# Characteristics of Open Circuit Voltage Relaxation in Lithium-Ion Batteries for the Purpose of State of Charge and State of Health Analysis

^{*}

## Abstract

**:**

## 1. Introduction

- How does the VR magnitude and curve shape change as a function of SOC value?
- What period of time is necessary for the battery to be considered fully rested at SS-OCV?
- For estimating SOC, how complex must a model be, and how short can the VR period be, to accurately determine SS-OCV?
- For estimating SOH, how complex must a model be, and how short can the VR period be, at a sensitive SOC position, to accurately capture VR magnitude and curve shape?

- 5.
- The model should capture the dynamic behavior of the voltage curve while being computationally simple to curve fit.
- 6.
- VR for SOH evaluation will likely be implemented during charge as most applications have specific demands during discharge that prohibit long open circuit conditions. For example, EV and mobile devices are often left unattended while charging thus presenting a VR opportunity.
- 7.
- The model must function over the complete SOC operating range.
- 8.
- A consistent SOC for SOH estimation must be chosen to maximize VR magnitude sensitivity and curve shape changes for estimating SOH.

## 2. Materials and Methods

#### 2.1. Cells

- Panasonic NCR18650B (Nickel Cobalt Aluminum [NCA] positive electrode);
- LG Chem INR18650B4 (Nickel Manganese Cobalt [NMC532] positive electrode, henceforth NMC refers to 532 composition unless otherwise stated.);
- Lithium Werks APR18650m1B (Lithium Iron Phosphate [LFP] positive electrode; “nanophosphate”).

#### 2.2. Test Equipment

#### 2.3. Test Procedure

#### 2.4. Battery Model Architecture and Coefficient Determination

_{0}, which is responsible for the IR voltage drop. The ideal voltage sources are fit with the RC values for each VR period. In open circuit conditions the RC pairs in the VR model will always be charging, whereas the standard model will always have them discharging. These changes are primarily for SS-OCV determination which has impacts on curve fitting accuracy. For each model, the definition of ${V}_{s}$ is different. In the standard model it represents the SS-OCV, whereas in the VR model it represents the terminal voltage at open circuit. Both models have the same initial equation for ${V}_{\mathrm{terminal}}$, the voltage that appears at the battery terminals, Equation (1). Note that t = 0 is when the open circuit condition is applied and thus ${R}_{0}$ does not factor into the voltage equation for relaxation, which starts at t = 0

^{+}. Therefore, it was removed in the VR model:

^{+}. For the VR model the ideal voltage source polarity is reversed. With the original model the RC pairs will operate via Equation (2) compared to the VR model where they operate via Equation (3):

^{+}instead of the SS-OCV. This makes the SS-OCV value the sum of all RC pairs voltage, ${V}_{\mathrm{RC},p}$, and the series voltage source, ${V}_{s}$. This was shown graphically in Figure 3. This way, when curve fitting SS-OCV can be larger or smaller than the final value of a dataset since it is not assumed to be ${V}_{s}$ and dependent on multiple coefficients. This is the primary reason that the standard model was changed to the VR model for better curve fitting on VR datasets.

_{s}, V

_{1}, ${\tau}_{1}$, V

_{2}, ${\tau}_{2}$, …, V

_{n}, ${\tau}_{n}$} are iterated. The Levenberg–Marquardt algorithm was chosen as it can perform fitting over the large time constant ranges while being quick to solution. This fitting technique for non-linear functions minimizes error based on the sum of the residuals squared. It is an interpolation between the Gauss–Newton algorithm and gradient descent method. The Levenberg–Marquardt algorithm is dependent on initial values and can be susceptible to ending in local minima should the initial value be poorly estimated. To alleviate this concern 5 initial values were used ranging from 0 to half VR magnitude for the voltage terms and spanning 3 orders of magnitude for the time constants. These values were estimated by scaling previous estimations that produced a low error.

#### 2.5. Assessing Model Inaccuracy Impacts on SOC and SOH

## 3. Results and Discussion

#### 3.1. VR Magnitude vs. SOC

#### 3.2. VR Shape and SS-OCV

#### 3.3. Model Complexity

_{RC,p}refers to voltage magnitude attributable to a specific RC pair of the model.

_{RC,1}V

_{RC,2}and V

_{RC,3}line up with the 3 linear regions observed on the semi-log plot, explaining why there is such significant improvement to the accuracy from the 1 to 2 to 3 RC models. It also explains the marginal improvements after this point as most of the behavior is captured with 3 RCs. These marginal improvements better represent the real electrochemical function which dictate the profile as it is not simply exponential decay functions.

_{RC,5}and V

_{RC,4}appear to resolve into a single V

_{RC}. This new V

_{RC}has a time constant between the previous ones, and a higher V

_{n}to account for the magnitude of both. When changing from the 4 RC to 3 RC model, the lower time constant circuits V

_{RC,3}and V

_{RC,2}appear to resolve. During these changes all RCs time constants migrate closer together to account for the simplification of the crossover period.

#### 3.4. Quantifying Inaccuracy Impacts on SOC and SOH

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Standard equivalent circuit model and proposed VR equivalent circuit model of an LIB with its response to open circuit conditions.

**Figure 5.**VR from operational profile as a function of SOC with values post discharge current shown in gray and values post charge current shown in red.

**Figure 6.**Twenty-four hour relaxation shown on a linear and logarithmic scale of NMC, NCA, and LFP chemistries.

**Figure 7.**Equivalent circuit model curve fit for models employing 1, 3, or 5 RC circuits, along with the overall RMSD error for models ranging from 1–6 RC circuits.

**Figure 8.**Impact of each RC circuit in a model on the VR profile for NMC batteries for 3, 4, and 5 RC circuit models.

Specification | NCA | NMC | LFP |
---|---|---|---|

Capacity (Ah) | 3.2 | 2.6 | 1.1 |

Nominal voltage (V) | 3.6 | 3.7 | 3.3 |

Voltage range (V) | 2.50–4.20 | 2.75–4.20 | 2.00–3.60 |

Max Continuous Discharge Rate (C-rate; hour rate) | 2; 0.5 | 2; 0.5 | 27; 0.037 |

Max Continuous Charge Rate (C-rate; hour rate) | 0.5; 2 | 1; 1 | 3.6; 0.28 |

Cycles to 80% of original capacity | 250 | 300 | 4000 |

Step | Type | Control Parameters | Description |
---|---|---|---|

1 | Rest | 30 s and cell temperature ≤ 30 °C | Initial data |

2 | CC Charge | 2 h rate to high voltage limit | As defined by specification sheet |

3 | CV Charge | hold voltage for 1 h | 1 h results in lower currents than specification sheet requires |

4 | Rest | 30 s and cell temperature ≤ 30 °C | |

5 | CC Discharge | 2 h rate for 10% rated coulombic capacity | Discharge 10% SOC |

6 | Rest | 3 h rest | Collecting VR data |

7 | Jump | To Step 5 until low voltage value is reached | Discharging to low SOC for 9 data points |

8 | CC Charge | 2 h rate for 10% rated capacity | Charge 10% SOC |

9 | Rest | 3 h rest | Collecting VR data |

10 | Jump | To Step 8 until high voltage limit is reached | Charging to high SOC for 9 data points |

11 | Rest | 30 s rest | End rest |

Step | Type | Control Parameters | Description |
---|---|---|---|

1 | Rest | 30 s and cell temperature ≤ 30 °C | Initial data |

2 | CC Discharge | 2 h rate to low voltage limit | As defined by spec sheet |

3 | Rest | 30 s and cell temperature ≤ 30 °C | |

4 | CC Charge | 2 h rate for optimal rated capacity | Charge to optimal SOC (LFP = 66%; NCA, NMC = 45%) |

5 | Rest | 30 min rest | Collecting VR data |

6 | Rest | 23 h 30 min rest | Collecting VR data |

n RC Model | NCA | NMC | LFP |
---|---|---|---|

1 | 5.26% | 6.67% | 20.53% |

2 | 0.57% | 0.99% | 6.80% |

3 | <0.5% | <0.5% | 4.87% |

4 | <0.5% | <0.5% | 2.27% |

5 | <0.5% | <0.5% | 2.27% |

6 | <0.5% | <0.5% | 2.27% |

Chemistry: | NCA | NMC | LFP | |||
---|---|---|---|---|---|---|

n RC Model | RMSD | EST | RMSD | EST | RMSD | EST |

1 | 9.59% | 0.15 h | 7.84% | 0.16 h | 4.99% | 0.13 h |

2 | 1.14% | 1.22 h | 0.94% | 2.26 h | 1.43% | 0.34 h |

3 | 0.62% | 4.73 h | 0.36% | 2.98 h | 0.62% | 2.36 h |

4 | 0.40% | 5.36 h | 0.16% | 3.43 h | 0.45% | 4.04 h |

5 | 0.23% | 85.35 h | 0.15% | 6.95 h | 0.36% | 5.36 h |

6 | 0.23% | 83.90 h | 0.37% | 26.60 h | 0.37% | 44.34 h |

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**MDPI and ACS Style**

Theuerkauf, D.; Swan, L.
Characteristics of Open Circuit Voltage Relaxation in Lithium-Ion Batteries for the Purpose of State of Charge and State of Health Analysis. *Batteries* **2022**, *8*, 77.
https://doi.org/10.3390/batteries8080077

**AMA Style**

Theuerkauf D, Swan L.
Characteristics of Open Circuit Voltage Relaxation in Lithium-Ion Batteries for the Purpose of State of Charge and State of Health Analysis. *Batteries*. 2022; 8(8):77.
https://doi.org/10.3390/batteries8080077

**Chicago/Turabian Style**

Theuerkauf, David, and Lukas Swan.
2022. "Characteristics of Open Circuit Voltage Relaxation in Lithium-Ion Batteries for the Purpose of State of Charge and State of Health Analysis" *Batteries* 8, no. 8: 77.
https://doi.org/10.3390/batteries8080077