A State-Space-Based Prognostics Model for Lithium-Ion Battery Degradation

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1 A State-Space-Based Prognostics Model for Lithium-Ion Battery Degradation Xin Xu a, Nan Chen a, a Department of Industrial and Systems Engineering, National University of Singapore Abstract This paper proposes to analyze the degradation of lithium-ion batteries with the sequentially observed discharging profiles. A general state-space model is developed in which the observation model is used to simulate the discharging profile of each cycle, the corresponding simulation parameter vector is treated as the hidden state, and the state-transition model is used to track the evolution of the parameter vector as the battery ages. The EM and EKF algorithms are adopted to estimate and update the model parameters and states jointly. Based on this model, we construct prediction on the end of discharge times for unobserved cycles and the remaining useful cycles before the battery failure. The effectiveness of the proposed model is demonstrated using a real lithium-ion battery degradation data set. Keywords: Degradation, PHM, state-space model, EKF, EM, RUC 1. Introduction Lithium-ion (Li-ion) batteries have undergone rapid development since they were commercialized in Nowadays, due to their great advantages, they have been the most promising rechargeable batteries and applied as the main power sources in more and more fields, from the daily used mobile device industry, to the rising electric vehicle (EV) industry, even to the crucial marine and space system (van Schalkwijk and Scrosati 2002). However, regardless of the type and design of Li-ion batteries, the degradation caused by aging occurs throughout life in every condition (Barr et al. 2013). Two principle phenomena to identify the degradation are capacity fade and impedance raise. Capacity fade means that the Corresponding author addresses: xuxin@nus.edu.sg (Xin Xu), isecn@nus.edu.sg (Nan Chen) Preprint submitted to Reliability Engineering & System Safety September 17, 2016

2 maximal usable energy which can be stored in Li-ion batteries becomes less and less as the charge-discharge cycle increases. Impedance raise determines the reduction of the maximum of available power. As batteries degrade, they will be unable to supply sufficient energy or power for systems finally. This kind of functional failures announce the end of life for batteries (Zhang and Lee 2011). In order to prevent Li-ion battery failures from occurring, and to optimize battery maintenance and replacement schedule, developing a Prognostics and Health Management (PHM) approach for Li-ion batteries, with emphasis on detecting underlying degradation and predicting remaining useful cycle (RUC), achieves more and more attention (Zhang and Lee 2011). The common performance data for Li-ion batteries include voltage, current, impedance and capacity. Among them, both impedance and capacity have been widely used for degradation prognostics, because they are inherent battery properties and it is easy to identify and extract degradation features from their measurements. For example, Saha et al. (2009) applied the proposed Bayesian learning framework on Li-ion battery prognostics based on internal impedance measurements. Zheng and Fang (2015) developed a novel method using unscented Kalman filter (UKF) with relevance vector regression to predict the RUC. There are some other noted algorithms proposed for capacity or impedance degradation, such as artificial neutral network (Kozlowski 2002), relevance vector machine (Wang et al. 2013), and so on (Xing et al. 2013; Guo et al. 2015). If given accurate impedance or capacity measurements, the above approaches are easily to implement and can present good prediction accuracy. The problem lies in that both impedance and capacity of Li-ion batteries cannot be measured simply and efficiently. In literature, the most widely used experimental technique for impedance measurement is electrochemical impedance spectroscopy (EIS) test (Seaman et al. 2014). This test is timeconsuming and cost-ineffective to take regularly. It needs be conducted with bulk equipment and has strict requirements on experimental environment (Xing et al. 2013). The battery capacity can be estimated by the Coulomb counting method which integrates discharging current from a fully charged state to a fully discharged state (Zhang and Lee 2011). However, the capacity can only be obtained at the end of entire discharge process. Moreover, the measured capacity of each discharge cycle depends on the cut-off voltage greatly. 2

3 Capacity fade as cycle aging Discharging voltages in different cycles Capacity Voltage Cut off Voltage cycle 100 cycle 1 cycle 50 (a) Capacity fades Time (b) Voltage drops Figure 1: Illustration of battery operation within each discharge cycle and its degradation across multiple cycles. Different with impedance and capacity measurements, current and voltage can be easily obtained by the sensor technology in real applications. There have been some attempts to use the charging/discharging profiles for Li-ion battery degradation investigation. Some authors propose new capacity estimation methods based on the collected charging/discharging profiles. For example, Lu et al. (2014) proposed using the four geometric features extracted from charging/discharging profiles to estimate battery capacity. Tao et al. (2015) developed an approach named dynamic spatial time warping to recognize the similarities of current or voltage curves and further estimate battery capacity. These new capacity estimation methods can estimate the SoH, but cannot predict the RUC. Some authors try to extract features from charging/discharging profiles, and use these features as the health indicators to conduct the SoH estimation directly. For example, Widodo et al. (2011) introduced the sample entropy of discharging profiles into Li-ion battery degradation prognostics. Liu et al. (2015) proposed a general framework for the health indicator extraction and optimization. However, to better utilize these extracted features, accurate capacity measurements should be used for model training or feature transformation before on-line application. Other authors rely on the equivalent circuit model (ECM) and derive circuit components from discharging profiles for the SoH estimations. For example, Jonghoon Kim once developed a framework 3

4 for SoC and SoH joint estimation with the discharging profiles based on a designed ECM model (Kim and Cho 2011; Kim et al. 2012). Nevertheless, these ECM based methods can only provide the SoH estimation. The RUC prediction can not be achieved. In this paper, we propose to develop a data-driven method for Li-ion battery degradation prognostics with the sequentially observed discharging profiles. On the one hand, we don t choose to extract features from the discharging profiles, because we think the discharging profile of a single cycle can be used to predict how long the battery can be used and what is the current releasable capacity given any cut-off voltage. These predictions are important for battery operation in a single cycle. One the other hand, we don t stop at the SoH estimation, but aim at the prediction of RUC. Based on the above considerations, a state-space-based prognostics model is adopted in this paper. In more detail, we build a observation model to depict the discharging profile of each cycle and use a state-transition model to track the evolution of the parameter vector in observation model. With the orderly observed discharging profiles, we adopt the expectation maximization (EM) and extended Kalman filtering (EKF) algorithms to estimate the model parameters and states jointly. About the (unobserved) future cycles, we can predict the possible state-transition paths, simulate the sequence of discharging profiles and finally construct prognostics on end of discharging (EoD) time of each cycle and the RUC. The remainder of this paper is organized as follows. Section 2 formulates a general state space model for Li-ion battery degradation. Section 3 introduces the model estimation and updating via EM and EKF algorithm. Section 4 presents the prognostics based on the well learned state-space model. Section 5 uses a real case study to demonstrate the effectiveness of the proposed model. Section 6 concludes the paper with discussions on future research directions. 2. A state-space model for Li-ion battery degradation As presented in last section, we propose to take full use of the observed discharging profiles (current and voltage) for Li-ion battery degradation prognostics. In this section, we start with the simplest battery operation case in which the battery is discharged in a steady mode, i.e., loading current, ambient temperature, depth of discharge, and etc. are 4

5 held constant. In this way, we focus on extracting the degradation pattern from the voltage profiles of different cycles. As shown in Figure 1b, the output voltage in each cycle drops as the discharge proceeds. This drop is mainly caused by the existence of internal impedance and the motion of active lithium ions between the battery electrodes along with electro-chemical reactions. This means the path of voltage drop is not random. For a specific type of Li-ion batteries, the voltage profiles would always present a similar decline trend during discharging processes. Therefore, we can use the same parametric family to characterize all of the observed voltages profiles. Here, we denote U i (t) as the measured voltage at time t in the i th discharging cycle. The discharging profile can be expressed by the observation model U i (t) = h(t; θ i ) + ν i (t), t 0, (1) where h(t; θ i ) is the profile characterizing function with parameter vector θ i, and ν i (t) is the measurement error which is assumed to be independent and identically distributed with ν i (t) N(0, σ 2 ) at any time point t. For any cycle, once θ i is known, the output voltage at any time t can be estimated, and the entire discharging profile can be depicted easily. Moreover, given a cut-off voltage ζ i, both EoD time T i and releasable capacity Q i can be figured out. As the charge-discharge cycle increases, the internal impedance will increase, and the amount of active lithium ions and other electrode materials will decrease. These changes make the voltage profiles of different cycles present variation. In another word, this variation can be used to identify the battery degradation. Based on the observation model, the transition of {θ i, i = 1, 2,...} decides the variation of their discharging profiles and further stands for the battery degradation. Thus, we treat the parameter vectors {θ i, i = 1, 2,...} of different cycles as a series of hidden states in battery degradation process. The state transition model can be expressed as θ i = f i (θ i 1 ) + ω i, (2) 5

6 where f i ( ) is the state-transition function, and ω i is the transition noise which is assumed to be zero mean multivariate Gaussian noise with covariance Q i. Considering that the battery is operated in a steady mode, we assume the state transition function f i ( ) and covariance Q i are the same for different i. For simplicity, we use the simplest linear function f(θ i 1 ) = Bθ i 1 + b (3) to clarify the following prognostics approach and parameter estimation procedure. Combining the observation model (1) and state-transition model (2), a general statespace model for Li-ion battery degradation is established. Different with conventional statespace models, the outputs are not simple value-type observations (e.g. impedance and capacity) (Pedregal and Carnero 2006; Carnero and Pedregal 2010) but curve-type (voltage profiles). With this state space model, we can use the observed voltages profiles to estimate their hidden state θ, predict state evolution trajectory for future cycles, depict the future discharging profiles and construct RUC prognostics on a Li-ion battery. Remark : The above model is built under the steady mode without considering the effects of impacting factors like temperature and discharging current, which can not only influence the discharging profiles but also the degradation of Li-ion batteries. To incorporate the effects caused by these impacting factors, we can treat them as known inputs u i to our state-space model. Then the model can be written as θ i = f i (θ i 1, u i ) + ω i, U i (t) = h(t; θ i, u i ) + ν i (t). (4) As a consequence, the function f i ( ) and h( ) could be more complex, and the corresponding prognostics and inference could be more difficult. 6

7 3. Model estimation and updating In last section, we built a general state-space model which is summarized as following, θ i = Bθ i 1 + b + ω i U i = h(θ i ) + ν i, (5) where U i = [U i (t i1 ),..., U i (t imi )] is the observed voltage profile, h(θ i ) = [h(t i1 ; θ i ),..., h(t imi ; θ i )] is the approximation of profile, θ i is the hidden degradation state, ω i and ν i are the process and observation noises which are both assumed to be zero mean multivariate Gaussian noises with covariance Q and σ 2 I respectively. Suppose we have observed the first k discharging voltage profiles. The observed discharging profiles and their hidden degradation states until current cycle k can be denoted as U 1:k = {U 1, U 2,, U k }, θ 1:k = {θ 1, θ 2,, θ k } (6) The model parameter can be denoted as Φ = [B, b, Q, σ 2 ] Hidden state estimation With these obervations, if we have known the model parameters, it is easy to estimate the hidden degradation states {θ i, i = 1,..., k} by EKF algorithm. For convenience, we define ˆθ i j = E(θ i U 1:j ) and V i j = V ar(θ j U 1:j ) as the expectation and variance of θ i which are conditional on the first j discharging profiles respectively. Considering that the discharging profiles are in nonlinear form apparently, to apply EKF, we first linearize the observation function h(θ i ) at ˆθ i i 1 as h(θ i ) = h(ˆθ i i 1 ) + H i (θ i ˆθ i i 1 ) (7) where H i = h θ ˆθi i 1. The general EKF algorithm is composed of two steps as follows. Extend Kalman filtering algorithm For i = 1,..., k, the filtering procedure can be summarized as 7

8 Step 1. Predict: ˆθ i i 1 = Bˆθ i 1 i 1 + b V i i 1 = BV i 1 i 1 B T + Q (8) Step 2. Update: ( K i = V i i 1 H T i Hi V i i 1 H T i + σ 2 I ) 1 ˆθ i i = ˆθ ) i i 1 + K i (U i h(ˆθ i i 1 ) (9) V i i = (I K i H i ) V i i 1 Through the EKF algorithm, we can only estimate the hidden degradation state conditioning on the observed profiles by current cycle (i.e. ˆθi i and V i i ). If we want to get the estimation of the hidden degradation states conditioning on all observed profiles (i.e. θ i 1 k, V i 1 k and V i,i 1 k ), we should construct backward smoothing after forward filtering. Backward smoothing For i = k, k 1,..., 1, the backward recursion can be summarized as below, J i 1 = V i 1 i 1 B T V i i 1 θ i 1 k = ˆθ i 1 i 1 + J i 1 (ˆθi k ˆθ i i 1 ) (10) V i 1 k = V i 1 i 1 + J i 1 ( V i k V i i 1 ) J T i 1 with the initial state ˆθ k k and V k k derived from the forward filtering. The covariance V i,i 1 k can also be computed for i = k 1, k 2,..., 1, as V i,i 1 k = V i i J(i 1) + J i ( V i+1,i k BV i i ) J T i 1 (11) where the recursion is initialized with V k,k 1 k = [I K k H k ] BV k 1 k 1 (12) The combination of EKF and backward smoothing is called exerted Kalmal smoothing (EKS). Details of EKS can be found in Borkar et al. (2010) and Schön et al. (2011). We 8

9 only present the final formulations Parameter estimation and updating However, in more cases, the primary estimation of model parameters according to historical data are not accurate. As more and more discharging profiles are available, the model parameters should be revised for prognostics. Thus, it is important to provide a method for model parameter estimation and updating. Due to the existence of hidden degradation states θ 1:k, it is difficult to estimate the parameter Φ directly by maximum-likelihood estimation (MLE) approach. Hence we employ expectation-maximum (EM) algorithm in which θ 1:k are treated as missing data and (U 1:k, θ 1:k ) are complete data. By iteratively computing and maximizing the conditional expectation of log-likelihood function consisting on the complete data, the EM algorithm can generate a sequence of parameter estimates which converge to the MLE of the parameter (Bučar et al. 2004). The joint log-likelihood function of both the measurement sequence U 1:k and the degradation state sequence θ 1:k can be formulated as [ k L (U 1:k, θ 1:k Φ) = log i=1 p (U i θ i, Φ) ] k i=1 p (θ i θ i 1, Φ) = const k i=1 m i log σ k log Q 2 1 2σ 2 k i=1 (U i h(θ i )) T (U i h(θ i )) 1 2 k i=1 (θ i Bθ i 1 b) T Q 1 (θ i Bθ i 1 b). (13) Let ˆΦ (j) be the estimation of the parameter Φ from the j th iteration of EM algorithm and E ˆΦ(j) { U 1:k } be the conditional expectation operator with respect to a probability density function determined by ˆΦ (j). Then the conditional expectation of the complete data loglikelihood function can be expressed as ( Q Φ, ˆΦ (j)) = E {L (U ˆΦ(j) 1:k, θ 1:k Φ) U 1:k } = const k i=1 m i log σ k log Q 2 } {(H i θ i a i ) T (H i θ i a i ) U 1:k 1 k 2σ 2 i=1 E ˆΦ(j) } {(θ i Bθ i 1 b) T Q 1 (θ i Bθ i 1 b) U 1:k (14) 1 2 k i=2 E ˆΦ(j) where a i = U i h(ˆθ i i 1 ) + H iˆθi i 1 and H i is the derivative of h(θ i ) at θ = ˆθ i i 1 defined ( in (7). To calculate Q Φ, ˆΦ (j)), we must derive the conditional expectation of each term on 9

10 the right-hand side of (14). With this in mind, for i = 1, 2,..., k, we first define the following quantities: ˆθ i k = E {θ ˆΦ(j) i U 1:k } { } ˆP i k = E θi θ T ˆΦ(j) i U 1:k { } ˆP i,i 1 k = E θi θ T i 1 U ˆΦ(j) 1:k ˆV i k = ˆP i ˆθ i kˆθt i k ˆV i,i 1 k = ˆP i,i 1 ˆθ i kˆθt i 1 k. And these quantities can be obtained by the above EKS algorithm. Considering Φ is comprised of parameters B, b, Q and σ 2, it is difficult to directly derive ( the Φ (j+1) which can maximizes Q Φ, ˆΦ (j)), i.e., (15) ( ˆΦ (j+1) = arg max Q Φ, ˆΦ (j)). (16) Φ Therefore, w choose to use the EM equation where one maximizes each parameter matrix in Φ one by one. In this case, the parameters that are not being maximized are set at their ( iteration j values, and we take the derivative of Q Φ, ˆΦ (j)) with respect to the parameter of interest and solve for its value by setting the partial derivative to zero. The final update equations for each parameter in Φ are summarized as following. The related details about derivation are put in Appendix. (ˆθi k Bˆθ i 1 k ) b (j+1) = 1 k k i=1 ( B (j+1) k ( )) ( = i=1 ˆP i,i 1 k bˆθ T k ˆP ) 1 i 1 k i=1 i 1 k Q (j+1) ( = 1 k k i=1 ˆP i k ˆP i,i 1 k B T B ˆP i,i 1 k ˆθ i k b T bˆθ T i k+ B ˆP ) i 1 k B T + Bˆθ i 1 k b T + bˆθ T i 1 kb T + bb T ( ( ) ) T ) σ 2(j+1) 1 = k k i=1 m i=1 T r H i ˆV i k H T i + (H iˆθi k a i (H ) iˆθi k a i i (17) From the above derivations, we can observe that, once the new observation data (discharging profiles) are available, we first utilize the EM algorithm to update the model parameters. This process takes little time. Subsequently, we estimate and update the degradation state at the current cycle by EKF. In this way, the model parameters and hidden degradation 10

11 states can be timely updated and prepared for prognostics. 4. State-space based prognostics Through the model estimation and updating, we are able to obtain the PDF of θ k conditional on the observed discharging profiles up to the k th cycle, i.e., θ k MN ) (ˆθk k, V k k. For the future cycles, since the corresponding discharging profiles {U k+1, U k+2,...} have not been collected, we project all possible state degradation paths to construct prognostics. In general, the recursive representation of state prediction for cycle k + j, j = 1, 2,... is given by where θ k+j 1 k is a random sample from MN ˆθ k+j k = B θ k+j 1 k + b V k+j k = BV k+j 1 k B T + Q (ˆθk+j 1 k, V k+j 1 k ). Suppose we generated g = 1,..., G predicted state transition paths { θ (g) (g) k+1 k, θ k+2 k,...}. With these predicted states, we can easily plot the corresponding discharging profiles by the observation model (1). Denote the predefined cut-off voltage for discharge as ζ. The EoD time of the i th cycle is defined mathematically as (18) T i = inf{t : U i (t) ζ}. (19) By the definition (19), the CDF of T k+j conditional on U 1:k can be computed through P r(t k+j t U 1:k ) 1 G G I g=1 ( ) (g) h(t; θ k+j k) ζ, (20) where I( ) is the indicator function which equals one when the condition is true and equals zero otherwise. Consequently, the probability density function (PDF) of EoD in the (k +j) th cycle can be obtained through kernel smoothing or numerical differentiation of (20). ( ) We can figure out the EoD times for future cycles {T (g) k+1, T (g) k+2,...} by solving h (g) t; θ k+j k = ζ, g = 1,, G; j = 1, 2,. Since the discharging current I is assumed to be constant in this paper, it is easy to obtain capacity fade paths {Q (g) k+1, Q(g) k+2,...}. Once the estimated capacity is equal to or smaller than the predefined threshold Q for capacity fade, the battery 11

12 will be considered to be failed and has to be replaced. The RUC can be defined as L k = inf{j : Q k+j Q Q k > Q }. (21) According to definition (21), the probability mass function (PMF) of RUC can be estimated through P r(l k = J U 1:k ) = P r(l k J U 1:k ) P r(l k J 1 U 1:k ) 1 ) G g=1 (Q G I (g) k+1 > Q,..., Q (g) k+j 1 > Q, Q (g) k+j Q. (22) So far, the general procedure of Li-ion battery degradation prognostics with our model can be summarized as the following Table 1. In the next section, we will construct a case study to illustrate the general procedure, and demonstrate the effectiveness of the proposed state-space-based prognostics model. Table 1: General procedure of Li-ion battery prognostics with the state-space model 1. Data acquisition: measuring the discharging voltage profiles; 2. Empirical analysis: find the observation function f( ) to fit the voltage curves; find the parameters of observation functions in different cycles; 3. Model building: formulate the observation model and state transition model; 4. Parameter estimation: use EM algorithm estimate the model parameters; 5. State estimation: use EKF algorithm estimate the model hidden state; 6. Prediction: predict the states for future cycles, and its EoD time; predict the capacity for future cycles, and RUC; Once new discharging profile is available, repeat the step 4 to Case study In this section, a real case study is presented to demonstrate the feasibility and efficiency of the proposed approach for the Li-ion battery degradation prognostics. The data set we used are from the NASA Ames Prognostics Center of Excellence. The battery we chose for testing is Battery 5 (denoted as B5) in this data set. In the rest of this section, we will follow the procedure shown in Table 1 to conduct the Li-Ion battery prognostics. 12

13 According to the measurement scheme, the chosen testing Li-ion battery went through three different operation profiles (charging, discharging and impedance test) at constant ambient temperature of 24 C. In each discharging process, the load current is 2 A and the preset cut-off voltage is 2.7 V. The threshold for capacity degradation is Q = 1.3 Ahr. Figure 2 shows the acquired discharging voltage profiles. Voltage Figure 2: Voltage profiles during discharging of a Li-ion battery in 138 cycles. Time 5.1. Empirical analysis and model specification As we proposed in Section 2, we build a state-space model to characterize the discharging profiles and substantial battery degradation. In real application, the first step is to find a suitable characterization function h(t; θ) for the observation model. From Figure 2, we can easily observe that all discharging processes can be simply divided into three stages. The first stage is the quick and transient voltage drop (IR drop) at the beginning of discharge. It is caused by the internal impedance on application of load current. After that, the second stage spans majority of the discharging process where the output voltage goes down slowly. In the third stage which happens around the end of discharge, the voltage decreased significantly again until it reaches the predefined cut-off value. Based on above observations, we adopt a smoothing spline with a single knot as the approximation 13

14 Voltage(V) real data fitting curve cycle 100 cycle 1 cycle Time( 10 3 s) Figure 3: Fitting the discharging profiles with the single knot smoothing splines. function h(t; θ). The single knot is used to represent the changing point between the second and third stages. a 0 + a 1 t + a 2 t 2 + a 3 t 3, h(t; θ) = b 0 + b 1 (t τ) + b 2 (t τ) 2 + b 3 (t τ) 3, t τ t > τ, (23) where τ is the change point (knot) of the spline model and a i, b i, i = 0, 1, 2, 3 are the corresponding coefficients of the splines. Especially a 0 represents the instantaneous IR drop volume. To ensure the smoothness of h(t; θ), we impose that it is second order continuous at each point t, particularly at τ. Therefore, the parameters must satisfy b 0 = a 0 + a 1 τ + a 2 τ 2 + a 3 τ 3 b 1 = a 1 + 2a 2 τ + 3a 3 τ 2. (24) b 2 = a 2 + 3a 3 τ Therefore, the parameter vector is θ = [a 0, a 1, a 2, a 3, b 3, τ] in this case. As shown in Figure 3, the proposed single knot smoothing spline function can well approximate the discharging profiles. 14

15 The second step is to find a suitable state transition function. We keep on fitting all 138 discharging profiles and obtain the corresponding states θ 1:138 by the least square estimation method. Each panel in Figure 4 shows the evolution path for one element in θ. Obviously, the hidden states θ i are not stationary as the battery degrades. If we use the simplest linear model (2) as the state-transition model, the long-term/multi-step prediction of future states will be diverging. As a result, we choose to build a stationary model between θ i and θ i 1, i.e., θ i = B θ i 1 + b + ω i, where θ i = θ i θ i 1. In this case, we denote X i = θ i 1 as the new state vector. θ i state-transition model can be expressed as θ i 1 θ i = I I θ i B θ i 1 b The corresponding 0. (25) ω i Accordingly, the observation equation (1) can be written as U i (t) = h(t; θ i 1 + θ i ) + ν i (26) In this way, the newly state-space model can be re-constructed as following, X i = BX i 1 + b + ω i U i = h(t, X i ) + ν i. (27) The proposed prognostics and inference methods are still applicable Results of Prognostics In this part, we demonstrate the efficiency and accuracy of the prognostics in both EoD prediction and RUC prediction. We use the first k = 100 discharging profiles as observations to predict θ i and U i in future cycles (i > 100). Not surprisingly, the model has better prediction accuracy for near future cycles (e.g.,i = 110) than that for far future cycles (e.g., i = 120). Figure 5 shows some of the predicted discharging curves of a few cycles conditioning on the first 100 observed profiles. As can be seen, the real discharging profiles can be approximated well by the posterior mean 15

16 a a (a) (b) a a (c) (d) b τ (e) (f) Figure 4: Evolution of elements in parameter vector θ 16

17 Voltage(V) observed profile predicted profiles mean profile Voltage(V) Time( 10 3 s) (a) i = Time( 10 3 s) (b) i = 120 Figure 5: Predicted discharging curves of future cycles based on the first k = 100 profiles. The black line is the real observations. The green bands are the 95% confidence band based on predicted curves. The red dashed line is the mean of predicted profiles. of the predictions. In addition, they are well covered by the 95% confidence band. We can also notice that the prediction accuracy decreases when the discharging time is longer. This is because the predicted value has larger variance and less accurate mean. However, we find that the third stage of the discharging process is more dynamic and changes more rapidly. Therefore, the prediction accuracy is not as good as the prediction for the second stage discharging. With these predicted discharging curves, given the cut-off voltage, we can predict the EoD times for these cycles which are concluded in the Table 2. Table 2: EoD prediction results Cut-off voltage Current cycle Prediction cycle True EoD ( 10 3 EoD estimation ( 10 s) 3 s) Mean STD V As we mentioned earlier, one advantage of our approach is that our prognostic model does not depend on the cut-off voltage ζ in each cycle. Based on the predicted profile and different ζ, we can even find the distribution of the corresponding EoD times. Figure 6 shows the distributions of EoD times of cycle i = 110 when the cut-off voltages are ζ = 2.7V and 3.0V respectively. It demonstrates that regardless of the chosen ζ, the distribution of EoD 17

18 Density EoD density real EoD Density EoD time( 10 3 s) (a) ζ = 2.7V EoD time( 10 3 s) (b) ζ = 3.0V Figure 6: Conditional density of EoD times, n = 100 and i = 110. The black solid line is the density, and the red dashed line indicates the real EoD value. can be accurately predicted satisfactorily. Aside from predicting EoD, we can also predict RUC with ease according to (22). In the analysis, the voltage cut-off is set to ζ = 2.7 V. Given the threshold for capacity degradation Q = 1.3 Ahr and the constant discharging current 2 A, the corresponding EoD threshold is T = s before the battery failure. Figure 7 shows the capacity degradation assessment and prediction based on the first k = 90 and k = 110 observed voltage profiles. The black line stands for the real capacity fading path, the lightblue lines stand for predicted degradation paths and the red dash line stands for the mean degradation path. We can find that with more observed profiles, prediction of the capacity (EoD) degradation becomes more precise. Not only the variance becomes smaller, but also the mean path gets closer to the real one. At the mean time, we can compute the distribution of RUC according to (22). The density curves of the RUC distributions at different prediction times are also compared for reference. The quantitative results of RUC prediction are presented in the Table Conclusion This paper proposes a novel data-driven prognostics method for Li-ion batteries degradation. Different with previous works, this method don t rely on the measurements of battery 18

19 Capacity(Ahr) real capacity fade predicted capacity mean of capacity Capacity(Ahr) (a) Capacity prediction when k = 90 (b) Capacity prediction when k = 110 Density RUC density real RUC Density RUC (c) RUC when k = RUC (d) RUC when k = 110 Figure 7: Capacity degradation assessment and RUC prediction Table 3: RUC prediction results RUC estimation Capacity Threshold Current cycle True RUC Mean STD Ahr

20 capacity or impedance, but take full use of the easy-to-measure voltage profiles. In summary, we use the same parametric function to approximate the discharging voltage curves. The changing parameters of different cycles are used to reflect the battery degradation. Based on this idea, we build a state-space model and adopt the EM and EKF algorithm to estimate the model parameters and states jointly. With this model, we can predict the voltage profiles of future cycles. Given any cut-off voltage, we can further estimate the corresponding end of discharge (EoD) times and assess the releasable capacities. If the capacity degradation threshold is defined, we can finally predict the remaining useful cycles (RUC). The proposed method is suitable for various types of lithium-ion batteries just with suitable observation and state-transition models. This approach is also adaptive in real applications because the parameters and states can be efficiently updated once the new observations are available. Moreover, the control inputs in state-space model allows incorporating the influence of degradation impacting factors, like environmental temperature, discharging current, and etc.. However, there still remains considerable room for improvements. In this paper, we limit the use of lithium-ion batteries under a steady mode. Actually, in many applications, the discharging current and the environmental conditions could be fluctuated. In those situations, the discharging profiles for different cycles could be hard to characterized by data-driven methods efficiently. What s more, the hidden state in our model is not real physical properties of lithium-ion batteries. This is not helpful for understanding the nature of lithium-ion battery degradation. In the future, we consider to incorporate a physical model into the PHM approach and track the degradation of real battery properties. Appendix A1. Testing results on Battery 7 In the case study section, we tested the proposed model on Battery 5 (denoted as B5) in the NASA Battery Data Set. Besides this sample, we also applied the model on Battery 7 (denoted as B7). Figure 8a shows the different capacity fade paths of B5 and B7. Figure 8b shows the discharging voltage profiles of the two batteries. The cut-off voltage for B7 is 2.2 V, and the cut-off voltage for B5 is 2.7 V. 20

21 Vo age p o es o B5 Vo age p o es o B Capacity(Ahr) Capacity(Ahr) 1.8 Capacity fade of B5 Capacity fade of B (a) (b) F gure 8 Compar son between B5 and B7 The testing results on B5 have been presented in the case study section. Considering the voltage profiles of B5 and B7 are similar, we put the testing results of B7 here. The testing results can be divided into two parts similarly. Part 1. Given the first 100 voltage profiles, we predict the possible voltage profiles of the 110th and 120th cycles. As shown in Figure 9, the black lines are the real discharging voltage profiles, the lightblue lines are the predicted possible voltage profiles, and the red dashed lines are the mean of predictions Vo age d op V Vo age d op V obse ved p o e p ed c ed p o es mean p o e T me 10 s T me 10 s (a) (b) F gure 9 Test ng resu ts of B7 Part 1 vo tage profi es pred ct on 21 25

22 Part 2. Given the first 90 or 110 voltage profiles, we predict the possible capacity degradation paths. Here we assume the cut-off voltages keep the same for different cycles, and equals 2.2 V. As shown in Figure 10, the black dot lines are the real capacity fade paths, the lightblue lines are the predicted possible capacity fade paths, and the red dashed lines are the mean of predictions. Capacity(Ahr) real capacity fade predicted capacity fade mean of capacity fade Capacity(Ahr) (a) (b) Figure 10: Testing results of B7, Part 2: capacity degradation prediction A2. Simple comparison of RUC prediction We compare our method with a commonly used exponential capacity degradation model (He et al. 2011) C = α 1 exp(α 2 i) + α 3 exp(α 4 i); (28) Based on the capacity measurements in historical cycles, parameters in (28) can be estimated, and future capacities can be predicted. Figure 11a and 11b compare the predicted capacities by two models based on the data of the first 90 and 110 cycles, respectively. Red dash lines are the mean predictions of our method and blue solid lines are the predictions of (28). We can observe that the predictions are very similar to each other. Unfortunately, we do not have sufficient batteries to conduct a comprehensive comparison. In this paper, we included this result, but also highlighted that no statistical conclusion can be made about the two methods. 22

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