4.2. Multi-Blade Coordinate Transformation
It is known that loads on the turbine blades have the largest component at the 1P harmonic frequency. Thus, a load control system is designed focusing on the harmonic components of the load at 1P and neighboring frequencies. The multi-blade coordinate transformation (MBC) [
6] is used to extract the one-per-revolution (1P) harmonic load components from the blade root moment sensors [
8]. The MBC assumes that the three turbine blades are identical to each other and located 120 degrees apart. The MBC transformation requires the azimuth angle of the rotor (
) for the calculation of harmonic load components.
Figure 6 shows the wind turbine augmented with the MBC transformations.
The forward MBC transformation is used to map the flapwise root bending moments of the three blades,
,
and
, from the rotating frame to
and
, the corresponding loads in a fixed frame. The forward MBC transformation can be written as:
where
is the azimuth angle (in radians) of the turbine rotor. Under steady wind conditions, the forward MBC transforms the 1P harmonic loads in the rotating frame into constant loads in a fixed frame.
The right inverse of the forward MBC transformation, denoted as “Inverse MBC” block in
Figure 6, is utilized to calculate the sectional-lift actuator commands for the three blades, i.e., the rotating frame signals
,
and
, by transforming the control commands in the fixed frame
and
. The fixed frame control commands are calculated by a feedback controller described in
Section 4.6. The inverse MBC transformation can be written as follows:
The inverse MBC transformation commands the sectional lift actuator of each blade with 1P cosine and sine carrier signals modulated by the control signals and . When these latter signals are steady (constant), and the rotor frequency is constant, the rotating frame control signals are pure harmonics at 1P. Otherwise, the rotating frame control signals will have spectral content at frequencies other than 1P as well.
The configuration shown in
Figure 6 is useful for developing an SLC system that reduces blade-root loads in the rotating frame at 1P and its neighboring frequencies; that is, reduce the blade-root loads in the frequency range 1P
, with
P. In order to reduce the 1P harmonic loads using the MBC transformations, one would need to design a control system to lower the fixed-frame components of the load in the frequency interval
. Recall that the MBC transforms a 1P rotating-frame frequency component to a 0 rad/s fixed-frame frequency component [
8,
11].
4.3. Design Model via System Identification
A model for the dynamic system with fixed frame input
and fixed frame outputs
is obtained for the design of the SLC. This system is shown in
Figure 6. To determine a mathematical model for this system, it is assumed that the wind speed is constant and uniform over the rotor disk. Under this assumption, the open-loop plant model may be assumed (approximately) time invariant, albeit depending on the constant wind speed
V impinging on the rotor.
The primary focus of the work described in this article is on designing and utilizing SLC in above-rated wind speeds, when the loads are relatively large. The rated and cut-out wind speeds for the 3.4 MW are 9.8 and 25 m/s, respectively [
26]. The design model for developing a robust SLC solution is obtained at
m/s, which is close to the midpoint between rated and cut-out wind speeds. A linear state-space model of the open-loop plant at
m/s is obtained using the nonlinear simulation data from the modified OpenFAST model of the turbine. The simulations assume uniform wind without any shear or turbulence.
To identify the plant model, it is necessary to excite the plant with appropriate inputs and record the response. Filtered pseudo-random binary signals (PRBS) are chosen as the input to the open-loop plant in
Figure 6. A second-order low-pass Butterworth filter with a 3-dB bandwidth of 20 rad/s given by
is used. The frequency response of the low-pass filter is shown in
Figure 7. Note that the largest 1P frequency of the rotor is 1.22 rad/s, which corresponds to the rated rotor speed. Thus, a filter with 20 rad/s bandwidth is suitable to identify the relevant open-loop dynamics.
Two independent PRBS signals are filtered by
and the resulting signals applied at the input
of the open-loop plant in
Figure 6. The OpenFAST simulations are run for a total of 3000 s, and the first 500 s (98 revolutions) are discarded to remove any initialization effects, resulting in 2500 s (490 revolutions) of usable data. A time step of 0.01 s is chosen for the simulation, which allows estimation of frequency responses up to 50 Hz or 314 rad/s, theoretically.
Figure 8 shows the filtered PRBS inputs signals. The left plots show a small interval (15 s) of the time series signals at the two plant inputs. The corresponding plots on the right show the magnitude of the Fourier transform of the signals. The magnitude of the transfer function of the low pass filter
is shown with a red line. This magnitude is calibrated to match the signal spectrum at low frequency. It can be seen that the frequency content of the signals is focused in the range of 0–10 rad/s and slowly tapers off at higher frequencies as desired, following the magnitude of the low-pass filter.
Table 3 shows the structural degrees of freedom enabled in the OpenFAST model for system identification. Other degrees of freedom and aerodynamic behavior options like wind shear and tower-blade interaction are also relevant to the control problem but are not enabled in system identification. These latter effects are considered unmodeled disturbances/dynamics and are handled with the robust control method selected for the design of the feedback controller.
The outputs
are logged and the prediction error method (
pem function in MATLAB [
40]) is used to identify a state-space model from the simulation data. The 2500-s window of usable data is divided into two sets, a 2000-s long “training” dataset and a 500-s long “validation” dataset. The training dataset is used to obtain models of varying orders, which are evaluated for best fit to the validation dataset. The model mismatch is measured with the normalized root mean square error (NRMSE), defined by
where
is the time series validation data,
is the time series predicted by the model, and the
norm of the time series
x is defined by
The NRMSE for models of different orders, for the case of
m/s, is shown in
Table 4. The model with
states is selected as it has the smallest NRMSE with the validation data.
Further insight into the model’s ability to capture essential dynamics at low frequency (up to 3–5 rad/s, approximately) can be obtained from
Figure 9. This figure shows the frequency response of the transfer function from the input
to the output
(see block diagram in
Figure 6). The frequency response calculated using the sixth order model shows good fit of the empirical transfer function obtained from the time-series data. The empirical transfer function is obtained using the
etfe command provided by the System Identification Toolbox [
40] in MATLAB.
Let denote the 2×2 transfer matrix from the input to the output for the LTI model identified at m/s. Note that the units of are kN·m.
We define the four entries of this matrix as follows:
The four entries of the transfer matrix
are shown in
Appendix A.
Figure 10 shows the Bode plots of the entries of
. It follows from the figure that
and
, for frequencies
rad/s.
In addition, in this frequency range, we have the imaginary part of
as approximately zero. As a result of this structure, the frequency response of
is essentially a rotation matrix modulo of a non-unitary magnitude. This structure implies that the two singular values of
approximately satisfy
in the interval
rad/s. This, in turn, implies that in this frequency range, inverting the frequency response matrix is a fairly robust operation as its condition number is nearly one.
Figure 11 depicts the two singular values of the frequency response matrix
. The Euclidean norm of the first column of the frequency response matrix
, defined in the right-hand side of Equation (
7), is also shown with a red dashed line.
The condition number (ratio of maximum to minimum singular value) is shown in
Figure 12. These data suggest that inverting the frequency response matrix does not pose a challenge. The inverse of the frequency response matrix is used in the development of the SLC algorithm presented in
Section 4.6. More specifically, this transfer matrix is inverted at
to decouple the control loops at low frequencies.
4.6. Synthesis of the Control Algorithm
To specify the control algorithm in
Figure 15, it is necessary to choose the frequency
, the robust controller, the saturation management scheme and the anti-windup gain
. The selection of
and, if necessary, the robust controller is done using a loop shaping design procedure [
35] whereby the augment plant
is pre- and post-compensated with weights
and
, respectively, so that the frequency shaped plant
has desired closed-loop properties as measured by the frequency response of its (two) singular values. In our method, these weights are taken as
where
I denotes the
identity matrix. The weight
can be seen in
Figure 15 following the robust controller block, while the weight
is shown following the output of the augmented plant. After a desirable loop shape for
is achieved by selecting
, the resulting loop is tested for robust stability.
If robust stability is satisfactory, then the robust controller block is taken to be identity, and the design of the linear elements of the control algorithm is complete. Otherwise, a robust controller with transfer matrix
is obtained by solving the following
optimization problem
where the minimization is done over all controllers that stabilize
. The controller
is generally of high-order, and its order can and should be reduced to obtain the final robust controller in
Figure 15.
Let us elaborate on the selection of the weights
and
. The post-compensator
renders the transfer matrix
non-dimensional and equal to the identity matrix at
. That is, post-compensation is used to decouple and normalize the two control channels at DC. The calculation of
, and its use as a post-compensator, is numerically stable and remarkably robust given that the condition number of
is close to one at low frequencies (cf.
Figure 12). The singular values of
are shown in
Figure 16. Note that they are exactly one at
, and remain close to one for low frequencies. Thus, in order to reduce the fixed-frame moments
and
for frequencies ranging from DC to a low-frequency
, it suffices to select an integral pre-compensator
, as shown in Equation (
9). For all practical purposes,
is the cross-over frequency of magnitude for
. Hence, if robust stability can be achieved with the robust controller
,
will be the closed-loop bandwidth. That is,
defines the frequency range where load reduction of fixed-frame moments will occur. Given this argument, we recommend taking
, where
is the rated turbine rotor frequency.
To complete the design method, we need to define the criteria for robust stabilization of the shaped plant
. We use the normalized coprime factor stability margin [
34] defined by
Based on experience, we take the minimum acceptable stability margin to be 0.4. That is, if
, the robust controller block in
Figure 15 is set to be the identity matrix. Otherwise, we design the robust controller from Equation (
11) and possibly include a controller order reduction step. Note that our threshold for the stability margin exceeds the recommended value
[
41] to increase robustness to modeling uncertainty.
The significance of the normalized coprime factor stability margin in Equation (
12) may be understood with the aid of the block diagram in
Figure 17. In this figure,
and
are dynamic perturbations that represent inverse multiplicative uncertainty at plant output and direct multiplicative uncertainty at plant input. It is well-known that the block
can capture modeling uncertainty and/or parametric variations, e.g., pole migrations due to changes in operating conditions such as the wind speed. On the other hand, the block
may capture unmodeled dynamics, such as unsteady aerodynamic effects due to sectional lift actuation using plasma-based devices. See [
41] for further details on uncertainty modeling. It turns out that the feedback loop in
Figure 17 is (internally stable) if and only if
Loosely speaking, if
, then the feedback system in
Figure 17 can tolerate modeling uncertainty as large as 57% at all frequencies without losing stability.
Further insight on the interpretation of
follows from the bounds it provides for classical single-input single-output (SISO) stability margins. While this is not our case, these SISO bounds are informative. More specifically, a given normalized coprime stability margin
provides the bounds on the gain and phase margin shown in Equation (
14) [
34].
For example, a coprime stability margin guarantees a gain margin of 2.33 and a phase margin of 47 degrees, which are acceptable values for SISO loops.
Now, we provide the details of our SLC design for the 3.4 MW turbine. The design model is given by the augmented plan as identified at 18 m/s wind speed. The choice of the bandwidth is based on the reduction in the damage equivalent load of the flapwise root bending moments on the turbine blades. In order to choose the bandwidth, controllers were designed and tested with turbulent wind. The value that provides the most load reduction is selected. Based on this analysis, the bandwidth is chosen to be rad/s.
Once a numerical value for
is known, the shaped plant
in Equation (
8) is known and we may compute its stability margin using Equation (
12). For our plant, we obtained
. This implies that the control system is robust and
loop-shaping based modifications are not needed. Thus, the final controller is a MIMO integral controller of the form
Figure 18 shows the singular value plots of the open-loop transfer matrix
for the control loop broken at plant output, the closed-loop sensitivity
and complementary sensitivity
. Load reduction is expected to occur for frequencies up to 0.4–0.5 rad/s because the sensitivity is less than one in this range of frequencies. The peak value of the sensitivity is
, and the peak value of the complementary sensitivity is
, well below recommended values [
34].
The linear component of the control algorithm in
Figure 15 was designed using the plant model identified at a wind speed of 18 m/s. The same design process can be used to design linear controllers for plants at other wind speeds. The controllers thus obtained can be utilized to obtain an SLC scheduled on wind speed.
Section 5.4 demonstrates the results of this approach. No significant benefits are observed with the scheduled SLC in terms of load reduction or stability margin or performance to justify the added complexity of scheduling control parameters on wind speed. Therefore, the results presented in the rest of the paper are based on the controller designed at 18 m/s, as shown in Equation (
15).
Saturation Management
In this paper, we assume that the actuator limits are
. These limits on the sectional lift coefficient (i.e., actuator) are within reach of future plasma technology [
7,
25]. Following Reference [
42], nonlinear scaling is applied to the control signals calculated by the controller, as shown on
Figure 15. The purpose of the scaling is to modify the output of the controllers so that they remain within the actuator limits while providing suitable load reduction. Denote the inputs to the nonlinear scaling block as
and
, while the outputs are
and
. The scaling can be described as follows
It should be noted that the scaled inputs always have the following property.
The constraint shown in Equation (
17) on the fixed frame control signal implies that the corresponding control signals in the rotating frame are always within the range of
, which satisfies the actuator constraint. This scaling method is non-conservative (i.e., it uses the full actuator authority) because the magnitude of each actuator command satisfies
. Furthermore, scaling the control signals in the fixed frame, prior to applying the MBC transformation to obtain the actual actuator commands, ensures that at every time instant, the sum of all the actuator commands is zero, which is necessary to avoid undesirable changes on the collective lift on the blades.
When the output of the scaling function saturates, the system in
Figure 15 becomes open-loop. The two integrators in the controller would windup unless this saturation is detected and acted upon. A simple back-calculation anti-windup scheme [
43] is implemented to ensure the integrator outputs
and
track the scaled signals
and
to avoid integrator windup. The anti-windup gain
is non-dimensional and determines the bandwidth of the anti-windup loops. A unit value for
implies a bandwidth comparable to the SLC. Typically, this gain would need to be much larger than one. After trial and error, we take
. The parameters of the final SLC controller are shown in
Table 5.