Remaining useful life assessment of lithium-ion batteries in implantable medical devices

Journal of Power Sources 375 (2018) 118–130

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Remaining useful life assessment of lithium-ion batteries in implantable medical devices

T

Chao Hua,b,∗, Hui Yec, Gaurav Jainc, Craig Schmidtc a

Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA c Medtronic Energy and Component Center, Brooklyn Center, MN 55430, USA b

H I G H L I G H T S data-driven/model-based method to lithium-ion battery prognostics is proposed. • AIt employs sparse Bayesian learning (model-based) to infer capacity from features. • It adopts particle filters (data-driven) to predict remaining useful life (RUL). • RUL prediction involves the use of single or multiple capacity fade models. •

A R T I C L E I N F O

A B S T R A C T

Keywords: Capacity Health monitoring Prognostics Remaining useful life Lithium-ion battery

This paper presents a prognostic study on lithium-ion batteries in implantable medical devices, in which a hybrid data-driven/model-based method is employed for remaining useful life assessment. The method is developed on and evaluated against data from two sets of lithium-ion prismatic cells used in implantable applications exhibiting distinct fade performance: 1) eight cells from Medtronic, PLC whose rates of capacity fade appear to be stable and gradually decrease over a 10-year test duration; and 2) eight cells from Manufacturer X whose rates appear to be greater and show sharp increase after some period over a 1.8-year test duration. The hybrid method enables online prediction of remaining useful life for predictive maintenance/control. It consists of two modules: 1) a sparse Bayesian learning module (data-driven) for inferring capacity from charge-related features; and 2) a recursive Bayesian filtering module (model-based) for updating empirical capacity fade models and predicting remaining useful life. A generic particle filter is adopted to implement recursive Bayesian filtering for the cells from the first set, whose capacity fade behavior can be represented by a single fade model; a multiple model particle filter with fixed-lag smoothing is proposed for the cells from the second data set, whose capacity fade behavior switches between multiple fade models.

1. Introduction

longevity, refers to the available service time left before the capacity fade reaches an unacceptable level [4]. Accurately tracking these parameters allows battery management system (BMS) to perform predictive maintenance/control of a cell through concurrent estimation of the cell SOH (diagnostics) and prediction of the cell RUL (prognostics). Recent literature reports a variety of approaches to estimating the capacity of a Li-ion battery cell in operation. In general, these approaches can be categorized into 1) adaptive filtering approaches [1,2,5–11], 2) coulomb counting approaches [12–15], 3) neural network approaches [16–18], and 4) kernel regression approaches [19–21]. The capacity estimation of a cell by most of these existing approaches only requires readily available measurements (i.e., voltage, current and temperature) acquired from the cell. A more recent

Lithium-ion (Li-ion) batteries are widely used in consumer electronics, such as cell phones and laptops, and in transportation applications, such as hybrid and electric vehicles. Recently, Li-ion batteries have found use in implantable medical devices such as neurostimulators for the relief of chronic pain and deep brain stimulators for the treatment of Parkinson's disease. As a Li-ion battery cell ages, the decrease of capacity and the increase of internal resistance degrade the electrical performance of the cell by means of energy and power losses [1]. Capacity, which quantifies the total amount of energy stored in a fully charged cell, is an important indicator of the state of health (SOH) of the cell [1–3]; remaining useful life (RUL), also called remaining ∗

Corresponding author. 2026 Black Engineering Bldg., Iowa State University, Ames, IA 50011, USA. E-mail addresses: [email protected], [email protected] (C. Hu).

https://doi.org/10.1016/j.jpowsour.2017.11.056 Received 24 August 2017; Received in revised form 11 November 2017; Accepted 17 November 2017 0378-7753/ © 2017 Elsevier B.V. All rights reserved.

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medical device, it is very important to be able to track the capacity fade of the battery and assess its RUL throughout the lifetime. This can provide information to the patient and his/her health care provider regarding when and how a replacement of the device might be needed. The information could be crucial for ensuring device operation and minimizing therapy interruptions. The need for predictable capacity fade models and ability to predict RUL is particularly significant given long targeted lifetime of implanted devices (typically 10 years; up to 25 years in some cases) and large variation in use conditions (many cycles versus few cycles over a certain calendar time) depending on the therapy needs. Examples of implantable medical devices that may be powered by a Li-ion battery include neurological stimulators, spinal stimulators, cardiac stimulators such as pacemakers and defibrillators, and diagnostic devices such as cardiac monitors. In this paper, a hybrid data-driven/model-based method is employed for online RUL assessment of Li-ion batteries in implantable medical devices. The hybrid method integrates sparse Bayesian learning (data-driven) with recursive Bayesian filtering (model-based) to enable real-time inference of capacity from charge-related features and prediction of RUL from recursive updating and extrapolation of capacity fade models. A generic particle filter is adopted to implement recursive Bayesian filtering for batteries whose capacity fade behavior can be represented by a single fade model, and a multiple model particle filter (MMPF) with fixed-lag smoothing is proposed for batteries whose capacity fade behavior switches between multiple fade models. The effectiveness of the proposed method is demonstrated by leveraging daily cycling data from eight fresh cells from Medtronic, PLC (hereafter referred to as the MDT cells) as well as eight fresh/post-explant cells from Manufacturer X (hereafter referred to as the Mfg. X cells). A large difference in the capacity fade behavior is seen between the two manufacturers' data sets used in this paper and hence these data sets serve as excellent test cases for developing robust prognostic techniques applicable to wide variety of fade characteristics. The reminder of this paper is organized as follows. Section 2 presents the fundamentals of the proposed method. The method is applied to online capacity estimation and RUL prediction of Li-ion batteries used in implantable applications. Section 3 discusses the experimental results of this application. The paper is concluded in Section 4.

development in the kernel regression category was a sparse capacity estimator based on sparse Bayesian learning [22], and the estimator is a kernel regression model that approximates the relationship between the capacity of a battery cell and five characteristic features extracted from a capacity versus voltage function, Q(V), or charge curve [22,23]. The capacity estimator, as a highly sparse regression model, was applied to infer the capacity of Li-ion battery designed for use in implantable medical devices, and achieved satisfactory estimation accuracy on lab and post-explant Li-ion cells cycled with a nominally weekly discharge rate [22]. Extensive research has been conducted on RUL assessment of a general engineered system with an emphasis on modeling the RUL distribution. In general, three categories of approaches have been developed that enable continuous updating of system health condition and RUL distribution: (i) model-based approaches [24–26], (ii) datadriven approaches [27–29], and (iii) hybrid approaches [30,31]. These approaches, although not developed specifically for Li-ion battery prognostics, can generally be adapted for RUL assessment of Li-ion battery. Research devoted to developing new approaches for Li-ion battery prognostics was mainly conducted by researchers in the prognostics and health management (PHM) society. A Bayesian framework with particle filter was proposed for RUL prediction of Li-ion battery based on impedance measurement and by updating an empirical capacity fade model that employs a single exponential function [32]. A similar attempt with impedance measurement was later made with the use of recurrent neural network [33]. In order to eliminate the reliance of battery prognostics on impedance measurement equipment, researchers developed various model-based approaches that predict RUL by extrapolating a capacity fade model [14,34–39]. An empirical capacity transition model was created to capture the degradation (via the use of coulombic efficiency) and self-recharge (via the use of an exponential function) of a battery cell, and the capacity transition model was updated using particle filter for RUL prediction [34]. Two interesting attempts on battery prognostics were made to improve the accuracy of the single exponential function in capacity fade modeling [35,36]. The first attempt developed a new empirical model consisting of two exponential functions and applied the new model to enable accurate RUL prediction with particle filter [35]. The second attempt employed relevance vector machine (RVM) to assist an empirical model (i.e., a sum of exponential and power functions) with accurately representing the capacity fade behavior of Li-ion battery [36]. Particle filter often directly treats the transition prior (i.e., without the use of prior measurements) as the proposal importance density used for drawing new particles. This treatment makes the implementation of particle filter convenient and computationally efficient but may cause a rapid loss of particle diversity, known as particle degeneracy. To mitigate particle degeneracy and ensure effective model updating and accurate RUL prediction, researchers have made attempts to derive better proposal importance densities by incorporating recent measurements of cell capacity. These attempts generated proposal importance densities by employing unscented Kalman filter [37], Gauss-Hermite Kalman filter [14], and spherical cubature integration-based Kalman filter [38]. In particular, the integration of Gauss-Hermite Kalman filter with particle filter resulted in the so-called Gauss-Hermite particle filter, which was applied to predict the RUL distribution of implantable Li-ion battery [14]. This model-based prognostics approach produced accurate prediction of how long a Li-ion battery cell will perform in an implantable application before the cell capacity fades to an unacceptable level. More recently, sigma-point Kalman filter was proposed to update empirical capacity fade models for RUL prediction in the presence of additive Gaussian (process and measurement) noises [39] whose variances may differ from one cell to another. Under this Gaussian assumption, the use of particle filter that would require more computational effort than Kalman filter was shown to be unnecessary and may lead to less accurate RUL prediction [39]. When a Li-ion battery is used as the power source in an implantable

2. Technical approach Given the current and voltage signals measured from a cell operating under a typical use condition and a discrete-time state space model that describes the capacity fade behavior of the cell, we aim at estimating the capacity of the cell at every charge/discharge cycle and predicting its RUL, i.e., how long the cell is expected to operate before its capacity falls below an unacceptable level (or a capacity threshold). The subsequent sections present our proposed prognostic method to accomplish this online task. As shown in Fig. 1, the proposed prognostic method consists of two essential modules: 1) sparse Bayesian learning, which automatically learns (from a training data set) a mapping from charge-related features to capacity measurement; and 2) recursive Bayesian filtering, which recursively updates an empirical capacity fade model with the capacity measurement and extrapolates the model for prediction of RUL. In what follows, the two modules will be explained in further detail. Section 2.1 describes the sparse Bayesian learning scheme for capacity estimation; and Section 2.2 presents the recursive Bayesian filtering technique for RUL prediction. 2.1. Sparse Bayesian learning of capacity (module 1) Sparse Bayesian learning or RVM [22,40] will be employed to train a sparse capacity estimator that learns the complex mapping from the feature (z) space to the capacity measurement (y) space (see Fig. 2). Suppose we have a training data set {zj, yj}, j = 1, 2, …, M, consisting of M input-output pairs from training cells. The sparse measurement 119

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Fig. 1. Flowchart of the proposed method for Li-ion battery prognostics.

predictions of RUL to the fade behavior of these individual cells. 2.2.1. Capacity fade models This study involves the use of two empirical models, denoted as Models 1 and 2, that can be used to represent the capacity fade behavior of Li-ion battery. These two models are described as follows. Model 1 (Exponential/linear hybrid model): Model 1 assumes capacity loss can be expressed as a hybrid of exponential and linear functions of cycle number, and takes the following form

ck =

estimator can be built in the form of a linear kernel regression model to approximate this data set, expressed as [40]. M

∑ ωj K (ztk , zj) = ωΤK j=0

(2)

where Ck is the capacity at the kth charge/discharge cycle, C0 is the initial capacity, α is the coefficient of the exponential component of capacity fade, λ is the exponential capacity fade rate, β is the coefficient of the linear component of capacity fade, and ck is the normalized capacity at the kth cycle. It was reported that the exponential function captures the active material loss [41], and the hybrid of linear and exponential functions was shown to provide a good fit to up to 10 years' cycling data acquired from LiCoO2/graphite cells [22,42]. In recursive Bayesian filtering with Model 1, we treat the parameters of the fade model, α, λ, and β, as the state variables, x, and recursively update the posterior distribution of x with the capacity measurement, y (see Section 2.2.2). Model 2 (Linear model): Model 2 is of a simpler form than Model 1, and assumes capacity loss has a linear relationship with the cycle number. In this model, the normalized capacity ck of a battery cell at the cycle k is formulated as

Fig. 2. Approximation of mapping from feature space to state space by sparse Bayesian learning.

yk ≈ h (ztk ) =

Ck = 1 − α [1 − exp(−λk )] − βk C0

(1)

zkt

is the feature vector from the kth charge cycle of a testing cell where whose capacity is unknown and needs to be estimated, ω = (ω0, …, ωM)T is a kernel weight vector, and K(zkt, zj) is a kernel function. Training the model in Eq. (1) using sparse Bayesian learning determines the posterior distribution of the weight vector ω. The posterior mean vector and covariance matrix of the weights can be respectively expressed as: μω = σ−2ΣΦy and Σω = (σ−2ΦTΦ+A) −1, where y = (y1, …, yM)T, Φ is an M × M design matrix constructed with the training vectors, and Φij = K(zi, zj), and A is a diagonal matrix with the diagonal entries being M+1 hyper-parameters. In this study, the output of a RVM regression model is the capacity measurement y of a battery cell and the inputs are five charge-related features z of the cell [22]. The capacity measurement (both the mean and standard deviation of the estimate) will be fed into recursive Bayesian filtering, which updates the parameters of capacity fade models using the measurement.

ck = γ − βk

(3)

where γ and β are the intercept and negative of the slope of the linear model, respectively. This model is capable of representing that linear fade behavior, where capacity decreases linearly with cycle. In Model 2, the model parameters that need to be updated are γ and β, and these are considered as the state variables, x. 2.2.2. State estimation problem and generic particle filter To make our discussion more concrete, let us consider the following discrete state-space model

2.2. Recursive Bayesian filtering of fade model parameters (module 2) Upon the estimation of capacity measurements, the proposed method employs a recursive Bayesian filtering technique, namely particle filters, to recursively update the parameters of empirical capacity fade models and extrapolate the updated models for prediction of RUL [14]. The updating of a capacity fade model enables adapting the model to track the fade trends of individual battery cells and tailoring the

Transition: x k = f (x k − 1) + uk Measurement: yk = g (x k − 1) + vk

(4)

where xk is the vector of system states at the kth measurement step, uk is the vectors of process noise for states, yk is the vector of system observations (or measurements), vk is the vectors of measurement noise, 120

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and f(•) and g(•) are the state transition and measurement functions, respectively. The objective of the tracking task in Eq. (4) is to recursively estimate the system states x from the noisy measurements y. More specifically, we aim to obtain filtered estimates of xk at the current step (k) based on all available measurements up to the step, y1:k. In our task of battery prognostics, we treat the fade model parameters (i.e., α, λ and β in Model 1 or γ and β in Model 2) as the system states x, and the normalized capacity, y, as the system measurement. The system transition and measurement functions can then be rewritten for Model 1 as [14].

according to the dynamics described by the state transition equations in Eq. (4) (e.g., Eq. (5) for Model 1 and Eq. (6) for Model 2 in the task of battery prognostics). The procedure consists of the following steps.

• Particle Initialization: A set of N •

Transition: αk = αk − 1 + u1, k , λk = λk − 1 + u2, k , βk = βk − 1 + u3, k Measurement: yk = 1 − αk [1 − exp( −λk k )] − βk k + vk  

•

ck

(5) and for Model 2 as

Transition: γk = γk − 1 + u4, k , βk = βk − 1 + u5, k Measurement: yk = γk − βk k + vk  ck

•

(6)

Here, yk is the capacity measurement (or the mean of the estimate by the sparse capacity estimator) at the kth charge/discharge cycle, and u1, u2, u3, u4, u5 and v are the Gaussian noise variables with zero means, and the standard deviation of v is the standard deviation of the capacity estimate by sparse Bayesian learning (see Section 2.1). The inference tasks described in Eqs. (5) and (6) are to learn on the fly the capacity fade behavior of a cell by tracking the evolution of the fade parameters (or states), x, based on the capacity estimate (or measurement), y, from the cell. In a Bayesian framework, the posterior probability distribution functions (PDF) of the states given the past measurements, p(xk | y1:k), constitutes a statistical solution to this inference problem and properly captures the uncertainty of the states. Recursive Bayesian filtering enables a continuous update of the posterior PDF with new measurements. It is often difficult or even impossible to derive an analytical form of the updated posterior p(xk | y1:k). It is, however, easier to draw samples (or particles) from the true posterior (or a distribution proportional to the posterior) and devise an approximation to the posterior based on the particles. This is the basic idea of particle filters, which implement recursive Bayesian filtering by sequential Monte Carlo simulation [43,44]. In particle filters, the state posterior is built based on a set of particles and their associated weights, expressed as

p (x k y1: k) ≈

2.2.3. Fixed-lag multiple model particle filter The multiple model particle filter (MMPF) has been proposed in several earlier studies [45–47] on nonlinear filtering with switching dynamic models. In these studies, the MMPF aimed at addressing hybrid state estimation problems where the (augmented) state vector consists of both a vector of state variables (e.g., the fade model parameters in the task of battery prognostics) and a vector of model variables (e.g., the fade model in effect). Let us define the augmented state vector as ψk = [xkT, mk]T, where mk ∈ {1, 2, …, S} with S being the number of candidate fade models (S = 2 in this study). The main difference between the MMPF and the generic particle filter lies in that the former allows for the use of multiple transition/measurement models that better captures the complex dynamics of a system than the use of a single model. At each measurement step, a transition/measurement model is weighed according to how well the observations predicted by the model match the actual measurements. For example, if the fade model mk at the kth charge/discharge cycle matches the best with the capacities measured from a battery cell, then the fade behavior of the cell is deemed to be best represented by the model mk at cycle k. Given the past and current measurements y1:k at step k and D future measurements yk+1:k+D, a fixed-lag smoothing step estimates the posterior distribution of the states p(xk | y1:k+D) [48,49]. Here, D is a fixed time delay that is often predefined based on prior knowledge about the system dynamics. The underlying assumption of the smoothing step is that measurements at future steps provide additional information about the current states xk and can thus help build a more accurate approximation of xk. In our task of battery prognostics, the main objective of updating the parameters of a fade model is not to match the modelpredicted capacity at the current cycle with the actual measurement. Rather, it is for the updated model to capture the capacity fade trend including and beyond the current cycle (i.e., the model shall match the capacity measurements at multiple cycles starting from the current cycle). To properly update and evaluate multiple fade models for battery prognostics, this study integrates the fixed-lag smoothing step with the MMPF, and the procedure of the resulting fixed-lag MMPF is described explicitly in the pseudo-code in Table 2. The basic ideas of the fixed-lag MMPF procedure are that 1) the posterior distribution of the states x at step k is estimated using measurements up to a future step k + D, and 2) a resampling step is used to reassign particles to different

NP

∑ j=1 wkj δ (xk − xkj)

{x kj} j = 1: NP

(7)

{wkj} j = 1: NP

and are respectively the particles and weights where estimated at the kth measurement step, NP is the number of particles, and δ is the Dirac delta function. If the particles are drawn from a proposal importance density q(xk | xk–1, yk), the weight of the jth particle at the kth step takes the following form

wkj ∝ wkj− 1

p (yk x kj) p (x kj x kj − 1) q (x kj x kj − 1, yk)

P initial particles are randomly drawn from an initial probability distribution of the states x, p0(x), and these particles are assigned equal weights 1/NP (Line 1). State Transition: Each of the particles is evolved forward independently for the time period between the (k–1)th and kth measurements according to the state transition equations in Eq. (4) (Line 5). The evolved particles form the prior distribution of the states at the kth step before the measurement at the step is used. Weight Evaluation and Normalization: The weight of each particle is computed based on its previous weight and its likelihood at the current step (Line 6). The likelihood of each particle is calculated based on the model-predicted distribution of measurement (see the measurement equation in Eq. (4)) and the actual measurement. The particle weights are normalized such that their sum equals to 1 (Lines 8, 10 and 11). Resampling: Particles having high importance weights are multiplied and particles having low importance weights are eliminated (Line 13). As a result, the filter draws more particles from the ones that fits better to the measurement. In the task of battery prognostics, these resampled particles are used as the updated parameters of a fade model at the current step (k) for prediction of RUL (see Section 2.2.4).

(8)

For the purpose of convenient implementation, the so-called transition prior, p(xk | xk–1), is often used as the proposal importance density, i.e., q(xk | xk–1, yk) = p(xk | xk–1). Substituting this transition prior, q(xk j | xk–1j, yk) = p(xk j | xk–1j) into Eq. (8) yields a simplified form of the weight, wkj ∝ wk–1j p(yk | xkj), which is the most common choice of weighting scheme. The pseudo-code of a generic particle filter with a single measurement model is summarized in Table 1. The basic ideas of the filtering procedure are that 1) the posterior distribution of the states x at any time can be approximated with a finite number (NP) of Monte Carlo samples (or particles) and their associated weights, and 2) the chronological behavior of these particles can be modeled sequentially 121

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Table 1 Pseudo-code of generic particle filter (single transition/measurement model).

the smoothing are multiplied and particles having low importance weights are eliminated (Line 20), and as a results of resampling, the filter draws more particles (states and models) from the ones that fit better to the D+1 measurements. In the task of battery prognostics, these resampled particles are used as the updated parameters and their fade models at the current step (k) for prediction of RUL (see Section 2.2.4).

measurement models, allowing a better model to possess a larger number of particles. To preserve diversity, each model has a minimum of NPmin particles. The procedure consists of the following steps.

• Particle Initialization: A set of N • •

•

•

P initial particles are randomly drawn from an initial probability distribution of the augmented states p0(ψ), and these particles are assigned equal weights 1/NP (Line 1). Model Transition: The model variable in each of the particles is evolved forward for the time period between the (k–1)th and kth measurements (Line 5). The transition simply uses the updated model variables from the (k–1)th step as the prior distribution of the model variable at the kth step. State Transition: The state variables in each of the particles are evolved forward independently for the time period between the (k–1)th and kth measurements according to the evolved model variables and the state transition equations in Eq. (4) (Line 6). The evolved particles form the prior distribution of the augmented states at the kth step before the use of the measurements at the step and its subsequent steps. Fixed-Lag Smoothing: For each of the D lagged steps, the state and model variables in each of the particles are evolved forward in time (Lines 10 and 11), and the weight of each particle is computed based on its previous weight and its likelihood at the lagged step (Line 12). The likelihood at each lagged step is calculated based on the modelpredicted distribution of measurement and the actual measurement at that step. Then, the particle weights are normalized such that their sum equals to 1 (Lines 14, 16 and 17). Thus, the state xkj and model mkj are evaluated for the additional D steps during the smoothing, and the particle weights at the end of the smoothing, wkj(D+1), are calculated by updating the prior weights wk–1j with the current measurement yk and D future measurements yk+1:k+D. Resampling: Particles having high importance weights at the end of

2.2.4. RUL prediction with updated model parameters Once the posterior PDFs of the fade model parameters are updated at the kth charge cycle, the prediction of the normalized capacity forward by l cycles can be performed and explicitly expressed as

p (ck + l y1: k ) ≈

NP

∑ j=1 wkj δ (ck +l − ckj+l)

(9)

where

ckj+ l =

j j j j ⎧1 − αk [1 − exp(−λk (k + l))] − βk (k + l), if mk = 1 j j ⎨ γ − β (k + l), if mkj = 2 k ⎩ k

(10)

j

For example, if mk = 1, and we define the normalized capacity being 75% as the failure threshold, then the RUL (in cycles) can be obtained for each particle as the number of cycles between the current cycle k and the end-of-service (EOS) cycle

Lkj = root[αkj [1 − exp(−λkj k )] + βkj k = 0.25] − k

(11)

Finally, the RUL distribution can be built based on these particles, expressed as

p (Lk y1: k ) ≈

NP

∑ j=1 wkj δ (Lk − Lkj)

(12)

This completes a generic derivation of the RUL distribution approximated by the generic particle filter or fixed-lag MMPF. 122

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Table 2 Pseudo-code of fixed-lag MMPF.

3. Experimental results

performance of these cells. Section 3.2 reports the feature extraction and capacity estimation results on the MDT cells. The RUL prediction results on the MDT and Mfg. X cells are presented and discussed in Section 3.3.

The performance of the proposed method is evaluated based on data from daily cycling of two sets of Li-ion prismatic cells in implantable applications: 1) eight fresh MDT cells whose rates of capacity fade appear to gradually decrease over a 10-year test duration; and 2) eight fresh/post-explant Mfg. X cells whose rates appear to sharply increase after some period over a 1.8-year test duration. In this evaluation, the capacity estimate of a cell at a specific charge/discharge cycle is obtained through feature extraction and sparse Bayesian learning (see Section 2.1), and the RUL of the cell is predicted through recursive Bayesian filtering and model projection (see Section 2.2). The daily cycling data used in this evaluation were gathered by the authors at MDT, and the evaluation was conducted by the author at the Iowa State University (ISU). This section reports the results of this evaluation. Section 3.1 presents the procedure of the cycling tests and the cycling

3.1. Test procedure and cycling data This study involved testing two sets of eight Li-ion cells with hermetically sealed prismatic cases: 1) eight fresh MDT cells; and 2) two fresh Mfg. X cells and six post-explant Mfg. X cells. These two sets of cells were subject to full depth of charge/discharge cycling with a nominally daily discharge rate under 37 °C in the lab. This represents a high end use condition for typical implantable neurostimulator devices. The cycling tests were conducted with the following parameter settings for the MDT (Mfg. X) cells: (i) the charge rate for the constant current charge was C/4 (C/2); (ii) the constant current charge cutoff voltage 123

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Fig. 3. Capacity fade performance of MDT and Mfg. X cells cycled with a nominally daily discharge rate: (a) MDT cells 1–4, (b) MDT cells 5–8, (c) Mfg. X cells 1 and 2 (fresh cells), and (d) Mfg. X cells 3–8 (post-explant cells). For the ease of visualization, capacity measurements are plotted every 100 (10) daily cycles for MDT (Mfg. X) cells; and the capacity fade data from either set of cells are plotted in two separate graphs. Although the BOL capacities of the post-explant Mfg. X cells are unknown, these cells in general show a similar trend of capacity fade as the two fresh Mfg. X cells.

250 cycles (stage 3). This degradation pattern appears to be similar with the three stages of degradation reported in several prior simulation and experimental studies [50–52]. This pattern contrasts with the almost continuous decrease in the rate of capacity fade (i.e., capacity fade slows down with ageing) observed from the MDT cells (see Fig. 3(a) and (b)). Third, the six post-explant Mfg. X cells (Mfg. X cells 3–8) also show a three-stage degradation pattern (see Fig. 3(d)), similar to that observed from the fresh Mfg. X cells but with shorter durations of stages 1 and 2 (i.e., the cells entered stage 3 of capacity fade at around 50 cycles). The shorter durations of stages 1 and 2 are expected as the postexplant cells might have undergone significant portions of the first two stages of degradation while in use when implanted. As the BOL capacities of these post-explant cells are unknown, the normalized capacity shown in Fig. 3(d) should not be viewed as the percent capacity relative to the BOL capacity but rather that relative to the capacity at a later point of the lifetime (i.e., the beginning of the cycling tests). Although the true normalized capacities of the post-explant cells are unknown (these are expected to be slightly lower than those shown in Fig. 3(d)), these six cells in general show a similar trend of capacity fade as the two fresh cells.

(Vmax) was 4.075 V (4.10 V); (iii) the time duration of the constant voltage charge was 60 min (30 min); (iv) the discharge rate was C/24, i.e., a nominally daily discharge rate; and (v) the discharge cutoff voltage (Vmin) was 3.575 V (2.7 V). These test parameters, except the discharge rate, were set in accordance with the cell specification of the respective manufacturer. The six Mfg. X field cells from explanted devices, after being extracted from the devices, were cycled using the test procedure described above along with the two cells from fresh devices. All of these samples showed capacities and voltage characteristics close to the normal values expected per the Mfg. X specification. This provided some evidence that all these cells are representative samples and had not experienced any damage due to handling. The discharge capacities of these cells are plotted against the cycle number in Fig. 3. Note that, for confidentiality reasons, the discharge capacity of a cell in Fig. 3 and in the discussions thereafter is presented after being normalized by the beginning-of-life (BOL) discharge capacity of the cell. The eight fresh MDT cells were continuously cycled with the daily discharge rate for an extended duration (i.e., around 10 years), while the eight Mfg. X cells were tested under a similar cycling condition for a shorter term (i.e., around 1.8 years for the fresh cells and around 1.3 years for the post-explant cells). A few observations can be made from these plots. First, the eight fresh MDT cells appear to have consistently decreasing rates of capacity fade (i.e., more stable capacity fade as the cell ages) over the 10-year test duration, as shown in Fig. 3(a) and (b), and these cells maintained 70%-75% of their initial capacities after the 10 years of repeated cycling. Second, the two fresh Mfg. X cells (Mfg. X cells 1 and 2) exhibit a three-stage fade behavior (see Fig. 3(c)), that is, the rate of capacity fade was initially high (stage 1), but slowed down quickly after around 50 cycles (stage 2), and at around 150 cycles, started to increase and stayed constant after around

3.2. Feature extraction and capacity estimation results 3.2.1. Feature data generation The 10 years' cycling data from the eight fresh MDT cells were used to verify the effectiveness of the proposed method in capacity estimation. In order to simulate actual use cases where a cell starts charge at a partially discharged state and ends up being fully charged, we generated a partial charge curve from a full charge curve by truncating the full charge curve below a pre-assigned initial charge voltage [22,23]. 124

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Table 3 Numbers of charge/discharge cycles, charge cycles used for feature data generation and feature vectors from eight fresh MDT cells.

No. of cycles No. of cycles used for feature generation No. of feature vectors

Cell 1

Cell 2

Cell 3

Cell 4

Cell 5

Cell 6

Cell 7

Cell 8

3183 64

3141 63

3206 65

3148 63

3329 67

3283 66

3261 66

3329 67

192

189

195

189

201

198

198

201

Table 4 Capacity estimation results of MDT cells from eight CV trials.

3.2.2. Definition of error measures RVM training was implemented to learn the posterior distributions of the kernel weights ω from the generated feature data and the corresponding capacity values (see Section 3.2.1). The accuracy of a trained RVM regression model was evaluated by using the k-fold cross validation (CV) [22]. Let Il = {i: xi∈Xl}, l = 1, 2, …, 8, denote the index set of the data points that construct the subset Xl. Then the CV root mean square (RMS) error is computed as the root square of the average error over all the eight CV trials, expressed as [22].

1 U

l = 1 i ∈ Il

(13)

where U is the number of feature vectors for the CV, μh,X\\Xl is the mean predicted capacity by the RVM regression model built with the complete data set X excluding the subset Xl, and y(xi) is the measured capacity of xi. The CV maximum (Max) error is computed as the maximum absolute error over all the eight CV trials, expressed as [22].

(

εMax = max max μh, X \\ Xl (x i) − y (x i) 1≤l≤8

i ∈ Il

)

CV trial 2

CV trial 3

CV trial 4

CV trial 5

CV trial 6

CV trial 7

CV trial 8

Overall

RMS (%) Max (%)

0.49 2.48

0.55 2.41

0.63 2.32

0.61 2.08

0.88 2.40

0.77 2.22

0.77 1.65

0.51 1.87

0.67 2.48

3.2.4. Sparsity property of RVM model As aforementioned in Section 2.1 and reported in the prior study [22], one of the desirable properites that RVM possesses is sparsity, i.e., only a subset of training data points is retained in the trained regression model. Those retained data points are termed relevance vectors that form the final sparse solution. The distribution of relevance vectors in a twodimensional feature space (z1 and z2) is shown in Fig. 5 for CV trials 1 and 2, where the normalized feature vectors from cells 2–8 and those from cells 1 and 3–8 were used as the training data sets, respectively. Among the 1371 (1374) feature vectors in the training data sets for CV trial 1 (CV trial 2), only 21 (20), i.e., 1.4% (1.5%) of all the feature vectors, were chosen as relevance vectors. Similar sparsity was observed on the other CV trials, and the ratio of relevance vectors to feature vector appeared to be consistently less than 2.0%. As stated in the prior study [22], this suggests that the sparsity property of RVM enables the generation of a reduced-scale regression model that utilizes only a small fraction of the training data set, thereby improving computational efficiency in capacity estimation.

8

∑ ∑ (μh,X \\ Xl (xi) − y (xi) )2

CV trial 1

capacity estimator is capable of producing accurate capacity estimation under a wider range of the initial charge level (i.e., 11%–45% SOC). The good accuracy is consistent with that reported in the prior study [22], and can likely be attributed to two facts: (i) the MDT cells exhibit a somewhat homogeneous capacity fade behavior (see Fig. 3(a) and (b)) throughout the cycling test; and (ii) since the fade behavior is fairly consistent among the lab cells, a training data set, which carries the information about the ageing behavior of 7 training cells, is capable of capturing the ageing behavior of the testing cell. The capacity estimation results of the first two MDT cells (cells 1 and 2) are shown in Fig. 4. The measured capacity for a charge/discharge cycle was calculated using the coulomb counting method, which integrates the discharge current over time for the entire discharge cycle. It can be observed from Fig. 4 that the sparse Bayesian learning method closely tracks the capacity fade trend throughout the 10 years' daily cycling test. For most of the cycles shown in the figure, the intervals of ± three standard deviation around the mean capacity estimates (i.e., μ ± 3σ error bars) contain the measured capacities. This suggests that the proposed method, as a statistical learning technique, is capable of archiving robust capacity estimation by accounting for the uncertainty in both the feature data and kernel weight estimates.

The initial charge voltage pre-assigned to a charge cycle was randomly generated from a uniform distribution between a lower bound VLB (= 3.8 V and roughly 11% SOC) and an upper bound VUB (= 3.9 V and roughly 45% SOC). This range was selected to cover a wide range of the initial SOC at which a cell starts charge. The five charge-related features detailed in Section 2.1 were then extracted from the partial charge curves that had been generated. The feature data were generated using the charge curves at every 50 charge/discharge cycles from each of the eight MDT cells. Table 3 summarizes the numbers of charge/discharge cycles, charge cycles used for feature data generation and extracted feature vectors from the MDT cells. Before applying the RVM regression method, the feature data were normalized to [0,1].

εRMS =

Errors

(14)

The error formulae in Eqs. (13) and (14) indicate that all the U feature vectors in the complete data set X are used for both training and testing, and each feature vector is used for testing exactly once and for training seven times.

3.3. RUL prediction results 3.2.3. Capacity estimation results In the RVM regression model, the Gaussian kernel function was employed as the kernel function with the kernel width r set to 0.8, and the maximum number of RVM training iterations was set to 500. Following the prior study on the sparse capacity estimator [22], the kernel type and the value of the kernel width were empirically determined based on the accuracy of a trained RVM model in capacity estimation. It was observed that the chosen kernel type (Gaussian kernel) and width (0.8) resulted in a trained RVM model with a minimum CV RMS error in capacity estimation. Table 4 summarizes the capacity estimation errors (RMS and Max) by the trained sparse capacity estimator. It can be observed that the RMS errors are less than 1.00% for all the eight MDT cells, and the Max errors are less than 3.00%. These results suggest that the sparse

3.3.1. RUL prediction of MDT cells using estimated capacities Based on the capacity estimates by the sparse capacity estimator, the generic particle filter was used to update the parameters of the first capacity fade model (Model 1) in Eq. (2), and the updated model was projected to an EOS value (or a capacity threshold) for RUL prediction. Here two settings were considered for the capacity threshold: (i) Setting 1–High capacity threshold (75% of the BOL discharge capacity, i.e., cth = 0.75); and (ii) Setting 2–Low capacity threshold (50% of the BOL discharge capacity, i.e., cth = 0.5). Note that Setting 2 represents the scenario where a higher degree of capacity fade is tolerable for a specific population of patients/physicians, as compared to the scenario in Setting 1, where this population of patients/physicians is tolerant about a lower degree of capacity fade. 125

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Fig. 4. Capacity estimation results of first two MDT cells: cell 1 (a) and cell 2 (b). Results are plotted every 100 daily cycles for the ease of visualization. Error bars indicate intervals of ± three standard deviations around mean capacity estimates.

unknown, without collecting more data, how far ahead the model maintains its predictive accuracy to the extent suitable for life prediction. Table 5 quantitatively summarizes these life predictions on all eight MDT cells. The predicted EOS time (in years) of a cell under the second setting of capacity threshold (cth = 0.5) was estimated by extrapolating a linear time vs. cycle model, built with all available data collected from the cell, to the predicted EOS cycle. The results show that the MDT cells delivered 2550-3050 cycles (or 7.6–8.9 years) of useful life under a daily rate cycling condition before their capacities degraded to 75% of the respective BOL capacities. The life predictions at cycle 3000 suggest that these cells are expected to deliver 6726-7627 cycles (or 18.4–20.7 years) of useful life before they reach the 50% BOL capacity threshold. Since more data are needed to verify the capacity fade behavior of these cells beyond their last cycles of measurement, the EOS cycle numbers for cth = 0.5 in Table 5 should only be treated as predictions whose accuracy remains largely unknown.

The RUL predictions at cycle 500 (or at 1.7 years) and cycle 1000 (or at 3.2 years) under the first setting of capacity threshold (cth = 0.75) are shown in Fig. 7(a) and (b), respectively. The predicted PDF of the life at cycle 500 suggests an early prediction (see Fig. 6(a)), i.e., a shorter-than-true predicted RUL (see Fig. 6(b)). This is mainly because the fade at the first 500 cycles is more rapid than the subsequent cycles, and thus the capacity fade model updated with these initial fade data cannot capture the slower fade at the later cycles. As shown in Fig. 6(b), the prediction at cycle 1000 yields a more accurate prediction of life, a slightly conservative solution that includes the true EOS cycle (i.e., 2780 cycles or approximately 8.2 years). Fig. 6(c) plots the RUL predictions (means ± three standard deviations) under the first setting of capacity threshold at multiple cycles throughout the lifetime. The graph shows that the RUL prediction keeps being updated over time, and that as the battery approaches its EOS cycle, the mean prediction tends to converge to the true value. The error bars at each cycle indicate the intervals of ± three standard deviation around the mean RUL prediction (i.e., μ ± 3σ), and under the assumption of a Gaussian distribution, the intervals cover 99.7% of the predicted RULs at the cycle. Fig. 6(d) plots the mean predictions of the EOS cycle at multiple cycles under both settings of capacity threshold (cth = 0.75 and cth = 0.5). Under the first setting of capacity threshold, the predicted life closely matches the true life after cycle 2000. This suggests the updated capacity fade model accurately represents the fade behavior of the cell at least up to the EOS cycle (i.e., cycle 2780). It is likely that the model maintains this accuracy of representation beyond the EOS cycle, and produces a reasonably accurate prediction of the EOS time under the second setting of capacity threshold (cth = 0.5). However, it is

3.3.2. RUL prediction of Mfg. X cells using measured capacities Based on the capacity measurements from the eight Mfg. X cells, the fixed-lag MMPF was used to update the parameters of the empirical capacity fade models (Models 1 and 2) in Eqs. (5) and (6), and the updated models were projected to a capacity threshold for RUL prediction. Again, two settings (cth = 0.75 and cth = 0.5) were considered for the capacity threshold. The capacity projections based on the updated particles at cycle 200 (or at 0.8 years) and cycle 400 (or at 1.5 years) are shown as 2 Gy curves in Fig. 7(a), where the first setting of capacity threshold (cth = 0.75) is considered. The updated particles at cycle 200 yield a shorter predicted life than the true life (i.e., 420 cycles

Fig. 5. Distributions of relevance vectors in two-dimensional feature space (z1 and z2) in CV trial 1 (a) and CV trial 2 (b). Here, training data refer to 1371 (1374) normalized feature vectors from cells 2–8 (cells 1 and 3–8) for CV trial 1 (2). The 21 (20) relevance vectors for CV trial 1 (2) are marked with red squares. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 6. RUL prediction results of MDT cell 1 using single model particle filter. Figures (a) and (b) plot the RUL predictions at cycles 500 and 1000, respectively, under the first setting (high capacity threshold); (c) plots the RUL predictions at multiple cycles under the first setting; and (d) plots the mean life predictions at multiple cycles under both settings (high and low capacity thresholds). Results are plotted every 100 cycles for the ease of visualization. Error bars in Figure (c) indicate intervals of ± three standard deviations around mean RUL predictions.

versus 519 cycles). This is mainly because the dominant fade model at cycle 200, Model 1, projects an exponential fade whose rates accelerate over cycle (see the capacity projection in Fig. 7(a)), and this projected fade over-predicts the capacity fade at the later cycles. In contrast, the dominant model at cycle 400, Model 2, better captures the linear fade at the later cycles, and the extrapolation of the model yields a predicted life much closer to the true life (i.e., 516 cycles versus 519 cycles). Fig. 7(b) plots the RUL predictions under the first setting of capacity threshold at multiple cycles throughout the test duration. The graph shows that the RUL prediction keeps being updated over time, and at around cycle 380 when the dominant model transitions from Model 1 to Model 2 (see Fig. 7(d)), the mean prediction converges to the true line. The error bars at each cycle indicate the three-standard-deviation interval around the mean prediction, i.e., μ ± 3σ. Fig. 7(c) plots the mean predictions of the EOS life at multiple cycles under both settings of capacity threshold (cth = 0.75 and cth = 0.5). Again, the EOS life prediction starts to follow the true line at around cycle 380 when Model 2 becomes dominant. It should be emphasized that the capacity projection from cycle 400 was performed based on the linear fade model in Eq. (6), and that it is nearly impossible to determine, without collecting more data, whether the linear model can still capture the fade behavior of the cell beyond the last cycle of measurement. Table 6 quantitatively summarizes the life predictions on all eight Mfg. X cells with Model 2 being the most probable model. Similar to the analysis on the MDT cells, the predicted EOS time (in years) of a Mfg. X cell in the absence of data at the time was estimated by extrapolating a linear time vs. cycle model, built with all available data collected from the cell, to the predicted EOS cycle. The life prediction results suggest the capacities of the Mfg. X cells are expected to fade by 50% after 693-1414 cycles (or 2.0–4.4 years) of useful life under a daily rate cycling condition. The EOS cycle numbers for cth = 0.5 in Table 6 should only be treated as

Table 5 Life predictions (means) on MDT cells using estimated capacities. MDT cell number

Current cycle [year]

High capacity threshold (cth = 0.75)

Low capacity threshold (cth = 0.5)

Predicted (cycle [year])

True (cycle [year])

Predicted (cycle [year])

True (cycle [year])

2780 [8.2] – 2965 [8.7] – 2750 [8.2] – 3050 [8.9] – 2650 [7.9] – 2765 [8.2] – 2740 [8.1] – 2550 [7.6] –

6829 [18.6]

–

7001 [19.1] 7228 [19.6]

– –

7228 [19.6] 6858 [18.7]

– –

7023 [19.1] 6866 [18.7]

– –

7627 [20.7] 6657 [18.2]

– –

6726 [18.4] 6919 [18.8]

– –

7101 [19.3] 6657 [18.2]

– –

7015 [19.1] 5980 [16.4]

– –

6734 [18.4]

–

Cell 1

2000 [6.1]

2777 [8.2]

Cell 2

3000 [8.8] 2000 [6.1]

– 2928 [8.6]

Cell 3

3000 [8.8] 2000 [6.1]

– 2704 [8.0]

Cell 4

3000 [8.8] 2000 [6.1]

– 2926 [8.6]

Cell 5

3000 [8.8] 2000 [6.1]

– 2563 [7.7]

Cell 6

3000 [8.8] 2000 [6.1]

– 2664 [7.9]

Cell 7

3000 [8.8] 2000 [6.1]

– 2648 [7.9]

Cell 8

3000 [8.8] 2000 [6.1]

– 2351 [7.1]

3000 [8.8]

–

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Fig. 7. RUL prediction results of Mfg. X cell 1 using multiple model particle filter. Figure (a) plots the mean capacity projections starting from cycles 200 and 400, respectively, under the first setting (high capacity threshold); (b) plots the RUL predictions at multiple cycles under the first setting; (c) plots the mean life predictions at multiple cycles under both settings; and (d) plots the most probable fade models at multiple cycles. Results are plotted every 20 cycles for the ease of visualization. Error bars in Figure (b) indicate intervals of ± three standard deviations around mean RUL predictions.

Table 6 Life predictions (means) on Mfg. X cells using measured capacities (most probable model: Model 2). Mfg. X cell number

Cell Cell Cell Cell Cell Cell Cell Cell

1 2 3 4 5 6 7 8

Current cycle [year]

400 400 320 320 395 320 385 320

[1.5] [1.5] [1.2] [1.1] [1.4] [1.1] [1.4] [1.1]

High capacity threshold (cth = 0.75)

Low capacity threshold (cth = 0.5)

Predicted (cycle [year])

True (cycle [year])

Predicted (cycle [year])

True (cycle [year])

516 732 326 334 408 362 458 372

519 – 328 335 – 356 – 366

974 [2.9] 1414 [4.4] 693 [2.0] 705 [2.0] 779 [2.3] 719 [2.1] 846 [2.5] 728 [2.1]

– – – – – – – –

[1.8] [2.5] [1.2] [1.2] [1.4] [1.3] [1.5] [1.3]

[1.8] [1.2] [1.2] [1.2] [1.3]

Fig. 8. Capacity projections for MDT cell 1 and Mfg. X cell 1. The capacity fade data measured from both cells are also included on this figure. The capacity projections for MDT cell 1 and Mfg. X cell 1 were made from cycle 3000 and cycle 400, respectively.

predictions whose accuracy still needs to be confirmed with the collection of longer-term data.

Mfg. X cells is more complicated and the modeling of this behavior required the use of a more sophisticated approach (i.e., the fixed-lag MMPF described in Section 2.2.3). An incorrect prediction could be made if extrapolation from limited data is conducted or if the extrapolation uses a modeling approach that does not give enough consideration to the varying fade mechanisms and fade profiles. Fig. 8 shows the capacity projections for MDT cell 1 and Mfg. X cell 1, along with the lower setting of capacity threshold (cth = 0.5). For the ease of visualization, only one cell from each set was selected to be included in the plot. The capacity projections on both cells were performed based on the assumption that the capacity losses of these cells

4. Discussion The data sets in this paper provide two distinct examples of fade behavior and the challenges associated with predicting RUL for longlife applications. For the MDT cells, the use of particle filter with a single exponential model (i.e., the generic single model particle filter described in Section 2.2.2) resulted in good fits and accurate predictions of RUL (see Table 5). This was likely due to a single key mechanism dominating the fade characteristics. The fade behavior of the 128

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would continue to follow the trends predicted by the respective capacity fade models. It can be seen that the two cells have distinct trends of capacity fade, likely due to the aforementioned difference in their fade mechanisms, and the proposed prognostic method is capable of adapting the respective models to the distinct trends of capacity fade. Similar observations could be made when performing such projections on other MDT and Mfg. X cells.

lithium-ion battery SOC and capacity estimation, Appl. Energy 92 (2012) 694–704. [12] K.S. Ng, C.S. Moo, Y.P. Chen, Y.C. Hsieh, Enhanced coulomb counting method for estimating state-of-charge and state-of-health of lithium-ion batteries, Appl. Energy 86 (9) (2009) 1506–1511. [13] W. Waag, D.U. Sauer, Adaptive estimation of the electromotive force of the lithiumion battery after current interruption for an accurate state-of-charge and capacity determination, Appl. Energy 111 (2013) 416–427. [14] C. Hu, G. Jain, P. Tamirisa, T. Gorka, Method for estimating capacity and predicting remaining useful life of lithium-ion battery, Appl. Energy 126 (2014) 182–189. [15] Hu C., and Jain G., 2014, Method for estimating capacity and predicting remaining useful life of Li-Ion battery,” us patent application No. 61/973,601. [16] A. Eddahech, O. Briat, N. Bertrand, J. Deletage, J. Vinassa, Behavior and state of health monitoring of Li-ion batteries using impedance spectroscopy and recurring neural networks, Int. J. Electr. Power & Energy Syst. 42 (1) (2012) 487–494. [17] J. Kim, S. Lee, B.H. Cho, Complementary cooperation algorithm based on DEKF combined with pattern recognition for SOC/capacity estimation and SOH prediction, IEEE Trans. Power Electron. 27 (1) (2012) 436–451. [18] G. Bai, P. Wang, C. Hu, M. Pecht, A generic model-free approach for lithium-ion battery health management, Appl. Energy 135 (2014) 247–260. [19] B. Pattipati, C. Sankavaram, K. Pattipati, System identification and estimation framework for pivotal automotive battery management system characteristics, IEEE Trans. Syst. Man, Cybern. Part C Appl. Rev. 41 (6) (2011) 869–884. [20] A. Nuhic, T. Terzimehic, T. Soczka-Guth, M. Buchholz, K. Dietmayer, Health diagnosis and remaining useful life prognostics of lithium-ion batteries using datadriven methods, J. Power Sources 239 (2013) 680–688. [21] A. Widodo, M.C. Shim, W. Caesarendra, B.S. Yang, Intelligent prognostics for battery health monitoring based on sample entropy, Expert Syst. Appl. 38 (9) (2011) 11763–11769. [22] C. Hu, G. Jain, C. Schmidt, C. Strief, M. Sullivan, Online estimation of lithium-ion battery capacity using sparse bayesian learning, J. Power Sources 289 (2015) 105–113. [23] C. Hu, G. Jain, P. Zhang, C. Schmidt, P. Gomadam, T. Gorka, Data-driven method based on particle swarm optimization and k-nearest neighbor regression for estimating capacity of lithium-ion battery, Appl. Energy 129 (2014) 49–55. [24] N. Gebraeel, M. Lawley, R. Li, J.K. Ryan, Residual-life distributions from component degradation signals: a bayesian approach, IIE Trans. 37 (2005) 543–557. [25] J. Luo, K.R. Pattipati, L. Qiao, S. Chigusa, Model-based prognostic techniques applied to a suspension system, IEEE Trans. Syst. Man Cybern. Part A 38 (5) (2008) 1156–1168. [26] N. Gebraeel, J. Pan, Prognostic degradation models for computing and updating residual life distributions in a time-varying environment, IEEE Trans. Reliab. 57 (4) (2008) 539–550. [27] X.S. Si, W. Wang, C.H. Hu, M.Y. Chen, D.H. Zhou, A Wiener-process-based degradation model with a recursive filter algorithm for remaining useful life estimation, Mech. Syst. Signal Process. 35 (2013) 219–237 n1–2. [28] T. Wang, J. Yu, D. Siegel, J. Lee, A Similarity-based Prognostics Approach for Remaining Useful Life Estimation of Engineered Systems, International Conference on Prognostics and Health Management, Denver, CO, Oct 6–9, (2008). [29] F.O. Heimes, Recurrent Neural Networks for Remaining Useful Life Estimation, IEEE, International Conference on Prognostics and Health Management, Denver, CO, Oct 6–9, 2008. [30] K. Goebel, N. Eklund, P. Bonanni, Fusing Competing Prediction Algorithms for Prognostics,” Proceedings of 2006 IEEE Aerospace Conference, New York, 2006. [31] J. Liu, W. Wang, F. Ma, Y.B. Yang, C.S. Yang, A data-model-fusion prognostic framework for dynamic system state forecasting, Eng. Appl. Artif. Intell. 25 (4) (2012) 814–823. [32] B. Saha, K. Goebel, S. Poll, J. Christophersen, Prognostics methods for battery health monitoring using a Bayesian framework, IEEE Transaction Instrum. Meas. 58 (2) (2009) 291–296. [33] Liu J., Saxena A., Goebel K., Saha B., and Wang W., An Adaptive Recurrent Neural Network for Remaining Useful Life Prediction of Lithium-ion Batteries, Annual Conference of the Prognostics and Health Management Society, Portland, Oregon, October 2010. [34] B. Saha, K. Goebel, Modeling Li-ion Battery Capacity Depletion in a Particle Filtering Framework, in Proceedings of Annual Conference of the PHM Society, San Diego, CA, Sep. 27–Oct. 1, (2009). [35] W. He, N. Williard, M. Osterman, M. Pecht, Prognostics of lithium-ion batteries based on dempster–shafer theory and the bayesian Monte Carlo method, J. Power Sources 196 (23) (2011) 10314–10321. [36] D. Wang, Q. Miao, M. Pecht, Prognostics of lithium-ion batteries based on relevance vectors and a conditional three-parameter capacity degradation model, J. Power Sources 239 (2013) 253–264. [37] Q. Miao, L. Xie, H. Cui, W. Liang, M. Pecht, Remaining useful life prediction of lithium-ion battery with unscented particle filter technique, Microelectron. Reliab. 53 (6) (2013) 805–810. [38] D. Wang, F. Yang, K.L. Tsui, Q. Zhou, S.J. Bae, Remaining useful life prediction of lithium-ion batteries based on spherical cubature particle filter, IEEE Trans. Instrum. Meas. 65 (6) (2016) 1282–1291. [39] D. Wang, F. Yang, Y. Zhao, K.-L. Tsui, Prognostics of lithium-ion batteries based on state space modeling with heterogeneous noise variances, Microelectron. Reliab. 75 (2017) 1–8. [40] M.E. Tipping, Sparse Bayesian learning and the relevance vector machine, J. Mach. Learn. Res. 1 (2001) 211–244. [41] Kohei Honkura, Ko Takahashi, Tatsuo Horiba, Capacity-fading prediction of lithium-ion batteries based on discharge curves analysis, J. Power Sources 196 (23) (2011) 10141–10147.

5. Conclusion This paper presented a hybrid data-driven/model-based method for online RUL assessment of Li-ion batteries in implantable medical devices. The hybrid method integrates a data-driven machine learning method with a model-based filtering technique to form a complete solution for online capacity estimation and RUL prediction. The integration enables the fusion of historical data and domain knowledge (of capacity fade) to infer battery health and predict battery RUL. The hybrid method was applied to prognostics of two battery designs (MDT cells and Mfg. X cells) that exhibit different capacity fade characteristics under nominally daily rate cycling conditions. The capacity fade trend from the MDT cells shows a decreasing rate of fade over time, while the trend from the Mfg. X cells shows an initial rapid decrease in the capacity, immediately followed by a decrease in the rate of fade, and then a faster linear fade (i.e., capacity decreases linearly at a faster rate). It was demonstrated that the proposed method is capable of adapting capacity fade models to changes in the rate of capacity fade over the lifetime of a single cell, as well as to variations in the trend of capacity fade across different cells from different manufacturers. Accurate prediction of battery lifetime often necessitates a profound understanding of the underlying fade mechanisms and the collection of long-term fade data under conditions similar to actual use. Our future work will investigate the fade mechanisms of the Mfg. X cells as well as verify the prognostics results on these cells through the collection of longer-term data. Acknowledgement This research was in part supported by the US National Science Foundation (NSF) Grant Nos. CNS-1566579 and ECCS-1611333. Any opinions, findings or conclusions in this paper are those of the authors and do not necessarily reflect the views of the sponsoring agency. References [1] G.L. Plett, Extended Kalman filtering for battery management systems of LiPB-based HEV battery packs Part 3. State and parameter estimation, J. Power Sources 134 (2) (2004) 277–292. [2] G.L. Plett, Sigma-point Kalman filtering for battery management systems of LiPBbased HEV battery packs Part 2: simultaneous state and parameter estimation, J. Power Sources 161 (2) (2006) 1369–1384. [3] L. Lu, X. Han, J. Li, J. Hua, M. Ouyang, A review on the key issues for lithiumion battery management in electric vehicles, J. Power Sources 226 (2013) 272–288. [4] J. Zhang, J. Lee, A review on prognostics and health monitoring of Li-ion battery, J. Power Sources 196 (15) (2011) 6007–6014. [5] M. Verbrugge, Adaptive, multi-parameter battery state estimator with optimized time-weighting factors, J. Appl. Electrochem. 37 (5) (2007) 605–616. [6] S. Lee, J. Kim, J. Lee, B.H. Cho, “State-of-charge and capacity estimation of lithiumion battery using a new open-circuit voltage versus state-of-charge,”, J. Power Sources 185 (2) (2008) 1367–1373. [7] A.P. Schmidt, M. Bitzer, A.W. Imre, L. Guzzella, Model-based distinction and quantification of capacity loss and rate capability fade in Li-ion batteries, J. Power Sources 195 (2010) 7634–7638. [8] Y.-H. Chiang, W.-Y. Sean, J.-C. Ke, Online estimation of internal resistance and open-circuit voltage of lithium-ion batteries in electric vehicles, J. Power Sources 196 (8) (2011) 3921–3932. [9] W. He, N. Williard, C. Chen, M. Pecht, State of charge estimation for electric vehicle batteries using unscented Kalman filtering, Microelectron. Reliab. 53 (6) (2013) 840–847. [10] R. Xiong, F. Sun, Z. Chen, H. He, A data-driven multi-scale extended Kalman filtering based parameter and state estimation approach of lithium-ion polymer battery in electric vehicles, Appl. Energy 113 (2014) 463–476. [11] C. Hu, B.D. Youn, J. Chung, A multiscale framework with extended kalman filter for

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Journal of Power Sources 375 (2018) 118–130

C. Hu et al.

Sonar Navig. 150 (2003) 344–349. [48] A. Doucet, N.J. Gordon, V. Krishnamurthy, Particle filters for state estimation of jump Markov linear systems, IEEE Trans. signal Process. 49 (3) (2001) 613–624. [49] A. Doucet, A.M. Johansen, A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later. Handbook of Nonlinear Filtering vol. 12, (2009), pp. 656–704. [50] R. Spotnitz, Simulation of capacity fade in lithium-ion batteries, J. Power Sources 113 (1) (2003) 72–80. [51] Q. Zhang, R.E. White, Capacity fade analysis of a lithium ion cell, J. Power Sources 179 (2) (2008) 793–798. [52] X. Lin, J. Park, L. Liu, Y. Lee, A.M. Sastry, W. Lu, A comprehensive capacity fade model and analysis for Li-ion batteries, J. Electrochem. Soc. 160 (10) (2013) A1701–A1710.

[42] Jason Brown, Erik Scott, Craig Schmidt, William Howard, A Practical Longevity Model for Lithium-ion Batteries: De-coupling the Time and Cycle-dependence of Capacity Fade, 208th ECS Meeting, Abstract #239, (2006). [43] S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for on-line non-linear/non-gaussian bayesian tracking, IEEE Transaction Signal Process. 50 (2) (2002) 174–188. [44] O. Cappe, S.J. Godsill, E. Moulines, An overview of existing methods and recent advances in sequential Monte Carlo, IEEE Proc. 95 (5) (2007) 899–924. [45] S. McGinnity, G.W. Irwin, Multiple model bootstrap filter for maneuvering target tracking IEEE Trans, Aerosp. Electron. Syst. 36 (2000) 1006–1012. [46] Y. Bores, J.N. Driessen, Hybrid. state Estim. a target Track. Appl. Autom. 38 (2002) 2153–2158. [47] Y. Bores, J.N. Driessen, Interacting multiple model particle filter IEE Proc, Radar

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