ECEN 667 Power System Stability Lecture 21: Modal Analysis

Transcription

1 ECEN 667 Power System Stablty Lecture 21: Modal Analyss Prof. Tom Overbye Dept. of Electrcal and Computer Engneerng Texas A&M Unversty, 1

2 Announcements Read Chapter 8 Homework 7 s posted; due on Tuesday Nov 28 Fnal s as per TAMU schedule. That s, Frday Dec 8 from 3 to 5pm 2

3 Sngle Machne Infnte Bus A qute useful analyss technque s to consder the small sgnal stablty assocated wth a sngle generator connected to the rest of the system through an equvalent transmsson lne Drvng pont mpedance lookng nto the system s used to calculate the equvalent lne's mpedance The Z value can be calculated qute quckly usng sparse vector methods Rest of the system s assumed to be an nfnte bus wth ts voltage set to match the generator's real and reactve power njecton and voltage 3

4 Small SMIB Example As a small example, consder the 4 bus system shown below, n whch bus 2 really s an nfnte bus GENCLS Bus 4 X=0.1 Bus 1 Bus 2 X=0.2 Infnte Bus slack Deg pu X=0.1 Bus 3 X= Deg pu 4.46 Deg pu 0.00 Deg pu To get the SMIB for bus 4, frst calculate Z Ybus j Z44 j Z 44 s Z th n parallel wth jx' d,4 (whch s j0.3) so Z th s j0.22 4

5 Example: Bus 4 wth GENROU Model The egenvalues can be calculated for any set of generator models Ths example replaces the bus 4 generator classcal machne wth a GENROU model There are now sx egenvalues, wth the domnate response comng from the electro-mechancal mode wth a frequency of 1.83 Hz, and dampng of 6.92% 5

6 Example: Bus 4 wth GENROU Model and Excter Addng an relatvely slow EXST1 excter adds addtonal states (wth K A =200, T A =0.2) As the ntal reactve power output of the generator s decreased, the system becomes unstable Case s saved as B4_GENROU_Sat_SMIB 6

7 Example: Bus 4 wth GENROU Model and Excter Wth Q 4 = 25 Mvar the egenvalues are And wth Q 4 =0 Mvar the egenvalues are 7

8 Example: Bus 4 wth GENROU Model and Excter Graph shows response followng a short fault when Q4 s 0 Mvar gfedcb Rotor Angle_Gen Bus 4 #1 Ths motvates tryng to get addtonal nsght nto how to ncrease system dampng, whch s the goal of modal analyss 8

9 Modal Analyss - Comments Modal analyss (analyss of small sgnal stablty through egenvalue analyss) s at the core of SSA software In Modal Analyss one looks at: Egenvalues Egenvectors (left or rght) Partcpaton factors Mode shape Goal s to determne how the varous parameters affect the response of the system Power System Stablzer (PSS) desgn n a multmachne context s done usng modal analyss method. 9

10 Egenvalues, Rght Egenvectors For an n by n matrx A the egenvalues of A are the roots of the characterstc equaton: det[ A li] A li 0 Assume l 1 l n as dstnct (no repeated egenvalues). For each egenvalue l there exsts an egenvector such that: Av l v v s called a rght egenvector If l s complex, then v has complex entres 10

11 Left Egenvectors For each egenvalue l there exsts a left egenvector w such that: w A w l t t Equvalently, the left egenvector s the rght egenvector of A T ; that s, t A w lw 11

12 Egenvector Propertes The rght and left egenvectors are orthogonal.e. w v t 0, w v 0 ( j) t j We can normalze the egenvectors so that: w v t 1, w v 0 ( j) t j 12

13 Egenvector Example l 4 A, l A I 3 2 l 2 l l l1,2 2 3 (3) 4(10) , Rght Egenvectors l 1 5 v v 4v 5v Av1 5v1 v1 v 3v 2v 5v Smlarly, 4 l2 2 v choose v 1 v 1 v

14 Egenvector Example Left egenvectors t t l w1a w1 w11 w21 w11 w [ ] 5[ ] 3 2 w11 3w21 5w11 Let w21 4, then w11 3 4w 2w 5w w l w v v 3 w 4 w 1 Verfy w v 7, w v 7, w v 0, w v 0 t t t t t wv 1. We would lke to make Ths can be done n many ways. 14

15 1 3 1 Let W Then T W V Egenvector Example I Verfy It can be verfed that W T =V -1. The left and rght egenvectors are used n computng the partcpaton factor matrx. 15

16 Modal Matrces The devaton away from an equlbrum pont can be defned as Δx AΔx From ths equaton t s dffcult to determne how parameters n A affect a partcular x because of the varable couplng To decouple the problem frst defne the matrces of the rght and left egenvectors (the modal matrces) V [ v, v... v ] & W [ w, w,... w ] 1 2 n 1 2 AV VΛ when Λ Dag ( l ) n 16

17 It follows that Modal Matrces To decouple the varables defne z so Then 1 V AV Λ x Vz x Vz AΔx AVz 1 z V AVz WAVz Λz Snce s dagonal, the equatons are now uncoupled wth z l z So x( t) Vz( t) 17

18 Example Assume the prevous system wth 1 4 A V V AV

19 Modal Matrces Thus the response can be wrtten n terms of the ndvdual egenvalues and rght egenvectors as n 1 t x( t) v z ( 0) e l Furthermore wth 1 T Δx= VZ z V x W x Note, we are requrng that the egenvalues be dstnct! So z(t) can be wrtten as usng the left egenvectors as t z( t) W x( t) [ w w... w ] 1 2 n t x1() t xn () t 19

20 Modal Matrces We can then wrte the response x(t) n terms of the modes of the system t z ( t) w x( t) t z ( 0) w x( 0) c so x( t) n 1 v c e l t l1t l2t Expandng x ( t) v c e v c e... v c e n n So c s a scalar that represents the magntude of exctaton of the th mode from the ntal condtons l t n 20

21 Numercal example x1 0 1 x1 1, ( 0) x x x 2 4 Egenvalues are l 4, l Egenvectors are v1, 2 4 v Modal matrx V Normalze so V

22 Numercal example (contd) Left egenvector matrx s: W T V T z = W AVz z1 4 0z1 z 0 2 z

23 Numercal example (contd) 1 z 4z, z( 0) V x( 0) 1 1 z1( 0) z2 2z2, z2( 0) 0 4t 2t T z1( t) z1( 0) e ; z2( t) z2( 0) e, C W x( 0) 0 x = Vz x1( t) 1 1z1( t) x2( t) 4 2 z2( t) c1 z1() t c Because of the ntal condton, the 2 nd mode does not get excted 2 z2( t) c vz ( 0) e l 1 t 23

24 Mode Shape, Senstvty and Partcpaton Factors So we have t x( t) Vz( t), z( t) W x( t) x(t) are the orgnal state varables, z(t) are the transformed varables so that each varable s assocated wth only one mode. From the frst equaton the rght egenvector gves the mode shape.e. relatve actvty of state varables when a partcular mode s excted. For example the degree of actvty of state varable x k n v mode s gven by the element V k of the the rght egenvector matrx V 24

25 Mode Shape, Senstvty and Partcpaton Factors The magntude of elements of v gve the extent of actvtes of n state varables n the th mode and angles of elements (f complex) gve phase dsplacements of the state varables wth regard to the mode. The left egenvector w dentfes whch combnaton of orgnal state varables dsplay only the th mode. 25

26 Egenvalue Parameter Senstvty To derve the senstvty of the egenvalues to the parameters recall Av = l v ; take the partal dervatve wth respect to A kj by usng the chan rule A v l v v A v l A kj Akj Akj Akj Multply by w t A v l v w v w A w v w l t t t t Akj Akj Akj Akj t A t v t l w v w [ A l I] w v A A A kj kj kj 26

27 Egenvalue Parameter Senstvty t Ths s smplfed by notng that w ( A l I) by the defnton of w beng a left egenvector Therefore w t A A kj v Snce all elements of j th column s 1 Thus l WV k j A kj l A kj A A kj 0 are zero, except the k th row, 27

28 Senstvty Example In the prevous example we had A, l1,2 5, 2,, 3 2 V 1 3 W Then the senstvty of l 1 and l 2 to changes n A are l WV l k j, l Akj A A For example wth Aˆ 1 4, ˆ l1,2 5.61, 1.61,

29 Egenvalue Parameter Senstvty t Ths s smplfed by notng that w ( A l I) by the defnton of w beng a left egenvector Therefore w t A A kj v Snce all elements of j th column s 1 Thus l WV k j A kj l A kj A A kj 0 are zero, except the k th row, 29

30 Egenvalue Parameter Senstvty To derve the senstvty of the egenvalues to the parameters recall Av = l v ; take the partal dervatve wth respect to A kj by usng the chan rule A v l v v A v l A kj Akj Akj Akj Multply by w t A v l v w v w A w v w l t t t t Akj Akj Akj Akj t A t v t l w v w [ A l I] w v A A A kj kj kj 30

31 Egenvalue Parameter Senstvty t Ths s smplfed by notng that w ( A l I) by the defnton of w beng a left egenvector Therefore w t A A kj v Snce all elements of j th column s 1 Thus l WV k j A kj l A kj A A kj 0 are zero, except the k th row, 31

32 Senstvty Example In the prevous example we had A, l1,2 5, 2,, 3 2 V 1 3 W Then the senstvty of l 1 and l 2 to changes n A are l WV l k j, l Akj A A For example wth Aˆ 1 4, ˆ l1, ,

33 Partcpaton Factors The partcpaton factors, P k, are used to determne how much the k th state varable partcpates n the th mode P V W k k k The sum of the partcpaton factors for any mode or any varable sum to 1 The partcpaton factors are qute useful n relatng the egenvalues to portons of a model 33

34 Partcpaton Factors For the prevous example wth P k = V k W k and A,, 3 2 V 1 3 W We get P

35 PowerWorld SMIB Partcpaton Factors The magntudes of the partcpaton factors are shown on the PowerWorld SMIB dalog The below values are shown for the four bus example wth Q 4 = 0 Case s saved as B4_GENROU_Sat_SMIB_QZero 35

36 Measurement Based Modal Analyss Wth the advent of large numbers of PMUs, measurement based SSA s ncreasng used The goal s to determne the dampng assocated wth the domnant oscllatory modes n the system Approaches seek to approxmate a sampled sgnal by a seres of exponental functons (usually damped snusodals) Several technques are avalable wth Prony analyss the oldest Method, whch was developed by Gaspard Rche de Prony, dates to 1795; power system applcatons from about 1980's Here we'll consder a newer alternatve, based on the varable projecton method 36

37 Some Useful References J.F. Hauer, C.J. Demeure, and L.L. Scharf, "Intal results n Prony analyss of power system response sgnals," IEEE Trans. Power Systems, vol.5, pp 80-89, Feb 1990 D.J. Trudnowsk, J.M. Johnson, and J.F. Hauer, "Makng Prony analyss more accurate usng multple sgnals," IEEE Trans. Power Systems, vol.14, pp , Feb 1999 A. Borden, B.C. Leseutre, J. Gronqust, "Power System Modal Analyss Tool Developed for Industry Use," Proc North Amercan Power Symposum, Manhattan, KS, Sept

38 Varable Projecton Method (VPM) Idea of all technques s to approxmate a sgnal, y org (t), by the sum of other, smpler sgnals (bass functons) Bass functons are usually exponentals, wth lnear and quadratc functons also added to detrend the sgnal Propertes of the orgnal sgnal can be quantfed from bass functon propertes (such as frequency and dampng) Sgnal s consdered over an nterval wth t=0 at the begnnng Approaches work by samplng the orgnal sgnal y org (t) Vector y conssts of m unformly sampled ponts from y org (t) at a samplng value of T, startng wth t=0, wth values y j for j=1 m Tmes are then t j = (j-1)t 38

39 Varable Projecton Method (VPM) At each tme pont j, where t j = (j-1)t the approxmaton of y j s yˆ ( α) b ( t, α) j j 1 where α n s a vector wth the real and magnary j egenvalue components, wth ( t, α) e for a correspondng to a real egenvalue, and t j j +1 j 1 j ( t, α) e cos( t ) and ( α) e sn( +1t j ) for a complex egenvector value j t t 39

40 Varable Projecton Method (VPM) Error (resdual) value at each pont j s r ( t, α) y yˆ ( t, α) j j j j j s the vector contanng the optmzaton varables Functon beng mnmzed s 1 1 α r α 2 2 m 2 ( y ˆ j y j ( t j, )) ( ) j1 2 2 r() s the resdual vector Method teratvely changes to reduce the mnmzaton functon 40

41 Varable Projecton Method (VPM) A key nsght of the varable projecton method s that yˆ( α) Φ( α) b And then the resdual s mnmzed by selectng b Φ( α) y where Φα ( ) s the m by n matrx wth values j t j ( α) e f corresponds to a real egenvalue, t j j α e 1tj j1 j and ( ) cos( ) and ( α) e sn( t ) for a complex egenvalue; t j 1 T Fnally, Φ( α) s the pseudonverse of Φ( α) j t 1 j 41