A real-time predictive controller for automatic voltage regulation of a synchronous generator K Y Lee and S A Hossain Department of Ele, ctrical Engineerinq, University of Houston, TX 77004, USA

The use ~?./a predictive controller in atttomatic voltage regulation (A VR) o]a s.wlcfmmous generator is proposed ti,r real-time control purposes. The controller consists o f a last-time linearized 2nd-order model o f a n alternator and control logic. The fast-time mudcl is used to predict the time response o./ the controlled system (generator voltage outpztt), am[ the control h~gic is responsib[e for choosing the ncccssm3' input t~ the controlled s.vstem. The principles o f t ime-optimal control and linear regulator arc used in contro[ler design.

The effectiveness o f such a controller is examined by digital simulation study jor the response o f a synchronous machine, comtected to an infinite bus under various input a~td initial conditkms. The eJ]~,cts on the stability o f the rn,erall system arc analysed.

I.

O) R ]:'FI) H D

Ut U d Ct r t

Kz:

IQ4

K r Tt: Um a x Umin

KI = ~ P e A6 ;:q'

Nomenclature rotor angle rotor speed field voltage machine-inertia constant damping factor direct-axis transient open-circuit time constant control signal control signal to the fast model output reference input error exciter constant related to self-excited field exciter time constant regulator gain regulator amplifier time constanl regulator stabilizing-circuit gain regulator stabilizing-circuit time constant maximum value (31" regulator voltage minimum value of regulator wJtage

change in electrical power for a change in rotnr angle with constant flux linkages in the direct axis

This work is supported in part by the Bonneville Power Administra- t ton, US Department of Energy, under contract 14-03-7260N

= APe K2 ~l :q ~,

X' o + X e K 3 -

Xd + Xe

I At:'q' K 4 -

K 3 At}

K s /~5 t:'t t'

~..~L" t

change in electrical power for a change in direct-axis flux linkages vJith cons/ant rotor angle

demagnetizing effect of a change in rotor angle

chan,,e~ in terminalwflta-e= with change in rotor ang[e for constant Eq'

change in termmat wfltage with change in t~q' for COllSlal)t r o t o r angle

II. Introduction The basic principle of any predictive control technique can be roughly summarized as follows. A "fast model" is set up. which corresponds to the plant, but with all time constants reduced by some factor. At tile start of each contputat ion, the state of tim model is set to correspond to the present state of the plant, and the model is then allowed to run, predicting the future behaviour of tile plant under some given input. From this behaviour, a logical decision is made in determining the required present plant input, and the model is then reset to start the next iteration.

One of tile first major applications of this technique was presented by Chestnut el al. l to control the landing of an aircraft via a digital computer. Fallside and Thedchamorthy 2 used the predictive control scheme in controlling an iso- lated synchronous generator. In their work, the technique of an adaptive fast model was developed. Nehrir 3 applied this method in tile automatic voltage regulation of a DC generator. In references 2 and 3, a control scheme was adopted which incorporates a small linear band around the origin to prevent the limit cycle.

One of the main advantages of the predictive controller is that it aw)ids the need for calculation and storage of com- plicated switching surfaces used ill other methods of opt inmm switching. Another mare advantage of tim pre- dictive controller is its ability to operate with inexact models which makes it highly desirable for use in dynamic

Vol 1 No 3 October 1979 0142-061 5/79/030181-06 ~02.00 © 1979 IPC Business Press 181

systems where system parameters change with operating condit ions.

In this paper, the predictive control technique is tised to control the output voltage of a synchronous generator connecied to an infinite bus and subjected to load dislurb- ances. A 2nd-order fast model is used t(/control the 4tb- order exciter-generator system. To awfid tile limit cycle, tile principle of a linear regulator is used when tile state of tile system reaches a predefined rectangular zone around the origin of tile state space. This predictive control scheme has resulled in a stable system under all operating condi- tions and thus eliminated the stability problem that com- monly arises when the conventional AVR is used.

I I I . Predict ive control scheme I f a system to be controlled is of die 2nd-order, tile state of the system is then described by tile output and its derivative. Let tile system input be u(t). As in all physical systems, the input is bounded such that U--~ u(t) ~< U.

(gt)nsidei tlie error and error-rate phase plane of the systenl as shown in Figure 1, where the systeln error e(t) is equal 1o the system output c(t) less the desired reference r(t). As is kllowu l'i-on/the Pontryagin's maxilrluln principle, 4 the fastest way to get from ally point P in the phase plane to origin is to let tlte system input be + U o r U as required, and it is only necessaiy to switch once (at point S ill Figure 1) between tile two levels of input. This allows lhe state of the system to move along I" 1", which is the only trajectt)ry with +U or g ti lat goes through the origin.

Tile switching point S is found by letting the fast model of die system make a repetitive prediction 1o see where tile system response will end up if the input is switched at the present time. It is noted that the switching point may only occur in the second or fourth quadrant of tile phase plane where error and error-rate have different polarities. When the system trajectory is in the f]rst or third quadrant, the system response is going away from tile desired reference. and tile input is chosen to drive the trajectory to the fourth or second quadrant, respectively, which are tile prediction quadrants.

Consider Figure 2, where the system is at point a initially. The system is switched to restore its error to zero, in this case to u(t) = +U. The system will proceed along a, b, c, . . . . and if this actuating or control signal is maintained tile

b(t)

F

u z U

e ( t ) ~ c ( t ) - r ( t J

Figure 1 Switching curves in error and error-rate phase plane

I e

d \ i ~, c D p i ~ e d i r t r l i ' : , : ! ~2

,!, - ¢

I }

Figure 2 Phase plane trajectories of plant ( - - - ) and prediction ( - - - )

system wi l l eventually overshoot. Tile purpose ~)1 lifts scheme is I{) find tile correct instant at which to switch lhe controller to u([) = U to drive error and error-rate t<> zel<> simultaneously. Accordingly, when tile system is in :l p~e- dict ion quadrant, tile fast model is set to tile current st:lie of the system, and then a trial of the model is carried <)til with u(Q) = U(for tile second quadrallt), where l/ ic_'prc- seats a faster time scale. Tlle fast model runs faster than ihc original model due to reduced time constants. Suppose [hi> switching occurs at point b and the fast model predicts the subsequent behaviour of tile system output. It is seen lhat this switching is prenlature and e(ts) ¢ d(t:) = O. aild so lhe fast model is reset to the new present state o f tile planl, and ;t further lr ial is made at point c. This procedure ~d resel- l ing lhe fast model is repeated unt i l , tit the trial switchm7 point j ] lhe fast model predicts t l lai the plant olrcir and error-rate wi l l be zero sinlultaneousiy. At this instanl (sl ight ly later than fb0cause o i tile conlput ing time el the fast inodel), lhe control signal is reversed to u ( / ) = U. As the switching occurs slightly later thai) f, tile syslell/ does not pass through the or igin; instead it l 'oill ls a stead\ slate l imit cycle in the neigllbourhoocl o f tbe origii l .

1"o avoid tile limit cycle, a small rectangular regitm is Ct)ll- sidered around the origin, hi this region, tile principle ~I linear regulah)r is used as follows:

Consider the system defined as

2(t) = Ax([ ) + Bu( t )

and tile quadratic cost function given as

7)

J = IxT(t) Qx(t) + uT"(t)Ru(t)lUt

0

where Q is an n x #/real symmetric positive-semidel]nitc matrix and R is an r x r real symmetric positive-del]nite matrix. The weighting matrices Q and R need to be cll~)~et~ by tire engineering judgement, often by trial and ern)r.

t / )

(2)

file optimunl control that mininlizes the co)st lt,ncliol; is then given by 4

14*( . t ) = R 1 B T / ' £ ( t ) X ( t )

where the n x n matrix K(t) satisfies a matrix Riccati differential equation

/~.'(t) = K(t)A ATK( t ) Q+K(t)BR 1/77K(t)

3 )

!4t

182 Electrical Power & Energy Systems

For a time inwu-iant system, this equation provides a steady stale solution K which is COllstallt and not a [tlnctioll ol" t. Thus for a 2nd-order system the optimal controller is of the lorna

u * ( t ) = M x i ( t ) iVx2(t) (5)

whore x i ailcl x 2 are coi~q~oilents o f the slate vector x, and M and N are r vectors of COilSlalltS.

The basic control scheme of the predictive contro l is shown in Figure 3 and the control logic for this scheme is shown in Fi?ure 4.

IV . S imula t ion of the contro l led system As a practical application of the predictive controller, the regulation of the w)ltage output of a synchronous machine connected to an infinite bus through an external impedance is a t tempted. Such a system is shown in a block diagram form in Figure 5.

Input ~ Co!trol r (t) logic

F Actuating s,c~nal - uf t ) !

u( t f )

c(tf )

Fast-time I model of plant I

Figure 3 Basic scheme of a predictive controller in a physical system

I t

I e : c R ] I

i ~tYes Ins~dereg,on Irectangularl~No

Me÷Ne >>0 Fes ~ - -

[- t , Let the modell predict once I No Yes V - e(t f )1 e ( t H_ ©>0

I I _L l-

[Apply u(t) to actuator I Figure4 Control logic flowchart of a predictive controller

I+ K 3 T-doS I

Figure5 Linearized generator model

The constants K l K 6 in the block diagram are defined ill Section 1. These constants depend upon the network para- meters, the quiescent operating conditions, and the infinite bus wfltage.

The transfer function of the system can be given by refer- ence 5

V t K3K 6 (2Hs 2 +Ds +KlCO R ) CORK2K3K s - N ( s ) -

I:'I:D (1 + K3T~'h,s)( 2Hs 2 + Ds + KtCOR)

CORK2K3K4

We now add to this generator model an excitation system related 13 3 ,

F F D 1

Ut KK + TE s

The overall transfer function of the system then hecomes

Vt K3K6(2Hs2+Ds+KlooR) coRK2K3Ks

U t (K E + TEs) {(I +K3T,'t ,s)(2Hs 2 +Ds +KlCO R)

CORKzK 3K4}

(6)

Tiffs 4th-order system is simulated to represent the physical system.

In order to obtain a simple switching curve for the pre- dictive controller, the fast-time model must be reduced to the 2nd-order. Most of the existing methods 6,'L8 reducing high-order linear systems are based on one of the following principles: that the low performance terms can be discarded and the high performance terms should be re ta ined:or that the sum of squares of the errors between the responses of the real system and those of the approximate model at the sampling instant is minimized to obtain the parameters of the approxi- mate model. Here, the method of the second Cauer form for linear system reduction 6 is used, which reduces equation (6) to a 2nd-order approximation represented as

Vt K

U t s 2 + a l s + a 2 (7)

where a l , a2, and K are constants obtained by the pro- cedure and depend upon the values of parameters in tile system model in equation (6).

Vol 1 No 3 October 1979 183

The fast-lime model to he used for the predictive controller is then represented by

Vr. r _ K . . . . . . . (8 )

i;i i;) + a I + a 2

where p is the time scale factor of the model and p ~ I.

The model in equation (7) contains the essential property of oscillation for tire exciter-generator system. In simulation, the fast-time model in equation (8) is initialized at each sampling time to the actual state of the system which is obtained from the simulation of the higher-order model in equation (6). This makes the predictive controller robust to the order of the approximate fast-time model, and thus makes the 2nd-order nrodel sufl'icient. The fast-time model in equation (8) is used for both the predictive controller and the optimal regulator when the system enters the rectangular region in Figure 2. The size of the rectangular region is +0.01 for e ( t ) and +0.05 for 0(t).

Since the K parameters of the linear model in equation 16) wiry considerably with operating points and cause the dynamic behaviour of the machine to be quite different, three different operating conditions were simulated to cover the wide range of parameter values. These parameters were computed from values found in reference 5 and are sun> marized in Table I. Case 1 represents the light loading con- dition of a synchronous machine connected to the infinite bus: lhe loading is heavier for case 2 and heaviest for case 3.

V. S imula t ion results "fhe controlled system is sinmlated on an IBM 360. A slep Vr~. r= 0.1 p.u. is applied at the input and the different output variables are shown in Figures 7 12. The system's output variables are then compared with those resulting lrom the generator controlled by the conventional AVR modelled by the IEEE type 1 regulator with parameters given in Table 1. The control signal of the predictive controller followed by the optimal linear regulator is shown in Figure 6.

It can be seen l]om the simulation results thai the AVR system has a laraer overshoot and lon,,er settlin,, time and, above all, is unstable at severe operating conditions such as

cz

o L u~

m

Y Y o

( )

UCRL1X i

I i !

i

~/TT]ID

!

? 1 3

T~me ( s )

L t .

4 % 6

Figure6 Case 1" control signal to the plant whenastep input is applied to the controller

, 1. I

/

/

)) ~) ( 7 ~ - - F r-(?(J : i ,~' ' ,"

/

o ,) 0241 /

( ) L : L • i , 1 2 3 .1 ~ ;

[irTqe ( S J Figure7 Case 1: response of the controlled system whena step input is applied

o I - - P red ic t i ve : o n t r q i

O . C ) 6 0 ~ . . . . . A V R

• 0 12Oi I

~o I

' ) 1 8 0 / J r~ ,( , _'r , , .

0 ~40 . ~ i .

a : , . . e (s:,

- - ~ ) r E ! ~ , : t , / . ' 0 0 1 -

i !" r, r

24 q

i i

~d

1 i 1 : : i

1 2 3 4 % :i b T ,me (s)

F i g u r e 8 f o l l o w i n g

Case 1: a angle deviation; b frequency deviation step increase at the input

r

': ' ~ ) 2 1

) ")(k~

c.

o o 3(3

cJ: ¢'t J ' :' (-)(; 4 /

! i 1

j / - - - _ _ _

/ AV P

1

[ IT'C! [ % ,

Figure9 Case 1: voltage response of the controlled system following a simulated disturbance causing an error of

0.10 p.u. voltage deviation

184 Electrical Power & Energy Systecns

I /~_\ / \ ~. 0 o g G ] ""

E [ / ~ oo72~, i / ',, - . . . . . AVR /

i ; / ", /

gr ! I ',

I 17 \l 1 2 3 4 5 6

T ~ m e ( s )

Figure 10 Case 3: voltage response of the controlled system when a step input is applied

-0.10o

~ 0 2 o c I

- O 3 0 0 ]

1 2 3 4 % 6

a T i m e ( s )

0.002;

' 5 1 2 3 4 5 6

b T i m e (s)

Figure 11 Case 3: a angle deviation; b frequency deviation of a synchronous generator controlled By predictive controller following a step change in Vre f

i l l case 3. For the AVR to be stable in case 3, the regulator gain must be reduced considerably st) that the combined exciter-regulator gain is below the permissible maximum value, s The other time responses, like angle and fiequency deviations, are more or less comparable except that a cun- ventional AVR systenr has more deviations in the first swing. The instability in case 3 with the AVR system is clearly seen in Figure 12. It is observed that in the case of the AVR system a step input causes a continuous drop in the output w)llage aml contiriutms increase it] angle and frequency deviations. These results are listed in Table 2.

Allhough the predictive controller should yield the fiBtest responses possible because oF the bang bang principle, lhe opposite occurs. This is because of the delay caused by lhe computa t ion time of" the fast-time model and the size of tile rectangular region I'or the optimal regulator. The delay Call be minimized by optimizing the sampling interval and the size of tile rectangular region.

VI . Effects on machine stabil i ty At severe operating conditions, the paranreter K s becomes negative (cases 2 and 3). It is well known s,9 that in the case

D

uo

/ /

/ /

5 0 / i

/

i I

3 0 / /

/ /

/ i

J /

lO

1 2 3 4 5 G

a T i m e ( s )

0 . 0 1 2

O.OO6

{3_

3 <:l o

/J / - J

/ t J

/#

J \ j

0 . 0 0 3 - - [ . . . . ~ . . . . . . . . . . . . . [ _ _ ± 1 2 3 4 5 6

b T~me ( s )

Figure 12 Case 3: a angle deviation; b frequency deviatlon of a synchronous generator controlled by A V R following a step change in the Vre f

Table 1 Computed constants for the |inearized syn- chronous machine

Generator

Exciter

Regulator*

Case 1 Case 2 Case 3 Constants (p.u.) (p.u.) (p.u.)

K 1 1.448 1.076 0.865 K 2 1.317 1.258 1.131 K3 0.307 0.307 0.3072 K4 }.805 1.712 1.5409 Ks 0.029 0.041 - 0.0428 K 6 0.526 0.497 0.497 2H 4.74 4.74 4.74 D 2.0 2.0 2.0 T'~,, 5.9 5.9 5.9 COR 377.0 377.0 377.0

K/c 1.0 1.0 1.0 T/: 0.8 0.8 0.8

K,i 400.0 400.0 400.0 T A 0.02 0.02 0.02 KF 0.03 0.03 0.03 TF 1 .o 1 .o 1 .o Urea x 7.3 7.3 7.3 U,ni,~ 7.3 7.3 7.3

* The IEEE type 1 regulator model is used only for the simulation of the conventional AVR and not for the predictive controller

Vol 1 No 3 October 1979 185

Table 2 Transient performance characteristics of pre- dictive controller and AVR under similar operating conditions

Case Type of no. system

Case 1 Predictive control ler AVR

(7:.tse 2 Predictive control ler A V R

Case 3 Predictive control ler AVR

Overshoot c;~ in 5~ of Peak Time Settl ing tile final time delay time vahle (T,.,L s (7-'d). s (Ts), s

8.07 1.55 0.45 I._>~- 13.04 1.30 0.30 2.5

5.55 1.45 0.45 1.55 15.84 "~ " _._ 1.,_ 0._~0 " ">

5.11 1.60 0.45 1.65 Unstable :it this operat ing point

o1" generators with :a fast excitatic, n system, this ileg:.ltive value c, f K s causes naiur.:ll dumping of lhe system it:, be reduced, which somet imes cont r ibutes increasip<g oscilla- ti,,m leading to instabil i ty. This is seen ill c:.tse _-, v,,here a step hlput apF, lied to the AVR system causes tile system angle deviation to increase indefini tely, tnakmg the systetn unstable+ The reason is apparent in Table 1 which shuv.,'s thai the gener:.itor operating p<+iltt plays a significant role in syslonl perlorlnallce. Tile loading seeins to int]uence the vahles o | / \ " i and/~5 nit)re than ,.)tiler constallts. At heaviui loads, lhe ramies of cOilstallts change in sHch a i/l~.lllilei {IS to I,,Y,,ver the F, ermissible inaximunl vahie ,.~1 tile exciter- i-egtll:.lt,.',t+ gain Kc for lt~e convent ional A V R system. 5 Fol the proMem under s tudy, the heavier load conditi,.ms c,l cases 2 and 3 al low lower l imits for K+ than for the less sew.'re condit ion of case I.

f h u s tile s.:une exci ter-regulator gain which drove the A V R systeln It, zero error in cases I and 2 causes inst:.lbilit), in case 3, as il exceeds the Ynaxinltnn permissible value under that loading condit ion.

This is not the case with tile predictive control ler. With the proposed control ler, tile system is driven quickly to n0s+ir )ere error condit ion by the bang bang hlpt i l . Within the rect:.n3gnlar region, the linear regulator nleihod is tisetl. wl i ich ininitnizes the cost tunc l ion, clefined to give zero errtu +tit final thne. Beyond lhc rectzlllgtllLir region, the iilpttt iS hail 7 hLiilg., :.tnd tie feedback loop exists which may,a._'ause the systenl lo be trustable, unless the syn- chrtmc+us_generator itself is titist:.lble at that operat ing ptfint.

V I I . Conclusions ~l'[l~" pLIper ]l+tts p r c s e i l | c d t h e {lSC (/ILI ] ' J lCdiv t i \ c ' Lt)ll i ~ ~

It>t re~u]atiug the v,,+llage output t>l u s\nchlt+tit+u, +flay l+it ,' c tmnected Ic, di/ i i / l ' i n i t e t'll.IS. ] ' h e p/ol'lll21+l , , i lill)il , \ , {C

which is in l lctcnl ~ i l h pledicl ive conltoIlet~ Jl:> hCCll e l i t l l i l / a l e d b V t h e u s e ~+t LI liile+tlr t Q 2 t t l a [ ~ l ;it t h e li l l l t i

cycle/<,no :u<+uml tile ~>/igii/. t!nl ike the c<,uvc~/tl,+md re t2t l i ,+i tor:-, , t ~ t l f plopt~sed conlrol le i ctocs l l O I c+;ttl,,,e ;il l ',,

sl;lbilit'+ ploblcn/ )el l'aulis al severe ho+tivy It>actitl 2.

The w t , k presented in this p:.q+el i~ ~l prel imimu) s iud\ I<+ demu, n'+,tt',ue the leasibility and the pt+tentkd <',t the i-,~ ,> p,+sed ctmtiol ler . Tile method can I+e ex tended It+ tlic u>,c of 3rd- ,+1 4th<ucter fast models. Work still retnuil/s t~, bc done m explorh/g the capabili t ies ,.fl this contrc41cl t>n lnt+le detuile,.I an,..] cmiltinlachhle systems.

V I I I . References 1 Chestnut, H, Sollecito, W E and Troutman, P H 'Pre-

dictive control system application' AIEE Trans. AppL Ind. Vol 55 (July 1961) pp 128-139

Fallside, F and Thedchamorthy, N 'Predictive control using an adaptive fast model' Prec. lEE Vol 114 No 11 (November 1967) pp 1761-1771

Nehrir, M H 'A predictive controller for automatic voltage regulation of DC generators - a hybrid simulation study' IEEE Trans. Ind. Electron. Control Instrum. Vol IECI-22 No 1 (February 1975) pp 4 3 - 4 6

4 Kirk, E Optima/control theory Prent ice-Hal l , USA (1970)

5 Anderson, P M and Fouad, A A Power system control and stability Iowa State University Press, USA (1977)

Chen, C F and Shieh, L S 'A novel approach to linear model simplification' Int. J. Control Vol 8 No 6 (1968) pp 561 570

Davison, E J 'A method for simplifying linear dynamic systems' IEEE Trans. Autom. Control Vol AC-11 (January 1966) pp 93--101

Sinha, N K and Bereuzuai, G T 'Optimum approxima tion of high-order systems by low-order models' Int. J. Control Vo114 No 5 (1971) pp 9 5 1 - 9 5 9

Demello, F P and Concordia, C 'Concepts of syn- chronous machine stability as affected by excitation control' IEEE Trans. Power Appar. & Syst. Vol PAS-88 No 4 pp 3 1 6 - 3 2 9

186 Electrical Power & Energy Systems