Foundation models for the dynamic response of offshore wind turbines

Transcription

1 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. Foundation models for the dynamic response of offshore wind turbines. M.B. Zaaijer, MSc Delft University of Technology, The Netherlands SYNOPSIS Due to the variety of excitation frequencies and the larger influence of the foundation on the wind turbine response, modelling of the dynamic behaviour of the foundation becomes a more pronounced issue for offshore wind turbines. This paper investigates the sensitivity of the support structure s natural frequency to variation in models for pile foundations of a monotower, tripod and lattice tower. The use of a stiffness matrix at mudline provides a significant reduction of complexity and results in acceptable loss of accuracy. Uncoupled springs and an effective fixity depth model are discouraged. For the tripod and lattice structure of this study lateral flexibility of the foundation is more important than axial flexibility. INTRODUCTION An important aspect of extreme- and fatigue loading of the support structure of an offshore wind energy converter (OWEC) is its dynamic response. Its dynamic behaviour differs in some important aspects from that of platforms for the offshore oil industry and of onshore wind energy converters. The natural frequency of an OWEC is wedged between different excitation frequencies, whereas the natural frequency of a platform for the offshore oil industry is usually designed to be above all main excitation frequencies. The geometry and dimensions of offshore foundations differ from typical onshore solutions, resulting particularly in a larger influence of foundation stiffness. Due to the variety of excitation frequencies and the larger influence of the foundation on the wind turbine response, modelling of the dynamic behaviour of the foundation becomes a more pronounced issue for offshore wind turbines. This paper investigates the sensitivity of the support structure s natural frequency to variation in models for pile foundations. The assessment is performed for a monotower, a tripod and a lattice tower. These support structures are designed for North Sea conditions with water depths between 15 and 5 m and for a 3 MW turbine. The monotower and lattice tower are based on the design solutions of the Opti-OWECS project [4]. The tripod is designed by Heerema and was used in a study of the simulation of offshore wind turbines under stochastic loading [5]. The main dimensions of the support structures are given in Table 1. Table 1 Main properties of analysed support structures Monotower Tripod Lattice tower Hub height (m) Rotor-nacelle mass (kg) 130, , ,000 Water depth (m) Piles Cross section (m x m) 3.5 x x x 0.0 Penetration (m) Tower Pile distance to centre (m) Braces (m x m) x Diameter (m) (one transition) (in steps) (different members) Wall thickness (m) (in steps) (in steps) (different members) 1 st Natural frequency (Hz) nd Natural frequency (Hz) Authors Biographies Michiel Zaaijer is presently research scientist in offshore wind energy at Delft University of Technology. His research focuses on dynamic behaviour of offshore turbines and integrated wind farm design methods. After his M.Sc. in physics in 1993 he has worked for several years as a researcher in the field of aircraft navigation.

2 The foundation piles of all structures are larger than strictly required to provide sufficient bearing. However, these piles have the same stiffness as infinitely long piles and therefore the dynamic behaviour of the support structures is insensitive to pile penetration depth. For the analysis of the natural frequencies of the support structures a finite element model of the construction is made in ANSYS. For the foundation, a finite element model of the pile with distributed springs for pile-soil interaction is used as a reference model (see Section Description of foundation models ). PHYSICAL BACKGROUND FOR FREQUENCY ANALYSIS A structure in free vibration does not exchange energy with the environment. Thus, during each vibration cycle kinetic and strain energy in the structure are exchanged. The kinetic and strain energy can be calculated for an assumed vibration, z() t Ψ( s), where z gives the time dependency and Ψ the vibration shape parameterised with s. For a harmonic vibration, equation of the maximum kinetic and strain energy leads to the following expression for the vibration frequency, ω : with: k () s = distributed stiffness, m () s = distributed mass, s ( ()) k () s f ( Ψ() s ) m() s Ψ() s ω =, (1) f Ψ = local displacement or curvature, depending on the type of strain energy. The lowest possible frequency that can be obtained in this way is the first natural frequency of the system and the corresponding vibration shape is the first mode shape. This method is called after Rayleigh and is described for instance in [3]. To get more insight, Rayleigh s method is used to obtain an expression for the sensitivity of the first natural frequency, ω, to changes in a physical parameter, x. Derivation of Equation 1 leads to: 1 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. ( Ψ) k m f Ψ f ( Ψ) ω Ψ k ω m ω = + m Ψ m Ψ. () Because the first natural frequency is the minimum value obtained with Equation 1, by varying the vibration shape Ψ, the derivative of Equation 1 with fixed k and m must be zero around the first mode shape, Ψ 1, resulting in: ω Ψ=Ψ1 = ( Ψ) f k m Ψ ω Ψ m Ψ=Ψ1 = 0. (3) Combining Equations and 3 for ω = ω1 and Ψ = Ψ1 leads to: ω1 = k f ( Ψ ) ω x m Ψ1 1 m Ψ 1. (4) k m Substituting = a1 k and = a m in Equation 4 shows that the distribution of strain energy, k f ( Ψ), and kinetic energy, ω m Ψ, are useful parameters to analyse both the absolute natural frequency of Equation 1 and the sensitivity of Equation 4. The latter substitution is based on the idea that relevant changes in stiffness and mass are several percent of the absolute stiffness and mass. For each of the support structures these parameters are plotted in Figure 1, scaled with the total energy of the vibration. Table gives the integrated relative energy values for parts of the structure.

3 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September Kinetic rotor-nacelle: 0.78 Kinetic rotor-nacelle: 0.6 Kinetic rotor-nacelle: Height above seabed (m) Kinetic (structure) Strain (structure) Strain (soil - lateral) Strain (soil - axial) Relative energy (per m) Monotower Tripod Lattice tower Figure 1 Relative energy distribution of support structures For all support structures the kinetic energy is concentrated in the rotor-nacelle assembly, with a small contribution of the tower near the top. The jagged character of the distributions is caused by the transitions in the structure and meshing of the finite element model. The monotower strain energy increases where the tower diameter decreases at approximately 1 m, but experiences a dip at the very rigid grouted connection between 16 and 1 m. Strain energy decrease toward the top, where the bending curvature decreases and the largest bending in the pile occurs a few meter below the seabed. Strain energy in the tripod largely occurs near the joint of the braces at 5 m. Below the joint the high stiffness of the tripod results in very small deformations. The large, but narrow, peak of strain energy near the seabed is caused by the accumulated strain energy of the base members. The strain energy distribution of the lattice tower is very jagged, due to changes in the geometry and in member dimensions. Apparently the section around 5 m is fairly rigid. For all structures the strain energy associated with the lateral compression of the soil is concentrated in the first 10 m below the seabed. The strain energy associated with the axial shear of the soil, which is only relevant for the tripod and lattice tower, is very small. Table Relative energy of parts of the support structures Relative kinetic energy (-) Relative strain energy (-) Monotower Tripod Lattice tower Rotor-nacelle Tower Pile Tower Pile Soil - lateral Soil - axial Table shows that contribution of the kinetic energy of the pile is negligible, which is due to the small displacements. Mass of the soil is not included in the finite element model, but it is expected that kinetic energy of the soil is similarly negligible. For the monopile the strain energy in the pile exceeds the strain energy in the soil, which indicates dominance of pile stiffness over soil stiffness. The difference is less pronounced for the tripod and the lattice tower. The results for the relative strain energy of the foundations lead to expect the largest sensitivity to foundation parameters and model for the monopile. The tripod and particularly the lattice tower have larger strain energy in lateral compression of the soil than in axial shear. This indicates larger sensitivity to lateral behaviour, especially because pile strain energy is expected to be contained mainly in lateral bending.

4 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. DESCRIPTION OF FOUNDATION MODELS Introduction The following models for the foundation are analysed in this paper: finite element model with distributed springs for lateral and axial soil stiffness, effective fixity length of the foundation pile, stiffness matrix of foundation behaviour at mudline, uncoupled lateral, axial and rotational springs at mudline. The finite element model is used as reference. The models are illustrated in Figure, which also shows two different approaches to obtain the lateral and rotational spring stiffness. Pile wall F k = M k xx θx k k xθ θθ u θ F u M θ u F θ M. Stiffness matrix Ignore θ Ignore u Applied force/moment Ignore M Ignore F Forced displacement/rotation Reference: Distributed springs 1. Fixity length 3. Uncoupled springs 3.a Force method 3.b Displacement method Figure Foundation models Reference model The reference model with distributed springs is commonly applied in the offshore oil and gas industry. The stiffness of the springs in the reference model is determined according to the recommendations in [1] for cyclic loading. The springs represent lateral resistance, internal and external shaft friction and pile tip and plug resistance. In accordance with Winkler s assumption the springs are considered to be independent. The springs have a non-linear stress-strain relation and are linearised around unloaded conditions. Effective fixity length A simple model of the clamping effect of the soil is replacement of the soil by rigid clamping of the pile at an effective depth below the seabed. This model is sometimes used for (preliminary) dynamic analysis of structures for the offshore oil and gas industry, using tabulated values for the effective fixity length. The values proposed by in [] are given in Table 3 as a function of the soil type and pile diameter, D, along with values obtained from analysis of an offshore wind turbine support structure [6]. Table 3 Effective fixity lengths, suggested in literature Configuration Stiff clay Very soft silt General calculations From measurement of an offshore turbine (500 kw) Effective fixity length 3.5 D D 7 D - 8 D 6 D 3.3 D D Stiffness matrix For pile foundations a stiffness matrix can express the stiffness of the pile-soil system at the seabed. The stiffness matrix gives the forces, F, and moments, M, for displacements and rotations of the pile head. The relevant degrees of freedom of a laterally loaded pile are the horizontal translation, u, in the plane of interest and the rotation, θ, about the horizontal axis perpendicular to this plane. Two methods are used to obtain the values for the elements of the stiffness matrix, as described below:

5 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. Load displacement analysis with p-y curves The unknown elements of the stiffness matrix can be solved from the set of algebraic equations that is obtained from two load case analyses of the reference model. For the applied load cases the deflection of the foundation is in the linear region of the stress-strain curves, which is representative for small vibrations. The second load case is sufficiently different from the first one to give independent equations. For the tripod and lattice tower the stiffness matrix is combined with a spring for the vertical degree of freedom. The spring stiffness is determined from an axial load case of the reference model. Randolph elastic continuum model Randolph performed dimension analysis and finite element analysis of piles in an elastic continuum to obtain an expression for the pile head flexibility [7]. This resulted in parameterised flexibility matrices for piles in an elastic continuum with a constant soil shear modulus and with a linearly increasing soil shear modulus. The latter is the preferred model for sandy soil, where the soil shear modulus varies with depth and generally increases due to increasing effective vertical soil pressure. The stiffness matrix for Randolph s approach is the inverse of the flexibility matrix he obtained in his study. The rate of change of the soil shear modulus is obtained from a linear fit to the actual modulus. For the tripod and lattice tower the stiffness matrix is not combined with a spring for the vertical degree of freedom. Uncoupled springs In this model the coupled stiffness of the pile head is simplified to independent springs for each relevant degree of freedom. In Figure the springs are illustrated for lateral translation and rotation. For the tripod and lattice tower this is combined with a spring for the vertical degree of freedom. To obtain the stiffness of the spring elements the two approaches of Figure are applied, as described below: Force method In the Force method the stiffness is obtained by application of a force or moment on the pile head in the reference model, without constraining the response. The response will consist of a rotation and a translation of the pile head, but only the corresponding degree of freedom will be considered. Displacement method In the Displacement method a deflection is applied to the pile head in the reference model and the corresponding required force is used. The force corresponding to the other degree of freedom is ignored. The resulting stiffness is equivalent to the diagonal terms of the stiffness matrix for the same load conditions. COMPARISON OF FOUNDATION MODELS For each of the support structures and foundation models the first and second natural frequency are determined and normalised with the results for the reference model. The results are plotted in Figure st natural frequency (normalised) 1.0 Monotower Tripod Lattice tower nd natural frequency (normalised) Winkler FEM (reference) 8D 6D 4D D (Seabed) 0D FEM-based Randolph Force method Displacement method Only axial springs Effective fixity depth Stiffness matrix Springs Only lateral stiffness matrix Only rotation (Force method) Winkler FEM (reference) 8D 6D 4D D (Seabed) 0D Effective fixity depth FEM-based Randolph Force method Displacement method Only axial springs Stiffness matrix Springs Only lateral stiffness matrix Only rotation (Force method) Figure 3 Predicted natural frequencies for several foundation models

6 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. Since the effective fixity depth cannot be determined a priori in a rigorous way, a variety of clamping depths is analysed. Additionally, the effect of constraining some degrees of freedom is determined to assess the relative importance of lateral and rotation flexibility of the foundation. Note that only the Randolph stiffness matrix and the effective fixity depth model are independent of the reference model, which is used for pre-analysis of the other foundation models. Several observations from Figure 3 are highlighted below: Effective fixity depth Both first and second natural frequency of the tripod and lattice tower correspond with the reference value for an effective fixity depth of approximately 6 times the pile diameter. This is in agreement with the suggested value for general calculations for offshore platforms. This can be expected, since the vibration shape and pile diameter are similar to that of fixed space frame type offshore structures. The effective fixity depth of the tubular tower gives better results for lower values, in the order of 4 times the pile diameter. This is in agreement with earlier studies of monopile behaviour. The difference between the appropriate values may partly be contributed to the larger diameter of the pile, but the different mode shape of the vibration is probably also an important aspect. The results of this model are very sensitive to the selected effective fixity depth, particularly for the tubular tower. The tabulated values of the effective fixity depth as shown in Table 3, show a large variation as a function of soil conditions. Therefore, large inaccuracies of natural frequency must be anticipated for a priori assumed fixity depths. Randolph s linear elastic model The first natural frequencies obtained with Randolph s model correspond within.5 with the finite element model. Similar results were found for two other locations, with slightly different soil conditions. It must be noted that Randolph s model assumes that the piles are longer than a critical pile length and thus have the same behaviour as infinitely long piles. The lengths of the piles used in this study are also selected to show the same behaviour as infinitely long piles. When the pile length is below the critical pile length the increased flexibility of the pile will not be revealed by Randolph s model. Furthermore, the natural frequency with the finite element foundation model is obtained for the linear region of the stress-strain curves. When large deflections are expected the non-linear effect will not be revealed with Randolph s model, since this is a linear model. Note that part of the error made with Randolph s model for the tripod and lattice tower is caused by the omission of axial pile flexibility. Uncoupled lateral and rotational springs For the tubular tower the springs determined with the force method give closer correspondence of the first natural frequency than those obtained with the displacement method. This can be expected, since the shapes of the pile deflections of the first approach are close to the mode shape of the vibration, whereas those of the second approach are very different. For a similar reason the displacement method gives better results for the first natural frequency of the tripod and lattice tower. For higher natural frequencies the mode shapes of the vibration tend to deviate from the static deflection shapes and give worse results. Lateral behaviour and rotation For the monotower use of a rotational spring only does not result in a large difference with the results with uncoupled springs. However, part of the horizontal flexibility is implicitly present when the stiffness of the rotational spring is determined with the force method. For the tripod and lattice tower, the use of only a lateral stiffness matrix gives less deviation than the model with only axial springs. This indicates that the natural frequencies of the tripod and lattice tower are dominated by the lateral behaviour of the piles. Due to the lateral flexibility of the piles, which is at least an order of magnitude higher than the axial flexibility, the horizontal translation of the tower is a more important contribution to the vibration than its rotation. SENSITIVITY FOR FOUNDATION PARAMETERS A previous study focussed on the sensitivity of the natural frequencies to changes in physical parameters of the system, see [8]. The main results for foundation parameters are repeated here, to assess the differences found between different foundation models against the background of expected variation and inaccuracy. A distinction is made between the uncertainty of a parameter, its variation during the lifetime of the wind turbine and its variation within the wind farm. For several parameters that characterise the foundation, an indication of its variation is obtained from a literature survey and for this range the natural frequency has been determined with the finite element model. The results are summarised in Table 4.

7 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. Table 4 Sensitivity of first natural frequency to typical variation in physical parameter Parameter Monotower Tripod Lattice (Cohensionless soil) Uncertainty Lifetime Farm Uncertainty Lifetime Farm Uncertainty Lifetime Farm Effective soil unit weight Friction angle Coefficient of lateral earth pressure Modulus of subgrade reaction General scour Local scour Postholing gap Maximum As expected, the monotower is more sensitive to variation of soil parameters than the tripod and the lattice tower. Considering the low relative strain energy in the soil for all support structures (see Table ), the sensitivity is rather high. This is due to the very large variations in soil stiffness that occur. Scour results in a significant sensitivity, because the soil strain energy is concentrated in the upper soil layers, whose supporting function vanishes. DISCUSSION AND CONCLUSIONS The foundation models of this study are based on three different bases. The effective fixity depth model uses tabulated values of the clamping depth. The linear elastic model developed by Randolph uses an analytical expression for the elements of the stiffness matrix, based on pile and soil parameters. The reference model uses a finite element representation of the pile and non-linear pile-soil interaction. The other models share the finite element basis of the reference model, using a static pre-analysis to obtain model parameters. Suitable values for the effective fixity depth for a monopile and for the tripod and lattice tower differ, probably due to the different mode shapes of the vibration. Besides, the appropriate value of the fixity depth depends on both pile stiffness and soil properties and this dependency is not rigorously represented in tabulated values. Due to the large sensitivity of the predicted natural frequency to the effective fixity depth this model is strongly discouraged as an a priori model for analysis beyond an initial guess of support structure behaviour. An effective fixity depth could be determined from a reference model or from measurements to reduce complexity of the foundation model. However, a sensitivity study is always recommended when this model is applied. The results obtained with Randolph s linear elastic model are similar to the results of the reference model. However, the soil of this case study is nearly uniform, which makes it particularly suitable for the elastic model. Randolph s method will not reveal the influence of pile length and loading conditions and is therefore less suitable for piles with a penetration less than a critical pile length and for deflections outside the linear region of the soil-structure interaction. The first and second natural frequency obtained with a stiffness matrix with coupled lateral behaviour gives very good correspondence with the finite element foundation model for all assessed support structure types. The observed difference is far less than expected uncertainties in foundation behaviour. This stiffness matrix has far less degrees-offreedom than the comprehensive finite element model and will therefore reduce computations in dynamic analyses. Uncoupled springs for lateral displacement, rotation and axial displacement give larger errors, which can exceed the expected errors due to uncertainties. Since no significant reduction of degrees-of-freedom is obtained, the use of uncoupled springs is not recommended. For the monotower the expected dominance of rotation flexibility is confirmed by the results. For the tripod and lattice tower the lateral flexibility of the piles appear to be much more important than the axial flexibility. However, the relative importance of lateral and axial foundation behaviour depends mainly on lateral and axial foundation stiffness, spacing between the piles and the mass distribution of the tower and turbine. It requires further analysis to assess if the conclusions for these cases are generally valid. Note that the finite element model, which has been used as a reference throughout this study, will also differ from reality. This model is known to generally underestimate soil stiffness, resulting in a low prediction of the natural frequency. Additionally, errors can be made by imperfections in or incompleteness of the information of the turbine and support structure and by other modelling aspects. ACKNOWLEDGEMENT The author would like to acknowledge the contributions made by the partners in the OWTES project to this paper. The project Design methods for Offshore Wind Turbines at Exposed Sites is being funded by the European Commission under contract number JOR3-CT98-084, with co-funding of NOVEM under contract

8 Marine Renewable Energy Conference (MAREC), Newcastle, UK, September 00. REFERENCES 1. API, RP A-LRFD: API Recommended Practices for Planning, Designing and Constructing Fixed Offshore Platforms Load and Resistance Factor Design, July 1, Barltrop, N.D.P., Adams, A.J., Dynamics of Fixed Marine Structures, Butterworth-Heinemann Ltd, Oxford, Clough, R. W., Penzien, J., Dynamics of Structures, McGraw-Hill, New York, Ferguson, M.C. (editor); Kühn, M.; Bussel, G.J.W. van; Bierbooms, W.A.A.M.; Cockerill, T.T.; Göransson, B.; Harland, L.A.; Vugts, J.H.; Hes, R., Opti-OWECS Final Report Vol. 4: A Typical Design Solution for an Offshore Wind Energy Conversion System, Institute for Wind Energy, Delft, Kühn, M., Simulation of Offshore Wind Turbines Under Stochastic Loading, Proceedings of the 5 th European Wind Energy Association Conference and Exhibition, Greece, October Kühn, M., Overall Dynamics of Offshore Wind Energy Converters, Opti-OWECS Final Report Vol. : Methods Assisting the Design of Offshore Wind Energy Conversion Systems, Part D, Institute for Wind Energy, Delft, Randolph, M.F., The Response of Flexible Piles to Lateral Loading, Géotechnique, Vol. 31, no., Zaaijer, M.B., Vugts, J.H., Sensitivity of Dynamics of Fixed Offshore Support Structures to Foundation and Soil Properties, Proceedings of the European Wind Energy Conference and Exhibition 001, Denmark, July 001.