The problem of parameter estimation of the proton exchange membrane fuel cell (PEMFC) model plays a significant role in the simulation and optimization of a PEMFC system. In the current research, a moth flame optimization algorithm (MFOA) is used to identify the best parameters of PEMFC. Two different PEMFCs, Nedstack PS6, 6 kW, and SR-12 PEM 500 W are used to demonstrate the accuracy of the MFOA. Throughout the optimization process, seven unidentified parameters (_{1}, _{2}, _{3}, _{4}, λ, ℛ, and B) of PEMFC are appointed to be decision variables. The fitness function, which needed to be minimum, is represented by the root-mean-square error between the calculated voltage of the PEMFC and the experimental dataset. The results attained by the MFOA are compared with the sine cosine algorithm (SCA) and particle swarm optimization (PSO). The following metrics are considered in the comparison: best value, worst value, average value, standard deviation, and efficiency. The main findings verified the supremacy of the MFOA in estimating the best parameters of the PEMFC model in comparison with PSO and SCA. For the Nedstack PS6, the efficiency values are 24.50458%, 79.477%, and 38.84747% for MFOA, PSO and SCA, respectively. For PEMFC (SR-12 PEM), the average efficiency values are 14.965%, 79.933% and 31.87% for MFOA, PSO, and SCA, respectively.

The increasing global demand for low carbon-emitting fuel cell-based vehicles and growing adoption of clean sources of energy production have augmented the need for developments in fuel cell technology [

An accurate model of a PEMFC depends on the availability of parameters that are unavailable in the manufacturing datasheet. It is therefore mandatory to determine the unknown PEMFC parameters to provide an accurate model closely matching the experimental data under various operational conditions [

The output voltage of a PEMFC relies on parameters related to the chemical procedures occurring inside the PEMFC [

In this work, a moth flame optimization algorithm (MFOA) is used to identify the best parameters of a PEMFC. Two different PEMFCs (NedStack PS6 and SR-12 PEM 500 W) are used to demonstrate the accuracy of the MFOA. Throughout the optimization process, seven unidentified parameters (_{1}, _{2}, _{3}, _{4}, λ, ℛ, and B) of a PEMFC are appointed to be decision variables. The objective function, which needs to be the minimum, is represented by the root-mean-square error (RMSE) between the calculated voltage of a PEMFC and the experimental dataset. The results attained by the MFOA are compared with the sine cosine algorithm (SCA) and particle swarm optimization (PSO).

The contributions of the current work are briefly summed as follows:

MFOA is applied to estimate the best parameters of the PEMFC model.

A comprehensive comparison is made with PSO and SCA.

The superiority and reliability of the MFOA-based strategy in solving the PEMFC parameter identification problem is validated.

The rest of the paper is organized as follows. Section 2 presents a general description of the parameter identification problem of PEMFCs. A brief overview of the core idea of the MFOA, PSO, and the SCA are introduced in Section 3. Section 4 discusses the obtained results. Finally, the main findings are outlined in Section 5.

Details of the steady-state characteristics of PEMFC are described using electrochemical relations [_{1}, _{2}, _{3}, and _{4} denote the semi-empirical parametric factors of the PEMFC, with λ representing a scalar factor varying from 10 to 24, ℛ denoting the contact resistance (Ohm), and

The output voltage of a PEMFC versus the current density is plotted in

where

The thermodynamic potential (

where

Next, the activation loss can be estimated as follows:

where

_{f} represents the cell current.

The ohmic loss is calculated follows:

where

ℛ is the contact resistance.

The concentration loss is defined by

where _{max} denotes the maximum current density.

The PEMFC stack includes several series-connected cells to increase the total voltage. Therefore, the total stack voltage of PEMFC (

The main purpose of this work is to estimate the best parameters of the PEMFC model using three different optimizers: MFOA, PSO, and SCA. As mentioned above, the seven undetermined parameters of the PEMFC model are designated during the optimization process as decision variables. The fitness function, which needs to be the minimum, is represented by the RMSE between the calculated voltage and the experimental dataset. The optimization problem can be defined using the following relationships [

where

Subject to the following constraints:

where

In this paper, three optimization algorithms are considered: MFOA, PSO, and SCA.

The original version of MFOA was proposed by Mirjalili [^{1} An updating process for MFOA can be carried out based on the following. The logarithmic spiral is defined for MFOA by the following relationship.

where

_{i} denotes the distance between the moth and the flame.

_{i} denotes the

_{j} denotes the

Originally suggested by Kennedy and Eberhart [^{2}. The updating process for velocity and location of particles can be defined as follows.

_{best} is the best solution, _{best} is the global best.

_{1} and _{2} denote cognitive and social factors. The values of _{1} and _{2} are 1.5 and 2, respectively. _{1} and _{2} random values in range of [0 1]

The original version of SCA was proposed by Mirjalili [^{3} The updating process for SCA can be executed based on the following.

where

_{1} can be estimated using the following relationship.

where

_{2}, _{3,} and _{4} are random numbers, which can be defined as follows:

The parameters of two PEMFCs, Nedstack PS6 and SR-12 PEM 500 W, are determined in this current research using MFOA, PSO, and SCA. The specifications of the considered PEMFCs are presented in

Nedstack PS6 | SR-12 PEM 500 W | |
---|---|---|

65 | 32 | |

^{2}) |
240 | 64 |

178 | 178 | |

0.5-5 | 1 | |

0.5-5 | 0.2095 | |

343 | 333 | |

_{a} |
100% | |

_{c} |
100% |

To be fair in the comparison, the population size and maximum number of iterations for MFOA, PSO, and SCA are set to 30 (population size) and 100 (maximum number of iterations). During the optimization procedure, the RMSE between the calculated cell voltage and measured data is used in the fitness function, which needs to be the minimum. The unknown parameters of the PEMFC model are designated as decision variables. The lower and upper limits of these parameters are given in

Design variables | ℛ | ||||||
---|---|---|---|---|---|---|---|

Min limit | –1.19969 | 0.001 | 10 | 0.0136 | |||

Max limit | 0.8532 | 0.005 | 24 | 0.5 |

The best-identified parameters of PEMFC model applying MFOA, PSO, and SCA are shown in

Parameter | PSO | MFOA | SCA | PSO | MFOA | SCA |
---|---|---|---|---|---|---|

RMSE | 0.14714 | 0.12222 | 0.15325 | 0.13358 | 0.13038 | 0.13444 |

–0.03121 | –1.19969 | –0.62064 | –0.33498 | –0.47151 | –0.42172 | |

0.00101 | 0.00398 | 0.00213 | 0.00135 | 0.00115 | 0.001 | |

9.80E-05 | 6.85E-05 | 5.84E-05 | 8.07E-05 | 3.82E-05 | 3.88E-05 | |

–0.0001 | –0.0001 | –0.0001 | –0.0001 | –0.0001 | –0.0001 | |

24 | 13.44758 | 14.04721 | 24 | 24 | 24 | |

ℛ | 0.00021 | 0.0001 | 0.00012 | 0.00015 | 0.0001 | 0.00013 |

0.06647 | 0.0136 | 0.0136 | 0.15546 | 0.15652 | 0.15463 |

Metric | PSO | MFOA | SCA | PSO | MFOA | SCA |
---|---|---|---|---|---|---|

Best | 0.147139 | 0.122221 | 0.153249 | 0.133575 | 0.130382 | 0.134436 |

Worst | 68.17925 | 0.310127 | 1.112778 | 7.898515 | 0.361609 | 1.002483 |

Mean | 5.984966 | 0.161541 | 0.425626 | 2.179344 | 0.185512 | 0.526204 |

STD | 12.62254 | 0.040965 | 0.24436 | 1.581097 | 0.05995 | 0.244456 |

Efficiency | 24.50458 | 79.47701 | 38.84747 | 14.965278 | 75.93332 | 31.87294 |

Considering the data presented in

Run | NedStack PS6 | SR-12 PEM | ||||
---|---|---|---|---|---|---|

PSO | MFOA | SCA | PSO | MFOA | SCA | |

8.53752 | 0.12222 | 0.59146 | 2.7477 | 0.18716 | 0.13444 | |

68.17925 | 0.14028 | 0.64454 | 1.59351 | 0.16196 | 0.3564 | |

13.10421 | 0.13881 | 0.26335 | 0.89402 | 0.13046 | 0.37401 | |

0.14714 | 0.20258 | 0.25355 | 1.08206 | 0.16293 | 0.96376 | |

0.78561 | 0.16184 | 0.39115 | 2.97972 | 0.17523 | 0.41433 | |

4.33816 | 0.13112 | 0.68531 | 0.37819 | 0.13038 | 0.38245 | |

1.30115 | 0.13067 | 0.18082 | 0.49815 | 0.24294 | 0.22363 | |

1.55999 | 0.15046 | 0.93359 | 2.74813 | 0.13038 | 0.36692 | |

0.44023 | 0.13126 | 0.35411 | 3.07753 | 0.17278 | 0.77 | |

1.28099 | 0.31013 | 0.45313 | 2.72414 | 0.20917 | 1.00248 | |

2.51893 | 0.15771 | 0.16048 | 4.29613 | 0.243 | 0.74271 | |

0.15375 | 0.12225 | 0.43063 | 2.62047 | 0.15505 | 0.81628 | |

0.17825 | 0.12901 | 0.20219 | 1.35494 | 0.18259 | 0.55223 | |

6.39235 | 0.15189 | 0.21838 | 0.13358 | 0.17815 | 0.37905 | |

13.19668 | 0.12232 | 0.16008 | 0.16722 | 0.1597 | 0.4522 | |

0.24448 | 0.21845 | 0.33771 | 2.58381 | 0.15502 | 0.95112 | |

10.352 | 0.14725 | 0.15325 | 3.60592 | 0.13039 | 0.85662 | |

1.10253 | 0.2151 | 0.63501 | 1.2615 | 0.36161 | 0.56315 | |

0.39591 | 0.13824 | 1.11278 | 1.75191 | 0.13046 | 0.46332 | |

0.23317 | 0.18123 | 0.38336 | 3.80309 | 0.14095 | 0.46224 | |

7.0298 | 0.13965 | 0.31761 | 2.09392 | 0.24204 | 0.85917 | |

1.16663 | 0.19211 | 0.24015 | 2.25593 | 0.19283 | 0.23096 | |

0.15046 | 0.12489 | 0.9113 | 7.89851 | 0.18377 | 0.78393 | |

0.22919 | 0.15975 | 0.15605 | 0.32859 | 0.15152 | 0.50418 | |

0.23 | 0.19273 | 0.51706 | 1.98493 | 0.27217 | 0.30953 | |

19.53852 | 0.18237 | 0.24877 | 3.69387 | 0.36152 | 0.49601 | |

0.74391 | 0.15 | 0.60842 | 1.06736 | 0.17812 | 0.51333 | |

12.79506 | 0.13382 | 0.46462 | 1.68612 | 0.17723 | 0.36374 | |

0.22866 | 0.22634 | 0.36081 | 3.42854 | 0.13042 | 0.18419 | |

2.99446 | 0.14176 | 0.39912 | 0.64083 | 0.13543 | 0.31375 |

A comparison between the estimated and measured data of NedStack PS6 is introduced in

A comparison between the calculated and experimental dataset of SR-12 PEM is introduced in

Data | NedStack PS6 | Data | SR-12 PEM | ||||
---|---|---|---|---|---|---|---|

Measured | Estimated | Error | Measured | Estimated | Error | ||

61.64 | 62.2194 | –0.57936 | 43.01 | 43.4037 | –0.3935 | ||

59.57 | 59.6484 | –0.07839 | 41.071 | 41.3214 | –0.25 | ||

58.94 | 58.9186 | 0.02144 | 40.026 | 40.0289 | –0.0034 | ||

57.54 | 57.3712 | 0.16883 | 38.98 | 39.0124 | –0.0328 | ||

56.8 | 56.5958 | 0.20424 | 37.985 | 38.0115 | –0.0268 | ||

56.13 | 55.9258 | 0.20424 | 36.964 | 37.0556 | –0.0913 | ||

55.23 | 55.0436 | 0.18641 | 36.02 | 36.1267 | –0.1063 | ||

54.66 | 54.5104 | 0.14964 | 35.204 | 35.1721 | 0.032 | ||

53.61 | 53.5297 | 0.08032 | 34.056 | 34.2575 | –0.2014 | ||

52.86 | 52.8459 | 0.01411 | 33.163 | 33.2908 | –0.1276 | ||

51.91 | 51.354 | 0.556 | 32.092 | 32.245 | –0.1531 | ||

51.22 | 50.9452 | 0.27485 | 31.199 | 31.1657 | 0.0333 | ||

49.66 | 49.3514 | 0.30861 | 29.719 | 30.0124 | –0.293 | ||

49 | 48.5679 | 0.43211 | 29.056 | 28.8237 | 0.2324 | ||

48.15 | 47.9776 | 0.17235 | 28.036 | 27.4336 | 0.6022 | ||

47.52 | 47.5869 | –0.06692 | 26.301 | 25.8882 | 0.4128 | ||

47.1 | 47.0039 | 0.09612 | 24.158 | 24.1009 | 0.0573 | ||

46.48 | 46.2162 | 0.26379 | 21.352 | 21.9208 | –0.5687 | ||

45.66 | 45.4207 | 0.2393 | |||||

44.85 | 44.8128 | 0.0372 | |||||

44.24 | 43.9971 | 0.24295 | |||||

42.45 | 42.9606 | –0.51057 | |||||

41.66 | 42.1074 | –0.44744 | |||||

40.68 | 41.0063 | –0.32634 | |||||

40.09 | 40.3356 | –0.2456 | |||||

39.51 | 39.6396 | –0.12961 | |||||

38.73 | 38.6923 | 0.03773 | |||||

38.15 | 37.9655 | 0.18452 | |||||

37.38 | 36.9564 | 0.42358 |

In this research, a moth flame optimization algorithm (MFOA) has been used to estimate the seven unknown parameters (_{1}, _{2}, _{3}, _{4}, λ, ℛ, and