Estimation and Control of Wind Turbine Tower Vibrations Based on Individual Blade-Pitch Strategies
Summary (3 min read)
Introduction
- Moreover, typical tower damping control strategies provide an additional blade pitch signal collectively to all the blades in response to the tower velocity [13], that is inevitably coupled with the rotor speed regulation loop, thus, affecting the power output of the turbine.
- On the other hand, well-designed IPCs are largely decoupled from the CPC, thus there are potential benefits to designing an IPC-based tower damping controller.
- The space R denotes the space of proper real-rational transfer function matrices and ẋ represents the time derivative of x.
II. MODELLING
- The damping ratio of the blade and tower are ζb, ζt ∈ R and ωb, ωt ∈ R are the respective natural frequencies of the blade and tower.
- Thus, the fore-aft rotational velocity of the tower-top can be approximated as ˙̃ϕfa(t) ≈ 23h ˙̃xfa(t).
- Notice that the system matrix A ∈ Rnx×nx is timedependent owing to the time-varying nature of the azimuth angle.
III. TRANSFORMATION TO AN LTI SYSTEM AND OBSERVABILITY ANALYSIS
- For a linear time-varying (LTV) system (3), there exist techniques for observability analysis and estimator design (e.g. [17]).
- The problem of establishing the observability proof and synthesising an estimator for the LTV system (3) can be greatly simplified by reformulating (3) as an LTI system.
- The same also applies to the wind speed ṽi.
- Rnξ is the projection of the states associated with the blade dynamics upon a non-rotating reference frame (19) and the states of the tower dynamics (20).
IV. DESIGN OF THE ESTIMATOR AND CONTROLLER
- Figure 2 depicts the architecture of the proposed estimation and control system, where the tower motion estimator produces an estimate ˆ̇xfa(t) of the fore-aft velocity of the tower-top based on Coleman-transformed blade moment measurements M̃col(t), M̃tilt(t), M̃yaw(t) and pitch signals θ̃col(t), θ̃tilt(t), θ̃yaw(t).
- The individual pitch-based tower controller subsequently employs this estimate to provide additional referred blade pitch signals upon the tilt axis θ̃tilt(t) for attenuating the tower motion.
- Note that this architecture is deliberately chosen so as to augment, rather than replace the existing turbine controllers.
A. Estimator design
- The system (5) is driven by the wind-induced disturbance, which consists of slow-moving mean wind speeds and fastchanging turbulence.
- The authors consider these wind speed disturbances as coloured noise.
- Rnd is defined as follows: ξ̇w(t) = Awξw(t) +Bww(t), dcm(t) = Cwξw(t), (6) where the system matrices {Aw, Bw, Cw} are determined by fitting the spectra of the model output to the known spectra of the wind speed disturbances.
- Rny is the prediction error between the plant and model output.
B. Estimation-based controller design
- Typically, a tower controller provides an additional collective blade pitch signal on top of the CPC loop in response to the tower fore-aft velocity, in order to dampen the foreaft structural mode.
- The key challenge is to separate the existing IPC loop and the tower damping control loop, which is particularly important since the tower estimate is also dependent upon the blade load measurements.
- To see this, firstly consider the LTI system (5) in its transfer function form: ycm(s) = Gcm(s)ucm(s).
- To examine the coupling between the existing IPC and the proposed tower controller, Figure 3 shows the closed-loop sensitivity functions of the original IPC controller S(s) := (I + GcmKipc(s)) −1 and the coupled controller structure Sm(s) := (I + GcmK m ipc(s)) −1.
V. NUMERICAL RESULTS AND DISCUSSION
- This section presents simulation results to demonstrate the performance of the proposed estimator and estimation-based controller for the tower fore-aft motion.
- The turbine model employed in this work is the NREL 5MW turbine [18] and the simulations are conducted on FAST [21].
- This turbine model is of much greater complexity than the linear model (7).
- All degrees-of-freedom were enabled, including flap-wise and edge-wise blade modes, in addition to the tower and shaft dynamics.
A. Estimator Performance
- The proposed observer (7) was compared with a typical double-integrator Kalman-filter design based on measurements from the tower fore-aft accelerometers (e.g. [13]), subsequently referred to as the baseline design.
- True signal Proposed method Baseline method (b) Tower fore-aft velocity.
- Proposed method Baseline method (d) Auto-correlations of the errors.
- Test case (i) with the above-rated wind conditions, also known as Fig. 4.
- The improvements in estimation error are obtained in the low frequency range, as shown in Figure 4e.
B. Controller Performance
- To showcase the use of the tower estimate, a novel individual-pitch-based tower damping control strategy is proposed that uses θ̃tilt as an input.
- Simulations were conducted under a wind case, shown in Figure 7a, with a mean wind speed of 18 ms−1 and turbulence intensity of 5%.
- The key benefit of the individual-pitch-based design is that it is decoupled from the existing CPC loop, owing to the inherent properties of the Coleman transforms.
- Compared to the collective pitch-based design, the individual-pitch-controller imposed slightly larger tilt and yaw loads at the tower resonant frequency, upon the non-rotating turbine structure.
- Relative to the peak loads, these were insignificant.
VI. CONCLUSION
- No tower controller Individual-pitch-based design collective-pitch-based design (d) Rotational speed of the rotor.
- The coupling between states in both rotating and fixed frames of reference led to an initial system model that was linear but time-varying, and so the Coleman Transforms were employed to manipulate this into a simpler LTI model.
- Having verified observability, a state estimator was synthesised that produced good estimates of the tower fore-aft motion, based solely upon the blade-load measurements.
- This was subsequently used in a novel individual pitch-based tower damping controller.
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References
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"Estimation and Control of Wind Turb..." refers background in this paper
...At present, most tower damping control strategies assume a direct measurement of tower motion, typically from a nacellemounted accelerometer [4], [13]....
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...The IPC provides additional pitch demand signals to each blade in order to balance the loads across the rotor plane, typically in response to the measurements of the flapwise blade root bending moments [2]–[4], whereas tower damping control provides a further adjustment to the collective blade pitch angle in order to reduce excessive tower vibrations, in response to tower fore-aft velocity measurements [5]–[8]....
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29 citations
"Estimation and Control of Wind Turb..." refers background in this paper
...6 Hz) [20], whereas tower loads occur mainly at the tower resonant frequency Fig....
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28 citations
"Estimation and Control of Wind Turb..." refers background in this paper
...2833064 Typically, and for reasons of simplicity of implementation favored by the industry, IPCs and tower damping controllers are designed separately from the CPC and carefully in order to avoid cross-excitation [9]–[12]....
[...]
27 citations
"Estimation and Control of Wind Turb..." refers background in this paper
...If so, this indicates redundancy in the information provided by the tower motion sensor that can either be exploited in terms of a reduction in sensor count, or for fault-tolerant control purposes [14]–[16]....
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Frequently Asked Questions (15)
Q2. What is the simplest way to model the blade and tower?
Owing to variable blade geometry, the wind-induced forces are not uniformly distributed on the blades and to model such forces, blade element/momentum theory is adopted [13], where the blade is discretised into small elements.
Q3. What is the key benefit of the individual pitch-based design?
The key benefit of the individual-pitch-based design is that it is decoupled from the existing CPC loop, owing to the inherent properties of the Coleman transforms.
Q4. What is the key challenge of the tower damping system?
The key challenge is to separate the existing IPC loop and the tower damping control loop, which is particularly important since the tower estimate is also dependent upon the blade load measurements.
Q5. What is the contribution of the paper?
The contributions of this paper lay in the extraction of useful additional information from existing blade load sensors and7 100 105 110 115 120 125 130Time t [s]161718192021W in d S p ee d v (t ) [ms− 1 ](a) Hub-height wind speed.
Q6. What is the heuristic switching policy used to overcome the transition between wind conditions?
The following heuristic switching policy was employed to overcome the transition between wind conditions:ẋfa(t) = ∑κρκ(t)ẋfa,κ(t), ∑κρκ(t) = 1, (14)6where κ ∈ {1, 2} is the index of observers designed in the below-rated and above-rated wind conditions, whilst ρκ ∈ R denotes the weighting on the tower estimate of the κ-th estimator.
Q7. What is the spectra of the model output?
Rnd is defined as follows:ξ̇w(t) = Awξw(t) +Bww(t), dcm(t) = Cwξw(t), (6)where the system matrices {Aw, Bw, Cw} are determined by fitting the spectra of the model output to the known spectra of the wind speed disturbances.
Q8. What is the bending moment of the blade and tower?
This work implicitly assumes the tower is a prismatic beam so that the ratio between rotation and displacement is 23h , where h ∈ R is the height of the tower [3].
Q9. What is the simplest way to model the aerodynamic forces of the blade and tower?
The nonlinear aerodynamic forcing functions on the blade and tower are typically linearised around the operating wind conditions to obtain the perturbation forces, f̃M (θ̃i, ṽi) : R × R → R and f̃x(θ̃col, ṽcol) : R × R → R, defined as follows:f̃M (θ̃i, ṽi) = dfMdθ∣ ∣ ∣ ∣θ∗,v∗ θ̃i(t) +dfMdv∣ ∣ ∣ ∣θ∗,v∗ ṽi,(t), (1c)f̃x(θ̃col, ṽcol) = dfxdθ∣ ∣ ∣ ∣θ∗,v∗ θ̃col(t) +dfx dv∣ ∣ ∣ ∣θ∗,v∗ ṽcol(t), (1d)where dfM dθ , dfx dθ ∈ R and dfM dv , dfx dv ∈ R are the variations of the forcing with respect to the pitch angle and apparent wind speed.
Q10. What is the correct treatment of the Coleman Transforms?
Theorem 3.1: Assuming a fixed rotor speed and Coleman transformations (4), the linear time-varying system (3) can be transformed into the following LTI form:ξ̇(t) = Aξξ(t) +Bξucm(t) +Bξddcm(t),ycm(t) = Cξξ(t), (5)where ycm(t) = [M̃col(t), M̃tilt(t), M̃yaw(t)]
Q11. What are the natural frequencies of the blade and tower?
The damping ratio of the blade and tower are ζb, ζt ∈ R and ωb, ωt ∈ R are the respective natural frequencies of the blade and tower.
Q12. What is the angle of the blade and tower?
The azimuthal angle of each blade is defined as [φ1(t), φ2(t), φ3(t)] := [φ(t), φ(t) + 2π 3 , φ(t) + 4π 3 ], where φ(t) is the angle of the first blade from the horizontal yaw axis with respect to the clockwise direction.
Q13. What is the error magnitude of the proposed method?
In addition, in Figure 6d, it is clearly seen that the error auto-correlation of the proposed method was closer to zero, suggesting its residual was almost white noise.
Q14. What is the effect of the wind on the blade and tower?
Since the focus of this work is on the blade disturbance induced by the wind, the effect of the wind perturbations upon the blade, ṽi(t) in (1), can be approximated by averaging the apparent wind speed perturbations ṽi,l(t) along the blade, as follows:ṽi(t) ≈ 1L∑lṽi,l(t),= ṽ∞,i(t)− ˙̃xfa(t) + kϕ ˙̃xfa(t) sin ( φi(t) ) .
Q15. What is the effect of the individual-pitch-based design on the turbine structure?
Compared to the collective pitch-based design, the individual-pitch-controller imposed slightly larger tilt and yaw loads at the tower resonant frequency, upon the non-rotating turbine structure.