Gain-scheduled control of wind turbine exploiting inexact wind speed measurement for full operating range

Journal Pre-proof Gain-scheduled control of wind turbine exploiting inexact wind speed measurement for full operating range Mohammad J. Yarmohammadi, Arash Sadeghzadeh, Mostafa Taghizadeh PII:

S0960-1481(19)31404-1

DOI:

https://doi.org/10.1016/j.renene.2019.09.148

Reference:

RENE 12591

To appear in:

Renewable Energy

Received Date: 16 March 2019 Revised Date:

30 July 2019

Accepted Date: 15 September 2019

Please cite this article as: Yarmohammadi MJ, Sadeghzadeh A, Taghizadeh M, Gain-scheduled control of wind turbine exploiting inexact wind speed measurement for full operating range, Renewable Energy (2019), doi: https://doi.org/10.1016/j.renene.2019.09.148. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Gain-scheduled control of wind turbine exploiting inexact wind speed measurement for full operating range Mohammad J. Yarmohammadia , Arash Sadeghzadehb , Mostafa Taghizadeha a Faculty

of Mechanical Engineering, Shahid Beheshti University, Tehran, Iran of Electrical Engineering, Shahid Beheshti University, Tehran, Iran

b Faculty

Abstract This note tackles the problem of variable-speed variable-pitch wind turbine control in the whole wind speed range. Gain-scheduled (GS) control strategy is exploited to provide maximum power point tracking operation and to achieve effective pitch angle regulation. The wind turbine is modeled in the form of a linear parameter varying (LPV) system using wind speed as the scheduling parameter to cope with the varying system dynamics. The wind speed measurement errors are systematically considered in the design procedure. Modeling error is handled through employing an induced L2 -gain performance criterion. The proposed method results a common gain-scheduled controller for the whole operating region. A benchmark example illustrates the advantages of the presented method. Keywords: variable-speed variable-pitch; wind turbine; gain-scheduled output feedback; linear parameter varying; induced L2 gain performance

1. Introduction

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The significant increase of wind turbine (WT) sizes over the last decades leads to more complicated wind turbine control problems to efficiently convert the wind energy into electricity. The overriding WT control objectives are the optimization of the wind energy conversion in low wind speed and the reduction of dynamic loads on the mechanical structure in high wind speed [1]. Due to nonlinear aeroelastic properties of blades and complicated structural dynamics, the WT is usually modeled as a nonlinear time-varying system which in turn complicates the control problem. The WT operation is decomposed into different operating zones depending on the wind speed range. According to Figure 1, in low wind speed range, called partial-load region (Region 1 in Figure 1), the task of wind energy system is to maximize the extracted power. Alternatively, in high wind speed range,

Email address: [email protected] (Arash Sadeghzadeh)

Preprint submitted to Journal of LATEX Templates

November 11, 2019

Figure 1: Wind turbine operating regions

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called full-load region (Region 2 in Figure 1), the electric power and rotational speed of the rotor have to be regulated at their nominal values. Additionally, the transition between these regions (region 1 21 ) should also be appropriately addressed to attain a gentle switching behavior in order to decrease both the loss of performance and the transition load on the mechanical structure. In order to fulfill these requirements, various control strategies have previously been suggested for the WT control. In most of the available approaches, the controllers are designed around an specific operating point which directly leads to the fact that they are only valid for a narrow operating range but not for the whole operating one [2]. To take advantages of the relative maturity of the linear controller design techniques, the strategy of interpolating between some linear controllers may also be considered [3]. However, note that the interpolation of linear controllers will not even guarantee the closed-loop stability. It is worth mentioning that the WT control strategies for the entire wind speed envelope can be categorized in two distinct groups: using two separate controllers for the partial-load and full-load regions, employing a common controller for the whole operating range [4]. Obviously, the latter methodology leads to more desirable performance since it also factors in the transition between the regions methodically [5, 6]. Sophisticated gain-scheduling technique exploiting linear parameter varying (LPV) model of WT has received great attention in the recent years [4, 7, 8]. However, majority of the presented methods are based on the two-controller topology. For instance in [9], an H∞ gain-scheduled PI controller is designed for the generator torque and pitch angle control. In [2], a switching control strategy is proposed for the whole operating range. A robust LPV control method for full-load region is considered in [10] where the dynamic model uncertainties are handled. A robust and fault tolerant LPV controller in full-load region is investigated in [11] in which the model uncertainties in the aerodynamic model are taken into account in the synthesis procedure. In [4], an anti-windup LPV controller synthesis procedure is proposed to improve the transition operation between the two operating regions. The methods which

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consider the design of a common controller for the whole operating range are scarce. Nonetheless, in a study conducted in [6], an LPV controller is proposed for the partial- and full-load operating regions considering damage on mechanical subsystems. H∞ state feedback controller synthesis for the whole wind speed envelop is developed in [12]. In [5] an LPV method is presented for the pitch and generator torque control introducing appropriate weighting functions to assure a smooth transition between the operating regions. It is worthwhile to mention that in the relevant literature, it is assumed that the wind speed value is exactly available. In fact, the wind speed considered in dynamics of the WT is the mean value of the wind speed imposed on the rotor area, which is called as the effective wind speed. However, in practice, the wind speed is commonly obtained by an anemometer witch can only measure the wind speed at a single point. Thus, there always exists an uncertainty in the effective wind speed value due to this type of measurement which in turn leads to performance losses and deterioration [13, 14]. Additionally, unknown sensor bias and slight sensor failures may lead to inaccurate wind speed value. To the best of our knowledge, the wind speed value inaccuracy has not been previously considered in the WT control for the gain-scheduled controller synthesis. More recently, observes like Kamlan filter [15] and sliding mode [16]-[17] are deployed to estimate the effective wind speed. Nonetheless, this may lead to high computation time and complexity in practice. Motivated by the above drawbacks, gain-scheduled controller synthesis for a variable speed variable pitch (VS-VP) WT is tackled in this note. The design of a common controller for the whole operating range is considered. One of the contribution of the proposed method is to consider the inexactly measured wind speed as the scheduling parameter to deal with a more realistic situation. The WT is modeled in an LPV framework . An induced L2 -gain performance criterion is considered to cope with the external disturbances and modeling errors. The proposed method is implemented on a 5 MW WT benchmark to demonstrate its benefits. Numerical comparison with the available approaches clearly reveals the superiority of the new method at the whole operating range. This paper is organized as follows. In Section 2, nonlinear dynamics of 5MW wind turbine benchmark is presented. To design gain-scheduled output feedback (GSOF) controller, the nonlinear dynamic equation of the wind turbine is turned into the LPV form using Jacobian method. Section 3 presents the gain scheduled output feedback controller design procedure. Numerical studies are conducted in order to evaluate the effectiveness of the proposed GSOF control scheme in Section 4. First the presented LPV form of the wind turbine is evaluated and compared with the Fast nonlinear code [18]. Subsequently, the designed controller is implemented on the wind turbine under different wind speed uncertainty conditions. Moreover, to demonstrate the effectiveness of the proposed method, the obtained GSOF controller is compared with some of the previously available control schemes. Finally, some conclusions are drawn in the last section.

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2. Wind turbine model

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The wind turbine as a flexible mechanical structure with various aerodynamical and mechanical subsystems is complicated in modeling. It is common in the related literature of wind turbines to design the controller based on the dominant vibration modes of the drive-train and pitch mechanism [4]. In this work, a model with two subsystems is investigated which consists of a static aerodynamic subsystem and a subsystem with the first mode of the drive-train and pitch mechanism. Subsequently, to obtain an appropriate LPV model, the nonlinear model is linearized along the desired operating trajectory of the WT. In what follows, the procedure to obtain the appropriate control-oriented LPV model of the WT is provided. 2.1. Aerodynamic Subsystem The aerodynamic interaction of the wind and the blades produces a torque which rotates the rotor, and a thrust force acting on the turbine nacelle. The captured power (Pr ) and low speed shaft torque (Ta ) of the rotor depend on the mean wind speed passing through the rotor (V ), the air density (ρ), and the non-dimensional power coefficient (CP ) as follows [19]: Pr (V, β, ωr ) =

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πρR2 CP (λ, β)V 3 2

(1)

πρR3 Cq (λ, β)V 2 (2) 2 where, β is the pitch angle, ωr is the rotor rotational speed, R is the rotor radius. Note that the non-dimensional torque coefficient (Cq ) and tip-speed-ratio (TSR) (λ) are respectively given by Ta (V, β, ωr ) =

Cq (λ, β) =

CP (λ, β) λ

λ = ωr R/V 110

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(3)

(4)

Figure 2 illustrates the power coefficient (CP ) related to a typical 5 MW WT benchmark [20] as a function of the pitch angle (β) and TSR (λ). 2.2. Structural Subsystem Usually, the drive-train is modelled as a series of inertias linked by flexible shafts with linear friction. The formulation of the drive-train dynamics is given by [8] θ˙ = ωr − ωg /Ng (5) Jr ω˙r = Ta − Tsh

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Figure 2: Power coefficient (CP ) of a typical WT with respect to pitch angle and TSR (λ)

Jg ω˙ g = Tsh /Ng − Tg

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where ωg is the generator rotational speed, Jr and Jg are inertia of the rotor and the generator, respectively, Ng is the gearbox ratio of the drive-train, Tg is the generator torque and θ describes the difference between the high-speed shaft and the low speed shaft angle. Also the high speed shaft torque is given by Tsh = Ks θ + Bs ωr − Bs ωg /Ng (8) where Ks and Bs are stiffness coefficient and friction coefficient, respectively. In the VS-VP WT, the electrical machine is interfaced by a full- or partialpower converter that controls the generator torque. Because the electrical dynamics is much faster than the mechanical subsystem, without loss of generality, it can be assumed that Tg coincides with the output torque of the WT. Finally, the pitch actuator is usually modeled as a first-order dynamic system as follows 1 1 β˙ = − β + βd (9) τ τ where τ is the time constant of the pitch actuator and βd represents the pitch angle command. 2.3. Linear parameter varying model In order to obtain an appropriate LPV model, the highly nonlinear expression of the low speed shaft torque in Eq.(2) is linearized along the desired op¯ ω erating trajectory (V¯ , β, ¯ r ) as follows: T˜a = −kw ω ˜ r + kv V˜ + kβ β˜

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(10)

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where bar and tilde over the variables denote the values of the desired operating trajectory and their deviations, respectively. Thus, we have V = V¯ + V˜ ,

˜ β = β¯ + β,

ω=ω ¯ +ω ˜

(11)

The parameters kw , kv and kβ can be computed as follows: ∂Ta ¯ ω kw (V¯ , β, ¯r ) = − | ¯ ¯ ∂ωr (V ,β,¯ωr ) ∂Ta ¯ ω kv (V¯ , β, ¯r ) = | ¯ ¯ ∂V (V ,β,¯ωr ) ∂Ta ¯ ω kβ (V¯ , β, ¯r ) = | ¯ ¯ ∂β (V ,β,¯ωr )

(12)

To proceed further, note that the previously mentioned two main operating regions of WT can be identified as follows [19]: Region 140 1- The objective in this region is to maximize the energy extraction, which implies keeping the pitch angle at β0 and the TSR at the optimum value λ0 , so that the power coefficient reaches at its maximum value CP (λ0 , β0 ) = CP,max . Region 2- The objective in the full-load region is customarily to keep the WT oper145 ating at its nominal power (P = PN ) and rated rotational speed. With this aim, the electrical torque is kept constant at the rated value, while the rotational speed is regulated using the pitch actuator in order to reduce the captured power of the rotor.

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Based on the aforementioned interpretation, the desired TSR and pitch angle trajectories are illustrated in Figure 3. Taking into account the desired TSR and pitch angle trajectories, kw , kv , and kβ given by Eq.(12), can numerically be computed. Figure 4 illustrates the related parameter values for a 5 MW WT benchmark. Now considering Eqs. (2)-(12), the following state-space LPV description for the WT is obtained: ( x(t) ˙ = Ap (V¯ )x(t) + B1p (V¯ )w(t) + B2p (V¯ )u(t) G(V¯ ) =: (13) y(t) = Cp (V¯ )x(t)

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Figure 3: (a) Desired pitch angle (β), (b) desired TSR (λ) with respect to wind speed

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Figure 4: (a) The partial derivative of Ta with respect to ωr (kw ), (b) The partial derivative of Ta with respect to β (Kβ ), (c) The partial derivative of Ta with respect to V (Kv )

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where 

 0 1 −1/Ng 0  −Ks /Jr −Bs /Jr −Bs /Jr Ng 0   Ap (V¯ ) =  2 Ks /Jg Ng Bs /Jg Ng −Bs /Jg Ng 0  0 0 0 −1/τ     0 0 0 0 0 0 0 0 0 −1/Jr 0 0    + kβ 0 0 0 1/Jr  + kw  0 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0    T   0 0 0 0 0  0   1/Jr  0   Cp = 0 0  B2p =  B1p (V¯ ) = kv  −1/Jg     0  0 1 0 0 1/τ 0 1 0

(14)

˜ω ˜ T is the state vector, u = [T˜g , β˜d ]T is the control input Here x = [θ, ˜r , ω ˜ g , β] ˜T ¯ vector, w = V − V = V˜ is the effective wind speed disturbance, and y = [˜ ωg , β] is the output vector. 160

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3. Gain-scheduled output feedback Controller Design Method To effectively track the desired operating trajectories and to cope with the modeling errors and exogenous disturbances, gain-scheduled induced L2 -gain performance control methodology is utilized for the WT control. The modeling error arises from the linearizion of the original nonlinear system along the desired trajectories which are computed based on the assumption that the effective wind speed is available. Specifically, we consider reducing the impact of the deviation of the effective wind speed on the closed-loop performance of the WT. Since the wind speed value as the scheduling parameter is measured, thus, inherent inaccuracies in the available values for the wind speed are inevitable. By resort to the methods in which the inexact scheduling parameters are routinely exploited for the gain-scheduled controllers, one can compensate the wind speed measurement inaccuracy. The induced L2 -gain performance controller is designed according to the control setup shown in Figure 5. In order to track the desired power curve, attenuate high frequency components of the control efforts and the plant output, and to provide smooth transition between two operating regions, varying scaling high-pass frequency weighting functions are employed for We and Wu . Some rules of thumb are given by [5] for choosing the weighting functions. Based on the control setup in Figure 5 and using the varying scaling frequencydependent weights (they also depend on the mean wind speed value) the aug-

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We Z Wu

Reference

Tg +-

K(Θ)

βd

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G(Θ)

β

Figure 5: Augmented system for the controller design

mented LPV model can be represented as follows: x˙ = A(Θ)x + B1 (Θ)w + B2 (Θ)u z = C1 (Θ)x + D1 (Θ)w + D2 (Θ)u y = C2 (Θ)x + Dy (Θ)w

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(15)

where x ∈ Rn , u ∈ Rp , w ∈ Rm , z ∈ Rr and y ∈ Rq respectively indicate the state vector, the control input, the external input (disturbance), the performance output and the measured output of the LPV system and Θ is the time varying scheduling parameter vector. Due to the fact that the weighting functions and the state space matrices of the WT LPV model depend on the wind speed, the state space matrices of the augmented LPV system also depend on the scheduling parameters. To address the gain-scheduled controller design for an LPV system with polynomially parameter dependency on the scheduling parameters, the presented method by [21] with some slight modifications to cope with the arbitrarily fast scheduling parameter variations is employed. In [21] the problem of gainscheduled output feedback control design for continuous-time LPV systems is investigated where the scheduling parameters of the LPV system are supposed to be measured in real-time with absolute uncertainties. The control design problem is formulated in terms of solutions to a set of parameter-dependent

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Linear Matrix Inequalities (LMIs) where an upper bound on the induced L2 gain performance criterion is minimized. In this paper the goal is to design a GSOF controller as follows: ( ˆ c + Bc (Θ)y ˆ x˙ c =Ac (Θ)x ˆ K(Θ) := (16) ˆ c + Dc (Θ)y ˆ u =Cc (Θ)x 200

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ˆ , [θˆ1 , θˆ2 , · · · , θˆk ]T is the measured value of the real scheduling pawhere Θ rameter vector and may be different from the actual scheduling parameter, xc ∈ Rn is the controller state vector and Ac , Bc , Cc and Dc are the to-bedesigned parameter-dependent controller matrices. In order to handle the measurement inaccuracies of the scheduling parameters, θˆi is assumed to be sum of the exact value (θ¯i ) and the absolute uncertainty of the measured scheduling parameter (δi ). To proceed further, let us define Ξi := {θˆi | θˆi = θi + δi , −θi ≤ θˆi ≤ θi , −δ i ≤ δi ≤ δ i } Λtii := {(θi , θˆi )| − θi ≤ θi ≤ θi , θˆi ∈ Ξi } n o ˆ ∈ Rk × Rk | (θi , θˆi ) ∈ Λti , i = 1, · · · , k Λt := (Θ, Θ) i

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where θi , θi , δ i , and δ i are all a priori known positive numbers. The admissible region for Λtii is depicted in Figure 6. The considered control objective is to design a controller such that stabilizes the closed-loop system and satisfies the induced L2 -gain performance constrain ˆ denoted by over all admissible trajectories of Θ and Θ sup||w||2 6=0,w∈L2

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(17)

||z||2 <γ ||w||2

(18)

In the case that either B2 or C2 , given in Eq.(15), are not parameterdependent, the required gain-scheduled controller can be obtained by the theorem presented in Appendix A. Where parameter-dependent matrices of the GSOF controller in Eq. 16 are obtained as follows:  ˆ = L1 (Θ) ˆ − Y Ap (Θ)X( ˆ ˆ − Y B2 (Θ)L ˆ 3 (Θ) ˆ − L2 (Θ)C ˆ 2 (Θ)X( ˆ ˆ Ac (Θ) Θ) Θ)  −1 ˆ 4 (Θ)C ˆ 2 (Θ)X( ˆ ˆ ˆ − Y X(Θ) ˆ +Y B2 (Θ)L Θ) S(Θ) ˆ =L2 (Θ) ˆ − Y B2 (Θ)L ˆ 4 (Θ) ˆ Bc (Θ)   −1 ˆ = L3 (Θ) ˆ − L4 (Θ)C ˆ 2 (Θ)X( ˆ ˆ ˆ − Y X(Θ) ˆ Cc (Θ) Θ) S(Θ) ˆ =L4 (Θ) ˆ Dc (Θ)

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(19)

θˆi

(θi − δ i , θi )

(θi , θi ) (θi , θi − δ i )

θi (−θi , −θi + δ i ) (−θi , −θi ) (−θi + δ i , −θi )

Figure 6: Admissible region for the pair (θi , θˆi )

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ˆ B2 (Θ), ˆ and C2 (Θ) ˆ are respectively obtained by replacing Θ by Θ ˆ where Ap (Θ), in Ap (Θ), B2 (Θ), and C2 (Θ). For the two cases related to s = 1 or s = 2, one should search for the optimal values of scalar parameters α1 and α2 to obtain a GSOF controller which assures as far as possible the least guaranteed cost value γ on the induced L2 -gain performance. Remark 1. In a special case that the exact values of the scheduling parameters ˆ Eq.(A.2) presented in Appendix A is changed to are available, i.e. Θ = Θ, ˆ |ˆ > 0 Ψ1 (θ, θ) θ=θ 4. Simulation results for a benchmark problem

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In this section numerical studies are conducted in order to evaluate the effectiveness of the proposed GSOF controller design method. For the simulations, we consider NREL 5 MW, offshore WT benchmark. First of all, an LPV model given in Eq.(13) is computed for this WT. To evaluate the resulted LPV dynamic model, the open-loop responses of the LPV model are compared with that of FAST nonlinear code[18]. Then two scenarios are considered for the controller synthesis and the results are compared with the available approaches. 4.1. LPV model validation Using the proposed method in Section 2.3, an LPV model given in Eq.(13) is derived for the considered WT benchmark. Subsequently, the open-loop re12

Table 1: Numerical values of the benchmark 5M W wind turbine

symbol S R N Ng PN ωN Bls Bhs Kdt λopt Jg Jr

Parameter Name Sweep Area of Rotor Radius of Rotor Number of Blades Gear Ratio Nominal Power Rotor nominal speed Low speed shaft damping High speed shaft damping Drivetrain equivalent stiffness Optimal Tip-speed ratio Generator inertia Rotor inertia

Value 12445.3 m2 63 m 3 97 5.297 M W 12.1 rpm 1000 N.m/(rad/s) 1 N.m/(rad/s) 867,637,000 N.m/rad 14.1 534.2 kgm2 38,759,227 kgm2

sponses of this model is compared with the related FAST code (see Figure 7). The wind turbine’s parameters are listed in Table.1. Pitch Angle Command Generator Torque command

Rotor Speed

FAST Block Diagram

Wind Speed

High Speed Shaft Torque

Rotor Speed

LPV Model

High Speed Shaft Torque

Figure 7: Openloop block diagram wind turbine

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The open-loop responses for the obtained LPV model are computed for the mean wind speed step from 14m/s to 18m/s (full-load region) considering pitch angle β = 12◦ and the generator torque Tg = 41kN.m. Moreover, by taking into account a random sin-shaped wind speed for the partial-load region and for the case that the pitch angle β = 12◦ and generator torque Tg = 12kN.m are considered, the open-loop responses are calculated (see block diagram in Figure 7). In Figure 8 the obtained results are compared with those of the FAST code. 13

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Figure 8: Comparisons of the open-loop responses related to the LPV model and the FAST code in partial- and full-load regions. Right column figures: partial-load region results. Left column figures: full-load region results. Solid line: LPV model results. Dashed line: FAST code results

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It is readily observed that the full-load and the partial-load responses of the LPV model have a high compatibility with FAST code. It should be emphasized that in the FAST code simulation the effect of tower deflection on Ta is taken into account where it is ignored in Eq.(13). The discrepancy between the obtained results by the LPV model and FAST code is to some extent attributed to this fact. 4.2. Gain-scheduled controller design Based on the presented method in Section 3, a gain-scheduled controller is designed for the 5 MW benchmark. Inspired by the method presented in [5], the following frequency-dependent varying scaling parametric weighting functions are employed for the gain-scheduled controller design. # " (5s+1) 0 (0.1s+1) We = (5s+1) 0 (0.1s+1) (20) " # (5s+1) k1 (V¯ ) (0.1s+1) 0 Wu = 0 k2 (V¯ ) (5s+1) (0.1s+1)

where

1000 ) 1 + e(−10(V¯ −12)) (21) −10 k2 (V¯ ) =10 + ( ) ¯ 1 + e(−10(V −12)) For the reference, the state space parameter-dependent matrices of the general LPV model in Eq.(15) are given in Appendix B. Note that the matrices have polynomial parameter-dependency on the scheduling parameters Θ = [θ1 θ2 θ3 θ4 θ5 ]. Bear in mind that the scheduling parameters are nonlinear functions of the effective wind speed (see Eq.(21) and Figure 4), i.e. we have k1 (V¯ ) =10 + (

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θ1 = kw (V¯ ), θ2 = kv (V¯ ), θ3 = kβ (V¯ ), θ4 = k1 (V¯ ), θ5 = k2 (V¯ )

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To coincide with the definition of scheduling parameter ranges in Eq.(17), define θ10 , · · · , θ50 the mean values of θ1 , · · · , θ5 and replace θ1 → θ10 + θ1 , Then we have

θ2 → θ20 + θ2 ,

θ3 → θ30 + θ3 ,

θ4 → θ40 + θ4 ,

θ1 = θ1 = 1.6 × 107 ,

θ10 = 2 × 107

θ2 = θ2 = 0.5 × 106 ,

θ20 = 1 × 106

θ3 = θ3 = 7.5 × 105 ,

θ30 = −7.5 × 105

θ4 = θ4 = 505,

θ40 = 505

θ5 = θ5 = 5,

θ50 = 5

θ5 → θ50 + θ5

The aforementioned parameter values are related to 5 ≤ V¯ ≤ 30 m/s2 .

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It should be emphasized that in practice the wind speed V is measured using an appropriate sensor. Thus, due to the measurement errors, the actual values of V¯ (the wind speed around which the LPV model is computed) are not exactly available. In the proposed method, we can systematically take into account these inaccuracies. It is assumed that the measured value of the wind speed, denoted by Vˆ , is related to V¯ as follows: Vˆ = V¯ (1 + ξ),

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|ξ|≤ ξ

where ξ represents the wind speed measurement error limited by a priori known bound ξ. Taking into account that the scheduling parameters are nonlinear functions of the mean wind speed, the related uncertainties on the scheduling parameters can be computed. For example, considering ξ = 0.05 results the following lower and upper bounds for the scheduling parameter inaccuracies: δ 1 = 8 × 105 ,

δ 1 = 8.4 × 105

δ 2 = 2.1 × 104 , δ 2 = 2.6 × 104 δ 3 = 3.7 × 103 , δ 3 = 4.1 × 103 δ 4 = δ 4 = 25 δ 5 = δ 5 = 0.25

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The proposed theorem in Appendix A is employed for the controller design, where we consider all parameter-dependent matrices to be affine with respect to the scheduling parameters. The optimization problems are solved by YALMIP [22] interface for the LMI solver SeDuMi [23]. Since the LMI conditions are polynomially parameter-dependent on the scheduling parameters, a specialized parser ROLMIP (Robust LMI Parser) [24] is employed for the controller design to obtain tractable finite-dimensional LMI problems. At this point, let us consider the gain-scheduled controller synthesis for the WT benchmark considering ξ = 0.05. In order to obtain a controller with the least possible γ, the search for the scalar values α1 and α2 is performed with 8 points linearly gridded over a logarithmic scale in [10−8 10−1 ]. Both of the cases s = 1 and s = 2 in the presented theorem in Appendix A are considered where the case s = 1 yields lower values for γ. The results are given in Table 2. The obtained performance by this controller is investigated in the next section. 4.2.1. Evaluating the wind speed measurement inaccuracies on the closed-loop performance To reveal the impact of the mean wind speed measurement inaccuracy on the closed-loop performance of the WT control system, we design gain-scheduled controllers for three different cases ξ = 0.05, ξ = 0.1, and ξ = 0.2. The guaranteed cost values γ obtained by the theorem presented in Appendix A is given in Table 3. Subsequently, a time-domain simulation is performed using the closedloop control scheme shown in Figure 9. The associated closed-loop performances are illustrated in Figure 10. 16

Table 2: The resulted γ for different values of α1 and α2 for the case s = 1 (NaN implies that the optimization problem is not feasible).

α1 α2

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1

2.199 3.726 1.161 12.925 15.16 63.75 NaN NaN

2.268 4.125 0.912 12.598 18.354 71.42 NaN NaN

2.357 5.154 1.274 11.111 20.812 76.11 NaN NaN

2.635 5.332 1.349 9.129 32.15 89.12 NaN NaN

3.415 5.621 2.011 7.548 36.316 NaN NaN NaN

2.046 9.798 2.976 5.421 NaN NaN NaN NaN

NaN NaN NaN NaN NaN NaN NaN NaN

NaN NaN NaN NaN NaN NaN NaN NaN

Table 3: RMSE values related to different wind speed inaccuracies

Wind speed inaccuracy (ξ) γ RMSE

Pitch angle (β) Generator rotational speed (wg )

0.05 0.912 0.268 0.861

0.1 1.12 0.864 2.521

0.2 2.09 0.968 3.64

For the quantitative evaluation of the closed-loop performance, let us define the Root Mean Squared Error (RMSE) criterion as follows: RMSE =

305

310

1 T

Z

!1/2

T

(y − yref )2 dt

0

where y represents either β or ωg ; yref denotes the corresponding desired values; T is the simulation time. The defined RMSE values represent the tracking errors between the desired values and the actual ones in the time range of T = 0 to T = 40. The RMSE values related to three different wind speed uncertainties are given in Table 3. It is easily seen that the proposed GSOF controllers result good tracking performance in the presence of the wind speed measurement inaccuracies. From Figure 10, one can see that the wind speed uncertainty has less impact on the pitch dynamics in comparison with the output power. That is due to the fact that the wind speed value (V ) does not directly appear in Eq. 9 representing pitch actuator dynamics.

17

315

320

325

4.2.2. Comparison with the available approaches For the comparison purposes with the available methods, the designed gainscheduled controller in this paper is compared with both the anti-windup controller in [4] and the proposed gain-scheduled controller in [5]. The controllers are designed and implemented on the 5 MW WT benchmark under the same wind speed conditions. To have a fair comparison, since the proposed methods in [4, 5] only investigate the case that the exact scheduling parameters are available, we also design a gain-scheduled controller by considering ξ = 0 using Remark1. Time domain simulations are performed, and the obtained results are depicted in Figure 11. For the quantitative evaluation, RMSE values related to different control strategies are computed and reported in Table.4. One can readily see that the proposed gain-scheduled controller leads to better closed-loop performance in the whole operating range. To evaluate the transition perfor-

𝑉

Wind Speed Measurement Sensor

Sensor Uncertainty

𝑉

𝑉

Scheduling Parameter

Desired Input

𝜃 𝜔𝑔

𝑉

𝜔𝑔

-

Desired Output

𝛽

𝑇𝑔

𝑇𝑔

𝛽𝑑

𝛽

Figure 9: Closed-loop control scheme

18

+

𝑇𝑔

𝜔𝑔

+

GSOF Controller

+

𝛽𝑑

𝛽

𝛽𝑑 Wind Turbine Nonlinear Model

1300 ω g (rpm)

Wind speed (m/s)

20 15 10

1200 1100 1000

0

5

10

15

20

25

30

35

40

0

5

10

15

Time(Seconds) (a)

β d (deg)

β (deg)

30

35

40

25

30

35

40

25

30

35

40

15

10 5 0 0

5

10

15

20

25

30

35

×104

10 5 0

40

0

Time(Seconds) (c)

5

10

15

4.2

20

Time(Seconds)

×106 Power(W)

T g (N.m)

25

(b)

15

4.4

20

Time(Seconds)

(d)

5 4.5 4

4 0

5

10

15

20

25

30

35

40

0

5

10

15

20

Time(Seconds)

Time(Seconds)

(e)

(f)

Figure 10: Closed-loop response of WT for several wind speed uncertainty (a) wind speed (b) generator rotational speed (c) pitch angle (d) pitch angle command (e) generator torque (f) output power. Dotted line: ξ = 0.05 uncertainty; Dashed line: ξ = 0.1 uncertainty; Solid line: ξ = 0.2 uncertainty

19

1250 1200 ω g(rpm)

Wind Speed (m/s)

20

15

1150 1100 1050

10

1000 0

5

10

15

20

25

30

35

40

0

5

10

15

Time(Seconds) (a)

20

30

35

40

15 β d (deg)

β (deg)

25

(b)

20

15 10

10 5

5

0

0 0

5

4.4

10

15

20

25

30

35

0

40

5

10

15

20

25

30

25

30

35

40

Time(Seconds)

Time(Seconds)

×104

(c)

(d)

×106

4.3

5 Power (W)

T g (N.m)

20

Time(Seconds)

4.2 4.1

4.5 4

4 0

5

10

15

20

25

30

35

40

3.5 0

Time(Seconds)

5

10

15

20

Time(Seconds) (f)

(e)

Figure 11: Time domain closed-loop response of wind turbine for several control strategies (a) wind speed (b) generator rotational speed (c) pitch angle (d) pitch angle command (e) generator torque (f) output power. Dashed line: GSOF controller; Solid line: Method in Ref. [5]; Dotted line: Method in Ref. [4]

20

35

40

Table 4: RMSE comparison of tracking error for different control strategies

Method Pitch angle (β) Generator rotational speed (wg )

330

335

Proposed method 0.217 1.09

[4] 1.641 4.448

[5] 3.072 6.821

mance of the controllers between partial- and full-load regions, the pitch angle and the generator rotational speed are depicted versus the wind speed in Figure 12 using the proposed controller, the anti-windup controller [4], and the gain-scheduled controller presented in [5]. Obviously, the results indicate the superiority of the proposed method in comparison with the available ones. Note that contrary to the available methods in which the gain-scheduled controller is designed exploiting a griding method, in the proposed method the stability and performance of the closed-loop system is analytically guaranteed, in addition to more precisely tracking the desired outputs. 5. Conclusion

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This paper presents a novel method for designing a common gain-scheduled controller covering both partial- and full-load operating conditions that directly leads to a smooth transition between different operation regions. Wind speed measurement inaccuracy is systematically considered in the controller synthesis procedure. Stability and performance of the closed-loop system is analytically guaranteed. Due to employing an induced L2 -gain performance criterion, the effect of the modeling errors and disturbances is markedly reduced on the system performance. Numerical studies reveal the superiority of the new presented method. References [1] F.D.Bianchi, R.J.Mantz, C.F.Christiansen, Control of variable speed wind turbines by LPV gain scheduling, Wind Energy 7 (1) (2004) 1–8.

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[2] F. Lescher, J. Y. Zhao, P. Borne, Switching LPV controllers for a variable speed pitch regulated wind turbine, The Proceedings of the Multiconference on Computational Engineering in Systems Applications, 2007, pp. 1334– 1340. [3] S. Wang, P. J. Seiler, Gain scheduled active power control for wind turbines, 32 ASME Wind Energy Symposium (2014) 1–15. [4] F. A. Inthamoussou, H. D. Battista, R. J. Mantz, LPV-based active power control of wind turbines covering the complete wind speed range, Renewable Energy 99 (2016) 996–1007.

21

1200

ω g (rpm)

1150

1100

1050 Partial Load Region

1000

Full Load Region

950 9

10

11

12

13

14 Wind Speed(m/s)

12

13

14 Wind Speed(m/s)

15

16

17

18

19

16 14 12

Partial Load Region

β (deg)

10

Full Load Region

8 6 4 2 0 9

10

11

15

16

Figure 12: Closed-loop wind speed domain response of wind turbine for several control strategies. Dashed-dot line: GSOF controller; Solid line: Method in Ref. [5]; Dotted line: Method in Ref. [4]; Dashed line: desired trajectory

22

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[5] G. Cao, K. Grigoriadis, Y. D. Nyanteh, LPV control for the full region operation of a wind turbine integrated with synchronous generator, Hindawi Publishing Corporation the Scientific World Journal 2015. [6] K. Z. Ostegaard, J. Stoustrup, P.Brath, Linear parameter varying control of wind turbines covering both partial load and full load conditions, International Journal of Robust and Nonlinear Control 19 (2008) 92–116.

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[7] F. A. Inthamoussou, F. D. Bianchi, H. D. Battista, R. J. Mantz, LPV wind turbine control with anti-windup features covering the complete wind speed range, IEEE Transactions on Energy Conversion 29 (1) (2014) 259–266. [8] F. A. Shirazi, K. M. Grigoriadis, D.Viassolo, Wind turbine integrated structural and LPV control design for improved closed-loop performance, International Journal of Control 85 (8) (2012) 1178–1196. [9] E. Muhando, T. Senjyu, A. Uehara, T. Funabashi, Gain-scheduled H∞ control for WECS via LMI techniques and parametrically dependent feedback part II: Controller design and implementation, IEEE Transactions on Industrial Electronics 58 (1) (2010) 57–65.

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[10] F. D. Bianchi, H. D. Battista, R. J.Mantz, Robust multi-variable gain scheduled control of wind turbines for variable power production, International Journal of Systems Control 1 (2010) 103–112. [11] C. Sloth, T. Esbensen, J. Stoustrup, Robust and fault-tolerant linear parameter-varying control of wind turbines, Mechatronics 21 (4) (2011) 645659. [12] X.-J. Yao, C.-C. Guo, Y. Li, LPV H-infinity controller design for variablepitch variable-speed wind turbine, in: 2009 IEEE 6th International Power Electronics and Motion Control Conference, 2009, pp. 2222–2227.

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[13] W. Meng, Q. Yang, Y. Ying, Y. Sun, Z. Yang, Y. Sun., Adaptive power capture control of variable-speed wind energy conversion systems with guaranteed transient and steady-state performance, IEEE Transactions on Energy Conversion 28 (3) (2013) 716–725. [14] B. Boukhezzar, H. Siguerdidjane, H. M.Hand, Nonlinear control of variable speed wind turbines for generator torque limiting and power optimization, Journal of Solar Energy Engineering 128 (4) (2006) 516–530. [15] E. Gauterin, P. Kammerer, M. Khn, H. Schulte, Effective wind speed estimation: Comparison between Kalman Filter and TakagiSugeno observer techniques, ISA Transactions 62 (2016) 60–72.

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[16] F. Golnary, H. Moradi, Dynamic modelling and design of various robust sliding mode controls for the wind turbine with estimation of wind speed, Applied Mathematical Modelling 65 (2019) 566–585.

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[17] F. Golnary, H. Moradi, Design and comparison of quasi continuous sliding mode control with feedback linearization for a large scale wind turbine with wind speed estimation, Renewable Energy 127 (2018) 495–508. 400

[18] J. M. Jonkman, M. L, B. Jr., FAST Users Guide, Tech. rep., Technical Report, NREL/EL-500-38230 (2005). [19] F. D. Bianchi, H. D. Battista, R. J. Mantz, Wind turbine control systems: Principles, modelling and gain scheduling design, Springer-Verlag, London, UK, 2006.

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[20] J. Jonkman, S. Butterfield, W. Musial, G. Scott, Definition of a 5-MW reference wind turbine for offshore system development, Tech. rep., National Renewable Energy Laboratory (2009). [21] A. Sadeghzadeh, Gainscheduled continuous-time control using polytopebounded inexact scheduling parameters, International Journal of Robust and Nonlinear Control 28 (2018) 5557–5574. [22] J. L¨ ofberg, YALMIP: A toolbox for modeling and optimization in MATLAB, in: CACSD Conference, 2004. URL http://control.ee.ethz.ch/~joloef/yalmip.php

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[23] J. F. Sturm, Using SeDuMi 1.02, a Matlab toolbox for optimization over symmetric cones, Optimization Methods and Software 11 (1-4) (1999) 625– 653. [24] C. M. Agulhari, R. C. L. F. de Oliveira, P. L. D. Peres, Robust LMI Parser: a computational package to construct LMI conditions for uncertain systems, XIX Brazilian Conference on Automation (CBA 2012), Campina Grande, PB, Brazil, 2012, pp. 2298–2305.

Appendix A. Gain scheduled output feedback controller design theorem

425

430

Theorem. [21] For a positive value γ, if there exist positive scalers α1 , α2 ˆ ∈ Rn×n , L2 (Θ) ˆ ∈ Rn×r , L3 (Θ) ˆ ∈ and parameter dependent matrices L1 (Θ) p×n p×r n×n n×n ˆ ˆ ˆ R , L4 (Θ) ∈ R , X(Θ) ∈ R , S(Θ) ∈ R and matrices Y ∈ Rn×n , n×n n P2 ∈ R and symmetric matrices P1 ∈ S , P3 ∈ Sn such that Eq.(A.1) ˆ ∈ Λt , then the controller and Eq.(A.2) satisfy for all combinations of (Θ, Θ) ˆ K(Θ) whose the state space matrices are given in Eq.(19) makes the closedloop system exponentially stable and satisfies the induced L2 -gain performance ˆ constraint Eq.(18) for all admissible trajectories of θ and θ. 

P1 P2

? P3

24

 > 0,

(A.1)

"

ˆ Ψ1 (Θ, Θ) ˆ − α1 Ψ2 (Θ, Θ) ˆ T Ψ3 (Θ, Θ)

#

?

> 0,

2α1 I

(A.2)

where Ψ2 and Ψ3 are ˆ = Ψ2 (Θ, Θ)

s Y

Φi ,

3 Y

ˆ = Ψ3 (Θ, Θ)

i=1

Φj

(A.3)

j=s+1

The real positive number s can be either 1 or 2. " Φ1 =

0 T

T

ˆ B2 (Θ) L4 (Θ)

YT ˆ T L2 (Θ)

0

0n×(2n+m)

T

#T

T

ˆ D2 (Θ) L4 (Θ) 0r×(2n+m)   ˆ ˆ Ap (Θ) − Ap (Θ) B2 (Θ) − B2 (Θ) Φ2 = , (A.4) ˆ C2 (Θ) − C2 (Θ) 0   ˆ ˆ −X(Θ) 0 0n×q 0 −α2 X(Θ) 0 Φ3 = ˆ ˆ 2 (Θ) 0p×q L4 (Θ)D ˆ y (Θ) −α2 L3 (Θ) ˆ ˆ 2 (Θ) −L3 (Θ) −L4 (Θ)C −α2 L4 (Θ)C and also ˆ = Ψ1 (Θ, Θ)



Φ4 Φ5

? Φ6

 ,

where Φ4 = [φ4ij ] , Φ5 = [Φ5ij ] ,   ˆ α2 hX(Θ)i ? Φ6 = ˆ + α2 I α2 hY i α2 S(Θ) with Φ4ij and Φ5ij as follows ˆ + B2 (Θ)L3 (Θ)i ˆ φ411 = −hAp (Θ)X(Θ) ˆ − Ap (Θ)0 − C2 (Θ)0 L4 (Θ) ˆ 0 B2 (Θ)0 φ421 = φ412 T = −L1 (Θ) ˆ 2 (Θ)i φ422 = −hY Ap (Θ) + L2 (Θ)C ˆ − D2 (Θ)L3 (Θ) ˆ φ431 = φ413 T = −C1 (Θ)X(Θ) ˆ 2 (Θ) φ432 = φ423 T = −C1 (Θ) − D2 (Θ)L4 (Θ)C ˆ 0 B2 (Θ)0 φ441 = φ414 T = B1 (Θ)0 + Dy (Θ)0 L4 (Θ) ˆ 0 φ442 = φ424 T = B1 (Θ)0 Y 0 + Dy (Θ)0 L2 (Θ) ˆ 0 D2 (Θ)0 φ443 = φ434 T = D1 (Θ)0 + Dy (Θ)0 L4 (Θ) φ444 = γ 2 I

φ433 = I

25

(A.5)

ˆ − P1 − α2 X(Θ) ˆ 0 Ap (Θ)0 − α2 L3 (Θ) ˆ 0 B2 (Θ)0 φ511 = X(Θ) ˆ 0 φ512 = I − P2 − α2 L1 (Θ) ˆ 0 C1 (Θ)0 − α2 L3 (Θ) ˆ 0 D2 (Θ)0 φ513 = −α2 X(Θ) ˆ − P20 − α2 Ap (Θ)0 − α2 C2 (Θ)0 L4 (Θ) ˆ 0 B2 (Θ)0 φ521 = S(Θ) ˆ 0 φ522 = Y − P3 − α2 Ap (Θ)0 Y 0 − α2 C2 (Θ)0 L2 (Θ) ˆ 0 D2 (Θ)0 φ523 = −α2 C1 (Θ)0 − α2 C2 (Θ)0 L4 (Θ) φ514 = 0

435

φ524 = 0

Appendix B. State space matrices of an LPV model for a 5 MW WT benchmark



Ap (Θ) −Be Cp 0

       C1 (Θ) D1 (Θ) D2 (Θ)  =     −De Cp   C2 (Θ) Dy (Θ) 0  Cp 

Ap (Θ) B1 (Θ) B2 (Θ)



0 Ae 0 Ce 0 0

0 0 Au

B1p (Θ) 0 0

0 Cu (Θ)

0 0

0

B2p 0 Bu



      0  Du (Θ)  

0

where    0 1 −1/Ng 0 0 0 0 0   −Ks /Jr  −Bs /Jr −Bs /Jr Ng 0   + θ1 0 −1/Jr 0 0 Ap (Θ) =  Ks /Jg Ng Bs /Jg Ng −Bs /Jg Ng2 0 0  0 0 0 0 0 0 0 0 0 0 −1/τ   0 0 0 0 0 0 0 1/Jr   + θ3  0 0 0 0  0 0 0 0    T   0 0 0 0 0  0   1/Jr  0   Cp = 0 0  B2p =  B1p (Θ) = θ2  −1/Jg     0  0 1 0 0 1/τ 0 1 0        −10 0 1 0 −490 0 50 Ae = Au = Be = Bu = Ce = De = 0 −10 0 1 0 −490 0     −490θ4 0 50θ4 0 Cu (Θ) = Du (Θ) = 0 −490θ5 0 50θ5 

26

 0 50

Highlights:

• • • •

Gain-scheduled controller synthesis using a linear parameter varying model Employing inexact wind speed measurement as the scheduling parameter A common controller for both partial- and full-load operating regions Stability and performance are analytically guaranteed