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High order subsynchronous resonance models and multi-mode stabilization 1977

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HIGH ORDER SUBSYNCHRONOUS RESONANCE MODELS AND MULTI-MODE STABIL IZATION by K i n g K u i T se B . S c . (Hon . ) » N o r t h e a s t e r n U n i v e r s i t y , 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPL IED SCIENCE i n THE FACULTY OF GRADUATE STUDIES in t h e Depa r tmen t o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June 1977 0 King Kui Tse, 1977 In p resent ing t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements fo r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and study. I f u r t h e r agree tha t permiss ion fo r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n p e r m i s s i o n . Department of E l e c t r i c a l Eng inee r ing The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date JuTv 6. 1977. ABSTRACT Subsynchronous resonance (SSR) occurs i n a s e r i e s - c a p a c i t o r - compensated power system when a mechanical mass-spring mode coincides with that of the e l e c t r i c a l system. In t h i s t h e s i s , a complete high order model incl u d i n g mass-spring system, series-capacitor-compensated trans- mission l i n e , synchronous generator, turbines and governors, e x c i t e r and voltage regulator i s derived. Eigenvalue analysis i s used to f i n d the e f f e c t of capacitor compensation, conventional lead-lag s t a b i l i z e r , loading and dampers on SSR. F i n a l l y , c o n t r o l l e r s are designed to s t a b i l i z e m u l t i - mode subsynchronous resonance simultaneously over a wide range of capacitor compensation. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENT v i i NOMENCLATURE . . v i i i 1. INTRODUCTION 1 1.1 Subsynchronous Resonance 1 1.2 The Scope of the Thesis 2 2. A COMPLETE POWER SYSTEM MODEL FOR SUBSYNCHRONOUS RESONANCE STUDIES • 4 2.1 Introduction 4 2.2 The Steam Turbines and Generator Multi-Mass Torsional System 4 2.3 The Turbine Torques and Speed Governor 8 2.4 The Synchronous Generator 10 2.5 The E x c i t e r and Voltage Regulator . . 14 2.6 State Equations f o r the Complete System 16 3. EIGENVALUE ANALYSIS OF THE SSR MODEL 1 8 3.1 Introduction , 18 3.2 The E f f e c t of Capacitor Compensation I 8 3.3 The E f f e c t of Conventional S t a b i l i z e r 18 3.4 The E f f e c t of Loading I 9 3.5 The E f f e c t of Dampers I 9 4. MULTI-MODE TORSIONAL OSCILLATIONS STABILIZATION WITH LINEAR OPTIMAL CONTROL 2 9 4.1 State Equations with Measurable Variables 29 4.2 State Equations i n Canonical Form 31 4.3 Linear Optimal Control Design 34 i i i Page 4.4 S t a b i l i z a t i o n of SSR . . 35 5. CONCLUSIONS 42 REFERENCES 43 i v LIST OF TABLES Table Page 3-1 Data f o r SSR Model 20 3-2 Eigenvalues of SSR model at d i f f e r e n t degrees of capaci- tor compensation without conventional s t a b i l i z e r . . . . 21 3- 3 E f f e c t of damper winding for zero t o t a l reactance . . . 28 4- 1 Eigenvalues of o r i g i n a l system and reduced order models without c o n t r o l l e r at 30% compensation 37 4-2 Eigenvalues of reduced 22nd order model with/without c o n t r o l l e r and o r i g i n a l system with the c o n t r o l l e r at 30% compensation . 38 4-3 Eigenvalues of reduced 22nd order model with/without c o n t r o l l e r and o r i g i n a l system with the c o n t r o l l e r at 50% compensation . 39 4-4 Eigenvalues of reduced 19th order model with/without c o n t r o l l e r and o r i g i n a l system with the c o n t r o l l e r at 30% compensation 40 v LIST OF ILLUSTRATIONS Figure Page 2-1 A f u n c t i o n a l block diagram of the complete system for subsynchronous resonance studies . . . 6 2-2 Mechanical mass and shaft system ^ 2-3 Torques of a mass-shaft system ^ 2-4 A speed governor model for the steam turbine system . . 8 2-5 A l i n e a r model of the steam turbine system 9 2-6 A synchronous machine model . . . . ^ 2-7 A s i n g l e l i n e representation of the transmission l i n e . ^ 2-8 E x c i t e r and voltage regulator model ^ 2- 9 A supplementary e x c i t a t i o n control . . . ^ 3- 1 The e f f e c t of capacitor compensation without s t a b i l i z e r . 22 3-2 Enlarged portion of F i g . 3-1 23 3-3 The e f f e c t of capacitor compensation with s t a b i l i z e r . . 24 3-4 Enlarged portion of F i g . 3-3 25 3-5 The e f f e c t of loading without s t a b i l i z e r 26 3- 6 The e f f e c t of loading with s t a b i l i z e r 27 4- 1 The e f f e c t of capacitor compensation with c o n t r o l l e r . . 41 v i ACKNOWLEDGEMENT I w i s h to express my s i n c e r e g r a t i t u d e to my s u p e r v i s o r s , Dr . Y . N . Yu and Dr . M.D. Wvong, f o r t h e i r p a t i e n c e , guidance, many hours o f c o n s u l t a t i o n and v a l u z b l e adv ice du r ing the course of the resea rch work and w r i t i n g o f t h i s t h e s i s . F i n a n c i a l support from a B r i t i s h Columbia Telephone Company S c h o l a r s h i p , a U n i v e r s i t y of B r i t i s h Columbia Summer Research F e l l o w s h i p and a t each ing a s s i s t a n t s h i p i s g r a t e f u l l y acknowledged. Thanks are a l s o due to Mary E l l e n Flanagan and Sann i fe r L o u i e f o r t y p i n g t h i s t h e s i s . I am g r a t e f u l to my parents and Mui Ha i n Hong Kong f o r t h e i r p a t i e n t unders tanding and encouragement throughout my u n i v e r s i t y ca ree r . ^NOMENCLATURE General X state vector of immeasurable model Z state vector of measurable model Y state vector of canonical A system matrix of X-model F system matrix of Z-model F system matrix of Y-model o 1 B control matrix of X-model G control matrix of Z-model G control matrix of Y-model o U control vector M transformation matrix for Z-model T transformation matrix for Y-model X eigenvalue j complex operator, Mass-Spring System M i n e r t i a c o e f f i c i e n t = 2H H i n e r t i a constant K shaft s t i f f n e s s D damping Q rotor angle a) rotor speed oj synchronous speed Synchronous Machine i instantaneous value of current V instantaneous value of voltage v i i i ib flux-linkage R resistance X reactance 6 torque angle, rad. to angular v e l o c i t y , rad./s T e e l e c t r i c a l torque i terminal voltage P+jQ generator output power Transmission Network Xfc,Rt reactance and resistance of transformer X ,R reactance and resistance of the l i n e e e X c reactance of capacitor V i n f i n i t e bus voltage o Exciter and Voltage Regulator regulator gain T^ regulator time constant, s T exciter time constant, s E V - reference voltage ref. ° Governor and Turbine System K g actuator gain T r T 2 actuator time constant T 3 servomotor time constant a change i n actuator signal PGV power at. gate outlet TCH steam chest time constant T RH reheater time constant T CO cross-over time constant i x F high pressure turbine power fraction intermediate pressure turbine power fraction F^p^ low pressure turbine 1 power fra c t i o n F^p2 l° w pressure turbine 2 power fraction T high pressure turbine torque T-j-p intermediate pressure turbine torque TLp^,TLp2 low pressure turbine torque Subscripts d,q d i r e c t - and quadrature-axis stator quantities f f i e l d c i r c u i t quantities D,Q,G d i r e c t - and quadrature-axis damper quantities c quantities associate with capacitor a armature phase quantities Superscripts -1 inverse of a matrix t transpose of a matrix d i f f e r e n t i a l operator P r e f i x l i n e a r i z e d quantities d i f f e r e n t i a l operator d i f f e r e n t i a l operator x 1. 1. INTRODUCTION 1.1 Subsynchronous Resonance [1] To increase the power transfer capability of a power system, the use of series-capacitor-compensated transmission lines i s the best alternative to the addition of transmission lines because of environmental considerations and the limited a v a i l a b i l i t y of right-of-way. They are also more economical than other methods such-as HVDC. However, subsyn- chronous resonance (SSR) may occur and shaft damage may res u l t . Two turbine shafts were severely damaged [2] at the Mohave generating station of the Southern^ C a l i f o r n i a Edison Company because of the excessive tor- sional o s c i l l a t i o n s caused by interaction between the e l e c t r i c a l reson- ance of the series-capacitor-compensated system and the natural modes of the multi-mass generator turbine mechanical system. Subsynchronous resonance may occur i n a system i n the steady- state or transient state due to a system f a u l t or major switching. The former may be called the steady-state subsynchronous res- onance and the l a t t e r the transient subsynchronous resonance. The main v. problems are the s e l f e x c i t a t i o n , the torsional interaction, and the transient torques [3]. When SSR occurs, the synchronous machine i s '••'self-' excited and behaves l i k e an induction generator. I f the negative r e s i s - " tance of the machine, as an induction generator, exceeds the t o t a l r e s i s - tance of the external e l e c t r i c a l system, s e l f excitation of SSR occurs. Torsional o s c i l l a t i o n i s due to the mechanical modes of the multi-mass turbine-generator system. The torsional frequencies are i n the subsynchronous range. I f the e l e c t r i c a l resonant frequency i s equal or close to a torsional mode, the rotor o s c i l l a t i o n s and the induced voltages w i l l b u i l d up and the interaction between the e l e c t r i c a l and 2. mechanical systems ensues [1,4]. Transient torques are caused by system disturbances on a series- capacitor-compensated l i n e and the energy stored i n the series capacitor produces large subsynchronous currents i n the l i n e s . When the frequency of the current coincides with the natural torsional frequency, transient torque results. After the reported turbine shaft f a i l u r e s [3], corrective measures have been proposed. Some of them are under serious consideration-and others already put into practice. Without too much modification to the ex i s t i n g system, the simplesttway to avoid the subsynchronous resonance i s to reduce the degree of capacitor compensation. Another suggestion i s the i n s t a l - l a t i o n of passive f i l t e r units i n series with the generator transformer neutral at the high voltage side. Each f i l t e r unit i s a high-Q p a r a l l e l resonant c i r c u i t tuned to block the subsynchronous current at a par t i c u l a r frequency corresponding to one of the mechanical modes. Additional amor- tisseur windings on the pole faces can reduce the effe c t i v e negative res- istance [5]. Supplementary excitation control i s being considered and the s t a b i l i z i n g signals are derived from rotor speed. F i n a l l y a subsyn- chronous overcurrent relay has been developed for the automatic protec- tion of generating units i n case of sustained subsynchronous o s c i l l a t i o n s . 1.2 Scope of the thesis The widely accepted method for subsynchronous resonance studies i n engineering practice consists of a two-step analysis [5]. The e l e c t r i - c a l and mechanical modes are determined separately. The transient e l e c - t r i c a l torque from the e l e c t r i c a l system i s calculated f i r s t and then applied to the mechanical system as a forcing function. In this thesis, a complete model including the e l e c t r i c a l , mechanical and control systems 3 . w i l l b e d e v e l o p e d a n d p r e s e n t e d i n C h a p t e r 2 . B y u s i n g e i g e n v a l u e a n a l - y s i s , t h e e f f e c t o f v a r i o u s d e g r e e s o f c o m p e n s a t i o n , l o a d i n g c o n d i t i o n s a n d c o n v e n t i o n a l s u p p l e m e n t a r y e x c i t a t i o n c o n t r o l o n s u b s y n c h r o n o u s r e s - o n a n c e w i l l b e e x a m i n e d i n C h a p t e r 3 . F o r b r o a d - b a n d f r e q u e n c y m u l t i - m o d e s u b s y n c h r o n o u s r e s o n a n c e c o n t r o l , l i n e a r o p t i m a l c o n t r o l l e r s w i l l b e d e s i g n e d i n C h a p t e r 4 . A s u m m a r y o f a l l i m p o r t a n t r e s u l t s a n d c o n c l u - s i o n s w i l l b e p r e s e n t e d i n C h a p t e r 5 . 4. 2. A COMPLETE POWER SYSTEM MODEL FOR SUBSYNCHRONOUS RESONANCE STUDIES 2.1 Introduction For any dynamic or transient s t a b i l i t y study of a power system, an accurate model of the system i s required. In addition to the in d i v i d u a l e f f o r t s [2,7,12], a benchmark model has been proposed by the IEEE Subsyn- chronous Resonance Working Group for SSR studies [20], In this chapter, a complete subsynchronous resonance model i s presented, including steam turbines and generator multi-mass t o r s i o n a l system, the turbine torques and speed governor, the synchronous generator, the capacitor-compensated transmission l i n e s , and the exciter and voltage regulator. A functional block diagram of the complete system i s shown i n Fig. 2-1. 2.2 The Steam Turbines and Generator Multi-Mass Torsional System Assume that the steam turbine-generator set consists of one high-pressure steam turbine, one intermediate-pressure turbine, 'two low- pressure turbines, one generator rotor and one exciter, a l l mechanically coupled on the same shaft as shown i n Fig. 2-2. They comprise a six-mass torsional system. For the purpose of analysis [13], they are considered to have concentrated masses and to be coupled by shafts of negligible mass and known torsional s t i f f n e s s . Each mass i s denoted by a c i r c u l a r disc, as i n Fig. 2-3, with an i n e r t i a constant M_̂, a positive torsional torque K . ( 9 . , , - 9.) on the l e f t and a negative torque - K . .,(8. - 8. ,) on the x 1+1 1 i - l 1 i - l right. There i s an external torque T̂  applied to the mass inaa positive d i r e c t i o n , an accelerating torque M ^ O K i n the same direction and a damping torque D̂. OK i n the opposite d i r e c t i o n . The net accelerating torque becomes M.u). l l = T . - D . O ) . + K . ( e . , , - e . ) - K . . ( e . - e . , ) i i i i i + i i i - i i i - i (2-1) 5. th where M. = the i n e r t i a constant of i rotor 1 th 0^ = the r o t a t i o n a l displacement for i rotor = damping c o e f f i c i e n t for 1 ^ rotor th K. .... = the t o r s i o n a l s t i f f n e s s of the shaft between the i x,x+l rotor and the i+l*"* 1 rotor By applying equation (2-1) to the s i x mass turbine-generator system, twelve d i f f e r e n t i a l equations are obtained: V K X) T „ . . _ 56 . K56 a 66 . , HP Hxgh Pressure p ^ = — 6 - — ^ - — % + — (2-2) 6 6 6 6 p9g = tog 0)0" (2-3) K56 a ( K45 + K 56 } M K45 Q °55 .. A T I P Intermedxate pa, = — ^ B +—Q (2-4) Pressure 5 5 5 5 5 p 9 5 = w 5 w ° ( 2 - 5 ) K 4 5 ( K 3 4 + K 4 5 ) K 3 4 ° 4 4 T L P 1 Low Pressure 1 po>4 = g — 65 g- 9 4 + M ~ 6 3 _ MT" W 4 + ^ ~ ( 2 _ 6 ) % J 4 M, p8, = 03. wo (2-7) 4 4 K , . . (K__ .+.K_,) K„„ D , T Low Pressure 2 ^ = ^ 6 , - 63 + ^ 6 _ _ M ^ + _LP2 ( 2 _ 8 ) p6 3 = O J 3 OJO (2-9). K23 n ( K12 + K 2 3 } ^ K12 Q D22 Te i n , Generator p «o = g - 83 g- 6 + M ~ °1 " " " ( 2 " 1 0 ) p 6 = OJ OJO (2-11) K12 K12 °11 E x c i t e r po), = — — 6 - — — 6, - — — 0).. (2-12) r 1 1 . 1 p6^ = 0)^ OJO (2-13) SR Governor 6 o) etc Steam Turbines i=3^6 K e±,u±, i=3^6 Torsional System 6,u Generator E l e c t r i c a l Network FD e 1,o) 1 Exciter Voltage Regulator IL 6 o) ... etc Fig. 2-1 A functional block diagram of the complete system for subsynchronous resonance studies. 7. The generator has an e l e c t r i c torque output Te, and the exciter e l e c t r i c torque i s neglected. Note that while angles are i n radians, the speed i s i n p.u.; 03o = 1 p.u. = 377 e l e c t r i c a l radian/second High Intermediate Low Low Generator Exciter pressure pressure pressure 1 pressure 2 0 £ , toA 6 , 0 ) , - 0 ^ , 0 ) ^ 9^ , o)^ 6 , 0 ) 0 ^ , 0 ) ^ D. .0). i i l Fig. 2-3 Torques of a mass-shaft system 8. 2.3 The Turbine Torques and Speed Governor The steam turbine and speed governor representation i s based on an IEEE committee report [14]. Usually the speed i s , sensed.between the low- pressure turbine and the generator rotor. Combined with the speed refer- ence, the speed deviation or error s i g n a l i s derived and relayed through the actuator to activate the servomotor, which i n turn opens or closes the steam valves. A block diagram [14] i s shown i n Fig. 29-4. Foraa l i n - ear study, the system equations may be written; K 1 p A a = Ato - — Aa (2-14) 1 1 P AP G V - |- Aa - |- AP G V (2-15) Ato \ '- K. . • : S Aa + ( (1 + ST^ \ 1 S AP GV Fig. 2-4 A speed governor model for the steam turbine system 9. AT. HP AT IP AT. LPl'. AT. LP 2 HP F PIP LPl] "LP 21 A P GV 1 + ST CH 1 + ST RH 1 + ST CO Fig. 2-5 AA Linear model of the steam turbine system Fig. 2-5 shows a standard turbine representation for s t a b i l i t y studies [14]. The system consists of one high-pressure, one intermediate- pressure and two low-pressure turbines. Their output torques are denoted by T̂ -., T T T >, T T i , T T respectively. There i s a reheater between high-rir i r L i r i LarZ pressure and intermediate-pressure stages, and crossover pipings between intermediate-pressure and low-pressure stages. The steam into the turbines flow through the governor-controlled valves at the i n l e t of the steam chest. The time constants of the steam chest, the reheater and the cross-'s over piping are denoted by T^, T ^ a n d T p n, respectively. F^, F T p , CO HP IP ! F_ _ . j and F 0 represent fractions of the t o t a l power developed i n the various stages. Therefore F,. p A T HP HP CCH A PGV " A THP (2-16) "IP "HP X TRH P A T I P = F,m x~ „„A THP --T-, ATT LPl p A T L p l = „ A T RH 1 IP m x T c o " ~ I P T c o ' T l p 1 (2-17) (2-18) 10. A T. PL2 LP2 A T. LP1 (2-19) 2.4 The Synchronous Generator The synchronous generator i s assumed to have s i x windings. In addition to the d and q armature windings on the respective axes, there i s a f i e l d winding f, a damper winding D on the d-axis and two damper wind- ings Q and G on the q-axis. They are schematically shown i n Fig. 2-6. d-axis rqrraXlS •-fM) + v. Fig. 2-6 A synchronous machine model I'l. The voltage equations i n the li n e a r form are A V, = pAip , - OJOAIJJ — Au - R Ai q qo a d A V = vAty + OJOAIJ ; , + IJJ, Ao) - R Ai q v r q r d rdo a q A V f = pAi|jf + R A i f 0 = pA^ + A i D 0 = pA*Q + RQ A i Q PA*G + RG A H where the f l u x linkages are Aijj. Aip. Air D _1 uo - x d 0 Xad X A ad 0 -X q 0 0 " Xad 0 X f Xad " Xad 0 Xad 0 -X aq 0 0 0 -X aq 0 0 X X aq 0 0 aq 0 0 X aq X X r aq G Ai . Ai Ai, A i , Ai G (2-20) (2-21) anand the tf/s, X's, R's and i ' s are the per unit f l u x linkages, reactances, resistances and currents respectively. The saturation i n the iron c i r c u i t i s neglected. The stator transient voltages p i ^ an P ^ j although normally neglected i n s t a b i l i t y studies [15,16] are retained i n this study because the capacitor compen- sated transmission l i n e s , to which the armature windings are connected i n series, must be described by d i f f e r e n t i a l equations. 12. R- X„ ct F i g . 2-7 A single l i n e representation of the transmission l i n e In F i g . 2-7, and are d-q components of the terminal v o l t - age i s the voltage across the capacitor and V i s the terminal voltage at the capacitor. The transformer i s represented by a reactance X and a resistance R and the transmission l i n e by a reactance X and a l i n e t e resistance R . e Let the terminal voltage equations i n a-b-c phase coordinates be [v t] a b c = [ R ] [ I t ] + [ L ] dt [ I t ] + 1 V c ] ' ' a,b,c a,b,b a,b,c + [ V J a,b, c (2-22) where [R] = a resistance matrix: R^ + R t e X + X [L] = an inductance matrix: t e 0)O Let Park's transformation matrix be [T] = cos 6 cos(0-120) cos(6+120) - s i n -sin( 6-120.) -sin(6+120) o 1 1 1 (2-23) 13. and the transformations are [V] , = [T][V]. and [I] , = [ T ] [ I ] , n n (2-24) L Ja,b,c Jd,q,o Ja,b,c d,q,o Then we have ^ d , q , 0 = [ R ] [ I ] d , q , 0 + ^ d f ^d.q.o + ^ ' ^ d . q + [V!] + [v ] c , o , d,q,o d,q,o (2-25) Note that [T] ^ [T] = 0 1 0 -1 0 0 0 0 1 d e d t (2-26) The terminal voltage equation i n d-q coordinates when li n e a r i z e d , becomes AV AV R +R t e X +X t e -X -X t e R +R t e A i , d Ai q , : ( xt + V + ~ — : — P A i Ai + AV cd AV cq + V cos 6 o o -V s i n 6 o o A<S (2 where V ^ and V are the d-q components of the voltage across the capaci- tor, and V q the i n f i n i t e bus voltage. The zero component equation i s or- thogonal to the other two equations and i s usually neglected except for asymmetric loading. The capacitor equations may be written After transformation, i t becomes (2-28) a,b, c [I1d,q,o " [ C ] ^ [ V c ] + [ C ] [ T ] _ 1 ^ [ T ] [ V c ] < 2" 2 9 ) d,q,o d,q,o which when li n e a r i z e d , gives AI AI to X 1 o c AV cd AV cq X -AV cq AV cd (2-30) 14. 2.5 The Exciter and Voltage Regulator The exciter and voltage regulator model i n th i s thesis i s based on an IEEE committee report [18] with some s i m p l i f i c a t i o n . The regulator input f i l t e r time constant, the saturation function and the s t a b i l i z i n g feedback loop are neglected. 1 + ST, R 1 + STT 4 E m a FD u; E voltage regulator exciter Fig. 2-8 Exciter and Voltage Regulator Model In Fig. 2-8, V i s the generator terminal voltage, . . U"E the supplementary control, the voltage regulator gain, T^ i t s time constant, T £ the exciter time constant and E p D a per unit output voltage of the exciter. Although the voltage l i m i t s are shown i n the figure, they w i l l be neglected i n linear analysis. Mathematically we have K. K, . p AV„ = AV + —— u - -=• AV„ v R T. t T. T. R A A A (2-31) P A EFD = T l A VR " TT A EFD hi Ci (2-32) where the li n e a r i z e d terminal voltage AV V, V AV. + =32. A V V_ d V̂ _ q (2-33) to to 15. Substituting \AV','AV;f rom equation (2-27) into equation (2-33) and the results into equation (2-31), we have V doWV . . , W W Â̂do . P A VR " V T. o)Q P A l d + V, TAao)o p q T X T * V c d to A o to A o A to K.V K + m T T q° AV + m „ [V, (R +R ) + V (X +X ) ] A i T,V cq T. V t L do v t e qo t e' d A to A to ^ + i V ^ d o W> + V qo<W ] M q + Y7 "\ A to A A K V + m ° [V, cos6 - V sinS ] AS (2-34) T.V L do o qo o A to ^ Fig. 2-9 shows a supplementary excitation control of the lead- lag compensation type [19]. Aoi K ST s b 1 + ST X c 1 + ST X 1 + ST 1 + ST y 1 + ST y Fig. 2-9 A supplementary excitation control Mathematically, K s T pAd) - T p b = b (2-35) T p . b - T p c = \c - b (2-36) x y -T p u + T p c = \u - • . c (2-37) y x "~ 16. 2.6 State Equations for the Complete System The component system equations previously derived can be combined into a single set of state equations i n the form of X = [A] X (2-38) where X i s the state variable vector and [A] the system matrix. Equation (2-38) can be conveniently partitioned X. II A I , I | A I , II A. II, I II,II X II (2-39) where X^ contains the state variables of the mechanical system and X ^ those of the electrical system; namely Xj. = [ V e i r o ) , 6, v e 3 , v e 4 , y e 5 , v e f i,- a, P G V , T H P , T I P , T L P L ] X I I = [ ± d ' \> ± f V V ±G' Vcd' Vcq' V V A_ represents the coupling between the two systems where the interaction occurs through the e l e c t r i c a l torque Since T = ( i K i - ib i,) e d q q d per unit (2-40) AT ={ (ij;, A i + i A^j - ib A i , - i , Aib >}/ u> e Ydo q qo T d Tqo d do Tq o = { (X - X , ) i A i , + [(X - X , ) i , + X ,i| ] Ai + i X ,Ai^ 1 q d qo d q d do ad fo q qo ad f + i X , Ai - i , X Ai - i , X Ai } /j oo (2-41) qo ad D do aq Q do aq G o Next, the partitioned matrices A.̂  ^ and A^ ^ of the e l e c t r i c a l system s h a l l be f i r s t assembled i n the form of B Xfl - C1X1 + CZ1 XIT (2-42) 17. Then we have Ki - B _ 1 c i x i + B _ 1 c u x n ( 2 - 4 3 ) x n • A n , i x i + A I I , I I x n where A ^ = B ' 1 4 ; A ^ - B " 1 (2-44) Thus we have completed the derivation of the state equations for the overall system. 18. 3. EIGENVALUE ANALYSIS OF THE SSR MODEL 3.1 Introduction Eigenvalue analysis technique is useful in investigating the stability of systems. The complex eigenvalues are associated with oscil- latory modes of the system and the real! part of the eigenvalues provide the information on system damping. When an eigenvalue has a positive real part, instability of the system is indicated. In this thesis the effect of capacitor compensation, conventional stabilizer and dampers will be investigated using the data taken from the benchmark model [20] Table 3-1. 3.2 The Effect of Capacitor Compensation Fig. 3-1 and 3-2 show the eigenvalues of the system with various degrees of capacitor compensation, at a particular loading. The pair of eigenvalues corresponding to A6 and Ato of the synchronous machine have positive real parts when the compensation is 20% or less. The natural frequencies of the multi-mass torsional system are approximately 298, 203, 160, 127 and 99 radians/second which correspond to 47.4, 32.3, 25.5, 20.2, and 16 Hz respectively. By changing the degree of compensation, the nat- ural oscillation frequency of the transmission system changes. When the frequency of the electrical mode is closedto a mechanical mode, SSR may occur. At 50% and 60% compensation, two mechanical modes are excited simultaneously. 3.3 The Effect of Conventional Stabilizer Fig. 3-3 and 3-4 repeat the study of the effectoof capacitor compensation, but with the addition of a conventional stabilizer of the lead-lag type. Whereas the pair of eigenvalues corresponding to A6 and 19. Au of the synchronous machine were unstable for compensation below 30%, Fig. 3-1 shows they are substantially moved to the l e f t half of the com- plex plane with the supplementary excitation control, Fig. 3-3. However, the lowest mechanical mode of 99 rad/sec i s always excited and shifted to the right - h a l f plane. This i s i n agreement with other findings [21, 22]. The damping of other mechanical modes i s decreased s l i g h t l y . 3.4 The Effect of Loading The effect of different loading with and without s t a b i l i z e r on SSR i s shown i n F i g . 3-5 and 3-6 respectively. Most of the eigenvalues do not change except those corresponding to the generator mechanical model Generally the system becomes more unstable with more leading power factor. Most u t i l i t i e s operate their systems between 0.9 power factor lagging and unity power factor. For this reason, 0.9 power factor lagging i s chosen for the studies here. 3.5 The Effect of Dampers As reported [5] the addition of an amortisseur winding can reduce the p o s s i b i l i t y of SSR. For this investigation, the additional damper effect i s represented by decreased damper impedance. When the t o t a l re- actance of l i n e and transformer i s zero, SSR occurs. The result i s shown i n Table 3-2. The excited mode i s damped out by decreasing the damper impedance which agrees with previous results [5]. Table 3-1 Numerical Values of Model i n p.u. system Mass-spring System Parameters M l = 0.068433 K12 = 2.822 D l l = 0.1 M 2 = 1.736990 R23 = 70.858 D22 = 0.1 k 3 = 1.768430 K34 = 52.038 D33 = 0.1 M4 = 1.717340 K45 " 34.929 D44 = 0.1 M M5 = 0.311178 K56 = 19.303 D55 = 0.1 M6 = 0.185794 D66 = 0.1 Synchronous Machine Parameters X, = 1.79 a X , = 1.66 ad X = 1.71 q X = 1.58 aq R = 0.0015 a X f h XQ 1.6999 1.6657 1.6845 1.8250 R f % \ R„ = 0.00105 = 0.00371 = 0.00526 = 0.01820 Exciter and Voltage Regulator K, = 50 A = 0.002 T, = 0.01 A Transmission Line Parameters X =0.14 R = 0.02 e Governing and Turbine System R = 0.01 X varies from c X = 0.56 e 0.056 - 0.56 (10% - 100%) K =25 g T 3 = 0.3 TC0 = °'2 FLP1= °' 2 2 T 1 =0.2 TCH = °-3 FHP " °'3 FLP2= °' 2 2 RH IP 0 7.0 0.26 S t a b i l i z e r parameters K =20 s T = 3.0 T =0.125 x T = 00.05 y 21. 20% 30% 50% -0.1817 + j298.18 -0.1818 + J298.18 -0.1818 + j298.18 -0.2104 + j203.20 +0.1541 + J204.35 +0.1560 + J202.68 Shaft modes -0.2266 + j160.66 -0.2496 + J160.72 +0.9100 + J161.42 -0.6679 + J127.03 -0.6706 + J127.03 -0.6799 + J127.08 -0.2660 + j 99.13 -0.2877 + j 99.21 -0.3545 + j 99.79 -6.9800 + J512.30 -7.0224 + J542.80 -7.0800 + J591.15 Stator/Network -6.0717 + J241.01 -6.1984 + j209.20 -6.8387 + J161.47 -8.5681 -8.4404 -8.1277 Synchronous -31.578 -31.920 -32.808 Machine Rotor -25.397 -25.404 -25.423 -2.0196 -1.9830 -1.9070 Exciter and -499.98 -499.97 -499.97 Voltage Regulator -101.97 -101.91 -101.76 X & a) +0.0415 + J8.0234 -0.0479 + J8.4801 -0.2674 + J9.5459 -0.1416 -0.1417 -0.1418 Turbine and -4.6679 -4.6160 -4.0496 Governor -2.9271 -3.0336 -3.3335 -4.7039 + jO.7567 -4.6732 + J0.6269 -4.7939 + J0.3198 Table 3.2 Eigenvalues of SSR model at different degrees of capacitor compensation without conventional s t a b i l i z e r for P = 0.9 p.u. at 0.9 power factor lagging. 70D JFD "R - 6 0 0 -100-70 cd cq 5 3 2 3 4 rzSJ I I " 20-10 -9 -8 ^3> e3 to, 6 f-600 500 400 •300 250 h!50 100 50 r 25 '87 15 T T HP IP t- 5 1 l L P l -1 • I " ' I " 1 2 N 3 F i g . 3-1 The e f f e c t of capacitor compensation without s t a b i l i z e r f o r P = 0.9 p.u. at 0.9 power factor lagging. (The symbols 1,2,3,...9 respectively correspond to 10,20,30,...90% compensation) -300 -250 V ,,V cd cq 2 1 !S|§0 4 014,6^ 4 3 a aa i 6 •150 2 98 <S3>'e3 0),5 B 7 6 5 4321 5 • 4 3 -1.0 -o.a -0.6 -0.4 -0.2 100 h 50 2 1 • i • • •»i • • • • i • • • • i 1 2 3 4 5 Fig. 3-2 Enlarged portion of Fig. 3-1. (The symbols 1,2,3,...9 respectively correspond to 10,20,30,...90% compensation) IjO 700 EFD VR rB • • • g r— -600 -100-70 Q -3H i d ' 1 J H P ' T I P -20-10 -8 -S -4 -3 •1 0 "I Ml" 2 3 r TT r r 4 Fig. 3=3' The'effect of capacitor compensation-with stabilizer for P = 0.9 p.u. at 0.9 power factor lagging. (The symbols 1,2,3...9 respectively correspond to 10,20,30...90% compensation) S3 cd cq . c,u 1 2 3 4 5 67B9 9 8 7 6 5 4-321 6 ,td p-rrr - 1 0 - a - 6 u 6 » 6 6 "5 , e 5 2 1 !Sg, 0 3} Wit, o it 2 4 3 7 m 5 t&3 ,63 -4 -1.4 -1.2 -1.0 -0.B -0.6 -0.4 -0.2 •300 250 6 150 6 7 h 50 1 • 1 • 1 1 " 1 • 1" " 1" •' 1 1 2 3 4 5 F i g . 3-4 Enlarged portion of F i g . 3-3 (The symbols 1,2,3,...9 respectively correspond to 10,20,30...90% compensation) •700 - Q H » B - | d q V c d ' V c q a,P obit, 0^ 0 ) 1 , 0 1 ^ 3 . 0 3 o) > 6 CD G V & T H P ' T I P CD F-500 ' F-MO -250 •ISO 100 so zs IS LLP1 -20 -10 0 I M . . X . . . | J . . . M . . H J l j l . * . . i . j ; . a 6 J D i » i - 4 ' - 3 TTCTT ' 1 i " 5 The e f f e c t of loading without s t a b i l i z e r /.* ... (The symbols© = 0.8 p.f. leading, A= 0.9 p.f. leading, +.= unity power fa c t o r X = 0.9 p.f. lagging, <J> = 0.8 p.f. lagging) JFLD R |B' • -HJO -100 -70 d q Vcd' Vcq OA JO-IS c,u 6 ,co X o F SOO Pap L 06» 66 . 0 5 . 6 5 ^ 0)4, 0^ co 1 > 8 j B " 3 . 6 3 , -250 HlSO 0 *- SO C-2S 5 r 1 a'PGV + THP' TIP XD b ' T L P l 1 i O 111 »n 11 -g - a »j"KlyiD|"i' S g n P( i i 8 m i i | i i i i i i i i i ^ n i i i i i i i ^ i i Fig. 3-6 The effect of loading with stabilizer (The symbols CD = 0.8 p.f. leading, A = OV-9 >.f. leading, + =•'unity power factor X = 0.9 p.f. lagging, 0 = 0.8 p.f. lagging) 28. o r i g i n a l system damper impedance x 0.6 damper impedance x 1.5 -0.1818 ± J298.18 -0.1818 ± J298.18 -0.1818 ± J298.18 -0.0288 ± J202.87 -0.0296 ± J202.87 -0.0278 ± J202.87 Shaft modes -0.1536 ± J160.52 -0.1543 ± j160.52 -0.1528 ± J160.52 -0.6521 ± J126.98 -0.6522 ± J126.98 -0.6518 ± J126.98 -0.0238 ± j 98.47 -0.0285 ± j 98.50 -0.0163 i 3 98.42 Stator/Network -7.1913 ± J715.78 +0.3604 ± j 37.21 -7.1841 ± J718.50 -0.2149 ± j 30.47 -7.1873 ± J712.71 +0.9064 ± j 42.09 -5.8109 -5.1370 -7.2552 Synchronous -43.39 -31.63 -55.88 Machine Rotor -25.60 -25.51 -25.72 -0.5040 -0.4187 -0.5597 Exciter and -499.96 -499.98 -499.94 Voltage Regulator -100.68 -100.43 -100.95 A 6 co -3.8086 ± j 20.07 -4.1469 ± j 23.81 -3.7030 ± j 18.24 -0.1406 -0.1404 -0.1407 Turbine -4.1296 -4.8816 -3.8373 Governor -3.1202 -3.0225 -3.1684 -4.5440 ± J0.1525 -3.7438 ± J-.5651 -4.7916 ± j-,2688 Table 3. 3 Effect of Damper Winding i n zero t o t a l reactance and P = 0.9 p.u. at 0.9 power factor lagging. 29. 4. MULTI-MODE TORSIONAL OSCILLATIONS STABILIZATION WITH LINEAR OPTIMAL CONTROL Linear optimal control theory has been applied to the stabilizer design of power systems [23,8,24]. For practical applications, the state variables used in the design must be measurable. Another problem of opti- mal control design is the choice of the weighting matirices Q and R in the cost index. A simple procedure was proposed [8] which requires thesstate equations in the canonical form. The Q/R ratio in the procedure can be judiciously chosen. 4.1 State Equations With Measurable Variables The state equations of the system was written in Chapter 2 in the form X = A X + Bu (4-1) where A was given as (2-39). For an excitation control, K t B = [0 0 0 ... 0 -^] r (4-2) A as in (2-31). Let Z = M X (4-3) where Z is the measurable variable vector. Then Z = MX = MA M _ 1 Z + M B u or Z = F Z + G u (4-4) where F = M A M_1 and G = M B Assume that a l l mechanical system variables, such-,as angles and speeds of every turbine rotor and that of generator and exciter; torques output from each stage of turbine and governor system and the electrical system vari- ables such as generator power and current, voltage across the capacitor and to ground (generator side), damper currents and voltage output from voltage regulator and exciter j are-measurable,, then we have 30. For electrical power AP = V_, Ai + V Ai + i , AV, + i AV (4-5) do d qo q do d qo q By substituting V^, V^ from equation (2-25) AP = in,. Ai, +m._ Ai + m. . AV , + m. _ AV + nx., A6 (4-6) 11 d 12 q 14 cd 15 cq 16 jre m._ = Vj + (R +R ) i , + (X +Xji 11 do e t do e t qo whei ix m12 = Vqo " ^ e + V ^ o + ( R e + R t ) : L qo 14 do rn,- = i 15 qo m = V [ i , cos 6 - i sin 6] 16 o do qo For terminal current i , i A i = _do j o A . . ( 4 _ 7 ) or M t = m21 M d + m22 A ± q i x do _ qo where m = — , m - ^ to to For voltage across the capacitor or A V = l£do Icqo A V ( 4 _ 8 ) c t Vco C d Vco C q AV = m..AAV , + rn,- AV c 44 cd 45 cq V , V cdo _ cqo where m,, = ^ — , ^ - y co co For voltage at the terminal of the capacitor. As shown in Fig. 2-7, V is the voltage at the generator side of the capacitor with respect to ground 31. AV ctdo AV , + ctqo AV ct V 'ctd V . 'ctq cto cto (4-9) where V and V are the d, q components of V and can be expressed i n ctd ctq ct terms of the voltage across the capacitor and i n f i n i t e bus voltages. AV , = AV , + V cos 6 A 6 ctd cd o (4-10) or AV = AV - V s i n 6 A 6 ctq cq o AV = mc. AV , + m c c AV + m A <S ct 54 cd 55 cq 56 (4-11) where m 54 V ctdo v , cto m 55 V ^ cto mc, = [V cos 6 - V ^ s i n 6] — 56 ctdo ctqo V cto Besides m,,, m„„, m.. and m r c, the other main diagonal elements 11 22 ra44 55 are unity. Other off-diagonal elements are zero except those already derived. 4.2 State Equations i n Canonical Form A design procedure has been developed u t i l i s i n g state equations i n canonical form [8]: 0 1 0 - a 1 - a 2 Let Z = T Y 0 1. 0; -a .-a n-1 n (4-12) (4-13) 32. we s h a l l have Y = T " 1 Z = F Y + G U (4-14) o o where F = T _ 1 F T (4-15) o and G = T G = [ 0 0 0 ... 1 ] ' (4-16) o The transformation matrix T can be found as follows: Since the eigenvalues remain unchanged with s i m i l a r i t y transformation, we s h a l l have | AI - F | = Q-A^ (A-A2) ... (\-*n) = 0 (4-17) and IAI - F I = A n + d A n _ 1 + a ,A n~ 2 + ... + a, = U0 (4-18) .i o1 n n-1 1 where A^, X^, ... A n are the eigenvalues of the system. The a's can be determined from (4-16) and (4=17), n a = - I X. 1=1 a .. = A.. A„ + A,A„ + ... + A„A„ + ... + A J n-1 1 2 1 3 2 3 n-1 n ot „ = — A A _ A „ — A A. A „ A . — ... — A _ A .A n-2 1 2 3 1 2 4 n-2 n-1 n n a, = ( - l ) n ir A. (4-19) 1 i - l " 1 Let the transformation T matrix be written as T = [ T R T 2 , T 3 , . . . T q ] (4-20) where T., , T 0 , T - , . . . T are the column vectors of T matrix. From (4-15) 1 2 3 n and (4-20), we have T F = F T ( o 33. or [ T r T 2, T 3 > ... T Q ] F Q = F [ T 1 5 1^ ?y ... T j (4-21) From (4-16) and (4-21), we have T = G (4-22) n Hence, we can compute T^, T 2, 1^ ... by using the following recursive formula T . = FT • i - i + A M I G i ==1, 2, 3, ... n-1 (4-23) n - i n-i+1 n-i+1 and F T : + = 0 (4-24) The condition of (4-24) may not be met due to the accumulated computation errors. Let T^ T 2 > T 3, ... T^ be the computed results and T^ T 2 > T 3 > ... T be correct values n F T 1 + a G = 0 F T + o 1 G = ee (4-25) and the error T - T1 = n ± (4-26) Then = - F - 1 e (4-27) Si m i l a r l y , F T 2 + a 2 G = T x F T 2 + a 2 G = T1 (4-28) r - l n 2 = r Therefore, and n 2 = F - 1 ^ (4-29) n. = F _ 1 n. . i = 2, 3, 4, ... n (4-30) l i - l A. T. = n. + T. l i l 34. 4.3 Linear Optimal Control Design The system equations i n canonical form were Y = F Y + G U (4-14) o o The chara c t e r i s t i c equation of the open loop system i s IXI - F I = Xn + ct A n _ 1 + a . XU~2 + . . . + a . • (4-31) 1 o1 n n-1 1 •\ y\ ^ Let the desired eigenvalues of the closed-loop system be A^, X^, A^, ... X^ The new ch a r a c t e r i s t i c equation w i l l be ( A - A . ) (A - A ) (A - A J ... (A - A ) 1 z - - 3 n = A n + a A11"1 + a . A n" 2 + . . . + a, = 0 (4-32) n n-1 1 Since char a c t e r i s t i c equation of the closed loop system i s |AI - (F - G S )| 1 o o o 1 = A n + (a + 3 ) A n - 1 + (a _ + 3 J A n~ 2 + ... + (a, + B j n n n-x n-x X l = 0 (4-33) where U = -S Q Y (4-34) Equating (4-32) to (4-33) gives a. - a. = 3. i = 1, 2, 3, ... n (4-35) X X X and SN = [B-, 3 , , 3o, ••• 3 J (4-36) o 1 2. 3J> n F i n a l l y , the l i n e a r optimal controller i n measurable state variables U I -S Y o or U = -S T"1 Z (4-37) o 35. 4.4 S t a b i l i z a t i o n of SSR Because of the number of state variables which can be measured, the 22nd order and 19th reduced order models are used for the li n e a r optimal control design. Eigenvalue analysis shows that a l l the important mechanical and e l e c t r i c a l eigenvalues are e s s e n t i a l l y unchanged; Table 4-1. Single mechanical mode s t a b i l i z a t i o n At 30% compensation and 0.9 power factor lagging of the reduced 22nd order system without s t a b i l i z e r , the 204 rad./sec. or 32.5 hertz mechanical mode i s excited and has negative damping (eigenvalues with positive r e a l p art). By u t i l i z i n g the design procedure described i n th i s chapter, an optimal controller can be designed to s h i f t the eigenvalues from +0.1541 ± j204.35 to -6.500 ± J204.35 and another mechanical mode which i s barely stable from -0.08805 ± j8.4938 to -6.000 ± j8.4938. A l l eigenvalues are s t a b i l i z e d as shown i n Table 4-2. The controller i s 7 . " (7.823Aco1, 0.0964AG1, -183.005Aco, -8.801A6, 192.487Aco3, -3.398A63, 65 .534AC0-, 7.448A6.v -1.336A6,-, -47.478Aco^, -2.738A6,, -29.746AP, 4 4 5 6 6 28.645A1 , 1 . 4 5 4 A i £ , 1 .475A1 , -5.931Ai„, -5.929Ai„, -60.094AV , t f D Q G c 35.987AV , -0.000260AV,,, -0.00259AE 1 7J1 c t ' R FD S t a b i l i z a t i o n of two mechanical modes simultaneously For the same system but with 50% compensation, two mechanical modes were excited +0.1560 ± j202.68 and +0.9101 ± j161.42 were excited simultaneously. Another optimal controller i s designed to s h i f t the two mechanical modes to -6.500 ± J202.68 and -3.500 ± J161.42 as shown i n Table 4-3. The controller i s (1.124Aco1, -4.959^^ -23.848Aio, 189.462A6, 13.843Aco3> -321 .30lAe 3 , 24.286Aco4, 147.568A64, -6.018Aa)5> 4.467A05, -9.927Au6, -25.064AG&, -26.719AP, 29.533Ai f c, -1.030Ai f, -1.007Ai D, - 6 . 7 1 l A i n , -6.712Ai„, -22.422AV , 31.169AV , -0.000278AVR, -0.00264AE„n). 36. Low Order Stabilization Design Although the two controllers designed by the procedure presented in this chapter have been proved to be effective in stabilizing the system, the damper currents are not directly measurable. S t i l l another linear optimal controller is designed for the system, without.the need for damper currents. The equations associated with the damper windings are dropped, resulting in a 19th order system. The controller is (1.84Aiô , l.OlAO^, -41.51Au), -30.63AS, 54.5lAco3, 39.93A63, 7.37Au4, -5.77A04> -6.41AOJ5> -3.32A05, -7!46Au>6, -2.16A06, -1.66AP, 2.63Ait, -0.872Aif, -2.26AVc, -2.35AV. , -0.000295AV̂ , -0.00274AEO, and the eigenvalues of the system tc K £D with and without the controller are shown in Table 4-4. Finally the con- troller is tested on the original system for various degrees of compensa- tion. The results are plotted in Fig. 4-1. It is found that the con- troller designed for the 19th order model with 30% compensation, not only can stabilize the original 27th order system for 30% compensation but also can stabilize the original system from 10 to 70% compensation. This proves the effectiveness of such controller design in wide-range- compensation multi-mode SSR stabilization. 37. o r i g i n a l system reduced 22nd model reduced 19th model -0.1818 ± J298.18 -0.1818 ± j298.18 -0.1818 ± J298.18 +0.1541 ± j204.35 +0.1541 ± J204.35 -0.2290 ± J203.22 Shaft modes -0.2496 ± J160.72 -0.2496 ± J160.72 -0.2273 ± J160.66 -0.6706 ± J127.03 -0.6706 ± j127.03 -0.6677 ± J127.03 -0.2877 ± j 99.21 -0.2877 ± j 99.21 -0.2627 ± j 99.14 X <5 co -0.0479 ± J8.4801 -0.0881 ± J8.4938 -0.2266 ± j7.9054 Stator/Network -7.0224 ± J542.80 -6.1984 ± j209.20 -7.0224 ± J542.80 -6.1984 ± j209.20 -4.8208 ± j514.02 -3.6580 ± J238.75 -8.4404 -8.4858 -8.0056 Synchronous -31.920 -31.920 Machine Rotor -25.404 -1.9830 -25.404 -2.1855 Exciter and -499.97 -499.97 -499.52 Voltage Regulator -101.91 -101.91 -93.682 -0.1417 Turbine and -4.6160 Governor -3.0336 -4.6732 ± jO.6269 Table 4-1 Eigenvalues of o r i g i n a l system and reduced order models without controller at 30% compensation and P = 0.9 p.u. at 0.9 power factor lagging. 38. reduced 22nd order model without controller reduced 22nd order model with controller o r i g i n a l system with controller -0.1818 ±:j298.18 -0.1818 ± J298.18 -0.1818 ± j298.18 +0.1541 ± j204.35 -6.5000 ± J204.35 -6.5000 ± J204.35 Shaft modes -0.2496 ± J160.72 -3.5000 ± J160.72 -3.5000 ± J160.72 -0.6706 ± J127.03 -0.6706 ± J127.03 -0.6706 ± J127.03 -0.2877 ± j 99.21 -0.2877 ± j 99.21 -0.2877 ± j 99.21 X 6 to -0.0881 ± j8.4938 -6.0000 ± J8.4938 -6.2367 ± j8.4158 Stator/Network -7.0224 ± j542.80 -6.1984 ± j209.20 -7.0224 ± J542.80 -6.1984 ± J209.20 -7.0224 ± J542.80 -6.1984 ± j209.20 -8.4858 -8.4858 -9.6038 Synchronous -31.920 -31.920 -31.923 Machine Rotor -25.404 -25.404 -25.404 -2.1855 -2.1855 -1.5570 Exciter and -499.97 -499.97 -499.97 Voltage Regulator -101.91 -200.00 -200.00 -0M404 Turbine and -4.8741 Governor -2.8538 -3.9883 ± J2.9898 Table 4-2 Eigenvalues of reduced 22nd order model with/without controller and o r i g i n a l system with the controller at 30% compensation and P =0.9 p.u. at 0.9 power factor lagging. 39. reduced 22nd order model without c o n t r o l l e r reduced 22nd order model with c o n t r o l l e r o r i g i n a l system with c o n t r o l l e r -0.1818 ± j298.18 -0.1818 ± J298.18 -0.1818 ± J298.18 +0.1560 ± J202.68 -6.5000 ± J202.68 -6.5000 ± J202.68 Shaft modes +0.9101 ± j161.42 -3.5000 ± j161.42 -3.5000 ± J161.42 -0.6799 ± J127.08 -0.6799 ± J127.08 -0.6799 ± j127.08 -0.3545 ± j 99.49 -0.3545 ± j 99.49 -0.3545 ± j 99.49 A 6 co -0.2958 ± J9.5621 -6.0000 ± J9.5621 -6.4682 ± J9.7544 Stator/Network -7.0799 ± J591.16 -6.8387 ± J161.47 -7.0799 ± J591.15 -6.8387 ± J161.47 -7.0799 ± J591.15 -6.8387 ± J161.47 -8.1783 -8.1783 -10.953 Synchronous -32.808 -32.808 -32.793 Machine Rotor -25.423 -25.423 -25.424 -2.1030 -2.1030 -1.0891 E x c i t e r and -499.97 -499.97 -499.97 Voltage Regulator -101.76 -200.00 -200.00 -0.1384 Turbine and -4.8909 Governor -3.1454 -2.9767 ± J4.0177 Table 4-3 Eigenvalues of reduced 22nd order model with/without c o n t r o l l e r and o r i g i n a l system with the c o n t r o l l e r at 50% compensation and P =0.9 p.u. at 0.9 power factor lagging. 40. reduced 19th reduced 19th o r i g i n a l order model order model system without with with controller controller controller -0.1818 + J298.18 -0.1817 + J298.18 -0.1818 + J298.18 -0.2290 + J203.22 -6.5000 + J203.22 -0.4968 + J203.19 Shaft modes -0.2273 + jl60v66 -3.5000 + j160.66 -0.2790 + J160.52 -0.6677 + j127.03 -0.6676 + j127.03 -0.6697 + J127.04 -0.2627 + j 99.14 -0.2628 + j 99.14 -0.2770 + j 99.22 A 6 oo -0.2266 + j7.9054 -6.0000 + j7.9054 -0.1092 + j8.7874 -4.8208 + J514.02 -4.8208 + J514.02 -7.1255 + J542.54 Stator/Network -3.6580 + j238.75 -3.6582 + J238.75 -5.9600 + J209.43 -8.0056 -8.0056 -25.4025 Synchronous -2.8136 + J0.2572 Machine Rotor -2.8136 - J0.2572 -1.6473 + J391.91 Exciter and -499.52 -499.52 -773.59 Voltage Regulator -93.682 -200.00 -1.6473-,— J391.91 -0.1401 Turbine and -4.6414 Governor -0.2592 -4.7914 ± J0.9552 Table 4-4 Eigenvalues of reduced order model with/without controller and o r i g i n a l system with the controller at 30% compensation and P = 0.9 p.u. at 0.9 power factor lagging. 7D0 4 3 2 cd cq 4 5 6 7 w6, 6 u5, 65 % ° 4 co co, 6 £-500 500 t-400 •300 [-250 32S-2D0 1,°1 u3, 63 s •150 I 50 25 15 ^300- TO a,P -20-10 TrrrTT -9 i i i | i i i ir I I u | i i -7 -6 HP TfrrJJ; 11 i i | i n -S - 4 t- 5 r- 1 ^ !. j | 111 iTrfr 5?!' ' ' '"' l ' 0 1 -3 -2 -1 3 4 4-1-The effect of capacitor compensation with controller for P = 0.9 p.u. at 0.9 power (SHy^S^S....? respectively correspond to 10,20,30,.. . 70% compensation) 42. 5. CONCLUSIONS A high-order power system model for subsynchronous resonance studies i s developed. The model includes mass-spring system, synchronous machine, series capacitor compensated transmission l i n e s , turbines and governor, voltage regulator and exciter. The transient terms p^^ and p i p q are included. From eigenvalue analysis, i t i s found that by changing the degree of compensation the frequency of the e l e c t r i c a l mode w i l l be changed and that, i n some cases, even more than one mechanical mode can be excited at the same time. When a conventional lead-lag supplementary excitation control for the s t a b i l i z a t i o n of small o s c i l l a t i o n s i s included, i t has an adverse effect on the other mechanical modes close to the small o s c i l l a t i o n mode. Such finding i s i n agreement with other previous work [22], When the damper impedance i s decreased, i t does reduce the possi- b i l i t y of SSR under i d e a l conditions [5]. Linear optimal coritrollers-.h^^d:/upon--aa-'-earlier-.-:develpped method [8] are designed. Two controllers are designed with a reduced 22nd order model and one with a reduced 19th order model and the l a t t e r con- t r o l l e r not only can s t a b i l i z e the o r i g i n a l 27th order system for 30% compensation, but also can s t a b i l i z e the system for wide-range compensa- tion and multi-mode SSR. 43. REFERENCES [1] IEEE Task Force, "Analysis and Control of Subsynchronous Resonance", IEEE Publication 76CH1066-0-PWR, IEEE, New York, 1976. [2] L.A. Kilgore, L.C E l l i o t t and E.R. Taylor, "The Prediction and Control of Self-Excited O s c i l l a t i o n s due to Series Capacitors i n Power Systems", IEEE Transactions on Power Apparatus and Systems, Vol. PAS 90, pp. 1305-1311, May/June 1971. [3] M.C. H a l l and D.A. Hodges, "Experience with 500 KV Subsynchronous Resonance and Resulting Turbine Generator Shaft Damage at Mohave Generating Station", IEEE Publication 76CH1066-0-PWR, pp. 22-25, 1976. [4] IEEE Committee, "Proposed Terms and Definitions for Subsynchronous Resonance i n Series Compensated Transmission Systems", IEEE Publica- tion 76CH1066-0-PWR, pp.55-58, 1976. [5] R.G. Farmer, A.L. Schwalb and E l i Katz, "Navajo Project Report on Subsynchronous Resonance Analysis and Solutions", IEEE Publication 76CH1066-0-PWR, pp.55-58, 1976. [6] L.A. Kilgore, D.G. Ramey and M.C. H a l l , "Simplified Transmission and Generation Station System Analysis Procedures for Subsynchronous Resonance Problems", IEEE Publication 76CH1066-0-PWR, pp.6-11, 1976. [7] Colin E.J. Bowler and Donald N. Ewart, "Self-Excited Torsional Fre- quency Osc i l l a t i o n s with Series Capacitors", IEEE Trans, on PAS, Vol. PAS 92, pp. 1688-1695, Sept./Oct. 1973. [8] B. Habibullah and Yao-Nan Yu, "Physical Realizable Wide Power Range Optimal Controllers for Power Systems", IEEE Trans, on PAS, Vol. 93, pp. 1498-1506, Sept./Oct. 1974. [9] IEEE Task Force, "Symposium on Adequacy and Philosophy of Modelling: Dynamic System Performance", IEEE Publication 75CH0970-4-PWR, IEEE, New York, 1975. [10] F.P. deMello, "Power System Dynamics - overview", IEEE Publication 75CH0970-4-PWR, pp. 5-15, 1975. [11] Charles Concordia and Richard P. Schulz, "Appropriate Component Representation for the Simulation of Power System Dynamics", IEEE Publication 75CH0970-4-PWR, pp. 16-23, 1975. [12] P.L. Dandeno, " P r a c t i c a l Application of Eigenvalues Techniques i n the Analysis of Power System Dynamic S t a b i l i t y Problems", Canadian E l e c t r i c a l Engineering Journal, Vol. 1, No. 1, pp. 35-46, 1976. [13] W.A. Tuplin, "Torsional Vibration", Pitman, England, 1966. [14] IEEE Committee Report, "Dynamic Models for Steam and Hydro Turbines i n Power System Studies", IEEE Trans, on PAS, Vol. PAS 92, pp. 1904- 1915, Nov./Dec. 1973. 44. [15] W. Janischewskyj and P. Kunder, "Simulation of the Non-Linear Dynamic Response of the Interconnected Synchronous Machines", IEEE frans^con PAS, Vol. PAS 91, pp. 2064-2077, Sept./Oct. 1972. [16] P.L. Dandeno, P. Kunder and R.P. Schulz, "Recent Trends and Progress in Synchronous Machine Modeling in the Electric Utility Industry", Proceeding of IEEE, July 1974. [17] E.W. Kimbark, "Power System Stability", (Vol. I l l , pp. 57-60), Wiley, New York, 1956. [18] IEEE Committee Report, "Computer Respresentation of Excitation Systems", IEEE Trans. PAS, Vol. PAS 87, pp. 1460-1470, June/July 1968. [19] F.P. deMello and C. Concordia, "Concepts of Synchronous Machine Stability as Affected by Excitation Control", IEEE Trans. PAS, Vol. PAS 88, pp. 316-329, Mar./Apr. 1969. [20] IEEE Task Force, "First Benchmark Model for Computer Simulation of Subsynchronous Resonance", IEEE Publication F 77 102-7. [21] W. Watson and M.E. Coultes, "Static Exciter Stabilizing Signals on Large Generators - Mechanical Problems", IEEE Trans. PAS, Vol. 92, pp. 204-211, Jan./Feb. 1973. [22] V.M1.' Raina, W.J. Wilson and J.H. Anderson, "The Control of Rotor Torsional Oscillations Excited by Supplementary Exciter Stabiliza- tion", IEEE Publication A76 457-2. [23] H.A.M. Moussa and Y.N. Yu, "Optimal Power System Stabilization Trough Excitation and/or Governor Control", IEEE Trans. PAS, Vol. PAS 91, pp. 1166-1174, May/June 1972. [24] Y.N. Yu, K. Vongsuriya and L.N. Wedman, "Application of an Optimal Control Theory to a Power System", IEEE Trans. PAS, Vol. PAS 89, pp. 55-62, Jan./Feb. 1970. [25] J.A. Anderson and V.M. Raina, "Power System Excitation and Governor Design Using Optimal Control Theory", Int. Journal of Control, Vol.' 12, pp; 289-308,1972. '

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