# Multi-Objective Non-Dominated Sorting Genetic Algorithm Optimization for Optimal Hybrid (Wind and Grid)-Hydrogen Energy System Modelling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hybrid-Hydrogen Energy System Modelling

#### 2.1. Wind-Energy System

#### 2.1.1. Site Description and Data Acquisition

#### 2.1.2. Weibull Distribution Model

#### 2.1.3. Wind Power and Energy Density

#### 2.1.4. Wind Turbine

- idle (v < ${v}_{ci}$):
- -
- wind turbine barely produces useful power ${P}_{w}={P}_{n}=0$,

- startup (${v}_{ci}$ < ${v}_{r}$):
- -
- wind turbine starts generating useful power at low speeds. ${P}_{w}={P}_{s}$,

- power generation (${v}_{r}$ ≤ v < ${v}_{co}$):
- -
- wind turbine produces rated power, i.e., maximum output power, ${P}_{w}={P}_{r}$,

- shutdown (v ≥ ${v}_{co}$):
- -
- wind turbine shuts down and ceases to produce any power at maximum speed, ${P}_{w}={P}_{n}=0$.

#### 2.2. Grid-Energy System

#### 2.3. Energy Management System

#### 2.4. Proton Exchange Membrane Electrolyzer

#### 2.5. Cost of Electricity Model

#### 2.6. Efficiency Model

## 3. Optimization Problem Definition Model

- •
- objective functions;
- •
- constraint functions;
- •
- design variables.

- find the design variables:$${x}_{i},{x}_{(i+1)},\dots ,\phantom{\rule{1.em}{0ex}}i\phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}\mathbb{N},\phantom{\rule{3.33333pt}{0ex}}i\le I,$$$${f}_{j}({x}_{i},{x}_{(i+1),\dots}),{f}_{(j+1)}({x}_{i},{x}_{(i+1)},\dots ),\dots ,\phantom{\rule{1.em}{0ex}}j\phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}\mathbb{N},\phantom{\rule{3.33333pt}{0ex}}j\le J,$$$${g}_{k}({x}_{i},{x}_{(i+1),\dots}),{g}_{(k+1)}({x}_{i},{x}_{(i+1)},\dots ),\dots ,\phantom{\rule{1.em}{0ex}}k\phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}\mathbb{N},\phantom{\rule{3.33333pt}{0ex}}k\le K,$$$${x}_{i}^{l}\le {x}_{i}\le {x}_{i}^{u}.$$

## 4. Application, Results and Discussion

#### 4.1. Hybrid-Hydrogen Energy System Application and Results

#### 4.2. Optimization Application and Results

## 5. Conclusions

- -
- The wind REDZs have a high average wind speed (above 6 m/s), therefore, this is an indication of the high potential for green H${}_{2}$ production.
- -
- The knowledge of wind speed characteristics in conjunction with the appropriate wind turbine model successfully present the energy production potential of the wind REDZs.
- -
- The developed H-HES model is successfully demonstrated.
- -
- The open-source Pymoo module is a powerful tool that can be used in conjunction with NSGA-II to obtain optimal solutions free of charge.
- -
- The proposed H-HES interface models ensured that NSGA-II successfully converged to a Pareto front of the cost of electricity and efficiency for all the wind REDZs.
- -
- The optimal cost of electricity and efficiency successfully determine the required optimal variables to model the H-HES.
- -
- The derived optimal variables found for each wind REDZs greatly influence the choice of an appropriate wind turbine model for the W-ES and efficiencies of the H-HES, resulting in an optimal H-HES model.
- -
- The proposed optimal H-HES model can be adapted for green hydrogen production to ensure minimum cost (from the G-ES) at maximized efficiency with appropriate wind turbine model selection.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Ideal wind turbine power curve [35].

**Figure 4.**Transmission zones for loads [38].

**Figure 5.**Low- and high-demand seasons’ TOU periods [38].

Variable | Range |
---|---|

${\eta}_{r}$ | $0.45$–$0.50$ |

${\eta}_{gb}$ | $0.85$–$0.95$ |

${\eta}_{ge}$ | $0.90$–$0.95$ |

Wind REDZs | Overberg | Komsberg | Cookhouse | Stormberg | Springbok | Beaufort West |
---|---|---|---|---|---|---|

Wind Masts | Napier | Sutherland | Humansdrop | Butterworth | Pofadder | Beaufort West |

Wind REDZs | ||||||
---|---|---|---|---|---|---|

Overberg | Komsberg | Cookhouse | Stormberg | Springbok | Beaufort West | |

${E}_{d}$ (kJ) | $5028.10$ | $3411.02$ | $3367.70$ | $2623.00$ | $3004.42$ | $2593.17$ |

Wind Turbine Model | ${\mathit{P}}_{\mathit{r}}$ (kW) | ${\mathit{v}}_{\mathit{ci}}$ (m/s) | ${\mathit{v}}_{\mathit{r}}$ (m/s) | ${\mathit{v}}_{\mathit{co}}$ (m/s) | ${\mathit{r}}_{\mathit{d}}$ (m) | ${\mathit{h}}_{\mathit{h}}$ (m) |
---|---|---|---|---|---|---|

N43 | 600 | $2.5$ | $15.0$ | $25.0$ | $43.0$ | $60.0,78.0$ |

D$4/48$ | 600 | $2.5$ | $11.5$ | $22.0$ | $48.0$ | $60.0,70.0$ |

N50 | 800 | $2.5$ | $15.0$ | $25.0$ | $50.0$ | $46.0,50.0,60.0,70.0$ |

G52 | 800 | $4.0$ | $13.0$ | $25.0$ | $52.0$ | $44.0,55.0,65.0$ |

V54 | 850 | $4.0$ | $14.0$ | $25.0$ | $52.0$ | $60.0,65.0,70.0,75.0,85.0$ |

G58 | 850 | $3.0$ | $12.5$ | $25.0$ | $58.0$ | $44.0,55.0,65.0$ |

N54 | 1000 | $3.5$ | $14.0$ | $25.0$ | $54.0$ | $60.0,70.0$ |

Wind REDZs | ||||||
---|---|---|---|---|---|---|

Overberg | Komsberg | Cookhouse | Stormberg | Springbok | Beaufort West | |

s (-) | $0.16$ | $0.13$ | $0.22$ | $0.11$ | $0.15$ | $0.11$ |

Actual Design Variables |
---|

$40.0$ ≤ ${r}_{d}$ ≤ $60.0$ |

$60.0$ ≤ ${h}_{h}$ ≤ $90.0$ |

$2.5$ ≤ ${v}_{ci}$ ≤ $4.0$ |

12 ≤ ${v}_{r}$ ≤ $15.0$ |

$23.0$ ≤ ${v}_{co}$ ≤ $25.0$ |

$0.40$ ≤ ${\eta}_{r}$ ≤ $0.50$ |

$0.85$ ≤ ${\eta}_{gb}$ ≤ $0.95$ |

$0.90$ ≤ ${\eta}_{ge}$ ≤ $0.96$ |

$0.90$ ≤ ${\eta}_{em}$ ≤ $0.98$ |

$0.90$ ≤ ${\eta}_{c}$ ≤ $0.95$ |

$0.70$ ≤ ${\eta}_{e}$ ≤ $0.90$ |

**Table 7.**Urban Megaflex non-authority tariff charges (excluding VAT) [38].

Active Energy Charges (South African Rand (ZAR)/kWh) | |||||||
---|---|---|---|---|---|---|---|

High Demand Season (Jun–Aug) | Low Demand Season (Sep–May) | ||||||

Transmission Zone | Voltage | Peak | Standard | Off Peak | Peak | Standard | Off Peak |

>600 km and ≤900 km | ≥500 V and ≤66 kV | $4.5935$ | $1.3917$ | $0.7557$ | $1.4984$ | $1.0314$ | $0.6543$ |

>900 km | ≥500 V and ≤66 kV | $4.6392$ | $1.4052$ | $0.7628$ | $1.5131$ | $1.0412$ | $0.6607$ |

**Table 8.**Specifications of a 2 MW H-TEC PEM system [40].

Specification | Value |
---|---|

Electrical nominal power (${P}_{L}$) | 2 MW |

H${}_{2}$ production | 900 kg/day (420 Nm${}^{3}$/h) |

H${}_{2}$ purity | $99.9$% |

Energy consumption | $4.8$ kWh/Nm${}^{3}$ H${}_{2}$ |

System efficiency | 74% |

Wind REDZs | Cost of Electricity (ZAR million) |
---|---|

Overberg, Komsberg and Springbok | $22.8$ |

Cookhouse, Stormberg and Beaufort West | $22.6$ |

Optimization Parameters |
---|

Population size: 500 |

Number of offspring: 1000 |

Number of generations: 500 |

Sampling: Random |

Mutation: Polynomial (Probability= $0.5$, index = 20) |

Crossover: Simulated binary (Probability = $0.9$, index = 15) |

Wind REDZs | ||||||
---|---|---|---|---|---|---|

Overberg | Komsberg | Cookhouse | Stormberg | Springbok | Beaufort West | |

${C}_{e}$ (ZAR million) | $1.423$ | $1.081$ | $1.824$ | $1.714$ | $0.813$ | $1.675$ |

$\eta $ (-) | $0.393$ | $0.475$ | $0.452$ | $0.510$ | $0.540$ | $0.526$ |

Design Variables | Wind REDZs | |||||
---|---|---|---|---|---|---|

Overberg | Komsberg | Cookhouse | Stormberg | Springbok | Beaufort West | |

${r}_{d}$ (m) | $40.0$ | $40.8$ | $40.0$ | $45.0$ | $40.6$ | $42.0$ |

${h}_{h}$ (m) | $60.3$ | $65.4$ | $85.0$ | $89.0$ | $79.0$ | $68.8$ |

${v}_{ci}$ (m/s) | $2.5$ | $2.5$ | $2.5$ | $2.5$ | $2.5$ | $2.5$ |

${v}_{r}$ (m/s) | $15.0$ | $15.0$ | $15.0$ | $15.0$ | $15.0$ | $15.0$ |

${v}_{co}$ (m/s) | $24.0$ | $25.0$ | $25.0$ | $25.0$ | $24.0$ | $24.0$ |

${\eta}_{r}$ (-) | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ |

${\eta}_{gb}$ (-) | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ |

${\eta}_{ge}$ (-) | $0.92$ | $0.96$ | $0.96$ | $0.96$ | $0.95$ | $0.96$ |

${\eta}_{em}$ (-) | $0.93$ | $0.98$ | $0.97$ | $0.98$ | $0.98$ | $0.98$ |

${\eta}_{c}$ (-) | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ |

${\eta}_{e}$ (-) | $0.90$ | $0.90$ | $0.90$ | $0.90$ | $0.90$ | $0.90$ |

Wind REDZs | ||||||
---|---|---|---|---|---|---|

Overberg | Komsberg | Cookhouse | Stormberg | Springbok | Beaufort West | |

${C}_{e}$ (ZAR million) | $1.442$ | $1.212$ | $1.9028$ | $1.824$ | $0.8211$ | $2.088$ |

$\eta $ (-) | $0.325$ | $0.40$ | $0.38$ | $0.42$ | $0.445$ | $0.47$ |

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**MDPI and ACS Style**

Mukoni, E.; Garner, K.S. Multi-Objective Non-Dominated Sorting Genetic Algorithm Optimization for Optimal Hybrid (Wind and Grid)-Hydrogen Energy System Modelling. *Energies* **2022**, *15*, 7079.
https://doi.org/10.3390/en15197079

**AMA Style**

Mukoni E, Garner KS. Multi-Objective Non-Dominated Sorting Genetic Algorithm Optimization for Optimal Hybrid (Wind and Grid)-Hydrogen Energy System Modelling. *Energies*. 2022; 15(19):7079.
https://doi.org/10.3390/en15197079

**Chicago/Turabian Style**

Mukoni, Esmeralda, and Karen S. Garner. 2022. "Multi-Objective Non-Dominated Sorting Genetic Algorithm Optimization for Optimal Hybrid (Wind and Grid)-Hydrogen Energy System Modelling" *Energies* 15, no. 19: 7079.
https://doi.org/10.3390/en15197079