# Online State of Charge and State of Health Estimation for a Lithium-Ion Battery Based on a Data–Model Fusion Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Battery Modeling and Identification

#### 2.1. Battery Modeling

_{s}is the ohmic resistance. The polarization resistance (R

_{p}) and capacitance (C

_{p}) construct an RC network to simulate the transient dynamics of an LIB.

_{t}and V

_{p}are the terminal and polarization voltage, respectively.

_{2}(NMC) cell (ICR18650-26F, Samsung, Seoul, Korea), which has a nominal capacity of 2.2 Ah, to determine the correlation between SOC and OCV. The cell was first fully charged with the constant-current-constant-voltage (CCCV) method until the upper cut-off voltage of 4.2 V was reached, where the SOC was defined as 100%. Afterwards, the cell was discharged with a series of current pulses until the lower cut-off voltage of 3 V was reached, where the SOC was defined as 0%. The cell was left in an open-circuit condition for 5 h for depolarization at the end of each pulse current discharge, then the terminal voltages were measured and treated as discharge OCVs. At the same time, the corresponding SOCs were calibrated with coulombic counting. The same procedures were executed for the charging process to obtain charge OCVs. The real OCVs were averaged from the discharge and charge OCVs. As a long depolarization time is applied, the hysteresis voltage is found to be very small based on the calibration result. The hysteresis effect is therefore not considered in the modeling for the purpose of simplification. In this paper, the SOC-OCV function is determined by polynomial fitting to the offline tested SOC-OCV correction as:

_{p}is the order of polynomial fitting (n

_{p}= 5 here), and c

_{i}is the polynomial coefficient obtained using least-squares-based curve fitting. The experimentally determined and curve-fitted SOC-OCV relations are shown in Figure 2.

#### 2.2. Online Identification of Model Parameters

_{t}− V

_{oc}. Then, the transfer function of Equation (1) can be expressed as:

_{s}/(q + 1), Equation (4) can be re-written as:

_{s}is the onboard sampling interval. From Equation (5), the following discrete-time expression can be written as:

**θ**

_{k}= [a

_{1,k}b

_{0,k}b

_{1,k}]

**,**

^{T}**φ**

_{k}= [−y

_{k−}

_{1}I

_{k}I

_{k}

_{−1}]

**. Then, the model identification problem boils down to solving the regression model represented by Equation (7).**

^{T}#### 2.3. Adaptive Forgetting Recursive Least Squares

**P**and

**L**; thus, the estimates tend to be uncertain. In contrast, a large λ potentially causes a loss of tracking capability for fast-varying parameters. This can be explained by analyzing Equation (10): under sufficient excitation, the term on the right-hand side inside the square brackets decays faster than it is inflated by the multiplier 1/λ, resulting in the gradual decay of the covariance matrix.

**P**

_{k}

_{−1}

**φ**

_{k}in this case is close to zero; thus, Equation (10) becomes

**P**

_{k}=

**P**

_{k}/λ, indicating that the covariance matrix grows exponentially. When the excitation recovers, the covariance matrix and gain have been very large and cause large fluctuations in the estimation.

**I**represents the unit matrix, and ε is the estimation residual calculated by:

**P**is updated as [45]:

## 3. Co-Estimation of SOC and SOH

#### 3.1. H-Infinity Filter (HIF)

**x**

_{k},

**u**

_{k}, and y

_{k}are the system state, input, and measurement, respectively;

**w**

_{k}and v

_{k}are, respectively, the process and measurement noises with covariance matrices

**Q**and R;

**δ**

_{k}is a linear combination of different system states, and

**h**

_{k}is a user-defined matrix. The state-space model represented by Equation (16) aims to obtain the optimized estimate of

**δ**

_{k}. It is needed to set

**h**

_{k}=

**I**if

**x**

_{k}is estimated directly.

**x**

_{0},

**w**

_{k}, and v

_{k}. The measure of performance is then given by the following cost function:

**x**

_{0}, and

**S**

_{k}and

**P**

_{0}are user-defined symmetric positive matrices. The following operation regarding an arbitrary matrix

**M**and vector

**n**is defined as follows to clarify Equation (17):

**δ**

_{k}among all possible estimates should satisfy:

**δ**

_{k}plays against the exogenous inputs, i.e.,

**x**

_{0},

**w**

_{k}, and v

_{k}. Then, the optimization criterion of HIF can be expressed as:

_{k}can be obtained by using Equation (16) if the optimal estimates of

**x**

_{0}and

**w**

_{k}are determined. Therefore, Equation (20) can be alternatively rewritten as:

#### 3.2. OCV Observation

_{k}are the OCV estimate and the estimation residual, respectively, which can be expressed as:

#### 3.3. Joint Estimaiton of SOC and Capacity

_{p}) can be ruled out from the state vector as it has no correlation with the OCV. Therefore, the system input and measurement are defined as I and ${\widehat{V}}_{oc}$, respectively, while the state vector is defined as

**x**= [z, 1/Q]

**. The following state-space formula can then be formulated:**

^{T}**x**

_{k},

**u**

_{k}) is the function of OCV with regard to the system state and input, which can be determined by the calibrated SOC-OCV function expressed by Equation (3); η is the coulombic efficiency describing the ratio of the total charge extractable from the battery to the total charge that can be injected into the battery over a full cycle. It is calibrated to be 99.2% in this paper.

## 4. Simulation Study

#### 4.1. Data Acquisition

#### 4.2. Simulation Results

_{s}= R

_{p}= 10 mΩ, C

_{p}= 1 kF, SOC

_{0}= 60%, Q

_{0}= 1.8 Ah.

## 5. Experimental Study

#### 5.1. Experimental Setup

#### 5.2. Reference Data Extraction

_{s}is calibrated by the instantaneous voltage jump following a step change of current, i.e., R

_{s}= ΔV

_{t}/ΔI. The time constant, R

_{p}, and C

_{p}can be calibrated by observing both the percentile and absolute change of voltage in terms of time. However, this method is performed offline based on a certain hybrid pulse characterization. The calibrated parameters are optimal for the offline testing condition, but may deviate from real values in real experiments, as the experimental condition, especially for the current pattern, can be substantially different from the testing environment. Alternatively, in this paper, these values are determined from the pure modeling perspective by minimizing the prediction error. As OCVs can be known from reference SOCs, R

_{p}and C

_{p}are determined by fitting the voltage responses to real measurements. In this way, the model parameters are obtained from real experiments instead of offline testing; thus, the problem of load pattern mismatch can be avoided. The selection of the calibration method will be made on a case-by-case basis depending on the real application.

#### 5.3. Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Load current and terminal voltage of the simulation study: (

**a**) hybrid pulse test (HPT); (

**b**) Federal Urban Dynamic Schedule (FUDS).

**Figure 5.**Results of SOC and capacity joint estimation under the HPT simulation study: (

**a**) estimation of SOC; (

**b**) estimation of capacity; (

**c**) error of SOC estimation; (

**d**) relative error of capacity estimation.

**Figure 7.**Estimation results of SOC and capacity under the FUDS simulation study: (

**a**) estimation of SOC; (

**b**) estimation of capacity; (

**c**) error of SOC estimation; (

**d**) relative error of capacity estimation.

**Figure 9.**Estimation results of SOC and capacity for the experimental study: (

**a**) estimation of SOC; (

**b**) estimation of capacity; (

**c**) error of SOC estimation; (

**d**) relative error of capacity estimation.

Definition:${\widehat{A}}_{k}={\frac{\partial F}{\partial x}|}_{{x}_{k}={\widehat{x}}_{k}^{+}},{\widehat{C}}_{k}={\frac{\partial G}{\partial x}|}_{{x}_{k}={\widehat{x}}_{k}^{+}}$ | |

Initialization:${\widehat{x}}_{0}^{+}$, ${P}_{0}^{+}$, Q, R, S_{0}, τFor k = 1, 2, … | |

Update of priori state: | ${\widehat{x}}_{k}^{-}=F\left({\widehat{x}}_{k-1}^{+},{u}_{k-1}\right)$ |

Update of priori error covariance: | ${P}_{k}^{-}={\widehat{A}}_{k-1}{P}_{k-1}^{+}{\widehat{A}}_{k-1}^{T}+Q$ |

Update of symmetric positive matrix: | ${M}_{k}={h}_{k}^{T}{S}_{k}{h}_{k}$ |

Update of gain matrix: | ${K}_{k}={\widehat{A}}_{k}{P}_{k}^{-}{\left(I-\tau {M}_{k}{P}_{k}^{-}+{\widehat{C}}_{k}^{T}{R}_{k}^{-1}{\widehat{C}}_{k}{P}_{k}^{-}\right)}^{-1}{\widehat{C}}_{k}^{T}{R}_{k}^{-1}$ |

Update of posteriori state: | ${\widehat{x}}_{k}^{+}={\widehat{x}}_{k}^{-}+{K}_{k}\left[{y}_{k}-G\left({\widehat{x}}_{k}^{-},{u}_{k}\right)\right]$ |

Update of posteriori error covariance: | ${P}_{k}^{+}={P}_{k}^{-}{\left(I-\tau {M}_{k}{P}_{k}^{-}+{\widehat{C}}_{k}^{T}{R}_{k}^{-1}{\widehat{C}}_{k}{P}_{k}^{-}\right)}^{-1}$ |

Measure | HPT | FUDS |
---|---|---|

MAE | 0.26% | 0.23% |

RMSE | 0.33% | 0.27% |

Measure | HPT | FUDS |
---|---|---|

MRE | 1.70% | 1.16% |

RMSE | 2.32% | 1.95% |

Measure | SOC | Capacity |
---|---|---|

MAE | 0.45% | 0.045 Ah (MRE = 2.10%) |

RMSE | 0.46% | 0.073 Ah (MRE = 3.38%) |

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## Share and Cite

**MDPI and ACS Style**

Wei, Z.; Leng, F.; He, Z.; Zhang, W.; Li, K. Online State of Charge and State of Health Estimation for a Lithium-Ion Battery Based on a Data–Model Fusion Method. *Energies* **2018**, *11*, 1810.
https://doi.org/10.3390/en11071810

**AMA Style**

Wei Z, Leng F, He Z, Zhang W, Li K. Online State of Charge and State of Health Estimation for a Lithium-Ion Battery Based on a Data–Model Fusion Method. *Energies*. 2018; 11(7):1810.
https://doi.org/10.3390/en11071810

**Chicago/Turabian Style**

Wei, Zhongbao, Feng Leng, Zhongjie He, Wenyu Zhang, and Kaiyuan Li. 2018. "Online State of Charge and State of Health Estimation for a Lithium-Ion Battery Based on a Data–Model Fusion Method" *Energies* 11, no. 7: 1810.
https://doi.org/10.3390/en11071810