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Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Inverse Load Calculation of Wind 

Turbine Support Structures – 

A Numerical Verification Using 

the Comprehensive Simulation 

Code FAST

Preprint

Thomas Pahn and Raimund Rolfes

Leibniz Universität Hannover

Jason Jonkman and Amy Robertson

National Renewable Energy Laboratory

Presented at the 53rd Structures, Structural Dynamics, and 

Materials Conference

Honolulu, Hawaii

April 23 – 26, 2012

Conference Paper

NREL/CP-5000-54675

Revised November 2012
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ERRATA SHEET

NREL REPORT/PROJECT NUMBER:NREL/CP-5000-54675 

TITLE: Inverse Load Calculation of Wind Turbine Support Structures - A Numerical 

Verification Using the Comprehensive Simulation Code FAST: Preprint 

AUTHORS: Thomas Pahn and Raimund Rolfes, Leibniz Universitat Hannover, and 

Jason Jonkman and Amy Robertson, NREL

ORIGINAL PUBLICATION DATE: April 2012 

DATE OF CORRECTIONS: November 2012 

The following corrections were made to this report/document:

A previous version of this paper, available until November 28, 2012, incorrectly 

calculated the differences between the dynamic components of the inverse load and the 

rotor thrust in Figures 5, 8, and 11. These figures have been replaced and now illustrate 

the correct calculations. 
1

Inverse load calculation of wind turbine support structures –

a numerical verification using the comprehensive simulation 

code FAST

Thomas Pahn1

Leibniz Universität Hannover (LUH), Hannover, Germany

Jason Jonkman2

National Renewable Energy Laboratory (NREL), Golden CO, United States of America

Raimund Rolfes3

Leibniz Universität Hannover (LUH), Hannover, Germany

Amy Robertson4

National Renewable Energy Laboratory (NREL), Golden CO, United States of America

Physically measuring the dynamic responses of wind turbine support structures enables 

the calculation of the applied loads using an inverse procedure. In this process, inverse 

means deriving the inputs/forces from the outputs/responses. This paper presents results of a 

numerical verification of such an inverse load calculation. For this verification, the 

comprehensive simulation code FAST is used. FAST accounts for the coupled dynamics of 

wind inflow, aerodynamics, elasticity and turbine controls. Simulations are run using a 5-

MW onshore wind turbine model with a tubular tower. Both the applied loads due to the 

instantaneous wind field and the resulting system responses are known from the simulations. 

Using the system responses as inputs to the inverse calculation, the applied loads are 

calculated, which in this case are the rotor thrust forces. These forces are compared to the 

rotor thrust forces known from the FAST simulations. The results of these comparisons are 

presented to assess the accuracy of the inverse calculation. To study the influences of turbine 

controls, load cases in normal operation between cut-in and rated wind speed, near rated 

wind speed and between rated and cut-out wind speed are chosen. The presented study 

shows that the inverse load calculation is capable of computing very good estimates of the 

rotor thrust. The accuracy of the inverse calculation does not depend on the control activity 

of the wind turbine.

Nomenclature

BE = viscous modal damping matrix

D = damping matrix

Di = modal damping ratio

F = Fourier transform of the force vector

Finv = inversely calculated load

f (t) = force vector

f (t) = force signal

 1 Dipl.-Ing., Institut für Statik und Dynamik (ISD), Appelstraße 9A 30167 Hannover. 2 Ph.D., National Wind Technology Center (NWTC), 1617 Cole Blvd Golden CO 80410-3393, AIAA Professional 

Member. 3 Prof. Dr.-Ing. habil., Institut für Statik und Dynamik (ISD), Appelstraße 9A 30167 Hannover. 4 Ph.D., National Wind Technology Center (NWTC), 1617 Cole Blvd Golden CO 80410-3393.
2

f = frequency

0i f = eigenfrequency in Hz

H( jω) = frequency response function (FRF) matrix

Hg ( jω) = generalized FRF matrix

i = number of vibration mode

M = mass matrix red Mg = generalized mass matrix of the reduced system

mgi = entry of the generalized mass matrix of the reduced system

K = stiffness matrix red Kg = generalized stiffness matrix of the reduced system

k = stiffness

RotThrust = rotor thrust from FAST simulation

t = time

U0 = modal matrix

Y( jω) = Fourier transform of the displacement vector

y (t) = displacement vector

y(t) = displacement signal

y (t) = velocity vector

y (t) = acceleration vector

ω = angular frequency

ω0i = eigenfrequency in s-1

I. Introduction

nowing the loads acting on wind turbine structures is essential for different reasons. First of all, load 

assumptions used for the structural design need to be verified, as stated by the International Electrotechnical 

Commission Technical Standard (IEC TS) 61400-131

. Also, knowing the acting loads is important in terms of 

structural health monitoring, especially for offshore wind turbines. At least 10 % of all offshore wind turbines in 

German waters will be equipped with monitoring systems to ensure the reliability of the support structures. This 

requirement is demanded by a standard published for the BSH (Federal Maritime and Hydrographic Agency, 

Germany)2

. Finally, from the means of realistic load values, lifetime predictions can be calculated.

To determine the applied loads, inverse load calculation procedures can be used. To apply such a procedure, the 

structural-dynamic behavior of the wind turbine needs to be accurately modeled. Then, measured dynamic structural 

responses such as accelerations can be used to calculate the applied force values. This concept is not new and has 

been successfully used in various fields – for instance in aviation3,4, automotive industry5,6, dynamically operating 

machines7

, and even in geophysical sciences8. Recently, these procedures have been applied to wind turbines. Ref. 9

and 10 describe laboratory tests of simple structures under stochastic loads. These tests were aimed mainly at 

verifying the numerical accuracy of the inverse calculation procedure and thus do not account for the coupled 

dynamics of the blades and structure, nor do they allow the inclusion of wind turbine control. In Ref. 11 and 12, the 

results of inverse load calculations of two operating 5-MW wind turbines are shown. However, although all authors 

present excellent results, a verification of the calculated forces is not possible due to the fact that real load values 

hardly can be measured.

Hence, an important intermediate step has not been investigated yet: what effects do these coupled dynamics and 

turbine control have in relation to the inverse load calculation? These aspects are investigated using the 

comprehensive simulation code FAST13

. FAST accounts for these effects. Furthermore, aside from the calculated 

structural responses, the loads are known. This approach provides the basis for a discussion about the accuracy of 

inverse load calculation when applied to wind turbine structures.

K
3 

II. Procedure of the inverse load calculation

Different numerical methods exist to calculate loads inversely. Generally, these methods can be divided into 

time-domain and frequency-domain approaches. Both have their advantages and drawbacks. Evaluative summaries 

are given in Ref. 6, 7 and 14. This paper seeks to verify the frequency-based procedures presented in Ref. 12 and 15. 

The fundamental theoretical principles of this approach are described briefly below.

The equation of motion for a discrete linear dynamical system can be described by the following second order 

ordinary differential equation in the time domain.

My Dy Ky f   ( ) ( ) ( ) ( ) t t tt ++ = (1)

The spatial system is defined by the stiffness matrix K and the mass matrix M, both of dimension n x n. D is the 

viscous damping matrix (n x n), y(t) the vector of outputs containing the displacements (with its derivatives in time

y ( )t and y( )t ), and f(t) represents the vector of forces/inputs. The solution of the forward problem that determines 

the responses based on its excitation leads in the frequency domain to

Y HF ( ) ( ) ( ) j jj ω ωω = ⋅ (2)

with ω as the angular frequency and Y(jω) as Fourier transform of the displacement response vector. F(jω) is the 

Fourier transform of the forces and H(jω) represents the n x n frequency response function (FRF) matrix. The FRF 

matrix H(jω) defines the dynamic characteristics of the system. 

To solve the inverse problem, equation (2) needs to be inverted so that

( ) ( ) ( ) −1 FH Y j jj ω ωω = ⋅ . (3)

As can be seen, an inversion of the FRF matrix is required. Equation (3) represents a determined system, in which 

the number of known responses equals to the number of unknown forces. So, the FRF matrix is square. As long as it 

is also nonsingular, equation (3) has a unique solution. However, it has to be taken into account that the inverse 

solution usually is ill-conditioned in the vicinity of the system resonances.

In many practical cases, it is desirable to consider a different number of known responses n than unknown forces

m. Choosing n > m leads to an over-determined system that enables the elimination of random errors by applying 

e.g., a least squares approach. A solution can be obtained in terms of the pseudo-inverse H+

(jω) (Ref. 16 and 8). The 

equation to solve the inverse problem results in

m,1 ( ) ( ) ( ) m n, n,1 j jj ω ωω + F HY = ⋅  (4)

where F (jω) represents an approximate of F(jω). The pseudo-inverse, needed in equation (4), is calculated 

according to 1

, ,, ,

T T

mn mn nm mn

− + =⋅ ⋅     H HH H (5)

whereas superscript T denotes the Hermetian transpose (the transpose of the complex conjugate) of the FRF matrix

H. 

Equations (1) to (5) are given in spatial coordinates. Usually, the system properties are known in a modal 

representation, such as when they are derived from system identification. Ref. 17 describes the identification of a 5-

MW onshore wind turbine with a jacket foundation structure. To connect the spatial to the modal coordinates, the 

scaled modal matrix U0 is required. U0 results from an eigenvalue analysis of the undamped system, which is 

described by the stiffness and the mass matrix given in equation (1). The modal matrix consists of the eigenvectors 

of the system. Considering a FRF matrix in modal coordinates denoted as Hg(jω), the relation between the modal 

and the spatial domain is given by

( ) ( ) ( ) 1 11 1

0 0

− −− − =⋅ ⋅ T

g H UH U j j ω ω . (6)

Finally, the inversely calculated load can be transformed into the time domain using the inverse Fourier 

transform applied to the vector of forces. 

( ) ( ) { }−1

f F t j = ω (7)

III. Structure, load cases and simulation

To run the time-domain simulations needed to verify the inverse procedure, the comprehensive aeroelastic code 

FAST is used. FAST uses a nonlinear combined modal and rigid multibody formulation for modeling the dynamics 

of two- or three-bladed horizontal-axis wind turbines13. For this verification study, the NREL 5-MW reference wind 

turbine model is used, which is a three-bladed, upwind, horizontal-axis model, with a tubular steel tower and rigid 
4

foundation. The turbine has a hub height of 90 m and rotor-blade lengths of 63 m. The structure is described in Ref. 

18 in detail. Within the simulation, 16 degrees of freedom (DOFs) are enabled, so that the dynamics of the following 

components are represented: First and second flapwise blade modes (6 DOFs), 1st edgewise blade modes (3 DOFs), 

drivetrain rotational-flexibility (1 DOF), generator (1 DOF), yaw compliance (1 DOF), 1st and 2nd tower bending 

modes in fore-aft and side-side direction (4 DOFs).

To focus on the aeroelastic interaction and turbine control influences, a simple structure is desired. For that 

reason the tubular tower structure with a rigid foundation is chosen.

In terms of turbine control, the full-span collective blade-pitch control and variable-speed generator-torque 

control are enabled. FAST also provides nacelle yaw control, which is not used in this study because the influence of 

a changing yaw angle is not of high interest in the verification of the inverse load calculation. Only an additional 

coordinate transformation would be required.

Using TurbSim19, stochastic turbulent wind fields are generated as input for the simulations. Load cases defined 

in IEC 61400-1 Ed. 3: 200520 under normal wind conditions for power production are chosen. Such load conditions 

are dominant contributions of fatigue, which the inverse load calculation aims to predict. An overview of the load 

cases (LCs) is given in Table 1. All of the load cases represent mean wind speeds at hub height between cut-in and 

rated wind speed (middle of region 2), near rated wind speed and between rated and cut-out wind speed (middle of 

region 3). Thus, conditions in which there are few expected control activities (load case 1) and high expected control 

activities (load case 3) are generated. In addition, load case 2 represents the transition between controllers where 

maximum thrust occurs.

The FAST simulations are run in a way that they correspond as closely as possible to practical field test 

conditions. The simulation time is set to 600 s, to match requirements from the IEC design process. Also, the timeTable 1: Load cases

Load case No. Mean wind speed at hub height Turbulence type Turbulence characteristic

1 7 m/s

NTM 1) B 2) 2 12 m/s

3 18 m/s

1) normal turbulence model 20

2) medium turbulence characteristic – 14 % turbulence intensity at 15 m/s 20

Tower top:

acceleration, displacement

Approx. half tower height:

acceleration

Sensors locations 

Rotor thrust force 

RotThrust

wind load

Inversely calculated force 

Finv

Figure 1. FAST simulation output parameter
5

to frequency-domain transforms are sufficiently accurate for this simulation length. Output parameters of the wind 

motion and the turbine control are used for plausibility checks. Through an additional simulation – a linearization –

the system matrices can be computed. The linearization simulation in FAST linearizes the nonlinear equation of 

motion about a given operating point and ouputs the system matrices of the full system. For the inverse calculation, 

the system matrices are reduced to the fore-aft tower bending modes via a dynamic reduction method. Getting 

measurement data for the rotating blades is much more complicated than measuring the motions of fixed parts, such 

as the tower. Moreover, structural information of the blades is oftentimes confidential and consequently cannot be 

considered known data.

Parameters used in the inverse calculation are depicted in Fig. 1. The system responses at the tower top 

(maximum amplitude of first tower bending mode) and near half tower height (maximum amplitude of second tower 

bending mode) are needed. At both locations, acceleration time series are used to calculate the dynamic part of the 

load inversely. Although FAST also calculates tower displacements, accelerations are used because field tests 

usually use accelerometers to record structural motions.

In terms of the inverse load calculation, the acceleration time series are double-integrated to obtain the 

displacements. The integration is done using a frequency-domain approach21. For the double-integration, the 

frequency spectra need to be divided by -(2πf)2, which has the advantage that only one calculation step is needed.

This approach causes a mathematical singularity around ω = 0, which is eliminated by applying a high-pass filter to 

the obtained displacements. With the displacements, the dynamic component can be calculated using equation (4), 

which is depicted in Fig. 2 by the dark grey box. In addition, displacements at the tower top in combination with 

known stiffness properties from the stiffness matrix K are used to calculate a static/quasi-static component (light 

grey box in Fig. 2). To ensure compatibility to the dynamic component, the tower top displacement is low-pass 

filtered, so that the mean value and the low-frequency content remains. This approach is chosen due to the load 

characteristics with respect to the requirements of the inverse load calculation procedure and the numerical 

integration of the signals. The described components are superimposed to obtain the inverse load (Fig. 2). To 

distinguish between the quasi-static and the dynamic components, a cut-off frequency has to be chosen. This 

frequency is chosen at a spectral gap, so that neither exciting frequencies nor eigenfrequencies are influenced. In this 

study, the cut-off frequency is chosen to be 0.15 Hz.

The result of the inverse load calculation is an equivalent rotor thrust force. For verification purposes, this force 

will be compared to the rotor thrust force known from the FAST simulation (see Fig. 1). The rotor thrust output 

from FAST does not represent the applied aerodynamic thrust, but the force transmitted between the rotor and the 

low-speed shaft. This force includes both the applied aerodynamic thrust as well as the rotor inertia forces from the 

turbine vibration.

IV. Results

The inverse calculation uses the over-determined approach, such as mentioned in equation (4). The limiting 

factors for the system of equations are the number of vibration modes of the system and the unknown forces that are 

intended to be calculated inversely. The chosen approach exclusively uses vibration modes of the support structure, 

which are the tower bending modes of the used model. Only the fore-aft direction is assumed to be loaded 

significantly of which the first and second fore-aft tower bending modes are available. Because the number of 

response signals must not exceed the number of vibration modes, a maximum of two response signals can be used. 

This study examines the inverse calculation of the applied loads for an onshore wind turbine. These loads are 

Inverse load

Dynamic component

(above cut-off)

Static/quasi-static component

(mean value and below cut-off)

f y (tkt ) = ⋅ ( ) ( ) { ( ) ( )} 1 ft F j j ω ω − + = H Y⋅

Figure 2. Overview of superposition of the components
6

dominantly caused by the instantaneous wind field at the swept rotor area. A reasonable simplification is the 

description of the wind loads by the rotor thrust. Therefore, the goal of the inverse load calculation of an onshore 

wind turbine is to determine the rotor thrust force. According to equation (4), the system of equations appears in the 

following form.

1,1 ( j jj ω ωω ) 1,2 ( ) 2,1 ( ) + F HY = ⋅  (8)

The two acceleration responses are measured at the locations depicted in Fig. 1. The acceleration time series are 

double-integrated to obtain the displacements. In this way, the response vector Y(jω) is obtained.

The properties of the system dynamics are gained by a linearization. During linearization, FAST extracts 

linearized representations of the complete nonlinear aeroelastic wind turbine. This capability allows for developing 

the system matrices and the full-system mode shapes. The eigenvalue analysis of the system matrices gives the 

eigenfrequencies of the fore-aft tower bending modes as f01 = 0.32 Hz (first mode) and f02 = 2.92 Hz (second mode). 

Additionally, the modal matrix U0 that contains the eigenvectors column-wise is known.

To reduce the system to the fore-aft tower bending modes, a modal reduction is used. The modal reduction is 

based on the multiplication of the mass matrix and the stiffness matrix with a transformation matrix22. The 

transformation matrix contains the eigenvectors of the vibration modes that shall remain in the reduced system – in 

this case the eigenvectors of the fore-aft tower bending modes. The reduced system has the same eigenfrequencies 

as the full system. The reduced stiffness matrix red Kg and mass matrix red Mg in modal space are:

6

1

6

1.06 10 0

0 0.92 10

red

g in Nm−   ⋅ =   ⋅  

K (9)

5

3

2.56 10 0

0 2.73 10

red

g in kg   ⋅ =   ⋅  

M . (10)

The damping is assumed to be viscous modal damping. The linearization gives damping ratios that are D1 = 0.04 

and D2 = 0.01, which enables the determination of the damping matrix BE.

BE = ⋅⋅⋅ diag m D (2 gi i i ω0 ) (11)

For the calculation of BE, the modal damping ratios Di, the entries of the reduced mass matrix mgi, and the 

eigenfrequencies ω0i = 2πf0i are needed, with i being the number of the vibration mode. Having the system 

description in modal space, the inverse FRF matrix in modal space can be calculated23.

( ) 1 2 red red

g g Eg j j ωω ω − H M BK =− + + (12)

Using equation (6), the FRF matrix can be transformed back to the spatial domain. The FRF matrix still is a 

square matrix, since the system matrices of the reduced system are square as well. Truncating the last column of H

leads to the dimension (2 x 1). That means, the first mode and coupling between the first and the second mode are 

present. Now, the pseudo-inverse is calculated according to equation (5), which allows for the solution of equation 

(8). In this way, the inverse load is calculated in the frequency domain. Applying equation (7) gives the time-domain 

representation.

A comparison of the inversely calculated force (Finv) and the rotor thrust force from FAST (RotThrust) serves for 

the verification of the accuracy of the inverse calculation.

For each load case, the control activities that occur during the simulation are described briefly. Additionally, the 

inversely calculated force is compared graphically to the rotor thrust that serves as the reference value. To estimate 

the similarity between both forces, the quasi-static and the dynamic components are compared seperately, each in 

the time domain and frequency domain.

A. Load case 1

Load case 1 (see Table 1) represents a mean wind speed at hub height between cut-in and rated wind speed. Load 

case 1 is chosen because little control activity is expected, since the variable-speed controller operates in region 2.

The FAST output channels that contain the main external conditions and control activities during the simulation give 

the following information. The mean wind speed at hub height is about 7 m/s, which indeed represents the middle of 

region 2. The nacelle yaw holds a constant position. The blade-pitch angle is constant at zero, which means the three 

blades do not show blade-pitch control activity. The generator-torque is proportional to the square of the rotor speed

in the active region.
7

Figure 3 shows the inversely calculated load compared to the rotor thrust force that is computed by FAST. The 

inversely calculated force follows the simulated force qualitatively.

Figure 4 gives a comparison of the quasi-static force components in the time domain and the frequency domain. 

Again, a very good agreement can be stated. To gain a better insight of the pure dynamic force component, Fig. 5 

shows a comparison of these components in the time domain and the frequency domain. As is shown in the time 

domain, the inverse calculation has higher amplitudes than the simulation. Comparing the frequency spectra reveals 

that the differences are mainly related to the amplitudes. The frequencies of the FAST rotor thrust force also appear 

in the inversely calculated force, which leads to qualitatively similar frequency spectra. This observed increase 

amplitude of the high frequency content in the inverse load will lead to an overestimation of fatigue damage.

Time in s

Force in kN

Frequency in Hz

|Force| in kNs

Figure 4. Quasi-static components – LC 1

Finv

RotThrust

Finv

RotThrust

Time in s

Force in kN

Finv

RotThrust

Figure 3: Rotor thrust and inversely calculated force – LC 1
8 

Figure 5: Dynamic components – LC 1

B. Load case 2

Load case 2 (see Table 1) represents a mean wind speed at hub height near rated wind speed. This load 

case represents the transition between controllers, where maximum thrust occurs. The mean wind speed at hub 

height is approximately 12 m/s, which is near rated wind speed. The generator torque is no longer proportional to the 

rotor speed. Again, the nacelle yaw position is constant around zero. Now, blade-pitch control shows activity during 

the simulation. Since collective blade-pitch control is enabled, all three blades follow exactly the same control 

algorithm. The blade pitch angle varies between 0° and 10°.

Figure 6 shows the result of the inverse calculation. As were found in load case 1, the inversely calculated load 

and the rotor thrust are compared. Again, a good match between both signals is visible.

The quasi-static components are depicted in Fig. 7, both in the time domain and frequency domain. Both graphs 

match nearly perfect. An analog depiction is given in Fig. 8 for the dynamic components. The comparison of the 

dynamic components in the time domain and the frequency domain leads to similar results as gained for LC 1. The 

inverse calculation shows higher amplitudes than the reference rotor thrust in the time-domain comparison. Indeed, 

the characteristics of the frequency spectrum are calculated well by the inverse procedure. The peaks of the rotor 

thrust are very similar, but the amplitude of the inversely calculated rotor thrust is higher than the amplitude of the 

rotor thrust from FAST. Force in kN |Force| in kNs

Inverse load

Rotor thrust

Inverse load

Rotor thrust

Time in s

Frequency in Hz

Time in s

Force in kN

Finv

RotThrust

Figure 6: Rotor thrust and inversely calculated force – LC 2
9 

Figure 8. Dynamic components – LC 2

C. Load case 3

Load case 3 (see Table 1) represents a mean wind speed at hub height between the rated and cut-out wind speeds 

(middle of region 3). Load case 3 is chosen because high control activity is expected. The main external conditions 

and control activities of this load case are the following. The mean wind speed at hub height is at 18 m/s, which 

indicates the middle of region 3. As for load cases 1 and 3, the yaw position is constant at zero. The collective bladeTime in s Force in kN

Frequency in Hz

|Force| in kNs

Figure 7. Quasi-static components – LC 2

Finv

RotThrust

Finv

RotThrust

Force in kN |Force| in kNs

Inverse load

Rotor thrust

Inverse load

Rotor thrust

Time in s

Frequency in Hz
10

pitch angle ranges between 10° to 20°. While the torque controller operates in region 3, relative to region 2, the 

control strategy has changed from “optimal power” in region 2 to “constant power” in region 3.

An illustration of the result of the inverse load calculation for load case 3 is shown in Fig. 9. The inversely 

calculated load and the rotor thrust of the FAST simulation are depicted. As visible, both signals show a very good 

agreement, as is the case for load cases 1 and 2 as well.

The quasi-static components are depicted in Fig. 10, both in the time domain and the frequency domain. The 

excellent agreement of both force components is visible. Figure 11 shows the time-domain and frequency-domain 

depiction of the dynamic component. As already seen for the load cases 1 and 2, the inverse process calculates

similar frequencies as occur for the FAST rotor thrust force. Again, the amplitudes of the inversely calculated force 

are higher than the amplitudes of the rotor thrust, which is true for both the time-domain and the frequency-domain 

comparisons. 

Time in s

Force in kN

Finv

RotThrust

Figure 9: Rotor thrust and inversely calculated force – LC 3

Time in s

Force in kN

Frequency in Hz

|Force| in kNs

Finv

RotThrust

Finv

RotThrust

Figure 10. Quasi-static components – LC 3
11

Figure 11. Dynamic components – LC 3

D. Summary of the load cases

All three load cases show similar characteristics in terms of the accuracy of the results for the inverse 

calculation. The static/quasi-static components are calculated with a very good agreement. The dynamic components 

contribute most to the observed differences between the inversely calculated force and the rotor thrust. The inversely 

calculated load partly has twice the amplitude. Hence, the reason for the differences is assumed to be part of the 

inverse calculation of the dynamic parts. The mathematics of the underlying equations produce exact results, as can 

be shown using simple numerical examples. But, the inverse calculation is based on an important assumption. The 

system used for the inverse calculation is a reduced system. The reasons for the reduction are described in section 

III. So, the rotor thrust force is derived using the full system description, whereas the inverse load is calculated with 

a reduced system. The reduction only affects the inverse calculation of the dynamic components where the main 

contribution to the differences occur. Thus, the system reduction is assumed to be the main reason for the observed 

differences. Because the differences are considered acceptable, the system reduction is an appropriate assumption.

As indicated by the results of the three presented load cases, the inverse calculation tends to produce force 

amplitudes that are always slightly higher than the amplitudes of the reference force.

The resulting quality of the inverse load calculation is similar for the three load cases. Consequently, the inverse 

calculation does not depend on the control activities of the wind turbine. The three load cases were set up to 

represent different operating conditions with different control activities. Because the differences do not vary 

significantly, the stated conclusion is demonstrated.

V. Conclusion and Outlook

This paper presents a numerical verification study of an inverse load calculation procedure. The inverse load 

calculation is done for a 5-MW onshore wind turbine structure using a model of the system dynamics and system 

responses, both simulated with the comprehensive simulation code FAST. This approach accounts for coupled 

dynamics and turbine control. For verification, the inversely calculated load is compared to the known rotor thrust

from the FAST simulation. A concept for superimposing the inverse load by a static/quasi-static component and a 

dynamic component is demonstrated. The theoretical bases of the dynamic inverse load calculation are discussed.

As shown, the inverse load calculation is capable of generating good estimates of the applied load, which is 

shown for simulations that account for coupled dynamics of wind inflow, aerodynamics, elasticity and turbine 

control. Force in kN |Force| in kNs

Inverse load

Rotor thrust

Inverse load

Rotor thrust

Time in s

Frequency in Hz
12

Three different load cases are simulated. The load cases are set up so that little control activity, maximum rotor 

thrust, and high control activity occur. The results show that the accuracy of the inverse load calculation does not 

depend on the control activity of the wind turbine.

The presented study shows the ability of the inverse load calculation to derive the applied loads for onshore wind 

turbines, represented by the rotor thrust. In practice, the rotor thrust can be measured. Consequently, the application 

of the inverse load calculation is useful if the measurement of the rotor thrust is impeded e.g., by prohibited/hindered 

accessibility. Another potential application for the inverse load calculation would be for systems with more 

complicated load conditions, such as those caused by combined wind and wave loads.

For that reason, further studies will investigate the capability of the inverse load calculation to compute 

combined wind and wave loads for an offshore wind turbine. Again, a numerical study that uses the comprehensive 

simulation code FAST will be performed. 

Eventually, the inverse load calculation is intended to be used for lifetime predictions using measurement data of 

offshore wind turbines. The fatigue-strength analysis mainly depends on the range of stresses. Assuming a linear 

dependency between the loads and the stresses, as is done for steel under normal operational conditions, the quasistatic and the dynamic component of the inverse load will be used for the fatigue analysis. The quasi-static 

component shows an excellent agreement and the dynamic component contains amplified amplitudes. Consequently, 

using inversely calculated loads for fatigue analysis is a safe assumption. This conclusion is also valid in case of 

nonlinear load-to-stress relations, as occurs for reinforced concrete. Then, the mean values of the stresses have to be 

taken into account additionally, which depends on the static component that shows an excellent accuracy as well.

Acknowledgments

The presented research was done during a scientific research exchange of T. Pahn at the National Renewable 

Energy Laboratory (NREL) in Golden CO, USA. T. Pahn cordially thanks the NREL staff for their remarkable 

support. T. Pahn also thanks the German Academic Exchange Service that funded the scientific exchange.

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