Robust Active Disturbance Rejection Control Approach to Maximize Energy Capture in Variable-Speed Wind Turbines

Abstract

This paper proposes an alternative robust observer-based linear control technique to maximize energy capture in a 4.8 MW horizontal-axis variable-speed wind turbine. The proposed strategy uses a generalized proportional integral (GPI) observer to reconstruct the aerodynamic torque in order to obtain a generator speed optimal trajectory. Then, a robust GPI observer-based controller supported by an active disturbance rejection (ADR) approach allows asymptotic tracking of the generator speed optimal trajectory. The proposed methodology controls the power coefficient, via the generator angular speed, towards an optimum point at which power coefficient is maximum. Several simulations (including an actuator fault) are performed on a 4.8 MW wind turbine benchmark model in order to validate the proposed control strategy and to compare it to a classical controller. Simulation and validation results show that the proposed control strategy is effective in terms of power capture and robustness.

1. Introduction

The use of wind energy has a history of over a hundred years. Its applications include agriculture, milling, water pumping, and power production. In the 1970s, this technology started developing as an experimental technology. Nowadays, the conversion of wind energy into electrical energy by wind turbines is a mature technology that exhibits the highest growth rates among the renewable energy sources [1] and can be considered as the most promising option for replacing a significant part of the electricity produced by conventional sources [2].

The main objective of wind turbines is to convert efficiently wind energy into electrical power. There are wind turbines available in a different number of configurations (vertical axis, horizontal axis, fixed speed, variable speed, etc.). The most used type for large-scale power production is the variable-speed horizontal-axis wind turbine (HAWT) with a two- or three-blade rotor [3]. HAWTs are commonly equipped with blade-pitch actuators, generator torque control and many sensors for use in real time control [4]. HAWTs operate in different control regions, and the region considered in the present work is a low-to-medium wind speeds operation (also called partial load operation or operation in region 2) where the main objective is to extract the maximum power from the wind.

Modern wind turbines are machines that require big efforts when maximizing wind energy capture, not only because of their highly nonlinear aerodynamics, but also because of the high efficiency required even when model uncertainties, external perturbations, or system faults are present. As a consequence, the efficiency of both power capture and power generation is strongly dependent on the selected control method [5]. This situation provides a motivation to consider new alternative control techniques that improve the performance of HAWT without any structural change.

A large number of control schemes to find the best way of solving the energy capture maximization problem for wind turbines at low-to-medium wind speeds have been proposed (see, e.g., [6–12]). The control techniques range from standard torque control [6], disturbance tracking control [7], maximum power point tracking [8], and aerodynamic torque feedforward [9] to complex nonlinear strategies [5, 10–12]. Most of these techniques deal with the wind turbine complexity using linearization techniques or nonlinear control. Following a different approach, some of the active-disturbance-rejection- (ADR-) based techniques allow linear control solutions for some class of uncertain complex nonlinear systems and could offer a linear, simpler, and robust solution to the wind energy capture maximization problem. This is the case of the ADR philosophy-based technique called generalized proportional integral (GPI) Control [13] and its GPI observer-based control extensions [14, 15].

Generalized proportional integral (GPI) control technique was started in 2000 by Fliess et al. [13, 16] and involves in its design the active rejection of disturbances. The dual counterpart of the generalized proportional integral controller, called GPI observer, was introduced in [15], in the context of sliding mode observers for flexible robotics systems. The nonsliding version appears in [14] applied to chaotic systems synchronization. The GPI control strategies have been adapted, extended, and applied successfully in areas other than wind energy, such as induction motor control [17], chaotic systems control [18], and power converters control [19]. Therefore, it is interesting to adapt, evaluate, and determine the scope of this control method in partial load wind turbine operation.

GPI observer-based control of nonlinear uncertain systems is very much related to methodologies known as disturbance accommodation control (DAC) and active disturbance rejection control (ADRC). These approaches deal with the problem of cancelling, from the controller’s actions, endogenous and exogenous unknown additive disturbance inputs affecting the system. Perturbation effects are made available via a suitable linear or nonlinear estimation. The reader is invited to read the works by Johnson [20], Han [21], and Gao et al. [22, 23].

This work presents an alternative linear control technique based on GPI observers to maximize wind energy capture in variable-speed wind turbines operating at partial load. The proposed strategy uses a GPI observer to reconstruct the aerodynamic torque in order to provide a generator speed optimal trajectory to a robust GPI observer-based controller that regulates the power coefficient, via the generator torque, towards an optimum point at which the power coefficient is maximum. It is expected that the proposed GPI observer-based control technique adds robustness to the system and solves the control problem through linear active estimation and rejection of nonlinearities and perturbations of the wind energy conversion system (WECS).

This paper is organized as follows. In Section 2, a wind turbine dynamic model for control purposes is presented. Section 3 formulates the control problem and presents the proposed methodology to solve it. Section 4 describes the benchmark and the tests model used to validate the proposed methodology. Section 5 presents the simulations and validations results of the proposed control strategy. Finally, Section 6 contains the conclusions and suggestions for further work.

2. Wind Turbine Model

Consider a wind turbine represented by a two-mass mechanical system as shown in Figure 1(a), where the aerodynamic power captured by the rotor is given by [24] 𝑃_𝑎(𝑡)=12𝜌𝜋𝑅²𝐶_𝑝𝜆,𝛽𝜐(𝑡)³, (1) where𝑅is the rotor radius,𝜌is the air density, 𝜐(𝑡) is the wind velocity, and 𝐶_𝑝(𝜆,𝛽) is the power coefficient which denotes the aerodynamic efficiency of the wind energy conversion system. The power coefficient curve 𝐶_𝑝(𝜆,𝛽) of the wind turbine considered in this work is shown in Figure 1(b). This curve was taken from the benchmark model published in [25]. The power coefficient 𝐶_𝑝(𝜆,𝛽) depends on both the blade pitch angle𝛽and the tip-speed ratio 𝜆. The latter is defined as 𝜔𝜆=^𝑟(𝑡)𝑅𝜐(𝑡), (2) where 𝜔_𝑟(𝑡) is the rotor angular speed. The aerodynamic power 𝑃_𝑎(𝑡) can also be expressed in function of both the aerodynamic torque 𝜏_𝑎(𝑡) and the rotor speed 𝜔_𝑟(𝑡) as follows [24]: 𝑃_𝑎(𝑡)=𝜔_𝑟(𝑡)𝜏_𝑎(𝑡). (3) The aerodynamic torque 𝜏_𝑎(𝑡) is given by 𝜏_𝑎(𝑡)=12𝜌𝜋𝑅³𝐶_𝑞𝜆,𝛽𝜐(𝑡)², (4) where 𝐶_𝑞 is the torque coefficient and is defined as follows: 𝐶_𝑞=𝐶𝜆,𝛽^𝑝𝜆,𝛽𝜆. (5) The mechanical system of the wind turbine is modeled using Newton’s laws, and the following system is derived [25]: 𝑑𝐱𝑑𝑡^𝐰𝐭(𝑡)=𝐀_𝐰𝐭𝐱_𝐰𝐭(𝑡)+𝐁_𝐰𝐭𝜏_𝑔(𝑡)+𝐅_𝐰𝐭𝜏_𝑎(𝑡),𝑦(𝑡)=𝐂_𝐰𝐭𝐱_𝐰𝐭(𝑡) (6) with 𝐱_𝐰𝐭(𝑡)=𝜔_𝑟(𝑡)𝜔_𝑔(𝑡)𝜃_Δ(𝑡),𝐀_𝐰𝐭=−𝐵_𝑑𝑡−𝐵_𝑟𝐽_𝑟𝐵_𝑑𝑡𝐽_𝑟𝑁_𝑔−𝐾_𝑑𝑡𝐽_𝑟𝐵_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔−𝐵_𝑔+𝐵_𝑑𝑡𝜂_𝑑𝑡/𝑁²_𝑔𝐽_𝑔𝐾_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔1−1𝑁_𝑔0,𝐁_𝐰𝐭=0−1𝐽_𝑔0,𝐅_𝐰𝐭=1𝐽_𝑟00,𝐂_𝐰𝐭=010, (7) where 𝜔_𝑔(𝑡) is the generator side angular speed, 𝜃_Δ(𝑡) is the torsion angle of the drive train, 𝐽_𝑟 is the moment of inertia of the low speed shaft, 𝐽_𝑔 is the moment of inertia of the high speed shaft, 𝐾_𝑑𝑡 is the torsion stiffness of the drive train, 𝐵_𝑑𝑡 is the torsion damping coefficient of the drive train, 𝐵_𝑟 is the viscous friction of the low speed shaft, 𝐵_𝑔 is the viscous friction of the high speed shaft, 𝑁_𝑔 is the gear ratio, 𝜂_𝑑𝑡 is the efficiency of the drive train, and 𝜏_𝑔(𝑡) is the generator torque.

(a)

(b)

Figure 1 

Mechanical and aerodynamical characteristics of the wind turbine.

The power converter and generator dynamics are given by [25] 𝑑𝜏𝑑𝑡^𝑔(𝑡)=−𝛼_gc𝜏_𝑔(𝑡)+𝛼_gc𝜏_𝑔,ref(𝑡),𝑃_𝑔(𝑡)=𝜂_𝑔𝜔_𝑔(𝑡)𝜏_𝑔(𝑡), (8)(9) respectively, where 𝜏_𝑔,ref(𝑡) is the desired generator torque (control input), 𝑃_𝑔(𝑡) is the produced power by the generator, 𝛼_gc denotes the dynamic coefficient of the generator/converter, and 𝜂_𝑔 is the generator efficiency.

3. Active Disturbance Rejection Design for Wind Turbine Control

The following assumptions in relation to the system (1)–(9) are stated.

Assumption 1. All the parameters of the WECS are known.

Assumption 2. The pair (𝐀_𝐰𝐭,𝐂_𝐰𝐭) is completely observable.

Assumption 3. The generator angular speed 𝜔_𝑔(𝑡) as well as the generated generator torque 𝜏_𝑔(𝑡) is available to be used in the control system.

Assumption 4. For sufficiently large positive integer 𝑝, the disturbance input 𝜏_𝑎(𝑡) exhibits uniformly absolute bounded time derivative of order 𝑝. This condition assures the existence of an unknown but finite constant, 𝐾_𝜏𝑎, such that sup_𝑡≥0|𝜏_𝑎^(𝑝)(𝑡)|≤𝐾_𝜏𝑎.

3.1. Problem Formulation

For a partial load-operating regime (operation in region 2), the main control objective is the maximization of wind power capture. This objective has a strong relation with the wind turbine power coefficient curve 𝐶_𝑝(𝜆,𝛽), which has a unique maximum point that corresponds to the optimal capture of the wind power: 𝐶_𝑝𝜆_opt,𝛽_opt=𝐶_𝑝opt, (10) where 𝜆_opt=𝜔_𝑔opt(𝑡)𝑅𝑁_𝑔𝜐(𝑡). (11) Accordingly, in order to maximize wind power capture, the blade pitch angle 𝛽 is fixed to its optimal value 𝛽_opt, and in order to maintain 𝜆 at its optimal value 𝜆_opt, the generator speed must be adjusted to track the optimal reference 𝜔_𝑔opt(𝑡), given by 𝜔_𝑔opt(𝑡)=𝑁_𝑔𝜆_opt𝑅𝜐(𝑡). (12) Then, it is desired to force the output 𝜔_𝑔(𝑡) to accurately track the given trajectory 𝜔_𝑔opt(𝑡), independently of the aerodynamic torque input and possible unmodeled perturbation inputs in the WECS, using the desired generator torque 𝜏_𝑔,ref(𝑡) as the control input and the generator angular speed 𝜔_𝑔(𝑡) as the feedback signal.

3.2. GPI Observer Design for Aerodynamic Torque Estimation

In order to obtain the optimal reference 𝜔_𝑔opt(𝑡), (4), (5), and (12) are combined. The wind velocity 𝜐(𝑡) is easily represented as a function of 𝜏_𝑎(𝑡) and 𝐶_𝑝 using (4) and (5): 𝜐(𝑡)=2𝜆𝜏_𝑎(𝑡)𝜌𝜋𝑅³𝐶_𝑝.𝜆,𝛽 (13) Then, by replacing (13) in (12), setting 𝜆 and 𝐶_𝑝 to its optimal values, and changing 𝜏_𝑎(𝑡) to its estimated version 𝜏_𝑎(𝑡), the following expression is obtained for the optimal reference trajectory: 𝜔_𝑔opt(𝑡)=𝑁_𝑔𝜆_opt𝑅2𝜆_opt𝜏_𝑎(𝑡)𝜌𝜋𝑅³𝐶_𝑝opt. (14) According to (14), it is necessary to estimate the aerodynamic torque𝜏_𝑎(𝑡). For that purpose, an extended Luenberger-like linear observer is developed, here referred as GPI observer. The proposed observer uses an approximated internal model of the unknown input disturbance to compose an augmented model for the plant and the disturbance input. Inherent to this kind of observer, a state estimation is also provided. This estimation will be used in the establishment of the GPI observer-based control for 𝜔_𝑔opt(𝑡) tracking.

Given a positive integer 𝑝, the unknown input perturbation 𝜏_𝑎(𝑡) can be modeled by the approximation of its internal model given by 𝑑^𝑝𝜏_𝑎(𝑡)𝑑𝑡^𝑝≈0. (15) Consider the following disturbance states, related to (15): 𝐱_𝐝(𝑡)=𝜏_𝑎(𝑡)𝜏̇_𝑎(𝑡)⋯𝜏_𝑎^(𝑝−2)(𝑡)𝜏_𝑎^(𝑝−1)(𝑡)^𝑇, (16) where its corresponding dynamics is given by 𝑑𝐱𝑑𝑡^𝐝(𝑡)=𝐀_𝐝𝐱_𝐝(𝑡)+𝐁_𝐝𝜏_𝑎^(𝑝)(𝑡),𝜏_𝑎(𝑡)=𝐂_𝐝𝐱_𝐝(𝑡) (17) with 𝐀_𝐝=,𝐁010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0_𝐝=00⋮01,𝐂_𝐝=100⋯0, (18) where 𝐱_𝐝(𝑡)∈ℝ^𝑝×1, 𝐀_𝐝∈ℝ^𝑝×𝑝, 𝐁_𝐝∈ℝ^𝑝×1, and 𝐂_𝐝∈ℝ^1×𝑝.

Now, the disturbance states 𝐱_𝐝(𝑡) can be added to the system state vector𝐱_𝐰𝐭(𝑡)to form the following augmented system: 𝑑𝐱(𝑡)(𝑡)𝑑𝑡=𝐀𝐱+𝐁𝜏_𝑔(𝑡)+𝐁_𝐚𝜏_𝑎^(𝑝)(𝑡),𝑦(𝑡)=𝐂𝐱(𝑡) (19) with 𝐱(𝑡)=𝐱_𝐰𝐭(𝑡)𝐱_𝐝(𝑡)𝐀,𝐀=^𝐰𝐭𝐅_𝐰𝐭𝐂_𝐝𝟎𝐀_𝐝,𝐁𝐁=^𝐰𝐭𝟎,𝐁_𝐚=𝟎𝐁_𝐝,𝐂=𝐂_𝐰𝐭𝟎, (20) where 𝐱(𝑡)∈ℝ^(𝑝+3)×1,𝐀∈ℝ^{(𝑝+3)×(𝑝+3)},𝐁,𝐁_𝐚∈ℝ^(𝑝+3)×1, and𝐂∈ℝ^1×(𝑝+3).

The next step is to design a GPI observer for the composite system in (19) regarding the approximated internal model given in (15). The estimated augmented state vector𝐱(𝑡)contains a real-time estimate of 𝐱_𝐝(𝑡), which is used along with𝐂_𝐝to recover𝜏_𝑎(𝑡).

Remark 5. ADR-GPI observer-based controllers use an internal model approximation of the perturbation functions to reconstruct and reject the perturbations. Under this disturbance model approximation setting, several authors have applied it to different areas. Parker and Johnson used a first-order perturbation approximation to model wind speed perturbations in a wind turbine operating in region 3 [26]. Freidovich and Khalil [27] used a first-order perturbation model approximation to estimate the model uncertainty and disturbance on a nonlinear system. Zhao and Gao also used a first-order internal model disturbance approximation to estimate the resonance in two-inertia systems [28] and a first- and second-order approximation to estimate the nonlinearities of an actuator [29]. Zheng et al. also used disturbance model approximation applied to disturbance decoupling control [30].

Remark 6. The parameter 𝑝 is related to the complexity of the signal to estimate, as in the case of Taylor polynomial approximation. A first-order perturbation model approximation means that the internal model approximation is capable to converge towards a constant disturbance. Equation (15) is a more generalized extension of the internal model perturbation function which provides extra information and increases the ability to track different types of disturbances. For example, 𝑝=2 allows convergence to a disturbance with a constant derivative, 𝑝=3 allows convergence to a disturbance with a constant acceleration, and so forth.

Theorem 7. Supposing that Assumptions 1–4 are valid, for the augmented WECS represented by (19) and (20), the following GPI observer is proposed: 𝑑𝐱(𝑡)𝐱(𝑡)𝑑𝑡=𝐀+𝐁𝜏_𝑔(𝑡)+𝜶𝑦−𝐂𝐱(𝑡), (21) where 𝐱(𝑡)=𝐱_𝐰𝐭(𝑡)^𝐓𝐱_𝐝(𝑡)^𝐓𝑇=𝜔_𝑟(𝑡)𝜔_𝑔(𝑡)𝜃_Δ(𝑡)𝜏_𝑎(𝑡)𝜏̇_𝑎(𝑡)⋯𝜏_𝑎^(𝑝−2)(𝑡)𝜏_𝑎^(𝑝−1)(𝑡)^𝑇 is the augmented system state estimation and 𝜶=𝛼_𝑝+3𝛼_𝑝+2⋯𝛼₂𝛼₁^𝑇are the observer gains. The GPI observer (21) asymptotically and exponentially reconstructs the system states 𝜔_𝑟(𝑡),𝜔_𝑔(𝑡),𝜃_Δ(𝑡), and the perturbation inputs 𝜏_𝑎(𝑡),𝜏̇_𝑎(𝑡),…,𝜏_𝑎^(𝑝−1) forcing the state estimation error𝐞̃_𝐱(𝑡)=𝐱(𝑡)−𝐱(𝑡)to converge towards the interior of a disk centered in the origin of the corresponding estimation error phase space, as long as the set of coefficients {𝛼_𝑝+3,…,𝛼₂,𝛼₁} is chosen in such a way that the characteristic polynomial defined by det(𝑠𝐈−𝐀+𝜶𝐂)=0 (22) is a Hurwitz polynomial, with roots located to the left of the imaginary axis of the complex plane.

Proof. By subtracting the proposed observer (21) from the augmented system state equation (19), the following estimation error dynamics is obtained: ̇̃𝐞_𝐱(𝑡)=(𝐀−𝜶𝐂)𝐞̃_𝐱(𝑡)+𝐁_𝐚𝜏_𝑎^(𝑝)(𝑡)=𝐀_𝐚𝐞̃_𝐱(𝑡)+𝐁_𝐚𝜏_𝑎^(𝑝)(𝑡), (23) where the eigenvalues of 𝐀_𝐚=(𝐀−𝜶𝐂) can be placed as desired by selecting the gain vector 𝜶.
In order to obtain an ultimate bound for𝐞̃_𝐱(𝑡), let𝐐∈ℝ^{(𝑝+3)×(𝑝+3)}be a constant, positive definite symmetric matrix. The proper stable character of the matrix𝐀_𝐚implies the existence of a positive definite matrix 𝐏∈ℝ^{(𝑝+3)×(𝑝+3)}such that 𝐀^𝐓_𝐚𝐏+𝐏𝐀_𝐚=−𝐐. Consider the Lyapunov function candidate𝑉(𝐞̃_𝐱)=(1/2)𝐞̃^𝐓_𝐱(𝑡)𝐏𝐞̃_𝐱(𝑡). The time derivative of𝑉satisfies 𝑉̇𝐞̃_𝐱,𝑡=12𝐞̃^𝐓_𝐱(−𝐐)𝐞̃_𝐱+𝐁^𝐓_𝐚𝐏𝐞̃_𝐱𝜏_𝑎^(𝑝)(𝑡). (24) For𝑄=𝐼, that is, an identity matrix,𝑉̇(𝐞̃_𝐱,𝑡)satisfies 𝑉̇𝐞̃_𝐱,𝑡=12𝐞̃^𝐓_𝐱(−𝐐)𝐞̃_𝐱+𝐁^𝐓_𝐚𝐏𝐞̃_𝐱𝜏_𝑎^(𝑝)(𝑡)1≤−2𝐞̃_𝐱²₂+𝐁^𝐓_𝐚2‖𝐏‖₂𝐞̃_𝐱2𝐾_𝜏𝑎<0. (25) Given that‖𝐁^𝐓_𝐚‖₂=1and according to (25),𝑉̇(𝐞̃_𝐱,𝑡)is strictly negative if 𝐞̃_𝐱2>2𝐾_𝜏𝑎‖𝐏‖₂. (26) Therefore,𝑉̇(𝐞̃_𝐱,𝑡)is strictly negative outside the following disc: 𝐷_𝑥=𝐞̃_𝐱(𝑡)∈ℝ^𝑝+3,𝐞̃_𝐱2≤2𝐾_𝜏𝑎‖𝐏‖₂. (27) Consequently, a uniform ultimate bounded (UUB) result was obtained regarding the estimation error phase variables𝐞̃_𝐱(𝑡).

Remark 8. GPI observers are bandwidth limited by the roots location of the estimation error characteristic polynomial. Generally, the larger the observer bandwidth is, the more accurate the estimation will be. However, a large observer bandwidth will increase noise sensitivity. Then, the selection of the roots of the estimation error characteristic polynomial affects the bandwidth of the GPI observer and also the influence of measurement noises on the estimations. Therefore, GPI observers are usually tuned in a compromise between disturbance estimation performance (set by the internal model approximation degree) and noise sensitivity.

3.3. Robust GPI Observer-Based Control Design for Energy Capture Maximization

Based on (6), the generator angular speed𝜔_𝑔(𝑡)satisfies the following dynamics: 𝑑𝜔_𝑔(𝑡)=𝐵𝑑𝑡^𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔𝜔_𝑟(𝑡)−𝐵_𝑔+𝐵_𝑑𝑡𝜂_𝑑𝑡/𝑁²_𝑔𝐽_𝑔𝜔_𝑔(𝑡)+𝐾_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔𝜃_Δ(𝑡)−1𝐽_𝑔𝜏_𝑔(𝑡). (28) Then, by reorganizing and lumping together some terms of (28), the following simplified system (typical of the ADR philosophy) can be obtained: 𝜔̇_𝑔(𝑡)=𝜅𝜏_𝑔(𝑡)+Δ(𝑡)+𝜑(𝑡) (29) with 1𝜅=−𝐽_𝑔,𝜑(𝑡)=𝐵_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔𝜔_𝑟(𝑡)−𝐵_𝑔+𝐵_𝑑𝑡𝜂_𝑑𝑡/𝑁²_𝑔𝐽_𝑔𝜔_𝑔(𝑡)+𝐾_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔𝜃_Δ(𝑡), (30) where 𝜅 is a known constant, 𝜑(𝑡) is a state dependent input perturbation, and Δ(𝑡) is an input perturbation function that lumps together all the uncertainty associated to the system. The perturbationΔ(𝑡)contains the rest of the system dynamics (actuator), including some unmodeled dynamics, disturbances of additive nature, actuator faults, parameter variations, and nonlinear effects of the WECS.

In relation to the simplified system (29), the following assumption is stated.

Assumption 9. For a sufficiently large positive integer 𝑚, the disturbance inputΔ(𝑡)exhibits a uniformly absolute bounded time derivative of order 𝑚. This condition assures the existence of an unknown but finite constant,𝐾_Δ, such that sup_𝑡≥0|Δ^(𝑚)(𝑡)|≤𝐾_Δ.

Assumption 10. The unknown input perturbationΔ(𝑡)can be modeled by the approximation of its internal model given by 𝑑^𝑚Δ(𝑡)𝑑𝑡^𝑚≈0. (31) Consider the following disturbance states, related to (31), 𝐱_Δ(𝑡)=Δ(𝑡)Δ̇(𝑡)⋯Δ^(𝑚−2)(𝑡)Δ^(𝑚−1)(𝑡)^𝑇, (32) where their corresponding dynamics is given by: 𝑑𝐱𝑑𝑡^Δ(𝑡)=𝐀_Δ𝐱_Δ(𝑡)+𝐁_ΔΔ^(𝑚)(𝑡),Δ(𝑡)=𝐂_Δ𝐱_Δ(𝑡) (33) with 𝐀_Δ=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0,𝐁_Δ=00⋮01,𝐂_Δ=100⋯0, (34) where 𝐱_Δ(𝑡)∈ℝ^𝑚×1, 𝐀_Δ∈ℝ^𝑚×𝑚, 𝐁_Δ∈ℝ^𝑚×1, and 𝐂_Δ∈ℝ^1×𝑚.
Then, it is possible to augment the simplified system in (29) with the unknown input perturbation state vector 𝐱_Δ(𝑡); thus, 𝑑𝐱𝑑𝑡^𝑐(𝑡)=𝐀_𝑐𝐱_𝑐(𝑡)+𝐁_𝑐1𝜏_𝑔(𝑡)+𝐁_𝑐2𝜑(𝑡)+𝐁_𝑐3Δ^(𝑚)(𝑡),𝑦(𝑡)=𝐂_𝑐𝐱_𝑐(𝑡) (35) with 𝐱_𝑐(𝑡)=𝜔_𝑔(𝑡)𝐱_Δ(𝑡),𝐀_𝑐=0𝐂_Δ0𝐀_Δ,𝐁_𝑐1=𝜅𝟎,𝐁_𝑐2=1𝟎,𝐁_𝑐3=0𝐁_Δ,𝐂_𝑐=1𝟎, (36) where 𝐱_𝑐(𝑡)∈ℝ^(𝑚+1)×1,𝐀_𝑐∈ℝ^{(𝑚+1)×(𝑚+1)},𝐁_𝑐1,𝐁_𝑐2,𝐁_𝑐3∈ℝ^(𝑚+1)×1, and𝐂_𝑐∈ℝ^1×(𝑚+1).
It is desired that the generator angular speed𝜔_𝑔(𝑡)accurately tracks the optimal reference trajectory 𝜔_𝑔opt(𝑡), with tracking error defined by 𝑒_𝜔(𝑡)=𝜔_𝑔(𝑡)−𝜔_𝑔opt(𝑡)absolutely bounded by a small quantity 𝜀; that is, sup_𝑡≥0|𝑒_𝜔(𝑡)|≤𝜀. Then, based on (29), (31), and (35), the following GPI observer-based control is proposed.

Theorem 11. Given Assumptions 9 and 10, the estimation of the perturbation functionΔ(𝑡), denoted asΔ(𝑡), is given by the following GPI observer: 𝑑𝐱𝑑𝑡^𝑐(𝑡)=𝐀_𝑐𝐱_𝑐(𝑡)+𝐁_𝑐1𝜏_𝑔(𝑡)+𝐁_𝑐2𝜑(𝑡)+𝝈𝑦(𝑡)−𝐂_𝑐𝐱_𝑐(𝑡,Δ(𝑡)=)0𝐂_Δ𝐱_𝑐(𝑡) (37)(38) with, 𝜑(𝑡)=𝐵_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔𝜔_𝑟(𝑡)−𝐵_𝑔+𝐵_𝑑𝑡𝜂_𝑑𝑡/𝑁²_𝑔𝐽_𝑔𝜔_𝑔(𝑡)+𝐾_𝑑𝑡𝜂_𝑑𝑡𝐽_𝑔𝑁_𝑔𝜃_Δ(𝑡), (39) where 𝐱_𝑐(𝑡)=𝜔_𝑔(𝑡)Δ(𝑡)Δ̇(𝑡)⋯Δ^(𝑚−2)(𝑡)Δ^(𝑚−1)(𝑡)^𝐓 is the estimated system state vector, 𝝈=𝜎_𝑚+1𝜎_𝑚⋯𝜎₂𝜎₁^𝑇is the observer gains, and𝜑(𝑡) is the estimation of𝜑(𝑡)reconstructed by using the states of the aerodynamic torque observer given in (21) that asymptotically and exponentially reconstructs the perturbation input Δ(𝑡), forcing the state estimation error 𝐞̃_𝐱𝑐(𝑡)=𝐱_𝑐(𝑡)−𝐱_𝑐(𝑡) to converge towards the interior of a disk centered in the origin of the corresponding estimation error phase space, provided the set of coefficients: {𝜎_𝑚+1,…,𝜎₂,𝜎₁}, which are chosen in such a way that the polynomial 𝑃_Δ(𝑠), in the complex variable𝑠, defined by 𝑃_Δ(𝑠)=𝑠^𝑚+1+𝜎_𝑚+1𝑠^𝑚+⋯+𝜎₂𝑠+𝜎₁ (40) is a Hurwitz polynomial, with roots located to the left of the imaginary axis of the complex plane.

Proof . By subtracting the proposed GPI observer (37) from the augmented system state equation (35), the following estimation error dynamics is obtained: ̇̃𝐞_𝐱𝐜(𝑡)=𝐀_𝐜−𝝈𝐂_𝐜𝐞̃_𝐱𝐜(𝑡)+𝐁_𝐜𝟑Δ^(𝑚)(𝑡)=𝐀_𝐚𝐜𝐞̃_𝐱𝐜(𝑡)+𝐁_𝐜𝟑Δ^(𝑚)(𝑡), (41) where the roots ofdet(𝑠𝐈−𝐀_𝐚𝐜)=𝑠^𝑚+1+𝜎_𝑚+1𝑠^𝑚+⋯+𝜎₂𝑠+𝜎₁can be placed as desired by selecting the gain vector 𝝈.
Following the same idea of the proof of Theorem 7, let𝐐_𝐜=𝐈∈ℝ^{(𝑚+1)×(𝑚+1)}be a constant, positive definite symmetric matrix; then, a positive definite matrix𝐏_𝐜∈ℝ^{(𝑚+1)×(𝑚+1)}exists such that𝐀^𝐓_𝐚𝐜𝐏_𝐜+𝐏_𝐜𝐀_𝐚𝐜=−𝐐_𝐜. Consider the Lyapunov function candidate𝑉(𝐞̃_𝐱𝐜)=(1/2)𝐞̃^𝐓_𝐱𝐜(𝑡)𝐏_𝐜𝐞̃_𝐱𝐜(𝑡). The time derivative of𝑉, that is, 𝑉̇(𝐞̃_𝐱𝐜,𝑡), is strictly negative outside the disc: 𝐷_𝑥𝑐=𝐞̃_𝐱𝐜(𝑡)∈ℝ^𝑚+1,𝐞̃_𝐱𝐜2≤2𝐾_Δ𝐏_𝐜2. (42) Consequently, a uniform ultimate bounded (UUB) result was obtained regarding the estimation error phase variables𝐞̃_𝐱𝐜(𝑡).

Theorem 12. Assume that there is an accurate estimation of 𝜑(𝑡), Δ(𝑡), 𝜏_𝑎(𝑡), and 𝜏̇_𝑎(𝑡); then, for the simplified system (29), the following control law is proposed: 𝜏_𝑔(𝑡)=1𝜅𝜔̇_𝑔opt(𝑡)−𝑘_𝑐𝜔_𝑔(𝑡)−𝜔_𝑔opt(𝑡)−𝜑(𝑡)−Δ(𝑡) (43) with 𝜔̇_𝑔opt(𝑡)=𝑁_𝑔𝜆_opt2𝑅2𝜆_opt𝜌𝜋𝑅³𝐶_𝑝opt1𝜏_𝑎(𝑡)𝜏̇_𝑎(𝑡), (44) where𝜏_𝑎(𝑡)and𝜏̇_𝑎(𝑡)are provided by the aerodynamic torque observer given in (21),𝜑(𝑡) is reconstructed by the estimated states of the aerodynamic torque observer given in (21), andΔ(𝑡)is provided by the GPI disturbance observer given in (37) and (38).
Such law asymptotically and exponentially forces the closed loop system tracking error𝑒_𝜔(𝑡)to converge towards the interior of a disk of radius as small as desired centered in zero, provided that the coefficient is𝑘_𝑐>0.

Proof. By replacing (43) in (29), the following dynamics is obtained: 𝜔̇_𝑔(𝑡)−𝜔̇_𝑔opt(𝑡)+𝑘_𝑐𝜔_𝑔(𝑡)−𝜔_𝑔opt(𝑡𝜑(𝑡)−Δ(𝑡)(𝑡)(𝑡).)=−+Δ+𝜑 (45) Then, by defining some estimation errors: 𝑒̃_𝜔opt=𝜔_𝑔opt(𝑡)−𝜔_𝑔opt(𝑡),𝑒_Δ̃(𝑡)=Δ(𝑡)−Δ(𝑡), and 𝑒_𝜑̃(𝑡)=𝜑(𝑡)−𝜑(𝑡) and by replacing them in (45), the following control system tracking error dynamics is obtained: 𝜔̇_𝑔(𝑡)−𝜔̇_𝑔opt(𝑡)+𝑘_𝑐𝜔_𝑔(𝑡)−𝜔_𝑔opt(𝑡̇𝑒̃)=−_𝜔opt(𝑡)−𝑘_𝑐𝑒̃_𝜔opt(𝑡)+𝑒_Δ̃(𝑡)+𝑒_𝜑̃(𝑡),𝑒_𝜔̇(𝑡)+𝑘_𝑐𝑒_𝜔(𝑡)̇=−𝑒̃_𝜔opt(𝑡)−𝑘_𝑐𝑒̃_𝜔opt(𝑡)+𝑒_Δ̃(𝑡)+𝑒_𝜑̃(𝑡). (46) Therefore, as long as𝑘_𝑐>0and the estimation errors 𝑒̃_𝜔opt(𝑡), 𝑒_Δ̃(𝑡), and 𝑒_𝜑̃(𝑡) are ultimately bounded by the GPI observers (21) and (37), the tracking error dynamics𝑒_𝜔(𝑡)will remain stable and bounded since the right side of (46) is also bounded.

Since the relative order of the WECS is one and system order is three, zero dynamics come into play and in consequence are analyzed. Considering the third-order system dynamics defined in (7), the zero dynamics is given by𝜔_𝑟(𝑡)and𝜃_Δ(𝑡)with𝜔_𝑔(𝑡)set to zero: 𝜔̇_𝑟(𝑡)𝜃̇_Δ(𝑡)=−𝐵_𝑑𝑡+𝐵_𝑟𝐽_𝑟−𝐾_𝑑𝑡𝐽_𝑟10_𝐴zero𝜔_𝑟(𝑡)𝜃_Δ(𝑡)+1𝐽_𝑟0𝜏_𝑎(𝑡). (47) Then, the internal dynamics is now given by the eigenvalues of 𝐴_zero (48). In this work, the eigenvalues are stable since the parameters of the WECS 𝐵_𝑑𝑡,𝐵_𝑟,𝐾_𝑑𝑡, and𝐽_𝑟are positive: 𝑠𝐼−𝐴_zero=𝐵𝑠+^𝑑𝑡+𝐵_𝑟𝐽_𝑟𝐾_𝑑𝑡𝐽_𝑟−1𝑠=𝑠²+𝐵_𝑑𝑡+𝐵_𝑟𝐽_𝑟𝐾𝑠+^𝑑𝑡𝐽_𝑟. (48) Some troubles may arise in the control system when the internal dynamics is poorly damped; however, in the studied case, such internal dynamics does not lead to any problem. In cases where the internal dynamics became problematic, some methods are used [31].

Remark 13. Note that for the GPI observer-based control strategy defined in (43) and (37), the energy capture maximization depends on the accurate reconstruction of the optimal reference trajectory 𝜔_𝑔opt(𝑡).

4. Benchmark Model

The simulations are carried out using a benchmark model for wind turbine control implemented in MATLAB/Simulink. This benchmark model was published by Odgaard et al. in [25] and can be used to evaluate both fault tolerant and classic control schemes in any region of operation of a wind turbine. The test bench model is based on a realistic nonlinear generic three-bladed horizontal-axis variable-speed wind turbine, containing sensors, actuators, system faults, tower shadow and wind shear effects, full converter coupling, and rated power at 4.8 MW.

For wind speeds between 0 and 12.5 m/s, the turbine is controlled to operate in region 2. The wind profile used has an average hub-height wind speed of 8.68 m/s and a turbulence intensity of 12%. The test bench defines a standard torque control strategy for the operation in region 2 with the following control law: 𝜏_𝑔,ref(𝑡)=𝑘_opt𝜔²_𝑔(𝑡) with 𝑘_opt=1.2171, 𝐶_𝑝opt=0.4554, and 𝜆_opt=8.0. The converter model has the following constraints: max torque gradient =1.25×10⁴ N·m/s, min torque gradient =−1.25×10⁴ N·m/s, max torque =3.6×10⁴ N·m, and min torque =0 N·m.

5. Results and Discussion

The proposed aerodynamic torque GPI observer (21) and the GPI observer-based control (37)–(43) were implemented and tested on the nonlinear wind turbine benchmark model described previously. Figure 2 shows the system scheme of the proposed control strategy used for simulations. The parameters of the GPI observers and the control strategy were selected as follows:GPI observer for aerodynamic torque estimation:𝑚=3,𝜎₁=1000, and 𝜎₂=1028.22, 𝜎₃=1028.62, 𝜎₄=0.39, 𝜎₅=234.81, 𝜎₆=28.46.GPI observer-based control:𝑝=3, 𝑘_𝑐=1, 𝛼₁=39.38, 𝛼₂=61.69, 𝛼₃=35.30, and 𝛼₄=8.66.

Figure 2 

Closed loop system schema of the proposed control strategy.

5.1. Aerodynamic Torque Estimation

The estimate of the aerodynamic torque obtained by the GPI observer in (21) is shown in Figure 3(a). The aerodynamic torque estimation error is shown in Figure 3(c). The estimation of the generator speed optimal reference trajectory𝜔_𝑔opt(𝑡)is shown in Figure 3(b), and its estimation error 𝑒̃_𝜔opt=𝜔_𝑔opt(𝑡)−𝜔_𝑔opt(𝑡) can be detailed in Figure 3(d). Note that this estimation error has a small offset of −1.0811 rad/s, which in turn will cause a slight offset in the aerodynamic power coefficient control.

(a)

(b)

(c)

(d)

Figure 3 

Aerodynamic torque estimation results: (a) aerodynamic torque, (b) generator speed optimal trajectory, (c) aerodynamic torque estimation error, and (d) generator speed trajectory estimation error.

5.2. Energy Capture Maximization

The simulation results of the proposed control strategy are shown in Figure 4. Under nominal conditions, the control system tracks the optimal reference trajectory to force the WECS power coefficient𝐶_𝑝close to its optimal value (see Figures 4(b) and 4(e)). Despite the external disturbances and nonlinearities of the benchmark model, the tracking error of the control system is near to zero as shown in Figure 4(c). In order to achieve a compromise between energy capture and dynamic loads on the low speed shaft, a medium performance on the control gain𝑘_𝑐was selected. Most of the fast fluctuations of the aerodynamic torque transferred to the generator speed optimal trajectory are not tracked (see Figure 4(b)).

(a)

(b)

(c)

(d)

(e)

Figure 4 

Simulation results of the proposed GPI observer-based control: (a) wind profile, (b) generator angular speed, (c) tracking error, (d) aerodynamic power captured, and (e) power coefficient.

Figures 4(d) and 4(e) detail the aerodynamic power captured by the GPI observer-based control and the evolution in time of the WECS power coefficient, respectively. It is observed that the captured aerodynamic power with the proposed control is greater than the power captured by the standard torque control defined in the benchmark. In addition, it is noted that the proposed control strategy forces the power coefficient to stay close to its optimal value 𝐶_𝑝opt=0.4554, which allows better power capture.

The performance of each control system is compared using an aerodynamic efficiency index𝜂_aero[24]. It is defined as follows: 𝜂_aero(%)=∫^𝑡𝑓_𝑡𝑖𝑃_𝑎(𝑡)𝑑𝑡∫^𝑡𝑓_𝑡𝑖𝑃_𝑎opt(𝑡)𝑑𝑡,𝑃_𝑎opt(𝑡)=12𝜌𝜋𝑅²𝐶_𝑝opt𝜐(𝑡)³. (49) The evaluation of the criteria defined in (49) stated that the aerodynamic efficiency obtained by the proposed control technique is 98.8%, while the efficiency of the classical controller (benchmark) is 95.6%.

5.3. Active Disturbance Rejection Evaluation

The benchmark model [25] contains faults which require the control system to be reconfigured to continue power generation, as well as very severe faults which require a safe and fast shutdown of the wind turbine. In this work, in order to evaluate the active disturbance rejection capability of the proposed GPI observer-based control strategy, a typical malfunction in the internal power converter control loops is used. As a consequence, this nonsevere fault must be accommodated in some way, and the wind turbine must continue its operation.

The fault considered is an offset, denoted asΔ𝜏_𝑔, on the generator torque, which can be caused by an error in the initialization of the converter controller [32]. The converter offset is configured to Δ𝜏_𝑔=5000 N·m.

Figure 5 shows the closed loop performance of both the standard torque control and the proposed GPI observer-based control approach under the actuator fault. The fault occurs from 200 s to 400 s as seen in Figure 5(d). As the fault is active, the GPI observer (37), via the observer stateΔ(𝑡), estimates the perturbation function on line (see Figure 5(c)) and actively rejects the internal perturbation applying the GPI observer-based control law (43).

(a)

(b)

(c)

(d)

Figure 5 

Simulation results on power converter fault.

It is observed in Figure 5(a) that the perturbation is rejected by the proposed control and any captured aerodynamic power lost is appreciated. Figure 5(b) shows that the power coefficient of the WECS is still close to its optimal value. On the other hand, the standard torque control of the benchmark cannot handle the actuator fault, and much of the aerodynamic power is lost. The evaluation of the criteria defined in (49) stated that the aerodynamic efficiency obtained by the proposed control is 98.65%, while the efficiency of the classical controller (benchmark) is 85.88%.

6. Conclusions

In this paper, a linear active disturbance rejection control strategy based on two GPI observers for maximum wind energy capture of variable-speed wind turbines operating at partial load has been proposed. In order to create the generator speed optimal trajectory towards an optimum point at which the WECS power coefficient is maximum, an ADR philosophy-based GPI observer was developed to estimate the aerodynamic torque and its first derivative. Then, an ADR philosophy-based GPI observer-based controller was designed, and it was able to absolutely and arbitrarily bound the generator speed tracking error.

The proposed design strategy solved the control problem based on linear active estimation of possible nonlinearities and perturbations of the WECS, and these accurate estimations were used by a simplified linear control law, in which the captured wind energy was maximized.

It was shown through simulation tests on a fully nonlinear benchmark model that the proposed dual GPI observer control strategy maximized the captured wind energy even when an actuator fault was applied. This is a demonstration of some robustness added by the GPI observer-based control.

Since performance of wind turbines is significantly affected by the used control strategy, considering new alternative linear control strategies that can improve the performance of the WECS is a motivation to use, adapt, and evaluate linear GPI observer-based controllers to operate wind turbines at partial load.

It is worth noting that the proposed control strategy is related to exact feedback linearization, but there are some important differences between both strategies which give advantages to GPI observer-based control, such as the following: (a) GPI observer-based control does not require system state measurements, (b) any mismatch between the system model and the real system is lumped together in a perturbation function that is estimated and rejected on line, (c) GPI observers are capable of estimating a certain number of perturbation function derivatives (useful to determine 𝜔̇_𝑔opt), and (d) ADR philosophy plays a very important role in GPI observer-based control since the internal model of the perturbation functions is taken into account in the design process.