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Electric Power Systems Research, 8 (1984/85) 261 - 274 261 Theoretical Study of a Shunt Reactor Subsynchronous Resonance Stabilizer for a Nuclear Powered Generator M. A. LAHOUD and R. G. HARLEY Department of Electrical Engineering, University of Natal, Durban 4001 (South Africa) (Received August 4, 1984) i SUMMARY The paper presents the results of a theoreti- cal investigation into the suppression of subsynchronous resonance (SSR) instability of a nuclear powered turbogenerator by using a signal derived from the generator speed to modulate the reactance of a shunt reactor situated at the generator terminals. The analysis and design of such a shunt reactance controller are carried out by considering the full two-axis representation of the generator and network. :Predicted results are presented to illustrate the ability of this controller to suppress subsynchronous resonance even when the system is subjected to severe large disturbances. I. INTRODUCTION In order to improve the power transfer capability of an EHV transmission line, sev- eral methods of line compensation, including series capacitors, are currently used. However, the use of series capacitors leads to instability and subharmonic oscillations known as sub- synchronous resonances or SSRs. The two turbogenerators to be installed at Koeberg nuclear power station in the Western Cape will be connected through 1400 km of 400 kV series capacitor compensated trans- mission line to the larger Transvaal grid. A previous investigation [1] has shown that, unless precautionary steps are taken, unstable SSR oscillationscould occur at Koeberg if the transmission system is to be operated with anything but a modest level of seriescapacitor compensation. Various schemes could be considered as possible solutions to the Koeberg SSR problem. These include twin transmission lines, an HVDC link, series static filters, auxiliary excitation control and optimal control. Although the first method would work, the additional transmission line would be expensive and detrimental to the environ- ment. While the second technique would also work, it in turn would involve expensive ter- minal equipment [2], and would not permit convenient connections to other loads or generating centres at points along the route without further expensive terminal equip- ment. The third method, comprising static filters, would be successful only under certain conditions [3]. The fourth possibility is based on the use of an auxiliary excitation control scheme; the principles of such a stabilizer have been described by others [4-6]. Although optimal control is the most sophisti- cated scheme, it suffers from the same problems as the auxiliary excitation control method owing to the limitation introduced by the characteristics of the rotating diode exciter, and the relatively large field time constant peculiar to the Koeberg generators. A further problem with the optimal control solution is the difficulty of measuring 22 state variables. This paper deals with the possibility of applying suitable shunt reactance control [7-9] to the Koeberg system in order to suppress SSR over a wide range of compensa- tion levels. Figure 1 shows the generator and controller system; the mechanical system is represented in Fig. 2. A modal analysis [ 1 ] of this shaft system results in the mode diagrams and modal frequencies in Fig. 3. The now familiar mathematical models of the generator and the distributed mass system appear elsewhere [1, 4]. However, this paper illustrates the use of an alternative generator 0378-7796/85/$3.30 © Elsevier Sequoia/Printed in The Netherlands
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Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

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Page 1: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

Electric Power Systems Research, 8 (1984/85) 261 - 274 261

Theoretical Study of a Shunt Reactor Subsynchronous Resonance Stabilizer for a Nuclear Powered Generator

M. A. LAHOUD and R. G. HARLEY

Department of Electrical Engineering, University of Natal, Durban 4001 (South Africa)

(Received August 4, 1984)

i

SUMMARY

The paper presents the results o f a theoreti- cal investigation into the suppression o f subsynchronous resonance (SSR) instability o f a nuclear powered turbogenerator by using a signal derived from the generator speed to modulate the reactance o f a shunt reactor situated at the generator terminals. The analysis and design o f such a shunt reactance controller are carried out by considering the full two-axis representation o f the generator and network. :Predicted results are presented to illustrate the ability o f this controller to suppress subsynchronous resonance even when the system is subjected to severe large disturbances.

I. INTRODUCTION

In order to improve the power transfer capability of an EHV transmission line, sev- eral methods of line compensation, including series capacitors, are currently used. However, the use of series capacitors leads to instability and subharmonic oscillations known as sub- synchronous resonances or SSRs.

The two turbogenerators to be installed at Koeberg nuclear power station in the Western Cape will be connected through 1400 km of 400 kV series capacitor compensated trans- mission line to the larger Transvaal grid. A previous investigation [1] has shown that, unless precautionary steps are taken, unstable SSR oscillations could occur at Koeberg if the transmission system is to be operated with anything but a modest level of series capacitor compensation.

Various schemes could be considered as possible solutions to the Koeberg SSR

problem. These include twin transmission lines, an HVDC link, series static filters, auxiliary excitation control and optimal control. Although the first method would work, the additional transmission line would be expensive and detrimental to the environ- ment. While the second technique would also work, it in turn would involve expensive ter- minal equipment [2], and would not permit convenient connections to other loads or generating centres at points along the route without further expensive terminal equip- ment. The third method, comprising static filters, would be successful only under certain conditions [3]. The fourth possibility is based on the use of an auxiliary excitation control scheme; the principles of such a stabilizer have been described by others [ 4 - 6 ] . Although optimal control is the most sophisti- cated scheme, it suffers from the same problems as the auxiliary excitation control method owing to the limitation introduced by the characteristics of the rotating diode exci ter , and the relatively large field time constant peculiar to the Koeberg generators. A further problem with the optimal control solution is the difficulty of measuring 22 state variables.

This paper deals with the possibility of applying suitable shunt reactance control [ 7 - 9 ] to the Koeberg system in order to suppress SSR over a wide range of compensa- tion levels. Figure 1 shows the generator and controller system; the mechanical system is represented in Fig. 2. A modal analysis [ 1 ] of this shaft system results in the mode diagrams and modal frequencies in Fig. 3.

The now familiar mathematical models of the generator and the distributed mass system appear elsewhere [1, 4]. However, this paper illustrates the use of an alternative generator

0378-7796/85/$3.30 © Elsevier Sequoia/Printed in The Netherlands

Page 2: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

262

~ , _ . R 1 L 1 C R 2 L 2 I II

I I

Shunt Reactor

I

L3~=

\ /

Fig. 1. ' rurbogenerator with shunt reactance stabilizer.

I K(,) I I-

bus I rr~ Vb L

J1 J2 J3 J4 J5

Fig. 2. Koeberg turbogenerator system.

e~ , J6

7 L? ol I o o o o ~ MODE 0

OJ 1.05 HZ -1 • MODEl

8 0 6.87 HZ

- I --o o

8 0 12.5 HZ -1 • 1] / / / D , . ~ MODE 3

8 0 ~ 16,1 HZ

• 1 ~ MODE 4 6 o- 17.5 HZ

-1 +1]- ~ MO DE 5

8 0 1 ] ~ ~ ~ ~ ~ / " 92.6 HZ

Fig. 3. Mode diagrams and modal frequencies of the Koeberg turbogenerator.

representation which is useful in the presence of a shunt load such as a reactance, and par- ticularly in the case of a multimachine SSR investigation. It also describes the design of the shunt reactance controller (SRC) by examining the eigenvalue behaviour of the linearized mathematical model. The ability of

a specific SRC to reduce torsional interaction is then verified by simulation of the full non- linear description of the generator, mechani- cal system, transmission network, automatic voltage regulator (AVR), governor and SRC, when subjected to a severe disturbance.

2. THEORY

A previous investigation [7] used a phasor representation of the generator and a number of approximations in the derivation of design equations for the reactance controller. How- ever, this section presents the two-axis equa- tions of the generator in a form which allows their easy combination with the two-axis equations of the transmission system and simplifies the choice of the network state variables.

2.1. Generator model The two-axis mathematical model for the

Koeberg generator provides for two q-axis damper windings, one d-axis damper winding, and one d-axis field winding. These nonlinear two-axis equations can be manipulated [10] into an alternative description (see Appendix A) in order to initially represent the generator as an equivalent circuit in ABC phase variables in Fig. 4 for the purpose of orderly choosing the minimum number of total system state variables. This is a particularly useful tech- nique to cast the equations of a large multi- machine network into two-axis form.

In Appendix B the network d, q equations are derived and linked to those of the genera- tor which are also in a synchronously rotating reference frame.

Page 3: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

263

v c i 2

----4-'-3 or~ R 2 L 2

R 3

v t

Infinite bus Ra

1 i 3

" i g

Fig. 4. System configuration when the generator is represented by an equivalent AC circuit.

The generator state variables are chosen as the d- and q-axis components i2D and i2q of the stator current i2 (flowing through L* in Fig. 4) in addition to the rotor flux linkages ~rd and ~rq where:

- - [ [Rrd] [Lrd] - I [ [o]

+ [ [Rrd] [Lrd]- IMdT [ 0

[0] [Lrq]_l] [ ~rd [R,.] ~,q ]

0 [Rrq][Lrq]-lMqTl [T] [ i2D ] i2Q J

(I)

2.2. Ne twork description The chosen network states in Fig. 4 are the

d, q components of vc, i3 and i2. The two-axis components of i2 are also the generator stator state variables i2D and i2Q.

2.3. Turbogenerator shaft system The Koeberg turbine and generator me-

chanical system can be modelled [1] by six lumped inertias and five torsional intercon- necting shafts as illustrated in Fig. 2. The

torque balance equation is given by the fol- lowing set of second-order differential equa- tions:

0 = [J]p2b + [D]p5 + [K]~ + r (2)

where b is the vector of angular deviations, [J] is a diagonal matrix of inertias, [D] is a diagonal matrix of damping coefficients, [K] is a symmetric matrix of shaft stiffnesses and T is a forcing torque vector.

2.4. The turbine and governor The turbine and governor mathematical

models illustrated by the block diagram in Fig. 5 were simplified in order to reduce the complexity of the total model for the purpose of this investigation. These simplifications are justified, since the relatively slow turbine and governing system have no major effect on the suppression of SSR. Numerical values of the parameters in Fig. 5 appear in Appendix C.

2.5. Exciter and automatic voltage regulator The rotating diode exciter and AVR

mathematical models closely describe the actual equipment. Their block diagram ap- pears in Fig. 6 and their parametric values in Appendix C. The adjustable AVR gain Kay is chosen according to the methods presented in ref. 11.

2.6. Shunt reactance controller (SRC) The SRC proposed in this paper uses a con-

ditioned generator speed deviation signal to modulate the shunt reactance. The controller function K(s) in Fig. 1 may, in general, have a lag-lead or lead-lag form depending on the values of W 1 and W2, hence:

K(s) = K(s + WI)/(s + W2) (3)

The state space decomposition of eqn. (3) is given by:

p x = pag -- W2x (4 )

Yl = (xW1 + p x ) K (5)

I Pref phase advance servo entrained compensation saturation motor steam reheater

L! , / I,,!" + q "+ I v .

Fig. 5. Simplified block diagram of the Koeberg governor and turbine.

Page 4: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

2 6 4

where p6g is the speed deviation of the gener- ator's inertia and y ~ is the modulating param- eter for the shunt reactance. In other words, if the shunt inductance has some nominal value L 3 nora, the actual modulated inductance L 3 is given by:

L 3 = (1 + Y l ) L a n o m (6)

The choice of the parameters W1, W2, K and Lahore is based upon the eigenvalue loci in Figs. 7 and 8.

2.7. Linearized form of the nonlinear state equations

The entire system discussed so far is there- fore described by 31 differential equations (some of which are nonlinear), which are

made up as follows: ten for the network and generator, twelve for the shaft system, eight for the AVR, governor and turbine, and one for the shunt reactance controller.

In order to use eigenvalue techniques during the design of this controller, any non- linear equations are linearized about some steady operating point to produce a set of 31 linearized equations, which can be sum- marized as follows:

[P] Apz = [E] Az + [S] Au (7)

where

Z = [ ~ J r d , ~ / r q , p 6 , 6 , i2DQ, i3DQ,

VcDQ, X, A,/G] T (Sa)

U = [VbDQ, bDQ, Old, Pt] T (8b)

Vref

Input filter I compensation

d l mm I+ Ue2

Exciter Uel

1 Vfma Vfd

saturation

Fig. 6. Block diagram of the Koeberg AVR and excitation system.

125.00

tO \ [3 <Z r r

>- n ~ ,,( Z

0

1 0 0 . 0 0 ,

75. 0 0 .

50. 0 0 .

25.00.

0. 0 0 -1. 50

M 4

M 3

MO

M1 -

M2 C---7

t L ~_--J

I /-

77 .0 ] - I 0 . 8 0 0.I~:

I 1 i i =

- . 0 0 - . 5 0 0 . 0 0 . 5 0 1 . 0 0

REAL I/S

Fig. 7. Selected eigenvalue loci as X3 nora varies from 6 to 10 p.u. for the generator without an SRC.

Page 5: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

\ 0 < n~

265

1 0 1 . 00 .

>- [E , ( Z

C3 <

• • • • " Wl

1 0 0 . 0 0 .

gg. 0 0 .

gs. 0 0 - I . 25

102 . 0 0 ,

t - . 75 5 0 - 1 . 0 0

(a) REAL 1 / S

7 g . 5 0

I - . 25 0 .

CJl \ 0

7 g . 0 0 .

• K W l 78 . 5 0 .

7 8 . 00 .

>-

< Z

[..9~'~ 7 7 . 50. " I :~~"~'i. < ~r

77. 00 - 1 . 0 0 - . 5 0 0 . 0 0 . 5 0 1.

(b) REAL 1 / S

Fig. 8. Selected eigenvalue loci of the Koeberg generator fitted with an SRC for various values of K, WI and W 2 : (a) mechanical mode 3 eigenvalue; (b) mechanical mode 2 eigenvalue.

The eigenvalues o f the matrix [ P ] - I [ E ] are also the eigenvalues o f the system. The loci in Figs. 7 - 12 are produced by successive calcu- lat ion o f these eigenvalues fo l l owing the adjustment o f a particular parameter. How- ever, o f the 31 eigenvalues, o n l y those critical loc i which direct ly inf luence the particular aspect o f the system behaviour are p lot ted; in the case o f c o m p l e x conjugate pairs, o n l y the posit ive imaginary parts are drawn.

3. RESULTS

3.1. Choice of shunt reactance controller parameters

The control ler was designed around the steady operat ing po in t o f

Vb = 1.0 p.u. , Pm = 0 . g p . u . , Vm = 1.12 p.u.

and a c o m p e n s a t i o n level o f 60%, where

% c o m p e n s a t i o n _~ IOOXc/(X 1 + X2 + X*) (9)

Page 6: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

266

09 \

0 < Of

>- rY < Z

(3 <

=E

125. 0 0

100. 0 0

75. 0 0 .

5 0 . 0 0 .

25 . 0 0 .

0 . 0 0 . - 1 . 5 0

MO

M4

M1 6 o

7

M3

10

M2 O • 7

I

I

I I

F

I 10 I

I

I I

I

b

F

I

- 1 . 0 0 - , 50 0 . 0 0 5 0

REAL I/S

Fig. 9. Selected eigenvalue loci as X3nom varies from 6 to 10 p . u . for the generator with an S R C .

tO \ 0 < O~

>-

< Z

t_3 <

125. 0 0 .

1 0 0 . 0 0 _

75. 00 .

5 0 . 0 0 .

25. 00_

M 4 - ~ X c = 0 . 4 9

i M3

M2

M1

0 . 0 0 - 2 . 0 0 0 . 0 0

MO

- 1 t. 00

REAL

Fig. 10. Selected eigenvalue loci as X c varies from 0.1 to 1.1

1 I. 021 2.

1 / s

p.u. for the generator without an SRC.

The parameters of the Koeberg turbo- generators (including those assumed for the mechanical damping) appear elsewhere [1]. The transmission system parameters are given in Appendix C.

Without the SRC loop closed, but with X3,om connected to the generator terminals, the critical eigenvalue loci calculated at the above operating point appear in Fig. 7 for the values of X3,om varying from 6 to 10 p.u. A low value of X3no m gives the most damping to mode 2 (M2), but requires a large steady state

reactive current , thus increasing its MVA rating. On the other hand, a large value of X3nom requires less reactive current but pro- vides less damping to M2. For the purpose of this paper, a value of 7 p.u. was chosen in order to proceed with the design of the con- troller parameters K, WI and W 2 .

With X3nom = 7 p.u. the controller param- eters K, W1 and W 2 of eqn. (3) were scanned and the critical resulting eigenvalue loci are shown in Fig. 8. Along locus K the gain K is varied from 0 to 0.15 while W n and W2 remain

Page 7: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer
Page 8: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

268

the SRC from the unstable locus in Fig. 7 to the partially stable one in Fig. 9; it has also become more sensitive since in Fig. 7 it moved from 0.81 to 0.87 and in Fig. 9 from --0.43 to 0.06.

This design technique differs from that in ref. 7 (which develops a sizing relationship for the SRC reactance which depends on the angular deviation expected from the rotor), since the eigenvalue loci indicate the relative damping of each mechanical mode. These loci can be recalculated for different combinations of K, W 1, W 2 and X3nom to ensure some 'best' design depending on the restrictions of the parameters.

However, the remainder of this paper evalu- ates the behaviour of the system with an SRC which has the previously selected parametric values. Calculated results are now presented in order to illustrate the effects of compensation level and system frequency upon the system's small-signal stability and then to demonstrate the damping effect of the shunt reactance controller when the system is subjected to large disturbances.

3.2. Variation of compensation level 3.2.1. SRC loop open (K = O) For this particular test the AVR and

governor control loops are open and the initial operating conditions are as stated earlier. Figure 10 shows a part of the locus of eigenvalues as the capacitive line reactance is increased from 0.1 to 1.1 p.u. (a compensa- tion level of 7% to 77%). Mode 4 (locus M4) is unaffected but modes 1 to 3 (M1, M2, M3) all exhibit considerable interaction. M3 is the first to go unstable when X¢ reaches 0.49 p.u. or a compensation level of 34%. Although not shown in Fig. 10, mode 5 is unaffected by the variation in compensation level. If Xc were to be increased above 1.3 p.u. the induction generator effect would cause an electrical locus (not shown in Fig. 10) to become un- stable [3].

3.2.2. SRC loop operating This section investigates the ability of the

SRC to prevent the above instabilities of modes 1 to 3. Figure 11 shows only the sen- sitive eigenvalue loci for the Koeberg system when equipped with its AVR and governor as well as with the SRC, while Xc is varied over

the same range as in Fig. 10. The initial con- ditions remain as stated earlier.

A comparison of the results in Figs. 10 and 11 show that the addition of the SRC

(a) has a negligible effect on mode 4 (locus M4) since the generator is situated on a node of this mode,

(b) stabilizes modes 2 (M2) and 3 (M3) and increases their damping,

(c) increases the damping of mode 0 (M0), and

(d) stabilizes mode 1 (M1) up to X c = 1.0 p.u. (compensation level of 70%) as it was designed to do.

This SRC does not suffer from the tow compensation problem from which the static filter SSR solution suffers [3].

3.2.3. Variation in the power system fre- quency During the normal operation of a power

system, there will be minor variations in the system frequency as load is shed or picked up. For a 60% compensated system equipped with SRC and the same initial conditions as before, the system eigenvatues are calculated successively while the frequency is varied from 48.75 to 51.25 Hz (i.e. 50 Hz + 2.5%).

The resulting eigenvalue loci in Fig. 12 indicate that these changes in system fre- quency (unlike the power filter scheme [3]) do not cause the SRC equipped generator to go unstable, although there is a slight decrease in the damping of the 78 rad s-I mode (locus M2).

3.3. Transient behaviour of the SRC equipped system

In order to test the ability of the SRC to suppress the dangerous transient torques in the turbogenerator shafts during SSR condi- tions, this section evaluates the predicted transient behaviour of the Koeberg generator at a capacitive compensation level of 60%. The 31 nonlinear differential equations referred to earlier are numerically integrated step by step. The initial conditions remain as stated earlier.

3.3.1. Temporary 20% drop in infinite bus voltage For purposes of comparison, Fig. 13 is in-

cluded to illustrate how 8SR sets in on the Koeberg system without its SRC; the distur- bance is a temporary 20% reduction in the

Page 9: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

infinite bus voltage for 100 ms. However, for the same disturbance, the addition of the SRC stabilizes the system as illustrated in Fig. 14

269

in which X 3 is regulated around a nominal value of 7 p.u. between a maximum of about 9.5 p.u. and a minimum of about 4.6 p.u.

1.14 £

l. ll

l.¢e

O. ee .50

TIME S

B. F m.

2. 00

I. 00

0. 00

- I. OR I B. 0o

TERMINAL VOLTAGE

I. 00 I. = 2. 00

TORQUE LP-GEN3 , i

.50 1. I I 1.50 2 . 0 0 TIME S

57. 5|

47. 5

45. pl g. g e

ELECTRIC TORQUE I. 21

1. Ba

• O g .

• 4 0

T I N E S

GEN. LOAD ANGLE 0¢. 0¢ : i i

2. I I

L= L= L= z.= TIME S

Fig. 13. Predicted response o f the Koeberg generator (without an SRC) following a 20% drop in infinite bus volt- age for 100 ms.

1.1|

I.E

I. 0¢

¢t BB

I. 7R

1.5~I

1.25

1.01

• 75

.51

• 25

I . I I I. 00

TERNINAL VOLTAGE i i

%0 ;.= L= 2.,0 T I M E S

TORQUE LP3-GEN i o

i t. SI I. II ~. SI

T I N E S | , II

5 ¢ F a

58. 0 0

~ 07.00

55- 0 B

$4. ~

GEN. LOAD ANGLE : i i

55. I I

5 2 . P ¢ 0 0 I . ' m

1 0 m SHUNT REACTANCE

L m L = 2.00 T I M E S

%0 L= L= S

.~ 0. m

£

7. U

X~ 8. m

.% 08

4. P i . l l I

T I N E 2.1111

Fig. 14. Predicted response o f the Koeberg generator equipped with an SRC fol lowing a 20% drop in infinite bus voltage for 100 ms.

Page 10: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

270

In another investigation [3] it was found that an auxiliary excitation controller (AEC), using generator speed feedback to the AVR, was unable to stabilize the Koeberg system after this same disturbance. This was due to the limited exciter ceiling voltages and the large time constants of the rotating diode exciter and field windings of these machines.

3.3.2. Three-phase short-circui t When the generator and its SRC are sub-

jected to a 100 ms three-phase short-circuit at the infinite bus, the SRC control action demands that the shunt reactance X 3 be regulated to instantaneous values of less than zero unless a limit is placed on this minimum value (X3min) of X 3. The results in Fig. 15 show the faulted system behaviour when X3mm = 0, and the terminal voltage collapse illustrates that this type of control can be viewed as a repetitive short-circuit at the machine terminals whenever X 3 = X3 ram. As a consequence the SRC is unable to dampen out the SSR instability.

However, if the value of X 3 mm is raised to 10% (0.7 p.u.) of its nominal value (7.0 p.u.), the SRC is able to stabilize the faulted system as shown in Fig. 16.

4. CONCLUSIONS

This paper has considered a scheme to con- trol subsynchronous resonance (SSR) at the Koeberg power station in South Africa by using a shunt reactance at the terminals of the generator; the reactance is modulated by a signal derived from the generator speed. The equations for the generator and the network were used in a form particularly useful for multimachine studies and it has been shown how, after linearization, their eigenvalue loci can be used to design the shunt reactance controller (SRC}.

The main conclusions regarding the effec- tiveness of this particular SRC to dampen SSR at Koeberg are as follows:

(a) Torsional interaction can be suppressed up to high values of compensation (60%} even when the system is subjected to a three-phase short-circuit.

(b) This controller scheme is insensitive to system frequency changes within the range 50 Hz + 2.5%.

(c) The excitation system does not require any special characteristics, since the controller action takes place on the stator side and is therefore not affected by the exciter ceiling

311. m

2, gm T E R M I N A L VOLTAGE

I, 5E

• 5 1

$ , a . 5 t l

TIME

TORQUE LP3-GEN I

2B. Bi

£ lB . BI

B. me

- l B . m

-2B. Bg

-3B. m B. BB

1. BB I. 5$ 2, BB

S

i

'. sB L BB ;. 5B 2 BB

T I M E S

10B. Bg

5B. EB

B. Ba

-5m. Bm

GEN. LOAD ANGLE

- I B L m e. ee ' .se 1. se ~.s~ 2. ee

TIME s

SHUNT REACTANCE

B ~ . . :

B. g l . s~ I. BO 1.50 2. ~

TIME S

Fig. 15. Predicted response of the Koeberg generator equipped with an SRC following a temporary short-circuit

and X3 Bin = 0.0.

Page 11: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

271

] . 4 F T E R M I N A L V O L T A G E o i i

£ 1 o N

• 8g

• Oil

• 4 i l

. t : ~ i / i il. u . '*il ~. u 1. s i l 2. m

TIME S

TORQUE LP3-QEN 4. ilil I I

.~ 3. il £

2.

- 2 . il~ il. ilil '. sil ~. M ~. sil 2. u

TIME S

7 ~ M - GEN. LOAD A N G L E

Oil. i l a

55. i l l

SiS. i lB

45. i l l

4 i l . i l i l L N

2i l .

5 a~ 15- i l |

l i l . Bg

5. ile

'.~, l.- t.se 2. u T I M E S

S H U N T R E A C T A N C E i u

' . ~ l .m I . ~ 2. ilil T I M E S

El. BE i l . i l e

Fig. 16. Predicted response of the Koeberg generator equipped with an SRC following a temporary short-circuit and X3min ffi 0.7 p.u.

limits and the long exciter and field time Pm constants peculiar to a rotating diode exciter Pt system. VbDQ

Although only a radial network has been investigated, the techniques described in this Vb paper are also applicable to more complex Vm networks of which the possible SSR hazards X3nom will form the subject of another investigation. X*

real power at terminals of generator power supplied to turbine d, q components of infinite bus voltage r.m.s, infinite bus voltage r.m.s, generator terminal voltage = 090L3nom

= ~ 0 L *

ACKNOWLEDGEMENTS

The authors acknowledge the assistance of R.C.S. Peplow, D. C. Levy and H. L. Nattrass in the Digital Processes Laboratory of the Department of Electronic Engineering, Uni- versity of Natal. They are also grateful for financial support received from the CSIR and the University of Natal.

N O M E N C L A T U R E

bDQ

P

d, q components of generator volt- age bg AVR and exciter state variables governor and turbine state vari- ables = d / d t , derivative operator

~g

A (~J0

load angle of generator small change operator speed of synchronously reference frame

rotating

R E F E R E N C E S

1 D. J. N. Limebeer, R. G. Harley and S. M. Schuck, Subsynchronous resonance of the Koeberg turbogenerators and of a laboratory micro-alternator system, Trans. S. AfT. Inst. Electr. Eng., 70 (1979) 278 - 297.

2 S. Svensson and K. Mor tensen , Damping of sub- synch ronous osci l lat ions by an HVDC link. An HVDC s imula tor s tudy , IEEE Trans., PAS-IO0 (1981) 1431 - 1439.

3 D. J. N. Limebeer , R. G. Harley and M. A. Lahoud , Suppressing subsynch ronous resonance wi th stat ic filters, Proc. Inst. Electr. Eng., Part C, 128 (1981) 33 - 44.

Page 12: Theoretical Study of a Shunt Reactor Sub Synchronous Resonance Stabilizer

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4 SSR Task Force, Analysis and Control o f Sub- synchronous Resonance, IEEE Publ. No. 76 CH 1066-0-PWR, 1976.

5 A. A. Fouad and K. T. Khu, Damping torsional oscillations in power systems with series com- pensated lines, IEEE Trans., PAS-97 (1978) 744 - 753.

6 E. T. Ooi and M. M. Sartawi, Concepts on field excitation control of subsynchronous resonance in synchronous machines, IEEE Trans., PAS-97 (1978) 1637 - 1645.

7 T. H. Putman and D. G. Ramey, Theory of the modulated reactance solution for subsynchronous resonance, IEEE Trans., PAS-101 (1982) 1527 - 1535.

8 D. G. Ramey, J. W. Dorney, D. S. Kimmel and F. H, Kroenig, Dynamic stabilizer verification tests at the San Juan station, IEEE Syrup. on Countermeasures for Subsynchronous Resonance, IEEE Publ. No. 81 TH 0086-9-PWR.

9 O. Wasynczuk, Damping subsynchronous reso- nance using reactive power control, IEEE Trans., PAS-IO0 (1981) 1096 - 1104.

10 N. Jaleeli, E. Vaahedi and D. C. Macdonald, Multimachine system stability, IEEE PICA Conf. Proc., Toronto, May 1977, IEEE Publ. NO. 77 CH ll31-2-PWR, pp. 51 - 58.

11 D. J. N. Limebeer and R. G. Harley, Synchronous machine stability using composite governor and voltage regulator models, Electr. Power Syst. Res., 1 (1978) 68 - 75.

APPENDIX A: THE GENERATOR EQUIVALENT CIRCUIT

This A p p e n d i x shows br ief ly h o w the two- axis equa t ions o f the s y n c h r o n o u s m a c h i n e m o d e l are deve loped into an equ iva len t AC circui t m o d e l consis t ing o f a res is tance , an i nduc t ance and a non l inear vol tage source [10 ] .

The two-axis equa t ions in a r o t o r r e fe rence f r a m e are:

Vgd = P ~ g d + [Rgd]igd + Ugd (A-l)

Vgq = p ~ g q + [Rgq]igq + Ugq

where

[Rgd] =diag{Ra, Rf, Rkd}

[Rgq] = d i a g { R a , Rkql , Rkq2}

Vgd = [Vd, Ufd , 0] T, Vgq = [Vq, 0, 0] T

Vrd : [Ufd , 0] T, Vrq---- [0, 0] T

i g d = [id, ifd, ikd] T, igq= [iq, ik~tl, ikq2] T

ird = [ifd, ikd] T, irq = [ikql, /kq2] T

Ugd= [~qpO, O, 0] T

Ugq----[--~dPO, O, 0] T

0 = COot + 5

Fol lowing m i n o r man ipu la t ions :

[vd] 0}[ d l vq 0 Lq* iq

+ pO - -Ld* 0 iq

i~ b d (A-2)

where

Ld* = L d - - Ma T [L~d]- 1Md

Lq* = Lq - - M q T [ L~q]- ' M q

Md = [Lind, Lind] T, Mq

[ L~fd Lmd ]

[L~d] = Lma Lkkd ]

[ Lkkql Lmq ]

[Lrq ] = Lm q Lkkq2

= [Lmq , Lmq] T

0

0 + pO

--MdT[Lrd] - '

The r o t o r f lux l inkages are:

@rd = Mdid + [Lrd] ird

~rq = Mqiq + [L~q]i~q

(A-3)

(A-4)

Using eqns. ( A - l ) and (A-4):

lord _ [ [R~j [L~J- ,

[ 0

0

(A-5)

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If a synchronously rotating reference frame is chosen for the network, the transformation between rotor reference frame quantities Ydq and synchronous reference frame quantities )'DQ is as follows:

Ydq = [T]YDQ (A-6)

where

[ ] cos ~ --sin

[T] = sin 6 cos

Applying this transformation to the stator currents, the derivative of stator currents and the terminal voltage in eqn. (A-2) yields

+ R a [iD + ,°1 [::] (A-7)

where

L* = (Ld* + Lq*)/2, Ly = (Ld* -- Lq*)/2

+ Ly [--sin(2~) --cos(2~i)J

~ _ [sin(25) cos(25) ] [ in ]

+ L Y ( ~ ° + z P S ) [ c o s ( 2 5 ) --sin(2~)] i e

(A-8)

The instantaneous phase A variables bg, vt, i2 corresponding to bD, b e and VD, Ve and iD, iQ are found in eqn. (A-9) by applying the relevent inverse Park transformation to eqn. (A-7):

bg = b D cos(o.~0t ) + b e sin(o~ot ) (A-Oa)

vt = VD COS(~Oot) + V e sin(o~0t) (A-9b)

i2 = i2D COS((-'J0t) + i2e sin(e~0t) (A-9c)

where i D in the stator is also i2D in the net- work. Similarly iQ = i2e.

This means that the generator can be repre- sented by an equivalent circuit in ABC phase variables consisting of a nonlinear voltage source bg, an inductor L* and the stator phase

273

resistance Ra in series with the network as shown in Fig. 4.

APPENDIX B: THE NETWORK EQUATIONS

By examining the network in Fig. 4, the following four basic equations in ABC phase variables are obtained:

V b = R l i I + L l P i I + v c + R 3 i 3 + L 3 p i 3 (B-l)

R3i3 + L3Pi3 = (R 2 + Ra)i 2 + (L2 + L*)Pi2 + bg

(B-2)

i I = i 2 + i a (B-3)

PVc = i l / C (B-4)

Rearrangement of eqns. (B-l) to (B-4) yields the following state variables:

Pi2 = --{(R2 ÷ Ra)i2 - - R 3 i 3 - - L3[Vb -- R l i 2

- - i3(R 1 + R3) -- Vc]/(L 1 + L3) + bg}/Lj

(B-5)

p i 3 = ((R 2 + Ra)i 2 - R 3 i 3 + (L* + L2)[v b

- - R l i 2 - - i3(R , + R3) - - Vc]/L 1 + bg ) / L k

(B-6)

pve = (i2 + i3)/C (B-7)

where

L| = L 3 L I / ( L 1 + L3) + L* + L2

Lk -- (L* + L2)(LI + L3)/L1 + L3

Equations (B-5) to (B-7) are now recast into equivalent d, q equations in a synchronous reference frame by application of the appro- priate Park transformation:

Pi2D Q = - - { ( R 2 + Ra) i2v Q - - R 3i3VQ - - L 3 [VbV Q

- -RI i2DQ - - i3DQ(R1 + R3)

- - VcDQ]/(L, + L3) + bDQ} /L j

- - [ 06.~0 O0]/2DQ (B-8)

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274

P/3DQ = ( ( R 2 + R a ) i 2 D Q - - R 3i3DQ

+ (L* + L2)[VbD Q - - R I i 2 D Q

--i3DQ(R1 + R3) - - VcDQ]/LI

0 COo] + b D Q } / L k - - --CO 0 0 i3DQ

(S-9)

pVcD Q - i2DQ + i3DQ [ 0 090 ]

C --(D O 0 VcDQ

(B-10)

The n e t w o r k and the m a c h i n e are t h e r e f o r e descr ibed by t h e ten d i f fe ren t ia l equa t ions cons is t ing o f fou r in eqn. (A-5), and six in eqns. (B-8) to (B-10).

APPENDIX C: SYSTEM PARAMETERS

Turbine and governor system parameters Governor gain, Kg 0.05 Phase advance compensation, Tgl 0.30 s Phase advance compensation, Tg2 0.03 s Maximum turbine output, Pmax 1.2 p.u. Servo time constant, T¢3 0.15 s Entrained steam delay, Tg4 0.62 s Steam reheat time constant, Tgs 2.56 s Shaft output ahead of reheater, F 0.337 p.u.

Exciter and A VR parameters AVR gain, Kay 0.0055 AVR compensator time

constants, Tvl 0.62 s Tv2 2.18 s Tv3 0.20 s Tv4 0.04 s

Input filter time constant, Tvs 0.02 s Exciter time constant, T x 0.49 s Exciter ceiling, Yfm a 0.0122 p.u. AVR ceiling, Ema 0.0335 p.u. AVR ceiling, Emi --0.0248 p.u. Pfd for rated armature voltage

at no-load 0.62 × 10 -3 p.u.

The exc i te r s a tu ra t ion f ac to r Se is given by:

Se = 0 .6093 exp(0.2165V~d)

Transmission system parameters ( refer to Fig. 4)

R 1 = 0 .0844 p.u. R 2 = 0 .001 p.u.

X l = 1 .1547 p.u.

R 3 = 0.01 p.u.

X 3 = 7.0 p .u .

X2 = 0 .005 p .u .

X c = 0 .8463 p.u. (60%

c o m p e n s a t i o n )