2.2 Modeling of the asynchronous generator

For the induction generator, the Park model is the model that is commonly used [16]. After applying the synchronously rotating reference frame transformation to the stator and rotor fluxes equations of the generator, the following differential equations describe the dynamics of the rotor and stator fluxes [17]:

$$\begin{cases} \dot{\Phi}\_{dr} = w\_b \left( v\_{dr} + (w\_s - w\_r) \Phi\_{qr} - R\_r i\_{dr} \right) \\ \dot{\Phi}\_{qr} = w\_b \left( v\_{qr} - (w\_s - w\_r) \Phi\_{dr} - R\_r i\_{qr} \right) \\ \dot{\Phi}\_{ds} = w\_b \left( v\_{ds} + w\_s \Phi\_{qs} - R\_s i\_{ds} \right) \\ \dot{\Phi}\_{qr} = w\_b \left( v\_{qs} - w\_s \Phi\_{ds} - R\_s i\_{qs} \right) \end{cases} \tag{5}$$

3. Estimation algorithms

Advanced Monitoring of Wind Turbine DOI: http://dx.doi.org/10.5772/intechopen.84840

where x∈Rn, u∈ Rm, y ∈R<sup>s</sup>

Eq. (12) in the following form:

K1

δ<sup>1</sup> >0; ⋯; δ<sup>p</sup> > 0∈ N <sup>∗</sup> � � such as:

2 6 4

⋱

Kp

3 7

ii. σμkþ<sup>v</sup> ¼ σμkþv�<sup>1</sup> þ δr, k ¼ 1, ⋯, p, v ¼ 1, ⋯, η<sup>k</sup> � 1

<sup>∂</sup>zj 6¼ 0 ) σ<sup>i</sup> ⩾σj, i, j ¼ 1, ⋯, n, j 6¼ μk, k ¼ 1, …, p

\_

And there exists T1 such as, for all T, 0 < T < T1.

S Sð Þ¼ ; δ

Let K ¼

matrix.

iii. <sup>∂</sup>φ<sup>i</sup>

So,

With,

27

systems. Consider the following nonlinear system:

.

The observer must satisfy the following theorem [20]:

i. The function φ is globally Lipschitz uniformly to u.

This observer class is applied for nonlinear system classes of the form Eq. (12). Its applications are so large [18, 19]. We briefly present the developed survey in [20] that points up the synthesis of observers adapted to the observable nonlinear

x\_ ¼ f xð Þþ g xð Þu

First, the system Eq. (12) must be uniformly locally observable, and then it will be possible to make the variable change z = Γ(x) that will transform the system

z\_ ¼ Az þ φð Þ u; z

<sup>5</sup> an adequate size matrix such as, for every Ki block, the

<sup>θ</sup> K Cð Þ ^z � y (14)

3 7 5 (12)

(13)

<sup>y</sup> <sup>¼</sup> h xð Þ �

<sup>y</sup> <sup>¼</sup> Cz �

Ak � KkCk should give all its eigenvalues with negative real part: Let's suppose that there exists two integer sets f g σ1,⋯, σ<sup>n</sup> ∈Z and

^<sup>z</sup> <sup>¼</sup> <sup>A</sup>^<sup>z</sup> <sup>þ</sup> <sup>φ</sup>ð Þ� ^z; <sup>u</sup> <sup>S</sup>�<sup>1</sup>

<sup>S</sup><sup>δ</sup>1<sup>Δ</sup> <sup>S</sup><sup>δ</sup><sup>1</sup> � �

1

S

⋱

⋱

S<sup>δ</sup>pΔ S<sup>δ</sup> � �<sup>p</sup>

Sηθ�<sup>1</sup>

is an exponential observer for the system Eq. (13) as well.

2 6 4

Δθð Þ¼ S

3.1 High gain observer

where ws = 1 is the synchronous angular speed in the synchronous frame and wb = 2πf rad/s is the base angular speed, with f = 60 Hz. With additional variables stator-rotor mutual flux Φdm and Φqm, rotor current idr and iqr and stator current ids and iqs can be expressed as:

$$\begin{cases} \dot{i}\_{dr} = \frac{\Phi\_{dr} - \Phi\_{dm}}{L\_{lr}} \\\\ \dot{i}\_{qr} = \frac{\Phi\_{qr} - \Phi\_{qm}}{L\_{lr}} \end{cases} \tag{6}$$

$$\begin{cases} \dot{i}\_{ds} = \frac{\Phi\_{ds} - \Phi\_{dm}}{L\_{ls}} \\\\ \dot{i}\_{qs} = \frac{\Phi\_{qs} - \Phi\_{qm}}{L\_{ls}} \end{cases} \tag{7}$$

where

$$\begin{aligned} \Phi\_{dm} &= L\_{ad} \left( \frac{\Phi\_{dr}}{L\_{lr}} + \frac{\Phi\_{ds}}{L\_{ls}} \right) \\ \Phi\_{qm} &= L\_{aq} \left( \frac{\Phi\_{qr}}{L\_{lr}} + \frac{\Phi\_{qs}}{L\_{ls}} \right) \end{aligned} \tag{8}$$

are the stator-rotor mutual flux.

Where constants Lad and Laq are the (d-q) mutual flux factors, expressed as:

$$L\_{ad} = L\_{aq} = \frac{1}{\frac{1}{L\_m} + \frac{1}{L\_d} + \frac{1}{L\_{lr}}} \tag{9}$$

The relationship between mechanical torque Tm, electrical torque Te and rotor speed wr can be shown by the following differential equation,

$$
\dot{w}\_r = \frac{1}{2H}(T\_m - T\_e - Fw\_r) \tag{10}
$$

where constant F is the friction factor and H is the generator inertia, and Te, the electrical torque which can be expressed as:

$$T\_e = \Phi\_{ds} i\_{qs} - \Phi\_{qs} i\_{ds} \tag{11}$$

These equations are derived in [4] and all parameters are defined in per unit based on the generator ratings and synchronous speed.
