Chapter 7 machine equations in operational impedances and time constants

Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek

R.H Park [1] , in his original paper, did not specify the number of rotor circuits Instead, he expressed the stator fl ux linkages in terms of operational impedances and

a transfer function relating stator fl ux linkages to fi eld voltage In other words, Park recognized that, in general, the rotor of a synchronous machine appears as a distributed parameter system when viewed from the stator The fact that an accurate, equivalent lumped parameter circuit representation of the rotor of a synchronous machine might require two, three, or four damper windings was more or less of academic interest until digital computers became available Prior to the 1970s, the damper windings were seldom considered in stability studies; however, as the capability of computers increased,

it became desirable to represent the machine in more detail

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.

MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES

AND TIME CONSTANTS

7

The standard short-circuit test,which involves monitoring the stator short-circuit currents, provides information from which the parameters of the fi eld winding and one

damper winding in the d -axis can be determined The parameters for the q -axis damper

winding are calculated from design data Due to the need for more accurate parameters, frequency–response data are now being used as means of measuring the operational impedances from which the parameters can be obtained for any number of rotor wind-ings in both axes

In this chapter, the operational impedances as set forth by Park [1] are described The standard and derived synchronous machine time constants are defi ned and their relationship to the operational impedances established Finally, a method of approxi-mating the measured operational impedances by lumped parameter rotor circuits is presented

7.2 P ARK ’ S EQUATIONS IN OPERATIONAL FORM

R.H Park [1] published the original qd 0-voltage equations in the form

v s r i s s p

b s

where

ψqs r

q qs r

X p i

ψds r

d ds r

fd r

X p i G p v

In these equations, positive stator current is assumed out of the machine, the operator

X q ( p ) is referred to as the q -axis operational impedance, X d ( p ) is the d -axis operational

impedance, and G ( p ) is a dimensionless transfer function relating stator fl ux linkages

per second to fi eld voltage

With the equations written in this form, the rotor of a synchronous machine can be considered as either a distributed or lumped parameter system Over the years, the elec-trical characteristics of the rotor have often been approximated by three lumped param-eter circuits, one fi eld winding and two damper windings, one in each axis Although this type of representation is generally adequate for salient-pole machines, it does not suffi ce for a solid iron rotor machine It now appears that for dynamic and transient stability considerations, at least two and perhaps three damper windings should be used in the

q -axis for solid rotor machines with a fi eld and two damper windings in the d -axis [2]

OPERATIONAL IMPEDANCES AND G(p) 273

7.3 OPERATIONAL IMPEDANCES AND G ( p ) FOR A SYNCHRONOUS

MACHINE WITH FOUR ROTOR WINDINGS

In Chapter 5 , the synchronous machine was represented with a fi eld winding and one

damper winding in the d -axis and with two damper windings in the q -axis It is helpful

to determine X q ( p ), X d ( p ), and G ( p ) for this type of rotor representation before deriving

the lumped parameter approximations from measured frequency-response data For this purpose, it is convenient to consider the network shown in Figure 7.3-1 It is helpful

in this and in the following derivations to express the input impedance of the rotor circuits in the form

Since it is customary to use the Laplace operator s rather than the operator p , Laplace

notation will be employed hereafter In (7.3-1)

qa lkq

b kq

X r

qb lkq

b kq

X r

Qa lkq lkq

From Figure 7.3-1

Z s sX

q b

ls b

Solving the above equation for X q ( s ) yields the operational impedance for two damper

windings in the q -axis, which can be expressed

q

b kq lkq mq

1 1 11

=

τω

q

b kq lkq mq

2

2 21

=

τω

q

b kq lkq

mq lkq lkq mq

q

b kq lkq

q

b kq lkq

q

b kq lkq

The d -axis operational impedance X d ( s ) may be calculated for the machine with a fi eld

and a damper winding by the same procedure In particular, from Figure 7.3-2 a

da lfd

b fd

X r

τω

db lkd

X r

OPERATIONAL IMPEDANCES AND G(p) 275

Da lfd lkd

b fd kd

ed da kd db fd

r r R

to zero and following the same procedure, as in the case of the q -axis

The fi nal expression is

11

=

τω

d

b kd lkd md

21

=

τω

d

b kd lkd

md lfd lfd md

r X

X X

31

d

b fd lfd

=

τω

d

b kd lkd

=

τω

d

b kd lkd

The transfer function G ( s ) may be evaluated by expressing the relationship between stator

fl ux linkages per second to fi eld voltage, v′fd r , with ids r equal to zero Hence, from (7.2-5)

where τ db is defi ned by (7.3-17)

7.4 STANDARD SYNCHRONOUS MACHINE REACTANCES

It is instructive to set forth the commonly used reactances for the four-winding rotor synchronous machine and to relate these reactances to the operational impedances

whenever appropriate The q - and d -axis reactances are

These reactances were defi ned in Section 5.5 They characterize the machine during balanced steady-state operation whereupon variables in the rotor reference frame are

constants The zero frequency value of X q ( s ) or X d ( s ) is found by replacing the operator

s with zero Hence, the operational impedances for balanced steady-state operation are

STANDARD SYNCHRONOUS MACHINE REACTANCES 277

Similarly, the steady-state value of the transfer function is

r

md fd

Although X q has not been defi ned previously, we did encounter the d -axis transient ′

reactance in the derivation of the approximate transient torque-angle characteristic in Chapter 5

The q - and d -axis subtransient reactances are defi ned as

The high-frequency response of the machine is characterized by these reactances It is

interesting that G ( ∞ ) is zero, which indicates that the stator fl ux linkages are essentially

insensitive to high frequency changes in fi eld voltage Primes are used to denote transient and subtransient quantities, which can be confused with rotor quantities referred to the stator windings by a turns ratio Hopefully, this confusion is minimized by the fact that

X d and ′ X q are the only single-primed parameters that are not referred impedances ′

Although the steady-state and subtransient reactances can be related to the tional impedances, this is not the case with the transient reactances It appears that the

d -axis transient reactance evolved from Doherty and Nickle ’ s [3] development of an approximate transient torque-angle characteristic where the effects of d -axis damper

windings are neglected The q -axis transient reactance has come into use when it became desirable to portray more accurately the dynamic characteristics of the solid iron rotor machine in transient stability studies In many of the early studies, only one damper winding was used to describe the electrical characteristics of the

q -axis, which is generally adequate in the case of salient-pole machines In our earlier

development, we implied a notational correspondence between the kq 1 and the fd windings and between the kq 2 and the kd windings In this chapter, we have associated the kq 1 winding with the transient reactance (7.4-6) , and the kq 2 winding with the subtransient reactance (7.4-8) Therefore, it seems logical to use only the kq 2 winding

when one damper winding is deemed adequate to portray the electrical characteristics

of the q axis It is recalled that in Chapter 5 , we chose to use the kq 2 winding rather than the kq 1 winding in the case of the salient-pole hydro turbine generator

It is perhaps apparent that the subtransient reactances characterize the equivalent reactances of the machine during a very short period of time following an electrical disturbance After a period, of perhaps a few milliseconds, the machine equivalent reactances approach the values of the transient reactances, and even though they are

not directly related to X q ( s ) and X d ( s ), their values lie between the subtransient and

steady-state values As more time elapses after a disturbance, the transient reactances give way to the steady state reactances In Chapter 5 , we observed the impedance of the machine “changing” from transient to steady state following a system disturbance Clearly, the use of the transient and subtransient quantities to portray the behavior of the machine over specifi c time intervals was a direct result of the need to simplify the machine equations so that precomputer computational techniques could be used

7.5 STANDARD SYNCHRONOUS MACHINE TIME CONSTANTS

The standard time constants associated with a four-rotor winding synchronous machine are given in Table 7.5-1 These time constants are defi ned as

′

τqo and τ′do are the q - and d -axis transient open-circuit time constants

qo and ′′ ′′do are the q - and d -axis subtransient open-circuit time constants

′

τq and τ′d are the q - and d -axis transient short-circuit time constants

q′′ and ′′d are the q - and d -axis subtransient short-circuit time constants

In the above defi nitions, open and short circuit refers to the conditions of the stator circuits All of these time constants are approximations of the actual time constants, and when used to determine the machine parameters, they can lead to substantial errors

in predicting the dynamic behavior of a synchronous machine More accurate sions for the time constants are derived in the following section

7.6 DERIVED SYNCHRONOUS MACHINE TIME CONSTANTS

The open-circuit time constants, which characterize the duration of transient changes

of machine variables during open-circuit conditions, are the reciprocals of the roots

of the characteristic equation associated with the operational impedances, which, of course, are the poles of the operational impedances The roots of the denominators

of X q ( s ) and X d ( s ) can be found by setting these second-order polynomials equal to zero

From X ( s ), (7.3-7)

DERIVED SYNCHRONOUS MACHINE TIME CONSTANTS 279

τ τ

TABLE 7.5-1 Standard Synchronous Machine Time Constants

Open-Circuit Time Constants

′′ =

′ + ′

It can be shown that

In most cases, the right-hand side of (7.6-5) and (7.6-6) is much less than unity Hence,

the solution of (7.6-3) with 4 c / b 2

≪ 1 and c / b ≪ b is obtained by employing the

binomial expansion, from which

Similarly, the d -axis open-circuit time constants are

′′ =+

DERIVED SYNCHRONOUS MACHINE TIME CONSTANTS 281

TABLE 7.6-1 Derived Synchronous Machine Time Constants

Open-Circuit Time Constants

X X

X X X

1 1

The short-circuit time constants are defi ned as the reciprocals of the roots of the numerator of the operational impedances Although the stator resistance should be included in the calculation of the short-circuit time constants; its infl uence is generally

The roots are of the form given by (7.6-3) and, as in the case of the open-circuit time

constants, 4 c / b 2 ≪ 1 and c / b ≪ b Hence

′′ =+

and

In the lumped parameter approximation of the rotor circuits, r kd is generally much larger ′

than r fd, and therefore the standard d -axis time constants are often good approximations ′

of the derived time constants This is not the case for the q -axis lumped parameter

approximation of the rotor circuits That is, r kq2 is seldom if ever larger than ′ r kq1, hence ′

the standard q -axis time constants are generally poor approximations of the derived

time constants

PARAMETERS FROM SHORT-CIRCUIT CHARACTERISTICS 283

7.7 PARAMETERS FROM SHORT-CIRCUIT CHARACTERISTICS

For much of the twentieth century, results from a short-circuit test performed on an

unloaded synchronous machine were used to establish the d -axis parameters [5]

Alter-native techniques have for the most part replaced short-circuit characterization Despite being replaced, many of the terms, such as the short-circuit time-constants, have roots

in the analytical derivation of the short circuit response of a machine Thus, it is useful

to briefl y describe the test herein

If the speed of the machine is constant, then (7.2-1) – (7.2-6) form a set of linear differential equations that can be solved using linear system theory Prior to the short circuit of the stator terminals, the machine variables are in the steady state and the stator terminals are open-circuited If the fi eld voltage is held fi xed at its prefault value, then the Laplace transform of the change in v′fd r is zero Hence, if the terms involving r s2 are neglected, the Laplace transform of the fault currents (defi ned positive out of the

machine), for the constant speed operation ( ω r = ω b ), may be expressed

b s qs r d

It is clear that the 0 quantities are zero for a three-phase fault at the stator terminals It

is also clear that ω r , ω b , and ω e are all equal in this example

Initially, the machine is operating open-circuited, hence

The three-phase fault appears as a step decrease in vqs r

to zero Therefore, the Laplace transform of the change in the voltages from the prefault to fault values are

If (7.7-6) and (7.7-7) are substituted into (7.7-1) and (7.7-2) , and if the terms involving

r s are neglected except for α , wherein the operational impedances are replaced by

their high-frequency asymptotes, the Laplace transform of the short-circuit currents becomes

i s X s

qs r

q b

ds r

d b

Replacing the operational impedances with their high frequency asymptotes in α is

equivalent to neglecting the effects of the rotor resistances in α

If we now assume that the electrical characteristics of the synchronous machine can be portrayed by two rotor windings in each axis, then we can express the operational impedances in terms of time constants It is recalled that the open- and short-circuit time constants are respectively the reciprocals of the roots of the denominator and numerator of the operational impedances Therefore, the reciprocals of the operational impedances may be expressed

Bs s

Ds s

PARAMETERS FROM SHORT-CIRCUIT CHARACTERISTICS 285

ττ

ττ

q q

s s

ττ

⎛⎛

d d

s s

1 (7.7-18) Although the assumption that the subtransient time constants are much smaller than the

transient time constants is appropriate in the case of the d -axis time constants, the ference is not as large in the case of the q -axis time constants Hence, (7.7-17) is a less

dif-acceptable approximation than is (7.7-18) This inaccuracy will not infl uence our work

in this section, however Also, since we have not restricted the derivation as far as time constants are concerned, either the standard or derived time constants can be used in the equations given in this section However, if the approximate standard time constants are used, (τqo′ /τ′q)( /1 X q) and (τdo′ /τ′d)( /1 X d) can be replaced by 1 /X q′ and 1 /X d′, respectively

If (7.7-17) and (7.7-18) are appropriately substituted into (7.7-8) and (7.7-9) , the fault currents in terms of the Laplace operator become

⎞⎞

⎠⎟

′+ ′

⎡

⎣

⎢+

τ

ττ

q q

q qo

q q

q q

s s

s s

τ

ττ

d d

d do

d d

d d

s s

s s

1

Equations (7.7-19) and (7.7-20) may be transformed to the time domain by the following

inverse Laplace transforms If a and α are much less than ω b , then

If (7.7-21) is applied term by term to (7.7-19) with a set equal to zero for the term 1/ X q

and then 1 /τ′q and 1 / ′′q for successive terms, and if (7.7-22) is applied in a similar manner to (7.7-20) , we obtain [6]