Electric Power Systems Research, 10 (1986) 25 - 33 25

O p t i m a l O u t p u t F e e d b a c k D e s i g n o f a Shunt Reactor C o n t r o l l e r f o r D a m p i n g

T o r s i o n a l O s c i l l a t i o n s

J. C. BALDA, E. EITELBERG and R. G. HARLEY Department of Electrical Engineering, University of Natal, King George V Ave., Durban 4001 (Republic of South Africa)

(Accepted September 4, 1985)

SUMMARY

Suppression o f the torsional oscillations o f a turbogenerator by means o f a shunt reactor placed at the turbogenerator terminals has been proposed by others and would appear to be the most promising o f all the counter- measures suggested so far. This paper investi- gates the damping o f subsynchronous reso- nance (SSR ) oscillations by using such a shunt reactor stabilizer and placing it at the high voltage instead o f the low voltage side o f the generator step-up transformer. It shows how the shunt reactor controller (SRC) is designed by employing optimal ou tpu t feedback tech- niques. A digitally computed step-by-step solution o f the 31 non-linear differential equations yields the transient response o f the non-linear system and illustrates the stabiliz- ing effect o f the shunt reactor controller, even for an initially unstable plant.

INTRODUCTION

A power station sited in the Western Cape area in South Africa has recently been com- missioned and is transmitting power to the national grid via a 1400 km, 400 kV series capacitor compensated AC line. The proba- bility of unstable torsional oscillations oc- curring in this system, as well as a number of possible countermeasures, have been investi- gated and reported elsewhere [1 - 5], together with a complete set of data for the system.

This paper extends the work started in ref. 4 by also investigating SSR damping using a thyristor-controlled shunt reactor but con- necting it to the high voltage side of the generator step-up transformer as shown in

Fig. 1. Instead of using the classical eigenvalue approach [4], the linearized controller design is carried ou t by employing optimal ou tpu t feedback techniques, based on deterministic interpretation by Eitelberg [6]. The well- known linear quadratic performance index is minimized around a chosen operating point by applying gradient and line searching tech- niques [7] to the system's linearized differ- ential equations. The closed-loop transient response of the non-linear turbogenerator system wi thout and with the designed linear optimal controller is then predicted by digital simulation for a 100 ms three-phase short- circuit at the infinite bus. Finally, the im- portance of the shunt reactor location is emphasized by comparing the results obtained in ref. 4 (shunt reactor located at the turbo- generator terminals) with those obtained when the shunt reactor is placed at the high voltage side of the step-up transformer for an unstable level of series compensation (60%).

RI ×l ×c

~r • r~ > TRANSFORMER

I I x 2

TURBI~ STAGES LFJ

~ ~ J l a~ J~ ,~ Js ~Yi

SHUNT REACTOR CQNTROLLER

Fig, 1. I n f i n i t e h u s - t u r b o g e n e r a t o r sys tem.

0378-7796/86/$3.50 © Elsevier Sequoia/Printed in The Netherlands

26

MATHEMATICAL MODEL OF THE SYSTEM

The dynamic behaviour of the system in Fig. 1 is described by the differential equa- tions which represent:

(a) the two-axis model for the turbo- generator [4, 8] ;

(b) the rotational movement of the six turbogenerator masses with their inertias, damping and spring constants [4] ;

(c) the dynamics of the AC network [4] ; (d) the turbogenerator automatic voltage

regulator (AVR) and governor [4]. During the optimization task the AVR loop cannot be omit ted because some of the AVR time con- stants are small enough to influence the total behaviour of the turbogenerator. The re- sponse of the governor loop is such that it has no major influence on the suppression of SSR and therefore its representation is neglected for optimization purposes.

These 26 differential equations are lin- earized around a steady state operating point and re-arranged into the following state variable form for the deviations from this operating point:

, 52 , : [Ap] ,SX, + [Bp] ,SU, (1)

,SY, : [C,] ,SX, (2)

By assuming that the change in the generator voltage, by action of the shunt reactor, is negligible, the modulat ion of the thyristor currents is equivalent to modulat ion of the inductance of the shunt reactor around some nominal value L3nom. Under these conditions the relationship between the controller out- put and the shunt reactor inductance can be written as [4]

L3 = (1 + ,SYl)L3nom (3)

The controller function SRC(s) in Fig. 1 was chosen to be

,SY 1 K~r(s + W1) - (4 )

,5~G s + W2 where L 3 is the shunt reactor inductance; ,Sy.~ the controller ou tpu t = ,SU~; and ,5co~ = ,56G is the generator rotor speed deviation.

The shunt reactor controller is represented by the following linearized state-space equa- tion:

Ak = ACoG -- W2 Az = ,5U2 (5)

and

,SYl = K~r[,SC0G + (Wl -- Wz) ,Sz] : ,SUl (6)

The plant and controller equations (eqns. (1 ) - (6)) are combined into an augmented model [6]. The state, input and ou tpu t vectors of the augmented system are, respec- tively,

A X = Az ] ,SU= ,SUz ,Sz J

(7)

and the state, input and ou tpu t matrices are re-written as follows:

[Ap] [01 1 [A] = [01 [0]

[BI = [ [B,] [011 [01 U]

[c] = [[cp] [01] [o] [;1

(8)

The linearized augmented system equations become

A ) ( = [ A ] A X + [B] AU

AY = [CI AX (9)

OPTIMAL OUTPUT FEEDBACK CONTROL

This theory is needed to find those values of K~r, W1 and Wz in the SRC which will ensure optimal system response after an impulse-type disturbance, but nevertheless also stabilize the system during steady state operation.

The problem to solve is: find the feedback gain matrix [K0] for the dynamic controller which minimizes the well-known linear quadratic performance index given by

I = f ( A X w [Q] AX + AU w [R] AU) dt (10) t o

according to the dynamic ou tpu t feedback control law

AU = [K0] ̀ SY (11)

where [Q] and JR] are weighting matrices of the state and input variables and [K0] is the feedback gain matrix for the dynamic con- troller. [Q] is a positive semidefinite matrix and [R] is a positive definite matrix.

The solution of eqn. (9) with eqn. (11) as a function of the transition matrix [0] is

z~X = [O(t, to)] AX0

= exp{([A] + [B] [Ko] [C])(t -- to)) AXo

(12)

Substi tut ion of eqns. (11) and (12) into eqn. (10) yields

I = AXoWf [0(t, to)]W([Q] t o

-4- [C T] [Ko T] [RI [Ko] [C] )[¢(t , t0)] dt AX 0

Equation (13) is re-written as

I = AXo w [P(to)] AXo

(13)

(14)

where [P(to)] is a symmetric positive semi- definite matrix defined by

[P(to)] = J[cp(t, to)lW([Q] t o

+ [CTKoTRKoC])[O(t, to)] dt

= [P]

where [P] is the solution of the Lyapunov equation

[P]([A] + [BKoC]) + ([A] + [SKoC])v[P]

+ ([Q] + [CTKoTRKoC]) = [0] (15)

for [P] constant. From an analysis of eqn. {14), it is noted

that the performance index depends upon the unknown initial states of the linearized state vector. To overcome this problem it has been suggested that a new performance index, defined as its expected mean value [9], i.e.

I = tr{[P] [Qo]} (16)

where tr means trace and [Qo] = E{AXo AXo w), should be minimized. If nothing is known

27

about AXo, it is advisable to consider [Qo] equal to the identity matrix [I].

A necessary condition for the gain matrix [Ko] to be a minimum is t h a t the gradient of the performance index with respect to any element of [Ko] will be zero. The i,jth ele- ment of the gradient can be written as

- tr [Q0] = 0 (17) ~K0 ij

By differentiating eqn. (15), solving the resulting Lyapunov equation to find (O[P]/

[K0]] and making use of some trace proper- ties, the gradient of the performance index becomes

- 2 .0 ( [ B T][P] + [ R ] [ K o ] [ C ] ) [ M ] [ C T] ~[K0]

(18)

where the matrix [M] is the solution of the Lyapunov equation given by

([A] + [B][Ko][C])[M] + [M]([A]

+ [s][go][C]) T+ [Qo] = [0] (19)

From eqns. (17) and (18) the feedback gain matrix [K0] is of the form

[K0] = - - [ R ] - I [ B w] [P] [M] [C T]

X ( [ C ] [M] [ c T ] ) -1 (20)

A more detailed description of the method outl ined above appears in ref. 6.

The sufficient optimali ty conditions for the ou tpu t feedback case are not ye t known [9]. Furthermore, the coupled non-linear eqns. (15) and (19) cannot be solved analyti- cally. Therefore, a numerical gradient search technique to minimize the performance index is employed. This gradient search technique can be interpreted as a recursive method:

(a) guess a stable value for the feedback gain matrix [K0] ;

(b) solve the Lyapunov eqns. (15) and (19);

(c) calculate the gradient of the perfor- mance index given by eqn. (18);

(d) by using gradient information a local minimum for the performance index I is calculated by the quadratic convergence des- cent method presented in ref. 7. This method searches for the minimum of the function along the line direction defined by

28

[S] = --[H] - - (21) [K0]

where [H] is a symmetric positive definite matrix which is initially chosen to be the identi ty matrix.

The feedback gains are then modified ac- cording to

[K0] k+l = [K0] k + fl[S]

where 13[S] is the step size matrix and k is the iteration number.

By applying a convenient stopping criterion a stable minimum in the direction of search is located.

(e) Check if the local minimum has been found. If this is so, stop the program, other- wise return to step (b).

Looking again at eqn. (17) it is worth men- tioning that the optimization task can be done with respect to any number of elements in [K0]. The designer is free to fix the values of some parameters and optimize with respect to the others.

CONTROLLER DESIGN

The design of the shunt reactor controller (SRC) is carried out around a typical steady state operating point corresponding to turbo- generator active power PG = 0.9 p.u., terminal voltage VG = 1.12 p.u., infinite bus voltage Vb = 1.0 p.u. and a level of series compensa- tion K = (Xc/XL) × 100.0% = 30%, where Xc is the series capacitor reactance and XL is the sum of the subtransient reactance of the generator and the inductive reactances of the step-up transformer and the transmission line.

After linearization around the above oper- ating point, the complete system is repre- sented by 27 linear differential equations with only two inputs, as defined in eqns. (5) and (6). The feedback gain matrix for this case has the following form:

[ Ksr Ks"(W1-- W2) l (22) [K°] = 1.0 --W 2

In order to find an optimal controller design that reduces the peak of the transient torques and cancels torsional oscillations and to establish how the weighting of a particular element affects the optimal values, several

cases were considered by weighting only one diagonal factor in [Q] at a time while the rest of the diagonal elements were zero and the two elements in [R ] were given a small value.

A small weighting value for the generator speed factor in [Q] produces a non-minimum phase opt imum controller with a small nega- tive gain value. By increasing this weighting factor, stable opt imum controllers are found and the controller gain becomes positive and increases. By adding more weight to the diagonal factors corresponding to the gener- ator angle and currents, the controller gain is increased and the pole and zero move closer to the origin. Weighting of the shunt reactor current states produces opt imum parameters with a zero on the right-hand side plane and a negative gain. It was found that weighting those diagonal elements corresponding to generator flux linkages and shunt reactor con- troller states produced opt imum parameters with a controller zero on the right-hand side plane and a small negative gain value. Some of the results from the optimization programme are shown in Table 1.

The controller design cannot only be done by considering results obtained from the optimization programme since these results were obtained by linearizing the system around a steady state operating point. System optimal behaviour has been achieved around this point but did not take account of non- linearities. Therefore several transient cases were digitally simulated for different sets of SRC parameters. From these results (not shown here) it was noted that small values of Ksr slow the system response down, whilst higher values of Ksr speed it up. By increasing Ksr the peak of the transient torques is not significantly reduced, instead, the electrical torque starts to oscillate due to the control action of damping SSR and reaches peaks (equivalent to those obtained when the fault was initially applied) before it finally settles down.

After considering all these factors the SRC parameters were chosen as

Ksr = 0.17; W~ = 3384.0; W2 = 655.2

which corresponds to a generator angle weighting factor of 0.1 and represents a com- promise between fast response and SRC effort.

TABLE 1

Op t imum feedback parameters for only one non-zero weight ing factor in [Q] and [R] = diag[0.1, 0 .1]

29

State Weight in [Q]

0.1 1.0 10.0 100.0

Gen. speed Ks~ WI W2

Gen. angle

Ksr Wl W2

Gen. current

Ksr Wl W2

Shunt cur ren t K~ Wl W2

SRC(s) g~ W1 W2

Flux linkages Ks~ wl w2

- -8 .95 X 10 -2 1.35 1.28 1.30 - -4450 .0 502.0 100.00 95.50

1100.0 231.0 8.10 4.34

0.17 1.53 1.74 1.75 3384.0 101.9 89.80 89.63

655.2 10.4 0.26 8.3 X 10 -2

0.17 1.45 1.47 1.453 974.0 97.10 92.01 91.60 337.5 5.21 0.71 0.27

- -0 .32 - -0 .45 - -0 .36 - -0 .31 - -33 .0 - -111 .3 - - 204.2 - -232 .3 1274.0 787.0 430.4 345.5

- -3 .4 X 10 -2 - -3 .4 x 10 -2 - -3 .2 x 10 -2 - -3 .4 X 10 -2 - -18 .5 - -20 .3 - -21 .2 - -18 .4 1634.0 1640.5 1697.5 1633.3

- -3 .4 x 10 -2 - -3 .4 x 10 -2 - -6 .6 x 10 -2 - -2 .74 X 10 -2 - -19 .5 - -31 .8 - -79 .4 - -1664 .70

1633.2 1632.6 1626.0 1593.3

RESULTS

This section presents the transient response of the turbogenerator, with and without the SRC, when the system was subjected to a 100 ms temporary three-phase short-circuit at the infinite bus for the initial conditions stated earlier. The turbogenerator was also equipped with the AVR and governor shown in ref. 4. The complete system is described by 31 non-linear differential equations which are integrated by a 5th-order Kutta-Merson algo- rithm.

Figure 2 shows the system transient re- sponse when the SRC is out of service for the initial conditions stated earlier; the system remains stable and electrical torque oscilla- tions reach a peak of 1.7 p.u. Figure 3, which contains the transient response of the turbo- generator with the SRC for the same distur- bance, shows that the LP3-GEN torque is

better damped (than in Fig. 2) after 0.5 s since the subsynchronous oscillations took about that long to start building up in Fig. 2.

In order to examine the behaviour of the system when the level of series compensation is large enough (60%) to correspond to an unstable case, Fig. 4 is used to illustrate the turbogenerator response when the SRC is out of service; subsynchronous oscillations start to build up and grow with time. Figure 5 shows the system behaviour when the shunt reactor with the SRC designed in ref. 4 is placed at the generator terminals. The system is stabilized and torque oscillations in the LP3- GEN shaft section reach a peak of 4.0 p.u. Subsynchronous oscillations are still present after 3.0 s but they have considerably de- creased. The ability of the SRC to stabilize the system and cancel subsynchronous oscilla- tions when the shunt reactor is located at the high voltage side of the generator step-up

30 GEN SPEED DEV

7. S0 I I

0: s. oi

0 . 0 0

~.~ - z s t -S .O

0. 00

OR

~ 00

1 . 0 0

0 . 0 0

-1.08 0 . 0 0

'. 50 L 00 L so ' ~ so 0. 00 2 , 0 0 . 3 . 0 0 . 5 0 1 . 0 0 " 1 . 5 0 2 . 0 0 ~ SO

TIME SEC. TIME SEC.

TORQUE LP3-QEN ELECTRIC TORQUE I i I I i I i I I i 2. 0 0

£~ 1.s I.

1.0 - t

0 . 00.

(3

p--

- 1 . 0 i '. so L 00 L ~o ~. 00 ~. so ~. 00 0. 00 '. so L 00 L so 2. 00 ~. ~o TIME SEC. TIME SEC.

3. 0 0

3. 0 0

Fig. 2. Simulated t ransient response of the tu rbogenera to r (wi thout an SRC) fol lowing a 100 ms three-phase short-circui t at the infinite bus at a series compensa t i on level of 30%.

5. 00 r

[

4. O~t-

QEN SPEED DEV SHUNT CURRENT 1. SO F

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0. 00 . 5 0 i. 00 1. 50 2. 00 2. 50 TIME SEO.

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TIME

3. 00 0. 00 . 5 0 1 .00 1. SO 2. 00 2. 50 3. I~ TIME SEC.

ELECTRIC TORQUE - 2. 00 I i I I I ,

1 .0 i l ' ` " " i ua ' " " " ' 1 I I ~ , . . . . . . . . . . . . . . . . . . " . . . ' : " . . . ' ' "

1. 50 2 . 00 2 . 50 3 . 00 0 . 00 . 50 1 .00 I . 50 2 . 00 2 . 50 3 . 0;

SEC. TIME SEC.

Fig. 3. Simulated t ransient response of the tu rbogenera to r (with an SRC) following a 100 ms three-phase short- circuit at the infini te bus at a series c o m p e n s a t i o n level o f 30%.

31

GEN SPEED DEV 10. 0~ l

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SEC.

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3.0B . 5 0 1. B0 1.50 2 .00 2. 50 3.00 TIME SEC.

ELECTRIC TORQUE

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T I M E SEC. T I M E S E C .

3 . 0 0

Fig. 4. Simulated t ransient response of the tu rbogenera to r (wi thout an SRC) fol lowing a 100 ms three-phase short-circui t at the inf ini te bus at a series c o m p e n s a t i o n level o f 60%.

GEN SPEED DEV

5

0. 00

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m

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TIME SEC.

TORQUE LP3-GEN

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0. 0 0 . 5 0 1 . 0 0 1.50 2. 0 0 2. 50 3. 0 0

TIME SEC.

SHUNT CURRENT 1. OR ~

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-. 5E

-1.0 I

-1.5 i / i i i

TIME SEC.

FI FKTRTC Tn l~Ol l~

0. 00 . 5 0 1 .00 1 .50 2 .00 2-50 3-$ TIME SEC.

Fig. 5. Simulated t ransient response of the tu rbogenera to r and the shunt reactor placed at the genera tor terminals (with an SRC) fol lowing a 100 ms three-phase shor t -c i rcui t at the infini te bus at a series compensa t i on level o f 60%.

32

5.0£

2. 52 i

0. 05

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, , 00

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GEN SPEED DEV i i

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TORQUE LP3-QEN

SEE.

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I.$~ £

~.BB

u

-1.5

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SHUNT CURRENT

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TIME SEE. TIME SEC.

Fig. 6. Simulated transient response o f the turbogenerator and the shunt reactor placed at the high voltage side of the step-up transformer (wi th an SRC) fo l l owing a 100 ms three-phase short-circuit at the inf inite bus at a series c o m p e n s a t i o n level of 60%.

transformer is shown in Fig. 6. The peak transient torque in the LP3-GEN section is 3.4 p.u. and the torsional oscillations are mostly eliminated after 2.0 s.

CONCLUSIONS

This paper has shown that subsynchronous resonance can be counteracted in an efficient way by means of a shunt reactor connected at the high voltage side of the generator step-up transformer. From an analysis of the optimal programme results together with the system transient responses, values for the controller parameters were chosen which reduce the peak of the transient torques and eliminate unstable torsional oscillations. It has also illustrated that the SRC controller is capable of damping subsynchronous oscillations, even when the level of series compensation is as high as 60% and in the face of a severe dis- turbance.

ACKNOWLEDGEMENTS

The authors acknowledge the assistance of R. Peplow, H. L. Natrass and D. C. Levy of

the Digital Processes Laboratory of the Department of Electronic Engineering, Uni- versity of Natal. They are also grateful for the financial support received from the CSIR and the University of Natal. 3. C. Balda is grateful to Rotary International for financial support.

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 turbo-generators and of a laboratory micro-al ternator sy s t e m, Trans. S. Aft . Inst. Electr. Eng., 70 (1979) 278 - 297.

2 D. J. N. Limebeer , R. G. Harley and M. A. Lahoud, Suppressing subsynchronous resonance wi th static filters, Proc. Inst. Electr. Eng., Part C, 128 (1981) 33 - 44.

3 D. J. N. Limebeer , R. G. Harley and M. A. Lahoud , The suppress ion of subsynchronous resonance wi th the aid o f an auxi l iary exc i ta t ion contro l signal, Trans. S. Afr. Inst. Electr. Eng., 74 (1983) 198 - 209.

4 M. A. Lahoud and R. G. Harley, Theoret ical s tudy of a shunt reactor s u b s y n c h r o n o u s reso- nance stabil izer for a nuclear p o w e r e d generator, Electr. Power Syst. Res., 8 (1985) 261 - 274.

5 R. G. Harley and J. C. Balda, Subsynchronous resonance damping by special ly control l ing a parallel H V D C link, Proc. Inst. Electr. Eng., Part C, 132 (1985) ]54 - 159.

6 E. Eitelberg, Perturbationstechniken bei der Op- timierung grosser linearer Regulungssysteme, VDI-Verlag, Dusseldorf, 1983, ISBN-3-18- 146006-7.

7 R. Fletcher and M. J. D. Powell, A rapidly con- vergent descent method for minimization, Comput. J., 6 (1963) 163 - 168.

33

8 N. Jaleeli, E. Vaahedi and D. C. MacDonald, Multimachine system stability, Proc. PICA Conf., Toronto, May 1977, IEEE Publ. 77 CH 1131-2- PWR, pp. 51 - 58.

9 B. D. O. Anderson and J. B. Moore, Linear Op- timal Control, Prentice-Hall, Englewood Cliffs, NJ, 1971.