Keywords

1 Introduction

Wind turbines are highly complex dynamical systems. They are flexible, mechanical structures subjected to spatially and temporally distributed disturbances, with interconnected dynamics, poorly damped, with physical constraints, etc. Additionally, they are operated and controlled in different modes depending on the wind speed. The operational region of wind turbines can be divided into three regions. On the one side, at low wind speeds one finds the partial-load region, also called region 1, where the main control objective is energy capture maximization. A complementary objective in this region is to reduce, or at least not to amplify, the aerodynamic loads [1]. At the opposite side of the operational wind speed range there is region 3, the full-load region. The objective there is to keep the turbine at its rated operating point. In this region, mitigation of aerodynamic and mechanical losses is crucial for the useful life of the wind turbine. In between, there exists a transition region (region 2) where the objective is to achieve a smooth transition between power tracking and regulation. Therefore, controller performance is mainly assessed in terms of loads alleviation.

Wind turbine control has been traditionally addressed in two ways. In one way, a multivariable controller is designed to guarantee satisfactory performance along the whole wind speed envelope. Some problems like low controllability and poorly damped oscillations make this task very complicated, leading to solutions that are too conservative. In the other approach, two different controllers are designed to fulfill the control objectives for partial- and full-load, whereas a bumpless or anti-windup compensation avoids undesirable responses after controller switching. This is the control structure implemented in commercial wind turbines.

For many years, the focus has been on improving controller performance in low and high wind speeds. Less attention has been given to the transition region where there was not a clear control objective (see for instance [2–5]). However, the detrimental effects of loading is increasing as wind turbine size grows exponentially, thus moving the attention to load mitigation. That is why operation and controller performance in the transition zone, where the low controllability problems appear, is now receiving special interest in the wind industry and academy.

In this chapter, the two-controller approach with anti-windup compensation is revisited. A robust, gain-scheduling control scheme designed in the \(H_{\infty }\) optimal control framework is discussed in detail, with special focus on the performance in the transition zone. The turbine is controlled through the generator torque in the partial-load region, and through the pitch angle in the full-load one. An optimal anti-windup strategy is also included to achieve smooth operation in the transition region.

2 Wind Turbine Modeling

The energy captured by a wind turbine is a function of the rotor radius R, the wind speed V, the rotor speed \(\Omega_{r}\) and the pitch angle β. More precisely, the rotor power can be expressed as

$$P_{r} (V,\beta ,\Omega_{r} ) = \frac{{\pi \rho R^{2} }}{2}C_{P} (\lambda ,\beta )V^{3} ,$$
(4.1)

where ρ is the air density and \(\lambda = \Omega_{r} R/V\) is the tip-speed ratio. The efficiency of the energy capture is characterized by the power coefficient \(C_{P} ( \cdot )\). Figure 4.1 shows the power coefficient of the 5 MW NREL wind turbine benchmark reported in [6].

Fig. 4.1
figure 1

Power coefficient C P for a 5 MW variable-speed variable-pitch wind turbine [6]

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The rotor torque results from dividing the captured power by the rotational speed:

$$T_{r} (V,\beta ,\Omega_{r} ) = P_{r} (V,\beta ,\Omega_{r} )/\Omega_{r} .$$
(4.2)

Modern wind turbines are complex mechanical systems exhibiting coupled translational and rotational movements. This complex dynamic behavior is in general well-captured by aeroelastic simulation codes such as the FAST (Fatigue, Aerodynamics, Structures, and Turbulence) code developed by the National Renewable Energy Laboratory (NREL) [7]. However, these models are not suitable for control design purposes. Simpler models including only a few oscillation modes are enough to design a control law. Here, for the sake of clarity, a two-mass model capturing just the first drive-train mode is used, whereas the unmodeled dynamics will be covered by additive uncertainty. The dynamical equations describing this model are

$$\begin{aligned} {\dot\Theta} & = \Omega_{r} - \Omega_{g} /N_{g} , \\ J_{t} {\dot\Omega}_{r} & = T_{r} - T_{\text{sh}} , \\ J_{g} {\dot\Omega}_{g} & = T_{\text{sh}} /N_{g} - T_{g} , \\ \end{aligned}$$
(4.3)

where the state variables are the torsion angle \(\Theta\), the rotor speed \(\Omega_{r}\) and the generator speed \(\Omega_{g}\). The model variables \(T_{g}\) and T sh = K s Θ + B s (Ω r  − Ω g ) are the generator and shaft torques, respectively. The model parameters are the inertia J t combining the hub and the blades, the generator inertia J g , the gear box ratio N g , and the shaft stiffness K s and friction B s coefficients. A representation of this two-mass model can be observed in Fig. 4.2.

Fig. 4.2
figure 2

Two-mass model describing the first drive-train mode

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In variable-speed wind turbines, the electrical generator is interfaced by a full or partial power converter that controls the generator torque T g and decouples the rotational speed from the electrical grid. Since the power converter and generator dynamics are much faster than the mechanical subsystem, it can be assumed for the purpose of this work that the torque reference of the power converter coincides with the electrical torque T g imposed to the wind rotor. That is, it can be assumed that T g is a control input.

The pitch actuator is a highly nonlinear mechanic and hydraulic system [5]. For control-oriented purposes, it is usually modeled as a first-order low-pass filter with saturation in the amplitude β and rate of change \(\dot{\beta }\). The pitch system is showed in Fig. 4.3. In the linear zone, the pitch actuator dynamics can be described by

$$\dot{\beta } = - \frac{1}{\tau }\beta + \frac{1}{\tau }\beta_{r} ,$$
(4.4)

where τ is the time constant and β r the pitch angle command.

Fig. 4.3
figure 3

Pitch actuator model

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The drive-train dynamics (Eq. 4.3) is highly nonlinear. This nonlinearity comes mainly from the aerodynamic torque (Eq. 4.2). For optimal control design, a linear representation of the system dynamics is necessary. With this aim, the aerodynamic torque is linearized around the operating point:

$$\hat{T}_{r} (\bar{V},\bar{\beta },{\bar\Omega}_{r} ) = B_{r} (\bar{V},\bar{\beta },{\bar\Omega}_{r} ){\hat\Omega}_{r} + k_{V} (\bar{V},\bar{\beta },{\bar\Omega}_{r} )\hat{V} + k_{\beta } (\bar{V},\bar{\beta },{\bar\Omega}_{r} )\hat{\beta },$$
(4.5)

where

$$\begin{aligned} B_{r} (\bar{V},\bar{\beta },{\bar\Omega}_{r} ) & = \left. {\frac{{\partial T_{r} }}{{\partial \Omega_{r} }}} \right|_{{(\bar{V},\bar{\beta },{\bar\Omega}_{r} )}} ,\quad k_{V} (\bar{V},\bar{\beta },{\bar\Omega}_{r} ) = \left. {\frac{{\partial T_{r} }}{\partial V}} \right|_{{(\bar{V},\bar{\beta },{\bar\Omega}_{r} )}} , \\ k_{\beta } (\bar{V},\bar{\beta },{\bar\Omega}_{r} ) & = \left. {\frac{{\partial T_{r} }}{\partial \beta }} \right|_{{(\bar{V},\bar{\beta },{\bar\Omega}_{r} )}} , \\ \end{aligned}$$

The bar over the variables denotes the corresponding values at the operating point, whereas the hat denotes deviations with respect to the operating point.

Substituting the linearized expression of the aerodynamic torque (Eq. 4.5) in the two-mass model (Eq. 4.3) and adding the linear model of the pitch system, the wind turbine becomes described, locally around a given operating point, by

$$\begin{aligned} \dot{x} = & \left[ {\begin{array}{*{20}c} 0 & 1 & { - 1/N_{g} } & 0 \\ { - K_{s} /J_{r} } & {(B_{r} (\bar{V},\bar{\beta },{\bar\Omega}_{r} ) - B_{s} )/J_{r} } & {B_{s} /J_{r} N_{g} } & {k_{\beta } (\bar{V},\bar{\beta },{\bar\Omega}_{r} )/J_{r} } \\ {K_{s} /J_{g} N_{g} } & {B_{s} /J_{g} N_{g} } & { - B_{s} /J_{g} N_{g}^{2} } & 0 \\ 0 & 0 & 0 & { - 1/\tau } \\ \end{array} } \right]x \\ & + \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ {k_{V} (\bar{V},\bar{\beta },{\bar\Omega}_{r} )/J_{r} } & 0 & 0 \\ 0 & { - 1/J_{g} } & 0 \\ 0 & 0 & {1/\tau } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\widehat{V}} \\ {T_{g} } \\ {\beta_{r} } \\ \end{array} } \right], \\ \end{aligned}$$
(4.6)

where \(x = [{\hat{\Theta}\;\hat{\Omega}}_{r}\;{\hat\Omega}_{g}\;\hat{\beta}]^{T}\) is the state. The signal \(\hat{V}\) is the wind speed acting as disturbance, and T g and β r are the control inputs.

3 Objectives and Control Scheme

A wind turbine normally works in different operating modes along the wind speed range [2]. These operating modes are illustrated in the power-wind speed curve of Fig. 4.4. The control objectives in these regions are substantially different. Below rated wind speed V N (region 1), the objective is to capture as much energy as possible. In this case, the pitch angle is kept constant at its optimum value, whereas the rotational speed is varied in proportion to the wind speed by properly controlling the generator torque. Above rated wind speed (region 3), the objective is to regulate the rotational speed and power at their rated values in order to protect the wind turbine from high mechanical loads and excessive electrical currents. In this region, the rotational speed is regulated acting on the pitch angle, whereas the generator torque is kept constant at its rated value. Between these two regions, there is a transition zone (region 2), in which the objectives and control structure exhibit significant changes.

Fig. 4.4
figure 4

Wind turbine operating regions

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The operating locus of the wind turbine corresponding to the power-wind strategy of Fig. 4.4 can be conveniently represented on a torque-rotational speed-pitch angle space. This graph is shown in Fig. 4.5. In this figure, the projection of this 3D curve onto the torque-rotational speed plane is also depicted. The three operating regions can be easily identified.

Fig. 4.5
figure 5

Typical torque-rotational speed curve for a variable-speed variable-pitch wind turbine

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  • In region 1, the objective is energy capture maximization. Both the tip-speed ratio and the pitch angle should be maintained as close as possible to their optimum values: \(C_{P} (\lambda_{o} ,\beta_{o} ) = C_{{P_{\hbox{max} } }}\). To this end, the generator torque is conventionally chosen to follow a quadratical relationship with the rotational speed, i.e.,

$$T_{g} = \left( {\frac{{\pi \rho R^{5} }}{{2\lambda_{o}^{3} }}C_{P\hbox{max} } } \right)\Omega_{g}^{2} = k_{t} \cdot \Omega_{g}^{2} .$$
(4.7)

  • The transition region 2 commonly comprises two subregions. Once the rotational speed reaches the lower limit \(\underline{\Omega}_{ \lim }\), the torque increases proportionally until the rated value \(T_{N}\). Beyond the upper limit \(\overline{\Omega}_{ \lim }\), the torque is maintained constant at the rated value. The objective in this region is to decouple as much as possible the control laws for regions 1 and 3.

  • In region 3, the generator torque is kept constant at the rated value whereas the pitch angle is used to regulate the rotational speed.

In the literature, several control schemes covering the entire operating range have been proposed. Basically, two approaches can be identified among the proposals. In one of them, two different controllers for low and high wind speeds together with a switching strategy are designed (see for example, [8, 9]). Simple controllers can be used in each region at the cost of a bumpless or anti-windup compensation to reduce undesirable transients in the transition region. The other approach consists of just one control law (generally nonlinear) covering the whole operating envelope (see for example, [2, 5]). This approach yields more complex controllers, hence requiring more advanced control techniques. Furthermore, these controllers are usually more conservative and quite complex to implement.

The control scheme covering the entire operating range presented in this chapter is based on the first approach. It is sketched in Fig. 4.6. As previously mentioned, the generator torque follows the law in Eq. 4.7 below rated wind speeds. This control law is implemented as a look-up table, the LUT block in Fig. 4.6. While the rotational speed evolves well below rated (\(\Omega_{N}\)), the pitch angle remains saturated at its lower limit \(\beta_{o}\). Only after speed reaches \(\Omega_{N}\) or approaches it fast enough, the pitch control becomes active. This pitch control is designed here using H ∞ optimal control tools. A gain scheduling technique is used to deal with the nonlinearity of the aerodynamic torque. In addition, anti-windup compensation is also included to smooth the transients in the transition region. This compensation is only active when the pitch angle saturates in order to recover optimally and softly the non saturated loop condition. The damping filter is commonly added to increase the damping of the drive-train oscillation mode. This filter is active in the three regions and it must be considered when the pitch controller is designed.

Fig. 4.6
figure 6

Control scheme covering the entire operating range

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Figure 4.7 depicts the coefficients B r , k V and k β as functions of wind speed over the operational wind speed range of the 5 MW wind turbine [6]. In particular, the intrinsic damping B r and pitch gain k β are of special interest since they affect the stability and performance of the closed loop system. To compensate for the nonlinearity in the pitch action, the inverse of the coefficient k β (k −1 β ) is inserted in the loop. Besides, the intrinsic damping B r is considered an uncertain parameter instead of a nonlinear function of the operating point. In this way, by canceling the nonlinearity in k β and covering with uncertainty the nonlinearity in B r , the rotational speed can be regulated with a linear time-invariant (LTI) controller. That is, an LTI controller designed for a given operating point can be used for the whole operating locus. To deal with the parameter uncertainty, H ∞ optimal control tools will be used.

Fig. 4.7
figure 7

Values of the linearization coefficients B r , k V and k β in the entire wind speed range

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To implement this control strategy, it is necessary to invert k β , which is parameterized by wind speed. Recall, however, that wind speed in not measurable. To overcome this problem, the one-to-one correspondence between wind speed and pitch angle that exists in region 3 of the operating locus of Fig. 4.5 can be exploited. In fact, this one-to-one correspondence means that \(\bar{\beta }\) suffices to uniquely determine the operating point in this region. Particularly, it means that k β can be parameterized just in terms of \(\bar{\beta }\). This allows computing \(k_{\beta }^{ - 1}\) as function of a measurable variable. To simplify this computation, the coefficient k β can be approximated by

$$k_{\beta } (\bar{\beta }) = k_{\beta 0} k_{\text{gs}} (\bar{\beta })$$

The gain \(k_{\beta 0}\) is \(k_{\beta }\) evaluated at operating point where the LTI controller is designed, whereas \(k_{\text{gs}} ( \cdot )\) is a second order polynomial of the form

$$k_{\text{gs}} (\bar{\beta }) = c_{2} \bar{\beta }^{2} + c_{1} \bar{\beta } + c_{0}$$

computed by curve fitting. The values of \(k_{\beta } /k_{\beta 0}\) and the approximation k gs for the 5 MW NREL wind turbine can be seen in Fig. 4.8. To avoid loop interactions, the scheduling parameter \(\bar{\beta }\) is obtained by passing the pitch angle through a low pass filter (LPF).

Fig. 4.8
figure 8

Values of \(k_{\beta } /k_{\beta 0}\) for several wind speeds and the second order polynomial approximation

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Before describing the control design, a brief review of H ∞ optimal control concepts is provided in the next section.

4 H ∞ Optimal Control Background

Consider an LTI system with the following state space realization

$$\begin{aligned} \dot{x} & = Ax + B_{w} w + B_{u} u \\ z & = C_{z} x + D_{\text{zw}} w + D_{\text{zu}} u \\ y & = C_{y} x + D_{\text{yw}} w + D_{\text{yu}} u \\ \end{aligned}$$
(4.8)

where \(A \in {\mathbb{R}}^{n \times n}\), \(D_{zw} \in {\mathbb{R}}^{{n_{z} \times n_{w} }}\) and \(D_{yu} \in {\mathbb{R}}^{{n_{y} \times n_{u} }}\). The signal u is the control input and w is the disturbance. The signal y is the controlled variable and z is a fictitious output signal that serves to state the control objectives. The signal z is commonly named as the performance output.

It is assumed that system in Eq. 4.8 is stabilizable and detectable. That is, there exists a control law \(u = K(s)y\) that stabilizes the closed loop system

$$T_{zw} (s) = G_{zw} (s) + G_{zu} (s)K(s)(I + G_{yu} (s)K(s))^{ - 1} G_{yw} (s)$$

with I the identity matrix and G(s) the transfer function of the system in Eq. 4.8 partitioned as

The γ-suboptimal H ∞ synthesis problem consists in finding an internally stabilizing control law \(u = K(s)y\) that guarantees an \(\infty\)-norm of the closed-loop transfer function from the disturbance w to the performance output z lower than γ. Being \(T_{zw} (s)\) being the closed-loop transfer function from w to z, the control objective can be formalized as

$${\parallel }T_{zw} {\parallel }_{\infty } < \gamma ,$$
(4.9)

where \({\parallel } \cdot {\parallel }_{\infty }\) denotes the infinity norm. For a stable system with transfer function G(s), the \(\infty\)-norm is defined as

$${\parallel }G(s){\parallel }_{\infty } = \mathop {\hbox{max} }\limits_{\omega } \sigma_{\hbox{max} } (G(j\omega ))$$

where \(\sigma_{\hbox{max} }\) is the maximum singular value and ω is the frequency [10]. In other words, the \(\infty\)-norm is basically the maximum gain of the frequency response of the transfer function G(s).

There are several solutions for the γ-suboptimal H ∞ synthesis problem, but the most popular nowadays is the formulation as an optimization problem with linear matrix inequalities (LMIs) constraints [11]. Considering a controller with a state-space realization

the controller matrices can be found by solving the following optimization problem:

  • minimize \(\gamma (R,S,\hat{B}_{k} ,\hat{C}_{k} ,D_{k} )\),

  • subject to

$$\begin{aligned} \left[ {\begin{array}{*{20}c} {AR + B_{u} \hat{C}_{k} + ({ \star })} & { \star } & { \star } \\ {(B_{w} + B_{y} D_{k} D_{yw} )^{T} } & { - \gamma I_{{n_{w} }} } & { \star } \\ {C_{z} R + D_{zu} \hat{C}_{k} } & {D_{zw} + D_{zu} D_{k} D_{yw} } & { - \gamma I_{{n_{z} }} } \\ \end{array} } \right] &<0, \hfill \\ \left[ {\begin{array}{*{20}c} {(SA + \hat{B}_{k} C_{y} ) + ({ \star })} & { \star } & { \star } \\ {(SB_{w} + \hat{B}_{k} D_{yw} )^{T} } & { - \gamma I_{{n_{w} }} } & { \star } \\ {C_{z} + D_{zu} D_{k} C_{y} } & {D_{zw} + D_{zu} D_{k} D_{yw} } & { - \gamma I_{{n_{z} }} } \\ \end{array} } \right] &<0, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \left[ {\begin{array}{*{20}c} R & I \\ I & S \\ \end{array} } \right] &>0, \hfill \\ \end{aligned}$$

with “>” and “<” denoting positive and negative definite matrices, respectively, and \({ \star }\) represents the matrices needed to obtain a symmetric matrix.

After finding the positive definite matrices R and S and matrices \(\hat{B}_{k}\), \(\hat{C}_{k}\) and \(D_{k}\), the controller matrices can be computed from

$$\begin{aligned} A_{k} & = - (A + B_{u} D_{k} C_{y} )^{T} + \left[ {\begin{array}{*{20}c} {SB_{w} + \hat{B}_{k} D_{{yw}} } & {C_{z} + D_{{zu}} D_{k} C_{y} } \\ \end{array} } \right] \\ & \quad \left[ {\begin{array}{*{20}c} { - \gamma I} & {(D_{{zw}} + D_{{zu}} D_{k} D_{{yw}} )^{T} } \\ {D_{{zw}} + D_{{zu}} D_{k} D_{{yw}} } & { - \gamma I} \\ \end{array} } \right]^{-1}\left[ {\begin{array}{*{20}c} {(B_{w} + B_{y} D_{k} D_{{yw}} )^{T} } \\ {C_{z} R + D_{{zu}} \hat{C}_{k} } \\ \end{array} } \right] \\ B_{k} & = N^{ - 1} (\hat{B}_{c} - SB_{u} D_{c} ), \\ C_{k} & = (\hat{C}_{k} - D_{k} C_{y} R)M^{ - T} , \\ \end{aligned}$$

with \(MN^{T} = I - RS\).

The optimization problem involved in the H ∞-synthesis can be effectively solved with available software such as Sedumi [12] and YALMIP [13]. It is also available as a command in the Robust Control Toolbox for Matlab. Therefore, the design process of an H ∞ optimal control requires only to put the control specifications in terms of the minimization of the norm in Eq. 4.9, i.e., to construct the augmented plant (Eq. 4.8) by selecting the performance output z and the disturbance w. More details about this process will be given in the next following sections.

One of the most common uses of H ∞-synthesis is the design of robust controllers. The dynamics of a system can be described by a model, but this description is always an approximation. There always exists discrepancy between system behavior and response predicted by the model, caused for example for changes in some parameters or unmodeled phenomena. To consider this situation, the system is described as a family of models in the form

$$\tilde{G}(s) = G_{o} (s) + W_{\varDelta } (s)\varDelta ,\quad {\parallel }\varDelta {\parallel }_{\infty } < 1,$$
(4.10)

where G o is the nominal model and \(W_{\varDelta }\) is a filter describing the modeling error at different frequencies. \(\varDelta\) is an unknown LTI system with \(\infty\)-norm bounded by 1. It can be proved that the closed loop comprising system \(\tilde{G}(s)\) and controller K(s) (Fig. 4.9) is stable for all \({\parallel }\varDelta {\parallel }_{\infty } < 1\) if the \(\infty\)-norm of the transfer function from r to u is lower than 1, i.e.,

$${\parallel }K(s)(I + G(s)K(s))^{ - 1} {\parallel }_{\infty } < 1.$$
(4.11)
Fig. 4.9
figure 9

Closed loop system with uncertain plant

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When a controller K(s) ensures stability for all plants in the set in Eq. 4.10, it is said that the closed loop is robustly stable and Eq. 4.11 is the robust stability condition associated to the uncertainty representation in Eq. 4.10. Notice that there are other uncertainty representations, but Eq. 4.10 is one of the most commonly used (for more details see for example, [10]).

5 Wind Turbine Control Design

In this section, we present an H ∞ pitch controller. As said before, an anti-windup controller was added to avoid undesirable behavior caused by pitch saturation in the transition from region 1 to region 3. The controller is designed following a two-step process. First, the H ∞ pitch controller is designed disregarding the pitch saturation. Then, the anti-windup controller is designed for a suitable performance in the transition region.

5.1 H ∞ Optimal Pitch Control

As described in the previous section, the first step in the design of an H ∞ optimal control is to state the augmented plant. This implies stating the specifications in terms of the norm-minimization of the mapping between the performance output z and the disturbance w. Model uncertainty can be covered by properly choosing these signals, thus ensuring robust stability. Therefore, before describing the augmented plant, we find the model uncertainty representation to cover the variation of the parameter B r and other errors caused by the approximation of the wind turbine behavior with a low-order model. The use of the scheduling gain \(k_{\text{gs}}^{ - 1} (\bar{\beta })\) allows canceling the variation of the linearized aerodynamic torque with respect to the operating conditions. However, the variations in B r are not so simple to compensate for, since they affect the eigenvalues of the linear model.

For control design purposes, the wind turbine can be modeled, after applying the scheduling gain in the control input, by the following parameter dependent transfer function

Figure 4.10 shows the frequency response of this transfer function for several operating points in region 3. These results correspond to the 5 MW wind turbine previously mentioned. It can be seen that these variations can be covered by additive uncertainty of the form in Eq. 4.10 with the weighting function \(W_{\varDelta } (s)\) shown in the right plot of Fig. 4.10. In the uncertainty set, the scheduling gain errors caused by the polynomial approximation and the high-frequency unmodeled oscillation modes can also be covered.

Fig. 4.10
figure 10

Left frequency response of G(s) corresponding to several operating points in region 3. Right weighting function \(W_{\varDelta }\) of the additive uncertainty representation Eq. 4.10

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In high wind speeds, the control objectives are the regulation of the rotational speed close to the rated value \(\Omega_{N}\) and the reduction of the pitch activity to avoid high mechanical stresses. In the H ∞ optimal control framework, these objectives lead to the augmented plant of Fig. 4.11. In this case, the control input u is the pitch command and the controlled signal y is the rotational speed error \(e = \Omega_{N} - \Omega_{g}\). Then, the performance signal is \(z = [\tilde{e},\;\tilde{\beta }_{r} ]^{T}\) and the disturbance is the rotational set-point \(w = \Omega_{N}\). Notice that the wind speed could also be considered as a disturbance. However, this would not improve the performance, but only increase the controller complexity.

Fig. 4.11
figure 11

Controller design setup augmented with weighting functions

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The speed regulation results in the minimization of the low frequency components of the speed error \(\tilde{e} = W^{\prime}_{e} (s)(\Omega_{N} - \Omega_{g} )\), where

$$W^{\prime}_{e} (s) = M(s)W_{e} (s)$$
(4.12)

with W e a stable transfer function. If M includes integral action, the weight W e can be a simple constant, such as,

$$M(s) = 1/s,\quad{W}_{e} = k_{e} .$$

The integral action ensures a zero steady-state error and penalizes the rotational speed error in low frequencies. Tracking error in high frequencies is not recommended because it would expose the wind turbine to excessive mechanical loads. The factorization in Eq. 4.12 is necessary for satisfying stabilizability conditions [10]. The controller actually applied to the wind turbine is

$$K(s) = M(s)\tilde{K}(s)$$

with \(\tilde{K}(s)\) the controller produced by the optimization algorithm.

The weighting function W u (s) is a high pass filter such as

$$W_{u} (s) = k_{u} \frac{{s/0.1\omega_{u} + 1}}{{s/\omega_{u} + 1}}$$

with k u and ω u design parameters. This transfer function penalizes the high frequency components of the control signal, thus limiting the pitch activities. High frequency control action must be also avoided to prevent from rate pitch saturation. Since the modeling errors are covered by additive uncertainty, robust stability and the limitation of the pitch activity can be expressed as a constraint on the same closed loop transfer function. Therefore, the weighting function \(W_{u}^{{\prime }} (s)\) in Fig. 4.11 is the most restrictive function between \(W_{u} (s)\) and \(W_{\varDelta } (s)\).

Notice that if the drive-train damping controller is considered, the plant \(G(s)\) in Fig. 4.11 must include its dynamics in order to guarantee close loop stability.

Since the pitch controller is only active in region 3, an anti-windup controller is necessary to ensure good behavior during the transition from region 1 to region 3. With this purpose an optimal anti-windup controller is added. This controller is not shown in Fig. 4.6 for a matter of clarity. The anti-windup controller connection is shown in Fig. 4.12.

Fig. 4.12
figure 12

Anti-windup compensation scheme

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5.2 Anti-windup Compensation

In the control scheme of Fig. 4.6, the pitch controller will be in saturation during the low wind speed intervals. So, anti-windup compensation is absolutely necessary. To achieve high performance compensation, the anti-windup scheme proposed by [14, 15] is adopted here. It can be seen in Fig. 4.12. This scheme offers a good trade-off between stability margin and smooth recovery from saturation.

As can be seen in Fig. 4.12, the anti-windup compensator produces two terms: y d acting on the controller input and another u d acting on the control output. It can be proved, after some system manipulations, that the block diagram in Fig. 4.13 is equivalent to the scheme in Fig. 4.12 by defining

$$\left[ {\begin{array}{*{20}c} {u_{d} (s)} \\ {y_{d} (s)} \\ \end{array} } \right] = T_{\text{aw}} (s)\check{u} (s) = \left[ {\begin{array}{*{20}c} {X(s) - I} \\ {Y(s)} \\ \end{array} } \right]\check{u}(s),$$
(4.13)

where X and Y are the coprime factors of G, i.e., \(G = X^{ - 1} Y\). Therefore, the anti-windup compensator can be expressed as

(4.14)

where F is chosen for \(A + B_{u} F\) to be Hurwitz.

Fig. 4.13
figure 13

Equivalent representation of the anti-windup compensation scheme in Fig. 4.12

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In this way X must be designed to ensure the closed-loop stability of \(X - I\) and the deadzone nonlinear operator. At the same time, X must be designed to minimize the effect of y d on the controlled variable. It can be proved that using the Lyapunov function \(V(x_{\text{aw}} ) = x_{\text{aw}}^{T} Px_{\text{aw}} > 0\) and forcing

$$\dot{V}(x_{\text{aw}} ) + y_{d}^{T} y_{d} - \nu u^{T} u < 0,$$
(4.15)

with \(x_{\text{aw}}\) the state of the system \(T_{\text{aw}}\), the previously mentioned objectives are satisfied. Because of the sector boundedness of the dead-zone nonlinearity, the following condition is satisfied

$$2\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{u}^{T} U^{ - 1} (u - Fx_{\text{aw}} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{u} ) \ge 0,$$
(4.16)

with U a diagonal matrix.

After some mathematical manipulations, the problem of finding an anti-windup compensator is reduced to a state feedback F satisfying the following optimization problem with LMI constraints:

  • minimize \(\nu (Q,U,L)\),

  • subject to

$$\begin{gathered} \left[ {\begin{array}{*{20}c} {(AQ + B_{u} L) + ({ \star })} & {B_{u} U - QF^{T} } & 0 & {QC_{y}^{T} + L^{T} D_{yu}^{T} } \\ { \star } & { - 2U} & I & {UD_{yu}^{T} } \\ { \star } & { \star } & { - \nu I} & 0 \\ { \star } & { \star } & { \star } & { - \nu I} \\ \end{array} } \right] < 0, \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad Q = Q^{T} > 0, \hfill \\ \end{gathered}$$

where \({ \star }\) are inferred by symmetry. The state feedback gain is then obtained as \(F = LQ^{ - 1}\) (see [16] for more details).

6 Results

The system behavior was evaluated by simulation on the 5 MW NREL wind turbine benchmark [6]. Simulations were performed in the \({\text{FAST/Simulink}}^{\circledR} / {\text{Matlab}}^{{\circledR }}\) environment. A more complete 16 degrees-of-freedom model available in FAST [7] was used as a way to assess the robustness of the proposed control scheme against unmodeled dynamics. The wind turbine data are given in Table 4.1, whereas the limit values of the operating locus are listed in Table 4.2.

Table 4.1 Wind turbine parameters
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Table 4.2 Operating curve values
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The pitch controller was designed according to the control setup in Fig. 4.11 with

$$W^{\prime}_{e} (s) = M(s)W_{e} (s) = \frac{1}{s}k_{e} ,W^{\prime}_{u} = k_{u} \frac{{s/0.1\omega_{u} + 1}}{{s/\omega_{u} + 1}}$$

where \(k_{e} = 0.3\), \(\omega_{u} = 50\) and \(k_{u} = 0.25\). The frequency response of the weights \(W^{\prime}_{e}\), \(W_{\varDelta }\) and \(W_{u}\) are shown in Fig. 4.14. Remember that \(W^{\prime}_{u}\) must be more demanding than \(W_{u}\) and \(W_{\varDelta }\) at every frequency. So, as can be seen in Fig. 4.14, it suffices to choose \(W^{\prime}_{u} = W_{u}\).

Fig. 4.14
figure 14

Weight functions used in the pitch control design

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The \(\infty\)-norm of the closed loop transfer function \(T_{zw}\) resulted in 0.977. In particular, the norm of the transfer function from \(\Omega_{N}\) to the control signal \(\beta\), i.e.,

$$||K(I + KG)^{ - 1} ||_{\infty } = 0.972.$$

As the norm is lower than 1, stability against covered modeling errors is guaranteed.

After designing the pitch controller, the anti-windup compensation was computed by solving the optimization problem described in Sect. 5.2. The optimization problems for obtaining the H ∞ controller and the anti-windup compensation were solved using the Robust Control Toolbox for Matlab, Sedumi [12] and YALMIP [13].

For the sake of comparison, a classical gain-scheduled PI controller was also implemented and simulated. This PI, widely used in the literature as benchmark controller, was designed following the guidelines in [6, 17]. Basically, the PI controller was tuned as in [6] after linearizing the wind turbine model at the operating point \((\bar{V},\bar{\beta },{\bar\Omega}_{r} ) = (11.4\,{\text{m/s}},\;0,\;12.1\,{\text{rpm}})\). The controller gains were calculated to achieve appropriate damping (0.7) and natural frequency (0.6 rad/s) [17]. As the controller ensures the desired behavior only at the design operating point, a pitch-dependent gain is applied to compensate for the nonlinear rotor torque. This gain is a function of \(\beta\) obtained by fitting the values of \(k_{\beta }\) along the operating locus. In addition, a classical anti-windup compensation was added to improve the transient between regions 1 and 3. The PI tuning constants are \(K_{P} (\beta = 0) = 0.01882681\) s, \(K_{I} (\beta = 0) = 0.008068634\). The function that makes the gain scheduling is \(f(\beta ) = 1/(1 + \beta /\beta_{k} )\), where \(\beta\) is the pitch angle and \(\beta_{k} = 6.30236\) is the blade pitch angle when the rotor power has doubled. The generator torque control is the same as in the \(H_{\infty }\) approach.

Three scenarios were simulated to assess the control performance. The first scenario is aimed at evaluating the performance of the controllers in region 3. To this end, the wind gust suggested in the IEC 61400-1 standard was used. This gust is very pitch demanding. The second scenario is mainly intended to assess the performance of the anti-windup compensator, i.e., the performance in the transition region. To this end, the wind rise proposed also in IEC 61400-1 standard was used. The last scenario illustrates the behavior of the controller under a realistic wind speed profile.

The simulation results obtained in the first scenario are presented in Fig. 4.15. As can be seen, the pitch controllers are always active. The H ∞ controller achieves better speed regulation with lower pitch activity than the PI controller. The speed overshoot is 16.86 % in the H ∞ case and 19.25 % in the PI case. Additionally, the H ∞ controller offers smoother responses in output power and shaft torque.

Fig. 4.15
figure 15

System response to a wind gust profile. Gray lines PI controller, black lines H ∞ controller

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The results obtained in the second scenario are shown in Fig. 4.16. The wind profile rising from 6 to 13 m/s in 10 s makes the wind turbine to operate along the three regions. Again, it can be seen that the H ∞ controller achieves a better speed regulation with less pitch activity. Note that H ∞ with the anti-windup starts pitching the blades a bit before the PI with classical anti-windup does. The speed overshoot is 25.49 % in the H ∞ case and 32.96 % in the PI case. Like in the first scenario, the H ∞ controller offers smoother responses in output power and shaft torque.

Fig. 4.16
figure 16

System response to a wind rise profile. Gray lines PI controller, black lines H ∞ controller

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The results for the last scenario are shown in Fig. 4.17. The 10 min wind speed field was generated with Turbsim [18]. The 8 m/s mean wind speed was selected so that the wind turbine operates in all the three operating regions, but most of the time in the transition one. It can be seen in the figure that the H ∞ controller achieves better speed regulation with significant less pitch activity.

Fig. 4.17
figure 17

System response to a realistic wind profile. Gray lines PI controller, black lines H ∞ controller

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7 Conclusion

This chapter presents a robust H ∞ pitch control design for variable-pitch variable-speed wind turbines operating along the entire wind speed range.

The main nonlinearities of the wind turbine dynamics are either canceled by inversion or covered with uncertainty so that an LTI H ∞ controller designed for a given operating point can be applied to the whole operating locus. The controller design guarantees robustness against unmodeled dynamics, cancelation errors and parameter uncertainties.

The system behavior was assessed by numerical simulations of a high-order wind turbine benchmark under three very demanding scenarios. Under these scenarios the H ∞ controller achieves better speed regulation with lower pitch activity than classical PI control. This lower pitch activity leads to lower mechanical stress spreading the wind turbine lifetime and also resulting in softer output power.

8 Future Research

The conservativeness inherent to the applied linearization, particularly due to the low-frequency uncertainty, can be eliminated by replacing the LTI design with the linear parameter varying (LPV) one. At the cost of a bit more complex implementation, better results particularly in speed regulation can be obtained. The complexity of the controller can be increased in order to consider more specifications as attenuating additional oscillation modes by adding more sensing signals.