EXCITATION CONTROL OF TORSIONAL OSCILLATIONS DUE TO SUBSYNCHRONOUS RESONANCE. Andrew {^<a n. B.S.E.E. University of Texas at Arlington, 1977

Transcription

1 EXCITATION CONTROL OF TORSIONAL OSCILLATIONS DUE TO SUBSYNCHRONOUS RESONANCE by Andrew {^<a n B.S.E.E. University of Texas at Arlington, 1977 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 April, Andrew Yan, 1979

2 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. ( Andrew Yan ) Department of Electrical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 n.tp 9 A? r E-6 BP I E

3 ABSTRACT Subsynchronous resonance phenomena. in a power system with series-capacitor-compensated transmission lines may cause damaging torsional oscillations in the shafts of the turbine generator. In this thesis a high-order power system model for subsynchronous resonance studies is derived. An excitation procedure for control of torsional oscillations is presented. The excitation control is of the linear optimal type synthesized from the system's output signals. Dynamic performance tests of the excitation controller on the nonlinear model show that all mode oscillations can be stabilized simultaneously within the range of dynamic stability for a wide range of capacitor compensation, but not for a severe transient fault. i i

4 TABLE OF CONTENTS ABSTRACT Page i i TABLE OF CONTENTS i i i LIST OF TABLES ',. v LIST OF ILLUSTRATIONS v i ACKNOWLEDGMENT NOMENCLATURE viii ix 1. INTRODUCTION Subsynchronous Resonance Problems and Corrective Measures... ± 1.2 Recent Work on Excitation Control of SSR Scope of the Thesis 3 2. MODELLING A POWER SYSTEM FOR SUBSYNCHRONOUS RESONANCE STUDIES Introduction Modelling the Mechanical System Modelling the Electrical System Linearized Model Reduced Order Model Eigenvalue Analysis LINEAR OPTIMAL EXCITATION CONTROL DESIGN Introduction.. ; Linear State Regulator Problem Solution to the Matrix Riccati Equation Wide Range SSR Stabilizer and the Choice of Weighting Matrices EFFECT OF EXCITATION CONTROLLER ON STEADY STATE AND TRANSIENT SSR Introduction Controller for Steady State SSR Controller for Transient SSR CONCLUSION 56 i i i

5 Page REFERENCE 57 APPENDIX I 59 APPENDIX II 60 iv

6 LIST OF TABLES Table Page 2.1 Eigenvalues of various order SSR model at 30% capacitor compensation for P=0.9 p.u. at 0.9 power factor lagging Eigenvalues of various order SSR model at 50% capacitor compensation for P=0.9 p.u. at 0.9 power factor lagging Eigenvalues of various order SSR model at 70% capacitor compensation for P=0.9 p.u. at 0.9 power factor lagging Eigenvalues of various order SSR model at 80% capacitor compensation for P=0.9 p.u. at 0.9 power factor lagging Eigenvalues of the system. with linear optimal control at various degrees of capacitor compensation for P=0.9 p.u. power factor = 0.9 lagging Shaft modes- of the system with and. without linear optimal control at various degrees of capacitor compensation for P= 1.25 p.u. at 0.9 power factor lagging Shaft modes of the system with and without linear optimal control at various degrees of capacitor compensation for P=0.5 p.u. at 0.9 power factor leading 31 v

7 LIST OF ILLUSTRATIONS Figure - Page 2.1 Functional block diagram of a power system for SSR study Model of the steam turbine system The generating unit mass-spring system Modelling of the mass-spring system in the vicinity of the i rotational mass The speed governor model The transmission system Voltage regulator and exciter model A six-winding generator model Equivalent circuit of a five-winding generator model from a six-winding model Armature current responses of the six-winding and the fivewinding generator model Q-axis damper current of the five-winding generator model Q-axis damper currents of the six-winding generator model Dynamic responses of the system without control when subjected to 10% load change Dynamic responses of the power system without control when subjected to a pulse torque disturbance Dynamic responses of the power system with control when subjected to a pulse torque disturbance Torsional oscillation of the generator-exciter shaft at 80% compensation (without control) Torsional oscillations of the generator-exciter shaft at 80% compensation (with control) Electrical network for the simulation of subsynchronous resonance Dynamic responses of the power system without control when sujected to a three-phase fault at the remote end (X =0.01 p.u.) 46 ii 4.8 Dynamic responses of the power system with control when subjected to a three-phase fault at the remote end (Xg=0.01 p.u.) 50 vi

8 Figure Page 4.9 Dynamic responses of the power system without control when subjected to a three-phase fault(x =0.06 p.u.) Dynamic response of the power system with control when subjected to a three phase fault (X =0.06 p.u.) 55 vii

9 ACKNOWLEDGEMENT I would like to express my most grateful thanks and deepest gratitude to Dr. Yao-nan Yu and Dr. M.D. Wvong, supervisors of this project, for their continued interest, encouragement and guidance during the research work and writing of this thesis. I also wish to express my deep appreciation to Mr. El-Sharkawi for his interest and timely advice. The financial support of the Natural Sciences and Engineering Research Council of Canada and the University of British Columbia is gratefully acknowledged. I am grateful to my parents, members of my family, and Mr. Frank Wong for their encouragment throughout my university career. viii

10 NOMENCLATURE General A system matrix B control matrix u x y H control vector state vector of the unmeasurable model state vector of the measurable model transformation matrix Q symmetric positive semi-definite weighting matrix R symmetric positive definite weighting matrix K Riccati matrix G closed loop system matrix M composite matrix as defined in (3.14) A,X eigenvalue matrix,eigenvector matrix of M o subscript-denoting initial condition x time derivative of x t superscript denoting transpose -1 superscript denoting inverse s differential operator A prefix denoting a linearized variable j Mass-spring complex operator, S-T system M inertia coefficient K shaft stiffness constant D damping 0 rotor angle to rotor speed CJ^ synchronous speed ix

11 Synchronous Machine i instantaneous value of current V instantaneous value of voltage T R X flux-linkage resistance reactance 6 torque angle electric torque i terminal current V terminal voltage jq^ generator output power d,q subscript denoting direct- and quadrature-axis stator quantities f subscript denoting field circuit quantities D,Q,S subscript denoting direct- and quadrature-axis damper quantities &,L subscript denoting leakage impedance in damper and stator c subscript denoting quantities associate with capacitor a subscript denoting armature phase quanities Transmission Line X,R reactance and resistance of the transformer X,R reactance and resistance of the transmission line e' e X reactance of capacitor c V infinite bus voltage o Excitor and Voltage Regulator regulator gain T. regulator time constant T exciter time constant L V reference voltage ref x

12 Governor and Steam Turbine System K actuator gain g T^,T2 actuator time constant T^ servomotor time constant a actuator signal P power at gate outlet VJV T LH steam chest time constant T reheater time constant Kri T cross-over time constant F high pressure turbine power fraction rlr F^.p intermediate pressure turbine power fraction F LPA low pressure turbine A power fraction F^pg low pressure turbine B power fraction T^p high pressure turbine torque T^p intermediate pressure turbine torque T LPA' T LPB ^ O W P r e s s u r e turbine torque xi

13 1 1. INTRODUCTION 1.1 Subsynchronous Resonance Problems and Corrective Measures The increasing electric power demand, the unavailability of generation sites at heavy load centres, and the difficulties in obtaining the right of way to build new transmission lines due to environmental considerations have frequently made long series-capacitor-compensated transmission lines necessary for transferring bulk power. However, the use of series capacitors for transmission line compensation may cause electrical system resonance and damage. These oscillations are the result of interaction between resonance phenomena in the electrical system and the mechanical oscillation of the turbine-generator mass-spring system. The hazards of these oscillations were not seriously considered until after the two shaft failures at Mohave generating station in 1970 and 1971 [ 1 ]. The term "Subsynchronous Resonance (SSR)" is used to designate the general phenomemon encompassing the oscillatory attributes of electrical and mechanical variables associated with the turbine-generators when coupled to a seriescapacitor-compensated transmission system. Despite the potentially damaging SSR effect, utilities still favor the use of series capacitors to increase the power transfer capability as an alternative to additional transmission lines and more capital investment. Hence, in order to overcome the problems due to SSR, extensive effort has been made in analyzing the two shaft failures. Problems are identified as induction generator effect, torsional interaction, and transient torques [ 2 ]. SSR phenomenon may occur in two different forms: self-excited or steady state SSR and transient SSR. Self-excited SSR involves spontaneous oscillations that are either sustained or increased in magnitude with time.

14 2 Two mechanisms that produce these oscillations are the induction generator effect where the nagative resistance of the generator at subsynchronous frequencies exceeds the resistance of the transmission system; and the bilateral coupling known as torsional interaction between the mechanical modes of the mass-spring system and the natural oscillation mode of the electrical network. Transient SSR involves the transient torques on segments of the turbine-generator shaft. These torques result from subsynchronous oscillation currents in the electrical network caused by faults or switching operations. After analyzing and identifying the SSR problems, corrective measures have been proposed [2 ], e.g., additional amortisseur windings to reduce the induction generator effect ; static high-q filters to block the subsynchronous currents at critical frequencies ; supplementary excitation control to provide additional system damping; a capacitor dual gap flashing scheme to minimize the magnitude of the transient torques in the shaft; and finally, a subsynchronous overcurrent relay to protect the generating unit in case of sustained subsynchronous oscillations. Most of these proposals have already been put into practice. 1.2 Recent Work on Excitation Control of SSR Although a perfectly tuned high-q blocking filter can provide adequate damping to the system, when detuned, the high-q filter resistance at a mode frequency is sharply reduced resulting in reduced damping performance. Detuning occurs because of ambient temperature variation, capacitor failure, and changes in system frequencies during swing conditions. In view of the shortcomings of static filter, supplemental excitation control may provide a better alternative to suppress the steady state SSR phenomon.

15 3 Saito et.al. [ 3 ] proposed a Negative Damping Stabilizer which is an excitation controller using reactive power as the control signal ; Hamdam and Hughes [ 4 ] proposed a similar type controller using reactive or real power feedback. In examining their approach in designing these controls, it is found that their aim is to suppress the electrical resonance phenomena whereas SSR is much more complicated due to the electromechanical interaction. To account for the electromechanical interaction, a unified electromechanical system model should be used in analysis and design. Fouad and Khu [ 5 ] proposed a controller using speed signal where the pick-up speed deviation will be produced solely by the mode to be controlled. Yu, Wvong and Tse [ 6 ] proposed a wide range linear optimal controller using measurable state feedback as the control signal. El-Serafi and Shaltout [ 7 ] suggested a multi-loop excitation controller by using filtered and phase shifted terminal voltage as signals. In the analog studies by Fouad and Khu, excitation voltage ceiling limits are neglected. In the last two papers [6,7], the results were not verified on the non-linear power system. Hence, the lack of nonlinear testing and voltage ceiling limits may have led to:.. optimistic conclusions' about excitation" control of " SSR Scope of the Thesis In this thesis, an excitation controller of the linear optimal type using measurable state variables for the feedback will be developed and tested on both linear and nonlinear models,[21], In Chapter 2, a unified nonlinear power system model is developed; the model is then linearized and its order is reduced to facilitate the controller design; and the eigenvalues of the system at various degrees of capacitor compensation are studied. In Chapter 3, a wide-capacitor-range linear optimal controller is designed and

16 4 its effectiveness is verified by eigenvalue analysis. In Chapter 4, the controller is further tested on the original nonlinear power system model developed in Chapter 2. The dynamic responses of the system with and without control are obtained. Finally, a summary of important findings is given in Chapter 5.

17 5 2. MODELLING A POWER SYSTEM FOR SUBSYNCHRONOUS RESONANCE STUDIES 2.1 Introduction The component representation of a power system for subsynchronous resonance studies is quite different from that of a conventional power system stability study. In order to account for the torsional resonance of the mechanical mass-spring system due to SSR, factors that are usually neglected in conventional stability studies are now important: namely the generator and turbine shaft torsional stiffnesses [8], generator rotor amortisseur parameters [9.], armature transients and network transients. In the first part of this chapter, a detailed high order mathematical model is derived. The model consists of the generator turbine mass-spring system, governor, steam turbine torques, generator, exciter, and capacitor-compensated transmission line. A functional block diagram of the model is shown in Figure 2.1. In the second part, the system equations describing the high order model are linearized about a nominal operating point and the system's eigenvalues are examined. Finally a reduced order model for further investigation is obtained. U Exciter and Voltage Regulator Governor" Steam Turbine J fd m Transmission System P Q V e, e, t Generator Electromagnetic Dynamics Mass-spring System Figure 2.1 Functional block diagram of a power system for SSR studies

18 6 2.2 Modelling the Mechanical System Steam turbine : The steam turbine system model is shown in Figure 2.2. It is a tandem compound, single reheat system. The model is mainly based on the IEEE committee report [ 10 ]. 1 + st CH 1 + st RH 1 + st CO HP IP LPA LPB T HP f T IP T LPA T LPB Figure 2.2 Model of the steam turbine system The corresponding equations are HP T CH G V T CH H P ( 2.1 ) "IP IP FHP T RH H P T IP RH ( 2.2 ) LPA LPA F T IP IP CO Tco LPA ( 2.3 ) LPB LPB F LPA L P A ( 2.4 )

19 7 Generating Unit Mass-spring System : The generating unit torsional system consists of one high pressure turbine (HP), one intermediate pressure turbine (IP), two low pressure turbines (LPA,LPB), one generator (G), and one exciter (EX) which are all mechanically coupled together by shafts as shown in Figure 2.3. To make the analysis simple, the following assumptions are made : (a) Each mass-spring element has a lumped mass of inertia constant M. (b) The mass of the shaft between any two elements is negligible and behaves like a linear torsional spring. (c) Only mechanical damping is considered.' Figure 2.4 illustrates the various torsional forces experienced by the i*"* 1 element in the system - a positive torsional torque ±+±(' ±+± 0. ) on the left, a negative torque -K.,.,.( , ) on the right, 1 ' l+l, x x x-1 ' an external torque T_^,a positive accelerating torque M^io^, and a negative damping torque ~ ± <ii ±. A general equation of motion of the i ^ rotor is as follows M.u>. = T. - D.u. + K.... ( )-K. ( ) (2.5) XX X XX 1,1+1 V 1+1 X 1,1-1 X x-1 where M. : inertia constant of the i*"^ rotor x th 6^ : rotation displacement of the i rotor K..,.. : torsional stiffness constant of the shaft between i*"* 1 and x th l+l rotor

20 Figure 2.4 Modelling of the mass-spring system in the vicinity of the i rotational mass

21 9 Applying equation (2.5) to the six mass torsional system, twelve differential equations are obtained : High pressure turbine L = K 12 M 1 2 K12 a 1 ^ T HP S7~ e i " MT w i M7" (2.6) (2.7) Intermediate u) = K 12 pressure turbine 2 M < ^ * 2 + M, D T 2 ^ IP ^ u 2 + sr (2.8) (2.9) Low pressure turbine A (D = 23 M 3 2 K 23 + K 34 K 34 D T 3, LPA (2.10) (2.11) Low pressure turbine B to. = 34 M. 3 4 r K 34 + K 45. K 45 D T 4 LPB M, 4 M. 4 4 (2.12) V w4 " % } (2.13) Generator OJ. = 45 M 5 4, K 45 + K K 56,. M 56 ( ) 6 + D c T 5 e M 5 " " M 5 (2.14) 6 = 0), ( 0) D (2.15) Exciter O J, = 56 K 6-56 M, 6 M, (2.16) V W6 " u o } (2.17) where 6 : electrical angular displacement in electrical radian which is equal, to the mechanical radian for a two pole machine. til O K : speed of the i rotor in per unit. 03- q : synchronous speed which is equal to 1 per unit. o), : base speed which is equal to 377 radian/second, b 6 : mechanical angular displacement in radian. : electric torque across the air gap in per unit,

22 10 Speed Governor: The speed governor model shown in Figure 2.5 is based on an IEEE committee report [10]. The initial power P q is the load reference which combines with the increments due to speed deviation to obtain the total power, P,,, subject to the time lag, To, introduced by the servomotor mechanism. Figure 2.5 The speed governor The corresponding state equations are P < P < P GV. - GV - GV mm max 2.3 Modelling the Electrical System Transmission System: The transmission system is represented by a single line connected to an infinite bus as shown in Figure 2.6. The resistance and reactance of the transformer are represented by R and X t respectively. The resistance and reactance of the transmission line are denoted by R and X. Voltage across the capacitor is V c, whereas V represents the terminal voltage at

23 11 the capacitor. f" R Gen }-4-^-A/"\A^~ t X t \ -/yvv ct Figure 2.6 The transmission system The general voltage equation for the system shown in Figure 2.6 is : [v],_ - r R 1 [ I 1, + [ L ] [ I ] + f V ], 1 t J phase L J 1 t J phase L J dt L t phase ' c phase (2.20) o phase All the quantities in phase coordinate are transformed into Park's coordinate [ 11 ] by a transformation matrix cos cos(9-120) cos ( ) [ T ] = -sine -sin(9-120) -sin( ) (2.21) Since balance operation is assumed, the o-component of the d-q-o coordinate is zero. Hence the equations of the transmission line are : Terminal voltage X + X V, = (. ) I, - ( X_ + X ) I + ( R + R ) I, + V d to d t e q t e d cd + V sinfi o (2.22) V = ( X t + X e ) I +(X +X ) I j + ( R +R ) I +V q q t e d t e 7 q cq + V cos6 o (2.23)

24 12 Capacitor voltage V cd = V + a) X I, cq c d (2.24) - SL_ = _ V, + co X I cd c q (2.25) Voltage Regulator and Exciter The exciter and voltage regulator model is shown in Figure 2.7. It is a continuous acting type which is based on an IEEE committee report [12] with some simplifications - the regulator input filter time constant, the saturation, and the stabilizing feedback loop are neglected. max J fd ref 1 + st R min Figure 2.7 Voltage regulator and exciter model where U is the supplementary control signal E K is the voltage regulator gain T^ is the voltage regulator time constant T is the exciter time constant E E^ D is the output voltage of the exciter V. R min and V. R max are the regulator ceiling voltage The corresponding equations are: K A V = (V - V + U ) - 1 R I, ref t V A 7 Kmin "R "R max (2.26)

25 13 E f D = -Yl\ ' -T-hv ( 2 ' 2 7 ) V = / V 2 + V 2 (2.28) t d q The synchronous generator In previous work [13], a six-winding synchronous generator model, as shown in Figure 2.8, is assumed. Since the damper windings are permanently short circuited, the question is : should the two damper windings on the Q-axis be represented by one equivalent winding? By taking the parallel equivalent of the two Q-damper leakage impedance, as shown in Figure 2.9, a five winding- model is derived. The dynamic responses of the six-winding and five-winding models when connected to. the rest of the system are shown in Figure 2.10 to Figure It is found that the armature currents of the two models are the same, and i of the five winding model is simply the sum of i and i 0 Q b of the six-winding model. In other words, the electric torque produced by the two models across the air gap is the same. Hence, a five-winding model representation is sufficient. The voltage equations of the rotor circuit are : V, = f. - uf - R i (2.29) d d q a d V + ml' - R i (2.30) q q d a q V = - R i (2.31) 0 - % " ; R D S> ( 2' 3 2 ) o - * Q - V " (2 33) Q The electric torque equation in per unit is T = i - y i, (2.34) e d q q d

26 q-axis d-axis D (TO d f Figure 2.8 A six-winding generator model d-axis r i j Si V X X mq T ql R a -AA/V q-axis i i Figure 2.9 Equivalent circuit of a five-winding generator model from a six-winding model ( solid linedamper linkage impedance of the five-winding model; dotted line-damper leakage impedances of the six-winding model)

27 Figure 2.10 Armature current response of the six-winding and five-winding generator model 15

28 Figure 2.12 Q-axis damper currents of the six-winding generator model 16

29 17 The flux linkage equation is : '- x d X md X md ± A q -X q X mq i q = md X f X md ± f (2.35) X md X md X D 1 Q J -X mq X Q.. V 2.4 Linearized Model The model derived above is nonlinear. Excluding the ceiling limits imposed on the voltage regulator and governor, the other nonlinearities are : the quadratic terms of the currents in the electric torque equation, the trigonmetric functions due to Park's transformation, and the speed voltage terms, Stability analysis, is quite complex for nonlinear systems. It is, however, expected that the stability criteria for linear systems could be applied to nonlinear systems if the deviations from the: equilibrium state are sufficiently small so that the nonlinearity has only a minor effect [ T4 ]. By neglecting the ceiling limits, the original system of equations is linearized about a nominal operating point as described in [ 15 ] which results in a set of standard state variable equations in the form of [ x ] = [ A ] [ x ] (2.36) Equation (2.36) can be partitioned into electric and mechanical subsystems as follows f X X I II Vi V I I A A II.I II,II f x I 1 X I IIJ (2.37)

30 18 where [ x^. ] are the state variables of the mechanical system [ X-J-J] are the state variables of the electrical system 2.5 Reduced Order Model For system design purposes, it is desirable to have a low order model to approximate the high order model so as to minimize the computational effort. Deciding on which component to neglect mainly depends on the time span of study, component values, and degree of coupling of the component to the rest of the system. However, one principle is that the lower order model should retain to a certain extent all the dominant properties of the high order model. Since the exciter mass spring constant is small in comparison to the rest of the mass-spring system, it is neglected. Furthermore, the governor and steam turbines are neglected because of its large time constant. Thus the 26*"^ order th model is reduced to a 19 order model. 2.6 Eigenvalue Analysis In modern analysis,systems are usually described by a set of differential equations in state space form. Hence, the stability of a linear time invariant system can easily be determined by examining the eigenvalues of the system matrix. The real part of the eigenvalues discloses the stability of the system : stable if all its eigenvalues have negative real parts ; unstable if any of its..eigenvalues have a positive real part. The imaginary part of the eigenvalues indicates the system's natural frequencies. For the system under consideration here, most of the eigenvalues do not change with different loadings [ 15 ], so the eigenvalue analysis is confined to the following conditions : P q = 0.9 p.u., P.F. = 0.9 lagging, V FC = 1.0 p.u. The eigenvalues of the 27^ order model [ 15 ], 26*"^ order model, th and 19 order model for various degrees of compensation are listed in

31 19 Table Table 2.4. It is found that the system is unstable when the natural frequency of the electrical mode is close to a mechanical mode, and as the degree of compensation increases, two mechanical modes are excited simultanously. From the Tables, we can see that the corresponding eigenvalues of the 27 t ' 1 order and 26^ model are close, so it provides further evidence that t h 26 order model is sufficient. Furthermore, the dominant eigenvalues of the 26 t ' 1 order model are retained in the 19^ order model. Hence, it. is decided that the 19 order model can be used for stabilizer design.

32 20 uth, 27 t h order 26 order reduced 19^ model model order model ± J ± J ±j " ± j ± j ± j shaft modes ± J ± ± j ± J ± J ± j ± j99: ± jlol ± j ± j ± J Turbine and Governor ± jo ± J Stator ± j ± j ± J and Network ± j ± j ± J Machine rotor Exciter and Voltage Regulator Table 2.1 Eigenvalues of various order SSR model.at 30% capacitor compenation for P = 0.9 p.u. at 0.9 power factor lagging.

33 21 27 t h order 26 order reduced 19 ^ model model order model J ± J J J ± J J Shaft modes J ± J J J ± J J ± J jlol J ± J J Turbine and Governor jo ± J Stator and Network J ± J J J ± J J Machine rotor Exciter and Voltage Regulator Table 2.2 Eigenvalues of various order SSR model at 50% capacitor compensation for P = 0.9 p.u. at 0.9 power factor lagging.

34 22 ^th 27 t h order 26 order reduced 19^ model model order model ± J ± J ± J ± J ± J ± J Shaft modes ^ ± j ± J ± J ± J ± J ± J ± J ± J ± jlo ± jlo ± jlo.980 Turbine and Governor ± jo ± jo ± jo ± jo.1672 Stator J ± J ± J and Network ± il ± J ± J Machine rotor Exciter and Voltage Regulator Figure 2.3 Eigenvalues of various order S.SRimodel at 70% capacitor compensation for P = 0.9 p.u. at 0.9 power factor lagging.

35 23 27*"' 1 model order 26 order model reduced 19^ order model ± J ± j ± J i j ± J ± J Shaft modes ± ± j j ± J j ± J ± J ± J ± J ± j ± jll ± jll Turbine and Governor ± jo ± jo ± jo ± jo.0801 Stator and Network ± ± J J ± ± J J ± J ± J Machine rotor , Exciter and Voltage Regulator Figure 2.4 Eigenvalues of various order SSR model at 80% capacitor compensation for P = 0.9 p.u. at 0.9 power factor lagging.

36 24 3. LINEAR OPTIMAL EXCITATION CONTROL DESIGN 3.1 Introduction With the advent of large digital computers and numerical analysis techniques, some modern control theories can readily be applied to electrical, economic, and other large systems. A certain class of linear optimal control theory, known as the state regulator problem, has been applied to the stabilizer design to improve the dynamic response of power systems, e.g., the low frequency system oscillation [16-19] and the SSR [6 ]. It was found that the linear optimal controller not only can provide good damping to the system, but also can stabilize the system over a wide-power-range operation [18,19]. As found in Chapter 2, multiple unstable eigenmodes may exist in the power system due to subsynchronous resonance. Because of the merits of linear optimal control, a linear optimal excitation controller will be designed to stabilize the subsynchronous resonance in the system. For the design, a reduced 19^ order model with all measurable state variables is chosen. But the control is applied to the 26 order full model for a linear test in this Chapter, where the eigenvalues of the closed loop system at th various operating and compensation conditions will be examined. 3.2 Linear State Regulator Problem The linear optimal regulator may be formulated as follows: Consider the linearized system state equations x Ax + Bu (3.1) y Hx (3.2) Find the optimal performance function J = 1 oo. t t 9- / ( y Q y + u R u) dt (3.3) 0 subject to the system dynamic contriant (3.1) and (3.2)

37 25 substituting (3.2) into (3.3) gives 1 t t t J = ^ / [ x (HQ H) x + u R u] dt (3.4) 2 0 A Hamiltonian was formed by appending (3.1) to (3.4), H = [ x t (H t Q H) x + u fc R u] + p C (Ax + Bu) (3.5) where p fc is the costate vector or Lagrange multipliers. The optimal control can be found from 8f//8u, resulting or u = - R 1 B t K x (3.6) u = - R ^ K H _ 1 y (3.7) where in the particular LOC design in this thesis, H is a invertable square matrix. K of (3.6) and (3.7) is the Riccati matrix which satifies the nonlinear matrix algebraic equation K A + A t K - K B R _ 1 B t K = - H fc Q H (3.8) In (3.8), Q is a positive semi-definite matrix and R is a positive definite matrix. With u decided, the closed loop system equation becomes x = Gx (3.9) where G = (A-B R _ 1 B t K) (3.10) For the particular case K A B = [ ] (3.11) T A x = [ Ao^.Ae^ Au> 2, A0 2, Aw A9 3, Ao> 4, A6 4, Atii, A6 ; Ai,,Ai, Ai., Ai, Ai., AV,, AV, AV n, AE d' q' f D' Q' cd' cq' R' fd J t (3.12) y = [ Aw 1,Ae i, Aw 2, A6 2, Aw A0^, Au^, AQ^, Aco, A6 ; AP e,aq e, AV t, Ai a,ai f, AV c, AV^.AV R,AE f l ) ] t (3.13)

38 26 The transformation matrix H is shown in the Appendix I. 3.3 Solution to the Matrix Riccati Equation The nonlinear algebraic matrix equation (3.8) is solved by the composite matrix method [20]. For the continuous control system of equation (3.1), (3.2), and cost function of equation (3.4), the 2n x 2n composite matrix [ M ] is given by TM ] -H t Q H -B R-V -A (3.14) The 2n eigenvalues of matrix [ M ] are symmetrically distributed on the right and the left parts of the complex plane. Let the eigenvalue matrix be [ A ] = ( A. II (3.15) and the corresponding eigenvector matrix be [ X ] I III X II x iv (3.16) where [ A^ ] constitutes n stable eigenvalues of [ M ] which are the eigenvalues of the closed loop system G in (3.9), and [ A ] = - [ ]. Then the Riccati matrix [ K ] may be computed from [ K ] = [ X T I ] [ X I ]" (3.17)

39 Wide Range SSR Stabilizer and the Choice of Weighting Matrices The properties of the composite matrix [ M ] suggest that for any chosen weighting matrices [Q] and [R] of the specified type, there will always be an optimal control which will stabilize the system. Although the above information suggests that controllers for various degrees of capacitor compsation can always be designed, it is desirable to have a single controller that will stabilize the system for a wide range of capacitor compensation and operating conditions. The solution [K] for equation (3.7) is unique for a given set of weighting matrices [Q] and [R], so the amount of damping achieved in the closed loop system and the goal of wide range stabilization of the system depend mainly upon the choice of the weighting matrices. Although iterative scheme [18] can be used to obtain [Q] for the prescribed closed loop eigenvalues, choice of elements of [Q] according to the results of previous work [16-18] may also give sufficient damping to the system. Common practice for series capacitor compensation in most utilities is up to 70%. Within that compensation range, results from Chapter 2 disclose that stability of the system under consideration is the worst at 50% compensation; there are two unstable modes. Hence, the controller design will be based on 50% capacitor compensation. Previous work [16-18] has suggested that a speed deviation be weighted more than an angle deviation, and a mechanical variable deviation more than an electrical variable deviation. Accordingly, the final weighting matrices are chosen as follows: [R] = 1 [Q] = diag[100,10,5000,50,5000,50,50000,50,5000,50; 150,150,1,1,0,1,1,0,0] (3.15)

40 28 In this case, a special weight is given to the speed deviation of the low pressure turbine. With this choice, the undesirable eigenvalues can be effectively shifted to the left without excessive individual gains. With the weighting matrices chosen above, a controller designed for 50% compensation has the following gain : For [x] model -[ R _ 1 B t K] = [ , , , , , , , , , ; , , , , , , , , ] (3.16) For [y] model -[ R~ 1 B t K H _ 1 ] = [ , , , , , , , , , ; , , , , , , , , ] (3.17) The control is then applied to the linearized 26 1 "* 1 order model -L and the eigenvalues of the controlled system, at a compensation (X / x ) ranging from 10% to 90% and for three different loading conditions (P = 0.9, 1.25, and 0.5 p.u.), are examined. Typical results are given in Tables It is found that for P = 0.9 p.u. and 1.25 p.u., the controlled system is stable within the prescribed range of compensation. However, for light load (P =0.5 p.u.) the system is only stable up to 70% compensation. Thus the linear optimal excitation control has proved to be effective in stabilizing SSR within the linear-operation-range of the system. But, this may not be valid for the nonlinear range. So the linear optimal controller will be further tested on the original nonlinear system as derived in Chapter 2, and presented in Chapter 4.

41 29 capacitor 30% 50% 80% compensation compensation compensation compensation ± J ± ± J ± J ± J ± J Shaft modes ± J ± J ± J ± J ± J ± J ± J ± J ± J ± J ± J ± jll Turbine and Governor ± jo ± j2i ± jl ± jo.2262 Stator and Network ± j ± J ± J ± J ± j ± j Machine rotor ± j ± J ± J Exciter and Voltage Regulator Table 3.1 Eigenvalues of the system with linear optimal control at various degrees of capacitor compensation. Operating condition: P=0.9 p.u., P.F. =0.9 lagging, V = 1.0 p.u.

42 30 30% 50% 70% 80% compensation compensation compensation compensation ± J ± J ± J ± J ± J ± J ± j ± j ± J ± j ' j ± J ± J ± J ± J ± J ± J ± J ± j ± J ± J ± J ± jlo ± j (a) 30% 50% 70% 80% compensation compensation compensation compensation ± J ± j ± J ± J ± J ± J ± J ± J ± J ± jl63.ll ± jl60.' ± J ± J ± J ± J ± J ± J ± J ± jloo ± jloo ± j ± J ± J ± J (b) Table 3.2 (a) Shaft modes< of the system at various degrees of compensation at P=1.25 p.u., P.F.=0.9 lagging V =1.0 p.u. (without control) (b) Shaft modes of the system with linear optimal control at various degrees of compensation at P=1.25 p.u., P.F. =0.9 lagging, V =1.0 p.u.

43 31 30% 50% 70% 80% compensation compensation compensation compensation ± J ± J ± j ± J ± J ± j ± j ± J ± J ± J ± J ± j ± J ± jl27.ll ± jl ± J ± J ± J ± jloo ± j ± J ± jlo ± jll ± J (a) 30% 50% 70% 80% compensation compensation compensation compensation ± j ± J ± J ± J ± j ± j ± J ± J ± J ± J ± J ± J ± ± J ± J ± J ± J ± J ± j ± J ± J ± J ± jll ± J (b) Table 3.3 (a) Shaft modes Gf the system at various degrees of compensation at P=0.5 p.u., P.F=0.9 leading,. V =1.0 p.u. (without control) (b) Shaft modes of the system with linear optimal control at various degrees of compensation at P=0.5 p.u., P.F.=0.9 leading, V =1.0 p.u.

44 32 4. EFFECT OF EXCITATION CONTROLLER ON STEADY STATE AND TRANSIENT SSR 4.1 Introduction It has been shown that linear optimal controller designed in Chapter 3 can suppress the SSR phenomenon. However, it must be noted that all tests were performed on the linearized system without considering exciter voltage limits and other nonlinearities. In some cases, the disturbances have caused the exciter voltage to swing well beyond any reasonable ceiling voltage. In this Chapter, the dynamic performance of the system, with and without supplementary excitation control, designed in Chapter 3, will be investigated.on the nonlinear system model described by 26 first order differential equations. All governor and exciter physical limits are included. Transient torques on the shafts of the generator mass-spring system and the dynamic responses of the generator with and without control will be examined when the system is subjected to various disturbances. 4.2 Controller for Steady State SSR In daily operation, a power system is frequently exposed to disturbance such as load change; such disturbance may not be a problem of steady state stability but self-excited SSR may arise. Figure 4.1 shows the dynamic responses of the system when it is subjected to a 10% load change; the system is initially operating at P of 0.9 p.u. and the 10% load change is represented as follows T = { T + O.lt 0 < t < 1 sec. mo - - T t < 1 sec. mo - - In Figure 4.1, growing shaft torques due to SSR are observed.

45 TIME(SEC) XT Figure 4.1 Dynamic responses of the system without control when subjected to 10% load change.

46 34 For further studies, a more severe disturbance, i.e., a pulse torque of 20% for 0.2 second is assumed for various operating conditions. Typical result for 0.9 per unit generator loading and 50% capacitor compensation are shown in Figures 4.2 and 4.3. With control, the system has no excessive response and the disturbed system returns to its normal operating condition in a well-damped manner as shown in Figure 4.3. In contrast, Figure 4.2 shows sustained oscillations and growing torsional shaft torques in the system without control. The same disturbance is applied to the system with 80% capacitor compensation and typical results are shown in Figure 4.4 and Figure 4.5. Without control, Figure 4.4, the torsional oscillations on the shafts exceed their once-in-a-lifetime limit so that the shafts would be damaged. However, as shown in Figure 4.5, the dangerous torsional oscillations due to SSR can be alleviated when the system has the linear optimal excitation control. Although a pulse torque input to the system is a pseudo disturbance, not normally encountered in practice, it can be taken as an extreme case in the daily power system operation.hence, as shown in the above results, the linear optimal excitation control can efficiently control the steady state SSR problem for a wide range of capacitor compensation.

47 CO CO U J a in. in I I TIME(SEC) 4.0 I 5.0 <o. X 0_cn I 1.0 ~ TIME(SEC) TIME(SEC) Figure 4.2 Dynamic responses of the power system at 0.9 p.u. generator loading, 0.9 power facter lagging, and 50% capacitor compensation when subjected to a pulse torque disturbance (without control) Co

48 36 in o 0.0 ) TIME(SEC) Figure 4.2 ( continued )

49 Figure 4.2 (continued ) 37

50 38 in TIME(SEC) Figure 4.2 (continued )

51 Figure 4.3 Dynamic responses of the power system with control at 0.9 p.u. generator loading, 0.9 power factor lagging, and 50% capacitor compensation when, subjected to a pulse torque disturbance. CO

52

53 TORQUE (LPB-GEN) P.U S TORQUE CLPR-LPB) P.U S o

54 in n i I TIME(SEC) 4.0 Figure 4.3 (continued)

55 Figure 4.5 Torsional oscillations of the generator-exciter shaft at 80% compensation (with control) 43

56 Controller for Transient SSR The transient torques experienced by the turbogenerator shafts as a result of electrical transients caused by transmission faults are now considered. In order to evaluate the effectiveness of the designed excitation controller, a simultanous three fault was applied at bus B in Figure 4.6 at time t=0 and then removed after second. The fault impedance and series capacitor are assumed to be 0.04 p.u. and 0.28 p.u. respectively. The fault location, hence the impedance between bus B and the infinite bus, Xg, is also varied. Figure 4.6 Electrical network for the simulation of subsynchronous resonance.

57 45 With the fault located at the remote end of the transmission line, i.e., Xg equal to 0.01 p.u., the responses of the system without and with control are shown in Figures 4.7 and 4.8 respectively. Figure 4.7 shows clearly the torsional interaction between the electrical and mechanical system. The forcing frequency of the electrical torque ( P = T^ in p.u. ) is close to a torsional mode frequency of the mass-spring system so that the mode is excited, and growing torsional shaft torques are observed. But no excessive oscillations are observed in Figure 4.8 for the system with excitation control. As the fault location is moved closer to the generator terminal, the disturbance to the system becomes more severe. The system is subjected to the standard test as discribed in the Benchmark Model [13] with X equal to 0.06 p.u. and capacitor equal to 0.28 p.u.. Typical results are shown in Figures 4.9 and The fault effect seems more severe than the previous case that the shafts experience larger torsional stress and it takes a longer time for oscillations to decay. Furthermore, when the control is tested on the system at 80% capacitor compensation and the system is subjected to the same disturbance, it is unstable and the shaft torque increases as those shown in Figure 4.4, because of the excitation voltage ceilings and other limitations that are not considered in a linear controller. To summarize, the linear optimal excitation control can provide sufficient damping to stabilize the system in most cases. But it is not recommended for transient stability control of a power system with a very severe fault.

58 Figure 4.7 Dynamic responses of the power system without control at 0.9 p.u. generator loading, 0.9 power factor lagging,and at 50% capacitor compensation when subjected to a three-phase fault at the remote end (X =0.01 p.u.)

59 Figure 4.7 (continued ) 47

60 Figure 4.7 (continued ) 48

61 49 o o 1 D_(D o o H 1 r i i i 0.0 l.o TIMECSECJ Figure 4.7 (continued )

62 _ o" I Xo" ^o 'o CL -,.o. LU '!! i t 41 J, ill "A firry i iiw mi -i o Oo. LU 1 LUm O n TIME(SEC) C0 0".. i 0.0 ~i 1.0 i TIMEtSEC) co 01 o- cn CL, CC LU rsj Oo' CL. U3 o" ~T TIMEtSEC) I 4.0! n 1 I I I TIMEtSEC) 4.0 ~1 5.0 Figure 4.8 Dynamic responses of the power system with control at 0.9 p.u. generator loading, 0.9 power factor lagging, and 50% capacitor compensation when subjected to a three-phase fault at the remote end (Xg=0.01 p.u.) o

63 51 -JtD " 4 CL. CD oh i i i i i i TIME(SECJ Figure 4.8 (continued)

64 52 in o tm o l TIME(SEC) Figure 4.8 (continued)

65 53 Figure 4.8 (continued)

66 54 in CM" 3. CVJ D_ LU LD I a Q J 'in LU " ZD O a cn -. 1 "'1 ti'feli ijliifliifjlliilj J;?'U'l'jjj,.Vd. " "I''!!!'"' : '!!'! iiiiihi! lllii J K! 31 r in i 1 r TIME(SEC) CD CM CL CL I I -< ID :!!:, i III 'fe I'lll III' I 1 I/it ill JJ iijil i n I TIME(SEC) Figure 4.9 Dynamic responses of the power system without control for a three-phase fault (Xg = 0.06 p.u.) at 50% compensation.

67 55 Figure 4.10 Dynamic responses of the power system with control for a three-phase fault (X B =0.06 p.u.) at 50% compensation.

68 56 5. CONCLUSION A high-order nonlinear power system model for studying the torsional oscillations due to subsynchronous resonance has been developed. From eigenvalue analysis of the linearized model, it is found that more than one mechanical mode can be excited simultanously for a high degree of th capacitor compensation. Eigenvalue analysis of the reduced 19 order model revealed that these unstable mechanical modes remain even though the exciter mass is neglected. Hence, the potential dangers of subsynchronous resonance cannot be neglected in a turbine generator unit even with a static exciter. A linear optimal controller has been designed and its effectiveness is initially tested by eigenvalue analysis. It not only stabilizes the system for capacitor compensation ranging from 10% to 90% at normal operating conditions, but also improves the stability considerably at other operating conditions. Dynamic performance tests using a high-order nonlinear model further show that the system without control exhibits growing shaft torques not only in the section between the generator and the exciter, but also in other sections such as the shafts on either side of the intermediate pressure turbine. With moderate disturbance, the system with excitation control shows damped responses. However, with more severe disturbance, such as a three-phase fault close to the generator terminal, the linear optimal controller is no longer effective. In summary, an excitation controller of the linear optimal type can be designed to effectively stabilize the torsional oscillations within the range of dynamic stability, but it is not recommended for torsional oscillations of the severe transient type.

69 57 REFERENCES [1] M.C. Hall and D.A. Hodges, "Experience with 500kV Subsynchronous Resonance and Resulting Turbine Generator Shaft Damage at Mohave Generating Station", IEEE Publication 76CH PWR, pp , [2] R.G. Farmer, A.L. Schalb and Eli Katz, "Navajo Project Report on Subsynchronous Resonance Analysis and Solutions", IEEE Publication 76Chl066-0-PWR, pp , [3] 0. Saito, H. Mukae and K. Murotani, "Suppression of Self Excited Oscillations in Series-compensated Transmission Lines By Excitation Control of Synchronous Machine", IEEE Trans, on PAS, Vol. PAS 94, pp , Sept/Oct [4] H.M.A. Hamdan and F.M. Hughes, "Excitation Controller Design For The Damping of Self Excited Oscillations in Series Compensated Lines", Paper A , IEEE PES Summer Meeting, Los Angeles, July [5] A.A. Fouad and K.T. Khu, "Damping of Torsional Oscillations in Power Systems With Series-compensated Lines",IEEE Trans, on PAS, Vol. PAS 97, pp , May/June [6] Yao-nan Yu, M.D. Wvong and K.K. Tse, "Multi-Mode Wide-Range Subsynchronous Resonance Stabilization", Paper A , IEEE PES Summer Meeting, Los Angeles, July [7] A.M. El-Serafi and A.A. Shaltout, "Control of Subsynchronous Resonance Oscillations By Multi-loop Excitation Controller" Paper A , IEEE PES Winter Meeting, New York City, Feb [8] R.A. Hedin, R.C. Dancy and K.B. Stump, "An Analysis of the Subsynchronous Interaction of Synchronous Machine and Transmission Networks", Proceeding of the American Power Conference, Vol. 35, 1973, pp [9] C. Concordia and R.P. Schulz, "Appropriate Component Representation for the Simulation of Power Sysyem Dynamics", IEEE Publication 75CH PWR, pp , [10]. IEEE Committee Report, "Dynamic Models for Steam and Hydro Turbines in Power System Studies", IEEE Trans, on PAS, Vol. PAS 92, pp , Nov/Dec [11] E.W. Kimbark, " Power System Stability", Vol. III., Wiley, New York [12] IEEE Committee Report, "Computer Representation of Excitation Systems", IEEE Trans, on PAS, Vol PAS 87, pp , June/July 1968.