# A Variational Bayesian and Huber-Based Robust Square Root Cubature Kalman Filter for Lithium-Ion Battery State of Charge Estimation

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## Abstract

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## 1. Introduction

## 2. Battery Modeling and Parameter Identification

#### 2.1. Battery Modeling

#### 2.2. OCV–SOC Relationship Determination

- (1)
- First, fully charge the battery and rest for 1 h to finish the process of depolarization. Then the measured terminal voltage is assumed to be the discharge OCV value.
- (2)
- Discharge the battery at a constant current of $1A$ until $10\%$ of the maximum available capacity is consumed, and measure the OCV after resting for 1 h.
- (3)
- Repeat step (2) until the battery reaches its lower cut-off voltage.

#### 2.3. Online Parameter Identification

#### 2.4. Model Validation

## 3. Variational Bayesian-Based Adaptive Square Root Cubature Kalman Filter

**Step****1.**- Initialize state and parameter estimation: ${\widehat{x}}_{0}$, ${S}_{0}$, ${Q}_{0}$, ${\upsilon}_{0}$, ${V}_{0}$.
**Step****2.**- Predict ($k=1,2,3,\dots $).
**Step****2.1**- Calculate the cubature points$${X}_{i,k-1}={S}_{k-1}{\xi}_{i}+{\widehat{x}}_{k-1}\phantom{\rule{1.em}{0ex}}i=1,2,\dots ;2n,$$
**Step****2.2**- Propagate the cubature points through the process equation and calculate the predicted state values:$${\chi}_{i,k|k-1}=f({X}_{i,k-1},{I}_{L,k-1}),$$$${\widehat{x}}_{k|k-1}=\frac{1}{2n}\sum _{i=1}^{2n}{\chi}_{i,k|k-1}.$$
**Step****2.3**- Calculate the square root of the covariance of the predicted state$${S}_{k|k-1}=Tria({[{\chi}_{k|k-1}^{\ast}\phantom{\rule{1.em}{0ex}}{S}_{Q,k-1}]}^{T}),$$$${\chi}_{k|k-1}^{\ast}=\frac{1}{\sqrt{2n}}[{\chi}_{1,k|k-1}-{\widehat{x}}_{k|k-1},\dots ,\phantom{\rule{0.166667em}{0ex}}{\chi}_{2n,k|k-1}-{\widehat{x}}_{k|k-1}].$$
**Step****2.4**- Calculate the parameters of the IW distribution of measurement noise covariance$${\upsilon}_{k|k-1}=\rho ({\upsilon}_{k-1}-n-1)+n+1,$$$${V}_{k|k-1}=B{V}_{k-1}{B}^{T},$$

**Step****3.**- Update: the update of VB-ASRCKF utilizes an iterate filtering framework.
**Step****3.1**- First set ${\widehat{x}}_{k}^{\left(0\right)}={\widehat{x}}_{k|k-1}$, ${S}_{k}^{\left(0\right)}={S}_{k|k-1}$, ${V}_{k}^{\left(0\right)}={V}_{k|k-1}$, and ${\upsilon}_{k}=1+{\upsilon}_{k|k-1}$.
**Step****3.2**- Calculate the cubature points of the predicted state$${X}_{i,k|k-1}={S}_{k|k-1}{\xi}_{i}+{\widehat{x}}_{k|k-1}.$$
**Step****3.3**- Propagate the cubature points through the measurement equation and calculate the predicted measurement value$${Z}_{i,k|k-1}=h\left({X}_{i,k|k-1}\right),$$$${\widehat{z}}_{k|k-1}=\frac{1}{2n}\sum _{i=1}^{2n}{Z}_{i,k|k-1}.$$
**Step****3.4**- Calculate the covariance of the state and the measurement$${P}_{xz,k|k-1}={\chi}_{k|k-1}{Z}_{k|k-1}^{T},$$$${\chi}_{k|k-1}=\frac{1}{\sqrt{2n}}[{X}_{1,k|k-1}-{\widehat{x}}_{k|k-1},\dots ,\phantom{\rule{0.166667em}{0ex}}{X}_{2n,k|k-1}-{\widehat{x}}_{k|k-1}],$$$${Z}_{k|k-1}=\frac{1}{\sqrt{2n}}[{Z}_{1,k|k-1}-{\widehat{z}}_{k|k-1},\dots ,\phantom{\rule{0.166667em}{0ex}}{Z}_{2n,k|k-1}-{\widehat{z}}_{k|k-1}].$$
**Step****3.5**- For $j=1:N$, iterate the following N (N denotes iterated times) steps.
**Step****3.5.1**- Calculate the measurement covariance$${R}_{k}^{\left(j\right)}={({\upsilon}_{k}-n-1)}^{-1}{V}_{k}^{(j-1)}.$$
**Step****3.5.2**- Calculate the square root of the innovation covariance$${S}_{zz,k|k-1}^{\left(j\right)}=Tria({[{Z}_{k|k-1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}_{R,k}^{\left(j\right)}]}^{T}),$$
**Step****3.5.3**- Calculate the filter gain$${K}_{k}^{\left(j\right)}=\frac{{P}_{xz,k|k-1}/{\left[{S}_{zz,k|k-1}^{\left(j\right)}\right]}^{T}}{{S}_{zz,k|k-1}^{\left(j\right)}}.$$
**Step****3.5.4**- Calculate the state estimate and the square root of its covariance$${\widehat{x}}_{k}^{\left(j\right)}={\widehat{x}}_{k|k-1}+{K}_{k}^{\left(j\right)}({z}_{k}-{\widehat{z}}_{k|k-1}),$$$${S}_{k}^{\left(j\right)}=Tria\left({[{\chi}_{k|k-1}-{K}_{k}^{\left(j\right)}{Z}_{k|k-1}\phantom{\rule{1.em}{0ex}}{K}_{k}^{\left(j\right)}{S}_{R,k}^{\left(j\right)}]}^{T}\right).$$
**Step****3.5.5**- Calculate the updated parameter of the IW distribution of measurement noise covariance$${X}_{i,k}^{\left(j\right)}={S}_{k}^{\left(j\right)}{\xi}_{i}+{\widehat{x}}_{k}^{\left(j\right)},$$$${V}_{k}^{(j)}={V}_{k|k-1}+\frac{1}{2n}\sum _{i=1}^{2n}({z}_{k}-h({X}_{i,k}^{(j)})){({z}_{k}-h({X}_{i,k}^{(j)}))}^{T}.$$

**Step****3.6**- Until $j=N$, set ${\widehat{x}}_{k}={\widehat{x}}_{k}^{\left(N\right)}$, ${S}_{k}={S}_{k}^{\left(N\right)}$, ${V}_{k}={V}_{k}^{\left(N\right)}$, and end for. Then one cycle of the VB-ASRCKF algorithm is finished.

## 4. Variational Bayesian and Huber-Based Robust Square Root Cubature Kalman Filter

**Step****1.**- Initialize: ${\widehat{x}}_{0}$, ${S}_{0}$, ${Q}_{0}$, ${\upsilon}_{0}$, ${V}_{0}$.
**Step****2.****Step****3.**- Update.
**Step****3.1****Step****3.2**- Set ${\widehat{x}}_{k}^{\left(0\right)}={\widehat{x}}_{k|k-1}$, ${S}_{k}^{\left(0\right)}={S}_{k|k-1}$, ${V}_{k}^{\left(0\right)}={V}_{k|k-1}$, and ${\upsilon}_{k}=1+{\upsilon}_{k|k-1}$
**Step****3.3**- For $j=1:N$, iterate the following N (N denotes iterated times) steps.

**Step****3.3.1****Step****3.3.2****Step****3.3.3**- Replace ${z}_{k}$ with ${\tilde{z}}_{k}$ in (22)–(25). That is, proceed with the following calculations:$$\begin{array}{cc}\hfill {\widehat{x}}_{k}^{\left(j\right)}& ={\widehat{x}}_{k|k-1}+{K}_{k}^{\left(j\right)}({\tilde{z}}_{k}-{\widehat{z}}_{k|k-1}),\hfill \\ \hfill {S}_{k}^{(j)}& =Tria({[{\chi}_{k|k-1}-{K}_{k}^{(j)}{Z}_{k|k-1}\phantom{\rule{1.em}{0ex}}{K}_{k}^{(j)}{S}_{R,k}^{(j)}]}^{T}),\hfill \\ \hfill {X}_{i,k}^{\left(j\right)}& ={S}_{k}^{\left(j\right)}{\xi}_{i}+{\widehat{x}}_{k}^{\left(j\right)},\hfill \\ \hfill {V}_{k}^{\left(j\right)}& ={V}_{k|k-1}+\frac{1}{2n}\sum _{i=1}^{2n}({\tilde{z}}_{k}-h({X}_{i,k}^{(j)})){({\tilde{z}}_{k}-h({X}_{i,k}^{(j)}))}^{T},\hfill \end{array}$$

**Step****3.4**- Until $j=N$, set ${\widehat{x}}_{k}={\widehat{x}}_{k}^{\left(N\right)}$, ${S}_{k}={S}_{k}^{\left(N\right)}$, ${V}_{k}={V}_{k}^{\left(N\right)}$ and end for. Then one cycle of the VB-HASRCKF algorithm is finished.

## 5. Experimental Verification and Analysis

#### 5.1. Experimental Settings

**Case a**: without any outliers or mistuning;

**Case b**: with mistuned measurement noise covariance;

**Case c**: with outliers in the measurements;

**Case d**: with both mistuned measurement noise covariance and outliers in the measurements.

#### 5.2. SOC Estimation Experimental Results under a 1A Constant-Current Discharge Test

#### 5.3. SOC Estimation Experimental Results under UDDS Test

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ACKF | Adaptive Cubature Kalman Filter |

ALS | Autocovariance Least Squares |

ASRCKF | Adaptive Square Root Cubature Kalman Filter |

ANN | Artificial Neural Network |

BESS | Battery Energy Storage System |

BMS | Battery Management System |

CC | Coulomb Counting |

CE | Coulomb Efficiency |

CKF | Cubature Kalman Filter |

ECM | Equivalent Circuit Model |

EKF | Extended Kalman Filter |

EV | Electric Vehicle |

FL | Fuzzy Logic |

IW | Inverse Wishart |

KF | Kalman Filter |

KL | Kullback–Leibler |

MM | Multiple Model |

OCV | Open-Circuit Voltage |

PF | Particle Filter |

RC | Resistor–Capacitor |

SOC | State Of Charge |

SVM | Support Vector Machine |

SRCKF | Square Root Cubature Kalman Filter |

UDDS | Urban Dynamometer Driving Schedule |

UKF | Unscented Kalman Filter |

VB | Variational Bayesian |

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**Figure 4.**Experimental terminal voltage results in an urban dynamometer driving schedule (UDDS) test.

**Figure 6.**The outlying voltage and current measurements in a constant-current discharge test under cases c and d.

**Figure 7.**SOC estimation results of four filters in a constant-current discharge test under cases a and b.

**Figure 8.**SOC estimation results of four filters in a constant-current discharge test under cases c and d.

**Table 1.**The maximum and mean absolute estimation errors of four filters in a constant-current discharge test under four cases. MaE = maximum absolute error, MeE = mean absolute error.

Case a | Case b | Case c | Case d | |||||
---|---|---|---|---|---|---|---|---|

MaE | MeE | MaE | MeE | MaE | MeE | MaE | MeE | |

SRCKF | $1.52\%$ | $0.69\%$ | $5.51\%$ | $2.24\%$ | $6.92\%$ | $3.50\%$ | $8.15\%$ | $4.84\%$ |

HSRCKF | $1.84\%$ | $0.76\%$ | $5.51\%$ | $2.24\%$ | $6.68\%$ | $2.47\%$ | $7.76\%$ | $4.25\%$ |

VB-ASRCKF | $2.00\%$ | $0.67\%$ | $2.02\%$ | $0.67\%$ | $7.00\%$ | $2.85\%$ | $7.30\%$ | $2.85\%$ |

VB-HASRCKF | $1.97\%$ | $0.70\%$ | $2.01\%$ | $0.70\%$ | $6.62\%$ | $1.91\%$ | $6.73\%$ | $1.91\%$ |

Constant Current Discharge Test | UDDS Test | |
---|---|---|

SRCKF | 12.1816 s | 30.1736 s |

HSRCKF | 12.4577 s | 31.1709 s |

VB-ASRCKF | 13.1058 s | 32.7837 s |

VB-HASRCKF | 14.1003 s | 35.4377 s |

**Table 3.**The maximum and mean absolute estimation errors of four filters (after 10 min) in the UDDS test under four cases.

Case a | Case b | Case c | Case d | |||||
---|---|---|---|---|---|---|---|---|

MaE | MeE | MaE | MeE | MaE | MeE | MaE | MeE | |

SRCKF | $1.46\%$ | $0.91\%$ | $4.87\%$ | $4.49\%$ | $6.72\%$ | $4.14\%$ | $7.88\%$ | $4.63\%$ |

HSRCKF | $1.80\%$ | $1.28\%$ | $5.68\%$ | $5.38\%$ | $2.49\%$ | $1.44\%$ | $4.77\%$ | $3.85\%$ |

VB-ASRCKF | $2.85\%$ | $1.03\%$ | $2.88\%$ | $1.03\%$ | $4.15\%$ | $2.71\%$ | $4.22\%$ | $2.71\%$ |

VB-HASRCKF | $2.85\%$ | $1.03\%$ | $2.84\%$ | $1.03\%$ | $2.80\%$ | $1.18\%$ | $2.96\%$ | $1.18\%$ |

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**MDPI and ACS Style**

Hou, J.; He, H.; Yang, Y.; Gao, T.; Zhang, Y.
A Variational Bayesian and Huber-Based Robust Square Root Cubature Kalman Filter for Lithium-Ion Battery State of Charge Estimation. *Energies* **2019**, *12*, 1717.
https://doi.org/10.3390/en12091717

**AMA Style**

Hou J, He H, Yang Y, Gao T, Zhang Y.
A Variational Bayesian and Huber-Based Robust Square Root Cubature Kalman Filter for Lithium-Ion Battery State of Charge Estimation. *Energies*. 2019; 12(9):1717.
https://doi.org/10.3390/en12091717

**Chicago/Turabian Style**

Hou, Jing, He He, Yan Yang, Tian Gao, and Yifan Zhang.
2019. "A Variational Bayesian and Huber-Based Robust Square Root Cubature Kalman Filter for Lithium-Ion Battery State of Charge Estimation" *Energies* 12, no. 9: 1717.
https://doi.org/10.3390/en12091717