Optimal regulation with order reduction of a power system

Comput. & Elect. Engng, Vol. 1, pp. 425-452. Pergamon Press, 1973. Printed in Great Britain.

OPTIMAL REGULATION WITH ORDER REDUCTION OF A POWER SYSTEM

GURDIAL SINGH a n d RICHARD A. YACKEL

Department of Electrical Engineering, University of Illinois at Urbana Champaign, Urbana, Illinois 61801, U.S.A.

(Received 17 September 1973)

Abstract--A method of order reduction is applied to optimally regulate a hydropower plant with one synchronous machine. A fifteenth order dynamic model is developed for the power plant and optimal as well as near-optimal feedback regulators are designed by the method of temporal decoupling. The resulting systems are simulated and their dynamic responses are compared.

1. I N T R O D U C T I O N

The Linear Regulator Theory of optimal control has been applied to study the stabilization and optimization of power systems[I-5]. Due to the high dynamic order of such systems large amounts of computer time and memory capacities are required for adequate solution of even the simplest of cases. Consequently, it is a common procedure to restrict the model size by compromising on the accuracy of the representation of the synchronous machine. The small time constants like those which represent the armature and amortisseur windings are generally neglected.

Dynamic decoupling methods have recently been developed for optimization of large scale linear systems [6-9]. The basic approach involves decomposition of a large system into two or more smaller subsystems which are successively optimized. The decomposition is performed spatially, resulting in dynamically independent subsystems; or the decomposition is performed temporally, yielding two low order subsystems which represent the "fast" and "slow" parts of the original system. Since the optimization is performed only at subsystem levels, considerable saving in computational time and memory requirements can be achieved. This permits the optimization of the full system without neglecting small but important dynamics which may sometimes play a crucial role in the dynamic response of the system.

In this paper the operation of an isolated "run of river" type of hydropower plant with one synchronous machine connected to an infinite bus-bar through a transmission line is considered. The configuration of the system and the parameters correspond to one machine of the Bajina-Bagta hydroelectric power plant in Yugoslavia.~" A fifteenth order nonlinear dynamic model is developed, without simplifications which are usually adopted to describe the synchronous machine, its hydraulic turbine and associated control equipment. Only the case of regulating the system around a nominal operating point following a disturbance is considered. The system equations are linearized about this nominal pofnt and an optimal

tThe authors are indebted to Dr. Milan C~ilovi6 of the Serbian Electric Company, Belgrade, Yugoslavia for providing the numerical data used in this paper.

425

426 GURDIAL SINGH and RICHARD A. YACKEL

as well as suboptimal linear regulators are designed. The optimal regulator is obtained by sol~ ing the Riccati equation of the full system while the near optimal regulators are obtained by using approximations to the Riccati equation obtained by the method of temporal decoupling[8]. The resulting optimal and near optimal feedback systems are simulated and their dynamic responses are compared.

2. ORDER REI)t (TION O1- TIlE LINEAR STATE REGt LA7OR

Consider the linear time invariant system

dx ....... A x + Bu ll) dl

where x is the n-dimensional state vector and u is the r-dimensional control vector. If a scalar parameter 2 is introduced into equation (1), such that it multiplies small time constants, this equation can be partitioned into the form

dt2] = 1 I

d.\- ,421/2 A22/2j Lx2] k d t J I ] + it

B2/.),

12)

where

J = ~ ( x ' Q x + u ' R u ) d t (3)

Q

Under the usual conditions of the control law is

u* = - R- 1B'/£*x (4)

where/~* is the unique positive definite solution of the Riccati equation

- K A - A ' K + K S g - ( 2 = O 15)

with

S = B R - 1 B '.

If 2 is sufficiently small and if /£*(2) satisfies certain continuity and differentiability conditions at 2 = 0, a first approximation to /(*(2) can be obtained by setting 2 = 0 in

[(/11 Q121. Q21 Q22 linear regulator problemE10 ], for ,i > 0. the optimal

where x l , x2 are nt, n2-dimensional vectors respectively, and n I + n 2 = n.

In the model of a physical system 2 would generally multiply small inertial elements such as masses, inductances, etc. Such a system when partitioned appropriately will appear in

dx 2 the above form. In equation (2), if/l is small, the derivative -~- becomes large causing these

states to experience a fast transient behavior. It is a common practice to neglect the system parameters multiplied by 2, thus eliminating the corresponding "fast" states, x2.

For the system of equation (2) it is desired to minimize the performance index

Optimal regulation with order reduction of a power system 427

equation (5). Further improvement can be obtained by expanding/(*(2) in a power series about 2 = 0[8]. However, because of the terms A21/,~,... in equation (2) the dependence of this equation on 2 at 2 = 0 is not regular. Hence, it is not clear whether/(*(2) at 2 = 0 satisfies the above mentioned conditions. To investigate the behavior of/(*(2), it is partitioned in a form which is commonly used in Singular Perturbation of the Riccati equation,

g = [ Kll ~'K121 (6)

L2K~2 2 K 2 2 ] "

In terms of the newly defined Riccati gains K11, K12, K22, the partitioned form of equation (5) is

- K I I A l l - A~IKli - K12A21 - A'21K'12 + K l l S 1 1 K l l + K11S12K'12

+ K12S'12Kll + K12S22K'12 - Qll = 0 (7)

- K 1 1 A 1 2 - K12A22 - 2A'llK12 - A'21K22 + 2K11S11K12 + Kl lS12K22

+ 2K12S'12K12 + K12S22K22 - Q12 = 0 (8)

-2K'22A12 - 2A~2K12 - K22A22 - A'22K22 + 22K'12SllK12 + 2K'12S12K22

+ 2K22S'12K12 + K22S22K22 - Q22 = 0 (9)

where $11 = B1R-1B'I , $22 = B2R-1B'2, $12 = BIR-1B'2. At 2 = 0 equations (7-9) reduce to

- - K l l A l l -- A ' l l K l l - K12A21 - A'21K'12 + K l l S l l K 1 1 + KllS12K'12

+ K12S'12Kll + K12S22K'12 - Qll = 0 (10)

- K 1 1 A 1 2 - KI2A22 - A'21K22 + K l l S I 2 K 2 2 + K12S22K22 - Q12 = 0 (11)

--K22A22 -- A22K22 + K22S22K22 - Q22 = 0. (12)

An important property of the system of equations (10-12) is that equation (12) is a n2(n 2 + 1)/2 dimensional Riccati equation independent of the equations (10) and (11). Under appropriate conditions on A22 , B22 , Q22 [8] it can be solved for K~2(0 ).

Using K'E(0), equation (11) is solved for K12 to yield

g12 = K l l E 1 - E2 (13)

where

E1 = (S12K~2(0) __ /112)(A22 _ S22K~2(0))-1

E 2 = (A~IK~2(0) + Q12)(A22 - S22K~2(0)) -1.

Substitution of K12 from equation (13) into equation (10) gives

- .4 'K11 - K l l . 4 + K l l B R - 1 B ' K l l - (~ = 0

where

,,~ -- All + ELM21 + $12E' 2 + EIS22E'2,

]~ = B 1 + EIB2,

= Q l l - - E 2 A 2 1 - - Ar21EP2 -4- EES22E' 2

(14)

(15)

(16)

428 ( JURDIAL SINGH a n d R I C H A R D A . YACKEL

and S is defined as S=BR '/~'

= Sll + S12E'I + EIS'12 + EIS22E' 1. (17)

Equation (15) is a nl(nl + 1)/2 dimensional Riccati equation which, under appropriate conditions[8] can be solved for K*1(0 ). K]'2(0 ) is then evaluated from equation (13).

The solution of the Riccati equation (5h/~*(2), requires solving 011 + t12)(,1 + H2 + 11/2 nonlinear algebraic equations. For large order systems, the computational requirements for solving equation (5) can be excessively large. On the other hand, the solution of equations (10)-(12) requires solving two decoupled lower order Riccati equations• K22 is obtained by solving the nz(n 2 + 1)/2 dimensional Riccati equation (12). This gain can be interpreted as the optimal solution to a "fast" regulator governing the dynamics of states x2 with the weighting matrices R and Q22. Similarly the gain K 1 l is obtained by solving the iz~(nl + 1)/2 dimensional system Riccati equation (15). This gain represents the optimal solution of a "'slow" regulator governing the dynamics of the states x~ with the weighting matrices R and Q.

The following theorem, proved in [8], justifies the use of/~*(0) as a first approximation to /£*(2), and a truncated power series of/£*(),) in 2 about 2 = 0 as the higher order approximations:

Theorem. Under the following conditions

(i) R is a symmetric positive definite matrix,

(ii) Q is a symmetric positive semidefinite matrix,

(iii) the pairs (A22, B2) and (/L/~) are completely controllable,

(iv) the pairs (A22, C22) and (/i, (~) are completely observable,

where C 2 2 and (7 are any matrices satisfying C 2 2 C 2 2 = Q22, (~'C -~ 0 respectiveb, and for sufficiently small ),,/~*(2) is analytic in ), about )~ = 0.

The mth order approximation for the Riccati gain is

/~*(2) ~- [ P]~(2) )oP12(}~) 1 (18)

L).P1 2(2) ')'P2 2(')-)J

where P~j is the truncated sum )/

P~)(2) = ~ O/K*] 1! (19) /=0 =

for ij = 11, 12, 22. In order to calculate the higher order terms in equation (19) define

X=0 = 6qJK12 63JK2-2 ~JKll M~2 M½2 = (20) M{ 1 = ~2 j - ' ~)~J ), = 0 ' o '~ J )• = 0

and differentiating equations (7-91 with respect to ). and setting 2 = 0 yields • , / j~

0 = M~l~CJll -t-.v/i'lmJ'l + M{2,~2, + .~'~,M,2 + H{, (21)

0 = MJ2.~J22 + Mi~,~/,2 + .~/~1M½2 + H{2 (22)

0 = M'i2,(:~t22 + .~/22M{2 + H J2 (23)


where H{l, H{ 2, H J2 do not depend on M{1, M{2, M~2, and

~11 = - -Al l + SllKTI(0) + $12KT2(0), ~¢12 = --A12 + S12K~2(0),

d21 = --A21 + St12K~l(0) + $22K~2(0),~22 = -A22 + S22K'2(0) -

Equations (21-23) is a set of linear decoupled Lyapunov type equations, which can be solved for as many derivatives as are needed in the expansion. The near optimal control is given by substituting equation (18) into equation (4)

u(2) = - R - l [ ( B ' l P l l + B'2P'12)x1 + (2B'2P,2 + B'2P22)x2]. (24)

3. A COMPREHENSIVE STATE SPACE MODEL OF A SINGLE MACHINE POWER SYSTEM[I 1-14]

In this section the dynamic equations of a single-machine power system are developed. Figure 1 shows the schematic representation of the system under consideration. All equations are directly written in the per unit system. This system of units and the symbols used are explained in Appendix A.

Synchronous machine equations[11,121

Consider a three-phase synchronous machine with the following assumptions. (i) The stator windings are sinusoidally distributed along the air gap as far as all mutual

effects with the "rotor are concerned. (ii) Eddy currents, hysteresis and saturation are neglected. (iii) Harmonics in inductances due to stator slots are neglected. The dynamic equations of the machine in d-q-O variables are:

d dt (tP°) = (Co + rio)w °

d (~Pd) = (ca + ~PqW + eia)w °

d (Wq) = (e~ - O?aW + riq)w °

d (~:JSd) = ( e r a - - Rfalfa) wO

d dt(Wla) = ( -RlaI~a)w °

d dt (W lq) = ( - R~qI ~)w °

with qJo = - loio

Oil d ~ - --lai d + L~fdlfd + Lalalld

qJq = - l q i q + L,,lqllq

= -Laf i + Ls f t f . +

tPld = --L,,ldia + l-,ylalfd + Ltlalxd

tPlq = --L,,lqiq + L l l j 1 q.

(25)

(26)

430 GURDIAL S1NGH and RICHARD A. YACKEL

s p e e d

C

M e c h a n i c a l

P o w e r

t , t n s m x s s l ;

G e n e r a t o ~

: x ; : i a i c r

flux linkages. If

and

!fa = (B) ~Pfa =

LI,~J LV,d

k l:,Gq, ql =

bll b12 b 1 3 - I i t P a 1

b21 b22 b231%d I, b31 b32 b33 L_t~ 1 ,l/

C21 ('22d tlJl

i o = bo~ o

is the solution of equation (26) for the currents, then

and

1 V L l l q c = X-~t_Lolq

1VL~Id -- Lf fL1 ld

B = - - | L . l d L f l d -- L~:eL1 la Ad l

kL:ldLo:d -- LoxdLss

1 bo ~ m ,

lo

- L a l q i 7 with Aq = L2alq -- L q L 1 lq, - L q J

L,¢aLl l d -- L a l d L f l d

l d L 1 1 a - L21a

L , , l aL , , fd - laLfl a

LrfL~ i d -- Lfl dLafd -

L~ i dLay'd -- laL f l d

IA.ss - L~s~ with Ad = 2L , laL:laL,:a + laL::Ll la - IdL~Id -- L::L~,la -- LI ldL5, I .

Terminal conditions

Consider the terminal configuration of Fig. 2. Assuming the static network to be in a steady state we can write for any one phase

(27)

128)

129)

(30)

(31)

(32)

Control I ~ u2 Excitation System J

Fig. 1. Block diagram of a single machine power system

Choosing flux linkages as states, equation (26) is solved for the currents in terms of the


Transformer

G e n e r a t o r

v t

1

R Z x Z

vVV~

Infinite

Vb I Bus

~-B2

B 2

Fig. 2. Circuit diagram of the transmission line interconnection.

1 f~t(jB1) + (bt - ~b) - - - { (33)

Rt + jXt

where the bar over the variables denotes phasor representation. Thus 7= ia + jiq, ~b = ibb(sin 6 + j COS fi), Vt = ed + jeq and Vb = vb(sin 6 + j cos 6) where 3 is the angle between the bus voltage Vb and the induced voltage in the machine.

Substituting ~t, ~b, [from above, into equation (33) yields

1 (e,~ + jeq)(jB1) + ((ed + je, 7) -- (Vb sin 3 + vb cos 6)) Rt + jXt ia + jiq. (34)

Equating the real and imaginary parts, substituting for id, iq from equations (27) and (28) and solving for ed and eq yields

[ ed] l [ R t - -Xj jz lVb, lWd+b12Wfd+b13tP,a] eq R, jL I

I [ RiB, I - x, ,ibcos q (35)

+ ~ 1 - XIB1 -RIB1 1Lvbsin61

where

Xaz = X l - B1Z~

Z 2 = R 2 + X 2

A e = 1 + B21Ztl 2 - 2BIX l. (36)

If equations (35) are substituted in the machine performance equations (25) the following equations result.

d dt (Vd)

d d~ (tp~)

= 6 + (1 - X,B,)sinb)vb + w°Re,bl,Vd + w [---~fJc, ,q~q W~Ae (RzB1 cos ol XBz~

XBz~ + w°tPq w + w°Re,blEWfd + w°Re~blaWld + w ° - ~ f ) Cl2Wlq

wO = ~e(( 1 -- XIB1) c°s 6 + (-RiBt)s in6)vb + w ° bllWd + wO(--~Fd w)

o Xsz XBZ + W°RerCll~/q + W A~ -b12~l/fd + W O ~ e blatI'tld + w°RerClE~lq

432 C, URDIAL SINGH and RICHARD A. YACKEL

d d [ ( % d ) = wOCfd - - w ° R f d b 2 1 t~ld - - w O R f d b 2 2 t l l f d -- w°Rrab23 t l l , d

d (]t(tlJhl) = -W°Rlab31 t ] Jd - w°Rldb32 tp .n l - - w°R1,/B33t]~l,/

d ~ ( t I J l q ) ~--- - -W°Rlq ( . . 21 t l Jq - - W ° R l q C 2 2 ~ lq

where

137)

R l R,,~ = ~, , + r.

D y n a m i c equa t ions o f the rotor

The power transferred across the air gap is[12]

p,, = (t~diq -- ~9qid)w (38)

and the e lect romagnet ic torque is

m~ = ~diq -- (%id. (39)

The equat ion of mot ion of rota t ing masses for the generat ing set in terms of torque and absolute units is

j df~m d t = M t - M, , . (40)

In terms of power and in per unit equat ion (40) is

d w T(; ~ft = ~ P ' - p ' ' (41)

The equat ion relating angular posit ion and angular velocity is

dO = ww ° (42)

dt

where 0 is in electrical radians. If we define 0 ° = w°t and 6 = 0 - 0 ° then equat ion (42) becomes

d6 = (w - 1)w '1. 143)

dt

Equat ions (41) and (43) form the set of ro ta t ing mass equat ions to be used for linearization.

E x c i t a t i o n s y s t e m

The excitat ion system is as shown in Fig. 3, The dynamic equat ions of each element are

d - 1 c a,,k,, ~,~, 44) Main Exciter: (It (9) = T,,,~ :r + 7/~.~


Fig. 3. The excitation system.

Amplidyne :

Pilot Exciter:

; (v,) = -; u, + g uo2

a2 a2

&,,, = -f 0,2 + $ II,1

al Ul

$(D.d = -f U,l + kp Lx,-v P Tp p

Magnetic Amplifier:

where

0, = t&f + u2

v2 = e$ + e2 t 4’

and u2 is a control input to the excitation system. The input to the main field of the generator is similarly

(49)

(50)

Hydraulic installation [ 131

efd = cievs. (51)

Consider the case of a run-of-river hydro-plant (Fig. 4) operating with a low head hydro turbine and a transient speed droop governor. There is no tunnel or surge tank and losses in the penstock will be neglected. The final equations will be in linearized form.

Hqldro turbine equations

The performance of a hydraulic turbine depends on turbine flow Q,, speed of rotation N, efficiency I?,, output P,, net head H, and gate opening A. The main variables Qt, nt, Pt can

_ _ - _ _ - ____ - __-_ -- _ __-- --_--_ - _____ ---_--___-- ------_------___--__- ----- --_-_ -- ____ -___

-----_-_ _ ___-___ -_-----_--_____

Fig. 4. A “run of river” type power plant.

GURDIAL SINGH and RICHARD A, YACKEL 434

be cons idered as nonl inear functions of H, A and N,

Qt = .]'q(H, A, N)

'h = [;,(H, A, N)

P, = ./p(H, ,4, N). (52)

F o r small d i sp lacements abou t an opera t ing poin t (0) we can l inearize the above equat ions,

which become after no rma l i za t i on :

Aqt = e l A h + e 2 A a + eTA~

Arh = e3Ah + e 4 A a + el 1A.]

Apt = e~Ah + % A a + e,~AJ (531

where A denotes small changes of the variables , A ] = An = Aw = per unit f requency

dev ia t ion f rom nomina l ,

el = (?q,/(~h)o, N , A = const .

C 2 = ((gqt/~gh)o , N, A = const., etc. (54)

Due to the in te rdependence a m o n g the e-coefficients, it is sufficient to ca lcula te el , e2, e3 and e4 from the turb ine per formance curves. The o ther coefficients are then given by

es = 1 + el + e3

e-, : ] - - 2el

e 9 = 1 -- 2e I -- 2e3

e l l = - -2e 3. (55)

Also, define

C 8 = C 2 C 5 - - C 1 C 6

elo = ese7 - eleg. (56)

Pens tock equation

For the type of p lant considered, if pens tock losses are neglected

1 A0, , = . . . . Ah (57)

where

qc = flow th rough pens tock

= qt, flow th rough tu rb ine in our case.


Combining equations:

equation (57) with equation (53) yields the turbine penstock dynamic

Aq t = 1 e 2 - e lTAq , + ~ITAa + eT Aw (58) el

Ap, = e-! A q , - e ~ A a - elo Aw. e l e l e l

The mechanical power input to the generator is

Ap,. = ~sAp,.

(59)

(60)

Governor equations For small displacements the equations of a transient speed droop governor are

1 1 1 Aa = (rg + r')Aa + ~ Aw T A x , - +

r' 1 A:~g = ~ Aa - -~ Axg

(61)

(62)

where u 1 is a control input to the governor. Figure 5 shows the turbine-penstock equations (58), (59) and the governor equations

(61), (62) in block diagram form.

Linearization

Note that the equations for the Hydraulic Installation, equations (58-62), and the Excitation System, equations (44-50), are already in linearized form. Thus only equations (37), (41) and (43) need to be linearized. Define x, the state vector representing the linearized

1

PENSTOCK AND HYDI%O-TURBINE

.............................................

GOVE I~OR

u 1

A f

Fig. 5. Block diagram of the linearized hydraulic system.

AP t

436 OURDIAL SINGH and RICHARD A. YACKEL

states of the system and u, the control vector as

X ' = [AO Aw Atl~fd Aa Ax~ Aqt At'.f Av~ Av~2 AWd AWq

A % . A~lq hr.1 Av.]

J A ' = [/d 1 122]. (63)

Linearizing the system abou t a nominal point (0) and collecting coefficients results in a fifteenth order linear dynamic system, in matr ix no ta t ion

2 = A x + Bu (64)

where x is the 15 vector of states and u is the 2 vector of controls defined in equat ion (63), A is a 15 x 15 constant matr ix and B is a 15 x 2 cons tant matrix. The nonzero elements of A and B are

A(1, 2) = w °

1 ~d) A(2, 2) = _ T~o (Voio o.o

A(2, 3) = L b l z U r ~° T~

0~ s e 8 A(2, 4) . . . . .

TG el

A ( 2 , 6 ) - cq e 5 TG el

1 A(2, 10) = --~-£(i ° - b l ,W °)

1 _ io ) A(2, 11) = - ~ c ( W ° c l ~

l 0 A(2, 12) = ~Gbl3~q

1 A(2, 13) = - - - - c 1 2 W °

TG

A(3, 3) = - w°Ridbz2

A(3, 7) = w°c~

A(3, 10) = - w°Rfdb21 A(3, 12) = - w ° R f a b 2 3

1 A (4, 2) . . . .

r~

A(4,4) = - (rg + r')

(X s t?10

T~ e,


1 A(4, 5) = - -

T~ l,,¢

A(5, 4) = - - T~

1 A(5, 5) -

T~

1 e 7 A(6, 2) -

T~e~

1 e 2 A(6, 4) -

T~ea

1 A(6, 6) -

T~el

1 A(7, 7) -

T~

~ake A(7, 8) -

T~.

1 A(8, 8) -

To~

A(8, 9) k. T.2

1 A(9, 9) -

A(9, 10) = ~ T~

W o A(10, 1) = ~(-R,Bx sin 6 ° + (1 - X , B i ) cos 60)Vb

A(10, 2) o o = w W q

A(10, 3) = w°R~,bt2 A(10, 10) = w°Re,bll

A(10, 1 1 ) = w ° I 1 - ~--~BflCls

A(10, 12) = w°Re~b13

A(10, 13) = - w ° XBz - e C12

438 GURDIAL S1NGH and RICHARD A. YACKEL

W o A(11, 1) = ~ (-R~B1 cos 6 ° - (1 - X~B~) sin 6°)vh

A (11 , 2) = - w ° h o °

A(ll , 3 ) = w°X~Zb Ae 12

A ( l l , 1 0 ) = w ° XSz ~ - b l l - 1

A(l l , 11) = W°CllRe,

A(11,12)--- oXnz W Ae h i3

A(ll , 13) = w°Rercl2 A(12,3) = -w°R1ab32

A(12, 10) -- -w°Rlab31 A(12, 12) = -w°Rldb33 A(13, 11) - - - --W°RlqC21 A(13, 13) = -W°RlqC22

1 A(14, 14) = Tv

A(14, 15) = ~mk~e

1 A(15, 15) -

T,.

1 B ( 1 , 4 ) = - -

km B(2, 15) = - - '

Tm

The numerical values of the parameters are listed in Appendix B and Table 5 shows the A matrix in numerical form.

4. THE C O N T R O L P R O B L E M A N D N U M E R I C A L R E S U L T S

The system of equations (64) represents a small signal model of the nonlinear power plant about a nominal operating point. The control problem is to design a feedback controller which will optimally regulate the system under small load disturbances. The initial conditions of the system variables reflect the application of this disturbance. The feedback regulator is designed by application of the linear regulator theory and the method of order reduction described in Section 2.


For this system two different cases of weighting matrices are considered.

Case a

R = diagonal E1

Q = diagonal [1

Case b

R = diagonal [1

Q = diagonal [1

11

1 1 1 1 1 1 1 1 1 1 1 1 1 1]

1]

1 1 1 1 1 1 1 1 100 100 100 100 100 1001

The full Riccati equation is solved by the eigenvector method E15, 16] for both cases. This requires the solution of a system of 120 nonlinear algebraic equations.

In each case the method of order reduction described in Section 2 is also used to generate a near optimum solution. The system is partitioned as shown in equation (2) with n 1 = 9 and n 2 = 6. The linear regulator is singularly perturbed by artificially introducing the parameter 2 on the right-hand side of the last six state equations. These six equations are chosen since the state variables associated with them exhibit a faster transient behavior than the first nine variables. These "fast" variables are:

Xx0 = A~Pd direct axis armature flux

Xll = AqJq quadrature axis armature flux

xlz = A~PI~ direct axis amortisseur flux

x~3 = A~P~q quadrature axis amortisseur flux

x14 = AVal pilot exciter output voltage

x a5 = Avp magnetic amplifier output voltage.

These variables are often neglected when a power plant is modeled, so that the system order remains within workable limits.

The zeroth order approximation of/~* in the order reduction method is obtained by solving two lower order Riccati equations, equations (12) and (15) consisting of 21 and 45 variables respectively. In addition a 6th order inverse has to be performed to solve equation (13). The first order terms of the approximation to/~* are obtained by solving equations (21 - 23) with j = 1. Equations (21) and (23) are Lyapunov type equations of dimensions 9 and 6 respectively and equation (22) is a linear matrix equation requiring a 6th order inversion. The approximations of/~* are given by equations (18) and (19) with 2 = 1.

Tables 3 and 4 list the different Riccati gains for cases a and b respectively. Only the upper halves of the matrices are shown since they are symmetric. The tables list the optimal, zeroth order near optimal and first order near optimal feedback gains. The entries for each element of the matrix are arranged in the order; optimal, zeroth order near optimal, first order near optimal.

The eigenvalues of the optimal closed-loop system are listed in Tables 1 and 2 for cases a and b respectively. The eigenvalues of the near optimal closed loop systems are also listed in Tables 1 and 2. The near optimal closed loop system is obtained by use of equations (24) for the control u, instead of equation (4). The open loop eigenvalues representing the system with u = 0 are also in Tables 3 and 4 for comparison purposes.

440 GURD1AL SINGH and RICHARD A. YACKEL

Table 1. Eigenvalues, case a

Closed loop Closed loop zeroth C l o s e d l o o p t irst Open loop optimal order appxn, o r d e r a p p x n

1 40-6 + j425 40.6 * /425 40-6 + j425 4o.6 ~ j425 2 - 4 0 . 6 - , j 4 2 5 - 4 0 . 6 /425 4(I -6 - / 4 2 5 40.5 )425 3 - 2 5 - 3 3 . 2 5 + j10.5 ~7 4 j14.9 v[ .~

4 - 1 5 . 7 - 3 3 . 2 5 /10.5 37 j14.9 1 .~; i32-4

5 - -14-28 - 1 5 . 7 15.58 I . - , , Q 4 6 - 1 0 14.25 ~ /25 .3 10.4 ~ j2.1 15.-:

7 - 1 0 14.25 - /25.3 10-4 [2.1 l l.,)~

8 - - 5 . 2 + j 0 . 4 7 5 - 1 1 - 2 1 8.64 <).42 ~ /I.2~ 9 - -5 .2 .j0.475 8.53 ÷4 .37 Jr j 24 .7 9 4 2 [121

10 - 4 6.71 ~ 4 - 3 7 - j 2 4 . 7 3.29 ~ j3.67

11 2-03 - 2 . 5 7 + j 9 . 0 4 . 2.95 + j 9 . 9 2 ~.29 i3.67

12 - -0 .9 + j8.21 - 2 . 5 7 - j 9 - 0 4 2.95 j 9 .92 2-59 i7.56 13 - -0 .9 -- j8 .21 2.01 - 0.89 + ]0.22 2.59 + /7-56

14 -- 0.311 - 0 .957 - 0.89 - .]0.22 (!.924

15 - 0 .065 -- 0-655 0.646 0.658

The free response of the system with u = 0 is shown in Figs. 6 11 for some typical variables. The initial conditions used in the simulation represent the steady state with a one percent shift in the reference input to the governor. The optimal, zeroth order near optimal and first order near optimal closed loop system responses for the same variables in case b are shown in Figs. 12 29.

As can be seen from Tables 1 and 2 and Figs. 6--29 the response of the optimally con- trolled power system is better than that of the open- loop system in that the system rapidly settles down to a new steady state condition.

In case b the use of the zeroth and first order approximations result in stable systems which are comparable to the optimal system. In case a the use of the zeroth order approximation results in an unstable system while the use of the first order approximation results in a system that is comparable to the optimal system. A close examination of Tables 3 and 4 show that the first order approximation over-corrects for the difference between the

Table 2. Eigenvalues, case b

Closed loop Closed loop zeroth Closed loop first Open loop optimal order appxn, order appxn.

1 - 4 0 . 6 + j 4 2 5 - 4 0 . 6 + j 4 2 5 - 4 0 . 6 + j 4 2 5 40.6 + . j 4 2 5

2 - 4 0 . 6 - j 4 2 5 - 4 0 . 6 - j 4 2 5 - 4 0 . 6 - j 4 2 5 - 4 0 - 6 - .i425 3 - 2 5 - 1 1 7 - 120.11 118.48

4 - 15.7 - 31.07 - 30.629 - 23.83 + / 6 . 8 3 5 - 14.25 - 17.84 + j 7 .99 - 16.39 + j2 .61 - 2 3 - 8 3 - / 6 . 8 3 6 - 10 - 17.84 - j 7 . 99 -- 16.39 - j 2 . 6 t 19.7 + jO.119 7 - 1 0 - 1 4 - 2 6 + j 2 . 3 6 - -12-64 - j 1 2 . 2 6 - 19.7 jO.119

8 - 5-2 + j0 .475 14.26 - j2 -36 - 12-64 + j l 2.26 l 5.87 9 - 5.2 - .]0.475 13.93 12.51 - 9.24 + j l 6,1

10 - 4 - - 6 . 6 4 + .]14.29 8-22 + j 1 6 . 3 6 9.24 ~-- /16-1

11 - 2-03 - 6.64 - j l 4-29 - 8-22 - j l 6.36 ~ 6.36 12 - 0 . 9 + j8 .21 - 6.66 7.07 2.55 13 - 0 - 9 - j 8 . 2 1 - 1 . 7 2 + j 0 . 2 5 7 1-96 - 1.92 + i2.31 14 - 0 . 3 1 1 - 1.72 j 0 . 2 5 7 I. 19 1,92 /2.31 15 - 0 .065 - 0.663 0.662 0.665

Optimal regulat ion wi th order reduct ion o f a p o w e r sys tem 441

Table 3. Riccat i gains, case a

D

+ + + 5- + + + -+ .48109 1.935 .2363 -.0283 .0110 -,5009 .02310 ,0073 .0481 .898 9.048 .4248 .087 .0165 -.8725 .0374 .0059 .0197 .09166 -5.003 ,i028 ,0255 .0062 -'.2365 .0199 .0195 .i008

+ + + 4. + + + -t 778.24 -8.331 -4.274 -,2263 19,44 .4793 -,1292 -,768 1327.8 10.91 -7.523 .384 28,53 -.423 ,9513 -.1702 357,2 8.366 -1.646 -.1444 17,66 .6847 -.23884-1.078

4- + + + + + + 1,646 -.0045 - ' .0039 -I,089 .ISO .0602 .405 1,78 -.'3498 ,0004 -1.479 .[717 ,0258 .086 1.631 -.0549 -.0054 -I,031 ,10q5 ,i053 .509

+ + + + + + -+ + ,i14 .0937 .0556 -.0081 -.3029 -.020 .1367 .0939 .9473 - .0083 - . 9913 -.004 . i 001 .0937 .03007 - .0097 - .0056 .027

+ + + 4- + + + + 1.497 -,0072 .0009 .000 36 .9025 1.497 .0221 -,00028 -.00005 -.0012

u

+ 4 + + * + .0036 .0088 00284 .0048 .0193 .00827 .00515 .0092 ÷.0267 - . 1 8 7 ,0183 .0201 .00094 .0085 .0413 .251 .0045 -.00068

+ 4 4 -{' 4"_.279 dr -.398 .169 -1.58 -8.23 -.~123 -.626 .266 -3.1 -10.4 -.149 -.1636

4--'0961 4. .0764 4.'5328 .+-.484 -~-'493 ~1.-'0291

.0007 .0014 ,.0377 .120 .170 .0780

.0038 .0004 .0364 .083 .0806 .0885 • -.00251 .0021 .0473 .119 .2621 ,0742

+ .oo264..oon-- + 4- "+ + .0072 .016 -.0088 -.0041 .0035 -.0016 .0171 .059 -.0045 -.005 .0019 -.0006 .00011 -.9209 -.0147 -.9048

+ 4- -+ 4- 4 4- .00006 -.00003 .00075 .003 -,00117 -.0005 .00018 - .00007 .30088 .003 - .0002 - .0002

.+.i.497 --.0044 .~-.001094.-100035 .00161_-.00004_~.00001 .00109 .0069 -,00109 -.0007

+ + + + '+-.26o -I- 2.23 -.1068 -.0346 -.2288 -.0058 .00193 ,0629 -.0939 "%L-.0418 3.164 -,i196 - 0188 -.0621 -.0136 .0045 -.0693 -.226 -.0578 -.063 2.040

+ + + + +

+ + + + +

+ + + + +

.3413 2.04!i

+ + + + + + + +

t + + + ~- + + +

+ + 4- + + + + +

+ + + + + + + -~

+ + + + + + + ' +

4- + + + 4. + + +

+ + + + + + + +

-.1289 -.07201 -.0352 .002I ,00009 -.0789 -.4623 -.1723 -.0321

+ + -, + -+ + + 4, + .1235 -.387 .2517 .09032 .00014 .9027 .0103 .1016 .0046 .I039 .0161 .0525 ,000209..00007 ,0013 .0035 ,0487 .0537 .1428 .0659 .3143 .000105 .0008 .9039 .0189 .1580 .0383

+ + + + 4 -+ 4- ~ -+ .0462 .2791 -.000905 .000056 ,00801 .00322 ..i045 .0429 .0213 .9650 .00002 .900014 .00019 ,0004 ,0685 .0664

~- -~ ,0832 "~" ,389 "~ ,000044 .00007 -~ .001964" ,0042 -I" ,1799 "~ .0135

2.28 -.00008 .00039 .005 .0209 1.022 .4632 .00038 .00005 ,90059 .0013 .317 .3485 .0002 .0003 .0046 ,0163 1.4(5 1.168

-t- + 4. + 4- 4 .9130 -.00029 .?0013 -.0003 -.0003 -.00003 ,0129 -.00019 .00064 -.00]8 0.0 0.0 .Ol3l -.0003 .090~3 ,002 .00007 .00008

• + + 4- -+ 4- .+ .012 :00116 --.00148 -.00017 .00008 .q12 .0014 -.00046 0.0 0.0 ,'312 .00084 -,0026 .00005 .0005

4- 4- -I- + 4. + .0353 .3113 .00198 .0008 ,0317 -.00047 O.O 0.0 .0388 .0236 .0005 .0006

• + + .+ -+ -4- 4. .1136 .0C841 .0036 .3743 0.0 0.0 .1566 .001] .001

+ + 4- + + + .619 .3459 .0426 .0319 .338 .356

4" + + + 4. 4- .284 .0796 .435

+ + + + + +

reduced solutions and the optimal solutions. This indicates that additional terms of the expansion may be necessary to yield a suitable result.

The computation time for the optimal and zeroth and first order approximations of the Riccati gain are comparable when solved by the eigenvector method. However, for larger problems one might expect that the near-optimal solution could be computed faster. The variable storage requirement for the optimal solution is about three times larger than that needed for the reduced solution and the first order approximation. The main contribution of the order reduction method for solving large scale problems of this kind is the reduction in the variable storage requirement.

5. C O N C L U D I N G R E M A R K S

The results of our studies in this paper show the possible application of optimal control theory to improve the transient response of power systems and the feasibility of using the

4 4 2 G U R D I A L S 1 N G H a n d R I C H A R D A . Y A K E L

T a b l e 4. R i c c a t i ga ins , c a s e b

14.19 +1~9.0~, %.496 ±.8~ +<~0, +-,. ,~0,+ .. . . . . 4.1710 + ........ +.5,,3~, ! .... ±.:,44 +4~.>~ + ..... + . . . . . 2 9 . 3 7 0 3 0 4 . 6 3 4 . 8 4 1 7 1 . 3 7 2 8 .02261 - 9 . / 4 8 1 . 5933 • 1 9 t 6 1 .4938 ,7999 •75406 1 .395 1 4 , 0 6 8 9 1 2 1 •16034 1.6207 117.67 3.5495 ,7809 319!12 -8.411 { .;Sq] .20231 1.6531] 4'79~ .803436 .19909 2.14658 .706( .1141

-h7618.~+-]83.8 +-~o.,,~4- ,7 :8 +83; L.,~+- ~ . ) : ; # ~ 4 . 8 2 , + . ~ 3 0 + - s ~0786.t~o: +-,:.,, 44-2~].6~+.8.21~+-9 v;~5 2 3 0 6 4 . 5 4 0 8 . 7 9 - 4 1 . ~51 2 . 0 2 5 9 3 2 . 8 5 - ) 7 . 7 8 - I 3 . 3 5 . 0104 - 1 0 . 3 ~ ~ .938 5 6 . 3 6 - 178 .06 6 ] . 1 9 ~ - 1 ! . 2 5 5 19075 .3 4 0 5 . 3 6 - ] 8 . J9 1 , 8 4 3 8 9 1 . 8 6 0 - 4 1 . 5 7 1 -16 .t~71~ 135 .011 3 . 6 9 1 0 8 2 . 7 0 0 1 - . 3 9 5 ~ - . 0 5 7 4 9 -62.3[* 10 ,30

I r -{-18.374 ~ - - . 2 7 1 3 , 4 - . ~ 1 6 1 6 4 - - 2 9 . , - 9 8 " ~ , 0 6 , ! 1 -[- .8205 4 7 6 . 8 4 9 8 ~ - , 0 1 3 6 4 - k 0 4 0 0 4 4 7 . 0 ) 6 a 4 ~ . : 5 1 - ~ / . 3 0 a O - ~ • 3 ] i ~ 18 .834 - . 2 0 8 2 . ,14403 3 1 . 6 [ . 9 3 2 . 7069 5 . 5 9 2 8 . ,)546 .02587 1 .36$ 2 . 7 5 2 3 3 .415 .601112 1 9 . 5 2 2 - . 2 8 3 7 . ,F,09 i ' . .97 '4 2 . [ 9 ! 6 9 . 89681 7 . 3 3 2 / . 9 0 4 9 8 .04121 , 1.843~" 4 . 4 8 5 4 ~. 35~*'~ .57~3

4- 4- 4-.e,~496 4".,91,,, 4-.~;<,,~ q- o95~64:..o1923"~.169o54-.'n~oo +-.qo7o4m1'-.o~,:~'~ 4 : 2 , : +- o~'<:+-.~>~ . 3 1 3 u 0 9 7 8 5 .21811 . 0387 - . 0 1 5 0 8 - . 1 2 4 ~ ,0196 - . 0 0 9 0 4 . ,}94112 .12 ,6.' . 742,' . 0138~ . }748', , ] 9 7 3 8 . 4 5 1 8 ! ) 4 6 5 . 0 ! .]657~, .0 !271{ - . 0 1 0 2 0 ~ .029 15 - 3,',g , ] , 9 t - . , ] ! 3 4

4 - Jl- ~ - 4 - ';uC5 - ~ - - . 0 8 8 ' , ~ 4 - - . 0 0 2 6 9 8 4 . 0 0 0 8 7 1 9 -~--U06 1 c ' ~ - . 0 0 : 1 6 5 t - } - - - 0 0 0 2 9 8 " ~ . 1 0 4 / ~ 0 - ~ - 3 1 , 8 8 ' 4 - }. , -~-.CO036:19 1 .4495 .](128 , 0 0 3 3 . 0 0 1 0 3 . 0 0 7 1 3 .00060 . 000387 .8047{) 8 . ) l b O ] .00435q ,00076b I :~993 - . 0 9 7 2 I . r )03177 . 0 0 i l l . 008194 . 000038 .00029 12 .005777 1 2 i 8 8 , ~ 0 . 7 .00042

~- 4 - ~ - -~- 4"~.3.317 ~ ' - ' . 3 8 ' 4 - -1 .2472 -~ -10 ,297J1" - . 2819 4 - . [ 2 , ' 1 - ~ 2 . t , ' ; 3 4 ~ J . ; . 9 "4-~.,,11 4 - ~ • 8 1 9 [ b -1/•" " , ' i - 6 , 7 9 2 2 - . 4 4 5 7 •11166 % 2 7 5 h . ]827 3,3* 89 . 9 4 u

6 ].11,45 i.,~ U:,5 ,4 -11 , ]08 - . 3 1 7 2 0 .12949 ? ~ ' ; , 5 6 .1 ; , 8

4- 4- 4- 4- + 4- ~,. + 4-t.],5,84-.0134 +-.004~:4.c'~,:,'. 4 :,,:,~ 4- . ' ,~ +.,9~; . 1209 . 1 3 8 i .~0123 .01808 -.006058 .091 'B .29 ~'~ 5 , ) 2 9 , 8 9 6 7 3 , ~395 ;I . 1 1 4 8 1 . 2 0 8 . 0 ] 4 9 6 , 004253 1 i 6 4 1 ~ 2 7 ~ , 7 . ff)40q

• 148915 4-

. 0 9 2 8 4 ,9591 . ) 0 6 3 9 2 - . 0 0 2 1 2 3 .'322 ;5' I ; } : , * ~4]:. . '.191)022

.87, '1~ . / ¢ / 5 " . 903597 - , 0 0 2 0 5 7 . - ] , ~ a . ~),1 ,i P ~.72[ . n , ] 8 9

4 - 4 - 4 - 4'- "4- - ~ 4 £ 0 9 9 ~ 4 T . 5 4 1 ~ + . 0 4 1 8 2 J I - - . 0 1 4 2 7 ~ - . i . . . . . . - ~ . ; ' ! i -~-, X,4! - ~ . ~ R , 0 4.946 ~ . 0 4 9 9 8 -.016532 . 25404 8 2 8 2 , O2 .6 ~0~} 6 , 7 4 2 7 , 0 4 5 2 1 3 - , ) 1 1 8 0 0 1 . ~ ~ Y, t ,.P~ < 59~ , H S ,

+ + + + 4 + 4- 4- +~ 3878 -b.02065+- ..... 4- ~,:~., + , 1 , , 4-,]0~2, ] . 2 8 8 - . 0 1 9 9 , ,oa~2 .2~ ,m > : o . 1 1 2 9 1 6 - . 0 2 1 8 ~ , ,~¢1: . : i ~ , .0~);~

1.2 1 4 3 ' , .146616 ~l , }

+ 4- 4- + 4- 4- 4- 4- 4- + + . , . . , + .~ : , 4- :7i,~ + ] e ~ ,

:}7, :e>~, i : : , ] , ~ ;.:

+ + + + + + + + + 4- + + s . ~ , , 4 - . , s .v - .+~>~

: ~m,, . 0884

+ -b q- + + + + 4 + q" + + + ~:,,, + 8o~,2 , 1 2 : .3(171

4- 4- + + + + + + 4- q- + + 4- + 9',01~ 86188

11~'7

+ + + + + + + + + + + + + +

order reduction method to obtain near optimum solutions to large linear regulator

problems. It has been shown that the transient response of a single machine power system under

small disturbances can be considerably improved by using an optimum regulator. Although single machine power systems rarely operate isolated it is hoped that the basic features of the results do extend to the cases of multimachine power systems.

~0 Free response fo rX I X

8 . 0 0 -

% x F r e e r e s p o n s e for X 4

7 . 5 8 .

6 . 7 5 ~

6 . 1 3

5 . 5 0

4 . 8 8

4 .25

5"65

3 ' 0 0 0-13 0"25 0"38 0 5 0 0"63 0"75 0"88 I()0 xlO I

Fig. 6. Free response of A6.

I ' 00

0-92

0-85

0 .77

0 ' 7 0

0-62

0"55

0"47

0 4 0


015 0 '2~ 0 " :~ 0"50 0"63 0.75 0-88 I'O0 x l O t

Fig. 9. Free response of Aa.

w Free response for X2 6 .00

4"25

2 5 0

0 -75

-I .00

-2 "75

-4 -50

- 6 . 2 5

-6 . OOJ

0"63 0"75 0"88 I '00x lO I

Fig. 7. Free response of Ato.

o Free response f o r X l O

- i - 5 0 ,

H 7 0 •

- 1 . 9 0 -

- 2 . 1 0 -

- 2 5 0 -

- 2 " 5 0

- 2 , 7 0

- 2 - 9 0

- 3 . I0 0 O.t3 0 2 5 0-38 0"50 0 .63 0 -75 0.88 1.00 x lO l

Fig. 10. Free response o fAq ' a.

% x Free response for X3

-1.70

- I . 8 4

-I. 9 7

-2.11

-2 .25

2 . 5 9

2 . 5 2 -

2"66-

2 '80 - 0 015 0"25 0'38 0 ' 5 0 0"63 0"75 0"68 I ' 0 0 X I0 I

Fig. 8. Free response of A~I~fd.

' 0

Free response for X 12 -I 5 0

-~. 6 7

- I . 8 5

-2"02

-2 .20

-2 "37

- 2 " 5 6

-2 '72

-2"90 0 015 0 , 2 5 0 .38 0 . 5 0 0 .63 0 .75 0 .68 I .OOxlO I

Fig. I 1. Free response of AqJtd.

4 4 4 ( JURDIAL SINGH a n d R I C H A R D A . YACKEL

% O p t i m a l response for X[

7 O0

6 1 3 ~ !

525I i

4"38 i

I 35o! ,,

2 6 3 ~

1 75 ~ k

0 8 8 ~

0 0 63 I 25 188 2 5 0 3 13 3 75 4 38 5 0 0

Fig. 12. Optimal response of A4, case h.

* O p h m a l response for X4 4 0

Z2 /

05 I

C tit

0 70 I

0 52

() :35

0 i 7

o' 063 rEs 18~ zso s',s 3% ~28 ~---Too

IIig. 15. Optimal response of Aa. case t}

Opt ima l r esponse fo rX2 I o 0 0 0 .63 125 f : ~ 2 . 5 0 - ' ~ 5 . 1 3 3.75 4.38 ~ 0 0

- 0 . 5 0 ~[ /

/

/

- [ 0 0 !

- I . 5 0 ~

i , - 2 0 0 j

J 2 5 0 ~ i

'1 11'

-3o0~

3.5o !

- 4 0 0

Fig. 13. Optimal response of Ac.x case b.

x O p t i m a l response for XIO 200]

i 1 " 3 6 i

0 7 5 ,

o ~3- o

- 0 50)

- I 1 3 "

"X

0 6 3 I 25 i 8 8 2.50 313 375 4.38 6 0 0

- : 75 J I

- - 2 38 ! t I

3CC

Fig 6. Optimal response o l A ~ . , case hi

T o_ i ~oOPtimal~_ response for X 5

/ Q ,

1 1 9 ;

0 9 7 "

0 7 6 ,

0 5 5 ~

0 3 4 ~

0 124

0 9 0 _ 0 6 3 125 I 88 2 50 513 375 4 3 8 5 0 0 0

0 30 ) '

Fig. 14. Optimal response of A~PI,. case/}.

? ©

O p t i m a l response for X I2 3 0 0 ,

2 2s!

! 5 0 ~ /

J 0 75 i / ""- ,~ ,

i I

O o ~ . 6 3 125 1.88 2.50 313 3.78 4 . ~ - 5.00 i

- - 0 7 5 ~ /

ii I -I 50d I

i

-- 2 25-- / i

3 o o J

i i g I7. O p t i m a l r c s p o n w o I .Xl ~,~.ca~ch.


x Sub-optimal response for Xl 7.00

6.15

525

4 36

5 50 i

265 J

] 75]

0 8 8

0 0163 125 188 2'50 5/19 3'75 4'.B8 500

% "; Sub-optimal responsefor X4 3.00 I

1-88

150 ~ ~

1"13

0.75

0.58

0 063 125 188 2 50 313 375 438 500

Fig. 18. Zeroth order near-optimal response of A6, case b. Fig. 21. Zeroth order near-optimal response of Aa, case b.

%

I'00

-0 25

-E "50

-2'75

-4-00

-525

-6.50

-7'75

-900 ]

Sub-optimal response forX2

2 5 0 " • 3-13 3.75 4:38 500

'O × Sub-optimal response for XlO

3.00

2,25

1.50

0.75

O 0 O0

-0'75 r

-I "50

-225

-500

Fig. 19. Zeroth order near-optimal response of Ae~, case b. Fig. 22. Zeroth order near-optimal response of AtYd, case b.

% Sub-optimal response forX3

18o 1

I 2 7 ,

I OI

0", '5

0-49

0"2~i t O ~ '63 1.25 1.'88 2.50 3.13 3.75 4.38 5.00

× Sub-optimal r e s p o n s e f o r X l 2

° - - 0 /0.63 ,.z~ 1.88 2:5o3.,3 s:75 4:s8 5-00

- o , 5 t / ~ -I.50! /

-2 251;

- 3 . 0 0 ~

Fig. 20. Zeroth order n•ar-optimal response of A~ia, case b. Fig. 23. Zeroth order near-optimal response of AtI'la, case b.


% -~ Sub-op t ima l response for XI

I 0 0 -

0 86-

0 72-

0 5 9

0 .45 '

0.31

0 1 7

O'O4 ] O

- 0 IOJ 25__ 1 .88~250 313 375 4"38 5 0 0

'b × Sub-opt imal response for X4

dO0!

0 821

065],[

O~21

-0 23 i ~'~. r

0 4 0 j

3r5 4 ~%7oo

Fig. 24. First o rder nea r -op t imal r e sponse of Aft, case b. F ig 27. I irsl order near -opt imal response of A,, case h,

% S u b - o p t i m a l response forX2

8 0 0

6.25

4 5 0

2-75.

I 00.

0]63 1.88 2.50 313 3'75 4 3 8 6.00

-4" 25 ]

- 6 O0

F i g . 25. F i r s t o r d e r n e a r - o p t i m a l r e s p o n s e o f A~u, case b.

? ©

Sub-op t ima l response for XIO

P 001

O. 38 /" " /

-0 -25 ' 0 ~ 3 ~ 2 5 [ 8 8 - - 2 5 0 ~ 3 i 3 5 75 4.38 5.00

- 0 . 8 8 / i i'

- I 50 ~ I i

- 2 1 3 4 l' I I

- 2 7 5 I

-338

-4 O0

Fig. 28. First order near -op t imal response of A~,z, case h.

% S u b - o p t i m a l response forX3

8 0 0 -

66,3.

5"25'

3 . 8 8

2 5 0

t 13

- 0 2 5 p 50r3:13 3.75 4.~B 5.00

- - 6 3 1

-300 j

Fig. 26. First o rder nea r -op t ima l response of ARise, case b.

× Sub-op t ima l response fo rX I2 1 . 0 0 -

/. • /

038~

U~3 ,~5 ,~3-2.~of3,3-- 3~5 438 500 - 0 2 5 s /

,/ - o 8 8 i ,I

/ - • 5 0 4 ,i'

/ -2 13j I

I'

- 2 7 5 ~ / _338iV - 4 0 0 '

Fig. 29. First order near -op t imal responsc o1 A~I,J, case h.


Acknowledgements--The authors are grateful to Professor P. V. Kokotovi6 and to Dr. Milan C~ilovi6 for their helpful discussions and comments. Also to Professor B. C. Kuo and the Control Systems Research Laboratory of the University of Illinois for the support given to this work.

REFERENCES

1. Y. N. Yu, K. Vongsuriya and L. N. Wedman, Application of an optimal control theory to a power system, IEEE Trans. Power Apparatus Syst. PAS-89, 55-62 (1970).

2. O. 1. Elgerd and C. E. Fosha, Jr., Optimum Megawatt-frequency control of multiarea electric energy systems, IEEE Trans. Power Apparatus Syst. PAS-$9, 556-563 (1970).

3. J. H. Anderson, The control of a synchronous machine using optimal control theory, Proc. IEEE 59, 25-35 (1971).

4. E. J. Davison and N. S. Rau, The optimal output feedback control of a synchronous machine, IEEE Trans. Power Apparatus Syst. PAS-90, 2123-2134 (1971).

5. M. C~dovid, Linear regulator design for a load and frequency control, IEEE Trans. Power Apparatus Syst. PAS-91, 2271-2285 (1972).

6. P. Sannuti and P. V. Kokotivi6, Near-optimum design of linear systems by a singular perturbation method, IEEE Trans. Automatic' Control AC-14, 15-22 (1969).

7. P. V. Kokotovid and R. A. Yackel, Singular perturbation of linear regulators--basic theorems, IEEE Trans. Automatic Control AC-17, 29-37 (1972).

8. R. A. Yackel and G. Singh, Order reduction in the design of the time invariant linear regulator, to be published. Results are also contained in [14].

9. P. V. Kokotovi6, W. R. Perkins, J. B. Cruz, Jr. and G. D'Ans, e-Coupling method for near-optimum design of large-scale linear systems, Proc. IEE 116, 889-892 (1969).

10. R. E. Kalman, Contributions to the theory of optimal control, Bol. Soc. Mat. Mex. 102-119 (1960). 11. R. H. Park, Two-reaction theory of synchronous machines--I. Generalized method of analysis, AIEE Trans.

48, 716-730 (1929). 12. C. Concordia, Synchronous Machines--Theory and Performance. Wiley, New York (1951). 13. M. Cfilovi6, Dynamic state space models of electric power systems, ALCOA Foundation Prof Rep., Depart-

ment of Electrical and Mechanical Engineering, University of Illinois, Urbana, Illinois (1971). 14. G. Singh, Optimization and decoupling of large dynamic systems with applications to power systems,

Ph.D. Thesis, Rep. R-531, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 (September 1972).

15. J. E. Potter, Matrix quadratic solutions, J. S IAM Appl. Math. 14, 496-501 (1966). 16. A. F. Fath, Computational aspects of the linear optimal regulator problem, IEEE Trans. Automatic Control

AC-14, 547-550 (1969). 17. A. W. Rankin, Per-unit impedances of synchronous machines, AIEE Trans. 64, 569-573 (1945). 18. C. Concordia, Rotating electric machine time constants at low speeds, AIEE Trans. 65, 882-886 (1946).

APPENDIX A PER UNIT SYSTEM AND NOTATION AND N O M E N C L A T U R E

A.1 P e r un i t s y s t e m

The per uni t system of values used in this paper is consis tent with that used in power

system analysis. The value of each var iable is expressed as a ra t io of its absolute value and

its base value. The me thod of choos ing base values used here is explained in [173. N o t e

that the base value is no t to be confused with the nomina l or opera t ing point value. They

are the same only in the case of the variable w denot ing angular f requency and have been used in terchangeably in that context.

A.2 N o t a t i o n and n o m e n c l a t u r e

N o t a t i o n

The symbols used in this paper are systematized as fol lows:

U p p e r case c h a r a c t e r s - - a b s o l u t e values of variables and parameters .

Lower case c h a r a c t e r s - - p e r uni t or relat ive values of variables and parameters .


Sign A--var ia t ion of variables around nominal value superscript by index O.

determinant of matrices when used with subscript.

d Sign • - time derivative.

dt

The only exceptions to this notation are the R, L, X, B, Z which may represent per unit values of various types of impedances.

Nomenclature

A, u

A B1, B2 B,b bo B C, c

ea, eb, ec

e d, eq, c 0

e l , e 2 , . . . , e l l

f H H,h ~., ih, ic ld, iq, i 0

(rd l id

Ilq

lbb

Ira, lie, llq

J

.J ke k~ kp k,, kp~ lii l o Li i Li.i ld

turbine gate opening system matrix susceptances of transmission line constants relating flux linkages and current in the direct axis constant relating flux linkages and current in the zero sequence axis control distribution matrix constants relating flux linkages and current in the quadrature axis voltages of phases a, b, c respectively voltages of direct, quadrature and zero sequence axes respectively turbine performance coefficients field winding voltage frequency inertia constant water head current in phases a, b, c respectively current in direct, quadrature and zero ~equence axes respectively current in field winding current in direct axis amortisseur current in quadrature axis amortisseur generator output current per phase current at the bus bar per phase current in field, direct axis amortisseur, quadrature axis amortisseur ~caled

by 2:3 moment of inertia imaginary unit gain constant of main exciter gain constant of amplidyne gain constant of pilot exciter gain constant of magnetic amplifier gain constant of potential transformer self inductance of ith circuit mutual inductance between ith and jth circuits self inductance of rotor circuit i scaled by 3/2 mutual inductance between ith and j th rotor circuits scaled by 3/2 direct axis synchronous inductance quadrature axis synchronous inductance direct axis transient inductance

Opt ima l regu la t ion wi th order r educ t ion o f a power sys tem 449

t~ M e , me

Mr, mt N , n

B,p, P, Pe,Pe Qt, qt qc Qe, qe Q r

r:a r id

r lq

R:d, Rla, Rlq Rz rg

r I

R S, s S

t

r~ r~ re rG r~x r~l, r~ r , rm

HI~ /'12

Ut

Db U pt

I)ref I) m I)p

Ual

l)a2

l) e

v: X~

direct axis subtransient inductance quadrature axis subtransient inductance generator electromagnetic torque turbine torque speed of rotation turbine power output generator power output turbine water flow penstock water flow generator reactive power output weighting matrix for state variables resistance of phases a, b, c resistance of field winding resistance of direct axis amortisseur resistance of quadrature axis amortisseur resistance of corresponding rotor circuit scaled by 3/2 resistance of transmission line in per unit permanent speed droop of governor transient speed droop of governor weighting matrices for control variables voltamperes Laplace domain operator time coordinate penstock time constant servomotor time constant dashpot time constant acceleration time constant of generating unit main exciter time constant

. amplidyne time constants pilot exciter time constant magnetic amplifier time constant direct axis transient open circuit time constant direct axis transient short circuit time constant direct axis subtransient open circuit time constant quadrature axis subtransient open circuit time constant control inputs terminal voltage of generator, per phase bus bar voltage per phase output voltage of potential transformer reference voltage input voltage of magnetic amplifier input voltage of pilot exciter input voltage of amplidyne stage 1 input voltage of amplidyne stage 2 input voltage of main exciter output voltage of main exciter auxiliary governor variable

C.a~.E.E., Vol, I, No. 3 1

450 GURD1AL SINGH and RICHARD A. YACKEL

X l

O~ s

O~ e

(X a

~ p

O~ m

O~ pt

6 0 0 o

0,, Oh,Oc W

W o

w o

q'a, Ut'b, qJc ~ d, Ut'q, U? O %d

utJ l q

reactance of transmission line in per unit ratio of turbine power base to generator power base ratio of generator to main exciter voltage bases ratio of main exciter to amplidyne voltage bases ratio of amplidyne to pilot exciter voltage bases ratio of magnetic amplifier to pilot exciter voltage bases ratio of potential transformer to magnetic amplifier voltage bases difference of rotor phase angle and the reference phase angle rotor phase angle reference or nominal phase angle phase angles of phases a, b, c respectively per unit angular velocity nominal electrical angular velocity, same as base value nominal mechanical angular velocity efficiency of turbine flux linkages in phases a, b, c respectively flux linkages in direct, quadrature and zero sequence axes respectively flux linkages in field flux linkages in direct-axis amortisseur flux linkages in quadrature axis amortisseur.

All other nontypical variables are explained in the text when used for the first time. As a rule the M.KS. system of units is used.

A P P E N D I X B N L I M E R I C A L D A T A

1. Generator, transJormer and transmission line ratings

The synchronous generator is rated to operate at 100 MVA, 95 MW, 31 MVAR, 0-95 powerfactor, 15.65 kV, 136.36 rpm and 50 Hz. The unsaturated reactances and time constants of the generator are: ld = 0-9, 15 = 0.3, l~ = 0.2; lq = 0.58, I~ = 0.22; Tdo = 7.2 sec, Tj~ = 0-05 sec, Tq'; = 0.1 sec. The field resistance is 0.166 f~ and acceleration constant T 6 is 8.15 sec.

The transformer is rated 100 MVA, 15.65 kV primary, 242 kV secondary and has a reactance lr = 0.125.

The transmission line is 100 miles long and is rated for 220 kV operation.

2. Generator, transjbrmer and transmission line data Jor the linearized system

The following base values are chosen for the generator, transformer and transmission line system: power base, 100 MVA; voltage base, 220 kV (h.v. side), 14.23 kV (1.v. side); field current base, 758 amperes; frequency base, 314 radians/sec. A saturation factor of 0.7 is used to reduce the generator direct axis unsaturated reactances to their saturated values. The expressions in [t 8] are used and the transformer reactance is absorbed with the appropriate generator reactances. The following numerical values result for the generator, transformer and transmission line :

/d = 0-915, / o = 0-854, l¢r ~ = 1.5

/11" = 2.1, 111 q -~ 5-6, l,f a = 0.875

Optimal regulation with order reduction of a power system

l,,ln = 0.875, l,,lq = 1.55, lyld = 0.875

Rid = 0.0679, R lq = 0.0679, Rid = 0.000953

r = 0.005, TG = 8.15

R t = 0.04, X I --- 0.129, Bt = 0.097, B2 = 0.097.

T h e n o m i n a l va lues a re

Vb = 1 " / 0 °, iOb = 0-684/ - - 18"2 °, 6 ° = 0.534, W ° = 314,

• 0 = 0.544, V ° = 1.044/4"34 °, i ° = 0 .642 / - - 1"43, i ° = 0.34, tq

0 0"936, ~ 0 = 0.939, ~pO = _ 0.459, e ° = 0.462, eq =

~P~d = 1.95, W° d = 1, qjoq = 0.85, pO = 0.667,

qO = 0.0674.

451

3. Hydraulic installation parameters

T h e n u m e r i c a l d a t a for the h y d r a u l i c tu rb ine , p e n s t o c k a n d g o v e r n o r is

e l = 0.6, e 2 = 0.95, e a = 0.07, e4 = 0.18,

e5 = 1.67, e 6 = 1.13, e7 = - 0 . 2 , e8 = 0.91,

e 9 = - 0 . 3 4 , e lo = - 0 . 1 3 , e l l = - 0 - 1 4 ,

as = 0.669

= 0 . 8

rg = 0-05, r ' = 0.4, T~ = 0.1, Te = 3"0.

4. Excitation system parameters

Since we wish to syn thes ize a f eedback c o n t r o l l e r for o u r sys tem the usua l vo l t age feed-

b a c k ga in is t a k e n to be zero . T h e n u m e r i c a l d a t a for t he m a i n exci ter , p i lo t exci ter , a m p l i -

d y n e a n d m a g n e t i c ampl i f i e r is

Tex = 0-25, ke = 2"5, ~% = 0"0025

Tp = O" 1, kp = 3"3, ~tp = 0-56

Tax = 0.04, To2 = 0.07, ka - 18"05, ~a = 0.76

T,, = 0.1, km= 1-2, ct,, = 0.714

~pt = 2200.

T h e sys tem m a t r i x A, c o r r e s p o n d i n g to the a b o v e d a t a is s h o w n in T a b l e 5. T h e n o n - z e r o e l emen t s o f t he c o n t r o l m a t r i x B are

B(4 , 1) = I0.

B(15, 12) = 12.


Table 5. A-Matrix of linearized system

4- ) . 2 2 7 ~

+

+

+

-2.~}83

+

314.

+ + • t -0 10638 - ) .©78 -i , i24

+ + + +

- 0 . 4 4 8

+ ~ t t

- l O . O0 -4., 1 ] . 6c,

+ ~ t + +

t t • t

-0. 416 i. )7'

• + • + t

+ + + ~

+ + + t t

+ t + T +

273.06 144.12 20.0~7

+ + + ~ +

-162.91 -294.8 2~,.9%

+ t + ~ f

0.,~677

t + + + +

+ + + ~ 4-

+ + t 4-

+ + + + +

# + +

~ t

• + +

+ p

t t

t t t

4 . ~ [ ~ 7 . 6 O 0

+ + +

-14.2~ 257 - '

- 2 % , 0 ~ '

+ + +

-f. +

+ t + +

+ + + +

+ t + ~-

+ + + +

+ + + ~ #

+ t + ~ + .1~77 9 . 3 1 3 ~ , t;4Li -., 7 ~

, 4 1 (~ . ~ ~

+ t +

T + --

t T Jr

t t r

t ~ ~

4r t t ,~

- 4 4 . 8 £ 411.i~ ~ : ~ - 2 : , ,

-P t ~ -t +

-441.2 -34.27 2 9 . 2 6 . 5

1 5 . 2 4 i , . ) r

14.054 7. ~12'~

J r - ~ t t

b

+

+

,+

L4 .~,o

÷

+

+

r

+

-, o ,,

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Optimal regulation with order reduction of a power system