Dynamic soil-structure interaction issues of offshore wind turbines

Transcription

1 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: ; ISBN: Dynamic soil-structure interaction issues of offshore wind turbines Laszlo Arany 1, Subhamoy Bhattacharya 2, S. J. Hogan 3, John Macdonald 4 1 Department of Engineering Mamatics, University of Bristol, Office 1.80 Queens Building, University Walk Clifton BS8 1TR, PH (+44) , laszlo.arany@bristol.ac.uk 2 Department of Civil and Environmental Engineering, University of Surrey, Guildford GU27 XH, UK 3 Department of Engineering Mamatics, University of Bristol, Office 2.26, Merchant Venturers Building, University Walk Clifton BS8 1TR, PH (+44) (0) , s.j.hogan@bristol.ac.uk 4 Department of Civil Engineering, University of Bristol, Office 2.36 Queens Building, University Walk Clifton BS8 1TR, (+44) (0) ABSTRACT: One of important design drivers for offshore wind turbine (OWT) structures is fatigue life. In order for such structures to make worthwhile investments, y need to be in operation for years after installation. wind turbine and foundation are subject to fatigue damage from environmental loading (wind, waves) as well as from cyclic loading imposed through rotational frequency (1P) through mass and aerodynamic imbalances and from blade passing frequency (3P) of wind turbine. Through dynamic amplification and resonance, fatigue damage suffered by structure can severely increase if natural frequency of wind turbine gets close to frequency of excitation, reby reducing service lifetime of OWT. refore, predicting first natural frequency is of paramount importance. In this paper a mechanical and mamatical model is presented, which provides a good initial estimate of natural frequency of OWTs for conceptual design. soil-structure interaction (SSI) is modelled through a set of springs, which also includes crosscoupling between lateral and rotational stiffness of foundation. Approximate analytical formulae are given to approximate natural frequency. results are compared to measured data as well as results from similar software. sensitivity of natural frequency of structure to stiffness parameters of foundation are analysed and discussed. KEY WORDS: Offshore wind turbine; Euler-Bernoulli beam ory; Soil-structure interaction; Natural frequency; Cross stiffness; Sensitivity analysis. 1 INTRODUCTION Offshore wind farms (OWF) are expected to become significant contributors to electricity production in future in Europe and worldwide. To make m a cost-effective alternative to fossil fuel power plants, offshore wind turbines (OWTs) are usually designed to be operational for at least years. OWTs are subjected to intensive dynamic loading in a wide frequency band during ir lifetime. main dynamic loads are environmental loading from wind turbulence and wave loading, and mechanical loading from aerodynamic- and mass imbalance of rotating rotor (1P frequency band) and blade passage (3P frequency band) in front of tower. structures need to survive a large number of load cycles and refore fatigue damage is an important design driver in OWT technology. Offshore wind turbines are slender columns with a heavy mass on top: y are dynamically sensitive structures [1] [3]. refore, it is essential that structure is designed such that its natural frequency is reasonably far from frequency bands of excitations in order to minimise fatigue damage and achieve a long service lifetime. Furr details on loading and frequency bands associated with loadings can be found in [1]. Designing support structure and foundation to fit se criteria is a challenging task. It requires estimation of stiffness of foundation, which involves soil-structure interaction (SSI), a source of uncertainty. Furrmore, re are also dynamic issues related to soil stiffness properties, which may change over time due to cyclic/dynamic excitation, as was demonstrated in [4] [6] Change in natural frequency of an OWT over time was reported in [7]. Measured natural frequencies at Walney site were reported to be 6-7% higher than design value [8]. Depending on natural frequency of wind turbine structure, three forms of design are adopted: soft-soft, softstiff and stiff-stiff. Among se, soft-stiff is current preferred design option whereby natural frequency is designed to be within 1P (rotational frequency) and 3P (blade passing frequency). It is to be noted here that neir underestimation nor overestimation of natural frequency of OWT is conservative, as fatigue damage may increase due to dynamic amplification with frequency change in any direction. Some cases of fatigue type failure of OWTs (specifically failure of grouted connection between tower and transition piece) have also been reported [9]. A posteriori changes in an offshore environment are very expensive, however, and are to be avoided. In this paper an attempt is made to provide a simple and quick method to estimate first natural frequency of an OWT for conceptual design phase in order to provide a means for incorporating fatigue in early stages of design. In this formulation only basic information about particular wind turbine and site is required. Furrmore, analytical formulae are provided to analyse sensitivity of natural frequency to changing soil parameters. 2 MODEL OF THE OWT CONSIDERING SSI INCLUDING CROSS-COUPLING TERM A typical offshore wind turbine supported on a monopile foundation is shown in Figure 1. main structural elements of an OWT are rotor, nacelle, tower, substructure and 3611

2 foundation. slender columns are typically connected to substructure via a transition piece (TP). most common foundations are monopiles, gravity base and jacket structures [10], although floating turbines are also being tested. In this paper a simplified mechanical model is used, whereby rotor-nacelle assembly is modelled as a top head mass with rotational inertia, tower is modelled as an Euler- Bernoulli beam, and foundation stiffness is modelled by three springs (lateral-, rotational- and cross springs). omitted here and only final results are given; details can be found in detail in [1], [2], [12]. equivalent bending stiffness for calculation of non-dimensional axial force is given in Equation 2-4. (2) (3) 1 1 (4) where and are bottom and top diameters of tower, respectively, is area moment of inertia of tower cross section at bottom. equivalent stiffness for non-dimensional stiffness parameters is given in Equation 5-6. (5) (6) where is bending stiffness at top. rotor-nacelle assembly is modelled as a lumped mass on top of tower, with mass moment of inertia (or rotational inertia). Due to gravity, this mass also exerts an axial force along tower. Table 1. Parameters of natural frequency calculation. Figure 1. Model of offshore wind turbine. For simplicity, several parameters are introduced in Table 1 (for derivation see see [1]). three springs model of foundation stiffness is described by Equation 1. (1) where, and are lateral, rotational and cross stiffness, respectively, and are displacement and slope at foundation, respectively, and and are reactions (see Figure 1 for coordinates). In absence of more detailed information and formulae, we used Eurocode 8 Part 5 [11] Some methods for obtaining soil stiffness parameters are given in [1]. tower of length is modelled as an Euler-Bernoulli beam, using mass per length, equivalent bending stiffness. Equivalent bending stiffness needs to be calculated because tower is tapered. calculations are Dimensionless group Non-dimensional lateral stiffness Non-dimensional rotational stiffness Non-dimensional cross stiffness Non-dimensional axial force Formula Mass ratio (top head mass / tower mass) Non-dimensional rotary inertia Frequency scaling parameter Non-dimensional rotational frequency ( is rotational frequency) Ω Euler-Bernoulli beam equation with compressive axial force and without excitation is written in Equation 7.,,, 0 (7) where is displacement in direction, is time, is equivalent bending stiffness of tower, is 3612

3 mass per length of tower, is axial force (see Figure 1). Assuming constant, and, using parameters introduced in Table 1, separating variables with, and using non-dimensional coordinate /, Euler-Bernoulli beam equation can be simplified to form given in Equation 8 Ω 0 (8) Looking for a solution in form following characteristic equation can be written: Ω 0 (9) Replacing with we get following: Ω 0 (10), Ω Ω, Ω (11) solution can be written in following form: cos sin cosh with Using vector notation: cos sin cosh sinh sinh (12) and (13) P P boundary conditions are also formulated with nondimensional parameters introduced in Table 1. derivations are given in [1] in detail. I. sum of shear forces at bottom is zero: (14) II. sum of bending moments at bottom is zero: 0, 0, 0, 0 (15) III. sum of shear forces at top is zero: 1 1 Ω 1 0 (16) IV. sum of bending moments at top is zero: 1 Ω 1 0 (17) Substituting solution for given in Equation 12 into boundary conditions and looking for non-trivial solutions, one obtains matrix shown in Equation 18. sinh αω cosh cosh Ω sinh cosh αω sinh sinh Ω cosh (18) determinant of this matrix is set to zero to find natural frequency. det 0 (19) This determinant produces a non-linear transcendental equation, which has to be solved numerically. 3 APPROXIMATE FORMULAE In order to study dependency of natural frequency on foundation stiffness parameters, analytical approximations are formulated to fit solutions calculated by numerically solving transcendental equation given in Equation 19. natural frequency is expressed in terms of six main parameters,,,,, as defined in Table 1 (see Equation 20.),,,,, (20) First fixed base value of natural frequency is calculated, which is natural frequency on a perfectly stiff foundation with,, 0. For this calculation some initial values of axial force, mass ratio and rotational inertia parameters are selected:,,. Using se values fixed base natural frequency is expressed as:,, 0,,, (21) This fixed base frequency is practically calculated for a vertical cantilever beam carrying an end mass. This frequency can be calculated by standard formulae for uniform beams [13]: and (22) where is mass of rotor-nacelle assembly, is mass of tower, is stiffness of 1 degree of freedom system, is length of tower and is equivalent bending stiffness of tower. More accurate estimate of natural frequency may be obtained by using equivalent stiffness and equivalent mass formulae given in [3]. dependency of natural frequency on parameters are determined by separation of variables. flexible foundation is taken into account by coefficients and as given in Equation 23. (23) Once fixed base natural frequency is available, se coefficients are calculated to incorporate effects of a flexible foundation, using three springs model shown in Equation 1. two coefficients and represent rotational and lateral stiffness dependency, respectively. Both coefficients are, however, dependent on all three variables, as shown in Equation 24.,,,,. (24) expressions given in Equation 25 and 26 were found to approximate numerically calculated curves well. 3613

4 ,, 1,, 1 ; conditions set in Equation 25 and 26 are necessary to capture effect of cross coupling stiffness. Referring to Figures 2 and 3, natural frequency approaches zero as lateral or rotational stiffnesss value decreases. In case without cross coupling (short dashed lines in Figure 2-3.) natural frequency tends to zero as stiffness values tend to zero. inclusion of cross coupling term shifts this point from 0 to / and / for rotational and lateral stiffness, respectively. In se low stiffness regions stability of system is not guaranteed and thus it is to be avoided. In Equations 25 and 26 coefficients and are determined empirically as 0.6, 0.5 (27) se values were calculated using one particular wind turbine, Siemens SWT y proved to be reasonably accurate for or wind turbines as well (see case studies in Section 4) (seee turbine D in Section 4). However, range of validity of and is constrained by conditions set in Equations 25 and 26; close to limit values errors increase. A practical rule for validity of constants may be given as 1.2 and 1.5 (28) In se cases error is below 1%. Figure 2 and 3 show accuracy of approximation for and, respectively. It also includes curves for no cross coupling case and shows typical values of and calculated for Siemens SWT wind turbine at Walney 1 site [14] (see wind turbine D in Section 4.) ; (25) (26) 4 NATURAL FREQUENCY RESULTS natural frequency of several wind turbines with data available in literature was determinedd to address accuracy of analytical formulation given in Section 2. following four OWTs are used: A: Lely A2 NedWind 40/500 2 bladed 500kW study purposes wind turbine (Nerlands) B: North Hoyle, Vestas V66 3 bladed 2MW industrial wind turbine (UK) C: Irene Vorrink, NordTank NTK 600/43 3 bladed 600kW wind turbine (Nerlands) D: Walney 1, Siemens SWT bladed 3.6MW industrial wind turbine (UK) Figure 3. Numerically calculated and approximated curve of natural frequency as a function of non-dimension nal lateral stiffness. calculated non-dimensional parameters of OWTs are presented in Table 2 ( non-dimensional rotational inertia was taken as zero in all cases due to lack of data). calculations and data are found in [1], [3], [6]. soil stiffness parameters, and are taken based on formulae taken from Eurocode 8 [11] keeping in mind limitations. Table 2. Data of several Offshore Wind Turbines Parameter A 2698 B C 5880 D Figure 2. Numerically calculated and approximated curve of natural frequency as a function of non-dimensional rotational stiffness

5 Proceedings of 9th International Conference on Structural Dynamics, EURODYN 2014 natural frequency results computed using Euler- software using methodology of Section 2. results given by software are compared to approximations provided by formulae of Section 3 in Table 3. Bernoulli beam ory are calculated by authors accuracy of results is acceptable for preliminary design. It is important to note that analysis requires minimal data about offshore wind turbine or site, and that re is highh uncertainty in parameters especially in soil stiffness values. Also, as it was mentioned before, frequency may deviate from its initial value with time as well [4] [6]. Table 3. Natural frequency results A. B. C. D. Measured N/A However, necessary adjustments needs to be made to take into account tapered shape of column and flexible foundation [1], [3], [6]. It must be added that a wind turbine will be very far from bucking at all times. refore, approximating critical load with formula of Equation 32 is reasonably accurate. Equally, curve can be approximated by following linear relationship (33) where is determined empirically as numerically calculated and approximation curves are shown in Figure 4, as well as simplified expressions of Equation 32 and 33. Calculated Error % 11.7% N/A Approximation % 5.4% METHODOLOGY FOR SENSITIVITY ANALYSIS method of sensitivity analysis adopted in this paper is based on determining approximate analytical expressions for dependence of natural frequency on different variables. Using se approximate formulae partial derivatives with respect to each variable can be calculated, which describe slope of tangential line at initial point for each variable, giving a good indicator of sensitivity. approximate formula to incorporate stiffness parameters was developed in Section 3. soil stiffness parameters are subject to high uncertainty and can also change over time, refore sensitivity of se parameters is most important to analyse. However, dependence on or three main parameters,, is also investigated in this section as it may be useful for conceptual design. In se cases coefficients are expressed in terms of initial values,, that were used to determine fixed base natural frequency (see Equation 21). Using se values natural frequency is expressed as (29) Substituting initial values,, into formulae for coefficients,,, y are all unity and in thatt case Equation 29 is equivalent to Equation 24. following expression fits results well for changing non-dimensional axial force: (30) where is non-dimensional axial force at buckling expression for critical force a uniform cross section cantilever column is given by [13] (31) (32) Figure 4. Numerically calculated and approximated curves of natural frequency as a function of non-dimension nal axial force. dependencee on masss ratio is approximated with coefficient. constant is determined empirically to be 4. numerically calculated and approximated curves shown in Figure 5, good fit is apparent. (34) are Figure 5. Numerically calculated and approximated curve of natural frequency as a function of mass ratio 3615

6 Finally, dependence on non-dimensional rotational inertia is expressed with. (35) parameter d is determined empirically and its value was found to be 2.2. curves are compared in Figure 6. 1 With this partial derivative can be written as: 1 (37) (38) 6.2 Non-dimensional laterall stiffness Using Equation 37 as a basis, partial derivative is given similarly: C 6.3 Non-dimensional cross stiffness partial derivative with respect to can be expressed as: (39) ( (40) Figure 6. Numerically calculated and approximated curve of natural frequency as a function of non-dimensional rotational inertia. natural frequency can now be expressed as a function of fixed base natural frequency and six parameters using Equation 29 with coefficients given in Equations 25, 26, 30, 34, 35. constants,,,, in coefficients were determined empirically. In this work y were calculated for Siemens SWT wind turbine (denoted by D in Table 2). values are given in Equation , 0.5, 4, 2.2,0.075 (36) initial values for this wind turbine are 0.043, 0.9, 0. non-dimensional axial force at buckling is SENSITIVITY ANALYSIS Having obtained an approximate analytical formula for natural frequency as a function of six parameters,,,,,, sensitivity of natural frequency to se parameters can be analysed. partial derivatives of this approximate expression with respect to different variables gives slope of curves at initial point. This slope is used to characterise sensitivity of natural frequency to each parameter. partial derivative of with respect to a parameter is determined by assuming that coefficients in Equation 24 thatt do not contain given parameter are constant. 6.1 Non-dimensional rotational stiffness For partial derivative with respect to Equation 29 can be written with,,. 6.4 Non-dimensional axial force partial derivative can be written using n Orwise, using simple linear expression of Equation 33, slope is simply. 6.5 Mass ratio partial derivative is written using and result is 6.6 Non-dimensional rotational inertia partial derivative is written using and result is,,,,,,,, 1,,,, 1 ( (41) (42) (43) (44) (45) (46) Table 4 shows partial derivatives at parameter values given in Table 2, that is, slope of tangential lines at marked values in Figures 2-6. As an example, column A in table refers to Lely A2 500kW wind turbine. row corresponding to contains value , which means that for a unit change in (±1), natural frequency 36166

7 increases by Hz. Similarly, row corresponding to for example mass ratio contains negative value of , which corresponds to decreasing natural frequency with increasing mass ratio. Clearly, slopes are meaningful only for small variations in parameters, for significant changes graphs in Figures 2-6 are to be used. According to Kühn [7], soil stiffness parameters had an uncertainty of - 20% to +40% in case of Lely A2 wind farm. Table 4. Sensitivity of natural frequency for parameters. Sensitivity of OWT Parameter A B C D 8.32E E E E E E E E E E E E E E E E E E E E E E E E-01 It is important to note it is more meaningful to incorporate orders of magnitude of parameters in order to compare sensitivity. For example, one can look at change in natural frequency if parameters change by 1%. Table 5 shows result of this analysis for soil stiffness parameters. Table 5. Frequency change with 1% parameter change. Sensitivity of OWT Parameter A B C D It is clear from Table 5 that natural frequency is most sensitive to rotational stiffness, and refore overturning moment resistance is most important task. lateral and cross stiffness parameters show equal sensitivity, but both are generally an order of magnitude smaller than rotational stiffness. 7 CONCLUSION In this paper a methodology was presented for calculating natural frequency of an offshore wind turbine structure on flexible foundation using Euler-Bernoulli beam ory and a three spring model to take into account flexible foundation and soil-structure interaction. analysis yielded a non-linear transcendental equation that needs to be solved numerically. For preliminary design it is useful to formulate a simplified expression for natural frequency, refore analytical formulae were derived to approximate numerically calculated natural frequencies. approximation curves incorporate dependence of natural frequency on lateral, rotational and cross stiffness parameters, as well as axial force in column, mass ratio of rotor-nacelle assembly and tower, and rotational inertia of rotornacelle assembly. analytical formulae were found to approximate numerically obtained results reasonably well. approximate formulae were also useful to analyse sensitivity of natural frequency to each parameter. It was shown that parameters with likely changes and high uncertainty are foundation stiffness parameters. Among se soil stiffness parameters rotational stiffness ranks as most important variable with highest sensitivity. REFERENCES [1] S. Adhikari and S. Bhattacharya, Dynamic analysis of wind turbine towers on flexible foundations, Shock Vib., vol. 19, no. 1, pp , [2] S. Bhattacharya, D. Lombardi, and D. Muir Wood, Similitude relationships for physical modelling of monopile-supported offshore wind turbines, Int. J. Phys. Model. Geotech., vol. 11, no. 2, pp , Jun [3] S. Adhikari and S. Bhattacharya, Vibrations of wind-turbines considering soil-structure interaction, Wind Struct., vol. 14, no. 2, pp , [4] D. Lombardi, S. Bhattacharya, and D. Muir Wood, Dynamic soil structure interaction of monopile supported wind turbines in cohesive soil, Soil Dyn. Earthq. Eng., vol. 49, pp , Jun [5] S. Bhattacharya, N. Nikitas, J. Garnsey, N. A. Alexander, J. Cox, D. Lombardi, D. Muir Wood, and D. F. T. Nash, Observed dynamic soil structure interaction in scale testing of offshore wind turbine foundations, Soil Dyn. Earthq. Eng., [6] S. Bhattacharya and S. Adhikari, Experimental validation of soil structure interaction of offshore wind turbines, Soil Dyn. Earthq. Eng., vol. 31, no. 5 6, pp , May [7] M. Kühn, Soft or stiff: A fundamental question for designers of offshore wind energy converters, in Proc. European Wind Energy Conference EWEC 97, [8] D. Kallehave and C. L. Thilsted, Modification of API p-y Formulation of Initial Stiffness of Sand, in Offshore Site Investigation and Geotechnics: Integrated Geotechnologies - Present and Future, [9] I. Lotsberg, Structural mechanics for design of grouted connections in monopile wind turbine structures, Mar. Struct., vol. 32, pp , Jul [10] European Wind Energy Association, European offshore wind industry - key trends and statistics 2012, [11] European Committee for Standardization, Eurocode 8: Design of Structures for earthquake resistance - Part 5: Foundations, retaining structures and geotechnical aspects, [12] S. Bhattacharya, SDOWT: USER MANUAL (Simplified Dynamics of Wind Turbines), Bristol, [13] R. D. Blevins, Formulas for Natural Frequency and Mode Shape. Krieger Publishing Company, [14] DONG Energy, Walney Offshore Wind Farm - Facts of project, [Online]. Available: project/ pages/facts.aspx. [Accessed: 06-May-2013]. 3617

8