Prediction of remaining useful life for lithium-ion battery with multiple health indicators

Lithium-ion (Li-ion) battery is the key power unit for a variety of engineering equipments, including portable computers, electric automobiles, satellites, and spacecrafts, etc. With the increased cycles of charge-discharge, the battery’s capacity will deteriorate gradually, and when the capacity is inferior to a given threshold, failure may occur. This phenomenon will reduce the equipment’s reliability and even lead to catastrophic failures[31]. Therefore, it is crucial to timely forecast Li-ion battery’s capacity as well as its remaining useful life (RUL). For Li-ion battery, its RUL can be defined as the number of remaining charge-discharge cycles before the battery’s capacity deteriorates to a predetermined failure threshold [1]. The prediction methods for battery RUL can be categorized into three types, i.e. model-based approach, data-driven approach and hybrid approach [23]. Model-based approaches adopt mathematical representation or failure physics model to describe the degradation process of battery capacity, including electrochemical model [14, 16, 25], equivalent circuit model [6] and empirical model [33], etc. Considering the interactions among the various factors and the calculation of the parameters, it is not easy to establish a reliable and accurate prediction model for the electrochemical model and equivalent circuit model. Conversely, by fitting a large amount of degradation data, empirical model is easier to be established [32]. Based on the battery’s historical operational data, data-driven approach can extract the feature information and obtain the inherent degradation tendency. As a kind of data-driven approach, time series analysis has been widely used in RUL prediction, including autoregressive (AR) [12], auto-regressive and moving average (ARMA) [18], auto-regressive integrated moving average (ARIMA) [39] and the improved models [13, 35]. In recent years, artificial intelligence (AI) algorithm becomes a hot point in data-driven approaches due to its powerful ability in self-learning and data mining. The frequentlyused AI algorithms include artificial neural network (ANN) [11, 21, 30], support vector machine (SVM) [15, 26], relevance vector machine (RVM) [36], etc. By combining two or more model-based or data-driven approaches, hybrid approach can overcome the limitations of a single method and thus improve the accuracy and efficiency of the prediction [5, 37]. Hybrid approach can be a combination of model-based approach and data-driven approach, and it can also be a combination of two or more types of data-driven approaches [1, 22]. In most existing studies, the Li-ion battery’s RUL is predicted with static capacity data. In fact, the battery’s remaining capacity can only be measured when it is out of service, meanwhile it also requires strict Keywords

cordingly.They are calculated resistance (CR), rate of temperature change (TR), duration time of equal discharging voltage difference (DTEDVD) and sample entropy of discharge voltage (SampEn), respectively.

Step 2: Estimating the remaining capacity.A generalized regression neural network (GRNN) with cross-validation is applied to estimate the battery's remaining capacity.To reduce the interference and improve the accuracy of the estimation, the wavelet denoising is performed for two times and with different thresholds.

Step 3: Predicting RUL.With the estimated remaining capacity, the battery's RUL is predicted via the non-linear autoregressive (NAR) method.


Fig. 1.
Process of the proposed approach

The remainder of this article is organized as follows.Section 2 introduces the four HIs and their calculation methods respectively.Section 3 illustrates the basic theory of GRNN, NAR and wavelet denoising.Based on the Li-ion battery data from NASA, Section 4 conducts a case study of Li-ion battery RUL predict on, and the effectiveness of the proposed approach is illustrated.Conclusions and future works are given in Section 5.


Health indicators

To illustrate the discharging characteristics of the battery changing with the number of the charge cycles, in this study four HIs are extracted, they are CR, TR, DTEDVD and SampEn respectively.The value of the above four HIs will change with the increase of cycles, and the corresponding calculation methods are described as the follows.


Calculation of CR

For Li-ion battery, its internal resistance is closely related to the battery's remaining capacity.Saha et al [19] found that the remaining capacity of Li-ion batteries is linearly related to the sum of their electrolyte impedance and the charge transfer impedance.

CR is the measured internal resistance, which can be obtained with battery's terminal voltage and current during its operating sta

.Fig. 2 depicts th
voltage and current's change at the instant of battery discharge.When a constant DC current passes through the battery, the terminal voltage changes ΔU n at the time t 1 .At that time, the polariza-tion effect has not occurred in the battery, therefore the voltage change is caused by ohm resistance.In this paper, the CR refers to the ohm

resistance R n .A
cording to Ohm's Law, it can be obtained as [18]:
1, 2, , n n n U R n N I ∆ = = … (1)
where ΔU n is the voltage change caused by ohmic resistance of cycle n; I n is the load current in cycle n; and N is the total cycle life of the battery.


Fig. 2. Pulse discharge r sponse of the battery


Calculation of TR

During the discharging process, the battery's internal temperature will change.Here we define TR as the rate that the temperature changes in the discharge process.TR will increase with the increase of cycles.In cycle n, TR can be calculated as follows:

(

) / 1, 2, ,
n n n e s n TR TP TP t n N = − = … (2)
where TR n denotes the rate of temperature change in cycle n; TP e n is the end temperature of cycle n; TP s n is the initial temperature of cycle n; and t n is the duration time of cycle n's discharge process.


Calc y, the time for full discharge will be shortened gradually with the increase of the cycles.Therefore, the duration time of equal discharging voltage differ

ce (DTEDVD) will a
so be shortened with the increase of the cycles' number.For a constant voltage interval [U L , U H ]. We define n D t as the duration time in cycle n when the voltage is reduced from U H to U L , as shown in Fig. 3.


1, 2, ,
n n n D H L t t t n N = − = …(3 culation of SampEn

Sample entropy (SampEn) was firstly proposed by Richman and Mooman to measure the complexity of time series [17].In this study, we regard the discharging process as a time series, and the Sam

n of discharge voltage
is used as an HI for the Li-ion battery.()
m V α is defined as the number of [ ( ), ( )] ( ) m m m d X X r α β α β ≤ ≠
, and the definition of the function is as the follows [28]: .Hence, the sample entropy can be defined as: 1
( ) ( ) 1 1, 2,..., 1 1

m B r V
( ) ( ) - T m m m B r B r T m α α − = = ∑(5)( ) ( , , ) ln ( ) m m B r SampEn r m T B r +   = −    (6)

Methodology

In this study, the battery's remaining capacity is estimated with the proposed HIs.On this basis, by analyzing t

degradation trend of
he remaining capacity, the battery's RUL is predicted.

GRNN is used to establish the mapping relationship between the HIs and the battery's capacity, and hereafter to estimate the remaining capacity.Here, NAR model is used to demonstrate the degeneration of nce of the estimated capacity.Here, we illustrate the basic t RNN model has strong ability for non-linear mapping.Therefore, it is suitable for solving non-linear problems [ lly, there are four layers in GRNN, i.e. input

Fig. 4.  In
ut layer: Among it, each neuron is a simple distribution (1) unit, which will transfer the input variable directly to the pattern layer.In addition, the number of the input neurons in a GRNN eq als to the dimension of the input vector e.


The input of the network is

Pattern layer: The number of neurons equals to the number (2) of learning sample m.Each neuron corresponds to a different sample, and the neurons transfer function can be calculated as [36]:
( ) ( ) 2 exp 1, 2, ,2T i i i p i m σ   − − = − = …       X X X X (7)
where σ is the smoothing parameter; X is the input variable; a

X i is the cor
esponding learning sample of neuron i.

Summation layer: It sums the output of the pattern layer.In (3) this layer, two types of neurons are used for sum ation.The simple summation of the pattern outputs is determined with:
1 m D i i S p = = ∑(8)
And, the weighted summation of the pattern output can be determined by:
1 1, 2, , m Nj ij i i S y p j k = = = … ∑(9)
where y ij is the weight of the i-th neuron in the pattern layer, which is connected to the summation layer.

Output layer: The results calculated in the summation layer (4) will be sent to the output layer.The number of neurons is equal to the dimension k of the output vector in the sample.The 10)
where y i is the output of the j-th node in the output layer.In this study, the cross-validation is used to gain th optimized parameters.


Theory of NAR

Here, by combining the variable's information during the early time, NAR model is used to describe the variable information at a particular time.I ies [8]: The structure of NAR-NN is shown as in Fig. 5.It is consisted o output layer and the delay function.


Fig. 5. Structure of NAR-NN

In Fig. 5, ( 1)
u x u p ≤ ≤
is the input y(k) is the output; W 1 is the connection weight between the input layer and the hidden layer; W 2 is the connection weight between the hidden layer and the output layer; and p is the delay time of the output.

The output of the node v 2, , p v uv u v u h g w x c v V = = + = ∑ (12)
where 1 () g ⋅ is the activation function of the hidden nodes; V is the number of the hidde

nodes; u x is
he u-th delay output of the output signal y; uv w is the connection weight between the hidden node u and the delay node v; and v c is the v-th neural threshold of the hidden layer.
The output of NAR is:
2 1 ( ) V v v v y g w h d = = + ∑ (13)
where v w is the connection weight between the hidden node v and the output node; and d is the neural threshold of the output layer.


Wavelet denoising

Wavelet analysis is a kind of for sign ch can separate the noise signals and extract effective data from the noisemixed data.In wavelet analysis, the data can be decomposed into two parts with the expansion and translation of the mother wavelet, i.e. approximate art and detail part [24].

Let H(x) denote the measured signal mixed wi b c a b H x x H x a a ψ ψ +∞ −∞ − = = ∫ (14)
where a and b are the expansion and translation parameters of the mother wavelet respect

ely; and c(a, b) i
the wavelet coefficient of H(x).

During the process of wavelet denoising, the selection of wavelet denoising threshold is closely related to the consequence of noise reduction.In this study, a twice denoising method with different thresh-olds is applied [3,4].The first denoising threshold is made by the Sqtwol g method: where N is the number of signal H(x).In this paper, it equals to the cycle life of the battery.

The thresholds of t method is larger than that of the minimax method.Therefore, the first wavelet denoising can remove the lager noise signal, while the second wa elet denoising can remove the smaller noise signal.By using the twice wavelet denoising with different thresholds, the interference among the measured data can be reduced effectively.


Implementation of the proposed approach

Here, the training batteries are used to establish the mapping relationship between the HIs and the battery capacity.The basic steps for estimating battery RUL are described as follows:

Based on the historical operation data, the HIs of the training (1) batteries and testing batteries are calculated respectively.By using the training batteries, train a GRNN model to estab- (2) lish the mapping relationship between the HIs and the battery's capacity.

With the HIs data of testing batteries in the previous cycles, the (3) capacity of the testing batteries is estimated on the basis of the well-trained GRNN model.The wavelet denoising is applied to deduce the noising of the (4) estimated capacity.

With the est

lished capacity sequence and by using th
NAR (5) model, the remaining capacity and RUL of the testing battery can be predicted.


Prognostics Experiment


Experiment Data and the Procedure

In this section, a case study is cond cted to demonstrate the effectiveness and efficiency of the proposed approach, where the Liion battery data come from the data repository of the NASA Ames Prognostics Center of Excellence (PCoE) [20].Here, three groups of batteries are taken as examples, i.e.B05, B06 a d B07.Among them, B05 and B06 are used as the training batteries, and B07 is used as the target battery.The experimental parameters of the discharge process are listed as in Table 1.The capacity's degradation trends with the cycles are shown as in Fig. 6.Obviously, the battery's capacity shows a non-monotonic decline with the increase of the cycles.Meanwhile, the degradation is accompanied

city self-regeneration as well as
ocal fluctuations.Therefore, it is essential to develop an effective prognostic approach to accurately estimate the capacity and predict its RUL by overcoming the uncertainty fluctuation.


Calculating HIs

In this section, by using the battery's historical operation data (including current, voltage and temperature) during the discharging process, we calculate the HIs for batteries of B05, B06 and B07 respectively.Fig. 7 shows the variation of the four HIs with the increase of cycles respec ively.

Next, we calculate the Pearson correlation coefficient between the four HIs and the original capacity of Li-ion batteries, as shown in Table 2. Obviously, the HIs have a close correlation with the capacity.Furthermore, the DTEDVD has a negative correlation with the capacity, while the other three HIs are in positive correlation.


Establishing GRNN model

With the obtained HIs data, GRNN model is used to train and establish the mapping relationship bet

en the HIs (inpu
) and the capacity (output).In this study, the data of B05 and B06 batteries are used for training.On that basis, the well-trained GRNN model is used to estimate B07's capacity with B07's HIs.The predicted results are shown in Fig. 8.

As shown in Fig. 8, the proposed method has a relatively high prediction accuracy concerned with the capacity in the long-term trend.However, the predicted result is not so good for the first 40 points.The reason is that the HIs are not so stable during the early stage.Therefore, the first 40 points can be removed to obtain a more accurate prediction result.

Moreover, the indicators, includ

g root means square erro
(RMSE), the maximum estimation error and the maximum covariance (cov) value, are utilized to evaluate the accuracy of the estima-  3. Obviously, the capacity estimation is with high accuracy.


RUL prediction

To remove the influence of self-regeneration phenomena and local fluctuations, a twice wavelet denoising is implemented with different threshold .The result is shown as in Fig. 9.

In this case, the threshold of Sqtwolog method is larger than that of minimax method.As shown in Fig. 9, the first denoising can only eliminate the large noise signal, and the second denoising can elimi-nate the smaller noise signal.Thus, with the proposed twice wavelet denoising, the noise can be eliminated more thoroughly an a more accurate prediction can be obtained.

Based on the historical operation data, we can estimate the remaining capacity for each past cycle with the above steps.In addition, by selecting the obtained data points as the input set, the future degraded cap

ity can be pred
cted with the NAR model.On that basis, we can obtain the corresponding value of the RUL.Here, the number of the hidden layer is set as 1, the delay time is 20.It is supposed that th failure threshold of the battery capacity is 1.45Ah, and the total cycle life of B07 is 142.Here, we predict the B07's RUL when the operated cycles (M) are 110, 120, 130 respectively.In addition, considering that there is randomness in the training model, the predicted RUL will change with the experiments, thus we repeat the experiments for 100 times.Fig. 10 and Table 4 s ow some of the predicted results.Set the range of the 100 predicted results as the RUL bound (RUB).For the battery of B07, the RUB is shown as in Table 5.

With the above predicted results, we can conclude that the proposed methods can obtain the decline tendency of the Li-ion battery capacity with a high accuracy.However, the RUL's predicted result is not so stable.The possible reason is that the tested life cycle of the battery is not long enough.Therefore, the training data are not sufficient.Furthermore, considering that the capacity change during each cycle is too small, and a small error in capac ty prediction may lead to a larger error in the RUL's prediction.It can also be found that the range of RUB will be narrowed down with the increase of the input cycles.It shows that the RUL prediction is sensitive to the number of input data.Therefore, by increasing the number of the training samples, the stability and robustness of the predictions can be improved.

ARIMA model is also widely used in the battery's RUL prediction [21].To compare the characteristics of different prediction models, here we make a comparison for RUL prediction with the ARIMA model.For the same input, the ARIMA method has a unique prediction result.Thus, the prediction results do not have a RUB.The RUL predicted results with ARIMA model are shown as in Table 6.

By comparing the three predicted results of different M, it can be found that or different M, the errors keep the same.It means that the prediction accuracy is not sensitive to the number of input cycles.

Comparing Table 4 and Table 6, it can be found that the predicted results of NAR model are more accurate than the results of ARIMA model.However, the predicted results are not as stable as the results of ARIMA model.Furthermore, since NAR model is sensitive to the number of the input data, the proposed NAR method is more suitable for large data set.


Conclusions

In this study, an innovative hybrid data-driven method is proposed to predict Li-ion battery's RUL which is based on multiple HIs, including CR, TR, DTEDVD and SampEn.Furthermore, the GRNN model, NAR model and twice wavelet denoising are integrated.Case study shows that the proposed method can achieve Li-ion battery's RUL prediction with a high accuracy.Compared with ARIMA model, NAR model is more sensitive to the size of training samples.Meanwhile

the proposed
method is established on the basis of battery historical data, it can overcome the limitation of online capacity measurement approaches.Moreover, the proposed RUL prediction method does not concern the physical and chemical reactions in the battery, thus it is well suitable for other kinds of batteries.

The future works can be focused on the following aspects: (1) to improve the robustness of the RUL prediction results, the NAR model can be improved further.(2) new HIs can be searched for more stable prediction, especially for the initial stage.(3) new RUL prediction methods can be developed by considering various operating environments of the Li-ion batteries.



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