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High order subsynchronous resonance models and multi-mode stabilization Tse, King Kui 1977-12-31

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HIGH ORDER SUBSYNCHRONOUS RESONANCE MODELS AND MULTI-MODE STABILIZATION by King Kui Tse B.Sc. (Hon.)» Northeastern University, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES in the Department of Electrical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1977 0 King Kui Tse, 1977 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Electrical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date JuTv 6. 1977. ABSTRACT Subsynchronous resonance (SSR) occurs in a series-capacitor-compensated power system when a mechanical mass-spring mode coincides with that of the electrical system. In this thesis, a complete high order model including mass-spring system, series-capacitor-compensated trans mission line, synchronous generator, turbines and governors, exciter and voltage regulator is derived. Eigenvalue analysis is used to find the effect of capacitor compensation, conventional lead-lag stabilizer, loading and dampers on SSR. Finally, controllers are designed to stabilize multi-mode subsynchronous resonance simultaneously over a wide range of capacitor compensation. ii TABLE OF CONTENTS Page ABSTRACT ii TABLE OF CONTENTS iiLIST OF TABLES v LIST OF ILLUSTRATIONS vi ACKNOWLEDGEMENT viNOMENCLATURE . . vii1. 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 System2.3 The Turbine Torques and Speed Governor 8 2.4 The Synchronous Generator 10 2.5 The Exciter and Voltage Regulator . . 14 2.6 State Equations for the Complete System 6 3. EIGENVALUE ANALYSIS OF THE SSR MODEL 18 3.1 Introduction , 18 3.2 The Effect of Capacitor Compensation I3.3 The Effect of Conventional Stabilizer 18 3.4 The Effect of Loading I9 3.5 The Effect of Dampers4. MULTI-MODE TORSIONAL OSCILLATIONS STABILIZATION WITH LINEAR OPTIMAL CONTROL 29 4.1 State Equations with Measurable Variables 29 4.2 State Equations in Canonical Form 31 4.3 Linear Optimal Control Design 4 iii Page 4.4 Stabilization of SSR . . 35 5. CONCLUSIONS 42 REFERENCES 3 iv LIST OF TABLES Table Page 3-1 Data for SSR Model 20 3-2 Eigenvalues of SSR model at different degrees of capaci tor compensation without conventional stabilizer .... 21 3- 3 Effect of damper winding for zero total reactance ... 28 4- 1 Eigenvalues of original system and reduced order models without controller at 30% compensation 37 4-2 Eigenvalues of reduced 22nd order model with/without controller and original system with the controller at 30% compensation . 38 4-3 Eigenvalues of reduced 22nd order model with/without controller and original system with the controller at 50% compensation . 39 4-4 Eigenvalues of reduced 19th order model with/without controller and original system with the controller at 30% compensation 40 v LIST OF ILLUSTRATIONS Figure Page 2-1 A functional 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 system2-4 A speed governor model for the steam turbine system . . 8 2-5 A linear model of the steam turbine system 9 2-6 A synchronous machine model .... ^ 2-7 A single line representation of the transmission line . ^ 2-8 Exciter and voltage regulator model ^ 2- 9 A supplementary excitation control ... ^ 3- 1 The effect of capacitor compensation without stabilizer. 22 3-2 Enlarged portion of Fig. 3-1 23 3-3 The effect of capacitor compensation with stabilizer . . 24 3-4 Enlarged portion of Fig. 3-3 25 3-5 The effect of loading without stabilizer 26 3- 6 The effect of loading with stabilizer 7 4- 1 The effect of capacitor compensation with controller . . 41 vi ACKNOWLEDGEMENT I wish to express my sincere gratitude to my supervisors, Dr. Y.N. Yu and Dr. M.D. Wvong, for their patience, guidance, many hours of consultation and valuzble advice during the course of the research work and writing of this thesis. Financial support from a British Columbia Telephone Company Scholarship, a University of British Columbia Summer Research Fellowship and a teaching assistantship is gratefully acknowledged. Thanks are also due to Mary Ellen Flanagan and Sannifer Louie for typing this thesis. I am grateful to my parents and Mui Ha in Hong Kong for their patient understanding and encouragement throughout my university career. ^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 inertia coefficient = 2H H inertia constant K shaft stiffness D damping Q rotor angle a) rotor speed oj synchronous speed Synchronous Machine i instantaneous value of current V instantaneous value of voltage viii ib flux-linkage R resistance X reactance 6 torque angle, rad. to angular velocity, rad./s Te electrical 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 line e e Xc reactance of capacitor V infinite 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 TrT2 actuator time constant T3 servomotor time constant a change in actuator signal PGV power at. gate outlet TCH steam chest time constant T RH reheater time constant T CO cross-over time constant ix F high pressure turbine power fraction intermediate pressure turbine power fraction F^p^ low pressure turbine 1 power fraction 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 direct- and quadrature-axis stator quantities f field circuit quantities D,Q,G direct- 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 differential operator Prefix linearized quantities differential operator differential 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 is the best alternative to the addition of transmission lines because of environmental considerations and the limited availability 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 result. Two turbine shafts were severely damaged [2] at the Mohave generating station of the Southern^ California Edison Company because of the excessive tor sional oscillations caused by interaction between the electrical reson ance of the series-capacitor-compensated system and the natural modes of the multi-mass generator turbine mechanical system. Subsynchronous resonance may occur in a system in the steady-state or transient state due to a system fault or major switching. The former may be called the steady-state subsynchronous res onance and the latter the transient subsynchronous resonance. The main v. problems are the self excitation, the torsional interaction, and the transient torques [3]. When SSR occurs, the synchronous machine is '••'self-' excited and behaves like an induction generator. If the negative resis- " tance of the machine, as an induction generator, exceeds the total resis tance of the external electrical system, self excitation of SSR occurs. Torsional oscillation is due to the mechanical modes of the multi-mass turbine-generator system. The torsional frequencies are in the subsynchronous range. If the electrical resonant frequency is equal or close to a torsional mode, the rotor oscillations and the induced voltages will build up and the interaction between the electrical and 2. mechanical systems ensues [1,4]. Transient torques are caused by system disturbances on a series-capacitor-compensated line and the energy stored in the series capacitor produces large subsynchronous currents in the lines. When the frequency of the current coincides with the natural torsional frequency, transient torque results. After the reported turbine shaft failures [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 existing system, the simplesttway to avoid the subsynchronous resonance is to reduce the degree of capacitor compensation. Another suggestion is the instal lation of passive filter units in series with the generator transformer neutral at the high voltage side. Each filter unit is a high-Q parallel resonant circuit tuned to block the subsynchronous current at a particular frequency corresponding to one of the mechanical modes. Additional amor-tisseur windings on the pole faces can reduce the effective negative res istance [5]. Supplementary excitation control is being considered and the stabilizing signals are derived from rotor speed. Finally a subsyn chronous overcurrent relay has been developed for the automatic protec tion of generating units in case of sustained subsynchronous oscillations. 1.2 Scope of the thesis The widely accepted method for subsynchronous resonance studies in engineering practice consists of a two-step analysis [5]. The electri cal and mechanical modes are determined separately. The transient elec trical torque from the electrical system is calculated first and then applied to the mechanical system as a forcing function. In this thesis, a complete model including the electrical, mechanical and control systems 3. will be developed and presented in Chapter 2. By using eigenvalue anal ysis, the effect of various degrees of compensation, loading conditions and conventional supplementary excitation control on subsynchronous res onance will be examined in Chapter 3. For broad-band frequency multi-mode subsynchronous resonance control, linear optimal controllers will be designed in Chapter 4. A summary of all important results and conclu sions will be presented in Chapter 5. 4. 2. A COMPLETE POWER SYSTEM MODEL FOR SUBSYNCHRONOUS RESONANCE STUDIES 2.1 Introduction For any dynamic or transient stability study of a power system, an accurate model of the system is required. In addition to the individual efforts [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 is presented, including steam turbines and generator multi-mass torsional system, the turbine torques and speed governor, the synchronous generator, the capacitor-compensated transmission lines, and the exciter and voltage regulator. A functional block diagram of the complete system is shown in 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, all mechanically coupled on the same shaft as shown in 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 stiffness. Each mass is denoted by a circular disc, as in Fig. 2-3, with an inertia constant M_^, a positive torsional torque K.(9.,, - 9.) on the left and a negative torque -K. .,(8. - 8. ,) on the x 1+1 1 i-l 1 i-l right. There is an external torque T^ applied to the mass inaa positive direction, an accelerating torque M^OK in the same direction and a damping torque D^. OK in the opposite direction. The net accelerating torque becomes M.u). l l = T. - D.O). + K.(e.,, - e.) - K. .(e. - e. ,) i ii i i+i i i-i i i-i (2-1) 5. th where M. = the inertia constant of i rotor 1 th 0^ = the rotational displacement for i rotor = damping coefficient for 1^ rotor th K. .... = the torsional stiffness of the shaft between the i x,x+l rotor and the i+l*"*1 rotor By applying equation (2-1) to the six mass turbine-generator system, twelve differential 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" (2-3K56 a (K45 + K56} M K45 Q °55 .. A TIP Intermedxate pa, = — ^ B +—Q (2-4) Pressure 5 5 5 5 5 p95 = w5 w° (2-5) K45 (K34 + K45) K34 °44 TLP1 Low Pressure 1 po>4 = g— 65 g- 94 + M~ 63 _ MT" W4 + ^~ (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) p63 = OJ3 OJO (2-9). K23 n (K12 + K23} ^ K12 Q D22 Te in, Generator p «o = g- 83 g- 6 + M~ °1 " " " (2"10) p 6 = OJ OJO (2-11) K12 K12 °11 Exciter po), = —— 6 - —— 6, - —— 0).. (2-12) r 1  . 1 p6^ = 0)^ OJO (2-13SR Governor 6 o) etc Steam Turbines i=3^6 K e±,u±, i=3^6 Torsional System 6,u Generator Electrical Network FD e1,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 electric torque output Te, and the exciter electric torque is neglected. Note that while angles are in radians, the speed is in p.u.; 03o = 1 p.u. = 377 electrical 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). ii 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 is based on an IEEE committee report [14]. Usually the speed is, sensed.between the low-pressure turbine and the generator rotor. Combined with the speed refer ence, the speed deviation or error signal is derived and relayed through the actuator to activate the servomotor, which in turn opens or closes the steam valves. A block diagram [14] is shown in Fig. 29-4. Foraa lin ear study, the system equations may be written; K 1 p A a = Ato - — Aa (2-14) 1 1 P APGV - |- Aa - |- APGV (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 stability studies [14]. The system consists of one high-pressure, one intermediate-pressure and two low-pressure turbines. Their output torques are denoted by T^-., TTT>, TTi,TT respectively. There is a reheater between high-rir ir Liri 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 inlet of the steam chest. The time constants of the steam chest, the reheater and the cross 's over piping are denoted by T^, T^and Tpn, respectively. F^, FTp, CO HP IP! F_ _.j and F 0 represent fractions of the total power developed in the various stages. Therefore F,. p A T HP HP CCH A PGV " A THP (2-16) "IP "HP X TRH P A TIP = F,m x~T„„A THP --T-, ATT LPl p A TLpl = „ A T RH 1 IP m x Tco" ~IP Tco ' Tlp1 (2-17) (2-18) 10. A T. PL2 LP2 A T. LP1 (2-19) 2.4 The Synchronous Generator The synchronous generator is assumed to have six windings. In addition to the d and q armature windings on the respective axes, there is a field 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 in Fig. 2-6. d-axis rqrraXlS •-fM) + v. Fig. 2-6 A synchronous machine model I'l. The voltage equations in the linear form are A V, = pAip , - OJOAIJJ — Au - R Ai q qo ad A V = vAty + OJOAIJ; , + IJJ, Ao) - R Ai q v rq rd rdo a q A Vf = pAi|jf + R A if 0 = pA^ + A iD 0 = pA*Q + RQ A iQ PA*G + RG A H where the flux linkages are Aijj. Aip. Air D _1 uo -xd 0 Xad X A ad 0 -X q 0 0 "Xad 0 Xf 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 Xr aq G Ai . Ai Ai, Ai, Ai G (2-20) (2-21) anand the tf/s, X's, R's and i's are the per unit flux linkages, reactances, resistances and currents respectively. The saturation in the iron circuit is neglected. The stator transient voltages pi^ an P^j although normally neglected in stability studies [15,16] are retained in this study because the capacitor compen sated transmission lines, to which the armature windings are connected in series, must be described by differential equations. 12. R- X„ ct Fig. 2-7 A single line representation of the transmission line In Fig. 2-7, and are d-q components of the terminal volt age is the voltage across the capacitor and V is the terminal voltage at the capacitor. The transformer is represented by a reactance X and a resistance R and the transmission line by a reactance X and a line t e resistance R . e Let the terminal voltage equations in a-b-c phase coordinates be [vt] a b c = [R] [It] + [L] dt [It] + 1Vc] ' ' a,b,c a,b,b a,b,c + [VJ 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) -sin -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 + ^df ^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 de dt (2-26) The terminal voltage equation in d-q coordinates when linearized, becomes AV AV R +R t e X +X t e -X -X t e R +R t e Ai, d Ai q , :(xt+V + ~—:— P Ai Ai + AV cd AV cq + V cos 6 o o -V sin 6 o o A<S (2 where V^ and V are the d-q components of the voltage across the capaci tor, and Vq the infinite bus voltage. The zero component equation is or thogonal to the other two equations and is usually neglected except for asymmetric loading. The capacitor equations may be written After transformation, it becomes (2-28) a,b, c [I1d,q,o " [C]^[Vc] + [C][T]_1^[T][Vc] <2"29) d,q,o d,q,o which when linearized, 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 in this thesis is based on an IEEE committee report [18] with some simplification. The regulator input filter time constant, the saturation function and the stabilizing feedback loop are neglected. 1 + ST, R 1 + STT 4Em a FD u; E voltage regulator exciter Fig. 2-8 Exciter and Voltage Regulator Model In Fig. 2-8, V is the generator terminal voltage, . . U"E the supplementary control, the voltage regulator gain, T^ its time constant, T£ the exciter time constant and EpD a per unit output voltage of the exciter. Although the voltage limits are shown in the figure, they will be neglected in 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 AEFD = Tl AVR " TT AEFD hi Ci (2-32) where the linearized terminal voltage AV V, V AV. + =32. AV V_ d V^_ 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 VdoWV . . , WW ^A^do . PAVR " V T. o)Q P Ald + V, TAao)o p q TXT * Vcd to A o to A o A to K.V K + m TTq° AV + m „ [V, (R +R ) + V (X +X ) ] Ai T,V cq T. Vt L dov t e qo t e' d A to A to ^ + iV ^do W> + Vqo<W] Mq + 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, Ks T pAd) - T p b = b (2-35) Tp.b-Tp c= \c - b (2-36x 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 in the form of X = [A] X (2-38) where X is the state variable vector and [A] the system matrix. Equation (2-38) can be conveniently partitioned X. II AI, I | AI, 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 eiro), 6, v e3, v e4, y e5, v efi,- a, PGV, THP, TIP, TLPL] XII = [±d' \> ±f V V ±G' Vcd' Vcq' V V A_ represents the coupling between the two systems where the interaction occurs through the electrical torque Since T = (iKi - ib i,) e d q q d per unit (2-40) AT ={ (ij;, Ai + i A^j - ib Ai, - i, Aib >}/ u> e Ydo q qo Td Tqo d do Tq o = { (X -X,)i Ai, + [(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 electrical system shall be first assembled in the form of B Xfl - C1X1 + CZ1 XIT (2-42) 17. Then we have Ki - B_1 ci xi+ B_1 cu xn (2-43) xn • An,i xi+ AII,II xn 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 left half of the com plex plane with the supplementary excitation control, Fig. 3-3. However, the lowest mechanical mode of 99 rad/sec is always excited and shifted to the right-half plane. This is in agreement with other findings [21, 22]. The damping of other mechanical modes is decreased slightly. 3.4 The Effect of Loading The effect of different loading with and without stabilizer on SSR is shown in Fig. 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 utilities operate their systems between 0.9 power factor lagging and unity power factor. For this reason, 0.9 power factor lagging is chosen for the studies here. 3.5 The Effect of Dampers As reported [5] the addition of an amortisseur winding can reduce the possibility of SSR. For this investigation, the additional damper effect is represented by decreased damper impedance. When the total re actance of line and transformer is zero, SSR occurs. The result is shown in Table 3-2. The excited mode is damped out by decreasing the damper impedance which agrees with previous results [5]. Table 3-1 Numerical Values of Model in p.u. system Mass-spring System Parameters Ml = 0.068433 K12 = 2.822 Dll = 0.1 M2 = 1.736990 R23 = 70.858 D22 = 0.1 k3 = 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 Xf h XQ 1.6999 1.6657 1.6845 1.8250 Rf % \ 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 T3 = 0.3 TC0 = °'2  FLP1= °'22 T1 =0.2 TCH = °-3  FHP " °'3  FLP2= °'22 RH IP 0 7.0 0.26 Stabilizer 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 stabilizer for P = 0.9 p.u. at 0.9 power factor lagging. 70D JFD "R -600 -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 lLPl -1 •I" 'I" 1 2 N3 Fig. 3-1 The effect of capacitor compensation without 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) -300 -250 V ,,V cd cq 2 1 !S|§0 4 014,6^ 43a 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 id'1 JHP'TIP -20-10 -8 -S -4 -3 •1 0 "I Ml" 2 3 rTTrr 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 -10 -a -6 u6»66 "5 ,e5 2 1 !Sg,0 3} Wit, o it 2437 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 •11"1 • 1" " 1" •' 1 1 2 3 4 5 Fig. 3-4 Enlarged portion of Fig. 3-3 (The symbols 1,2,3,...9 respectively correspond to 10,20,30...90% compensation) •700 -QH»B-| d q Vcd'Vcq a,P obit, 0^ 0)1,01 ^3.03 o) > 6 CD GV & THP'TIP CD F-500 ' F-MO -250 •ISO 100 so zs IS LLP1 -20 -10 0IM..X...|J...M..HJ ljl.*..i.j;.a6JDi»i -4 ' -3 TTCTT '1i" 5 The effect of loading without stabilizer /.* ... (The symbols© = 0.8 p.f. leading, A= 0.9 p.f. leading, +.= unity power factor 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 L06»66 .05.65^ 0)4, 0^ co 1 > 8 j B "3.63 , -250 HlSO 0 *- SO C-2S 5 r 1 a'PGV + THP'TIP XD b'TLPl 1 iO 111 »n 11 -g -a »j"KlyiD|"i' Sgn P( ii8mii|iiiiiiiii^niiiiiii^ii 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. original 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 in zero total 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 (4-3where 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 all 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 + (Re+Rt):Lqo 14 do rn,- = i 15 qo m = V [i, cos 6 - i sin 6] 16 o do qo For terminal current i, i Ai = _do jo A. . (4_7) or Mt = m21 Md + m22 A±q i x do _ qo where m = — , m -^ to to For voltage across the capacitor or AV = l£do Icqo AV (4_8) ct Vco Cd Vco Cq AV = m..AAV , + rn,- AV c 44 cd 45 cq V , V cdo _ cqo where m,, = ^— , ^ - y co cFor 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 in ctd ctq ct terms of the voltage across the capacitor and infinite bus voltages. AV , = AV , + V cos 6 A 6 ctd cd o (4-10) or AV = AV - V sin 6 A 6 ctq cq o AV = mc. AV , + mcc 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 ^ sin 6] — 56 ctdo ctqo V cto Besides m,,, m„„, m.. and mrc, 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 in Canonical Form A design procedure has been developed utilising state equations in canonical form [8]: 0 1 0 -a1 -a2 Let Z = T Y 0 1. 0; -a .-a n-1 n (4-12) (4-13) 32. we shall have Y=T"1Z=FY+GU (4-14) o o where F = T_1 F T (4-15o and G = TG = [ 000 ... 1]' (4-16) o The transformation matrix T can be found as follows: Since the eigenvalues remain unchanged with similarity transformation, we shall have | AI - F | = Q-A^ (A-A2) ... (\-*n) = 0 (4-17) and IAI - F I = An + d An_1 + a ,An~2 + ... + a, = U0 (4-18) .i o1 n n-1 1 where A^, X^, ... An 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 12 13 2 3 n-1 n ot „ = — A A _ A „ — A A. A „ A . — ... — A _ A .A n-2 12 3 12 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 = [TR T2, T3, ... Tq] (4-20) where T., , T0, 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 [Tr T2, T3> ... TQ] FQ = F[T15 1^ ?y ... Tj (4-21) From (4-16) and (4-21), we have T = G (4-22) n Hence, we can compute T^, T2, 1^ ... by using the following recursive formula T . = FT • i-i + A MI 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^ T2> T3, ... T^ be the computed results and T^ T2> T3> ... T be correct values n F T1 + a G = 0 F T + o1 G = ee (4-25) and the error T - T1 = n± (4-26) Then = -F-1 e (4-27Similarly, F T2 + a2 G = Tx F T2 + a2 G = T1 (4-28) r-l n2 = r Therefore, and n2 = F-1^ (4-29n. = F_1 n. . i = 2, 3, 4, ... n (4-30) l i-l A. T. = n. + T. l il 34. 4.3 Linear Optimal Control Design The system equations in canonical form were Y = F Y + G U (4-14) o o The characteristic equation of the open loop system is IXI - F I = Xn + ct An_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 characteristic equation will be (A-A.) (A - A ) (A - AJ ... (A - A ) 1 z - - 3 n = An + a A11"1 + a . An"2 + .. . + a, = 0 (4-32) n n-1 1 Since characteristic equation of the closed loop system is |AI - (F - G S )| 1 o o o 1 = An + (a + 3 ) An-1 + (a _ + 3 J An~2 + ... + (a, + Bj n n n-x n-x X l =0 (4-33) where U = -SQ Y (4-34) Equating (4-32) to (4-33) gives a. - a. = 3. i = 1, 2, 3, ... n (4-35) XXX and SN = [B-, 3,, 3o, ••• 3J (4-36) o 1 2. 3J> n Finally, the linear optimal controller in measurable state variables U I -S Y o or U = -S T"1 Z (4-37) o 35. 4.4 Stabilization 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 linear optimal control design. Eigenvalue analysis shows that all the important mechanical and electrical eigenvalues are essentially unchanged; Table 4-1. Single mechanical mode stabilization At 30% compensation and 0.9 power factor lagging of the reduced 22nd order system without stabilizer, the 204 rad./sec. or 32.5 hertz mechanical mode is excited and has negative damping (eigenvalues with positive real part). By utilizing the design procedure described in this chapter, an optimal controller can be designed to shift the eigenvalues from +0.1541 ± j204.35 to -6.500 ± J204.35 and another mechanical mode which is barely stable from -0.08805 ± j8.4938 to -6.000 ± j8.4938. All eigenvalues are stabilized as shown in Table 4-2. The controller is 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 , 1.454Ai£, 1.475A1 , -5.931Ai„, -5.929Ai„, -60.094AV , t f D Q G c 35.987AV , -0.000260AV,,, -0.00259AE17J1 ct' R FD Stabilization 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 is designed to shift the two mechanical modes to -6.500 ± J202.68 and -3.500 ± J161.42 as shown in Table 4-3. The controller is (1.124Aco1, -4.959^^ -23.848Aio, 189.462A6, 13.843Aco3> -321.30lAe3, 24.286Aco4, 147.568A64, -6.018Aa)5> 4.467A05, -9.927Au6, -25.064AG&, -26.719AP, 29.533Aifc, -1.030Aif, -1.007AiD, -6.71lAin, -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. Still 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 (1.84Aio^, 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. , -0.000295AV^, -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. original 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 original 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 original 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 original 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 controller reduced 22nd order model with controller original system with controller -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 Exciter 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 controller and original system with the controller at 50% compensation and P =0.9 p.u. at 0.9 power factor lagging. 40. reduced 19th reduced 19th original 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 original 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 |iii ir II u |i i -7 -6 HP TfrrJJ; 11 i i | i n -S -4 t- 5 r- 1 ^ !. j | 111 iTrfr5?!''''"' 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 is developed. The model includes mass-spring system, synchronous machine, series capacitor compensated transmission lines, turbines and governor, voltage regulator and exciter. The transient terms p^^ and pipq are included. From eigenvalue analysis, it is found that by changing the degree of compensation the frequency of the electrical mode will be changed and that, in 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 stabilization of small oscillations is included, it has an adverse effect on the other mechanical modes close to the small oscillation mode. Such finding is in agreement with other previous work [22], When the damper impedance is decreased, it does reduce the possi bility of SSR under ideal 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 latter con troller not only can stabilize the original 27th order system for 30% compensation, but also can stabilize 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 Elliott and E.R. Taylor, "The Prediction and Control of Self-Excited Oscillations due to Series Capacitors in Power Systems", IEEE Transactions on Power Apparatus and Systems, Vol. PAS 90, pp. 1305-1311, May/June 1971. [3] M.C. Hall 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. 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