# Long-Term Degradation Trend Prediction and Remaining Useful Life Estimation for Solid Oxide Fuel Cells

^{*}

## Abstract

**:**

## 1. Introduction

- To improve the durability, a dynamic model of the ohmic ASR is proposed, which can accurately evaluate the performance degradation characteristics of the SOFC;
- Based on the established dynamic model, the PF algorithm is proposed to achieve the long-term prediction of the degradation trend and the accurate estimation of the RUL of the SOFC.

## 2. Nonlinear Dynamic Model of the SOFC

#### 2.1. Energy Balance Sub-Model

#### 2.2. Electrochemical Sub-Model

## 3. Degradation Trend Prediction and RUL Estimation of the SOFC

#### 3.1. Particle Filter Algorithm

#### 3.2. Prediction of the Degradation Trend Based on the PF Algorithm

#### 3.3. Estimation of the RUL

Algorithm 1 RUL estimation algorithm |

$\mathrm{For}\text{}k=1:Length$ $\mathrm{Set}\text{}P=0,\text{}Alpha=R\left(k\right),Beta=\lambda \left(k\right)\times t,$ $and\text{}Alph{a}_{\mathrm{max}}={R}_{EOL},RU{L}_{k}=0$ $\mathrm{While}\text{}Alpha\le Alph{a}_{\mathrm{max}}$ $Alpha=Alpha+Beta$ $P=P+1$ End While $RUL(k)=P$ End |

## 4. Results and Discussion

#### 4.1. ASR Prediction Results of the Degradation Trend

^{3}h [26], and in this study we use the ASR change during 1600 h to characterize the long-term performance degradation trend of the SOFC. Firstly, the initial values of the PF algorithm i.e., covariance matrix P

_{0}, state matrix X

_{0}, process and measurement error covariance matrix Q and R are, respectively, set as follows:

^{2}) are used as performance indicators. Different from RMSE and MAPE, the nearer to 1 the R

^{2}is, the better the predictive effect is. The R

^{2}is defined as:

^{2}value for the PF and KF algorithms are separately 0.9866 and 0.8895. These further indicate that the PF algorithm has better prediction accuracy than that of the KF approach.

#### 4.2. RUL Prediction Results of the Degradation Trend

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**The degradation trend prediction results (Experimental results (Hou et al. [23])).

Symbol | Definition | Value |
---|---|---|

$N$ | number of cells | 5 |

$K$ | stack lumped thermal capacity | 5500 J·K^{−1} |

$A$ | active area | 100 cm^{2} |

${x}_{{\mathrm{H}}_{2}}^{in}$ | the initial mole fraction of hydrogen | 0.97 |

${x}_{{\mathrm{O}}_{2}}^{in}$ | the initial mole fraction of oxygen | 0.21 |

${x}_{{\mathrm{H}}_{2}\mathrm{O}}^{in}$ | the initial mole fraction of water vapor | 0.03 |

${V}_{EOL}$ | minimal nominal acceptable voltage | 4 V |

$R$ | ideal gas constant | 8.3142 J·mol^{−1}·K^{−1} |

$F$ | Faraday constant | 96,485 C·mol^{−1} |

${E}_{ox}$ | activation energy for the scale growth | 220 K J·mol^{−1} |

${E}_{el}$ | activation energy for the oxide scale conductivity | 75.2 K J·mol^{−1} |

${k}_{p}$ | rate constant for the thickness growth of the scale | 0.0126 cm^{2}·s^{−1} |

${\delta}_{ox}^{0}$ | conductivity constant | 3.2 × 10^{5} S·cm^{−1} |

PF | KF | |
---|---|---|

RMSE | 0.0008 | 0.0126 |

MAPE | 0.0430 | 1.4582 |

R^{2} | 0.9866 | 0.8895 |

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**MDPI and ACS Style**

Cui, L.; Huo, H.; Xie, G.; Xu, J.; Kuang, X.; Dong, Z.
Long-Term Degradation Trend Prediction and Remaining Useful Life Estimation for Solid Oxide Fuel Cells. *Sustainability* **2022**, *14*, 9069.
https://doi.org/10.3390/su14159069

**AMA Style**

Cui L, Huo H, Xie G, Xu J, Kuang X, Dong Z.
Long-Term Degradation Trend Prediction and Remaining Useful Life Estimation for Solid Oxide Fuel Cells. *Sustainability*. 2022; 14(15):9069.
https://doi.org/10.3390/su14159069

**Chicago/Turabian Style**

Cui, Lixiang, Haibo Huo, Genhui Xie, Jingxiang Xu, Xinghong Kuang, and Zhaopeng Dong.
2022. "Long-Term Degradation Trend Prediction and Remaining Useful Life Estimation for Solid Oxide Fuel Cells" *Sustainability* 14, no. 15: 9069.
https://doi.org/10.3390/su14159069