State-of-Charge Estimation of Lithium-Ion Batteries Based on Dual-Coefficient Tracking Improved Square-Root Unscented Kalman Filter

Abstract

Accurate state of charge (SOC) estimation is helpful for battery management systems to extend batteries’ lifespan and ensure the safety of batteries. However, due to the pseudo-positive definiteness of the covariance matrix and noise statistics error accumulation, the SOC estimation of lithium-ion batteries is usually inaccurate or even divergent using Kalman filters, such as the unscented Kalman filter (UKF) and the square-root unscented Kalman filter (SRUKF). To resolve this problem, an SOC estimation method based on the dual-coefficient tracking improved square-root unscented Kalman filter for lithium-ion batteries is developed. The method is composed of an improved square-root unscented Kalman filter (ISRUKF) and a dual-coefficient tracker. To avoid the divergence of SOC estimation due to the covariance matrix with pseudo-positive definiteness, an ISRUKF based on the QR decomposition covariance square-root matrix is presented. Moreover, the dual-coefficient tracker is designed to track and correct the state noise error of the battery, which can reduce the SOC estimation error caused by the accumulation of the battery model error using the ISRUKF. The accuracy and robustness of the SOC estimation method using the developed method are validated by the comparison with the UKF and SRUKF. The developed algorithm shows the highest SOC estimation accuracy with the SOC error within 1.5%.

1. Introduction

Lithium-ion batteries have been widely used in electric vehicles and battery energy storage [1,2] due to the advantages of a high energy density, long cycle life, and low self-discharge rate [3,4,5]. The battery management system is indispensable to the monitoring and estimation of the critical internal states of lithium-ion batteries [6]. Furthermore, state of charge (SOC) estimation is one of the battery management system’s most important state parameters [7,8,9]. Accurate SOC estimation helps the battery management system to optimize the operation characteristics, eliminating potential safety risks and prolonging the batteries’ lifespan [10]. However, it is difficult to accurately estimate the SOC in complex driving environments due to the nonlinear electrochemical performance of lithium-ion batteries and the SOC immeasurability [11].

Recently, other literature has been published to accurately estimate the SOC for lithium-ion batteries. In general, the SOC estimation method can be classified into four groups: the looking-up table-based method, ampere-time integration method, data-driven method, and model-based method [12]. In the looking-up table-based method, the initial SOC value is determined by the open-circuit voltage (OCV) collected using the battery management system, and the SOC value is attained by the SOC-OCV correspondence table via the subsequent OCV measurement value [13]. However, it is more suitable for application in a laboratory environment because of its drawbacks, such as being time-consuming and susceptible to external environment influences in batteries’ temperature and aging conditions. The ampere-time integration method estimates the SOC by integrating the currents flowing into and out of the batteries over time. The SOC estimation based on this method depends on SOC initial errors and current measurement errors, which results in increasing SOC estimation errors [14,15,16]. Data-driven methods—such as the neural network method [17], support vector machine method [18], and Bayesian network method [19,20]—are studied to accurately estimate the SOC for a nonlinear system.

These methods are not suitable for practical SOC estimation online, though, due to the requirements of an enormous amount of training data and the high calculation cost. Model-based methods consisting of the batteries’ equivalent circuit model (ECM) and a filter—such as a Kalman filter (KF) and particle filter (PF)—are attracting extensive attention and becoming one of the most popular algorithms to accurately estimate the SOC for lithium-ion batteries [21].

Compared with the PF, which requires many particles to calculate the posterior probability density, the KF can reduce the number of sampled particles and ensure the accuracy requirements with the help of deterministic sampling. The standard KF is a widely used linear filter but has poor adaptability to nonlinear time-varying systems [22]. As an improved method of the KF, the extended Kalman filter (EKF) can be used in nonlinear time-varying systems. However, the SOC estimation accuracy using the EKF is degraded due to its requirements of the Jacobian matrix calculation and a linearized approximation of the nonlinear time-varying function using the first-order terms of the Taylor formula, which does not consider the higher-order terms of the Taylor formula [23].

To resolve the disadvantages of the KF and the EKF noted above, the unscented Kalman filter (UKF) is presented to estimate the SOC [24]. The UKF based on unscented transform does not need to linearize system equations without accounting for the higher-order terms of the Taylor formula, and the UKF shows higher SOC estimation accuracy than the EKF. SOC estimation using the UKF will be divergent because of its drawbacks, such as the pseudo-positive definiteness of the covariance matrix caused by the error accumulation in the UKF. To resolve these shortcomings, the square-root unscented Kalman filter (SRUKF) is presented to estimate the batteries’ SOC [25,26,27,28,29]. Compared to the UKF, the SRUKF uses the state covariance square-root matrix instead of the state covariance matrix, which can overcome the problem of the pseudo-positive definiteness of the covariance matrix. The SOC estimation accuracy based on the SRUKF will be reduced because the SRUKF needs to use the Cholesky decomposition method to achieve the state covariance square-root matrix.

Moreover, the SOC estimation accuracy based on the KF mentioned above is greatly degraded by unknown or inaccurate noise statistics, such as the measurement and model noise covariance of the batteries [30]. In the iteration process of these KFs, the calculation accuracy of the system covariance matrix is dependent on the accurate noise statistics. The error accumulation of noise statistics will be generated in the iteration process, which leads to the divergence of the KFs. An adaptive EKF based on the combination of adaptive modification and the EKF is proposed to avoid the divergence of the algorithm error [31]. Based on the UKF and an adaptive noise estimator established by noise covariance matching, an adaptive unscented Kalman filter (AUKF) is developed to estimate the SOC. The SOC estimation accuracy using the AUKF is improved for the battery system with uncertain noise statistics [32], but its accuracy is still degraded because of not considering the state noise error accumulative calculation in the iteration process of the AUKF.

As mentioned above, the discharging and charging process of lithium-ion batteries is a nonlinear electrochemical reaction process. It is difficult to accurately attain the SOC of lithium-ion batteries due to their nonlinear operating characteristics. Model-based methods consisting of the KF and its improved algorithms, such as the UKF, the AUKF, and the SRUKF, are widely used to estimate the SOC of the batteries. However, the estimated SOC using these KFs will be divergent due to their drawbacks, such as the pseudo-positive definiteness of the covariance matrix in the UKF and the Cholesky decomposition in the SRUKF. Moreover, the SOC estimation accuracy is degraded by the inaccurate model noise covariance and its accumulative error of the batteries using these model-based methods, including the UKF, the SRUKF, and the AUKF. To solve these problems, a dual-coefficient tracking ISRUKF is developed to achieve accurate SOC estimation for lithium-ion batteries with uncertain noise statistics. The innovation of the developed method includes the following: (1) To avoid the divergence of SOC estimation due to the pseudo-positive definiteness of the state covariance matrix, the ISRUKF based on the QR decomposition method of the state covariance square-root matrix is presented. (2) A dual-coefficient tracker based on the strong tracking filter (STF) is designed to track and correct the state noise error accumulation of the battery, which can reduce the SOC estimation error caused by the batteries’ model error accumulation using the UKF and the ISRUKF. (3) An SOC estimation method based on the ISRUKF and the dual-coefficient tracker is developed for lithium-ion batteries with uncertain noise statistics.

The rest of this paper is organized as follows: Section 2 introduces the construction of the ECM and its state space equation of lithium-ion batteries; Section 3 presents the developed SOC estimation method based on dual-coefficient tracking ISRUKF, including the ISRUKF and the dual-coefficient tracker; Section 4 shows the accuracy and robustness of the developed SOC estimation method by the comparison of the simulation results and the experimental data in different conditions; and Section 5 discusses the conclusions.

2. Equivalent Circuit Model and State Space Equation of Lithium-Ion Batteries

2.1. Equivalent Circuit Model of the Batteries

Since it can accurately describe the dynamic characteristics of the batteries [33], an ECM based on a two-order RC circuit in this paper is shown in Figure 1. $U_{OC}$ describes the open circuit voltage of the batteries; $R_0$ represents the batteries’ internal resistance; $R_L$ illustrates the concentration resistance; $C_L$ is the concentration capacitance; $R_S$ represents the electrochemical resistance; $C_S$ means the electrochemical capacitance; $I_t$ is the current of the batteries; and $U_t$ is the terminal voltage of the batteries that is connected to the loads.

As shown in Figure 1, according to Kirchhoff’s voltage law, the functional relationship between the terminal voltage $U_t$ and the current of the batteries $I_t$ can be expressed as:

$$U_t=U_{OC}-U_S-U_L-I_t{R}_0$$

(1)

where $U_S$ represents the electrochemical voltage of the batteries, and $U_L$ means the concentration voltage of the batteries.

In the ECM of the batteries, according to the experimental data of the batteries in the charging and discharging process, the nonlinear functional relationship between $R_0$, $R_L$, $C_L$, $R_S$, $C_S$, and the SOC can be achieved by the least-squares-error curve-fitting method, which can be represented as:

$$U_{OC}(t)=a_0{e}^{-a_{1}SOC\left(t\right)}+a_2+a_{3}SOC(t)-a_{4}SO{C}^2(t)+a_{5}SO{C}^3(t)$$

(2)

$$R_0(t)=b_0{e}^{-b_{1}SOC\left(t\right)}+b_2+b_{3}SOC(t)-b_{4}SO{C}^2(t)+b_{5}SO{C}^3(t)$$

(3)

$$R_L(t)=c_0{e}^{-c_{1}SOC(t)}+c_2$$

(4)

$$C_L(t)=d_0{e}^{-d_{1}SOC(t)}+d_2$$

(5)

$$R_S(t)=e_0{e}^{-e_{1}SOC(t)}+e_2$$

(6)

$$C_S(t)=f_{0}e+f_{1}SOC(t)+f_2$$

(7)

where $a_0~a_5$, $b_0~b_5$, $c_0~c_2$, $d_0~d_2$, $d_0~d_2$, and $f_0~f_2$ are the coefficients, which can be obtained by the least-squares-error curve-fitting method via the experimental data.

The SOC, which represents the percentage of the current battery capacity in the total battery capacity, can be expressed as:

$$SOC\left(t\right)=SO{C}_0-\frac{\int \eta I_{t}dt}{Q_N}$$

(8)

where ${SOC}_0$ means the initial value of $SOC$, $\eta$ shows coulomb efficiency, and $Q_N$ denotes the nominal capacity.

2.2. State Space Equation of the Battery Model

To describe the battery model in state space equations, the SOC, the electrochemical voltage $U_S$ and the concentration voltage $U_L$ are selected as battery state variables. The $I_t$ is chosen as the input variables of the batteries, and the $U_t$ is assumed to be the output variables of the batteries. According to the SOC expression in (8) and the batteries’ ECM as shown in Figure 1, the state space equation of the batteries in discrete time can be written as:

$$\left[\begin{array}{c}SO{C}_{k+1}\\ U_{S,k+1}\\ U_{L,k+1}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{(-\Delta t/{\tau }_1)}& 0\\ 0& 0& e^{(-\Delta t/{\tau }_2)}\end{array}\right]\left[\begin{array}{c}SO{C}_k\\ U_{S,k}\\ U_{L,k}\end{array}\right]+\left[\begin{array}{c}-\eta \Delta t/Q_N\\ R_S[1-e^{(-\Delta t/{\tau }_1)}]\\ R_L[1-e^{(-\Delta t/{\tau }_2)}]\end{array}\right]I_{t,k}+w_k$$

(9)

where $\Delta t$ represents the sampling time; ${\tau }_1~{\tau }_2$ illustrates the time constant, ${ \tau }_1=R_S{C}_S$, and ${\tau }_2=R_L{C}_L$; and $w_k$ is the state noise.

Moreover, the measurement equation of the batteries in discrete time can be expressed as:

$$U_{t,k+1}=U_{OC,k+1}-U_{S,k+1}-U_{L,k+1}-I_{t,k+1}{R}_{0,k+1}+v_{k+1}$$

(10)

where $v_k$ is the measurement noise.

3. SOC Estimation Based on the Dual-Coefficient Tracking ISRUKF

Due to not considering the higher-order terms of the Taylor formula, the SOC estimation based on the EKF is inaccurate. Compared to using the EKF, the SOC estimation accuracy is higher using the UKF since it does not require the Jacobian matrix calculation. However, the UKF’s SOC estimation accuracy and stability are degraded given the pseudo-positive definiteness of the covariance matrix and the inaccurate noise statistics. The SRUKF based on the Cholesky decomposition method can partly reduce the influence of the pseudo-positive definiteness of the covariance matrix in the iteration process, but the SRUKF does not completely resolve the problem of pseudo-positive definiteness due to its nonlinear state calculation.

In this paper, a dual-coefficient tracking ISRUKF is developed to improve SOC estimation accuracy in two ways. First, the ISRUKF based on the QR decomposition method can overcome the problem of the pseudo-positive definiteness of the covariance matrix in the UKF and the SRUKF. Second, a dual-coefficient tracker based on the STF is designed to track and correct the batteries’ state noise error. In the dual-coefficient tracker, the tracking coefficient of state noise ${\delta }_k$ is used to adaptively track and correct the state noise error, and the fading factor ${\lambda }_k$ of the STF is used to track and adjust the state covariance square-root matrix. The diagram of the developed ISRUKF is shown in Figure 2.

3.1. Improved Square-Root Unscented Kalman Filter (ISRUKF)

3.1.1. Standard SRUKF

For a nonlinear discrete time system, the system state equation and the system measurement equation can be described as:

$$\left\{\begin{array}{c}{x}_k=f(x_{k-1},u_k)+w_k\\ y_k=g(x_k,u_k)+v_k\end{array}\right.$$

(11)

where $x_k$ is the system state vector; $y_k$ is the system measurement vector; $f(·)$ and $g(·)$ denote the nonlinear state and measurement models, respectively; $u_k$ is the system input vector; $w_k$ and $v_k$ are the system state noise and measurement noise, respectively, and their statistics characteristics can be expressed as:

$$\left\{\begin{array}{l}E\left[w_k\right]=q,\mathit{cov}({\omega }_k,{\omega }_j)=Q\\ E\left[v_k\right]=r,\mathit{cov}(v_k,v_j)=R\\ \mathit{cov}({\omega }_k,v_j)=0\end{array}\right.$$

(12)

where r and q denote the mean value of $v_k$ and $w_k$ separately. R and Q denote the covariance value of $v_k$ and $w_k$, respectively. The detailed steps of the SRUKF can be summarized as follows:

(1)

Initialize the mean (x₀) and the state covariance square root ($S_0$) of the system state:

$$\left\{\begin{array}{c}\stackrel{\wedge }{x_0}=E(x_0)\\ S_0=\sqrt{E[(x_0-\stackrel{\wedge }{x_0})(x_0-\stackrel{\wedge }{x_0}{)}^T]}\end{array}\right.$$

(13)

where $E (·)$ is the expectation mean value, and ($·$)^T is the matrix transpose operation.

(2)

Assign weights and obtain sampling points:

$$\left\{\begin{array}{c}{x}_{0,k}=\stackrel{\wedge }{x_k}\\ x_{i,k}=\stackrel{\wedge }{x_k}+\sqrt{(n+\lambda )S_{k|k}}\\ x_{i+n,k}=\stackrel{\wedge }{x_k}-\sqrt{(n+\lambda )S_{k|k}}\end{array}\right.$$

(14)

where n represents the dimension of the state vector, and λ means a scale that can be presented as:

$$\lambda ={\alpha }^2(n+h)-n$$

(15)

where $\alpha$ is a scaling parameter, its range is set as 0~1, h is the column factor, and h = 3 − n.

The weighted coefficients can be expressed as

$$\left\{\begin{array}{c}{w}_0^m=\frac{\lambda }{n+\lambda }\\ w_0^c=\frac{\lambda }{n+\lambda }+(1+\beta -{\alpha }^2)\\ w_i^m=w_i^c=\frac{1}{2(n+\lambda )}\end{array}\right.$$

(16)

where $w^m$ is the variance weight factor; $w^c$ is the mean weight factor; and $\beta$ is the error magnitude of the higher-order term.

(3)

Time update for the system states:

a.

Update the sample point:

$$x_{i,k|k-1}=f(x_{i,k-1})$$

(17)

b.

Estimate the system state:

$${\stackrel{\wedge }{x}}_{k|k-1}=\sum _{i=0}^{2n}{w}_i^m{x}_{i,k|k-1}+q_k$$

(18)

c.

Update the covariance of the estimated state:

$$S_{k|k-1}^{*}=qr\left\{\left[\sqrt{w_i^c}\left(x_{i=1:2n,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}\right),\sqrt{Q_{k-1}}\right]\right\}, i=\mathrm{1,2},\dots ,2n$$

(19)

$$S_{k|k-1}=cholupdate\{S_{k|k-1}^{*},x_{i,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1},w_0^c\}$$

(20)

where $S_{k|k-1}^{*}$ is the updated state calculation value; $qr$($·$) is the QR decomposition; S_k|k−1 is the state covariance square-root matrix; and $cholupdate$($·$) is the Cholesky decomposition.

(4)

Measurement update:

a.

Attain the measurement:

$$y_{i,k|k-1}=g(x_{i,k-1})$$

(21)

b.

Update the measurement:

$${\stackrel{\wedge }{y}}_{k|k-1}={\sum }_{i=0}^{2n}{w}_i^m{y}_{i,k|k-1}+r_k$$

(22)

(5)

Calculate the SRUKF gain matrix L:

$$L_k=P_{xy}/S_z{S}_z^T$$

(23)

where $P_{xy}$ is the mutual covariance, as shown in Formula (24), and $S_z$ is the measurement covariance square-root matrix, as shown in Formula (26).

$$P_{xy}=\sum _{i=0}^{2n}{w}_i^c(x_{i,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}){(y_{i,k|k-1}-{\stackrel{\wedge }{y}}_{k|k-1})}^T$$

(24)

$$S_z^{*}=qr\{[\sqrt{w_i^c}({\stackrel{\wedge }{y}}_{i=1:2n,k|k-1}-{\stackrel{\wedge }{y}}_{k|k-1}),\sqrt{R_{k-1}}]\}$$

(25)

$$S_z=cholupdate\{S_z^{*},y_{i,k|k-1}-{\stackrel{\wedge }{y}}_{k|k-1},w_0^c\}$$

(26)

where $S_z^{*}$ is the updated calculation measurement.

(6)

Measurement correction:

a.

Update the estimated state:

$${\stackrel{\wedge }{x}}_{k|k}={\stackrel{\wedge }{x}}_{k|k-1}+L_k(y_k-{\stackrel{\wedge }{y}}_{k|k-1})$$

(27)

b.

Update the propagated covariance:

$$S_{k|k}=cholupdate\{S_{k|k-1},L_k{S}_z,-1\}$$

(28)

where $S_{k|k}$ is the state covariance square-root optimal estimation matrix.

3.1.2. The ISRUKF Based on the QR Decomposition Method

As seen from Formula (20) in the standard SRUKF, the SRUKF requires the Cholesky decomposition to obtain the state covariance square-root matrix $S_{k|k-1}$. Due to the matrix computation errors in the Cholesky decomposition, the SRUKF estimation accuracy will be degraded by the pseudo-positive definiteness of the $S_{k|k-1}$. To solve this problem, the ISRUKF based on the QR decomposition method is developed in this paper. The detailed core concept of the ISRUKF based on the QR decomposition method is presented as follows.

The formula of the state covariance matrix $P_{k|k-1}$ can be described as

$$P_{k|k-1}=\sum _{i=0}^{2n}{w}_i^c\left[x_{i,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}\right]{\left[x_{i,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}\right]}^T+Q_{k-1}$$

(29)

Then, the formula of the state covariance matrix P_k/k−1 and the state covariance square-root matrix $S_{k|k-1}$ is presented as

$$P_{k|k-1}=S_{k|k-1}{S}_{k|k-1}^T=[\sqrt{w_i^c}(x_{i,k|k-1}{\stackrel{\wedge }{x}}_{k|k-1}),\sqrt{Q_{k-1}}][\sqrt{w_i^c}(x_{i,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}),\sqrt{Q_{k-1}}{]}^T$$

(30)

we set $S_{k|k-1}={({q_{k}r}_k)}^T$; then

$$P_{k|k-1}={\left({q_{k}r}_k\right)}^T\left({q_{k}r}_k\right)=r_k^T{q}_k^{T}q{r}_k=r_k^T{r}_k$$

(31)

where q_k is the upper triangular matrix, and $r_k$ is the orthogonal matrix.

By combining Formulas (30) and (31), the following formula can be obtained

$$\left\{\begin{array}{c}{r}_k=qr\{[\sqrt{w_i^c}(x_{i=1:2n,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}),\sqrt{Q_{k-1}}{]}^T\}\\ S_{k|k-1}=r_k^T\end{array}\right.$$

(32)

Substitute Formula (32) with Formula (20), which can effectively avoid the pseudo-positive qualitative problem existing in the standard SRUKF.

Similarly, Formulas (26) and (28) can be replaced by

$$\left\{\begin{array}{c}{r}_z=qr\{[\sqrt{w_i^c}(y_{i=1:2n,k|k-1}-{\stackrel{\wedge }{y}}_{k|k-1}),\sqrt{R_{k-1}}{]}^T\}\\ S_Z=r_k^T\end{array}\right.$$

(33)

$$\left\{\begin{array}{c}{r}_{k|k}=qr\left\{{[\left(e-L_k{H}_k\right)S_{k|k-1},L_k\sqrt{R_{k-1}} ]}^T\right\}\\ S_{k|k}=r_{k|k}^T\end{array}\right.$$

(34)

where e is the identity matrix of order n, and $H_k$ is the coefficient matrix of the measurement function, which can be expressed as

$$H_k=(P_{xy}{)}^T(S_{k|k-1}{S}_{k|k-1}^T)$$

(35)

3.2. The Dual-Coefficient Tracker Based on the Strong Tracking Filter

3.2.1. Strong Tracking Filter

The STF is used to attain the fading factor to update the state covariance square-root matrix in the iterative of the ISRUKF. The fading factor can be attained as

$$\left\{\begin{array}{c}{S}_{k|k-1}={\lambda }_k{S}_{k|k-1}{\lambda }_k^T+\sqrt{Q_{k-1}}\\ { \lambda }_k=diag({\mu }_0,{\mu }_k)\end{array}\right.$$

(36)

where ${\lambda }_k$ is the fading factor, which is also the tracking coefficient of the state covariance square-root matrix,${ \lambda }_k=diag({\mu }_0,{\mu }_k)$,${ \mu }_0=[\mathrm{1,1},1...1]$, whose dimension is n columns, and ${ \mu }_k=[u_k,u_k,u_k,...u_k]$, which is described as

$$u_k=\left\{\begin{array}{c}\frac{trace(C_k-H_k{Q}_{k-1}{H}_k^T-R_{k-1})}{trace(P_{k|k-1}-Q_{k-1})H_k^T{H}_k} {\mu }_k\ge 1\\ \begin{array}{c}\\ 1 {\mu }_k<1\end{array}\end{array}\right.$$

(37)

where trace($·$) is the trace of the corresponding matrix, and $C_k$ is the covariance of the residual sequence of outputs and can be expressed as

$$C_k=\left\{\begin{array}{c}{\epsilon }_1{\epsilon }_1^T \mathrm{k}=1\\ \frac{\rho C_{k-1}+{\epsilon }_k{\epsilon }_k^T}{1+\rho } \mathrm{k}\ge 2\end{array}\right.$$

(38)

where $\rho$ is the forgetting factor, and its range is set as 0.95~0.99; ${\epsilon }_k$ is the output residual sequence, and ${\epsilon }_k=y_k-{\stackrel{\wedge }{y}}_{k|k-1}$.

3.2.2. The State Noise Tracking Coefficient

In Formula (32), the state covariance square-root matrix $S_{k|k-1}$ is directly affected by the $Q_{k-1}$. If the $Q_{k-1}$ diverges, the $S_{k|k-1}$ will further diverge. In this paper, the variation of the matrix elements of the $S_{k|k-1}$ based on its historical data is used to reflect the variation trend of $Q_{k-1}$. The updated state covariance square-root matrix $S_{k|k-1}^c$ can be expressed as

$$\left\{\begin{array}{c}{S}_{k|k-1}^c={\lambda }_k{S}_{k|k-1}{\lambda }_k^T+{\delta }_k\sqrt{Q_{k-1}}{\delta }_k^T\\ {\delta }_k=diag[m_k,m_k,m_k,...,m_k]\end{array}\right.$$

(39)

where $S_{k|k-1}^c$ is the corrected state covariance square-root matrix at time k, and ${\delta }_k$ is the state noise tracking coefficient, whose dimension is n columns. $m_k$ is the matrix element of the state noise tracking coefficient. Its value is attained by comparing the element mean value of the state covariance square-root matrix in the iteration process. The assignment process of $m_k$ is described as follows:

(1)

Initialize the $m_k$ and its dimension to n columns.

(2)

$P_{k|k-1}^{(\mathrm{i})}$ is set as an element in the $\mathrm{i}th$ row and $\mathrm{i}th$ column of the state covariance square-root matrix at time k. We set the mean of the elements in m adjacent square-root matrices as a parameter and definite A and B as the pairwise comparisons between adjacent parameters, which can be described as

$$\mathrm{A}=\frac{1}{m}{\sum }_{k=m}^{k=2m-1}{S}_{k|k-1}^{(i)}-\frac{1}{m}{\sum }_{k=2m}^{k=3m-1}{S}_{k|k-1}^{(i)}$$

(40)

$$\mathrm{B}=\frac{1}{\mathrm{m}}{\sum }_{\mathrm{k}=2\mathrm{m}}^{\mathrm{k}=3\mathrm{m}-1}{\mathrm{S}}_{\mathrm{k}|\mathrm{k}-1}^{(\mathrm{i})}-\frac{1}{\mathrm{m}}{\sum }_{\mathrm{k}=3\mathrm{m}}^{\mathrm{k}=4\mathrm{m}-1}{\mathrm{S}}_{\mathrm{k}|\mathrm{k}-1}^{(\mathrm{i})}$$

(41)

By comparing A and B, six outcomes will be attained:

$$\left\{\begin{array}{c}\mathrm{A}<\mathrm{B}<0 (\mathrm{Ⅰ})\\ \mathrm{A}<0<\mathrm{B} (\mathrm{Ⅱ})\\ \mathrm{B}<\mathrm{A}<0 (\mathrm{Ⅲ})\\ \mathrm{B}<0<\mathrm{A} (\mathrm{Ⅳ})\\ 0<\mathrm{A}<\mathrm{B} (\mathrm{Ⅴ})\\ 0<\mathrm{B}<\mathrm{A} (\mathrm{Ⅵ})\end{array}\right.$$

(42)

(3)

Attain the updated $m_k$. If the outcome does not match outcome (II) or outcome (V), the $m_k$ should not be modified. If the calculated result matches outcome (II) or outcome (V), the $m_k$ should be reassigned. If the value of $Q_0$ is higher than 2% of the $U_{OC}$, the $Q_0$ is judged to be too large, and the state noise tracking coefficient should be shrunk, and then the value $m_k=[1-\gamma ,1-\gamma ,1-\gamma ,...,1-\gamma ]$ should be assigned. If the $Q_0$ value is less than 2% of the $U_{OC}$, the $Q_0$ is judged to be too small, and the state noise tracking coefficient should be amplified, and then the value $m_k=[1+\gamma ,1+\gamma ,1+\gamma ,...,1+\gamma ]$ should be assigned, where $\gamma$ is the correction coefficient.

(4)

According to $m_k$ and Formula (39), we can get the state noise tracking coefficient ${\delta }_k$.

(5)

The ${\delta }_k$ and ${\lambda }_k$ are used as inputs to form the dual-coefficient tracker.

(6)

The dual-tracking coefficient ${\lambda }_k$ and ${\delta }_k$ are put into the ISRUKF to calculate the updated state covariance square-root matrix $S_{k|k-1}^c$, which can be illustrated as

$$\left\{\begin{array}{c}{r}_k=qr\{[{\lambda }_k\sqrt{w_i^c}(x_{i=1:2n,k|k-1}-{\stackrel{\wedge }{x}}_{k|k-1}),{\delta }_k\sqrt{Q_{k-1}}{]}^T\}\\ S_{k|k-1}^c=r_k^T\end{array}\right.$$

(43)

Figure 3 shows the diagram of the dual-coefficient tracker.

3.3. Battery SOC Estimation Procedure Based on Dual-Coefficient Tracking ISRUKF

According to the above analysis, the flow diagram of the battery SOC estimation based on the developed method shown in Figure 4 can be expressed as follows:

Step 1: Initialize the mean (x₀) and the state covariance square root ($S_0$) by Formula (13), and initialize the ${\lambda }_0$ and ${\delta }_0$;

Step 2: Assign weights $w^m$ and $w^c$ and obtain sampling points $x_{i,k}$ by Formulas (14) and (16);

Step 3: Time update for the state estimation $x_{i,k|k-1}$, ${\stackrel{\wedge }{x}}_{k|k-1}$, and $S_{k|k-1}$ by Formulas (17), (18) and (32);

Step 4: Measurement update $y_{i,k|k-1}$, ${\stackrel{\wedge }{y}}_{k|k-1}$, and $S_Z$ by Formulas (21), (22), and (33);

Step 5: Calculate the mutual covariance $P_{xy}$ by Formula (24) and estimate the coefficient matrix of measurement function $H_k$ by Formula (35);

Step 6: Calculate the ${\lambda }_k$ by Formulas (36), (37) and (38); attain the mean values of the state covariance square-root matrix by Formulas (40) and (41) and compare A and B by Formula (42) to determine the assignment of ${\delta }_k$;

Step 7: The ${\lambda }_k$ and ${\delta }_k$ are substituted into Formula (43) to update the state covariance square-root matrix $S_{k|k-1}^c$;

Step 8: Use $S_{k|k-1}^c$ to update the coefficient matrix of the measurement function by Formula (35);

Step 9: Calculate the filtering gain matrix $L_k$ by Formula (23) and the output estimation of the SOC by Formula (27);

Step 10: Attain the state covariance square-root optimal estimation matrix $S_{k|k}$ by the updated state covariance square-root matrix and the update coefficient matrix of the measurement function via Formula (34) and start the next iteration process.

4. Simulation and Experimental Results and Analysis

4.1. Test Platform and Experiment Parameters

The test platform shown in Figure 5 consists of an Arbin-BT2000 battery test system, lithium-ion batteries, and a computer. The battery test system is used for the testing of the batteries to attain battery parameters, such as the battery current, voltage, charge capacity, discharge capacity, internal resistance, etc. The single-channel voltage measurement range of the battery test system is 0~5 V, and the resolution of the measured voltage is ±0.01% of the full scale. The computer can record and process experimental data including the battery voltage, current, internal resistance, etc. Lithium-ion batteries are connected to the battery test system and can be continuously charged and discharged.

To validate the SOC estimation accuracy using the developed ISRUKF, three test schedules were conducted via some comparisons of simulation results with experimental data in the discharging process. In each test, the battery simulation and experiment were carried out at the temperature of 25 °C. The batteries were discharged with the initial SOC of 0.9 and were ended when the SOC was 0.2. The simulation and experimental parameters of batteries are listed in

Table 1. The coefficients of battery performance parameters.

Normal Voltage		3.7 V	Battery Capacity		860 mAh
Upper Cut-Off Voltage		4.2 V	Lower Cut-Off Voltage		3.2 V
a0	−0.915	a1	40.867	a2	3.632
a3	0.537	a4	0.499	a5	0.522
b0	0.1463	b1	30.27	b2	0.1037
b3	0.0584	b4	0.1747	b5	0.1288
c0	0.1063	c1	62.49	c2	0.0437
d0	−200	d1	138	d2	300
e0	0.0712	e1	61.4	e2	0.0288
f0	−3083	f1	180	f2	5088

.

4.2. Comparison of SOC Estimation Accuracy Using the Developed ISRUKF, SRUKF, and UKF with Different Covariance $Q_k$

In this section, the comparison of SOC estimation accuracy between the developed ISRUKF, SRUKF, and UKF is carried out when the $Q_k$ is 0.005 and 0.01, respectively, to verify the accuracy of the developed ISRUKF. Figure 6 shows the comparative results of the SOC and battery voltage using the three algorithms when the $Q_k$ is set as 0.005.

Figure 6a,b show the SOC experiment and estimation results and the corresponding absolute errors by the different methods. The UKF, SRUKF, and developed ISRUKF can be used to estimate the SOC with different estimation accuracy. Compared to the UKF and the SRUKF, the developed ISRUKF shows the highest SOC estimation accuracy in the discharging process.

As shown in Figure 6b, the SOC estimation errors using the developed ISRUKF are constantly kept below 0.01 given its dual-coefficient tracker, which can reduce state noise accumulative calculation errors. The SOC estimation errors using the UKF and SRUKF gradually increase from 0.01 to 0.06, even to divergence. Figure 6c,d show the battery voltage experiment and estimation results and the corresponding absolute errors by the different methods. Similar to the SOC estimation results, the estimated battery voltage using the UKF, SRUKF, and developed ISRUKF can effectively capture the experiment results. However, the estimated battery voltage based on the developed ISRUKF shows the lowest voltage errors within 0.01 V, as shown in Figure 6d. Moreover, under the same conditions of achieving 270,000 sampling points in the calculation step of 0.01 s, the calculation time of the SOC using the UKF, the SRUKF, and the developed method is 7.5 s, 7.6 s, and 7.7 s, respectively. Compared to the UKF and the SRUKF, the developed method spends almost an equal amount of time to estimate the SOC, but it can achieve the highest SOC estimation accuracy within 1.5% error, and the SOC errors of the UKF and the SRUKF are 6.7% and 5.5%, respectively, as shown in Figure 6b and Figure 7b.

Figure 7 illustrates the SOC estimation results using the UKF, SRUKF, and developed ISRUKF when the $Q_k$ is set as 0.01. Figure 7a,b present the SOC experiment and estimation results and the corresponding absolute errors by the different methods. Figure 7c,d show the battery voltage estimation results and the corresponding absolute errors using the three methods.

The estimation accuracy of the SOC and battery voltage using the developed ISRUKF is the highest compared to the UKF and the SRUKF. For example, the SOC errors are constantly controlled below 0.01, as shown in Figure 7b, and the batteries’ voltage errors are kept within 0.015 V, which is about 0.5% of the batteries’ nominal voltage, as shown in Figure 7d.

Moreover, with the help of the QR decomposition method (which can overcome the problem of the pseudo-positive definiteness of the covariance matrix), the errors of the estimated SOC and battery voltage estimation using the developed ISRUKF are stable or not divergent. However, the errors of the estimated SOC and battery voltage estimation using the UKF and the SRUKF gradually increase and even diverge in the discharging process shown in Figure 6b,d and Figure 7b,d. It is further proven that the developed ISRUKF based on a dual-coefficient tracker can accurately estimate the SOC for lithium-ion batteries and effectively avoid SOC divergence.

4.3. Comparison of SOC Estimation Results Using Different ISRUKFs, the ISRUKF with Standard STF and the Developed ISRUKF

In this section, a comparison of SOC estimation accuracy using different ISRUKFs (including the developed ISRUKF and the ISRUKF with the standard STF) is carried out to further validate the effectiveness of the developed ISRUKF. Figure 8 shows the SOC estimation accuracy using different ISRUKFs when the $Q_k$ is set as 0.005. Figure 8a compares the estimated SOC using the developed ISRUKF (in red) versus the ISRUKF with the standard STF (in green). It is shown that the SOC based on the ISRUKF with the standard STF can consistently follow the experimental results at the beginning of the discharging process, but this SOC deviates further and further from the experiment results as the discharging process moves forward. Figure 8b shows the corresponding SOC absolute error using the different ISRUKFs. Due to the dual-coefficient tracker reducing state noise accumulative calculation errors, the SOC absolute error using the developed ISRUKF (in red) is always lower than the SOC absolute error using the ISRUKF with the standard STF.

Without the ability of adjusting the state noise tracking coefficient ${\delta }_k$, the ISRUKF with the standard STF cannot reduce the state noise error accumulation in the iteration process of the UKF, which leads to the SOC absolute error based on this ISRUKF increasing continuously. For example, the SOC absolute error using the developed ISRUKF is constantly kept within 0.01, but the SOC absolute error using the ISRUKF with the standard STF gradually increases from 0.01 to 0.05. Figure 8c,d describe the battery voltage estimation results and the corresponding absolute errors using the different ISRUKFs. It is found that the battery voltage profile based on the developed method can track the experimental results accurately in the whole discharging process with a lower absolute error compared to using the ISRUKF with the standard STF. Therefore, compared to the ISRUKF with the standard STF, the developed method can achieve higher SOC and voltage estimation accuracy.

4.4. SOC Estimation by the Developed ISRUKF with Different Covariance $Q_k$

To verify the effectiveness and adaptation of the developed ISRUKF, a comparison of SOC estimation accuracy is performed when the $Q_k$ is different. The $Q_k$ is randomly set to 0.001, 0.005, and 0.01, respectively. Figure 9 shows SOC estimation results using the developed ISRUKF with different $Q_k$. Figure 9a,b illustrate the SOC estimation accuracy and its corresponding absolute error by the developed method with various noise statistics. It is noted that the estimated SOC using the developed method can exactly capture the experimental results, and the corresponding SOC absolute errors are stable and small despite the various noise statistics. As shown in Figure 9b, when the noise statistics are set to 0.001, 0.005, and 0.01, respectively, the highest SOC absolute error is controlled below 0.015, and the lowest SOC absolute error is controlled below 0.005 (means 0.5%).

Figure 9c,d show the battery voltage estimation and the corresponding absolute error using the developed ISRUKF with different noise statistics. With the help of the dual-coefficient tracker, the developed method can reduce state noise accumulative calculation errors; in turn, this contributes to gradually reducing the battery voltage deviation from the experimental results and improving its estimation accuracy. Therefore, the developed ISRUKF can not only precisely estimate the SOC of lithium-ion batteries but also shows good robustness when $Q_k$ varies.

4.5. Discussions

As a nonlinear system of the battery system, it is difficult to estimate the SOC accurately due to the influence of batteries’ incorrect noise statistics, such as the battery model error and the round-off error of the filter. To verify the SOC estimation precision using the developed method based on the ISRUKF with a dual-coefficient tracker, three test schedules are carried out. With the help of the dual-coefficient tracker, which can track and correct the batteries’ model noise error accumulation, the SOC estimation errors using the developed ISRUKF show the highest precision (below 1.5%) compared with the UKF (6.7%) and the SRUKF (5.5%) when the $Q_k$ is 0.005 and 0.01, respectively. Moreover, because the ISRUKF based on the QR decomposition method can overcome the problem of the pseudo-positive definiteness of the batteries’ model covariance, the estimated SOC errors and battery voltage errors using the developed method are kept within a stable range. However, the estimated SOC errors based on the UKF and the SRUKF gradually increase and even diverge.

However, due to not considering the residual sequence of the measurement noise in the dual-coefficient tracker, the SOC estimation accuracy using the developed method based on the dual-coefficient tracker is degraded. In addition, because of the addition of the tracking coefficient calculation, it takes more time for the developed method to accurately estimate the SOC of batteries compared to the UKF and the SRUKF. Some measures should be taken to optimize the iteration process to reduce the calculation time of the developed method.

5. Conclusions

Precise SOC estimation can contribute to protecting and extending the life of lithium-ion batteries. However, the SOC estimation accuracy estimated by the UKF and SRUKF is degraded when the noise statistics arise. Therefore, an SOC estimation method based on a dual-coefficient tracking improved square-root unscented Kalman filter (ISRUKF) was developed in this paper for lithium-ion batteries. Compared to the SRUKF and the UKF, the developed ISRUKF shows the highest SOC estimation accuracy with an SOC error below 1.5% when the $Q_k$ is 0.005 and 0.01, but the SOC error using the UKF and the SRUKF is 6.7% and 5.5%, respectively. At the same time, with the help of its dual-coefficient tracker, which can reduce state noise accumulative calculation errors, the SOC absolute error using the developed ISRUKF is always lower than the SOC absolute error using the ISRUKF with the standard STF when the $Q_k$ is constant (set as 0.005). For example, the SOC error using the ISRUKF with the standard STF gradually increases from 0.01 to 0.05, and its maximum error is about 5%, while the developed ISRUKF can keep the SOC error under 1.1%. Moreover, the developed ISRUKF can not only precisely estimate the SOC of lithium-ion batteries with the lowest SOC absolute error of 0.5% but also shows good robustness when the state noise statistics are set to 0.001, 0.005, and 0.01, respectively.

Future work includes the investigation of optimizing the dual-coefficient tracker. The SOC estimation accuracy of lithium-ion batteries is reduced using the developed method because of ignoring the measurement noise and corresponding noise covariance of the batteries, such as the measurement errors of the terminal voltage and current of the batteries. The optimization of the batteries’ measurement noise in the developed method will be performed in future works. Additionally, further research will also be carried out to explore reducing the calculation time of the developed ISRUKF. It is important for online SOC estimation to reduce the calculation time of the ISRUKF and other algorithms.