# Estimation and Control of Wind Turbine Tower Vibrations Based on Individual Blade-Pitch Strategies

## Summary (3 min read)

### Introduction

- Index Terms—State estimation of dynamical systems, Kalman filter, active damping control, wind energy.
- 1Wai Hou Lio is with Department of Wind Energy, Technical University of Denmark, DK-4000 Roskilde, Denmark.
- 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).
- 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) ≈ 1 L ∑ l ṽi,l(t), = ṽ∞,i(t)− ˙̃xfa(t) + kϕ ˙̃xfa(t) sin ( φi(t) ) .
- By substituting (2) into (1), the state-space representation of (1) can be formulated as follows: ẋ(t) = A(t)x(t) +Bu(t) +Bdd(t), y(t) = Cx(t), (3) where u(t) := [θ̃1(t), θ̃2(t), θ̃3(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 typical Coleman transform Tcm ( φ(t) ) ∈ R3×3 is 3 defined as follows (e.g. [10] and references therein): [M̃col(t), M̃tilt(t), M̃yaw(t)] , (4b) where M̃col(t), M̃tilt(t), M̃yaw(t) denote the perturbation on the collective, tilt and yaw referred flap-wise blade rootbending moments, respectively. , (4d) where θ̃col(t), θ̃tilt(t), θ̃yaw(t) represent the perturbations on the collective pitch and the referred pitch signals upon the tilt and yaw axis, respectively.
- 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.
- Combining the LTI system (5) and the wind disturbance model (6), the authors yield the proposed tower observer as follows: ẋa(t) = Aaxa(t) +Baucm(t) + Le(t), ycm(t) = Caxa(t), (7) where xa(t) = [ξ(t), ξw(t)].
- 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).
- (9) Referring to Figure 2, together with the proposed tower controller Kt ∈ R and the observer Γob ∈ R1×(nu+ny), the pitch signal θtilt on the tilt axis becomes: θ̃tilt(s) = K (1,1) ipc (s)M̃tilt(s) +K (1,2) ipc (s)M̃yaw(s) +Kt(s)X̂fa(s), (10) where the estimate of the tower-top fore-aft velocity X̂fa ∈ R can be expressed as follows: X̂fa(s) = Γob(s)[ucm(s), ycm(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

- The contributions of this paper lay in the extraction of useful additional information from existing blade load sensors and 7 100 105 110 115 120 125 130 Time t [s].
- 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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### Cites background from "Estimation and Control of Wind Turb..."

...A fixed sensor output fault means that the measured sensor output maintains a constant value βfixed, described as: βm(t) = βfixed (4) The total sensor fault is modelled as: βm(t) = 0 (5) The multiplicative sensor fault with gain f (f = 1 means fault-free) is defined as: βm(t) = fβ(t) + ds(t) = β(t) + (f − 1)β(t) + ds(t) (6)...

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...The comparison between three pitch measurements in the faulty blade (blade 1) pitch corresponding (1) Sensor bias (2) Fixed output (3) Total failure (4) Multiplicative fault are shown in Fig....

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##### References

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### "Estimation and Control of Wind Turb..." refers methods in this paper

...(8) Secondly, consider the existing Coleman transform-based IPC controller Kipc ∈ R2×2, adopted from [11], [19]:...

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### "Estimation and Control of Wind Turb..." refers background or methods in this paper

...Corollary 1: Assuming the model parameters obtained from linearizing the baseline turbine [18], the system (5) is observable....

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...The turbine model employed in this brief is the NREL 5MW turbine [18] and the simulations are conducted on FAST [21]....

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3,598 citations

### "Estimation and Control of Wind Turb..." refers background in this paper

...[13]) and (ii) a nominal case with no tower controller....

[...]

...[13]), subsequently referred to as the baseline design....

[...]

...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....

[...]

...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....

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[...]

1,131 citations

### "Estimation and Control of Wind Turb..." refers methods in this paper

...The turbine model employed in this work is the NREL 5MW turbine [18] and the simulations are conducted on FAST [21]....

[...]

...The turbine model employed in this brief is the NREL 5MW turbine [18] and the simulations are conducted on FAST [21]....

[...]

604 citations

### "Estimation and Control of Wind Turb..." refers background in this paper

...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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