An interpretable BRB model with interval optimization strategy for lithium battery capacity prediction

Accurately predicting the capacity of lithium battery is conducive to improving its safety. Affected by complex internal electrochemical reaction and external use conditions, the prediction accuracy is difficult to guarantee; at the same time, the existing prediction methods are unexplainable, resulting in the inability to trace the prediction process. Therefore, a capacity prediction model based on belief rule base with interpretability and interval optimization strategy is proposed in this paper. First, the reasoning process is designed according to interpretability modeling criteria. Second, to achieve a balance of accuracy and interpretability, based on the whale optimization algorithm, a model parameter optimization method using interval optimization strategy is proposed. Finally, through a case study, the model's effectiveness is verified. Comparison with other models shows that the proposed model has certain advantages in accuracy and interpretability.


| INTRODUCTION
Lithium battery has been applied extensively in large equipment of aerospace, rail transit, and military communications due to its excellent performance of high power, long service time, and low maintenance cost. 1,2 The performance is directly related to the normal operation of the whole equipment. Capacity degradation has a great effect on battery performance, not only results in decreased battery life but also lead to an increase in costs. In cases where the capacity is less than 70%, the battery must be replaced immediately, or serious accidents will occur, 3 such as explosion, spontaneous combustion. Once the battery system in large equipment fails, the consequences are immeasurable. Therefore, from the perspective of safe operation and economy, the accurate prediction of battery capacity cannot be ignored, which is helpful to make predictive adjustments to the use of battery in advance, 4 so that the equipment can be in safe and stable operation and extend the life. 5 Battery capacity degradation is a complex and nonlinear process. Before a prediction is made, the reasons for its complexity should be analyzed. From the point of view of lithium batteries, since the different materials used in lithium batteries are different, so will capacity decay speed. From the perspective of practical application, temperature, environmental stability, charge-discharge rate, and discharge depth will make the capacity change mechanism more complex. It is necessary and difficult to make an accurate analysis of capacity degradation under many complicated factors.
To realize lithium battery capacity prediction, various methods have been developed, which can be categorized into three types according to the difference in the way prediction is implemented: physical model, data-driven model, and hybrid model. 6 The physical method is constructed by mathematical or physical models of the internal chemical processes and model parameters of the battery, combining changes in battery capacity with the internal parameters, and therefore, the existing observation data can be substituted into the model to obtain the capacity variation. Saha et al. built a Bayesian learning framework, incorporated features in the electrochemical impedance spectrum, and then applied the particle filter (PF) algorithm, 7 this method deals well with the uncertainties, sensor errors and noise. Afshari et al. achieved online estimation based on a smooth variable structure filtering method, 8 which has the very strong robustness. Chen et al. incorporated double exponential model, using the second-order central difference PF, 9 and the particle degradation phenomenon of PF can be solved. He et al. simulated the trajectory of battery capacity decay and incorporated PF, 10 and the availability of data increases the model's reliability. Li et al. put forward a low-order single-particle model, 11 which provided the information about SEI layer formation and crack propagation.
The data-driven method analyzes the monitored data, excavates the correlation characteristic, and then deduces the running rule. The prediction model of this method is established through nonlinear modeling or statistical methods. Huang et al. proposed a GA-Elman battery capacity prediction model with genetic algorithm optimization, 12 which has a good adaptability to different batteries. Liu et al. constructed a correlation vector machine prediction model, integrating multiple kernel functions, and the Drosophila algorithm was used in this model, 13 reducing the mean absolute error. Zhang et al. applied the ant colony algorithm-optimized BP neural network, 14 so the problem of sensitive initial value is solved, but the effect of temperature is not fully considered. Guo et al. adopted charging health feature extraction to construct model, 15 the accuracy of which needs further improvement. Zhao et al. used a deep belief network (DBN), and a support vector machine (SVM), 16 and this method has stable performance.
There are always drawbacks to using one of the two models alone, so the hybrid method is proposed. It is usually a combination of physical method and data-driven method or multi-data-driven methods; therefore, their advantages are maximized, and the accuracy of the prediction is improved. 17 Dong et al. put forward support vector regression and the PF method. 18 Yu proposed a multiscale method combining logistic regression and Gaussian process regression. 19 Liu et al. proposed an interacting multiple model and unscented PF for the state of health estimation. 20 Tian et al. realized an online prediction using the artificial fish swarm algorithm, and PF. 21 Hu et al. proposed a weighted integration of data-driven methods such as SVM, relevance vector machine, recurrent neural networks, and Bayesian linear regression models. 22 Currently, capacity prediction has become an important research direction of lithium battery because it can ensure the normal operation of the equipment. 23 It is worthwhile to answer whether modeling strategies among the above three methods is optimal to predict capacity. Therefore, three methods are compared: the physical method has strong adaptability and satisfactory prediction accuracy. But it needs a large number of parameters, and with different parameters having different trends in the process of capacity decay, the importance of parameters is not the same, so the process of parameter estimation is very difficult, leading to the computational complexity and cost of physical models being very high. [24][25][26][27] It can be seen that the method is not suitable for many practical applications. The data-driven method does not necessarily focus on the physical mechanism, 28 which is easier to implement than the physical model, and the concept of parameter error does not exist, but the lack of physical constraints results in a reduction in the effectiveness of prediction. 29 It needs to acquire a lot of data for training, 30 and the accuracy is highly dependent on data completeness and certainty. As a black box model, its accuracy is closely related to the data sample, and the interpretability of the modeling process is limited. To sum up, this method has limitations. However, the hybrid method takes full advantage of both methods. Not only does it has higher accuracy, but it also avoids the construction of complex model according to degradation mechanism. The method can be a plausible solution to maintain accuracy and interpretability. Therefore, there is a growing consensus that hybrid method is a promising approach to predicting the capacity.
In the capacity prediction model for lithium battery, two issues need to be considered. First, due to the influence of many complex factors, there is great uncertainty in the capacity prediction. Therefore, the constructed capacity prediction model must be able to deal with uncertainty, and the model accuracy has become harder to guarantee. Second, the capacity prediction model for lithium batteries will be used in many large-scale equipment, so the model needs to be reliable and easy for users to understand and use. [31][32][33] The above problems can be well addressed by belief rule base (BRB), and it has received widespread attention due to its nonlinear modeling ability and strong advantages in processing the uncertainty. BRB is a rule-based modeling method that is easily understood by humans. At the same time, BRB has good robustness because of its ability to integrate expert knowledge. It is widely used in decisionmaking, prediction, reliability evaluation, fault diagnosis and other fields. Therefore, the best prediction result can be obtained by BRB, and a BRB model for predicting capacity is proposed in this paper. This model belongs to the ante-hoc interpretability model, which has the built-in interpretability, making it possible to understand the working mechanism of the model without additional approaches. 34 The opposite of ante-hoc interpretability is post-hoc interpretability, which is interpreting the original weak interpretability model by designing a high-fidelity interpretation method or constructing a high-precision interpretation model, showing how it works understandably.
However, the lithium battery capacity prediction model based on BRB faces three problems. First, Cao et al. proposed a set of eight criteria for the interpretation of BRB. 35 Based on these criteria, establishing interpretability criteria for predicting lithium battery capacity is the first problem to be considered. Second, how to build the capacity prediction model is the second problem. Finally, the lithium battery capacity prediction system has two important indicators: interpretability and accuracy. Expert knowledge is an important embodiment of interpretability, but the randomness of the optimization algorithm leads to the loss of the optimized expert knowledge, which decreases the interpretability of the model. Therefore, how to balance the interpretability and accuracy of the model is the third problem. 36 The interpretability of the lithium battery capacity prediction model makes the model easy to understand, integrates capacity decay mechanism and experience knowledge, and keeps the rationality and transparency of the prediction process. To solve these problems, this paper proposes an interpretable BRB capacity prediction model. The model uses interval optimization strategy to balance accuracy and interpretability.
The main contributions of the paper include the following: (1) A new BRB-I model is proposed to predict the capacity of lithium battery in an interpretable manner. (2) An interval optimization strategy is designed to ensure the accuracy and interpretability.
The remainder of this paper is organized as follows. In Section 2, all notations are summarized and problems of lithium battery capacity prediction are described. In Section 3, the construction, reasoning and optimization process based on the interpretability criteria are defined. In Section 4, the model's validity is tested through a case. In Section 5, the content of this paper is summarized.

| PROBLEM DESCRIPTION
In this section, all notations are described in Section 2.1, problems of the prediction model in Section 2.2.

| Notations
All notations are defined in Table 1.

| Problems with the prediction model
Four problems must be taken into account in constructing a BRB-I model for predicting lithium battery capacity: Problem 1. Interpretability increases the effect of capacity prediction for lithium batteries. It is therefore necessary to establish interpretable criteria for predicting lithium battery capacity. Meanwhile, the definition of BRB interpretation is equally important. The description is made as follows: where c represents the interpretable definition criterion and n denotes the number of criteria.
Problem 2. How to build the prediction model of lithium battery capacity. When model parameters and reasoning are established, the rationality of calculation and the causal relation between input and output must be fully considered. The steps are as follows: where x denotes the indicator set of the capacity prediction model for lithium batteries, a denotes the parameter set of the inference process, y represents attributes of lithium battery capacity prediction, and g ( )  represents the inference function.
Problem 3. How to achieve the balance of accuracy and interpretability in optimization. When the parameters of BRB are optimized, the conflict between the newly generated belief degree and interpretability often affects interpretability. The interpretability and accuracy cannot always be improved simultaneously. Therefore, reasonable optimization design under the limit of interpretability is indispensable. In this paper, an interval optimization strategy is proposed, and a reasonable optimization design is carried out under the limit of interpretability. This process can be described by the following nonlinear mapping relationships: where D represents the parameter set when optimizing, D* is the optimal parameter set, and optimize( )  is the optimization function.

| THE BRB-I MODEL FOR LITHIUM BATTERY CAPACITY PREDICTION
The BRB-I capacity prediction model of lithium batteries is modeled in the following steps: (1) Establish a BRB capacity prediction model; (2) A parameter optimization model is established for higher accuracy, and the parameters are optimized and updated based on the interval optimization strategy; (3) Use the trained model to achieve capacity prediction for lithium batteries.
Remark. Research on how to improve the performance and safety in use of lithium batteries is ongoing. 37,38 To The set of reference values corresponding to the attributes The belief level corresponding to the N th consequence under the kth belief rule The position vector of the best solution by far r r , 1 2 Random numbers within 0-1 t A number changing within −1 to 1 The distance between the humpback whale and its prey u The number used to describe the shape of a spiral A rand The vector of randomly selected humpback whale positions guarantee the interpretability and accuracy in a good level, factors such as completeness of the rule base have an important impact on the accuracy. 39 In condition of considering interpretability constraints, an interpretable BRB model is built with the modified whale optimization algorithm (WOA).
The prediction model is defined in Section 3.1; the interpretability of the model in Section 3.2; the reasoning process of the model in Section 3.3; the optimization process in Section 3.4.

| Definition of the prediction model
The structure of the BRB-I model for lithium battery capacity prediction is shown in Figure 1. The model relies on belief rules, including the relationship between the inputs and outputs of the model, and it has many simple if-then rules, 40 of which the kth BRB rule is as follows 41 : With a rule weight and attribute weight , , …, where R k represents the kth belief rule of the BRB model, represents the attribute weight of each prerequisite attribute.

| Interpretability definition of the prediction model
To get a credible and easy-to-understand model, one of the research goals of the study is to guarantee and promote the interpretability. In the current study of BRB, Cao et al. have established a series of BRB interpretability criteria to ensure the interpretability of BRB, 35 mainly discussing interpretability from the following three aspects: knowledge base, inference engine, and model optimization. 34

| The reasoning process of the BRB-I model
The interpretable BRB has three parts: knowledge base, inference engine, and optimization model. 34 To get the final output of the system, rule inference uses the evidential reasoning (ER) analysis algorithm to perform combined inference on belief rules. 41 First, the activation weight needs calculating, and then ER is used. The The BRB-I model for predicting capacity. algorithm integrates the activation rules, and the specific steps are divided into the following four steps 31 : (1) It is necessary to calculate the degree of matching between the input sample information and the belief rule, which is the degree of rule adaptation. The rule fitness of the kth rule under the ith input is calculated as follows 31 : (2) The activation weight can be calculated as follows 31 : where δ i M ( = 1, …, ) i represents the attribute weight of the ith index.
(3) The final belief degree is obtained by using the rule inference of the ER analysis algorithm, which can be calculated as follows 31 : (4) Calculate the expected utility value and obtain the final output. 31 where S ( )  represents the set consisting of belief distributions, A′ represents the actual input vector, μ H ( ) n represents the utility of H n , and μ S A ( ( ′)) is the final expected utility. The final belief distribution y is described as follows:

| The optimization process of the BRB-I model
When constructing the lithium battery capacity prediction model, an increase in accuracy tends to decrease interpretability. The accuracy and interpretability can be improved by optimizing the parameters with the optimization algorithm. Therefore, this paper proposes an interval optimization strategy to achieve optimization and adopts the modified WOA with interpretability constraints, as shown in Figure 3. The WOA is a new metaheuristic optimization method inspired by nature, 42 and it can construct a bubble net search strategy by simulating the hunting behavior of humpback whales. 27 As shown in Figure 4, the interval optimization strategy proposed in this paper can fully cover the entire feasible region, and diffusion starts from expert knowledge. The step length L of each diffusion is 0.01. The number of strategies is set to I. The belief degree, rule weight and attribute weight of the BRB model are all in the range of 0-1. In Figure 4, x 1 and x 2 represent F I G U R E 2 The interpretability criteria for the interpretability prediction model.
The optimization process of BRB-I model.
two of the parameters. In the process of I increasing, the area of image cross-section is also increasing, which means the feasible region space of interval optimization strategy is increasing. When I reaches 100, the feasible region space of the interval optimization strategy at this time is equal to the entire feasible region of the original algorithm.
On the basis of the original WOA, adding interpretability constraints can have an impact on keeping the interpretability when optimizing. The modified WOA steps are as follows: Step 1: Initialization parameters. Let the population of whales as O, the number of iterations as q, and the dimension size of the search space as S s .
Step 2: Sprinkle operation. Each whale was randomly sprinkled, 42 and the ith whale can be described as: where f is a random number whose value range is [0, 1]. The range of X i is lb ub [ , ]. Ek is expert knowledge. lb is the parameter boundary minimum value, and it is closely related to the value of I. ub is the parameter boundary maximum value.
Step 3: Calculate adaptive value. Mean square error (MSE) is the objective function, and a is the parameter set of the inference process.
Step 4: Constraint operation. Selecting only WOA to optimize the parameters does not guarantee the interpretability of the model. Considering that WOA has no constraints, the following constraint is designed to prevent the interpretability from being destroyed in optimization.
The interpretability of BRB model reflects in the belief rule. The optimized rules should be realistic; otherwise they violate the eighth criterion of interpretability in Section 3.2.
where W k is the interpretability constraint of the belief distribution under the kth rule.
Step 5: Move location operation. 43 Step 5.1: Surrounding prey. Humpback whales circle their prey to catch, which can be described as 44 : where the coefficient vectors are represented by H and J ; the position vector of the current best solution is represented by a * q , which changes with iteration; and the position vector is represented by a q . H and J are obtained in the following formulas 44 : where a decreases linearly from 2 to 0, and r 1 and r 2 are random numbers in the range of 0-1.
Step 5.2: Spiral bubble net foraging. 45 Humpback whales swim toward their prey in a spiral motion. This behavior can be described as follows: a a Ze πt = * + cos ( 2 ) , F I G U R E 4 Interval optimization strategy.
where the record of the best solution obtained is a* g ; Z denotes the distance between the humpback whale and its prey; t is a number changing within −1 to 1; and the u is a number used to describe the shape of a spiral.
Step 5.3: Hunt for prey. In addition to surrounding prey and feeding prey with spiral bubble nets, humpback whales also randomly search for prey, and the process is as follows: where the vector of randomly selected humpback whale positions is denoted as a rand .

| CASE STUDY
In Section 4.1, the dataset of the case is introduced. In Section 4.2, the construction of the initial BRB model is described. In Section 4.3, the optimization process of the model is shown. In Section 4.4, the model interpretability is verified. In Section 4.5, the expert knowledge reliability is verified. In Section 4.6, the comparative experiments with different methods and the B0018 lithium battery are carried out to demonstrate the advantages of the proposed model.

| Dataset introduction
Using the aging dataset of 18650 LiCoO 2 battery from the National Aeronautics and Space Administration Prognostics Center of Excellence, the rated capacity of this battery is 2 A h. In the charge-discharge cycle test experiment, the first step is to charge, charging at a constant current (CC) of 1.5 A until the voltage reaches 4.2 V, then charging at a constant voltage (CV) of 4.2 V until the current drops to 20 mA, at this time into discharge mode, the CC is 2 A. When the battery capacity reaches the lowest level that can be safely used, that is, when it is 70% of the rated capacity, the cycle experiment is terminated.

| Construction of the initial BRB model
As shown in Figures 5 and 6, the periodic analysis of the operating state of the lithium battery by experts can give the general belief distribution trend. For example, when the attribute state of the PC time-CC stage is VL and the attribute state of the PC time-CV stage is also VL, the state of the attribute is all the best state, so the belief distribution of the lithium battery capacity prediction model can be set to {1, 0, 0, 0}. In addition, when the PC time-CC stage attribute is in the S state, the PC time-CV stage attribute is also in the L state, the model belief distribution can be set to {0.00, 0.00, 0.20, 0.80}. The belief distribution of the initial model is described in Table A1; rule weight and attribute weight are 1, and the initial model based on expert knowledge is constructed. Based on the mechanism analysis of lithium battery capacity decline and the long-term practice accumulation, the typical reference point is obtained. The model defines two attributes for the capacity of lithium batteries and assigns four semantics to each attribute: very long (very long, VL), longer (long, L), normal (normal, N), and short (short, S). The reference value is shown in Table 2. When lithium batteries are applied to engineering, four reference points are selected to indicate the health status of them, as shown in Table 3: completely safe (CS), safe (S), and little bad (little bad, LB), very bad (very bad, VB). The reference value is selected according to the judgment of expert knowledge. 46 Among them, by observing the charging habits of users, according to the relationship between voltage and electricity, select 3.8 V voltage as the starting voltage, 4.2 V cut-off voltage as the termination voltage. Thus, the required charging time from voltage 3.8 V to 4.2 V serves as a health factor to characterize battery health and is recorded as the part charging time of the CC Stage (represented by PC time-CC). In the practical application of batteries, there is a situation of incomplete charging, failing to reach the cut-off current value. Select part of CV stage, that is, select 1.5 A current value as the starting current, 0.5 A current as the ending current. Thus, the charging time required during the CV stage when the current value drops from 1.5 to 0.5 A is another health factor that characterizes battery health, recorded as the part charging time of the CV stage (denoted by PC time-CV). 47

| Model optimization
The effectiveness of the interval optimization strategy is demonstrated, as shown in Figure 7. It is easily seen that Part A and Part B have the same accuracy variation. In Part A, the BRB-I model can find a solution vector with the same accuracy as the global optimal solution. However, the results after optimization for different intervals differ in the Euclidean distance from expert knowledge. In Figure 7, the Euclidean distances of the C part and the D part are quite different. The Euclidean distance represents how similar the optimized solution is to expert knowledge. Therefore, the solution of Part C has a high similarity with the expert knowledge and retains more information of the expert knowledge, which is more interpretable. Cao et al. proposed that the optimization of the model is a local optimization space based on expert knowledge. 35 Thus, the model adopts an interval optimization strategy, which optimizes from near to far from expert knowledge, realizing the optimization of different intervals. Moreover, the BRB-I model allows for both accuracy and interpretability.

| Verification of model interpretability
The higher the similarity between the belief degree of the lithium battery capacity prediction model after real-time data correction and the expert knowledge is, the richer the features of the retained expert knowledge will be and the stronger the model interpretability. Figure 8 demonstrates that the belief distribution of the BRB-I model is very close to expert knowledge, which means that the rules of the two are consistent. Based on the same accuracy, the interpretability of the BRB-I model is stronger. In contrast, as a BRB model without interpretability constraints, the belief distribution of the WOA-BRB is quite different from the initial judgment of experts, which shows that expert knowledge is rarely applied in the WOA-BRB rules, and even in the belief distribution of the WOA-BRB rules, wrong rules are leading to a conflict of belief degree, as shown in Rule 3. When the system is in the state of Rule 3, expert knowledge gives a high belief degree to the CS state; however, the belief degree of the WOA-BRB output is extremely low, which is incompatible with the actual system.
Different optimization intervals affect different belief distributions. When the MSE is the same, the closer the distance is, and the higher the similarity is, the more effective information of expert knowledge can be retained, so the interpretability is stronger. 48 For example, in Rule 1, 2, 3, 9, 10, 11 of Figure 9, the curve of I = 20 is closer to expert knowledge, and the similarity is stronger than that of I = 98, so the interpretability is higher; that is, the BRB-I model (I = 20) can achieve a good balance between accuracy and interpretability, and meet practical engineering needs better.

| Verification of expert knowledge reliability
Because of the complex factors affecting the capacity degradation of lithium battery, the expert system can only get the trend of the capacity degradation, and the accuracy is limited. Figure 10 shows that the prediction trend of the expert system is consistent with the actual value, which shows that the prediction result of the model based on expert knowledge is consistent with the law of capacity degradation; that is to say, the expert knowledge is reliable, and can provide the basis for the prediction. In addition, BRB-I model is an optimization process based on expert knowledge, and reliable expert knowledge lays the foundation for the model's effectiveness. The prediction result of the model is close to the actual value, which proves that the model has high accuracy and the validity of the model is proved.

| Other comparative experiments
a) Comparative experiment of B0006 battery.
As shown in Table 4, this paper compares and analyzes the new BRB-I model with BRB-WOA model, backpropagation neural network (BPNN) model, radial basis function (RBF), long short-term memory (LSTM), DBN, decision tree, and SVM model, which demonstrates the performance of the proposed modified WOA. In Table 4, Part 1 is a comparison of BRB parts, and Part 2 is a comparison of other data-driven models. The BRB-I model is more accurate than other models, which proves that the modified WOA method is more effective in parameter optimization of the BRB model proposed in this paper. It has little difference in accuracy with BRB-WOA model, BPNN model, and RBF model, but the BRB model in this paper has the following advantages over other models: (1) The BRB-I model has interpretability, and its belief distribution is close to reality. The output can be reasonably interpreted. However, the optimized belief degree distribution of the BRB-WOA model is contrary to the real lithium battery system. (2) The BRB-I model is a rule-based modeling method which can depict the battery operation process in a way that human can understand easily. In addition, it also incorporates expert knowledge and system mechanism, while models of BPNN and RBF do not have such capabilities. (3) The reasoning engine of the BRB-I model uses the ER analysis algorithm, the reasoning process is clear and transparent, and the results can be traced back. However, models of BPNN and DBN are black-box models, and their internal structures are invisible. All things considered, the BRB-I model has better practical value.

b) Comparative experiment of B0018 battery
The B0018 lithium battery was verified for the applicability of the BRB-I model. 49 Table 5 shows the results of the comparative tests. Obviously, the BRB-I model has not only strong interpretability, but also has good accuracy. As shown in Figure 11, because of the internal chemical reaction of lithium batteries, expert F I G U R E 10 Experimental result data comparison. knowledge is limited, and predicting the capacity of B0018 lithium batteries has become more difficult. The BRB-I model is an optimization process based on expert knowledge, which can better retain expert knowledge information and allow for both interpretability and accuracy.

| SUMMARY
In this paper, a new BRB-I model is proposed and optimized using a modified WOA. First, the reasoning process is effectively designed through the interpretability criteria. Second, this paper uses an interval optimization strategy for optimization. There are two innovations in this paper: (1) The BRB-I model for lithium battery capacity prediction is constructed through the interpretability criteria, and the reasoning process is effectively designed. Moreover, an initial model is established by converting the expert knowledge into the parameters by means of belief degree. (2) Design an interval optimization strategy. By changing the optimal feasible region space, interpretability is guaranteed while maintaining accuracy. The proposed model is verified with good effectiveness through a case study of a lithium battery, so the balance between accuracy and interpretability has been achieved. In future research, methods of ensuring the interpretability of the BRB model when optimized deserve more discussion. BRB interpretability research is significant and can provide a better mechanism for users to understand logical reasoning. As more and more researchers are aware of the importance of interpretability, in the future research, how to ensure the interpretability of BRB model and evaluate it are worthy of further study. In addition, in the research of lithium battery capacity prediction, how to extract more lithium battery performance and introduce more types of lithium battery data sets is also worthy of further study.