Chapter 8 alternative forms of machine equations

299

8.1. INTRODUCTION

There are alternative formulations of induction and synchronous machine equations

that warrant consideration since each has a specifi c useful purpose. In particular, (1)

linearized or small-displacement formulation for operating point stability issues; (2)

neglecting stator electric transients for large-excursion transient stability studies; and

(3) voltage-behind reactance s ( VBR s) formulation convenient for machine-converter

analysis and simulation. These special formulations are considered in this chapter.

Although standard computer algorithms may be used to automatically linearize

machine equations, it is important to be aware of the steps necessary to perform lineariza￾tion. This procedure is set forth by applying Taylor expansion about an operating point.

The resulting set of linear differential equations describe the dynamic behavior during

small displacements or small excursions about an operating point, whereupon basic linear

system theory can be used to calculate eigenvalues. In the fi rst sections of this chapter,

the nonlinear equations of induction and synchronous machines are linearized and the

eigenvalues are calculated. Although these equations are valid for operation with stator

voltages of any frequency, only rated frequency operation is considered in detail.

Over the years, there has been considerable attention given to the development of

simplifi ed models primarily for the purpose of predicting the dynamic behavior of

electric machines during large excursions in some or all of the machine variables.

Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk,

Scott Sudhoff, and Steven Pekarek.

© 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc.

ALTERNATIVE FORMS OF

MACHINE EQUATIONS

8

300 ALTERNATIVE FORMS OF MACHINE EQUATIONS

Before the 1960s, the dynamic behavior of induction machines was generally predicted

using the steady-state voltage equations and the dynamic relationship between rotor

speed and torque. Similarly, the large-excursion behavior of synchronous machines was

predicted using a set of steady-state voltage equations with modifi cations to account

for transient conditions, as presented in Chapter 5 , along with the dynamic relationship

between rotor angle and torque. With the advent of the computer, these models have

given way to more accurate representations. In some cases, the machine equations are

programmed in detail; however, in the vast majority of cases, a reduced-order model

is used in computer simulations of power systems. In particular, it is standard to neglect

the electric transients in the stator voltage equations of all machines and in the voltage

equations of all power system components connected to the stator (transformers, trans￾mission lines, etc.). By using a static representation of the power grid, the required

number of integrations is drastically reduced. Since “neglecting stator electric tran￾sients” is an important aspect of machine analysis especially for the power system

engineer, the theory of neglecting electric transients is established and the voltage equa￾tions for induction and synchronous machines are given with the stator electric tran￾sients neglected. The large-excursion behavior of these machines as predicted by these

reduced-order models is compared with the behavior predicted by the complete equa￾tions given in Chapter 5 and Chapter 6 . From these comparisons, not only do we

become aware of the inaccuracies involved when using the reduced-order models, but

we are also able to observe the infl uence that the electric transients have on the dynamic

behavior of induction and synchronous machines.

Finally, in an increasing number of applications, electric machines are coupled to

power electronic circuits. In Chapter 4 , Chapter 5 , and Chapter 6 , a great deal of the

focus was placed upon utilizing reference-frame theory to eliminate rotor-dependent

inductances (or fl ux linkage in the case of the permanent magnet machine). Although

reference-frame theory enables analytical evaluation of steady-state performance and

provides the basis for most modern electric drive controls, it can be diffi cult to apply

a transformation to some power system components, particularly power electronic

converters. In such cases, one is forced to establish a coupling between a machine

modeled in a reference frame and a power converter modeled in terms of physical

variables. As an alternative, it can be convenient to represent a machine in terms of

physical variables using a VBR model. In this chapter, the derivation of a physical

variable VBR model of the synchronous machine is provided, along with explanation

of its potential application and advantages over alternative model structures. In addi￾tion, approximate forms of the VBR model are described in which rotor position￾dependent inductances are eliminated, which greatly simplifi es the modeling of

machines in physical variables.

8.2. MACHINE EQUATIONS TO BE LINEARIZED

The linearized machine equations are conveniently derived from voltage equations

expressed in terms of constant parameters with constant driving forces, independent of

MACHINE EQUATIONS TO BE LINEARIZED 301

time. During steady-state balanced conditions, these requirements are satisfi ed, in the

case of the induction machine, by the voltage equations expressed in the synchronously

rotating reference frame, and by the voltage equations in the rotor reference frame in

the case of the synchronous machine. Since the currents and fl ux linkages are not

independent variables, the machine equations can be written using either currents or

fl ux linkages, or fl ux linkages per second, as state variables. The choice is generally

determined by the application. Currents are selected here. Formulating the small￾displacement equations in terms of fl ux linkages per second is left as an exercise for

the reader.

Induction Machine

The voltage equations for the induction machine with currents as state variables may

be written in the synchronously rotating reference frame from (6.5-34) by setting

ω = ωe as

v

v

v

v

r p X X p qs

e

ds

e

qr

e

dr

e

s

b

ss

e

b

ss

′

′

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

=

+

ω

ω

ω ω

ω

ω

ω

ω ω

ω

ω ω

ω

ω

ω ω

b

M

e

b

M

e

b

ss s

b

ss

e

b

M

b

M

b

M

e

b

M r

X X

X r p X X p X

p X sX r p

− +−

′+ b

rr

e

b

rr

e

b

M

b

M

e

b

rr r

b

rr

X sX

s X p X sX r p X

′ ′

− − ′ ′+ ′

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

ω

ω

ω

ω ω

ω

ω ω

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

′

′

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

i

i

i

i

qs

e

ds

e

qr

e

dr

e

(8.2-1)

where s is the slip defi ned by (6.9-13) and the zero quantities have been omitted since

only balanced conditions are considered. The reactances X ss and X′

rr are defi ned by

(6.5-35) and (6.5-36) , respectively.

Since we have selected currents as state variables, the electromagnetic torque is

most conveniently expressed as

T X ii ii e M qse dr

e ds

e qr

e = ( ) ′ − ′ (8.2-2)

Here, the per unit version of (6.6-2) is selected for compactness. The per unit relation￾ship between torque and speed is (6.8-10) , which is written here for convenience

T Hp T e

r

b

= + 2 L

ω

ω

(8.2-3)

Synchronous Machine

The voltage equations for the synchronous machine in the rotor reference frame may

be written from (5.5-38) for balanced conditions as