# A State-of-Charge Estimation Method Based on Multi-Algorithm Fusion

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## Abstract

**:**

## 1. Introduction

## 2. Modelling and Parameter Identification

#### 2.1. Modelling for Lithium-Ion Batteries

#### 2.2. OCV-SOC Curve

- The battery is fully charged through the standard constant current and constant voltage (CC-CV) charging method. After standing for 5 h, the terminal voltage was measured. This value is regarded as the open circuit voltage value of SOC = 100%.
- Discharge with standard current and constant current. The cutoff condition is that the discharge capacity reaches 5% of the maximum available capacity, or the battery voltage drops to the discharge cutoff voltage. After standing for 5 h, measure the terminal voltage.
- Repeat step 2 until the power battery reaches the discharge cutoff voltage.

_{0}= 3.04342712, α

_{1}= 0.69965544, α

_{2}= −0.70823233, α

_{3}= 0.31435882, α

_{4}= −0.014878876, α

_{5}= −0.10513395, α

_{6}= −0.01276305.

#### 2.3. Parameter Identification

- Fully charge the two batteries with the CC-CV charging method.
- Let stand for 5 h.
- Load the mixed pulse current excitation sequence, discharge the battery with constant current for a certain period of time, and then let it stand for 1 h. (Constant current discharge of battery to ensure 10% SOC interval between two times).
- Repeat step 3 until the discharge reaches the cutoff voltage.

## 3. State of Charge Estimation

#### 3.1. State of Charge Definition

_{t}is the present SOC; SOC

_{0}is the SOC’s initial value; i

_{L}is the instantaneous load current (assumed positive for discharge, negative for charge); η

_{i}is the Coulomb efficiency, which is the function of the current and the temperature; and Ca is the present maximum available capacity, which may be different from the rated capacity for the age effect.

#### 3.2. Extended Kalman Filter Estimation Method

#### 3.3. SOC Estimation Algorithm with Adaptive Extended Kalman Filter Method

#### 3.4. H Infinity Filter SOC Estimation Algorithm

#### 3.5. Multi-Algorithm Fusion SOC Estimation

- 1.
- Import the terminal voltage residuals of the previous three algorithms.
- 2.
- Calculate the residual mean ${\overline{r}}_{k,i}$ and variance ${s}_{k,i}$ of each algorithm. ($i$ = 1,2,3, corresponding to the three algorithms).$$\{\begin{array}{l}{\overline{r}}_{k,i}=\frac{1}{M}{\displaystyle {\displaystyle \sum}_{j=k-M-1}^{k}}{r}_{j,i}\\ {s}_{k,i}=\frac{1}{M}{\displaystyle {\displaystyle \sum}_{j=k-M-1}^{k}}({r}_{j,i}-{\overline{r}}_{k,i})\end{array}$$
- 3.
- Calculate the conditional probability density function at time k for each algorithm.$${f}_{{Y}_{k}|{\alpha}_{\mathrm{i}},{Y}_{k-1}}=\frac{1}{\sqrt{2\pi {s}_{k,i}}}\mathrm{exp}(-\frac{1}{2}{\overline{r}}_{k,i}{}^{T}{s}_{k,i}{\overline{r}}_{k,i})$$
- 4.
- Calculate the weight ${\omega}_{k}$ of each algorithm at time k, where n is the number of algorithms.$${\omega}_{k,i}=(1-\frac{{f}_{{Y}_{k}|{\alpha}_{\mathrm{i}},{Y}_{k-1}}{\mathrm{s}}_{k,i}}{{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{f}_{{Y}_{k}|{\alpha}_{\mathrm{i}},{Y}_{k-1}}{\mathrm{s}}_{k,i}})/(n-1)$$
- 5.
- Obtain the SOC estimated value of the fusion algorithm according to the weight.$${Z}_{F,\mathrm{k}}={\displaystyle \sum}_{i=1}^{n}{\omega}_{k,i}{Z}_{k,i}$$

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

EV | Electric vehicle |

SOC | State of charge |

EKF | Extended Kalman filter |

AEKF | Adaptive extended Kalman filter |

HIF | H infinite filter |

BMS | Battery management system |

ECM | Equivalent circuit model |

KF | Kalman filter |

PF | Particle filter |

NARXNN | Nonlinear autoregressive algorithm with exogenous neural network |

AWCPF | Adaptive weighted volume particle filter |

DP | Dual polarization |

AIC | Akaike Information Criterion |

CC-CV | Constant current and constant voltage |

HPPC | Hybrid Pulse Power Characterization |

DST | Dynamic Stress Test |

OCV | Open circuit voltage |

ME | Mean error |

MAE | Mean absolute error |

RMSE | Root mean square error |

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**Figure 4.**(

**a**) SOC estimates under DST; (

**b**) Local graph of SOC estimates under DST; (

**c**) Local graph of SOC estimates under DST.

**Figure 5.**(

**a**) SOC estimates under HPPC; (

**b**) Local graph of SOC estimates under HPPC; (

**c**) Local graph of SOC estimates under HPPC.

SOC | R_{i} | RD | CD |
---|---|---|---|

0.9 | 0.0200678 | 0.0280174 | 6005.6549 |

0.8 | 0.0198582 | 0.0295043 | 5975.013 |

0.7 | 0.0198537 | 0.0291771 | 6244.846 |

0.6 | 0.0198369 | 0.0250574 | 6376.551 |

0.5 | 0.0198402 | 0.0272475 | 6077.899 |

0.4 | 0.0201856 | 0.0312714 | 45,796.440 |

0.3 | 0.0203209 | 0.0322185 | 65,557.043 |

0.2 | 0.0206306 | 0.0360988 | 55,199.546 |

0.1 | 0.020996 | 0.0429594 | 44,303.6293 |

Establish the Linear Discretization Equation of Thevenin Model. | |
---|---|

Initialization | Set the Initial Value of the State Observer: ${\mathit{x}}_{0},{\mathit{P}}_{0},\mathit{Q},\mathit{R},\mathit{\lambda},\mathit{S}$ |

- ①
- $\mathrm{from}\mathit{k}-{\mathbf{1}}^{+}\phantom{\rule{0ex}{0ex}}\mathrm{to}{\left(\mathit{k}\right)}^{-}$
| System state estimation: ${\stackrel{\wedge}{x}}_{{}_{k}}^{-}=f({x}_{k-1},{u}_{k-1})$ |

HIF feature matrix estimation: ${P}_{k}^{-}={A}_{k-1}{P}_{k-1}{A}_{k-1}{}^{T}+Q$ | |

- ②
- $\mathrm{from}{\left(\mathit{k}\right)}^{-}\phantom{\rule{0ex}{0ex}}\mathrm{to}{\left(\mathit{k}\right)}^{+}$
| Innovation matrix: ${e}_{k}={y}_{k}-h({\stackrel{\wedge}{x}}_{{}_{k}}^{-},{u}_{k})$ |

Gain matrix: ${K}_{k}={A}_{k}{P}_{k}^{-}{(1-\lambda S{P}_{\mathrm{k}}^{-}+{C}_{k}^{T}{R}^{-1}{C}_{k}{P}_{\mathrm{k}}^{-})}^{-1}{C}_{k}^{T}{R}^{-1}$ | |

System status correction: ${\stackrel{\wedge}{x}}_{{}_{k}}^{+}={\stackrel{\wedge}{x}}_{{}_{k}}^{-}+{K}_{k}{e}_{k}$ | |

Feature matrix correction: ${P}_{k}^{+}={P}_{k}^{-}{(1-\lambda S{P}_{k}^{-}+{C}_{k}^{T}{R}^{-1}{C}_{k}{P}_{k}^{-})}^{-1}$ | |

- ③
- Time scale update
| Take the state and covariance matrix at time ${\left(k\right)}^{+}$ as the final output, prepare the state estimate at time (k + 1). |

_{0}is the initial value of the state quantity, ${P}_{0}$ is the initial state error covariance matrix, $\lambda $ is the performance boundary, S is the emphasis matrix; if there is a high degree of attention to a certain state quantity, the value in the matrix corresponding to the state quantity is greater than the value in the matrix corresponding to another state quantity.

Algorithms | ME (%) | MAE (%) | RMSE (%) | Run Time (ms) |
---|---|---|---|---|

EKF | 0.97 | 0.27 | 0.30 | 75 |

HIF | 0.58 | 0.29 | 0.30 | 141 |

AEKF | 0.25 | 0.20 | 0.20 | 150 |

FUSE | 0.46 | 0.23 | 0.24 | 103 |

Algorithms | ME (%) | MAE (%) | RMSE (%) | Run Time (ms) |
---|---|---|---|---|

EKF | 1.41 | 0.82 | 0.92 | 180 |

HIF | 0.97 | 0.42 | 0.52 | 430 |

AEKF | 1.01 | 0.45 | 0.55 | 455 |

FUSE | 1.12 | 0.53 | 0.64 | 382 |

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**MDPI and ACS Style**

Tang, A.; Gong, P.; Li, J.; Zhang, K.; Zhou, Y.; Zhang, Z. A State-of-Charge Estimation Method Based on Multi-Algorithm Fusion. *World Electr. Veh. J.* **2022**, *13*, 70.
https://doi.org/10.3390/wevj13040070

**AMA Style**

Tang A, Gong P, Li J, Zhang K, Zhou Y, Zhang Z. A State-of-Charge Estimation Method Based on Multi-Algorithm Fusion. *World Electric Vehicle Journal*. 2022; 13(4):70.
https://doi.org/10.3390/wevj13040070

**Chicago/Turabian Style**

Tang, Aihua, Peng Gong, Jiajie Li, Kaiqing Zhang, Yapeng Zhou, and Zhigang Zhang. 2022. "A State-of-Charge Estimation Method Based on Multi-Algorithm Fusion" *World Electric Vehicle Journal* 13, no. 4: 70.
https://doi.org/10.3390/wevj13040070