A new state‐of‐health estimation method for Li‐ion batteries based on interpretable belief rule base with expert knowledge credibility

State‐of‐health (SOH) estimation methods for Li‐ion batteries are important for the safe operation of the entire system. However, it is often challenging due to the uncertainty within the batteries and the criteria for model interpretability. Belief rule base (BRB) is a rule‐based expert system that has certain advantages for both aspects. However, several problems with BRB interpretability need to be solved urgently. First, expert knowledge credibility is often given subjectively, while objective information is neglected to be considered. Second, BRB interpretability is easily ignored or corrupted in the optimization process. Third, expert knowledge is assumed to be completely reliable information to be used as interpretability evaluation criterion. Therefore, a new SOH estimation method for Li‐ion batteries based on interpretable BRB with expert knowledge credibility (IBRB‐c) is proposed. In the IBRB‐c, the calculation method of expert knowledge credibility is given. Then, an optimization algorithm with interpretability strategies is used. Finally, the concept of the fuzzy interpretable interval is proposed to design the interpretable evaluation criterion. The effectiveness of the proposed method is verified by using the experiment of NASA Li‐ion battery as a case study.

2][3] However, as Li-ion batteries age and undergo internal chemical reactions, they tend to enter an unstable state.If these batteries are used continuously under such unstable conditions, it will lead to low battery voltage and affect the battery life. 4It becomes critical to develop accurate and interpretable estimation techniques.The state of health (SOH) estimation of batteries refers to the evaluation of batteries' current condition and their ability to perform within specified parameters.SOH estimation can provide important information about battery capacity degradation.By estimating the SOH, potential risks can be mitigated, and necessary preventive measures can be taken in time.
Many SOH estimation methods for Li-ion batteries have been developed by researchers and can be mainly classified into model-driven methods and data-driven methods.
Model-driven methods rely on the development of mathematical and physical models that describe the underlying physics and chemistry of the battery system. 5hese models take into account various parameters, such as battery voltage, current, temperature, and other physical characteristics.Dong et al. introduced a new capacity degradation parameter for battery health state detection and lifetime prediction using a filter algorithm supporting vector regression particle filter. 6Xiong et al. extracted health indicators based on partial battery charging voltage profiles and constructed a linear aging model for battery health estimation using a moving window approach. 7Dong et al. proposed a probabilistic approach for battery degradation modeling and health prediction using a particle filter inference algorithm based on dynamic Bayesian network. 8ata-driven methods typically employ techniques such as machine learning algorithms.These algorithms learn from historical data and identify patterns associated with different SOH.Adnan et al. used support vector machines with load ensembles for training and testing data processing to perform health diagnosis and remaining life prediction of Li-ion power batteries. 9Bai et al. integrated an artificial neural network with a double extended Kalman filter algorithm for Li-ion battery health management. 10Liu et al. applied a long shortterm memory submodel to estimate residuals with a Gaussian process regression submodel to fit intrinsic mode functions. 11owever, the limitation of the model-driven methods is that they tend to oversimplify the complexity of the battery system, leading to inaccurate estimation results.These models may not fully capture the complex interactions and degradation mechanisms occurring within the battery, thus compromising the accuracy of the estimation. 12On the other hand, data-driven methods rely on processing large sets of observational data to extract patterns and relationships.While datadriven methods can provide accurate estimation based on available data, they often lack interpretability and transparency.These models lack insight into the internal structure and mechanism of Li-ion batteries, and the estimation process may not be easily traceable. 13iven the above analysis, the belief rule base (BRB) approach emerges as a promising alternative for Li-ion battery health estimation.The BRB approach combines expert knowledge that encompasses the theoretical foundation and ensures that the model meets the requirements of the specific domain.At the same time, it utilizes datadriven components to improve adaptability and accuracy, thus enabling it to effectively handle real-world variability.In essence, this approach combines model-driven interpretability with data-driven flexibility, resulting in a robust framework capable of solving intricate real-world problems accurately and reliably. 14lthough BRB models are inherently interpretable, ensuring that this potential is effectively utilized in realworld applications still requires careful attention. 15Much work has already been carried out to enhance the interpretability of BRBs.The importance of expert knowledge as a key component of BRB interpretability has been emphasized in previous work by Yang et al. 16 Based on this concept, Feng et al. proposed a new optimization method, which aims to ensure that the optimized parameters are closely integrated with expert knowledge while retaining as much expert knowledge as possible. 17Han et al. proposed an interpretable BRBbased capacity prediction model for Li-ion batteries, which uses an interval optimization strategy to test the model performance under different intervals.The objective is to find an optimal interval parameter to balance accuracy and interpretability. 18Then, Han et al. proposed a new evaluation criterion in the literature, 4 considering both the accuracy and interpretability of BRB.The credibility of expert knowledge is the degree of trustworthiness and reliability that is associated with a body of knowledge.However, the expert knowledge in the proposed criteria is assumed to be completely reliable.Therefore, Han et al. pointed out that the evaluation criterion should be improved in the future by considering the influence of expert knowledge credibility.
After extensive research, the study is carried out for the interpretability of BRB, and a new SOH estimation method for Li-ion batteries based on the interpretable BRB with expert knowledge credibility (IBRB-c) is proposed.
The main contributions are summarized as follows: (1) The calculation method of expert knowledge credibility is proposed with full consideration of subjective and objective information.(2) Two interpretability strategies are designed to guarantee the interpretability of the optimization process.(3) A new model evaluation criterion is proposed to evaluate the comprehensive performance of the model.It is also introduced into the optimization objective function to achieve a balance between accuracy and interpretability.
The remainder of this paper is organized as follows.In Section 2, the problem of the SOH estimation method for Li-ion batteries based on BRB is formulated and the initial IBRB-c model is constructed.In Section 3, a new model evaluation criterion is proposed.In Section 4, inference and optimization of the model are introduced.In Section 5, the estimation modeling method is described.In Section 6, an experimental study is conducted.In Section 7, the paper is concluded.

| PROBLEM FORMULATION AND THE INITIAL CONSTRUCTION OF THE IBRB-C MODEL
In Section 2.1, the problem of the BRB-based SOH estimation method for Li-ion batteries is presented.In Section 2.2, the initial IBRB-c model is constructed.

| Problem formulation
The following issues need to be addressed in the SOH estimation method.Problem 1.How to consider more objective information to calculate expert knowledge credibility.
Experts contribute valuable knowledge to the estimation of complex systems with their deep experience.However, the process of knowledge generation is influenced by numerous factors that make it challenging to accurately predict the output of these complex systems, for example, familiarity with the system, expertise, and personal bias. 19Therefore, by taking into account the uncertainty of expert knowledge, the reliability and validity of estimation can ultimately be improved.
where E c denotes the expert knowledge credibility.x t ( ) denotes the observed data.F denotes the trust factor.Ω denotes the nonlinear function to calculate expert knowledge credibility.
Problem 2. How to design evaluation criteria that takes into account expert knowledge credibility to achieve a balance between accuracy and interpretability.
The evaluation criterion of a model plays an important role in the whole inference process as a benchmark for evaluating its model performance.Mean squared error (MSE) is a commonly used evaluation criterion for model performance and has been used in various methods to assess the accuracy of the method.However, MSE considering only accuracy can no longer meet real-world needs, especially in complex and critical domains, which require highly interpretable and reliable models. 4Therefore, it is necessary to develop an evaluation criterion for BRB that considers both accuracy and interpretability.
where E denotes a new evaluation criterion that takes into account accuracy and interpretability.
Problem 3. How to build an interpretable SOH estimation model based on BRB.
First, the observed data are extracted based on the degradation mechanism, and the initial estimation model is built based on the knowledge provided by the expert.
where x i denotes observed data of the ith attribute.N is the attribute number.E k denotes expert knowledge.Ξ denotes the model structure.
Second, the initial parameters are determined by experts to make a general judgment about the system, but due to the complex system mechanism, the model parameters ϖ need to be fine-tuned to find the optimal solution.
where ϖ optimal is the optimal set of parameters.Ψ denotes the optimization process.P is the parameter set of the optimization algorithm.C is the interpretability strategy, which is used to enhance the interpretability of the optimization process.
Finally, the SOH results of the Li-ion batteries are estimated.
where y SOH denotes the SOH estimation result.Φ denotes the inference process of the result.

| The initial construction of the IBRB-c model
The BRB is a rule-based model that incorporates expert knowledge to make decisions or predictions.It consists of a set of IF-THEN rules, each of which specifies conditions and corresponding conclusions. 20,21The belief level associated with each rule represents the degree of belief in the validity of that rule.The BRB provides interpretability by allowing humans to understand and interpret the reasoning behind the decisions or predictions made by the model.
The kth of these rules is described as follows: ), ( , ), …, ( , )}, with a rule weight and attribute weights , , …, , where are the set of reference values corresponding to the attributes.
are the belief levels represented by the M results.
, are the belief distribution of the result.θ k is the rule weight of the kth rule.
are the weight of the N attribute.

| Calculation of expert knowledge credibility
Expert knowledge credibility plays an important role in assessing the accuracy and reliability of the information provided by experts.It recognizes that experts, despite their expertise, are subject to various factors, such as distraction, fatigue, or cognitive limitations, leading to the possibility that their knowledge may deviate from the true underlying mechanisms of the system. 22To address this challenge, this paper presents an approach to calculating expert knowledge credibility that aims to illustrate the impact of uncertainty on expert knowledge.
The calculation method proposed takes into account the objective information and the subjective trust factor.The calculation method is as follows: where E c denotes the expert knowledge credibility.y i ( ) denotes the estimation value obtained from the knowledge provided by the experts through the BRB expert system.y i ( ) denotes the true SOH value of Li-ion batteries.F denotes the trust factor.n denotes the training data set.

| Evaluation criteria for BRB interpretability considering expert knowledge credibility
Both accuracy and interpretability are crucial considerations when evaluating an interpretable BRB model.MSE is used in the current study to assess the accuracy of predictions, but it is important to note that accuracy alone is not sufficient when constructing an interpretable model, and interpretability should also be an important performance criterion.
In the solution space, interpretability is expressed as the similarity between the parameters and the expert knowledge, because it retains valuable information from expert insights. 4Here, the credibility of the expert knowledge is taken into account.If expert knowledge credibility E c = 1, which represents the full credibility of the expert, then the similarity of the newly generated population to the real mechanism of the system, that is, to the knowledge provided by the expert.If E c = 0.5, it means that the expert is not completely credible, then a fuzzy interpretability interval is generated around the expert knowledge, and then the similarity of the new population to this interval is calculated.
Then, the method of determining the evaluation criteria is discussed.First, it is necessary to determine the fuzzy interpretability interval in the parameter space.The parameters in this interval are considered to be similar to the expert's judgment.
where v 0 is the parameter point given by expert.
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| 4725 It is important to note that the range of parameters must be guaranteed to be within the [0, 1].
Next, the Euclidean distance is calculated to measure the similarity of the parameters to the expert knowledge.
low up (10)   where υ p denotes the intermediate variable for calculat- ing the Euclidean distance.v p denotes a certain reference point. where

| THE INFERENCE AND OPTIMIZATION OF THE IBRB-C MODEL
The inference of the IBRB-c model is introduced in Section 4.1.The interpretability strategy of the IBRB-c model is introduced in Section 4.2.In Section 4.3, the optimization algorithm is improved.

| The inference of the IBRB-c model
Evidential reasoning (ER) is a decision framework that combines multiple sources of evidence to derive conclusions.Based on evidence theory, uncertainty and incomplete information can be handled. 23,24To obtain the final SOH estimation results for Li-ion batteries, belief rule fusion inference is performed using the ER algorithm.The steps of the algorithm are shown in Figure 1.
The specific inference process of the ER algorithm is shown below: (1) Conversion of input information.
The input evidence is converted into a belief distribution, which is a measure of the degree of endorsement for various outcomes.This transformation process is executed through the following calculation: where α is the matching degree.x i is the input data.X i l+1 and X i l represent the l + 1th reference value and the lth reference value in the ith attribute, respectively.T represents the reference value number.(2) Calculation of activation weights for rules.
After processing the observed data, the belief rules in the SOH estimation model will be activated accordingly.The activation weight is essentially a measure of the degree to which the inputs satisfy the belief rule.The activation weights are calculated as follows: where The inference procedure of IBRB-c.ER, evidential reasoning.
where w k represents the activation weight of the belief rule, and its value depends on the input characteristics of the system.δ′ i denotes the relative weight of the ith attribute.K is the rule number.(3) Aggregation of activated rules.
After calculating the activation weights of the rules, rule synthesis is implemented using the ER analysis algorithm, and the final belief degree is calculated: where β m is the belief level of the health state generated by the established SOH estimation model.(4) Calculation of expected utility value.
Calculated from the above equation, the final belief distribution is described below: where A* is the input vector.O A ( *) denotes the belief distribution of the final health state estimated by the model.
Finally, the expected utility value can be calculated.  where μ D ( ) m denotes the utility of D m and y denotes the final utility value.

| Interpretability strategies
(1) The initial optimization step size is controlled by expert knowledge credibility.
A new control operation is used along with expert knowledge.It makes a clear correlation between expert knowledge credibility and initial step size.When the expert knowledge credibility is low, the initial step size is relatively large, and the parameters are updated more quickly.On the contrary, the smaller the initial step size is. 25The formula for calculating the initial step size is as follows: where F ε denotes the step factor, the value of which was derived by synthesizing empirical data.In this paper, an appropriate value of 0.2 for this factor was determined based on the specifics of the experimental study.ε 0 denotes the initial optimization step size.
(2) Reasonable belief distribution strategy.Without considering the model interpretability has led to the development of many erroneous rules that are inconsistent with the distributional properties of the actual states. 15Such distributions are impractical and contradictory because a well-founded belief rule should not assign a high belief level to contradictory health states at the same time.Such a problem is easily neglected in the current application of BRB.
A reasonable belief distribution should be expressed as follows:

| The improved optimization algorithm
For the optimization of BRB, the essence is the process of fine-tuning the parameters. 15,26As one of the most effective algorithms, the projection covariance matrix adaptive evolution strategy (P-CMA-ES) is frequently applied in optimization studies of BRBs.In this paper, the improvement of the P-CMA-ES algorithm is achieved by setting new evaluation criteria and interpretability strategies.The objective function of IBRB-c is constructed as follows: where ψ θ δ β ( , , ) denotes the DMSEc value.
Step 1: Set the initial parameters.
The initial covariance matrix C°, population size λ, and offspring population size τ.The step size ε is used to adjust the search range determined by the expert knowledge credibility.
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where ψ i u+1 denotes the ith individual in the u + 1th generation.w u denotes the mean of the optimal overall in the uth generation.denotes the normal distribution.
This operation makes the data satisfy the equation constraint by projecting the projection parameters onto the feasible region hyperplane as follows: The hyperplane used to represent the feasible region of the equation constraint can be expressed as where a e denotes the number of parameters contained in the constraint.j and A e denote the equation constraint number and the vector of equation parameters, respectively.
Select τ optimal populations to update the mean value to make the optimization direction point to the optimal solution: where r i denotes the weight of the ith solution.ψ i λ u :

+1
denotes the ith of the λ populations.
The covariance matrix is updated: where c 1 and c 2 are learning rates.p c is the evolutionary path of covariance, and the evolutionary rule is as follows: where c c is the parameter of the optimized path.The step size is updated as follows: where c σ denotes the evolutionary path parameter.d σ denotes the damping coefficient.p σ u+1 denotes the conjugate evolutionary path.E I ‖ (0, )‖ denotes the expected length of p σ .

Repeat
Step 2 until the maximum number of evolutions is completed.

| SOH ESTIMATION MODELING METHOD FOR LI-ION BATTERIES BASED ON IBRB-C
The SOH estimation modeling method is introduced in this section, which is divided into two parts: (1) The training part of the model plays a key role in resolving the inherent uncertainty associated with expert knowledge.It aims to improve the performance of the model by using the observed data to determine the optimal parameters.Furthermore, by explicitly considering the constraints and limitations of the training process, the model can effectively improve its overall performance on specific Problem (2)  The testing part is used to test the performance of IBRB-c.
The modeling process of IBRB-c is shown in Figure 2, and the modeling steps are as follows: Step 1: Extract relevant attributes according to the research experiment.
Step 2: Introduce expert knowledge to build the initial model.
Step 3: Determine the training data and test data.
Step 4: Obtain expert knowledge credibility according to the calculation method proposed.
Step 6: Train the model using P-CMA-ES with interpretability strategies and other parameter constraints.
Step 7: Test the performance of the IBRB-c model using the ER method.

| EXPERIMENTAL RESEARCH
In this section, the Li-ion battery experiment at the NASA Ames Prognostics Center of Excellence 27 is used as an example to illustrate the validity of the proposed method.

| Data set description
The experimental data used in this study were obtained from the NASA Ames Prediction Center of Excellence battery data set. 27This data set was derived from in-depth experiments conducted on 18650 Li-ion batteries with a 2Ahr rating.
A subset of the battery data, No. B0006 was selected for this paper.The battery exhibited significant degradation characteristics under controlled room temperature conditions (24°C).The experimental procedure was as follows: (1) Charging process: The charging process started with a constant current of 1.5 A until the voltage reached 4.2 V. Continue charging at a constant voltage of 4.2 V until the current drops to 20 mA.
(2) Discharge: The battery was discharged at a constant current of 2 A until the voltage dropped to 2.5 V.
To construct an SOH estimation model based on battery capacity, the relevant capacity values were extracted from the discharge data in the raw data set.The battery data set can be used to study the complexities of Li-ion battery health and to accurately and reliably predict its SOH.

| The initial construction of the IBRB-c model
The time interval of equal discharging voltage difference (TIEDVD) (i.e., the time required for the battery voltage to drop by a fixed value during discharge) and the mean temperature (MT) during the TIEDVD are chosen as the characteristics of the battery health indicator, as shown in Figure 3. TIEDVD is a practical and effective method for characterizing battery behavior during the phases of the discharge cycle.However, the feature extraction process must be flexible to adapt to the specific characteristics of the battery under study.This adaptability is key to ensuring the validity and accuracy of the feature extraction process for different types and behaviors. 28Temperature within a battery affects various electrochemical processes that can affect the overall health of the battery.In addition, in practice, battery systems often experience temperature variations during operation.Understanding how temperature variations, even subtle ones, interact with other factors that affect capacity degradation is critical to practical battery health assessment and predictive maintenance. 29The battery capacity was chosen as the health indicator.The performance degradation process of the Li-ion battery is shown in Figure 4.
Three semantic values of short time (S), medium time (M), and long time (L) are set for TIEDVD, and three semantic values of low temperature (L), medium temperature (M), and high temperature (H) are set for MT.Their reference values are determined by experts, as shown in Tables 1 and 2, respectively.The health states of the Li-ion battery are defined as very healthy (VH), healthy (H), bad (B), and very bad (VB), and their reference values are shown in Table 3.The health value is indicated by the capacity of the battery, which is the basic parameter indicating the overall condition of the battery.There are 3 × 3 = 9 rules for the combination of the two attributes.Their initial reference values are shown in Table 4.The rule weights are determined by experts.First, experts have indepth knowledge of the system, so the relative importance of different rules can be determined based on their real- The modeling process of IBRB-c.BRB, belief rule base; P-CMA-ES, projection covariance matrix adaptive evolution strategy; SOH, State-of-health.
world impact.Second, patterns and trends observed by the experts from historical data inform the weighting process.Therefore, experts make full use of domain knowledge and empirical data to determine the rule weights.It is a fundamental component of the BRB model, designed to provide accurate, interpretable results for complex systems.

| The procedures of SOH estimation
A total of 169 sets of data were extracted and randomly divided into 112 sets of training data and 57 sets of testing data.In the P-CMA-ES algorithm, the number of iterations is set to 100 rounds.The calculated expert knowledge credibility is 0.8964.Therefore, the step size calculated is 0.2072.From the activation of the rules in Figure 5, it can be seen that all rules have been activated.Therefore, all the rules are involved in the optimization process.Finally, the optimized parameters are shown in Table 5.The belief distributions of the estimation results are shown in Figure 6, where the different levels of support for each result can be seen.The estimation results are shown in Figure 7.The calculated DMSEc value is 0.0437.

| Comparative study
To illustrate the effectiveness of the proposed method.This paper consists of two main parts of comparison experiments.
• One part is the comparison of three different BRBs, which are the BRB expert system driven by initial expert knowledge (represented by BRB0), classical BRB (represented by BRB1), and the proposed IBRB-c in this paper (represented by BRB2).• The other part is the comparison of different popular algorithms, which are the BP neural network and fuzzy inference.

| Comparison of the three BRBs (1) Accuracy analysis
As shown in Figure 8, the initial parameter settings make it difficult to accurately estimate the SOH due to the complexity inside the battery, and the optimized BRB1 and BRB2 effectively improve the accuracy of the model.Since No.

Rule weights
The initial belief distribution | 4731 BRB2 is optimized with both accuracy and interpretability as criteria, BRB2 is not as good as BRB1 in terms of estimation accuracy.However, as can be seen in Figure 8 and Table 6, BRB2 still maintains a high accuracy.
Figure 9 shows the nine rule weights with two attribute weights.BRB1 does not consider the initial judgments of the experts and randomly seeks the optimal parameters.Therefore, the optimized weights deviate severely from the original judgments of the experts.In contrast, BRB2, which considers the interpretable optimization criterion, maintains consistency with expert knowledge.
The belief distributions of the rules determined by the experts and optimized by BRB1 and BRB2 are shown in Figure 10.It can be seen that BRB2 maintains a similar belief distribution to BRB0.In addition, BRB1 without the interpretability strategies even has some incorrect rules, such as rules 3, 4, and 6.The belief distributions of these rules conflict with human common sense.The interpretability and reliability of the model are destroyed.
Stochasticity is an inherent characteristic of optimization algorithms, and it is crucial to control its direction effectively in the optimizing process of interpretable models.DMSEc as an objective function brings obvious benefits in terms of improved interpretability.
T A B L E 5 The optimized rules.

Rule weights
The belief distribution The optimized DMSEc value of 0.0041 by BRB1 and 0.0437 by BRB2 proved the validity of the proposed interpretable evaluation criteria.The curves in Figure 11 represent the variation of the Euclidean distance of the parameters from the expert knowledge during the optimization process for BRB1 and BRB2.It can be seen that the blue curve representing BRB1 shows an increasing trend, indicating a deviation from the expert knowledge and a decrease in interpretability.In contrast, the red curve representing BRB2 shows a decreasing trend, which implies that the optimization parameters pursue consistency with expert knowledge.

| Comparison of the three different algorithms
This section compares the classical BRB, IBRB-c, BP neural network, and fuzzy inference, and the corresponding MSE values are shown in Table 7.It can be concluded that the classical BRB obtains a higher accuracy with small sample data.Fuzzy inference, as a typical white-box model, obtains lower accuracy.And BP neural network as a black-box model achieved higher accuracy than IBRB-c.However, the IBRB-c proposed in this paper shows a strong advantage in estimating the SOH of Li-ion batteries.First, unlike BP neural networks that rely on exact values, IBRB-c can effectively handle incomplete or uncertain information by incorporating belief degrees and rule-based inference.Second, IBRB-c can provide high interpretability.The knowledge in IBRB-c is given by human experts in the form of rules, and the consistency with the knowledge is easily understood and verified.In contrast, the inner workings of BP neural networks are often challenging to interpret due to their complex network structure and numerical weights.Furthermore, while both IBRB-c and fuzzy inference are somewhat transparent, IBRB-c can provide a more explicit and transparent operation of the decision process.And IBRB-c can make informed decisions based on data-driven learning and expert knowledge.

| CONCLUSIONS
In this paper, a new SOH estimation method for Li-ion batteries based on the IBRB-c model is proposed, which ensures the accuracy and interpretability of the estimation results.Finally, comparative experiments show that the proposed method has advantages in SOH estimation for Li-ion batteries.
The contributions of this paper are as follows.First, expert knowledge credibility is calculated by combining subjective and objective information.Second, a fuzzy interpretability interval concept is proposed to design a new model evaluation criterion.The evaluation criteria balances the accuracy and interpretability of the BRB model, in which expert knowledge credibility is introduced.In addition, two interpretability strategies are designed to improve interpretability.
The proposed method makes a positive exploration of interpretable SOH estimation methods for Li-ion batteries.In the future, the proposed method can be tested on other types of batteries to verify the generalization capability of the model.In addition, how to improve the model accuracy by considering the effect of expert knowledge credibility is also an urgent issue to be addressed.

F I G U R E 3
The two attributes extracted for Li-ion battery No. B0006.F I G U R E 4 The capacity of Li-ion battery No. B0006.T A B L E 1 Referential values of TIEDVD.

T A B L E 2 3 4
Referential values of MT.The health states of the Li-ion battery.The initial rules.

FF I G U R E 9
I G U R E 7 Evaluation results of the proposed IBRB-c method.F G U R E 8 Comparison of health estimated by three BRBs.T A B L E 6 Comparison of the accuracy of the three BRBs.The comparison of weights.F I G U R E 10 Comparison of the belief distribution.F I G U R E 11 Comparison of Euclidean distance by MSE and DMSEc as evaluation criterion.
MSE E K denotes the MSE value calculated by the BRB expert system with the knowledge framework provided by the expert.
T A B L E 7 MSE comparison of different algorithms.