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Synthesis of optimal controller for synchronous power system
Electric Power Systems Research, 2 (1979) 27 ° 46
27
© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands
Synthesis of Optimal Controller for Synchronous Power System N. T. KHOBRAGADE* and D. G. TAMASKAR
Visvesvaraya Regioaal College of Engineering, Nagpur-440011 (India) (Received October 27, 1978)
SUMMARY
During the last decade research work has been in progress in the area of system optimisation by control system engineers. Power system engineers have been endeavouring to apply the optimisation technique to obtain an optimal controller to improve the transient response of a synchronous power system. In this paper an a t t e m p t has been made to select the best control quantities with a view to extending stability boundaries and improving transient response. The transient response corresponding to different combinations of control quantities was considered when selecting the most effective control law. The optimal control law obtained on the basis of the quadratic index of performance for one particular condition has been tested for operating conditions over a wide range, and the possibility of adopting the same controller for all other operating conditions has been investigated. The influence of local load on the optimal control law and transient response has also been examined, and that of governor and turbine dynamics on the transient response has been assessed.
1. INTRODUCTION
Stability and transient response have been studied by several investigators by applying the optimisation technique. The results obtained by these investigators show that the stability boundaries are extended and better transient response can be obtained using this technique than the control obtained using a voltage regulator or other control quantities.
*On leave of absence from the College of Engineering, Pune-5, India.
In most of these investigations the emphasis has been on obtaining the best transient performance. Formulation of state-space models [1 - 6] for the synchronous generator is well established. Habibullah and Yu [7] have proposed an all-state optimal controller in terms of measurable power system variables. Newton and Hogg [8] have given the design of a linear optimal regulator for the micro-machine system to control both the power input and field voltage. In this paper an effort has been made to identify the best choice of control quantities and a corresponding optimal feedback law which would cover a wide range of power system operating conditions and give an acceptable transient response. Investigations made in this paper correspond to a round rotor machine with conventional wound rotor and are at first confined to excitation control. Later on the investigations are extended to both the excitation and governor control. The state-space models proposed in this paper have state vectors with measurable state variables. An all-state optimal controller has been designed. The transient response of the system has been determined for an initial perturbation of /x8 = 0.095 radians. 2. DEVELOPMENT OF DYNAMIC MODELS OF A POWER SYSTEM
The synchronous machine has been treated as a dynamical system with the main intention of obtaining a model in the state variable form. The application of theorems of modern control theory then takes care of the generation of an optimal controller. The state-space models developed retain Park's transformation and are based on the following assumptions.
28
(i) The influence of damper windings is at first neglected and a (4 × 4) model formed. The influence of damper windings is later considered and (6 × 6) models formed. (ii) P S a and p 6 q terms (transformer voltages) in Park's equations are neglected in all cases. (iii) Single-axis excitation and the voltage regulator are included in the modelling process. (iv) Voltages due to incremental variations in speed are n o t neglected. (v) The governor and prime mover dynamics are n o t considered in the first phase of the investigations. Later on t he y are considered. (Performance equations for a synchronous machine are given in the Appendix §A.2.)
3. THE POWER SYSTEM
3.1. Block diagram of power system Figure 1 shows the block diagram of the p o w e r system considered in this paper. Installed capacity of power station = 1000.0 MW Nominal voltage 13.8 kV Nominal p o w e r factor = 0.9 lag The parameters of the generator and the details regarding the transformers, transmission lines and shunt reactors are given in the Appendix § A.1. Vg
2.1. Formation o f the state-space model (i) Influence o f d a m p e r windings is neglected and the (4 X 4) model is formed: ~[ = AX + B u 1
(1)
where X = [A8 Aco At~f AEfd] t. In the above state vector, the state / ~ f is n o t observable. Hence a transformation is a d o p ted to obtain an observable model with a state vector Z = [AS Azo AV t AEfd] t (ii) Influence of damper windings is considered, the size of the state-space model becoming (6 × 6): X = [A(~ AOd A ~ f A ~ D A ~ Q A E f d ] t
(2)
The state-space model with this state vector is referred to as a flux linkage model. F o u r state-space models are derived with the help o f appropriate t r ans f or m at i on matrices from the flux linkage model. (The f or m ul at i on of the state-space model is given in the Appendix §A.3.)
Lenglh oi Ir'a :7Sc: OXM~
Fig. 1. Block diagram of the power system.
3.2. Operating conditions Operating conditions are specified in Table 1. For all operating conditions: Vg = 1.05 p.u., V-- 1.0 p.u.
4. SOLUTION OF MATRIX RICCATI EQUATION
4.1. General philosophy of matrix Riccati equation The quadratic cost functional has two desirable properties [ 9 ] : (i) it is mathematically tractable; (ii) it leads to optimal feedback systems which are linear. This is the reason that people say that "Quadratic criteria fit linear systems like a glove". We have, therefore, used the quadratic perform ance index
= FZ + Gu 1 (general structure of model) J = J (XTQX + u T R l u l ) d t
Z 1 = [A8 Aco A V t Air A P A E f d ] t Z 2 = [AS Aco A V t A i f A i b A E f d ] t
(3) Z 3 ---- [AS ACO A V t A V t A p A E f d ]
(4)
0
t
Z 4 = [AS Aco AV t A P A P A E f d ] t F o r various models the plant matrices are calculated for different operating conditions.
Q is a positive definite real sym m et ri c matrix. This determines how the instantaneous errors are to be weighted during the transient process. R 1 is a positive definite real symmetric matrix. This represents the weight of control cost with respect to minimising error. u I = - K X (a linear feedback law)
(5)
29 TABLE 1 Operating conditions Operating condition No.
Active power Pg Units at generator terminals connected* (p.u.)
System angle 5 (deg)
1 2 3 4
0.6 0.8 1.0 0.2
76.0 89.0 100.5 30.5
I II II III
Reactive power at generator terminals (p.u.) 0.1598 0.1253 0.3328 --0.022
Reactive power at infinite bus (p.u.) --0.0945 ....0.1992 --0.3995 - 0.1230
*I: three generators connected through double circuit line and three step-up transformers; II: four generators operating through double circuit line and four step-up transformers; III: two generators operating through double circuit line and two step-up transformers.
K = R~-IBT Pz
(6)
P1 is the s o l u t i o n o f the m a t r i x Riccati equation. T h e m a t r i x Riccati e q u a t i o n is given b e l o w [6, 9 ] : P1 -- P I B R 1 1 B T P 1 + Q + P1A + ATp1 = 0 (7) In the s t e a d y state I)1 = 0, and t h e r e f o r e the m a t r i x Riccati e q u a t i o n assumes the following form: ATP1 + P1A - - P I B R { - 1 B T p 1 + Q = 0
(8)
T h e s o l u t i o n o f eqn. (8) is o b t a i n e d b y an iterative p r o c e d u r e o n a digital c o m p u t e r (IBM-360). 4.2. Weigh ting ma trices Since the o p t i m a l c o n t r o l law is sensitive to the choice o f weighting matrices R 1 and Q, these matrices s h o u l d be chosen p r o p e r l y . T h e choice o f R 1 and Q is m o r e or less arbitrary. T o m a k e a reasonable choice o f these matrices, a p r e l i m i n a r y s t u d y was m a d e o n a (6 × 6) m o d e l with a state v e c t o r Z = [A6 Aco AV t A V t A]b AEfd] t, the influence o f weighting m a t r i x Q o n the o p t i m a l c o n t r o l law and transient response being observed. R 1 was assumed t o be a u n i t a r y m a t r i x and the m a t r i x Riccati e q u a t i o n was solved f o r d i f f e r e n t " Q " matrices. T h e o p t i m a l c o n t r o l laws o b t a i n e d were studied. Table 2 indicates the m a n n e r in which the weighting o n d i f f e r e n t states is varied. T h e settling times and the first o v e r s h o o t s are r e c o r d e d in the same Table. T h e transient response curves are p l o t t e d in Fig. 2. T h e results t a b u l a t e d in Table 2 indicate t h a t rela-
tively large weighting on the A5 state (emphasis o n r o t o r angle deviation) helps in arriving at a b e t t e r o p t i m a l c o n t r o l l e r which w o u l d settle the system m u c h faster t h a n o n e with m o r e weighting o n any o t h e r state. In the light o f the above s t u d y the following weighting matrices have been c h o s e n in this analysis. In all cases: R 1 = 1.0 In the case o f the (4 × 4) m o d e l : Q = Diag(50, 10, 10, 10) In the case o f the (6 × 6) m o d e l : Q = Diag(100, 1, 1, 1, 1, 1) T h e o p t i m a l c o n t r o l laws o b t a i n e d are given in Table 3. Figure 3 shows the b l o c k diagram with o p t i m a l r e g u l a t o r [ 1 0 ] . T h e gains m a r k e d as K l l , K12, K13 , etc. in Fig. 3 correspond t o the gains on the various states.
5. SELECTION OF CONTROL QUANTITIES T h e solution o f the m a t r i x Riccati e q u a t i o n is used t o o b t a i n the linear state variable feedback law. T h e result!ng closed-loop system can be described b y X = (A -- BK)X. Eigenvalue analysis is carried o u t and the t r a n s i e n t response is t h e n p l o t t e d . Figure 4 shows the t r a n s i e n t response f o r f o u r o p e r a t i n g conditions with five d i f f e r e n t models. T h e comb i n a t i o n o f c o n t r o l quantities is d i f f e r e n t for each m o d e l . T h e response o f the system was d e t e r m i n e d f o r an initial p e r t u r b a t i o n o f A5 = 0 . 0 9 5 radians. 5.1. State vector Z = [A~ Aco AVt AEfd] t In this m o d e l the influence o f d a m p e r windings has been ignored. T h e t r a n s i e n t
11.54
20.58
54.00
20.41
Q5 = (1, 1, 1, 1, 20, 1)
Q6 = (20, 20, 20, 20, 20, 1)
Q7 = (100, 1, 1, 1, 1, 1)
9.722
43.69
8.841
0.53
1.659
2.247
0.6539
0.6423
1.141
0.6535
Optimal control law
Q4 = (1, 1, 1, 20, 1, 1)
Q3 = (1, 1, 20, 1, 1, 1)
Q2 : (1, 20, 1, 1, 1, 1)
Q1 = (1, 1, 1, 1, 1, 1)
Weighting matrix Q All diagonal matrices
58.95
115.2
41.17
23.75
23.00
87.29
18.64
1.823
3.811
1.262
1.096
0.7262
2.687
--0.693
-1.266
--0.278
--0.41
--9.268
--1.025
--0.216
/~Yt ~ ~kEfdit
0.59
Operating condition: Vg = 1.05, V = 1.0, Pg = 0.8 p.u., Z = [AS Ac0 A V t
Influence of weighting matrix on transient response
TABLE 2
1.47
1.98
1.55
1.28
1.21
1.644
1.174
4.0
5.8
>10.0
>10.0
5.6
>9.0
>9.0
Settling time (s)
1.25 × applied step size
0.936 × applied step size
2.0 × applied step size
1.474 × applied step size
1.58 × applied step size
0.821 × applied step size
1.67 × applied step size
Overshoot
Transient response
Gains reasonable. Overshoot moderate. Settling time lowest
O v e r s h o o t less. Settling t i m e also less, but gains large which discourages choice of this weighting matrix
O v e r s h o o t large. Response less oscillatory but very sluggish
O v e r s h o o t reduced but settling time large
First o v e r s h o o t large
Response less oscillatory but quite sluggish. Gains high
First o v e r s h o o t large
Remarks
5o
31
O, L6
O-lJl
o.o 8
® o .o~
,'~.
i
,
,
,
. .:
~: /,"
i:i,~.:'I" I
'~..-
" ' " ~ i
,v"
-o"
tl'il
o
0= o,,G,,.,.,.,o.,.,,
-o..j
'
(~)
Q =
D]AG(IO0,1
1 1 1 1)
Fig. 2. Influence of weighting matrix Q on transient response.
r- . . . . . . . .
~, X2
~
Oplimal
Regulator
~
_A-_MZ'2_Ry_u'at°:___
,- . . . . r
"I
:
T
7
F,12 I~
_
~ ~~,~ xsXs "[--Z~---'~~
t I
I I I
i "
I
ER~f /
i
Iv, i
I
I
' , Ii
"~ K16 I'-" ---I I Summer .......... z j
L
'
.......
I'
~
r-Synchronous \Machine
l']
[.nfinlte
Bus
Z-Ex~eT" m a~or damping loop
Fig. 3. Block diagram with optimal regulator.
response is found to be oscillatory, the oscillatory process continuing for a relatively long time. The influence of damper windings is then considered and various state vectors are proposed to find the best combination of stabilising signals. The settling times in each case are recorded in Table 3. The quality of the transient response is discussed below.
5.2. State vector Z = [AS A¢~ A Vt Air &P
AEta ] t In this model, the transient response obtained for operating conditions No. 1, 2, and 4 is n o t oscillatory. The settling times are quite reasonable, n o t exceeding 5.0 s for any of the operating conditions. However, when the system is operating at Pg = 1.0 p.u. and then is subjected to a small perturbation, it
0.6 0.8 1.0 0.2 0.6 0.8 1.0 0.2 0.6 0.8 1.0 0.2
1 2 3 4
1 2 3 4
1 2 3 4
Z = [/k~ A y t Aif ~l~ ~ E f d ] t
Z = [A~ /~G9 AV t A~rt ~ f d ]
--4.13 --8.97 --15.74 --0.11
14.44 20.41 29.89 3.68
11.3 --0.5 37.18 4.63
--12.12 --20.61 --34.41 0.0184
14.98 27.84 35.12 --85.02
=
--0.41 --0.353 --1.29 --0.25
0.817 0.53 --0.10 --0.40
1.32 8.98 --5.3 0.002
0.07 --0.21 --1.179 --0.116
--0.589 --0.224 --0.365 --18.76
O p t i m a l c o n t r o l law K R-1 BTp
7.37 0.69 --10.8 9.58
48.37 58.95 76.47 32.46
40.44 19.54 85.80 15.2
13.01 1.34 --19.0 17.31
64.92 67.94 69.09 185.2
22.36 30.68 49.95 11.61
1.12 1.82 2.72 0.46
--0.43 --2.57 1.59 --0.04
5.7 7.22 10.15 3.12
3.389 3.366 3.379 3.966
0.23 0.27 0.37 0.25
--0.33 --0.69 --1.19 --0.09
--2.3 --13.4 9.15 --0.2
24.97 37.35 66.98 7.05
1.34 1.48 1.62 1.2
1.4 1.5 1.6 1.3
1.45 2.277 1.74 1.21
1.37 1.46 1.60 1.205
3.5 5.0 >I0.0 3.0 3.5 4.25 3.25 3.25
2.3 2.6 4.1 3.75
3.5 3.8 10.0 2.3 3.5 1.75 2.75 3.75
2.3 2.25 3.10 3.25
4.8 3.6 4.5 Long t i m e
4.8 6.5 >10.0
Settling t i m e w i t h s u b o p t i m a l law (s)
4.8 3.6 Long t i m e 3.6
4.8 5.8 6.0 4.8
Settling t i m e w i t h o p t i m a l law (s)
*The s u b o p t i m a l law c h o s e n is t h e same as t h e o p t i m a l c o n t r o l law c o r r e s p o n d i n g to o p e r a t i n g c o n d i t i o n No. I in each case.
t
0.6 0.8 1.0 0.2
1 2 3 4
Z = [A5 ACO A V t &if L~P z~LEfd]t
Z = [A(~ A(.O/kyt ~ P ~ f d ]
0.6 0.8 1.0 0.2
1 2 3 4
Z = [A5 ACO A F t L2kEfdIt
t
Pg (p.u.)
Operating condition
No.
State v e c t o r
Optimal and s u b o p t i m a l c o n t r o l laws a n d settling t i m e (for all o p e r a t i n g c o n d i t i o n s Vg = 1.05, V = 1.0 p.u.)*
TABLE 3
¢.D bO
33
P6
=0"6
,:#
_
J I
# I
i"
°"
p<3 +_ 1 - o .
i!
f
ii /l i t/
;
>.o4
,, !
O.0
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!
•
+
.
,X',.Jlr" l
i~i" i\ ,
,
r\
e
I
~
i i
i
.
.
.
.~.
.
'
l\lil# i'v''l
o.o~
.
~
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__
i
'.,, v
•
~1',;
i
,
,,. __
i !
.......
~.:,--~.+ , , o
.,v,+~,,,:A,~+:t
"= z:.,~,A,~.,,,,,.,,~>~
I~-+^,-,,,Lv'~,,:A,~o''-"o.2~6 ~ o.ssE=
~1
~= L+'t"o+,+'v~, +,P, ",~m.t~ +
I
o. ---m,..
i.
2. "I"IME,. IN I I £ C O N ~
3.
Fig. 4. T r a n s i e n t r e s p o n s e o f d i f f e r e n t m o d e l s .
4'.
~
o.
i. ~
"l't~
z. 3. iN 43s.cot4~,~
~i.o
34 is found that the transient response is quite oscillatory and the transient process would continue for a long time. 5.3. State vector Z = [AS Ao~AVt/XifAP AEfd] t In this model, the transient process is characterised by large deviations with a maxim u m overshoot of 2.5 times the applied step size for an operating condition corresponding to Pg = 1.0 p.u. The settling times are quite low, but as the response in general is oscillatory with remarkable overshoots, this combination of stabilising signals is unacceptable. S t a t e v e c t o r Z = [A5 A~o A V t A V t /~P AEfd] t In this model, the response in general is oscillatory but the m a x i m u m overshoot is limited to 1.6 times the applied step size corresponding to operating condition No. 3 (Pg -- 1.0 p.u.). The oscillatory process terminates quickly. 5.4.
5.5.
State
vector
Z
=
[AS Aw AVt AP A/~
AEfd] t The transient response obtained in this case is characterised by monotonic convergence for operating conditions No. 1 and 2 and is slightly oscillatory in nature for operating conditions 3 and 4. Settling times are very much reduced. The profiles of the transient response drawn in Fig. 4 are indicative of these observations. It is, therefore, concluded here that the combination of stabilising signals given by this state vector will give the best transient response with minimum settling time over a wide range of operating conditions. Therefore, the best group of control quantities which should be adopted for stabilising the synchronous machine is given by a state vector
should be made to identify the best choice of optimal control law which will give reasonably good performance over the wide range of operating conditions. To explore the possibility of using the same controller to cover a wide spectrum of operating conditions, the optimal control law obtained for operating condition No. 1 (P~ = 0.6 p.u.), which is the median condition between the extremes of heavy load and light load, has been tested for other operating conditions. Table 3 gives the information about the suboptimal law and the settling times obtained with the application of this law. Using the models proposed in this paper, eigenvalues obtained with the control law chosen have negative real parts, indicating that the suboptimal controller can maintain the system stable. This analysis clearly indicates the adoption of the same controller to cover a wide range of operating conditions. Figure 5 shows the transient response obtained with optimal and suboptimal control laws in the case of a (6 X 6) model with state vector Z = [AS Aw AV t APAPAEfd] t. The transient response curves indicate that the suboptimal control law gives a transient performance almost identical to the one with an optimal control law, except for the fact that the settling time is increased to some extent as compared with the settling time with the respective optimal law. In the case of a (6 × 6) model, with state vector Z = [A5 ~w A V t A i f A P A E f d ] t, it has been observed that the suboptimal controller gives a better transient response with a lower settling time for operating condition No. 4 (Fig. 6). This indicates that the optimal control law obtained with definite weighting matrices need not be the best, though it is optimal. It is possible to have a better transient response with a lower settling time with some other control law.
Z = [AS Aw AV t AP APAEfd] t 7. INFLUENCE OF LOCAL LOAD 6. POSSIBILITY OF ADOPTING THE SAME OPTIMAL CONDTROLLER TO COVER A WIDE RANGE OF OPERATING CONDITIONS In practice, it is difficult to adjust the gains of the optimal controller with change in operating conditions. Therefore, an effort
When the local load is taken into consideration and an optimal controller is designed, it is observed that the quality of the transient response is improved, particularly in terms of the settling time (Fig. 7). It has also been observed that the feedback gains for all states are less than the correspon-
35
pe-o.§
_P¢=o-8
8 u 8 - oP-rIMAL F~r~wAcg,
,..f,.Cv.~.)
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I
/
',
O ]Wlr"I I ~ A J.
,,
0.04
,
~.o
\
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t l
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• #
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i I
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! I
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~ t two'
-o-0|
(v.R.)
Per I-o
PG= O-~
~rgn- oPqr'IMAA. F£J~BACK
~u 8-01~'IMAL F £r-~'SAC K
p~A- F~'F,DB AcK
~,....,.~ e"
8"""t ,,.\
i
o.
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J.
TIM,F.
o p ' r l MAC.
b
"-..'"
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~.
.~,
H',I $£~0/',,t,V~
o.
l. _--- " n M L
P_ ~
K
4.
SLooP.
Fig. 5. T r a n s i e n t r e s p o n s e w i t h o p t i m a l a n d s u b o p t i m a l c o n t r o l laws f o r p o w e r - p o w e r derivative m o d e l .
ding gains in an optimal controller which is designed without considering the local load.
Table 4 gives an idea of the changes in the optimal law when the local load is considered.
36 PG :0.2
t
0 10 ,~
0.08[ /fOptimal feedbqc~< 9.04 ~
~.
008
powz~
// / / / /~
AT GZN
TZ,eMIN.4L
= • ¢p~
No Local iood
fs:35 sec OsciltQtor\
I.
~Sub oplinl@: !eedLq, k
I
~OOS
o
o
o
c 0.0l, ,-,o <]
002
008 [
0
1 2 3 4 I ime in sec Fig. 6. Transient response with optimal and subo p t i m a l c o n t r o l laws for current d e r i v a t i v e - p o w e r derivative m o d e l .
P[=0.1. P~
[ fs:2.3 sec
t
1I, ! Monotonic /[ it Local[ood /1 1 ; PL=O2'Ts:251~ec Osci[Iotory ~J//
\\\
~
--
--
-
1 2 3 L Time in sec Fig. 7. Influence o f local load.
TABLE 4 I n f l u e n c e o f local load Operating c o n d i t i o n Pg (p.u.)
Local load PL (p.u.)
F e e d b a c k law Z = [/k~ /kCO A V t z~P ~/~ A E f d ] t
1.0 1.0 1.0
0.0 0.2 0.4
--15.74 ---14.76 --13.68
The above analysis clearly indicates that while designing an optimal controller for a synchronous power system, due consideration should be given to the presence of local load. If this factor is ignored for some reason, the designer will have to propose the optimal controller with relatively higher gains.
--1.291 --0.845 --0.635
--10.80 --9.565 --8.824
49.95 40.53 36.36
0.37 0.36 0.28
1.62 1.57 1.57
8. C O M P A R I S O N O F P E R F O R M A N C E O F A N ALL-STATE FEEDBACK CONTROLLER WITH VOLTAGE REGULATOR
With any of the all-state feedback controllers discussed above, the transient response is relatively much superior to the transient response obtained with the voltage regulator only. This can be seen in Fig. 5. It is also seen from Fig. 5 that the all-state feedback with optimal or suboptimal controller extends the stability limit beyond that obtained with only the voltage regulator in action.
9. I N F L U E N C E O F G O V E R N O R A N D T U R B I N E DYNAMICS ON TRANSIENT RESPONSE
~ime
Stele
: Isec/mch,
AmplHude
O 1
v/inch
Fig. 8. Governor and turbine response to unit ster input (TRH = 8.0 s).
The mathematical model given in Fig. 7-E of the IEEE Committee's Report [11] has been simulated on an analogue computer. The response curves of mechanical power {Pro) v s . time (t) have been obtained for different reheat time constants (TRH). Figure 8 shows the nature of the response obtained. Since the
37
response curve is reasonably exponential, it is considered adequate to represent the governor and turbine dynamics by a single time constant to reduce the complexity of the state-space model. The combined equivalent time constant of the governor and turbine has been taken as Teq = 0.5 s which would correspond to the closing stroke, and T~q = 4.0 s which would correspond to the opening stroke. The order of the state-space model is increased by one, becoming a (7 × 7) model, when the governor and turbine dynamics are considered. The state vector is Z = [AS A ~
AV t APAPAEfd
K=
0.607
.~004 •~- 0
1
2
3
4
- 004 - 0.08
l
~(~)
PG :0.~
~
PG : 1,0
_0 . 0 4 ~
The plant matrix and the control distribution matrix have been calculated for different operating conditions. The suboptimal law is used for assessing the eigenvalue spectrum and the transient response of the system. The following control law is used: --0.417
fs =44 sec
ATm] t
The control vector is u = [ul, ug]t
I[--4.129 3.657
PG=O.8 0.08
7.374 0.0
22.36 0.0
- 00
1.345
0.0
0.0
(i) Teq = 0.5 s The transient response obtained with the (7 × 7) model is m o n o t o n i c for all operating conditions. There is remarkable improvement in the quality of response as compared with the response obtained only with the excitation control for operating conditions No. 3 (Pg = 1.0 p.u.) and 4 (Pg = 0.2 p.u.). For operating conditions No. 1 (Pg = 0.6 p.u.) and 2 (Pg = 0.8 p.u.), the response obtained with excitation control only and with excitation and governor control are more or less alike. (ii) Teq = 4.0 s With the increase in the equivalent time constant, it has been observed that, for operating conditions No. 1 (Pg = 0.6 p.u.) and 4 (Pg = 0.2 p.u.), the response is still
~
Fig. 9. Transient response with conventional controls (curve 1);excitation and governor control, Teq = 0.5 s (curve 2); excitation control alone (curve 3); excitation and governor control, Tea = 4.0 s (curve 4).
0.235
This control law maintains the system stable under all operating conditions considered. The profiles of the transient responses are given in Fig. 9.
5
--11.21]
!
11.40]
monotonic, but it becomes oscillatory for operating condition No. 3 (Pg = 1.0 p.u.). In this case also the transient process terminates after a few oscillations only. The above analysis suggests that the governor and turbine dynamics must be considered when designing an optimal or suboptimal controller for a synchronous power system as also for determining the transient response of the controlled power system.
10. CONCLUSIONS
(i) The choice of weighting matrix Q is important to arrive at a better and physically realisable control law. (ii) The choice of the quadratic performance index which is most often used does n o t guarantee the best transient performance. (iii) The quality of the transient response, particularly in terms of the settling time, is improved considerably with the inclusion of
38 stabilising signals in the form of active power and rate of change of active power. (iv) The quality of the transient response, particularly in terms of the settling time, is improved and the gains of the stabilising signals are reduced as the local load is increased. (v) The governor and turbine dynamics should be considered when designing an optimal controller for a synchronous power system as it influences the transient response considerably. (vi) It is possible to obtain a suitable control law to ensure stable operation of a p o w e r system, and also good transient performance over a wide range of operating conditions of the power system. A proper combination of control quantities together with the proper choice of weighting matrices can generate a control law which will ensure considerably improved transient performance of the power system over a complete range of operating conditions.
TA
KE TE KF TF
Control system X Z A F B G uI M K R1 Q
NOMENCLATURE
P1
Synchronous machine and transmission system Vd, Vq Vf, if
Efd S d, ~q ~f
~D, ~Q id, iq ~o ~0 R +jX V or V o Vg or V t Tm P, Pg
direct axis and quadrature axis voltage field circuit voltage, field circuit current air gap line open-circuit voltage direct axis and quadrature axis flux linkages direct axis field flux linkages direct axis and quadrature axis damper flux linkages direct axis and quadrature axis components of terminal current instantaneous angular frequency (rad/s) synchronous angular frequency (rad/s) equivalent series impedance of transmission system infinite bus bar voltage generator terminal voltage mechanical torque torque angle (rad) active power at generator terminals
Excitation system KA
exciter amplifier gain
exciter amplifier time constant (s) exciter gain exciter time constant (s) gain of stabilising loop time constant of stabilising loop (s)
A
state vector of flux linkage model state vector of transformed model system matrix of X-model (n X n) system matrix of Z-model (n X n) control state distribution matrix of X-model (n X m) control state distribution matrix of Z-model (n X m) control vector transformation matrix optimal gain matrix (m X n ) positive definite real symmetric matrix (m X m) another positive definite real symmetric weighting matrix (n X n) positive definite symmetric matrix obtained as the solution of matrix Riccati equation (n X n) prefix denoting a linearised variable
ACKNOWLEDGEMENTS The authors would like to record their thanks to the Government of Maharashtra and the Government of India for sponsoring the candidate under the Quality Improvement Programme in the year 1976. The facilities extended by the Visvesvaraya Regional College of Engineering, Nagpur, are gratefully acknowledged.
APPENDIX A. 1. System parameters The parameters of the generator correspond to the parameters of the micro-alternator which will be used at Visvesvaraya Regional College of Engineering, Nagpur, for experimental work.
39
M icroalterna tor p arame ters xd
= 2.013p.u. 0 . 2 7 3 2 p.u. Xd" = 0 . 1 8 4 p.u. xq = 1 . 9 1 3 p.u. x q " = 0 . 2 3 2 5 p.u. Xal = 0 . 1 1 3 p.u. x~ = 2 . 0 7 5 p.u. X d '
Ra R~ co o Td0'
0.006p.u. 0 . 0 1 2 p.u. 3 1 4 . 0 radians 5.0 s Tdo"= 0 . 0 0 9 1 s n Tqo = 0 . 0 7 3 s H = 2.057 s
----"
= = = =
Control complex parameters K A = 50.0 K E = 1.0 g F = 0.1
TA TE TF
= 0.05s =0.05s = 0.2 s
Details o f step-up transformer F o r each step-up t r a n s f o r m e r Nominal capacity = 300.0 MVA T r a n s f o r m a t i o n ratio = 1 3 . 8 / 4 0 0 kV Leakage r e a c t a n c e = 0.15 p.u.
Details o f transmission line T h e transmission lines consist o f t w o 4 0 0 kV circuits. L e n g t h o f line = 755 km R e a c t a n c e o f line = 0 . 3 0 5 ~2 p H -1 k m -1 Surge i m p e d a n c e = 2 9 5 . 0 ~2
Shunt reactors O p e r a t i n g c o n d i t i o n No. 1 (Pg = 0.6 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 3 2 9 4 p.u. O p e r a t i n g c o n d i t i o n No. 2 (Pg = 0.8 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 1 6 4 7 p.u. O p e r a t i n g c o n d i t i o n No. 3 (Pg = 1.0 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 1 6 4 7 p.u. O p e r a t i n g c o n d i t i o n No. 4 (Pg = 0.2 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 2 9 8 8 p.u.
A.2. P a r k ' s e q u a t i o n s f o r a s y n c h r o n o u s m a c h i n e have the following f o r m [3, 12, 13] : 1
09
Vd -
pt# d . . . . °90
t~q - - R a i d 090
1
09
Vq = ---P~d
+ --
090
~ d .... R a i q
090
1 V~ = - - p ~
+ R~i~
(A1)
090
1 0
-
0
=--p~Q 090
090
P ~ D + R D iD
1
+RQiQ
The t o r q u e e q u a t i o n s are p5 = 0 9 - - 0 9 0
(A2)
090 ( T m __ Te ) P09= 2H where Te = ~diq - - Sqi d
A s s u m i n g a solid-state e x c i t e r o f negligible time c o n s t a n t and a voltage r e g u l a t o r with single t i m e c o n s t a n t [ 7 ] , we get AE~d -
--K A (1 + P T A ) (Vt - - Vre~ - - u l )
R e a r r a n g e m e n t o f this e q u a t i o n will give us 1
PA E~d -
KA
TA AE~d -- -~A (Vt - - Vre~ - - u l )
(A3)
40 F l u x linkage equations:
id
~d
= --Xd
+ Xad if + Xad
~q
= --Xq iq + Xaq iQ
~f
=--Xadi d + x f i f
iD (A4)
+Xad/D
~/D =--Xadid +Xadif +XDiD ~O
= --Xaq i q +XQiQ
A.3. Formulation of flux linkage model E q u a t i o n s (A1) - (A4) are linearised and t h e n arranged in the following m a t r i x f o r m : A5
A~f
=
/~]D
-0
1
0
0
0
0 -
0
0
d23
d24
d25
0
0
0
d33
d34
0
d36
0
0
d43
d44
0
0
A~ D
0
0
0
0
ds,~ 0
A~Q
0
d62
d65
d64
d65
i
AS Aw +
d66
0
0
m
0
21 ]'/22
Ih31
0
~41
0
0
h52
0
~
+
0 [ul]
(A5)
61 h62 L
"J
or = D X + HI + Bu 1
(A5a)
T o eliminate the c u r r e n t v e c t o r f r o m eqn. (A5), f r o m Park's e q u a t i o n s we can o b t a i n AV d +RaAi d-xq''Aiq=-o4A$
1 q + - - (xq"iq--O4~Q)ACz (DO (A6)
1 AVq +RaAi q +Xd"Ai d = a 2 A ~ f + O3A~ D + - - ( o 2 ~ COO
+ a3~D--Xd"id)A~
Considering one m a c h i n e c o n n e c t e d t o an infinite bus as s h o w n in Fig. A1 [ 1 ] , the e q u a t i o n s r e p r e s e n t i n g the transmission s y s t e m can be w r i t t e n as: AVd = (K1 "7 + K2"o)Aid + (K2"~ - - K 1 .o)Aiq -- Vo(K2sin5 - - K l c o s S ) A 5 (A7) AVq = (K 1.0 - - K 2 . ~ ) A i d + (K 2.o + K 1.~)Aiq - - Vo(K2cos5 + K 1sinS)AS w h e r e A V d and A Vq are the d-axis and q-axis c o m p o n e n t s o f Vg as viewed f r o m the e x t e r n a l circuit.
VI
V2
Vo
X~
Fig. A1. One machine connected to an infinite bus.
41
From eqns. (A6) and (A7), the expressions for current I can be written as A8 A~9
I:
/dl = FCll
C12 C13 C14 C15 C161
A~f
iq]
C22 C23 624
ASD
(A8)
[621
625
626~
ASQ
&El d or
I = CX
(A8a)
From eqns. (A5a) and (A8a) :K =
(D + HC)X
+ Bu 1
or = AX + Bu 1
Expressions for the various coefficients are given below: d23 -
d24 -
d25 -
d3 4 -
dss
¢Oo (Xd--Xd')(Xd"--X~,) 2H (xd -- Xal)(Xd' -- Xal) 090 Xd t --Xd" 2 H Xd I - - Xal
1
(Xd' --Xd")(Xd --X~,)
Too' (Xd' Td0,
--
d55 -
i +
--
x~,)
(Xd' -- X j ) ( X d (xd'
-
-- Xd')]
~,)(x~' -~,~ J
TdO' X d - - Xd'
1
Xd __ Xdt
Td0" Xd ~ Xal
1 Td0"
1 Tq0"
Vo
d63 -
X~,)(Xd'
X d ~ Xal
1
d44-
id
2 H Xq - - Xal
1[
d43 -
iq
CO0 Xq - - X q "
-
d36 -
iq
VQ
) KA
K A (Xd--Xd')(Xd"--Xal) VtTA (Xd __ Xal)(Xd, __ Xal) Vq
(A9)
42 KA
d64 -
Xd' - - Xd" Vq
Vt TA Kh
d65 -
X d ' __ Xal
Xq - - Xq " Vd
V t T A Xq - - X a l 1
d66 =
__ _ _
TA
rt
K s
-G) 0
-
Xq"iq
--
~ Q
Xq - - Xal
K 9 = ( 0 2 ~ f + 0"3~ D --Xd"id)/CO 0 0" 1 = C O o / 2 H
(X d - - X d ' ) ( X d " - - Xal ) 02 (X d ' - - X a l ) ( X d - - X a l ) Xd' - - Xd" 03 X d ' - - Xal tr
Xq - - X q 0 4 -Xq - - Xal
h21
pr
= .O'I[(X
~t
q
- - X d )iq - - o'4t~Q ]
(X d - - Xa l ) ( x d ' ' - - X a l )
Tdo,(X d . . . . Xal )
h3a = __
h 4 1 --
(X d' - - Xal ) Td0" KA
/261 -
(RaV d + Xd"Vq) Vt TA rl
h22 = --Ol[(X q
--Xd")i
d + @ f o 2 + @DO3]
1 h52 -
,, ( X q - - Xal )
Tqo KA h62 -
Vt TA
(Xq" V d - - R a VQ)
= 1 + RG -- XB = XG
+ RB
7? = R - - ( X t X G
+ XtBR)
o =(X t +X+XtRG
)-xtxB
43 D e t e r = (K1 ",7 + K2"a + Ra) 2 - - ( K 2 " 7 --K1 "0--xq")(K, " 0 - K2"? + Xd" ) 1 + R-G - - X ' B
X-G + R ' B K2 -
0/2 + ~2
K 3 = Vo(K2sin5--K 1 cosS)
K4 = Vo(K2 cos 5 - - K 1 sinS) K5 = (K1 ",7 + K2"o + R a ) / D e t e r K 6 = (K 1 .a + Xq" - - K2 ",7)/ D e t e r K v = - - ( K 1 .o + Xd" - - K 2",7)/Deter
C n =KsK3 + K s K 4 K5 C12
--
O)o
C13 -- (/2
..
(Xq lq - - O 4 ~ Q ) +
Ks O9o
( 0 2 ~ f + O3~D - - X d " i d )
"K6
.C14 -- 03 "K6 C15 = - - 0 4 .K 5 C16 = 0.0 C21 -- K7 "K3 + K~ "K4
K7 C22 C23 =
090
,,.
(Xq lq - - 0 4 ~ q )
K5
+ ~ ( G 2 " ~ f + O 3 " ~ D --Xd"id) 090
o2"K5
C2 4 = 0 3"K5 C2 5 --
--04
.K 7
C26 = 0.0 I f the stabilising l o o p is c o n s i d e r e d (refer t o Fig. 3) [ 1 0 ] , KA/T A in eqn. (AS) should b e replaced by
(KAKE)/(T A + T~ + TF + KAKEKF)
Formation o f appropriate transformation matrix (voltage-power-power derivative model) Let Z = MX Therefore
(A10)
44 2 = MX
(All)
= M(AX + Bu 1) = M(AM-lZ + Bul) _- M A M - l Z + MBu 1 = FZ + Gu 1
(A12)
where
X:
[AS A(~) A~f A~D A~Q AEfd] t
Z = [AS Ao) A V t A P A [ g A E f d ] t T h e terminal voltage
z~V t - Vd AV d + -Vq - AVq
Wt
(A13)
Yt
= m31A5 +m32Ac~ +m33A@f +m34A@D + m35A@Q +m36AE~d where
/7/31 --
Vd
,, Vq ,, (Xq C21 - - R a C l l ) + ~ (--x d e l l --RAG21 )
Vd Vq /7232 = --~t (Ks +xq"C22 --RAG12) + ~tt (K9 --Xd"C12 --RAG22) Vd Vu m33 = ~tt (--R~C13) + ~tt (a2 --Xd"C13) Vd
rrt34
--
Ytt
Vd
(--RaC14)
+
.
Vq "-Yt (03 --Xd"C14)
Vq
m35 = Vtt (xq C25 -- at) + -Vt (--RaC25) ma6 = 0.0 Consider an e x p r e s s i o n f o r electrical p o w e r o u t p u t : P = V d id + Vq iq
(A14)
Linearising eqn. (A14), AP = VdAi d + idAV d + VqAiq + iqAVq
= m41A5 + m42Aco + m 4 3 A ~ f + m 4 4 A ~ D + m 4 5 A ~ q +m46AEfd where m41 = V d C l l + VqC21 + i d ( x q " C 2 1 - - R a C l l ) - - i q ( X d " C l l +RaC21) m r 2 = VdC12 + /d(Xq"C22 - - R , C 1 2 + K s ) + VqC22 + iq (~Xd"C12 - - R ~ C 2 2 + Kg)
m43 = YdC13 + id(--RaC13 ) + iq(a 2 --Xd"C13) m44 = VdC14 + id(--RaC14} + iq(a~ --Xd"C14 )
(A15) (A16)
45 m45 = id(Xq"C25 - - 04) + YqC25 + iq(--RaC25)
m46 = 0.0 Differentiating eqn. (A16) :
+
+
+
+
= m51A5 + m52Aw + m s a A ~ f + m54A~D + rn55A~b Q + rn56AEfd
(A17)
where m51 = m42A21 + m43A31 + rn44Aal + m a s A s l m52 = m 4 1 + m 4 2 A 2 2 + m 4 3 A 3 2 + m 4 4 A 4 2 + m 4 5 A 5 2 m53 = m42A23 +m43A33 + m 4 4 A 4 3
rn54 = m42A24 + m43A34 + rn44A44 m55 = m42A25 + m45A55
ms~ = m43A36 ( H e r e A21 , A22, etc. are the elements of matrix A in eqn. (A9).)
Now the Z-vector can be co-related with the X-vector in the following manner:
1 Im
Aco ' AVt
AP. AP
10
=
31
0
0
0
0
0
1
0
0
0
0
m32 m33 m34 m35 0 m42 rn43 m44 m45 0
[i41 51 m52 m53 msa m55 m~ 0
.AEf,
0
0
0
1
A5
1
A~o
I
A~,f A~D I
(A18)
.AEfdJ
Formulation of a (4 × 4) model is similar to that of a (6 X 6) flux linkage model. The procedure delineated in getting a voltage-power--power derivative model can be adopted to obtain any other equivalent model.
Nomenclature
Xafd Xakd
id , iq iD, iQ X d ~ Xq
XD XQ Xf
direct axis and quadrature components of terminal currents direct axis and quadrature axis damper currents direct axis and quadrature axis reactances of synchronous generator damper circuit reactance on direct
Xad
Xaq
axis
R a
damper circuit reactance on quadrature axis total reactance in the field
R~ Xd r
mutual reactance between armature and field windings mutual reactance between armature and damper windings mutual reactance between generator field and d-axis armature winding (all rotor-stator and r o t o r - r o t o r reactances have been assumed to be equal, i.e. Xatd = Xakd = Xtkd) generator quadrature axis magnetising reactance armature resistance field resistance direct axis transient reactance of synchronous generator
46 r~
Xd Xq
It
Td0' Td0" Tq0" H
To
d i r e c t axis s u b t r a n s i e n t r e a c t a n c e o f synchronous generator q u a d r a t u r e axis s u b t r a n s i e n t reactance of synchronous generator d i r e c t axis t r a n s i e n t o p e n - c i r c u i t time constant d i r e c t axis o p e n - c i r c u i t s u b t r a n s i e n t time constant q u a d r a t u r e axis o p e n - c i r c u i t s u b transient time constant inertia constant electrical torque
REFERENCES 1 J. M. Undrill, Power system stability studies by the method of Liapunov: state-space approach to synchronous machine modelling. IEEE Trans., PAS-86 (1967) 791 - 801. 2 J. H. Anderson, Matrix methods for the study of a regulated synchronous machine. IEEE Proc., 57 (1969) 2122 - 2136. 3 M. A. Laughton, Matrix analysis of dynamic stability in synchronous multi-machine system. Proc. Inst. Electr. Eng., 113 (1966) 325 - 336.
4 S. Raman and S. C. Kapoor, Synthesis of optimal regulator for synchronous machine. Proc. Inst. Electr. Eng., 119 (1972) 1383 - 1390. 5 Y. N. Yu and B. Habibullah, Improving supplemental excitation control design using accurate model. IEEE PES Winter Meeting, New York, Jan. 28 - Feb. 2, 1973, C-73, 219-3. 6 C. E. Fosha and O. I. Elgerd, The megawattfrequency control problem: a new approach via optimal control theory. IEEE Trans., PAS 89 (1970) 563- 577. 7 B. Habibullah and Y. N. Yu, Physically realisable wide power range optimal controller for power systems. IEEE Trans., PAS-93 (1974) 1498 - 1506. 8 M. E. Newton and B. W. Hogg, Optimal control of micro-alternator system. IEEE Trans., PAS-95 (1976) 1822 - 1833. 9 M. Athans and P. L. Falb, Optimal Control, McGraw-Hill, New York, 1966. 10 IEEE Committee Report, Computer representation of excitation systems. IEEE Trans., PAS-87 (1968) 1460 - 1464. 11 IEEE Committee Report, Dynamic models for steam and hydro turbines in power system studies. IEEE Trans., PAS-92 (1973) 1904 - 1915. 12 C. Concordia, Synchronous Machines, Wiley, New York, 1951. 13 B. Adkins, The General Theory of Electrical Machines, Chapman and Hall, 1959.
27
© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands
Synthesis of Optimal Controller for Synchronous Power System N. T. KHOBRAGADE* and D. G. TAMASKAR
Visvesvaraya Regioaal College of Engineering, Nagpur-440011 (India) (Received October 27, 1978)
SUMMARY
During the last decade research work has been in progress in the area of system optimisation by control system engineers. Power system engineers have been endeavouring to apply the optimisation technique to obtain an optimal controller to improve the transient response of a synchronous power system. In this paper an a t t e m p t has been made to select the best control quantities with a view to extending stability boundaries and improving transient response. The transient response corresponding to different combinations of control quantities was considered when selecting the most effective control law. The optimal control law obtained on the basis of the quadratic index of performance for one particular condition has been tested for operating conditions over a wide range, and the possibility of adopting the same controller for all other operating conditions has been investigated. The influence of local load on the optimal control law and transient response has also been examined, and that of governor and turbine dynamics on the transient response has been assessed.
1. INTRODUCTION
Stability and transient response have been studied by several investigators by applying the optimisation technique. The results obtained by these investigators show that the stability boundaries are extended and better transient response can be obtained using this technique than the control obtained using a voltage regulator or other control quantities.
*On leave of absence from the College of Engineering, Pune-5, India.
In most of these investigations the emphasis has been on obtaining the best transient performance. Formulation of state-space models [1 - 6] for the synchronous generator is well established. Habibullah and Yu [7] have proposed an all-state optimal controller in terms of measurable power system variables. Newton and Hogg [8] have given the design of a linear optimal regulator for the micro-machine system to control both the power input and field voltage. In this paper an effort has been made to identify the best choice of control quantities and a corresponding optimal feedback law which would cover a wide range of power system operating conditions and give an acceptable transient response. Investigations made in this paper correspond to a round rotor machine with conventional wound rotor and are at first confined to excitation control. Later on the investigations are extended to both the excitation and governor control. The state-space models proposed in this paper have state vectors with measurable state variables. An all-state optimal controller has been designed. The transient response of the system has been determined for an initial perturbation of /x8 = 0.095 radians. 2. DEVELOPMENT OF DYNAMIC MODELS OF A POWER SYSTEM
The synchronous machine has been treated as a dynamical system with the main intention of obtaining a model in the state variable form. The application of theorems of modern control theory then takes care of the generation of an optimal controller. The state-space models developed retain Park's transformation and are based on the following assumptions.
28
(i) The influence of damper windings is at first neglected and a (4 × 4) model formed. The influence of damper windings is later considered and (6 × 6) models formed. (ii) P S a and p 6 q terms (transformer voltages) in Park's equations are neglected in all cases. (iii) Single-axis excitation and the voltage regulator are included in the modelling process. (iv) Voltages due to incremental variations in speed are n o t neglected. (v) The governor and prime mover dynamics are n o t considered in the first phase of the investigations. Later on t he y are considered. (Performance equations for a synchronous machine are given in the Appendix §A.2.)
3. THE POWER SYSTEM
3.1. Block diagram of power system Figure 1 shows the block diagram of the p o w e r system considered in this paper. Installed capacity of power station = 1000.0 MW Nominal voltage 13.8 kV Nominal p o w e r factor = 0.9 lag The parameters of the generator and the details regarding the transformers, transmission lines and shunt reactors are given in the Appendix § A.1. Vg
2.1. Formation o f the state-space model (i) Influence o f d a m p e r windings is neglected and the (4 X 4) model is formed: ~[ = AX + B u 1
(1)
where X = [A8 Aco At~f AEfd] t. In the above state vector, the state / ~ f is n o t observable. Hence a transformation is a d o p ted to obtain an observable model with a state vector Z = [AS Azo AV t AEfd] t (ii) Influence of damper windings is considered, the size of the state-space model becoming (6 × 6): X = [A(~ AOd A ~ f A ~ D A ~ Q A E f d ] t
(2)
The state-space model with this state vector is referred to as a flux linkage model. F o u r state-space models are derived with the help o f appropriate t r ans f or m at i on matrices from the flux linkage model. (The f or m ul at i on of the state-space model is given in the Appendix §A.3.)
Lenglh oi Ir'a :7Sc: OXM~
Fig. 1. Block diagram of the power system.
3.2. Operating conditions Operating conditions are specified in Table 1. For all operating conditions: Vg = 1.05 p.u., V-- 1.0 p.u.
4. SOLUTION OF MATRIX RICCATI EQUATION
4.1. General philosophy of matrix Riccati equation The quadratic cost functional has two desirable properties [ 9 ] : (i) it is mathematically tractable; (ii) it leads to optimal feedback systems which are linear. This is the reason that people say that "Quadratic criteria fit linear systems like a glove". We have, therefore, used the quadratic perform ance index
= FZ + Gu 1 (general structure of model) J = J (XTQX + u T R l u l ) d t
Z 1 = [A8 Aco A V t Air A P A E f d ] t Z 2 = [AS Aco A V t A i f A i b A E f d ] t
(3) Z 3 ---- [AS ACO A V t A V t A p A E f d ]
(4)
0
t
Z 4 = [AS Aco AV t A P A P A E f d ] t F o r various models the plant matrices are calculated for different operating conditions.
Q is a positive definite real sym m et ri c matrix. This determines how the instantaneous errors are to be weighted during the transient process. R 1 is a positive definite real symmetric matrix. This represents the weight of control cost with respect to minimising error. u I = - K X (a linear feedback law)
(5)
29 TABLE 1 Operating conditions Operating condition No.
Active power Pg Units at generator terminals connected* (p.u.)
System angle 5 (deg)
1 2 3 4
0.6 0.8 1.0 0.2
76.0 89.0 100.5 30.5
I II II III
Reactive power at generator terminals (p.u.) 0.1598 0.1253 0.3328 --0.022
Reactive power at infinite bus (p.u.) --0.0945 ....0.1992 --0.3995 - 0.1230
*I: three generators connected through double circuit line and three step-up transformers; II: four generators operating through double circuit line and four step-up transformers; III: two generators operating through double circuit line and two step-up transformers.
K = R~-IBT Pz
(6)
P1 is the s o l u t i o n o f the m a t r i x Riccati equation. T h e m a t r i x Riccati e q u a t i o n is given b e l o w [6, 9 ] : P1 -- P I B R 1 1 B T P 1 + Q + P1A + ATp1 = 0 (7) In the s t e a d y state I)1 = 0, and t h e r e f o r e the m a t r i x Riccati e q u a t i o n assumes the following form: ATP1 + P1A - - P I B R { - 1 B T p 1 + Q = 0
(8)
T h e s o l u t i o n o f eqn. (8) is o b t a i n e d b y an iterative p r o c e d u r e o n a digital c o m p u t e r (IBM-360). 4.2. Weigh ting ma trices Since the o p t i m a l c o n t r o l law is sensitive to the choice o f weighting matrices R 1 and Q, these matrices s h o u l d be chosen p r o p e r l y . T h e choice o f R 1 and Q is m o r e or less arbitrary. T o m a k e a reasonable choice o f these matrices, a p r e l i m i n a r y s t u d y was m a d e o n a (6 × 6) m o d e l with a state v e c t o r Z = [A6 Aco AV t A V t A]b AEfd] t, the influence o f weighting m a t r i x Q o n the o p t i m a l c o n t r o l law and transient response being observed. R 1 was assumed t o be a u n i t a r y m a t r i x and the m a t r i x Riccati e q u a t i o n was solved f o r d i f f e r e n t " Q " matrices. T h e o p t i m a l c o n t r o l laws o b t a i n e d were studied. Table 2 indicates the m a n n e r in which the weighting o n d i f f e r e n t states is varied. T h e settling times and the first o v e r s h o o t s are r e c o r d e d in the same Table. T h e transient response curves are p l o t t e d in Fig. 2. T h e results t a b u l a t e d in Table 2 indicate t h a t rela-
tively large weighting on the A5 state (emphasis o n r o t o r angle deviation) helps in arriving at a b e t t e r o p t i m a l c o n t r o l l e r which w o u l d settle the system m u c h faster t h a n o n e with m o r e weighting o n any o t h e r state. In the light o f the above s t u d y the following weighting matrices have been c h o s e n in this analysis. In all cases: R 1 = 1.0 In the case o f the (4 × 4) m o d e l : Q = Diag(50, 10, 10, 10) In the case o f the (6 × 6) m o d e l : Q = Diag(100, 1, 1, 1, 1, 1) T h e o p t i m a l c o n t r o l laws o b t a i n e d are given in Table 3. Figure 3 shows the b l o c k diagram with o p t i m a l r e g u l a t o r [ 1 0 ] . T h e gains m a r k e d as K l l , K12, K13 , etc. in Fig. 3 correspond t o the gains on the various states.
5. SELECTION OF CONTROL QUANTITIES T h e solution o f the m a t r i x Riccati e q u a t i o n is used t o o b t a i n the linear state variable feedback law. T h e result!ng closed-loop system can be described b y X = (A -- BK)X. Eigenvalue analysis is carried o u t and the t r a n s i e n t response is t h e n p l o t t e d . Figure 4 shows the t r a n s i e n t response f o r f o u r o p e r a t i n g conditions with five d i f f e r e n t models. T h e comb i n a t i o n o f c o n t r o l quantities is d i f f e r e n t for each m o d e l . T h e response o f the system was d e t e r m i n e d f o r an initial p e r t u r b a t i o n o f A5 = 0 . 0 9 5 radians. 5.1. State vector Z = [A~ Aco AVt AEfd] t In this m o d e l the influence o f d a m p e r windings has been ignored. T h e t r a n s i e n t
11.54
20.58
54.00
20.41
Q5 = (1, 1, 1, 1, 20, 1)
Q6 = (20, 20, 20, 20, 20, 1)
Q7 = (100, 1, 1, 1, 1, 1)
9.722
43.69
8.841
0.53
1.659
2.247
0.6539
0.6423
1.141
0.6535
Optimal control law
Q4 = (1, 1, 1, 20, 1, 1)
Q3 = (1, 1, 20, 1, 1, 1)
Q2 : (1, 20, 1, 1, 1, 1)
Q1 = (1, 1, 1, 1, 1, 1)
Weighting matrix Q All diagonal matrices
58.95
115.2
41.17
23.75
23.00
87.29
18.64
1.823
3.811
1.262
1.096
0.7262
2.687
--0.693
-1.266
--0.278
--0.41
--9.268
--1.025
--0.216
/~Yt ~ ~kEfdit
0.59
Operating condition: Vg = 1.05, V = 1.0, Pg = 0.8 p.u., Z = [AS Ac0 A V t
Influence of weighting matrix on transient response
TABLE 2
1.47
1.98
1.55
1.28
1.21
1.644
1.174
4.0
5.8
>10.0
>10.0
5.6
>9.0
>9.0
Settling time (s)
1.25 × applied step size
0.936 × applied step size
2.0 × applied step size
1.474 × applied step size
1.58 × applied step size
0.821 × applied step size
1.67 × applied step size
Overshoot
Transient response
Gains reasonable. Overshoot moderate. Settling time lowest
O v e r s h o o t less. Settling t i m e also less, but gains large which discourages choice of this weighting matrix
O v e r s h o o t large. Response less oscillatory but very sluggish
O v e r s h o o t reduced but settling time large
First o v e r s h o o t large
Response less oscillatory but quite sluggish. Gains high
First o v e r s h o o t large
Remarks
5o
31
O, L6
O-lJl
o.o 8
® o .o~
,'~.
i
,
,
,
. .:
~: /,"
i:i,~.:'I" I
'~..-
" ' " ~ i
,v"
-o"
tl'il
o
0= o,,G,,.,.,.,o.,.,,
-o..j
'
(~)
Q =
D]AG(IO0,1
1 1 1 1)
Fig. 2. Influence of weighting matrix Q on transient response.
r- . . . . . . . .
~, X2
~
Oplimal
Regulator
~
_A-_MZ'2_Ry_u'at°:___
,- . . . . r
"I
:
T
7
F,12 I~
_
~ ~~,~ xsXs "[--Z~---'~~
t I
I I I
i "
I
ER~f /
i
Iv, i
I
I
' , Ii
"~ K16 I'-" ---I I Summer .......... z j
L
'
.......
I'
~
r-Synchronous \Machine
l']
[.nfinlte
Bus
Z-Ex~eT" m a~or damping loop
Fig. 3. Block diagram with optimal regulator.
response is found to be oscillatory, the oscillatory process continuing for a relatively long time. The influence of damper windings is then considered and various state vectors are proposed to find the best combination of stabilising signals. The settling times in each case are recorded in Table 3. The quality of the transient response is discussed below.
5.2. State vector Z = [AS A¢~ A Vt Air &P
AEta ] t In this model, the transient response obtained for operating conditions No. 1, 2, and 4 is n o t oscillatory. The settling times are quite reasonable, n o t exceeding 5.0 s for any of the operating conditions. However, when the system is operating at Pg = 1.0 p.u. and then is subjected to a small perturbation, it
0.6 0.8 1.0 0.2 0.6 0.8 1.0 0.2 0.6 0.8 1.0 0.2
1 2 3 4
1 2 3 4
1 2 3 4
Z = [/k~ A y t Aif ~l~ ~ E f d ] t
Z = [A~ /~G9 AV t A~rt ~ f d ]
--4.13 --8.97 --15.74 --0.11
14.44 20.41 29.89 3.68
11.3 --0.5 37.18 4.63
--12.12 --20.61 --34.41 0.0184
14.98 27.84 35.12 --85.02
=
--0.41 --0.353 --1.29 --0.25
0.817 0.53 --0.10 --0.40
1.32 8.98 --5.3 0.002
0.07 --0.21 --1.179 --0.116
--0.589 --0.224 --0.365 --18.76
O p t i m a l c o n t r o l law K R-1 BTp
7.37 0.69 --10.8 9.58
48.37 58.95 76.47 32.46
40.44 19.54 85.80 15.2
13.01 1.34 --19.0 17.31
64.92 67.94 69.09 185.2
22.36 30.68 49.95 11.61
1.12 1.82 2.72 0.46
--0.43 --2.57 1.59 --0.04
5.7 7.22 10.15 3.12
3.389 3.366 3.379 3.966
0.23 0.27 0.37 0.25
--0.33 --0.69 --1.19 --0.09
--2.3 --13.4 9.15 --0.2
24.97 37.35 66.98 7.05
1.34 1.48 1.62 1.2
1.4 1.5 1.6 1.3
1.45 2.277 1.74 1.21
1.37 1.46 1.60 1.205
3.5 5.0 >I0.0 3.0 3.5 4.25 3.25 3.25
2.3 2.6 4.1 3.75
3.5 3.8 10.0 2.3 3.5 1.75 2.75 3.75
2.3 2.25 3.10 3.25
4.8 3.6 4.5 Long t i m e
4.8 6.5 >10.0
Settling t i m e w i t h s u b o p t i m a l law (s)
4.8 3.6 Long t i m e 3.6
4.8 5.8 6.0 4.8
Settling t i m e w i t h o p t i m a l law (s)
*The s u b o p t i m a l law c h o s e n is t h e same as t h e o p t i m a l c o n t r o l law c o r r e s p o n d i n g to o p e r a t i n g c o n d i t i o n No. I in each case.
t
0.6 0.8 1.0 0.2
1 2 3 4
Z = [A5 ACO A V t &if L~P z~LEfd]t
Z = [A(~ A(.O/kyt ~ P ~ f d ]
0.6 0.8 1.0 0.2
1 2 3 4
Z = [A5 ACO A F t L2kEfdIt
t
Pg (p.u.)
Operating condition
No.
State v e c t o r
Optimal and s u b o p t i m a l c o n t r o l laws a n d settling t i m e (for all o p e r a t i n g c o n d i t i o n s Vg = 1.05, V = 1.0 p.u.)*
TABLE 3
¢.D bO
33
P6
=0"6
,:#
_
J I
# I
i"
°"
p<3 +_ 1 - o .
i!
f
ii /l i t/
;
>.o4
,, !
O.0
•
!
•
+
.
,X',.Jlr" l
i~i" i\ ,
,
r\
e
I
~
i i
i
.
.
.
.~.
.
'
l\lil# i'v''l
o.o~
.
~
•
i
__
i
'.,, v
•
~1',;
i
,
,,. __
i !
.......
~.:,--~.+ , , o
.,v,+~,,,:A,~+:t
"= z:.,~,A,~.,,,,,.,,~>~
I~-+^,-,,,Lv'~,,:A,~o''-"o.2~6 ~ o.ssE=
~1
~= L+'t"o+,+'v~, +,P, ",~m.t~ +
I
o. ---m,..
i.
2. "I"IME,. IN I I £ C O N ~
3.
Fig. 4. T r a n s i e n t r e s p o n s e o f d i f f e r e n t m o d e l s .
4'.
~
o.
i. ~
"l't~
z. 3. iN 43s.cot4~,~
~i.o
34 is found that the transient response is quite oscillatory and the transient process would continue for a long time. 5.3. State vector Z = [AS Ao~AVt/XifAP AEfd] t In this model, the transient process is characterised by large deviations with a maxim u m overshoot of 2.5 times the applied step size for an operating condition corresponding to Pg = 1.0 p.u. The settling times are quite low, but as the response in general is oscillatory with remarkable overshoots, this combination of stabilising signals is unacceptable. S t a t e v e c t o r Z = [A5 A~o A V t A V t /~P AEfd] t In this model, the response in general is oscillatory but the m a x i m u m overshoot is limited to 1.6 times the applied step size corresponding to operating condition No. 3 (Pg -- 1.0 p.u.). The oscillatory process terminates quickly. 5.4.
5.5.
State
vector
Z
=
[AS Aw AVt AP A/~
AEfd] t The transient response obtained in this case is characterised by monotonic convergence for operating conditions No. 1 and 2 and is slightly oscillatory in nature for operating conditions 3 and 4. Settling times are very much reduced. The profiles of the transient response drawn in Fig. 4 are indicative of these observations. It is, therefore, concluded here that the combination of stabilising signals given by this state vector will give the best transient response with minimum settling time over a wide range of operating conditions. Therefore, the best group of control quantities which should be adopted for stabilising the synchronous machine is given by a state vector
should be made to identify the best choice of optimal control law which will give reasonably good performance over the wide range of operating conditions. To explore the possibility of using the same controller to cover a wide spectrum of operating conditions, the optimal control law obtained for operating condition No. 1 (P~ = 0.6 p.u.), which is the median condition between the extremes of heavy load and light load, has been tested for other operating conditions. Table 3 gives the information about the suboptimal law and the settling times obtained with the application of this law. Using the models proposed in this paper, eigenvalues obtained with the control law chosen have negative real parts, indicating that the suboptimal controller can maintain the system stable. This analysis clearly indicates the adoption of the same controller to cover a wide range of operating conditions. Figure 5 shows the transient response obtained with optimal and suboptimal control laws in the case of a (6 X 6) model with state vector Z = [AS Aw AV t APAPAEfd] t. The transient response curves indicate that the suboptimal control law gives a transient performance almost identical to the one with an optimal control law, except for the fact that the settling time is increased to some extent as compared with the settling time with the respective optimal law. In the case of a (6 × 6) model, with state vector Z = [A5 ~w A V t A i f A P A E f d ] t, it has been observed that the suboptimal controller gives a better transient response with a lower settling time for operating condition No. 4 (Fig. 6). This indicates that the optimal control law obtained with definite weighting matrices need not be the best, though it is optimal. It is possible to have a better transient response with a lower settling time with some other control law.
Z = [AS Aw AV t AP APAEfd] t 7. INFLUENCE OF LOCAL LOAD 6. POSSIBILITY OF ADOPTING THE SAME OPTIMAL CONDTROLLER TO COVER A WIDE RANGE OF OPERATING CONDITIONS In practice, it is difficult to adjust the gains of the optimal controller with change in operating conditions. Therefore, an effort
When the local load is taken into consideration and an optimal controller is designed, it is observed that the quality of the transient response is improved, particularly in terms of the settling time (Fig. 7). It has also been observed that the feedback gains for all states are less than the correspon-
35
pe-o.§
_P¢=o-8
8 u 8 - oP-rIMAL F~r~wAcg,
,..f,.Cv.~.)
0.o@ II '~
I
/
',
O ]Wlr"I I ~ A J.
,,
0.04
,
~.o
\
L
,,
[
~,
;
,
,, |
t l
t
'
• #
,
/
t
I t
-0.1~
I
I #
I
I
i
t
i I
I
'
/
/ i
! I
i
I
~ t two'
-o-0|
(v.R.)
Per I-o
PG= O-~
~rgn- oPqr'IMAA. F£J~BACK
~u 8-01~'IMAL F £r-~'SAC K
p~A- F~'F,DB AcK
~,....,.~ e"
8"""t ,,.\
i
o.
I
J.
TIM,F.
o p ' r l MAC.
b
"-..'"
Z.
~.
.~,
H',I $£~0/',,t,V~
o.
l. _--- " n M L
P_ ~
K
4.
SLooP.
Fig. 5. T r a n s i e n t r e s p o n s e w i t h o p t i m a l a n d s u b o p t i m a l c o n t r o l laws f o r p o w e r - p o w e r derivative m o d e l .
ding gains in an optimal controller which is designed without considering the local load.
Table 4 gives an idea of the changes in the optimal law when the local load is considered.
36 PG :0.2
t
0 10 ,~
0.08[ /fOptimal feedbqc~< 9.04 ~
~.
008
powz~
// / / / /~
AT GZN
TZ,eMIN.4L
= • ¢p~
No Local iood
fs:35 sec OsciltQtor\
I.
~Sub oplinl@: !eedLq, k
I
~OOS
o
o
o
c 0.0l, ,-,o <]
002
008 [
0
1 2 3 4 I ime in sec Fig. 6. Transient response with optimal and subo p t i m a l c o n t r o l laws for current d e r i v a t i v e - p o w e r derivative m o d e l .
P[=0.1. P~
[ fs:2.3 sec
t
1I, ! Monotonic /[ it Local[ood /1 1 ; PL=O2'Ts:251~ec Osci[Iotory ~J//
\\\
~
--
--
-
1 2 3 L Time in sec Fig. 7. Influence o f local load.
TABLE 4 I n f l u e n c e o f local load Operating c o n d i t i o n Pg (p.u.)
Local load PL (p.u.)
F e e d b a c k law Z = [/k~ /kCO A V t z~P ~/~ A E f d ] t
1.0 1.0 1.0
0.0 0.2 0.4
--15.74 ---14.76 --13.68
The above analysis clearly indicates that while designing an optimal controller for a synchronous power system, due consideration should be given to the presence of local load. If this factor is ignored for some reason, the designer will have to propose the optimal controller with relatively higher gains.
--1.291 --0.845 --0.635
--10.80 --9.565 --8.824
49.95 40.53 36.36
0.37 0.36 0.28
1.62 1.57 1.57
8. C O M P A R I S O N O F P E R F O R M A N C E O F A N ALL-STATE FEEDBACK CONTROLLER WITH VOLTAGE REGULATOR
With any of the all-state feedback controllers discussed above, the transient response is relatively much superior to the transient response obtained with the voltage regulator only. This can be seen in Fig. 5. It is also seen from Fig. 5 that the all-state feedback with optimal or suboptimal controller extends the stability limit beyond that obtained with only the voltage regulator in action.
9. I N F L U E N C E O F G O V E R N O R A N D T U R B I N E DYNAMICS ON TRANSIENT RESPONSE
~ime
Stele
: Isec/mch,
AmplHude
O 1
v/inch
Fig. 8. Governor and turbine response to unit ster input (TRH = 8.0 s).
The mathematical model given in Fig. 7-E of the IEEE Committee's Report [11] has been simulated on an analogue computer. The response curves of mechanical power {Pro) v s . time (t) have been obtained for different reheat time constants (TRH). Figure 8 shows the nature of the response obtained. Since the
37
response curve is reasonably exponential, it is considered adequate to represent the governor and turbine dynamics by a single time constant to reduce the complexity of the state-space model. The combined equivalent time constant of the governor and turbine has been taken as Teq = 0.5 s which would correspond to the closing stroke, and T~q = 4.0 s which would correspond to the opening stroke. The order of the state-space model is increased by one, becoming a (7 × 7) model, when the governor and turbine dynamics are considered. The state vector is Z = [AS A ~
AV t APAPAEfd
K=
0.607
.~004 •~- 0
1
2
3
4
- 004 - 0.08
l
~(~)
PG :0.~
~
PG : 1,0
_0 . 0 4 ~
The plant matrix and the control distribution matrix have been calculated for different operating conditions. The suboptimal law is used for assessing the eigenvalue spectrum and the transient response of the system. The following control law is used: --0.417
fs =44 sec
ATm] t
The control vector is u = [ul, ug]t
I[--4.129 3.657
PG=O.8 0.08
7.374 0.0
22.36 0.0
- 00
1.345
0.0
0.0
(i) Teq = 0.5 s The transient response obtained with the (7 × 7) model is m o n o t o n i c for all operating conditions. There is remarkable improvement in the quality of response as compared with the response obtained only with the excitation control for operating conditions No. 3 (Pg = 1.0 p.u.) and 4 (Pg = 0.2 p.u.). For operating conditions No. 1 (Pg = 0.6 p.u.) and 2 (Pg = 0.8 p.u.), the response obtained with excitation control only and with excitation and governor control are more or less alike. (ii) Teq = 4.0 s With the increase in the equivalent time constant, it has been observed that, for operating conditions No. 1 (Pg = 0.6 p.u.) and 4 (Pg = 0.2 p.u.), the response is still
~
Fig. 9. Transient response with conventional controls (curve 1);excitation and governor control, Teq = 0.5 s (curve 2); excitation control alone (curve 3); excitation and governor control, Tea = 4.0 s (curve 4).
0.235
This control law maintains the system stable under all operating conditions considered. The profiles of the transient responses are given in Fig. 9.
5
--11.21]
!
11.40]
monotonic, but it becomes oscillatory for operating condition No. 3 (Pg = 1.0 p.u.). In this case also the transient process terminates after a few oscillations only. The above analysis suggests that the governor and turbine dynamics must be considered when designing an optimal or suboptimal controller for a synchronous power system as also for determining the transient response of the controlled power system.
10. CONCLUSIONS
(i) The choice of weighting matrix Q is important to arrive at a better and physically realisable control law. (ii) The choice of the quadratic performance index which is most often used does n o t guarantee the best transient performance. (iii) The quality of the transient response, particularly in terms of the settling time, is improved considerably with the inclusion of
38 stabilising signals in the form of active power and rate of change of active power. (iv) The quality of the transient response, particularly in terms of the settling time, is improved and the gains of the stabilising signals are reduced as the local load is increased. (v) The governor and turbine dynamics should be considered when designing an optimal controller for a synchronous power system as it influences the transient response considerably. (vi) It is possible to obtain a suitable control law to ensure stable operation of a p o w e r system, and also good transient performance over a wide range of operating conditions of the power system. A proper combination of control quantities together with the proper choice of weighting matrices can generate a control law which will ensure considerably improved transient performance of the power system over a complete range of operating conditions.
TA
KE TE KF TF
Control system X Z A F B G uI M K R1 Q
NOMENCLATURE
P1
Synchronous machine and transmission system Vd, Vq Vf, if
Efd S d, ~q ~f
~D, ~Q id, iq ~o ~0 R +jX V or V o Vg or V t Tm P, Pg
direct axis and quadrature axis voltage field circuit voltage, field circuit current air gap line open-circuit voltage direct axis and quadrature axis flux linkages direct axis field flux linkages direct axis and quadrature axis damper flux linkages direct axis and quadrature axis components of terminal current instantaneous angular frequency (rad/s) synchronous angular frequency (rad/s) equivalent series impedance of transmission system infinite bus bar voltage generator terminal voltage mechanical torque torque angle (rad) active power at generator terminals
Excitation system KA
exciter amplifier gain
exciter amplifier time constant (s) exciter gain exciter time constant (s) gain of stabilising loop time constant of stabilising loop (s)
A
state vector of flux linkage model state vector of transformed model system matrix of X-model (n X n) system matrix of Z-model (n X n) control state distribution matrix of X-model (n X m) control state distribution matrix of Z-model (n X m) control vector transformation matrix optimal gain matrix (m X n ) positive definite real symmetric matrix (m X m) another positive definite real symmetric weighting matrix (n X n) positive definite symmetric matrix obtained as the solution of matrix Riccati equation (n X n) prefix denoting a linearised variable
ACKNOWLEDGEMENTS The authors would like to record their thanks to the Government of Maharashtra and the Government of India for sponsoring the candidate under the Quality Improvement Programme in the year 1976. The facilities extended by the Visvesvaraya Regional College of Engineering, Nagpur, are gratefully acknowledged.
APPENDIX A. 1. System parameters The parameters of the generator correspond to the parameters of the micro-alternator which will be used at Visvesvaraya Regional College of Engineering, Nagpur, for experimental work.
39
M icroalterna tor p arame ters xd
= 2.013p.u. 0 . 2 7 3 2 p.u. Xd" = 0 . 1 8 4 p.u. xq = 1 . 9 1 3 p.u. x q " = 0 . 2 3 2 5 p.u. Xal = 0 . 1 1 3 p.u. x~ = 2 . 0 7 5 p.u. X d '
Ra R~ co o Td0'
0.006p.u. 0 . 0 1 2 p.u. 3 1 4 . 0 radians 5.0 s Tdo"= 0 . 0 0 9 1 s n Tqo = 0 . 0 7 3 s H = 2.057 s
----"
= = = =
Control complex parameters K A = 50.0 K E = 1.0 g F = 0.1
TA TE TF
= 0.05s =0.05s = 0.2 s
Details o f step-up transformer F o r each step-up t r a n s f o r m e r Nominal capacity = 300.0 MVA T r a n s f o r m a t i o n ratio = 1 3 . 8 / 4 0 0 kV Leakage r e a c t a n c e = 0.15 p.u.
Details o f transmission line T h e transmission lines consist o f t w o 4 0 0 kV circuits. L e n g t h o f line = 755 km R e a c t a n c e o f line = 0 . 3 0 5 ~2 p H -1 k m -1 Surge i m p e d a n c e = 2 9 5 . 0 ~2
Shunt reactors O p e r a t i n g c o n d i t i o n No. 1 (Pg = 0.6 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 3 2 9 4 p.u. O p e r a t i n g c o n d i t i o n No. 2 (Pg = 0.8 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 1 6 4 7 p.u. O p e r a t i n g c o n d i t i o n No. 3 (Pg = 1.0 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 1 6 4 7 p.u. O p e r a t i n g c o n d i t i o n No. 4 (Pg = 0.2 p.u.) I n d u c t i v e s u s c e p t a n c e = 0 . 2 9 8 8 p.u.
A.2. P a r k ' s e q u a t i o n s f o r a s y n c h r o n o u s m a c h i n e have the following f o r m [3, 12, 13] : 1
09
Vd -
pt# d . . . . °90
t~q - - R a i d 090
1
09
Vq = ---P~d
+ --
090
~ d .... R a i q
090
1 V~ = - - p ~
+ R~i~
(A1)
090
1 0
-
0
=--p~Q 090
090
P ~ D + R D iD
1
+RQiQ
The t o r q u e e q u a t i o n s are p5 = 0 9 - - 0 9 0
(A2)
090 ( T m __ Te ) P09= 2H where Te = ~diq - - Sqi d
A s s u m i n g a solid-state e x c i t e r o f negligible time c o n s t a n t and a voltage r e g u l a t o r with single t i m e c o n s t a n t [ 7 ] , we get AE~d -
--K A (1 + P T A ) (Vt - - Vre~ - - u l )
R e a r r a n g e m e n t o f this e q u a t i o n will give us 1
PA E~d -
KA
TA AE~d -- -~A (Vt - - Vre~ - - u l )
(A3)
40 F l u x linkage equations:
id
~d
= --Xd
+ Xad if + Xad
~q
= --Xq iq + Xaq iQ
~f
=--Xadi d + x f i f
iD (A4)
+Xad/D
~/D =--Xadid +Xadif +XDiD ~O
= --Xaq i q +XQiQ
A.3. Formulation of flux linkage model E q u a t i o n s (A1) - (A4) are linearised and t h e n arranged in the following m a t r i x f o r m : A5
A~f
=
/~]D
-0
1
0
0
0
0 -
0
0
d23
d24
d25
0
0
0
d33
d34
0
d36
0
0
d43
d44
0
0
A~ D
0
0
0
0
ds,~ 0
A~Q
0
d62
d65
d64
d65
i
AS Aw +
d66
0
0
m
0
21 ]'/22
Ih31
0
~41
0
0
h52
0
~
+
0 [ul]
(A5)
61 h62 L
"J
or = D X + HI + Bu 1
(A5a)
T o eliminate the c u r r e n t v e c t o r f r o m eqn. (A5), f r o m Park's e q u a t i o n s we can o b t a i n AV d +RaAi d-xq''Aiq=-o4A$
1 q + - - (xq"iq--O4~Q)ACz (DO (A6)
1 AVq +RaAi q +Xd"Ai d = a 2 A ~ f + O3A~ D + - - ( o 2 ~ COO
+ a3~D--Xd"id)A~
Considering one m a c h i n e c o n n e c t e d t o an infinite bus as s h o w n in Fig. A1 [ 1 ] , the e q u a t i o n s r e p r e s e n t i n g the transmission s y s t e m can be w r i t t e n as: AVd = (K1 "7 + K2"o)Aid + (K2"~ - - K 1 .o)Aiq -- Vo(K2sin5 - - K l c o s S ) A 5 (A7) AVq = (K 1.0 - - K 2 . ~ ) A i d + (K 2.o + K 1.~)Aiq - - Vo(K2cos5 + K 1sinS)AS w h e r e A V d and A Vq are the d-axis and q-axis c o m p o n e n t s o f Vg as viewed f r o m the e x t e r n a l circuit.
VI
V2
Vo
X~
Fig. A1. One machine connected to an infinite bus.
41
From eqns. (A6) and (A7), the expressions for current I can be written as A8 A~9
I:
/dl = FCll
C12 C13 C14 C15 C161
A~f
iq]
C22 C23 624
ASD
(A8)
[621
625
626~
ASQ
&El d or
I = CX
(A8a)
From eqns. (A5a) and (A8a) :K =
(D + HC)X
+ Bu 1
or = AX + Bu 1
Expressions for the various coefficients are given below: d23 -
d24 -
d25 -
d3 4 -
dss
¢Oo (Xd--Xd')(Xd"--X~,) 2H (xd -- Xal)(Xd' -- Xal) 090 Xd t --Xd" 2 H Xd I - - Xal
1
(Xd' --Xd")(Xd --X~,)
Too' (Xd' Td0,
--
d55 -
i +
--
x~,)
(Xd' -- X j ) ( X d (xd'
-
-- Xd')]
~,)(x~' -~,~ J
TdO' X d - - Xd'
1
Xd __ Xdt
Td0" Xd ~ Xal
1 Td0"
1 Tq0"
Vo
d63 -
X~,)(Xd'
X d ~ Xal
1
d44-
id
2 H Xq - - Xal
1[
d43 -
iq
CO0 Xq - - X q "
-
d36 -
iq
VQ
) KA
K A (Xd--Xd')(Xd"--Xal) VtTA (Xd __ Xal)(Xd, __ Xal) Vq
(A9)
42 KA
d64 -
Xd' - - Xd" Vq
Vt TA Kh
d65 -
X d ' __ Xal
Xq - - Xq " Vd
V t T A Xq - - X a l 1
d66 =
__ _ _
TA
rt
K s
-G) 0
-
Xq"iq
--
~ Q
Xq - - Xal
K 9 = ( 0 2 ~ f + 0"3~ D --Xd"id)/CO 0 0" 1 = C O o / 2 H
(X d - - X d ' ) ( X d " - - Xal ) 02 (X d ' - - X a l ) ( X d - - X a l ) Xd' - - Xd" 03 X d ' - - Xal tr
Xq - - X q 0 4 -Xq - - Xal
h21
pr
= .O'I[(X
~t
q
- - X d )iq - - o'4t~Q ]
(X d - - Xa l ) ( x d ' ' - - X a l )
Tdo,(X d . . . . Xal )
h3a = __
h 4 1 --
(X d' - - Xal ) Td0" KA
/261 -
(RaV d + Xd"Vq) Vt TA rl
h22 = --Ol[(X q
--Xd")i
d + @ f o 2 + @DO3]
1 h52 -
,, ( X q - - Xal )
Tqo KA h62 -
Vt TA
(Xq" V d - - R a VQ)
= 1 + RG -- XB = XG
+ RB
7? = R - - ( X t X G
+ XtBR)
o =(X t +X+XtRG
)-xtxB
43 D e t e r = (K1 ",7 + K2"a + Ra) 2 - - ( K 2 " 7 --K1 "0--xq")(K, " 0 - K2"? + Xd" ) 1 + R-G - - X ' B
X-G + R ' B K2 -
0/2 + ~2
K 3 = Vo(K2sin5--K 1 cosS)
K4 = Vo(K2 cos 5 - - K 1 sinS) K5 = (K1 ",7 + K2"o + R a ) / D e t e r K 6 = (K 1 .a + Xq" - - K2 ",7)/ D e t e r K v = - - ( K 1 .o + Xd" - - K 2",7)/Deter
C n =KsK3 + K s K 4 K5 C12
--
O)o
C13 -- (/2
..
(Xq lq - - O 4 ~ Q ) +
Ks O9o
( 0 2 ~ f + O3~D - - X d " i d )
"K6
.C14 -- 03 "K6 C15 = - - 0 4 .K 5 C16 = 0.0 C21 -- K7 "K3 + K~ "K4
K7 C22 C23 =
090
,,.
(Xq lq - - 0 4 ~ q )
K5
+ ~ ( G 2 " ~ f + O 3 " ~ D --Xd"id) 090
o2"K5
C2 4 = 0 3"K5 C2 5 --
--04
.K 7
C26 = 0.0 I f the stabilising l o o p is c o n s i d e r e d (refer t o Fig. 3) [ 1 0 ] , KA/T A in eqn. (AS) should b e replaced by
(KAKE)/(T A + T~ + TF + KAKEKF)
Formation o f appropriate transformation matrix (voltage-power-power derivative model) Let Z = MX Therefore
(A10)
44 2 = MX
(All)
= M(AX + Bu 1) = M(AM-lZ + Bul) _- M A M - l Z + MBu 1 = FZ + Gu 1
(A12)
where
X:
[AS A(~) A~f A~D A~Q AEfd] t
Z = [AS Ao) A V t A P A [ g A E f d ] t T h e terminal voltage
z~V t - Vd AV d + -Vq - AVq
Wt
(A13)
Yt
= m31A5 +m32Ac~ +m33A@f +m34A@D + m35A@Q +m36AE~d where
/7/31 --
Vd
,, Vq ,, (Xq C21 - - R a C l l ) + ~ (--x d e l l --RAG21 )
Vd Vq /7232 = --~t (Ks +xq"C22 --RAG12) + ~tt (K9 --Xd"C12 --RAG22) Vd Vu m33 = ~tt (--R~C13) + ~tt (a2 --Xd"C13) Vd
rrt34
--
Ytt
Vd
(--RaC14)
+
.
Vq "-Yt (03 --Xd"C14)
Vq
m35 = Vtt (xq C25 -- at) + -Vt (--RaC25) ma6 = 0.0 Consider an e x p r e s s i o n f o r electrical p o w e r o u t p u t : P = V d id + Vq iq
(A14)
Linearising eqn. (A14), AP = VdAi d + idAV d + VqAiq + iqAVq
= m41A5 + m42Aco + m 4 3 A ~ f + m 4 4 A ~ D + m 4 5 A ~ q +m46AEfd where m41 = V d C l l + VqC21 + i d ( x q " C 2 1 - - R a C l l ) - - i q ( X d " C l l +RaC21) m r 2 = VdC12 + /d(Xq"C22 - - R , C 1 2 + K s ) + VqC22 + iq (~Xd"C12 - - R ~ C 2 2 + Kg)
m43 = YdC13 + id(--RaC13 ) + iq(a 2 --Xd"C13) m44 = VdC14 + id(--RaC14} + iq(a~ --Xd"C14 )
(A15) (A16)
45 m45 = id(Xq"C25 - - 04) + YqC25 + iq(--RaC25)
m46 = 0.0 Differentiating eqn. (A16) :
+
+
+
+
= m51A5 + m52Aw + m s a A ~ f + m54A~D + rn55A~b Q + rn56AEfd
(A17)
where m51 = m42A21 + m43A31 + rn44Aal + m a s A s l m52 = m 4 1 + m 4 2 A 2 2 + m 4 3 A 3 2 + m 4 4 A 4 2 + m 4 5 A 5 2 m53 = m42A23 +m43A33 + m 4 4 A 4 3
rn54 = m42A24 + m43A34 + rn44A44 m55 = m42A25 + m45A55
ms~ = m43A36 ( H e r e A21 , A22, etc. are the elements of matrix A in eqn. (A9).)
Now the Z-vector can be co-related with the X-vector in the following manner:
1 Im
Aco ' AVt
AP. AP
10
=
31
0
0
0
0
0
1
0
0
0
0
m32 m33 m34 m35 0 m42 rn43 m44 m45 0
[i41 51 m52 m53 msa m55 m~ 0
.AEf,
0
0
0
1
A5
1
A~o
I
A~,f A~D I
(A18)
.AEfdJ
Formulation of a (4 × 4) model is similar to that of a (6 X 6) flux linkage model. The procedure delineated in getting a voltage-power--power derivative model can be adopted to obtain any other equivalent model.
Nomenclature
Xafd Xakd
id , iq iD, iQ X d ~ Xq
XD XQ Xf
direct axis and quadrature components of terminal currents direct axis and quadrature axis damper currents direct axis and quadrature axis reactances of synchronous generator damper circuit reactance on direct
Xad
Xaq
axis
R a
damper circuit reactance on quadrature axis total reactance in the field
R~ Xd r
mutual reactance between armature and field windings mutual reactance between armature and damper windings mutual reactance between generator field and d-axis armature winding (all rotor-stator and r o t o r - r o t o r reactances have been assumed to be equal, i.e. Xatd = Xakd = Xtkd) generator quadrature axis magnetising reactance armature resistance field resistance direct axis transient reactance of synchronous generator
46 r~
Xd Xq
It
Td0' Td0" Tq0" H
To
d i r e c t axis s u b t r a n s i e n t r e a c t a n c e o f synchronous generator q u a d r a t u r e axis s u b t r a n s i e n t reactance of synchronous generator d i r e c t axis t r a n s i e n t o p e n - c i r c u i t time constant d i r e c t axis o p e n - c i r c u i t s u b t r a n s i e n t time constant q u a d r a t u r e axis o p e n - c i r c u i t s u b transient time constant inertia constant electrical torque
REFERENCES 1 J. M. Undrill, Power system stability studies by the method of Liapunov: state-space approach to synchronous machine modelling. IEEE Trans., PAS-86 (1967) 791 - 801. 2 J. H. Anderson, Matrix methods for the study of a regulated synchronous machine. IEEE Proc., 57 (1969) 2122 - 2136. 3 M. A. Laughton, Matrix analysis of dynamic stability in synchronous multi-machine system. Proc. Inst. Electr. Eng., 113 (1966) 325 - 336.
4 S. Raman and S. C. Kapoor, Synthesis of optimal regulator for synchronous machine. Proc. Inst. Electr. Eng., 119 (1972) 1383 - 1390. 5 Y. N. Yu and B. Habibullah, Improving supplemental excitation control design using accurate model. IEEE PES Winter Meeting, New York, Jan. 28 - Feb. 2, 1973, C-73, 219-3. 6 C. E. Fosha and O. I. Elgerd, The megawattfrequency control problem: a new approach via optimal control theory. IEEE Trans., PAS 89 (1970) 563- 577. 7 B. Habibullah and Y. N. Yu, Physically realisable wide power range optimal controller for power systems. IEEE Trans., PAS-93 (1974) 1498 - 1506. 8 M. E. Newton and B. W. Hogg, Optimal control of micro-alternator system. IEEE Trans., PAS-95 (1976) 1822 - 1833. 9 M. Athans and P. L. Falb, Optimal Control, McGraw-Hill, New York, 1966. 10 IEEE Committee Report, Computer representation of excitation systems. IEEE Trans., PAS-87 (1968) 1460 - 1464. 11 IEEE Committee Report, Dynamic models for steam and hydro turbines in power system studies. IEEE Trans., PAS-92 (1973) 1904 - 1915. 12 C. Concordia, Synchronous Machines, Wiley, New York, 1951. 13 B. Adkins, The General Theory of Electrical Machines, Chapman and Hall, 1959.