ECE 4/5 Power System Operatios & Plaig/Power Systems Aalysis II : 6 - Small Sigal Stability Sprig 014 Istructor: Kai Su 1
Refereces Kudur s Chapter 1 Saadat s Chapter 11.4 EPRI Tutorial s Chapter 8 Power Oscillatios
Power Oscillatios The power system aturally eters periods of oscillatio as it cotiually adjusts to ew operatig coditios or experieces other disturbaces. Typically the amplitude of the oscillatios is small ad their lifetime is short. Whe the amplitude of the oscillatios becomes large or the oscillatios are sustaied, a respose may be required. A system operator may have the opportuity to respod ad elimiate harmful oscillatios or, less desirably, protective relays may activate to trip system elemets. 3
Small Sigal Stability Small sigal stability (also referred to as small-disturbace stability or steady-state stability) is the ability of a power system to maitai sychroism whe subjected to small disturbaces I this cotext, a disturbace is cosidered to be small if the equatios that describe the resultig respose of the system may be liearized for the purpose of aalysis It is coveiet to assume that the disturbaces causig the chages disappear (the details of the disturbace is ot importat) The system is stable if it returs to its origial state, i.e. a stable equilibrium poit. Such a behavior ca be determied i the liearized model of the power system 4
Small Sigal Stability of a Sigle-Machie-Ifiite-Bus System Because of the relative size of the large system to which the machie is supplyig power, dyamics associated with the machie will cause virtually o chage i the voltage ad frequecy of Thevei s E B The large system is ofte referred to as a Ifiite Bus (costat voltage ad costat frequecy) The voltage phasor at the ifiite bus ca be the referece of the rotor agle, i.e. E B 0 5
Cosider the Classical Model (T e ) P+jQ P t +jq t PB +jq B (T m ) (P e =T e ) Liearize swig equatios at δ=δ 0 : Complex power behid X d : = T m - T e With resistace (R T ) eglected: T e = P e = P= P t = P B = Defie sychroizig torque coefficiet EE B KS cos0 Pmax cos0 X T P max = E E B /X T 6
d K D d K T dt H dt s 0 0 m H H Apply Laplace Trasform: K D Ks0 0T s s H H H Characteristic equatio: m Note: K S =P S ad K D =Dω 0 i Saadat s book d D d P T dt 0 s 0 0 m H dt H H s D P s H H 0 0 s 0 7
Compared to the geeral form of a d order system (0<ζ<<1 for a geerator): s s 0 EE B ( KS cos0 Pmax cos0) X T Udamped atural frequecy: Dampig ratio: Two cojugate complex roots: s, s j 1 j 1 Note: resistace R T is igored here, whose effect is to result i more dampig d Damped oscillatio frequecy: Whe H or K S, oscillatio frequecy (ω or ω d ) (e.g. whe X T or δ 0 ) 8
High vs. Low Frequecy Oscillatios i Realistic Systems Whe power flows, I R losses occur. These eergy losses help to reduce the amplitude of the oscillatio. The higher the frequecy of the oscillatio, the faster it is damped. High frequecy (>1.0 HZ) oscillatios are damped more rapidly tha low frequecy (<1.0 HZ) oscillatios. Usually, i realistic systems: Power system operators do ot wat ay oscillatios. However, it is better to have high frequecy oscillatios tha low frequecy. The power system ca aturally dampe high frequecy oscillatios. Low frequecy oscillatios are more damagig to the power system, which may exist for a log time, become sustaied (udamped) oscillatios, ad eve trigger protective relays to trip elemets 9
Blackout Evet o August 10, 1996 1. Iitial evet (15:4:03): Short circuit due to tree cotact Outages of 6 trasformers ad lies,100 MW loss 970 MW loss. Vulerable coditios (miutes) Low-damped iter-area oscillatios Outages of geerators ad tie-lies 11,600 MW loss 15,80MW loss 3. Blackouts (secods) Uitetioal separatio Loss of 4% load 1500 1400 1300 100 1100 0.76 Hz oscillatios Dampig>7% Mali-Roud Moutai #1 MW 15:4:03 0.64 Hz oscillatios 3.46% Dampig 00 300 400 500 600 700 800 Time i Secods 10 15:47:36 0.5 Hz oscillatios Dampig 1% Oscillatio frequecy (ω or ω d ) whe H or K S (e.g. whe X T or δ 0 ) 15:48:51 System isladig ad blackouts
System Respose after a Small Disturbace Y() s = X() s = ( si A) x(0) B U() s 1 [ + ] x1 x / r 0 T u m H x1 0 0 x 1 0Tm x 0 x 1 H x() t = Ax() t + B ut () 1 0 y() t x() t 0 1 x x 1 s 0 0 s s X () s (0) U() s s x + B s 0 0 ( s) 0 s (0) ( ) u r( s) s s r(0) s Zero-iput Zero-state Zero-iput Zero-state sx() s x(0) = AX() s + B U() s u U() s s Usually ω r (0) 0 followig a disturbace 11
s 0 0 ( s) 0 s (0) ( ) u r( s) s s r(0) s Zero-iput respose E.g. whe the rotor is suddely perturbed by a small agle δ(0) 0 ad assume ω r (0)=0 ( s ) (0) () s s s (0) / r () s s 0 s Takig iverse Laplace trasforms Note: Zero-state respose E.g. whe there is a small icrease i mechaical torque T m (= P m i pu) δ () s = ω () s = r / ( ) / ω u 0 ( + ζωs+ ω) ss s r u H T m u + ζω s+ ω 0 r 0 0 (0) e t si( ) dt 1 r (0) 1 0 e t si d t δ ω u 1 1 e si 1 ζ ( ω t θ) 0 ζω = t d + ω ω = r ω u 1 ζ e ζω t siω t d θ = cos 1 ζ 1 4H τ = = (Respose time costat) ζω K D 1
s 1 0 ( s) s (0) ( ) u ( s) s s (0) s Note: Saadat s book defies 0 0 u H T m Zero-iput respose E.g. whe the rotor is suddely perturbed by a small agle δ(0) 0 ad assume ω r (0)=0 ( s ) (0) () s s s (0) () s s s Takig iverse Laplace trasforms Zero-state respose E.g. whe there is a small icrease i mechaical torque T m (= P m i pu) δ () s = ω() s = ss u + s+ ( ζω ω) s u + ζω + ω s (0) e t si( ) dt 1 (0) e 1 t si d t u 1 ζω δ = 1 e t si ( ω ) dt+ θ ω 1 ζ ω = ω u 1 ζ e ζω t siω t d θ = cos 1 ζ 1 4H τ = = (Respose time costat) ζω K D 13
Saadat s Example 11. ad Example 11.3 H=9.94 MJ/MVA, D=0.138, P=0.6 pu with 0.8 power factor. Obtai the zero-iput ad zero-state resposes for the rotor agle ad the geerator frequecy: (1) δ(0)=10 o =0.1745 rad () P=0.pu δ(0)=16.79+10 o =6.79 o 14
Zero-iput respose: δ(0)=10 o Zero-state respose: P=0.pu δ(0)=16.79+10=6.79 o δ( )=16.79+5.76=.55 o 15
Cosider the Excitatio System (see Kudur s 1.4 for a ij ad b 1 ) K K [ K (1 st ) K G ( s)] T e 3 4 R 5 ex fd s TT 3 R s( T3TR )1 K3K6Gex ( s) 16
The effect of the AVR o dampig ad sychroizig torque compoets is primarily iflueced by G ex (s) ad K 5 With K 5 <0, the AVR may itroduce a positive sychroizig torque K K [ K (1 st ) K G ( s)] T e 3 4 R 5 ex fd s TT 3 R s( T3TR)1 K3K6Ge x( s) For a give oscillatio frequecy s=jω: K T K K jk fd = K K I 0 e R I R r S( ) D( ) r fd fd K R ad K I are respectively the real ad imagiary parts of the coefficiet of δ j j/ s/ r 0 / Sychroizig ad dampig torque coefficiets due to ψ fd 17
Example o effects of differet AVR settigs Steady-state sychroizig torque coefficiet: The effect of the AVR is to icrease the sychroizig torque compoet at steady state Dampig ad sychroizig torque compoets at rotor oscillatio frequecy 10 rad/s (s=jω=j10) 18
T e = T S + T D = K S δ+k D ω K S = K S( ψfd)+ K S(ge & etwork) K D = K D( ψfd) +K D(ge & etwork) Usually, K S (ge & etwork) >0 K D(ge & etwork) >0 Costat field voltage (K A =0): K D >0 Perhaps, K S =K S(ge & etwork) + K S( ψfd) <0 With excitatio cotrol (large K A ) K S >0 Perhaps, K D =K D(ge & etwork) + K D( ψfd) <0 19
Power System Stabilizer The basic fuctio of a power system stabilizer (PSS) is to add dampig to the geerator rotor oscillatios by cotrollig its excitatio usig o-voltage auxiliary stabilizig sigal(s) If the trasfer fuctio from PSS s output to T e was pure gai, a direct feedback of ω r would create a positive dampig torque compoet. 0
However, actual geerators ad exciters exhibit frequecy depedet gai ad phase-lag characteristics Therefore, G PSS (s) should provide phase-lead compesatio to create a torque i phase with ω r Gs () 1 1 s G(s) G 1/τ 1 1/τ 1 1 PSS () s ( 1) 1 s s 1
PSS Model Stabilizer gai K STAB determies the amout of dampig itroduced by PSS Sigal washout block: High-pass filter with T W log eough (typically 1~0s) to allow sigals associated with oscillatios i ω r to pass uchaged. However, if it is too log, steady chages i speed would cause geerator voltage excursios Phase compesatio block: Provides phase-lead compesatio over the frequecy rage of iterest (typically, f=0.1~.0 Hz, i.e. ω=0.6~1.6 rad/s) Two or more first-order blocks, or eve secod-order blocks may be used. Geerally, some uder-compesatio is desirable so that the PSS results i a slight icrease of the sychroizig torque as well
Kudur s Example 1.4 3
PSS i NERC Itercoectios The WECC ad the MRO Regios have operatig requiremets that madate the use of PSS. All uits with fast excitatio systems must be equipped with well tued PSS i these Regios. PSS are typically istalled i the majority of geeratig uits i the problem area. Several uits i the PJM system are equipped with PSS to address local oscillatory stability cocers 4
Characteristics of Small-Sigal Stability Problems Local or machie-system modes (0.7~Hz): oscillatios ivolve a small part of the system Local plat modes: associated with rotor agle oscillatios of a sigle geerator or a sigle plat agaist the rest of the system; similar to the sigle-machie-ifiite bus system Iter-machie or iterplat modes: associated with oscillatios betwee the rotors of a few geerators close to each other Iter- or itra-area modes (0.1~0.7Hz): machies i oe part of the system swig agaist machies i other parts Iter-area model (0.1~0.3Hz): ivolvig all the geerators i the system; the system is essetially split ito two parts, with geerators i oe part swigig agaist machies i the other parts. Itra-area mode (0.4~0.7Hz): ivolvig subgroups of geerators swigig agaist each other. Cotrol or torsioal modes: Due to iadequate tuig of the cotrol systems, e.g. geerator excitatio systems, HVDC coverters ad SVCs, or torsioal iteractio with power system cotrol 5
Homework Problems 11.10~11.13 i Saadat s book (3 rd ed), due by April 15 (Tuesday) i class or by email 6