Remaining useful life prediction of lithium-ion batteries with adaptive unscented kalman filter and optimized support vector regression

Remaining Useful Life Prediction of Lithium-ion Batteries with Adaptive Unscented Kalman Filter and Optimized Support Vector Regression

Communicated by Dr. Ma Lifeng Ma

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Remaining Useful Life Prediction of Lithium-ion Batteries with Adaptive Unscented Kalman Filter and Optimized Support Vector Regression Zhiwei Xue, Yong Zhang, Cheng Cheng, Guijun Ma PII: DOI: Reference:

S0925-2312(19)31342-6 https://doi.org/10.1016/j.neucom.2019.09.074 NEUCOM 21329

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Neurocomputing

Received date: Revised date: Accepted date:

24 June 2019 14 September 2019 24 September 2019

Please cite this article as: Zhiwei Xue, Yong Zhang, Cheng Cheng, Guijun Ma, Remaining Useful Life Prediction of Lithium-ion Batteries with Adaptive Unscented Kalman Filter and Optimized Support Vector Regression, Neurocomputing (2019), doi: https://doi.org/10.1016/j.neucom.2019.09.074

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Remaining Useful Life Prediction of Lithium-ion Batteries with Adaptive Unscented Kalman Filter and Optimized Support Vector Regression Zhiwei Xuea,b , Yong Zhanga,b,∗, Cheng Chengc , Guijun Mad a School

of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan 430081, China Research Center for Metallurgical Automation and Measurement Technology of Ministry of Education, Wuhan 430081, China c School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China. d School of Mechanical science and engineering, Huazhong University of Science and Technology, Wuhan 430074, China. b Engineering

Abstract To solve the problem of the inaccurate prediction on remaining useful life (RUL) for lithium-ion battery, we proposed an integrated algorithm which combines adaptive unscented kalman filter (AUKF) and genetic algorithm optimized support vector regression (GA-SVR). Firstly, the state space model with double exponential is established to describe the degradation of lithium battery. Then, the AUKF algorithm is introduced to update adaptively both the process noise covariance and the observation noise covariance. Next, the genetic algorithm is utilized to optimize the key parameters of SVR which realizes multi-step prediction. The effectiveness of the proposed method is verified by simulation experiments with NASA of battery dataset. Simulation results show that the proposed AUKF-GA-SVR achieves better prediction accuracy than existed methods such as unscented kalman filter, extended kalman filter, adaptive extended kalman filter (AEKF), adaptive unscented kalman filter, unscented kalman filter and relevance vector regression and AEKF-GA-SVR. Keywords: Remaining useful life prediction; Adaptive unscented kalman filter; Genetic algorithm; Support vector regression. 1. Introduction

parameters need to be represented by other physical parameters. Compared with the measured physical parameters, battery caAs a kind of lightweight and high energy density power pacity reveals the degradation process of batteries and has been sources, lithium-ion batteries are widely used in spacecraft, airthe focus of researchers as the definitive battery degradation incraft, electric vehicles and portable electronic devices [1]. With dicator [4]. Therefore, capacity-based approach in this paper is regard to the practical application of lithium battery, it inevitably chosen as the key parameter to achieve satisfactory prognosis degrades until the failure with the increase of consumption time. accuracy. To avert the accidents caused by battery degradation, it is necRecently, a variety of literatures [2, 3, 4] report the predicessary to monitor the state of charge (SOC), evaluate the state tion of RUL for lithium-ion batteries, and the main approaches of health (SOH) and predict the remaining useful life (RUL) of can be categorized into model-based, data-driven and data-modellithium-ion battery [2]. Among these problems, how to predict fusion methods. Similar to model-based methods of fault dithe RUL of battery is a vital and challenging issue in battery agnosis [5, 6], the mathematical model with priori knowledge management system. Especially, due to complex internal elecof a battery’s life cycle to describe the physical mechanism of trochemical reaction and various external uncertain factors, it lithium battery. One of the typical techniques are filter-based is difficult to predict the RUL accurately on account of irregumethods which include kalman filter (KF), extended kalman lar degradation process. However, an accurate RUL prediction filter (EKF) and unscented kalman filter. These techniques are value of battery is conducive to make correct judgment about powerful algorithms to estimate the battery SOH and predict the aging level and forecast the healthy working hours, and then the RUL, and the main difference is that KF is used to describe the optimal maintenance strategy with ample time can be dethe linear degrade process, while EKF [7, 8] and UKF [9] are signed so as to save resources, reduce costs and ensure safety. employed to describe the nonlinear degrade model. Hu [7] proRecently, more and more researchers began to pay their attenposes a multiscale framework with EKF for SOC and capacity tions to the prognosis of RUL for lithium battery [3]. estimation. In [8], the expectation-maximization algorithm and To predict efficiently the RUL, appropriate features are seEKF algorithm are adopted to estimate and update the model lected firstly, which including physical parameters (e.g.,temperature,parameters and states jointly, and the degradation of lithiumcurrent and voltage) and virtual parameter (e.g., capacity). Usuion batteries are forecasted. Considering the error of EKF with ally, physical parameters can be measured directly, while virtual Jacobian matrix may be high because of with high-order term, the prediction with UKF may be proper choice. In [9], an online recursive least square algorithm and UKF are employed to ∗ Corresponding author estimate the system matrices and SOC at every prediction point. Email address: [email protected] (Yong Zhang) Preprint submitted to Elsevier

October 8, 2019

Although model-based method has achieved some successes in RUL prediction, an accurate mathematical model is usually unobtainable, especially when the batteries operate under noisy and/or uncertain environments. Data-driven methods often extract typical features from the degradation data and then employ machine learning techniques [10, 11, 12] to construct the mapping relationship between degraded data and health state, and then estimate the battery degradation and forecast its RUL [13]. Support vector machine (SVM), as a typical representative of artificial intelligence algorithm, is widely used in the RUL prediction of batteries ([14, 15, 16, 17]). Patil [15] extracted key features from voltage and temperature curves, then utilized support vector regression (SVR) to achieve accurate prediction of RUL. Nuhic [14] combined with a new method for training and testing data processing based on load collectives, which solved the defect that RUL estimation is susceptible to environmental and load conditions. Li [16] introduced the variation characteristics of terminal voltage and voltage derivative during charging process, and uses them as training vectors to predict the residual life of lithium batteries by utilizing SVM algorithm. Zhang [17] extracted the time interval of equal voltage difference under charging and discharging as health indexes, and the new method of SVR and Feature vector selection (FVS) is selected to achieve RUL prediction. For data-driven prediction methods, the accuracy of prediction often depends on the established mapping relationships which are sensitive to the quality and quantity of battery data. Data-model hybrid method can not only reflect the degradation mechanism of the battery, but also obtain the information and evolution law of the battery health state from data set. Therefore, the hybrid between SVR and KF have aroused the attention of some scholars [18, 19, 20]. Zheng [18] developed a novel method with UKF and relevance vector regression (RVR) to realize short-term capacity prediction of batteries. In [19], a hybrid method based on error correction is proposed to predict the RUL of the battery which combining UKF, CEEMD and RVR. Baptista [18] investigated the prognostics performance of the Kalman filter on five data-driven approaches of RUL. It is worth pointing out that the key parameters of SVR have obvious influence on prediction accuracy, and intelligent optimization algorithms [21, 22, 23, 24] can be used to find these optimal parameters on certain performance constraints. Inspired by [19], the hybrid data-model driven method in this paper is proposed for the prediction of RUL for lithium-ion batteries. To update adaptively both the process noise covariance and the observation noise covariance which standard unscented kalman filtering can not be realized, adaptive unscented kalman filtering (AUKF) is introduced. On the other hand, the genetic algorithm (GA) is adopted to optimize the key parameters of kernel function for SVR, so that higher prediction accuracy of RUL can be achieved. Finally, the effectiveness of the proposed method is verified by using NASA’s lithium battery dataset. The major contributions of this paper lie in the following several aspects: 1) the AUKF algorithm is utilized to renovate dynamically the noise covariances; 2) the SVR method is adopted to realize multi-step prediction and GA is employed to 2

optimize the parameters of SVR; 3) the hybrid data-model integrating AUKF and optimized SVR is developed, more accurate prediction of RUL can be achieved. 2. Related theory 2.1. Degradation model of lithium battery As verified in [25], the exponential model usually has a better global regression performance than the polynomial model and other typical models, therefore, we choose the following double exponential model as the capacity degradation pattern of lithium battery: yk = a ∗ exp(b · k) + c ∗ exp(d · k)

(1)

where yk denotes battery capacity, and k is the number of cycles. a, b, c, d represent the model parameters which need to be initialized with actual data. Due to the complexity and timeliness of battery degradation, the model parameters ak , bk , ck , dk in this paper are assumed to time-varying, which is more in line with the actual situation of battery degradation. Denote xk , col{ak , bk , ck , dk }, and consider the influence of external noise and uncertainty, we establish the following state space model:    xk+1 = xk + $k (2)   y = a ∗ exp(b · k) + c ∗ exp(d · k) + υ k k k k k k

where $k and υk represent the process noise and observation noise, which are assumed as uncorrelated zero-mean white noise with covariance Qk and Rk , respectively. By generalizing (2), one obtains the nonlinear system as follows:    xk+1 = f (xk ) + $k (3)   y = h(x ) + υ k k k

where f (∗) and g(∗) are nonlinear function endowing the mapping relation between capacity and the parameters of degrade model (1). Furthermore, set P0 as the initialization covariance of initial state x0 , Q0 and R0 are the initial covariance, x¯ and P are the mean and variance of state x(k) with 4 dimension, respectively. 2.2. Adaptive Unscented Kalman Filtering Algorithm

In order to make full use of the obtained mechanism model (3) and data of battery degradation, the data-model hybrid model is adopted to predict the RUL of lithium battery. In this paper, the prognosis process is divided into two phases: 1) AUKF algorithm is used to ascertain the degradation model of battery with training set data; 2) AUKF and GA-SVR are employed to realize the multi-step prediction of RUL. Instead of EKF which employs linearization to achieve the estimation of the nonlinear stochastic system, the UKF [26] is adopted because it utilizes unscented transform (UT) to deal with the nonlinear transfer of mean and variance, so that higher calculation accuracy and better stability can be obtained. Especially, the proposed adaptive unscented kalman filter (AUKF) Algorithm 1 in this part can

Algorithm 1 Adaptive Unscented Kalman Filter

adaptively update the process noise covariance and the observation noise covariance, therefore, more accurate estimate can be achieved. Before introducing the AUKF algorithm, UT is used to firstly calculate the following 2n + 1 sigma points:  (0)  i=0 xk−1|k−1 = xˆk−1|k−1 ,     p   (i) xk−1|k−1 = xˆk−1|k−1 + ( (n + λ)P0 ), i = 1, 2, ..., n (4)     p    x(i) = xˆ − ( (n + λ)P ), i = n + 1, ..., 2n k−1|k−1

k−1|k−1

Input: Initialization parameters P0 , x0 , Q0 , R0 Iteration: while k < N do Step 1. Prediction process: 1.1). Using UT transform to get sigma point set R x = (i) }, i = 0, · · · , 2n {xk−1|k−1 (i) (i) ) = f (xk−1|k−1 1.2). xk|k−1 P2n (i) (i) 1.3). xˆk|k−1 = i=0 ωm xk|k−1

0

and then compute the weights of sigma points  λ   ω(0)  m =   n + λ      λ  (0) ωc = + (1 − α2 + β)    n + λ      λ  i   ω(i) i = 1, ..., 2n m = ωc = 2(n + λ)

(i) ˆk|k−1 1.4). e(i) x = xk|k−1 − x P2n (i) (i) (i) T 1.5). Pk|k−1 = i=0 ωc e x (e x ) + Qk−1 1.6). Using new prediction point xˆk|k−1 and UT transformation again. we can obtain the new sigma point sets (i) R xˆ = {xk|k−1 }, i = 0, · · · , 2n

(5)

(i) (i) ) = h(xk|k−1 1.7). yk|k−1 P2n (i) (i) 1.9). yˆ k|k−1 = i=0 ωm yk|k−1 Step 2. Update process: (i) 2.1). e(i) ˆ k|k−1 y = yk|k−1 − y P2n (i) (i) (i) T 2.2). Pyy = i=0 ωc ey (ey ) + rk−1 P (i) (i) (i) T 2.3). P xy = 2n i=0 ωc e x (ey ) −1 2.4). Kk = P xy Pyy 2.5). ek = yk − yˆ k|k−1 2.6). xˆk|k = xˆk|k−1 + Kk ek 2.7). Pk|k = Pk|k−1 − Kk Pyy KTk 2.8). yˆ k|k = h( xˆk|k ) Step 3. Covariance updating process: 3.1). rk(i) = y(i) − yˆ k|k k|k−1 1 Pk 3.2). Gk = L i=k−L+1 rk(i) · (rk(i) )T 3.3). Qk = Kk Gk KTk 3.4). r¯k = yk − yˆ k|k P (i) T 3.5). Rk = 2n i=0 ωc r¯k r¯k + Gk end while Output: yˆ k|k − Capacity estimate at time k.

where n is the dimension of state vector, λ is the scaling parameter, α is the control value for the distribution of sampling points, β is the weight coefficient, L is the window size for covariance matching. 2.3. Support Vector Regression Based on the obtained residual data generated by AUKF Algorithm 1, the ε-SVR method is selected to achieve the multistep prediction, and the basic idea of ε-SVR prediction can be described as follows. For the obtained data {ek } in step 2.5 and its mapping {zk }, which makes up the observation sample set Z = {(e1 , z1 ), (e2 , z2 ), · · · , (en , zn )} (en ∈ Rn , zn ∈ R). The mapping relationship of observation set Z can be expressed as z = g(e) = wT e + q, w ∈ Rn

(6)

where w and q are the weight vector and the offset constant, respectively. Then the mapping (6) can be further transformed to the following optimization problem: n X  kwk2  +C · (ξi + ξi∗ ) 2 i=1  T  z − w e − q ≤ ε + ξi  i i     T s.t.  w ei + q − zi ≤ ε + ξi      ξi , ξ∗ ≥ 0, i = 1, ..., n i

min

solution with parameters w, αi , α∗i can be written as

(7)

g(e) =

n X (αi − α∗i )K(ei , e) + b

(9)

i=1

(8)

where K(ei , e) is a kernel function, which has a significant impact on the regression performance of SVR. In this paper, we choose the radial basis kernel function with following form

where C is a preset penalty coefficient, ε is the deviation between the training data and the actual observation data, ξi and ξi∗ are slack variables. Usually, penalty parameter C is used to control the balance between generalization ability and classification accuracy. Similar to [27], by introducing Lagrange multiplier αi and α∗i , equation (7) can be turned into a dual problem, and optimal

K(ei , e) = exp{

−|ei − e|2 } 2σ2

(10)

where σ is the parameter of kernel function, which can affect the complexity of SVR algorithm. In order to achieve more effective prognosis, the Genetic algorithm (GA) [23] is introduced to optimize the parameters C and σ. GA is an adaptive global optimization search method which imitates the evolutionary law of biology. Compared with 3

traditional optimization algorithm, GA does not depend on specific mathematical equation and derivative expression, it has the advantages of strong global search ability, high efficiency and fast search speed, therefore, GA is an appropriate choice to optimize SVR for RUL.

Initialization: x0 , P0 , Q0 , R0

i=1

(αi − α∗i )K(ζi , ζt ) + p

= =

g(ζt+1−m ) TX −m i=1

(αi −

α∗i )K(ζi , ζt+1−m )

+p

xˆk |k = xˆk |k −1 + K k ek yˆ k |k = h( xˆk |k )

Weights calculation ωm(i ) , ωc(i )

Covariance update

Qk , Rk

Prior state estimation

xˆk |k −1

Filtering(AUKF)

k = T,s =1 State transition equation xˆk( i+)s|k + s −1 = f ( xˆk( i+)s −1 )

(11)

xˆk + s|k + s −1 =

8 i =0

Prognostics(GA-SVR) Genetic algorithm optimize parameters C,σ

ωm( i ) xˆk( i+)s|k + s −1

Measurement equation yˆ = h ( xˆ k( i+) s|k + s −1 ) (i ) k + s |k + s −1

where ζt = [et , et+1 , · · · , et+m−1 ] and ηt = et+m (t = 1, 2, ..., T − m). Relying on the past m set of data ζt+1−m = [et+1−m , · · · , et ], the one-step prediction relation at time t = T can be established as e˜ t+1

State estimation updation ek = yk − yˆ k |k −1

Sigma points sampling xk(i−)1|k −1

3.1. Multi-steps prediction with GA-SVR In this part, the multi-step prediction method with residual data is adopted to obtain higher prediction accuracy. Firstly, the starting point of prediction is supposed as T and the residual data e1:T from 1 to T can be computed with Algorithm 1. Next, the achieved residual data e1:T is utilized to build SVR prediction model which can be expressed as: TX −m

Qk −1 , Rk −1

State vector xk −1

3. Integrated algorithm for battery RUL prediction

ηt = g(ζt ) =

Measurement yk

Covariance

8

yˆ k + s|k + s −1 =

i=0

ω m( i ) yˆ k( i+) s|k + s −1

SVR iterative prediction model get the residual eˆk + s

State updation xˆk + s = xˆk + s|k + s −1 + K k + s eˆk + s s=s+1

Output updation yˆ k + s = h( xˆk + s )

(12)

Covariance update

where e˜ t+1 represents the residual. To improve the prognosis effect of one-step prediction (12), the genetic algorithm is added to optimize the kernel function K(ζi , ζt+1−m ), and the optimization goal is the SVR parameter C and σ. On the other hand, by performing the one-step prediction (12), p-step prediction can be obtained:  eˆ t+1 = g(ζt+1−m ) = g(et+1−m , · · · , et )        eˆ t+2 = g(ζt+2−m ) = g(et+2−m , · · · , et , eˆ t+1 )       . .. (13)         eˆ t+p = g(ζt+n−m )      = g(et+n−m , · · · , et , eˆ t+1 , · · · , eˆ t+p−1 )

Qk , Rk Yes If yˆ k + s < threshold

Yes

EOL

No

If s< p No

END

Figure 1: Flow chart of the integrated algorithm

Actually, the AUKF algorithm can not only achieve the adaptive update of process noise covariance and observation noise covariance, correct the defect of standard unscented kalman filter, but also improve the filter accuracy. On the other hand, SVR is adopted to implement multi-step prediction, and GA algorithm is added to modify the parameters (C, σ) in SVR so that the prediction accuracy of SVR algorithm can be improved. Therefore, the proposed hybrid AUKF-GA-SVR algorithm is a more interesting method which can be verified by NASA battery dataset.

After the new prediction residual sequences {ˆet+i (i = 1, 2, ..., p)} is obtained by the GA-SVR, and the updated residual data is added to the update process with the AUKF Algorithm 1. Consequently, the state vector value ak , bk , ck , dk can be updated each time, then adaptive character of the exponential model (1) can be acquired, finally the multi-step prediction can be realized. 3.2. Integrated RUL prediction algorithm Based on the AUKF and GA-SVR algorithm above, we propose a hybrid RUL prediction method which combines AUKF and GA-SVR algorithm. The flow chart of AUKF-GA-SVR method is shown in Figure 1, the specific prediction process is summarized as Algorithm 2.

4. Experimental results 4.1. Experimental data set description In this section, we utilize the National Aeronautics and Space Administration (NASA) battery dataset [28] to verify the effec4

Battery capacity 2.1

Algorithm 2 Hybrid multistep prediction method Step 1. State estimation: Set the time point of starting prediction as T , the real capacity data {xi , yi }Ti=1 of the lithium battery from 1 to T is used to determine the empirical model (2) parameters ak , bk , ck , dk with AUKF Algorithm 1. Step 2. Residual data acquisition: Utilize the empirical model obtained in Step 1 and Algorithm 1 to acquire the residual data {ei }Ti=1 . Step 3. Training the GA-SVR model: Firstly, the residual data {ei }Ti=1 are filtered with AUKF Algorithm 1, and the outliers are removed. Consequently, take the obtained data as a training set and recur to the genetic algorithm to optimize the SVR model (12), then acquire the multi-step prediction result p {ˆeT +s } s=1 by the iterative calculation method (13). Step 4. State update: Employ the obtained residual data p and the AUKF Algorithm 1 to produce {ˆeT +s } s=1 p the state estimation value { xˆT +s } s=1 . Utilize the measurement equation (2) to obtain the predicted p capacity {ˆyT +s } s=1 . Step 5. RUL prediction: If the predicted battery capacity reaches the defined threshold, then output the RUL by calculating the interval between the prediction starting cycle T and the battery failure threshold point. Else, go to Step 3.

B0005 capacity B0006 capacity B0007 capacity B0018 capacity

2 1.9 1.8

Capacity

1.7 1.6 1.5 1.4 1.3 1.2 1.1

0

20

40

60

80 100 Cycle

120

140

160

180

Figure 2: The capacity decay curves of four batteries.

tiveness of the proposed method. By analyzing the characteristics and experimental conditions of battery capacity attenuation data, we choose B0005, B0006, B0007 and B0018 as experimental data. The capacity decay curves of the four batteries are shown in the figure 2. The above four batteries were set to run through 3 different operational profiles (charge, discharge and impedance) at room temperature (24 ◦ C). The parameters including ambient temperature (AT), charge current (CC), discharge current (DC), end-of-discharge (EOC), and end of-life criteria (EOLC) are detailed in [28]. The rated capacity is 2 Ah (only slightly different cyclic discharge conditions). Due to the experimental conditions of four batteries are different, the threshold values are chosen as 1.35 Ah, 1.35 Ah, 1.5 Ah and 1.45 Ah respectively.

predicted battery capacity, y¯ k represents the average of actual battery capacity, Rt is the true RUL result and R p describes the RUL prediction value. Specifically, for the indicators ERMS E , E MAE and ERUL , if they are close to 0, then the capacity prediction accuracy are higher. As far as R2 and ERA , which are more close to 1, the more accurate RUL prediction results can be achieved.

4.2. Evaluation matrices To assess the prediction performance the proposed integrated approach, we adopt the traditional matrices such as root mean square error ERMS E , mean absolute error E MAE and R2 coefficient as the evaluation matrices. Furthermore, the prediction error ERUL and its relative error ERA are also introduced to indicate the accuracy of RUL prediction. The specific form of these matrices can be described as follows

ERMS E

=

E MAE

=

R2

=

ERUL

=

ERA

=

v t

4.3. Simulation result analysis In this part, we carry out firstly a set of simulation experiments to verify that the double exponential model and genetic algorithm have higher prediction accuracy. The comparison of related indicators are shown in Tables 1 and 2. Among them, AUKF-GASVR-P represents polynomial model, AUKF-GASVRE marks double exponential model, PSO is particle swarm optimization algorithm, FA is firefly optimization algorithm and GAS is genetic algorithm. Now, it’s time to utilize the proposed AUKF-GA-SVR method to predict the RUL for four batteries data with different estimation moments. At the same time, in order to verify the superiority of the proposed technology, several good methods are introduced as the comparison algorithms such as unscented kalman filter (UKF), extended kalman filter (UKF), adaptive extended kalman filter (AEKF), adaptive unscented kalman filter (AUKF), unscented kalman filter and relevance vector regression (RVR-UKF) [18] and adaptive extended kalman filter and genetic algorithm optimized support vector regression (AEKF-GA-SVR).

n

1X (yk − yˆ k )2 n k=1 n

1 X (ˆyk − yk ) | | n k=1 yk Pn (yk − yˆ k )2 1 − Pk=1 n ¯ k )2 k=1 (yk − y |Rt − R p | |Rt − R p | 1− Rt

where yk indicates the actual battery capacity, yˆ k denotes the 5

Table 1: Comparison of two models based on B0005 battery data

model AUKF-GASVR-P

Cycle 60 80 60 80

AUKF-GASVR-E

Rt 80 60 80 60

Rp 87 64 83 59

ERUL 7 4 3 1

ERMS E 0.0216 0.0186 0.0171 0.0153

R2 0.8760 0.9023 0.9132 0.9432

E MAE 0.0198 0.0176 0.0156 0.0152

ERA 0.9125 0.9333 0.975 0.983

Table 2: Comparison of three algorithms based on B0006 battery data

algorithm AUKF-FASVR

Cycle 60 80 60 80 60 80

AUKF-PSOSVR AUKF-GASVR

Rt 65 45 65 45 65 45

Rp 55 39 57 40 59 42

ERUL 10 6 8 5 6 3

ERMS E 0.0501 0.0395 0.0478 0.0370 0.0444 0.0324

B0005 10−step−prediction in 60 cycles

R2 0.81 0.8243 0.8250 0.8536 0.8760 0.8850

E MAE 0.0350 0.0347 0.0346 0.0312 0.0325 0.0287

B0005 10−step−prediction in 80 cycles

2

Battery capacity data UKF RVR−UKF AEKF AEKF−GASVR AUKF AUKF−GASVR

Battery capacity data

1.9

UKF

1.9

RVR−UKF AEKF

1.8

1.8

AEKF−GASVR AUKF

1.6

1.7

AUKF−GASVR

1.4

Capacity

Capacity

1.7

1.35 1.5

1.45

1.6

1.4 1.5 1.35

1.3 1.4

ERA 0.846 0.867 0.877 0.889 0.907 0.933

130

140

1.4

150

120

130

1.3 1.3

prediction

estimation

prediction

estimation

140

1.2 1.2 20

40

60

80

100

120

140

160

1.1

180

0

20

40

60

80

100

120

140

160

180

Cycle

Cycle

Figure 3: Ten-step-prediction for battery B0005 with start point 60 cycles.

Figure 4: Ten-step-prediction for battery B0005 with start point 80 cycle.

On the other hand, some key factors such as the number of data cycles, initial capacity and discharge conditions of each battery group are different, and which have an important effect on the RUL prediction. Therefore, different parameters including prediction start points, steps of multi-step prediction and threshold of battery failure, which are selected in the simulation to verify the prediction effect of developed approach. As far as prediction step is concerned, five-step-prediction and ten-stepprediction are selected for B0018 and B0005, respectively, and the prediction step of other two data sets are chosen as twentystep-prediction. As a result, the RUL prediction curve for four kinds of batteries with different starting prediction points are acquired in Figures 3 – 10, and their quantitative results of prediction accuracy can be obtained in Tables 3–6 with five kind of performance matrices. Specifically, the degradation model of the battery capacity is established by using the real data before the prediction point, and the integrated algorithm proposed in this paper is used to predict the RUL after the starting prediction point. The comparison between the real capacity and the predicted capacity of different prediction methods are shown in Figures 3 – 10

and Tables 3–6. Compared with UKF, RVR-UKF, AEKF and AEKF-GA-SVR techniques, AUKF and AUKF-GA-SVR have better prediction for RUL from above simulation results. Four methods, such as UKF, can hardly complete the RUL prediction task. It can be seen from Figure 3 that AEKF, AEKF-GA-SVR, AUKF and AUKF-GA-SVR algorithms have much better filtering effect than UKF and RVR-UKF methods because adaptive characteristic reduce efficiently the prediction and estimation error. On the other hand, the amount of degraded data also has an important effect on the prediction accuracy. For example, the 60th and 80th cycle is chosen as the prediction starting point for B0006 battery, Figures 5-6 and Table 4 verify that the later starting points result in higher prediction accuracy. By analyzing the above simulation results, we can conclude that the higher prediction accuracy may be caused by the following reasons: 1) the UKF algorithm has not involved the calculation of its Jacobian matrixit by ignoring the high-order term; 2) the adaptive characteristics is used to adjust dynamically the estimation and prediction error; 3) SVR algorithm is introduced to offset the shortcoming of AUKF which can only do one-step-predict; 4) genetic algorithm is employed to 6

Table 3: Prediction effect comparison of B0005 battery

Method UKF RVR-UKF AEKF AEKF-GASVR AUKF AUKF-GASVR

Cycle 60 80 60 80 60 80 60 80 60 80 60 80

Rt 80 60 80 60 80 60 80 60 80 60 80 60

Rp 92 76 88 74 80 79 80 78 79 56 80 57

ERUL 12 16 8 14 0 19 0 18 1 4 0 3

ERMS E 0.0399 0.0385 0.0389 0.0381 0.0350 0.0306 0.0334 0.0304 0.0265 0.0196 0.0230 0.0192

E MAE 0.0285 0.0279 0.0280 0.0277 0.0256 0.0221 0.0227 0.0218 0.0187 0.0136 0.0148 0.0125

R2 0.6745 0.7006 0.7001 0.7116 0.7554 0.8127 0.7723 0.8156 0.8235 0.9086 0.8753 0.9099

E MAE 0.0465 0.0994 0.0453 0.0985 0.0429 0.0468 0.0423 0.0459 0.0395 0.0371 0.0392 0.0368

R2 0.6613 -1.1996 0.6651 -1.1973 0.7245 0.5240 0.7259 0.5265 0.7731 0.6885 0.7743 0.6888

ERA 0.8 0.578 0.817 0.622 0.8 0.8 0.817 0.822 0.817 0.822 0.833 0.844

E MAE 0.0446 0.0424 0.0453 0.0411 0.0193 0.0467 0.0186 0.0461 0.0094 0.0092 0.0091 0.089

R2 0.0735 -0.3636 0.0777 -0.2954 0.8373 -0.2266 0.8399 -0.2232 0.9530 0.9317 0.9547 0.9343

ERA 0.754 0.887 0.754 0.889 0.883 0.756 0.883 0.8 0.933 1 0.95 1

ERA 0.85 0.733 0.9 0.767 1 0.683 1 0.7 0.9875 0.933 1 0.95

Table 4: Prediction effect comparison of B0006 battery

Method UKF RVR-UKF AEKF AEKF-GASVR AUKF AUKF-GASVR

Cycle 60 80 60 80 60 80 60 80 60 80 60 80

Rt 65 45 65 45 65 45 65 45 65 45 65 45

Rp 53 26 54 28 53 36 54 37 54 37 55 38

ERUL 12 19 11 17 12 9 11 8 11 8 10 7

ERMS E 0.0617 0.1275 0.0603 0.1265 0.0573 0.0599 0.0565 0.0593 0.0527 0.0489 0.0510 0.0483

Table 5: Prediction effect comparison of B0007 battery

Method UKF RVR-UKF AEKF AEKF-GASVR AUKF AUKF-GASVR

Cycle 60 80 60 80 60 80 60 80 60 80 60 80

Rt 65 45 65 45 65 45 65 45 65 45 65 45

Cycle 40 60 40 60 40 60 40 60 40 60 40 60

Rt 39 19 39 19 39 19 39 19 39 19 39 19

Rp 49 39 49 40 72 34 72 36 69 45 68 45

ERUL 16 6 16 5 7 11 7 9 4 0 3 0

ERMS E 0.0658 0.0625 0.0656 0.0606 0.0293 0.0697 0.0286 0.0687 0.0142 0.0136 0.0134 0.0124

Table 6: Prediction effect comparison of B0018 battery

Method UKF RVR-UKF AEKF AEKF-GASVR AUKF AUKF-GASVR

Rp 38 33 38 32 38 35 38 35 39 23 39 23

ERUL 1 14 1 13 1 16 1 16 0 4 0 4

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ERMS E 0.0666 0.0335 0.0657 0.0326 0.0640 0.0315 0.0637 0.0306 0.0556 0.0246 0.0547 0.0233

E MAE 0.0450 0.0295 0.0442 0.0287 0.0438 0.0273 0.0434 0.0267 0.0388 0.0125 0.0382 0.0121

R2 0.5105 0.7036 0.5130 0.7089 0.5200 -0.7159 0.5220 0.7188 0.5606 0.8355 0.5614 0.8389

ERA 0.9744 0.641 0.9744 0.667 0.9744 0.59 0.9744 0.59 1 0.933 1 0.933

B0006 20−step−prediction in 60 cycles 2.5

B0007 20−step−prediction in 60 cycles

2

Battery capacity data RVR−UKF

Battery capacity data

prediction

estimation

UKF

UKF RVR−UKF

1.9

AEKF

AEKF

AEKF−GASVR

AEKF−GASVR

AUKF

1.8

AUKF

AUKF−GASVR

AUKF−GASVR

1.7

2 1.5

Capacity

Capacity

1.45 1.4 1.35

1.6

1.5

1.3 1.5

100

110

120

1.4

130

1.55 1.3

estimation 1

0

20

40

60

1.5 1.45

1.2

prediction

80

100

120

140

160

1.1

180

120 0

20

40

60

80

Cycle

120

140

160

180

Figure 7: Twenty-step-prediction for battery B0007 with start point 60 cycle.

B0006 20−step−prediction in 80 cycles 2.2

B0007 20−step−prediction in 80 cycles

2

Battery capacity data RVR−UKF

prediction

Battery capacity data

prediction

estimation

UKF

2

140

Cycle

Figure 5: Twenty-step-prediction for battery B0006 with start point 60 cycle.

estimation

130 100

UKF RVR−UKF

1.9

AEKF

AEKF

AEKF−GASVR

AEKF−GASVR

AUKF

1.8

AUKF

AUKF−GASVR

AUKF−GASVR

1.8

1.6

Capacity

Capacity

1.7

1.4 1.5

1.5

1.4

1.55

1.3

1.5

1.45

1.2

1.4 1.35 1

1.3

0

1.45

1.2 100

0.8

1.6

20

110 40

120 60

110

130 80

100

120

140

160

1.1

180

0

20

120 40

Cycle

130 60

80

100

120

140

160

180

Cycle

Figure 6: Twenty-step-prediction for battery B0006 with start point 80 cycle.

Figure 8: Twenty-step-prediction for battery B0007 with start point 80 cycle.

optimize SVR parameters. Therefore, the AUKF-GA-SVR algorithm including above advantages is adopted to predict precisely the RUL.

6. Acknowledgement This work was supported in part by the National Natural Science Foundation of China [Grant 61873197, 51905197], and in part by the Primary Research and Development Plan of Jiangsu Province [Grant BE2017002].

5. Conclusion This paper studied the RUL prediction problem for lithium battery with hybrid algorithm of AUKF and GA-SVR. According to the characteristic of NASA data set, the battery capacity is extracted as the degradation index and the state space model is established. To predict the RUL efficiently, we employ AUKF to reduce dynamically the influence of noise, and utilize SVR to improve the disadvantage that the AUKF algorithm can only forecast one step. At the same time, GA is introduced to optimize the key parameters of SVR. The experiment results with NASA dataset show that the proposed method can improve effectively the prediction accuracy of RUL. Future research will focus on further improving the accuracy of prediction by extracting the proper health indicators, exploring the right degradation model and developing the appropriate optimization algorithm.

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B0018 5−step−prediction in 40 cycles 2

Battery capacity data UKF

1.9

estimation

RVR−UKF

prediction

AEKF AEKF−GASVR

1.8

AUKF AUKF−GASVR

1.7

Capacity

1.6 1.5 1.4 1.55 1.3 1.5 1.2

1.45

1.1 1

1.4 70 0

20

40

75

80

85

60

90

80

100

120

140

Cycle

Figure 9: Five-step-prediction for battery B0018 with start point 40 cycle. B0018 5−step−prediction in 60 cycles 2 1.9

Battery capacity data UKF RVR−UKF AEKF AEKF−GASVR AUKF AUKF−GASVR

prediction

estimation

1.8 1.7

Capacity

1.6 1.5 1.55

1.4

1.5

1.3

1.45 1.2 1.4 1.1 1

75 0

20

80

85

90 40

95 60

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Figure 10: Five-step-prediction for battery B0018 with start point 60 cycle.

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Dear editors: We are no conflict of interest form. Thank you and best regards. Yours sincerely, Yong

Biography

Yong Zhang received the B.Sc. degree in mathematics from Jiangsu Normal University, Xuzhou, China, in 2001, and the M.Sc. degree in applied mathematics from Three Gorges University, Yichang, China, in 2007. and the Ph.D. degree in control theory and control engineering from Huazhong University of Science and Technology, Wuhan, China, in 2010. From 2014 to 2015, he was a Visiting Scholar with the Department of Information Systems and Computing, Brunel University London, Uxbridge, U.K. He is currently an Associate Professor with the School of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan, China. He has authored over 10 papers in refereed international journals. His current research interests include remaining useful life prediction of key equipment, fault diagnosis and faulttolerant control of networked systems. Dr. Zhang is a very active Reviewer for many international journals.

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