# State of Charge Estimation for Power Battery Base on Improved Particle Filter

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## Abstract

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## 1. Introduction

- (1)
- The PSO algorithm is used to identify the parameters of the second-order RC model to obtain a battery model with high accuracy.
- (2)
- This paper designs self-mutation and mutual learning of particles to drive particles to high-likelihood regions, which effectively suppresses the degradation of particle weights without increasing the number of particles and increasing particle diversity.
- (3)
- Compared with the PF algorithm, the IPSO-PF algorithm has better particle diversity and much higher estimation accuracy in any SOC stage under different driving conditions.

## 2. Battery Modeling and Parameter Identification

#### 2.1. Battery Modeling

#### 2.2. Parameter Identification

## 3. SOC Estimation Based on IPSO-PF Algorithm

#### 3.1. PF Algorithm

- Prediction:$${\left\{{\tilde{x}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}=f\left({\left\{{x}_{k-1}^{\left(i\right)}\right\}}_{i=1}^{N},{u}_{k}\right)$$$${\left\{{\tilde{y}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}=h\left({\left\{{\tilde{x}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N},{u}_{k}\right)$$
- Update:$${w}_{k}^{\left(i\right)}=\left(\frac{1}{\sqrt{2\mathsf{\pi}{R}_{k}}}\right){\mathrm{e}}^{-\frac{{\left({y}_{k}-{\tilde{y}}_{k}^{\left(i\right)}\right)}^{2}}{2{R}_{k}}}$$$${\tilde{w}}_{\mathrm{k}}^{\left(\mathrm{i}\right)}=\frac{{w}_{\mathrm{k}}^{\left(\mathrm{i}\right)}}{{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{N}}}{w}_{\mathrm{k}}^{\left(\mathrm{i}\right)}}$$
- Resample: obtain the posterior state sets ${\left\{{x}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}$, ${\left\{{w}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}=1/N$ after resampling the particle sets ${\left\{{\tilde{x}}_{k}^{\left(i\right)},{\tilde{w}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}$.
- Estimation:$${\hat{x}}_{k}={\displaystyle \sum}_{i=1}^{\mathrm{N}}{w}_{k}^{\left(i\right)}{x}_{k}^{\left(i\right)}$$

#### 3.2. IPSO Algorithm

#### 3.3. SOC Estimation

- Prediction.$${\left\{{\tilde{x}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}=f\left({\left\{{x}_{k-1}^{\left(i\right)}\right\}}_{i=1}^{N},{u}_{k}\right)$$$${\left\{{\tilde{y}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}=h\left({\left\{{\tilde{x}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N},{u}_{k}\right)$$
- IPSO optimization.
- Initialize the population.$${\left\{{x}_{\left(i\right)}\right\}}_{i=1}^{N}={\left\{{\tilde{y}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}$$$${\left\{{v}_{\left(i\right)}\right\}}_{i=1}^{N}=rand$$$${\left\{{P}_{\mathrm{best}\left(i\right)}\right\}}_{i=1}^{N}={\left\{fitness\left(i\right)\right\}}_{i=1}^{N}$$$${G}_{\mathrm{best}}=\mathrm{max}\left({\left\{fitness\left(i\right)\right\}}_{i=1}^{N}\right)$$
- Set the fitness function.$$fitness\left(i\right)={\mathrm{e}}^{-\frac{1}{2{R}_{k}}\left({x}_{\mathrm{t}}-{x}_{\mathrm{p}\left(i\right)}\right)}{}^{2}$$
- Update the speed and position.$${v}_{\left(i\right)}=w{v}_{\left(i\right)}+{c}_{1}{r}_{1}\left({P}_{\mathrm{best}\left(i\right)}^{}-{x}_{\left(i\right)}\right)+{c}_{2}{r}_{2}\left({G}_{\mathrm{best}}^{}-{x}_{\left(i\right)}\right)$$$${x}_{\left(i\right)}={x}_{\left(i\right)}+{v}_{\left(i\right)}$$
- Output the optimal value.$${\left\{{\tilde{y}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}={\left\{{x}_{\left(i\right)}\right\}}_{i=1}^{N}$$

- Update:$${w}_{k}^{\left(i\right)}=\left(\frac{1}{\sqrt{2\mathsf{\pi}{R}_{k}}}\right){\mathrm{e}}^{-\frac{{\left({y}_{k}-{\tilde{y}}_{k}^{\left(i\right)}\right)}^{2}}{2{R}_{k}}}$$$${\tilde{w}}_{\mathrm{k}}^{\left(\mathrm{i}\right)}=\frac{{w}_{\mathrm{k}}^{\left(\mathrm{i}\right)}}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{N}}}{w}_{\mathrm{k}}^{\left(\mathrm{i}\right)}}$$
- Resample: obtain the posterior state sets ${\left\{{x}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}$, ${\left\{{w}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}=1/N$ after resampling the particle sets ${\left\{{\tilde{x}}_{k}^{\left(i\right)},{\tilde{w}}_{k}^{\left(i\right)}\right\}}_{i=1}^{N}$.
- Estimation:$${\hat{x}}_{k}={\displaystyle \sum}_{i=1}^{\mathrm{N}}{w}_{k}^{\left(i\right)}{x}_{k}^{\left(i\right)}$$

## 4. Experimental Results and Analysis

#### 4.1. Model Verification

#### 4.2. SOC Estimation Verification

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

SOC | state of charge |

PF | particle filter |

PSO | particle swarm optimization |

IPSO | improved particle swarm optimization |

KF | Kalman filter |

EKF | extended Kalman filter |

UKF | unscented Kalman filter |

RMSE | root mean squared error |

MAE | mean absolute error |

ME | max error |

OCV | open circuit voltage |

DST | dynamic street test |

FUDS | federal urban driving schedule |

US06 | intense Driving Schedule |

BJDST | Beijing dynamic stress test |

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Type | Normal Voltage | Normal Capacity | Cutoff Voltage |
---|---|---|---|

INR 18650-20R | 3.6 V | 2.0 Ah | 2.5 V/4.2 V |

R_{0}/Ω | R_{p}/Ω | R_{d}/Ω | C_{p}/F | C_{d}/F |
---|---|---|---|---|

0.0687 | 0.0131 | 0.0035 | 1359.7 | 432.6 |

Model | Computation Cost | RMSE (%) |
---|---|---|

first-order | 3.55 | 0.8351 |

second-order | 8.24 | 0.8337 |

third-order | 15.88 | 0.8338 |

Method | RMSE/% | MAE/% | ME/% |
---|---|---|---|

EKF | 1.09 | 1.74 | 1.99 |

PF | 0.80 | 0.76 | 1.37 |

PSO-PF | 0.56 | 0.47 | 1.47 |

IPSO-PF | 0.39 | 0.33 | 0.99 |

Method | RMSE/% | MAE/% | ME/% |
---|---|---|---|

EKF | 1.13 | 0.94 | 2.65 |

PF | 0.71 | 0.67 | 1.23 |

PSO-PF | 0.64 | 0.52 | 1.54 |

IPSO-PF | 0.25 | 0.21 | 0.68 |

Method | RMSE/% | MAE/% | ME/% |
---|---|---|---|

EKF | 0.89 | 0.73 | 1.88 |

PF | 0.62 | 0.57 | 1.11 |

PSO-PF | 0.44 | 0.36 | 0.95 |

IPSO-PF | 0.34 | 0.26 | 0.90 |

Method | RMSE/% | MAE/% | ME/% |
---|---|---|---|

EKF | 0.92 | 0.75 | 2.04 |

PF | 0.74 | 0.70 | 1.39 |

PSO-PF | 0.49 | 0.41 | 1.07 |

IPSO-PF | 0.33 | 0.28 | 0.86 |

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## Share and Cite

**MDPI and ACS Style**

Liu, X.; Fan, X.; Wang, L.; Wu, J.
State of Charge Estimation for Power Battery Base on Improved Particle Filter. *World Electr. Veh. J.* **2023**, *14*, 8.
https://doi.org/10.3390/wevj14010008

**AMA Style**

Liu X, Fan X, Wang L, Wu J.
State of Charge Estimation for Power Battery Base on Improved Particle Filter. *World Electric Vehicle Journal*. 2023; 14(1):8.
https://doi.org/10.3390/wevj14010008

**Chicago/Turabian Style**

Liu, Xingtao, Xiaojie Fan, Li Wang, and Ji Wu.
2023. "State of Charge Estimation for Power Battery Base on Improved Particle Filter" *World Electric Vehicle Journal* 14, no. 1: 8.
https://doi.org/10.3390/wevj14010008