# SOC and SOH Joint Estimation of the Power Batteries Based on Fuzzy Unscented Kalman Filtering Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model for the Lithium Battery

#### 2.1. Setup of Equivalent Circuit Model for the Lithium Battery

_{res}is the remaining battery capacity after the discharge of partial electric quantity and Q

_{N}is the nominal battery capacity.

_{R}is the battery SOH, which defined based on the ohmic resistance R

_{0}; R

_{0}(end) is the ohmic resistance when the actual maximum battery capacity drops to 80% of the nominal battery capacity; R

_{0}(t) is the ohmic resistance of the battery at t; and R

_{0}(0) is the ohmic resistance upon the battery delivery from the factory.

_{p}and current-controlled current source (CCCS) on the left side characterize the battery capacity, SOC and running time. The second-order RC circuit on the right side simulates the internal polarization characteristics of the battery, R1 and C

_{1}describe the concentration polarization characteristics of the battery, while R

_{2}and C

_{2}describe the electrochemical polarization characteristics of the battery. The voltage-controlled voltage source (VCVS) simulates the nonlinear relationship between the open-circuit voltage and U

_{soc}, which links the circuit parts on both sides.

_{1}U

_{2}]

^{T}as the state variable to obtain the following continuous state space equation:

_{N}is the nominal battery capacity and η is the charge–discharge efficiency of the battery. The discretized state equation and observation equation are as follows:

#### 2.2. Open-Circuit Voltage and SOC Setting Experiments

#### 2.3. Parameter Identification of the Lithium Battery Model

_{OC}(k) − U

_{L}(k), and the final derivation of the Bayesian identification algorithm is as follows.

_{θ}(0) is a·I, among which a is a small positive number and I is a 5-order unit matrix. Use the recursion Formula (9) of the Bayesian identification algorithm to estimate the model parameters and then calculate the resistance and capacitance values of the model via Equation (8). In practical applications, it is necessary to consider the amount of calculation and the length of time for parameter identification. The joint estimation algorithm designed in this paper has a large amount of computation. Therefore, the mean value of the online identification result was selected as the parameter identification result. The results are shown in Table 1.

## 3. SOC and SOH Joint Estimation Based on F-UKF

#### 3.1. Unscented Kalman Filtering Algorithm

#### 3.2. Fuzzy Unscented Kalman Filtering Algorithm

_{y}(k|k − 1) and then update the Kalman filter gain and state-error covariance matrix with the updated $\hat{P}$

_{y}(k|k − 1).

#### 3.3. Design and Implementation of the SOC and SOH Joint Estimation Algorithm

_{x}(k) is the process noise, with the mean value of zero and variance E

_{x}(k); v

_{x}(k) is the observation noise, with the mean value of zero and variance V

_{x}(k); and E

_{x}(k) and V

_{x}(k) are irrelevant.

_{0}(k) is the state variable; y(k) is the predicted terminal voltage of the battery; e

_{R}(k) is the process noise, with the mean value of zero and variance E

_{R}(k); V

_{R}(k) is the observation noise, with the mean value of zero and variance V

_{R}(k); and E

_{R}(k)and V

_{R}(k) are irrelevant.

- (1)
- Parameter initialization. First, initialize the corresponding parameters of the F-UKF algorithm for the battery SOC estimation; then, initialize the corresponding parameters of the F-UKF algorithm for the ohmic resistance estimation, and the ohmic resistance should be close to the actual value to ensure the fast convergence of the battery SOC.
- (2)
- Obtain the terminal voltage U
_{L}(k) and working current i(k) of the battery at the time k through the voltage-current acquisition module. - (3)
- Obtain the estimated value of the battery SOC at the time k through the recursion formula using the F-UKF algorithm based on the above terminal voltage and working current at the time k.
- (4)
- Obtain the estimated value of ohmic resistance at the time k through the recursion formula using the F-UKF algorithm based on the estimated value of battery SOC and working current at the time k.
- (5)
- Take the value of SOC(k) obtained from step (3) into the nonlinear functions of open-circuit voltage and the battery SOC to obtain the open-circuit voltage U
_{OC}(k) at the time k; repeat the steps (2), (3), (4) and (5) for the real-time estimation of the battery SOC and ohmic resistance.

## 4. Experimental Verification and Result Analysis

#### 4.1. Sensitivity Verification of the F-UKF Algorithm against Initial Values

#### 4.2. Joint Simulation Verification of UDDS Driving Cycles

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**State of charge (SOC) estimation curve based on the fuzzy unscented Kalman filtering algorithm (F-UKF) under discharge conditions with constant-current pulses.

**Figure 10.**SOC estimation error curve based on F-UKF under discharge conditions with constant-current pulses.

**Figure 11.**(

**a**) SOC estimation curve of urban dynamometer driving schedule (UDDS) driving cycles based on the F-UKF and UKF algorithms. (

**b**) SOC estimation error curve of UDDS driving cycles based on the F-UKF and UKF algorithms.

**Figure 12.**(

**a**) SOC estimation curve of UDDS driving cycles based on the F-UKF and joint estimation algorithms. (

**b**) SOC estimation error curve of UDDS driving cycles based on the F-UKF and joint estimation algorithms.

**Figure 13.**Ohmic resistance estimation curve of UDDS driving cycles based on the joint estimation algorithm.

**Figure 14.**State of health (SOH) estimation curve of UDDS driving cycles based on the joint estimation algorithm.

**Figure 15.**SOH estimation error curve of UDDS driving cycles based on the joint estimation algorithm.

Model Parameter | Maximum Value | Minimum Value | Average Value |
---|---|---|---|

Ohmic internal resistance R_{0} (mΩ) | 1.704 | 0.923 | 1.278 |

Concentration polarization internal resistance R_{1} (mΩ) | 0.0603 | 0.1189 | 0.0927 |

Concentration polarization capacitor C_{1} (KF) | 6.017 | 3.021 | 3.821 |

Electrochemical polarization internal resistance R_{2} (mΩ) | 0.248 | 0.176 | 0.219 |

Electrochemical polarization capacitance C_{2} (KF) | 3.281 | 2.683 | 2.746 |

Input fuzziness | Input Small (IS) | Input Middle (IM) | Input Big (IB) |

Output fuzziness | Output Small (OS) | Output Middle (OM) | Output Big (OB) |

Nominal Capacity (Ah) | 40 | |

Battery voltage (V) | Charge cutoff voltage | 3.6 |

Discharge cutoff voltage | 2.0 | |

Cycle life (times) | 80% DOD | ≥2000 |

70% DOD | ≥3000 | |

Standard charge–discharge current (A) | 0.3C | |

Maximum charge current (A) | 3C | |

Maximum discharge current (A) | 4C | |

Operating temperature (°C) | −25–55 |

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**MDPI and ACS Style**

Zeng, M.; Zhang, P.; Yang, Y.; Xie, C.; Shi, Y.
SOC and SOH Joint Estimation of the Power Batteries Based on Fuzzy Unscented Kalman Filtering Algorithm. *Energies* **2019**, *12*, 3122.
https://doi.org/10.3390/en12163122

**AMA Style**

Zeng M, Zhang P, Yang Y, Xie C, Shi Y.
SOC and SOH Joint Estimation of the Power Batteries Based on Fuzzy Unscented Kalman Filtering Algorithm. *Energies*. 2019; 12(16):3122.
https://doi.org/10.3390/en12163122

**Chicago/Turabian Style**

Zeng, Miaomiao, Peng Zhang, Yang Yang, Changjun Xie, and Ying Shi.
2019. "SOC and SOH Joint Estimation of the Power Batteries Based on Fuzzy Unscented Kalman Filtering Algorithm" *Energies* 12, no. 16: 3122.
https://doi.org/10.3390/en12163122