Using active tuned mass dampers with constrained stroke to simultaneously control vibrations in wind turbine blades and tower

Abstract

Simultaneous control of wind turbine blades and tower vibrations is studied in this article. Four active tuned mass dampers have been incorporated into each blade and tower to reduce vibrations. A decentralized constrained H_∞ velocity output feedback which restricts the tuned mass damper stroke as a hard constraint is proposed by solving linear matrix inequality. Each active tuned mass damper is driven individually by the output of the corresponding velocity signal. Considering the structural dynamics subjected to gravity, variable rotor speed, and aerodynamic loadings, a model describing dynamics of rotating blades coupled with tower, including the dynamics of active tuned mass dampers, was developed by Euler–Lagrangian formulation. A numerical simulation is carried out to verify the effectiveness of the proposed decentralized control scheme. Investigations show promising results for the active tuned mass damper in simultaneous control blade vibrations and tower vibrations by decentralized control approach. Numerical results demonstrate that the decentralized control has the similar performance compared to centralized control and effectively reduce the displacement of vibrations.

Keywords

active tuned mass damper blade vibrations decentralized control tower vibrations

Introduction

With the increasing length of blades, uncontrolled vibrations can cause structural damage which could decrease the life of the blades (Larsen and Hansen, 2004). To suppress the vibrations, researchers have studied two methodologies. In the first methodology, the pitch control is used (Corradini et al., 2017) for changing the lift and drag forces to control aerodynamic loads. Noting that the pitch control is coupled with power regulation or speed regulation, this may be not properly suited for load reduction (Rice and Verhaegen, 2010). The second method is active structural control used in civil engineering to directly restrain the blade or the nacelle vibrations. In the study by Staino and Basu (2015), a dual control strategy decoupling pitch control and vibration control was proposed. Previous work on structural control focused on the damping devices. Tuned mass damper (TMD), which is successfully applied in the vibration control of mechanical engineering (Rahman et al., 2016) and structural engineering (Yazdi et al., 2016), dissipated vibration energy by tuning to a structural resonance frequency. Due to the hollow inside of blades and the nacelle, the damper installed inside of these components can independently on speed or power regulation. For mitigation of a wind turbine tower, Murtagh et al. (2005) proposed a passive TMD placed on the top of the tower for reduction vibration. Lackner and Rotea (2011) modified the simulation code of wind turbine called FAST by adding two TMDs within side-side and fore-aft direction into the nacelle, thus adding structural control capacity to FAST. The control structure interaction in active control and actuator dynamics was considered in Stewart and Lackner (2013) implementing this modified version. Brodersen et al. (2017) demonstrated that the active tuned mass damper (ATMD) can be used to further reduce the tower displacement of wind turbines without an increase in damper mass, where the additional actuator force is controlled using velocity feedback. For suppressing blade vibration, there are few studies regarding structural control. Arrigan et al. (2011) used the semi-active TMDs to control flapwise vibrations. Dinh et al. (2016) used semi-active TMD to control the vibrations of blades, nacelle, and spar in an offshore wind turbine. Because the active control has better control effect. Fitzgerald et al. (2013) proposed utilizing active TMD for suppressing edgewise vibrations. It was noted that the active TMDs can provide better reduction than the passive TMDs. Fitzgerald and Basu (2014) further proposed a cable connected active TMD to suppress edgewise vibrations. Because the aerodynamic damping in the edgewise direction is low, in this article, we concentrate on suppressing the edgewise vibrations. Inspired by the works of Fitzgerald et al. (2013), Fitzgerald and Basu (2014), and Dinh et al. (2016), this article explores simultaneous control of the blade and tower vibrations by ATMDs. In previous articles, the authors focus on the feasibility of damper design and the control strategy using a state feedback control. In practical implementation, full knowledge of the state vector is rarely available. An available approach is an output feedback control. Another flaw in applying the linear quadratic regulator (LQR) approach is that the control design cannot directly handle the constraint as the TMD stroke must be restricted due to the working space. Meanwhile, the centralized control strategy needs a large number of sensors and communication network, and this may have a poor reliability. For large-scale structures, decentralized control is a more practical approach. Compared to dynamic output feedback control, static output feedback control has a simple dynamic (Sadabadi and Peaucelle, 2016). Thus, a decentralized constrained $H_{\infty}$ static output feedback control with hard constraint was implemented, in which each TMD is controlled individually by the velocity signal and the limitation of the TMD stroke is modeled as a hard constraint. A reduced 8-degree-of-freedom (DOF) model in which each blade and nacelle equipped a TMD has been considered. Different to Fitzgerald et al. (2013), Fitzgerald and Basu (2014), and Dinh et al. (2016), the force induced by variable rotor speed is also introduced. Based on the coupled system model, the control design can guarantee the stability of the full system dynamic. To verify the proposed control scheme, the National Renewable Energy Laboratory (NREL) 5-MW wind turbine is simulated using MATLAB.

Theoretical model for the wind turbine with decentralized ATMD system

In this article, we focus on control of the vibrations in edgewise direction of blades and make the basic assumption that the coupling between the edgewise and flapwise can be ignored, that is, the pre-twist angle is zero; this has a little effect on the accuracy according to the conclusion in Basu et al. (2016). Hence, the design and analysis of the decentralized control are fully based on the edgewise vibrations coupled with the nacelle side-side vibrations. A reduced 8-DOF wind turbine model is presented in which each blade and nacelle is equipped with one ATMD. Figure 1 illustrates the schematic representation of blades coupling with nacelle/tower. To derive a theoretical model, the motion of each blade is described in a co-rotating blade coordinate system $(x_{b}, y_{b}, z_{b})$ which the origin is at the center of the hub and rotational speed $Ω rad / s$ . While a ground coordinate $(X, Y, Z)$ is used to describe the motion of nacelle. Through the modal expansion, the flexible vibration characteristic can be captured by the first mode (Jonkman and Buhl, 2005). Therefore, we only control the fundamental mode. Denoting the blade modal coordinates $q_{i}$ and mode shape functions $ϕ_{b}$ , the edgewise motion is represented as $u_{i} = ϕ_{b} (x) q_{i} (t), i = 1, 2, 3$ , while the side-side motion of the nacelle is described by the degree of freedom $u_{ss} = ϕ_{t} (1) q_{ss} (t)$ representing the tower top displacement in the Y-direction, where $ϕ_{t}$ is the tower mode shape functions.

Figure 1.

A schematic diagram of ATMDs in blade and nacelle.

Denoting the kinetic energy and potential energy as T and V, respectively, by the classical Euler–Lagrange method, the equations of motion are given by

\frac{d}{dt} (\frac{\partial (T (q, \overset{\cdot}{q}) - V (q, \overset{\cdot}{q}))}{\partial \overset{\cdot}{q}}) - \frac{\partial (T (q, \overset{\cdot}{q}) - V (q, \overset{\cdot}{q}))}{\partial q} = Q

(1)

where q and Q are the modal coordinate and the modal loading acting on the blade or the nacelle for the corresponding degree of freedom, respectively.

The governing differential equations of vibration are obtained by applying the Euler–Lagrangian approach. Based on the methodology developed by Fitzgerald et al. (2013), Fitzgerald and Basu (2014), and Dinh et al. (2016), the ATMD system can be enrolled in the differential equations of motion. To represent the blade motions, a position vector in the blade frame consists of a vector of a radius x and a vector of nacelle motion; this can be represented by the unit vector $[i, j, k]$ in rotating frame and unit vector $[I, J, K]$ in fixed frame as follows

\begin{matrix} {\vec{r}}_{m} = [i, j, k] [x, ϕ_{b} q_{m}, 0]^{T} \\ + [I, J, K] T_{ϕ, m} [0, q_{ss}, 0]^{T}, m = 1, 2, 3 \end{matrix}

(2)

where the transformation matrix is

T_{Φ, m} = [\begin{matrix} \cos ψ_{m} & 0 & - \sin ψ_{m} \\ 0 & 1 & 0 \\ \sin ψ_{m} & 0 & \cos ψ_{m} \end{matrix}]

where $ψ_{i} = \int_{0}^{t} Ω d τ + (i - 1) 2 / 3 π, i = 1, 2, 3$ is the azimuth angle of each blade.

In the process of modeling, each blade is assumed as a Bernoulli–Euler cantilever beam which rotates about the hub’s horizontal axis. The geometric parameters are non-uniform with distributed mass density $μ (x)$ and distributed stiffness $EI (x)$ per unit length. As developed in Fitzgerald et al. (2013), by differentiating the position vector with respect to time, the total kinetic energy of three blades and the nacelle are

T = \frac{1}{2} (\sum_{i = 1}^{3} (\int_{0}^{l} μ (x) {\overset{\cdot}{\vec{r}}}_{i}^{2} dx) + m_{ss} {\overset{\cdot}{q}}_{ss}^{2})

(3)

where $m_{ss}$ is the mass of nacelle/tower.

The blades’ potential energy consists of the potential energy caused by bending stiffness, the potential energy generated by centrifugal stiffness, and the potential energy from gravitation. Adding the potential energy of nacelle, the total potential energy is

V = \frac{1}{2} \sum_{i = 1}^{N} ((k_{e} + k_{c} + k_{g} \cos (ψ_{i})) q_{i}^{2} + \frac{1}{2} k_{ss} q_{ss}^{2}

(4)

where $k_{ss}$ is the nacelle side-side stiffness calculated by numeral integration, $k_{e}$ is the bending stiffness, $k_{c}$ denotes the centrifugal force arising from the rotation of blades, and $k_{g}$ represents the stiffness contributed to the gravity force of the blade. These terms are given as follows

\begin{matrix} k_{e} = \int_{0}^{L} EI (x) ϕ_{b}^{″ 2} dx \\ k_{c} = Ω^{2} \int_{0}^{L} (\int_{x}^{L} μ (ξ) ξ d ξ) ϕ_{b}' dx \\ k_{g} = - \int_{0}^{L} (\int_{x}^{L} μ (ξ) d ξ) ϕ_{b}^{' 2} dx \end{matrix}

In this system, the total generalized external forces acting on the blade and the nacelle consist of the modal aerodynamic load $Q_{a} = [Q_{a 1}, Q_{a 2}, Q_{a 3}, Q_{ass}]$ and the modal gravity load $Q_{g} = [Q_{g 1}, Q_{g 2}, Q_{g 3}, 0]$ , which can be calculated from the corresponding local load and mode shapes. The modal aerodynamic load for the fundamental mode is calculated as

Q_{ai} = \int_{0}^{L} p_{N} (x, t) ϕ (x) dx, i = 1, 2, 3

(5)

where $p_{N} (x, t)$ is the variable local wind load intensity at a distance x from the hub. And it can be calculated by blade element momentum (BEM) theory (Hansen, 2008). The BEM method couples momentum theory with blade element theory, the local events taking place at each blade element. In this theory, the blade is assumed to be discredited into N sections and no aerodynamic interactions between different components. The local edgewise loads are given by

p_{N} (x, t) = \frac{1}{2} ρ V_{rel}^{2} (x, t) c (r) C_{N} (α)

(6)

where $ρ$ is the density of air, $V_{rel}$ is the relative wind speed, and $c (r)$ in the length of local chord

c_{N} (x, t) = C_{l} (α) \cos (ψ) + C_{d} (α) \sin (ψ)

(7)

where $α$ is the local angle of attack, and $C_{l} (α)$ and $C_{d} (α)$ are the drag and lift coefficients which can be obtained by look-up table, respectively. The detailed process can be found in Hansen (2008).

The aerodynamic load on the nacelle is

Q_{ss} = \sum_{i = 1}^{3} \int_{0}^{L} p_{N} (x, t) dxcos (ψ_{i})

(8)

The gravity load on the nacelle is zero and on each blade is

Q_{gi} = gsin (ψ_{i}) \int_{0}^{L} μ (x) ϕ (x) dx, i = 1, 2, 3

(9)

where g is the gravity acceleration.

The generalized coordinates is denoted by $q (t) = [q_{1}, q_{2}, q_{3}, q_{ss}]^{T}$ . Substituting the kinetic energy, potential energy, and external force into equation (1), the second-order matrix differential equation of 4-DOF open-loop system can be written as

M_{o} \overset{\cdot\cdot}{q} (t) + C_{o} \overset{\cdot}{q} (t) + K_{o} q (t) = Q (t)

(10)

The details of the matrices are given in Appendix 1.

In the closed-loop system, four mass dampers were incorporated into the model, three placed inside each blade to control the response of the blade with mass $m_{bd}$ and stiffness $k_{bd}$ , and one placed inside the nacelle with mass $m_{nd}$ and stiffness $k_{nd}$ . Figure 1 illustrates the typical blade and nacelle equipped with ATMD. An ideal active actuation device is assumed to drive the TMD. The external force $F = B_{c} U_{a}$ is given by the vector of actuator control input forces $U_{a} = [u_{1}, u_{2}, u_{3}, u_{ss}]^{T}$ , and

B_{c} = {[\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]}^{T}

The position vector of the TMD which is located at a location $r_{t}$ in blade is indicated by

\begin{matrix} {\vec{r}}_{d, m} = & [i, j, k] [r_{t}, ϕ_{b} q_{m} + d_{m} + y_{0} {, 0]}^{T} \\ + [I, J, K] T_{ψ, m} {[0, q_{ss}, 0]}^{T}, m = 1, 2, 3 \end{matrix}

(11)

where $y_{0}$ is the original position of TMD from the chord’s center of the blade, and $d_{i}$ is the distance from $y_{0}$ representing the relative motion of the TMD. The velocity vector of each TMD, $v_{d}$ , is obtained by differentiating $r_{dm} (t)$ about time.

The kinetic energy and potential energy of all TMDs are

T = \frac{1}{2} \sum_{i = 1}^{3} (m_{bd} v_{d, i}^{2}) + \frac{1}{2} m_{nd} {\overset{\cdot}{d}}_{n}^{2}

(12)

V = \frac{1}{2} \sum_{i = 1}^{3} (k_{bd} d_{i}^{2}) + \frac{1}{2} k_{nd} d_{n}^{2}

(13)

where $d_{n}$ is the relative motion of the TMD in nacelle. The generalized coordinates is denoted by $q (t) = [q_{1}, d_{1}, q_{2}, d_{2}, q_{3}, d_{3}, q_{ss}, d_{n}]^{T}$ . Similar to the derivation of equation (10), substituting the kinetic energy, potential energy, and external force into equation (1), the second-order matrix differential equation of 8-DOF closed-loop system can be written as

M_{c} \overset{\cdot\cdot}{q} (t) + C_{c} \overset{\cdot}{q} (t) + K_{c} q (t) = Q (t)

(14)

The details of the matrices are given in Appendix 1. In Fitzgerald et al. (2013), Fitzgerald and Basu (2014), and Dinh et al. (2016), the rotor speed is assumed as constant. When the wind turbine is working at variable rotor speed, the forces act on the blade and the TMD induced by variable rotor speed is derived from the kinetic energy term and added to the external force. This is calculated by

\begin{matrix} Q_{Ω b} = - \overset{\cdot}{Ω} \int_{0}^{L} μ (x) x ϕ_{b} (x) dx - \overset{\cdot}{Ω} m_{d} r_{t} \\ Q_{Ω d} = - \overset{\cdot}{Ω} m_{d} r_{t} \end{matrix}

(15)

The actuators of the TMD are ideal actuation devices which are modeled as an external force F acting on the system by the principle of virtual work. Thus, the total generalized external force is given by

Q (t) = F + Q_{a} + Q_{g} + Q_{Ω}

(16)

Because of the space limitation to incorporate the TMD, the stroke of TMD must be restricted. This can be represented as $d_{i} < z_{\max, i}$ , where $z_{\max, i}$ is the work space in the blade or the nacelle. When the TMD reaches the bound, a stopping force will be generated. Taking the model from the FAST software model (Lackner and Rotea, 2011), the stopping forces in the blade and the nacelle are expressed as equation (17), here $k_{stop}$ , $c_{stop}$ are the stiffness and damping coefficients obtained based on experience. A constrained output is defined to model the working space of TMD as follows

f_{stop, i} = - {\begin{matrix} k_{stop} Δ d_{i}, : (d_{i} > d_{max} \land {\overset{\cdot}{d}}_{i} \leq 0) \lor (d_{i} < d_{min} \land {\overset{\cdot}{d}}_{i} \geq 0) \\ k_{stop} Δ d_{i} + c_{stop} {\overset{\cdot}{d}}_{i} : (d_{i} > d_{max} \land {\overset{\cdot}{d}}_{i} > 0) \lor (d_{i} < d_{min} \land {\overset{\cdot}{d}}_{i} < 0) \\ 0 : otherwise \end{matrix}

(17)

z_{2} = [\begin{matrix} C_{z} & 0_{8 \times 8} \end{matrix}] x (t) = C_{2} x (t)

(18)

where $C_{z} = diag (0, 1, 0, 1, 0, 1, 0, 1)$ .

The control design is to guarantee that the closed-loop system is stable under disturbance and satisfy the hard constraint (18).

Decentralized H_∞ static output feedback control

By modifying equation (14) according to the structural constraints, we consider the system with decentralization input and output pairs depicted as follows

\begin{matrix} \overset{\cdot}{x} = Ax + Ew + \sum_{i = 1}^{N} (B_{i} u_{i}) \\ y_{i} = c_{yi} x \\ z_{1} = C_{1} x \\ z_{2} = C_{2} x \end{matrix}

(19)

where x denotes the state; N is the number of the input and output pairs; u and w denote the control input and the disturbance input, respectively; $z_{1}$ and y denote the control output and measurement output, respectively; and $z_{2}$ is constraint output in the previous section.

The controller design is a two-step procedure. First, a state-feedback controller is

u = Gx

(20)

Thus, the closed-loop system can be rewritten as

\begin{matrix} \overset{\cdot}{x} = (A + BG) x + Ew \end{matrix}

(21)

From the Bounded Real Lemma (Gao et al., 2010), $H_{\infty}$ performance $∥ z ∥_{2} < τ ∥ w ∥_{2}$ , for $τ > 0$ , is equivalent if there exists a matrix G and symmetric positive-definite matrix X satisfies the following matrix inequality

[\begin{matrix} sym ((A + BG) X) + E E^{T} & * \\ C_{1} X & - I \end{matrix}] < 0

(22)

Denoting the variables $Y = GX, η = τ^{- 2}$ , equation (22) is transformed into the following linear matrix inequality (LMI)

[\begin{matrix} sym (AX + BY) + η E E^{T} & * \\ C_{1} X & - I \end{matrix}] < 0

(23)

It is noted that the value of $τ$ represents the level of disturbance attenuation. Hence, the optimal control design is converted into the convex optimization problem given by the following expression

\max_{X > 0, η > 0} η, subject to LMI ((23))

(24)

If the problem (24) is feasible with respect to X and Y, then the state-feedback gain is $G = Y X^{- 1}$ .

For the output case, the aim is to find a gain matrix F which satisfies $G = F C_{y}$ . Substitute $G = F C_{y}$ in equation (23). In this case, equation (23) will become a bilinear matrix inequality (BMI). The BMI problem (23) can be indirectly solved by the commercial or the free open-source BMI solvers (Sadabadi and Peaucelle, 2016). Another method for solving BMI is transformed to the LMI. Following the transformations in Palacios-Quiñonero et al. (2014): $X = Q X_{Q} Q^{T} + R X_{R} R^{T}$ , $Y = Y_{R} R^{T}$ . Here, matrix $R = C_{1}^{T} (C_{1} C_{1}^{T})^{- 1}$ , the columns of matrix Q are the base vector of kernel space of matrix C. Therefore, equation (23) can be converted into

[\begin{matrix} sym (A (Q X_{Q} Q^{T} + R X_{R} R^{T}) + \\ B Y_{R} R^{T}) + τ^{- 2} E E^{T} & * \\ C_{1} (Q X_{Q} Q^{T} + R X_{R} R^{T}) & - I \end{matrix}] < 0

(25)

which is similar to the derivation in Gao et al. (2010). Since the closed-loop system satisfies the $H_{\infty}$ performance, thus $x^{T} X^{- 1} x < ρ$ , where $ρ = τ^{2} w_{\max}$ , the constraint can be modeled as following inequality

\begin{matrix} \max | z_{2} | = \max ‖ x^{T} C_{2}^{T} C_{2} x ‖_{2} \\ \leq ρ λ (X^{- \frac{1}{2}} C_{2}^{T} C_{2} X^{- \frac{1}{2}}) \end{matrix}

(26)

Using Cauchy–Schwarz inequality and Schur complement, the constraint can be written by equation (27)

[\begin{matrix} - z_{\max, i} (Q X_{Q} Q^{T} + R X_{R} R^{T}) & \sqrt{ρ} C_{2, i}^{T} \\ * & - I \end{matrix}] < 0

(27)

To obtain an optimal level of disturbance attenuation, the constrained $H_{\infty}$ output feedback control design is solving the convex optimization given as

\max_{X > 0, η > 0} η, subject to LMI ((25)) ((27))

(28)

If the problem (28) is feasible with respect to $X_{R}$ and $Y_{R}$ , then $F = Y_{R} X_{R}^{- 1}$ is the constrained $H_{\infty}$ output feedback control matrix. Therefore, the problem is reduced to solving the optimization problem (28) with respect to $X_{R}$ and $Y_{R}$ . The decentralized output feedback controller is a special case of output feedback control which constrains the LMI variables $X_{R}$ and $Y_{R}$ to a zero-nonzero form. In this article, for a given decentralized static output feedback control $u_{i} = F_{i} y_{i}$ , the matrices are defined as $F_{d} = diag (F_{1}, \dots, F_{N}), C_{y} = [\begin{matrix} C_{y 1} & \dots & C_{yN} \end{matrix}]^{T}, B = [\begin{matrix} B_{1} & \dots & B_{N} \end{matrix}]$ . By constraining a specific diagonal formulation on the LMI variables $X_{R}$ and $Y_{R}$ , decentralized controller $F_{d}$ is obtained; otherwise, the centralized controller $F_{c}$ is obtained.

Decentralized control of simultaneously control blades and tower vibrations

In this section, the decentralized control to simultaneously suppress blades and tower vibrations are checked out. We simulated a wind turbine in MATLAB using the data from NREL 5-WM reference wind turbine. Table 1 provides the data of this wind turbine. More details of the parameters are given in Jonkman et al. (2009).

Table 1.

Properties of NREL 5-MW wind turbine (Jonkman et al., 2009).

Item	Value
Rated power	5000 kW
Nacelle mass	240,000 kg
Blade mass	17,740 kg
Tower mass	347,460 kg
Blade length	61.5 m
Hub height	90 m
Blade damping ratio	$0.48 %$
Tower damping ratio	$1 %$

NREL: National Renewable Energy Laboratory.

The mode shapes are a six-order polynomial of distance $\bar{x} = r / L$ which is calculated from wind turbine data using mode code (Buhl, 2005). The first mode shapes of blade and tower considered are provided in Appendix 1. For the model in this work, the dynamic equation (14) has a state space realization. Thus, the system state equation matrices are given by

\begin{array}{l} A = [\begin{matrix} 0_{8 \times 8} & I_{8 \times 8} \\ - M_{c}^{- 1} K_{c} & - M_{c}^{- 1} C_{c} \end{matrix}], \\ E = [\begin{matrix} 0_{8 \times 8} \\ M_{c}^{- 1} \end{matrix}], B = [\begin{matrix} 0_{8 \times 4} \\ M_{c}^{- 1} B_{c} \end{matrix}] \end{array}

Due to rotating, the uncertainties in the damp and stiffness matrices caused by rotating are incorporated into the disturbance. Thus, w can be represented as

w (t) = M_{c}^{- 1} (Q_{a} + Q_{g}) + M_{c}^{- 1} Δ C \overset{\cdot}{q} (t) + M_{c}^{- 1} Δ Kq (t)

The additional actuator force is controlled using output feedback from the relative velocity of the damper and the measured velocity signal of the blade or the nacelle where the damper installed. Thus, the measures are given as velocity matrices. In this article, the blade TMDs are located 45 m along the blades which in this location has a chord $3 m$ , while the nacelle TMD is located at the bottom of the nacelle. To optimize the TMD parameters, there are many expressions in the literature. For simplicity, the TMD parameters are assumed to be optimal. The mass ratio of TMD $μ_{mass}$ and damp tuning ratio $υ$ are taken from Fitzgerald and Basu (2014) and Fitzgerald et al. (2013); the damping ratio is given by $η = 0.5 \sqrt{μ_{mass}}$ .

Centralized and decentralized controllers

The controller was designed in MATLAB. Following the design procedure described as aforementioned, the matrix Q is given by MATLAB command null(), R is obtained by calculating the pseudo inverse of $C_{1}$ . The optimization problem is given by constraining the LMI variables $X_{R}$ and $Y_{R}$ to a zero-nonzero form and can be solved with the optimization tools of MATLAB LMI Toolbox (Balas et al., 2011). After solving equation (24), the state feedback $H_{\infty}$ control matrix is obtained. For the centralized output feedback control case, addressing the optimization problem (28), the centralized output feedback controller is given. When we set a diagonal-type matrix as mentioned above (28), the decentralized controller is obtained. For comparison, we consider a control scheme based on the LQR investigated in Fitzgerald et al. (2013) and Fitzgerald and Basu (2014). In the LQR approach, the control objective is to consume as little energy as possible to achieve the minimum displacement, velocity, and acceleration of the blades and tower. Therefore, the overall performance index of the LQR control is determined to be

J = \int_{0}^{\infty} [x^{T} Rx + u^{T} Gu] dt

(29)

The control feedback is given by

u (t) = K_{L} x (t)

(30)

where $K_{L}$ is solved via a Riccati equation.

Simulation results

In this section, numerical simulations were performed to illustrate the efficacy of the decentralized control strategy. The wind turbine is operated at rated rotor angular velocity $Ω = 12.1 r / \min$ . To demonstrate the control performance at different wind speeds. The local aerodynamic loads of wind speed 12 m/s with turbulence intensity 10% and wind speed 22 m/s with turbulence intensity 30% were calculated using BEM theory (see Hansen (2008) for more details) and numerical integration along the blade. Airfoil files containing the coefficients of lift and drag are given in Jonkman et al. (2009). Figure 2 depicts the wind speed and aerodynamic loads acting on blades and the nacelle.

Figure 2.

Wind speed on the hub height and aerodynamic loads action on blades and nacelle.

In Table 2, a numerical comparison between the uncontrolled 4-DOF model with wind turbine software FAST (Jonkman and Buhl, 2005) is carried out which enables the corresponding DOF in the FAST to verify the accuracy of the model in this article. The response of blade 1 tip under wind speed $12 m / s$ is plotted in Figure 3. It observed a high coincident between the 4-DOF model and the results of FAST. Although the model established in this article is slightly different from the frequency of the FAST model, this difference is caused by the modal mass and stiffness as well as the aerodynamic loading which are obtained from the integration due to the distribution parameters of the blade. The numerical accuracy of the integration leads to calculation errors.

Table 2.

Results compared the 4-DOF uncontrolled model with FAST.

Item	FAST	4-DOF
First blade in-plane mode frequency	1.08 Hz	1.10 Hz
First tower side-side mode frequency	0.33 Hz	0.34 Hz

DOF: degree of freedom.

Figure 3.

The blade 1 tip displacement between 4-DOF model and FAST.

In this simulation, the initial states are selected to be $x_{0} = 0_{16 \times 1}$ . When analyzing the response, we compared uncontrolled displacements and the one under LQR (Fitzgerald and Basu, 2014; Fitzgerald et al., 2013) and decentralized control. Figure 4 shows the response of wind turbine including the blade and the nacelle under the condition of wind speed $12 m / s$ . One can see that the robust decentralized control efficiently reduces the structural vibrations of both blade and nacelle. As mentioned above, the stroke of TMD needed to be limited inside the blade and nacelle. The top of Figure 5 illustrates the blade damper displacement of the different control scheme, while the control force of the different control scheme is shown at the bottom of Figure 5. We can conclude from this figure that the robust decentralized control with the stroke constraints can effectively utilize the blade space and better dissipate the vibration energy.

Figure 4.

Response of the wind turbine (Wind speed 12 m/s).

Figure 5.

Response of the blade TMD (Wind speed 12 m/s).

The vibration control effect at a high load is crucial when the wind speed is rapidly changing. When considering an extreme wind speed with a $22 m / s$ mean speed with turbulence intensity I = 30%, the displacement response is shown in Figure 6, and the corresponding TMD stroke and control force is illustrated in Figure 7. Robust decentralized control shows more improving results than LQR.

Figure 6.

Response of the wind turbine (Wind speed 22 m/s).

Figure 7.

Response of the blade TMD (Wind speed 22 m/s).

The root mean square (RMS) values of response displacement obtained by different control algorithms including the uncontrolled system, LQR, and the decentralized $H_{\infty}$ output feedback control are given in Table 3. By comparing the RMS value in blade 1, it is noticed that centralized LQR control showed an RMS reduction of $12.9 %$ compared to the uncontrolled case. Decentralized control reduced RMS $16.2 %$ compared to the uncontrolled system when wind speed is $22 m / s$ . In addition, the nacelle RMS under decentralized control reduced $23.1 %$ compared to the open loop. As the load increases, the decentralized $H_{\infty}$ control shows more excellent performance which decreases an RMS response $21.6 %$ than obtained by the control scheme of LQR in blade. Moreover, from Table 4, less control force is applied. This result further verified that the proposed decentralized control strategy is feasible. No doubt that the vibration suppresses robust control better than that of LQR. Analyzing the results from Figures 5 and 7, the cut-off occurs when the TMD reaches the bound of the blade. This is caused by the stop forces. According to equation (17), this force can be treated as an external disturbance. Due to no capacity to robustness against this disturbance and handle output constraints on the TMD in LQR approach, the TMD will reach the bound of the blade and have a poor control performance. The constraint decentralized $H_{\infty}$ control approach can improve robustness against disturbance and effectively utilize the blade space. In addition, the decentralized control is based on local signals. Compared to the full-state feedback which requires a large number of sensor networks, the structure of decentralized control is simple and the reliability is higher.

Table 3.

The RMS value of displacements in different control algorithms.

Term	Wind speed (12 m/s)
Term	Uncontrolled	LQR	Decentralized
Blade 1	0.5914 m	0.5147 m	0.4955 m
Blade 2	0.5873 m	0.5257 m	0.4873 m
Blade 3	0.5652 m	0.5018 m	0.4958 m
Nacelle	0.0297 m	0.0185 m	0.0155 m
Term	Wind speed (22 m/s)
	Uncontrolled	LQR	Decentralized
Blade 1	1.0637 m	0.7652 m	0.5993 m
Blade 2	1.0159 m	0.7649 m	0.6282 m
Blade 3	1.0392 m	0.7197 m	0.6060 m
Nacelle	0.0827 m	0.0431 m	0.0367 m

RMS: root mean square; LQR: linear quadratic regulator.

Table 4.

The RMS value of control forces.

Term	Wind speed (12 m/s)		Wind speed (22 m/s)
Term	LQR	$H_{\infty}$	LQR	$H_{\infty}$
Blade 1	2244.3 N	1158.9 N	2388.2 N	1688.2 N
Blade 2	2306.6 N	1304.0 N	2531.7 N	2429.6 N
Blade 3	2197.7 N	1113.8 N	2324.7 N	1542.6 N
Nacelle	2976.3 N	306.5 N	6714.4 N	780.3 N

RMS: root mean square; LQR: linear quadratic regulator.

To analyze the frequency response, the power spectrum density (PSD) for the blade tip displacement and the nacelle are shown in Figure 8. It is noticed that under control action from both centralized state-feedback control and decentralized output feedback control, the peak of the first edgewise mode frequency (about $1.1 Hz$ ) is reduced. This is because the TMD is tuned to the fundamental resonance frequency. The other two blades get the similar conclusion. Additionally, the PSD of nacelle illustrates the first vibration mode frequency response of the tower (about $0.3 Hz$ ) and the frequency response coupled by the blade (about $1.1 Hz$ ) is also reduced.

Figure 8.

Comparison of power spectrum density in frequency 0–3 Hz.

Conclusion

In this article, simultaneously control vibrations for wind turbine blades and tower by ATMD with constrained stroke are studied in the article. A decentralized constrained $H_{\infty}$ control was applied simultaneously for control blades and tower vibrations. The controller is designed by solving linear matrix inequalities which are based on the variable substitution. In order to investigate a general performance of the proposed decentralized control scheme, simulations have been carried out on the NREL 5-MW wind turbine. Numerical simulations demonstrate the decentralized control can achieve the improving closed-loop response compared to the centralized LQR state feedback controls. Furthermore, in decentralized control approach, each ATMD is controlled individually by the corresponding velocity signal. Meanwhile, the TMD stroke constraint is guaranteed by modeling as hard constraint in control design process. Hence, it improved the system reliability and is more available than the LQR approach in practice.

Footnotes

Appendix 1

\begin{matrix} ϕ_{b} (\bar{x}) = - 0.6952 {\bar{x}}^{6} + 2.3760 {\bar{x}}^{5} - 3.5772 {\bar{x}}^{4} + 2.5337 {\bar{x}}^{3} + 0.3627 {\bar{x}}^{2} \\ ϕ_{t} (\bar{x}) = 0.5357 {\bar{x}}^{6} - 2.2395 {\bar{x}}^{5} + 3.0871 {\bar{x}}^{4} - 1.7684 x^{3} {\bar{x}}^{3} + 1.3850 {\bar{x}}^{2} \end{matrix}

System matrices for second matrix differential equation:

\begin{array}{l} M_{o} = [\begin{matrix} m_{2} & 0 & 0 & a_{1} \\ 0 & m_{2} & 0 & a_{2} \\ 0 & 0 & m_{2} & a_{3} \\ a_{1} & a_{2} & a_{3} & m_{n} \end{matrix}], & C_{o} = [\begin{matrix} c_{b} & 0 & 0 & 0 \\ 0 & c_{b} & 0 & 0 \\ 0 & 0 & c_{b} & 0 \\ d_{1} & d_{2} & d_{3} & c_{s s} \end{matrix}], & K_{o} = [\begin{matrix} k_{1} & 0 & 0 & 0 \\ 0 & k_{2} & 0 & 0 \\ 0 & 0 & k_{3} & 0 \\ f_{1} & f_{2} & f_{3} & k_{s s} \end{matrix}] \end{array}

where

\begin{matrix} m_{0} = \int_{0}^{L} μ (x) dx, m_{1} = \int_{0}^{L} μ (x) ϕ_{b} dx, \\ m_{2} = \int_{0}^{L} μ (x) ϕ_{b}^{2} dx, m_{ss} = \int_{0}^{L} μ_{t} (x) ϕ_{t} dx, m_{n} = 3 m_{0} + m_{ss} \\ a_{i} = m_{1} \cos (ψ_{i}), i = 1, 2, 3 \\ d_{i} = - 2 Ω m_{1} \sin (ψ_{i}), i = 1, 2, 3 \\ f_{i} = - Ω^{2} m_{1} \cos (ψ_{i}), i = 1, 2, 3 \\ k_{i} = k_{e} + k_{c} - k_{g} \cos (φ_{i}) - Ω^{2} m_{2}, i = 1, 2, 3 \\ c_{b} = \frac{2 ξ_{b}}{ω_{b}} k_{b}, c_{ss} = \frac{2 ξ_{n}}{ω_{n}} k_{ss} \end{matrix}

\begin{matrix} M_{c} = [\begin{matrix} t_{1} & v_{1} & 0 & 0 & 0 & 0 & a_{1} & 0 \\ v_{1} & m_{bd} & 0 & 0 & 0 & 0 & b_{2} & 0 \\ 0 & 0 & t_{2} & v_{2} & 0 & 0 & a_{2} & 0 \\ 0 & 0 & v_{2} & m_{bd} & 0 & 0 & b_{2} & 0 \\ 0 & 0 & 0 & 0 & t_{3} & v_{3} & a_{3} & 0 \\ 0 & 0 & 0 & 0 & v_{3} & m_{bd} & b_{3} & 0 \\ a_{1} & b_{1} & a_{2} & b_{2} & a_{3} & b_{3} & N_{m} & m_{nd} \\ 0 & 0 & 0 & 0 & 0 & 0 & m_{nd} & m_{nd} \end{matrix}] \\ C_{c} = [\begin{matrix} c_{b} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c_{bd} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{b} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{bd} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{b} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{bd} & 0 & 0 \\ d_{1} & e_{1} & d_{2} & e_{2} & d_{3} & e_{3} & c_{ss} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{nd} \end{matrix}], \\ K_{c} = [\begin{matrix} k_{1} & k_{d} & 0 & 0 & 0 & 0 & 0 & 0 \\ k_{d} & k_{dt} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & k_{2} & k_{d} & 0 & 0 & 0 & 0 \\ 0 & 0 & k_{d} & k_{dt} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{3} & k_{d} & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{d} & k_{dt} & 0 & 0 \\ f_{1} & g_{1} & f_{2} & g_{2} & f_{3} & g_{3} & k_{ss} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & k_{nd} \end{matrix}] \end{matrix}

where

\begin{matrix} N_{m} = 3 m_{0} + 3 m_{bd} + m_{ss} + m_{nd} \\ t_{i} = m_{2} + m_{bd} ϕ_{b} (r_{i})^{2}, i = 1, 2, 3 \\ v_{i} = m_{bd} ϕ_{b} (r_{i}), i = 1, 2, 3 \\ a_{i} = m_{1} \cos (ψ_{i}) + m_{bd} ϕ_{b} (r_{i}) \cos (ψ_{1}), i = 1, 2, 3, \\ b_{i} = m_{bd} \cos (ψ_{i}), i = 1, 2, 3 \\ d_{i} = - 2 Ω m_{1} \sin (ψ_{i}) - 2 Ω m_{bd} \sin (ψ_{i}), i = 1, 2, 3 \\ e_{i} = - 2 Ω m_{bd} ϕ_{b} (r_{i}) \sin (ψ_{i}), i = 1, 2, 3 \\ k_{i} = k_{e} + k_{c} - k_{g} \cos (φ_{i}) - Ω^{2} m_{2} - Ω^{2} m_{bd} ϕ (r_{i}), i = 1, 2, 3 \\ k_{d} = - Ω^{2} m_{bd} ϕ (r_{t}) \\ k_{dt} = - Ω^{2} m_{bd} + k_{bd} \\ f_{i} = - Ω^{2} m_{1} \cos (ψ_{i}) - Ω^{2} m_{bd} ϕ (r_{i}) \cos (ψ_{i}) \\ - \overset{\cdot}{Ω} m_{1} \sin (ψ_{i}) - \overset{\cdot}{Ω} m_{bd} ϕ (r_{i}) \sin (ψ_{i}), i = 1, 2, 3 \\ g_{i} = - Ω^{2} m_{bd} \cos (ψ_{i}) - \overset{\cdot}{Ω} m_{bd} \sin (ψ_{i}), i = 1, 2, 3 \end{matrix}

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Cong Cong

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Using active tuned mass dampers with constrained stroke to simultaneously control vibrations in wind turbine blades and tower

Abstract

Keywords

Introduction

Theoretical model for the wind turbine with decentralized ATMD system

Decentralized H∞ static output feedback control

Decentralized control of simultaneously control blades and tower vibrations

Centralized and decentralized controllers

Simulation results

Conclusion

Footnotes

Appendix 1

Declaration of Conflicting Interests

Funding

ORCID iD

References

Decentralized H_∞ static output feedback control