DISCUSIÓN PARTICULAR.

3.2.2.3.1 Generator Electrical Model

For modelling the IG electrical system, standard transformations were used to map the 3-phase stator and rotor windings into direct and quadrature axis reference frame models with axes rotating at synchronous speed. When deriving the model, the q-axis was assumed to be 90” ahead of the d-axis in the direction of rotation. The dq-model is often used to reduce the abc-model complexity. A generalized 5th _{order mathematic model was}

used here for modelling the DFIG as shown in Fig. 3.13.

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The equations describing an asynchronous machine in terms of phase variables were derived to develop the model with all rotor variables referred to the stator. Assume that the stator current is positive when flowing from the grid to the generator, and then the voltage expressions can be represented as [77]:

3.18 3.19

3.20

3.21

where , , and are the dq stator and rotor voltages, , , and are the dq- stator and rotor currents, , , , are the dq-stator and rotor fluxes, , are the stator and rotor resistance, is the synchronous electrical angular speed and is the generator electrical angular speed. Sign ( ′ ) to indicate these parameters

are in stator side. The terms of the stator and rotor flux can be expressed in the stator and rotor currents as:

3.22 3.23

3.24

3.25 where and represent the stator and rotor self-inductances, respectively, and is the mutual inductance between the stator and the rotor. Both of the stator and rotor self- inductances can be defined as:

3.26 3.27 with and are the stator and rotor leakage inductances. The electromagnetic torque ( ) expression of the induction machine is given by:

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3.28

where is the number of pole pairs. The stator and rotor active and reactive powers are: 3.29 3.30 3.31 3.32

where and represent the stator and rotor active powers while and represent

the stator and rotor reactive powers.

3.2.2.3.2 Converter Model

In order to be able to feed and control a DFIG from a variable frequency and voltage source, the DFIG was connected to a back-to-back converter consisting of two voltage source inverters separated by a DC link. The DC link separates the two inverters and therefore can be controlled independently of each other. Therefore, only the RSI has been considered in the model while the GSI and DC link have been replaced by a DC voltage source. In a real WT the GSI controls the reactive power exchange with the grid and the DC Link voltage. An ideal lossless representation of the RSI as depictedin Fig. 3.14 was assumed. It has sixIGBTs (T1 to T6) where each one is equipped with anti-parallel diode. A PWM signals or gating signals ( to ), are generated in the controller, switch on and off the transistors. The duty cycle of the transistor and the diode determines whether the transistor or a diode is conducting in a transistor leg (e.g., T1 and T4).

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3.2.2.3.3 The RSI Controller Model

The RSI is used to regulate the active and reactive power exchanged between the generator and the grid. In a real WT, the active power is controlled in order to be adapted to the wind speed and the reactive power control allows obtaining a unitary power factor between the stator and the grid. Based on the SFOVC, the controller has been designed and modelled in the system as shown in Fig. 3.15, where the torque and consequently active power can be regulated only through the q-rotor current component and the reactive power can be regulated by the d-rotor current component.

Fig. 3.15: Schematic diagram of the active & reactive power control of DFIG

The active power comprises 3 cascaded loops; q-rotor current control (PI), torque control and speed control while the reactive power has only d-rotor current control (PI). The relationship between the variables in this model can be expressed in the following forms as: 3.33 3.34 3.35 3.36 3.37

where is the generator reference torque, is the optimal power tracking factor, is the magnitude of the stator phase voltage, is the turn ratio between stator and the rotor, and are the d- and q-rotor reference currents, and are

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the d- and q-rotor error currents, and are the d- and q-rotor reference

voltage, , , and are the d- and q-current control parameters. The design of

this controller and the derivative of the above equations will be described in details in the next chapter.

3.2.2.3.4 Grid Model

A simple model has been used to simulate the grid in the MATLAB model. The main components of this model is the grid voltage ( ) and the grid impedance ( ), as shown in Fig. 3.16.

Fig. 3.16: Grid model

The grid voltage comprises three phase voltages ( , and ) with same magnitude ( ) and shifted by from each other as given by:

3.38

where is the grid or supply frequency. Meanwhile, the grid impedance has been modelled with using 3-pahse impedance. The grid impedance for each phase can be written as:

3.39 where and are resistance and inductance of the grid. The impedance parameters have been estimated to have small values in order to generate some harmonics in the generator variables.

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