Fractional Poisson Process Predicts Remaining Useful Life of Lithium Battery by Walk Sequences

1. Introduction

The capacity of lithium-ion batteries (LIBs) has a degradation trend with charging and discharge cycles. This results in the reduction of the end of life (EOL). A recommended practice to replace the battery on the capacity basis is estimated below 80% of the manufacturer’s rating [IEEE 1188-1996]. Widespread applications of LIBs demand accurate prediction of RUL of LIBs.

Xiong et al. [1] deduces mathematical equations through electrochemical theory and applies finite element analysis and numerical methods to solve equations to build a predicting model. Li et al. [2] proposed to simulate the battery load response under specific temperature conditions through a coupling model, which is suitable for the temperature range of 25℃ to 40℃ in daily life. Using circuit theory to explore and analyze the performance of batteries [3], the Rint model and impedance-capacitor (RC) network model [4] are classic examples of such methods. 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 is modeled, and then a polynomial and exponential decay model is derived [5]. Most empirical models rely on statistical random filtering technology to track battery decay trends and optimize model parameters, such as particle filtering, Kalman filtering, etc. [6]. The prediction accuracy is highly dependent on the of model construction. Focusing on the research of data-driven methods, Chen et al. [7] combined variable modal decomposition and GPR, first decompose the capacity degradation data of lithium batteries into relatively stable components, and then conduct RUL prediction based on these components.

Li et al. [8] proposed a GPR prediction model with an automatic selection mechanism of kernel function. Jia et al. [9] mined key features from charge and discharge data, and then used GPR to estimate SOH and RUL, and effectiveness is proved. Peikun et al. [10] combined with gray correlation analysis and GPR method of multi-island genetic algorithm to optimize the prediction of SOH accuracy. these methods need high standards for data collection.

Xiao et al. [11] proposed a prediction technology combining RVM and three-parameter capacity decay model, Feng et al. [12] explored health indicator extracted from surface temperature changes when battery discharged, and Wang et al. [13] extracted 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 RVM for prediction may lead to instability of predicting result [14].

Deep learning such as recurrent neural network (RNN) is used for battery RUL prediction. Researchers [15] extracted key features of degradation data through empirical modal decomposition and grey relationship analysis, and used these features to deep RNNs to simulate cell RUL.

The current methods can only predict the volatility of the degradation trend. The fPp process can simulate irregular jumps caused by emergencies such as overcharge, short circuit or temperature abnormalities, and capacity regeneration, and can capture the continuous trend-long correlation and random fluctuations in the data.

The above methods can only predict the undulatory property of degradation trend, without considering the long-range dependence of 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 jump.

The organization of this paper is as follows. Section 2 discuses Long-range Dependence and Poisson Distribution, Section 3 Definition and characteristic of fractional Poisson process(fPp) Section 4 constructs a stochastic differential equation with adaptive jump intensity based on the fPp process. Section 5 is parameter estimation, section 6 is case study, section 7 is conclusion.

2. Long-Range Dependence and Poisson Distribution

2.1. Long-range Dependence

An autocorrelation function of a random series is a measure of the dependence of time series under different time interval(i.e. delays). Given a random sequence $\left\{X\left(t\right),0<t<\infty \right\}$, its autocorrelation function $R\left(\tau \right)$see equation (1):

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

(1)

Because $R\left(\tau \right)$decay is very slow, it means that $\left\{X\left(t\right)\right\}$ is significant correlation between sequence values even after a long lag. $R\left(\tau \right)$decays obeys power law, 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 $0<\eta <1$ and c are a constant, H is the Hurst exponent, and its range is$H\in \left(0,1\right)$, which is used to determine whether a random process has the LRD characteristic. When $H=0.5$, the autocorrelation function can be integrated or summed, meaning that the random sequence is not autocorrelation characteristics; when $H\ne 0.5$ , the autocorrelation function cannot be integrated or summed, revealing that the random sequence exhibits correlation, where $0<H<0.5$ shows a short correlation, and when $0.5<H<1$, the random process exhibits LRD characteristics.

2.2. Estimation of the Hurst exponent

A number of estimation algorithms of the Hurst exponent were proposed. Here, the heavy scale extreme difference method is used, namely the R/S method, which is as follow:

Divide the random sequence $\left\{X_1,X_2,\dots ,X_n\right\}$ into $h$ subsequence of length $a$. The average value of these subsequenece is as follows (3):

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

(3)

Calculate the cumulative deviation, as shown in Equation (4):

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

(4)

Calculate the extreme difference value, as shown in 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)

Calculate the standard deviation, as shown in Equation (6)

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

(6)

Calculate the standard deviation, as shown in Equation (7)

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

(7)

Next, repeat the above steps, and on the premise that the formula (6) is satisfied, take different $a$, get $h\times a=n$. Finally, the re-scaling extreme difference ${\displaystyle \frac{R}{S}}\left(a\right)$ takes the logarithm and fits the straight line using the least squares method, as shown in 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. Jump Characteristics of Poisson Distribution

For any time length $t$, the probability of occurring n events obeys the Poisson distribution, as follows (9)

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

(9)

Where $\lambda$ is the average times number occurs in unit time, and is recorded as $N_t\sim Poisson\left(\lambda t\right)$.

The Poisson process is suitable for simulating events that occur randomly and relatively sparsely within a given time or space area. As shown Figure 2, when $\lambda =3$, PDF of occurring different sparse events $k$ is shown as Figure 2.

2.4. Random walks characteristics of Poisson distribution

A Poisson process ${\tilde{N}}_t$, defines $\left\{Q_k^n,k=0,\dots ,n\right\}$, as a random walk approximation of ${\tilde{N}}_t$ is described as in Equation (10):

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

(10)

Where ${\eta }_1^n,\dots ,{\eta }_n^n$ is a random variable with independent and homogeneous distribution. For $\forall k$, there is the probability of formula (11):

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

(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. When $\lambda$ increases, the irregularity is stronger.

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 $N^H=\left\{N_t^H,t\ge 0\right\}$ is given as follows (12):

$$\begin{array}{c}{N}_t^H\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(H-\frac{1}{2}\right)}}{\int }_0^t{u}^{{\displaystyle \frac{1}{2}}-H}\left({\int }_u^t{\tau }^{H-{\displaystyle \frac{1}{2}}}\right({\tau -u)}^{H-{\displaystyle \frac{1}{2}}}d\tau \left)dq\right(u)\end{array}$$

(12)

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

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 is shown in Figure 4, and the Hurst ranges is 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 sequences $\left\{X_t,t>0\right\}$, 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 $\mu$ and $\delta$ are the drift and diffusion coefficients, respectively; $B_t$ is the Brownian motion, and is replaced by fBm, thus the equation (13) has LRD characteristics, shown as equation (14), which can more accurately simulate the remaining useful life decay process of lithium batteries.

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

(14)

Where $B_{t,H}$ is the fBm process, $N_{t,H}$ is the fPp process, $\eta$ is the jump amplitude, and discretization can obtain the discrete equation (15).

$$\Delta X_t=\mu X_t\Delta t+\delta X_t\Delta B_{t,H}+\eta X_t\Delta N_{t,H}$$

(15)

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

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

(16)

let $\mu =2.7$，$\delta =0.3$，$H=0.85$，$\Delta t=2$，$\lambda =0.05$，$q_1=0.0453$，$q_2=0.9752$，substituing $X_t=1$ into equation (16)，get random sequence$\left\{X_t,t>0\right\}$ simulation, as shown Figure 5.

5. Parameter estimation

5.1. Estimation of drift and diffusion coefficients

For model (16), the maximum likelihood method is applied to estimate the parameters. let sample $\left\{X_1,X_2,\dots ,X_n\right\}$, the joint probability density is as follows:

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

(17)

Two side takes logarithm(18)

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

(18)

Take the partial derivative for parameter $\mu$ and ${\sigma }^2$, and let them equal zero, respectively, get equation (19) , (20) and(21)

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

(19)

5.2. Estimation Parameter $\lambda$

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

(20)

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

(21)

First calculate the difference for $\left\{X_1,X_2,\dots ,X_n\right\}$, shown as equation (22)

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

(22)

Where $x_i$ is the battery capacity of the ith times charge or discharage period, here let 95^th percentile as jump threshold, shown as equation (23)

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

(23)

Record jump times:

$$N_{shocks}=\left|\left\{\Delta x_i:\Delta x_i>p_{95}\right\}\right|$$

(24)

When given a Poisson process, the probability of occur $N$ times events during $T$ follows:

$$\begin{gathered} {\displaystyle \frac{\partial l}{\partial {\sigma }^2}}=-{\displaystyle \frac{n}{2{\sigma }^2}}+{\displaystyle \frac{1}{2{\left({\sigma }^2\right)}^2}}{\displaystyle \sum _{i=1}^n{\left(X_i-\mu \right)}^2=0} \\ P\left(N=n\right)={\displaystyle \frac{e^{-\lambda T}{\left(\lambda T\right)}^n}{n!}} \end{gathered}$$

(25)

Construct the logarithmic 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:

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

(28)

where $I\left(\bullet \right)$represents the exponent function for $\Delta x_i\le p_{95}$, is equal to 1, otherwise it is equal to zero.

6. Case Study

This experiment used the RW12 degradation walk data set of four 18650 lithium batteries from NASA[17], which operated continuously at charge and discharge currents between -4.5A and 4.5A. This type of charging and discharging operation is called random walking (RW) operation. Each load period lasts for 5 minutes, 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 lithium battery during the charge and discharge cycle are shown in Figure 6:

The capacity of RW12 battery is degraded as shown in Figure 7. The battery fails when the capacity EOL is set to 1.4Ah 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 8(a) is obtained. Four points exceed the 95% threshold, so there is a situation that the fBm model cannot predict. As shown in Figure 8(b), 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 result of our simulation is shown in Figure 9 (a); the gray area in Figure 9 (b) 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 1.

Comparing with the fBm model, LSTM model, fLsm model and Wiener model, Through the error analysis of MAE, MAPE, RMSE and $R^2$, the prediction error evaluation of the fPp model is the smallest,see Table 2.

Table 2 are visualized as a histogram, see Figure 10. From Figure 10 (a), (b), and (c), it is evident that the fPp model has the lowest prediction error. Furthermore, Figure 10 (d) clearly shows that the fPp model's prediction result is close to 1, it is mean that the fPp degradation model has high accuracy.