Brief Review of Linear System Theory

Transcription

1 Brief Review of Liear Sytem heory he followig iformatio i typically covered i a coure o liear ytem theory. At ISU, EE 577 i oe uch coure ad i highly recommeded for power ytem egieerig tudet. We have developed a model that appear a x Ax We may write thi more compactly a where the i implied. x aig the LaPlace traform, with iitial coditio x0, we have: X x0 AX X AX x0 Factorig out the vector X reult i: I A X x0 where I i the idetity matrix of ame dimeio a A. Pre-multiplyig both ide by [I-A] -, we get: X L- ad taig the ivere-laplace traform lead to x t L I A x0 L-2a Note that i the above, by expreig [I-A] -, we implicitly aume that it i ivertible ad therefore o-igular thi require that our ytem ha o-zero determiat. Recall that a matrix ivere i the adjoit divided by the determiat, i.e., K - =AdjK/detK. Applyig thi to eq. L-, we have: Ax I A x0

2 2 he determiat of a matrix i a calar quatity, ad i thi cae, it i a calar polyomial i the LaPlace variable o that: 0... det a a a A I Such a polyomial may alway be factored i the form: L-2b where the, =,, are the root of the polyomial. herefore, L-3 Eq. L-3 expree the -dimeioal vector X a a fuctio of. he matrix Adj[I-A], 2. he vector x0 3. he factored polyomial Note that the umerator i the product of a matrix ad a vector ad therefore it i, which i the dimeio of the righthad-ide ad thu the vector X. hi i a it hould be, ice X i the vector of tate, ad there hould be tate. If oe of the root, =,, are repeated, it will be poible to ue partial fractio expaio to expre eq. L-3 i the followig way: L-4 0 det Adj x A I A I X det 2 0 a a a A I... 0 Adj 0 det Adj 2 x A I x A I A I X R... R R 2 2 X

3 where each R i a vector. he ivere LaPlace traform will the appear a: 2 x t R t e t R2 t e t... R t t e he, =,, are, i geeral, complex, uch that = +j. he, =,, are called the ytem eigevalue. We ee that the ytem eigevalue, =,, dictate the ature of the ytem i term of the ytem modal repoe, where each correpod to a ytem mode. hee mode may be ocillatory or o-ocillatory, damped or udamped.. Ocillatory: Ay mode with 0 i ocillatory. If there exit a = +j uch that 0, the there will exit a correpodig = -j. hee two eigevalue correpod to the ame ytem mode. Ay mode with =0 i o-ocillatory. 2. Dampig: Ay mode = j, a. if >0, the mode i egatively damped utable b. if <0, the mode i poitively damped table c. if =0, the mode i margially damped. If repeated root occur i the factorizatio of L-2b, the thee root will have time-domai expreio lie t r- e -λt, ad will therefore have the followig effect: a. if >0, the mode i egatively damped utable b. if <0, the mode i poitively damped table; however, the effect of the t coefficiet might iitially domiate the effect of the expoetial ad caue very large ocillatio that could dirupt the ytem. c. with =0, the effect of the t coefficiet will reult i growig repoe utable I practice, it i very uliely to ee repeated root for power ytem. herefore, we afely aume there are o repeated root. 3

4 Right eigevector: For each eigevalue,, =,,, there exit a -elemet colum vector p, called a right eigevector, uch that Ap p Sice there are eigevalue, there are right eigevector. We may form a matrix of thee right eigevector a follow: P p... p he above matrix, P, i called the modal matrix. Left eigevector: For each eigevalue,, =,,, there exit a -elemet colum vector q, called a left eigevector, uch that Sice there are eigevalue, there are left eigevector. We may form a matrix of thee left eigevector a follow: Q q q Some propertie: For ay two eigevalue, j,, the For j, q j ad p are orthogoal, i.e., their dot product i 0: For j=, q A q q q p j p j j 0 c where c j i a cotat. A imple calig of either the right or the left eigevector will provide that q j p j j 4

5 Now coider, baed o the above propertie, we will get: q p q p q p... q p q p q q p q p q p... q p 0 q p Q P p... p q p q p q p... q p 0 0 q p q q p q p q p... q p q p We ca go a tep further if the calig i performed: Q Pot-multiplyig both ide by P - reult i Q P I P Note that: PP - =I [Q ] - Q =I We ca illutrate calculatio of the right ad left eigevector uig the ample ytem give i the boo fig. 2.9, ad example 3.2, havig tate-pace model of Oberve the eigevalue i able

6 Alo oberve the relative rotor agle plot of fig. 3.3-b, for the cae whe a mall load wa added to bu #8. Here we ee that oe mode ca be clearly oberved havig a period of about 0.7 ec f=.4hz, ω=2πf=8.8 rad/ec. he other mode 2.Hz i ot readily obervable, although it preece i probably repoible for the ditortio ee i the 3 plot. Uig matlab, we ue [P,D]=eigA where A i the matrix give above. 6

7 he the matrix of eigevalue D i give by i i i i Ad the matrix of right eigevector P i give by i i i i i i i i i i i i i i i i Ad the matrix of left eigevector Q i give by P -, which i: i i i i i i i i i i i i i i i i Note that here, the eigevector are alog the row. aig trapoe, we get Q, which i i i i i i i i i i i i i i i i i I the above, the left eigevector are the colum. Note alo that the colum of right or left eigevector correpodig to complex cojugate eigevalue are complex cojugate eigevector. he umerator of eq. L-4 Let retur to eq. L-4, which i retated here for coveiece: R R 2 R X

8 What are thee R, =,,? o awer thi, let retur to eq. L-, which i: Let pre-multiple the right-had ide by PP - ad pot-multiply the right-had-ide by [Q ] - Q. hi i acceptable, ice both of thee product yield the idetity. hi reult i: X PP Bracet the ier product: We ca how that: where he proof i below: X X P P I A x0 I A Q Q x0 Q x0 I A Q I P I A Q diag he, we have that: 8

9 X P I Q x 9 0 *# wo commet are relevat at thi poit:. he matrix beig iverted i a diagoal matrix. herefore, the matrix ivere i obtaied by ivertig each diagoal elemet. 2. Recall the orthogoality property p i q j =0 for i j. Uig thee commet, we ca maipulate *# to obtai: X p q x 0 q x0 p aig the ivere LaPlace traform, we obtai: x t t q x0 e p L-5 hi i a very importat relatiohip. It how how we ca ue the right eigevalue to determie the hape of the th mode. Ipectig eq. L-5, we ee that the right eigevector p determie the relative ditributio of the mode through the tate variable. o ee thi, ote that q, p, ad xt are all vector, with elemet i correpodig to the i th tate variable. q x t i calar ad multiplie every elemet of p ; 0 e therefore it doe ot ditiguih ay tate differetly tha aother tate p i therefore the oly thig that ditiguihe oe tate from aother i term of the mode dyamic. If the tate are limited to oly the geerator iertial tate ad, the each elemet of p give the relative ditributio of the mode i a particular geerator agle or peed. Oe cautio: he right eigevector doe NO tell you how much the tate ifluece the mode.

10 he right eigevector doe tell you the relative phae of each tate i that mode. If you plot each elemet a complex umber ad thu iterpretable a a vector correpodig to each tate oe for each geerator i the right eigevector p, you ca ee which geerator are wigig agait oe aother. hi i called mode hape. he relative phae ca be oberved i the time domai imulatio. Some iteretig way of illutratig the relative phae of each a determied by the p are how i the followig. Klei, Roger, ad Kudur, A fudametal tudy of iterarea ocillatio i power ytem. See page 95-96, attached below. Fig. 2 how the mode hape where ge,2 wig agait ge,2, ad i the time domai imulatio, Fig. 3. 0

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12 Wag, Howell, Kudur, Chug, ad Xu, A tool for mall-igal ecurity aemet of power ytem. See mode hape, Fig. 5. 2

13 Y. Maour, Applicatio of eigeaalyi to the Weter North America Power ytem, able 4, 5, ad 6, each table for a certai coditio, give eigevector elemet for peed deviatio at each of a umber of geerator. Figure, 2, ad 3 how, for three coditio, geographical plot of the mode hape for 4 differet mode. 3

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15 Fially, below i ome wor recetly doe reflectig mode hape i the Southweter WECC ytem for a certai mode. 5

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