# Efficient Two-Step Parametrization of a Control-Oriented Zero-Dimensional Polymer Electrolyte Membrane Fuel Cell Model Based on Measured Stack Data

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## Abstract

**:**

## 1. Introduction

## 2. Fuel Cell Model

#### 2.1. Model Description

#### 2.2. Model Reduction

## 3. Two-Step Parametrization Method

#### 3.1. Key Idea

#### 3.2. Parameter Sensitivity Analysis

#### 3.3. Procedure

- Thermodynamic submodel
- (a)
- Parameter sensitivity analysis with respect to the thermodynamic parameters ${\theta}_{\mathrm{th}}$ yields a subset with the most significant parameters ${\theta}_{\mathrm{th},\mathrm{ms}}$, where ${\theta}_{\mathrm{th},\mathrm{ms}}\phantom{\rule{3.33333pt}{0ex}}\subseteq \phantom{\rule{3.33333pt}{0ex}}{\theta}_{\mathrm{th}}$ holds.
- (b)
- Parametrization with respect to the most significant parameters ${\theta}_{\mathrm{th},\mathrm{ms}}$ yields the optimized parameters ${\theta}_{\mathrm{th},\mathrm{opt}}$. The least significant parameters ${\theta}_{\mathrm{th},\mathrm{ls}}$ are kept constant at their initial values.

- Electrochemical submodel
- (a)
- Solve thermodynamic submodel using the optimized thermodynamic parameters ${\theta}_{\mathrm{th},\mathrm{opt}}$ and store the resulting model states $\mathbf{x}$ for further usage.
- (b)
- Parameter sensitivity analysis with respect to the electrochemical parameters ${\theta}_{\mathrm{el}}$ yields a subset with the most significant parameters ${\theta}_{\mathrm{el},\mathrm{ms}}$, where ${\theta}_{\mathrm{el},\mathrm{ms}}\phantom{\rule{3.33333pt}{0ex}}\subseteq \phantom{\rule{3.33333pt}{0ex}}{\theta}_{\mathrm{el}}$ holds.
- (c)
- Parametrization with respect to the most significant parameters ${\theta}_{\mathrm{el},\mathrm{ms}}$ yields the optimized parameters ${\theta}_{\mathrm{el},\mathrm{opt}}$. The least significant parameters ${\theta}_{\mathrm{el},\mathrm{ls}}$ are kept constant at their initial values.

#### 3.4. Validation of Method

## 4. Experimental Setup

#### 4.1. Media Supply

#### 4.2. Cooling Circuits

#### 4.3. Test Bench Control System

#### 4.4. Experimental Tests and Operating Conditions

## 5. Results and Discussion

#### 5.1. Results

#### 5.2. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FC | Fuel cell |

FIM | Fisher information matrix |

ODE | Ordinary differential equation |

PEMFC | Polymer electrolyte membrane fuel cell |

SVD | Singular value decomposition |

## Nomenclature

Subscripts | ||

0 | Initial | |

$\mathrm{act}$ | Activation | |

$\mathrm{an}$ | Anode | |

$\mathrm{atm}$ | Atmosphere | |

$\mathrm{bp}$ | Backpressure | |

$\mathrm{ca}$ | Cathode | |

$\mathrm{cm}$ | Center manifold | |

$\mathrm{el}$ | Electrochemical | |

$\mathrm{em}$ | Exit manifold | |

${\mathrm{H}}_{2}$ | Hydrogen | |

$\mathrm{in}$ | Inflow | |

$\mathrm{leak}$ | Leakage | |

$\mathrm{liq}$ | Liquid water | |

$\mathrm{ls}$ | Least significant | |

$\mathrm{max}$ | Maximum | |

$\mathrm{min}$ | Minimum | |

$\mathrm{ms}$ | Most significant | |

$\mathrm{m}$ | Membrane | |

${\mathrm{N}}_{2}$ | Nitrogen | |

$\mathrm{norm}$ | Normalized | |

${\mathrm{O}}_{2}$ | Oxygen | |

$\mathrm{opt}$ | Optimized | |

$\mathrm{out}$ | Outflow | |

$\mathrm{perm}$ | Permeability | |

$\mathrm{purge}$ | Purge | |

$\mathrm{reci}$ | Recirculation | |

$\mathrm{sm}$ | Supply manifold | |

$\mathrm{th}$ | Thermodynamic | |

$\mathrm{vap}$ | Vapour | |

i | Running index for parameters | |

k | Sampling instant | |

l | Running index for singular values | |

Symbols | ||

$\alpha $ | Valve position | 1 |

$\mathsf{\Psi}$ | Output parameter sensitivity matrix | ${\mathbb{R}}^{4\times {n}_{\theta}}$ |

$\psi $ | Output parameter sensitivity vector | ${\mathbb{R}}^{4\times 1}$ |

$\sum $ | Singular value matrix | ${\mathbb{R}}^{{n}_{\theta}\times {n}_{\theta}}$ |

${\sum}_{\mathbf{e}}$ | Prediction error covariance matrix | ${\mathbb{R}}^{4\times 4}$ |

$\theta $ | Parameter vector | ${\mathbb{R}}^{25\times 1}$ |

${\theta}_{\mathrm{el}}$ | Parameter vector of the electrochemical submodel | ${\mathbb{R}}^{8\times 1}$ |

${\theta}_{\mathrm{th}}$ | Parameter vector of the thermodynamic submodel | ${\mathbb{R}}^{17\times 1}$ |

$\xi $ | State parameter sensitivity vector | ${\mathbb{R}}^{9\times 1}$ |

${\u03f5}_{2}$ | Membrane conductivity parameter | K |

$\gamma $ | Threshold | 1 |

${\lambda}_{\mathrm{Air}}$ | Excess air ratio | 1 |

$\mathbf{F}$ | Fisher information matrix | ${\mathbb{R}}^{{n}_{\theta}\times {n}_{\theta}}$ |

$\mathbf{f}$ | System function of the reduced model | ${\mathbb{R}}^{9\times 1}$ |

${\mathbf{f}}_{\mathrm{nr}}$ | System function of the non-reduced model | ${\mathbb{R}}^{12\times 1}$ |

${\mathbf{f}}_{\mathrm{th}}$ | System function of the thermodynamic submodel | ${\mathbb{R}}^{9\times 1}$ |

$\mathbf{g}$ | Output function of the reduced model | ${\mathbb{R}}^{4\times 1}$ |

${\mathbf{g}}_{\mathrm{nr}}$ | Output function of the non-reduced model | ${\mathbb{R}}^{4\times 1}$ |

${\mathbf{g}}_{\mathrm{th}}$ | Output function of the thermodynamic submodel | ${\mathbb{R}}^{3\times 1}$ |

${\mathbf{Q}}_{\mathbf{y}}$ | Output weighting matrix | ${\mathbb{R}}^{4\times 4}$ |

${\mathbf{Q}}_{\theta}$ | Regularization matrix | ${\mathbb{R}}^{25\times 25}$ |

$\mathbf{U}$ | Left singular vector matrix | ${\mathbb{R}}^{{n}_{\theta}\times {n}_{\theta}}$ |

$\mathbf{u}$ | Input vector | ${\mathbb{R}}^{8\times 1}$ |

$\mathbf{V}$ | Right singular vector matrix | ${\mathbb{R}}^{{n}_{\theta}\times {n}_{\theta}}$ |

$\mathbf{v}$ | Right singular vector | ${\mathbb{R}}^{{n}_{\theta}\times 1}$ |

$\mathbf{x}$ | State vector of the reduced model | ${\mathbb{R}}^{9\times 1}$ |

${\mathbf{x}}_{\mathrm{nr}}$ | State vector of the non-reduced model | ${\mathbb{R}}^{12\times 1}$ |

${\mathbf{x}}_{\mathrm{th}}$ | State vector of the thermodynamic submodel | ${\mathbb{R}}^{9\times 1}$ |

$\mathbf{y}$ | Output vector | ${\mathbb{R}}^{4\times 1}$ |

${\mathbf{y}}^{*}$ | Measured output vector | ${\mathbb{R}}^{4\times 1}$ |

${\mathbf{y}}_{\mathrm{th}}$ | Output vector of the thermodynamic submodel | ${\mathbb{R}}^{3\times 1}$ |

$\sigma $ | Singular value | |

${\sigma}_{{\theta}_{i}}$ | Total information of parameter ${\theta}_{i}$ | |

$\tau $ | Time constant | s |

$\theta $ | Parameter | |

$\phi $ | Relative humidity | 1 |

a | Water activity | 1 |

$CD$ | Combined diffusion coefficient | mol/s |

E | Energy | J |

${g}_{\mathrm{el}}$ | Output function of the electrochemical submodel | ${\mathbb{R}}^{1\times 1}$ |

I | Current | A |

J | Objective function | ${\mathbb{R}}^{1\times 1}$ |

K | Intrinsic exchange current parameter | $\mathrm{A}/{\mathrm{m}}^{2}$ |

k | Nozzle or mass flow coefficient | $\mathrm{kg}/\left(\mathrm{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}\right)$ |

${k}_{\mathrm{cond}}$ | Condensation coefficient | 1/s |

${k}_{\mathrm{evap}}$ | Evaporation coefficient | $1/\left(\mathrm{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{Pa}\right)$ |

m | Mass | kg |

${n}_{k}$ | Number of sample instants (${n}_{k}+1$) | 1 |

${n}_{\theta}$ | Number of parameters | 1 |

p | Pressure | Pa |

R | Mass-specific gas constant | $\mathrm{J}/\left(\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}\right)$ |

${R}_{\mathrm{c}}$ | Ohmic contact resistance | |

T | Fuel cell temperature | K |

t | Time | s |

U | Voltage | V |

V | Volume | ${\mathrm{m}}^{3}$ |

v | Right singular vector component | |

${y}_{\mathrm{el}}$ | Output of the electrochemical submodel | ${\mathbb{R}}^{1\times 1}$ |

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**Figure 1.**Schematic overview of the model structure of the lumped, transient FC stack model [10].

**Figure 2.**Plots (

**a**,

**b**): Total information of thermodynamic ${\sigma}_{{\theta}_{\mathrm{th},i}}$ (obtained from ${\mathbf{F}}_{\mathrm{th},\mathrm{norm}}$) and electrochemical parameters ${\sigma}_{{\theta}_{\mathrm{el},i}}$ (obtained from ${\mathbf{F}}_{\mathrm{el},\mathrm{norm}}$), respectively. Plots (

**c**,

**d**): Relative distribution (100 independent estimations) of optimized most significant parameters of thermodynamic ${\theta}_{\mathrm{th},\mathrm{ms},\mathrm{opt},i}$ and electrochemical submodel ${\theta}_{\mathrm{el},\mathrm{ms},\mathrm{opt},i}$, respectively. Plot (

**e**): Relative distribution (100 independent estimations) of optimized parameters of entire model ${\theta}_{\mathrm{opt},i}$ with usual “single-step” approach. ${\theta}_{0,i}$ denotes true parameter value, and red + symbol indicates outliers in the box plots.

**Figure 4.**Plots (

**a**,

**e**): Cathode supply manifold pressure ${p}_{\mathrm{ca},\mathrm{sm}}$ (blue, yellow and red), and air mass flow ${\dot{m}}_{\mathrm{ca},\mathrm{in}}$ (purple). Plots (

**b**,

**f**): Cathode exit manifold pressure ${p}_{\mathrm{ca},\mathrm{em}}$ (blue, yellow and red), and FC temperature T (purple). Plots (

**c**,

**g**): Anode exit manifold pressure ${p}_{\mathrm{an},\mathrm{em}}$ (blue, yellow and red), and anode supply manifold pressure ${p}_{\mathrm{an},\mathrm{sm}}$ (purple). Plots (

**d**,

**h**): Stack voltage U (blue, yellow and red), and stack current I (purple). The plots depict parameterization and validation data, respectively, and $\mathbf{y}$ denotes output and $\mathbf{u}$ input.

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

Standard stack voltage range | 60–120 VDC |

Continuous stack current | 120–400 A |

Air compressor pressure ratio at 400 A | 1.64 (closed throttle valve) |

Standard excess air ratio (${\lambda}_{\mathrm{Air}}$) | 1.5 |

Air inlet temperature at cathode | 40 °C |

Anode pressure | 1700 mbar |

H2 pump speed | 4000 RPM |

Stack coolant inlet temperature | 55 °C |

Ambient temperature | 23 °C |

Ambient pressure | 1000 mbar |

Relative humidity of ambient air | 50% |

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## Share and Cite

**MDPI and ACS Style**

Du, Z.P.; Steindl, C.; Jakubek, S.
Efficient Two-Step Parametrization of a Control-Oriented Zero-Dimensional Polymer Electrolyte Membrane Fuel Cell Model Based on Measured Stack Data. *Processes* **2021**, *9*, 713.
https://doi.org/10.3390/pr9040713

**AMA Style**

Du ZP, Steindl C, Jakubek S.
Efficient Two-Step Parametrization of a Control-Oriented Zero-Dimensional Polymer Electrolyte Membrane Fuel Cell Model Based on Measured Stack Data. *Processes*. 2021; 9(4):713.
https://doi.org/10.3390/pr9040713

**Chicago/Turabian Style**

Du, Zhang Peng, Christoph Steindl, and Stefan Jakubek.
2021. "Efficient Two-Step Parametrization of a Control-Oriented Zero-Dimensional Polymer Electrolyte Membrane Fuel Cell Model Based on Measured Stack Data" *Processes* 9, no. 4: 713.
https://doi.org/10.3390/pr9040713