ECE 4 Power Sytem Operatio & Plaig 6 Small Sigal Stability Sprig 15 Itructor: Kai Su 1
Referece Saadat Chapter 11.4 EPRI Tutorial Chapter 8 Power Ocillatio Kudur Chapter 1
Power Ocillatio The power ytem aturally eter period of ocillatio a it cotiually adjut to ew operatig coditio or experiece other diturbace. Typically the amplitude of the ocillatio i mall ad their lifetime i hort. Whe the amplitude of the ocillatio become large or the ocillatio are utaied, a repoe may be 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. 3
Small Sigal Stability Small igal tability (alo referred to a mall diturbace tability or teady tate 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 It i coveiet to aume that the diturbace cauig the chage diappear (the detail of the diturbace i ot importat) The ytem i table if it retur to it origial tate, i.e. a table equilibrium poit. Such a behavior ca be determied i the liearized model of the power ytem 4
Coider the Claical Model (T e ) (T m ) P+jQ P t +jq t PB +jq B Liearize wig equatio at = : (P e =T e ) Complex power behid X d : =T m -T e With reitace (R T ) eglected: Defie ychroizig torque coefficiet EE B KS = cod = Pmaxcod X T T e = P e = P= P t = P B = P max = E E B /X T 5
d Dd K D ddd K w w DT dt + H dt + m D d = H H Apply Laplace Traform: K D Kw wdt D d+ D d+ D d= H H H Characteritic equatio: m Note: P S =K S ad D =K D i Saadat book d Dd Dw ddd Pw w DT dt + m + D d = H dt H H D w P w + + = H H 6
EE B ( KS = cod = Pmaxco d) X T Compared to the geeral form of a d order ytem (<<<1 for a geerator): + zw + w = Dampig ratio Nature frequecy Two cojugate complex root:, =- jw =-zw jw 1-z 1 d Zero iput repoe i damped iuoidal ocillatio: - D d( t) = Ae t i( w t+ q) d = zw Natural frequecy: w pf w = KS = PS rad/ H H Dampig ratio: 1 K D D z = = K Hw S p f HP S Note: tramiio reitace i igored here, o the actual dampig i higher tha Whe H or K S, ocillatio frequecy ( or d ) How doe chage whe X T or icreae? 7
Sytem Repoe after a Small Diturbace Y() X() ( IA) x() BU() 1 + x1 =Dd x =D w =Dd/ w r é ù é ù ê ú ë û êë úû DT D u = m H x1 é w ù x é 1 ùdtm = + x ê-w w -zw ú x ê 1 ú H ë û ë û x() t Ax() t Bu() t é1 ù y() t = x() t =ê ú éx ê ù ú 1 ê 1 êx ú ë ú û ë û 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 Zero-iput Zero-tate ù ú Zero-iput Zero-tate X() x() AX() BU() Du D U() = Uually r () followig a diturbace 8
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 ) Dd() d() = + + zw w w Dd() / w D wr () =- + + zw w Takig ivere Laplace traform D w =D d/ w = ( w -w )/ w i pu r D D u = 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( w ) dt+ q 1-z Dw r wdd() i rad/ =- e 1-z -zw t i w d t i rad r T 1 1 e i 1 m t d H T m i rad/ t i H 1 e t t d co 1 1 4H (Repoe time cotat) K D 9
Saadat Example 11. ad 11.3 H=9.94 MJ/MVA, D=.138, P=.6 pu with.8 power factor. Obtai the zero iput ad zero tate repoe for the rotor agle ad the geerator frequecy: (1) ()=1 o =.1745 rad () P=.pu ()=16.79+1 o =6.79 o 1
Zero iput repoe: ()=1 o Zero tate repoe: P=.pu ()=16.79+1=6.79 o ()=16.79+5.76=.55 o 11
Characteritic of Small Sigal Stability Problem Local or machie ytem mode (.7~Hz): ocillatio ivolve a mall part of the ytem 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 machieifiite bu ytem Iter machie or iterplat mode: aociated with ocillatio betwee the rotor of a few geerator cloe to each other Iter or itra area mode (.1~.7Hz): machie i oe part of the ytem wig agait machie i other part 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. Cotrol or torioal mode: Due to iadequate tuig of the cotrol ytem, e.g. geerator excitatio ytem, HVDC coverter ad SVC, or torioal iteractio with power ytem cotrol 1
High v. Low Frequecy Ocillatio i Realitic Sytem Whe power flow, I R loe occur. Thee eergy loe help to reduce the amplitude of the ocillatio. The higher the frequecy of the ocillatio, the fater it i damped. High frequecy (>1. HZ) ocillatio are damped more rapidly tha low frequecy (<1. HZ) ocillatio. Uually, i realitic ytem: Power ytem operator do ot wat ay ocillatio. However, it i better to have high frequecy ocillatio tha low frequecy. 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 13
Blackout Evet o Augut 1, 1996 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 15,8MW lo 3. Blackout (ecod) Uitetioal eparatio Lo of 4% load 15 14 13 1 11.76 Hz ocillatio Dampig>7% Mali-Roud Moutai #1 MW 15:4:3.64 Hz ocillatio 3.46% Dampig 15:47:36.5 Hz ocillatio Dampig 1% 15:48:51 Sytem iladig ad 3 4 5 6 7 8 Time i Secod Ocillatio frequecy ( or d ) whe H or K S (e.g. whe X T or ) blackout 14