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A multi-objective optimization method for the structural design of horizontal-axis wind turbine (HAWT) blades is presented. The main goal is to minimize the weight and cost of the blade which uses glass fiber reinforced plastic (GFRP) coupled with carbon fiber reinforced plastic (CFRP) materials. The number and the location of layers in the spar cap, the width of the spar cap and the position of the shear webs are employed as the design variables, while the strain limit, blade/tower clearance limit and vibration limit are taken into account as the constraint conditions. The optimization of the design of a commercial 1.5 MW HAWT blade is carried out by combining FEM analysis and a multi-objective evolutionary algorithm under ultimate (extreme) flap-wise load and edge-wise load conditions. The best solutions are described and the comparison of the obtained results with the original design is performed to prove the efficiency and applicability of the method.

The blade is one of the most important components of wind turbines. A successful structural design of Horizontal-Axis Wind Turbine (HAWT) blades must satisfy a wide range of objectives, such as minimization of weight and cost, resistance to extreme and fatigue loads, restricted tip deflections, and avoiding resonances, but some of these objectives are in conflict [

In general, the weight and cost of the turbine are the keys to making wind energy competitive with other sources of power [

A number of papers have recently described how to deal with the structural design problems of blades using numerical models and optimization techniques. Liao [

In these papers, optimization methods are described where a single objective function is taken into account at each time with the presence of constraints. The problem involving multiple objectives is addressed using a single objective function where the multiple objectives are combined by means of appropriate weights. Thus, these methods do not have the capability to obtain the real set of trade-off solutions among multiple objectives. In many circumstances, however, the designer is interested in knowing the complete set of optimal blade configurations which correspond to the desired objectives.

This paper considers a two-objective optimization strategy of minimizing the weight as well as the material cost of a 1.5 MW HAWT blade. The main characteristics of the blade structure, namely, the number and the location of layers in the spar cap, the position of the shear webs and the width of the spar cap are employed as design variables. The optimization design for the blade is carried out under the action of ultimate flap-wise load, edge-wise load and their combination conditions using a non-dominated sorting genetic algorithm (NSGA) II and a FEM model of the blade for structural analysis.

The original blade, with a length of 37 m and a weight of 6580.4 kg, is composed of three parts: root, skin and shear webs.

The finite element method (FEM) has traditionally been used in the development of wind turbine blades mainly to investigate the structural performance in terms of global stress/strain levels, tip deflections and frequencies [

The FEM model of the blade is a parametric model, which means that the main structural parameters of the blade can be modified to create various blade models. The SHELL91 and the SHELL99 types are used for modeling the thick sandwich structures and the other parts of the blade, respectively. In order to simplify the model, the shear webs and the spar cap are connected directly without considering the effect of the adhesives. A regular quadrilateral mesh generation method is used to generate elements with low aspect ratios to prevent producing erroneous results. In this work, a few changes in the spar cap of the model are made for a better parameterization, and the blade weight of the modified FEM model is 6555.2 kg. The FEM model of the blade is shown in

The loading on a wind turbine blade is stochastic and has components from the following sources: aerodynamic, gravitational, inertial (centrifugal and gyroscopic) and operational (gridfailure, braking,

The spar cap of the blade consists of laminates made by primarily unidirectional fibers to carry the flap-wise and edge-wise bending loads. Its thickness is typically large in comparison to those of the shear webs and the outer shells [

Eight discrete control points are used to define the layup of the selected region, and the number of layers changes linearly between the control points, as shown in _{1}, _{2} are used to define the positions of the shear webs and the width of the spar cap, as shown in

Twenty variables in total are defined in this paper, which can be expressed in the following form:
_{1} to _{7} are the number of GFRP layers in the spar cap, _{8} to _{14} are the number of GFRP layers in the spar cap, _{15} to _{18} are the location of layers in the spar cap, _{19} is the position of the shear webs, and _{20} is the width of the spar cap.

The purpose of the present work is to improve the structural characteristics of the blade, reducing both its overall weight and its cost. Therefore, a weight function and a function that combines the main materials cost of the blade represent the objective functions of the problem. The first objective function _{w}_{0}, in formula:
_{i}_{i}

The second objective function _{c}_{0} of the original blade, in formula:
_{i}_{i}_{c}

The structural design of the blade is a multi-criteria constrained optimization problem [

The strain constraint: the strain generated by the loads cannot exceed associated permissible strain [_{max} is the maximum strain of the parts use both GFRP and CFRP, while
_{CFRP}_{GFRP}_{S1} is the strain safety factor, _{4}_{a}_{4}_{b}_{4}_{a}_{4}_{b}

The tip deflection constraint: in order to avoid the risk of blade/tower collisions, the maximum tip deflection should be less than the set value. The Germanischer Lloyd (GL) regulations specify that the quasi-static tip deflection under the extreme unfactored operational loading is not to exceed 50 percent of the clearance without blade deflection. The International Electrotechnical Commission (IEC) 61400-1 specifications, on the other hand, require no blade/tower contact when the extreme loads are multiplied by the combined partial safety factors for loads and the blade material [_{max} is the maximum tip deflection; _{a}_{S}_{2} is the tip deflection safety factor.

The vibration constraint: a good design philosophy for reducing vibration is to separate the natural frequencies of the blade from the harmonic vibration associated with rotor rotation, which would avoid resonance where large amplitudes of vibration could severely damage the blade. This is expressed in the inequality form:
_{blade} is the first natural frequency of the blade, _{rotor} is the frequency of the rotor rotation and Δ is the associated allowable tolerance.

The buckling constraint: since the blade is a thin-walled structure, and it is subjected to large flap-wise bending moments, the surface panels near the blade root are particularly vulnerable to elastic instability, so the buckling problem must be addressed [_{1} is a ratio of the buckling load to the maximum ultimate load, called lowest buckling load factor; γ_{S}_{3} is the buckling safety factor. λ_{1} is calculated using a nonlinear buckling analysis in ANSYS under the above ultimate load conditions.

The lifetime fatigue constraint: the durability requirement for the turbine blades is typically defined as a minimum 20-year fatigue life (which corresponds roughly to 10^{8} cycles) when subjected to stochastic wind-loading conditions and cyclic gravity-induced edge-wise bending loads in the presence of thermally fluctuating and environmentally challenging conditions [_{max} is the maximum stress of the blade; σ_{Y}_{S}_{4} is the lifetime safety factor; _{0} is the number of allowable cycles.

In addition, considering the manufacturing maneuverability and the continuity of the material layup, the design variables should be satisfied with the following inequality form:
^{L}^{U}

The lower and upper bounds of the variables and the constraint conditions are shown in

The NSGA II [

The method randomly generates an initial parent population _{0} of size _{0} with the same size as the parent population is created through recombination based on binary tournament selection and by inducing variations using mutation operators. From the first generation onward, the procedure is different. First, a combined population _{t}_{t}_{t}_{t}_{+1} is formed by adding solutions from the first front till the size becomes _{t}_{+1} of size

The NSGA II parameters used in this paper are listed in

To better explain the formation of the Pareto front, three optimized solutions extracted at different positions on the Pareto front are analyzed, as marked in

The position of the shear webs and the width of the spar cap both decrease after optimization. In order to find the effect of the position of the shear webs on the structural performance of the blade, a sensitivity analysis has been carried out. The result shows that the moving the shear webs to the centerline of the spar cap can reduce the maximum equivalent strain,

The structural performance of the blade is presented in

The decrease in total number of layers leads to a degradation in the fatigue strength of the blade, thus the fatigue lifetime gradually approaches the critical value after optimization. Since the blade A has both lower cost and less weight than the original blade, it seems to be a more desirable result than the other blades by the consideration of the present two optimization objectives.

This paper illustrates a two-objective optimization method that uses NSGA II and a FEM model for the structural design of HAWT blades. The method is used to obtain the best trade-off solutions between the blade weight and the cost. The NSGA II handles the design parameters chosen for optimization and searches for the group of optimal solutions following the basic principles of Genetic Programming and Pareto concepts. The FEM model utilizes ANSYS to determine the structural performance of a HAWT blade, while it measures the fitness functions of the optimization as well.

The method has been applied successfully to a 1.5 MW commercial HAWT blade, and a set of trade-off solutions are obtained. The results indicate that the minimization of mass requires more CFRP, while the minimization of cost requires a good arrangement of GFRP combined with CFRP. Satisfactory results to reduce the weight as well as the cost of the blade are achieved and significant improvements in the structural performance of the blade are obtained by rearranging the original material layup in the spar cap, the width of the spar cap and the positions of the shear webs, which can be advantageous from the production and manufacturing requirements point of views.

This work was supported by the Jiangsu Collaborative Innovation Center for Coastal Development and Protection (Jiangsu Government Office Document [2013] No. 56) and the Science and Technology Plan to Guide the Project of Nantong (2013400303).

The authors declare no conflict of interest.

The geometry and a typical structural cross section of the blade.

FEM model of the blade.

(

Original material layup of the spar cap.

Parametric of the material layup of the spar cap.

Flowchart of the NSGA II.

Pareto front of two objectives.

(

Material properties of the glass fiber and carbon fiber.

_{1} (GPa) |
_{2} (GPa) |
_{12} (GPa) |
_{12} |
^{3}) |
^{3}) | |
---|---|---|---|---|---|---|

GFRP | 42.19 | 12.53 | 3.52 | 0.24 | 1910 | 1 |

CFRP | 130.00 | 10.30 | 7.17 | 0.28 | 1540 | 10 |

Lower and upper bounds of the variables and the constraint conditions.

_{1} |
0 | 38 | - |

_{2} |
0 | 48 | - |

_{3} |
0 | 58 | - |

_{4} |
0 | 65 | - |

_{5} |
0 | 55 | - |

_{6} |
0 | 45 | - |

_{7} |
0 | 40 | - |

_{8} |
0 | 30 | - |

_{9} |
0 | 35 | - |

_{10} |
0 | 40 | - |

_{11} |
0 | 45 | - |

_{12} |
0 | 40 | - |

_{13} |
0 | 35 | - |

_{14} |
0 | 30 | - |

_{15} |
0.17 | 0.23 | - |

_{16} |
0.25 | 0.33 | - |

_{17} |
0.41 | 0.49 | - |

_{18} |
0.53 | 0.59 | - |

_{19} |
0.13 | 0.25 | m |

_{20} |
0.50 | 0.70 | m |

_{max} |
- | 5,000 | μ |

_{max} |
- | 3,180 | μ |

_{max} |
- | 5.5 | m |

_{blade} |
≤0.94 or ≥0.96 | Hz | |

λ_{1} |
1.30 | - | - |

1.20 × 10^{8} |
- | - |

Implemented NSGA II parameters.

Number of individuals | 20 |

Number of iterations | 50 |

Probability of crossover | 0.8 |

Probability of mutation | 0.01 |

Values of the design variables.

_{1} |
33 | 30 | 20 | 5 | - |

_{2} |
43 | 39 | 27 | 6 | - |

_{3} |
53 | 45 | 29 | 9 | - |

_{4} |
62 | 53 | 31 | 10 | - |

_{5} |
53 | 43 | 27 | 8 | - |

_{6} |
43 | 37 | 25 | 6 | - |

_{7} |
33 | 29 | 20 | 3 | - |

_{8} |
0 | 2 | 4 | 12 | - |

_{9} |
0 | 2 | 11 | 21 | - |

_{10} |
0 | 3 | 13 | 27 | - |

_{11} |
0 | 3 | 15 | 30 | - |

_{12} |
0 | 2 | 8 | 25 | - |

_{13} |
0 | 1 | 5 | 20 | - |

_{14} |
0 | 1 | 3 | 14 | - |

_{15} |
0.19 | 0.19 | 0.20 | 0.20 | - |

_{16} |
0.28 | 0.30 | 0.30 | 0.28 | - |

_{17} |
0.46 | 0.45 | 0.44 | 0.45 | - |

_{18} |
0.55 | 0.55 | 0.56 | 0.55 | - |

_{19} |
0.188 | 0.146 | 0.143 | 0.142 | m |

_{20} |
0.62 | 0.58 | 0.55 | 0.54 | m |

Structural performance of the blades.

| ||||||||
---|---|---|---|---|---|---|---|---|

Original design | 1.000 | 1.000 | 4074 | 4286 | 4.59 | 1.027 | 2.02 | 2.28 × 10^{8} |

Blade A | 0.936 | 0.958 | 4859 | 3143 | 5.13 | 1.016 | 1.45 | 1.73 × 10^{8} |

Blade B | 0.831 | 1.294 | 4253 | 3056 | 3.77 | 1.210 | 1.87 | 1.61 × 10^{8} |

Blade C | 0.752 | 1.768 | 4685 | 3131 | 3.01 | 1.328 | 2.33 | 1.32 × 10^{8} |