An overview of modeling approaches for complex offshore wind ...

An overview of modeling approaches for complex offshore wind turbine support structures P.L.C. van der Valk 1,2 , S.N. Voormeeren 1,2 1 Delft University of Technology, Faculty of 3ME, Section Engineering Dynamics, Mekelweg 2, 2628CD, Delft, The Netherlands e-mail: [email protected], [email protected] 2

Siemens Wind Power, Offshore Center of Competence Prinses Beatrixlaan 800, 2595BN, The Hague, The Netherlands

Abstract

The current trend in offshore wind energy is to install ever larger turbines in deeper waters farther offshore. As a result, monopile support structures become uneconomical and more complex types of support structures (e.g. jackets, tripods) are preferred. For certification purposes thousands of aero-elastic simulations are performed, taking into account the global dynamics of the complete structure. However, dynamic models of complex support structures require many more degrees of freedom (DoF) than monopile models, leading to excessive computation times. In this paper several modeling approaches for including these offshore support structures in aero-elastic simulations are discussed. All methods are based on the concept of dynamic substructuring, where the turbine model and support structure model are taken as separate substructures. Four different methods for including the support structure in the simulation are addressed: Guyan reduction, the (Augmented) Craig-Bampton method and the Impulse Based Substructuring method. These methods are applied to a numerical case study on the 5 MW NREL wind turbine on an offshore jacket. In order to resemble the practical design procedure for offshore wind turbines, a split is made between the wind turbine manufacturer and support structure designer. In practice, the former is responsible for the wind turbine tower and the coupled simulations of the total system while the latter takes responsibility for the detailed design of the offshore foundation. It is shown that the approach using Guyan reduction could lead to significant errors, as it neglects the internal dynamics of the support structure and thereby also the dynamic coupling with the wind turbine. Since the Craig-Bampton and Impulse Based Substructuring methods are both able to describe the internal dynamics of the jacket, these methods give much more reliable results.

1 Introduction In an effort to come to a more sustainable society, more renewable ways of generating electricity are being implemented throughout the world. As a result of this, we have seen an exponential increase in the number of wind turbines installed, both onshore and offshore. As many of the favorable onshore locations are usually also densely populated and therefore not suited for large wind farms, many of these farms will have to be built offshore. In addition to this, offshore locations offer a number of other advantages, such as a higher and more uniform wind (hence, less turbulence and wind shear) and, in theory, there are no size constraints for offshore wind turbines. In order to bring down the levelized cost of offshore wind energy, larger turbines are being developed and installed. Currently, these offshore wind farms are being installed with ever larger turbines on sites with deeper waters, which are also farther offshore. As a result, traditional monopile support structures become uneconomical for many of these offshore sites and more complex types of support structures (e.g. jackets, tripods) are preferred. 4437

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As the environmental conditions are usually different for every offshore site, offshore wind turbines are custom engineered for each offshore site. There is usually a clear design split when designing an offshore wind turbine. The turbine manufacturer is responsible for the detailed design of the rotor nacelle assembly (RNA) and the tower, whereas the foundation designer’s responsibility is the detailed design of the offshore foundation. For certification purposes thousands of aero-elastic simulations are performed by the turbine manufacturer, taking into account the global dynamics of the complete structure. These results are then used by both parties to verify the designs and update the initial designs. This loop is repeated until a satisfactory design is found. As the detailed structural models of complex foundations, created by the foundation designer, consist of large numbers of degrees of freedom (DoF), they cannot be used directly in the aero-elastic simulations. Therefore, one generally applies different kinds of modeling or reduction strategies to reduce the number of DoF in order to be able to include a compact model of the foundation in the coupled aero-elastic simulations. In this paper several modeling approaches for including these offshore support structures in aero-elastic simulations are discussed. All methods are based on the concept of dynamic substructuring, where the turbine model and support structure model are taken as separate substructures. Four different substructuring methods for including the support structure in the simulation are addressed: Guyan reduction, the (Augmented) CraigBampton method and the Impulse Based Substructuring method. These methods are applied to a representative case study of a 5 MW reference wind turbine installed on an offshore jacket structure. To resemble the practical design procedure for offshore wind turbines, a split is made between the wind turbine manufacturer and support structure designer. Firstly, the coupled simulations are performed using the different types of compact foundation models. Secondly, the interface forces between the tower and foundation are computed from these results and the jacket response is reconstructed by expanding the modal DoF. Afterwards the interface forces are applied to the full finite element model of the foundation in both a static and a dynamic analysis, thereby comparing the different modeling approaches with a reference (or “truth”) solution. The paper is organized as follows. Firstly, in section 2 the current practises for designing offshore wind turbines is discussed. Section 3 presents several methods for obtaining compact foundation models, which can be used in aero-elastic simulations. These methods are then demonstrated using a predefined test case and the results are compared to each other and a reference solution in section 4. Finally, in section 5, some conclusions and recommendations are given.

2 Current practices for designing an offshore wind turbine Due to possible large differences in environmental and site conditions, each offshore wind farm is custom designed. Even in a single wind farm, soil conditions and depth can vary to such an extent that multiple designs per wind farm may be required. The rotor nacelle assembly (RNA) is, in this design process, an of-the-shelf component, but the entire support structure needs to be customized in order to ensure a 20 year lifetime of the entire offshore wind turbine. The three most important design criteria are in general the fatigue lifetime, ultimate loading, buckling and first natural frequency, which needs to be in between the 1P (rotor speed) and 3P (blade passing) frequency [1]. In general, the design of the offshore foundation is performed by an additional party, the foundation designer. Hence, in the design process of the support structure, as shown in figure 1, two parties are involved that need to interface in order to finalize the design. Starting from the expected wind turbine loading, the turbine manufacturer creates an initial tower design. From the initial estimated loads and tower design and the environmental and site specific conditions, the foundation designer comes up with an initial foundation design. By including the designs of both the tower and foundation into the aero-elastic model, the required simulations can be performed.

W IND TURBINE VIBRATIONS

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Turbine manufacturer Foundation model

Initial tower design

Update tower design

Foundation designer

Aero-elastic simulations

Interface forces/ displacements

Simplify model

Initial foundation design

Check detailed foundation design

Update detailed foundation design

Check detailed tower design

Figure 1: Design process for the support structure of an offshore wind turbine In general, thousands of load cases need to be simulated using the aero-elastic model. Hence, in order to keep computation times acceptable, the aero-elastic model has to remain compact. Therefore, the models of the tower and foundation have to be coarse and one cannot include the detailed finite element models of these structures. From the computed turbine responses, the detailed tower is used to verify if the requirements are met and to optimize the design. The interface forces and/or displacements are used by the foundation designer to reconstruct the response of the detailed foundation models due to the operational loading. For estimating the lifetime of the different components of the support structure, the severity of each load case and its likelihood is taken into account. After evaluation of these results, it is known whether or not the design requirements are met or if an update on the design is required. With these updated models a new design iteration is started. In the end, after a number of iterations, a satisfactory design of the support structure is obtained.

3

Overview of different modeling approaches

As the models of complex foundations are in general too large to include in most aero-elastic codes, one often uses approximate or pseudo models. These models could be as simple as an approximation of the equivalent stiffness (and mass) at the interface, or could be more sophisticated models generated using classical spatial reduction methods, also known as Component Mode Synthesis (CMS) [2]. In this section, several methods for modeling the foundation are briefly discussed. Firstly, one of the best known CMS methods, Guyan reduction, will be discussed, follow by the classical Craig-Bampton method. After this, a method for augmenting the Craig-Bampton reduction basis is briefly touched upon. Finally, an entirely different method, named Impulse Based Substructuring, is outlined in section 3.3.

3.1 Superelement modeling: Guyan reduction Since component model reduction methods exploit the principle of modal superposition, they have traditionally been derived for linear structures. As a starting point we therefore take the linear equations of motion of a discretized dynamic component model s: ¨ (s) (t) + C (s) u˙ (s) (t) + K (s) u(s) (t) = f (s) (t) M (s) u

(1)

Here M (s) , C (s) and K (s) are the component’s mass, damping and stiffness matrix, respectively. Vec¨ (s) (t) its respective first and second time tor u(s) (t) denotes the degrees of freedom, with u˙ (s) (t) and u (s) derivatives, and f (t) the external force vector. For the sake of simplicity we will omit the explicit time

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dependency and component notation in subsequent equations. Furthermore, damping will not be considered here but can be included in the developments. In order to reduce the system of equations, the degrees of freedom u are split into internal DoF (u[i] ) and interface (or boundary) DoF (u[b] ). By assuming no external forces are applied to the internal DoF u[i] and neglecting the substructure’s internal inertia forces, the internal DoF can be condensed on the boundary DoF. # " −1 −K[ii] K[ib] u[i] ΨC = u[b] = u[b] = Ru[b] , (2) u[b] I I where Ψ C represent the so called static constraint modes. The reduction basis R is thus formed by the set of constraint modes and corresponds to the Guyan-Irons reduction technique [3, 4]. Substitution of (2) into (1), leads to the reduced dynamic equations, which are now only a function of the boundary DoF. Hence, the substructure is reduced from N DoF to only N[b] DoF, where N is the size of the full finite element model and N[b] is the number of interface DoF, such that N[b] N . ˜u ˜ [b] = f˜[b] , ¨ [b] + Ku M where:  −1 T ˜   K = R KR = K[bb] − K[bi] K[ii] K[ib] ˜ = RT M R = M[bb] − M[bi] K −1 K[ib] − K[bi] K −1 M[ib] + K[bi] K −1 M[ii] K −1 K[ib] M [ii] [ii] [ii] [ii]   ˜ f[b] = RT f

(3)

(4)

Note that the reduction basis and reduced equations are per substructure. By identification of the interface DoF (u[b] ) one is able to assemble these reduced component models to other (reduced) component models to obtain the total model of the structure. Since in the derivation of the condensed stiffness the inertia forces are neglected, the exact solution is found if this technique is applied to static problems. If it is applied to dynamic problems, an approximate solution is found. The accuracy of the approximation depends on spectral and spatial content of the applied excitation. If the spectral content of the excitation is well below the first (flexible) eigenfrequency and the spatial content has a high correlation with the reduction basis, one will get an accurate solution. However, if these conditions are not met, erroneous results can be obtained.

3.2 Superelement modeling: The Craig-Bampton method In order to find a better approximation of the solution for dynamic models, the Craig-Bampton method [5] was introduced at the end of the 1960’s. In this method, fixed interface modes are added to the original Guyan reduction basis (2), in order to include the dynamic response of the internal DoF. Ψ C Φ[i] RCB = , (5) I 0 where Φ[i] are the fixed interface normal modes, which are found by performing an eigenmode analysis on the fixed interface finite element model, i.e. solving: K[ii] − ωr2 M[ii] φ[i],r = 0 (6) Again, by projecting the full FE model (1) on the reduction basis (5),  ˜ = RT KRCB  K CB ˜ = RT M RCB M CB  ˜ T f f = RCB the system of equations is reduced from N to N[b] + NΦ , where NΦ N[i] .

(7)


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3.2.1 Augmenting the Craig-Bampton reduction basis A Craig-Bampton superelement is able to accurately represent the response of the full model, as long as the loading is applied at the retained interface DoF. However, in engineering practice this is often not the case and large parts of structures can be subject to external forces. For instance, the offshore foundation is excited by waves and currents on a large part of the structure. If this is now seen from a modeling perspective, this area will consist of a large number of nodes in a finite element model. As these DoF need to be retained as interface DoF, the size of the reduced matrices would grow quadratically with respect to the size of the force loaded area. In order to overcome this issue, we describe these forces as a summation of spactial distributions of the force (Pj (x)), multiplied by their corresponding force amplitudes (αj (t)): f[i] (x, t) =

np X

pj (x)αj (t) = P α

(8)

j=1

As can be seen, separation of variables is applied to the force f (x, t), which is dependent on both time (t) and location (x). In order to be able to ensure that the external forces are captured well by the reduced models, we can compute the static response of the structure to these force shapes in space and augment the reduction basis with these pseudo modes [6, 7]. This is often referred to as Modal Truncation Augmentation (MTA), hence these pseudo modes are also known as MTA’s. For numerical stability, we require that the MTA’s are orthogonal to the fixed interface modes, which are already included in the reduction basis

where:

¯ M T A = T K −1 P , Φ [ii]

(9)

T = I − Φ[i] ΦT[i] M[ii] .

(10)

To further improve the numerical robustness and retain the sparsity of the reduced matrices, the MTA vectors are mutually orthogonalized by solving an eigenvalue problem defined as: 2 ¯T ¯T ¯ ¯ Φ K y = σ Φ Φ M y (11) Φ M T A M T A [ii] M T A [ii] MT A Here σ 2 is a diagonal matrix containing the “pseudo-frequencies” corresponding to the MTAs. These frequencies are always higher than those of the fixed interface vibration modes and provide an indication of the frequency range in which the MTA’s provide a dynamic correction. The orthonormalized MTAs are computed by ¯ M T A y, ΦM T A = Φ (12) and have the following properties:

ΦTM T A M[ii] ΦM T A = I ΦTM T A K[ii] ΦM T A = σ 2

(13)

After including the MTA vectors in the reduction basis, the solution of the internal DoF using the CraigBampton method can now be represented by     u u[b] u[i] Ψ C Φ[i] ΦM T A  [b]  η [i] = = RACB  η [i]  . (14) u[b] I 0 0 ζ ζ By again projecting the full model on the reduction basis, similar to Eq. (7), the set of equations of motion is reduced to N[b] + Nφ + NM T A . In our example, Proper Orthogonal Decomposition (POD) is applied to the times series of wave forces in order to find the space distributions of the force (Pj (x)), as used in Eq. (8).

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3.3


Impulse Based Substructuring as an alternative for classical superelements

An alternative method for modeling a linear finite element model in a more compact manner, is to include it in the simulations as a set of Impulse Response Functions (IRFs); methods for doing so are known as Impulse Based Substructuring (IBS) [8]. IRFs represent the response of a system due to a unit impulse and therefore contain the full dynamic response of the system. As such, they can be considered as a “superelement in time”. Hence, the response of a substructure, initially at rest, due to a time varying loading can be written as (s)

u

Zt (t) =

Y (s) (t − τ )f (s) (τ )dτ ,

(15)

τ =0

where Y (s) (t) is the set of IRFs. Solving this convolution product (also known as the Duhamel integral), the structural response in time is obtained. 3.3.1 Coupling Impulse Response Functions and Finite Element Models Coupling the IRFs to a general (possibly nonlinear) finite element model, denoted as being substructure (r), gives the following set of coupled equations [9, 10],  Zt    (s) (s) (s)T (s)   Y (t − τ ) f (τ ) − B λ(t) dτ u (t) =    τ =0 (16)  (r) (r) (r) (r) (r) (r) (r) (r)T  ¨ (t) + p u (t) u M u˙ (t), u (t) = f (t) − B λ(t)       Bu(t) = 0 , where λ(t) are Lagrange multipliers that enforce the compatibility, representing the coupling force intensities and B (?) are signed (substructure) Boolean matrices, denoting the interface DoF [11]. Note that (r) (r) (r) p u˙ (t), u (t) is a non linear force vector, containing the internal elastic and damping forces. Discretizing the Duhamel integral with a piecewise linear force approximation and the nonlinear FE model using the Newmark relations [12], where the time derivatives are related by the finite differences (Eq. (18)), a set of equations is obtained that can be solved numerically  n−1   dt X (s) (s) (s) (s)T (s)   Y f + f − B λ + λ u = i i+1  n−i i i+1  n 2  i=0 (17) T (r) (r) (r) (r) (r) (r) (r)  ¨ ˙ + B (r) λn − fn(r) = 0 r = M u u + p u , u  n n n n      Bu = 0 . n

Here, r (r) is the error on the equilibrium and can be interpreted as a residual force that needs to be minimized in order to find the set of displacements, velocities and accelerations of the nonlinear finite element model at each time step. 1 (r) (r) (r) (r) 2 1 ˙ ¨ ¨ (r) u − u + dt u + dt ( − β) u u = n n n−1 n−1 n−1 dt2 β 2 (18) γ (r) (r) (r) (r) (r) (r) (r) 2 1 ¨ n−1 + ¨ n−1 u˙ n = u˙ n−1 + (1 − γ)dtu un − un−1 + dtu˙ n−1 + dt ( − β)u , dtβ 2 where γ and β are parameters of the scheme. The details for solving this coupled set of discretized equations of motion will not be discussed in this paper, but can be found in [9] and [10]. Note that, as the IRFs are


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computed from the full FE model, no reduction in the spatial sense is applied (as is the case for the Guyan and Craig-Bampton methods) and the only error introduced is a result of the time discretization. Hence, if the IRFs are computed with the same time discretization as the final coupled simulations, the numerically exact answers can be found. The advantage of this approach, described by Eq. (17), is that the dynamics of substructure (s) is precomputed as an IRF, and used in the coupling through a convolution product, so that if substructure (r) is modified the time response of (s) does not need to be recomputed fully. Also the scheme in Eq. (17) makes co-simulation, namely handling different substructures in different codes, easy since the code handling substructure r (the wind turbine in our case) only needs to interact through the interface IRFs of the attached substructures (the support structure). 3.3.2 Using an impulse response due to a distributed load In order to compute the response to a set of external forces, we need to compute the IRFs of all force loaded DoF. In the case of wave loading on the jacket, this involves a large number of DoF, and the number of entries in the matrix containing the required IRFs would grow quadratically with respect to the size of the force loaded area. Similar to the superelement modeling described in section 3.2.1, we can apply Eq. (8) and describe these forces as a summation of force shape distributions in space (Pj (x)), multiplied by their corresponding force amplitudes (αj (t)), in order to overcome this issue. By substituting Eq. (8) into Eq. (15), the impulse response describes the response of a structure due to a distributed load. Zt u(t) =

Y (t − τ )P (x)α(τ )dτ

(19)

0

Suppose that the foundation contains a large number (Nin ) of (wave) force loaded DoF (also referred to as inputs) with only Nout response dof of interest (also referred to as outputs, in our case the interface DoF of the foundation), such that Nout Nin . In Eq. (19) Y (t) is then a Nout × Nin × nmax sized matrix (where nmax is the number of time steps), hence computing the needed IRFs and evaluating the convolution integral will be computationally expensive. Using the spatial distributions of the load to reduce the size of the impulse response function, leads to: Z t (20) u(t) = YP (t − τ )(x)α(τ )dτ 0

where,

YP (t) = Y (t)P .

(21)

This step reduces the size of the impulse response function from Nout × Nin × nmax to Nout × Np × nmax , where Np equals the number of force shape distribution functions and Np Nin . Note that, as the applied force is spatially reduced, one will find an approximate solution of the original problem. Again, Proper Orthogonal Decomposition (POD) is applied to the times series of wave forces in order to find the spatial distributions of the force (Pj (x)), as used in Eq. (19).

4 Numerical test case In order to review the effect of the different modeling strategies a test case is defined here. The goal is to show the effect of the different modeling strategies in the normal support structure design process of an offshore wind turbine, as discussed in section 2. Firstly, the setup and technical details of the numerical test case are discussed in section 4.1. In section 4.2 the results are shown and discussed afterwards.

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4.1


Setup of the numerical test case

Reference model

In order to emulate a support structure design process, the following test case has been created and is shown in figure 2. All the data and computations in the hands of the turbine manufacturer are shown in the purple blocks and all the data and computations in the hands of the foundation designer are shown in the green blocks. Note that the block containing “Wind loads, wave loads” has both colors and the blue blocks represent the comparison between the reference finite element model and the reduced models.

Turbine model

Assemble models

Reduced models

Complex foundation model

Create superelement/ IRFs

Assemble models

Pre-processing

Coupled simulations

Compute interface forces

Wind loads, Wave loads

Compare interface forces

Compare foundation responses

Coupled simulations

Compute interface forces

Foundation response (simulations)

Simulations

Post-processing

Figure 2: Abstract view of the test case As one can see, the test case consists of three stages: Pre-processing, Simulations and Post-processing, which are all discussed in more detail in the rest of this section. 4.1.1 Pre-processing: Building the simulation models The numerical test case is started with the models of the NREL 5MW (offshore) reference turbine [13] and the UpWind reference jacket [14], which are shown in figure 3. The wind turbine is modeled as a nonlinear finite element model, where the nonlinearity arises from a nonlinear damper that is attached to the hub-node, which is added for simulating an aerodynamic coupling. Obviously the damper is a very simple representation of reality, but is used here just to show simulate the nonlinear character of the turbine model. It should be noted that the damping matrix is tuned, using the Rayleigh damping parameters, such that the first two modes of the full structure have a damping ratio of 1% critical damping, which is evenly divided over the wind turbine and the jacket foundation. In order to compare all the different modeling strategies, a Model name GR CB20 CB1010 IRF20

# of constraint modes 6 6 6 -

# of fixed interface modes 0 20 10 -

# of force shapes/MTA’s 0 0 10 20

Matrix size 6×6 26 × 26 26 × 26 6 × 26 × nmax

Table 1: Summary of models used in the numerical test case reference solution is needed, which is computed from the reference model, as indicated in figure 2. To obtain this reference model, the full finite element models of the two substructures are assembled through 6 DoF (3 translation and 3 rotations) on the interface point.


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Figure 3: Models of the Upwind reference jacket (left) and the NREL 5MW turbine (right) In addition to the reference model, the reduced models are built using the different techniques discussed in section 3 and the details for the different models are given in table 1. Note that for some of the techniques, such as the Augmented Craig-Bampton (CB1010) and the IBS method (IRF20), the time series of wave loading are required in order to compute the force shape vectors, as briefly mentioned in section 3.2.1. The different simulation models of the offshore wind turbine are now created by assembling the turbine model with each of the reduced models and also assembling a turbine model with the foundation described in its IRFs. 4.1.2 Coupled simulations and the defined load case In addition to the different models of the offshore wind turbine, a set of loads is required in order to perform the coupled simulations (as shown in figure 4). Note that the waves on the reduced foundation models are also reduced using a the associated spatial reduction basis and are thus indicated as f˜waves . In the test case fwind (t)

fwind (t)

f˜waves (t) fwaves (t)

Figure 4: Simulations using the given wind and wave input forces the turbine is excited by both wind and waves from the same direction. The wind loading is represented by a first order low-pass filtered (cut-off frequency of 2.5 Hz) random force and a constant force over the entire rotor. Note that a simplified aerodynamic coupling is added by a nonlinear damper, as was mentioned in section 4.1.1. In addition to this, the jacket structure is excited by a series of sea waves, which are assumed to be irregular (linear) waves with a significant wave height of 9.4 m, and a current with speeds up to 1.2 m/s at the mean sea level. The wave forces are computed beforehand, which is done by assuming small

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velocities of the structure, hence discarding the relative velocity terms in the Morison equation [15]. Note that the added mass effect of the water is taken into account in the structural model of the jacket. Simulations are performed for 100 s, with time steps of 0.04 s, using the constant average acceleration Newmark method (γ = 1/2, β = 1/4). 4.1.3 Post-processing: Reconstructing the response of the jacket The final step in this test case is the post-processing step (figure 5). Firstly, the internal (or interface) forces between the tower and jacket will be computed from the results of the coupled simulations. Secondly, the response of the detailed jacket model is reconstructed using different techniques. Finally, the results from the reconstruction are compared with the reference solution. In this case, the full jacket response will be reconstructed using three different approaches. fwind (t)

1

Reconstructing jacket response

2

3

fwind (t)

Quasi-static

f˜int (t)

qjacket (t)

Dynamic

f˜waves (t)

fint (t)

Modal expansion

Compare

fwaves (t)

Figure 5: Post-processing: 1) Computing interface forces (left), 2) Reconstructing the response of the jacket using different methods, 3) Compare with the full reference solution (right). Reconstruction of the response by quasi-statically applying the interface forces An often used approach in the industry, is to assume a quasi-static response of the foundation [16, 17]. Hence, the deformation of the foundation is reconstructed by computing its static response due to the computed interface forces and wave loads, at each time step. Note that by doing so, one neglects the contribution of the foundation’s inertia, which could lead to significant errors when structure responds dynamically. Dynamic simulations using the time series of interface forces The second approach used is to compute the dynamic response of the jacket, by applying both the wave loads and interface forces. Although this appears to be counter-intuitive at first, as the dynamics seem to be accounted for twice, it can be proven to be theoretically exact. By applying not the primal, but the dual formulation of the coupled problem [11], we obtain  T ¨ (s) (t) + C (s) u˙ (s) (t) + K (s) u(s) (t) − f (s) (t) = −B (s) λ(t)  M (s) u T (22) ¨ (r) (t) + C (r) u˙ (r) (t) + K (r) u(r) (t) − f (r) (t) = −B (r) λ(t) M (r) u  Bu(t) = 0, where λ(t) are Lagrange multipliers, representing the interface force intensities and B (?) are signed (substructure) Boolean matrices, selecting the interface DoF. Hence, if the coupled simulations are performed


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and we have obtained the Lagrange multipliers over time, the dynamic response of substructure s can simply be reconstructed by applying these as an external force. T

¨ (s) (t) + C (s) u˙ (s) (t) + K (s) u(s) (t) = f (s) (t) − B (s) λ(t) M (s) u

(23)

Such that one obtains the exact same response as in Eq. (22). Note however, that the coupled simulations are performed with an approximate foundation model, after which the interface forces are applied to the full FE model of the foundation. One can imagine that, if the approximate model is a poor representation of the full model, this could lead to significant errors. Expansion of the modal coordinates computed in the coupled simulations The third and simplest option presented in figure 5 is the modal expansion of the reduced set of DoF, obtained from the coupled simulations " (s) # u[i] (t) = R(s) q (s) (t), (24) (s) u[b] (t) where, q (s) (t) is the array containing the reduced set of DoF pertaining to the foundation substructure, as shown in section 3. Again, one can imagine that if the reduced basis gives a poor representation of the response of the full model, this could lead to errors with respect to the reference solution. In theory, one could use the full set of IRFs of the jacket expand the results of the IBS analysis using the convolution product, but as this offers no practical benefits, it is not discussed here and its results are not shown for our test case.

4.2 Results of the numerical test case All the methods will be compared with the reference solution from the full model. The comparison will be made using the Potential energy (V) of the system at each time step, T

(s) (s) Vn(s) = u(s) n K un ,

(25)

where superscript s denotes the foundation substructure. The Potential energy can be seen as a quantitative measure for the total stress and deformation of the foundation and is thus a good parameter for comparing the results, since stress and deformation are important indicators in the evaluation of the design. Note that the results presented in this section, are the results after the Post-processing step as shown in figure 2 and discussed in section 4.1.3. 4.2.1 Computed interface forces between the foundation and tower As a first step of the post-processing part, the interface forces between the tower and foundation are computed and are compared in figure 6. It is clear that all methods, except the Guyan reduced model, result in an almost perfect match with the interface force computed in the reference case. The force computed using the Guyan reduced model is able to follow the general trend, but shows some clear differences with the force from the reference model. Since the flexibility resulting from the internal dynamics of the jackets is not taken into account in the Guyan reduced foundation, it will behave stiffer at higher frequencies. Due to this, the resulting interface forces will also be higher at these higher frequencies, as can be observed. 4.2.2 Reconstructing the foundation responses Using the interface forces computed in section 4.2.1 and the modal amplitudes obtained from the coupled simulations, the results for the different approximate models can be computed using the different postprocessing methods.

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P ROCEEDINGS OF ISMA2012-USD2012 5

x-component of the interface forces(N)

6

x 10

4

2

0

−2

−4 30

reference

GR

31

33

32

CB20 34

35 time (s)

CB1010 36

IBS 37

38

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Figure 6: x-component of the interface forces, computed using the different coupled models Results using a Guyan reduced foundation The first approximate model that was used in the coupled simulations, is the Guyan reduced foundation model (see table 1 and section 3.1). The jacket response is reconstructed using the different methods and shown in figure 7. It can clearly be seen that the slight 4

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Figure 7: Results from the analysis using a Guyan reduced foundation model differences in the interface forces, can result in significant errors when reconstructing the response on the full foundation model. In this particular case, the higher interface forces (as explained in section 4.2.1) excite one of the eigenmodes of the full jacket model, hence overestimating the response. The expanded solution slightly underestimates the total deformation of the foundation. This can be explained from the fact that the reduction basis of the Guyan reduction is not able to capture all the wave loading in the coupled simulations, hence discarding a part of the wave forces on the structure, while also discarding the contribution of the internal dynamics of the structure. Results using a Craig-Bampton reduced foundation The same simulations were performed using a Craig-Bampton reduced model, the results are shown in figure 8. Again, it can be seen that the quasi-static reconstruction of the jacket response is well off, as it underestimates the total deformation of the structure.


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It can also be observed that the Potential energy of the expanded results is again slightly lower than the potential energy computed from the reference solution. This is similar to the results from the Guyan reduced 4

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Figure 8: Results from the analysis using a Craig-Bampton (20 fixed interface modes) reduced foundation model superelement and can be explained from the fact that the reduction basis of the Craig-Bampton superelement is not able to capture all the wave loading. The dynamic reconstruction gives in this case very good results, which is also expected from the results of figure 6. Results using the Augemented Craig-Bampton reduced foundation Again, from figure 9 it can be seen that the quasi-static reconstruction of the response is inaccurate, but both the dynamic reconstruction and 4

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Figure 9: Results from the analysis using a Augmented Craig-Bampton (20 fixed interface modes, 10 MTA’s) reduced foundation model the expansion of the set of modal DoF given excellent results. As the reduced basis also contains pseudomodes that are able to capture the deformations due to the wave loading very well, the overall (reconstructed) response matches perfectly with the reference solution.

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Results using the Impulse Response Functions of the foundation Also here, the quasi-static reconstruction gives significant errors. The dynamic reconstruction is a perfect match with the reference solution, as can be seen in figure 10. This is due to the fact that the applied wave force is captured perfectly using the 20 force shape modes in the coupled simulation. 4

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Figure 10: Results from the analysis using IBS for the coupled simulations

5

Conclusions

In this paper several modeling methods for including complex foundation models in aero-elastic simulations where shown. All the methods are able to significantly reduce the number of DoF of the full finite element model of the foundation, such that they can be assembled with aero-elastic models of wind turbines. Even though the methods were all tested using a simplified model and load case, a number of conclusions can still be made: • Firstly, it can be concluded that if the foundation superelement used for the coupled simulations, does not represent the internal dynamics of the structure, one is not able to correctly compute the interface forces and/or reconstruct the detailed structural response of the jacket. • Secondly, it showed that, regardless of the method used, the quasi-static reconstruction of the jacket response led to significant errors in the computed deformations of the jacket. • Finally, if the reduced basis is unable to capture the effect of the wave loads on the structure, the response of the foundation will be underestimated. Note that the severity of these errors will differ from case to case and that in some specific cases using (over) simplified models could still lead to the correct responses. But if one is unable to verify the simplified modeling approaches, significant errors can be made in the design process. Hence, in order to achieve cost savings for future projects, it is advisable to be “over-conservative” in the model refinement for the offshore foundation models for aero-elastic simulations, instead of being overconservative when adding extra safety margins to the design in order to cope with the uncertainties resulting from poor approximate models.


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