# Research on State of Power Estimation of Echelon-Use Battery Based on Adaptive Unscented Kalman Filter

^{1}

^{2}

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## Abstract

**:**

_{k}, R

_{k}and the processes noise equation q

_{k}, Q

_{k}, and (b) the ohmic internal resistance (OIR) and actual capacity (AC) are estimated based on the aforementioned algorithm, which uses the observation noise equation γ

_{θ,k}, R

_{θ,k}and the process noise equation q

_{θ,k}, Q

_{θ,k}. Third, the working voltage and OIR are predicted using optimal estimation, and the SOP of the EULB is estimated. MATLAB simulation results show that EULB symmetry capacity decays to 80%, 60%, 40%, and 20% of rated capacity, the proposed algorithm is adaptive regardless of whether the initial SOE value is consistent with the actual value, and the estimation error of the EULB’s SOP is less than 3.28%, showing high accuracy. The results of this study can provide valuable reference for estimating EULB parameters, and help to understand the usage behavior of retired batteries.

## 1. Introduction

- (1)
- Owing to the degradation of EULB, the SOE initial value is not clear, thus parameter estimation of EULB is discussed.
- (2)
- Aiming at unclear SOE of EULB, UT is adopted and an adaptive method used, that can be adjusted adaptively to estimate the accuracy based on the observation noise equation γ
_{k}, R_{k}and the processes noise equation q_{k}, Q_{k}. - (3)
- To improve the accuracy of SOP estimation, UT and ADUKF are adopted to estimate the parameters of EULB based on the observation noise equation γ
_{θ,k}, R_{θ,k}and the process noise equation q_{θ,k}, Q_{θ,k}aimed at performance attenuation of the EULB and the inaccuracy of AC and OIR. - (4)
- It provides a theoretical basis for the effective utilization of the whole life cycle of a lithium-ion battery.

## 2. SRCSEM of EULB

## 3. Model Parameter Identification

#### 3.1. OCV

- (1)
- Charging current is 36 A and cut-off voltage is 3.65 V.
- (2)
- Discharging current is 36 A and cut-off voltage is 2.5 V. Record the discharge voltage ${U}_{disarge}$.
- (3)
- Charging current is 36 A and cut-off voltage is 3.65 V. Record the charge voltage ${U}_{disarge}$.
- (4)
- The OCV U_OC is as follows:$${U}_{OC}=\frac{{U}_{discharge}+{U}_{charge}}{2}$$

#### 3.2. OIR

#### 3.3. Polarization Resistance

## 4. SOP Estimation

#### 4.1. UT

#### 4.2. SOE Estimation Based on ADUKF

- Step 1: Inialize $x$ is as follows:$${\widehat{x}}_{0}=E\left({x}_{0}\right).$$$${\widehat{P}}_{0}=E[\left({x}_{0}-{\widehat{x}}_{0}\right){\left({x}_{0}-{\widehat{x}}_{0}\right)}^{T}$$
- Step 2: Generating sigma points:$${\chi}_{i}=\left[\overline{x},\overline{x}+{\left(\sqrt{\left(n+\lambda \right){P}_{x}}\right)}_{i},\overline{x}-{\left(\sqrt{\left(n+\lambda \right){P}_{x}}\right)}_{i-n}\right]$$
- Step 3: Time update of the system state $x$ is as follows$${\chi}_{i.k}=f\left({\chi}_{i.k-1}\right)$$$${x}_{k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{m}{\chi}_{i.k}+{q}_{k}$$$${P}_{k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{c}\left[{\chi}_{i.k}-{x}_{k}\right]{\left[{\chi}_{i.k}-{x}_{k}\right]}^{T}+{Q}_{k}$$$${y}_{i,k}=g\left({\chi}_{i.k-1}\right)$$$${y}_{k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{m}{y}_{i.k}+{\gamma}_{k}$$$${P}_{y,k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{c}\left[{y}_{i.k}-{y}_{k}\right]{\left[{y}_{i.k}-{y}_{k}\right]}^{T}+{R}_{k}$$$${P}_{xy,k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{c}\left[{\chi}_{i.k}-{x}_{k}\right]{\left[{y}_{i.k}-{y}_{k}\right]}^{T}$$
- Step 4: Status update of the system state $x$ is as follows:The Kalman gain is as follows:$${K}_{k}={P}_{xy,k}{P}_{y,k}^{-1}$$The optimal estimation of state variables is as follows:$${\widehat{x}}_{k}={x}_{k}+{K}_{k}\left[{y}_{k}-{\widehat{y}}_{k}\right]$$The optimal estimate of the covariance is as follows:$${\widehat{P}}_{k}={P}_{k}-{K}_{k}{P}_{y,k}{K}_{k}^{T}$$
- Step 5: Process noise mean and covariance equation is as follows:$${q}_{k}=\left(1-{d}_{k}\right){q}_{k-1}+{d}_{k}\left[{\widehat{x}}_{k}-{{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{m}f\left({\chi}_{i.k-1},{u}_{k},{\theta}_{k}\right)\right]$$$${Q}_{k}=\left(1-{d}_{k}\right){Q}_{k-1}+{d}_{k}\left[{K}_{k}({\widehat{y}}_{k}-{y}_{k}\right){({\widehat{y}}_{k}-{y}_{k})}^{T}{K}_{k}^{T}+{P}_{k}-{A}_{k-1}{\widehat{P}}_{k-1}{A}_{k-1}^{T}]$$
- Step 6: Observe the noise mean and covariance equation as follows:$${\gamma}_{k}=\left(1-{d}_{k}\right){\gamma}_{k-1}+{d}_{k}\left[{y}_{k}-{{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{i}^{m}g\left({\chi}_{i.k-1}\right)\right]$$$${R}_{k}=\left(1-{d}_{k}\right){R}_{k-1}+{d}_{k}\left[({\widehat{y}}_{k}-{y}_{k}\right){({\widehat{y}}_{k}-{y}_{k})}^{T}-{C}_{k}{P}_{k}{C}_{k}^{T}]$$

#### 4.3. OIR and AC Estimation Based on AUKF

- Step 1: Initialize $\theta $ as follows:$${\widehat{\theta}}_{0}=E\left({\theta}_{0}\right)$$$${P}_{\theta ,0}=E\left[\left({x}_{0}-{\widehat{x}}_{0}\right){\left({x}_{0}-{\widehat{x}}_{0}\right)}^{T}\right]$$
- Step 2: Generating sigma points:$${\chi}_{\theta ,i}=\left[\overline{x},\overline{x}+{\left(\sqrt{\left(n+\lambda \right){P}_{\theta}}\right)}_{i},\overline{x}-{\left(\sqrt{\left(n+\lambda \right){P}_{\theta}}\right)}_{i-n}\right]$$
- Step 3: Time update of the system state $\theta $ is as follows:$${\chi}_{\theta i.k}=f\left({\chi}_{\theta i.k-1}\right)$$$${\theta}_{k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{\theta i}^{m}{\chi}_{\theta i.k}+{q}_{\theta ,k}$$$${P}_{\theta ,k}={\widehat{P}}_{\theta ,k-1}+{Q}_{\theta ,k}$$$${y}_{\theta i,k}=g\left({\chi}_{\theta i.k-1}\right)$$$${y}_{\theta ,k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{\theta i}^{m}{y}_{\theta i.k}+{\gamma}_{\theta ,k}$$$${P}_{\theta y,k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{\theta i}^{c}\left[{y}_{\theta i.k}-{y}_{\theta ,k}\right]{\left[{y}_{\theta i.k}-{y}_{\theta ,k}\right]}^{T}+{R}_{\theta ,k}$$$${P}_{\theta xy,k}={{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{\theta i}^{c}\left[{\chi}_{\theta i.k}-{\theta}_{k}\right]{\left[{y}_{\theta i.k}-{y}_{\theta ,k}\right]}^{T}$$
- Step 4: Status update of the system state $\theta $ is as follows:The Kalman gain is as follows:$${K}_{\theta ,k}={P}_{\theta xy,k}{P}_{\theta y,k}^{-1}$$The optimal estimation of state variables is as follows:$${\widehat{\theta}}_{k}={\theta}_{k}+{K}_{\theta ,k}\left[{y}_{k}-{\widehat{y}}_{k}\right]$$The optimal estimate of the covariance is as follows:$${\widehat{P}}_{\theta ,k}={P}_{\theta ,k}-{K}_{\theta ,k}{P}_{\theta y,k}{K}_{\theta k}^{T}$$
- Step 5: Process noise mean and covariance equation is as follows:$${q}_{\theta ,k}=\left(1-{d}_{\theta ,k}\right){q}_{\theta ,k-1}+{d}_{\theta ,k}\left[{\widehat{\theta}}_{k}-{{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{\theta i}^{m}f\left({\chi}_{\theta i.k-1},{u}_{k},{\theta}_{k}\right)\right]$$$${Q}_{\theta ,k}=\left(1-{d}_{\theta ,k}\right){Q}_{\theta ,k-1}+{d}_{\theta ,k}\left[{K}_{\theta ,k}({\widehat{y}}_{k}-{y}_{k}\right){({\widehat{y}}_{k}-{y}_{k})}^{\mathrm{T}}{K}_{\theta ,k}^{\mathrm{T}}+{P}_{\theta ,k}-{A}_{k-1}{\widehat{P}}_{\theta ,k-1}{A}_{k-1}^{\mathrm{T}}]$$
- Step 6: Observe the noise mean and covariance equation as follows:$${\gamma}_{\theta ,k}=\left(1-{d}_{\theta ,k}\right){\gamma}_{\theta ,k-1}+{d}_{\theta ,k}\left[{y}_{\theta ,k}-{{\displaystyle \sum}}_{i=0}^{2n}{\omega}_{\theta i}^{m}g\left({\chi}_{\theta i.k-1}\right)\right]$$$${R}_{\theta ,k}=\left(1-{d}_{\theta ,k}\right){R}_{\theta ,k-1}+{d}_{\theta ,k}\left[({\widehat{y}}_{k}-{y}_{k}\right){({\widehat{y}}_{k}-{y}_{k})}^{\mathrm{T}}-{C}_{k}{P}_{\theta ,k}{C}_{k}^{\mathrm{T}}]$$

#### 4.4. SOP Estimation Based on AUKF

#### 4.5. The Flow Chart of SOP Estimation Based on ADUKF

## 5. Simulation and Discussion

#### 5.1. SOE Estimation Based on ADUKF

#### 5.2. Capacity Decays to 80% and SOE Starts at 70%

#### 5.3. Capacity Decays to 80% and SOE Starts at 40%

#### 5.4. Capacity Decays to 60% and SOE Starts at 70%

#### 5.5. Capacity Decays to 60% and SOE Starts at 40%

#### 5.6. Capacity Decays to 40% and SOE Starts at 70%

#### 5.7. Capacity Decays to 40% and SOE Starts at 40%

#### 5.8. Capacity Decays to 20% and SOE Starts at 70%

#### 5.9. Capacity Decays to 20% and SOE Starts at 40%

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EULB | Echelon-use lithium-ion battery |

AUKF | Adaptive unscented Kalman filter |

SRCSEM | Second-order resistor-capacitance symmetry equivalent model |

SOC | State of charge |

SOH | State of health |

SOP | State of power |

SOE | State of energy |

OCV | Open circuit voltage |

ADUKF | Adaptive dual unscented Kalman filter |

UT | Unscented transformation |

EVs | Electric vehicles |

UKF | Unscented Kalman filter |

USABC | United States Advanced Battery Consortium |

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**Figure 3.**Battery testing system. (

**a**) The lithium iron echelon-use battery. (

**b**) The charge/discharge experiment.

**Figure 4.**The simulation verification curve when capacity decays to 80% and SOE starts at 70%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

**Figure 5.**The simulation verification curve when capacity decays to 80% and SOE starts at 40%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

**Figure 6.**The simulation verification curve when capacity decays to 60% and SOE starts at 70%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

**Figure 7.**The simulation verification curve when capacity decays to 60% and SOE starts at 40%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

**Figure 8.**The simulation verification curve when capacity decays to 40% and SOE starts at 70%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

**Figure 9.**The simulation verification curve when capacity decays to 40% and SOE starts at 40%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

**Figure 10.**The simulation verification curve when capacity decays to 20% and SOE starts at 70%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) of SOP.

**Figure 11.**The simulation verification curve when capacity decays to 20% and SOE starts at 40%. Estimation and error curves of (

**a**) SOE, (

**b**) working voltage, and (

**c**) SOP.

Items | Parameter | Remarks |
---|---|---|

capacity | 72 Ah | 72 A |

nominal voltage | 3.2 V | |

working voltage | 2.5–3.65 V | |

charging time | 3 h | 24 A |

charging temperature | 0–45 °C | |

discharging temperature | −20–55 °C |

Estimation Error | SOE | Working Voltage | SOP | |
---|---|---|---|---|

capacity decays to 80% | SOE = 70% | −2.29% to 2.34% | 4.28% | 3.25% |

SOE = 40% | −2.31% to 2.34% | 4.28% | 3.26% | |

capacity decays to 60% | SOE = 70% | −2.23% to 2.35% | 4.28% | 3.26% |

SOE = 40% | −2.25% to 2.35% | 4.29% | 3.26% | |

capacity decays to 40% | SOE = 70% | −2.36% to 2.34% | 4.32% | 3.27% |

SOE = 40% | −2.38% to 2.34% | 4.33% | 3.28% | |

capacity decays to 20% | SOE = 70% | −2.37% to 2.34% | 4.32% | 3.28% |

SOE = 40% | −2.39% to 2.34% | 4.33% | 3.28% |

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## Share and Cite

**MDPI and ACS Style**

Hou, E.; Xu, Y.; Qiao, X.; Liu, G.; Wang, Z.
Research on State of Power Estimation of Echelon-Use Battery Based on Adaptive Unscented Kalman Filter. *Symmetry* **2022**, *14*, 919.
https://doi.org/10.3390/sym14050919

**AMA Style**

Hou E, Xu Y, Qiao X, Liu G, Wang Z.
Research on State of Power Estimation of Echelon-Use Battery Based on Adaptive Unscented Kalman Filter. *Symmetry*. 2022; 14(5):919.
https://doi.org/10.3390/sym14050919

**Chicago/Turabian Style**

Hou, Enguang, Yanliang Xu, Xin Qiao, Guangmin Liu, and Zhixue Wang.
2022. "Research on State of Power Estimation of Echelon-Use Battery Based on Adaptive Unscented Kalman Filter" *Symmetry* 14, no. 5: 919.
https://doi.org/10.3390/sym14050919