**3.1. Wind turbine model**

The wind turbine mechanical power may be calculated as:

$$P\_{\rm mac} = 0.5 C\_p \rho \, r^2 \mathcal{U}\_W^3 \tag{1}$$

Where r is the radius of the wind turbine rotor, Uw is the average wind speed (m/s), *ρ*is the air-specific mass (kg m3) and CP is the wind turbine power coefficient [12]. CP is a function of the tip speed ratio λ and the blade pitch angle β, and can be expressed as:

$$\mathcal{C}\_{P}(\mathcal{k}, \mathcal{J}) = 0.73(\frac{115}{\lambda\_{\text{l}}} - 0.58 \,\beta^{2.14} - 13.2)e^{\frac{-18.4}{\lambda\_{\text{l}}}} \tag{2}$$

$$\lambda = \frac{1}{\frac{1}{\lambda} - 0.02\,\beta + \frac{0.003}{\beta^3 + 1}}\tag{3}$$

**Figure 1.** DFIGs Scheme

To represent the electrical and mechanical interaction between the electrical generator and wind turbine in transient stability studies, the global mass model is presented:

$$\frac{d\overline{o}\_r}{dt} = \frac{1}{2H\_T} \left( T\_m - \overline{T}\_c - \overline{D}\overline{o}\_r \right) \tag{4}$$

Where *<sup>D</sup>*¯ is the damping coefficient; and *HT* is the inertia constant, in seconds.

#### **3.2. The DFIG model**

For power system stability studies, the generator may be modeled as an equivalent voltage source behind a transient impedance. Since the stator dynamic are very fast, when com‐ pared with the rotor ones, it is possible to neglect them. The differential equations of the in‐ duction generator rotor circuits with equivalent voltage behind transient impedance as state variables can be given in a d-q reference frame rotating at synchronous speed. For adequate‐ ly representing the DFIG dynamics the second order model of the induction generator is used in the following per-unit form [14,15]:

$$
\overline{\boldsymbol{\sigma}}\_{ds} = -\overline{\boldsymbol{\mathcal{R}}}\_{s}\overline{\boldsymbol{i}}\_{ds} + \overline{\boldsymbol{X}}\,\overline{\boldsymbol{i}}\_{qs} + \overline{\boldsymbol{e}}\_{d}^{\cdot} \tag{5}
$$

$$
\overline{\boldsymbol{\upsilon}}\_{qs} = -\overline{\boldsymbol{\mathsf{R}}}\_{s}\overline{\boldsymbol{\mathsf{i}}}\_{qs} - \overline{\boldsymbol{\mathsf{X}}}'\overline{\boldsymbol{\mathsf{i}}}\_{ds} + \overline{\boldsymbol{\mathsf{e}}}'\_{q} \tag{6}
$$

$$\frac{d\overline{\overline{e\_d}}}{dt} = -\frac{1}{T\_o} \cdot \left[ \overline{e\_d} - \left( \overline{X} - \overline{X} \right) \overline{I}\_{q\*} \right] + s\alpha \rho\_s \overline{e\_q} - \alpha \rho\_s \frac{\overline{L}\_m}{\overline{L}\_{rr}} \overline{\sigma}\_{qr} \tag{7}$$

$$\frac{d\overline{\boldsymbol{e}}\_{q}^{\cdot}}{dt} = -\frac{1}{T\_{o}} \cdot \left[\overline{\boldsymbol{e}}\_{q}^{\cdot} + \left(\overline{\boldsymbol{X}} - \overline{\boldsymbol{X}}^{\cdot}\right)\overline{\boldsymbol{i}}\_{ds}\right] - s\boldsymbol{o}\_{s}\overline{\boldsymbol{e}}\_{d}^{\cdot} + \boldsymbol{o}\_{s}\overline{\frac{\overline{\boldsymbol{L}}\_{m}}{\overline{\boldsymbol{L}}\_{rr}}}\overline{\boldsymbol{v}}\_{d\boldsymbol{\nu}}\tag{8}$$

Where:

*V*ds, *V*qs: d and q axis stator voltages;

*R*s: stator resistance;

X´, X: transient reactance and the open circuit reactance;

*i*ds, *i*qs: d and q axis stator currents;

*ed* ' ,*eq* ' : d-axis and q-axis components of the internal voltage; *To* ' : open circuit time constant in seconds;

ωS: synchronous speed;

*L*m: mutual inductance;

*L*rr: rotor inductance;

*V*dr, *V*qr: d and q axis rotor voltages;

The components of the internal voltage behind the transient reactance are defined as:

$$
\overline{\mathcal{e}\_d} = -\frac{\overline{\alpha}\_s \overline{L}\_m}{\overline{L}\_{rr}} \cdot \overline{\mathcal{A}}\_{qr} \tag{9}
$$

$$
\overline{\mathbf{e}}\_q' = \frac{\overline{a}\_s \times \overline{L}\_m}{\overline{L}\_{rr}} \cdot \overline{\mathcal{A}}\_{dr} \tag{10}
$$

where λqr and λdr are d and q rotor fluxes. The reactances and transients open-circuit time constant are given:

$$\overline{X}' = \overline{\alpha}\_s \left( \overline{L}\_{ss} - \frac{\overline{L}\_m^2}{\overline{L}\_{rr}} \right) = \overline{X}\_s + \frac{\overline{X}\_r}{\overline{X}\_r + \overline{X}\_m} \tag{11}$$

$$
\overline{T\_o} = \frac{\overline{L}\_r + \overline{L}\_{su}}{\overline{R}\_r} = \frac{\overline{L}\_{rr}}{\overline{R}\_r} \tag{12}
$$

$$
\overline{X} = \overline{\alpha}\_s \overline{L}\_{ss} \tag{13}
$$

$$T\_o = \frac{\overline{L}\_{rr}}{2\pi f\_{base}\overline{R}\_r} \tag{14}$$

#### **3.3. DFIG converter model**

( ) <sup>1</sup>

For power system stability studies, the generator may be modeled as an equivalent voltage source behind a transient impedance. Since the stator dynamic are very fast, when com‐ pared with the rotor ones, it is possible to neglect them. The differential equations of the in‐ duction generator rotor circuits with equivalent voltage behind transient impedance as state variables can be given in a d-q reference frame rotating at synchronous speed. For adequate‐ ly representing the DFIG dynamics the second order model of the induction generator is

' '

' '

*d qs s q s qr o rr de <sup>L</sup> e XX i se v dt <sup>T</sup> <sup>L</sup>* =- × - - + - é ù

*q ds s d s dr o rr de <sup>L</sup> e XX i se v dt <sup>T</sup> <sup>L</sup>* =- × + - - + é ù

w w

w w

'' '

'' '

*<sup>q</sup>* 1 *<sup>m</sup>*

1 *d m*

*ds s ds qs d v Ri Xi e* =- + + (5)

*qs s qs ds q v Ri Xi e* =- - + (6)

ê ú ë û (7)

ê ú ë û (8)

'

: open circuit time constant in

*me r*

w

(4)

2 *r*

*dt H* w

( ) '

( ) '

'

'

X´, X: transient reactance and the open circuit reactance;

: d-axis and q-axis components of the internal voltage; *To*

**3.2. The DFIG model**

228 Advances in Wind Power

Where:

*ed* ' ,*eq* '

seconds;

*R*s: stator resistance;

ωS: synchronous speed;

*L*m: mutual inductance;

*L*rr: rotor inductance;

used in the following per-unit form [14,15]:

*V*ds, *V*qs: d and q axis stator voltages;

*i*ds, *i*qs: d and q axis stator currents;

*V*dr, *V*qr: d and q axis rotor voltages;

*T <sup>d</sup> T TD*

Where *<sup>D</sup>*¯ is the damping coefficient; and *HT* is the inertia constant, in seconds.

= --

In this study the converters are modeled according to reference [16], as show the diagrams presented in Figs. 2 and 3. The Vd1 component is used to the capacitor voltage and Vq1 is used to fix at zero the reactive power absorbed by the rotor side converter. This component may be used to provide additional reactive power support to the system.

The component Iq2 of the rotor current is used to control the rotor speed and as a conse‐ quence, the active power supplied by the machine. The component Id2 of the rotor current is used to control the generator terminal voltage or power factor.

#### **3.4. OEL model**

The objective of the over excitation limiter is to protect the generator from thermal overload. The OEL model adopted in this study is the same of reference [14] and the model is present‐ ed in Fig. 4. The OEL detects the over-current condition, and after a time delay, acts reduc‐ ing the excitation by reducing the field current to a value of 100% to 110% of the nominal value. Once the OEL acts, the field current no longer increases, limiting the reactive power supplied by the machine to a minimum value, overloading the other generators, contribu‐ ting significantly to the voltage instability.

**Figure 2.** Control loop of the stator side converter (SSC) a) Component Vd1 b) Component Vq1.

**Figure 3.** Control loop of the rotor side converter (RSC) a) Component Iq2 b) Component Id2.

**Figure 4.** OEL model.

a)

b)

**Figure 2.** Control loop of the stator side converter (SSC) a) Component Vd1 b) Component Vq1.

230 Advances in Wind Power

**Figure 3.** Control loop of the rotor side converter (RSC) a) Component Iq2 b) Component Id2.

#### **4. Test system**

The test system used in this study is shown in Fig. 5 and is based on the system developed in [14] for voltage stability analysis. To conduct this study the original system was modified, adding a transformer between buses 8 and 12 to connect a 212.5 MW wind farm at bus 12, consisting of 250 turbines of 850kW. Four generation sources are modeled: G1, G2 and G3 are synchronous generators and the wind farm is based on DFIG. The OEL device is instal‐ led in generator G3 and the OLTC between buses 10 and 11. The OLTC model used is pre‐ sented in reference [17].

**Figure 5.** Test system diagram.

Table 1 shows the generation and load scenarios considered. The penetration level of the wind farm is 17.6%. Scenario 1 considers the load at bus 8 modeled as constant impedance for both active and reactive components and the load at bus 11 modeled as 50% constant im‐ pedance and 50% constant current for both active and reactive components. Scenario 2 con‐ siders the active power component of load at bus 8 represented as an equivalent of 450 induction motors which parameters are presented in the Appendix, and all other compo‐ nents are modeled as in scenario 1.


**Table 1.** Load Scenarios


**Table 2.** Generation Scenarios

The intermittent characteristic of wind generation is considered following the wind regime presented on Fig. 6, with the initial wind speed of 12 m/s.

**Figure 6.** Wind speed regime.
