

Step 1: Calculate the net joint item of the target level

Extract the joint evidence items of the target level $n$, subtract the joint items of local ignorance shared across all levels, and obtain the net joint items of the target level.

$${\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i{\beta }_{i,n}+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})]}-{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})}]$$

(27)

Step 2: Calculate the net joint term after conflict correction

Multiply the net joint term by ${\mu }_{\mathrm{j}}$ to eliminate the interference of evidence conflict.

$${\mu }_{\mathrm{j}}[{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i{\beta }_{i,n}+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})]}-{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})}]]$$

(28)

Step 3: Fix global unreliability

$$1-{\mu }_{\mathrm{j}}[{\displaystyle \prod _{i=1}^T(1-r_i)}]$$

(29)

Step 4: Calculate the aggregate confidence

$$s_{\mathrm{j},n}={\displaystyle \frac{{\mu }_{\mathrm{j}}[{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i{\beta }_{i,n}+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})]}-{\displaystyle \prod _{i=1}^T[(1-r_i)+{\omega }_i(1-{\displaystyle \sum _{n=1}^N{\beta }_{i,n}})}]]}{1-{\mu }_{\mathrm{j}}[{\displaystyle \prod _{i=1}^T(1-r_i)}]}}$$

(30)

where $0\le s_{j,n}\le 1,{\displaystyle {\sum }_{n=1}^N{s}_{j,n}=1}$, ${\beta }_{i,n}$ is the abbreviation for ${\beta }_{i,j_i,n}$, ${\mu }_{\mathrm{j}}$ represents the normalization factor.

The predicted output is calculated as follows:

$$y_{\mathrm{j}}={\displaystyle \sum _{n=1}^N{s}_{\mathrm{j},n}{d}_n}$$

(31)

where $d_n$ is the representative value of the output level.

The final output is the weighted average of all activation combination outputs; the formula is as follows:

$$y={\displaystyle \frac{{\displaystyle {\sum }_{\mathrm{j}\in ℂ}{w}_{\mathrm{j}}{y}_{\mathrm{j}}}}{{\displaystyle {\sum }_{\mathrm{j}\in ℂ}{w}_{\mathrm{j}}}}}, if{\displaystyle {\sum }_{\mathrm{j}\in ℂ}{w}_{\mathrm{j}}}=0, \mathrm{then} \mathrm{y}=0$$

(32)

4.2. Optimization Model of the New Wind Power Generation Forecasting Model

In the engineering practice of complex system modeling and prediction, the main goal of model optimization is to improve prediction accuracy. Given that the projection covariance matrix adaptation evolution strategy (P-CMA-ES) algorithm demonstrates fast convergence, high optimization precision, and strong robustness in continuous space optimization problems, it can effectively enhance the efficiency of parameter tuning for the BRB-f model. Therefore, this algorithm is used to optimize the BRB-f model.

To build an optimization model, the function to be optimized must first be clarified. From the engineering requirement of minimizing prediction error, the difference between the BRB-f model’s predicted values and the actual values is the main optimization metric, represented as $MSE(.)$. Based on this, the objective function to be optimized is defined as a global minimization function, with the expression as follows:

$$\begin{array}{l}\min MSE(\beta ,\vartheta ,\theta )\\ st.{\displaystyle \sum _{n=1}^N{\beta }_{n,k}=1,k=1,\dots ,L}\\ 0\le {\beta }_{n,k}\le 1,n=1,\dots ,N,k=1,\dots ,L\\ 0\le r_k\le 1,k=1,\dots ,L\\ 0\le {\omega }_i\le 1,i=1,\dots ,L\end{array}$$

(33)

In the above formula, the calculation method of $MSE$ is as follows.

$$MSE(\beta ,r,\omega )={\displaystyle \frac{1}{T_{train}}}{\displaystyle \sum _{k=1}^{T_{train}}{(y-y^{*})}^2}$$

(34)

where $T_{train}$ represents the total number of training sample data, $y$ is the predicted output value of the BRB-f model, and $y^{*}$ is the actual value of the complex system. Based on this objective function, the optimization process of the P-CMA-ES algorithm is as follows:

First, parameter initialization. In the BRB-f method, the parameters to be optimized directly affect the model’s prediction performance and stability, mainly including key parameters such as belief degrees, rule reliabilities, and rule weights. To facilitate efficient algorithmic search, the set of parameters to be optimized is represented in vector form as follows:

$${\Omega }^0={\sigma }^0$$

(35)

$${\sigma }^0=\{{\beta }_{1,1},\dots ,{\beta }_{N,L},r_1,\dots ,r_L,{\omega }_1,\dots ,{\omega }_L\}$$

(36)

Second, the sampling operation. Each generation of candidate parameters is generated through normal distribution sampling, ensuring that the samples cover the parameter space while also considering search efficiency, which can be expressed as

$${\sigma }_i^{s+1}\sim {\Omega }^s+{\epsilon }^s\mathrm{N}(0,Q^s),i=1,\dots ,h$$

(37)

where ${\sigma }_i^{s+1}$ is the $i_{th}$ solution in the ${(s+1)}_{th}$ generation optimization. ${\epsilon }^s$ is the step size. ${\Omega }^s$ is the mean of the search distribution in the $s_{th}$ generation. $Q^s$ is the covariance matrix. $\mathrm{N}(.)$ represents the normal distribution. $h$ is the number of offspring.

Third, constrained projection. Since the parameters of the BRB-f model have physical meaning constraints, candidate solutions need to be projected onto a feasible hyperplane to prevent invalid parameter combinations from causing the model to fail. The specific implementation is as follows:

$$\begin{array}{c}{\sigma }_i^{s+1}(1+{\eta }_e\times (\tau -1):{\eta }_e\times \tau )\\ ={\sigma }_i^{s+1}(1+{\eta }_e\times (\tau -1):{\eta }_e\times \tau )-V^T\times {(V\times V^T)}^{-1}\\ \times {\sigma }_i^{s+1}(1+{\eta }_e\times (\tau -1):{\eta }_e\times \tau )\times V\end{array}$$

(38)

where $V={[1,\dots ,1]}_{1\times N}$ represents a parameter vector with all ones. ${\eta }_e=1,\dots ,N$ is the number of variables with constraints. $\tau =1,\dots ,N+1$ is the count of equality constraints.

Fourth, mean update. Update the mean of the next generation of parameters through a weighted average to accelerate the convergence speed. The specific operation is as follows:

$${\Omega }^{s+1}={\displaystyle \sum _{i=1}^{\varphi }{\omega }_i{\sigma }_{i:h}^{s+1}}$$

(39)

where ${\omega }_i$ represents the weight coefficient. $\varphi$ is the population size of the offspring. ${\sigma }_{i:h}^{s+1}$ is the $i_{th}$ solution among the $h$ solutions in the ${(s+1)}_{th}$ generation.

Fifth, updating the covariance matrix. Update the covariance matrix based on the information from the population’s evolution, so that the search region contracts towards the optimal solution. The specific steps are as follows.

$$\begin{array}{l}{Q}^{s+1}=(1-e_1-e_2)Q^s+e_1{P}_e^{s+1}{(P_e^{s+1})}^T+e_2{\displaystyle \sum _{i=1}^{\upsilon }{\omega }_i}({\displaystyle \frac{K_{i:h}^{s+1}-{\theta }^s}{{\vartheta }^s}})\\ \hspace{1em}\hspace{1em} \times ({\displaystyle \frac{K_{i:h}^{s+1}-{\theta }^s}{{\vartheta }^s}}{)}^T\end{array}$$

(40)

where ${\vartheta }^s$ represents the step size in the $s_{th}$ generation. In the ${(s+1)}_{th}$ generation, $P_e^{s+1}$ is the evolutionary path, $e_1$ and $e_2$ is the learning rate. in the $s_{th}$ generation, ${\theta }^s$ is the population of offspring. $K_{i:h}^{s+1}$ represents the $i_{th}$ solution vector among $h$ solution vectors in the ${(s+1)}_{th}$ generation.

Finally, recursively perform the above steps until the optimization is complete.

Based on the aforementioned engineering optimization process, a BRB-f model optimization structure that balances prediction accuracy with engineering feasibility can be obtained, as shown in Figure 5.

4.3. Modeling Process of the New Wind Power Generation Forecasting Model

Step 1: Construct the initial BRB-f model. Build the initial framework for wind power generation forecasting based on BRB-f. Model parameters are determined by engineering experts based on the characteristics of wind power generation data, including the division of attribute reference values and the initialization of belief degrees.

Step 2: Introduce fuzzy membership. Calculate the membership of each attribute input to each attribute reference value using fuzzy membership functions. This eliminates the boundary jumps of traditional interval matching, thereby achieving a smooth transition.

Step 3: Rule activation and weight calculation. Determine the contribution weight of each rule based on the degree of matching, preparing a weighted set of evidence for subsequent reasoning, thus avoiding bias from a single rule.

Step 4: Evidence reasoning and rule fusion. Synthesize evidence considering rule reliabilities, perform weighted averaging, and generate the final prediction result that comprehensively accounts for all evidences.

Step 5: Belief degree distribution calculation and parameter optimization. Calculate the belief distribution based on the activated rules, and introduce optimization algorithms to train and optimize the main model parameters, enhancing the model’s prediction accuracy and robustness.

Step 6: Model performance validation. Conduct a comprehensive performance evaluation of the optimized BRB-f model using a wind turbine power generation forecasting, focusing on the accuracy and stability of wind power generation forecasting.

The modeling process of the BRB-f model is shown in Figure 6.

4.4. Computational Complexity and Scalability Analysis

The proposed BRB-f framework inevitably incurs additional computational overhead due to the introduction of fuzzy membership, multiple rule activation and fusion, and parameter optimization. To systematically evaluate the practicality of this method, this section will analyze its computational cost, space complexity, and scalability in large-scale application scenarios.

• Fuzzy membership calculation

Let the number of reference values for a single feature be $K$, the computational complexity for a single sample with $T$ features is $O(TK)$, and the total complexity for $n$ samples is $O(nTK)$. Memory consumption increases linearly with the number of samples, feature dimensions, and the number of reference values.

2.

Multi-rule activation and fusion

Combine activation rules for multiple attributes and calculate the output belief degree using the ER algorithm. Each attribute input value can activate up to 2 rules, and the number of rule combinations for a single sample is $2^T$. For a dataset containing $T$ attributes and $N$ evaluation results, the computational complexity of a single inference using the ER algorithm is $O(TN)$, and the total complexity is $O(nTN{2}^T)$. The complexity increases exponentially with the number of features in the dataset.

3.

Parameter optimization

BRB-f optimizes the belief degrees, rule reliabilities, rule weights, and other parameters through the P-CMA-ES algorithm. For a dataset with $L$ rules and $N$ evaluation results, the dimension of optimized parameters is $D$, where $D=L\times N+L+L$, the total population size is $\lambda$, the complexity of a single iteration is $O(\lambda D^2)$, and the total complexity is $O(G\lambda D^2)$.

4.

Overall computational burden

The overall complexity of this framework is polynomial and is dominated by multi-rule activation and the iterative optimization phase. Although BRB-f requires more resources than IBRB to compute multi-rule activation, this part of the overhead mainly depends on the number of features in the dataset, and the number of features can be reduced through methods such as feature selection or principal component analysis (PCA). During the iterative optimization process, the evaluation of candidate parameters is independent of each other and does not need to be performed sequentially, making it suitable for parallel execution.

5.

Scalability and feasibility

Despite the additional overhead, this framework can still be scaled to medium- and large-scale datasets. The complexity grows exponentially with the number of dataset features, but the number of features can be reduced and dimensions optimized through mainstream dimensionality reduction methods such as feature selection and principal component analysis (PCA). The dimension of optimized parameters grows linearly with the number of rules, but the BRB-f framework only requires setting a small number of attribute reference values and rules. The parameter optimization module supports parallel computing to improve efficiency, making this method suitable for practical engineering systems. Future work will further explore lightweight strategies and distributed deployment to enhance its applicability to large-scale scenarios.

5. Case Study

Taking wind turbine power generation forecasting as an example to verify the effectiveness and accuracy of the BRB-f method. The definition of the experiment is presented in Section 5.1 and Section 5.2; the effectiveness of the method is further demonstrated through the analysis of wind turbine power generation forecasting. The training and testing of the model is involved in Section 5.3. A comparative study of the experimental results is provided in Section 5.4. A generalization analysis of the method is provided in Section 5.5.

5.1. Experimental Definitions

This section conducts experimental validation using a wind turbine power generation forecasting in actual operation as the research object. The dataset is selected from an open-source dataset on Kaggle, which contains hourly meteorological data and power generation measurements collected from January 2013 to December 2016. The data include variables such as temperature, relative humidity, wind speed at different heights, and power generation at different locations. To control variables, from 30,540 data points at the same location, 2000 samples were selected using simple random sampling, where wind speed and direction were measured at 10 m above the ground. To serve as the experimental dataset, the dataset has been preprocessed.

The specific preprocessing steps are as follows:

Step 1: Outlier handling

Calculate the mean $\mu$ and standard deviation (Std) $\sigma$ of the data; the formula is as follows:

$$\mu ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i}$$

(41)

$$\sigma =\sqrt{{\displaystyle \frac{1}{n-1}}{\displaystyle \sum _{i=1}^n{(x_i-\mu )}^2}}$$

(42)

where $n$ represents the total number of samples, $x_i$ represents the value of the $i_{th}$ sample.

If $\left|x_i-\mu \right|>3\sigma$, then $x_i$ is an outlier. $x_i$ is replaced with the mean $\mu$, that is, $x_i=\mu$.

Step 2: Min-Max normalization

Map data linearly to [0, 1] intervals, the formula is as follows:

$$x_i^{\prime }={\displaystyle \frac{x_i-\min (X)}{\max (X)-\min (X)}}$$

(43)

where $x_i$ represents raw data, $\min (X)$ represents the minimum value of the dataset, $\max (X)$ represents the maximum value of the dataset. $x_i^{\prime }(0\le x_i^{\prime }\le 1)$ represents standardized data.

The dataset has a total of six attributes, they are air temperature, relative humidity, dew point temperature, wind speed, wind direction, and wind gust. Among them, wind speed is the core driving factor affecting power generation; its magnitude directly determines the intensity of wind power input. Wind gust, as the instantaneous maximum wind speed, not only causes transient fluctuations in power output but also shows a significant positive correlation with the average wind speed, jointly affecting the stability of power generation. Air temperature, relative humidity, and dew point temperature indirectly affect energy capture efficiency at the same wind speed by altering air density through coupling effects. Wind direction determines the angle between the airflow and the wind turbine rotor, with the prevailing wind direction corresponding to higher wind energy utilization efficiency. In specific terrains, wind direction can also synergize with wind speed, further impacting the level of power output.

In conclusion, all six attributes of the dataset were selected. The reference values for each attribute and the result values were determined to both cover the actual variation range of each attribute and focus on regions with dense data. The initial belief distribution was given based on correlation analysis between each attribute and the result. In the initial belief distribution setting, the core basis is the Pearson correlation coefficient between each attribute and the output, as shown in Figure 7. To ensure the rationality of the rules, the initial belief degree is set according to the following principles: (1) if the correlation between an attribute and the outcome is negative, the belief degree decreases as the evaluation level rises; if the correlation is positive, the belief degree increases as the evaluation level rises. (2) The larger the absolute value of the correlation coefficient between an attribute and the outcome, the faster the belief degree increases or decreases in each rule; the smaller the absolute value of the correlation coefficient between an attribute and the outcome, the slower the belief degree increases or decreases in each rule. Specific parameter settings are provided in Section 5.2, and both initial rule reliabilities and rule weights were set to 1. The dataset was divided into seven different training-test split ratios: 10% training, 90% testing; 20% training, 80% testing; 30% training, 70% testing; 40% training, 60% testing; 50% training, 50% testing; 60% training, 40% testing; and 70% training, 30% testing.

The evaluation metrics [23,24] for the model are selected as follows:

• mean squared error (MSE):

  $$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

  (44)
• root mean squared error (RMSE):

  $$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}$$

  (45)
• mean absolute error (MAE):

  $$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|}$$

  (46)
• mean absolute percentage error (MAPE):

  $$MAPE={\displaystyle \frac{100\%}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}$$

  (47)
• symmetric mean absolute percentage error (SMAPE):

  $$SMAPE={\displaystyle \frac{100\%}{n}}{\displaystyle \sum _{i=1}^n\frac{\left|y_i-{\widehat{y}}_i\right|}{\left(\left|y_i\right|+\left|{\widehat{y}}_i\right|\right)/2}}$$

  (48)

where $n$ represents the total number of samples, $y_i$ represents the true value of the $i_{th}$ sample, and ${\widehat{y}}_i$ represents the predicted value of the $i_{th}$ sample.

In this study, the introduction of the t-test is aimed at quantifying the performance differences between different models under the same training/test split ratio: by testing the significance of the differences in performance metrics across multiple rounds of experiments, it distinguishes between ‘the inherent performance advantages of the method’ and ‘random experimental fluctuations,’ thereby enhancing the reliability of the conclusions and improving the scientific rigor and comparability of the results within the field.

The specific steps of the t-test are as follows:

Step 1: Calculate the difference

Calculate the difference in performance metrics between Model A and Model B in the same round of experiments; the formula is as follows:

$${\Delta }_i=A_i-B_i$$

(49)

where $i$ represents the $i_{th}$ experiment, $A_i$ represents the performance metric value of model $A$ in the $i_{th}$ experiment, and $B_i$ represents the performance metric value of model $B$ in the $i_{th}$ experiment.

Step 2: Calculate the mean

Calculate the mean of the differences for all rounds. The formula is as follows:

$$\overline{\Delta }={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\Delta }_i}$$

(50)

where $n$ represents the number of experiment rounds, ${\Delta }_i$ represents the difference in performance metrics between model $A$ and model $B$ in the $i_{th}$ experiment.

Step 3: Calculate the standard deviation

Calculate the sample standard deviation of the differences for all rounds. The formula is as follows:

$$s_{\Delta }=\sqrt{{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{({\Delta }_i-\overline{\Delta })}^2}}{n-1}}}$$

(51)

where $\overline{\Delta }$ represents the average of all ${\Delta }_i$ in $n$ rounds of experiments.

Step 4: Calculate the $t$ statistic

The calculation formula for the $t$ statistic is as follows:

$$t={\displaystyle \frac{\overline{\Delta }}{s_{\Delta }/\sqrt{n}}}$$

(52)

where $s_{\Delta }$ represents the sample standard deviation of the differences for all rounds.

Step 5: Calculate the degrees of freedom

$$df=n-1$$

(53)

Step 6: Calculate the $p$ value

The $p$ value is the probability of observing the current t statistic under the null hypothesis that there is no difference in the performance metrics between the two models. The formula is as follows:

$$p=2\times [1-F_{t(df)}(\left|t\right|)]$$

(54)

where $F_{t(df)}(·)$ represents the cumulative distribution function of the $t$ distribution with the degrees of freedom is $df$. $t$ represents the absolute value of the $t$ statistic, a coefficient of $2$ corresponds to a two-tailed test.

5.2. Construction of the Wind Power Generation Forecasting Model Based on BRB-f

To address the issues of insufficient smoothness and limited single-rule activation expression capability in the IBRB model for wind power generation forecasting, a new wind power generation forecasting method based on BRB-f was developed. This model achieves more accurate wind power generation forecasting by incorporating fuzzy membership degrees, multi-rule-weighted fusion, and optimization algorithms, providing efficient technical support for grid scheduling.

In the constructed wind power generation forecasting model, six input attributes were selected: air temperature (AT), relative humidity (RH), dew point temperature (DPT), wind speed (WS), wind direction (WD), and wind gust(WG). The predicted power generation values were divided into five evaluation levels, corresponding to wind power values of 0, 0.25, 0.50, 0.75, and 1, respectively. Based on the inference mechanism of the BRB-f, the $k_{th}$ belief rule in the BRB-f model can be expressed as follows:

$$\begin{array}{c}Rul{e}_k:\\ If \mathrm{AT} \mathrm{is} A_{1,j_1}\vee RH \mathrm{is} A_{2,j_2}\vee DPT \mathrm{is} A_{3,j_3}\vee \\ WS \mathrm{is} A_{4,j_4}\vee WD \mathrm{is} A_{5,j_5}\vee WG \mathrm{is} A_{6,j_6}\\ Then \mathrm{y} is\{(D_1,{\beta }_{1,k}),(D_2,{\beta }_{2,k}),(D_3,{\beta }_{3,k}),(D_4,{\beta }_{4,k}),(D_5,{\beta }_{5,k})\}\\ with rule reliability{ \mathrm{r}}_k and rule weight {\omega }_k\\ k\in \{1,2,\dots ,L\},{\displaystyle {\sum }_{n=1}^5}{\beta }_{n,k}\le 1\end{array}$$

(55)

where $L$ represents the total number of rules, $A_{i,j_i}(i=1,\dots ,6,j_i=1,\dots ,J_i)$ represents the $j_{i th}$ reference value corresponding to the $i_{th}$ characteristic of the actual system, $J_i$ represents the total number of reference values for the $i_{th}$ characteristic. In the constructed wind power generation forecasting model, five reference values have been set for all six attributes, so $J_1=J_2=J_3=J_4=J_5=5$. $D_n(n=1,\dots ,5)$ is the prediction level of BRB-f, $N$ represents the number of prediction levels. ${\beta }_{n,k}(n=1,\dots ,5)$ represents the belief degree of each level under the $k_{th}$ rule, $r_k$ is the rule reliability, ${\omega }_k$ is the rule weight. For a more intuitive presentation,

Table 2. Table of attributes and their actual meanings.

Attribute	Actual Meaning
AT	air temperature
RH	relative humidity
DPT	dew point temperature
WS	wind speed
WD	wind direction
WG	wind gust

is drawn as follows.

The distribution of attribute values is shown in Figure 8, the distribution of result values is shown in Figure 9, and the distribution of the 30 initial belief rules is shown in Figure 10.

It should be noted that the number of reference values and the granularity of the division directly determine the complexity and prediction accuracy of the BRB-f model. Therefore, in this experiment, based on the measured data distribution characteristics of six input attributes and output, five reference points were set for each attribute and power generation evaluation level: very low (VL), low (L), medium (M), high (H), and very high (VH). The specific reference values for AT, RH, DPT, WS, WD, WG, and output level are shown in Figure 11.

Based on six input attributes and the reference values for each attribute, the initial parameters of the wind power generation forecasting model are shown in

Table A1. Initial parameter table.

ID	Attribute	Referential Value	Rule Reliability	Rule Weight	Output Belief Degree
1	1	0	1	1	{0.43, 0.34, 0.15, 0.06, 0.02}
2	1	0.33	1	1	{0.41, 0.32, 0.16, 0.07, 0.04}
3	1	0.55	1	1	{0.39, 0.30, 0.17, 0.08, 0.06}
4	1	0.76	1	1	{0.37, 0.28, 0.18, 0.09, 0.08}
5	1	1	1	1	{0.35, 0.26, 0.19, 0.11, 0.09}
6	2	0	1	1	{0.30, 0.27, 0.22, 0.15, 0.06}
7	2	0.30	1	1	{0.29, 0.26, 0.22, 0.16, 0.07}
8	2	0.53	1	1	{0.28, 0.25, 0.22, 0.17, 0.08}
9	2	0.77	1	1	{0.27, 0.24, 0.22, 0.18, 0.09}
10	2	1	1	1	{0.26, 0.23, 0.22, 0.19, 0.10}
11	3	0	1	1	{0.43, 0.34, 0.15, 0.06, 0.02}
12	3	0.29	1	1	{0.41, 0.32, 0.16, 0.07, 0.04}
13	3	0.52	1	1	{0.39, 0.30, 0.17, 0.08, 0.06}
14	3	0.75	1	1	{0.37, 0.28, 0.18, 0.09, 0.08}
15	3	1	1	1	{0.35, 0.26, 0.19, 0.11, 0.09}
16	4	0	1	1	{0.01, 0.02, 0.05, 0.28, 0.64}
17	4	0.26	1	1	{0.02, 0.03, 0.08, 0.31, 0.56}
18	4	0.50	1	1	{0.03, 0.04, 0.11, 0.34, 0.48}
19	4	0.75	1	1	{0.04, 0.05, 0.14, 0.37, 0.40}
20	4	1	1	1	{0.05, 0.07, 0.19, 0.21, 0.48}
21	5	0	1	1	{0.04, 0.12, 0.20, 0.28, 0.36}
22	5	0.25	1	1	{0.03, 0.11, 0.19, 0.29, 0.38}
23	5	0.50	1	1	{0.02, 0.10, 0.18, 0.30, 0.40}
24	5	0.75	1	1	{0.01, 0.09, 0.17, 0.31, 0.42}
25	5	1	1	1	{0.01, 0.08, 0.17, 0.32, 0.42}
26	6	0	1	1	{0.01, 0.03, 0.18, 0.33, 0.45}
27	6	0.27	1	1	{0.02, 0.04, 0.19, 0.34, 0.41}
28	6	0.51	1	1	{0.03, 0.05, 0.20, 0.35, 0.37}
29	6	0.76	1	1	{0.04, 0.07, 0.22, 0.30, 0.37}
30	6	1	1	1	{0.03, 0.05, 0.20, 0.35, 0.37}

. In the initial BRB-f model, the reliability of each belief rule is set to 1, and the weight of each belief rule is set to 1.

5.3. Training and Validation of the Wind Power Generation Forecasting Model

In this section, based on the training data and the constructed optimization model, the wind power generation forecasting model can be optimized. The internal parameter settings of the optimizer are as follows: population size of 25, number of parents 12, number of iterations is 500, using logarithmic weights. As shown in

Table A2. Optimized parameter table.

ID	Attribute	Referential Value	Rule Reliability	Rule Weight	Output Belief Degree
1	1	0	0.4754	0.9780	{0.0153, 0.0344, 0.1261, 0.2037, 0.6205}
2	1	0.33	0.4448	0.5008	{0.4888, 0.2278, 0.1625, 0.1005, 0.0203}
3	1	0.55	0.7164	0.3823	{0.0319, 0.0741, 0.5361, 0.3268, 0.0311}
4	1	0.76	0.2743	0.6574	{0.1516, 0.1790, 0.1827, 0.2113, 0.2754}
5	1	1	0.1825	0.4209	{0.4488, 0.3382, 0.1733, 0.0427, 0}
6	2	0	0.6629	0.4619	{0.1012, 0.1880, 0.4578, 0.1969, 0.0561}
7	2	0.3	0.4227	0.0085	{0.0855, 0.1170, 0.1867, 0.1963, 0.4145}
8	2	0.53	0.5761	0.0623	{0.5515, 0.2321, 0.1302, 0.0711, 0.0151}
9	2	0.77	0.0763	0.6600	{0.0085, 0.1066, 0.4057, 0.3784, 0.1008}
10	2	1	0.1983	0.5793	{0.1142, 0.1825, 0.2041, 0.2072, 0.2921}
11	3	0	0.0575	0.3691	{0.5246, 0.2523, 0.1127, 0.0685, 0.0418}
12	3	0.29	0.2412	0.7416	{0.1448, 0.1935, 0.2854, 0.2426, 0.1338}
13	3	0.52	0.0203	0.2529	{0.0937, 0.1337, 0.1543, 0.2773, 0.3410}
14	3	0.75	0.5128	0.5898	{0.3794, 0.2742, 0.1356, 0.1310, 0.0797}
15	3	1	0.2429	0.7389	{0.0444, 0.1436, 0.5549, 0.1996, 0.0576}
16	4	0	0.2521	0.0523	{0.0734, 0.1147, 0.1710, 0.3167, 0.3242}
17	4	0.26	0.9147	0.7380	{0.5301, 0.2589, 0.1230, 0.0852, 0.0029}
18	4	0.5	0.9235	0.1695	{0.1605, 0.2216, 0.3089, 0.2910, 0.0180}
19	4	0.75	1.0000	0.8125	{0.0562, 0.0811, 0.0965, 0.3678, 0.3985}
20	4	1	0.6276	0.4548	{0.0274, 0.1413, 0.4677, 0.2395, 0.1241}
21	5	0	0.1982	0.0192	{0.0669, 0.1719, 0.1768, 0.2263, 0.3581}
22	5	0.25	0.8557	0.4051	{0.4130, 0.2765, 0.2315, 0.0671, 0.0118}
23	5	0.5	0.8859	0.6244	{0.0379, 0.0602, 0.4463, 0.4445, 0.0110}
24	5	0.75	0.7971	0.7259	{0.0455, 0.1069, 0.1909, 0.3264, 0.3303}
25	5	1	0.7428	0.1878	{0.1280, 0.3338, 0.0767, 0.3956, 0.0659}
26	6	0	1.0000	0.9263	{0.9464, 0.0108, 0.0012, 0.0267, 0.0151}
27	6	0.27	0.3443	0.2959	{0.4224, 0.1450, 0.2688, 0.1640, 0}
28	6	0.51	0.6974	0.2875	{0.0595, 0.0669, 0.1198, 0.1807, 0.5731}
29	6	0.76	0.3392	0.7484	{0.1373, 0.1116, 0.3986, 0.0258, 0.3268}
30	6	1	0.4469	0.1291	{0.2776, 0.0864, 0.2154, 0.2160, 0.2045}

, the new wind power generation forecasting model has significantly improved prediction accuracy compared to other models. The optimized wind power generation forecasting model still consists of 30 belief rules and 210 parameters. Although some observation points cannot be accurately predicted, the errors are all within an acceptable range, making it still useful as a reference for decision-making in engineering practice.

To verify the effectiveness and convergence characteristics of the P-CMA-ES algorithm in the parameter optimization process of the BRB-f model, this paper introduces an iterative optimization process diagram to monitor the algorithm’s operational status in real time, as shown in Figure 12. Figure 12 illustrates that under seven different segmentation ratios, as the number of iterations gradually increases, the objective function value (MSE) tends to stabilize. It provides a more intuitive reflection of the trend of the objective function value. This trend is associated with the number of iterations during the optimization process. Meanwhile, it offers important support for evaluating the algorithm’s performance.

The three-dimensional distribution of the multi-attribute fuzzy membership function is shown in Figure 13.

5.4. Comparative Study

In order to demonstrate the effectiveness of the BRB-f model proposed in this paper for wind power generation forecasting, a comparative study is conducted in this section. The main innovation of BRB-f lies in introducing a fuzzy membership function to achieve stability in power generation forecasting, combined with multi-rule-weighted fusion and optimization algorithms to improve prediction accuracy.

Compared with the IBRB model, the BRB-f model proposed in this paper shows significant advantages in the context of wind power generation forecasting. First, it replaces fixed interval matching with reference value fuzzy matching, making the model output continuous and smooth, thereby enhancing the stability of power generation forecasting. Second, through multi-rule collaborative activation and weighted fusion, it captures the complex nonlinear relationships in wind power generation systems more finely. And prediction accuracy is enhanced significantly when combined with optimization algorithms.

Based on the above improvements, BRB-f has better stability and accuracy compared to IBRB. To verify BRB-f’s advantage in smoothness for avoiding boundary mutations, we selected 124 data from the test set that had at least one input attribute value near the reference value boundary, with the threshold near the reference value set at 0.005. The comparison of prediction errors between IBRB and BRB-f is shown in Figure 14. It can be seen that the BRB-f performs much better than IBRB in predicting values at the reference value boundary. In order to more intuitively demonstrate the advantages of BRB-f, the mean, variance (Var), and standard deviation (Std) of the prediction errors of IBRB and BRB-f are shown in

Table 3. Table of comparison of error statistical features between IBRB and BRB-f.

Model	Mean	Var	Std
IBRB	4.73 × 10−2	8.64 × 10−4	2.94 × 10−2
BRB-f	2.93 × 10−2	4.31 × 10−4	2.08 × 10−2

.

This section compares the BRB-f model with IBRB-i, IBRB-new1, DNN, RF, SVM [25], Attention [26] and Transformer [27] methods, and the training and test sets are divided into seven different split ratios.

Table A3. Performance comparison of each model at different split ratios.

Ratio	Model	MSE	RMSE	MAE	MAPE	SMAPE
1:9	BRB-f	1.116 × 10−3	3.34 × 10−2	2.80 × 10−2	7.66%	7.13%
	IBRB-i	2.767 × 10−3	5.26 × 10−2	4.42 × 10−2	11.52%	10.60%
	IBRB-NEW1	2.378 × 10−3	4.88 × 10−2	3.84 × 10−2	10.76%	9.81%
	DNN	3.749 × 10−3	6.12 × 10−2	4.75 × 10−2	13.08%	12.69%
	RF	2.378 × 10−3	4.88 × 10−2	3.84 × 10−2	10.76%	9.81%
	SVM	2.184 × 10−3	4.67 × 10−2	3.59 × 10−2	10.85%	9.38%
	Attention	3.772 × 10−3	6.14 × 10−2	5.59 × 10−2	11.52%	10.76%
	Transformer	3.920 × 10−3	6.26 × 10−2	5.70 × 10−2	11.94%	11.10%
2:8	BRB-f	1.069 × 10−3	3.27 × 10−2	2.76 × 10−2	7.65%	7.03%
	IBRB-i	2.714 × 10−3	5.21 × 10−2	4.40 × 10−2	11.35%	10.58%
	IBRB-NEW1	2.278 × 10−3	4.77 × 10−2	3.77 × 10−2	10.36%	9.60%
	DNN	2.358 × 10−3	4.86 × 10−2	3.87 × 10−2	10.66%	9.93%
	RF	1.945 × 10−3	4.41 × 10−2	3.50 × 10−2	10.72%	9.20%
	SVM	1.652 × 10−3	4.06 × 10−2	3.19 × 10−2	9.28%	8.32%
	Attention	3.575 × 10−3	5.98 × 10−2	5.23 × 10−2	10.58%	9.90%
	Transformer	3.810 × 10−3	6.17 × 10−2	5.33 × 10−2	10.49%	9.84%
3:7	BRB-f	1.063 × 10−3	3.26 × 10−2	2.75 × 10−2	7.83%	7.04%
	IBRB-i	2.725 × 10−3	5.22 × 10−2	4.40 × 10−2	11.36%	10.72%
	IBRB-NEW1	2.198 × 10−3	4.69 × 10−2	3.76 × 10−2	10.37%	9.59%
	DNN	2.282 × 10−3	4.78 × 10−2	3.75 × 10−2	11.13%	9.59%
	RF	1.754 × 10−3	4.19 × 10−2	3.31 × 10−2	10.19%	8.67%
	SVM	1.446 × 10−3	3.80 × 10−2	3.06 × 10−2	9.07%	8.02%
	Attention	3.542 × 10−3	5.95 × 10−2	5.39 × 10−2	11.14%	10.41%
	Transformer	3.540 × 10−3	5.95 × 10−2	5.35 × 10−2	10.99%	10.29%
4:6	BRB-f	1.024 × 10−3	3.20 × 10−2	2.70 × 10−2	7.67%	6.93%
	IBRB-i	2.735 × 10−3	5.23 × 10−2	4.42 × 10−2	11.67%	10.80%
	IBRB-NEW1	2.176 × 10−3	4.67 × 10−2	3.71 × 10−2	10.51%	9.51%
	DNN	2.075 × 10−3	4.56 × 10−2	3.59 × 10−2	10.11%	9.16%
	RF	1.527 × 10−3	3.91 × 10−2	3.10 × 10−2	9.79%	8.26%
	SVM	1.285 × 10−3	3.58 × 10−2	2.91 × 10−2	8.67%	7.64%
	Attention	3.533 × 10−3	5.94 × 10−2	5.45 × 10−2	11.62%	10.80%
	Transformer	3.212 × 10−3	5.67 × 10−2	5.15 × 10−2	11.01%	10.27%
5:5	BRB-f	1.050 × 10−3	3.24 × 10−2	2.72 × 10−2	7.85%	6.98%
	IBRB-i	2.714 × 10−3	5.21 × 10−2	4.41 × 10−2	11.75%	10.77%
	IBRB-NEW1	2.212 × 10−3	4.70 × 10−2	3.77 × 10−2	10.91%	9.62%
	DNN	1.696 × 10−3	4.12 × 10−2	3.28 × 10−2	9.80%	8.57%
	RF	1.489 × 10−3	3.86 × 10−2	3.08 × 10−2	9.88%	8.20%
	SVM	1.276 × 10−3	3.57 × 10−2	2.93 × 10−2	9.00%	7.71%
	Attention	3.166 × 10−3	5.63 × 10−2	5.08 × 10−2	10.69%	9.99%
	Transformer	3.186 × 10−3	5.64 × 10−2	4.96 × 10−2	10.11%	9.50%
6:4	BRB-f	9.920 × 10−4	3.15 × 10−2	2.65 × 10−2	7.81%	6.87%
	IBRB-i	2.683 × 10−3	5.18 × 10−2	4.39 × 10−2	11.99%	10.85%
	IBRB-NEW1	2.126 × 10−3	4.61 × 10−2	3.71 × 10−2	10.92%	9.69%
	DNN	1.557 × 10−3	3.95 × 10−2	3.10 × 10−2	9.11%	8.34%
	RF	1.472 × 10−3	3.84 × 10−2	3.03 × 10−2	10.19%	8.21%
	SVM	1.223 × 10−3	3.50 × 10−2	2.87 × 10−2	9.10%	7.67%
	Attention	3.001 × 10−3	5.48 × 10−2	4.97 × 10−2	10.40%	9.80%
	Transformer	2.887 × 10−3	5.37 × 10−2	4.70 × 10−2	9.62%	9.04%
7:3	BRB-f	1.018 × 10−3	3.19 × 10−2	2.70 × 10−2	7.49%	6.88%
	IBRB-i	2.756 × 10−3	5.25 × 10−2	4.48 × 10−2	12.62%	11.18%
	IBRB-NEW1	2.197 × 10−3	4.69 × 10−2	3.77 × 10−2	11.63%	9.97%
	DNN	1.372 × 10−3	3.70 × 10−2	2.99 × 10−2	8.93%	7.80%
	RF	1.454 × 10−3	3.81 × 10−2	3.03 × 10−2	10.57%	8.24%
	SVM	1.203 × 10−3	3.47 × 10−2	2.83 × 10−2	9.39%	7.60%
	Attention	2.906 × 10−3	5.39 × 10−2	4.76 × 10−2	9.66%	9.09%
	Transformer	2.542 × 10−3	5.04 × 10−2	4.54 × 10−2	9.39%	8.89%

compares various indicators of the wind power generation forecasting using the BRB-f model developed in this study with different methods, such as IBRB-i, IBRB-new1, DNN, RF, SVM, Attention and Transformer; the experimental results are the average of 10 repeated experiments. The dataset was divided into different split ratios. The mean and Std of the MSE values from 10 repeated experiments are used to characterize the central tendency and the fluctuation degree of the model performance index, respectively. The $p$ value is obtained from the paired samples t-test. It denotes the probability that the observed performance difference between the two models arises from random sampling error, with a smaller $p$ value indicating a lower probability of randomness and a more statistically significant difference. The specific results are shown in

Table A4. Statistical indicators comparison of each model at different split ratios.

Ratio	Model	Mean of MSE	Std of MSE	p
1:9	BRB-f	1.116 × 10−3	1.243 × 10−5	p < 0.001
	IBRB-i	2.767 × 10−3	1.358 × 10−5	p < 0.001
	IBRB-NEW1	2.378 × 10−3	1.314 × 10−5	p < 0.001
	DNN	3.749 × 10−3	1.288 × 10−5	p < 0.001
	RF	2.378 × 10−3	1.199 × 10−5	p < 0.001
	SVM	2.184 × 10−3	1.199 × 10−5	p < 0.001
	Attention	3.772 × 10−3	1.180 × 10−5	p < 0.001
	Transformer	3.920 × 10−3	1.341 × 10−5	p < 0.001
2:8	BRB-f	1.069 × 10−3	1.288 × 10−5	p < 0.001
	IBRB-i	2.714 × 10−3	1.310 × 10−5	p < 0.001
	IBRB-NEW1	2.278 × 10−3	1.172 × 10−5	p < 0.001
	DNN	2.358 × 10−3	1.362 × 10−5	p < 0.001
	RF	1.945 × 10−3	1.334 × 10−5	p < 0.001
	SVM	1.652 × 10−3	1.210 × 10−5	p < 0.001
	Attention	3.575 × 10−3	1.204 × 10−5	p < 0.001
	Transformer	3.810 × 10−3	1.205 × 10−5	p < 0.001
3:7	BRB-f	1.063 × 10−3	1.229 × 10−5	p < 0.001
	IBRB-i	2.725 × 10−3	1.273 × 10−5	p < 0.001
	IBRB-NEW1	2.198 × 10−3	1.254 × 10−5	p < 0.001
	DNN	2.282 × 10−3	1.226 × 10−5	p < 0.001
	RF	1.754 × 10−3	1.290 × 10−5	p < 0.001
	SVM	1.446 × 10−3	1.196 × 10−5	p < 0.001
	Attention	3.542 × 10−3	1.226 × 10−5	p < 0.001
	Transformer	3.540 × 10−3	1.241 × 10−5	p < 0.001
4:6	BRB-f	1.024 × 10−3	1.259 × 10−5	p < 0.001
	IBRB-i	2.735 × 10−3	1.325 × 10−5	p < 0.001
	IBRB-NEW1	2.176 × 10−3	1.208 × 10−5	p < 0.001
	DNN	2.075 × 10−3	1.271 × 10−5	p < 0.001
	RF	1.527 × 10−3	1.286 × 10−5	p < 0.001
	SVM	1.285 × 10−3	1.177 × 10−5	p < 0.001
	Attention	3.533 × 10−3	1.290 × 10−5	p < 0.001
	Transformer	3.212 × 10−3	1.202 × 10−5	p < 0.001
5:5	BRB-f	1.050 × 10−3	1.181 × 10−5	p < 0.001
	IBRB-i	2.714 × 10−3	1.358 × 10−5	p < 0.001
	IBRB-NEW1	2.212 × 10−3	1.361 × 10−5	p < 0.001
	DNN	1.696 × 10−3	1.330 × 10−5	p < 0.001
	RF	1.489 × 10−3	1.229 × 10−5	p < 0.001
	SVM	1.276 × 10−3	1.188 × 10−5	p < 0.001
	Attention	3.166 × 10−3	1.305 × 10−5	p < 0.001
	Transformer	3.186 × 10−3	1.256 × 10−5	p < 0.001
6:4	BRB-f	9.920 × 10−4	1.192 × 10−5	p < 0.001
	IBRB-i	2.683 × 10−3	1.267 × 10−5	p < 0.001
	IBRB-NEW1	2.126 × 10−3	1.175 × 10−5	p < 0.001
	DNN	1.557 × 10−3	1.350 × 10−5	p < 0.001
	RF	1.472 × 10−3	1.220 × 10−5	p < 0.001
	SVM	1.223 × 10−3	1.301 × 10−5	p < 0.001
	Attention	3.001 × 10−3	1.230 × 10−5	p < 0.001
	Transformer	2.887 × 10−3	1.272 × 10−5	p < 0.001
7:3	BRB-f	1.018 × 10−3	1.277 × 10−5	p < 0.001
	IBRB-i	2.756 × 10−3	1.205 × 10−5	p < 0.001
	IBRB-NEW1	2.197 × 10−3	1.362 × 10−5	p < 0.001
	DNN	1.372 × 10−3	1.323 × 10−5	p < 0.001
	RF	1.454 × 10−3	1.356 × 10−5	p < 0.001
	SVM	1.203 × 10−3	1.347 × 10−5	p < 0.001
	Attention	2.906 × 10−3	1.288 × 10−5	p < 0.001
	Transformer	2.542 × 10−3	1.352 × 10−5	p < 0.001

. Its initial belief rule base was determined by domain experts according to wind power generation characteristics, and the inputs are measured data from six influencing factors. The total number of rules is the sum of the reference values of each attribute. The prediction performance of the BRB-f model compared with IBRB-i and IBRB-new1 is shown in Figure 15.

Compared with the aforementioned models, the BRB-f model proposed in this paper can effectively integrate observational data and expert knowledge. As shown in Figure 15, under conditions of significant fluctuations in power generation, only the BRB-f model maintains good stability. This indicates that the BRB-f model is better suited for practical engineering applications in wind power generation forecasting. To more intuitively compare the accuracy of each model under different split ratios, Figure 16 is plotted as follows.

In summary, among the five main metrics for measuring prediction accuracy: MSE, RMSE, MAE, MAPE, and SMAPE, the BRB-f model consistently maintains the lowest values across seven different training/testing split ratios, and there is a significant quantitative gap compared to other models, as shown in

Table 4. Comparison of mean values of each indicator.

Model	Mean Value of MSE	Mean Value of RMSE	Mean Value of MAE	Mean Value of MAPE	Mean Value of SMAPE
BRB-f	1.047 × 10−3	3.24 × 10−2	2.73 × 10−2	7.71%	6.98%
IBRB-i	2.728 × 10−3	5.22 × 10−2	4.42 × 10−2	11.75%	10.79%
IBRB-NEW1	2.223 × 10−3	4.71 × 10−2	3.76 × 10−2	10.78%	9.69%
DNN	2.156 × 10−3	4.58 × 10−2	3.62 × 10−2	10.40%	9.44%
RF	1.717 × 10−3	4.13 × 10−2	3.27 × 10−2	10.30%	8.66%
SVM	1.467 × 10−3	3.81 × 10−2	3.05 × 10−2	9.34%	8.05%
Attention	3.356 × 10−3	5.79 × 10−2	5.21 × 10−2	10.80%	10.11%
Transformer	3.300 × 10−3	5.73 × 10−2	5.11 × 10−2	10.51%	9.85%

.

Using the IBRB models (IBRB-i, IBRB-NEW1), black-box models (DNN, RF, SVM), and frontier models (Attention, Transformer) in the field of wind power generation forecasting as benchmarks, the average accuracy improvement of the BRB-f model across seven experimental groups was calculated ((benchmark model metric − BRB-f metric)/benchmark model metric × 100%), and the results are shown in

Table 5. Average accuracy improvement of the BRB-f compared to other models.

Model	Upgrade Rate of MSE	Upgrade Rate of RMSE	Upgrade Rate of MAE	Upgrade Rate of MAPE	Upgrade Rate of SMAPE
IBRB-i	61.60%	38.05%	38.29%	34.40%	35.28%
IBRB-NEW1	52.89%	31.37%	27.56%	28.49%	27.94%
DNN	51.41%	29.40%	24.68%	25.91%	26.06%
RF	39.00%	21.60%	16.64%	25.15%	19.36%
SVM	28.60%	15.05%	10.77%	17.43%	13.26%
Attention	68.81%	44.02%	47.59%	28.62%	30.94%
Transformer	68.27%	43.45%	46.53%	26.62%	29.12%

.

From the results of quantitative analysis, the BRB-f model demonstrates the following advantages in wind power generation forecasting tasks: all five main error metrics remain the lowest across seven experiments with different split ratios, indicating high accuracy. Even in the extreme scenario where the training set accounts for only 10%, it can still maintain excellent performance (MSE = 1.116 × 10⁻¹). Considering the five main error metrics and the performance in seven scenarios with different split ratios, the BRB-f model not only addresses the issues of insufficient accuracy and smoothness in IBRB models but also overcomes the shortcomings of black-box models, which are highly dependent on data and with poor stability. This results in a unique advantage of “optimal accuracy, strong stability, and high practicality,” providing a better model choice for actual wind power generation forecasting and possessing significant engineering application value.

5.5. Generalization Analysis

To comprehensively evaluate the generalization ability and robustness of the proposed BRB-f method, this study adopts a 5-fold cross-validation [28] method. The dataset of 2000 samples was randomly divided into five mutually exclusive subsets of 400 samples each. In each iteration, four subsets were used for model training, and the remaining subset was used for testing, repeating this process five times to ensure that each subset served as the test set once. This evaluation strategy effectively avoids the potential bias caused by a single data split and provides a more reliable estimation of model performance. The results of the five-fold cross-validation experiments are shown in

Table 6. Performance comparison of the BRB-f based on five-fold cross-validation.

	MSE	RMSE	MAE	MAPE	SMAPE
Fold 1	1.003 × 10−3	3.17 × 10−2	2.65 × 10−2	7.99%	7.14%
Fold 2	1.068 × 10−3	3.27 × 10−2	2.76 × 10−2	7.65%	7.03%
Fold 3	1.015 × 10−3	3.19 × 10−2	2.72 × 10−2	6.87%	6.70%
Fold 4	1.046 × 10−3	3.23 × 10−2	2.72 × 10−2	7.85%	6.98%
Fold 5	1.032 × 10−3	3.21 × 10−2	2.74 × 10−2	7.50%	7.23%
Mean	1.033 × 10−3	3.21 × 10−2	2.72 × 10−2	7.57 × 10−2	7.02 × 10−2
Std	2.559 × 10−5	3.98 × 10−4	4.23 × 10−4	4.35 × 10−3	2.02 × 10−3

.

To further validate the cross-dataset generalizability of the method, experiments were conducted on three datasets from different sources, namely the spacecraft flywheel system dataset, the lithium-ion battery dataset, and the diesel engine dataset. The performance of the BRB-f method on prediction tasks is shown in

Table 7. Performance comparison of different models on three datasets.

Dataset	Model	MSE	RMSE	MAE	MAPE	SMAPE
Flywheel	BRB-f	5.10 × 10−5	7.12 × 10−3	3.66 × 10−3	1.81%	1.56%
	IBRB-i	2.44 × 10−2	1.56 × 10−1	1.22 × 10−1	39.82%	22.79%
	IBRB-NEW1	2.42 × 10−2	1.56 × 10−1	1.05 × 10−1	17.47%	17.65%
	DNN	4.84 × 10−4	2.20 × 10−2	7.53 × 10−3	1.17%	1.18%
	RF	8.47 × 10−3	9.20 × 10−2	4.89 × 10−2	13.56%	10.08%
	SVM	4.95 × 10−3	7.03 × 10−2	5.89 × 10−2	14.03%	11.75%
Lithium-ion battery	BRB-f	1.67 × 10−3	4.09 × 10−2	2.34 × 10−2	1.38%	1.40%
	IBRB-i	8.38 × 10−3	9.15 × 10−2	6.74 × 10−2	4.26%	4.32%
	IBRB-NEW1	9.41 × 10−3	9.70 × 10−2	6.54 × 10−2	4.08%	4.16%
	DNN	8.93 × 10−4	2.99 × 10−2	1.58 × 10−2	0.96%	0.97%
	RF	2.82 × 10−3	5.31 × 10−2	3.49 × 10−2	2.23%	2.25%
	SVM	8.87 × 10−3	9.42 × 10−2	8.18 × 10−2	5.65%	5.56%
Diesel engine	BRB-f	3.91 × 10−2	1.98 × 10−1	1.17 × 10−1	5.80%	5.54%
	IBRB-i	4.84 × 10−1	6.95 × 10−1	6.52 × 10−1	36.30%	33.00%
	IBRB-NEW1	5.93 × 10−2	2.44 × 10−1	1.65 × 10−1	9.53%	9.14%
	DNN	5.26 × 10−2	2.29 × 10−1	1.15 × 10−1	5.69%	5.33%
	RF	7.88 × 10−2	2.81 × 10−1	1.59 × 10−1	7.96%	7.43%
	SVM	7.96 × 10−2	2.82 × 10−1	1.42 × 10−1	7.15%	6.79%

. As seen from the results in Table 7, BRB-f demonstrates strong generalization ability across multiple datasets. Compared with models such as IBRB-i, IBRB-NEW1, DNN, RF, SVM, Attention and Transformer, BRB-f consistently maintains high prediction accuracy, performing excellently in key metrics such as MSE, RMSE, MAE, MAPE, and SMAPE. Only on the lithium-ion battery dataset does the accuracy slightly lag behind DNN, but it still maintains a high level of precision, and DNN performs worse than BRB-f on the other two datasets. The research results indicate that the proposed BRB-f method has potential application value in engineering prediction tasks and broad engineering applicability.

6. Conclusions

In response to the characteristics of strong nonlinearity [29], volatility [30], and continuity in wind power generation forecast data in wind power engineering practice, as well as the insufficient smoothness and poor flexibility of the IBRB when dealing with multi-attribute problems, a new wind power generation forecasting method based on BRB-f is proposed. This model, through key innovations such as fuzzy membership functions, multi-rule collaborative activation, and weighted fusion, successfully overcomes the limitations of IBRB. It can accurately capture complex nonlinear relationships, providing stable output power for wind farms, while offering reliable input for grid scheduling, improving the safety of grid integration and enhancing the scientific basis of power dispatch decisions.

Regarding the issues faced by the IBRB model in wind power generation forecasting, the applicability of the proposed BRB-f model is mainly reflected in the following aspects:

Firstly, to address the strong nonlinearity of wind power generation data, the BRB-f model introduces fuzzy membership functions to quantify the matching degree between each attribute input and the reference value, solving the problem of IBRB being unable to accurately capture information within intervals. Secondly, for the volatility of wind power generation data, its multi-rule collaborative activation and weighted fusion mechanisms enhance the ability to capture fluctuating features, reducing prediction errors. Finally, to tackle the continuity of wind power generation data, fuzzy reference value matching avoids the jump problem caused by fixed interval matching in IBRB, ensuring stable output of predicted power in line with actual physical variations.

The BRB-f model has already achieved significant results in current research. By introducing fuzzy membership functions, multi-rule collaborative activation, and weighted fusion, it has successfully overcome the limitations of the IBRB model, showing excellent performance in wind power generation forecasting and bringing important technological innovations to wind power engineering. However, the actual wind power engineering environment is far more complex than the factors considered in current research. In addition to the six main factors mentioned above, there are numerous complex interfering factors that are interwoven and jointly influence wind energy generation and variation. Therefore, enhancing the generalization capability of the BRB-f model to better adapt to various complex and changing scenarios will be a key direction for future exploration.