Title
Author(s)
Stabilization of Synchronous Machine through Integral Action
Yamashita, Katsumi; Taniguchi, Tsuneo
Editor(s)
Citation
Issue Date
URL
Bulletin of University of Osaka Prefecture. Series A, Engineering and nat
ural sciences. 1980, 29(1), p.1-9
1980-10-31
http://hdl.handle.net/10466/8325
Rights
http://repository.osakafu-u.ac.jp/dspace/
1
Stabilization of a Synchronous Maehine through
Integral Action '
Katsumi YAMAsHITA* and Tsuneo TANiGucHI**
(Received June 16, 1980)
Integral control proposed by C.D. Johnsoni) takes a feedback fbrm of all state'
variables. In power systems, however, it'is diMcult to measure all state variables,
and then his controller is not practically acceptable. It is the purpose of the present
paper to design an integral controller based on quantities easily measurable at the
generator location, using the a!gorithm of C.D. Johnson for optimization. This
controller will be physically realizable and practical to implement.
List o￡ Prineipal Symbols
M ==machine (generator) inertia constant, second. .
M. ==machine (induction motor) inertia constant, second.
xd ==p.u. direct axis synchronous reactance.
x2 ==p.u. direct axis transient reactance.
X'
.
===Xll-X?2!X22. ''
Ei =:p.u. internal voltage behind transient reactance x2.
.
jE3 = p.u. internal voltage behind transient reactance x'.
Vt =p.u. machine (generator) terminal voltage.
TSo ==direct axis field time constant, second. ･
7b ==xza1(worza).
too ==2nj5 (A=60Hz).
A ==incrementaloperator.
1. Introduction
In recent years, considerable works2)'3) have been done on the application of optimal
control theory to the design of regulators for linear machine models. A great part of
the previous works, however, are based on proportional feedback of the state variables,
and the equilibrium condition cannot be maintained by such a method of feedback, if
a constant external disturbance is present.
C.D. Johnsoni) Proposed the integral eontroller to elfninate this defect, but his controller was not usually accepted in power systems, because it assumed that all state variables of the systems were avaiiable for control.
This paper presents the method for designing on integral controller based on only
op Spe￠iality Student, Department of Electrical Engineering, College of Engineering.
*ts Department of Eie￠tricai Engineering, College of Engineering. ･ '
Katsumi YAMAsHm and Tsuneo TA}q!GueHi
2
those variables that are easily measurable at the generator location, using the algorithn
of C.D. Johnson for optimization. The proposed controller is applied to sitabilization
of a synchronous generator with dynarniC'' behavior of load. The results are then compared with those given by the optimal control in the conventional linear regulator problenf), and the proposed controller is found to be usefu1 practically in power systems.
2. Suboptimal Contro1
Consider a linearized power system model described by the state variabie forrn
'
'
.t t
'
'
:li ==
Ax+Bu+Dw,
(1)
y ==( t,
where x is the n-dimensional state vector, g is the m-dimensional measurable output
vector, u is the r-dimensional control vector, w is a p-dimensional constant disturbance
vector, and A, B, C, Dare constant matrices of appropriate dimensions. It is assumed
that B has the maximal rank r and the range of D is contained in the range of B, Further the system description is assumed to be completely controllable.
'
Optimal control can be achived by minimizing a quadratic cost' function
of the form
'
tt
lu === E[i,OO(xTQx+ddUt'R:Uildt], (2)
subject to the system control constraint Eq. (1), where E is the expectation operator
over the random initial state, Q is an nxn symmetric positive semi-definite constant
matrix and R is an rxr symmetric positive definite constant matrix.
Now, to introduce integral action on the input variablesi), we rewrite Eq'. (1) as
du
-== Ax+Bu+Brw, -
y == Cx ,
where
r = (BTB)'iBTD,
and define a new state variable vector Vas
The fo11owing augmented system may then be obtained:
f, [u+Xrw] me [3 g][u+Xr.]"[9]V' (s)
where 1 is the r × r identity matrix.
Next, assume the new variable in Eq. (5) as fbllows:
Stabilixation ofa S-vnehronotts Mbchine, through intagral Action 3
V==---Ky,' ･, .. -.
(6)
where Kis an rxm feedback gain matrixl '" '' ''' ' '' '･
'' On the basis of this assumption, and by defining the augmented state vector
'' i' Z A" '[.+Xri ] '.'. .. . , ,' ..･' 1,',1 '' '' '(7.)
the, plant equation becomes ,
#---(A-DKb)z, .,. .,.,, ,.(s)
'
and the perforrnance index is also expressed as .,,'
.'
t ttt1 ,'tt .
/, . , . , '
pl -k E[j,oozr(o+6TKTRKe)zdt], (g)
where
･A=-[3 g],.Bl[9],
b=-[c'' o], o=-[Q. O.].
. t .t .t
'
Further, by substituting Eq. (8) into Eq. (9), the performance index becomes
Pl'-E[Z(O)T.P40)], (10)
where P is the solption of the Lyapunov Matrix equation
(A-hKO)Tp+p(A-bkb)+bTKTRKe+d =- o. (ii)
On assuming now that Z(O) is uniformly distributed over the surface of a hypersphere5), the performance index becomes
where tr P is the sum of the diagonal terms of P.
Therefore K for the suboptimal controller must be determined in such a manner
that P can be minimized subject to the constraint given by Eq. (11). For minimizing
IV, the Hamiltonian .
tt
L = tr P+ tr[S' {(A- BKTC )TP+?(A--- BKC) .
AA
+CTKTRKC+Q}] .. (13)
is chosen and the necessary conditions for minimization of L
60KL =o, 6apL =o, aOsL ==-;o, '". (i4)
yield the fo11owing solutions: '''''
tt tt
't.'t.
4 .Katsymi .YAMAsHiM and T$uneo TANIGucEii'
A AAA ,
K == R-iBTpsaT(cscT)'i
AAA AAA
(A-BKC)S+S(A-BKC)T+I==O,
,, .
. t..t
AA
AA
(A-BKb)p+p(A-BK6)+bTKTRKb+di-=o.i-.-....
(ls)
By solving Eq. (15) simultaneously with respect to K S and P,'and then making
use of Eqs. (4) and (6), the suboptimal control is found to be･ -'
u== -K jlyde+u,, '' '' ' (16)
tt
where uo is the initial value of' u. ' '
This controller with integral action will put the state x back in its normal value
even under the presence of a constant disturbance such as w. ' '
ttt
3. Example
A synchronous machine connected to an infinite bus as shown in Fig. 1 is used to
evaluate the effectiveness of the foregoing controUer. In this system, an induction machine and a shunt impedance are connected in parallel at local terminal.
- .t. t-
I, 275 KV 3SO km 154 KV 25 km l,
Trirl
T
T
.bL
-T-
54eMWp.t==O.95
G
'condenser
4Q%
li,
Ppwer
60%
ZLoAD
I.M.
Fig.1. Modelsystem.
4
4
4
--
Zi X2
k, %,
k, k,
X3
,t
.
Yli,
.
-g,
E,
E,
E,
th model is as fo11ows ;
The nodale quation which gives the current-voltage relation of is
(17)
.
Then, the electro-mechanical oscillation of the synchronous generator can be written
as
d6
dt
-=w'
dgo - wo
J2ii- - M-(Pm-Pe) ,
(18)
5
Stabilixatien ofa th,nchronOnts Machine thro'ngh integ inl Actien
' ･ - ･ . '
･ ''
.P,-Real
[Ei-tt]･ '
'
-
'
be written
The electromagnetic oscillation of the power system can
the time lag i
'
tt.
p the exciter,,aS . . ,,1 .,
tttt
f:-Illti -=･u-Tl,,6 {(xd-x3)lk+EinvEes} ,. - .'.i.･･･..'.,,../.
''
'
tt
Where
Ik .. I. [E,4/ IE, 1]･ ' ' i ''' i ' '
. t.t
'
'
tt
(20)
de %
193 :Emd+LiEme,
P, = Real (Rbi5] ,
' Tablel. SystemconditiQns.
O.6051
,Xd
Transfbrmer
'-O.2017
T2e
'
6.0
M
14.4
7)i
O.0756
Z2
O.0722
Reactance
Tr3
Trans. Iine
(1 route)
Induction
motor
Impedance
1rs
O.0290
2rs
O.O087
3rs
O.O058
350 km
O.0622+jO.5581
25 km
O.O166+jO.1070
rll
1ri
O.O135
r22
2rs
O.O13S
oo11
1rs
1.2001
X22
2rpt
1.1923
X12
1.IS97
ua
7.26
(250 KV, 2oo MVA base) .
･ , (19)
'
･. dE3 =, -.-,istu,R,--IL'L' {E3--:1'(xn-x')iS} , '
Generator
.･
ttt t/tt
S-S, -= th.(p.-q) ' ..
ecd
'
tt t tt
t tt tt t tt
'
Further, for the induction machine6), the equations are;
by neglecting
Katsumi YAMAsmTA and Tsuneo TANIGvcHi
6
Table2. Initialconditions.
---LLsLLELLial-ethti2alLl
Q,S. 643
o.o
1.3452
1.15
O.0241
1.3. .458
O.9032
,-
re.2Q64
Therefore, linearizing Eqs. (18), (19) and (2e) about the steady state operating point
and taking the presence of a constant disturbance i.ntQ account, we can obtain the first
order linear differential equation given by Eq. (1).
The output variables considered for the design of a suboptimal controller is formed
in terms of the angular velocity, terminal voltage and electrical power of the synchronous
generator, which are quantities easily measurable at the generator location.
The system conditions and initial conditions are shown in Tables 1 and 2. Then,
' ' ''
by solving Eq. (15), the integral control u is expressed as
u :-O.225 j: Ato dt-1.379 I:AV, dt-O.769jl 3P, dt. - (21)
On the other hand, the optimal contro14) in the conventional undisturbed linear
regulator problem with
Z/F-Ax+Bu ･/ ･ ･(22)
tt t 'tt
and
'
Ill=S,"O(xTQx+ru.2)dt,'
''' (23)
s
Aee
(rad,Isec)
'
'
tS st-'
ttv}I
No, 3
t
No.1
-t
s
!
tttt
tttl
tltl
1lt1
tttl
t11t
tett
l
lJtletttt
OA6(ra
tts
1tt
lt1'
lltts.t
,',tttttttt
Fig. 2.
5
tt
t,l,l
t,tl,ltttt
Jtll
ttt
tlt
No,2
t..
Nt
tt
st
st
-t
--
Relation between-A6 and da).
5
Stabilixation of a bynthronous Mathine throttgh lbetagial Action
7
5
AS (rad) O
.
t-
.
l's t t N-l
l".
.
5.0
No. 3
-s
.
.
---
10.0
t (sec)
--
-
-
No. 2
No. 1
-10
-15
Relation between d6 and t.
Fig. 3.
s
No. 3
l'x,
No. 1
A
il
p,
tt :
Aw
(rad,,s￠c)
No. 2
:11
t
:
{
: II
tl :
;I ll
o
,
'
llttt
t'
,
t
t
t
1
t
'
t
,
lt
tt
-t
tt
l
,
1 tsl
ll
･1i
11 tt
el
1t}
lt
:t
tt
lt
v
t
t
:
t
t
ttt
N
ti 1
1
'
t,, ;t s.ek/ . .,:
NII
SI
s
tt
t
'
t
'
'
'
ls-Jt
..'
10.0
i (sec)
'l
t
'
'
-t
st
-- 5
Fig. 4.
Relation between Aa) and t. .
O.5
AEi (p･u.)
o
e- -.
5.0
No. 1
No. 2
Np. 3
-O.5
Fig. 5.
Relation between ziEi and t.
t (sec)
10.0
Katsumi YAMAsHIca and Tguneo TA!glGucH!
8
'
O.O05
No. 3
ASO
No. 2
o. 1
,
- --""- 5.0
--
t (sec)
10.0
-
-O.O05
Fig. 6. Relation between dS and t.
o.os
No. 1
No. 3
AEmd
(p･u･)
o
No. 2
;zL.-
tt " ''--- -- .-
5.0
Fig. 7.
10,O
t (sec)
Relation between dE.d and t.
AEmq
(p. u.)
tt
t" s
o
;t
t
'
rN
x
tl tt
tS s itss l's tN
tt
s
Il
tt
iis it
v
tl
ll tt Sytt
v
'
Ss ts.o N./ NNtt kl
.
X[lllN. 3
t
Sl
No. 2
-No.1
-O.O05
Fig. 8.
Relation between dE;ne and t.
-
t" N
A
..t
S IO.O
t (sec)
--
Stabiligation qfa Synchrenous Machine throtrgh integraI Action 9
ls found to be
u = -5.157Aa+O.S19dca"5.633AE,
-12.68dS-O.446dE.,+e.050AE.,. (24)
The responses of uncontrolled system (No. 1), optimal system given by Eq. (24)
(No. 2) and proposed suboptimal systern given by Eq. (21) (No. 3) fbra value of w= O.e7
are shown in Fig. 2AJ Fig. 8.
The results, shown in Fig. 2tvFig. 8, indicate that the proposed controller with
integral action puts the state x back in its normal value even under the presence of a
constant disturbance.
4. Conclusion
The present paper has considered the development of an integral controller based
on measurement of only the avilable output quantities, using the algorithm of C.D. John-
son fbr optimization. The proposed controlier has been applied to stabilization of a
synchronous machine and found to result in good control.
Aeknowledgment
The authors acknowledge Mr. K. Okano, Student of University of Osaka Prefecture,
for his assistance of numerical calculation. Numerical calculation were performed by
using ACOS-600 at the computer center, Univ. of Osaka Prefecture.
Referenees
2)3)4) 5)6)
1)
C.D. Johnson. IEEE Trans. Autom. Control, AC-13, 416 (1968)
Y.N. Yu et al, IEEE Trans. Power Apparatus Syst, RAS89, 55 (1970)
C.E. Fosha et al, IEEE Trans. Power Apparatus Syst, PAS-89, 563 (1970)
M. Atans et al, Optimal Control: An Introduction to the Theory and Its Applications,
McGraw-Hill, New York (1966)
W.S. Levine et al, IEEE Trans. Autom. Control, AC-15, 44 (1970)
T. Tanigtichi et al, Trans. of IEE of Japan, vol. 92-B, 323 (1972)