Fan / Miao Modeling and Analysis of Doubly Fed Induction Generator Wind Energy Systems

1. Auflage 2015
ISBN: 978-0-12-802986-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 154 Seiten

Wind Energy Systems: Modeling, Analysis and Control with DFIG provides key information on machine/converter modelling strategies based on space vectors, complex vector, and further frequency-domain variables. It includes applications that focus on wind energy grid integration, with analysis and control explanations with examples. For those working in the field of wind energy integration examining the potential risk of stability is key, this edition looks at how wind energy is modelled, what kind of control systems are adopted, how it interacts with the grid, as well as suitable study approaches. Not only giving principles behind the dynamics of wind energy grid integration system, but also examining different strategies for analysis, such as frequency-domain-based and state-space-based approaches. - Focuses on real and reactive power control - Supported by PSCAD and Matlab/Simulink examples - Considers the difference in control objectives between ac drive systems and grid integration systems

Autoren/Hrsg.

Fan, Lingling

Miao, Zhixin

Leseproben

Chapter 2

AC Machine Modeling

Abstract

This chapter focuses on the dynamic modeling of induction machines. Two types of induction machine models are presented: space vector-based model and complex vector-based model. Type-3 wind generator, or doubly-fed induction generator, is an induction generator with rotor side connected to a converter. This chapter focuses on the modeling of the electric machines. Two examples are discussed in the chapter. In the first example, complex vector-based model is used for dynamic simulation of free acceleration of an induction generator. In the second example, space-vector-based model is used to examine the effect of stator voltage dip on rotor voltages.

Keywords

Space vector

Complex vector

Induction machine

2.1 SPACE VECTOR AND COMPLEX VECTOR EXPLANATION   8

2.1.1 Examples of Space Vector   12

2.2 DERIVATION OF AN INDUCTION MACHINE MODELING IN SPACE VECTOR AND COMPLEX VECTOR   13

2.2.1 Induction Machine Modeling in Space Vector   13

2.2.1.1 Per Unit System   15

2.2.2 Induction Machine Modeling in Complex Vector   16

2.2.2.1 Model in Per-Unit System   18

2.2.3 Swing Equation   18

2.3 DFIG MODELING   21

2.3.1 Wind Turbine Aerodynamics Model   21

2.4 EXAMPLES   23

2.4.1 Example 1: Free Acceleration   23

2.4.1.1 Induction Machine Simulink Model   25

2.4.2 Example 2: DFIG Stator Voltage Drop   27

REFERENCES   33

In this chapter, analytical models of induction machines are first described. Analytical models of DFIG are then presented. The models are related to electric machine only. Converter controls will be discussed in Chapter 3. Two examples are given to demonstrate the usage of the analytical models to simulate free acceleration of an induction machine or analyze the consequences of DFIG stator voltage dip.

2.1 Space Vector and Complex Vector Explanation

The type of induction machines to be discussed is three-phase induction machines where both the stator and the rotor have abc windings. The key dynamics of the magnetic field can be described by Faraday’s law that the change of flux field induces electromagnetomotive force (EMF). To model a three-phase system, a direct approach is to express the EMF phase by phase. However, this approach will lead to the use of phase-coupled and time-varied inductances. The contribution of Park’s transformation is to develop analytical models based on a dq rotating reference frame. When the rotating speed of the reference frame is the same as the speed of the rotating flux, the models will have a much simpler form and the inductances of the dq-axis are decoupled and constant.

In this chapter, however, we will not use Park’s transformation to derive the analytical models. Instead, we rely on the concept of space vector to derive the analytical models due to its insightful and simple explanation of physics. In this section, let us first define and explain space vector as well as complex vector.

With the assumptions such as sinusoidal distribution of air gap flux and a uniform air gap, three-phase balanced stator currents in a nonsalient two-pole AC machine (Fig. 2.1) will lead to an magnetomotive force (MMF) which can be viewed as a traveling waveform with a constant magnitude in the air gap (Fig. 2.2) or a rotating MMF—a space vector. This MMF leads to a rotating flux in the air gap (Fig. 2.1). The rotating flux in the air gap then induces sinusoidal EMFs in the three-phase stator and rotor windings.

Figure 2.1 A three-phase AC machine.

Figure 2.2 MMF distribution in the air gap.

The concept of space vector comes from this physical mechanism: three-phase currents lead to a rotating MMF. For each phase current, at a random position (defined as angle a referring to the a-axis) in the air gap, the resulting MMF can be found based on Ampere’s law. The mathematical expressions are listed as follows:

a(a)=NiacosaFb(a)=Nibcosa-2p3Fc(a)=Niccosa+2p3

where a is the general angle in the air gap referring to the a-axis, F is MMF and N is the number of windings in each phase.

From the above equations, we can find that Fa reaches maximum when a = 0. Accordingly, Fb is maximum when =2p3; Fc is maximum when =4p3. Consider the current i as a sinusoidal function of time t:

a(t)=Imcos(?st+?a)ib(t)=Imcos(?st+?a-2p3)ic(t)=Imcos(?st+?a+2p3)

where Im is the amplitude of the current, ?s is the angular frequency of the current and ?a is the initial angle at t = 0.

Then the total MMF can be found as

(a,t)=NImcos(?st+?a)cosa+cos?st+?a-2p3cosa-2p3+cos?st+?a+2p3cosa+2p3=32NImcosa-?st-?a

  (2.1)

Remarks

It is obvious that the total MMF due to a set of three-phase balanced currents in a nonsalient AC machine has a constant magnitude. The total MMF in the air gap is a traveling waveform. As time evolves, the peak of the waveform moves forward. Figure 2.2 illustrates the MMFs due to per-phase currents and the total MMF at an instant in the air gap. This MMF can be viewed as a space vector notated by a magnitude and an angle where the peak occurs. The MMF can be notated as

?(t)=32NImej(?st+?a)=32NImej?aej?st

  (2.2)

It can be observed that three-phase balanced currents result in a rotating MMF space vector with a constant magnitude and a constant speed. Based on the physical mechanism, we can now define a space vector through a similar procedure.

The space vector is defined as

?(t)=23ej0fa(t)+ej2p3fb(t)+ej4p3fc(t)

  (2.3)

Should fa(t), fb(t), and fc(t) be balanced sinusoidal waveforms with an angular speed ? and initial angle ?0, and an amplitude of fm, the space vector can be further expressed as follows:

?(t)=|F|ej(?t+?0)=|F|ej?0ej?t=fa-jfß

  (2.4)

where |F| = fm (the amplitude of the waveforms).

The space vector can be further expressed in real and imaginary components fa and fß, where

afß=231cos2p3cos2p30-sin2p3sin2p3fafbfc

2.1.1 Examples of Space Vector

Example 1

Find the space vector for a three-phase voltage expressed as follows:

va(t)=v^pcos(?t+?p)+v^ncos(?t+?n)+v^0cos(?t+?0)vb(t)=v^pcos(?t+?p-2p3)+v^ncos(?t+?n+2p3)+v^0cos(?t+?0)vc(t)=v^pcos(?t+?p+2p3)+v^ncos(?t+?n-2p3)+v^0cos(?t+?0)

  (2.5)

Solution

The three-phase voltage consists of positive-, negative-, and zero-sequence components. Based on the definition of space vector, we can find the space vectors corresponding to the positive-, negative-, and zero-sequence components, respectively.

vp?=v^pej?pej?tvn?=v^ne-j?ne-j?tv0?=0

  (2.6)

The resulting space vector is...