Brief Review of Linear System Theory

Transcription

1 Brief Review of Liear System heory he followig iformatio is typically covered i a course o liear system theory. At ISU, EE 577 is oe such course ad is highly recommeded for power system egieerig studets. We have developed a model that appears as Δ& x = AΔx We may write this more compactly as x & = Ax where the Δ is implied. aig the LaPlace trasform, with iitial coditios x(0), we have: s x(0) = A s A = x(0) Factorig out the vector X(s) results i: [ s I A] = x(0) where I is the idetity matrix of same dimesio as A. Pre-multiplyig both sides by [si-a] -, we get: = [ si A] x(0) (L-) ad taig the iverse-laplace trasform leads to x( t) = L {[ si A] x(0) } (L-2a) Note that i the above, we implicitly assumed that [si-a] is osigular (this requires that our system has o-zero determiat). Recall that a matrix iverse is the adjoit divided by the determiat, i.e., K - =Adj(K)/det(K). Applyig this to eq. (L-), we have:

2 {[ si A] } {[ si A] } Adj = x(0) det he determiat of a matrix is a scalar quatity, ad i this case, it is a scalar polyomial i the LaPlace variable s so that: {[ si A] } = a s + a s a0 det Such a polyomial may always be factored i the form: {[ ]} det s I A = a s + a s a0 = ( s λ )( s λ 2 )...( s λ ) L-2b where the λ, =,, are the roots of the polyomial. X ( s ) = Adj det {[ s I A ]} {[ s I A ]} x(0) = {[ si A] } Adj x(0 ) ( s λ )( s λ )...( s λ ) 2 (L-3) Eq. (L-3) expresses the -dimesioal vector X(s) as a fuctio of. he matrix Adj[sI-A], 2. he vector x(0) 3. he factored polyomial (s-λ )(s-λ ) (s-λ ) Note that the umerator is the product of a matrix ad a vector ad therefore it is, which is the dimesio of the righthad-side ad thus the vector X(s). his is as it should be, sice X(s) is the vector of states, ad there should be states. If oe of the roots λ, =,, are repeated, it will be possible to use partial fractio expasio to express eq. (L-3) i the followig way: R ( s ) X ( s ) = ( s λ ) + R 2( s) ( s λ ) R ( s) ( s λ ) (L-4) 2

3 where each R (s) is a vector. he iverse LaPlace trasform will the appear as: λt λ t ( t) e + x2 ( t) e x( t) = x x ( t) e he λ, =,, are, i geeral, complex, such that λ =σ +jω. he λ, =,, are called the system eigevalues. We see that the system eigevalues λ, =,, dictate the ature of the system i terms of the system modal respose, where each λ correspods to a system mode. hese modes may be oscillatory or o-oscillatory, damped or udamped.. Oscillatory: Ay mode with ω 0 is oscillatory. If there exists a λ =σ +jω such that ω 0, the there will exist a correspodig λ =σ -jω. hese two eigevalues correspod to the same system mode. Ay mode with ω =0 is o-oscillatory. 2. Dampig: Ay mode λ =σ ±jω, a. if σ >0, the mode is egatively damped (ustable) b. if σ <0, the mode is positively damped (stable) c. if σ =0, the mode is margially damped. If repeated roots occur i the factorizatio of (L-2b), the these roots will have time-domai expressios lie t r- e -λt, ad will therefore have the followig effects: a. if σ >0, the mode is egatively damped (ustable) b. if σ <0, the mode is positively damped (stable); however, the effects of the t coefficiet might iitially domiate the effects of the expoetial ad cause very large oscillatios that could disrupt the system. c. with σ =0, the effects of the t coefficiet will result i growig respose (ustable) I practice, it is very uliely to see repeated roots for power systems. herefore, we safely assume there are o repeated roots. 3 λ t

4 Right eigevectors: For each eigevalue, λ, =,,, there exists a N-elemet vector p, called a right eigevector, such that Ap = λ p Sice there are eigevalues, there are right eigevectors. We may form a matrix of these right eigevectors as follows: [ p ] P =... he above matrix, P, is called the modal matrix. Left eigevectors: For each eigevalue, λ, =,,, there exists a N-elemet colum vector q, called a left eigevector, such that q A = λ q Sice there are eigevalues, there are left eigevectors. We may form a matrix of these left eigevectors as follows: Q q = M q Some properties: For ay two eigevalues, λ j, λ, the For j, q j ad p are orthogoal, i.e., For j=, q p j = 0 q p = c j j where c is a costat. A simple scalig of either the right or the left eigevector will provide that q j p j = p 4

5 From the above two properties, we have that: Uscaled versio: Q P = ci Scaled versio: Q P = I Post-multiplyig both sides by P - results i Uscaled versio: Q = cp Scaled versio: Q = P Note that: PP - =I [Q ] - Q =I We ca illustrate calculatio of the right ad left eigevectors usig the sample system give i the boo (fig. 2.9, ad example 3.2), havig state-space model of Δ & δ 0 3 Δ & δ 0 23 = Δ & ω Δ & ω Observe the eigevalues i able Δδ3 Δδ 23 0 Δω3 0 Δω23 Also observe the relative rotor agle plots of fig. 3.3-b, where we see that oe mode ca be clearly observed havig a period of about 0.7 sec (f=.4 Hz, ω=2πf=8.8 rad/sec). 5

6 he other mode (2. Hz) is ot readily observable, although its presece is probably resposible for the distortio see i the δ 3 plot. Usig matlab, we use [P,D]=eig(A) where A is the matrix give above. he the matrix of eigevalues D is give by i i i i Ad the matrix of right eigevectors P is give by i i i i i i i i i i i i i i i i 6

7 Ad the matrix of left eigevectors Q is give by P -, which is: i i i i i i i i i i i i i i i i Note that here, the eigevectors are alog the rows. aig traspose, we get Q, which is i i i i i i i i i i i i i i i i I the above, the left eigevectors are the colums. Note also that the colums of right (or left) eigevectors correspodig to complex cojugate eigevalues are complex cojugate eigevectors. he umerators of eq. (L-4) Let s retur to eq. (L-4), which is restated here for coveiece: R( s) R 2( s) R ( s) = ( s + λ ) ( s + λ2) ( s + λ ) What are these R, =,,? o aswer this, let s retur to eq. (L-), which is: = [ si A] x(0) Let s pre-multiple the right-had side by PP - ad post-multiply the right-had-side by [Q ] - Q. his is acceptable, sice both of these products yield the idetity. his results i: = PP Bracet the ier products: = P P [ si A] [ Q ] Q x(0) { } Q x(0) [ si A] [ Q ] 7

8 We ca show that: [ s I Λ] = P [ si A] [ Q ] where Λ = diag( λ ) he proof is below: he, we have that: } } { [ ] } } } = P si Λ Q x(0) Note carefully that the matrix beig iverted is a diagoal matrix. herefore, 678 p ( q x(0)) ( q x(0)) p = = = s λ = s λ aig the iverse LaPlace trasform, we obtai: x( t) λ t [ q x(0) e ] = = 8 p (L-5) his is a very importat relatioship. It shows how we ca use the right eigevalue to determie the shape of the th mode.

9 Ispectig eq. (L-5), we see that the right eigevector p determies the relative distributio of the mode through the state variables. o see this, ote that p ad x(t) are both vectors, with elemet i correspodig to the i th state variable. λ t q x( 0) e is scalar ad multiplies every elemet of p ; therefore it does distiguish ay state ay differetly tha aother state p is therefore the oly thig that distiguishes oe state from aother i terms of the mode dyamics. If the states are limited to oly the geerator iertial states Δδ ad Δω, the each elemet of p gives the relative distributio of the mode i a particular geerator s agle or speed. Oe cautio: he right eigevector does NO tell you how much the state iflueces the mode. he right eigevector does tell you the relative phase of each state i that mode. If you plot each elemet (a complex umber ad thus iterpretable as a vector) correspodig to each Δω state (oe for each geerator) i the right eigevector p, you ca see which geerators are swigig agaist oe aother. his is called mode shape. he relative phases ca be observed i the time domai simulatios. Some iterestig ways of illustratig the relative phase of each Δω as determied by the p s are show i the followig. Klei, Rogers, ad Kudur, A fudametal study of iterarea oscillatios i power systems. See page 95-96, attached below. Fig. 2 shows the mode shape where ges,2 swig agaist ges,2, ad i the time domai simulatio, Fig. 3. 9

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11 Wag, Howell, Kudur, Chug, ad Xu, A tool for small-sigal security assessmet of power systems. See mode shape, Fig. 5.

12 Y. Masour, Applicatio of eigeaalysis to the Wester North America Power system, ables 4, 5, ad 6, each table for a certai coditio, give eigevector elemets for speed deviatio at each of a umber of geerators. Figures, 2, ad 3 show, for three coditios, geographical plots of the mode shapes for 4 differet modes. 2

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