SNratio loglo f MB

The third till the sixth column show the deviation of the four parameters from their real values after optimization. The deviation is defined as real parameter - optimized parameter A = —-real papameter p- %

The seventh column shows the number of iterations it took to converge to a local or to the global minimum. Finally, the eighth column displays the final value of the objective function Fobj.

Run

Additive noise simulations: "Beamlsd"

S/N-ratio

[dB]

A Czl [%]

[%]

A C23 [%]

[%]

N, H

Fobj [-]

1

oo

0.00

0.00

0.00

0.00

293

8.54 -10"28

2

50.1

0.00

-0.01

0.00

0.01

663

3.183 -10^

3

30.1

-0.01

-0.14

0.00

0.08

554

3.183-10"2

4

10.1

-0.06

-1.43

0.04

0.75

586

3.183

5

-3.87

-0.29

-6.73

0.20

2.68

665

79.585

6

-9.89

-0.52

-12.59

0.39

2.38

587

318.39

Table 5.3: Model updating results for Beamlsd. Case 3: Nelder-Mead (additive noise).

Table 5.3: Model updating results for Beamlsd. Case 3: Nelder-Mead (additive noise).

It can be concluded that noise causes the optimization to converge more slowly (compare run 1 with run 2-6), or even fails to converge to the real values. Especially the damping parameters Keq12 and Keq23 are hard to estimate from noise contaminated data. The deviations from the real values are acceptable up to a signal-to-noise ratio of 30 dB.

The previous three cases have shown the ability of the proposed model parameter updating procedure to reproduce the known parameter values from simulated data. This observation gives confidence in its application to data generated by more complex models.

As a next step in the verification of the model parameter updating procedure, the mechanical model SDLW1 describing the structural dynamics of the complete

Lagerwey LW-50/750 wind turbine is used to generate the "measured" data. In addition, the results will be used to determine the required magnitude of input force and the excitation/observation point(s) on the Lagerwey LW-50/750 wind turbine that would excite all modes of interest and result in adequate response levels.

5.4.2 SDLW1

In this subsection the mechanical model SDLW1 is used to generate "measured" data. This 18-DOF model (exclusive pitch and azimuth) describes the structural dynamics of the complete Lagerwey LW-50/750 wind turbine. Both the tower and rotor blades are approximated by one superelement. A schematic of this module is depicted in Fig. 5.7. Additional information can be found in Appendix I.2.3.

Substructure Superelement

Superelement approximation (SDLW1)

Lagerwey LW-50/750 wind turbine

Superelement approximation (SDLW1)

Figure 5.7: Superelement approximation of the Lagerwey LW-50/750 wind turbine. Both the tower and rotor blades are approximated by one superelement (Towerlzf and Bladelxz respectively). The flexibility of the foundation is approximated by a torsional spring. Sensor locations S1 to S7 are marked with a ■ .

Again it is assumed that the mass, inertia and length of the rigid bodies within the superelements are estimated fairly accurately, but that calculation of the stiffness and damping parameters may be difficult. Consequently, Cfz, Kfz (foundation spring and damper), Cz1T, Kz1T, Cz3T, Kz3T (tower springs and dampers), Czi, Kzi, Cz3, Kz3 (rotor blade springs and dampers) are the tunable parameters.

The input is a stepwise change in the force at the tower top in positive x-direction (step time 0 s, initial value of 40000 and final value of 0 N) producing an initial tower top deflection of 0.0405 m. To ensure that the simulation starts in steady-state, the equilibrium point has been determined using the SIMULINK® function trim and the resulting state vector has been saved in a MAT-file. This state vector is used as initial condition in the performed simulations. The outputs are the accelerations measured at 7 locations on the structure (marked with a ■ in Fig. 5.7).

The objective function is defined as the sum of squares of the difference between the measured and simulated outputs. Because the measured accelerations have strongly different orders of magnitude, they are scaled to ensure that no information contained in the signals will be lost. Time histories of 1001 points are generated and the stiff solver Linsim is used to numerically integrate the differential equations with a fixed step size of 0.01 s.

It is assumed that flexibility of the foundation can be approximated by a torsional spring. The spring constant is experimentally determined by Jacobs [116] and is equal to

Both the tower and the rotor blades of the Lagerwey LW-50/750 wind turbine are approximated by one superelement. The torsional spring constants for the superelements can be derived directly from the physical data supplied by the manufacturer using the automated structural modeling procedure outlined in Section 3.4. The resulting tower torsional spring constants are

and the rotor blade torsional spring constants are

Viscous damping, sufficient to produce a damping ratio of about 1 %, has been added to the model by specifying the following coefficients of viscous damping

Kfz =

5 • 107

[kg/s]

Kz1T =

1 • 107

[kg/s]

Kz3T =

1 • 106

[kg/s]

Kzi =

9 • 103

[kg/s]

Kz3 =

1 • 103

[kg/s]

The above mentioned constants are used to generate the "measured" response. In the sequel, these constants will be referred to as "real values". The objective is depicted in Fig. 5.8 as a function of the percentage of variation in the parameter vector 9 (i.e. 100% = 90, and 110% means that the numerical values of all ten tunable parameters are set to 110 % of their real values). Obviously, both the Nelder-Mead method and the least-squares method should be able to locate the global minimum without much difficulty.

Objective function

Objective function

Figure 5.8: Objective function Fobj of SDLW1 with o calculated values.

Figure 5.8: Objective function Fobj of SDLW1 with o calculated values.

The optimization is started by invoking NMstart.m which sets the initial values of the tunable parameters a certain percentage of their real values and subsequently performs successive simulations on SDLW1. The Nelder-Mead algorithm changes the tunable parameters in an attempt to minimize the error between "measured" and simulated response. In this case, the optimization time can be reduced from 10 to 1 second by multiplying the objective function by the time vector t. The explanation is as follows: because t is small in the early stages of the response, it weights early errors less heavily than late errors allowing a reduction of the simulation time.

The numerical results are given in Table 5.4. The first column indicates the run number. The second till the sixth column specify the initial values of the tunable parameters. For example, Cz\T = 110 % means that the initial value of the torsional spring constant between the first and the second rigid body of the tower is set to 110% of the real value (i.e. 1.72901 • 109). The seventh column shows the number of iterations it took to converge to a local or to the global minimum. It can be concluded that the torsional spring and damper constants can be identified uniquely using perfect data provided that the initial guess is sufficiently close to the real values.

The region of convergence can be increased substantially by first applying the aforementioned least-squares method and subsequently finetuning the results by using the Nelder-Mead direct search method. The results of this case are summarized in Table 5.5.

Run

Noise-free simulations: "SDLW1"

CzlT > Kzir

CZ3T > ^ziT

CZ„KZ 3

Minimum

NMO

60%

60%

60%

60%

60%

7199

local

NM 1

65%

65%

65%

65%

65%

6795

global

NM 2

80%

80%

80%

80%

80%

6301

global

NM 3

90%

90%

90%

90%

90%

3297

global

NM 4

100 %

100%

100%

100 %

100 %

579

global

NM 5

110%

110%

110%

110%

110%

3259

global

NM 6

120 %

120 %

120%

120 %

120 %

3368

global

NM 7

125%

125%

125%

125%

125%

4026

local

NM 8

120 %

80%

120%

80%

120 %

3683

global

Table 5.4: Model updating results for SDLW1. Case 1: Nelder-Mead (noise-free)

Run

Noise-free simulations: "SDLW1"

t\r n AI

Cz3T>Kz3T

Ca,Ka

cz3,Kz3

Minimum

LS-NM 0

15%

185%

15%

185 %

15%

2681

local

LS-NM 1

20%

180%

20%

180 %

20%

4333

global

LS-NM 2

25%

25%

25%

25%

25%

6445

local

LS-NM 3

30%

30%

30%

30%

30%

2592

global

LS-NM 4

40%

40%

40%

40%

40%

3544

global

LS5

60%

60%

60%

60%

60%

4038

global

LS-NM 6

80%

80%

80%

80%

80%

1520

global

LS 7

100 %

100%

100 %

100 %

100%

236

global

LS-NM 8

120 %

120 %

120 %

120 %

120%

2883

global

LS-NM 9

140 %

140 %

140 %

140 %

140%

3527

global

LS-NM 10

145 %

145 %

145 %

145 %

145%

3110

local

LS 11

125 %

75%

125 %

75%

125%

1985

global

LS-NM 12

130 %

70%

130 %

70%

130%

3680

local

Table 5.5: Model updating results for SDLW1. Case 2: least-squares and Nelder-Mead (noise-free).

Table 5.5: Model updating results for SDLW1. Case 2: least-squares and Nelder-Mead (noise-free).

The ability of the presented off-line model parameter updating procedure to reproduce the known parameter values gives confidence for its application to field data obtained from a modal test.

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