# Aquila Optimization Algorithm for Wind Energy Potential Assessment Relying on Weibull Parameters Estimation

^{1}

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## Abstract

**:**

^{2}), mean absolute error, and wind production deviation. The simulation results declare that the proposed AO optimization algorithm offers greater accuracy than several optimization algorithms in the literature for estimating the Weibull parameters. Furthermore, statistical analysis of the compared methods demonstrates the high stability of the AO algorithm. Thus, the proposed AO has greater accuracy and more stability in the obtained outcomes for Weibull parameters and wind energy calculations.

## 1. Introduction

^{2}), root mean square error (RMSE), mean absolute error (MAE), correlation coefficient (R), standard deviation, wind energy error (WEE), and Monte Carlo simulation [15,23,24]. Where iterative features have been used, the analytical approaches still exhibit their limitations. The wind speed distribution cannot always be represented efficiently due to analytical approaches. As a result, the optimum Weibull distribution parameters may be determined using heuristic optimization methods that employ the goal function to obtain the best fitting values. In this context, the mean squared, RMSE, MAE, and MSE can be used as objective functions to obtain optimal values of Weibull parameters.

- Analysis of the distribution of real data obtained for Zafarana and Shark El-Ouinate sites, Egypt.
- Implementation of MLM, EM, MM, and EPFM as analytical methods for optimal fitting based on R
^{2}, RMSE, MAE, and wind production deviation (WPD). - Analysis of novel intelligence optimization methods called PSO and AO for measuring the accuracy of fitting by statistical tests.

## 2. Wind Speed Distribution Functions

#### 2.1. Analytical Methods

#### 2.2. Heuristic Optimization Methods

#### 2.3. Aquila Optimizer Model

- 1.
- Initialization process: the solution of any optimization process starts with random solutions of candidate solution (X
_{ij}) in the range between upper (up_{j}) and lower limits (lb_{j}) which is expressed as follows:$${X}_{ij}=rand\times \left(u{p}_{ij}-l{p}_{ij}\right)+l{p}_{j},$$ - 2.
- Firstly, the Aquila explores any prey in the search space. This exploration process is accomplished at high soar, which is called expanded exploration in the search space. Once the Aquila explores prey, it drops with a vertical stoop to grab the prey. This behavior is mathematically as follows:$${X}_{1}\left(t+1\right)={X}_{b}\left(t\right)\times \left(1-\left(\frac{t}{T}\right)\right)+\left({X}_{M}\left(t\right)-{X}_{b}\left(t\right)\times rand\right),$$$${X}_{M}\left(t\right)=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{X}_{i}\left(t\right),$$
- 3.
- The second process is the most used by Aquila, and the hunting process is at contour flight with a short glide attack. Therefore, the Aquila is close to the hounded prey, which leads to narrow exploration in the search space. This process is modeled as:$${X}_{2}\left(t+1\right)={X}_{b}\left(t\right)\times Levy\left(D\right)+{X}_{R}\left(t\right)+\left(y-x\right)\times rand,$$$$Levy\left(D\right)=s\times \frac{u\times \omega}{{\left|\upsilon \right|}^{1/\beta}},$$$$\omega =\left(\frac{\mathcal{T}\left(1+\beta \right)\times \mathrm{sin}\left(\frac{\pi \beta}{2}\right)}{\mathcal{T}\left(\frac{1+\beta}{2}\right)\times \beta \times {2}^{\left(\frac{\beta -1}{2}\right)}}\right),$$$$y=r\times \mathrm{cos}\left(\theta \right),$$$$x=r\times \mathrm{sin}\left(\theta \right),$$$$r={r}_{1}+U\times {D}_{1},$$$$\theta =-W\times {D}_{1}+{\theta}_{1,}$$$${\theta}_{1}=3\times \frac{\pi}{2},$$
- 4.
- The third process is the expanded exploitation (X
_{3}) that belongs to any flooring prey that have a slow escape response. Aquila is flying at a low level, and once it selects its prey, it grabs progressively on the prey with a slow decent attack. This step is mathematically presented as in the following equation:$${X}_{3}\left(t+1\right)=\left({X}_{b}\left(t\right)-{X}_{M}\left(t\right)\right)\times \alpha -rand+\left(rand\times \left(up-lp\right)+lp\right)\times \delta ,$$ - 5.
- The fourth process (X
_{4}) is when the Aquila walks on the land and captures prey by pulling it. It is used for dealing with large prey, which is described as a narrowed exploitation step. This step is mathematically modeled as:$${X}_{4}\left(t+1\right)=QF\times {X}_{b}\left(t\right)-\left({G}_{1}\times X\left(t\right)\times rand\right)-{G}_{2}\times Levy\left(D\right)+rand\times {G}_{1},$$$$QF\left(t\right)={t}^{\frac{2\times rand()-1}{{\left(1-T\right)}^{2}},}$$$${G}_{1}=2\times rand()-1,$$$${G}_{2}=2\times \left(1-\frac{t}{T}\right).$$

## 3. Assessment of Fit Performance

#### 3.1. Root Mean Square Error (RMSE)

#### 3.2. Determination Coefficient (R^{2})

^{2}), which measures the variance or the efficiency of the method. R

^{2}is defined by:

#### 3.3. Mean Absolute Error (MAE)

#### 3.4. Wind Production Deviation (WPD)

## 4. Results and Discussion

#### 4.1. Wind Speed Data

#### 4.2. Goodness of Fit

^{2}of 0.9913570. On the other side, the other heuristic optimizer, PSO, finds an R

^{2}of 0.9910771. Nevertheless, the analytical methods such as MLM, EMP, MM, and EPF find R

^{2}of 0.990696, 0.98849, 0.972545, and 0.958787, respectively. Added to that, the AO algorithm provides the least value of RMSE of 0.0065105. On the other side, the other heuristic algorithm, PSO, finds an RMSE of 0.0065151. Nevertheless, the analytical methods such as MLM, EMP, MM, and EPF find RMSE of 0.006755, 0.007513, 0.011604, and 0.014217, respectively.

^{2}of 0.942592. On the other side, the other heuristic algorithm, PSO, finds an R

^{2}of 0.941294. Nevertheless, the analytical methods such as EPF, EMP, MLM, and MM find R

^{2}of 0.896893, 0.892958, 0.888387, and 0.892456, respectively.

#### 4.3. Statistical Results for PSO and AO for Estimating Weibull Parameters

^{−5}and 2.51561 × 10

^{−6}, respectively.

^{−5}and 2.4 × 10

^{−6}, respectively.

`has`the fastest convergence compared to the other since it provides its best solution through less than 40% of the total number of iterations.

## 5. Power Curve Modeling

_{1}–g

_{9}) are considered to find the best model for each site. In order to compare the performance of these nine models in estimating the energy from a wind turbine, the error ($\mathsf{\epsilon}$) between the actual energy density (${E}_{actual}$) and estimated energy (${E}_{fitted}$) is calculated by the following expression [21]:

_{2}) provides the minimum error of energy density of 5.728%. After that, the fourth (g

_{4}) and seventh (g

_{7}) models come with an error of energy density of 6.005 and 6.167%, respectively. On the other side, all the models illustrate close errors where there is no great divergence in the results, as the worst model provides an error of energy density of 6.593%. From Figure 10, at the Shark El-Ouinate site, the eigth model (g

_{8}), the cubic form, provides the minimum error of energy density of 13.2%. After that, the ninth (g

_{4}) and seventh (g

_{7}) models come with an error of energy density of 13.6 and 16.6%, respectively.

## 6. Conclusions

^{2}, the least value of RMSE, and the least value of MAE for tuning Weibull PDF parameters. Additionally, the mean objective obtained by the AO algorithm is better than the others. Further, the AO algorithm is the most robust technique as it achieves the least standard deviation and standard error compared to the others. Furthermore, the AO algorithm demonstrates the greatest stability characteristics as its histogram of the obtained objectives is better than the PSO algorithm. Moreover, the AO algorithm presents the fastest convergence compared to the other since it provides its best solution through the first and second site parameters calculation, respectively. Furthermore, different models are analyzed to deduce the nonlinear relationship between the wind output power and the speed, where the error of wind energy density between actual and estimated is greatly minimized using the cubic model. A suggestion for future work is to obtain a rank for each method, where the mean wind speed and its standard deviation are estimated for a greater number of periods.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AO | Aquila optimizer |

EM | Empirical method |

EPFM | Energy pattern factor method |

LSM | Least square method |

MAE | Mean absolute error |

MLM | Maximum likelihood method |

MM | Moment method |

MMOM | Modified method of moment |

PD | Power density method |

Probability density function | |

PSO | Particle warm optimization |

R^{2} | Determination coefficient |

RMSE | Root mean square error |

WEE | Wind energy error |

WEIM | Wind energy intensification method |

WPD | Wind production deviation |

R | Correlation coefficient |

k | Shape factor |

c | Scaling factor |

$f\left(v\right)$ | Weibull probability distribution function |

$F\left(v\right)$ | Weibull cumulative distribution function |

n | Number of bins performed |

${v}_{i}$ | Peed in time step i from time series data |

$\sigma $ | Standard deviation of wind speed |

${y}_{i}$ | Probability distribution of Weibull |

${x}_{i}$ | Probability distribution of actual wind speed |

$\overline{y}$ | Probability of the mean wind speed |

$\rho $ | Air density in (kg/m^{3}) |

STE | Standard error |

STD | Standard deviation |

${a}_{1}$, ${a}_{2}$, ${a}_{3}$, ${a}_{4}$, and ${a}_{5}$ | Polynomial coefficients |

${v}_{in}$ | Cut-in speed |

${v}_{r}$ | Rated speed |

${v}_{out}$ | Cut-out speed |

${E}_{actual}$ | Actual energy density |

${E}_{fitted}$ | Estimated energy |

${p}_{m}\left({v}_{i}\right)$ | Wind output power of the model (m) at wind speed $\left({v}_{i}\right)$ |

$A$ | Swept area |

${f}_{actual}\left({v}_{i}\right)$ | Frequency of wind speed from histogram at speed $\left({v}_{i}\right)$ |

${f}_{fitted}\left({v}_{i}\right)$ | Weibull distribution of wind speed $\left({v}_{i}\right)$ |

T | Maximum number of iterations |

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Site | Zafarana | Shark El-Ouinat |
---|---|---|

Wind Turbine Type | Nordex N43 | Nordex-N100 |

Rated power (P_{r}) (kw) | 600 | 2500 |

Hub height (m) | 55 | 100 |

Cut-in wind speed (m/s) | 2.5 | 3 |

Cut-off wind speed (m/s) | 25 | 25 |

Rated wind speed (m/s) | 15 | 12.5 |

Parameters | Zafarana (m/s) | Shark El-Ouinat (m/s) |
---|---|---|

Mean | 7.1468 | 6.4966 |

Maximum | 15.897 | 13.508 |

Minimum | 1.8 | 0.077 |

Standard deviation | 1.8666 | 2.5139 |

Variance | 3.4844 | 6.3198 |

**Table 3.**Statistical analysis: Zafarana site, 2019 [34].

Method | C | K | R^{2} | RMSE | MAE | WPD |
---|---|---|---|---|---|---|

EPF | 7.938554 | 3.537746 | 0.958787 | 0.014217 | 0.01132 | −0.79318 |

MM | 7.822509 | 4.589375 | 0.972545 | 0.011604 | 0.007059 | −0.8114 |

EMP | 7.852397 | 4.297248 | 0.98849 | 0.007513 | 0.005485 | −0.80762 |

MLM | 7.852799 | 4.198748 | 0.990696 | 0.006755 | 0.005216 | −0.80692 |

PSO | 7.8428736 | 4.1682588 | 0.9910771 | 0.0066151 | 0.0051341 | −0.80742 |

AO | 7.8381818 | 4.1138222 | 0.9913570 | 0.0065105 | 0.0051603 | −0.80716 |

Method | C | K | R^{2} | RMSE | MAE | WPD |
---|---|---|---|---|---|---|

MM | 7.295115 | 2.807504 | 0.892456 | 0.014719 | 0.011650 | −0.71425 |

MLM | 7.266921 | 2.790555 | 0.888387 | 0.014996 | 0.011920 | −0.71669 |

EMP | 7.295451 | 2.804119 | 0.892958 | 0.014685 | 0.011638 | −0.71404 |

7.298281 | 2.775332 | 0.896893 | 0.014413 | 0.011540 | −0.71221 | |

PSO | 7.794983 | 2.685606 | 0.941294 | 0.010875 | 0.009005 | −0.64316 |

AO | 7.715474 | 2.662966 | 0.942592 | 0.010755 | 0.008761 | −0.65233 |

STD | STE | Best | Worst | Mean | |
---|---|---|---|---|---|

PSO | 0.005307261 | 0.00075056 | 0.000656396 | 0.022794503 | 0.006801911 |

AO | 1.7788 × 10^{−5} | 2.51561 × 10^{−6} | 0.000635809 | 0.000717186 | 0.000649055 |

STD | STE | Best | Worst | Mean | |
---|---|---|---|---|---|

PSO | 0.004278 | 0.000605 | 0.001774 | 0.020585 | 0.006692 |

AO | 1.69 × 10^{−5} | 2.4 × 10^{−6} | 0.001735 | 0.001827 | 0.001749 |

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**MDPI and ACS Style**

Abou El-Ela, A.A.; El-Sehiemy, R.A.; Shaheen, A.M.; Shalaby, A.S.
Aquila Optimization Algorithm for Wind Energy Potential Assessment Relying on Weibull Parameters Estimation. *Wind* **2022**, *2*, 617-635.
https://doi.org/10.3390/wind2040033

**AMA Style**

Abou El-Ela AA, El-Sehiemy RA, Shaheen AM, Shalaby AS.
Aquila Optimization Algorithm for Wind Energy Potential Assessment Relying on Weibull Parameters Estimation. *Wind*. 2022; 2(4):617-635.
https://doi.org/10.3390/wind2040033

**Chicago/Turabian Style**

Abou El-Ela, Adel A., Ragab A. El-Sehiemy, Abdullah M. Shaheen, and Ayman S. Shalaby.
2022. "Aquila Optimization Algorithm for Wind Energy Potential Assessment Relying on Weibull Parameters Estimation" *Wind* 2, no. 4: 617-635.
https://doi.org/10.3390/wind2040033