# Potential Failure Prediction of Lithium-ion Battery Energy Storage System by Isolation Density Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Isolation Density

_{i}is an anomalous one. At first, a random value in a randomly selected dimension is generated, by which the samples are divided into two parts. Then, the density of the subsets that contain and do not contain x

_{i}, i.e., d and d’, are calculated. This process is repeated until x

_{i}is fully isolated, or the maximum isolation depth is reached; the process is just like that shown in Figure 1a–f. As d

_{i}is the sequence of the density generated in the isolation process of x

_{i}, it is called the isolated density. The central assumption of the proposed method is that d will always be greater than d’ in the statistical sense.

_{i}is isolated independently by several binary trees, and the final isolation density of x

_{i}can be expressed as:

_{i}on the isolation path, V is the hyper-cube volume on the isolation path, L is the length of the edge of the hyper-cube, r is the ratio of the edge length to the original length of the hyper-cube on the isolation path, and V

_{0}is the original volume of the hyper-cube. Superscript t denotes the tth tree, and subscripts I, j, and f denote the ith sample, the jth isolation operation, and the fth feature, respectively.

_{i}is the total estimated on the whole isolation path. That is, the consideration of the “outlier” of the algorithm includes not only the sample itself, but also its domain samples. At this point, the definition of “outlier” is introduced in this paper: the sample with a lower probability density, either by itself or in a subset including some neighborhood samples, is considered as an outlier.

## 4. Algorithm Validation by Artificial Datasets

#### 4.1. Circle Distribution Dataset

#### 4.2. Double-Moon Distribution Dataset

## 5. Algorithm Validation by Public Datasets

#### 5.1. Evaluation Criteria of Algorithm Performance

_{1}, the Matthews correlation coefficient MCC, and mean score f

_{m}. Their formulas are described as follows:

#### 5.2. Wisconsin Breast Cancer Dataset

#### 5.3. Pen Handwriting Dataset

#### 5.4. Statlog (Shuttle) Dataset

#### 5.5. KDD Cup 1999 Dataset

#### 5.6. Banknote Authentication Dataset

#### 5.7. Multi-distribution Dataset

## 6. Anomaly Detection of Lithium-ion Batteries

#### 6.1. Description of Lithium-ion Battery Dataset

#### 6.2. Anomaly Detection of Lithium-ion Batteries

#### 6.3. Result Analysis

## 7. Conclusions

- (1)
- A new anomaly detection method based on isolation density was proposed in this paper and was fully verified by manual datasets and public datasets containing different types of anomalies.
- (2)
- Isolation density can be viewed as the sparse degree or probability density of the battery, from the aspects of density or statistics, respectively. As it inherently involves the idea of ensemble learning, without any prior assumption about the data distribution, the method is characterized by high adaptation and can effectively be used for the detection of many kinds of anomalies.
- (3)
- The voltage variations during a whole discharge process of the batteries are taken as the features of the work condition. Then, through the proposed method and time series data of the voltages, the batteries with different abnormal discharge states are effectively detected.
- (4)
- The abnormal discharge states can be divided into three classes: violent changes, sawtooth wave oscillations, and smooth deviations from normal values.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Abbreviations | |

AUC | area under the curve |

ESS | energy storage system |

FN | false negative |

FP | false positive |

FPR | false positive rate |

LOF | local outlier factor |

ROC | receiver operating characteristic |

LIB | lithium-ion battery |

RE | renewable energy |

SOC | state of charge |

TN | true negative |

TP | true positive |

TPR | true positive rate |

Mathematical Symbols | |

d_{i} | the isolation density of sample x_{i} |

${d}_{i}^{t}$ | the insolation density of sample x_{i} on the tth insolation tree |

${d}_{i,j}^{t}$ | the density of sample x_{i} on the jth isolation step of the tth binary tree |

k | the corresponding weight of insolation density |

L_{f} | the interval length of feature |

n | the total number of features |

nt | number of binary trees |

n’ | depth of isolation path |

p | Precision |

V | the hyper-cube volume |

x_{i} | the ith sample |

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**Figure 1.**Anomaly detection of two-dimensional dataset by isolation density: (

**a**) The first isolation operation; (

**b**) The second isolation operation; (

**c**) The third isolation operation; (

**d**) The fourth isolation operation; (

**e**) The fifth isolation operation; (

**f**) The sixth isolation operation.

**Figure 3.**The detection results of four methods: (

**a**) detection result of HBOS; (

**b**) detection result of iForest; (

**c**) detection result of LOF; (

**d**) detection result of the proposed method.

**Figure 5.**The detection results of four methods: (

**a**) detection result of HBOS; (

**b**) detection result of iForest; (

**c**) detection result of LOF; (

**d**) detection result of the proposed method.

Methods | a | r | p | f_{1} | fm | mcc | auc |
---|---|---|---|---|---|---|---|

HBOS | 0.931507 | 0.830000 | 0.732910 | 0.773643 | 0.781455 | 0.739036 | 0.973333 |

iForest | 0.941096 | 0.740000 | 0.819343 | 0.771269 | 0.779672 | 0.742948 | 0.987619 |

LOF | 0.956164 | 0.870000 | 0.834545 | 0.846955 | 0.852273 | 0.825007 | 0.987143 |

Our method | 0.960274 | 0.890000 | 0.849261 | 0.860406 | 0.869630 | 0.843569 | 0.993016 |

Methods | a | r | P | f_{1} | fm | mcc | auc |
---|---|---|---|---|---|---|---|

HBOS | 0.537888 | 0.220000 | 0.828515 | 0.347421 | 0.524257 | 0.225380 | 0.784804 |

iForest | 0.749689 | 0.612222 | 0.911301 | 0.731437 | 0.761762 | 0.550313 | 0.935055 |

LOF | 0.954037 | 0.958889 | 0.959321 | 0.958873 | 0.959105 | 0.907339 | 0.991252 |

Our method | 0.970186 | 0.982222 | 0.965763 | 0.973698 | 0.973993 | 0.940066 | 0.992113 |

Methods | a | r | P | f_{1} | fm | mcc | auk |
---|---|---|---|---|---|---|---|

HBOS | 0.980532 | 0.960934 | 0.852222 | 0.903246 | 0.906578 | 0.894501 | 0.989560 |

iForest | 0.984245 | 0.976310 | 0.872326 | 0.921358 | 0.924318 | 0.914444 | 0.996031 |

LOF | 0.985353 | 0.986902 | 0.874377 | 0.927207 | 0.930640 | 0.921199 | 0.998799 |

Our method | 0.986128 | 0.999772 | 0.879417 | 0.933989 | 0.939595 | 0.929800 | 0.999225 |

Methods | a | r | p | f_{1} | fm | mcc | auc |
---|---|---|---|---|---|---|---|

HBOS | 0.916730 | 0.858935 | 0.971104 | 0.910599 | 0.915020 | 0.840357 | 0.992173 |

iForest | 0.765162 | 0.551711 | 0.962672 | 0.701344 | 0.757191 | 0.586455 | 0.952720 |

LOF | 0.951331 | 0.922148 | 0.979244 | 0.947828 | 0.950696 | 0.906384 | 0.990327 |

Our method | 0.992158 | 1.000000 | 0.984612 | 0.992233 | 0.992306 | 0.984464 | 0.999962 |

Methods | a | R | p | f_{1} | Fm | mcc | auc |
---|---|---|---|---|---|---|---|

HBOS | 0.791616 | 0.84918 | 0.867229 | 0.857918 | 0.858205 | 0.4678 | 0.829684 |

iForest | 0.895018 | 0.978852 | 0.890506 | 0.932472 | 0.934679 | 0.716576 | 0.907079 |

LOF | 0.975091 | 0.982787 | 0.98367 | 0.983179 | 0.983228 | 0.935519 | 0.994111 |

Our method | 0.983961 | 0.99459 | 0.983991 | 0.989248 | 0.989291 | 0.958014 | 0.997232 |

Methods | A | r | p | f_{1} | fm | mcc | auc |
---|---|---|---|---|---|---|---|

HBOS: | 0.978833 | 0.781081 | 0.868213 | 0.820983 | 0.824647 | 0.811803 | 0.991721 |

iForest | 0.950500 | 0.275676 | 0.788182 | 0.406651 | 0.531929 | 0.446883 | 0.978098 |

LOF | 0.992667 | 0.978378 | 0.9118 | 0.943145 | 0.945089 | 0.940343 | 0.999726 |

Our method | 0.993167 | 0.978378 | 0.919928 | 0.947388 | 0.949153 | 0.944749 | 0.999141 |

Batteries | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{60} | |
---|---|---|---|---|---|---|---|

battery 1 | 3.246 | 3.244 | 3.242 | 3.242 | 3.241 | … | 3.047 |

battery 2 | 3.248 | 3.245 | 3.243 | 3.242 | 3.242 | … | 3.035 |

battery 3 | 3.230 | 3.227 | 3.224 | 3.224 | 3.223 | … | 3.027 |

battery 4 | 3.247 | 3.244 | 3.242 | 3.242 | 3.241 | … | 3.033 |

battery 5 | 3.227 | 3.224 | 3.223 | 3.222 | 3.221 | … | 2.979 |

… | … | … | … | … | … | … | … |

battery 216 | 3.249 | 3.248 | 3.246 | 3.246 | 3.245 | … | 3.054 |

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**MDPI and ACS Style**

Zhu, Y.; Liu, M.; Wang, L.; Wang, J.
Potential Failure Prediction of Lithium-ion Battery Energy Storage System by Isolation Density Method. *Sustainability* **2022**, *14*, 7048.
https://doi.org/10.3390/su14127048

**AMA Style**

Zhu Y, Liu M, Wang L, Wang J.
Potential Failure Prediction of Lithium-ion Battery Energy Storage System by Isolation Density Method. *Sustainability*. 2022; 14(12):7048.
https://doi.org/10.3390/su14127048

**Chicago/Turabian Style**

Zhu, Yong, Mingyi Liu, Lin Wang, and Jianxing Wang.
2022. "Potential Failure Prediction of Lithium-ion Battery Energy Storage System by Isolation Density Method" *Sustainability* 14, no. 12: 7048.
https://doi.org/10.3390/su14127048