Cross Trajectory Gaussian Process Regression Model for Battery Health Prediction

Jianshe Feng; Xiaodong Jia; Haoshu Cai; Feng Zhu; Xiang Li; Jay Lee

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Cross Trajectory Gaussian Process Regression Model for Battery Health Prediction PDF

- ORCID：
Jianshe Feng
✉
- ORCID：
Xiaodong Jia
✉
- ORCID：
Haoshu Cai
✉
- ORCID：
Feng Zhu
✉
- ORCID：
Xiang Li
✉
- ORCID：
Jay Lee
✉

NSF I/UCR Center for Intelligent Maintenance Systems, Department of Mechanical Engineering, University of Cincinnati, Cincinnati, USA

Updated：2021-09-27

DOI：10.35833/MPCE.2019.000142

Abstract

Accurate battery capacity prediction is important to ensure reliable battery operation and reduce the cost. However, the complex nature of battery degradation and the presence of capacity regeneration phenomenon render the prediction task very challenging. To address this problem, this paper proposes a novel and efficient algorithm to predict the battery capacity trajectory in a multi-cell setting. The proposed method is a new variant of Gaussian process regression (GPR) model, and it utilizes similar trajectories in the historical data to enhance the prediction of desired capacity trajectory. More importantly, the proposed method adds no extra computation cost to the standard GPR. To demonstrate the effectiveness of the proposed method, validation tests on two different battery datasets are implemented in the case studies. The prediction results and the computation cost are carefully benchmarked with cutting-edge GPR approaches for battery capacity prediction.

Keywords

Prognostic; lithium-ion battery; Gaussian process regression; state of health

I. Introduction

STATE of health (SoH) prediction is increasingly important in battery prognostics since it is a key enabler of appropriate battery management strategy to avoid catastrophic failure, to enhance battery durability and to optimize the cost [

1]. Also, due to the widespread applications of lithium-ion battery (LiB) in many industrial sectors, the research of SoH prediction holds great academic value and economic impact. In general, SoH is defined as a performance index to describe the degree of battery degradation. In the present study, SoH prediction mainly predicts future battery capacity deterioration.

In past decades, various methods have been proposed for battery SoH prediction. Similarity-based method [

2], fuzzy logic [3], and neural networks [4] are explored in several early studies and the satisfactory prediction results are reported. However, a common shortcoming of these methods is the need for abundant historical data to describe the prediction uncertainties. To address this problem, Gaussian process regression (GPR), particle filters (PFs) and relevance vector machines (RVMs) are discussed recently as non-parametric model methods due to its flexibility to yield uncertainty distributions [5]-[7], which are suitable to obtain the battery SoH prediction and confidence interval. Although many recent studies can effectively describe the prediction uncertainty, most of these methods still predict the SoH in a single-cell setting and fail to characterize the capacity regeneration phenomenon (CRP) very well.

The CRP, which is also known as a self-recharging phenomenon, describes the situation that the battery performance will recover relative to the previous cycle [

8], [9]. The presence of such CRP will interrupt the battery degradation trend and make the prediction task challenging. Reference [10] proposes a PF-based framework for battery prognostics which can simultaneously detect and isolate the CRP within the life-cycle model. Another study employs sparse Bayesian predictive method to model the correlation between the capacity loss and the sample entropy of short voltage sequence, and then to predict the future capacity loss [1]. Although this paper reports the improved results, it is still unable to depict the confidence interval of SoH prediction over the operation cycles. Based on these aforementioned studies, improved prediction accuracies are reported in a more recent study using the multi-task Gaussian process (MTGP) [11]. In their study, the superiority of MTGP for battery prognostics, especially for long-term prediction, is well demonstrated and the improvements in the results are contributed by exploiting the cross-correlations between current SoH trajectory and historical trajectories. Although the results are promising, this method is computationally expensive since it requires to compute the inverse of large kernel matrix, and this renders MTGP less practical in real applications.

To present an efficient solution for battery SoH prediction in a multi-cell setting, this paper aims to simplify the computation cost of MTGP while keeping all its advantages. The proposed method demonstrates a simpler way to exploit the cross-trajectory correlations without adding any computation complexity to the standard GPR model. More importantly, the proposed method can also predict CRP very well. To further make the proposed method practical, a systematic methodology is described considering the important preprocessing steps and the trajectory selection strategy.

The rest of this paper is organized as follows. A review of Gaussian process (GP) for battery prognostics is stated in Section II. Section III highlights the novelty based on previous works. In Section IV, case studies are demonstrated. The conclusion remarks are stated in Section V.

II. Gaussian Process for Battery Health Prediction

GPR is a non-parametric method that can be used to model complex systems and it is preferred in battery prognostics due to its flexibility to provide uncertainty representations [

12]. GPR models the battery degradation as a real process

f (x)

that is characterized by the mean function (MF)

m (x)

and a covariance function (CovF)

k (x, x')

, as described below:

y = f (x) ~ N (m (x), k (x, x'))

(1)

where x and y are the input and output pairs in the training dataset $T = \{x, y\}$ ; $f (x)$ is the underlying regression function; and $m (x)$ is the MF which is often set to be 0. The CovF $k (x, x')$ which describes the similarity between data points, is the key ingredient in GPR since data points with similar input x are likely to have similar target value y [

13]. By choosing CovF properly, GPR can model arbitrarily complex systems. According to [13], a large class of CovFs is available, among which the squared exponential (SE) and Matern (MA) kernel functions (KerFs) are commonly studied in battery prognostics [11], as shown below:

k_{S E} (d) = θ_{1}^{2} e x p (- \frac{d^{2}}{2 θ_{2}^{2}})

(2)

k_{M A} (d) = σ^{2} \frac{2^{1 - ν}}{Γ (ν)} {(\frac{2^{1 - ν}}{ρ})}^{ν} K_{ν} (\frac{\sqrt[]{2 ν}}{ρ} d)

(3)

where $d$ is the Euclidean distance between two indexes and $d = {‖x - x'‖}_{2}$ ; $K_{ν} (\cdot)$ is the modified Bessel function; $Γ (ν)$ is the Gamma function; and $θ_{1}$ , $θ_{2}$ , $ρ$ , $σ$ , $ν$ are the hyperparameters that need to be estimated iteratively.

During the model training, the aim of GPR is to learn a regression model $y = f (x) + ε$ , where ε~N(0, $σ_{n}^{2}$ ) is the random noise, $σ_{n}^{2}$ is the noise variance of the signal. The hyperparameters in the MF and CovF are obtained by minimizing the negative log marginalized likelihood (NLML) in (4) based on its partial derivatives, which can be used in conjunction with a numerical optimization routine such as gradient descent optimization to find optimal hyperparameter settings. More details about partial derivatives of NLML and hyperparameter selection can be found in Appendix A.

\begin{array}{l} L = - l g (p (y | x, θ)) = \\ \frac{1}{2} l g | K + σ_{n}^{2} I | + \frac{1}{2} y^{T} (K + σ_{n}^{2} {I)}^{- 1} y + \frac{n}{2} l g (2 π) \end{array}

(4)

where $p (\cdot)$ is the marginalized likelihood; $K$ is the kernel matrix; $I$ is the unit matrix which shares the same size with $K$ ; and $θ$ is the hyperparameter in likelihood function.

In the model testing phase, the predictive distribution of GPR at input data point $x_{*}$ can be described as:

f_{*} |_{x, y, x_{*}} ~ N ({\bar{f}}_{*}, c o v (f_{*}))

(5)

{\bar{f}}_{*} = m (x_{*}) + K (x, x_{*}) (K (x, x) + σ_{n}^{2} {I)}^{- 1} (y - m (x))

(6)

c o v ({\bar{f}}_{*}) = K (x_{*}, x_{*}) - K (x_{*}, x) {(K (x, x) + σ_{n}^{2} I)}^{- 1} K (x, x_{*})

(7)

where ${\bar{f}}_{*}$ is the mean of prediction output and $c o v ({\bar{f}}_{*})$ can be employed to describe the confidence interval.

For battery SoH prediction, three different GPR models are employed in current literature as shown in Fig. 1. The single input single output (SISO)-GPR in Fig. 1(a) is discussed in [

14] to predict the battery internal resistance and subsequently transfer the predicted values to capacity domain. Though the result is satisfactory, the extrapolation accuracy deteriorates very fast as the prediction horizon increases. To make GPR more useful for long-term prediction, [8] proposes to use a linear or quadratic mean function to improve the prediction accuracy. However, the assumption in MF is too subjective. Another study in [15] proposes a mixture of GP model to predict the battery capacity in a multi-cell setting. This model employs the multiple input single output (MISO)-GPR model in Fig. 1(c). The unknown parameters are updated recursively using a PF and the GPR is mainly employed to initialize the parameter of the PF.

Fig. 1 Three GPR models for battery SoH prediction. (a) SISO-GPR. (b) MIMO-GPR. (c) MISO-GPR.

A more recent study using MTGP model for battery capacity prediction demonstrates significant improvements in the long-term prediction and also in the prediction of CRP [

13]. The MTGP model is originally proposed in [16], [17] and also demonstrates great potential in physiological time series prediction [16]. The MTGP model employs the multiple input multiple output (MIMO)-GPR model in Fig. 1(b) since it can predict multiple outputs (tasks) at the same time. In comparison, models in Fig. 1(a) and (c) are referred to single-task GPR model. The uniqueness of the MTGP is that it models the cross-trajectory correlation by constructing a novel CovF as:

k_{M T G P} (x, x^{'}, l, l^{'}) = k_{c} (l, l^{'}) k_{t} (x, x^{'})

(8)

where $k_{c}$ and $k_{t}$ are the models of the cross-trajectory correlation and the covariance for one trajectory, respectively; $l, l^{'} \in \{1,2, . . ., m\}$ are the indices of tasks and there are $m$ tasks in total; and $x$ and $x^{'}$ are the time indices for the task $l$ and $l^{'}$ , respectively. Based on the Kerf in (8), one can thus construct the covariance matrix for MTGP as:

K_{M T G P} (X, L, θ_{c}, θ_{t}) = K_{C} (L, θ_{c}) \otimes K_{t} (X, θ_{t})

(9)

where $K_{M T G P}$ is a $(\sum_{t} n^{t}) \times (\sum_{t} n^{t})$ matrix, and $n^{t}$ is the number of data points for t^th task; $K_{C}$ and $K_{t}$ are the $m \times m$ and $n^{t} \times n^{t}$ matrices, respectively; $\otimes$ is the Kronecker product; $X = {x_{t}^{j} | j = 1,2, . . ., m, i = 1,2, . . ., n^{t}}$ is the training index of all tasks; $L = {l | l = 1,2, . . ., m}$ is the task index; and $θ_{c}$ and $θ_{t}$ are the hyperparameters for MTGP model.

The improvements of MTGP in battery SoH prediction are mainly contributed by exploiting the cross-correlation among historical trajectories, as stated in [

11]. The shortcoming of MTGP is also obvious since it requires to inverse large kernel matrix in (9) when historical data are abundant. If we assume

n^{j}

is the same for different training trajectories and

n^{j} = n

, the computation complexity for evaluating MTGP is

O (m^{3} n^{3})

compared with

m \times O (n^{3})

for standard GPR [16].

By comparing the three model structures in Fig. 1, it can be found that the SISO-GP is over-simplistic by predicting battery capacity in a single-cell setting. However, the MIMO-GP is over-complex for battery prediction since only one prediction output is desired in most scenarios. Inspired by this idea, this paper aims to propose a novel MISO-GP model to achieve two main purposes: ① to enhance the prediction performance by predicting the battery SoH in a multi-cell setting; ② to simplify the MTGP model while keeping all its advantages for battery SoH prediction.

III. Proposed Methodology

Based on the statement above, the main contributions of this paper are highlighted as follows: ① a novel MISO-GPR model is proposed to predict the battery capacity in a multi-cell setting; ② the prediction accuracy is significantly improved compared with the standard GPR; ③ the computation complexity is significantly reduced compared with the MTGP; ④ a systematic methodology is then proposed to make the proposed GPR model more practical for implementation.

A. Prediction of Trajectory Difference Using GPR

The initial idea of the proposed method is predicting the trajectory difference using GPR instead of predicting the battery capacity directly. By visualizing the capacity trajectories T₁-T₃ in Fig. 2 and the corresponding trajectory differences in Fig. 3, the benefits of predicting trajectory difference can be explained in two aspects: ① trajectory differences tend to have fewer variations comparing with the original trajectories, especially when there are similar trajectories in the historical data, for example, the trajectory T₁ and T₂ in Fig. 2; ② the CRP can be better predicted by utilizing the historical trajectories as references and the long-term prediction accuracy can be enhanced accordingly as shown in Fig. 4.

Fig. 2 National Aeronautics and Space Administration (NASA) battery dataset.

Fig. 3 Prediction of trajectory differences of T₁-T₂ and T₁-T₃.

Fig. 4 Prediction of T₁. (a) Use standard GPR model with $M F = 0$ , $A E B = 11.51$ , $M S E = 4.74 \times 10^{- 3}$ . (b) Predict T₁-T₂ and then use T₂ as reference trajectory and $A E B = 4.46$ , $M S E = 8.77 \times 10^{- 5}$ . (c) Predict T₁-T₃ and then use T₃ as reference trajectory and $A E B = 15.18$ , $M S E = 2.68 \times 10^{- 3}$ .

Figure 3 demonstrates the prediction of trajectory differences T₁-T₂ and T₁-T₃, respectively. The training-testing sample ratio is set to be 6:4 in this case and the MF is set to be 0. It is found that the prediction accuracy for T₁-T₂ is much higher than that for T₁-T₃, and this is mainly because the trajectories T₁ and T₂ in Fig. 2 are similar. For all the prediction results in Fig. 4, the covariance function is set as:

K = k_{S E} + k_{M A 5} + k_{M A 3}

(10)

where $k_{S E}$ is the SE kernel in (2); $k_{M A 5}$ is the MA kernel with $ν = 5$ ; and $k_{M A 3}$ is the MA kernel with $ν = 3$ .

Figure 4 compares the prediction of T₁ using different approaches and different reference trajectories. The prediction results in Fig. 4(b) and (c) are obtained by adding T₂ and T₃ with the predicted trajectory differences in Fig. 4(b). Figure 4(a) predicts the future capacity using standard GPR model in a single-cell setting. The $A E B$ and $M S E$ in Fig. 4 represent the values of area of error bound (AEB) and mean square error (MSE), respectively, which are only based on the testing data. The AEB is calculated as the area of 3-Sigma confidence interval that is shadowed in the figures.

Comparison of the results in Fig. 4 reveals two important observations: ① it is better to predict the battery SoH in a multi-cell setting so that the historical data can be utilized to enhance the prediction result; ② the prediction accuracy can be greatly enhanced, and the CRP can be well predicted if similar trajectory in the historical data can be employed as reference. Although the results in Fig. 4 are promising, there are still two unaddressed problems in this initial study. One is that the trajectory difference prediction model in this section fails to include multiple historical trajectories. Another is that a practical solution is still needed to find the highly similar trajectory from the historical data.

B. Cross Trajectory GP (CTGP)

The trajectory difference prediction model in the previous subsection can be further generalized below to include more historical trajectories.

y = f (x) = \underset{C r o s s - c o r r e l a t i o n t e r m}{\underset{︸}{α_{1} T_{1} + α_{2} T_{2} + . . . + α_{m} T_{m}}} + \underset{R e s i d u a l t e r m}{\underset{︸}{R}}

(11)

where $T_{1}, T_{2}, . . ., T_{m}$ are the reference trajectories that cover the whole life cycle of the battery; and $α_{1}, α_{2}, \dots, α_{m}$ describe the similarity (or correlation) between the target trajectory and the reference trajectories. The cross-correlation term in (11) models the target trajectory as a linear combination of reference trajectories so that the useful information in the historical data can be effectively utilized. In the setting of GPR mode, the cross-correlation term in (11) can be described as MF and the residual term can be modeled by GPR. The generalized model can be described as:

y = f (x) ~ N (α_{1} T_{1}^{'} + α_{2} T_{2}^{'} + . . . + α_{m} T_{m}^{'}, k (x, x^{'}))

(12)

where $T_{1}^{'}, T_{2}^{'}, . . ., T_{m}^{'}$ are the time synchronized reference trajectories. During the model training, the parameters in MF and CovF are estimated by maximizing the NLML in (4). To simplify the discussion, the model proposed in (12) will be mentioned as CTGP in the following context.

The major superiority of CTGP over standard GPR involves predicting battery capacity in a multi-cell setting, so that the long-term prediction can be more accurate than standard GPR. Compared with MTGP, the major advantage of CTGP is the reduced model complexity. The biggest difference between CTGP and MTGP involves how historical data are utilized in the model. In MTGP, the historical information is introduced by constructing a large kernel matrix as in (9). However, this large kernel matrix is avoided in CTGP by using the novel mean function in (12). If the number of samples for different training trajectories is denoted as $n$ , the complexity of evaluating CTGP is $m \times O (n^{3})$ which is the same as standard GPR.

Prediction results of T₁ based on CTGP are demonstrated in Fig. 5. It is observed that CTGP can achieve better prediction accuracy than the results in Fig. 4 and the prediction uncertainty is greatly reduced. The results in Fig. 5(a)-(c) demonstrate that the CTGP model can predict the future capacity accurately if similar trajectories in historical data are included in the model. Comparison of the results in Fig. 5(b) and (c) shows that the prediction based on less similar historical trajectory using CTGP can still give good accuracy but with higher uncertainty.

Fig. 5 Prediction of T₁ based on CTGP using mean function. (a) $m (x) = α_{2} T_{2} + α_{3} T_{3}$ , $A E B = 2.51$ , $M S E = 8.62 \times 10^{- 6}$ . (b) $m (x) = α_{2} T_{2}$ , $A E B = 2.67$ , $M S E = 1.44 \times 10^{- 5}$ . (c) $m (x) = α_{3} T_{3}$ , $A E B = 7.60$ , $M S E = 2.21 \times 10^{- 5}$ .

To better demonstrate the improvements made by the proposed CTGP model, the prediction results of MTGP in the same experimental setting are benchmarked in Fig. 6, where the mean function is set to be 0. As shown in Fig. 6, although the MTGP predicts the CRP very well, the prediction accuracy and uncertainty is less promising than that of the CTGP models in Fig. 5. The result in Fig. 6(b) indicates that the prediction model fails to converge well in the training phase. Comparison of the results through Figs. 4-6 shows that using a less similar historical trajectory as reference might result in larger prediction uncertainty. Based on these observations, we find that it is necessary to preselect the historical trajectories to further enhance the prediction accuracy.

Fig. 6 Prediction of capacity trajectory T₁ based on MTGP. (a) Use $T_{2}, T_{3}$ as reference trajectory and $A E B = 5.46$ , $M S E = 1.37 \times 10^{- 4}$ . (b) Use $T_{2}$ as reference trajectory and $A E B = 3.84$ , $M S E = 9.32 \times 10^{- 4}$ . (c) Use $T_{3}$ as reference trajectory and $A E B = 9.86$ , $M S E = 2.36 \times 10^{- 4}$ .

The limitation of CTGP model in (12) involves its inability to extrapolate to the time points where reference trajectory is unavailable. To solve this problem, we propose to modify (12) as:

y = f (x) ~ N (α_{1} {\bar{T}}_{1} + α_{2} {\bar{T}}_{2} + . . . + α_{m} {\bar{T}}_{m}, k (x, x^{'}))

(13)

where ${\bar{T}}_{i}$ is the i^th extrapolated reference trajectory based on original trajectory $T_{i}$ . ${\bar{T}}_{i}$ can be obtained by regression model and it consists of two major parts: the interpolation part and the extrapolation part. The time indices of the interpolation part are within the time horizon of $T_{i}$ while those for extrapolation are beyond time horizon of $T_{i}$ . Normally, the interpolation part resembles the reference trajectory, but extrapolation part may experience larger uncertainty.

Figure 7(a) demonstrates the capability of CTGP to fuse the extrapolation of multiple historical trajectories. It is found that the CRP can be well predicted only when using interpolated historical trajectories as references and the prediction result for this part is almost the same as that in Fig. 5(a). In comparison, Fig. 7(b) shows the best prediction result using standard GPR and the exponential function below is employed as MF:

E M F (x) = a + b e x p (c x + d)

(14)

Fig. 7 Prediction of capacity trajectory T₁. (a) CTGP in (13) uses $T_{2}, T_{3}$ as reference and the reference trajectories are pre-trained based on standard GPR with EMF. (b) Use standard GPR model with EMF. (c) Use MTGP with EMF.

where a, b, c, d are the unknown parameters in the exponential mean function (EMF). Figure 7(c) shows the extrapolation of MTGP with EMF. Comparison of these results indicates that the prediction in Fig. 7(a) is quite reasonable and the CTGP model in (13) can be effectively employed to predict battery SoH with extrapolated references.

C. Systematic Methodology

An overview of the proposed methodology is demonstrated in Fig. 8. The first key step in the flow chart is the preparation of the reference trajectories. Due to the requirement of GPR, the time indices for the reference trajectory and the target trajectory must be synchronized. This can be achieved either by interpolating the reference trajectories or by extrapolating based on regression models. When the extrapolation of the reference trajectory is needed, standard GPR is recommended. If not, simple interpolation or some off-the-shelf sampling method [

18], [19] can meet the requirement. In the model training phase, trajectory selection is strongly recommended to reduce the uncertainty of the prediction. Therefore, the objective of this step is to locate similar trajectories in the historical data. There are two key steps in the trajectory selection as described in Fig. 9, where “Reference” represents the referenced trajectories; “Target” represents the target trajectory which needs the prediction of further behaviors;

x_{1}, x_{2}, . . ., x_{k}

are the aligned time indices for all trajectories. For each trajectory, the color goes from light to dark red, which represents the capacity degradation process. The first key step is to rank the historical trajectories based on similarity, and the trajectory similarity can be assessed by Euclidean distance between the current trajectory and each historical trajectory. The Euclidean distance can be calculated based on the training data as in Fig. 9. After ranking the trajectories by similarity, a forward search can be utilized to obtain the best combination of reference trajectories. As described in Fig. 9, the validation error can be calculated by training a GPR model based on the validation train data and then predicting the validation test data. The validation error equals to the prediction error, i.e., MSE or root mean square error (RMSE), of the portion of validation test data with available ground truth.

Fig. 8 Overview of proposed methodology.

Fig. 9 Overview of trajectory selection strategy.

IV. Case Study

A. Case 1: NASA Battery Dataset

The LiB data under study are provided by NASA Prognostics Center of Excellence (CoE). In the dataset, the LiB is charged and discharged at different temperatures, and the degradation of capacity is assessed by measuring the impedance. The details of dataset and experiments are available in [

20]. There are 34 experiments in total performed in the dataset, but many of them only have very limited samples. As most of the related works [8], [11], batteries 5, 6, and 7 are chosen in this paper since they have sufficient data samples for prognostics and they are marked as T₁, T₂, T₃ in Fig. 2, respectively.

As presented in Fig. 9, it is essential to select the similar trajectories before the model training. Take T₂ as an example, in the pre-processing stage, the battery with very few samples are ruled out. The remaining batteries and their capacity degradation curves are shown in Fig. 10. The similarity between T₂ and reference trajectories are ranked in Fig. 11. In Fig. 11(a), the Euclidean distance between trajectories are employed to evaluate the similarity. Therefore, a smaller value means better similarity. Prior to the similarity ranking, necessary interpolation or extrapolation steps are taken for synchronizing the time indices.

Fig. 10 Capacity trajectories with sufficient samples in Case 1.

Fig. 11 Similarity between T₂ and reference trajectories. (a) Ranking of trajectory similarity. (b) Forward search for optimal reference combination in Case 1.

Based on the similarity ranking, the forward search is then implemented to select the optimal combination of reference trajectories. The results in Fig. 11(b) shows that the combination of the first 2 trajectories gives the best validation error. To select the similar trajectories in the past, the training-testing sample ratio for validation is set to be 6:4 in Fig. 9, so that the validation error in Fig. 11(b) can be evaluated. Similar phenomena can be found for T₁ and T₃, that is, for T₁, T₂, and T₃, the combination of rest 2 battery trajectories gives the most optimal reference.

The RMSE tabulated in Table I indicates that the proposed CTGP method is comparable to the MTGP in terms of prediction accuracy. In Table I, the best RMSE under each test condition is marked in bold character. In this paper, five different GPR models are benchmarked as in Table I, where the 0MF means the MF always equals to 0 and the $M T G P + E M F$ denotes the MF is set to be the EMF in (14). The standard GPR with EMF is not listed because the prediction results are not very stable under some test conditions. Besides standard GPR, MTGP with different mean functions, sparse Gaussian process regression (SGPR) model is also benchmarked. As a recent variant GP model, SGPT shows its capability to reduce the computational complexity while keeping good performances in battery capacity prediction [

1], [21]. By comparing the results, it is found that CTGP is significantly better than the standard GPR and SGPR, which implies the proposed method can be effectively employed for battery prognostics. In the meantime, the results of CTGP are similar to the MTGP, which indicates useful information in the reference trajectories is effectively utilized in the proposed method.

Table I Benchmark of RMSE for Different GPR Models

Target	Reference	Model setting	RMSE under different training-testing split ratios
Target	Reference	Model setting	5:5	6:4	7:3	8:2	9:1
T₁	T₂, T₃	Standard $G P R + 0 M F$	0.1228	0.0127	0.0089	0.0068	0.0048
		CTGP	0.0072	0.0030	0.0029	0.0026	0.0016
		$M T G P + 0 M F$	0.0064	0.0117	0.0019	0.0026	0.0028
		$M T G P + E M F$	0.0074	0.0082	0.0033	0.0339	0.0019
		SGPR	0.1442	0.0255	0.0231	0.0152	0.0126
T₂	T₁, T₃	Standard $G P R + 0 M F$	0.1439	0.0509	0.0438	0.0117	0.0061
		CTGP	0.0166	0.0088	0.0056	0.0041	0.0020
		$M T G P + 0 M F$	0.0137	0.0068	0.0056	0.0025	0.0021
		$M T G P + E M F$	0.0205	0.0031	0.0064	0.0031	0.0018
		SGPR	0.1592	0.0172	0.0329	0.0210	0.0163
T₃	T₁, T₂	Standard $G P R + 0 M F$	0.1370	0.0615	0.0392	0.0245	0.0269
		CTGP	0.0361	0.0077	0.0142	0.0145	0.0046
		$M T G P + 0 M F$	0.0159	0.0197	0.0425	0.0152	0.0050
		$M T G P + E M F$	0.1162	0.0426	0.0125	0.0081	0.0076
		SGPR	0.1250	0.0671	0.0258	0.0212	0.0192

Although the prediction accuracies are similar, the proposed CTGP is much more efficient than MTGP when it comes to the computation cost, as in Fig. 12. Based on Fig. 12, two important things can be observed: ① the average iteration time (AIT) for CTGP is similar to standard GPR and SGPR, and it is almost 10 times smaller than MTGP; ② the AIT for MTGP increases very fast as more data samples are utilized for training purpose.

Fig. 12 Benchmark of AIT using different GP models in Case 1.

B. Case 2: NASA Randomized Battery Usage Data

The battery usage data in this case study are also provided by NASA Prognostics CoE. In this dataset, the capacity degradation is charged and discharged using a randomized current at different temperatures to test the battery degradation behavior under a more practical battery usage condition. This makes SoH prediction very challenging since the battery degradation is randomized [

11]. In the experiment, 7 groups of experimental conditions are considered, and 4 LiBs are tested under each condition [22]. Therefore, there are 28 different degradation trajectories in this dataset. In the present case study, data groups 1, 2, 3 and 5 are chosen since they have sufficient data samples for modeling and prediction.

In Fig. 13, the name of each trajectory is started with “RW” in the dataset, which represents “randomized walk”. RW10 is chosen as the prediction target and its similarity with reference trajectories are ranked in Fig. 13(a). To justify the similarity ranking results, the top-ranking trajectories are plotted in Fig. 14. Based on the similarity ranking, the forward search is then implemented to select the optimal combination of reference trajectories. The results in Fig. 13(b) show that the combination of first 7 trajectories gives the best prediction. Figure 15 compares CTGP and MTGP by using different combinations of reference trajectories, and the training-testing sample ratio is 7:3 for all the results.

Fig. 13 Ranking of trajectory similarity and forward search. (a) Ranking of trajectory similarity. (b) Forward search for optimal reference combination in Case 2.

Fig. 14 Top-ranking capacity trajectories that are similar to RW10 in Case 2.

Fig. 15 Prediction results of CTGP and MTGP based on different combinations of reference trajectories in Case 2. (a) CTGP with RW9 as reference trajectory. (b) MTGP with RW9 as reference trajectory. (c) CTGP with RW11 as reference trajectory. (d) MTGP with RW11 as reference trajectory. (e) CTGP with RW9, RW11, RW8, RW2, RW16, RW15, RW7 as reference trajectories. (f) MTGP with RW9, RW11, RW8, RW2, RW16, RW15, RW7 as reference trajectories.

It is observed that the prediction accuracies of CTGP are slightly better than MTGP under all the test conditions and the prediction uncertainties of CTGP are smaller as well.

More importantly, the main contribution of CTGP is justified in Fig. 16 by comparing its AIT with standard GPR, SGPR, and MTGP. It is well demonstrated that the AIT of CTGP is very similar to standard GPR and SGPR, and the value does not rise drastically as the number of references increases.

Fig. 16 Benchmark of AIT in Case 2.

Table II presents further comparisons of standard GPR, CTGP, MTGP, and SGPR. Four battery cells from 4 groups are chosen as capacity prediction target, and the rest battery capacity curves from the same group are used as reference trajectories. By comparing the results in Table II, we can draw a similar conclusion in Case 1: ① CTGP has a much better performance than the standard GPR and SGPR in battery prognostics; ② the prediction accuracies of CTGP and MTGP are quite similar, but the efficiency of CTGP is significantly higher than the MTGP, especially when abundant historical data samples are available. Therefore, the superiority of CTGP over standard GPR, SGPR, and MTGP is justified.

Table II Benchmark of RMSE for Different GPR Models

Target	Reference	Model setting	RMSE under different training-testing sample ratios
Target	Reference	Model setting	5:5	6:4	7:3	8:2	9:1
RW9	RW10, RW11, RW12	Standard $G P R + 0 M F$	0.0961	0.0578	0.0564	0.0670	0.0572
		CTGP	0.0286	0.0183	0.0127	0.0118	0.0078
		MTGP	0.0279	0.0199	0.0183	0.0160	0.0044
		SGPR	0.0933	0.0899	0.0573	0.0513	0.0504
RW3	RW4, RW5, RW6	Standard $G P R + 0 M F$	0.0240	0.0166	0.0268	0.0417	0.0264
		CTGP	0.0137	0.0082	0.0126	0.0163	0.0121
		MTGP	0.0609	0.0159	0.0122	0.0222	0.0167
		SGPR	0.0462	0.0182	0.0264	0.0559	0.0241
RW17	RW18, RW19, RW20	Standard $G P R + 0 M F$	0.0366	0.0097	0.0147	0.0402	0.0220
		CTGP	0.0075	0.0104	0.0070	0.0181	0.0181
		MTGP	0.0090	0.0068	0.0077	0.0185	0.0144
		SGPR	0.0273	0.0296	0.0159	0.0451	0.0257
RW1	RW2, RW7, RW8	Standard $G P R + 0 M F$	0.0477	0.0235	0.0267	0.0327	0.0389
		CTGP	0.0146	0.0252	0.0376	0.0191	0.0225
		MTGP	0.0316	0.0234	0.0154	0.0200	0.0081
		SGPR	0.0952	0.0266	0.0286	0.0323	0.0451

V. Conclusion

In this paper, a novel and efficient GPR model, i.e., CTGP, is proposed to predict battery capacity degradation in a multi-cell setting, and the effectiveness of the proposed method is demonstrated on two different battery datasets. The uniqueness of the proposed method involves contributing an efficient way to integrate historical information to enhance the prediction outcome. Benchmarking with the standard GPR, the prediction accuracy of CTGP is significantly improved. In comparison with MTGP, although both methods can predict the capacity regeneration phenomenon very well, the proposed CTGP is much more efficient than MTGP as shown in the case studies. Therefore, we conclude that the proposed method can be an effective and efficient tool to predict battery capacity in multi-cell setting.

Appendix

APPENDIX A

A. Model Specification

\begin{array}{l} y = f (x) ~ N (α_{1} T_{1} + α_{2} T_{2} + . . . + α_{m} T_{m}, k_{S E} (x, x') + \\ k_{M A 5} (x, x') + k_{M A 3} (x, x') + σ_{n}^{2}) = N (m (x), k (x, x') + σ_{n}^{2}) \end{array}

(A1)

B. Hyperparameter Estimation

The NLML of proposed Gaussian process is given by:

\begin{matrix} L = - l g (p (y | x, θ)) = \frac{1}{2} l g | K + σ_{n}^{2} I | + \frac{1}{2} (y - m {(x))}^{T} \cdot \\ (K + σ_{n}^{2} {I)}^{- 1} (y - m (x)) + \frac{n}{2} l g (2 π) \end{matrix}

(A2)

The hyperparameter selection is obtained by minimizing the NLML based on its partial derivatives which are evaluated:

\{\begin{array}{l} \frac{\partial L}{\partial θ_{m f}} = (y - m {(x))}^{T} (K + σ_{n}^{2} {I)}^{- 1} \frac{\partial m (x)}{\partial θ_{m f}} \\ \frac{\partial L}{\partial θ_{c f}} = \frac{1}{2} t r a c e ((K + σ_{n}^{2} {I)}^{- 1} \frac{\partial (K + σ_{n}^{2} I)}{\partial θ_{c f}}) - \frac{1}{2} (y - m {(x))}^{T} \cdot \\ (K + σ_{n}^{2} {I)}^{- 1} \frac{\partial (K + σ_{n}^{2} I)}{\partial θ_{c f}} (K + σ_{n}^{2} {I)}^{- 1} (y - m (x)) \end{array}

(A3)

where $θ_{m f} = (α_{1}, α_{2}, . . ., α_{m})$ and $θ_{c f}$ are the hyperparameters of the mean and covariance functions, respectively. A gradient based optimizer developed in [

13], [23] is adopted to minimize the NLML.

C. Posterior Distribution

The joint distribution of both training stage and testing stage can be written as:

[\begin{matrix} y \\ f_{*} \end{matrix}] ~ N ([\begin{matrix} m (x) \\ m (x_{*}) \end{matrix}], [\begin{matrix} K (x, x) + σ_{n}^{2} I & K (x, x_{*}) \\ K (x_{*}, x) & K (x_{*}, x_{*}) \end{matrix}])

(A4)

where $m (x_{*}) = α_{q} T_{*}$ is the testing mean vector, and $T_{*}$ is the reference trajectories at prediction time horizon.

Based on (A4), the posterior distribution of the proposed method can be obtained as conditional distribution of $f_{*}$ given $y$ , which is as follows:

f_{*} | x, y, x_{*} ~ N ({\bar{f}}_{*}, c o v (f_{*}))

(A5)

{\bar{f}}_{*} = \sum_{q = 1}^{m} α_{q} {T_{*}}_{q} + K (x, x_{*}) (K (x, x) + σ_{n}^{2} {I)}^{- 1} (y - \sum_{q = 1}^{m} α_{q} T_{q})

(A6)

c o v ({\bar{f}}_{*}) = K (x_{*}, x_{*}) - K (x_{*}, x) {(K (x, x) + σ_{n}^{2} I)}^{- 1} K (x, x_{*})

(A7)

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