Transient approximation of a bilinear two-area interconnected power system

Electric Power Systems Research, 26 (1993) 11 19

11

Transient approximation of a bilinear two-area i n t e r c o n n e c t e d power system S. A1-Baiyat, A. S. F a r a g and M. Bettayeb Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261 (Saudi Arabia)

(Received June 25, 1992)

Abstract A balancing model reduction scheme for high order bilinear systems, similar to the linear balanced reduction algorithm, is applied to power system modelling. An original 17th-order two-area interconnected bilinear power system is reduced, using both linear and bilinear balancing algorithms, to a 10th-order reduced model. Overall superiority of the bilinear reduction scheme over linear balancing is observed in the performed simulation study. The reduction scheme also leads to a very acceptable approximate response compared with that of the original system. Keywords: two-area interconnected power system, bilinear system balanced model reduction.

1. I n t r o d u c t i o n L i n e a r systems t h e o r y is well established. Modelling, identification, a p p r o x i m a t i o n and model reduction, optimal control, filtering a n d estimation of physical l i n e a r systems made i m p o r t a n t a d v a n c e s in the last two to three decades. Non-linear systems analysis and synthesis tools are l a c k i n g due to t h e i r g e n e r a l i t y a n d complex behaviour. However, a special class of non-linear systems, called bilinear systems, naturally arises in m a n y physical systems such as power systems, n u c l e a r power plants, chemical plants, biomedical systems and other areas of engineering. M a t h e m a t i c a l modelling of these physical systems often leads to high order bilinear models. Simplification of such h i g h order bilinear systems by lower order bilinear models n a t u r a l l y leads to better approximations t h a n lower order linear models. Bilinear model reduction is desirable as it leads to simplified analysis, s i m u l a t i o n and design of bilinear systems. Design and i m p l e m e n t a t i o n of schemes r e c e n t l y developed for the control, filtering a n d estimation of bilinear systems require r e a s o n a b l y low order bilinear models [1-5]. Unlike model r e d u c t i o n of linear systems, little a t t e n t i o n has been paid in the l i t e r a t u r e to the model r e d u c t i o n of bilinear systems. Recently, the model r e d u c t i o n problem for bilinear systems has been i n v e s t i g a t e d using three different ap0378-7796/93/$6.00

proaches. In the first a p p r o a c h [6], a necessary and sufficient condition for a bilinear system to admit an aggregated reduced order bilinear model was given. In the second a p p r o a c h [7], a method was developed for finding r e d u c e d order models with a specified s t r u c t u r e which gives the best approximation, in the least square sense, of state-affine systems over a finite interval. Bilinear systems were considered as a special case of the proposed work in ref. 7. Finally, some prelimi n a r y work concerning the model r e d u c t i o n problem of bilinear systems using b a l a n c i n g was reported by Hsu et al. [8]. This paper is organized as follows: §2 gives the b a l a n c e d model r e d u c t i o n a l g o r i t h m for bilinear systems; the general state-space r e p r e s e n t a t i o n of a power system is given in §3; in §4 the bilinear model of a two-area i n t e r c o n n e c t e d power system is presented; in §5 s i m u l a t i o n of an original 17thorder system and the linear a n d bilinear reduced order models is given.

2. Bilinear m o d e l r e d u c t i o n a l g o r i t h m B a l a n c i n g of linear systems [9] rests on two i m p o r t a n t quantities, the c o n t r o l l a b i l i t y and observability Grammians. A l i n e a r system is said to be b a l a n c e d if the c o n t r o l l a b i l i t y a n d observability G r a m m i a n s are equal and diagonal. The diagonal entries of these Grammians, called the (C 1993 ElsevierSequoia. All rights reserved

12

singular values, measure the degree of controllability and observability of the states in this representation, and therefore directly affect the input o u t p u t dynamic response of the system. States corresponding to large singular values of the Grammians are strongly controllable and strongly observable and are therefore r e t u r n e d in the b a l a n c e d r e d u c t i o n scheme. W e a k l y controllable and w e a k l y observ~ible states with small singular values have the least effect on the response and are therefore omitted. The concept of balancing for linear systems can be extended to bilinear systems by defining similar operators. Bilinear systems are those systems which are linear s e p a r a t e l y with respect to the state and the control, b u t not jointly. They can be characterized by the following state variable equations: x(t) = Ax(t) + ~ i=1

y(t)

(1)

0

Zh

0

1

f

~

Jl ,J2 .....

Jm = 1

exp[A(t-rl)]

0

× Nil e x p [ A ( r , - r2)]Nj2... × N ~ exp[A(zk , - r k ) ]

Bujl(rl).

• •

Qi_,Nm

(2)

B

i =2,3 .... The controllability and observability G r a m m i a n matrices are then defined as P = i=,~ f ' ' "

f PiP* dtl...dti

It can be shown [11] t h a t the Grammians will satisfy the following generalized algebraic Lyap u n o v equations: A P + P A T + ~ N~PN w + B B T = 0

exp(At~ ) NmP~ _ 1]

(7)

i=l

and ATQ+QA+

~ N~QNi + C T C = 0 i-1

(8)

(9)

G = (A®I +I®A

•

+ N , @N,

+N2®N~+'"+Nm

(10)

®Nm)

and p = vec(P) = (p~,, P21,

• • • ,

Pnl, P12,P22. . . . .

P,n, ~. . . .

Pnn)

T

c = vec( - B B T)

P~ ( t , , . . . , t~ ) = [exp(At~ ) N, Pi-1, 1,

(5)

and

P n2 . . . . .

N2Pi_

(4)

exp(Ati)

where

To develop the idea of balancing bilinear systems consider the following: let {P,, P2, P~, • • • } and {Q1, Q2, Q~ . . . . } be infinite sequences of vectors defined recursively from the matrices A, N1 . . . . , Nm, B, and C, of eqn. (1) as

exp(Ati)

ti ) =

exp(Ati )

Gp = c

× u~m(rk) d r , . . , drk

Pl(t) = exp(At)

Qi (t, . . . . .

1 N1

Qi-, N2 exp(Ati )

The s i n g l e - i n p u t - s i n g l e - o u t p u t version of eqns. (7) and (8) was first r e p o r t e d in ref. 8 w i t h o u t proof. Clearly, if Ni = 0 for i = 1, 2 . . . . , m, then eqns. (7) and (8) reduce to the normal L y a p u n o v equations of linear systems, as expected• E q u a t i o n (7) can be written in a K r o n e c k e r p r o d u c t linear matrix equation as

= t C exp[A(t - r~)] BU(rl) dr1 J 0

+ k~=2 f f " "

Qi-

f

= Cx(t)

t ~:1

Q, (t) = C exp(At)

Nix(t ) U i (t) ~- Bu(t)

where x(t) is an n × 1 state vector, u(t) is an m × 1 input vector; ui is the ith c o m p o n e n t of u(t), y(t) is a p × 1 o u t p u t vector, and A, Na, N 2, . . . , Nm, B, and C are real matrices of appropriate size. It is well k n o w n t h a t the i n p u t - o u t p u t repres e n t a t i o n of system (1) is given by [10] fbt

y(t)

and

•

•

,

i=2,3 ....

(3)

The existence and uniqueness of eqn. (9) is discussed in ref. 2. Equation (8) is t r e a t e d similarly. Once P and Q have been determined, the balanced realization of system (1) can be obtained

13 by applying the state-space b a l a n c i n g transformation

Bb = T 1B

(22)

Ch = CT

(23)

xb(t) = T ~x(t)

To obtain a reduced order model, let the balanced system given in eqn. (12) be p a r t i t i o n e d as

(11)

to eqn. (1). The state-space r e p r e s e n t a t i o n of the new system is

Xb2J

Xb(t) = AbXb(t) + ~ Nb~Xb(t)u~(t) + BbU(t)

i 1

y(t) = Cb Xb(t)

LA21 A22Jkxb2J

~- ~ I Nlli N12i~[Xb1~ui + [BI 1 u

(12)

i = 1 LN21i

N22iJLXb2J

B2

(24)

where y = [cl

Nbi=T

Ab = T-1AT, Bb = T - I B ,

Cb = CT

The c o n t r o l l a b i l i t y and observability G r a m m i a n s of the new system are given by Pb = T 1pT-T

(13)

Qb = TTQ T

(14)

Moreover, these G r a m m i a n s are equal and diagonal with the following special a r r a n g e m e n t :

Pb = Qb = 1~ = diag[al, a2 . . . . . a,]

(15)

a~ >~a2>~..- >~a, > 0 The a~, called the H/inkel s i n g u l a r values of the system, are d e t e r m i n e d by

ai = [2~(PQ)] ~/',

(16)

where )~i (PQ) denotes the i t h eigenvalue of PQ. An efficient a l g o r i t h m for the c o m p u t a t i o n of a b a l a n c e d r e p r e s e n t a t i o n for linear systems developed by L a u b et al. [13] will be used in this paper to find a b a l a n c e d bilinear system. The a l g o r i t h m is summarized as follows. Step 1. Use eqns. (7) and (8) to find the controllability and observability Grammians. Step 2. Compute the Cholesky factors of the Grammians: let Lr a n d L o denote the lower triangular Cholesky factors of G r a m m i a n s P and Q, t h a t is, e = L r L rT,

Q = LoLoy

(17)

Step 3. Compute the s i n g u l a r value decomposition of the p r o d u c t of the Cholesky factors: LoTL~= UZV T

(18)

Step 4. Form the b a l a n c i n g t r a n s f o r m a t i o n T

= LrV?E

(19)

1/2

Step 5. Form the b a l a n c e d state-space matrices Ab = T - ~AT Nhi = T - 1 N i T

c21 xu2

1NiT

(19) i = 1, 2, . . . , m

(21)

where the vector Xbl e R r c o n t a i n s the most controllable and observable states and the vector Xb2 e R n- r c o n t a i n s the least controllable and observable states. Also, let Z be p a r t i t i o n e d in a similar way:

where Z1 = diag[el, . . . , arl and I~2 = diag[o"r . 1, . . . . a,~]. If O'r/O'r+ 1 ~ 1, t h e n the subsystem given by

i 1NlliXbrUi(t ) + Blu(t)

3~br(t) = A H X b r ( t ) + ~

(27)

~(t) = ClXbr(t )

(28)

is the reduced order model of the full order balanced system which will c o n t a i n only the most controllable and most observable parts of the system.

3. D y n a m i c s t a t e - s p a c e m o d e l s o f electric power systems An electric power system, composed of sets of power plants, t r a n s m i s s i o n lines, t r a n s f o r m e r stations and substations, secondary d i s t r i b u t i o n n e t w o r k s and consumer loads, is a complex dynamic system. The t r a n s i e n t performance of such a system can be described by a set of differential equations of the form ~(t) = fix(t), u(t), t]

(29)

where

x

= [V, (",f,]',p, q, v, ~, ~]

state vector

u = [fm, Vm]T control vector V = [V1, V2 . . . . .

f

Vn] w subvector of system voltages

= [fl,f2 . . . . . fn] w subvector of system frequencies

14

P

= [Pl,P2

s u b v e c t o r of reactive powers in system

q =[ql,q2,-.-,qn] T v

=[V~,V~,...,Vn]

(ii) the process of reactive power and system voltage behaviour. I n t e r a c t i o n b e t w e e n the p h e n o m e n a described by these two models is such that, on the whole, the second has a small influence on the first, which implies t h a t the model describing the system power and frequency b e h v i o u r can be assumed to be i n d e p e n d e n t of the p h e n o m e n o n dealing with system voltages and reactive powers. The model presented deals mainly with the p h e n o m e n o n of e l e c t r o m e c h a n i c a l transients and system models suitable for the study of primary and s e c o n d a r y load and frequency control in electric p o w e r systems [14-18].

s u b v e c t o r of real powers in system

. . . . ,p,]T

s u b v e c t o r of phase angles of system nodes

T

2 = [21, 22 . . . . . 2,~]I" s u b v e c t o r of other, nonliste'd, system state variables L

:

[ f m l , fro2 . . . . .

fmn] T

Ym = [Vml, IJm2, " " " , Vmn ] W

s u b v e c t o r of real power control s u b v e c t o r of reactive p o w e r control

n = total n u m b e r of nodes in system

4. B i l i n e a r p o w e r s y s t e m m o d e l

With the a s s u m p t i o n of small v a r i a t i o n s of all variables a r o u n d an arbitrarily chosen reference operating point, the system model can be bilinearized and described by a set of first-order differential equations, with c o n s t a n t coefficient matrices and variables t a k e n as variations a r o u n d reference values [14]. The overall system model can be decomposed into submodels, describing two separate phenomena: (i) the process of real p o w e r and system frequency behaviour;

U n d e r the assumptions adopted, the system frequency is a common variable for all the elements of any p o w e r system, as shown in the schematic r e p r e s e n t a t i o n of one p o w e r system in Fig. 1. This scheme, based on physical relationships among different elements, will be used for further development of m a t h e m a t i c a l models of different elements and the power system as a whole. The elements to be considered are: (i)

f - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . .

I Hydro Plant 1 , , Contr~ . ," I-- - ~ - - ~ T ~ r

__J Hydraulic I I nste

~,

II

Ins=n.

~

I ~

Tu~..

~

i

.

: i T~.~n.

~ Opening [. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

I~1 Generator

'

~

-;;,~-~;r

I

j

I

I

',

/

C..n..., I

JI-

II

i I

I I i t

F~

I i

,

'

........

_--

Transmission Network With System Inertia

I

'

.

i

i - I IV~ I I Opening L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Control

Generator

System Fr~.~ueM"

_o~_,_%. . . . . . . . . . . . . . . . . . . . . . . . . .

--I ~ovemer

Control

Head Water r~w

Tud~ne

[_u. . . . . . . . . .

Control

~ ~

.... --]--;~£

~ i Turbine - i r,.~overner

I ~

I

I

I

.... ~jJ----~;-~~

_L'~'t Tudoine

I"--~l~Generator I I

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

Fig. 1. Electric p o w e r s y s t e m - - s i m p l i f i e d s c h e m e .

)lant utputs

....

r~1~l C"Qnerlllt° r

Opening

i

II i I

Flow I I

Tie line With Another Power System

15

t u r b i n e governors for steam and hydro units, (ii) hydraulic installations and turbines, (iii) steam turbines with the installations for steam generation, (iv) p o w e r plants, (v) consumers, (vi) transmission n e t w o r k with system inertia, and (vii) tie-line. A model for two i n t e r c o n n e c t e d p o w e r systems [14], each area having one steam and one hydro unit, can be r e p r e s e n t e d as follows.

Tsw, AaT1 - - -

A f 1 -[- I~/,ST1AfmT1

A/~(1)_ Kt~ 1 - - t l -- T~ 1 AaT1 -- ~ AP{~I) A]~(1) -

~t2

~H2 Kq2

+ ~

T= e2

1

~

Aq2 - ~ Af2 + ~ AP~2 - ~ APL~

r~ AV2 = G

-- Vr 1 ~ - - t l

1 AaH2 -- G

AV2

AdH2 _ -- eT2 G 2 T~2 a p ( 2 ) ~--tl T2 TSH2

ADO)____I AP(~)

l

emKw2 Aam

T~

1

rT1

--

+ ew2(1 - Cw)(1 - Cs2) A p ~ )

AO2 = 1 hall2 1 1 Tw2 -T~q2 Aq2 + ~ Af2

System I AdT~ --

System H a f2 - ~ Cv~ nP{p + eT~(1 -- Cw)G2 AP[~) T~ T~

eT2 ( 1 -- Cv2)Cs2 Ta2 A ID(2)

~--t2

T2 TSH2

_ eT2(1 -- Cv2)(1 -- Cs2)Ta2 Ap{~)

Vr I ~ t 2

T~Ts.~ Ap(,)_

1 APO)_

--t3 - T n l Ad m --

--

~-t2

1 AP(~)

+

Tnn; ~-t3

1

8H2

~SH2(

Tw2Ta2 T2 rHz--r2) AaH2

GT1 Cv1 Tal Ap{1)_ ~T1 ( 1 - Cv, )Cs, T ~ ~--t2aP°)

T1 TSH1

1

T1 TSH1

~H2Kq2 Ta2 e2 Ta2 -- T2 Af2 T2 TSH2 Aq2-4- T2 TSH2

h

+ TSH2

V2

gT1(1 -- CV1)( 1 -- Csl)Tal kp{~ ) T1 TSH1

1

(gHIKwlT~I

q-TsHI\

T1 TSH1 Aqi

_ Tal _

T~T~.I aP~ + ~

rl 1

A/1

-

1

+

- - T, T, Tsm Af,

Ab(Z)= 1 A P ( m _

e I Ta!

Tal

afm.~ + T~T~.~ API~

Te, AV, 1

1= Ap(2 ) A- t 21 ~51- t , (A p (22 ) _) Tr2 t~

Kt2 AP~) = Tu2 Aaw2

1 Ap(2 )

~

~--tl

AdT2 -- -- rT2 AaT2 -- 1 Af2 + TST2 1'ST2 1

Aa~ - Tq~ Aq~ + Tnn Af~

ET1 Cv1 AD(1 ) _~_GTI(1 -- Cv1)Cs1 A jg(1) ~--t2 T, ~--tl T1 A- 8T1(1 -- C v l ) ( 1 -- CS1) ~--t3Ap(1) eHIKw~l A a m T1 T1 gill Kql

+~Aql-~l

el All -

1

Ta2

AfmH2 + T2 TSH~ APL~

1 A/:)(2)

1

A V ' = Tel AaHl

AO~ = ~

FHI--F'IAaH1

em & l Tal A V1

T~H2

)

~

1 if- ~

0~Ta2

7'2 TSH2 AP12 +

1 1 ~2 AP12 - ~-~1APL~

Disturbance and tie-line APL~ = K~,(AaH, Af~T,) + K~2(Af~ Af~m)

1

AfmT2

The variables and p a r a m e t e r s in the above equations are defined in Appendix A. The state-space variables of the two interconnected mixed p o w e r systems are Xl ~ AaT1,

Ap(1) X2 ~ ~ - - t l ,

AlO(1) -- A iD(1) X3 = /~/-t2 ~ X4 -- ~--t3

xs=Aam,

x6=AV1,

xv=Aql,

xs=Afl

xg=AP12,

Xlo=Af2,

xn=Aq2,

x12=AV2

X13 : AaH2,

X14 = AD(2) ~ t 3 ~ X15 --- AD(2) ~ z t 2 ~ X16 : Ap(2) ~tl

x17 = Aaw2,

ul = Afmwl, U2 = Afmm

APL~ = K~I(Af2 AfmH2) + K~2(AaH2 AfmT2)

Ua = AfmH2, U4 = Afmw2

AP,2 = m~)(Af~ - Af2)

Z 1 :

klX5U1-{-k2xsU2,

Z2

:

k3xloU3 q- k4x13u4

16

The coefficients of the state equation can be simplified and the state and output equations can be written in the form icl = A~xl + A2x4 + BlUl, 23 = A~x2 + A6x3,

ic,~=A3x~ + A 4 x 2

24 = ATx3 + Asx~

ing changes in system frequency and tie-line power, evaluated for step changes in both areas, are s h o w n in Figs. 2 ( a ) - ( e ) . For a step variation in steam admission to one unit in area 1, the different responses to frequency variations in both areas and the tie-line power are

2 0 = n 9 x 2 + nlox~ + n l l x 4 + AleX5 + A13x6 + A14x7 + A15xs + A16x9 + B2u2 + ~/lzl 2 6 = A17X5 + a l s x 6 ,

X7 =

algx 5 + A20x7 + a21x8

A27xs + A2sx9

bbal.

0

+ ~/2zl

-0.05

29 = A29xs + A3oxlo 210 = A31x9

orig.

0.1 6. 0.05

2s = A22x2 + A23x3 + A24x4 + A25x~ + A26x7 +

0.15

I0 20 TIME(sec)

+ A32x~o + A33x. + A34x13

+ A35x14 + A36x15 + A37x16 + ~3Z2

30

Fig. 2(a). F r e q u e n c y v a r i a t i o n in a r e a 1 d u e to a step v a r i a t i o n i n s t e a m u n i t no. 1.

2 H = A38x10 + A39Xll + A40x13 0.15

3C12= A41x12 + nn2x13

orig.

X13 = a43x9 + a44x10 + a45x11 -~- A46x12 + a47x13

+ n48x14 + nn9x15 + A50x16 + B3u3 + ~3z2 214 = A51x14 + n52x15,

215 = A53x15 +

X16 = A55X16 + A56x17,

X17 =

Yl=Xs,

Y2=X9,

n54x16

A57Xlo+ A58x17

¢£ 0.05

/

0 -0.05 0

I0 20 TIME(sec)

y~=X~o

Note that the bilinear terms z~ and z2 in the above 17th-order model are present in the 5th, 8th, 10th and 13th dynamic state equations. The values of the parameters in the above equations are listed in Appendix B.

bbak

0.1

30

Fig. 2(b). F r e q u e n c y v a r i a t i o n i n a r e a 2 d u e to a step v a r i a t i o n i n s t e a m u n i t no. 1.

1

0.8

orig...~ / ~ / ~ . _

0.6 '~ 0.4

lbal.

5. S i m u l a t i o n r e s u l t s a n d c o n c l u s i o n s 0.2

An electric power system, composed of sets of steam and hydraulic power plants, transmission and distribution networks, is a complex dynamic system. System frequency and tie-line power under various operating conditions are used as variation p h e n o m e n a to evaluate different models of power systems to reduce the computation time used in very large power system simulation. Different techniques have been incorporated in system reductions for a multi-area power system using linearized models. The i n p u t - o u t p u t behaviour of a two-area power system consisting of different steam and hydraulic generating units is compared using the original 17th-order model (orig.), the reduced balanced linear (lbal.), and the reduced biliaear (bbal.) 10th-order models. The results of compar-

0 0

10 20 TIME(sec)

30

Fig. 2(c). T i e - l i n e v a r i a t i o n d u e to a s t e p v a r i a t i o n i n s t e a m u n i t no. 1.

0.2 0.1 / :5 0 "J

/ ~

/orig. ~ff/bbal. ~lbal.

-0.1 -0.2

10 20 TIME(sec)

30

Fig. 2(d). F r e q u e n c y v a r i a t i o n in a r e a 1 d u e to a s t e p v a r i a t i o n in h y d r o c o n t r o l u n i t no. 1.

17 0.2

0.5

_

ofig.

0.1 cL

0

bbal.

0

,~ -0.5

-0A

-i.5

-0.2

orig. bbal.\

A

JIA 10 20 TIME (sec)

30

Fig. 2(e). F r e q u e n c y v a r i a t i o n in a r e a 2 due to a step v a r i a t i o n in hydro control u n i t no. 1.

1.5 [ /

0

V 10 20 TIME (sec)

30

Fig. 2(i). Tie-line v a r i a t i o n due to a step v a r i a t i o n in hydro control u n i t no. 2.

0.1 Ol'lg.

0

't A A A bba ';

lbal. , , , /

-0.1 6, -0.2 -0.3

-0.5 0

10 20 TIME (sec)

30

Fig. 2(f). Tie-line v a r i a t i o n due to a step v a r i a t i o n in hydro control u n i t no. 1.

-0.4

20 TIME (sec)

30

10

Fig. 2(j). F r e q u e n c y v a r i a t i o n in area 1 due to a step v a r i a t i o n in steam u n i t no. 2.

0.15

0.8

orig.

bbal.,,~

0.6 0.1

,~ 0.4 0.2

0.05

0 -0,2

10 20 TIME (sec)

30

,

10

20

30

TIME (sec)

Fig. 2(g). F r e q u e n c y v a r i a t i o n in area 1 due to a step v a r i a t i o n in hydro control u n i t no. 2.

0.8

0

Fig. 2(k). F r e q u e n c y v a r i a t i o n in area 2 due to a step v a r i a t i o n in s t e a m u n i t no. 2.

0.15

,

0.6

, orig.

, bbal. ,~

0.4 d, 0.2

0.05

0 -0.2

10 20 TIME (sec)

30

0

10

2'0

30

TIME (sec)

Fig. 2(h). F r e q u e n c y v a r i a t i o n in area 2 due to a step v a r i a t i o n in hydro control u n i t no. 2.

Fig. 2(1), Tie-line v a r i a t i o n due to a step v a r i a t i o n in steam u n i t no. 2.

shown in Figs. 2(a) (c) for the three different models. Results for frequency v a r i a t i o n s and tieline power for a step c h a n g e in the hydro control u n i t in area 1 are shown in Figs. 2(d)-(f). Results for step changes in hydro control u n i t 2 on frequency v a r i a t i o n s and tie-line power are shown in

Figs. 2(g)-(i), while results for a step change in steam u n i t 2 are shown in Figs. 2(j)-(1). From the Figures, it is clear t h a t the reduced bilinear model gives better a p p r o x i m a t i o n s t h a n the reduced linear model for a step v a r i a t i o n in the frequency of the fast response steam units. In the

18

case of the slow response hydro units, the reduced bilinear and linear models give approximately the same responses. This is in accordance with the nature of the dynamics of the physical units. It is clear from these results that the 9th-order linear balanced model can substitute and present fully the original model of the two-area power system presented by a 17th-order model. The linear balanced model can be used in some cases and may not be adequate for other cases. In conclusion, the 10th-order bilinear model can represent well the actual 17th-order power system. Comparison of bilinear and linear models for the same order level shows the merit of the proposed bilinear model reduction techniques over linear model reduction.

Appendix A Aam, Aa,2 AaT~, AaT2 CS1, CS2

C V, 1Cv2 e l , e2

Af~, Af2 AfmH1, AfmH2 Arrow1, AfmW2

g~l, gt2

gwl, gw~

K~I, K~2

variations of hydro turbine gate opening variations of steam turbine valve openings in two systems fractions of total power generated downstream from reheater by intermediate pressure turbines fractions of total power generated by high pressure turbines consumer and turbine system self-regulation coefficients frequency variations hydro unit control input variations steam unit control input variations constants derived from steadystate operation of hydroturbines, relating gate opening as function of water flow through turbine proportionality factors connecting control valve position variation and turbine output variation in steady state, p.u. constants derived from steadystate operation of hydroturbines, relating water head as function of water flow through turbine constants relating load disturbances to variations of steam turbine valves and control input variations

KI~

constants relating load disturbances to variations of hydroturbine gate opening and control input variations constant relating frequency ms variation to tie-line exchange tie-line exchange variations AP12 AP2~ = - ¢( AP~2 change in tie power area load disturbances mixed AP[~, APL~ from variations of gate openings and frequency variations An(l) Ap(2) variations of steam turbine, Ftl , ~tl high pressure output An(l) Ap~) variations of steam turbine, in/'-t2 termediate pressure output A r}(1) AD(2) variations of steam turbine, low /~t3 ~ ~ a t 3 pressure output Aql, Aq2 variations of hydro turbine flow permanent speed droop of hyFH1 ~ FH2 droturbines, p.u. permanent speed droops of two FT1, rT2 systems, p.u. ! ! transient speed droops, p.u. El, r2 T,,T2 system acceleration time constant Ta,, Ta~ time constants of accelerators time constants of dashpots To,, To2 T~, T~ time constants of penstock when net head is constant time constants in crossover pipTni, Tn~ ing time constants of penstock when T.1, net head and gate opening are constant time constants in reheat piping Trl ~ Tr2 time constants of system pilot Ts.,, Ts.2 valve servo motor turbine gates for hydro system time constants of system pilot TST2 TSTI• valve servo motor turbine gates time constants of turbines, Tul, Tu2 characterizing time delay between control valves and turbine nozzles time constants of penstock when Twl ~ Tw2 gate opening is constant A V1, A V2 variations of dashpot piston position relative to level of permanent speed droop of hydroturbine governor participation coefficient of hy~HI~ ~H2 dro generating unit in total system load participation coefficient of ~T1, ~T2 steam generating unit in total system load K~2,

19 Appendix

B

A, = -2.0

A2 = -4.0

A 3 -- 4 . 8

A 4 =

A6 = - 0.2 A , 1 -- 0 . 1 1

A 7 = 2.0 A , 2 -- - 4 . 0

A 8 = - 2.0 A13 = 1 0 . 0

A 9 = - 0.08 A14 = - - 0 . 9 3

A,6 = 0.67

A17 = 0 . 2

A18

A19 = 1 . 3 2

A20 = - 1 . 4

A21 -- - 0 . 2 8

A22 = 0 . 0 1

A23 = 0 . 0 1

A24 = 0 . 0 1 4

A25 = - 0 . 0 6

A26 -- 0 . 1 2

A27 = - 0 . 1 1

A28 = - 0 . 0 8

A29 = 2 2 . 2

A30 -- - 2 2

A31 A3~ A41 A46 As, As~

-- 0 . 0 8 = 0.01

A~2 = - 0 . 1

A33 = 0 . 0 9

A34 = - 0 . 0 5

A35 = 0 . 0 2

A37 = 0 . 0 1

A38 = - 0 . 2 8

A39 = - 1.4

A40 = 1 . 3 2

=

A42 A47 A52 A57

A43 A4s A~3 A58

A44 = - - 0 . 9 1 A49 = - 0 . 0 9 A54 = 0 . 1 7

A45 = - - 0 . 7 4 A59 = - 0 . 1 0 A ~ -- - 5 . 0

- 0.5

= 10 = - 2.0 = 4.8

fll = 4 . 0 ,'1 = 0 . 0 6 7

= = -=

0.2 - 4.1 2.0 - 4.0

f12 = 1 0 . 0 ~'2 -- - 0 . 0 0 8

=

= = = =

--0.5

--

0.67

0.14 0.17 2.0

f13 = 1 0 . 0 ~'3 = - 0 . 0 0 8

The

authors University

A~ = 0.2 Ale = - 0.08 A15 -- - 9 . 1

f14 = 4 . 0 74 = 0 . 0 6 7

Proc. 22nd IEEE Conf. Decision and Control, San Antonio, TX, USA, 1983, pp. 783 788.

Acknowledgement

Fahd

-5.0

acknowledge of Petroleum

the support and

of King

Minerals.

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