## Mathematical model of DFIG

The configuration of a DFIG, with corresponding static converters and controllers is given in Fig.1. Two converts are connected between the rotor and grid, following a back to back scheme with a dc intermediate link. Fig.2 gives the reference frames, where a, b and c indicate stator phase a, b and c winding axes; A, B and C indicate rotor phase A, B and C winding axes, respectively; x-y is the synchronous rotation coordinate system in the grid side; d is the angle between q axis and x axis.

Applying Park's transformation, the voltage equations of a DFIG in the d-q coordinate system rotating at the synchronous speed a>s, in accordance with generator convention, which means that the stator and rotor currents are positive when flowing towards the network, and real and reactive powers are positive when fed into grid, can be deducted as follows in a per unit system.

1 dWds ms dt

Pg+jQg

Pg+jQg

8^8

h controller ¡4-

Fig. 2. Schematic diagram of DFIG with converters and controllers

Fig. 3. Reference coordinates for DFIG

1 dW

qr dt

Pg = Ps + Pr = (Udslds + UqSIqS ) + (UdrIdr + Uq„Iq„ ) (10)

Qg = QS + Qr = (UqsIds - UdsIqs ) + (UqrIdr - UdrIqr ) (11)

Where U, I, W denote the voltage, current and flux linkage; P and Q denote the real and reactive power outputs of wind generator, respectively; Tm and Te denote the mechanical and electromagnetic torques of wind generator, respectively; R and X denote resistance and reactance, respectively; the subscripts r and s denote the stator and rotor windings, respectively; the subscript g means generator; H is the inertia constant, and t stands for time; s is the slip of speed.

The reactances Xs and Xr can be calculated in following equations.

Where Xsr7 and Xrr7 are the leakage reactances of stator and rotor windings, respectively; Xm is the mutual reactance between stator and rotor.

The aforementioned equations describe the electrical dynamic performance of a wind turbine, namely, the asynchronous machine. However, these equations are not suitable for small signal analysis directly. It is necessary and imperative to deduce the simplified and practical model. The following assumptions are presented to model the DFIG.

a. Magnetic saturation phenomenon is not considered during modelling;

b. For the wind turbine equipped with DFIG, all rotating masses are represented by one element, which means that a so-called 'lumped-mass' or 'one-mass' representation is used;

c. The stator transients and stator resistance are negligible, i.e.—— = 0, —— = 0, and

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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