# A Hybrid Method for Performance Degradation Probability Prediction of Proton Exchange Membrane Fuel Cell

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## Abstract

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## 1. Introduction

- (1)
- With consideration of the uncertainties in the degradation process, a Wiener process model is established to describe the overall degradation trend of PEMFC and multiple kinds of variability sources are adequately considered in the model.
- (2)
- To overcome the disadvantage of LSTM in parallel processing, a degradation prediction model is established by transformer, which is used to predict the degradation trend and capture the local fluctuation information.
- (3)
- The MC-dropout is added in transformer network to quantify the uncertainty of the prediction results in order to provide more effective decision support for practical engineering applications.

## 2. Degradation Modeling of PEMFC

## 3. State of Health Estimation and Parameter Estimation

Algorithm 1 The procedures of the UKF algorithm |

1. Initialization (k = 0):${\widehat{\mathbf{X}}}_{0}=\mathbf{E}({\mathbf{X}}_{\mathbf{0}})$, ${\mathbf{P}}_{0}=\mathbf{E}\left[({\mathbf{X}}_{0}-{\widehat{\mathbf{X}}}_{0}){({\mathbf{X}}_{0}-{\widehat{\mathbf{X}}}_{0})}^{T}\right]$ 2. Time update:${\widehat{\mathbf{X}}}_{k|k-1}=\mathbf{A}{\widehat{\mathbf{X}}}_{k-1|k-1}$, ${\mathbf{P}}_{k|k-1}=\mathbf{A}{\mathbf{P}}_{k-1|k-1}{\mathbf{A}}^{T}+\mathbf{\Phi}$ 3. Sigma points and weights calculation:(1) $\{{\mathbf{X}}_{k|k-1}^{(0)}={\widehat{\mathbf{X}}}_{k|k-1},\text{}i=0$; ${\mathbf{X}}_{k|k-1}^{(i)}={\widehat{\mathbf{X}}}_{k|k-1}+(\sqrt{{(n+\lambda )\mathbf{P})}_{i}},\text{}i=1\dots n$; ${\mathbf{X}}_{k|k-1}^{(i)}={\widehat{\mathbf{X}}}_{k|k-1}-(\sqrt{{(n+\lambda )\mathbf{P})}_{i}},\text{}i=n+1\dots 2n$ where ${(\mathbf{P})}^{T}(\mathbf{P})=\mathbf{P}$, ${(\mathbf{P})}_{i}$ is the ith column of the square root of the matrix $\mathbf{P}$. (2) ${w}_{m}^{(0)}=\lambda /n+\lambda $; ${w}_{c}^{(0)}=\lambda /n+\lambda +(1-{\tau}^{2}+\beta )$; ${w}_{m}^{(i)}={w}_{c}^{(i)}=\lambda /2(n+\lambda ),\text{}i=1\dots 2n$, where $\lambda ={\tau}^{2}(n+\kappa )$ is the scaling parameter, and the other parameters are generally set to ${\tau}^{2}=0.01,\text{}\beta =0,\text{}(n+\lambda )\mathbf{P}$ = 3 4. Measurement update${V}_{k|k-1}^{(i)}=h\left[{\mathbf{X}}_{k|k-1}^{(i)}\right]$ ${\overline{V}}_{k|k-1}={\sum}_{i=0}^{2n}{w}_{m}^{(i)}{V}_{k|k-1}^{(i)}$ ${P}_{{V}_{k}{V}_{k}}={\sum}_{i=0}^{2n}{w}_{c}^{(i)}[{V}_{k|k-1}^{(i)}-{\overline{V}}_{k|k-1}]{[{V}_{k|k-1}^{(i)}-{\overline{V}}_{k|k-1}]}^{T}+R$ ${\mathbf{P}}_{{\mathbf{X}}_{k}{V}_{k}}={\sum}_{i=0}^{2n}{w}_{c}^{(i)}[{\mathbf{X}}_{k|k-1}^{(i)}-{\widehat{\mathbf{X}}}_{k|k-1}]{[{V}_{k|k-1}^{(i)}-{\overline{V}}_{k|k-1}]}^{T}$ ${\mathbf{K}}_{k}={\mathbf{P}}_{{\mathbf{X}}_{k}{V}_{k}}{P}_{{V}_{k}{V}_{k}}^{-1}$ ${\widehat{\mathbf{X}}}_{k|k}={\widehat{\mathbf{X}}}_{k|k-1}+{\mathbf{K}}_{k}[{V}_{k}-{\overline{V}}_{k}]$ ${\mathbf{P}}_{k|k}={\mathbf{P}}_{k|k-1}-{P}_{{V}_{k}{V}_{k}}{\mathbf{K}}_{k}{\mathbf{K}}_{k}^{T}$ |

## 4. Method of Degradation Prediction

#### 4.1. Problem Description

#### 4.2. The Transformer Structure

#### 4.2.1. Data Input

#### 4.2.2. Encoder

#### 4.2.3. Decoder

#### 4.2.4. Output

#### 4.3. The Method of MC-Dropout

#### 4.4. The Hybrid Prediction Method for Performance Degradation

## 5. Experimental Study

#### 5.1. Experimental Dataset

#### 5.2. State Estimation

#### 5.3. Performance Degradation Prediction Results

#### 5.4. Verification with LSTM

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameters Setting | Value |
---|---|

Window length | 10 |

Batch size | 50 |

Epochs | 50 |

Dropout | 0.1 |

Multi-Head | 10 |

Learning rate | 0.001 |

Training Set | MAPE (%) | RMSE (%) | MAE (%) |
---|---|---|---|

40% | 4.0261 | 1.0416 | 0.9380 |

50% | 1.5429 | 0.4826 | 0.3630 |

60% | 1.2410 | 0.3845 | 0.2923 |

Training Set | MAPE (%) | RMSE (%) | MAE (%) |
---|---|---|---|

40% | 2.9276 | 0.9675 | 0.7665 |

50% | 1.1153 | 0.4408 | 0.2851 |

60% | 0.9896 | 0.4117 | 0.2515 |

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## Share and Cite

**MDPI and ACS Style**

Hu, Y.; Zhang, L.; Jiang, Y.; Peng, K.; Jin, Z.
A Hybrid Method for Performance Degradation Probability Prediction of Proton Exchange Membrane Fuel Cell. *Membranes* **2023**, *13*, 426.
https://doi.org/10.3390/membranes13040426

**AMA Style**

Hu Y, Zhang L, Jiang Y, Peng K, Jin Z.
A Hybrid Method for Performance Degradation Probability Prediction of Proton Exchange Membrane Fuel Cell. *Membranes*. 2023; 13(4):426.
https://doi.org/10.3390/membranes13040426

**Chicago/Turabian Style**

Hu, Yanyan, Li Zhang, Yunpeng Jiang, Kaixiang Peng, and Zengwang Jin.
2023. "A Hybrid Method for Performance Degradation Probability Prediction of Proton Exchange Membrane Fuel Cell" *Membranes* 13, no. 4: 426.
https://doi.org/10.3390/membranes13040426