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Steady-state and transient stability analyses of multimachine power systems using five different catastrophe… Sum, Kin Ming 1998

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S T E A D Y - S T A T E AND/TRANSffiNT STABILITY A N A L Y S E S OF MULTIMACHINE POWER SYSTEMS USING FIVE DIFFERENT C A T A S T R O P H E M O D E L S BY KIN MING S U M B . A . S c , UNIVERSITY OF BRITISH COLUMBIA, 1988  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E M A S T E R D E G R E E OF APPLIED SCIENCE IN T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF E L E C T R I C A L ENGINEERING  We accept this thesis as confirming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A May 1998 ©Kin Ming Sum, 1998  In  presenting this  degree at the  thesis  in  partial  fulfilment  of  the  requirements  University  of  British  Columbia,  I agree that the Library shall make it  freely available for reference and study. I further agree that copying  of  department  this thesis for scholarly or  by  his  or  her  purposes may be  representatives.  It  is  for  an  permission for extensive  granted by the understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of The University of British Columbia Vancouver, Canada  DE-6 (2/88)  advanced  head of my copying  or  my written  ABSTRACT  Steady-state and transient stability analyses are important in planning and operation of electric power systems.  For large power systems, such analyses are very time consuming.  On-line  stability assessment is necessary for secure and reliable operation because power systems are being operated close to their, maximum limits. In the last three decades, research work has been done in the area of fast on-line assessment by direct methods in order to minimize computational time.  In these methods, major difficulties  are power system modeling, stability system assessment, and adaptation to system operation. Catastrophe theory was applied to study power system stability by Deng and Zhang for steadystate stability, assessment and by Wvong, Mihiring, and Parsi-Feraidoonian for transient stability assessment.  Although the cusp catastrophe was proposed by Deng and Zhang to study the  steady-state stability assessment of power systems, no detailed formulation or specific results were presented.  The swallowtail catastrophe was proposed to study the transient stability of  power systems by Wvong, Mihiring and Parsi-Feraidoonian, but research did not identify the critical clearing angle values. In this thesis, further research is done on using catastrophe theory for steady-state and transient stability of power systems.  In this thesis,  different catastrophe models such as the fold, cusp,  swallowtail, butterfly, and wigwam catastrophes assessment.  are derived for steady-state stability  The accuracy and limitations of these different catastrophe models on two test  systems (three-machine WSCC system and seven-machine CIGRE system) are discussed.  Five  catastrophe models are also derived for transient stability assessment of power systems.  The  'ii  critical clearing angle of the critical group of machines for two test systems for various balanced three-phase faults are then determined using the cusp catastrophe model.  iii  TABLE OF CONTENTS Page 1.  ABSTRACT  2.  T A B L E OF CONTENTS  3.  LIST OF TABLES  vl  4.  LIST OF FIGURES  vi?  5.  ACKNOWLEDGEMENT  vii  6.  CHAPTER 1  Introduction..  1  7.  CHAPTER 2  Application of Catastrophe Theory to Steady-State Analysis of Multimachine Power Systems  4  8.  9.  CHAPTER 3  CHAPTER 4  10. R E F E R E N C E S 11. APPENDIX A  ii *v  ;  Application of Catastrophe Theory to Transient Stability Analysis of Multimachine Power Systems  33  Discussion And Conclusions  51 :  Derivation of One-Machine Infinite Bus Dynamic Equivalents of Multimachine Power Systems ..  54  56  12. APPENDLX B  Detailed Derivation of Control Parameters of Wigwam Catastrophe Model (Steady-State Stability Analysis) ... 59  13. APPENDLX C  Three-Machine WSCC Test System Catastrophe Models Simulation Results (Steady-State Stability Analysis) 61  14. APPENDIX D  Seven-Machine CIGRE Test System Catastrophe Models Simulation ' Results (Steady-State Stability Analysis) 72  15. APPENDIX E  Three-Machine WSCC Test System Catastrophe Model Simulation Results (Transient Stability Analysis) 83  16. APPENDIX F  Seven-Machine CIGRE Test System Catastrophe Model Simulation Results (Transient Stability Analysis) , 87  LIST OF TABLES  T A B L E 2.2.1:  Single State Space Dimension Catastrophes  T A B L E 2.4.2:  Steady-State Catastrophe Models - Manifold and Control Parameters  T A B L E 2.5.1:  Three-Machine WSCC System - Pre-disturbance System Data  T A B L E 2.5.2:  Seven-Machine CIGRE System - Pre-disturbance System Data  T A B L E 2.5.3  Three-Machine WSCC System - Critical Mechanical Power Input Change (Steady-State Stability) Determined by (a) Different Catastrophe Methods (b) The E E A C Method  T A B L E 2.5.4:  Seven-Machine CIGRE System - Critical Mechanical Power Input Change (Steady-State Stability) Determined by (a) Different Catastrophe Methods (b) The E E A C Method  T A B L E 3.2.1  Transient Stability Catastrophe Models - Manifold and Control Parameters  T A B L E 3.3.1:  Three-Machine WSCC System - Critical Clearing Angle (Transient Stability) Determined by (a) The Cusp Catastrophe Methods (b) The E E A C Method  T A B L E 3.3.2:  Seven-Machine CIGRE System - Critical Clearing Angle (Transient Stability) Determined by (a) The Cusp Catastrophe Methods (b) The E E A C Method  LIST OF FIGURES  FIGURE 2.4.1:  Steady-State Analysis - A P Vs. Clearing Angle,  FIGURE 2.5.1:  Three-Machine WSCC System Configuration (three machine, nine buses)  FIGURE 2.5.2:  Seven-Machine CIGRE System Configuration (seven machine, seventeen  m  buses) FIGURE 2.5.3:  The Cusp Catastrophe Bifurcation Set For Steady-State Stability (Plot of w Vs. x Parameters)  FIGURE 2.5.4:  The Swallowtail Catastrophe Bifurcation Set For Steady-State Stability (Plot of w Vs. x Parameters)  FIGURE 2.5.5:  The Butterfly Catastrophe Bifurcation Set For Steady-State Stability (Plot of w Vs. x Parameters)  FIGURE 2.5.6:  The Wigwam Catastrophe Bifurcation Set For Steady-State Stability (Plot of w Vs. x Parameters) - Generator  FIGURE 2.5.7:  The Wigwam Catastrophe Bifurcation Set For Steady-State Stability (Plot of w Vs. x Parameters) - Motor  FIGURE 3.2.1:  Multimachine System for Balanced Three-Phase Fault (OMIB)  FIGURE 3.3.1:  The Cusp Bifurcation Set for Transient Stability (Plot of w Vs. x Parameters)  ACKNOWLEDGEMENT  I would like to thank Dr. M.D. Wvong for his continual encouragement, invaluable and patient guidance throughout the course of this research.  I would also like to thank my wife, Laura, for her encouragement and patience throughout my graduate program.  vii  CHAPTER ONE INTRODUCTION Power system stability refers to the ability of synchronous machines to move from one steadystate operating point following a disturbance to another steady-state operating point, without losing synchronism.  There are three types of power system stability, namely, steady-state,  transient, and dynamic stability [13] . Steady-state stability involves slow or gradual changes in operating points.  Steady-state  stability studies are required to ensure that phase angles are not too large, that bus voltages are close to nominal values, and that generators, transmission lines, transformers and other equipment are not overloaded. Dynamic stability involves an even longer time period, typically several minutes.  It is possible  for controls to affect dynamic stability even though transient stability is maintained.  The action  of turbine-governors, excitation systems, tap changing transformers, and controls from a power . system dispatch centre can interact to stabilize or destabilize a power system several minutes after a disturbance has occurred. Transient stability involves major disturbances such as loss of generation, line switching operations,.faults, and sudden load changes.  Following a disturbance, synchronous machines  frequencies undergo transient deviations from synchronous frequency (60Hz), and machine power angles change.  The objective of a transient study is to determine whether or not the  machines will return to synchronous frequency with new steady-state power angles. in power flows and bus voltages are also of concern.  Changes  In many cases, transient stability is  determined during the first swing of machine power angles following a disturbance.  During  the first swing, which typically lasts about one second, the mechanical output power and the internal voltage of a generating unit are often assumed constant. In large scale interconnected power systems, the greatest concern is security of the when subjected to disturbances.  system  Hence, power system stability becomes an increasingly  important consideration in system planning and operation.  l  Extensive stability studies are  required to ensure system stability before a planning or operating decision is made.  Each  contingency for each disturbance considered requires a large number of stability studies to determine the critical clearing angle or system stability limits. A typical steady-state and transient stability study consists of obtaining time solution to power system differential and algebraic equations with initial system conditions.  The power system  equations should include all significant parameters that influence stability such as generator controls, stability controls and protective devices. Although the time solution of stability analysis is very reliable and accurate, it has the following limitations: 1.  The process is very time consuming in the system planning stage where a large number of cases need to be considered.  2.  In systems where immediate operational decisions need to be made, time solutions may not provide fast enough on-line assessment.  3.  The power system operating conditions change during the course of the day and the time of the year, while stability studies are done off-line for certain severe cases.  This leads to  improper decisions in some cases and hence may increase expenditures. Therefore, fast and reliable assessment methods should be provided for operators to make prompt on-line decisions. to be studied off-line.  Also, these fast direct methods will help reduce the number of cases  The desired method for fast analysis of transient stability should satisfy  the following criteria: 1.  Provide a fast and reliable answer to indicate whether the system is stable or not when a specified disturbance is encountered.  2.  Provide the necessary information to indicate the degree of system stability so that the operators can ensure system security.  3.  If fast methods are to be used for on-line purposes, it must be adaptable to changes in operating conditions, different disturbances and stability controls.  2  Extensive research has been conducted in this area, but little has been achieved.  Catastrophe  theory was proposed in [7,14,15,18] as an alternative fast on-line method to determine power system stability.  The cusp catastrophe was proposed [7] to study the steady-state stability of  power systems, but no specific result arid formulation were presented.  In [14,15,18], the  swallowtail catastrophe was used to study transient stability of power systems, but no work has been done to determine the  critical clearing angle values.  The motivation of this research is to exterid catastrophe theory as an alternative for fast on-line method to determine steady-state and transient stability of power systems. In Chapter Two, different steady-state catastrophe models of the fold, cusp, swallowtail, butterfly, and wigwam catastrophes will be developed to study stability limits of power systems by determining their maximum mechanical power input change for two test power systems; namely, the three-machine WSCC system [11] and the seven-machine CIGRE system [12]. Stability limits obtained from the Extended Equal Area Criteria (EEAC) method [1] will be used as a bench mark to evaluate results obtained from the catastrophe models. accuracy of these catastrophes will be discussed.  Limitations and  Note that the E E A C method determines the  stability limit when the potential energy which can be absorbed by the post-disturbance power system equals the kinetic energy generated by accelerating power during the disturbance period. In Chapter Three,  different catastrophe models is formulated for transient stability in power  systems with balanced three-phase faults.  Critical clearing angles of the system will be  determined by the cusp catastrophe and the E E A C method for the same two test power systems (WSCC and CIGRE systems) .  Accuracy of catastrophe result can be determined by  comparing with the result of E E A C method. Chapter Four concludes the achievement of this project and gives suggestions for future research.  CHAPTER TWO APPLICATION OF CATASTROPHE THEORY TO STEADY-STATE OF MULTMACHINE POWER SYSTEMS 2.1 Introduction Any physical system that is designed to perform certain pre-assigned tasks in steady-state must remain stable at all times for sudden disturbances with an adequate safety margin.  In  a large physical system such as a modern interconnected power system, analytical techniques are required to interpret the region of system stability. Since the famous blackout in north-eastern. U.S.A. in 1965, considerable research work has been done in power systems to prevent future recurrence and ensure secure and reliable operation. stability [1].  Much work has been done in the area of direct and fast assessment of transient Promising results have been achieved with energy functions [2,3] and pattern  recognition [4]. Catastrophe theory has been applied to the study of various dynamic systems [5] and in recent years to the steady-state stability analysis of power systems [6,7].  An attractive  feature of catastrophe theory is that the stability regions are defined in terms of the catastrophe control parameters bounded by lines of stability limits.  Deng and Zhang [7]  proposed using the cusp catastrophe to study the steady-state stability of power systems, but did not show any particular formulation and specific results. In this chapter, the cusp, swallowtail, butterfly, and wigwam catastrophe models of interconnected multimachine power systems are proposed for the study of steady-state stability subjected to change in mechanical power input.  The critical mechanical power  input change (maximum mechanical power input before system instability) will be obtained from these catastrophes and the results will be compared with that obtained from using the extended equal area criterion (EEAC)[1].  The E E A C method uses the following criterion  to determine the maximum mechanical power input change of power systems, the system remains stable if the kinetic energy generated by changing of mechanical power input of the  4  system is less than or equal to the potential energy available which can be absorbed during the post disturbance period (Specifically, area A l is smaller or equal to area A2 in Figure 2.4.1).  .  The structure of this chapter is briefly described as follows. First, a brief review of catastrophe theory will be presented in Section 2.2  followed by formulation of  multimachine power system dynamic equivalents in Section 2.3.  In Section 2.4, five  catastrophe models for steady-state analysis of multimachine power systems will be introduced.  These catastrophe models will be applied to the three-machine WSCC system  [11] and to the seven-machine CIGRE system [12]. test systems will be discussed in Section 2.5. Section 2.6.  5  Results and observations of the two Finally, conclusions will be stated in  Catastrophe Theory Catastrophe theory was originally presented by Professor Rene Thorn and published in his book "Structural Stability and Morphogenesis" [8]. explain sudden changes in morphogenesis.  Thorn used differential topology to  This theory explores the region of  sudden  changes in dynamic systems and deals with the properties of discontinuities directly.  It  has been defined as the study from a qualitative point of view of the ways the solutions to differential equations may change [9],  Natural phenomena such as the sudden collapse of  bridges and the phase change of water from liquid to solid can be well described by use of catastrophe theory. Catastrophe theory can be briefly described as follows.  Consider a system whose  behaviour is usually smooth but which exhibits some discontinuities.  Suppose the system  has a smooth potential function to describe the system dynamics and has "n" state variables and "m" control parameters. following:  Given such a system, catastrophe theory tells us the  The number of qualitative different configurations of discontinuities that can  occur depends not on the number of state variables but on the number of control parameters. four,  Specifically,  if the number of the control parameters is not greater than  there are seven basic or elementary catastrophes, and in none of these are more than  two state variables involved [10].' Consider a continuous potential function V(Y,C) which represents the system behaviour, where Y are the state variables and C are the control parameters.  The potential function  V(Y,C) can be mapped in terms of its control variables C to define the continuous region. Let the potential function be represented by V(Y,C):M®R  (  where M , C are manifolds in the state space R and the control space R respectively. n  r  We now define the catastrophe manifold M as the equilibrium surface that represents all critical points of V(Y,C).  It is the subset of R X R n  . VYV (Y) = 0  r  defined by  ,  C  .  (2.2)  where V ( Y ) = V ( Y , C ) and W i s the partial derivative with respect to Y . C  Equation 2.2 is  the set of all critical points of the function V(Y,C). Next, we find the singularity set, S, which is the subset of M that consists of all degenerate critical points of V. V VC(Y) = 0  These are the points at which and V Y V ( Y ) 2  Y  C  =0  (2.3)  The singularity set, S, is then projected down onto the control space R by eliminating the r  state variables Y using Equations (2.2) and (2.3), to obtain the bifurcation set, B.  The  bifurcation set provides a projection of the stability region of the: function V bounded by the degenerate critical point at which the system exhibits sudden changes when it is subject to small changes. Let us illustrate this process by considering the cusp manifold (one of the elementary catastrophe).  The cusp manifold equation is represented as  V V ( Y ) = y + wy + x = 0  (2.4)  3  Y  C  where y is the only state variable and w , x are the control parameters and therefore V y Vc(Y) = 3 y + w = 0 2  (2 5)  2  By algebraic manipulation of Equations (2.4) and (2.5), the cusp bifurcation set may be put in the form of 4 w + 27x = 0 3  (2.6)  2  7  Equation (2.6) describes system stability boundary.  Table 2.2.1 summarizes the single  state space dimension of different catastrophe manifolds  Catastrophe Control Space Fold 1  State Space  Function  Catastrophe Manifold  1  y /3+xy=0  y +x=0  3  2  Cusp  2  1  y /4+wy /2+xy=0  y +wy+x=0  Swallowtail  3  1  y /5+vy /3+wy /2+xy=0  y -r-vy +wy-r-x=0  Butterfly  4  1  y /6+ y /4+vy /3+wy /2+xy=0  y +uy +vy -rwy+x=0  y /7+ty /5+uy /4+vy /3+wy /2+xy=0  y +ty -ruy +vy +wy+x=0  4  2  5  3  6  4  3  2  3  4  2  U  Wigwam  5  1  7  5  4  3  2  2  5  6  y is state variable and t, u, v, w, x are control parameters  3  4  3  .  Table 2.2.1: Single State Space Dimension Catastrophes  8  2  2  2.3 Dynamic Equivalents of Multimachine Power System The swing equation of generator i in a power system of n-machines is given by:  m; 8; + d; 8;  Pa - Pm;" Pe? ;  i = 1,2,3,  ,n  (2.7)  where n  E [E; Ej (gy cos 8ij + bij sin Sy)]  Pe  ;  (2.8)  j=i 8;  =  internal rotor angle of generator i  m;  =  inertia constant of generator i  d;  =  damping coefficient (assume zero for simulation purpose)  p .  =  mechanical power input of generator i  p.  =  electrical power output of generator i  p.  =  accelerating power of generator i  m  e  a  '  gij,by =  real and imaginary parts of reduced nodal admittance matrix .  8jj  8; - Sj  =  Ei, Ej =  internal voltages of generator i, j.  Under steady-state conditions, Pa; equals zero and 8; is constant.  When a system is  subject to a disturbance, Pa; becomes different from zero and Equation (2.7) describes the behaviour of 8; with time.  For generator i to be stable, 8i must assume a constant value  and Pa; must be zero. When a disturbance occurs in a large system, only a few machines are affected and these tend to oscillate against the rest of the system. machines; the other machines are non-critical.  9  These machines are called critical  The group of critical machines, j = 1, 2', 3, ..., k may be represented by a single equivalent machine with an inertia constant and rotor angle, respectively, of:  M  k  6k  =  k £ mj j=i  (2.9)  =  I k — £ mj8j M .  (2.10)  j = 1  k  Similarly, the group of non-critical machines, j = k+1, k+2, k+3, . . n may be represented by another single equivalent machine with an inertia constant and rotor angle, respectively, of: Mo  n  Z  -  nij  (2.H)  j=k+l  8  0  V  I n = — £ mj5j • Mo J k-l  '  (2.12)  By suitable algebraic manipulation, the swing equation for the group of critical machines can be put in the form of:  M¥k=P -Pc-TkSin(y +a ) m  k  k  (2.13)  where Mo M M  k  =  (2.14) M +M 0  k  Mo-Mk V  =  (2.15) Mo + M  ^k  =  5 -5 k  k  (2.16)  0  10  Mo Pm  =  M  k  —  • E.Pmj  M +M 0  j  „  k  " Mo+Mk  1  k  EPnij  (217)  j = k + 1  M„ M P= [ E E EiEj (gy cosSy + by sin5,j)]-[E Mo+M '.. ' M + M k  k  k  n  n  £ EEj (g cosSy + by sinSy)] (2.18)  c  i = l j = 1  k  T  k  =  0  VAk + B 2  k  (2.19)  2 k  kn E E [ E i Ej (u  Ak  y  i = k + l j = k + 1  g i j  cos (8 - Te) + b sin (8 - T ) ) ] U  y  y  (2.20)  k  i-lj-k+l  B  a  k  k  kn E E [ E ; Ej (by cos (8ij - T ) - u i=lj=k+l  •=  k  =  g i j  sin (8 - Tk)) ]  (2.21)  ;j  tan - (Ak /B )  (2.22)  1  k  or by defining Mi 8;-8  i = 1,2,3,.'.  =  • (p; =  k  8i - 8  0  , k critical machines  (2.23)  i = k+l,k+2,...., n non-critical machines  (2.24)  P , Ak and B may be represented in the form of: c  k  Mo k k M =[ — £ EEiEj (gijCOSTiij+bijSinriijJl-f £ Mo+M . Mo+Mk k  p  c  ,=,J=1  n  n  £ EEjfejCos co ij+bySin (p )] (2.25) ;j  ,=k+lj=k+1  k  Ak  =  kn E E [ E i E j ( p g j C O s ( r i i - ( / ) j ) + bijSin(rii-(p ))] i=lj=k+l  (2.26)  Bk  ='  kn E E [ E ; Ej (by cos fa - <?J) - U g sin fa - 0j))] i=lj=k+l  (2.27)  i  j  ;j  Detailed derivation of the above is presented in Appendix A.  n  2.4. Multimachine Power System Steady-state Stability Catastrophe Theory Model  /  p  0  /M/  /  - P +TkSinOF + a ) 0  k  k  7  0 m  "  / o k  i  1  m c ik  ^k  \  V  u  Angle, Tk  Figure 2.4.1: Steady-State Analysis - A P Vs. Clearing Angle, *¥ m  k  With reference to Figure 2.4.1 and using the extended equal area stability criterion (EEAC) of Equation (2.13), we have:  S [ P ° + A P - P - T k sin(T +a )]dTk + S [ P ° + A P - P c - T s i n ( ^ + a k ) ] d ^ k = 0 (2.28) m  m  c  k  k  m  m  k  k  for the limiting case of the system to be stable, i.e. (P ° + A P m  -P)W  m  - TO + T cos ( ¥  c  k  c k  + at) - T cosCFk + a ) 0  k  k  + (P^ + A P - P c ) ( ^ k - T O + TkCOs(Tk + a k ) - T c o s ( ^ k + ak) = 0 u  u  m  c  k  (2.29)  After algebraic manipulation, Equation (2.29) may be put in the form of: (P  o m  + A P - P c ) ( % - T O + Tk[cos(^ u  m  u k  + ak)-cos( Pk + a )] = 0 v  O  k  (2.30)  where P °- P 1  ^  =  sin  m  A c  a  1  T  k  k  12  (2.31)  Pm° + A P - P m  T  =  c k  sin  :  1  c  •  a  k  P ° + AP -P  c  T  k  m  W  m  TT-sin —  =  (2.32)  =  1  T  a  (2.33)  k  k  P °  =  initial value of the mechanical power input  (2.34)  P  =  new value of the mechanical power input  (2.35)  AP =  P  -P °  (2.36)  T  W+ATk  (2.37)  m  n m  m  n m  m  Let =  u k  Equation (2.30) may be represented in the form: (P  0 r a  + A P - P ) A F + T [ c o s C F ° + a ) ( c o s A P - l ) - s i n ( ¥ + a k ) s i n A ¥ k ] = 0 (2.38) v  m  c  x  k  k  k  k  0  k  k  Grouping like terms of A *Fk, Equation (2.38) becomes : K AT 1  + K cos(A Pk) + K 3 s i n ( A T ) - K 2 = 0  (2.39)  v  k  2  k  where K,  =  (Pm°+AP -P )  K  2  =  T cos(T ° + a )  (2.41)  K  3  =  - Tk sin(T ° + at)  (2.42)  m  k  k  (2.40)  c  k  k  Note that parameter K i is the only parameter which is dependent upon of A P . m  13  2.4.1Series Expansion of sin( A T k ) and cos( A T k ) AT  is unknown in Equation (2.39) and would normally require an iterative approach to  k  solve.  The iterative approach can be avoided by use of catastrophe theory after series  expansion of sin( A T ) and cos( A Tk) to get: k  AT K,AT +K k  1  2  AT  2 k  AT  4 k  AT  6 k  }+K { A T  +  3  * 2!  4!  AT  3 k  5 k  AT  7 k  } - K = 0(2.43)  k  2  6!  3!  5!  7!  Equation 2.43 can be put in the form of AT (K!+K )-K 3  AT  k  K  2  2!  AT  2 k  +K  3  AT  3 k  +K  2  3!  AT  4 k  K  3  4!  5!  2  AT  5 k  — '-K 6!  6 k  +.. = 0 (2.44)  3  7!  Let AT  y + 15  k  (2.45)  so that (AT )  2  = (y + fi)  = y +2fly+A  (AT )  3  = (y + fi)  = y +3By +3fl y+^  (ATk)  4  = (y + fi)  = y +4fiy +6ft y +4Ji y+^  (AT )  5  = (y + Ji)  5  = y +5fiy +10B y +10fiy+5Ji y+J5  (AT )  6  = (y + Ji)  6  = y ^&fn5&y+26&Y+l5&¥+6E> y+E>  (AT )  7  k  2  k  k  4  = (y + fi)  7  -  2  3  3  k  k  2  2  4  3  5  4  2  2  2  2  (2.47)  3  3  (2.48)  4  3  4  6  (2.49)  5  5  (2.50)  6  = y +7fiy +21B y +35l5y+35B y +21B y +7B y+ft 7  6  2  (2.46)  5  4  3  5  2  6  7  (2.51)  By selecting as many terms of A Tk in Equation (2.44) as needed and choosing p such that the coefficient of the appropriate term in the catastrophe manifold equation becomes zero, different orders of catastrophe manifolds may be derived. This procedure is detailed in the following sections.  14  2.4.2Wigwam Catastrophe Selecting terms of A Y k to the sixth degree and normalizing A Y k , Equation (2.44) may 6  be put in the following form: K2  AT + 7  K.2  6  k  K.3  A ^-42 A ¥^-210  K2  K1+K3  A P +840A F +2520 A ^-5040 K3 K3 K3 x  3  k  x  2  k  - = 0 (2.52)  P, expand (y+P) where n = 1, 2,..,6, and make the coefficient n  Substitute A ^ = y +  of y  zero so that: K  2  K  3  6p + 7  = 0,  7K  or  (2.53)  2  P=  (2-54) 6K  3  Hence we get the wigwam manifold equation y + ty + uy + vy + wy + x = 0 6  4  3  ' '  2  (2.55)  and it may be shown that (see Appendix B) t  =  - 4 2 - 15p  (2.56)  u  =  4p(3-10p )  v  =  840 + P (288 - 45p )  w  -  - 1 2 p ( 4 0 - 3 1 p + 2p )  x  =  - p ( 1320 - 138p + 5 p ) - 5040  2  (2.57)  2  2  (2.58)  2  2  .  4  (2.59) K, + K  2  2  3  (2.60)  4  Note that K i is the only parameter which varies as A P varies; K and K remain constant. m  2  3  Hence, P remains constant and x is the only control parameter which varies with A P . m  15  5  2.4.3Butterfly Catastrophe Selecting terms of A T k to the fifth degree and normalizing A T k , Equation (2.44) may 5  be put in the following form: K3 AT  5 k  K3  - 6-  AT K  -30 A T  4 k  3 k  +120  AT K  2  Substitute A T k = y + p, expand  (y+P)  n  K1+K3 2 k  +360 A T  k  -720  =0  (2.61)  K2  2  where n = 1, 2,..,5, and make the coefficient of y  4  zero so that:  P  -  6K  3  (2.62) 5K  2  Hence we get the butterfly manifold equation y + uy + vy + wy + x = 0 5  3  (2.63)  2  where u  =  -30-10p  (2.64)  v  • =  w  =  5p (22 - 3 p ) + 360  x  =  -p ( 240 - 70p + 4p ) - 720  2  10P(l-2p )  (2.65)  2  2  (2.66)  2  K, 2  (2.67)  4  K  . .  2  Note again that K i is the only parameter which varies as A P varies; K and K remain m  constant.  2  3  Hence, P remains constant and x is the only control parameter which varies  with A P . m  16  2.4.4SwaIIowtail Catastrophe Selecting terms of A  to the fourth degree and normalizing A ^k , Equation (2.44) may 4  be put in the following form:  K  K  2  A^k + 5:  3  Substitute A ^ k = y +  A Y k + 120—:  2  K3  K1+K3  2  A ^ k - 20 A ^ k - 60  4  K3  =  0  (2:68)  K3  P, expand (y+P) where n = 1, 2,..,4, and make the coefficient n  of y  zero so that:  5K P  2  =  (2,69)  4K  '  3  Hence we get the swallowtail manifold equation 4 y  + vy + wy + x = 0  (2.70)  2  where v  -  - 20 - 6p  (2.71)  w  =  8P ( 1 - p )  2  (2.72)  2  K, + K x  =  3  P (28-3P )+ 1 2 0 2  2  (2.73) K  3  Note again that K\ is the only parameter which varies as A P varies; K and K remain m  constant.  2  3  Hence, p remains constant and x is the only control parameter which varies  with A P . m  17  3  2.4.5Cusp Catastrophe Selecting terms of A T to the third degree and normalizing A T k , Equation (2.44) may 3  k  be put in the following form:  K3 AT  K1+K3  - 4 -- —  3 k  AT  2 2 k k  - 1 2 A TTk ++ 24 24-— k  K? Substitute A T  k  =0  (2.74)  K 9  =  y + P, expand (y+P) where n = 1, 2,3, and make the coefficient n  of  y  zero so that: . 4K  3  (2.75) 3K  2  .  Hence we get the cusp manifold equation (2.76)  y + wy + x = 0 3  where  w  =  -12-3p  (2.77)  x  =  2P (3 - p ) + 24  2  Ki .  (2.78)  2  K  2  Note again that K i is the only parameter which varies as A P varies; K and K remain m  constant.  2  3  Hence, P remains constant and x is the only control parameter which varies  with A P . m  18  2  2.4.6FoId Catastrophe Selecting terms of A^Pk to the second degree, normalizing A ^ k , Equation (2.44) may 2  be put in the following form:  K AY  2 k  K1+K3  2  A T -6— k-6 AV  +3  —  k  K3  K  Substitute A Y k = y +  =0  (2.79)  3  P, expand (y+P) where n = 1, 2, and make the coefficient n  of y zero,  we find P as follows: 3K P  3  =  (2.80) 2K  -  2  Hence we get the fold manifold equation y +x=0  (2.81)  2  where K, + K .  x  =  3  - P - 6—  .  2  K  (2.82)  3  Note : The fold catastrophe will not be used for simulation tests because x is the only control parameter of this catastrophe and would require plotting of x control parameter and state variable y  ;  19  2.4.7Extended Equal Area Criterion (EEAC) Method The E E A C method will be used as the reference method to determine the accuracy of the simulation results obtained from the cusp, swallowtail, butterfly, and wigwam catastrophes for two test systems (three-machine WSCC and.seven-machine O G R E systems).  Critical  mechanical power input change, A P , of power systems can be determined when area A l m  is equal to area A2 in Figure 2.4.1. Mathematically, area A l and A2 can be evaluated as follows: T \ k  AreaAl  = $ [ P ° + A P - P - T sin(T +a )]dT m  m  c  k  k  k  (2.83)  k  T ° k  AreaA2  = - $ [P °+AP -P -T sin( P +a )]dT v  m  m  c  k  k  k  (2.84)  k  After algebraic manipulation, Area (A2 - A l ) can be determined by : Area (A2-A1) = T  [cos(W + a ) - c o s ( T ° + a ) ] - ( P ° + A P - P ) ( T - T ) (2.85) u  k  k  k  k  m  m  c  k  p  k  From energy standpoint, area A l is the kinetic energy generated by accelerating power due to mechanical power input change, APm and area A2 is the required potential energy which can be absorbed by the post disturbance system.  2.4.8Sunimary of catastrophe manifold and control parameters All control parameters of the cusp, swallowtail, butterfly, and wigwam catastrophes' manifold are summarized in Table 2.4.2.  20  ro ro + '  /-—s m ©  o  ^—\ CO.  o a  +  +  CN  c a ro  •  i  1  00 CS  o CS c a  /  c a • CO c a CS  r-H CN  Cd. 1  CN'  i  i—i *-—' '  CN  CN  o a o t-  © CN ro  CS +  o CS  CN  CQ. 00 ro  c a  CU +•>  co.  <u  e.  CO..  o  +  CO +  CN  o a *—i  o a  1  O  i—l  >  CN  CS CS  c a CN  .  si  •  c a CO  ro  >he Models: Manifold and Control Pan  :ATASTROPHE CONTROL PARAMETERS  ro  /-^  1  '  CO  o CS  CN c a CO 1  . CS 1  CN  o a  .ca  00  CN  c a  m  CN  t  c a  1  1—1  • o CS  c a o  CN  c a + O  CN  c a CS  00 00 <N  1  -  00  /—\  a  c a ©  i—i • ro  c a •*  O  l. •«-> V) « «  CN  c a o  t-H 1  o CO  u  . 1  cu  CS  CN  c a  • >ri  1  * J 1  CS  « cu CZ5 N-<  ca  i4 CN  • r14 . •  CN  >n  CS  CN  i4  ND  CO  •  1  CATAS TROPHE MODELS  -  o H  "II "w PH  II  i X  >  o  II X  o  +  >  II  + •  X +  +  >  >  +  l':  CU  «  o  c  fN  cu  s pa  .  ©  II X +  II X +  "3 :S  a.  o  • CJ  t/3  o  S  •2 "o  2.5 Simulation Results The catastrophe models described in Section 2.4 are applied to the three-machine WSCC system [11] and the seven-machine O G R E system [12].  These catastrophe models were  used to determine the critical mechanical power input, A P , to power systems before the m  system becomes unstable.  Results obtained from these catastrophes were compared with  that obtained from the E E A C method.  The E E A C method use the following criterion to  determine system stability: when area A1 exceeds area A2 in Figure 2.4.1, system will become unstable.  Critical A P  m  is determined when area A l is equal to area A2.  Refer  to Section 2.4.6 for detailed formulation of the E E A C method. Since no definitive method exists for assigning machines to be critical [1], exhaustive combination ( single, double, etc.) of machines were considered as the critical group in the simulation. Configurations of the three-machine WSCC and seven-machine CIGRE systems are shown in Figure 2.5.1 and 2.5.2.  18kV  System data are shown in Table 2.5.1 and 2.5.2.  230kV  LoadC  230kV  13.8kV  ®  ©  ®  Load A  LoadB •©  ©  Fig. 2.5.1 - Three-Machine WSCC System Configuration (three machines, nine buses)  22  Bus No.  Volt. Mag. /p.u.  1 2 3 4. 5 6 ' 7 8 9  1.04 1.02 1.02 1.02 0.99 1.01 1.02 1.01 1.03  .  Volt. Ang 0.00 9.30 4.70' -2.200 -4.000 -3.700 3.70 0.70 2.00  '  Real Gen. Power / p.u. 0.71 1.63 0.85 0.00 0.00 • 0.00 • 0.00 0.00 0.00  Imag.Gen Pwr /p.u. 0.27 • 0.06 -0.109 0.00 0.00 0.00 0.00 0.00 0.00  Generator Data  RealLoad Power /p.u. 0.00 0.00 0.00 0.00 1.25 0.90 0.00 1.00 0.00  Imag. Load Power /p.u.  •  0.00 0.00 0.00 0.00 0.50 0.30 0.00 0.35 0.00  R+X'd /p.u. 0.000 0.000 0.000  0.060 0.119 0.181  •  _ _  _  _  •  _  _ _  • _  _  Transimission Line Bu From  •  1 2 3 4 4 5 6 7 8  Admittance To 4 . •7 9 5 6 7 9 8 9  0.00 0.00 0.00 1.365 1.942 1.187 1.28 1.617 1.155  Shunt/2  . • _  -  _  -  0.00 0.00 0.00 0.00 0.00 0.00  _ _  .• 0.08 0.07 0.15 0.17 0.074 0.104  Table 2.5.1: Three-Machine WSCC System - Pre-Disturbance System Data  Fig. 2.5.2  Seven-Machine CIGRE System Configuration ( seven machines, seventeen buses )  23  CIGRE system data before disturbance (7 machines 17 buses) Generator Bus Pbase X ' M Pm E Angle (MVA) (%) > (MWsVrad) (MW) (D.U.) (dee) 1 100 7.4 6.02 217 1.106 7.9 2 100 11.8 4.11 120 1.156 -0.2 3 100 6.2 7.59 256 1.098 6.5 4 100 ' 4.9 9.54 300 3.9 •1.11 5 100 7.4 6.02 230 1.118 7 6 100 7.1 6.77 160 1.039 3.6 7 100 8.7 5.68 174 1.054 7.9 Loads Bus-" p O Bus P O (MW) (Mvar) (MW) (Mvar) 17 200 120 9 100 50 - 13 650 405 . 11 230 140 10 80 30 • 15 • 90 45 8 90 40 Transmission Line Data • Bus ' R X wC/2 From To (ohm) (ohm) (micro S) 16 12 5 24.5 . 200 16 13 5 24.5 100 17 12 22.8 62.6 200 17 15 8.3 32.3 •• 300 12 13 6 39.5 300 12 11 5.8 28 200 13 14 2 10 200 13 10 • 3.8 10 ' 1200 13 11 24.7 97 200 13 15 8.3 33 300 10 ,9.5 31.8 9 200 8 9 •6 39.5 300 9 11 24.7 97 200 Note: These values include the transformer's reactances and are expressed on a 100 MVA ( 1  (1)  Table 2.5.2: Seven-Machine CIGRE System - Pre-Disturbance System Data  24  2.5.1Three-Machine WSCC System For the WSCC system, different combinations of machine(s) are grouped together to form the critical machine group and the non-critical machine group. These combination groups are applied for the cusp, swallowtail, butterfly, and wigwam catastrophe models and a typical simulation result is summarized in Appendix C. Mechanical Power Input is the implicit state variable for catastrophe manifolds and followings are the observation: 1.  Catastrophe bifurcation set describes a definite envelope for system stability region boundary.  Different catastrophe has its own bifurcation envelope for control  parameters w and x, and these envelopes are graphed in Figures 2.5.3, 2.5.4, 2.5.5, 2.5,6, and 2.5.7.  0  / \  C u s p Catastrophe Bifurcation Envelope  \  / ^PM  •%  S  M o t o r Stable Region  increase  /  —  \  ^  jBenerator Stable R e g i o n /  >.  X  Fig 2.5.3: The Cusp Catastrophe Bifurcation Set For Steady-State Stability (Plot of w vs. x Parameters)  25  0  w  CtXnerator R egion N  s  Pm increase  \  (  ^X^x^^^V^  ^ ' ^  ^  >Ss  >^ s  S w alio w ta il C a tastrophe B ifurcation Envelope  ^ ^ ^ ^ ^**>>^  X  Fig 2.5.4:The  Swallowtail  Catastrophe Bifurcation Set For Steady-State  Stability (Plot of w vs. x Parameter)  .  / ^^^*s«^/  Generator Regions Pirymcrease  /  /  r  Butterfly Catastrophe Bifurcation —H)C Envelope ^ ^ C / ^  X  Fig 2.5.5: The Butterfly Catastrophe Bifurcation Set For Steady-State Stability (Plot of w vs. x Parameter)  26  •  X-  ^ Ji/  Pm increase \ Generator  Wigwam Catastrophe Bifurcation . Envelope  \ o  Fig 2.5.6: The Wigwam Catastrophe Bifurcation Set For Steady-State Stability (Plot of w vs. x Parameter) - Generator  o  w Motor stability Pjn-4rrCfease  / \^ \^  /  Wigwam Catastrophe Bifurcation Envelope  X  Fig 2.5.7: The Wigwam Catastrophe Bifurcation Set For Steady-State Stability (Plot of w vs. x Parameter) - Motor  27  2.  For the three-machine WSCC system, there are six possible critical machine group combinations, namely three (3) one-machine and three (3) two-machines groups.  The  combination with the least mechanical power input change prior to reaching the unstable region is taken to be the critical machine group for one-machine-infinite-bus (OMIB) model. From our test, Machine 2 as the sole critical machine is the proper combination choice. 3.  If Machine 2 is used as the critical machine for OMIB model and acts as a generator, then the choice using Machines 1 and 3 as critical machines would have mirror performance in the bifurcation plane but acting as an equivalent motor.  4.  The cusp catastrophe bifurcation plane (x-w plane) is described in Figure 2.5.3 which shows that: a.  Generator lies onrighthalf plane (i.e. x > 0) while motor lies on in the left half plane.  b.  Bifurcation set divides the x-w plane into stable and unstable region.  Shaded area in  Figure. 2.5.3 denotes the stable region. c.  When mechanical power input of the system increases,  value of 'w' parameter  remains constant and that of x parameter increases. d.  Bifurcation curve in the 'x-w' plane is symmetrical at line x equals zero.  e.  If w parameter remains constant, two possible x limit values can be found from the bifurcation curve; these values determine a transition value of a system from stable operation to unstable operation.  5.  The swallowtail catastrophe bifurcation plane is described in Figure 2.5.4 which shows that: a.  Generator lies on the lower half of 'x-w' plane (i.e. w < 0) while motor lies on the upper half plane.  b.  Similar to the cusp catastrophe, the swallowtail bifurcation set also defines a stable envelope for system stability; shaded area defines the stable region for generators.  28  c.  For a particular critical machine group, u and v parameters remain constant and x parameter varies as mechanical power input of the system varies (In the 'x-w' plane, if w parameter is greater than zero, x parameter increases as mechanical power input of the system increases. However, if w parameter is less than zero, x parameter will decrease as mechanical power input of the system increases.).  d.  Bifurcation curve in 'x-w' plane is symmetrical at line w equals zero.  e.  Three possible limit values of x parameter can be found if that of w parameter remains constant.  The butterfly catastrophe bifurcation plane is described in Figure 2.5.5, which shows that: a.  Generator lies on the left half of the 'x-w' plane (i.e. w < 0) while motor lies on the right half plane.  b.  Butterfly bifurcation curve also defines a stable region for system stability.  Shaded  area defines the stable region generators. c.  For a particular critical machine group, values of u, v and w parameters remain constant and that of x parameter varies as mechanical power input of the system varies;  specifically, x parameter decreases as mechanical power input of the system  increases. d.  Butterfly stability envelope is similar to that of the cusp envelope except that the left half plane and the right half plane of the bifurcation curve is not symmetrical at x equals zero (Note also that if v parameter changes sign, bifurcation curve in the left half of 'x-w' plane will sits on the right half plane while that on the right half plane will sits on the left).  e.  Four possible limit values of x parameter can be found from the bifurcation curve if w parameter remains constant.  29  7.  The wigwam catastrophe bifurcation plane is described in Figure 2.5.6 and 2.5.7, which shows that: Shaded area in Figure 2.5.6 denotes generator stable region (if parameter u is negative) and shaded area in Figure 2.5.7 denotes motor stable region (if parameter u is positive). For a particular critical machine group, values of u, v and w parameters remain constant and that of x varies as mechanical power input of the system varies; specifically, x parameter value increases (if u is negative) or decreases (if u is positive) as mechanical power input of the system increases. c.  Five possible limit values of x parameter can be found from the bifurcation curve if w parameter remains constant.  8.  Different catastrophe bifurcation envelopes are used to determine the maximum change of mechanical power input in power systems before instability occurs and results are compared with that obtained from the E E A C method.  Table 2.5.3. summarizes the results.  CRITICAL MACHINES  2  3  2 and 3  EEAC (AP )  1.387184 p.u  1.460789 p.u.  1.484128 p.u.  m  CATASTROPHE MODELS  AP .  % Error  AP .  % Error  AP .  % Error  Cusp  1.27761 p.u.  7.899%  1.3396 p.u.  8.296%  1.36613 p.u.  7.951%  Swallowtail  1.3342 p.u.  3.820%  1.3811 p.u.  5.455%  1.428 p.u.  3.782%  Butterfly  1.3993 p.u.  -0.873%  1.4779 p.u.  -1.171%  1.4974 p.u.  -0.894%  Wigwam  1.3914 p.u.  -0.304%  1.4696 p.u.  -0.603%  1.494 p.u.  -0.665%  m  m  m  Table 2.5.3: Three-Machine WSCC System-Critical Mechanical Power Input change (Steady-State Stability) Determined by (a) Different Catastrophe Models (b) The E E A C Method Note: %Error = [(EEAC Value - Catastrophe Value)  30  E E A C Value ) X 100%  With reference to Figure 2.5.3, one can conclude that the wigwam catastrophe has a highest accuracy while the cusp has the least accuracy when compare with values obtained by the E E A C method.  Although the butterfly and the wigwam catastrophes give a better accuracy,  their values are greater than that of the E E A C solution. wigwam is used for stability assessment.  This is unsafe if the butterfly or the  However, if a bias value is added to the butterfly or  the wigwam catastrophe for safety margin, these envelopes may provide better accuracy for stability assessment.  2.5.2Seven-machine CIGRE System a.  The procedure is repeated for the seven-machine CIGRE system to determine the maximum allowable mechanical power input change of the system. The cusp catastrophe is applied to different critical machine combinations to determine the proper critical machine group.  Minimum change in mechanical power input of the system is used as  selection criterion.  It is found that Machine 7 is the sole critical machine group.  The  wigwam, butterfly, swallowtail and cusp catastrophes are then applied to this particular critical group for detailed study.  Table 2.5.4 summarizes the test results.  (See  Appendix D) . CRITICAL M A C H I N E  7  E E A C Solution ( A P )  2.80473 p.u.  m  CATASTROPHE MODELS  AP  Cusp  2.57149 p.u.  Swallowtail  2.62972213 p.u.  6.240%  Butterfly  2.84079 p.u.  -1.286%.  Wigwam  2.82713 p.u.  -0.799%  % Error  m  .8,316%  Table: 2.5.4: Seven-Machine CIGRE System-Critical Mechanical Power Input change (Steady-State Stability) Determined by (a) Different Catastrophe Models (b) The E E A C Method Note: %Error = [(EEAC Value - Catastrophe Value)  31  E E A C Value ) X 100%  2.6 Conclusions 1.  Catastrophe theory has been shown in the thesis to define a steady-state stability region of multimachine power systems subjected to mechanical power input change.  2.  Different catastrophe models can be applied in power systems to determine the critical mechanical power input change before system instability occur.  Results show good  agreement with that obtained from EEAC method. 3.  Using EEAC method as bench mark of comparison, the thesis also concluded that higher order catastrophe such as the wigwam and the butterfly show better accuracy. However, without overshooting the stability value,  the swallowtail catastrophe  proved to be adequate for stability assessment, and the cusp catastrophe provides a clear envelope in visualizing power system stability region.  32  CHAPTER T H R E E APPLICATION OF CATASTROPHE T H E O R Y M O D E L T O TRANSIENT STABILITY ANALYSIS OF MULTIMACHINE POWER SYSTEM  3.1 Introduction Catastrophe theory model for steady-state analysis of multimachine power systems with variation of mechanical power input has been described in Chapter Two.  In this chapter,  catastrophe models of interconnected multimachine power system is proposed for the study of transient stability with balance three-phase faults at different power system locations. Wvong, Mihiring and Parsi-Feraidoonian [14,15,16] proposed to use the swallowtail catastrophe to study transient stability of power systems, but research did not identify the critical clearing angles.  The cusp catastrophe will be developed in this chapter to  determine the transient stability of power systems byfindingthe critical clearing angle.  In Section 3.2, catastrophe theory models for transient stability analysis of multimachine power systems for balanced three-phase faults will be developed.  In Section 3.3 , the  cusp catastrophe will be applied to the three-machine WSCC and seven-machine CIGRE systems.  Exhaustive simulation were made assuming various machines to be critical in  single-machine, two-machine, etc. because no definitive method exists for determining criticality [1].  Test results and observation will be discussed in Section 3.4 ( typical  simulation result is presented in Appendix E and F). Section 3.5.  33  Conclusions will be stated in  3.2 Catastrophe Theory Model for Three Phase Fault  A n g L i n R i d . n .  Figure 3.2.1: Multimachine Power System with BalancedThree-Phase Fault (OMTB) During the transient period, an exchange of energy takes place between the rotor of a critical machine (or a group of critical machine group) and the post-fault power system network.  The kinetic energy generated by the accelerating power during the fault-on  period must be fully absorbed by the post-fault network in order to maintain stability.  The  kinetic energy and the potential energy which can be absorbed by the post fault network can be well described by Figure 3.2.1 With reference to Figure 3.2.1, and using the extended equal-area stability priterion, of Equation (2.13), we have:  S [Pm""- P Y ° k  flt c  -T  flt k  s i n ( Y + c O ] d ¥ + S [P ^ k  k  c  k  34  pre m  -Pc -T pos  pos k  sin(Y +a k  pos k  )] d ¥ = 0 (3.1) k  For the limiting case of the system to be stable, i.e. (PnT - P ) ( T - T ° ) + T cos ( T + <x ) - T cos(T ° + a ) + O V " - P c H (Tk" - T ) + IV cosCF + a ) - IV cos(T + aiT) = 0 (3.2) flt  c  c  .  flt  k  k  c  k  flt  k  c  flt  k  03  u  k  pos  k  flt  k  k  k  08  k  c  k  By substitution of W = Tk + A T , and with some algebraic manipulation, Equation (3.2) c  k  may be put in the form of: Ki A T + K cos( A T ) + K sin( A Y ) + K4 .= 0 k  2  k  3  (3.3)  k  where pro  p  W  =  sin  ;  1  Tkprc  pre :  a  =P  K  2  =T  K  3  = - Tk" sin(T + c O  pre  pos k  (3.5)  08  cos(T  +a  c k  03  K4  (3.4)  pre k  - •'  Ki  m  -Pc"  p  pos k  )  (3.6) (3.7)  c  k  =(P  pre m  + T  -Pe )(T -HV) m  c  k  flt k  cos ( T  c k  + a ) -T flt  k  flt k  cos(T ° +a ) - T V cos(T + c O flt  k  k  05  c  k  (3.8)  and the superscript pre  =  pre-fault value  (39)  fit  =  during fault value  (3.10)  pos  =  post fault value  (3.11)  Note that parameters K , K , and K 4 varies, and Ki remains constant, with clearing angle, 2  3  Tk . c  35  3.2.1Series Expansion of sin(A Fk) and cosfATk) ,  A Y k is unknown in Equation (3.3) and would normally require an iterative approach to solve.  The iterative approach can be avoided by use of catastrophe theory after series  expansion of s i ^ A ^ k ) and cos(AYk) to get: A*F K,AT +K k  AHV A ¥  k  A¥  6 k  +K  2  2!  .  AT  3  +  k  6!  4!  A¥  3 k  3!  5 k  +.  + BC, = 0  (3.12)  5!  Let AT  : k  y + fi>  (3.13)  so that (A^k)  2  = (y + ii)  (AY,)  3  = (y + fi)  (A*F ) k  4  2  3  = (y +  Ii)  4  (A¥ )  5  = (y + fi)  (A¥ )  6  = (y + fi)  k  k  5  6  = y +2fly+fl 2  (3.14)  2  = y +3fiy +3fl y+fj 3  2  2  3  (3.15)  •= y +4i5y +6fl y +4fi y+fi 4  3  2  2  3  = y +5fiy +10B y +l 5  4  2  3  (3.16)  4  OBy+SIiV+fi  5  Q.IT)  = y +6fiy +l 5fi y +20I5 y +l 5 B y + 6 f i y + B 6  5  2  4  3  3  5  6  (3.18)  By selecting as many terms of ( A Y k ) in Equation (3.12) as needed and choosing p such that the coefficient of the appropriate term in the catastrophe manifold equation becomes zero, different catastrophe models may be derived.  36  3.2.2Wigwam Catastrophe Selecting terms of A T k to the sixth degree and normalizing A T  6 k  , Equation (3.12) may  be put in the following form: K3  K3  A»Pk -6  A ^ k -30ATk +120  6  5  K i + K3  AT  4  3 k  +360A*F -720 k  K-2  K2  K2+K4  A f -720 r-  2  "='0 (3.19)  k  K.2  K.2  Substitute A T k = y + p\ expand (y+P) where n = 1, 2,..,6, and make the coefficient of y n  5  zero so that: K  3  K  2  - = 0,  6 0-6  K  3  K  2  or  .  ,  • (3.20)  (3.21)  Hence, we get the wigwam manifold equation y + ty + uy + vy + wy + x = 0 6  4  3  (3.22)  2  and it may be shown that t  =  -15 (2+p )  (3.23)  u  =  - 40 p  (3.24)  v  =  45(8 + 4 p - p )  w  =  24p ( 10 - p ) - 720  2  3  2  ,  4  (3.25)  K, 3  (3.26)  2  K  2  K + K,P —-)• K 4  x  =  -5P (72-18p 2  2  + p )-720( 1+ 4  (3.27)  2  Note that P is a function of K and K ; K , K ,and K 4 are functions of clearing angle ( T ) ; c  2  3  2  3  k  Therefore, the wigwam control parameters t, u, v, w, and x varies with T . c  k  37  3.2.3Butterfly Catastrophe Selecting terms of A^Fk to the fifth degree and normalizing A ^ k , Equation (3.12) may 5  be put in the following form: K2  K.2 AT  5 k  +5  A T^-20 A 4^-60  A ¥^+120  K  K3  Substitute A ¥  K2+K4  K.1+K3 A¥ +120  — =0  k  K  3  3  (3.28)  K3  = y + p\ expand (y+P) where n = 1, 2,..,5, and make the coefficient of y n  k  zero so that: K  2  P  (3.29) K  3  Hence, we get the butterfly manifold equation y + uy + vy +wy + x = 0 5  3  (3.30)  2  where u  =  - 10(2 + p )  (3.31)  -20P  (3.32)  2  3  v  K, w  =  15p (4-p )+120(1+^—) K ; 2  (3.33)  2  3  K4+K1P x  =  4P ( 1 0 - p ) + 120 3  (3.34)  2  K  •  3  Note that P is a function of K and K ; K , K ,and K 4 are functions of clearing angle (^k ). 0  2  3  2  3  Therefore, the butterfly control parameters u, v, w, and x varies with %  38  4  3.2.4SwalIowtaiICatastrophe Selecting terms of A T to the fourth degree and normalizing A Tk , Equation (3.12) may 4  k  be put in the following form: K -4— A T  K1+K3  3  AT  4 k  K Substitute A T  k  + 24 ——  2 k  K  2  =y+  k  - 12AT  3  K +K4 • 2  -AT  k  + 24  K  2  =0  (3.35)  2  P, expand (y+P) where n = 1, 2,..,4, and make the coefficient n  of y  zero so that: K p  3  =  (3.36) K  2  Hence we get the swallowtail manifold equation y + vy + wy + x = 0 4  (3.37)  2  where v  =  -6(2+p )  (3.38)  w  =  - 8p + 24 K, / K  x  =  3p ( 4 - p ) + 24( 1 +—• K  2  (3.39)  2  2  K4 + K,p 2  )  2  .  (3.40)  2  Note that P is a function of K and K ; K , K ,and K 4 are functions of clearing angle (Tk ). C  2  3  2  3  Therefore, the swallowtail control parameters v, w, and x varies with Tk . c  39  3  3.2.5Cusp Catastrophe Selecting terms of A T  to the third degree and normalizing A T k , Equation (3.12) may 3  k  be put in the following form: K1+K3  K.2  ATk + 3  ATk -6  3  K  K3  Substitute A T k = y +  K2+K4  A T  2  k  - 6 — — K  3  =0  (3.41)  3  P, expand (y+P) where n = 1, 2,.3, and make the coefficient n  of y  zero, we find P as follows: K p  2  =  (3.42) K  3  Hence we get the cusp manifold equation y +wy + x = 0  (3.43)  3  where  w  =  -3p + 6 ( l + - — ) K3  x  =  2p -6  (3.44)  2  K4+K1P (3.45)  3  K  3  Note that P is a function of K and K ; K , K ,and K 4 are functions of clearing angle (Tk°). 2  3  2  3  Therefore, the cusp control parameters w and x varies with Tk . C  In order to visualize  transient.stability in two dimensional study, only the cusp catastrophe will be applied in the two test systems.  The wigwam, butterfly and swallowtail catastrophes will not be used  since they require more than two dimensional data.  40  2  3.2.6Fold Catastrophe Selecting terms of A * F to the second degree and normalizing A^k, k  Equation (3.12)  may be put in the following form: K1+K3 A^Fk -2•  K.2+K4 A4V2-  2  K2  =0  (3.46)  K2  Substitute A ^ k = y +  P, expand (y+P) where n = 1, 2, and make the coefficient of y zero n  sot that:  K,+K  3  p  =  (3.47)  K  2  Hence we get the fold manifold equation y +x=0  (3.48)  2  where  x  :  - p - 2( 1 + - — )  (3.49)  2  K  2  Note : The fold catastrophe will not be used on the test systems because x is the only control parameter of this catastrophe and would require plotting of x control parameter and the state variable y.  •  41  3.2.7Extended Equal Area Criterion (EEAC) Method The E E A C method will be used as the reference method to determine the accuracy of the cusp catastrophe simulation result.  Refer to Figure 3.2.1, power systems will remain  stable if area 1 is less thanor equal to area 2.  Critical clearing angle of the system can be  determined when area 1 is equal to area 2. Mathematically, area 1 and 2 can be evaluated as follows:  Areal  = S[P„r - P  Area 2  = - $ [PnT + Pc" - T ^ sin(^ +a )]dT  flt c  -T  flt k  sin^+a^d^  (3.50)  4V 05  8  k  k  k  (3.51)  Refer to Section 3.2 for symbol  T ^ c o s W + k  +T  flt k  cO-Tk^ro^ cos(4V + a VT^^ f l  k  (3.52)  From energy standpoint, area 1 can be described as kinetic energy generated by the accelerating power during the fault on period and area 2 can be described as the required potential energy which can be absorbed by post fault power systems.  When the post fault  potential energy of the system is larger than the kinetic energy generated during fault, the system must be stable.  3.2.8Summary of Catastrophe Models Different catastrophe models and control parameters for transient stability analysis are summarized in Table 3.3.1  42  CO. c a  / \ CQ.  o  +  /-  N  +V  o CN  CN  ca + ca oo  ca • o  CN  ca  +  CN  +  ca CN  u  <u  ca CO  E  CQ.  C3 J-  e s P k o -M  H  ©  CN  . o  r-  CN +  CQ.  CO.  "*  e U •o a  +  4  ca oo  +  es  TS  ca CO  CN  a es  H cn  O  ca +  ca o CN  ca +  •a o  CN  00  a o  u cn  ca o  es  ca +  e s  CN  u  Q . es  (Z)  c  cu es  u  ca  *4  i4  Ui +  H CN  ro  UJ Q  +  O.  +•  es  o  H •  +  w  en  O  .If  + +  +  +  H H  >  + + .  x +  s  o  •v.  •a  ti C s  ea  o CO  U  3.3 Simulation Result Catastrophe models as described in Section. 3,2 were developed for transient stability analysis of a multimachine power system.  The stability boundary of different catastrophe  manifold described in the previous section are dependent upon the parameters K i , K , K , 2  3  K 4 and only K i is independent of the change in the clearing angle , 4 V . Therefore, for the sake of clarity,  only the cusp catastrophe model is used to study the three-machine  W S C C and seven-machine CIGRE system.  Configurations of these test systems were  shown in Figure 2.5.1, Figure 2.5.2 , and system data were listed in Table 2.5.1 , and Table 2.5.2.  Note that critical clearing angles of the system obtained by the cusp catastrophe  will be compared with that obtained from the E E A C method.  Refer to Section 2.4.7 for  the E E A C method.  3.3.1.Three-Machine WSCC System Different combinations of machine(s) are grouped together to form the critical group and non-critical group.  These combinations are applied for the cusp catastrophe model and a  typical test result are summarized in Appendix E .  Clearing angle, 4 V , is the implicit state  variable for this manifold and following are the observations:  1.  The cusp catastrophe bifurcation set defines an envelope for the stability region. A typical stability envelope is shown in Figure 3.3.1.  44  x Parameter  Figure 3.3.1: The Cusp Bifurcation Set for Transient Stability (Plot of w vs. x Parameters) Comparison of Simulation Result with the E E A C solution Fault Line Node Open 5 4-5 5 4-5 5 4-5 5 5-7 5 5-7 5 5-7 6 6-9 6 6-9 6 6-9 6 4-6 6 4-6 6 4-6 8 7-8 ' 8 7-8 8 7-8 8 8-9 8 8-9 8 . 8-9  Critical Machine 2 3 2&3 2 3 2&3 2 3 2&3 2 3 2&3 2 3 2&3 2 3 2&3  EEAC solution (Chg. In Clearing Angle) 2.0184 Stable 1.6260 1.8448 Stable 1.1513 Stable Stable 1.3494 Stable Stable 1.6706 1.1212 2.2542 1.4955 1.3480 .2.1667 1.5258  Cusp Catastrophe Percentage (Chg. In Clearing Angle) Error 2.0149 0.1737% Stable Stable 1.6069 1.1886% 1.8411 0.2010% Stable Stable 1.1428 0.7438% Stable Stable Stable Stable 1.3372 0.9124% Stable Stable Stable Stable 1.6568 0.8329% 1.1121 0.8183% 2.2510 0.1422% 1.4783 1.1635% 1.3304 1.3229% 2.1638 0.1340% 1.5084 1.1535%  Note: Stable means never unstable Table 3.3.1: Three-Machine WSCC System - Critical Clearing Angle . (Transient Stability) Determined by (a) The Cusp Catastrophe (b) The E E A C Method  45  3.  A typical cusp bifurcation envelope is drawn in Figure 3.3.1 to define a post fault transient stability region of the system.  The locus of catastrophe control variables (x  and w) change as increment of clearing angle  - 4V).  The system becomes  unstable when the locus crosses the cusp envelope.  Critical clearing angle obtained  from the cusp result is compared with that of E E A C method and results are summarized in Table 3.3.1  In general, critical clearing angle obtained from the cusp  catastrophe is smaller than that of E E A C method and the percentage error is in the order of one (1) percentage. 4.  If the critical group of power systems can be identified successfully, the cusp catastrophe can be used successfully to determine the post fault stability in term of critical clearing angle.  Note that the cusp catastrophe provide a graphical tool for  operator to determine post fault instability may occur.  From engineering standpoint,  the cusp catastrophe can even provide a better safety margin that that of E E A C method.  3.4.2Seven-Machine CIGRE System 1.  The cusp catastrophe is then applied to the seven-machine CIGRE system to determine the maximum allowable clearing angle for balanced three-phase faults at different buses.  Simulation results are compared with that obtained from E E A C  method and results are summarized in Table 3.3.2. Fault Line Node Open 9 9 9 9 9 9 9 9 9 9 9  Critical Machine  9-10 ,9-10 9-10 9-10 9-10 9-10  1 2 3 4 5 ' 6  9-10  7  9-11 9-11 9-11 9-11  1  E E A C solution Cusp Catastrophe Percentage Chg. In Clearing Angle (Chg. In Clearing Angle) Error Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable Stable 1.638007 1.609833 1.7501% Stable Stable  2 3 4  Stable Stable  46  Stable  Stable  Stable Stable Stable  Stable Stable Stable  9  9-11  9 9  9-11 9-11  10  9-10  10 10  9-10  5 6  Stable  Stable  Stable  7  . Stable 1.922813  Stable 1.902927  Stable 1.0450%  1  Stable  Stable  Stable  9-10  2 3  Stable Stable  Stable Stable  10  9-10  . 4  Stable  Stable  Stable Stable Stable  10 10 10  9-10  5  9-10 9-10  6 7  Stable 2.283991 Stable  Stable 2.270053 Stable  Stable .0.6140% Stable  10 10 10  10-13 10-13 10-13 10-13 10-13 10-13 10-13  1 2 3  Stable Stable Stable  4 5 6  Stable Stable 1.955282  Stable Stable Stable Stable Stable 1.927501  Stable Stable Stable Stable Stable 1.4413%  7  Stable  Stable  Stable  Stable Stable 2.093896  Stable Stable 2.077922  Stable Stable 0.7687%  11-12 11-12 11-12  1 2 3 4 5 6 7  Stable Stable Stable Stable  Stable Stable Stable Stable  Stable Stable  12 12  12-16 12-16  1 2  12-16  3  Stable Stable 2.007387  Stable Stable  12  Stable Stable 2.025550  0.9048%  12 12  12-16 12-16  4 5  Stable Stable  Stable  Stable Stable  12  12-16  Stable  12  12-16  6 7  Stable Stable  Stable  Stable  Stable  12 12 12  12-17 12-17 12-17 12-17 12-17 12-17  1 2 3 4 5  Stable Stable 2.081358  Stable  Stable  Stable 2.064949  Stable .0.7946%  Stable Stable Stable Stable  Stable Stable  10 10 10 10 12 12 12 12 12 12 12  12 ' 12 12 12 13  11-12 11-12 11-12 11-12  Stable Stable Stable  . .  Stable Stable  Stable  12-17 10-13  6 7 1  13 13 13  10-13 10-13 10-13  2 3 4  Stable 2.396633  Stable 2.387382 .  Stable Stable 0.3875%  13  10-13  5  2.328945  2.323067  0.2530%  Stable Stable Stable  47  Stable Stable  Stable Stable Stable  13 13  10-13  6 7  13  10-13 10-13  13  11-13  13 13 13 13 13 13 13 13 13  11-13 11-13 11-13  13 13 13  12-13 12-13 12-13 12-13 12-13 12-13 13-16 13-16 13-16 13-16 13-16 13-16  Stable Stable 2.299119  Stable Stable 2.284525  Stable Stable Stable 2.414661 2.348639 Stable Stable 2.352596  Stable Stable Stable 2.406559 2.343071 Stable Stable 2.341330  Stable Stable  Stable Stable  3 4 5  Stable 2.411632 2.342330  Stable 2.403241 2.336671  Stable Stable 2.344713 Stable Stable Stable 2.402353 2.331417  Stable Stable 2.332718 Stable Stable Stable 2.393656 2.325673  Stable Stable 0.5142%  Stable  13-16 13-16  6 7 4&5 1 2 3 4 5 6 7 4&5  Stable Stable 2.318102  Stable Stable 0.5542%  16 16  12-16 12-16  1 2  1.931665 Stable  1.911999  1.0286%  Stable  Stable  16  12-16 12-16  3  Stable  16  4  Stable  Stable Stable  Stable Stable  16  12-16  5  Stable  Stable  Stable  16 16  12-16  6 7  Stable  Stable  Stable  Stable 1.901050  Stable 1.880496  Stable 1.0930%  Stable Stable Stable Stable Stable  Stable Stable  Stable  13 13 13 13 13 13 13 13 13 13 13  16 16 16 16 16 16 16 17. 17  1.1-13 11-13 11-13 11-13 12-13 12-13  12-16 13-16  4& 5 1 2 3 4 5 . 6 7 4&5 1 2  13-16 13-16  1 2 3  13-16 13-16 13-16 13-16  4 5 6 7  12-17  1 2  12-17  Stable 2.330948  -  48  Stable  Stable Stable Stable Stable  Stable 2.148111  Stable 2.121857  Stable Stable 0.6388% Stable Stable Stable 0.3367% 0.2376% Stable Stable 0.4812% Stable Stable Stable 0.3492% 0.2422%  Stable Stable Stable 0.3633% 0.2470%  Stable Stable Stable Stable Stable Stable 1.2373%  17 17. 17  12-17  .3 4 5  Stable  1.2-17  6 7  17  15-17  1  17 17  15-17 15-17  2 3  17 17  15-17 15-17  4 5  17 17  15-17 15-17  6 7  17 17  12-17 12-17 12-17  Stable Stable Stable  Stable  Stable Stable  Stable Stable  Stable Stable  Stable 2.243431  Stable 2.216019  Stable 1.2370%  Stable  Stable  Stable Stable  Stable Stable  Stable Stable Stable  Stable Stable  Stable Stable  Stable Stable  Stable Stable  Stable Stable  Note: Stable means never unstable Table 3.3.2: Three-Machine WSCC System - Critical Clearing Angle (Transient Stability) Determined by (a) The Cusp Catastrophe (b) The E E A C Method  3.4. Conclusions 1.  Catastrophe theory can be used to define the transient stability region of a multimachine power system provided that proper combination of critical machine(s) can be identified.  Further research is required to identify critical machine(s) in order  for the E E A C and catastrophe model to applied successfully. 2.  As observed from the simulation result, the cusp manifold of transient catastrophe model shows a close agreement with the extended equal area critical in identifying (1) the stability region, and (2) critical clearing angle.  Once the critical angle can  identified, clearing time can be calculated as stated in [1]. 3.  Since K , K ,and K 4 of the catastrophe energy equation developed in this paper are 2  3  dependent on the system clearing angle. catastrophe in a two-dimensional study.  This restricts our research to the cusp Although the application of catastrophe is  only.applied in the cusp catastrophe, it shows a very good agreement with result obtained from E E A C method.  Hence, the cusp catastrophe can provide a alternative  means to visualize transient stability of power systems.  49  Further research can be done in this area to explore whether some alternative formulation of catastrophe envelope can be found to provide two dimensional study for the wigwam, butterfly, swallowtail catastrophes.  50  CHAPTER 4 CONCLUSIONS In this thesis, the energy function was efficiently used with catastrophe theory to define comprehensive steady-state and transient stability regions for power systems.  Explicit  boundaries of the stability regions were identified by the bifurcation set of catastrophe manifolds.  For steady-state stability application, different catastrophe models were  successfully used to define system stability regions and to determine the critical mechanical power inputs of the system before instability occur.  For three-phase fault application, the  cusp catastrophe was developed not only to find transient stability boundaries but also to determine critical clearing angle of power systems.  Critical mechanical input power of the  steady-state study and critical clearing angle of the transient stability study were also found by the E E A C method (refer to Section 2.47, and Section 3.2.7  for details).  Results obtained  from E E A C were used to compared with to that obtained from catastrophe results. catastrophe result shows good agreement with the E E A C method.  The  The catastrophe method  is made practical if the concept of critical machines for a specified fault can be identified successfully.  ,  This thesis investigated the application of catastrophe theory to stability problem of multimachine power systems.  The first half of the thesis studied the application of  catastrophe theory to power system subjected to variation of mechanical power input.  By  suitable selection of Taylor series expansion of balanced energy equation, different orders of catastrophe manifold such as the cusp, swallowtail, butterfly, and wigwam can be developed to study the steady-state stability regions.  The study shows that catastrophe theory can be  successfully used to define the stability region and to determine maximum mechanical power input of power systems before system instability occur.. Higher order catastrophes, in general, show better accuracy than that of lower order.  The swallowtail catastrophe,  however, provide adequate result without overshooting the stability boundary. In the second half of the thesis, the cusp catastrophe was applied to study transient stability for symmetrical three phase fault.  Because the values of catastrophe parameters K , K , and 2  51.  3  K 4 are dependent upon system clearing angle, only study.  the cusp catastrophe was used in the  Transient stability limits can be successfully described by the bifurcation set of the  cusp catastrophe.  Critical clearing angle can be found at the intersection between the  clearing angle locus and cusp catastrophe's bifurcation envelope.  The result is in good  agreement (in the order of one percent) with those obtained by the E E A C method.. The major contributions and conclusions of this research are the followings: 1.  For the first time, different orders of catastrophe manifold are applied to the transient, stability problem.  The cusp, swallowtail, butterfly, and wigwam catastrophe can be  used to define transient stability regions (power systems with mechanical input as loading disturbance). 2.  The thesis concluded that higher order catastrophes have better accuracy than the lower order catastrophe (steady-state application). may give results above the actual limits.  However, the higher order catastrophe  Overshooting depends on the formulation of  Taylor series expansion of balanced energy equation.  The swallowtail catastrophe  developed in this thesis for the steady-state study proved to have the best accuracy without overshooting the actual value.  From engineering standpoint, The swallowtail  catastrophe would be the best choice for steady-state stability study. 3.  A new energy balance equation was developed in this thesis to study transient stability of systems with balanced three-phase faults. Since all catastrophe control parameters varies as clearing angle of the system, only the cusp catastrophe is practical for transient stability analysis of power systems in two dimensional graphics. The thesis concluded that the cusp catastrophe can be used to define comprehensive transient stability regions and find critical clearing angle of the system with good accuracy.  The result is  important to ensure power system security for any disturbance considered.  Hence,  enable power system planners to design proper stability controls to prevent system instability.  52  Therefore, catastrophe models can be used for fast on-line and off-line stability assessment of power systems. 1.  However, more research is needed in the following:  Critical machine(s) identification is the key factor of this thesis catastrophe model.  [3]  has suggested a way to identify the critical machine(s) in multimachine power system but does not appear to be definitive. 2.  The thesis only used clearing angle as stability criteria for power system study. Clearing time is essential for operators to determine when to open a breaker for system instability. A method has been suggested in [3] to convert clearing angle to clearing time but is not definitive.  3.  This research considered only single disturbances and balanced three-phase faults. Multiple disturbances and single-phase faults should also be considered in future research.  53  REFERENCES 1.  M . Ribbens-Pavella and F.J. Evans, " Direct methods for studying dynamics of large scale electric power systems - a survey", Automatica, 21, 1-21, Jan. 1985.  2.  A.A. Fouad and S.E. Stanton, "Transient stability of multimachine power systems", IEEE Transactions on Power Apparatus and Systems, PAS-100, 3408-3424, August 1981.  3.  M . Ribbens-Pavella, Th. Van Cutsem, R. Dhifaoui and B, Toumi, "Energy type Lyapunovlike direct criteria for rapid transient stability analysis", Proc. O f International Symposium on Power System Stability, Ames, Iowa, May 1985, 135-146.  4.  S. Tamashira, T. Koike and A. El-Albiad, "Fast transient security assessment and enhancement using pattern recognition", Proc. Of 8 PSCC, August 1984. th  5.  T. Poston and I. Stewart, "Catastrophe Theory and Its Applications", Pitman Publishing Co., London, 1979.  6.  A . A . Sallam and J.L. Dineley, "Catastrophe Theory as a Tool for Determining Synchronous Power System Dynamic Stability", LEEE Trans. PAS, vol. 102, pp. 622-630, March 1983.  7.  Deng Jixiang and Zhang Changjian, "Steady-State Stability Assessment of Multimachine Power Systems Using Catastrophe Theory" LEEE Tencon '93, Beijing.  8.  R. Thorn, "Structural Stability and Morphogenesis", Benjamin-Addison Wesley, New York, 1975.  9.  I. Stewart, "Elementary Catastrophe Theory", LEEE Trans. On Circuits and Systems, Vol CAS-30, pp. 578-586, August 1983.  10. P.T. Saunders, "An Introduction to Catastrophe Theory", Cambridge University press, 1980. 11. P.M. Anderson and A. A. Fouad, "Power System Control and Stability", Vol.1, The Iowa State University Press, 1977.  54  12. M . A . Pai, "Power System Stability", North-Holland systems and control series vol. 3, North-Holland Publishing Company. 13. Duncan Glover and Mulukutla Sarma, "Power System Analysis and Design", PWS Publishing Company, Boston, 1994. 14. M . D . Wvong and A . M . Mihirig, " Application of Catastrophe Theory to Transient Stability Analysis of Power Systems", Proceedings of IAS T E D International Conference on High . - Technology in the Power Industry, Bosman, Montana, August 1986. 15. M . D . Wvong and A . M . Mihirig, "Transient Stability Analysis of Multimachine Power Systems by Catastrophe Theory", Paper submitted for publication to IEE proceeding C. 16. T. Poston and I. Stewart, " Catastrophe Theory and Its Applications", Pitman Publishing Co., London, 1979. 17. A:A. Sallam and J.L. Dineley, "Catastrophe theory as a tool for determining synchronous Power System Dynamic Stability", IEEE Trans. PAS, Vol. 102, pp. 622-630, March 1983. 18. M . D . Wvong, Xiang-lin Sun and R. Parsi-Feraidoonian, "Catastrophe Theory Model of Multimachine Power Systems For Transient Stability Studies", IEEE Tencon '93, Beijing.  55  APPENDIX A DERIVATION OF ONE-MACHINE INFINITE BUS DYNAMIC EQUIVALENTS OF MULTrMACHINE SYSTEM A l . Derivation of Dynamic Equivalents Equation (2.13) can derived by multiplying Equation (2.16) by M , i.e  MY* =  M5 -M5 k  Mo  (A.1)  0  .. Mk 5  M  k  Mo 8  k  Mo+Mk  0  Mo+Mk  Mo  k k n E Pm " E [ E E; Ej (gy cos Sij + by sin 8y) ] i=l r1 j 1 ;  Mo+Mk M  k  E Pm - E [' E E; Ej (gy cos 6g + by sin 8y) ] i=k+l i=k+lj=l ;  Mo+Mk  MoEPmi-MkEPmi i=l i=k+l  .Mo+Mk Mo Mo+Mk M  | k n E [ E E; Ej (gy cos Sy + by sin 6y)] 1  i = 1 j = 1  k  +.  Mo+Mk  E [ E E; Ej (gy cos Sij + by sin 5y)] =k+l j=l  56  Mo P  n  i E [(E + E ) E; Ej ( i=l j=l j=k+l  m  Mo+Mk M  k  I k  I  k  •+  „  k  cos 5y + by sin 8y)]  g i j  cos 8ij + by sin 8y) ]  n  \ E [(E + E ) Ei Ej ( i=k+l j=l j=k+l  Mo+Mk  g i j  rk n —< E [ E Ei Ej (gij cos 6ij + by sin.By)] i=1 j = k + 1 Mo+Mk Mo  P - Pc m  L  M  j  k  n  k  +> —< E [ E E; Ej (gy cos 5y + by sin 8y)] i=k+l j=l Mo+Mk M„  k n E [ E E; Ej (gy cos 5y + by sin 8y)] | i=l j=k+l  P -P Mo+Mk M  I  k  + Mo+Mk Mo P -P m c  n  k n E [ E Ei Ej ( gy COS( gy-Tk+Tk ) + by sin( 6y-T +T ))] i=l j=k+l ' k  A  Mo+Mk  A  M +  k  ^ ,E [ E E; Ej (gy cos 5y - by sin 6y)] li=l j=k+l  k  —  -<  Mo+M  k  k  k n E [ E E; Ej (gy cos(8y-T +T ) - by sin( 8y - T + T ) ) ] i=l j=k+l k  k  k  k  p -p A  m -- c 1  Mo Mo+Mk M  k  + Mo+Mk  k  Ei Ej gy [cos(8ij - Tk) cosTk - sin(8y-T ) sinT ]  n  k  k  E E i i=l j=k+l + Ei Ej by [sin(8y - T ) cosT + cos(8y - T ) sinT ] k  k  k  k  k  Ej Ej gy [cos(8y - T ) cosT - sin(8y - T ) sinT ]  n  k  E E i=l j=k+l  k  k  k  - Ei Ej by [sin(8y - T ) cosT + cos(8y - T ) sinT ] k  57  k  k  k  Pm-P  c  Mo Ej Ej [gy cos(8ij-Y ) + ,bij sinCSij-^k) ] k  >  COS^k  -Mk E; Ej [gy cos(8ij-¥k) - by sinCSy-^fc) ]  +  _J  Mo Ei Ej [-gy sinCSij-^k) + by cos(5y-¥k) ]  k „  sin^k  Mo+M  i=1 k  J=? i ^ +  - M Ei Ej [-gy sinCSy-^k) - by cos^y-^) ] k  P -P  E E E; Ej [ .11 gy cosCSy-Yk) + by sin(8 - P ) ] COS T i=lj=k+l ,  s  k  k  k  n  + E E E; Ej [ by cos(8y-T ) - u gij sinCSy-^k) ] sin W i=lj=k+l k  P  m  k  - P - [ T sin ak cos *P + T cos a sin Y ] c  k  k  Pm-Pc-[T sinCPk + a ) ] k  k  58  k  k  k  APPENDIX B CATASTROPHE CONTROL PARAMETER DERIVATION (STEADY-STATE STABILITY ANALYSIS^ BI. Detailed Calculation of Wigwam Catastrophe Control Parameters a.  Parameter t ( coefficient of y ) 4  = 15 p + 7 ( K / K ) ( 5 p ) - 4 2 2  2  3  = 1 5 p + (-6P)(5p)-42 2  = - 42 - 15 p b.  2  Parameter u (coefficient of y ) 3  = 20 p + 7 (K / K ) (10p ) - 42 (4P)-210(K / K ) 3  2  2  3  2  3  . = 20 P + (-63) (10P ) - 168P - 30(-6P) 3  2  = 20p -60p -168p+180p 3  3  = -40p +12p 3  = 4p(3-10p ) 2  c.  Parameter v (coefficient of y ) 2  = 15 p + 7 (K / K ) (10p ) - 42 (6p ) -210(K / K ) (3p) + 840 4  3  2  2  3  2  3  = 15 P + (-6p)(10p ) - 42 (6P ) - 30(-6p) (3p) + 840 4  3  2  = -45 P - 252 P + 540 p +840 4  2  2  = -45 p + 288 p +840 4  2  = 840 + p (288 -45p ) 2  2  59  Parameter w ( coefficient of y )  = 6 p + 7 (K / K ) (5p ) - 42 (4p ) -210(K / K ) (3p ) + 840(2P) +2520(K / K ) 5  4  2  3  2  3  2  3  2  3  = 6 p + (-6p)(5p ) - 42 (4P ) - 30(-6p) (3p ) + 840(2p) + 360 (-6P) 5  4  3  2  --24p - 168p + 540p + 1680P - 2160P 5  3  3  =-24P + 372p - 480p 5  3  = -12P(40-31p + 2p ) 2  4  Parameter x (constant term)  = P +7(X /K )(p )-42(P )-210(X^ 6  5  2  4  3  = p + (-6p)(p ) - 42(p ) - 30(-6p)(p ) + 840(P) + 360 (-6p)P - 5040(K,+K )/K 6  5  4  2  3  3  = -5p - 42(p ) + 180p + 840(p ) - 2160p - 5040(K +K )/K 6  4  4  2  2  1  = -5p + 138P - 1320p - 5040(Ki+K )/K 6  4  2  3  3  =-p (1320 - 138P + 5p ) - 5040(K +K )/K 2  2  4  1  3  3  P = 6 K / 5 K ( B y setting coefficient of y to zero) 5  3  2  60  3  3  3  APPENDIX C THREE-MACHINE WSCC TEST  SYSTEM  CATASTROPHE MODELS SIMULATION (STEADY-STATE STABILITY  RESULTS  ANALYSIS^  (CRITICAL M A C H I N E =  2)  C A S E 00:  SYSTEM DATA AND RESULT  C A S E CI:  CUSP  C A S E C2:  SWALLOWTAIL  C A S E C3:  BUTTERFLY CATASTROPHE  C A S E C4:  WIGWAM  SUMMARY  CATASTROPHE  61  CATASTROPHE  CATASTROPHE  C A S E 00: WSCC System'Data and Result Summary  NofCm Cm Mk Mo Pm° PC Tk Alpha Delk Delo  Table 00: WSCC System Data of One Machine Infinite Bus (3 Machines, 9 Buses) 1 1 1 2 2 1 2 3 1 &2 1&3 0.1254 0.034 0.016 0.1594 0.1414 0.0499 0.1414 0.1594 0.016 0.034 -1.5711 1.0113 0.5598 -0.5598 -1.0113 -0.5911 0.0497 0.0769 -0.0769 -0.0497 2.9998 2.8487 2.4813 2.4813 2.8487 -0.0643 0.0608 0.0701 -0.0701 -0.0608 0.0396 0.3447 0.2304 0.1046 0.0612 0.3082  0.0612  0.1046  0.2304  0.3447  Table CI: Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Cusp Catastrophe Method # of Critical Machine(s) Critical Machine(s) PM Limit (1) PM Limit (2) % Error  1 2 1.38718 1.27761 7.90%  1 3 1.460789 1.3396 8.30%  2 2&3 1.484128 1.36613 7.95%  Table C2 : Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Swallowtail Catastrophe Method for WSCC test System # of Critical Machine(s) 1 1 2 Critical Machine(s) 2 3 2&3 PM Limit (1)  1.38718  PM Limit (2) % Error  1.3342 3.82%  1.460789 1.3811 5.46%  1.484128 1.428 3.78%  Table C3: Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Butterfly Catastrophe Method # of Critical Machirie(s) Critical Machine(s)  1 2  3  2 2&3  PM Limit (1) PM Limit (2) % Error  1.38718  1.460789  1.484128  1.3993 -0.87%  1.4779 -1.17%  1.4974 -0.89%  1  Table C4: Comparision of Changes in Mechanical Power (1) Solving the EEAC equation and (2) Wigwam Catastrophe Method # of Critical Machine(s) Critical Machine(s) PM Limit (1)  1 2 1.38718  1 3 1.460789  2&3 L484128  PM Limit (2)  1.3914  1.4696  1.494  % Error  -0.30%  -0.60%  -0.67%  62  2  2 2&3 0.0499 0.1254 1.5711 0.5911 2.9998 0.0643 0.3082 0.0396  Table 01 : EEAC Solution for Steady-State Stability for WSCC System Critical Machine = 2 Chg, PM Thgc Thgu Area 1(A1) Area 2(A2) A2- Al 0 0.283521 2.736472 0.000000 3.004233 3.004233 0.1 0.321075 2.698917 0.001873 2.762684 2.760811 .0.2 0.359205 2.660788 0.007530 2.528703 2.521173 0.3 0.397996 2.621997 0.017032 2.302412 2.285380 0.4 0.437545 2.582448 0.030450 2.083953 2.053503 0.5 0.477965 2.542028 0.047866 1.873489 1.825623 0.6 0.519386 2.500606 0.069372 1.671207 1.601835 0.7 0.561965 2.458027 0.095078 1.477322 1.382245 0.8 0.605889 2.414104 0.125106 1.292084 1.166978 0.9 0.651387 2.368605 .0.159604 1.115784 0.956181 1 0.698749 2.321243 0.198741 0.948765 0.750024 1.1 0.748347 2.271646 0.242724 0.791434 0.548711 1.2 0.800672 2.219320 0.291797 0.644287 0.352489 1.3 0.856406 2.163586 0.346267 • 0.507931 0.161664 1.31 0.862201 2.157791 0.352025 0.494917 0.142892 1.32 0.868041 2.151951 0.357841 0.482020 0.124179 1.33 0.873927 2.146065 0.363716 0.469240 0.105524 1.34 0.879861 2.140132 0.369649 0.456577 0.086928 1.35 0.885842 2.134150 0.375643 0.444034 0.068392 1.36 0.891874 2.128118 0.381696 0.431612 0.049916 1.37 0.897958 2.122035 0.387810 0.419310 0.031500 1.38 0.904095 2.115898 0.393985 0.407130 0.013146 1.381 0.904711 2.115281 0.394606 0:405919 , 0.011313 1.382 0.905329 2.114664 0.395227 0.404709 0.009482 1.383 0.905946 2.114046 0.395849 0.403501 0.007651 1.384 0.906565 2.113428 0.396472 0.402293 0.005821 1.385 0.907184 2.112809 0.397095 0.401087 0.003991 1.386 0.907803 2.112190 0.397719 0.399882 0.002162 1.387 0.908423 2.111570 0:398344 0.398678 0.000334 1.3871 0.908485 2.111508 0.398406 0.398558 0.000151 1.38711 0.908491 2.111501 0.398413 0.398546 0.000133 1.38712 0.908497 2.111495 0.398419 0.398534 • 0.000115 1.38713 0.908504 2.111489 0.398425 0.398522 0.000096 1.38714 0.908510 2.111483 0.398431 0.398510 0.000078 1.38715 0.908516 2.111477 0.398438 0.398498 0.000060 1.38716 0.908522 2,111470 0.398444 0.398486 • 0.000042 1.38717 0.908528 2.111464 0.398450 0.398473 0.000023 1.38718 0.908535 2.111458 0.398456 0.398461 0.000005 1.387181 0.908535 2.111457 0.398457 0.398460 0.000003 1.387182 0.908536 2.111457 0.398458 0.398459 0.000001 1 387183 0 908537 2 111456 0.398458 0 398458 0.000000 1.387184 0.908537 2.111455 0.398459 0.398457 -0.000002  63  C A S E CI: Cusp Catastrophe  U of CM CMs W X Limit PM Chg. 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.040 1.050 1.060 1.070 . 1.080 1.090 1.100 1.110 1.120 1.130 1.140 1.150 1.160 1.170 1.180 1.190 1.200 1.210 1.220. 1.230 1.240 1.250 1.260 1 270 1 2S0 1.290 1.300 1.310 1.320 1 330 1 340 1.350 1 360 I 370 1.380 1.390  Table CIA: WSCC System Steady-State Stab lity Analysis Cusp Catastrophe Numerical Result 1 2 3 -12.6859 -12.21 . 17.3913 16.4217 x , cm = 2 cm = 3 5.9563 3.212 6.8514 4.1981 7.7464 5.1842 8.6414 6.1703 9.5364 .7.1564 10.4315 8.1425 11.3265 9.1286 12.2215 10.1146 13.1166 11.1007 14.0116 12.0868 14.9066 13.0729 15.2646 13.4673 15.3541 13.5659 15.4436 13.6646 15.5331 . 13.7632 15.6226 13.8618 15.7121 13.9604 15.8016 14.059 15.8912 14.1576 15.9807 14.2562 16.0702 14.3548 16.1597 14.4534 . 16.2492 < 14.552 16.3387 14.6506 16.4282 14.7493 16.5177 14.8479 16.6072 14.9465 16.6967 15.0451 16.7862 15.1437 16.8757 15.2423 . 16.9652 15.3409 17.0547 15.4395 17.1442 • 15.5381 ' 17.2337 15.6367 17 3232 15.7353 17.4127 15.8339 17.5022 • 15.9326 17.5917 16.0312 17.6812 16.1298 17.7707 16.2284 17.8602 te 327 . 179497 16 4256 18.0392 16.5242 18.1287 16.6228 18.2182 16.7214 18.3077 16.82 18.3972 16.9186 :  64  2 2&3 > -12.6373 17.2913 cm = 2,3 5.7267 6.5733 7.4198 8.2663 9.1128 9.9593 10.8059 11.6524 12.4989 13.3454 14.1919 14.5305 14.6152 14.6998 14.7845 14.8691 14.9538 15.0384 15.1231 . .15.2078 15.2924 15.3771 15.4617 15.5464 15.631 15.7157 15.8003 . 15.885 15.9696 16.0543 16.1389 16.2236 16.3082 . 16.3929 ' 16.4775 16.5622 16.6468 16.7315 16.8161 16.9008 16.9854 17.0701 17.1547 17.2394 17.3241 17.4087 17.4934  J O J D I U B J B J AV  CASE C 2 : Swallowtail Catastrophe  #ofCM CMs V W X Limit 1 X Limit 2 X Limit 3 PM Chg. 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 0.910 0.920 0.930 0.940 0.950 0.960 0.970 0.980 0.990 1.000 1.100 1.200 1.210 1.220 1.230 1.240 1.250 1.260 1:270 1.280 1.290 1.300 1.310 1.320 1.130 1.340 1.350 1.360 1.370. 1.380 L390 1.400 1.410 1.420 1,430 1.440  Table C2A:WSCC System Steady-State Stablity Analysis Swallowtail Catastrophe Numerical Result •2 -92.9012 -310.9311 -269.1737 190.5849 4393.9056  3 -258.0824 -1949.2561 -3141.8005 -3955.6934 40400.7569  2 2&3 -98.4546 -349.3318 -321.3642 154.5994 5013.4189  X  cm = 2 -102.6763 -115.1555 -127.6348 -140.114 -152.5933 -165.0725 -177.5518 -190.031 -202.5103 -214.9896 -216.2375 -217.4854. -218.7333 -219.9813 -221.2292 -222.4771 -223.725 -224.973 -226.2209 -227.4688 -239.9481 -252.4273 -253.6752 -254.9232 -256.1711 -257.419 -258.6669 -259.9149 -261.1628 -262.4107 -263.6586 -264.9066 -266.1545 -267.4024 -268 6503 -269 898^ -271.1462 -272.3941 -273.642 -274.89 -276.1379 -277.3858 -278.6337 -279.8817 -281.1296 -282.3775  cm = 3 -3612.5506 -3637.397 . -3662.2433 -3687.0897 -3711.9361 -3736.7825 -3761.6289 -3786.4753 -3811.3217 -3836.1681 -3838.6527 -3841.1374 -3843.622 -3846.1066 -3848.5913 -3851.0759 -3853.5606 -3856.0452 -3858.5298 -3861.0145 -3885.8609 -3910.7073 -3913.1919 -3915.6765 -3918.1612 -3920.6458 -3923.1304 . -3925.6151 -3928.0997 -3930.5844 -3933.069 -3935.5536 . . -3938.0383 -3940.5229 -3943.0076 -3945.4922 -3947.9768 -3950.4615 -3952.9461 -3955 4308  M^m'i  -3960.4 -3962.8847 -3965.3693 -3967.8539 -3970.3386  66  cm = 2,3 -146.8055 -159.0497 -171.2939 -183.5381 -195.7823 -208.0265 -220.2707 -232.5149 -244.7591 -257.0033 -258.2277 -259.4521 -260.6765 -261.9009 -263.1254 -264.3498 -265.5742 -266.7986 -268.023 -269.2475 -281.4917 -293.7359 -294.9603 -296.1847 -297.4091 -298.6335 -299.858 -301.0824 -302.3068 -303.5312 -304.7556 -305.9801 -307.2045 -308.4289 -309.6533 -310.8777 -312.1022 -313.3266 -314.551 -315.7754 -316.9998 -318.2243 -319.4487 -320 6731  '4mwi  -323.1219  J3)3UIGJBJ M  1 _ ml  C A S E C3: Swallowtail Catastrophe  #ofCM CMs u v w X Limit 1 X Limit 2 X Limit 3 X Limit 4 PM Chg.  .  0.000 0.100 ,. 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.150 1.200 1.210 1.220 1.230 1.240 1.250 1.260 1.270 1.280 1.290 1.300 1.310 1.320 1.330 1.340 1.350. 1.360 1.370 1.380 I 390 1 400 1.410 1.420 1.430 1.440 1.450 1.460 1.470 1.480  1.490 1 500  Table C3A:WSCC System Steady-State Stablity Analysis Butterfly Catastrophe Numerical Result 1 3 2 -30.567 -31.8518 -2.7095 -2.1112 379.8557 366.1891 521.1786 522.0345 566.7675 548.7188 -536.1656 -523.8765 -448.2173 -469.4369 X cm = 2 cm = 3 -160.4373 -86.6668 -116.2494 -187.2882 -214.139 -145.832 -240.9899 -175.4146 -267.8407 -204.9972 -294.6916 -234.5798 -321.5424 -264.1624 -348.3933 -293.7451 -375.2442 -323.3277 -402.095 -352.9103 -428.9459 -382.4929 -469.2222 -426.8668 -482.6476 -441.6581 -485.3327 -444.6164 -488.0178 -447.5747 -490.7029. -450.5329 -493.3879 -453.4912 , -496.073 -456.4494 -498.7581 -459.4077 -501.4432 -462.366 -504.1283 -465.3242 -506.8134 -468.2825 -509.4985 •471.2408 -512.1835 -474.199 -514.8686 -477.1573 -517.5537 480.1155 -520.2388 -483.0738 -522.9239 -486.0321 -525.609 -488.9903 -528.2941 -491.9486. -530.9791 -494.9068 -533 6642 497.8651 -536.3493 -500.8234 -539.0344 . -503.7816 -541.7195 . -506.7399 -544.4046 -509.6982 -547.0897 -512.6564 -549.7747 -515.6.147 -552.4598 -518.5729 -555.1449 -521.5312 -557.83 -524.4895  2 2&3 -31.7207 -2.7206 378.484 522.4602 565.4026 -534.5544 -449.7495 cm = 2,3 -154.282 -179.6775 -205.0731 -230.4687 -255.8643 -281.2599 -306.6554 -332.051 -357.4466 -382.8422 -408.2378 -446.3311 -459.0289 -461.5685 -464.108 -466.6476 -469.1872 •471.7267 -474.2663 -476.8058 -479.3454 •481.885 -484.4245 -486.9641 -489.5036 -492.0432 -494.5827 .-497.1223 -499.6619 -502.2014 -504.741 r507.2805 -509.8201 -512.3596 -514.8992 -517.4388 -519.9783 -522.5179 -525.0574 -527.597 -530.1366 ' .  -560.5151  -527.4477  -532.6761  . -563.2002  -530.406  -535.2157  68  008- '  — J»)D1URJBJ M  CASE C4: Wigwam Catastrophe  #ofCM CMs t u V  w x Limit 1 x Limit 2 x Limit 3 PM Chg. 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000 1.050 . 1.100 1.150 1.200 1.250 1.270 1.280 1.290 1.300 1.310 1.320 1.330 1.340 1.350 1.360 1.370 1.380 1390 I 400 1.410 1.420 1.430 1.440 1.450 1 4o0 1.47(1 1.480 1 490 1.500  Table C4A:WSCC System Steady-State Stablity Analysis Wigwam Catastrophe Numerical Result 1 2 2 3 2&3 -200.7626 -560.4905 -212.8567 -1338.3129 -8058.3963 -1497.1980 -1152.8688 -42971.4657 -1717.9517 2500.8998 -95814.1378 2171.5658 -786.6426 -71906.9151 -492.5274 2852.3826 -75600.0695 3136.1450 4086299.7255 133355984.4960 5063398.1860 X  cm = 2 -4440.142 -3916.0135 -3391.8849 -2867.7563 -2343.6278 -1819.4992 -1295.3706 -771.242 . -247.1135 277.0151 801.1437 1063.208 1325.2723 1587.3365 1849.4008 2111.4651 2216.2908 2268.7037 2321.1165 2373.5294 2425.9423 2478.3551 2530.768 2583.1808 2635.5937 2688.0065 2740.4194 2792.8323 2845 2451 2897*65$ 2950.0708 3002.4837 . 3054.8965 3107.3094 3159.7223 3212.1351 3264.548 3316.9608 3369.3737  cm = 3 -87242.6306 -86199.0822 -85155.5338 -84111.9854 -83068.4371 -82024.8887 -80981.3403 -79937.7919 -78894.2435 -77850.6951 -76807.1468 -76285.3726 -75763.5984 -75241.8242 -74720.05 -74198.2758 -73989.5661 -73885.2113 -73780.8564 -73676.5016 -73572.1468 -73467.7919 -73363.4371 -73259.0823 -73154.7274 -73050.3726 -72946.0177 -72841.6629 -72737.3081 -72632.9532 -72528.5984 -72424.2436 -72319.8887. -72215.5339 -72111.179 -72006 8242 -71902.4694 -71798.1145 -71693.7597  cm = 2,3 -4520.0676 -4005.8112 -3491.5547 -2977.2983 -2463.0419 -1948.7854 -1434.529 -920.2726 -406.0161 108.2403 622.4967 879.625 1136.7532 1393.8814 1651.0096 1908.1378 2010.9891 2062.4148 2113.8404 2165.266 2216.6917 2268.1173 ,2319.543 2370.9686 2422.3943 2473.8199 . 2525.2455 2576.6712 2628.0968 2679.5225 2730.9481 2782.3738 2833.7994 2885.2251 2936.6507 2988.0763 3039.502 3090.9276 3  3421.7865  -71589.4048  3193.77*9  70  APPENDIX D SEVEN-MACHINE CIGRE TEST S Y S T E M C A T A S T R O P H E M O D E L S SIMULATION R E S U L T S ( S T E A D Y - S T A T E STABILITY ANALYSIS^) (CRITICAL MACHINE = 7) C A S E DO:  S Y S T E M D A T A AND RESULT S U M M A R Y  CASED1:  CUSP C A T A S T R O P H E  C A S E D2:  SWALLOWTAIL C A T A S T R O P H E  C A S E D3:  B U T T E R F L Y CATASTROPHE  C A S E D4:  WIGWAM CATASTROPHE  72  C A S E DO: CIGRE System Data and Result Summary  Mk 0.0568  Table DO: CIGRE System Data of One Machine Infinite Bus (7 Machines, 17 Buses) No of Critical Machine = 1, Critical Machine = 7 Mo Pmo PC Tk Alpha Delk 0.4005 -0.0684 . -0.6525 4.4313 0.0869 0.1304 Table D l : Comparison of Changes of Mechanical Power Values (1) Solving the EEAC equation and (2) Catastrophe Method _ _ _ _ for CIGRE Test System Catastrophe Cusp Swallowtail Butterfly Wigwam Critical Machine 7 7 7 7 PM Limit (1) 2.80473 2.80473 2,80473 2.80473 PM Limit (2) 2.57149 2.62972213 2.84079 2.82713 % Error 9.07% 6.65% -1.27% -0.79%  73  Delo 0.0851  Table DO A: EEAC Solution of Steady-State Stablity forCIGRE System Critical Machine: machine 7 Chg,PM 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 .1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.71 2.72 2.73 2.74 • 2.75 2.76 2.77 ' 2.78 . 2.79 2.8 2.802 2.803 2.804731 2.X047336 2.8047346  Thgc 0.045297 0.068099 0.090982 0.113960 0.137047 0.160255 0.183599 0.207096 , 0.230762 0.254613 0.278668 0.302947 0.327470 0.352261 0.377344 0.402746 0.428497 0.454629 0.481179 . 0.508187 0.535697 0.563763 0.592441 0.621800 0.651917 0.682885 0.714812 0.747829 0.751196 0.754576 0.757969 0.761375 0.764795 0.768227 0.771673 0.775134 0.778608 0.782096 0.782795 0.783145 0.783751 0.783752 0.783753'  Thgu 2.922496 2.899694 2.876810 2.853832 2.830746 2.807538 2.784193 2.760696 2.737031 2.713180 2.689125 2.664846 2.640323 2.615532 2.590449 2.565047 2.539296 2.513164 2.486614 2.459606 2.432095 2.404030 2.375351 2.345993 2.315875 2.284908 2.252981 2.219964 2.216597 2.213216 2.209823 ' 2.206417 2.202998 2.199565 2.196119 2.192659 2.189185 2.185697 2.184997 2.184647 2.184041 2.1X4040 2.184040  74  Area 1(A1) 0.000000 0.001139 0.004563 0.010280 0.018299 . 0.028634 0.041295 0.056299 0.073661 0.093398 0.115531 0.140080 0.167069 0.196523 0.228471 0.262943 0.299973 0.339596 0.381853 0.426788 0.474448 0.524886 0.578161 0.634338 0.693487 0.755690 0.821037 0.889629 0.896672 0.903747 0.910857 0.918001 0.925179 0.932391 0.939637 0.946919 0.954234 0.961585 0.963059 0.963797 0.965074 0.965076 0.965077  Area 2(A2) 7.104699 6.819258 6.538386 6.262099 5.990419 5.723367 5.460971 5.203259 4.950262 4.702017 4.458563 4.219941 3.986199 3.757388 . 3.533564 3.314788 3.101127 2.892654 2.689448 2.491598 2.299198 2.112355 1.931185 1.755818 1.586397 1.423083 1.266057 1.115522 1.100834 1.086214 1.071662 1.057177 1.042761 1.028413 1.014134 0.999924 0.985784 0.971713 0.968907 0.967505 0.965080 0 965077 0.965075  A2-A1 7.104699 6.818119 6.533823 6.251820 5.972119 5.694734 5.419676 5.146960 4.876601 4.608619 4.343032 4.079861 3.819130 3.560865 3.305093 3.051845 2.801155 2.553058 2.307595 2,064810 1.824750 1.587469 1.353024 1.121480 0.892910 0.667393 0.445020 0.225893 0.2Q4163 0.182467 0.160805 0.139176 0.117582 0.096022 0.074497 0.053006 0.031550 0.010128 0.005848 0.003709 0.000006 0.000000  -0.000002  CASE DI: Cusp Catastrophe  #ofCM CMs w x Limit 1 x Limit 2 , PM Chg. 0.000 0.050 0.100 . 0.150 0.200 0.250 0.300 0.350 . 0.360 0.370 0.380 0.390 0.400 0.410 0.420 0.430 0.440 0.450 0.460 0.470 0.480 0.490 0.500 0.510 0.520 0.530 • . 0.540 0.550 0.560 0.570 0.580 0.590 0.600 0.650 0.700 0.750 0.800 0.850 0.900 . 0.950  Table DI A: CIGRE System Steady-State Stablity Analysis Cusp Catastrophe Numerical Result 1 . •7 ' . •• . ' ' -12,0943 16.18897003 -16.18897003 X PMChg X PMChg 2.1389 1.000 • 7.6026 2.000 2.4121 1.050 .7.8758 2.050 2.6853 1.100 8.1489 2.100 2.9585 1.150 8.4221 2.150 3.2317 1.200 8.6953 2.200 3.5048 1.250 8.9685 2.250 3.778 1.300 9.2417 2.300 4.0512 1.350 9.5149 2.350 4.1058 1.360 9.5695 2.360 4.1605 1.370 9.6241 2.370 4.2151 1.380 9.6788 2.380 4.2697 1.390 9.7334 2.390 4.3244 1.400 9.788 2.400 4.379 1.410 9.8427 2.410. 4.4337 1.420 9.8973 2.420 4.4883 1.430 . 9.9519 2.430 4.5429 . 1.440 10.0066 2.440 4.5976 1.450 10.0612 2.450 4.6522 1.460 10.1159 2.460 4.7068 1.470 10.1705 2.470 4.7615 1.480 10.2251 2.480 4.8161 1.490 10.2798 2.490 4.8707 1.500 10.3344 2.500 4.9254 1.5.10 10.389 2.510 4.98 1.520 10.4437 2.520 5.0347 1.530 10.4983 2.530 5.0893 1.540 10.553 2.540 5.1439 1.550 10.6076 2.550 .5.1986 1.560 10.6622 2.560 5.2532 1.570 10.7169 2 570 5.3078 1.580 10.7715 2 580 5.3625 1.590 10.8261 2.590 5.4171 1.600 10.8808 2.600 5.6903 1.650 11.154 2.650 5.9635 1.700 11.4271 2.700 6.2367 1.750 11.7003 2.750 6.5098 1.800 11.9735 2.800 6.783 1.850 12.2467 2.850 . 7.0562 1.900 12.5199 2.900 7.3294 1.950 12.793 2.950  75  X  13.0662 13.3394 13.6126 13.8858 14.159 14.4321 14.7053 14.9785 15.0331 15.0878 15.1424 15.1971 15.2517 15.3063 15.361 15.4156 15.4702 15.5249 15.5795 15.6342 15.6888 15.7434 15.7981 15.8527 15.9073 15.962 16.0166 16.0712 16.1259 164805 16,2352 16.2898 ' 16.3444 16.6176 16.8908 17.164 17.4372 17.7103 17.9835 18.2567  <  S  II  3 1f  S ta "«  5  l i s  sS CO  O.  II '  3 31  |  S 31 •= S °" M u  «  I  U  II  J3JJIUBJFJ .«  CASE D2: Swallowtail Catastrophe  Table D2: CIGRE System Steady-State Stablity Analysis #ofCM . CMs •. y w X Limit 1 X Limit 2 PM Chg. 0.000 0.100 0.200 0.210 0.220 0.230 0.240 0.250 0.260 0.270 0.280 0.290 0.300 0.310 0.320 0.330 0.340 0.350 0.360 0.370 0.380 0.390 0.400 0.410 0.420 0.430 0.440 0.450 0.460 0.470 0.480 0.490 0.500 0.510 0.520 0.530 0.540 0.550 0.560 0.570 0.580 0.590 0.600 0.610 0.620 . 0.630  X -20943.715 -21)^04.2^73 -20984.7996 -20986.8538 -20990.9622 -2(jyy3.U165 -20995.0707 -20997.1249 -20999.1791 -21001.2334 -21003.2876 -21005.3418 -21007.396 -21UUy.45U3 -21U11.5U45 -21U13.5587 -21U15.bl3 -21017.6672 ,-21Uiy.7214 -21021.7756 -21U23.82yy -21U25.SS41 -21U27.y3«3 -2iu2y.yy25 -21U32.U46S -21U34.1U1 -21U36.1552 -21U38.2Uy4 -21U4U.2637 -21U42.31/y -21U44.3721 -21U4t>.4263 -21U48.48U6 -21Um^34« -21U52.!)«y -21UM.6433 -211D8.7M7 -21U6U.8U5y -21Ub2.8b(J2 -21Ub4.yi44 -21U66.y686 -21Uby.U22« -21U71.U771 -2111/3.1313  Swallowtail Catastrophe Numerical Result 1 7 -550.1042 -6568.4261 -20,275.68197 -21483.91942 PMChg X 1.000 -21149.1377 1.100 -21169.6799 1.200 -21190.2222 1.210 -21192.2764 1.220 -21194.3306 1.230 -21196.3849 1.240 -21198.4391 1.250 -21200.4933 1.260 -21202.5475 1.270 \ -21204.6018 1.280 • -21206.656 1.290 -21208.7102. 1.300 -21210.7644 1.310 -21212.8187 1.320 -21214.8729 •• 1.330 -21216.9271 1.340 -21218.9814 1.350 -21221.0356 1.360 -21223.0898 1.370 -21225.144 1.380 -21227.1983 1.390 -21229.2525 1.400 -21231.3067 1.410 -21233.3609 1.420 -21235.4152 1.430 -21237.4694 1.440 -21239.5236 1.450 -21241.5778 1.460 -21243.6321 1.470 -21245.6863 1.480 -21247.7405 1.490 -21249.7947 1.500 -21251.849 •1.510 , -21253.9032 1.520 -21255.9574 1.530 -21258.0116 1.540 • -21260.0659 1.550 -21262.1201 1.560 • -21264.1743 1.570 -21266.2286 1.580 . -21268.2828 . 1.590 -21270.337 1.600 -21272.3912 1.610 -21274.4455 1.620 -21276.4997 1.630 -21278.5539  77  PMChg 2.000 2.100 2.200 2.210 2.220 2.230 2.240 2.250 2.260 2.270 2.280 2.290 2.300 2.310 2.320 2.330 2.340 2.350 2.360 2.370 2.380 2.390 2.400 2.410 2.420 2.430 2.440 2.4502.460 2.470 2.480 2.490 2.500 2.510 2.520 2.530 • 2.540 2.550 2.560 2.570 2.580 2.590 2.600 2.610 2.620 2.630  X -21354.5603 -21375.1025 -21395.6448 -21397.699 • -21399.7533 -21401.8075 -21403.8617 -21405.9159 -21407.9702 -21410.0244 -21412.0786 -21414.1328 • -21416.1871 -21418.2413 -21420.2955 -21422.3497 -21424.404 -21426.4582 -21428.5124 -21430.5667 -21432.6209 -21434.6751 -21436.7293 -21438.7836 -21440.8378 -21442.892 -21444.9462 -21447.0005 -21449.0547 -21451.1089 -21453.1631 -21455.2174 -21457.2716 -21459.3258 -21461.38 -21463.4343 -21465.4885 -21467.5427 -21469.597 -21471.6512 -21473.7054 -21475.7596 -21477.8139 -21479.8681 -2148L9223 -21483.9765  CASE D3: Swallowtail Catastrophe  #ofCM CMs U V w X Limit 1 X Limit 2 X Limit 3 X Limit 4 PMChg. 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.610 0.620 0.630 0.640 0.650 0.660 0.670 0.680 0.690 0.700 0.710 0.720 0.730 0.740 0.750 0.760 0.770 0.780 0.790 0.800 0.810 0.820 0.830 0.840 0.850 0.900 0.950  Table D3: CIGRE System Steady-State Stablity Analysis Butterfly Catastrophe Numerical Result  ' x -57.7339 -65.9294 -74.1249 -82.3204 -90.5158 -98.7113 -106.9068 -115.1023 -123.2978 -131.4933 -139.6887 -147.8842 -156.0797 -157.7188 -159.3579 -160.997 -162.6361 -164.2752 : -165.9143 -167.5534 -169.1925 -170.8316 -172.4707 -174.1098 -175.7489 -177.388 -179.0271 -180.6662 -182.3052 -183.9443 -185.5834 .. -187.2225 -188.8616 • -190.5007 -192.1398 -193.7789 -195.418 -197.0571 -205.2526 -213.4481  PM Chg. 1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.350 1.400 " 1.450 1.500 1.550 1.600 1.610 1.620 1.630 1.640 1.650 1.660 1.670 1.680 1.690 1.700 1.710 1.720 1.730 1.740 1.750 1.760 1.770 1.780 1.790 1.800 1.810 1.820. 1.830 1.840 1.850 1.900 1.950  79  1 . 7 -30.2547 ' -1.5145 • 362.79160 '516.045393 541.380387 -523.36753 -479.07337 . . X -221.6436 -229.8391 -238.0345 -246.23 -254.4255 . -262.621 -270.8165 -279.0119 -287.2074 -295.4029 -303.5984 -311.7939 -319.9894 -321.6285 -323.2676 -324.9067 -326.5458 -328.1848 . -329.8239 ' -331.463 -333.1021 -334.7412 -336.3803 -338.0194 -339.6585 -341.2976 -342.9367 -344.5758 -346.2149 -347.854 -349.4931 -351.1322 -352.7713 -354.4104 -356.0495 -357.6886 -359.3277 -360.9668 -369.1623 -377.3577  •  PM Chg. 2.000 2.050 2.100 2.150 2.200 2-250. 2.300 2.350 2.400 2.450 2.500 2.550 2.600 2.610 2.620 2.630 2.640 2.650 2.660 2.670 2.680 2.690 2.700 2.710 2.720 2.730 2.740 2.750 2.760 2.770 2.780 2.790 2.800 2.810 2.820 2.830 2 8-10 2.85!) 2.900 2.950  X -385.5532 -393.7487 . 401.9442 410.1397 418.3352 426.5306 434.7261 442.9216 451.1171 459.3126 467.5081 475.7035 483.899 485.5381 487.1772 488.8163 490.4554 492.0945 493.7336 . 495.3727 497.0118 498.6509 -500.29 -501.9291 -503.5682 -505.2073 -506.8464 • -508.4855 -510.1246 -511.7637 -513.4028 , -515.0419 -516.681 -518.3201 -519.9592 -521.5983 -523.2373 -524 876-t -533.0719 -541.2674  CASE D4: Swallowtail Catastrophe Table D4: CIGRE System Steady-State Stablity Analysis Wigwam Catastrophe Numerical Result 1 7 -1196.449 -26902.2797 -243,545.09750 -1000194.528 -1,595,397.046960 -1,595,509.396349 -1,539,178.044549 • . -1,543,208.741521 1,445,059,990.943130 X PM Chg. X PM Chg. -1563569.865 1.000 -1554942.115 2.000 -1563138.478 1.050 -1554510.727 2.050 . -1562707.09 1.100 -1554079.34 2.100 -1562275.703 1.150 -1553647.952 2.150 -1561844.315 1.200 -1553216.565 2.200 -1561412.927 1.250 -1552785.177 2.250 -1560981.54 1.300 -1552353.79 2.300 -1560550.152 1.350 -1551922.402 2.350 . -1560118.765 . 1.400 -1551491.015 2.400 -1559687.377 1.450 -1551059.627 2.450 -1559255.99 1.500 -1550628.24 2.500 -1558824.602 1.550 -1550196.852 2.550 -1558393.215 1.600 -1549765.465 2.600 -1557961.827 1.650 -1549334.077 2.650 -1557875.55 1.660 -1549247.8 2.660. -1557789.272 1.670 -1549161.522 2.670 -1557702.995 1.680 -1549075.245 2.680 -1557616.717 1.690 -1548988.967 2.690 -1557530.44 1.700 -1548902.69 2.700 -1557444.162 1.710 -1548816.412 2.710 -1557357.885 1.720 -1548730.135 2.720 -1557271.607 1.730 -1548643.857 2.730 -1557185.33 1.740 -1548557.58 2.740 -1557099.052 1.750 -1548471.302 2.750 -1557012.775 1.760 -1548385.025 2.760 -1556926.497 1.770 -1548298.747 2.770 -1556840.22 1.780 -1548212.47 2.780 -1556753.942 1.790 -1548126.192 2.790 -1556667.665 1.800 -1548039.915 2.800 -1556581.387 1.810 -1547953.637 2.810 -1556495.11 1.820 -1547867.36 2.820 -1556408.832 1.830 -1547781.082 2 830 -1556322.555 1.840 -1547694.805 2.840 -1556236.277 1.850 -1547608.527 2.850 -1555804.89 1.900 -1547177.14 2.900 -1555373.502 1.950 -1546745.752 2.950 :  \.  #ofCM CMs t u V w x Limit 1 x Limit 2 x Limit 3 x Limit 4 x Limit 5 PM Chg. 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650. 0.660 0.670 0.680 0.690 0.700 0.710 0.720 0.730 0.740 0.750 0.760 0.770 0.780 •. 0.790 0.800 0.810 0.820 0.830 0.840 0.850 0.900 0.950  81  -  X -1546314.365 -1545882.977 -1545451.59 -1545020.202 -1544588.815 -1544157.427 -1543726.04 -1543294.652 • -1542863.265 -1542431.877 -1542000.49 -1541569.102 -1541137.715 -1540706.327 -1540620.05 -1540533.772 -1540447.495 -1540361.217 -1540274.94 -1540188.662 -1540102.384 -1540016.107 -1539929.829 -1539843.552 -1539757.274 . -1539670.997 -1539584.719 -1539498.442 -1539412.164 -1539325.887 -I53Y2 39 6(^1 -I5VJ1S3 332 -1539067.054 -1538980.777 -1538549.389 . -1538118.002  00O009I-  0000S91-  APPENDIX  E  THREE-MACHINE WSCC TEST SYSTEM CATASTROPHE MODELS SIMULATION RESULTS (TRANSIENT STABILITY  ANALYSIS)  ( C R I T I C A L M A C H I N E = 2) AT  FAULT BUS = 5  AND  LINE OPEN B E T W E E N NODE 4 AND 5  83  ©  o co  NO NO  ON  ©  ON ON  cs  o  ON  SO  NO OS  I—H  ON  O  VO CS  O  CO  00  w> ON  NO CO  ro  OS  SO  ro  -*  r~ o  CN OS SO  o  cs o , f>  ro  r~ r-~ ro Ti o o ©  VO  o ©  s  cs o o  ON  vo  O  CO  CS  ON ON ON  SO  ©  OS SO  OS  00  ««  SO  CS  o  SO T3 O  6  o w >>  00 •/"> |  O ' •O  </"> Ti  00  NO ON  00  00  CS  00  r—(  3  IS'  3  m  00  Ti Os O O  OS  •n  r~ o  •/->  I  CN •*  CS CS  SO  o o ©  SO  00  ©  o o  <L>  a  60  e 79 o  ft  P  © ' CS CS  .  o  <n  1—1  TI  CN  00 SO  ca  CO VO  o ro >n  Os  rOs  C SON  ro I co ©  CS  I SO I  so CS  o  Tl SO OS  S0  12  <U /~N 60 CS^  ©  if  *°  ©  o  o  SO  o ©  CNOS SO  rro  ro  •a  CS  00 o  4 •—>  CS'  « '5. 1 8 3 •g a  £^  u  NO  1—I  S  <N 1  •8  cs  ON  3  OS  o  CO OS  sg  &  Ti ©  '2  vo OS  OS  O  00  OS CS SO SO  o  ©  ON ON T r-H  i—I  00  Tl CN  00 o  o o  CO  I*  e  s  l l  1 I  ,3, Ifel Si o O PL, a,  ro r~ cs o  3  3  1  SO "* Ti-  03  Ii,  1  PL, CS ft  I  ca  to  ca  ca  3  4  O  ft  00  cs a o w 6 o 00  6 0 | 6p| C W  •a o  p cs  w  oo  21  00  oo.  8 ft  •S o  Ti  Q E  Tl Tl  Tl  OS  o ©  I' 3  T> CS CS  00  m  8 •8 o  ro  IT) CS CS  VO ON  II  OS  00  <o  2  r»  CS NO NO ©•  I©I  ro ro  PQ  ro  so CS  OS CS  CN "O fN CO  00  OS  •a t~>l•c. O 1*1  O 60  u 60  g  1^'  NO Tf Tf  cs co p  vi CS NO CO CS CS CS  vo co vo  ON CO ON CS  O  ON Tf O  vo vo co Tf  CN CS 00 ON ON ON  wo  CO  co cs  co  co cs  cs cs fcs r~ t~ p  NO  t> o vi  00 Tf Tf  cs r-  *n  cs cs © co cs 00 Tf  o  ON  o  t*» cs  cs  wo"  vi  ON ON CO ON  cs  Vl  00  WN  CS W O CN O ON  co  cs  cs  ON  c-  Vl >o Tf  co Tf ON 00  Tf ON NO ON  o >o  c-^  ' vo co  ON  _| vi  Tf 00  *o  wo  VI  •* o ro cs  00  r-  Tf 00 Tf CO  CO NO  >o o V) CO  r-  00  t> NO  Vl  cs  Tf 00 NO  CO  CS  rNO  Tf  CO  Tf  Vl •o  o  00  00 00  00 Tf  t> cs p  r~ Vl  TICS C-^  Vl  Tf CS *ri ON  o o  CO NO  —' ON 00 00 Tf ON  d Tf  o  wo 00 Tf  cs cs o  ON O  o  CO  cs  ON 00 00 Tf ON  Vl d  ON CO CO  •>!ON VI  d  cs Vl  CO  Vl  CO  Tf  c~ Vl  vi  00 NO  "f 00  o  00  tt-~ -  ' —' CS  r~ p cs  r-  CO  NO  ON ON  NO  rcs cs  CS CS  VO OO  Vl  CO CO  r-  >o o  *o  w-i  o d  VI NO CS  cs  t-; CO CS  VN vi  CO CO NO Tf 00  ON  VN  00 Tf  ON  v> co o o ON CO  O  VN Tf NO  ON ON  o  Tf  NO ON 00  rVN  CO Tf  NO  •—* d—> d  d  d  oo  00  vo  VO ON VO 00  00 NO ON NO 00  cs  cs  CS  d  cs d  CO 00 Tf  •0CO 00  vi d  00 NO ON  *o 00  r~ cs  VN ON 00 Tf CO ON CS  Tf  ON CS 00 CO  cs cs  cs CO  ON  cs  vo *r\  VO CO wo 0 00 ON Tf 00  CO  CO  w-i CS  rON CS  00  ON.  Tf  T?:  cs  VO  r-  WO  CO  • — < ON  r~  Tf ON  rcs  CO  O  i  "t  NO ON Tf CS  CO WO 00 TjON  Vl  CO ON  cs  o o  cs  CS NO Tf  ON  NO 00 CO CO NO  NO  Tf  1 — 1  ^ CO VO  ON  00 00  00 ON  00  CO  wo  s> s> vi CO CO  r~ o r~ o  Tf  o  Tf Tf  VN OO  o  CO  NO  o  cs *o  ON Tf CO  NO Tf NO 00  o  NO 00 Tf  VI d  r-  ON  00 00  v» cs  c~  ON f~  r~  O  d  O ON I>  Tf CO 0 ON ON VO  ON  cs  00 NO Tf  00 ON Tf O 00 NO ON  CO NO  f~  cs  NO NO ON  ON Tf  >o  Tf  ^7*  t~  ON NO  Tf  CO CO CO 00 00  ON CO Tf CO  00 00 Tf ON VN  d CO  r-  ON 00 00 Tf ON  VN d ON CS  r~ NO 00  CO  vt  -* VI  CO  Tf ON NO 0 CS  CO CO  CS Tf  cs cs  0  ON  Tf 00  vi  ON  v.  VN  CO  Tf  d  d  CO 00  r~ d  00 NO ON NO 00  c~ cs  VI d  «n  00  cs r>  0 00 CO  r-  r-  cs  CO  vo  Tf 0  p  CO CO ON  VO ON ON 00 O CS  d  cs cs cs  ON  ON 00 00 Tf ON  d  wo d  CO  Tf  vo co vq  cs CO  r-  WO  00 CO CO  Tf  00  0 00  «n vo  r-  —  O r-  0 CO  cs d  O r»  Tf CO  cs d  >Ti  WO  CO 00 00  CO 00 ON  CO CO 0  d  O  00 NO ON NO 00  00 VO ON VO 00  Tf  vi  t-; cs'  rt Tf *r\ wo  rcs  ^  00, VO ON VO 00  cs  WO  NO  d  m  d ON 00 00  —*  VI Tf  wo 00  T"  00. CO  cs  MO  •9 9  Tf  vi  cs  «o cs  d  Vl  00 NO ON NO 00 C~  0  d  CS  d  Tf  CO  ON  cs d  CO 00 NO  c-  o  d Tf  0  0 ON *Ti  ~* o cs  0  vo  ^ ~ ~ ON ON 00 00 Tf ON  cs 0  cs  r-  Vl  o  CO  Tf  »n  ON  *!•  o  vi  vo  o t-  t>  Tf  cs d  CO  ^ ON ON CS  vi  CO 00  P~ cs  00 Tf  vi  V~i ON  Tf WO CO 00 CO  wo  r-  d  wo  Tf  o  '  ^  © o o o o d  wi  00  CO ON  o  co cs o cs t-  ~*  cs wo *o «o  ON  "fr  tri  ON  cs  Vl  cs  cs  r~  cs  co cs rs co rs  00  00  vi  vo  cs cn cs  VI cs  Tf NO  9  ON CO CN  ON  r-  d  CS  rTf C~  ON  V~t 00  o  00  ct-  00 ON tf  CO  cs  vi  00  wo ©  r~  ON  SO  ON 00 00 Tf ON  CO  NO CS 00  00  00 "Ti VD TT CO Tf OS 00  CO 00  CO  00 ID Tf  vo  UJ  CO ON O  >o  Tf  a  t~  d  d  Tf  J3)3U1BJBJ M  APPENDIX F SEVEN-MACHINE CIGRE TEST S Y S T E M C A T A S T R O P H E M O D E L S SIMULATION RESULTS (TRANSIENT STABILITY ANALYSIS') (CRITICAL M A C H I N E = 7) AT  FAULTBUS=9  AND  LINE OPEN B E T W E E N N O D E 9 A N D 10  87  SO  oo  3  ja  2 2 60 t> ,  so  O II 60 H "  a  E 6g I 3 w CJ  a O Q  a  3 !* e  3 m  GO PL,  5  6 M  Xi  U S • • OTJ  « .a  o o  e  a  p S &l «  »2 •f ^ O CO  O «  X  CQ  ON  00  3  II  x  

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