Modeling and Analysis with Induction Generators, 3rd Edition

39

3

Transient Model of

Induction Generators

3.1 SCOPE OF THIS CHAPTER

This chapter will explain how steady-state terminal voltage builds up during the self-

excitation process and during the recovery of voltage during perturbations across the

terminal voltage and stator current due to load changes. A set of state equations are

derived to obtain the instantaneous output voltage and current in the self-excitation

process. In the example in this chapter, the equations describe an induction machine

with the following rated data: 220/380 V, 26/15 A, 7.5 kW, and 1765 rpm. We also

show that the equations in this chapter can be easily used to calculate output voltage,

including the variation of mutual inductance with the magnetizing current discussed

further in Chapter 4. Furthermore, we will present a general matrix equation for

simulation of the parallel aggregation of induction generators (IG). In any case, for

every machine at a given speed, we show that there is a minimum capacitance value

causing self-excitation in agreement with the steady state case shown in Chapter 2.

We will discuss rotor parameter variation and demonstrate that it has little effect

on the accuracy of these calculations and that comparable results can be achieved

using the standard open circuit and locked rotor parameters. Saturation effects are

also considered in this chapter by taking into account the nonlinear relationship

between magnetizing reactance and the magnetizing current of a machine; using this

approach, the mutual inductance, M, will be shown to vary continually.

3.2 INDUCTION MACHINE IN TRANSIENT STATE

During self-excitation, asynchronous or IGs exhibit transient phenomena that are

very difcult to model from an operational point of view; the impact of self-excitation

is more pronounced in generators with heavier loads.

1–3

A crucial problem to be

avoided is the demagnetization of the IG. To analyze an IG in a transient state, we

will apply the Park transformation (also known as the Blondel or Blondel-Park trans-

formation), which is associated with the general theory of rotating machines used in

this chapter.

The Park transformation species a two-phase primitive machine with xed sta-

tor windings and rotating rotor windings to represent xed stator windings (direct

axis) and pseudo-stationary rotor windings (quadrature axis) (see Figure 3.1). Any

machine can be equivalent to a primitive machine with an appropriate number of

coils on each axis.

4–6

40 Modeling and Analysis with Induction Generators

Saturation effects are also considered in this chapter by taking into account a non-

linear relationship between the magnetizing reactance and the magnetizing current

of the machine. Under this approach, the mutual inductance M varies continuously.

3.3 STATE SPACE–BASED INDUCTION GENERATOR MODELING

Figure 3.1 represents a primitive machine through a Park transformation as applied

to the induction generator. Currents i

ds

and i

qs

refer to the stator currents, and I

dr

and

I

qr

to the rotor currents, in the direct and quadrature axis, respectively. The angular

speed ω = dθ/dt is the mechanical rotor speed.

As can be seen in Figure 3.1, no external voltage is applied across the rotor or

the stator windings. This is the standard form of stationary reference axis used in

the machine theory texts.

7–10

However, there is an additional component: the self-

excitation capacitor C.

3.3.1 no-load InductIon Generator

A self-excited induction generator (SEIG) with a capacitor is initially considered

to be operating at no load.

1,2

The relationships between the resulting voltages and

currents, direct and quadrature, can be obtained from Figures 3.1 and 3.2 and

Equation 3.1. That expression represents the ordinary symmetrical three-phase

machine connected to a three-phase bank of identical parallel capacitors. The

reference position is put on the stator under every normal operating condition,

C

i

ds

i

dr

i

qr

C

i

qs

q

d

a

c

b

ω

FIGURE 3.1 Representation of an unloaded SEIG in d–q axis.