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3630

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 10 October 1982

A NOVEL APPROACH TO THE TORSIONAL INTERACTION AND ELECTRICAL DAMPING OF THE SYNC

HRONOUS MACHINE Part I: Theory
I. M. Canay BBC BROWN, BOVERI & COMPANY Limited, Baden, Switzerland

Abstract - The torsional interaction phenomenon between the electrical and mechanical system is explained with the aid of the newly introduced complex torque coefficients ke(jX) and km(jA). The frequency response ke(jA) characterizes the behaviour of the electrical system including the electrical damping with and without series capacitors, whereas km(jX) describes the mechanical system. Multiplication of these by the phasor C, which denotes the generator rotor oscillation, yields the electrical and the mechanical torques, respectively.
It is shown that all questions concerning the torsional interaction phenomenon can be answered by comparing ke(jA) and km(jA). Examples demonstrate the significance of the proposed method and illustrate that under certain circumstances even the electrical systems natural frequencies, which deviate from the torsional

network N2, which carry approximately 1% of the total current transmitting hardly any active power. At time t=0.033s these lines are disconnected. This small change leads to an interaction effect during which the torque in the shaft section Gen_LP2 begins to pulsate and the oscillations of frequency fi=31.4 Hz increase with a time constant of 1.08s. At t=3.8s the disconnected lines are reconnected, whereupon the torsional oscillations immediately begin to decay with a time constant of 2.27s.

120

T2[%]

modal frequencies interaction
machines.

of the shaft, can also initiate an effect., The proposed method can also be applied to prob -n_Generato_(reactance_in_p.u._an_time_in_s A crt lems of free or forced oscillations of synchronous
80

INTRODUCTION

Hp

MP

LPI

LP2

G

Ex

T2

~~~~~~~~0,021 +j 0,2724 -jO ,1346
FN

Induction generator effect he , i.e. the interplay torsional interaction The induction effect is the dominant factor in system disturbances and reducesvthe total resistance of the stator circuit for subsynchronous currents. As a result the time constants of decay of the exciting torques become greater. In contrast to the induction effect, the torsional interaction, i.e. the interplay between the electrical and mechanical systems, is the most significant factor during steady-state operation. Small oscillations caused by even slight changes in the operating condition can be amplified by torsional inthat the exciting electrical teraction. as This an cndqutmeans antcmesaetengtv ing~~~~~~~~~ torque of the generator with the frequency fr can exhibit a negative damping during interaction. Consequently, the mechanical shaft torques with a negative time constant is also-amplified if the mechanical damping is inadequate and cannot compensate the negative da'mping of the electrical system.
-

In power systems the subsynchronous resonance phenomenon caused by series capacitors is based on two physical effects [1].
effect, T

x'0032 0,1407 N0
ra=0.0036

0,0076 .j 0,2794

---===2-]
0,0076 tjh ,2794

-

IN2

Interaction effect Generator (reactances in p.u. and time in s) Ms 3 4 5 1= 2 =1.
x

Fig. 1

d
q

x11=0.235, =1.858, x'=0.29,
q q

5in

T

6

T"=O0.027 T'=0.223 q q

r 0 x Shaft train

x

Mass No.
Hs]

1
021

2
077

3
15

4
.8

5
024

6
014
.0

K[p.u.
Dpus

32.1
. .0
.0

86.2

113.6
.0
.0

105.7
.0
.0

39.5
.0
.0

Dpus

.0

An example of the torsional interaction phenomenon is demonstrated in Fig.l. A generator operating on full load feeds three lines, the uncompensated lines to the

seen that

From the data given below Fig.l, it can be easily during single line operation the electrical natural frequency referred to the rotor
f
r
=

x
n

n W x

+xTr+xL

ce

=

32.6

Hz

-deviates from the second torsional modal frequency of
82 WM 006-5 A paper recomnended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering. Society for presentation at the-IEEE PES 1982 Winter Meeting, New York, New York, January 31February 5, 1982. Manuscript submitted January 26, 1981; made available for printing October 23, 1981.

the shaft fm=3l.lBHz by about 1.4Hz, i.e. fr1fm' and also that the oscillation frequency fi during interaction is neither exactly equal to fr nor to fm. Thus, it can be asked: - Can torsional interaction also occur if fr were to deviate even more from fm (e.g. fr=33.SHz, fm=31.18 Hz)?
? 1982 IEEE

0018-9510/82/1000-3630$O0.75-

3631

- To what extent is the interaction effect influenced by the generator preload conditions ? - What influence is exerted by the ohmic resistances and the network configuration ? - How can the electrical damp in_ be precisely calcu-

controlled such that it follows the rotor oscillation ?. (4) Au = g(p)?

How far can the negative electrical damping be suppressed by additional field excitation control ?

_aEea r ~ ~~~~~~~~~~~~~~~------UN
XT. rT

U

T

L

C

Fig. 3

Simplified
network con-

XL.rL ,

xce

It is the object of this paper to answer these and other allied questions and to propose a general physically interpretative method within the scope of the two-axis theory without any simplifications.
In the second part of this paper it will be shown how this method can be applied to a highly complicated network having numerous nodes.

r*-rT+rL

x.wXT+XL ,Xce

figuration

STATOR CURRENT DURING ROTOR OSCILLATION
A synchronous machine connected to a power system can, under circumstances, exhibit small rotor oscillations of amplitude e. To an observer rotating at syn-

Fig.3 shows the single line basic network configuration, comprising a net resistance re, a net inductance xe and a capacitance xce between the machine terminals and the infinite bus of voltage UN. Using the voltage equations as a basis we can derive for the current components Aid, Aiq in the p-domain of the Laplace transformation the following relationship (Appendix I):

Ai

d

[

qo

de

-x

qe

p-l >d

?

(5)

qe de do chronous speed the generator rotor angle pulsates abou a mean value 10 between t0-9 and %0+9 with relative o the machine flux for the c T o complifrequency A (Fig.2). (N.B. To avoid of . . . given ~~~~The . unnecessary oscillating components in time t is cation in the relevant equations the given in App. I in * Pdeov 'Pqeo are, p.u., i.e. ~~~~~~ p.u.). ~~boundary conditions p u., i.e. ls=27Tfpu) The transfer functions Zde(p) xde(p), - P*Ui *eq. (A.20). * n zqe(P), xqe(p) and the established xd(p), x (p), G(p) are given in eqs. (A.19) and (A.6), (A.7), A.8), reUNqO+AUtJ q-

UNqo

UENq

UNjqo-

UNqo~t<; Il ,
/

spectively.

Re[c]
~

I'

N

. t+-

I<aat ,/
UNd- AUNd

/%

B/1t

?

e

e'g

r -f/ AUN+ UNO+

f

n

The first term in eq.5 gives the components iqoE -idos of the constant stator current i0 in the os(l) cillating rotor coordinates. It should be 'noted that these do not yield any pulsating component in the fixed 'stator frame, as in the case of the constant network voltage uN (eq.2, Fig.2). Therefore, it is the second wto / U \term which denotes the true response of the machine to
the oscillations.

Fig. 2

Eq. (5) can be considerably simplified by substitu- ting Xd(p) =xq(p) - and by ne lebtihg the ohmic resistances. Such simplification, however, is nOt recommendable, since the influence of certain parameters-can Assuming small oscillations, the d-q-components of then no longer be apprehended. For example, in a netthe nonoscillating network voltage uN would now seem to work without series compensation the well known hunting oscillate about their constant values uNdo, uNqo with phenomenon of a synchronous machine having a high reand frequency fr=Afn. sistance cannot be predicted. AuNd, AUNq u 'c -u AUAu.uN .., .. = ANd + ' UNd, UNd - UNdo Nqo? with frequency Xf. the inNd - Nqo Ndo + AuNd, For rotor oscillations (2) in the t-domain is obof eq.(5) transformation verse Au = -u E u + Au., uNqNq Ndo Nq Nq Nqo tained by substituting p=jA. The deviations are then the ~~~~~represented correspodn phsrids Ai odnphsrby ~~~~~~~~~~~~~~~~~~~~~ Aq and E. by the phasorm cresponding rers en The quantities with the prefix A represent devia_ s e tions which are of 'a pulsating nature and which are Ai d = Re[Ai] i c Ai caused by the small rotor oscillation . Identical ex-=d ] -d =-idc(6) Re[ai ai i E Ai pressions involving pulsating components, although unq q -q = -qi known at the present stage, are also valid for the machine quantities: The additional stator current phasor in the rotor frame is now + Au u =u (7) Terminal voltage: U = uo + Au Ai = Re[Ai ] + j Re[Ai ]

Schematic representation of rotor oscillation

Stator

current:

i
-

i

+

qo

do

Ai
h

By

taking

the

rotor

mtion

into

account

this

cur-

d

q

Flux linkage:

d=

q

pqo +tq

do + + Ald A'

rent phasor can bve transformed in the fixed-stator coordinate system. To this end the resultant current phasor SD+Ai must first be multiplied by the factor

ej(tt+c)

eJt (l+c)

=

eit[l+ .i(+s*)]3

The pulsating components Aud. ..AlPq are dependent on e. For the purpose of generalization, it can additionally be assumed that the excitation voltage uf is

and then linearized by neglecting the teXrms involving c2. The steady-state current (ido+iiqo)eJt must now be

3632

already has a small negative value in the oscillation frequency range Xfn> 20Hz (note the log scale). The compensation by the series capacitors results in a large increase in this negative damping of the 1 -dE -i +j(i Ai ai +i )] Rei(l+X)t (8) -S - 2 qo -qc do generator in the frequency range about the network natural frequency This effect becomes more pro1 . '1 Et + the degree of compensation and nounced the higherfr. [i -i ÷+j(i* +i e)] the oscillation frequency moves towards the lower The above expression clearly demonstrates that the range. stator current induced by the rotor oscillation has two - Fig.4b illustrates the influence of uncompensated components: lines connected in parallel with the compensated The of result is a heavy (computation reduction in in thePart negative The supersychronou componntoffequencone. (l+X)fn damping the generator II).
-i +j(i +i )| e ~ t 2 1|i -de qo -qe do super
( (8 1)

substracted from this to obtain the additional component Alt of the stator current phasor. This yields

The subsynchronous component of frequency (l-X)fn

2!dj_-i -oij(i

+id )

='Sub

(8.2)

. If the machine were a passive element (no excitaand tion only an impedance xd=xq=xd(p)=xq(p)in eq.(5)), then idc=iqo l.e=-ido and isubtsuperO, as expected.

COMPLEX TORQUE COEFFICIENT
A synchronous machine develops an additional torque ATe in response to small rotor oscillations (Appendix II, Te >0 for braking torque).

Te

[ufo-(xd-xq(P)ido]Aiq+[xq-xd(p)liqoAid

(9)

For rotor oscillations with frequency Xfn we can substitute p=jX. This yields ATe in the established complex form of representation. AT = k aT = ReLAT ] (10) -e -e e e with k e (JA) - Lufo -(x d -x q (j))i do ]i A)i -qc +(x q x( d3qJ-de

(jX)?,
K
+

e

j XD

(10.1)

e

. . . \ the "comWe shall denote the coefficient k,(jX) plex torque coefficient". In the case without a series capacitor, k5(jX) is known in the european technical literature as the "complex synchronizing coefficient" I [2]. ke( tsuper and i b can easily be computed, even with only a desk top calculator.

The real part Ke of ke(jX) represents the electrical spring constant and Dew the imaginary part after dividing by X, the electrical damping constant. Ke and De include the influence of both the current components isub and Isuper as well as the additional excitation control with the transfer function g(p). On highly simPlifying the eq.(10.1) De is obtained in the known form as given by Kilgore et al [3]. But such simplifications are not necessary, and moreover, as indicated earlier, can constrain the general validity of e.

.~~~~~~~~~~~~~~-

Db-thprlacndio

-

and D e are dependent upon the Ke e power system configuration, xce etc. - preload condition (operating point) - type of additional excitation control - number of machines operating in parallel, etc. Fig.4 illustrates these influences. The following comments should be noted: - The two curves denoted by "0%o" in Fig.4a refer to -the turbogenerator when connected to an uncompen.sated long line with xe=O.4l3l. Even in this case De

Accordingly, in a highly meshed grid the negative damping tends towards a lower value. Thus, power system meshing also has the effect of increasing the stability in cases of interaction. HVDC transmission lines [4] exert the same influence and stabilize the generator terminal voltage. This type of stabilization is a highly useful aid for decreasing the nega~~~~~~~tive damping or even for its elimination, - The dependency of De on the total armature resistance ra+re is demonstrated in Fig.4c. It is very interesting to note that for small values of ra+re the generator damping is positive even in the vicinity of the natural frequency. As ra+re now exceeds a critical value (in this example 0.005), De immediately assumes a high negative value in a narrow frequency range. From its highest value, De decreases almost linearly with increasing resistance, whereby the pertinent frequency range becomes wider. The influence of the preload condition can be especially pronounced in cases involving machines with anisotropic rotors and is moreover dependent on the network configuration. - Several machines operating in parallel will have the effect of shifting the De-curves to the left, i.e. towards the lower frequencies (Fig.8). - In Fig.4d and 4e the possibility of influencing the electrical damping De by an additional excitation control (SEDC) is demonstrated for two network configurations and for two elementary types of control, these being: - proportional control Auf = -2000? ~~~~~~~" Auf = -2000 PE - differential d Despite the very high gain, in the case of a single line (Fig.4d) the negative damping can hardly be affected in the vicinity. of the resonance frequency. However, a certain influence is experienced away from the resonance frequency. The presence of parallel lines results in the effect of SEDC being much greater (Fig. 4e). It is quite clear that these examples cannot explain the effect of SEDC to the full extent. As is known, the transfer function g(p) can influence the results of such studies [5]. Therefore, a comprehensive study must also take into account ..the influence of the technically feasible transfer function of the excitation control g(p) cein oltae - the preload condition - the true magnetic coupling between the field and the damper circuit L6] - saturation influence, which could totally nullify the effect of SEDC in highly utilized modern machines.
Remark: To determine the effect of SEDC it is absolu-

necessary that when ascertaining the function G(p) ~~~~~~~~~~~~tely

of the generator (see Appendix I, eq.(A.8)) account must be taken of the true magnetic coupling between the and the damper circuit. This coupling is given by ~~~~~~~~~~~~field the characteristic reactance XC of the synchronous machine L6].As an example, for the generator in Fig.l the actual function is

3633
100
10-

49.4%
---

De

\

15%

Ke Spring Const.
0.1
/

----0%

10-

0.01-20

-0.01

3D

40

fn

Hz

1

1

;

<

10 40 20 30 01 ~~~~~~~~~~~~~~~~~-o -0,01

Afn50 2030 10 60OOt

4
50 tnX6

Damping Const

-10

\49.4%

/

1

-1001
10M
1of

a
I

-10

-ioo
I

d
De
~

Ke: Spring Const.

100-

t~~~~~-- 5-s-------\

V

~~~~~~~~~~10

0.1
1

'1 0 20

/

//

4

-- N N
10

\ ,I

\

.1

-0.01 -0.01

0.01

_____Hz__0,1__\'__
3

3,440 * fn00,01
-~~~

20

1l

30

40

50

60

-1

~

~

~

~ fnA' ~~~~~~~~~~~~~~0,01

-100

Damping Const.

I

b

-10
~~~~~~Fig.4Complex
torque coefficient

e
ke(JX)=Ke+jXDe
line only

100 -De -10

\ 4\ |\ I: Dampngon0.00
1
\
/1 \

and network data as in Fig.l ~~~~~~~~~~~~Generator iewihcmesto asnl

-0.1~
0,1

~

\\ \> / i \ Influence of parallel lines ~~~~~~~b: ~ ~ ~~~~~~~~~~~~~~~~~0
\

with the compensated ~~~~~~~~~~~Netwlork

\ \Xce/XL-49.4 \ 0 \ ..... \A== ~~~II: with an additional parallel line
40

0.0 F 0.01 10 -0.01-

0.0055l
20
30

o1|

\

aol0.01
\

50 f

-lI

60Hz

/

c:
e:

(O.0O76+jO.2794) III: with additional parallel lines two'te nl omwisthou SieDC Netorkwih (0.0038+jO.1397)

Influence of ohmic

resistance,

Network I

.0

\ \ 11//

d: Influence of SEDC, Network I
Influence of SEDCr Network III
and e:a witorhd II:

0.11

ceL=494D -proportional
a

control

-sDO-

Au0=-20Opc=-5 .03A76[ 1/s7] A, B see Remark. ~~~~~~~C

differential control

3634

G(p) = (

lop 2131)-(l+p 8.64)

l+p

1.1

(curve A in Fig.4e)

whereas, the conventional theory, i.e. utilizing Xa instead of xc, yields

the generator rotor executes an oscillation as illustrated in Fig.2, i.e. At2=e. Eliminating At1, A%3, At4, etc. in eq.(ll) and substituting p in the expression for frequency response by jX as is customary, we obtain

system that

G(p)

(+p

21314.1p

8.64)

(curve B in Fig.4e)

Thus a comparison between these two curves shows that the result obtained on the basis of the conventional theory can lead to unreasonably favourable results.

C = (12) [km(jX) + Herein the second term is known as the electrical torque as given in eq.(10). This is the basic equation of oscillation in the interaction phenomenon.

kejOX)]

TORSIONAL INTERACTION

KeKe
1

During interaction and due to the negative damping of the generator, the shaft will tend to oscillate at its modal frequencies when this motion is not counteracted by the mechanical damping of the shaft itself. In principle,-only a study of the equation of motion,which includes the pertinent electrical torquet can predict whether or not interaction could occur. The interrelationships between the n-masses of the shaft train are given in eq.(ll) in matrix form. To simplify the representation, the second mass is assumed to be the genera-. tor rotor (Fig.l).

De

~~~Ke( j A)sKe+iADe

electrical system

1 ( -- dKm

K,-O

-Hi
Km

Km(jX)-Km+j.Dm
mechanical sy

Km2A2HI-KI
Dm 'Di
l

Dm Dm

H1(p) -K1(p)
-K
L
____ J

A

1
-AT

I
Shaft train simulation

(p4 -K (p) LK2(p) H3(p) -K3(p

AID

Fig. 5
L i - -(11)

l____ j _____[ ~ Kn l(P)j Hn(P)I
K D ..)p+K.+ )p+K +K )+0
D

with H Hi() (p) 2+(D D D P - 2H iP(

(111)

Ki(p)

)p+K

(11.2)

The frequency response now characterizes the complete behaviour of shaft dynamics from the point of view of the generator rotor only (Fig.5). Let us now as the "complex designate analogous to torque coefficient of the mechanical system". This can be determined from

km(jX)

km(jX),

ke(jX),

AtO denotes the angle of deviation of the ith mass from the linear rotary motion. Moreover, all the quantities, i.e. inertia constants Hi, spring constants Ki and the damping constants Di for the mass i and Di(i+l) for the shaft between the masses i and i+l are in p.u.
In the case of a single transmission 'line and a n-mass shaft train, eq.(11) can be transformed into a system of first-order linear equations comprising a toeigenvalue pairs talofnn+5 talof equation, th ine

km (jX)
with A

Km+jXDm

H2( X) K.(jX) i

H

1

K2(jX)A2(jX)

(13) (14)

iJK )X)A i+l

i+l

i+

These equations can easily be handled even by a desk top calculator.

nfollowing
1
.

wherdeu

Pi

to the absence of damping Dm=O. The real part Km intersects the abscissa at the points known as the torsional modal frequencies (note log scale).

is first computed without mechanical damping km(JX) (Di=Di(i+l)=O), and is shown plotted in Fig.6a. Owing

For the shaft train shown in Fig.l the value of

A negative time constant Ti would then signify that the pertinent oscillation of the frequency Xifn will be am~ there ~ plified, whereas, if all Ti values are positive will be no interaction. This mathematically sophisticated method yields a result without a physical interpretation of the 'problem. Moreover it is not feasible to apply this method to a highly'meshed network without certain simplifications. Therefore, this paper proposes a general method which can be applied to any network and where physic-al understanding is preferred to mere mathematical sophistication.1
The basic idea underlying this approach is that it must be possible ~to describe the interaction effect1 simply by the equation of motion - or by the energy balance - of the generator rotor (here the 2nd mass). Based on this, and similar to the way in which the complex torque coefficient for the' electrical torque ATe is determined,' it is also assumed for the mechanical

theIn principle, Km must represent the basic form of th equation of oscillation for each of these modal frequencies: K 2 + (15) m - ii + i
The gradients of the curves at the point of intersection with abscissa Km=O yield the pertinent shaft modal inertia constants Hi. dK

H1 = - 1g
values:

dKm)

at Km

=0°

(16)
yields the following

By way of an example,

Fig. 6a

H-l68 H1 - 168

H2 =21s

.9sH-2 8sH-2304 3 - 2391 4 -7804

3635

Fig.6b illustrates the behaviours of Km and Dm

-1000-

Ii Km t
| /
/

-100-

/ '

Bl

-10- /
-1l

; / f ,X / l /i 1o l l ll 2 3

) ___

taking mechanical damping into account. The

|
: t

previous case. Simiues of the modal inertias (H1=l.6776, H2=2.1987s). In to the previous case, the points of interseccontrast | tion yielding frequencies f3 and f4 no longer exist for the generator, although these frequencies do exist in the case of eigenvalue analysis. The reason for this

ly

intersects the abscissa only at two points yielding frequencies f, and f2, these having remained practical-

curve

Km

larly, damping has

unaltered with

respect

an insignificant effect on the val-

to the

Mode 1
-0.1

4

discrepancy is that the generator rotor now forms a nodal point for the frequency modes f3 and f4.

-0.1d-826.95!
0.01-10

dKm

1717.19

21963.8
136.44
40 l\

l' 4.0181x7

By introducing km(jX), eq.(12) can be written in the form

0.1-

20 30 /0 /l 31.18: fm= 19.61

A~ fn

50 50

58.4BHz l l
I ,

Hz/li

60
l
I !

K m + K e + jX

(Dm +D e )=

(12.1)

Where there is no damping (Dm+De=O) the frequencies of shaft oscillation must satisfy only the following con-

dition

Il tt 1l1
10
10

I
1

0 K O ~~ + K~~~~~~~~~~m e
l

(interaction frequencies)

(17)

//
/| /

l

X } |

|

100
/

/00

.000

a

i.e. at these frequencies the curves K e and -K m intersect. Since the values of Ke are relatively small,these points of intersection must lie close to the modal frequencies (Fig.6a and 6b). In the case of a resultant damping (Dm+DeO),as explained earlier, the oscillation frequency will deviate only insignificantly from the undamped case. If the resulting damping is positive, then the oscillation would decay. Therefore an interaction between the electrical and mechanical systems occurs only if the resultant damping for the frequencies satisfying the eq.(17) is negative.
Km + K e

-10001
-100 /

II1000t

--

_e
11
or

Dm + De < 0

(18)

,
(

/

-', II
Ii

/

7

A ,'The rigorous solution of the eq.(12) or (12.1) in the form

-10 -1 1
-0.1
-0.01 t 0.01
0.1

/\
0

pi
1

' Km

l
10 20 30 I 40

50 ~~~Hz 60

fnA

lies only in having the knowledge whether the interaction process is fast or slow. On the ground of additional space requirement the exact solution will be deduced in a later treatise, which most probably will be published in "Electrical Machines and Electromechanics". For the present the equation as used in the literatures will be given.
T

yields the frequency Xi as well as the time constant Ti of interaction. However, with regard to Ti the interest

T

X

T

I t

I I

I I
/

1 |
mn {

|

Dm

(log.Dec. = ~~~~~~~~~~~~~~i D-miDe-fnu-is
- i

4 H.

(19)

r

10~~~~~~~~~~~~~~~~~~~~~~~~~~~~e lo.. \ / e
\ /t , \ / 11
/o 100
1000

I
b

Fig. 6

Complex torque coefficient of the shaft km(jX) =Km + iXDm (shaft train in Fig.l) a) withqut mechanical damping, Dm b) with mechanical damping (see Fig.l)

an approximate expression. This is no more fully valid, in case e.g. the -Km-curve intersects the Ke~ \-/| \curve in a very steep region (Fig.4b between the regions P,P',i.e. in the vicinity of the maximum negative electrical damping). In such cases for the same value of Dm+De the time constant Ti will be influenced not only by the shaft modal inertia constant Hi alone, but also by the network. This method is applied to the three different network configurations shown in Fig.4b and the result is illustrated in Fig.7. This leads to the following inferences:

only

However it should be mentioned here that this is

- The points of intersection A and B between the curw m - ~~ves K and ~Km correspond to f11=19.78Hz and fi2= 3l.45Gz (refer to result in Fig. 1). These deviate

3636

-K

10b
1

/----/
Ke

I

-Km
//

|

\

/' /
1 ffA

, ,,/ I,1 'I

__between Fi (30.5Hz) and F (34.5Hz) negative. Since the point of intersection 7 lies within this region,
Damping: Time constant:

an interaction will result which is characterized by:

D +D = 3.80 - 11.85 = -8.05
T.

B 1/ Ke I; t!1
~ .--I

4 -.05
-8.05

2.190
1

= -1.088s

---

__1 Zt 11

f

I

neg.log.Dec

-1.088 ~~~~~~~~~~~~~~~~31.45

= 0.0292

0.1

M Mode
0.01

2
40 40

0.01-_l_| A fn -0.01 10 20 -0.013a
20 10

3b

5b

50

60 Hz 60

-0.1 -1
-10

\

/ /C:>

De
\ D Dm

/z B \ / \ iF1m Fd

\\ )t /

/B

D

-100|
-1000
-1000-

1

Fig. 7

Searching the region of interaction k e (jX\) - 0 J + km a (JA) = (Fig. 4b) + (Fig. 6b) II, III: Correspond to the different network configurations as given in Fig. 4b

network I the electrical natural _ For frequency of configuration the system referred to the rotor is 32.6Hz. This frequency corresponds to point D, which exhibits the highest value of negative damping. Although the exciting electrical frequency of 32.6Hz deviates by about 1.4Hz from the torsional modal frequency of 31.18Hz, interaction still occurs. However, if the line compensation by the series capacitor is 45.9% (xL=0.2724, xce=O.l25), then the electrical system frequency referred to the rotor is shifted and assumes the value 33.5Hz. The new De-curve (not shown in Fig.7 for simplification) consequently yields another value of resultant damping Dm+De=3.8O-5.OO=-l.2. Despite the relatively large deviation between the natural frequency of the electrical system fr=33.5Hz and the torsional modal frequency of the shaft line fm=31.18Hz, a slow interaction exhibiting a time constant Ti=-4-2.19/1.2= -7.3s will now take place. _ If two identical turbosets running in parallel are connected to the network configuration I, then the De-curve will be shifted to the left, i.e. towards the lower frequencies (Fig.8). Consequently, the insince the negative electeraction ceases to exist, trical damping occurs in a region of much higher mechanical damping (note the shape of the curve DM). However, the result may be totally different in the case of other turbogenerators or networks. It follows that the curves De and Dm can be used to answer all questions pertaining to torsional interaction phenomenon, since not a single frequency, but the entire region is dealt with.

-0.01

10

20

30

-0.1

40 1, / {, B A

50

Afn

~~Hz

60

CONCLUSION
Y The conventional method for calculation of the interaction phenomenon is based on a comparison between the mechanical damping Dm and the electrical damping De at the shaft modal frequencies. In the proposed method this comparison is made for the entire frequency range or in the vicinity of the shaft modal frequencies. With this type of analysis it is possible to study more closely the danger of interaction and also its countermeasures (e.g. SEDC) taking the possible deviations of the network and shaft natural frequencies into consideration. (SEDC: Suppl. Exc. Damp. Control)

-1
-10 -100

De \

\

/' / /r a ,__I\/ "/D, ~ m

Fig. 8

The electrical and mechanical behaviours of the system can be described by the "torque coefficients" ke(jX) and km(jX). In ke(jX) the effects of sub- and supersynchronous frequencies are included. The pertinent equations are deduced rigorously without any simplification whatsoever. On the basis of the above mentioned coefficients it is possible to calculate the very little from the modal frequencies of the shaft electrical damping De as well as the spring constant Ke line. For other modal frequencies there is no inter- the imaginary and real parts of ke(jX) - and thus action effect (no further points-of intersection), formulate the equation of oscillation for the interac- For the network configurations II and III (Fig.4b) tion phenomenon. el of he quinsdrvd he ub the mechanical damping has a higher value than theWihte and suesnhroou comonnt negative electrical damping of the generator at the of the h statos eivdtor curren points A and B. Consequently there is no interacan Uesnhno cmnnt ofhe ttrcuet . .~~~~~~~can .. be calculated as a function of the rotor oscillation. Only in the case of the network configuration tin. Iis the resultant damping in the frequency region tos

Influence of generato-rs in parallel A: one generator only B: two generators in parallel Network configuration I in Fig. 4b

3637

The application of this approach in the case of an arbitrary network with innumerable nodes is given in Part II.
This method can also be applied to solve problems related to free or forced oscillations of synchronous machines operating in power systems without series capacitor.

Inclusion of an external impedance:
The external impedance at nominal frequency is given by

Ze

=

re + j(x e-x ce)

(A.ll)

represents the reactance of the series capacitor. The equations for the voltage drop between the phasors u of the machine and !N of the network voltage are: di
u

xce

APPENDIX I

General voltage equation for rotor oscillation
The rotor speed of an oscillating synchronous machine is described by
n =1+

du

-N
c
x

ece-

e

dt

(A.12)

dt

i

dt

-de
+
-

=

j jt Re[e [

(A.1)

The application of Park's transformation, the rotor oscillation (O = t + s) into account
u = Cu + j u)
*=

taking

The common voltage equations are

Ud
q

ud -ri

=raid
raiq

n*q
nld

~~~diP dt
dt
t

i =

d

(A.2)
d

q et+)(A.13) (d+ J Lq e +)
+

u = -rai

Ud

yields:

.
e d

de

By linearizing these about an operating point (substituting ud = udo + Aud etc.) we obtain:
Aud
Au

du

Nd
Nq Nq

1+dt Xeq
e

+dt
-

ed
q

d
+

de qo dt
=

d

uqce

u

rAid

q

Auq

=

-Wdo

dt

raAiq

de
-

-

d

A4'd

dt

d

(A.3) Xd

APq

d

d ucd dt~~~~~~
d

+ (1+ re e iq -t) xe d
(

d

-t xe

d

uc cq
(A.14)

d+ dt

u

cq

and in the

p-domain of Laplace transformation
(A.4)

Xceq dt

xi

ucq+

+(1

+ dt

d)u

Ucd

Au d = p E i - r a Ai d + At - p Ap 4'd qo q Au q = -PE:* do - rAi q A4 d -p A4'_ q
a

Linearization (ud = udo + Aud etc.) and subsequent application of the Laplace transformation leads to

with

Aud
=

-AuxNd i ere+Px e <
-

Ald xd(p)Aid G(P)Auf,
-

A4q = xq(p)Aiq

(A.5)
~~0ud

Auq AuNq
0

Xedo
u cqo

e

-xe re PXe
0
-x
ce

1
0

0
1

Aid
A

q

and

impedance operators
=

-x ce
do_

p

-1 Au cd
P A

xd(P)
x

__d (l+pT)(l+pT.6)

U+PT~~~)(1+PT11)

0

1

(A.15)

(P)

=

xq

(

(l+pT' )(l+pT")
q qo

qo
,, '

(A.7)
dx c XI x'd (A.8 8) TddcTd x'-x d-d c x"

equation

After eliminating ucd, ucq and rearranging, this can be written as

G(p)

=

__

d__T____itx- ) (l+pT'0)(l+pT do do

l+Pd

Aud-u qo qo
AuudoE
with
z

Au N-u Nq Nd Nq

E ()A ze()x e Ai d iqo z

x()

-

(A.16)

Xc in eq.(A.8) denotes the characteristic reactance of the synchronous machine and can be determined from a sudden short circuit test. Definition in [6].
Elimination of Ad and Aip
in

Au +uNdoe
p (xe+
x
ce

Ze(P)

Ai

d

eq.(A.4) yields
q

(p)

=

re Of
-

+

Ox

(A.17)

=9pE+pe+
4'do
where

xt

(p)Au -|

(t

d()~
z

(r)129

(A.9)
(A.10)

e- 7 Eq.(A.16) describes mathematically oscillating d-q coordinates.

xe(

an

impedance

in

zd(p) = ra + P xd(p),

z (p) = ra + P x (p) q q

~~~~~~write

eq,(4) and AuNd, AUNq from eq.(2) in eq.(A.9) we can

By substituting AUd, Auq from eq.(A.16), Auf from

3638
P -1
+ deo
z de

(p)

1p
where

L:L: L 9 Xde(P| qe(P) I tiq+idoE
qEp

_ _ .e=

-x qe

(P) dj [i-i dqo
(p)
.

xde(p)

zq

IAiq+1idosl

(A.18)

Technical literature [1] often shows that the torque equation (A.21) is directly transformed in the p-domain, yielding the form

Te
t

q(p) id(P) - Wd(P) 'q
id(T)
dT

)

zde(p)

+re+p
=

d (p)+xe
x

p2+1

x ce

This expressionZ however is not correct. According to the Convolution (Faltung) theorem, e.g. the product q(P) id(p) in the t-domain is [7]

xde(P)
z

Xd(P)+Xe~ cei
x

(A. 19)

0

f 4q(t-T)

qe

(p) = r +r +p(x (p)+x + ae
q

ep2+1

Ae19

and not 4q id.

xqe(p)
tdp_o
=

x (p)+x

=xce p2+
[x q-xd (p) i q

Therefore to obtain the torque equation in the p-domain eq.(A.21) must be linearized prior to the transformation and not afterwards.

and the oscillating components of the machine flux for boundary conditions 4dso' 4qeo are given by

G(p) g(p)
4)do

qo -u

- (p)g()x [xPxd(p)] do [xdfxd(p)]
q

+

BIBLIOGRAPHY AND LITERATURE
[1] IEEE Committee Report: "A Bibliography for Study of Subsynchronous Resonance between Rotating Machines and Power Systems" IEEE-Trans., Vol. PAS-95, No.1, 1976, pp.216-218, and First Supplement: Vol. PAS-98, No.6, 1979,

Lq0

(A.20)

The solution of eq.(A.18) leads to eq.(5).

pp.1872-1875

APPENDIX II
Electrical torque due to rotor oscillation
The general expression for torque is T e =4)q i d - 4 d i q

[2] Th. Laible, "Die Theorie der Synchronmaschine im nichtstationaren Betrieb" Springer-Verlag Berlin 1952

(A.21)

[3] L.A. Kilgore, D.G. Ramey, M.C. Hall, "Simplified Transmission and Generation System Analysis Procedures for Subsynchronous Resonance IEEE, PES Winter Meeting 1976

Problems"

where Te > 0 for the braking torque (coordinate system in Fig.2). For small deviations about an operating point, this equation can be linearized as follows
e

[4] S. Svensson, K. Mortensen, "Damping of Subsynchro-

Wqo hd
=

+ do

Qq
q

-

Vdo "iq
x

-

'AT A*d iqO

Ai

-(A.2280 (.2
+i

nous Oscillations by an HVDC Link. An HVDC Simulator Study" SM 668-4, IEEE PES Summer Meeting, Minneapolis

It is now possible to use the Laplace transformation.

Ad

d

x (p) qi ,

d

d

q

(p) qi q

(A.23) .
(A.24)

[5] El-Serafi, A.M. Shaltout, "Damping of SSR oscillations by Multi-Loop Excitation Controller" IEEE, PES Summer Meeting 1979, Vancouver [61 I.M. Canay, "Extended Synchronous-Machine Model for
the Calculation of Transient Processes and Stability" Electric Machines and Electromechanics, Vol.1, No.2, pp.137-150
R.V.

so that we obtain

ATe

[)qod.qoxd ()] Aid - [4doidoxq(p)] Aiq

The substitution

[7]

Mc Graw-Hill Book Co. New York 1958

Churchill, "Operational

Mathematics"

do

=Xd ido ufo,

4)qo

xq iqo

(A.25)
For combined discussions see page 3644.

leads to eq.(9).


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