# Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production

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*Applied Sciences*: Invited Papers in Electrical, Electronics and Communications Engineering Section)

## Abstract

**:**

## Featured Application

**This article provides the strategy to find the best cell design and operating condition of a diphasic electrochemical cell using an evolutionary algorithm.**

## Abstract

## 1. Introduction

_{2}produced by the industrial activities including the production of cement and steel. The renewable energies generated from various systems such as wind turbines and photovoltaic panels do not produce greenhouse gases but suffer from time intermittency and geographic limitations. In order to resolve this challenge, energy produced at off-peak hours must be stored. Batteries can store this energy, but their energy density is low, and they discharge the stored energy over time. Thus, if the energy needs to be stored more than a month, another storage mean must be provided. Hydrogen storage can store (renewable) energy with a high density for a long period and its use (by combustion and by reduction) only produces water. Most hydrogen is currently produced via a process that needs fossil fuels and which produces greenhouse gases. However, although there exist several other processes that can generate hydrogen without generating pollutants, water electrolysis is one of these processes and is very promising. Nevertheless, the high operation cost hampers its development. One way to decrease the cost is to choose the cheapest water electrolysis technology: Alkaline water electrolysis. There are two other technologies: PEM electrolysis and high temperature electrolysis. They both are theoretically more efficient than alkaline water electrolysis but they need noble and expensive material (PEM) or suffer from low life duration (high temperature electrolysis). Although alkaline electrolysis is the cheapest technology among all the electrolysis technology, it is still needed to decrease the hydrogen cost. Like most electrochemical processes, the cost of the hydrogen production is divided into two parts: CAPEX and OPEX. CAPEX represents the capital cost divided by the total hydrogen production. The OPEX is the production process cost. Thus, the higher the cell voltage increase, the higher the OPEX is.

^{2}and, $j$ the current density in A m

^{−2}.

## 2. Physics and Economy of the Alkaline Water Electrolysis Cell

#### 2.1. Theory

^{−1}.

^{3}s

^{−1}, S the electrode surface in m², ${v}_{mol}$ the molar volume m

^{3}mol

^{−1}, $F$ = 96,500 C mol

^{−1}the Faraday’s constant in C mol

^{−1}, ${R}_{G}$ = 8.314 J kg K

^{−1}the ideal gas constant in J mol

^{−1}K

^{−1}, $T$ the temperature in K, $p$ the pressure in Pa.

^{−1}and $\theta $ the surface coverage ratio.

#### 2.2. Physics of the Void Fraction

#### 2.2.1. Mathematical and Numerical Tools

**Finite Volume Model (CFD)**

- The flow is Newtonian, viscous and incompressible;
- the flow is considered isothermal;
- ions distributions are neglected;
- the flow is considered laminar;
- bubble diameter is constant for a given operating condition; and
- The current density is taken as constant.

_{2}, H

_{2}) or liq, ρ is the density in kg m

^{−3}, $V$ the velocity in m s

^{−1}, and ${S}_{k}$ is the term source in kg m

^{−3}s

^{−1}

^{−2}, and ${\overrightarrow{F}}_{k}$ is the exchange term in N m

^{−3}

**Artificial Neural Network (ANN)**

^{3}–10

^{4}] A m

^{−2}, an electrode height ${H}_{elec}$ in [0.01–0.1] m, a temperature $T$ in [293–350] K, a pressure $p$ in [1–30] bar, a mass fraction $Y$ in [0.2–0.4], a electrolyte thickness $h$ in [1.5 × 10

^{−4}–1.5 × 10

^{−3}] m.

#### 2.2.2. Sensitivity Analysis of Void Fraction and Ohmic Resistance

^{−3}, the electrode height ${H}_{elec}$ has been set to 5 × 10

^{−2}.

^{−2}and reached a plateau. The value of the plateau depends on the geometry of the cell especially the ratio electrolyte thickness/electrode height. However, the kinematic viscosity $\nu $ seems to play a role also.

#### 2.3. Economy

^{−1}, ${U}_{cell}$ the cell voltage in V, $F$ Faraday’s constant in A s mol

^{−1}, $EC$ the energy cost in € kWh

^{−1}, $I{C}_{S}$ the initial capital cost per unit area in € m

^{−2}, $t$ the electrolyzer lifespan in s, ${M}_{{H}_{2}}$ is the molar mass in kg mol

^{−1}.

^{−1}and in Germany about 0.30 € kWh

^{−1}. The effect of these two costs will be studied in the optimization section.

^{−1}. According to Grigoriev et al. [18], the electrolyzer power density is between 4 and 10 kW m

^{−2}. Thus, $I{C}_{S}$ will be explored between 800 and 14,000 € m

^{−2}. The Figure 8 presents the $OPEX$, $CAPEX$, and total hydrogen cost for an electrolyzer depending on the current density for a naïve design (${h}_{{H}_{2}}$ = ${h}_{{O}_{2}}$ = 1.5 mm, ${H}_{elec}=10$ cm, $p$ = 1 bar, $T$ = 353 K, $Y$ = 0.30, ${b}_{an}=$ 0.15 V dec

^{−1}, ${b}_{cath}=$ 0.15 V dec

^{−1}, $IC=$ 14,000 € m

^{−2}, and $EC=$ 0.17 € kWh

^{−1}) optimum current density is 2923 A m

^{−2}and the minimum cost is 12.48 € kg

^{−1}

_{.}

## 3. Set up of the Genetic Algorithm

#### 3.1. Optimization Problem

^{−1}.

#### 3.2. Genetic Optimization Algorithm (GA)

#### 3.3. Coding

#### 3.4. Evaluation

#### 3.5. Operators

#### 3.5.1. Initial Random Population

#### 3.5.2. Roulette Wheel Selection

#### 3.5.3. Single Point Crossing-Over

#### 3.5.4. Bitwise Mutation

#### 3.5.5. Replacement Generator (Generational)

#### 3.5.6. Presentation of the Evolution Parameter

- ${k}_{gen}$ the number of generations
- ${k}_{pop}$ the number of populations per generation
- ${p}_{m}$ probability of mutation
- ${p}_{c}$ probability of crossing-over is definitively set to 50%

#### 3.6. Validation of the Algorithm

- $I{C}_{s}$ = 800 € m
^{−2} - $EC$ = 17c€ kWh
^{−1} - ${b}_{an}$ = ${b}_{cath}$ = 0.15 V dec
^{−1}

_{2}and O

_{2}) converge to the smallest available value. This can be explained by the Ohm’s law in the monophasic model.

- The simplified model does not take into account the two-phase phenomena, which can greatly influence the hydrogen costs (${F}_{c}$).
- The evolution parameters used in this first try are not well adapted to the problem (missing convergence phase to achieve evolution).

## 4. Sensitivity Study of the Results to the Evolution Parameters

- $I{C}_{s}$ = 800 € m
^{−2} - $EC$ = 17 c€ kWh
^{−1} - ${b}_{an}$ = ${b}_{cath}$ = 0.15 V dec
^{−1} - ${R}_{memb}$ = 3.23 × 10
^{−5}Ohm m^{2} - $r$ = 25 × 10
^{−6}m

#### 4.1. Impact of ${k}_{gen}$ and ${k}_{pop}$

#### 4.2. Impact of p_{m}

#### 4.3. Optimization of the Naïve Solution

^{−1}to 12.30 € kg

^{−1}

^{−1}(1% of decrease) and an increase the current density of 290 A m

^{−2}(10% of increase).

^{−1}at $j$ = 10

^{4}A m

^{−2}.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

N° | $\overline{\mathit{R}{\mathit{e}}_{{\mathit{V}}_{\mathit{G}}}}$ | $\overline{\mathit{F}{\mathit{r}}_{{\mathit{V}}_{\mathit{G}}}}$ | $\overline{{\mathit{h}}^{*}}$ | $\overline{{\mathit{r}}^{*}}$ | ${\mathit{\epsilon}}_{\mathit{f}\mathit{l}\mathit{u}\mathit{e}\mathit{n}\mathit{t}}$ |
---|---|---|---|---|---|

1 | −0.375 | 1 | 0.625 | −0.25 | 2.07 × 10^{−4} |

2 | −0.875 | −0.5 | 0.75 | 0.125 | 3.52 × 10^{−4} |

3 | −0.75 | −0.125 | −0.875 | −0.5 | 1.95 × 10^{−3} |

4 | −0.625 | 0.25 | −0.375 | 1 | 3.868 × 10^{−4} |

5 | 0.5 | 0.875 | −0.125 | −0.75 | 3.972 × 10^{−4} |

6 | 1 | −0.375 | −0.25 | 0.625 | 4.543 × 10^{−4} |

7 | 0.25 | −0.625 | 1 | −0.375 | 3.884 × 10^{−4} |

8 | 0.125 | 0.75 | 0.5 | 0.875 | 1.737 × 10^{−4} |

9 | 0 | 0 | 0 | 0 | 3.624 × 10^{−4} |

10 | 0.375 | −1 | −0.625 | 0.25 | 1.477 × 10^{−1} |

11 | 0.875 | 0.5 | −0.75 | −0.125 | 8.782 × 10^{−4} |

12 | 0.75 | 0.125 | 0.875 | 0.5 | 1.897 × 10^{−4} |

13 | 0.625 | −0.25 | 0.375 | −1 | 5.931 × 10^{−4} |

14 | −0.5 | −0.875 | 0.125 | 0.75 | 7.161 × 10^{−4} |

15 | −1 | 0.375 | 0.25 | −0.625 | 5.067 × 10^{−3} |

16 | −0.25 | 0.625 | −1 | 0.375 | 3.289 × 10^{−3} |

17 | −0.125 | −0.75 | −0.5 | −0.875 | 1.515 × 10^{−3} |

18 | 1 | −1 | −1 | 1 | 1 |

19 | 1 | 1 | 1 | 1 | 1.219 × 10^{−4} |

20 | −1 | −1 | 1 | 1 | 1.856 × 10^{−1} |

21 | 1 | −1 | 1 | −1 | 2.096 × 10^{−2} |

22 | −1 | 1 | −1 | 1 | 1.063 × 10^{−1} |

23 | −1 | 1 | 1 | −1 | 2.935 × 10^{−3} |

24 | 1 | 1 | −1 | −1 | 3.986 × 10^{−3} |

25 | −1 | −1 | −1 | −1 | 1 |

26 | −0.9325 | −0.98 | −0.95 | −1 | 4.058 × 10^{−2} |

27 | −0.9325 | −1 | −0.875 | 1 | 6.778 × 10^{−1} |

28 | −0.99 | −0.99 | −0.95 | 1 | 9.828 × 10^{−2} |

29 | −0.875 | −0.99 | −0.875 | −1 | 2.301 × 10^{−2} |

30 | −0.99 | −0.98 | −0.875 | 0 | 4.274 × 10^{−2} |

31 | −0.875 | −1 | −0.95 | 0 | 8.424 × 10^{−1} |

32 | −0.946 | −0.990 | −0.728 | −0.427 | 1.460 × 10^{−2} |

33 | −0.983 | −0.997 | −0.704 | −0.200 | 4.440 × 10^{−2} |

34 | −0.973 | −0.996 | −1.023 | −0.578 | 4.539 × 10^{−1} |

35 | −0.964 | −0.994 | −0.925 | 0.328 | 6.174 × 10^{−2} |

36 | −0.881 | −0.991 | −0.876 | −0.729 | 2.390 × 10^{−2} |

37 | −0.844 | −0.997 | −0.900 | 0.101 | 3.623 × 10^{−2} |

38 | −0.900 | −0.998 | −0.655 | −0.503 | 1.993 × 10^{−2} |

39 | −0.909 | −0.992 | −0.753 | 0.252 | 1.350 × 10^{−2} |

40 | −0.918 | −0.995 | −0.851 | −0.276 | 2.892 × 10^{−2} |

41 | −0.890 | −0.999 | −0.974 | −0.125 | 8.122 × 10^{−1} |

42 | −0.853 | −0.993 | −0.999 | −0.352 | 1.044 × 10^{−1} |

43 | −0.863 | −0.994 | −0.679 | 0.026 | 1.155 × 10^{−2} |

44 | −0.872 | −0.996 | −0.778 | −0.881 | 2.310 × 10^{−2} |

45 | −0.955 | −0.999 | −0.827 | 0.177 | 9.900 × 10^{−2} |

46 | −0.992 | −0.993 | −0.802 | −0.654 | 6.528 × 10^{−2} |

47 | −0.937 | −0.992 | −1.048 | −0.050 | 6.000 × 10^{−1} |

48 | −0.927 | −0.998 | −0.95 | −0.805 | 1.431 × 10^{−1} |

49 | −0.8 | −0.87 | −0.85 | −0.875 | 5.296 × 10^{−3} |

50 | −0.8 | −0.97 | −0.55 | −0.125 | 3.516 × 10^{−3} |

51 | −0.9 | −0.92 | −0.85 | −0.125 | 6.844 × 10^{−3} |

52 | −0.7 | −0.92 | −0.55 | −0.875 | 2.807 × 10^{−3} |

53 | −0.9 | −0.97 | −0.7 | −0.875 | 7.510 × 10^{−3} |

54 | −0.7 | −0.87 | −0.7 | −0.125 | 2.482 × 10^{−3} |

55 | −0.9 | −0.87 | −0.55 | −0.5 | 2.279 × 10^{−3} |

56 | −0.7 | −0.97 | −0.85 | −0.5 | 1.068 × 10^{−2} |

57 | −0.8 | −0.92 | −0.7 | −0.5 | 3.584 × 10^{−3} |

58 | −0.8 | −0.87 | −0.85 | −1 | 1.740 × 10^{−2} |

59 | −0.8 | −0.97 | −0.55 | −0.9 | 9.525 × 10^{−3} |

60 | −0.7 | −0.92 | −0.85 | −0.9 | 1.082 × 10^{−2} |

61 | −0.9 | −0.92 | −0.55 | −1 | 1.636 × 10^{−2} |

62 | −0.7 | −0.97 | −0.7 | −1 | 3296 × 10^{−2} |

63 | −0.9 | −0.87 | −0.7 | −0.9 | 3899 × 10^{−3} |

64 | −0.7 | −0.87 | −0.55 | −0.95 | 7417 × 10^{−3} |

65 | −0.9 | −0.97 | −0.85 | −0.95 | 1508 × 10^{−2} |

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**Figure 3.**Results of the training of the artificial neural network (ANN), the results of void fraction of the ANN ${\epsilon}_{ANN}$ are compared to the void fraction calculated to ${\epsilon}_{fluent}$. The blue points represent the learning samples and the red points the validation samples. The red line is ${\epsilon}_{fluent}\text{}+\text{}$ 40% and the blue line ${\epsilon}_{fluent}$ − 40%.

**Figure 4.**Void fraction ${\epsilon}_{ANN}$ (

**a**) and ohmic resistance $R$ (

**b**) depending on the pressure in [1–30] bar.

**Figure 5.**Void fraction ${\epsilon}_{ANN}$ (

**a**) and ohmic resistance $R$ (

**b**) depending on the electrolyte thickness $h$ in [0.3;1.5] mm.

**Figure 6.**Void fraction ${\epsilon}_{ANN}$ (

**a**) and ohmic resistance $R$ (

**b**) depending current density in [1;10] kA m

^{−2}.

**Figure 7.**Void fraction ${\epsilon}_{ANN}$ (

**a**) and ohmic resistance $R$ (

**b**) depending on electrode height ${H}_{elec}$.

**Figure 8.**Investment cost (CAPEX), operating cost (OPEX), and Cost function for an electrochemical cell with ${h}_{{H}_{2}}$ = ${h}_{{O}_{2}}$ = 1.5 mm, ${H}_{elec}=10$ cm, $p$ = 1 bar, $T$ = 353 K, $Y$ = 0.30, ${b}_{an}=$ 0.15 V dec

^{−1}, ${b}_{cath}=$ 0.15 V dec

^{−1}, $r=$ 25 µm, $I{C}_{s}=$ 14,000 € m

^{−2}, and $EC=$ 0.17 € kWh

^{−1}.

**Figure 10.**Averaged over 100 evolutions (

**a**) fitness and (

**b**) cost function evolution according to the generation.

**Figure 11.**Design parameters evolution for the best $DP$ set solution (in full $DP$ space and for a single evolution) of the best candidate depending on the generation (

**a**) for the anolyte and catholyte thickness ${h}_{H2},{h}_{o2}$ (

**b**) for electrolyte temperature T.

**Figure 12.**Design parameters evolution for the best $DP$ set (in full $DP$ space and for a single evolution) of the best candidate depending on the generation (

**a**) current density $p$, (

**b**) electrode height ${H}_{elec}$, (

**c**) pressure $p$ and (

**d**) KOH mass fraction $Y$.

**Figure 14.**Result of the sensitivity study of the fitness to the mutation rate. In (

**a**) the whole research space has been plotted and (

**b**) only the mutation rate between 0 and 2% has been plotted.

**Figure 15.**Averaged over 100 evolutions (

**a**) Fitness and (

**b**) cost function evolution according to the generation with ${k}_{gen}$ = 118, ${k}_{pop}$ = 17, ${p}_{m}$ = 0.5% ($I{C}_{s}$ = 800 € m

^{−2}, $EC$ = 0.17 € kWh

^{−1}, ${b}_{an}={b}_{cath}=$ 0.15 V dec

^{−1}).

**Figure 16.**Averaged over 100 evolutions (

**a**) Fitness and (

**b**) cost function evolution according to the generation with ${k}_{gen}$ = 118, ${k}_{pop}$ = 17, ${p}_{m}$ = 0.5% ($I{C}_{s}$ = 14,000 € m

^{−2}, $EC$ = 0.17 € kWh

^{−1}, ${b}_{an}={b}_{cath}=$ 0.15 V dec

^{−1}).

**Figure 17.**CAPEX, OPEX, and total cost for the naïve solution (dashed line) and optimum solution (solid line) depending on the current density ($I{C}_{s}$ = 14,000 € m

^{−2}, $EC$ = 0.17 € kWh

^{−1}, ${b}_{an}={b}_{cath}=$ 0.15 V dec

^{−1}).

**Figure 18.**(

**a**) Cell voltage of the naïve solution (dashed line) and optimum solution (solid line) (

**b**) proportion of the different voltage of the cell voltage depending on the current density.

Dimensionless Parameter | ${\mathit{x}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\frac{{\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}+{\mathit{x}}_{\mathit{m}\mathit{i}\mathit{n}}}{2}$ | $\frac{{\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{x}}_{\mathit{m}\mathit{i}\mathit{n}}}{2}$ |
---|---|---|---|---|

$R{e}_{{V}_{G}}$ | 1 × 10^{−2} | 300 | 150.005 | 149.995 |

$F{r}_{{V}_{G}}$ | 5.85 × 10^{4} | 5.30 × 10^{10} | 2.65 × 10^{10} | 2.65 × 10^{10} |

${r}^{*}$ | 1.14 × 10^{−5} | 1.50 × 10^{−3} | 7.75 × 10^{−2} | 7.25 × 10^{−2} |

${h}^{*}$ | 5 × 10^{−3} | 1.50 × 10^{−1} | 7.557 × 10^{−4} | 7.442 × 10^{−4} |

i = 0 | i = 1 | i = 2 | i = 3 | ||
---|---|---|---|---|---|

${H}_{1,i}$ | ${w}_{0,i}$ | 1.3965 | 1.3532 | 1.4242 | 1.4926 |

${c}_{0,i}$ | 0.61711 | 0.6445 | 0.5920 | 0.1461 | |

${H}_{2,i}$ | ${w}_{1,i}$ | 41.63 | −0.848 | −0.444 | – |

${w}_{1,\left(i+3\right)}$ | −2.6544 | −0.837 | 114 | – | |

${w}_{1,\left(i+6\right)}$ | −1.5778 | −9.13 | −0.015 | – | |

${w}_{1,\left(i+9\right)}$ | 0.2068 | −0.154 | −0.347 | – | |

${c}_{1,i}$ | 29.78 | −13.57 | 81.04 | – | |

${H}_{3,i}$ | ${w}_{2,0}$ | −0.4 | – | – | – |

${w}_{2,1}$ | −0.959 | – | – | – | |

${w}_{2,2}$ | 12.54 | – | – | – | |

${c}_{2,0}$ | 19.84 | – | – | – | |

${H}_{4,i}$ | ${w}_{3,0}$ | 0.99 | – | – | – |

${c}_{3,0}$ | −2.03 | – | – | – | |

${H}_{5,i}$ | ${w}_{4,0}$ | 2.30 | – | – | – |

Variables | Min | Max |
---|---|---|

$T$ (K) | 293 | 363 |

$Y$ (–) | 0.2 | 0.3 |

$h$ (m) | 4 × 10^{−4} | 10^{−3} |

${H}_{elec}$ (m) | 5 × 10^{−2} | 10^{−1} |

$j$ (A m^{−2}) | 10^{3} | 10^{4} |

$p$ (bar) | 1 | 30 |

Design Parameter | ${\mathit{x}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{Q}}_{\mathit{w}\mathit{t}\mathit{d}}$ |
---|---|---|---|

${h}_{{H}_{2}}$ (m) | 1.5 × 10^{−4} | 1.5 × 10^{−3} | 1 × 10^{−4} |

${h}_{{H}_{2}}$ (m) | 1.5 × 10^{−4} | 1.5 × 10^{−3} | 1 × 10^{−4} |

$p$ (bar) | 1 | 30 | 1 |

${H}_{elec}$ (m) | 5 × 10^{−2} | 10^{−1} | 5 × 10^{−3} |

$T$ (K) | 298 | 358 | 10 |

$j$ (A m^{−2}) | 10^{3} | 10^{4} | 10^{2} |

$Y$ (-) | 0.2 | 0.4 | 2.5 × 10^{−2} |

Design Parameter | ${\mathit{Q}}_{\mathit{w}\mathit{t}\mathit{d}}$ | ${\mathit{n}}_{\mathit{b}}$ | ${\mathit{Q}}_{\mathit{o}\mathit{b}\mathit{t}}$ | Example | Coded Example | Decoded Example |
---|---|---|---|---|---|---|

${h}_{{H}_{2}}$ (m) | 10^{−4} | 4 | 9 × 10^{−5} | 3 × 10^{−4} | [0 0 0 1 1] | 2.765 × 10^{−4} |

${h}_{{O}_{2}}$ (m) | 10^{−4} | 4 | 9 × 10^{−5} | 3 × 10^{−4} | [0 0 0 1 1] | 2.765 × 10^{−4} |

$p$ (bar) | 1 | 7 | 7. 79 × 10^{−1} | 3 | [0 0 0 0 0 1 0] | 2.54 |

${H}_{elec}$ (m) | 5 × 10^{−3} | 4 | 3.33 × 10^{−3} | 6 × 10^{−2} | [0 0 1 1] | 5.94 × 10^{−2} |

$T$ (K) | 10 | 3 | 7.85 | 3.08 × 10^{2} | [0 0 1] | 3.04 × 10^{2} |

$j$ (A m^{−2}) | 10^{2} | 7 | 7.08 × 10^{1} | 2 × 10^{3} | [0 0 0 1 1 1 0] | 1.98 × 10^{3} |

$Y$ (-) | 2.5 × 10^{−2} | 4 | 1.33 × 10^{−2} | 0.3 | [0 1 1 1] | 2.87 × 10^{−1} |

$\mathit{E}\mathit{C}\text{}\mathbf{(}\mathbf{\u20ac}\text{}{\mathbf{kWh}}^{-1})$ | |||||||
---|---|---|---|---|---|---|---|

0.17 | 0.3 | ||||||

$\mathit{O}\mathit{P}\mathit{E}{\mathit{X}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{O}\mathit{P}\mathit{E}{\mathit{X}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathit{O}\mathit{P}\mathit{E}{\mathit{X}}_{\mathit{m}\mathit{i}\mathit{n}}$ | $\mathit{O}\mathit{P}\mathit{E}{\mathit{X}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ||||

9.48 | 13.46 | 16.74 | 23.76 | ||||

$I{C}_{s}$ (€ m^{−2}) | 800 | $CAPE{X}_{min}$ | 0.02 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.70 $f$ = 0.04 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.70 $f$ = 0.02 |

$CAPE{X}_{max}$ | 0.24 | $\overline{F}$ = 0.98 $f$ = 0.92 | $\overline{F}$ = 0.69 $f$ = 0 | $\overline{F}$ = 0.99 $f$ = 0.96 | $\overline{F}$ = 0.69 $f$ = 0 | ||

3000 | $CAPE{X}_{min}$ | 0.09 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.12 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.07 | |

$CAPE{X}_{max}$ | 0.89 | $\overline{F}$ = 0.92 $f$ = 0.77 | $\overline{F}$ = 0.67 $f$ = 0 | $\overline{F}$ = 0.95 $f$ = 0.86 | $\overline{F}$ = 0.68 $f$ = 0 | ||

6000 | $CAPE{X}_{min}$ | 0.17 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.20 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.13 | |

$CAPE{X}_{max}$ | 1.78 | $\overline{F}$ = 0.85 $f$ = 0.61 | $\overline{F}$ = 0.63 $f$ = 0 | $\overline{F}$ = 0.91 $f$ = 0.74 | $\overline{F}$ = 0.66 $f$ = 0 | ||

9000 | $CAPE{X}_{min}$ | 0.27 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.27 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.18 | |

$CAPE{X}_{max}$ | 2.68 | $\overline{F}$ = 0.80 $f$ = 0.50 | $\overline{F}$ = 0.60 $f$ = 0 | $\overline{F}$ = 0.88 $f$ = 0.65 | $\overline{F}$ = 0.64 $f$ = 0 | ||

14,000 | $CAPE{X}_{min}$ | 0.42 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.35 | $\overline{F}$ = 1 $f$ = 1 | $\overline{F}$ = 0.71 $f$ = 0.25 | |

$CAPE{X}_{max}$ | 4.17 | $\overline{F}$ = 0.72 $f$ = 0.37 | $\overline{F}$ = 0.56 $f$ = 0 | $\overline{F}$ = 0.82 $f$ = 0.53 | $\overline{F}$ = 0.61 $f$ = 0 |

**Table 7.**Design of experiments for the two variables ${P}_{GP}$ and ${R}_{GP}$. The optimal number of generation ${k}_{gen}$ and population size ${k}_{pop}$ are in bold.

${\mathit{P}}_{\mathit{G}\mathit{P}}$ | ||||||||
---|---|---|---|---|---|---|---|---|

500 | 1000 | 1500 | 2000 | 2500 | 3000 | |||

${R}_{GP}$ | 1 | ${k}_{pop}$ | 22 | 32 | 38 | 44 | 50 | 54 |

${k}_{gen}$ | 23 | 31 | 40 | 45 | 50 | 56 | ||

3 | ${k}_{pop}$ | 13 | 19 | 23 | 26 | 29 | 32 | |

${k}_{gen}$ | 38 | 53 | 66 | 76 | 85 | 93 | ||

5 | ${k}_{pop}$ | 10 | 14 | 17 | 20 | 22 | 24 | |

${k}_{gen}$ | 50 | 70 | 87 | 100 | 112 | 122 | ||

7 | ${k}_{pop}$ | 8 | 12 | 15 | 17 | 19 | 21 | |

${k}_{gen}$ | 59 | 84 | 102 | 118 | 132 | 145 | ||

10 | ${k}_{pop}$ | 7 | 10 | 12 | 14 | 16 | 17 | |

${k}_{gen}$ | 71 | 100 | 123 | 141 | 158 | 173 |

Value | |
---|---|

${k}_{pop}$ | 17 |

${k}_{gen}$ | 118 |

${p}_{c}$ | 50% |

${p}_{m}$ | 0.5% |

${R}_{GP}$ | 7 |

${P}_{GP}$ | 200 |

**Table 9.**Optimum $DP$ with their associated hydrogen cost for $I{C}_{s}$ = 800 € m

^{−2}, EC = 17 c€ kWh

^{−1}, $r$ = 25 µm and ${R}_{memb}$ = 3.23 × 10

^{−5}Ohm m

^{2}.

Value | |
---|---|

${h}_{H2}$(m) | 3 × 10^{−4} |

${h}_{O2}$ (m) | 2 × 10^{−4} |

${H}_{elec}$(m) | 5 × 10^{−2} |

$j$ (A m^{−2}) | 1000 |

$Y$ (-) | 0.2 |

$T$ (K) | 350 |

$p$ (bar) | 1 |

$CAPEX$ (€ kg ^{−1}) | 0.22 |

$OPEX$ (€ kg ^{−1}) | 9.85 |

${F}_{c}$ (€ kg ^{−1}) | 10.07 |

**Table 10.**Naïve and Optimum $DP$ with their associated hydrogen cost for EC = 17 c€ kWh

^{−1}, $r$ = 25 µm and ${R}_{memb}$ = 3.23 × 10

^{−5}Ohm m

^{2}.

Naïve | GA1 | GA2 | |
---|---|---|---|

$I{C}_{s}$ (€ m^{−2}) | 1.4 × 10^{4} | 1.4 × 10^{4} | 8 × 10^{2} |

${h}_{H2}$ (m) | 1.5 × 10^{−3} | 4 × 10^{−4} | 3 × 10^{−4} |

${h}_{O2}$ (m) | 1.5 × 10^{−3} | 4 × 10^{−4} | 2 × 10^{−4} |

${H}_{elec}$ (m) | 10^{−1} | 5 × 10^{−2} | 5 × 10^{−2} |

$j$ (A m^{−2}) | 2923 | 3214 | 1000 |

$Y$ (–) | 0.3 | 0.23 | 0.2 |

$T$ (K) | 350 | 350 | 350 |

$p$ (bar) | 1 | 1 | 1 |

$CAPEX$ (€ kg^{−1}) | 1.42 | 1.30 | 0.22 |

$OPEX$ (€ kg^{−1}) | 11.06 | 11 | 9.85 |

${F}_{c}$ (€ kg^{−1}) | 12.48 | 12.30 | 10.07 |

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**MDPI and ACS Style**

Le Bideau, D.; Chocron, O.; Mandin, P.; Kiener, P.; Benbouzid, M.; Sellier, M.; Kim, M.; Ganci, F.; Inguanta, R.
Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production. *Appl. Sci.* **2020**, *10*, 8425.
https://doi.org/10.3390/app10238425

**AMA Style**

Le Bideau D, Chocron O, Mandin P, Kiener P, Benbouzid M, Sellier M, Kim M, Ganci F, Inguanta R.
Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production. *Applied Sciences*. 2020; 10(23):8425.
https://doi.org/10.3390/app10238425

**Chicago/Turabian Style**

Le Bideau, Damien, Olivier Chocron, Philippe Mandin, Patrice Kiener, Mohamed Benbouzid, Mathieu Sellier, Myeongsub Kim, Fabrizio Ganci, and Rosalinda Inguanta.
2020. "Evolutionary Design Optimization of an Alkaline Water Electrolysis Cell for Hydrogen Production" *Applied Sciences* 10, no. 23: 8425.
https://doi.org/10.3390/app10238425