Modelling the Wind Turbine by Using the Tip-Speed Ratio for Estimation and Control

Abstract

The development of dynamic models for control purposes is characterised by the challenge of finding a compromise between the minimum necessary information about the system dynamics contained in the model and a model with a low level of complexity such that the model-based control system design becomes comfortable. To achieve this balance, a modified dynamic model for the drivetrain of a wind turbine is proposed in this contribution. The main idea is to introduce the tip-speed ratio as a state variable so that an interval observer can be designed in such a way that its estimates can be used in the torque control during the partial load operation as well as for the estimation of the effective wind speed. During the runtime, the observer’s matrix gain is recalculated to adapt the behaviour to the current operational state, which changes all the time with the wind speed. Besides the theoretical formulation, a numerical example of a 20 MW reference wind turbine illustrates the utility of the method. The results show good control performance concerning the tip-speed ratio control loop and a satisfactory estimation of the effective wind speed.

1. Introduction

To perform optimally, today’s very large wind turbines with flexible structures need a mathematical description of the dynamics using nonlinear partial differential equations on one hand, and sophisticated control algorithms on the other. However, such mathematical representation is not convenient for automatic control system design. To overcome this dilemma, much work has been devoted to building particularly simple models that can be used for control [1,2].

In fact, model development activities are presented in the majority of papers presenting the application of model-based control algorithms. For example, a two-mass model of the drivetrain and a standard nonlinear expression for the aerodynamic torque are frequently used in modelling approaches. The power coefficient C_p is included in the form of an empirical nonlinear function, and its coefficients are adjusted using curve fitting algorithms. This kind of modelling approach can be found in [3,4,5]. In addition, Refs. [6,7] include a linear second-order differential equation to describe the fore-aft displacement of the tower. In [8], the model of the tower is significantly more complex but still simple. Some control applications using similar complex models are reported in [9,10,11].

There are also some contributions centred on the development of models for wind turbines directly for control design objectives (see, for instance, [12,13,14]). A summary of specific modelling aspects of wind turbines, which are important from the control-engineering point of view, is provided in [15].

The attention of the present work is focused on the rotating subsystem which, as a rule, is described as a multi-mass subsystem. Three-mass models and their applications are examined in [16,17]. Two-mass models are proposed, for example, in [3,18]. A comparative study of the available multi-mass dynamic models for the drivetrain is presented in [19].

The tip-speed ratio, defined as the quotient between the tip speed and the wind speed, is a significant variable from both the control and estimation points of view. For instance, one control strategy in partial load operation is to just maintain the tip-speed ratio at its maximum value (see [20,21,22] for details). On the other hand, the tip-speed ratio is also used to estimate the unmeasurable effective wind speed. Despite this, the tip-speed ratio is not considered in the models of all the above-mentioned references.

Because effective wind speed cannot be measured, numerous estimation methods have been presented in the literature. For instance, an interesting compendium is described in [23]. Some of them are the immersion and invariance estimator [24], the observer for an unknown input [25], and methods based on observers [26] as well as Kalman filters [27]. Some of them require the in-between estimation of the tip-speed ratio.

The idea of the present work is to formulate the model of the drivetrain in terms of the tip-speed ratio, i.e., to define the tip-speed ratio as a state variable of the model. Hence, a state observer is designed that estimates the generator speed and the tip-speed ratio, which directly leads to an estimate of the effective wind speed. The estimate of the tip-speed ratio can also be used as a feedback variable for the control in partial load operation. The rest of the paper is organised as follows: First, the state-space model is derived in Section 2, followed by the steady-state model in Section 3. In the sequel, the observer design is the subject of Section 4. Section 5 is devoted to describing two potential applications for the model, which are exemplified and analysed in Section 6, based on a numerical example using a 20 MW reference wind turbine. Finally, conclusions are drawn in Section 7.

2. Dynamic Modelling of the Drivetrain

By using decomposition and coordination techniques for large-scale systems [28,29,30], it is possible to isolate a modular representation of the drivetrain of a wind turbine. The rotor of the generator turns as a result of the wind turbine’s rotation. This rotation has the opposite effect, in that the produced electromagnetic torque at the generator causes a reaction that affects the turbine rotation [31]. Thus, although it is possible to find representations of the drivetrain that uses only one rotating mass [32], the assumption of more than one mass is the most common case (see [19] for different modelling approaches). A very simple model considers two rotating masses, one for the rotor and one for the generator [3,10]. In addition, the low-speed shaft is assumed to be flexible, the fast-speed shaft to be rigid, and finally, the gearbox to be only considered as a transmission ratio. The flexibility of the low-speed shaft is modelled by a spring-damper element. The whole scheme is presented in Figure 1.

The drivetrain is represented by the differential equations

$$\begin{array}{l}{J}_r {\dot{\omega }}_r+(B_r+D_r) {\omega }_r-(D_r/n_x) {\omega }_g+K_r\Delta \theta -T_a=0\\ J_g {\dot{\omega }}_g-(D_r/n_x) {\omega }_r+(D_r/n_x^2+B_g) {\omega }_g-(K_r/n_x) \Delta \theta +T_g=0,\\ \Delta \dot{\theta }-{\omega }_r+{\omega }_g/n_x=0\end{array}$$

(1)

where J, B, D, K, and n stand for mass second moments of inertia, viscous friction coefficient, damping coefficient, stiffness coefficient, and the gearbox ratio, respectively. Rotation speed, rotation angle, and torque are variables denoted by ω, θ, and T, respectively. Finally, the subscripts are r for the rotor, g for the generator, and x for the gearbox. The variable Δθ is the twist angle in the flexible low-speed shaft. The modelling process follows the idea introduced in [33] for the definition of the tip-speed ratio λ, i.e.,

$$\lambda =\frac{R {\omega }_r}{v_w}=\frac{ {\omega }_r}{\mu },$$

(2)

where μ is defined by μ = v_w/R, with R as the rotor diameter. Rewriting ω_r as

$${\omega }_r=\mu \lambda ,$$

(3)

and introducing it in (1), the model becomes

$$\begin{array}{l}\mu J_r \dot{\lambda }+(B_r+D_r) \mu \lambda -(D_r/n_x) {\omega }_g+K_r\Delta \theta -T_a=0\\ J_g {\dot{\omega }}_g-(D_r/n_x)\mu \lambda +(D_r/n_x^2+B_g) {\omega }_g-(K_r/n_x)\Delta \theta +T_g=0\\ \Delta \dot{\theta }-\mu \lambda +{\omega }_g/n_x=0\end{array}$$

(4)

Hence, the equations can also be rewritten into the form

$$\begin{array}{l} \dot{\lambda }=-\frac{B_r+D_r}{J_r} \lambda +\frac{D_r}{\mu n_x{J}_r} {\omega }_g-\frac{K_r}{\mu J_r}\Delta \theta +\frac{1}{\mu J_r}{T}_a\\ {\dot{\omega }}_g=\frac{D_r}{n_x{J}_g}\mu \lambda -\frac{D_r+n_x^2{B}_g}{n_x^2 J_g} {\omega }_g+\frac{K_r}{n_x{J}_g}\Delta \theta -\frac{1}{J_g}{T}_g.\\ \Delta \dot{\theta }=\mu \lambda -\frac{1}{n_x}{\omega }_g\end{array}$$

(5)

Using x₁ = λ, x₂ = ω_g, and x₃ = Δθ as state variables, (5) changes to

$$\begin{array}{l} {\dot{x}}_1=-\frac{B_r+D_r}{J_r} x_1+\frac{D_r}{\mu n_x{J}_r} x_2-\frac{K_r}{\mu J_r}{x}_3+\frac{1}{\mu J_r}{T}_a\\ {\dot{x}}_2=\frac{D_r}{n_x{J}_g}\mu x_1-\frac{D_r+n_x^2{B}_g}{n_x^2 J_g} x_2+\frac{K_r}{n_x{J}_g}{x}_3-\frac{1}{J_g}{T}_g\\ {\dot{x}}_3=\mu x_1-\frac{1}{n_x}{x}_2\end{array}$$

(6)

which can also be described in the matrix form

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right] =\left[\left[\begin{array}{ccc}-\frac{B_r+D_r}{J_r}& \frac{D_r}{n_x{J}_r}& -\frac{K_r}{J_r}\\ \frac{D_r}{n_x{J}_g}& -\frac{D_r+n_x^2{B}_g}{n_x^2 J_g}& \frac{K_r}{n_x{J}_g}\\ 1& -1/n_x& 0\end{array}\right]\otimes \left[\begin{array}{ccc}1& \frac{1}{\mu } & \frac{1}{\mu }\\ \mu & 1& 1\\ \mu & 1& 1\end{array}\right]\right] \left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right] + \left[\begin{array}{c}\frac{1}{\mu J_r}{T}_a\\ \frac{-1}{J_g}{T}_g\\ 0\end{array}\right].$$

(7)

The symbol ⊗ denotes element-wise multiplication. In compact form, it is represented by

$$\dot{x} =A(\mu )x+f(\mu ).$$

(8)

The aerodynamic torque T_a can be described by [34]

$$T_a =\frac{\pi }{2} {\rho }_a{R}^5\frac{C_p(\lambda ,\beta )}{{\lambda }^3}{\omega }_r^2=\frac{\pi }{2} {\rho }_a{R}^5\frac{C_p(\lambda ,\beta )}{\lambda }{\mu }^2=K C_p(\lambda ,\beta )\frac{{\mu }^2}{\lambda }.$$

(9)

In this case, the function f(μ) in (8) becomes

$$f(\mu )= {\left[\begin{array}{ccc}\frac{K}{J_r}\frac{\mu C_p(x_1,\beta )}{x_1}& \frac{-1}{J_g}{T}_g& 0\end{array}\right]}^{\mathrm{T}}.$$

(10)

The pitch angle β and the electromagnetic torque T_g are the inputs, which are defined in the vector form by u = [β T_g]^T. The output is selected from

$$y(t)={\omega }_g=\left[\begin{array}{ccc}0& 1& 0\end{array}\right] x(t).$$

(11)

3. Steady-State Modelling

The steady-state model is constituted by a set of algebraic equations that describe the system resting on the equilibrium point [35,36]. However, in the case of wind turbines, this equilibrium depends on the wind speed [37]. Thus, the wind turbine is characterised by a locus of equilibrium points, and, therefore, there is a steady-state model for each wind speed value [38]. Setting (5) to zero yields the steady-state model for a given wind speed, i.e.,

$$\begin{array}{l}-(B_r+D_r) {\lambda }_{\infty }+\frac{D_r}{\mu n_x} {\omega }_{g\infty }-\frac{K_r}{\mu }\Delta {\theta }_{\infty }+\frac{1}{\mu }{T}_{a\infty }=0\\ D_r\mu {\lambda }_{\infty }-\frac{D_r+n_x^2{B}_g}{n_x^2} {\omega }_{g\infty }+K_r\Delta {\theta }_{\infty }-T_{g\infty }=0,\\ \mu {\lambda }_{\infty }-\frac{1}{n_x}{\omega }_{g\infty }=0\end{array}$$

(12)

where the notation x_∞ means x(t) for t→∞ and represents the steady-state value of variable x. The equivalent friction coefficient B_e is derived as

$$B_e=B_r+n_x^2{B}_g=\frac{n_x(T_{a\infty }-n_x{T}_{g\infty })}{{\omega }_{g\infty } }$$

(13)

by integrating the third equation of (12) into the other two. The first equation and (13) yield

$$\Delta {\theta }_{\infty }=\frac{n_x(n_x{B}_g{T}_{a\infty }+B_r{T}_{g\infty })}{B_e{K}_r }$$

(14)

and, finally,

$${\lambda }_{\infty }=\frac{T_{a\infty }-n_x{T}_{g\infty }}{\mu B_e }=\frac{R {\omega }_{g\infty }}{v_{w\infty } }.$$

(15)

The wind speed v_w∞ has to be understood as a mean value assumed to be constant over time. The friction coefficients B_r and B_g are normally unknown. From (13), an equivalent value can be computed, but at least one coefficient must be given a priori to determine the other. Moreover, both parameters are positive. Hence, the given coefficient must always satisfy one of the following two conditions

$$0<B_g<\frac{T_{a\infty }-n_x{T}_{g\infty }}{n_x {\omega }_{g\infty } } \mathrm{or} 0<B_r<\frac{n_x (T_{a\infty }-n_x{T}_{g\infty })}{{\omega }_{g\infty } }.$$

(16)

Consequently, either the first element or the second element (depending on the chosen coefficient) in the diagonal of the system matrix A of (8) is also variable. If it is assumed that both parameters have the same value, they can be found using

$$B_r=B_g=\frac{n_x(T_{a\infty }-n_x{T}_{g\infty })}{(1+n_x^2) {\omega }_{g\infty } },$$

(17)

and, in this case, both parameters are variable with the wind speed.

4. Observer Design Using the Model Based on the Tip-Speed Ratio

4.1. Observer Configuration

The model derived in the previous section is used to design an observer to estimate non-measurable variables [39] such as, for instance, effective wind speed. The scheme is presented in Figure 2.

The observer has two main characteristics: on the one hand, it is nonlinear in the inputs [40]; on the other hand, the matrix A has variable parameters [41]. A block diagram of the observer is shown in Figure 3.

4.2. Determination of the Observer Gain

The equations that describe the observer are given by

$$\begin{array}{l}\dot{\hat{x}}(t) =A(\mu )\hat{x}(t)+f[\mu ,\hat{x}(t),u(t)]+L(\mu ) [y(t)-\hat{y}(t)],\\ \widehat{y}(t)=c^{\mathrm{T}}x(t)\end{array}$$

(18)

which can be rewritten as

$$\dot{\hat{x}}(t) =[A(\mu )-L(\mu ) c^{\mathrm{T}}] \widehat{x}(t)+f[\mu ,\widehat{x}(t),u(t)]+L(\mu )y(t),$$

(19)

where the observer gain is L = [l₁ l₂ l₃]^T, and the system matrix is defined by

$$A(\mu )-L(\mu ) c^{\mathrm{T}}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& 0\end{array}\right]\otimes \left[\begin{array}{ccc}1& 1/\mu & 1/\mu \\ \mu & 1& 1\\ \mu & 1& 1\end{array}\right]+\left[\begin{array}{ccc}0& l_1& 0\\ 0& l_2& 0\\ 0& l_3& 0\end{array}\right]=\left[\begin{array}{ccc}{a}_{11}& a_{12}/\mu +l_1& a_{13}/\mu \\ \mu a_{21}& a_{22}+l_2& a_{23}\\ \mu a_{31}& a_{32}+l_3& 0\end{array}\right]$$

(20)

such that the characteristic equation is defined by the polynomial

$$P(s)=\det [sI-A(\mu )+L(\mu ) c^{\mathrm{T}}]=\det \left[\begin{array}{ccc}s-a_{11}& -a_{12}/\mu +l_1(\mu )& -a_{13}/\mu \\ -\mu a_{21}& s-a_{22}+l_2(\mu )& -a_{23}\\ -\mu a_{31}& -a_{32}+l_3(\mu )& s\end{array}\right]=0,$$

(21)

which can also be represented by

$$P(s)=R(s)+\mu T(s)=0,$$

(22)

where R, T, and P are the polynomials

$$\begin{array}{l}R(s)=s^3+r_1 s^2+r_2 s+r_3,\\ T(s)=t_0 s+t_1, \mathrm{and}\\ P(s)=s^3+p_1 s^2+p_2 s+p_3.\end{array}$$

(23)

with the coefficients

$$\begin{array}{l}{r}_1 =l_2-a_{11}-a_{22},\\ r_2=-a_{11}{l}_2+a_{23}{l}_3+a_{11}{a}_{22}-(a_{13}{a}_{31}+a_{23}{a}_{32}+a_{12}{a}_{21}),\\ r_3=-a_{13} a_{31} l_2+(a_{13} a_{21}-a_{11} a_{23}) l_3+a_{13} a_{31} a_{22}+a_{23} a_{32} a_{11}-(a_{12} a_{23} a_{31}+a_{13} a_{32} a_{21}),\\ t_0 =a_{21} l_1, \mathrm{and}\\ t_1=a_{23} a_{31} l_1.\end{array}$$

(24)

The idea behind the observer design is to keep the positions of the observer poles constant for all values of μ. Hence, pole placement [42] is an adequate design method. The given poles specify the coefficients of the polynomial P(s), which are always constant. The coefficient comparison leads to

$$\begin{array}{l}{p}_1=r_1,\\ p_2=r_2+\mu t_0, \mathrm{and}\\ p_3=r_3+\mu t_1.\end{array}$$

(25)

Finally, the observer gain is obtained from

$$\left[\begin{array}{c}{l}_1(\mu )\\ l_2(\mu )\\ l_3(\mu )\end{array}\right]={\left[\begin{array}{ccc}0& 1& 0\\ \mu a_{21}& -a_{11}& a_{23}\\ \mu a_{23} a_{31}& -a_{13} a_{31}& a_{13} a_{21}-a_{11} a_{23}\end{array}\right]}^{-1}\left[\begin{array}{l}{p}_1+a_{11}+a_{22}\hfill \\ p_2+a_{13}{a}_{31}+a_{23}{a}_{32}+a_{12}{a}_{21}-a_{11}{a}_{22}\hfill \\ p_3+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}-(a_{13} a_{22} a_{31}+a_{11} a_{23} a_{32})\hfill \end{array}\right].$$

(26)

The values of the wind speed cut-in (v_ci) and cut-out (v_co) are the span limits for the operation of the wind turbine. Consequently, the variable μ is also constrained in light of

$${\mu }_{min}=\frac{v_{ci}}{R}\le \mu \le \frac{v_{co}}{R}={\mu }_{max}.$$

(27)

In addition, ω_g∞, T_a∞, and T_g∞ are also bounded. This implies the boundness of a₁₁ and a₂₂, which are variable due to (16). The consequence of a variable matrix A is that (26) must be continuously recomputed in the runtime.

4.3. Stability and Convergence

The dynamic representation of the observer (19) is now rewritten as

$$\dot{\hat{x}}(t) =[A(\mu )-L(\mu ) c^{\mathrm{T}}] \hat{x}(t)+g[\mu ,\hat{x}(t),u(t),y(t)],$$

(28)

where function g is

$$g[\mu ,\widehat{x}(t),u(t),y(t)] =f[\mu ,\widehat{x}(t),u(t)]+L(\mu ) y(t).$$

(29)

Under the consideration of the model parameters and an optimal control law for the generator control, the function g can be expressed by

$$g(\mu ,x,u,y)= \left[\begin{array}{c}\frac{K}{J_r}\frac{\mu C_p(x_1,\beta )}{x_1}+l_1(\mu ) {\omega }_g\\ \frac{-K_{opt}}{n_x{J}_g}{\omega }_g^2+l_2(\mu ) {\omega }_g\\ l_3(\mu ) {\omega }_g\end{array}\right].$$

(30)

The conventional analysis considers the stability of [A(μ)−L(μ) c^T] and imposes the condition

$$\left|\right|g(\mu ,x,u,y)\left|\right| \le \gamma \left|\right|x\left|\right|$$

(31)

with γ > 0 for g (see, e.g., [43,44,45]). For the stability of [A(μ) − L(μ) c^T], the characteristic Equation (22) is analysed. It is important to note that due to (27) and (16), P(s) is an interval polynomial with

$$[p_0^{-}{p}_0^{+}]=[1 1], [p_1^{-}{p}_1^{+}]=[r_{1,min} r_{1,max}], [p_2^{-}{p}_2^{+}]=[(r_{2,min}+{\mu }_{min}{t}_0) (r_{2,max}+{\mu }_{max}{t}_0)], \mathrm{and} [p_3^{-}{p}_3^{+}]=[(r_{3,min}+{\mu }_{min}{t}_1) (r_{3,max}+{\mu }_{max}{t}_1)].$$

(32)

This means that the observer is an interval observer [43,45,46]. The stability requires that the four Kharitonov’s polynomials [47]

$$\begin{array}{c}{P}_1(s)=s^3+p_1{s}^2+p_2^{-}s+p_3^{-}, P_2(s)=s^3+p_1{s}^2+p_2^{+}s+p_3^{+}, \\ P_3(s)=s^3+p_1{s}^2+p_2^{-}s+p_3^{+}, \mathrm{and} P_4(s)=s^3+p_1{s}^2+p_2^{+}s+p_3^{-}\end{array}$$

(33)

are Hurwitz.

According to the definition of the state variables, ||x|| is given by

$$||x||=\sqrt{{\lambda }^2+{\omega }_g^2+\Delta {\theta }^2},$$

(34)

and for g, one has

$$||g||=\sqrt{{\left[\frac{K C_p(\beta ,x_1)\mu }{x_1}+l_1 {\omega }_g\right]}^2+{\left(\frac{K_{opt}{\omega }_g^2}{J_g}+l_2 {\omega }_g\right)}^2+l_3^2 {\omega }_g^2}.$$

(35)

C_p, λ, and μ are always bounded. For partial load operation, ω_g < ω_g,rated and, by assuming perfect pitch control in full load operation, it follows ω_g = ω_g,rated. Hence, it is always possible to find a finite γ such that

$$\gamma >\sqrt{\frac{{\left(K C_{p,max}{\mu }_{max}/{\lambda }_{min}+l_1 {\omega }_{g,rated}\right)}^2+\left[{(K_{opt}/J_g+l_2)}^2+l_3^2\right]{\omega }_{g,rated}^2}{{\lambda }_{min}^2+{\omega }_{g,min}^2+\Delta {\theta }_{g,min}^2}},$$

(36)

where the right side is the upper bound for the system and (31) holds.

5. Applications

The proposed model can be used for several objectives. However, only two applications are presented in this work, namely, the control during the partial load operation, which is achieved by using tip-speed ratio control, and the estimation of the effective wind speed. Both approaches are described in the following.

5.1. Tip-Speed Ratio Control in Partial Load Operation

In the partial load operation, the tip-speed ratio is maintained constant at the value of λ* in order to maximise the power coefficient C_p and, consequently, the energy conversion. There are several approaches to this objective (see, e.g., [21,48,49]), but one of them is the tip-speed ratio control (TSRC), which uses a closed-loop system with λ* as the set point (see Figure 4).

The control law is given by

$$T_{gi} =T{ }_{g,rated}-f_{con}({\lambda }^{*}-\widehat{\lambda }),$$

(37)

where f_con is the controller function, as, for example, the PI control law (proportional-integral), which in such a case can be expressed in the parallel decoupled form as

$$f_{con}({\lambda }^{*}-\widehat{\lambda }) =K_m[K_p({\lambda }^{*}-\widehat{\lambda })+K_i{\displaystyle \int ({\lambda }^{*}-\widehat{\lambda }) }dt]$$

(38)

with K_p and K_i as proportional and integral gains, respectively. K_m is an adaptive adjustment factor defined as

$$K_m =K_1+K_2\mathrm{sech}(K_3({\lambda }^{*}-\widehat{\lambda }))$$

(39)

such that K_m = K₁ for very large ${\lambda }^{*}-\widehat{\lambda }$ and K_m = K₁ + K₂ for small ${\lambda }^{*}-\widehat{\lambda }$.

Notice that when the control error is equal to zero, the system reaches the rated values, and the linear controller provides a control variable equal to zero. However, the input torque may not be zero. Thus, this situation is avoided in the control law (37), because, when the controller output is zero, the torque is equal to its rated value.

5.2. Estimation of Effective Wind Speed

From (2), it follows that the wind speed can be calculated by

$$v_w =\frac{R}{\lambda }{\omega }_r,$$

(40)

where ω_r and λ are not measurable variables, but they can be estimated by the observer, i.e.,

$${\widehat{v}}_w =\frac{R}{\widehat{\lambda }}{\widehat{\omega }}_r.$$

(41)

This idea is schematised in Figure 5, where all state variables provided by the observer are brought into play to estimate the effective wind speed.

6. Numerical Example

The effectiveness of the proposed modelling approach is demonstrated by a numerical example regarding the wind speed estimation (Figure 5) as well as the control in the partial load operation (Figure 4). The numerical example is carried out by using a 20 MW reference wind turbine from [50], which has been modified and studied for modelling and control in [51,52].

The controller and observer designs, the simulation of both, and the evaluation of the results are carried out using MATLAB^® and Simulink^®. The simulation of the wind turbine is undertaken using OpenFAST (formerly known as FAST) [53].

6.1. Parameters and Rated Values

The relevant parameters of the reference wind turbine for model (5) are presented in

Table 1. Essential parameters of the 20 MW reference wind turbine.

Parameter	Notation	Values	Units
Equivalent shaft spring constant	Kr	6.94 × 109	Nm/rad
Equivalent shaft damping constant	Dr	4.97 × 107	Nm/(rad/s)
Gearbox ratio	nx	164	--
Gearbox efficiency, generator efficiency	ηx, ηg	97.8, 96.1	%
Rotor radius	R	138	m
Rotor mass moment of inertia	Jr	2919.66 × 106	kg m2
Mass moment of inertia of the generator	Jg	7248.32	kg m2

, and the rated values are summarised in

Table 2. Rated values for the 20 MW wind turbine.

Variable	Notation	Values	Units
Rated mechanical power	Pm,rated	21.191	MW
Rated rotor speed	ωr,rated	7.1567	rpm
Rated generator speed	ωg,rated	1173.7	rad/s
Cut-in, rated, and cut-out wind speed	vci, vw,rated, vco	4.48, 10.92, 25	m/s
Rated aerodynamic torque (on the low-speed shaft)	Ta,rated	28434.7	kNm
Rated generator torque (on the high-speed shaft)	Tg,rated	169.76	kNm
Maximum value of the power coefficient Cp	Cp,max	0.4812	--
Optimal tip-speed ratio	λ*	10.115	--
Density of air	ρa	1.225	kg/m3

.

6.2. Model Calibration

It is shown in Section 3 that the steady-state model provides important information to obtain unknown parameters as well as variables in the stationary state. Steady-state variables depend on the wind speed. Thus, the steady-state values for the rated wind speed are summarised in

Table 3. Steady-state values of the 20 MW wind turbine for the rated wind speed.

Variable	Notation	Values	Units
Rotor speed	ωr∞	0.7495	rad/s
Generator speed	ωg∞	122.912	rad/s
Aerodynamic torque (on the low-speed shaft)	Ta∞	28,225.215	kNm
Generator torque	Tg∞	168.424	kNm

. Steady-state values for other wind speeds are shown in the form of curves, in particular, in Figure 6.

From (17), it follows

$$B_r=B_g=\frac{n_x(T_{a\infty }-n_x{T}_{g\infty })}{(1+n_x^2) {\omega }_{g\infty } }=0.03,$$

(42)

which leads to

B_e = 806.003.

(43)

Applying these values to (14) and (15), Δθ_∞ and λ_∞ are computed to be Δθ_∞ = 4.067 × 10⁻⁶ rad and λ_∞ = 9.4428. The range for μ is

$$0.0325<\mu <0.1816,$$

(44)

and the steady-state value μ_∞ is 0.0793.

These stationary values dependent on the wind speed are obtained at the overrated wind for the corresponding value of the pitch angle according to the pitch angle locus of Figure 6a. In Figure 6b, the variable parameter B_e, with the axis on the left, and the steady-state values of μ, with the axis on the right are presented against the wind speed changing from 0 to 25 m/s. Finally, the stationary twist angle of the low-speed shaft (Δθ_∞), with the axis on the left, and the tip-speed ratio, with the axis on the right are shown in Figure 6c against a wind speed changing from 0 to 25 m/s.

6.3. Observer Design

As mentioned previously, the observer is designed by using the pole placement approach to maintain the poles always in the same position independently of the wind speed changes (or changes in the variable μ, which also varies with the wind speed).

The selection of the observer’s poles is carried out, taking into consideration reaction time, convergence, and estimation errors. Compromising all these criteria, the poles have been chosen after several runs of the simulation code. The final decision yields p_o1 = −50.0, p_o2 = −115.0, and p_o3 = −10.0.

The observer matrix gain is computed in the runtime for each value of the wind speed. Thus, the three elements of the matrix gain are obtained in the form of time series, which are portrayed in Figure 7.

It can be observed that while l₁, which is strongly related to λ, presents important variability between 125.8 and 241.09, l₂ and l₃ remain practically constant for the whole time with average values of 174.73 and 1.19, respectively.

6.4. Tip-Speed Ratio Control in Partial Load Operation

The control system design for partial load operation has been presented in the previous section and is schematised in Figure 4. From Table 2, it follows that the set point for the tip-speed ratio is λ^* = 10.115. The aim of the control system in partial load operation is to maximise power extraction. The metric used to evaluate the control performance is the converted average energy expressed in kWh. For the comparison, the classic optimal torque control law given by

$$T_{gi} =\frac{\pi }{2} {\rho }_a{R}^5\frac{C_{p,max}}{n_x^2{({\lambda }^{*})}^3}{\omega }_g^2=K_{opt} {\omega }_g^2$$

(45)

is used, and, according to the information presented in Table 1 and Table 2, the optimal gain K_opt is computed to be 1132.9. The PI controller of the TSRC has been tuned but not optimized, and the corresponding parameters are K_p = 1536.2 and K_i = –0.1732. In addition, an anti-windup mechanism based on the back calculation concept [54] has been activated with a gain of K_a = 10.5. Parameters K₁, K₂, and K₃ in the control law (39) are set to 0.43, 0.57, and 0.1, respectively.

The power curves obtained in the simulation under both control laws are portrayed in Figure 8. It corresponds to a calculated energy of 3385 kWh for optimal torque control and 3667 kWh for tip-speed ratio control.

It is very important to emphasise that the intention is not to compare the performances of both control algorithms but to show that the tip-speed ratio control in the proposed configuration can be implemented with satisfactory and comparable results by using the described approach.

6.5. Estimation of State Variables

The most important state variable to be estimated is the tip-speed ratio since the model has been modified in order to obtain it. On the other hand, a good estimation of this variable is needed for successful generator control in partial load operation, as well as for wind speed estimation.

The simulation results comparing the real tip-speed ratio obtained from the simulation of the high-definition model with the observer output are presented in Figure 9, where the delay manifested in its reaction can be appreciated. Moreover, the observer also has a smoothing characteristic, which is not only visible in Figure 9, but is more evident in the frequency spectrum of Figure 10. Hence, the qualitative analysis indicates a satisfactory result. A more detailed evaluation is provided by using a quantitative analysis in 6.6.

The variable μ, which is dependent on the unknown wind speed as shown in (2), also affects the tip-speed ratio λ. This dependence corresponds to an R-squared value of 0.9695, and is shown in Figure 11. Hence, it is also necessary to estimate μ, whose result is depicted in Figure 12.

The twist angle of the low-speed shaft Δθ_r is more difficult to verify because it is not directly obtainable from the high-resolution model. However, it can be studied from two points of view. The first one is to calculate Δθ from measurable data. This is shown in Figure 13. The smoothing feature of the observer is also clearly exposed in the frequency spectrum of Figure 14.

The other way is to use the rotor speed. Notice that the rotor speed is obtained from the third equation of (1) by using the derivative of the twist angle and the generator speed. The generator speed has a very high degree of agreement, because it is a measurable variable used as an input to the observer. Hence, the estimated rotor speed would be incorrect if the estimated derivative of the twist angle was erroneous. However, this is not the case, as can be appreciated in Figure 15 as well as in

Table 4. Quantitative evaluation of the model and observer performance.

Parameter	λ	ωg	Δθr	ωr	μ	vw
MAE (Mean Absolute Error)	0.2676	0.0026	9.6978 × 10−6	4.7753 × 10−4	0.1646	0.0012
MSE (Mean Square Error)	0.2438	1.555 × 10−5	3.3227 × 10−10	8.8550 × 10−7	0.0925	4.8028 × 10−6
R-squared (Square of the correlation)	0.9336	1.0000	0.5130	1.0000	0.9528	0.9535
RMSE (Root Mean Squared Error)	0.4938	0.0039	1.8228 × 10−5	9.4101 × 10−4	0.3041	0.0592
NRMSE (Normalized RMSE)	0.0713	4.4283 × 10−5	0.0647	0.0017	0.0597	0.0022

.

In the case of the generator speed, a very high agreement between the real signal and estimates is expected since this is a measurable output of the system and an input to the observer. This is confirmed in Figure 16 as well as in Table 4, where the cross-correlation factor equal to one supports the expectation.

Finally, the wind speed is estimated by using Equation (41), which is an equation that is based on two estimated state variables, and therefore, a poor agreement might be anticipated. However, the result can be considered highly satisfactory, as evidenced by Figure 17 and the last column of Table 4.

6.6. Quantitative Analysis of the Observer Performance

To quantify the performance of the model and the designed observer, several metrics for estimation goodness are computed and assessed. Table 4 provides a summary of the findings.

For R-squared (R²), the closer the value is to 1, the better the observer’s behaviour is. For all other metrics, the best results are characterised by values closer to 0. The advantage of MSE over MAE is that MSE punishes significant errors more severely. On the other hand, RMSE maintains the same units as the original signal, which does not happen with MSE. Finally, RMSE provides the offset difference between both signals in the same unit (0 for no offset, i.e., full coincidence), and R-squared highlights the temporal shifting between both signals (1 for full coincidence). Satisfactory results from the point of view of the SME (Subject Matter Experts, [55,56]) are given for a threshold above 0.90 for R-squared, and a threshold below 0.50 for RMSE.

Under such conditions, the results of Table 4 are satisfactory in general, with acceptable values of RMSE and R-squared for λ. The R-squared value of Δθ_r is low, but the variable is not particularly important for the general control system.

7. Conclusions

This contribution describes a new modelling idea that includes the tip-speed ratio as a state variable in the dynamic model of the drivetrain. This concept leads to a model with interval parameters, for which an interval observer is designed.

The advantage of this model lies in the fact that the effective wind speed, which is normally unavailable, is not necessary to compute the tip-speed ratio. Contrarily, the estimates provided by the observer can be used to estimate the effective wind speed.

Another application is the torque control in partial load operation by using a tip-speed ratio control loop, where the tip-speed ratio estimated by the interval observer is fed back to the controller.

Both applications, i.e., effective wind speed estimation and tip-speed ratio control, are studied using a 20 MW reference wind turbine. The simulation results show the practical applicability of the model. It is important to note that the high-definition model of the reference wind turbine is reduced to a very simple third-order model. Despite this, the results have been very satisfactory. It is therefore to be expected that by using models of higher order, the outcomes can be even better.

Hence, the next step in the investigation is to consider models that include additional degrees of freedom. In particular, three-mass models, which include an additional degree of freedom for the blades, have been considered to date. Moreover, control algorithms will be implemented for testing in real-time operation using a Hardware-in-the-Loop (HiL) facility [57].