System Identification for Electric Power Systems

SYSTDl IDENTIFICATION FOR ELECTRIC POWER SYSTDffi

F. C. Schweppe Nassachuse tt s Ins titut e of Technology Cambr idge, Massachusetts

R. D. Masiello Leeds & Northrup Company North Wa l es , Pennsylvania

I NTRODUCTION

tern loads were represent ed as static nonlinear loads and rotating dynamic machinery. The simulation was thus a sophisticated one (mor e than 100 states are involved) and its use in th e study is quite distinct from the identified model.

Several studies in r ecent years have achieved i mprovements in the dynamic stability of electric power genera ting equipment via the application of linea r r~gula~or design to exc itation supplementary controls. \1-4)

HYPOTHESIZE LOW ORDER LINEA R MODEL STRUCTURE

However, for multiple machine problems the practical difficulties of modeling the system can be severe . These include high dimensionality, imbedded nonlinea rities, and th e presence of unknown load dynamics and interconnect ed mac hines.

The most critical step in the identification process was the hypothesiz ing of a low order, linear model structure whose parameters were to be estimated. The goals in this process were: the model must adequately represent typical behavior in th e relevant dynamic range and frequency spectrum, it must retain a direct correspondence with engineering mode ls of the phys ical sys tem elements, and all known parameters of the latter should be used as such. These conditions proscribed the use of a ca nonic a 1 form.

This pa per describes th e des i gn of a multimachine stabiliz e r. using an identifi ed model. Four steps in the procedure a re detailed and flowcharted in Figur e 1:

(1 ) hypothesize a low ord e r approxima te model on e nginee ring grounds;

Th e model used for each of th e internal s ys tem generators included th e electro-mechanical rotor dynam ics (swing equation) and the transient electrical time constant of the rotor direct axis field winding. Thi s leads to th e simplest model which is customarily used in studying th e dynamics of excitation sys tems. The linea riz ed model for th e exciter c orr~ sp onds directly to th e I EEE type 1 exciter mode l (7) used in tra nsie nt s tability studies, appropriately linearized. This is a fairly detailed mode l: th e rea s on for retaining so much detail is that th e mode l parame ters r eprese nt design values for e l ec tronic or electromechanical components which are known with confidence. Also, the exciters are the prime sources of control and the use of an accurate representation should provide some benefits. The governor-turbine is mode led by a single unknown lag.

(2 ) fit unknown paramet ers in the model to coll ect ed data from the system (simul a ted, for the purpose of the study); (3 ) using the identifi ed model, design a linear r egul a tor to achieve th e desired stabilization; (4 ) verify the design by t es t (again, using the s i mu lation) • The desired form of th e final s t ab ilizer is a centra liz ed s tructur e : one central controller senses s i gna l s from the mach ine s of th e controlled system (real and reactive power, field a nd terminal voltages , frequency) a nd int e rcha nge flows to the int e rconnect ed system. It ge nerat es s uppleme ntary control signals sent to th e controlled exciters. Figures 2 and 3 show the sys tem vs. th e model and th e controller. Th e parame ter identific a tion a lgorithm permits id entification of individual elements of a system ma trix description in ge ner a l phys ical form. It is related to maximum like lihood identification a nd is a n extension of the rec ent work of Y. Bar-Shalom.(S) The simulation employed in th e study was a careful representation of two interconnec t ed powe r systems eac h with several ge ner a torb. The relevant dynamics and nonlinearities of turbines, governors, exciters, alternators, and th e transmission sy~tem ,,,ere a ll represented u s ing standard forms. (6) Sys-

A similar model structure i s used for one generator which represents the entire external system, with t wo changes : the regulator and exciter were represented by a single lag a nd the rotor electromechanics were referred to the reference frames of the interna l system machines. The un'«nown external system equivalent generator thus became a " dynamic infinite bus." Elimination of the dependent variables, bus complex voltages, involves linea rization using the transfer admittances of the system network: in general, th e se linearizations produce unknown coeffici ents

176

"1.. .. . ....

' ••

0

-: . . -_ ...

1

' ••

r.. '.::::.. .1::

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

J. I

..

-J•• ,

I::

;!~---C~---1~---7~--~.,_

.;:.

- ;

177

' •. 'A~'

becaus e of th e loads, the int e rconne ct ed sys t em, and the stea dy st a t e conditions . Thus in th e lineariz a tion of the model e quations the " inpu t s" to the exciters includ ed unk nown ga ins multiply ing th e rotor angl e and th e transi ent voltage a t ea ch ge ne rator. The same was tru e for th e eq uations des cribing th e rotor dy nam ic s a nd th e fi e ld tr a ns i ent volta ge .

ca se in thi s sec tion. Pr ob l em Fo rmul a tion a nd Not a ti on Th e sys t em i s desc rib ed by :

(1 ) ~ (0) i s N(O, ! )

~

Finally , th e load disturbance s driving th e sys t em were mode l e d as white nois e dri ve n, coupl e d, dampe d harmonic o sc illators coupling into th e excitor a nd rotor s ub sys t ems v i a unknown ga ins.

~(n) i s N(O, g )

Iv

::::(n) i s N (O,~)

~,

~ (n) i s knmVIl

~

Th e final lineariz e d mod e l y i e ld ed a mod e l of the form with a continuous time tra ns ition matrix-of th e form of Figur e 4 , with known and unknown e l ement s a s s hown. A continuous time C matrix des cribing ob s ervations wa s simila rly d;rived. Since the gain and tim e cons t a nt of th e re gul a tors a re known and s inc e th e ma chine t e rmina l volt ages a re both ob servations a nd exciter inputs, th e line ari zed coe ffici e nt s of th e t e rmina l volta ges to machine el e ctrical va riabl es a ppea r ed in rows of the A and C matric e s multipli e d by known cons t a nt s . The s; rows-in th e A a nd C ma tric es ar e the r e for e r e l a t ed to e ach other by-multiplica tive cons t a nts in each ca se .

~,

The B matrix simply ins e rts white di s turbanc es w into-the disturbanc e sub system a nd th e F ma trixcouple s control s i gnal s to the k nown machine exciters. An important r e sult of augme nting th e s tate with the colore d di s turbance i s tha t th e s trong corre lation b e t wee n the proces s and ob se rvation di s turb a nce is now r e presented. In orde r to prese rve th e s pa rs e s tructur e and known e lement s o f th e sys t em, a n Eul e r rul e di s cr etization is pr e f erre d. Unfortunat e l y , th e ex citer and rotor s ub syst ems cont a in time cons t a nts too s mall to pe rmit th e us e of Euler rul e di s cre tization. Consequ ently, exponentia l di sc r e ti z ation was a pplie d to the individua l exc it e r s ubs ys tems a nd to th e five -st a t e s ub sys t em compri sed of th e thr ee rotor s p eed s and t wo ang l es . A 0 .1 second sampling time was u se d.

v d i me n s ion Nz d i me ns i on Nu

g, y a re nons ingu l a r a nd th e s to chas ti c va ria bl e s a r e u ncor re l a t ed .

x= Ax

The r e asonable approx i mation th a t rotor e l e ctric torqu e i s the s ame as t e rmina l e l e ctric powe r pe rmits a similar st a t ement as r e gards th e row in C corre sponding to th e obs e rvation of t e rminal r eal powe r a nd th e inputs to th e trans ition e qua tion for th e rotor fr e quency in A. The s e r e lations a r e noted in Figur e 4.

d i me ns i o n Nx d i me ns i o n N,,,

Le t p (~)

de not e th e prob a bility of

~

p(~,~)

th e proba bilit y o f

~

a nd b

p (~ /~)

t he p roba bilit y of

~

g iven b

a col umn vec t o r whi c h co nt a ins (:::: Cl); :::: (2 ); . ::::( N) )

..

th e op tima l es tima t e (o r pred iction) of x Ci) g i ve n (~ (l)-~ (j) = ~j

(2 ) The ide ntification probl em cons id e r e d her e i s as follo ws : l e t a b e a vec tor o f unknown pa r ame t e r s in th e ma tric es A, F, a nd C. Find a n a th a t opt imi zes th e joint proba bility of zN a nd th e corres ponding ~N; p(~,~), thu s th e ; ax i mum a pos t e riori es t imat e of a . Optimiz a tion of the Maximum a Po s t e riori Probability Function p(~,~) can be writt en in t e rm s of the r es idua l s :

(3 ) wh e r e ~N i s th e sequ e nc e ~(n) de t e rmine d by ~N a nd ~ . Si milarly,

~

a nd

~

de t e rmine

in t e rms of s = B w ~(i )

~ (i) - ~ ~(i ,/N )

N

Bar- Sha lom has de t a il e d a new framewo,k for forming pa r ame t e r es timat es of LTI sys t ems :(S) th e joint probability of th e ob s erva tions a nd the optima l int e rpolated es timat es x(n/ N), or p(z n, xn) is max imiz e d. It wa s s hown that th e n e c essary conditions for optima lity could b e se para t ed into a p a ir of coupl ed linea r al ge braic e quations which could b e it e r a tive l y s olved in turn to y i e ld a num eric a l pro cedur e for a chi eving th e e s timat es . For s c a lar sy st ems , all th e prop erti es of max imum lik e lihood es tima t es we r e cl a ime d. The a l gorithmic portion of th ese re s ult s a r e ext e nde d to a more ge ne r a l

exPtl/2 p(~)

so th a t

th e n:

l (i) = ~(i+l /N ) - ~ ~(i1N )

PARAMETER IDENTIFICATION

~N'

I

- I ~(i)

~(i)T~_l~(i)J

i =l [ ( 2/T )NZ IRI 1 / 2 JN

(4 )

N

eXP[-1 ! 2 p (iN )

178

~ !(i)[!l. g

!l.Trl i(i)]

i=l [(2 /T )Ns I!l. g !l.Tl l / 2 JN

(S)

P (~-N'§,) can nm" be ",r itt e n as

P(~"'~N )

=

~

C

V~.J

C

x

By us e of a set of ma trix identities (9) and some ma nipulation

exp L-l /2(J v +J",+J x )]

(6) (17 )

N

Jv

)

•T

A,

'{ \ i)

R- I ~(i )

----'

N

L ~(i/N) ~(i/N)T

i=l N

J",

J

~ i =l

i(i)T

x = ~(O )T

[.!? 9. .!?T J- I i

i ~l

(8 )

(i)

i -I i(O)

[(27T ) NZ 1~ll /2JN [(2;r)Nx I.!?

(18 )

(9 ) (10 )

g .!?Tll / 2 ]N

If Rand y(i) are defined as:

(ll ) p

(2rr )Nx 10/ 11 / 2

(1 2 )

The necessary conditions ar e a t the optimum:

then

(13.1 ) (13.2) An optimal interpolator(8) handily finds the optima l §, for a given a. If a means of easily satisfying (13.2) can be obtained, then th e problem solution can iteratively proceed as (a) Guess

~YNYN

(14 )

-

de note elements

-

J v can be written as

t

tr

If

~

=

H:

No:" X Nay

om

~

=

~ QV

+ L

~°:

O

= 0

No:" X 1

(20)

is C im jm(~ th jm th e l eme nt of ~) then

(~_l [~(iH: ~(i /N)J[~(i)-~ ~(i !N)JT)

i =l a nd J w as

(19 )

For the special case where the e ntire C matrix is unknown, the solution of (14) for ~ is-trivial and not dependent upon R. However, if the dimension of z is less than that of x (as is usually the case) a uniquely convergent answer is not available unless a is restricted. Equation (14) can be solved if a contains only certain individual elements of the various matrices. For an QV containing e lements of C, (14) takes th e form: -

dJ v

9W

(i+l / N) 2:: (i)T

i =l

Since J x , Cv, and c,,, do not depend on Q1 the necessary condition reduces to

and

y(i) yT(i)

L~

Qf-I

(d) If convergence is reached quit, if not, go to (b)

~

I

N

~XN+lYN

(c) Solve (13.2) for ~i, given ~~

Let QV denote elements in in A and F.

=

i =l

a.

(b) Solve (13.1) fnr ~~, given

N-l

(21 )

(15) (22 )

N

Jw =

L

tr

([.!? 9. .!?TJ-I [~ (i+l /N )-~ ~(i1N)

i =l

L\ (16)

179

= 1

~ = 0

(23 )

At each iteration:

Applications to the Power System Problem The high order nonlinear simulation was used to generate observations. The amplitudes of the disturbances used in the simulation were fairly small, representing peak values of about 1-5% at the load busses with a small sinusoidal content. The periods, phases, and bandwidth of the load disturbance varied frcm bus to bus. Since most of what is usually considered power system meter noise is information for this problem, no observation noise was simulated. Data from the simulation corresponding to the observations z were collected for 30 seconds of simulated time.

(24 ) where CO is C with all a set to zero; and

Then (20) is solved for a at that iteration. The process for satisfying (14) for elements of ~ proce eds similarly. If the unknown parameters are related a solution may still be possibl e . The special case 9; = ! .§., used for this problem, produces

Presentation of the numerical identification results is omitted. Since the hypothesized structure did not correspond to the simulation, there is no " correct" answer and errors cannot be plotted. Also, there would be considerable difficulty in presenting the approximately 150 matrix entries identified.

Computational Considerations The derivations have assumed that Rand [B Q BT] were nonsingular. This may not be-the ca;e-; but for computational purposes a reasonable approach is to modify the •. true" ~ and / or [~g ~T]. Thus, if R differs from th e identity only by R11 = 0, it may suffice to let R1l ", 10- 4.

CONTROL DESIGN The identified linear model was used to design and test regulators. The design process involved choosing the weighting matrices on the 22 states and two controls for use in solving the Ricatti equation associated with the linear quadratic regulator problem.

It should be pointed out that if (14) could somehow be solved rigorously taking a singular R or [B Q BT] into account, the algorithm will break down. -ifthe transistion equation for the first state variable is exactly and completely Xl (i+I)",a12 x 2(i), then the optimal smoother, (but not the filter) will produce estimates which exactly solve that equation, and the optimal estimate of a12 at that iteration "il1 be unchanged. Relation to Maximum Likelihood Identification The difference between the described algorithm and maximum likelihood lies in the difference between optimizing the terms of P(~N'~N) and those of p(~). It can be shown(9)(10) that in the Gaussian case

"here C (9 ) is a deterministic scalar 'vhich depends on th e statistical properties of ~N (i.e., on 3), but not on the values of observations. Thus, the optimality condition for 3 used here (13.1, 13.2) is not the same as the maximum likelihood condition

The difference is the effect of the deterministic or bias term dcjd3 (9 ).

The design considerations were to achieve stabilization and smooth behavior without the use of excessive control action. Proper weightings on the controls provide the latter, once the varying per-unit gains of the known machines are taken into account. As the generating units must respond to disturbances the stabilization must damp out intermachine electro-mechanical oscillations but not general system response -- terms in the state cost matrix were chosen accordingly. Additionally, the method of modifying the system transition matrix used in solving the linear regulator problem was partially used. This method involves adding a term dA to each element of the transition matrix to produce a new artificial transition matrix. The eigenvalues of the controlled system(ar~ then guaranteed to be shifted leftward by dA. 11) This method has the drawback of generating unnecessary control action to achieve a uniform eigenvalue shift. The approximation was made that, by adding dA only to diagonals in A associated with rotor speeds and angles, primarily-those variables would be affected. In general this result would not be guaranteed, but the results were satisfactory in this instance. CONTROL RESULTS

For many applications, the bias term is not important, so that the proposed method is very like maximum likelihood identification and estimation. This has been born out by experimental data. The function of the bias term is more pronounced if the structure and dimension of the identified model is also open to question and identification.

The nonlinear, high order simulation was run with the linear feedback regulator designed using the low order, linear identified model. It was also run without any stabilizing control and with conventional rate feedback supplementary control. For these tests the disturbance was altered to generate

180

tem Optimization ,~ith Predet ermined Degree of Stability," Proc. of IEEE (GB), Vo1. 116, No. 12, pp.2083-2086, December 1969.

a large and growing destabilizing influence. The periodic components of load disturbance were increased to 100 to 20), of load and then relative phases chosen to excite inter-generator oscillations. These disturbances had pronounced effects on the unstabilized systems, but had much less pronouncedeffects on the system stabilized by the linear feedback control as seen in Figure 5.

4.

Concordia, C. and DeHe110, F. P., "Concepts of Synchronous Hachine Stability as Affected by Excitation Control, " IEEE TRANS. on Pm~er Apparatus and Systems, Vol. PAS-88, No. 4, pp.3l6329, April 1969.

5.

Bar-Shalom, Y., "Optimal Simultaneous Estimation and Parameter Identification in Linear Discrete Systems," IEEE TRANS. on Automatic Control, Vol. AC-17, No. 2, pp.308-3l9, June 1972.

6.

Dornmel, H. W. and Sato, N., Notes on Transient Stability in Pm~er Systems, Bonnevi11e Pm~er Administration, Portland, Oregon, 1969.

7.

IEEE \.Jorking Group Report, " Computer Representation of Excitation Systems, " IEEE TRANS. on Power Apparatus and Systems, vo1. PAS-87, No. 6, pp. 1460-1464, June 1968.

8.

Meditch, J. S., Stochastic Optimal Linear Estimation and Control, McGraw Hill, New York, N.Y., 1969.

9.

Sch,~eppe, F. C., Uncertain Dynamic Systems, Prentice Hall, New York, N. Y., 1973.

CONCLUS IONS An application has been described for a new algorithm for system identification which effectively demonstrates the unique advantages of the ne,~ identification approach. The algorithm is shown to be suitable for use on high dUlensional multi-inputmulti-output models ,~ ith known structure and knmffi parameters or parameter relations. ACKNOWLEDGES The Bonneville Power Administration sponsored this work. Mr. A. W. Brooks and Mr. A. R. Benson of BPA provided much useful data and many valuable comments. REFERENCES 1.

2.

3.

Mousou, H.A.M. and Yu, Y. N., action of Multi Machine Power tion Control, " IEEE TRANS. on and Systems, Vol. PAS-74, No. July 1974.

" Dynamic InterSystem & ExcitaPower Apparatus 4, pp.1150-ll58,

10. Masiello, R. D., "Adaptive Modeling and Control of Electric Power Systems," MIT Electric Power Sy stems Engineering Laboratory Report 40, Cambrid~e, Mass., 1972.

Outhred, H. R. and Evans, F. J., "A Model Reference Controller for Turbo-Alternators in Large Power Systems, " Proc. of 4th Power System Computation Conference, Grenoble, 1972.

11. Middlestadt, W. A., Cormnents to " Optimal Power System Stabilization Through Excitation Control," IEEE TRANS. on Power Apparatus and Systems, Vo1. PAS-91, No. 3, pp.1182, May 1972.

Anderson, B. O. and Morey, J. B., "Linear Sys-

181