Yuqiao Zheng， Lu Zhang， Fugang Dong and Bo Dong

（School of Mechanical and Electronic Engineering, Lanzhou University of Technology, Lanzhou 730050, China）

Abstract: A multi-objective optimization process for wind turbine steel towers is described in present work. The objective functions are tower top deformation and mass. The tower's height, radius and thickness are considered as design variables. The mathematical relationships between objective functions and variables were predicted by adopting a response surface methodology (RSM).Furthermore, the multi-objective non-dominated sorting genetic algorithm-II (NSGA-II) is adopted to optimize the tower structure to achieve accurate results with the minimum top deformation and total mass. A case study on a 2MW wind turbine tower optimization is given, which computes the desired tower structure parameters. The results are compared with the original tower: a reduction of tower top deformation reduction by about 16.5% and a reduction of a mass by about 1.5% could be achieved for such an optimization process.

Key words: wind turbine tower；statics analysis；experiment design；multi-objective optimization

Towers must carry the accessories of wind turbines and withstand the influence of the external environment. As the technology matures,tower heights improve and installation ranges enlarge continuously. Besides the wind, they must bear many other complicated loads, like earthquakes and waves. Towers are essential to provide safe and reliable operation of wind turbine systems under a variety of load conditions[1-2].

In recent decades, numerical studies have been developed to meet the demands of optimum tower structural design. Bukala[3]combined finite element analysis, economic evaluation, aeroelastic coupling analysis, and genetic algorithms to optimize the height, diameter, mast size and other parameters of a lattice wind turbine to improve the economic efficiency of the tower, and finally proposed a global optimization scheme for the tower under complex conditions. Gencturk[4]combined the different height of towers with turbines of different power generation capacities for seismic regions and applied the Taboo search algorithm to minimize the total cost of the tower and the foundation. Perelmuter[5]solved the optimization problem of steel conic shell towers of wind turbines by using an improved gradient method, plotted the relationship between the tower’s height and mass, and produced a capacity of wind turbines. Dai[6]used BP-ANN theory and NASG-II algorithm to minimize mass and tower top displacement. Gentils[7]established the nonlinear mathematical model between the structural parameters of the tower and the stress, the top deformation, and the natural frequency of tower based on the BP artificial neural network theory to solved the multi-objective optimization problem. Nikos[8]proposed two formulations which are examined differing on the shape of the wind tower along the height to optimize the construction cost of the tower, and it was proved that structural optimization represents a very significant component of the economic and technological analysis. Kaveh[9]performed an optimization design of offshore wind turbine jacket support structures, considering the diameter and thickness of the catheter as the design variables, then employed the meta-heuristic algorithm of collision body optimization to seek the minimum mass of the support structure. Schafhirt[10]employed two-stage local optimization for offshore wind turbine support structures; supposing that changing dimensions of a structural member does not affect other members of the structure, the optimization process saved analysis time by roughly 40%.

The aim of this work is to optimize the tower top deformation and mass. Response surface models of the tower are set up based on the optimal latin hypercube experiment; then, the elitist non-dominated sorting genetic algorithm(NSGA-II) is developed to solve the optimization problem of the tower.

1 Wind Turbine Tower Structure

The 77.865m vertical steel tower is assembled with thin-wall cylindrical and conical pieces of varying diameters (2 100 mm at the base to 1 502.5 mm at the top) and wall thicknesses (42 mm at the base to 14 mm at the top).

1.1 Materials

It is usually made from hot-rolled steel, welded together circumferentially and longitudinally,with welded flanges at either tower end. The tower in this case is made of Q345, with a yield strength σ=345 MPa, a modulus of elasticity E=2.06×1011MPa, and density ρ=7 850 kg/m3.

1.2 Finite element model

To estimate the effect of the total structural parameters on the deflection of the tower, a simplified approach is used, which assumes a rigid foundation and that the structural details such as flanges, tower sections, and door frames have not been involved in the finite element model. In order to optimize the parameters of the tower, the four sections are considered as a whole. A concentrated mass point is adopted instead of the nacelle and rotor. A finite element model of the tower is created in ANSYS using Solid shell elements of a size of 350 mm, generating 8 262 nodes and 4 009 elements. The jacket is modelled as a rigid region and free elsewhere. The finite element model global coordinate system is shown in Fig.1.

Fig.1 FE model global coordinate system

1.3 Ultimate loads

The tower mass and extreme load are considered. Three loadings (DLC1.5, DLC3.2 and DLC6.1) are derived from GH-Bladed to analyze the statics characteristic of the tower. It is concluded that the top deformation is the largest under DLC6.1. So, the extreme loading case DLC6.1 is applied to the tower structure optimization, and the values are listed in Tab. 1.

Tab. 1 Ultimate loads

2 Approximate Model Establishment

2.1 Tower parametric model

This instance is performed without the door,and the thickness of the tower increases linearly from top to bottom. The tower’s height(h), radius(r), top thickness(t1), and bottom thickness(t2) directly determine the structural performance and cost of the tower. So, taking the above four parameters as design variables, the parametric model of the tower is established, which is shown in Fig.2.

Fig.2 Tower parametric model

2.2 Sample point acquisition

The tower is an important component of a wind turbine, accounting for 15%–20% of the total cost. Under DLC6.1, the tower top deformation is 641.84 mm, accounting for 0.82% of the tower height, which exceeds 0.8% of the engineering experience requirements. Therefore, this paper mainly discusses the influence of tower structure parameters on tower mass and tower top deformation.

25 sets of experiments for different top tower thicknesses, bottom tower thicknesses, tower heights and base tower radii were carried out by the Optimal Latin hypercube design (Opt LHD)[11]. Tab. 2 shows the detail of different combinations of independent variables and response values (m is tower mass and w is tower top deformation).

Tab. 2 Parameters values in the DOE

2.3 Sensitivity analysis

Fig.3 Pareto graphs for responses

Fig.3 only displays the effect of single factors on responses by plotting the relationship as determined by regression analysis of the experimental data. The tower mass is positively correlated with all factors, and the contribution to mass from greatest to least is t2(29.9%), h(28.15%), r (20.41%), and t1(9.13%), respectively (Fig.3a). Only h has a positive effect on top deformation (Fig.3b) with a contribution rate of 29.89%; t1, t2, and r all have negative effects on top deformation with contribution rates of 2.35%, 11.62%, and 24.42%, respectively. It is difficult to ensure that all objectives achieve the optimal solution of optimal design simultaneously. This work aims to achieve the best tradeoff (Pareto front) of the tower mass and top deformation.

2.4 Response surface establishment

The response surface model (RSM) is an approximate model widely applied currently. It replaces a complex model with higher-order functional relations to reduce huge computational burden owing to the large number of dimensions and improve computational efficiency[12-13].

The second-order response surface model of tower mass and its top deformation are established, applying the sample points obtained from the above experiment as follows.

There are two indices to evaluate the accuracy of the approximate model: the complex correlation coefficient R2and the root mean square error (RMSE)[14]. In order to obtain satisfactory approximation models, R2should be close to 1 and the RMSE should be close to 0. The above two indicators are shown by

Here, yiis the actual response value obtained by FEA analysis of the structure; y?iis the predicted value obtained by the approximate model; μ(y) is the mean and standard variance of the sample points; n is the number of sample points.

Cross-validation method is applied to verify its fitting accuracy. The values of R2for tower mass and top deformation are 0.998 and 0.997,respectively, which are close to 1. The values of RSME for tower mass and top deformation are 0.002 and 0.026, which are close to 0. The results prove that the accuracy of the model established in this work meets the requirements.

3 Multi-Objective Optimization Description

This section describes the multi-objective optimization strategy used in the examples of Section 5. NSGA-II is a desirable solution of multiobjective optimization problem[15].

The multi-objective non-dominated sorting genetic algorithm-II (NSGA-II) is improved based on the non-dominated sorting genetic algorithm (NSGA)[16-18]. Compared to NSGA, the advantages of NSGA-II are as follows.

① The complexity of the sorting is reduced to O (MN2), and the computational efficiency is improved. ② The elite strategy is added to the algorithm to preserve the best individuals. ③ The sharing function is replaced by a crowded-comparison obtained by calculating the sum of the distance difference of two adjacent individuals on each sub-objective function.

The main steps of NSGA-II are as follows.

① First, generate an initial population with a size of N randomly, and the first offspring population is obtained through selection, crossover,and mutation after the non-dominant sorting.

② Second, from the second generation,merge the parent population with the offspring population and implement fast non-dominant sorting. Meanwhile, calculate the crowding distance of individuals and select appropriate individuals to form a new parent population.

③ Finally, generate new offspring populations by the essential operation of the genetic algorithm, and repeat until meeting the conditions for program termination.

4 Optimization Problem

4.1 Objective functions

Based on the main goal of cost reduction and security enhancement, the objective functions are the total mass of structure (m) and top deformation on the tower (w). This purpose can be achieved by choosing the corresponding tower heights (h), radii (r), and thicknesses (t1, t2).The basic formulation of the problem is expressed as

The expressions of m and w are shown in Eqs. (2)(3).

4.2 Design constraints

① Strength constraint

To ensure the tower safety, the maximum tower stress σmaxunder loads should be less than the material yield

where σ is von Mises stress, [σ] is allowable stress, [σ]=σs/ns, σsis the material yield, and nsis the safety factor, whose value is 1.1[19].

② Stiffness constraints

The action of various loadings leads to overlarge deformation at the top of the tower. Furthermore, severe vibration generates and destroys the regular operation of the wind turbine system.

To ensure the wind turbine’s operation, the tower must satisfy its stiffness conditions.

Here fmaxis the maximum tower top deformation and [f ] is the allowable deflection. According to the specifications, the allowable deflection should be 0.5%–0.8% of the total height of the tower[20].

③ Frequency constraint

Resonance between wind rotor and tower not only shortens the wind turbine life, but also leads to the collapse of the whole system. To prevent it, the natural frequency of the tower is specified to avoid 1P (rotor rotation frequency, ω1P)and 3P (blade passing frequency, ω3P) turbine excitation frequency ranges[21].

The maximum rotor speed is assumed to be 15.35 r/min. The corresponding rotor rotation frequency and blade passing frequency are 0.256 Hz and 0.768 Hz, respectively. So, the first natural frequency constraint of the tower for the example can be described as

④ Size constraint

Due to the limitation of road transportation condition, the radius of the tower bottom has significantly reached the maximum. The size constraints of design variables are expressed as

5 Result Discussion

Because of conflicts between objectives, one solution that is best on one objective may be poor on other objectives. A Pareto front with a set of non-dominated solutions should be calculated which has the least objective conflicts compared to other solutions, providing a better choice for designers[15]. The Pareto front of the tower mass and tower top deformation are illustrated in Fig.4.

Fig.4 Pareto solutions of w/m

Fig.4 presents that the objective functions of tower mass and tower top deformation are contradictory to each other and cannot be minimized at the same time. Therefore, for engineering applications, the designers can choose the optimal solution according to the design needs. One of the goals in this task is reducing tower top deformation. Three groups of solutions with better performance were selected from a series of solutions, and the comparison with the original tower are listed in Tab. 3.

Tab. 3 Three groups of solutions

In Tab.3, it can be noticed that, tower 1 has the lightest mass and tower 2 has the least top deformation. In fact, after optimization, the tower height also changes accordingly. It is not possible to purely take the minimum top deformation as the criterion, but instead the minimum ratio of deformation to tower height. Tower 3 has a similar scheme with a ratio of deformation to height of 0.072%. So, tower 3 is determined to be the optimal scheme. Finite element analysis of tower 3 is performed; then, the verified result is compared with optimal result, as listed in Tab. 4.

In Tab.4, the results of the tower 3 are presented and compared with the original structure. After optimization, the tower top thickness,height, and tower bottom radius are reduced, and tower bottom thickness is increased. The optimized tower mass is reduced by 1.5% and the tower top deformation is reduced by 16.5%.

The comparison between the tower optimization value and the finite element validation value shows that the error between the optimal value of tower mass and the verified value is 0.26%,and the error between the optimal value of tower top deformation and the verified value is 2.4%. It is proved that the optimal solution obtained by establishing the approximate model has high accuracy.

Since the tower size is also closely related to the maximum stress and frequency of the tower,the two responses of the optimized tower have been optimized.

Tab. 4 Final solution and verification

6 Conclusions

The 2MW wind turbine tower was taken as a structure optimization case. The influence between design variables and objective functions was explored by Latin hypercube experiment.Subsequently, a second-order response surface approximate model of tower mass and tower top deformation was established as the basis of multiobjective optimization of the tower. The RME models showed a good agreement with experimental results.

The multi-objective non-dominated sorting genetic algorithm-II (NSGA-II) was employed to find the optimal solution of Pareto for tower mass and tower top deformation. Compared to the initial design, the optimized tower mass was reduced by 1.5% and the tower top deformation was reduced by 16.5%, which is less than 0.8% of the tower height. The results indicate that the proposal is desired. The error between the optimal tower and verified tower is less than 2.5%which prove that the optimal solution obtained by establishing the approximate model has high accuracy.