*2.2. Aerodynamic Model*

The wind power extracted by a WT is represented as [34,35]

$$P\_W = \frac{1}{2} \rho \pi R^2 V^3 \mathcal{C}\_\mathbb{P}(\mathcal{J}, \lambda) \tag{1}$$

$$
\lambda = \frac{R\omega\_{\rm m}}{V} \tag{2}
$$

where *β* is the pitch angle, *ρ* is the air density, *V* is the wind speed, *R* is the radius of WT, *C*p is the power coefficient, *λ* is the tip speed ratio, and *ω*m is the mechanical rotation speed. The *C*p can be defined as a function of *β* and *λ*

$$C\_{\rm P} = 0.22(\frac{116}{\lambda\_i} - 0.4\beta - 5)e^{-\frac{-12.5}{\lambda\_i}}\tag{3}$$

$$\frac{1}{\lambda\_i} = \frac{1}{\lambda + 0.08\beta} - \frac{0.035}{\beta^3 + 1} \tag{4}$$

A hydraulic/mechanical actuator can vary the blade pitch. The following first order linear model represents a simplified model of the dynamics:

$$
\dot{\beta} = -\frac{\beta}{\tau\_{\beta}} + \frac{\beta\_{\rm r}}{\tau\_{\beta}} \tag{5}
$$

where *β*r is required pitch angle, and *τβ* is the actuator time constant.

The state-space model of the PMSG-WT is given as [35]:

$$
\dot{\mathbf{x}} = f(\mathbf{x}) + \mathbf{g}\_1(\mathbf{x})\boldsymbol{\mu}\_1 + \mathbf{g}\_2(\mathbf{x})\boldsymbol{\mu}\_2 + \mathbf{g}\boldsymbol{\varepsilon}(\mathbf{x})\boldsymbol{\mu}\_3 \tag{6}
$$

where

*f*(*x*) = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ − *β τβ* − *R*s *L*md *i*md + *<sup>ω</sup>*e*L*mq *L*md *<sup>i</sup>*mq − *R*s *<sup>L</sup>*mq *<sup>i</sup>*mq − 1*<sup>L</sup>*mq *<sup>ω</sup>*e(*<sup>L</sup>*md*i*md + *Ke*) 1 *<sup>J</sup>*tot(*<sup>T</sup>*<sup>e</sup> − *T*m − *T*f − *<sup>B</sup>ω*m) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ , *g*1(*x*)=[− *βτβ* 000]T, *g*2(*x*)=[<sup>0</sup> 1*L*md 0 0]T, *g*3(*x*)=[<sup>0</sup> 0 1*<sup>L</sup>*mq 0]T, *x* = [*β i*md *<sup>i</sup>*mq *<sup>ω</sup>*m]T, *u* = [*<sup>u</sup>*1, *u*2, *<sup>u</sup>*3]*<sup>T</sup>* = [*β*r, *V*md, *<sup>V</sup>*mq]*<sup>T</sup>*, *y* = [*y*1, *y*2, *<sup>y</sup>*3]*<sup>T</sup>* = [*h*1(*x*), *h*2(*x*), *<sup>h</sup>*3(*x*)]*<sup>T</sup>* = [*<sup>ω</sup>*m, *i*md, *<sup>i</sup>*mq]*<sup>T</sup>*

where *x* ∈ *R*<sup>4</sup> , *u* ∈ *R*<sup>3</sup> and *y* ∈ *R*<sup>3</sup> are state vector, input vector and output vector, respectively; *f*(*x*), *g*(*x*) and *h*(*x*) are smooth vector fields. *V*md and *<sup>V</sup>*mq are the d, q axis stator voltages, *i*md and *<sup>i</sup>*mq are the d, q axis stator currents, *L*md and *<sup>L</sup>*mq are d, q axis stator inductances, *R*s is the stator resistance, *p* is the number of pole pairs, *Ke* is the field flux given by the magnet, *J*tot is the total inertia of the drive train, *B* is the friction coefficient of the PMSG, *<sup>ω</sup>*e(= *p<sup>ω</sup>*m) is the electrical generator rotation speed, and *T*m, *T*f and *T*e are the WT mechanical torque, static friction torque and electromagnetic torque, respectively.

The electromagnetic torque is expressed as:

$$T\_{\mathbf{e}} = p\left[ (L\_{\rm mdd} - L\_{\rm mq}) i\_{\rm md} i\_{\rm mq} + i\_{\rm mq} K\_{\mathbf{e}} \right] \tag{7}$$
