When a wind turbine operates above the rated wind speed, the blade pitch may be governed by a basic single-input–single-output PI controller, with the shaft speed as input. The performance of the wind turbine depends upon the tuning of the gains and filters of this controller. Rules of thumb, based upon pole placement, with a rigid model of the rotor, are inadequate for tuning the controller of large, flexible, offshore wind turbines. It is shown that the appropriate controller tuning is highly dependent upon the characteristics of the aeroelastic model: no single reference controller can be defined for use with all models. As an example, the ubiquitous National Renewable Energy Laboratory (NREL) 5 MW wind turbine controller is unstable when paired with a fully flexible aeroelastic model. A methodical search is conducted, in order to find models with a minimum number of degrees of freedom, which can be used to tune the controller for a fully flexible aeroelastic model; this can be accomplished with a model containing 16–20 states. Transient aerodynamic effects, representing rotor-average properties, account for five of these states. A simple method is proposed to reduce the full transient aerodynamic model, and the associated turbulent wind spectra, to the rotor average. Ocean waves are also an important source of loading; it is recommended that the shaft speed signal be filtered such that wave-driven tower side-to-side vibrations do not appear in the PI controller output. An updated tuning for the NREL 5 MW controller is developed using a Pareto front technique. This fixes the instability and gives good performance with fully flexible aeroelastic models.
Much of the research on wind energy systems is based on reference wind turbines, including descriptions of the aerodynamics, structures, and controls. These reference turbines are implemented in a variety of models, from high-resolution 3-D geometry for CFD/FEM to models containing just a few degrees of freedom for electrical grid analysis. A consistent implementation of the controls is of the utmost importance: few aspects of wind turbine or wind power plant dynamics can be studied without considering the controls. Yet there is a sensitive interdependence between the controller and the aeroelastic properties of the wind turbine model. In general, the same controller will not produce the same closed-loop dynamic response on models of different fidelities. If the responses differ in important respects such as power fluctuations, rotor loads, pitch activity, and stability, then the models are, in essence, not of the same wind turbine.
It is often taken for granted that a 1 or 2 degree-of-freedom
drivetrain model, with pole-placement techniques, will provide a reasonable
gain tuning for the controller of a wind turbine. Hansen et al. (2005) describe
such a gain tuning and scheduling approach where the target for the rotor
speed control mode – that is, the mode which appears above the rated
wind speed, where pitching of the blades is used to hold the rotor speed near
a constant target value – has a natural frequency of 0.1 Hz and damping
ratio of 0.66. The gains are scheduled on the basis of a single parameter:
The flexibility and aerodynamic response of a real wind turbine have a strong influence on the rotor speed control mode. If the gains are not adapted accordingly, the actual mode will have a response which differs significantly from the target frequency and damping ratio. This fact is well-known, also to those authors who have employed the simplistic gain-tuning approaches. The performance of the wind turbine is subsequently verified by aeroelastic simulations, and if the control system performs reasonably from an engineering standpoint, it may be considered a successful design.
There are problems with this approach, though. One or two degree-of-freedom models provide little insight into the true dynamics of the system: essentially, the controller tuning is being conducted blindly. The resulting behavior of the rotor speed control mode depends strongly upon the properties of the aeroelastic model, so the same controller may function well or poorly, in a given application. Users of the controller may not understand, or acknowledge, the limitations.
For example, the NREL 5 MW proportional-integral (PI) controller is often adopted as the baseline for comparison against advanced control algorithms: see Schlipf et al. (2013), Spencer et al. (2013), Jafarnejadsani and Pieper (2014), and Yang et al. (2015) for some recent examples. Yet this controller is unstable when paired with a fully flexible aeroelastic model which includes elastic twisting of the blades. For a fair comparison, the reference PI controller should be tuned according to the same model and criteria that were used to demonstrate the performance of the optimal control algorithms; failure to do so weakens the scientific basis of the results.
Controller tuning does not need to be based on a reduced model. Tibaldi et al. (2012) optimized the gains of blade pitch and generator torque controllers using full aeroelastic load simulations, combined with a component cost model. The optimization, which ran through seven iterations, was noted to require 4000 h computing time, which limits practicability of the method. Nonetheless, the approach of Tibaldi et al., considering the influence of loads and energy production on lifetime cost, is the proper way to evaluate the overall performance of a wind turbine control system.
A practical model for control design contains a minimal number of degrees of freedom. It is reasonable to use a low-fidelity model, since a well-designed controller will be robust to small inaccuracies associated with neglected higher-order effects. The model must be of sufficient resolution to capture the important first-order effects. In particular, the frequencies and damping ratios of the control-dominated closed-loop modes within the full model should be preserved in the simple model.
There is not a perfect consensus on which degrees of freedom must be
included in a model for control design. Among older publications, Leithead
and Connor (2000) is a good place to start, as they conclusively
demonstrated that rigid-body models of the drivetrain are inadequate. They
included the response of the rotor aerodynamic torque to perturbations in
the rotational speed, blade pitch angle, and rotor-average wind speed:
Wright (2004) conducted a methodical investigation into the structural degrees of freedom necessary to obtain a stable control tuning. The 600 kW, 42.6 m diameter CART (Controls Advanced Research Turbine) was used as a reference case. A brief, initial investigation demonstrated the importance of drivetrain flexibility and actuator dynamics for a reference PI controller. A more extensive degree-of-freedom study was conducted with a disturbance-accommodating control (DAC) strategy, which is a state-feedback control algorithm where additional states are used to model, and eventually cancel, disturbances such as turbulence. Though DAC and PI controllers are not identical, lessons learned about the influence of structural flexibility on a DAC controller can likely be applied to PI tuning as well.
Wright progressively activated one structural degree of freedom at a time: drivetrain torsion, collective blade flap, and tower fore–aft. For each set of active degrees of freedom, a controller was synthesized and subsequently evaluated by a brief time-domain simulation with a stepped wind speed profile. It was found that the first blade-flap-wise modes have the potential to destabilize the first drivetrain mode and must be included in models for control design. The tower modes, principally the first fore–aft mode, were found not to have a significant influence on the behavior of the rotor speed control.
The effective above-rated blade pitch and generator torque controller of the NREL 5 MW wind turbine, for small perturbations about a mean operating state.
Sønderby and Hansen (2014) revisited the question of which
degrees of freedom should be included in a control tuning model, in the
context of an onshore version of the NREL 5 MW wind turbine. The controller
was not specified; rather, the investigation was based on open-loop transfer
functions between the actuated degrees of freedom – collective blade pitch
and generator torque – and generator speed. Particular emphasis was placed
on how the poles (indicating frequency and damping properties) associated
with the structural modes changed with the activated structural and
aerodynamic degrees of freedom. Capturing the non-minimum phase
zeros Define an actuator-to-output transfer function
In contrast with Wright, Sønderby and Hansen found that the control design model should include blade-flap-wise and edgewise modes as well as tower fore–aft and side-to-side modes. Quasi-steady aerodynamics may be used at low frequencies (below the first tower modes), though blade torsional flexibility should be included when linearizing the aerodynamic forces.
The discrepancy between the conclusions of Wright and Sønderby and Hansen can partly be attributed to the increased flexibility of large, multi-MW wind turbines, in comparison with the older CART turbine. However, it is also the case that Sønderby and Hansen, in selecting degrees of freedom, applied criteria which were too rigorous in the context of tuning a typical PI blade pitch controller. Though the poles associated with the structural modes are of concern, as are non-minimum phase zeros, it is not the case that the resulting controller performance is sensitive to each pole and each zero.
Thus, there is the need to revisit simple models for the design and tuning of PI controllers for highly flexible offshore wind turbines. This is addressed in Sect. 2, where a study like that of Wright is conducted, incrementally adding degrees of freedom. In contrast with Wright, focus is placed on a PI blade pitch controller, which is evaluated in terms of the pole (frequency and damping ratio) associated with the rotor speed control mode. A simplified method is also implemented to account for the dynamic wake effect, which may be relevant when gains are low. For basic controller tuning, it is recommended to use, at minimum, a model with elastic driveshaft, blade flap, and torsion, and tower fore–aft modes, as well as a dynamic wake. This is applicable to the basic control of rotor speed; auxiliary control functions like active damping may require additional degrees of freedom.
Since the NREL 5 MW wind turbine controller is used as a baseline in so many
studies, including the present one, it is critical to characterize its
performance. Dunne et al. (2016) have recently found that the effective behavior
of this controller is not what one would expect from the reported
Ocean waves may excite tower resonant vibrations; Sect. 4 shows that these may appear in the primary blade pitch control path, which is not desirable. In the context of the NREL 5 MW controller, this constrains the cutoff frequency of the low-pass filter on the shaft speed measurement.
The NREL 5 MW wind turbine controller is retuned in Sect. 6, according to simple metrics of system performance established in Sect. 5. The selected tuning is compared to a revised pole-placement approach, using an appropriate aeroelastic model. The fundamental, unavoidable tradeoff is between the fluctuations in rotor speed and the pitch activity; a Pareto front illustrates this explicitly.
The most important finding, deserving of special emphasis, is this: the appropriate controller tuning is highly dependent on the aeroelastic model; therefore, no single reference PI controller can be defined for use with all models.
Consider the NREL 5 MW wind turbine, mounted atop the OC3 monopile foundation, as described by Jonkman and Musial (2010). The wind turbine is operating in its steady-state condition at a uniform, above-rated wind speed. Fluctuations in the wind speed are, at present, limited to small perturbations about the mean. In this case, the rotor speed and generator power output are controlled as shown in Fig. 1. (All speeds are given in reference to the low-speed shaft. The gain scheduling in Fig. 1 differs from the controller described by Jonkman et al. (2009), for reasons which are made clear in Sect. 3.)
It is desirable to ask some basic questions about this controller. How well does it perform? Could the gains and low-pass filter be chosen differently, to improve the performance? Is the same controller tuning also applicable to an offshore wind turbine? These are the topics of Sects. 4–6. In order to arrive at the answers, a model of the closed-loop system dynamics is needed. This could be a high-resolution model. Yet there are advantages in adopting a simple model. A simple model is computationally efficient, aids understanding of the system behavior, and can form the basis for more advanced state-space control algorithms. In light of inconsistencies in the literature regarding which degrees of freedom are needed, it is worthwhile to establish some minimum requirements for a model of the closed-loop system dynamics.
Closed-loop transfer functions of collective blade pitch (gray
lines) and rotor speed (black lines) with respect to a uniform sinusoidal
perturbation in the axial wind speed. Angular units are radians. Note the
different
Magnitudes (left) and phases (right) of transfer functions between axial wind speed and blade pitch, rotor speed, and tower mud line bending moments. Three gains are shown: 0.5, 1.0 (thick lines), and 1.5 times the baseline gains from Fig. 1. The natural frequency (Hz) and damping ratio of the rotor speed control (1) and dynamic wake (2) modes are also shown, tabulated as a function of the gain multiple.
With the use of a multiblade coordinate transform, a three-bladed wind turbine
operating under normal conditions (balanced rotor, no extreme excursions)
can be represented as a linear time-invariant system, with state and output
equations of the form
In the discussion that follows we must distinguish between two categories of
modes. STAS employs modal reduction of each body (tower, nacelle, driveshaft,
and the three blades) prior to assembling the bodies, via constraint
equations, into the full wind turbine. For instance, the amplitudes of the
first fore–aft and side-to-side modes of the tower body (including the
foundation and soil
On the left: normalized spectra of the collective component of
rotationally sampled axial turbulence (
The second class of modes is the eigenvectors of the equations of motion of the assembled structure, including systems such as the generator, pitch actuators, and controls. These system modes may be dominated by one body mode – for instance, there is an obvious “first tower fore–aft” system mode – or they may have complicated shapes which are not so easily described.
Representing the wind turbine in the form of Eq. (1), the modal properties of the system can be computed. An examination of the system modes reveals one primary and one secondary mode, which, within reasonable bounds of the gain tuning, contain the dominant action of the controller. The primary mode can be called the “rotor speed control” mode, as it represents the fluctuation in the rotational speed of the wind turbine rotor, under the combined control actions of the generator and blade pitch actuators. The secondary mode is associated with the influence of dynamic wake effects on the rotor speed control; this will be called the “dynamic wake” mode. It is most active when control gains are set to comparatively low values.
There is overlap between the rotor speed control and dynamic wake modes. The rotor speed control mode contains the dominant rotor speed and blade pitch responses, but the states associated with the dynamic wake – the induced velocities – also participate. The dynamic wake mode contains the dominant response of the rotor-wide collective induced velocities, but these are driven by changes in the rotor speed and blade pitch, which also appear in this mode. Thus, the participation of the dynamic wake in the rotor speed control mode is not to be confused with the influence of the dynamic wake mode on the rotor speed and blade pitch response. The former is a dominant effect, which is addressed in Sect. 2.2. The latter, it will be shown shortly, is not so relevant, except when control gains are lower than usual. The salient point is that a dynamic wake model may be needed, even if the dynamic wake mode makes little contribution to the response of the relevant control variables.
The rotor speed control mode is clearly visible in transfer functions between axial wind speed and rotor speed. Figure 2 shows these transfer functions, as well as those for blade pitch, at four wind speeds between rated and cut-out. These results were obtained for a full (ca. 600 states) model of the NREL 5 MW wind turbine on a flexible tower and foundation, including soil flexibility.
Figure 2 also lists the natural frequency and damping ratio of the rotor speed control mode. The natural frequency is associated with the peak in the rotor speed transfer function, while the damping ratio indicates to some extent the sharpness of the peak. Although the rotor speed control mode is dominant, several other system modes also participate in the response.
To keep things simple, the discussion of model fidelity is focused on the two system modes with the greatest contribution to the low-frequency rotor speed response. For the baseline gains of Fig. 1, typical participation factors (Kundur, 1994) associated with the rotor rotational degree of freedom are 0.5 for the dominant rotor speed control mode and 0.2 for the secondary dynamic wake mode. These two modes serve as surrogates for the full transfer function: the properties of the transfer function, within the region influenced by the control tuning, can be inferred from the properties of the modes.
As an example, let the NREL 5 MW turbine, on the OC3 monopile foundation, be
operating at a mean wind speed of 16 m s
There is evidently a minimum in the peak sensitivity of rotor speed to fluctuating winds. At high gains, the blade pitch responds aggressively, in a manner which reduces the damping of the rotor speed control mode, while at low gains, the blade pitch response is so passive that it does not promptly arrest perturbations to the rotor speed.
Within reasonable bounds, gain tuning has little influence on the resonant response of the tower. This is mainly due to the non-minimum phase zero at 0.236 Hz. The presence of this zero is associated with the first tower fore–aft body mode. The nacelle moves in such a manner that the measured fluctuation in shaft speed is near zero, and there is thus no control response. The particular characteristics of the zero are influenced by other body modes, as well as where in the drivetrain the shaft speed is measured. In the most basic case where the only elastic degree of freedom is the tower fore–aft motion, the zero is caused by nacelle fore–aft motion which nearly cancels the fluctuating wind speed. When all the elastic degrees of freedom are included, the motion at the frequency of the zero defies such a simple description, but the outcome is similar.
The controller influence at higher frequencies is suppressed by the low-pass filter, with a corner frequency of 0.25 Hz.
The response of the wind turbine depends on both the input–output transfer
functions, as in Fig. 2, and the characteristics of the environmental
inputs. Typical spectra of rotationally sampled atmospheric turbulence (the
collective component at an outboard blade station) and ocean wave forces are
plotted on the left side of Fig. 4. Most of the energy in the turbulence is
concentrated at low frequencies, while that of the ocean waves is in the
vicinity of
The right-hand side of Fig. 4 shows spectra of the tower mud line bending
moments, for three values of the gain multiple
To sum up: if we know the natural frequency and damping ratio of the rotor speed control and dynamic wake modes, we can infer much about the response of the wind turbine to the control actions. For a reduced model to be useful in tuning gains, a minimal requirement is that it is able to correctly predict the properties of the rotor speed control mode. If low gains are to be evaluated – for instance, if the rotor speed control mode might be placed below the ocean wave frequency band – then it is also needed to predict the properties of the dynamic wake mode.
Closed-loop transfer functions of collective blade pitch (thin lines) and rotor speed (thick lines) with respect to a uniform sinusoidal perturbation in the axial wind speed. Nonlinear time-domain computations using FAST v8 are compared to equivalent results from a linear state-space model, obtained using equilibrium and dynamic wakes. The FAST results should be compared to the equilibrium-wake (black) curves.
The above statements are valid in the context of basic rotor speed control, for a wind turbine operating above the rated wind speed. Additional control functions – say, active damping of tower or drivetrain resonance – may require that additional system modes are also correctly predicted.
Aerodynamic forces on the blades are subject to transients as conditions
change, with a particularly strong effect associated with the blade pitch
angle. The transients can be grouped into the categories of circulation lag
(Theodorsen), associated with the development of lift along the blade;
dynamic stall, connected with movement of the chordwise location of flow
separation; and dynamic wake (or dynamic inflow), related to the downstream
convection of vorticity in the wake, which governs the induced velocity at
the rotor plane. In an analysis with the blade element momentum method, these
phenomena can be represented by a set of linear differential equations,
associated with each blade element. The equations employed here are based on
the circulation lag method described by Leishman (2002) and also Hansen et al. (2004); the Merz et al. (2012) variant of the Øye (1990) dynamic stall model;
and Øye's dynamic wake model, as documented by Snel and Schepers (1995).
Neglecting the tangential component of induced velocity, the aerodynamic
state equations associated with a given blade element are
The influence of transient circulation on the closed-loop transfer functions of collective blade pitch and rotor speed with respect to wind speed.
An examination of the
Considering first the dynamic wake, Fig. 5 shows transfer functions of rotor speed and blade pitch with respect to rotor-average wind speed. The solid curves were computed in the frequency domain, based upon a linear state-space model. Two cases are shown, one with the dynamic wake model active, and another with it inactive, such that the induced velocities are always in equilibrium with the airfoil forces.
Nonlinear time-domain results were obtained by defining a spatially uniform
wind field, whose axial velocity component varied in time about the mean,
with a prescribed frequency, and an amplitude of 0.5 m s
The FAST v8 program (Jonkman and Jonkman, 2016), with the BeamDyn blade
module and AeroDyn v15 aerodynamic module, was used for the time-domain
calculations. This version of AeroDyn included the Beddoes–Leishman model of
transient circulation, but it was limited to an equilibrium wake; thus, the
time-domain results should be compared to the black curves. The FAST
model did not include the elastic properties of the seabed; these were
stiffened in the linear model, in order that the tower natural frequencies
should match. (For other analyses, the seabed properties have been
represented by
The principal effect of the dynamic wake is that a blade pitch action
results in an initially large change, or overshoot, in the airfoil forces.
If the pitch angle is subsequently held steady, the forces decay to their
quasi-steady values over a timescale of roughly
A sketch of the matrix operations used to reduce the number of
aerodynamic states. The input
Above the rated wind speed, mean induced velocities decrease with the wind speed, so the dynamic wake becomes less significant at higher mean wind speeds.
Transient circulation has a moderate influence on rotor speed control, reducing the damping, as illustrated by the transfer functions in Fig. 6.
If one is interested in the bulk flow characteristics across the rotor, as would be relevant for the tuning of a collective pitch controller, it is not desirable to retain element-by-element resolution over the span.
A simple method is suggested to “collapse” the transient aerodynamics into a
set of equations associated with a single blade element. The transients of
state space Eq. (3) are computed according to a representative blade element
at
It is perhaps easiest to explain this operation by sketching the process by
which the aerodynamic states are reduced, as in Fig. 7. For simplicity, this
is presented as if there were only one aerodynamic state associated with
each blade element; the process is identical for each of the five types of
states in Eq. (3). In the state vector, there is a group
The process is repeated for each of the five types of aerodynamic states in Eq. (3), and the result is that five states represent the collective, transient aerodynamics of the rotor.
There are other, more formal methods by which the number of aerodynamic states could be reduced. A low-order series representation of rotor induction, such as the acceleration potential method described by Burton et al. (2001), is one possibility. Another is the modal reduction approach of Sønderby (2013) though as derived, this did not include a dynamic wake.
Nonetheless, the above ad hoc matrix reduction method works well for the present purpose of control tuning, where low-frequency, rotor-average wind inputs are of greatest concern. The reduced model is validated, in Table 1 and Sect. 5, against the original matrices employing the full radius-dependent Øye model of Eq. (3).
A series of models was constructed, progressing from the simplest case with
only rigid-body rotation of the rotor, through to the full case with the
elastic structure represented by 110 modal degrees of freedom. The reduced
models employed either quasi-steady aerodynamics or the five-state
transient model of Sect. 2.3, whereas the full model employed a blade
element momentum method with transients computed for each element. For the
reduced models, the number of states
Table 1 lists the models. For each model, the natural frequency and damping
ratio of the rotor speed control mode were computed at three above-rated
wind speeds: 12, 16, and 20 m s
The conclusion is that models which do not include at least tower fore–aft, blade-flap-wise, and torsional flexibility may give misleading estimates of the rotor speed control response and are therefore unfit for the purpose of tuning the controller. With blade flap and torsion, tower fore–aft, and a five-state transient aerodynamic model, the rotor speed control and dynamic wake modes are well-predicted. Adding blade edge and tower side-to-side flexibility makes little difference. Model 5D is therefore recommended as a minimal model for controller gain tuning.
Drivetrain torsional flexibility is expected to have little influence on the basic control tuning, provided that the low-pass filter frequency is reasonably low; however, this degree of freedom is retained in the models as it is common practice. It is indeed important to evaluate the damping of the first drivetrain mode, as this can potentially be destabilized by the generator torque control. Model 7D is recommended for evaluating the first drivetrain torsional resonance mode, as this is influenced by the flexibility of the blade edgewise and tower side-to-side modes.
Natural frequencies and damping ratios of the rotor speed control mode (using the baseline gains) and dynamic wake mode (at half the baseline gains), obtained from models with various degrees of freedom (DOFs). Results obtained with the reference high-fidelity model are highlighted in bold. Model 5D is recommended as a minimum model for basic controller gain tuning. Aero indicates whether the aerodynamics were quasi-steady (QS) or included transient dynamics (Dyn). BEM: blade element momentum method. For other abbreviations, see the table in Appendix A.
R: rigid rotor and blade pitch; d: driveshaft
torsion; e: blade edgewise; f: blade-flap-wise; t: blade torsion; F: tower
fore–aft; S: tower side to side. Notes:
On the left, the equivalent functions of the NREL 5 MW turbine controller during normal, non-saturated operation. The integral pathway (dashed box) contains the scheduled gain outside the integrated speed error. On the right, an alternate integral pathway with the scheduled gain inside the integral of the speed error.
It is emphasized that other control functions which are not shown in Fig. 1 may require additional degrees of freedom. Incorporating environmental inputs such as misaligned ocean waves may also require additional degrees of freedom, at least those of Model 7D.
As an alternative to an incremental study like that of Table 1, formal model-reduction methods could be employed, for instance Zhou et al. (1996).
The nonlinear, gain-scheduled NREL 5 MW wind turbine controller, during
operation above the rated wind speed, is shown in Fig. 8. The rate limits,
pitch angle limits, and integral gain saturation are omitted. With the
exception of the 0
The critical feature to note is that the scheduling of the integral gain
happens outside the integral of the speed error,
It will be shown that this leads to a misleading definition of proportional
and integral gain. An alternative is to schedule the integral gain inside
the integral,
Dunne et al. (2016) identified the fact that by scheduling the integral gain
outside of the accumulated speed error, the effective gains, for small
perturbations about a mean operating point
For a steady-state operating point, with zero speed error, the integral
pathway provides the mean blade pitch angle set point. This is clearly
illustrated by observing the behavior of the controller while the turbine
starts up in a condition of above-rated wind speed; Fig. 9 is an example. In
this simulation, using the FAST v8 program, the wind speed was a constant 15 m s
Thus, the integrated error
The integral pathway also acts, together with the low-pass filter, to determine the lag between fluctuations in the rotational speed and the blade pitch angle, which in turn influences the response of the system. This is best illustrated in the frequency domain, as in the following section, where the concept of phase can be applied.
Startup of a simulation at a wind speed of 15 m s
Let the NREL 5 MW wind turbine be operating in a uniform, steady,
above-rated wind. Let there be a small perturbation to the shaft speed,
That is, the signal coming from the integral pathway contributes both integral and proportional effects. This is confusing, to say the least.
The behavior of the two versions of the controller can be visualized in the
frequency domain, using phasors, as shown in Fig. 10. This particular phasor
diagram was generated using Model 5Q of Sect. 2.4, during normal operation
at a mean wind speed of 15 m s
In the present example, all quantities are given in reference to the
low-speed shaft, with
The phasor diagram is interpreted as follows. The low-pass filter on the
shaft speed fluctuation
Comparing Eqs. (8b) and (9b), it is evident that when the gain is scheduled
outside the integral term, the controller behaves as though the baseline
gains
A phasor diagram of the controller dynamics for gain scheduling (on the left) outside the integral and (on the right) inside the integral. The magnitudes and phases are normalized with respect to the shaft speed input; the dashed gray line indicates the unit circle.
The factor giving the effective gains of the NREL 5 MW controller, with respect to the nominal values, which have been scheduled outside the integral.
The gain factor
The natural frequency and damping ratio of the rotor speed
control mode, where the integral gain has been properly scheduled inside the
integrator, comparing the performance of the original published gains with
the case where the gains are reduced by the
A FAST v8/BeamDyn time-domain analysis of the NREL 5 MW turbine
with baseline controller, showing unstable behavior at wind speeds below 11.9 m s
Transfer functions (above) of waterline wave forces to rotor
speed and spectra (below) for a wave state of
According to Fig. 12, the rotor speed control mode of the NREL 5 MW turbine,
with its baseline controller, is unstable in the vicinity of the rated
wind speed. Whether the instability is present in a given analysis depends
upon the degrees of freedom implemented in the aeroelastic model. Blade
torsional flexibility is of particular importance; at a wind speed of 11.5 m s
The instability is confined to a narrow range of operation. On the
low-wind speed side, it is bounded by the control mode transition from rated
power and speed to maximum
Yet the instability is significant. Near the rated wind speed, the controller is driven through a greater number of mode transitions than necessary, which leads to more variability in the power production. The blade pitch is more active than necessary, which is reflected in both the pitch actuator duty cycle and the fluctuating loads on the blades, drivetrain, and support structure.
The instability can be demonstrated in the time domain. Fig. 13 shows the
response to a uniform wind which decreases in steps, at intervals of 30 s,
from 12.5 to 11.6 m s
It is concluded that the baseline NREL 5 MW controller is workable if the blades are modeled as rigid in torsion, but only because the inaccuracies associated with the simple rigid-shaft model used for gain tuning were counterbalanced by the effect of scheduling the gains outside the integral. If run with a fully flexible model, the baseline controller is unstable in an interval just above the rated wind speed. As a consequence, the many wind turbine control studies which have used the NREL 5 MW controller as a baseline have compared it against a PI controller whose tuning is somewhat arbitrary.
The essence of this conclusion is not unique to the NREL 5 MW controller. The appropriate controller tuning is highly dependent on the aeroelastic model; therefore, no single reference PI controller can be used with all models.
Ocean waves excite tower motions, and this can influence the rotor speed and control actions. As seen in Fig. 1, the cutoff frequency of the low-pass filter frequency of the NREL 5 MW wind turbine controller is 0.25 Hz, which is above the wave frequency band and nearly the same as the first tower natural frequency. This means that the controller will respond to wave-driven motions of the structure if these perturb the rotor speed measurement.
There are two ways in which structural motion could influence the rotor speed measurement. One is via a change in the relative wind speed, and thus the aerodynamic forces on the rotor. The other is by causing rotation of the nacelle about the axis of the driveshaft. A speed measurement at the low-speed shaft, as on a direct-drive wind turbine, is particularly susceptible to the latter effect.
The influence of ocean waves was determined by examining the closed-loop
transfer functions between waterline wave forces and rotor speed,
Figure 14 plots the transfer functions and also the spectra of rotor speed
fluctuations for ocean waves with significant wave height
To avoid the situation where the controller responds to wave-driven tower resonance, it is recommended to set the low-pass filter cutoff frequency to a value well below the first natural frequency of the tower. It is acceptable – and likely unavoidable – that the cutoff frequency then lies within the wave-frequency band of roughly 0.05–0.20 Hz.
While it is desired to minimize the response of the primary PI control path to tower motions, this does not rule out active damping of the tower. Active damping can be implemented via an auxiliary control path, where the phase is adjusted to maximize the damping effect.
When tuning the gains and filters of a wind turbine controller, it makes sense to implement some indicative performance metrics, in addition to stability criteria. The reason is that the environmental load inputs are highly nonuniform in terms of the spectra or frequency content of the signals; Fig. 4 illustrates this point. The response of the system depends on the properties of the modes – in particular the rotor speed control mode – in relation to the inputs.
Above the rated wind speed, the primary functions of the controller are to keep the rotor speed near the rated value and the generator producing the rated power, while preventing the generator from exciting drivetrain torsional resonance. The pitch actuator duty cycle is also of concern, and the pitch action has a strong influence on the structural response.
A set of simple metrics could then be the standard deviations of the rotor
speed,
In order to compute the stochastic response of the wind turbine using the simplified transient aerodynamic method of Sect. 2.3, the turbulent wind field must be reduced to a single rotor-average wind speed input. The starting point is the full matrix of rotationally sampled turbulence cross spectra between blade elements. Merz et al. (2012, 2015c) describe the methods used to generate this spectral matrix. Velocity cross spectra between each pair of rotating blade elements are computed analytically using isotropic turbulence theory, together with the Von Karman spectrum. The resulting spectral matrix is transformed into multiblade coordinates, giving the characteristic 3 nP signals in the ground-fixed frame.
The collective multiblade components of the spectral matrix are retained,
and the cosine and sine components are discarded. Then an averaging
procedure is performed, weighting the contribution at each blade element
according to its swept area
In cases with ocean waves, the wave force spectrum is derived by running a time-domain hydrodynamic analysis, summing the forces to a point on the tower at the waterline and computing the spectrum from the time series of forces.
Spectra of rotor speed, nacelle fore–aft displacement, and blade
pitch acceleration in small-amplitude (
Values of the standard deviation metrics, derived from Figs. 15 and 16.
Model 7D, with Eq. (13), is capable of approximating the standard deviation
metrics, local to an operating point, from a full linear model of the wind
turbine. Figure 15 shows the spectra of rotor speed, nacelle fore–aft
displacement, and blade pitch acceleration for small stochastic
perturbations
The results of Fig. 16, repeated with
Model 7D, with the single turbulence input of Eq. (13), is compared to a full (572-state) model, with a full 3-D input turbulence field. Corresponding results were also generated with a nonlinear time-domain model. The blade pitch angle was kept in the vicinity of the operating point by the low value of turbulence intensity, such that the influence of gain scheduling was negligible.
Model 7D provides an accurate estimate of the rotor speed and tower fore–aft displacement from the full linear model. The estimate of blade pitch acceleration is not precise, but it is reasonable. The same can be said for the comparison between the linear model and FAST: the agreement is not precise, but it is reasonable, seen in the light of the variability typically encountered in code-to-code comparisons. The trends in the spectra give confidence that the relevant physical phenomena are represented.
There is one exception: tower side-to-side resonance – visible in the rotor
speed and blade pitch responses – which is much more pronounced in the
nonlinear analysis. This discrepancy is curious, since the fore–aft response
matches nicely; it is likely attributable to the low side-to-side damping.
At a wind speed of 16 m s
Though not as crucial to controller tuning at a given operating point, it is
also of interest to evaluate how well a linear model can predict stochastic
fluctuations under realistic operating conditions. The analyses were
repeated with
In the simulation with a mean wind speed of 16 m s
Based on these results, it can be expected that Model 7D provides the correct trends in the performance metrics, and is useful for stability analysis and gain tuning at an operating point.
Points on the Pareto front, plotted according to the objectives
(on the left) and the control tuning parameters (on the right). The wind speed is 16 m s
The baseline tuning of the NREL 5 MW controller is workable when used in combination with simplified aeroelastic models which do not include blade torsional flexibility. For more advanced aeroelastic models, a different controller tuning is required in order to eliminate the instability and improve the overall performance near the rated wind speed.
The retuning could be as simple as reapplying the pole-placement strategy
described in the introduction, using Model 5D (or 7D or 8D) instead of Model
R. Selecting
An alternative, in the manner of Tibaldi et al. (2012), is to tune the controller gains and filters based upon an evaluation of system performance. Here we use the metrics of Sect. 5, together with Model 7D. This model runs quickly enough that a complete mapping of the tuning parameters, within reasonable bounds, is feasible. Sophisticated optimization techniques are not needed.
A comparison of the gains, scheduled as a function of the pitch angle.
The pole-placement tuning and resulting metrics. The low-pass filter
frequency
As an example of one possible approach for tuning the controller, consider
the case with a mean wind speed of 16 m s
Figure 17 plots the Pareto front, in terms of the objectives, i.e., minimization
of
The curved lower boundary visible in the left-hand plot represents the fundamental tradeoff between pitch activity and rotor speed; tightly limiting fluctuations in rotor speed requires rapid pitch action and hence high pitch accelerations. Extremes in either direction – very aggressive or very passive control – are associated with more severe structural loads. A balanced tuning is preferred. It is possible to further reduce structural loads by straying from the lower boundary, sacrificing some performance in terms of pitch activity and rotor speed.
The selected Pareto tuning and resulting metrics. The low-pass
filter frequency
Modal frequency and damping properties of the pole-placement tuning. For abbreviations, including subscripts, please see the table in Appendix A.
Modal frequency and damping properties of the selected Pareto tuning. For abbreviations, including subscripts, please see the table in Appendix A.
An example of how tighter speed control is obtained by increasing
the frequency of the rotor speed control mode. The wind speed is 16 m s
The plot on the right illustrates that, among the Pareto-optimal points, a high filter frequency and high proportional gain are associated with a high integral gain; likewise for low values. The trends are summarily explained: cases with a high filter frequency and high gains lie on the Pareto front because they minimize the deviations in rotor speed and cases with a low filter frequency and low gains minimize the pitch activity. Other cases represent varying degrees of tradeoffs between rotor speed, pitch activity, and structural response.
To pick one tuning from the Pareto front requires some assumptions; there is no single correct solution. Let us propose some guidelines: (1) comparatively tight speed control and responsive pitch control is desired in the vicinity of the mode transition at the rated wind speed; (2) the metrics are more important near the rated wind speed, where the turbine will be operating most often; (3) in comparison with the pole-placement tuning, we wish to trade a somewhat increased pitch activity for tighter speed control and reduced structural motions.
Extended to the full range of wind speeds between rated and cutout, these
guidelines suggest that the gains
A low-pass filter frequency of 0.17 Hz was found to be reasonable over the
entire wind speed range between rated and cutout. The best metrics, according
to the chosen performance criteria, were obtained within
Tables 3 and 4 list the gains, together with the primary metrics
The “preferred” damping ratio of the rotor speed control mode is roughly
0.3, in contrast with the value of 0.6 chosen for pole placement. Tighter
control of the rotor speed is achieved not by increasing the damping but
rather by increasing the frequency. This moves the peak in the
There are limits to where the pole of the rotor speed control mode can be
placed by varying
The natural frequency of the rotor speed control mode tends to increase at wind speeds approaching cutout. The pole-placement technique, holding the frequency at 0.1 Hz, requires a comparatively high integral gain (Table 3) in relation to the proportional gain – the integral path acts as a negative stiffness on the speed fluctuations.
There is no single reference control tuning which performs well with all types of wind turbine models. It is therefore incumbent upon the analyst to understand the properties and limitations of the model and select a control tuning that gives the desired behavior. The aspects of behavior relevant to control tuning can be largely understood in terms of the rotor speed control mode of the closed-loop system.
Rule-of-thumb methods, using pole placement on a rigid rotor, are inadequate. The NREL 5 MW wind turbine controller was tuned in this manner, and it is unstable near the rated wind speed when paired with a fully flexible aeroelastic model. This calls into question some of the comparisons between optimal and PI controllers which have been published over the last decade.
Simple models, with 16–20 states, can be used to tune the controller for use with a fully flexible aeroelastic model. The degrees of freedom must be selected with care: at a minimum, a model for control tuning requires tower fore–aft, driveshaft, blade-flap-wise, and blade torsional flexibility. A dynamic wake model is also needed.
The control tuning models can be generated in an automated manner from full linearized models, which are commonly available as output from aeroelastic codes. The matrices are partitioned, retaining only the rows and columns associated with the selected states. It is acceptable, for purposes of control tuning, to retain only one set of transient aerodynamic equations, which then represents the rotor-average aerodynamic response. The appropriate representation of the turbulent wind spectrum then becomes a challenge. A simple solution is to use the swept-area-weighted average of the collective, rotationally sampled turbulence components associated with each blade element. The spectral matrix is thereby reduced to a single scalar input.
The proper architecture of a gain-scheduled PI controller places the scheduled integral gain inside the integral of the error.
Ocean waves, especially when misaligned with respect to the wind, drive tower side-to-side vibrations, to which the pitch controller may respond. It is recommended to low-pass filter the shaft speed measurement with a cutoff frequency that is well below the frequency of the first tower fore–aft mode.
A revised controller tuning was developed for the NREL 5 MW wind turbine, reducing the low-pass filter cutoff frequency and rescheduling the gains. A Pareto optimization approach was used in order to identify a set of gains which give a stable, balanced performance in terms of minimizing the rotor speed error, the blade pitch accelerations, and the structural motions.
The data files associated with the figures and tables are archived by SINTEF Energy Research. Requests for access to the data may be sent to the corresponding author.
This work has been funded by NOWITECH, the Norwegian Research Centre for
Offshore Wind Technology (
Many thanks to Fiona Dunne of the University of Colorado, Boulder, for providing a prepublication version of a manuscript addressing the gain scheduling issue of Sect. 3. Thanks also to Jason Jonkman of NREL for facilitating this exchange. Edited by: M. Muskulus Reviewed by: two anonymous referees