Fractional Poisson Process for Estimation of Capacity Degradation in Li-Ion Batteries by Walk Sequences

Abstract

Each charging/discharging cycle leads to a gradual decrease in the battery’s capacity. The degradation of capacity in lithium-ion batteries represents a non-monotonous process with random jumps. Earlier studies claimed that the instantaneous degradation value of a lithium-ion battery is influenced by the historical dataset with long-range dependence. The existing methods ignore large jumps and long-range dependences in degradation processes. In order to capture long-range-dependent behavior with random jumps, we refer to the fractional Poisson process. We also outline the relationship between the long-range correlation and the Hurst index. The connection between random jumps in capacitance and long-range dependence of the fractional Poisson process is proven. In order to construct the fractional Poisson predictive model, we included fractional Brownian motion as the diffusion term and the fractional Poisson process as the jump term. The proposed approach is implemented on NASA’s dataset for Li-ion battery degradation. We believe that the error analysis for the fractional Poisson process is advantageous compared with that of the fractional Brownian motion, the fractional Levy stable motion, the Wiener model, and the long short-term memory model.

1. Introduction

The capacity of lithium-ion batteries (LIBs) has a degradation trend with charging and discharge cycles. This results in a reduction in the end of life (EOL). A recommended practice is to replace the battery when the capacitance drops below 80% of its original capacity [1]. Widespread applications of LIBs demand accurate prediction of the RUL of LIBs.

Due to potential economic benefits, predictive algorithms for estimation of the remaining useful life (RUL) of LIBs have recently garnered attention from the scientific community. For example, Li J. et al. [2] used the concepts of mathematical equations for electrochemical theory and applied finite element analysis and numerical methods to solve equations with the goal of building a predictive model. The study by Cho K.et al. [3] offers a simulation of the battery load response under specific temperature conditions through a coupling model, which is suitable for the temperature range of 25 °C to 40 °C in daily life. A classical Rint model represents a simplified equivalent circuit model [4], and it is efficient for certain applications [5]. The model consists of an ideal voltage source representing the open-circuit voltage and a resistor in series, symbolizing the internal resistance of the battery. Through a large amount of historical degradation data, based on the capacity transfer relationship of the battery at adjacent time points, and applying Kulun’s law, the capacity loss of the battery was modeled, and then a polynomial and exponential decay model was derived [6]. Most empirical models rely on statistical random filtering technology to track battery decay trends and optimize model parameters, such as particle filtering or Kalman filtering [7]. The prediction accuracy highly depends on the model construction. Focusing on the research of data-driven methods, Chen et al. [8] combined variable modal decomposition and GPR, first decomposing the capacity degradation data of lithium batteries into relatively stable components and then conducting RUL prediction based on these components.

Li et al. [9] proposed a GPR prediction model with an automatic selection mechanism of the kernel function. Jia et al. [10] mined key features from charge and discharge data and then used the GPR model to estimate the SOH and RUL, and the effectiveness of their method was confirmed. Peikun et al. [11] combined gray correlation analysis and the GPR method of multi-island genetic algorithm to optimize the prediction accuracy of the SOH. These methods need high standards for data collection.

Xiao et al. [12] proposed a prediction technology, combining RVM and a three-parameter capacity decay model; Feng et al. [13] studied a health indicator extracted from surface temperature changes when the battery was discharged, and Wang et al. [14] extracted a new health indicator from the current curve during constant-voltage charging of the battery. The RVM model is an emerging data-driven method, but its inherent high sparsity means that direct use of the RVM for prediction may lead to instability of the result [15].

The ability of recurrent neural networks (RNNs) to handle time series data is well studied for predicting the RUL of batteries. The authors of study [8] employed an RNN to extract key features of the degradation data through empirical modal decomposition and gray relationship analysis.

The current methods only predict the volatility of the degradation trend. The fPp process is capable of catching and simulating irregular jumps due to unexpected events, e.g., short circuits or temperature abnormalities.

The above methods only predict the undulatory property of the degradation trend, without consideration of the long-range dependence of the lithium battery degradation process and the irregular jumps caused by capacity regeneration due to overcharge, short circuit, or abnormal temperature emergencies. This paper proposes that the fPp prediction model can capture the long-range dependence property and random jumps.

Our article is organized as follows: In Section 2 we discuss long-range dependence and Poisson distribution. Characteristics of the fractional Poisson process (fPp) are discussed in Section 3. In Section 4 we use the fPp to construct a stochastic differential equation with adaptive jump intensity. Section 5 is Parameter Estimation; Section 6 is a case study, and Section 7 is the conclusion.

2. Long-Range Dependence and Poisson Distribution

2.1. Long-Range Dependence

An autocorrelation function $R\left(\tau \right)=0$ measures the correlation between a time series $\left\{X\left(t\right),0<t<\infty \right\}$ and a lagged version of itself. In other words, the function quantifies how similar a series is to a time-shifted version of itself, revealing patterns and dependencies in sequential data (see Equation (1)):

$${\displaystyle {\int }_0^{\infty }R\left(\tau \right)d\tau =\infty },$$

(1)

The autocorrelation function of the time series has a gradual decay, indicating that, despite a lengthy latency, there is a substantial connection between sequence values. It is clear that the decay of the autocorrelation function is expressed as the power law tail; see Equation (2):

$$\left\{\begin{array}{l}R\left(\tau \right)\sim c{\left|\tau \right|}^{-\eta }\\ \eta =2-2H\end{array}\right.,$$

(2)

where $0<\eta <1$ is the power law exponent, and $c$ is a constant; $H\in \left(0,1\right)$ is the Hurst exponent—a measure used in time series analysis to quantify the LRD or persistence of a process. For $H=0.5$, the signal resembles Brownian motion (i.e., a Wiener process), i.e., a random walk whose increments stem from an uncorrelated Gaussian white noise process. When $H\ne 0.5$, the autocorrelation function cannot be integrated or summed, revealing that the random sequence exhibits correlation. For $0<H<0.5$, the series has a short correlation, and the random process exhibits LRD characteristics for $0.5<H<1$.

2.2. Estimation of the Hurst Exponent

There exist a number of estimation algorithms of the Hurst exponent. For our one-dimensional data, we will use rescaled range (R/S) analysis. The trend-corrected method is based on the statistical self-similarity in the signal. Let us discretize a random sequence $\{X_1,X_2,\dots ,X_n\}$ into $h$ segments of length $a$. The mean value of each segment is calculated as follows (3):

$${〈X〉}_a={\displaystyle \frac{1}{a}}{\displaystyle \sum _{i=1}^h{X}_i},$$

(3)

Calculation of the cumulative deviation, given by Equation (4),

$$X\left(i,a\right)={\displaystyle \sum _{i=1}^h\left(X_i-{〈X〉}_a\right)},$$

(4)

enables us to create a range series of extreme difference values (Equation (5)):

$$R\left(a\right)=\underset{1\le i\le a}{\max }X\left(i,a\right)-\underset{1\le i\le a}{\min }X\left(i,a\right)$$

(5)

Then the standard deviation reads as follows:

$$S\left(a\right)=\sqrt{{\displaystyle \frac{1}{a}}}{\displaystyle \sum _{i=1}^h{\left(X_i-{〈X〉}_a\right)}^2}$$

(6)

As a result, the rescaled range series is calculated:

$${\displaystyle \frac{R}{S}}\left(a\right)={\displaystyle \frac{1}{h}}\times {\displaystyle \frac{R\left(a\right)}{S\left(a\right)}}$$

(7)

The above steps are repeated with the premise that Formula (6) is satisfied. Then we take different $a$ and calculate $h\times a=n$. In the following, the logarithm of the re-scaling extreme difference ${\displaystyle \frac{R}{S}}\left(a\right)$ fits the straight line using the least squares method (see Equation (8)):

$$\log {\displaystyle \frac{R}{S}}\left(a\right)=\log \left(c\right)+H\times \log a+\omega$$

(8)

The slope ratio is the desired $H$; see Figure 1.

2.3. Sudden-Jump Characteristics of Poisson Distribution

The Poisson distribution is used to predict a dependent variable that consists of “count data” given one or more independent variables. For any time length $t$, the probability of $n$ events occurring obeys the following distribution (9):

$$P(N(t)=n)={\displaystyle \frac{{(\lambda t)}^n{e}^{-\lambda t}}{n!}}$$

(9)

where $\lambda$ is the rate of the process (the number of events per unit interval), and it is recorded as $N_t\sim Poisson\left(\lambda t\right)$.

The Poisson regression is used to analyze count data, i.e., to answer questions such as what factors can predict the frequency of an event. As shown Figure 2, when $\lambda =3$, the PDF of different sparse events occurring is $k$.

2.4. Characteristics of a Random Walk

A Poisson process $\widetilde{N_t}$, defining $\{Q_n^k,k=0,\dots ,n\}$ as a random walk approximation of $\widetilde{N_t}$, is described as in Equation (10):

$$Q_0^n=0;Q_k^n={\displaystyle \sum _{i=1}^k{\eta }_i^n}$$

(10)

where ${\eta }_1^n,\dots ,{\eta }_n^n$ are random variables with an independent and homogeneous distribution. For $\forall k$, there is the probability of Formula (11):

$$P({\eta }_k^n=k_n-1)=1-P({\eta }_k^n=k_n)=k_n$$

(11)

where $k_n=e^{{\displaystyle \frac{-\lambda }{n}}}$. Let $Q_t^n={\displaystyle \sum _{i=1}^{⌊nt⌋}{\eta }_i^n}$; $Q_t$ is the Poisson random walk process; see Figure 3. If the rate of the process increases, the random walk Q_t becomes irregular and unpredictable.

3. Definition and Characteristics of the Fractional Poisson Process

According to Wang et al. [16], the fPp is defined as a class of non-Gaussian processes with stationary increments, and the definition of $N_t^H=\{N_t^H,t\ge 0\}$ is given as follows (12):

$$N_t^H(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(H-\frac{1}{2})}}{\displaystyle {\int }_0^t{u}^{\frac{1}{2}-H}}\left({\displaystyle {\int }_u^t{\tau }^{H-\frac{1}{2}}}{\left(\tau -u\right)}^{H-{\displaystyle \frac{1}{2}}}d\tau \right)dq(u)$$

(12)

where $q(u)={\displaystyle \frac{N(u)}{\sqrt{\lambda }}}-\sqrt{\lambda }u$.

According to Equation (12), when $H\in \left(0,0.5\right)$, $N_t^H$ represents short-term correlation, when $H=0.5$ is a typical Poisson process. When $H\in \left(0.5,1\right)$, fPp has LRD properties. Obviously, the Poisson process is actually only a special case of fPp. The curve of the Hurst exponent with time log is shown in Figure 4, and the Hurst exponent ranges from 0.5 to 0.1. The higher the surface curve, the stronger the long-range dependence.

4. Fractional Poisson Process Predictive Model

For any random sequence $\{X_t,t>0\}$, using the Black–Scholes formula, we can obtain a random differential equation based on Brownian motion, such as Equation (13):

$$d{X}_t=\mu X_{t}dt+\delta X_{t}d{B}_t,$$

(13)

where B_t is a standard Brownian motion (we will replace it with fBm); μ is the drift rate of B_t; and δ is the diffusion coefficient. As a result, Equation (13) with LRD characteristics accurately simulates the remaining useful life decay process:

$$d{X}_t=\mu X_{t}dt+\delta X_{t}d{B}_{t,H}+\eta X_{t-}d{N}_{t,H}$$

(14)

where $B_{t,H}$ is the fBm process; $N_{t,H}$ is the fPp process; and $\eta$ is the jump amplitude. Then we discretize Equation (14) and obtain the following expression:

$$\mathsf{\Delta }{X}_t=\mu X_t\mathsf{\Delta }t+\delta X_t\mathsf{\Delta }{B}_{t,H}+\eta X_t\mathsf{\Delta }{N}_{t,H}$$

(15)

the fPp difference prediction degradation model is deduced, as shown in Equation (16):

$$X_{t+1}=X_t+\mu X_t\mathsf{\Delta }t+\delta X_t{w}_t{\left(\mathsf{\Delta }t\right)}^H+\eta X_t{q}_t{\left(\mathsf{\Delta }t\right)}^H.$$

(16)

Let $\mu =2.7$, $\delta =0.3$, $H=0.85$, $\mathsf{\Delta }t=2$, $\lambda =0.05$, $q_1=0.0453$, $q_2=0.9752$; substituting $X_t=1$ into Equation (16), we get a random sequence $\{X_t,t>0\}$, which is shown in Figure 5.

5. Parameter Estimation

5.1. Estimation of Drift and Diffusion Coefficients

The maximum likelihood method was applied to estimate the parameters of the Equation (16). Assume that we have a sample $\{X_1,X_2,\dots ,X_n\}$. The corresponding joint probability density reads as follows [17]:

$$L(\mu ,{\sigma }^2\mid X)={\displaystyle \prod _{i=1}^n\frac{1}{\sqrt{2\pi {\sigma }^2}}}\exp \left(-{\displaystyle \frac{{\left(X_i-\mu \right)}^2}{2{\sigma }^2}}\right)$$

(17)

Applying the logarithm to both sides of Equation (17), we obtain the following:

$$l\left(\mu ,{\sigma }^2\left|X\right.\right)=-{\displaystyle \frac{n}{2}}\log \left(2\pi \right)-{\displaystyle \frac{n}{2}}\log \left({\sigma }^2\right)-{\displaystyle \frac{1}{2{\sigma }^2}}{\displaystyle \sum _{i=1}^n{\left(X_i-\mu \right)}^2}$$

(18)

Here, l is small, while in (17) L is used.

Minimization of the partial derivatives with respect to the drift rate $\mu$ and ${\sigma }^2$ gives us the following identities:

$${\displaystyle \frac{\partial l}{\partial \mu }}={\displaystyle \frac{1}{{\sigma }^2}}{\displaystyle \sum _{i=1}^n\left(X_i-\mu \right)}=0$$

(19)

$${\displaystyle \frac{\partial l}{\partial {\sigma }^2}}=-{\displaystyle \frac{n}{2{\sigma }^2}}+{\displaystyle \frac{1}{2{\left({\sigma }^2\right)}^2}}{\displaystyle \sum _{i=1}^n{\left(X_i-\mu \right)}^2}=0$$

(20)

where

$$\widehat{\mu }={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{X}_i},{\widehat{\sigma }}^2={\displaystyle \frac{1}{n-1}}{\displaystyle \sum _{i=1}^n{\left(X_i-\mu \right)}^2}$$

(21)

5.2. Estimation of the Rate of the Process

In our further considerations, we calculate the difference for $\{X_1,X_2,\dots ,X_n\}$, shown as Equation (22):

$$\mathsf{\Delta }x=x_{i+1}-x_i$$

(22)

where $x_i$ is the capacitance of the battery after the i-th cycle of charge–discharge. Let the 95th percentile be the jump threshold, shown as Equation (23):

$$p_{95}=prctile\left(difference,95\right)$$

(23)

Capacitance jumps are recorded as follows:

$$N_{shocks}=\left|\left\{\mathsf{\Delta }{x}_i:\mathsf{\Delta }{x}_i>p_{95}\right\}\right|.$$

(24)

Recall that the probability of a discrete number of events N occurring within a specific interval T is described by the following Poisson distribution:

$$P\left(N=n\right)={\displaystyle \frac{e^{-\lambda T}{\left(\lambda T\right)}^n}{n!}}$$

(25)

where $\lambda$ is the rate of the process. Taking natural logarithms on both sides of (25), we obtain the log-likelihood function:

$$l\left(\lambda \right)={\displaystyle \sum _{i=1}^N\log \left(\frac{e^{-\lambda T}{\left(\lambda T\right)}^{n_i}}{n_i!}\right)}$$

(26)

By using the properties of logarithm, Equation (26) can be expanded as follows:

$$l\left(\lambda \right)=-\lambda NT+\left({\displaystyle \sum _{i=1}^N{n}_i}\right)\log \left(\lambda \right)+\log \left(T\right){\displaystyle \sum _{i=1}^N{n}_i}-{\displaystyle \sum _{i=1}^N\log \left(n_i!\right)}$$

(27)

Then we obtain the following:

$$\widehat{\lambda }={\displaystyle \frac{{\displaystyle \sum _{i=1}^N\mathsf{\Delta }{x}_{i}I\left(\mathsf{\Delta }{x}_i>p_{95}\right)}}{NT}}$$

(28)

where $I()$ represents the exponent function for $\mathsf{\Delta }{x}_i\le p_{95}$ and is equal to 1 and 0 elsewhere.

6. Case Study

This experiment used the RW12 degradation walk dataset of four 18,650 lithium batteries from NASA [18], which operated continuously at charge and discharge currents between −4.5 A and 4.5 A. This type of charging and discharging operation is called a random walking (RW) operation. Each load period lasts for 5 min, and after 1500 cycles (approximately 5 days), a series of reference charge and discharge cycles are performed to provide a reference standard for the health of the battery. The voltage, current, and temperature changes of the lithium battery during the charge and discharge cycle are shown in Figure 6.

The capacity of the RW12 battery is degraded as shown in Figure 7. The battery fails when the capacity EOL is set to 1.4 Ah in this article. Therefore, the RW12 battery fails after the 56th cycle.

By degrading the data difference of RW12, the jump situation as shown in Figure 8a is obtained. Four points exceed the 95% threshold, so there is a situation that the fBm model cannot predict. As shown in Figure 8b, the large jump point has both capacity regeneration and significant decline in capacity, which is suitable for RUL prediction using the fPp degradation model.

The degradation data of battery RW12 meet the LRD property. In this paper, the prediction starting point is selected as the 30th cycle, and the prediction starting point is selected at an interval of every two cycles. The results of parameter estimation are shown in

Table 1. Parameter estimation results of RW12.

Prediction Starting Point	$\mathit{H}$	$\mathit{\lambda }$	95 Percentile	$\mathit{\mu }$	$\mathit{\delta }$	${\mathit{q}}_1$	${\mathit{q}}_2$
30	0.6257	0.0345	0.0531	−0.0154	0.0385	−2.2254	0.2007
32	0.6230	0.0645	0.0477	−0.0145	0.0410	−1.8797	0.0000
34	0.6167	0.0606	0.0430	−0.0139	0.0420	−1.8619	0.0342
36	0.6321	0.0571	0.0382	−0.0134	0.0424	−1.8504	0.0000
38	0.6446	0.0541	0.0335	−0.0132	0.0418	−1.8450	0.0067
40	0.6592	0.0513	0.0287	−0.0129	0.0413	−1.8413	0.0003

.

The result of our simulation is shown in Figure 9a; the gray area in Figure 9b represents that the different $\lambda$ values are within the error of 10%, i.e., the predicted values have an acceptable error.

The error analysis of the fPp RUL prediction for battery RW12 is shown in

Table 2. The fPp model prediction results and error analysis of RW12.

Start Point	Actual RUL	Predicted RUL	AE	RE
30	26	26	0	0.0000
32	24	23	1	0.0417
34	22	21	1	0.0455
36	20	18	2	0.1000
38	18	17	1	0.0556
40	16	15	1	0.0625

.

The error analysis based on the MAE, MAPE, RMSE, and $R^2$ models demonstrates that the performance of the fPp model is considerably better than that of the models based on the Wiener, fBm, LSTM, and fLsm models (see

Table 3. Comparative analysis between the fPp model and other models.

Battery	Model	MAE	MAPE (%)	RMSE	${\mathit{R}}^2$
RW12	fPp	1.0000	5.0863	1.1547	0.9823
	fBm	2.0000	9.7122	2.0817	0.6577
	LSTM	2.0000	9.2484	2.1602	0.9725
	fLsm	1.5000	7.2959	1.7795	0.9292
	Wiener	2.0000	10.1128	2.1602	0.9468

).

The results given in Table 3 are visualized as a histogram in Figure 10. It is clear that the fPp model has the lowest prediction error. Furthermore, Figure 10d clearly shows that the fPp model’s prediction result is close to 1; it means that the fPp degradation model has a high accuracy.

7. Conclusions

By using the NASA degradation walk dataset, the effectiveness of the fPp differential degradation model with the capacity regeneration jumps was verified, and the $\lambda$ parameter can describe the jumps. The walk degradation data satisfies the LRD and determines the EOL. By selecting different prediction starting points, the results of parameter estimation are fed into the fPp degradation model to obtain the PDF and RUL predictions. Finally, the fBm model, LSTM model, fLsm model, and Wiener model were used as comparison models to verify the effectiveness of the fPp degradation model.

Future research directions will focus on multimodal factors because the practice operating conditions are random multimodal.