A synchronous generator stabilizer based on a universal model

Electrical Power & Energy Systems, Vol. 20, No. 6, pp. 435-442, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain ELSEVIER

PII: S0142-0615(97)O0063-X

0142-0615/98/$19.00+0.00

A synchronous generator stabilizer based on a universal model Youn~l-Moon Park a, Jin-Ho Lee a, Seung-Ho Hyun" and Kwang Y. Lee c aElectrical Engineering School, Seoul National University, Seoul, 151-742, South Korea bKorea Railroad Research Institute, Kyungki-do, 437-050, South Korea CElectrical Engineering Department, The Pennsylvania State University, University Park, PA 16802, USA parameter optimization techniques are employed to obtain the PSS parameter values [3-5]. Although conventional PSSs have shown efficient controls, they may encounter problems when the system operating conditions meet severe changes. Adaptive control techniques were introduced to the PSS design for wide range operations [6-8]. They are simple to implement and, in particular, they have the advantage of adaptation to the changing operating conditions of a power system. Recently, alternative control schemes have been proposed to deal with nonlinearities and uncertainties in power systems. These are intelligent control schemes such as artificial neural networks or fuzzy logic, which showed promising results in the PSS problem [9-11]. In this paper, a new approach is proposed for power system stabilization. A parameter-free model, or a universal model, is newly developed using input-output data from the controlled system. The universal model has very few parameters, so the proposed PSS based on the proposed model has the advantage of using very few design parameters compared with conventional PSS. Any conventional methods for controller design based on the universal model can be adapted, and in this paper, the LQR (Linear Quadratic Regulator) method is adapted for PSS design based on the universal model. The conventional LQR controller requires a state observer because all of state variables are not available in the power system. The proposed controller does not need a state observer because state variables are defined as inputoutput data and they are directly obtained from inputoutput data in this paper. The basic ideas of the universal model and the controller design procedure are developed in Section II. Application of the proposed approach to the PSS design is demonstrated numerically in Section Ilk and conclusions are drawn in Section IV.

A power system stabilizer (PSS) based on a universal model is developed. The uniw'rsal model is proposed to approximate an arbitrary system using the input-output data only rather than a mathematical model The conventional LQR (Linear Quadratic Regulator) method is adapted for PSS design based on a universal model. A state observer is not needed in the proposed PSS by defining state variables as input-output data. The proposed PSS has the advantage of using very few design ,parameters compared with conventional PSS, which are designed based on mathematical models of controlled systems. The proposed PSS is applied to a typical single-mackine infinite-bus power system. Simulation results under various operating conditions show that the proposed PSS is comparable to conventional PSS in loading conditions and system failures. © 1998 Elsevier Science Ltd. All rights reserved. Keywords: power system stabilizer, universal model, linear quadratic regulator

I. Introduction In many situations a power system stabilizer (PSS) is needed for damping of low-frequency oscillations caused by load disturbances or short-circuit faults, etc. [ 1,2]. Since deMello and Concordia's work [3], the use of supplementary excitation control through Power System Stabilizer (PSS) has been recognized as an effective means for power system stability. A power system i,; inherently nonlinear. The usual methods for PSS design are based on linearized models of the power system equations around a nominal operating point. In these model-based approaches, the gains and time constants have f:xed values, which are valid only when the system operates near the operating point of interest. Lead-lag compensation methods, optimal control theory or

435

A synchronous generator stabilizer based on a universal model: Y-M. Park et al

436

II. Universal m o d e l

Moving Average (ARMA) form:

II. 1 Universal model representation A new representation of dynamic system is derived under minor assumptions in this paper. To derive the representation, we make the following definition [12]:

y(k+l)=

~

aiy(k-i+l)+

~. b i u ( k - i + l )

i:l

If IA/y(k)l and IAJu(k)l are negligibly small for some then AN+ly(k + 1) is approximated by

Definition 1: The backward differences, A ~¢(k), are defined as

b{u(k)-N~ ' i=0

follows:

A"f(k)----An-lf(k)-An-lf(k - 1), n--- 1, n is an integer

(3)

i=1

i,j > N,

Aiu(k-- 1)}

where b is

(1)

a°f(k) =f(k)

i=l

where f(.) is a discrete-time sequence, and k is an integer representing the time index. The backward difference operator will be used to system output and input. If the system is single input/single output (SISO), then the system output can be represented as

Proof of proposition 1 is given in Appendix B. With assumption 1, the system output equation (2) can be approximated by the universal model defined as follows:

~(~+ 1)= y. ~iy(k)+b u(k)- }-" Z~u(~- 1)

N

y(k+l):

Z

i=0

Aiy(k)+AN+ly(k+I)

i=0

(2)

(4)

i=0

where y(k) and u(k) are measured system output and input at time k, and ~(k + 1) is the estimated system output at time k + 1. It should be noted that the universal model is simple and has only one parameter, b, which contains the system information, so it has a handling advantage for that reason. The dimension of the universal model, N, should be chosen so that Assumption 1 is valid. It should be noted that the order Nhas no relationship to the system order of the original system in the state space. The justification of Assumption 1 has been made for a linear time invariant (LTI) discrete-time system. But the universal model can be used for a nonlinear system. In adaptive control literature, the ARMA model can be used for identification of a nonlinear system. Parameters of the ARMA model such as a~ and bi are updated at every sampling time and they change as time varying since the identified system is nonlinear. The universal model has only one parameter, b, that is the summation of b i. It can be seen that b changes as time varying and it should be updated at every sampling time. The validity of Assumption 1 and the universal model is illustrated in the following nonlinear system [131

where A i, i = 0, 1..... N + 1, are the backward difference operators defined in equation (1), and N is an arbitrary positive integer representing the order of the representation. Proof of equation (2) is given in Appendix A. It should be noted that equation (1) holds for any system output, whether the system is linear or not, since no assumptions on linearity are made in the proof. The system output is determined from the previous system output history and the previous input history. The representation equation (2) is made of two terms, N

Z

Aiy(k)

i=0

and AN+ly(k + 1). The first term is not affected by the system input u(k), but only the second term, AN+ly(k + 1), is affected by u(k). The relationship between AN+ly(k + 1) and u(k) will be the focal point in developing the proposed universal model representation.

Assumption 1: If IA iy(k) l and IAJu(k) l are negligibly small for some i,j > N, then AN+ly(k + 1) is approximated by

- ~ sin y -

b u(k)- y . Aiu(k - 1)

y2

=

u

(5)

where y is the system output and u is the system input. System input and output data are obtained with a sampling period of 0.1 s. Input and output data at time from 2.7 to 3.2 s, and their differences, Ayi(k) and Aui(k), are given in Table 1. Note that as the order increases the differences are decreasing. Figure 1 shows the trajectories of the system output and the universal model output. The dimension of the universal

i=O

for some real constant b. The justification of Assumption 1 is made for a linear time invariant (LTI) discrete-time system.

Proposition 1: Consider a linear time invariant (LTI) discrete-time system, represented in the Auto Regressive Table 1. I/O data and their difference

i 0

u(k - i)

- 6.2284

Aiu(k)

--

y ( k - i) Aiy(k)

6.2284 1.4485 1.4485

1 - 4.8304 - 1.3980 1.1953 0.2531

2

3

4

- 3.4083 0.0241 0.9322 - 0.0099

- 1.9923 0.0301 0.6650 - 0.0059

- 0.6079 0.0046 0.3987 - 0.0008

5 0.7211 0.0028 0.1383 - 0.0005

A synchronous generator stabilizer based on a universal model: Y-M. Park et al

437

2.5

1.5

1 0.5 0 -0.5 -1 -1.5 -2 -2.5

0.5

i

1.'5

2

2:5

3

3.5

4

4:5

5

Time(sec) Figure 1. Comparison of the system output and the universal model output

model, N, is set to be 5 since values of ASy(k) and ASu(k) are negligibly small. We see that the dimension N has no relationship to the sysl:em dimension, 2. The parameter b can be calculated from the previous I/O data; i.e. if the values of y(k),

"

/

~(k + 1) = ~ . aiAiy(k) + b u(k) -

N

Z

the designer. Any conventional controller design scheme can be applied to the universal model; however, the LQR (Linear Quadratic Regulator) design method is applied in this paper. We rewrite the modified universal model as follows:

Aiy(k- 1)

i=0

i=O

: Z

Z

ci Aiu(k - 1)

/

i=0

N

and ANu(k - 1) are given, then from equation (4), b is calculated as follows:

"-1

N-1

aiAiy(k) +bOu(k) + ~

i=0

bi+lAiu(k- 1)

i=0

(9)

{ y ( k ) - ~ A} i y/( k=- 1 ) o b=

(6)

ANu(k- 1)

The comparison of trajectories between the system output and the universal model output is shown in Figure 1. The universal model approximates the system output value very closely. The above example is the case when Assumption 1 is satisfied. Linear systems and most nonlinear systems with small sampling times satisfy the assumption. However, this may not be the case for general nonlinear systems. In this case, the universal model is extended to the following modified universal model:

,

~(k+ 1)= Z

aiAiy( k ) + b

i=0

{

,1

u(k)- ~ . ciAiu(k - 1)

/

i=0

(7) where the coefficients ai, ci are determined so as to decrease the effect of higher arder differences. In this paper, the coefficients are determined as follows: a i = ( e l y ) i,

c i = (Olu) i

(8)

where b0 = b, b i + 1 : - bci, i >- O. We now introduce the following transformation that converts the universal model to a linear model so that the LQR design method can be applied:

xl (k) = y(k) x2(k)=Ay(k)+~u(k-1) x3(k) : A2y(k) + l~2u(k - 1) +/~lAu(k -- 1)

XN+ l(k) = ANy(k) + 13NU(k- 1) + t3N_ 1Au(k - 1) + . . . +/31 AN- l u ( k - 1) where xi, i = 1,2 ..... N + 1, are the state variables for the new coordinate system. The transformation equation (10) transforms the universal model equation (9) into the following

linear system: x(k + 1) :Ax(k) + Bu(k)

(11)

y(k)=Cx(k)

where 0 < C~y,a , < 1. This choice of coefficients limits the number of parameters 1:o only 2 instead of 2N + 1. 11.2 Universal model-based controller design If a system has a universal model representation, a controller can be designed for the universal model instead of the original system model that may or may not be known to

(1 O)

A=

[ oil a0

al

. ..

ao- l

a 1

...

a N

)

i

ao

1 al-1

".

...

a

438

A synchronous generator stabilizer based on a universal model: Y-M. Park et al

0

]

1.0041.00~"

bo -~-~l

B=

1.002-

b0 +/3]

-t-/32-+-

1.001-

-" +/3N

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

iiiI

......................... t ......... t.------~- .o.o-"

t ........t ............... -/--

1"

c=[1

0

...

0]

where the values of/3i

~r 0.999"

given by

iama2 aN]Ii] [bl a2

a3

are

0

/32

b2

x

iN!0 a

=

0.9980.997"

(12)

0.996

0

01s

i

l'.s

2:s

~

3:s

~

4:s

s

time (sec)

0

The derivations of equations (11) and (12) are given in Appendix C. Given the transformed linear system model, the LQR design method can now be applied with the associated quadratic cost function:

'[Y

J = 2 Lk= 1 y2(k) +Ru2(k)

]

Figure 2. Comparison of frequency trajectory due to disturbance at power angle

65-

(13)

60"

where R is a positive weighting factor and M represents the time horizon. The LQR controller, which minimizes the cost function, is in the following feedback form:

o'~ 55-

(14)

u(k) = - Kx

_o

where K is obtained from the steady-state discrete Ricatti equation [14]. It should be noted that an observer is not necessary since the state variables are defined in terms of measured system output and input data, as shown in equation

iiiiii-tilfll . iiii .......i i ii.--iiii iiiii i i i iiiiii i......i i i i i i.

50" 45-

0 ~-

40-

(11). 3O

III. PSS design using the universal model A typical single-machine infinite-bus (SM1B) power system is chosen for analysis of the proposed PSS. Representations and parameters for the simulated synchronous generator system are given in Appendix D. The control input is supplementary exciter input Uess and the output is Ao~. The performances of the UMBC PSS and the LQR PSS are compared for different case studies. The dimension of the universal model, N, is chosen to be 5. The parameter b in the universal model is determined by the collected I/O data from the system. Other constants, O/y and ~x,, are set to be 0.9 and 0.65, respectively, by trial and error. The time horizon in the cost function, M, is chosen to be 6, and the weight factor R is 0.000035. The design of the conventional LQR PSS is based on the linearized model which is shown in Ref [2]. The operating conditions, which are used for the linearized model, are as follows: Peo = 0 . 7

Oeo=0.015

Wto

--~

1.05 (p.u.)

0~5

1

115

2Js

~

~5

~

41s

time (see)

Figure 3. Comparison of power angle trajectory due to disturbance at power angle

0.05

"~

n

(15) O

where P~o, Q~o, and Vto are real power, reactive power, and terminal voltage of generator respectively. LQR PSS uses a Luenberger observer to obtain state variables that are not measurable, but the proposed PSS does not need an observer. First, the studied power system is operating near the condition as represented in equation (15). A disturbance in the initial condition for the rotor angle deviation is assumed to be 0.3 (radian). The proposed PSS is implemented to damp low frequency oscillation and the performance of the

0

-0.05-

41.1-

-0.15-

-0.2-

-0.25

o

o15

;

lls

£s

~

31s

~

41s

s

time (see)

Figure 4. Comparison of PSS output due to disturbance at power angle

A synchronous generator stabilizer based on a universal model: Y-M. Park et al 0.04

1.0041.003 ..................................................... Z ~ - l w i t h o u t

1 .I)01. 1 0.999

!

PSSt~

0.02-

.......

0-

=

g

i

1:5

~

2:s

ii:ioi:ii! !i!iii:

-0.04-

i iiiii

-0.~0

-0.~-0.1-

0.997 ..................................................................................................................

o:s

-0.02-

0_

-.

0.998"

0.9~

439

~.12~.14

3

o

o:5

;

1:5

time (see)

2is

~

315

~

4J5

5

time (sec)

Figure 5. Comparison of frequency trajectory at heavy load condition with disturbance at power angle controller is shown i~ Figures 2 and 3, where the control performance of the LQR PSS is also given for comparison. The dynamic performance of the proposed PSS is comparable to that of the LQR PSS. The output of each PSS is compared in Figure 4. It can be seen that they are similar in magnitude. The load condition is changed from the nominal to heavy load, i.e. the re,al power load is increased from 0.7 to 1,1 p.u. A disturbance in the initial condition for the rotor angle deviation is assumed to be 0.2 (radian). Frequency and power angle trajectories are compared for the LQR PSS and the proposed PSS in Figures 5 and 6, respectively. The control efforts of each PSS are compared in Figure 7. A line fault is assuraed. The transition line met a ground fault at time 0.5 s and the fault is cleared at time 0.7 s. The control results of the LQR PSS and the proposed PSS are compared in Figures 8 and 9. It can be seen that the damping effect of the proposed PSS compares quite well with that of the LQR PSS. The control efforts of each PSS are compared in Figure 10.

Figure 7. Comparison of PSS output at heavy load condition with disturbance at power angle

1.003

1.0041.003 ........... =

1.002

...........

1.001 ...........

0 . 9 9 9 ...........

0.998 0

time (sec)

Figure 8. Comparison of frequency trajectory due to disturbance at line fault

90

.~.-,=F------ Iwith°ut PSS

75-

85-

/,~ .dt~'t~

Iwith°ut PSS I

70-

== 80Q)

© 6575~

60-

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

e-

~

55-

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

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

655060

o

0:5

1:5

~

2:5

time (see)

45

o

~ tlme (sec)

Figure 6. Compari-¢;on of power angle trajectory at heavy load condition with disturbance at power angle

Figure 9. Comparison of power angle trajectory due to disturbance at line fault

440

A synchronous generator stabilizer based on a universal model: Y-M. Park et al 0.4

¸

03.....................tiE-

11.

7S !

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

(FARMA) model. 1EEE Transactions on Energy Conservation, 1996, 11(2), 442-448. Zhang, Y., Malik, O. P., Hope, G. H. and Chen, G. P., Application of an inverse input/output mapped ANN as a power system stabilizer. IEEE Transactions on Energy Conversion, 1994, 9(3), 433-441. Burden, R. L. and Faires, J. D., Numerical Analysis, Fourth Edition. PWS-KENT, Boston, Massachusetts, 1989. Slotine, J.-J. E. and Li, W., Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, NJ, 1991. Houpis, C. H. and Lamont, G. B., Digital Control Systems: Theory, Hardware, Software, Second Edition. McGraw-Hill, New York, 1992.*.

0.2.................................................................................................................... ~.-..-.-.-. 12. 13. 14. ~

-0.1" -0.2-0.3" -0.4

o

Appendix o2s

i

,'.s

:~

2'.s

time (see)

A

A. 1 Proof of equation (2) Applying the backward difference operation repeatedly,

Figure 10. Comparison of PSS output trajectory due to disturbance at line fault

IV. Conclusions A universal model-based controller is developed for power system stabilization. The controller is inherently designed for general systems with only the input-output data rather than a mathematical model of the system. The controller based on a universal model has the advantage of using no or very few design parameters compared with conventional model-based controllers, which are designed based upon mathematical models of controlled systems. The conventional LQR method is applied to design PSS using a universal model. Since the state variables are defined in terms of measured system output and input data, an observer is not necessary for the controller. The proposed PSS was tested in a typical single-machine infinite-bus power system. The performance of the UMBC PSS was consistent for a wide range of operations, demonstrating the robustness against parameter uncertainties and system failure. The parameters of the proposed PSS are determined by trial and error in this paper, but analytical research on the effect of the parameters is in progress.

V. References 1. Anderson, P. M. and Fouad, A. A., Power System Control and Stability. IEEE Press, New York, 1993. 2. Yu, Y. N., Electric Power System Dynamics. Academic Press, New York, 1983. 3. deMello, F. P. and Concordia, C., Concepts of synchronous machine stability as affected by excitation control. 1EEE Transactions on Power Apparatus and Systems, 1969, PAS-88(4), 316-329. 4. Hwang, T. L., Hwang, T. Y. and Yang, W. T., Two-level optimal output feedback stabilizer design. IEEE Transactions on Power Systems, 1991, 6(3), 1042-1047. 5. Khaldi, M. R., Sarkar, A. K., Lee, K. Y. and Park, Y. M., The modal performance measure for parameter optimization of power system stabilizer. IEEE Transactions on Energy Conversion, 1993, 8(4). 6. Pierre, D. A., A perspective on adaptive control of power systems. IEEE Transactions on Power Systems, 1987, PWRS-2(2), 387-396. 7. Ghosh, A., Ledwich, G., Malik, O. P. and Hope, G. S., Power system stabilizer based on adaptive control techniques. IEEE Transactions on Power Apparatus and Systems, 1984, PAS-103(8), 1983-1989. 8. Gu, W. and Bollinger, K. E., A self-tuning power system stabilizer for wide-range synchronous generator operation. IEEE Transactions on Power Systems, 1989, 4(3), 1191-1199. 9. Park, Y. M., Hyun, S. H. and Lee, J. H., A synchronous generator stabilizer design using neuro inverse controller and error reduction network. IEEE Transactions on Power Systems, 1996, 16(11). 10. Park, Y. M., Moon, U. C. and Lee, K. Y., A self-organizing power system stabilizer using fuzzy auto-regressive moving average

y(k+ 1) -y(k) = Ay(k+ 1) Ay(k + 1) - Ay(k) = A2y(k+ 1) A2y(k+ 1) - A2y(k) = A3y(k+ 1)

(A.1)

ANy(k+ 1) -- ANy(k)= AN+ ly(k + 1) If we add all terms, then we get

y(k +

N

Aiy(k)4- AN+ ty(k +

1) ---- Z

1)

(A.2)

i=0

A.2 A.2.1 Proof o f Proposition 1 We take the backward difference operator with order N on b o t h s i d e s of ARMA m o d e l in e q u a t i o n (3).

ANy(k+I) = ~. aiANy(k-i+l)+ ~. biANu(k-i+l) i=1

i=1

(B.1)

By expressing y(k - i + 1) and (k) and A'u(k), respectively,

ANy(k+ 1) =

~.

i-l~.

ai AN ~

(-

u(k - i + 1) in terms of Aiy l)i ( 1i - )

Aiy(k)

j=0

i=1

+

y (-ii=l

aJu(k)

j=0

(B.2)

By rewriting,

ANy(k4- 1) = n-l{

+ jY= l

(-1)J(i-~)ai] ×AN+jy(k)} i =jX+l

n--1 {

+Y. j=l

(-1)J(i- 1)ai] ×AN+Ju(k)} i =j~+ l

(B.3)

441

A synchronous generator stabilizer based on a universal model: Y-M. Park et al With the assumption that IA iy(k) l and IAJu(k)[ are negligibly small for some i j > N, ANy(k + 1) is approximated by

=

An-2y(k+l)+

Hn - i- 2A iu(k) i=0

ANy(k+l)~

(~=~1 a , ) A N y ( k ) + ( i = ~

b~)ANu(k)

(B.4)

-- { An-2y(k)+

Representing AUy(k + 1) as E

Aiy(k)

~ Mn--2--i Ai+lutka~.)

i=0

i=0

The state equation of nth state variable is defined as:

we rewrite equation (0) as follows:

n-2

xn(k+ 1 ) = A " - l y ( k +

N-I

y(k + 1)-~ Z

1)

n-3

=An-lY(k+l)+ ~.

N--1

y(k+ 1)-

Hn-i-2Aiu(k--

i=0

1)+ E

Hn -- i-- 1A i u(k)

(C.4)

i=0

AiY(k)

Inserting equation (0) in equation (0), we get the following state equation:

i=0

+(i=~ 1 ai'~)ANy(k)+(i=~l

bi)ANu(k)

(B.5)

x , ( k + 1) = x , _ l ( k + 1) - x , _ l(k) + H n - l U ( k ) = x n _ 2(k + 1) - x, _ 2(k) - x n _ 1(k) + Hn - 2u(k)

If sampling period is small,

+ 3, - I u(k) = x, _ 3(k + 1) - x, _ 3(k) - x, _ 2(k) ~ . ai

- x, _ 1(k) + H, - 3u(k) + Hn - 2u(k) + Hn -- 1u(k)

i=l

is nearly 1, and we siraplify equation (0) as follows: = xl (k + 1) - xz(k) - x3(k) - . . . - xn- 1(k)

y(k + 1) ~ X

AiY(k) +

i=0 / or

b i AUu(k)

(B.6)

+ 31u(k) + H2u(k) -}-... + Hn-lu(k)

i=l

N N-1

n-1

n-1

= Z aixi+,(k)- ~_. xi(k)+bou(k)+ E Hiu(k)

)

i=0

AN+ ly(k + 1) ~ b ( \ u ( k ) - i=oZ Aiu(k)

i=2

i=1

(c.5) A.2.2

APPENDIX A.2.2.2:

A P P E N D I X A.2.2.l: Derivation of equation (10) and equation (11) We rewrite the modified universal model in equation (9). N

N-1

aiAiy(k)+bou(k)+ E

y(k+ 1)= Z i=0

bi+ 1Aiu(k- 1)

i=0

(C.1) Solving equation (10) for Aiy(k) and substituting it in equation (0), U Xl (k + 1) = Z aixi + 1(k) -'}-bou(k)

APPENDIX A.2.2.2.1: SMIB model and parameters used in simulations The following set of equations and Figure 1 1 describe a single-machine synchronous generator-steam turbine system connected to an infinite bus: d~/dt = o:B(60 - o:0)

(D. 1)

do:/dt = I [ T m - Te - D(o: - o:0)]

(D.2)

deq'/dt = ~

1,

[EFD -- eq' + (x d' --Xd)id]

(D.3)

i=0

+ (b 1 - alH 1 - azH 2 - . . . - aN3N)u(k -- 1) + (b2 - a::H1 - a 3 H 2 - . . . - aNHN-1)Au(k-- 1) + . . . + (bN -- aNH1) AN- lu(k - 1)

(C.2)

If we choose Hi so th~Lt coefficients of Aiu(k - 1) are zero, then Hi c a n be represented by equation (11). The state equation of first state variable xl is N

x~ (k + 1) = Z

[~

aixi + 1(k) + bou(k)

i=0

Z : R+jX Y = G+jB

from equation (0). If we derive ~ n - l ( k + 1) from equation (10), we can see that it is described as: Z~kXn -- 1 (k

+ 1) = x.

_ 1( '~ +

1) -- X. _ 1(k)

(C.3)

t

Figure 11. A SMIB power system

A synchronous generator stabilizer based on a universal model: Y-M. Park et al

442

(D.4)

A V R and exciter:

l,'q = eq -- x d t id

(D.5)

dEFD/dt = ~A[KA(Pr -- ~'t) q- KAUpss -- EFD]

2 ~ = ~ + ~q

(D.6)

GOV:

r e ~ Pdid q- pqiq

(D.7)

dUgmt = ~ [ K g ( ~ r - ~) - ug]

1 dTm/dt = ~c[FhpUg -- Tin]

(D.8)

Pd = Xqiq

1 t

1

(D. 10)

(D. 1 l)

Generator parameters in p.u.: xa = 0 . 9 7 3 xa' = 0 . 1 9 Xq = 0.55 Tao' = 7 . 7 6 M = 9.26

Network equation:

D = 0 . 0 1 Fhp = 1 T c = O . 1

X

71I/ ]

(D.12)

Transmission line parameters in p.u.:

iq

R = - 0.034 X = 0.997 G = 0.249 B = 0.262

(D. 13)

A V R and GOV parameters: C2

C1

Pq

K A = 5 0 ~rA= 0.05 /~g= 10 ~ = 0 . 1

k cos

C1 = 1 + R G - X B C 2 = X G + RB

(D.9)

(D.14)