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Abstract

This article is devoted to the study of the vibration control for blades and tower in a wind turbine. Based on the Euler–Lagrangian method, a multi-body dynamic model including three blades with distributed parameter, tower, and their coupling is obtained. Multi active tuned mass dampers have been utilizing as damping devices. Therefore, the dynamics of the tuned mass dampers are also considered in modeling. The influence of extreme wind, and grid dynamics on the vibration of the blade was analyzed. Moreover, the nonlinearity induced by space constraints, which impact on vibration control, is introduced. For active control, the constrained decentralized control strategy is designed via linear matrix inequality which tuned mass dampers stroke constraints are modeled as hard constraints. A doubly fed induction generator connected to an infinite bus including the detailed electrical and structural model was performed on MATLAB/Simulink. Simulation results show that the control strategy can effectively reduce the vibration of the blade while the damper stroke satisfies the working space permitted by the blade. Investigations demonstrate promising results for decentralized constrained control in simultaneous control blade vibrations and tower vibrations. Each actuator is driven separately from the output of the corresponding sensor so that only local feedback control is achieved; this improves the system reliability.

Keywords

Wind turbines structural control coupled mechano-electric model decentralized control active tuned mass damper

Introduction

Vibrations in wind turbines can affect the safe operation and influence on power production.¹ To reduce the impacts of vibrations, structural control is the ideal control approach for wind turbines.² Due to the hollow nature of the blade and the nacelle, the damper installed inside of these components can independently on speed or power regulation. Previous work by many researchers focused on the feasible of the damping devices. The tuned mass damper which consists of a mass, spring, and damping, is the most common device in vibration control.³ For mitigating of a tower of the wind turbine, a passive tuned mass damper (TMD) placed on the top of the tower for reduction vibration is proposed.⁴ Lackner and Rotea⁵ modified the FAST wind turbine computer-aided engineering software tool, by incorporating two independent TMDs into the nacelle. Thus, add structural control capacity to Fatigue, Aerodynamics, Structures, and Turbulence (FAST). Implementing this modified version of FAST, Tong et al.⁶ proposed a new optimal method for designing TMD which is used to control the tower vibration. In the study of Brodersen et al.,⁷ the TMD parameter estimation is conducted under different wind and wave conditions. Zuo et al.⁸ investigate applying multiple TMD to reduce offshore wind turbine vibrations under multiple hazards. For suppressing blade vibrations, there are few studies regarding structural control. The motion of the blade is more complicated than the tower. No simulation software can be used to design and simulate blade structural control. In the literature,^9–11 the TMDs are used to control the blade vibrations which are mainly based on the established simplified dynamic equation. The semi-active TMDs to control edgewise vibrations is presented in Arrigan et al.⁹ In Fitzgerald et al.,¹⁰ the authors investigated using active TMD for mitigating edgewise vibrations. It was found that the active TMD can provide better reduction than the passive TMD. Fitzgerald and Basu¹¹ extends the result¹⁰ by proposed a cable connected active TMD to suppress edgewise vibrations. However, these research focuses on the feasibility of dampers. The control algorithm uses simple linear optimal regulator state feedback and cannot guarantee the stroke constraints in theoretically. Nevertheless, in their design process, the nonlinearity due to space constraints is not considered. In the FAST-SC (the modified version of FAST⁵) simulation, it has been demonstrated that the stop forces have a strong influence on vibration control.¹² Moreover, in the model, the rotor speed is assumed as constant and does not take into account the dynamic due to the electrical dynamic. The grid-connected wind turbines often have a voltage variation in actual operation. Due to the coupling of the mechanical part and the electrical part, the electrical disturbance will affect the mechanical characteristics of the wind turbine and cause mechanical vibration.^13–16 Therefore, detailed mechanical model and electrical model of wind turbines must be considered.

In a practical implementation, full knowledge of the state vector is rarely available. An available approach is an output feedback control. Meanwhile, the centralized control strategy needs a large number of sensors and communication network; this may have poor reliability.¹⁷ For large-scale structures, decentralized static output feedback control is a more practical approach. Solving the static output feedback with information constraints is a popular research topic.¹⁸ An linear matrix inequality (LMI) variable transformations were obtained and a two-step procedure is proposed to improve feasibility issues of solving LMI optimization problem.^19–22 In this article, based on these fore-mentioned considerations in the existing article.^{10,13,14,17,18} The major contributions of this article are as follows: (1) a closed-loop 12-degree-of-freedom (DOF) structural model established in Fitzgerald et al.¹⁰ is extended to consider the grid-connected dynamic, in which the rotor speed is variable due to the change of grid voltage. Moreover, the mechano-electrical coupling model of grid-connected doubly fed induce wind turbines was developed in this article. (2) The dynamic equation established in Fitzgerald et al.¹⁰ is extended to modeling the TMD stroke constraint. Due to the space limitations of the blade and the nacelle, the stroke of the TMD must be limited, the stopping forces are incorporated into the model when the TMD displacement exceeds the allowable stroke. (3) A decentralized static output feedback controller was implemented, which each TMD is controlled individually by the corresponding velocity signal. To the authors’ knowledge, decentralized control of wind turbines has not been studied in existing literature. To fill this gap, the present paper explores decentralized structural control of the wind turbines with TMD subjected to wind load where the damper stroke is modeled as the constraint output. The designed controllers can be easily solved via LMI optimization problem.

Electrical model of wind turbine system

Electrical disturbances in the electrical system will vary the rotor speed and electromagnetic torque of the wind turbine, therefore affecting the structural dynamics of the wind turbines.^13–15 More precisely, the variation of the rotor velocity mainly affects the in-plane vibration of the blade and the lateral vibration of the tower. A grid-connected doubly fed induction generator (DFIG) system and its model are shown in Figures 1 and 2. The entire system model includes the generator, drive train, and mechanical model. Generator models are studied in Prajapat et al.,¹⁶ and for the sake of document integrity, a brief description is given here. The wind turbine is linked to the single machine-infinity bus system via a transmission line. Via back-to-back converters, the rotor is connected to the grid. These back-to-back converters consist of two series control loops. The outer loop is current control which controls the stator active power and reactive power; inner loop is voltage control which regulates the rotor voltage. Meantime, the grid-side converter maintains direct-current (DC) bus voltage stability and network-side power factor adjustment. The present paper focuses on the structural control, although there have many attempts to design advanced control approach in the literature. In this context, all controllers in the electrical model are using a simple proportional integral (PI) control. Due to space limitations, the detailed control strategy can be found in Prajapat et al.¹⁶

Figure 1.

Schematic diagram of the grid-connected DFIG.

Figure 2.

The overall simulation and control structure of wind turbine system.

The dynamic of the DFIG is represented in synchronously rotating d-q reference frame as follows

\begin{matrix} {\overset{\cdot}{φ}}_{qs} = v_{qs} - ω_{e} φ_{ds} + \frac{R_{s}}{σ L_{s}} (\frac{L_{m}}{L_{r}} φ_{qr} - φ_{qs}) \\ {\overset{\cdot}{φ}}_{ds} = v_{ds} - ω_{e} φ_{qs} + \frac{R_{s}}{σ L_{s}} (\frac{L_{m}}{L_{r}} φ_{dr} - φ_{ds}) \\ {\overset{\cdot}{φ}}_{qr} = v_{qr} - (ω_{e} - ω_{r}) φ_{dr} + \frac{R_{s}}{σ L_{r}} (\frac{L_{m}}{L_{s}} φ_{qs} - φ_{qr}) \\ {\overset{\cdot}{φ}}_{dr} = v_{dr} - (ω_{e} - ω_{r}) φ_{qr} + \frac{R_{s}}{σ L_{r}} (\frac{L_{m}}{L_{s}} φ_{ds} - φ_{dr}) \end{matrix}

(1)

\begin{matrix} i_{qs} = \frac{1}{σ L_{s}} φ_{qs} - \frac{L_{m}}{σ L_{s} L_{r}} φ_{qr} \\ i_{ds} = \frac{1}{σ L_{s}} φ_{ds} - \frac{L_{m}}{σ L_{s} L_{r}} φ_{dr} \\ i_{qr} = \frac{1}{σ L_{s}} φ_{qr} - \frac{L_{m}}{σ L_{s} L_{r}} φ_{qs} \\ i_{dr} = \frac{1}{σ L_{s}} φ_{dr} - \frac{L_{m}}{σ L_{s} L_{r}} φ_{ds} \end{matrix}

(2)

where $φ_{r}, φ_{s}$ are rotor flux and stator flux, respectively; $L_{s}, L_{r}, L_{m}$ denote stator self-inductance, rotor self-inductance, and mutual inductance, respectively; $R_{s}, R_{r}$ represent stator resistance and rotor resistance, respectively; $v_{s}, v_{r}$ signify stator voltage and rotor voltage, respectively; $i_{s}, i_{r}$ characterize stator current and rotor current, respectively. ω_e, ω_r are the stator angular electrical frequency and the rotor angular electrical speed.

The generator electromagnetic torque is

T_{e} = \frac{1.5 p L_{m}}{σ L_{s} L_{r}} (φ_{qs} φ_{dr} - φ_{ds} φ_{qr})

(3)

The electrical system and the mechanical system are coupled by a drive train, which is modeled as a third-order flexible shaft model, as follows

\begin{matrix} \overset{\cdot}{Ω} = \frac{1}{I_{rot}} (T_{shaft} - ϕ K_{shaft} - \overset{\cdot}{ϕ} B_{shaft}) \\ \overset{\cdot}{ω} = \frac{1}{I_{gen}} (- T_{gen} + \frac{1}{N} (ϕ K_{shaft} + \overset{\cdot}{ϕ} B_{shaft})) \\ \overset{\cdot}{θ} = Ω - \frac{1}{N} ω \end{matrix}

(4)

where $K_{shaft}$ , $C_{shaft}$ are the friction coefficient and the spring coefficient, respectively. $ω$ is the high-speed shaft angular speed; $θ$ is the main shaft twist angle.

Nonlinear mechanical model for the wind turbine system with active TMDs

A reduced 12-DOF nonlinear wind turbine structural model was proposed which each blade and nacelle are equipped with one active tuned mass damper (ATMD) to control the in-plane vibration.¹⁰ Figure 3 illustrates the schematic representation of vibration mode which blades are coupling with nacelle/tower including the attached ATMDs. The tower is modeled as Bernoulli cantilever beam described in the fixed coordinate system $(X, Y, Z)$ which blades and nacelle are modeled by a lumped mass. While the motion of each blade is described in a co-rotating coordinate system $(x, y, z)$ . This coordinate system is rotating with blade. The azimuth angle $ψ_{j} (t)$ of each blade, which is positive when rotating clockwise as observed from z position, is given by

ψ_{i} = \int_{0}^{t} Ω d τ + (i - 1) 2 / 3 π, i = 1, 2, 3

(5)

Figure 3.

A schematic diagram of ATMDs in blade and nacelle.

Due to the coupled mechanical structure in wind turbine, modeling process will be complicated. As the method developed in previous (Fitzgerald et al.),¹⁰ the governing differential equations of wind turbines with ATMD can be obtained by applying the Euler–Lagrangian approach. Variable $q_{j, in}$ represents the modal coordinate in edgewise direction and $q_{j, out}$ describes the modal coordinate in flapwise direction. $ϕ_{in}$ , $ϕ_{out}$ denote the corresponding mode shape functions. $q_{ss}$ , $q_{fa}$ are the modal coordinate of tower side-side and fore-aft motions, respectively. Variables $d_{1}$ , $d_{2}$ , and $d_{3}$ characterize the DOF of the ATMD in each blade and $d_{n}$ signifies the DOF of the ATMD in the nacelle. By the modal expansion, each mode of motion is described by the fundamental modal coordinates and mode shape functions. An energy formulation is used in the Euler–Lagrange method, given as

\frac{d}{d t} (\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

(6)

where $q_{i} (t)$ , $Q_{ext, i} (t)$ are the generalized coordinate and the generalized loading acting on blade or tower for the degree of freedom i, respectively. T is the kinetic energy and V is the potential energy.

Kinetic and potential energy

A position vector ${\vec{r}}_{j} (t)$ which the hub is the reference point, is used for representing the blade motion. This can be represented by the unit vector $[\vec{i}, \vec{j}, \vec{k}]$ in the rotating frame as follows

\begin{matrix} {\vec{r}}_{j} = & (x + q_{ss} \sin (ψ_{i})) \vec{i} \\ + (ϕ_{in} q_{j, in} + q_{ss} \cos (ψ_{i})) \vec{j} \\ + (ϕ_{out} q_{j, out}) \vec{k}, j = 1, 2, 3 \end{matrix}

(7)

The position vector ${\vec{r}}_{d, j}$ is used for representing the ATMD motion, given by

\begin{matrix} {\vec{r}}_{d, j} = & (r_{t} + q_{ss} \sin (ψ_{i})) \vec{i} \\ + (ϕ_{in} q_{j, in} + q_{ss} \cos (ψ_{i}) + d_{j} + y_{0}) \vec{j} \\ + (ϕ_{out} q_{j, out}) \vec{k}, j = 1, 2, 3 \end{matrix}

(8)

where $r_{t}$ is the location of the ATMD; $y_{0}$ is the reference position from the centerline of the blade along the longitudinal axis.

The velocity vector of the blade and the ATMD are obtained by differentiating ${\vec{r}}_{j}$ and ${\vec{r}}_{d, j}$ with respect to time. Therefore, the kinetic energy is given by

\begin{matrix} T = & \frac{1}{2} (\sum_{j = 1}^{3} (\int_{0}^{L} μ (x) \overset{\cdot}{\vec{r_{j}^{2}}} dx + m_{d} \overset{\cdot}{\vec{r_{d, j}^{2}}} dx) \\ + m_{fa} {\overset{\cdot}{q}}_{fa}^{2} dx + m_{ss} {\overset{\cdot}{q}}_{ss}^{2} dx + m_{nd} {({\overset{\cdot}{d}}_{n} + {\overset{\cdot}{q}}_{ss})}^{2} dx) \end{matrix}

(9)

where $μ (x)$ is the density of blade, $m_{d}$ denotes the mass of the blade damper, $m_{nd}$ represents the mass of the nacelle damper, and $m_{ss}$ and $m_{fa}$ are the mass of tower in side-side and fore-aft direction, respectively.

The elasticity force, centrifugal force, and the gravity force together cause the total potential energy, given by

\begin{matrix} V = & \frac{1}{2} [\sum_{j = 1}^{3} [\int_{0}^{L} (k_{io} \frac{\partial}{\partial x} u_{j, in} \frac{\partial}{\partial x} u_{j, out} \\ + k_{in} {(\frac{\partial^{2}}{\partial x^{2}} u_{j, in})}^{2} + k_{out} {(\frac{\partial^{2}}{\partial x^{2}} u_{j, out})}^{2} \\ - \frac{1}{2} g \cos (φ_{j}) \int_{x}^{L} μ (ξ) \frac{\partial}{\partial x} {(u_{j, out})}^{2} d ξ \\ - \frac{1}{2} g \cos (φ_{j}) \int_{x}^{L} μ (ξ) \frac{\partial}{\partial x} {(u_{j, out})}^{2} d ξ \\ + Ω \int_{x}^{L} μ (ξ) ξ d ξ \frac{\partial}{\partial x} {(u_{j, out})}^{2} \\ + Ω \int_{x}^{L} μ (ξ) ξ d ξ \frac{\partial}{\partial x} {(u_{j, in})}^{2}) dx \\ + k_{bd} d_{j}^{2}] + k_{ss} q_{ss}^{2} + k_{ss} q_{fa}^{2} + k_{nd} d_{n}^{2}] \end{matrix}

(10)

where $k_{in}, k_{out}$ are the distributed stiffness of blade, $k_{bd}$ denotes the stiffness of the blade’s damper, and $k_{nd}$ describes the stiffness of the nacelle’s damper.

The external force action on blades and nacelle

The total generalized external force vector consists of the modal aerodynamic load $Q_{a}$ , modal gravity load $Q_{g}$ , the force induced by variable rotor speed $Q_{Ω}$ , and the stop forces $F_{stop}$

Q_{ext} (t) = Q_{a} + Q_{g} + Q_{Ω} + F_{stop}

(11)

The first modal aerodynamic load on the blade j is calculated as

\begin{matrix} Q_{j, in} = \int_{0}^{L} p_{N, j} (x, t) ϕ_{in} (x) dx \\ Q_{j, out} = \int_{0}^{L} p_{T, j} (x, t) ϕ_{out} (x) dx \end{matrix}

(12)

where $p_{N, j} (x, t)$ , $p_{T, j} (x, t)$ are the variable local wind load intensity in the in-plane and out-plane direction which can be calculated by blade element momentum (BEM) theory.²³

The aerodynamic force on tower is

Q_{ss} = \sum_{j = 1}^{3} Q_{j, in} \cos (ψ (j)), Q_{fa} = \sum_{j = 1}^{3} Q_{j, in}

(13)

The gravity load on the tower is zero and on the blade is

Q_{g} = g \sin (ψ_{j}) \int_{0}^{L} μ (x) ϕ_{in} (x) dx

(14)

In previous studies, the rotor speed is assumed constant.^10,11 When the wind turbine is working at variable rotor speed, the force acts on the blade and the TMD induced by variable rotor speed is derived 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)

Due to the limitation of the space, the damper displacement should be completely contained inside the blade and the nacelle, and the stroke of the ATMD must be limited. The maximum allowable displacement is expressed as $| d_{i} | < Z_{i, max}$ . When the ATMD stroke reaches the maximum allowable displacement, it is subjected to the stopping force. Referring to the modeling method of the TMD in the nacelle used in the FAST software model,¹² the stopping forces in the blade and the nacelle are expressed as equation (16); here $k_{stop}$ , $c_{stop}$ are the stiffness and damping coefficient, obtained based on experience

f_{s t o p, i} = - {\begin{array}{l} k_{s t o p} Δ d_{i}, : (d_{i} > d_{\max} \land {\dot{d}}_{i} \leq 0) \lor (d_{i} < d_{\min} \land {\dot{d}}_{i} \geq 0) \\ k_{s t o p} Δ d_{i} + c_{s t o p} {\dot{d}}_{i} : (d_{i} > d_{\max} \land {\dot{d}}_{i} > 0) \lor (d_{i} < d_{\min} \land {\dot{d}}_{i} < 0) \\ 0 : o t h e r w i s e \end{array}

(16)

The ATMD is assumed driven by ideal active actuation devices which active control force is modeled as an external force $f_{j}$ into the equations of motion. The vector of the active control forces u is given by $u = [f_{1} f_{2} f_{3} f_{n}]^{T}$ .

Second-order matrix differential equation

Substitute the kinetic energy and the potential energy into Euler–Lagrangian equation (6), and the external force vector is obtained from the principle of virtual work. The final second-order matrix differential equation of the coupled system can be written as

M (t) \overset{\cdot\cdot}{q} + C (t) \overset{\cdot}{q} + K (t) q = Q_{ext} (t) + Bu (t)

(17)

where the matrices of the model are given in Appendix 1.

Denoting the state $x = [q, \overset{\cdot}{q}]^{T}$ . The system (equation (17)) can be described as a state space expression

\begin{matrix} \overset{\cdot}{x} = Ax + Bu + Ew \end{matrix}

(18)

where the system state equation matrices are given as

\begin{matrix} A = [\begin{matrix} 0_{12 \times 12} & I_{12 \times 12} \\ - M^{- 1} K & - M^{- 1} C \end{matrix}], E = [\begin{matrix} 0_{12 \times 12} \\ - M^{- 1} \end{matrix}] \\ B = [\begin{matrix} 0_{12 \times 4} \\ - M^{- 1} \bar{B} \end{matrix}], w = Q_{ext} \\ \bar{B} = {[\begin{matrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} \end{matrix}

The uncertainties in the stiffness and damping matrices are presented as $Δ K$ and $Δ C$ . Thus, the disturbance input is represented as $w = Q_{ext} + Δ Kq + Δ C \overset{\cdot}{q}$ .

Decentralized H_∞ static output feedback control

By modifying equation (18) according to the structural constraints, 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 i} = C_{2 i} x \end{matrix}

(19)

where $z_{1}$ is control output and $z_{2 i}$ is constraint output.

For a given decentralized static output feedback control $u_{i} = F_{i} y_{i}$ . Define matrices: $F_{d} = diag (F_{1}, \dots, F_{N}), C_{y} = [C_{y 1}, \dots, C_{yN}]^{T}$ , $B = [B_{1}, \dots, B_{N}]$ . A decentralized static output feedback control is obtained by the following theorem.

Theorem 1

The closed-loop system with control law $u = Kx$ satisfies the performance $∥ z_{1} ∥_{2} < γ ∥ w ∥_{2}$ , for $γ > 0$ , if there exists a matrix Y and symmetric positive-definite matrix X satisfies the following matrix inequality²⁰

[\begin{matrix} sym (AX + BY) + γ^{- 2} E E^{T} & * \\ C_{1} X & - I \end{matrix}] < 0

(20)

\begin{matrix} [\begin{matrix} 1 / {z_{\max}}_{i} & \sqrt{ρ} (C_{12, i} X + D_{22, i} Y) \\ * & X \end{matrix}] > 0 \end{matrix}

(21)

where $λ$ denotes the eigenvalue.

The optimal constraint $H_{\infty}$ state feedback control law $K = Y^{- 1} X$ is obtained by the optimization problem given by the following expression

\min_{X > 0, γ > 0} γ, subject to LMI (20) and (21)

(22)

Proof

From the Bounded Real Lemma,²⁰ solving a standard state feedback $H_{\infty}$ control problem is equivalent to equation (20). Similar to the derivation in Li et al.,²⁴ since the closed-loop system satisfies the $H_{\infty}$ performance, it yields $x^{T} X^{- 1} x < ρ$ , where $ρ = γ^{2} w_{\max}$ . The constraint can be modeled as the 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}

(23)

Using Cauchy–Schwarz inequality and Schur complement, the constraint can be written as equation (21). In next step, to obtain a static output feedback control satisfy $K = F C_{y}$ , the following theorem is proposed.

Theorem 2

Consider the system (equation (19)) with decentralization input and output pairs, there exist constants $γ > 0$ such that there exists a symmetric matrix $X_{R} > 0$ and a general matrix $Y_{R}$ satisfy equations (24) and (25)

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

(24)

\begin{matrix} [\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 \end{matrix}

(25)

If LMI (equations (24) and (25)) is solvable, the system (equation (19)) is stabilizable with disturbance attenuation $γ$ via decentralized static output feedback control of $F = Y_{R}^{- 1} X_{R}$ .

Proof

To find a preassigned matrix satisfies $K = F C_{y}$ , it follows the transformations proposed by Rubió-Massegú et al.¹⁹: $X = Q X_{Q} Q^{T} + R X_{R} R^{T}$ , $Y = Y_{R} R^{T}$ . Here, matrix $R = C_{1}^{T} (C_{1} C_{1}^{T})^{- 1} + L$ , the columns of matrix Q are the base vector of kernel space of matrix C. Equation (20) is converted into equation (24), and equation (21) is converted into equation (25).

Theorem 3

To obtain a optimal $γ$ , denoting $τ = γ^{- 2}$ , the constrained $H_{\infty}$ output feedback control design is solving the convex optimization given as

\max_{X_{Q} > 0, X_{R} > 0, τ > 0} τ, subject to LMI (24) and (25)

(26)

If the problem (equation (26)) is feasible with respect to $X_{R}$ and $Y_{R}$ , then $F_{d} = Y_{R} X_{R}^{- 1}$ is the constrained $H_{\infty}$ output feedback control matrix. Therefore, the problem is reduced to solving the optimization problem (equation (26)) with respect to $X_{R}$ and $Y_{R}$ . By constraint a specific formulation on the LMI variables $X_{R}$ and $Y_{R}$ , the decentralized controller $F_{d}$ is obtained; otherwise, the centralized controller $F_{c}$ is obtained.

Remark 1

The orthogonal basis of $Ker (C_{y})$ can be obtained with the MATLAB command null() and the Moore–Penrose can be obtained with the MATLAB command pinv(). When these matrices are obtained, the optimal problem (equation (26)) is solved via MATLAB Robust Control Toolbox.

Remark 2

In order to obtain a preassigned decentralized structure on the control gain matrix, in our control problem, the LMI variables $Y_{R}$ and $X_{R}$ should be chosen as

Y_{R} = [\begin{matrix} • & • & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & • & • & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & • & • & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & • & • \end{matrix}] < 0

(27)

X_{R} = [\begin{matrix} • & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & • & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & • & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & • & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & • & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & • & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & • & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & • \end{matrix}] < 0

(28)

where the dot denotes the non-zero term to be designed.

Remark 3

According to the discuss in Palacios-Quiñonero et al.,²⁰ the problem (equation (26)) may be infeasible. In this situation, a perturbed state matrix is used to overcome this problem. If the perturbation state matrix optimization problem (equation (26)) is infeasible, a method to obtain a feasible solution is proposed in present paper. The optimal γ-value is obtained by solving equation (22). As discussed in the literature, $γ_{s} < γ_{oc} < γ_{od}$ .¹⁹ A substitute procedure was proposed by Algorithm 1.

Algorithm 1. The process of design decentralized static output feedback control.
1: Solve the optimization problem (22) in Theorem 1. 2: If there is a solution and obtain a value of $γ_{s}$ in Step 1, the matrix L can be designed as follows $L = Q^{†} X C^{T} (CX C^{T})^{- 1}$ (29) 3: Solve the optimization problem (26) in Theorem 3. 4: If there is a solution go to Step 7, else go to Step 5. 5: Set $γ > γ_{s}$ and solve the feasible problem of equations (24) and (25) in Theorem 2. 6: If there is a solution, go to Step 7, else increase $γ$ and return to Step 5. 7: Return $F_{d} = Y_{R} X_{R}^{- 1}$ .

Algorithm 1. The process of design decentralized static output feedback control.

1: Solve the optimization problem (22) in Theorem 1.
2: If there is a solution and obtain a value of

γ_{s}

in Step 1, the matrix L can be designed as follows

L = Q^{†} X C^{T} (CX C^{T})^{- 1}

(29)

3: Solve the optimization problem (26) in Theorem 3.
4: If there is a solution go to Step 7, else go to Step 5.
5: Set

γ > γ_{s}

and solve the feasible problem of equations (24) and (25) in Theorem 2.
6: If there is a solution, go to Step 7, else increase

γ

and return to Step 5.
7: Return

F_{d} = Y_{R} X_{R}^{- 1}

Numerical simulations

In this section, the decentralized control to simultaneously suppress vibrations of the blade and the tower is checked out. A numerical model using the data from a 5WM wind turbine designed by the National Renewable Energy Laboratory (NREL-5WM) is established and simulated in MATLAB.Table 1 lists the fundamental data of the wind turbine. While the distributed parameters of the blade and tower, and the airfoils are given in Jonkman et al.²⁵ Some parameters with regard to induction generator and grid connection are provided in Table 2.

Table 1.

Properties of NREL5-MW wind turbine.²⁵

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%

Table 2.

Induction generator and grid parameters.²⁶

Item	Value
Rated power	$5000 kW$
Rated mechanical torque	$40.809 kN m$
Rated stator line-to-line voltage	$950 V (rms)$
Rated stator phase voltage	$548.48 V (rms)$
Frequency	$50 Hz$
Number of pole pairs	3
Stator winding resistance	$1.552 m Ω$
Rotor winding resistance	$1.446 m Ω$
Stator leakage inductance	$1.2721 mH$
Rotor leakage inductance	$1.1194 mH$
Magnetizing inductance	$5.5182 mH$

In this article, the TMD in the blade and nacelle are tuned to the blade’s first in-plane mode and the tower’s first side-side mode, respectively. All parameters of TMD are assumed to be optimal, where the TMD mass ratio, tuning ratio, and the damping ratio are taken as Fitzgerald et al.¹⁰ The blade dampers are located at 45 m distance from the hub where an allowable damper stroke is approximately $3 m$ . Meantime, the allowable damper stroke in nacelle is approximately $8 m$ .

Load simulation

The control effect is analyzed in two different wind conditions. The vibration control effect at a high load is crucial when the wind speed is rapidly changing. In the first simulation performed, an extreme wind speed of a mean value $22 m / s$ with turbulence intensity $I = 30 %$ was considered. In the second simulation, to emphasize the impact of the grid dynamic, a stationary wind speed of a $12 m / s$ with turbulence intensity $I = 10 %$ is performed. The time series of turbulence is simulated by the spectral representation (SR) method which was applied for simulation multi-dimensional random process.²³ The wind speed and aerodynamic loads acting on blades and the nacelle of this two case are shown in Figure 4.

Figure 4.

Wind speed and aerodynamic load in in-plane direction action on wind turbine.

Simulation results

In this section, the effect in the proposed model was illustrated. The measured output is the velocity of the TMD and the blade or the nacelle. In all simulation, the initial states are selected to be $x_{0} = 0_{24 \times 1}$ . According to Algorithm 1, the control design is via a two-step procedure. In the first step, solving the optimal problem (equation (22)) obtained $τ_{s} = 1.2071 \times 10^{- 4}$ . In the second step, when directly solving the optimal problem (equation (26)), the problem is reported to be infeasible. Therefore, set a value greater than $γ_{s}$ , and solve the feasible problem about $X_{R}, Y_{R}$ in Theorem 2. The associated $γ_{od} = 2.4141 \times 10^{- 4}$ and the gain matrix is obtained as equation (30)

F_{d} = [\begin{matrix} - 1578 & - 12 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - 2223 & - 12 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - 1544 & - 11 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - 12881 & - 35747 \end{matrix}] (30)

(30)

As mentioned above, the stroke of the TMD must be incorporated inside the blade and the nacelle. It is subject to a stop force when the damper reaches the max allowable displacement. These stop forces are treating as disturbance input in the control design. The closed-loop response of blade and TMD using model and control method in Fitzgerald et al.¹⁰ under wind speed of $22 m / s$ with turbulence intensity $I = 30 %$ is given in Figure 5. One can see that in the first wind speed file (extreme wind speed), the displacement of the damper exceeds the maximum allowable displacement. Due to no stroke constraint and stop forces considered, this leads to the model fails. It is necessary to consider the stop forces to improve the accuracy of the model. Figure 5 also illustrates the TMD displacement obtained for the closed-loop system model with considering stroke constraint. When the stroke constraint is introduced, the displacement of the damper is always kept within the maximum allowable stroke.

Figure 5.

Compared response of TMD with and without stroke constraint (Blade1).

In Fitzgerald et al.,¹⁰ the linear quadratic regulator (LQR) is used as control law, which is essentially a centralized control approach. For comparison, the results of LQR control law are also present in this article. To show the effectiveness of the proposed control strategies. A 200 s simulation result which is compared the uncontrolled system, LQR controlled system, and decentralized controlled system was shown in Figures 6 and 7. As shown in these figures, the vibration displacements of both the blade and the nacelle are reduced under the decentralized control action. When compared with the uncontrolled system. A $45 %$ peak-to-peak reduction in the blade and $79 %$ reduction in the nacelle are obtained in the decentralized controlled case. Moreover, when compared with the LQR, a $9 %$ peak-to-peak reduction in the blade and $41 %$ reduction in the nacelle are obtained in the decentralized controlled case. The TMD stroke was also shown in Figures 8 and 9. One can find that the stroke in the case of using the constraint robust decentralized control is more effective. Under the decentralized robust control action, vibration energy is dissipated which the blade space is used effectively. According to Figure 4, when the wind speed changes drastically, the load changes rapidly and the robust control can effectively reduce the vibration due to the good performance on disturbance suppression for the load change and stop forces.

Figure 6.

Response of closed-loop system using decentralized control (Blade1).

Figure 7.

Evolution of the ATMD (Blade1).

Figure 8.

Response of closed-loop system using decentralized control (Nacelle).

Figure 9.

Evolution of the ATMD (Nacelle).

In order to quantify the amount of vibration displacement, it is usually using the root mean square (RMS) value. The RMS value of a vector x is calculated as

x_{RMS} = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} x_{j}^{2}}, j = 1, \dots, n

(31)

The RMS values of response displacement obtained by different control algorithm including the uncontrolled system, centralized LQR¹⁰ approach, and decentralized $H_{\infty}$ control approach (Algorithm 1) were analyzed. The decentralized control achieves RMS reductions of $70.4 %$ in the blade compared with the uncontrolled system and $31.3 %$ compared with LQR. Meanwhile, an RMS reduction of $82.4 %$ in the nacelle compared with the uncontrolled system and $57.1 %$ compared with LQR is achieved.

To analyze the frequency response, the power spectrum density (PSD) for blade tip displacement is shown in Figure 10. It is noted that under the control action, the peak of the first edgewise mode frequency of $1.08 Hz$ is reduced. Also, the PSD for nacelle displacement is depicted in Figure 11; from this figure one can obtain that the first modal vibration of $0.33 Hz$ in tower side-side direction is suppressed; moreover, the peak in $1.08 Hz$ due to the coupling with the blade is also reduced. This verifies the effectiveness of the control algorithm proposed in this article.

Figure 10.

Comparison of power spectrum density (Blade 1).

Figure 11.

Comparison of power spectrum density (Nacelle).

When considering the influence of electrical dynamics on the mechanical vibration, the system is initialized at its equilibrium point. To emphasize the impact of the grid, a stationary wind speed of $12 m s^{- 1}$ with turbulence intensity $I = 10 %$ is used. At t = 30 s, a three-phase grounding short-circuit fault occurs on the infinity bus, and the fault duration is five cycles. This causes a voltage dip, resulting in an unbalanced power between the input mechanical power and the electromagnetic power. The voltage, the torque of the induction generator, and the rotor speed are shown in Figure 12. As can be seen from this figure, the voltage dip changes the torque of the generator and affects the rotational speed through the coupled mechano-electric model. Because the main purpose of this article is to design the structural control scheme. The influence of the grid voltage on the rotation speed and the influence of the vibration are not analyzed, these can be found in the literature.^13,14 To show the control performance under electrical fault, the wind turbine is under the wind speed case 2 and a three-phase ground short-circuit fault occurs on the infinite bus as illustrated in Figure 12. The closed-loop displacement response of the in-plane vibration of the blade is shown in Figure 13. From the analysis of this case, it can be concluded that the voltage oscillations of the grid aggravate the mechanical vibrations of the blades through the coupled mechano-electrical dynamic. The control strategy proposed in this article can effectively reduce the blade vibrations. To achieve this aim, the active control force required is also shown in Figure 13. This result further verified that the proposed decentralized control strategy is feasible.

Figure 12.

System dynamic under voltage disturbance.

Figure 13.

Blade closed-loop response.

Conclusion

In this article, a decentralized control strategy was proposed for control blades and nacelle vibrations in a wind turbine. A grid-connected wind turbine model with detailed mechanical structure, including the blade and the tower lateral vibration, is established. The dynamic equation of TMD established in Fitzgerald et al.¹⁰ is extended to modeling the influence of variable speed and considering the TMD stroke constraint. The constrained decentralized $H_{\infty}$ static output feedback control is applied which the designed controllers are via solving LMI problem. Numerical simulations demonstrate the improved closed-loop response compared to LQR. The applied decentralized output feedback control can effectively suppress the in-plane vibration of the blades and the tower side-side vibration. At meantime, the stroke of TMD is constrained in the allowed stroke. Furthermore, each actuator is driven separately from the output of the corresponding sensor so that only local feedback control is achieved; this improves the system reliability.

Footnotes

Appendix 1

System matrices for second matrix differential equation

(32)

q (t) = {[\begin{matrix} q_{1, in} & d_{1} & q_{2, in} & d_{2} & q_{3, in} & d_{3} & q_{ss} & d_{n} & q_{1, out} & q_{2, out} & q_{3, out} & q_{fa} \end{matrix}]}^{T}

(33)

Q = {[\begin{matrix} Q_{1}, f_{stop, 1}, Q_{2}, f_{stop, 2}, Q_{3}, f_{stop, 3}, Q_{ss}, f_{stop, 4}, Q_{1, out}, Q_{2, out}, Q_{3, out}, Q_{fa} \end{matrix}]}^{T}

where $Q_{j} = Q_{j, in} + Q_{Ω} + Q_{g, j} - f_{stop, j}$

(34)

M (t) = [\begin{matrix} t_{1, in} & v & 0 & 0 & 0 & 0 & a_{1} & 0 & 0 & 0 & 0 & 0 \\ v & m_{bd} & 0 & 0 & 0 & 0 & b_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & t_{1, in} & v & 0 & 0 & a_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & v & m_{bd} & 0 & 0 & b_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & t_{1, in} & v & a_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & v & m_{bd} & b_{3} & 0 & 0 & 0 & 0 & 0 \\ a_{1} & b_{1} & a_{2} & b_{2} & a_{3} & b_{3} & m_{ss} & m_{nd} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & m_{nd} & m_{nd} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t_{1, out} & 0 & 0 & c_{1} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t_{1, out} & 0 & c_{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & t_{1, out} & c_{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{1} & c_{2} & c_{3} & m_{fa} \end{matrix}]

(35)

C (t) = [\begin{matrix} c_{in} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & c_{bd} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & c_{in} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c_{bd} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & c_{in} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & c_{bd} & 0 & 0 & 0 & 0 & 0 & 0 \\ d_{1} & e_{1} & d_{2} & e_{2} & d_{3} & e_{3} & c_{ss} & - c_{nd} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{nd} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{out} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{out} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{out} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{fa} \end{matrix}]

(36)

K (t) = [\begin{matrix} k_{1, in} & k_{d} & 0 & 0 & 0 & 0 & 0 & 0 & k_{io} & 0 & 0 & 0 \\ k_{d} & k_{bd} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & k_{1, in} & k_{d} & 0 & 0 & 0 & 0 & 0 & k_{io} & 0 & 0 \\ 0 & 0 & k_{d} & k_{bd} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{1, in} & k_{d} & 0 & 0 & 0 & 0 & k_{io} & 0 \\ 0 & 0 & 0 & 0 & k_{d} & k_{bd} & 0 & 0 & 0 & 0 & 0 & 0 \\ f_{1} & g_{1} & f_{2} & g_{2} & f_{3} & g_{3} & k_{ss} & - k_{nd} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & k_{nd} & 0 & 0 & 0 & 0 \\ k_{io} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & k_{b, out} & 0 & 0 & 0 \\ 0 & 0 & k_{io} & 0 & 0 & 0 & 0 & 0 & 0 & k_{b, out} & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{io} & 0 & 0 & 0 & 0 & 0 & k_{b, out} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & k_{fa} \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 \end{matrix}

\begin{matrix} 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}

Handling Editor: Ali Kazemy

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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Decentralized control of vibrations in wind turbines using multiple active tuned mass dampers with stroke constraint

Abstract

Keywords

Introduction

Electrical model of wind turbine system

Nonlinear mechanical model for the wind turbine system with active TMDs

Kinetic and potential energy

The external force action on blades and nacelle

Second-order matrix differential equation

Decentralized H∞ static output feedback control

Theorem 1

Proof

Theorem 2

Proof

Theorem 3

Remark 1

Remark 2

Remark 3

Numerical simulations

Load simulation

Simulation results

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References

Decentralized H_∞ static output feedback control