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Aging characteristics-based health diagnosis and remaining useful life prognostics for lithium-ion batteries
eTransportation 1 (2019) 100004
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Aging characteristics-based health diagnosis and remaining useful life prognostics for lithium-ion batteries YongZhi Zhang a, b, Rui Xiong a, *, HongWen He a, Xiaobo Qu b, Michael Pecht c a
National Engineering Laboratory for Electric Vehicles, Department of Vehicle Engineering, School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China b Department of Architecture and Civil Engineering, Chalmers University of Technology, Gothenburg, SE-412 96, Sweden c Center for Advanced Life Cycle Engineering (CALCE), University of Maryland, College Park, MD, 20742, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2019 Received in revised form 11 June 2019 Accepted 19 June 2019 Available online xxx
This paper developed methods for improving the practicability of battery health diagnosis and remaining useful life prognostics. Battery state of health was estimated using a feature extraction-based method based on the charging voltage curve. Battery remaining useful life was predicted by identifying recognizable aging stages. Acceleration aging test data for 9 cells at constant current rates including 0.5C, 1C, 1.5C, and 2C, and dynamic current rates were used to validate the developed methods. The capacity estimates were accurate with estimation errors less than 1% at most cycles. The remaining useful life was predicted within 0.3 s at dynamic current rates, with the prediction errors at most cycles less than 10 after 300 cycles and the 95% confidence intervals covering about 20 cycles for each prediction. © 2019 Elsevier B.V. All rights reserved.
Keywords: Electric vehicles Lithium-ion battery Aging characteristics State of health diagnosis Remaining useful life prognostics
1. Introduction Lithium-ion batteries are widely used in electric vehicles (EVs) owing to their high energy/power density and long cycle lifetime [1]. A battery's performance degrades during its working lifetime, and it reaches the end of life (EOL) once its capacity passes 80% of the initial value. Battery health should be monitored for reliability evaluation and maintenance purposes. State of health (SOH) diagnosis and remaining useful life (RUL) prognostics of lithium-ion batteries can help EV users monitor battery health in real time and maintain the battery pack in advance of failures. SOH diagnosis methods can be divided into model-based methods and feature extraction-based methods. Model-based methods estimate battery capacities based on equivalent circuit models [2] or electrochemical models [3] combined with advanced filter techniques including Kalman filter and particle filter (PF). These methods generally require the relationship of stage of charge (SOC)eopen circuit voltage (OCV) covering the entire battery
* Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (R. Xiong), [email protected] (H. He), [email protected] (X. Qu), [email protected] (M. Pecht). https://doi.org/10.1016/j.etran.2019.100004 2590-1168/© 2019 Elsevier B.V. All rights reserved.
lifetime, and the construction of this relationship is timeconsuming. The constructed models based on these methods are often computation-intensive for on-board battery capacity estimations for EVs [4]. The feature extraction-based methods were developed to reduce the on-board computation burden of battery capacity estimations. These methods estimate battery capacities by constructing a data-driven model describing the relationship of health indictors (HIs) and capacities based on offline training data. Because the models that require intensive computations were constructed offline, the on-line computation for capacity estimation can be fast. The HIs were generally extracted from battery terminal voltage curves. Ref. [5] extracted battery HIs based on the discharging voltage curve that was collected at a constant discharging current rate. Lithium-ion batteries for EVs work at variable discharging current rates, whereas the charging current was generally constant. Therefore, the extracted HIs based on the charging voltage curve can be effective for EVs. Li et al. [6] and Wang et al. [7] extracted the differential voltage (DV) based on the charging voltage and capacity to indicate battery SOH. The DV was calculated at a constant current rate of 1/20C, whereas the charging current rate for EVs was generally larger than 1/5C. Hu et al. [8] extracted five charge-related HIs covering the whole voltage range,
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whereas the charging for EVs often covered partial voltage ranges since batteries were not discharged to the lower cut-off voltages. Therefore, HIs extracted from the charging voltage curve at a common range and charging current rate were developed in this paper to estimate battery capacities for EVs. RUL prediction methods can be divided into model-based methods and data-driven methods. Model-based methods predict battery RUL based on a nonlinear model, such as the two-term exponential to describe battery capacity degradations, combined with advanced filter techniques, such as PF to estimate model parameters. He et al. [9] developed a method for battery RUL prediction using the DempstereShafer theory and the Bayesian Monte Carlo (BMC) method. Model parameters were initialized offline by combining sets of training data based on the DempstereShafer theory. The BMC was then used to update the model parameters and predict the RUL based on available capacity data online. To prevent the particle degeneracy of PF, Miao et al. [10] predicted battery RUL using the unscented particle filter (UPF), which obtained a proposal distribution provided by the unscented Kalman filter (UKF) for particle sampling. However, Liu et al. [11] developed a framework of improved particle learning by typically resampling state particles first, considering the current measurement information, and then propagating them. Experimental results showed that both developed methods in Refs. [10,11] predicted battery RUL more accurately than those using standard PF. Su et al. [12] developed an interacting multiple model particle filter (IMMPF) for battery RUL prediction by combining the predictions of different models including the polynomial model, exponential model, and Verhulst model. RUL predictions using the IMMPF method in Ref. [12] were more accurate and precise than those using standard PF. Zhang et al. [13] transformed the nonlinear capacity degradation into a linear degradation, where a linear aging model was developed for battery RUL prediction. Model-based methods predict battery RUL depending on offline training data, which was collected under the same working conditions as those of online cells. In this case, it was difficult to design acceleration aging tests to collect effective offline training data since lithium-ion batteries for EVs work at dynamic conditions that are difficult to simulate in tests. Data-driven methods predict battery RUL using machine learning techniques, where some offline or online training data is required. Zhang et al. [14] used long-short-term memory (LSTM) recurrent neural network (RNN) to learn long-term dependencies of capacity degradations. Patil et al. [15] developed a multistage support vector machine (SVM) method for lithium-ion battery RUL prediction. In Refs. [14,15], a large amount of offline training data at different temperatures and current rates was required to train the LSTM RNN and SVM for accurate online RUL predictions at various working conditions. Nuhic et al. [16] used SVM to predict battery RUL by learning previous online capacity degradations independent of offline training data. Liu et al. [5] used relevance vector machine (RVM) combined with a gray model (GM) to improve the long-term prediction capability of RVM and RUL prediction accuracy, where only previous online capacity data was required to train the combined model. The machine learning methods [5,16] used data mining techniques to predict battery RUL rather than consider battery aging characteristics. Therefore, time-consuming and computation-intensive training was required for the SVM and RVM to mine all available online data for each RUL prediction, which can lead to inaccurate and unreliable prediction results. This paper developed a RUL prediction method based on battery aging characteristics analysis. The developed method required offline training data at one current rate (0.5C), and the training was conducted offline. The battery RUL was predicted online in a recursive way with a low computation burden. The rest of this paper is organized as follows: Section 2 introduces the acceleration
aging test of lithium-ion batteries. Sections 3 and 4 describe the theory and related algorithms for battery health diagnosis and RUL prognostics, respectively. Section 5 shows the battery capacity estimation and RUL prediction results using the developed methods, followed by the conclusions in Section 6. 2. Battery acceleration aging tests Fig. 1 shows the equipment used to conduct the acceleration aging tests for lithium-ion batteries. It includes an Arbin BT-5HC test system to charge/discharge batteries, three thermal chambers to control battery temperatures, and a computer for data monitoring and storage. The acceleration aging tests used the 18650 lithium-ion cells manufactured by Panasonic. The cell was labelled NCR18650BD, and its specifications are listed in Table 1. The cell had a rated capacity of 3 Ah. It consisted of Li(NiCoAl)O2 on the cathode and carbon on the anode. The allowed terminal voltage range was from 2.5 V to 4.2 V. The acceleration aging test conditions are summarized in Table 2. The cycle test had two test conditions, under which the cells were charged under a constant current (CC)-constant voltage (CV) process with a CC of 0.3C until the upper cut-off voltage of 4.2 V, then followed by a CV until the cut-off current of 0.02C. After a rest of 30 min, the cells were discharged at a CC until the lower cut-off voltage of 2.5 V. Under test condition 1, the cells were discharged at 0.5C, 1C, 1.5C, and 2C, respectively, and two cells were cycled at each current rate. The discharging current rates of lithium-ion batteries for EVs change dramatically. Therefore, one cell was discharged at different current rates at different cycles under test condition 2, where the cell was cycled at 0.5C, 1C, 1.5C, and 2C. Under this test condition, the discharging current was constant at one single cycle and changed to another rate at the next single cycle. The ambient temperature in the cycle test was set at 30 C by the thermal chamber. After the cells were cycled by 50 or 100 cycles, the characteristic tests including the capacity calibration test, the urban dynamometer driving schedule (UDDS) test, and the dynamic stress test (DST) were conducted to investigate the dynamic response characteristics of lithium-ion batteries. Note that the characteristic results were not used in this paper. 3. Battery health diagnosis and remaining useful life prognostics This section introduces the theory for realizing battery health diagnosis, aging stage identification, and remaining useful life prediction. The lithium-ion battery SOH, which is used to indicate
Fig. 1. Battery test equipment.
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3.2. Aging stage identification
Table 1 Cell specifications of the NCR18650BD battery. Rated capacity
3 Ah
Material Maximum continuous discharge current Allowed voltage range Temperature range Charge Discharge Storage
Li(NiCoAl)O2/carbon 10 A 2.5e4.2 V 10e45 C 20e60 C 20e50 C
the battery health, is represented by the present capacity divided by the initial capacity, which is expressed as:
SOH ¼
Cp $100% Ci
(1)
where Ci represents the initial capacity and Cp represents the present capacity [17]. 3.1. Health diagnosis Dubarry et al. [18] and Gao et al. [19] showed that the battery aging mechanisms can be identified by deriving the peak area from the incremental capacity (IC) curve of the cell. This peak area indicates the accumulated capacity from the peak that covers a specific voltage range. However, the IC curve is generally plotted at a very low charging current rate, typically at 1/20C, and the peak of the curve can disappear at a commonly used charging rate, typically at 0.3C. Besides, the peak of the IC curve can occur at a low voltage range which does not cover the commonly used voltage ranges of the lithium-ion batteries for EVs. Therefore, the incremental capacity analysis (ICA) is unable to be used to identify on-board capacities of lithium-ion batteries for EVs. The peak area of the IC curve is extracted to indicate the battery aging mechanisms based on the principle that the required capacity to make the same change on the battery voltage is different at different battery SOHs. Based on this principle, this paper extracted six accumulated capacities covering six voltage ranges from the charging voltage curve to indicate battery SOH. The charging current rate and voltage range are both commonly used values, which are, respectively, 0.3C and 3.6 Ve4.2 V. The six voltage ranges are 3.6e3.7 V, 3.7e3.8 V, 3.8e3.9 V, 3.9e4.0 V, 4.0e4.1 V, and 4.1e4.2 V, and the six accumulated capacities are extracted from each voltage range as HIs. Fig. 2 shows the feature extraction results of one cell at 0.5C. Fig. 2(a) shows the charging voltage curves at different cycles, which indicates that the charging time until the upper cut-off voltage shortens as the cell ages. Fig. 2(b) shows the extracted capacities at different voltage ranges, where the feature numbers in Fig. 2(b) correspond to the six voltage ranges from 3.6e3.7 V to 4.1e4.2 V. The extracted capacities at each voltage range generally decrease as the cell ages, and the cell at different SOHs shows different HI patterns. The relationship between the HI patterns and battery SOHs was constructed using machine learning techniques.
Fig. 3 shows the capacity degradation trajectories of cells under different test conditions. The capacity at each cycle was the discharging capacity at the corresponding discharging current rate. Fig. 3(a and b) show the capacity degradations under test condition 1, where Fig. 3(a) shows the capacity degradations at 0.5C and 1C, and Fig. 3(b) shows the capacity degradations at 1.5C and 2C. Fig. 3(c) shows the capacity degradations under test condition 2, where the capacity degradations at the specific discharging current rate of each cycle are presented. Fig. 3 shows that the capacity degradation trajectory can generally be divided into two stages, with a fast capacity degradation in the first stage and a slow capacity degradation in the second stage. This capacity degradation trajectory is similar to that in Ref. [20], where the same type of cell was cycled. The first stage covered about 100 cycles, and during which, the new battery started degrading, and the capacity degradation rate was relatively high. During the second stage, the capacity degradation rate decreased, and a nearly linear capacity degradation was observed. The aging mechanisms behind these two aging stages should be different [26], which however cannot be furhter identified based on the avaliable experimental data. For other commonly used commercial lithium-ion batteries, such as the LiMn2O4- or LiFePO4-based batteries, the capacity also degrades linearly near the battery EOL [21,22]. Fig. 3(b and c) show that the cell capacity at 1.5C or 2C degraded at a high rate during the second stage, which was mainly caused by the increasing internal resistance as the cell aged. The internal resistance caused a voltage drop with discharging current loaded on the cell. There was still capacity in the cell owing to this voltage drop when the cell discharged to the lower cut-off voltage. The higher the internal resistance and the discharging current rate were, the larger the voltage drop was, which led to more remaining capacities in the cell and thus less discharging capacities. Therefore, the capacity degardation rate at 1.5C or 2C was higher during the second stage than that at 0.5C or 1C owing to the increasing internal resistance as the cell aged. For the same reason, Fig. 3(c) shows that the differences of the discharging capacities at 1.5C or 2C between those at 0.5C or 1C increased as the cell aged, indicating an increasing capacity degrdation rate at 1.5C or 2C during the second stage owing to the increasing internal resistance. The battery capacity should be calibrated at a small discharging current rate, such as 0.5C or 1C, to reduce the influence of the increasing resistance as the battery ages on capacity calibration [23,24]. In this paper, the discharging capacity at 0.5C was used to indicate the present battery capacity in Eq. (1). 3.3. Remaining useful life prediction Section III. B shows that the battery EOL fell within the second degradation stage of capacities, which was a nearly linear degradation at 0.5C. Therefore, battery RUL can be predicted based on available capacities at the second stage using a linear aging model. When the battery degraded into the second stage, a linear aging
Table 2 Cell test conditions of the NCR18650BD battery. Test Category
Test Condition 1
Test Condition 2
Cycle tests
Charge: CCCV, charged at 0.3C rate up to 4.2 V, 0.02C rate cutoff Charge rest: 30 min Discharge: 0.5C, 1C, 1.5C and 2C rate, 2.5 V cutoff Discharge rest: 30 min Temperature: 30 C Capacity calibration-UDDS test-DST test
Charge: CCCV, charged at 0.3C rate up to 4.2 V, 0.02C rate cutoff Charge rest: 30 min Discharge: 0.5Ce1C-1.5Ce2C rate, 2.5 V cutoff Discharge rest: 30 min Temperature: 30 C Capacity calibration-UDDS test-DST test
Characteristic tests
3
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Fig. 2. Feature extraction results of one cell at 0.5C: (a) the charging voltage curves and (b) the extracted capacities at different cycles.
Fig. 3. Capacity degradation trajectories of cells at: (a) 0.5C and 1C; (b) 1.5C and 2C; (c) dynamic rate.
model was constructed based on the collected data at this stage. The input and output of the model were the cycle number and SOH. The linear model was extrapolated, and the battery EOL was predicted once the extrapolated SOH was lower than 80%. 4. Algorithms This section introduces related algorithms, including the neural network (NN), for battery health diagnosis, the random forest (RF) for battery aging stage identification, and the recursive least squares (RLS) algorithm with a forgetting factor for battery RUL prognostics. 4.1. Neural network The NN, which is able to build complex nonlinear relationships between inputs and outputs, was used to describe the relationship between the extracted HIs and battery capacities. The commonly used NN, whose structure is shown in Fig. 4, has three layers, including one input layer, one hidden layer, and one output layer. The input layer included six HIs in this paper, which were the extracted capacities from six voltage ranges and were represented by HI1 to HI6. The hidden layer included 50 sigmoid neurons [25], which were all input into the output layer including one linear neuron. The output layer calculated the battery capacity. 4.2. Random forest algorithm RFs are ensembles of tree-type classifiers, where each classifier is trained with a different subset of the training set (“bagging”), thereby improving the generalization ability of the classifier. Samples are processed along a path from the root to a leaf in each
Fig. 4. Structure of the neural network.
tree by performing a binary test at each internal node along this path. The binary test compares a certain feature with a threshold. Training a forest amounts to identifying the set of tests at each node that best separate the data into the different training classes. RF generally performs pattern classification problems better than NN [26] and was thus selected to identify the battery aging stage in this paper. In this paper, the inputs of the RF were six extracted HIs and the outputs were the corresponding aging stages. The RF used 100 trees to construct the relationship between the HIs and the corresponding aging stages [26e28].
4.3. Recursive least squares algorithm The RLS method is a recursive form of the standard least squares
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method, which inputs all historical data sets to construct a linear model [29]. RLS operates in an iterative way and only inputs the present data set to identify the model parameters, thus, it needs less memory space and is more computation-efficient in practice than the standard least squares method. A linear model based on capacity data at aging stage 2 (Fig. 3) can predict erroneous battery RULs owing to two problems: first, the capacity degradation at aging stage 2 is not strictly linear, and second, RF can erroneously identify starting cycles in stage 2 that actually belong in stage 1. The RLS method with a forgetting factor can construct a linear model by forgetting old data, and was thus used to identify the model parameters in this paper. The following linear model was constructed to capture the battery capacity degradation at stage 2:
SOHk ¼ b0 þ b1 $k þ εk ; εk N 0; s2
b q 0 ¼ EðqÞ
(3)
i h q 0 Þðq bq 0 ÞT P0 ¼ E ðq b
(4)
where q ¼ [b0, b1]T is the model parameter vector to be identified and P is the estimation-error covariance of q. In the paper, q0 ¼ [0, 0]T, and P0 ¼ 106I because no information about q was available at the beginning. I is a unity matrix. Step 2, for i ¼ 1, 2, …, update the estimate of q and the estimation-error covariance P as follows:
1 Ki ¼ Pi1 4Ti 4i Pi1 4Ti þ m
(5)
b q i ¼ bq i1 þ Ki ðyi 4i bq i1 Þ
(6)
1
ðI Ki 4i ÞPi1
5. Experimental results The experimental data at constant current rates, including 0.5C, 1C, 1.5C, and 2C, and dynamic current rates were used to evaluate the performance of the developed method. The simulation was performed based on Matlab 2016b, which operated on a laptop with an Intel Core i7-6700 HQ processor (6 MB cache, up to 3.50 GHz). 5.1. Health diagnosis
Step 1, initialize the estimator as follows,
m
relationship the starting cycles for stage 2 were identified. The capacity data in stage 2 was used to construct a linear aging model using RLS with a forgetting factor in step 3. This linear model with uncertainties sampled using MC simulation was extrapolated to predict the battery RUL PDF in step 4.
(2)
where k indicates the cycle number; b0 and b1 are model parameters to be identified, and εk are random errors that are independent and normally distributed with a zero mean and variance of s2. The linear model parameters are identified using RLS with a forgetting factor in the following two-step process [30].
Pi ¼
5
(7)
where K represents the estimator gain matrix; 4 ¼ [1, k] represents the model input vector, and y ¼ SOH represents the model output; m is the forgetting factor ranging from 0 to 1, and m ¼ 1 means forgetting no old data. Once the model parameters were identified using RLS, the future battery SOH could be predicted by extrapolating Eq. (2), and the battery EOL was obtained if the predicted SOH at cycle k was lower than 0.8. The EOL prediction uncertainties were generated using Monte Carlo (MC) simulation. The model uncertainties existed in random errors εk, from which a total of 103 samples were sampled. Eq. (2) was extrapolated to predict one battery EOL with one sampled ε value. At each cycle, there were 103 predicted EOLs which generated the EOL probability distribution function (PDF). Fig. 5 shows four steps for battery health diagnosis and RUL prediction. In the first step, the NN was used to build a relationship between the extracted battery HIs and capacities, and the battery SOH was diagnosed based on on-board battery data and this relationship. Step 2 used RF to build a relationship between the extracted HIs and the corresponding aging stages, and based on this
Fig. 6 shows the estimated capacities at different cycles using the health diagnosis method. The NN was trained to construct a relationship between the HIs and capacities based on the aging data of one cell at each test condition, and then the trained NN was used to estimate the capacity data of another cell at the same test condition. Herein, the capacities were obtained at the corresponding discharging current rate. Fig. 6 shows the capacity estimation results of cells at four discharging current rates including 0.5C, 1C, 1.5C, and 2C. The estimated capacities were all close to the real values with the estimation errors mostly within 1%. The capacity estimation errors at 0.5C, for example, were close to zero and within 1% at most cycles. The root mean square errors (RMSEs) of capacity estimation at 0.5C, 1C, 1.5C, and 2C were 0.40%, 0.53%, 0.63%, and 0.53%, respectively. In practice, the extracted HIs from the charging voltage curve of lithium-ion batteries for EVs are input to an NN, which is trained based on the acceleration aging test data at 0.5C, to estimate the onboard capacities. The test data at dynamic discharging current rates was used to verify the capacity estimation performance of the NN for lithium-ion batteries for EVs. Fig. 7 shows the capacity estimation results based on the test data at dynamic discharging current rates, where an NN was constructed based on the test data of two cells at 0.5C to describe the relationship between the HIs and capacities. The HIs were extracted from the charging voltage curve at each cycle of the test data at dynamic current rates and input to this NN to estimate capacities. The capacities at 0.5C of the test data at dynamic current rates were considered reference data in Fig. 7(a), where the capacities at 1C, 1.5C, and 2C were all replaced by the interpolated values based on the capacities at 0.5C. Fig. 7(a) shows that the estimated capacities were close to the reference data with capacity estimation errors within 1% at most cycles (Fig. 7(b)), and the RMSE was 0.80%, indicating the high on-board capacity estimation accuracy and feasibility of the method for lithium-ion battery capacity estimations for EVs. 5.2. Aging stage identification In order to identify the starting cycle of aging stage 2 of capacities at 0.5C, a linear model was used to fit the capacities of one cell at 0.5C from different cycles using Matlab's curve fitting tool. Table 3 shows the fitting results, where the performance was indicated by R-square and RMSE. R-square is the square of the correlation between the real values and estimated values, and can evaluate the linear correlation degree of capacities. R-square can take on any value between 0 and 1, with a value closer to 1 indicating a stronger linear degradation of capacities. RMSE indicates the fitting accuracy. Table 3 shows that the R-square increases from 0.9956 to 0.9978 with the starting cycle number increasing from 60
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Fig. 5. Steps for battery SOH diagnosis and RUL prognostics.
Fig. 6. Capacity estimation results based on the NN method at: (aeb) 0.5C; (ced) 1C; (eef) 1.5C; (geh) 2C.
Fig. 7. Capacity estimation results based on the NN method at dynamic current rates.
degradation rate was large at stage 1 with a degradation rate close
Table 3 Fitting results of capacities from different cycles at 0.5C Starting cycle
R-square
RMSE ( 103 Ah)
60 80 100 120 160
0.9956 0.9965 0.9978 0.9976 0.9973
8.83 7.33 5.49 5.38 5.42
to 100, whereas the RMSE decreases from 8.83 103 Ah to 5.49 103 Ah. After 100 cycles, the R-square decreases as the cycle number increases, whereas the RMSE decreases a little from 5.49 103 Ah to 5.38 103 Ah, then increases to 5.42 103 Ah. Therefore, the 100th cycle was considered the starting cycle of stage 2 of capacity degradations at 0.5C. Fig. 8 shows the average capacity degradation rate per cycle before and after 100 cycles of the cell at 0.5C. The capacity
Fig. 8. Average capacity degradation rate per cycle at 0.5C.
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to 0.25%, whereas at stage 2, the capacity degradation rate was low and close to 0.1%. The test data of the cell at 0.5C with identified aging stages was used to train the RF model. The input was the extracted HIs at each cycle, and the output was the stage pattern, with 0 indicating stage 1 and 1 indicating stage 2. The trained RF model was then used to identify the stages of capacities of other cells at different C rates. The test data of one cell at each rate was used to verify the RF model. Fig. 9 shows the stage identification results of capacities at 0.5C (Fig. 9(a)), 1C (Fig. 9(b)), 1.5C (Fig. 9(c)), and 2C (Fig. 9(d)), where the identification results using the NN method are also shown for comparison. Failed classifications indicated cycles that failed to be classified to the right stages. Fig. 9 shows that the RF always identified aging stages more efficiently than the NN with fewer failed classifications. Fig. 9(b), for example, shows that the NN failed to classify 40 cycles' data with varying pattern values between 0 and 1, whereas the RF failed to classify 1 cycle's data. The identified starting cycle number of stage 2 of capacities at 0.5C, 1C, 1.5C, and 2C using RF were, respectively, 105, 110, 104, and 129. These started cycle numbers for cells at 0.5C, 1C, and 1.5C varied between 100 and 110, whereas the cycle number of the cell at 2C was 129, indicating slower capacity aging at stage 1 and 2C than other C rates. Fig. 10 shows the average capacity degradation rate per cycle at different current rates. The capacity degradation rates of stage 1 were larger than those of stage 2 at all current rates. The degradation rate of stage 1 at 2C was the lowest and close to 0.17%, whereas the degradation rates at 0.5C, 1C, and 1.5C were all higher than 0.2%. This lower capacity degradation rate at 2C than other C rates led to a later starting cycle of stage 2 in Fig. 9. The capacity degradation rate of stage 2 at 2C was the largest owing to the increasing internal resistance as the cell aged, whereas the capacity degradation rates at 0.5C, 1C, and 1.5C were all close to 0.1%. Fig. 11 shows the aging stage identification results at dynamic current rates, where an RF model was trained based on the test data of two cells at 0.5C. The extracted HIs from the charging voltage curve at each cycle of the cell at dynamic current rates were input to
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Fig. 10. Average capacity degradation rate per cycle of aging stage 1 and 2 at different current rates.
Fig. 11. Aging stage identification results of the cell at dynamic current rates.
the trained RF to identify the starting cycle number of stage 2 of capacities, which was 107 with 3 misclassified cycles.
Fig. 9. Aging stage identification based on RF and NN at: (a) 0.5C; (b) 1C; (c) 1.5C; (d) 2C.
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5.3. Remaining useful life prediction The RLS with a forgetting factor was used to predict the battery RUL based on the on-board capacity estimation and aging stage identification results. The RLS started when 50 cycles’ capacity data from the starting cycle of stage 2 was collected, and continued to predict battery RUL every 10 cycles. Fig. 12 shows the battery RUL prediction results at different current rates using RLS with different forgetting factors, including m ¼ 1, 0.99, 0.97, 0.95, and a variable forgetting factor. The variable forgetting factor was defined as
mi ¼ ami1 þ ð1 aÞ
(8)
where a determines the change rate of forgetting factor and i indicates the sampling moment. In this paper, a ¼ 0.995 and m0 ¼ 0.95. Fig. 13 shows the variable m trajectories with different a values. m converges faster to 1 with a smaller a value. The upper and lower error thresholds in Fig. 12 represented RUL values that were, respectively, 50 cycles larger and smaller than the real data. RUL predictions were considered accurate once they fell within the error thresholds. Tables 4 and 5 lists the RUL prediction performance at different current rates and m values, where the starting cycle indicates when the predicted RUL fell within the error thresholds, and the RMSEs of RUL predictions were calculated after this starting cycle. In Tables 4 and 5, overstriking indicates the earliest starting cycle and lowest RMSE of RUL predictions at each current rate. RUL predictions with a smaller m captured the newest information more efficiently and were thus less robust against data variations. For this reason, Fig. 12 shows that RUL predictions with m ¼ 0.95 varied most strongly at each current rate. Table 5 shows that RMSEs with m ¼ 0.95 were generally the largest at each current rate except the one at 1.5C, which was the lowest. This lowest RMSE was mainly caused by small variations of the estimated capacity data at 1.5C rate after 300 cycles (Fig. 6(e)). Fig. 12 shows that stable RULs with m ¼ 1 and 0.99 were predicted, and the RMSEs of RUL predictions at each current rate except 1.5C were generally the
Fig. 13. Variable m trajectories at different a values.
Table 4 Starting cycle of RUL predictions at different current rates and m values. Current Rate/m value
1
0.99
0.97
0.95
Variable
0.5C 1C 1.5C 2C
155 180 374 299
155 180 344 289
155 180 314 279
155 180 294 269
155 180 324 279
Table 5 RMSE of RUL predictions at different current rates and m values. Current Rate/m value
1
0.99
0.97
0.95
Variable
0.5C 1C 1.5C 2C
10 14 27 17
10 13 21 18
12 14 17 23
17 19 16 27
11 14 16 24
lowest in Table 5. At 1.5C, the estimated capacities in the beginning
Fig. 12. RUL prediction results based on RLS method with different m values at: (a) 0.5C; (b) 1C; (c) 1.5C; (d) 2C.
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Fig. 14. RUL prediction results at dynamic current rates using RLS with a variable forgetting factor.
of stage 2 degraded at a lower rate than the capacities did later. Therefore, the RLS with a large m, which greatly depended on data in the beginning, always predicted larger RULs than those using the RLS with a small m. These large predicted RULs caused the large RMSEs of predictions with m ¼ 1 and 0.99 at 1.5C. For the same reason, the RUL predictions with m ¼ 1 and 0.99 fell within the error thresholds at larger cycles than those with other m values did at 1.5C and 2C. The RLS with m ¼ 0.95, which forgot the data in the beginning quickly, captured the capacity variations efficiently and thus predicted RULs falling within the error thresholds early. To improve the prediction performance, the RLS was expected to possess a small forgetting factor in the beginning for capturing capacity degradations efficiently and a large forgetting factor later for robust RUL predictions against capacity variations. The RLS with a variable forgetting factor increasing from 0.95 in the beginning to 1 in the end was thus developed. Table 5 shows that the RLS with a variable m generally predicted one of the lowest RMSEs at each current rate, and predicted RULs falling within the error threshold at one of the earliest starting cycles at each current rate. RUL predictions with a variable m fell into the error threshold at a starting cycle number that was, respectively, 50 and 20 at 1.5C, and 20 and 10 at 2C smaller than those with m ¼ 1 and 0.99 at the corresponding current rate. The RLS with a variable forgetting factor was used to predict the battery RUL at dynamic current rates owing to its high prediction performance. Fig. 14 shows the RUL prediction results at dynamic current rates using the RLS with a variable m. The RUL was predicted based on the on-board estimated capacities in Fig. 7. Fig. 14 shows that the predicted RUL fell within the error thresholds after 200 cycles, and the RUL prediction errors at most cycles were less than 10 after 300 cycles. The RUL PDF was generated using MC simulation. The 95% confidence interval of RUL predictions at 257, 357, and 457 cycles were, respectively, [265, 285], [181, 197], and [88, 108] cycles, indicating precise RUL predictions. The calculation time for each RUL prediction was around 0.3 s, indicating a low on-board computation burden. 6. Conclusions Lithium-ion battery health for electric vehicles (EVs) must be monitored and prognosed in real time for reliability evaluation and maintenance purposes. This paper aims at developing practical and effective battery state of health (SOH) diagnostic and remaining useful life (RUL) prognostic methods based on the aging characteristics analysis. Battery health was indicated by six health indicators (HIs), which were extracted from the charging voltage curve within a commonly used range. A neural network (NN) and random forest
9
(RF) model were constructed to describe the relationships between the HIs and battery capacities, and the HIs and aging stages of capacity degradations, respectively, based on offline training data. The battery capacities and aging stages at different current rates were estimated and identified online by inputting the extracted battery HIs into the constructed NN and RF, respectively. Battery RUL was predicted based on the capacity estimation and aging stage identification results. The recursive least squares (RLS) method with a forgetting factor predicted the battery RUL based on available on-board capacity data of aging stage 2, combined with Monte Carlo (MC) simulation to generate the RUL probability distribution function (PDF). Experimental data for 9 cells at constant current rates, including 0.5C, 1C, 1.5C, and 2C, and dynamic current rates was used to validate the developed methods for battery SOH diagnosis and RUL prognostics. Experimental results showed that the capacity estimates were accurate, with estimation errors less than 1% at most cycles at both constant current rates and dynamic current rates. Based on the capacity estimation and aging stage identification results, the RLS method with a variable forgetting factor predicted accurate RULs at an early life stage. Accurate and precise RULs can be predicted at dynamic current rates after 200 cycles, with the prediction errors at most cycles less than 10 after 300 cycles and the 95% confidence intervals at 257, 357, and 457 cycles covering about 20 cycles for each prediction. The calculation time for each RUL prediction was about 0.3 s. The experimental results at dynamic current rates indicate that, in practice, the developed methods can predict on-board battery SOH and RUL for EVs accurately with a low computation burden. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51877009) and Beijing Municipal Natural Science Foundation (Grant No. 3182035). The systemic experiments of the lithium-ion batteries were performed at the Advanced Energy Storage and Application (AESA) Group, Beijing Institute of Technology. References [1] Xiong R, Zhang Y, He H, Zhou X, Pecht M. A double-scale, particle-filtering, energy state prediction algorithm for lithium-ion batteries. IEEE Trans Ind Electron Feb. 2018;65(2):1526e38. [2] Deng Z, Yang L, Cai Y, Deng H, Sun Liu. Online available capacity prediction and state of charge estimation based on advanced data-driven algorithms for lithium iron phosphate battery. Energy Sep. 2017;201:257e69. [3] Bartlett A, Marcicki J, Onori S, Rizzoni G, Yang XG, Miller T. Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery. IEEE Trans Control Syst Technol Mar. 2017;24(2): 384e99. [4] Farmann A, Waag W, Marongiu A, Sauer DU. Critical review of on-board capacity estimation techniques for lithium-ion batteries in electric and hybrid electric vehicles. J Power Sources May. 2015;281:114e30. [5] Liu D, Zhou J, Liao H, Peng Y, Peng X. A health indicator extraction and optimization framework for lithium-ion battery degradation modeling and prognostics. IEEE Transactions on Systems, Man, and Cybernetics: Systems Jun. 2015;45(6):915e28. [6] Li X, Jiang J, Wang LY, Chen D, Zhang Y, Zhang C. A capacity model based on charging process for state of health estimation of lithium ion batteries. Appl Energy Sep. 2016;177:537e43. [7] Wang L, Pan C, Liu L, Cheng Y, Zhao X. On-board state of health estimation of LiFePO4 battery pack through differential voltage analysis. Appl Energy Apr. 2016;168:465e72. [8] Hu C, Jain G, Schmidt C, Strief C, Sullivan M. Online estimation of lithium-ion battery capacity using sparse Bayesian learning. J Power Sources Sep. 2015;289:105e13. [9] He W, Williard N, Osterman M, Pecht M. “Prognostics of lithium-ion batteries based on DempstereShafer theory and the Bayesian Monte Carlo method. J Power Sources Dec. 2011;196(23):10314e21. [10] Miao Q, Xie L, Cui H, Liang W, Pecht M. Remaining useful life prediction of lithium-ion battery with unscented particle filter technique. Microelectron
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Reliab Jun. 2013;53(6):805e10. [11] Liu Z, Sun G, Bu S, Han J, Tang X, Pecht M. Particle learning framework for estimating the remaining useful life of lithium-ion batteries. IEEE Transactions on Instrumentation and Measurement Feb. 2017;66(2):280e93. [12] Su X, Wang S, Pecht M, Zhao L, Ye Z. Interacting multiple model particle filter for prognostics of lithium-ion batteries. Microelectron Reliab Mar. 2017;70: 59e69. [13] Zhang Y, Xiong R, He H, Pecht M. “Lithium-ion battery remaining useful life prediction with BoxeCox transformation and Monte Carlo simulation. IEEE Trans Ind Electron Feb. 2019;66(2):1585e97. [14] Zhang Y, Xiong R, He H, Pecht M. Long short-term memory recurrent neural network for remaining useful life prediction of lithium-ion batteries. IEEE Trans Veh Technol Jul. 2018;67(7):5695e705. [15] Patil MA, Tagade P, Hariharan KS, Kolake SM, Song T, Yeo T, Doo S. A novel multistage support vector machine based approach for Li ion battery remaining useful life estimation. Appl Energy Dec. 2015;159:285e97. [16] Nuhic A, Terzimehic T, Soczka-Guth T, Buchholz M, Dietmayer K. Health diagnosis and remaining useful life prognostics of lithium-ion batteries using data-driven methods. J Power Sources Oct. 2013;239:680e8. [17] Berecibar M, Garmendia M, Gandiaga I, Crego J, Villarreal I. State of health estimation algorithm of LiFePO4 battery packs based on differential voltage curves for battery management system application. Energy May 2016;103: 784e96. [18] Dubarry M, Truchot C, Liaw BY, Gering K, Sazhin S, Jamison D, Michelbacher C. Evaluation of commercial lithium-ion cells based on composite positive electrode for plug-in hybrid electric vehicle applications. Part II. Degradation mechanism under 2 C cycle aging. J Power Sources Dec. 2011;196(23): 10336e43. [19] Gao Y, Jiang J, Zhang C, Zhang W, Ma Z, Jiang Y. Lithium-ion battery aging mechanisms and life model under different charging stresses. J Power Sources Jul. 2017;356:103e14. [20] Watanabe S, Kinoshita M, Nakura K. Capacity fade of LiNi(1-x-y)CoxAlyO2 cathode for lithium-ion batteries during accelerated calendar and cycle life
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test. I. Comparison analysis between LiNi(1-x-y)CoxAlyO2 and LiCoO2 cathodes in cylindrical lithium-ion cells during long term storage test. J Power Sources Feb. 2014;247:412e22. Dubarry M, Truchot C, Cugnet M, Liaw BY, Gering K, Sazhin S, Jamison D, Michelbacher C. Evaluation of commercial lithium-ion cells based on composite positive electrode for plug-in hybrid electric vehicle applications. Part I: initial characterizations. J Power Sources Dec. 2011;196:10328e35. Jin X, Vora A, Hoshing V, Saha T, Shaver G, García RE, Wasynczuk O, Varigonda S. Physically-based reduced-order capacity loss model for graphite anodes in Li-ion battery cells. J Power Sources Feb. 2017;342:750e61. Ecker M, Nieto N, K€ abitz S, Schmalstieg J, Blanke H, Warnecke A, Sauer Dirk Uwe. Calendar and cycle life study of Li(NiMnCo)O2-based 18650 lithium-ion batteries. J Power Sources Feb. 2014;248:839e51. Saxena S, Hendricks C, Pecht M. Cycle life testing and modeling of graphite/ LiCoO2 cells under different state of charge ranges. J Power Sources Sep. 2016;327:394e400. Wu J, Zhang C, Chen Z. An online method for lithium-ion battery remaining useful life estimation using importance sampling and neural networks. Appl Energy Jul. 2016;173:134e40. Liu M, Wang M, Wang J, Li D. Comparison of random forest, support vector machine and back propagation neural network for electronic tongue data classification: application to the recognition of orange beverage and Chinese vinegar. Sensor Actuator B Chem Feb. 2013;177:970e80. Breiman L. Random forests. Mach Learn 2001;45(1):5e32. Mahapatra D. Analyzing training information from random forests for improved image segmentation. IEEE Trans Image Process Apr. 2014;23(4): 1504e12. Badoni M, Singh A, Singh B. Variable forgetting factor recursive least square control algorithm for DSTATCOM. IEEE Trans Power Deliv Oct. 2015;30(5): 2353e61. Li Z, Xiong R, Mu H, He H, Wang C. A novel parameter and state-of-charge determining method of lithium-ion battery for electric vehicles. Appl Energy Dec. 2017;207:363e71.
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Aging characteristics-based health diagnosis and remaining useful life prognostics for lithium-ion batteries YongZhi Zhang a, b, Rui Xiong a, *, HongWen He a, Xiaobo Qu b, Michael Pecht c a
National Engineering Laboratory for Electric Vehicles, Department of Vehicle Engineering, School of Mechanical Engineering, Beijing Institute of Technology, Beijing, 100081, China b Department of Architecture and Civil Engineering, Chalmers University of Technology, Gothenburg, SE-412 96, Sweden c Center for Advanced Life Cycle Engineering (CALCE), University of Maryland, College Park, MD, 20742, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2019 Received in revised form 11 June 2019 Accepted 19 June 2019 Available online xxx
This paper developed methods for improving the practicability of battery health diagnosis and remaining useful life prognostics. Battery state of health was estimated using a feature extraction-based method based on the charging voltage curve. Battery remaining useful life was predicted by identifying recognizable aging stages. Acceleration aging test data for 9 cells at constant current rates including 0.5C, 1C, 1.5C, and 2C, and dynamic current rates were used to validate the developed methods. The capacity estimates were accurate with estimation errors less than 1% at most cycles. The remaining useful life was predicted within 0.3 s at dynamic current rates, with the prediction errors at most cycles less than 10 after 300 cycles and the 95% confidence intervals covering about 20 cycles for each prediction. © 2019 Elsevier B.V. All rights reserved.
Keywords: Electric vehicles Lithium-ion battery Aging characteristics State of health diagnosis Remaining useful life prognostics
1. Introduction Lithium-ion batteries are widely used in electric vehicles (EVs) owing to their high energy/power density and long cycle lifetime [1]. A battery's performance degrades during its working lifetime, and it reaches the end of life (EOL) once its capacity passes 80% of the initial value. Battery health should be monitored for reliability evaluation and maintenance purposes. State of health (SOH) diagnosis and remaining useful life (RUL) prognostics of lithium-ion batteries can help EV users monitor battery health in real time and maintain the battery pack in advance of failures. SOH diagnosis methods can be divided into model-based methods and feature extraction-based methods. Model-based methods estimate battery capacities based on equivalent circuit models [2] or electrochemical models [3] combined with advanced filter techniques including Kalman filter and particle filter (PF). These methods generally require the relationship of stage of charge (SOC)eopen circuit voltage (OCV) covering the entire battery
* Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (R. Xiong), [email protected] (H. He), [email protected] (X. Qu), [email protected] (M. Pecht). https://doi.org/10.1016/j.etran.2019.100004 2590-1168/© 2019 Elsevier B.V. All rights reserved.
lifetime, and the construction of this relationship is timeconsuming. The constructed models based on these methods are often computation-intensive for on-board battery capacity estimations for EVs [4]. The feature extraction-based methods were developed to reduce the on-board computation burden of battery capacity estimations. These methods estimate battery capacities by constructing a data-driven model describing the relationship of health indictors (HIs) and capacities based on offline training data. Because the models that require intensive computations were constructed offline, the on-line computation for capacity estimation can be fast. The HIs were generally extracted from battery terminal voltage curves. Ref. [5] extracted battery HIs based on the discharging voltage curve that was collected at a constant discharging current rate. Lithium-ion batteries for EVs work at variable discharging current rates, whereas the charging current was generally constant. Therefore, the extracted HIs based on the charging voltage curve can be effective for EVs. Li et al. [6] and Wang et al. [7] extracted the differential voltage (DV) based on the charging voltage and capacity to indicate battery SOH. The DV was calculated at a constant current rate of 1/20C, whereas the charging current rate for EVs was generally larger than 1/5C. Hu et al. [8] extracted five charge-related HIs covering the whole voltage range,
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whereas the charging for EVs often covered partial voltage ranges since batteries were not discharged to the lower cut-off voltages. Therefore, HIs extracted from the charging voltage curve at a common range and charging current rate were developed in this paper to estimate battery capacities for EVs. RUL prediction methods can be divided into model-based methods and data-driven methods. Model-based methods predict battery RUL based on a nonlinear model, such as the two-term exponential to describe battery capacity degradations, combined with advanced filter techniques, such as PF to estimate model parameters. He et al. [9] developed a method for battery RUL prediction using the DempstereShafer theory and the Bayesian Monte Carlo (BMC) method. Model parameters were initialized offline by combining sets of training data based on the DempstereShafer theory. The BMC was then used to update the model parameters and predict the RUL based on available capacity data online. To prevent the particle degeneracy of PF, Miao et al. [10] predicted battery RUL using the unscented particle filter (UPF), which obtained a proposal distribution provided by the unscented Kalman filter (UKF) for particle sampling. However, Liu et al. [11] developed a framework of improved particle learning by typically resampling state particles first, considering the current measurement information, and then propagating them. Experimental results showed that both developed methods in Refs. [10,11] predicted battery RUL more accurately than those using standard PF. Su et al. [12] developed an interacting multiple model particle filter (IMMPF) for battery RUL prediction by combining the predictions of different models including the polynomial model, exponential model, and Verhulst model. RUL predictions using the IMMPF method in Ref. [12] were more accurate and precise than those using standard PF. Zhang et al. [13] transformed the nonlinear capacity degradation into a linear degradation, where a linear aging model was developed for battery RUL prediction. Model-based methods predict battery RUL depending on offline training data, which was collected under the same working conditions as those of online cells. In this case, it was difficult to design acceleration aging tests to collect effective offline training data since lithium-ion batteries for EVs work at dynamic conditions that are difficult to simulate in tests. Data-driven methods predict battery RUL using machine learning techniques, where some offline or online training data is required. Zhang et al. [14] used long-short-term memory (LSTM) recurrent neural network (RNN) to learn long-term dependencies of capacity degradations. Patil et al. [15] developed a multistage support vector machine (SVM) method for lithium-ion battery RUL prediction. In Refs. [14,15], a large amount of offline training data at different temperatures and current rates was required to train the LSTM RNN and SVM for accurate online RUL predictions at various working conditions. Nuhic et al. [16] used SVM to predict battery RUL by learning previous online capacity degradations independent of offline training data. Liu et al. [5] used relevance vector machine (RVM) combined with a gray model (GM) to improve the long-term prediction capability of RVM and RUL prediction accuracy, where only previous online capacity data was required to train the combined model. The machine learning methods [5,16] used data mining techniques to predict battery RUL rather than consider battery aging characteristics. Therefore, time-consuming and computation-intensive training was required for the SVM and RVM to mine all available online data for each RUL prediction, which can lead to inaccurate and unreliable prediction results. This paper developed a RUL prediction method based on battery aging characteristics analysis. The developed method required offline training data at one current rate (0.5C), and the training was conducted offline. The battery RUL was predicted online in a recursive way with a low computation burden. The rest of this paper is organized as follows: Section 2 introduces the acceleration
aging test of lithium-ion batteries. Sections 3 and 4 describe the theory and related algorithms for battery health diagnosis and RUL prognostics, respectively. Section 5 shows the battery capacity estimation and RUL prediction results using the developed methods, followed by the conclusions in Section 6. 2. Battery acceleration aging tests Fig. 1 shows the equipment used to conduct the acceleration aging tests for lithium-ion batteries. It includes an Arbin BT-5HC test system to charge/discharge batteries, three thermal chambers to control battery temperatures, and a computer for data monitoring and storage. The acceleration aging tests used the 18650 lithium-ion cells manufactured by Panasonic. The cell was labelled NCR18650BD, and its specifications are listed in Table 1. The cell had a rated capacity of 3 Ah. It consisted of Li(NiCoAl)O2 on the cathode and carbon on the anode. The allowed terminal voltage range was from 2.5 V to 4.2 V. The acceleration aging test conditions are summarized in Table 2. The cycle test had two test conditions, under which the cells were charged under a constant current (CC)-constant voltage (CV) process with a CC of 0.3C until the upper cut-off voltage of 4.2 V, then followed by a CV until the cut-off current of 0.02C. After a rest of 30 min, the cells were discharged at a CC until the lower cut-off voltage of 2.5 V. Under test condition 1, the cells were discharged at 0.5C, 1C, 1.5C, and 2C, respectively, and two cells were cycled at each current rate. The discharging current rates of lithium-ion batteries for EVs change dramatically. Therefore, one cell was discharged at different current rates at different cycles under test condition 2, where the cell was cycled at 0.5C, 1C, 1.5C, and 2C. Under this test condition, the discharging current was constant at one single cycle and changed to another rate at the next single cycle. The ambient temperature in the cycle test was set at 30 C by the thermal chamber. After the cells were cycled by 50 or 100 cycles, the characteristic tests including the capacity calibration test, the urban dynamometer driving schedule (UDDS) test, and the dynamic stress test (DST) were conducted to investigate the dynamic response characteristics of lithium-ion batteries. Note that the characteristic results were not used in this paper. 3. Battery health diagnosis and remaining useful life prognostics This section introduces the theory for realizing battery health diagnosis, aging stage identification, and remaining useful life prediction. The lithium-ion battery SOH, which is used to indicate
Fig. 1. Battery test equipment.
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3.2. Aging stage identification
Table 1 Cell specifications of the NCR18650BD battery. Rated capacity
3 Ah
Material Maximum continuous discharge current Allowed voltage range Temperature range Charge Discharge Storage
Li(NiCoAl)O2/carbon 10 A 2.5e4.2 V 10e45 C 20e60 C 20e50 C
the battery health, is represented by the present capacity divided by the initial capacity, which is expressed as:
SOH ¼
Cp $100% Ci
(1)
where Ci represents the initial capacity and Cp represents the present capacity [17]. 3.1. Health diagnosis Dubarry et al. [18] and Gao et al. [19] showed that the battery aging mechanisms can be identified by deriving the peak area from the incremental capacity (IC) curve of the cell. This peak area indicates the accumulated capacity from the peak that covers a specific voltage range. However, the IC curve is generally plotted at a very low charging current rate, typically at 1/20C, and the peak of the curve can disappear at a commonly used charging rate, typically at 0.3C. Besides, the peak of the IC curve can occur at a low voltage range which does not cover the commonly used voltage ranges of the lithium-ion batteries for EVs. Therefore, the incremental capacity analysis (ICA) is unable to be used to identify on-board capacities of lithium-ion batteries for EVs. The peak area of the IC curve is extracted to indicate the battery aging mechanisms based on the principle that the required capacity to make the same change on the battery voltage is different at different battery SOHs. Based on this principle, this paper extracted six accumulated capacities covering six voltage ranges from the charging voltage curve to indicate battery SOH. The charging current rate and voltage range are both commonly used values, which are, respectively, 0.3C and 3.6 Ve4.2 V. The six voltage ranges are 3.6e3.7 V, 3.7e3.8 V, 3.8e3.9 V, 3.9e4.0 V, 4.0e4.1 V, and 4.1e4.2 V, and the six accumulated capacities are extracted from each voltage range as HIs. Fig. 2 shows the feature extraction results of one cell at 0.5C. Fig. 2(a) shows the charging voltage curves at different cycles, which indicates that the charging time until the upper cut-off voltage shortens as the cell ages. Fig. 2(b) shows the extracted capacities at different voltage ranges, where the feature numbers in Fig. 2(b) correspond to the six voltage ranges from 3.6e3.7 V to 4.1e4.2 V. The extracted capacities at each voltage range generally decrease as the cell ages, and the cell at different SOHs shows different HI patterns. The relationship between the HI patterns and battery SOHs was constructed using machine learning techniques.
Fig. 3 shows the capacity degradation trajectories of cells under different test conditions. The capacity at each cycle was the discharging capacity at the corresponding discharging current rate. Fig. 3(a and b) show the capacity degradations under test condition 1, where Fig. 3(a) shows the capacity degradations at 0.5C and 1C, and Fig. 3(b) shows the capacity degradations at 1.5C and 2C. Fig. 3(c) shows the capacity degradations under test condition 2, where the capacity degradations at the specific discharging current rate of each cycle are presented. Fig. 3 shows that the capacity degradation trajectory can generally be divided into two stages, with a fast capacity degradation in the first stage and a slow capacity degradation in the second stage. This capacity degradation trajectory is similar to that in Ref. [20], where the same type of cell was cycled. The first stage covered about 100 cycles, and during which, the new battery started degrading, and the capacity degradation rate was relatively high. During the second stage, the capacity degradation rate decreased, and a nearly linear capacity degradation was observed. The aging mechanisms behind these two aging stages should be different [26], which however cannot be furhter identified based on the avaliable experimental data. For other commonly used commercial lithium-ion batteries, such as the LiMn2O4- or LiFePO4-based batteries, the capacity also degrades linearly near the battery EOL [21,22]. Fig. 3(b and c) show that the cell capacity at 1.5C or 2C degraded at a high rate during the second stage, which was mainly caused by the increasing internal resistance as the cell aged. The internal resistance caused a voltage drop with discharging current loaded on the cell. There was still capacity in the cell owing to this voltage drop when the cell discharged to the lower cut-off voltage. The higher the internal resistance and the discharging current rate were, the larger the voltage drop was, which led to more remaining capacities in the cell and thus less discharging capacities. Therefore, the capacity degardation rate at 1.5C or 2C was higher during the second stage than that at 0.5C or 1C owing to the increasing internal resistance as the cell aged. For the same reason, Fig. 3(c) shows that the differences of the discharging capacities at 1.5C or 2C between those at 0.5C or 1C increased as the cell aged, indicating an increasing capacity degrdation rate at 1.5C or 2C during the second stage owing to the increasing internal resistance. The battery capacity should be calibrated at a small discharging current rate, such as 0.5C or 1C, to reduce the influence of the increasing resistance as the battery ages on capacity calibration [23,24]. In this paper, the discharging capacity at 0.5C was used to indicate the present battery capacity in Eq. (1). 3.3. Remaining useful life prediction Section III. B shows that the battery EOL fell within the second degradation stage of capacities, which was a nearly linear degradation at 0.5C. Therefore, battery RUL can be predicted based on available capacities at the second stage using a linear aging model. When the battery degraded into the second stage, a linear aging
Table 2 Cell test conditions of the NCR18650BD battery. Test Category
Test Condition 1
Test Condition 2
Cycle tests
Charge: CCCV, charged at 0.3C rate up to 4.2 V, 0.02C rate cutoff Charge rest: 30 min Discharge: 0.5C, 1C, 1.5C and 2C rate, 2.5 V cutoff Discharge rest: 30 min Temperature: 30 C Capacity calibration-UDDS test-DST test
Charge: CCCV, charged at 0.3C rate up to 4.2 V, 0.02C rate cutoff Charge rest: 30 min Discharge: 0.5Ce1C-1.5Ce2C rate, 2.5 V cutoff Discharge rest: 30 min Temperature: 30 C Capacity calibration-UDDS test-DST test
Characteristic tests
3
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Fig. 2. Feature extraction results of one cell at 0.5C: (a) the charging voltage curves and (b) the extracted capacities at different cycles.
Fig. 3. Capacity degradation trajectories of cells at: (a) 0.5C and 1C; (b) 1.5C and 2C; (c) dynamic rate.
model was constructed based on the collected data at this stage. The input and output of the model were the cycle number and SOH. The linear model was extrapolated, and the battery EOL was predicted once the extrapolated SOH was lower than 80%. 4. Algorithms This section introduces related algorithms, including the neural network (NN), for battery health diagnosis, the random forest (RF) for battery aging stage identification, and the recursive least squares (RLS) algorithm with a forgetting factor for battery RUL prognostics. 4.1. Neural network The NN, which is able to build complex nonlinear relationships between inputs and outputs, was used to describe the relationship between the extracted HIs and battery capacities. The commonly used NN, whose structure is shown in Fig. 4, has three layers, including one input layer, one hidden layer, and one output layer. The input layer included six HIs in this paper, which were the extracted capacities from six voltage ranges and were represented by HI1 to HI6. The hidden layer included 50 sigmoid neurons [25], which were all input into the output layer including one linear neuron. The output layer calculated the battery capacity. 4.2. Random forest algorithm RFs are ensembles of tree-type classifiers, where each classifier is trained with a different subset of the training set (“bagging”), thereby improving the generalization ability of the classifier. Samples are processed along a path from the root to a leaf in each
Fig. 4. Structure of the neural network.
tree by performing a binary test at each internal node along this path. The binary test compares a certain feature with a threshold. Training a forest amounts to identifying the set of tests at each node that best separate the data into the different training classes. RF generally performs pattern classification problems better than NN [26] and was thus selected to identify the battery aging stage in this paper. In this paper, the inputs of the RF were six extracted HIs and the outputs were the corresponding aging stages. The RF used 100 trees to construct the relationship between the HIs and the corresponding aging stages [26e28].
4.3. Recursive least squares algorithm The RLS method is a recursive form of the standard least squares
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method, which inputs all historical data sets to construct a linear model [29]. RLS operates in an iterative way and only inputs the present data set to identify the model parameters, thus, it needs less memory space and is more computation-efficient in practice than the standard least squares method. A linear model based on capacity data at aging stage 2 (Fig. 3) can predict erroneous battery RULs owing to two problems: first, the capacity degradation at aging stage 2 is not strictly linear, and second, RF can erroneously identify starting cycles in stage 2 that actually belong in stage 1. The RLS method with a forgetting factor can construct a linear model by forgetting old data, and was thus used to identify the model parameters in this paper. The following linear model was constructed to capture the battery capacity degradation at stage 2:
SOHk ¼ b0 þ b1 $k þ εk ; εk N 0; s2
b q 0 ¼ EðqÞ
(3)
i h q 0 Þðq bq 0 ÞT P0 ¼ E ðq b
(4)
where q ¼ [b0, b1]T is the model parameter vector to be identified and P is the estimation-error covariance of q. In the paper, q0 ¼ [0, 0]T, and P0 ¼ 106I because no information about q was available at the beginning. I is a unity matrix. Step 2, for i ¼ 1, 2, …, update the estimate of q and the estimation-error covariance P as follows:
1 Ki ¼ Pi1 4Ti 4i Pi1 4Ti þ m
(5)
b q i ¼ bq i1 þ Ki ðyi 4i bq i1 Þ
(6)
1
ðI Ki 4i ÞPi1
5. Experimental results The experimental data at constant current rates, including 0.5C, 1C, 1.5C, and 2C, and dynamic current rates were used to evaluate the performance of the developed method. The simulation was performed based on Matlab 2016b, which operated on a laptop with an Intel Core i7-6700 HQ processor (6 MB cache, up to 3.50 GHz). 5.1. Health diagnosis
Step 1, initialize the estimator as follows,
m
relationship the starting cycles for stage 2 were identified. The capacity data in stage 2 was used to construct a linear aging model using RLS with a forgetting factor in step 3. This linear model with uncertainties sampled using MC simulation was extrapolated to predict the battery RUL PDF in step 4.
(2)
where k indicates the cycle number; b0 and b1 are model parameters to be identified, and εk are random errors that are independent and normally distributed with a zero mean and variance of s2. The linear model parameters are identified using RLS with a forgetting factor in the following two-step process [30].
Pi ¼
5
(7)
where K represents the estimator gain matrix; 4 ¼ [1, k] represents the model input vector, and y ¼ SOH represents the model output; m is the forgetting factor ranging from 0 to 1, and m ¼ 1 means forgetting no old data. Once the model parameters were identified using RLS, the future battery SOH could be predicted by extrapolating Eq. (2), and the battery EOL was obtained if the predicted SOH at cycle k was lower than 0.8. The EOL prediction uncertainties were generated using Monte Carlo (MC) simulation. The model uncertainties existed in random errors εk, from which a total of 103 samples were sampled. Eq. (2) was extrapolated to predict one battery EOL with one sampled ε value. At each cycle, there were 103 predicted EOLs which generated the EOL probability distribution function (PDF). Fig. 5 shows four steps for battery health diagnosis and RUL prediction. In the first step, the NN was used to build a relationship between the extracted battery HIs and capacities, and the battery SOH was diagnosed based on on-board battery data and this relationship. Step 2 used RF to build a relationship between the extracted HIs and the corresponding aging stages, and based on this
Fig. 6 shows the estimated capacities at different cycles using the health diagnosis method. The NN was trained to construct a relationship between the HIs and capacities based on the aging data of one cell at each test condition, and then the trained NN was used to estimate the capacity data of another cell at the same test condition. Herein, the capacities were obtained at the corresponding discharging current rate. Fig. 6 shows the capacity estimation results of cells at four discharging current rates including 0.5C, 1C, 1.5C, and 2C. The estimated capacities were all close to the real values with the estimation errors mostly within 1%. The capacity estimation errors at 0.5C, for example, were close to zero and within 1% at most cycles. The root mean square errors (RMSEs) of capacity estimation at 0.5C, 1C, 1.5C, and 2C were 0.40%, 0.53%, 0.63%, and 0.53%, respectively. In practice, the extracted HIs from the charging voltage curve of lithium-ion batteries for EVs are input to an NN, which is trained based on the acceleration aging test data at 0.5C, to estimate the onboard capacities. The test data at dynamic discharging current rates was used to verify the capacity estimation performance of the NN for lithium-ion batteries for EVs. Fig. 7 shows the capacity estimation results based on the test data at dynamic discharging current rates, where an NN was constructed based on the test data of two cells at 0.5C to describe the relationship between the HIs and capacities. The HIs were extracted from the charging voltage curve at each cycle of the test data at dynamic current rates and input to this NN to estimate capacities. The capacities at 0.5C of the test data at dynamic current rates were considered reference data in Fig. 7(a), where the capacities at 1C, 1.5C, and 2C were all replaced by the interpolated values based on the capacities at 0.5C. Fig. 7(a) shows that the estimated capacities were close to the reference data with capacity estimation errors within 1% at most cycles (Fig. 7(b)), and the RMSE was 0.80%, indicating the high on-board capacity estimation accuracy and feasibility of the method for lithium-ion battery capacity estimations for EVs. 5.2. Aging stage identification In order to identify the starting cycle of aging stage 2 of capacities at 0.5C, a linear model was used to fit the capacities of one cell at 0.5C from different cycles using Matlab's curve fitting tool. Table 3 shows the fitting results, where the performance was indicated by R-square and RMSE. R-square is the square of the correlation between the real values and estimated values, and can evaluate the linear correlation degree of capacities. R-square can take on any value between 0 and 1, with a value closer to 1 indicating a stronger linear degradation of capacities. RMSE indicates the fitting accuracy. Table 3 shows that the R-square increases from 0.9956 to 0.9978 with the starting cycle number increasing from 60
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Fig. 5. Steps for battery SOH diagnosis and RUL prognostics.
Fig. 6. Capacity estimation results based on the NN method at: (aeb) 0.5C; (ced) 1C; (eef) 1.5C; (geh) 2C.
Fig. 7. Capacity estimation results based on the NN method at dynamic current rates.
degradation rate was large at stage 1 with a degradation rate close
Table 3 Fitting results of capacities from different cycles at 0.5C Starting cycle
R-square
RMSE ( 103 Ah)
60 80 100 120 160
0.9956 0.9965 0.9978 0.9976 0.9973
8.83 7.33 5.49 5.38 5.42
to 100, whereas the RMSE decreases from 8.83 103 Ah to 5.49 103 Ah. After 100 cycles, the R-square decreases as the cycle number increases, whereas the RMSE decreases a little from 5.49 103 Ah to 5.38 103 Ah, then increases to 5.42 103 Ah. Therefore, the 100th cycle was considered the starting cycle of stage 2 of capacity degradations at 0.5C. Fig. 8 shows the average capacity degradation rate per cycle before and after 100 cycles of the cell at 0.5C. The capacity
Fig. 8. Average capacity degradation rate per cycle at 0.5C.
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to 0.25%, whereas at stage 2, the capacity degradation rate was low and close to 0.1%. The test data of the cell at 0.5C with identified aging stages was used to train the RF model. The input was the extracted HIs at each cycle, and the output was the stage pattern, with 0 indicating stage 1 and 1 indicating stage 2. The trained RF model was then used to identify the stages of capacities of other cells at different C rates. The test data of one cell at each rate was used to verify the RF model. Fig. 9 shows the stage identification results of capacities at 0.5C (Fig. 9(a)), 1C (Fig. 9(b)), 1.5C (Fig. 9(c)), and 2C (Fig. 9(d)), where the identification results using the NN method are also shown for comparison. Failed classifications indicated cycles that failed to be classified to the right stages. Fig. 9 shows that the RF always identified aging stages more efficiently than the NN with fewer failed classifications. Fig. 9(b), for example, shows that the NN failed to classify 40 cycles' data with varying pattern values between 0 and 1, whereas the RF failed to classify 1 cycle's data. The identified starting cycle number of stage 2 of capacities at 0.5C, 1C, 1.5C, and 2C using RF were, respectively, 105, 110, 104, and 129. These started cycle numbers for cells at 0.5C, 1C, and 1.5C varied between 100 and 110, whereas the cycle number of the cell at 2C was 129, indicating slower capacity aging at stage 1 and 2C than other C rates. Fig. 10 shows the average capacity degradation rate per cycle at different current rates. The capacity degradation rates of stage 1 were larger than those of stage 2 at all current rates. The degradation rate of stage 1 at 2C was the lowest and close to 0.17%, whereas the degradation rates at 0.5C, 1C, and 1.5C were all higher than 0.2%. This lower capacity degradation rate at 2C than other C rates led to a later starting cycle of stage 2 in Fig. 9. The capacity degradation rate of stage 2 at 2C was the largest owing to the increasing internal resistance as the cell aged, whereas the capacity degradation rates at 0.5C, 1C, and 1.5C were all close to 0.1%. Fig. 11 shows the aging stage identification results at dynamic current rates, where an RF model was trained based on the test data of two cells at 0.5C. The extracted HIs from the charging voltage curve at each cycle of the cell at dynamic current rates were input to
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Fig. 10. Average capacity degradation rate per cycle of aging stage 1 and 2 at different current rates.
Fig. 11. Aging stage identification results of the cell at dynamic current rates.
the trained RF to identify the starting cycle number of stage 2 of capacities, which was 107 with 3 misclassified cycles.
Fig. 9. Aging stage identification based on RF and NN at: (a) 0.5C; (b) 1C; (c) 1.5C; (d) 2C.
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5.3. Remaining useful life prediction The RLS with a forgetting factor was used to predict the battery RUL based on the on-board capacity estimation and aging stage identification results. The RLS started when 50 cycles’ capacity data from the starting cycle of stage 2 was collected, and continued to predict battery RUL every 10 cycles. Fig. 12 shows the battery RUL prediction results at different current rates using RLS with different forgetting factors, including m ¼ 1, 0.99, 0.97, 0.95, and a variable forgetting factor. The variable forgetting factor was defined as
mi ¼ ami1 þ ð1 aÞ
(8)
where a determines the change rate of forgetting factor and i indicates the sampling moment. In this paper, a ¼ 0.995 and m0 ¼ 0.95. Fig. 13 shows the variable m trajectories with different a values. m converges faster to 1 with a smaller a value. The upper and lower error thresholds in Fig. 12 represented RUL values that were, respectively, 50 cycles larger and smaller than the real data. RUL predictions were considered accurate once they fell within the error thresholds. Tables 4 and 5 lists the RUL prediction performance at different current rates and m values, where the starting cycle indicates when the predicted RUL fell within the error thresholds, and the RMSEs of RUL predictions were calculated after this starting cycle. In Tables 4 and 5, overstriking indicates the earliest starting cycle and lowest RMSE of RUL predictions at each current rate. RUL predictions with a smaller m captured the newest information more efficiently and were thus less robust against data variations. For this reason, Fig. 12 shows that RUL predictions with m ¼ 0.95 varied most strongly at each current rate. Table 5 shows that RMSEs with m ¼ 0.95 were generally the largest at each current rate except the one at 1.5C, which was the lowest. This lowest RMSE was mainly caused by small variations of the estimated capacity data at 1.5C rate after 300 cycles (Fig. 6(e)). Fig. 12 shows that stable RULs with m ¼ 1 and 0.99 were predicted, and the RMSEs of RUL predictions at each current rate except 1.5C were generally the
Fig. 13. Variable m trajectories at different a values.
Table 4 Starting cycle of RUL predictions at different current rates and m values. Current Rate/m value
1
0.99
0.97
0.95
Variable
0.5C 1C 1.5C 2C
155 180 374 299
155 180 344 289
155 180 314 279
155 180 294 269
155 180 324 279
Table 5 RMSE of RUL predictions at different current rates and m values. Current Rate/m value
1
0.99
0.97
0.95
Variable
0.5C 1C 1.5C 2C
10 14 27 17
10 13 21 18
12 14 17 23
17 19 16 27
11 14 16 24
lowest in Table 5. At 1.5C, the estimated capacities in the beginning
Fig. 12. RUL prediction results based on RLS method with different m values at: (a) 0.5C; (b) 1C; (c) 1.5C; (d) 2C.
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Fig. 14. RUL prediction results at dynamic current rates using RLS with a variable forgetting factor.
of stage 2 degraded at a lower rate than the capacities did later. Therefore, the RLS with a large m, which greatly depended on data in the beginning, always predicted larger RULs than those using the RLS with a small m. These large predicted RULs caused the large RMSEs of predictions with m ¼ 1 and 0.99 at 1.5C. For the same reason, the RUL predictions with m ¼ 1 and 0.99 fell within the error thresholds at larger cycles than those with other m values did at 1.5C and 2C. The RLS with m ¼ 0.95, which forgot the data in the beginning quickly, captured the capacity variations efficiently and thus predicted RULs falling within the error thresholds early. To improve the prediction performance, the RLS was expected to possess a small forgetting factor in the beginning for capturing capacity degradations efficiently and a large forgetting factor later for robust RUL predictions against capacity variations. The RLS with a variable forgetting factor increasing from 0.95 in the beginning to 1 in the end was thus developed. Table 5 shows that the RLS with a variable m generally predicted one of the lowest RMSEs at each current rate, and predicted RULs falling within the error threshold at one of the earliest starting cycles at each current rate. RUL predictions with a variable m fell into the error threshold at a starting cycle number that was, respectively, 50 and 20 at 1.5C, and 20 and 10 at 2C smaller than those with m ¼ 1 and 0.99 at the corresponding current rate. The RLS with a variable forgetting factor was used to predict the battery RUL at dynamic current rates owing to its high prediction performance. Fig. 14 shows the RUL prediction results at dynamic current rates using the RLS with a variable m. The RUL was predicted based on the on-board estimated capacities in Fig. 7. Fig. 14 shows that the predicted RUL fell within the error thresholds after 200 cycles, and the RUL prediction errors at most cycles were less than 10 after 300 cycles. The RUL PDF was generated using MC simulation. The 95% confidence interval of RUL predictions at 257, 357, and 457 cycles were, respectively, [265, 285], [181, 197], and [88, 108] cycles, indicating precise RUL predictions. The calculation time for each RUL prediction was around 0.3 s, indicating a low on-board computation burden. 6. Conclusions Lithium-ion battery health for electric vehicles (EVs) must be monitored and prognosed in real time for reliability evaluation and maintenance purposes. This paper aims at developing practical and effective battery state of health (SOH) diagnostic and remaining useful life (RUL) prognostic methods based on the aging characteristics analysis. Battery health was indicated by six health indicators (HIs), which were extracted from the charging voltage curve within a commonly used range. A neural network (NN) and random forest
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(RF) model were constructed to describe the relationships between the HIs and battery capacities, and the HIs and aging stages of capacity degradations, respectively, based on offline training data. The battery capacities and aging stages at different current rates were estimated and identified online by inputting the extracted battery HIs into the constructed NN and RF, respectively. Battery RUL was predicted based on the capacity estimation and aging stage identification results. The recursive least squares (RLS) method with a forgetting factor predicted the battery RUL based on available on-board capacity data of aging stage 2, combined with Monte Carlo (MC) simulation to generate the RUL probability distribution function (PDF). Experimental data for 9 cells at constant current rates, including 0.5C, 1C, 1.5C, and 2C, and dynamic current rates was used to validate the developed methods for battery SOH diagnosis and RUL prognostics. Experimental results showed that the capacity estimates were accurate, with estimation errors less than 1% at most cycles at both constant current rates and dynamic current rates. Based on the capacity estimation and aging stage identification results, the RLS method with a variable forgetting factor predicted accurate RULs at an early life stage. Accurate and precise RULs can be predicted at dynamic current rates after 200 cycles, with the prediction errors at most cycles less than 10 after 300 cycles and the 95% confidence intervals at 257, 357, and 457 cycles covering about 20 cycles for each prediction. The calculation time for each RUL prediction was about 0.3 s. The experimental results at dynamic current rates indicate that, in practice, the developed methods can predict on-board battery SOH and RUL for EVs accurately with a low computation burden. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 51877009) and Beijing Municipal Natural Science Foundation (Grant No. 3182035). The systemic experiments of the lithium-ion batteries were performed at the Advanced Energy Storage and Application (AESA) Group, Beijing Institute of Technology. References [1] Xiong R, Zhang Y, He H, Zhou X, Pecht M. A double-scale, particle-filtering, energy state prediction algorithm for lithium-ion batteries. IEEE Trans Ind Electron Feb. 2018;65(2):1526e38. [2] Deng Z, Yang L, Cai Y, Deng H, Sun Liu. Online available capacity prediction and state of charge estimation based on advanced data-driven algorithms for lithium iron phosphate battery. Energy Sep. 2017;201:257e69. [3] Bartlett A, Marcicki J, Onori S, Rizzoni G, Yang XG, Miller T. Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery. IEEE Trans Control Syst Technol Mar. 2017;24(2): 384e99. [4] Farmann A, Waag W, Marongiu A, Sauer DU. Critical review of on-board capacity estimation techniques for lithium-ion batteries in electric and hybrid electric vehicles. J Power Sources May. 2015;281:114e30. [5] Liu D, Zhou J, Liao H, Peng Y, Peng X. A health indicator extraction and optimization framework for lithium-ion battery degradation modeling and prognostics. IEEE Transactions on Systems, Man, and Cybernetics: Systems Jun. 2015;45(6):915e28. [6] Li X, Jiang J, Wang LY, Chen D, Zhang Y, Zhang C. A capacity model based on charging process for state of health estimation of lithium ion batteries. Appl Energy Sep. 2016;177:537e43. [7] Wang L, Pan C, Liu L, Cheng Y, Zhao X. On-board state of health estimation of LiFePO4 battery pack through differential voltage analysis. Appl Energy Apr. 2016;168:465e72. [8] Hu C, Jain G, Schmidt C, Strief C, Sullivan M. Online estimation of lithium-ion battery capacity using sparse Bayesian learning. J Power Sources Sep. 2015;289:105e13. [9] He W, Williard N, Osterman M, Pecht M. “Prognostics of lithium-ion batteries based on DempstereShafer theory and the Bayesian Monte Carlo method. J Power Sources Dec. 2011;196(23):10314e21. [10] Miao Q, Xie L, Cui H, Liang W, Pecht M. Remaining useful life prediction of lithium-ion battery with unscented particle filter technique. Microelectron
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