# Evolutionary Multi-Objective Energy Production Optimization: An Empirical Comparison

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## Abstract

**:**

_{2}emissions. The Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) is also adopted in the comparison so as to enrich the empirical evidence by contrasting the NSGA-II versions against a non-Pareto-based approach. Spacing and Hypervolume are the chosen metrics to compare the performance of the algorithms under study. The obtained results suggest that there is no significant improvement by using the variant of the NSGA-II over the original version. Nonetheless, meaningful performance differences have been found between MOEA/D and the other two algorithms.

## 1. Introduction

_{2}emissions, production costs and production adequacy, using the aforementioned algorithm. Wang and Zhou [15] utilized the same algorithm to optimize the emissions and energy-savings of a wind power system. Liu and Dongdong [16] optimized a multiple-source power system considering its production cost and the amount of emissions it produces. Zhou and Sun [17] utilized NSGA-II to optimize a hybrid energy system consisting of solar and wind power. All these works demonstrate the positive impact that NSGA-II has in these type of energy-based problems. For this study, the results yielded by NSGA-II, as well as the results obtained by MOEA/D, are compared with a modified version of this popular algorithm, called L-NSGA-II, proposed by Liu et al. [18].

_{2}emissions. These functions depend on four real-domain variables: hours of operation of solar power (${h}_{s}$), wind power (${h}_{w}$), and natural gas power (${h}_{g}$), and the amount of natural gas power to be produced (${P}_{g}$). The hours of operation are constrained to a minimum of 240 h and a maximum of 672 h each, equivalent to 10 days and 28 days, respectively. Natural gas power production is bounded to the range of 4.88 MW to 7.07 MW. More detail of the system configuration is provided in Section 6.

## 2. Energy Sources

#### 2.1. Solar Power

^{2}.

#### 2.2. Wind Power

^{3}and V is the wind speed, measured in m/s.

#### 2.3. Capacity Factor

#### 2.4. Production Cost

#### 2.5. CO_{2} Emissions Rate

_{2}). The National Aeronautics and Space Administration (NASA) states that CO

_{2}is one of the most important contributors to global warming [23].

_{2}they produce per Watt or Watt-hour. These emissions are referred as CO

_{2}emissions rate. For this work, the emission rates are measured in gr/KWh. Emissions rates are shown in Table 3.

## 3. Optimization Problem

_{2}emissions. Four decision variables are contemplated in this study. The first three variables are the hours of operation of each system, referred as ${h}_{s}$, ${h}_{w}$, and ${h}_{g}$. The fourth variable is the amount of energy produced by the natural gas system alone, ${P}_{g}$.

#### 3.1. Power Production Function

#### 3.2. Production Cost Function

#### 3.3. CO_{2} Emissions Function

_{2}emissions caused by the three energy sources. This objective function is to be minimized, as formally formulated in Equation (5).

_{2}emissions rates for solar, wind, and natural gas systems, respectively (see Table 3).

_{2}and SO

_{2}emissions for a wind energy system. Their three objective functions were constrained with respect to the wind power output. The authors of [16] proposed a multi-energy system to be optimized where two objective functions were introduced: (1) daily operation costs and (2) emissions. Their energy system takes into account natural gas power and electricity. In [17], the authors optimized the power generation costs of solar power and wind power, where these two power sources are adversaries between each other. Finally, the work in [18] optimizes production wasting and power consumption as the objective functions, in a process of natural gas and oil power generation. Considering the fact that these power systems are not equal as the system we proposed in this research, the objective functions used in this paper are significantly different as well.

## 4. Climate Model

^{2}. In order to match with the previously commented units, measurements are converted to Watts, by multiplying the data by the equivalent area of all the available solar panels and dividing it by the total number of seconds in each month. Twenty thousand solar panels are considered for this study, each having an area of 1.65 m

^{2}, which is a common commercial surface. The number of seconds per month is calculated depending on whether a month has 28, 30 or 31 days. The wind speed is measured in miles per hour and only needs to be converted to m/s.

#### 4.1. Solar Radiation Prediction

#### 4.2. Wind Speed Prediction

## 5. Multi-Objective Optimization Evolutionary Algorithm

Algorithm 1: NSGA-II |

## 6. Experiments

## 7. Results and Discussion

#### 7.1. Spacing Results Analysis

#### 7.2. Hypervolume Data Analysis

## 8. Conclusions and Future Work

_{2}emissions. Solar and wind energies highly rely on weather, then a climate model was constructed using previous data from Oklahoma.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MOEA | Multi-Objective Evolutionary Algorithm |

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**Figure 1.**The proposed power production system for the State of Oklahoma. Each sub-system contributes in the generated power P, the production cost C and the CO

_{2}emissions E.

**Figure 2.**Basic configuration of a wind turbine. The circular surface covered by the three blades is called swept area.

Power Source | Capacity Factor (%) |
---|---|

Solar Power | 33 |

Wind Power | 43 |

Natural Gas Power | 87 |

Power Source | Production Cost ($/MWh) |
---|---|

Solar Power | 48.2 |

Wind Power | 33 |

Natural Gas Power | 15.5 |

Power Source | CO_{2} Emissions Rate (gr/KWh) |
---|---|

Solar Power | 49.91 |

Wind Power | 34.11 |

Natural Gas Power | 572 |

${\mathit{h}}_{\mathit{s}}$ | ${\mathit{h}}_{\mathit{w}}$ | ${\mathit{h}}_{\mathit{g}}$ | ${\mathit{P}}_{\mathit{g}}$ | |
---|---|---|---|---|

Maximum | 672 h | 672 h | 672 h | 7.07 MW |

Minimum | 240 h | 240 h | 240 h | 4.88 MW |

**Table 5.**Spacing and Hypervolume statistical results, including Average, Standard Deviation and Deviation percentage with respect to the average.

Algorithm | Spacing | Hypervolume | ||||
---|---|---|---|---|---|---|

Average | Std. Dev. | Deviation % | Average | Std. Dev. | Deviation % | |

NSGA-II | $6.8981\times {10}^{8}$ | $1.5647\times {10}^{8}$ | 22.68 | $2.1821\times {10}^{14}$ | $4.2554\times {10}^{12}$ | 1.95 |

L-NSGA-II | $7.5259\times {10}^{8}$ | $3.5883\times {10}^{8}$ | 47.67 | $2.1832\times {10}^{14}$ | $4.3454\times {10}^{12}$ | 1.99 |

MOEA/D | $1.5612\times {10}^{8}$ | $2.2108\times {10}^{7}$ | 14.16 | $1.8262\times {10}^{14}$ | $4.1948\times {10}^{12}$ | 2.29 |

NSGA-II | L-NSGA-II | MOEA/D | |
---|---|---|---|

Time | $1.007\pm 0.231$ | $0.541\pm 0.074$ | $0.395\pm 0.022$ |

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**MDPI and ACS Style**

Vargas-Hákim, G.-A.; Mezura-Montes, E.; Galván, E. Evolutionary Multi-Objective Energy Production Optimization: An Empirical Comparison. *Math. Comput. Appl.* **2020**, *25*, 32.
https://doi.org/10.3390/mca25020032

**AMA Style**

Vargas-Hákim G-A, Mezura-Montes E, Galván E. Evolutionary Multi-Objective Energy Production Optimization: An Empirical Comparison. *Mathematical and Computational Applications*. 2020; 25(2):32.
https://doi.org/10.3390/mca25020032

**Chicago/Turabian Style**

Vargas-Hákim, Gustavo-Adolfo, Efrén Mezura-Montes, and Edgar Galván. 2020. "Evolutionary Multi-Objective Energy Production Optimization: An Empirical Comparison" *Mathematical and Computational Applications* 25, no. 2: 32.
https://doi.org/10.3390/mca25020032