# Effective Realization of Multi-Objective Elitist Teaching–Learning Based Optimization Technique for the Micro-Siting of Wind Turbines

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## Abstract

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## 1. Introduction

- To check the effectiveness of the multi-objective teaching–learning based optimization technique.
- To determine the optimal locations of the wind turbines in a given land area of $2\mathrm{km}\times 2\mathrm{km}$ to achieve the maximum energy production while minimizing the total cost.
- To find the optimal layouts along with the optimal number of turbines in a given land area of $2\mathrm{km}\times 2\mathrm{km}$ with minimum fitness evaluation.

## 2. Materials and Methods

#### 2.1. The Analytical Model: Jensen’s Wake Modelling

#### 2.2. Fitness Evaluation

#### 2.2.1. Estimation of Wind Farm Cost

#### 2.2.2. Estimation of Wind Farm Power

#### 2.2.3. Evaluation of Fitness Function

#### 2.2.4. Calculation of Efficiency

#### 2.3. Elitist Teaching–Learning Based Optimization Algorithm

#### 2.3.1. Teacher Phase

#### 2.3.2. Learner Phase

#### 2.3.3. Evaluation of Fitness Function

#### 2.4. Wind Scenarios

## 3. Results and Discussion

#### 3.1. Mosetti et al. vs. WFAO-ETLBO

#### 3.1.1. Sensitivity Analysis for Initial Population

#### 3.1.2. Sensitivity of Teaching Factor

#### 3.2. Comparison Results of the MO-ETLBO

#### 3.2.1. Numerical Results for Scenario-I

^{−3}/kW as well as the power of 20495 kW obtained by 41 turbines and efficiency of 95.49%. Additionally, the saving in the total cost over the compared algorithms is shown in Table 4. On the other hand, the optimal configurations for the wind farm layout reported by prominent studies, Mosetti [1], Grady [2], and Mittal [3], versus those achieved by the MO-TLBO algorithms are visualized in Figure 11a–d. Figure 11e shows convergence plot for proposed optimal number of turbines for Scenario-I. The plot shows the convergence of the optimal number of turbines for each learner iteration by iteration. For randomly selected iteration numbers, at iteration 1, it can be seen that all the learners have different fitness values; at iterations 10, 20, and 30 the fitness values of each learner are improved but do not satisfy the convergence criteria, while at iteration number 41, all the learners are approaching the same fitness value and successfully satisfying the convergence criteria.

#### 3.2.2. Numerical Results of Scenario-II

^{−3}/kW as well as the power of 16913.86 kW obtained by 36 turbines and efficiency of 87.76%. Additionally, the saving in the total cost over the compared algorithms is shown in Table 7, where the distribution achieved by the MO_ETLBO algorithm finds the better layout as regarding the total cost compared to the other methods by 20.89%, 3.99%, 0.02%, 1.99%, 3.99%, 6.59%, 1.09%, 1.59%, 1.09%, 3.77%, 0.7%, and 9.19%, for Mosetti [1], Grady [2], Mittal [3], Feng [20], SSA [25], SBO [25], DE [25], GWO [25], WCA [25], BPSO-TVAC [11], L-SHADE [26], and Gao, respectively. On the other hand, the optimal configurations for the wind farm layout reported by prominent literatures, Mosetti [1], Grady [2], and Mittal [3], versus those achieved by the MO-ETLBO are visualized in Figure 12a–d. Figure 12e shows convergence plot for proposed optimal number of turbines for Scenario-II. The plot shows the convergence of the optimal number of turbines for each learner, iteration by iteration. For randomly selected iteration numbers, at iteration 1, it can be seen that all the learners have significantly different fitness values, and at iterations 43, 86, and 129 the fitness values of each learner are improved but do not satisfy the convergence criteria. At iteration number 172, all the learners are approaching the same fitness value and successfully satisfying the convergence criteria.

#### 3.2.3. Numerical Results of Scenario-III

^{−4}/kW as well as power of 31,086.86 kW obtained by 38 turbines and efficiency of 87.59%. Additionally, the saving in the total cost over the compared algorithms is shown in Table 5, where the distribution achieved by the proposed EOPS algorithm finds the layout which can save the total cost compared to the other methods by 16.89%, 1.51%, 1.86%, 1.38%, 2.28%, 2.08%, 1.08%, 1.18%, 2.71%, and 0.7% for Mosetti [1], Grady [2], Mittal [3], Feng [20], SSA [25], SBO [25], DE [25], GWO [25], WCA [25], BPSO-TVAC [11], and L-SHADE [26], respectively. For further validation, the fitness function’s behavior is illustrated by depicting the convergence behavior over the searching period as in Figure 13e. Additionally, the optimal wind turbine positioning obtained by Mosetti [1], Grady [2], and Mittal [3] versus that achieved by the MO-ETLBO algorithms is visualized in in Figure 13a–d. Figure 13e shows convergence plot for proposed optimal number of turbines for Scenario-III. The plot shows the convergence of the optimal number of turbines for each learner iteration by iteration. For randomly selected iteration numbers, at iteration 1, it can be seen that all the learners have different fitness values; at iterations 16, 33, and 49 the fitness values of each learner are improved but do not satisfy the convergence criteria. At iteration number 66, all the learners are approaching the same fitness value and successfully satisfying the convergence criteria. Finally, it can be concluded that the proposed MO-ETLBO algorithm has a high capability in solving the challenging combinatorial wind farm layout problems.

## 4. Conclusions

- It has been efficiently implemented to solve the wind farm layout discrete optimization problem by considering the multiple objectives optimization.
- The MO-ETLBO provides superior and promising results for a single scenario of the WFL-DO problem.
- It can achieve a better convergence and well-distributed set of non-dominated solutions when dealing with multiple objectives of the wind farm layout discrete optimization problem. These characteristics are useful and can provide reasonable layouts for the decision-maker to extract the best compromise solution or operating solution from the available finite alternatives based on decision-making regulations or total cost obligations.
- It can be concluded that the present study can provide a twofold contribution. The first one is incorporating the pattern search technique, and the second one is caused by simultaneously considering many objectives of the wind farm layout optimization problem.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviation

TLBO | Teaching–learning Based Optimization |

ETLBO | Elitist Teaching–learning Based Optimization |

${u}_{0}$ | undisturbed/freestream wind speed |

${C}_{T}$ | thrust coefficient |

A | interference coefficient/induction/perturbation coefficient |

${r}_{r}$ | rotor radius |

${r}_{1}$ | downstream rotor radius |

D | rotor diameter |

${z}_{0}$ | surface roughness |

$Z$ | hub/axis height of turbine |

X | wind downstream distance |

$\theta $ | wake spread angle |

$\alpha $ | entrainment constant |

K.E. | kinetic energy |

N | number of turbines |

P | actual power of wind turbine |

${P}_{ideal}$ | ideal power of wind turbine |

$\rho $ | density |

${u}_{wmwe}$ | velocity of the turbine with multiple wake effect |

$\eta $ | efficiency of wind farm |

${P}_{tot,wmwe}$ | total power with multiple wake effects |

${P}_{tot,wowe}$ | total power without wake effects |

L_{j,k,i} | ${k}^{\mathrm{th}}$ learner of ${j}^{\mathrm{th}}$ subject at ${i}^{\mathrm{th}}$ iteration |

f (L_{j,k,i}) | fitness value of ${k}^{\mathrm{th}}$ learner of ${j}^{\mathrm{th}}$ subject at ${i}^{\mathrm{th}}$ iteration |

T_{F} | teaching factor |

L^{′}_{j,k,i} | fitness value of ${k}^{\mathrm{th}}$ modified learner of ${j}^{\mathrm{th}}$ subject at ${i}^{\mathrm{th}}$ iteration in the teacher phase |

Difference_Mean_{j,k,i} | difference mean of ${k}^{\mathrm{th}}$ learner of ${j}^{\mathrm{th}}$ subject at ${i}^{\mathrm{th}}$ iteration |

L^{″}_{j,k,i} | fitness value of ${k}^{\mathrm{th}}$ modified learner of ${j}^{\mathrm{th}}$ subject at ${i}^{\mathrm{th}}$ iteration in the student phase |

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**Figure 3.**Proposed layout and its binary representation by Mosetti et al. [1].

**Figure 6.**Wind rose (frequency distribution) for (

**a**) Wind Scenario-I; (

**b**) Wind Scenario-II; (

**c**) Wind Scenario-III.

**Figure 7.**Difference plot with coarse grid meshing using 26 numbers of turbines (

**left**); convergence plot with coarse grid meshing using 26 numbers of turbines (

**right**).

Number of Learners | Power (KW) | Fitness Value | Efficiency (%) |
---|---|---|---|

20 | 12,229 | 0.0016366 | 90.73 |

50 | 12,361 | 0.0016196 | 91.71 |

100 | 12,495 | 0.0016014 | 92.71 |

150 | 12,427 | 0.0016132 | 92.20 |

Teaching Factor | Initial No. of Learners | Power (KW) | Fitness Value | Efficiency (%) | Converged No. of Iterations |
---|---|---|---|---|---|

1 | 100 | 12,374 | 0.0016152 | 91.89 | 30 |

2 | 12,267 | 0.0016309 | 91.05 | 50 | |

Rand [1 2] | 12,371 | 0.0016168 | 91.81 | 56 |

Elite Size (%) | Optimal Fitness Value | Efficiency (%) | Power (KW) | Optimal No. of Turbines | Converged No. of Iterations | Simulation Time (Sec) |
---|---|---|---|---|---|---|

0 | 0.0014256 | 92.99 | 19,284 | 40 | 44 | 6.3 |

5 | 0.0014285 | 92.11 | 20,056 | 42 | 23 | 3.3 |

10 | 0.0014329 | 93.38 | 18,395 | 38 | 16 | 2.9 |

15 | 0.0014262 | 92.95 | 19,275 | 40 | 14 | 1.8 |

20 | 0.0014539 | 92.04 | 18,131 | 38 | 13 | 1.6 |

Statistical Measures | TLBO |
---|---|

Minimum | 1.3315 × 10^{−3} |

Mean | 1.3152 × 10^{−3} |

Median | 1.3136 × 10^{−3} |

Worst | 1.3532 × 10^{−3} |

Std. Dev. | 4.3342 × 10^{−3} |

Algorithm | Number of Turbines | Fitness Value × 10^{−3} ($/KW) | Total Power (KW/Year) | Efficiency (%) | Converged Number of Iterations | Saving by Present Study (%) | Average Power Produced by Unit Turbine (KW/Year) |
---|---|---|---|---|---|---|---|

Mosetti [1] | 26 | 1.619 | 12,352 | 91.65 | 400 | 17.75 | 475.08 |

Grady [2] | 30 | 1.5436 | 14,310 | 92.015 | 1203 | 13.74 | 477.00 |

Mittal [3] | 44 | 1.3602 | 21,936 | 96.17 | NA | 2.11 | 498.55 |

MO-ETLBO | 41 | 1.315 | 20,495 | 95.49 | 41 | - | 499.88 |

Statistical Measures | TLBO |
---|---|

Minimum | 1.5271 × 10^{−3} |

Mean | 1.5352 × 10^{−3} |

Median | 1.5336 × 10^{−3} |

Worst | 1.5452 × 10^{−3} |

Std. Dev. | 6.2507 × 10^{−3} |

Algorithm | Number of Turbines | Fitness Value × 10^{−3} ($/KW) | Total Power (KW/Year) | Efficiency (%) | Converged Number of Iterations | Saving by Present Study (%) | Average Power Produced by Unit Turbine (KW/Year) |
---|---|---|---|---|---|---|---|

Mosetti [1] | 19 | 1.736 | 9244 | NA | 350 | 12.03 | 486.53 |

Grady [2] | 39 | 1.567 | 17,220 | 85.174 | 3000 | 2.55 | 441.54 |

Mittal [3] | 38 | 1.5273 | 17259 | 87.61 | NA | 0.01 | 454.18 |

Feng [20] | 39 | 1.547 | 17,406 | NA | NA | 1.29 | 446.31 |

SSA [25] | 39 | 1.567 | 17,175 | 85 | NA | 2.55 | 440.38 |

SBO [25] | 40 | 1.593 | 17,254 | 82 | NA | 4.14 | 431.35 |

DE [25] | 40 | 1.538 | 17,877 | 86 | NA | 0.71 | 446.93 |

GWO [25] | 40 | 1.543 | 17,817 | 86 | NA | 1.03 | 445.43 |

WCA [25] | 40 | 1.538 | 17,878.32 | 86.22 | NA | 0.71 | 446.96 |

BPSO-TVAC [22] | 35 | 1.5648 | 15,796 | 87.06 | NA | 2.41 | 451.31 |

L-SHADE [26] | 40 | 1.5341 | 17,920 | 86.42 | NA | 0.46 | 448.00 |

Gao [15] | 39 | 1.619 | 15,333 | 77.83 | NA | 5.68 | 393.15 |

MO-ETLBO | 36 | 1.5271 | 16,913.86 | 87.76 | 41 | - | 469.83 |

Statistical Measures | MO-ETLBO |
---|---|

Minimum | 8.2517 × 10^{−4} |

Mean | 8.2782 × 10^{−4} |

Median | 8.2716 × 10^{−4} |

Worst | 8.2064 × 10^{−4} |

Std. Dev. | 2.0059 × 10^{−6} |

Algorithm | Number of Turbines | Fitness Value × 10^{−3} ($/KW) | Total Power (KW/Year) | Efficiency (%) | Converged Number of Iterations | Saving by Present Study (%) | Average Power Produced by Unit Turbine (KW/Year) |
---|---|---|---|---|---|---|---|

Mosetti [1] | 15 | 0.99405 | 13,460 | NA | 400 | 16.99 | 897.33 |

Grady [2] | 39 | 0.8403 | 31,850 | 86.619 | 1203 | 1.80 | 816.67 |

Mittal [3] | 41 | 0.84379 | 21,936 | 86.729 | NA | 2.21 | 535.02 |

Feng [25] | 39 | 0.839 | 32,096 | NA | NA | 1.65 | 822.97 |

SSA [25] | 41 | 0.848 | 33,099 | 85 | NA | 2.69 | 807.29 |

SBO [25] | 40 | 0.846 | 32,501.28 | 85 | NA | 2.46 | 812.53 |

DE [25] | 40 | 0.836 | 32,901.41 | 86 | NA | 1.30 | 822.54 |

GWO [25] | 38 | 0.837 | 31,498 | 86 | NA | 1.41 | 828.89 |

WCA [25] | 40 | 0.833 | 33,005 | 87 | NA | 0.94 | 825.13 |

BPSO-TVAC [22] | 46 | 0.8523 | 36,433 | 82.76 | NA | 3.18 | 792.02 |

L-SHADE [26] | 39 | 0.8322 | 32,351 | 86.68 | NA | 0.84 | 829.51 |

MO-ETLBO | 38 | 0.82517 | 31,086.86 | 87.59 | 66 | - | 839.13 |

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## Share and Cite

**MDPI and ACS Style**

Hussain, M.N.; Shaukat, N.; Ahmad, A.; Abid, M.; Hashmi, A.; Rajabi, Z.; Tariq, M.A.U.R.
Effective Realization of Multi-Objective Elitist Teaching–Learning Based Optimization Technique for the Micro-Siting of Wind Turbines. *Sustainability* **2022**, *14*, 8458.
https://doi.org/10.3390/su14148458

**AMA Style**

Hussain MN, Shaukat N, Ahmad A, Abid M, Hashmi A, Rajabi Z, Tariq MAUR.
Effective Realization of Multi-Objective Elitist Teaching–Learning Based Optimization Technique for the Micro-Siting of Wind Turbines. *Sustainability*. 2022; 14(14):8458.
https://doi.org/10.3390/su14148458

**Chicago/Turabian Style**

Hussain, Muhammad Nabeel, Nadeem Shaukat, Ammar Ahmad, Muhammad Abid, Abrar Hashmi, Zohreh Rajabi, and Muhammad Atiq Ur Rehman Tariq.
2022. "Effective Realization of Multi-Objective Elitist Teaching–Learning Based Optimization Technique for the Micro-Siting of Wind Turbines" *Sustainability* 14, no. 14: 8458.
https://doi.org/10.3390/su14148458