# State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Method Research

#### 2.1. Fractional-Order Calculus

#### 2.2. Fractional-Order Model

#### 2.3. Model Parameter Identification and Validation

## 3. Fuzzy Controller

## 4. SOC Estimation

- 1
**Initialization**- (1)
- Give the initial state ${x}_{o}$, Q, R and state error covariance P.

- 2
**Time updating**- (1)
- Calculate sigma points using the singular value decomposition:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \phantom{\rule{4pt}{0ex}}{P}_{k-1|k-1}& ={U}_{k-1}{S}_{k-1}{V}_{k-1}^{T},\hfill \\ \hfill {x}_{0,k-1|k-1}& ={\widehat{x}}_{k-1|k-1},\hfill \\ \hfill {x}_{i,k-1|k-1}& ={\widehat{x}}_{k-1|k-1}+\rho {U}_{i}\sqrt{{s}_{i}},\hfill \\ \hfill i& =1,2,\cdots ,n\hfill \\ \hfill {x}_{i,k-1|k-1}& ={\widehat{x}}_{k-1|k-1}-\rho {U}_{i}\sqrt{{s}_{i}},\hfill \\ \hfill i& =n+1,n+2,\cdots ,2n,\hfill \end{array}\right.\end{array}$$$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\omega}_{m}^{0}& =\frac{\lambda}{n+\lambda},\hfill \\ \hfill {\omega}_{c}^{0}& =\frac{\lambda}{n+\lambda}+(1-{\alpha}^{2}+\beta ),\hfill \\ \hfill {\omega}_{m}^{i}& ={\omega}_{c}^{i}=\frac{1}{2(n+\lambda )},i=1,2,\cdots ,2n,\hfill \end{array}\right.\end{array}$$

- (2)
- Transform the sigma sampling points using the nonlinear function $f(\xb7)$:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\varphi}_{i,k-1|k-1}& =f({x}_{i,k-1|k-1},{u}_{k-1}),i=0,1,...2n,\hfill \\ \hfill {D}^{\eta}{\widehat{x}}_{k|k-1}& =\sum _{i=0}^{2n}{\omega}_{m}^{i}{\varphi}_{i,k-1|k-1}.\hfill \end{array}\right.\end{array}$$

- (3)
- Update the prior states estimation. The mean and covariance of ${D}^{\eta}{x}_{k}$ and ${x}_{k}$ can be calculated by:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {P}_{k|k-1}^{\Delta \Delta}& =Cov\left[{D}^{\eta}{x}_{k|{y}_{k-1}}\right],\hfill \\ & =\sum _{i=0}^{2n}{\omega}_{c}^{j}({\varphi}_{i,k-1|k-1}-{D}^{\eta}{\widehat{x}}_{k|k-1}),\hfill \\ & \times {({\varphi}_{i,k-1|k-1}-{D}^{\eta}{\widehat{x}}_{k|k-1})}^{T}+Q,\hfill \\ \hfill {P}_{k|k-1}^{\Delta \Delta}& =Cov[{x}_{k-1},{D}^{\eta}{x}_{k|{y}_{k-1}}],\hfill \\ & =\sum _{i=0}^{2n}{\omega}_{c}^{j}({\varphi}_{i,k-1|k-1}-{D}^{\eta}{\widehat{x}}_{k|k-1}),\hfill \\ & \times {({\varphi}_{i,k-1|k-1}-{D}^{\eta}{\widehat{x}}_{k|k-1})}^{T},\hfill \end{array}\right.\end{array}$$$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\widehat{x}}_{k|k-1}& ={D}^{\eta}{\widehat{x}}_{k|k-1}-\sum _{j=1,}^{k}{(-1)}^{j}{\gamma}_{j}{\widehat{x}}_{k-j|k-j},\hfill \\ \hfill {P}_{k|k-1}& ={P}_{k|k-1}^{\Delta \Delta}+{\gamma}_{1}{P}_{k|k-1}^{x\Delta}\hfill \\ & +{P}_{k|k-1}^{\Delta x}{\gamma}_{1}+\sum _{j=1}^{k}{\gamma}_{j}{P}_{k-j|k-j}{\gamma}_{j}.\hfill \end{array}\right.\end{array}$$

- 3
**Observation updating**- (1)
- Calculate sigma points using the singular value decomposition. The weight of the sigma points is obtained using (20):$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill \phantom{\rule{4pt}{0ex}}{P}_{k|k-1}& ={U}_{k-1}{S}_{k-1}{V}_{k-1}^{T},\hfill \\ \hfill {x}_{0,k-1|k-1}& ={\widehat{x}}_{k-1|k-1},\hfill \\ \hfill {x}_{i,k-1|k-1}& ={\widehat{x}}_{k-1|k-1}+\rho {U}_{i}\sqrt{{s}_{i}},\hfill \\ \hfill i& =1,2,\cdots ,n\hfill \\ \hfill {x}_{i,k-1|k-1}& ={\widehat{x}}_{k-1|k-1}+\rho {U}_{i}\sqrt{{s}_{i}},\hfill \\ \hfill i& =n+1,n+2,\cdots ,2n.\hfill \end{array}\right.\end{array}$$

- (2)
- Transform the sigma sampling points using the nonlinear function $h(\xb7)$:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {\theta}_{i,k|k-1}& =h\left({x}_{i,k|k-1}\right),i=0,1,\cdots ,2n,\hfill \\ \hfill {\widehat{y}}_{k|k-1}& =\sum _{i=0}^{2n}{\omega}_{m}^{i}{\theta}_{i,k|k-1}.\hfill \end{array}\right.\end{array}$$

- (3)
- Estimate the observation-error covariance matrix:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {P}_{k|k-1}^{yy}& =Cov\left[{y}_{k}\right|{y}_{k-1}],\hfill \\ & =\sum _{i=0}^{2n}{\omega}_{c}^{j}({\theta}_{i,k|k-1}-{\widehat{y}}_{k|k-1}),\hfill \\ & \times {({\theta}_{i,k|k-1}-{\widehat{y}}_{k|k-1})}^{T}+R,\hfill \\ \hfill {P}_{k|k-1}^{xy}& =Cov[{x}_{k},{y}_{k}|{y}_{k-1}],\hfill \\ & =\sum _{i=0}^{2n}{\omega}_{c}^{j}({\theta}_{i,k|k-1}-{\widehat{y}}_{k|k-1}),\hfill \\ & \times {({\theta}_{i,k|k-1}-{\widehat{y}}_{k|k-1})}^{T}.\hfill \end{array}\right.\end{array}$$

- (4)
- Calculate the theoretical and actual covariances:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {M}_{k}& =\sum _{i=0}^{2n}{\omega}_{c}^{j}({\theta}_{i,k|k-1}-{\widehat{y}}_{k|k-1}),\hfill \\ & \times {({\theta}_{i,k|k-1}-{\widehat{y}}_{k|k-1})}^{T}+R,\hfill \\ \hfill {N}_{k}& =\frac{1}{n}\sum _{i}^{k}[{y}_{i}-{y}_{i|i-1}]{[{y}_{i}-{y}_{i|i-1}]}^{T}.\hfill \\ \hfill i& =k-n+1.\hfill \end{array}\right.\end{array}$$

- (5)
- Update the observation noise variance:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {G}_{k}& ={M}_{k}-{N}_{k},\hfill \\ \hfill \widehat{{V}_{k}}& ={\mu}_{k}{V}_{k},\hfill \end{array}\right.\end{array}$$

- (6)
- Update the posterior states estimation:$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {K}_{k}& ={P}_{k|k-1}^{xy}{\left({P}_{k|k-1}^{yy}\right)}^{-1},\hfill \\ \hfill {\widehat{x}}_{k|k}& ={\widehat{x}}_{k|k-1}+K({y}_{k}-{\widehat{y}}_{k|k-1}),\hfill \\ \hfill {P}_{k|k}& ={P}_{k|k-1}-{K}_{k}{P}_{k|k-1}^{yy}{K}_{k}^{T},\hfill \end{array}\right.\end{array}$$

## 5. Numerical Verification and Discussion

#### 5.1. Experimental Results at 25 °C

#### 5.2. Experimental Results at 0 °C

#### 5.3. Experimental Results at 45 °C

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**The SOC estimation curves under BJDST and FUDS at $25{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 11.**The SOC estimation error curves under BJDST and FUDS at $25{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 12.**The SOC estimation curves under BJDST and FUDS at $0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 13.**The SOC estimation error curves under BJDST and FUDS at $0{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 14.**The SOC estimation curves under BJDST and FUDS at $45{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

**Figure 15.**The SOC estimation error curves under BJDST and FUDS at $45{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$C.

${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{R}}_{0}$ |

$2.4877$ | $1.8243$ | $0.6608$ | $1.1131$ | $-3.2348$ | $0.0687$ |

${R}_{1}$ | ${C}_{1}$ | ${R}_{2}$ | ${C}_{2}$ | $\alpha $ | $\beta $ |

0.5975 | 264.25 | 1.2679 | 448.54 | 0.4325 | 0.4380 |

Input fuzziness | NB | NS | Z | PS | PB |

Output fuzziness | NB | NS | Z | PS | PB |

RMSE | EKF | FUKF | FFUKF |
---|---|---|---|

FUDS | 0.87% | 0.67% | 0.20% |

BJDST | 1.95% | 0.68% | 0.13% |

RMSE | EKF | FUKF | FFUKF |
---|---|---|---|

FUDS | 0.88% | 0.85% | 0.20% |

BJDST | 1.49% | 1.04% | 0.32% |

RMSE | EKF | FUKF | FFUKF |
---|---|---|---|

FUDS | 1.49% | 1.08% | 0.51% |

BJDST | 2.18% | 1.20% | 0.58% |

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**MDPI and ACS Style**

Chen, L.; Chen, Y.; Lopes, A.M.; Kong, H.; Wu, R.
State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter. *Fractal Fract.* **2021**, *5*, 91.
https://doi.org/10.3390/fractalfract5030091

**AMA Style**

Chen L, Chen Y, Lopes AM, Kong H, Wu R.
State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter. *Fractal and Fractional*. 2021; 5(3):91.
https://doi.org/10.3390/fractalfract5030091

**Chicago/Turabian Style**

Chen, Liping, Yu Chen, António M. Lopes, Huifang Kong, and Ranchao Wu.
2021. "State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter" *Fractal and Fractional* 5, no. 3: 91.
https://doi.org/10.3390/fractalfract5030091