Critical Mode Identification and its Application to Load Frequency Control in Power Systems

Copyright © IFAC Large Scale Systems, Beijing, PRC, 1992

CRITICAL MODE IDENTIFICATION AND ITS APPLICATION TO LOAD FREQUENCY CONTROL IN POWER SYSTEMS R.L. Kleln, Jlanmlng She and A. Fellachl Department of Electrical and CompuJer Engineering, West Virginia University, Morgantown, WV 26505. USA

Abstract Based on the theoretical results in [1] and [2], a load frequency control problem in multi-area power system has been modeled including its full order and two reduced order models. The ranking indices proposed are shown to be very useful for identifying the critical modes. The methods employed in this paper may be applied to large scale system mode ranking and model order reduction generally. Keywords Critical mode, Load Frequency Control, Model reduction, Reduced Order system

1.

Introduction

Large scale systems theory and several existing physical examples such as electric power, communications, etc. are well known. Dealing with the problem of mode ling such systems and their large number of eigenvalues and, in turn, their relative importance to the overall behavior of the system continues to be an important problem. For some applications such as output feedback control design, reduced order mode ling and simulation, only a few of these eigenvalues are critical. In order to identify the critical modes of large scale systems, good eigenvalue ranking procedures are needed although eigenvalue ranking can be accomplished through costly simulation. Many investigations have used the eigenstructure of the system to rank eigenvalues, specifically, research has focused on the coupling between state variables and eigenstructure. Most of this work was done in the context of model reduction [3-5]. In [1] the eigenvalues have been ranked according to their contributions to a quadratic performance measure that represents the output energy and a ranking index is presented. It is shown in 129

[2] that the indices presented in [1] might not be accurate as each term contains joint contributions from the other eigenvalues. The essential or critical modes are defined as the ones that contribute the most to a prespecified quadratic performance index, e.g. output energy function. The other two ranking methods called the Weighting Factor Ranking Index and Remainder Ranking Index are proposed for continuous LTI systems in [2].

In this paper we summarize the recent results on critical mode identification and give the problem formulation in section 2. Then we apply these eigenvalue ranking techniques to a problem in interconnected power systems. The particular problem addressed is the controller design for load frequency control. In section 3, An example multiarea interconnected power system is modeled in detail. The model incl udes turbine, generator with speed governor and the tie-lines. The overall state-space model is derived from the system block diagrams and based on these models the load frequency control problem is formulated. As the example chosen illustrates, in general, a multi-area interconnected power system model is

large in dimension and therefore it is properly viewed as a large scale system. In section 4 the critical mode identification criteria are applied to the example model to effectively reduce the system order while maintaining good system behavior fidelity relating to the full order model. Several simulation results are given which illustrate the quality of the reduced order system behavior.

2. Critical Criteria

Mode

Identification

A linear, time-invariant, continuous system is described by the following state space representation x(t) = Ax + Bu x(O)=O y=Cx where A: Rn~Rn, B : Rm~Rn, c: Rn ~ Rk are linear maps and x(t)ERn, u(t)ERm, y(t)ERk. The objective here is to rank the eigenvalues of A according to their contributions to the following quadratic index:

Rl1 S !!

•

M:

R 12S21

A2· + Al R31 S 13

A2 + A2 A2 + A3 R33 S 33 R32 S 23

A2 + An R3nSn~

•

+ Al

RnlSl n An

•

+ Al

•

A3

•

J.: IRe< RilSli I • Ai

+ A2 A3

R n 2 S 2n An

•

•

+ A3

•

A3

R n 3 S 3n

+ . A2 An

•

+ A3

•

+ An

RnnSnn An

•

+ An

+

+ Al

S Ri2 s 2i * +Ri3 . 3i + ... +Rinsni . Ai + A2 Ai + A3 Ai + An

)1 .

Such an equal sharing of joint contribution does not actually reflect true roles of the mode contributions in the Ji. So in [2] two more reliable critical mode ranking indices are proposed: (II) Weighing factor ranking index Let us denote 2M 11 " 2M jj " _ W Wji = IJ-Mii + Mjj' Mjj + Mii Then we dffine a new ranling index as

Fi = Re(

~ WijMij) j=l

= R ( p, M ii Mij ) e L M '. +M " j=l 11 JJ (Ill) Remainder ranking index We can also rank the critical modes by their remainder contribution. The remainder contribution of the mode associated with Ai is defined as

~ ~ WkjMkj) k= 1 j=l k:;ti This quantity indicates the the remainder output energy with the mode of Ai being absent. Ri = Re(

~

T

3. in

Ql. G·G ·T. where P=o eA tReAtdt·,S= L I I, i=l Gi=WBi. If A is stable then P satisfies the following Lyapunov matrix equation A*P + PA =-R

Automatic Generation Power Systems

Control

We first present a generator model for load frequency control with the following block diagram,

Rij

where

•

(I) The ranking index used in [1];

{G? feATtReAtdt Gil i= 1 0 = trace (PS) oof

RlnSnl

• Al + An R2 n Sn2

A3

A = diag[AI, 1.2, ... ,An] = WAV where W = V-I. then J=

•

• Al + A2 Al + A3 R23 S 32 R22 S 22

00

J = ~ f [yi(-t)]TQyi(-t) d't i= 1 0 where yi(t) = Cxi(t) x i(t) is the state response to an impulse function through the i-th control channel, i.e., Ui(t) = 8(0) and Uj(t) = 0 where j:;ti. We assume that A is a simple matrix and denote

R13 S 31

Al + Al R21 S 12

Pij = - Ai* + Aj

We can write J in an explicit form with symmetry, J = -UMUT where U = [1 1 .. .... 1] and

130

~D

M=Xl

Governor Fig. 3.1

+

Turbine

Single Generator Block Diagram

~PVl~Tl

IGovernor 1RTurbine IFf

Generator

l.......:l~~ t--r----...,

Generator 2 .......:I...=:~

Generator 3 ............=:~ t--L...---.....

Fig. 32 Multi-area power system block diagram

automatic generation control are the following: to regulate the power frequency to its scheduled value to while maintaining the generation level at its economical set point.

Coosi
Modem power systems are highly interconnected multi-area systems and the modeling of such systems is the one of the important practical needs motivating the development of large scale system theory. Based on the above generator block diagram, we will study an AGe 131

problem on a multi-area interconnected power system and use it as a practical setting in which to apply the critical mode identification method to accomplish model order reduction and computer simulation. We use a three-area interconnected power system as follows:

regularized system model as full order system model for rest of this paper. The results of top ten and fifteen eigenvalues by the ranking indices given in section 2 are: 4.-~= . ~~ . ~~.~~ . ==~ . ~~ . ~~ . ~-,

.

. . ..

.

.

.

.

..

.

.

3 .... . .;...... , .. ... .: ..... .;..... . , . .. . ..;. . .. . .:..... .

2 •••••• ~ •... . : ..... . :. .. ... : . . ... : ...•• ~ •• 0 0 0 0:0 • • 0 0.. .B ~

t J

o'

T '·: .:. . . . .

1 0••• 0 0:0 0 0 ••• : .

•

o' o' .;: ...

. .

0 0 0:0 0 0 0 0 0: 0 0 0 0':

•

...

0. ' :0 0" 0 0

,:.. -,~

:~I:::: : :~:::::::::: : :[::::::;:::::::::: :~ ::::: :\: : : : : ~

: o,

0 0 oo~ 0 0 000: 00 0 ooo:~~L~_~j~:"'oo 0

-8

-s

-6

-7

-4

-3

-2

-I

0

Fig.3.2 Three-Area Power Systems

:I ""'"'__."' ..........'. ''''''.,., ,.. 1

According to the connection of Fig.3 .2. we obtain the multi-area power system block diagram which is shown in Fig.3.3:

20 0 oB

Based on the block diagram Fig.3.3. a state-space model is found as (For area # 1): TpXI =-DXI+K7X2+KsX3+K3X4+TpX5-~PD l- X 2ZX23 T7 X2 =-X2+ X3 T6X3 =-X3+X4 T5 X4 =-X4+ X5

.

!2.

T 4XS =-xs+(1 -T 2ff 3) x6 + T 3 X7 T3 X6 =-x6+ x7 .

1

T I X7 =-R: XI-X7+~PCI where X22 = .1PI2. X23 = .1PI3. Similar equations can be obtained for area 2 (xg - X14) and area 3 (XIS - X21). The connection equations are: X22 = 27tT 12(X I - Xg) X23 = 21tT 13 (XI - XIS) X24 = 21tT23(xg - XIS) Adding three state variables to enforce zero steady-state frequency deviation: X2S = Xl x26 = Xg x27 = xIS , this three-area interconnected power system becomes 27th order from which we can envision the dimensional complexity of even more realistic multiarea interconnected power system.

~

1 j

.

0. ..

.

0

.

..

~

00 000

0

1 ...... : .. 0 • 0 •••• 0••••• 0 • • •••• • " •• 0 0 0 •• • ••

o·

-:- . .

o.

0

o. 0••• ~

. -·:. ·. · 00

00' 0 .

-I

00

- - ~ : _. - - -, . . ~.

-2

_3 00. 0 .. 000 00 0000 -4-8~--_77----6~--_~S--~-4--~ _ 3--~ _2--~_I~~O Real Axis

Fig. 4.2, The eigenvaJue Distribution of the reduced Order Systems Based on these eigenvalue ranking indices we identify the critical modes to use for reduced order models. We will employ the Chidanbara approach to obtain the reduced order system with the desired critical eigenvalues. For the reduced order model we design optimal controllers for eliminating the effect for load step disturbance to the system. For comparison. the full order system response for a step disturbance caused by .1PD2 = -500MW are shown. Under optimal control the frequency deviation responses of the full order system (25th) are shown in the following figures :

4. Model Reduction by the Identified Critical Modes With the assigned values of the coefficients remove the and obtain a 25th order. We will refer to this 132

Full order: Loss of Load in area #2 =500MW, with Optimal Controller

0. 8.---------------------------------~----------~

·

. . . ; : : : : : : 0.6 -/- \- ...... . ':' . . . . .. .... ~ .. .... ..... : . . . . ... . . . ':' .......... ':' . . . . .... . . ~ . ..... .. . .. ~ ·· .. .. .. .. . .. .. .. . . .. . . . ··· ... . . . . . 0.4 .. ... ... .. :... ... ......:....... ... . ; ... ........ :........... .;.... ...... ..;...... .. ... ;.... ... .... . · ... .. .. ... ... ... ··· .. .. .. ... .. . ·· . . . .. . . .

f'I

0.2 .. - ...... -;- . ... ... ... ~ ........ .. . : .. ... ...... :-.. ...... .. -;- ... ... .... ~ .. ....... .. ~ ... -... ... .

i

I

-

.. .

···

:

/ " ,\ :

oj

.. .

.'. . \(.:~:;.7·'1· .-+ ..

.. . :

:

.. .

.. . :

:

..i···.. ·__ ...,·..·-:·...,·..·_ .. ·M·T·· . w ······...

.. . :

i··. ·. . .--·. ·. i·'M.M. ·····

.. :.. .. . . ..... .: ........ ... ; ... ........ :.... . . . .. . . -: ... . . . ... . . . .. -. . . . . . . :.. . . ... . . .. . · . . .. ... ... ... ··· ... .. .. . . . . ·· .. .. .. .. .. ..

-0.2

~

-O.4~----~ · ------~ . ------~ . ------.~----~.~----~.~----~.~----~

o

2

4

6

8

10

12

14

16

Time (sec) df1: ... , df2:_, df3:--Fig. 4.3, The Responses of Frequency Deviations of the Full Order System

The frequency deviations for a 15th order reduced order model are shown in Fig.4.4:

15th R.O.M.: Loss of Load in area 2 = 500MW, with Optimal Controller

0.3

0.25 0.2

§ 0.15

'4:1

.... ~:>-. ~

0.1

g 0.05 ·· ."' ..... .. ... .... .... : .... .......:... ..... ... ; .... ... .... : ... .. .. .... :...... ... ........ ... ... .

g

".\

0"

I

:

' \

:

:

:

...·.•. ····T·········T··········C··············..·····t····· • ••• .

~ -O.o:" .~ .·

-0.1~----~----~----~----~----~----~----~--~

o

2

4

6

8

10

12

Time (sec) df1: ... , df2:_, df3:--Fig. 4.4, The Responses of Frequency and Deviations of the 15th order Reduced Order System

The frequency deviations for a 10th order reduced order model are shown in Fig.4.5 :

133

14

16

10th RO.M.: Loss of Load in area 2 = 500MW, with Optimal Controller

0.3

:

:

:

:

:

:

:

0.2511 ··········:···········~···········:············:· ...... .... : .......... :.................. .... . ... ... .. . ; ........... ~ ....... .. .. :........ ... ~ ........... ~ ......... :... .........:..... ..... . :. :. :. .: .: ...... ... :....... ... . ~... .. ....... :.. ..... .... .:........... ~ ........... :...... ... .. -: .. ........ .

0.2

.g 0.15 ~ 0.11: .......... ;.. ..... .... ; ...........:......... .. .-. .......... , ........... :............:.......... .

I.::::I~.•~E:-:::·:::· :.: : : ::,::. :::.::.t::.::::::.l::::.::.:::.:!::::.:::::: >.

.

:

:

:

:

:

:

:

1

. . . . . . . . -0.10L---~--~--~--~-~=-------:-':----:-'1'":-4------:'16

··· ·

2

4

6

8

10

12

Time (sec) df1: ... , df2:_, df3:--Fig. 4.5, The Responses of Frequency and Tie Line Power Deviations of the 10th order Reduced Order System

Consequently, employing the ranking indices presented, we have reduced the original system order 60% yet still have faithful system responses. Recalling the fact that the computation cost for matrices increases exponentially with dimension, by identifying critical modes used proposed criteria and methods, significant computation time and space can be saved 5. Conclusions A critical mode ranking method based on mode contributions to the output energy were derived in [1] and even more accurate ranking indices were obtained in [2]. Based on these theoretical results, a load frequency control problem in multi-area power system has been modeled including its full order and two reduced order models. The ranking indices proposed are shown to be very useful for identifying the critical modes. Both a 15th and a 10th reduced order models for the 25th full order system are given and shown to yield good reduced order models . The methods employed in this paper may be applied to large scale system mode

ranking and model order reduction generally . References: [1] A. Feliachi, Identification of Critical Modes in Power Systems. IEEE Transactions on Power System, Vo1.5,pp.783-787,August 1990. [2] 1. She, A. Feliachi, On Eigenvalue Ranking Indices For Continuous Linear Time Invariant Systems. Proc. of the IEEE International Conf. on Systems Engineering, pp.291-294,August 1991. [3] L. Litz, Order Reduction of Linear State-Space Models via Optimal Approximation of the Non-dominant Modes. Large Scale systems, Vo1.2, No.3,pp . 171-184,1981. [4] R. Skelton, Cost Decomposition of Linear Systems with Application to Model Reduction . Int. 1. Control, Vol.32, No.6,pp.1031-1055, 1980. [5] C. Commault, Optimal Choice of Modes for Aggregation. Automatica, Vol.17, No.2, pp.397-399,1981. [6] G. Aly, Y.L.Abdel-Magid, M.A.Wali, Load Frequency Control of Interconnected Power System via Minimum Variance Regulators, Electric Power Systems Research, 7, pp. I-It. 1984. [7] A. Feliachi, X.Zhang, C.S. Sims, Power System Stablizers Design Using Optimal Reduced Order Models, Part I : Model Reduction . IEEE Trans. on Power Systems, Vo1.3, No.44, pp.1670-1675, Nov. 1988.

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