Battery Remaining Useful Life Prediction at Different Discharge Rates

Microelectronics Reliability 78 (2017) 212–219 Contents lists available at ScienceDirect Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel Battery remaining useful life prediction at different discharge rates Dong Wang, Fangfang Yang ⁎, Yang Zhao, Kwok-Leung Tsui Department of Systems Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China a r t i c l e i n f o Article history: Received 6 October 2016 Received in revised form 24 August 2017 Accepted 6 September 2017 Available online xxxx Keywords: Remaining useful life Lithium-ion batteries Particle filter Prognostics and health management Different discharge rates a b s t r a c t Lithium-ion batteries are widely used in hybrid electric vehicles, consumer electronics, etc. As of today, given a room temperature, many battery prognostic methods working at a constant discharge rate have been proposed to predict battery remaining useful life (RUL). However, different discharge rates (DDRs) affect both usable battery capacity and battery degradation rate. Consequently, it is necessary to take DDRs into consideration when a battery prognostic method is designed. In this paper, we propose a discharge-rate-dependent battery prognostic method that is able to track usable battery capacity affected by DDRs in the process of battery degradation and to predict RUL at DDRs. An experiment was designed to collect accelerated battery life testing data at DDRs, which are used to investigate how DDRs influence usable battery capacity, to design a discharge-rate-dependent state space model and to validate the effectiveness of the proposed battery prognostic method. Results show that the proposed battery prognostic method can work at DDRs and achieve high RUL prediction accuracies at DDRs. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Lithium-ion batteries are widely used in hybrid electric vehicles, consumer electronics, etc. Considering several significant battery capacity degradation factors [1], including storage voltage, environment temperature, discharge rate, depth of discharge, etc., one needs to take these factors into consideration in battery prognostics and health management [2], especially battery remaining useful life (RUL) prediction. Here, battery RUL can be regarded as how many charge/discharge cycles are left before battery capacity fails to provide reliable power for electric systems and products [3]. As of today, many battery prognostic methods have been proposed to predict battery RUL at a constant discharge rate. Among these battery prognostic methods, particle filter (PF) based battery prognostic methods [4–9] have attracted lots of attention because PF provides a way to solve numerical integration required in non-linear state space models. Moreover, PF based methods have been demonstrated to be effective in diagnostics and prognostics of other critical components, such as bearing [10], gear [11], carrier plate [12], gas turbine [13], aluminum electrolytic capacitors [14], fatigue crack [15,16], etc. For PF based battery prognostics, Saha et al. [17] proposed to combine relevance vector machine and PF so as to predict battery RUL at a constant discharge rate. In their further comparison study [18], they experimentally demonstrated that the PF based prognostic method has higher RUL prediction ⁎ Corresponding author. E-mail addresses: dongwang4-c@my.cityu.edu.hk (D. Wang), fangfyang2-c@my.cityu.edu.hk (F. Yang), yangzhao9-c@my.cityu.edu.hk (Y. Zhao), kltsui@cityu.edu.hk (K.-L. Tsui). http://dx.doi.org/10.1016/j.microrel.2017.09.009 0026-2714/© 2017 Elsevier Ltd. All rights reserved. accuracies than autoregressive integrated moving average and extended Kalman filter based prognostic methods. Following the work done by Saha et al., He et al. [19] used a bi-exponential function as an empirical battery degradation model so as to fit battery degradation data at a constant discharge rate and they experimentally found that the bi-exponential function has good ability to fit the battery degradation data. Based on the empirical battery degradation model, they built a state space model at a constant discharge rate and used PF to posteriorly estimate parameters distributions for battery RUL prediction at a constant discharge rate. To better fit local battery degradation behavior, Xing et al. [20] combined an exponential function and a polynomial function with an order of 2 to form an ensemble empirical battery degradation model and they experimentally demonstrated that the new empirical battery degradation model is able to predict battery RUL at a constant discharge rate better than the bi-exponential function based prognostic method. Since then on, many other researchers have tried to improve battery RUL prediction accuracies at a constant discharge rate by enhancing the performance of PF, including its particle diversity [21,22], model adaptation [23] and its importance function [24–26]. Even though the aforementioned battery prognostic methods had good RUL prediction accuracies at a constant discharge rate, these prognostic methods did not consider the influence of discharge rate on battery degradation. Actually, given a room temperature, discharge rate is one of the most significant factors to influence battery capacity degradation [27]. Normally, the higher a discharge rate, the faster a capacity degradation rate. Moreover, discharge rate affects usable capacity. The higher a discharge rate, the smaller a usable capacity. And, when a discharge rate is changed from a high rate to a low rate, most ‘lost’ capacity caused by the high rate is revoked [28]. This is the reason why we use D. Wang et al. / Microelectronics Reliability 78 (2017) 212–219 usable capacity instead of capacity in this paper to distinguish capacity influenced by different discharge rates (DDRs). Consequently, it is necessary to take DDRs into consideration when a battery prognostic method is designed. In this paper, a discharge-rate-dependent battery prognostic method is proposed. The main contributions of this paper are highlighted as follows. Firstly, an experiment was designed to collect four battery degradation data at DDRs. The design of the experiment aims to investigate how DDRs affect usable battery capacity. Even though only four battery degradation samples at DDRs are available for our analyses, it took one year to collect them and the collection of battery degradation at DDRs is time-consuming. Secondly, because DDRs influence the value of usable capacity, it is difficult to directly use some empirical battery degradation models working at a constant discharge rate, such as the exponential function [17,18,23,25], the bi-exponential function [19], the ensemble function [20], etc. to describe battery degradation at DDRs. It is necessary to develop a more general battery degradation model working at DDRs. In this paper, we take the exponential function as an example and extend it to a more general empirical battery degradation model working at DDRs by discovering the relationship between the amplitude and slope of the exponential function and DDRs. According to our preliminary analyses, the exponential function is good enough in this paper to describe a battery degradation curve at a specific discharge rate. If one parameter has a linear relationship with discharge rate, only four hidden states are required in the state space modeling of battery degradation at the DDRs, which can be efficiently and posteriorly updated by using PF. Thirdly, based on the more general empirical battery degradation model, a discharge-rate-dependent state space model is proposed to track usable capacity degradation data at DDRs. More interestingly, given a constant discharge rate, the dischargerate-dependent state space model can be reduced to the state space model used in [17,18,23,25]. Fourthly, we illustrate how to use PF to posteriorly estimate the parameter distributions of the discharge-ratedependent state space model. Once the parameter distributions of the discharge-rate-dependent state space model are determined, we are able to predict battery RUL at DDRs by extrapolating the established state space model to a discharge-dependent soft failure threshold. Here, the discharge-rate-dependent soft failure threshold is taken as 80% of initial usable capacity values at DDRs. The main reason why we are interested in predicting RUL at DDRs is that we are concerned about how many charge/discharge cycles are left if the current discharge rate is changed to another concerned discharge rate. Battery RUL prediction at DDRs is able to suggest users when they are not allowed to use a higher discharge rate instead of the current discharge rate. According to our literature review, this new idea related to battery RUL prediction at DDRs is new and seldom reported. The rest of this paper is outlined as follows. An experiment was designed in Section 2 to collect usable capacity degradation data at DDRs. In Section 3, to mathematically describe usable capacity degradation data at DDRs and predict battery RUL at DDRs, the discharge-rate-dependent battery prognostic method is proposed. In Section 4, the effectiveness of the proposed prognostic method is experimentally validated. Conclusions are drawn in the last section. 2. Design of an experiment for collection of usable lithium-ion battery capacity degradation data at DDRs Before the discharge-rate-dependent battery prognostic method is detailed in Section 3, an experiment was designed to collect lithiumion battery degradation data at DDRs. In the experiment, four cylindrical BAK 18650 battery samples and their specifications are tabulated in Table 1. The room temperature was kept at an environment temperature of 25 °C. The battery test bench, composed of an Arbin BT2000 tester for loading and sampling the batteries, a host computer with an Arbin MITS Pro Software for on-line experiment control and data recording, and a computer with Matlab R2012b Software for preliminary 213 Table 1 The specifications of the four battery samples used in the experiment. Cathode Anode Rated capacity Upper/lower cut-off voltage End-of-charge current Max continuous discharge current LiFePO4 Graphite 1 Ah 3.6 V/2 V 0.01C 10C data analysis, is shown in Fig. 1(a). A profile comprising a sequence of repetitive 0.5C, 1C, 3C, and 5C constant current discharge regimes was implemented to collect usable capacity degradation data in the course of four-cycle rotation aging. The four batteries were recharged with a schedule recommended by manufacturer, which comprised a 1C constant current charging step followed by a constant voltage charging step until a cutoff current of C/100 was reached. Fig. 1(b) shows the measured current and voltage profile in a four-cycle rotation interval. In the Arbin testing system, the discharge and charge currents were respectively represented by negative and positive values. The accumulated usable discharge capacity was calculated by integrating the current over time. To show the rated capability of a lithium-ion battery, the relationship between the discharge curve and the accumulated usable capacity is plotted in Fig. 1(c). Clearly, the maximum releasable capacity at 5C is only 0.92 Ah, which is less than those at the lower discharge rates, especially 0.5C. Thus, the deliverable usable capacity is reduced if the battery is discharged at a very high rate. The usable capacity degradation data of one of the four batteries are plotted in Fig. 2(a), where the peculiar increasing usable capacity data at some initial four-cycle rotations have been artificially removed for the sake of RUL prediction because these usable capacity are not sufficiently useful in describing usable capacity degradation. It was observed that the DDRs have significant impacts on usable capacity degradation data. The higher a discharge rate, the smaller usable capacity. Additionally, when a battery was discharged at a high rate of 5C, it is observed that its associated usable capacity is more fluctuated. This phenomenon is explained by a high battery surface temperature with a high variation at a discharge rate of 5C. By attaching a thermocouple to the surface of a battery with a sampling rate of 1 s, we plot the averages of the battery surface temperatures in every discharging process in Fig. 2(b), in which it is observed that the higher a discharge rate, the higher a battery surface temperature. 3. The proposed discharge-rate-dependent prognostic method for battery RUL prediction at DDRs In this section, we propose a discharge-rate-dependent prognostic method which is able to track usable capacity degradation data at DDRs and to predict battery RUL at DDRs. In the proposed dischargerate-dependent prognostic method, one of the most important key steps is to construct a discharge-rate-dependent state space model so as to describe usable capacity degradation data at the DDRs including 0.5C, 1C, 3C and 5C as shown in Fig. 2(a). Here, batteries 1 to 3 are used to provide the historical batteries data and battery 4 is used to provide the testing degradation data. Using the historical degradation data at the DDRs have the following three purposes. Firstly, because a physical battery degradation model is seldom reported, it is necessary to use the historical battery degradation data at the DDRs to establish an empirical battery degradation model. Secondly, it is necessary to extend the empirical battery degradation model to a more general empirical battery degradation model working at the DDRs. Thirdly, based on the more general empirical battery degradation model, a discharge-rate-dependent state space model is constructed. Moreover, its parameters are initialized by the historical battery degradation data. To achieve the first purpose, a discharge-rate-dependent soft failure threshold xthreshold = 1.093 − 0.01665 × D is firstly defined. Here, D is the discharge rate. The discharge-rate-dependent soft failure threshold 214 D. Wang et al. / Microelectronics Reliability 78 (2017) 212–219 Fig. 1. Design of an experiment for collection of lithium-ion battery degradation data at DDRs: (a) the battery test bench; (b) the measured current and voltage in a four-cycle rotation; (c) the discharge curves with accumulated discharge capacity over a four-cycle rotation. is estimated by fitting four soft failure thresholds at the discharge rates of 0.5C, 1C, 3C and 5C. Each of the four soft failure thresholds is defined as 80% of the initial usable capacity of a battery. The goodness-of-fit of the linear function is 0.996 by using the R-squared measure. Then, only the usable capacity degradation data ranged from the initial usable capacity to the usable capacity reaching the discharge-rate-dependent soft failure threshold are used to establish an empirical battery capacity degradation model. In this paper, we found that an exponential function a exp(b × i) with the amplitude a and slope b is able to respectively fit all the usable capacity degradation data. Here, i is the ith four-cycle rotation. To validate this statement, the goodness-of-fit is used again. The boxplots of the estimated parameters of the fitted exponential functions for all the historical data and their associated R-squared values are plotted in Fig. 3(a) to (c), respectively, where it is connoted that almost all data are well fitted because the R-squared values are larger than 0.95. The main reason why two usable capacity degradation data are not well fitted is caused by the severe fluctuations of the data at the discharge rate of 5C. To achieve the second purpose, a discharge-rate-dependent exponential function is proposed to correlate the amplitude a and slope b of the exponential function with the DDRs. For each of the three historical batteries, the estimated amplitudes and slopes of the fitted exponential functions at the DDRs are plotted in Fig. 4(a) to (f), where we observed that the amplitude and the slope have a linear relationship with the discharge rate D, respectively. The specific formulas for clarifying the relationship between the amplitude a and slope b of the empirical battery degradation model and the DDRs for each of the three historical batteries are shown in Fig. 4(a) to (f), respectively, where Fig. 2. Degradation of a lithium-ion battery at the DDRs including 0.5C, 1C, 3C and 5C: (a) degradation data; (b) the surface temperature of the battery. Fig. 3. Boxplots of the amplitudes and slopes of the fitted exponential functions and their associated R-squared values for lithium-ion batteries 1, 2 and 3 at the DDRs: (a) the fitted amplitude; (b) the fitted slope; (c) the R-squared value. D. Wang et al. / Microelectronics Reliability 78 (2017) 212–219 215 a(D) = (a1 + a2 × D) is used to express the linear relationship between the amplitude a of the exponential function and the discharge rate D, and b(D) = (b1 + b2 × D) is used to express the linear relationship between the slope b of the exponential function and the discharge rate D. Here, a1 , a2 ,b1 and b2 are four parameters used in the aforementioned two linear functions a(D) and b(D). Therefore, the discharge-rate-dependent exponential function is formulated as (a1 +a2 × D)× exp((b1 + b2 × D) × i). To achieve the third purpose and describe the variance among the three batteries and, the discrete-time discharge-rate-dependent state space model is developed as follows: a1;i ¼ a1;i−1 þ v1 a2; i ¼ a2; i−1 þ v2 b1; i ¼ b1; i−1 þ v3 b2; i ¼ b2; i−1 þ v4


xi ¼ a1; i þ a2; i  Di  exp b1; i þ b2; i  Di  i þ v5 ; ð1Þ where Di is the discharge rate at four-cycle rotation i; each of the v1, v2, v3, v4 and v5 follows a Gaussian distribution with a zero mean and its associated deviation σ1, σ2, σ3, σ4 and σ5, respectively. xi is the current usable capacity degradation data at four-cycle rotation i. The initialization of Eq. (1) can be established by taking the mean of the estimated amplitudes and slopes as shown in Fig. 4. Specifically, a1,0, a2,0, b1,0 and b2,0 were respectively equal to 1.1026, − 0.0231, − 2.6503 × 10− 4 and − 4.19 × 10−5. According to the scale of each of the parameters used in Eq. (1), the σ1, σ2, σ3, σ4 and σ5 were empirically set to 10−4, 10−4, 10−5, 10−5, and 5 × 10−3, respectively.For a testing battery, given its measurements up to date xM = {x1, x2, …, xM} with a length of M and a sequence of different discharge rates DM = {d1 , d2 , … , dM} at fourcycle rotations from 1 to M, the parameters ϕM = [a1,M, a2,M, b1,M, b2,M] and their distributions can be posteriorly estimated by using dynamic Bayesian inference. Because the nature of Kalman filter is the optimal linear filter in cases where usable capacity degradation data xi must be a linear function of four hidden states a1,i, a2,i, b1,i and b2,i in Eq. (1) and the optimal Kalman filter gain is mathematically derived only when all formulas in Eq. (1) are linear, it is impossible to directly use Kalman filter to estimate the posterior distributions of the ϕ. Therefore, a numerical integration based dynamic Bayesian inference method called PF is employed. The main idea of the PF is to use an amount of random particles drawn from an importance function and their associated weights to approximate parameter distributions. For the fundamental theory of the PF, please refer to [29,30]. How the proposed discharge-rate-dependent prognostic method works at the DDRs is des tailed as follows. Initially, Ns = 2000 random particles {ak1,0}N k = 1, k Ns k Ns k Ns {a2,0}k=1, {b1,0}k=1 and {b2,0}k=1 are drawn from the Gaussian distributions N(a1,0, σ21), N(a2,0, σ22), N(b1,0, σ23) and N(b2,0, σ24), respectively. All random particles have the equal weights. Suppose that the posterior distributions of a1,i − 1, a2,i − 1, b1,i − 1 and b2,i − 1 at four-cycle rotation k Ns s i-1 are available. Ns random particles {a k1,i − 1 }N k = 1 , {a 2,i − 1 } k = 1 , k Ns k Ns {b1,i − 1}k = 1 and {b2,i − 1}k = 1 are drawn from the Gaussian distributions N(a1,i − 1, σ21), N(a2,i − 1, σ22), N(b1,i − 1, σ23) and N(b2,i − 1, σ24), res spectively. Moreover, their associated weights are {ω ik− 1 }N k = 1. Given a new measurement xi, the weight of each random particle is updated by: 0    2 1
k k k k B− xi − a1; i−1 þ a2; i−1  di  exp b1; i−1 þ b2; i−1  di  i C ωki ∝ωki−1 exp@ A; 2σ 25 ð2Þ k ¼ 1; 2; …; N s : Fig. 4. The amplitudes and slopes of the fitted exponential functions at the DDRs in the case of batteries 1, 2 and 3: (a) the relationship between the amplitude a of the exponential function and the DDRs for battery 1; (b) the relationship between the slope b of the exponential function and the DDRs for battery 1; (c) the relationship between the amplitude a of the exponential function and the DDRs for battery 2; (d) the relationship between the slope b and the DDRs for battery 2; (e) the relationship between the amplitude a of the exponential function and the DDRs for battery 3; (f) the relationship between the slope b of the exponential function and the DDRs for battery 3. Then, each of the weights is normalized by dividing the sum of all the weights. From Eq. (2), it is not difficult to find that only random particles which produce better estimates for the measurement xi have higher weights at four-cycle rotation i. After several iterations, most of the weights will become negligible. To alleviate such degenerative problem, a systematic resampling algorithm [29] is employed to replace particles 216 D. Wang et al. / Microelectronics Reliability 78 (2017) 212–219 with small weights with particles with large weights when the following condition is satisfied: probability density function (PDF) of the battery RUL as follows: pðκ jxM ; DM ; D ¼ ddÞ ¼ Ns  2 ∑ ωki k¼1 !−1 k¼1 b0:9  Ns : ð3Þ Increase i = i + 1 and repeat the aforementioned steps until i N M. The posterior distributions of ϕMat four-cycle rotation Mare represented as follows:
p a1; M jxM ; DM ¼
p a2; M jxM ; DM ¼
p b1; M jxM ; DM ¼
Ns  X p b2; M jxM ; DM ¼ Ns X k¼1 Ns X k¼1 Ns X k¼1 Ns X k¼1   ωkM δ a2; M −a2; kM ð5Þ    k k exp b1; M þ b2; M  dd  j ≤xthreshold ðddÞÞ þ MÞÞ: Consequently, the estimate of battery RUL and its lower and upper bounds are the 50th, 5th, and 95th percentiles of Eq. (5), respectively. At last, the failure probability density function (FPDF) of the battery is calculated by: pðψjxM ; DM ; D ¼ ddÞ ¼   a1; M −a1; kM ; ωkM δ     ωkM δ κ− inf j∈ int : a1; kM þ a2; kM  dd  Ns  X k¼1     ωkM δ ψ− inf j∈ int : a1; kM þ a2; kM  dd  ð6Þ    k k exp b1; M þ b2; M  dd  j ≤ xthreshold ðddÞÞÞÞ: ð4Þ   k ωkM δ b1; M −b1; M ;   k ωkM δ b2; M −b2; M : Because the DDRs affect the usable capacity degradation data, the discharge-rate-dependent threshold xthreshold(D) defined at the beginning of Section 3 should be used in RUL prediction. Moreover, it is interesting to infer RUL at DDRs. Considering this issue, given Eq. (4) and a concerned discharge rate D = dd, Eq. (5) aims to extrapolate the exponential function established by Eq. (4) to reach the discharge-rate-dependent soft failure threshold for the first time and calculates the The flowchart of the proposed discharge-rate-dependent prognostic method for battery RUL prediction at DDRs is summarized in Fig. 5. 4. A case study of the proposed discharge-rate-dependent prognostic method for lithium-ion battery RUL prediction at DDR In this section, the usable capacity degradation data of battery 4 at the DDRs are used to validate the effectiveness of the proposed discharge-rate-dependent prognostic method. The raw capacity degradation data of battery 4 at the DDRs are similar with those reported in Fig. 2(a) and they were collected according to the same collection procedure reported in Section 2. However, in practice, batteries usually do not run from a fully charged state to a fully discharged state in each cycle. To test the robustness of the proposed method in the case of Fig. 5. A flowchart of the proposed discharge-rate-dependent prognostic method for battery RUL prediction at DDRs. D. Wang et al. / Microelectronics Reliability 78 (2017) 212–219 217 incomplete degradation data, we artificially generated a random sequence of battery degradation data at the DDRs. Here, we assumed that in each four-cycle rotation only one capacity data was available. Firstly, in the first four-cycle rotation, we randomly generated a discharge rate from the four DDRs including 0.5C, 1C, 3C and 5C. Then, we picked up the usable capacity value of battery 4 at the generated discharge rate in the first four-cycle rotation. Secondly, at the second fourcycle rotation, we randomly generated another discharge rate from the four DDRs and then picked up another usable capacity value of battery 4 at the new generated discharge rate in the second four-cycle rotation. Repetitively, we artificially constructed the sequence of the DDRs in Fig. 6(a) and its corresponding usable capacity degradation data shown in Fig. 6(b). It should be noted that in our case study, we only picked up one usable capacity value from one four-cycle rotation even though one four-cycle rotation contained four cycles. Consequently, we made battery RUL prediction at a ‘cycle’ with a length of a fourcycle rotation, which is not a big issue in practice because capacity can be measured at each cycle rather than a four-cycle rotation. Here, again, the design of our four-cycle rotation mainly aims to accelerate battery degradation because battery degradation at the DDRs is timeconsuming. The actual life was defined as the lifetime from a four-cycle rotation corresponding to an initial usable capacity to the four-cycle rotation related to the usable capacity reaching the discharge-rate-dependent soft failure threshold. Then, we made battery RUL predictions at the 30% to the 90% of the actual life with an increment of 10%. Moreover, we are interested in RUL prediction at the DDRs. It means that once the current discharge rate and the latest usable capacity measurement are used to update the proposed prognostic method, Eq. (5) is used to predict battery RUL at more other rates besides the current discharge rate. For example, in Fig. 6(c), we made RUL prediction at the 60% of the actual life. To visualize the prognostic results at the DDRs, the raw data, the predicted usable capacity data by using the proposed discharge-rate-dependent prognostic method, the FPDFs at the DDRs, the predicted future battery capacity degradation paths at the DDRs are plotted in Fig. 6(c), respectively. It was observed that the predicted capacity degradation data highlighted by the cyan color and bold lines are well matched with the raw usable capacity degradation data highlighted by the purple color and dotted lines. At the prediction time, the current discharge rate is 1C and the RUL prediction is located in Fig. 6(c). The predicted RUL results at more DDRs are tabulated in Table 2, where the results are the RUL predictions associated with the current discharge rate and the other results are the RUL predictions associated with other alternative DDRs. For the RUL predictions at the current discharge rate, it is observed that the proposed discharge-rate-dependent prognostic method achieves high RUL prediction accuracies even though the proposed method works at the DDRs. Another interesting RUL prediction result can be found in Table 2 when the RUL prediction is made at the 80% and 90% of the actual life. At those prediction times, if the RUL predictions at the current discharge rate of 0.5C are changed to the RUL Fig. 6. Battery RUL predictions at the DDRs by using the proposed DDRs-dependent prognostic method in the case of battery 4: (a) the sequence of the DDRs; (b) the usable battery capacity degradation data corresponding to the sequence of the DDRs; (c) the battery RUL prediction at the DDRs and 60% of the actual life. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 Battery RUL predictions at DDRs by using the proposed DDRs-dependent prognostic method (unit: four-cycle rotation). The current Prediction time (the discharge rate D percentage of the actual life) 3C 5C 3C 1C 1C 0.5C 0.5C 30% 40% 50% 60% 70% 80% 90% The percentile of predicted The percentile of predicted The percentile of predicted The percentile of predicted RUL and error at the discharge RUL and error at the discharge RUL and error at the discharge RUL and error at the discharge rate of 0.5C rate of 1C rate of 3C rate of 5C True RUL 5% 50% 95% Error True RUL 5% 50% 95% Error True RUL 5% 50% 95% Error True RUL 5% 50% 95% Error 249 214 178 142 107 71 36 194 130 96 103 67 50 15 212 166 140 118 89 61 25 191 132 103 102 74 23 1 208 165 140 116 86 50 15 171 142 124 67 15 1 1 193 164 142 110 79 14 1 138 133 95 28 1 1 1 173 157 139 98 64 1 1 235 209 193 135 111 72 36 37 48 38 24 18 10 11 234 199 163 127 92 56 21 227 203 183 129 99 82 54 26 34 23 11 6 6 6 197 162 126 90 55 19 −16 223 191 164 172 197 182 227 4 -2 −16 −20 −24 5 −17 176 141 105 69 34 −2 −37 223 188 209 220 350 280 326 3 −16 −34 −29 −30 −3 −38 218 D. Wang et al. / Microelectronics Reliability 78 (2017) 212–219 Table 3 Battery RUL predictions at DDRs by using the traditional PF based prognostic method (unit: four-cycle rotation). The current discharge rate D 3C 5C 3C 1C 1C 0.5C 0.5C Prediction time (the percentage of the actual life) 30% 40% 50% 60% 70% 80% 90% True RUL 197 141 126 127 92 71 36 The percentile of RUL and error 5% 50% 95% Error 127 67 84 76 66 45 15 131 71 92 84 77 57 24 139 76 101 91 90 70 34 66 70 34 43 15 14 12 predictions at the rates of 3C and 5C, the 5th percentile and the 50th percentile of the PDF of the battery RUL are equal to 1, which connotes that the shape of the PDF of the battery RUL concentrates on 1 and the battery will suffer quick soft failure. The traditional PF based prognostic method was used here to compare the proposed discharge-rate-dependent prognostic method. The traditional PF based prognostic method was the pioneered prognostic method reported in reference [19] and its corresponding state space model is given in Eq. (7), where the exponential function of the state space model in Eq. (7) does not consider the correlation of the amplitude a and slope b with the DDRs. Consequently, the state space model in Eq. (7) is a traditional model that was designed for battery prognostics at a constant discharge rate. a1;i ¼ a1;i−1 þ v1 b1; i ¼ b1; i−1 þ v2
xi ¼ a1; i  exp b1; i  i þ v3 : ð7Þ All the settings used in the traditional PF based prognostic method are the same with those used in the proposed discharge-rate-dependent prognostic method. Moreover, the posterior parameters of Eq. (7) are estimated by using the same PF procedure reported in the last section. For a fair comparison, the discharge-rate-dependent soft failure threshold is also used in the traditional PF based prognostic method. The prediction results by using the traditional PF based prognostic method are tabulated in Table 3, where it is observed that the RUL prediction accuracies provided by the traditional PF based prognostic method re not as high as those provided by the proposed discharge-rate-dependent prognostic method in Table 2. To visualize the RUL predictions of the traditional PF based prognostic method, the predictions are plotted in Fig. 7. Inaccurate capacity tracking of the traditional PF based prognostic method working lowers its RUL prediction accuracies. 5. Conclusion In this paper, a discharge-rate-dependent prognostic method was proposed to predict battery RUL at the DDRs. Firstly, an exponential function was used to fit the historical degradation data. Secondly, the discharge-rate-dependent empirical battery degradation model was built to correlate the exponential model with the DDRs. Further, a discharge-rate-dependent state space model was constructed to consider the individual variances among the historical degradation data. Thirdly, given the measurements and the DDRs up to date, the particle filer was introduced to posteriorly estimate the parameter distributions of the discharge-rate-dependent state space model. Then, the extrapolations of the established state space model to the discharge-rate-dependent soft failure threshold were used to predict the battery RUL at the DDRs. At last, the data provided by battery 4 were used to validate the effectiveness of the proposed discharge-rate-dependent prognostic method. The results showed that the proposed prognostic method is able to accurately track the usable degradation data affected by the DDRs. Moreover, the proposed prognostic method is able to predict the battery RUL at other discharge rates. Compared with the traditional PF based prognostic method, the proposed prognostic method can produce higher RUL prediction accuracies at the current discharge rate. In the near future, temperature will be incorporated into the proposed discharge-rate-dependent battery prognostic method to deal with the case of a large environment temperature variance. Acknowledgement This research work was partly supported by National Natural Science Foundation of China (Project Nos. 51505307 and 11471275), General Research Fund (Project No. CityU 11216014), and the Research Grants Council Theme-based Research Scheme under Project T32-101/ 15-R. The authors would like to thank three anonymous reviewers and the handling editor for their valuable and constructive comments on our manuscript. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.microrel.2017.09.009. References Fig. 7. Battery RUL predictions at the 60% of the actual life by using the traditional PF based prognostic method in the case of battery 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) [1] S.M. 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