# Variable-Order Equivalent Circuit Modeling and State of Charge Estimation of Lithium-Ion Battery Based on Electrochemical Impedance Spectroscopy

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## Abstract

**:**

## 1. Introduction

## 2. Characteristic Analysis of Lithium-Ion Battery Based on EIS

#### 2.1. Electrochemical Impedance Spectroscopy

- (1)
- Ultra-high frequency region (above 10 kHz), mainly affected by the wiring and windings, which is inductive, is represented by R
_{L}//L parallel circuit, earmarked by purple rectangle in the figure. - (2)
- Ohmic internal resistance, the intersection of impedance spectrum and real axis, represented by R
_{0}. - (3)
- High frequency region, characterized by a semicircle related to the diffusion and migration of lithium ions through the insulating layer on the surface of active electrode material particles, is represented by R
_{S}//C_{S}parallel circuit, earmarked by red rectangle in the figure. Among them, R_{S}is the resistance of lithium ion diffusion migration through the surface film on the positive and negative electrodes, which is approximately equal to the semicircle diameter [15], and capacitance C_{S}describes the capacitance effect of surface films. - (4)
- Middle frequency region, characterized by a semicircle related to the charge transfer process in the electrode reaction, is represented by R
_{ct}//C_{dl}parallel circuit, earmarked by green rectangle in the figure. R_{ct}is the charge transfer resistance, which is approximately equal to the semicircle diameter, and the capacitance C_{dl}is the electric double layer capacitor. - (5)
- Low frequency region, characterized by an oblique line associated with the solid diffusion process of lithium ions inside the active material particles, is represented by a Warburg impedance, earmarked by cyan rectangle in the figure.

#### 2.2. The Experiment of EIS

_{S}’s insensitivity to the change of SOC [19], so in the middle section of the SOC, the two semicircles merge into one semicircle when the electrochemical reaction time constants are close and overlap [20].

_{S}is larger [21], making it easy to distinguish the semicircle in the high frequency band, therefore, the two semicircles appear in the impedance spectrum. High temperatures will destroy the surface film and decrease R

_{S,}which leads to the fact that at higher temperatures, the time constants of two semicircles are close to each other and overlap [22], thus causing the two semicircles to merge into one.

## 3. Variable Order Equivalent Circuit Model

#### 3.1. RC Equivalent Circuit Model

_{OCV}represents the open circuit voltage, which is the function of SOC, that is, U

_{OCV}= f (SOC), where f can be the mixture of polynomial and exponential function. The expression of SOC is:

_{Q}represents the available capacity of the battery at present, which is the function of temperature and number of cycles, that is:

_{N}represents the standard capacity, and the factor 3600 is introduced for the transition of capacity from Ah to As. f

_{1}and f

_{2}are cycle life and temperature correction factors respectively. Battery aging is not taken into consideration in this paper, so, f

_{1}= 1. The temperature correction factor, f

_{2}, can correct the actual capacity under different temperature so as to improve the accuracy of model parameter identification and SOC estimation.

_{2}is established according to the well-known Arrhenius formula [23]:

_{T}and b

_{T}are coefficients to be identified, which are related to the activation energy and molar gas constant [24], T represents the working temperature of the battery. To identify the coefficients a

_{T}and b

_{T}, the constant current pulse (CCP) experiments at different temperatures are carried out to acquire the actual capacity of the battery at specific temperature and the necessary data as well to establish to establish the variable order equivalent circuit model (VOEM) in the next section. The experiment steps are shown in Table 2.

_{T}and b

_{T}in the Equation (5) can be identified by fitting the actual capacities under different temperatures and the results are shown in Table 3. It can be seen that the fitting error is small.

#### 3.2. The Determination of the Order of Model

#### 3.2.1. Bayesian Information Criterion

^{2}is the mean square error of the model output, that is:

#### 3.2.2. Result and Analysis of the Determination of Model Order

## 4. SOC Estimation Method Based on VOEM-AR-UKF

#### 4.1. Autoregressive Equation

_{OC}(k) − V

_{L}(k), V

_{L}(k) and I(k) be the terminal voltage and current value of the kth sampling point, respectively. Then the difference form can be deduced from the transfer function of V and I by Laplace inverse transformation [27]:

_{1}, a

_{2}, a

_{3}, b

_{1}and b

_{2}can be identified by forgetting factor recursive least squares (FFRLS) [27] online, and then, the state equation and observation equation based on AR equation are written in the form of (11):

#### 4.2. VOEM-AR-UKF Algorithm

- (1)
- Off line phase: the 3D drawing for order determination is generated according to BIC and the characteristic of EIS. The 1-RC and 2-RC model for order selection can be established by hybrid pulse power characteristic [30] (HPPC).
- (2)
- Online phase: input the real-time temperature, current and voltage values, and the model order is determined by looking up the 3D drawing, and then the order determination result can be arranged into corresponding autoregressive form, whose coefficients can be identified by FFRLS online. Finally, the UKF algorithm is adopted to estimate SOC, which is fed back to determine the model order at the next moment. Figure 12 shows the flow chart of the proposed method.

## 5. Results and Analysis

#### 5.1. Model Validation

#### 5.1.1. Constant Current Pulse Test

#### 5.1.2. BJDST Test

#### 5.2. SOC Estimation

#### 5.2.1. Constant Current Pulse Test

_{total}is adopted to present the efficiency of the these methods.

#### 5.2.2. DST Test

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 15.**SOC estimation results of three methods under constant current pulse discharge experiment.

Step | Content |
---|---|

1 | Charge with constant current of 1.5 A at room temperature until the terminal voltage reaches 4.2 V |

2 | Charge at constant voltage of 4.2 V until the current is less than 150 mA, that is, the battery is fully charged. |

3 | Adjust the incubator to T °C and let the battery stand for 3 h |

4 | EIS measurement, frequent range: 0.01 Hz—10 kHz |

5 | The battery is discharged at constant current 1C to 90% SOC, 70% SOC, 50% SOC, 30% SOC, 20% SOC and 10% SOC respectively, and then repeat steps 3–5 until EIS under all SOC is measured. |

Step | Content |
---|---|

1 | Charge with constant current of 1.5 A at room temperature until the terminal voltage reaches 4.2 V |

2 | Charge at constant voltage of 4.2 V until the current is less than 150 mA, that is, the battery is fully charged. |

3 | Standing for 3 h at −10 °C |

4 | The discharge at constant current of 1C is suspended when the battery SOC drops by 0.1, and the OCV is recorded after standing for 1 h, and then continues until the battery is empty. Data were collected every 0.2 s. |

5 | Standing for 3 h and repeat steps 1–2 |

6 | Change the temperature T = 0 °C, 25 °C and 45 °C respectively, and then repeat the test steps 4–6 |

a_{T} | b_{T} | SSE | R-Square |
---|---|---|---|

17.5860 | 844.1388 | 0.00162 | 0.99164 |

1RC-UKF | 2RC-UKF | VOEM-AR-UKF | |
---|---|---|---|

MAE | 0.0114 | 0.0107 | 0.0008 |

RMSE | 0.0128 | 0.0123 | 0.0015 |

T_{total}/s | 6.83 | 8.59 | 6.32 |

1RC-UKF | 2RC-UKF | VOEM-AR-UKF | |
---|---|---|---|

MAE | 0.0169 | 0.0103 | 0.0007 |

RMSE | 0.0176 | 0.0115 | 0.0055 |

T_{total}/s | 4.35 | 5.22 | 4.16 |

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**MDPI and ACS Style**

Zhang, J.; Wang, P.; Liu, Y.; Cheng, Z.
Variable-Order Equivalent Circuit Modeling and State of Charge Estimation of Lithium-Ion Battery Based on Electrochemical Impedance Spectroscopy. *Energies* **2021**, *14*, 769.
https://doi.org/10.3390/en14030769

**AMA Style**

Zhang J, Wang P, Liu Y, Cheng Z.
Variable-Order Equivalent Circuit Modeling and State of Charge Estimation of Lithium-Ion Battery Based on Electrochemical Impedance Spectroscopy. *Energies*. 2021; 14(3):769.
https://doi.org/10.3390/en14030769

**Chicago/Turabian Style**

Zhang, Ji’ang, Ping Wang, Yushu Liu, and Ze Cheng.
2021. "Variable-Order Equivalent Circuit Modeling and State of Charge Estimation of Lithium-Ion Battery Based on Electrochemical Impedance Spectroscopy" *Energies* 14, no. 3: 769.
https://doi.org/10.3390/en14030769