# Thermal and Electrical Parameter Identification of a Proton Exchange Membrane Fuel Cell Using Genetic Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}and the stack temperature. The identification process used different random steps signals as inputs. In [19] a PEMFC dynamic model that included the polarization curve characteristics and a double layer charge effect is proposed. The model input was a typical current demand of a DC-DC or a DC-AC. In [20] a NARMAX model to represent the MIMO relations and to identify the coefficients satisfying the PEMFC voltage simulation is used. Also a NARMAX model is used by [21] to represent a PEMFC and used a GA to the model identification, however, the model only represents the fuel cell temperature. Buchlozt and Krebs [22] splits the PEMFC model into a dynamic part and a static part. The static model was identified with neural networks whereas the dynamic model was developed with a mix of transfer functions and linear state-space models. The model inputs were: current density, oxygen stoichiometry, gas supply pressure, and gasses relative humidity; other values as stoichiometry of oxygen and stack temperature were set to constant. The model output was the sum of the dynamic and the static voltage. The authors exposed that the split model allows to reduce the computational time and to improve the accuracy. A split model was also presented in [23]. Regarding the dynamic part, the inputs were the current and the cathode pressure. All these works get deeper in the different relationships between input and output signals, so they model cell voltage responses to gas pressures and current variations. Nevertheless, PEMFC operation produces heat that changes the cell temperature. The temperature affects the cell performance and features as open circuit voltage, internal gas pressures, gas humidity, and internal resistances. Therefore, the use of temperature as an input variable will give more accuracy to the model despite the fact that the complexity and nonlinearity are increased.

## 2. The PEMFC Model

^{®}(2010, National Instruments, Austin, TX, USA) environment. The equations were grouped into electrical and thermal sets. The most remarkable equation in the electrical set is the cell potential E

_{cell}(t) which is calculated with the Nernst’s equation. Equation (1) is a simplification of the Nernst’s due the assumptions mentioned above. E

_{d,cell}(t) represents the electrical effect of gas pressure changes during load transients and classical voltage drops:

_{0}(t) is the reference potential at standard conditions (298 K, 1 atm); p

_{H2}*(t) is the H

_{2}effective partial pressure; p

_{O2}*(t) is the O

_{2}partial pressure. E

_{d,cell}(t) is initially modelled in Laplace domain as Equation (2) and implemented in the time domain in Equation (3):

_{cell}is the convective heat transfer coefficient (W/m

^{2}·K) of the stack; N

_{cell}is the number of cells in the stack; A

_{cell}is the cell area (cm

^{2}). The control system of a Nexa includes the operation of a fan and cooling system, providing oxygen inlet and keeping the temperature under a limit to keep operation conditions and avoid membrane damage. A

_{f}(t) is a coefficient to adjust the temperature related to the cooling system.

- c_APCD is a parameter related to the cell current density.
- c_APa is a parameter related to the distance between the anode channel and the catalyst surface.
- c_Apc1 is a parameter related to the distance between the cathode channel and the catalyst surface.
- c_Apc2 is a parameter that fits the pressure of saturated H
_{2}O curve in function of the temperature.

- c_Act1 is a parameter related to the activation voltage drop that only depends on temperature.
- c_Act2 is a parameter related to the activation voltage drop, that depends on current and temperature.
- c_Ohm1 is the parameter related to ohmic losses that depends on current and temperature.
- c_Ohm2 is a parameter related to ohmic losses that only depends on cur-rent.
- c_Conc is a parameter related to the voltage concentration drop.

- c_Pot1 is a value that adjusts the internal electric potential of the cell.
- c_Pot2 is a parameter related to the free Gibbs energy (∆G).

- c_TDD is the gasses delay time constant during load transients.
- c_TDDG represents a gain that affects the delay by load transients.

- c_HLh is a gain that affect the overall heat loss.
- c_HLaf is the parameter fitting the thermal loss associated to the cathode side. It is included in the stack thermal loss.
- c_HLfan is a gain associated with the cooling fan system and it is included in the stack thermal loss.

- c_PEMh is related to the total mass of stack and its overall specific heat capacity.

^{k}is the population of the k-th iteration:

_{j}

^{k}is j

^{th}parameter set of the kth population and the i

^{th}model parameter will be noted as c

^{k}

_{i,j}. For example, c

^{7}

_{1,2}corresponds with the value of parameter 1 c_APCD in parameter set 2 of the 7th population.

## 3. Parameter Identification

^{®}environment, achieving a modular and versatile programming structure. Figure 5 shows the identification process. The process begins with the estimation of the initial coefficients. The second step is the creation of a first population using a random function starting from the initial coefficients set. In the third step, each coefficient set is simulated in the model with a real data input file. At least, outputs from simulated and real data are compared in order to calculate the error. The optimization process ends when a stop condition is met. The stop condition can be specified as a threshold on the error or as a maximum number of iterations. If the stop condition is not fulfilled, the OA creates a new population by using a genetic algorithm. This new population is evaluated again in Step 3, thus repeating the process until the optimization ends.

_{1}

^{1}as indicated in Equation (5):

_{d}is a value to generate initial dispersion (some GAs include special criteria to create this first population). Each $\epsilon P{S}_{j}^{1}$ is simulated and comparing the real output with simulated outputs, to calculate the error $\epsilon P{S}_{j}^{1}$ with Equation (6):

_{V}and RMSE

_{T}stand for the root mean square error between real and simulated output voltage signals and stack temperature signals, respectively. FSV and FST stand for the device full scales related to the output voltage signal and stack temperature signal, respectively:

_{R}(t) and Out

_{S}(t) are the real and simulated output signal values at time t, respectively. Therefore, the goal is to minimize $\epsilon P{S}_{j}^{k}$.

_{i}

^{GBest}is de i coefficient belonging to the global best solution until iteration k − 1, z is a random number in the range [−1; 1], and n is a perturbation value. This proposal is named PSOp because the use of perturbations.

_{n}is a value in the range [0; 1] which represents the percent of scouts in the population. The offspring population is calculated as:

_{j}

^{GBest}is the coefficient set achieving the best solution until iteration k, z

_{j}is a random number in the range [−1; 1] which modifies all values in one set, and v

_{os}is the spread value of offspring which modifies the whole coefficient set.

_{i}

^{GBest}is the coefficient i of the global best solution until iteration k. z

_{i}is a random number in the range [−1; 1] which affects only the i

^{th}coefficient, and v

_{Sc}is the spread scout value.

## 4. Results

_{d}) was set to 0.5 to create enough diversity. The maximum iteration number (k) was set to 200 in order to give the same opportunity to each OA.

## 5. Conclusions and Future Works

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 8.**Simulation with the identified parameters using the profile of current 1. (

**a**) Stack temperature; (

**b**) output voltage.

**Figure 9.**Simulation with the identified parameters using the second current profile. (

**a**) Stack temperature; (

**b**) output voltage.

Power | 1200 W |

Operating voltage range | 22–50 V |

Current | 55 A |

Hydrogen consumption | 18.5 slpm |

Air flow | 90 slpm |

Temperature | 80 °C |

Cooling air flow | 3600 slpm |

Criteria/Algorithm | PSO | PSOp | HADE | EA | ScGA | |
---|---|---|---|---|---|---|

Precision (%) | Value | 79.6 | 26.8 | 10.9 | 5.95 | 3.08 |

Score | 2.97 | 1 | 4.07 | 2.22 | 1.15 | |

Optimization velocity (iteration) | Value | 42 | 190 | 194 | 193 | 63 |

Score | 1 | 4.52 | 4.62 | 4.60 | 1.50 | |

Computational time (ms) | Value | 18.9 | 19.1 | 22.8 | 11.6 | 10.8 |

Score | 1.75 | 1.77 | 2.11 | 1.07 | 1 | |

Total score | 5.72 | 7.29 | 10.80 | 7.89 | 3.65 |

# | Coefficient | Initial Value | Identified Value |
---|---|---|---|

1 | c_APCD | 5.00 × 10^{−1} | 6.46 × 10^{−1} |

2 | c_APa | 1.65 | 3.39 |

3 | c_APc1 | 4.19 | 2.46 |

4 | c_APc2 | 1.00 × 10^{2} | 4.39 × 10^{1} |

5 | c_Act1 | 1.30 | 9.37 × 10^{−1} |

6 | c_Act2 | 1.30 | 7.76 × 10^{−1} |

7 | c_Ohm1 | −1.30 | −1.13 |

8 | c_Ohm2 | 3.00 × 10^{−5} | 7.58 × 10^{−6} |

9 | c_Conc | −2.60 | −3.87 × 10^{−1} |

10 | c_Pot1 | 1.58 × 10^{−2} | 4.50 × 10^{−3} |

11 | c_Pot2 | 1.63 × 10^{−1} | 5.24 × 10^{−2} |

12 | c_TDDG | 1.60 × 10^{−1} | 1.26 × 10^{−1} |

13 | c_TDD | 8.00 × 10^{1} | 3.13 × 10^{1} |

14 | c_HLh | 9.50 | 2.25 |

15 | c_HLaf | 5.16 | 1.14 |

16 | c_HLfan | 7.67 | 5.22 × 10^{−3} |

17 | c_PEMh | 3.42 × 10^{4} | 2.09 × 10^{4} |

Current Profile | εV (%) | εT (%) | ε (%) |
---|---|---|---|

1 | 2.21 | 1.97 | 2.09 |

2 | 2.75 | 2.22 | 2.48 |

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**MDPI and ACS Style**

Ariza, H.E.; Correcher, A.; Sánchez, C.; Pérez-Navarro, Á.; García, E. Thermal and Electrical Parameter Identification of a Proton Exchange Membrane Fuel Cell Using Genetic Algorithm. *Energies* **2018**, *11*, 2099.
https://doi.org/10.3390/en11082099

**AMA Style**

Ariza HE, Correcher A, Sánchez C, Pérez-Navarro Á, García E. Thermal and Electrical Parameter Identification of a Proton Exchange Membrane Fuel Cell Using Genetic Algorithm. *Energies*. 2018; 11(8):2099.
https://doi.org/10.3390/en11082099

**Chicago/Turabian Style**

Ariza, H. Eduardo, Antonio Correcher, Carlos Sánchez, Ángel Pérez-Navarro, and Emilio García. 2018. "Thermal and Electrical Parameter Identification of a Proton Exchange Membrane Fuel Cell Using Genetic Algorithm" *Energies* 11, no. 8: 2099.
https://doi.org/10.3390/en11082099