# State of Charge Estimation of Lithium-Ion Battery Based on Improved Adaptive Unscented Kalman Filter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Battery State–Space Model

#### 2.1. Second-Order Thevenin Equivalent Circuit Model

_{o(SOC)}denotes the open-circuit voltage (OCV) of the battery and its value changes with the SOC; U

_{L}denotes the terminal voltage; I

_{c}denotes the charging/discharging current, and it is a positive number in the discharge process in this paper; R

_{p}and R

_{s}denote the polarization resistance; C

_{p}and C

_{s}denote the polarization capacitance; R

_{p}C

_{p}parallel network, and R

_{s}C

_{s}parallel network that simulates the concentration polarization and the electrochemical polarization of the battery. The polarization voltage across C

_{p}and C

_{s}are U

_{p}and U

_{s}, respectively.

_{p}, and U

_{s}of the two RC networks are selected as system state variables. Equation (1) is discretized to obtain the discrete battery state–space equation and output equations as follows:

_{p}U

_{s}]

^{T}; k represents the time step; x

_{k}and I

_{c,k}denote the state variable of the system and the charging/discharging current at time step k, respectively; τ

_{p}and τ

_{s}are time parameters, τ

_{p}= R

_{p}C

_{p}, τ

_{s}= R

_{s}C

_{s}; $\mathsf{\Delta}\mathrm{t}$ denotes the time interval of discretization; η denotes the Coulombic efficiency, which is assumed to be 1 for lithium-ion batteries; and C

_{N}denotes the nominal capacity of the battery. It should be noted that the analogous definition will apply throughout the paper if not otherwise stated.

_{k}denotes the measurement variable, and u

_{k}denotes the input variable.

_{k}denotes the process noise of the system, whose mean value is q

_{k}and covariance value is Q

_{k}, and v

_{k}denotes the measurement noise of the system, whose mean value is r

_{k}and covariance value is R

_{k}.

#### 2.2. Relationship between OCV and SOC

#### 2.3. Parameters Identification of Battery Model

_{o}, R

_{p}, R

_{s}, C

_{p,}and C

_{s}in the second-order RC equivalent circuit model of the battery. As one of the most commonly used methods in parameters identification, the least squares method has the characteristics of small computational effort and no requirement for prior statistical knowledge. In this paper, the offline recursive least squares method [22] is applied to identify the model parameters R

_{o}, R

_{p}, R

_{s}, C

_{p,}and C

_{s}under dynamic stress test (DST) [23]. In order to reduce the error of model parameters identification, this paper takes the average values of multiple battery parameters obtained by the offline recursive least squares as the final identification results. The identification results are shown in Table 1.

_{o}, R

_{p}, R

_{s}, C

_{p,}and C

_{s}identified in Section 2.3 are substituted into Equation (8) to obtain the state–space equation and output equation of 18,650 power lithium battery. Based on the state–space equation and output equation, the IAUKF can realize the real-time optimal estimation of the battery SOC.

## 3. State of Charge Estimation

#### 3.1. Singular Value Decomposition

_{1}, s

_{2}, … , s

_{r}) is a diagonal matrix, which is composed of singular values of matrix P; and r is the rank of matrix P, and s

_{1}≥ s

_{2}≥ … ≥ s

_{r}> 0.

#### 3.2. Description of the IUKF

- (1)
- Initialize the mean value ${\overline{\mathrm{x}}}_{0}$ of the system state variable x
_{0}, the error covariance P_{0,}and the weighted coefficient.$$\{\begin{array}{l}{\overline{\mathrm{x}}}_{0}{=\mathrm{E}(\mathrm{x}}_{0})\\ {\mathrm{P}}_{0}{=\mathrm{E}[(\mathrm{x}}_{0}-{\overline{\mathrm{x}}}_{0}{)(\mathrm{x}}_{0}-{\overline{\mathrm{x}}}_{0}{)}^{\mathrm{T}}]\end{array}$$$$\{\begin{array}{l}{\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}=\frac{\mathsf{\lambda}}{\mathrm{n}+\mathsf{\lambda}},\mathrm{i}=0\\ {\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}=\frac{\mathsf{\lambda}}{\mathrm{n}+\mathsf{\lambda}}+(1-{\mathsf{\alpha}}^{2}+\mathsf{\beta}),\mathrm{i}=0\\ {\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}{=\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}=\frac{1}{2(\mathrm{n}+\mathsf{\lambda})},\mathrm{i}=1~2\mathrm{n}\end{array}$$^{−4}~1; λ is a composite coefficient, which can be expressed as$${\mathsf{\lambda}=\mathsf{\alpha}}^{2}(\mathrm{n}+\mathsf{\kappa})-\mathrm{n}$$ - (2)
- Calculate Sigma sampling points of the state variable at time step k − 1 using SVD.$${\mathrm{P}}_{\mathrm{k}-1}{=\mathrm{U}}_{\mathrm{k}-1}\left(\begin{array}{cc}{\mathrm{S}}_{\mathrm{k}-1}& 0\\ 0& 0\end{array}\right){\mathrm{G}}_{\mathrm{k}-1}{}^{\mathrm{T}}$$$$\{\begin{array}{l}{\mathrm{x}}_{\mathrm{i},\mathrm{k}-1}{=\hat{\mathrm{x}}}_{\mathrm{k}-1},\text{}\mathrm{i}=0\\ {\mathrm{x}}_{\mathrm{i},\mathrm{k}-1}{=\hat{\mathrm{x}}}_{\mathrm{k}-1}+(\sqrt{(\mathrm{n}+\mathsf{\lambda})}{\mathrm{U}}_{\mathrm{k}-1}\sqrt{{\mathrm{S}}_{\mathrm{k}-1}}{)}_{\mathrm{i}},\text{}\mathrm{i}=1~\mathrm{n}\\ {\mathrm{x}}_{\mathrm{i},\mathrm{k}-1}{=\hat{\mathrm{x}}}_{\mathrm{k}-1}-{(\sqrt{(\mathrm{n}+\mathsf{\lambda})}{\mathrm{U}}_{\mathrm{k}-1}\sqrt{{\mathrm{S}}_{\mathrm{k}-1}})}_{\mathrm{i}},\text{}\mathrm{i}=\mathrm{n}+1~2\mathrm{n}\end{array}$$
_{i,k−}_{1}denotes the Sigma sampling point; the symbol “^” denotes the estimated value and ${\hat{\mathrm{x}}}_{\mathrm{k}-1}$ is the optimal estimate of the state variable at time step k − 1; ${\mathrm{P}}_{\mathrm{k}-1}$ is the error covariance at time step k − 1; and (·)_{i}denotes the ith column of the matrix. - (3)
- Time update for the mean and covariance of the state variable: use 2n + 1 Sigma sampling points obtained in step (2) to calculate the predicted mean and error covariance of the state variable in time step k.$${\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1}={\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}{\mathrm{f}(\mathrm{x}}_{\mathrm{i},\mathrm{k}-1}{,\mathrm{u}}_{\mathrm{k}})}{+\mathrm{q}}_{\mathrm{k}-1}$$$${\mathrm{P}}_{\mathrm{k}|\mathrm{k}-1}={\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}{(\hat{\mathrm{x}}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1})}{{(\hat{\mathrm{x}}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1})}^{\mathrm{T}}{+\mathrm{Q}}_{\mathrm{k}-1}$$$${\hat{\mathrm{x}}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}{=\mathrm{f}(\mathrm{x}}_{\mathrm{i},\mathrm{k}-1}{,\mathrm{u}}_{\mathrm{k}}),\mathrm{i}=0~2\mathrm{n}$$
- (4)
- Use SVD again to obtain new Sigma sampling points based on the predicted mean and error covariance of the state variable obtained in step (3).$${\mathrm{P}}_{\mathrm{k}|\mathrm{k}-1}{=\mathrm{U}}_{\mathrm{k}|\mathrm{k}-1}(\begin{array}{cc}{\mathrm{S}}_{\mathrm{k}|\mathrm{k}-1}& 0\\ 0& 0\end{array}){\mathrm{G}}_{\mathrm{k}|\mathrm{k}-1}{}^{\mathrm{T}}$$$$\{\begin{array}{l}{\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}{=\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1},\text{}\mathrm{i}=0\\ {\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}{=\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1}+(\sqrt{(\mathrm{n}+\mathsf{\lambda})}{\mathrm{U}}_{\mathrm{k}|\mathrm{k}-1}\sqrt{{\mathrm{S}}_{\mathrm{k}|\mathrm{k}-1}}{)}_{\mathrm{i}},\text{}\mathrm{i}=1~\mathrm{n}\\ {\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}{=\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1}-{(\sqrt{(\mathrm{n}+\mathsf{\lambda})}{\mathrm{U}}_{\mathrm{k}|\mathrm{k}-1}\sqrt{{\mathrm{S}}_{\mathrm{k}|\mathrm{k}-1}})}_{\mathrm{i}},\text{}\mathrm{i}=\mathrm{n}+1~2\mathrm{n}\end{array}$$
- (5)
- Time update for the mean of the measurement variable: calculate the predicted values of the measurement variable at time step k based on regained Sigma points in step (4).$${\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1}={\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}{\mathrm{g}(\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}{,\mathrm{u}}_{\mathrm{k}})}{+\mathrm{r}}_{\mathrm{k}-1}$$
- (6)
- Calculate the IUKF gain matrix K
_{k}.$${\mathrm{K}}_{\mathrm{k}}{=\mathrm{P}}_{\mathrm{xy},\mathrm{k}}{\mathrm{P}}_{\mathrm{y},\mathrm{k}}^{-1}$$$$\{\begin{array}{l}{\mathrm{P}}_{\mathrm{y},\mathrm{k}}={\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}{(\widehat{\chi}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1})}{{(\widehat{\chi}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1})}^{\mathrm{T}}{+\mathrm{R}}_{\mathrm{k}-1}\\ {\mathrm{P}}_{\mathrm{xy},\mathrm{k}}={\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}{(\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}-{\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1})}{{(\widehat{\chi}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1})}^{\mathrm{T}}\end{array}$$$${\widehat{\chi}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}{=\mathrm{g}(\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}{,\mathrm{u}}_{\mathrm{k}})$$ - (7)
- Measurement correction of the state variable: calculate the corrected state variable estimated value and the optimal covariance matrix at time step k.$${\hat{\mathrm{x}}}_{\mathrm{k}}{=\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1}{+\mathrm{K}}_{\mathrm{k}}{(\mathrm{y}}_{\mathrm{k}}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1})$$$${\mathrm{P}}_{\mathrm{k}}{=\mathrm{P}}_{\mathrm{k}|\mathrm{k}-1}-{\mathrm{K}}_{\mathrm{k}}{\mathrm{P}}_{\mathrm{y},\mathrm{k}}{\mathrm{K}}_{\mathrm{k}}{}^{\mathrm{T}}$$

#### 3.3. Description of Sage–Husa Adaptive Filter

- (1)
- Calculate the estimated mean value of the system process noise.$${\hat{\mathrm{q}}}_{\mathrm{k}}=(1-{\mathrm{d}}_{\mathrm{k}}{)\hat{\mathrm{q}}}_{\mathrm{k}-1}{+\mathrm{d}}_{\mathrm{k}}{(\hat{\mathrm{x}}}_{\mathrm{k}}-{\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}{\mathrm{f}(\mathrm{x}}_{\mathrm{i},\mathrm{k}-1}{,\mathrm{u}}_{\mathrm{k}})})$$
_{k}can be expressed as$${\mathrm{d}}_{\mathrm{k}}=\frac{1-\mathrm{b}}{1-{\mathrm{b}}^{\mathrm{k}+1}}$$ - (2)
- Calculate the estimated covariance value of the system process noise.$${\hat{\mathrm{Q}}}_{\mathrm{k}}=(1-{\mathrm{d}}_{\mathrm{k}}){\hat{\mathrm{Q}}}_{\mathrm{k}-1}{+\mathrm{d}}_{\mathrm{k}}[{\mathrm{K}}_{\mathrm{k}}{\mathrm{e}}_{\mathrm{k}}{\mathrm{e}}_{\mathrm{k}}^{\mathrm{T}}{\mathrm{K}}_{\mathrm{k}}^{\mathrm{T}}{+\mathrm{P}}_{\mathrm{k}}-{\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}{(\hat{\mathrm{x}}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1})}{{\hat{\mathrm{x}}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{x}}}_{\mathrm{k}|\mathrm{k}-1})}^{\mathrm{T}}]$$
_{k}is a residual, which can be expressed as$${\mathrm{e}}_{\mathrm{k}}{=\mathrm{y}}_{\mathrm{k}}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1}$$ - (3)
- Calculate the estimated mean value of the system measurement noise.$${\hat{\mathrm{r}}}_{\mathrm{k}}=(1-{\mathrm{d}}_{\mathrm{k}}){\hat{\mathrm{r}}}_{\mathrm{k}-1}{+\mathrm{d}}_{\mathrm{k}}{(\mathrm{y}}_{\mathrm{k}}-{\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}{\mathrm{g}(\mathrm{x}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}^{-}{,\mathrm{u}}_{\mathrm{k}})})$$
- (4)
- Calculate the estimated covariance value of the system measurement noise.$${\hat{\mathrm{R}}}_{\mathrm{k}}=(1-{\mathrm{d}}_{\mathrm{k}}{)\hat{\mathrm{R}}}_{\mathrm{k}-1}{+\mathrm{d}}_{\mathrm{k}}{[\mathrm{e}}_{\mathrm{k}}{\mathrm{e}}_{\mathrm{k}}^{\mathrm{T}}-{\displaystyle \sum _{\mathrm{i}=0}^{2\mathrm{n}}{\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}{(\widehat{\chi}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1})}{{(\widehat{\chi}}_{\mathrm{i},\mathrm{k}|\mathrm{k}-1}-{\hat{\mathrm{y}}}_{\mathrm{k}|\mathrm{k}-1})}^{\mathrm{T}}]$$

#### 3.4. Estimate SOC Using IAUKF

_{0}of error covariance, the initial mean value ${\hat{\mathrm{q}}}_{0}$ and covariance value ${\hat{\mathrm{Q}}}_{0}$ of process noise, the initial mean value ${\hat{\mathrm{r}}}_{0}$, and covariance value ${\hat{\mathrm{R}}}_{0}$ of measurement noise. Additionally, the weighted coefficients ${\mathsf{\omega}}_{\mathrm{m}}^{(\mathrm{i})}$ and ${\mathsf{\omega}}_{\mathrm{c}}^{(\mathrm{i})}$ are also calculated. Thirdly, according to IAUKF calculation steps, the optimal estimated value of system state variable at time step k is obtained through Equations (15)–(27). The estimated SOC at time step k can be separated from the state variable optimal estimated value ${\hat{\mathrm{x}}}_{\mathrm{k}}$. It should be noted that when estimating the SOC of the battery at time step k, it is necessary to replace ${\mathrm{q}}_{\mathrm{k}-1}$ in Equation (17), ${\mathrm{Q}}_{\mathrm{k}-1}$ in Equation (18), ${\mathrm{r}}_{\mathrm{k}-1}$ in Equation (22) and ${\mathrm{R}}_{\mathrm{k}-1}$ in Equation (24) with ${\hat{\mathrm{q}}}_{\mathrm{k}-1}$, ${\hat{\mathrm{Q}}}_{\mathrm{k}-1}$, ${\hat{\mathrm{r}}}_{\mathrm{k}-1}$ and ${\hat{\mathrm{R}}}_{\mathrm{k}-1}$, which are calculated based on the principle of Equation (35) using the relevant parameters at time step k − 1. Fourthly, based on the relevant parameters at the time step k, ${\hat{\mathrm{q}}}_{\mathrm{k}}$, ${\hat{\mathrm{Q}}}_{\mathrm{k}}$, ${\hat{\mathrm{r}}}_{\mathrm{k}}$ and ${\hat{\mathrm{R}}}_{\mathrm{k}}$ are calculated through Equation (35) for the SOC estimation at time step k + 1; finally, determine whether the battery discharges to the lower cutoff voltage. If the battery has discharged to the lower cutoff voltage, then, end the SOC estimation. Otherwise, continue the SOC estimation.

## 4. Experimental Simulation and Verification

_{c}and terminal voltage U

_{L}curves of the lithium-ion battery under the FUDS test are shown in Figure 4 and Figure 5. In Figure 4, the plus value represents the discharging current, and the minus value represents the charging current.

^{T}, P

_{0}= 10

^{−1}× E

_{3×3}, ${\hat{\mathrm{q}}}_{0}=0$, ${\hat{\mathrm{Q}}}_{0}={10}^{-6}\times {\mathrm{E}}_{3\times 3}$, ${\hat{\mathrm{r}}}_{0}=0$, ${\hat{\mathrm{R}}}_{0}=0.1$, where E

_{3×3}denotes the unit matrix of degree 3.

_{k}denotes the theoretical value of SOC; ${\widehat{\mathrm{S}}}_{\mathrm{k}}$ denotes the estimated value of SOC; L is the number of samples; |·| denotes the absolute value; and max (·) represents the maximum value of all samples.

_{0}= −10

^{−1}× E

_{3×3}, and the other initial parameters are unchanged. In this case, the UKF cannot carry out the Cholesky factor decomposition due to the non-positive error covariance, which leads to the failure of SOC estimation. However, the IUKF and IAUKF can still complete the SOC estimation. When the error covariance is negative definite and positive definite, the SOC estimation error comparative curves of the IUKF are shown in Figure 8, and the SOC estimation error comparative curves of the IAUKF are shown in Figure 9.

## 5. Conclusions

- (1)
- Replacing the Cholesky decomposition in traditional UKF with SVD will not decrease the SOC estimation accuracy, and the IUKF and UKF can achieve almost the same SOC estimation results. The MAE of the traditional UKF and IUKF both are 2.4%, and the RMSE of the two methods both are 0.0094 when the character length limits of the computer are taken into consideration.
- (2)
- With the fantastic numerical stability of SVD, the IUKF can suppress the non-positive definiteness of the error covariance, so as to improve the stability of SOC estimation, and this characteristic can also be found in IAUKF, namely, whether the initial error covariance is positive definite or negative definite, both the IUKF and IAUKF can still complete the SOC estimation of the battery. However, when the initial error covariance is negative definite, the traditional UKF fails to estimate the SOC because the Cholesky decomposition cannot be carried out any longer.
- (3)
- The IAUKF is formed by combining IUKF with the Sage–Husa adaptive filter. With the help of the Sage–Husa adaptive filter, the IAUKF can realize the adaptive estimation of the system process and measurement noise during the process of SOC estimation.
- (4)
- The influence of the unknown or inaccurate system noise statistics on SOC estimation precision can be reduced by the IAUKF; thereby the SOC estimation accuracy can be improved. The MAE of the IAUKF is 1.92%, and the RMSE is 0.005; the two values are all smaller than those of IUKF and traditional UKF.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**SOC estimation error comparative curves of FUDS test when the initial error covariance matrix is positive and negative definite of the IUKF.

**Figure 9.**SOC estimation error comparative curves of FUDS test when the initial error covariance matrix is positive and negative definite of the IAUKF.

Parameters | Value |
---|---|

R_{o}/mΩ | 70.6 |

R_{p}/mΩ | 18 |

C_{p}/F | 223.74 |

R_{s}/mΩ | 44.9 |

C_{s}/F | 1261.7 |

Estimation Method | MAE (%) | RMSE |
---|---|---|

IAUKF | 1.92 | 0.005 |

IUKF | 2.4 | 0.0094 |

UKF | 2.4 | 0.0094 |

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**MDPI and ACS Style**

Xing, J.; Wu, P.
State of Charge Estimation of Lithium-Ion Battery Based on Improved Adaptive Unscented Kalman Filter. *Sustainability* **2021**, *13*, 5046.
https://doi.org/10.3390/su13095046

**AMA Style**

Xing J, Wu P.
State of Charge Estimation of Lithium-Ion Battery Based on Improved Adaptive Unscented Kalman Filter. *Sustainability*. 2021; 13(9):5046.
https://doi.org/10.3390/su13095046

**Chicago/Turabian Style**

Xing, Jie, and Peng Wu.
2021. "State of Charge Estimation of Lithium-Ion Battery Based on Improved Adaptive Unscented Kalman Filter" *Sustainability* 13, no. 9: 5046.
https://doi.org/10.3390/su13095046