A new hybrid method for the prediction of the remaining useful life of a lithium-ion battery

Applied Energy xxx (xxxx) xxx–xxx

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

A new hybrid method for the prediction of the remaining useful life of a lithium-ion battery ⁎

Yang Changa,b, Huajing Fanga,b, , Yong Zhangc a b c

School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China National Key Laboratory of Science and Technology on Multispectral Information Processing, Wuhan 430074, China School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan 430081, China

H I G H L I G H T S proposed prognostic method can make full use of historical information. • The method of obtaining historical error data is discussed in detail. • The experiments based on data-driven and model-based methods are performed. • Comparative • Battery working with different discharging currents is considered.

A R T I C L E I N F O

A B S T R A C T

Keywords: Lithium-ion battery Remaining useful life Unscented Kalman filter Relevance vector machine Complete ensemble empirical mode decomposition Error-correction

The lithium-ion battery has become the main power source of many electronic devices, it is necessary to know its state-of-health and remaining useful life to ensure the reliability of electronic device. In this paper, a novel hybrid method with the thought of error-correction is proposed to predict the remaining useful life of lithium-ion battery, which fuses the algorithms of unscented Kalman filter, complete ensemble empirical mode decomposition (CEEMD) and relevance vector machine. Firstly, the unscented Kalman filter algorithm is adopted to obtain a prognostic result based on an estimated model and produce a raw error series. Secondly, a new error series is constructed by analyzing the decomposition results of the raw error series obtained by CEEMD method. Finally, the new error series is utilized by relevance vector machine regression model to predict the prognostic error which is adopted to correct the prognostic result obtained by unscented Kalman filter. Remaining useful life prediction experiments for batteries with different rated capacities and discharging currents are performed to show the high reliability of the proposed hybrid method.

1. Introduction With the advantages of high energy density, low self-discharge rate, long lifetime and light weight, the lithium-ion battery has become an important and widely used power source in numerous fields, ranging from communication system, electric vehicle and military equipment to space system. However, as the number of charging and discharging increases, the battery ages gradually and becomes failure in the end. An aged battery can affect the function of device, reduce the reliability of system, even result in catastrophe, therefore, prognostics for lithiumion battery is very necessary. Generally, the prognostics of battery is a process of predicting its state-of-life (SOH) and remaining useful life (RUL) [1,2]. RUL of a lithium-ion battery is defined as the number of remaining charge-discharge cycles before the performance deteriorates

⁎

to the rated failure threshold. The battery degradation can be characterized by health indicator (HI), such as current, voltage, impedance and capacity. In the existing literature, method for battery prognostics can be broadly categorized into data-driven approach, model-based approach and hybrid approach [3]. The data-driven prognostics depend on the historical data and mine the degradation information via various data analysis methods [4]. One of the classical data-driven methods for battery prognostics is the time series analysis, this method has the advantages of simple calculation and low complexity, which has been widely used in real applications [5]. Because of the powerful learning and data mining capabilities, intelligence algorithms have become the most active data-driven approaches in recent years, including artificial neural network (ANN) [6], support vector machine (SVM) [2,7], relevance vector machine (RVM)

Corresponding author at: School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China. E-mail addresses: [email protected] (Y. Chang), [email protected] (H. Fang), [email protected] (Y. Zhang).

http://dx.doi.org/10.1016/j.apenergy.2017.09.106 Received 26 May 2017; Received in revised form 16 September 2017; Accepted 18 September 2017 0306-2619/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Chang, Y., Applied Energy (2017), http://dx.doi.org/10.1016/j.apenergy.2017.09.106

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Nomenclature

x́k ýk

f (·) h (·) y̆k ek̂ pT̂ EOM

k Qk TEOL TEOM xkerror

the states produced by the estimated state transition equation the measurements produced by the estimated measurement equation the state transition equation for augmented state vector the measurement equation for augmented state vector the predicted measurements of the hybrid method the predicted error the estimated parameter vector

xk∗ yk∗ CEEMD EOL EOM HI IMF RBF RUL RVM UKF UKF-PE

hybrid

̂ TEOL xT̂ EOM ∼ xk

the end of life cycle predicted by the hybrid method the estimated states the augmented state vector comprising states and parameters hybrid  RULTEOM the remaining useful life predicted by the hybrid method ek the reconstructed error f (·) the state transition equation f (·,pT̂ EOM ) the estimated state transition equation h (·) the measurement equation h (·,pT̂ EOM ) the estimated measurement equation

the cycle index the battery capacity at cycle k the end of life cycle the end of monitoring cycle the raw error between real measurements and the measurements produced by the estimated measurement equation at estimation phase the predicted states of the unscented Kalman filter the predicted measurements of the unscented Kalman filter complete ensemble empirical mode decomposition end of life end of monitoring health indicator intrinsic mode function radial basis function remaining useful life relevance vector machine unscented Kalman filter prognostic error produced by unscented Kalman filter

a number of filtering algorithms have been employed for battery prognostics [16,18–23]. Generally speaking, the process of model-based prognostic method is divided into two phases: estimation phase and prediction phase. In the estimation phase, the prognostic method is adopted to identify the parameters and states of the proposed model based on the available measurements. In the prediction phase, the battery degradation process is predicted based on the estimated model, since there are no measurements available, the parameters of the model cannot be updated any more. Compared with the data-driven method, the model of the model-based method has been established in advance, only the parameters need to be estimated, therefore, the model-based method can achieve a good prognostic result even though the amount of historical data is not large. However, the majority of existing pure filtering-based methods keep the model parameters unchanged during the prediction phase [24], which results in the inability to depict the uncertainty of degradation, thus limiting the online application. Both the pure data-driven method and pure model-based method have their own limitations when applied to lithium-ion battery prognostics. It is worth pointing out that most of the related research mainly focuses on developing various algorithms to build an accurate model for RUL prediction. It may achieve good result for short-term prognostics, when the prediction horizon becomes longer, the performance will deteriorate due to the degradation uncertainty. Therefore, a new kind of approach named model-data hybrid method is proposed for lithiumion battery prognostics. Liu et al. [25] put forward a model-data hybrid method where the data-driven part was adopted to predict the battery future measurements, which were fed into particle filter (PF) for longterm RUL prediction. Zheng et al. [26] adopted a relevance vector regression (RVR) model to predict the filtering error of unscented Kalman filter (UKF), the filtering error was incorporated into UKF to predict the RUL iteratively. These methods utilized both the pre-established model of model-based method and uncertainty description ability of data-

[8,9], and some other methods [10,11]. Compared with the time series analysis method, the intelligence algorithms can solve the more complicated prognostic problems. Patil et al. [12] proposed a multistage SVM method, it integrated classification model and regression model to develop an efficient RUL estimation algorithm which can achieve a result for battery at varying operating conditions with sufficient accuracy. Ng et al. [3] proposed a naive Bayes prognostic method, the method can achieve good performance under different usage conditions and ambient temperatures. The data-driven method only relies on historical data and it can construct a model without knowing any complex physics knowledge about battery degradation. However, when the quantity of the historical data is too small or the quality is too poor, the application of data-driven method cannot achieve a satisfactory result. Unlike a data-driven method, the application of model-based method is highly dependent on the analysis of degradation process and failure mechanisms of the lithium-ion battery. The model used in model-based method can be an electrochemical model [13–16], an equivalent circuit model [16,17] or an empirical model [18]. The electrochemical model is built with considering the effects of many factors, there are many parameters in the model that need to be estimated, which makes it difficult to apply in reality. The equivalent circuit model can approximate the dynamic characteristics of battery by constructing a circuit model, it is more applicable than electrochemical model, but still too complicated to use for battery prognostics online. Actually, the empirical model is the most commonly used model, it is empirically established by fitting a large amount of HI degradation data [18]. Compared with the other two models, the empirical model is easier to construct and applicable to a wider range of applications. Nowadays, many kinds of empirical models have been proposed, such as the resistance-based exponential growth model [19] and linear parameter-varying electrical model [20], the capacity-based exponential model [21] and polynomial model [18]. Based on the model,

Fig. 1. Three parts of the proposed hybrid method.

2

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3. Prognostics result based on unscented Kalman filter method

driven method, which can improve the accuracy of prognostic result to a certain extent. However, the coupling between the model-based part and the data-based part is so strong that a low-quality data-driven model can severely affect the function of model-based part, and vice versa, thus resulting in a bad prognostic result. To overcome the limitations of the model-data hybrid method, this paper proposes a new hybrid prognostic method based on the thought of error-correction to predict the SOH and RUL of lithium-ion battery. The hybrid method combines UKF algorithm, RVM algorithm as well as complete ensemble empirical mode decomposition (CEEMD) algorithm, and its implementation can be summarized as the following three phases: in the first phase, the prognostic result is achieved by using the UKF algorithm and a series of raw error data is the by-product; in the next phase, after decomposing the error data with CEEMD algorithm, the dominant mode can be obtained with the aid of statistical analysis, which is regarded as the new error data, i.e. the so-called prognostic error training data; in the last phase, the prognostic error is predicted by employing the RVM algorithm so that the UKF-based prognostic result can be corrected and the final RUL prediction can be acquired. Fig. 1 shows the relationship between the three phases of the proposed hybrid method. The comparative experiments about RUL prediction of lithium-ion battery with different discharging currents and rated capacities are conducted and the results show that compared with UKF method and RVM method, the proposed hybrid method can achieve more accurate and robust RUL prediction result. Unlike the hybrid methods proposed in [25,26], the coupling between the model-based part and the data-driven part is very loose in the hybrid method proposed in this paper, and the historical data used for prognostic error prediction has been optimized, which can ensure the accuracy and robustness of the final RUL prediction result. The remainder of this paper is organized as follows: Section 2 provides some basic conditions for describing the proposed method. The detailed implementations of the proposed method are discussed at Sections 3–5. Sections 6 and 7 conduct RUL prediction experiments of lithium-ion battery and present some indicators to analyze the results. Conclusions and future works are discussed in Sections 8 and 9.

3.1. Joint estimation of states and parameters based on unscented Kalman filter UKF is a nonlinear filter based on discrete system and has a strong ability of dealing with uncertainty and nonlinearity. UKF approximates the distribution of model states using sigma points which are obtained by unscented transformation (UT) [27]. Compared with extended Kalman filter (EKF), UKF can obtain a higher estimation accuracy, and it has lower computational complexity compared with PF. The implementation of UKF algorithm contains two update processes, namely state update process (prediction) and measurement update process (correction). Apart from prognostics of lithium-ion battery [26], UKF were also applied to prognostics of the high power white light emitting diodes [28], estimation of state of charge (SOC) of the lithium-ion battery [29,30]. For UKF-based prognostic method, in order to obtain an accurate model which describes the battery degradation process, both the states and unknown parameters are needed to be estimated and updated. The joint estimation of states and model parameters can be achieved by treating model parameters as states and constructing an augmented state vector, the augmented state vector is labeled as ∼ xk = [xk ,pk ]T . Ideally, the parameters of the model keep unchanged during the battery degradation process, but in fact the parameters cannot be constant due to the uncertainty. Therefore, the evolution of parameters need to be considered, which can be described by adding a Gaussian noise with zero mean [24,31]:

pk + 1 = pk + wkwn

And transition of the augmented state vector can be represented by:

f (xk ,pk ) ⎤ ⎡ wk ⎤ ∼ + xk + 1 = ⎡ ⎢ pk ⎦ ⎥ ⎣ wkwn ⎦ ⎣

2. Basic conditions In order to introduce the detailed process of the proposed hybrid method, we consider the following conditions are known:

• The mathematical model describing battery degradation process: In this work, we assume that the battery degradation is a first-order Markov process, which can be represented by:

⎧ xk + 1 = f (xk ,pk ) + wk ⎨ ⎩ yk = h (xk ) + vk

• •

(2)

(1)

where f (·) and h (·) represent the state transition equation and the measurement equation of the mathematical model, respectively. xk ∈ n is the unobservable state vector at cycle k ,yk ∈ m is the measurement data which is treated as the HI, pk is the parameter vector, wk ∈ n is adopted to indicate the system process noise and vk ∈ m is used to indicate the system measurement noise, they are both uncorrelated zero-mean Gaussian noise with covariance Sk and Rk . ThresholdEOM and TEOM : Monitor the degradation process until the measurement data reaches the end of monitoring (EOM) threshold, the EOM threshold is labeled as ThresholdEOM , the EOM cycle is labeled as TEOM . ThresholdEOL and TEOL : The end of life (EOL) threshold is defined as the value when the HI exceeds it the battery can be seen as failure, the EOL threshold is labeled as ThresholdEOL , the EOL cycle is labeled as TEOL . Fig. 2. Flowchart of the proposed hybrid method.

3

(3)

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quality of the raw error data is so poor that it cannot be employed directly for predicting UKF-PE. It is necessary to reduce the uncertainty of raw error data while retain its evolution information as much as possible. In this work, we reconstruct a new error series based on the decomposition results of CEEMD method.

Therefore, the mathematical model (1) turns into:

∼ ∼ ∼ ⎧ xk + 1 = f (xk ) + wk ∼ ⎨ yk = h (xk ) + vk ⎩

(4)

where f (·) and h (·) are obtained by transforming the original state transition equation and measurement equation, respectively. The initial ∼ covariance of the augmented state vector is P0 =diag[P0,P0wn ] and the ∼ covariance of system process noise is Sk =diag[Sk,Skwn ]. The UKF algorithm for the system described in (4) is summarized in Appendix A.

4. Construct the new error series 4.1. Decompose the raw data by CEEMD method The CEEMD method is an improved empirical mode decomposition (EMD) method. EMD has the ability of processing nonlinear and nonstationary data [32]. The method assumes that there are many different modes in a time series at the same time, and the modes can be extracted and decomposed into a finite number of independent intrinsic mode function (IMF) components and a smooth trend component step-by-step by sifting process. However, EMD method has the disadvantage of mode mixing, in order to solve this problem, a noise-assisted data analysis method named ensemble empirical mode decomposition (EEMD) was proposed [33], where the true components are the mean of an ensemble of trials, each trial is the EMD result of white noise-added data. Nevertheless, as the average number of times increases, the realization of EEMD method becomes time-consuming, in addition, different implementations of white noise-added data can produce different amounts of components. Therefore, [34] put forward the CEEMD method, which added a particular adaptive white noise to the residue of each decomposition stage and it can provide a better decomposition result with lower computational cost. The CEEMD method has been applied to many practical problems, like fault detection [35], seismic attenuation estimation [36] and biomedical image processing [37]. The CEEMD algorithm is described in Appendix B.

3.2. Obtain the UKF-based degradation prediction result Estimate and update the states and model parameters iteratively until the EOM cycle TEOM . Since there are no new measurements at prediction phase, the state update based on measurements cannot be conducted any more, furthermore, we don’t know too much about the uncertainty of the degradation process. Supposing that there is no noise during the prediction phase and the model parameters stay unchanged, then the degradation process at prediction phase obtained by UKF method is: ∗ ∗ ⎧ xk + 1 = f (xk ,pT̂ EOM ) ⎨ yk∗ = h (xk∗) ⎩

xk∗

(5)

is the predicted state vector, yk∗ is the predicted measurement cycle is k=TEOM + 1,…, and xT∗EOM = xT̂ EOM , the notations xT̂ EOM

where data, the and pT̂ EOM are the estimated values of the model state vector and parameter vector at cycle TEOM , respectively. 3.3. Obtain the raw error series Since the effect of uncertainty on degradation is ignored, there is a certain error in the UKF-based prognostic result. In order to obtain a more accurate result, the UKF-based prognostic error (UKF-PE) need to be taken into account. However, the physical principle of UKF-PE is too complex to analyze, which makes it difficult to construct the error evolution model, therefore, it is impractical to adopt the model-based method to predict the error. For the sake of predicting the UKF-PE, we choose data-driven method as long as sufficient historical error data can be collected. Because the UKF-based prognostic result is obtained based on the model (5), in other words, the UKF-PE is produced by this model, we can also obtain the historical data for UKF-PE prediction based on this model, namely, use the error between the real measurements and the measurements generated by model (5) at the estimation phase to predict the error between the two at the prediction phase:

4.2. Analyze components and reconstruct new error series When a time series is decomposed into components of different scales by CEEMD method, there is a dominant mode that can reflect the long-term trend of the time series. The dominant mode may be a single component or composition of components (summing up more than one IMF is called “composition”) and it has been used to estimate the impacts of extreme events on crude oil price [38]. In this work, we choose the dominant mode as the new training data for UKF-PE prediction, because it comprises the long-term trend of the error evolution and neglects the local fluctuations caused by uncertainty, which can reduce the effect of uncertainty on UKF-PE prediction. In order to find the dominant mode, we apply statistical method to analyze the components, there are two statistical variables need to be calculated [38]: the variance and correlation coefficient. The variance can reflect the contribution of each component to total fluctuations of the raw error series and the correlation coefficient is adopted to measure the correlation between each component and the original series. When we find the dominant mode of the CEEMD components, label it as ek (k = 1,…,TEOM ) , then we can use it to construct a nonlinear RVM regression model to predict the UKF-PE series.

⎧ x1́ = f (x 0̂ ,pT̂ EOM ) ⎨ y0́ = h (x 0̂ ) ⎩ ⎧ x2́ = f (x1́ ,pT̂ EOM ) ⎨ y1́ = h (x1́ ) ⎩

(6)

⋮ ⎧ xT́ EOM + 1 = f (xT́ EOM ,pT̂ EOM ) ⎨ yT́ EOM = h (xT́ EOM ) ⎩

5. Get the final prognostic result

where x́k and ýk are the state vector and measurement data produced by model (5) at estimation phase, respectively. It is worth noting that the parameter vector is pT̂ EOM , which is constant during estimation phase.

xkerror = yk −ýk

5.1. Relevance vector machine for regression Given a training data set G = {xk ,tk }kN= 1 with the input {xk }kN= 1 and target {tk }kN= 1, the data-driven regression prediction model can be generally described as [25,26]:

(7)

xkerror

where is the raw error data used as the training data for datadriven method to predict the UKF-PE, the cycle is k = 0,1,…,TEOM . After collecting the UKF-PE historical data, the next step is to predict its evolution over time based on data-driven method. However, the

tk = g (x1: k ,t1: k − 1) + wk

(8)

where t1: k − 1 is the target vector, g (·) is the nonlinear prediction reasoning between the input and the target values, wk is considered as 4

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A1 A2 A3 A4

0.9 0.85 0.8 0.75 0.7 0.65

Fig. 3. Capacity degradation curves of the four type batteries.

Type B

Capacity(Ah)

Capacity(Ah)

Type A

1

0.8

0.6 0

100

200

300

0

500

Cycles

1000

Cycles

Type C

Type D 2

Capacity(Ah)

Capacity(Ah)

1 0.8 0.6 0.4

1.8 1.6 1.4

0.2 0

0

500

1000

1.2

0

50

Cycles

100

The discharge capacity of type C

200

noise that represents the uncertainty. RVM is a kind of supervised machine learning algorithm that based on sparse Bayesian learning theory [39]. Compared with SVM, which is another supervised machine learning algorithm, RVM has the characteristics of higher sparsity and stronger generalization capability, besides, RVM is not constrained by the Mercer’s condition when choosing kernel function. The regression output of RVM can be expressed as:

1

Capacity(Ah)

150

Cycles

0.8 0.6

t = y (x ,ω) + ε

(9)

0.4 N

y (x ,ω) =

0.2 0 0

ωi K (x ,x i ) + ω0

(10)

i=1

(0,σ 2),K

200

400 600 Cycles

(x ,x i ) is the kernel function and ωi is the the noise term is ε ∼ N corresponding weight value. Therefore, the regression model of RVM can be simplified as:

800

(11)

t = Φω + ε

The maximum discharge capacity of type C

]T

where ω = [ω0,…,ωN is the vector of weights, the kernel matrix is Φ = [ϕ (x1),ϕ (x2),…,ϕ (xN )]T , and ϕ (x i ) = [1,K (x i,x1),…,K (x i,xN ]T . Generally speaking, the most commonly used kernel function is the radial basis function (RBF), which is defined as:

1

Capacity(Ah)

∑

0.8

K (x ,x i ) = exp ⎛− ⎝ ⎜

0.6

(x −x i )2 ⎞ δ2 ⎠ ⎟

(12)

0.4

δ is the scale parameter and it determines the sparsity and accuracy of the RVM regression model.

0.2

Table 1 The initial values of the model parameters.

0 0

200

400 600 Cycles

800

Fig. 4. The discharge capacity and the maximum discharge capacity of type C.

5

Battery ID

a

b

c

d

A1 A2 A3 A4

−0.07682 −0.0001895 −2.401e−5 4.319

0.007858 0.0255 0.077 0.006634

0.9866 0.9063 0.9 −3.395

0.00116 −0.0006111 −0.0012249 0.01236

B C D

−0.004787 0.0298 2.408

0.007249 −0.019113 −0.00451

1.063 1.077 −0.5899

3.13e−06 −1.886e−4 −0.02314

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The maximum discharge capacity

[40] proposed a method where the discharging voltage difference of equal time interval (DVD_ETI) was used to estimate the capacity, the extracted data was optimized by Box-Cox transformation, the method can be used at satellite system. A similar method was proposed by Zhou et al. [41], they utilized the mean voltage falloff (MVF) of the discharging voltage to estimate the capacity, the MVF was also optimized by Box-Cox transformation. Both the two methods achieved good estimate results. Therefore, we treat capacity as the HI to conduct the experiments. Seven lithium-ion battery data sets which can be divided into four types based on the rated capacity and discharging current are adopted to verify the effectiveness of the proposed hybrid prognostic method. The first three types (type A, type B and type C) we utilized are from the Center for Advanced Life Cycle Engineering (CALCE) of University of Maryland, they have a graphite anode and a lithium cobalt oxide cathode. Under room temperature, the Arbin BT2000 battery testing system was adopted to conduct cycling of the batteries through multiple charge-discharge tests [9,21]. The rated capacity of type A is 0.9 Ah with constant discharging current 0.45 A. The type B and type C have the same rated capacity of 1.1 Ah, but different discharging currents, type B was discharged at a constant current of 1.1 A, while type C was discharged with varying discharging current altering between 0.11 A, 0.22 A, 0.55 A, 1.1 A, 1.65 A, 2.2 A. Charging was conducted under the constant-current voltage for the three types, the constant current step was performed at 0.5 C (0.45 A for type A and 0.55 A for type B and type C) until the voltage reached 4.2 V, then the constant voltage step was performed until the current dropped to below 0.05 A. The data set of type D is obtained from NASA Prognostic Center of Excellence (PCoE) and has a rated capacity of 2 Ah, the battery was tested under the room temperature, charging was carried out in a constant current mode at 1.5 A until the battery voltage reached 4.2 V, discharge was conducted at a constant current of 2 A until the battery voltage reached 2.7 V [42]. Fig. 2 shows the flowchart of the proposed hybrid method and Fig. 3 shows the capacity degradation curves of the seven lithium-ion batteries. It is worth noting that the batteries do not always run from fully charged to fully discharged, this is why the four batteries of type A have different degradation rates [4]. For the reason that the battery of type C was tested with different discharging currents, its degradation process becomes very complex, the discharge capacity varies over a wide range, it is a challenge to predict the battery RUL. Here we put forward a solution, where we treat the maximum discharge capacity as the HI to characterize the SOH of the battery. From the curve we can see that no matter how wide range the discharge capacity varies with discharging current, the maximum discharge capacity is the upper bound of the range, and the trend of the maximum discharge capacity constantly degrades as the cycle increases. When the maximum discharge capacity degrades to a certain degree, the battery cannot meet the needs of users (see Fig. 4).

Original data Filled data

1.1

Capacity(Ah)

1.05 1 0.95 0.9 0.85 0.8 200

400 Cycles

600

800

Fig. 5. The maximum discharge capacity of type C.

5.2. Predict the error series and obtain the prognostic result of hybrid method Train the newly obtained error data ek and build a RVR iterative prediction model [26], which is:

ek̂ = g (ek − 1,ek − 2,…,ek − i,…,ek − d )

(13)

where d is the length of data set, cycle is k = TEOM + 1,… and ek̂ is the one step forward prediction result of UKF-PE. When k−i is greater than TEOM ,ek − i is equal to ek̂ − i . Predict the UKF-PE iteratively and add the predicted value to the UKF-based prognostic result, the result of hybrid prognostic method is:

yk̆ = yk∗ + ek̂

(14)

where cycle is k=TEOM + 1,…, and y̆k is the prediction of HI of the proposed hybrid method, yk∗ is the prediction of HI of the UKF-based method defined at model (5). When the HI predicted by the proposed hybrid method reaches the hybrid ̂ ThresholdEOL , we can get the EOL cycle TEOL , the RUL of the lithiumion battery predicted by hybrid method at cycle TEOM is: hybrid hybrid  ̂ RULTEOM = TEOL −TEOM

(15)

6. Experimental verification 6.1. Battery data sets The capacity is an important HI for lithium-ion battery, compared with other battery HIs, it can indicate the battery SOH intuitively. Although the capacity is difficult to measure directly on-line, it can be estimated by analyzing other available battery parameters. Liu et al. Table 2 CEEMD results of type A: EOM threshold is 0.8 Ah. A1

A2

A3

A4

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 Trend

3.4292 0.0914 1.1667 2.1917 1.0317 6.2664 33.529

0.2964 0.2741 0.3035 0.3035 0.5031 0.4925 0.7935

7.5388 2.9806 1.2593 3.9317 1.3281 13.047 15.071

0.3726 0.3326 0.3113 0.4205 0.3440 0.4877 0.5147

15.331 4.1807 5.6795 6.2578 3.8319 39.564 0.95245

0.3716 0.3902 0.3673 0.3828 0.6954 0.7581 0.4174

16.711 5.1840 6.2886

0.4761 0.4128 0.7235

73.239

0.8680

Original

60.026

48.120

112.84

6

147.11

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Table 3 CEEMD results of type A: EOM threshold is 0.765 Ah. A1

A2

A3

A4

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 Trend

3.6038 0.15178 0.95963 2.1867 0.74916 7.0409 16.489

0.3248 0.2853 0.3248 0.3681 0.4661 0.5482 0.7040

6.3910 2.4078 1.0144 2.7850 1.1899 7.7261 20.180

0.3533 0.3049 0.2863 0.3754 0.3256 0.4556 0.6778

15.907 3.3773 5.2365 5.5302 1.7809 50.346 0.99264

0.3692 0.4172 0.3834 0.3022 0.2973 0.7591 0.7019

13.990 2.6817 3.4910

0.2643 0.3471 0.4653

36.512

0.8734

Original

39.779

47.698

106.28

251.02

capacity and the EOL threshold ThresholdEOL is set to be 75% of the rated capacity.

6.2. Before experiment The degradation process of lithium-ion battery can be represented by an exponential model [21], which is described as:

x = x k + wk ⎧ k+1 ⎨ ⎩Qk = ak exp(bk ·k ) + ck exp(dk ·k ) + vk

For the data set of type D, because there is a difference between the initial capacity and the rated capacity, the EOM threshold is set to be 80% of the initial capacity, and the EOL is set to be 70% of the initial capacity [42]. When the ThresholdEOM is determined, the data used to estimate the model parameters can be obtained. The initial values of the model parameters are listed in Table 1: For the maximum discharge capacity data extracted from the data set of type C, there is no capacity data in some cycles. We can fill these cycles with data using the model (16), the values of the model parameters are listed in Table 1, and the variance of vk is set to be 0.005 [16]. The curve is showed in Fig. 5.

(16)

where xk = [ak ,bk ,ck,dk ] is the model parameter vector, ak and bk are related to the internal impedance, while ck and dk are the aging parameters [42], Qk is the battery capacity at cycle k, both the wk = [wa,wb,wc,wd] and vk are Gaussian noise with zero mean. Next, we define the EOM threshold and EOL threshold for the batteries of four types, respectively. In order to test the robustness and accuracy of the results produced by proposed method, we define three sets of EOM threshold and EOL threshold combinations for the RUL prediction experiments of type A:

6.3. UKF-based prediction

(1) The EOM threshold ThresholdEOM is set to be 90% of the rated capacity and the EOL threshold ThresholdEOL is set to be 80% of the rated capacity. (2) The EOM threshold ThresholdEOM is set to be 90% of the rated capacity and the EOL threshold ThresholdEOL is set to be 75% of the rated capacity. (3) The EOM threshold ThresholdEOM is set to be 85% of the rated capacity, and the EOL threshold ThresholdEOL is set to be 75% of the rated capacity.

In model (16), only the model parameters need to be estimated at estimation phase. In order to reduce the effect of uncertainty on determining TEOM , we assume that when the capacities of three consecutive cycles, for example, Q (k ), Q (k + 1), Q (k + 2) , are less than ThresholdEOM , the model estimation phase ends and the prediction of battery capacity starts, the EOM cycle is TEOM = k−1. Without obtaining new capacity, the model used at prediction phase is:

For the data sets of type B and C, we define two sets of EOM threshold and EOL threshold combinations:

̂ ⎧ xk̂ = xEOM ∗ ̂ ·k ) + cEOM ̂ ·k ) ⎨Qk = aEOM ̂ exp(bEOM ̂ exp(dEOM ⎩

(1) The EOM threshold ThresholdEOM is set to be 90% of the rated capacity and the EOL threshold ThresholdEOL is set to be 80% of the rated capacity. (2) The EOM threshold ThresholdEOM is set to be 90% of the rated

̂ , cEOM ̂ ]T is ̂ ̂ ,bEOM ̂ ,dEOM = [aEOM where k = TEOM + 1,TEOM + 2,… and xEOM the estimated parameter vector at cycle TEOM , Qk∗ is the UKF-based predicted capacity. Based on the Eqs. (6) and (7), the UKF-based capacity prediction kerror is: error Q

(17)

Table 4 CEEMD results of type B, type C, type D. B

C

D

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

Variance (×10−6)

Correlation coefficient

IMF1 IMF2 IMF3 IMF4 IMF5 IMF6 IMF7 IMF8 Trend

9.0697 2.0457 4.2056 3.0020 5.2308 8.3289 9.5958 47.696 127.54

0.1814 0.1596 0.2069 0.2430 0.2284 0.3569 0.5749 0.7345 0.7955

17.352 2.9976 1.3442 0.87466 0.63743 0.80237 0.15875

0.5858 0.3971 0.3273 0.2313 0.1780 0.1894 0.1969

64.293 15.130 39.752 40.852 69.315 63.285

0.2003 0.3892 0.4505 0.5310 0.5953 0.5344

21.648

0.6447

36.638

0.3321

Original

361.66

50.599

432.20

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A2 0.02

0.01

0.01

Capacity(Ah)

Capacity(Ah)

A1 0.02

0 −0.01 −0.02 −0.03

Original Reconstructed Trend

20

40

60

0 −0.01 Original Reconstructed Trend

−0.02

80

100

120

140

−0.03

160

20

40

Cycles

60

0.01

0

0 −0.01 −0.02 −0.03

−0.05

Original Reconstructed Trend

20

100

120

140

30

35

A4 0.01

Capacity(Ah)

Capacity(Ah)

A3 0.02

−0.04

80

Cycles

Original Reconstructed Trend

−0.01 −0.02 −0.03 −0.04

40

60

80

−0.05

100

5

10

Cycles

15

20

25

Cycles

Fig. 6. Reconstructed error series vs. the original error series of type A (EOM threshold is 0.8 Ah).

A1

A2

0.015

0.03

Original Reconstructed Trend

0.02

0.005

Capacity(Ah)

Capacity(Ah)

0.01

0 −0.005 −0.01 −0.015 −0.02

Original Reconstructed Trend

−0.025

0.01 0 −0.01

50

100

−0.02

150

100

50

Cycles

A4 0.01

0.01

0

0

−0.01

Capacity(Ah)

Capacity(Ah)

A3 0.02

−0.01 −0.02 −0.03 −0.04 −0.05

150

Cycles

Original Reconstructed Trend

−0.02 −0.03 −0.04

Original Reconstructed Trend

20

40

−0.05 60

80

100

−0.06

120

Cycles

10

20

30

Cycles

Fig. 7. Reconstructed error series vs. the original error series of type A (EOM threshold is 0.765 Ah).

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B 0.08 0.06 0.04 0.02

−0.02

50

100 150 200 250 300 350 400

0.06

0.01 0 −0.01

0

D 0.08

0.02

Capacity(Ah)

Capacity(Ah)

C 0.03

Original Reconstructed Trend

−0.02

0.04

Capacity(Ah)

0.1

Original Reconstructed Trend+IMF7

100

0.02 0 Original Reconstructed Trend

−0.02 200

Cycles

300

−0.04

400

20

40

Cycles

60

80

100

Cycles

Fig. 8. Reconstructed error series vs. the original error series of type B, C, D.

kerror = Qk−Q́k Q

represent not only the long-term trend of the error evolution, but also its large-range fluctuations.

(18)

where k=1,2,…,TEOM ,Q́k is the estimated capacity.

6.5. Final prognostic result 6.4. Construct new error data So next we construct the RVR recursive prediction model using the error data ek and then predict the evolution of the error series. The kernel function used for constructing RVR model is the RBF, the scale parameter δ equals to 0.333. After obtaining the predicted error series, we add it to the UKFbased capacity prediction result, based on the Eq. (15), RUL prediction result of the proposed hybrid method is obtained. With the purpose of displaying the high performance of the proposed hybrid method, comparative experiments based on RVM method and UKF method are also conducted.

Apply CEEMD method to the raw error series, where the standard deviation (SD) of white Gaussian noise is εi = 0.2 , then calculate the variance and correlation coefficient of the components. The results are listed in Tables 2–4. In the tables, Variance represents the variance of each component, Correlation Coefficient represents the correlation coefficient between the component and the original error series. Considering that: (1) the trend component can reflect the long-term trend of error evolution; (2) the last IMF component is highly correlated with the original error series and accounts for a relatively high variance ratio, furthermore, its frequency is the lowest one between the IMFs which can reflect the relatively large range of fluctuations in the error evolution; (3) the high frequency IMFs only show the local fluctuations of error evolution, which have little significance for long-term prediction of error evolution, we treat the sum of trend component and last IMF as the dominant mode for type A and B. For type C, the variance and correlation coefficient of last IMF is very small compared with trend component, so the trend component is chosen as the dominant mode. For type D, value of the statistical variables for each component has little difference, in order to avoid the effect of high frequency components on the prediction, we choose the sum of trend component and the last IMF as the dominant mode. The dominant mode is labeled as ek (k = 1,2,…,TEOM ). The sum of the remaining components contains the major uncertainty of original data, which represents the short-term fluctuations. In Figs. 6–8, Original represents the raw error series, Trend represents the trend component, Reconstructed is the reconstructed error series, i.e. the dominant mode. Fig. 8 also shows the curve of the sum of trend component and the last IMF for type C. From the figures we can find that compared with trend component, the dominant mode can

7. Prognostic results 7.1. Evaluation indicators We define some indicators to evaluate the steadiness and accuracy of each prognostic result [24]. Error indicator (EI) EI is used to show the error between the predicted RUL and real RUL:

EI = RULk − RULk where RULk is the real RUL value at cycle k, and  RULk is the predicted value. Accuracy indicator (AI) AI is adopted to evaluate the relative error of the predicted RUL, which is defined as:

RULk − RULk ⎫ AI = ⎧1− × 100% ⎨ ⎬ RUL k ⎩ ⎭

Table 5 Prediction results of three methods for A1 and A2. Comb.

Method

A1

A2

TEOM

RULT

 RULT

EI

AI

SI

TEOM

RULT

 RULT

EI

AI

SI

1

UKF RVM Hybrid

152 152 152

56 56 56

39 51 54

17 5 2

69.64% 91.07% 96.43%

0.0137 0.0094 0.0068

133 133 133

55 55 55

70 38 59

15 17 4

72.73% 69.09% 92.73%

0.0129 0.0095 0.0036

2

UKF RVM Hybrid

152 152 152

69 69 69

54 59 66

15 10 3

78.26% 85.51% 95.65%

0.0241 0.0125 0.0083

133 133 133

66 66 66

85 42 73

19 24 10

71.21% 63.64% 84.85%

0.0181 0.0196 0.0071

3

UKF RVM Hybrid

177 177 177

44 44 44

41 49 45

3 5 1

93.18% 88.64% 97.72%

0.0145 0.0101 0.0059

166 166 166

33 33 33

41 24 35

8 9 2

75.76% 72.73% 93.94%

0.0195 0.0126 0.0037

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Table 6 Prediction results of three methods for A3 and A4. Comb.

Method

A3

A4

TEOM

RULT

 RULT

EI

AI

SI

TEOM

RULT

 RULT

EI

AI

SI

1

UKF RVM Hybrid

93 93 93

38 38 38

29 30 34

9 8 4

76.32% 78.95% 89.47%

0.0175 0.0073 0.0058

32 32 32

15 15 15

9 13 13

6 2 2

60% 86.67% 86.67%

0.0274 0.0157 0.0141

2

UKF RVM Hybrid

93 93 93

45 45 45

35 34 41

10 11 6

77.78% 75.56% 86.67%

0.0243 0.0190 0.0116

32 32 32

19 19 19

14 15 18

5 4 1

73.68% 78.95% 94.74%

0.0419 0.0209 0.0157

3

UKF RVM Hybrid

120 120 120

18 18 18

17 12 17

1 6 1

94.44% 66.67% 94.44%

0.0045 0.0106 0.0048

42 42 42

9 9 9

11 8 8

2 1 1

77.78% 88.89% 88.89%

0.0140 0.0105 0.0070

the meaning of the parameters in AI function is same as the ones in EI function. Steady indicator (SI) SI is used to character the deviation between the prediction and the actual value:

SI =

1 N

N

∑

Table 8 Prediction results of the three methods for type D. Method

UKF RVM Hybrid

k )2 (Qk−Q

k=1

where Qk is the real capacity at cycle k ,Qk̂ is the predicted capacity at cycle k. A smaller SI value indicates a more stable result.

D

TEOM

RULT

 RULT

EI

AI

SI

98 98 98

63 63 63

43 21 59

20 42 4

68.25% 33.33% 93.65%

0.0257 0.0257 0.0129

some abnormal results whose AI values become larger in combination 2, which means that their prognostic errors become smaller. The appearance of these abnormal results is due to the local fluctuations caused by degradation uncertainty. In addition, the prediction accuracy of the hybrid method is much higher than the other two methods. For type B and type C, since the historical data used for constructing RVR model is sufficient, performance of the predictions obtained by RVM are almost the same as the result get by hybrid method, and even for type B, the prediction result of RVM is more stable and accurate, which is highlighted in bold. However, as the prediction horizon becomes longer, the result performance of RVM method deteriorates quicker than the proposed hybrid method’s, therefore, the proposed method can provide an accurate and more robust prediction result. For type C, the performance of UKF-based prognostics is so poor that it cannot get a result in the given cycles, however, by correcting the UKF-based prognostic result with the predicted error, the proposed hybrid method can achieve a good RUL prediction result. Compare the results of type A with type B and type C, the results of hybrid method of type A are less robust. It is because that the error prediction is obtained by RVM method, the amount of the data has a certain effect on the robustness of the proposed method. However, the reconstructed error data has high quality, which avoids quick deterioration as the prediction horizon becomes longer.

7.2. Results analysis The prediction results of three different methods are shown in Tables 5–8 and Figs. 9–15. Comb. represents the combination of EOM threshold and EOL threshold mentioned above. TEOM is the start cycle of RUL prediction, RULT is the real RUL at TEOM ,  RULT is the predicted value of RULT , EI, AI, SI are the evaluation indicators. 7.2.1. Impacts of the amount of historical data on the prognostic results First, we analyze the results of type A. Compared with combination 1 and 2, batteries in combination 3 have more historical data and shorter prediction horizon. From the indictors listed in tables we can see that although there exist some abnormal results which are caused by degradation uncertainty, the prediction performance of each method in combination 3 becomes better for the most of results. Moreover, the results of hybrid method are more accurate than the other two methods. 7.2.2. Impacts of the length of prediction horizon on the prognostic results For the same batteries of type A in combination 1 and 2, they have same EOM threshold but different EOL thresholds. In other word, each battery in different combinations has same amount of historical data, but in combination 2, the prediction horizon is longer, therefore, the prediction accuracy and steadiness should decrease. However, there are Table 7 Prediction results of three methods for type B and type C. Comb.

Method

B

C

TEOM

RULT

 RULT

EI

AI

SI

TEOM

RULT

 RULT

EI

AI

SI

1

UKF RVM Hybrid

389 389 389

144 144 144

207 124 167

63 20 23

56.25% 86.11% 84.03%

0.0308 0.0107 0.0182

449 449 449

319 319 319

– 268 359

– 51 40

– 84.01% 87.46%

0.0191 0.0082 0.0090

2

UKF RVM Hybrid

389 389 389

218 218 218

258 150 224

40 68 6

81.65% 68.81% 97.24%

0.0371 0.0199 0.0177

449 449 449

397 397 397

– 284 447

– 113 50

– 71.53% 87.40 %

0.0254 0.0107 0.0097

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EOM threshold

0.8 0.75

0.85 0.8

0.7

EOM threshold

0.75

0.85 0.8 EOM threshold

0.75

0.7

0

0.7 EOL threshold

EOL threshold

EOL threshold

50

100

150

0.65 0

200

Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

0.85

A1:EOM threshold is 0.765 Ah,EOL threshold is 0.675 Ah

Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

0.9

Capacity(Ah)

A1:EOM threshold is 0.8 Ah,EOL threshold is 0.675 Ah

Measurements UKF method RVM method Hybrid method

50

Cycles

100

150

0.65 0

200

50

Cycles

100

150

200

Cycles

Fig. 9. The prediction results of the three methods for battery A1. A2:EOM threshold is 0.8 Ah,EOL threshold is 0.72 Ah

0.85

EOM threshold

0.75

EOM threshold

0.8 0.75

0.85 0.8 EOM threshold

0.75

0.7

0.7 EOL threshold

EOL threshold

EOL threshold

0.7 0

50

100

150

0.65

200

Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

0.85

A2:EOM threshold is 0.765 Ah,EOL threshold is 0.675 Ah

Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

Capacity(Ah)

0.9

0.8

A2:EOM threshold is 0.8 Ah,EOL threshold is 0.675 Ah

Measurements UKF method RVM method Hybrid method

0

50

100

Cycles

150

0.65

200

50

100

Cycles

150

200

Cycles

Fig. 10. The prediction results of the three methods for battery A2.

Capacity(Ah)

0.9

0.85 0.8 EOM threshold

A3:EOM threshold is 0.8 Ah,EOL threshold is 0.675 Ah Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

Measurements UKF method RVM method Hybrid method

0.75

0.85 0.8

EOM threshold

0.75

A3:EOM threshold is 0.765 Ah,EOL threshold is 0.675 Ah

0.85 0.8 EOM threshold

0.75

0.7

0.7 0

20

0.7 EOL threshold

EOL threshold

EOL threshold

40

60

80

100

120

0.65 0

140

Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

A3:EOM threshold is 0.8Ah,EOL threshold is 0.72Ah

50

Cycles

100

0.65 0

150

50

Cycles

100

150

Cycles

Fig. 11. The prediction results of the three methods for battery A3.

Measurements UKF method RVM method Hybrid method

0.85

0.8

EOM threshold

0.75

0.85 0.8

EOM threshold

0.75

A4:EOM threshold is 0.765 Ah,EOL threshold is 0.675 Ah

0.85 0.8 EOM threshold

0.75

10

0.7 EOL threshold

EOL threshold

20

30

40

50

0.65 0

Measurements UKF method RVM method Hybrid method

0.9

0.7 EOL threshold

0.7 0

Measurements UKF method RVM method Hybrid method

0.9

Capacity(Ah)

Capacity(Ah)

0.9

A4:EOM threshold is 0.8 Ah,EOL threshold is 0.675 Ah

Capacity(Ah)

A4:EOM threshold is 0.8 Ah,EOL threshold is 0.72 Ah

10

20

Cycles

30

40

50

60

0.65 0

Cycles

10

20

30

40

50

60

Cycles

Fig. 12. The prediction results of the three methods for battery A4.

7.2.4. Summary Through the above analyses, it is concluded that: (1) the uncertainty does have a certain impact on lithium-ion battery prognostics; (2) no matter what the rated capacity or discharging current is, the proposed hybrid method can achieve a robust and accurate RUL prediction result. The realization of hybrid method is based on the thought of error-

7.2.3. The wide applicability of the hybrid method The first three data sets are obtained from CALCE, the data set D is from NASA, however, the result of hybrid method has better performance than the results of other two methods, which shows the wide range of applicability and good performance of the hybrid method.

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B:EOM threshold is 0.99 Ah,EOL threshold is 0.88 Ah Measurements UKF method RVM method Hybrid method

Measurements UKF method RVM method Hybrid method

1.1

Capacity(Ah)

1.1

Capacity(Ah)

B:EOM threshold is 0.99 Ah,EOL threshold is 0.825 Ah

1.05 1 EOM threshold 0.95

1.05 1 EOM threshold 0.95 0.9

0.9 EOL threshold 0.85 0

0.85 200

400

600

EOL threshold

0.8 0

800

200

400

Cycles

600

800

Cycles

Fig. 13. The prediction results of the three methods for Battery B.

C:EOM threshold is 0.99 Ah,EOL threshold is 0.88 Ah

Capacity(Ah)

1.05 1 EOM threshold 0.95

Measurements UKF method RVM method Hybrid method

1.1

Capacity(Ah)

Measurements UKF method RVM method Hybrid method

1.1

C:EOM threshold is 0.99 Ah,EOL threshold is 0.825 Ah

1.05 1 EOM threshold 0.95 0.9

0.9 EOL threshold 0.85 0

0.85

200

400

600

EOL threshold

0.8 0

800

200

400

Cycles

600

800

Cycles

Fig. 14. The prediction results of the three methods for Battery C.

D:EOM threshold is 1.5 Ah,EOL threshold is 1.3 Ah Measurements UKF method RVM method Hybrid method

1.8

Capacity(Ah)

based prognostic result. In order to verify the high performance of the proposed hybrid method, pure UKF method, RVM method and the proposed hybrid method are applied to the RUL prediction experiments of lithium-ion battery. Seven sets of battery data are adopted for the experiments, according to the discharging current and rated capacity, data sets can be divided into four types. For type A, B and D, because the discharging current is constant, the capacity degrades without complex varying and it can be treated as the HI for the battery. For type C, since the discharge capacity is very complex, we use the maximum discharge capacity to characterize the health state of the battery. The experimental results show that the robustness of the hybrid method is affected by the amount of historical data. However, compared with the UKF method and the RVM method, the proposed hybrid method can achieve more accurate and robust results, which proves the effectiveness and high performance of the proposed hybrid method. When the discharging current does not change too much in each cycle, the battery capacity can be treated as the HI, and the proposed hybrid method can be applied in some online applications, for example, satellite system where the payloads power consumption is relatively stable in-orbit [40]. On the other hand, from the experimental results of type C we can find that for the complex application, if we can find an indicator to characterize the battery health, the proposed hybrid method becomes applicable. It can be concluded that the proposed method can be applied as a potential online RUL prediction method for lithium-ion battery. The main contributions of this work can be concluded as: (1) propose a high performance hybrid prognostic method based on the thought of error-correction, which makes full use the historical information, including the measurements, degradation model as well as model-based prognostics error and it has wide range of applicability;

1.7 1.6 1.5

EOM threshold

1.4 1.3

EOL threshold

1.2 0

50

100

150

Cycles Fig. 15. The prediction results of the three methods for Battery D.

correction, even though the result of model-based part is not so accurate and has a large prognostic error due to the inaccurate model or poor measurements, the hybrid prognostic method can obtain a relatively accurate prediction by correcting the error. 8. Conclusion This paper proposes a model-data hybrid method for the RUL prediction of lithium-ion battery, which is based on the thought of errorcorrection. The proposed method fuses the approaches of UKF, RVM and CEEMD, where the CEEMD and RVM methods are adopted to predict the UKF-PE, the predicted error is employed to correct UKF12

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(2) propose methods to produce raw error data and reduce its uncertainty; (3) for the case that lithium-ion battery works with different discharging currents, we put forward a solution to achieve RUL prediction.

or constructing a new HI which is less susceptible to the complex operation conditions.

9. Future work

This work is supported by National Natural Science Foundation of China (Grant No. 61473127). We would like to thank the CALCE of University of Maryland and PCoE at NASA for providing the experimental data sets of lithium-ion battery.

Acknowledgement

Future works include exploring a new method for constructing the error data which can make the hybrid method more robust, and finding Appendix A. UKF method

+ ∼+ The mean and covariance of state ∼ xk − 1 are assumed known, which are labeled as ∼ x k̂ − 1 and P k − 1. With the purpose of propagating from time k−1 to time k, the computation procedures of UKF are as follows [43]:

1. State update process Construct 2n + 1 sigma points and the corresponding weights: +

χk(i−) 1

⎧∼ x k̂ − 1, i=0 ⎪ + + ∼ ∼ i ( ) ̂ = x k − 1 + ( (n + κ ) P k − 1 ) , i = 1,…,n ⎨ ∼+ ⎪ ∼+̂ x −( (n + κ ) P k − 1 )(i), i = n + 1,…,2n ⎩ k−1

(A.1)

κ

ω(i) =

⎧n + κ, i = 0 ⎨ κ ,i = 1,…,2n ⎩ 2(n + κ )

χk(i)

Predict the sigma point using the constructed point ∼− the covariance P k at time k:

χk(i) = f (χk(i−) 1 ),

(A.2)

χk(i−) 1,

− then combine the predicted sigma points to obtain the a priori state estimate ∼ x k̂ and

i = 0,1,…,2n

(A.3)

2n

− ∼ x k̂ =

∑

ω(i) χk(i)

(A.4)

i=0

2n

∼− Pk =

∑

− − ∼ ω(i) (χk(i) −∼ x k̂ )(χk(i) −∼ x k̂ )T + Sk

(A.5)

i=0

2. Measurement update process Use the predicted sigma point χk(i) to compute the measurement point γk(i) , and then combine the measurement points to obtain the predicted measurement data yk̂ :

γk(i) = h (χk(i) ), i = 0,1,…,2n

(A.6)

2n

yk̂ =

∑

ω(i) γk(i)

(A.7)

i=0 − Compute the covariance of the predicted measurement data yk̂ and the cross covariance between ∼ x k̂ and yk̂ , i.e. Pkyy and Pkxy :

2n

Pkyy =

∑

ω(i) (γk(i)−yk̂ )(γk(i)−yk̂ )T + Rk

(A.8)

i=0

2n

Pkxy =

∑

−

ω(i) (χk(i) −∼ x k̂ )(γk(i)−yk̂ )T

(A.9)

i=0

After obtaining the new measurement data yk , the update of estimated state can be performed by employing the normal Kalman filter equations:

Kk = Pkxy (Pkyy )−1

(A.10)

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Y. Chang et al. + − ∼ x k̂ = ∼ x k̂ + Kk (yk −yk̂ )

(A.11)

∼+ ∼− P k = P k −Kk Pkyy KkT

(A.12)

Appendix B. CEEMD method We define that εi is the standard deviation (SD) of white Gaussian noise wi (i = 1,…,I ), the operator Ej (·) produces the j-th EMD mode, IMFn (t ) is the n-th IMF of CEEMD. For the given raw error data, for convenience, define x (k ) = xkerror (k = 1,…,TEOM ), the procedures of decomposing raw error data using CEEMD algorithm are described as follows: 1. Apply EMD method to the new series X (k ) = x (k ) + ε0 wi (k ) to get their first modes IMF i (k ) , and calculate the first CEEMD IMF by:

IMF1 (k ) =

1 I

I

∑

IMF i (k ) (B.1)

i=1

2. The first residue is r1 (k ) = x (k )−IMF1 (k ) , then the realizations r1 (k ) + ε1 E1 (wi (k )) are decomposed by EMD method until their first IMF components are obtained and the second CEEMD IMF is:

IMF2 (k ) =

1 I

I

∑

E1 [r1 (k ) + ε1 E1 (wi (k ))]

(B.2)

i=1

3. For n = 2,…,N , the n-th residue is rn (t ) = rn − 1 (k )−IMFn (k ) , calculate the first EMD IMF of realizations rn (k ) + εk En (wi (k )) and get the (n + 1) -th CEEMD IMF by:

IMFn + 1 (k ) =

1 I

I

∑

En [rn (k ) + εn En (wi (k ))]

(B.3)

i=1

4. Repeat the step 3 until the number of extrema in the last obtained residue is not more than two, the raw error data can be expressed as: N

xkerror = x (k ) =

∑

IMFn (k ) + r (k )

(B.4)

n=1

where r (k ) is the final residue, i.e. the trend component of CEEMD results.

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