Structural parameters multi-objective optimisation and dynamic characteristics analysis of large-scale wind turbine towers.
Since traditional sources of energy such as coal, oil and natural gas are finite and non-renewable, there is a strong emphasis in the development of technologies and capabilities in the generation of renewable energy around the world. Currently, wind energy is the world's fastest growing renewable energy. There are mainly two types of wind turbines, namely vertical axis type, and horizontal axis type. Horizontal axis wind turbines dominate in wind power industry. It mainly consists of the wind rotor, main transmission system, nacelle, generator, converter, tower and some other parts. Wind turbine tower supports the nacelle and the wind rotor to the required height. Thus, it bears complex loads in operation process. Furthermore, with the increase of single machine capacity of wind turbines, wind turbine hub height need increase corresponding, which also causes the increase of the tower height. For example, V164-8.0 MW wind turbine (Vestas Wind System A/S) has 131 m high tower, E-126 wind turbine (German Enercon Company) has 135 m high tower and XE/DD115-5.0 MW wind turbine (China XEMC) has 100 m high tower. Moreover, the quality of the tower of large-scale wind turbines is generally more than a hundred tons. The material and manufacturing cost of the tower is usually 15-20% of the total cost of a wind turbine. For a long time, many researchers have carried out analysis and design of wind turbines towers from different angles. For instance, in order to optimise wind turbine tower structure, five optimisation strategies were developed and tested (Negm and Maalawi 2000). The minimum tower cost, including material and manufacturing cost was selected to be the optimised objective and the design procedure was also proposed (Uys et al. 2007). The behaviour and the load capacity of a prototype steel tower were studied using a refined finite element and other simplified models (Bazeos et al. 2002). A 1 MW wind turbine tower was designed, and two different finite element models were presented to simulate the tower response (Lavassas et al. 2003). Based on the SCADA data, the relationship between the tower vibration and some wind turbine parameters was discussed (Kusiak and Zhang 2010, Zhang and Kusiak 2012). Considering the aerodynamic loads and wave loads, the dynamic response of the offshore wind turbine tower was investigated by modelling the tower using space beam element(Li et al. 2009). In the previous work, tower structure parameter optimisation was also researched, but only for single objective (Dai et al. 2013). Referring to the above-mentioned work, especially deepening and developing the research by Dai et al. (2013), a multi-objective optimisation method of tower structural parameters is researched in this paper. The influence of tower structure parameter before and after optimisation on aeroelastic coupling effect is also analysed.
2. Multi-objective optimisation model of tower structural parameters
2.1. Design variable and objective function
Usually, large-scale wind turbine towers have a variable cross-section cylindrical structure by a certain taper. For the convenience of transportation, the tower is usually divided into several segments. As an example, a three-segment structure tower (illustrated in Figure 1) is used to demonstrate the presented optimisation method and process. Thus, structural parameters mainly include the bottom diameter, top diameter, segment length and wall thickness. Since the height of the tower is mainly determined by wind turbine capacity, wind resource characteristic and geographical location, the height value can be regarded as a constant when optimising tower structural parameters. In addition, taking into account, the matching of the tower top and the nacelle, the tower top diameter can also be regarded as a constant. Therefore, the tower design variables x are defined as the upper segment wall thickness [[delta].sub.1], the middle segment wall thickness [[delta].sub.2], the lower segment wall thickness [[delta].sub.3], the lower segment length [H.sub.1], the middle segment length [H.sub.2] and the tower bottom diameter D, i.e. (Dai et al. 2013),
x = [[[delta].sub.1], [[delta].sub.2], [[delta].sub.3], [H.sub.1], [H.sub.2], D] (1)
In order to reduce the material cost, the tower mass should be selected to be an optimisation objective for tower structural parameters. On the other hand, considering the aeroelastic coupling effect caused by the structure flexibility in wind turbine operation, the minimum tower top displacement could be selected to be another optimisation objective. In this way, the tower structure parameter optimisation objective can be written as
Min F(x) = [[f.sub.m](x), [f.sub.d](x)] (2)
where, [f.sub.m](x) is the tower mass function, [f.sub.d](x) is the tower top displacement function.
2.2. Constraint conditions
In order to meet the strength requirements of the tower structure and to avoid the tower resonance frequency, both the maximum tower stress and natural frequency are selected to be the constraints. In addition, considering the specific circumstances of the tower design, manufacture and operation, structural parameters should be within a certain range. Therefore, constraint conditions include variable boundary constraints, strength constraints and natural frequency constraints.
2.2.1. Variable boundary constraints
To determine the variable boundary constraint conditions, the reference values of variables [[delta].sub.1] [[delta].sub.2], [[delta].sub.3], [H.sub.1], [H.sub.2] and D should be firstly determined. In other words, reference values corresponding to these variables are firstly determined based on prior knowledge, which can be expressed as [circ.[delta].sub.1], [circ.[delta].sub.2], [circ.[delta].sub.3], [circ.H.sub.1] [circ.H.sub.2] and [circ.D]. Then, a range for each variable is given to carry out a numerical simulation to obtain the influence of each variable on stress, top displacement and natural frequency in ANSYS software. For the example used in this paper, [circ.[delta].sub.1], [circ.[delta].sub.2], [circ.[delta].sub.3], [circ.H.sub.1], [circ.H.sub.2] and [circ.D] are set to be 0.013, 0.023, 0.031, 17, 25, 4.1 m, respectively. The relationship between the variable structural parameters and the top displacement for the case is illustrated in Figure 2. Since the values of different variables are different, transverse coordinates have been normalised in Figure 2 for the convenience of analysis. Figure 2 shows the relationship between structural parameters and the tower top displacement. Through ANSYS numerical simulation, it can be known that both the upper segment wall thickness [[delta].sub.1] and bottom diameter D have a greater influence on the stress, tower top displacement and natural frequency. Therefore, when determining the parameter range, the upper segment wall thickness [[delta].sub.1] and bottom diameter D should not be too large, and other parameters can be larger. Finally, the variable bounds could be determined based on the analysis results, which are expressed as (Dai et al. 2013)
[mathematical expression not reproducible] (3)
For the example used in this paper, the variable bounds of [[delta].sub.1], [circ.[delta].sub.2], [[delta].sub.3, [H.sub.1], [H.sub.2] and D (SI unit) are selected to be [0.012, 0.018], [0.018, 0.026], [0.026, 0.045], [15, 20], [20, 35]and [3.8, 4.4], respectively.
2.2.2. Strength constraints
To ensure the tower safety, the maximum tower stress [[sigma].sub.max] under loads should be less than the material yield stress, i.e.
[[sigma].sub.max] * [n.sub.st] < [[sigma].sub.b] (4)
where, [n.sub.st] is the safety factor; [[sigma].sub.b] is the material yield stress.
2.2.3. Natural frequency constraints
If wind rotor rotation frequency or blade passing frequency is close to the natural frequency of the tower, resonance phenomenon will appear and cause serious security problems. An appropriate interval is necessary between the natural frequency of the tower [f.sub.0,n] and the excitation frequencies [f.sub.R] and [f.sub.R*m]. The relationship between them has been introduced in "Specification for wind turbines" (promulgated by the China Classification Society, 2008), that is (Dai et al. 2013).
[mathematical expression not reproducible] (5)
where [f.sub.R] is the maximum rotational frequency of wind rotor in normal operating range; [f.sub.0,1] is the first natural frequency of the tower; [f.sub.R,m] is the passing frequency of the mth wind turbine blade; [f.sub.0,n]] is the n-order natural frequency of the tower.
For the example used as above, the maximum rotor speed of the turbine is assumed to be 22.5 r/min. The corresponding rotor rotation frequency and blade passing frequency are 0.375 and 1.125 Hz, respectively. So, the first natural frequency constraint of the tower for the example can be obtained as
0.395 [less than or equal to] f [less than or equal to] 1.071 (6)
3. Relationship modelling between tower structural parameters and the stress, top displacement and natural frequency
The solving process of tower structure parameter optimisation model is an iterative process. The tower top displacement, the maximum stress and the natural frequency will be used in the iteration process. To accelerate the optimisation process, the non-linear relationship between structural parameters and the stress, top displacement and natural frequency should be understood and a rapid alternative model (RAM) should be established. With the rapid development of artificial intelligence (AI) theory, the artificial neural network, the support vector machine and some other AI theories have been widely used in the complex non-linear mathematical modelling. After comparing these theories, the BP artificial neural network is employed. It is a kind of multilayer network using a non-linear differentiable function to train weight value. With simple structure and strong plasticity, it has a significant advantage in dealing with problems of multi-dimension, complex and nonlinear (Dai et al. 2012). Figure 3 shows the structure of BP neurons; Equation (7) is the corresponding mathematical expression. X = [[x.sub.1], [x.sub.2], ... [x.sub.n0]] is the input matrix, W = [[w.sub.1], [w.sub.2], ... [w.sub.n0]] is the weight matrix, a is the weighted sum of all inputs (a = [SIGMA] [w.sub.i][x.sub.i]), f is the activation function relative to a and threshold [theta]. Then, RAM (alternative model) is established by three layers BP neural network (as shown in Figure 4). In the established RAM, the input is variable structural parameters of the tower, the output is the maximum stress, top displacement and natural frequency. It should be pointed out that since BP neural network is sensitive to the neuron number in the hidden layer. So, it is rather important to define a reasonable neuron number. If the number is too short, the network will be trained without any result. Contrary, the network will take a lot of time to train and the fitting curve would oscillate between different sample points.
[mathematical expression not reproducible] (7)
In order to obtain the required sample data for modelling, the mixed experimental design method which combines the orthogonal experimental design method and the random experimental design method is employed to determine the experimental scheme. By employing the finite element numerical simulation, sample values can be obtained. Variable structural parameters of the tower are set to be the six factors of the orthogonal experiment. For the used example, the experimental factor level is shown in Table 1. According to the orthogonal experimental method, 25 experiments are carried out. In addition, 200 experiments are randomly selected to carry out from the total 15625 experimental schemes of permutation and combination, too. Therefore, there are 225 sample data. Then, according to the maximum, the sample data are normalised. Through trial computation, the Levenberg-Marquardt rule is adopted to be the training algorithm. The hyperbolic tangent sigmoid transfer function and the pure linear transfer function are employed to be the hidden layer and output layer transfer functions, respectively. The numbers of neurons in the hidden layer is determined to be 12, the network training accuracy is 0.00004, the maximum number of cycle is 100000, the learning rate is 0.1.
Table 2 shows the output results of the tower stress, top displacement and natural frequency by RAM and finite element analysis (FEA) software, respectively. It can be seen that the results from the RAM and FEA software are very close, but the computation time of the rapid alternative model (RAM) is significantly shortened. So, in structure parameter optimisation, if the optimisation algorithms or the optimisation results are not satisfactory, it has obvious advantages to repeat the optimisation calculation using RAM.
4. Solution of optimisation model for structure parameters
Unlike the single-objective optimisation, the optimal solution of each objective is generally different in multi-objective optimisation. Furthermore, the objectives are usually conflicting and are difficult to achieve the optimal values simultaneity. The traditional weighting method, constraint method and the mixed method are to seek the balance among multiple objectives according to a certain strategy, convert the multi-objective problem into a single-objective optimisation problem, and use the optimal solution set of multiple single-objective optimisation problems to approximate the Pareto optimal set of the multi-objective optimisation problem. In order to obtain the Pareto optimal solution of multi-objective optimisation better, the special multi-objective optimisation algorithm should be used. In all kinds of multi-objective optimisation algorithms, NSGA-II (Non-dominated Sorting GA) algorithm, which uses elitist strategy and biodiversity protection method, has excellent performance for 2 or 3 optimisation objectives. For the used example, the population size is set to be 100, the evolutional generation is 100. The obtained Pareto optimisation solution set is shown in Figure 5. From the distribution of the solution set, it can be seen that the tower mass and the top displacement are often in conflict, which means that if the tower mass reduces, the top displacement could increase in some cases. Considering cost, performance property, manufacturing, transportation and installation comprehensively, a set of satisfactory solution from the Pareto solution set is selected. Tower structure parameters, mass, the maximum stress, top displacement and natural frequency are shown in Table 3. After optimisation, the tower mass decreases 7601 kg, accounting for about 7.3% of the tower mass. The increase in the natural frequency after optimisation is conducive to the tower to avoid resonance. Though in this selected solution set, the tower top displacement has a slight increase after optimisation, it is acceptable according to the wind turbine characteristics.
5. Dynamic characteristics analysis
The dynamic characteristics of wind turbine towers can be expressed approximately using two-order harmonic damping system, i.e.
[mathematical expression not reproducible] (8)
where [x.sub.T](t) is the tower displacement, [F.sub.T] is loads on the tower, [m.sub.T] is the generalised mass of the tower, nacelle and rotor; [k.sub.T] is the mode stiffness coefficient ([k.sub.T] = [m.sub.T][[omega].sub.T.sup.2], [[omega].sub.T] is the tower frequency), [c.sub.T] is tower structural and aerodynamic damping.
For wind turbines in a land-based wind farm, variable tower loads come mainly from the aerodynamic thrust. When the rotor speed and the pitch angle are fixed, the aerodynamic thrust is mainly determined by the wind speed. However, to calculate the apparent local relative velocity, the structural motion speed should be subtracted which is determined by the load for a specific type of wind turbine. So, the structural and aerodynamic coupling will form complex aeroelastic problems. It should be pointed out that the blade and tower are the two main vibration parts of wind turbines. Moreover, the blade is mounted on the tower, and the blade movement is also based on the tower movement. Thus, the effects of the blade should also be considered when analysing the tower vibration characteristics.
Figure 6 shows the fundamental shape of vibration coupling of the blade and the tower (Burton et al. 2005). In the figure, [mu](r) is the first blade mode shape for the rigid tower, [[mu].sub.TJ](r) is the normalised rigid body deflection of blade J resulting from the tower first mode excitation, L is the distance between the hub center and transverse plane through the intersection of the tangent to the top of the deflected tower and the un-deflected tower axis.
In Equation (8), [m.sub.T], [F.sub.T] can be calculated based on Equation (9) (Burton et al. 2005).
[mathematical expression not reproducible] (9)
where [m.sub.T](z) is the tower mass for each unit high, [m.sub.N] and [m.sub.R] are the nacelle mass and rotor mass, respectively, [I.sub.R] is the inertia of the rotor about the horizontal axis in its plane, [[mu].sub.T] is the first mode shape of the tower, q(z, t) is the load on the hub which is fed back from the blade.
In order to analyse the dynamic characteristics of the tower, both ANSYS and BLADED software are employed. In ANSYS software, loads acting on the tower top, is set to be F = 195000 + 10000 sin(2t) (N). The corresponding tower top displacement, velocity and acceleration at fore-and-aft direction are shown in Figure 7. It can be seen that before and after optimisation, tower top displacements under the same loads vary slightly, the top displacement after optimisation is slightly less than that before optimisation, and both the difference of vibration velocity and the difference of acceleration are small before and after optimisation.
In order to compare with ANSYS analysis results, the tower parameters before and after optimisation are inputting the BLADED software to carry out simulation analysis, respectively. The wind turbine used to simulate has three blades using NACA63-4xx series airfoil, rotates clockwise; rotor diameter is 82 m. Both the pitch angle and rotor speed can be regulated. Cut-in and cut-out wind speeds are 3.5 and 25 m/s, respectively. Tower height is 62 m, the tower material Young's modulus is set to be 2.1 x [10.sup.11] N/[m.sup.2] and the material density is set to be 7850 kg/[m.sup.3]. When simulating, the wind speed at the hub is given as 13 m/s. Figure 8 shows the tower bending stiffness distribution curves before and after optimisation, the bending stiffness of the lower part (50 m) of the tower after optimisation is larger than that before optimisation, the bending stiffness of the upper part (12 m) after optimisation is less than that before optimisation. It is easy to see that the overall stiffness after optimisation is larger than that before optimisation.
Rotor speed curves are shown in Figure 9. It can be seen that the speed curves are basically the same before and after optimisation. Both of them fluctuate in the range of 2.093 to 2.096 rad/s. Since the rotor speed of the turbine is set to be constant in simulation, the fluctuation amplitude of the rotor speed is very small (only 0.14%), where the fluctuation amplitude is defined as the value of the difference between the maximum value and minimum value divided by the maximum value. Figure 10 shows the thrust curves acting on the hub at fore-and-aft direction. Since the real-time status of aeroelastic coupling is different before and after optimisation, the thrust amplitudes do not coincide in the running process. The load fluctuation ranges are slightly different, which is in the range of 183.5 to 204.9 kN before optimisation and 181.8 to 204.7 kN after optimisation.
Figure 11(a) shows nacelle (tower top) displacement changes at the fore-and-aft direction before and after optimisation. Since the change of tower structure parameters will affect the aeroelastic coupling state of the turbine, the dynamic displacement after optimisation is different with that before optimisation in BLADED. The displacement fluctuation is in the range of 0.149 to 0.185 m before optimisation and 0.141 to 0.173 m after optimisation. The maximum value has reduced by 6.5%. In Figure 11(b), the vibration velocity fluctuation is in the range of -0.046 to 0.055 m/s before optimisation and -0.053 to 0.049 m/s after optimisation. Obviously, the changes of structure parameter have an influence on the aeroelastic coupling state of the turbine to a certain extent. In Figure 11(c), the maximum vibration acceleration of the nacelle (tower top) before optimisation is -0.37 m/[s.sup.2] (minus sign denotes direction); after optimisation, it is -0.43 m/[s.sup.2].
The purpose of the study is to present a multi-objective optimisation method of wind turbine tower, demonstrate the specific optimisation process and analyse the dynamic response characteristics. It is expected to provide a reference idea for the development of the actual wind turbine tower. Through the study, the following conclusions could be obtained. (1) Aiming at the minimum mass and tower top displacement, it is feasible using NSGA-II algorithm to solve multi-objective optimisation model of tower structure parameters. (2) The non-linear mathematical model, which describes the relationships between tower structure parameters and stress, top displacement and natural frequency based on BP artificial neural network, meets the precision requirement. So, it has obvious advantages in iterative optimisation calculation. (3) Changes of wind turbine aeroelastic state before and after parameters optimisation result in changes of load on the tower, tower top vibration displacement, velocity and acceleration to a certain extent.
No potential conflict of interest was reported by the authors.
This work is supported by Hunan Provincial Natural Science Foundation of China [grant number 2016JJ5024]; Scientific Research Fund of Hunan Provincial Education Department [grant number 15B084].
Notes on contributors
J. C. Dai received PhD degree in mechanical engineering at Central South University (China) in 2011 and is a visiting scholar in Newcastle University (UK) in 2016. He is currently an associate professor in Hunan university of Science and Technology in China. His main research interests are wind power technology and equipment.
Z. Q. Liu received MS degree in mechanical engineering at Hunan university of Science and Technology (China).
X. Liu received MS degree in mechanical engineering at Hunan university of Science and Technology (China).
S. Y. Yang received PhD degree in mechanical engineering at China University of Mining. She is a professor in Hunan university of Science and Technology. Her main research interests are Mechanical dynamics and optimal design.
X. B. Shen is an engineer in XEMC Windpower Co., Ltd. He received MS degree in mechanical engineering at China University of Geosciences.
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J. C. Dai, Z. Q. Liu, X. Liu, S. Y. Yang, and X. B. Shen
School of Electromechanical Engineering, Hunan University of Science and Technology, Xiangtan, China
CONTACT J.C. Dai firstname.lastname@example.org
Received 16 January 2015
Accepted 9 February 2017
Table 1. Factor level table of tower structure parameters. (Dai et al. 2013). A B C Factor [[delta].sub.1]/m [[delta].sub.2]/m [[delta].sub.3]/m Level 0.008 0.018 0.025 0.01 0.021 0.030 0.012 0.025 0.035 0.014 0.028 0.040 0.016 0.032 0.045 D E F Factor [H.sub.1]/m [H.sub.2]/m D/m Level 8 25 3.6 12 28 3.8 16 31 4.0 20 34 4.2 24 38 4.4 Table 2. computation error and time for different models. Stress Top displacement Output RAM FEA RAM FEA Error/% -6-8 - -5-7 - Time/s 0.68-1.2 3.43-3.86 0.64-1.05 3.43-3.86 Natural frequency Output RAM FEA Error/% -2-2 - Time/s 0.65-1.13 9.95-10.81 Table 3. Results of tower structure parameter optimisation. Parameter Before optimisation After optimisation [[delta].sub.1]/m 0.014 0.013 [[delta].sub.2]/m 0.022 0.020 [[delta].sub.3]/m 0.028 0.026 [H.sub.1]/m 17 16 [H.sub.2]/m 25 34 D/m 4.2 4.3 Top displacement/m 0.537 0.544 Mass/kg 104000 96399 The maximum stress/ 125 128 MPa Natural frequency/Hz 0.4788 0.4791
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|Author:||Dai, J.C.; Liu, Z.Q.; Liu, X.; Yang, S.Y.; Shen, X.B.|
|Publication:||Australian Journal of Mechanical Engineering|
|Date:||Mar 1, 2018|
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