When performing modal decomposition and expansion (MDE), modal truncation is needed due to a limited number of sensors. Furthermore, a finite number of Ritz vectors can be included to assess the quasi-static part of the response. Hence, it is important to have an overview of the different governing loads to be accounted for in the response estimates. This section gives an example of how diverse operational and environmental conditions can impact the DELs of the IEA 15 MW RWT and hence contribute differently to lifetime damage. Specifically, statistical values of relevant operational parameters and the tower base fore–aft (FA) and side–side (SS) section moments are considered during normal power production (DLC 1.2).

In Fig. , the statistics (minimum, mean, maximum) of the operational parameters (rotor speed, electrical power, generator torque, thrust, and pitch angle) and the wave amplitude are presented, while Fig.  shows the associated statistics of the tower base FA and SS section moments and the 1 Hz DELs for the individual HAWC2 time series (evaluated by Eq. ) for DLC 1.2. The operational parameters in Fig.  are compared with steady-wind rotor performance values from , generated by the Wind-plant Integrated System Design and Engineering Model (WISDEM), which uses the aeroelastic code OpenFAST.

Figure a–e show that the mean values generally coincide well with the WISDEM output, and Fig. f verifies that the minimum and maximum wave amplitudes follow the development of the input significant wave height. The greatest discrepancies are observed for the thrust in Fig. d and the pitch angle in Fig. e. The discrepancies in the thrust and pitch angle are due to steady versus turbulent operation and the ElastoDyn beam model used in the WISDEM calculation not including a torsional degree of freedom . The generally good match between the models indicates that the HAWC2 model may be used for further analysis.

The statistical values for the tower base FA moment presented in Fig. a follow the thrust curve from Fig. d as expected. The DELs associated with the tower base FA moment presented in Fig. c generally increase with both the wind speed and turbulence. However, they plateau at wind speeds from approximately 12–16 m s⁻¹, in which range the blades start to pitch (see Fig. e). This illustrates that the DELs in the FA direction at the tower base are primarily governed by quasi-static wind loading, while operational parameters (e.g. the pitch angle) also affect the damage. Similarly to the statistical values of the tower base FA moment, the mean values of the tower base SS moment presented in Fig. b follow the generator torque curve in Fig. c. The minimum and maximum values of the tower base SS moment are symmetric around the mean value with increasing amplitudes for increasing wind speeds. The associated DELs in Fig. d also increase with the wind speed and turbulence. Furthermore, Fig. d shows that the variance in the DELs increases with the wind speed up to rated wind speed, from where it is rather significant.

To assess the cause of the high variance, the time histories of the tower base SS moment, wind speed (in the SS direction), and wave height associated with the minimum and maximum DELs for the wind speed of 14 m s⁻¹ are presented in Fig. . Considering the moment time series in Fig. a and the related PSD in Fig. b, it is concluded that DELs are mainly driven by the first tower SS mode. There is not a significant difference in the frequency content of the wind around the natural frequencies of the first-order tower bending modes. However, the mean wind speed in the SS direction is significantly higher for the maximum DEL than for the minimum DEL due to the -10° yaw error. Furthermore, the waves have an angle of attack of -32.5° for the maximum DEL, whereas it is 0° for the minimum DEL. Thus, the variation in DEL magnitude is caused by the excitation of the first tower SS mode occurring for the maximum DEL, while not for the minimum DEL, likely due to the difference in the excitation forces resulting from the varying angle of attack of the wind and waves between the two time series.

In conclusion, the present section underlines that the DELs calculated for the IEA 15 MW RWT are indeed influenced by environmental parameters such as turbulence, which govern the quasi-static response, and wave direction. Furthermore, operational parameters such as pitch angles and yaw errors can, in some cases, contribute to the excitation of the dynamic modes, which significantly impacts the DELs. Thus, the MDE configuration presented in Sect.  is required to accurately capture both quasi-static and dynamic responses for varying operational and environmental conditions.

The present section investigates the lifetime damage of the IEA 15 MW RWT caused by the individual design load cases presented in Sect. , thereby giving an overview of which operating scenarios are significant for the fatigue damage in the support structure.

According to , the relative lifetime damage caused in a given structure by a load case i is given as 1di,rel=ni(ΔPeq,i)mnT(ΔPeq)m, where ΔPeq,i represents the 1 Hz DEL ranges for the individual load case i, m is the Wöhler coefficient, ni is the number of 1 Hz cycles for load case i, nT is the total number of 1 Hz cycles in the structure's lifetime, and ΔPeq is the lifetime DEL range.

In the present analysis, a similar approach to that of in Eq. () is used for the evaluation of the relative lifetime damage for individual DLCs. By adding the 1 Hz DELs from the HAWC2 simulations contained in a DLC, the relative damage of the individual DLCs is calculated as 2dDLC,rel=∑s∈DLCns(ΔPeq,s)mnT(ΔPeq)m, where 3ns=p(DLC,V,θyaw,θwwm)nTnseed is the number of 1 Hz cycles during the lifetime of the IEA 15 MW RWT, p() is the joint probability of the input parameters for the operational and environmental conditions (DLC, wind speed (V), yaw error (θyaw), and wind–wave misalignment (θwwm)) used for the simulation s, and nseed is the number of simulations that share these operational and environmental conditions. Note that the number of summations in Eq. () refers to the number of (converged) simulations in Table  for a given DLC at MWL equal to MSL. Finally, in Eq. () the 1 Hz DEL range for the individual HAWC2 simulations is evaluated as 4ΔPeq,s=∑njΔPjmneq1m, where neq is the number of 1 Hz cycles in the time series s, while ΔPj and nj are the binned load ranges and corresponding number of load cycles identified from the individual time series using the rainflow counting method from . In the present work, a single slope S–N curve with a Wöhler coefficient of m=5 is used for the support structure. This is based on m1 of the S–N curves for welded and non-welded circular hollow sections from Chapter 8 in , which is not representative of the damage at all locations in the support structures but still considered sufficiently accurate for the assessment of the impact of the individual DLCs.

The relative damage for the individual DLCs, dDLC,rel, is presented in Fig.  for the FA and SS directions of the IEA 15 MW RWT support structure.

From these relative damage plots in Fig. , it is observed that there is a big resemblance in the distribution of damage across the height of the IEA 15 MW RWT for DLCs 1.2 and 2.4, which is expected as these load cases are both for operation in normal conditions. A similar expected resemblance is found for DLCs 3.1 and 4.1, as these load cases represent start-up and shut-down, respectively. Figure a shows that approximately 99 % of the damage in the FA direction is caused by DLC 1.2 (power production in normal conditions), DLC 6.4 (parked – idle rotor in normal conditions), and DLC 7.2 (fault – locked rotor in normal conditions). In the SS direction, shown in Fig. b, the damage from DLC 6.4 falls below 1%, so only DLCs 1.2 and 7.2 are considered significant for the damage in the SS direction. As presented in Table , DLC 1.2 is significantly more frequent than DLCs 6.4 and 7.2, and the significant damage contribution of this DLC is associated with the large duration, whereas for DLCs 6.4 and 7.2, the substantial damage contribution is associated with the large DELs (see Fig. ).

The relative damage in the FA direction in Fig. a is dominated by DLC 1.2 at the tower top (≈100–144 m above the MSL). This is due to 3P effects (tower shadow, wind shear, and turbulence), which are significant contributors to damage in the tower top, as the varying forces on the blades and uneven loading on the rotor result in a significant moment at the hub. In the remainder of the free-standing support structure (≈-30–100 m), the relative damage in the FA direction is dominated by DLC 7.2. In this area, the section moments are governed to a higher degree by the global bending of the support structure caused by the thrust loading (for DLC 1.2) and especially the first tower FA mode (for DLCs 6.4 and 7.2). The tower bending modes in the FA direction for DLC 1.2 are subject to significant aerodynamic damping arising from the operating rotor, thus explaining the smaller contribution to the relative damage from this DLC and the larger contribution from DLCs 6.4 and 7.2, where the rotor does not operate and the aerodynamic damping is effectively negligible. Below MSL the relative damage contribution from DLCs 1.2 and 7.2 approaches each other and balances out at the mud line. This is likely due to the influence of wave loads, which increase with water depth and are less affected by the aerodynamic damping present for DLC 1.2.

The relative damage in the SS direction in Fig. b is dominated by DLC 7.2 at the tower top (≈120–144 m above MSL), while DLC 1.2 dominates the damage below this area. Unlike the FA response, the SS response for DLC 1.2 is not significantly affected by aerodynamic damping, the 3P effects, or the thrust load variations. Consequently, the damage in both DLC 1.2 and DLC 7.2 is primarily driven by ambient excitation at the turbine's resonant frequencies. However, the locked rotor condition in DLC 7.2 particularly influences damage at the tower top. Because the rotor is fixed in rotation and the blades are pitched 90°, the blades are more susceptible to turbulence-induced excitation, which creates a moment at the blade root. This, in turn, excites the second tower SS mode and possibly different rotor modes, resulting in DLC 7.2's dominant contribution to damage in the upper part of the support structure. In the lower part, the damage patterns are more governed by the first-order tower bending modes, which are similar for DLCs 1.2 and 7.2. However, the significantly longer duration of DLC 1.2 (90 % of the turbine's lifetime) results in it being dominant below 120 m. This effect is visible in Fig. b, where the distribution of relative damage from DLCs 1.2 and 7.2 remains rather constant in the support structure below 100 m, with dDLC,rel for the two DLCs varying between 69%–80% and 19%–29%, respectively.

In conclusion, the damage in the support structure of the IEA 15 MW RWT is governed by both normal operation conditions and conditions where the rotor is idling or locked, whereas start-up and shut-down of the wind turbine and operation with yaw error are less critical. However, in a real operating scenario, shut-down and start-up may have a larger influence on the lifetime damage, as they occur more frequently than described by due to, for example, curtailment. This has not been accounted for in the present paper. The damage contribution across the elevation of the support structure arises from different local and global effects caused by different environmental and operational scenarios, e.g. turbulence, 3P effects, wave loads, and inherent dynamical properties. It should be emphasised that the durations used in this analysis for the DLCs are estimated, and scenarios can occur where the durations are differently distributed between the DLCs. Therefore, it is also relevant to evaluate DELs for individual DLCs, without accounting for their specific durations, when assessing how the different operational scenarios impact lifetime damage, as done in Sect. .

Modal decomposition and expansion (MDE) is a well-established process model in virtual sensing (see Sect. ). The formulation used in the present work is described in . MDE assumes that the displacement vector u(t) of an undamped dynamic system can be decomposed and written as a linear combination of the system's mode shapes and modal coordinates on the matrix form: 5u(t)=Φq(t). The mode shape matrix Φ=[φ1,φ2,…,φn] contains the n mode shapes (φj) included to describe the dynamical system, while the modal coordinate vector q(t)=[q1(t),q2(t),…,qn(t)]T collects the corresponding modal coordinates (qj) at each time instant t. The mode shapes of the system φj can be derived from e.g. experimental or operational modal analysis, while in the present work, the vectors φj are derived from an FE model representing the dynamic system in Sect. . Assuming that the FE model is an accurate representation of the considered dynamic system, it follows that 6Φ=ΦFE, which applies in the remainder of the paper. If the total number of degrees of freedom (DOFs) in the FE model is ndof, the modal matrix Φ becomes an ndof×n array. The nodal displacement vector u(t) in Eq. () is conveniently partitioned as 7u(t)=um(t)up(t)=ΦmΦpq(t), where the first nm DOFs in um(t) represent those that are measured by physical sensors, while the remaining np DOFs in up(t) are those that are predicted by the MDE, i.e. the virtual sensors. By direct comparison of Eqs. () and (), the mode shape matrix is similarly partitioned into 8Φ=ΦmΦp, in which the nm×n array Φm refers to the mode shape amplitudes associated with the measured DOFs, while, correspondingly, the np×n array Φp accounts for the remaining DOFs that are used for the subsequent prediction procedure. From the above partitioning in Eqs. () and (), it is seen that the total number of DOFs in the FE model is ndof=nm+np, i.e. the sum of measured and predicted DOFs.

MDE utilises the fact that the displacements in um(t) are available from measurements, while the remaining DOFs in up(t) are predicted simultaneously once the modal matrix in Eq. () can be obtained from the underlying FE model with sufficient accuracy. It follows from Eq. () that the predicted nodal displacements can be expressed by the modal representation 9up(t)=Φpq(t). The modal coordinates in q(t), used for the extrapolation in Eq. (), are determined by the corresponding relation 10um(t)=Φmq(t) for the measured DOFs in um(t). The inversion of this relation requires that the dynamic displacement field can be represented by at most n modes, where n must be less than or equal to the number of measured DOFs nm. Hereby, the modal coordinates can be determined as 11q(t)=Φm†um(t) using the Moore–Penrose pseudo-inverse depicted by the commonly used ()† symbol. The predicted nodal displacements are then obtained by substitution of Eq. () into Eq. (), which then takes its final form 12up(t)=ΦpΦm†um(t).

In virtual sensing, one of the objectives is to minimise the number of physical sensors nm by introducing virtual sensors. Hence, the condition n≤nm poses a challenge, as this limits the number of modes n that can be included to describe the dynamic system. Furthermore, for low frequencies, it can be desirable to perform MDE using only a subset of the measurements ũm(t) to minimise the noise introduced in the estimates or to introduce Ritz vectors containing static deflection shapes to predict the response up(t) in frequency ranges not dominated by resonant response (see Sect. ). The introduction of multi-band virtual sensing in utilises the fact that the nodal displacement vector u(t) can be divided into separate bands Bi in the frequency domain, which, when combined by summation, reattains the original nodal displacement vector 13u(t)=∑i=1Nui(t)=∑i=1NBi(u(t)), where ui is the nodal displacement vector band-pass filtered in the band Bi, and i=1,2…N denotes the individual frequency bands (see Fig. ). Similarly, the predicted nodal displacements up(t) can be calculated in individual bands and combined by summation as 14up(t)=∑iui,p(t)=∑iΦ̃i,pΦ̃i,m†ũi,m(t), now only including the modes and Ritz vectors Φ̃i and the measurements ũm(t) that are relevant for the band Bi. This representation assumes that the energy content of up(t) is fully captured by the sum of its filtered components in the bands Bi.

Section  has explained how modal decomposition and expansion can be used to predict displacement response at virtual sensor locations. The present section extends the MDE to predict internal forces based on the predicted nodal displacement vector up(t).

The section forces to be predicted by the proposed method are specific to the element of the applied FE representation, e.g. bending moments for the planar beam elements used to describe the dynamics of the present support structure. Let the nodal forces be contained in the nodal element vector 15re(t)=rA(t)rB(t)e=fxA(t),fyA(t),mA(t),fxB(t),fyB(t),mB(t)eT=-NA(t),VA(t),-MA(t),NB(t),-VB(t),MB(t)eT for a planar (2D) beam element e between two nodes A and B, with fx, fy, and m representing the nodal normal force, shear force and moment, respectively. As shown in Eq. (), the corresponding section forces N, V, and M are derived from the nodal force by appropriate sign changes.

For a given element (subscript) e, the element nodal force vector in Eq. () can be determined by the element stiffness matrix ke. The element stiffness relation can thus be written as 16re(t)=keTeup(t), where Te is a 6×np array that both collects and rotates the 6 DOFs from the global vector up(t) into the local coordinate system for element e=1,2…Ne. Elimination of the response in the predicted DOFs up(t) by Eq. () gives the compact representation 17re(t)=keTeΦpΦm†um(t)=Deum(t), where 18De=keTeΦpΦm† defines the section force matrix that predicts the section forces re(t) from the measured nodal displacements in um(t). For a model with vertical beam elements, as in the present case, the transformation matrix Te is an all-zero 6×np matrix, except for ±1 entries in the 6×6 block associated with the specific element e.

The prediction FE model from which the mode shapes and Ritz vectors used in the MDE are obtained is a 3D linear elastic beam model with the rotor nacelle assembly (RNA) and transition piece modelled as lumped inertias. The FE model is created using non-commercial Python-based FE software. The beam model is presented schematically in Fig. . The geometrical properties and the mass and stiffness input parameters for the prediction FE model are extracted from the HAWC2 model of the IEA 15 MW RWT described in Sect.  and presented in Appendix .

The beam element stiffness is established according to , which combines the element stiffness matrix developed from the Timoshenko beam theory Kbeam,e with a so-called geometric stiffness term Kg,e, expressing the total element stiffness matrix as 19Ke=Kbeam,e+Kg,e, thus accounting for the stiffness contribution adhering from the normal forces causing Euler buckling in bending, although omitting the stiffness terms associated with torsion, i.e. loads causing lateral buckling in static analysis.

The monopile foundation support conditions are modelled using lateral linear elastic soil springs in the embedded part of the monopile. The stiffness of the individual springs ksoil,n varies with the embedment depth, as presented in Table . The bottom node in the beam model restrains torsion and vertical translation.

The mass contributing to the modal mass of the prediction FE model includes the distributed mass of the tower, transition piece, and monopile presented in Appendix ; the nodal mass of the transition piece MTP located at the top of the transition piece; and the eccentric nodal mass and inertia tensor of the RNA, MRNA and IRNA, located at the distances ax, ay, and az relative to the top of the tower. The input parameters for the nodal masses for the TP and RNA and the mass moments and mass products of inertia included in the inertia tensor (Ixx,Iyy,Izz,Ixy,Ixz,Iyz,) are presented in Table .

In addition to the mass contributions already presented, an external mass contribution, referred to as the hydrodynamic mass mhydro, arises when a body moves in a fluid. According to , the hydrodynamic mass per unit length of a free circular cylinder can be expressed as 20mhydro=ρCmA if the current is disregarded. Here, the fluid density is ρ=1027 kg m⁻³, Cm=1 is the hydrodynamic mass coefficient for a cylinder, and A=πr2 is the fluid-displaced area for the monopile with radius r.

The first three tower bending mode shapes used for the MDE configuration in Sect.  have been calculated using the FE model presented above. They are shown in Fig.  for displacements and bending moments in the FA and SS directions.

This section presents the basis for the MDE performed for the IEA 15 MW RWT support structure in terms of sensor type and placement (i.e. the HAWC2 output channels in um(t)), band separation used in the frequency domain, and the choices of Ritz vectors and mode shapes used within the individual bands (Φ̃i).

As presented in Sect. , it is widely accepted in the literature that the dynamic part of the response up(t) can be predicted based on measured accelerations. From these accelerations, displacements are obtained through double integration. However, for the quasi-static part of the response, the displacements are often inaccurate because measurement noise in the acceleration measurements is amplified during low-frequency integration. To overcome this challenge, use strain gauge measurements as input to the MDE for the quasi-static response estimation. Alternatively, use the low-pass-filtered (vertical) accelerations obtained from DC accelerometers relative to the gravitational acceleration to estimate rotations. This has the advantage that no double integration must be performed, and no additional sensors must be installed. In the present work, um(t) therefore contains displacements and rotations for the prediction of dynamic and quasi-static responses, respectively (see Fig. ).

Obviously, the location of the accelerometers will impact the quality of the virtual sensors. Different methods have been used to optimise the sensor placement . However, in practical applications, accessibility is just as relevant for the installation of sensors, since maintenance and replacement of structural health monitoring systems play a central role in the robustness of the overall system. Thus, in the present work, the physical sensors are placed at locations where internal platforms are most likely installed inside the tower (see Fig. ).

As presented in Fig. , the multi-band MDE in Eq. () is performed by separating the response of the IEA 15 MW RWT into four individual bands (B1 to B4) before combining them into the total predicted response up(t).

The rationale for the band separation depends on case-specific factors, including the frequency distribution of the external loads, the dynamic properties of the considered structure, and the properties of the sensors available in the monitoring system. Thus, the frequency bands should be selected such that the response is predicted accurately without exceeding the inherent sensor limitations of the MDE. The justification of the present band separation is given below for the MDE configuration summarised in Table :

B1 is defined with an upper limit of 0.05 Hz. According to , accurate displacements cannot be obtained from measured accelerations at frequencies below 0.05 Hz. Hence, the measured DOFs in Φm are defined in terms of rotations in B1, and the boundary represents a practical limitation of the sensors. B1 represents the quasi-static domain of the response, primarily driven by turbulence. Thus, the Ritz vectors included for the prediction in this band are obtained from the nodal force and moment in Fig. a and b. Furthermore, the wind is assumed to act as a distributed load across the tower, whereby the first tower bending mode shapes in Fig. a are also included in the MDE.

Fatigue DELs reduce a load history to a single equivalent load range ΔPeq, which is defined as the constant amplitude 1 Hz sinusoidal load causing the same amount of fatigue damage as the original load history. The same applies to fatigue DESs ΔSeq, making DELs and DESs convenient measures of comparing fatigue contributions across load cases with different durations . Thus, in the present section, the DELs and DESs combined for the individual DLCs presented in Sect.  are compared and discussed. Furthermore, the MDE performance is assessed: first for DELs and DESs calculated for the individual DLCs and, subsequently, in Sect. , for the DESs calculated for the individual HAWC2 section moment time histories. In both cases, the comparison is performed in all nodes of the IEA 15 MW RWT HAWC2 model.

The DEL for a single load history ΔPeq,s can be calculated as in Eq. (), where neq is the number of 1 Hz cycles in the considered time series. Similarly, the DEL for the individual DLCs can be calculated as 26ΔPeq,DLC=∑s∈DLCneq(ΔPeq,s)mneq,DLC1m, where 27neq,DLC=neqnseed,DLC is the total number of 1 Hz cycles in the simulations contained in the individual DLCs, with nseed,DLC being the simulation seeds for the individual DLC (i.e. the number of (converged) simulations in Table  for a given DLC at MWL equal to MSL). Inserting Eq. () in Eq. () yields the more compact representation 28ΔPeq,DLC=∑s∈DLC(ΔPeq,s)mnseed,DLC1m. As the DEL retains the unit of the load, the DES ΔSeq,s can be obtained by applying Navier's stress distribution formula to the DEL ΔPeq,s for the individual nodes of interest in the support structure. However, the elements in the IEA 15 MW RWT are not consistent in terms of bending stiffness across the nodes, whereby Navier's formula will produce discontinuous stresses at the nodes. Thus, only the DES associated with the maximum nodal stresses in the monopile and tower circumference are considered for each node. Furthermore, only the contributions arising from the bending moments are included in the DESs, which are calculated as 29ΔSeq,DLC=∑s∈DLC(ΔSeq,s)mnseed,DLC1m for the individual DLCs.

Figures  and show the DELs and DESs related to the FA and SS section moments obtained from the HAWC2 simulations directly (solid line) and predicted using the multi-band MDE configuration from Sect.  (dashed line).

As illustrated in Fig. , the DELs generally look similar to the moment curve from the first tower bending modes or the thrust load (see Figs.  and ), with overlying effects from other loads and modes. In the FA direction (a), the operating DLCs 1.2 and 2.4 generally induce lower DELs compared to DLCs 3.1, 4.1, 6.4, and 7.2, with DLC 6.4 resulting in the maximum DEL across all DLCs and directions (FA and SS) at the mud line. The lower DELs of DLCs 1.2 and 2.4 can be attributed to the significant aerodynamic damping provided by the operating rotor, as discussed in Sect. . However, within the tower top region, specifically from around 120–144 m, the operating DLCs show higher DELs due to uneven loading of the rotor and 3P effects, as discussed in Sect. .

In the SS direction (b), in which the aerodynamic damping, the effects from thrust load variations, and the 3P effects have less influence, the differences in DEL between operating and non-operating DLCs are generally smaller than those observed in the FA direction (a). It is worth noting that DLC 7.2 results in significantly higher DELs than all other DLCs at elevations above approximately 75 m. As discussed in Sect. , this can be attributed to tower top moment arising from the blade vibrations, which are enabled by the locked rotor configurations specific to this DLC.

An inherent problem of the DELs in Fig.  is that they do not explicitly account for changes in cross-section dimensions, whereby small DELs might still cause large stresses in regions with small tower diameters. Thus, in Fig. , the DESs have large values in the tower top region, where the corresponding DELs in Fig.  are small. This indicates that the accuracy of the MDE cannot be ignored in the tower top region. For the present analysis in Fig. , this is especially important for DLCs 1.2 and 2.4 in the FA direction (a) and DLC 7.2 in the SS direction (b), which have their DES maxima in the tower top region.

Figures  and show that the MDE underestimates the DELs in a ±15 m zone around the MSL for all DLCs in both the FA and the SS directions. Furthermore, for the DESs estimated by MDE in Fig. , it is seen that the multi-band MDE performs poorly at the tower top, where it overestimates the DESs of the operating DLCs (1.2 and 2.4) in both the FA and the SS directions while underestimating the DESs for the standstill DLCs (6.4 and 7.2) in the FA direction.

Figures  and are based on a combined DEL and DES calculated for the individual DLCs for each elevation z along the IEA 15 MW RWT support structure. Thus, it corresponds to an averaged or mean error, conveniently used for assessing long-term MDE performance, although inherently sensitive to bias errors. Therefore, to assess the short-term performance of the MDE in the individual HAWC2 simulations, the relative error of the DESs is calculated for the individual HAWC2 simulations as 30εMDE=ΔSeq,s,MDEΔSeq,s,HAWC2-1, where ()HAWC2 denotes the DESs calculated from the HAWC2 time series of the FA and SS section moments, and ()MDE denotes the DESs calculated from the corresponding MDE estimate. Figure  presents the relative error εMDE of the DESs, related to the FA and SS section moment and calculated for each elevation z along the IEA 15 MW RWT support structure.

It is observed in Fig.  that the error εMDE is predominantly in the range of ±5 %, except at the tower top, where the MDE performs inconsistently for the various DLCs. The error generally shows a dependency on the wind speed, which can be attributed to the operational and environmental variability of the IEA 15 MW RWT, arising from the varying rotor speeds, changing turbulence, and changing wave loads, which cannot be captured by the MDE, assuming a linear and time-invariant response.

In the following sections, the tower top MDE error observed in Figs.  and is discussed for the individual operating scenarios (operating, idle, and locked rotor). Furthermore, the MDE error observed at ±15 m around MSL in Figs.  and is discussed. To contextualise the behaviour observed in Figs.  to and enhance insight into the discrepancies between the DELs and DESs obtained from HAWC2 and MDE, selected sample moment histories are considered in the discussion. These are chosen based on the error metric εMDE; however, for DLC 7.2, the sample moment histories have been chosen based on the absolute error obtained from ΔSeq,s,MDE-ΔSeq,s,HAWC2. Intervals of the selected moment histories discussed in the following sections are presented in Appendix 

When combining the conclusions from the above discussion, it is assessed that the MDE used in the present work generally performs well, except at the tower top, where it performs inconsistently across the different considered DLCs. Furthermore, in the ±15 m range around MSL, the MDE consistently underestimates the DES, resulting in an error of up to 6 %. This may be improved by further enhancing the MDE as described below, although, in practice, a 5 %–6 % error level may be overshadowed by other uncertainties. Hereby, the main challenges associated with the present use of MDE the following:

capturing the local effects of the flexible and dynamic response of the rotor and blades