Deep belief rule based photovoltaic power forecasting method with interpretability

Accurate prediction of photovoltaic (PV) output power is of great significance for reasonable scheduling and development management of power grids. In PV power generation prediction system, there are two problems: the uncertainty of PV power generation and the inexplicability of the prediction result. The belief rule base (BRB) is a rule-based modeling method and can deal with uncertain information. Moreover, the modeling process of BRB has a certain degree of interpretability. However, rule explosion and the inexplicability of the optimized model limit the modeling ability of BRB in complex systems. Thus, a PV output power prediction model is proposed based on a deep belief rule base with interpretability (DBRB-I). In the DBRB-I model, the deep BRB structure is constructed to solve the rule explosion problem, and inefficient rules are simplified by a sensitivity analysis of the rules, which reduces the complexity of the model. Moreover, to ensure that the interpretability of the model is not destroyed, a new optimization method based on the projection covariance matrix adaptation evolution strategy (P-CMA-ES) algorithm is designed. Finally, a case study of the prediction of PV output power is conducted to illustrate the effectiveness of the proposed method.

extendability can cause expert knowledge to be reluctant to be redesigned when new attribute information is used. Therefore, the first question is how to build a new BRB model, which has good extendability and can avoid the combination explosion problem.
where x is the input data of the PV power generation system,C is the set of interpretability constraints, is the model parameter,Ek is the expert knowledge introduced into the model, and y is the predicted result of PV power generation systems.
Problem 2 Interpretability constraints are designed. BRB is a rule-based modelling method that can easily understand the modelling process of the model. However, the interpretability of the model optimization process will be destroyed due to the randomness of the optimization algorithm. Thus, the second problem is how to design effective interpretability constraints to ensure that the interpretability of the model is not destroyed.
where P is the set of parameters in the optimization process.
Problem 3 Improve the simplicity of the rules base. The rule base in BRB is generated by the attribute reference value in the form of Descartes. Some redundant rules and inefficient rules exist in the rules base of BRB, which will reduce the accuracy and readability of the model. Therefore, the third question is how to reasonably and transparently remove redundant rules and inefficient rules in the BRB rule base.
where g is the error fluctuation of the model,MSE SA is the accuracy of the model after sensitivity analysis, and MSE (initial) is the accuracy of the model built with expert knowledge.

Construction of the DBRB-I model.
BRB is composed of a series of belief rules, and the kth rule can be described as follows: where x 1 , x 2 , ..., x T k is the antecedent attribute of the PV power prediction system.A k 1 , A k 2 , ..., A k T k is a series of reference values for the antecedent attribute x 1 , x 2 , ..., x T k .T k is the number of attributes in the kth rule.D 1 , D 2 , ..., D N are the consequences, and β 1,k , β 2,k , ..., β N,k are their corresponding belief degrees.θ k is the weight of the kth belief rule.δ 1 , δ 2 , ..., δ i is the weight of the ith attribute.L is the number of rules.C 1 , C 2 , ..., C n is the interpretability constraint of the model. The DBRB-I model is composed of multiple Sub-BRBs in a deep structure. The modelling process of the DBRB-I model based on the PV power generation system is shown in Fig. 1. First, the attributes of PV power generation systems are trended analysed and sorted by correlation to changes in results. Second, the rules of the initial BRB constructed from expert knowledge are simplified through sensitivity analysis. Finally, each Sub-BRBn of DBRB-I is optimized by an optimization algorithm with interpretability.

The DBRB-I model interpretability
To maintain the interpretability of the DBRB-I model, it is crucial to construct interpretability constraints for PV power generation systems. Cao et al. constructed a general BRB interpretability criterion 23 . Thus, the DBRB-I model should conform to the general interpretability BRB criterion. Moreover, to make the DBRB-I model more interpretable, this paper focuses on the interpretability criteria7, 8. The interpretability of the DBRB-I model is shown in Fig. 2.
Criterion 7: The simplicity of the rule base.
The modelling method based on IF-THEN rules can make the model structure clear and easy to understand. However, the number of rules grows exponentially in complex systems of PV power generation, which leads to the explosion of rules in the BRB system. Too many rules can reduce the readability of the model and reduce model interpretability.
Thus, to improve the interpretability and readability of the DBRB-I model, the rules in the system should be reasonably reduced. A reasonable number of rules can improve the accuracy of the BRB model 24 . The initial BRB constructed by expert knowledge can be effectively analysed to determine which rules are redundant and (1) y = f (x, C, �, Ek) (2) Interpretability : {C|C 1 , C 2 , ...C m } (3) � = optimize(x, y, P, C)  www.nature.com/scientificreports/ Thus, in a PV power generation system, to ensure the reasonableness of the prediction results, the interpretability constraints are as follows: where U k is the interpretability constraint in the kth rule. For different systems, the interpretability constraints will be different, but it should be noted that the operating mechanism and common sense of the actual system need to be satisfied 25 . Moreover, a reasonable belief distribution shape should be monotonic or convex.
BRB is a rule-based modelling method, and the relationship between the input and output of the model is traceable. Thus, the interpretability of the model structure is the intrinsic feature of BRB. Due to limited expert knowledge, the initial BRB model built by experts does not meet the needs of the actual system, and the model needs to be optimized by observation data. However, optimization algorithms have randomness, which destroys the interpretability of BRB models. Therefore, the following constraints are designed to preserve the interpretability of the BRB model, and the feasible region for DBRB-I model optimization is shown in Fig. 4. Expert knowledge is obtained through the analysis of the actual system and the experience of long-term accumulation. It is one of the important sources of model interpretability 26 . The optimization process of the interpretable BRB model is a local search process based on the initial judgement of experts 22 . Thus, expert knowledge is converted into parameters and brought into the initial population of the optimization algorithm, which can provide guidance for the optimization process and effectively extract useful information from the search space 27 .   www.nature.com/scientificreports/ Constraint 2: The optimized parameters meet the judgement of experts.
Compared with the black box model, such as the backpropagation neural network (BPNN) and support vector machine (SVM), the parameters of the BRB model are of practical significance, which makes users more trust this model. However, the physical significance of the optimized BRB model parameters may be lost. For example, the initial rule weight is given 0.9 by experts, but the optimized rule weight is 0.005. Experts believe that this rule is crucial, but the results of the model after optimization are inconsistent with the judgment of the expert. This will lead to the reduction of experts' trust in the model. Thus, the key parameters of the BRB model need to be guaranteed to be used effectively. To solve this problem, the parameters of the BRB model are constrained as follows: where P lp is the minimum value of the parameter and P up is the maximum value of the parameter. Effective constraints for the parameters of the BRB model can avoid modelling accuracy reduction due to overoptimization. Constraint 3: Local optimization process based on expert knowledge.
The interpretability of the optimization process reflects the optimization in a local search domain judged by experts for interpretability BRBs 22 . Thus, to enhance the interpretability of the model, an interpretability constraint is designed.
Interpretability constraints on individuals of the initial population by introducing Euclidean distance. The Euclidean distance reflects the straight-line distance between two points in space. Constrain the distance between the individual and the expert knowledge of the algorithm, which further realizes the optimization of the local search domain based on the initial judgment of the expert and enhances the interpretability of the optimization.
ρ(x n , x ′ n ) is the Euclidean distance between the individuals of the initial population and expert knowledge. d is the parameter of the distance, and the value is determined by the expert.
BRB is a rule-based modeling method that conforms to the method of human knowledge expression. Through expert knowledge and observational data, the reasoning process and modelling process of the BRB model can be easily understood. Moreover, BRB has a transparent reasoning process based on the ER algorithm, and decision makers and users can directly touch and access the model. Thus, interpretability is an intrinsic feature of BRB models. However, any algorithm has randomness, and this randomness destroys the interpretability of the BRB model. Therefore, it is necessary to design the above interpretability constraints to protect the interpretability of the BRB model 23 .

Sensitivity analysis
An appropriate BRB model structure and parameters are crucial to improve the accuracy and interpretability of the predicted PV output power. In the literature 24 , Zhang et al. demonstrated that a suitable number of rules can improve the accuracy of the BRB model. Moreover, the simplicity of the BRB rule base can improve the readability of the model 23 . Therefore, there are many methods of rule reduction 28,29 . However, reasonable and transparent reduction rules are important for interpretable BRB models. Sensitivity analysis is a great way to simplify the rule base in an interpretable way.
Sensitivity analysis (SA) refers to analysing the direct influence of model input parameters and model results 30 . SA can identify key parameters of the model, which can help users improve the prediction accuracy of the model. There are two main methods of sensitivity analysis: global sensitivity analysis (GSA) and local sensitivity analysis (LSA). GSA refers to the study of the effect of two or more parameters changing together on the model parameters. However, GAS is computationally expensive 30 . LSA refers to exploring changes in the response of a model by changing one parameter of the model while keeping the other parameters constant. The advantages of local sensitivity analysis are simplicity and ease of understanding.
Thus, this paper uses LSA to simplify the structure of the BRB model. The input parameter of the LSA is the rule weight, and the sensitivity of the rule is reflected by the fluctuation of the model error g; that is, the greater the mean square error (MSE), the greater the sensitivity of the rule. Through LSA, users can learn which rules are important and which rules are inefficient. Finally, the inefficient rules are simplified, which simplifies the BRB model and improves the model readability.

Reasoning process and optimization process of the DBRB-I model
In "The reasoning process of the DBRB-I model" section, the reasoning process of the DBRB-I model is introduced. Then, in "Optimization process of the DBRB-I model" section, the optimization process of the DBRB-I model is introduced.
The reasoning process of the DBRB-I model. The reasoning process of BRB is based on the evidential reasoning (ER) algorithm. The ER algorithm with transparency and reliability is described as follows: Step 1 Different forms of input information are transformed into belief distributions.
where a i,j is the matching degree between the input information and the reference value A i,j .
Step 2 The activation weight of the kth belief rule is calculated.
Step 3 The belief degree of the inference output is generated by the analytical ER algorithm.
Step 4 The final belief distribution of the inference output is expressed as follows: where A ′ is the input vector of the actual system,u(D j ) is the utility of the D j .u(S(A ′ )) is the final expected utility.
Optimization process of the DBRB-I model. In current research, the projection covariance matrix adaptation evolution strategy (P-CMA-ES) optimization algorithm is one of the effective algorithms and has also been applied to the research of different BRBs 23 . The P-CMA-ES optimization algorithm has the following advantages: (1) It has good optimization performance. (2) The algorithm has a fast convergence speed and strong robustness.
(3) It has the advantages of rotation invariance and spread rotation invariance. Thus, the P-CMA-ES algorithm is used to optimize the DBRB-I model in this paper. However, the P-CMA-ES algorithm generates new solutions that will destroy DBRB-I model interpretability. Therefore, to maintain the interpretability of the DBRB-I model, interpretability constraints are added to the original P-CMA-ES algorithm. The newly modified P-CMA-ES optimization algorithm is shown in Fig. 5. The pseudocode of the optimization method is given in Algorithm 1, and the specific process is as follows: Step 1 (Construct the objective function) To improve the prediction accuracy of the DBRB-I model, the parameters of the model are optimized through the training data. Therefore, the objective optimization function of the DBRB-I model is described as follows: www.nature.com/scientificreports/ where MSE(·) is the degree of difference between the predicted value of the PV system and the true value, which can be described as follows: where n is the number of training data.y i is the true value of the PV power generation system. y i is the predicted value of the DBRB-I model.
Step 3 (Sampling operation) Through interpretability constraints 1, 2, and 3, generate the initial population by Interpretability constraint 1 incorporates expert knowledge into the initial population of the model, and expert knowledge can play a guiding role in the model optimization process, which improves the model optimization process. Moreover, interpretability constraint 1 enables the starting point of the optimization to be close to the optimal solution of the model.
Interpretability constraint 2 guarantees that the parameters do not lose their physical meaning during optimization, which maintains the interpretability of the model. www.nature.com/scientificreports/ Interpretability constraint 3 guarantees that the optimization process is a local optimization based on expert knowledge for the interpretability model, which further enables the optimized parameters to have good similarity to the expert knowledge.
Step 4 (Constraint operation) Through interpretability criterion 8, the rules that do not meet the actual system are adjusted.
where � (g+1) k is the kth solution in the (g + 1)th generation, which may not satisfy the belief distribution of the actual system.β (g+1) k is the newly generated belief distribution satisfying the interpretability criterion 8.⇐ is the replacement operation.
Step 5 (Projection operation) To satisfy the equality constraint, the projection operation maps candidates back into the feasible region hyperplane.
The projection operation is implemented as follows: Step 6 (Selection operation) Calculate the MSE value of population individuals and sort them. The process is described as follows: The optimal subgroup is updated as follows: Step 7 (Adapting operation) Update the search covariance matrix, the evolution path of the covariance matrix, the search step size and the evolution path through the most subgroup strategy. www.nature.com/scientificreports/

Case study
In "Description of the dataset" section, the dataset is described. The initial DBRB-I model is constructed in "Construct the initial BRB by expert knowledge" section. Then, in "Sensitivity analysis of the initial DBRB-I model" section, a sensitivity analysis of the rules of the DBRB-I model is presented. The optimized DBRB-I model is described in "The optimized DBRB-I model" section. In "Discussion of the interpretability of the DBRB-I model" section, the interpretability of the DBRB-I model is discussed. The increase in the proportion of PV power generation will increase the difficulty of power grid scheduling. When the proportion exceeds 15%, it may cause paralysis of the grid system 31 . An interpretable model can provide grid operators with some reference for grid scheduling and management. Thus, it is very important to predict PV power in a reliable, safe and interpretable way 31 .
Description of the dataset. The PV power dataset is obtained from AI Studio. The dataset is the operation data of PV modules in China in 2018. The data collection interval was 10 min, and the data were desensitized. This paper selects a week of data from January 1, 2018, to January 7, 2018, for the experiment, as shown in Fig. 6. A total of 280 data samples are used for training, and 140 data samples are used for testing. Construct the initial BRB by expert knowledge. PV power generation systems use solar energy to generate electricity, and their output power is strongly affected by solar irradiance. At the same time, voltage, ambient temperature and module temperature are important factors that affect PV output efficiency 32 . Then, the dataset is normalized, and the relationship between each attribute and output power is trend analysed, as shown in Fig. 7. The irradiance has the largest relationship with the output power, while the ambient temperature has the smallest relationship with the output power, and the most obvious part is marked with a black curve. Thus, the order of importance to the output power attributes of the PV power generation system is as follows: irradiance, voltage, module temperature, and ambient temperature. The initial DBRB-I model is shown in Fig. 8.
In practical engineering, the selection of the reference value requires the judgment of expert knowledge to select the interval range with typical significance and then combine the data statistics to give accurate results 33 . Thus, normalized data usually use four semantic values to describe the attributes and the state of the system, that is, "Excellent", "Good", "Middle", and "Low". The reference values are given in Tables 1 and 2. Moreover, the beliefs of the initial models Sub-BRB1, Sub-BRB2 and Sub-BRB3 are shown in "Appendix A".

Sensitivity analysis of the initial DBRB-I model.
To meet the needs of the PV power generation system, the model error g = 0.001. A sensitivity analysis of the rules for the initially constructed Sub-BRB1, Sub-BRB2 and Sub-BRB3 is shown in Figs. 9, 10, and 11, respectively. Rules 3,4,5,7,8,9,12,13,14, and 16 in Sub-BRB1 have little effect on the system. These rules can be considered inefficient rules of the system. However, rules 1, 2, 6, 10, 11, and 15 have a large impact on the system and satisfy the model error g.
The purpose of sensitivity analysis is to remove inefficient rules, which will reduce the complexity of the model and increase the readability of the model by describing the PV system with fewer rules. Moreover, the difficulty of optimization is reduced, and the effect of optimization is improved. Table 3 shows the effective rules of each sub-BRB of DBRB-I.
To further prove the redundancy of inefficient rules, the rule activation weights of DBRB-I are analyzed. Figures 12, 13, and 14 show the activation weights of each rule for Sub-BRB1, Sub-BRB2, and Sub-BRB3, respectively. As seen from Fig. 11 (the blue curve represents that the activation weight of the rule is too small, the green   www.nature.com/scientificreports/   www.nature.com/scientificreports/ curve represents that the rule is not activated, the black curve represents that the rule is very important, and the purple curve represents that the rule has little effect on the model), the activation weight of rules 3, 12, and 14 is too small, it is difficult for experts to judge whether rule 3 has an effect on the system, which reduces the user's understanding of the model, and rule 3 will reduce the interpretability of the model. Thus, rule 3 is classified as an inefficient rule. Moreover, rules 4, 8, 9, and 13 have no effect on the model, which shows that these rules are redundant. Reducing these rules does not affect the accuracy of the model but improves the interpretability of the model. The number of times rule 6 is activated in the system shows that rule 6 is very important to the system. Rule 6 is activated many times in the system, which shows that rule 6 has a great influence on the system. Finally, although the activation weight of rule 16 is relatively large, the number of activations is too small, and it has little effect on the model, so rule 16 is also classified as an inefficient rule. To prove the reliability and rationality of expert knowledge and whether the DBRB-I model can correctly reduce inefficient rules, the sensitivity analysis of the Sub-BRB1, Sub-BRB2 and Sub-BRB3 rules after DBRB-I model training is shown in Figs. 15, 16 and 17, respectively. Table 4 gives the effective rules for the trained DBRB-I model to satisfy the threshold g. Through the comparison of Tables 3 and 4, the inefficient rules of the DBRB-I model after training are consistent with the judgment of expert knowledge. Reasonable rule reduction for the initial DBRB-I can reduce the complexity of the model and improve the optimization process of the model. In this paper, the initial DBRB-I is constructed from expert knowledge, and the expert knowledge is reliable. Only rule 12 in Sub-BRB2 is inconsistent with the judgment of the initial DBRB-I model. However, in Fig. 10, rule 12 has a certain effect on the initial model but does not meet the threshold g. Moreover, as shown in Fig. 13, the partial activation weight of rule 12 is too small, and experts cannot judge whether this rule works, which will reduce the interpretability of the model. Therefore, rule 12 is considered an inefficient rule in the initial model.  www.nature.com/scientificreports/     The DBRB-I model conforms to expert judgment. The belief distribution of each rule of Sub-BRB3 is shown in Fig. 19. In rules 2, 3, 5, 6, 7, 8, and 9, the optimized belief distribution is close to the expert knowledge, which shows that Sub-BRB3 improves accuracy while maintaining interpretability. In rules 1 and 4, the overall belief distribution of each rule is consistent with the actual system and can be trusted by users. Moreover, it can be seen that the belief distributions in the optimized DBRB-I (20) are close to the initial DBRB-I (20), which indicates that these parameters are locally optimized based on the initial judgement of experts.

Number of initial rules Number of efficient rules Efficient rule
The test data 64th value is 0.3653. www.nature.com/scientificreports/ is that the rules with less influence of the model are simplified. An effective way to improve the accuracy of the DBRB-I(20) model is to lower the threshold g for rule reduction and allow more rules to participate in the model. To prove that reducing the rule reduction threshold g can improve the accuracy of the model, therefore, the threshold g = 0, only the rules that have no effect on the system are reduced for the DBRB-I model. Table 5 gives the number of rules for DBRB-I(41). As shown in Fig. 21, the DBRB-I(41) model can accurately predict the PV power generation system in an interpretable manner. However, DBRB-I(41) has 41 rules, and DBRB-I(20) has 20 rules; that is, the DBRB-I(20) model is more readable than DBRB-I(41).

Sub
To further prove that removing inefficient rules can improve the accuracy of the model, a comparison between the DBRB(48) model and the DBRB(41) model is shown in Fig. 22. There are 7 fewer rules and 35 fewer optimization parameters in the entire optimization process, which saves more computing resources for DBRB(41) and improves the optimization process of the DBRB(41) model. The MSEs of DBRB(41) and DBRB(48) are 2.81E-5 and 4.83E-4, respectively. Figure 22 shows the comparison between DBRB(41), DBRB(48) and the atcual value. It can be seen that the prediction effect of DBRB(41) is better than that of DBRB(48). This proves that reducing inefficient rules can improve the accuracy of the model. When inefficient rules exist in the system, it will increase www.nature.com/scientificreports/   www.nature.com/scientificreports/ the burden on the optimization process of the model and reduce the interpretability of the model. Therefore, choosing appropriate rules is crucial for the model. Fig. 23. The blue and red curves are the modified P-CMAES algorithm and the original P-CMAES algorithm, respectively. Compared with the original P-CMAES algorithm, the modified P-CMAES algorithm is more suitable for the optimization process of the interpretability model. The starting point of the optimization process of the modified algorithm starts from the vicinity of the expert knowledge so that the initial population of the algorithm will carry some characteristics of the expert knowledge information. Moreover, the optimization process of the DBRB-I model is a local search process based on the initial judgment of expert knowledge. This proves that the optimization process of the modified algorithm can maintain a good balance between model interpretability and modelling accuracy. For the DBRB-I model, the expert knowledge accumulated from the PV power generation system plays a crucial role in the interpretability of the model. The parameters obtained by expert knowledge are closer to the optimal point in the feasible domain search space; that is, expert knowledge can provide a guiding direction for the optimization process. Therefore, putting expert knowledge into the initial population of the algorithm can improve the optimization process of the model. Furthermore, interpretability constraint 3 further realizes that the optimization process of the DBRB-I model is a local search domain based on expert judgment.

Analysis of interpretability constraints of the DBRB-I model. The effectiveness of interpretability constraints 1 and 3 is demonstrated in
Belief rule base is an intelligent expert system that combines the characteristics of expert system and datadriven model 21 . Experts can build the initial BRB model according to their experience and domain knowledge, which effectively reduces the difficulty of parameter optimization and improves the interpretability of the model 23 . Moreover, the model parameters are further optimized by observation data, which enables the BRB model to achieve good accuracy. Thus, the DBRB-I model is a hybrid model that combines knowledge and data. The initial DBRB-I model is reasonably constructed by expert knowledge, so that the model has good interpretability. Through observational data, an optimization algorithm with interpretability constraints is used for optimization, enabling fine-tuning of parameters based on expert knowledge. Therefore, the DBRB-I model is able to maintain a good balance between interpretability and accuracy.  Appendix Table B4.     Table 7. The MSE standard deviation of DBRB-I is smaller than that of LSTM and DBN, which means the robustness of the DBRB-I model is stronger than LSTM and DBN. In practical engineering, the DBRB-I model is more suitable for reliable and safe systems 23 .
Analysis of DBRB-I model skill scores. To further verify the prediction effect of the DBRB-I model, benchmarking is also performed by skill score, which is defined as follows 37 : where error proposed is the error of the proposed model and error reference is the error of the reference model.     1) The DBRB-I model is a rule-based modeling method, which can describe the modeling process of the system in a language semantic way. (2) The DBRB-I model has a transparent inference engine, which makes the internal structure clear and transparent and can be directly accessed by users. (3) The DBRB-I model can incorporate expert knowledge and system mechanisms, which can better help people understand and trust the model.

Discussion of the interpretability of the DBRB-I model.
As an interpretable model, DBRB-I has the characteristics of a transparent reasoning process, process interpretability, and traceability of results. Power grid operators can directly access the model, but the internal structure of the data-driven black-box model is invisible. Moreover, DBRB-I can identify key parameters of the PV power generation system, and experts can further improve their expert knowledge by analyzing the key parameters. The sensitivity analysis for each rule of Sub-BRB3 is shown in Fig. 27. The parts drawn with black circles clearly show that there are jumps in the system, which can have a huge impact on the PV system when the rule weights are within a certain interval. Therefore, to ensure that the system can maintain a balance between accuracy and interpretability, interpretability constraint 2 is necessary. For the higher sensitivity rules 1, 2, 4, 5, and 7, grid operators should analyze the hidden mechanisms in detail as feedback on the actual situation 38 . For the less sensitive rules 3, 6, 8, and 9, the modelling process should be further improved, which can improve the modelling accuracy 38 .
The reasoning process and modelling process of the DBRB-I(20) model can be accessed by decision makers and users. Therefore, DBRB-I as an interpretability model can effectively analyse the system. Each rule of Sub-BRB1, Sub-BRB2, and Sub-BRB3 of DBRB-I(20) is shown in Fig. 28. As shown in Figure a, Sub-BRB1 has low sensitivity to the belief level G but high sensitivity to E and L. The reason for this is that the irradiance, voltage and output power data of the model are not well fitted at the belief level G, as shown in Fig. 6, which is consistent with reality. As shown in Figure b, Sub-BRB2 has low sensitivity to belief level E but high sensitivity to G, L, and M. The reason for this is that the module temperatures of the system do not reach the belief level of E, as shown in Fig. 6. As shown in Figure c, Sub-BRB3 combines Sub-BRB1 and Sub-BRB2 and is sensitive to belief levels E, G, M, and L. This further demonstrates the effectiveness of the DBRB-I(20) model.

Conclusion
In this paper, a new PV power generation prediction model based on deep belief rule base with interpretability (DBRB-I) is proposed. In the DBRB-I model, first, the attributes of the PV power generation system are trended toward analysis, and the Sub-BRB is constructed through the correlation between the attributes and the results. www.nature.com/scientificreports/ Second, sensitivity analysis of the initial DBRB-I model constructed by experts, which reduces inefficient rules and redundant rules to reduce model complexity. Finally, the simplified DBRB-I model is optimized by an interpretability optimization algorithm. There are three innovations in this paper. A PV power generation prediction model is proposed based on the DBRB-I model. The DBRB-I model consists of multiple Sub-BRBs in a deep structure, which effectively solves the problem of rule explosion and weak scalability of BRBs. To ensure that the interpretability of the model after optimization is not destroyed, a new optimization method is designed. Moreover, to improve the readability of PV power generation prediction models, a transparent and reliable rule reduction method sensitivity analysis is proposed. A case study of a PV power generation system is used to verify the validity of the proposed model. The results show that the DBRB-I model can maintain a good balance between interpretability and accuracy.
However, the local sensitivity analysis method used in this paper has limitations. This method neglects the mutual influence between uncertain parameters, which will interfere with the decision-making results. Therefore, how to use global sensitivity analysis deserves further study. Moreover, more benchmark datasets should be used and how to adequately analyze and interpret the sensitivity of a model in future research. www.nature.com/scientificreports/

Data availability
The datasets generated and analysed during the current study are available in the AI Studio repository, https:// aistu dio. baidu. com/ aistu dio/ datas etdet ail/ 147402.