Participation Factors. However, it does not give the influence of each state on the mode.

Transcription

1 Particiation Factors he mode shae, as indicated by the right eigenvector, gives the relative hase of each state in a articular mode. However, it does not give the influence of each state on the mode. We would lie to be able to obtain the influence of states on modes because then we will now which states (machines) to control in order to increase daming of a certain roblem mode. Let s define a new state variable ( xi ) as follows: x Notice that is a scalar since the transose of the left eigenvector is a 1n and the state vector is an n1. hus, is a combination of all of the states, but the manner in which all of the other states is combined is through the left eigenvector elements of the th mode. An imortant attribute of, and the reason why it is of great interest here, is that it is a state which is associated with the th mode and no other mode. We can rove this as follows. Start with the system state euations: Pre-multily both sides by x Ax Recall that the left eigenvector is defined as x. x Ax (P-1) A (P-2) Notice that the left-hand-side of e. (P-2), is on the right-hand-side of e. (P-1). Substituting the right-hand-side of e. (P-2) into the right-hand-side of e. (P-1), we obtain: 1

2 x (P-3) x Returning to the definition of our new state variable, which is x x, we note the left-hand-side of e. (P-3) is while the right hand side of e. (P-3) is. Maing these substitutions, e. (P-3) becomes: (P-4) Now e. (P-4) is a time-domain exression. So let s tae the LaPlace transform to obtain: s s () s () s () () aing the inverse LaPlace transform, we find that ) t ( t) ( e (P-5) his confirms that is associated with only the th mode and no other mode. hat is, the only dynamics associated with are e ()t. What is the imlication of this fact? he state variables that influence are the state variables that influence the th mode. So we can study to learn about the th mode. Let s see if we can determine which state variables influence. Recall from the notes called Linear system theory e. (L-5), which was: 2

3 x( t) t x() e n 1 (L-5) I will change the summation index so as to not confuse with the index () used reviously in these notes: x( t) jt x() e j n j 1 Now substitute e. (P-6) into the definition of : n jt x() e j j (P-6) x (P-7) j j 1 Note that the art of the summation in bracets is a scalar. herefore we can move the as to re-multily j inside the summation, beyond the bracets, so. his results in: n jt x() e j x (P-8) j j 1 Recalling that the matrices P and Q are orthogonal, we now that for j (P-9) j herefore, there is only one non-zero term in the summation of e. (P-8), and that is the term for which =j. As a result, e. (P-8) is: Rearranging slightly, we have, t x() e (P-1) But note: t x() e (P-11) 3

4 2 n (P-12) We may exress the above vector roduct as a summation: n j 1 j j n (P-13) Substitution of e. (P-13) into e. (P-11), we obtain: n t x( ) j j e (P-14) j1 Now we are in a osition to mae a definition: Particiation factor: (P-15) j j j Substitution of e. (P-15) in e. (P-14) results in: n t x() j e j1 Observe that if all (P-16) j =, then the th mode would not exist, an observation that leads to a conclusion that existence, or revalence, of a mode deends on the magnitudes of the various. he articiation factor indicates the articiation (influence) of the j th state in the th mode. j j 4

5 he articiation factor is extremely useful. Consider that you learn through eigenvalue calculation and/or time-domain simulation that mode is a roblem mode, i.e., it is marginally damed or negatively damed. hen the way one can identify what to do about this roblem mode (which state to control) is by insecting articiation factors for it. here will be n articiation factors for this mode, j, j=1,,n (where n is the number of states). he states having the larger articiation factors (in terms of magnitudes) are the states which should be most strongly considered to control in order to affect this roblem mode. Note: in contrast to x(), is less deendent of the initial j conditions and therefore serves as more of a universal indicator of articiation than does x(). So let s loo at the big icture. How does one generally roceed in a small-signal analysis study? 1. Comute eigenvalues and eigenvectors for a certain oerating condition. 2. Choose a ; If any = j has <, or >, then this is a roblem mode at that oerating condition. 3. Identify right and left eigenvectors for mode. a. Identify grous of generators based on mode shae using (use the angles of the elements of corresonding to the seed deviation states). b. For each grou, identify the seed deviation states (and thus the generators) most heavily articiating (influencing) the mode, based on. j 4. Install, or retune the ower system stabilizer (PSS) on the generators identified in ste 3-b using seed deviation as a control signal so that they increase daming of the th mode. 5

6 Last comment: his is for linearized (small-signal) analysis, not for large-signal (fault) analysis. Going bac to our examle (see notes on linear system theory), we recall that Observe the eigenvalues in table Also observe the relative rotor angle lots of fig. 3.3-b, where we see that one mode can be clearly observed having a eriod of about.7 sec (f=1.4 hz). he other mode is not readily observable, although its resence is robably resonsible for the distortion seen in the 31 lot. 6

7 Using matlab, we use [P,D]=eig(A) where A is the matrix given above. hen the matrix of eigenvalues D is given by i i i i And the matrix of right eigenvectors P is given by i i i i i i i i i i i i i i i i And the matrix of left eigenvectors Q is given by P -1, which is: i i i i i i i i i i i i i i i i 7

8 Note that here, the eigenvectors are along the rows. aing transose, we get Q, which is i i i i i i i i i i i i i i i i Now I comute the articiation matrix below. a=[ 1 ; 1; ; ]; [P, D]=eig(a); Q=inv(P); Q=Q'; j=1; % j is index on columns (modes) % i is index on rows (states) while j<5, i=1; while i<5, f(i,j)=q(i,j)*p(i,j); i=i+1; end j=j+1; end f his gives DELA i i i i DELA i i i i OMEGA i i i i OMEGA i i i i MODE 1 MODE 1 MODE 2 MODE2 From this, we see that ω 23 articiates most heavily in mode 1. ω 13 articiates most heavily in mode 2. his is the information that we would use to decide where to lace a PSS to enhance daming of a articular mode, although there is some ambiguity regarding whether ω 1 is a state associated with unit 1 or unit. Returning to Fig. 3.3 in the boo (and given above), we observe that although mode 1 is clearly visible in both lots, mode 2 is only visible in the ω 13 lot. his is consistent with the indication from the articiation factors. 8

9 Recall that with uniform daming, we were able to eliminate one seed deviation state. In general, this is not ossible, and so you end u with one seed generation state for each generator, a develoment which solves the ambiguity roblem mentioned above. As an examle, the aer by Mansour rovides articiation factors for several cases, as indicated below. Note that for all articiation vectors the articiation factor is a normalized magnitude. 9

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