Dynamic System Simplification and Stability Using Vector Lyapunov Functions

DYNAMIC SYSTEM SIMPLIFICATION AND STABILITY USING ViCTOR LYAPUNOV FUNCTIONS

c.

LakShminarayana

Department of Blectrical Engineering, College of Engineering, OSmania University, Hyderabad,. India

AbStract: In this paper a new concept of vector Lyapunov functions is introducE!d as a tool for investigating the dynamic' stability behaviour of a large scale power .system. This concept is then extend'ed to derive reduced ord er equivalents 'of the . interconnected power system. It is shown that the behavior of this equivalent is similar to that of the original system and is obtained with reduced comp~tational effort. The method is illustrated with reference' to two numerical examples. Advantages of such an analysis Of stability and e
IN'l'ROn UCT ION Dynamic stability of a power system is defined as the stability of the 'system under smail perturbations with the action ~ regulating devices considered. Under these conditions the differentialeq~tions defining the system behaviour-- Can belinearised about the operating point in the post fault state and can be cast in the state space form of the type c:

A!

+

BQ

(1 )

where X is the state vector of the proper-order, A is .the system matrix, B is tne control matrix and .J:! is the vector of forcing functions. The dynamic response of the svstem, is then obtained by a·solution of {lJ. For complex interconnected systems,' however, the system matrix A is too la1'ge t.o be.handled even by the present daY computers, thus involVing consiaerable computational problems. . In the past; considerable attention has therefore been devoted to'approximate the linearsys tern (1) by an equivalent model of a lower order. Most of these methods involve the determination of the eigenvalues.of the system which itself- is formidable for a large sized system. Once these e:LgenvaloJes 'ar'e obtained,. a reduced order equivalent is constructed. The respons e of· this sys'tem determines the stability of the original system. In this paper, we present a new tech-

nique of oynamj,c stability investigations, based on a recently developed concept of vector Lyapunov functions and the decomposition and al2:~regation methods. The method consists in representing the large syst em in the form of a number of low order subsystems and their interconnections, Standard techn~ques fOr constructing scalar Lyapunov functions are applied to each of these subsystems. Then an aggregated model involving a vector Lyapunov function is Co~structed for assessing the stability of the overall systems. This new concept of vector Lyapunov, functions was first proposed by Bellman but Bailey demonstrated its USefulness in the analysis of large Scale systems. Recent works of Siljak and others have opened new lineS of research in the applicability of Lyapunov functiotls for the analysis of large scale systems. However, applications of this technique have been few. The early works of. Pai am Narayana and re!=ent work of Jocic have demonstrated the.feasibility of the approaCh in the transient stability analysis of large scale systems. In this paper, we propose this -new method to the dynamic stability in-. vestigations of complex power systems. Dynamic stability analysis of large Systems in a quantitative 'way has hi-. therto been carried out by the USe of 183

reduced order models of the power sy .. stem. These methods, as stated earlier, involve the determination of .the eigenvalues of the large system an:i eliminating certain of' the undesired modes according to the state of the system. An alternative technique of obtaining a simplified mod el was proposed by Sarkar and Rao using quadratiC type Lyapunov functions. The method involves the modelling of an nth order system by an mth order. system so that the Lyapunov surfaces of the model an:i the system match closely. Although elegant in approach, the method involves considerable computational complexities. In this paper, we present a method based on vector Lyapunov functions that enables . one to apply the method of Sarkar and Rao with reduced computational burden. SOM~ ASP ECT oS OF VSCTa R LtAPUNOV FUNCTIONS

·Consider a free linear dynamical system S desCribed by

(2) where the state 'n', the system rum. It will be decomposed ·into t elllS Si d ef ined

.

i, • ~

A· .X. + H-~

vector! is· of order matrix A is of order assumed that (2) is 'm' dynamical subsysby

..... L.e· ·A·· j.llj~~ ~J

~

~J

for all i, j are the properly partitioned submatrices of A. Clearly the state vector l; is given by

!

.

[

!l

.

'!2

T

. •.

TJT

Ii

'"

Aii!i

'?U (Il!i ll J ~ vi l!i)

~ '? i2(I/!i 11)

"i(!i) ~ -YJi3(11!i U)

T •

where vi(!i) '" (Grad Vi) !i is the total time derivative of Vi Of i) alonB the sol ut ions of (6) arrl (Grad vi) is the gradient vector. "'111 and '1i2 are constants given by the lowest arrl highest eigenvalues of P .. and 'YI1 . 3 1.1. 1. is the lowest 'eigenvalue of Q. In (8), vi l!i) .. V ;/2{!.i J. To infer the stability of overall system, i t will be assumed that the interacting functions are bounded and satisfy the following conditions:

t

I![grad v i O~i)J hi (!i)1I ~ • 1,2 •. m,

j

I-

me

.L 5 ijU !jU

J=l

L

(9)

where hi (!oil is the interacting function defined in (J) and ~,. are . 1.J nonnegative numbers. '\lie now d er ine a vector Lyapunov function Yy(!)suCh that

Y.v (!)

..

[VI (!l)'

v,2{!o2) .. vm(!m)]T

!.m

(la)

~ch term in the summation term on the right hand side of DJ is defined as an interacting function that exhibits the action of SUbsystem 'j' on sUbsystem'i'. The constant eij assumes values of 1 or a depend ing on whether Or not subsystem oS, intt!racts with subsystem SJ. If the summation terms are equated to zero, then the system given by

.

11.

where Qii is any ~rbitrary positive definite matrix. Then V.1. (X.) also -1 satisfies the follo.wing cond i tions:

j

-J i=l,Z, ••• m

The matrix p ..

is symmetric and satisfies the Lyapunov-matrix equation T A;; P , + P.1.1.. ,Ai'1. .. -Q.. (7) ...... 1.1. i1

X.

U) th where A' is the state pf the i subsystem ~f order n,. The matriCes A. ,

T

is positive definite.

(5)

is referred to as an 'isolated or free' subsystem. The system ~iof order n. is asymptotically stable i f ~ . there exists a quadratic Lyapunov function V,~ (X. Jof the type -~ . T V·(l,} .. -1 X, P'i.i (6) ~~. '1

Taking the time derivative of (l0) along the solutions of ()) and using (8) and (9), one obtains the following aggregated model of the overall system in terms of the vectpr Lyapunov function Yv {!):

where the elements of the aggregate matrix R of order rnxm are: -I

- '7i'l. "'';3 J.

i .. j

-I

'$;.("1;,

i

f. j

(12)

The stability of the overall system is established by confirming that all the eigenvalues of R have negative real parts, which is eqUivalent to R satisfying the following conditions: 184

1"12

So

0

(-1)

m

rn

rim/

r12

. .

Proof of this statement is contained in re£erence 10. Thus the maximum order of the matrix to be deal't with is equal to the number of sUbsystems derived from the overall system and hence is always of a much lower order than that of the system \2)".· .

r2l

r22

rml

rm2

T

Autu

Consider the continuous time linear dynamical system (2) of order In' given by

Decomposition of (14) into systems yields

'm'

SUb-

.

r2m

(lJ)

r nm - Qu

-t Pi i Aa •

(18)

~

The time derivative by Vi (!i)

V

of V is given

• - 1.T Q.. 1. -~

DYNAM! C S'tSTEM SIMPLIF leAT ION

The vector Lyapunov functions defined in (10) Can al~o be used to obtain a reduce1 order equivalent of the overall system describea by equation (2), when one is interested in a quantitative behaviour of certain modes in the system. The method is based on the fact that the value of the Lyapunov function for the overall system does not get affected due to cer'tain modes in the system. Such of these modes can there£ore be eliminated.For large syste!!lS, however, the cOMtruCtion of a Lyapunov function involves the solution of a large number of algebraic equations.. Use of the decomposition procedure outlined in the previous section will be taken to avoid this complexity. The modelling of the system wilt be done, th~rerore at the subs'stem level in a manner outlined in reference 15. These subsystems are later unified to give a reduced order equivalent.

>0

(19)

~~-~

(18) may now be written as 'f

PiiAii-tQii/2 • -AiiPii-Qli/2 • S

(20)

Hence

Now, let it be assumed that !i

•

[!tr !!1eJT

(22)

Where !ir and !.ie represent the states to be retained and eliminated respectively. Then (17) and (19r Can be ca~t in the form

!I"ii!.i· [!.1r

!.I~rp :1 p12Jr!irJ ~

iJjl~ie

i2 P TT··

· ! i l il!ir -tIil U!ie -t 'l' T T X. P'21. -t X. P'2X, -~e ~ -~r -~r ~ -~e

• [!Ir !i.]rQ~l

l

Qi

2

(2)

Qi21[!ir] Qi:d !ie

... ·!IeQi)!ie ... irt whiCh the iaolated aubsystem '1' is given by (16)

(16) is asymptotically stable if tbe quadratic function

(17) is positive definite and the symmetric matrix p. satisfies the Lyapunov' matrix equation

XT Q X -ir 12-ie

('H)

....

where Pij 's ani Q ij ' s are the properly partitioned sl,lbmatrices of Pi1 _I'd Qi1 respectively. In order that the derivatives of the Lyapunov functions' for the original al'd reduced subsystems are to· be the same, last J terms on the right hand a1de of (24) are equated to zero.This implies .1.85

4 Q i2

Q iJ

•

0

•

(25)

"r:

'whicn yields, Qi i

.NUMsRICAL

u : ] (261

With this choice of Q.. , if the value 1.J. of the Lyapunov function is to be unaffected, the last three terms or (2)) will be equated to zero resulting in

T

!i

..

P iili

y'T. p

air

X i l -ir

Pi; (Sm - Qil/ 2 )

(28)

where Sm is obtained by retainin..~ the rows and columns of the matrix S corresponding to the original system. We. now. take the derivative of V.J. (X.) -J. along the solutions of (15). This gi';' .

v~

iXample-l: The system given by

· 0 1·.2 -0.21 ·3 = 0 Xl

.A-

.0

1.4-

a.

..... i j ·

A

l

p-11

". 'C ij

('0) .,

If, however,tohe 1 th subsystem is to be retained in its form but the jth subsystem is reduced 1n ord'er, ihen the n"W interconnection matriX Aij is obtained f r o m ' . A

Aij

•

1 Pii Qij

(31)

Where, .Qij is the matrix resulting from the deletion of the columns of Pli~j product, ·correspoR1ing to the modes to be eliminated in the jth

1

0

0

Xl

-1

1

0

.](2

0

0

1

1..3

-20

-12

1.

-1

(2)

4

is the model of a single machine system with a double time constant ~ove­ rning system. The initial conditions !(O) on (32) are

!(O)

In order that this ·de~ivative is again to be· the same 'iri the ith reQuced am the unreduced subsystem5, the row correspomin~ to the mode to be eliminated 1:5 d'eleted from the product PUAiJ,.·Let this .be given by Qt·. J Then, the ~ew interconnection matrix Afj exhibiting the interaction between the reduced i th subsystem and tl!.e j th subsystem is obtained using P11· and Qfj from the following relation:

EXA.MPL~

. The above procedure for assessing stability and obtaining a dynamic equivalent is illustrated with r~fp.rence to two numerical examples. The responses of the equivalent derived by the prp.sent method are then compared with those obtained .by the other me-' thods. .

(27)

Having now obtained P i1 and Qi1' equ~ ation (21) Can be used to build the reduced order matrix Air of subsystem 'i' and is of the form~ Air"

SUbsystem •.

.. [0.02

00

DJ}

Stabil1t~:

It is now desired to d et ermi ne the stability of system (32).' We divide (32) into two subsy st ems ~l and 52 such that S1 is

1 Xi] ro . 1][1. [i 2 -Lo21 X2] [01 -1

+

0] [X;]

0

X (34) 4

The isolated subsystems are

Sl:[~jf21~J[~~f2t~~:O-l~[2 (36)

It <:an be' shown that each of' the free subsystems in
. [-1 0]

QU •

0

-1

and

"22 'C

.r-L ol

0J

-1

186

5 and using equation (7).

Pn ·f2.9S6

This giv~

2'.3811; P22 -11.175

~.381 2.88~

0.025l

US)

0.0438J

LO.025

The estimates on villi)' i .. 1,2 are 0.683 2

1;10 ~ vI '11 )

~

2 .3 2 411

~lll

.

O.20781!211 ~ v2(!2) ~ 1.08411 ~211 and

(39J

;l l !l) ~ -0 .215 ~ • • (40j v2(!2)~ -0.461 ~ !211 ,

!lll

.a;stimates of ~ ij 's are computed in a manner outlined in reference 12. The state variables of the subsystem are assigned their values at the initial point and using (9), one obtains

f I2

.. 1.}-17

i

and

21 .. 0.001

•

v2 {!2 J

~ - 0.42534 v2+ 0.001 v 1

-a

O. 0926

C

0.001

]

-0.5 is

We' now determine A~l and All. Taking the derivative v (!l) along the I solutions of (34) and extracting P11 A12 , we ~et,

We delete the second column yielding Q'ij as

Q12

(42)

Aij

(43)

2~881JT

.. [2.381

Premultiplying' by

is given by

6.12

..

Then the coefficient matrix A 2r

- 0.O?26 v l + 1.917 v 2 '

HenCe the mat'ru-

R ..

Sm

to the equa-

(41)

We now take the time derivative of v i l~i) along the sol utions of the subsystems (4) and· DS)' and using (39) -(41) we get, .

~1(11) ~

acco~ing

'rhe matrix 5 m tion (20J is

.. [0

Pii

(50)

we get

1JT

(51)

Thus the subsystem 1 with the effect of mode X eliminated gives 4

-0.4254

It can be shown that R satisfies conditions (13J. Hence the system is dynamiCallY stable. b. system simplification:

Similarly one obtains the reduced orof subsystem 2 as

It is desired to eliminate mode X in 4 subsystem 2. Hence we construct a Lyapunov.function with Q22 given by

de~

:i ..

-1.464X.3+

[0 -0.0732][X1] '.

X

(53)

2

Combining (52) and (53), the reduc ed systsu; equivalent to (32) is given by

o~er

Then the matrix

P22 in the function

V2l~2) is given by

P22 .. ['3,4 1 7

.025

.02 5

J

(45)

.00281

.;iince the mode 1 is to be eliminated 4 the seconi row ani the secont column' in (45) will be d'!leted giving

.

Xl

. x·

3

0

1 -1

o

The eigenvalues of the system (54) together with those obtained in refe-: rences 4 and 15, are shown in table-l 187

6 systems, yielding

Figur ·-1 givea the tree re::sponses of this model with the initial conditi-' ons of (33). It is se~n that the model with the pre~ent method compares very well with tbe original response.

Pl1 ~ P22 •

. Table-I:

-

No.

Co~a:rison -

P33 • [1.5008

0."808J

.

0.9808

0.48.08

P44 • [1.175

0.025 ]

0.0.25

0.0438

of lUgenvalue::s· and

Sigenvalue 'by the method of rre5ent Ref. 15 method Ref. 2 -

1

-0.258

-0.264

-0.270

2

-0.869

-a.8b8

-0.810

3

-L873

-1.424

-1.378

QJJ

Ql l •

Q22 •

'L~ _:]

Q•• •

C-~

_:]

( 57)

Tne initial conditions are given by

I(O) • ~.02 . 0

5cample- 2 :

The matrix

R

o

°

°

0.02 (5~) 0 'j ~

is given by

This exa1flJ1e pertains to the description of a' two machine system of reference2. The system model developed in this reference is of the form

Xl

1

o

o

o

a

o

o

-1.04 ':'·1

1·

a

1.04

0

o

o

0

o

1

o

0

o

a

-1

-20

-12

0

o

o

0

o

1

1.04· 0

o

°

o

1.

0

o

i;.8

o o

o o o

o o

0

o

o

1:

.2 X

o

o X4 o = o ~5· 3

.1.6

7

o

0

o

-1 .

°1

1

-1.04 -1

°

-12

-20

a. Stability: th Decomposing this 8 order system into 4 subsystems. we get SUbsystem 1,

[~j{:.04 j[~~+~ :J[~~t04 :J[:~+~

[iiJ1

:J[:j

0J[1 51 ro ' 0J[1..7J' -~ 1..~\~O -l~lxJ+ o. 1..6J+Lo 8 o 01 [1..~ro 0]r1..31Jo 1 1[X 51 ra °l[X7] Subsystem 3: [X 51r iJl.04 ~ 1..~1? 0 LxJ L.1.04 -~ 1..J+Ll 0 18 '

SUbsystem 2:

J {0'

Ol[X~ re

llP'31 [0

0

SUbsys t em.' [ : : } [ :

0

~ [:~+~

0

X

:Y:j+[: _:][::1:0 _:J[::]

The free SUbsystems Can be easily identified from (56J. We now construct Lyapunov t-unctions for the free sUb-

(56)

7

a ..

-0.5'

0.38

0.001 -0.425

0 •.39

0

0

0

0.39

0

-0.5

0.)8

0

0

0.001

-0.42

ACKNOWLED(H~MENTS

( 59)

It is easily shown that R satisf ies nence it is concl ai ed that (lJ) the syste mi~ asymp totica lly stabl e.

am

'The autho r wishe s to thank Dr.B.N . Garud achar, Head, ilectr ical Sngin eering Depar tment, Osroan ia Unive rsity, India , for provid ing the neces sary facil ities during the COUrse of this work. REF.&RENCBS

Kimba rk; S.W., (1948 ). Power system st(a book) , Wiabilit y - Vol.I II York. Nett ley, Ad opting the proced ure outlin ed earlKuppu rajulu ,A and Elang ovart, S.,(19 71l X,;;: ier, elimin at ion of modes )(4 and Simpl ified power system model s for th ' v ord6 a s yield ic stabi lity studie s! JI.IEg g (55) dynam system in the is matriX cient coJl'i Power 'App.& Syst., 90, 11-~3. er model . The Undri l,J.M. and Turne r,A.E .(}971 ). given by 'Cons tructi on of power system elec0 0 0 trome chanic al equiv alents , Jl.Igg E 0 1 0 App.& Syst. ,90, 2049-2 059. Power 0 0 L04 1 -1.04 -:I. Radha krishn a, C.Lak shmin arayan a, C.and 0 0 Reddy; i~i.i. 1l978) . Simpl ified modal -0.074 , -1.47 u 1 ol for dynam iC stabi lity stucontr 0 1 0 0 0 0 dies, ISU PES Summer Meeti ng, 1 Paper No .A78 609-0 . -1.04 -1 . 0 '1.04 0 NeSs, J .'r;. and Zimme r,H.an d Cultu , Van -1.47 0 -0.074 0 0 0 , 14.(19 73). Reduc tion of dynam ic mo(60) dels of power systeD \s, PICA Proc. 105-1 1~ • trauss ,P.O. and Altalib,Y.(1976)~ DyThis model clOSe ly agree s with that namic .equiv alents by combi nation obtain ed in r~'erences 2 and 4. of reduce d OI1l er mcnel s of system compo nents, 'Jl.lS EE Power App.& CON CL us lOSS . :iyst. , 95, 1535- 1544. Lyapu nov f'ur Vecto ). (1962 R. an, catiBellm appli the in s pment develo t Recen nction SIAM .J.Con trol, I, 32-34 . . on of Lyapu nov functi ons to the staBaile y,F.h. {1966 /. The appli cation of bility analy sisof large scale power podecom , Lyapu nov's secon i method to interthe s toward ed system s direct equconne cted system s, S+:AM.J. ContrO l, . am ds sition - aggre gation metho J, 443-4 62. ivalen cing techni ques have opened ,out Jiljak ,D.D. (1973 ). On stabi lity of newer lines of resea rch. In this paplarge scale system s under struc er a new conce pt 01 vecto r Lyapu nov, , , pertu rbatio ns, Ii&E Trans . of tural iples princ the ing utiliz 'funct ions 415-417. .3, bejM~, has gation aggre decom positi on am Gruji c, Lj.T.a nd Siljak ,D.J>. (l97.3 ). en presen ted to solv'e the dynam ic stASym ptotic stabi lity and instab iabilit y proble m of a· large power syslity of large SCale system s, 1&81 method t em. 'l'he eff ective ness of the 'frans .Auto. Contr . ,18, 636-6 45. depen ds upon the degre e of decou pling l1A.N .(1974 ). Stabi lity analy sis larMiche henCe and betwe en the subsy stems nterco nnecte d system s, ;:HAM.J. of g'9ly depen ds upon the method of deco. ol, 12, 554-5 79. coContr in is usion concl .This mposi tion. (191~) C~L. ana, Naray ar¥:i M-.A. .. Pai posidecom nfirm iW,w ith the gener al .:stab ility of large scale power sytion princ iple as applie d to the ~ta­ stem, Froe. 6th !FAC Congr .,No.) 1.6 s. system iC dynam of sis bility analy 1-10. The metho d, howev er, Is system atic Naray ana,C .L.(19 77). Vecto r Lyapu nov and 'can be suitab ly progra mmed for funct ions tor power system stabi lcompu ter usage . -41so prese nted is a ity stUdi es, Natio nal Syst.C ohf., method that Can USe Lyapu nov functi on batore . ord I,;oim reduce a ing obtain for as a tool JOCiC iLj.B'. a.Ribb ens Pavel la,M. ard der model of the origin al system . Re;3i jak, u.D.(1 978). Multim achine sults obtain ed an a numbe r of system s systeru :J: stabi lity &.decompofairly power at-a s model the tnclic ate tnat, and aggre gation , .lug Trans n .sitio aCCur ate. Moreo ver, the compu tation ' ., 2.3, ')25- 3:32.. Contr . Autom der-' consi d reduce are s lexitie al comp 72). .D.(19 Rao,lf ar¥:i K. kar,A. low DeSar very with ably as one has to deal qamic sy~tem simpl ificat ion and an order subsy stems at a time. This procation to power system stabily appli suitab ,be also can method posed c • !'BE, 7 ,904-9 10 • studie s,Fro lity . cation progra mmed for compu t:'er appli .. b. Sxster o simpl ificat iqn:

'PY-.

.~

189

, _

secs

FiA!G.:71m.e. t"fJfSf OJLS6 o/ ....ode- xl

6

- - - Ori.~J':rt.a.t S'ys~1'I'L

_.-.~

rd.4 .

----- J're~n;t. ~ m..c.~~.

-.-a-

°ref.15

190