Lithium-Ion Battery State-of-Health Estimation Method Using Isobaric Energy Analysis and PSO-LSTM

Te precise estimation of the state of health (SOH) for lithium-ion batteries (LIBs) is one of the core problems for battery management systems. To address the problem that it is difcult to accurately evaluate SOH because of the LIB capacity re-generation phenomenon, this paper proposes an approach for LIB SOH estimation using isobaric energy analysis and improved long short-term memory neural network (LSTM NN). Specifcally, at frst, the isobaric energy curve is plotted by analyzing the battery energy variation during the constant current charging stage. Ten, the mean peak value of the isobaric energy curve is extracted as a health factor to characterize the battery SOH aging. Eventually, the LIB SOH estimation model is developed using the improved LSTM NN. In this regard, the improved LSTM NN refers to the selection of the number of hidden layers and the learning rate of the LSTM NN using the particle swarm algorithm (PSO). To verify the precision of the proposed method, validation experiments are performed based on four battery aging data with diferent charging multipliers. Te experimental results indicate that the proposed method can efectively estimate the LIB SOH. Meanwhile, the proposed method is compared with other conventional machine learning algorithms, which demonstrates that the proposed method has better estimation performance.


Introduction
Lithium-ion batteries (LIBs) have been implemented in a variety of application scenarios, from smart grids to aerospace and electric vehicles.Tis is attributed to its merits such as high energy density, long cycle life, and low selfdischarge rate [1][2][3].However, the LIB state of health (SOH) continues to decrease during the continued service of LIBs.As a result, the LIB SOH can be used as an important indicator to diagnose the degree of battery aging.Nevertheless, due to the complexity and diversity of the aging parameters of LIBs [4], it is challenging to accurately predict the LIB SOH.In recent years, many scholars have devoted themselves to SOH estimation techniques for LIBs.Tese techniques can be broadly classifed into model-based and datadriven techniques [5].
Te model-based methodology implies constructing a simulation model of LIB as a way to simulate the chemical reactions occurring inside the battery during its operation.Tree commonly accepted equivalent models for LIBs are the electrochemical mechanism model, the equivalent circuit model, and the empirical degradation model [6][7][8].For example, Xiong et al. [9] proposed a simplifed pseudo-two-dimensional model using a fnite analysis method.Mevawalla et al. [10] developed a simplifed electrochemical thermal model involving a large number of partial diferential equation calculations.Yang et al. [11] proposed an equivalent circuit model and improved the accuracy of the model by using parallel connections.Zheng et al. [12] proposed a feedforward empirical model and combined it with a feedback neural network to predict the battery capacity.However, owing to the complex computational process and parameters of the model-based approach, it is often not easy to implement in real-time monitoring of the battery management system.
Instead of analyzing the internal electrochemical reactions of LIBs, the data-driven approach investigates the historical data from experimental measurements [13] and then builds a LIB SOH estimation model from the extracted current and voltage characteristics.Commonly employed data-driven methods include support vector machine (SVM) [14], extreme learning machine (ELM) [15], relevance vector regression (RVR) [16], and long short-term memory neural network (LSTM NN) [17].Among them, the input weights of the ELM algorithm are random and cannot be fne-tuned for changes in data features, which is less controllable and the output results of the model are unstable [18].Te SVM method is difcult to divide large-scale data samples, and the computational cost will be greatly increased, which requires the selection of regularization parameters, kernel function, and kernel function parameters [15].Although the accuracy of the RVR approach is similar to the SVM in regression and classifcation tasks, the training time is shorter.In addition, the RVR approach can be more generalized and more suitable for online prediction scenarios [19].LSTM NN can well learn historical and future information so as to overcome the difculty of accurate model of LIB SOH caused by the capacity regeneration problem in practical applications of LIB.In addition, the gradient disappearance of the traditional recurrent neural network is addressed in long-term sequence prediction, which efectively improves the generalization and robustness of the network model [20].
Although the data-driven approach can deal with the nonlinearity problem well, the model estimation accuracy depends on the analysis and extraction of features.By analyzing the relationship between battery aging characteristics and charging time, Lin et al. [21] estimated SOH accurately.Te voltage diference within a fxed time diference was extracted by Liu and Chen [22] during constant current and constant voltage charging stages as a health characteristic for SOH estimation.Goh et al. [23] split the discharge process into various phases based on the curvature of the discharge curve and extracted many HIs with a high correlation to battery SOH in the discharge platform stage of the discharge curve.Shu et al. [24] proposed to simultaneously achieve the estimation of state of health and optimize the healthy features therein, which are excavated based on the charging voltage curves within a fxed range.
Te above methods extract features from charging curves, and the extracted features cannot indicate the internal violent reaction process of the battery [25,26].In recent years, incremental capacity analysis (ICA) has been widely used to analyze battery aging characteristics and extract features.Guo et al. [27] used the maximum value of the IC curve, the corresponding voltage, and the energy and the capacity of a constant current charging interval determined by the maximum value of the IC curve to estimate the battery SOH.Based on calendar aging results collected during 11 months of testing, Stroe and Schaltz [28] were able to relate the capacity fade of the studied batteries to the evolution of four metric points, which were obtained using the ICA.Zhang et al. [29] proposed the ICA method and improved the broad learning system network-based SOH estimation technology for lithium-ion batteries.Te experimental results demonstrate that the proposed method can efectively evaluate the SOH with strong robustness.Li et al. [30] used Gaussian fltering to extract the feature of the IC curve after smoothing the static charging curve and then derived the mapping relationship, but there is a large measurement error noise interference resulting in poor SOH estimation accuracy.
According to the research on the above methods, this paper proposes a LIB SOH estimation method based on isobaric energy analysis and the modifed LSTM NN.First, the isobaric energy curve is derived by analyzing the battery energy variation during the constant current charging phase.Ten, the average peak is extracted as the health factor to characterize the aging of LIB SOH.Particle swarm optimization (PSO) is utilized to adjust the learning rate and the number of hidden layers of the LSTM NN in order to build an accurate SOH estimation model.Te SOH estimation capability of the proposed method is validated based on diferent aging data.Experimental results indicate that the method has good estimation capability and stability for battery SOH with four diferent charging and discharging rates.In addition, the proposed method is compared with the widely used machine learning algorithms to increase the convincing power of the excellent estimation performance for the proposed method.Specifcally, several key contributions are presented below.
(1) Te isobaric energy analysis method is developed, which can extract efcient health factors.(2) Te optimized LSTM network is constructed through the PSO algorithm fltrates the learning rate and the number of hidden layers of LSTM NN. (3) Te proposed method evidently can be easily generalized to the SOH estimation of other types of batteries without understanding the battery mechanism.
Tis paper is organized as follows.Te isobaric energy analysis and the PSO-LSTM approaches, respectively, are presented in Sections 2 and 3. Te background of the experimental setup is provided in Section 4. Lastly, Section 5 summarizes the paper.

Isobaric Energy Analysis
Isobaric energy analysis refers to the process of extracting a more efective characterization of battery aging by observing the energy change brought about by charging an equal amount of voltage during constant current charging.To observe the energy change more visually, it is necessary to plot the isobaric energy curve according to where E and V are the charging energy and voltage, respectively, and f(•) means the function relationship between the energy and voltage.

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As an example, take a domestic brand 18650 ternary LIB and draw the isobaric energy curve as shown in Figure 1.It is clear that throughout the constant current charging process, the battery has a signifcant number of wave peaks.As the wave is being charged and discharged, the wave's position steadily falls.Tis demonstrates a signifcant relationship between the battery SOH and the height of wave peaks.As a result, the isobaric energy curve's wave peaks can be used to characterize the battery's aging process.However, as shown in Figure 1, there may be multiple wave peaks in the isobaric energy curve.All of these wave peaks are correlated with the battery SOH.Accordingly, in order to characterize battery aging more accurately, this paper proposes to extract the mean value of the wave peaks as a new health factor to characterize battery aging.

SOH Estimation Methodology
3.1.LSTM NN.LSTM NN is designed to overcome the phenomenon of gradient disappearance or gradient explosion in recurrent neural networks for long-time series prediction problems [31].By combining the nonlinear and data-dependent control units into the recurrent neural network units, LSTM NN keeps the objective function associated with the state signal from vanishing in gradient.Figure 2 depicts the basic structure of the LSTM NN.
Te LSTM NN mainly consists of input gate, output gate, and forget gate.While the input gate removes the key information to be retained in the internal state, the forget gate discards the redundant information, and it is the output gate that decides the information to be output.Accordingly, the LSTM NN preserves and updates the key information effciently over a longer period of time.It is represented below for each gate function and state transfer process in the LSTM module [32].
where f, i, h, and o are the forget gate, input gate, hidden layer, and output gate, respectively; w represents the input weight; u indicates the recurrent weight; b refers to the bias; σ is the sigmoid function which activates the three gates; and tanh expresses the hyperbolic tangent function. ( Te learning rate of LSTM NN has a great impact on the model, which determines the learning process of the model and afects whether the objective function converges to the optimal solution.A large learning rate can easily lead to oscillations in the output of the algorithm, while a low learning rate results in overftting and slow conversion.Terefore, an appropriate learning rate is very important [33].Te hidden layer count directly afects the model accuracy.PSO, a parametric optimization technique that is frequently used, is efective in locating the best solution to typical situations.Consequently, LSTM NN's learning rate and the number of hidden layers are selected using the PSO algorithm.

PSO Algorithm.
Te PSO algorithm is a general optimization technique built on the foundation of group search.In many science and engineering felds, it is used to solve nonlinear, nonconvex, or combinatorial optimization issues [34].It can reach the global optimal solution by the defned ftness function F, accomplished by updating the  generations for the velocity and position of particles.Te basic fows of the PSO algorithm in D-dimensional space are as follows [35].
Step 1. Initialize the PSO algorithm including the size of the swarm N and randomly produce the velocity v i and position x i of each particle i; v i and x i of particle i represent the D-dimensional vector and are defned as Step 2. Calculate the ftness value F[i] of particle i by the ftness function F.
Step 3. Generate the optimal solution of each particle and global solution by where p best (i) is the optimal position of the particle i and g best represents the optimal position of the particle.
Step 4. Update the velocity and position of particles by where r and s are random numbers; k � 1, 2, . . ., H represents the index of the iteration; c 1 and c 2 are constants; and w is referred to as the inertia constant.
Step 5. Repeat steps (2)-( 4) until the terminated criterion is met and then exit the program.

PSO-LSTM Model.
Te ftness function is the mean absolute error (MAE), which represents the diference between the estimated and true values of the LIB's SOH.Te steps of PSO optimization of LSTM NN learning rate and hidden layers number are described as follows: (1) Initialize PSO algorithm parameters, such as population size, particle index, number of iterations, particle velocity, position, and end conditions.Meanwhile, experience determines the LSTM NN's learning rate and the hidden layer count.Te PSO algorithm stops when the error between the true and estimated SOH is less than 1% more than 10 times in a row.(2) Calculate the ftness value for each particle based on a custom ftness function.Te ftness function takes the average absolute error between the estimated and true LIB SOH values.(3) Generate the local optimal solution and the global optimal solution for each particle, where the local optimal solution is obtained by comparing the actual ftness value of each particle with the historical ftness value.At the same time, the global optimal solution is obtained by comparing the current ftness values of the particles with all the historical ftness values.( 4) Apply ( 6) and (7), respectively, to update the velocity and location.(5) Keep going through steps ( 2) through (4) to attain the algorithm termination condition.( 6) Output the learning rate and hidden layer number of LSTM NN.

Experiment Setting and Results
4.1.Experiment Data.Te cycle aging experiment was conducted in a laboratory environment using four cylindrical 18650 LIBs with the same specifcations.Te specifc description is as follows: ( Each battery's observed aging data are shown in Figure 3, and it is obvious that as a result of repeated charging and discharging cycles and local regeneration events, the battery's discharging capacity gradually declines.Tis makes it challenging to estimate the LIB SOH.

Experiment Procedures. SOH estimation experiments
were executed with four battery aging curves of diferent charging multipliers.Figure 4 represents the SOH estimation steps, which are explained as follows: (1) Plot the isobaric energy curve after analyzing the LIB energy variation in charging stage.( 2 where P 1 P 2

Isobaric Energy Analysis Results.
Based on the constant current charging stage data, the isobaric energy curve is calculated as shown in Figure 5.It can be observed that two obvious peaks appear in the isobaric energy curve for 0.5 C, but four more obvious peaks appear in the isobaric energy curves for 0.3 C, 0.2 C, and 0.1 C, which are due to the increase in charging multiplier, making the peaks less likely to appear.With the aging test, the peak of the isobaric energy curve for the four diferent multiples shifts downward, which indicates a strong correlation between the peak value and the LIB SOH.Terefore, this paper adopts the average peak as the health factor to estimate LIB SOH.

Results of PSO-Optimized LSTM Hyperparameters.
Te process of PSO algorithm optimizing the learning rate and the hidden layer count of LSTM NN for four diferent charging rates of aging data is shown in Figure 6.Te optimized LSTM NN learning rates and hidden layer count for the diferent charging multipliers curves are shown in Table 1.Ten, the PSO algorithm applies the LSTM NN to Based on the battery energy data in the constant current charging phase, plot the isobaric energy curve.
Obtain the average peak and the SOH data to generate the processed dataset.
Divide the processed dataset equally into training set and testing set.
Apply the PSO algorithm to optimize the learning rate and the number of hidden layers of the LSTM.
Establish the SOH estimation model based on training set.
Estimate lithium-ion batteries SOH using the established model based on testing set.Journal of Electrical and Computer Engineering build the LIB SOH estimation model after optimizing the LSTM learning rates and the number of hidden layers.

SOH Estimation Results and Analysis.
Te established estimation model is applied to evaluate LIB SOH after discharging within the corresponding charging-discharging cycle, using the average peak of the isobaric energy curve obtained during the charge phase within the chargingdischarging cycle as input.Te experimental results are shown in Figure 7. Te proposed method is very close to the real value for diferent charging multipliers of LIBs and has excellent estimation performance in the results.However, the estimation error gradually increases due to the accumulation of errors.In order to quantify the estimation efect of presented algorithm, evaluation indicators including    8.Meanwhile, the comparison experiment errors are recorded in Table 3.By comparing the proposed method with the commonly used SVM and RVR algorithms, it can be concluded that the proposed algorithm outperforms both SVM and RVR for battery aging data with four diferent charging multipliers.It is because the LSTM NN has excellent time series prediction ability.In this paper, the average peak of the isobaric energy curve is proposed as the health factor, and the learning rate     Journal of Electrical and Computer Engineering   Journal of Electrical and Computer Engineering curve with the strongest nonlinear characteristics, both MAE and RMSE are 0.6076% and 0.6720% of the proposed method, respectively, while the estimation errors of SVM and RVR are around 1.5%, which illustrates that the proposed method's estimation accuracy is much better than that of the commonly used SVM and RVR.As for the 0.2 C charging curve with severe abrupt changes, the proposed method is able to capture the battery SOH variation much faster than the RVR and SVM.Tis indicates that the method is well adapted to the LIB capacity regeneration phenomenon.For the 0.1 C and 0.3 C charging curves where the aging process is relatively smooth, it has excellent estimation accuracy, with MAE and RMSE no higher than 0.5%.In summary, the LIB SOH estimation algorithm presented in this paper can efectively estimate the SOH of LIBs with good robustness and generalizability.

Conclusions
A method for LIB SOH estimation using isobaric energy analysis and PSO-LSTM method has been developed in this paper.Firstly, the evolution pattern of battery energy during the constant-current charging stage has been analyzed and the isovoltaic voltage curves have been plotteds.Ten, the variation law of isobaric energy curve with battery degradation has been analyzed, and the average peak has been extracted as the new health factor of battery SOH.Finally, LSTM NN has been optimized in terms of the hidden layer quantities and learning rates using the PSO method, which established the LIB SOH estimation model.Te robustness and stability of the proposed method have been verifed based on cyclic aging data with four diferent charging/discharging ratios.
Experimental results have illustrated that the developed method can precisely estimate the LIB SOH with an average estimation error of 1%.It has a superior ability to capture the sudden change of LIB capacity phenomenon.Meanwhile, a comparison experiment has been designed based on the commonly used LIB SOH estimation algorithms, in which the proposed method has been compared with RVR and SVM algorithms.Te results indicate that the proposed method has a better estimation performance and good robustness and generalizability.

Figure 2 :
Figure 2: Te basic unit of the LSTM.

1 )
Constant current charging stage: four LIBs are charged with 0.5 C, 0.3 C, 0.2 C, and 0.1 C current, respectively, until the battery terminal voltage reaches the maximum cutof voltage of 4.2 V. (2) Constant voltage charging stage: following constant current charging, the batteries are charged at a constant voltage of 4.2 V until the current drops to 0.1 A. (3) Discharging stage: the four batteries are discharged at 1 C current until the terminal voltage drops to 3 V.
) Utilize SOH data corresponding to the charging and discharging cycle as the training target and the average peak as the training sample.Ten, use the PSO algorithm to maximize the learning rate and the hidden layer count in the LSTM NN. (5) Create the SOH estimate model with the optimal LSTM NN. (6) Based on the testing set, apply the established LIB SOH estimation model to evaluate the LIB SOH.
and the number of hidden layers of the traditional LSTM NN are preferred by PSO algorithm.Tis further enhances the nonlinear regression modeling capability of the LSTM NN.Meanwhile, the error statistics demonstrate that the proposed algorithm has good SOH estimation capability and high robustness and stability.Regarding the 0.5 C charging

Table 2 :
SOH estimation errors of the proposed method.

Table 3 :
Estimation errors of the comparative experiment.