# Real-Time Estimation of PEMFC Parameters Using a Continuous-Discrete Extended Kalman Filter Derived from a Pseudo Two-Dimensional Model

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Literature Review

#### 1.3. Study Aim

#### 1.4. Nomenclature and Constants

## 2. Physical Two-Dimensional Model

#### 2.1. Geometry and Assumptions

- Oxygen transport in the cathode catalyst layer (CCL) is fast;
- The oxygen transport in the GDL is purely 1D in the x-direction;
- The oxygen transport in the channel is assumed to be a plug-flow with an averaged velocity and the choice of plug flow in fuel cell microchannels is justified by the small dimensions of the channels. It was shown in previous studies [29,30] that the laminar velocity profile in rectangular microchannels is flat at the center. Thus, at the first order and given the high mass diffusivity of air in the cathode channel, one can assume that the average velocity of a plug flow is able to give a good description of the mass transfer in the fuel cell microchannels;
- The model is isothermal;
- The CL is supposed to be infinitely small;
- The voltage potential is equal along the CL in the y-dimension;
- All channels of the cell are supposed to be operated in the same condition such as temperature, humidity, etc.; and
- The concentrations in the canal and the GDL are supposed to be equal ${\mathrm{C}}^{\mathrm{ref}}$ (concentration on the channel inlet) before any current is applied to the FC.

#### 2.2. General Equations

#### 2.3. Normalized Adimensional Equations

## 3. Finite-Difference Discretization

## 4. Description of the Experimental Test Bench and Considered Fuel Cell

^{2}single cell was specially designed to validate the proposed model. It includes two parallel channels of 1.5$\mathrm{mm}$ width and $200\mathrm{mm}$ in length. The flow field was machined directly in the copper current collector to enable an optical access inside the channel via a transparent Plexiglas plate on the top of the channel (see Figure 2). The current collectors were also gold-plated to ensure minimal ohmic resistance. Between the current collector, a membrane electrode assembly (MEA) was inserted; it was made of a $15\text{\xb5}\mathrm{m}$ thick Gore PEM with $0.5\mathrm{mg}/{\mathrm{cm}}^{2}$ platinum loading in the CL. The thin membrane and high platinum loading were specially chosen to ensure low membrane resistance and high current density. The MEA was sandwiched between two 10BC GDL from SGL

^{®}compressed to a width of $250\mu \mathrm{m}$ using two rigid spacers. Finally, two aluminum end-plates were used to assemble all the fuel cell components. A water-cooling circuit connected to a temperature control system was drilled inside the endplates to keep the fuel cell temperature constant.

^{®}to control the air and hydrogen flow rates, temperatures, humidity, and pressure. The current and voltage were also recorded through the FCT 50 test station.

## 5. Experimental Validation of PEMFC Model

_{2}(1.1) and air (5), with a relative humidity of 15%.

#### 5.1. Current Profile with Step-Up

#### 5.2. Current Profile with Forward/Backward Sweeps

## 6. Estimating the PEMFC Parameters Using an Extended Kalman Filter

## 7. Results of the EKF Observer and Discussion

#### 7.1. Current Profile with Forward/Backward Sweeps

#### 7.2. Current Profile with Step-Up/Down

#### 7.3. Modified PEMFC Cell with Step-Up/Down Profile

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

- Du, Z.; Liu, C.; Zhai, J.; Guo, X.; Xiong, Y.; Su, W.; He, G. A Review of Hydrogen Purification Technologies for Fuel Cell Vehicles. Catalysts
**2021**, 11, 393. [Google Scholar] [CrossRef] - Madsen, R.T.; Klebanoff, L.E.; Caughlan, S.A.M.; Pratt, J.W.; Leach, T.S.; Appelgate Jr, T.B.; Kelety, S.Z.; Wintervoll, H.-C.; Haugom, G.P.; Teo, A.T.Y. Feasibility of the Zero-V: A Zero-Emissions Hydrogen Fuel-Cell Coastal Research Vessel. Int. J. Hydrogen Energy
**2020**, 45, 25328–25343. [Google Scholar] [CrossRef] - Klebanoff, L.; Pratt, J.; Johnson, T.; Arienti, M.; Shaw, L.; Moreno, M. Analysis of H
_{2}Storage Needs for Early Market Non-Motive Fuel Cell Applications; Technical Report SAND2012-17392012; Sandia National Laboratories: Livermore, CA, USA, 2012. [Google Scholar] - Ursua, A.; Gandia, L.M.; Sanchis, P. Hydrogen Production from Water Electrolysis: Current Status and Future Trends. Proc. IEEE
**2012**, 100, 410–426. [Google Scholar] [CrossRef] - Gray, E.M.; Webb, C.J.; Andrews, J.; Shabani, B.; Tsai, P.J.; Chan, S.L.I. Hydrogen Storage for Off-Grid Power Supply. Int. J. Hydrogen Energy
**2011**, 36, 654–663. [Google Scholar] [CrossRef] - Petrovic, S.; Hossain, E. Development of a Novel Technological Readiness Assessment Tool for Fuel Cell Technology. IEEE Access
**2020**, 8, 132237–132252. [Google Scholar] [CrossRef] - Larminie, J.; Dicks, A.; McDonald, M.S. Fuel Cell Systems Explained; J. Wiley: Chichester, UK, 2003; Volume 2. [Google Scholar]
- Zhong, Z.-D.; Zhu, X.-J.; Cao, G.-Y.; Shi, J.-H. A Hybrid Multi-Variable Experimental Model for a PEMFC. J. Power Source
**2007**, 164, 746–751. [Google Scholar] [CrossRef] - Zhong, Z.-D.; Zhu, X.-J.; Cao, G.-Y. Modeling a PEMFC by a Support Vector Machine. J. Power Source
**2006**, 160, 293–298. [Google Scholar] [CrossRef] - Husar, A.; Strahl, S.; Riera, J. Experimental Characterization Methodology for the Identification of Voltage Losses of PEMFC: Applied to an Open Cathode Stack. Int. J. Hydrogen Energy
**2012**, 37, 7309–7315. [Google Scholar] [CrossRef] [Green Version] - Tang, Y.; Yuan, W.; Pan, M.; Li, Z.; Chen, G.; Li, Y. Experimental Investigation of Dynamic Performance and Transient Responses of a KW-Class PEM Fuel Cell Stack under Various Load Changes. Appl. Energy
**2010**, 87, 1410–1417. [Google Scholar] [CrossRef] - Shi, Y.; Janßen, H.; Lehnert, W. A Transient Behavior Study of Polymer Electrolyte Fuel Cells with Cyclic Current Profiles. Energies
**2019**, 12, 2370. [Google Scholar] [CrossRef] [Green Version] - Zhang, G.; Wu, L.; Qin, Z.; Wu, J.; Xi, F.; Mou, G.; Wang, Y.; Jiao, K. A Comprehensive Three-Dimensional Model Coupling Channel Multi-Phase Flow and Electrochemical Reactions in Proton Exchange Membrane Fuel Cell. Adv. Appl. Energy
**2021**, 2, 100033. [Google Scholar] [CrossRef] - Barragán, A.J.; Enrique, J.M.; Segura, F.; Andújar, J.M. Iterative Fuzzy Modeling of Hydrogen Fuel Cells by the Extended Kalman Filter. IEEE Access
**2020**, 8, 180280–180294. [Google Scholar] [CrossRef] - Zhao, J.; Jian, Q.; Luo, L.; Huang, B.; Cao, S.; Huang, Z. Dynamic Behavior Study on Voltage and Temperature of Proton Exchange Membrane Fuel Cells. Appl. Therm. Eng.
**2018**, 145, 343–351. [Google Scholar] [CrossRef] - Abbou, A.; El Hasnaoui, A.; Khan, S.S.; Yamin, F. Analysis of the Novel Dynamic Semiempirical Model of Proton Exchange Membrane Fuel Cell by Incorporating Ambient Condition Variations. Int. J. Energy Environ. Eng.
**2021**, 12, 1–16. [Google Scholar] [CrossRef] - Lajnef, T.; Abid, S.; Ammous, A. Modeling, Control, and Simulation of a Solar Hydrogen/Fuel Cell Hybrid Energy System for Grid-Connected Applications. Adv. Power Electron.
**2013**, 2013, 1–9. [Google Scholar] [CrossRef] [Green Version] - San Martín, I.; Ursúa, A.; Sanchis, P. Modelling of PEM Fuel Cell Performance: Steady-State and Dynamic Experimental Validation. Energies
**2014**, 7, 670–700. [Google Scholar] [CrossRef] [Green Version] - Sousa, R.; Gonzalez, E.R. Mathematical Modeling of Polymer Electrolyte Fuel Cells. J. Power Source
**2005**, 147, 32–45. [Google Scholar] [CrossRef] - Luna, J.; Ocampo-Martinez, C.; Serra, M. Nonlinear Predictive Control for the Concentrations Profile Regulation under Unknown Reaction Disturbances in a Fuel Cell Anode Gas Channel. J. Power Source
**2015**, 282, 129–139. [Google Scholar] [CrossRef] [Green Version] - Petrone, R.; Zheng, Z.; Hissel, D.; Péra, M.-C.; Pianese, C.; Sorrentino, M.; Becherif, M.; Yousfi-Steiner, N. A Review on Model-Based Diagnosis Methodologies for PEMFCs. Int. J. Hydrogen Energy
**2013**, 38, 7077–7091. [Google Scholar] [CrossRef] - Chevalier, S.; Auvity, B.; Olivier, J.C.; Josset, C.; Trichet, D.; Machmoum, M. Detection of Cells State-of-Health in PEM Fuel Cell Stack Using EIS Measurements Coupled with Multiphysics Modeling. Fuel Cells
**2014**, 14, 416–429. [Google Scholar] [CrossRef] - Fouquet, N.; Doulet, C.; Nouillant, C.J.J.; Dauphin-Tanguy, G.; Ould-Bouamama, B. Model Based PEM Fuel Cell State-of-Health Monitoring via Ac Impedance Measurements. J. Power Source
**2006**, 159, 905–913. [Google Scholar] [CrossRef] - Jouin, M.; Gouriveau, R.; Hissel, D.; Péra, M.-C.; Zerhouni, N. Prognostics of PEM Fuel Cell in a Particle Filtering Framework. Int. J. Hydrogen Energy
**2014**, 39, 481–494. [Google Scholar] [CrossRef] [Green Version] - Jouin, M.; Gouriveau, R.; Hissel, D.; Péra, M.-C.; Zerhouni, N. Remaining Useful Life Estimates of a PEM Fuel Cell Stack by Including Characterization-Induced Disturbances in a Particle Filter Model. In Proceedings of the Conference Internationale Discussion on Hydrogen Energy and Applications, IDHEA’14, Nantes, France, 14 January 2014; pp. 1–10. [Google Scholar]
- Luna, J.; Usai, E.; Husar, A.; Serra, M. Distributed Parameter Nonlinear State Observer with Unmatched Disturbance Estimation for PEMFC Systems. In Proceedings of the 6th International Conference on “Fundamentals & Development of Fuel Cells”, Tolouse, France, 27 May 2015. [Google Scholar]
- Luna, J.; Usai, E.; Husar, A.; Serra, M. Observation of the Electrochemically Active Surface Area in a Proton Exchange Membrane Fuel Cell. In Proceedings of the IECON 2016-42nd Annual Conference of the IEEE Industrial Electronics Society, Florence, Italy, 24–27 October 2016; pp. 5483–5488. [Google Scholar]
- Chevalier, S.; Josset, C.; Bazylak, A.; Auvity, B. Measurements of Air Velocities in Polymer Electrolyte Membrane Fuel Cell Channels Using Electrochemical Impedance Spectroscopy. J. Electrochem. Soc.
**2016**, 163, F816–F823. [Google Scholar] [CrossRef] - Bruus, H. Theoretical Microfluidics; Oxford University Press Inc.: New York, NY, USA, 2008; Volume 18. [Google Scholar]
- Chevalier, S. Semianalytical Modeling of the Mass Transfer in Microfluidic Electrochemical Chips. Phys. Rev. E
**2021**, 104, 035110. [Google Scholar] [CrossRef] [PubMed] - Mainka, J. Local Impedance in H
_{2}/Air Proton Exchange Membrane Fuel Cells (PEMFC): Theoretical and Experimental Investigations. Ph.D. Thesis, Université Henri Poincaré-Nancy, Nancy, France, 2011. [Google Scholar] - Chevalier, S.; Olivier, J.-C.; Josset, C.; Auvity, B. Polymer Electrolyte Membrane Fuel Cell Operating in Stoichiometric Regime. J. Power Source
**2019**, 440, 227100. [Google Scholar] [CrossRef] - Abdin, Z.; Webb, C.J.; Gray, E. PEM Fuel Cell Model and Simulation in Matlab–Simulink Based on Physical Parameters. Energy
**2016**, 116, 1131–1144. [Google Scholar] [CrossRef] - Youssef, M.E.; Amin, R.S.; El-Khatib, K.M. Development and Performance Analysis of PEMFC Stack Based on Bipolar Plates Fabricated Employing Different Designs. Arab. J. Chem.
**2018**, 11, 609–614. [Google Scholar] [CrossRef] [Green Version] - Diab, Y.; Auger, F.; Schaeffer, E.; Wahbeh, M. Estimating Lithium-Ion Battery State of Charge and Parameters Using a Continuous-Discrete Extended Kalman Filter. Energies
**2017**, 10, 1075. [Google Scholar] [CrossRef] [Green Version] - Xiong, R.; He, H.; Sun, F.; Zhao, K. Evaluation on State of Charge Estimation of Batteries with Adaptive Extended Kalman Filter by Experiment Approach. IEEE Trans. Veh. Technol.
**2013**, 62, 108–117. [Google Scholar] [CrossRef] - Zhang, C.P.; Liu, J.Z.; Sharkh, S.M.; Zhang, C.N. Identification of Dynamic Model Parameters for Lithium-Ion Batteries Used in Hybrid Electric Vehicles. High Technol. Lett.
**2010**, 16, 6–12. [Google Scholar] [CrossRef] - He, H.; Xiong, R.; Zhang, X.; Sun, F.; Fan, J. State-of-Charge Estimation of the Lithium-Ion Battery Using an Adaptive Extended Kalman Filter Based on an Improved Thevenin Model. IEEE Trans. Veh. Technol.
**2011**, 60, 1461–1469. [Google Scholar] - Kulikov, G.Y.; Kulikova, M.V. Accurate Numerical Implementation of the Continuous-Discrete Extended Kalman Filter. IEEE Trans. Autom. Control
**2014**, 59, 273–279. [Google Scholar] [CrossRef] - Axelsson, P.; Gustafsson, F. Discrete-Time Solutions to the Continuous-Time Differential Lyapunov Equation with Applications to Kalman Filtering. IEEE Trans. Autom. Control
**2015**, 60, 632–643. [Google Scholar] [CrossRef] [Green Version] - Auger, F.; Hilairet, M.; Guerrero, J.M.; Monmasson, E.; Orlowska-Kowalska, T.; Katsura, S. Industrial Applications of the Kalman Filter: A Review. IEEE Trans. Ind. Electron.
**2013**, 60, 5458–5471. [Google Scholar] [CrossRef] [Green Version] - Mazzoni, T. Computational Aspects of Continuous–Discrete Extended Kalman-Filtering. Comput. Stat.
**2008**, 23, 519–539. [Google Scholar] [CrossRef] [Green Version] - Guihal, J.-M.; Auger, F.; Bernard, N.; Schaeffer, E. Efficient Implementation of Continuous-Discrete Extended Kalman Filters for State and Parameter Estimation of Nonlinear Dynamic Systems. IEEE Trans. Ind. Inform.
**2021**, 18, 3077–3085. [Google Scholar] [CrossRef] - Lee, C.-Y.; Lee, Y.-M.; Lee, S.-J. Local Area Water Removal Analysis of a Proton Exchange Membrane Fuel Cell under Gas Purge Conditions. Sensors
**2012**, 12, 768–783. [Google Scholar] [CrossRef] [Green Version] - Pérez-Page, M.; Pérez-Herranz, V. Effect of the Operation and Humidification Temperatures on the Performance of a PEM Fuel Cell Stack. Ecs Trans.
**2009**, 25, 733. [Google Scholar] [CrossRef] - Pauchet, J.; Prat, M.; Schott, P.; Pulloor Kuttanikkad, S. Performance Loss of Proton Exchange Membrane Fuel Cell Due to Hydrophobicity Loss in Gas Diffusion Layer: Analysis by Multiscale Approach Combining Pore Network and Performance Modelling. Int. J. Hydrogen Energy
**2012**, 37, 1628–1641. [Google Scholar] [CrossRef] [Green Version] - Emerson, A.; Montville, L. Electrochemical Characterization and Water Balance of a PEM Fuel Cell; Worcester Polytechnic Institute: Worcester, MA, USA, 2010. [Google Scholar]
- Niu, H.; Ji, C.; Wang, S.; Liang, C. Quantitative Analysis on Cold Start Process of a PEMFC Stack with Intake Manifold. Int. J. Hydrogen Energy
**2021**, 47, 2647–2661. [Google Scholar] [CrossRef] - Promislow, K.S. Phase Change and Hysteresis in PEMFCs. In Device and Materials Modeling in PEM Fuel Cells; Springer: New York, NY, USA, 2009; pp. 253–295. [Google Scholar]
- Laribi, S.; Mammar, K.; Sahli, Y.; Koussa, K. Air Supply Temperature Impact on the PEMFC Impedance. J. Energy Storage
**2018**, 17, 327–335. [Google Scholar] [CrossRef] - Saleh, M.M.; Okajima, T.; Hayase, M.; Kitamura, F.; Ohsaka, T. Exploring the Effects of Symmetrical and Asymmetrical Relative Humidity on the Performance of H
_{2}/Air PEM Fuel Cell at Different Temperatures. J. Power Source**2007**, 164, 503–509. [Google Scholar] [CrossRef] - Gaumont, T.; Maranzana, G.; Lottin, O.; Dillet, J.; Guétaz, L.; Pauchet, J. In Operando and Local Estimation of the Effective Humidity of PEMFC Electrodes and Membranes. J. Electrochem. Soc.
**2017**, 164, F1535. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) PEMFC half-cell geometry reprinted from [28]. (

**b**) A simple fuel cell electrical circuit.

**Figure 2.**(

**a**) Three-dimensional view of the straight channel fuel cell reprinted from [32]. (

**b**) Fuel cell benchmark setup in LTeN, Nantes.

**Figure 4.**(

**a**) Simulation and experimental curves of current. (

**b**) Relative error of simulated current.

**Figure 7.**(

**a**) Simulation and experimental curves of current. (

**b**) Relative error of simulated current.

**Figure 8.**(

**a**) Measured current and current estimated by EKF. (

**b**) Zoomed-in image of both curves between 2177 and 2187 s. (

**c**) Relative error of estimated current.

**Figure 9.**(

**a**) Concentration ${\mathrm{C}}_{\mathrm{c}}\left(\mathrm{y}=\frac{\mathrm{k}}{\mathrm{K}}{\mathrm{L}}_{\mathrm{c}},\mathrm{t}\right)={\mathrm{C}}_{\mathrm{cn}}\left({\mathrm{y}}_{\mathrm{n}},\mathrm{t}\right){\mathrm{C}}^{\mathrm{ref}}$ as a function of time and $\mathrm{k}$. (

**b**) Concentration ${\mathrm{C}}_{\mathrm{g}}\left(\mathrm{x}=\frac{\mathrm{d}}{\mathrm{D}}{\mathrm{h}}_{\mathrm{d}},\mathrm{y}=\frac{\mathrm{k}}{\mathrm{K}}{\mathrm{L}}_{\mathrm{c}},\mathrm{t}\right)={\mathrm{C}}_{\mathrm{gn}}\left({\mathrm{x}}_{\mathrm{n}},{\mathrm{y}}_{\mathrm{n},},\mathrm{t}\right){\mathrm{C}}^{\mathrm{ref}}$ as a function of time and $\mathrm{d}$, and two values of $\mathrm{k}$ =$0$ and$\mathrm{k}=\mathrm{K}=20$.

**Figure 10.**EKF results: (

**a**) potential $\mathsf{\eta}={\mathsf{\eta}}_{\mathrm{n}}\mathrm{b}$; (

**b**) capacitances ${\mathrm{C}}_{\mathrm{DL}}$; (

**c**) resistance ${\mathrm{r}}_{\Omega}$; (

**d**) Tafel slope $\mathrm{b}$; (

**e**) exchange current density ${\mathrm{i}}_{\mathrm{c}}$; and (

**f**) diffusion coefficient ${\mathrm{D}}^{\mathrm{eff}}$.

**Figure 12.**(

**a**) Measured current and current estimated by EKF. (

**b**) Experimental voltage of a new PEMFC cell for step-up/down profile. (

**c**) Relative error of the estimated current.

**Figure 13.**EKF results: (

**a**) potential $\mathsf{\eta}={\mathsf{\eta}}_{\mathrm{n}}\mathrm{b}$; (

**b**) capacitances ${\mathrm{C}}_{\mathrm{DL}}$; (

**c**) resistance ${\mathrm{r}}_{\Omega}$; (

**d**) Tafel slope $\mathrm{b}$; (

**e**) exchange current density ${\mathrm{i}}_{\mathrm{c}}$; and (

**f**) diffusion coefficient ${\mathrm{D}}^{\mathrm{eff}}$.

Geometry and Model Parameters | |||||
---|---|---|---|---|---|

Parameter | Unit | Signification | ${\mathrm{V}}_{\mathrm{d}}$ | m/s | Constant |

$\mathrm{b}$ | V | Tafel slope | $\mathrm{x}$ | M | Spatial coordinate |

${\mathrm{C}}_{\mathrm{c}}$ | mol/m^{3} | Oxygen concentration in the channel | $\mathrm{y}$ | M | Spatial coordinate |

${\mathrm{C}}_{\mathrm{g}}$ | mol/m^{3} | Oxygen concentration in the GDL | ${\mathrm{x}}_{\mathrm{n}},{\mathrm{y}}_{\mathrm{n}}$ | - | Normalized spatial coordinates |

${\mathrm{C}}^{\mathrm{ref}}$ | mol/m^{3} | Reference concentration on the channel inlet | $\mathsf{\eta}$ | V | Cathode potential |

${\mathrm{C}}_{\mathrm{cn}}$ ${\mathrm{C}}_{\mathrm{gn}}$ | - | Normalized oxygen concentrations | ${\mathsf{\eta}}_{\mathrm{n}}$ | - | Normalized potential |

${\mathrm{C}}_{\mathrm{DL}}$ | F/m^{2} | Double-layer capacity | Discretization parameters | ||

${\mathrm{D}}^{\mathrm{eff}}$ | m^{2}/s | Effective diffusivity of the GDL | $\mathrm{d}$ | - | $\mathrm{Variable}\mathrm{related}\mathrm{to}\mathrm{x}=\frac{d}{D}{h}_{d}$ |

$\mathrm{e}$ | V | Cell voltage | $\mathrm{D}$ | - | Number of grid points in the x-axis |

${\mathrm{e}}^{\mathrm{ref}}$ | V | Fuel cell potential reference | $\mathrm{k}$ | - | $\mathrm{Variable}\mathrm{related}\mathrm{to}\mathrm{y}=\frac{k}{K}{L}_{c}$ |

$\mathrm{F}$ | C/mol | Faraday constant | $\mathrm{K}$ | - | Number of grid points in the y-axis |

${\mathrm{h}}_{\mathrm{c}}^{\mathrm{eff}}$ | m | Channel depth | n | - | Number of state variables |

${\mathrm{h}}_{\mathrm{d}}$ | m | GDL thickness | Observer parameters | ||

${\mathrm{i}}_{\mathrm{c}}$ | A/m^{2} | Exchange current density | $\mathrm{a}$ | - | Constant |

$\mathrm{i}$ | A | Fuel cell current | $\mathrm{H}$ | - | Observation matrix |

$\mathrm{j}$ | A/m^{2} | Local current density | ${\mathrm{K}}_{\mathrm{N}}$ | - | Kalman gain |

${\mathrm{j}}_{\mathrm{n}}$ | - | Normalized Local current density | $\mathrm{F}$ | - | Dynamic matrix |

${\mathrm{L}}_{\mathrm{c}}$ | m | Channel length | $\mathrm{N}$ | - | Time index |

${\mathrm{l}}_{\mathrm{c}}$ | m | Channel width | $\mathrm{q}$ | - | White noise covariance |

${\mathrm{j}}_{\mathrm{lim}}$ | A/m^{2} | Limiting current density | $\mathrm{P}$ | - | Covariance matrix |

${\mathrm{r}}_{\mathsf{\Omega}}$ | Ω m^{2} | Ohmic resistance | ${\mathrm{T}}_{\mathrm{s}}$ | s | Sampling period |

$\mathrm{t}$ | s | Temporal coordinate | $\mathrm{R}$ | - | Measurement noise covariance |

$\mathrm{T}$ | °C | Temperature | $\mathrm{v}$ | - | Process noise |

${\mathrm{u}}_{\mathrm{c}}$ | m/s | Air velocity | $\mathrm{x}$ | - | State variables |

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

$\mathrm{b}$ | $0.03$ V | ${\mathrm{h}}_{\mathrm{c}}^{\mathrm{eff}}$ | $1.5\times {10}^{-3}$ m |

${\mathrm{C}}_{\mathrm{DL}}$ | $200$ F/m^{2} | ${\mathrm{h}}_{\mathrm{d}}$ | $250\times {10}^{-6}$ m |

${\mathrm{C}}^{\mathrm{ref}}$ | $7.42$ mol/m^{3} | ${\mathrm{i}}_{\mathrm{c}}$ | $300$ A/m^{2} |

${\mathrm{D}}^{\mathrm{eff}}$ | ${10}^{-5}$ m^{2}/s | ${\mathrm{r}}_{\mathsf{\Omega}}$ | $2.2\times {10}^{-5}$ Ωm^{2} |

${\mathrm{e}}^{\mathrm{ref}}$ | $0.95$ V | $\mathrm{T}$ | $70$ °C |

$\mathrm{F}$ | 96,487 C/mol | ${\mathrm{u}}_{\mathrm{c}}$ | $0.42$ m/s |

Period of Real-Time Measurement (s) | Number of Samples | ${\mathbf{T}}_{\mathbf{s}}\left(\mathbf{s}\right)$ | Period of Simulation Time (s) |
---|---|---|---|

3979 | 23,471 | 0.1695 | 2340 |

6000 | 81,455 | 0.073 | 5188 |

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**MDPI and ACS Style**

Diab, Y.; Auger, F.; Schaeffer, E.; Chevalier, S.; Allahham, A.
Real-Time Estimation of PEMFC Parameters Using a Continuous-Discrete Extended Kalman Filter Derived from a Pseudo Two-Dimensional Model. *Energies* **2022**, *15*, 2337.
https://doi.org/10.3390/en15072337

**AMA Style**

Diab Y, Auger F, Schaeffer E, Chevalier S, Allahham A.
Real-Time Estimation of PEMFC Parameters Using a Continuous-Discrete Extended Kalman Filter Derived from a Pseudo Two-Dimensional Model. *Energies*. 2022; 15(7):2337.
https://doi.org/10.3390/en15072337

**Chicago/Turabian Style**

Diab, Yasser, Francois Auger, Emmanuel Schaeffer, Stéphane Chevalier, and Adib Allahham.
2022. "Real-Time Estimation of PEMFC Parameters Using a Continuous-Discrete Extended Kalman Filter Derived from a Pseudo Two-Dimensional Model" *Energies* 15, no. 7: 2337.
https://doi.org/10.3390/en15072337