Optimal Output Feedback Design of a Shunt Reactor Controller for Damping Torsional Oscillations

1. Electric Power Systems Research, 10 (1986) 25 - 33
Optimal Output Feedback
Torsional Oscillations
25
D e s i g n o f a Shunt Reactor C o n t r o l l e r
for Damping
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 countermeasures suggested so far. This paper investigates the damping o f subsynchronous resonance (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 h o w
the shunt reactor controller (SRC) is designed
by employing optimal o u t p u t feedback techniques. A digitally c o m p u t e d 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 stabilizing effect o f the shunt reactor controller, even
for an initially unstable plant.
Fig. 1. Instead of using the classical eigenvalue
approach [4], the linearized controller design
is carried o u t by employing optimal o u t p u t
feedback techniques, based on deterministic
interpretation by Eitelberg [6]. The wellknown linear quadratic performance index is
minimized around a chosen operating point
b y applying gradient and line searching techniques [7] to the system's linearized differential equations. The closed-loop transient
response of the non-linear turbogenerator
system w i t h o u t and with the designed linear
optimal controller is then predicted by digital
simulation for a 100 ms three-phase shortcircuit at the infinite bus. Finally, the importance o f the shunt reactor location is
emphasized by comparing the results obtained
in ref. 4 (shunt reactor located at the turbogenerator 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%).
INTRODUCTION
RI
A power station sited in the Western Cape
area in South Africa has recently been commissioned and is transmitting p o w e r to the
national grid via a 1400 km, 400 kV series
capacitor compensated AC line. The probability of unstable torsional oscillations occurring in this system, as well as a number of
possible countermeasures, have been investigated 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 b u t connecting it to the high voltage side of the
generator step-up transformer as shown in
0378-7796/86/$3.50
×l
×c
~r•
r~
TURBI~ T G S
SA E
~ a~J ,
~ ~ J
~
> TRANSFORMER
II x2
LJ
F
l J ~i
s Y
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
system.
© Elsevier Sequoia/Printed in The Netherlands

2. 26
and
MATHEMATICAL MODEL OF THE SYSTEM
The dynamic behaviour of the system in
Fig. 1 is described by the differential equations which represent:
(a) the two-axis model for the turbogenerator [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
omitted because some of the AVR time constants are small enough to influence the total
behaviour of the turbogenerator. The response 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 linearized around a steady state operating point
and re-arranged into the following state
variable form for the deviations from this
operating point:
,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 o u t p u t
vectors of the augmented system are, respectively,
AX=
Az
]
,SU=
,SUz
,Sz J
(7)
and the state, input and o u t p u t matrices are
re-written as follows:
[Ap]
[A] = [01
[01 1
[0]
[BI = [ [B,]
[011
U]
[c] = [[cp]
[o]
[01]
[01
(8)
[;1
, 5 2 , : [Ap] ,SX, + [Bp] ,SU,
(1)
The linearized augmented system equations
become
,SY, : [C,] ,SX,
(2)
A ) ( = [ A ] A X + [ B ] AU
By assuming that the change in the generator
voltage, by action of the shunt reactor, is
negligible, the modulation of the thyristor
currents is equivalent to modulation of the
inductance of the shunt reactor around some
nominal value L3nom. Under these conditions
the relationship between the controller output 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
,5~G
-
K~r(s + W1)
s + W2
(4)
where L 3 is the shunt reactor inductance;
,Sy.~ the controller o u t p u t = ,SU~; and ,5co~ =
,56G is the generator rotor speed deviation.
The shunt reactor controller is represented
by the following linearized state-space equation:
Ak = ACoG -- W2 Az = ,5U2
(5)
(9)
AY = [CI AX
OPTIMAL OUTPUT F E E D B A C K 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, b u t 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)
to
according to the dynamic o u t p u t feedback
control law
AU = [K0] `SY
(11)

3. 27
where [Q] and JR] are weighting matrices of
the state and input variables and [K0] is the
feedback gain matrix for the dynamic controller. [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
a b o u t 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 element of the gradient can be written as
z~X = [O(t, to)] AX0
= exp{([A] + [B] [Ko] [C])(t -- to)) AXo
(12)
Substitution of eqns. (11) and (12) into eqn.
(10) yields
I = AXoWf [0(t, to)]W([Q]
~K0 ij
[Q0]
= 0
(17)
By differentiating eqn. (15), solving the
resulting Lyapunov equation to find (O[P]/
[K0]] and making use of some trace properties, the gradient of the performance index
becomes
~[K0]
to
- tr
- 2.0 ( [ B T][P] + [ R ] [ K o ] [ C ] ) [ M ] [ C T]
(18)
-4- [C T] [Ko T] [RI [Ko] [C] )[¢(t, t0)] dt AX 0
(13)
([A] + [B][Ko][C])[M] + [M]([A]
Equation (13) is re-written as
I = AXo w[P(to)] AXo
where the matrix [M] is the solution of the
L y a p u n o v equation given by
(14)
where [P(to)] is a symmetric positive semidefinite matrix defined by
+ [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]
[P(to)] =
J[cp(t, to)lW([Q]
X ( [ C ] [M] [ c T ] ) -1
(20)
to
+ [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 u p o n the
u n k n o w n 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
A more detailed description of the m e t h o d
outlined above appears in ref. 6.
The sufficient optimality conditions for
the o u t p u t feedback case are not y e t k n o w n
[9]. Furthermore, the coupled non-linear
eqns. (15) and (19) cannot be solved analytically. 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 L y a p u n o v eqns. (15) and
(19);
(c) calculate the gradient of the performance index given by eqn. (18);
(d) by using gradient information a local
minimum for the performance index I is
calculated by the quadratic convergence descent m e t h o d presented in ref. 7. This m e t h o d
searches for the minimum of the function
along the line direction defined by

4. 28
[S] = --[H] - -
[K0]
(21)
where [H] is a symmetric positive definite
matrix which is initially chosen to be the
identity matrix.
The feedback gains are then modified according 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 m i n i m u m in the direction of search is
located.
(e) Check if the local minimum has been
found. If this is so, stop the program, otherwise return to step (b).
Looking again at eqn. (17) it is worth mentioning t h a t 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 turbogenerator 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 compensation 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 operating point, the complete system is represented 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
[K°] =
1.0
Ks"(W1--W2)l
(22)
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 optimum controller with a small negative gain value. By increasing this weighting
factor, stable o p t i m u m controllers are found
and the controller gain becomes positive and
increases. By adding more weight to the
diagonal factors corresponding to the generator 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 o p t i m u m 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 controller states produced optimum 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 nonlinearities. 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
--W 2
Ksr = 0.17;
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
W~ = 3384.0;
W2 = 655.2
which corresponds to a generator angle
weighting factor of 0.1 and represents a compromise between fast response and SRC
effort.

5. 29
TABLE 1
O p t i m u m f e e d b a c k p a r a m e t e r s for o n l y o n e n o n - z e r o weighting factor in [Q] and [ R ] = diag[0.1,
State
Weight in
[Q]
0.1
Gen. s p e e d
Ks~
WI
W2
Gen. angle
Ksr
Wl
W2
Gen. c u r r e n t
Ksr
Wl
W2
0.1]
1.0
- - 8 . 9 5 X 10 -2
--4450.0
1100.0
10.0
100.0
1.35
502.0
231.0
1.28
100.00
8.10
1.30
95.50
4.34
0.17
3384.0
655.2
1.53
101.9
10.4
1.74
89.80
0.26
1.75
89.63
8.3 X 10 -2
0.17
974.0
337.5
1.45
97.10
5.21
1.47
92.01
0.71
1.453
91.60
0.27
--0.32
--33.0
1274.0
--0.45
--111.3
787.0
--0.36
- - 204.2
430.4
Shunt current
K~
Wl
W2
--0.31
--232.3
345.5
SRC(s)
g~
W1
W2
F l u x linkages
Ks~
wl
w2
- - 3 . 4 X 10 -2
--18.5
1634.0
- - 3 . 4 x 10 -2
--20.3
1640.5
- - 3 . 2 x 10 -2
--21.2
1697.5
- - 3 . 4 X 10 -2
--18.4
1633.3
- - 3 . 4 x 10 -2
--19.5
1633.2
- - 3 . 4 x 10 -2
--31.8
1632.6
- - 6 . 6 x 10 -2
--79.4
1626.0
- - 2 . 7 4 X 10 -2
--1664.70
1593.3
RESULTS
This section presents the transient response
of the turbogenerator, with and w i t h o u t 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 algorithm.
Figure 2 shows the system transient response when the SRC is out of service for the
initial conditions stated earlier; the system
remains stable and electrical torque oscillations reach a peak of 1.7 p.u. Figure 3, which
contains the transient response of the turbogenerator with the SRC for the same disturbance, shows t h a t 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 LP3GEN shaft section reach a peak of 4.0 p.u.
Subsynchronous oscillations are still present
after 3.0 s but they have considerably decreased. The ability of the SRC to stabilize
the system and cancel subsynchronous oscillations when the shunt reactor is located at the
high voltage side of the generator step-up

6. 30
GEN
7. S0
0:
SPEED
I
DEV
I
s. oi
0. 00
~.~ - z
st
-S.O
0. 00
'. 50
L 00
L so
2,00
'
TIME
TORQUE LP3-QEN
OR
I
~. so
0. 00
3.00
.50
SEC.
i
I
I
1.00
"1.50
2.00
TIME
2.
i
ELECTRIC TORQUE
00
I
~ SO
3. 0 0
SEC.
i
I
I
i
£~1.s
I.
~ 00
1.0
1.00
0.
0.00
-t
00.
(3
p--
-1.08
'. so
0.00
L 00
L ~o
~. 00
TIME
~. so
~. 00
-1.0
0. 00
'. so
L 00
L so
i
2. 00
TIME
SEC.
~. ~o
3. 0 0
SEC.
Fig. 2. Simulated t r a n s i e n t r e s p o n s e o f the t u r b o g e n e r a t o r ( w i t h o u t an SRC) following a 100 ms t h r e e - p h a s e
short-circuit at t h e infinite bus at a series c o m p e n s a t i o n level o f 30%.
QEN SPEED DEV
5. 0 0
SHUNT CURRENT
r
1.
SOF
[
4. O~t-
0. 00
.50
i. 00
1. 50
2. 00
TIME
3. 0
2. 50
3. 00
0. 00
.50
SEO.
TORQUE LP3-QEN
~
1.00
1. SO
2. 00
TIME
-
2. 00
ELECTRIC TORQUE
I
2. 50
3. I~
SEC.
i
1.0
I
II
I
I
il ' ` " " i
u a ' " "" ' 1
~ , ..................
"...':"...
,
''
"
1.0~
(3
0. 00.
-1. 00,
0. 00
I
.50
TIME
i
1.00
1. 5 0
2. 00
2. 50
SEC.
3. 00
0. 00
.50
TIME
1.00
I. 50
2.
00
2. 50
3 . 0;
SEC.
Fig. 3. Simulated transient r e s p o n s e o f the t u r b o g e n e r a t o r ( w i t h an SRC) following a 100 m s t h r e e - p h a s e shortcircuit at t h e infinite bus at a series c o m p e n s a t i o n level o f 30%.

7. 31
GEN SPEED DEV
10. 0~
I
l
i
I
L 00
L s,
~. 00
~ 0~
-5. 0 0
ffl
-10. 0 0
i
0. 0 0
.50
6.00
~. so
3.0B
.50
TIME
0. 0 0
SEC.
TIME
TORQUE L P 3 - G E N
q
I
I
t
1.50
2.00
~.50
2.00
2. 50
SEC.
3.00
ELECTRIC TORQUE
2. 5¢
I
1. B0
2. 00
d
4. e a
1.52
2.00 ~
1. OR
• 5R
0. 0 ~
0. 0~
~ -2. OZ
D
-. 5R
-4. 0~
-1.0~
1.50
0. 0 0
~.00
~.SB
2• 0 0
TIME
~50
3-0B
¢.50
0. 0 0
SEC.
~.00
TIME
2i . 5 0
3. 00
SEC.
Fig. 4. Simulated t r a n s i e n t r e s p o n s e o f t h e t u r b o g e n e r a t o r ( w i t h o u t an SRC) following a 100 ms t h r e e - p h a s e
short-circuit at t h e infinite bus at a series c o m p e n s a t i o n level o f 60%.
GEN S P E E D DEV
1.O
R
SHUNT CURRENT
~
5
0.Of
0.00
- . 5E
, 0 -5. 00
,
-1.0I
m
- 1 0 . 0~,
0. 0o
. so
TIME
I. o~
I. s0
i 00
£ s0
SEC.
3. 0o
i
/
i
i
TIME
FI FKTRTC
TORQUE LP3-GEN
.
-1.5
i
SEC.
Tnl~Oll~
4. o
0. 0 0
.50
TIME
1.00
1.50
2. 0 0
2. 50
SEC.
3. 0 0
0.00
.50
TIME
1.00
1.50
2.00
2-50
SEC.
3-$
Fig. 5. S i m u l a t e d t r a n s i e n t r e s p o n s e o f t h e t u r b o g e n e r a t o r and t h e s h u n t r e a c t o r placed at t h e g e n e r a t o r terminals
(with an SRC) following a 100 m s t h r e e - p h a s e s h o r t - c i r c u i t at t h e infinite bus at a series c o m p e n s a t i o n level o f
60%.

8. 32
GEN SPEED
i
DEV
1.5~
i
5.0£
2. 52
i
0. 05
L.oJo
-2. 5EL
£
/
SHUNT C U R R E N T
I.$~
~.BB
u
-1.5
-7. 5
~.~
, , 00
TIME
SEE.
~1 ~r'T~TF"
TORQUE LP3-QEN
4.
TIME
~.~
SEC.
Tn#f311~
0~
1
3.,
I. ,J
pTIME
SEE.
TIME
SEC.
Fig. 6. S i m u l a t e d transient response o f the turbogenerator and the shunt reactor placed at the high voltage side o f
the step-up transformer (with an S R C ) f o l l o w i n g a 100 ms three-phase short-circuit at the infinite bus at a series
c o m p e n s a t i o n level o f 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 disturbance.
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, University 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.
REFERENCES
1 D. J. N. L i m e b e e r , R. G. Harley and S. M.
Schuck,
S u b s y n c h r o n o u s r e s o n a n c e o f the
K o e b e r g turbo-generators and o f a laboratory
micro-alternator s y s t e m , Trans. S. Aft. Inst.
Electr. Eng., 70 (1979) 278 - 297.
2 D. J. N. L i m e b e e r , R. G. Harley and M. A.
L a h o u d , Suppressing s u b s y n c h r o n o u s resonance
w i t h static filters, Proc. Inst. Electr. Eng., Part C,
128 (1981) 33 - 44.
3 D. J. N. L i m e b e e r , R. G. Harley and M. A.
L a h o u d , The suppression o f s u b s y n c h r o n o u s
resonance w i t h the aid o f an auxiliary e x c i t a t i o n
control signal, Trans. S. Afr. Inst. Electr. Eng., 74
(1983) 198 - 209.
4 M. A. L a h o u d and R. G. Harley, Theoretical
s t u d y o f a shunt reactor s u b s y n c h r o n o u s resonance stabilizer for a nuclear p o w e r e d generator,
Electr. Power Syst. Res., 8 ( 1 9 8 5 ) 261 - 274.
5 R. G. Harley and J. C. Balda, S u b s y n c h r o n o u s
resonance damping by specially controlling a
parallel H V D C link, Proc. Inst. Electr. Eng., Part
C, 132 (1985) ] 5 4 - 159.

9. 33
6 E. Eitelberg, Perturbationstechniken bei der Op-
timierung
grosser
linearer
Regulungssysteme,
VDI-Verlag,
Dusseldorf,
1983,
ISBN-3-18146006-7.
7 R. Fletcher and M. J. D. Powell, A rapidly convergent descent
method for minimization,
Comput. J., 6 (1963) 163 - 168.
8 N. Jaleeli, E. Vaahedi and D. C. MacDonald,
Multimachine system stability, Proc. PICA Conf.,
Toronto, May 1977, IEEE Publ. 77 CH 1131-2PWR, pp. 51 - 58.
9 B. D. O. Anderson and J. B. Moore, Linear Optimal Control, Prentice-Hall, Englewood Cliffs,
NJ, 1971.