Application of an Optimal Control Theory to a Power System

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-89, NO. 1, JANUARY 1970

Application of an Optimal Control Th

to a Power System \I

YAO-NAN YU, SENIOR MEMBER, IEEE, KHIEN VONGSURIYABER, IEAND LEONARD N. WEDMAN, MEMBER, IEEE l

Abstract-In recent years important research has been done inthe area of system optimization by control engineers. Many the-oretical results have been published but application examples havemainly been on low-order systems. An attempt is made to apply acertain class of optimal control theory, known as the state regulatorproblem, to obtain an optimal controller to improve the dynamicresponse of a power system. The system differential equationsare written in the first-order state variable form. A cost functional isthen chosen, and the matrix Riccati equation is solved. Pui's andGruver's method is applied for the numerical computation, and thesystem is made initially stable by shifting the system eigenvalues.

INTRODUCTION

FOR THE INTEREST of most power engineers who mightnot be familiar with the optimal control technique an intro-

duction to the state regulator problem seems in order. For thestudy, the synchronous machine and controller equations arewritten

x = Ax + Bu (1)where x is the state variable vector. Each variable, for example,torque angle or speed, represents a deviation from an initialstate under investigation; u is the control vector, A and B arematrices. For notations see the Nomenelature.A cost functionial, usually of the quadratic form,

1 TJ = - {[x'Qx] + [u'Ru]} dt (2)

2o

For an optimal control u, p and x must be the solution of

OH dHXt=aH p = _a

Op Ox

which leads to

x=Ax-+-Bu, P= -Qx-A'p.

It is also necessary that

(4)

(5)

(6a)cu

which gives

u = -R-1B'p.By assuming a solution for p,

p = Kx

the optimal control becomes

u = R'-B'Kx.

Time derivative of (7) with substitution of (5) gives

K = -A'K - KA + KBR'BK - Q

(6)

(7)

(8)

(9)

which is the matrix Riceati equation well known to controlengineers. K is symmetric because K' satisfies the same (9).For a time invariant system, A, B, and K are constant matrices,where K is the solution of the nonlinear matrix algebraic equation

is then chosen, and Pontryagin's maximum principle is applied tofind the optimal control. Engineering experience is useful inchoosing Q and R matrices. But, once chosen, the results of theoptimal control are unique. Since u is the control and x is thedeviation of the state variable x(T) = 0 for T = o for a stablesystem, the purpose of minimizing J is to minimize the controleffort and error response of the system after a disturbance.The solution of the problem begins with the introduction of -a

costate variable vector p and a Hamiltonian H of the followingform

H = 2 [x'Qx] + - [u'Ru] + p'[Ax + Bu].2 2

(3)

Paper 69 TP 104-PWR, recommended and approved by the PowerSystem Engineering Committee of the IEEE Power Group forpresentation at the IEEE Winter Power Meeting, New York, N. Y.,January 26-31, 1969. Manuscript submitted February 9, 1968;made available for printing November 22, 1968. This work wassupported in part by the National Research Council of Canada underGrant A3626.

Y. N. Yu and L. Wedman are with the Department of ElectricalEngineering, University of British Columbia, Vancouver 8, B. C.,Canada.K. Vongsuriya is with the National Energy Authority, Bangkok,

Thailand.

-A'K - KA + KBR-'B'K - Q = 0.

The system equation with the optimal control becomes

x = (A - BR-'B'K)x.

(10)

(11)

For more details, see [3].

SYSTEM EQUATIONSThe power system under investigation consists of a syn-

chronous machine unit connected to an infinite bus through atransmission line (Fig. 1). It has both voltage regulator andspeed governor controls. The machine equations are written

d6'= n

dr(12)

d =_ {[(g + 1.5h) - Don]

rVO Sin 6 QF (Xd I- X4) vo2 1_ 0 + S(n 26- Po] (13)TXdPO 2Xd'Xq

dIF VF Xd 'F (Xd - Xd')Vo COS (

d- + . 14)

d,r ,B XdfTd0' OXd I

55

IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, JANUARY 1970

Equations (12)-(19) consist of a complete set of systenm equa-tions. The a', n, VF, vs, g, gf, and h are the state variables, andu1 and u2 are the controls. The system is nonlinear because of thenonlinear terms of (13), (14), and (15). Although a nonlinearcontrol is under study [5], this paper confines itself to the linear-ized system.When linearized, the system equations can be written in the

standard form (1). The resulting A matrix is given in (20):

Fig. 1. Synchronous machine infinite bus system.

+ Ale VF

~~~~I J+ T P _Ps1+Up

+ [5P I

Fig. 2. Exciter-voltage regulator system.

Fig. 3. Governor-hydraulic system.

The details are given in the Appendix.

A control signal u1 is fed into the summing junction of the

exciter-voltage regulator system through a transfer function

/.L/(1 + Tsp) (Fig. 2). The exciter of the machine has a transfer

function Me6/(l + Tep) and is shown in the forward branch of the

figure. The system equations may be written

dVF VF ±(e)

dTX Te Atre

dv VS As

dr (3r. OiT3

(15)

(16)

Since the machine terminal voltage vt can be calculated fromvd and v0,

Vt =Vd2 + V2 (15a)

and its deviation from the initial state may be approximated by

Vt = Vd + Vq .

Vto Vto(15b)

The expressions for vd and v, in terms of state variables are givenin the Appendix and the initial values of vdo and vqo can bedetermined from the initial load of the system.A control signal u2 is fed into the summing junction of the

governor system through a servo motor, Fig. 3. The motor has a

transfer function /ia/(1 + rap). The governor system transferfunction 1/(a + rTp) and the hydraulic operator (1 -

(1 + 0.5 rwp) are shown in the forward branch of the figure.The governor system equations may be written

dg=

og 1 l-ondr 1Tg (T kwo

dgf _ gf Aadr r3Ta f3Ta

dh 2 dg 2h.

dr d-r 3rw

(17)

(18)

0 1 0 0D

a2l -FM a23 0

1a3l 0 a33

1a4l 0 a43 --

3Te

0 0 0 0

0I

LoOTg

0

0 0

01

02M

0 0

0 1.5g2M

0 0 0 0

_ He

(Te

_ 1

OTs

0 0 0

0 0 0

0 0

f7r0 O r,7

0 0 0 0 0 0 -1

0 2 0 0 0 2o 2wo0Tq T0 rO

where

a21 = { -cos 2a0

2M Xd'TdO' Xd'X q

vo sin 60a23 0B2MXd'Tdo'

(Xd - Xd')VO sin 80a31 = Xd

Xda33 = -

AeVo qx0vdo cos s0 xd4vI0 sin 6oO3TeVto X Xq Xd J

I-le XVqOa43 =

TeVAtO Xd Tdo

The state variables are

x = [8 ,n,pFf,VF,vS,g,gf,h]'.The B matrix may be written

B=0 0 0 1 0 0 0]'

0 1

and the controls are

u = [u "U ]

0

0

(20w)(20)

(20a)

(21)

(22)

(23)where

A MsUl A MaU2UV = u =

I3Ts /3Ta

v,

SALIEN r POL ESYNCHRONOUSMACHINE

INFINIrE BUS

56

(19) (23a)

YU el al.: OPTIMAL CONTROL THEORY AND POWER SYSTEMS

Numerical Example

Puri and Gruver's procedure [4] is followed to compute theK matrix. Equation (10) is written

A1l(k)K(k) + K(k)As(k) + Ql(k) = 0 (24)

where

.3 75 .75 .715

*2 .*1 .J

.05 .05 055

Al(k) = A -SK(k-1)

Ql (k) = Q + K(k-1)SK(k-1)

and

(24a) ¾a ° NC

0 c ,

-l -.05 -.05 -.05

S = BRI1B'.

For a fixed system structure, S is constant. A method has beendeveloped to compute K(k) of (24). To begin the computation,a K(°) matrix is required so that A,(') is stable. K(0) can be foundby shifting all the eigenvalues of the initial system to the lefthalf of the complex plane through an eigensystem sensitivitymethod [11]The system under investigation has the following parametric

values:transmission line

x = 0.7417, B = 0.1339

synchronous machine unit

Xd = 1.000, xd' = 0.270, Xq = 0.600

TO= 9.000, M = 0.2122, D = 0.005 37

exciter and voltage regulator system

Me = 10.00, re = 1.000, rs = 0.500

governor and hydraulic system

o- = 0.045, T-= 0.100,

Ta = 0.010, Ir, = 1.600.

The numerical values of M,a and M,a are not needed as they are

included in u, and u,, respectively [see (23a) ].

For an initial load and a terminal voltage of the synchronousmachine

P0 = 0.735, Qo = 0.034, VeO = 1.050.

The initial currents, voltages, flux linkages, and torque angle are

ido = 0.286, iqo = 0.640, vdo = 0.384

v5o = 0.977, VF = 1.263, vo = 1.058

{FO= 9.491, = 0.887.

The time is scaled in computation by a factor: = 7.308. Sub-stituting these values into (20), the numerical values of the co-

efficient matrix A is obtained:

0

-0.683-0.08320.1300

0

0

0

1

-0.03460

0

0

-0.0265

0

0 .0531

0

-0.0816-0.0254-0.107

0

0

0

0

0

0

0.13-0.13

0

0

0

0

- 2 .1 --I

-.3 -.15 - 45

-4 - 2 2

Fig. 4. Initial system responses without controller (5', n, 4fl, vF).

.15 .75 .75 *3

*05 05 .05

0 0 *0 *0

X *.5 X

-*05 - -05 -.05 -7

_-1.1 -. -- - 2

-.75 --75 -- 15 --3

-2 - 2 -*2 -*4

Fig. 5. Initial system responses without controller (v., g, gf, h).

The system is oscillatory (see Figs. 4 and 5). The responsesare obtained from an integration of (1) using the A matrix of theinitial system (25) without control. Being a linearized system,the initial condition for the computation is found from therelease of a load perturbation P,'.An eigenvalue-shift technique is then applied to find a K(°),

and the eigenvalues are shifted to the left of the left half complexplan as much as possible. The optimal control is finally found

o o 0 00 0.882 0 1.320

7 0 0 0 017 -1.370 0 0 (25)

-0.274 0 0 00 -0.0616 -1.370 00 0 -13.70 00 0.123 2.74 -0.171J

57


.3 .15 .75 .75

2 *7 .1

.1 .05 .55 .55

NO .5N12.5N2.1

10 c Zb ,~i

-41 - 05 -.o5 -.c

- 2 -4t -41 -*

- 3 -415 -415 --K

--4 - 2 - 2 - 24.0

E-SEC50 6-0 7.0

Fig. 6. System responses with optimal controller (6', n, 4,p Vp).

.3 .06 .3 .75

*2 54 2 1

7 02 71 -05

-4 -4

- 2 - 04 -2

3.3-Q06 -.3

0

.3 .75 .75 .75

2 .7 .l *1

-4 o0 -05 -55

--2 -7 --I --I

-.3 -415 -.75 -15

-4 - 2 - 2 -2 o 1.0 250 3-0 4 0 5s0 60 75TIME- SEC

Fig. 8. System responses with optimal controller but withoutemphasis on 6' and n (6', n, 4F, VF).

*3 .3 .5 31

2 *2 *7 .2

.* *05

_N N_ _

N *0 o *0S *~0 .5*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

_. -4 --05 -

--2 - 2 -7 -*2

.3 -3 45 -3

-.4 - 4 - 2 - 4J0 20 30 4 0

TIME-SEC

Fig. 7. System responses with optimal controller (v,, g9f, sh). Fig. 9. System responses with optimal controller but withoutemphasis on 6' and n (v., g, gf, h).

from (24) through successive approximations. The K matrixthus found is as in (26):

24.669 6.09076.0907 43.5492.1174 -2.55980.4597 -0.0206

-0.3313 -0.1065-48.945 7.9589-1.1545 4.6784

_-29.572 29.720

2.1174-2.55988.24351.2263

-0.8016-26.410-0.3098-14.742

0.4597-0.02061.12631.3983

-0.9763-4.47100.0023

-2.1996

-0.3313-0.1065-0.8016-0.97631.66193.0856

-0.02141.4128

-48.9457.9589

-26.410-4.47103.0856535.66

-1.6837256.90

and the Q and R matrices are

Q=diag [10 10 1 1 1 1 1 1]

R = diag [1 1].

The optimal controls are found from (8) and (23a) and A1 fromA -SK. The responses of the system with the optimal control

(27) are found from an integration of x = Aix and are shown in Figs.6 and 7.

-1.15454.6784

-0.30980.0023

-0.0214-1.68371.15815.0112

-29.57229.720

-14.742-2.19961.4128256.905.0112153.67

(26)

58

YU et al.: OPTIMAL CONTROL THEORY AND POWER SYSTEMS

CONCLUSIONAn optimal control theory has been applied to a power system.

An initially unstable system can be stabilized with an optimalcontroller. It is found that with emphasis on the torque and speeddeviations in choosing Q matrix of the cost functional, the sys-tem settles much faster than that with Q as a unit matrix (Figs.8 and 9). There are more works remaining to be done in applyingoptimal control theory to power systems with nonlinearities andwith control constraints. It is hoped that the results in thispaper will be vindicated by power system model tests.

APPENDIXSTATE VARIABLE FORM SYNCHRONOUS MACHINE EQUATIO]Neglecting armature resistance R, time derivative of

linkage pvd and p{,, and subtransient time constants, Paequations [1] become

Vd = - q

Vq = C {d

where

Utilizing (28), (29a), (30), and (31a), i4, Vd, t, and v1 can besuccessively solved,

vO sin axc

Cd xvO sin8

Xa (36)

(1 - xB)F Vo COs 5

Xd'' Xd'

X/F Xd Vo Cos aVa I

, +Xd 'rdo Xd

where

Xd = (1 - xB)xd' + X

Xa = (1 -xB)x1 + x.

(28) Substituting id of (36) into (31a), the solution of VFR is

Xd/'F (Xd - Xda')Vo COS aVFR Xddt Xd I

(29a)CO2=q -XqiqVF I +7rdoP1: {d = - Xdid-

The last equation may be written

C fi = VFR - Xdid

where

VFR =VF + PrdO (Xd - Xd') id

1 + dO'P

which may be written

(29b)

(30)

(30a)

where

Xd = (1 - xB)Xd + X.

In computation the time is scaled

r = ft.

Hence

n=-.I3

(36a)

(37)

(37a)

(38)

(39)

Dividing through (31) and (32) by : and (33) by /2 and substitut-ing (36) and (37) into the results, one has

PF = VF - VFR (31)

where

VF = TdO [VFR - (Xd - X)d]) - (31a)

In this paper, iF is chosen as one of the state variables and (31)as one of the synchronous machine equations. The other twoequations are

pa' = co' (32)

PLO = M[F' -D'w-P t] (33)

The last equation is obtained from

P1' = (D + Mp)w' + Pew (34)

where

Pe = (i,V, + idVd) - Po- (34a)Since VP, a', anid w' are chosen as the state variables, other

variables in (31), (32), and (33) must be eliminated. From themachine terminal condition (Fig. 1), one has

vt(jB) + (Vt- vo)/jx = i. (35a)

In dq components

(Vd + jv1)(jB) + [(Vd + jv2) - (v0 sin a + jCo cos a)]/jx

= id + ji. (35b)

da'=n

dr

dn - 1 [P' Dn - P,']

dVF VF Xd1F/' (Xd - Xd)VO cos adr / /Xd'rdo' ± Xd

where PF' is approximated by

P/= g + 1.5h

p,/ vo sin a/F + (Xd' -X)Vo2 sin 2 -PoXd ±dO 2Xd'X s

(12)

(13a)

(14)

(13b)

(13C)

Thus (12), (13), and (14) consist of the state variable formsynchronous machine equations for the optimal control study,where VF iS a state variable chosen for the exciter-voltage regula-tor system, and g and h are state variables of the governor andhydraulic system.

NOMENCLATURE

Optimal Control

A

A,BH

matrix coefficient of state variables of systemwithout control

same, but with the optimal controlmatrix coefficient of controlsHamiltonian

59


cost functional, a scalarK Riccati matrixQ positive definite matrixR positive definite matrixx state variable vectorp costate variable vectoru control vectorU1,U2 control signalsU,, u, see (23a)I "transpose" of a matrix or a vector.

System Parameters

B transmission line susceptancex transmission line reactanceD damping coefficientM inertia constantXd synchronous reactance d axisXd' transient reactance d axisxq synchronous reactance q axisXd, Xd', X, see (36a) and (37a)TdOl open-circuit time constant of the field're exciter time constant

voltage control feedback loop time constant(Fig. 2)

Tg gate time constantTa governor actuator time constant (Fig. 3)Irw water time constantH1e exciter gainAs voltage control feedback loop gain (Fig. 2)Ita governor actuator gain (Fig. 3)af permanent droop.

General (all in per unit unless otherwise stated)

Pi mechanical power inputPe power of the electromechanical energy con-

versionP generator power outputQ generator reactive power outputVO infinite bus voltage, assumed constantvt generator terminal voltageVd d component of vt

q component ofvta field voltagesee (30a)armature currentd component of i, i, = q componentd-axis flux linkageq-axis flux linkagea field flux linkage, see (31a)torque angle (radians)synchronous speed (377 rad/s)speed (rad/s)speed (rad/s), see (39)gate opening

governor feedback loop signal (Fig. 3)water head-d/dt, 1/stime-scaling factorscaled time (seconds)"initial value" of a variable (subscript)"deviation" from the initial value of a variable.

ACKNOWLEDGMENTThe author wishes to thank G. Dawson for his assistance in

computation and Dr. F. Noakes for his interest in the project.

REFERENCES[1] R. H. Park, "Two-reaction theory of synchronous machines,

pt.I: generalized method of analysis," AIEE Trans., vol. 48,pp. 716-730,July 1929.

[2] M. Athans, "The status of optimal control theory and applica-tions for deterministic systems," IEEE Trans. AutomaticControl, vol. AC-11, pp. 580-596, July 1966.

[3] M. Athans and P. L. Falb, Optimal Control. New York:McGraw-Hill, 1966, ch. 9.

[4] N. N. Puri and W. A. Gruver, "Optimal control design viasuccessive approximations," Preprints, 1967 JACC (Phila-dephia, Pa.), pp. 335-344.

[5] K. Vongsuriya, "The application of Liapunov function topower system stability analysis and control," Ph.D. diserta-tion, Dept. of Elec. Engrg., University of British Columbia,Vancouver, Canada, 1968.

[6] L. Wedman, "Optimal control of power system with sensitivityconsiderations," M.A.Sc. thesis, Dept. of Elec. Engrg., Uni-versity of British Columbia, Vancouver, Canada, 1968.

Discussion

W. J. Smolinski (Urniversity of New Brunswick, Fredericton,N. B. Canada): One can only agree with the authors that "manytheoretical results (on optimal control systems) are published, butapplication examples are mainly on low-order systems." The authorsare to be complimented on applying the results of optimal controltheory to an eight-order system of considerable significance topower-system engineers. A couple of observations on their work willalso be made.

It is noted that the optimal control law achieved by the authors isU. = K5-X and U, = K7- X, where K5 and K7 are the fifth andseventh rows of K and the dot notation means a scalar or innerproduct. Several remarks may be made about this optimal controllaw: it is linear in the control vector (as assumed and required tosatisfy the Hamilton-Jacobi equation), it contains each of the termsof the state vector, and it is a feedback law. These are well-knownproperties of an optimal control law for a linear system with a qua-dratic performance measure of the type used by the authors. Cer-tain commercial excitation system controllers U, appear to be linearonly in one or two of 5, n, and 4pF of the state vector. These controllersare often referred to as speed and power stabilizers. It would beinteresting to know if the authors have compared their optimalcontroller to an optimized speed or power stabilizer of typical com-mercial design. To do this, an optimal control law should be foundfor a one-dimensional control vector, i.e., U, = 0. It would also beinteresting to know if the authors have investigated the case of aone-dimensional optimal controller, with U, - 0. That is, how muchdoes the turbine governor contribute to the overall control effortof this optimal controller? A comparison of gate opening g in Figs.5 and 7 shows an actual reversal in the direction of gate openingalthough the speed change is in the same direction in both caseswhich would indicate a considerable change in governor performance.Is this perhaps because the governor is that of a hydraulic turbinebut has no temporary droop compensation? One of the disadvan-tages of the optimal controller proposed by the authors is that itrequires that all the state variables be measurable and available ateach of two summing points to form the optimal control law. Havethe authors investigated the feasibility of this?The seconid observation has to do with one of the more thought

provoking aspects of optimal control theory, that is, with themultiplicity of optimal control laws that are possible for a given

Manuscript received January 14, 1969.

vQVF

VFR

i

id~d

OF'acoo

CdO

n99fhp

0IT

0

60

YU et al.: OPTIMAL CONTROL THEORY AND POWER SYSTEMS

system depending on the performance measure or criterion to beoptimized, the nature of the model of the system, and the natureof the constraints on the system and its control vector. The authorsemphasis on torque angle and speed deviations produces a well-damped machine and is therefore a good choice of performancemeasure, penalizing large deviations of torque angle and speed as itdoes. This can therefore be considered to be a form of controller toproduce "optimal transient removal" in power systems as describedin [71. However, the authors' controller is linear in the state vector,whereas the controller described in [7] is of the bang-bang type.The difference is of course that the bang-bang controller recognizesconstraints on the control vector whereas the linear controller doesnot. Bang-bang control theory in fact results from this difference inthe nature of the control vector (which cannot be ignored in excita-tion systems with definite ceiling voltages and under large distur-bances to the generator). The authors indicate that further work isto be done in applying optimal control theory to power systems withcontrol constraints. It would be interesting to know if the authorsare considering bang-bang controllers under these circumstances orif they are considering some modification to the optimal linear con-troller described in this paper. The authors are to be complimentednot only for this new application of optimal control theory to powersystems but for the questions they raise regarding heretofore estab-lished commercial practices in power system control such as theuse of speed and power stabilizers and the need for temporary droopcompensation for hydraulic turbine governors if such an optimallinear controller is available.

REFERENCES[7] 0. J. M. Smith, "Optimal transient removal in a power sys-

tem," IEEE Trans. Power Apparatus and Systems, vol. PAS-84, pp. 361-374, May 1965.

W. A. Mittelstadt (Bonneville Power Administration, Portland,Ore.): This paper brings an important control theory to bear on apower system stability application. The ease with which the feed-back control law is established makes this method particularlyattractive. The quadratic cost functional employed is superior tothe optimal time criteria since excursions in the state and controlvectors are penalized.An important question to ask is can this control method be applied

in a practical sense to generators operating in-a multimachine powersystem? If a direct application of the theory is attempted, eachcontrol signal (U, and U2) would require state measurements fromall system generators. A practical scheme may be possible, however,if the infinite bus of Fig. 1 can be replaced by a dynamic equivalentrepresenting the remainder of the system, or if the system differentialequations can be adequately decoupled.Another problem which appears in the multimachine case is that of

quickly estimating postfault target values for generator power andtorque angle.Do the authors have any suggestions for overcoming these dif-

ficulties?

Manuscript received February 10, 1969.

where T is the specified terminal time. This is certainly agreeableas F can be chosen equal to zero, and as the authors indicate x( T) =

0 as T -X co and have as their stated purpose the minimization ofthe control effort and error of the system following a disturbance.Since the system state equation is

x = Ax + BU (41)

the use of the cost functional

2= { [X QX] + [u'Ru]} dt (42)

implies that we wish to keep the state x near zero.However, controlability of the system implies that we can find a

control that will accomplish the afore-mentioned objective, acontrol that drives x near zero. The question arises whether ornot the system of (41) should be controllable in order to obtain anoptimal solution as presented by the authors. It would not benecessary for the system of (41) to be controllable provided thatthe interval of control to T is finite because the contribution of theuncontrollable states to the cost functional would be finite.However, the authors have indicated that the terminal time

T = c, which requires that the system be controllable for the costof control to be finite. Have the authors tested the system forcontrollability, or was it assumed that the system tested was con-trollable?

I agree with the authors that once the positive definite matricesQ and R are chosen, the optimal control found is unique. However,I had a question as to what techniques were used to pick Q and R?What engineering experience would one bring to bear in this selec-tion process?

I would also like to know if the authors placed emphasis onvariables other than torque and speed deviations in their selectionof the Q matrix? If so, what are the relative effects produced bysuch variation in emphasis, and what did the control signals U1and U2 look like?

N. S. Rau (British Columbia Hydro and Power Authority, Van-couver 2, B. C., Canada): The authors have written an excellentpaper applying optimal control theory to power system problems.The system equations are linearized and are valid for small variationsin the speed of the machine.

It will be very interesting to see the control signals ul and U2plotted as a function of time for the cases shown in Figs. 6 and 7.Also, do the authors feel that such a signal could be derived fromone or more of the machine quantities?

Furthermore, the analysis indicates the optimum signals u, and U2for a step change in power. Will the optimum signals ul and U2remain the same if small changes in power are random in nature, asin any power system. The initial value of the state vectors X(o)will be different for successive power variations that occur duringthe random power changes. Can the authors comment on thefeasibility of a design for a controller incorporating the featuresdescribed.


John D. Morgan (University of Missouri, Rolla, Mo. 65401): Theauthors are to be commended for their work in applying optimalcontrol theory to the power system problem. In reviewing the paperI noticed some points that were neglected in the presentation thatraised some questions in my mind. First, the cost function chosenneglects the terminal cost portion (40); normally associated with thestate-regulator problem.

1=i - [x(T)'Fx(T)] (40)2

Edward J. Hyland (Arizona Public Service Company, Phoenix,Ariz.): The authors have provided one additional step towardgreater understanding of the role played by control devices inmaintaining stability of the electrical system. This particularapplication is rather limited with respect to the scope of the systemand the sophistication of machine representation, and yet it clearlypoints toward a method of optimizing controls on a system. The


61



assumptions made concerning the system size, machine representa-tion, and its linearization lead to the following questions.

1) Given an external system which is time variant, as is themachine itself, will the eigenvalues become functions of the externalsystem and thus optimize control parameters on a given machine as atime variable of system performance?

2) It appears that the matrix K(°) is a function of initial conditionsonly plus shift of eigenvalues. Can the authors amplify their state-ment "A method has been developed to compute K(k) of (24)?"

3) Have the authors tested the change in control parameters byusing a conventional dynamic stability program with machinerepresentation and input of the two different sets of control pa-rameters?

4) The authors recognize potential inaccuracies due to treating anonlinear system with linear methods. Have they conducted anyerror analysis of eigenvalues (and shifts) due to inaccuracies in thelinear simulation?

Y. N. Yu, K. Vongsuriya, and L. N. Wedman: The authors appreciate very much the comments and questions raised in thediscussions which certainly will stimulate research in the future.

Prof. Smolinsky presented a comprehensive discussion. For years,people have done an excellent job of developing stabilizing signalsfor excitation control of power systems from the damping concept[8]-[10]. While they are searching the signal by adjusting controlparameters one at a time, we are trying to find a general stabilizingsignal by adjusting all of the control parameter simultaneouslyfrom optimal control theory with least system error response andminimum control signal excursion. We have not made direct com-parison of the two methods, but in one example we have foundthat a governor controller is even more effective than a linear excita-tion controller. This area certainly deserves further investigation.The fact that the gate reverses its direction of motion from a

system with an optimal control compared to that from a systemwithout an optimal control is not due to the lack of temporary droop.We found the same phenomenon from a system with the droop.Normally, the hydraulic turbine responds to the gate openingresulting from a system speed drop with a torque decrease whichaggravates the speed drop. The optimal control reverses the gateopening to minimize the torque decrease. There is no unique way ofchoosing the system variables as the state variables. We did notinvestigate the feasibility problem of the optimal controller butwere able to eliminate certain state variables for the design of asuboptimal controller from a sensitivity study [11].The most searching question being asked by Prof. Smolinsky and

other discussers is probably the validity of the results based on thestudy of a linearized model. First, we feel that a linearized model isthe first approximation of a system which must be thoroughlyinvestigated even if one wishes to pursue a detailed nonlinearanalysis, e.g., by Lyapunov's direct method. Second, the linearmodel approach is the practical approach. There are still manydifficulties in working nonlinear computation and optimal control.This, however, does not mean that the problem has been overlooked.Analysis and computation of a system with braking resistance andforced excitation after fault are still in progress.

Manuscript received April 17, 1969.

Mr. Mittelstadt's discussion is very useful. The difficulty inapplying the techniques developed in this paper to a multimachinesystem could be the computation itself. Few people have really trieda very high-order dynamic system. The suggestion of replacing aninfinite bus by a dynamic equivalent should be a great improvement.The best modeling of a large power system could be a four- or five-machine system with each machine represented in detail.We are indebted to Prof. Morgan for helping to clarify some point

of our paper. We did not include J1 because we wanted to keep xnear zero as T - -. In this study the initial control K(°) is so chosenthat it will make the initial system stable. This is necessary toguarantee the convergency of the Riccati matrix solution. Thefirst K(°) is arbitrarily chosen and then is improved from the studyof eigenvalue sensitivity with respect to the K elements themselves.The eigenvalues are shifted to the left in the left half complex planeas much as possible before starting the Riccati matrix solution.Therefore, the system is controllable.There is no unique way of selecting Q matrix, and the technique

can be improved only through experience. We obtained bettersystem response by emphasizing 6 and n in choosing Q. R must bechosen on the basis of the physical controller. Making R elementvalue larger than Q's will most likely depreciate the performance butmay result in a cheaper controller.Mr. Rau's discussion is interesting. The control signal of our paper

is derived directly from the state variables. We did not investigate apower system with random power change, and the optimal control isderived from a given state of operation.We appreciate Mr. Hyland's discussion. It is true that the eigen-

values vary with different system configuration of line switching orchange in load. However, we found that the optimal controller isvery insensitive to the operating condition [111. K(°) is first chosenas a null matrix. If the system is unstable, then the eigenvalues areshifted into the left half complex plane to the left as much as possible.The final value of K(°) is used to solve the Riccati matrix. Thedetailed technique of computing K(k) is given in [111. As mentionedbefore, we found that the optimal controller is very insensitive tooperating conditions. We did not compare our results with thatusing conventional dynamic stability programs. They might notrepresent the synchronous machine in detail. We did not investigatethe inaccuracy of the model parameters themselves. However, thereshould be no inaccuracy in the eigenvalue analysis itself althoughthere is a potential inaccuracy in the linearized model. We feel thatthe most important step to be taken in the stability and controlstudy in the future is to investigate a multimachine nonlinear powersystem.

REFERENCES[8] R. H. Shier and A. L. Blythe, "Field tests of dynamic stability

using a stabilizing signal and computer program verification,"IEEE Trans. Power Apparatus and Systems, vol. PAS-87,pp. 315-322, February 1968.

[9] F. P. DeMello and C. Concordia, "Concepts of synchronousmachine stability as affected by excitation control," IEEETrans. Power Apparatus and Systems, vol. PAS-88, pp. 316-329, April 1969.

[10] F. R. Schleif, H. D. Hunkins, E. E. Hattan, and W. B. Gish,"Control of rotating exciters for power system damping: pilotapplications and experience," IEEE Trans. Power Apparatu8and Systems, vol. PAS-88, pp. 1259-1266, August 1969.

[11] L. N. Wedman and Y. N. Yu, "Computation techniques forthe stabilization and optimization of high-order power sys-tems,"(Proc. 1969 IEEE PICA Conf. (Denver, Colo.), pp. 324-343.

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Application of an Optimal Control Theory to a Power System

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