An ensemble model for predicting the remaining ... - Semantic Scholar
Microelectronics Reliability 53 (2013) 811–820
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Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
An ensemble model for predicting the remaining useful performance of lithium-ion batteries Yinjiao Xing a,b,⇑, Eden W.M. Ma a, Kwok-Leung Tsui a,b, Michael Pecht c a
Centre for Prognostics and System Health Management, City University of Hong Kong, Kowloon, Hong Kong Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong c Center for Advanced Life Cycle Engineering (CALCE), University of Maryland, College Park, MD 20740, USA b
a r t i c l e
i n f o
Article history: Received 16 June 2012 Received in revised form 23 October 2012 Accepted 7 December 2012 Available online 11 January 2013
a b s t r a c t We developed an ensemble model to characterize the capacity degradation and predict the remaining useful performance (RUP) of lithium-ion batteries. Our model fuses an empirical exponential and a polynomial regression model to track the battery’s degradation trend over its cycle life based on experimental data analysis. Model parameters are adjusted online using a particle filtering (PF) approach. Experiments were conducted to compare our ensemble model’s prediction performance with the individual results of the exponential and polynomial models. A validation set of experimental battery capacity data was used to evaluate our model. In our conclusion, we presented the limitations of our model. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In battery-powered systems, lithium-ion batteries are being widely used in consumer electronics, electric vehicles, and even space systems [1]. Compared with lead-acid, nickel–cadmium, and nickel–metal-hydride batteries, lithium-ion batteries offer a higher energy density, longer cycle life, and lower self-discharge rate [2,3]. However, an inevitable problem is that battery performance degrades with cycling and aging. To use a battery in an electric vehicle as an analogy, it is similar to a ‘‘fuel’’ tank that shrinks with use so that the driver can never know what a full or partial fuel reading actually means with respect to miles that would be achievable. Since battery degradation cannot be measured directly, there is a need to estimate its maximum available performance and give users an accurate estimate of battery capacity ahead of time so that decisions for battery replacement can be made. Battery capacity is widely used as an indicator of battery status. Battery performance is represented by the state of health (SOH), which is a ratio of the present capacity to the initial capacity of a battery [4–6]. In the battery community, end of performance (EOP) is usually set as the time that the maximum available capacity is reduced to 80% of its initial value [7,8]. Remaining useful performance (RUP) is defined as the length of time from the time of observation to the time when the EOP criterion is reached. In the battery management systems (BMSs), estimating battery state of charge (SOC) has been conducted and implemented in the form ⇑ Corresponding author at: Centre for Prognostics and System Health Management, City University of Hong Kong, Kowloon, Hong Kong. E-mail addresses: [email protected] (Y. Xing), [email protected] (E.W.M. Ma), [email protected] (K.-L. Tsui), [email protected] (M. Pecht). 0026-2714/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.microrel.2012.12.003
of the remaining time or range [9–13]. However, RUP prediction is still at the premature stage. There are three main challenges in RUP prediction. First, uncertainties, including operational and environmental factors, unit-tounit variation, and measurement noise, will impose the difficulties on RUP prediction. Second, it is difficult to estimate the maximum releasable capacity of a battery because a battery is usually not fully discharged in real applications. Furthermore, there is still controversy over the definition of battery SOH. More factors, such as internal impedance, self-discharge, and cycle counting, are suggested for evaluating SOH [1,11,14]. The uncertainty of SOH estimation will also influence RUP prediction. Predictions of battery RUP have been performed using both physics of failure (PoF) model and data-driven approaches. Most practitioners have adopted electrochemical impedance spectroscopy (EIS) measurements to establish physics-based models by extracting the internal parameters, which are able to characterize varying aging and fault processes of the battery. Based on these battery models, data-driven methods, such as auto-regressive moving average, fuzzy logic, neural network, support vector machine, and nonlinear filtering methods, are combined to predict the RUP of batteries [4,11,15]. However, online applications of EIS measurements suffer from size limitations and high cost. Moreover, it is time-consuming and difficult to quantify the internal impedance at low frequency [11]. Another key problem is that the noises caused by other integrated components of an online system have an impact on the accuracy of the EIS measurements due to the slight excitation signal required for EIS measurement. Prediction based on the empirical regression models has been recommended to address these challenges in [16–18]. Taking into account the limitations of EIS techniques, He et al. [16] introduced
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The developed model is then validated to demonstrate its robustness using a different battery set. Our conclusions are presented in Section 5.
an empirical exponential model to conduct curve fitting for capacity fade data. The model parameters were initialized via Dempster– Shafer theory, which is effective for fusing varieties of battery data and apply in the parametric initialization. The parameters were updated step-by-step (each cycle) through particle filtering (PF). Micea et al. [17] adopted an empirical second-order polynomial model as a function of cycle number to estimate the stored maximum capacity. The proposed least square algorithm was implemented to estimate the capacity function. Other new prognostics-related methods and models can be found in [19–21]. In this paper, we presented an ensemble degradation model that integrates two regression models: an exponential model and a polynomial model [16,17]. The ensemble model is a more accurate parametric model than each of the above two models because it takes into account onboard applications and global and local regression characteristics. Since the model parameters should vary with the dynamic characteristics of battery degradation, PF is used to estimate and adjust the model parameters and hence to track the battery aging process with nonlinear and non-Gaussian characteristics. The predicted results of RUP are presented with a narrow probability distribution. Moreover, we used another battery set with a different rated capacity to validate the performance of our model. The effectiveness and robustness of our model was verified through comparisons with the other two regression models. This paper is organized as follows: Section 2 conducts the regression analysis of capacity degradation based on three different models: the exponential model, the polynomial model and our model. Our new model is presented based on our experimental data and comparative results. Section 3 introduces the PF approach for state estimation. Then, the procedure for battery state estimation and prediction is presented in terms of integrating the regression model with the PF approach. Section 4 shows the RUP results and performance comparison according to the above framework.
Cell #01
2. Empirical models 2.1. Capacity measurement Capacity indicates the amount of charge that comes out of a battery from the full to empty state, corresponding to the upper and lower cut-off voltages preset by manufacturers [14]. Capacity is calculated by integrating the current over time [22]. The SOH indication used in this paper is based on the maximum available capacity, which can be obtained by discharging from full charge (100%) to empty charge (0%) in each cycle. To develop the degradation model for SOH estimation, six batteries under test went through the full charge and discharge procedure. The rated capacity of the tested lithium-ion battery was 1.35 A h. Using the battery test equipment, Arbin BT2000, the batteries were fully charged under the constant-current/constant-voltage mode, which is widely used in the industry. Then, they were discharged at a 1C rate, which meant 1.35 A for the battery discharge in this case. The test was run at room temperature, which was approximately 25 °C. The discharge capacity was recorded after each full charge–discharge process. The capacity fade is shown in Fig. 1, while the EOP was defined as 80% of its initial capacity. 2.2. Regression analysis based on different models As mentioned in the introduction, an exponential model (model A) [16] and a polynomial model (model B) [17] have been empirically established through fitting a large amount of battery degrada-
Cell #02
1.2
1.1
1.3
Real data Model A
Capacity (Ah)
1.3
Real data Model A
Capacity (Ah)
Capacity (Ah)
Real data Model A
Cell #03
1.4
1.4
1.4
1.2
1.1 End of Performance
1.3 1.2 1.1 End of Performance
End of Performance
1
0
200
400
600
1
800
0
200
Cycle
400
600
1
800
0
200
Cycle
400
600
800
Cycle
(a) Cell #01
Cell #02
1.4
Real data Model B
1.2
1.1
1.3
Real data Model B
Capacity (Ah)
Capacity (Ah)
Capacity (Ah)
1.3
Cell #03
1.4
1.4 Real data Model B
1.2
1.1 End of Performance
1.3 1.2 1.1 End of Performance
End of Performance
1
0
200
400
600
Cycle
800
1
0
200
400
600
800
Cycle
(b) Fig. 1. Curve fitting based on (a) model A, and (b) model B.
1
0
200
400
600
Cycle
800
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tion data. Model A is made up of two exponential functions, while model B is a polynomial regression model. Model A (exponential model):
C Ak ¼ C Ak1 þ C Ak2 ¼ a1 expða2 kÞ þ a3 expða4 kÞ
ð1Þ
Model B (polynomial model): 2
C Bk ¼ C Bk1 þ C Bk2 ¼ b1 k þ b2 k þ b3
ð2Þ
where CAk and CBk are the capacity of the battery and k is the cycle number; a1, a2, a3, a4 and b1, b2, b3 are the model parameters to be estimated in Eqs. (1) and (2), respectively. The capacity data over the whole life (to 80% of the initial capacity) were used to evaluate the goodness-of-fit of these two parametric models. Fig. 1a and b shows curve fittings for Cell_#01#03 based on models A and B, respectively. The model parameters of models A and B were fitted in a MATLAB environment. With respect to each model’s characteristics, model A was fitted using the nonlinear least square method, while model B was estimated using a linear least square. Both criteria aim to minimize the sum of the squares of the errors [23]. The more accurate the model is, the better the prediction performance will be. As shown in Table 1, the adjusted R-square (R2adj ) and root mean square error (RMSE) are two indices to compare the goodness-of-fit of these two models. R2adj is a modification of R-square to compensate for the extra variables included in the model. The best fit is indicated by ‘‘1’’ in R2adj and ‘‘0’’ in RMSE. Model A showed a better global regression performance than model B using all the capacity data before EOP. However, the fitting result cannot characterize the local regression performance of both models. In fact, the capacity fade of a lithium-ion battery could have several stages with different degradation trends. Spotnitz [7] partitioned the capacity fade into four regions according to manufacturers’ information, while Zhang and White [24] divided the degradation into three stages based on experimental data. In this paper, all the capacity data before hitting the EOP are partitioned into three identical portions. This partition does not distinguish different fade patterns, but is used to analyze the local regression characteristics of models A and B, and employ an ensemble model that provides the potential for better fits to the whole life data. The regression results when the models are fitted in the different portions are shown in Table 2.
Table 1 Goodness-of-fit of models A and B. R2adj
No. of cell
Cell_#01 Cell_#02 Cell_#03 Cell_#04 Cell_#05 Cell_#06
RMSE
Model A
Model B
Model A
Model B
0.9832 0.9497 0.9803 0.9609 0.9749 0.9712
0.9795 0.9442 0. 9793 0.9517 0.9664 0.9603
0.0101 0.0144 0.0097 0.0130 0.0101 0.0108
0.0112 0.0152 0.0100 0.0144 0.0118 0.0127
Table 2 Goodness-of-fit of models A and B under different data portions. No. of cell
R2adj
(the first one-
third)
Cell_#01 Cell_#02 Cell_#03 Cell_#04 Cell_#05 Cell_#06
R2adj
(the second
one-third)
R2adj
For the first one-third of the data, larger R2adj values were obtained from model A. However, for the second portion, model B presents a better fit for Cell_#03#06. For the last portion, model A only has a better goodness-of-fit than B and C on only three cells (i.e., Cell_#01, #04, and #06). It can be seen that model A is suitable for tracking degradation earlier in the battery life. Thus, a poor-fitting on the particular data offers an opportunity to develop a more accurate model, which not only has a better global regression, but also fits the local data well, especially the last 1/3 of life data. It is worth noting that the poor-fitting are presented for Cell_#01 and #02 were due to the large fluctuation in capacity. It can also be found in Fig. 1. Thus, to evaluate the performance of the prognostic approaches integrated with our developed model, Cell_#03#06 were used as training samples to establish the degradation model and estimate model parameters while Cell_#01 and #02 were used for testing in Section 4. 2.3. Model selection and validation 2.3.1. Model selection Since both models A and B only fit part of the data well, we developed an ensemble model to fuse these two models in order to achieve a high global goodness-of-fit as well as a more accurate prediction from the middle and late stages of degradation. Although a weighting factor can be used to form a sophisticated model that combines these two models, more parameters need to be estimated and probably lead to over-fitting. Through fitting the experimental capacity data, the ensemble model (model C) is developed as follows: In model A in Eq. (1), only one of the exponential parts (CAk2) plays a vital role in the regression process. Taking the training samples (Cell_#0306) as examples, CAk1 can be ignored compared to the scale of CAk2 (Table 3). In model B in Eq. (2), the quadratic and linear items are compared to reduce the possible number of model parameters. Since both CBk1 and CBk2 refer to the variable k, a comparison is conducted among three models: CAk2 + CBk1 + CBk3 (model C1), CAk2 + CBk2 + CBk3 (model C2), and CAk2 + CBk1 + CBk2 + CBk3 (model C3). When comparing regression models that use the same dependent variables and the same estimation period, the fact that RMSE goes down is equivalent to when R2adj goes up. In Table 4, R2adj is kept to quantify the regression performance here. In addition, the Akaike information criterion (AIC) is introduced to make auxiliary assessment for model selection in order to impose a heavier penalty on model complexity. 2.3.2. Model validation Model accuracy and prediction performance will be validated in this section. Two aspects will be considered: (1) the accuracy of the new models in terms of the different types and the number of parameters; and (2) the over-fitting issue due to the increased number of parameters. AIC is measured to provide validation among a series of nonlinear models and is widely used for model selection to consider model complexity. AIC is suitable for selecting the preferred model as a supplement to other indicators (i.e., R2adj , RMSE) and includes a penalty for increased model complexity [23].
AIC ¼ 2m InðLÞ;
ð3Þ
(the last one-
third)
Model A
Model B
Model A
Model B
Model A
Model B
0.2915 0.7632 0.9259 0.8824 0.8984 0.9160
0.2244 0.6849 0.8669 0.8273 0.8361 0.8485
0.8866 0.5153 0.6050 0.7417 0.8284 0.7427
0.8852 0.5075 0.6621 0.8018 0.8698 0.7799
0.9234 0.7925 0.9028 0.7302 0.6986 0.7926
0.8603 0.8011 0.9354 0.5517 0.7188 0.6950
Table 3 Variation of CAk1 and CAk2 (A h) over k in the whole life. No. of cell
Variation of CAk1
Variation of CAk2
Cell_#03 Cell_#04 Cell_#05 Cell_#06
0.0278–1.066e17 0.091–6.843e05 0.0682–2.865e06 0.0766–8.495e06
1.336–1.104 1.260–1.097 1.290–1.106 1.279–1.106
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Table 4 Comparison among three models.
Cell_#01 Cell_#02 Cell_#03 Cell_#04 Cell_#05 Cell_#06
CAk2 + CBk1 + CBk3 (model C1)
CAk2 + CBk2 + CBk3 (model C2)
CAk2 + CBk1 + CBk2 + CBk3 (model C3)
R2adj
AIC
R2adj
AIC
R2adj
AIC
0.9796 0.9525 0.9863 0.9609 0.9747 0.9716
5216 3659 5246 3716 4081 3852
0.9790 0.9510 0.9793 0.9609 0.9468 0.9713
5191 3639 4925 3716 4086 3846
0.9797 0.9531 0.9864 0.9609 0.9749 0.9717
5217 3663 5245 3714 4084 3851
Model C1 1.35 1.25 1.2 1.15 1.1 End of Performance 1.05
Model C3
Real data Fitted Curve Predicted Curve
1.35
C apacity (A h)
1.3
Capacity (Ah)
Model C2
Real data Fitted Curve Predicted Curve
1.3 1.25 1.2 1.15 1.1 End of Performance 1.05
1.3 1.25 1.2 1.15 1.1 End of Performance 1.05
1
1
1
0.95
0.95
0.95
0
100 200 300 400 500 600 700 800 900
0
100 200 300 400 500 600 700 800 900
Cycle
Real data Fitted Curve Predicted Curve
1.35
C apacity (A h)
No. of cell
0
100 200 300 400 500 600 700 800 900
Cycle
Cycle
Fig. 2. Prediction based on models C1, C2, and C3 individually under the first 75% of capacity data.
where m is the number of parameters (a penalty) and In(L) is the maximum value of the log-likelihood of the estimated model.
3. Particle filtering (PF) for RUP prediction
" ! # PN ^ 2 N i¼1 ðyi ypi Þ þ1 InðLÞ ¼ In 2p 2 N
Accurate state estimation and prediction not only rely on an accurate model, but also depend on the adjustments of model parameters to track the variation in capacity fade. In order to characterize the dynamic capacity fade, the particle filtering (PF) approach is used to estimate the current capacity. The core idea of PF is based on Bayesian filtering and Monte Carlo simulation. Through Bayes’ Theorem, PF is able to provide a probability estimation for nonlinear and non-Gaussian systems [26]. In order to approximate the marginal distribution, Monte Carlo simulation is used to generate a large amount of random particles and estimate the posterior density function through accumulating these particles with associated weights. Fig. 3 shows the flowchart of RUP prediction.
ð4Þ
^pi calculates the residuals bewhere N is the sample size, while yi y tween the true value and the estimated value. The lower the AIC is, the better the model will be [25]. As shown in Table 4, the goodness-of-fit based on models C1 and C3 is generally better than that of model C2. However, AIC is not sufficient to reflect the model over-fitting that could be triggered by estimating more parameters. Therefore, the prediction performance of the model needs to be validated. In this study, predicting was conducted to extrapolate the estimated curve based on the previous fitting using the first 75% of the capacity data. Model C3 could possibly reduce the training errors because of having greater complexity, but would pose a risk of over-fitting and cause a poor prediction at the same time. Fig. 2 presents an over-fitting result under model C3. Thus, model C1 is regarded as an ensemble model in terms of a good fitting result and predicted validation. Later, the ensemble model will be represented as model C: 2
C Ck ¼ c1 expðc2 kÞ þ c3 k þ c4
ð5Þ
3.1. Recursive Bayes The state-space model is used to solve the tracking problem using the nonlinear process function, fk, and measurement function, hk:
c1;k1 3 2 #1;k1 3 7 6 7 6c 6 2;k1 7 6 #2;k1 7 xk ¼ fk ðxk1 ; #k1 Þ ¼ 6 7þ6 7 $ pðxk jxk1 Þ 4 c3;k1 5 4 #3;k1 5 2
c4;k1
ð6Þ
#4;k1
where CCk is the capacity of the battery and k is the cycle number. Table 5 Prediction results for Cell_#01 and #02 at different cycle numbers. Real failure cycle time
a b
Prediction cycle (% of life)
Model A
Model B
Model C
Error
STD
Error
STD
Error
STD
1a 49 1
284.2 167 163.3
27b 103 103
88.6 46.4 25.8
82 129 119
30.1 17.3 7.6
9 14 10
29 62.9 30.7
Cell _#01
849
283 (1/3) 425 (1/2) 566 (2/3)
325 202 191
167.6 57.4 56.8
Cell _#02
643
214 (1/3) 322 (1/2) 428 (2/3)
304 133 42
50.5 22.8 65.2
Negative values mean the prediction result is earlier than the actual end of performance. 1 Means that the prediction does not converge in Fig. 4.
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Model A
(a)1.4
Real Data Estimated Values Predicted Values
1.35
Capacity (Ah)
1.3 1.25 1.2 1.15 1.1
Prediction Time End of Performance
1.05 1
0
200
400
600
800
Cycle Number Model B
(b)1.4
Real Data Estimated Values Predicted Values
1.35
Capacity (Ah)
1.3 1.25 1.2
RUP pdf
1.15 Prediction Time
1.1 End of Peformance
1.05 1
0
200
400
600
800
Cycle Number Model C
(c)1.4
Real Data Estimated Values Predicted Values
1.35
Capacity (Ah)
1.3 1.25 1.2 1.15
Fig. 3. Flowchart of implementation of RUP prediction.
yk ¼ hk ðxk ; uk Þ $ pðyk jxk Þ
ð7Þ
where fxk ; k 2 Ng is the state vector that is assumed as the unobserved Markov process; fyk ; k 2 Ng are the observations or measurements that are conditionally independent given the process; #k is an i.i.d. process noise sequence; uk is an i.i.d. measurement noise sequence; k is the time index; p(xk|xk1) is the transition distribution; and p(yk|xk) is the observation distribution. Corresponding to our case, the state xk is the vector of model parameters [c1k, c2k, c3k, c4k]T to be estimated, while #k1 is distributed as Nð0; r2i IÞ; i ¼ 1; 2; 3; 4 . . . yk is the capacity sequence that is measured after a full discharge through Coulomb counting methods. The recursive Bayesian method provides a generic way to estimate the posterior probability density function (pdf) p(xk|y1:k), given the capacity and the posterior expectation I(fk) based on the sequence of measured capacity.
Iðfk Þ ¼
Z
fk ðxk Þpðxk jy1:k Þdxk
RUP pdf
1.1
ð8Þ
Through recursive Bayesian filtering, prediction and update will be recursively implemented in two steps. The first step is to achieve the prior pdf p(xk|y1:k1), which means that the state xk is
Prediction Time End of Performance
1.05 1
0
200
400
600
800
Cycle Number Fig. 4. Prediction results at cycle 283 (reaching 1/3 of life) for Cell_#01 based on three models. (a) Prediction result at 1/3 of life based on model A. The intersection of the projection with the EOP exceeds the length of the range of test data. (b) Prediction result at 1/3 of life based on model B. The intersection of the projection with the EOP exceeds the length of the range of test data. (c) Prediction result at 1/3 of life based on model C. The predicted RUP is 27 cycles earlier than the real one. The STD of the predicted RUP is 88.6 cycles.
inferred from the measurements y1:k1 = {yn, n = 1, . . . , k 1}. The second step is to obtain the posterior pdf p(xk|y1:k) in terms of the current measurement:
pðxk jy1:k1 Þ ¼
pðxk jy1:k Þ ¼
Z
pðxk jxk1 Þpðxk1 jy1:k1 Þdxk1
pðxk jy1:k1 Þpðyk jxk Þ pðyk jy1:k1 Þ
ð9Þ
ð10Þ
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Model A
(a) 1.4
Real Data Estimated Values Predicted Values
1.1
1.3
Capacity (Ah)
Capacity (Ah)
1.35
Model A
(a)
Real Data Estimated Values Predicted Values
1.25 Prediction Time
1.2 1.15
RUP pdf
1.1 End of Performance
1 RUP pdf
0.95 0.9 End of Performance
1.05 1
Prediction Time
1.05
0.85
0
200
400
600
800
0.8
Cycle Number
0
200
400
600
800
Cycle Number
Model B
(b)1.4
Real Data Estimated Values Predicted Values
1.1
1.3 Prediction Time
1.25
1.05
Capacity (Ah)
Capacity (Ah)
1.35
Model B
(b)
Real Data Estimated Values Predicted Values
RUP pdf
1.2 1.15 1.1 End of Peformance
1 0.95 Prediction Time
0.9
1.05 1
End of Performance
0.85
0
200
400
600
800 0.8
Cycle Number
Prediction Time
1.25
Prediction Time
1.05
1.2 RUP pdf
1.15
800
Real Data Estimated Values Predicted Values
1.1
Capacity (Ah)
Capacity (Ah)
600
Model C
(c)
1.3
1.1 End of Performance
1 0.95 RUP pdf
0.9
1.05
End of Performance
0
200
400
600
0.85
800
Cycle Number
0.8
Fig. 5. Prediction results at cycle 428 (reaching 2/3 of life) for Cell_#02 based on three models. (a) Prediction result at 2/3 of life based on model A. The predicted RUP is 42 cycles later than the real one. The STD of the predicted RUP is 65.2 cycles. (b) Prediction result at 2/3 of life based on model B. The predicted RUP is 119 cycles earlier than the real one. The STD of the predicted RUP is 7.6 cycles. (c) Prediction result at 2/3 of life based on model C. The predicted RUP is 10 cycles later than the real one. The STD of predicted RUP is 30.7 cycles.
where the normalizing constant is:
Z
400
Cycle Number Real Data Estimated Values Predicted Values
1.35
pðyk jy1:k1 Þ ¼
200
Model C
(c)1.4
1
0
pðxk jy1:k1 Þpðyk jxk Þdxk
0
200
400
600
800
Cycle Number Fig. 6. Prediction result at cycle 211 (reaching 1/3 of life) for Cell_#V4 based on three models. (a) Prediction result at 1/3 of life based on model A. The predicted RUP is 83 cycles later than the real one. The STD of the predicted RUP is 23.7 cycles. (b) Prediction result at 1/3 of life based on model B. The projection diverges, but does not converge to the EOP. (c) Prediction result at 1/3 of life based on model C. The predicted RUP is 121 cycles earlier than the real one. The STD of the predicted RUP is 19.0 cycles.
ð11Þ
where p(yk|xk) is determined by (7). 3.2. Monte Carlo simulation for posterior pdf Often, there is no analytical solution to Eqs. (9) and (10). PF therefore utilizes Monte Carlo (MC) simulation to approximate the probability density with a large amount of weighted samples,
which are the weighted particles. Thus, fxi0:k ; xi0:k g is a random measured pair that characterizes the posterior pdf, where fxi0:k ; i ¼ 0; . . . ; N s g is the associated weights of a set of particles fxi0:k ; i ¼ 0; . . . ; N s g, and Ns is the total number of particles. The posterior pdf at the kth cycle can be approximated as:
pðxk jy1:k Þ
Ns X
xik dðxk xik Þ
i¼1
ð12Þ
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Y. Xing et al. / Microelectronics Reliability 53 (2013) 811–820
Model A
(a)
Real Data Estimated Values Predicted Values
1.1
where d() is a Dirac function. The weights are normalized P i i as i xk ¼ 1. The sample xk is drawn from importance density q(xk|y1:k) based on the importance sampling [27,28]. Through recursive relation, the weights are updated as follows:
Capacity (Ah)
1.05
xik / xik1
1
Prediction Time
0.9 End of Performance
0.85 0.8
0
200
400
600
800
Model B Real Data Estimated Values Predicted Values
1.1
Capacity (Ah)
1 0.95 RUP pdf
Prediction Time
0.9 End of Performance
0.85
0
200
400
600
800
Cycle Number
^IN ðfk Þ ¼ s
XN s
f ðxi Þ i¼1 k 0:k
xik
ð15Þ
Real Data Estimated Values Predicted Values
1.1
Prediction of the expectation and the posterior pdf % p(xk+j|y1:k) of the state at the j-step (j = 2, . . . , T k e N) ahead is expected. T is the time horizon related to the number of cycles of the battery. Since the new measurement yk+j has not been collected, the present or measured state vector and its current posterior distribution p(xk|y1:k) would be projected among all possible future paths. Each capacity trajectory can be calculated by model C:
C ikþj ¼ ci1;k expðci2;k ðk þ jÞÞ þ ci3;k ðk þ jÞ2 þ ci4;k
ð16Þ
The posterior distribution of the capacity at the current cycle (cycle k) is as follows:
Model C
(c)
pðC kþj jC 1:k Þ
Ns X
xik dðC kþj C ikþj Þ
ð17Þ
i¼1
The expectation of posterior capacity is calculated according to (15):
1.05
Capacity (Ah)
ð14Þ
3.3. Prognostics of RUP
1.05
0.8
^ eff ¼ P 1 N Ns i 2 i¼1 ðxk Þ
The expectation of the state can be estimated after resampling fxik ; xik g:
Cycle Number
(b)
ð13Þ
Since the degeneracy of the particles will be aroused by sequential importance sampling [26], the effective sample size Neff is introduced to compare with the threshold NT. Once Neff falls below NT, the resampling will be started.
RUP pdf
0.95
pðyk jxik Þpðxik jxi1 k Þ qðxik jxi0:k1 ; y1:k Þ
1 0.95
^ kþj ¼ C
RUP pdf
Ns X
xik C ikþj
ð18Þ
i¼1
Prediction Time
0.9 End of Performance
0.85 0.8
0
200
400
600
800
Cycle Number Fig. 7. Prediction result at cycle 422 (reaching 2/3 of life) for Cell_#V4 based on three models. (a) Prediction result at 2/3 of life based on model A. The predicted RUP is 208 cycles later than the real one. The STD of the predicted RUP is 19.1 cycles. (b) Prediction result at 2/3 of life based on model B. The predicted RUP is 80 cycles earlier than the real one. The STD of the predicted RUP is 27.7 cycles. (c) Prediction result at 2/3 of life based on model C. The predicted RUP is 1 cycles later than the real one. The STD of the predicted RUP is 83.0 cycles.
In this case, Ck represents a battery health indicator, and the RUP is the remaining time before it hits the pre-defined performance threshold, which is 80% of its initial capacity. For each cycle where k + l projects l steps ahead of the current cycle k, the esti^ðRUP P ljC 1:k Þ is equal to p ^ðC kþl 6 0:8C init jC 1:k Þ. As time mate p proceeds, the estimate is updated with the new measurements collected {Cj, j = k + 1, . . . , k + l}. The posterior pdf of RUP at cycle k can be estimated by:
pðRUPk jC 1:k Þ
Ns X
xik dðRUPk RUPik Þ
ð19Þ
i¼1
The expectation of RUP is approximated by:
Table 6 Prediction results for Cell_#V4 at different cycle numbers. Real failure cycle time
Cell_#V4
633
Prediction time (cycle K1)
211(1/3) 317(1/2) 422(2/3)
Model A
Model B
Model C
Error
STD
Error
STD
Error
STD
83 78 208
23.7 15.9 19.1
1 1 80
N/A N/A 27.7
121 145 1
19.0 21.2 83.0
Note: (1) Negative values mean the prediction result is earlier than the actual end of performance; and (2) 1 means that the prediction does not converge in Fig. 6.
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^ k¼ RUP
Ns X
RUPik xik
ð20Þ
i¼1
The procedure of prognostics based on PF proceeds as follows: At cycle k = 0 Step 0: Initialization Sample xi0 pðx0 Þ, set xi0 ¼ 1=N s ; i ¼ 1; . . . ; N s At cycle k P 1, for k ¼ 1; . . . ; T Step 1: Importance sampling For i = 1, . . ., Ns, Propagate particles: xik qðxk jxk1 ; yk Þ ~ ik according to (13) Assign importance weights: x Step 2: Weight calculation ~i x
Normalize weights: xik ¼ PNs k i¼1
~i Þ ðx k
; i ¼ 1; . . . ; N s
Step 3: Resampling ^ eff < N T IF N Select Ns particle index ji 2 f1; Ng According to weights fxjk1 g16j6N Set fxik1 ¼ xjk1 g, and xjk1 ¼ 1=N; i ¼ 1; . . . ; N s Otherwise Set xik1 ¼ xjk1 ; i ¼ 1; . . . ; N s Step 4: State estimation Substitute xik ; xik into (16) and (18), respectively Step 5: RUP prediction Predict RUP according to (19) and (20)
4. The prognostic results of RUP Six battery cells were tested under the same charge mode and discharged at a 1C rate. According to the data analysis in Section 2.1, Cells_#03#06 were used as the training samples to initialize model parameters, while Cells_#01#02 were used to test the prognostic performance due to the large fluctuation in capacity fade. A comparison was conducted among models A, B, and C, as mentioned in Section 2. In addition, another battery set, Cells_#V1#V4, with a different rated capacity was tested under similar charge/discharge conditions. The purpose was to validate the developed model and verify its robustness through model comparison. Parameter initializations for the different models include initializing the model parameters and their corresponding variance. The model parameters are initialized using the average value through curve fitting based on the battery training samples. Nonlinear least square fitting is performed to initialize the parameters of models A and C, while linear least square is implemented in model B throughout the whole battery life. The initialization of parameters’ variance is time-consuming. In order to make the prediction comparable among these three models, a simple method is employed. Here, taking model C as an example, the parameter vector [c1, c2, c3, and c4]T is estimated according to the curve fitting for Cells_#03#06. The variance of c1 can be approximated on the assumption that the range of [min(c1), max(c1)] is set as 6rc1 . This approach is also employed by the other parameters of each model, P while all the initial variances w of the capacity are set equal to 1e05. Table 5 shows the predicted results. The prediction time K1 is selected when cycling is conducted for up to 1/3, 1/2, and 2/3 of the battery life. The prediction error and standard deviation (STD) are two indicators, in terms of number of cycles, for evaluating the prediction performance.
In Table 5, both error and STD are taken into consideration when assessing the prediction results. A small STD means a narrow RUP pdf and, thus, a high confidence level for prediction. For the assessment of the prediction result, the premise is that the actual EOP falls within the 95% confidence interval. If this is true, the smaller the absolute values of the error and STD, the more accurate the prediction. Meanwhile, a result predicted later with a smaller STD should be more effective and credible than an early warning with a large STD, even if the earlier result is expected. For Cell_#01, the absolute values of the prediction error and STD from model C are smaller than those of model A. Comparing the prediction results at 1/2 of life, model B presents a smaller error than the other models, but its broad prediction distribution leads to low confidence level. Moreover, when predicting at 1/3 of life and 2/3 of life, the projection under model B has no intersection with the EOP. For Cell_#02, predicted results are more effective and credible based on model C. Although the prediction based on model B is in advance of the actual EOP, the error is so large that the real value falls outside of the 95% confidence interval. Thus, model C is regarded as a more accurate capacity degradation model to characterize the prediction performance. By measuring more capacity data, a more accurate RUP prediction is also obtained. Taking two prediction results as examples, Fig. 4 presents the earlier prediction for Cell_#01 when measuring up to the first 1/3 of life (at cycle 283), while Fig. 5 is the prediction for Cell_#02 when predicting the degradation trend at 2/3 of life (at cycle 428). Another battery set with a rated capacity of 1.1 A h was used to validate the effectiveness and robustness of the developed model. These batteries were run similar to Cell_#01#06 with the same charge–discharge mode. According to the definition of the EOP, Cells_#V1, #V2, and #V3 failed at the 469th, 472nd, and 434th cycles, respectively, while Cell_#V4 failed at cycle 633 because of a large dynamic change of capacity data. Thus, the first three cells were trained to average the values of the initial model parameters as discussed above, and #V4 was used for testing. The RUP of Cell_#V4 was also predicted at different prediction cycles: 1/3, 1/ 2, and 2/3 of capacity data. Figs. 6 and 7 show the RUP prediction for Cell_#V4 at different cycles. The prediction results are in line with Table 6, which indicates that model C is better for tracking the capacity fade of the battery than the other two models. When predicting the RUP at cycle 211 (1/3 of the capacity data), both prediction errors based on model C and model A were large (Fig. 6). However, an early estimated result before the actual EOP is more acceptable than a late expected value. Meanwhile, the extrapolation based on model B cannot converge to the EOP, even measuring up to 1/2 of the capacity data. In fact, the dynamic characteristics of battery aging are difficult to predict for a non-rigorous monotonic decreasing. For example, there are data jumps at around cycles 210 and 420, which have a large adjustment on the model parameters, and therefore affect the estimated trajectory and degradation trend. By measuring more capacity data and adjusting the model parameters online, model C is able to track the aging trend and, accordingly, obtain a smaller error and a higher probability of the predicted results than model A and B. Fig. 8 shows the predicted results when measuring up to 80% of the data (corresponding to cycle 507). The objective is to verify and compare the prediction performance in the latter stages of degradation. The estimated errors based on models A, B, and C are 141, 542, and 24 cycles, respectively. By adjusting the parameters to track the changes of the aging dynamics, neither model A nor model B can track the variation well compared to model C, which offers a more accurate predicted result ahead of the actual EOP with a narrower STD. It is worth mentioning that model B’s projection cannot converge to the EOP when the predictions are at cycles 211 and 317. Even if the prediction at cycle 507 converges to the EOP, the pre-
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Model B
Model A 1.1
1.1
Capacity (Ah)
Capacity (Ah)
1.05 1 RUP pdf
0.95 Prediction Time
0.9
1.05 1 0.95 Prediction Time
0.9
200
1.05 RUP pdf
1 0.95 Prediction Time
0.9 End of Performance
0.85 0
Real Data Estimated Values Predicted Values
1.1
End of Performance
End of Performance
0.85 0.8
Model C Real Data Estimated Values Predicted Values
Capacity (Ah)
Real Data Estimated Values Predicted Values
400
600
Cycle Number
800
0.8
0.85 0
200
400
600
800
0.8
0
200
Cycle Number
400
600
800
Cycle Number
Fig. 8. Prediction at the 507th cycle (reaching 80% of life) for Cell_#V4 based on three models.
diction is meaningless because of a large prediction error beyond the measured scope. For model B, a brief explanation is that a strong constraint should be imposed on the algebraic relationship of the model parameters to ensure a decreasing trend because of the second-order polynomial regression. However, the issue of how to use constraints to improve the prediction performance refers to the area related to parameter optimization, is beyond the scope of this paper. Therefore, from the perspective of the degradation model, the ensemble model (model C) presents a more effective and robust characteristic for the prediction of capacity fade of the battery. 5. Conclusions Capacity fade can indicate the SOH of lithium-ion batteries. The estimation of the remaining useful performance (RUP) of a battery depends on the accuracy of the degradation model and the online model parameter adjustment. In this study, an ensemble model modified from two existing empirical models was developed to depict the capacity data that were obtained through continuous full charge and discharge cycles. This ensemble model has a better regression characteristic over the whole battery life than the existing empirical models. Taking into account the uncertainties in the degradation process, particle filtering was used to adjust the model parameters and calculate the estimated capacity with associated weights up to the current observed cycles. RUP was predicted by extrapolating the degradation model to EOP, while its probability distribution was estimated in terms of the associated weights bound up with the current discharge capacity. Through comparing the predicted results from the ensemble model to the results from the exponential and polynomial models at different prediction times, the ensemble model demonstrated better prediction performance (smaller prediction errors and a narrower standard deviation). This is because this model balanced the global and local regression performance. Moreover, the developed model was also validated using another battery set with a different rated capacity. For these two kinds of battery samples, a credible and reliable prediction result can be achieved using a wide range of initialization. The effectiveness and robustness of the developed model in the prediction of capacity degradation was demonstrated. This methodology can be propagated into other degradation-related applications. There are some limitations to applying the developed model. In some cases, it is difficult to quantify the actual maximum capacity because the battery is usually not fully discharged in every cycle. One practical solution to this issue is to map the capacity of the partial discharge into the equivalent fully discharged capacity before using the developed model. The transform relation can then
be explored by measuring the different voltages and finding the interaction between the random cut-off discharge and fully discharged voltage. Furthermore, temperature effects on battery degradation should also be considered in model modification and validation in future work. Acknowledgements The work presented in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (CityU8/CRF/09). The authors would like to thank the members of the Center for Advanced Life Cycle Engineering at the University of Maryland for their support of this work. References [1] Rufus F, Lee S, Thakker A. Health monitoring algorithms for space application batteries; 2008. p. 1–8. [2] Huggins RA. Advanced batteries: materials science aspects. Springer Verlag; 2008. [3] Kim ILS. A technique for estimating the state of health of lithium batteries through a dual-sliding-mode observer. Power Electron, IEEE Trans on 2010;25:1013–22. [4] Saha B, Goebel K, Poll S, Christophersen J. Prognostics methods for battery health monitoring using a Bayesian framework. Instrum Meas, IEEE Trans on 2009;58:291–6. [5] Verbrugge M, Koch B. Generalized recursive algorithm for adaptive multiparameter regression. J Electrochem Soc 2006;153:A187. [6] Schmidt AP, Bitzer M, Imre ÁW, Guzzella L. Model-based distinction and quantification of capacity loss and rate capability fade in Li-ion batteries. J Power Sources 2010;195:7634–8. [7] Spotnitz R. Simulation of capacity fade in lithium-ion batteries. J Power Sources 2003;113:72–80. [8] Saha B, Goebel K, Poll S, Christophersen J. An integrated approach to battery health monitoring using bayesian regression and state estimation; 2007. p. 646-53. [9] Bo C, Zhifeng B, Binggang C. State of charge estimation based on evolutionary neural network. Energy Convers Manage 2008;49:2788–94. [10] Santhanagopalan S, White RE. State of charge estimation using an unscented filter for high power lithium ion cells. Int J Energy Res 2010;34:152–63. [11] Pattipati B, Sankavaram C, Pattipati K. System identification and estimation framework for pivotal automotive battery management system characteristics. IEEE Trans Syst Man and Cyber – Part C Appl Rev 2011;41:869. [12] Plett GL. Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs: Part 3. State and parameter estimation. J Power Sources 2004;134:277–92. [13] Xing Y, Ma EWM, Tsui KL, Pecht M. Battery management systems in electric and hybrid vehicles. Energies 2011;4:1840–57. [14] Andrea D. Battery management systems for large lithium battery packs. Artech House Publishers; 2010. [15] Kozlowski JD. Electrochemical cell prognostics using online impedance measurements and model-based data fusion techniques; 2003. p. 3257–270. [16] He W, Williard N, Osterman M, Pecht M. Prognostics of lithium-ion batteries based on Dempster–Shafer theory and the Bayesian Monte Carlo method. J Power Sources 2011;196:10314–21. [17] Micea MV, Ungurean L, Cârstoiu GN, Groza V. Online state-of-health assessment for battery management systems. Instrum Meas, IEEE Trans on 2011;60:1997–2006.
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[18] Bergveld HJ, Kruijt WS, Notten PHL. Battery management systems: design by modelling, vol. 1. Springer; 2002. [19] Dai J, Das D, Pecht M. Prognostics-based risk mitigation for telecom equipment under free air cooling conditions. Appl Energy 2012. [20] Sun J, Zuo H, Wang W, Pecht MG. Application of a state space modeling technique to system prognostics based on a health index for condition-based maintenance. Mech Syst Signal Process 2011. [21] Si X-S, Wang W, Hu C-H, Zhou D-H. Remaining useful life estimation – a review on the statistical data driven approaches. Eur J Oper Res 2011;213:1–14. [22] Zhang J, Lee J. A review on prognostics and health monitoring of Li-ion battery. J Power Sources 2011;196:6007–14. [23] Friedman J, Hastie T, Tibshirani R. The elements of statistical learning. Springer; 2008.
[24] Zhang Q, White RE. Capacity fade analysis of a lithium ion cell. J Power Sources 2008;179:793–8. [25] Spiess AN, Neumeyer N. An evaluation of R2 as an inadequate measure for nonlinear models in pharmacological and biochemical research: a Monte Carlo approach. BMC Pharmacol 2010;10:6. [26] Arulampalam MS, Maskell S, Gordon N, Clapp T. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. Signal Process, IEEE Trans on 2002;50:174–88. [27] Doucet A, Godsill S, Andrieu C. On sequential Monte Carlo sampling methods for Bayesian filtering. Statist Comput 2000;10:197–208. [28] Lee DJ. Nonlinear Bayesian filtering with applications to estimation and navigation. Texas A&M University; 2005.
Contents lists available at SciVerse ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
An ensemble model for predicting the remaining useful performance of lithium-ion batteries Yinjiao Xing a,b,⇑, Eden W.M. Ma a, Kwok-Leung Tsui a,b, Michael Pecht c a
Centre for Prognostics and System Health Management, City University of Hong Kong, Kowloon, Hong Kong Department of Systems Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong c Center for Advanced Life Cycle Engineering (CALCE), University of Maryland, College Park, MD 20740, USA b
a r t i c l e
i n f o
Article history: Received 16 June 2012 Received in revised form 23 October 2012 Accepted 7 December 2012 Available online 11 January 2013
a b s t r a c t We developed an ensemble model to characterize the capacity degradation and predict the remaining useful performance (RUP) of lithium-ion batteries. Our model fuses an empirical exponential and a polynomial regression model to track the battery’s degradation trend over its cycle life based on experimental data analysis. Model parameters are adjusted online using a particle filtering (PF) approach. Experiments were conducted to compare our ensemble model’s prediction performance with the individual results of the exponential and polynomial models. A validation set of experimental battery capacity data was used to evaluate our model. In our conclusion, we presented the limitations of our model. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In battery-powered systems, lithium-ion batteries are being widely used in consumer electronics, electric vehicles, and even space systems [1]. Compared with lead-acid, nickel–cadmium, and nickel–metal-hydride batteries, lithium-ion batteries offer a higher energy density, longer cycle life, and lower self-discharge rate [2,3]. However, an inevitable problem is that battery performance degrades with cycling and aging. To use a battery in an electric vehicle as an analogy, it is similar to a ‘‘fuel’’ tank that shrinks with use so that the driver can never know what a full or partial fuel reading actually means with respect to miles that would be achievable. Since battery degradation cannot be measured directly, there is a need to estimate its maximum available performance and give users an accurate estimate of battery capacity ahead of time so that decisions for battery replacement can be made. Battery capacity is widely used as an indicator of battery status. Battery performance is represented by the state of health (SOH), which is a ratio of the present capacity to the initial capacity of a battery [4–6]. In the battery community, end of performance (EOP) is usually set as the time that the maximum available capacity is reduced to 80% of its initial value [7,8]. Remaining useful performance (RUP) is defined as the length of time from the time of observation to the time when the EOP criterion is reached. In the battery management systems (BMSs), estimating battery state of charge (SOC) has been conducted and implemented in the form ⇑ Corresponding author at: Centre for Prognostics and System Health Management, City University of Hong Kong, Kowloon, Hong Kong. E-mail addresses: [email protected] (Y. Xing), [email protected] (E.W.M. Ma), [email protected] (K.-L. Tsui), [email protected] (M. Pecht). 0026-2714/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.microrel.2012.12.003
of the remaining time or range [9–13]. However, RUP prediction is still at the premature stage. There are three main challenges in RUP prediction. First, uncertainties, including operational and environmental factors, unit-tounit variation, and measurement noise, will impose the difficulties on RUP prediction. Second, it is difficult to estimate the maximum releasable capacity of a battery because a battery is usually not fully discharged in real applications. Furthermore, there is still controversy over the definition of battery SOH. More factors, such as internal impedance, self-discharge, and cycle counting, are suggested for evaluating SOH [1,11,14]. The uncertainty of SOH estimation will also influence RUP prediction. Predictions of battery RUP have been performed using both physics of failure (PoF) model and data-driven approaches. Most practitioners have adopted electrochemical impedance spectroscopy (EIS) measurements to establish physics-based models by extracting the internal parameters, which are able to characterize varying aging and fault processes of the battery. Based on these battery models, data-driven methods, such as auto-regressive moving average, fuzzy logic, neural network, support vector machine, and nonlinear filtering methods, are combined to predict the RUP of batteries [4,11,15]. However, online applications of EIS measurements suffer from size limitations and high cost. Moreover, it is time-consuming and difficult to quantify the internal impedance at low frequency [11]. Another key problem is that the noises caused by other integrated components of an online system have an impact on the accuracy of the EIS measurements due to the slight excitation signal required for EIS measurement. Prediction based on the empirical regression models has been recommended to address these challenges in [16–18]. Taking into account the limitations of EIS techniques, He et al. [16] introduced
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The developed model is then validated to demonstrate its robustness using a different battery set. Our conclusions are presented in Section 5.
an empirical exponential model to conduct curve fitting for capacity fade data. The model parameters were initialized via Dempster– Shafer theory, which is effective for fusing varieties of battery data and apply in the parametric initialization. The parameters were updated step-by-step (each cycle) through particle filtering (PF). Micea et al. [17] adopted an empirical second-order polynomial model as a function of cycle number to estimate the stored maximum capacity. The proposed least square algorithm was implemented to estimate the capacity function. Other new prognostics-related methods and models can be found in [19–21]. In this paper, we presented an ensemble degradation model that integrates two regression models: an exponential model and a polynomial model [16,17]. The ensemble model is a more accurate parametric model than each of the above two models because it takes into account onboard applications and global and local regression characteristics. Since the model parameters should vary with the dynamic characteristics of battery degradation, PF is used to estimate and adjust the model parameters and hence to track the battery aging process with nonlinear and non-Gaussian characteristics. The predicted results of RUP are presented with a narrow probability distribution. Moreover, we used another battery set with a different rated capacity to validate the performance of our model. The effectiveness and robustness of our model was verified through comparisons with the other two regression models. This paper is organized as follows: Section 2 conducts the regression analysis of capacity degradation based on three different models: the exponential model, the polynomial model and our model. Our new model is presented based on our experimental data and comparative results. Section 3 introduces the PF approach for state estimation. Then, the procedure for battery state estimation and prediction is presented in terms of integrating the regression model with the PF approach. Section 4 shows the RUP results and performance comparison according to the above framework.
Cell #01
2. Empirical models 2.1. Capacity measurement Capacity indicates the amount of charge that comes out of a battery from the full to empty state, corresponding to the upper and lower cut-off voltages preset by manufacturers [14]. Capacity is calculated by integrating the current over time [22]. The SOH indication used in this paper is based on the maximum available capacity, which can be obtained by discharging from full charge (100%) to empty charge (0%) in each cycle. To develop the degradation model for SOH estimation, six batteries under test went through the full charge and discharge procedure. The rated capacity of the tested lithium-ion battery was 1.35 A h. Using the battery test equipment, Arbin BT2000, the batteries were fully charged under the constant-current/constant-voltage mode, which is widely used in the industry. Then, they were discharged at a 1C rate, which meant 1.35 A for the battery discharge in this case. The test was run at room temperature, which was approximately 25 °C. The discharge capacity was recorded after each full charge–discharge process. The capacity fade is shown in Fig. 1, while the EOP was defined as 80% of its initial capacity. 2.2. Regression analysis based on different models As mentioned in the introduction, an exponential model (model A) [16] and a polynomial model (model B) [17] have been empirically established through fitting a large amount of battery degrada-
Cell #02
1.2
1.1
1.3
Real data Model A
Capacity (Ah)
1.3
Real data Model A
Capacity (Ah)
Capacity (Ah)
Real data Model A
Cell #03
1.4
1.4
1.4
1.2
1.1 End of Performance
1.3 1.2 1.1 End of Performance
End of Performance
1
0
200
400
600
1
800
0
200
Cycle
400
600
1
800
0
200
Cycle
400
600
800
Cycle
(a) Cell #01
Cell #02
1.4
Real data Model B
1.2
1.1
1.3
Real data Model B
Capacity (Ah)
Capacity (Ah)
Capacity (Ah)
1.3
Cell #03
1.4
1.4 Real data Model B
1.2
1.1 End of Performance
1.3 1.2 1.1 End of Performance
End of Performance
1
0
200
400
600
Cycle
800
1
0
200
400
600
800
Cycle
(b) Fig. 1. Curve fitting based on (a) model A, and (b) model B.
1
0
200
400
600
Cycle
800
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tion data. Model A is made up of two exponential functions, while model B is a polynomial regression model. Model A (exponential model):
C Ak ¼ C Ak1 þ C Ak2 ¼ a1 expða2 kÞ þ a3 expða4 kÞ
ð1Þ
Model B (polynomial model): 2
C Bk ¼ C Bk1 þ C Bk2 ¼ b1 k þ b2 k þ b3
ð2Þ
where CAk and CBk are the capacity of the battery and k is the cycle number; a1, a2, a3, a4 and b1, b2, b3 are the model parameters to be estimated in Eqs. (1) and (2), respectively. The capacity data over the whole life (to 80% of the initial capacity) were used to evaluate the goodness-of-fit of these two parametric models. Fig. 1a and b shows curve fittings for Cell_#01#03 based on models A and B, respectively. The model parameters of models A and B were fitted in a MATLAB environment. With respect to each model’s characteristics, model A was fitted using the nonlinear least square method, while model B was estimated using a linear least square. Both criteria aim to minimize the sum of the squares of the errors [23]. The more accurate the model is, the better the prediction performance will be. As shown in Table 1, the adjusted R-square (R2adj ) and root mean square error (RMSE) are two indices to compare the goodness-of-fit of these two models. R2adj is a modification of R-square to compensate for the extra variables included in the model. The best fit is indicated by ‘‘1’’ in R2adj and ‘‘0’’ in RMSE. Model A showed a better global regression performance than model B using all the capacity data before EOP. However, the fitting result cannot characterize the local regression performance of both models. In fact, the capacity fade of a lithium-ion battery could have several stages with different degradation trends. Spotnitz [7] partitioned the capacity fade into four regions according to manufacturers’ information, while Zhang and White [24] divided the degradation into three stages based on experimental data. In this paper, all the capacity data before hitting the EOP are partitioned into three identical portions. This partition does not distinguish different fade patterns, but is used to analyze the local regression characteristics of models A and B, and employ an ensemble model that provides the potential for better fits to the whole life data. The regression results when the models are fitted in the different portions are shown in Table 2.
Table 1 Goodness-of-fit of models A and B. R2adj
No. of cell
Cell_#01 Cell_#02 Cell_#03 Cell_#04 Cell_#05 Cell_#06
RMSE
Model A
Model B
Model A
Model B
0.9832 0.9497 0.9803 0.9609 0.9749 0.9712
0.9795 0.9442 0. 9793 0.9517 0.9664 0.9603
0.0101 0.0144 0.0097 0.0130 0.0101 0.0108
0.0112 0.0152 0.0100 0.0144 0.0118 0.0127
Table 2 Goodness-of-fit of models A and B under different data portions. No. of cell
R2adj
(the first one-
third)
Cell_#01 Cell_#02 Cell_#03 Cell_#04 Cell_#05 Cell_#06
R2adj
(the second
one-third)
R2adj
For the first one-third of the data, larger R2adj values were obtained from model A. However, for the second portion, model B presents a better fit for Cell_#03#06. For the last portion, model A only has a better goodness-of-fit than B and C on only three cells (i.e., Cell_#01, #04, and #06). It can be seen that model A is suitable for tracking degradation earlier in the battery life. Thus, a poor-fitting on the particular data offers an opportunity to develop a more accurate model, which not only has a better global regression, but also fits the local data well, especially the last 1/3 of life data. It is worth noting that the poor-fitting are presented for Cell_#01 and #02 were due to the large fluctuation in capacity. It can also be found in Fig. 1. Thus, to evaluate the performance of the prognostic approaches integrated with our developed model, Cell_#03#06 were used as training samples to establish the degradation model and estimate model parameters while Cell_#01 and #02 were used for testing in Section 4. 2.3. Model selection and validation 2.3.1. Model selection Since both models A and B only fit part of the data well, we developed an ensemble model to fuse these two models in order to achieve a high global goodness-of-fit as well as a more accurate prediction from the middle and late stages of degradation. Although a weighting factor can be used to form a sophisticated model that combines these two models, more parameters need to be estimated and probably lead to over-fitting. Through fitting the experimental capacity data, the ensemble model (model C) is developed as follows: In model A in Eq. (1), only one of the exponential parts (CAk2) plays a vital role in the regression process. Taking the training samples (Cell_#0306) as examples, CAk1 can be ignored compared to the scale of CAk2 (Table 3). In model B in Eq. (2), the quadratic and linear items are compared to reduce the possible number of model parameters. Since both CBk1 and CBk2 refer to the variable k, a comparison is conducted among three models: CAk2 + CBk1 + CBk3 (model C1), CAk2 + CBk2 + CBk3 (model C2), and CAk2 + CBk1 + CBk2 + CBk3 (model C3). When comparing regression models that use the same dependent variables and the same estimation period, the fact that RMSE goes down is equivalent to when R2adj goes up. In Table 4, R2adj is kept to quantify the regression performance here. In addition, the Akaike information criterion (AIC) is introduced to make auxiliary assessment for model selection in order to impose a heavier penalty on model complexity. 2.3.2. Model validation Model accuracy and prediction performance will be validated in this section. Two aspects will be considered: (1) the accuracy of the new models in terms of the different types and the number of parameters; and (2) the over-fitting issue due to the increased number of parameters. AIC is measured to provide validation among a series of nonlinear models and is widely used for model selection to consider model complexity. AIC is suitable for selecting the preferred model as a supplement to other indicators (i.e., R2adj , RMSE) and includes a penalty for increased model complexity [23].
AIC ¼ 2m InðLÞ;
ð3Þ
(the last one-
third)
Model A
Model B
Model A
Model B
Model A
Model B
0.2915 0.7632 0.9259 0.8824 0.8984 0.9160
0.2244 0.6849 0.8669 0.8273 0.8361 0.8485
0.8866 0.5153 0.6050 0.7417 0.8284 0.7427
0.8852 0.5075 0.6621 0.8018 0.8698 0.7799
0.9234 0.7925 0.9028 0.7302 0.6986 0.7926
0.8603 0.8011 0.9354 0.5517 0.7188 0.6950
Table 3 Variation of CAk1 and CAk2 (A h) over k in the whole life. No. of cell
Variation of CAk1
Variation of CAk2
Cell_#03 Cell_#04 Cell_#05 Cell_#06
0.0278–1.066e17 0.091–6.843e05 0.0682–2.865e06 0.0766–8.495e06
1.336–1.104 1.260–1.097 1.290–1.106 1.279–1.106
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Table 4 Comparison among three models.
Cell_#01 Cell_#02 Cell_#03 Cell_#04 Cell_#05 Cell_#06
CAk2 + CBk1 + CBk3 (model C1)
CAk2 + CBk2 + CBk3 (model C2)
CAk2 + CBk1 + CBk2 + CBk3 (model C3)
R2adj
AIC
R2adj
AIC
R2adj
AIC
0.9796 0.9525 0.9863 0.9609 0.9747 0.9716
5216 3659 5246 3716 4081 3852
0.9790 0.9510 0.9793 0.9609 0.9468 0.9713
5191 3639 4925 3716 4086 3846
0.9797 0.9531 0.9864 0.9609 0.9749 0.9717
5217 3663 5245 3714 4084 3851
Model C1 1.35 1.25 1.2 1.15 1.1 End of Performance 1.05
Model C3
Real data Fitted Curve Predicted Curve
1.35
C apacity (A h)
1.3
Capacity (Ah)
Model C2
Real data Fitted Curve Predicted Curve
1.3 1.25 1.2 1.15 1.1 End of Performance 1.05
1.3 1.25 1.2 1.15 1.1 End of Performance 1.05
1
1
1
0.95
0.95
0.95
0
100 200 300 400 500 600 700 800 900
0
100 200 300 400 500 600 700 800 900
Cycle
Real data Fitted Curve Predicted Curve
1.35
C apacity (A h)
No. of cell
0
100 200 300 400 500 600 700 800 900
Cycle
Cycle
Fig. 2. Prediction based on models C1, C2, and C3 individually under the first 75% of capacity data.
where m is the number of parameters (a penalty) and In(L) is the maximum value of the log-likelihood of the estimated model.
3. Particle filtering (PF) for RUP prediction
" ! # PN ^ 2 N i¼1 ðyi ypi Þ þ1 InðLÞ ¼ In 2p 2 N
Accurate state estimation and prediction not only rely on an accurate model, but also depend on the adjustments of model parameters to track the variation in capacity fade. In order to characterize the dynamic capacity fade, the particle filtering (PF) approach is used to estimate the current capacity. The core idea of PF is based on Bayesian filtering and Monte Carlo simulation. Through Bayes’ Theorem, PF is able to provide a probability estimation for nonlinear and non-Gaussian systems [26]. In order to approximate the marginal distribution, Monte Carlo simulation is used to generate a large amount of random particles and estimate the posterior density function through accumulating these particles with associated weights. Fig. 3 shows the flowchart of RUP prediction.
ð4Þ
^pi calculates the residuals bewhere N is the sample size, while yi y tween the true value and the estimated value. The lower the AIC is, the better the model will be [25]. As shown in Table 4, the goodness-of-fit based on models C1 and C3 is generally better than that of model C2. However, AIC is not sufficient to reflect the model over-fitting that could be triggered by estimating more parameters. Therefore, the prediction performance of the model needs to be validated. In this study, predicting was conducted to extrapolate the estimated curve based on the previous fitting using the first 75% of the capacity data. Model C3 could possibly reduce the training errors because of having greater complexity, but would pose a risk of over-fitting and cause a poor prediction at the same time. Fig. 2 presents an over-fitting result under model C3. Thus, model C1 is regarded as an ensemble model in terms of a good fitting result and predicted validation. Later, the ensemble model will be represented as model C: 2
C Ck ¼ c1 expðc2 kÞ þ c3 k þ c4
ð5Þ
3.1. Recursive Bayes The state-space model is used to solve the tracking problem using the nonlinear process function, fk, and measurement function, hk:
c1;k1 3 2 #1;k1 3 7 6 7 6c 6 2;k1 7 6 #2;k1 7 xk ¼ fk ðxk1 ; #k1 Þ ¼ 6 7þ6 7 $ pðxk jxk1 Þ 4 c3;k1 5 4 #3;k1 5 2
c4;k1
ð6Þ
#4;k1
where CCk is the capacity of the battery and k is the cycle number. Table 5 Prediction results for Cell_#01 and #02 at different cycle numbers. Real failure cycle time
a b
Prediction cycle (% of life)
Model A
Model B
Model C
Error
STD
Error
STD
Error
STD
1a 49 1
284.2 167 163.3
27b 103 103
88.6 46.4 25.8
82 129 119
30.1 17.3 7.6
9 14 10
29 62.9 30.7
Cell _#01
849
283 (1/3) 425 (1/2) 566 (2/3)
325 202 191
167.6 57.4 56.8
Cell _#02
643
214 (1/3) 322 (1/2) 428 (2/3)
304 133 42
50.5 22.8 65.2
Negative values mean the prediction result is earlier than the actual end of performance. 1 Means that the prediction does not converge in Fig. 4.
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Model A
(a)1.4
Real Data Estimated Values Predicted Values
1.35
Capacity (Ah)
1.3 1.25 1.2 1.15 1.1
Prediction Time End of Performance
1.05 1
0
200
400
600
800
Cycle Number Model B
(b)1.4
Real Data Estimated Values Predicted Values
1.35
Capacity (Ah)
1.3 1.25 1.2
RUP pdf
1.15 Prediction Time
1.1 End of Peformance
1.05 1
0
200
400
600
800
Cycle Number Model C
(c)1.4
Real Data Estimated Values Predicted Values
1.35
Capacity (Ah)
1.3 1.25 1.2 1.15
Fig. 3. Flowchart of implementation of RUP prediction.
yk ¼ hk ðxk ; uk Þ $ pðyk jxk Þ
ð7Þ
where fxk ; k 2 Ng is the state vector that is assumed as the unobserved Markov process; fyk ; k 2 Ng are the observations or measurements that are conditionally independent given the process; #k is an i.i.d. process noise sequence; uk is an i.i.d. measurement noise sequence; k is the time index; p(xk|xk1) is the transition distribution; and p(yk|xk) is the observation distribution. Corresponding to our case, the state xk is the vector of model parameters [c1k, c2k, c3k, c4k]T to be estimated, while #k1 is distributed as Nð0; r2i IÞ; i ¼ 1; 2; 3; 4 . . . yk is the capacity sequence that is measured after a full discharge through Coulomb counting methods. The recursive Bayesian method provides a generic way to estimate the posterior probability density function (pdf) p(xk|y1:k), given the capacity and the posterior expectation I(fk) based on the sequence of measured capacity.
Iðfk Þ ¼
Z
fk ðxk Þpðxk jy1:k Þdxk
RUP pdf
1.1
ð8Þ
Through recursive Bayesian filtering, prediction and update will be recursively implemented in two steps. The first step is to achieve the prior pdf p(xk|y1:k1), which means that the state xk is
Prediction Time End of Performance
1.05 1
0
200
400
600
800
Cycle Number Fig. 4. Prediction results at cycle 283 (reaching 1/3 of life) for Cell_#01 based on three models. (a) Prediction result at 1/3 of life based on model A. The intersection of the projection with the EOP exceeds the length of the range of test data. (b) Prediction result at 1/3 of life based on model B. The intersection of the projection with the EOP exceeds the length of the range of test data. (c) Prediction result at 1/3 of life based on model C. The predicted RUP is 27 cycles earlier than the real one. The STD of the predicted RUP is 88.6 cycles.
inferred from the measurements y1:k1 = {yn, n = 1, . . . , k 1}. The second step is to obtain the posterior pdf p(xk|y1:k) in terms of the current measurement:
pðxk jy1:k1 Þ ¼
pðxk jy1:k Þ ¼
Z
pðxk jxk1 Þpðxk1 jy1:k1 Þdxk1
pðxk jy1:k1 Þpðyk jxk Þ pðyk jy1:k1 Þ
ð9Þ
ð10Þ
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Y. Xing et al. / Microelectronics Reliability 53 (2013) 811–820
Model A
(a) 1.4
Real Data Estimated Values Predicted Values
1.1
1.3
Capacity (Ah)
Capacity (Ah)
1.35
Model A
(a)
Real Data Estimated Values Predicted Values
1.25 Prediction Time
1.2 1.15
RUP pdf
1.1 End of Performance
1 RUP pdf
0.95 0.9 End of Performance
1.05 1
Prediction Time
1.05
0.85
0
200
400
600
800
0.8
Cycle Number
0
200
400
600
800
Cycle Number
Model B
(b)1.4
Real Data Estimated Values Predicted Values
1.1
1.3 Prediction Time
1.25
1.05
Capacity (Ah)
Capacity (Ah)
1.35
Model B
(b)
Real Data Estimated Values Predicted Values
RUP pdf
1.2 1.15 1.1 End of Peformance
1 0.95 Prediction Time
0.9
1.05 1
End of Performance
0.85
0
200
400
600
800 0.8
Cycle Number
Prediction Time
1.25
Prediction Time
1.05
1.2 RUP pdf
1.15
800
Real Data Estimated Values Predicted Values
1.1
Capacity (Ah)
Capacity (Ah)
600
Model C
(c)
1.3
1.1 End of Performance
1 0.95 RUP pdf
0.9
1.05
End of Performance
0
200
400
600
0.85
800
Cycle Number
0.8
Fig. 5. Prediction results at cycle 428 (reaching 2/3 of life) for Cell_#02 based on three models. (a) Prediction result at 2/3 of life based on model A. The predicted RUP is 42 cycles later than the real one. The STD of the predicted RUP is 65.2 cycles. (b) Prediction result at 2/3 of life based on model B. The predicted RUP is 119 cycles earlier than the real one. The STD of the predicted RUP is 7.6 cycles. (c) Prediction result at 2/3 of life based on model C. The predicted RUP is 10 cycles later than the real one. The STD of predicted RUP is 30.7 cycles.
where the normalizing constant is:
Z
400
Cycle Number Real Data Estimated Values Predicted Values
1.35
pðyk jy1:k1 Þ ¼
200
Model C
(c)1.4
1
0
pðxk jy1:k1 Þpðyk jxk Þdxk
0
200
400
600
800
Cycle Number Fig. 6. Prediction result at cycle 211 (reaching 1/3 of life) for Cell_#V4 based on three models. (a) Prediction result at 1/3 of life based on model A. The predicted RUP is 83 cycles later than the real one. The STD of the predicted RUP is 23.7 cycles. (b) Prediction result at 1/3 of life based on model B. The projection diverges, but does not converge to the EOP. (c) Prediction result at 1/3 of life based on model C. The predicted RUP is 121 cycles earlier than the real one. The STD of the predicted RUP is 19.0 cycles.
ð11Þ
where p(yk|xk) is determined by (7). 3.2. Monte Carlo simulation for posterior pdf Often, there is no analytical solution to Eqs. (9) and (10). PF therefore utilizes Monte Carlo (MC) simulation to approximate the probability density with a large amount of weighted samples,
which are the weighted particles. Thus, fxi0:k ; xi0:k g is a random measured pair that characterizes the posterior pdf, where fxi0:k ; i ¼ 0; . . . ; N s g is the associated weights of a set of particles fxi0:k ; i ¼ 0; . . . ; N s g, and Ns is the total number of particles. The posterior pdf at the kth cycle can be approximated as:
pðxk jy1:k Þ
Ns X
xik dðxk xik Þ
i¼1
ð12Þ
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Y. Xing et al. / Microelectronics Reliability 53 (2013) 811–820
Model A
(a)
Real Data Estimated Values Predicted Values
1.1
where d() is a Dirac function. The weights are normalized P i i as i xk ¼ 1. The sample xk is drawn from importance density q(xk|y1:k) based on the importance sampling [27,28]. Through recursive relation, the weights are updated as follows:
Capacity (Ah)
1.05
xik / xik1
1
Prediction Time
0.9 End of Performance
0.85 0.8
0
200
400
600
800
Model B Real Data Estimated Values Predicted Values
1.1
Capacity (Ah)
1 0.95 RUP pdf
Prediction Time
0.9 End of Performance
0.85
0
200
400
600
800
Cycle Number
^IN ðfk Þ ¼ s
XN s
f ðxi Þ i¼1 k 0:k
xik
ð15Þ
Real Data Estimated Values Predicted Values
1.1
Prediction of the expectation and the posterior pdf % p(xk+j|y1:k) of the state at the j-step (j = 2, . . . , T k e N) ahead is expected. T is the time horizon related to the number of cycles of the battery. Since the new measurement yk+j has not been collected, the present or measured state vector and its current posterior distribution p(xk|y1:k) would be projected among all possible future paths. Each capacity trajectory can be calculated by model C:
C ikþj ¼ ci1;k expðci2;k ðk þ jÞÞ þ ci3;k ðk þ jÞ2 þ ci4;k
ð16Þ
The posterior distribution of the capacity at the current cycle (cycle k) is as follows:
Model C
(c)
pðC kþj jC 1:k Þ
Ns X
xik dðC kþj C ikþj Þ
ð17Þ
i¼1
The expectation of posterior capacity is calculated according to (15):
1.05
Capacity (Ah)
ð14Þ
3.3. Prognostics of RUP
1.05
0.8
^ eff ¼ P 1 N Ns i 2 i¼1 ðxk Þ
The expectation of the state can be estimated after resampling fxik ; xik g:
Cycle Number
(b)
ð13Þ
Since the degeneracy of the particles will be aroused by sequential importance sampling [26], the effective sample size Neff is introduced to compare with the threshold NT. Once Neff falls below NT, the resampling will be started.
RUP pdf
0.95
pðyk jxik Þpðxik jxi1 k Þ qðxik jxi0:k1 ; y1:k Þ
1 0.95
^ kþj ¼ C
RUP pdf
Ns X
xik C ikþj
ð18Þ
i¼1
Prediction Time
0.9 End of Performance
0.85 0.8
0
200
400
600
800
Cycle Number Fig. 7. Prediction result at cycle 422 (reaching 2/3 of life) for Cell_#V4 based on three models. (a) Prediction result at 2/3 of life based on model A. The predicted RUP is 208 cycles later than the real one. The STD of the predicted RUP is 19.1 cycles. (b) Prediction result at 2/3 of life based on model B. The predicted RUP is 80 cycles earlier than the real one. The STD of the predicted RUP is 27.7 cycles. (c) Prediction result at 2/3 of life based on model C. The predicted RUP is 1 cycles later than the real one. The STD of the predicted RUP is 83.0 cycles.
In this case, Ck represents a battery health indicator, and the RUP is the remaining time before it hits the pre-defined performance threshold, which is 80% of its initial capacity. For each cycle where k + l projects l steps ahead of the current cycle k, the esti^ðRUP P ljC 1:k Þ is equal to p ^ðC kþl 6 0:8C init jC 1:k Þ. As time mate p proceeds, the estimate is updated with the new measurements collected {Cj, j = k + 1, . . . , k + l}. The posterior pdf of RUP at cycle k can be estimated by:
pðRUPk jC 1:k Þ
Ns X
xik dðRUPk RUPik Þ
ð19Þ
i¼1
The expectation of RUP is approximated by:
Table 6 Prediction results for Cell_#V4 at different cycle numbers. Real failure cycle time
Cell_#V4
633
Prediction time (cycle K1)
211(1/3) 317(1/2) 422(2/3)
Model A
Model B
Model C
Error
STD
Error
STD
Error
STD
83 78 208
23.7 15.9 19.1
1 1 80
N/A N/A 27.7
121 145 1
19.0 21.2 83.0
Note: (1) Negative values mean the prediction result is earlier than the actual end of performance; and (2) 1 means that the prediction does not converge in Fig. 6.
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Y. Xing et al. / Microelectronics Reliability 53 (2013) 811–820
^ k¼ RUP
Ns X
RUPik xik
ð20Þ
i¼1
The procedure of prognostics based on PF proceeds as follows: At cycle k = 0 Step 0: Initialization Sample xi0 pðx0 Þ, set xi0 ¼ 1=N s ; i ¼ 1; . . . ; N s At cycle k P 1, for k ¼ 1; . . . ; T Step 1: Importance sampling For i = 1, . . ., Ns, Propagate particles: xik qðxk jxk1 ; yk Þ ~ ik according to (13) Assign importance weights: x Step 2: Weight calculation ~i x
Normalize weights: xik ¼ PNs k i¼1
~i Þ ðx k
; i ¼ 1; . . . ; N s
Step 3: Resampling ^ eff < N T IF N Select Ns particle index ji 2 f1; Ng According to weights fxjk1 g16j6N Set fxik1 ¼ xjk1 g, and xjk1 ¼ 1=N; i ¼ 1; . . . ; N s Otherwise Set xik1 ¼ xjk1 ; i ¼ 1; . . . ; N s Step 4: State estimation Substitute xik ; xik into (16) and (18), respectively Step 5: RUP prediction Predict RUP according to (19) and (20)
4. The prognostic results of RUP Six battery cells were tested under the same charge mode and discharged at a 1C rate. According to the data analysis in Section 2.1, Cells_#03#06 were used as the training samples to initialize model parameters, while Cells_#01#02 were used to test the prognostic performance due to the large fluctuation in capacity fade. A comparison was conducted among models A, B, and C, as mentioned in Section 2. In addition, another battery set, Cells_#V1#V4, with a different rated capacity was tested under similar charge/discharge conditions. The purpose was to validate the developed model and verify its robustness through model comparison. Parameter initializations for the different models include initializing the model parameters and their corresponding variance. The model parameters are initialized using the average value through curve fitting based on the battery training samples. Nonlinear least square fitting is performed to initialize the parameters of models A and C, while linear least square is implemented in model B throughout the whole battery life. The initialization of parameters’ variance is time-consuming. In order to make the prediction comparable among these three models, a simple method is employed. Here, taking model C as an example, the parameter vector [c1, c2, c3, and c4]T is estimated according to the curve fitting for Cells_#03#06. The variance of c1 can be approximated on the assumption that the range of [min(c1), max(c1)] is set as 6rc1 . This approach is also employed by the other parameters of each model, P while all the initial variances w of the capacity are set equal to 1e05. Table 5 shows the predicted results. The prediction time K1 is selected when cycling is conducted for up to 1/3, 1/2, and 2/3 of the battery life. The prediction error and standard deviation (STD) are two indicators, in terms of number of cycles, for evaluating the prediction performance.
In Table 5, both error and STD are taken into consideration when assessing the prediction results. A small STD means a narrow RUP pdf and, thus, a high confidence level for prediction. For the assessment of the prediction result, the premise is that the actual EOP falls within the 95% confidence interval. If this is true, the smaller the absolute values of the error and STD, the more accurate the prediction. Meanwhile, a result predicted later with a smaller STD should be more effective and credible than an early warning with a large STD, even if the earlier result is expected. For Cell_#01, the absolute values of the prediction error and STD from model C are smaller than those of model A. Comparing the prediction results at 1/2 of life, model B presents a smaller error than the other models, but its broad prediction distribution leads to low confidence level. Moreover, when predicting at 1/3 of life and 2/3 of life, the projection under model B has no intersection with the EOP. For Cell_#02, predicted results are more effective and credible based on model C. Although the prediction based on model B is in advance of the actual EOP, the error is so large that the real value falls outside of the 95% confidence interval. Thus, model C is regarded as a more accurate capacity degradation model to characterize the prediction performance. By measuring more capacity data, a more accurate RUP prediction is also obtained. Taking two prediction results as examples, Fig. 4 presents the earlier prediction for Cell_#01 when measuring up to the first 1/3 of life (at cycle 283), while Fig. 5 is the prediction for Cell_#02 when predicting the degradation trend at 2/3 of life (at cycle 428). Another battery set with a rated capacity of 1.1 A h was used to validate the effectiveness and robustness of the developed model. These batteries were run similar to Cell_#01#06 with the same charge–discharge mode. According to the definition of the EOP, Cells_#V1, #V2, and #V3 failed at the 469th, 472nd, and 434th cycles, respectively, while Cell_#V4 failed at cycle 633 because of a large dynamic change of capacity data. Thus, the first three cells were trained to average the values of the initial model parameters as discussed above, and #V4 was used for testing. The RUP of Cell_#V4 was also predicted at different prediction cycles: 1/3, 1/ 2, and 2/3 of capacity data. Figs. 6 and 7 show the RUP prediction for Cell_#V4 at different cycles. The prediction results are in line with Table 6, which indicates that model C is better for tracking the capacity fade of the battery than the other two models. When predicting the RUP at cycle 211 (1/3 of the capacity data), both prediction errors based on model C and model A were large (Fig. 6). However, an early estimated result before the actual EOP is more acceptable than a late expected value. Meanwhile, the extrapolation based on model B cannot converge to the EOP, even measuring up to 1/2 of the capacity data. In fact, the dynamic characteristics of battery aging are difficult to predict for a non-rigorous monotonic decreasing. For example, there are data jumps at around cycles 210 and 420, which have a large adjustment on the model parameters, and therefore affect the estimated trajectory and degradation trend. By measuring more capacity data and adjusting the model parameters online, model C is able to track the aging trend and, accordingly, obtain a smaller error and a higher probability of the predicted results than model A and B. Fig. 8 shows the predicted results when measuring up to 80% of the data (corresponding to cycle 507). The objective is to verify and compare the prediction performance in the latter stages of degradation. The estimated errors based on models A, B, and C are 141, 542, and 24 cycles, respectively. By adjusting the parameters to track the changes of the aging dynamics, neither model A nor model B can track the variation well compared to model C, which offers a more accurate predicted result ahead of the actual EOP with a narrower STD. It is worth mentioning that model B’s projection cannot converge to the EOP when the predictions are at cycles 211 and 317. Even if the prediction at cycle 507 converges to the EOP, the pre-
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Model B
Model A 1.1
1.1
Capacity (Ah)
Capacity (Ah)
1.05 1 RUP pdf
0.95 Prediction Time
0.9
1.05 1 0.95 Prediction Time
0.9
200
1.05 RUP pdf
1 0.95 Prediction Time
0.9 End of Performance
0.85 0
Real Data Estimated Values Predicted Values
1.1
End of Performance
End of Performance
0.85 0.8
Model C Real Data Estimated Values Predicted Values
Capacity (Ah)
Real Data Estimated Values Predicted Values
400
600
Cycle Number
800
0.8
0.85 0
200
400
600
800
0.8
0
200
Cycle Number
400
600
800
Cycle Number
Fig. 8. Prediction at the 507th cycle (reaching 80% of life) for Cell_#V4 based on three models.
diction is meaningless because of a large prediction error beyond the measured scope. For model B, a brief explanation is that a strong constraint should be imposed on the algebraic relationship of the model parameters to ensure a decreasing trend because of the second-order polynomial regression. However, the issue of how to use constraints to improve the prediction performance refers to the area related to parameter optimization, is beyond the scope of this paper. Therefore, from the perspective of the degradation model, the ensemble model (model C) presents a more effective and robust characteristic for the prediction of capacity fade of the battery. 5. Conclusions Capacity fade can indicate the SOH of lithium-ion batteries. The estimation of the remaining useful performance (RUP) of a battery depends on the accuracy of the degradation model and the online model parameter adjustment. In this study, an ensemble model modified from two existing empirical models was developed to depict the capacity data that were obtained through continuous full charge and discharge cycles. This ensemble model has a better regression characteristic over the whole battery life than the existing empirical models. Taking into account the uncertainties in the degradation process, particle filtering was used to adjust the model parameters and calculate the estimated capacity with associated weights up to the current observed cycles. RUP was predicted by extrapolating the degradation model to EOP, while its probability distribution was estimated in terms of the associated weights bound up with the current discharge capacity. Through comparing the predicted results from the ensemble model to the results from the exponential and polynomial models at different prediction times, the ensemble model demonstrated better prediction performance (smaller prediction errors and a narrower standard deviation). This is because this model balanced the global and local regression performance. Moreover, the developed model was also validated using another battery set with a different rated capacity. For these two kinds of battery samples, a credible and reliable prediction result can be achieved using a wide range of initialization. The effectiveness and robustness of the developed model in the prediction of capacity degradation was demonstrated. This methodology can be propagated into other degradation-related applications. There are some limitations to applying the developed model. In some cases, it is difficult to quantify the actual maximum capacity because the battery is usually not fully discharged in every cycle. One practical solution to this issue is to map the capacity of the partial discharge into the equivalent fully discharged capacity before using the developed model. The transform relation can then
be explored by measuring the different voltages and finding the interaction between the random cut-off discharge and fully discharged voltage. Furthermore, temperature effects on battery degradation should also be considered in model modification and validation in future work. Acknowledgements The work presented in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (CityU8/CRF/09). The authors would like to thank the members of the Center for Advanced Life Cycle Engineering at the University of Maryland for their support of this work. References [1] Rufus F, Lee S, Thakker A. Health monitoring algorithms for space application batteries; 2008. p. 1–8. [2] Huggins RA. Advanced batteries: materials science aspects. Springer Verlag; 2008. [3] Kim ILS. A technique for estimating the state of health of lithium batteries through a dual-sliding-mode observer. Power Electron, IEEE Trans on 2010;25:1013–22. [4] Saha B, Goebel K, Poll S, Christophersen J. Prognostics methods for battery health monitoring using a Bayesian framework. Instrum Meas, IEEE Trans on 2009;58:291–6. [5] Verbrugge M, Koch B. Generalized recursive algorithm for adaptive multiparameter regression. J Electrochem Soc 2006;153:A187. [6] Schmidt AP, Bitzer M, Imre ÁW, Guzzella L. Model-based distinction and quantification of capacity loss and rate capability fade in Li-ion batteries. J Power Sources 2010;195:7634–8. [7] Spotnitz R. Simulation of capacity fade in lithium-ion batteries. J Power Sources 2003;113:72–80. [8] Saha B, Goebel K, Poll S, Christophersen J. An integrated approach to battery health monitoring using bayesian regression and state estimation; 2007. p. 646-53. [9] Bo C, Zhifeng B, Binggang C. State of charge estimation based on evolutionary neural network. Energy Convers Manage 2008;49:2788–94. [10] Santhanagopalan S, White RE. State of charge estimation using an unscented filter for high power lithium ion cells. Int J Energy Res 2010;34:152–63. [11] Pattipati B, Sankavaram C, Pattipati K. System identification and estimation framework for pivotal automotive battery management system characteristics. IEEE Trans Syst Man and Cyber – Part C Appl Rev 2011;41:869. [12] Plett GL. Extended Kalman filtering for battery management systems of LiPBbased HEV battery packs: Part 3. State and parameter estimation. J Power Sources 2004;134:277–92. [13] Xing Y, Ma EWM, Tsui KL, Pecht M. Battery management systems in electric and hybrid vehicles. Energies 2011;4:1840–57. [14] Andrea D. Battery management systems for large lithium battery packs. Artech House Publishers; 2010. [15] Kozlowski JD. Electrochemical cell prognostics using online impedance measurements and model-based data fusion techniques; 2003. p. 3257–270. [16] He W, Williard N, Osterman M, Pecht M. Prognostics of lithium-ion batteries based on Dempster–Shafer theory and the Bayesian Monte Carlo method. J Power Sources 2011;196:10314–21. [17] Micea MV, Ungurean L, Cârstoiu GN, Groza V. Online state-of-health assessment for battery management systems. Instrum Meas, IEEE Trans on 2011;60:1997–2006.
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