# Fast Impedance Spectrum Construction for Lithium-Ion Batteries Using a Multi-Density Clustering Algorithm

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## Abstract

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## 1. Introduction

- (1)
- A density-based spatial clustering of applications with a noise-based (DBSCAN-based) impedance data processing method is proposed to utilize the distribution properties of the measurement points by PRBS excitation.
- (2)
- A parameter setting workflow was designed to properly cluster the data points of the impedance measured by DBSCAN, where the key parameters, that is, the neighborhood radius and the minimum number of points within the specific neighborhood, can be adjusted according to the frequency ranges.
- (3)
- The fast measurement of a battery’s impedance by integrating the PRBS and the data processing method was verified at various temperatures and SOCs on a 3 Ah lithium-ion battery.

## 2. The Principle of Rapid Measurement of a Battery’s Impedance Spectrum

#### 2.1. PRBS Generation

_{n}are shifted one bit to the right register from D

_{n}

_{−1}to D

_{n}

_{−2}in each clock cycle f

_{gen}to produce a new bit a

_{n}. The complete MLS sequence is obtained after L × f

_{gen}cycles, where f

_{gen}is the clock period of the algorithm and L is the total length of the sequence. Moreover, f

_{gen}limits the upper range of the measurable frequency for the MLS to f

_{max}= 0.45 f

_{gen}. The register order l determines the length of the MLS as L = 2

^{l}− 1 and L limits the resolution of the measured frequency to f

_{min}= f

_{gen}/L.

#### 2.2. Calculation of the Battery’s Impedance

_{gen}and generate the MLS sequence offline according to the measurement bandwidth [f

_{min}, f

_{max}] for the battery to be tested. Afterward, the MLS sequence is injected into the lithium-ion battery as an excitation, and the battery’s impedance in the whole frequency domain is calculated by collecting the voltage and current measurements. Thus, a spectral curve of the impedance is obtained. The whole process is shown in Figure 2.

_{Re}($\omega $) and the imaginary part Z

_{Im}($\omega $) of the impedance can be obtained as follows:

## 3. Extraction of the Lithium-Ion Battery’s EIS Using Multi-Density Data Clustering

#### 3.1. Characterization of the Distribution of Batteries’ Impedance Based on PRBS Excitation

#### 3.2. Impedance Extraction Algorithm with DBSCAN

#### 3.2.1. Parameters Setting for DBSCAN

**MinPts**, and the neighborhood radius

**Eps**.

- (1)
**Minpts**

**MinPts**= 2·

**D**(

**D**is the dimension of the sample data) is usually set for DBSCAN. When the measurement data contain high noise,

**Minpts**should be appropriately increased on the basis of 2·

**D**. In this study, the

**MinPts**was selected as 5 for clustering the battery’s impedance data. This was because the battery’s impedance contained two dimensions of real and imaginary parts with extensive noise.

- (2)
**Eps**

**Eps**in DBSCAN is rather complicated, especially when dealing with large-scale and noisy battery impedance data. The parameter

**Eps**can be selected with the aid of constructing a k-distance curve characterizing the sparsity of the data. This curve is made up of the distances between the data points, where the distance d

_{1i}between a sample point (x

_{1},y

_{1}) and all other points (x

_{i},y

_{i}|i ≠ 1) in the dataset is calculated. The distance metric is based on Euclidean distance, which is calculated using Equation (8):

**Eps**.

#### 3.2.2. Clustering Impedance Data

**Eps**and the minimum number of points within the specific neighborhood

**MinPts**to distinguish the sparsity of the data. The structure of the algorithm is shown in Figure 4.

- (a)
- Determination of the cluster cores based on clustering parameters: if a data point has more than
**MinPts**points in its neighborhood with a radius of**Eps**, this data point is marked as a core point. We can then find all core points in the dataset. The circles of different colors in subfigures (a) and (b) represent the neighborhood of the point with radius**Eps**. - (b)
- Formation of a neighborhood chain: for each core point, find all points that are densely accessible from that core point, forming a chain of neighborhoods.
- (c)
- Labeling the cluster classes and noise points: All points contained in a neighborhood chain are labeled as a cluster class, boundary points are assigned to the core point cluster class to which they are connected, and points that are not assigned to any cluster class are labeled as noise. The entire impedance test dataset is scanned and then labeled with noise points and several data clusters consisting of core and boundary points.

#### 3.2.3. The Proposed Multi-Density Clustering Algorithm

**Eps**and

**MinPts**obtained from the calculation and divides the whole test sample set into multiple clusters. DBSCAN searches for clusters by examining the

**Eps**neighborhood of each point in the dataset, and if the

**Eps**neighborhood of a point p contains more points than

**MinPts**points, a cluster is created with p as the core object. DBSCAN then iteratively aggregates the objects directly density-reachable from these core objects, a process that may involve some merging of the density-reachable clusters. The process ends when no new points are added to any cluster and a cluster class is formed, followed by repeating the process until all the data have been labeled. The steps of the algorithm are shown in Algorithm 1, where the set of core points is defined as Ω, the neighborhood of a sample point x

_{j}is defined as the number of clustering clusters defined as k, the set of unvisited sample points is defined as ${}_{\mathrm{\Gamma}}$, the current set of unvisited sample points is defined as ${\mathrm{\Gamma}}_{old}$, a queue is defined as Q, any of the core objects in Ω are defined as o, the first sample point in queue Q is defined as q, and a clustering cluster is defined as C

_{k}.

Algorithm 1 The proposed multi-density clustering algorithm. |

Input: Dataset: D = {x_{1},x_{2}, …,x}, _{m}Eps:${}_{\epsilon}$, MinPts |

1. Ω ${=}_{\varnothing}$ |

2. for j = 1,2,…, m do |

3. if $\left|{N}_{\epsilon}\left({x}_{j}\right)\right|\ge MinPts$ then |

4. Ω = Ω ∪ {x_{j}} |

5. end if |

6. end for |

7. k = 0 |

8.${}_{\mathrm{\Gamma}}$ = D |

9. while ${\Omega}_{\ne \varnothing}$ do |

10.${\mathrm{\Gamma}}_{old}=\mathrm{\Gamma}$ |

11. Q = <o> |

12. $\mathrm{\Gamma}=\mathrm{\Gamma}\backslash \left\{o\right\}$ |

13. while ${\mathbf{\Omega}}_{\ne \varnothing}$ do |

14. if $\left|{N}_{\epsilon}\left(q\right)\right|\ge MinPts$ then |

15. $\mathrm{\Delta}={N}_{\epsilon}\left(q\right)\cap \mathrm{\Gamma}$ |

16. ${}_{\mathrm{\Gamma}}{}_{=}{}_{\mathrm{\Gamma}}{}_{\backslash}{}_{\mathrm{\Delta}}$ |

17. end if |

18. end while |

19.k = k+1, ${C}_{k}={\mathrm{\Gamma}}_{old}\backslash \mathrm{\Gamma}$ |

20. Ω = Ω\C_{k} |

21. end while |

Output: Clustering result: C = {C_{1},C_{2}, …,C_{k}} |

## 4. Experimental Verification

#### 4.1. Experimental Platform

#### 4.2. Validation of the Impedance Measurement

**Minpts1**was set to 5 to construct the k-distance (k = 5) curve for data in the high-frequency interval, and the curve’s inflection point was selected as

**Eps1**. The first clustering was carried out using the parameters above, and the results of the data discrimination are shown in Figure 7b.

**MinPts**caused two neighboring clusters to merge into a single one, causing the extraction algorithm to fail. Therefore, in the mid- and the low-frequency ranges,

**MinPts**was reduced appropriately. Table 2 shows the parameters of clustering in the three frequency ranges.

#### 4.3. Validation at Different SOCs and Temperatures

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Schematic diagram of the clustering algorithm. The process is as follows: (

**a**) selection of all core points; (

**b**) constructing the neighborhood chains; (

**c**) completing the classification of the clusters.

**Figure 7.**Process of clustering data from the high-frequency region. (

**a**) The parameters selected in the first round. (

**b**) The first clustering results. (

**c**) The parameters selected in the second round. (

**d**) The second set of clustering results. (

**e**) The parameters selected for the third round. (

**f**) The third set of clustering results.

**Figure 9.**Lithium-ion battery’s impedance under different test conditions: (

**a**) T = 15 °C, SOC = 50%; (

**b**) T = 25 °C, SOC = 50%; (

**c**) T = 35 °C, SOC = 50%; (

**d**) T = 25 °C, SOC = 80%.

Nominal Capacity | Voltage Range | Maximum Current |
---|---|---|

3000 mAh | 2.0–4.25 V | 30 A |

Frequencies | Times | MinPts | Eps |
---|---|---|---|

High frequency | 1 | 5 | 6.25 × 10^{−4} |

2 | 5.85 × 10^{−5} | ||

3 | 6.56 × 10^{−6} | ||

Mid-frequency | 1 | 4 | 2.65 × 10^{−4} |

2 | 6.30 × 10^{−5} | ||

3 | 9.64 × 10^{−6} | ||

Low frequency | 1 | 3 | 1.04 × 10^{−3} |

2 | 2.77 × 10^{−4} | ||

3 | 1.30 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Zhu, L.; Peng, J.; Meng, J.; Sun, C.; Cai, L.; Qu, Z.
Fast Impedance Spectrum Construction for Lithium-Ion Batteries Using a Multi-Density Clustering Algorithm. *Batteries* **2024**, *10*, 112.
https://doi.org/10.3390/batteries10030112

**AMA Style**

Zhu L, Peng J, Meng J, Sun C, Cai L, Qu Z.
Fast Impedance Spectrum Construction for Lithium-Ion Batteries Using a Multi-Density Clustering Algorithm. *Batteries*. 2024; 10(3):112.
https://doi.org/10.3390/batteries10030112

**Chicago/Turabian Style**

Zhu, Ling, Jichang Peng, Jinhao Meng, Chenghao Sun, Lei Cai, and Zhizhu Qu.
2024. "Fast Impedance Spectrum Construction for Lithium-Ion Batteries Using a Multi-Density Clustering Algorithm" *Batteries* 10, no. 3: 112.
https://doi.org/10.3390/batteries10030112