ECE 522 Power Systems Analysis II 3.2 Small Signal Stability

Transcription

1 ECE 5 Power Sytem Aalyi II 3. Small Sigal Stability Sprig 18 Itructor: Kai Su 1

2 Cotet 3..1 Small igal tability overview ad aalyi method 3.. Small igal tability ehacemet Referece: EPRI Dyamic Tutorial Chapter 1 ad 17 of Kudur Power Sytem Stability ad Cotrol Chapter 3 of Adero Power Sytem Cotrol ad Stability Joe H. Chow, Power Sytem Coherecy ad Model Reductio, Spriger, 13

3 3..1 Small Sigal Stability Overview ad Aalyi Method 3

4 Power Ocillatio The power ytem aturally eter period of ocillatio a it cotiually adjut to ew operatig coditio or experiece other diturbace. Typically, ocillatio have a mall amplitude ad do ot lat log. Whe the ocillatio amplitude become large or the ocillatio are utaied, a repoe i required: A ytem operator may have the opportuity to repod ad elimiate harmful ocillatio or, le deirably, protective relay may activate to trip ytem elemet. 4

5 Small Sigal Stability Small igal tability (alo referred to a mall diturbace tability) i the ability of a power ytem to maitai ychroim whe ubjected to mall diturbace I thi cotext, a diturbace i coidered to be mall if the equatio that decribe the reultig repoe of the ytem may be liearized for the purpoe of aalyi. k +P P() It i coveiet to aume that the diturbace cauig the chage already diappear ad detail o the diturbace are uimportat P The ytem i table oly if it retur to it origial tate, i.e. a table equilibrium poit (SEP). Thu, oly the behavior i a mall eighborhood of the SEP are cocered ad ca be aalyzed uig the liear cotrol theory. 5

6 Claic Model SMIB Sytem With all reitace eglected: T = P = P = P = P P e e t B max (T e ) = EE X T B max i d K S P e +jq e P t +jq t P B +jq B (P e =T e ) (T m ) dd d = w D wr dt dd wr H = Tm - Te - KD D wr dt =DT -DT -K Dw m e D r»dt -K Dd-K Dw m S D r Note: H i r i p.u. i rad. K D K S i p.u i p.u/rad Liearize wig equatio at = : T d e D Te» D d= KSD Sychroizig torque coefficiet: K S = P cod = EE B max XT d cod 6

7 State pace repreetatio d dt é w ù é Dd ù édd ù éù DT = K S K + ê D w ú ê r w ú ê r 1 ú ëd û - - D H ê ë û H H ú ë û ë û m K D Kw wdt D d+ D d+ D d= H H H Apply Laplace Traform: 1 wdt D d = KD Kw + + H H H m m Characteritic equatio: K K w + D + = H H 7

8 A harmoic ocillator F = M x=-k x-d x + D K M + M = H w K D d =-K D d- Dd (if DT = ) D w K K w + D + = H H m + zw + w = Dampig ratio Natural frequecy It ha two cojugate complex root ad it zero-iput repoe i a damped iuoidal ocillatio:, = jw=-zw jw 1-z : 1 1 t xt () = Ae i( wt+ j) -zwt = Ae - + i( wt 1 z j) The time of decayig to 1/e=36.8%: 1 1/ 8

9 Ocillatio Frequecy ad Dampig of a SMIB Sytem + zw + w = K K w + D + = H H, = jw=-zw jw 1-z 1 EE B ( KS = cod = Pmax co d) X T Note the uit: r i i p.u. i i rad. K D i i p.u K S i i p.u/rad w w EE cod w = + w = K = H HX B S T w K w = w 1- z = KS - H 16 H D z - 1 K = = = K D T D w K 8 HE EB co S Hw w + d 4 X =- zw =- KD H How do ad chage with the followig? if H (lower iertia) if X T (troger tramiio) if (lower loadig) 9

10 Sytem Repoe after a Small Diturbace d dt x1 é w ù Dd Dd DT = K S K + ê D w ú ê r w ú ê r 1 ú ëd û - - D H ê ë û H H ú ë û ë û é ù é ù é ù =Dd D u = D H T m x =D w =Dd / w r éx ù é 1 w ù é x ù é 1 ù = + Du ê x ú ê -w w -zw ú ê x ú ê 1ú ë û ë û ë û ë û x () t Ax() t Bu() t é1 ù y() t = Cx() t =ê ê 1ú éx êx 1 ù ú ë û ë û (Aumig the agle ad peed to be directly meaured) m Apply Laplace traform: X() x() AX() BU() D U() = Du Y() X() ( IA) x() BU() é + zw ê ê w w ú ë- û + zw+ w [ + U ] X () = () () x B D é + zw w ù é ù é Dd() ù ê-w w ú é Dd() ù = ë û ( + ) u ê wr() ú zw w ê D wr() ú ëd û + + ëd û êë úû w ù ú 1 + Zero-iput Zero-tate Zero-iput Zero-tate 1

11 é + zw w ù é ù é Dd() ù ê-w / w ú é Dd() ù = ë û ( + ) u ê wr() ú zw w ê D wr() ú ëd û + + ëd û êë úû Zero iput repoe E.g. whe the rotor i uddely perturbed by a mall agle () ad aume r ()= D ( zw) d() d() = + D + zw + w w Dd() / w D wr () =- + + zw w Ivere Laplace traform D w =D d / w = ( w -w )/ w i pu r D u = D H T m Zero tate repoe E.g. whe there i a mall icreae i mechaical torque T m (= P m i pu) ( ) r pu ( ) r u ( ) u Dd() -zw Dd i rad = e t i( wt+ q) 1-z Dw r wdd() -zwt i rad/ =- e i wt 1 - z i r rad i rad T 1 1 e i 1 m t H T m / t H 1 e t i t Dampig agle: co 1 11

12 Example Exp. 11. & 11.3 i Saadat book H=9.94, K D =.138pu, T m =.6 pu with PF=.8. Fid the repoe of the rotor agle ad frequecy uder thee diturbace (1) ()=1 o =.1745 rad () P e =.pu Zero iput repoe: ()=1 o ()= =6.79 o Zero tate repoe: P e =.pu ()= =.55 o 1

13 Small Sigal Stability of a Multi machie Sytem Iter area or itra area mode (.1.7Hz): machie i oe part of the ytem wig agait machie i other part Local mode (.7 Hz): ocillatio ivolve a mall part of the ytem Cotrol or torioal mode (Hz ) Iter area model (.1.3Hz): ivolvig all the geerator i the ytem; the ytem i eetially plit ito two part, with geerator i oe part wigig agait machie i the other part. Itra area mode (.4.7Hz): ivolvig ubgroup of geerator wigig agait each other. Local plat mode: aociated with rotor agle ocillatio of a igle geerator or a igle plat agait the ret of the ytem; imilar to the igle machie ifiite bu ytem Iter machie or iterplat mode: aociated with ocillatio betwee the rotor of a few geerator cloe to each other Due to iadequate tuig of the cotrol ytem, e.g. geerator excitatio ytem, HVDC coverter ad SVC, or torioal iteractio (ub ychroou reoace) with power ytem cotrol 13

14 High & Low Frequecy Ocillatio Wheever power flow, I R loe occur. Thee eergy loe help to reduce the amplitude of the ocillatio. High frequecy (>1. HZ) ocillatio are damped more rapidly tha low frequecy (<1. HZ) ocillatio. The higher the frequecy of the ocillatio, the fater it i damped. Power ytem operator do ot wat ay ocillatio. However, whe ocillatio occur, it i better to have high frequecy ocillatio tha low frequecy ocillatio. The power ytem ca aturally dampe high frequecy ocillatio. Low frequecy ocillatio are more damagig to the power ytem, which may exit for a log time, become utaied (udamped) ocillatio, ad eve trigger protective relay to trip elemet 14

15 Blackout o Augut 1, Iitial evet (15:4:3): Short circuit due to tree cotact Outage of 6 traformer ad lie,1 MW lo 97 MW lo. Vulerable coditio (miute) Low damped iter area ocillatio Outage of geerator ad tie lie 11,6 MW lo 3. Blackout (ecod) Uitetioal eparatio Lo of 4% load 15,8MW lo Hz ocillatio dampig ratio >7% Mali-Roud Moutai #1 MW 15:4:3.64 Hz ocillatio dampig ratio =3.46% 15:47:36.5 Hz ocillatio dampig ratio 1% 15:48: Time i Secod Traiet itability (blackout) 15

16 Ocillatio Mode of a Multi machie Sytem i the Claic Model Hi d i dt P P i 1,,, mi ei co i co P E G P E G EEY E G EE B G ei i ii ij i ii i j ij ij ij i ii i j ij ij ij ij j1 ji j1 ji j1 ji, B Y i, G Y co ij i j ij ij ij ij ij ij (Igorig dampig) Liearizatio at ij : ij ij ij i i co ij ij ij ij co co i ij ij ij ij H H i i d dt d dt i i K i 1,,, j 1 ji j 1 ji ij ij EE B co G i i j ij ij ij ij ij Sychroizig power coefficiet P K = E E B co G i ij ij ij i j ij ij ij ij ij compared to K S = EE X T B cod 16

17 Hi d dt i K i 1,,, j1 ji ij ij Note: There are oly ( 1) idepedet equatio becaue ij =, o we eed to formulate the ( 1) idepedet relative rotor agle equatio with oe referece machie, e.g., the -th machie. d i d K, 1,, 1 jj i dt dt H H 1 R K ijij i j1 j1 ji Coider each i = i - d K K K K, i 1,, 1 1 i ij i i j ij j dt Hi j 1 H j1h Hi ji ji d 1 i dt j1 i 1,,, 1 ij j ii Kij Ki ij Kj Kij Hi j1 H H Hi ji 17

18 State pace repreetatio Let x, x,, x,,, ad x, x,, x,,, x 1 1 x1 1 x x 1 x 1 x 1 x x x x 1 x x X 1 I X A It characteritic equatio I-A = ha (-1) imagiary root, which occur i (-1) complex cojugate pair A -machie ytem ha (-1) mode Read Adero Example 3. ad 3.3 about the liearizatio ad eige-aalyi o the IEEE 9-bu ytem 18

19 Formulatio of Geeral Multi machie State Equatio The liearized model of each dyamic device: x i Aixi Biv i Cx Yv i i i i x i Perturbed value of tate variable i i Curret ijectio ito etwork from the device v Vector of etwork bu voltage B i ad Y i have o-zero elemet correpodig oly to the termial voltage of the device ad ay remote bu voltage ued to cotrol the device i i ad v both have real ad imagiary compoet Such tate equatio for all the dyamic device i the ytem may be combied ito the form: x ADxBDv i C xy v D x i the vector of tate variable of the complete ytem A D ad C D are block diagoal matrice compoed of A i ad C i aociated with the idividual device Node equatio of the tramiio etwork: i= Y N v The overall ytem tate equatio: x A xb ( Y Y ) C x = Ax Read Kudur ec. 1.7 for other related iformatio, e.g. load model liearizatio ad electio of a referece rotor agle D 1 D D N D D 1 D D( N D) D AA B Y Y C 19

20 Modal aalyi (eige aalyi) o a dimeioal oliear ytem Equilibrium x (with u ): Liearizatio at the equilibrium x : coider a perturbatio at x ad u

21 Eige vector For ay λi, the colum vector i atifyig Ai λii a right eigevector of A aociated with λ Modal 1 AΦ ΦΛ Λ diag( λ1, λ, λ 1 Φ AΦ Λ A Similarly the row vector matrix A Φ T T T 1 j Left eigevector aociated with j j j j Ψ,,,,,, T ψ, ) if ha ditict eigevalue i called (uually true for a real-world ytem) The left ad right eigevector correpodig to differet eigevalue are orthogoal (row vector of -1 are left eigevector of A, o we may let = C -1 where C i a diagoal matrix or imply equal to I if ormalized) ΨΦ I if i j, or 1 i j i i i λ 1

22 Free (zero iput) repoe ad tability x Ax Liearized ytem without exteral forcig To elimiate the cro-couplig betwee the tate variable, coider a ew tate vector z i t z z 1 i izi zi t zi e t z t 1 z Λz = Φ AΦz Φz AΦz x(t) Φz(t) 1 izie i1 1 z t Φ x t Ψx t z t z x c i i i it i t xe c e x (c i i the magitude of the excitatio of the i th mode) i i i i i1 i1 it 1t t k ki i k1 1 k i1 x (t) c e c e... c e t Free repoe i a liear combiatio of mode t Each eigevalue = j A real eigevalue (=) correpod to a o-ocillatory mode. a decayig mode ha <; a mode with > ha aperiodic itability. Complex eigevalue () occur i cojugate pair; each pair correpod to oe ocillatory mode Frequecy of ocillatio i Hz: f= / Dampig ratio (rate of decay) of the ocillatio amplitude i

23 Mode Shape ad Mode Compoitio t t z (t), z (t),..., z (t) T x Φz 1 1 T T T T t t,,, x (t), x (t),..., x (t) z Ψ x 1 1 The variable x 1, x,, x are origial tate variable choe to repreet the dyamic performace of the ytem. Variable z 1, z,, z are traformed tate variable; each i aociated with oly oe mode. I other word, they are directly related to the mode. The right eigevector i give the mode hape of the i th mode, i.e. the relative activity of the origial tate variable whe the i th mode i excited: The k th elemet of i, i.e. ki, meaure the activity of tate variable x k i the i th mode The left eigevector i give the mode compoitio of the i th mode, i.e. what weighted compoitio of origial tate variable i eeded to cotruct the mode: The k th elemet of i, i.e. ik, weight the cotributio of x k activity to the i th mode T k ki i i1 x (t) z t z (t) x (t) i ik k k 1 3

24 Participatio factor ΨΦ I ΨΦ p 1 ii ik ki ik k1 k1 ki meaure the activity of x k i the i th mode ik weight the cotributio of thi activity to the mode Participatio factor p ki = ik ki meaure the participatio of the k th tate variable x k i the i th mode. p ki i dimeiole ad hece ivariat uder chage of cale o the variable Quetio: coiderig = -1, why do we have to defie the mode hape, mode compoitio, ad participatio factor, eparately? Lear Kudur Example 1. o a SMIB ytem 4