This work takes the bionic bamboo tower (BBT) of 2 MW wind turbine as the target, and the non-dominated sorting genetic algorithm (NSGA-II) is utilized to optimize its structural parameters. Specifically, the objective functions are deformation and mass. Based on the correlation analysis, the target optimization parameters were determined. Furthermore, the Kriging model of the BBT was established through the Latin Hypercube Sampling Design (LHSD). Finally, the BBT structure is optimized with multiple objectives under the constraints of strength, natural frequency, and size. The comparison shows that the optimized BBT has an advantage in the Design Load Case (DLC). This advantage is reflected in the fact that the overall stability of the BBT has increased by 2.45%, while the displacement of the BBT has decreased by 0.77%. In addition, the mass of the tower is decreased by 1.49%. Correspondingly, the steel consumption of each BBT will be reduced by 2789 Kg. This work provides a scientific basis for the structural design of the tower in service.

Tower is the principal supporting component of the wind turbine, which structural strength will directly affect the power generation efficiency of the wind turbine. As the tower tends to towering development, and the manufacturing cost of the tower keeps increasing, higher requirements are put forward for the structural design of the tower in service [

The strength and manufacturing cost of the tower is mainly determined by its structural parameters [

Similarly, combining different optimization algorithms with finite element analysis (FEA) can lead to reasonable optimization solutions [

This work presents an effective approach to minimize the mass of the BBT and improve its safety simultaneously. The task obtains a set of reasonable structural parameters through NSGA-II. Correspondingly, it is verified by using the FEA that the optimum proposals are possible and effective. Ultimately, this work achieved the multi-objective optimization of the BBT, which provided a reference for the tower's structural design.

In the previous work, our team proposed the BBT structure [

The BBT's main body is composed of 6 sections of the tower tube connected by flanges.

It is made from hot-rolled steel which is welded together circumferentially and longitudinally. Flanges connect the beginning and the end of each section. The BBT, in this case, is made of Q345FT, with a modulus of elasticity ^{11} MPa, a yield strength σ = 345 MPa, and density ρ = 7850 kg/m^{3}.

To investigate the effect of structural parameters on the deflection of BBT, its finite element model was established. Specifically, the mesh size was determined to be 400 mm by mesh convergence verification, and 70041 cells and 142,243 nodes were obtained. Hence, more computing resources are devoted to solving analytical problems. In addition, the Multi-point constraints are used at flange connection surfaces. Finally, a concentrated mass point is adopted instead of the caabin and wind wheel.

In general, the possibility of structural damage to the tower is higher under extreme operating conditions [

Conditions | DLC1.3 | DLC3.2 | DLC6.1 |
---|---|---|---|

_{x} |
−4924.9 | 2628.9 | −3328.4 |

_{y} |
−116.2 | −561.5 | −868.5 |

_{z} |
678.9 | 338.8 | 889.4 |

_{x} |
19.2 | −84.9 | −511.2 |

_{y} |
−13.3 | 429.8 | 8.32 |

_{z} |
1056.2 | −1231.7 | 1425.8 |

Obtaining the response of the BBT is a prerequisite for optimal design under the initial structural parameters. Furthermore, the structural parameters directly affect stress, displacement, inherent frequency, and fatigue life. Therefore, the present work applies FEA to investigate the stresses and displacements of the tower in extreme conditions. Ignoring the effects of their frequencies and fatigue.

Specifically, a Q345FT is assigned to the material properties of the BBT structure. The coordinate system of the top of the BBT is established [

Stress/MPa | Displacement/mm | |
---|---|---|

DLC1.3 | 54.99 | 129.30 |

DLC3.2 | 94.43 | 465.30 |

DLC6.1 | 116.98 | 715.76 |

In

In practical optimization, if all the structural parameters of the BBT are used as design variables, the number of sample points will be enormous. Thus, correlation analysis was used to screen the parameters that retained a significant degree of influence on the mass and displacement. The sample size can be reduced, and the solution accuracy can be improved by eliminating the parameters with less impact.

In the work, the input is the BBT's structural parameter and the output is the response of the stress and mass of the BBT. Specifically, the correlation coefficient is denoted as

In _{XY}_{X}_{Y}

The structural parameters include the diameter

The correlation coefficient reflects the degree of correlation between the parameter and the response. Based on the correlation analysis, 4 parameters were filtered out, the most relevant in each response, and the values were listed in

Parameter | Displacement | Parameter | Mass |
---|---|---|---|

_{2} |
−0.574 | _{1} |
0.329 |

_{1} |
−0.502 | _{1} |
0.323 |

_{3} |
−0.376 | _{8} |
0.260 |

_{4} |
−0.137 | _{7} |
0.248 |

According to _{2}, the top diameter of the second section _{1}, the top diameter of the fourth section _{3}, and the bottom thickness of the first section _{1} were used as the optimized design variables.

_{1}/mm |
_{1}/mm |
_{2}/mm |
_{3}/mm |
|||
---|---|---|---|---|---|---|

1 | 42.000 | 1926.522 | 1705.041 | 1703.562 | 166789.0 | 789.28 |

2 | 43.344 | 2010.270 | 1893.951 | 1690.974 | 175639.2 | 710.20 |

3 | 37.968 | 1950.450 | 1944.327 | 1615.446 | 166034.2 | 770.16 |

… | … | … | … | … | … | … |

21 | 46.032 | 1938.486 | 1931.733 | 1741.326 | 175754.5 | 715.36 |

22 | 39.984 | 1842.774 | 1730.229 | 1766.502 | 161758.6 | 848.09 |

23 | 44.016 | 1806.882 | 1843.575 | 1753.914 | 165831.3 | 847.38 |

The optimization variables were determined based on correlation analysis. Furthermore, the Kriging model, LHSD, and NSGA-II were combined to achieve multi-objective optimization of the BBT structure.

Kriging model consists of two parts: regression model and correlation model, which has high fitting accuracy, and its mathematical model is shown in

In which

Here

Based on the correlation analysis, _{1}, _{1}, _{2}, and _{3} were determined as the optimization variables. The Latin Hypercube Test was conducted with the mass (

The Kriging model is constructed based on ^{2}. To obtain a satisfactory Kriging model, the value of R^{2} should be close to 1. By solving, the R^{2} values of mass and displacement were obtained as 0.98 and 0.95, respectively, which are close to 1. This indicates that the accuracy of the model meets the requirements. However, there is a deviation in R^{2} [

It is shown that there is a high similarity between the predicted and the simulated values in the Kriging model (

The optimized design variables were determined based on correlation analysis.

The target is to minimize the total mass and displacement of the BBT.

Design constraints include the following 3 aspects:

Strength constraint

To ensure the BBT safety, the maximum stress σ max under external loads should be less than the material yield

In _{s}_{s}

Frequency constraint

The resonance between the wind rotor and the BBT when 1P (rotor rotation frequency, ω_{1P}) or 3P (blade passing frequency, ω_{3P}) is close to the natural frequency of the BBT, which lead to safety accidents. For this reason, the natural frequency of the BBT is specified to avoid 1P and 3P turbine excitation frequency ranges [

Size constraint

According to the actual working condition requirements, the BBT structure parameters should be taken within a certain range, and its parameter variation range is shown in

Parameter/mm | _{1} |
_{2} |
_{3} |
_{1} |
---|---|---|---|---|

Initial | 2001.0 | 1873.5 | 1732.0 | 42.0 |

Lower limit | 1800.9 | 1686.2 | 1558.8 | 37.8 |

Upper limit | 2100.0 | 2001.0 | 1873.5 | 46.2 |

NSGA-II was used to solve the Kriging model. Specifically, the initial population is 100, the genetic algebra is 600, the crossover probability is 0.9, and the mutation probability is 0.1. Further, the flowchart outlining the procedure for optimization is shown in

After optimization, a reasonable set of structural parameters was obtained and is listed in

Parameter/mm | _{1} |
_{2} |
_{3} |
_{1} |
---|---|---|---|---|

Initial | 2001 | 1873.5 | 1732 | 42 |

Optimal | 2004.8 | 1901.2 | 1776.4 | 37.9 |

_{1}, _{2}, and _{3} of the BBT after optimization slightly increased compared with the initial value, while the thickness of the first section of the tower _{1} decreased. To further explore the influence of the optimized structural parameters on the overall quality and the quality of each section of the BBT. Combined with

Initial/Kg | Optimal/Kg | Differentials/Kg | |
---|---|---|---|

1 | 52992 | 49208 | −3784 |

2 | 33237 | 33261 | 24 |

3 | 31279 | 31524 | 245 |

4 | 25216 | 25713 | 497 |

5 | 16909 | 17138 | 229 |

6 | 12719 | 12719 | 0 |

Total | 172352 | 169563 | −2789 |

What needs to be pointed out is that static strength refers to the power of the BBT structure to resist deformation and stress response under the external load, and it is the primary factor that reflects the structural performance of the BBT. To identify the advantages of the optimization tower, the FEA was applied to explore the static performance of BBT under DLC6.1 and compare it with the initial static performance, the results of which are shown in

Initial/mm | Optimal/MPa | |
---|---|---|

Maximum displacement | 715.76 | 710.28 |

Maximum stress | 116.98 | 127.10 |

Indeed, the maximum displacement of BBT decreases by 0.77%, and the maximum stress increases by 8.65% after optimization (

In practice, due to the influence of the tower's initial structural defects (geometric defects and structural defects), the structural instability is usually caused by the failure of the extreme point of the load. For this reason, nonlinear buckling analysis is used to explore the stability of the BBT and consider the structural nonlinearity and the effects of initial imperfections to obtain a realistic buckling load [

When the FEA is used for nonlinear buckling analysis, the gravity of the cabin, wind wheel, and tower is not considered, and axial load along the opposite direction of the z-axis is applied on the top of the BBT, and the bottom is fully constrained. First, set the mesh size to 400 mm, the bonded is used at the flange connection, and the flange connection surface is in the form of MPC. Secondly, turn on the large deflection and weak springs in the ANSYS environment to perform static analysis to solve the element stress stiffness matrix of the BBT. Next, add a nonlinear buckling analysis module, and expand the results after the study is completed to obtain the First-order buckling mode and buckling factor of the BBT. Finally, the First-order modes and buckling factors of the two towers are shown in

For

The 2 MW wind turbine BBT was taken as a structure optimization case. LHSD explored the influence between design variables and objective functions. Subsequently, a Kriging model of BBT mass and deformation was established as the basis of the multi-objective optimization of the tower. The NSGA-II was employed to find the optimal solution for mass and deformation. Based on the FEA, the static and dynamic performance advantages of the optimized BBT are considered comprehensively, and the following conclusions are drawn:

After optimization, the mass of the BBT decreased by 1.49% compared with the initial, and the corresponding steel consumption was reduced by 2789 Kg. Specifically, the reduced mass is concentrated in the first section of the tower, which provides a reference scheme for the tower's structural design in service.

Under DLC 6.1, the displacement of the optimized BBT decreases by 0.77%. On the other hand, the maximum stress reaches 127.10 MPa, which means that the maximum stress is increased by 8.65%, but there is still a large margin compared to the allowable material stress of 313.6 MPa. This indicates that the optimization of the BBT improves not only the static strength but also the efficiency of the material utilization.

The BBT stability after optimization is improved by 2.45%, which means that compared with the original BBT, the optimized BBT is more adaptable to the environment. Hence, there are more options for the siting range of the wind farm.