Modeling and Analysis with Induction Generators, 3rd Edition

183

8

Scalar Control for

Induction Generators

8.1 SCOPE OF THIS CHAPTER

Scalar control of induction motors and generators means control of the magnitude of

their voltage and frequency to achieve suitable torque and speed with an impressed

slip. Scalar control can be easily understood based on the fundamental principles of

induction-machine steady-state modeling. A power electronic system is used either

for a series connection of inverters and converters between the induction generator

and the grid or as a parallel path system capable of providing reactive power for

isolated operation. This chapter describes such principles, laying out foundations for

understanding the more complex vector-controlled systems.

8.2 SCALAR CONTROL BACKGROUND

Scalar control disregards the coupling effect on the generator; that is, the voltage will

be set to control the ux and the frequency in order to control the torque.

1

However,

ux and torque are functions of frequency and voltage, respectively. Scalar control

is different from vector control in which both magnitude and phase alignment of the

vector variables are directly controlled. Scalar control drives give an inferior per-

formance, but they are not difcult to implement and are very popular in machine

drives for pumping, several industrial applications, and large megawatt systems. The

importance of scalar control has diminished recently because of the superior per-

formance of the vector-controlled drives and the introduction of high-performance

inverters, but pulse-width modulation (PWM) transistor–based inverters have a

maximum power range. Scalar control is very useful with multilevel topologies

because of their inherent lack of modulation and its exibility when compared with

PWM, although for three-level inverters, it is possible to implement them also with

full PWM capabilities. High-performance inverters offer prices competitive to sca-

lar control–based inverters (V/Hz inverters), although they are cheap and widely

available.

The principles of scalar control are presented in this chapter along with a discus-

sion of some simple enhancements that may be retrot onto existing commercial

V/Hz systems.

The main constraint on the use of a scalar control method for induction motors

and generators is related to the transient response.

2

If shaft torque and speed are

bandwidth-limited and torque varies slowly (within hundreds of milliseconds up

to the order of almost a second), scalar control may be a good control approach.

Hydropower and wind power applications have slower mechanical dynamics for

184 Modeling and Analysis with Induction Generators

the scalar control method. Therefore, it seems that scalar control is a good control

approach for renewable energy applications.

As a background to understanding the scalar control method, a steady-state two-

inductance per-phase equivalent circuit for an induction motor can be used.

3

There

are two models: the Γ-stator equivalent model, presented in Figure 8.1, and the Γ-rotor

equivalent model, presented in Figure 8.2. The Γ-stator equivalent model is related to

the ordinary per-phase equivalent model by the transformation coefcient γ = X

s

/X

m

,

as shown in Table 8.1. The Γ-rotor equivalent model is related to the ordinary per-

phase equivalent model by the transformation coefcient ρ = X

m

/X

r

, as in Table 8.2.

In the Γ-stator equivalent model, the stator ux ψ

s

is considered to be approxi-

mated by the air-gap ux ψ

m

, while the stator resistance R

s

effects are usually

neglected. Considering that R

r,γ

/s = R

r,γ

ω

e

/ω

r

, the rotor current in the Γ-stator equiva-

lent model can be expressed as

I

R

r

m

r

r

rr

,

,

γ

γ

ψω

τω

=

()

+

2

1

(8.1)

where τ

r

= L

l,γ

/R

r,γ

, L

l,γ

is the total leakage inductance in this model; that is, L

l,γ

= X

l,γ

/ω.

I

s

I

r

R

s

R

r

s

jX

M

jX

L

V

s

jω

e

ψ

s

jω

e

ψ

r

I

M

FIGURE 8.1 Γ-stator equivalent model.

I

s

I

M

΄

I

r

΄

R

s

s

jX΄

L

jX΄

M

R΄

r

V

s

jω

e

ψ

s

jω

e

ψ΄

r

FIGURE 8.2 Γ-rotor equivalent model.

185Scalar Control for Induction Generators

When the resistive losses in the rotor (3R

r,γ

|I

r,γ

|

2

) is neglected, the electrical power

transferred from the mechanical shaft to the rotor circuit is approximated by

PR

IP

elec rr mech

=≈

3

2

,,γγ

ω

ω

(8.2)

The generator torque shaft can be calculated as the ratio of P

mech

to the mechani-

cal angular speed. For scalar voltage control, the angular shaft speed is leading the

stator angular speed by ω

m

= 2/Pω

e

(1 + s), and the developed electrical torque on the

shaft is

T

p

R

e

m

r

r

rr

=

()

+

3

2

1

2

2

ψω

τω

γ,

(8.3)

For ω

r

= 1/τ

r

the critical slip is s

critical

= 1/τ

r

ω

e

, and the maximum torque developed

is given by

T

p

L

M

m

l

,max

,

=

()

3

2

2

ψ

γ

TABLE 8.2

Γ-Rotor Equivalent Parameters

Rotor resistance referred to stator

R

r,ρ

= ρ

2

R

r

Magnetizing reactance

X

m,ρ

= ρX

m

Total leakage reactance

X

l,ρ

= X

ls

+ ρX

lr

Rotor current referred to stator

I

I

r

r

,ρ

ρ

=

Rotor ux

ψ

r,ρ

= ρψ

r

Angular frequency of rotor currents (rotor frequency)

ω

r

= sω

e

TABLE 8.1

Γ-Stator Equivalent Parameters

Rotor resistance referred to stator

R

r,γ

= γ

2

R

r

Magnetizing reactance

X

m,γ

= γX

m

= X

s

Total leakage reactance

X

l,γ

= γX

ls

+ γ

2

X

lr

Rotor current referred to stator

I

I

r

r

,γ

γ

=

Rotor ux

ψ

r,γ

= γψ

r

Angular frequency of rotor currents (rotor frequency)

ω

r

= sω

e

Rotor angular frequency related to the slip frequency

ωω

rs

l

p

=

2