# An Evolutionary Computation Approach for the Online/On-Board Identification of PEM Fuel Cell Impedance Parameters with A Diagnostic Perspective

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Fuel Cell Stack Models

#### 2.1. RC Model

#### 2.2. Fouquet Model

#### 2.3. Dhirde Model

## 3. The Evolutionary Algorithm

#### 3.1. The Algorithm

#### 3.2. The Objective Function

#### 3.3. The Sub-Populations

#### 3.4. Selection, Crossover and Mutation Operators

#### 3.5. Search Space and Parameter Correlations

## 4. Experimental Results

^{2}, whose results are presented in Reference [14]. The second set comes from experiments performed in the frame of the H2020 HEALTH-CODE project [3], characterized by various faults, such as those described in Reference [40].

#### 4.1. Validation by Literature Data with RC and Fouquet Model

#### 4.2. Validation by Experimental Dataset A with RC and Fouquet Model

#### 4.3. Validation by Experimental Dataset B with Dhirde Model

## 5. Identification Oriented to Diagnostics

**Fault Condition (FC)**= $(\Delta L<0)$ & $(\Delta {R}_{d}>0)$.

**Drying Condition (DC)**= (FC) & $(\Delta {Q}_{1}>0)$.

**Starvation Condition (SC)**= (FC) & $(\Delta {R}_{\mathrm{ct},1}>0)$.

**Cathode Drying (CDC)**= (DC) & $(\Delta {\tau}_{\mathrm{d}}<0)$,**Anode Drying (ADC)**= (DC) & $(\Delta {\tau}_{\mathrm{d}}>0)$.

**Air Starvation (ASC)**= (SC) & $(\Delta {Q}_{2}>0)$,**Fuel Starvation (FSC)**= (SC) & $(\Delta {Q}_{2}<0)$.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Fouquet experimental data [14]. Results achieved by the evolutionary algorithm (EA) in the identification of resistive-capacitive (RC) and Fouquet models.

**Figure 12.**Dataset B. EIS experimental spectra in various operating conditions. x-scale differs from y-scale to highlight the variation of the spectrum.

**Figure 13.**Parametric variations due to the analyzed faults. (

**a**) Parameter with small variation. (

**b**) Parameters with large variations.

**Table 1.**Identification on Fouquet data [14] by using the third order linear model.

Parameter | Best | Mean | $\mathit{\sigma}$ |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ (m$\mathsf{\Omega}$) | 4.514 | 4.515 | 3.17·10${}^{-3}$ |

${R}_{1}$ (m$\mathsf{\Omega}$) | 4.230 | 2.457 | 1.331 |

${R}_{2}$ (m$\mathsf{\Omega}$) | 5.296 | 4.987 | 1.233 |

${R}_{3}$ (m$\mathsf{\Omega}$) | 2.132 | 4.209 | 1.343 |

${C}_{1}$ (F) | 20.69 | 20.95 | 0.1829 |

${C}_{2}$ (F) | 1.7286 | 2.202 | 0.3345 |

${C}_{3}$ (F) | 0.8865 | 0.8830 | 2.430·10${}^{-3}$ |

Matlab time (s) | 3.87 | 15.87 | 12.07 |

BBB time (ms) | 0.258 | 0.765 | 0.438 |

${\mathcal{F}}_{\mathrm{obj}}$ (m$\mathsf{\Omega}$) | 9.075 | 9.861 | 1.250 |

**Table 2.**Identification on Fouquet data [14] by using the Fouquet model.

Parameter | Best | Mean | $\mathit{\sigma}$ |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ (m$\mathsf{\Omega}$) | 4.498 | 4.496 | 0.0490 |

${R}_{\mathrm{ct}}$ (m$\mathsf{\Omega}$) | 8.199 | 8.038 | 0.786 |

Q | 1.866 | 1.875 | 0.09742 |

$\varphi $ | 0.8366 | 0.8334 | 0.05016 |

${R}_{\mathrm{d}}$ (m$\mathsf{\Omega}$) | 3.501 | 3.669 | 0.8150 |

${\tau}_{\mathrm{d}}$ (s) | 0.1938 | 0.2089 | 0.01734 |

Matlab time (s) | 2.18 | 10.39 | 4.88 |

BBB time (ms) | 0.861 | 2.46 | 1.68 |

${\mathcal{F}}_{\mathrm{obj}}$ (m$\mathsf{\Omega}$) | 0.954 | 6.487 | 6.756 |

Parameter | Best | Mean | $\mathit{\sigma}$ |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ (m$\mathsf{\Omega}$) | 55.63 | 55.61 | 0.0584 |

${R}_{1}$ (m$\mathsf{\Omega}$) | 36.95 | 36.81 | 0.9730 |

${R}_{2}$ (m$\mathsf{\Omega}$) | 52.51 | 52.61 | 0.8150 |

${R}_{3}$ (m$\mathsf{\Omega}$) | 50.48 | 50.50 | 0.1580 |

${C}_{1}$ (F) | 0.0340 | 0.0340 | 0.0001 |

${C}_{2}$ (F) | 0.1483 | 0.1476 | 0.0056 |

${C}_{3}$ (F) | 1.934 | 1.9293 | 0.0115 |

Matlab time (s) | 9.880 | 27.48 | 9.500 |

BBB time (ms) | 0.4700 | 0.9830 | 0.5460 |

${\mathcal{F}}_{\mathrm{obj}}$ (m$\mathsf{\Omega}$) | 0.0294 | 0.0294 | 2.7$\xb7{10}^{-5}$ |

Parameter | Best | Mean | $\mathit{\sigma}$ |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ (m$\mathsf{\Omega}$) | 51.51 | 51.02 | 1.070 |

${R}_{\mathrm{ct}}$ (m$\mathsf{\Omega}$) | 101.3 | 103.6 | 4.815 |

Q | 0.1689 | 0.1807 | 0.0256 |

$\varphi $ | 0.7369 | 0.7258 | 0.0232 |

${R}_{\mathrm{d}}$ (m$\mathsf{\Omega}$) | 44.78 | 43.08 | 3.378 |

${\tau}_{\mathrm{d}}$ (s) | 0.2203 | 0.2222 | 0.0043 |

Matlab time (s) | 8.230 | 43.25 | 15.06 |

BBB time (ms) | 1.218 | 1.7680 | 0.3390 |

${\mathcal{F}}_{\mathrm{obj}}$ (m$\mathsf{\Omega}$) | 0.0392 | 0.0437 | 0.0031 |

**Table 5.**Experimental dataset B. Identification results in normal operative conditions with Dhirde model.

Parameter | Value | Parameter | Value |
---|---|---|---|

${R}_{\mathsf{\Omega}}$ (m$\mathsf{\Omega}$) | 54.20 | ${Q}_{1}$ | 2.953 |

${R}_{\mathrm{ct},1}$ (m$\mathsf{\Omega}$) | 46.30 | ${Q}_{2}$ | 0.1731 |

${R}_{\mathrm{ct},2}$ (m$\mathsf{\Omega}$) | 113.4 | ${\varphi}_{1}$ | 1.000 |

${R}_{\mathrm{d}}$ (m$\mathsf{\Omega}$) | 15.90 | ${\varphi}_{2}$ | 0.705 |

${\tau}_{\mathrm{d}}$ (s) | 7.8886 | L (mH) | 47.10 |

Matlab time (s) | 93.28 | ${\mathcal{F}}_{\mathrm{obj}}$ (m$\mathsf{\Omega}$) | 26.7 |

**Table 6.**Comparison among the parameters identified in various operating conditions at 40A nominal current. The number of arrows highlights, in a qualitative way, the percentage variation x with respect to nominal condition. $\uparrow :10\%\le x<50\%$, $\uparrow \uparrow :50\%\le x<100\%$, $\uparrow \uparrow \uparrow :x\ge 100\%$. ↓, $\downarrow \downarrow $, and $\downarrow \downarrow \downarrow $ are defined accordingly.

Parameter | Normal Condition | Cathode Drying | Anode Drying | Air Starvation | Fuel Starvation | ||||
---|---|---|---|---|---|---|---|---|---|

${R}_{\mathsf{\Omega}}$ ($\mathsf{\Omega}$) | 0.0542 | 0.0554 | 0.0549 | 0.0528 | 0.0546 | ||||

${R}_{\mathrm{ct},1}$ ($\mathsf{\Omega}$) | 0.0463 | 0.0584 | ↑ | 0.0477 | 0.0966 | $\uparrow \uparrow \uparrow $ | 0.0997 | $\uparrow \uparrow \uparrow $ | |

${R}_{\mathrm{ct},2}$ ($\mathsf{\Omega}$) | 0.1134 | 0.1063 | 0.1209 | 0.1142 | 0.1027 | ||||

${Q}_{1}$ | 2.9533 | 3.9406 | ↑ | 3.8822 | ↑ | 2.4108 | ↓ | 3.9387 | ↑ |

${Q}_{2}$ | 0.1731 | 0.1959 | ↑ | 0.2368 | ↑ | 0.2472 | ↑ | 0.1598 | $\downarrow \downarrow $ |

${\varphi}_{1}$ | 1.0000 | 0.8725 | ↓ | 0.9999 | 1.0000 | 0.7469 | |||

${\varphi}_{2}$ | 0.7050 | 0.7150 | 0.6719 | 0.7010 | 0.7321 | ||||

${R}_{\mathrm{d}}$ ($\mathsf{\Omega}$) | 0.0159 | 0.0316 | $\uparrow \uparrow $ | 0.0319 | $\uparrow \uparrow \uparrow $ | 0.0453 | $\uparrow \uparrow \uparrow $ | 0.0456 | $\uparrow \uparrow \uparrow $ |

${\tau}_{\mathrm{d}}$ (s) | 7.8861 | 5.9740 | ↓ | 8.9543 | ↑ | 6.0082 | ↓ | 6.5949 | ↓ |

L (H) | 0.0471 | 0.0023 | $\downarrow \downarrow $ | 0.0015 | $\downarrow \downarrow $ | 0.0028 | $\downarrow \downarrow $ | 0.0037 | $\downarrow \downarrow $ |

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**MDPI and ACS Style**

Zamboni, W.; Petrone, G.; Spagnuolo, G.; Beretta, D.
An Evolutionary Computation Approach for the Online/On-Board Identification of PEM Fuel Cell Impedance Parameters with A Diagnostic Perspective. *Energies* **2019**, *12*, 4374.
https://doi.org/10.3390/en12224374

**AMA Style**

Zamboni W, Petrone G, Spagnuolo G, Beretta D.
An Evolutionary Computation Approach for the Online/On-Board Identification of PEM Fuel Cell Impedance Parameters with A Diagnostic Perspective. *Energies*. 2019; 12(22):4374.
https://doi.org/10.3390/en12224374

**Chicago/Turabian Style**

Zamboni, Walter, Giovanni Petrone, Giovanni Spagnuolo, and Davide Beretta.
2019. "An Evolutionary Computation Approach for the Online/On-Board Identification of PEM Fuel Cell Impedance Parameters with A Diagnostic Perspective" *Energies* 12, no. 22: 4374.
https://doi.org/10.3390/en12224374