# Development of a Matlab/Simulink Model for Monitoring Cell State-of-Health and State-of-Charge via Impedance of Lithium-Ion Battery Cells

^{*}

## Abstract

**:**

## 1. Introduction

- Data-driven models

- Electrical equivalent circuit models (ECMs)

## 2. The Proposed Model

#### 2.1. SoC Model

#### 2.2. Cell Model

#### 2.2.1. OCV Measurement and Application

- Constant current (CC) charge: The cell is charged with a 1 C current until a high cut-off voltage (4.2 V) is reached.
- Constant voltage (CV) charge: When the cell reaches a voltage of 4.2 V, the charging current is reduced to maintain the voltage. Charging is terminated when the current becomes 1/10 C.
- Ninety minutes of relaxation is given to the cell. This step completes setting cell SoC to 100%.
- Discharge of 1 C for 5 min and relaxation for 10 min are repeated. The cell voltage at this time is shown by the solid line in Figure 4.
- When the cell voltage reaches a low cut-off voltage (2.8 V), the discharge is terminated and a final relaxation time of 10 min is given. At this step, the SoC of the cell becomes 0%.
- The cell voltage after each 10 min relaxation is collected as an OCV voltage. In Figure 4, the measured OCV is shown by the round marker.

#### 2.2.2. System Identification and Calculation of Cell Voltage

^{2}value is used. χ

^{2}calculates the distance between measured and simulated data, defined as Equation (6).

#### 2.3. Output of Continuous Cell Impedance

#### 2.4. Cell Temperature Model

## 3. Simulation Results and Validation

#### 3.1. Simulation of Continuous Cell Impedance during Discharge

#### 3.1.1. 1 Hz Impedance during Discharge

#### 3.1.2. 250 Hz Impedance at Each Cell SoH

#### 3.2. Simulation of Cell Impedance Discharging from Different Initial SoCs

#### 3.3. Simulation Results at Different Diagnostic Parameters

#### 3.3.1. Simulation of Continuous Cell Impedance from 1 to 100 Hz

#### 3.3.2. Simulation of Continuous Cell Impedance at Different Sampling Rates

## 4. Conclusions and Discussion

- As shown in Figure 10 and Figure 13, the estimation error of cell impedance gradually increases at DoD above about 90%. This is a problem due to the characteristics of the Li-ion battery cells. When the DoD of the cell increases above a certain level, the cell voltage decreases significantly nonlinearly, resulting in a large error in impedance measurement as shown in Figure 12. In addition, when estimating the cell voltage, the estimation error is relatively higher in the early stages of the discharge when the cell voltage is nonlinearly lowered. Despite the characteristics of Li-ion cells with this nonlinearity, this model shows competent overall estimation results.
- This model estimates the cell voltage and impedance only during discharge, not during charging. In the reference, to estimate the cell SoH, both cell impedance during discharge and charging are compared. Reference [9] only uses cell impedance while discharging to estimate cell SoC and mentions the difficulty of using impedance while charging. This model focuses on monitoring the cell state with impedance during discharge. Nevertheless, it is clear that application in a wider range will be possible when the proposed model can also estimate cell voltage and impedance during charging through further research.
- Last, cells are discharged in a temperature chamber adjusted to 25 °C in this paper. Therefore, the cell temperature is only considered from 25 to 30 °C. For wider conditions to be considered, the temperature range of the model will be increased.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Nyquist plots (

**a**) and Bode plots (

**b**) showing impedance at each cell SoC. In the Nyquist plot, the ‘+’ marker represents 1 Hz impedance, the ‘o’ marker represents 250 Hz impedance, and the ‘x’ marker represents 1 kHz impedance.

**Figure 7.**Input current (

**a**) and output voltage (

**b**) expressed in the time domain, and the output voltage expressed in the frequency domain (

**c**) as a result of the simulation model.

**Figure 8.**Simulation results of 1, 10, 100, and 1000 Hz impedance in the entire DoD range of the discharged cell.

**Figure 10.**(

**a**) Simulated and measured 1 Hz impedance of cells discharged at different C-rates (dashed line: measured impedance; solid line: simulated impedance). (

**b**) Percentage errors of the estimated impedance compared to the measured impedance at different C-rates.

**Figure 11.**Continuous 250 Hz impedance at each SoH (dashed line: measured impedance, solid line: simulated impedance).

**Figure 12.**(

**a**) Discharge curve of a cell in which 10 min of discharge and 60 min of relaxation are repeated (dashed line: measured voltage, solid line: simulated voltage). (

**b**) Percentage error of simulated cell voltage compared to the measured voltage.

**Figure 13.**(

**a**) Output of the cell temperature model when discharging for 10 min and relaxation for 60 min are repeated. (

**b**) Measured continuous 1 Hz impedance and simulated results (red line: before adjusting to temperature; green line: after adjusting to temperature). (

**c**) Percent error of simulated 1 Hz impedance considering the cell temperature compared to the measured impedance.

**Figure 14.**(

**a**) Continuous cell impedance from 1 Hz to 100 Hz simulated in the entire DoD range. (

**b**) The increased rate of impedance at each frequency.

**Figure 15.**(

**a**) Continuous cell impedance in the entire DoD range simulated at different sampling rates. (

**b**) Percentage errors of the simulation results between at the sampling rate of 2048 Hz and at each sampling rate of 16 Hz to 1024 Hz.

Parameter | Description |
---|---|

Product name | Samsung ICR 18650-26F |

Battery system | Li-ion |

Nominal voltage | 3.7 V |

Rated capacity | 2.6 Ah (0.2 C, to 2.75 V discharge) |

Wh rating | 9.62 Wh |

Anode | Based on intercalation graphite |

Cathode | Based on lithiated metal oxide ^{1} |

^{1}Consists of cobalt, nickel, and manganese.

C-Rate | Relative Capacity |
---|---|

1.5 | 0.92 |

1 | 0.92 |

0.5 | 0.94 |

0.25 | 0.97 |

0.125 | 1 |

Parameter | Description |
---|---|

EIS method | Galvanostatic |

Min. frequency | 200 mHz |

Max. frequency | 2 kHz |

DC bias | 0 A |

AC amplitude | 100 mA |

**Table 4.**Fitted element values for each SoC and ${\mathsf{\chi}}^{2}/N$ representing the fitting error.

SoC (%) | ${\mathit{\chi}}^{2}/\mathit{N}$ | L (H) | R0 (Ω) | R1 (Ω) | C1 (F) | R2 (Ω) | C2 (F) | R3 (Ω) | C3 (F) | R4 (Ω) | C4 (F) |
---|---|---|---|---|---|---|---|---|---|---|---|

100 | 1.77 × 10^{−3} | 3.41 × 10^{−7} | 3.82 × 10^{−2} | 1.81 × 10^{−3} | 1.75 × 10 | 5.53 × 10^{−3} | 5.38 × 10^{−1} | 5.23 × 10^{−3} | 7.82 × 10^{−2} | 3.01 × 10^{−3} | 2.16 × 10^{2} |

90 | 1.98 × 10^{−3} | 3.38 × 10^{−7} | 3.83 × 10^{−2} | 2.24 × 10^{−3} | 1.67 × 10 | 5.12 × 10^{−3} | 6.10 × 10^{−1} | 4.98 × 10^{−3} | 8.36 × 10^{−2} | 4.30 × 10^{−3} | 1.55 × 10^{2} |

80 | 2.12 × 10^{−3} | 3.36 × 10^{−7} | 3.85 × 10^{−2} | 2.32 × 10^{−3} | 1.57 × 10 | 5.06 × 10^{−3} | 6.22 × 10^{−1} | 5.02 × 10^{−3} | 8.33 × 10^{−2} | 4.41 × 10^{−3} | 1.48 × 10^{2} |

70 | 2.06 × 10^{−3} | 3.33 × 10^{−7} | 3.88 × 10^{−2} | 2.17 × 10^{−3} | 1.57 × 10 | 5.23 × 10^{−3} | 5.92 × 10^{−1} | 5.03 × 10^{−3} | 8.09 × 10^{−2} | 3.69 × 10^{−3} | 1.88 × 10^{2} |

60 | 2.05 × 10^{−3} | 3.27 × 10^{−7} | 3.95 × 10^{−2} | 1.81 × 10^{−3} | 1.67 × 10 | 5.70 × 10^{−3} | 5.47 × 10^{−1} | 5.19 × 10^{−3} | 7.78 × 10^{−2} | 2.20 × 10^{−3} | 3.24 × 10^{2} |

50 | 2.37 × 10^{−3} | 3.25 × 10^{−7} | 4.03 × 10^{−2} | 3.17 × 10^{−3} | 8.79 | 6.58 × 10^{−3} | 4.80 × 10^{−1} | 5.38 × 10^{−3} | 7.00 × 10^{−2} | 3.02 × 10^{−3} | 2.05 × 10^{2} |

40 | 2.32 × 10^{−3} | 3.20 × 10^{−7} | 4.07 × 10^{−2} | 4.25 × 10^{−3} | 7.03 | 6.96 × 10^{−3} | 4.82 × 10^{−1} | 5.62 × 10^{−3} | 6.80 × 10^{−2} | 3.93 × 10^{−3} | 1.54 × 10^{2} |

30 | 3.02 × 10^{−3} | 3.48 × 10^{−7} | 4.07 × 10^{−2} | 6.30 × 10^{−3} | 5.42 | 7.59 × 10^{−3} | 4.27 × 10^{−1} | 5.72 × 10^{−3} | 5.75 × 10^{−2} | 5.76 × 10^{−3} | 9.97 × 10 |

20 | 2.41 × 10^{−3} | 3.05 × 10^{−7} | 4.14 × 10^{−2} | 9.32 × 10^{−3} | 4.39 | 7.39 × 10^{−3} | 4.80 × 10^{−1} | 5.99 × 10^{−3} | 6.52 × 10^{−2} | 1.11 × 10^{−2} | 4.73 × 10 |

10 | 2.37 × 10^{−3} | 3.01 × 10^{−7} | 4.16 × 10^{−2} | 1.12 × 10^{−2} | 4.01 | 7.64 × 10^{−3} | 4.75 × 10^{−1} | 6.10 × 10^{−3} | 6.46 × 10^{−2} | 1.59 × 10^{−2} | 3.32 × 10 |

0 | 2.45 × 10^{−3} | 2.99 × 10^{−7} | 4.19 × 10^{−2} | 1.35 × 10^{−2} | 3.51 | 8.50 × 10^{−3} | 4.46 × 10^{−1} | 6.16 × 10^{−3} | 6.26 × 10^{−2} | 2.81 × 10^{−2} | 1.98 × 10 |

Frequency | Increased Rate |
---|---|

1 Hz | 113.15% |

10 Hz | 107.29% |

100 Hz | 104.73% |

1 kHz | 104.32% |

**Table 6.**Conditions for both simulation and measurement of the impedance of the cell during discharge.

Parameter | Description |
---|---|

DC offset | 1.5 C, 1 C, 0.5 C, 0.25 C and 0.125 C |

SoC | From 100% to 0% |

Test frequency | 1 Hz, 250 Hz |

Amplitude | 50 mA (each) |

C-Rate | Percent Error (%) |
---|---|

1.5 | 2.28 |

1 | 2.36 |

0.5 | 1.32 |

0.25 | 0.76 |

0.125 | 0.56 |

Total | 1.46 |

**Table 8.**Average values of percent errors of the estimated impedance relative to the measured impedance at different SoHs.

SoH (%) | Percent Error (%) |
---|---|

100 | 0.43 |

95 | 0.33 |

90 | 0.75 |

85 | 0.55 |

80 | 0.45 |

75 | 0.35 |

Mean | 0.48 |

**Table 9.**Conditions for both simulation and measurement of impedance of a cell discharged from different initial SoCs.

Parameter | Description |
---|---|

Test frequency | 1 Hz |

Amplitude | 50 mA |

$\mathrm{DC}\text{}\mathrm{bias}\text{}({\mathrm{I}}_{\mathrm{d}\mathrm{c}}$) | 2.6 A (1 C) |

Depth of discharge | From 0 to 100% |

Discharge time | 10 min. (each) |

Relaxation time | 60 min. (each) |

SoH | 95% |

Initial temperature | 25 °C |

Parameter | Description |
---|---|

Test frequency | 1 Hz to 100 Hz |

Depth of discharge | From 0 to 100% |

SoH | 95% |

$\mathrm{DC}\text{}\mathrm{bias}\text{}({\mathrm{I}}_{\mathrm{d}\mathrm{c}}$) | 1.3 A (0.5 C) |

Sampling rate | 2048 Hz |

Amplitude | 50 mA (each) |

**Table 11.**Simulation conditions for comparing continuous 1 Hz impedance at different sampling rates.

Parameter | Description |
---|---|

Sampling rate | 16 Hz to 2048 Hz |

Test frequency | 1 Hz |

Depth of discharge | From 0 to 100% |

SoH | 95% |

$\mathrm{DC}\text{}\mathrm{bias}\text{}({\mathrm{I}}_{\mathrm{d}\mathrm{c}}$) | 1.3 A (0.5 C) |

Amplitude | 50 mA (each) |

**Table 12.**Percent errors of the mean of the impedance simulated at a sampling rate of 2048 Hz and the mean of the impedance simulated at each different sampling rate.

Sampling Rate | Percent Error |
---|---|

16 Hz | 1.63% |

32 Hz | 0.48% |

64 Hz | 0.16% |

128 Hz | 0.06% |

256 Hz | 0.026% |

512 Hz | 0.01% |

1024 Hz | 0.00% |

2048 Hz | Reference |

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**MDPI and ACS Style**

Kim, J.; Kowal, J.
Development of a Matlab/Simulink Model for Monitoring Cell State-of-Health and State-of-Charge via Impedance of Lithium-Ion Battery Cells. *Batteries* **2022**, *8*, 8.
https://doi.org/10.3390/batteries8020008

**AMA Style**

Kim J, Kowal J.
Development of a Matlab/Simulink Model for Monitoring Cell State-of-Health and State-of-Charge via Impedance of Lithium-Ion Battery Cells. *Batteries*. 2022; 8(2):8.
https://doi.org/10.3390/batteries8020008

**Chicago/Turabian Style**

Kim, Jonghyeon, and Julia Kowal.
2022. "Development of a Matlab/Simulink Model for Monitoring Cell State-of-Health and State-of-Charge via Impedance of Lithium-Ion Battery Cells" *Batteries* 8, no. 2: 8.
https://doi.org/10.3390/batteries8020008