# Dynamic Modeling of a Proton-Exchange Membrane Fuel Cell Using a Gaussian Approach

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## Abstract

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## 1. Introduction

- Provides a novel FC model to estimate the output voltage behavior from the operating current of a fuel cell for steady-state and dynamic responses.
- The training complexity of the algorithm is medium, which makes it easily adaptable to different profiles for testing.
- The proposed FC model can be used in computer simulations and hardware emulators due to its simple implementation using an approximation to the exponential function.
- A commercial Nexa Fuel Cell Power Module is used to validate the proposed FC model.
- The results are compared using analytical and numerical techniques under the same data acquisition parameter to ensure a fair comparison between the models.

## 2. Gaussian Model

## 3. Unconstrained Nonlinear Optimization

- The electric current I is x,
- while the voltage v can be either ${f}_{\mathrm{Gauss}}\left(I\right)$ or the approximation ${\tilde{f}}_{\mathrm{Gauss}}\left(I\right)$.

Algorithm 1: Unconstrained nonlinear optimization procedure. |

Input: Measured dataset ${\left(\right)}_{(}^{{x}_{i}}$ |

1: Use the mathematical model defined by Equation (2) |

2: Determine the specific objective function $F(\Theta )$ to be minimized through Equation (3) |

3: Calculate the residual vector ${r}_{i}(\Theta )={y}_{i}-g({x}_{i},\Theta )$ |

4: Determine the Jacobian matrix ${J}_{r}(\Theta )$ |

5: Use a Non-linear Least Squares algorithm to estimate the optimal parameters as described in Equation (5) |

Output: The vector parameter $\Theta $ |

## 4. Experimental Results

#### 4.1. Training Models

#### 4.2. Validating Model

#### 4.3. Comparison of Gaussian Model with the Parameter Identification by Means of Evolution Strategy

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ARX | Auto regressive eXogenous |

ASO | Atom search optimization |

CHHO | Chaotic Harris Hawks optimization. |

ES | Evolution strategy. |

FC | Fuel cell. |

GOA | Grasshopper optimisation algorithm. |

GWO | Grey wolf optimizer. |

HGA | Hybrid genetic algorithm. |

KF | Kalman filter. |

MAE | Mean absolute error. |

MAEO | Modified Artificial Ecosystem Optimization. |

MPA | Marine Predators Algorithm. |

PEMFC | Proton exchange membrane fuel cell. |

PO | Political Optimizer. |

RE | Relative error. |

RLS | Recursive least square. |

SD | Standard deviation. |

VSDE | Vortex search differential evolution. |

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**Figure 1.**Resulting curve of the Gaussian peaks method when varying the values of the free parameter $\sigma $.

**Figure 2.**Experimental data adquisition configuration used for the Gaussian model training and validation.

**Figure 6.**Statistical results of proposed Gaussian model and the Diffusive global model for the profile shown in Figure 5.

**Figure 7.**Experimental Nexa FC data used for training: (

**a**) current load profile, (

**b**) output voltage simulated with parameters estimated by means of the ES, the diffusive global model and Gaussian model.

**Figure 8.**Experimental Nexa FC data used for validating: (

**a**) current load profile and (

**b**) output voltage simulated with parameters estimated by means of ES, the diffusive global model and the Gaussian model.

**Figure 9.**Statistical results of proposed Gaussian model, Diffusive global model and ES approach for the profile shown in Figure 8.

FC Model Strategy | Ref. | Static Model | V-I Dynamic Model | Variables Used to Evaluate the Model | Training Complexity | Implemen-Tation Cost | Tested with a Real FC |
---|---|---|---|---|---|---|---|

CHHO | [7] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | M | H | |||

GOA | [25] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | L | H | |||

GWO | [16] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | L | H | |||

HGA | [18] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | L | H | |||

Electrical circuit | [26] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | H | ||||

MAEO | [19] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | L | H | |||

VSDE | [20] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | M | H | |||

ASO | [21] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | H | H | |||

Electrical model | [6] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | H | ||||

MPA-PO | [27] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | M | H | |||

TS-KF | [28] | ${T}_{fc}$, ${i}_{fc}$ | H | H | |||

ARX-RLS | [10] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | L | H | |||

Bézier Curve | [22] | ${i}_{fc}$ | M | H | |||

ES | [8] | ${T}_{fc}$, ${i}_{fc}$, ${P}_{{H}_{2}}$, ${P}_{{O}_{2}}$, ${R}_{m}$ | H | ||||

Diffusive model | [24] | ${i}_{fc}$ | H | M | |||

This work | [-] | ${i}_{fc}$ | M | L |

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**MDPI and ACS Style**

González-Castaño, C.; Lorente-Leyva, L.L.; Alpala, J.; Revelo-Fuelagán, J.; Peluffo-Ordóñez, D.H.; Restrepo, C.
Dynamic Modeling of a Proton-Exchange Membrane Fuel Cell Using a Gaussian Approach. *Membranes* **2021**, *11*, 953.
https://doi.org/10.3390/membranes11120953

**AMA Style**

González-Castaño C, Lorente-Leyva LL, Alpala J, Revelo-Fuelagán J, Peluffo-Ordóñez DH, Restrepo C.
Dynamic Modeling of a Proton-Exchange Membrane Fuel Cell Using a Gaussian Approach. *Membranes*. 2021; 11(12):953.
https://doi.org/10.3390/membranes11120953

**Chicago/Turabian Style**

González-Castaño, Catalina, Leandro L. Lorente-Leyva, Janeth Alpala, Javier Revelo-Fuelagán, Diego H. Peluffo-Ordóñez, and Carlos Restrepo.
2021. "Dynamic Modeling of a Proton-Exchange Membrane Fuel Cell Using a Gaussian Approach" *Membranes* 11, no. 12: 953.
https://doi.org/10.3390/membranes11120953